LMFDB - Embedded newform 3549.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,2,Mod(1,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3549.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:	$28.3389076774$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + \cdots + 281 $ x^15 - 2x^14 - 27x^13 + 51x^12 + 290x^11 - 510x^10 - 1575x^9 + 2522x^8 + 4553x^7 - 6393x^6 - 6785x^5 + 7806x^4 + 4632x^3 - 3811x^2 - 1041x + 281
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.78394$ of defining polynomial
Character	$\chi$	$=$	3549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78394 q^{2} +1.00000 q^{3} +5.75034 q^{4} -1.59144 q^{5} -2.78394 q^{6} +1.00000 q^{7} -10.4407 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.78394 q^{2} +1.00000 q^{3} +5.75034 q^{4} -1.59144 q^{5} -2.78394 q^{6} +1.00000 q^{7} -10.4407 q^{8} +1.00000 q^{9} +4.43048 q^{10} +0.134788 q^{11} +5.75034 q^{12} -2.78394 q^{14} -1.59144 q^{15} +17.5657 q^{16} -7.07278 q^{17} -2.78394 q^{18} -5.94529 q^{19} -9.15132 q^{20} +1.00000 q^{21} -0.375242 q^{22} -7.87475 q^{23} -10.4407 q^{24} -2.46732 q^{25} +1.00000 q^{27} +5.75034 q^{28} +7.91488 q^{29} +4.43048 q^{30} +2.66844 q^{31} -28.0204 q^{32} +0.134788 q^{33} +19.6902 q^{34} -1.59144 q^{35} +5.75034 q^{36} +0.101235 q^{37} +16.5513 q^{38} +16.6158 q^{40} +3.72584 q^{41} -2.78394 q^{42} +0.825682 q^{43} +0.775076 q^{44} -1.59144 q^{45} +21.9228 q^{46} -2.58303 q^{47} +17.5657 q^{48} +1.00000 q^{49} +6.86886 q^{50} -7.07278 q^{51} +1.42870 q^{53} -2.78394 q^{54} -0.214507 q^{55} -10.4407 q^{56} -5.94529 q^{57} -22.0346 q^{58} +7.93495 q^{59} -9.15132 q^{60} -3.77939 q^{61} -7.42877 q^{62} +1.00000 q^{63} +42.8759 q^{64} -0.375242 q^{66} -4.56472 q^{67} -40.6709 q^{68} -7.87475 q^{69} +4.43048 q^{70} +10.6911 q^{71} -10.4407 q^{72} -1.49552 q^{73} -0.281832 q^{74} -2.46732 q^{75} -34.1874 q^{76} +0.134788 q^{77} +13.0892 q^{79} -27.9548 q^{80} +1.00000 q^{81} -10.3725 q^{82} +15.6291 q^{83} +5.75034 q^{84} +11.2559 q^{85} -2.29865 q^{86} +7.91488 q^{87} -1.40728 q^{88} +7.17174 q^{89} +4.43048 q^{90} -45.2824 q^{92} +2.66844 q^{93} +7.19100 q^{94} +9.46158 q^{95} -28.0204 q^{96} +13.9093 q^{97} -2.78394 q^{98} +0.134788 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 2 q^{2} + 15 q^{3} + 28 q^{4} + 9 q^{5} - 2 q^{6} + 15 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q - 2 * q^2 + 15 * q^3 + 28 * q^4 + 9 * q^5 - 2 * q^6 + 15 * q^7 - 9 * q^8 + 15 * q^9	\( 15 q - 2 q^{2} + 15 q^{3} + 28 q^{4} + 9 q^{5} - 2 q^{6} + 15 q^{7} - 9 q^{8} + 15 q^{9} + 21 q^{10} - 