LMFDB - Embedded newform 1849.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} + \cdots - 27 $ x^18 - 5x^17 - 15x^16 + 106x^15 + 47x^14 - 897x^13 + 364x^12 + 3855x^11 - 3223x^10 - 8851x^9 + 9909x^8 + 10400x^7 - 14483x^6 - 5118x^5 + 9945x^4 - 27x^3 - 2493x^2 + 513*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.31259$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31259 q^{2} -2.69989 q^{3} +3.34809 q^{4} -2.53973 q^{5} +6.24374 q^{6} +2.18729 q^{7} -3.11758 q^{8} +4.28939 q^{9} +O(q^{10})$	\(q-2.31259 q^{2} -2.69989 q^{3} +3.34809 q^{4} -2.53973 q^{5} +6.24374 q^{6} +2.18729 q^{7} -3.11758 q^{8} +4.28939 q^{9} +5.87337 q^{10} -3.12200 q^{11} -9.03946 q^{12} +3.44204 q^{13} -5.05832 q^{14} +6.85700 q^{15} +0.513513 q^{16} -2.33435 q^{17} -9.91963 q^{18} -3.38242 q^{19} -8.50325 q^{20} -5.90544 q^{21} +7.21991 q^{22} -6.58244 q^{23} +8.41711 q^{24} +1.45025 q^{25} -7.96004 q^{26} -3.48122 q^{27} +7.32324 q^{28} +5.41020 q^{29} -15.8574 q^{30} +2.13081 q^{31} +5.04761 q^{32} +8.42904 q^{33} +5.39840 q^{34} -5.55514 q^{35} +14.3613 q^{36} +3.47929 q^{37} +7.82215 q^{38} -9.29313 q^{39} +7.91782 q^{40} +2.13473 q^{41} +13.6569 q^{42} -10.4527 q^{44} -10.8939 q^{45} +15.2225 q^{46} -1.16939 q^{47} -1.38643 q^{48} -2.21576 q^{49} -3.35383 q^{50} +6.30248 q^{51} +11.5243 q^{52} +10.4019 q^{53} +8.05065 q^{54} +7.92904 q^{55} -6.81905 q^{56} +9.13214 q^{57} -12.5116 q^{58} +6.14435 q^{59} +22.9578 q^{60} +2.50141 q^{61} -4.92770 q^{62} +9.38216 q^{63} -12.7001 q^{64} -8.74187 q^{65} -19.4929 q^{66} +10.3393 q^{67} -7.81560 q^{68} +17.7718 q^{69} +12.8468 q^{70} -11.3311 q^{71} -13.3725 q^{72} +8.36206 q^{73} -8.04617 q^{74} -3.91551 q^{75} -11.3246 q^{76} -6.82872 q^{77} +21.4912 q^{78} +9.81573 q^{79} -1.30419 q^{80} -3.46928 q^{81} -4.93675 q^{82} +2.86913 q^{83} -19.7719 q^{84} +5.92862 q^{85} -14.6069 q^{87} +9.73307 q^{88} -14.9948 q^{89} +25.1932 q^{90} +7.52875 q^{91} -22.0386 q^{92} -5.75295 q^{93} +2.70433 q^{94} +8.59043 q^{95} -13.6280 q^{96} -4.22212 q^{97} +5.12414 q^{98} -13.3915 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9}+O(q^{10}) $ 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9	\( 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} - 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} - 25 q^{18} - 31 q^{19} - 25 q^{20} - 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} - 27 q^{26} - 23 q^{27} - 20 q^{28} - 37 q^{29} + 17 q^{30} - 12 q^{31} - 39 q^{32} - 38 q^{33} - 14 q^{34} + 16 q^{35} + 47 q^{36} + 19 q^{37} + 56 q^{38} - 46 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} + 7 q^{44} - 23 q^{45} + 47 q^{46} - q^{47} - 15 q^{48} - 6 q^{49} + 3 q^{50} - 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} - 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} - 28 q^{61} - 33 q^{62} - 26 q^{63} + 10 q^{64} - 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} - 7 q^{69} + 34 q^{70} - 86 q^{71} + 2 q^{72} + 27 q^{73} - 79 q^{74} - 31 q^{75} - 59 q^{76} - 43 q^{77} + 91 q^{78} + 17 q^{79} - 8 q^{80} - 10 q^{81} - 13 q^{82} - 12 q^{83} - 32 q^{84} - 28 q^{85} - 43 q^{87} + 23 q^{88} - 51 q^{89} + 10 q^{90} + 20 q^{91} + 18 q^{92} + 30 q^{93} + 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} + 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 - 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 - 25 * q^18 - 31 * q^19 - 25 * q^20 - 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 - 27 * q^26 - 23 * q^27 - 20 * q^28 - 37 * q^29 + 17 * q^30 - 12 * q^31 - 39 * q^32 - 38 * q^33 - 14 * q^34 + 16 * q^35 + 47 * q^36 + 19 * q^37 + 56 * q^38 - 46 * q^39 + 6 * q^40 - 7 * q^41 + q^42 + 7 * q^44 - 23 * q^45 + 47 * q^46 - q^47 - 15 * q^48 - 6 * q^49 + 3 * q^50 - 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 - 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 - 28 * q^61 - 33 * q^62 - 26 * q^63 + 10 * q^64 - 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 - 7 * q^69 + 34 * q^70 - 86 * q^71 + 2 * q^72 + 27 * q^73 - 79 * q^74 - 31 * q^75 - 59 * q^76 - 43 * q^77 + 91 * q^78 + 17 * q^79 - 8 * q^80 - 10 * q^81 - 13 * q^82 - 12 * q^83 - 32 * q^84 - 28 * q^85 - 43 * q^87 + 23 * q^88 - 51 * q^89 + 10 * q^90 + 20 * q^91 + 18 * q^92 + 30 * q^93 + 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 + 5 * q^98 + 38 * 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 $

$2$	−2.31259	−1.63525	−0.817625	−	0.575751i	$-0.804710\pi$
$2$	−2.31259	−1.63525	−0.817625	+	0.575751i	$0.804710\pi$
$3$	−2.69989	−1.55878	−0.779391	−	0.626538i	$-0.784471\pi$
$3$	−2.69989	−1.55878	−0.779391	+	0.626538i	$0.784471\pi$
$4$	3.34809	1.67404
$5$	−2.53973	−1.13580	−0.567902	−	0.823096i	$-0.692245\pi$
$5$	−2.53973	−1.13580	−0.567902	+	0.823096i	$0.692245\pi$
$6$	6.24374	2.54900
$7$	2.18729	0.826718	0.413359	−	0.910568i	$-0.364355\pi$
$7$	2.18729	0.826718	0.413359	+	0.910568i	$0.364355\pi$
$8$	−3.11758	−1.10223
$9$	4.28939	1.42980
$10$	5.87337	1.85732
$11$	−3.12200	−0.941318	−0.470659	−	0.882315i	$-0.655984\pi$
$11$	−3.12200	−0.941318	−0.470659	+	0.882315i	$0.655984\pi$
$12$	−9.03946	−2.60947
$13$	3.44204	0.954651	0.477325	−	0.878727i	$-0.341606\pi$
$13$	3.44204	0.954651	0.477325	+	0.878727i	$0.341606\pi$
$14$	−5.05832	−1.35189
$15$	6.85700	1.77047
$16$	0.513513	0.128378
$17$	−2.33435	−0.566163	−0.283081	−	0.959096i	$-0.591357\pi$
$17$	−2.33435	−0.566163	−0.283081	+	0.959096i	$0.591357\pi$
$18$	−9.91963	−2.33808
$19$	−3.38242	−0.775979	−0.387990	−	0.921664i	$-0.626830\pi$
$19$	−3.38242	−0.775979	−0.387990	+	0.921664i	$0.626830\pi$
$20$	−8.50325	−1.90138
$21$	−5.90544	−1.28867
$22$	7.21991	1.53929
$23$	−6.58244	−1.37253	−0.686267	−	0.727350i	$-0.740752\pi$
$23$	−6.58244	−1.37253	−0.686267	+	0.727350i	$0.740752\pi$
$24$	8.41711	1.71814
$25$	1.45025	0.290049
$26$	−7.96004	−1.56109
$27$	−3.48122	−0.669961
$28$	7.32324	1.38396
$29$	5.41020	1.00465	0.502325	−	0.864679i	$-0.332478\pi$
$29$	5.41020	1.00465	0.502325	+	0.864679i	$0.332478\pi$
$30$	−15.8574	−2.89516
$31$	2.13081	0.382705	0.191353	−	0.981521i	$-0.438713\pi$
$31$	2.13081	0.382705	0.191353	+	0.981521i	$0.438713\pi$
