LMFDB - Embedded newform 1875.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.13366265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 $ x^8 - x^7 - 12x^6 + 10x^5 + 41x^4 - 20x^3 - 48x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.69767$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69767 q^{2} +1.00000 q^{3} +5.27745 q^{4} -2.69767 q^{6} -3.56649 q^{7} -8.84149 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.69767 q^{2} +1.00000 q^{3} +5.27745 q^{4} -2.69767 q^{6} -3.56649 q^{7} -8.84149 q^{8} +1.00000 q^{9} -0.0695627 q^{11} +5.27745 q^{12} +4.85751 q^{13} +9.62122 q^{14} +13.2966 q^{16} +3.03430 q^{17} -2.69767 q^{18} +5.05259 q^{19} -3.56649 q^{21} +0.187658 q^{22} -7.43231 q^{23} -8.84149 q^{24} -13.1040 q^{26} +1.00000 q^{27} -18.8220 q^{28} +1.95440 q^{29} -5.63589 q^{31} -18.1868 q^{32} -0.0695627 q^{33} -8.18554 q^{34} +5.27745 q^{36} +6.21419 q^{37} -13.6302 q^{38} +4.85751 q^{39} -5.63577 q^{41} +9.62122 q^{42} +0.244040 q^{43} -0.367114 q^{44} +20.0499 q^{46} -3.23073 q^{47} +13.2966 q^{48} +5.71984 q^{49} +3.03430 q^{51} +25.6353 q^{52} +8.37482 q^{53} -2.69767 q^{54} +31.5331 q^{56} +5.05259 q^{57} -5.27233 q^{58} -1.60530 q^{59} +3.12503 q^{61} +15.2038 q^{62} -3.56649 q^{63} +22.4690 q^{64} +0.187658 q^{66} -2.94777 q^{67} +16.0133 q^{68} -7.43231 q^{69} +7.25925 q^{71} -8.84149 q^{72} +3.69960 q^{73} -16.7638 q^{74} +26.6648 q^{76} +0.248095 q^{77} -13.1040 q^{78} +4.70868 q^{79} +1.00000 q^{81} +15.2035 q^{82} -8.80783 q^{83} -18.8220 q^{84} -0.658341 q^{86} +1.95440 q^{87} +0.615038 q^{88} -3.55155 q^{89} -17.3243 q^{91} -39.2236 q^{92} -5.63589 q^{93} +8.71545 q^{94} -18.1868 q^{96} +8.80534 q^{97} -15.4303 q^{98} -0.0695627 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 9 * q^4 + q^6 + 12 * q^7 + 3 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9} + 12 q^{11} + 9 q^{12} + 14 q^{13} + 16 q^{14} + 15 q^{16} - q^{17} + q^{18} + 16 q^{19} + 12 q^{21} + 18 q^{22} - 4 q^{23} + 3 q^{24} - 34 q^{26} + 8 q^{27} - 21 q^{28} + 2 q^{29} + 13 q^{31} - 18 q^{32} + 12 q^{33} - 37 q^{34} + 9 q^{36} - 8 q^{37} - 24 q^{38} + 14 q^{39} - 12 q^{41} + 16 q^{42} + 20 q^{43} + 47 q^{44} + 33 q^{46} - 15 q^{47} + 15 q^{48} + 30 q^{49} - q^{51} - q^{52} - 4 q^{53} + q^{54} + 60 q^{56} + 16 q^{57} + 2 q^{58} + 14 q^{59} + 10 q^{61} + 4 q^{62} + 12 q^{63} + 41 q^{64} + 18 q^{66} + 19 q^{67} - 33 q^{68} - 4 q^{69} + 21 q^{71} + 3 q^{72} - 19 q^{73} - 9 q^{74} - q^{76} - 11 q^{77} - 34 q^{78} + 10 q^{79} + 8 q^{81} + 24 q^{82} - 27 q^{83} - 21 q^{84} + 42 q^{86} + 2 q^{87} + 53 q^{88} - 9 q^{89} - 12 q^{91} - 63 q^{92} + 13 q^{93} + 14 q^{94} - 18 q^{96} + 24 q^{97} - 24 q^{98} + 12 q^{99}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 9 * q^4 + q^6 + 12 * q^7 + 3 * q^8 + 8 * q^9 + 12 * q^11 + 9 * q^12 + 14 * q^13 + 16 * q^14 + 15 * q^16 - q^17 + q^18 + 16 * q^19 + 12 * q^21 + 18 * q^22 - 4 * q^23 + 3 * q^24 - 34 * q^26 + 8 * q^27 - 21 * q^28 + 2 * q^29 + 13 * q^31 - 18 * q^32 + 12 * q^33 - 37 * q^34 + 9 * q^36 - 8 * q^37 - 24 * q^38 + 14 * q^39 - 12 * q^41 + 16 * q^42 + 20 * q^43 + 47 * q^44 + 33 * q^46 - 15 * q^47 + 15 * q^48 + 30 * q^49 - q^51 - q^52 - 4 * q^53 + q^54 + 60 * q^56 + 16 * q^57 + 2 * q^58 + 14 * q^59 + 10 * q^61 + 4 * q^62 + 12 * q^63 + 41 * q^64 + 18 * q^66 + 19 * q^67 - 33 * q^68 - 4 * q^69 + 21 * q^71 + 3 * q^72 - 19 * q^73 - 9 * q^74 - q^76 - 11 * q^77 - 34 * q^78 + 10 * q^79 + 8 * q^81 + 24 * q^82 - 27 * q^83 - 21 * q^84 + 42 * q^86 + 2 * q^87 + 53 * q^88 - 9 * q^89 - 12 * q^91 - 63 * q^92 + 13 * q^93 + 14 * q^94 - 18 * q^96 + 24 * q^97 - 24 * q^98 + 12 * 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 $

$2$	−2.69767	−1.90754	−0.953772	−	0.300531i	$-0.902836\pi$
$2$	−2.69767	−1.90754	−0.953772	+	0.300531i	$0.902836\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.27745	2.63872
$5$	0	0
$5$	0	0
$6$	−2.69767	−1.10132
$7$	−3.56649	−1.34801	−0.674003	−	0.738729i	$-0.735427\pi$
$7$	−3.56649	−1.34801	−0.674003	+	0.738729i	$0.735427\pi$
$8$	−8.84149	−3.12594
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.0695627	−0.0209740	−0.0104870	−	0.999945i	$-0.503338\pi$
$11$	−0.0695627	−0.0209740	−0.0104870	+	0.999945i	$0.503338\pi$
$12$	5.27745	1.52347
$13$	4.85751	1.34723	0.673616	−	0.739082i	$-0.264740\pi$
$13$	4.85751	1.34723	0.673616	+	0.739082i	$0.264740\pi$
$14$	9.62122	2.57138
$15$	0	0
$16$	13.2966	3.32414
$17$	3.03430	0.735925	0.367963	−	0.929841i	$-0.380055\pi$
$17$	3.03430	0.735925	0.367963	+	0.929841i	$0.380055\pi$
$18$	−2.69767	−0.635848
$19$	5.05259	1.15914	0.579571	−	0.814922i	$-0.303220\pi$
$19$	5.05259	1.15914	0.579571	+	0.814922i	$0.303220\pi$
$20$	0	0
$21$	−3.56649	−0.778272
$22$	0.187658	0.0400087
$23$	−7.43231	−1.54974	−0.774872	−	0.632119i	$-0.782185\pi$
$23$	−7.43231	−1.54974	−0.774872	+	0.632119i	$0.782185\pi$
$24$	−8.84149	−1.80476
$25$	0	0
$26$	−13.1040	−2.56990
$27$	1.00000	0.192450
$28$	−18.8220	−3.55702
$29$	1.95440	0.362923	0.181461	−	0.983398i	$-0.441917\pi$
$29$	1.95440	0.362923	0.181461	+	0.983398i	$0.441917\pi$
$30$	0	0
$31$	−5.63589	−1.01224	−0.506118	−	0.862464i	$-0.668920\pi$
$31$	−5.63589	−1.01224	−0.506118	+	0.862464i	$0.668920\pi$
$32$	−18.1868	−3.21500
$33$	−0.0695627	−0.0121093
$34$	−8.18554	−1.40381
$35$	0	0
$36$	5.27745	0.879575
$37$	6.21419	1.02161	0.510803	−	0.859698i	$-0.329348\pi$
$37$	6.21419	1.02161	0.510803	+	0.859698i	$0.329348\pi$
$38$	−13.6302	−2.21112
$39$	4.85751	0.777824
$40$	0	0
$41$	−5.63577	−0.880159	−0.440079	−	0.897959i	$-0.645050\pi$
$41$	−5.63577	−0.880159	−0.440079	+	0.897959i	$0.645050\pi$
$42$	9.62122	1.48459
