LMFDB - Embedded newform 2450.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.41421 q^{3} +1.00000 q^{4} -3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.41421 q^{3} +1.00000 q^{4} -3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} -0.828427 q^{11} -3.41421 q^{12} +4.82843 q^{13} +1.00000 q^{16} -2.58579 q^{17} +8.65685 q^{18} +0.585786 q^{19} -0.828427 q^{22} +1.17157 q^{23} -3.41421 q^{24} +4.82843 q^{26} -19.3137 q^{27} -4.82843 q^{29} -2.82843 q^{31} +1.00000 q^{32} +2.82843 q^{33} -2.58579 q^{34} +8.65685 q^{36} +7.65685 q^{37} +0.585786 q^{38} -16.4853 q^{39} -3.07107 q^{41} +8.82843 q^{43} -0.828427 q^{44} +1.17157 q^{46} -5.17157 q^{47} -3.41421 q^{48} +8.82843 q^{51} +4.82843 q^{52} -6.48528 q^{53} -19.3137 q^{54} -2.00000 q^{57} -4.82843 q^{58} +8.58579 q^{59} +9.31371 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.82843 q^{66} -1.65685 q^{67} -2.58579 q^{68} -4.00000 q^{69} -4.48528 q^{71} +8.65685 q^{72} +9.41421 q^{73} +7.65685 q^{74} +0.585786 q^{76} -16.4853 q^{78} -6.82843 q^{79} +39.9706 q^{81} -3.07107 q^{82} +2.24264 q^{83} +8.82843 q^{86} +16.4853 q^{87} -0.828427 q^{88} +12.7279 q^{89} +1.17157 q^{92} +9.65685 q^{93} -5.17157 q^{94} -3.41421 q^{96} +7.75736 q^{97} -7.17157 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^6 + 2 * q^8 + 6 * q^9	$ 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 6 q^{9} + 4 q^{11} - 4 q^{12} + 4 q^{13} + 2 q^{16} - 8 q^{17} + 6 q^{18} + 4 q^{19} + 4 q^{22} + 8 q^{23} - 4 q^{24} + 4 q^{26} - 16 q^{27} - 4 q^{29} + 2 q^{32} - 8 q^{34} + 6 q^{36} + 4 q^{37} + 4 q^{38} - 16 q^{39} + 8 q^{41} + 12 q^{43} + 4 q^{44} + 8 q^{46} - 16 q^{47} - 4 q^{48} + 12 q^{51} + 4 q^{52} + 4 q^{53} - 16 q^{54} - 4 q^{57} - 4 q^{58} + 20 q^{59} - 4 q^{61} + 2 q^{64} + 8 q^{67} - 8 q^{68} - 8 q^{69} + 8 q^{71} + 6 q^{72} + 16 q^{73} + 4 q^{74} + 4 q^{76} - 16 q^{78} - 8 q^{79} + 46 q^{81} + 8 q^{82} - 4 q^{83} + 12 q^{86} + 16 q^{87} + 4 q^{88} + 8 q^{92} + 8 q^{93} - 16 q^{94} - 4 q^{96} + 24 q^{97} - 20 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^6 + 2 * q^8 + 6 * q^9 + 4 * q^11 - 4 * q^12 + 4 * q^13 + 2 * q^16 - 8 * q^17 + 6 * q^18 + 4 * q^19 + 4 * q^22 + 8 * q^23 - 4 * q^24 + 4 * q^26 - 16 * q^27 - 4 * q^29 + 2 * q^32 - 8 * q^34 + 6 * q^36 + 4 * q^37 + 4 * q^38 - 16 * q^39 + 8 * q^41 + 12 * q^43 + 4 * q^44 + 8 * q^46 - 16 * q^47 - 4 * q^48 + 12 * q^51 + 4 * q^52 + 4 * q^53 - 16 * q^54 - 4 * q^57 - 4 * q^58 + 20 * q^59 - 4 * q^61 + 2 * q^64 + 8 * q^67 - 8 * q^68 - 8 * q^69 + 8 * q^71 + 6 * q^72 + 16 * q^73 + 4 * q^74 + 4 * q^76 - 16 * q^78 - 8 * q^79 + 46 * q^81 + 8 * q^82 - 4 * q^83 + 12 * q^86 + 16 * q^87 + 4 * q^88 + 8 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 + 24 * q^97 - 20 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.41421	−1.97120	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−3.41421	−1.97120	−0.985599	+	0.169102i	$0.945913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.41421	−1.39385
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	8.65685	2.88562
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	−3.41421	−0.985599
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.58579	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$18$	8.65685	2.04044
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	0	0
$21$	0	0
$22$	−0.828427	−0.176621
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	−3.41421	−0.696923
$25$	0	0
$26$	4.82843	0.946932
$27$	−19.3137	−3.71692
$28$	0	0
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	1.00000	0.176777
$33$	2.82843	0.492366
$34$	−2.58579	−0.443459
$35$	0	0
$36$	8.65685	1.44281
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	0.585786	0.0950271
$39$	−16.4853	−2.63976
$40$	0	0
$41$	−3.07107	−0.479620	−0.239810	−	0.970820i	$-0.577085\pi$
$41$	−3.07107	−0.479620	−0.239810	+	0.970820i	$0.577085\pi$
$42$	0	0
$43$	8.82843	1.34632	0.673161	−	0.739496i	$-0.264936\pi$
$43$	8.82843	1.34632	0.673161	+	0.739496i	$0.264936\pi$
$44$	−0.828427	−0.124890
$45$	0	0
$46$	1.17157	0.172739
$47$	−5.17157	−0.754351	−0.377176	−	0.926142i	$-0.623105\pi$
$47$	−5.17157	−0.754351	−0.377176	+	0.926142i	$0.623105\pi$
$48$	−3.41421	−0.492799
$49$	0	0
$50$	0	0
$51$	8.82843	1.23623
$52$	4.82843	0.669582
$53$	−6.48528	−0.890822	−0.445411	−	0.895326i	$-0.646942\pi$
$53$	−6.48528	−0.890822	−0.445411	+	0.895326i	$0.646942\pi$
$54$	−19.3137	−2.62826
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	−4.82843	−0.634004
$59$	8.58579	1.11777	0.558887	−	0.829244i	$-0.311229\pi$
$59$	8.58579	1.11777	0.558887	+	0.829244i	$0.311229\pi$
$60$	0	0
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	−2.82843	−0.359211
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.82843	0.348155
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	−2.58579	−0.313573
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	8.65685	1.02022
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	7.65685	0.890091
