LMFDB - Embedded newform 6014.2.a.l.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6014,2,Mod(1,6014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6014 = 2 \cdot 31 \cdot 97 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6014.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:	$48.0220317756$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.56405 q^{3} +1.00000 q^{4} +2.83927 q^{5} +1.56405 q^{6} +2.44268 q^{7} -1.00000 q^{8} -0.553753 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.56405 q^{3} +1.00000 q^{4} +2.83927 q^{5} +1.56405 q^{6} +2.44268 q^{7} -1.00000 q^{8} -0.553753 q^{9} -2.83927 q^{10} +4.72484 q^{11} -1.56405 q^{12} +2.03248 q^{13} -2.44268 q^{14} -4.44075 q^{15} +1.00000 q^{16} +5.96221 q^{17} +0.553753 q^{18} -6.97372 q^{19} +2.83927 q^{20} -3.82047 q^{21} -4.72484 q^{22} -9.19659 q^{23} +1.56405 q^{24} +3.06143 q^{25} -2.03248 q^{26} +5.55824 q^{27} +2.44268 q^{28} -3.21045 q^{29} +4.44075 q^{30} +1.00000 q^{31} -1.00000 q^{32} -7.38988 q^{33} -5.96221 q^{34} +6.93543 q^{35} -0.553753 q^{36} +4.14097 q^{37} +6.97372 q^{38} -3.17889 q^{39} -2.83927 q^{40} +3.62949 q^{41} +3.82047 q^{42} +10.4575 q^{43} +4.72484 q^{44} -1.57225 q^{45} +9.19659 q^{46} -4.82756 q^{47} -1.56405 q^{48} -1.03330 q^{49} -3.06143 q^{50} -9.32518 q^{51} +2.03248 q^{52} +11.1373 q^{53} -5.55824 q^{54} +13.4151 q^{55} -2.44268 q^{56} +10.9072 q^{57} +3.21045 q^{58} -3.04935 q^{59} -4.44075 q^{60} +12.7623 q^{61} -1.00000 q^{62} -1.35264 q^{63} +1.00000 q^{64} +5.77075 q^{65} +7.38988 q^{66} -4.32274 q^{67} +5.96221 q^{68} +14.3839 q^{69} -6.93543 q^{70} -8.16703 q^{71} +0.553753 q^{72} -3.30698 q^{73} -4.14097 q^{74} -4.78823 q^{75} -6.97372 q^{76} +11.5413 q^{77} +3.17889 q^{78} -4.15419 q^{79} +2.83927 q^{80} -7.03210 q^{81} -3.62949 q^{82} +15.6956 q^{83} -3.82047 q^{84} +16.9283 q^{85} -10.4575 q^{86} +5.02130 q^{87} -4.72484 q^{88} +5.77053 q^{89} +1.57225 q^{90} +4.96470 q^{91} -9.19659 q^{92} -1.56405 q^{93} +4.82756 q^{94} -19.8002 q^{95} +1.56405 q^{96} -1.00000 q^{97} +1.03330 q^{98} -2.61640 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 38 q^{2} - 2 q^{3} + 38 q^{4} + 2 q^{5} + 2 q^{6} + 3 q^{7} - 38 q^{8} + 54 q^{9}+O(q^{10}) $ 38 * q - 38 * q^2 - 2 * q^3 + 38 * q^4 + 2 * q^5 + 2 * q^6 + 3 * q^7 - 38 * q^8 + 54 * q^9	\( 38 q - 38 q^{2} - 2 q^{3} + 38 q^{4} + 2 q^{5} + 2 q^{6} + 3 q^{7} - 38 q^{8} + 54 q^{9} - 2 q^{10} + 6 q^{11} - 2 q^{12} + 12 q^{13} - 3 q^{14} + 19 q^{15} + 38 q^{16} + 16 q^{17} - 54 q^{18} + 37 q^{19} + 2 q^{20} + 8 q^{21} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 66 q^{25} - 12 q^{26} - 5 q^{27} + 3 q^{28} + 3 q^{29} - 19 q^{30} + 38 q^{31} - 38 q^{32} + 12 q^{33} - 16 q^{34} - 16 q^{35} + 54 q^{36} + 5 q^{37} - 37 q^{38} + 36 q^{39} - 2 q^{40} + 7 q^{41} - 8 q^{42} + 7 q^{43} + 6 q^{44} + 45 q^{45} + 12 q^{46} - 10 q^{47} - 2 q^{48} + 111 q^{49} - 66 q^{50} - 13 q^{51} + 12 q^{52} + 5 q^{53} + 5 q^{54} + 56 q^{55} - 3 q^{56} - 5 q^{57} - 3 q^{58} + 14 q^{59} + 19 q^{60} + 54 q^{61} - 38 q^{62} - 3 q^{63} + 38 q^{64} + 8 q^{65} - 12 q^{66} - 9 q^{67} + 16 q^{68} + 45 q^{69} + 16 q^{70} + 13 q^{71} - 54 q^{72} + 65 q^{73} - 5 q^{74} - 14 q^{75} + 37 q^{76} - 22 q^{77} - 36 q^{78} - 11 q^{79} + 2 q^{80} + 46 q^{81} - 7 q^{82} - 42 q^{83} + 8 q^{84} + 18 q^{85} - 7 q^{86} - 19 q^{87} - 6 q^{88} + 74 q^{89} - 45 q^{90} + 14 q^{91} - 12 q^{92} - 2 q^{93} + 10 q^{94} - 10 q^{95} + 2 q^{96} - 38 q^{97} - 111 q^{98} + 15 q^{99}+O(q^{100}) \) 38 * q - 38 * q^2 - 2 * q^3 + 38 * q^4 + 2 * q^5 + 2 * q^6 + 3 * q^7 - 38 * q^8 + 54 * q^9 - 2 * q^10 + 6 * q^11 - 2 * q^12 + 12 * q^13 - 3 * q^14 + 19 * q^15 + 38 * q^16 + 16 * q^17 - 54 * q^18 + 37 * q^19 + 2 * q^20 + 8 * q^21 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 66 * q^25 - 12 * q^26 - 5 * q^27 + 3 * q^28 + 3 * q^29 - 19 * q^30 + 38 * q^31 - 38 * q^32 + 12 * q^33 - 16 * q^34 - 16 * q^35 + 54 * q^36 + 5 * q^37 - 37 * q^38 + 36 * q^39 - 2 * q^40 + 7 * q^41 - 8 * q^42 + 7 * q^43 + 6 * q^44 + 45 * q^45 + 12 * q^46 - 10 * q^47 - 2 * q^48 + 111 * q^49 - 66 * q^50 - 13 * q^51 + 12 * q^52 + 5 * q^53 + 5 * q^54 + 56 * q^55 - 3 * q^56 - 5 * q^57 - 3 * q^58 + 14 * q^59 + 19 * q^60 + 54 * q^61 - 38 * q^62 - 3 * q^63 + 38 * q^64 + 8 * q^65 - 12 * q^66 - 9 * q^67 + 16 * q^68 + 45 * q^69 + 16 * q^70 + 13 * q^71 - 54 * q^72 + 65 * q^73 - 5 * q^74 - 14 * q^75 + 37 * q^76 - 22 * q^77 - 36 * q^78 - 11 * q^79 + 2 * q^80 + 46 * q^81 - 7 * q^82 - 42 * q^83 + 8 * q^84 + 18 * q^85 - 7 * q^86 - 19 * q^87 - 6 * q^88 + 74 * q^89 - 45 * q^90 + 14 * q^91 - 12 * q^92 - 2 * q^93 + 10 * q^94 - 10 * q^95 + 2 * q^96 - 38 * q^97 - 111 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.56405	−0.903004	−0.451502	−	0.892270i	$-0.649112\pi$
$3$	−1.56405	−0.903004	−0.451502	+	0.892270i	$0.649112\pi$
$4$	1.00000	0.500000
$5$	2.83927	1.26976	0.634879	−	0.772611i	$-0.281050\pi$
$5$	2.83927	1.26976	0.634879	+	0.772611i	$0.281050\pi$
$6$	1.56405	0.638520
$7$	2.44268	0.923248	0.461624	−	0.887076i	$-0.347267\pi$
$7$	2.44268	0.923248	0.461624	+	0.887076i	$0.347267\pi$
$8$	−1.00000	−0.353553
$9$	−0.553753	−0.184584
$10$	−2.83927	−0.897855
$11$	4.72484	1.42459	0.712297	−	0.701878i	$-0.247655\pi$
$11$	4.72484	1.42459	0.712297	+	0.701878i	$0.247655\pi$
$12$	−1.56405	−0.451502
$13$	2.03248	0.563708	0.281854	−	0.959457i	$-0.409051\pi$
$13$	2.03248	0.563708	0.281854	+	0.959457i	$0.409051\pi$
$14$	−2.44268	−0.652835
$15$	−4.44075	−1.14660
$16$	1.00000	0.250000
$17$	5.96221	1.44605	0.723024	−	0.690823i	$-0.242752\pi$
$17$	5.96221	1.44605	0.723024	+	0.690823i	$0.242752\pi$
$18$	0.553753	0.130521
$19$	−6.97372	−1.59988	−0.799940	−	0.600079i	$-0.795136\pi$
$19$	−6.97372	−1.59988	−0.799940	+	0.600079i	$0.795136\pi$
$20$	2.83927	0.634879
$21$	−3.82047	−0.833696
$22$	−4.72484	−1.00734
$23$	−9.19659	−1.91762	−0.958811	−	0.284047i	$-0.908323\pi$
$23$	−9.19659	−1.91762	−0.958811	+	0.284047i	$0.908323\pi$
$24$	1.56405	0.319260
$25$	3.06143	0.612287
$26$	−2.03248	−0.398602
$27$	5.55824	1.06968
$28$	2.44268	0.461624
$29$	−3.21045	−0.596166	−0.298083	−	0.954540i	$-0.596347\pi$
$29$	−3.21045	−0.596166	−0.298083	+	0.954540i	$0.596347\pi$