$32$	5.04761	0.892299
$33$	8.42904	1.46731
$34$	5.39840	0.925818
$35$	−5.55514	−0.938990
$36$	14.3613	2.39354
$37$	3.47929	0.571991	0.285996	−	0.958231i	$-0.407676\pi$
$37$	3.47929	0.571991	0.285996	+	0.958231i	$0.407676\pi$
$38$	7.82215	1.26892
$39$	−9.29313	−1.48809
$40$	7.91782	1.25192
$41$	2.13473	0.333388	0.166694	−	0.986009i	$-0.446691\pi$
$41$	2.13473	0.333388	0.166694	+	0.986009i	$0.446691\pi$
$42$	13.6569	2.10730
$43$	0	0
$43$	0	0
$44$	−10.4527	−1.57581
$45$	−10.8939	−1.62397
$46$	15.2225	2.24444
$47$	−1.16939	−0.170573	−0.0852867	−	0.996356i	$-0.527181\pi$
$47$	−1.16939	−0.170573	−0.0852867	+	0.996356i	$0.527181\pi$
$48$	−1.38643	−0.200114
$49$	−2.21576	−0.316537
$50$	−3.35383	−0.474304
$51$	6.30248	0.882523
$52$	11.5243	1.59813
$53$	10.4019	1.42880	0.714402	−	0.699735i	$-0.246699\pi$
$53$	10.4019	1.42880	0.714402	+	0.699735i	$0.246699\pi$
$54$	8.05065	1.09555
$55$	7.92904	1.06915
$56$	−6.81905	−0.911234
$57$	9.13214	1.20958
$58$	−12.5116	−1.64285
$59$	6.14435	0.799926	0.399963	−	0.916531i	$-0.369023\pi$
$59$	6.14435	0.799926	0.399963	+	0.916531i	$0.369023\pi$
$60$	22.9578	2.96384
$61$	2.50141	0.320272	0.160136	−	0.987095i	$-0.448807\pi$
$61$	2.50141	0.320272	0.160136	+	0.987095i	$0.448807\pi$
$62$	−4.92770	−0.625819
$63$	9.38216	1.18204
$64$	−12.7001	−1.58751
$65$	−8.74187	−1.08430
$66$	−19.4929	−2.39942
$67$	10.3393	1.26315	0.631575	−	0.775315i	$-0.282409\pi$
$67$	10.3393	1.26315	0.631575	+	0.775315i	$0.282409\pi$
$68$	−7.81560	−0.947781
$69$	17.7718	2.13948
$70$	12.8468	1.53548
$71$	−11.3311	−1.34476	−0.672378	−	0.740208i	$-0.734727\pi$
$71$	−11.3311	−1.34476	−0.672378	+	0.740208i	$0.734727\pi$
$72$	−13.3725	−1.57597
$73$	8.36206	0.978705	0.489352	−	0.872086i	$-0.337233\pi$
$73$	8.36206	0.978705	0.489352	+	0.872086i	$0.337233\pi$
$74$	−8.04617	−0.935349
$75$	−3.91551	−0.452124
$76$	−11.3246	−1.29902
$77$	−6.82872	−0.778205
$78$	21.4912	2.43340
$79$	9.81573	1.10436	0.552178	−	0.833726i	$-0.313797\pi$
$79$	9.81573	1.10436	0.552178	+	0.833726i	$0.313797\pi$
$80$	−1.30419	−0.145812
$81$	−3.46928	−0.385475
$82$	−4.93675	−0.545173
$83$	2.86913	0.314928	0.157464	−	0.987525i	$-0.449668\pi$
$83$	2.86913	0.314928	0.157464	+	0.987525i	$0.449668\pi$
$84$	−19.7719	−2.15729
$85$	5.92862	0.643049
$86$	0	0
$87$	−14.6069	−1.56603
$88$	9.73307	1.03755
$89$	−14.9948	−1.58944	−0.794722	−	0.606973i	$-0.792383\pi$
$89$	−14.9948	−1.58944	−0.794722	+	0.606973i	$0.792383\pi$
$90$	25.1932	2.65560
$91$	7.52875	0.789227
$92$	−22.0386	−2.29768
$93$	−5.75295	−0.596553
$94$	2.70433	0.278930
$95$	8.59043	0.881360
$96$	−13.6280	−1.39090
$97$	−4.22212	−0.428691	−0.214346	−	0.976758i	$-0.568762\pi$
$97$	−4.22212	−0.428691	−0.214346	+	0.976758i	$0.568762\pi$
$98$	5.12414	0.517617
$99$	−13.3915	−1.34589
$100$	4.85555	0.485555
$101$	0.0929930	0.00925315	0.00462658	−	0.999989i	$-0.498527\pi$
$101$	0.0929930	0.00925315	0.00462658	+	0.999989i	$0.498527\pi$
$102$	−14.5751	−1.44315
$103$	−14.3782	−1.41672	−0.708361	−	0.705851i	$-0.750565\pi$
$103$	−14.3782	−1.41672	−0.708361	+	0.705851i	$0.750565\pi$
$104$	−10.7308	−1.05224
$105$	14.9982	1.46368
$106$	−24.0553	−2.33645
$107$	12.6274	1.22074	0.610369	−	0.792117i	$-0.291021\pi$
$107$	12.6274	1.22074	0.610369	+	0.792117i	$0.291021\pi$
$108$	−11.6554	−1.12154
$109$	14.4274	1.38189	0.690946	−	0.722906i	$-0.257194\pi$
$109$	14.4274	1.38189	0.690946	+	0.722906i	$0.257194\pi$
$110$	−18.3367	−1.74833
$111$	−9.39368	−0.891609
$112$	1.12320	0.106133
$113$	12.2090	1.14853	0.574265	−	0.818670i	$-0.305288\pi$
$113$	12.2090	1.14853	0.574265	+	0.818670i	$0.305288\pi$
$114$	−21.1189	−1.97797
$115$	16.7176	1.55893
$116$	18.1138	1.68183
$117$	14.7643	1.36496
$118$	−14.2094	−1.30808
$119$	−5.10590	−0.468057
$120$	−21.3772	−1.95146
$121$	−1.25313	−0.113921
$122$	−5.78474	−0.523725
$123$	−5.76352	−0.519679
$124$	7.13414	0.640665
$125$	9.01543	0.806364
$126$	−21.6971	−1.93293
$127$	6.77907	0.601545	0.300773	−	0.953696i	$-0.402755\pi$
$127$	6.77907	0.601545	0.300773	+	0.953696i	$0.402755\pi$
$128$	19.2749	1.70368
$129$	0	0
$130$	20.2164	1.77309
$131$	−2.62055	−0.228958	−0.114479	−	0.993426i	$-0.536520\pi$
$131$	−2.62055	−0.228958	−0.114479	+	0.993426i	$0.536520\pi$
$132$	28.2212	2.45634
$133$	−7.39833	−0.641516
$134$	−23.9107	−2.06557
$135$	8.84138	0.760944
$136$	7.27751	0.624041
$137$	3.76383	0.321566	0.160783	−	0.986990i	$-0.448598\pi$
$137$	3.76383	0.321566	0.160783	+	0.986990i	$0.448598\pi$
$138$	−41.0990	−3.49858
$139$	−18.3182	−1.55373	−0.776863	−	0.629670i	$-0.783190\pi$
$139$	−18.3182	−1.55373	−0.776863	+	0.629670i	$0.783190\pi$
$140$	−18.5991	−1.57191
$141$	3.15723	0.265887
$142$	26.2043	2.19901
$143$	−10.7460	−0.898630
$144$	2.20266	0.183555
$145$	−13.7405	−1.14108
$146$	−19.3380	−1.60043
$147$	5.98229	0.493411
$148$	11.6490	0.957538
$149$	−12.2136	−1.00058	−0.500288	−	0.865859i	$-0.666773\pi$
$149$	−12.2136	−1.00058	−0.500288	+	0.865859i	$0.666773\pi$
$150$	9.05497	0.739335
$151$	−8.48724	−0.690681	−0.345341	−	0.938477i	$-0.612237\pi$
$151$	−8.48724	−0.690681	−0.345341	+	0.938477i	$0.612237\pi$
$152$	10.5449	0.855308
$153$	−10.0129	−0.809498
$154$	15.7920	1.27256
$155$	−5.41169	−0.434678
$156$	−31.1142	−2.49113
$157$	19.9629	1.59321	0.796605	−	0.604500i	$-0.206627\pi$
$157$	19.9629	1.59321	0.796605	+	0.604500i	$0.206627\pi$
$158$	−22.6998	−1.80590
$159$	−28.0838	−2.22719
$160$	−12.8196	−1.01348
$161$	−14.3977	−1.13470
$162$	8.02303	0.630348
$163$	−15.4322	−1.20874	−0.604371	−	0.796703i	$-0.706576\pi$
$163$	−15.4322	−1.20874	−0.604371	+	0.796703i	$0.706576\pi$
$164$	7.14725	0.558106
$165$	−21.4075	−1.66657
$166$	−6.63514	−0.514986
$167$	14.5053	1.12246	0.561228	−	0.827661i	$-0.310329\pi$
$167$	14.5053	1.12246	0.561228	+	0.827661i	$0.310329\pi$
$168$	18.4107	1.42041
$169$	−1.15235	−0.0886420
$170$	−13.7105	−1.05155
$171$	−14.5085	−1.10949
$172$	0	0
$173$	−3.57561	−0.271849	−0.135924	−	0.990719i	$-0.543400\pi$
$173$	−3.57561	−0.271849	−0.135924	+	0.990719i	$0.543400\pi$
$174$	33.7799	2.56085
$175$	3.17211	0.239789
$176$	−1.60319	−0.120845
$177$	−16.5890	−1.24691
$178$	34.6768	2.59914
