LMFDB - Embedded newform 547.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [547,2,Mod(1,547)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(547, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("547.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	547.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:	$4.36781699056$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.924759$ of defining polynomial
Character	$\chi$	$=$	547.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.924759 q^{2} +2.22306 q^{3} -1.14482 q^{4} -3.95421 q^{5} +2.05580 q^{6} -3.08028 q^{7} -2.90820 q^{8} +1.94201 q^{9} +O(q^{10})$	\(q+0.924759 q^{2} +2.22306 q^{3} -1.14482 q^{4} -3.95421 q^{5} +2.05580 q^{6} -3.08028 q^{7} -2.90820 q^{8} +1.94201 q^{9} -3.65669 q^{10} -3.50276 q^{11} -2.54501 q^{12} +3.47242 q^{13} -2.84852 q^{14} -8.79046 q^{15} -0.399740 q^{16} -3.04988 q^{17} +1.79589 q^{18} +7.32400 q^{19} +4.52687 q^{20} -6.84766 q^{21} -3.23921 q^{22} -6.16987 q^{23} -6.46511 q^{24} +10.6358 q^{25} +3.21115 q^{26} -2.35198 q^{27} +3.52637 q^{28} -3.33808 q^{29} -8.12905 q^{30} +5.01284 q^{31} +5.44674 q^{32} -7.78685 q^{33} -2.82040 q^{34} +12.1801 q^{35} -2.22325 q^{36} -9.39929 q^{37} +6.77293 q^{38} +7.71940 q^{39} +11.4996 q^{40} -2.63973 q^{41} -6.33243 q^{42} +0.457015 q^{43} +4.01003 q^{44} -7.67911 q^{45} -5.70564 q^{46} -4.87863 q^{47} -0.888648 q^{48} +2.48812 q^{49} +9.83553 q^{50} -6.78008 q^{51} -3.97530 q^{52} +0.519948 q^{53} -2.17502 q^{54} +13.8506 q^{55} +8.95807 q^{56} +16.2817 q^{57} -3.08692 q^{58} +10.3773 q^{59} +10.0635 q^{60} +2.71700 q^{61} +4.63567 q^{62} -5.98193 q^{63} +5.83640 q^{64} -13.7307 q^{65} -7.20096 q^{66} -11.2680 q^{67} +3.49157 q^{68} -13.7160 q^{69} +11.2636 q^{70} +6.36651 q^{71} -5.64775 q^{72} -1.19812 q^{73} -8.69207 q^{74} +23.6440 q^{75} -8.38468 q^{76} +10.7895 q^{77} +7.13858 q^{78} -11.6933 q^{79} +1.58066 q^{80} -11.0546 q^{81} -2.44111 q^{82} -9.73676 q^{83} +7.83934 q^{84} +12.0599 q^{85} +0.422629 q^{86} -7.42077 q^{87} +10.1867 q^{88} +11.4691 q^{89} -7.10132 q^{90} -10.6960 q^{91} +7.06340 q^{92} +11.1439 q^{93} -4.51155 q^{94} -28.9607 q^{95} +12.1084 q^{96} +7.25071 q^{97} +2.30091 q^{98} -6.80238 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 27 q^{5} - 3 q^{6} - 11 q^{7} - 12 q^{8} + 14 q^{9}+O(q^{10}) $ 18 * q - 4 * q^2 - 10 * q^3 + 16 * q^4 - 27 * q^5 - 3 * q^6 - 11 * q^7 - 12 * q^8 + 14 * q^9	\( 18 q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 27 q^{5} - 3 q^{6} - 11 q^{7} - 12 q^{8} + 14 q^{9} - 5 q^{10} + 2 q^{11} - 32 q^{12} - 25 q^{13} - 7 q^{14} + 9 q^{15} + 8 q^{16} - 30 q^{17} - 10 q^{18} + 4 q^{19} - 41 q^{20} - 16 q^{21} - 24 q^{22} - 26 q^{23} - 12 q^{24} + 31 q^{25} - 18 q^{26} - 37 q^{27} - 16 q^{28} - 18 q^{29} + 8 q^{30} - 5 q^{31} - 28 q^{32} - 10 q^{33} + 5 q^{34} - 9 q^{35} + 31 q^{36} - 18 q^{37} - 45 q^{38} + 7 q^{39} + 7 q^{40} - 17 q^{41} + 4 q^{42} + 8 q^{43} + 12 q^{44} - 44 q^{45} + 30 q^{46} - 52 q^{47} - 7 q^{48} + 29 q^{49} + 13 q^{50} + 19 q^{51} - 14 q^{52} - 60 q^{53} + 11 q^{54} + 11 q^{55} + 7 q^{56} + 4 q^{57} + 14 q^{58} - 8 q^{59} + 86 q^{60} - 26 q^{61} + 4 q^{62} - q^{63} + 44 q^{64} - 6 q^{65} + 18 q^{66} + 12 q^{67} - 61 q^{68} - 38 q^{69} + 35 q^{70} - q^{71} + 28 q^{72} - 2 q^{73} + 16 q^{74} - 17 q^{75} + 66 q^{76} - 73 q^{77} + 115 q^{78} + 18 q^{79} - 32 q^{80} + 18 q^{81} + 44 q^{82} - 43 q^{83} + 41 q^{84} + 51 q^{85} + 4 q^{86} + 3 q^{87} - 17 q^{88} - 28 q^{89} + 58 q^{90} - q^{91} - 68 q^{92} - 60 q^{93} + 78 q^{94} - 18 q^{95} + 29 q^{96} - 34 q^{97} + 34 q^{98} + 15 q^{99}+O(q^{100}) \) 18 * q - 4 * q^2 - 10 * q^3 + 16 * q^4 - 27 * q^5 - 3 * q^6 - 11 * q^7 - 12 * q^8 + 14 * q^9 - 5 * q^10 + 2 * q^11 - 32 * q^12 - 25 * q^13 - 7 * q^14 + 9 * q^15 + 8 * q^16 - 30 * q^17 - 10 * q^18 + 4 * q^19 - 41 * q^20 - 16 * q^21 - 24 * q^22 - 26 * q^23 - 12 * q^24 + 31 * q^25 - 18 * q^26 - 37 * q^27 - 16 * q^28 - 18 * q^29 + 8 * q^30 - 5 * q^31 - 28 * q^32 - 10 * q^33 + 5 * q^34 - 9 * q^35 + 31 * q^36 - 18 * q^37 - 45 * q^38 + 7 * q^39 + 7 * q^40 - 17 * q^41 + 4 * q^42 + 8 * q^43 + 12 * q^44 - 44 * q^45 + 30 * q^46 - 52 * q^47 - 7 * q^48 + 29 * q^49 + 13 * q^50 + 19 * q^51 - 14 * q^52 - 60 * q^53 + 11 * q^54 + 11 * q^55 + 7 * q^56 + 4 * q^57 + 14 * q^58 - 8 * q^59 + 86 * q^60 - 26 * q^61 + 4 * q^62 - q^63 + 44 * q^64 - 6 * q^65 + 18 * q^66 + 12 * q^67 - 61 * q^68 - 38 * q^69 + 35 * q^70 - q^71 + 28 * q^72 - 2 * q^73 + 16 * q^74 - 17 * q^75 + 66 * q^76 - 73 * q^77 + 115 * q^78 + 18 * q^79 - 32 * q^80 + 18 * q^81 + 44 * q^82 - 43 * q^83 + 41 * q^84 + 51 * q^85 + 4 * q^86 + 3 * q^87 - 17 * q^88 - 28 * q^89 + 58 * q^90 - q^91 - 68 * q^92 - 60 * q^93 + 78 * q^94 - 18 * q^95 + 29 * q^96 - 34 * q^97 + 34 * q^98 + 15 * 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 $

$2$	0.924759	0.653903	0.326952	−	0.945041i	$-0.393979\pi$
$2$	0.924759	0.653903	0.326952	+	0.945041i	$0.393979\pi$
$3$	2.22306	1.28349	0.641743	−	0.766920i	$-0.278212\pi$
$3$	2.22306	1.28349	0.641743	+	0.766920i	$0.278212\pi$
$4$	−1.14482	−0.572411
$5$	−3.95421	−1.76838	−0.884188	−	0.467131i	$-0.845288\pi$
$5$	−3.95421	−1.76838	−0.884188	+	0.467131i	$0.845288\pi$
$6$	2.05580	0.839275
$7$	−3.08028	−1.16424	−0.582118	−	0.813104i	$-0.697776\pi$
$7$	−3.08028	−1.16424	−0.582118	+	0.813104i	$0.697776\pi$
$8$	−2.90820	−1.02820
$9$	1.94201	0.647336
$10$	−3.65669	−1.15635
$11$	−3.50276	−1.05612	−0.528061	−	0.849207i	$-0.677081\pi$
$11$	−3.50276	−1.05612	−0.528061	+	0.849207i	$0.677081\pi$
$12$	−2.54501	−0.734681
$13$	3.47242	0.963075	0.481538	−	0.876425i	$-0.340079\pi$
$13$	3.47242	0.963075	0.481538	+	0.876425i	$0.340079\pi$
$14$	−2.84852	−0.761298
$15$	−8.79046	−2.26969
$16$	−0.399740	−0.0999351
$17$	−3.04988	−0.739705	−0.369852	−	0.929091i	$-0.620592\pi$
$17$	−3.04988	−0.739705	−0.369852	+	0.929091i	$0.620592\pi$
$18$	1.79589	0.423295
$19$	7.32400	1.68024	0.840121	−	0.542399i	$-0.182484\pi$
$19$	7.32400	1.68024	0.840121	+	0.542399i	$0.182484\pi$
$20$	4.52687	1.01224
$21$	−6.84766	−1.49428
$22$	−3.23921	−0.690601
$23$	−6.16987	−1.28651	−0.643253	−	0.765653i	$-0.722416\pi$
$23$	−6.16987	−1.28651	−0.643253	+	0.765653i	$0.722416\pi$
$24$	−6.46511	−1.31969