$30$	4.44075	0.810766
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−7.38988	−1.28641
$34$	−5.96221	−1.02251
$35$	6.93543	1.17230
$36$	−0.553753	−0.0922922
$37$	4.14097	0.680772	0.340386	−	0.940286i	$-0.389442\pi$
$37$	4.14097	0.680772	0.340386	+	0.940286i	$0.389442\pi$
$38$	6.97372	1.13129
$39$	−3.17889	−0.509030
$40$	−2.83927	−0.448927
$41$	3.62949	0.566832	0.283416	−	0.958997i	$-0.408532\pi$
$41$	3.62949	0.566832	0.283416	+	0.958997i	$0.408532\pi$
$42$	3.82047	0.589512
$43$	10.4575	1.59475	0.797376	−	0.603483i	$-0.206221\pi$
$43$	10.4575	1.59475	0.797376	+	0.603483i	$0.206221\pi$
$44$	4.72484	0.712297
$45$	−1.57225	−0.234378
$46$	9.19659	1.35596
$47$	−4.82756	−0.704172	−0.352086	−	0.935968i	$-0.614528\pi$
$47$	−4.82756	−0.704172	−0.352086	+	0.935968i	$0.614528\pi$
$48$	−1.56405	−0.225751
$49$	−1.03330	−0.147614
$50$	−3.06143	−0.432952
$51$	−9.32518	−1.30579
$52$	2.03248	0.281854
$53$	11.1373	1.52983	0.764915	−	0.644132i	$-0.222781\pi$
$53$	11.1373	1.52983	0.764915	+	0.644132i	$0.222781\pi$
$54$	−5.55824	−0.756381
$55$	13.4151	1.80889
$56$	−2.44268	−0.326417
$57$	10.9072	1.44470
$58$	3.21045	0.421553
$59$	−3.04935	−0.396992	−0.198496	−	0.980102i	$-0.563606\pi$
$59$	−3.04935	−0.396992	−0.198496	+	0.980102i	$0.563606\pi$
$60$	−4.44075	−0.573298
$61$	12.7623	1.63404	0.817021	−	0.576607i	$-0.195624\pi$
$61$	12.7623	1.63404	0.817021	+	0.576607i	$0.195624\pi$
$62$	−1.00000	−0.127000
$63$	−1.35264	−0.170417
$64$	1.00000	0.125000
$65$	5.77075	0.715773
$66$	7.38988	0.909632
$67$	−4.32274	−0.528106	−0.264053	−	0.964508i	$-0.585059\pi$
$67$	−4.32274	−0.528106	−0.264053	+	0.964508i	$0.585059\pi$
$68$	5.96221	0.723024
$69$	14.3839	1.73162
$70$	−6.93543	−0.828942
$71$	−8.16703	−0.969248	−0.484624	−	0.874723i	$-0.661044\pi$
$71$	−8.16703	−0.969248	−0.484624	+	0.874723i	$0.661044\pi$
$72$	0.553753	0.0652604
$73$	−3.30698	−0.387053	−0.193527	−	0.981095i	$-0.561993\pi$
$73$	−3.30698	−0.387053	−0.193527	+	0.981095i	$0.561993\pi$
$74$	−4.14097	−0.481378
$75$	−4.78823	−0.552897
$76$	−6.97372	−0.799940
$77$	11.5413	1.31525
$78$	3.17889	0.359939
$79$	−4.15419	−0.467383	−0.233692	−	0.972311i	$-0.575081\pi$
$79$	−4.15419	−0.467383	−0.233692	+	0.972311i	$0.575081\pi$
$80$	2.83927	0.317440
$81$	−7.03210	−0.781344
$82$	−3.62949	−0.400811
$83$	15.6956	1.72282	0.861408	−	0.507914i	$-0.169583\pi$
$83$	15.6956	1.72282	0.861408	+	0.507914i	$0.169583\pi$
$84$	−3.82047	−0.416848
$85$	16.9283	1.83613
$86$	−10.4575	−1.12766
$87$	5.02130	0.538340
$88$	−4.72484	−0.503670
$89$	5.77053	0.611675	0.305838	−	0.952084i	$-0.401064\pi$
$89$	5.77053	0.611675	0.305838	+	0.952084i	$0.401064\pi$
$90$	1.57225	0.165730
$91$	4.96470	0.520442
$92$	−9.19659	−0.958811
$93$	−1.56405	−0.162184
$94$	4.82756	0.497925
$95$	−19.8002	−2.03146
$96$	1.56405	0.159630
$97$	−1.00000	−0.101535
$97$	−1.00000	−0.101535
$98$	1.03330	0.104379
$99$	−2.61640	−0.262958
$100$	3.06143	0.306143
$101$	−12.5626	−1.25002	−0.625011	−	0.780616i	$-0.714906\pi$
$101$	−12.5626	−1.25002	−0.625011	+	0.780616i	$0.714906\pi$
$102$	9.32518	0.923331
$103$	10.9925	1.08312	0.541562	−	0.840661i	$-0.317833\pi$
$103$	10.9925	1.08312	0.541562	+	0.840661i	$0.317833\pi$
$104$	−2.03248	−0.199301
$105$	−10.8473	−1.05859
$106$	−11.1373	−1.08175
$107$	15.6593	1.51385	0.756923	−	0.653504i	$-0.226702\pi$
$107$	15.6593	1.51385	0.756923	+	0.653504i	$0.226702\pi$
$108$	5.55824	0.534842
$109$	9.59290	0.918833	0.459416	−	0.888221i	$-0.348059\pi$
$109$	9.59290	0.918833	0.459416	+	0.888221i	$0.348059\pi$
$110$	−13.4151	−1.27908
$111$	−6.47668	−0.614739
$112$	2.44268	0.230812
$113$	7.96292	0.749089	0.374544	−	0.927209i	$-0.377799\pi$
$113$	7.96292	0.749089	0.374544	+	0.927209i	$0.377799\pi$
$114$	−10.9072	−1.02156
$115$	−26.1116	−2.43492
$116$	−3.21045	−0.298083
$117$	−1.12549	−0.104052
$118$	3.04935	0.280716
$119$	14.5638	1.33506
$120$	4.44075	0.405383
$121$	11.3242	1.02947
$122$	−12.7623	−1.15544
$123$	−5.67670	−0.511851
$124$	1.00000	0.0898027
$125$	−5.50410	−0.492302
$126$	1.35264	0.120503
$127$	−13.7171	−1.21720	−0.608599	−	0.793478i	$-0.708268\pi$
$127$	−13.7171	−1.21720	−0.608599	+	0.793478i	$0.708268\pi$
$128$	−1.00000	−0.0883883
$129$	−16.3560	−1.44007
$130$	−5.77075	−0.506128
$131$	12.7995	1.11830	0.559149	−	0.829067i	$-0.311128\pi$
$131$	12.7995	1.11830	0.559149	+	0.829067i	$0.311128\pi$
$132$	−7.38988	−0.643207
$133$	−17.0346	−1.47709
$134$	4.32274	0.373428
$135$	15.7813	1.35824
$136$	−5.96221	−0.511255
$137$	5.12149	0.437558	0.218779	−	0.975774i	$-0.429793\pi$
$137$	5.12149	0.437558	0.218779	+	0.975774i	$0.429793\pi$
$138$	−14.3839	−1.22444
$139$	20.2185	1.71491	0.857454	−	0.514560i	$-0.172045\pi$
$139$	20.2185	1.71491	0.857454	+	0.514560i	$0.172045\pi$
$140$	6.93543	0.586151
$141$	7.55054	0.635870
$142$	8.16703	0.685362
$143$	9.60314	0.803055
$144$	−0.553753	−0.0461461
$145$	−9.11533	−0.756987
$146$	3.30698	0.273688
$147$	1.61613	0.133296
$148$	4.14097	0.340386
$149$	21.4577	1.75789	0.878943	−	0.476927i	$-0.158249\pi$
$149$	21.4577	1.75789	0.878943	+	0.476927i	$0.158249\pi$
$150$	4.78823	0.390957
$151$	4.01564	0.326788	0.163394	−	0.986561i	$-0.447756\pi$
$151$	4.01564	0.326788	0.163394	+	0.986561i	$0.447756\pi$
$152$	6.97372	0.565643
$153$	−3.30159	−0.266918
$154$	−11.5413	−0.930024
$155$	2.83927	0.228055
$156$	−3.17889	−0.254515
$157$	−19.6713	−1.56994	−0.784970	−	0.619533i	$-0.787322\pi$
$157$	−19.6713	−1.56994	−0.784970	+	0.619533i	$0.787322\pi$
$158$	4.15419	0.330490
$159$	−17.4193	−1.38144
$160$	−2.83927	−0.224464
$161$	−22.4644	−1.77044
$162$	7.03210	0.552494
$163$	−16.6369	−1.30310	−0.651552	−	0.758604i	$-0.725882\pi$
$163$	−16.6369	−1.30310	−0.651552	+	0.758604i	$0.725882\pi$
$164$	3.62949	0.283416
$165$	−20.9819	−1.63343
$166$	−15.6956	−1.21821
$167$	0.317354	0.0245576	0.0122788	−	0.999925i	$-0.496091\pi$
$167$	0.317354	0.0245576	0.0122788	+	0.999925i	$0.496091\pi$
$168$	3.82047	0.294756
$169$	−8.86903	−0.682233
$170$	−16.9283	−1.29834
$171$	3.86172	0.295313
$172$	10.4575	0.797376
$173$	3.49659	0.265841	0.132921	−	0.991127i	$-0.457564\pi$
$173$	3.49659	0.265841	0.132921	+	0.991127i	$0.457564\pi$
$174$	−5.02130	−0.380664
$175$	7.47812	0.565292
