LMFDB - Embedded newform 2523.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,2,Mod(1,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, 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("2523.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2523 = 3 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2523.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:	$20.1462564300$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2523.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} -2.23607 q^{8} +1.00000 q^{9} +2.00000 q^{10} +0.236068 q^{11} +1.61803 q^{12} -5.47214 q^{13} +0.145898 q^{14} -3.23607 q^{15} +1.85410 q^{16} -3.00000 q^{17} +0.618034 q^{18} +7.23607 q^{19} -5.23607 q^{20} -0.236068 q^{21} +0.145898 q^{22} -7.70820 q^{23} +2.23607 q^{24} +5.47214 q^{25} -3.38197 q^{26} -1.00000 q^{27} -0.381966 q^{28} -2.00000 q^{30} -3.70820 q^{31} +5.61803 q^{32} -0.236068 q^{33} -1.85410 q^{34} +0.763932 q^{35} -1.61803 q^{36} -5.23607 q^{37} +4.47214 q^{38} +5.47214 q^{39} -7.23607 q^{40} -2.00000 q^{41} -0.145898 q^{42} -4.00000 q^{43} -0.381966 q^{44} +3.23607 q^{45} -4.76393 q^{46} -4.70820 q^{47} -1.85410 q^{48} -6.94427 q^{49} +3.38197 q^{50} +3.00000 q^{51} +8.85410 q^{52} +11.2361 q^{53} -0.618034 q^{54} +0.763932 q^{55} -0.527864 q^{56} -7.23607 q^{57} +4.47214 q^{59} +5.23607 q^{60} +5.23607 q^{61} -2.29180 q^{62} +0.236068 q^{63} -0.236068 q^{64} -17.7082 q^{65} -0.145898 q^{66} -13.1803 q^{67} +4.85410 q^{68} +7.70820 q^{69} +0.472136 q^{70} -5.23607 q^{71} -2.23607 q^{72} -6.76393 q^{73} -3.23607 q^{74} -5.47214 q^{75} -11.7082 q^{76} +0.0557281 q^{77} +3.38197 q^{78} +12.7639 q^{79} +6.00000 q^{80} +1.00000 q^{81} -1.23607 q^{82} +2.94427 q^{83} +0.381966 q^{84} -9.70820 q^{85} -2.47214 q^{86} -0.527864 q^{88} -5.00000 q^{89} +2.00000 q^{90} -1.29180 q^{91} +12.4721 q^{92} +3.70820 q^{93} -2.90983 q^{94} +23.4164 q^{95} -5.61803 q^{96} -18.6525 q^{97} -4.29180 q^{98} +0.236068 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9} + 4 q^{10} - 4 q^{11} + q^{12} - 2 q^{13} + 7 q^{14} - 2 q^{15} - 3 q^{16} - 6 q^{17} - q^{18} + 10 q^{19} - 6 q^{20} + 4 q^{21} + 7 q^{22} - 2 q^{23} + 2 q^{25} - 9 q^{26} - 2 q^{27} - 3 q^{28} - 4 q^{30} + 6 q^{31} + 9 q^{32} + 4 q^{33} + 3 q^{34} + 6 q^{35} - q^{36} - 6 q^{37} + 2 q^{39} - 10 q^{40} - 4 q^{41} - 7 q^{42} - 8 q^{43} - 3 q^{44} + 2 q^{45} - 14 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} + 9 q^{50} + 6 q^{51} + 11 q^{52} + 18 q^{53} + q^{54} + 6 q^{55} - 10 q^{56} - 10 q^{57} + 6 q^{60} + 6 q^{61} - 18 q^{62} - 4 q^{63} + 4 q^{64} - 22 q^{65} - 7 q^{66} - 4 q^{67} + 3 q^{68} + 2 q^{69} - 8 q^{70} - 6 q^{71} - 18 q^{73} - 2 q^{74} - 2 q^{75} - 10 q^{76} + 18 q^{77} + 9 q^{78} + 30 q^{79} + 12 q^{80} + 2 q^{81} + 2 q^{82} - 12 q^{83} + 3 q^{84} - 6 q^{85} + 4 q^{86} - 10 q^{88} - 10 q^{89} + 4 q^{90} - 16 q^{91} + 16 q^{92} - 6 q^{93} - 17 q^{94} + 20 q^{95} - 9 q^{96} - 6 q^{97} - 22 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9 + 4 * q^10 - 4 * q^11 + q^12 - 2 * q^13 + 7 * q^14 - 2 * q^15 - 3 * q^16 - 6 * q^17 - q^18 + 10 * q^19 - 6 * q^20 + 4 * q^21 + 7 * q^22 - 2 * q^23 + 2 * q^25 - 9 * q^26 - 2 * q^27 - 3 * q^28 - 4 * q^30 + 6 * q^31 + 9 * q^32 + 4 * q^33 + 3 * q^34 + 6 * q^35 - q^36 - 6 * q^37 + 2 * q^39 - 10 * q^40 - 4 * q^41 - 7 * q^42 - 8 * q^43 - 3 * q^44 + 2 * q^45 - 14 * q^46 + 4 * q^47 + 3 * q^48 + 4 * q^49 + 9 * q^50 + 6 * q^51 + 11 * q^52 + 18 * q^53 + q^54 + 6 * q^55 - 10 * q^56 - 10 * q^57 + 6 * q^60 + 6 * q^61 - 18 * q^62 - 4 * q^63 + 4 * q^64 - 22 * q^65 - 7 * q^66 - 4 * q^67 + 3 * q^68 + 2 * q^69 - 8 * q^70 - 6 * q^71 - 18 * q^73 - 2 * q^74 - 2 * q^75 - 10 * q^76 + 18 * q^77 + 9 * q^78 + 30 * q^79 + 12 * q^80 + 2 * q^81 + 2 * q^82 - 12 * q^83 + 3 * q^84 - 6 * q^85 + 4 * q^86 - 10 * q^88 - 10 * q^89 + 4 * q^90 - 16 * q^91 + 16 * q^92 - 6 * q^93 - 17 * q^94 + 20 * q^95 - 9 * q^96 - 6 * q^97 - 22 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	−0.618034	−0.252311
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	1.61803	0.467086
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0.145898	0.0389929
$15$	−3.23607	−0.835549
$16$	1.85410	0.463525
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0.618034	0.145672
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	−5.23607	−1.17082
$21$	−0.236068	−0.0515143
$22$	0.145898	0.0311056
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	2.23607	0.456435
$25$	5.47214	1.09443
$26$	−3.38197	−0.663258
$27$	−1.00000	−0.192450
$28$	−0.381966	−0.0721848
$29$	0	0
$29$	0	0
$30$	−2.00000	−0.365148
$31$	−3.70820	−0.666013	−0.333007	−	0.942925i	$-0.608063\pi$
$31$	−3.70820	−0.666013	−0.333007	+	0.942925i	$0.608063\pi$
$32$	5.61803	0.993137
$33$	−0.236068	−0.0410942
$34$	−1.85410	−0.317976
$35$	0.763932	0.129128
$36$	−1.61803	−0.269672
$37$	−5.23607	−0.860804	−0.430402	−	0.902637i	$-0.641628\pi$
$37$	−5.23607	−0.860804	−0.430402	+	0.902637i	$0.641628\pi$
$38$	4.47214	0.725476
$39$	5.47214	0.876243
$40$	−7.23607	−1.14412
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−0.145898	−0.0225126
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−0.381966	−0.0575835
$45$	3.23607	0.482405
$46$	−4.76393	−0.702403
$47$	−4.70820	−0.686762	−0.343381	−	0.939196i	$-0.611572\pi$
$47$	−4.70820	−0.686762	−0.343381	+	0.939196i	$0.611572\pi$
$48$	−1.85410	−0.267617
$49$	−6.94427	−0.992039
$50$	3.38197	0.478282
$51$	3.00000	0.420084
$52$	8.85410	1.22784
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	−0.618034	−0.0841038
$55$	0.763932	0.103009
$56$	−0.527864	−0.0705388
$57$	−7.23607	−0.958441
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	5.23607	0.675973
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	−2.29180	−0.291058