$75$	0	0
$76$	0.585786	0.0671943
$77$	0	0
$78$	−16.4853	−1.86659
$79$	−6.82843	−0.768258	−0.384129	−	0.923279i	$-0.625498\pi$
$79$	−6.82843	−0.768258	−0.384129	+	0.923279i	$0.625498\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	−3.07107	−0.339143
$83$	2.24264	0.246162	0.123081	−	0.992397i	$-0.460723\pi$
$83$	2.24264	0.246162	0.123081	+	0.992397i	$0.460723\pi$
$84$	0	0
$85$	0	0
$86$	8.82843	0.951994
$87$	16.4853	1.76741
$88$	−0.828427	−0.0883106
$89$	12.7279	1.34916	0.674579	−	0.738203i	$-0.264325\pi$
$89$	12.7279	1.34916	0.674579	+	0.738203i	$0.264325\pi$
$90$	0	0
$91$	0	0
$92$	1.17157	0.122145
$93$	9.65685	1.00137
$94$	−5.17157	−0.533407
$95$	0	0
$96$	−3.41421	−0.348462
$97$	7.75736	0.787641	0.393820	−	0.919187i	$-0.371153\pi$
$97$	7.75736	0.787641	0.393820	+	0.919187i	$0.371153\pi$
$98$	0	0
$99$	−7.17157	−0.720770
$100$	0	0
$101$	13.3137	1.32476	0.662382	−	0.749166i	$-0.269546\pi$
$101$	13.3137	1.32476	0.662382	+	0.749166i	$0.269546\pi$
$102$	8.82843	0.874145
$103$	14.8284	1.46109	0.730544	−	0.682865i	$-0.239266\pi$
$103$	14.8284	1.46109	0.730544	+	0.682865i	$0.239266\pi$
$104$	4.82843	0.473466
$105$	0	0
$106$	−6.48528	−0.629906
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	−19.3137	−1.85846
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	0	0
$111$	−26.1421	−2.48130
$112$	0	0
$113$	15.3137	1.44059	0.720296	−	0.693667i	$-0.244006\pi$
$113$	15.3137	1.44059	0.720296	+	0.693667i	$0.244006\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−4.82843	−0.448308
$117$	41.7990	3.86432
$118$	8.58579	0.790386
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	9.31371	0.843224
$123$	10.4853	0.945426
$124$	−2.82843	−0.254000
$125$	0	0
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	1.00000	0.0883883
$129$	−30.1421	−2.65387
$130$	0	0
$131$	−6.24264	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$131$	−6.24264	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$132$	2.82843	0.246183
$133$	0	0
$134$	−1.65685	−0.143130
$135$	0	0
$136$	−2.58579	−0.221729
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	−4.00000	−0.340503
$139$	19.8995	1.68785	0.843927	−	0.536459i	$-0.180238\pi$
$139$	19.8995	1.68785	0.843927	+	0.536459i	$0.180238\pi$
$140$	0	0
$141$	17.6569	1.48698
$142$	−4.48528	−0.376396
$143$	−4.00000	−0.334497
$144$	8.65685	0.721405
$145$	0	0
$146$	9.41421	0.779126
$147$	0	0
$148$	7.65685	0.629390
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−11.3137	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$151$	−11.3137	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$152$	0.585786	0.0475136
$153$	−22.3848	−1.80970
$154$	0	0
$155$	0	0
$156$	−16.4853	−1.31988
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	−6.82843	−0.543240
$159$	22.1421	1.75599
$160$	0	0
$161$	0	0
$162$	39.9706	3.14038
$163$	20.1421	1.57765	0.788827	−	0.614615i	$-0.210689\pi$
$163$	20.1421	1.57765	0.788827	+	0.614615i	$0.210689\pi$
$164$	−3.07107	−0.239810
$165$	0	0
$166$	2.24264	0.174063
$167$	−15.7990	−1.22256	−0.611281	−	0.791413i	$-0.709346\pi$
$167$	−15.7990	−1.22256	−0.611281	+	0.791413i	$0.709346\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	5.07107	0.387794
$172$	8.82843	0.673161
$173$	8.82843	0.671213	0.335606	−	0.942002i	$-0.391059\pi$
$173$	8.82843	0.671213	0.335606	+	0.942002i	$0.391059\pi$
$174$	16.4853	1.24975
$175$	0	0
$176$	−0.828427	−0.0624450
$177$	−29.3137	−2.20335
$178$	12.7279	0.953998
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−2.48528	−0.184730	−0.0923648	−	0.995725i	$-0.529443\pi$
$181$	−2.48528	−0.184730	−0.0923648	+	0.995725i	$0.529443\pi$
$182$	0	0
$183$	−31.7990	−2.35065
$184$	1.17157	0.0863695
$185$	0	0
$186$	9.65685	0.708075
$187$	2.14214	0.156648
$188$	−5.17157	−0.377176
$189$	0	0
$190$	0	0
$191$	10.1421	0.733859	0.366930	−	0.930249i	$-0.380409\pi$
$191$	10.1421	0.733859	0.366930	+	0.930249i	$0.380409\pi$
$192$	−3.41421	−0.246400
$193$	−5.65685	−0.407189	−0.203595	−	0.979055i	$-0.565262\pi$
$193$	−5.65685	−0.407189	−0.203595	+	0.979055i	$0.565262\pi$
$194$	7.75736	0.556946
$195$	0	0
$196$	0	0
$197$	25.7990	1.83810	0.919051	−	0.394139i	$-0.128957\pi$
$197$	25.7990	1.83810	0.919051	+	0.394139i	$0.128957\pi$
$198$	−7.17157	−0.509661
$199$	16.4853	1.16861	0.584305	−	0.811534i	$-0.301367\pi$
$199$	16.4853	1.16861	0.584305	+	0.811534i	$0.301367\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	13.3137	0.936749
$203$	0	0
$204$	8.82843	0.618114
$205$	0	0
$206$	14.8284	1.03315
$207$	10.1421	0.704927
$208$	4.82843	0.334791
$209$	−0.485281	−0.0335676
$210$	0	0
$211$	18.6274	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$211$	18.6274	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$212$	−6.48528	−0.445411
$213$	15.3137	1.04928
$214$	−9.65685	−0.660129
$215$	0	0
$216$	−19.3137	−1.31413
$217$	0	0
$218$	2.48528	0.168324
$219$	−32.1421	−2.17196
$220$	0	0
$221$	−12.4853	−0.839851
$222$	−26.1421	−1.75455