$176$	4.72484	0.356149
$177$	4.76933	0.358485
$178$	−5.77053	−0.432520
$179$	−8.75548	−0.654416	−0.327208	−	0.944952i	$-0.606108\pi$
$179$	−8.75548	−0.654416	−0.327208	+	0.944952i	$0.606108\pi$
$180$	−1.57225	−0.117189
$181$	−13.5524	−1.00734	−0.503672	−	0.863895i	$-0.668018\pi$
$181$	−13.5524	−1.00734	−0.503672	+	0.863895i	$0.668018\pi$
$182$	−4.96470	−0.368008
$183$	−19.9608	−1.47555
$184$	9.19659	0.677981
$185$	11.7573	0.864416
$186$	1.56405	0.114682
$187$	28.1705	2.06003
$188$	−4.82756	−0.352086
$189$	13.5770	0.987583
$190$	19.8002	1.43646
$191$	8.96313	0.648549	0.324274	−	0.945963i	$-0.394880\pi$
$191$	8.96313	0.648549	0.324274	+	0.945963i	$0.394880\pi$
$192$	−1.56405	−0.112875
$193$	−11.3792	−0.819093	−0.409546	−	0.912289i	$-0.634313\pi$
$193$	−11.3792	−0.819093	−0.409546	+	0.912289i	$0.634313\pi$
$194$	1.00000	0.0717958
$195$	−9.02573	−0.646346
$196$	−1.03330	−0.0738070
$197$	1.08595	0.0773706	0.0386853	−	0.999251i	$-0.487683\pi$
$197$	1.08595	0.0773706	0.0386853	+	0.999251i	$0.487683\pi$
$198$	2.61640	0.185939
$199$	−20.3458	−1.44227	−0.721137	−	0.692793i	$-0.756380\pi$
$199$	−20.3458	−1.44227	−0.721137	+	0.692793i	$0.756380\pi$
$200$	−3.06143	−0.216476
$201$	6.76097	0.476882
$202$	12.5626	0.883899
$203$	−7.84211	−0.550409
$204$	−9.32518	−0.652893
$205$	10.3051	0.719740
$206$	−10.9925	−0.765884
$207$	5.09264	0.353963
$208$	2.03248	0.140927
$209$	−32.9497	−2.27918
$210$	10.8473	0.748538
$211$	17.9632	1.23664	0.618319	−	0.785927i	$-0.287814\pi$
$211$	17.9632	1.23664	0.618319	+	0.785927i	$0.287814\pi$
$212$	11.1373	0.764915
$213$	12.7736	0.875235
$214$	−15.6593	−1.07045
$215$	29.6916	2.02495
$216$	−5.55824	−0.378190
$217$	2.44268	0.165820
$218$	−9.59290	−0.649713
$219$	5.17228	0.349510
$220$	13.4151	0.904445
$221$	12.1181	0.815149
$222$	6.47668	0.434686
$223$	−12.0807	−0.808981	−0.404491	−	0.914542i	$-0.632551\pi$
$223$	−12.0807	−0.808981	−0.404491	+	0.914542i	$0.632551\pi$
$224$	−2.44268	−0.163209
$225$	−1.69528	−0.113019
$226$	−7.96292	−0.529686
$227$	−12.2854	−0.815412	−0.407706	−	0.913113i	$-0.633671\pi$
$227$	−12.2854	−0.815412	−0.407706	+	0.913113i	$0.633671\pi$
$228$	10.9072	0.722349
$229$	−2.62118	−0.173213	−0.0866063	−	0.996243i	$-0.527602\pi$
$229$	−2.62118	−0.173213	−0.0866063	+	0.996243i	$0.527602\pi$
$230$	26.1116	1.72175
$231$	−18.0511	−1.18768
$232$	3.21045	0.210776
$233$	29.7135	1.94659	0.973297	−	0.229547i	$-0.0737246\pi$
$233$	29.7135	1.94659	0.973297	+	0.229547i	$0.0737246\pi$
$234$	1.12549	0.0735757
$235$	−13.7067	−0.894128
$236$	−3.04935	−0.198496
$237$	6.49736	0.422049
$238$	−14.5638	−0.944030
$239$	−22.5961	−1.46162	−0.730811	−	0.682580i	$-0.760858\pi$
$239$	−22.5961	−1.46162	−0.730811	+	0.682580i	$0.760858\pi$
$240$	−4.44075	−0.286649
$241$	25.0650	1.61458	0.807289	−	0.590156i	$-0.200934\pi$
$241$	25.0650	1.61458	0.807289	+	0.590156i	$0.200934\pi$
$242$	−11.3242	−0.727944
$243$	−5.67618	−0.364127
$244$	12.7623	0.817021
$245$	−2.93381	−0.187434
$246$	5.67670	0.361933
$247$	−14.1739	−0.901866
$248$	−1.00000	−0.0635001
$249$	−24.5487	−1.55571
$250$	5.50410	0.348110
$251$	11.3972	0.719385	0.359692	−	0.933071i	$-0.382882\pi$
$251$	11.3972	0.719385	0.359692	+	0.933071i	$0.382882\pi$
$252$	−1.35264	−0.0852086
$253$	−43.4524	−2.73183
$254$	13.7171	0.860689
$255$	−26.4767	−1.65803
$256$	1.00000	0.0625000
$257$	−3.09150	−0.192842	−0.0964211	−	0.995341i	$-0.530740\pi$
$257$	−3.09150	−0.192842	−0.0964211	+	0.995341i	$0.530740\pi$
$258$	16.3560	1.01828
$259$	10.1151	0.628521
$260$	5.77075	0.357887
$261$	1.77780	0.110043
$262$	−12.7995	−0.790756
$263$	−6.89498	−0.425163	−0.212581	−	0.977143i	$-0.568187\pi$
$263$	−6.89498	−0.425163	−0.212581	+	0.977143i	$0.568187\pi$
$264$	7.38988	0.454816
$265$	31.6218	1.94251
$266$	17.0346	1.04446
$267$	−9.02539	−0.552345
$268$	−4.32274	−0.264053
$269$	−17.9423	−1.09396	−0.546982	−	0.837144i	$-0.684223\pi$
$269$	−17.9423	−1.09396	−0.546982	+	0.837144i	$0.684223\pi$
$270$	−15.7813	−0.960421
$271$	3.38526	0.205640	0.102820	−	0.994700i	$-0.467213\pi$
$271$	3.38526	0.205640	0.102820	+	0.994700i	$0.467213\pi$
$272$	5.96221	0.361512
$273$	−7.76503	−0.469961
$274$	−5.12149	−0.309401
$275$	14.4648	0.872260
$276$	14.3839	0.865809
$277$	20.2196	1.21488	0.607440	−	0.794365i	$-0.292196\pi$
$277$	20.2196	1.21488	0.607440	+	0.794365i	$0.292196\pi$
$278$	−20.2185	−1.21262
$279$	−0.553753	−0.0331523
$280$	−6.93543	−0.414471
$281$	4.29185	0.256030	0.128015	−	0.991772i	$-0.459139\pi$
$281$	4.29185	0.256030	0.128015	+	0.991772i	$0.459139\pi$
$282$	−7.55054	−0.449628
$283$	−14.2986	−0.849961	−0.424980	−	0.905203i	$-0.639719\pi$
$283$	−14.2986	−0.849961	−0.424980	+	0.905203i	$0.639719\pi$
$284$	−8.16703	−0.484624
$285$	30.9685	1.83442
$286$	−9.60314	−0.567846
$287$	8.86571	0.523326
$288$	0.553753	0.0326302
$289$	18.5479	1.09105
$290$	9.11533	0.535270
$291$	1.56405	0.0916861
$292$	−3.30698	−0.193527
$293$	−16.3149	−0.953125	−0.476562	−	0.879141i	$-0.658117\pi$
$293$	−16.3149	−0.953125	−0.476562	+	0.879141i	$0.658117\pi$
$294$	−1.61613	−0.0942545
$295$	−8.65792	−0.504084
$296$	−4.14097	−0.240689
$297$	26.2618	1.52387
$298$	−21.4577	−1.24301
$299$	−18.6919	−1.08098
$300$	−4.78823	−0.276449
$301$	25.5443	1.47235
$302$	−4.01564	−0.231074
$303$	19.6485	1.12877
$304$	−6.97372	−0.399970
$305$	36.2355	2.07484
$306$	3.30159	0.188739
$307$	20.2899	1.15801	0.579003	−	0.815325i	$-0.303442\pi$
$307$	20.2899	1.15801	0.579003	+	0.815325i	$0.303442\pi$
$308$	11.5413	0.657627
$309$	−17.1928	−0.978065
$310$	−2.83927	−0.161260
$311$	22.3536	1.26755	0.633777	−	0.773516i	$-0.281504\pi$
$311$	22.3536	1.26755	0.633777	+	0.773516i	$0.281504\pi$
$312$	3.17889	0.179969
$313$	21.0459	1.18959	0.594793	−	0.803879i	$-0.297234\pi$
$313$	21.0459	1.18959	0.594793	+	0.803879i	$0.297234\pi$
$314$	19.6713	1.11012
$315$	−3.84052	−0.216389
$316$	−4.15419	−0.233692
$317$	−11.6298	−0.653193	−0.326597	−	0.945164i	$-0.605902\pi$
$317$	−11.6298	−0.653193	−0.326597	+	0.945164i	$0.605902\pi$
$318$	17.4193	0.976827
$319$	−15.1689	−0.849294
$320$	2.83927	0.158720
$321$	−24.4920	−1.36701
$322$	22.4644	1.25189
$323$	−41.5788	−2.31350
$324$	−7.03210	−0.390672
$325$	6.22230	0.345151