$43$	0.244040	0.0372158	0.0186079	−	0.999827i	$-0.494077\pi$
$43$	0.244040	0.0372158	0.0186079	+	0.999827i	$0.494077\pi$
$44$	−0.367114	−0.0553445
$45$	0	0
$46$	20.0499	2.95620
$47$	−3.23073	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$47$	−3.23073	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$48$	13.2966	1.91919
$49$	5.71984	0.817120
$50$	0	0
$51$	3.03430	0.424886
$52$	25.6353	3.55497
$53$	8.37482	1.15037	0.575185	−	0.818023i	$-0.304930\pi$
$53$	8.37482	1.15037	0.575185	+	0.818023i	$0.304930\pi$
$54$	−2.69767	−0.367107
$55$	0	0
$56$	31.5331	4.21378
$57$	5.05259	0.669231
$58$	−5.27233	−0.692291
$59$	−1.60530	−0.208993	−0.104496	−	0.994525i	$-0.533323\pi$
$59$	−1.60530	−0.208993	−0.104496	+	0.994525i	$0.533323\pi$
$60$	0	0
$61$	3.12503	0.400119	0.200060	−	0.979784i	$-0.435886\pi$
$61$	3.12503	0.400119	0.200060	+	0.979784i	$0.435886\pi$
$62$	15.2038	1.93088
$63$	−3.56649	−0.449335
$64$	22.4690	2.80862
$65$	0	0
$66$	0.187658	0.0230991
$67$	−2.94777	−0.360127	−0.180063	−	0.983655i	$-0.557630\pi$
$67$	−2.94777	−0.360127	−0.180063	+	0.983655i	$0.557630\pi$
$68$	16.0133	1.94190
$69$	−7.43231	−0.894745
$70$	0	0
$71$	7.25925	0.861514	0.430757	−	0.902468i	$-0.358247\pi$
$71$	7.25925	0.861514	0.430757	+	0.902468i	$0.358247\pi$
$72$	−8.84149	−1.04198
$73$	3.69960	0.433005	0.216503	−	0.976282i	$-0.430535\pi$
$73$	3.69960	0.433005	0.216503	+	0.976282i	$0.430535\pi$
$74$	−16.7638	−1.94876
$75$	0	0
$76$	26.6648	3.05866
$77$	0.248095	0.0282730
$78$	−13.1040	−1.48373
$79$	4.70868	0.529768	0.264884	−	0.964280i	$-0.414666\pi$
$79$	4.70868	0.529768	0.264884	+	0.964280i	$0.414666\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	15.2035	1.67894
$83$	−8.80783	−0.966785	−0.483393	−	0.875404i	$-0.660596\pi$
$83$	−8.80783	−0.966785	−0.483393	+	0.875404i	$0.660596\pi$
$84$	−18.8220	−2.05364
$85$	0	0
$86$	−0.658341	−0.0709908
$87$	1.95440	0.209534
$88$	0.615038	0.0655633
$89$	−3.55155	−0.376463	−0.188232	−	0.982125i	$-0.560276\pi$
$89$	−3.55155	−0.376463	−0.188232	+	0.982125i	$0.560276\pi$
$90$	0	0
$91$	−17.3243	−1.81608
$92$	−39.2236	−4.08934
$93$	−5.63589	−0.584415
$94$	8.71545	0.898930
$95$	0	0
$96$	−18.1868	−1.85618
$97$	8.80534	0.894047	0.447023	−	0.894522i	$-0.352484\pi$
$97$	8.80534	0.894047	0.447023	+	0.894522i	$0.352484\pi$
$98$	−15.4303	−1.55869
$99$	−0.0695627	−0.00699132
$100$	0	0
$101$	13.0891	1.30241	0.651207	−	0.758900i	$-0.274263\pi$
$101$	13.0891	1.30241	0.651207	+	0.758900i	$0.274263\pi$
$102$	−8.18554	−0.810490
$103$	18.8222	1.85461	0.927303	−	0.374312i	$-0.122121\pi$
$103$	18.8222	1.85461	0.927303	+	0.374312i	$0.122121\pi$
$104$	−42.9476	−4.21136
$105$	0	0
$106$	−22.5925	−2.19438
$107$	−6.15954	−0.595465	−0.297733	−	0.954649i	$-0.596230\pi$
$107$	−6.15954	−0.595465	−0.297733	+	0.954649i	$0.596230\pi$
$108$	5.27745	0.507823
$109$	6.18006	0.591943	0.295971	−	0.955197i	$-0.404357\pi$
$109$	6.18006	0.591943	0.295971	+	0.955197i	$0.404357\pi$
$110$	0	0
$111$	6.21419	0.589824
$112$	−47.4220	−4.48096
$113$	15.9751	1.50281	0.751407	−	0.659839i	$-0.229376\pi$
$113$	15.9751	1.50281	0.751407	+	0.659839i	$0.229376\pi$
$114$	−13.6302	−1.27659
$115$	0	0
$116$	10.3142	0.957653
$117$	4.85751	0.449077
$118$	4.33058	0.398662
$119$	−10.8218	−0.992031
$120$	0	0
$121$	−10.9952	−0.999560
$122$	−8.43031	−0.763245
$123$	−5.63577	−0.508160
$124$	−29.7431	−2.67101
$125$	0	0
$126$	9.62122	0.857127
$127$	19.9990	1.77463	0.887314	−	0.461166i	$-0.152569\pi$
$127$	19.9990	1.77463	0.887314	+	0.461166i	$0.152569\pi$
$128$	−24.2404	−2.14257
$129$	0.244040	0.0214865
$130$	0	0
$131$	9.50363	0.830336	0.415168	−	0.909745i	$-0.363723\pi$
$131$	9.50363	0.830336	0.415168	+	0.909745i	$0.363723\pi$
$132$	−0.367114	−0.0319532
$133$	−18.0200	−1.56253
$134$	7.95211	0.686958
$135$	0	0
$136$	−26.8277	−2.30046
$137$	−12.5608	−1.07314	−0.536571	−	0.843855i	$-0.680281\pi$
$137$	−12.5608	−1.07314	−0.536571	+	0.843855i	$0.680281\pi$
$138$	20.0499	1.70676
$139$	2.55389	0.216618	0.108309	−	0.994117i	$-0.465456\pi$
$139$	2.55389	0.216618	0.108309	+	0.994117i	$0.465456\pi$
$140$	0	0
$141$	−3.23073	−0.272076
$142$	−19.5831	−1.64338
$143$	−0.337902	−0.0282568
$144$	13.2966	1.10805
$145$	0	0
$146$	−9.98031	−0.825976
$147$	5.71984	0.471764
$148$	32.7950	2.69574
$149$	16.3046	1.33572	0.667861	−	0.744286i	$-0.267210\pi$
$149$	16.3046	1.33572	0.667861	+	0.744286i	$0.267210\pi$
$150$	0	0
$151$	−5.24543	−0.426867	−0.213434	−	0.976958i	$-0.568465\pi$
$151$	−5.24543	−0.426867	−0.213434	+	0.976958i	$0.568465\pi$
$152$	−44.6724	−3.62341
$153$	3.03430	0.245308
$154$	−0.669279	−0.0539320
$155$	0	0
$156$	25.6353	2.05246
$157$	8.73858	0.697415	0.348708	−	0.937232i	$-0.386621\pi$
$157$	8.73858	0.697415	0.348708	+	0.937232i	$0.386621\pi$
$158$	−12.7025	−1.01056
$159$	8.37482	0.664167
$160$	0	0
$161$	26.5072	2.08906
$162$	−2.69767	−0.211949
$163$	−20.7968	−1.62893	−0.814467	−	0.580210i	$-0.802971\pi$
$163$	−20.7968	−1.62893	−0.814467	+	0.580210i	$0.802971\pi$
$164$	−29.7425	−2.32250
$165$	0	0
$166$	23.7607	1.84419
$167$	24.9576	1.93128	0.965639	−	0.259889i	$-0.0836860\pi$
$167$	24.9576	1.93128	0.965639	+	0.259889i	$0.0836860\pi$
$168$	31.5331	2.43283
$169$	10.5954	0.815032
$170$	0	0
$171$	5.05259	0.386381
$172$	1.28791	0.0982022
$173$	−6.78592	−0.515924	−0.257962	−	0.966155i	$-0.583051\pi$
$173$	−6.78592	−0.515924	−0.257962	+	0.966155i	$0.583051\pi$
$174$	−5.27233	−0.399694
$175$	0	0
$176$	−0.924945	−0.0697204
$177$	−1.60530	−0.120662
$178$	9.58091	0.718120
$179$	9.73723	0.727795	0.363897	−	0.931439i	$-0.381446\pi$
$179$	9.73723	0.727795	0.363897	+	0.931439i	$0.381446\pi$
$180$	0	0
$181$	−7.24814	−0.538750	−0.269375	−	0.963035i	$-0.586817\pi$