$63$	0.236068	0.0297418
$64$	−0.236068	−0.0295085
$65$	−17.7082	−2.19643
$66$	−0.145898	−0.0179588
$67$	−13.1803	−1.61023	−0.805117	−	0.593115i	$-0.797898\pi$
$67$	−13.1803	−1.61023	−0.805117	+	0.593115i	$0.797898\pi$
$68$	4.85410	0.588646
$69$	7.70820	0.927959
$70$	0.472136	0.0564310
$71$	−5.23607	−0.621407	−0.310703	−	0.950507i	$-0.600565\pi$
$71$	−5.23607	−0.621407	−0.310703	+	0.950507i	$0.600565\pi$
$72$	−2.23607	−0.263523
$73$	−6.76393	−0.791658	−0.395829	−	0.918324i	$-0.629543\pi$
$73$	−6.76393	−0.791658	−0.395829	+	0.918324i	$0.629543\pi$
$74$	−3.23607	−0.376185
$75$	−5.47214	−0.631868
$76$	−11.7082	−1.34302
$77$	0.0557281	0.00635081
$78$	3.38197	0.382932
$79$	12.7639	1.43605	0.718027	−	0.696015i	$-0.245045\pi$
$79$	12.7639	1.43605	0.718027	+	0.696015i	$0.245045\pi$
$80$	6.00000	0.670820
$81$	1.00000	0.111111
$82$	−1.23607	−0.136501
$83$	2.94427	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$83$	2.94427	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$84$	0.381966	0.0416759
$85$	−9.70820	−1.05300
$86$	−2.47214	−0.266577
$87$	0	0
$88$	−0.527864	−0.0562705
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	2.00000	0.210819
$91$	−1.29180	−0.135417
$92$	12.4721	1.30031
$93$	3.70820	0.384523
$94$	−2.90983	−0.300126
$95$	23.4164	2.40247
$96$	−5.61803	−0.573388
$97$	−18.6525	−1.89387	−0.946936	−	0.321422i	$-0.895839\pi$
$97$	−18.6525	−1.89387	−0.946936	+	0.321422i	$0.895839\pi$
$98$	−4.29180	−0.433537
$99$	0.236068	0.0237257
$100$	−8.85410	−0.885410
$101$	7.47214	0.743505	0.371753	−	0.928332i	$-0.378757\pi$
$101$	7.47214	0.743505	0.371753	+	0.928332i	$0.378757\pi$
$102$	1.85410	0.183583
$103$	−4.94427	−0.487174	−0.243587	−	0.969879i	$-0.578324\pi$
$103$	−4.94427	−0.487174	−0.243587	+	0.969879i	$0.578324\pi$
$104$	12.2361	1.19985
$105$	−0.763932	−0.0745521
$106$	6.94427	0.674487
$107$	9.70820	0.938527	0.469264	−	0.883058i	$-0.344519\pi$
$107$	9.70820	0.938527	0.469264	+	0.883058i	$0.344519\pi$
$108$	1.61803	0.155695
$109$	9.47214	0.907266	0.453633	−	0.891189i	$-0.350128\pi$
$109$	9.47214	0.907266	0.453633	+	0.891189i	$0.350128\pi$
$110$	0.472136	0.0450164
$111$	5.23607	0.496986
$112$	0.437694	0.0413582
$113$	−19.0000	−1.78737	−0.893685	−	0.448695i	$-0.851889\pi$
$113$	−19.0000	−1.78737	−0.893685	+	0.448695i	$0.851889\pi$
$114$	−4.47214	−0.418854
$115$	−24.9443	−2.32607
$116$	0	0
$117$	−5.47214	−0.505899
$118$	2.76393	0.254441
$119$	−0.708204	−0.0649209
$120$	7.23607	0.660560
$121$	−10.9443	−0.994934
$122$	3.23607	0.292980
$123$	2.00000	0.180334
$124$	6.00000	0.538816
$125$	1.52786	0.136656
$126$	0.145898	0.0129976
$127$	−12.4721	−1.10672	−0.553362	−	0.832941i	$-0.686655\pi$
$127$	−12.4721	−1.10672	−0.553362	+	0.832941i	$0.686655\pi$
$128$	−11.3820	−1.00603
$129$	4.00000	0.352180
$130$	−10.9443	−0.959876
$131$	−8.70820	−0.760839	−0.380420	−	0.924814i	$-0.624220\pi$
$131$	−8.70820	−0.760839	−0.380420	+	0.924814i	$0.624220\pi$
$132$	0.381966	0.0332459
$133$	1.70820	0.148120
$134$	−8.14590	−0.703698
$135$	−3.23607	−0.278516
$136$	6.70820	0.575224
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	4.76393	0.405533
$139$	1.18034	0.100115	0.0500576	−	0.998746i	$-0.484060\pi$
$139$	1.18034	0.100115	0.0500576	+	0.998746i	$0.484060\pi$
$140$	−1.23607	−0.104467
$141$	4.70820	0.396502
$142$	−3.23607	−0.271565
$143$	−1.29180	−0.108025
$144$	1.85410	0.154508
$145$	0	0
$146$	−4.18034	−0.345967
$147$	6.94427	0.572754
$148$	8.47214	0.696405
$149$	−23.4164	−1.91835	−0.959173	−	0.282819i	$-0.908731\pi$
$149$	−23.4164	−1.91835	−0.959173	+	0.282819i	$0.908731\pi$
$150$	−3.38197	−0.276136
$151$	3.05573	0.248672	0.124336	−	0.992240i	$-0.460320\pi$
$151$	3.05573	0.248672	0.124336	+	0.992240i	$0.460320\pi$
$152$	−16.1803	−1.31240
$153$	−3.00000	−0.242536
$154$	0.0344419	0.00277540
$155$	−12.0000	−0.963863
$156$	−8.85410	−0.708896
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	7.88854	0.627579
$159$	−11.2361	−0.891078
$160$	18.1803	1.43728
$161$	−1.81966	−0.143409
$162$	0.618034	0.0485573
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	3.23607	0.252694
$165$	−0.763932	−0.0594720
$166$	1.81966	0.141233
$167$	−6.47214	−0.500829	−0.250414	−	0.968139i	$-0.580567\pi$
$167$	−6.47214	−0.500829	−0.250414	+	0.968139i	$0.580567\pi$
$168$	0.527864	0.0407256
$169$	16.9443	1.30341
$170$	−6.00000	−0.460179
$171$	7.23607	0.553356
$172$	6.47214	0.493496
$173$	−7.05573	−0.536437	−0.268219	−	0.963358i	$-0.586435\pi$
$173$	−7.05573	−0.536437	−0.268219	+	0.963358i	$0.586435\pi$
$174$	0	0
$175$	1.29180	0.0976506
$176$	0.437694	0.0329924
$177$	−4.47214	−0.336146
$178$	−3.09017	−0.231618
$179$	17.2361	1.28828	0.644142	−	0.764906i	$-0.277215\pi$
$179$	17.2361	1.28828	0.644142	+	0.764906i	$0.277215\pi$
$180$	−5.23607	−0.390273
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	−0.798374	−0.0591794
$183$	−5.23607	−0.387061
$184$	17.2361	1.27066
$185$	−16.9443	−1.24577
$186$	2.29180	0.168043
$187$	−0.708204	−0.0517890
$188$	7.61803	0.555602
$189$	−0.236068	−0.0171714
$190$	14.4721	1.04992
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0.236068	0.0170367
$193$	−8.47214	−0.609838	−0.304919	−	0.952378i	$-0.598629\pi$
$193$	−8.47214	−0.609838	−0.304919	+	0.952378i	$0.598629\pi$
$194$	−11.5279	−0.827652
$195$	17.7082	1.26811
$196$	11.2361	0.802576
$197$	−19.2361	−1.37051	−0.685257	−	0.728302i	$-0.740310\pi$
$197$	−19.2361	−1.37051	−0.685257	+	0.728302i	$0.740310\pi$
$198$	0.145898	0.0103685
$199$	−15.6525	−1.10957	−0.554787	−	0.831992i	$-0.687200\pi$
$199$	−15.6525	−1.10957	−0.554787	+	0.831992i	$0.687200\pi$
$200$	−12.2361	−0.865221
$201$	13.1803	0.929669
$202$	4.61803	0.324924
$203$	0	0
$204$	−4.85410	−0.339855
$205$	−6.47214	−0.452034
$206$	−3.05573	−0.212903
$207$	−7.70820	−0.535757
$208$	−10.1459	−0.703491
