LMFDB - Embedded newform 1001.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1001,2,Mod(1,1001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1001 = 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1001.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:	$7.99302524233$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23301.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2x + 1 $ x^4 - x^3 - 6x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.82757$ of defining polynomial
Character	$\chi$	$=$	1001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35366 q^{2} -2.82757 q^{3} -0.167605 q^{4} +0.473915 q^{5} +3.82757 q^{6} +1.00000 q^{7} +2.93420 q^{8} +4.99518 q^{9} +O(q^{10})$	\(q-1.35366 q^{2} -2.82757 q^{3} -0.167605 q^{4} +0.473915 q^{5} +3.82757 q^{6} +1.00000 q^{7} +2.93420 q^{8} +4.99518 q^{9} -0.641520 q^{10} -1.00000 q^{11} +0.473915 q^{12} -1.00000 q^{13} -1.35366 q^{14} -1.34003 q^{15} -3.63670 q^{16} -1.18123 q^{17} -6.76177 q^{18} +2.64152 q^{19} -0.0794304 q^{20} -2.82757 q^{21} +1.35366 q^{22} +0.986370 q^{23} -8.29667 q^{24} -4.77540 q^{25} +1.35366 q^{26} -5.64152 q^{27} -0.167605 q^{28} -5.28786 q^{29} +1.81394 q^{30} +0.0521702 q^{31} -0.945545 q^{32} +2.82757 q^{33} +1.59899 q^{34} +0.473915 q^{35} -0.837216 q^{36} -3.53972 q^{37} -3.57572 q^{38} +2.82757 q^{39} +1.39056 q^{40} -0.353660 q^{41} +3.82757 q^{42} +0.0473498 q^{43} +0.167605 q^{44} +2.36729 q^{45} -1.33521 q^{46} +2.98637 q^{47} +10.2830 q^{48} +1.00000 q^{49} +6.46427 q^{50} +3.34003 q^{51} +0.167605 q^{52} -6.84120 q^{53} +7.63670 q^{54} -0.473915 q^{55} +2.93420 q^{56} -7.46909 q^{57} +7.15796 q^{58} +7.94301 q^{59} +0.224595 q^{60} +13.5822 q^{61} -0.0706207 q^{62} +4.99518 q^{63} +8.55335 q^{64} -0.473915 q^{65} -3.82757 q^{66} +8.99036 q^{67} +0.197981 q^{68} -2.78903 q^{69} -0.641520 q^{70} +9.31030 q^{71} +14.6569 q^{72} -1.11543 q^{73} +4.79157 q^{74} +13.5028 q^{75} -0.442731 q^{76} -1.00000 q^{77} -3.82757 q^{78} -0.181235 q^{79} -1.72349 q^{80} +0.966280 q^{81} +0.478735 q^{82} +1.48754 q^{83} +0.473915 q^{84} -0.559805 q^{85} -0.0640955 q^{86} +14.9518 q^{87} -2.93420 q^{88} +17.0312 q^{89} -3.20451 q^{90} -1.00000 q^{91} -0.165320 q^{92} -0.147515 q^{93} -4.04253 q^{94} +1.25186 q^{95} +2.67360 q^{96} -5.00881 q^{97} -1.35366 q^{98} -4.99518 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 4 * q^7 + 3 * q^8 + q^9	\( 4 q - 2 q^{2} - q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + 13 q^{10} - 4 q^{11} - 5 q^{12} - 4 q^{13} - 2 q^{14} - 7 q^{15} + 20 q^{16} + 9 q^{17} - 8 q^{18} - 5 q^{19} - 4 q^{20} - q^{21} + 2 q^{22} + 9 q^{23} + 3 q^{24} + 5 q^{25} + 2 q^{26} - 7 q^{27} + 8 q^{28} - 9 q^{29} + 2 q^{30} + 14 q^{31} - 16 q^{32} + q^{33} + 15 q^{34} - 5 q^{35} - 31 q^{36} - 16 q^{37} + 10 q^{38} + q^{39} + 30 q^{40} + 2 q^{41} + 5 q^{42} - 5 q^{43} - 8 q^{44} + q^{45} + 12 q^{46} + 17 q^{47} + 10 q^{48} + 4 q^{49} - 19 q^{50} + 15 q^{51} - 8 q^{52} - 12 q^{53} - 4 q^{54} + 5 q^{55} + 3 q^{56} - 4 q^{57} - 18 q^{58} - q^{59} + 25 q^{60} + 32 q^{61} - 28 q^{62} + q^{63} + 31 q^{64} + 5 q^{65} - 5 q^{66} - 2 q^{67} + 18 q^{68} + 18 q^{69} + 13 q^{70} - 4 q^{71} + 21 q^{72} + 18 q^{73} + 35 q^{74} + 28 q^{75} - 40 q^{76} - 4 q^{77} - 5 q^{78} + 13 q^{79} - 40 q^{80} + 4 q^{81} + 14 q^{82} - 6 q^{83} - 5 q^{84} - 9 q^{85} - 26 q^{86} + 3 q^{87} - 3 q^{88} + 15 q^{89} - 32 q^{90} - 4 q^{91} - 18 q^{92} + 13 q^{93} + 8 q^{94} + 19 q^{95} + 4 q^{96} + 4 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 4 * q^7 + 3 * q^8 + q^9 + 13 * q^10 - 4 * q^11 - 5 * q^12 - 4 * q^13 - 2 * q^14 - 7 * q^15 + 20 * q^16 + 9 * q^17 - 8 * q^18 - 5 * q^19 - 4 * q^20 - q^21 + 2 * q^22 + 9 * q^23 + 3 * q^24 + 5 * q^25 + 2 * q^26 - 7 * q^27 + 8 * q^28 - 9 * q^29 + 2 * q^30 + 14 * q^31 - 16 * q^32 + q^33 + 15 * q^34 - 5 * q^35 - 31 * q^36 - 16 * q^37 + 10 * q^38 + q^39 + 30 * q^40 + 2 * q^41 + 5 * q^42 - 5 * q^43 - 8 * q^44 + q^45 + 12 * q^46 + 17 * q^47 + 10 * q^48 + 4 * q^49 - 19 * q^50 + 15 * q^51 - 8 * q^52 - 12 * q^53 - 4 * q^54 + 5 * q^55 + 3 * q^56 - 4 * q^57 - 18 * q^58 - q^59 + 25 * q^60 + 32 * q^61 - 28 * q^62 + q^63 + 31 * q^64 + 5 * q^65 - 5 * q^66 - 2 * q^67 + 18 * q^68 + 18 * q^69 + 13 * q^70 - 4 * q^71 + 21 * q^72 + 18 * q^73 + 35 * q^74 + 28 * q^75 - 40 * q^76 - 4 * q^77 - 5 * q^78 + 13 * q^79 - 40 * q^80 + 4 * q^81 + 14 * q^82 - 6 * q^83 - 5 * q^84 - 9 * q^85 - 26 * q^86 + 3 * q^87 - 3 * q^88 + 15 * q^89 - 32 * q^90 - 4 * q^91 - 18 * q^92 + 13 * q^93 + 8 * q^94 + 19 * q^95 + 4 * q^96 + 4 * q^97 - 2 * q^98 - 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 $

$2$	−1.35366	−0.957182	−0.478591	−	0.878038i	$-0.658852\pi$
$2$	−1.35366	−0.957182	−0.478591	+	0.878038i	$0.658852\pi$
$3$	−2.82757	−1.63250	−0.816251	−	0.577698i	$-0.803951\pi$
$3$	−2.82757	−1.63250	−0.816251	+	0.577698i	$0.803951\pi$
$4$	−0.167605	−0.0838024
$5$	0.473915	0.211941	0.105971	−	0.994369i	$-0.466205\pi$
$5$	0.473915	0.211941	0.105971	+	0.994369i	$0.466205\pi$
$6$	3.82757	1.56260
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.93420	1.03740
$9$	4.99518	1.66506
$10$	−0.641520	−0.202866
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.473915	0.136807
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.35366	−0.361781
$15$	−1.34003	−0.345994
$16$	−3.63670	−0.909175
$17$	−1.18123	−0.286492	−0.143246	−	0.989687i	$-0.545754\pi$
$17$	−1.18123	−0.286492	−0.143246	+	0.989687i	$0.545754\pi$
$18$	−6.76177	−1.59377
$19$	2.64152	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$19$	2.64152	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$20$	−0.0794304	−0.0177612
$21$	−2.82757	−0.617027
$22$	1.35366	0.288601
$23$	0.986370	0.205672	0.102836	−	0.994698i	$-0.467208\pi$
$23$	0.986370	0.205672	0.102836	+	0.994698i	$0.467208\pi$
$24$	−8.29667	−1.69355
$25$	−4.77540	−0.955081
$26$	1.35366	0.265475
$27$	−5.64152	−1.08571
$28$	−0.167605	−0.0316743
$29$	−5.28786	−0.981931	−0.490965	−	0.871179i	$-0.663356\pi$
$29$	−5.28786	−0.981931	−0.490965	+	0.871179i	$0.663356\pi$
$30$	1.81394	0.331179
$31$	0.0521702	0.00937004	0.00468502	−	0.999989i	$-0.498509\pi$
$31$	0.0521702	0.00937004	0.00468502	+	0.999989i	$0.498509\pi$
$32$	−0.945545	−0.167150