5 q^{11} + 28 q^{12} - 2 q^{14} + 9 q^{15} + 50 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} + 23 q^{20} + 15 q^{21} + 21 q^{22} + 4 q^{23} - 9 q^{24} + 50 q^{25} + 15 q^{27} + 28 q^{28} + 9 q^{29} + 21 q^{30} - 7 q^{31} - 35 q^{32} - 5 q^{33} + 2 q^{34} + 9 q^{35} + 28 q^{36} - 17 q^{37} - 12 q^{38} + 46 q^{40} + 22 q^{41} - 2 q^{42} + 36 q^{43} - 29 q^{44} + 9 q^{45} + q^{46} + 12 q^{47} + 50 q^{48} + 15 q^{49} - 53 q^{50} - q^{51} - 5 q^{53} - 2 q^{54} + 43 q^{55} - 9 q^{56} - 3 q^{57} - 29 q^{58} + 29 q^{59} + 23 q^{60} + 12 q^{61} + 14 q^{62} + 15 q^{63} + 95 q^{64} + 21 q^{66} + 12 q^{67} - 16 q^{68} + 4 q^{69} + 21 q^{70} - 36 q^{71} - 9 q^{72} + 29 q^{73} - 5 q^{74} + 50 q^{75} - 25 q^{76} - 5 q^{77} + 35 q^{79} + 89 q^{80} + 15 q^{81} + 51 q^{82} + 10 q^{83} + 28 q^{84} - 23 q^{85} - 19 q^{86} + 9 q^{87} + 73 q^{88} - 25 q^{89} + 21 q^{90} - 31 q^{92} - 7 q^{93} - 19 q^{94} - 7 q^{95} - 35 q^{96} + 26 q^{97} - 2 q^{98} - 5 q^{99}+O(q^{100}) \) 15 * q - 2 * q^2 + 15 * q^3 + 28 * q^4 + 9 * q^5 - 2 * q^6 + 15 * q^7 - 9 * q^8 + 15 * q^9 + 21 * q^10 - 5 * q^11 + 28 * q^12 - 2 * q^14 + 9 * q^15 + 50 * q^16 - q^17 - 2 * q^18 - 3 * q^19 + 23 * q^20 + 15 * q^21 + 21 * q^22 + 4 * q^23 - 9 * q^24 + 50 * q^25 + 15 * q^27 + 28 * q^28 + 9 * q^29 + 21 * q^30 - 7 * q^31 - 35 * q^32 - 5 * q^33 + 2 * q^34 + 9 * q^35 + 28 * q^36 - 17 * q^37 - 12 * q^38 + 46 * q^40 + 22 * q^41 - 2 * q^42 + 36 * q^43 - 29 * q^44 + 9 * q^45 + q^46 + 12 * q^47 + 50 * q^48 + 15 * q^49 - 53 * q^50 - q^51 - 5 * q^53 - 2 * q^54 + 43 * q^55 - 9 * q^56 - 3 * q^57 - 29 * q^58 + 29 * q^59 + 23 * q^60 + 12 * q^61 + 14 * q^62 + 15 * q^63 + 95 * q^64 + 21 * q^66 + 12 * q^67 - 16 * q^68 + 4 * q^69 + 21 * q^70 - 36 * q^71 - 9 * q^72 + 29 * q^73 - 5 * q^74 + 50 * q^75 - 25 * q^76 - 5 * q^77 + 35 * q^79 + 89 * q^80 + 15 * q^81 + 51 * q^82 + 10 * q^83 + 28 * q^84 - 23 * q^85 - 19 * q^86 + 9 * q^87 + 73 * q^88 - 25 * q^89 + 21 * q^90 - 31 * q^92 - 7 * q^93 - 19 * q^94 - 7 * q^95 - 35 * q^96 + 26 * q^97 - 2 * q^98 - 5 * 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 $

$2$	−2.78394	−1.96854	−0.984272	−	0.176658i	$-0.943471\pi$
$2$	−2.78394	−1.96854	−0.984272	+	0.176658i	$0.943471\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.75034	2.87517
$5$	−1.59144	−0.711714	−0.355857	−	0.934540i	$-0.615811\pi$
$5$	−1.59144	−0.711714	−0.355857	+	0.934540i	$0.615811\pi$
$6$	−2.78394	−1.13654
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−10.4407	−3.69135
$9$	1.00000	0.333333
$10$	4.43048	1.40104
$11$	0.134788	0.0406401	0.0203200	−	0.999794i	$-0.493531\pi$
