LMFDB - Embedded newform 5054.2.a.bm.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, 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("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} + \cdots + 1 $ x^12 - 3x^11 - 24x^10 + 68x^9 + 213x^8 - 555x^7 - 814x^6 + 1911x^5 + 1005x^4 - 2326x^3 + 570x^2 + 105*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.572050$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.572050 q^{3} +1.00000 q^{4} +2.21950 q^{5} +0.572050 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.67276 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.572050 q^{3} +1.00000 q^{4} +2.21950 q^{5} +0.572050 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.67276 q^{9} +2.21950 q^{10} +6.09289 q^{11} +0.572050 q^{12} +6.15016 q^{13} +1.00000 q^{14} +1.26966 q^{15} +1.00000 q^{16} -1.73887 q^{17} -2.67276 q^{18} +2.21950 q^{20} +0.572050 q^{21} +6.09289 q^{22} -3.38757 q^{23} +0.572050 q^{24} -0.0738264 q^{25} +6.15016 q^{26} -3.24510 q^{27} +1.00000 q^{28} -2.18857 q^{29} +1.26966 q^{30} -0.977436 q^{31} +1.00000 q^{32} +3.48544 q^{33} -1.73887 q^{34} +2.21950 q^{35} -2.67276 q^{36} +11.9256 q^{37} +3.51820 q^{39} +2.21950 q^{40} -8.25936 q^{41} +0.572050 q^{42} +3.28334 q^{43} +6.09289 q^{44} -5.93218 q^{45} -3.38757 q^{46} +9.95871 q^{47} +0.572050 q^{48} +1.00000 q^{49} -0.0738264 q^{50} -0.994722 q^{51} +6.15016 q^{52} +3.81835 q^{53} -3.24510 q^{54} +13.5232 q^{55} +1.00000 q^{56} -2.18857 q^{58} -3.92318 q^{59} +1.26966 q^{60} +1.40804 q^{61} -0.977436 q^{62} -2.67276 q^{63} +1.00000 q^{64} +13.6503 q^{65} +3.48544 q^{66} -1.83668 q^{67} -1.73887 q^{68} -1.93786 q^{69} +2.21950 q^{70} -16.2892 q^{71} -2.67276 q^{72} -8.95240 q^{73} +11.9256 q^{74} -0.0422324 q^{75} +6.09289 q^{77} +3.51820 q^{78} -15.3099 q^{79} +2.21950 q^{80} +6.16191 q^{81} -8.25936 q^{82} +11.7929 q^{83} +0.572050 q^{84} -3.85942 q^{85} +3.28334 q^{86} -1.25197 q^{87} +6.09289 q^{88} -13.2142 q^{89} -5.93218 q^{90} +6.15016 q^{91} -3.38757 q^{92} -0.559143 q^{93} +9.95871 q^{94} +0.572050 q^{96} +11.4955 q^{97} +1.00000 q^{98} -16.2848 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} + 12 q^{8} + 21 q^{9}+O(q^{10}) $ 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 + 12 * q^8 + 21 * q^9	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} + 12 q^{8} + 21 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} - 6 q^{13} + 12 q^{14} - 6 q^{15} + 12 q^{16} + 3 q^{17} + 21 q^{18} + 3 q^{20} + 3 q^{21} + 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} + 21 q^{27} + 12 q^{28} + 6 q^{29} - 6 q^{30} + 3 q^{31} + 12 q^{32} - 6 q^{33} + 3 q^{34} + 3 q^{35} + 21 q^{36} + 27 q^{37} + 27 q^{39} + 3 q^{40} - 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} + 12 q^{46} + 21 q^{47} + 3 q^{48} + 12 q^{49} + 33 q^{50} + 3 q^{51} - 6 q^{52} + 6 q^{53} + 21 q^{54} + 48 q^{55} + 12 q^{56} + 6 q^{58} - 27 q^{59} - 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} - 9 q^{65} - 6 q^{66} + 9 q^{67} + 3 q^{68} + 18 q^{69} + 3 q^{70} + 27 q^{71} + 21 q^{72} - 9 q^{73} + 27 q^{74} + 33 q^{75} + 3 q^{77} + 27 q^{78} - 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} + 3 q^{84} + 78 q^{85} + 30 q^{86} - 45 q^{87} + 3 q^{88} - 24 q^{89} + 30 q^{90} - 6 q^{91} + 12 q^{92} + 3 q^{93} + 21 q^{94} + 3 q^{96} - 18 q^{97} + 12 q^{98} + 48 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 + 12 * q^8 + 21 * q^9 + 3 * q^10 + 3 * q^11 + 3 * q^12 - 6 * q^13 + 12 * q^14 - 6 * q^15 + 12 * q^16 + 3 * q^17 + 21 * q^18 + 3 * q^20 + 3 * q^21 + 3 * q^22 + 12 * q^23 + 3 * q^24 + 33 * q^25 - 6 * q^26 + 21 * q^27 + 12 * q^28 + 6 * q^29 - 6 * q^30 + 3 * q^31 + 12 * q^32 - 6 * q^33 + 3 * q^34 + 3 * q^35 + 21 * q^36 + 27 * q^37 + 27 * q^39 + 3 * q^40 - 27 * q^41 + 3 * q^42 + 30 * q^43 + 3 * q^44 + 30 * q^45 + 12 * q^46 + 21 * q^47 + 3 * q^48 + 12 * q^49 + 33 * q^50 + 3 * q^51 - 6 * q^52 + 6 * q^53 + 21 * q^54 + 48 * q^55 + 12 * q^56 + 6 * q^58 - 27 * q^59 - 6 * q^60 + 18 * q^61 + 3 * q^62 + 21 * q^63 + 12 * q^64 - 9 * q^65 - 6 * q^66 + 9 * q^67 + 3 * q^68 + 18 * q^69 + 3 * q^70 + 27 * q^71 + 21 * q^72 - 9 * q^73 + 27 * q^74 + 33 * q^75 + 3 * q^77 + 27 * q^78 - 12 * q^79 + 3 * q^80 + 24 * q^81 - 27 * q^82 + 9 * q^83 + 3 * q^84 + 78 * q^85 + 30 * q^86 - 45 * q^87 + 3 * q^88 - 24 * q^89 + 30 * q^90 - 6 * q^91 + 12 * q^92 + 3 * q^93 + 21 * q^94 + 3 * q^96 - 18 * q^97 + 12 * q^98 + 48 * 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
$3$	0.572050	0.330273	0.165137	−	0.986271i	$-0.447193\pi$
$3$	0.572050	0.330273	0.165137	+	0.986271i	$0.447193\pi$
$4$	1.00000	0.500000
$5$	2.21950	0.992590	0.496295	−	0.868154i	$-0.334693\pi$
$5$	2.21950	0.992590	0.496295	+	0.868154i	$0.334693\pi$
$6$	0.572050	0.233539
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.67276	−0.890919
$10$	2.21950	0.701867
$11$	6.09289	1.83708	0.918538	−	0.395332i	$-0.129370\pi$
$11$	6.09289	1.83708	0.918538	+	0.395332i	$0.129370\pi$
$12$	0.572050	0.165137
$13$	6.15016	1.70575	0.852873	−	0.522118i	$-0.174858\pi$
$13$	6.15016	1.70575	0.852873	+	0.522118i	$0.174858\pi$
$14$	1.00000	0.267261
$15$	1.26966	0.327826
$16$	1.00000	0.250000
$17$	−1.73887	−0.421738	−0.210869	−	0.977514i	$-0.567629\pi$
$17$	−1.73887	−0.421738	−0.210869	+	0.977514i	$0.567629\pi$
$18$	−2.67276	−0.629975
$19$	0	0
$19$	0	0
$20$	2.21950	0.496295
$21$	0.572050	0.124832
$22$	6.09289	1.29901
$23$	−3.38757	−0.706358	−0.353179	−	0.935556i	$-0.614899\pi$
$23$	−3.38757	−0.706358	−0.353179	+	0.935556i	$0.614899\pi$
$24$	0.572050	0.116769
$25$	−0.0738264	−0.0147653
$26$	6.15016	1.20614
$27$	−3.24510	−0.624520
$28$	1.00000	0.188982
$29$	−2.18857	−0.406408	−0.203204	−	0.979136i	$-0.565135\pi$
$29$	−2.18857	−0.406408	−0.203204	+	0.979136i	$0.565135\pi$
$30$	1.26966	0.231808
$31$	−0.977436	−0.175553	−0.0877763	−	0.996140i	$-0.527976\pi$
$31$	−0.977436	−0.175553	−0.0877763	+	0.996140i	$0.527976\pi$
$32$	1.00000	0.176777
$33$	3.48544	0.606738
$34$	−1.73887	−0.298214
$35$	2.21950	0.375164
$36$	−2.67276	−0.445460
$37$	11.9256	1.96056	0.980279	−	0.197618i	$-0.0633205\pi$