$33$	2.82757	0.492218
$34$	1.59899	0.274225
$35$	0.473915	0.0801062
$36$	−0.837216	−0.139536
$37$	−3.53972	−0.581926	−0.290963	−	0.956734i	$-0.593976\pi$
$37$	−3.53972	−0.581926	−0.290963	+	0.956734i	$0.593976\pi$
$38$	−3.57572	−0.580058
$39$	2.82757	0.452774
$40$	1.39056	0.219867
$41$	−0.353660	−0.0552324	−0.0276162	−	0.999619i	$-0.508792\pi$
$41$	−0.353660	−0.0552324	−0.0276162	+	0.999619i	$0.508792\pi$
$42$	3.82757	0.590608
$43$	0.0473498	0.00722077	0.00361039	−	0.999993i	$-0.498851\pi$
$43$	0.0473498	0.00722077	0.00361039	+	0.999993i	$0.498851\pi$
$44$	0.167605	0.0252674
$45$	2.36729	0.352895
$46$	−1.33521	−0.196866
$47$	2.98637	0.435607	0.217803	−	0.975993i	$-0.430111\pi$
$47$	2.98637	0.435607	0.217803	+	0.975993i	$0.430111\pi$
$48$	10.2830	1.48423
$49$	1.00000	0.142857
$50$	6.46427	0.914186
$51$	3.34003	0.467698
$52$	0.167605	0.0232426
$53$	−6.84120	−0.939712	−0.469856	−	0.882743i	$-0.655694\pi$
$53$	−6.84120	−0.939712	−0.469856	+	0.882743i	$0.655694\pi$
$54$	7.63670	1.03922
$55$	−0.473915	−0.0639027
$56$	2.93420	0.392099
$57$	−7.46909	−0.989306
$58$	7.15796	0.939887
$59$	7.94301	1.03409	0.517046	−	0.855958i	$-0.327032\pi$
$59$	7.94301	1.03409	0.517046	+	0.855958i	$0.327032\pi$
$60$	0.224595	0.0289951
$61$	13.5822	1.73903	0.869514	−	0.493908i	$-0.164432\pi$
$61$	13.5822	1.73903	0.869514	+	0.493908i	$0.164432\pi$
$62$	−0.0706207	−0.00896884
$63$	4.99518	0.629333
$64$	8.55335	1.06917
$65$	−0.473915	−0.0587819
$66$	−3.82757	−0.471142
$67$	8.99036	1.09835	0.549174	−	0.835708i	$-0.314943\pi$
$67$	8.99036	1.09835	0.549174	+	0.835708i	$0.314943\pi$
$68$	0.197981	0.0240087
$69$	−2.78903	−0.335760
$70$	−0.641520	−0.0766763
$71$	9.31030	1.10493	0.552465	−	0.833536i	$-0.313687\pi$
$71$	9.31030	1.10493	0.552465	+	0.833536i	$0.313687\pi$
$72$	14.6569	1.72733
$73$	−1.11543	−0.130552	−0.0652759	−	0.997867i	$-0.520793\pi$
$73$	−1.11543	−0.130552	−0.0652759	+	0.997867i	$0.520793\pi$
$74$	4.79157	0.557009
$75$	13.5028	1.55917
$76$	−0.442731	−0.0507847
$77$	−1.00000	−0.113961
$78$	−3.82757	−0.433388
$79$	−0.181235	−0.0203905	−0.0101953	−	0.999948i	$-0.503245\pi$
$79$	−0.181235	−0.0203905	−0.0101953	+	0.999948i	$0.503245\pi$
$80$	−1.72349	−0.192692
$81$	0.966280	0.107364
$82$	0.478735	0.0528675
$83$	1.48754	0.163279	0.0816396	−	0.996662i	$-0.473984\pi$
$83$	1.48754	0.163279	0.0816396	+	0.996662i	$0.473984\pi$
$84$	0.473915	0.0517084
$85$	−0.559805	−0.0607194
$86$	−0.0640955	−0.00691160
$87$	14.9518	1.60300
$88$	−2.93420	−0.312787
$89$	17.0312	1.80531	0.902654	−	0.430366i	$-0.141616\pi$
$89$	17.0312	1.80531	0.902654	+	0.430366i	$0.141616\pi$
$90$	−3.20451	−0.337785
$91$	−1.00000	−0.104828
$92$	−0.165320	−0.0172358
$93$	−0.147515	−0.0152966
$94$	−4.04253	−0.416955
$95$	1.25186	0.128438
$96$	2.67360	0.272873
$97$	−5.00881	−0.508568	−0.254284	−	0.967130i	$-0.581840\pi$
$97$	−5.00881	−0.508568	−0.254284	+	0.967130i	$0.581840\pi$
$98$	−1.35366	−0.136740
$99$	−4.99518	−0.502034
$100$	0.800380	0.0800380
$101$	−11.8500	−1.17912	−0.589560	−	0.807724i	$-0.700699\pi$
$101$	−11.8500	−1.17912	−0.589560	+	0.807724i	$0.700699\pi$
$102$	−4.52126	−0.447672
$103$	13.4739	1.32762	0.663812	−	0.747899i	$-0.268937\pi$
$103$	13.4739	1.32762	0.663812	+	0.747899i	$0.268937\pi$
$104$	−2.93420	−0.287722
$105$	−1.34003	−0.130774
$106$	9.26067	0.899475
$107$	−20.5749	−1.98905	−0.994525	−	0.104501i	$-0.966675\pi$
$107$	−20.5749	−1.98905	−0.994525	+	0.104501i	$0.966675\pi$
$108$	0.945545	0.0909851
$109$	8.06979	0.772946	0.386473	−	0.922301i	$-0.373693\pi$
$109$	8.06979	0.772946	0.386473	+	0.922301i	$0.373693\pi$
$110$	0.641520	0.0611665
$111$	10.0088	0.949994
$112$	−3.63670	−0.343636
$113$	−10.4370	−0.981832	−0.490916	−	0.871207i	$-0.663338\pi$
$113$	−10.4370	−0.981832	−0.490916	+	0.871207i	$0.663338\pi$
$114$	10.1106	0.946946
$115$	0.467455	0.0435904
$116$	0.886270	0.0822881
$117$	−4.99518	−0.461805
$118$	−10.7521	−0.989814
$119$	−1.18123	−0.108284
$120$	−3.93192	−0.358933
$121$	1.00000	0.0909091
$122$	−18.3857	−1.66457
$123$	1.00000	0.0901670
$124$	−0.00874397	−0.000785232 0
$125$	−4.63271	−0.414362
$126$	−6.76177	−0.602387
$127$	8.08335	0.717282	0.358641	−	0.933476i	$-0.383240\pi$
$127$	8.08335	0.717282	0.358641	+	0.933476i	$0.383240\pi$
$128$	−9.68723	−0.856238
$129$	−0.133885	−0.0117879
$130$	0.641520	0.0562650
$131$	−0.566910	−0.0495311	−0.0247656	−	0.999693i	$-0.507884\pi$
$131$	−0.566910	−0.0495311	−0.0247656	+	0.999693i	$0.507884\pi$
$132$	−0.473915	−0.0412490
$133$	2.64152	0.229049
$134$	−12.1699	−1.05132
$135$	−2.67360	−0.230107
$136$	−3.46598	−0.297205
$137$	14.9342	1.27591	0.637957	−	0.770072i	$-0.279780\pi$
$137$	14.9342	1.27591	0.637957	+	0.770072i	$0.279780\pi$
$138$	3.77540	0.321384
$139$	21.0905	1.78887	0.894437	−	0.447193i	$-0.147576\pi$
$139$	21.0905	1.78887	0.894437	+	0.447193i	$0.147576\pi$
$140$	−0.0794304	−0.00671309
$141$	−8.44418	−0.711129
$142$	−12.6030	−1.05762
$143$	1.00000	0.0836242
$144$	−18.1660	−1.51383
$145$	−2.50600	−0.208112
$146$	1.50992	0.124962
$147$	−2.82757	−0.233214
$148$	0.593273	0.0487667
$149$	6.32393	0.518076	0.259038	−	0.965867i	$-0.416594\pi$
$149$	6.32393	0.518076	0.259038	+	0.965867i	$0.416594\pi$
$150$	−18.2782	−1.49241
$151$	−1.06808	−0.0869195	−0.0434598	−	0.999055i	$-0.513838\pi$
$151$	−1.06808	−0.0869195	−0.0434598	+	0.999055i	$0.513838\pi$
$152$	7.75075	0.628669
$153$	−5.90048	−0.477026
$154$	1.35366	0.109081
$155$	0.0247242	0.00198590
$156$	−0.473915	−0.0379436
$157$	−5.85484	−0.467267	−0.233633	−	0.972325i	$-0.575062\pi$
$157$	−5.85484	−0.467267	−0.233633	+	0.972325i	$0.575062\pi$
$158$	0.245330	0.0195174
$159$	19.3440	1.53408
$160$	−0.448108	−0.0354261
$161$	0.986370	0.0777368
$162$	−1.30801	−0.102767
$163$	1.01591	0.0795726	0.0397863	−	0.999208i	$-0.487332\pi$
$163$	1.01591	0.0795726	0.0397863	+	0.999208i	$0.487332\pi$
$164$	0.0592751	0.00462861
$165$	1.34003	0.104321
$166$	−2.01363	−0.156288
$167$	11.1475	0.862621	0.431310	−	0.902204i	$-0.358051\pi$
$167$	11.1475	0.862621	0.431310	+	0.902204i	$0.358051\pi$
$168$	−8.29667	−0.640102
$169$	1.00000	0.0769231
$170$	0.757785	0.0581195
$171$	13.1949	1.00904
$172$	−0.00793605	−0.000605118 0
$173$	−6.70960	−0.510122	−0.255061	−	0.966925i	$-0.582096\pi$
$173$	−6.70960	−0.510122	−0.255061	+	0.966925i	$0.582096\pi$