$223$	7.31371	0.489762	0.244881	−	0.969553i	$-0.421251\pi$
$223$	7.31371	0.489762	0.244881	+	0.969553i	$0.421251\pi$
$224$	0	0
$225$	0	0
$226$	15.3137	1.01865
$227$	−18.2426	−1.21081	−0.605403	−	0.795919i	$-0.706988\pi$
$227$	−18.2426	−1.21081	−0.605403	+	0.795919i	$0.706988\pi$
$228$	−2.00000	−0.132453
$229$	−16.1421	−1.06670	−0.533351	−	0.845894i	$-0.679068\pi$
$229$	−16.1421	−1.06670	−0.533351	+	0.845894i	$0.679068\pi$
$230$	0	0
$231$	0	0
$232$	−4.82843	−0.317002
$233$	−23.3137	−1.52733	−0.763666	−	0.645612i	$-0.776603\pi$
$233$	−23.3137	−1.52733	−0.763666	+	0.645612i	$0.776603\pi$
$234$	41.7990	2.73249
$235$	0	0
$236$	8.58579	0.558887
$237$	23.3137	1.51439
$238$	0	0
$239$	1.65685	0.107173	0.0535865	−	0.998563i	$-0.482935\pi$
$239$	1.65685	0.107173	0.0535865	+	0.998563i	$0.482935\pi$
$240$	0	0
$241$	−13.4142	−0.864085	−0.432043	−	0.901853i	$-0.642207\pi$
$241$	−13.4142	−0.864085	−0.432043	+	0.901853i	$0.642207\pi$
$242$	−10.3137	−0.662990
$243$	−78.5269	−5.03750
$244$	9.31371	0.596249
$245$	0	0
$246$	10.4853	0.668517
$247$	2.82843	0.179969
$248$	−2.82843	−0.179605
$249$	−7.65685	−0.485233
$250$	0	0
$251$	0.585786	0.0369745	0.0184873	−	0.999829i	$-0.494115\pi$
$251$	0.585786	0.0369745	0.0184873	+	0.999829i	$0.494115\pi$
$252$	0	0
$253$	−0.970563	−0.0610188
$254$	−2.82843	−0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	−30.1421	−1.87657
$259$	0	0
$260$	0	0
$261$	−41.7990	−2.58729
$262$	−6.24264	−0.385672
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	2.82843	0.174078
$265$	0	0
$266$	0	0
$267$	−43.4558	−2.65945
$268$	−1.65685	−0.101208
$269$	18.4853	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$269$	18.4853	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−2.58579	−0.156786
$273$	0	0
$274$	16.0000	0.966595
$275$	0	0
$276$	−4.00000	−0.240772
$277$	8.14214	0.489214	0.244607	−	0.969622i	$-0.421341\pi$
$277$	8.14214	0.489214	0.244607	+	0.969622i	$0.421341\pi$
$278$	19.8995	1.19349
$279$	−24.4853	−1.46590
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	17.6569	1.05145
$283$	2.24264	0.133311	0.0666556	−	0.997776i	$-0.478767\pi$
$283$	2.24264	0.133311	0.0666556	+	0.997776i	$0.478767\pi$
$284$	−4.48528	−0.266152
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	8.65685	0.510110
$289$	−10.3137	−0.606689
$290$	0	0
$291$	−26.4853	−1.55259
$292$	9.41421	0.550925
$293$	8.34315	0.487412	0.243706	−	0.969849i	$-0.421637\pi$
$293$	8.34315	0.487412	0.243706	+	0.969849i	$0.421637\pi$
$294$	0	0
$295$	0	0
$296$	7.65685	0.445046
$297$	16.0000	0.928414
$298$	−6.00000	−0.347571
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	−11.3137	−0.651031
$303$	−45.4558	−2.61137
$304$	0.585786	0.0335972
$305$	0	0
$306$	−22.3848	−1.27965
$307$	−14.9289	−0.852039	−0.426020	−	0.904714i	$-0.640085\pi$
$307$	−14.9289	−0.852039	−0.426020	+	0.904714i	$0.640085\pi$
$308$	0	0
$309$	−50.6274	−2.88009
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	−16.4853	−0.933295
$313$	−14.3848	−0.813076	−0.406538	−	0.913634i	$-0.633264\pi$
$313$	−14.3848	−0.813076	−0.406538	+	0.913634i	$0.633264\pi$
$314$	−6.48528	−0.365986
$315$	0	0
$316$	−6.82843	−0.384129
$317$	−10.4853	−0.588912	−0.294456	−	0.955665i	$-0.595138\pi$
$317$	−10.4853	−0.588912	−0.294456	+	0.955665i	$0.595138\pi$
$318$	22.1421	1.24167
$319$	4.00000	0.223957
$320$	0	0
$321$	32.9706	1.84024
$322$	0	0
$323$	−1.51472	−0.0842812
$324$	39.9706	2.22059
$325$	0	0
$326$	20.1421	1.11557
$327$	−8.48528	−0.469237
$328$	−3.07107	−0.169571
$329$	0	0
$330$	0	0
$331$	33.7990	1.85776	0.928880	−	0.370380i	$-0.120773\pi$
$331$	33.7990	1.85776	0.928880	+	0.370380i	$0.120773\pi$
$332$	2.24264	0.123081
$333$	66.2843	3.63236
$334$	−15.7990	−0.864482
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	10.3137	0.560992
$339$	−52.2843	−2.83969
$340$	0	0
$341$	2.34315	0.126888
$342$	5.07107	0.274212
$343$	0	0
$344$	8.82843	0.475997
$345$	0	0
$346$	8.82843	0.474619
$347$	3.17157	0.170259	0.0851295	−	0.996370i	$-0.472870\pi$
$347$	3.17157	0.170259	0.0851295	+	0.996370i	$0.472870\pi$
$348$	16.4853	0.883704
$349$	2.48528	0.133034	0.0665170	−	0.997785i	$-0.478811\pi$
$349$	2.48528	0.133034	0.0665170	+	0.997785i	$0.478811\pi$
$350$	0	0
$351$	−93.2548	−4.97757
$352$	−0.828427	−0.0441553
$353$	2.38478	0.126929	0.0634644	−	0.997984i	$-0.479785\pi$
$353$	2.38478	0.126929	0.0634644	+	0.997984i	$0.479785\pi$
$354$	−29.3137	−1.55801
$355$	0	0
$356$	12.7279	0.674579
$357$	0	0
$358$	4.00000	0.211407
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	−2.48528	−0.130623
$363$	35.2132	1.84821
$364$	0	0
$365$	0	0
$366$	−31.7990	−1.66216
$367$	−24.9706	−1.30345	−0.651726	−	0.758454i	$-0.725955\pi$
$367$	−24.9706	−1.30345	−0.651726	+	0.758454i	$0.725955\pi$
$368$	1.17157	0.0610725
$369$	−26.5858	−1.38400
$370$	0	0
$371$	0	0
$372$	9.65685	0.500685
$373$	30.4853	1.57847	0.789234	−	0.614093i	$-0.210478\pi$
$373$	30.4853	1.57847	0.789234	+	0.614093i	$0.210478\pi$
$374$	2.14214	0.110767
$375$	0	0