$37$	11.9256	1.96056	0.980279	+	0.197618i	$0.0633205\pi$
$38$	0	0
$39$	3.51820	0.563363
$40$	2.21950	0.350934
$41$	−8.25936	−1.28990	−0.644948	−	0.764227i	$-0.723121\pi$
$41$	−8.25936	−1.28990	−0.644948	+	0.764227i	$0.723121\pi$
$42$	0.572050	0.0882693
$43$	3.28334	0.500704	0.250352	−	0.968155i	$-0.419454\pi$
$43$	3.28334	0.500704	0.250352	+	0.968155i	$0.419454\pi$
$44$	6.09289	0.918538
$45$	−5.93218	−0.884318
$46$	−3.38757	−0.499471
$47$	9.95871	1.45263	0.726313	−	0.687364i	$-0.241232\pi$
$47$	9.95871	1.45263	0.726313	+	0.687364i	$0.241232\pi$
$48$	0.572050	0.0825684
$49$	1.00000	0.142857
$50$	−0.0738264	−0.0104406
$51$	−0.994722	−0.139289
$52$	6.15016	0.852873
$53$	3.81835	0.524491	0.262246	−	0.965001i	$-0.415537\pi$
$53$	3.81835	0.524491	0.262246	+	0.965001i	$0.415537\pi$
$54$	−3.24510	−0.441603
$55$	13.5232	1.82346
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.18857	−0.287374
$59$	−3.92318	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$59$	−3.92318	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$60$	1.26966	0.163913
$61$	1.40804	0.180281	0.0901405	−	0.995929i	$-0.471268\pi$
$61$	1.40804	0.180281	0.0901405	+	0.995929i	$0.471268\pi$
$62$	−0.977436	−0.124135
$63$	−2.67276	−0.336736
$64$	1.00000	0.125000
$65$	13.6503	1.69311
$66$	3.48544	0.429028
$67$	−1.83668	−0.224387	−0.112193	−	0.993686i	$-0.535788\pi$
$67$	−1.83668	−0.224387	−0.112193	+	0.993686i	$0.535788\pi$
$68$	−1.73887	−0.210869
$69$	−1.93786	−0.233291
$70$	2.21950	0.265281
$71$	−16.2892	−1.93317	−0.966584	−	0.256349i	$-0.917480\pi$
$71$	−16.2892	−1.93317	−0.966584	+	0.256349i	$0.917480\pi$
$72$	−2.67276	−0.314988
$73$	−8.95240	−1.04780	−0.523899	−	0.851780i	$-0.675523\pi$
$73$	−8.95240	−1.04780	−0.523899	+	0.851780i	$0.675523\pi$
$74$	11.9256	1.38632
$75$	−0.0422324	−0.00487658
$76$	0	0
$77$	6.09289	0.694350
$78$	3.51820	0.398358
$79$	−15.3099	−1.72250	−0.861252	−	0.508179i	$-0.830319\pi$
$79$	−15.3099	−1.72250	−0.861252	+	0.508179i	$0.830319\pi$
$80$	2.21950	0.248147
$81$	6.16191	0.684657
$82$	−8.25936	−0.912094
$83$	11.7929	1.29443	0.647217	−	0.762305i	$-0.275933\pi$
$83$	11.7929	1.29443	0.647217	+	0.762305i	$0.275933\pi$
$84$	0.572050	0.0624158
$85$	−3.85942	−0.418613
$86$	3.28334	0.354052
$87$	−1.25197	−0.134226
$88$	6.09289	0.649505
$89$	−13.2142	−1.40070	−0.700351	−	0.713798i	$-0.746973\pi$
$89$	−13.2142	−1.40070	−0.700351	+	0.713798i	$0.746973\pi$
$90$	−5.93218	−0.625307
$91$	6.15016	0.644712
$92$	−3.38757	−0.353179
$93$	−0.559143	−0.0579804
$94$	9.95871	1.02716
$95$	0	0
$96$	0.572050	0.0583846
$97$	11.4955	1.16719	0.583593	−	0.812046i	$-0.301646\pi$
$97$	11.4955	1.16719	0.583593	+	0.812046i	$0.301646\pi$
$98$	1.00000	0.101015
$99$	−16.2848	−1.63669
$100$	−0.0738264	−0.00738264
$101$	1.69905	0.169062	0.0845310	−	0.996421i	$-0.473061\pi$
$101$	1.69905	0.169062	0.0845310	+	0.996421i	$0.473061\pi$
$102$	−0.994722	−0.0984922
$103$	−2.28799	−0.225442	−0.112721	−	0.993627i	$-0.535957\pi$
$103$	−2.28799	−0.225442	−0.112721	+	0.993627i	$0.535957\pi$
$104$	6.15016	0.603072
$105$	1.26966	0.123907
$106$	3.81835	0.370871
$107$	0.576491	0.0557315	0.0278657	−	0.999612i	$-0.491129\pi$
$107$	0.576491	0.0557315	0.0278657	+	0.999612i	$0.491129\pi$
$108$	−3.24510	−0.312260
$109$	−7.45643	−0.714196	−0.357098	−	0.934067i	$-0.616234\pi$
$109$	−7.45643	−0.714196	−0.357098	+	0.934067i	$0.616234\pi$
$110$	13.5232	1.28938
$111$	6.82205	0.647520
$112$	1.00000	0.0944911
$113$	−14.3423	−1.34921	−0.674605	−	0.738179i	$-0.735686\pi$
$113$	−14.3423	−1.34921	−0.674605	+	0.738179i	$0.735686\pi$
$114$	0	0
$115$	−7.51872	−0.701124
$116$	−2.18857	−0.203204
$117$	−16.4379	−1.51968
$118$	−3.92318	−0.361158
$119$	−1.73887	−0.159402
$120$	1.26966	0.115904
$121$	26.1234	2.37485
$122$	1.40804	0.127478
$123$	−4.72477	−0.426018
$124$	−0.977436	−0.0877763
$125$	−11.2614	−1.00725
$126$	−2.67276	−0.238108
$127$	−0.780040	−0.0692174	−0.0346087	−	0.999401i	$-0.511018\pi$
$127$	−0.780040	−0.0692174	−0.0346087	+	0.999401i	$0.511018\pi$
$128$	1.00000	0.0883883
$129$	1.87824	0.165369
$130$	13.6503	1.19721
$131$	−6.42064	−0.560974	−0.280487	−	0.959858i	$-0.590496\pi$
$131$	−6.42064	−0.560974	−0.280487	+	0.959858i	$0.590496\pi$
$132$	3.48544	0.303369
$133$	0	0
$134$	−1.83668	−0.158665
$135$	−7.20250	−0.619893
$136$	−1.73887	−0.149107
$137$	10.1140	0.864098	0.432049	−	0.901850i	$-0.357791\pi$
$137$	10.1140	0.864098	0.432049	+	0.901850i	$0.357791\pi$
$138$	−1.93786	−0.164962
$139$	16.2063	1.37460	0.687300	−	0.726374i	$-0.258796\pi$
$139$	16.2063	1.37460	0.687300	+	0.726374i	$0.258796\pi$
$140$	2.21950	0.187582
$141$	5.69688	0.479764
$142$	−16.2892	−1.36696
$143$	37.4723	3.13359
$144$	−2.67276	−0.222730
$145$	−4.85753	−0.403396
$146$	−8.95240	−0.740906
$147$	0.572050	0.0471819
$148$	11.9256	0.980279
$149$	−23.4893	−1.92432	−0.962158	−	0.272493i	$-0.912152\pi$
$149$	−23.4893	−1.92432	−0.962158	+	0.272493i	$0.912152\pi$
$150$	−0.0422324	−0.00344826
$151$	5.57037	0.453311	0.226655	−	0.973975i	$-0.427221\pi$
$151$	5.57037	0.453311	0.226655	+	0.973975i	$0.427221\pi$
$152$	0	0
$153$	4.64758	0.375735
$154$	6.09289	0.490979
$155$	−2.16942	−0.174252
$156$	3.51820	0.281681
$157$	10.9616	0.874828	0.437414	−	0.899260i	$-0.355894\pi$
$157$	10.9616	0.874828	0.437414	+	0.899260i	$0.355894\pi$
$158$	−15.3099	−1.21799
$159$	2.18429	0.173226
$160$	2.21950	0.175467
$161$	−3.38757	−0.266978
$162$	6.16191	0.484126
$163$	0.541777	0.0424352	0.0212176	−	0.999775i	$-0.493246\pi$
$163$	0.541777	0.0424352	0.0212176	+	0.999775i	$0.493246\pi$
$164$	−8.25936	−0.644948
$165$	7.73593	0.602242
$166$	11.7929	0.915304
$167$	−5.89883	−0.456465	−0.228233	−	0.973607i	$-0.573295\pi$
$167$	−5.89883	−0.456465	−0.228233	+	0.973607i	$0.573295\pi$
$168$	0.572050	0.0441346
$169$	24.8244	1.90957
$170$	−3.85942	−0.296004
$171$	0	0
$172$	3.28334	0.250352
$173$	5.63686	0.428563	0.214281	−	0.976772i	$-0.431259\pi$
$173$	5.63686	0.428563	0.214281	+	0.976772i	$0.431259\pi$
$174$	−1.25197	−0.0949119
$175$	−0.0738264	−0.00558075
$176$	6.09289	0.459269
$177$	−2.24425	−0.168689
$178$	−13.2142	−0.990446