$179$	2.36482	0.176755	0.0883776	−	0.996087i	$-0.471832\pi$
$179$	2.36482	0.176755	0.0883776	+	0.996087i	$0.471832\pi$
$180$	−36.4738	−2.71860
$181$	−13.0290	−0.968436	−0.484218	−	0.874947i	$-0.660896\pi$
$181$	−13.0290	−0.968436	−0.484218	+	0.874947i	$0.660896\pi$
$182$	−17.4109	−1.29058
$183$	−6.75352	−0.499234
$184$	20.5213	1.51285
$185$	−8.83646	−0.649670
$186$	13.3042	0.975514
$187$	7.28783	0.532939
$188$	−3.91523	−0.285547
$189$	−7.61445	−0.553869
$190$	−19.8662	−1.44124
$191$	3.73795	0.270469	0.135234	−	0.990814i	$-0.456821\pi$
$191$	3.73795	0.270469	0.135234	+	0.990814i	$0.456821\pi$
$192$	34.2888	2.47458
$193$	−16.0669	−1.15652	−0.578260	−	0.815853i	$-0.696268\pi$
$193$	−16.0669	−1.15652	−0.578260	+	0.815853i	$0.696268\pi$
$194$	9.76404	0.701017
$195$	23.6021	1.69018
$196$	−7.41854	−0.529896
$197$	−20.2217	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$197$	−20.2217	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$198$	30.9690	2.20087
$199$	−9.28127	−0.657932	−0.328966	−	0.944342i	$-0.606700\pi$
$199$	−9.28127	−0.657932	−0.328966	+	0.944342i	$0.606700\pi$
$200$	−4.52126	−0.319701
$201$	−27.9150	−1.96897
$202$	−0.215055	−0.0151312
$203$	11.8337	0.830562
$204$	21.1012	1.47738
$205$	−5.42164	−0.378664
$206$	33.2508	2.31669
$207$	−28.2347	−1.96245
$208$	1.76753	0.122556
$209$	10.5599	0.730443
$210$	−34.6848	−2.39348
$211$	−10.0993	−0.695262	−0.347631	−	0.937631i	$-0.613014\pi$
$211$	−10.0993	−0.695262	−0.347631	+	0.937631i	$0.613014\pi$
$212$	34.8263	2.39188
$213$	30.5927	2.09618
$214$	−29.2021	−1.99621
$215$	0	0
$216$	10.8530	0.738451
$217$	4.66071	0.316389
$218$	−33.3647	−2.25974
$219$	−22.5766	−1.52559
$220$	26.5471	1.78981
$221$	−8.03492	−0.540488
$222$	21.7238	1.45800
$223$	−10.7183	−0.717753	−0.358876	−	0.933385i	$-0.616840\pi$
$223$	−10.7183	−0.717753	−0.358876	+	0.933385i	$0.616840\pi$
$224$	11.0406	0.737680
$225$	6.22068	0.414712
$226$	−28.2345	−1.87813
$227$	−16.9907	−1.12771	−0.563856	−	0.825873i	$-0.690683\pi$
$227$	−16.9907	−1.12771	−0.563856	+	0.825873i	$0.690683\pi$
$228$	30.5752	2.02489
$229$	−1.16971	−0.0772965	−0.0386482	−	0.999253i	$-0.512305\pi$
$229$	−1.16971	−0.0772965	−0.0386482	+	0.999253i	$0.512305\pi$
$230$	−38.6611	−2.54924
$231$	18.4368	1.21305
$232$	−16.8667	−1.10735
$233$	18.6761	1.22351	0.611754	−	0.791048i	$-0.290464\pi$
$233$	18.6761	1.22351	0.611754	+	0.791048i	$0.290464\pi$
$234$	−34.1438	−2.23205
$235$	2.96994	0.193738
$236$	20.5718	1.33911
$237$	−26.5014	−1.72145
$238$	11.8079	0.765390
$239$	15.8598	1.02589	0.512944	−	0.858422i	$-0.328555\pi$
$239$	15.8598	1.02589	0.512944	+	0.858422i	$0.328555\pi$
$240$	3.52116	0.227290
$241$	−3.07702	−0.198209	−0.0991043	−	0.995077i	$-0.531598\pi$
$241$	−3.07702	−0.198209	−0.0991043	+	0.995077i	$0.531598\pi$
$242$	2.89798	0.186289
$243$	19.8103	1.27083
$244$	8.37493	0.536150
$245$	5.62743	0.359523
$246$	13.3287	0.849806
$247$	−11.6424	−0.740789
$248$	−6.64297	−0.421829
$249$	−7.74634	−0.490904
$250$	−20.8490	−1.31861
$251$	−11.6194	−0.733412	−0.366706	−	0.930337i	$-0.619515\pi$
$251$	−11.6194	−0.733412	−0.366706	+	0.930337i	$0.619515\pi$
$252$	31.4123	1.97879
$253$	20.5504	1.29199
$254$	−15.6772	−0.983677
$255$	−16.0066	−1.00237
$256$	−19.1749	−1.19843
$257$	−24.8591	−1.55067	−0.775335	−	0.631550i	$-0.782419\pi$
$257$	−24.8591	−1.55067	−0.775335	+	0.631550i	$0.782419\pi$
$258$	0	0
$259$	7.61021	0.472876
$260$	−29.2685	−1.81516
$261$	23.2065	1.43645
$262$	6.06026	0.374404
$263$	3.69195	0.227655	0.113828	−	0.993501i	$-0.463689\pi$
$263$	3.69195	0.227655	0.113828	+	0.993501i	$0.463689\pi$
$264$	−26.2782	−1.61731
$265$	−26.4179	−1.62284
$266$	17.1093	1.04904
$267$	40.4842	2.47760
$268$	34.6170	2.11457
$269$	22.2263	1.35516	0.677581	−	0.735449i	$-0.263029\pi$
$269$	22.2263	1.35516	0.677581	+	0.735449i	$0.263029\pi$
$270$	−20.4465	−1.24433
$271$	7.29626	0.443216	0.221608	−	0.975136i	$-0.428869\pi$
$271$	7.29626	0.443216	0.221608	+	0.975136i	$0.428869\pi$
$272$	−1.19872	−0.0726830
$273$	−20.3268	−1.23023
$274$	−8.70421	−0.525841
$275$	−4.52767	−0.273029
$276$	59.5017	3.58158
$277$	7.37520	0.443133	0.221566	−	0.975145i	$-0.428883\pi$
$277$	7.37520	0.443133	0.221566	+	0.975145i	$0.428883\pi$
$278$	42.3624	2.54073
$279$	9.13989	0.547191
$280$	17.3186	1.03498
$281$	−19.1235	−1.14081	−0.570407	−	0.821362i	$-0.693215\pi$
$281$	−19.1235	−1.14081	−0.570407	+	0.821362i	$0.693215\pi$
$282$	−7.30138	−0.434791
$283$	19.7995	1.17696	0.588480	−	0.808512i	$-0.299727\pi$
$283$	19.7995	1.17696	0.588480	+	0.808512i	$0.299727\pi$
$284$	−37.9375	−2.25118
$285$	−23.1932	−1.37385
$286$	24.8512	1.46948
$287$	4.66927	0.275618
$288$	21.6512	1.27581
$289$	−11.5508	−0.679460
$290$	31.7761	1.86596
$291$	11.3992	0.668235
$292$	27.9969	1.63839
$293$	10.9778	0.641329	0.320664	−	0.947193i	$-0.396094\pi$
$293$	10.9778	0.641329	0.320664	+	0.947193i	$0.396094\pi$
$294$	−13.8346	−0.806851
$295$	−15.6050	−0.908558
$296$	−10.8469	−0.630466
$297$	10.8684	0.630647
$298$	28.2451	1.63619
$299$	−22.6570	−1.31029
$300$	−13.1095	−0.756875
$301$	0	0
$302$	19.6275	1.12944
$303$	−0.251071	−0.0144236
$304$	−1.73691	−0.0996189
$305$	−6.35291	−0.363766
$306$	23.1559	1.32373
$307$	−0.809531	−0.0462024	−0.0231012	−	0.999733i	$-0.507354\pi$
$307$	−0.809531	−0.0462024	−0.0231012	+	0.999733i	$0.507354\pi$
$308$	−22.8631	−1.30275
$309$	38.8194	2.20836
$310$	12.5150	0.710807
$311$	−10.8980	−0.617971	−0.308986	−	0.951067i	$-0.599989\pi$
$311$	−10.8980	−0.617971	−0.308986	+	0.951067i	$0.599989\pi$
$312$	28.9720	1.64022
$313$	16.0839	0.909113	0.454557	−	0.890718i	$-0.349798\pi$
$313$	16.0839	0.909113	0.454557	+	0.890718i	$0.349798\pi$
$314$	−46.1660	−2.60530
$315$	−23.8282	−1.34257
$316$	32.8639	1.84874
$317$	−1.34177	−0.0753611	−0.0376805	−	0.999290i	$-0.511997\pi$
$317$	−1.34177	−0.0753611	−0.0376805	+	0.999290i	$0.511997\pi$
$318$	64.9465	3.64202
$319$	−16.8906	−0.945694
$320$	32.2548	1.80310
$321$	−34.0926	−1.90286
$322$	33.2960	1.85552
$323$	7.89573	0.439330
$324$	−11.6154	−0.645302
$325$	4.99181	0.276896
$326$	35.6884	1.97660
$327$	−38.9523	−2.15407
$328$	−6.65517	−0.367471
$329$	−2.55780	−0.141016