$174$	−20.2397	−1.53437
$175$	−4.77540	−0.360987
$176$	3.63670	0.274127
$177$	−22.4595	−1.68816
$178$	−23.0545	−1.72801
$179$	−10.4787	−0.783217	−0.391609	−	0.920132i	$-0.628081\pi$
$179$	−10.4787	−0.783217	−0.391609	+	0.920132i	$0.628081\pi$
$180$	−0.396769	−0.0295734
$181$	−1.73622	−0.129052	−0.0645261	−	0.997916i	$-0.520554\pi$
$181$	−1.73622	−0.129052	−0.0645261	+	0.997916i	$0.520554\pi$
$182$	1.35366	0.100340
$183$	−38.4048	−2.83897
$184$	2.89421	0.213364
$185$	−1.67752	−0.123334
$186$	0.199685	0.0146416
$187$	1.18123	0.0863804
$188$	−0.500530	−0.0365049
$189$	−5.64152	−0.410360
$190$	−1.69459	−0.122938
$191$	7.81311	0.565337	0.282669	−	0.959218i	$-0.408780\pi$
$191$	7.81311	0.565337	0.282669	+	0.959218i	$0.408780\pi$
$192$	−24.1852	−1.74542
$193$	21.3729	1.53846	0.769228	−	0.638974i	$-0.220641\pi$
$193$	21.3729	1.53846	0.769228	+	0.638974i	$0.220641\pi$
$194$	6.78023	0.486792
$195$	1.34003	0.0959615
$196$	−0.167605	−0.0119718
$197$	−8.77457	−0.625162	−0.312581	−	0.949891i	$-0.601194\pi$
$197$	−8.77457	−0.625162	−0.312581	+	0.949891i	$0.601194\pi$
$198$	6.76177	0.480538
$199$	16.7201	1.18526	0.592629	−	0.805476i	$-0.298090\pi$
$199$	16.7201	1.18526	0.592629	+	0.805476i	$0.298090\pi$
$200$	−14.0120	−0.990797
$201$	−25.4209	−1.79305
$202$	16.0409	1.12863
$203$	−5.28786	−0.371135
$204$	−0.559805	−0.0391942
$205$	−0.167605	−0.0117060
$206$	−18.2391	−1.27078
$207$	4.92709	0.342457
$208$	3.63670	0.252160
$209$	−2.64152	−0.182718
$210$	1.81394	0.125174
$211$	23.6166	1.62583	0.812917	−	0.582379i	$-0.197878\pi$
$211$	23.6166	1.62583	0.812917	+	0.582379i	$0.197878\pi$
$212$	1.14662	0.0787501
$213$	−26.3256	−1.80380
$214$	27.8514	1.90388
$215$	0.0224398	0.00153038
$216$	−16.5533	−1.12631
$217$	0.0521702	0.00354154
$218$	−10.9238	−0.739850
$219$	3.15397	0.213126
$220$	0.0794304	0.00535519
$221$	1.18123	0.0794585
$222$	−13.5485	−0.909317
$223$	7.24279	0.485013	0.242507	−	0.970150i	$-0.422030\pi$
$223$	7.24279	0.485013	0.242507	+	0.970150i	$0.422030\pi$
$224$	−0.945545	−0.0631769
$225$	−23.8540	−1.59027
$226$	14.1282	0.939792
$227$	−18.2711	−1.21270	−0.606348	−	0.795199i	$-0.707366\pi$
$227$	−18.2711	−1.21270	−0.606348	+	0.795199i	$0.707366\pi$
$228$	1.25186	0.0829062
$229$	12.3729	0.817625	0.408813	−	0.912618i	$-0.365943\pi$
$229$	12.3729	0.817625	0.408813	+	0.912618i	$0.365943\pi$
$230$	−0.632776	−0.0417240
$231$	2.82757	0.186041
$232$	−15.5156	−1.01865
$233$	12.7770	0.837052	0.418526	−	0.908205i	$-0.362547\pi$
$233$	12.7770	0.837052	0.418526	+	0.908205i	$0.362547\pi$
$234$	6.76177	0.442031
$235$	1.41529	0.0923230
$236$	−1.33129	−0.0866593
$237$	0.512455	0.0332875
$238$	1.59899	0.103647
$239$	19.9615	1.29120	0.645600	−	0.763676i	$-0.276607\pi$
$239$	19.9615	1.29120	0.645600	+	0.763676i	$0.276607\pi$
$240$	4.87329	0.314569
$241$	16.0425	1.03339	0.516695	−	0.856169i	$-0.327162\pi$
$241$	16.0425	1.03339	0.516695	+	0.856169i	$0.327162\pi$
$242$	−1.35366	−0.0870166
$243$	14.1923	0.910438
$244$	−2.27645	−0.145735
$245$	0.473915	0.0302773
$246$	−1.35366	−0.0863062
$247$	−2.64152	−0.168076
$248$	0.153078	0.00972045
$249$	−4.20614	−0.266554
$250$	6.27111	0.396620
$251$	−11.1883	−0.706202	−0.353101	−	0.935585i	$-0.614873\pi$
$251$	−11.1883	−0.706202	−0.353101	+	0.935585i	$0.614873\pi$
$252$	−0.837216	−0.0527396
$253$	−0.986370	−0.0620125
$254$	−10.9421	−0.686569
$255$	1.58289	0.0991244
$256$	−3.99347	−0.249592
$257$	−7.13871	−0.445300	−0.222650	−	0.974898i	$-0.571471\pi$
$257$	−7.13871	−0.445300	−0.222650	+	0.974898i	$0.571471\pi$
$258$	0.181235	0.0112832
$259$	−3.53972	−0.219947
$260$	0.0794304	0.00492606
$261$	−26.4138	−1.63497
$262$	0.767403	0.0474103
$263$	5.97274	0.368295	0.184147	−	0.982899i	$-0.441048\pi$
$263$	5.97274	0.368295	0.184147	+	0.982899i	$0.441048\pi$
$264$	8.29667	0.510625
$265$	−3.24215	−0.199164
$266$	−3.57572	−0.219241
$267$	−48.1571	−2.94717
$268$	−1.50683	−0.0920441
$269$	15.2799	0.931633	0.465817	−	0.884881i	$-0.345761\pi$
$269$	15.2799	0.931633	0.465817	+	0.884881i	$0.345761\pi$
$270$	3.61915	0.220254
$271$	5.71613	0.347230	0.173615	−	0.984814i	$-0.444455\pi$
$271$	5.71613	0.347230	0.173615	+	0.984814i	$0.444455\pi$
$272$	4.29580	0.260471
$273$	2.82757	0.171133
$274$	−20.2158	−1.22128
$275$	4.77540	0.287968
$276$	0.467455	0.0281375
$277$	5.42174	0.325761	0.162881	−	0.986646i	$-0.447921\pi$
$277$	5.42174	0.325761	0.162881	+	0.986646i	$0.447921\pi$
$278$	−28.5494	−1.71228
$279$	0.260599	0.0156017
$280$	1.39056	0.0831019
$281$	15.4257	0.920222	0.460111	−	0.887861i	$-0.347810\pi$
$281$	15.4257	0.920222	0.460111	+	0.887861i	$0.347810\pi$
$282$	11.4306	0.680680
$283$	5.59417	0.332539	0.166269	−	0.986080i	$-0.446828\pi$
$283$	5.59417	0.332539	0.166269	+	0.986080i	$0.446828\pi$
$284$	−1.56045	−0.0925957
$285$	−3.53972	−0.209675
$286$	−1.35366	−0.0800436
$287$	−0.353660	−0.0208759
$288$	−4.72317	−0.278315
$289$	−15.6047	−0.917923
$290$	3.39227	0.199201
$291$	14.1628	0.830237
$292$	0.186952	0.0109405
$293$	−5.29503	−0.309339	−0.154669	−	0.987966i	$-0.549431\pi$
$293$	−5.29503	−0.309339	−0.154669	+	0.987966i	$0.549431\pi$
$294$	3.82757	0.223229
$295$	3.76431	0.219167
$296$	−10.3862	−0.603687
$297$	5.64152	0.327354
$298$	−8.56045	−0.495894
$299$	−0.986370	−0.0570432
$300$	−2.26314	−0.130662
$301$	0.0473498	0.00272920
$302$	1.44582	0.0831978
$303$	33.5068	1.92492
$304$	−9.60641	−0.550966
$305$	6.43683	0.368572
$306$	7.98724	0.456600
$307$	5.06410	0.289023	0.144512	−	0.989503i	$-0.453839\pi$
$307$	5.06410	0.289023	0.144512	+	0.989503i	$0.453839\pi$
$308$	0.167605	0.00955017
$309$	−38.0985	−2.16735
$310$	−0.0334682	−0.00190087
$311$	26.5781	1.50710	0.753552	−	0.657388i	$-0.228339\pi$
$311$	26.5781	1.50710	0.753552	+	0.657388i	$0.228339\pi$
$312$	8.29667	0.469706
$313$	28.3696	1.60354	0.801772	−	0.597631i	$-0.203891\pi$
$313$	28.3696	1.60354	0.801772	+	0.597631i	$0.203891\pi$
$314$	7.92546	0.447259
$315$	2.36729	0.133382
$316$	0.0303758	0.00170877
$317$	−34.1772	−1.91958	−0.959792	−	0.280712i	$-0.909429\pi$
$317$	−34.1772	−1.91958	−0.959792	+	0.280712i	$0.909429\pi$
$318$	−26.1852	−1.46839
$319$	5.28786	0.296063
$320$	4.05356	0.226601
$321$	58.1770	3.24713
$322$	−1.33521	−0.0744083
$323$	−3.12025	−0.173616