$11$	0.134788	0.0406401	0.0203200	+	0.999794i	$0.493531\pi$
$12$	5.75034	1.65998
$13$	0	0
$13$	0	0
$14$	−2.78394	−0.744040
$15$	−1.59144	−0.410908
$16$	17.5657	4.39142
$17$	−7.07278	−1.71540	−0.857701	−	0.514149i	$-0.828108\pi$
$17$	−7.07278	−1.71540	−0.857701	+	0.514149i	$0.828108\pi$
$18$	−2.78394	−0.656182
$19$	−5.94529	−1.36394	−0.681971	−	0.731379i	$-0.738877\pi$
$19$	−5.94529	−1.36394	−0.681971	+	0.731379i	$0.738877\pi$
$20$	−9.15132	−2.04630
$21$	1.00000	0.218218
$22$	−0.375242	−0.0800019
$23$	−7.87475	−1.64200	−0.820999	−	0.570930i	$-0.806583\pi$
$23$	−7.87475	−1.64200	−0.820999	+	0.570930i	$0.806583\pi$
$24$	−10.4407	−2.13120
$25$	−2.46732	−0.493463
$26$	0	0
$27$	1.00000	0.192450
$28$	5.75034	1.08671
$29$	7.91488	1.46976	0.734878	−	0.678199i	$-0.237239\pi$
$29$	7.91488	1.46976	0.734878	+	0.678199i	$0.237239\pi$
$30$	4.43048	0.808891
$31$	2.66844	0.479265	0.239633	−	0.970864i	$-0.422973\pi$
$31$	2.66844	0.479265	0.239633	+	0.970864i	$0.422973\pi$
$32$	−28.0204	−4.95336
$33$	0.134788	0.0234636
$34$	19.6902	3.37685
$35$	−1.59144	−0.269003
$36$	5.75034	0.958389
$37$	0.101235	0.0166429	0.00832145	−	0.999965i	$-0.497351\pi$
$37$	0.101235	0.0166429	0.00832145	+	0.999965i	$0.497351\pi$
$38$	16.5513	2.68498
$39$	0	0
$40$	16.6158	2.62719
$41$	3.72584	0.581878	0.290939	−	0.956742i	$-0.406032\pi$
$41$	3.72584	0.581878	0.290939	+	0.956742i	$0.406032\pi$
$42$	−2.78394	−0.429572
$43$	0.825682	0.125915	0.0629577	−	0.998016i	$-0.479947\pi$
$43$	0.825682	0.125915	0.0629577	+	0.998016i	$0.479947\pi$
$44$	0.775076	0.116847
$45$	−1.59144	−0.237238
$46$	21.9228	3.23235
$47$	−2.58303	−0.376773	−0.188387	−	0.982095i	$-0.560326\pi$
$47$	−2.58303	−0.376773	−0.188387	+	0.982095i	$0.560326\pi$
$48$	17.5657	2.53539
$49$	1.00000	0.142857
$50$	6.86886	0.971404
$51$	−7.07278	−0.990388
$52$	0	0
$53$	1.42870	0.196247	0.0981235	−	0.995174i	$-0.468716\pi$
$53$	1.42870	0.196247	0.0981235	+	0.995174i	$0.468716\pi$
$54$	−2.78394	−0.378847
$55$	−0.214507	−0.0289241
$56$	−10.4407	−1.39520
$57$	−5.94529	−0.787473
$58$	−22.0346	−2.89328
$59$	7.93495	1.03304	0.516521	−	0.856274i	$-0.327227\pi$
$59$	7.93495	1.03304	0.516521	+	0.856274i	$0.327227\pi$
$60$	−9.15132	−1.18143
$61$	−3.77939	−0.483902	−0.241951	−	0.970289i	$-0.577787\pi$
$61$	−3.77939	−0.483902	−0.241951	+	0.970289i	$0.577787\pi$
$62$	−7.42877	−0.943455
$63$	1.00000	0.125988
$64$	42.8759	5.35949
$65$	0	0
$66$	−0.375242	−0.0461891
$67$	−4.56472	−0.557670	−0.278835	−	0.960339i	$-0.589948\pi$
$67$	−4.56472	−0.557670	−0.278835	+	0.960339i	$0.589948\pi$