$330$	49.5069	2.72527
$331$	−31.0480	−1.70655	−0.853276	−	0.521459i	$-0.825388\pi$
$331$	−31.0480	−1.70655	−0.853276	+	0.521459i	$0.825388\pi$
$332$	9.60611	0.527204
$333$	14.9240	0.817832
$334$	−33.5449	−1.83550
$335$	−26.2591	−1.43469
$336$	−3.03252	−0.165438
$337$	17.5883	0.958094	0.479047	−	0.877789i	$-0.340982\pi$
$337$	17.5883	0.958094	0.479047	+	0.877789i	$0.340982\pi$
$338$	2.66491	0.144952
$339$	−32.9630	−1.79031
$340$	19.8495	1.07649
$341$	−6.65239	−0.360247
$342$	33.5523	1.81430
$343$	−20.1575	−1.08841
$344$	0	0
$345$	−45.1357	−2.43003
$346$	8.26894	0.444541
$347$	−10.3838	−0.557430	−0.278715	−	0.960374i	$-0.589908\pi$
$347$	−10.3838	−0.557430	−0.278715	+	0.960374i	$0.589908\pi$
$348$	−48.9053	−2.62160
$349$	3.01731	0.161513	0.0807564	−	0.996734i	$-0.474266\pi$
$349$	3.01731	0.161513	0.0807564	+	0.996734i	$0.474266\pi$
$350$	−7.33581	−0.392115
$351$	−11.9825	−0.639579
$352$	−15.7586	−0.839937
$353$	−27.2819	−1.45207	−0.726034	−	0.687659i	$-0.758638\pi$
$353$	−27.2819	−1.45207	−0.726034	+	0.687659i	$0.758638\pi$
$354$	38.3637	2.03901
$355$	28.7780	1.52738
$356$	−50.2039	−2.66080
$357$	13.7854	0.729598
$358$	−5.46887	−0.289039
$359$	−11.0931	−0.585470	−0.292735	−	0.956194i	$-0.594565\pi$
$359$	−11.0931	−0.585470	−0.292735	+	0.956194i	$0.594565\pi$
$360$	33.9626	1.78999
$361$	−7.55927	−0.397856
$362$	30.1307	1.58364
$363$	3.38331	0.177578
$364$	25.2069	1.32120
$365$	−21.2374	−1.11162
$366$	15.6181	0.816373
$367$	−27.3444	−1.42737	−0.713683	−	0.700469i	$-0.752974\pi$
$367$	−27.3444	−1.42737	−0.713683	+	0.700469i	$0.752974\pi$
$368$	−3.38017	−0.176203
$369$	9.15668	0.476678
$370$	20.4351	1.06237
$371$	22.7519	1.18122
$372$	−19.2614	−0.998656
$373$	1.07805	0.0558192	0.0279096	−	0.999610i	$-0.491115\pi$
$373$	1.07805	0.0558192	0.0279096	+	0.999610i	$0.491115\pi$
$374$	−16.8538	−0.871488
$375$	−24.3406	−1.25695
$376$	3.64567	0.188011
$377$	18.6221	0.959089
$378$	17.6091	0.905715
$379$	−3.02350	−0.155307	−0.0776534	−	0.996980i	$-0.524743\pi$
$379$	−3.02350	−0.155307	−0.0776534	+	0.996980i	$0.524743\pi$
$380$	28.7615	1.47543
$381$	−18.3027	−0.937677
$382$	−8.64436	−0.442284
$383$	20.1617	1.03022	0.515108	−	0.857125i	$-0.327752\pi$
$383$	20.1617	1.03022	0.515108	+	0.857125i	$0.327752\pi$
$384$	−52.0401	−2.65566
$385$	17.3431	0.883888
$386$	37.1562	1.89120
$387$	0	0
$388$	−14.1360	−0.717647
$389$	−9.10714	−0.461750	−0.230875	−	0.972983i	$-0.574159\pi$
$389$	−9.10714	−0.461750	−0.230875	+	0.972983i	$0.574159\pi$
$390$	−54.5820	−2.76387
$391$	15.3657	0.777077
$392$	6.90779	0.348896
$393$	7.07518	0.356896
$394$	46.7646	2.35596
$395$	−24.9293	−1.25433
$396$	−44.8358	−2.25309
$397$	−23.8136	−1.19517	−0.597586	−	0.801805i	$-0.703873\pi$
$397$	−23.8136	−1.19517	−0.597586	+	0.801805i	$0.703873\pi$
$398$	21.4638	1.07588
$399$	19.9747	0.999984
$400$	0.744721	0.0372361
$401$	5.89688	0.294476	0.147238	−	0.989101i	$-0.452962\pi$
$401$	5.89688	0.294476	0.147238	+	0.989101i	$0.452962\pi$
$402$	64.5561	3.21977
$403$	7.33434	0.365350
$404$	0.311349	0.0154902
$405$	8.81104	0.437824
$406$	−27.3665	−1.35818
$407$	−10.8623	−0.538425
$408$	−19.6485	−0.972744
$409$	28.0492	1.38694	0.693471	−	0.720485i	$-0.256081\pi$
$409$	28.0492	1.38694	0.693471	+	0.720485i	$0.256081\pi$
$410$	12.5380	0.619210
$411$	−10.1619	−0.501251
$412$	−48.1393	−2.37165
$413$	13.4395	0.661313
$414$	65.2953	3.20909
$415$	−7.28683	−0.357697
$416$	17.3741	0.851834
$417$	49.4570	2.42192
$418$	−24.4207	−1.19446
$419$	−18.6673	−0.911957	−0.455979	−	0.889991i	$-0.650711\pi$
$419$	−18.6673	−0.911957	−0.455979	+	0.889991i	$0.650711\pi$
$420$	50.2154	2.45026
$421$	29.7726	1.45103	0.725514	−	0.688208i	$-0.241602\pi$
$421$	29.7726	1.45103	0.725514	+	0.688208i	$0.241602\pi$
$422$	23.3555	1.13693
$423$	−5.01598	−0.243886
$424$	−32.4286	−1.57487
$425$	−3.38538	−0.164215
$426$	−70.7485	−3.42778
$427$	5.47131	0.264775
$428$	42.2777	2.04357
$429$	29.0131	1.40077
$430$	0	0
$431$	−5.00624	−0.241142	−0.120571	−	0.992705i	$-0.538473\pi$
$431$	−5.00624	−0.241142	−0.120571	+	0.992705i	$0.538473\pi$
$432$	−1.78765	−0.0860085
$433$	−11.6589	−0.560289	−0.280145	−	0.959958i	$-0.590382\pi$
$433$	−11.6589	−0.560289	−0.280145	+	0.959958i	$0.590382\pi$
$434$	−10.7783	−0.517376
$435$	37.0977	1.77870
$436$	48.3041	2.31335
$437$	22.2645	1.06506
$438$	52.2105	2.49472
$439$	−1.56426	−0.0746582	−0.0373291	−	0.999303i	$-0.511885\pi$
$439$	−1.56426	−0.0746582	−0.0373291	+	0.999303i	$0.511885\pi$
$440$	−24.7194	−1.17845
$441$	−9.50425	−0.452583
$442$	18.5815	0.883832
$443$	−11.1007	−0.527409	−0.263704	−	0.964604i	$-0.584944\pi$
$443$	−11.1007	−0.527409	−0.263704	+	0.964604i	$0.584944\pi$
$444$	−31.4509	−1.49259
$445$	38.0828	1.80530
$446$	24.7871	1.17371
$447$	32.9753	1.55968
$448$	−27.7788	−1.31242
$449$	−27.7667	−1.31039	−0.655196	−	0.755459i	$-0.727414\pi$
$449$	−27.7667	−1.31039	−0.655196	+	0.755459i	$0.727414\pi$
$450$	−14.3859	−0.678158
$451$	−6.66461	−0.313824
$452$	40.8769	1.92269
$453$	22.9146	1.07662
$454$	39.2926	1.84409
$455$	−19.1210	−0.896407
$456$	−28.4702	−1.33324
$457$	−26.9832	−1.26222	−0.631111	−	0.775692i	$-0.717401\pi$
$457$	−26.9832	−1.26222	−0.631111	+	0.775692i	$0.717401\pi$
$458$	2.70506	0.126399
$459$	8.12638	0.379307
$460$	55.9721	2.60971
$461$	25.9146	1.20696	0.603482	−	0.797377i	$-0.293780\pi$
$461$	25.9146	1.20696	0.603482	+	0.797377i	$0.293780\pi$
$462$	−42.6368	−1.98364
$463$	10.3116	0.479220	0.239610	−	0.970869i	$-0.422980\pi$
$463$	10.3116	0.479220	0.239610	+	0.970869i	$0.422980\pi$
$464$	2.77821	0.128975
$465$	14.6110	0.677567
$466$	−43.1901	−2.00074
$467$	−6.82555	−0.315849	−0.157924	−	0.987451i	$-0.550480\pi$
$467$	−6.82555	−0.315849	−0.157924	+	0.987451i	$0.550480\pi$
$468$	49.4321	2.28500
$469$	22.6151	1.04427
$470$	−6.86827	−0.316810
$471$	−53.8975	−2.48347
$472$	−19.1555	−0.881702
$473$	0	0
$474$	61.2869	2.81500
$475$	−4.90534	−0.225072
$476$	−17.0950	−0.783548
$477$	44.6177	2.04290
$478$	−36.6773	−1.67758
$479$	14.4974	0.662405	0.331203	−	0.943560i	$-0.392546\pi$
$479$	14.4974	0.662405	0.331203	+	0.943560i	$0.392546\pi$
$480$	34.6114	1.57979
$481$	11.9759	0.546052