$326$	16.6369	0.921434
$327$	−15.0038	−0.829709
$328$	−3.62949	−0.200405
$329$	−11.7922	−0.650125
$330$	20.9819	1.15501
$331$	19.4995	1.07179	0.535896	−	0.844284i	$-0.319974\pi$
$331$	19.4995	1.07179	0.535896	+	0.844284i	$0.319974\pi$
$332$	15.6956	0.861408
$333$	−2.29308	−0.125660
$334$	−0.317354	−0.0173648
$335$	−12.2734	−0.670568
$336$	−3.82047	−0.208424
$337$	−29.8120	−1.62396	−0.811980	−	0.583685i	$-0.801610\pi$
$337$	−29.8120	−1.62396	−0.811980	+	0.583685i	$0.801610\pi$
$338$	8.86903	0.482412
$339$	−12.4544	−0.676430
$340$	16.9283	0.918066
$341$	4.72484	0.255865
$342$	−3.86172	−0.208818
$343$	−19.6228	−1.05953
$344$	−10.4575	−0.563830
$345$	40.8397	2.19874
$346$	−3.49659	−0.187978
$347$	7.28533	0.391097	0.195548	−	0.980694i	$-0.437351\pi$
$347$	7.28533	0.391097	0.195548	+	0.980694i	$0.437351\pi$
$348$	5.02130	0.269170
$349$	23.4887	1.25732	0.628659	−	0.777681i	$-0.283604\pi$
$349$	23.4887	1.25732	0.628659	+	0.777681i	$0.283604\pi$
$350$	−7.47812	−0.399722
$351$	11.2970	0.602989
$352$	−4.72484	−0.251835
$353$	−26.6561	−1.41876	−0.709380	−	0.704826i	$-0.751025\pi$
$353$	−26.6561	−1.41876	−0.709380	+	0.704826i	$0.751025\pi$
$354$	−4.76933	−0.253487
$355$	−23.1884	−1.23071
$356$	5.77053	0.305838
$357$	−22.7785	−1.20556
$358$	8.75548	0.462742
$359$	−7.44387	−0.392872	−0.196436	−	0.980517i	$-0.562937\pi$
$359$	−7.44387	−0.392872	−0.196436	+	0.980517i	$0.562937\pi$
$360$	1.57225	0.0828650
$361$	29.6328	1.55962
$362$	13.5524	0.712300
$363$	−17.7115	−0.929614
$364$	4.96470	0.260221
$365$	−9.38941	−0.491464
$366$	19.9608	1.04337
$367$	−9.28377	−0.484609	−0.242304	−	0.970200i	$-0.577903\pi$
$367$	−9.28377	−0.484609	−0.242304	+	0.970200i	$0.577903\pi$
$368$	−9.19659	−0.479405
$369$	−2.00984	−0.104628
$370$	−11.7573	−0.611234
$371$	27.2050	1.41241
$372$	−1.56405	−0.0810921
$373$	6.06223	0.313891	0.156945	−	0.987607i	$-0.449835\pi$
$373$	6.06223	0.313891	0.156945	+	0.987607i	$0.449835\pi$
$374$	−28.1705	−1.45666
$375$	8.60868	0.444551
$376$	4.82756	0.248962
$377$	−6.52517	−0.336063
$378$	−13.5770	−0.698327
$379$	−13.9036	−0.714179	−0.357089	−	0.934070i	$-0.616231\pi$
$379$	−13.9036	−0.714179	−0.357089	+	0.934070i	$0.616231\pi$
$380$	−19.8002	−1.01573
$381$	21.4542	1.09913
$382$	−8.96313	−0.458593
$383$	−3.91970	−0.200287	−0.100144	−	0.994973i	$-0.531930\pi$
$383$	−3.91970	−0.200287	−0.100144	+	0.994973i	$0.531930\pi$
$384$	1.56405	0.0798150
$385$	32.7688	1.67005
$386$	11.3792	0.579186
$387$	−5.79087	−0.294366
$388$	−1.00000	−0.0507673
$389$	−17.5257	−0.888589	−0.444294	−	0.895881i	$-0.646546\pi$
$389$	−17.5257	−0.888589	−0.444294	+	0.895881i	$0.646546\pi$
$390$	9.02573	0.457035
$391$	−54.8320	−2.77297
$392$	1.03330	0.0521894
$393$	−20.0190	−1.00983
$394$	−1.08595	−0.0547093
$395$	−11.7949	−0.593464
$396$	−2.61640	−0.131479
$397$	35.1313	1.76319	0.881596	−	0.472005i	$-0.156470\pi$
$397$	35.1313	1.76319	0.881596	+	0.472005i	$0.156470\pi$
$398$	20.3458	1.01984
$399$	26.6429	1.33381
$400$	3.06143	0.153072
$401$	−12.6649	−0.632453	−0.316227	−	0.948684i	$-0.602416\pi$
$401$	−12.6649	−0.632453	−0.316227	+	0.948684i	$0.602416\pi$
$402$	−6.76097	−0.337207
$403$	2.03248	0.101245
$404$	−12.5626	−0.625011
$405$	−19.9660	−0.992118
$406$	7.84211	0.389198
$407$	19.5654	0.969823
$408$	9.32518	0.461665
$409$	29.7878	1.47291	0.736456	−	0.676486i	$-0.236498\pi$
$409$	29.7878	1.47291	0.736456	+	0.676486i	$0.236498\pi$
$410$	−10.3051	−0.508933
$411$	−8.01026	−0.395117
$412$	10.9925	0.541562
$413$	−7.44860	−0.366522
$414$	−5.09264	−0.250290
$415$	44.5640	2.18756
$416$	−2.03248	−0.0996504
$417$	−31.6227	−1.54857
$418$	32.9497	1.61162
$419$	−17.0653	−0.833695	−0.416847	−	0.908977i	$-0.636865\pi$
$419$	−17.0653	−0.833695	−0.416847	+	0.908977i	$0.636865\pi$
$420$	−10.8473	−0.529296
$421$	24.9960	1.21823	0.609114	−	0.793082i	$-0.291525\pi$
$421$	24.9960	1.21823	0.609114	+	0.793082i	$0.291525\pi$
$422$	−17.9632	−0.874435
$423$	2.67328	0.129979
$424$	−11.1373	−0.540876
$425$	18.2529	0.885396
$426$	−12.7736	−0.618884
$427$	31.1742	1.50863
$428$	15.6593	0.756923
$429$	−15.0198	−0.725162
$430$	−29.6916	−1.43186
$431$	−2.31221	−0.111375	−0.0556876	−	0.998448i	$-0.517735\pi$
$431$	−2.31221	−0.111375	−0.0556876	+	0.998448i	$0.517735\pi$
$432$	5.55824	0.267421
$433$	13.7386	0.660237	0.330119	−	0.943939i	$-0.392911\pi$
$433$	13.7386	0.660237	0.330119	+	0.943939i	$0.392911\pi$
$434$	−2.44268	−0.117253
$435$	14.2568	0.683562
$436$	9.59290	0.459416
$437$	64.1344	3.06797
$438$	−5.17228	−0.247141
$439$	21.1331	1.00863	0.504313	−	0.863521i	$-0.331746\pi$
$439$	21.1331	1.00863	0.504313	+	0.863521i	$0.331746\pi$
$440$	−13.4151	−0.639539
$441$	0.572192	0.0272472
$442$	−12.1181	−0.576397
$443$	7.71224	0.366420	0.183210	−	0.983074i	$-0.441351\pi$
$443$	7.71224	0.366420	0.183210	+	0.983074i	$0.441351\pi$
$444$	−6.47668	−0.307370
$445$	16.3841	0.776680
$446$	12.0807	0.572036
$447$	−33.5609	−1.58738
$448$	2.44268	0.115406
$449$	9.91773	0.468047	0.234023	−	0.972231i	$-0.424811\pi$
$449$	9.91773	0.468047	0.234023	+	0.972231i	$0.424811\pi$
$450$	1.69528	0.0799162
$451$	17.1488	0.807505
$452$	7.96292	0.374544
$453$	−6.28065	−0.295091
$454$	12.2854	0.576583
$455$	14.0961	0.660836
$456$	−10.9072	−0.510778
$457$	22.8877	1.07064	0.535321	−	0.844649i	$-0.320191\pi$
$457$	22.8877	1.07064	0.535321	+	0.844649i	$0.320191\pi$
$458$	2.62118	0.122480
$459$	33.1394	1.54681
$460$	−26.1116	−1.21746
$461$	−23.0172	−1.07202	−0.536008	−	0.844213i	$-0.680068\pi$
$461$	−23.0172	−1.07202	−0.536008	+	0.844213i	$0.680068\pi$
$462$	18.0511	0.839815
$463$	4.08864	0.190015	0.0950077	−	0.995477i	$-0.469712\pi$
$463$	4.08864	0.190015	0.0950077	+	0.995477i	$0.469712\pi$
$464$	−3.21045	−0.149041
$465$	−4.44075	−0.205935
$466$	−29.7135	−1.37645
$467$	6.65541	0.307975	0.153988	−	0.988073i	$-0.450788\pi$
$467$	6.65541	0.307975	0.153988	+	0.988073i	$0.450788\pi$
$468$	−1.12549	−0.0520259
$469$	−10.5591	−0.487573
$470$	13.7067	0.632244
$471$	30.7669	1.41766
$472$	3.04935	0.140358
$473$	49.4100	2.27187
$474$	−6.49736	−0.298434
$475$	−21.3496	−0.979586
$476$	14.5638	0.667530
$477$	−6.16733	−0.282383
$478$	22.5961	1.03352
$479$	−25.3562	−1.15855	−0.579277	−	0.815131i	$-0.696665\pi$