$179$	−2.87854	−0.215152	−0.107576	−	0.994197i	$-0.534309\pi$
$179$	−2.87854	−0.215152	−0.107576	+	0.994197i	$0.534309\pi$
$180$	−5.93218	−0.442159
$181$	−7.11191	−0.528624	−0.264312	−	0.964437i	$-0.585145\pi$
$181$	−7.11191	−0.528624	−0.264312	+	0.964437i	$0.585145\pi$
$182$	6.15016	0.455880
$183$	0.805470	0.0595421
$184$	−3.38757	−0.249735
$185$	26.4689	1.94603
$186$	−0.559143	−0.0409983
$187$	−10.5948	−0.774766
$188$	9.95871	0.726313
$189$	−3.24510	−0.236047
$190$	0	0
$191$	−4.59556	−0.332523	−0.166261	−	0.986082i	$-0.553170\pi$
$191$	−4.59556	−0.332523	−0.166261	+	0.986082i	$0.553170\pi$
$192$	0.572050	0.0412842
$193$	6.80042	0.489505	0.244752	−	0.969586i	$-0.421293\pi$
$193$	6.80042	0.489505	0.244752	+	0.969586i	$0.421293\pi$
$194$	11.4955	0.825325
$195$	7.80864	0.559188
$196$	1.00000	0.0714286
$197$	14.3681	1.02368	0.511841	−	0.859080i	$-0.328964\pi$
$197$	14.3681	1.02368	0.511841	+	0.859080i	$0.328964\pi$
$198$	−16.2848	−1.15731
$199$	−17.1660	−1.21686	−0.608432	−	0.793606i	$-0.708201\pi$
$199$	−17.1660	−1.21686	−0.608432	+	0.793606i	$0.708201\pi$
$200$	−0.0738264	−0.00522031
$201$	−1.05068	−0.0741090
$202$	1.69905	0.119545
$203$	−2.18857	−0.153608
$204$	−0.994722	−0.0696445
$205$	−18.3316	−1.28034
$206$	−2.28799	−0.159412
$207$	9.05417	0.629308
$208$	6.15016	0.426437
$209$	0	0
$210$	1.26966	0.0876152
$211$	3.13566	0.215868	0.107934	−	0.994158i	$-0.465577\pi$
$211$	3.13566	0.215868	0.107934	+	0.994158i	$0.465577\pi$
$212$	3.81835	0.262246
$213$	−9.31823	−0.638474
$214$	0.576491	0.0394081
$215$	7.28737	0.496994
$216$	−3.24510	−0.220801
$217$	−0.977436	−0.0663527
$218$	−7.45643	−0.505013
$219$	−5.12122	−0.346060
$220$	13.5232	0.911732
$221$	−10.6943	−0.719379
$222$	6.82205	0.457866
$223$	25.3696	1.69888	0.849438	−	0.527688i	$-0.176941\pi$
$223$	25.3696	1.69888	0.849438	+	0.527688i	$0.176941\pi$
$224$	1.00000	0.0668153
$225$	0.197320	0.0131547
$226$	−14.3423	−0.954035
$227$	3.22403	0.213987	0.106993	−	0.994260i	$-0.465878\pi$
$227$	3.22403	0.213987	0.106993	+	0.994260i	$0.465878\pi$
$228$	0	0
$229$	8.14777	0.538420	0.269210	−	0.963082i	$-0.413237\pi$
$229$	8.14777	0.538420	0.269210	+	0.963082i	$0.413237\pi$
$230$	−7.51872	−0.495769
$231$	3.48544	0.229325
$232$	−2.18857	−0.143687
$233$	11.8935	0.779171	0.389586	−	0.920990i	$-0.372618\pi$
$233$	11.8935	0.779171	0.389586	+	0.920990i	$0.372618\pi$
$234$	−16.4379	−1.07458
$235$	22.1033	1.44186
$236$	−3.92318	−0.255377
$237$	−8.75806	−0.568897
$238$	−1.73887	−0.112714
$239$	0.951552	0.0615508	0.0307754	−	0.999526i	$-0.490202\pi$
$239$	0.951552	0.0615508	0.0307754	+	0.999526i	$0.490202\pi$
$240$	1.26966	0.0819565
$241$	−6.48436	−0.417694	−0.208847	−	0.977948i	$-0.566971\pi$
$241$	−6.48436	−0.417694	−0.208847	+	0.977948i	$0.566971\pi$
$242$	26.1234	1.67927
$243$	13.2602	0.850644
$244$	1.40804	0.0901405
$245$	2.21950	0.141799
$246$	−4.72477	−0.301240
$247$	0	0
$248$	−0.977436	−0.0620673
$249$	6.74611	0.427517
$250$	−11.2614	−0.712230
$251$	−7.98176	−0.503805	−0.251902	−	0.967753i	$-0.581056\pi$
$251$	−7.98176	−0.503805	−0.251902	+	0.967753i	$0.581056\pi$
$252$	−2.67276	−0.168368
$253$	−20.6401	−1.29763
$254$	−0.780040	−0.0489441
$255$	−2.20778	−0.138257
$256$	1.00000	0.0625000
$257$	2.53675	0.158238	0.0791191	−	0.996865i	$-0.474789\pi$
$257$	2.53675	0.158238	0.0791191	+	0.996865i	$0.474789\pi$
$258$	1.87824	0.116934
$259$	11.9256	0.741021
$260$	13.6503	0.846553
$261$	5.84952	0.362077
$262$	−6.42064	−0.396668
$263$	11.7645	0.725429	0.362714	−	0.931900i	$-0.381850\pi$
$263$	11.7645	0.725429	0.362714	+	0.931900i	$0.381850\pi$
$264$	3.48544	0.214514
$265$	8.47483	0.520605
$266$	0	0
$267$	−7.55919	−0.462615
$268$	−1.83668	−0.112193
$269$	7.42578	0.452758	0.226379	−	0.974039i	$-0.427311\pi$
$269$	7.42578	0.452758	0.226379	+	0.974039i	$0.427311\pi$
$270$	−7.20250	−0.438330
$271$	−25.3145	−1.53775	−0.768873	−	0.639402i	$-0.779182\pi$
$271$	−25.3145	−1.53775	−0.768873	+	0.639402i	$0.779182\pi$
$272$	−1.73887	−0.105435
$273$	3.51820	0.212931
$274$	10.1140	0.611009
$275$	−0.449816	−0.0271249
$276$	−1.93786	−0.116646
$277$	14.8882	0.894545	0.447273	−	0.894398i	$-0.352395\pi$
$277$	14.8882	0.894545	0.447273	+	0.894398i	$0.352395\pi$
$278$	16.2063	0.971989
$279$	2.61245	0.156403
$280$	2.21950	0.132640
$281$	15.2413	0.909218	0.454609	−	0.890691i	$-0.349779\pi$
$281$	15.2413	0.909218	0.454609	+	0.890691i	$0.349779\pi$
$282$	5.69688	0.339244
$283$	14.5625	0.865649	0.432825	−	0.901478i	$-0.357517\pi$
$283$	14.5625	0.865649	0.432825	+	0.901478i	$0.357517\pi$
$284$	−16.2892	−0.966584
$285$	0	0
$286$	37.4723	2.21578
$287$	−8.25936	−0.487535
$288$	−2.67276	−0.157494
$289$	−13.9763	−0.822137
$290$	−4.85753	−0.285244
$291$	6.57598	0.385491
$292$	−8.95240	−0.523899
$293$	−15.3389	−0.896107	−0.448054	−	0.894007i	$-0.647883\pi$
$293$	−15.3389	−0.896107	−0.448054	+	0.894007i	$0.647883\pi$
$294$	0.572050	0.0333627
$295$	−8.70748	−0.506969
$296$	11.9256	0.693162
$297$	−19.7721	−1.14729
$298$	−23.4893	−1.36070
$299$	−20.8341	−1.20487
$300$	−0.0422324	−0.00243829
$301$	3.28334	0.189249
$302$	5.57037	0.320539
$303$	0.971943	0.0558367
$304$	0	0
$305$	3.12514	0.178945
$306$	4.64758	0.265685
$307$	−10.7228	−0.611980	−0.305990	−	0.952035i	$-0.598987\pi$
$307$	−10.7228	−0.611980	−0.305990	+	0.952035i	$0.598987\pi$
$308$	6.09289	0.347175
$309$	−1.30885	−0.0744576
$310$	−2.16942	−0.123215
$311$	31.8376	1.80535	0.902673	−	0.430327i	$-0.141602\pi$
$311$	31.8376	1.80535	0.902673	+	0.430327i	$0.141602\pi$
$312$	3.51820	0.199179
$313$	−9.09619	−0.514147	−0.257074	−	0.966392i	$-0.582758\pi$
$313$	−9.09619	−0.514147	−0.257074	+	0.966392i	$0.582758\pi$
$314$	10.9616	0.618597
$315$	−5.93218	−0.334241
$316$	−15.3099	−0.861252
$317$	−14.5668	−0.818152	−0.409076	−	0.912500i	$-0.634149\pi$
$317$	−14.5668	−0.818152	−0.409076	+	0.912500i	$0.634149\pi$
$318$	2.18429	0.122489
$319$	−13.3347	−0.746602
$320$	2.21950	0.124074
$321$	0.329782	0.0184066
$322$	−3.38757	−0.188782
$323$	0	0
$324$	6.16191	0.342328
$325$	−0.454044	−0.0251858
$326$	0.541777	0.0300062
$327$	−4.26545	−0.235880
$328$	−8.25936	−0.456047