$209$	1.70820	0.118159
$210$	−0.472136	−0.0325805
$211$	−10.9443	−0.753435	−0.376717	−	0.926328i	$-0.622947\pi$
$211$	−10.9443	−0.753435	−0.376717	+	0.926328i	$0.622947\pi$
$212$	−18.1803	−1.24863
$213$	5.23607	0.358769
$214$	6.00000	0.410152
$215$	−12.9443	−0.882792
$216$	2.23607	0.152145
$217$	−0.875388	−0.0594252
$218$	5.85410	0.396490
$219$	6.76393	0.457064
$220$	−1.23607	−0.0833357
$221$	16.4164	1.10429
$222$	3.23607	0.217191
$223$	15.1803	1.01655	0.508275	−	0.861195i	$-0.330283\pi$
$223$	15.1803	1.01655	0.508275	+	0.861195i	$0.330283\pi$
$224$	1.32624	0.0886130
$225$	5.47214	0.364809
$226$	−11.7426	−0.781109
$227$	−10.9443	−0.726397	−0.363198	−	0.931712i	$-0.618315\pi$
$227$	−10.9443	−0.726397	−0.363198	+	0.931712i	$0.618315\pi$
$228$	11.7082	0.775395
$229$	16.1803	1.06923	0.534613	−	0.845097i	$-0.320457\pi$
$229$	16.1803	1.06923	0.534613	+	0.845097i	$0.320457\pi$
$230$	−15.4164	−1.01653
$231$	−0.0557281	−0.00366664
$232$	0	0
$233$	17.4164	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$233$	17.4164	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$234$	−3.38197	−0.221086
$235$	−15.2361	−0.993891
$236$	−7.23607	−0.471028
$237$	−12.7639	−0.829106
$238$	−0.437694	−0.0283715
$239$	19.5967	1.26761	0.633804	−	0.773494i	$-0.281493\pi$
$239$	19.5967	1.26761	0.633804	+	0.773494i	$0.281493\pi$
$240$	−6.00000	−0.387298
$241$	10.4164	0.670980	0.335490	−	0.942044i	$-0.391098\pi$
$241$	10.4164	0.670980	0.335490	+	0.942044i	$0.391098\pi$
$242$	−6.76393	−0.434802
$243$	−1.00000	−0.0641500
$244$	−8.47214	−0.542373
$245$	−22.4721	−1.43569
$246$	1.23607	0.0788088
$247$	−39.5967	−2.51948
$248$	8.29180	0.526530
$249$	−2.94427	−0.186586
$250$	0.944272	0.0597210
$251$	9.18034	0.579458	0.289729	−	0.957109i	$-0.406435\pi$
$251$	9.18034	0.579458	0.289729	+	0.957109i	$0.406435\pi$
$252$	−0.381966	−0.0240616
$253$	−1.81966	−0.114401
$254$	−7.70820	−0.483656
$255$	9.70820	0.607951
$256$	−6.56231	−0.410144
$257$	−15.4164	−0.961649	−0.480825	−	0.876817i	$-0.659663\pi$
$257$	−15.4164	−0.961649	−0.480825	+	0.876817i	$0.659663\pi$
$258$	2.47214	0.153908
$259$	−1.23607	−0.0768055
$260$	28.6525	1.77695
$261$	0	0
$262$	−5.38197	−0.332499
$263$	22.8328	1.40793	0.703966	−	0.710234i	$-0.251411\pi$
$263$	22.8328	1.40793	0.703966	+	0.710234i	$0.251411\pi$
$264$	0.527864	0.0324878
$265$	36.3607	2.23362
$266$	1.05573	0.0647308
$267$	5.00000	0.305995
$268$	21.3262	1.30271
$269$	5.00000	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$270$	−2.00000	−0.121716
$271$	26.9443	1.63675	0.818374	−	0.574686i	$-0.194876\pi$
$271$	26.9443	1.63675	0.818374	+	0.574686i	$0.194876\pi$
$272$	−5.56231	−0.337264
$273$	1.29180	0.0781831
$274$	−2.18034	−0.131719
$275$	1.29180	0.0778982
$276$	−12.4721	−0.750734
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	0.729490	0.0437519
$279$	−3.70820	−0.222004
$280$	−1.70820	−0.102085
$281$	−21.4164	−1.27760	−0.638798	−	0.769375i	$-0.720568\pi$
$281$	−21.4164	−1.27760	−0.638798	+	0.769375i	$0.720568\pi$
$282$	2.90983	0.173278
$283$	7.41641	0.440860	0.220430	−	0.975403i	$-0.429254\pi$
$283$	7.41641	0.440860	0.220430	+	0.975403i	$0.429254\pi$
$284$	8.47214	0.502729
$285$	−23.4164	−1.38707
$286$	−0.798374	−0.0472088
$287$	−0.472136	−0.0278693
$288$	5.61803	0.331046
$289$	−8.00000	−0.470588
$290$	0	0
$291$	18.6525	1.09343
$292$	10.9443	0.640465
$293$	19.9443	1.16516	0.582578	−	0.812775i	$-0.302044\pi$
$293$	19.9443	1.16516	0.582578	+	0.812775i	$0.302044\pi$
$294$	4.29180	0.250303
$295$	14.4721	0.842600
$296$	11.7082	0.680526
$297$	−0.236068	−0.0136981
$298$	−14.4721	−0.838348
$299$	42.1803	2.43935
$300$	8.85410	0.511192
$301$	−0.944272	−0.0544269
$302$	1.88854	0.108673
$303$	−7.47214	−0.429263
$304$	13.4164	0.769484
$305$	16.9443	0.970226
$306$	−1.85410	−0.105992
$307$	12.6525	0.722115	0.361057	−	0.932544i	$-0.382416\pi$
$307$	12.6525	0.722115	0.361057	+	0.932544i	$0.382416\pi$
$308$	−0.0901699	−0.00513791
$309$	4.94427	0.281270
$310$	−7.41641	−0.421224
$311$	−28.7082	−1.62789	−0.813946	−	0.580940i	$-0.802685\pi$
$311$	−28.7082	−1.62789	−0.813946	+	0.580940i	$0.802685\pi$
$312$	−12.2361	−0.692731
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	1.23607	0.0697554
$315$	0.763932	0.0430427
$316$	−20.6525	−1.16179
$317$	1.47214	0.0826834	0.0413417	−	0.999145i	$-0.486837\pi$
$317$	1.47214	0.0826834	0.0413417	+	0.999145i	$0.486837\pi$
$318$	−6.94427	−0.389415
$319$	0	0
$320$	−0.763932	−0.0427051
$321$	−9.70820	−0.541859
$322$	−1.12461	−0.0626722
$323$	−21.7082	−1.20788
$324$	−1.61803	−0.0898908
$325$	−29.9443	−1.66101
$326$	3.70820	0.205378
$327$	−9.47214	−0.523810
$328$	4.47214	0.246932
$329$	−1.11146	−0.0612766
$330$	−0.472136	−0.0259902
$331$	20.7639	1.14129	0.570644	−	0.821197i	$-0.306693\pi$
$331$	20.7639	1.14129	0.570644	+	0.821197i	$0.306693\pi$
$332$	−4.76393	−0.261455
$333$	−5.23607	−0.286935
$334$	−4.00000	−0.218870
$335$	−42.6525	−2.33035
$336$	−0.437694	−0.0238782
$337$	8.18034	0.445612	0.222806	−	0.974863i	$-0.428478\pi$
$337$	8.18034	0.445612	0.222806	+	0.974863i	$0.428478\pi$
$338$	10.4721	0.569609
$339$	19.0000	1.03194
$340$	15.7082	0.851897
$341$	−0.875388	−0.0474049
$342$	4.47214	0.241825
$343$	−3.29180	−0.177740
$344$	8.94427	0.482243
$345$	24.9443	1.34295
$346$	−4.36068	−0.234432
$347$	25.2361	1.35474	0.677372	−	0.735641i	$-0.263119\pi$
$347$	25.2361	1.35474	0.677372	+	0.735641i	$0.263119\pi$
$348$	0	0
$349$	−13.4164	−0.718164	−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	0.798374	0.0426749
$351$	5.47214	0.292081
$352$	1.32624	0.0706887
$353$	−3.23607	−0.172239	−0.0861193	−	0.996285i	$-0.527447\pi$
$353$	−3.23607	−0.172239	−0.0861193	+	0.996285i	$0.527447\pi$
$354$	−2.76393	−0.146901
$355$	−16.9443	−0.899309
$356$	8.09017	0.428778
$357$	0.708204	0.0374821