$324$	−0.161953	−0.00899740
$325$	4.77540	0.264892
$326$	−1.37520	−0.0761654
$327$	−22.8179	−1.26183
$328$	−1.03771	−0.0572979
$329$	2.98637	0.164644
$330$	−1.81394	−0.0998544
$331$	−14.6222	−0.803711	−0.401855	−	0.915703i	$-0.631635\pi$
$331$	−14.6222	−0.803711	−0.401855	+	0.915703i	$0.631635\pi$
$332$	−0.249320	−0.0136832
$333$	−17.6815	−0.968941
$334$	−15.0899	−0.825685
$335$	4.26067	0.232785
$336$	10.2830	0.560986
$337$	5.85059	0.318702	0.159351	−	0.987222i	$-0.449060\pi$
$337$	5.85059	0.318702	0.159351	+	0.987222i	$0.449060\pi$
$338$	−1.35366	−0.0736294
$339$	29.5114	1.60284
$340$	0.0938259	0.00508843
$341$	−0.0521702	−0.00282517
$342$	−17.8614	−0.965832
$343$	1.00000	0.0539949
$344$	0.138934	0.00749080
$345$	−1.32177	−0.0711614
$346$	9.08252	0.488279
$347$	−22.8628	−1.22734	−0.613670	−	0.789563i	$-0.710307\pi$
$347$	−22.8628	−1.22734	−0.613670	+	0.789563i	$0.710307\pi$
$348$	−2.50600	−0.134335
$349$	32.5049	1.73995	0.869974	−	0.493098i	$-0.164135\pi$
$349$	32.5049	1.73995	0.869974	+	0.493098i	$0.164135\pi$
$350$	6.46427	0.345530
$351$	5.64152	0.301122
$352$	0.945545	0.0503977
$353$	12.0674	0.642285	0.321142	−	0.947031i	$-0.395933\pi$
$353$	12.0674	0.642285	0.321142	+	0.947031i	$0.395933\pi$
$354$	30.4025	1.61587
$355$	4.41229	0.234180
$356$	−2.85452	−0.151289
$357$	3.34003	0.176773
$358$	14.1846	0.749682
$359$	−3.91909	−0.206842	−0.103421	−	0.994638i	$-0.532979\pi$
$359$	−3.91909	−0.206842	−0.103421	+	0.994638i	$0.532979\pi$
$360$	6.94610	0.366092
$361$	−12.0224	−0.632757
$362$	2.35025	0.123526
$363$	−2.82757	−0.148409
$364$	0.167605	0.00878487
$365$	−0.528621	−0.0276693
$366$	51.9871	2.71741
$367$	−16.3095	−0.851347	−0.425674	−	0.904877i	$-0.639963\pi$
$367$	−16.3095	−0.851347	−0.425674	+	0.904877i	$0.639963\pi$
$368$	−3.58713	−0.186992
$369$	−1.76659	−0.0919653
$370$	2.27080	0.118053
$371$	−6.84120	−0.355178
$372$	0.0247242	0.00128189
$373$	24.8556	1.28698	0.643488	−	0.765456i	$-0.277487\pi$
$373$	24.8556	1.28698	0.643488	+	0.765456i	$0.277487\pi$
$374$	−1.59899	−0.0826818
$375$	13.0993	0.676447
$376$	8.76261	0.451897
$377$	5.28786	0.272339
$378$	7.63670	0.392789
$379$	−3.15227	−0.161921	−0.0809606	−	0.996717i	$-0.525799\pi$
$379$	−3.15227	−0.161921	−0.0809606	+	0.996717i	$0.525799\pi$
$380$	−0.209817	−0.0107634
$381$	−22.8563	−1.17096
$382$	−10.5763	−0.541131
$383$	−16.3664	−0.836284	−0.418142	−	0.908382i	$-0.637318\pi$
$383$	−16.3664	−0.836284	−0.418142	+	0.908382i	$0.637318\pi$
$384$	27.3914	1.39781
$385$	−0.473915	−0.0241529
$386$	−28.9317	−1.47258
$387$	0.236521	0.0120230
$388$	0.839500	0.0426192
$389$	20.7383	1.05147	0.525737	−	0.850647i	$-0.323790\pi$
$389$	20.7383	1.05147	0.525737	+	0.850647i	$0.323790\pi$
$390$	−1.81394	−0.0918527
$391$	−1.16513	−0.0589234
$392$	2.93420	0.148199
$393$	1.60298	0.0808596
$394$	11.8778	0.598394
$395$	−0.0858899	−0.00432159
$396$	0.837216	0.0420717
$397$	−31.1217	−1.56195	−0.780977	−	0.624560i	$-0.785278\pi$
$397$	−31.1217	−1.56195	−0.780977	+	0.624560i	$0.785278\pi$
$398$	−22.6334	−1.13451
$399$	−7.46909	−0.373922
$400$	17.3667	0.868336
$401$	−18.0995	−0.903847	−0.451923	−	0.892057i	$-0.649262\pi$
$401$	−18.0995	−0.903847	−0.451923	+	0.892057i	$0.649262\pi$
$402$	34.4113	1.71628
$403$	−0.0521702	−0.00259878
$404$	1.98612	0.0988131
$405$	0.457935	0.0227550
$406$	7.15796	0.355244
$407$	3.53972	0.175457
$408$	9.80031	0.485188
$409$	−0.425920	−0.0210604	−0.0105302	−	0.999945i	$-0.503352\pi$
$409$	−0.425920	−0.0210604	−0.0105302	+	0.999945i	$0.503352\pi$
$410$	0.226880	0.0112048
$411$	−42.2276	−2.08293
$412$	−2.25829	−0.111258
$413$	7.94301	0.390850
$414$	−6.66961	−0.327793
$415$	0.704970	0.0346056
$416$	0.945545	0.0463592
$417$	−59.6350	−2.92034
$418$	3.57572	0.174894
$419$	−4.16513	−0.203480	−0.101740	−	0.994811i	$-0.532441\pi$
$419$	−4.16513	−0.203480	−0.101740	+	0.994811i	$0.532441\pi$
$420$	0.224595	0.0109591
$421$	−13.7610	−0.670671	−0.335335	−	0.942099i	$-0.608850\pi$
$421$	−13.7610	−0.670671	−0.335335	+	0.942099i	$0.608850\pi$
$422$	−31.9689	−1.55622
$423$	14.9175	0.725311
$424$	−20.0735	−0.974853
$425$	5.64087	0.273623
$426$	35.6359	1.72656
$427$	13.5822	0.657291
$428$	3.44845	0.166687
$429$	−2.82757	−0.136517
$430$	−0.0303758	−0.00146485
$431$	−19.9214	−0.959579	−0.479789	−	0.877384i	$-0.659287\pi$
$431$	−19.9214	−0.959579	−0.479789	+	0.877384i	$0.659287\pi$
$432$	20.5165	0.987101
$433$	38.2295	1.83719	0.918595	−	0.395200i	$-0.129325\pi$
$433$	38.2295	1.83719	0.918595	+	0.395200i	$0.129325\pi$
$434$	−0.0706207	−0.00338990
$435$	7.08589	0.339742
$436$	−1.35253	−0.0647747
$437$	2.60552	0.124639
$438$	−4.26941	−0.204000
$439$	21.5927	1.03056	0.515282	−	0.857021i	$-0.327687\pi$
$439$	21.5927	1.03056	0.515282	+	0.857021i	$0.327687\pi$
$440$	−1.39056	−0.0662924
$441$	4.99518	0.237866
$442$	−1.59899	−0.0760562
$443$	23.9638	1.13855	0.569277	−	0.822145i	$-0.307223\pi$
$443$	23.9638	1.13855	0.569277	+	0.822145i	$0.307223\pi$
$444$	−1.67752	−0.0796118
$445$	8.07136	0.382619
$446$	−9.80428	−0.464246
$447$	−17.8814	−0.845760
$448$	8.55335	0.404108
$449$	−17.7232	−0.836411	−0.418206	−	0.908352i	$-0.637341\pi$
$449$	−17.7232	−0.836411	−0.418206	+	0.908352i	$0.637341\pi$
$450$	32.2902	1.52218
$451$	0.353660	0.0166532
$452$	1.74929	0.0822798
$453$	3.02009	0.141896
$454$	24.7329	1.16077
$455$	−0.473915	−0.0222175
$456$	−21.9158	−1.02630
$457$	14.4056	0.673865	0.336932	−	0.941529i	$-0.390611\pi$
$457$	14.4056	0.673865	0.336932	+	0.941529i	$0.390611\pi$
$458$	−16.7487	−0.782616
$459$	6.66396	0.311047
$460$	−0.0783477	−0.00365298
$461$	−12.6237	−0.587945	−0.293972	−	0.955814i	$-0.594977\pi$
$461$	−12.6237	−0.587945	−0.293972	+	0.955814i	$0.594977\pi$
$462$	−3.82757	−0.178075
$463$	−15.7488	−0.731908	−0.365954	−	0.930633i	$-0.619257\pi$
$463$	−15.7488	−0.731908	−0.365954	+	0.930633i	$0.619257\pi$
$464$	19.2304	0.892747
$465$	−0.0699096	−0.00324198
$466$	−17.2958	−0.801211
$467$	10.9390	0.506197	0.253099	−	0.967440i	$-0.418550\pi$
$467$	10.9390	0.506197	0.253099	+	0.967440i	$0.418550\pi$
$468$	0.837216	0.0387003
$469$	8.99036	0.415136
$470$	−1.91581	−0.0883700
$471$	16.5550	0.762813
$472$	23.3064	1.07276
$473$	−0.0473498	−0.00217715
$474$	−0.693690	−0.0318622
$475$	−12.6143	−0.578785
$476$	0.197981	0.00907442