$482$	7.11590	0.324121
$483$	38.8722	1.76875
$484$	−4.19559	−0.190709
$485$	10.7231	0.486909
$486$	−45.8132	−2.07813
$487$	−3.07378	−0.139286	−0.0696431	−	0.997572i	$-0.522186\pi$
$487$	−3.07378	−0.139286	−0.0696431	+	0.997572i	$0.522186\pi$
$488$	−7.79833	−0.353014
$489$	41.6652	1.88416
$490$	−13.0140	−0.587911
$491$	1.64163	0.0740855	0.0370428	−	0.999314i	$-0.488206\pi$
$491$	1.64163	0.0740855	0.0370428	+	0.999314i	$0.488206\pi$
$492$	−19.2968	−0.869966
$493$	−12.6293	−0.568795
$494$	26.9242	1.21138
$495$	34.0108	1.52867
$496$	1.09420	0.0491310
$497$	−24.7844	−1.11173
$498$	17.9141	0.802751
$499$	34.1945	1.53076	0.765378	−	0.643581i	$-0.222552\pi$
$499$	34.1945	1.53076	0.765378	+	0.643581i	$0.222552\pi$
$500$	30.1844	1.34989
$501$	−39.1628	−1.74966
$502$	26.8710	1.19931
$503$	−25.1640	−1.12201	−0.561004	−	0.827813i	$-0.689585\pi$
$503$	−25.1640	−1.12201	−0.561004	+	0.827813i	$0.689585\pi$
$504$	−29.2496	−1.30288
$505$	−0.236178	−0.0105098
$506$	−47.5246	−2.11273
$507$	3.11120	0.138173
$508$	22.6969	1.00701
$509$	5.14565	0.228077	0.114038	−	0.993476i	$-0.463621\pi$
$509$	5.14565	0.228077	0.114038	+	0.993476i	$0.463621\pi$
$510$	37.0168	1.63913
$511$	18.2903	0.809113
$512$	5.79385	0.256054
$513$	11.7749	0.519876
$514$	57.4891	2.53573
$515$	36.5167	1.60912
$516$	0	0
$517$	3.65084	0.160564
$518$	−17.5993	−0.773270
$519$	9.65376	0.423753
$520$	27.2535	1.19514
$521$	−23.9214	−1.04801	−0.524007	−	0.851714i	$-0.675563\pi$
$521$	−23.9214	−1.04801	−0.524007	+	0.851714i	$0.675563\pi$
$522$	−53.6672	−2.34895
$523$	−31.0987	−1.35985	−0.679925	−	0.733282i	$-0.737988\pi$
$523$	−31.0987	−1.35985	−0.679925	+	0.733282i	$0.737988\pi$
$524$	−8.77382	−0.383286
$525$	−8.56435	−0.373779
$526$	−8.53798	−0.372274
$527$	−4.97406	−0.216673
$528$	4.32842	0.188371
$529$	20.3285	0.883847
$530$	61.0939	2.65375
$531$	26.3555	1.14373
$532$	−24.7702	−1.07393
$533$	7.34782	0.318269
$534$	−93.6236	−4.05149
$535$	−32.0703	−1.38652
$536$	−32.2336	−1.39228
$537$	−6.38476	−0.275523
$538$	−51.4004	−2.21603
$539$	6.91759	0.297962
$540$	29.6017	1.27385
$541$	39.6765	1.70582	0.852912	−	0.522054i	$-0.174834\pi$
$541$	39.6765	1.70582	0.852912	+	0.522054i	$0.174834\pi$
$542$	−16.8733	−0.724769
$543$	35.1768	1.50958
$544$	−11.7829	−0.505186
$545$	−36.6417	−1.56956
$546$	47.0076	2.01174
$547$	−32.5243	−1.39064	−0.695319	−	0.718701i	$-0.744737\pi$
$547$	−32.5243	−1.39064	−0.695319	+	0.718701i	$0.744737\pi$
$548$	12.6016	0.538315
$549$	10.7295	0.457925
$550$	10.4707	0.446470
$551$	−18.2995	−0.779587
$552$	−55.4051	−2.35820
$553$	21.4699	0.912992
$554$	−17.0558	−0.724633
$555$	23.8575	1.01269
$556$	−61.3308	−2.60100
$557$	3.13152	0.132687	0.0663434	−	0.997797i	$-0.478867\pi$
$557$	3.13152	0.132687	0.0663434	+	0.997797i	$0.478867\pi$
$558$	−21.1369	−0.894794
$559$	0	0
$560$	−2.85264	−0.120546
$561$	−19.6763	−0.830735
$562$	44.2250	1.86552
$563$	−30.2154	−1.27343	−0.636713	−	0.771101i	$-0.719706\pi$
$563$	−30.2154	−1.27343	−0.636713	+	0.771101i	$0.719706\pi$
$564$	10.5707	0.445106
$565$	−31.0077	−1.30450
$566$	−45.7882	−1.92462
$567$	−7.58832	−0.318679
$568$	35.3256	1.48223
$569$	8.25304	0.345986	0.172993	−	0.984923i	$-0.444656\pi$
$569$	8.25304	0.345986	0.172993	+	0.984923i	$0.444656\pi$
$570$	53.6365	2.24658
$571$	−29.0155	−1.21426	−0.607131	−	0.794601i	$-0.707680\pi$
$571$	−29.0155	−1.21426	−0.607131	+	0.794601i	$0.707680\pi$
$572$	−35.9787	−1.50435
$573$	−10.0920	−0.421601
$574$	−10.7981	−0.450705
$575$	−9.54616	−0.398103
$576$	−54.4757	−2.26982
$577$	−16.3515	−0.680724	−0.340362	−	0.940295i	$-0.610550\pi$
$577$	−16.3515	−0.680724	−0.340362	+	0.940295i	$0.610550\pi$
$578$	26.7123	1.11109
$579$	43.3788	1.80276
$580$	−46.0043	−1.91022
$581$	6.27563	0.260357
$582$	−26.3618	−1.09273
$583$	−32.4746	−1.34496
$584$	−26.0694	−1.07876
$585$	−37.4973	−1.55032
$586$	−25.3871	−1.04873
$587$	−24.0047	−0.990781	−0.495390	−	0.868670i	$-0.664975\pi$
$587$	−24.0047	−0.990781	−0.495390	+	0.868670i	$0.664975\pi$
$588$	20.0292	0.825992
$589$	−7.20729	−0.296971
$590$	36.0880	1.48572
$591$	54.5963	2.24579
$592$	1.78666	0.0734312
$593$	41.5922	1.70799	0.853993	−	0.520285i	$-0.174174\pi$
$593$	41.5922	1.70799	0.853993	+	0.520285i	$0.174174\pi$
$594$	−25.1341	−1.03126
$595$	12.9676	0.531621
$596$	−40.8921	−1.67501
$597$	25.0584	1.02557
$598$	52.3965	2.14265
$599$	−16.7776	−0.685512	−0.342756	−	0.939424i	$-0.611360\pi$
$599$	−16.7776	−0.685512	−0.342756	+	0.939424i	$0.611360\pi$
$600$	12.2069	0.498344
$601$	3.26297	0.133099	0.0665496	−	0.997783i	$-0.478801\pi$
$601$	3.26297	0.133099	0.0665496	+	0.997783i	$0.478801\pi$
$602$	0	0
$603$	44.3495	1.80605
$604$	−28.4160	−1.15623
$605$	3.18262	0.129392
$606$	0.580625	0.0235863
$607$	35.6329	1.44630	0.723148	−	0.690693i	$-0.242694\pi$
$607$	35.6329	1.44630	0.723148	+	0.690693i	$0.242694\pi$
$608$	−17.0731	−0.692406
$609$	−31.9496	−1.29466
$610$	14.6917	0.594849
$611$	−4.02510	−0.162838
$612$	−33.5242	−1.35514
$613$	21.7996	0.880476	0.440238	−	0.897881i	$-0.354894\pi$
$613$	21.7996	0.880476	0.440238	+	0.897881i	$0.354894\pi$
$614$	1.87212	0.0755525
$615$	14.6378	0.590254
$616$	21.2891	0.857761
$617$	10.4546	0.420887	0.210443	−	0.977606i	$-0.432509\pi$
$617$	10.4546	0.420887	0.210443	+	0.977606i	$0.432509\pi$
$618$	−89.7735	−3.61122
$619$	−35.5916	−1.43055	−0.715273	−	0.698845i	$-0.753698\pi$
$619$	−35.5916	−1.43055	−0.715273	+	0.698845i	$0.753698\pi$
$620$	−18.1188	−0.727670
$621$	22.9149	0.919544
$622$	25.2027	1.01054
$623$	−32.7980	−1.31402
$624$	−4.77214	−0.191039
$625$	−30.1480	−1.20592
$626$	−37.1954	−1.48663
$627$	−28.5105	−1.13860
$628$	66.8374	2.66710
$629$	−8.12187	−0.323840
$630$	55.1049	2.19543
$631$	−2.50556	−0.0997447	−0.0498724	−	0.998756i	$-0.515881\pi$
$631$	−2.50556	−0.0997447	−0.0498724	+	0.998756i	$0.515881\pi$
$632$	−30.6013	−1.21725
$633$	27.2669	1.08376
$634$	3.10296	0.123234
$635$	−17.2170	−0.683237
$636$	−94.0271	−3.72842
$637$	−7.62673	−0.302182
$638$	39.0612	1.54645
$639$	−48.6036	−1.92273
$640$	−48.9532	−1.93504
$641$	−18.3108	−0.723232	−0.361616	−	0.932327i	$-0.617775\pi$
$641$	−18.3108	−0.723232	−0.361616	+	0.932327i	$0.617775\pi$