$376$	−5.17157	−0.266704
$377$	−23.3137	−1.20072
$378$	0	0
$379$	34.4853	1.77139	0.885695	−	0.464268i	$-0.153682\pi$
$379$	34.4853	1.77139	0.885695	+	0.464268i	$0.153682\pi$
$380$	0	0
$381$	9.65685	0.494736
$382$	10.1421	0.518917
$383$	32.4853	1.65992	0.829960	−	0.557823i	$-0.188363\pi$
$383$	32.4853	1.65992	0.829960	+	0.557823i	$0.188363\pi$
$384$	−3.41421	−0.174231
$385$	0	0
$386$	−5.65685	−0.287926
$387$	76.4264	3.88497
$388$	7.75736	0.393820
$389$	28.1421	1.42686	0.713431	−	0.700725i	$-0.247140\pi$
$389$	28.1421	1.42686	0.713431	+	0.700725i	$0.247140\pi$
$390$	0	0
$391$	−3.02944	−0.153205
$392$	0	0
$393$	21.3137	1.07513
$394$	25.7990	1.29973
$395$	0	0
$396$	−7.17157	−0.360385
$397$	−33.7990	−1.69632	−0.848161	−	0.529738i	$-0.822290\pi$
$397$	−33.7990	−1.69632	−0.848161	+	0.529738i	$0.822290\pi$
$398$	16.4853	0.826332
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	5.65685	0.282138
$403$	−13.6569	−0.680296
$404$	13.3137	0.662382
$405$	0	0
$406$	0	0
$407$	−6.34315	−0.314418
$408$	8.82843	0.437072
$409$	−10.5858	−0.523433	−0.261717	−	0.965145i	$-0.584289\pi$
$409$	−10.5858	−0.523433	−0.261717	+	0.965145i	$0.584289\pi$
$410$	0	0
$411$	−54.6274	−2.69457
$412$	14.8284	0.730544
$413$	0	0
$414$	10.1421	0.498459
$415$	0	0
$416$	4.82843	0.236733
$417$	−67.9411	−3.32709
$418$	−0.485281	−0.0237359
$419$	−20.8701	−1.01957	−0.509785	−	0.860302i	$-0.670275\pi$
$419$	−20.8701	−1.01957	−0.509785	+	0.860302i	$0.670275\pi$
$420$	0	0
$421$	17.3137	0.843819	0.421909	−	0.906638i	$-0.361360\pi$
$421$	17.3137	0.843819	0.421909	+	0.906638i	$0.361360\pi$
$422$	18.6274	0.906768
$423$	−44.7696	−2.17677
$424$	−6.48528	−0.314953
$425$	0	0
$426$	15.3137	0.741952
$427$	0	0
$428$	−9.65685	−0.466782
$429$	13.6569	0.659359
$430$	0	0
$431$	−22.3431	−1.07623	−0.538116	−	0.842871i	$-0.680864\pi$
$431$	−22.3431	−1.07623	−0.538116	+	0.842871i	$0.680864\pi$
$432$	−19.3137	−0.929231
$433$	−10.5858	−0.508720	−0.254360	−	0.967110i	$-0.581865\pi$
$433$	−10.5858	−0.508720	−0.254360	+	0.967110i	$0.581865\pi$
$434$	0	0
$435$	0	0
$436$	2.48528	0.119023
$437$	0.686292	0.0328298
$438$	−32.1421	−1.53581
$439$	−24.9706	−1.19178	−0.595890	−	0.803066i	$-0.703201\pi$
$439$	−24.9706	−1.19178	−0.595890	+	0.803066i	$0.703201\pi$
$440$	0	0
$441$	0	0
$442$	−12.4853	−0.593864
$443$	−3.02944	−0.143933	−0.0719665	−	0.997407i	$-0.522927\pi$
$443$	−3.02944	−0.143933	−0.0719665	+	0.997407i	$0.522927\pi$
$444$	−26.1421	−1.24065
$445$	0	0
$446$	7.31371	0.346314
$447$	20.4853	0.968921
$448$	0	0
$449$	−16.6274	−0.784696	−0.392348	−	0.919817i	$-0.628337\pi$
$449$	−16.6274	−0.784696	−0.392348	+	0.919817i	$0.628337\pi$
$450$	0	0
$451$	2.54416	0.119800
$452$	15.3137	0.720296
$453$	38.6274	1.81487
$454$	−18.2426	−0.856170
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−21.6569	−1.01306	−0.506532	−	0.862221i	$-0.669073\pi$
$457$	−21.6569	−1.01306	−0.506532	+	0.862221i	$0.669073\pi$
$458$	−16.1421	−0.754272
$459$	49.9411	2.33105
$460$	0	0
$461$	−12.8284	−0.597479	−0.298740	−	0.954335i	$-0.596566\pi$
$461$	−12.8284	−0.597479	−0.298740	+	0.954335i	$0.596566\pi$
$462$	0	0
$463$	16.9706	0.788689	0.394344	−	0.918963i	$-0.370972\pi$
$463$	16.9706	0.788689	0.394344	+	0.918963i	$0.370972\pi$
$464$	−4.82843	−0.224154
$465$	0	0
$466$	−23.3137	−1.07999
$467$	15.8995	0.735741	0.367870	−	0.929877i	$-0.380087\pi$
$467$	15.8995	0.735741	0.367870	+	0.929877i	$0.380087\pi$
$468$	41.7990	1.93216
$469$	0	0
$470$	0	0
$471$	22.1421	1.02026
$472$	8.58579	0.395193
$473$	−7.31371	−0.336285
$474$	23.3137	1.07083
$475$	0	0
$476$	0	0
$477$	−56.1421	−2.57057
$478$	1.65685	0.0757827
$479$	−17.1716	−0.784589	−0.392295	−	0.919840i	$-0.628319\pi$
$479$	−17.1716	−0.784589	−0.392295	+	0.919840i	$0.628319\pi$
$480$	0	0
$481$	36.9706	1.68571
$482$	−13.4142	−0.611001
$483$	0	0
$484$	−10.3137	−0.468805
$485$	0	0
$486$	−78.5269	−3.56205
$487$	−31.7990	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$487$	−31.7990	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$488$	9.31371	0.421612
$489$	−68.7696	−3.10987
$490$	0	0
$491$	32.2843	1.45697	0.728484	−	0.685062i	$-0.240225\pi$
$491$	32.2843	1.45697	0.728484	+	0.685062i	$0.240225\pi$
$492$	10.4853	0.472713
$493$	12.4853	0.562309
$494$	2.82843	0.127257
$495$	0	0
$496$	−2.82843	−0.127000
$497$	0	0
$498$	−7.65685	−0.343112
$499$	30.3431	1.35835	0.679173	−	0.733978i	$-0.262339\pi$
$499$	30.3431	1.35835	0.679173	+	0.733978i	$0.262339\pi$
$500$	0	0
$501$	53.9411	2.40991
$502$	0.585786	0.0261449
$503$	−17.6569	−0.787280	−0.393640	−	0.919265i	$-0.628784\pi$
$503$	−17.6569	−0.787280	−0.393640	+	0.919265i	$0.628784\pi$
$504$	0	0
$505$	0	0
$506$	−0.970563	−0.0431468
$507$	−35.2132	−1.56387
$508$	−2.82843	−0.125491
$509$	5.79899	0.257036	0.128518	−	0.991707i	$-0.458978\pi$
$509$	5.79899	0.257036	0.128518	+	0.991707i	$0.458978\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−11.3137	−0.499512
$514$	9.89949	0.436648
$515$	0	0
$516$	−30.1421	−1.32693
$517$	4.28427	0.188422