$479$	−25.3562	−1.15855	−0.579277	+	0.815131i	$0.696665\pi$
$480$	4.44075	0.202692
$481$	8.41643	0.383756
$482$	−25.0650	−1.14168
$483$	35.1353	1.59871
$484$	11.3242	0.514734
$485$	−2.83927	−0.128924
$486$	5.67618	0.257477
$487$	−36.1616	−1.63864	−0.819320	−	0.573337i	$-0.805649\pi$
$487$	−36.1616	−1.63864	−0.819320	+	0.573337i	$0.805649\pi$
$488$	−12.7623	−0.577721
$489$	26.0209	1.17671
$490$	2.93381	0.132536
$491$	17.0947	0.771472	0.385736	−	0.922609i	$-0.373948\pi$
$491$	17.0947	0.771472	0.385736	+	0.922609i	$0.373948\pi$
$492$	−5.67670	−0.255926
$493$	−19.1414	−0.862084
$494$	14.1739	0.637715
$495$	−7.42865	−0.333893
$496$	1.00000	0.0449013
$497$	−19.9495	−0.894856
$498$	24.5487	1.10005
$499$	35.9018	1.60719	0.803593	−	0.595179i	$-0.202919\pi$
$499$	35.9018	1.60719	0.803593	+	0.595179i	$0.202919\pi$
$500$	−5.50410	−0.246151
$501$	−0.496356	−0.0221756
$502$	−11.3972	−0.508682
$503$	−28.6046	−1.27542	−0.637708	−	0.770278i	$-0.720117\pi$
$503$	−28.6046	−1.27542	−0.637708	+	0.770278i	$0.720117\pi$
$504$	1.35264	0.0602515
$505$	−35.6685	−1.58723
$506$	43.4524	1.93170
$507$	13.8716	0.616059
$508$	−13.7171	−0.608599
$509$	−37.5062	−1.66243	−0.831216	−	0.555950i	$-0.812355\pi$
$509$	−37.5062	−1.66243	−0.831216	+	0.555950i	$0.812355\pi$
$510$	26.4767	1.17241
$511$	−8.07791	−0.357346
$512$	−1.00000	−0.0441942
$513$	−38.7616	−1.71137
$514$	3.09150	0.136360
$515$	31.2107	1.37531
$516$	−16.3560	−0.720033
$517$	−22.8095	−1.00316
$518$	−10.1151	−0.444431
$519$	−5.46884	−0.240056
$520$	−5.77075	−0.253064
$521$	10.7180	0.469563	0.234782	−	0.972048i	$-0.424562\pi$
$521$	10.7180	0.469563	0.234782	+	0.972048i	$0.424562\pi$
$522$	−1.77780	−0.0778121
$523$	−8.65004	−0.378240	−0.189120	−	0.981954i	$-0.560563\pi$
$523$	−8.65004	−0.378240	−0.189120	+	0.981954i	$0.560563\pi$
$524$	12.7995	0.559149
$525$	−11.6961	−0.510461
$526$	6.89498	0.300636
$527$	5.96221	0.259718
$528$	−7.38988	−0.321603
$529$	61.5772	2.67727
$530$	−31.6218	−1.37356
$531$	1.68859	0.0732785
$532$	−17.0346	−0.738543
$533$	7.37687	0.319528
$534$	9.02539	0.390567
$535$	44.4611	1.92222
$536$	4.32274	0.186714
$537$	13.6940	0.590940
$538$	17.9423	0.773549
$539$	−4.88217	−0.210290
$540$	15.7813	0.679120
$541$	18.7987	0.808220	0.404110	−	0.914710i	$-0.367581\pi$
$541$	18.7987	0.808220	0.404110	+	0.914710i	$0.367581\pi$
$542$	−3.38526	−0.145409
$543$	21.1966	0.909635
$544$	−5.96221	−0.255628
$545$	27.2368	1.16670
$546$	7.76503	0.332313
$547$	12.1270	0.518511	0.259256	−	0.965809i	$-0.416523\pi$
$547$	12.1270	0.518511	0.259256	+	0.965809i	$0.416523\pi$
$548$	5.12149	0.218779
$549$	−7.06716	−0.301619
$550$	−14.4648	−0.616781
$551$	22.3888	0.953794
$552$	−14.3839	−0.612220
$553$	−10.1474	−0.431510
$554$	−20.2196	−0.859051
$555$	−18.3890	−0.780570
$556$	20.2185	0.857454
$557$	35.2803	1.49488	0.747438	−	0.664331i	$-0.231284\pi$
$557$	35.2803	1.49488	0.747438	+	0.664331i	$0.231284\pi$
$558$	0.553753	0.0234422
$559$	21.2546	0.898974
$560$	6.93543	0.293075
$561$	−44.0600	−1.86022
$562$	−4.29185	−0.181041
$563$	−34.5099	−1.45442	−0.727210	−	0.686415i	$-0.759183\pi$
$563$	−34.5099	−1.45442	−0.727210	+	0.686415i	$0.759183\pi$
$564$	7.55054	0.317935
$565$	22.6089	0.951162
$566$	14.2986	0.601013
$567$	−17.1772	−0.721374
$568$	8.16703	0.342681
$569$	27.5160	1.15353	0.576765	−	0.816910i	$-0.304315\pi$
$569$	27.5160	1.15353	0.576765	+	0.816910i	$0.304315\pi$
$570$	−30.9685	−1.29713
$571$	1.49287	0.0624746	0.0312373	−	0.999512i	$-0.490055\pi$
$571$	1.49287	0.0624746	0.0312373	+	0.999512i	$0.490055\pi$
$572$	9.60314	0.401528
$573$	−14.0188	−0.585642
$574$	−8.86571	−0.370047
$575$	−28.1547	−1.17413
$576$	−0.553753	−0.0230731
$577$	−13.8253	−0.575557	−0.287778	−	0.957697i	$-0.592917\pi$
$577$	−13.8253	−0.575557	−0.287778	+	0.957697i	$0.592917\pi$
$578$	−18.5479	−0.771492
$579$	17.7976	0.739644
$580$	−9.11533	−0.378493
$581$	38.3394	1.59059
$582$	−1.56405	−0.0648319
$583$	52.6221	2.17939
$584$	3.30698	0.136844
$585$	−3.19557	−0.132121
$586$	16.3149	0.673961
$587$	39.9045	1.64703	0.823517	−	0.567291i	$-0.192009\pi$
$587$	39.9045	1.64703	0.823517	+	0.567291i	$0.192009\pi$
$588$	1.61613	0.0666480
$589$	−6.97372	−0.287347
$590$	8.65792	0.356441
$591$	−1.69847	−0.0698659
$592$	4.14097	0.170193
$593$	34.6300	1.42208	0.711042	−	0.703149i	$-0.248223\pi$
$593$	34.6300	1.42208	0.711042	+	0.703149i	$0.248223\pi$
$594$	−26.2618	−1.07754
$595$	41.3505	1.69520
$596$	21.4577	0.878943
$597$	31.8218	1.30238
$598$	18.6919	0.764367
$599$	−3.33972	−0.136457	−0.0682287	−	0.997670i	$-0.521735\pi$
$599$	−3.33972	−0.136457	−0.0682287	+	0.997670i	$0.521735\pi$
$600$	4.78823	0.195479
$601$	36.3143	1.48129	0.740645	−	0.671897i	$-0.234520\pi$
$601$	36.3143	1.48129	0.740645	+	0.671897i	$0.234520\pi$
$602$	−25.5443	−1.04111
$603$	2.39373	0.0974802
$604$	4.01564	0.163394
$605$	32.1523	1.30718
$606$	−19.6485	−0.798164
$607$	−2.30705	−0.0936402	−0.0468201	−	0.998903i	$-0.514909\pi$
$607$	−2.30705	−0.0936402	−0.0468201	+	0.998903i	$0.514909\pi$
$608$	6.97372	0.282822
$609$	12.2654	0.497021
$610$	−36.2355	−1.46713
$611$	−9.81191	−0.396947
$612$	−3.30159	−0.133459
$613$	6.18232	0.249701	0.124851	−	0.992176i	$-0.460155\pi$
$613$	6.18232	0.249701	0.124851	+	0.992176i	$0.460155\pi$
$614$	−20.2899	−0.818834
$615$	−16.1177	−0.649927
$616$	−11.5413	−0.465012
$617$	35.7624	1.43974	0.719870	−	0.694109i	$-0.244201\pi$
$617$	35.7624	1.43974	0.719870	+	0.694109i	$0.244201\pi$
$618$	17.1928	0.691596
$619$	9.99332	0.401665	0.200833	−	0.979626i	$-0.435635\pi$
$619$	9.99332	0.401665	0.200833	+	0.979626i	$0.435635\pi$
$620$	2.83927	0.114028
$621$	−51.1169	−2.05125
$622$	−22.3536	−0.896297
$623$	14.0956	0.564728
$624$	−3.17889	−0.127258
$625$	−30.9348	−1.23739
$626$	−21.0459	−0.841164
$627$	51.5350	2.05811
$628$	−19.6713	−0.784970
$629$	24.6893	0.984428
$630$	3.84052	0.153010
$631$	−18.9004	−0.752411	−0.376206	−	0.926536i	$-0.622771\pi$
$631$	−18.9004	−0.752411	−0.376206	+	0.926536i	$0.622771\pi$
$632$	4.15419	0.165245
$633$	−28.0953	−1.11669
$634$	11.6298	0.461877
$635$	−38.9466	−1.54555
$636$	−17.4193	−0.690721
$637$	−2.10015	−0.0832112
$638$	15.1689	0.600542
$639$	4.52252	0.178908