$25$	10.6358	2.12716
$26$	3.21115	0.629758
$27$	−2.35198	−0.452639
$28$	3.52637	0.666421
$29$	−3.33808	−0.619867	−0.309933	−	0.950758i	$-0.600307\pi$
$29$	−3.33808	−0.619867	−0.309933	+	0.950758i	$0.600307\pi$
$30$	−8.12905	−1.48415
$31$	5.01284	0.900333	0.450167	−	0.892945i	$-0.351365\pi$
$31$	5.01284	0.900333	0.450167	+	0.892945i	$0.351365\pi$
$32$	5.44674	0.962856
$33$	−7.78685	−1.35552
$34$	−2.82040	−0.483695
$35$	12.1801	2.05881
$36$	−2.22325	−0.370542
$37$	−9.39929	−1.54523	−0.772617	−	0.634873i	$-0.781053\pi$
$37$	−9.39929	−1.54523	−0.772617	+	0.634873i	$0.781053\pi$
$38$	6.77293	1.09872
$39$	7.71940	1.23609
$40$	11.4996	1.81825
$41$	−2.63973	−0.412256	−0.206128	−	0.978525i	$-0.566086\pi$
$41$	−2.63973	−0.412256	−0.206128	+	0.978525i	$0.566086\pi$
$42$	−6.33243	−0.977115
$43$	0.457015	0.0696942	0.0348471	−	0.999393i	$-0.488906\pi$
$43$	0.457015	0.0696942	0.0348471	+	0.999393i	$0.488906\pi$
$44$	4.01003	0.604535
$45$	−7.67911	−1.14473
$46$	−5.70564	−0.841251
$47$	−4.87863	−0.711621	−0.355811	−	0.934558i	$-0.615795\pi$
$47$	−4.87863	−0.711621	−0.355811	+	0.934558i	$0.615795\pi$
$48$	−0.888648	−0.128265
$49$	2.48812	0.355446
$50$	9.83553	1.39095
$51$	−6.78008	−0.949401
$52$	−3.97530	−0.551275
$53$	0.519948	0.0714203	0.0357101	−	0.999362i	$-0.488631\pi$
$53$	0.519948	0.0714203	0.0357101	+	0.999362i	$0.488631\pi$
$54$	−2.17502	−0.295982
$55$	13.8506	1.86762
$56$	8.95807	1.19707
$57$	16.2817	2.15657
$58$	−3.08692	−0.405333
$59$	10.3773	1.35101	0.675503	−	0.737357i	$-0.263927\pi$
$59$	10.3773	1.35101	0.675503	+	0.737357i	$0.263927\pi$
$60$	10.0635	1.29919
$61$	2.71700	0.347877	0.173938	−	0.984757i	$-0.444351\pi$
$61$	2.71700	0.347877	0.173938	+	0.984757i	$0.444351\pi$
$62$	4.63567	0.588731
$63$	−5.98193	−0.753652
$64$	5.83640	0.729550
$65$	−13.7307	−1.70308
$66$	−7.20096	−0.886376
$67$	−11.2680	−1.37661	−0.688304	−	0.725423i	$-0.741644\pi$
$67$	−11.2680	−1.37661	−0.688304	+	0.725423i	$0.741644\pi$
$68$	3.49157	0.423415
$69$	−13.7160	−1.65121
$70$	11.2636	1.34626
$71$	6.36651	0.755566	0.377783	−	0.925894i	$-0.376687\pi$
$71$	6.36651	0.755566	0.377783	+	0.925894i	$0.376687\pi$
$72$	−5.64775	−0.665594
$73$	−1.19812	−0.140229	−0.0701145	−	0.997539i	$-0.522336\pi$
$73$	−1.19812	−0.140229	−0.0701145	+	0.997539i	$0.522336\pi$
$74$	−8.69207	−1.01043
$75$	23.6440	2.73017
$76$	−8.38468	−0.961788
$77$	10.7895	1.22957
$78$	7.13858	0.808285
$79$	−11.6933	−1.31560	−0.657802	−	0.753191i	$-0.728513\pi$
$79$	−11.6933	−1.31560	−0.657802	+	0.753191i	$0.728513\pi$
$80$	1.58066	0.176723
$81$	−11.0546	−1.22829
$82$	−2.44111	−0.269576
$83$	−9.73676	−1.06875	−0.534374	−	0.845248i	$-0.679453\pi$
$83$	−9.73676	−1.06875	−0.534374	+	0.845248i	$0.679453\pi$
$84$	7.83934	0.855342
$85$	12.0599	1.30808
$86$	0.422629	0.0455732
$87$	−7.42077	−0.795590
$88$	10.1867	1.08591
$89$	11.4691	1.21572	0.607859	−	0.794045i	$-0.292028\pi$
$89$	11.4691	1.21572	0.607859	+	0.794045i	$0.292028\pi$
$90$	−7.10132	−0.748545
$91$	−10.6960	−1.12125
$92$	7.06340	0.736410
$93$	11.1439	1.15556
$94$	−4.51155	−0.465331
$95$	−28.9607	−2.97130
$96$	12.1084	1.23581
$97$	7.25071	0.736198	0.368099	−	0.929787i	$-0.380009\pi$
$97$	7.25071	0.736198	0.368099	+	0.929787i	$0.380009\pi$
$98$	2.30091	0.232427
$99$	−6.80238	−0.683665
$100$	−12.1761	−1.21761
$101$	−15.4716	−1.53948	−0.769741	−	0.638356i	$-0.779615\pi$
$101$	−15.4716	−1.53948	−0.769741	+	0.638356i	$0.779615\pi$
$102$	−6.26993	−0.620816
$103$	−11.1594	−1.09957	−0.549785	−	0.835306i	$-0.685290\pi$
$103$	−11.1594	−1.09957	−0.549785	+	0.835306i	$0.685290\pi$
$104$	−10.0985	−0.990238
$105$	27.0771	2.64245
$106$	0.480826	0.0467019
$107$	−15.8859	−1.53574	−0.767872	−	0.640603i	$-0.778684\pi$
$107$	−15.8859	−1.53574	−0.767872	+	0.640603i	$0.778684\pi$
$108$	2.69260	0.259096
$109$	2.66177	0.254951	0.127476	−	0.991842i	$-0.459313\pi$
$109$	2.66177	0.254951	0.127476	+	0.991842i	$0.459313\pi$
$110$	12.8085	1.22124
$111$	−20.8952	−1.98329
$112$	1.23131	0.116348
$113$	−3.96953	−0.373422	−0.186711	−	0.982415i	$-0.559783\pi$
$113$	−3.96953	−0.373422	−0.186711	+	0.982415i	$0.559783\pi$
$114$	15.0567	1.41019
$115$	24.3970	2.27503
$116$	3.82151	0.354818
$117$	6.74346	0.623433
$118$	9.59647	0.883427
$119$	9.39449	0.861191
$120$	25.5644	2.33370
$121$	1.26931	0.115392
$122$	2.51257	0.227478
$123$	−5.86828	−0.529125
$124$	−5.73881	−0.515360
$125$	−22.2851	−1.99324
$126$	−5.53184	−0.492815
$127$	−14.0151	−1.24364	−0.621822	−	0.783159i	$-0.713607\pi$
$127$	−14.0151	−1.24364	−0.621822	+	0.783159i	$0.713607\pi$
$128$	−5.49622	−0.485801
$129$	1.01597	0.0894515
$130$	−12.6976	−1.11365
$131$	21.9251	1.91560	0.957802	−	0.287430i	$-0.0928009\pi$
$131$	21.9251	1.91560	0.957802	+	0.287430i	$0.0928009\pi$
$132$	8.91455	0.775912
$133$	−22.5600	−1.95620
$134$	−10.4202	−0.900168
$135$	9.30024	0.800437
$136$	8.86967	0.760568
$137$	0.725655	0.0619969	0.0309984	−	0.999519i	$-0.490131\pi$
$137$	0.725655	0.0619969	0.0309984	+	0.999519i	$0.490131\pi$
$138$	−12.6840	−1.07973
$139$	2.05950	0.174685	0.0873423	−	0.996178i	$-0.472163\pi$
$139$	2.05950	0.174685	0.0873423	+	0.996178i	$0.472163\pi$
$140$	−13.9440	−1.17848
$141$	−10.8455	−0.913356
$142$	5.88749	0.494067
$143$	−12.1630	−1.01712
$144$	−0.776299	−0.0646916
$145$	13.1995	1.09616
$146$	−1.10797	−0.0916961
$147$	5.53126	0.456210
$148$	10.7605	0.884508
$149$	13.4893	1.10509	0.552544	−	0.833484i	$-0.313657\pi$
$149$	13.4893	1.10509	0.552544	+	0.833484i	$0.313657\pi$
$150$	21.8650	1.78527
$151$	0.695017	0.0565597	0.0282798	−	0.999600i	$-0.490997\pi$
$151$	0.695017	0.0565597	0.0282798	+	0.999600i	$0.490997\pi$
$152$	−21.2997	−1.72763
$153$	−5.92289	−0.478837
$154$	9.97766	0.804023
$155$	−19.8218	−1.59213
$156$	−8.83733	−0.707553
$157$	10.9574	0.874499	0.437249	−	0.899340i	$-0.355953\pi$
$157$	10.9574	0.874499	0.437249	+	0.899340i	$0.355953\pi$
$158$	−10.8135	−0.860277
$159$	1.15588	0.0916669
$160$	−21.5375	−1.70269
$161$	19.0049	1.49780
$162$	−10.2229	−0.803184
$163$	15.9789	1.25156	0.625780	−	0.779999i	$-0.284781\pi$
$163$	15.9789	1.25156	0.625780	+	0.779999i	$0.284781\pi$
$164$	3.02202	0.235980
$165$	30.7908	2.39706
$166$	−9.00415	−0.698858
$167$	13.3872	1.03594	0.517968	−	0.855400i	$-0.326689\pi$