$518$	0	0
$519$	−30.1421	−1.32309
$520$	0	0
$521$	19.0711	0.835519	0.417759	−	0.908558i	$-0.362816\pi$
$521$	19.0711	0.835519	0.417759	+	0.908558i	$0.362816\pi$
$522$	−41.7990	−1.82949
$523$	−23.8995	−1.04505	−0.522526	−	0.852623i	$-0.675010\pi$
$523$	−23.8995	−1.04505	−0.522526	+	0.852623i	$0.675010\pi$
$524$	−6.24264	−0.272711
$525$	0	0
$526$	−28.0000	−1.22086
$527$	7.31371	0.318590
$528$	2.82843	0.123091
$529$	−21.6274	−0.940322
$530$	0	0
$531$	74.3259	3.22547
$532$	0	0
$533$	−14.8284	−0.642290
$534$	−43.4558	−1.88052
$535$	0	0
$536$	−1.65685	−0.0715652
$537$	−13.6569	−0.589337
$538$	18.4853	0.796957
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	12.0000	0.515444
$543$	8.48528	0.364138
$544$	−2.58579	−0.110865
$545$	0	0
$546$	0	0
$547$	10.4853	0.448318	0.224159	−	0.974553i	$-0.428036\pi$
$547$	10.4853	0.448318	0.224159	+	0.974553i	$0.428036\pi$
$548$	16.0000	0.683486
$549$	80.6274	3.44109
$550$	0	0
$551$	−2.82843	−0.120495
$552$	−4.00000	−0.170251
$553$	0	0
$554$	8.14214	0.345926
$555$	0	0
$556$	19.8995	0.843927
$557$	−15.1716	−0.642840	−0.321420	−	0.946937i	$-0.604160\pi$
$557$	−15.1716	−0.642840	−0.321420	+	0.946937i	$0.604160\pi$
$558$	−24.4853	−1.03654
$559$	42.6274	1.80295
$560$	0	0
$561$	−7.31371	−0.308785
$562$	8.00000	0.337460
$563$	36.5858	1.54191	0.770954	−	0.636891i	$-0.219780\pi$
$563$	36.5858	1.54191	0.770954	+	0.636891i	$0.219780\pi$
$564$	17.6569	0.743488
$565$	0	0
$566$	2.24264	0.0942652
$567$	0	0
$568$	−4.48528	−0.188198
$569$	−29.3137	−1.22889	−0.614447	−	0.788958i	$-0.710621\pi$
$569$	−29.3137	−1.22889	−0.614447	+	0.788958i	$0.710621\pi$
$570$	0	0
$571$	−2.20101	−0.0921094	−0.0460547	−	0.998939i	$-0.514665\pi$
$571$	−2.20101	−0.0921094	−0.0460547	+	0.998939i	$0.514665\pi$
$572$	−4.00000	−0.167248
$573$	−34.6274	−1.44658
$574$	0	0
$575$	0	0
$576$	8.65685	0.360702
$577$	−6.10051	−0.253967	−0.126984	−	0.991905i	$-0.540530\pi$
$577$	−6.10051	−0.253967	−0.126984	+	0.991905i	$0.540530\pi$
$578$	−10.3137	−0.428994
$579$	19.3137	0.802650
$580$	0	0
$581$	0	0
$582$	−26.4853	−1.09785
$583$	5.37258	0.222510
$584$	9.41421	0.389563
$585$	0	0
$586$	8.34315	0.344652
$587$	−17.0711	−0.704598	−0.352299	−	0.935887i	$-0.614600\pi$
$587$	−17.0711	−0.704598	−0.352299	+	0.935887i	$0.614600\pi$
$588$	0	0
$589$	−1.65685	−0.0682695
$590$	0	0
$591$	−88.0833	−3.62326
$592$	7.65685	0.314695
$593$	−3.27208	−0.134368	−0.0671841	−	0.997741i	$-0.521401\pi$
$593$	−3.27208	−0.134368	−0.0671841	+	0.997741i	$0.521401\pi$
$594$	16.0000	0.656488
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−56.2843	−2.30356
$598$	5.65685	0.231326
$599$	−10.8284	−0.442438	−0.221219	−	0.975224i	$-0.571003\pi$
$599$	−10.8284	−0.442438	−0.221219	+	0.975224i	$0.571003\pi$
$600$	0	0
$601$	6.58579	0.268640	0.134320	−	0.990938i	$-0.457115\pi$
$601$	6.58579	0.268640	0.134320	+	0.990938i	$0.457115\pi$
$602$	0	0
$603$	−14.3431	−0.584098
$604$	−11.3137	−0.460348
$605$	0	0
$606$	−45.4558	−1.84652
$607$	−16.2843	−0.660958	−0.330479	−	0.943813i	$-0.607210\pi$
$607$	−16.2843	−0.660958	−0.330479	+	0.943813i	$0.607210\pi$
$608$	0.585786	0.0237568
$609$	0	0
$610$	0	0
$611$	−24.9706	−1.01020
$612$	−22.3848	−0.904851
$613$	12.3431	0.498535	0.249267	−	0.968435i	$-0.419810\pi$
$613$	12.3431	0.498535	0.249267	+	0.968435i	$0.419810\pi$
$614$	−14.9289	−0.602483
$615$	0	0
$616$	0	0
$617$	33.3137	1.34116	0.670580	−	0.741837i	$-0.266045\pi$
$617$	33.3137	1.34116	0.670580	+	0.741837i	$0.266045\pi$
$618$	−50.6274	−2.03653
$619$	29.0711	1.16846	0.584232	−	0.811586i	$-0.301396\pi$
$619$	29.0711	1.16846	0.584232	+	0.811586i	$0.301396\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	4.00000	0.160385
$623$	0	0
$624$	−16.4853	−0.659939
$625$	0	0
$626$	−14.3848	−0.574931
$627$	1.65685	0.0661684
$628$	−6.48528	−0.258791
$629$	−19.7990	−0.789437
$630$	0	0
$631$	−12.4853	−0.497031	−0.248516	−	0.968628i	$-0.579943\pi$
$631$	−12.4853	−0.497031	−0.248516	+	0.968628i	$0.579943\pi$
$632$	−6.82843	−0.271620
$633$	−63.5980	−2.52779
$634$	−10.4853	−0.416424
$635$	0	0
$636$	22.1421	0.877993
$637$	0	0
$638$	4.00000	0.158362
$639$	−38.8284	−1.53603
$640$	0	0
$641$	−24.6274	−0.972724	−0.486362	−	0.873757i	$-0.661676\pi$
$641$	−24.6274	−0.972724	−0.486362	+	0.873757i	$0.661676\pi$
$642$	32.9706	1.30124
$643$	−4.78680	−0.188773	−0.0943864	−	0.995536i	$-0.530089\pi$
$643$	−4.78680	−0.188773	−0.0943864	+	0.995536i	$0.530089\pi$
$644$	0	0
$645$	0	0
$646$	−1.51472	−0.0595958
$647$	23.1127	0.908654	0.454327	−	0.890835i	$-0.349880\pi$
$647$	23.1127	0.908654	0.454327	+	0.890835i	$0.349880\pi$
$648$	39.9706	1.57019
$649$	−7.11270	−0.279198
$650$	0	0
$651$	0	0
$652$	20.1421	0.788827
$653$	−4.34315	−0.169960	−0.0849802	−	0.996383i	$-0.527083\pi$
$653$	−4.34315	−0.169960	−0.0849802	+	0.996383i	$0.527083\pi$
$654$	−8.48528	−0.331801
$655$	0	0
$656$	−3.07107	−0.119905
$657$	81.4975	3.17952
$658$	0	0
$659$	−27.1716	−1.05845	−0.529227	−	0.848480i	$-0.677518\pi$