$181$	−7.24814	−0.538750	−0.269375	+	0.963035i	$0.586817\pi$
$182$	46.7352	3.46424
$183$	3.12503	0.231009
$184$	65.7126	4.84440
$185$	0	0
$186$	15.2038	1.11480
$187$	−0.211074	−0.0154353
$188$	−17.0500	−1.24350
$189$	−3.56649	−0.259424
$190$	0	0
$191$	−21.3284	−1.54327	−0.771635	−	0.636065i	$-0.780561\pi$
$191$	−21.3284	−1.54327	−0.771635	+	0.636065i	$0.780561\pi$
$192$	22.4690	1.62156
$193$	24.9564	1.79640	0.898201	−	0.439586i	$-0.144875\pi$
$193$	24.9564	1.79640	0.898201	+	0.439586i	$0.144875\pi$
$194$	−23.7539	−1.70543
$195$	0	0
$196$	30.1862	2.15615
$197$	18.0515	1.28611	0.643056	−	0.765819i	$-0.277666\pi$
$197$	18.0515	1.28611	0.643056	+	0.765819i	$0.277666\pi$
$198$	0.187658	0.0133362
$199$	10.9432	0.775744	0.387872	−	0.921713i	$-0.373210\pi$
$199$	10.9432	0.775744	0.387872	+	0.921713i	$0.373210\pi$
$200$	0	0
$201$	−2.94777	−0.207919
$202$	−35.3101	−2.48441
$203$	−6.97034	−0.489222
$204$	16.0133	1.12116
$205$	0	0
$206$	−50.7762	−3.53774
$207$	−7.43231	−0.516581
$208$	64.5882	4.47838
$209$	−0.351472	−0.0243118
$210$	0	0
$211$	−8.21731	−0.565703	−0.282852	−	0.959164i	$-0.591280\pi$
$211$	−8.21731	−0.565703	−0.282852	+	0.959164i	$0.591280\pi$
$212$	44.1977	3.03551
$213$	7.25925	0.497395
$214$	16.6164	1.13588
$215$	0	0
$216$	−8.84149	−0.601587
$217$	20.1003	1.36450
$218$	−16.6718	−1.12916
$219$	3.69960	0.249996
$220$	0	0
$221$	14.7391	0.991461
$222$	−16.7638	−1.12512
$223$	−1.92221	−0.128721	−0.0643604	−	0.997927i	$-0.520501\pi$
$223$	−1.92221	−0.128721	−0.0643604	+	0.997927i	$0.520501\pi$
$224$	64.8631	4.33385
$225$	0	0
$226$	−43.0957	−2.86668
$227$	7.29101	0.483922	0.241961	−	0.970286i	$-0.422209\pi$
$227$	7.29101	0.483922	0.241961	+	0.970286i	$0.422209\pi$
$228$	26.6648	1.76592
$229$	−17.0476	−1.12653	−0.563267	−	0.826275i	$-0.690456\pi$
$229$	−17.0476	−1.12653	−0.563267	+	0.826275i	$0.690456\pi$
$230$	0	0
$231$	0.248095	0.0163234
$232$	−17.2798	−1.13447
$233$	26.4383	1.73203	0.866015	−	0.500018i	$-0.166673\pi$
$233$	26.4383	1.73203	0.866015	+	0.500018i	$0.166673\pi$
$234$	−13.1040	−0.856634
$235$	0	0
$236$	−8.47190	−0.551474
$237$	4.70868	0.305862
$238$	29.1936	1.89234
$239$	11.7151	0.757784	0.378892	−	0.925441i	$-0.376305\pi$
$239$	11.7151	0.757784	0.378892	+	0.925441i	$0.376305\pi$
$240$	0	0
$241$	−8.71335	−0.561276	−0.280638	−	0.959814i	$-0.590546\pi$
$241$	−8.71335	−0.561276	−0.280638	+	0.959814i	$0.590546\pi$
$242$	29.6614	1.90670
$243$	1.00000	0.0641500
$244$	16.4922	1.05580
$245$	0	0
$246$	15.2035	0.969338
$247$	24.5430	1.56163
$248$	49.8297	3.16419
$249$	−8.80783	−0.558174
$250$	0	0
$251$	−21.3939	−1.35037	−0.675186	−	0.737648i	$-0.735937\pi$
$251$	−21.3939	−1.35037	−0.675186	+	0.737648i	$0.735937\pi$
$252$	−18.8220	−1.18567
$253$	0.517012	0.0325042
$254$	−53.9509	−3.38518
$255$	0	0
$256$	20.4547	1.27842
$257$	−0.433099	−0.0270160	−0.0135080	−	0.999909i	$-0.504300\pi$
$257$	−0.433099	−0.0270160	−0.0135080	+	0.999909i	$0.504300\pi$
$258$	−0.658341	−0.0409865
$259$	−22.1628	−1.37713
$260$	0	0
$261$	1.95440	0.120974
$262$	−25.6377	−1.58390
$263$	21.9239	1.35189	0.675943	−	0.736954i	$-0.263737\pi$
$263$	21.9239	1.35189	0.675943	+	0.736954i	$0.263737\pi$
$264$	0.615038	0.0378530
$265$	0	0
$266$	48.6121	2.98060
$267$	−3.55155	−0.217351
$268$	−15.5567	−0.950276
$269$	25.9465	1.58199	0.790993	−	0.611825i	$-0.209564\pi$
$269$	25.9465	1.58199	0.790993	+	0.611825i	$0.209564\pi$
$270$	0	0
$271$	−4.79789	−0.291451	−0.145725	−	0.989325i	$-0.546552\pi$
$271$	−4.79789	−0.291451	−0.145725	+	0.989325i	$0.546552\pi$
$272$	40.3457	2.44632
$273$	−17.3243	−1.04851
$274$	33.8850	2.04707
$275$	0	0
$276$	−39.2236	−2.36098
$277$	−2.42344	−0.145611	−0.0728053	−	0.997346i	$-0.523195\pi$
$277$	−2.42344	−0.145611	−0.0728053	+	0.997346i	$0.523195\pi$
$278$	−6.88955	−0.413208
$279$	−5.63589	−0.337412
$280$	0	0
$281$	−7.91275	−0.472035	−0.236017	−	0.971749i	$-0.575842\pi$
$281$	−7.91275	−0.472035	−0.236017	+	0.971749i	$0.575842\pi$
$282$	8.71545	0.518997
$283$	−17.0311	−1.01239	−0.506195	−	0.862419i	$-0.668949\pi$
$283$	−17.0311	−1.01239	−0.506195	+	0.862419i	$0.668949\pi$
$284$	38.3103	2.27330
$285$	0	0
$286$	0.911549	0.0539010
$287$	20.0999	1.18646
$288$	−18.1868	−1.07167
$289$	−7.79304	−0.458414
$290$	0	0
$291$	8.80534	0.516178
$292$	19.5244	1.14258
$293$	−6.65379	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$293$	−6.65379	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$294$	−15.4303	−0.899911
$295$	0	0
$296$	−54.9426	−3.19348
$297$	−0.0695627	−0.00403644
$298$	−43.9844	−2.54795
$299$	−36.1025	−2.08786
$300$	0	0
$301$	−0.870367	−0.0501671
$302$	14.1505	0.814268
$303$	13.0891	0.751949
$304$	67.1820	3.85315
$305$	0	0
$306$	−8.18554	−0.467936
$307$	−5.48149	−0.312845	−0.156423	−	0.987690i	$-0.549996\pi$
$307$	−5.48149	−0.312845	−0.156423	+	0.987690i	$0.549996\pi$
$308$	1.30931	0.0746047
$309$	18.8222	1.07076
$310$	0	0
$311$	31.9826	1.81357	0.906783	−	0.421597i	$-0.138530\pi$
$311$	31.9826	1.81357	0.906783	+	0.421597i	$0.138530\pi$
$312$	−42.9476	−2.43143
$313$	0.728573	0.0411814	0.0205907	−	0.999788i	$-0.493445\pi$
$313$	0.728573	0.0411814	0.0205907	+	0.999788i	$0.493445\pi$
$314$	−23.5739	−1.33035
$315$	0	0
$316$	24.8498	1.39791
$317$	−29.0469	−1.63144	−0.815718	−	0.578450i	$-0.803658\pi$
$317$	−29.0469	−1.63144	−0.815718	+	0.578450i	$0.803658\pi$
$318$	−22.5925	−1.26693
$319$	−0.135953	−0.00761193
$320$	0	0
$321$	−6.15954	−0.343792
$322$	−71.5079	−3.98498
$323$	15.3310	0.853042
$324$	5.27745	0.293192
$325$	0	0
$326$	56.1031	3.10726
$327$	6.18006	0.341758
$328$	49.8286	2.75132
$329$	11.5223	0.635248
$330$	0	0