$329$	9.95871	0.549041
$330$	7.73593	0.425849
$331$	−9.68807	−0.532504	−0.266252	−	0.963903i	$-0.585785\pi$
$331$	−9.68807	−0.532504	−0.266252	+	0.963903i	$0.585785\pi$
$332$	11.7929	0.647217
$333$	−31.8743	−1.74670
$334$	−5.89883	−0.322770
$335$	−4.07652	−0.222724
$336$	0.572050	0.0312079
$337$	9.16506	0.499253	0.249626	−	0.968342i	$-0.419692\pi$
$337$	9.16506	0.499253	0.249626	+	0.968342i	$0.419692\pi$
$338$	24.8244	1.35027
$339$	−8.20452	−0.445608
$340$	−3.85942	−0.209307
$341$	−5.95541	−0.322504
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.28334	0.177026
$345$	−4.30108	−0.231563
$346$	5.63686	0.303040
$347$	31.3135	1.68100	0.840500	−	0.541812i	$-0.182262\pi$
$347$	31.3135	1.68100	0.840500	+	0.541812i	$0.182262\pi$
$348$	−1.25197	−0.0671128
$349$	−26.3620	−1.41112	−0.705562	−	0.708648i	$-0.749305\pi$
$349$	−26.3620	−1.41112	−0.705562	+	0.708648i	$0.749305\pi$
$350$	−0.0738264	−0.00394619
$351$	−19.9579	−1.06527
$352$	6.09289	0.324752
$353$	−11.2622	−0.599428	−0.299714	−	0.954029i	$-0.596891\pi$
$353$	−11.2622	−0.599428	−0.299714	+	0.954029i	$0.596891\pi$
$354$	−2.24425	−0.119281
$355$	−36.1538	−1.91884
$356$	−13.2142	−0.700351
$357$	−0.994722	−0.0526463
$358$	−2.87854	−0.152136
$359$	8.23468	0.434610	0.217305	−	0.976104i	$-0.430273\pi$
$359$	8.23468	0.434610	0.217305	+	0.976104i	$0.430273\pi$
$360$	−5.93218	−0.312654
$361$	0	0
$362$	−7.11191	−0.373793
$363$	14.9439	0.784350
$364$	6.15016	0.322356
$365$	−19.8698	−1.04003
$366$	0.805470	0.0421026
$367$	−6.46885	−0.337671	−0.168836	−	0.985644i	$-0.554001\pi$
$367$	−6.46885	−0.337671	−0.168836	+	0.985644i	$0.554001\pi$
$368$	−3.38757	−0.176590
$369$	22.0753	1.14919
$370$	26.4689	1.37605
$371$	3.81835	0.198239
$372$	−0.559143	−0.0289902
$373$	20.9407	1.08427	0.542134	−	0.840292i	$-0.317616\pi$
$373$	20.9407	1.08427	0.542134	+	0.840292i	$0.317616\pi$
$374$	−10.5948	−0.547842
$375$	−6.44206	−0.332667
$376$	9.95871	0.513581
$377$	−13.4601	−0.693228
$378$	−3.24510	−0.166910
$379$	−15.0189	−0.771472	−0.385736	−	0.922609i	$-0.626052\pi$
$379$	−15.0189	−0.771472	−0.385736	+	0.922609i	$0.626052\pi$
$380$	0	0
$381$	−0.446222	−0.0228607
$382$	−4.59556	−0.235129
$383$	−19.0979	−0.975855	−0.487927	−	0.872884i	$-0.662247\pi$
$383$	−19.0979	−0.975855	−0.487927	+	0.872884i	$0.662247\pi$
$384$	0.572050	0.0291923
$385$	13.5232	0.689205
$386$	6.80042	0.346132
$387$	−8.77557	−0.446087
$388$	11.4955	0.583593
$389$	−10.1775	−0.516021	−0.258011	−	0.966142i	$-0.583067\pi$
$389$	−10.1775	−0.516021	−0.258011	+	0.966142i	$0.583067\pi$
$390$	7.80864	0.395406
$391$	5.89056	0.297898
$392$	1.00000	0.0505076
$393$	−3.67293	−0.185275
$394$	14.3681	0.723853
$395$	−33.9804	−1.70974
$396$	−16.2848	−0.818344
$397$	−5.17058	−0.259504	−0.129752	−	0.991546i	$-0.541418\pi$
$397$	−5.17058	−0.259504	−0.129752	+	0.991546i	$0.541418\pi$
$398$	−17.1660	−0.860453
$399$	0	0
$400$	−0.0738264	−0.00369132
$401$	14.6655	0.732361	0.366181	−	0.930544i	$-0.380665\pi$
$401$	14.6655	0.732361	0.366181	+	0.930544i	$0.380665\pi$
$402$	−1.05068	−0.0524029
$403$	−6.01138	−0.299448
$404$	1.69905	0.0845310
$405$	13.6764	0.679584
$406$	−2.18857	−0.108617
$407$	72.6615	3.60170
$408$	−0.994722	−0.0492461
$409$	−13.9231	−0.688451	−0.344225	−	0.938887i	$-0.611858\pi$
$409$	−13.9231	−0.688451	−0.344225	+	0.938887i	$0.611858\pi$
$410$	−18.3316	−0.905335
$411$	5.78572	0.285388
$412$	−2.28799	−0.112721
$413$	−3.92318	−0.193047
$414$	9.05417	0.444988
$415$	26.1742	1.28484
$416$	6.15016	0.301536
$417$	9.27082	0.453994
$418$	0	0
$419$	−5.94582	−0.290472	−0.145236	−	0.989397i	$-0.546394\pi$
$419$	−5.94582	−0.290472	−0.145236	+	0.989397i	$0.546394\pi$
$420$	1.26966	0.0619533
$421$	−26.6445	−1.29857	−0.649287	−	0.760543i	$-0.724933\pi$
$421$	−26.6445	−1.29857	−0.649287	+	0.760543i	$0.724933\pi$
$422$	3.13566	0.152642
$423$	−26.6172	−1.29417
$424$	3.81835	0.185436
$425$	0.128375	0.00622708
$426$	−9.31823	−0.451469
$427$	1.40804	0.0681398
$428$	0.576491	0.0278657
$429$	21.4360	1.03494
$430$	7.28737	0.351428
$431$	−7.15638	−0.344711	−0.172355	−	0.985035i	$-0.555138\pi$
$431$	−7.15638	−0.344711	−0.172355	+	0.985035i	$0.555138\pi$
$432$	−3.24510	−0.156130
$433$	−18.3923	−0.883877	−0.441939	−	0.897045i	$-0.645709\pi$
$433$	−18.3923	−0.883877	−0.441939	+	0.897045i	$0.645709\pi$
$434$	−0.977436	−0.0469184
$435$	−2.77875	−0.133231
$436$	−7.45643	−0.357098
$437$	0	0
$438$	−5.12122	−0.244701
$439$	20.5607	0.981309	0.490655	−	0.871354i	$-0.336758\pi$
$439$	20.5607	0.981309	0.490655	+	0.871354i	$0.336758\pi$
$440$	13.5232	0.644692
$441$	−2.67276	−0.127274
$442$	−10.6943	−0.508678
$443$	−9.17486	−0.435911	−0.217955	−	0.975959i	$-0.569939\pi$
$443$	−9.17486	−0.435911	−0.217955	+	0.975959i	$0.569939\pi$
$444$	6.82205	0.323760
$445$	−29.3289	−1.39032
$446$	25.3696	1.20129
$447$	−13.4370	−0.635550
$448$	1.00000	0.0472456
$449$	−25.3026	−1.19410	−0.597051	−	0.802203i	$-0.703661\pi$
$449$	−25.3026	−1.19410	−0.597051	+	0.802203i	$0.703661\pi$
$450$	0.197320	0.00930176
$451$	−50.3234	−2.36964
$452$	−14.3423	−0.674605
$453$	3.18653	0.149716
$454$	3.22403	0.151311
$455$	13.6503	0.639934
$456$	0	0
$457$	33.6647	1.57477	0.787384	−	0.616462i	$-0.211435\pi$
$457$	33.6647	1.57477	0.787384	+	0.616462i	$0.211435\pi$
$458$	8.14777	0.380720
$459$	5.64282	0.263384
$460$	−7.51872	−0.350562
$461$	−29.7527	−1.38572	−0.692862	−	0.721070i	$-0.743650\pi$
$461$	−29.7527	−1.38572	−0.692862	+	0.721070i	$0.743650\pi$
$462$	3.48544	0.162157
$463$	−26.2873	−1.22167	−0.610837	−	0.791756i	$-0.709167\pi$
$463$	−26.2873	−1.22167	−0.610837	+	0.791756i	$0.709167\pi$
$464$	−2.18857	−0.101602
$465$	−1.24102	−0.0575508
$466$	11.8935	0.550957
$467$	−3.03716	−0.140543	−0.0702716	−	0.997528i	$-0.522387\pi$
$467$	−3.03716	−0.140543	−0.0702716	+	0.997528i	$0.522387\pi$
$468$	−16.4379	−0.759841
$469$	−1.83668	−0.0848102
$470$	22.1033	1.01955
$471$	6.27057	0.288932
$472$	−3.92318	−0.180579
$473$	20.0050	0.919833
$474$	−8.75806	−0.402271
$475$	0	0
$476$	−1.73887	−0.0797011
$477$	−10.2055	−0.467280
$478$	0.951552	0.0435230
$479$	−4.49449	−0.205358	−0.102679	−	0.994715i	$-0.532742\pi$