$167$	13.3872	1.03594	0.517968	+	0.855400i	$0.326689\pi$
$168$	19.9144	1.53643
$169$	−0.942324	−0.0724865
$170$	11.1525	0.855355
$171$	14.2233	1.08768
$172$	−0.523201	−0.0398937
$173$	−17.9016	−1.36103	−0.680515	−	0.732734i	$-0.738244\pi$
$173$	−17.9016	−1.36103	−0.680515	+	0.732734i	$0.738244\pi$
$174$	−6.86242	−0.520239
$175$	−32.7612	−2.47651
$176$	1.40019	0.105544
$177$	23.0693	1.73400
$178$	10.6061	0.794962
$179$	24.0861	1.80028	0.900139	−	0.435602i	$-0.143464\pi$
$179$	24.0861	1.80028	0.900139	+	0.435602i	$0.143464\pi$
$180$	8.79121	0.655258
$181$	4.37140	0.324924	0.162462	−	0.986715i	$-0.448057\pi$
$181$	4.37140	0.324924	0.162462	+	0.986715i	$0.448057\pi$
$182$	−9.89123	−0.733187
$183$	6.04007	0.446495
$184$	17.9432	1.32279
$185$	37.1668	2.73255
$186$	10.3054	0.755627
$187$	10.6830	0.781218
$188$	5.58516	0.407340
$189$	7.24477	0.526979
$190$	−26.7816	−1.94294
$191$	−14.3169	−1.03593	−0.517967	−	0.855401i	$-0.673311\pi$
$191$	−14.3169	−1.03593	−0.517967	+	0.855401i	$0.673311\pi$
$192$	12.9747	0.936367
$193$	12.7297	0.916305	0.458152	−	0.888874i	$-0.348511\pi$
$193$	12.7297	0.916305	0.458152	+	0.888874i	$0.348511\pi$
$194$	6.70516	0.481402
$195$	−30.5241	−2.18588
$196$	−2.84846	−0.203461
$197$	13.4238	0.956407	0.478203	−	0.878249i	$-0.341288\pi$
$197$	13.4238	0.956407	0.478203	+	0.878249i	$0.341288\pi$
$198$	−6.29056	−0.447051
$199$	−1.56808	−0.111159	−0.0555793	−	0.998454i	$-0.517701\pi$
$199$	−1.56808	−0.111159	−0.0555793	+	0.998454i	$0.517701\pi$
$200$	−30.9310	−2.18715
$201$	−25.0495	−1.76686
$202$	−14.3075	−1.00667
$203$	10.2822	0.721671
$204$	7.76198	0.543447
$205$	10.4380	0.729024
$206$	−10.3198	−0.719012
$207$	−11.9819	−0.832802
$208$	−1.38807	−0.0962450
$209$	−25.6542	−1.77454
$210$	25.0398	1.72791
$211$	−16.3089	−1.12275	−0.561376	−	0.827561i	$-0.689728\pi$
$211$	−16.3089	−1.12275	−0.561376	+	0.827561i	$0.689728\pi$
$212$	−0.595247	−0.0408817
$213$	14.1532	0.969758
$214$	−14.6906	−1.00423
$215$	−1.80714	−0.123246
$216$	6.84004	0.465406
$217$	−15.4410	−1.04820
$218$	2.46149	0.166713
$219$	−2.66349	−0.179982
$220$	−15.8565	−1.06905
$221$	−10.5905	−0.712391
$222$	−19.3230	−1.29688
$223$	−6.07966	−0.407124	−0.203562	−	0.979062i	$-0.565252\pi$
$223$	−6.07966	−0.407124	−0.203562	+	0.979062i	$0.565252\pi$
$224$	−16.7775	−1.12099
$225$	20.6548	1.37698
$226$	−3.67086	−0.244182
$227$	19.1850	1.27336	0.636678	−	0.771130i	$-0.280308\pi$
$227$	19.1850	1.27336	0.636678	+	0.771130i	$0.280308\pi$
$228$	−18.6397	−1.23444
$229$	9.82576	0.649305	0.324652	−	0.945833i	$-0.394753\pi$
$229$	9.82576	0.649305	0.324652	+	0.945833i	$0.394753\pi$
$230$	22.5613	1.48765
$231$	23.9857	1.57814
$232$	9.70782	0.637350
$233$	−30.2087	−1.97904	−0.989519	−	0.144406i	$-0.953873\pi$
$233$	−30.2087	−1.97904	−0.989519	+	0.144406i	$0.953873\pi$
$234$	6.23607	0.407665
$235$	19.2911	1.25841
$236$	−11.8801	−0.773330
$237$	−25.9950	−1.68856
$238$	8.68763	0.563136
$239$	20.0414	1.29637	0.648186	−	0.761482i	$-0.275528\pi$
$239$	20.0414	1.29637	0.648186	+	0.761482i	$0.275528\pi$
$240$	3.51390	0.226821
$241$	−22.2442	−1.43287	−0.716436	−	0.697653i	$-0.754228\pi$
$241$	−22.2442	−1.43287	−0.716436	+	0.697653i	$0.754228\pi$
$242$	1.17381	0.0754551
$243$	−17.5192	−1.12386
$244$	−3.11048	−0.199128
$245$	−9.83857	−0.628563
$246$	−5.42674	−0.345997
$247$	25.4320	1.61820
$248$	−14.5784	−0.925726
$249$	−21.6454	−1.37172
$250$	−20.6083	−1.30338
$251$	−12.5368	−0.791313	−0.395656	−	0.918399i	$-0.629483\pi$
$251$	−12.5368	−0.791313	−0.395656	+	0.918399i	$0.629483\pi$
$252$	6.84824	0.431399
$253$	21.6116	1.35871
$254$	−12.9606	−0.813222
$255$	26.8099	1.67890
$256$	−16.7555	−1.04722
$257$	13.1786	0.822055	0.411028	−	0.911623i	$-0.365170\pi$
$257$	13.1786	0.822055	0.411028	+	0.911623i	$0.365170\pi$
$258$	0.939531	0.0584926
$259$	28.9524	1.79902
$260$	15.7192	0.974861
$261$	−6.48259	−0.401262
$262$	20.2754	1.25262
$263$	1.98374	0.122322	0.0611612	−	0.998128i	$-0.480520\pi$
$263$	1.98374	0.122322	0.0611612	+	0.998128i	$0.480520\pi$
$264$	22.6457	1.39375
$265$	−2.05598	−0.126298
$266$	−20.8625	−1.27916
$267$	25.4965	1.56036
$268$	12.8999	0.787985
$269$	−0.592176	−0.0361056	−0.0180528	−	0.999837i	$-0.505747\pi$
$269$	−0.592176	−0.0361056	−0.0180528	+	0.999837i	$0.505747\pi$
$270$	8.60047	0.523408
$271$	−21.4167	−1.30097	−0.650486	−	0.759518i	$-0.725435\pi$
$271$	−21.4167	−1.30097	−0.650486	+	0.759518i	$0.725435\pi$
$272$	1.21916	0.0739225
$273$	−23.7779	−1.43910
$274$	0.671056	0.0405400
$275$	−37.2546	−2.24653
$276$	15.7024	0.945172
$277$	−9.10027	−0.546783	−0.273391	−	0.961903i	$-0.588145\pi$
$277$	−9.10027	−0.546783	−0.273391	+	0.961903i	$0.588145\pi$
$278$	1.90454	0.114227
$279$	9.73498	0.582818
$280$	−35.4221	−2.11688
$281$	−2.45423	−0.146407	−0.0732036	−	0.997317i	$-0.523322\pi$
$281$	−2.45423	−0.146407	−0.0732036	+	0.997317i	$0.523322\pi$
$282$	−10.0295	−0.597246
$283$	−7.57111	−0.450056	−0.225028	−	0.974352i	$-0.572247\pi$
$283$	−7.57111	−0.450056	−0.225028	+	0.974352i	$0.572247\pi$
$284$	−7.28852	−0.432494
$285$	−64.3813	−3.81362
$286$	−11.2479	−0.665100
$287$	8.13110	0.479964
$288$	10.5776	0.623291
$289$	−7.69823	−0.452837
$290$	12.2063	0.716781
$291$	16.1188	0.944900
$292$	1.37163	0.0802685
$293$	−15.9522	−0.931935	−0.465967	−	0.884802i	$-0.654294\pi$
$293$	−15.9522	−0.931935	−0.465967	+	0.884802i	$0.654294\pi$
$294$	5.11508	0.298317
$295$	−41.0339	−2.38909
$296$	27.3350	1.58882
$297$	8.23843	0.478042
$298$	12.4744	0.722620
$299$	−21.4244	−1.23900
$300$	−27.0682	−1.56278
$301$	−1.40774	−0.0811405
$302$	0.642723	0.0369846
$303$	−34.3943	−1.97590
$304$	−2.92770	−0.167915
$305$	−10.7436	−0.615177
$306$	−5.47725	−0.313113
$307$	−10.0561	−0.573931	−0.286965	−	0.957941i	$-0.592646\pi$
$307$	−10.0561	−0.573931	−0.286965	+	0.957941i	$0.592646\pi$
$308$	−12.3520	−0.703822
$309$	−24.8081	−1.41128
$310$	−18.3304	−1.04110
$311$	−17.8541	−1.01241	−0.506207	−	0.862412i	$-0.668953\pi$
$311$	−17.8541	−1.01241	−0.506207	+	0.862412i	$0.668953\pi$
$312$	−22.4496	−1.27096
$313$	−11.0835	−0.626475	−0.313238	−	0.949675i	$-0.601414\pi$
$313$	−11.0835	−0.626475	−0.313238	+	0.949675i	$0.601414\pi$
$314$	10.1330	0.571837
$315$	23.6538	1.33274
$316$	13.3868	0.753065
$317$	4.47220	0.251184	0.125592	−	0.992082i	$-0.459917\pi$