$331$	2.95713	0.162539	0.0812693	−	0.996692i	$-0.474103\pi$
$331$	2.95713	0.162539	0.0812693	+	0.996692i	$0.474103\pi$
$332$	−46.4829	−2.55108
$333$	6.21419	0.340535
$334$	−67.3275	−3.68400
$335$	0	0
$336$	−47.4220	−2.58708
$337$	−26.6236	−1.45028	−0.725139	−	0.688602i	$-0.758225\pi$
$337$	−26.6236	−1.45028	−0.725139	+	0.688602i	$0.758225\pi$
$338$	−28.5830	−1.55471
$339$	15.9751	0.867650
$340$	0	0
$341$	0.392048	0.0212306
$342$	−13.6302	−0.737038
$343$	4.56568	0.246523
$344$	−2.15768	−0.116334
$345$	0	0
$346$	18.3062	0.984148
$347$	12.0229	0.645421	0.322710	−	0.946498i	$-0.395406\pi$
$347$	12.0229	0.645421	0.322710	+	0.946498i	$0.395406\pi$
$348$	10.3142	0.552901
$349$	32.1053	1.71856	0.859279	−	0.511507i	$-0.170913\pi$
$349$	32.1053	1.71856	0.859279	+	0.511507i	$0.170913\pi$
$350$	0	0
$351$	4.85751	0.259275
$352$	1.26512	0.0674314
$353$	10.8691	0.578504	0.289252	−	0.957253i	$-0.406593\pi$
$353$	10.8691	0.578504	0.289252	+	0.957253i	$0.406593\pi$
$354$	4.33058	0.230168
$355$	0	0
$356$	−18.7431	−0.993382
$357$	−10.8218	−0.572750
$358$	−26.2679	−1.38830
$359$	28.8562	1.52297	0.761486	−	0.648182i	$-0.224470\pi$
$359$	28.8562	1.52297	0.761486	+	0.648182i	$0.224470\pi$
$360$	0	0
$361$	6.52862	0.343611
$362$	19.5531	1.02769
$363$	−10.9952	−0.577096
$364$	−91.4279	−4.79212
$365$	0	0
$366$	−8.43031	−0.440659
$367$	6.78347	0.354095	0.177047	−	0.984202i	$-0.443345\pi$
$367$	6.78347	0.354095	0.177047	+	0.984202i	$0.443345\pi$
$368$	−98.8241	−5.15156
$369$	−5.63577	−0.293386
$370$	0	0
$371$	−29.8687	−1.55071
$372$	−29.7431	−1.54211
$373$	−5.09871	−0.264001	−0.132001	−	0.991250i	$-0.542140\pi$
$373$	−5.09871	−0.264001	−0.132001	+	0.991250i	$0.542140\pi$
$374$	0.569409	0.0294434
$375$	0	0
$376$	28.5644	1.47310
$377$	9.49351	0.488941
$378$	9.62122	0.494862
$379$	−0.991089	−0.0509088	−0.0254544	−	0.999676i	$-0.508103\pi$
$379$	−0.991089	−0.0509088	−0.0254544	+	0.999676i	$0.508103\pi$
$380$	0	0
$381$	19.9990	1.02458
$382$	57.5372	2.94386
$383$	−4.25182	−0.217258	−0.108629	−	0.994082i	$-0.534646\pi$
$383$	−4.25182	−0.217258	−0.108629	+	0.994082i	$0.534646\pi$
$384$	−24.2404	−1.23701
$385$	0	0
$386$	−67.3243	−3.42671
$387$	0.244040	0.0124053
$388$	46.4697	2.35914
$389$	9.50264	0.481803	0.240902	−	0.970550i	$-0.422557\pi$
$389$	9.50264	0.481803	0.240902	+	0.970550i	$0.422557\pi$
$390$	0	0
$391$	−22.5518	−1.14049
$392$	−50.5719	−2.55427
$393$	9.50363	0.479395
$394$	−48.6970	−2.45332
$395$	0	0
$396$	−0.367114	−0.0184482
$397$	0.796576	0.0399790	0.0199895	−	0.999800i	$-0.493637\pi$
$397$	0.796576	0.0399790	0.0199895	+	0.999800i	$0.493637\pi$
$398$	−29.5212	−1.47977
$399$	−18.0200	−0.902128
$400$	0	0
$401$	29.0334	1.44986	0.724928	−	0.688824i	$-0.241873\pi$
$401$	29.0334	1.44986	0.724928	+	0.688824i	$0.241873\pi$
$402$	7.95211	0.396615
$403$	−27.3764	−1.36372
$404$	69.0770	3.43671
$405$	0	0
$406$	18.8037	0.933212
$407$	−0.432276	−0.0214271
$408$	−26.8277	−1.32817
$409$	−21.1715	−1.04686	−0.523432	−	0.852067i	$-0.675349\pi$
$409$	−21.1715	−1.04686	−0.523432	+	0.852067i	$0.675349\pi$
$410$	0	0
$411$	−12.5608	−0.619579
$412$	99.3331	4.89379
$413$	5.72529	0.281723
$414$	20.0499	0.985401
$415$	0	0
$416$	−88.3426	−4.33135
$417$	2.55389	0.125064
$418$	0.948156	0.0463758
$419$	−5.61833	−0.274473	−0.137237	−	0.990538i	$-0.543822\pi$
$419$	−5.61833	−0.274473	−0.137237	+	0.990538i	$0.543822\pi$
$420$	0	0
$421$	−8.10655	−0.395089	−0.197545	−	0.980294i	$-0.563297\pi$
$421$	−8.10655	−0.395089	−0.197545	+	0.980294i	$0.563297\pi$
$422$	22.1676	1.07910
$423$	−3.23073	−0.157083
$424$	−74.0459	−3.59599
$425$	0	0
$426$	−19.5831	−0.948803
$427$	−11.1454	−0.539363
$428$	−32.5066	−1.57127
$429$	−0.337902	−0.0163141
$430$	0	0
$431$	13.8047	0.664947	0.332473	−	0.943113i	$-0.392117\pi$
$431$	13.8047	0.664947	0.332473	+	0.943113i	$0.392117\pi$
$432$	13.2966	0.639731
$433$	−39.7463	−1.91009	−0.955043	−	0.296466i	$-0.904192\pi$
$433$	−39.7463	−1.91009	−0.955043	+	0.296466i	$0.904192\pi$
$434$	−54.2242	−2.60284
$435$	0	0
$436$	32.6150	1.56197
$437$	−37.5524	−1.79637
$438$	−9.98031	−0.476878
$439$	17.2705	0.824277	0.412139	−	0.911121i	$-0.364782\pi$
$439$	17.2705	0.824277	0.412139	+	0.911121i	$0.364782\pi$
$440$	0	0
$441$	5.71984	0.272373
$442$	−39.7614	−1.89126
$443$	−26.9182	−1.27892	−0.639461	−	0.768824i	$-0.720842\pi$
$443$	−26.9182	−1.27892	−0.639461	+	0.768824i	$0.720842\pi$
$444$	32.7950	1.55638
$445$	0	0
$446$	5.18550	0.245541
$447$	16.3046	0.771179
$448$	−80.1354	−3.78604
$449$	−0.0865077	−0.00408255	−0.00204128	−	0.999998i	$-0.500650\pi$
$449$	−0.0865077	−0.00408255	−0.00204128	+	0.999998i	$0.500650\pi$
$450$	0	0
$451$	0.392039	0.0184604
$452$	84.3079	3.96551
$453$	−5.24543	−0.246452
$454$	−19.6688	−0.923102
$455$	0	0
$456$	−44.6724	−2.09198
$457$	−29.1547	−1.36380	−0.681899	−	0.731446i	$-0.738846\pi$
$457$	−29.1547	−1.36380	−0.681899	+	0.731446i	$0.738846\pi$
$458$	45.9888	2.14891
$459$	3.03430	0.141629
$460$	0	0
$461$	−9.80698	−0.456757	−0.228378	−	0.973572i	$-0.573342\pi$
$461$	−9.80698	−0.456757	−0.228378	+	0.973572i	$0.573342\pi$
$462$	−0.669279	−0.0311377
$463$	−26.8716	−1.24883	−0.624414	−	0.781093i	$-0.714662\pi$
$463$	−26.8716	−1.24883	−0.624414	+	0.781093i	$0.714662\pi$
$464$	25.9868	1.20641
$465$	0	0
$466$	−71.3219	−3.30392
$467$	4.06672	0.188185	0.0940926	−	0.995563i	$-0.470005\pi$
$467$	4.06672	0.188185	0.0940926	+	0.995563i	$0.470005\pi$
$468$	25.6353	1.18499
$469$	10.5132	0.485453
$470$	0	0
$471$	8.73858	0.402653
$472$	14.1933	0.653298
$473$	−0.0169761	−0.000780562 0
$474$	−12.7025	−0.583444
$475$	0	0
$476$	−57.1114	−2.61770
$477$	8.37482	0.383457
$478$	−31.6034	−1.44551