$358$	10.6525	0.563001
$359$	14.4721	0.763810	0.381905	−	0.924202i	$-0.375268\pi$
$359$	14.4721	0.763810	0.381905	+	0.924202i	$0.375268\pi$
$360$	−7.23607	−0.381374
$361$	33.3607	1.75583
$362$	−1.85410	−0.0974494
$363$	10.9443	0.574425
$364$	2.09017	0.109555
$365$	−21.8885	−1.14570
$366$	−3.23607	−0.169152
$367$	−13.5279	−0.706149	−0.353074	−	0.935595i	$-0.614864\pi$
$367$	−13.5279	−0.706149	−0.353074	+	0.935595i	$0.614864\pi$
$368$	−14.2918	−0.745011
$369$	−2.00000	−0.104116
$370$	−10.4721	−0.544420
$371$	2.65248	0.137710
$372$	−6.00000	−0.311086
$373$	−11.5279	−0.596890	−0.298445	−	0.954427i	$-0.596468\pi$
$373$	−11.5279	−0.596890	−0.298445	+	0.954427i	$0.596468\pi$
$374$	−0.437694	−0.0226326
$375$	−1.52786	−0.0788986
$376$	10.5279	0.542933
$377$	0	0
$378$	−0.145898	−0.00750419
$379$	−1.70820	−0.0877445	−0.0438723	−	0.999037i	$-0.513969\pi$
$379$	−1.70820	−0.0877445	−0.0438723	+	0.999037i	$0.513969\pi$
$380$	−37.8885	−1.94364
$381$	12.4721	0.638967
$382$	−7.41641	−0.379456
$383$	11.2361	0.574136	0.287068	−	0.957910i	$-0.407319\pi$
$383$	11.2361	0.574136	0.287068	+	0.957910i	$0.407319\pi$
$384$	11.3820	0.580834
$385$	0.180340	0.00919097
$386$	−5.23607	−0.266509
$387$	−4.00000	−0.203331
$388$	30.1803	1.53217
$389$	−17.3607	−0.880221	−0.440111	−	0.897944i	$-0.645061\pi$
$389$	−17.3607	−0.880221	−0.440111	+	0.897944i	$0.645061\pi$
$390$	10.9443	0.554185
$391$	23.1246	1.16946
$392$	15.5279	0.784276
$393$	8.70820	0.439271
$394$	−11.8885	−0.598936
$395$	41.3050	2.07828
$396$	−0.381966	−0.0191945
$397$	6.94427	0.348523	0.174262	−	0.984699i	$-0.444246\pi$
$397$	6.94427	0.348523	0.174262	+	0.984699i	$0.444246\pi$
$398$	−9.67376	−0.484902
$399$	−1.70820	−0.0855172
$400$	10.1459	0.507295
$401$	28.1803	1.40726	0.703630	−	0.710567i	$-0.251561\pi$
$401$	28.1803	1.40726	0.703630	+	0.710567i	$0.251561\pi$
$402$	8.14590	0.406280
$403$	20.2918	1.01081
$404$	−12.0902	−0.601508
$405$	3.23607	0.160802
$406$	0	0
$407$	−1.23607	−0.0612696
$408$	−6.70820	−0.332106
$409$	4.47214	0.221133	0.110566	−	0.993869i	$-0.464734\pi$
$409$	4.47214	0.221133	0.110566	+	0.993869i	$0.464734\pi$
$410$	−4.00000	−0.197546
$411$	3.52786	0.174017
$412$	8.00000	0.394132
$413$	1.05573	0.0519490
$414$	−4.76393	−0.234134
$415$	9.52786	0.467704
$416$	−30.7426	−1.50728
$417$	−1.18034	−0.0578015
$418$	1.05573	0.0516373
$419$	27.2361	1.33057	0.665284	−	0.746590i	$-0.268310\pi$
$419$	27.2361	1.33057	0.665284	+	0.746590i	$0.268310\pi$
$420$	1.23607	0.0603139
$421$	35.8885	1.74910	0.874550	−	0.484935i	$-0.161157\pi$
$421$	35.8885	1.74910	0.874550	+	0.484935i	$0.161157\pi$
$422$	−6.76393	−0.329263
$423$	−4.70820	−0.228921
$424$	−25.1246	−1.22016
$425$	−16.4164	−0.796313
$426$	3.23607	0.156788
$427$	1.23607	0.0598175
$428$	−15.7082	−0.759285
$429$	1.29180	0.0623685
$430$	−8.00000	−0.385794
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.85410	−0.0892055
$433$	6.65248	0.319698	0.159849	−	0.987142i	$-0.448899\pi$
$433$	6.65248	0.319698	0.159849	+	0.987142i	$0.448899\pi$
$434$	−0.541020	−0.0259698
$435$	0	0
$436$	−15.3262	−0.733994
$437$	−55.7771	−2.66818
$438$	4.18034	0.199744
$439$	13.2918	0.634383	0.317191	−	0.948362i	$-0.397260\pi$
$439$	13.2918	0.634383	0.317191	+	0.948362i	$0.397260\pi$
$440$	−1.70820	−0.0814354
$441$	−6.94427	−0.330680
$442$	10.1459	0.482591
$443$	3.76393	0.178830	0.0894149	−	0.995994i	$-0.471500\pi$
$443$	3.76393	0.178830	0.0894149	+	0.995994i	$0.471500\pi$
$444$	−8.47214	−0.402070
$445$	−16.1803	−0.767022
$446$	9.38197	0.444249
$447$	23.4164	1.10756
$448$	−0.0557281	−0.00263290
$449$	19.4721	0.918947	0.459473	−	0.888192i	$-0.348038\pi$
$449$	19.4721	0.918947	0.459473	+	0.888192i	$0.348038\pi$
$450$	3.38197	0.159427
$451$	−0.472136	−0.0222320
$452$	30.7426	1.44601
$453$	−3.05573	−0.143571
$454$	−6.76393	−0.317447
$455$	−4.18034	−0.195977
$456$	16.1803	0.757714
$457$	−1.47214	−0.0688636	−0.0344318	−	0.999407i	$-0.510962\pi$
$457$	−1.47214	−0.0688636	−0.0344318	+	0.999407i	$0.510962\pi$
$458$	10.0000	0.467269
$459$	3.00000	0.140028
$460$	40.3607	1.88183
$461$	15.8885	0.740003	0.370002	−	0.929031i	$-0.379357\pi$
$461$	15.8885	0.740003	0.370002	+	0.929031i	$0.379357\pi$
$462$	−0.0344419	−0.00160238
$463$	15.1803	0.705490	0.352745	−	0.935719i	$-0.385248\pi$
$463$	15.1803	0.705490	0.352745	+	0.935719i	$0.385248\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	10.7639	0.498630
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	8.85410	0.409281
$469$	−3.11146	−0.143674
$470$	−9.41641	−0.434347
$471$	−2.00000	−0.0921551
$472$	−10.0000	−0.460287
$473$	−0.944272	−0.0434177
$474$	−7.88854	−0.362333
$475$	39.5967	1.81682
$476$	1.14590	0.0525222
$477$	11.2361	0.514464
$478$	12.1115	0.553965
$479$	−14.4721	−0.661249	−0.330624	−	0.943762i	$-0.607259\pi$
$479$	−14.4721	−0.661249	−0.330624	+	0.943762i	$0.607259\pi$
$480$	−18.1803	−0.829815
$481$	28.6525	1.30644
$482$	6.43769	0.293229
$483$	1.81966	0.0827974
$484$	17.7082	0.804918
$485$	−60.3607	−2.74084
$486$	−0.618034	−0.0280346
$487$	36.9443	1.67410	0.837052	−	0.547123i	$-0.184277\pi$
$487$	36.9443	1.67410	0.837052	+	0.547123i	$0.184277\pi$
$488$	−11.7082	−0.530005
$489$	−6.00000	−0.271329
$490$	−13.8885	−0.627420
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	−3.23607	−0.145893
$493$	0	0
$494$	−24.4721	−1.10105
$495$	0.763932	0.0343362
$496$	−6.87539	−0.308714
$497$	−1.23607	−0.0554452
$498$	−1.81966	−0.0815409
$499$	3.29180	0.147361	0.0736805	−	0.997282i	$-0.476525\pi$
$499$	3.29180	0.147361	0.0736805	+	0.997282i	$0.476525\pi$
$500$	−2.47214	−0.110557
$501$	6.47214	0.289154
$502$	5.67376	0.253232
$503$	−27.5410	−1.22799	−0.613997	−	0.789309i	$-0.710439\pi$
$503$	−27.5410	−1.22799	−0.613997	+	0.789309i	$0.710439\pi$
$504$	−0.527864	−0.0235129
$505$	24.1803	1.07601