$642$	78.8424	3.11166
$643$	−39.3116	−1.55030	−0.775150	−	0.631778i	$-0.782326\pi$
$643$	−39.3116	−1.55030	−0.775150	+	0.631778i	$0.782326\pi$
$644$	−48.2048	−1.89953
$645$	0	0
$646$	−18.2596	−0.718415
$647$	−20.1378	−0.791697	−0.395848	−	0.918316i	$-0.629549\pi$
$647$	−20.1378	−0.791697	−0.395848	+	0.918316i	$0.629549\pi$
$648$	10.8157	0.424882
$649$	−19.1826	−0.752984
$650$	−11.5440	−0.452794
$651$	−12.5834	−0.493182
$652$	−51.6683	−2.02349
$653$	−17.4768	−0.683920	−0.341960	−	0.939714i	$-0.611091\pi$
$653$	−17.4768	−0.683920	−0.341960	+	0.939714i	$0.611091\pi$
$654$	90.0809	3.52244
$655$	6.65549	0.260052
$656$	1.09621	0.0427998
$657$	35.8682	1.39935
$658$	5.91515	0.230597
$659$	22.2409	0.866382	0.433191	−	0.901302i	$-0.357388\pi$
$659$	22.2409	0.866382	0.433191	+	0.901302i	$0.357388\pi$
$660$	−71.6743	−2.78992
$661$	−35.6536	−1.38676	−0.693382	−	0.720570i	$-0.743880\pi$
$661$	−35.6536	−1.38676	−0.693382	+	0.720570i	$0.743880\pi$
$662$	71.8014	2.79064
$663$	21.6934	0.842502
$664$	−8.94474	−0.347123
$665$	18.7898	0.728637
$666$	−34.5132	−1.33736
$667$	−35.6123	−1.37891
$668$	48.5651	1.87904
$669$	28.9383	1.11882
$670$	60.7267	2.34608
$671$	−7.80939	−0.301478
$672$	−29.8084	−1.14988
$673$	50.1296	1.93236	0.966178	−	0.257876i	$-0.0830227\pi$
$673$	50.1296	1.93236	0.966178	+	0.257876i	$0.0830227\pi$
$674$	−40.6745	−1.56672
$675$	−5.04863	−0.194322
$676$	−3.85815	−0.148391
$677$	19.7657	0.759657	0.379829	−	0.925057i	$-0.375983\pi$
$677$	19.7657	0.759657	0.379829	+	0.925057i	$0.375983\pi$
$678$	76.2301	2.92760
$679$	−9.23500	−0.354407
$680$	−18.4829	−0.708788
$681$	45.8730	1.75786
$682$	15.3843	0.589094
$683$	−35.3877	−1.35407	−0.677036	−	0.735949i	$-0.736736\pi$
$683$	−35.3877	−1.35407	−0.677036	+	0.735949i	$0.736736\pi$
$684$	−48.5758	−1.85734
$685$	−9.55913	−0.365236
$686$	46.6162	1.77981
$687$	3.15808	0.120488
$688$	0	0
$689$	35.8036	1.36401
$690$	104.381	3.97370
$691$	−3.97384	−0.151172	−0.0755860	−	0.997139i	$-0.524083\pi$
$691$	−3.97384	−0.151172	−0.0755860	+	0.997139i	$0.524083\pi$
$692$	−11.9715	−0.455087
$693$	−29.2911	−1.11268
$694$	24.0134	0.911537
$695$	46.5232	1.76473
$696$	45.5382	1.72612
$697$	−4.98319	−0.188752
$698$	−6.97781	−0.264114
$699$	−50.4232	−1.90718
$700$	10.6205	0.401418
$701$	46.4746	1.75532	0.877660	−	0.479284i	$-0.159104\pi$
$701$	46.4746	1.75532	0.877660	+	0.479284i	$0.159104\pi$
$702$	27.7107	1.04587
$703$	−11.7684	−0.443853
$704$	39.6496	1.49435
$705$	−8.01852	−0.301995
$706$	63.0919	2.37449
$707$	0.203403	0.00764975
$708$	−55.5416	−2.08738
$709$	−5.22052	−0.196061	−0.0980303	−	0.995183i	$-0.531254\pi$
$709$	−5.22052	−0.196061	−0.0980303	+	0.995183i	$0.531254\pi$
$710$	−66.5518	−2.49765
$711$	42.1036	1.57901
$712$	46.7474	1.75193
$713$	−14.0259	−0.525275
$714$	−31.8799	−1.19308
$715$	27.2921	1.02067
$716$	7.91763	0.295896
$717$	−42.8198	−1.59913
$718$	25.6538	0.957390
$719$	−33.9396	−1.26573	−0.632866	−	0.774261i	$-0.718122\pi$
$719$	−33.9396	−1.26573	−0.632866	+	0.774261i	$0.718122\pi$
$720$	−5.59417	−0.208482
$721$	−31.4492	−1.17123
$722$	17.4815	0.650594
$723$	8.30762	0.308964
$724$	−43.6221	−1.62120
$725$	7.84613	0.291398
$726$	−7.82422	−0.290384
$727$	11.6162	0.430821	0.215411	−	0.976524i	$-0.430891\pi$
$727$	11.6162	0.430821	0.215411	+	0.976524i	$0.430891\pi$
$728$	−23.4715	−0.869910
$729$	−43.0778	−1.59547
$730$	49.1135	1.81777
$731$	0	0
$732$	−22.6114	−0.835740
$733$	−25.0539	−0.925386	−0.462693	−	0.886519i	$-0.653117\pi$
$733$	−25.0539	−0.925386	−0.462693	+	0.886519i	$0.653117\pi$
$734$	63.2365	2.33410
$735$	−15.1934	−0.560418
$736$	−33.2256	−1.22471
$737$	−32.2794	−1.18903
$738$	−21.1757	−0.779488
$739$	−38.1121	−1.40198	−0.700988	−	0.713173i	$-0.747257\pi$
$739$	−38.1121	−1.40198	−0.700988	+	0.713173i	$0.747257\pi$
$740$	−29.5852	−1.08758
$741$	31.4332	1.15473
$742$	−52.6159	−1.93159
$743$	31.8156	1.16720	0.583600	−	0.812041i	$-0.301643\pi$
$743$	31.8156	1.16720	0.583600	+	0.812041i	$0.301643\pi$
$744$	17.9353	0.657539
$745$	31.0193	1.13646
$746$	−2.49309	−0.0912784
$747$	12.3068	0.450284
$748$	24.4003	0.892163
$749$	27.6199	1.00921
$750$	56.2900	2.05542
$751$	−5.66453	−0.206702	−0.103351	−	0.994645i	$-0.532956\pi$
$751$	−5.66453	−0.206702	−0.103351	+	0.994645i	$0.532956\pi$
$752$	−0.600498	−0.0218979
$753$	31.3712	1.14323
$754$	−43.0654	−1.56835
$755$	21.5553	0.784478
$756$	−25.4938	−0.927202
$757$	−34.1137	−1.23988	−0.619941	−	0.784648i	$-0.712843\pi$
$757$	−34.1137	−1.23988	−0.619941	+	0.784648i	$0.712843\pi$
$758$	6.99213	0.253966
$759$	−55.4837	−2.01393
$760$	−26.7813	−0.971461
$761$	−4.91016	−0.177993	−0.0889966	−	0.996032i	$-0.528366\pi$
$761$	−4.91016	−0.177993	−0.0889966	+	0.996032i	$0.528366\pi$
$762$	42.3268	1.53334
$763$	31.5569	1.14244
$764$	12.5150	0.452776
$765$	25.4302	0.919431
$766$	−46.6259	−1.68466
$767$	21.1491	0.763650
$768$	51.7700	1.86809
$769$	−30.4099	−1.09661	−0.548305	−	0.836278i	$-0.684727\pi$
$769$	−30.4099	−1.09661	−0.548305	+	0.836278i	$0.684727\pi$
$770$	−40.1076	−1.44538
$771$	67.1169	2.41715
$772$	−53.7933	−1.93606
$773$	−30.0965	−1.08250	−0.541248	−	0.840863i	$-0.682048\pi$
$773$	−30.0965	−1.08250	−0.541248	+	0.840863i	$0.682048\pi$
$774$	0	0
$775$	3.09020	0.111003
$776$	13.1628	0.472516
$777$	−20.5467	−0.737110
$778$	21.0611	0.755077
$779$	−7.22053	−0.258702
$780$	79.0218	2.82943
$781$	35.3757	1.26584
$782$	−35.5346	−1.27072
$783$	−18.8341	−0.673076
$784$	−1.13782	−0.0406364
$785$	−50.7004	−1.80957
$786$	−16.3620	−0.583614
$787$	−14.4153	−0.513850	−0.256925	−	0.966431i	$-0.582709\pi$
$787$	−14.4153	−0.513850	−0.256925	+	0.966431i	$0.582709\pi$
$788$	−67.7040	−2.41186
$789$	−9.96785	−0.354865
$790$	57.6514	2.05115
$791$	26.7047	0.949511
$792$	41.7490	1.48349
$793$	8.60995	0.305748
$794$	55.0712	1.95440
$795$	71.3255	2.52965
$796$	−31.0745	−1.10141
$797$	33.4165	1.18367	0.591837	−	0.806058i	$-0.298403\pi$
$797$	33.4165	1.18367	0.591837	+	0.806058i	$0.298403\pi$
$798$	−46.1933	−1.63522
$799$	2.72977	0.0965723
$800$	7.32028	0.258811
$801$	−64.3186	−2.27258
$802$	−13.6371	−0.481542
$803$	−26.1063	−0.921272
$804$	−93.4619	−3.29615