$479$	−0.636070	−0.0290628	−0.0145314	−	0.999894i	$-0.504626\pi$
$479$	−0.636070	−0.0290628	−0.0145314	+	0.999894i	$0.504626\pi$
$480$	0	0
$481$	30.1855	1.37634
$482$	23.5058	1.07066
$483$	26.5072	1.20612
$484$	−58.0264	−2.63756
$485$	0	0
$486$	−2.69767	−0.122369
$487$	0.591183	0.0267890	0.0133945	−	0.999910i	$-0.495736\pi$
$487$	0.591183	0.0267890	0.0133945	+	0.999910i	$0.495736\pi$
$488$	−27.6299	−1.25075
$489$	−20.7968	−0.940466
$490$	0	0
$491$	−17.4485	−0.787438	−0.393719	−	0.919231i	$-0.628812\pi$
$491$	−17.4485	−0.787438	−0.393719	+	0.919231i	$0.628812\pi$
$492$	−29.7425	−1.34089
$493$	5.93023	0.267084
$494$	−66.2090	−2.97888
$495$	0	0
$496$	−74.9380	−3.36481
$497$	−25.8900	−1.16133
$498$	23.7607	1.06474
$499$	13.5297	0.605673	0.302836	−	0.953043i	$-0.402066\pi$
$499$	13.5297	0.605673	0.302836	+	0.953043i	$0.402066\pi$
$500$	0	0
$501$	24.9576	1.11502
$502$	57.7138	2.57589
$503$	−3.92944	−0.175205	−0.0876025	−	0.996156i	$-0.527921\pi$
$503$	−3.92944	−0.175205	−0.0876025	+	0.996156i	$0.527921\pi$
$504$	31.5331	1.40459
$505$	0	0
$506$	−1.39473	−0.0620033
$507$	10.5954	0.470559
$508$	105.544	4.68275
$509$	1.93748	0.0858773	0.0429386	−	0.999078i	$-0.486328\pi$
$509$	1.93748	0.0858773	0.0429386	+	0.999078i	$0.486328\pi$
$510$	0	0
$511$	−13.1946	−0.583693
$512$	−6.69932	−0.296071
$513$	5.05259	0.223077
$514$	1.16836	0.0515342
$515$	0	0
$516$	1.28791	0.0566971
$517$	0.224738	0.00988397
$518$	59.7881	2.62694
$519$	−6.78592	−0.297869
$520$	0	0
$521$	31.0365	1.35973	0.679866	−	0.733336i	$-0.262038\pi$
$521$	31.0365	1.35973	0.679866	+	0.733336i	$0.262038\pi$
$522$	−5.27233	−0.230764
$523$	45.0692	1.97074	0.985369	−	0.170435i	$-0.0545173\pi$
$523$	45.0692	1.97074	0.985369	+	0.170435i	$0.0545173\pi$
$524$	50.1549	2.19103
$525$	0	0
$526$	−59.1435	−2.57878
$527$	−17.1010	−0.744930
$528$	−0.924945	−0.0402531
$529$	32.2392	1.40170
$530$	0	0
$531$	−1.60530	−0.0696642
$532$	−95.0995	−4.12309
$533$	−27.3758	−1.18578
$534$	9.58091	0.414607
$535$	0	0
$536$	26.0626	1.12573
$537$	9.73723	0.420193
$538$	−69.9953	−3.01771
$539$	−0.397888	−0.0171382
$540$	0	0
$541$	15.7996	0.679276	0.339638	−	0.940556i	$-0.389695\pi$
$541$	15.7996	0.679276	0.339638	+	0.940556i	$0.389695\pi$
$542$	12.9431	0.555956
$543$	−7.24814	−0.311047
$544$	−55.1842	−2.36600
$545$	0	0
$546$	46.7352	2.00008
$547$	−4.25467	−0.181917	−0.0909583	−	0.995855i	$-0.528993\pi$
$547$	−4.25467	−0.181917	−0.0909583	+	0.995855i	$0.528993\pi$
$548$	−66.2890	−2.83173
$549$	3.12503	0.133373
$550$	0	0
$551$	9.87477	0.420679
$552$	65.7126	2.79692
$553$	−16.7934	−0.714130
$554$	6.53766	0.277759
$555$	0	0
$556$	13.4780	0.571594
$557$	−12.2716	−0.519963	−0.259982	−	0.965614i	$-0.583717\pi$
$557$	−12.2716	−0.519963	−0.259982	+	0.965614i	$0.583717\pi$
$558$	15.2038	0.643628
$559$	1.18543	0.0501383
$560$	0	0
$561$	−0.211074	−0.00891155
$562$	21.3460	0.900427
$563$	−28.9344	−1.21944	−0.609721	−	0.792616i	$-0.708718\pi$
$563$	−28.9344	−1.21944	−0.609721	+	0.792616i	$0.708718\pi$
$564$	−17.0500	−0.717934
$565$	0	0
$566$	45.9442	1.93118
$567$	−3.56649	−0.149778
$568$	−64.1825	−2.69304
$569$	−20.8189	−0.872774	−0.436387	−	0.899759i	$-0.643742\pi$
$569$	−20.8189	−0.872774	−0.436387	+	0.899759i	$0.643742\pi$
$570$	0	0
$571$	−26.7522	−1.11954	−0.559771	−	0.828647i	$-0.689111\pi$
$571$	−26.7522	−1.11954	−0.559771	+	0.828647i	$0.689111\pi$
$572$	−1.78326	−0.0745618
$573$	−21.3284	−0.891008
$574$	−54.2230	−2.26322
$575$	0	0
$576$	22.4690	0.936208
$577$	−28.7636	−1.19744	−0.598722	−	0.800957i	$-0.704325\pi$
$577$	−28.7636	−1.19744	−0.598722	+	0.800957i	$0.704325\pi$
$578$	21.0231	0.874446
$579$	24.9564	1.03715
$580$	0	0
$581$	31.4130	1.30323
$582$	−23.7539	−0.984632
$583$	−0.582576	−0.0241278
$584$	−32.7099	−1.35355
$585$	0	0
$586$	17.9498	0.741498
$587$	−19.6227	−0.809916	−0.404958	−	0.914335i	$-0.632714\pi$
$587$	−19.6227	−0.809916	−0.404958	+	0.914335i	$0.632714\pi$
$588$	30.1862	1.24486
$589$	−28.4758	−1.17333
$590$	0	0
$591$	18.0515	0.742538
$592$	82.6273	3.39596
$593$	−20.8486	−0.856150	−0.428075	−	0.903743i	$-0.640808\pi$
$593$	−20.8486	−0.856150	−0.428075	+	0.903743i	$0.640808\pi$
$594$	0.187658	0.00769969
$595$	0	0
$596$	86.0465	3.52460
$597$	10.9432	0.447876
$598$	97.3928	3.98269
$599$	−21.6873	−0.886118	−0.443059	−	0.896492i	$-0.646107\pi$
$599$	−21.6873	−0.886118	−0.443059	+	0.896492i	$0.646107\pi$
$600$	0	0
$601$	−5.34431	−0.217999	−0.108999	−	0.994042i	$-0.534765\pi$
$601$	−5.34431	−0.217999	−0.108999	+	0.994042i	$0.534765\pi$
$602$	2.34797	0.0956960
$603$	−2.94777	−0.120042
$604$	−27.6825	−1.12639
$605$	0	0
$606$	−35.3101	−1.43438
$607$	33.9628	1.37851	0.689254	−	0.724520i	$-0.257939\pi$
$607$	33.9628	1.37851	0.689254	+	0.724520i	$0.257939\pi$
$608$	−91.8904	−3.72665
$609$	−6.97034	−0.282452
$610$	0	0
$611$	−15.6933	−0.634882
$612$	16.0133	0.647301
$613$	9.86615	0.398490	0.199245	−	0.979950i	$-0.436151\pi$
$613$	9.86615	0.398490	0.199245	+	0.979950i	$0.436151\pi$
$614$	14.7873	0.596766
$615$	0	0
$616$	−2.19353	−0.0883797
$617$	−11.1935	−0.450633	−0.225316	−	0.974286i	$-0.572342\pi$
$617$	−11.1935	−0.450633	−0.225316	+	0.974286i	$0.572342\pi$
$618$	−50.7762	−2.04252
$619$	37.0266	1.48822	0.744112	−	0.668054i	$-0.232873\pi$
$619$	37.0266	1.48822	0.744112	+	0.668054i	$0.232873\pi$
$620$	0	0
$621$	−7.43231	−0.298248
$622$	−86.2786	−3.45946
$623$	12.6665	0.507474
$624$	64.5882	2.58560
$625$	0	0
$626$	−1.96545	−0.0785553
$627$	−0.351472	−0.0140364
$628$	46.1174	1.84029
$629$	18.8557	0.751825
$630$	0	0
$631$	−29.7805	−1.18554	−0.592771	−	0.805371i	$-0.701966\pi$
$631$	−29.7805	−1.18554	−0.592771	+	0.805371i	$0.701966\pi$
$632$	−41.6317	−1.65602