$506$	−1.12461	−0.0499951
$507$	−16.9443	−0.752522
$508$	20.1803	0.895358
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	6.00000	0.265684
$511$	−1.59675	−0.0706360
$512$	18.7082	0.826794
$513$	−7.23607	−0.319480
$514$	−9.52786	−0.420256
$515$	−16.0000	−0.705044
$516$	−6.47214	−0.284920
$517$	−1.11146	−0.0488818
$518$	−0.763932	−0.0335652
$519$	7.05573	0.309712
$520$	39.5967	1.73643
$521$	−31.4164	−1.37638	−0.688189	−	0.725532i	$-0.741594\pi$
$521$	−31.4164	−1.37638	−0.688189	+	0.725532i	$0.741594\pi$
$522$	0	0
$523$	14.1246	0.617626	0.308813	−	0.951123i	$-0.400068\pi$
$523$	14.1246	0.617626	0.308813	+	0.951123i	$0.400068\pi$
$524$	14.0902	0.615532
$525$	−1.29180	−0.0563786
$526$	14.1115	0.615289
$527$	11.1246	0.484596
$528$	−0.437694	−0.0190482
$529$	36.4164	1.58332
$530$	22.4721	0.976127
$531$	4.47214	0.194074
$532$	−2.76393	−0.119832
$533$	10.9443	0.474049
$534$	3.09017	0.133725
$535$	31.4164	1.35825
$536$	29.4721	1.27300
$537$	−17.2361	−0.743791
$538$	3.09017	0.133227
$539$	−1.63932	−0.0706105
$540$	5.23607	0.225324
$541$	−8.18034	−0.351700	−0.175850	−	0.984417i	$-0.556267\pi$
$541$	−8.18034	−0.351700	−0.175850	+	0.984417i	$0.556267\pi$
$542$	16.6525	0.715285
$543$	3.00000	0.128742
$544$	−16.8541	−0.722614
$545$	30.6525	1.31301
$546$	0.798374	0.0341672
$547$	3.65248	0.156169	0.0780843	−	0.996947i	$-0.475120\pi$
$547$	3.65248	0.156169	0.0780843	+	0.996947i	$0.475120\pi$
$548$	5.70820	0.243842
$549$	5.23607	0.223470
$550$	0.798374	0.0340428
$551$	0	0
$552$	−17.2361	−0.733616
$553$	3.01316	0.128132
$554$	1.85410	0.0787732
$555$	16.9443	0.719244
$556$	−1.90983	−0.0809948
$557$	−25.4164	−1.07693	−0.538464	−	0.842649i	$-0.680995\pi$
$557$	−25.4164	−1.07693	−0.538464	+	0.842649i	$0.680995\pi$
$558$	−2.29180	−0.0970195
$559$	21.8885	0.925787
$560$	1.41641	0.0598542
$561$	0.708204	0.0299004
$562$	−13.2361	−0.558330
$563$	45.0689	1.89943	0.949713	−	0.313120i	$-0.101374\pi$
$563$	45.0689	1.89943	0.949713	+	0.313120i	$0.101374\pi$
$564$	−7.61803	−0.320777
$565$	−61.4853	−2.58671
$566$	4.58359	0.192663
$567$	0.236068	0.00991392
$568$	11.7082	0.491265
$569$	−35.2492	−1.47772	−0.738862	−	0.673857i	$-0.764637\pi$
$569$	−35.2492	−1.47772	−0.738862	+	0.673857i	$0.764637\pi$
$570$	−14.4721	−0.606171
$571$	15.4164	0.645157	0.322578	−	0.946543i	$-0.395450\pi$
$571$	15.4164	0.645157	0.322578	+	0.946543i	$0.395450\pi$
$572$	2.09017	0.0873944
$573$	12.0000	0.501307
$574$	−0.291796	−0.0121793
$575$	−42.1803	−1.75904
$576$	−0.236068	−0.00983617
$577$	40.9443	1.70453	0.852266	−	0.523108i	$-0.175228\pi$
$577$	40.9443	1.70453	0.852266	+	0.523108i	$0.175228\pi$
$578$	−4.94427	−0.205655
$579$	8.47214	0.352090
$580$	0	0
$581$	0.695048	0.0288355
$582$	11.5279	0.477845
$583$	2.65248	0.109854
$584$	15.1246	0.625861
$585$	−17.7082	−0.732144
$586$	12.3262	0.509192
$587$	−5.41641	−0.223559	−0.111780	−	0.993733i	$-0.535655\pi$
$587$	−5.41641	−0.223559	−0.111780	+	0.993733i	$0.535655\pi$
$588$	−11.2361	−0.463368
$589$	−26.8328	−1.10563
$590$	8.94427	0.368230
$591$	19.2361	0.791266
$592$	−9.70820	−0.399005
$593$	−15.5967	−0.640482	−0.320241	−	0.947336i	$-0.603764\pi$
$593$	−15.5967	−0.640482	−0.320241	+	0.947336i	$0.603764\pi$
$594$	−0.145898	−0.00598627
$595$	−2.29180	−0.0939545
$596$	37.8885	1.55198
$597$	15.6525	0.640613
$598$	26.0689	1.06604
$599$	−33.2918	−1.36027	−0.680133	−	0.733089i	$-0.738078\pi$
$599$	−33.2918	−1.36027	−0.680133	+	0.733089i	$0.738078\pi$
$600$	12.2361	0.499535
$601$	−8.18034	−0.333683	−0.166842	−	0.985984i	$-0.553357\pi$
$601$	−8.18034	−0.333683	−0.166842	+	0.985984i	$0.553357\pi$
$602$	−0.583592	−0.0237854
$603$	−13.1803	−0.536745
$604$	−4.94427	−0.201180
$605$	−35.4164	−1.43988
$606$	−4.61803	−0.187595
$607$	−1.41641	−0.0574902	−0.0287451	−	0.999587i	$-0.509151\pi$
$607$	−1.41641	−0.0574902	−0.0287451	+	0.999587i	$0.509151\pi$
$608$	40.6525	1.64868
$609$	0	0
$610$	10.4721	0.424004
$611$	25.7639	1.04230
$612$	4.85410	0.196215
$613$	5.58359	0.225519	0.112760	−	0.993622i	$-0.464031\pi$
$613$	5.58359	0.225519	0.112760	+	0.993622i	$0.464031\pi$
$614$	7.81966	0.315576
$615$	6.47214	0.260982
$616$	−0.124612	−0.00502075
$617$	−12.4721	−0.502109	−0.251055	−	0.967973i	$-0.580777\pi$
$617$	−12.4721	−0.502109	−0.251055	+	0.967973i	$0.580777\pi$
$618$	3.05573	0.122919
$619$	1.70820	0.0686585	0.0343293	−	0.999411i	$-0.489071\pi$
$619$	1.70820	0.0686585	0.0343293	+	0.999411i	$0.489071\pi$
$620$	19.4164	0.779782
$621$	7.70820	0.309320
$622$	−17.7426	−0.711415
$623$	−1.18034	−0.0472893
$624$	10.1459	0.406161
$625$	−22.4164	−0.896656
$626$	−6.79837	−0.271718
$627$	−1.70820	−0.0682191
$628$	−3.23607	−0.129133
$629$	15.7082	0.626327
$630$	0.472136	0.0188103
$631$	−23.6525	−0.941590	−0.470795	−	0.882243i	$-0.656033\pi$
$631$	−23.6525	−0.941590	−0.470795	+	0.882243i	$0.656033\pi$
$632$	−28.5410	−1.13530
$633$	10.9443	0.434996
$634$	0.909830	0.0361340
$635$	−40.3607	−1.60166
$636$	18.1803	0.720897
$637$	38.0000	1.50561
$638$	0	0
$639$	−5.23607	−0.207136
$640$	−36.8328	−1.45594
$641$	−38.3050	−1.51295	−0.756477	−	0.654020i	$-0.773081\pi$
$641$	−38.3050	−1.51295	−0.756477	+	0.654020i	$0.773081\pi$
$642$	−6.00000	−0.236801
$643$	0.708204	0.0279288	0.0139644	−	0.999902i	$-0.495555\pi$
$643$	0.708204	0.0279288	0.0139644	+	0.999902i	$0.495555\pi$
$644$	2.94427	0.116021
$645$	12.9443	0.509680
$646$	−13.4164	−0.527862
$647$	14.1803	0.557487	0.278743	−	0.960366i	$-0.410082\pi$
$647$	14.1803	0.557487	0.278743	+	0.960366i	$0.410082\pi$
$648$	−2.23607	−0.0878410
$649$	1.05573	0.0414410
$650$	−18.5066	−0.725888
$651$	0.875388	0.0343092
$652$	−9.70820	−0.380203
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	−5.85410	−0.228914
$655$	−28.1803	−1.10110
$656$	−3.70820	−0.144781
$657$	−6.76393	−0.263886
$658$	−0.686918	−0.0267788