$317$	4.47220	0.251184	0.125592	+	0.992082i	$0.459917\pi$
$318$	1.06891	0.0599413
$319$	11.6925	0.654654
$320$	−23.0783	−1.29012
$321$	−35.3153	−1.97111
$322$	17.5750	0.979415
$323$	−22.3373	−1.24288
$324$	12.6556	0.703088
$325$	36.9319	2.04861
$326$	14.7766	0.818399
$327$	5.91728	0.327226
$328$	7.67686	0.423884
$329$	15.0275	0.828495
$330$	28.4741	1.56745
$331$	28.5978	1.57187	0.785937	−	0.618306i	$-0.212181\pi$
$331$	28.5978	1.57187	0.785937	+	0.618306i	$0.212181\pi$
$332$	11.1469	0.611763
$333$	−18.2535	−1.00029
$334$	12.3800	0.677402
$335$	44.5561	2.43436
$336$	2.73729	0.149331
$337$	34.0126	1.85278	0.926392	−	0.376561i	$-0.122893\pi$
$337$	34.0126	1.85278	0.926392	+	0.376561i	$0.122893\pi$
$338$	−0.871422	−0.0473991
$339$	−8.82452	−0.479282
$340$	−13.8064	−0.748757
$341$	−17.5588	−0.950861
$342$	13.1531	0.711238
$343$	13.8978	0.750413
$344$	−1.32909	−0.0716599
$345$	54.2360	2.91997
$346$	−16.5546	−0.889982
$347$	−5.18016	−0.278086	−0.139043	−	0.990286i	$-0.544403\pi$
$347$	−5.18016	−0.278086	−0.139043	+	0.990286i	$0.544403\pi$
$348$	8.49546	0.455404
$349$	4.21519	0.225634	0.112817	−	0.993616i	$-0.464013\pi$
$349$	4.21519	0.225634	0.112817	+	0.993616i	$0.464013\pi$
$350$	−30.2962	−1.61940
$351$	−8.16706	−0.435926
$352$	−19.0786	−1.01689
$353$	22.6574	1.20593	0.602967	−	0.797766i	$-0.293985\pi$
$353$	22.6574	1.20593	0.602967	+	0.797766i	$0.293985\pi$
$354$	21.3336	1.13387
$355$	−25.1745	−1.33613
$356$	−13.1300	−0.695891
$357$	20.8845	1.10533
$358$	22.2738	1.17721
$359$	−5.93624	−0.313303	−0.156651	−	0.987654i	$-0.550070\pi$
$359$	−5.93624	−0.313303	−0.156651	+	0.987654i	$0.550070\pi$
$360$	22.3324	1.17702
$361$	34.6410	1.82321
$362$	4.04249	0.212469
$363$	2.82176	0.148104
$364$	12.2450	0.641814
$365$	4.73760	0.247978
$366$	5.58560	0.291964
$367$	16.1865	0.844930	0.422465	−	0.906379i	$-0.361165\pi$
$367$	16.1865	0.844930	0.422465	+	0.906379i	$0.361165\pi$
$368$	2.46635	0.128567
$369$	−5.12637	−0.266868
$370$	34.3703	1.78683
$371$	−1.60158	−0.0831501
$372$	−12.7577	−0.661458
$373$	−8.96744	−0.464316	−0.232158	−	0.972678i	$-0.574579\pi$
$373$	−8.96744	−0.464316	−0.232158	+	0.972678i	$0.574579\pi$
$374$	9.87919	0.510841
$375$	−49.5411	−2.55829
$376$	14.1880	0.731692
$377$	−11.5912	−0.596978
$378$	6.69966	0.344593
$379$	−32.2073	−1.65438	−0.827189	−	0.561924i	$-0.810061\pi$
$379$	−32.2073	−1.65438	−0.827189	+	0.561924i	$0.810061\pi$
$380$	33.1548	1.70080
$381$	−31.1566	−1.59620
$382$	−13.2397	−0.677400
$383$	32.7239	1.67211	0.836056	−	0.548644i	$-0.184856\pi$
$383$	32.7239	1.67211	0.836056	+	0.548644i	$0.184856\pi$
$384$	−12.2184	−0.623519
$385$	−42.6639	−2.17435
$386$	11.7719	0.599175
$387$	0.887527	0.0451156
$388$	−8.30077	−0.421408
$389$	−15.2384	−0.772616	−0.386308	−	0.922370i	$-0.626250\pi$
$389$	−15.2384	−0.772616	−0.386308	+	0.922370i	$0.626250\pi$
$390$	−28.2275	−1.42935
$391$	18.8174	0.951635
$392$	−7.23596	−0.365471
$393$	48.7408	2.45865
$394$	12.4138	0.625397
$395$	46.2379	2.32648
$396$	7.78751	0.391337
$397$	−31.8678	−1.59940	−0.799699	−	0.600401i	$-0.795008\pi$
$397$	−31.8678	−1.59940	−0.799699	+	0.600401i	$0.795008\pi$
$398$	−1.45010	−0.0726869
$399$	−50.1522	−2.51075
$400$	−4.25155	−0.212578
$401$	−7.87598	−0.393307	−0.196654	−	0.980473i	$-0.563007\pi$
$401$	−7.87598	−0.393307	−0.196654	+	0.980473i	$0.563007\pi$
$402$	−23.1647	−1.15535
$403$	17.4067	0.867088
$404$	17.7122	0.881216
$405$	43.7123	2.17208
$406$	9.50858	0.471903
$407$	32.9234	1.63195
$408$	19.7178	0.976178
$409$	−21.2120	−1.04886	−0.524432	−	0.851452i	$-0.675722\pi$
$409$	−21.2120	−1.04886	−0.524432	+	0.851452i	$0.675722\pi$
$410$	9.65267	0.476711
$411$	1.61318	0.0795721
$412$	12.7755	0.629405
$413$	−31.9649	−1.57289
$414$	−11.0804	−0.544572
$415$	38.5012	1.88995
$416$	18.9133	0.927303
$417$	4.57840	0.224205
$418$	−23.7239	−1.16038
$419$	−17.3991	−0.850003	−0.425001	−	0.905193i	$-0.639726\pi$
$419$	−17.3991	−0.850003	−0.425001	+	0.905193i	$0.639726\pi$
$420$	−30.9984	−1.51257
$421$	−13.3962	−0.652890	−0.326445	−	0.945216i	$-0.605851\pi$
$421$	−13.3962	−0.652890	−0.326445	+	0.945216i	$0.605851\pi$
$422$	−15.0818	−0.734171
$423$	−9.47433	−0.460658
$424$	−1.51211	−0.0734346
$425$	−32.4379	−1.57347
$426$	13.0883	0.634128
$427$	−8.36913	−0.405011
$428$	18.1865	0.879076
$429$	−27.0392	−1.30546
$430$	−1.67116	−0.0805907
$431$	32.3565	1.55856	0.779279	−	0.626677i	$-0.215586\pi$
$431$	32.3565	1.55856	0.779279	+	0.626677i	$0.215586\pi$
$432$	0.940183	0.0452346
$433$	8.08088	0.388342	0.194171	−	0.980968i	$-0.437798\pi$
$433$	8.08088	0.388342	0.194171	+	0.980968i	$0.437798\pi$
$434$	−14.2792	−0.685422
$435$	29.3433	1.40690
$436$	−3.04725	−0.145937
$437$	−45.1881	−2.16164
$438$	−2.46308	−0.117691
$439$	26.4717	1.26342	0.631712	−	0.775203i	$-0.282353\pi$
$439$	26.4717	1.26342	0.631712	+	0.775203i	$0.282353\pi$
$440$	−40.2804	−1.92029
$441$	4.83196	0.230093
$442$	−9.79362	−0.465835
$443$	−38.7060	−1.83898	−0.919488	−	0.393118i	$-0.871396\pi$
$443$	−38.7060	−1.83898	−0.919488	+	0.393118i	$0.871396\pi$
$444$	23.9213	1.13525
$445$	−45.3511	−2.14985
$446$	−5.62222	−0.266220
$447$	29.9876	1.41836
$448$	−17.9777	−0.849368
$449$	−4.41373	−0.208297	−0.104148	−	0.994562i	$-0.533212\pi$
$449$	−4.41373	−0.208297	−0.104148	+	0.994562i	$0.533212\pi$
$450$	19.1007	0.900414
$451$	9.24633	0.435393
$452$	4.54440	0.213751
$453$	1.54507	0.0725936
$454$	17.7415	0.832651
$455$	42.2943	1.98279
$456$	−47.3505	−2.21739
$457$	0.171589	0.00802659	0.00401329	−	0.999992i	$-0.498723\pi$
$457$	0.171589	0.00802659	0.00401329	+	0.999992i	$0.498723\pi$
$458$	9.08646	0.424582
$459$	7.17327	0.334819
$460$	−27.9302	−1.30225
$461$	−32.6067	−1.51864	−0.759322	−	0.650715i	$-0.774469\pi$
$461$	−32.6067	−1.51864	−0.759322	+	0.650715i	$0.774469\pi$
$462$	22.1810	1.03195
$463$	25.5148	1.18577	0.592887	−	0.805285i	$-0.297988\pi$
$463$	25.5148	1.18577	0.592887	+	0.805285i	$0.297988\pi$
$464$	1.33437	0.0619465
$465$	−44.0652	−2.04347
$466$	−27.9357	−1.29410
$467$	3.58271	0.165788	0.0828939	−	0.996558i	$-0.473584\pi$
$467$	3.58271	0.165788	0.0828939	+	0.996558i	$0.473584\pi$
$468$	−7.72006	−0.356860
$469$	34.7086	1.60270
$470$	17.8396	0.822881
$471$	24.3591	1.12241
$472$	−30.1792	−1.38911