$659$	−27.1716	−1.05845	−0.529227	+	0.848480i	$0.677518\pi$
$660$	0	0
$661$	−38.2843	−1.48909	−0.744543	−	0.667575i	$-0.767332\pi$
$661$	−38.2843	−1.48909	−0.744543	+	0.667575i	$0.767332\pi$
$662$	33.7990	1.31364
$663$	42.6274	1.65551
$664$	2.24264	0.0870313
$665$	0	0
$666$	66.2843	2.56846
$667$	−5.65685	−0.219034
$668$	−15.7990	−0.611281
$669$	−24.9706	−0.965418
$670$	0	0
$671$	−7.71573	−0.297862
$672$	0	0
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	10.3137	0.396681
$677$	−39.4558	−1.51641	−0.758206	−	0.652015i	$-0.773924\pi$
$677$	−39.4558	−1.51641	−0.758206	+	0.652015i	$0.773924\pi$
$678$	−52.2843	−2.00797
$679$	0	0
$680$	0	0
$681$	62.2843	2.38674
$682$	2.34315	0.0897237
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	5.07107	0.193897
$685$	0	0
$686$	0	0
$687$	55.1127	2.10268
$688$	8.82843	0.336581
$689$	−31.3137	−1.19296
$690$	0	0
$691$	1.75736	0.0668531	0.0334265	−	0.999441i	$-0.489358\pi$
$691$	1.75736	0.0668531	0.0334265	+	0.999441i	$0.489358\pi$
$692$	8.82843	0.335606
$693$	0	0
$694$	3.17157	0.120391
$695$	0	0
$696$	16.4853	0.624873
$697$	7.94113	0.300792
$698$	2.48528	0.0940693
$699$	79.5980	3.01067
$700$	0	0
$701$	2.48528	0.0938678	0.0469339	−	0.998898i	$-0.485055\pi$
$701$	2.48528	0.0938678	0.0469339	+	0.998898i	$0.485055\pi$
$702$	−93.2548	−3.51968
$703$	4.48528	0.169166
$704$	−0.828427	−0.0312225
$705$	0	0
$706$	2.38478	0.0897522
$707$	0	0
$708$	−29.3137	−1.10168
$709$	45.1127	1.69424	0.847121	−	0.531399i	$-0.178334\pi$
$709$	45.1127	1.69424	0.847121	+	0.531399i	$0.178334\pi$
$710$	0	0
$711$	−59.1127	−2.21690
$712$	12.7279	0.476999
$713$	−3.31371	−0.124099
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	−5.65685	−0.211259
$718$	28.2843	1.05556
$719$	41.4558	1.54604	0.773021	−	0.634380i	$-0.218745\pi$
$719$	41.4558	1.54604	0.773021	+	0.634380i	$0.218745\pi$
$720$	0	0
$721$	0	0
$722$	−18.6569	−0.694336
$723$	45.7990	1.70328
$724$	−2.48528	−0.0923648
$725$	0	0
$726$	35.2132	1.30688
$727$	3.51472	0.130354	0.0651768	−	0.997874i	$-0.479239\pi$
$727$	3.51472	0.130354	0.0651768	+	0.997874i	$0.479239\pi$
$728$	0	0
$729$	148.196	5.48874
$730$	0	0
$731$	−22.8284	−0.844340
$732$	−31.7990	−1.17532
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−24.9706	−0.921680
$735$	0	0
$736$	1.17157	0.0431847
$737$	1.37258	0.0505597
$738$	−26.5858	−0.978636
$739$	3.17157	0.116668	0.0583341	−	0.998297i	$-0.481421\pi$
$739$	3.17157	0.116668	0.0583341	+	0.998297i	$0.481421\pi$
$740$	0	0
$741$	−9.65685	−0.354753
$742$	0	0
$743$	−51.7990	−1.90032	−0.950160	−	0.311762i	$-0.899081\pi$
$743$	−51.7990	−1.90032	−0.950160	+	0.311762i	$0.899081\pi$
$744$	9.65685	0.354037
$745$	0	0
$746$	30.4853	1.11615
$747$	19.4142	0.710329
$748$	2.14214	0.0783242
$749$	0	0
$750$	0	0
$751$	−39.3137	−1.43458	−0.717289	−	0.696776i	$-0.754617\pi$
$751$	−39.3137	−1.43458	−0.717289	+	0.696776i	$0.754617\pi$
$752$	−5.17157	−0.188588
$753$	−2.00000	−0.0728841
$754$	−23.3137	−0.849035
$755$	0	0
$756$	0	0
$757$	−3.65685	−0.132911	−0.0664553	−	0.997789i	$-0.521169\pi$
$757$	−3.65685	−0.132911	−0.0664553	+	0.997789i	$0.521169\pi$
$758$	34.4853	1.25256
$759$	3.31371	0.120280
$760$	0	0
$761$	22.3848	0.811448	0.405724	−	0.913996i	$-0.367020\pi$
$761$	22.3848	0.811448	0.405724	+	0.913996i	$0.367020\pi$
$762$	9.65685	0.349831
$763$	0	0
$764$	10.1421	0.366930
$765$	0	0
$766$	32.4853	1.17374
$767$	41.4558	1.49688
$768$	−3.41421	−0.123200
$769$	19.5563	0.705220	0.352610	−	0.935770i	$-0.385294\pi$
$769$	19.5563	0.705220	0.352610	+	0.935770i	$0.385294\pi$
$770$	0	0
$771$	−33.7990	−1.21724
$772$	−5.65685	−0.203595
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	76.4264	2.74709
$775$	0	0
$776$	7.75736	0.278473
$777$	0	0
$778$	28.1421	1.00894
$779$	−1.79899	−0.0644555
$780$	0	0
$781$	3.71573	0.132959
$782$	−3.02944	−0.108332
$783$	93.2548	3.33266
$784$	0	0
$785$	0	0
$786$	21.3137	0.760235
$787$	1.27208	0.0453447	0.0226723	−	0.999743i	$-0.492783\pi$
$787$	1.27208	0.0453447	0.0226723	+	0.999743i	$0.492783\pi$
$788$	25.7990	0.919051
$789$	95.5980	3.40338
$790$	0	0
$791$	0	0
$792$	−7.17157	−0.254831
$793$	44.9706	1.59695
$794$	−33.7990	−1.19948
$795$	0	0
$796$	16.4853	0.584305
$797$	41.7990	1.48060	0.740298	−	0.672279i	$-0.234684\pi$
$797$	41.7990	1.48060	0.740298	+	0.672279i	$0.234684\pi$
$798$	0	0
$799$	13.3726	0.473088
$800$	0	0
$801$	110.184	3.89315
$802$	−6.00000	−0.211867
$803$	−7.79899	−0.275220
$804$	5.65685	0.199502
$805$	0	0
$806$	−13.6569	−0.481042
$807$	−63.1127	−2.22167
$808$	13.3137	0.468375
$809$	3.02944	0.106509	0.0532547	−	0.998581i	$-0.483040\pi$
$809$	3.02944	0.106509	0.0532547	+	0.998581i	$0.483040\pi$
$810$	0	0
$811$	32.5858	1.14424	0.572121	−	0.820169i	$-0.306121\pi$
$811$	32.5858	1.14424	0.572121	+	0.820169i	$0.306121\pi$
$812$	0	0
$813$	−40.9706	−1.43690
$814$	−6.34315	−0.222327
$815$	0	0
$816$	8.82843	0.309057
$817$	5.17157	0.180930
$818$	−10.5858	−0.370123
$819$	0	0
$820$	0	0
$821$	17.3137	0.604253	0.302126	−	0.953268i	$-0.402304\pi$