$659$	−24.5967	−0.958153	−0.479077	−	0.877773i	$-0.659028\pi$
$659$	−24.5967	−0.958153	−0.479077	+	0.877773i	$0.659028\pi$
$660$	1.23607	0.0481139
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	12.8328	0.498762
$663$	−16.4164	−0.637560
$664$	−6.58359	−0.255493
$665$	5.52786	0.214361
$666$	−3.23607	−0.125395
$667$	0	0
$668$	10.4721	0.405179
$669$	−15.1803	−0.586906
$670$	−26.3607	−1.01840
$671$	1.23607	0.0477179
$672$	−1.32624	−0.0511607
$673$	30.3050	1.16817	0.584085	−	0.811692i	$-0.301453\pi$
$673$	30.3050	1.16817	0.584085	+	0.811692i	$0.301453\pi$
$674$	5.05573	0.194739
$675$	−5.47214	−0.210623
$676$	−27.4164	−1.05448
$677$	0.416408	0.0160039	0.00800193	−	0.999968i	$-0.497453\pi$
$677$	0.416408	0.0160039	0.00800193	+	0.999968i	$0.497453\pi$
$678$	11.7426	0.450974
$679$	−4.40325	−0.168981
$680$	21.7082	0.832472
$681$	10.9443	0.419385
$682$	−0.541020	−0.0207167
$683$	10.8328	0.414506	0.207253	−	0.978287i	$-0.433548\pi$
$683$	10.8328	0.414506	0.207253	+	0.978287i	$0.433548\pi$
$684$	−11.7082	−0.447674
$685$	−11.4164	−0.436199
$686$	−2.03444	−0.0776754
$687$	−16.1803	−0.617318
$688$	−7.41641	−0.282748
$689$	−61.4853	−2.34240
$690$	15.4164	0.586893
$691$	−32.5967	−1.24004	−0.620019	−	0.784587i	$-0.712875\pi$
$691$	−32.5967	−1.24004	−0.620019	+	0.784587i	$0.712875\pi$
$692$	11.4164	0.433987
$693$	0.0557281	0.00211694
$694$	15.5967	0.592044
$695$	3.81966	0.144888
$696$	0	0
$697$	6.00000	0.227266
$698$	−8.29180	−0.313849
$699$	−17.4164	−0.658749
$700$	−2.09017	−0.0790010
$701$	−26.5410	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$701$	−26.5410	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$702$	3.38197	0.127644
$703$	−37.8885	−1.42899
$704$	−0.0557281	−0.00210033
$705$	15.2361	0.573824
$706$	−2.00000	−0.0752710
$707$	1.76393	0.0663395
$708$	7.23607	0.271948
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	−10.4721	−0.393012
$711$	12.7639	0.478685
$712$	11.1803	0.419001
$713$	28.5836	1.07046
$714$	0.437694	0.0163803
$715$	−4.18034	−0.156336
$716$	−27.8885	−1.04224
$717$	−19.5967	−0.731854
$718$	8.94427	0.333797
$719$	−50.2492	−1.87398	−0.936990	−	0.349356i	$-0.886400\pi$
$719$	−50.2492	−1.87398	−0.936990	+	0.349356i	$0.886400\pi$
$720$	6.00000	0.223607
$721$	−1.16718	−0.0434682
$722$	20.6180	0.767324
$723$	−10.4164	−0.387390
$724$	4.85410	0.180401
$725$	0	0
$726$	6.76393	0.251033
$727$	14.7639	0.547564	0.273782	−	0.961792i	$-0.411725\pi$
$727$	14.7639	0.547564	0.273782	+	0.961792i	$0.411725\pi$
$728$	2.88854	0.107057
$729$	1.00000	0.0370370
$730$	−13.5279	−0.500689
$731$	12.0000	0.443836
$732$	8.47214	0.313139
$733$	−18.4721	−0.682284	−0.341142	−	0.940012i	$-0.610814\pi$
$733$	−18.4721	−0.682284	−0.341142	+	0.940012i	$0.610814\pi$
$734$	−8.36068	−0.308598
$735$	22.4721	0.828897
$736$	−43.3050	−1.59624
$737$	−3.11146	−0.114612
$738$	−1.23607	−0.0455003
$739$	−6.18034	−0.227347	−0.113674	−	0.993518i	$-0.536262\pi$
$739$	−6.18034	−0.227347	−0.113674	+	0.993518i	$0.536262\pi$
$740$	27.4164	1.00785
$741$	39.5967	1.45462
$742$	1.63932	0.0601813
$743$	10.3475	0.379614	0.189807	−	0.981821i	$-0.439214\pi$
$743$	10.3475	0.379614	0.189807	+	0.981821i	$0.439214\pi$
$744$	−8.29180	−0.303992
$745$	−75.7771	−2.77626
$746$	−7.12461	−0.260851
$747$	2.94427	0.107725
$748$	1.14590	0.0418982
$749$	2.29180	0.0837404
$750$	−0.944272	−0.0344799
$751$	43.7771	1.59745	0.798724	−	0.601697i	$-0.205509\pi$
$751$	43.7771	1.59745	0.798724	+	0.601697i	$0.205509\pi$
$752$	−8.72949	−0.318332
$753$	−9.18034	−0.334550
$754$	0	0
$755$	9.88854	0.359881
$756$	0.381966	0.0138920
$757$	20.9443	0.761233	0.380616	−	0.924733i	$-0.375712\pi$
$757$	20.9443	0.761233	0.380616	+	0.924733i	$0.375712\pi$
$758$	−1.05573	−0.0383458
$759$	1.81966	0.0660495
$760$	−52.3607	−1.89932
$761$	28.1803	1.02154	0.510768	−	0.859718i	$-0.329361\pi$
$761$	28.1803	1.02154	0.510768	+	0.859718i	$0.329361\pi$
$762$	7.70820	0.279239
$763$	2.23607	0.0809511
$764$	19.4164	0.702461
$765$	−9.70820	−0.351001
$766$	6.94427	0.250907
$767$	−24.4721	−0.883638
$768$	6.56231	0.236797
$769$	−17.2361	−0.621549	−0.310774	−	0.950484i	$-0.600588\pi$
$769$	−17.2361	−0.621549	−0.310774	+	0.950484i	$0.600588\pi$
$770$	0.111456	0.00401660
$771$	15.4164	0.555208
$772$	13.7082	0.493369
$773$	−28.4721	−1.02407	−0.512036	−	0.858964i	$-0.671108\pi$
$773$	−28.4721	−1.02407	−0.512036	+	0.858964i	$0.671108\pi$
$774$	−2.47214	−0.0888591
$775$	−20.2918	−0.728903
$776$	41.7082	1.49724
$777$	1.23607	0.0443437
$778$	−10.7295	−0.384671
$779$	−14.4721	−0.518518
$780$	−28.6525	−1.02592
$781$	−1.23607	−0.0442300
$782$	14.2918	0.511074
$783$	0	0
$784$	−12.8754	−0.459835
$785$	6.47214	0.231000
$786$	5.38197	0.191968
$787$	−46.4721	−1.65655	−0.828276	−	0.560320i	$-0.810678\pi$
$787$	−46.4721	−1.65655	−0.828276	+	0.560320i	$0.810678\pi$
$788$	31.1246	1.10877
$789$	−22.8328	−0.812870
$790$	25.5279	0.908241
$791$	−4.48529	−0.159479
$792$	−0.527864	−0.0187568
$793$	−28.6525	−1.01748
$794$	4.29180	0.152310
$795$	−36.3607	−1.28958
$796$	25.3262	0.897665
$797$	−9.05573	−0.320770	−0.160385	−	0.987055i	$-0.551274\pi$
$797$	−9.05573	−0.320770	−0.160385	+	0.987055i	$0.551274\pi$
$798$	−1.05573	−0.0373724
$799$	14.1246	0.499693
$800$	30.7426	1.08692
$801$	−5.00000	−0.176666
$802$	17.4164	0.614995
$803$	−1.59675	−0.0563480
$804$	−21.3262	−0.752118
$805$	−5.88854	−0.207544
$806$	12.5410	0.441739
$807$	−5.00000	−0.176008
$808$	−16.7082	−0.587793
$809$	−36.3050	−1.27641	−0.638207	−	0.769865i	$-0.720324\pi$
$809$	−36.3050	−1.27641	−0.638207	+	0.769865i	$0.720324\pi$
$810$	2.00000	0.0702728
$811$	−23.6525	−0.830551	−0.415275	−	0.909696i	$-0.636315\pi$
$811$	−23.6525	−0.830551	−0.415275	+	0.909696i	$0.636315\pi$
$812$	0	0
$813$	−26.9443	−0.944977
$814$	−0.763932	−0.0267758
$815$	19.4164	0.680127