$473$	−1.60081	−0.0736055
$474$	−24.0391	−1.10415
$475$	77.8965	3.57414
$476$	−10.7550	−0.492955
$477$	1.00974	0.0462329
$478$	18.5335	0.847702
$479$	−27.2764	−1.24629	−0.623145	−	0.782106i	$-0.714146\pi$
$479$	−27.2764	−1.24629	−0.623145	+	0.782106i	$0.714146\pi$
$480$	−47.8793	−2.18538
$481$	−32.6382	−1.48818
$482$	−20.5705	−0.936959
$483$	42.2491	1.92240
$484$	−1.45313	−0.0660515
$485$	−28.6709	−1.30188
$486$	−16.2010	−0.734893
$487$	−32.4550	−1.47068	−0.735339	−	0.677700i	$-0.762977\pi$
$487$	−32.4550	−1.47068	−0.735339	+	0.677700i	$0.762977\pi$
$488$	−7.90159	−0.357688
$489$	35.5220	1.60636
$490$	−9.09830	−0.411019
$491$	8.45330	0.381492	0.190746	−	0.981639i	$-0.438909\pi$
$491$	8.45330	0.381492	0.190746	+	0.981639i	$0.438909\pi$
$492$	6.71813	0.302877
$493$	10.1808	0.458518
$494$	23.5185	1.05815
$495$	26.8981	1.20898
$496$	−2.00384	−0.0899749
$497$	−19.6106	−0.879657
$498$	−20.0168	−0.896974
$499$	−28.5526	−1.27819	−0.639094	−	0.769128i	$-0.720691\pi$
$499$	−28.5526	−1.27819	−0.639094	+	0.769128i	$0.720691\pi$
$500$	25.5124	1.14095
$501$	29.7607	1.32961
$502$	−11.5935	−0.517442
$503$	14.5826	0.650208	0.325104	−	0.945678i	$-0.394601\pi$
$503$	14.5826	0.650208	0.325104	+	0.945678i	$0.394601\pi$
$504$	17.3966	0.774908
$505$	61.1780	2.72238
$506$	19.9855	0.888463
$507$	−2.09485	−0.0930354
$508$	16.0448	0.711875
$509$	−16.1161	−0.714331	−0.357166	−	0.934041i	$-0.616257\pi$
$509$	−16.1161	−0.714331	−0.357166	+	0.934041i	$0.616257\pi$
$510$	24.7926	1.09784
$511$	3.69053	0.163260
$512$	−4.50233	−0.198977
$513$	−17.2259	−0.760543
$514$	12.1870	0.537545
$515$	44.1267	1.94445
$516$	−1.16311	−0.0512030
$517$	17.0887	0.751558
$518$	26.7740	1.17638
$519$	−39.7963	−1.74686
$520$	39.9315	1.75111
$521$	−7.13196	−0.312457	−0.156228	−	0.987721i	$-0.549934\pi$
$521$	−7.13196	−0.312457	−0.156228	+	0.987721i	$0.549934\pi$
$522$	−5.99483	−0.262386
$523$	−4.52473	−0.197853	−0.0989264	−	0.995095i	$-0.531541\pi$
$523$	−4.52473	−0.197853	−0.0989264	+	0.995095i	$0.531541\pi$
$524$	−25.1003	−1.09651
$525$	−72.8302	−3.17857
$526$	1.83448	0.0799870
$527$	−15.2886	−0.665981
$528$	3.11272	0.135464
$529$	15.0673	0.655099
$530$	−1.90129	−0.0825866
$531$	20.1527	0.874555
$532$	25.8272	1.11975
$533$	−9.16623	−0.397034
$534$	23.5781	1.02032
$535$	62.8160	2.71577
$536$	32.7697	1.41543
$537$	53.5449	2.31063
$538$	−0.547619	−0.0236096
$539$	−8.71529	−0.375394
$540$	−10.6471	−0.458179
$541$	−6.61816	−0.284537	−0.142268	−	0.989828i	$-0.545440\pi$
$541$	−6.61816	−0.284537	−0.142268	+	0.989828i	$0.545440\pi$
$542$	−19.8053	−0.850709
$543$	9.71790	0.417035
$544$	−16.6119	−0.712229
$545$	−10.5252	−0.450849
$546$	−21.9888	−0.941035
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−0.830745	−0.0354877
$549$	5.27644	0.225193
$550$	−34.4515	−1.46902
$551$	−24.4481	−1.04153
$552$	39.8889	1.69778
$553$	36.0188	1.53167
$554$	−8.41556	−0.357543
$555$	82.6240	3.50720
$556$	−2.35776	−0.0999914
$557$	−9.58172	−0.405991	−0.202995	−	0.979180i	$-0.565068\pi$
$557$	−9.58172	−0.405991	−0.202995	+	0.979180i	$0.565068\pi$
$558$	9.00251	0.381106
$559$	1.58695	0.0671207
$560$	−4.86887	−0.205747
$561$	23.7490	1.00268
$562$	−2.26957	−0.0957362
$563$	44.4878	1.87494	0.937468	−	0.348072i	$-0.113164\pi$
$563$	44.4878	1.87494	0.937468	+	0.348072i	$0.113164\pi$
$564$	12.4162	0.522815
$565$	15.6964	0.660351
$566$	−7.00145	−0.294293
$567$	34.0514	1.43002
$568$	−18.5151	−0.776876
$569$	−14.0592	−0.589394	−0.294697	−	0.955591i	$-0.595219\pi$
$569$	−14.0592	−0.589394	−0.294697	+	0.955591i	$0.595219\pi$
$570$	−59.5372	−2.49374
$571$	38.0971	1.59431	0.797156	−	0.603773i	$-0.206337\pi$
$571$	38.0971	1.59431	0.797156	+	0.603773i	$0.206337\pi$
$572$	13.9245	0.582213
$573$	−31.8273	−1.32961
$574$	7.51931	0.313850
$575$	−65.6214	−2.73660
$576$	11.3343	0.472264
$577$	7.80242	0.324819	0.162410	−	0.986723i	$-0.448073\pi$
$577$	7.80242	0.324819	0.162410	+	0.986723i	$0.448073\pi$
$578$	−7.11900	−0.296111
$579$	28.2990	1.17606
$580$	−15.1111	−0.627453
$581$	29.9919	1.24428
$582$	14.9060	0.617873
$583$	−1.82125	−0.0754285
$584$	3.48436	0.144184
$585$	−26.6651	−1.10246
$586$	−14.7519	−0.609395
$587$	−26.8275	−1.10729	−0.553644	−	0.832753i	$-0.686763\pi$
$587$	−26.8275	−1.10729	−0.553644	+	0.832753i	$0.686763\pi$
$588$	−6.33230	−0.261140
$589$	36.7141	1.51278
$590$	−37.9465	−1.56223
$591$	29.8420	1.22753
$592$	3.75728	0.154423
$593$	−34.0163	−1.39688	−0.698441	−	0.715667i	$-0.746123\pi$
$593$	−34.0163	−1.39688	−0.698441	+	0.715667i	$0.746123\pi$
$594$	7.61856	0.312593
$595$	−37.1478	−1.52291
$596$	−15.4428	−0.632564
$597$	−3.48595	−0.142670
$598$	−19.8124	−0.810187
$599$	22.3457	0.913021	0.456510	−	0.889718i	$-0.349099\pi$
$599$	22.3457	0.913021	0.456510	+	0.889718i	$0.349099\pi$
$600$	−68.7615	−2.80718
$601$	25.2733	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$601$	25.2733	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$602$	−1.30182	−0.0530580
$603$	−21.8826	−0.891127
$604$	−0.795671	−0.0323754
$605$	−5.01912	−0.204056
$606$	−31.8065	−1.29205
$607$	40.1847	1.63105	0.815524	−	0.578723i	$-0.196449\pi$
$607$	40.1847	1.63105	0.815524	+	0.578723i	$0.196449\pi$
$608$	39.8919	1.61783
$609$	22.8581	0.926255
$610$	−9.93524	−0.402266
$611$	−16.9406	−0.685344
$612$	6.78065	0.274092
$613$	44.1657	1.78384	0.891919	−	0.452196i	$-0.149359\pi$
$613$	44.1657	1.78384	0.891919	+	0.452196i	$0.149359\pi$
$614$	−9.29945	−0.375295
$615$	23.2044	0.935692
$616$	−31.3780	−1.26425
$617$	−22.9118	−0.922394	−0.461197	−	0.887298i	$-0.652580\pi$
$617$	−22.9118	−0.922394	−0.461197	+	0.887298i	$0.652580\pi$
$618$	−22.9415	−0.922841
$619$	26.6090	1.06951	0.534754	−	0.845008i	$-0.320404\pi$
$619$	26.6090	1.06951	0.534754	+	0.845008i	$0.320404\pi$
$620$	22.6925	0.911351
$621$	14.5114	0.582324
$622$	−16.5107	−0.662020
$623$	−35.3279	−1.41538
$624$	−3.08576	−0.123529
$625$	34.9409	1.39764
$626$	−10.2495	−0.409654
$627$	−57.0309	−2.27760
$628$	−12.5443	−0.500572
$629$	28.6667	1.14302
$630$	21.8741	0.871483
$631$	−10.4893	−0.417573	−0.208787	−	0.977961i	$-0.566951\pi$
$631$	−10.4893	−0.417573	−0.208787	+	0.977961i	$0.566951\pi$
$632$	34.0066	1.35271
$633$	−36.2557	−1.44104
$634$	4.13570	0.164250
$635$	55.4189	2.19923
$636$	−1.32327	−0.0524711