$479$	−4.49449	−0.205358	−0.102679	+	0.994715i	$0.532742\pi$
$480$	1.26966	0.0579520
$481$	73.3444	3.34422
$482$	−6.48436	−0.295354
$483$	−1.93786	−0.0881758
$484$	26.1234	1.18743
$485$	25.5141	1.15854
$486$	13.2602	0.601496
$487$	−29.2764	−1.32664	−0.663320	−	0.748336i	$-0.730853\pi$
$487$	−29.2764	−1.32664	−0.663320	+	0.748336i	$0.730853\pi$
$488$	1.40804	0.0637390
$489$	0.309924	0.0140152
$490$	2.21950	0.100267
$491$	23.1139	1.04312	0.521558	−	0.853216i	$-0.325351\pi$
$491$	23.1139	1.04312	0.521558	+	0.853216i	$0.325351\pi$
$492$	−4.72477	−0.213009
$493$	3.80565	0.171398
$494$	0	0
$495$	−36.1442	−1.62456
$496$	−0.977436	−0.0438882
$497$	−16.2892	−0.730669
$498$	6.74611	0.302300
$499$	16.2504	0.727467	0.363734	−	0.931503i	$-0.381502\pi$
$499$	16.2504	0.727467	0.363734	+	0.931503i	$0.381502\pi$
$500$	−11.2614	−0.503623
$501$	−3.37443	−0.150758
$502$	−7.98176	−0.356244
$503$	−36.2252	−1.61520	−0.807601	−	0.589729i	$-0.799235\pi$
$503$	−36.2252	−1.61520	−0.807601	+	0.589729i	$0.799235\pi$
$504$	−2.67276	−0.119054
$505$	3.77104	0.167809
$506$	−20.6401	−0.917566
$507$	14.2008	0.630681
$508$	−0.780040	−0.0346087
$509$	11.0110	0.488052	0.244026	−	0.969769i	$-0.421532\pi$
$509$	11.0110	0.488052	0.244026	+	0.969769i	$0.421532\pi$
$510$	−2.20778	−0.0977623
$511$	−8.95240	−0.396031
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.53675	0.111891
$515$	−5.07819	−0.223772
$516$	1.87824	0.0826847
$517$	60.6774	2.66859
$518$	11.9256	0.523981
$519$	3.22457	0.141543
$520$	13.6503	0.598604
$521$	−3.77025	−0.165177	−0.0825887	−	0.996584i	$-0.526319\pi$
$521$	−3.77025	−0.165177	−0.0825887	+	0.996584i	$0.526319\pi$
$522$	5.84952	0.256027
$523$	−15.7442	−0.688446	−0.344223	−	0.938888i	$-0.611858\pi$
$523$	−15.7442	−0.688446	−0.344223	+	0.938888i	$0.611858\pi$
$524$	−6.42064	−0.280487
$525$	−0.0422324	−0.00184317
$526$	11.7645	0.512956
$527$	1.69964	0.0740373
$528$	3.48544	0.151684
$529$	−11.5243	−0.501058
$530$	8.47483	0.368123
$531$	10.4857	0.455041
$532$	0	0
$533$	−50.7964	−2.20023
$534$	−7.55919	−0.327118
$535$	1.27952	0.0553185
$536$	−1.83668	−0.0793327
$537$	−1.64667	−0.0710591
$538$	7.42578	0.320148
$539$	6.09289	0.262440
$540$	−7.20250	−0.309946
$541$	28.0364	1.20538	0.602690	−	0.797975i	$-0.294096\pi$
$541$	28.0364	1.20538	0.602690	+	0.797975i	$0.294096\pi$
$542$	−25.3145	−1.08735
$543$	−4.06837	−0.174590
$544$	−1.73887	−0.0745535
$545$	−16.5495	−0.708904
$546$	3.51820	0.150565
$547$	−5.19672	−0.222196	−0.111098	−	0.993809i	$-0.535437\pi$
$547$	−5.19672	−0.222196	−0.111098	+	0.993809i	$0.535437\pi$
$548$	10.1140	0.432049
$549$	−3.76335	−0.160616
$550$	−0.449816	−0.0191802
$551$	0	0
$552$	−1.93786	−0.0824809
$553$	−15.3099	−0.651045
$554$	14.8882	0.632539
$555$	15.1415	0.642722
$556$	16.2063	0.687300
$557$	7.16210	0.303468	0.151734	−	0.988421i	$-0.451514\pi$
$557$	7.16210	0.303468	0.151734	+	0.988421i	$0.451514\pi$
$558$	2.61245	0.110594
$559$	20.1930	0.854075
$560$	2.21950	0.0937909
$561$	−6.06074	−0.255885
$562$	15.2413	0.642915
$563$	−30.1360	−1.27008	−0.635041	−	0.772479i	$-0.719017\pi$
$563$	−30.1360	−1.27008	−0.635041	+	0.772479i	$0.719017\pi$
$564$	5.69688	0.239882
$565$	−31.8327	−1.33921
$566$	14.5625	0.612106
$567$	6.16191	0.258776
$568$	−16.2892	−0.683478
$569$	7.48345	0.313722	0.156861	−	0.987621i	$-0.449862\pi$
$569$	7.48345	0.313722	0.156861	+	0.987621i	$0.449862\pi$
$570$	0	0
$571$	0.259851	0.0108744	0.00543720	−	0.999985i	$-0.498269\pi$
$571$	0.259851	0.0108744	0.00543720	+	0.999985i	$0.498269\pi$
$572$	37.4723	1.56679
$573$	−2.62889	−0.109823
$574$	−8.25936	−0.344739
$575$	0.250092	0.0104296
$576$	−2.67276	−0.111365
$577$	14.7288	0.613170	0.306585	−	0.951843i	$-0.400814\pi$
$577$	14.7288	0.613170	0.306585	+	0.951843i	$0.400814\pi$
$578$	−13.9763	−0.581338
$579$	3.89018	0.161670
$580$	−4.85753	−0.201698
$581$	11.7929	0.489250
$582$	6.57598	0.272583
$583$	23.2648	0.963531
$584$	−8.95240	−0.370453
$585$	−36.4839	−1.50842
$586$	−15.3389	−0.633643
$587$	25.4318	1.04968	0.524842	−	0.851200i	$-0.324124\pi$
$587$	25.4318	1.04968	0.524842	+	0.851200i	$0.324124\pi$
$588$	0.572050	0.0235910
$589$	0	0
$590$	−8.70748	−0.358481
$591$	8.21926	0.338095
$592$	11.9256	0.490140
$593$	−33.6590	−1.38221	−0.691104	−	0.722755i	$-0.742875\pi$
$593$	−33.6590	−1.38221	−0.691104	+	0.722755i	$0.742875\pi$
$594$	−19.7721	−0.811258
$595$	−3.85942	−0.158221
$596$	−23.4893	−0.962158
$597$	−9.81981	−0.401898
$598$	−20.8341	−0.851970
$599$	4.33047	0.176938	0.0884691	−	0.996079i	$-0.471803\pi$
$599$	4.33047	0.176938	0.0884691	+	0.996079i	$0.471803\pi$
$600$	−0.0422324	−0.00172413
$601$	−16.8879	−0.688871	−0.344435	−	0.938810i	$-0.611930\pi$
$601$	−16.8879	−0.688871	−0.344435	+	0.938810i	$0.611930\pi$
$602$	3.28334	0.133819
$603$	4.90901	0.199910
$604$	5.57037	0.226655
$605$	57.9808	2.35725
$606$	0.971943	0.0394825
$607$	0.473176	0.0192056	0.00960281	−	0.999954i	$-0.496943\pi$
$607$	0.473176	0.0192056	0.00960281	+	0.999954i	$0.496943\pi$
$608$	0	0
$609$	−1.25197	−0.0507325
$610$	3.12514	0.126533
$611$	61.2476	2.47781
$612$	4.64758	0.187867
$613$	20.5732	0.830944	0.415472	−	0.909606i	$-0.363616\pi$
$613$	20.5732	0.830944	0.415472	+	0.909606i	$0.363616\pi$
$614$	−10.7228	−0.432735
$615$	−10.4866	−0.422861
$616$	6.09289	0.245490
$617$	39.9095	1.60670	0.803348	−	0.595510i	$-0.203050\pi$
$617$	39.9095	1.60670	0.803348	+	0.595510i	$0.203050\pi$
$618$	−1.30885	−0.0526495
$619$	34.5062	1.38692	0.693460	−	0.720495i	$-0.256085\pi$
$619$	34.5062	1.38692	0.693460	+	0.720495i	$0.256085\pi$
$620$	−2.16942	−0.0871259
$621$	10.9930	0.441135
$622$	31.8376	1.27657
$623$	−13.2142	−0.529416
$624$	3.51820	0.140841
$625$	−24.6254	−0.985017
$626$	−9.09619	−0.363557
$627$	0	0
$628$	10.9616	0.437414
$629$	−20.7371	−0.826843
$630$	−5.93218	−0.236344
$631$	29.6170	1.17903	0.589517	−	0.807756i	$-0.299318\pi$
$631$	29.6170	1.17903	0.589517	+	0.807756i	$0.299318\pi$
$632$	−15.3099	−0.608997
$633$	1.79376	0.0712954
$634$	−14.5668	−0.578521
$635$	−1.73130	−0.0687044
$636$	2.18429	0.0866128
$637$	6.15016	0.243678
$638$	−13.3347	−0.527927