$821$	17.3137	0.604253	0.302126	+	0.953268i	$0.402304\pi$
$822$	−54.6274	−1.90535
$823$	−20.2843	−0.707065	−0.353533	−	0.935422i	$-0.615020\pi$
$823$	−20.2843	−0.707065	−0.353533	+	0.935422i	$0.615020\pi$
$824$	14.8284	0.516573
$825$	0	0
$826$	0	0
$827$	5.37258	0.186823	0.0934115	−	0.995628i	$-0.470223\pi$
$827$	5.37258	0.186823	0.0934115	+	0.995628i	$0.470223\pi$
$828$	10.1421	0.352464
$829$	−5.02944	−0.174680	−0.0873398	−	0.996179i	$-0.527837\pi$
$829$	−5.02944	−0.174680	−0.0873398	+	0.996179i	$0.527837\pi$
$830$	0	0
$831$	−27.7990	−0.964336
$832$	4.82843	0.167396
$833$	0	0
$834$	−67.9411	−2.35261
$835$	0	0
$836$	−0.485281	−0.0167838
$837$	54.6274	1.88820
$838$	−20.8701	−0.720944
$839$	−42.1421	−1.45491	−0.727454	−	0.686156i	$-0.759297\pi$
$839$	−42.1421	−1.45491	−0.727454	+	0.686156i	$0.759297\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	17.3137	0.596670
$843$	−27.3137	−0.940734
$844$	18.6274	0.641182
$845$	0	0
$846$	−44.7696	−1.53921
$847$	0	0
$848$	−6.48528	−0.222705
$849$	−7.65685	−0.262783
$850$	0	0
$851$	8.97056	0.307507
$852$	15.3137	0.524639
$853$	43.1716	1.47817	0.739083	−	0.673614i	$-0.235259\pi$
$853$	43.1716	1.47817	0.739083	+	0.673614i	$0.235259\pi$
$854$	0	0
$855$	0	0
$856$	−9.65685	−0.330064
$857$	4.92893	0.168369	0.0841846	−	0.996450i	$-0.473171\pi$
$857$	4.92893	0.168369	0.0841846	+	0.996450i	$0.473171\pi$
$858$	13.6569	0.466237
$859$	−7.21320	−0.246111	−0.123056	−	0.992400i	$-0.539269\pi$
$859$	−7.21320	−0.246111	−0.123056	+	0.992400i	$0.539269\pi$
$860$	0	0
$861$	0	0
$862$	−22.3431	−0.761011
$863$	4.97056	0.169200	0.0846000	−	0.996415i	$-0.473039\pi$
$863$	4.97056	0.169200	0.0846000	+	0.996415i	$0.473039\pi$
$864$	−19.3137	−0.657066
$865$	0	0
$866$	−10.5858	−0.359720
$867$	35.2132	1.19590
$868$	0	0
$869$	5.65685	0.191896
$870$	0	0
$871$	−8.00000	−0.271070
$872$	2.48528	0.0841622
$873$	67.1543	2.27283
$874$	0.686292	0.0232142
$875$	0	0
$876$	−32.1421	−1.08598
$877$	30.2843	1.02263	0.511314	−	0.859394i	$-0.329159\pi$
$877$	30.2843	1.02263	0.511314	+	0.859394i	$0.329159\pi$
$878$	−24.9706	−0.842716
$879$	−28.4853	−0.960785
$880$	0	0
$881$	−2.38478	−0.0803452	−0.0401726	−	0.999193i	$-0.512791\pi$
$881$	−2.38478	−0.0803452	−0.0401726	+	0.999193i	$0.512791\pi$
$882$	0	0
$883$	41.6569	1.40186	0.700932	−	0.713228i	$-0.252767\pi$
$883$	41.6569	1.40186	0.700932	+	0.713228i	$0.252767\pi$
$884$	−12.4853	−0.419925
$885$	0	0
$886$	−3.02944	−0.101776
$887$	55.1127	1.85050	0.925252	−	0.379354i	$-0.123854\pi$
$887$	55.1127	1.85050	0.925252	+	0.379354i	$0.123854\pi$
$888$	−26.1421	−0.877273
$889$	0	0
$890$	0	0
$891$	−33.1127	−1.10932
$892$	7.31371	0.244881
$893$	−3.02944	−0.101376
$894$	20.4853	0.685130
$895$	0	0
$896$	0	0
$897$	−19.3137	−0.644866
$898$	−16.6274	−0.554864
$899$	13.6569	0.455482
$900$	0	0
$901$	16.7696	0.558675
$902$	2.54416	0.0847111
$903$	0	0
$904$	15.3137	0.509326
$905$	0	0
$906$	38.6274	1.28331
$907$	−0.284271	−0.00943907	−0.00471954	−	0.999989i	$-0.501502\pi$
$907$	−0.284271	−0.00943907	−0.00471954	+	0.999989i	$0.501502\pi$
$908$	−18.2426	−0.605403
$909$	115.255	3.82276
$910$	0	0
$911$	36.2843	1.20215	0.601076	−	0.799192i	$-0.294739\pi$
$911$	36.2843	1.20215	0.601076	+	0.799192i	$0.294739\pi$
$912$	−2.00000	−0.0662266
$913$	−1.85786	−0.0614863
$914$	−21.6569	−0.716345
$915$	0	0
$916$	−16.1421	−0.533351
$917$	0	0
$918$	49.9411	1.64830
$919$	15.5147	0.511783	0.255892	−	0.966705i	$-0.417631\pi$
$919$	15.5147	0.511783	0.255892	+	0.966705i	$0.417631\pi$
$920$	0	0
$921$	50.9706	1.67954
$922$	−12.8284	−0.422482
$923$	−21.6569	−0.712844
$924$	0	0
$925$	0	0
$926$	16.9706	0.557687
$927$	128.368	4.21614
$928$	−4.82843	−0.158501
$929$	−17.2132	−0.564747	−0.282373	−	0.959305i	$-0.591122\pi$
$929$	−17.2132	−0.564747	−0.282373	+	0.959305i	$0.591122\pi$
$930$	0	0
$931$	0	0
$932$	−23.3137	−0.763666
$933$	−13.6569	−0.447105
$934$	15.8995	0.520247
$935$	0	0
$936$	41.7990	1.36624
$937$	−20.2426	−0.661298	−0.330649	−	0.943754i	$-0.607268\pi$
$937$	−20.2426	−0.661298	−0.330649	+	0.943754i	$0.607268\pi$
$938$	0	0
$939$	49.1127	1.60273
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	22.1421	0.721430
$943$	−3.59798	−0.117166
$944$	8.58579	0.279444
$945$	0	0
$946$	−7.31371	−0.237789
$947$	4.82843	0.156903	0.0784514	−	0.996918i	$-0.475002\pi$
$947$	4.82843	0.156903	0.0784514	+	0.996918i	$0.475002\pi$
$948$	23.3137	0.757194
$949$	45.4558	1.47556
$950$	0	0
$951$	35.7990	1.16086
$952$	0	0
$953$	−0.343146	−0.0111156	−0.00555779	−	0.999985i	$-0.501769\pi$
$953$	−0.343146	−0.0111156	−0.00555779	+	0.999985i	$0.501769\pi$
$954$	−56.1421	−1.81767
$955$	0	0
$956$	1.65685	0.0535865
$957$	−13.6569	−0.441463
$958$	−17.1716	−0.554788
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	36.9706	1.19198
$963$	−83.5980	−2.69391
$964$	−13.4142	−0.432043
$965$	0	0
$966$	0	0
$967$	37.4558	1.20450	0.602249	−	0.798308i	$-0.294271\pi$
$967$	37.4558	1.20450	0.602249	+	0.798308i	$0.294271\pi$