$68$	−40.6709	−4.93207
$69$	−7.87475	−0.948008
$70$	4.43048	0.529544
$71$	10.6911	1.26881	0.634403	−	0.773003i	$-0.281246\pi$
$71$	10.6911	1.26881	0.634403	+	0.773003i	$0.281246\pi$
$72$	−10.4407	−1.23045
$73$	−1.49552	−0.175038	−0.0875189	−	0.996163i	$-0.527894\pi$
$73$	−1.49552	−0.175038	−0.0875189	+	0.996163i	$0.527894\pi$
$74$	−0.281832	−0.0327623
$75$	−2.46732	−0.284901
$76$	−34.1874	−3.92156
$77$	0.134788	0.0153605
$78$	0	0
$79$	13.0892	1.47265	0.736323	−	0.676630i	$-0.236560\pi$
$79$	13.0892	1.47265	0.736323	+	0.676630i	$0.236560\pi$
$80$	−27.9548	−3.12544
$81$	1.00000	0.111111
$82$	−10.3725	−1.14545
$83$	15.6291	1.71552	0.857760	−	0.514050i	$-0.171856\pi$
$83$	15.6291	1.71552	0.857760	+	0.514050i	$0.171856\pi$
$84$	5.75034	0.627413
$85$	11.2559	1.22088
$86$	−2.29865	−0.247870
$87$	7.91488	0.848564
$88$	−1.40728	−0.150017
$89$	7.17174	0.760203	0.380101	−	0.924945i	$-0.375889\pi$
$89$	7.17174	0.760203	0.380101	+	0.924945i	$0.375889\pi$
$90$	4.43048	0.467014
$91$	0	0
$92$	−45.2824	−4.72102
$93$	2.66844	0.276704
$94$	7.19100	0.741695
$95$	9.46158	0.970737
$96$	−28.0204	−2.85982
$97$	13.9093	1.41227	0.706137	−	0.708076i	$-0.250436\pi$
$97$	13.9093	1.41227	0.706137	+	0.708076i	$0.250436\pi$
$98$	−2.78394	−0.281221
$99$	0.134788	0.0135467
$100$	−14.1879	−1.41879
$101$	12.2952	1.22342	0.611709	−	0.791083i	$-0.290482\pi$
$101$	12.2952	1.22342	0.611709	+	0.791083i	$0.290482\pi$
$102$	19.6902	1.94962
$103$	8.10004	0.798120	0.399060	−	0.916925i	$-0.369336\pi$
$103$	8.10004	0.798120	0.399060	+	0.916925i	$0.369336\pi$
$104$	0	0
$105$	−1.59144	−0.155309
$106$	−3.97742	−0.386321
$107$	−5.28792	−0.511202	−0.255601	−	0.966782i	$-0.582273\pi$
$107$	−5.28792	−0.511202	−0.255601	+	0.966782i	$0.582273\pi$
$108$	5.75034	0.553326
$109$	−0.0300910	−0.00288219	−0.00144110	−	0.999999i	$-0.500459\pi$
$109$	−0.0300910	−0.00288219	−0.00144110	+	0.999999i	$0.500459\pi$
$110$	0.597175	0.0569384
$111$	0.101235	0.00960879
$112$	17.5657	1.65980
$113$	−5.81240	−0.546785	−0.273392	−	0.961903i	$-0.588146\pi$
$113$	−5.81240	−0.546785	−0.273392	+	0.961903i	$0.588146\pi$
$114$	16.5513	1.55018
$115$	12.5322	1.16863
$116$	45.5132	4.22579
$117$	0	0
$118$	−22.0905	−2.03359
$119$	−7.07278	−0.648361
$120$	16.6158	1.51681
$121$	−10.9818	−0.998348
$122$	10.5216	0.952582
$123$	3.72584	0.335948
$124$	15.3444	1.37797
$125$	11.8838	1.06292
$126$	−2.78394	−0.248013
$127$	20.9759	1.86131	0.930654	−	0.365901i	$-0.119239\pi$
$127$	20.9759	1.86131	0.930654	+	0.365901i	$0.119239\pi$
$128$	−63.3232	−5.59703
$129$	0.825682	0.0726973
$130$	0	0