$639$	43.5370	1.72230
$640$	2.21950	0.0877334
$641$	25.4230	1.00415	0.502074	−	0.864824i	$-0.332570\pi$
$641$	25.4230	1.00415	0.502074	+	0.864824i	$0.332570\pi$
$642$	0.329782	0.0130155
$643$	−33.9370	−1.33834	−0.669172	−	0.743108i	$-0.733351\pi$
$643$	−33.9370	−1.33834	−0.669172	+	0.743108i	$0.733351\pi$
$644$	−3.38757	−0.133489
$645$	4.16874	0.164144
$646$	0	0
$647$	22.9460	0.902100	0.451050	−	0.892499i	$-0.351050\pi$
$647$	22.9460	0.902100	0.451050	+	0.892499i	$0.351050\pi$
$648$	6.16191	0.242063
$649$	−23.9035	−0.938294
$650$	−0.454044	−0.0178091
$651$	−0.559143	−0.0219145
$652$	0.541777	0.0212176
$653$	−5.74179	−0.224694	−0.112347	−	0.993669i	$-0.535837\pi$
$653$	−5.74179	−0.224694	−0.112347	+	0.993669i	$0.535837\pi$
$654$	−4.26545	−0.166792
$655$	−14.2506	−0.556817
$656$	−8.25936	−0.322474
$657$	23.9276	0.933504
$658$	9.95871	0.388231
$659$	4.94741	0.192724	0.0963619	−	0.995346i	$-0.469279\pi$
$659$	4.94741	0.192724	0.0963619	+	0.995346i	$0.469279\pi$
$660$	7.73593	0.301121
$661$	−17.3674	−0.675512	−0.337756	−	0.941234i	$-0.609668\pi$
$661$	−17.3674	−0.675512	−0.337756	+	0.941234i	$0.609668\pi$
$662$	−9.68807	−0.376537
$663$	−6.11770	−0.237592
$664$	11.7929	0.457652
$665$	0	0
$666$	−31.8743	−1.23510
$667$	7.41395	0.287069
$668$	−5.89883	−0.228233
$669$	14.5127	0.561094
$670$	−4.07652	−0.157490
$671$	8.57904	0.331190
$672$	0.572050	0.0220673
$673$	−44.9032	−1.73089	−0.865446	−	0.501002i	$-0.832965\pi$
$673$	−44.9032	−1.73089	−0.865446	+	0.501002i	$0.832965\pi$
$674$	9.16506	0.353025
$675$	0.239574	0.00922122
$676$	24.8244	0.954785
$677$	−4.12795	−0.158650	−0.0793250	−	0.996849i	$-0.525276\pi$
$677$	−4.12795	−0.158650	−0.0793250	+	0.996849i	$0.525276\pi$
$678$	−8.20452	−0.315093
$679$	11.4955	0.441155
$680$	−3.85942	−0.148002
$681$	1.84431	0.0706741
$682$	−5.95541	−0.228045
$683$	12.5746	0.481154	0.240577	−	0.970630i	$-0.422663\pi$
$683$	12.5746	0.481154	0.240577	+	0.970630i	$0.422663\pi$
$684$	0	0
$685$	22.4480	0.857694
$686$	1.00000	0.0381802
$687$	4.66094	0.177826
$688$	3.28334	0.125176
$689$	23.4835	0.894649
$690$	−4.30108	−0.163739
$691$	37.6206	1.43116	0.715578	−	0.698532i	$-0.246163\pi$
$691$	37.6206	1.43116	0.715578	+	0.698532i	$0.246163\pi$
$692$	5.63686	0.214281
$693$	−16.2848	−0.618610
$694$	31.3135	1.18865
$695$	35.9698	1.36441
$696$	−1.25197	−0.0474559
$697$	14.3620	0.543998
$698$	−26.3620	−0.997816
$699$	6.80370	0.257340
$700$	−0.0738264	−0.00279038
$701$	−18.4181	−0.695641	−0.347820	−	0.937561i	$-0.613078\pi$
$701$	−18.4181	−0.695641	−0.347820	+	0.937561i	$0.613078\pi$
$702$	−19.9579	−0.753262
$703$	0	0
$704$	6.09289	0.229635
$705$	12.6442	0.476209
$706$	−11.2622	−0.423859
$707$	1.69905	0.0638994
$708$	−2.24425	−0.0843443
$709$	−31.2604	−1.17401	−0.587005	−	0.809584i	$-0.699693\pi$
$709$	−31.2604	−1.17401	−0.587005	+	0.809584i	$0.699693\pi$
$710$	−36.1538	−1.35683
$711$	40.9198	1.53461
$712$	−13.2142	−0.495223
$713$	3.31114	0.124003
$714$	−0.994722	−0.0372265
$715$	83.1696	3.11037
$716$	−2.87854	−0.107576
$717$	0.544335	0.0203286
$718$	8.23468	0.307316
$719$	1.43857	0.0536496	0.0268248	−	0.999640i	$-0.491460\pi$
$719$	1.43857	0.0536496	0.0268248	+	0.999640i	$0.491460\pi$
$720$	−5.93218	−0.221079
$721$	−2.28799	−0.0852092
$722$	0	0
$723$	−3.70938	−0.137953
$724$	−7.11191	−0.264312
$725$	0.161574	0.00600072
$726$	14.9439	0.554619
$727$	5.24589	0.194559	0.0972797	−	0.995257i	$-0.468986\pi$
$727$	5.24589	0.194559	0.0972797	+	0.995257i	$0.468986\pi$
$728$	6.15016	0.227940
$729$	−10.9002	−0.403712
$730$	−19.8698	−0.735415
$731$	−5.70931	−0.211166
$732$	0.805470	0.0297710
$733$	−51.4997	−1.90218	−0.951092	−	0.308908i	$-0.900036\pi$
$733$	−51.4997	−1.90218	−0.951092	+	0.308908i	$0.900036\pi$
$734$	−6.46885	−0.238770
$735$	1.26966	0.0468323
$736$	−3.38757	−0.124868
$737$	−11.1907	−0.412216
$738$	22.0753	0.812602
$739$	−43.8368	−1.61256	−0.806282	−	0.591532i	$-0.798524\pi$
$739$	−43.8368	−1.61256	−0.806282	+	0.591532i	$0.798524\pi$
$740$	26.4689	0.973015
$741$	0	0
$742$	3.81835	0.140176
$743$	8.35100	0.306369	0.153184	−	0.988198i	$-0.451047\pi$
$743$	8.35100	0.306369	0.153184	+	0.988198i	$0.451047\pi$
$744$	−0.559143	−0.0204992
$745$	−52.1344	−1.91006
$746$	20.9407	0.766694
$747$	−31.5195	−1.15324
$748$	−10.5948	−0.387383
$749$	0.576491	0.0210645
$750$	−6.44206	−0.235231
$751$	13.1485	0.479796	0.239898	−	0.970798i	$-0.422886\pi$
$751$	13.1485	0.479796	0.239898	+	0.970798i	$0.422886\pi$
$752$	9.95871	0.363157
$753$	−4.56597	−0.166393
$754$	−13.4601	−0.490187
$755$	12.3634	0.449952
$756$	−3.24510	−0.118023
$757$	−19.8091	−0.719973	−0.359986	−	0.932958i	$-0.617219\pi$
$757$	−19.8091	−0.719973	−0.359986	+	0.932958i	$0.617219\pi$
$758$	−15.0189	−0.545513
$759$	−11.8072	−0.428574
$760$	0	0
$761$	−22.2730	−0.807397	−0.403698	−	0.914892i	$-0.632275\pi$
$761$	−22.2730	−0.807397	−0.403698	+	0.914892i	$0.632275\pi$
$762$	−0.446222	−0.0161649
$763$	−7.45643	−0.269941
$764$	−4.59556	−0.166261
$765$	10.3153	0.372951
$766$	−19.0979	−0.690033
$767$	−24.1281	−0.871217
$768$	0.572050	0.0206421
$769$	19.6398	0.708229	0.354114	−	0.935202i	$-0.384782\pi$
$769$	19.6398	0.708229	0.354114	+	0.935202i	$0.384782\pi$
$770$	13.5232	0.487341
$771$	1.45115	0.0522619
$772$	6.80042	0.244752
$773$	28.3161	1.01846	0.509230	−	0.860630i	$-0.329930\pi$
$773$	28.3161	1.01846	0.509230	+	0.860630i	$0.329930\pi$
$774$	−8.77557	−0.315431
$775$	0.0721606	0.00259208
$776$	11.4955	0.412663
$777$	6.82205	0.244740
$778$	−10.1775	−0.364882
$779$	0	0
$780$	7.80864	0.279594
$781$	−99.2482	−3.55138
$782$	5.89056	0.210646
$783$	7.10214	0.253810
$784$	1.00000	0.0357143
$785$	24.3292	0.868345
$786$	−3.67293	−0.131009
$787$	−12.0996	−0.431305	−0.215653	−	0.976470i	$-0.569188\pi$
$787$	−12.0996	−0.431305	−0.215653	+	0.976470i	$0.569188\pi$
$788$	14.3681	0.511841
$789$	6.72987	0.239590
$790$	−33.9804	−1.20897
$791$	−14.3423	−0.509953
$792$	−16.2848	−0.578656
$793$	8.65967	0.307514
$794$	−5.17058	−0.183497
$795$	4.84803	0.171942
$796$	−17.1660	−0.608432
$797$	−34.3005	−1.21499	−0.607493	−	0.794325i	$-0.707825\pi$