$968$	−10.3137	−0.331495
$969$	5.17157	0.166135
$970$	0	0
$971$	−33.3553	−1.07042	−0.535212	−	0.844718i	$-0.679768\pi$
$971$	−33.3553	−1.07042	−0.535212	+	0.844718i	$0.679768\pi$
$972$	−78.5269	−2.51875
$973$	0	0
$974$	−31.7990	−1.01891
$975$	0	0
$976$	9.31371	0.298125
$977$	12.6863	0.405870	0.202935	−	0.979192i	$-0.434952\pi$
$977$	12.6863	0.405870	0.202935	+	0.979192i	$0.434952\pi$
$978$	−68.7696	−2.19901
$979$	−10.5442	−0.336993
$980$	0	0
$981$	21.5147	0.686912
$982$	32.2843	1.03023
$983$	12.2010	0.389152	0.194576	−	0.980887i	$-0.437667\pi$
$983$	12.2010	0.389152	0.194576	+	0.980887i	$0.437667\pi$
$984$	10.4853	0.334259
$985$	0	0
$986$	12.4853	0.397612
$987$	0	0
$988$	2.82843	0.0899843
$989$	10.3431	0.328893
$990$	0	0
$991$	44.7696	1.42215	0.711076	−	0.703115i	$-0.248208\pi$
$991$	44.7696	1.42215	0.711076	+	0.703115i	$0.248208\pi$
$992$	−2.82843	−0.0898027
$993$	−115.397	−3.66201
$994$	0	0
$995$	0	0
$996$	−7.65685	−0.242617
$997$	18.2843	0.579069	0.289534	−	0.957168i	$-0.406500\pi$
$997$	18.2843	0.579069	0.289534	+	0.957168i	$0.406500\pi$
$998$	30.3431	0.960495
$999$	−147.882	−4.67879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bn.1.1		2
5.2	odd	4		2450.2.c.t.99.4		4
5.3	odd	4		2450.2.c.t.99.1		4
5.4	even	2		490.2.a.m.1.2	yes	2
7.6	odd	2		2450.2.a.bs.1.2		2
15.14	odd	2		4410.2.a.bt.1.2		2
20.19	odd	2		3920.2.a.bm.1.1		2
35.4	even	6		490.2.e.i.471.1		4
35.9	even	6		490.2.e.i.361.1		4
35.13	even	4		2450.2.c.w.99.2		4
35.19	odd	6		490.2.e.j.361.2		4
35.24	odd	6		490.2.e.j.471.2		4
35.27	even	4		2450.2.c.w.99.3		4
35.34	odd	2		490.2.a.l.1.1	✓	2
105.104	even	2		4410.2.a.by.1.2		2
140.139	even	2		3920.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.l.1.1	✓	2	35.34	odd	2
490.2.a.m.1.2	yes	2	5.4	even	2
490.2.e.i.361.1		4	35.9	even	6
490.2.e.i.471.1		4	35.4	even	6
490.2.e.j.361.2		4	35.19	odd	6
490.2.e.j.471.2		4	35.24	odd	6
2450.2.a.bn.1.1		2	1.1	even	1	trivial
2450.2.a.bs.1.2		2	7.6	odd	2
2450.2.c.t.99.1		4	5.3	odd	4
2450.2.c.t.99.4		4	5.2	odd	4
2450.2.c.w.99.2		4	35.13	even	4
2450.2.c.w.99.3		4	35.27	even	4
3920.2.a.bm.1.1		2	20.19	odd	2
3920.2.a.ca.1.2		2	140.139	even	2
4410.2.a.bt.1.2		2	15.14	odd	2
4410.2.a.by.1.2		2	105.104	even	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.41421 q^{3} +1.00000 q^{4} -3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.41421 q^{3} +1.00000 q^{4} -3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} -0.828427 q^{11} -3.41421 q^{12} +4.82843 q^{13} +1.00000 q^{16} -2.58579 q^{17} +8.65685 q^{18} +0.585786 q^{19} -0.828427 q^{22} +1.17157 q^{23} -3.41421 q^{24} +4.82843 q^{26} -19.3137 q^{27} -4.82843 q^{29} -2.82843 q^{31} +1.00000 q^{32} +2.82843 q^{33} -2.58579 q^{34} +8.65685 q^{36} +7.65685 q^{37} +0.585786 q^{38} -16.4853 q^{39} -3.07107 q^{41} +8.82843 q^{43} -0.828427 q^{44} +1.17157 q^{46} -5.17157 q^{47} -3.41421 q^{48} +8.82843 q^{51} +4.82843 q^{52} -6.48528 q^{53} -19.3137 q^{54} -2.00000 q^{57} -4.82843 q^{58} +8.58579 q^{59} +9.31371 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.82843 q^{66} -1.65685 q^{67} -2.58579 q^{68} -4.00000 q^{69} -4.48528 q^{71} +8.65685 q^{72} +9.41421 q^{73} +7.65685 q^{74} +0.585786 q^{76} -16.4853 q^{78} -6.82843 q^{79} +39.9706 q^{81} -3.07107 q^{82} +2.24264 q^{83} +8.82843 q^{86} +16.4853 q^{87} -0.828427 q^{88} +12.7279 q^{89} +1.17157 q^{92} +9.65685 q^{93} -5.17157 q^{94} -3.41421 q^{96} +7.75736 q^{97} -7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^6 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 6 q^{9} + 4 q^{11} - 4 q^{12} + 4 q^{13} + 2 q^{16} - 8 q^{17} + 6 q^{18} + 4 q^{19} + 4 q^{22} + 8 q^{23} - 4 q^{24} + 4 q^{26} - 16 q^{27} - 4 q^{29} + 2 q^{32} - 8 q^{34} + 6 q^{36} + 4 q^{37} + 4 q^{38} - 16 q^{39} + 8 q^{41} + 12 q^{43} + 4 q^{44} + 8 q^{46} - 16 q^{47} - 4 q^{48} + 12 q^{51} + 4 q^{52} + 4 q^{53} - 16 q^{54} - 4 q^{57} - 4 q^{58} + 20 q^{59} - 4 q^{61} + 2 q^{64} + 8 q^{67} - 8 q^{68} - 8 q^{69} + 8 q^{71} + 6 q^{72} + 16 q^{73} + 4 q^{74} + 4 q^{76} - 16 q^{78} - 8 q^{79} + 46 q^{81} + 8 q^{82} - 4 q^{83} + 12 q^{86} + 16 q^{87} + 4 q^{88} + 8 q^{92} + 8 q^{93} - 16 q^{94} - 4 q^{96} + 24 q^{97} - 20 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^6 + 2 * q^8 + 6 * q^9 + 4 * q^11 - 4 * q^12 + 4 * q^13 + 2 * q^16 - 8 * q^17 + 6 * q^18 + 4 * q^19 + 4 * q^22 + 8 * q^23 - 4 * q^24 + 4 * q^26 - 16 * q^27 - 4 * q^29 + 2 * q^32 - 8 * q^34 + 6 * q^36 + 4 * q^37 + 4 * q^38 - 16 * q^39 + 8 * q^41 + 12 * q^43 + 4 * q^44 + 8 * q^46 - 16 * q^47 - 4 * q^48 + 12 * q^51 + 4 * q^52 + 4 * q^53 - 16 * q^54 - 4 * q^57 - 4 * q^58 + 20 * q^59 - 4 * q^61 + 2 * q^64 + 8 * q^67 - 8 * q^68 - 8 * q^69 + 8 * q^71 + 6 * q^72 + 16 * q^73 + 4 * q^74 + 4 * q^76 - 16 * q^78 - 8 * q^79 + 46 * q^81 + 8 * q^82 - 4 * q^83 + 12 * q^86 + 16 * q^87 + 4 * q^88 + 8 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 + 24 * q^97 - 20 * q^99

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