$640$	−2.83927	−0.112232
$641$	−7.31925	−0.289093	−0.144547	−	0.989498i	$-0.546172\pi$
$641$	−7.31925	−0.289093	−0.144547	+	0.989498i	$0.546172\pi$
$642$	24.4920	0.966621
$643$	11.5094	0.453888	0.226944	−	0.973908i	$-0.427127\pi$
$643$	11.5094	0.453888	0.226944	+	0.973908i	$0.427127\pi$
$644$	−22.4644	−0.885219
$645$	−46.4391	−1.82854
$646$	41.5788	1.63589
$647$	−3.84468	−0.151150	−0.0755750	−	0.997140i	$-0.524079\pi$
$647$	−3.84468	−0.151150	−0.0755750	+	0.997140i	$0.524079\pi$
$648$	7.03210	0.276247
$649$	−14.4077	−0.565552
$650$	−6.22230	−0.244059
$651$	−3.82047	−0.149736
$652$	−16.6369	−0.651552
$653$	−16.8717	−0.660241	−0.330120	−	0.943939i	$-0.607089\pi$
$653$	−16.8717	−0.660241	−0.330120	+	0.943939i	$0.607089\pi$
$654$	15.0038	0.586693
$655$	36.3412	1.41997
$656$	3.62949	0.141708
$657$	1.83125	0.0714440
$658$	11.7922	0.459708
$659$	8.20434	0.319596	0.159798	−	0.987150i	$-0.448916\pi$
$659$	8.20434	0.319596	0.159798	+	0.987150i	$0.448916\pi$
$660$	−20.9819	−0.816717
$661$	15.6075	0.607062	0.303531	−	0.952822i	$-0.401834\pi$
$661$	15.6075	0.607062	0.303531	+	0.952822i	$0.401834\pi$
$662$	−19.4995	−0.757871
$663$	−18.9532	−0.736082
$664$	−15.6956	−0.609107
$665$	−48.3657	−1.87554
$666$	2.29308	0.0888549
$667$	29.5252	1.14322
$668$	0.317354	0.0122788
$669$	18.8947	0.730513
$670$	12.2734	0.474163
$671$	60.2998	2.32785
$672$	3.82047	0.147378
$673$	−12.6163	−0.486324	−0.243162	−	0.969986i	$-0.578185\pi$
$673$	−12.6163	−0.486324	−0.243162	+	0.969986i	$0.578185\pi$
$674$	29.8120	1.14831
$675$	17.0162	0.654954
$676$	−8.86903	−0.341117
$677$	−25.5838	−0.983265	−0.491633	−	0.870803i	$-0.663600\pi$
$677$	−25.5838	−0.983265	−0.491633	+	0.870803i	$0.663600\pi$
$678$	12.4544	0.478308
$679$	−2.44268	−0.0937416
$680$	−16.9283	−0.649171
$681$	19.2150	0.736320
$682$	−4.72484	−0.180924
$683$	−43.9514	−1.68175	−0.840877	−	0.541227i	$-0.817960\pi$
$683$	−43.9514	−1.68175	−0.840877	+	0.541227i	$0.817960\pi$
$684$	3.86172	0.147657
$685$	14.5413	0.555594
$686$	19.6228	0.749202
$687$	4.09966	0.156412
$688$	10.4575	0.398688
$689$	22.6364	0.862377
$690$	−40.8397	−1.55474
$691$	16.5253	0.628651	0.314325	−	0.949315i	$-0.398222\pi$
$691$	16.5253	0.628651	0.314325	+	0.949315i	$0.398222\pi$
$692$	3.49659	0.132921
$693$	−6.39103	−0.242775
$694$	−7.28533	−0.276547
$695$	57.4056	2.17752
$696$	−5.02130	−0.190332
$697$	21.6398	0.819666
$698$	−23.4887	−0.889059
$699$	−46.4733	−1.75778
$700$	7.47812	0.282646
$701$	−4.03454	−0.152383	−0.0761913	−	0.997093i	$-0.524276\pi$
$701$	−4.03454	−0.152383	−0.0761913	+	0.997093i	$0.524276\pi$
$702$	−11.2970	−0.426378
$703$	−28.8780	−1.08915
$704$	4.72484	0.178074
$705$	21.4380	0.807401
$706$	26.6561	1.00321
$707$	−30.6864	−1.15408
$708$	4.76933	0.179243
$709$	18.9412	0.711351	0.355676	−	0.934610i	$-0.384251\pi$
$709$	18.9412	0.711351	0.355676	+	0.934610i	$0.384251\pi$
$710$	23.1884	0.870244
$711$	2.30040	0.0862717
$712$	−5.77053	−0.216260
$713$	−9.19659	−0.344415
$714$	22.7785	0.852463
$715$	27.2659	1.01969
$716$	−8.75548	−0.327208
$717$	35.3414	1.31985
$718$	7.44387	0.277803
$719$	−38.6587	−1.44173	−0.720864	−	0.693077i	$-0.756255\pi$
$719$	−38.6587	−1.44173	−0.720864	+	0.693077i	$0.756255\pi$
$720$	−1.57225	−0.0585944
$721$	26.8512	0.999991
$722$	−29.6328	−1.10282
$723$	−39.2029	−1.45797
$724$	−13.5524	−0.503672
$725$	−9.82858	−0.365024
$726$	17.7115	0.657336
$727$	15.4334	0.572394	0.286197	−	0.958171i	$-0.407609\pi$
$727$	15.4334	0.572394	0.286197	+	0.958171i	$0.407609\pi$
$728$	−4.96470	−0.184004
$729$	29.9741	1.11015
$730$	9.38941	0.347518
$731$	62.3497	2.30609
$732$	−19.9608	−0.737773
$733$	37.1375	1.37170	0.685852	−	0.727741i	$-0.259430\pi$
$733$	37.1375	1.37170	0.685852	+	0.727741i	$0.259430\pi$
$734$	9.28377	0.342670
$735$	4.58862	0.169254
$736$	9.19659	0.338991
$737$	−20.4243	−0.752337
$738$	2.00984	0.0739834
$739$	18.5598	0.682734	0.341367	−	0.939930i	$-0.389110\pi$
$739$	18.5598	0.682734	0.341367	+	0.939930i	$0.389110\pi$
$740$	11.7573	0.432208
$741$	22.1687	0.814388
$742$	−27.2050	−0.998726
$743$	−14.3429	−0.526191	−0.263095	−	0.964770i	$-0.584743\pi$
$743$	−14.3429	−0.526191	−0.263095	+	0.964770i	$0.584743\pi$
$744$	1.56405	0.0573408
$745$	60.9242	2.23209
$746$	−6.06223	−0.221954
$747$	−8.69149	−0.318005
$748$	28.1705	1.03002
$749$	38.2508	1.39766
$750$	−8.60868	−0.314345
$751$	51.6139	1.88342	0.941709	−	0.336430i	$-0.109219\pi$
$751$	51.6139	1.88342	0.941709	+	0.336430i	$0.109219\pi$
$752$	−4.82756	−0.176043
$753$	−17.8258	−0.649607
$754$	6.52517	0.237633
$755$	11.4015	0.414942
$756$	13.5770	0.493792
$757$	−43.0275	−1.56386	−0.781931	−	0.623365i	$-0.785765\pi$
$757$	−43.0275	−1.56386	−0.781931	+	0.623365i	$0.785765\pi$
$758$	13.9036	0.505001
$759$	67.9617	2.46685
$760$	19.8002	0.718230
$761$	−5.87707	−0.213044	−0.106522	−	0.994310i	$-0.533971\pi$
$761$	−5.87707	−0.213044	−0.106522	+	0.994310i	$0.533971\pi$
$762$	−21.4542	−0.777205
$763$	23.4324	0.848310
$764$	8.96313	0.324274
$765$	−9.37410	−0.338921
$766$	3.91970	0.141624
$767$	−6.19774	−0.223787
$768$	−1.56405	−0.0564377
$769$	16.4347	0.592652	0.296326	−	0.955087i	$-0.404239\pi$
$769$	16.4347	0.592652	0.296326	+	0.955087i	$0.404239\pi$
$770$	−32.7688	−1.18091
$771$	4.83525	0.174137
$772$	−11.3792	−0.409546
$773$	−32.2546	−1.16012	−0.580058	−	0.814575i	$-0.696970\pi$
$773$	−32.2546	−1.16012	−0.580058	+	0.814575i	$0.696970\pi$
$774$	5.79087	0.208148
$775$	3.06143	0.109970
$776$	1.00000	0.0358979
$777$	−15.8205	−0.567556
$778$	17.5257	0.628327
$779$	−25.3111	−0.906863
$780$	−9.02573	−0.323173
$781$	−38.5879	−1.38079
$782$	54.8320	1.96079
$783$	−17.8445	−0.637709
$784$	−1.03330	−0.0369035
$785$	−55.8521	−1.99345
$786$	20.0190	0.714056
$787$	−28.7948	−1.02642	−0.513211	−	0.858262i	$-0.671544\pi$
$787$	−28.7948	−1.02642	−0.513211	+	0.858262i	$0.671544\pi$
$788$	1.08595	0.0386853
$789$	10.7841	0.383924
$790$	11.7949	0.419642
$791$	19.4509	0.691594
$792$	2.61640	0.0929696
$793$	25.9391	0.921123
$794$	−35.1313	−1.24676
$795$	−49.4581	−1.75410
$796$	−20.3458	−0.721137
$797$	8.37268	0.296576	0.148288	−	0.988944i	$-0.452624\pi$
$797$	8.37268	0.296576	0.148288	+	0.988944i	$0.452624\pi$
$798$	−26.6429	−0.943149