$477$	−34.1730	−1.56468
$478$	−27.0210	−1.23591
$479$	−2.35913	−0.107791	−0.0538956	−	0.998547i	$-0.517164\pi$
$479$	−2.35913	−0.107791	−0.0538956	+	0.998547i	$0.517164\pi$
$480$	1.26706	0.0578331
$481$	3.53972	0.161397
$482$	−21.7161	−0.989143
$483$	−2.78903	−0.126905
$484$	−0.167605	−0.00761840
$485$	−2.37375	−0.107786
$486$	−19.2116	−0.871455
$487$	−15.4915	−0.701985	−0.350993	−	0.936378i	$-0.614156\pi$
$487$	−15.4915	−0.701985	−0.350993	+	0.936378i	$0.614156\pi$
$488$	39.8530	1.80406
$489$	−2.87257	−0.129902
$490$	−0.641520	−0.0289809
$491$	33.6501	1.51861	0.759305	−	0.650735i	$-0.225539\pi$
$491$	33.6501	1.51861	0.759305	+	0.650735i	$0.225539\pi$
$492$	−0.167605	−0.00755620
$493$	6.24620	0.281315
$494$	3.57572	0.160879
$495$	−2.36729	−0.106402
$496$	−0.189727	−0.00851901
$497$	9.31030	0.417624
$498$	5.69369	0.255140
$499$	0.679994	0.0304407	0.0152204	−	0.999884i	$-0.495155\pi$
$499$	0.679994	0.0304407	0.0152204	+	0.999884i	$0.495155\pi$
$500$	0.776464	0.0347245
$501$	−31.5204	−1.40823
$502$	15.1452	0.675964
$503$	19.4626	0.867793	0.433897	−	0.900963i	$-0.357138\pi$
$503$	19.4626	0.867793	0.433897	+	0.900963i	$0.357138\pi$
$504$	14.6569	0.652868
$505$	−5.61590	−0.249904
$506$	1.33521	0.0593573
$507$	−2.82757	−0.125577
$508$	−1.35481	−0.0601099
$509$	−15.3208	−0.679083	−0.339542	−	0.940591i	$-0.610272\pi$
$509$	−15.3208	−0.679083	−0.339542	+	0.940591i	$0.610272\pi$
$510$	−2.14269	−0.0948801
$511$	−1.11543	−0.0493439
$512$	24.7803	1.09514
$513$	−14.9022	−0.657948
$514$	9.66338	0.426233
$515$	6.38549	0.281378
$516$	0.0224398	0.000987856 0
$517$	−2.98637	−0.131340
$518$	4.79157	0.210530
$519$	18.9719	0.832774
$520$	−1.39056	−0.0609801
$521$	−25.4651	−1.11565	−0.557823	−	0.829960i	$-0.688363\pi$
$521$	−25.4651	−1.11565	−0.557823	+	0.829960i	$0.688363\pi$
$522$	35.7553	1.56497
$523$	−8.05680	−0.352299	−0.176150	−	0.984363i	$-0.556364\pi$
$523$	−8.05680	−0.352299	−0.176150	+	0.984363i	$0.556364\pi$
$524$	0.0950167	0.00415083
$525$	13.5028	0.589311
$526$	−8.08506	−0.352525
$527$	−0.0616252	−0.00268444
$528$	−10.2830	−0.447512
$529$	−22.0271	−0.957699
$530$	4.38877	0.190636
$531$	39.6768	1.72182
$532$	−0.442731	−0.0191948
$533$	0.353660	0.0153187
$534$	65.1884	2.82098
$535$	−9.75075	−0.421562
$536$	26.3795	1.13942
$537$	29.6294	1.27860
$538$	−20.6838	−0.891743
$539$	−1.00000	−0.0430730
$540$	0.448108	0.0192835
$541$	37.4875	1.61171	0.805857	−	0.592110i	$-0.201705\pi$
$541$	37.4875	1.61171	0.805857	+	0.592110i	$0.201705\pi$
$542$	−7.73770	−0.332362
$543$	4.90929	0.210678
$544$	1.11691	0.0478872
$545$	3.82439	0.163819
$546$	−3.82757	−0.163805
$547$	−41.6968	−1.78283	−0.891414	−	0.453189i	$-0.850286\pi$
$547$	−41.6968	−1.78283	−0.891414	+	0.453189i	$0.850286\pi$
$548$	−2.50304	−0.106925
$549$	67.8458	2.89559
$550$	−6.46427	−0.275638
$551$	−13.9680	−0.595056
$552$	−8.18358	−0.348317
$553$	−0.181235	−0.00770689
$554$	−7.33920	−0.311813
$555$	4.74332	0.201343
$556$	−3.53487	−0.149912
$557$	33.6785	1.42700	0.713501	−	0.700654i	$-0.247108\pi$
$557$	33.6785	1.42700	0.713501	+	0.700654i	$0.247108\pi$
$558$	−0.352763	−0.0149337
$559$	−0.0473498	−0.00200268
$560$	−1.72349	−0.0728306
$561$	−3.34003	−0.141016
$562$	−20.8812	−0.880820
$563$	−1.55399	−0.0654929	−0.0327464	−	0.999464i	$-0.510425\pi$
$563$	−1.55399	−0.0654929	−0.0327464	+	0.999464i	$0.510425\pi$
$564$	1.41529	0.0595943
$565$	−4.94626	−0.208091
$566$	−7.57260	−0.318300
$567$	0.966280	0.0405800
$568$	27.3183	1.14625
$569$	−11.8653	−0.497421	−0.248711	−	0.968578i	$-0.580007\pi$
$569$	−11.8653	−0.497421	−0.248711	+	0.968578i	$0.580007\pi$
$570$	4.79157	0.200697
$571$	11.7587	0.492085	0.246042	−	0.969259i	$-0.420870\pi$
$571$	11.7587	0.492085	0.246042	+	0.969259i	$0.420870\pi$
$572$	−0.167605	−0.00700791
$573$	−22.0922	−0.922913
$574$	0.478735	0.0199820
$575$	−4.71032	−0.196434
$576$	42.7255	1.78023
$577$	−23.8250	−0.991849	−0.495925	−	0.868366i	$-0.665171\pi$
$577$	−23.8250	−0.991849	−0.495925	+	0.868366i	$0.665171\pi$
$578$	21.1234	0.878619
$579$	−60.4335	−2.51153
$580$	0.420017	0.0174402
$581$	1.48754	0.0617138
$582$	−19.1716	−0.794688
$583$	6.84120	0.283334
$584$	−3.27291	−0.135434
$585$	−2.36729	−0.0978754
$586$	7.16767	0.296094
$587$	−12.3650	−0.510358	−0.255179	−	0.966894i	$-0.582134\pi$
$587$	−12.3650	−0.510358	−0.255179	+	0.966894i	$0.582134\pi$
$588$	0.473915	0.0195439
$589$	0.137809	0.00567830
$590$	−5.09560	−0.209782
$591$	24.8108	1.02058
$592$	12.8729	0.529072
$593$	−19.6141	−0.805453	−0.402727	−	0.915320i	$-0.631938\pi$
$593$	−19.6141	−0.805453	−0.402727	+	0.915320i	$0.631938\pi$
$594$	−7.63670	−0.313338
$595$	−0.559805	−0.0229498
$596$	−1.05992	−0.0434160
$597$	−47.2774	−1.93493
$598$	1.33521	0.0546008
$599$	1.48272	0.0605825	0.0302912	−	0.999541i	$-0.490357\pi$
$599$	1.48272	0.0605825	0.0302912	+	0.999541i	$0.490357\pi$
$600$	39.6200	1.61748
$601$	7.06972	0.288380	0.144190	−	0.989550i	$-0.453942\pi$
$601$	7.06972	0.288380	0.144190	+	0.989550i	$0.453942\pi$
$602$	−0.0640955	−0.00261234
$603$	44.9085	1.82881
$604$	0.179016	0.00728406
$605$	0.473915	0.0192674
$606$	−45.3568	−1.84249
$607$	−11.8348	−0.480361	−0.240181	−	0.970728i	$-0.577207\pi$
$607$	−11.8348	−0.480361	−0.240181	+	0.970728i	$0.577207\pi$
$608$	−2.49768	−0.101294
$609$	14.9518	0.605878
$610$	−8.71328	−0.352790
$611$	−2.98637	−0.120816
$612$	0.988948	0.0399759
$613$	−5.79697	−0.234137	−0.117069	−	0.993124i	$-0.537350\pi$
$613$	−5.79697	−0.234137	−0.117069	+	0.993124i	$0.537350\pi$
$614$	−6.85506	−0.276648
$615$	0.473915	0.0191101
$616$	−2.93420	−0.118222
$617$	30.7443	1.23772	0.618859	−	0.785502i	$-0.287595\pi$
$617$	30.7443	1.23772	0.618859	+	0.785502i	$0.287595\pi$
$618$	51.5724	2.07455
$619$	11.4635	0.460756	0.230378	−	0.973101i	$-0.426004\pi$
$619$	11.4635	0.460756	0.230378	+	0.973101i	$0.426004\pi$
$620$	−0.00414390	−0.000166423 0
$621$	−5.56463	−0.223301
$622$	−35.9777	−1.44257
$623$	17.0312	0.682343
$624$	−10.2830	−0.411651
$625$	21.6815	0.867261
$626$	−38.4028	−1.53488
$627$	7.46909	0.298287
$628$	0.981298	0.0391581
$629$	4.18123	0.166717
$630$	−3.20451	−0.127671
$631$	−2.57007	−0.102313	−0.0511564	−	0.998691i	$-0.516291\pi$
$631$	−2.57007	−0.102313	−0.0511564	+	0.998691i	$0.516291\pi$
$632$	−0.531779	−0.0211530