$805$	36.5663	1.28879
$806$	−16.9614	−0.597438
$807$	−60.0085	−2.11240
$808$	−0.289913	−0.0101991
$809$	3.68957	0.129719	0.0648593	−	0.997894i	$-0.479340\pi$
$809$	3.68957	0.129719	0.0648593	+	0.997894i	$0.479340\pi$
$810$	−20.3763	−0.715952
$811$	11.1422	0.391257	0.195628	−	0.980678i	$-0.437325\pi$
$811$	11.1422	0.391257	0.195628	+	0.980678i	$0.437325\pi$
$812$	39.6202	1.39040
$813$	−19.6991	−0.690877
$814$	25.1201	0.880460
$815$	39.1936	1.37289
$816$	3.23641	0.113297
$817$	0	0
$818$	−64.8663	−2.26800
$819$	32.2938	1.12844
$820$	−18.1521	−0.633899
$821$	21.8105	0.761190	0.380595	−	0.924742i	$-0.375719\pi$
$821$	21.8105	0.761190	0.380595	+	0.924742i	$0.375719\pi$
$822$	23.5004	0.819671
$823$	−2.33873	−0.0815231	−0.0407616	−	0.999169i	$-0.512978\pi$
$823$	−2.33873	−0.0815231	−0.0407616	+	0.999169i	$0.512978\pi$
$824$	44.8250	1.56155
$825$	12.2242	0.425592
$826$	−31.0800	−1.08141
$827$	−0.335711	−0.0116738	−0.00583691	−	0.999983i	$-0.501858\pi$
$827$	−0.335711	−0.0116738	−0.00583691	+	0.999983i	$0.501858\pi$
$828$	−94.5321	−3.28522
$829$	−48.9122	−1.69879	−0.849395	−	0.527758i	$-0.823033\pi$
$829$	−48.9122	−1.69879	−0.849395	+	0.527758i	$0.823033\pi$
$830$	16.8515	0.584923
$831$	−19.9122	−0.690747
$832$	−43.7142	−1.51552
$833$	5.17235	0.179211
$834$	−114.374	−3.96044
$835$	−36.8397	−1.27489
$836$	35.3554	1.22279
$837$	−7.41783	−0.256398
$838$	43.1699	1.49128
$839$	9.35601	0.323005	0.161503	−	0.986872i	$-0.448366\pi$
$839$	9.35601	0.323005	0.161503	+	0.986872i	$0.448366\pi$
$840$	−46.7582	−1.61331
$841$	0.270269	0.00931962
$842$	−68.8519	−2.37279
$843$	51.6314	1.77828
$844$	−33.8132	−1.16390
$845$	2.92665	0.100680
$846$	11.5999	0.398814
$847$	−2.74096	−0.0941805
$848$	5.34149	0.183427
$849$	−53.4565	−1.83462
$850$	7.82901	0.268533
$851$	−22.9022	−0.785077
$852$	102.427	3.50910
$853$	1.88280	0.0644660	0.0322330	−	0.999480i	$-0.489738\pi$
$853$	1.88280	0.0644660	0.0322330	+	0.999480i	$0.489738\pi$
$854$	−12.6529	−0.432973
$855$	36.8478	1.26017
$856$	−39.3670	−1.34553
$857$	−47.0757	−1.60807	−0.804037	−	0.594579i	$-0.797319\pi$
$857$	−47.0757	−1.60807	−0.804037	+	0.594579i	$0.797319\pi$
$858$	−67.0956	−2.29060
$859$	22.8236	0.778731	0.389365	−	0.921083i	$-0.372694\pi$
$859$	22.8236	0.778731	0.389365	+	0.921083i	$0.372694\pi$
$860$	0	0
$861$	−12.6065	−0.429628
$862$	11.5774	0.394327
$863$	−5.93754	−0.202116	−0.101058	−	0.994881i	$-0.532223\pi$
$863$	−5.93754	−0.202116	−0.101058	+	0.994881i	$0.532223\pi$
$864$	−17.5718	−0.597806
$865$	9.08111	0.308767
$866$	26.9622	0.916213
$867$	31.1859	1.05913
$868$	15.6045	0.529650
$869$	−30.6447	−1.03955
$870$	−85.7919	−2.90862
$871$	35.5884	1.20587
$872$	−44.9785	−1.52316
$873$	−18.1103	−0.612942
$874$	−51.4888	−1.74164
$875$	19.7194	0.666636
$876$	−75.5885	−2.55390
$877$	−0.987134	−0.0333331	−0.0166666	−	0.999861i	$-0.505305\pi$
$877$	−0.987134	−0.0333331	−0.0166666	+	0.999861i	$0.505305\pi$
$878$	3.61751	0.122085
$879$	−29.6388	−0.999691
$880$	4.07167	0.137256
$881$	19.9638	0.672596	0.336298	−	0.941756i	$-0.390825\pi$
$881$	19.9638	0.672596	0.336298	+	0.941756i	$0.390825\pi$
$882$	21.9795	0.740087
$883$	1.27698	0.0429737	0.0214869	−	0.999769i	$-0.493160\pi$
$883$	1.27698	0.0429737	0.0214869	+	0.999769i	$0.493160\pi$
$884$	−26.9016	−0.904800
$885$	42.1318	1.41624
$886$	25.6713	0.862445
$887$	−6.66827	−0.223899	−0.111949	−	0.993714i	$-0.535709\pi$
$887$	−6.66827	−0.223899	−0.111949	+	0.993714i	$0.535709\pi$
$888$	29.2855	0.982758
$889$	14.8278	0.497308
$890$	−88.0700	−2.95211
$891$	10.8311	0.362855
$892$	−35.8859	−1.20155
$893$	3.95537	0.132361
$894$	−76.2585	−2.55047
$895$	−6.00602	−0.200759
$896$	42.1599	1.40846
$897$	61.1714	2.04245
$898$	64.2131	2.14282
$899$	11.5281	0.384484
$900$	20.8274	0.694246
$901$	−24.2815	−0.808936
$902$	15.4125	0.513181
$903$	0	0
$904$	−38.0626	−1.26594
$905$	33.0901	1.09995
$906$	−52.9921	−1.76054
$907$	−6.03846	−0.200504	−0.100252	−	0.994962i	$-0.531965\pi$
$907$	−6.03846	−0.200504	−0.100252	+	0.994962i	$0.531965\pi$
$908$	−56.8863	−1.88784
$909$	0.398884	0.0132301
$910$	44.2191	1.46585
$911$	−43.2321	−1.43234	−0.716172	−	0.697924i	$-0.754108\pi$
$911$	−43.2321	−1.43234	−0.716172	+	0.697924i	$0.754108\pi$
$912$	4.68947	0.155284
$913$	−8.95743	−0.296448
$914$	62.4012	2.06405
$915$	17.1521	0.567032
$916$	−3.91629	−0.129398
$917$	−5.73190	−0.189284
$918$	−18.7930	−0.620262
$919$	−13.6010	−0.448655	−0.224327	−	0.974514i	$-0.572018\pi$
$919$	−13.6010	−0.448655	−0.224327	+	0.974514i	$0.572018\pi$
$920$	−52.1185	−1.71830
$921$	2.18564	0.0720194
$922$	−59.9299	−1.97369
$923$	−39.0022	−1.28377
$924$	61.7279	2.03070
$925$	5.04583	0.165906
$926$	−23.8465	−0.783645
$927$	−61.6736	−2.02563
$928$	27.3086	0.896448
$929$	10.2628	0.336710	0.168355	−	0.985726i	$-0.446155\pi$
$929$	10.2628	0.336710	0.168355	+	0.985726i	$0.446155\pi$
$930$	−33.7892	−1.10799
$931$	7.49461	0.245626
$932$	62.5291	2.04821
$933$	29.4235	0.963282
$934$	15.7847	0.516492
$935$	−18.5091	−0.605314
$936$	−46.0288	−1.50450
$937$	−46.2018	−1.50935	−0.754673	−	0.656101i	$-0.772204\pi$
$937$	−46.2018	−1.50935	−0.754673	+	0.656101i	$0.772204\pi$
$938$	−52.2996	−1.70764
$939$	−43.4246	−1.41711
$940$	9.94363	0.324326
$941$	−23.4844	−0.765568	−0.382784	−	0.923838i	$-0.625035\pi$
$941$	−23.4844	−0.765568	−0.382784	+	0.923838i	$0.625035\pi$
$942$	124.643	4.06109
$943$	−14.0517	−0.457586
$944$	3.15520	0.102693
$945$	19.3387	0.629087
$946$	0	0
$947$	−39.6686	−1.28906	−0.644528	−	0.764580i	$-0.722946\pi$
$947$	−39.6686	−1.28906	−0.644528	+	0.764580i	$0.722946\pi$
$948$	−88.7289	−2.88178
$949$	28.7826	0.934321
$950$	11.3441	0.368050
$951$	3.62262	0.117471
$952$	15.9180	0.515906
$953$	22.5856	0.731619	0.365809	−	0.930690i	$-0.380792\pi$
$953$	22.5856	0.731619	0.365809	+	0.930690i	$0.380792\pi$
$954$	−103.182	−3.34066
$955$	−9.49340	−0.307199
$956$	53.1001	1.71738
$957$	45.6028	1.47413
$958$	−33.5267	−1.08320
$959$	8.23260	0.265845
$960$	−87.0845	−2.81064
$961$	−26.4596	−0.853537
$962$	−27.6953	−0.892931
$963$	54.1640	1.74541
$964$	−10.3021	−0.331810
$965$	40.8056	1.31358
$966$	−89.8956	−2.89234