$799$	−28.7829	−1.01827
$800$	−3.06143	−0.108238
$801$	−3.19545	−0.112906
$802$	12.6649	0.447212
$803$	−15.6250	−0.551394
$804$	6.76097	0.238441
$805$	−63.7823	−2.24803
$806$	−2.03248	−0.0715910
$807$	28.0627	0.987854
$808$	12.5626	0.441950
$809$	−28.4838	−1.00144	−0.500719	−	0.865610i	$-0.666931\pi$
$809$	−28.4838	−1.00144	−0.500719	+	0.865610i	$0.666931\pi$
$810$	19.9660	0.701534
$811$	−50.7358	−1.78158	−0.890788	−	0.454419i	$-0.849847\pi$
$811$	−50.7358	−1.78158	−0.890788	+	0.454419i	$0.849847\pi$
$812$	−7.84211	−0.275204
$813$	−5.29471	−0.185694
$814$	−19.5654	−0.685769
$815$	−47.2366	−1.65463
$816$	−9.32518	−0.326447
$817$	−72.9276	−2.55141
$818$	−29.7878	−1.04151
$819$	−2.74922	−0.0960655
$820$	10.3051	0.359870
$821$	−37.4285	−1.30626	−0.653132	−	0.757244i	$-0.726545\pi$
$821$	−37.4285	−1.30626	−0.653132	+	0.757244i	$0.726545\pi$
$822$	8.01026	0.279390
$823$	−14.3602	−0.500565	−0.250282	−	0.968173i	$-0.580523\pi$
$823$	−14.3602	−0.500565	−0.250282	+	0.968173i	$0.580523\pi$
$824$	−10.9925	−0.382942
$825$	−22.6236	−0.787654
$826$	7.44860	0.259170
$827$	9.70715	0.337551	0.168775	−	0.985655i	$-0.446019\pi$
$827$	9.70715	0.337551	0.168775	+	0.985655i	$0.446019\pi$
$828$	5.09264	0.176981
$829$	−24.9868	−0.867827	−0.433914	−	0.900954i	$-0.642868\pi$
$829$	−24.9868	−0.867827	−0.433914	+	0.900954i	$0.642868\pi$
$830$	−44.5640	−1.54684
$831$	−31.6245	−1.09704
$832$	2.03248	0.0704635
$833$	−6.16074	−0.213457
$834$	31.6227	1.09500
$835$	0.901052	0.0311822
$836$	−32.9497	−1.13959
$837$	5.55824	0.192121
$838$	17.0653	0.589511
$839$	−3.31156	−0.114328	−0.0571638	−	0.998365i	$-0.518206\pi$
$839$	−3.31156	−0.114328	−0.0571638	+	0.998365i	$0.518206\pi$
$840$	10.8473	0.374269
$841$	−18.6930	−0.644586
$842$	−24.9960	−0.861418
$843$	−6.71266	−0.231196
$844$	17.9632	0.618319
$845$	−25.1815	−0.866272
$846$	−2.67328	−0.0919091
$847$	27.6613	0.950454
$848$	11.1373	0.382457
$849$	22.3636	0.767518
$850$	−18.2529	−0.626070
$851$	−38.0828	−1.30546
$852$	12.7736	0.437617
$853$	16.6082	0.568653	0.284326	−	0.958728i	$-0.408230\pi$
$853$	16.6082	0.568653	0.284326	+	0.958728i	$0.408230\pi$
$854$	−31.1742	−1.06676
$855$	10.9645	0.374976
$856$	−15.6593	−0.535226
$857$	16.5654	0.565864	0.282932	−	0.959140i	$-0.408693\pi$
$857$	16.5654	0.565864	0.282932	+	0.959140i	$0.408693\pi$
$858$	15.0198	0.512767
$859$	25.8811	0.883053	0.441527	−	0.897248i	$-0.354437\pi$
$859$	25.8811	0.883053	0.441527	+	0.897248i	$0.354437\pi$
$860$	29.6916	1.01247
$861$	−13.8664	−0.472565
$862$	2.31221	0.0787541
$863$	13.8176	0.470356	0.235178	−	0.971952i	$-0.424433\pi$
$863$	13.8176	0.470356	0.235178	+	0.971952i	$0.424433\pi$
$864$	−5.55824	−0.189095
$865$	9.92776	0.337554
$866$	−13.7386	−0.466858
$867$	−29.0099	−0.985227
$868$	2.44268	0.0829101
$869$	−19.6279	−0.665831
$870$	−14.2568	−0.483351
$871$	−8.78587	−0.297698
$872$	−9.59290	−0.324856
$873$	0.553753	0.0187417
$874$	−64.1344	−2.16938
$875$	−13.4448	−0.454517
$876$	5.17228	0.174755
$877$	3.58645	0.121106	0.0605528	−	0.998165i	$-0.480714\pi$
$877$	3.58645	0.121106	0.0605528	+	0.998165i	$0.480714\pi$
$878$	−21.1331	−0.713206
$879$	25.5172	0.860675
$880$	13.4151	0.452223
$881$	0.0497925	0.00167755	0.000838776	−	1.00000i	$-0.499733\pi$
$881$	0.0497925	0.00167755	0.000838776		1.00000i	$0.499733\pi$
$882$	−0.572192	−0.0192667
$883$	−13.1727	−0.443296	−0.221648	−	0.975127i	$-0.571144\pi$
$883$	−13.1727	−0.443296	−0.221648	+	0.975127i	$0.571144\pi$
$884$	12.1181	0.407574
$885$	13.5414	0.455190
$886$	−7.71224	−0.259098
$887$	−40.3982	−1.35644	−0.678219	−	0.734860i	$-0.737248\pi$
$887$	−40.3982	−1.35644	−0.678219	+	0.734860i	$0.737248\pi$
$888$	6.47668	0.217343
$889$	−33.5066	−1.12377
$890$	−16.3841	−0.549195
$891$	−33.2256	−1.11310
$892$	−12.0807	−0.404491
$893$	33.6660	1.12659
$894$	33.5609	1.12245
$895$	−24.8592	−0.830950
$896$	−2.44268	−0.0816043
$897$	29.2350	0.976127
$898$	−9.91773	−0.330959
$899$	−3.21045	−0.107075
$900$	−1.69528	−0.0565093
$901$	66.4031	2.21221
$902$	−17.1488	−0.570992
$903$	−39.9526	−1.32954
$904$	−7.96292	−0.264843
$905$	−38.4789	−1.27908
$906$	6.28065	0.208661
$907$	17.4617	0.579807	0.289904	−	0.957056i	$-0.406377\pi$
$907$	17.4617	0.579807	0.289904	+	0.957056i	$0.406377\pi$
$908$	−12.2854	−0.407706
$909$	6.95656	0.230735
$910$	−14.0961	−0.467281
$911$	5.86759	0.194402	0.0972010	−	0.995265i	$-0.469011\pi$
$911$	5.86759	0.194402	0.0972010	+	0.995265i	$0.469011\pi$
$912$	10.9072	0.361175
$913$	74.1593	2.45431
$914$	−22.8877	−0.757058
$915$	−56.6741	−1.87359
$916$	−2.62118	−0.0866063
$917$	31.2651	1.03247
$918$	−33.1394	−1.09376
$919$	−37.3764	−1.23293	−0.616467	−	0.787381i	$-0.711437\pi$
$919$	−37.3764	−1.23293	−0.616467	+	0.787381i	$0.711437\pi$
$920$	26.1116	0.860873
$921$	−31.7344	−1.04568
$922$	23.0172	0.758030
$923$	−16.5993	−0.546373
$924$	−18.0511	−0.593839
$925$	12.6773	0.416828
$926$	−4.08864	−0.134361
$927$	−6.08714	−0.199928
$928$	3.21045	0.105388
$929$	53.7107	1.76219	0.881095	−	0.472939i	$-0.156807\pi$
$929$	53.7107	1.76219	0.881095	+	0.472939i	$0.156807\pi$
$930$	4.44075	0.145618
$931$	7.20593	0.236165
$932$	29.7135	0.973297
$933$	−34.9621	−1.14461
$934$	−6.65541	−0.217772
$935$	79.9836	2.61574
$936$	1.12549	0.0367878
$937$	−45.5727	−1.48879	−0.744397	−	0.667737i	$-0.767263\pi$
$937$	−45.5727	−1.48879	−0.744397	+	0.667737i	$0.767263\pi$
$938$	10.5591	0.344766
$939$	−32.9168	−1.07420
$940$	−13.7067	−0.447064
$941$	−44.7383	−1.45843	−0.729213	−	0.684286i	$-0.760114\pi$
$941$	−44.7383	−1.45843	−0.729213	+	0.684286i	$0.760114\pi$
$942$	−30.7669	−1.00244
$943$	−33.3790	−1.08697
$944$	−3.04935	−0.0992480
$945$	38.5488	1.25399
$946$	−49.4100	−1.60646
$947$	20.7662	0.674812	0.337406	−	0.941359i	$-0.390451\pi$
$947$	20.7662	0.674812	0.337406	+	0.941359i	$0.390451\pi$
$948$	6.49736	0.211024
$949$	−6.72137	−0.218185
$950$	21.3496	0.692672
$951$	18.1895	0.589836
$952$	−14.5638	−0.472015
$953$	−15.2582	−0.494262	−0.247131	−	0.968982i	$-0.579488\pi$
$953$	−15.2582	−0.494262	−0.247131	+	0.968982i	$0.579488\pi$
$954$	6.16733	0.199675
$955$	25.4487	0.823501
$956$	−22.5961	−0.730811
$957$	23.7249	0.766916
$958$	25.3562	0.819221
$959$	12.5102	0.403975
$960$	−4.44075	−0.143325