$633$	−66.7777	−2.65418
$634$	46.2643	1.83739
$635$	3.83082	0.152022
$636$	−3.24215	−0.128560
$637$	−1.00000	−0.0396214
$638$	−7.15796	−0.283387
$639$	46.5066	1.83977
$640$	−4.59092	−0.181472
$641$	−13.7626	−0.543590	−0.271795	−	0.962355i	$-0.587617\pi$
$641$	−13.7626	−0.543590	−0.271795	+	0.962355i	$0.587617\pi$
$642$	−78.7519	−3.10809
$643$	20.9175	0.824904	0.412452	−	0.910979i	$-0.364672\pi$
$643$	20.9175	0.824904	0.412452	+	0.910979i	$0.364672\pi$
$644$	−0.165320	−0.00651453
$645$	−0.0634501	−0.00249835
$646$	4.22376	0.166182
$647$	40.5363	1.59365	0.796824	−	0.604211i	$-0.206512\pi$
$647$	40.5363	1.59365	0.796824	+	0.604211i	$0.206512\pi$
$648$	2.83526	0.111380
$649$	−7.94301	−0.311790
$650$	−6.46427	−0.253550
$651$	−0.147515	−0.00578157
$652$	−0.170272	−0.00666837
$653$	−35.8595	−1.40329	−0.701645	−	0.712527i	$-0.747551\pi$
$653$	−35.8595	−1.40329	−0.701645	+	0.712527i	$0.747551\pi$
$654$	30.8877	1.20781
$655$	−0.268667	−0.0104977
$656$	1.28615	0.0502159
$657$	−5.57180	−0.217376
$658$	−4.04253	−0.157594
$659$	5.75779	0.224291	0.112146	−	0.993692i	$-0.464228\pi$
$659$	5.75779	0.224291	0.112146	+	0.993692i	$0.464228\pi$
$660$	−0.224595	−0.00874236
$661$	−17.1251	−0.666087	−0.333044	−	0.942911i	$-0.608076\pi$
$661$	−17.1251	−0.666087	−0.333044	+	0.942911i	$0.608076\pi$
$662$	19.7935	0.769298
$663$	−3.34003	−0.129716
$664$	4.36475	0.169385
$665$	1.25186	0.0485449
$666$	23.9348	0.927453
$667$	−5.21579	−0.201956
$668$	−1.86838	−0.0722896
$669$	−20.4795	−0.791785
$670$	−5.76749	−0.222818
$671$	−13.5822	−0.524337
$672$	2.67360	0.103136
$673$	19.9437	0.768771	0.384386	−	0.923173i	$-0.374413\pi$
$673$	19.9437	0.768771	0.384386	+	0.923173i	$0.374413\pi$
$674$	−7.91972	−0.305056
$675$	26.9405	1.03694
$676$	−0.167605	−0.00644634
$677$	23.6637	0.909470	0.454735	−	0.890627i	$-0.349734\pi$
$677$	23.6637	0.909470	0.454735	+	0.890627i	$0.349734\pi$
$678$	−39.9485	−1.53421
$679$	−5.00881	−0.192220
$680$	−1.64258	−0.0629900
$681$	51.6629	1.97973
$682$	0.0706207	0.00270421
$683$	−13.3715	−0.511645	−0.255822	−	0.966724i	$-0.582346\pi$
$683$	−13.3715	−0.511645	−0.255822	+	0.966724i	$0.582346\pi$
$684$	−2.21152	−0.0845596
$685$	7.07754	0.270419
$686$	−1.35366	−0.0516830
$687$	−34.9854	−1.33477
$688$	−0.172197	−0.00656495
$689$	6.84120	0.260629
$690$	1.78922	0.0681145
$691$	−15.8072	−0.601336	−0.300668	−	0.953729i	$-0.597210\pi$
$691$	−15.8072	−0.601336	−0.300668	+	0.953729i	$0.597210\pi$
$692$	1.12456	0.0427494
$693$	−4.99518	−0.189751
$694$	30.9485	1.17479
$695$	9.99511	0.379136
$696$	43.8716	1.66295
$697$	0.417755	0.0158236
$698$	−44.0006	−1.66545
$699$	−36.1280	−1.36649
$700$	0.800380	0.0302515
$701$	22.2221	0.839316	0.419658	−	0.907682i	$-0.362150\pi$
$701$	22.2221	0.839316	0.419658	+	0.907682i	$0.362150\pi$
$702$	−7.63670	−0.288229
$703$	−9.35023	−0.352650
$704$	−8.55335	−0.322366
$705$	−4.00182	−0.150717
$706$	−16.3352	−0.614784
$707$	−11.8500	−0.445666
$708$	3.76431	0.141471
$709$	−30.8556	−1.15881	−0.579403	−	0.815041i	$-0.696714\pi$
$709$	−30.8556	−1.15881	−0.579403	+	0.815041i	$0.696714\pi$
$710$	−5.97274	−0.224153
$711$	−0.905300	−0.0339514
$712$	49.9731	1.87282
$713$	0.0514591	0.00192716
$714$	−4.52126	−0.169204
$715$	0.473915	0.0177234
$716$	1.75629	0.0656355
$717$	−56.4425	−2.10788
$718$	5.30512	0.197985
$719$	28.9011	1.07783	0.538915	−	0.842360i	$-0.318835\pi$
$719$	28.9011	1.07783	0.538915	+	0.842360i	$0.318835\pi$
$720$	−8.60912	−0.320843
$721$	13.4739	0.501795
$722$	16.2742	0.605663
$723$	−45.3615	−1.68701
$724$	0.290999	0.0108149
$725$	25.2517	0.937823
$726$	3.82757	0.142055
$727$	−47.9941	−1.78000	−0.890001	−	0.455958i	$-0.849297\pi$
$727$	−47.9941	−1.78000	−0.890001	+	0.455958i	$0.849297\pi$
$728$	−2.93420	−0.108749
$729$	−43.0287	−1.59366
$730$	0.715573	0.0264846
$731$	−0.0559312	−0.00206869
$732$	6.43683	0.237912
$733$	7.04742	0.260302	0.130151	−	0.991494i	$-0.458454\pi$
$733$	7.04742	0.260302	0.130151	+	0.991494i	$0.458454\pi$
$734$	22.0775	0.814894
$735$	−1.34003	−0.0494277
$736$	−0.932657	−0.0343782
$737$	−8.99036	−0.331164
$738$	2.39137	0.0880275
$739$	−18.4916	−0.680225	−0.340112	−	0.940385i	$-0.610465\pi$
$739$	−18.4916	−0.680225	−0.340112	+	0.940385i	$0.610465\pi$
$740$	0.281161	0.0103357
$741$	7.46909	0.274384
$742$	9.26067	0.339970
$743$	24.0433	0.882064	0.441032	−	0.897491i	$-0.354612\pi$
$743$	24.0433	0.882064	0.441032	+	0.897491i	$0.354612\pi$
$744$	−0.432839	−0.0158686
$745$	2.99700	0.109802
$746$	−33.6461	−1.23187
$747$	7.43055	0.271870
$748$	−0.197981	−0.00723889
$749$	−20.5749	−0.751790
$750$	−17.7320	−0.647483
$751$	4.53091	0.165335	0.0826675	−	0.996577i	$-0.473656\pi$
$751$	4.53091	0.165335	0.0826675	+	0.996577i	$0.473656\pi$
$752$	−10.8605	−0.396043
$753$	31.6359	1.15288
$754$	−7.15796	−0.260678
$755$	−0.506181	−0.0184218
$756$	0.945545	0.0343892
$757$	−34.7322	−1.26236	−0.631182	−	0.775635i	$-0.717430\pi$
$757$	−34.7322	−1.26236	−0.631182	+	0.775635i	$0.717430\pi$
$758$	4.26710	0.154988
$759$	2.78903	0.101236
$760$	3.67319	0.133241
$761$	−34.3055	−1.24357	−0.621786	−	0.783187i	$-0.713593\pi$
$761$	−34.3055	−1.24357	−0.621786	+	0.783187i	$0.713593\pi$
$762$	30.9396	1.12082
$763$	8.06979	0.292146
$764$	−1.30951	−0.0473766
$765$	−2.79633	−0.101101
$766$	22.1545	0.800476
$767$	−7.94301	−0.286805
$768$	11.2918	0.407460
$769$	37.6709	1.35845	0.679223	−	0.733932i	$-0.262317\pi$
$769$	37.6709	1.35845	0.679223	+	0.733932i	$0.262317\pi$
$770$	0.641520	0.0231188
$771$	20.1852	0.726953
$772$	−3.58220	−0.128926
$773$	−29.0848	−1.04611	−0.523054	−	0.852300i	$-0.675207\pi$
$773$	−29.0848	−1.04611	−0.523054	+	0.852300i	$0.675207\pi$
$774$	−0.320169	−0.0115082
$775$	−0.249134	−0.00894915
$776$	−14.6968	−0.527586
$777$	10.0088	0.359064
$778$	−28.0726	−1.00645
$779$	−0.934200	−0.0334712
$780$	−0.224595	−0.00804180
$781$	−9.31030	−0.333149
$782$	1.57720	0.0564004
$783$	29.8316	1.06609
$784$	−3.63670	−0.129882
$785$	−2.77469	−0.0990331
$786$	−2.16989	−0.0773974
$787$	−22.1978	−0.791266	−0.395633	−	0.918409i	$-0.629475\pi$
$787$	−22.1978	−0.791266	−0.395633	+	0.918409i	$0.629475\pi$
$788$	1.47066	0.0523901
$789$	−16.8884	−0.601242
$790$	0.116266	0.00413655
$791$	−10.4370	−0.371097
$792$	−14.6569	−0.520809
$793$	−13.5822	−0.482320