$967$	43.7263	1.40614	0.703071	−	0.711119i	$-0.251811\pi$
$967$	43.7263	1.40614	0.703071	+	0.711119i	$0.251811\pi$
$968$	3.90673	0.125567
$969$	−21.3176	−0.684820
$970$	−24.7981	−0.796218
$971$	−1.74534	−0.0560106	−0.0280053	−	0.999608i	$-0.508916\pi$
$971$	−1.74534	−0.0560106	−0.0280053	+	0.999608i	$0.508916\pi$
$972$	66.3267	2.12743
$973$	−40.0671	−1.28449
$974$	7.10839	0.227768
$975$	−13.4773	−0.431620
$976$	1.28451	0.0411160
$977$	−31.6577	−1.01282	−0.506409	−	0.862293i	$-0.669027\pi$
$977$	−31.6577	−1.01282	−0.506409	+	0.862293i	$0.669027\pi$
$978$	−96.3546	−3.08108
$979$	46.8137	1.49617
$980$	18.8411	0.601858
$981$	61.8848	1.97583
$982$	−3.79641	−0.121148
$983$	−1.95919	−0.0624886	−0.0312443	−	0.999512i	$-0.509947\pi$
$983$	−1.95919	−0.0624886	−0.0312443	+	0.999512i	$0.509947\pi$
$984$	17.9682	0.572806
$985$	51.3577	1.63639
$986$	29.2064	0.930122
$987$	6.90578	0.219813
$988$	−38.9798	−1.24011
$989$	0	0
$990$	−78.6531	−2.49976
$991$	60.3980	1.91861	0.959303	−	0.282377i	$-0.0911230\pi$
$991$	60.3980	1.91861	0.959303	+	0.282377i	$0.0911230\pi$
$992$	10.7555	0.341487
$993$	83.8261	2.66014
$994$	57.3163	1.81796
$995$	23.5719	0.747281
$996$	−25.9354	−0.821795
$997$	−4.37427	−0.138534	−0.0692672	−	0.997598i	$-0.522066\pi$
$997$	−4.37427	−0.138534	−0.0692672	+	0.997598i	$0.522066\pi$
$998$	−79.0779	−2.50317
$999$	−12.1122	−0.383212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.n.1.3		18
43.14	even	21		43.2.g.a.24.1	yes	36
43.40	even	21		43.2.g.a.9.1	✓	36
43.42	odd	2		1849.2.a.o.1.16		18
129.14	odd	42		387.2.y.c.325.3		36
129.83	odd	42		387.2.y.c.181.3		36
172.83	odd	42		688.2.bg.c.353.1		36
172.143	odd	42		688.2.bg.c.497.1		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.9.1	✓	36	43.40	even	21
43.2.g.a.24.1	yes	36	43.14	even	21
387.2.y.c.181.3		36	129.83	odd	42
387.2.y.c.325.3		36	129.14	odd	42
688.2.bg.c.353.1		36	172.83	odd	42
688.2.bg.c.497.1		36	172.143	odd	42
1849.2.a.n.1.3		18	1.1	even	1	trivial
1849.2.a.o.1.16		18	43.42	odd	2

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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31259 q^{2} -2.69989 q^{3} +3.34809 q^{4} -2.53973 q^{5} +6.24374 q^{6} +2.18729 q^{7} -3.11758 q^{8} +4.28939 q^{9} +O(q^{10})\)	\(q-2.31259 q^{2} -2.69989 q^{3} +3.34809 q^{4} -2.53973 q^{5} +6.24374 q^{6} +2.18729 q^{7} -3.11758 q^{8} +4.28939 q^{9} +5.87337 q^{10} -3.12200 q^{11} -9.03946 q^{12} +3.44204 q^{13} -5.05832 q^{14} +6.85700 q^{15} +0.513513 q^{16} -2.33435 q^{17} -9.91963 q^{18} -3.38242 q^{19} -8.50325 q^{20} -5.90544 q^{21} +7.21991 q^{22} -6.58244 q^{23} +8.41711 q^{24} +1.45025 q^{25} -7.96004 q^{26} -3.48122 q^{27} +7.32324 q^{28} +5.41020 q^{29} -15.8574 q^{30} +2.13081 q^{31} +5.04761 q^{32} +8.42904 q^{33} +5.39840 q^{34} -5.55514 q^{35} +14.3613 q^{36} +3.47929 q^{37} +7.82215 q^{38} -9.29313 q^{39} +7.91782 q^{40} +2.13473 q^{41} +13.6569 q^{42} -10.4527 q^{44} -10.8939 q^{45} +15.2225 q^{46} -1.16939 q^{47} -1.38643 q^{48} -2.21576 q^{49} -3.35383 q^{50} +6.30248 q^{51} +11.5243 q^{52} +10.4019 q^{53} +8.05065 q^{54} +7.92904 q^{55} -6.81905 q^{56} +9.13214 q^{57} -12.5116 q^{58} +6.14435 q^{59} +22.9578 q^{60} +2.50141 q^{61} -4.92770 q^{62} +9.38216 q^{63} -12.7001 q^{64} -8.74187 q^{65} -19.4929 q^{66} +10.3393 q^{67} -7.81560 q^{68} +17.7718 q^{69} +12.8468 q^{70} -11.3311 q^{71} -13.3725 q^{72} +8.36206 q^{73} -8.04617 q^{74} -3.91551 q^{75} -11.3246 q^{76} -6.82872 q^{77} +21.4912 q^{78} +9.81573 q^{79} -1.30419 q^{80} -3.46928 q^{81} -4.93675 q^{82} +2.86913 q^{83} -19.7719 q^{84} +5.92862 q^{85} -14.6069 q^{87} +9.73307 q^{88} -14.9948 q^{89} +25.1932 q^{90} +7.52875 q^{91} -22.0386 q^{92} -5.75295 q^{93} +2.70433 q^{94} +8.59043 q^{95} -13.6280 q^{96} -4.22212 q^{97} +5.12414 q^{98} -13.3915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9}+O(q^{10}) \) 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9	\( 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} - 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} - 25 q^{18} - 31 q^{19} - 25 q^{20} - 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} - 27 q^{26} - 23 q^{27} - 20 q^{28} - 37 q^{29} + 17 q^{30} - 12 q^{31} - 39 q^{32} - 38 q^{33} - 14 q^{34} + 16 q^{35} + 47 q^{36} + 19 q^{37} + 56 q^{38} - 46 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} + 7 q^{44} - 23 q^{45} + 47 q^{46} - q^{47} - 15 q^{48} - 6 q^{49} + 3 q^{50} - 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} - 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} - 28 q^{61} - 33 q^{62} - 26 q^{63} + 10 q^{64} - 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} - 7 q^{69} + 34 q^{70} - 86 q^{71} + 2 q^{72} + 27 q^{73} - 79 q^{74} - 31 q^{75} - 59 q^{76} - 43 q^{77} + 91 q^{78} + 17 q^{79} - 8 q^{80} - 10 q^{81} - 13 q^{82} - 12 q^{83} - 32 q^{84} - 28 q^{85} - 43 q^{87} + 23 q^{88} - 51 q^{89} + 10 q^{90} + 20 q^{91} + 18 q^{92} + 30 q^{93} + 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} + 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 - 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 - 25 * q^18 - 31 * q^19 - 25 * q^20 - 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 - 27 * q^26 - 23 * q^27 - 20 * q^28 - 37 * q^29 + 17 * q^30 - 12 * q^31 - 39 * q^32 - 38 * q^33 - 14 * q^34 + 16 * q^35 + 47 * q^36 + 19 * q^37 + 56 * q^38 - 46 * q^39 + 6 * q^40 - 7 * q^41 + q^42 + 7 * q^44 - 23 * q^45 + 47 * q^46 - q^47 - 15 * q^48 - 6 * q^49 + 3 * q^50 - 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 - 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 - 28 * q^61 - 33 * q^62 - 26 * q^63 + 10 * q^64 - 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 - 7 * q^69 + 34 * q^70 - 86 * q^71 + 2 * q^72 + 27 * q^73 - 79 * q^74 - 31 * q^75 - 59 * q^76 - 43 * q^77 + 91 * q^78 + 17 * q^79 - 8 * q^80 - 10 * q^81 - 13 * q^82 - 12 * q^83 - 32 * q^84 - 28 * q^85 - 43 * q^87 + 23 * q^88 - 51 * q^89 + 10 * q^90 + 20 * q^91 + 18 * q^92 + 30 * q^93 + 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 + 5 * q^98 + 38 * q^99

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