$131$	8.26962	0.722520	0.361260	−	0.932465i	$-0.382347\pi$
$131$	8.26962	0.722520	0.361260	+	0.932465i	$0.382347\pi$
$132$	0.775076	0.0674617
$133$	−5.94529	−0.515522
$134$	12.7079	1.09780
$135$	−1.59144	−0.136969
$136$	73.8450	6.33215
$137$	10.0528	0.858868	0.429434	−	0.903098i	$-0.358713\pi$
$137$	10.0528	0.858868	0.429434	+	0.903098i	$0.358713\pi$
$138$	21.9228	1.86620
$139$	−12.8728	−1.09186	−0.545928	−	0.837832i	$-0.683823\pi$
$139$	−12.8728	−1.09186	−0.545928	+	0.837832i	$0.683823\pi$
$140$	−9.15132	−0.773428
$141$	−2.58303	−0.217530
$142$	−29.7635	−2.49770
$143$	0	0
$144$	17.5657	1.46381
$145$	−12.5961	−1.04605
$146$	4.16345	0.344570
$147$	1.00000	0.0824786
$148$	0.582134	0.0478512
$149$	−22.7196	−1.86126	−0.930630	−	0.365962i	$-0.880740\pi$
$149$	−22.7196	−1.86126	−0.930630	+	0.365962i	$0.880740\pi$
$150$	6.86886	0.560840
$151$	2.67518	0.217703	0.108852	−	0.994058i	$-0.465283\pi$
$151$	2.67518	0.217703	0.108852	+	0.994058i	$0.465283\pi$
$152$	62.0731	5.03479
$153$	−7.07278	−0.571801
$154$	−0.375242	−0.0302379
$155$	−4.24666	−0.341100
$156$	0	0
$157$	19.7016	1.57236	0.786178	−	0.618001i	$-0.212057\pi$
$157$	19.7016	1.57236	0.786178	+	0.618001i	$0.212057\pi$
$158$	−36.4395	−2.89897
$159$	1.42870	0.113303
$160$	44.5929	3.52538
$161$	−7.87475	−0.620617
$162$	−2.78394	−0.218727
$163$	−5.99748	−0.469759	−0.234879	−	0.972025i	$-0.575469\pi$
$163$	−5.99748	−0.469759	−0.234879	+	0.972025i	$0.575469\pi$
$164$	21.4248	1.67300
$165$	−0.214507	−0.0166994
$166$	−43.5106	−3.37708
$167$	5.89546	0.456204	0.228102	−	0.973637i	$-0.426748\pi$
$167$	5.89546	0.456204	0.228102	+	0.973637i	$0.426748\pi$
$168$	−10.4407	−0.805519
$169$	0	0
$170$	−31.3358	−2.40335
$171$	−5.94529	−0.454648
$172$	4.74795	0.362028
$173$	4.50959	0.342858	0.171429	−	0.985196i	$-0.445162\pi$
$173$	4.50959	0.342858	0.171429	+	0.985196i	$0.445162\pi$
$174$	−22.0346	−1.67044
$175$	−2.46732	−0.186511
$176$	2.36764	0.178468
$177$	7.93495	0.596428
$178$	−19.9657	−1.49649
$179$	6.50536	0.486234	0.243117	−	0.969997i	$-0.421830\pi$
$179$	6.50536	0.486234	0.243117	+	0.969997i	$0.421830\pi$
$180$	−9.15132	−0.682099
$181$	−11.5176	−0.856099	−0.428050	−	0.903755i	$-0.640799\pi$
$181$	−11.5176	−0.856099	−0.428050	+	0.903755i	$0.640799\pi$
$182$	0	0
$183$	−3.77939	−0.279381
$184$	82.2180	6.06119
$185$	−0.161109	−0.0118450
$186$	−7.42877	−0.544704
$187$	−0.953326	−0.0697141
$188$	−14.8533	−1.08329
$189$	1.00000	0.0727393
$190$	−26.3405	−1.91094
$191$	−22.6574	−1.63943	−0.819716	−	0.572771i	$-0.805868\pi$
$191$	−22.6574	−1.63943	−0.819716	+	0.572771i	$0.805868\pi$