$794$	42.1282	1.49507
$795$	9.16742	0.325135
$796$	−2.80237	−0.0993274
$797$	−4.82352	−0.170858	−0.0854289	−	0.996344i	$-0.527226\pi$
$797$	−4.82352	−0.170858	−0.0854289	+	0.996344i	$0.527226\pi$
$798$	10.1106	0.357912
$799$	−3.52760	−0.124798
$800$	4.51536	0.159642
$801$	85.0741	3.00595
$802$	24.5006	0.865146
$803$	1.11543	0.0393628
$804$	4.26067	0.150262
$805$	0.467455	0.0164756
$806$	0.0706207	0.00248751
$807$	−43.2051	−1.52089
$808$	−34.7703	−1.22322
$809$	−52.6654	−1.85162	−0.925808	−	0.377994i	$-0.876614\pi$
$809$	−52.6654	−1.85162	−0.925808	+	0.377994i	$0.876614\pi$
$810$	−0.619888	−0.0217806
$811$	55.0245	1.93217	0.966086	−	0.258220i	$-0.0831361\pi$
$811$	55.0245	1.93217	0.966086	+	0.258220i	$0.0831361\pi$
$812$	0.886270	0.0311020
$813$	−16.1628	−0.566854
$814$	−4.79157	−0.167944
$815$	0.481457	0.0168647
$816$	−12.1467	−0.425219
$817$	0.125075	0.00437583
$818$	0.576551	0.0201586
$819$	−4.99518	−0.174546
$820$	0.0280913	0.000980992 0
$821$	4.65203	0.162357	0.0811786	−	0.996700i	$-0.474132\pi$
$821$	4.65203	0.162357	0.0811786	+	0.996700i	$0.474132\pi$
$822$	57.1618	1.99375
$823$	19.5052	0.679910	0.339955	−	0.940442i	$-0.389588\pi$
$823$	19.5052	0.679910	0.339955	+	0.940442i	$0.389588\pi$
$824$	39.5352	1.37727
$825$	−13.5028	−0.470108
$826$	−10.7521	−0.374115
$827$	44.8347	1.55906	0.779528	−	0.626367i	$-0.215459\pi$
$827$	44.8347	1.55906	0.779528	+	0.626367i	$0.215459\pi$
$828$	−0.825804	−0.0286987
$829$	−5.90643	−0.205139	−0.102569	−	0.994726i	$-0.532706\pi$
$829$	−5.90643	−0.205139	−0.102569	+	0.994726i	$0.532706\pi$
$830$	−0.954289	−0.0331239
$831$	−15.3304	−0.531805
$832$	−8.55335	−0.296534
$833$	−1.18123	−0.0409274
$834$	80.7256	2.79530
$835$	5.28297	0.182825
$836$	0.442731	0.0153122
$837$	−0.294319	−0.0101732
$838$	5.63818	0.194768
$839$	−35.4274	−1.22309	−0.611546	−	0.791209i	$-0.709452\pi$
$839$	−35.4274	−1.22309	−0.611546	+	0.791209i	$0.709452\pi$
$840$	−3.93192	−0.135664
$841$	−1.03854	−0.0358117
$842$	18.6277	0.641954
$843$	−43.6174	−1.50226
$844$	−3.95826	−0.136249
$845$	0.473915	0.0163032
$846$	−20.1932	−0.694255
$847$	1.00000	0.0343604
$848$	24.8794	0.854362
$849$	−15.8179	−0.542870
$850$	−7.63583	−0.261907
$851$	−3.49147	−0.119686
$852$	4.41229	0.151163
$853$	−33.8898	−1.16037	−0.580183	−	0.814486i	$-0.697019\pi$
$853$	−33.8898	−1.16037	−0.580183	+	0.814486i	$0.697019\pi$
$854$	−18.3857	−0.629147
$855$	6.25324	0.213856
$856$	−60.3708	−2.06343
$857$	44.2992	1.51323	0.756616	−	0.653860i	$-0.226851\pi$
$857$	44.2992	1.51323	0.756616	+	0.653860i	$0.226851\pi$
$858$	3.82757	0.130671
$859$	8.35684	0.285132	0.142566	−	0.989785i	$-0.454465\pi$
$859$	8.35684	0.285132	0.142566	+	0.989785i	$0.454465\pi$
$860$	−0.00376101	−0.000128249 0
$861$	1.00000	0.0340799
$862$	26.9668	0.918492
$863$	6.61015	0.225012	0.112506	−	0.993651i	$-0.464112\pi$
$863$	6.61015	0.225012	0.112506	+	0.993651i	$0.464112\pi$
$864$	5.33431	0.181477
$865$	−3.17978	−0.108116
$866$	−51.7497	−1.75853
$867$	44.1234	1.49851
$868$	−0.00874397	−0.000296790 0
$869$	0.181235	0.00614797
$870$	−9.59189	−0.325195
$871$	−8.99036	−0.304627
$872$	23.6784	0.801851
$873$	−25.0199	−0.846795
$874$	−3.52698	−0.119302
$875$	−4.63271	−0.156614
$876$	−0.528621	−0.0178605
$877$	−46.8073	−1.58057	−0.790285	−	0.612739i	$-0.790068\pi$
$877$	−46.8073	−1.58057	−0.790285	+	0.612739i	$0.790068\pi$
$878$	−29.2292	−0.986437
$879$	14.9721	0.504996
$880$	1.72349	0.0580987
$881$	12.5751	0.423667	0.211834	−	0.977306i	$-0.432057\pi$
$881$	12.5751	0.423667	0.211834	+	0.977306i	$0.432057\pi$
$882$	−6.76177	−0.227681
$883$	−3.38256	−0.113832	−0.0569161	−	0.998379i	$-0.518127\pi$
$883$	−3.38256	−0.113832	−0.0569161	+	0.998379i	$0.518127\pi$
$884$	−0.197981	−0.00665881
$885$	−10.6439	−0.357790
$886$	−32.4389	−1.08980
$887$	−33.0928	−1.11115	−0.555574	−	0.831467i	$-0.687501\pi$
$887$	−33.0928	−1.11115	−0.555574	+	0.831467i	$0.687501\pi$
$888$	29.3678	0.985520
$889$	8.08335	0.271107
$890$	−10.9259	−0.366236
$891$	−0.966280	−0.0323716
$892$	−1.21393	−0.0406453
$893$	7.88855	0.263980
$894$	24.2053	0.809547
$895$	−4.96603	−0.165996
$896$	−9.68723	−0.323628
$897$	2.78903	0.0931232
$898$	23.9912	0.800598
$899$	−0.275869	−0.00920074
$900$	3.99804	0.133268
$901$	8.08107	0.269219
$902$	−0.478735	−0.0159401
$903$	−0.133885	−0.00445542
$904$	−30.6243	−1.01855
$905$	−0.822820	−0.0273515
$906$	−4.08817	−0.135820
$907$	28.3535	0.941462	0.470731	−	0.882277i	$-0.343990\pi$
$907$	28.3535	0.941462	0.470731	+	0.882277i	$0.343990\pi$
$908$	3.06233	0.101627
$909$	−59.1930	−1.96331
$910$	0.641520	0.0212662
$911$	2.92938	0.0970547	0.0485273	−	0.998822i	$-0.484547\pi$
$911$	2.92938	0.0970547	0.0485273	+	0.998822i	$0.484547\pi$
$912$	27.1629	0.899452
$913$	−1.48754	−0.0492306
$914$	−19.5003	−0.645011
$915$	−18.2006	−0.601694
$916$	−2.07376	−0.0685189
$917$	−0.566910	−0.0187210
$918$	−9.02073	−0.297729
$919$	−33.4804	−1.10441	−0.552207	−	0.833707i	$-0.686214\pi$
$919$	−33.4804	−1.10441	−0.552207	+	0.833707i	$0.686214\pi$
$920$	1.37161	0.0452206
$921$	−14.3191	−0.471831
$922$	17.0882	0.562770
$923$	−9.31030	−0.306452
$924$	−0.473915	−0.0155907
$925$	16.9036	0.555786
$926$	21.3185	0.700570
$927$	67.3046	2.21057
$928$	4.99991	0.164130
$929$	57.6855	1.89260	0.946301	−	0.323288i	$-0.104788\pi$
$929$	57.6855	1.89260	0.946301	+	0.323288i	$0.104788\pi$
$930$	0.0946339	0.00310317
$931$	2.64152	0.0865723
$932$	−2.14149	−0.0701469
$933$	−75.1515	−2.46035
$934$	−14.8077	−0.484523
$935$	0.559805	0.0183076
$936$	−14.6569	−0.479074
$937$	9.35919	0.305751	0.152876	−	0.988245i	$-0.451147\pi$
$937$	9.35919	0.305751	0.152876	+	0.988245i	$0.451147\pi$
$938$	−12.1699	−0.397361
$939$	−80.2171	−2.61779
$940$	−0.237208	−0.00773689
$941$	−58.2965	−1.90041	−0.950205	−	0.311625i	$-0.899127\pi$
$941$	−58.2965	−1.90041	−0.950205	+	0.311625i	$0.899127\pi$
$942$	−22.4098	−0.730151
$943$	−0.348840	−0.0113598
$944$	−28.8863	−0.940170
$945$	−2.67360	−0.0869722
$946$	0.0640955	0.00208392
$947$	−3.89068	−0.126430	−0.0632150	−	0.998000i	$-0.520135\pi$
$947$	−3.89068	−0.126430	−0.0632150	+	0.998000i	$0.520135\pi$
$948$	−0.0858899	−0.00278957
$949$	1.11543	0.0362085
$950$	17.0755	0.554003
$951$	96.6387	3.13372
$952$	−3.46598	−0.112333
$953$	23.8932	0.773976	0.386988	−	0.922085i	$-0.373515\pi$