$633$	−8.21731	−0.326609
$634$	78.3591	3.11204
$635$	0	0
$636$	44.1977	1.75255
$637$	27.7842	1.10085
$638$	0.366758	0.0145201
$639$	7.25925	0.287171
$640$	0	0
$641$	41.5447	1.64092	0.820459	−	0.571705i	$-0.193718\pi$
$641$	41.5447	1.64092	0.820459	+	0.571705i	$0.193718\pi$
$642$	16.6164	0.655798
$643$	16.6650	0.657202	0.328601	−	0.944469i	$-0.393423\pi$
$643$	16.6650	0.657202	0.328601	+	0.944469i	$0.393423\pi$
$644$	139.891	5.51246
$645$	0	0
$646$	−41.3582	−1.62722
$647$	−36.8673	−1.44940	−0.724702	−	0.689062i	$-0.758023\pi$
$647$	−36.8673	−1.44940	−0.724702	+	0.689062i	$0.758023\pi$
$648$	−8.84149	−0.347326
$649$	0.111669	0.00438340
$650$	0	0
$651$	20.1003	0.787794
$652$	−109.754	−4.29831
$653$	−28.3184	−1.10819	−0.554093	−	0.832455i	$-0.686935\pi$
$653$	−28.3184	−1.10819	−0.554093	+	0.832455i	$0.686935\pi$
$654$	−16.6718	−0.651919
$655$	0	0
$656$	−74.9363	−2.92577
$657$	3.69960	0.144335
$658$	−31.0835	−1.21176
$659$	35.8536	1.39666	0.698329	−	0.715777i	$-0.253927\pi$
$659$	35.8536	1.39666	0.698329	+	0.715777i	$0.253927\pi$
$660$	0	0
$661$	−34.8735	−1.35642	−0.678210	−	0.734868i	$-0.737244\pi$
$661$	−34.8735	−1.35642	−0.678210	+	0.734868i	$0.737244\pi$
$662$	−7.97738	−0.310050
$663$	14.7391	0.572420
$664$	77.8743	3.02211
$665$	0	0
$666$	−16.7638	−0.649586
$667$	−14.5257	−0.562437
$668$	131.712	5.09611
$669$	−1.92221	−0.0743170
$670$	0	0
$671$	−0.217386	−0.00839208
$672$	64.8631	2.50215
$673$	−7.43183	−0.286476	−0.143238	−	0.989688i	$-0.545751\pi$
$673$	−7.43183	−0.286476	−0.143238	+	0.989688i	$0.545751\pi$
$674$	71.8217	2.76647
$675$	0	0
$676$	55.9167	2.15064
$677$	−33.1734	−1.27496	−0.637478	−	0.770468i	$-0.720022\pi$
$677$	−33.1734	−1.27496	−0.637478	+	0.770468i	$0.720022\pi$
$678$	−43.0957	−1.65508
$679$	−31.4041	−1.20518
$680$	0	0
$681$	7.29101	0.279392
$682$	−1.05762	−0.0404983
$683$	46.0779	1.76312	0.881561	−	0.472070i	$-0.156493\pi$
$683$	46.0779	1.76312	0.881561	+	0.472070i	$0.156493\pi$
$684$	26.6648	1.01955
$685$	0	0
$686$	−12.3167	−0.470254
$687$	−17.0476	−0.650405
$688$	3.24490	0.123710
$689$	40.6808	1.54981
$690$	0	0
$691$	40.0004	1.52169	0.760843	−	0.648936i	$-0.224786\pi$
$691$	40.0004	1.52169	0.760843	+	0.648936i	$0.224786\pi$
$692$	−35.8123	−1.36138
$693$	0.248095	0.00942434
$694$	−32.4338	−1.23117
$695$	0	0
$696$	−17.2798	−0.654989
$697$	−17.1006	−0.647731
$698$	−86.6097	−3.27823
$699$	26.4383	0.999988
$700$	0	0
$701$	−28.1775	−1.06425	−0.532124	−	0.846666i	$-0.678606\pi$
$701$	−28.1775	−1.06425	−0.532124	+	0.846666i	$0.678606\pi$
$702$	−13.1040	−0.494578
$703$	31.3977	1.18419
$704$	−1.56300	−0.0589079
$705$	0	0
$706$	−29.3213	−1.10352
$707$	−46.6821	−1.75566
$708$	−8.47190	−0.318393
$709$	−38.6459	−1.45138	−0.725689	−	0.688023i	$-0.758479\pi$
$709$	−38.6459	−1.45138	−0.725689	+	0.688023i	$0.758479\pi$
$710$	0	0
$711$	4.70868	0.176589
$712$	31.4009	1.17680
$713$	41.8877	1.56871
$714$	29.1936	1.09254
$715$	0	0
$716$	51.3877	1.92045
$717$	11.7151	0.437507
$718$	−77.8446	−2.90514
$719$	14.1179	0.526508	0.263254	−	0.964727i	$-0.415204\pi$
$719$	14.1179	0.526508	0.263254	+	0.964727i	$0.415204\pi$
$720$	0	0
$721$	−67.1291	−2.50002
$722$	−17.6121	−0.655454
$723$	−8.71335	−0.324053
$724$	−38.2517	−1.42161
$725$	0	0
$726$	29.6614	1.10084
$727$	−6.18443	−0.229368	−0.114684	−	0.993402i	$-0.536586\pi$
$727$	−6.18443	−0.229368	−0.114684	+	0.993402i	$0.536586\pi$
$728$	153.172	5.67694
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.740491	0.0273880
$732$	16.4922	0.609569
$733$	33.8511	1.25032	0.625159	−	0.780498i	$-0.285034\pi$
$733$	33.8511	1.25032	0.625159	+	0.780498i	$0.285034\pi$
$734$	−18.2996	−0.675451
$735$	0	0
$736$	135.170	4.98243
$737$	0.205055	0.00755329
$738$	15.2035	0.559647
$739$	28.1239	1.03455	0.517277	−	0.855818i	$-0.326946\pi$
$739$	28.1239	1.03455	0.517277	+	0.855818i	$0.326946\pi$
$740$	0	0
$741$	24.5430	0.901609
$742$	80.5760	2.95804
$743$	−22.1429	−0.812346	−0.406173	−	0.913796i	$-0.633137\pi$
$743$	−22.1429	−0.812346	−0.406173	+	0.913796i	$0.633137\pi$
$744$	49.8297	1.82684
$745$	0	0
$746$	13.7547	0.503594
$747$	−8.80783	−0.322262
$748$	−1.11393	−0.0407294
$749$	21.9679	0.802690
$750$	0	0
$751$	−5.75539	−0.210017	−0.105009	−	0.994471i	$-0.533487\pi$
$751$	−5.75539	−0.210017	−0.105009	+	0.994471i	$0.533487\pi$
$752$	−42.9575	−1.56650
$753$	−21.3939	−0.779637
$754$	−25.6104	−0.932676
$755$	0	0
$756$	−18.8220	−0.684548
$757$	−18.4324	−0.669937	−0.334968	−	0.942229i	$-0.608726\pi$
$757$	−18.4324	−0.669937	−0.334968	+	0.942229i	$0.608726\pi$
$758$	2.67364	0.0971108
$759$	0.517012	0.0187663
$760$	0	0
$761$	−25.3526	−0.919031	−0.459515	−	0.888170i	$-0.651977\pi$
$761$	−25.3526	−0.919031	−0.459515	+	0.888170i	$0.651977\pi$
$762$	−53.9509	−1.95443
$763$	−22.0411	−0.797942
$764$	−112.560	−4.07227
$765$	0	0
$766$	11.4700	0.414429
$767$	−7.79777	−0.281561
$768$	20.4547	0.738095
$769$	30.3124	1.09309	0.546546	−	0.837429i	$-0.315942\pi$
$769$	30.3124	1.09309	0.546546	+	0.837429i	$0.315942\pi$
$770$	0	0
$771$	−0.433099	−0.0155977
$772$	131.706	4.74021
$773$	3.26668	0.117494	0.0587471	−	0.998273i	$-0.481289\pi$
$773$	3.26668	0.117494	0.0587471	+	0.998273i	$0.481289\pi$
$774$	−0.658341	−0.0236636
$775$	0	0
$776$	−77.8523	−2.79473
$777$	−22.1628	−0.795087
$778$	−25.6350	−0.919060
$779$	−28.4752	−1.02023
$780$	0	0
$781$	−0.504973	−0.0180694
$782$	60.8375	2.17554
$783$	1.95440	0.0698445
$784$	76.0542	2.71622
$785$	0	0
$786$	−25.6377	−0.914467
$787$	−2.44831	−0.0872728	−0.0436364	−	0.999047i	$-0.513894\pi$
$787$	−2.44831	−0.0872728	−0.0436364	+	0.999047i	$0.513894\pi$
$788$	95.2656	3.39370
$789$	21.9239	0.780511
$790$	0	0
$791$	−56.9751	−2.02580