$816$	5.56231	0.194720
$817$	−28.9443	−1.01263
$818$	2.76393	0.0966386
$819$	−1.29180	−0.0451390
$820$	10.4721	0.365703
$821$	25.4164	0.887039	0.443519	−	0.896265i	$-0.353730\pi$
$821$	25.4164	0.887039	0.443519	+	0.896265i	$0.353730\pi$
$822$	2.18034	0.0760481
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	11.0557	0.385145
$825$	−1.29180	−0.0449746
$826$	0.652476	0.0227025
$827$	−1.16718	−0.0405870	−0.0202935	−	0.999794i	$-0.506460\pi$
$827$	−1.16718	−0.0405870	−0.0202935	+	0.999794i	$0.506460\pi$
$828$	12.4721	0.433437
$829$	−30.2492	−1.05060	−0.525299	−	0.850917i	$-0.676047\pi$
$829$	−30.2492	−1.05060	−0.525299	+	0.850917i	$0.676047\pi$
$830$	5.88854	0.204394
$831$	−3.00000	−0.104069
$832$	1.29180	0.0447850
$833$	20.8328	0.721814
$834$	−0.729490	−0.0252602
$835$	−20.9443	−0.724806
$836$	−2.76393	−0.0955926
$837$	3.70820	0.128174
$838$	16.8328	0.581480
$839$	35.6525	1.23086	0.615430	−	0.788191i	$-0.288982\pi$
$839$	35.6525	1.23086	0.615430	+	0.788191i	$0.288982\pi$
$840$	1.70820	0.0589386
$841$	0	0
$842$	22.1803	0.764385
$843$	21.4164	0.737620
$844$	17.7082	0.609542
$845$	54.8328	1.88631
$846$	−2.90983	−0.100042
$847$	−2.58359	−0.0887733
$848$	20.8328	0.715402
$849$	−7.41641	−0.254530
$850$	−10.1459	−0.348001
$851$	40.3607	1.38355
$852$	−8.47214	−0.290251
$853$	−7.41641	−0.253933	−0.126966	−	0.991907i	$-0.540524\pi$
$853$	−7.41641	−0.253933	−0.126966	+	0.991907i	$0.540524\pi$
$854$	0.763932	0.0261412
$855$	23.4164	0.800824
$856$	−21.7082	−0.741971
$857$	−31.5967	−1.07932	−0.539662	−	0.841882i	$-0.681448\pi$
$857$	−31.5967	−1.07932	−0.539662	+	0.841882i	$0.681448\pi$
$858$	0.798374	0.0272560
$859$	36.8328	1.25672	0.628360	−	0.777923i	$-0.283727\pi$
$859$	36.8328	1.25672	0.628360	+	0.777923i	$0.283727\pi$
$860$	20.9443	0.714194
$861$	0.472136	0.0160904
$862$	−4.94427	−0.168403
$863$	37.0132	1.25994	0.629971	−	0.776618i	$-0.283067\pi$
$863$	37.0132	1.25994	0.629971	+	0.776618i	$0.283067\pi$
$864$	−5.61803	−0.191129
$865$	−22.8328	−0.776339
$866$	4.11146	0.139713
$867$	8.00000	0.271694
$868$	1.41641	0.0480760
$869$	3.01316	0.102214
$870$	0	0
$871$	72.1246	2.44385
$872$	−21.1803	−0.717257
$873$	−18.6525	−0.631291
$874$	−34.4721	−1.16604
$875$	0.360680	0.0121932
$876$	−10.9443	−0.369773
$877$	21.4164	0.723181	0.361590	−	0.932337i	$-0.382234\pi$
$877$	21.4164	0.723181	0.361590	+	0.932337i	$0.382234\pi$
$878$	8.21478	0.277235
$879$	−19.9443	−0.672704
$880$	1.41641	0.0477471
$881$	−22.5279	−0.758983	−0.379492	−	0.925195i	$-0.623901\pi$
$881$	−22.5279	−0.758983	−0.379492	+	0.925195i	$0.623901\pi$
$882$	−4.29180	−0.144512
$883$	27.4164	0.922636	0.461318	−	0.887235i	$-0.347377\pi$
$883$	27.4164	0.922636	0.461318	+	0.887235i	$0.347377\pi$
$884$	−26.5623	−0.893387
$885$	−14.4721	−0.486476
$886$	2.32624	0.0781515
$887$	−16.8197	−0.564749	−0.282374	−	0.959304i	$-0.591122\pi$
$887$	−16.8197	−0.564749	−0.282374	+	0.959304i	$0.591122\pi$
$888$	−11.7082	−0.392902
$889$	−2.94427	−0.0987477
$890$	−10.0000	−0.335201
$891$	0.236068	0.00790857
$892$	−24.5623	−0.822407
$893$	−34.0689	−1.14007
$894$	14.4721	0.484021
$895$	55.7771	1.86442
$896$	−2.68692	−0.0897636
$897$	−42.1803	−1.40836
$898$	12.0344	0.401595
$899$	0	0
$900$	−8.85410	−0.295137
$901$	−33.7082	−1.12298
$902$	−0.291796	−0.00971575
$903$	0.944272	0.0314234
$904$	42.4853	1.41304
$905$	−9.70820	−0.322712
$906$	−1.88854	−0.0627427
$907$	17.7771	0.590279	0.295139	−	0.955454i	$-0.404634\pi$
$907$	17.7771	0.590279	0.295139	+	0.955454i	$0.404634\pi$
$908$	17.7082	0.587667
$909$	7.47214	0.247835
$910$	−2.58359	−0.0856452
$911$	−10.8197	−0.358471	−0.179236	−	0.983806i	$-0.557362\pi$
$911$	−10.8197	−0.358471	−0.179236	+	0.983806i	$0.557362\pi$
$912$	−13.4164	−0.444262
$913$	0.695048	0.0230027
$914$	−0.909830	−0.0300945
$915$	−16.9443	−0.560160
$916$	−26.1803	−0.865023
$917$	−2.05573	−0.0678861
$918$	1.85410	0.0611945
$919$	58.0132	1.91368	0.956839	−	0.290619i	$-0.0938614\pi$
$919$	58.0132	1.91368	0.956839	+	0.290619i	$0.0938614\pi$
$920$	55.7771	1.83892
$921$	−12.6525	−0.416913
$922$	9.81966	0.323393
$923$	28.6525	0.943108
$924$	0.0901699	0.00296637
$925$	−28.6525	−0.942088
$926$	9.38197	0.308311
$927$	−4.94427	−0.162391
$928$	0	0
$929$	−35.5279	−1.16563	−0.582816	−	0.812604i	$-0.698049\pi$
$929$	−35.5279	−1.16563	−0.582816	+	0.812604i	$0.698049\pi$
$930$	7.41641	0.243194
$931$	−50.2492	−1.64685
$932$	−28.1803	−0.923078
$933$	28.7082	0.939864
$934$	7.41641	0.242672
$935$	−2.29180	−0.0749497
$936$	12.2361	0.399948
$937$	35.3607	1.15518	0.577592	−	0.816326i	$-0.303993\pi$
$937$	35.3607	1.15518	0.577592	+	0.816326i	$0.303993\pi$
$938$	−1.92299	−0.0627877
$939$	11.0000	0.358971
$940$	24.6525	0.804075
$941$	−10.1115	−0.329624	−0.164812	−	0.986325i	$-0.552702\pi$
$941$	−10.1115	−0.329624	−0.164812	+	0.986325i	$0.552702\pi$
$942$	−1.23607	−0.0402733
$943$	15.4164	0.502027
$944$	8.29180	0.269875
$945$	−0.763932	−0.0248507
$946$	−0.583592	−0.0189742
$947$	2.12461	0.0690406	0.0345203	−	0.999404i	$-0.489010\pi$
$947$	2.12461	0.0690406	0.0345203	+	0.999404i	$0.489010\pi$
$948$	20.6525	0.670761
$949$	37.0132	1.20150
$950$	24.4721	0.793981
$951$	−1.47214	−0.0477373
$952$	1.58359	0.0513245
$953$	41.8885	1.35690	0.678452	−	0.734645i	$-0.262651\pi$
$953$	41.8885	1.35690	0.678452	+	0.734645i	$0.262651\pi$
$954$	6.94427	0.224829
$955$	−38.8328	−1.25660
$956$	−31.7082	−1.02552
$957$	0	0
$958$	−8.94427	−0.288976
$959$	−0.832816	−0.0268930
$960$	0.763932	0.0246558
$961$	−17.2492	−0.556427
$962$	17.7082	0.570935
$963$	9.70820	0.312842
$964$	−16.8541	−0.542834
$965$	−27.4164	−0.882565
$966$	1.12461	0.0361838
$967$	−0.763932	−0.0245664	−0.0122832	−	0.999925i	$-0.503910\pi$
$967$	−0.763932	−0.0245664	−0.0122832	+	0.999925i	$0.503910\pi$
$968$	24.4721	0.786564
$969$	21.7082	0.697368