$961$	1.00000	0.0322581
$962$	−8.41643	−0.271357
$963$	−8.67142	−0.279432
$964$	25.0650	0.807289
$965$	−32.3086	−1.04005
$966$	−35.1353	−1.13046
$967$	23.1519	0.744514	0.372257	−	0.928130i	$-0.378584\pi$
$967$	23.1519	0.744514	0.372257	+	0.928130i	$0.378584\pi$
$968$	−11.3242	−0.363972
$969$	65.0312	2.08910
$970$	2.83927	0.0911634
$971$	−31.4816	−1.01029	−0.505146	−	0.863034i	$-0.668561\pi$
$971$	−31.4816	−1.01029	−0.505146	+	0.863034i	$0.668561\pi$
$972$	−5.67618	−0.182064
$973$	49.3873	1.58329
$974$	36.1616	1.15869
$975$	−9.73197	−0.311673
$976$	12.7623	0.408511
$977$	−18.7763	−0.600707	−0.300354	−	0.953828i	$-0.597105\pi$
$977$	−18.7763	−0.600707	−0.300354	+	0.953828i	$0.597105\pi$
$978$	−26.0209	−0.832058
$979$	27.2649	0.871389
$980$	−2.93381	−0.0937170
$981$	−5.31210	−0.169602
$982$	−17.0947	−0.545513
$983$	−18.0291	−0.575038	−0.287519	−	0.957775i	$-0.592830\pi$
$983$	−18.0291	−0.575038	−0.287519	+	0.957775i	$0.592830\pi$
$984$	5.67670	0.180967
$985$	3.08329	0.0982419
$986$	19.1414	0.609586
$987$	18.4436	0.587065
$988$	−14.1739	−0.450933
$989$	−96.1732	−3.05813
$990$	7.42865	0.236098
$991$	−52.0962	−1.65489	−0.827445	−	0.561546i	$-0.810207\pi$
$991$	−52.0962	−1.65489	−0.827445	+	0.561546i	$0.810207\pi$
$992$	−1.00000	−0.0317500
$993$	−30.4982	−0.967832
$994$	19.9495	0.632759
$995$	−57.7671	−1.83134
$996$	−24.5487	−0.777854
$997$	11.6228	0.368099	0.184049	−	0.982917i	$-0.441079\pi$
$997$	11.6228	0.368099	0.184049	+	0.982917i	$0.441079\pi$
$998$	−35.9018	−1.13645
$999$	23.0165	0.728211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6014.2.a.l.1.12	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6014.2.a.l.1.12	✓	38	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6014 = 2 \cdot 31 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.56405 q^{3} +1.00000 q^{4} +2.83927 q^{5} +1.56405 q^{6} +2.44268 q^{7} -1.00000 q^{8} -0.553753 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.56405 q^{3} +1.00000 q^{4} +2.83927 q^{5} +1.56405 q^{6} +2.44268 q^{7} -1.00000 q^{8} -0.553753 q^{9} -2.83927 q^{10} +4.72484 q^{11} -1.56405 q^{12} +2.03248 q^{13} -2.44268 q^{14} -4.44075 q^{15} +1.00000 q^{16} +5.96221 q^{17} +0.553753 q^{18} -6.97372 q^{19} +2.83927 q^{20} -3.82047 q^{21} -4.72484 q^{22} -9.19659 q^{23} +1.56405 q^{24} +3.06143 q^{25} -2.03248 q^{26} +5.55824 q^{27} +2.44268 q^{28} -3.21045 q^{29} +4.44075 q^{30} +1.00000 q^{31} -1.00000 q^{32} -7.38988 q^{33} -5.96221 q^{34} +6.93543 q^{35} -0.553753 q^{36} +4.14097 q^{37} +6.97372 q^{38} -3.17889 q^{39} -2.83927 q^{40} +3.62949 q^{41} +3.82047 q^{42} +10.4575 q^{43} +4.72484 q^{44} -1.57225 q^{45} +9.19659 q^{46} -4.82756 q^{47} -1.56405 q^{48} -1.03330 q^{49} -3.06143 q^{50} -9.32518 q^{51} +2.03248 q^{52} +11.1373 q^{53} -5.55824 q^{54} +13.4151 q^{55} -2.44268 q^{56} +10.9072 q^{57} +3.21045 q^{58} -3.04935 q^{59} -4.44075 q^{60} +12.7623 q^{61} -1.00000 q^{62} -1.35264 q^{63} +1.00000 q^{64} +5.77075 q^{65} +7.38988 q^{66} -4.32274 q^{67} +5.96221 q^{68} +14.3839 q^{69} -6.93543 q^{70} -8.16703 q^{71} +0.553753 q^{72} -3.30698 q^{73} -4.14097 q^{74} -4.78823 q^{75} -6.97372 q^{76} +11.5413 q^{77} +3.17889 q^{78} -4.15419 q^{79} +2.83927 q^{80} -7.03210 q^{81} -3.62949 q^{82} +15.6956 q^{83} -3.82047 q^{84} +16.9283 q^{85} -10.4575 q^{86} +5.02130 q^{87} -4.72484 q^{88} +5.77053 q^{89} +1.57225 q^{90} +4.96470 q^{91} -9.19659 q^{92} -1.56405 q^{93} +4.82756 q^{94} -19.8002 q^{95} +1.56405 q^{96} -1.00000 q^{97} +1.03330 q^{98} -2.61640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 38 q^{2} - 2 q^{3} + 38 q^{4} + 2 q^{5} + 2 q^{6} + 3 q^{7} - 38 q^{8} + 54 q^{9}+O(q^{10}) \) 38 * q - 38 * q^2 - 2 * q^3 + 38 * q^4 + 2 * q^5 + 2 * q^6 + 3 * q^7 - 38 * q^8 + 54 * q^9	\( 38 q - 38 q^{2} - 2 q^{3} + 38 q^{4} + 2 q^{5} + 2 q^{6} + 3 q^{7} - 38 q^{8} + 54 q^{9} - 2 q^{10} + 6 q^{11} - 2 q^{12} + 12 q^{13} - 3 q^{14} + 19 q^{15} + 38 q^{16} + 16 q^{17} - 54 q^{18} + 37 q^{19} + 2 q^{20} + 8 q^{21} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 66 q^{25} - 12 q^{26} - 5 q^{27} + 3 q^{28} + 3 q^{29} - 19 q^{30} + 38 q^{31} - 38 q^{32} + 12 q^{33} - 16 q^{34} - 16 q^{35} + 54 q^{36} + 5 q^{37} - 37 q^{38} + 36 q^{39} - 2 q^{40} + 7 q^{41} - 8 q^{42} + 7 q^{43} + 6 q^{44} + 45 q^{45} + 12 q^{46} - 10 q^{47} - 2 q^{48} + 111 q^{49} - 66 q^{50} - 13 q^{51} + 12 q^{52} + 5 q^{53} + 5 q^{54} + 56 q^{55} - 3 q^{56} - 5 q^{57} - 3 q^{58} + 14 q^{59} + 19 q^{60} + 54 q^{61} - 38 q^{62} - 3 q^{63} + 38 q^{64} + 8 q^{65} - 12 q^{66} - 9 q^{67} + 16 q^{68} + 45 q^{69} + 16 q^{70} + 13 q^{71} - 54 q^{72} + 65 q^{73} - 5 q^{74} - 14 q^{75} + 37 q^{76} - 22 q^{77} - 36 q^{78} - 11 q^{79} + 2 q^{80} + 46 q^{81} - 7 q^{82} - 42 q^{83} + 8 q^{84} + 18 q^{85} - 7 q^{86} - 19 q^{87} - 6 q^{88} + 74 q^{89} - 45 q^{90} + 14 q^{91} - 12 q^{92} - 2 q^{93} + 10 q^{94} - 10 q^{95} + 2 q^{96} - 38 q^{97} - 111 q^{98} + 15 q^{99}+O(q^{100}) \) 38 * q - 38 * q^2 - 2 * q^3 + 38 * q^4 + 2 * q^5 + 2 * q^6 + 3 * q^7 - 38 * q^8 + 54 * q^9 - 2 * q^10 + 6 * q^11 - 2 * q^12 + 12 * q^13 - 3 * q^14 + 19 * q^15 + 38 * q^16 + 16 * q^17 - 54 * q^18 + 37 * q^19 + 2 * q^20 + 8 * q^21 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 66 * q^25 - 12 * q^26 - 5 * q^27 + 3 * q^28 + 3 * q^29 - 19 * q^30 + 38 * q^31 - 38 * q^32 + 12 * q^33 - 16 * q^34 - 16 * q^35 + 54 * q^36 + 5 * q^37 - 37 * q^38 + 36 * q^39 - 2 * q^40 + 7 * q^41 - 8 * q^42 + 7 * q^43 + 6 * q^44 + 45 * q^45 + 12 * q^46 - 10 * q^47 - 2 * q^48 + 111 * q^49 - 66 * q^50 - 13 * q^51 + 12 * q^52 + 5 * q^53 + 5 * q^54 + 56 * q^55 - 3 * q^56 - 5 * q^57 - 3 * q^58 + 14 * q^59 + 19 * q^60 + 54 * q^61 - 38 * q^62 - 3 * q^63 + 38 * q^64 + 8 * q^65 - 12 * q^66 - 9 * q^67 + 16 * q^68 + 45 * q^69 + 16 * q^70 + 13 * q^71 - 54 * q^72 + 65 * q^73 - 5 * q^74 - 14 * q^75 + 37 * q^76 - 22 * q^77 - 36 * q^78 - 11 * q^79 + 2 * q^80 + 46 * q^81 - 7 * q^82 - 42 * q^83 + 8 * q^84 + 18 * q^85 - 7 * q^86 - 19 * q^87 - 6 * q^88 + 74 * q^89 - 45 * q^90 + 14 * q^91 - 12 * q^92 - 2 * q^93 + 10 * q^94 - 10 * q^95 + 2 * q^96 - 38 * q^97 - 111 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more