$792$	0.615038	0.0218544
$793$	15.1799	0.539053
$794$	−2.14890	−0.0762617
$795$	0	0
$796$	57.7523	2.04697
$797$	35.2494	1.24860	0.624299	−	0.781186i	$-0.285385\pi$
$797$	35.2494	1.24860	0.624299	+	0.781186i	$0.285385\pi$
$798$	48.6121	1.72085
$799$	−9.80298	−0.346805
$800$	0	0
$801$	−3.55155	−0.125488
$802$	−78.3226	−2.76567
$803$	−0.257354	−0.00908183
$804$	−15.5567	−0.548642
$805$	0	0
$806$	73.8526	2.60135
$807$	25.9465	0.913361
$808$	−115.727	−4.07126
$809$	−16.2002	−0.569570	−0.284785	−	0.958591i	$-0.591922\pi$
$809$	−16.2002	−0.569570	−0.284785	+	0.958591i	$0.591922\pi$
$810$	0	0
$811$	−47.4540	−1.66633	−0.833167	−	0.553021i	$-0.813475\pi$
$811$	−47.4540	−1.66633	−0.833167	+	0.553021i	$0.813475\pi$
$812$	−36.7856	−1.29092
$813$	−4.79789	−0.168269
$814$	1.16614	0.0408732
$815$	0	0
$816$	40.3457	1.41238
$817$	1.23303	0.0431384
$818$	57.1139	1.99694
$819$	−17.3243	−0.605358
$820$	0	0
$821$	34.4342	1.20176	0.600881	−	0.799338i	$-0.294816\pi$
$821$	34.4342	1.20176	0.600881	+	0.799338i	$0.294816\pi$
$822$	33.8850	1.18187
$823$	33.0865	1.15332	0.576661	−	0.816983i	$-0.304355\pi$
$823$	33.0865	1.15332	0.576661	+	0.816983i	$0.304355\pi$
$824$	−166.416	−5.79738
$825$	0	0
$826$	−15.4450	−0.537399
$827$	−22.4568	−0.780900	−0.390450	−	0.920624i	$-0.627681\pi$
$827$	−22.4568	−0.780900	−0.390450	+	0.920624i	$0.627681\pi$
$828$	−39.2236	−1.36311
$829$	4.99720	0.173560	0.0867800	−	0.996227i	$-0.472342\pi$
$829$	4.99720	0.173560	0.0867800	+	0.996227i	$0.472342\pi$
$830$	0	0
$831$	−2.42344	−0.0840683
$832$	109.143	3.78386
$833$	17.3557	0.601339
$834$	−6.88955	−0.238566
$835$	0	0
$836$	−1.85487	−0.0641521
$837$	−5.63589	−0.194805
$838$	15.1564	0.523570
$839$	−25.8577	−0.892705	−0.446353	−	0.894857i	$-0.647277\pi$
$839$	−25.8577	−0.892705	−0.446353	+	0.894857i	$0.647277\pi$
$840$	0	0
$841$	−25.1803	−0.868287
$842$	21.8688	0.753650
$843$	−7.91275	−0.272529
$844$	−43.3664	−1.49273
$845$	0	0
$846$	8.71545	0.299643
$847$	39.2141	1.34741
$848$	111.356	3.82399
$849$	−17.0311	−0.584504
$850$	0	0
$851$	−46.1857	−1.58323
$852$	38.3103	1.31249
$853$	−46.1376	−1.57972	−0.789860	−	0.613287i	$-0.789847\pi$
$853$	−46.1376	−1.57972	−0.789860	+	0.613287i	$0.789847\pi$
$854$	30.0666	1.02886
$855$	0	0
$856$	54.4595	1.86139
$857$	−17.8220	−0.608787	−0.304393	−	0.952546i	$-0.598454\pi$
$857$	−17.8220	−0.608787	−0.304393	+	0.952546i	$0.598454\pi$
$858$	0.911549	0.0311198
$859$	30.7896	1.05053	0.525263	−	0.850940i	$-0.323967\pi$
$859$	30.7896	1.05053	0.525263	+	0.850940i	$0.323967\pi$
$860$	0	0
$861$	20.0999	0.685003
$862$	−37.2405	−1.26842
$863$	−52.1333	−1.77464	−0.887319	−	0.461155i	$-0.847435\pi$
$863$	−52.1333	−1.77464	−0.887319	+	0.461155i	$0.847435\pi$
$864$	−18.1868	−0.618728
$865$	0	0
$866$	107.223	3.64357
$867$	−7.79304	−0.264666
$868$	106.079	3.60054
$869$	−0.327549	−0.0111113
$870$	0	0
$871$	−14.3188	−0.485174
$872$	−54.6410	−1.85038
$873$	8.80534	0.298016
$874$	101.304	3.42666
$875$	0	0
$876$	19.5244	0.659669
$877$	42.3249	1.42921	0.714605	−	0.699528i	$-0.246606\pi$
$877$	42.3249	1.42921	0.714605	+	0.699528i	$0.246606\pi$
$878$	−46.5903	−1.57234
$879$	−6.65379	−0.224427
$880$	0	0
$881$	6.21376	0.209347	0.104673	−	0.994507i	$-0.466620\pi$
$881$	6.21376	0.209347	0.104673	+	0.994507i	$0.466620\pi$
$882$	−15.4303	−0.519564
$883$	30.9588	1.04185	0.520924	−	0.853603i	$-0.325588\pi$
$883$	30.9588	1.04185	0.520924	+	0.853603i	$0.325588\pi$
$884$	77.7850	2.61619
$885$	0	0
$886$	72.6165	2.43960
$887$	−39.7363	−1.33422	−0.667108	−	0.744961i	$-0.732468\pi$
$887$	−39.7363	−1.33422	−0.667108	+	0.744961i	$0.732468\pi$
$888$	−54.9426	−1.84375
$889$	−71.3263	−2.39221
$890$	0	0
$891$	−0.0695627	−0.00233044
$892$	−10.1444	−0.339659
$893$	−16.3235	−0.546246
$894$	−43.9844	−1.47106
$895$	0	0
$896$	86.4530	2.88819
$897$	−36.1025	−1.20543
$898$	0.233370	0.00778765
$899$	−11.0148	−0.367363
$900$	0	0
$901$	25.4117	0.846586
$902$	−1.05759	−0.0352141
$903$	−0.870367	−0.0289640
$904$	−141.244	−4.69770
$905$	0	0
$906$	14.1505	0.470118
$907$	−17.1801	−0.570454	−0.285227	−	0.958460i	$-0.592069\pi$
$907$	−17.1801	−0.570454	−0.285227	+	0.958460i	$0.592069\pi$
$908$	38.4779	1.27694
$909$	13.0891	0.434138
$910$	0	0
$911$	−10.0621	−0.333371	−0.166685	−	0.986010i	$-0.553306\pi$
$911$	−10.0621	−0.333371	−0.166685	+	0.986010i	$0.553306\pi$
$912$	67.1820	2.22462
$913$	0.612697	0.0202773
$914$	78.6498	2.60150
$915$	0	0
$916$	−89.9676	−2.97261
$917$	−33.8946	−1.11930
$918$	−8.18554	−0.270163
$919$	12.2601	0.404424	0.202212	−	0.979342i	$-0.435187\pi$
$919$	12.2601	0.404424	0.202212	+	0.979342i	$0.435187\pi$
$920$	0	0
$921$	−5.48149	−0.180621
$922$	26.4560	0.871283
$923$	35.2619	1.16066
$924$	1.30931	0.0430730
$925$	0	0
$926$	72.4908	2.38220
$927$	18.8222	0.618202
$928$	−35.5443	−1.16680
$929$	2.29196	0.0751969	0.0375984	−	0.999293i	$-0.488029\pi$
$929$	2.29196	0.0751969	0.0375984	+	0.999293i	$0.488029\pi$
$930$	0	0
$931$	28.9000	0.947158
$932$	139.527	4.57035
$933$	31.9826	1.04706
$934$	−10.9707	−0.358972
$935$	0	0
$936$	−42.9476	−1.40379
$937$	39.3565	1.28572	0.642861	−	0.765983i	$-0.277747\pi$
$937$	39.3565	1.28572	0.642861	+	0.765983i	$0.277747\pi$
$938$	−28.3611	−0.926023
$939$	0.728573	0.0237761
$940$	0	0
$941$	−22.6288	−0.737678	−0.368839	−	0.929493i	$-0.620245\pi$
$941$	−22.6288	−0.737678	−0.368839	+	0.929493i	$0.620245\pi$
$942$	−23.5739	−0.768078
$943$	41.8868	1.36402
$944$	−21.3450	−0.694720
$945$	0	0
$946$	0.0457960	0.00148896
$947$	−44.2268	−1.43718	−0.718589	−	0.695435i	$-0.755212\pi$
$947$	−44.2268	−1.43718	−0.718589	+	0.695435i	$0.755212\pi$
$948$	24.8498	0.807084
$949$	17.9708	0.583358
$950$	0	0