$953$	23.8932	0.773976	0.386988	+	0.922085i	$0.373515\pi$
$954$	46.2587	1.49768
$955$	3.70275	0.119818
$956$	−3.34563	−0.108206
$957$	−14.9518	−0.483324
$958$	3.19345	0.103176
$959$	14.9342	0.482250
$960$	−11.4617	−0.369926
$961$	−30.9973	−0.999912
$962$	−4.79157	−0.154486
$963$	−102.775	−3.31189
$964$	−2.68880	−0.0866005
$965$	10.1289	0.326062
$966$	3.77540	0.121472
$967$	16.2079	0.521211	0.260606	−	0.965445i	$-0.416078\pi$
$967$	16.2079	0.521211	0.260606	+	0.965445i	$0.416078\pi$
$968$	2.93420	0.0943088
$969$	8.82275	0.283428
$970$	3.21325	0.103171
$971$	−47.8091	−1.53427	−0.767134	−	0.641487i	$-0.778318\pi$
$971$	−47.8091	−1.53427	−0.767134	+	0.641487i	$0.778318\pi$
$972$	−2.37870	−0.0762969
$973$	21.0905	0.676131
$974$	20.9702	0.671928
$975$	−13.5028	−0.432436
$976$	−49.3945	−1.58108
$977$	−37.9893	−1.21538	−0.607692	−	0.794173i	$-0.707905\pi$
$977$	−37.9893	−1.21538	−0.607692	+	0.794173i	$0.707905\pi$
$978$	3.88849	0.124340
$979$	−17.0312	−0.544321
$980$	−0.0794304	−0.00253731
$981$	40.3100	1.28700
$982$	−45.5509	−1.45359
$983$	−12.1682	−0.388105	−0.194052	−	0.980991i	$-0.562163\pi$
$983$	−12.1682	−0.388105	−0.194052	+	0.980991i	$0.562163\pi$
$984$	2.93420	0.0935389
$985$	−4.15840	−0.132498
$986$	−8.45524	−0.269270
$987$	−8.44418	−0.268781
$988$	0.442731	0.0140852
$989$	0.0467044	0.00148511
$990$	3.20451	0.101846
$991$	−29.5541	−0.938817	−0.469408	−	0.882981i	$-0.655533\pi$
$991$	−29.5541	−0.938817	−0.469408	+	0.882981i	$0.655533\pi$
$992$	−0.0493293	−0.00156621
$993$	41.3455	1.31206
$994$	−12.6030	−0.399742
$995$	7.92391	0.251205
$996$	0.704970	0.0223378
$997$	−19.4467	−0.615883	−0.307942	−	0.951405i	$-0.599640\pi$
$997$	−19.4467	−0.615883	−0.307942	+	0.951405i	$0.599640\pi$
$998$	−0.920481	−0.0291373
$999$	19.9694	0.631803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1001.2.a.g.1.2	✓	4
3.2	odd	2		9009.2.a.w.1.3		4
7.6	odd	2		7007.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1001.2.a.g.1.2	✓	4	1.1	even	1	trivial
7007.2.a.j.1.2		4	7.6	odd	2
9009.2.a.w.1.3		4	3.2	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:	\( N \)	\(=\)	\( 1001 = 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.35366 q^{2} -2.82757 q^{3} -0.167605 q^{4} +0.473915 q^{5} +3.82757 q^{6} +1.00000 q^{7} +2.93420 q^{8} +4.99518 q^{9} +O(q^{10})\)	\(q-1.35366 q^{2} -2.82757 q^{3} -0.167605 q^{4} +0.473915 q^{5} +3.82757 q^{6} +1.00000 q^{7} +2.93420 q^{8} +4.99518 q^{9} -0.641520 q^{10} -1.00000 q^{11} +0.473915 q^{12} -1.00000 q^{13} -1.35366 q^{14} -1.34003 q^{15} -3.63670 q^{16} -1.18123 q^{17} -6.76177 q^{18} +2.64152 q^{19} -0.0794304 q^{20} -2.82757 q^{21} +1.35366 q^{22} +0.986370 q^{23} -8.29667 q^{24} -4.77540 q^{25} +1.35366 q^{26} -5.64152 q^{27} -0.167605 q^{28} -5.28786 q^{29} +1.81394 q^{30} +0.0521702 q^{31} -0.945545 q^{32} +2.82757 q^{33} +1.59899 q^{34} +0.473915 q^{35} -0.837216 q^{36} -3.53972 q^{37} -3.57572 q^{38} +2.82757 q^{39} +1.39056 q^{40} -0.353660 q^{41} +3.82757 q^{42} +0.0473498 q^{43} +0.167605 q^{44} +2.36729 q^{45} -1.33521 q^{46} +2.98637 q^{47} +10.2830 q^{48} +1.00000 q^{49} +6.46427 q^{50} +3.34003 q^{51} +0.167605 q^{52} -6.84120 q^{53} +7.63670 q^{54} -0.473915 q^{55} +2.93420 q^{56} -7.46909 q^{57} +7.15796 q^{58} +7.94301 q^{59} +0.224595 q^{60} +13.5822 q^{61} -0.0706207 q^{62} +4.99518 q^{63} +8.55335 q^{64} -0.473915 q^{65} -3.82757 q^{66} +8.99036 q^{67} +0.197981 q^{68} -2.78903 q^{69} -0.641520 q^{70} +9.31030 q^{71} +14.6569 q^{72} -1.11543 q^{73} +4.79157 q^{74} +13.5028 q^{75} -0.442731 q^{76} -1.00000 q^{77} -3.82757 q^{78} -0.181235 q^{79} -1.72349 q^{80} +0.966280 q^{81} +0.478735 q^{82} +1.48754 q^{83} +0.473915 q^{84} -0.559805 q^{85} -0.0640955 q^{86} +14.9518 q^{87} -2.93420 q^{88} +17.0312 q^{89} -3.20451 q^{90} -1.00000 q^{91} -0.165320 q^{92} -0.147515 q^{93} -4.04253 q^{94} +1.25186 q^{95} +2.67360 q^{96} -5.00881 q^{97} -1.35366 q^{98} -4.99518 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 4 * q^7 + 3 * q^8 + q^9	\( 4 q - 2 q^{2} - q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + 13 q^{10} - 4 q^{11} - 5 q^{12} - 4 q^{13} - 2 q^{14} - 7 q^{15} + 20 q^{16} + 9 q^{17} - 8 q^{18} - 5 q^{19} - 4 q^{20} - q^{21} + 2 q^{22} + 9 q^{23} + 3 q^{24} + 5 q^{25} + 2 q^{26} - 7 q^{27} + 8 q^{28} - 9 q^{29} + 2 q^{30} + 14 q^{31} - 16 q^{32} + q^{33} + 15 q^{34} - 5 q^{35} - 31 q^{36} - 16 q^{37} + 10 q^{38} + q^{39} + 30 q^{40} + 2 q^{41} + 5 q^{42} - 5 q^{43} - 8 q^{44} + q^{45} + 12 q^{46} + 17 q^{47} + 10 q^{48} + 4 q^{49} - 19 q^{50} + 15 q^{51} - 8 q^{52} - 12 q^{53} - 4 q^{54} + 5 q^{55} + 3 q^{56} - 4 q^{57} - 18 q^{58} - q^{59} + 25 q^{60} + 32 q^{61} - 28 q^{62} + q^{63} + 31 q^{64} + 5 q^{65} - 5 q^{66} - 2 q^{67} + 18 q^{68} + 18 q^{69} + 13 q^{70} - 4 q^{71} + 21 q^{72} + 18 q^{73} + 35 q^{74} + 28 q^{75} - 40 q^{76} - 4 q^{77} - 5 q^{78} + 13 q^{79} - 40 q^{80} + 4 q^{81} + 14 q^{82} - 6 q^{83} - 5 q^{84} - 9 q^{85} - 26 q^{86} + 3 q^{87} - 3 q^{88} + 15 q^{89} - 32 q^{90} - 4 q^{91} - 18 q^{92} + 13 q^{93} + 8 q^{94} + 19 q^{95} + 4 q^{96} + 4 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 4 * q^7 + 3 * q^8 + q^9 + 13 * q^10 - 4 * q^11 - 5 * q^12 - 4 * q^13 - 2 * q^14 - 7 * q^15 + 20 * q^16 + 9 * q^17 - 8 * q^18 - 5 * q^19 - 4 * q^20 - q^21 + 2 * q^22 + 9 * q^23 + 3 * q^24 + 5 * q^25 + 2 * q^26 - 7 * q^27 + 8 * q^28 - 9 * q^29 + 2 * q^30 + 14 * q^31 - 16 * q^32 + q^33 + 15 * q^34 - 5 * q^35 - 31 * q^36 - 16 * q^37 + 10 * q^38 + q^39 + 30 * q^40 + 2 * q^41 + 5 * q^42 - 5 * q^43 - 8 * q^44 + q^45 + 12 * q^46 + 17 * q^47 + 10 * q^48 + 4 * q^49 - 19 * q^50 + 15 * q^51 - 8 * q^52 - 12 * q^53 - 4 * q^54 + 5 * q^55 + 3 * q^56 - 4 * q^57 - 18 * q^58 - q^59 + 25 * q^60 + 32 * q^61 - 28 * q^62 + q^63 + 31 * q^64 + 5 * q^65 - 5 * q^66 - 2 * q^67 + 18 * q^68 + 18 * q^69 + 13 * q^70 - 4 * q^71 + 21 * q^72 + 18 * q^73 + 35 * q^74 + 28 * q^75 - 40 * q^76 - 4 * q^77 - 5 * q^78 + 13 * q^79 - 40 * q^80 + 4 * q^81 + 14 * q^82 - 6 * q^83 - 5 * q^84 - 9 * q^85 - 26 * q^86 + 3 * q^87 - 3 * q^88 + 15 * q^89 - 32 * q^90 - 4 * q^91 - 18 * q^92 + 13 * q^93 + 8 * q^94 + 19 * q^95 + 4 * q^96 + 4 * q^97 - 2 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more