$637$	8.63980	0.342321
$638$	10.8127	0.428080
$639$	12.3638	0.489105
$640$	21.7332	0.859080
$641$	11.4048	0.450463	0.225232	−	0.974305i	$-0.427686\pi$
$641$	11.4048	0.450463	0.225232	+	0.974305i	$0.427686\pi$
$642$	−32.6581	−1.28891
$643$	−24.0029	−0.946581	−0.473291	−	0.880906i	$-0.656934\pi$
$643$	−24.0029	−0.946581	−0.473291	+	0.880906i	$0.656934\pi$
$644$	−21.7572	−0.857356
$645$	−4.01737	−0.158184
$646$	−20.6566	−0.812725
$647$	39.8641	1.56722	0.783609	−	0.621254i	$-0.213377\pi$
$647$	39.8641	1.56722	0.783609	+	0.621254i	$0.213377\pi$
$648$	32.1491	1.26294
$649$	−36.3491	−1.42683
$650$	34.1531	1.33959
$651$	−34.3262	−1.34535
$652$	−18.2929	−0.716407
$653$	−35.3964	−1.38517	−0.692584	−	0.721337i	$-0.743528\pi$
$653$	−35.3964	−1.38517	−0.692584	+	0.721337i	$0.743528\pi$
$654$	5.47205	0.213974
$655$	−86.6964	−3.38751
$656$	1.05521	0.0411989
$657$	−2.32675	−0.0907752
$658$	13.8968	0.541755
$659$	24.7728	0.965010	0.482505	−	0.875893i	$-0.339727\pi$
$659$	24.7728	0.965010	0.482505	+	0.875893i	$0.339727\pi$
$660$	−35.2500	−1.37211
$661$	−23.3350	−0.907626	−0.453813	−	0.891097i	$-0.649937\pi$
$661$	−23.3350	−0.907626	−0.453813	+	0.891097i	$0.649937\pi$
$662$	26.4460	1.02785
$663$	−23.5432	−0.914344
$664$	28.3164	1.09889
$665$	89.2069	3.45930
$666$	−16.8801	−0.654090
$667$	20.5955	0.797463
$668$	−15.3260	−0.592981
$669$	−13.5155	−0.522538
$670$	41.2036	1.59184
$671$	−9.51700	−0.367400
$672$	−37.2974	−1.43878
$673$	−5.98598	−0.230742	−0.115371	−	0.993322i	$-0.536806\pi$
$673$	−5.98598	−0.230742	−0.115371	+	0.993322i	$0.536806\pi$
$674$	31.4534	1.21154
$675$	−25.0152	−0.962835
$676$	1.07879	0.0414920
$677$	−27.5508	−1.05886	−0.529431	−	0.848353i	$-0.677595\pi$
$677$	−27.5508	−1.05886	−0.529431	+	0.848353i	$0.677595\pi$
$678$	−8.16055	−0.313404
$679$	−22.3342	−0.857109
$680$	−35.0725	−1.34497
$681$	42.6495	1.63433
$682$	−16.2376	−0.621771
$683$	41.1849	1.57590	0.787948	−	0.615741i	$-0.211143\pi$
$683$	41.1849	1.57590	0.787948	+	0.615741i	$0.211143\pi$
$684$	−16.2831	−0.622600
$685$	−2.86939	−0.109634
$686$	12.8521	0.490697
$687$	21.8433	0.833373
$688$	−0.182688	−0.00696490
$689$	1.80547	0.0687831
$690$	50.1552	1.90938
$691$	−6.87493	−0.261535	−0.130767	−	0.991413i	$-0.541744\pi$
$691$	−6.87493	−0.261535	−0.130767	+	0.991413i	$0.541744\pi$
$692$	20.4941	0.779069
$693$	20.9532	0.795948
$694$	−4.79040	−0.181841
$695$	−8.14370	−0.308908
$696$	21.5811	0.818029
$697$	8.05086	0.304948
$698$	3.89803	0.147543
$699$	−67.1558	−2.54007
$700$	37.5057	1.41758
$701$	15.9268	0.601548	0.300774	−	0.953695i	$-0.402755\pi$
$701$	15.9268	0.601548	0.300774	+	0.953695i	$0.402755\pi$
$702$	−7.55256	−0.285053
$703$	−68.8404	−2.59637
$704$	−20.4435	−0.770493
$705$	42.8854	1.61516
$706$	20.9526	0.788563
$707$	47.6569	1.79232
$708$	−26.4103	−0.992559
$709$	−8.19363	−0.307718	−0.153859	−	0.988093i	$-0.549170\pi$
$709$	−8.19363	−0.307718	−0.153859	+	0.988093i	$0.549170\pi$
$710$	−23.2804	−0.873696
$711$	−22.7086	−0.851637
$712$	−33.3544	−1.25001
$713$	−30.9286	−1.15828
$714$	19.3132	0.722777
$715$	48.0952	1.79866
$716$	−27.5743	−1.03050
$717$	44.5534	1.66388
$718$	−5.48959	−0.204870
$719$	13.0692	0.487397	0.243699	−	0.969851i	$-0.421639\pi$
$719$	13.0692	0.487397	0.243699	+	0.969851i	$0.421639\pi$
$720$	3.06965	0.114399
$721$	34.3741	1.28016
$722$	32.0346	1.19220
$723$	−49.4502	−1.83907
$724$	−5.00447	−0.185990
$725$	−35.5031	−1.31855
$726$	2.60944	0.0968455
$727$	−18.2107	−0.675397	−0.337699	−	0.941254i	$-0.609648\pi$
$727$	−18.2107	−0.675397	−0.337699	+	0.941254i	$0.609648\pi$
$728$	31.1062	1.15287
$729$	−5.78236	−0.214161
$730$	4.38114	0.162153
$731$	−1.39384	−0.0515531
$732$	−6.91480	−0.255578
$733$	−7.19358	−0.265701	−0.132851	−	0.991136i	$-0.542413\pi$
$733$	−7.19358	−0.265701	−0.132851	+	0.991136i	$0.542413\pi$
$734$	14.9686	0.552502
$735$	−21.8717	−0.806752
$736$	−33.6057	−1.23872
$737$	39.4691	1.45386
$738$	−4.74066	−0.174506
$739$	−0.780351	−0.0287057	−0.0143528	−	0.999897i	$-0.504569\pi$
$739$	−0.780351	−0.0287057	−0.0143528	+	0.999897i	$0.504569\pi$
$740$	−42.5493	−1.56414
$741$	56.5369	2.07694
$742$	−1.48108	−0.0543721
$743$	−13.0511	−0.478797	−0.239399	−	0.970921i	$-0.576950\pi$
$743$	−13.0511	−0.478797	−0.239399	+	0.970921i	$0.576950\pi$
$744$	−32.4086	−1.18816
$745$	−53.3396	−1.95421
$746$	−8.29272	−0.303618
$747$	−18.9089	−0.691839
$748$	−12.2301	−0.447178
$749$	48.9329	1.78797
$750$	−45.8136	−1.67287
$751$	−28.3735	−1.03536	−0.517681	−	0.855573i	$-0.673205\pi$
$751$	−28.3735	−1.03536	−0.517681	+	0.855573i	$0.673205\pi$
$752$	1.95019	0.0711159
$753$	−27.8700	−1.01564
$754$	−10.7191	−0.390366
$755$	−2.74824	−0.100019
$756$	−8.29396	−0.301649
$757$	−11.4037	−0.414476	−0.207238	−	0.978291i	$-0.566447\pi$
$757$	−11.4037	−0.414476	−0.207238	+	0.978291i	$0.566447\pi$
$758$	−29.7840	−1.08180
$759$	48.0438	1.74388
$760$	84.2234	3.05510
$761$	19.8526	0.719658	0.359829	−	0.933018i	$-0.382835\pi$
$761$	19.8526	0.719658	0.359829	+	0.933018i	$0.382835\pi$
$762$	−28.8123	−1.04376
$763$	−8.19899	−0.296823
$764$	16.3903	0.592980
$765$	23.4204	0.846765
$766$	30.2617	1.09340
$767$	36.0342	1.30112
$768$	−37.2485	−1.34409
$769$	−50.3300	−1.81495	−0.907474	−	0.420109i	$-0.861992\pi$
$769$	−50.3300	−1.81495	−0.907474	+	0.420109i	$0.861992\pi$
$770$	−39.4538	−1.42181
$771$	29.2967	1.05510
$772$	−14.5733	−0.524503
$773$	37.6442	1.35397	0.676984	−	0.735998i	$-0.263287\pi$
$773$	37.6442	1.35397	0.676984	+	0.735998i	$0.263287\pi$
$774$	0.820749	0.0295012
$775$	53.3155	1.91515
$776$	−21.0865	−0.756962
$777$	64.3631	2.30901
$778$	−14.0918	−0.505216
$779$	−19.3334	−0.692690
$780$	34.9447	1.25122
$781$	−22.3003	−0.797969
$782$	17.4015	0.622277
$783$	7.85112	0.280576
$784$	−0.994604	−0.0355216
$785$	−43.3280	−1.54644
$786$	45.0735	1.60772
$787$	−39.1131	−1.39423	−0.697116	−	0.716958i	$-0.745534\pi$
$787$	−39.1131	−1.39423	−0.697116	+	0.716958i	$0.745534\pi$
$788$	−15.3679	−0.547458
$789$	4.40997	0.156999
$790$	42.7589	1.52129
$791$	12.2273	0.434751
$792$	19.7827	0.702947
$793$	9.43456	0.335031
$794$	−29.4700	−1.04585
$795$	−4.57058	−0.162102
$796$	1.79518	0.0636284
$797$	−41.6279	−1.47454	−0.737269	−	0.675600i	$-0.763885\pi$
$797$	−41.6279	−1.47454	−0.737269	+	0.675600i	$0.763885\pi$