$970$	−37.3050	−1.19779
$971$	0.360680	0.0115748	0.00578738	−	0.999983i	$-0.498158\pi$
$971$	0.360680	0.0115748	0.00578738	+	0.999983i	$0.498158\pi$
$972$	1.61803	0.0518985
$973$	0.278640	0.00893280
$974$	22.8328	0.731611
$975$	29.9443	0.958984
$976$	9.70820	0.310752
$977$	39.7082	1.27038	0.635189	−	0.772357i	$-0.280922\pi$
$977$	39.7082	1.27038	0.635189	+	0.772357i	$0.280922\pi$
$978$	−3.70820	−0.118575
$979$	−1.18034	−0.0377238
$980$	36.3607	1.16150
$981$	9.47214	0.302422
$982$	4.94427	0.157778
$983$	4.94427	0.157698	0.0788489	−	0.996887i	$-0.474876\pi$
$983$	4.94427	0.157698	0.0788489	+	0.996887i	$0.474876\pi$
$984$	−4.47214	−0.142566
$985$	−62.2492	−1.98343
$986$	0	0
$987$	1.11146	0.0353780
$988$	64.0689	2.03830
$989$	30.8328	0.980427
$990$	0.472136	0.0150055
$991$	−26.0132	−0.826335	−0.413168	−	0.910655i	$-0.635578\pi$
$991$	−26.0132	−0.826335	−0.413168	+	0.910655i	$0.635578\pi$
$992$	−20.8328	−0.661443
$993$	−20.7639	−0.658923
$994$	−0.763932	−0.0242305
$995$	−50.6525	−1.60579
$996$	4.76393	0.150951
$997$	60.1378	1.90458	0.952291	−	0.305191i	$-0.0987204\pi$
$997$	60.1378	1.90458	0.952291	+	0.305191i	$0.0987204\pi$
$998$	2.03444	0.0643991
$999$	5.23607	0.165662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2523.2.a.c.1.2		2
3.2	odd	2		7569.2.a.k.1.1		2
29.28	even	2		87.2.a.a.1.1	✓	2
87.86	odd	2		261.2.a.b.1.2		2
116.115	odd	2		1392.2.a.q.1.2		2
145.28	odd	4		2175.2.c.k.349.3		4
145.57	odd	4		2175.2.c.k.349.2		4
145.144	even	2		2175.2.a.l.1.2		2
203.202	odd	2		4263.2.a.j.1.1		2
232.115	odd	2		5568.2.a.bs.1.1		2
232.173	even	2		5568.2.a.bl.1.1		2
348.347	even	2		4176.2.a.bn.1.1		2
435.434	odd	2		6525.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.a.a.1.1	✓	2	29.28	even	2
261.2.a.b.1.2		2	87.86	odd	2
1392.2.a.q.1.2		2	116.115	odd	2
2175.2.a.l.1.2		2	145.144	even	2
2175.2.c.k.349.2		4	145.57	odd	4
2175.2.c.k.349.3		4	145.28	odd	4
2523.2.a.c.1.2		2	1.1	even	1	trivial
4176.2.a.bn.1.1		2	348.347	even	2
4263.2.a.j.1.1		2	203.202	odd	2
5568.2.a.bl.1.1		2	232.173	even	2
5568.2.a.bs.1.1		2	232.115	odd	2
6525.2.a.ba.1.1		2	435.434	odd	2
7569.2.a.k.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2523 = 3 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2523.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} -2.23607 q^{8} +1.00000 q^{9} +2.00000 q^{10} +0.236068 q^{11} +1.61803 q^{12} -5.47214 q^{13} +0.145898 q^{14} -3.23607 q^{15} +1.85410 q^{16} -3.00000 q^{17} +0.618034 q^{18} +7.23607 q^{19} -5.23607 q^{20} -0.236068 q^{21} +0.145898 q^{22} -7.70820 q^{23} +2.23607 q^{24} +5.47214 q^{25} -3.38197 q^{26} -1.00000 q^{27} -0.381966 q^{28} -2.00000 q^{30} -3.70820 q^{31} +5.61803 q^{32} -0.236068 q^{33} -1.85410 q^{34} +0.763932 q^{35} -1.61803 q^{36} -5.23607 q^{37} +4.47214 q^{38} +5.47214 q^{39} -7.23607 q^{40} -2.00000 q^{41} -0.145898 q^{42} -4.00000 q^{43} -0.381966 q^{44} +3.23607 q^{45} -4.76393 q^{46} -4.70820 q^{47} -1.85410 q^{48} -6.94427 q^{49} +3.38197 q^{50} +3.00000 q^{51} +8.85410 q^{52} +11.2361 q^{53} -0.618034 q^{54} +0.763932 q^{55} -0.527864 q^{56} -7.23607 q^{57} +4.47214 q^{59} +5.23607 q^{60} +5.23607 q^{61} -2.29180 q^{62} +0.236068 q^{63} -0.236068 q^{64} -17.7082 q^{65} -0.145898 q^{66} -13.1803 q^{67} +4.85410 q^{68} +7.70820 q^{69} +0.472136 q^{70} -5.23607 q^{71} -2.23607 q^{72} -6.76393 q^{73} -3.23607 q^{74} -5.47214 q^{75} -11.7082 q^{76} +0.0557281 q^{77} +3.38197 q^{78} +12.7639 q^{79} +6.00000 q^{80} +1.00000 q^{81} -1.23607 q^{82} +2.94427 q^{83} +0.381966 q^{84} -9.70820 q^{85} -2.47214 q^{86} -0.527864 q^{88} -5.00000 q^{89} +2.00000 q^{90} -1.29180 q^{91} +12.4721 q^{92} +3.70820 q^{93} -2.90983 q^{94} +23.4164 q^{95} -5.61803 q^{96} -18.6525 q^{97} -4.29180 q^{98} +0.236068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9} + 4 q^{10} - 4 q^{11} + q^{12} - 2 q^{13} + 7 q^{14} - 2 q^{15} - 3 q^{16} - 6 q^{17} - q^{18} + 10 q^{19} - 6 q^{20} + 4 q^{21} + 7 q^{22} - 2 q^{23} + 2 q^{25} - 9 q^{26} - 2 q^{27} - 3 q^{28} - 4 q^{30} + 6 q^{31} + 9 q^{32} + 4 q^{33} + 3 q^{34} + 6 q^{35} - q^{36} - 6 q^{37} + 2 q^{39} - 10 q^{40} - 4 q^{41} - 7 q^{42} - 8 q^{43} - 3 q^{44} + 2 q^{45} - 14 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} + 9 q^{50} + 6 q^{51} + 11 q^{52} + 18 q^{53} + q^{54} + 6 q^{55} - 10 q^{56} - 10 q^{57} + 6 q^{60} + 6 q^{61} - 18 q^{62} - 4 q^{63} + 4 q^{64} - 22 q^{65} - 7 q^{66} - 4 q^{67} + 3 q^{68} + 2 q^{69} - 8 q^{70} - 6 q^{71} - 18 q^{73} - 2 q^{74} - 2 q^{75} - 10 q^{76} + 18 q^{77} + 9 q^{78} + 30 q^{79} + 12 q^{80} + 2 q^{81} + 2 q^{82} - 12 q^{83} + 3 q^{84} - 6 q^{85} + 4 q^{86} - 10 q^{88} - 10 q^{89} + 4 q^{90} - 16 q^{91} + 16 q^{92} - 6 q^{93} - 17 q^{94} + 20 q^{95} - 9 q^{96} - 6 q^{97} - 22 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9 + 4 * q^10 - 4 * q^11 + q^12 - 2 * q^13 + 7 * q^14 - 2 * q^15 - 3 * q^16 - 6 * q^17 - q^18 + 10 * q^19 - 6 * q^20 + 4 * q^21 + 7 * q^22 - 2 * q^23 + 2 * q^25 - 9 * q^26 - 2 * q^27 - 3 * q^28 - 4 * q^30 + 6 * q^31 + 9 * q^32 + 4 * q^33 + 3 * q^34 + 6 * q^35 - q^36 - 6 * q^37 + 2 * q^39 - 10 * q^40 - 4 * q^41 - 7 * q^42 - 8 * q^43 - 3 * q^44 + 2 * q^45 - 14 * q^46 + 4 * q^47 + 3 * q^48 + 4 * q^49 + 9 * q^50 + 6 * q^51 + 11 * q^52 + 18 * q^53 + q^54 + 6 * q^55 - 10 * q^56 - 10 * q^57 + 6 * q^60 + 6 * q^61 - 18 * q^62 - 4 * q^63 + 4 * q^64 - 22 * q^65 - 7 * q^66 - 4 * q^67 + 3 * q^68 + 2 * q^69 - 8 * q^70 - 6 * q^71 - 18 * q^73 - 2 * q^74 - 2 * q^75 - 10 * q^76 + 18 * q^77 + 9 * q^78 + 30 * q^79 + 12 * q^80 + 2 * q^81 + 2 * q^82 - 12 * q^83 + 3 * q^84 - 6 * q^85 + 4 * q^86 - 10 * q^88 - 10 * q^89 + 4 * q^90 - 16 * q^91 + 16 * q^92 - 6 * q^93 - 17 * q^94 + 20 * q^95 - 9 * q^96 - 6 * q^97 - 22 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more