$797$	−34.3005	−1.21499	−0.607493	+	0.794325i	$0.707825\pi$
$798$	0	0
$799$	−17.3169	−0.612628
$800$	−0.0738264	−0.00261016
$801$	35.3184	1.24791
$802$	14.6655	0.517858
$803$	−54.5460	−1.92489
$804$	−1.05068	−0.0370545
$805$	−7.51872	−0.265000
$806$	−6.01138	−0.211742
$807$	4.24792	0.149534
$808$	1.69905	0.0597724
$809$	−37.3465	−1.31303	−0.656516	−	0.754312i	$-0.727971\pi$
$809$	−37.3465	−1.31303	−0.656516	+	0.754312i	$0.727971\pi$
$810$	13.6764	0.480538
$811$	−32.6666	−1.14708	−0.573540	−	0.819178i	$-0.694430\pi$
$811$	−32.6666	−1.14708	−0.573540	+	0.819178i	$0.694430\pi$
$812$	−2.18857	−0.0768038
$813$	−14.4812	−0.507876
$814$	72.6615	2.54678
$815$	1.20247	0.0421208
$816$	−0.994722	−0.0348222
$817$	0	0
$818$	−13.9231	−0.486808
$819$	−16.4379	−0.574386
$820$	−18.3316	−0.640169
$821$	−29.2212	−1.01983	−0.509914	−	0.860226i	$-0.670323\pi$
$821$	−29.2212	−1.01983	−0.509914	+	0.860226i	$0.670323\pi$
$822$	5.78572	0.201800
$823$	21.9850	0.766349	0.383174	−	0.923676i	$-0.374831\pi$
$823$	21.9850	0.766349	0.383174	+	0.923676i	$0.374831\pi$
$824$	−2.28799	−0.0797059
$825$	−0.257318	−0.00895865
$826$	−3.92318	−0.136505
$827$	19.2733	0.670199	0.335100	−	0.942183i	$-0.391230\pi$
$827$	19.2733	0.670199	0.335100	+	0.942183i	$0.391230\pi$
$828$	9.05417	0.314654
$829$	16.1604	0.561273	0.280636	−	0.959814i	$-0.409454\pi$
$829$	16.1604	0.561273	0.280636	+	0.959814i	$0.409454\pi$
$830$	26.1742	0.908521
$831$	8.51680	0.295445
$832$	6.15016	0.213218
$833$	−1.73887	−0.0602483
$834$	9.27082	0.321022
$835$	−13.0924	−0.453083
$836$	0	0
$837$	3.17188	0.109636
$838$	−5.94582	−0.205395
$839$	19.5537	0.675067	0.337534	−	0.941313i	$-0.390407\pi$
$839$	19.5537	0.675067	0.337534	+	0.941313i	$0.390407\pi$
$840$	1.26966	0.0438076
$841$	−24.2102	−0.834833
$842$	−26.6445	−0.918231
$843$	8.71878	0.300291
$844$	3.13566	0.107934
$845$	55.0978	1.89542
$846$	−26.6172	−0.915119
$847$	26.1234	0.897609
$848$	3.81835	0.131123
$849$	8.33047	0.285901
$850$	0.128375	0.00440321
$851$	−40.3989	−1.38486
$852$	−9.31823	−0.319237
$853$	−30.3217	−1.03820	−0.519098	−	0.854715i	$-0.673732\pi$
$853$	−30.3217	−1.03820	−0.519098	+	0.854715i	$0.673732\pi$
$854$	1.40804	0.0481821
$855$	0	0
$856$	0.576491	0.0197041
$857$	−36.8137	−1.25753	−0.628766	−	0.777594i	$-0.716440\pi$
$857$	−36.8137	−1.25753	−0.628766	+	0.777594i	$0.716440\pi$
$858$	21.4360	0.731813
$859$	−15.6654	−0.534498	−0.267249	−	0.963628i	$-0.586115\pi$
$859$	−15.6654	−0.534498	−0.267249	+	0.963628i	$0.586115\pi$
$860$	7.28737	0.248497
$861$	−4.72477	−0.161020
$862$	−7.15638	−0.243747
$863$	−6.44962	−0.219548	−0.109774	−	0.993957i	$-0.535013\pi$
$863$	−6.44962	−0.219548	−0.109774	+	0.993957i	$0.535013\pi$
$864$	−3.24510	−0.110401
$865$	12.5110	0.425387
$866$	−18.3923	−0.624996
$867$	−7.99516	−0.271530
$868$	−0.977436	−0.0331763
$869$	−93.2819	−3.16437
$870$	−2.77875	−0.0942086
$871$	−11.2959	−0.382747
$872$	−7.45643	−0.252506
$873$	−30.7246	−1.03987
$874$	0	0
$875$	−11.2614	−0.380703
$876$	−5.12122	−0.173030
$877$	−23.9008	−0.807073	−0.403537	−	0.914963i	$-0.632219\pi$
$877$	−23.9008	−0.807073	−0.403537	+	0.914963i	$0.632219\pi$
$878$	20.5607	0.693890
$879$	−8.77461	−0.295960
$880$	13.5232	0.455866
$881$	−40.2251	−1.35522	−0.677609	−	0.735423i	$-0.736984\pi$
$881$	−40.2251	−1.35522	−0.677609	+	0.735423i	$0.736984\pi$
$882$	−2.67276	−0.0899965
$883$	−11.0160	−0.370718	−0.185359	−	0.982671i	$-0.559345\pi$
$883$	−11.0160	−0.370718	−0.185359	+	0.982671i	$0.559345\pi$
$884$	−10.6943	−0.359689
$885$	−4.98112	−0.167439
$886$	−9.17486	−0.308236
$887$	11.6706	0.391860	0.195930	−	0.980618i	$-0.437227\pi$
$887$	11.6706	0.391860	0.195930	+	0.980618i	$0.437227\pi$
$888$	6.82205	0.228933
$889$	−0.780040	−0.0261617
$890$	−29.3289	−0.983107
$891$	37.5439	1.25777
$892$	25.3696	0.849438
$893$	0	0
$894$	−13.4370	−0.449402
$895$	−6.38892	−0.213558
$896$	1.00000	0.0334077
$897$	−11.9182	−0.397936
$898$	−25.3026	−0.844357
$899$	2.13919	0.0713460
$900$	0.197320	0.00657734
$901$	−6.63963	−0.221198
$902$	−50.3234	−1.67559
$903$	1.87824	0.0625038
$904$	−14.3423	−0.477018
$905$	−15.7849	−0.524707
$906$	3.18653	0.105866
$907$	38.5007	1.27839	0.639197	−	0.769043i	$-0.279267\pi$
$907$	38.5007	1.27839	0.639197	+	0.769043i	$0.279267\pi$
$908$	3.22403	0.106993
$909$	−4.54116	−0.150621
$910$	13.6503	0.452502
$911$	32.3190	1.07077	0.535387	−	0.844607i	$-0.320166\pi$
$911$	32.3190	1.07077	0.535387	+	0.844607i	$0.320166\pi$
$912$	0	0
$913$	71.8527	2.37798
$914$	33.6647	1.11353
$915$	1.78774	0.0591008
$916$	8.14777	0.269210
$917$	−6.42064	−0.212028
$918$	5.64282	0.186241
$919$	17.6464	0.582102	0.291051	−	0.956707i	$-0.405995\pi$
$919$	17.6464	0.582102	0.291051	+	0.956707i	$0.405995\pi$
$920$	−7.51872	−0.247885
$921$	−6.13396	−0.202121
$922$	−29.7527	−0.979854
$923$	−100.181	−3.29750
$924$	3.48544	0.114663
$925$	−0.880425	−0.0289482
$926$	−26.2873	−0.863854
$927$	6.11525	0.200851
$928$	−2.18857	−0.0718434
$929$	−5.69051	−0.186700	−0.0933498	−	0.995633i	$-0.529757\pi$
$929$	−5.69051	−0.186700	−0.0933498	+	0.995633i	$0.529757\pi$
$930$	−1.24102	−0.0406945
$931$	0	0
$932$	11.8935	0.389586
$933$	18.2127	0.596258
$934$	−3.03716	−0.0993791
$935$	−23.5151	−0.769025
$936$	−16.4379	−0.537289
$937$	−38.4048	−1.25463	−0.627315	−	0.778766i	$-0.715846\pi$
$937$	−38.4048	−1.25463	−0.627315	+	0.778766i	$0.715846\pi$
$938$	−1.83668	−0.0599699
$939$	−5.20348	−0.169809
$940$	22.1033	0.720931
$941$	−29.6322	−0.965982	−0.482991	−	0.875625i	$-0.660450\pi$
$941$	−29.6322	−0.965982	−0.482991	+	0.875625i	$0.660450\pi$
$942$	6.27057	0.204306
$943$	27.9792	0.911128
$944$	−3.92318	−0.127689
$945$	−7.20250	−0.234297
$946$	20.0050	0.650420
$947$	−25.2166	−0.819431	−0.409715	−	0.912213i	$-0.634372\pi$
$947$	−25.2166	−0.819431	−0.409715	+	0.912213i	$0.634372\pi$
$948$	−8.75806	−0.284449
$949$	−55.0586	−1.78728
$950$	0	0
$951$	−8.33293	−0.270214
$952$	−1.73887	−0.0563572
$953$	−27.2031	−0.881193	−0.440597	−	0.897705i	$-0.645233\pi$
$953$	−27.2031	−0.881193	−0.440597	+	0.897705i	$0.645233\pi$
$954$	−10.2055	−0.330417
$955$	−10.1998	−0.330059
$956$	0.951552	0.0307754