$798$	−46.3787	−1.64179
$799$	14.8792	0.526389
$800$	57.9303	2.04815
$801$	22.2730	0.786979
$802$	−7.28338	−0.257185
$803$	4.19671	0.148099
$804$	28.6772	1.01137
$805$	−75.1495	−2.64867
$806$	16.0970	0.566992
$807$	−1.31644	−0.0463410
$808$	44.9945	1.58290
$809$	−1.32763	−0.0466770	−0.0233385	−	0.999728i	$-0.507430\pi$
$809$	−1.32763	−0.0466770	−0.0233385	+	0.999728i	$0.507430\pi$
$810$	40.4234	1.42033
$811$	−47.7210	−1.67571	−0.837856	−	0.545892i	$-0.816191\pi$
$811$	−47.7210	−1.67571	−0.837856	+	0.545892i	$0.816191\pi$
$812$	−11.7713	−0.413092
$813$	−47.6107	−1.66978
$814$	30.4462	1.06714
$815$	−63.1838	−2.21323
$816$	2.71027	0.0948785
$817$	3.34718	0.117103
$818$	−19.6159	−0.685855
$819$	−20.7717	−0.725823
$820$	−11.9497	−0.417301
$821$	18.2322	0.636307	0.318153	−	0.948039i	$-0.396937\pi$
$821$	18.2322	0.636307	0.318153	+	0.948039i	$0.396937\pi$
$822$	1.49180	0.0520325
$823$	10.6665	0.371810	0.185905	−	0.982568i	$-0.440478\pi$
$823$	10.6665	0.371810	0.185905	+	0.982568i	$0.440478\pi$
$824$	32.4538	1.13058
$825$	−82.8192	−2.88340
$826$	−29.5598	−1.02852
$827$	−3.91954	−0.136296	−0.0681478	−	0.997675i	$-0.521709\pi$
$827$	−3.91954	−0.136296	−0.0681478	+	0.997675i	$0.521709\pi$
$828$	13.7172	0.476705
$829$	34.1483	1.18602	0.593009	−	0.805196i	$-0.297940\pi$
$829$	34.1483	1.18602	0.593009	+	0.805196i	$0.297940\pi$
$830$	35.6043	1.23584
$831$	−20.2305	−0.701788
$832$	20.2664	0.702611
$833$	−7.58848	−0.262925
$834$	4.23392	0.146609
$835$	−52.9360	−1.83193
$836$	29.3695	1.01577
$837$	−11.7901	−0.407526
$838$	−16.0900	−0.555819
$839$	18.3406	0.633188	0.316594	−	0.948561i	$-0.397461\pi$
$839$	18.3406	0.633188	0.316594	+	0.948561i	$0.397461\pi$
$840$	−78.7456	−2.71698
$841$	−17.8572	−0.615765
$842$	−12.3882	−0.426927
$843$	−5.45591	−0.187912
$844$	18.6708	0.642675
$845$	3.72615	0.128183
$846$	−8.76147	−0.301226
$847$	−3.90983	−0.134343
$848$	−0.207844	−0.00713740
$849$	−16.8310	−0.577640
$850$	−29.9972	−1.02890
$851$	57.9924	1.98795
$852$	−16.2028	−0.555100
$853$	−24.8775	−0.851790	−0.425895	−	0.904773i	$-0.640041\pi$
$853$	−24.8775	−0.851790	−0.425895	+	0.904773i	$0.640041\pi$
$854$	−7.73942	−0.264838
$855$	−56.2418	−1.92343
$856$	46.1993	1.57906
$857$	7.01498	0.239627	0.119814	−	0.992796i	$-0.461770\pi$
$857$	7.01498	0.239627	0.119814	+	0.992796i	$0.461770\pi$
$858$	−25.0047	−0.853647
$859$	29.1462	0.994456	0.497228	−	0.867620i	$-0.334351\pi$
$859$	29.1462	0.994456	0.497228	+	0.867620i	$0.334351\pi$
$860$	2.06885	0.0705471
$861$	18.0759	0.616027
$862$	29.9220	1.01915
$863$	−21.2381	−0.722952	−0.361476	−	0.932381i	$-0.617727\pi$
$863$	−21.2381	−0.722952	−0.361476	+	0.932381i	$0.617727\pi$
$864$	−12.8106	−0.435827
$865$	70.7866	2.40682
$866$	7.47287	0.253938
$867$	−17.1136	−0.581210
$868$	17.6771	0.600001
$869$	40.9589	1.38944
$870$	27.1355	0.919978
$871$	−39.1272	−1.32578
$872$	−7.74095	−0.262142
$873$	14.0809	0.476568
$874$	−41.7881	−1.41350
$875$	68.6442	2.32060
$876$	3.04922	0.103024
$877$	−0.350498	−0.0118355	−0.00591773	−	0.999982i	$-0.501884\pi$
$877$	−0.350498	−0.0118355	−0.00591773	+	0.999982i	$0.501884\pi$
$878$	24.4799	0.826157
$879$	−35.4626	−1.19613
$880$	−5.53666	−0.186641
$881$	20.5547	0.692504	0.346252	−	0.938142i	$-0.387454\pi$
$881$	20.5547	0.692504	0.346252	+	0.938142i	$0.387454\pi$
$882$	4.46839	0.150459
$883$	38.4958	1.29549	0.647744	−	0.761858i	$-0.275713\pi$
$883$	38.4958	1.29549	0.647744	+	0.761858i	$0.275713\pi$
$884$	12.1242	0.407780
$885$	−91.2210	−3.06636
$886$	−35.7937	−1.20251
$887$	−32.5923	−1.09434	−0.547171	−	0.837021i	$-0.684295\pi$
$887$	−32.5923	−1.09434	−0.547171	+	0.837021i	$0.684295\pi$
$888$	60.7675	2.03922
$889$	43.1706	1.44789
$890$	−41.9388	−1.40579
$891$	38.7217	1.29723
$892$	6.96012	0.233042
$893$	−35.7311	−1.19570
$894$	27.7313	0.927472
$895$	−95.2415	−3.18357
$896$	16.9299	0.565588
$897$	−47.6277	−1.59024
$898$	−4.08164	−0.136206
$899$	−16.7333	−0.558087
$900$	−23.6460	−0.788201
$901$	−1.58578	−0.0528299
$902$	8.55062	0.284705
$903$	−3.12948	−0.104143
$904$	11.5442	0.383954
$905$	−17.2854	−0.574588
$906$	1.42881	0.0474692
$907$	−18.8155	−0.624757	−0.312379	−	0.949958i	$-0.601126\pi$
$907$	−18.8155	−0.624757	−0.312379	+	0.949958i	$0.601126\pi$
$908$	−21.9634	−0.728882
$909$	−30.0460	−0.996562
$910$	39.1120	1.29655
$911$	−6.57999	−0.218005	−0.109002	−	0.994041i	$-0.534766\pi$
$911$	−6.57999	−0.218005	−0.109002	+	0.994041i	$0.534766\pi$
$912$	−6.50846	−0.215517
$913$	34.1055	1.12873
$914$	0.158678	0.00524861
$915$	−23.8837	−0.789571
$916$	−11.2487	−0.371669
$917$	−67.5354	−2.23021
$918$	6.63354	0.218939
$919$	26.0037	0.857782	0.428891	−	0.903356i	$-0.358904\pi$
$919$	26.0037	0.857782	0.428891	+	0.903356i	$0.358904\pi$
$920$	−70.9513	−2.33919
$921$	−22.3553	−0.736632
$922$	−30.1533	−0.993046
$923$	22.1072	0.727667
$924$	−27.4593	−0.903345
$925$	−99.9688	−3.28695
$926$	23.5951	0.775382
$927$	−21.6717	−0.711791
$928$	−18.1817	−0.596843
$929$	8.06188	0.264502	0.132251	−	0.991216i	$-0.457780\pi$
$929$	8.06188	0.264502	0.132251	+	0.991216i	$0.457780\pi$
$930$	−40.7497	−1.33623
$931$	18.2230	0.597236
$932$	34.5836	1.13282
$933$	−39.6908	−1.29942
$934$	3.31314	0.108409
$935$	−42.2428	−1.38149
$936$	−19.6113	−0.641017
$937$	23.0454	0.752862	0.376431	−	0.926445i	$-0.377151\pi$
$937$	23.0454	0.752862	0.376431	+	0.926445i	$0.377151\pi$
$938$	32.0971	1.04801
$939$	−24.6393	−0.804072
$940$	−22.0849	−0.720330
$941$	37.3296	1.21691	0.608454	−	0.793589i	$-0.291790\pi$
$941$	37.3296	1.21691	0.608454	+	0.793589i	$0.291790\pi$
$942$	22.5263	0.733945
$943$	16.2868	0.530370
$944$	−4.14822	−0.135013
$945$	−28.6473	−0.931898
$946$	−1.48037	−0.0481309
$947$	−29.3980	−0.955306	−0.477653	−	0.878549i	$-0.658512\pi$
$947$	−29.3980	−0.955306	−0.477653	+	0.878549i	$0.658512\pi$
$948$	29.7597	0.966549
$949$	−4.16036	−0.135051
$950$	72.0355	2.33714
$951$	9.94198	0.322391
$952$	−27.3211	−0.885480
$953$	17.1464	0.555426	0.277713	−	0.960664i	$-0.410424\pi$
$953$	17.1464	0.555426	0.277713	+	0.960664i	$0.410424\pi$
$954$	0.933768	0.0302318
$955$	56.6120	1.83192
$956$	−22.9439	−0.742058
$957$	25.9932	0.840240
$958$	−25.2241	−0.814953
$959$	−2.23522	−0.0721790
$960$	−51.3046	−1.65585
$961$	−5.87140	−0.189400