$951$	−29.0469	−0.941910
$952$	95.6807	3.10103
$953$	−32.4366	−1.05072	−0.525362	−	0.850879i	$-0.676070\pi$
$953$	−32.4366	−1.05072	−0.525362	+	0.850879i	$0.676070\pi$
$954$	−22.5925	−0.731461
$955$	0	0
$956$	61.8256	1.99958
$957$	−0.135953	−0.00439475
$958$	1.71591	0.0554385
$959$	44.7980	1.44660
$960$	0	0
$961$	0.763274	0.0246217
$962$	−81.4306	−2.62543
$963$	−6.15954	−0.198488
$964$	−45.9842	−1.48105
$965$	0	0
$966$	−71.5079	−2.30073
$967$	−10.1870	−0.327591	−0.163796	−	0.986494i	$-0.552374\pi$
$967$	−10.1870	−0.327591	−0.163796	+	0.986494i	$0.552374\pi$
$968$	97.2136	3.12456
$969$	15.3310	0.492504
$970$	0	0
$971$	6.22735	0.199845	0.0999225	−	0.994995i	$-0.468141\pi$
$971$	6.22735	0.199845	0.0999225	+	0.994995i	$0.468141\pi$
$972$	5.27745	0.169274
$973$	−9.10840	−0.292002
$974$	−1.59482	−0.0511013
$975$	0	0
$976$	41.5521	1.33005
$977$	−8.75116	−0.279974	−0.139987	−	0.990153i	$-0.544706\pi$
$977$	−8.75116	−0.279974	−0.139987	+	0.990153i	$0.544706\pi$
$978$	56.1031	1.79398
$979$	0.247055	0.00789592
$980$	0	0
$981$	6.18006	0.197314
$982$	47.0703	1.50207
$983$	−0.807527	−0.0257561	−0.0128781	−	0.999917i	$-0.504099\pi$
$983$	−0.807527	−0.0257561	−0.0128781	+	0.999917i	$0.504099\pi$
$984$	49.8286	1.58848
$985$	0	0
$986$	−15.9978	−0.509474
$987$	11.5223	0.366760
$988$	129.524	4.12072
$989$	−1.81378	−0.0576749
$990$	0	0
$991$	6.58244	0.209098	0.104549	−	0.994520i	$-0.466660\pi$
$991$	6.58244	0.209098	0.104549	+	0.994520i	$0.466660\pi$
$992$	102.499	3.25434
$993$	2.95713	0.0938417
$994$	69.8428	2.21528
$995$	0	0
$996$	−46.4829	−1.47287
$997$	−33.4939	−1.06076	−0.530381	−	0.847760i	$-0.677951\pi$
$997$	−33.4939	−1.06076	−0.530381	+	0.847760i	$0.677951\pi$
$998$	−36.4987	−1.15535
$999$	6.21419	0.196608

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.o.1.1	yes	8
3.2	odd	2		5625.2.a.u.1.8		8
5.2	odd	4		1875.2.b.g.1249.1		16
5.3	odd	4		1875.2.b.g.1249.16		16
5.4	even	2		1875.2.a.n.1.8	✓	8
15.14	odd	2		5625.2.a.bc.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.8	✓	8	5.4	even	2
1875.2.a.o.1.1	yes	8	1.1	even	1	trivial
1875.2.b.g.1249.1		16	5.2	odd	4
1875.2.b.g.1249.16		16	5.3	odd	4
5625.2.a.u.1.8		8	3.2	odd	2
5625.2.a.bc.1.1		8	15.14	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69767 q^{2} +1.00000 q^{3} +5.27745 q^{4} -2.69767 q^{6} -3.56649 q^{7} -8.84149 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.69767 q^{2} +1.00000 q^{3} +5.27745 q^{4} -2.69767 q^{6} -3.56649 q^{7} -8.84149 q^{8} +1.00000 q^{9} -0.0695627 q^{11} +5.27745 q^{12} +4.85751 q^{13} +9.62122 q^{14} +13.2966 q^{16} +3.03430 q^{17} -2.69767 q^{18} +5.05259 q^{19} -3.56649 q^{21} +0.187658 q^{22} -7.43231 q^{23} -8.84149 q^{24} -13.1040 q^{26} +1.00000 q^{27} -18.8220 q^{28} +1.95440 q^{29} -5.63589 q^{31} -18.1868 q^{32} -0.0695627 q^{33} -8.18554 q^{34} +5.27745 q^{36} +6.21419 q^{37} -13.6302 q^{38} +4.85751 q^{39} -5.63577 q^{41} +9.62122 q^{42} +0.244040 q^{43} -0.367114 q^{44} +20.0499 q^{46} -3.23073 q^{47} +13.2966 q^{48} +5.71984 q^{49} +3.03430 q^{51} +25.6353 q^{52} +8.37482 q^{53} -2.69767 q^{54} +31.5331 q^{56} +5.05259 q^{57} -5.27233 q^{58} -1.60530 q^{59} +3.12503 q^{61} +15.2038 q^{62} -3.56649 q^{63} +22.4690 q^{64} +0.187658 q^{66} -2.94777 q^{67} +16.0133 q^{68} -7.43231 q^{69} +7.25925 q^{71} -8.84149 q^{72} +3.69960 q^{73} -16.7638 q^{74} +26.6648 q^{76} +0.248095 q^{77} -13.1040 q^{78} +4.70868 q^{79} +1.00000 q^{81} +15.2035 q^{82} -8.80783 q^{83} -18.8220 q^{84} -0.658341 q^{86} +1.95440 q^{87} +0.615038 q^{88} -3.55155 q^{89} -17.3243 q^{91} -39.2236 q^{92} -5.63589 q^{93} +8.71545 q^{94} -18.1868 q^{96} +8.80534 q^{97} -15.4303 q^{98} -0.0695627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 9 * q^4 + q^6 + 12 * q^7 + 3 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9} + 12 q^{11} + 9 q^{12} + 14 q^{13} + 16 q^{14} + 15 q^{16} - q^{17} + q^{18} + 16 q^{19} + 12 q^{21} + 18 q^{22} - 4 q^{23} + 3 q^{24} - 34 q^{26} + 8 q^{27} - 21 q^{28} + 2 q^{29} + 13 q^{31} - 18 q^{32} + 12 q^{33} - 37 q^{34} + 9 q^{36} - 8 q^{37} - 24 q^{38} + 14 q^{39} - 12 q^{41} + 16 q^{42} + 20 q^{43} + 47 q^{44} + 33 q^{46} - 15 q^{47} + 15 q^{48} + 30 q^{49} - q^{51} - q^{52} - 4 q^{53} + q^{54} + 60 q^{56} + 16 q^{57} + 2 q^{58} + 14 q^{59} + 10 q^{61} + 4 q^{62} + 12 q^{63} + 41 q^{64} + 18 q^{66} + 19 q^{67} - 33 q^{68} - 4 q^{69} + 21 q^{71} + 3 q^{72} - 19 q^{73} - 9 q^{74} - q^{76} - 11 q^{77} - 34 q^{78} + 10 q^{79} + 8 q^{81} + 24 q^{82} - 27 q^{83} - 21 q^{84} + 42 q^{86} + 2 q^{87} + 53 q^{88} - 9 q^{89} - 12 q^{91} - 63 q^{92} + 13 q^{93} + 14 q^{94} - 18 q^{96} + 24 q^{97} - 24 q^{98} + 12 q^{99}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 9 * q^4 + q^6 + 12 * q^7 + 3 * q^8 + 8 * q^9 + 12 * q^11 + 9 * q^12 + 14 * q^13 + 16 * q^14 + 15 * q^16 - q^17 + q^18 + 16 * q^19 + 12 * q^21 + 18 * q^22 - 4 * q^23 + 3 * q^24 - 34 * q^26 + 8 * q^27 - 21 * q^28 + 2 * q^29 + 13 * q^31 - 18 * q^32 + 12 * q^33 - 37 * q^34 + 9 * q^36 - 8 * q^37 - 24 * q^38 + 14 * q^39 - 12 * q^41 + 16 * q^42 + 20 * q^43 + 47 * q^44 + 33 * q^46 - 15 * q^47 + 15 * q^48 + 30 * q^49 - q^51 - q^52 - 4 * q^53 + q^54 + 60 * q^56 + 16 * q^57 + 2 * q^58 + 14 * q^59 + 10 * q^61 + 4 * q^62 + 12 * q^63 + 41 * q^64 + 18 * q^66 + 19 * q^67 - 33 * q^68 - 4 * q^69 + 21 * q^71 + 3 * q^72 - 19 * q^73 - 9 * q^74 - q^76 - 11 * q^77 - 34 * q^78 + 10 * q^79 + 8 * q^81 + 24 * q^82 - 27 * q^83 - 21 * q^84 + 42 * q^86 + 2 * q^87 + 53 * q^88 - 9 * q^89 - 12 * q^91 - 63 * q^92 + 13 * q^93 + 14 * q^94 - 18 * q^96 + 24 * q^97 - 24 * q^98 + 12 * q^99

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