$957$	−7.62814	−0.246583
$958$	−4.49449	−0.145210
$959$	10.1140	0.326598
$960$	1.26966	0.0409783
$961$	−30.0446	−0.969181
$962$	73.3444	2.36472
$963$	−1.54082	−0.0496523
$964$	−6.48436	−0.208847
$965$	15.0935	0.485877
$966$	−1.93786	−0.0623497
$967$	−24.8243	−0.798295	−0.399147	−	0.916887i	$-0.630694\pi$
$967$	−24.8243	−0.798295	−0.399147	+	0.916887i	$0.630694\pi$
$968$	26.1234	0.839637
$969$	0	0
$970$	25.5141	0.819210
$971$	52.2949	1.67822	0.839112	−	0.543958i	$-0.183075\pi$
$971$	52.2949	1.67822	0.839112	+	0.543958i	$0.183075\pi$
$972$	13.2602	0.425322
$973$	16.2063	0.519550
$974$	−29.2764	−0.938076
$975$	−0.259736	−0.00831821
$976$	1.40804	0.0450703
$977$	−36.2037	−1.15826	−0.579130	−	0.815235i	$-0.696608\pi$
$977$	−36.2037	−1.15826	−0.579130	+	0.815235i	$0.696608\pi$
$978$	0.309924	0.00991026
$979$	−80.5127	−2.57320
$980$	2.21950	0.0708993
$981$	19.9292	0.636291
$982$	23.1139	0.737594
$983$	−1.45212	−0.0463154	−0.0231577	−	0.999732i	$-0.507372\pi$
$983$	−1.45212	−0.0463154	−0.0231577	+	0.999732i	$0.507372\pi$
$984$	−4.72477	−0.150620
$985$	31.8899	1.01610
$986$	3.80565	0.121196
$987$	5.69688	0.181334
$988$	0	0
$989$	−11.1226	−0.353677
$990$	−36.1442	−1.14874
$991$	33.4036	1.06110	0.530550	−	0.847653i	$-0.321985\pi$
$991$	33.4036	1.06110	0.530550	+	0.847653i	$0.321985\pi$
$992$	−0.977436	−0.0310336
$993$	−5.54206	−0.175872
$994$	−16.2892	−0.516661
$995$	−38.0999	−1.20785
$996$	6.74611	0.213759
$997$	22.4043	0.709552	0.354776	−	0.934951i	$-0.384557\pi$
$997$	22.4043	0.709552	0.354776	+	0.934951i	$0.384557\pi$
$998$	16.2504	0.514397
$999$	−38.6998	−1.22441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bm.1.7		12
19.2	odd	18		266.2.u.d.99.2	yes	24
19.10	odd	18		266.2.u.d.43.2	✓	24
19.18	odd	2		5054.2.a.bl.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.u.d.43.2	✓	24	19.10	odd	18
266.2.u.d.99.2	yes	24	19.2	odd	18
5054.2.a.bl.1.6		12	19.18	odd	2
5054.2.a.bm.1.7		12	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:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.572050 q^{3} +1.00000 q^{4} +2.21950 q^{5} +0.572050 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.67276 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.572050 q^{3} +1.00000 q^{4} +2.21950 q^{5} +0.572050 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.67276 q^{9} +2.21950 q^{10} +6.09289 q^{11} +0.572050 q^{12} +6.15016 q^{13} +1.00000 q^{14} +1.26966 q^{15} +1.00000 q^{16} -1.73887 q^{17} -2.67276 q^{18} +2.21950 q^{20} +0.572050 q^{21} +6.09289 q^{22} -3.38757 q^{23} +0.572050 q^{24} -0.0738264 q^{25} +6.15016 q^{26} -3.24510 q^{27} +1.00000 q^{28} -2.18857 q^{29} +1.26966 q^{30} -0.977436 q^{31} +1.00000 q^{32} +3.48544 q^{33} -1.73887 q^{34} +2.21950 q^{35} -2.67276 q^{36} +11.9256 q^{37} +3.51820 q^{39} +2.21950 q^{40} -8.25936 q^{41} +0.572050 q^{42} +3.28334 q^{43} +6.09289 q^{44} -5.93218 q^{45} -3.38757 q^{46} +9.95871 q^{47} +0.572050 q^{48} +1.00000 q^{49} -0.0738264 q^{50} -0.994722 q^{51} +6.15016 q^{52} +3.81835 q^{53} -3.24510 q^{54} +13.5232 q^{55} +1.00000 q^{56} -2.18857 q^{58} -3.92318 q^{59} +1.26966 q^{60} +1.40804 q^{61} -0.977436 q^{62} -2.67276 q^{63} +1.00000 q^{64} +13.6503 q^{65} +3.48544 q^{66} -1.83668 q^{67} -1.73887 q^{68} -1.93786 q^{69} +2.21950 q^{70} -16.2892 q^{71} -2.67276 q^{72} -8.95240 q^{73} +11.9256 q^{74} -0.0422324 q^{75} +6.09289 q^{77} +3.51820 q^{78} -15.3099 q^{79} +2.21950 q^{80} +6.16191 q^{81} -8.25936 q^{82} +11.7929 q^{83} +0.572050 q^{84} -3.85942 q^{85} +3.28334 q^{86} -1.25197 q^{87} +6.09289 q^{88} -13.2142 q^{89} -5.93218 q^{90} +6.15016 q^{91} -3.38757 q^{92} -0.559143 q^{93} +9.95871 q^{94} +0.572050 q^{96} +11.4955 q^{97} +1.00000 q^{98} -16.2848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} + 12 q^{8} + 21 q^{9}+O(q^{10}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 + 12 * q^8 + 21 * q^9	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} + 12 q^{8} + 21 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} - 6 q^{13} + 12 q^{14} - 6 q^{15} + 12 q^{16} + 3 q^{17} + 21 q^{18} + 3 q^{20} + 3 q^{21} + 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} + 21 q^{27} + 12 q^{28} + 6 q^{29} - 6 q^{30} + 3 q^{31} + 12 q^{32} - 6 q^{33} + 3 q^{34} + 3 q^{35} + 21 q^{36} + 27 q^{37} + 27 q^{39} + 3 q^{40} - 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} + 12 q^{46} + 21 q^{47} + 3 q^{48} + 12 q^{49} + 33 q^{50} + 3 q^{51} - 6 q^{52} + 6 q^{53} + 21 q^{54} + 48 q^{55} + 12 q^{56} + 6 q^{58} - 27 q^{59} - 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} - 9 q^{65} - 6 q^{66} + 9 q^{67} + 3 q^{68} + 18 q^{69} + 3 q^{70} + 27 q^{71} + 21 q^{72} - 9 q^{73} + 27 q^{74} + 33 q^{75} + 3 q^{77} + 27 q^{78} - 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} + 3 q^{84} + 78 q^{85} + 30 q^{86} - 45 q^{87} + 3 q^{88} - 24 q^{89} + 30 q^{90} - 6 q^{91} + 12 q^{92} + 3 q^{93} + 21 q^{94} + 3 q^{96} - 18 q^{97} + 12 q^{98} + 48 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 + 12 * q^8 + 21 * q^9 + 3 * q^10 + 3 * q^11 + 3 * q^12 - 6 * q^13 + 12 * q^14 - 6 * q^15 + 12 * q^16 + 3 * q^17 + 21 * q^18 + 3 * q^20 + 3 * q^21 + 3 * q^22 + 12 * q^23 + 3 * q^24 + 33 * q^25 - 6 * q^26 + 21 * q^27 + 12 * q^28 + 6 * q^29 - 6 * q^30 + 3 * q^31 + 12 * q^32 - 6 * q^33 + 3 * q^34 + 3 * q^35 + 21 * q^36 + 27 * q^37 + 27 * q^39 + 3 * q^40 - 27 * q^41 + 3 * q^42 + 30 * q^43 + 3 * q^44 + 30 * q^45 + 12 * q^46 + 21 * q^47 + 3 * q^48 + 12 * q^49 + 33 * q^50 + 3 * q^51 - 6 * q^52 + 6 * q^53 + 21 * q^54 + 48 * q^55 + 12 * q^56 + 6 * q^58 - 27 * q^59 - 6 * q^60 + 18 * q^61 + 3 * q^62 + 21 * q^63 + 12 * q^64 - 9 * q^65 - 6 * q^66 + 9 * q^67 + 3 * q^68 + 18 * q^69 + 3 * q^70 + 27 * q^71 + 21 * q^72 - 9 * q^73 + 27 * q^74 + 33 * q^75 + 3 * q^77 + 27 * q^78 - 12 * q^79 + 3 * q^80 + 24 * q^81 - 27 * q^82 + 9 * q^83 + 3 * q^84 + 78 * q^85 + 30 * q^86 - 45 * q^87 + 3 * q^88 - 24 * q^89 + 30 * q^90 - 6 * q^91 + 12 * q^92 + 3 * q^93 + 21 * q^94 + 3 * q^96 - 18 * q^97 + 12 * q^98 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more