$962$	−30.1825	−0.973123
$963$	−30.8505	−0.994142
$964$	25.4656	0.820191
$965$	−50.3360	−1.62037
$966$	39.0703	1.25706
$967$	−34.1166	−1.09711	−0.548557	−	0.836113i	$-0.684823\pi$
$967$	−34.1166	−1.09711	−0.548557	+	0.836113i	$0.684823\pi$
$968$	−3.69141	−0.118646
$969$	−49.6573	−1.59522
$970$	−26.5136	−0.851301
$971$	−6.71553	−0.215512	−0.107756	−	0.994177i	$-0.534366\pi$
$971$	−6.71553	−0.215512	−0.107756	+	0.994177i	$0.534366\pi$
$972$	20.0563	0.643307
$973$	−6.34384	−0.203374
$974$	−30.0131	−0.961680
$975$	82.1018	2.62936
$976$	−1.08610	−0.0347651
$977$	−12.8585	−0.411380	−0.205690	−	0.978617i	$-0.565944\pi$
$977$	−12.8585	−0.411380	−0.205690	+	0.978617i	$0.565944\pi$
$978$	32.8493	1.05040
$979$	−40.1734	−1.28395
$980$	11.2634	0.359796
$981$	5.16917	0.165039
$982$	7.81726	0.249459
$983$	44.7830	1.42836	0.714178	−	0.699964i	$-0.246801\pi$
$983$	44.7830	1.42836	0.714178	+	0.699964i	$0.246801\pi$
$984$	17.0661	0.544049
$985$	−53.0806	−1.69129
$986$	9.41475	0.299827
$987$	33.4072	1.06336
$988$	−29.1151	−0.926274
$989$	−2.81973	−0.0896620
$990$	24.8742	0.790554
$991$	38.3443	1.21805	0.609024	−	0.793152i	$-0.291561\pi$
$991$	38.3443	1.21805	0.609024	+	0.793152i	$0.291561\pi$
$992$	27.3036	0.866892
$993$	63.5746	2.01748
$994$	−18.1351	−0.575211
$995$	6.20054	0.196570
$996$	24.7801	0.785189
$997$	−50.7982	−1.60880	−0.804398	−	0.594090i	$-0.797512\pi$
$997$	−50.7982	−1.60880	−0.804398	+	0.594090i	$0.797512\pi$
$998$	−26.4042	−0.835811
$999$	22.1070	0.699433

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.a.b.1.12	✓	18
3.2	odd	2		4923.2.a.l.1.7		18
4.3	odd	2		8752.2.a.s.1.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.12	✓	18	1.1	even	1	trivial
4923.2.a.l.1.7		18	3.2	odd	2
8752.2.a.s.1.2		18	4.3	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:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.a (trivial)

\(f(q)\)	\(=\)	\(q+0.924759 q^{2} +2.22306 q^{3} -1.14482 q^{4} -3.95421 q^{5} +2.05580 q^{6} -3.08028 q^{7} -2.90820 q^{8} +1.94201 q^{9} +O(q^{10})\)	\(q+0.924759 q^{2} +2.22306 q^{3} -1.14482 q^{4} -3.95421 q^{5} +2.05580 q^{6} -3.08028 q^{7} -2.90820 q^{8} +1.94201 q^{9} -3.65669 q^{10} -3.50276 q^{11} -2.54501 q^{12} +3.47242 q^{13} -2.84852 q^{14} -8.79046 q^{15} -0.399740 q^{16} -3.04988 q^{17} +1.79589 q^{18} +7.32400 q^{19} +4.52687 q^{20} -6.84766 q^{21} -3.23921 q^{22} -6.16987 q^{23} -6.46511 q^{24} +10.6358 q^{25} +3.21115 q^{26} -2.35198 q^{27} +3.52637 q^{28} -3.33808 q^{29} -8.12905 q^{30} +5.01284 q^{31} +5.44674 q^{32} -7.78685 q^{33} -2.82040 q^{34} +12.1801 q^{35} -2.22325 q^{36} -9.39929 q^{37} +6.77293 q^{38} +7.71940 q^{39} +11.4996 q^{40} -2.63973 q^{41} -6.33243 q^{42} +0.457015 q^{43} +4.01003 q^{44} -7.67911 q^{45} -5.70564 q^{46} -4.87863 q^{47} -0.888648 q^{48} +2.48812 q^{49} +9.83553 q^{50} -6.78008 q^{51} -3.97530 q^{52} +0.519948 q^{53} -2.17502 q^{54} +13.8506 q^{55} +8.95807 q^{56} +16.2817 q^{57} -3.08692 q^{58} +10.3773 q^{59} +10.0635 q^{60} +2.71700 q^{61} +4.63567 q^{62} -5.98193 q^{63} +5.83640 q^{64} -13.7307 q^{65} -7.20096 q^{66} -11.2680 q^{67} +3.49157 q^{68} -13.7160 q^{69} +11.2636 q^{70} +6.36651 q^{71} -5.64775 q^{72} -1.19812 q^{73} -8.69207 q^{74} +23.6440 q^{75} -8.38468 q^{76} +10.7895 q^{77} +7.13858 q^{78} -11.6933 q^{79} +1.58066 q^{80} -11.0546 q^{81} -2.44111 q^{82} -9.73676 q^{83} +7.83934 q^{84} +12.0599 q^{85} +0.422629 q^{86} -7.42077 q^{87} +10.1867 q^{88} +11.4691 q^{89} -7.10132 q^{90} -10.6960 q^{91} +7.06340 q^{92} +11.1439 q^{93} -4.51155 q^{94} -28.9607 q^{95} +12.1084 q^{96} +7.25071 q^{97} +2.30091 q^{98} -6.80238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 27 q^{5} - 3 q^{6} - 11 q^{7} - 12 q^{8} + 14 q^{9}+O(q^{10}) \) 18 * q - 4 * q^2 - 10 * q^3 + 16 * q^4 - 27 * q^5 - 3 * q^6 - 11 * q^7 - 12 * q^8 + 14 * q^9	\( 18 q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 27 q^{5} - 3 q^{6} - 11 q^{7} - 12 q^{8} + 14 q^{9} - 5 q^{10} + 2 q^{11} - 32 q^{12} - 25 q^{13} - 7 q^{14} + 9 q^{15} + 8 q^{16} - 30 q^{17} - 10 q^{18} + 4 q^{19} - 41 q^{20} - 16 q^{21} - 24 q^{22} - 26 q^{23} - 12 q^{24} + 31 q^{25} - 18 q^{26} - 37 q^{27} - 16 q^{28} - 18 q^{29} + 8 q^{30} - 5 q^{31} - 28 q^{32} - 10 q^{33} + 5 q^{34} - 9 q^{35} + 31 q^{36} - 18 q^{37} - 45 q^{38} + 7 q^{39} + 7 q^{40} - 17 q^{41} + 4 q^{42} + 8 q^{43} + 12 q^{44} - 44 q^{45} + 30 q^{46} - 52 q^{47} - 7 q^{48} + 29 q^{49} + 13 q^{50} + 19 q^{51} - 14 q^{52} - 60 q^{53} + 11 q^{54} + 11 q^{55} + 7 q^{56} + 4 q^{57} + 14 q^{58} - 8 q^{59} + 86 q^{60} - 26 q^{61} + 4 q^{62} - q^{63} + 44 q^{64} - 6 q^{65} + 18 q^{66} + 12 q^{67} - 61 q^{68} - 38 q^{69} + 35 q^{70} - q^{71} + 28 q^{72} - 2 q^{73} + 16 q^{74} - 17 q^{75} + 66 q^{76} - 73 q^{77} + 115 q^{78} + 18 q^{79} - 32 q^{80} + 18 q^{81} + 44 q^{82} - 43 q^{83} + 41 q^{84} + 51 q^{85} + 4 q^{86} + 3 q^{87} - 17 q^{88} - 28 q^{89} + 58 q^{90} - q^{91} - 68 q^{92} - 60 q^{93} + 78 q^{94} - 18 q^{95} + 29 q^{96} - 34 q^{97} + 34 q^{98} + 15 q^{99}+O(q^{100}) \) 18 * q - 4 * q^2 - 10 * q^3 + 16 * q^4 - 27 * q^5 - 3 * q^6 - 11 * q^7 - 12 * q^8 + 14 * q^9 - 5 * q^10 + 2 * q^11 - 32 * q^12 - 25 * q^13 - 7 * q^14 + 9 * q^15 + 8 * q^16 - 30 * q^17 - 10 * q^18 + 4 * q^19 - 41 * q^20 - 16 * q^21 - 24 * q^22 - 26 * q^23 - 12 * q^24 + 31 * q^25 - 18 * q^26 - 37 * q^27 - 16 * q^28 - 18 * q^29 + 8 * q^30 - 5 * q^31 - 28 * q^32 - 10 * q^33 + 5 * q^34 - 9 * q^35 + 31 * q^36 - 18 * q^37 - 45 * q^38 + 7 * q^39 + 7 * q^40 - 17 * q^41 + 4 * q^42 + 8 * q^43 + 12 * q^44 - 44 * q^45 + 30 * q^46 - 52 * q^47 - 7 * q^48 + 29 * q^49 + 13 * q^50 + 19 * q^51 - 14 * q^52 - 60 * q^53 + 11 * q^54 + 11 * q^55 + 7 * q^56 + 4 * q^57 + 14 * q^58 - 8 * q^59 + 86 * q^60 - 26 * q^61 + 4 * q^62 - q^63 + 44 * q^64 - 6 * q^65 + 18 * q^66 + 12 * q^67 - 61 * q^68 - 38 * q^69 + 35 * q^70 - q^71 + 28 * q^72 - 2 * q^73 + 16 * q^74 - 17 * q^75 + 66 * q^76 - 73 * q^77 + 115 * q^78 + 18 * q^79 - 32 * q^80 + 18 * q^81 + 44 * q^82 - 43 * q^83 + 41 * q^84 + 51 * q^85 + 4 * q^86 + 3 * q^87 - 17 * q^88 - 28 * q^89 + 58 * q^90 - q^91 - 68 * q^92 - 60 * q^93 + 78 * q^94 - 18 * q^95 + 29 * q^96 - 34 * q^97 + 34 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more