$192$	42.8759	3.09430
$193$	−12.8501	−0.924972	−0.462486	−	0.886627i	$-0.653042\pi$
$193$	−12.8501	−0.924972	−0.462486	+	0.886627i	$0.653042\pi$
$194$	−38.7226	−2.78012
$195$	0	0
$196$	5.75034	0.410738
$197$	−6.15529	−0.438546	−0.219273	−	0.975664i	$-0.570369\pi$
$197$	−6.15529	−0.438546	−0.219273	+	0.975664i	$0.570369\pi$
$198$	−0.375242	−0.0266673
$199$	−6.10499	−0.432771	−0.216386	−	0.976308i	$-0.569427\pi$
$199$	−6.10499	−0.432771	−0.216386	+	0.976308i	$0.569427\pi$
$200$	25.7605	1.82155
$201$	−4.56472	−0.321971
$202$	−34.2291	−2.40835
$203$	7.91488	0.555515
$204$	−40.6709	−2.84753
$205$	−5.92946	−0.414131
$206$	−22.5500	−1.57114
$207$	−7.87475	−0.547333
$208$	0	0
$209$	−0.801353	−0.0554308
$210$	4.43048	0.305732
$211$	3.35021	0.230638	0.115319	−	0.993329i	$-0.463211\pi$
$211$	3.35021	0.230638	0.115319	+	0.993329i	$0.463211\pi$
$212$	8.21550	0.564243
$213$	10.6911	0.732545
$214$	14.7213	1.00632
$215$	−1.31402	−0.0896157
$216$	−10.4407	−0.710401
$217$	2.66844	0.181145
$218$	0.0837716	0.00567373
$219$	−1.49552	−0.101058
$220$	−1.23349	−0.0831617
$221$	0	0
$222$	−0.281832	−0.0189153
$223$	14.5203	0.972350	0.486175	−	0.873861i	$-0.338392\pi$
$223$	14.5203	0.972350	0.486175	+	0.873861i	$0.338392\pi$
$224$	−28.0204	−1.87219
$225$	−2.46732	−0.164488
$226$	16.1814	1.07637
$227$	−14.8507	−0.985676	−0.492838	−	0.870121i	$-0.664041\pi$
$227$	−14.8507	−0.985676	−0.492838	+	0.870121i	$0.664041\pi$
$228$	−34.1874	−2.26412
$229$	21.6278	1.42920	0.714602	−	0.699531i	$-0.246608\pi$
$229$	21.6278	1.42920	0.714602	+	0.699531i	$0.246608\pi$
$230$	−34.8889	−2.30051
$231$	0.134788	0.00886840
$232$	−82.6370	−5.42539
$233$	7.71618	0.505504	0.252752	−	0.967531i	$-0.418664\pi$
$233$	7.71618	0.505504	0.252752	+	0.967531i	$0.418664\pi$
$234$	0	0
$235$	4.11073	0.268155
$236$	45.6286	2.97017
$237$	13.0892	0.850233
$238$	19.6902	1.27633
$239$	−3.84302	−0.248585	−0.124292	−	0.992246i	$-0.539666\pi$
$239$	−3.84302	−0.248585	−0.124292	+	0.992246i	$0.539666\pi$
$240$	−27.9548	−1.80447
$241$	4.88222	0.314491	0.157246	−	0.987560i	$-0.449739\pi$
$241$	4.88222	0.314491	0.157246	+	0.987560i	$0.449739\pi$
$242$	30.5728	1.96529
$243$	1.00000	0.0641500
$244$	−21.7328	−1.39130
$245$	−1.59144	−0.101673
$246$	−10.3725	−0.661328
$247$	0	0
$248$	−27.8604	−1.76914
$249$	15.6291	0.990456
$250$	−33.0838	−2.09240
$251$	11.3686	0.717581	0.358791	−	0.933418i	$-0.383189\pi$
$251$	11.3686	0.717581	0.358791	+	0.933418i	$0.383189\pi$
$252$	5.75034	0.362237
$253$	−1.06142	−0.0667310
$254$	−58.3956	−3.66407
$255$	11.2559	0.704873
$256$	90.5363	5.65852
$257$	25.0640	1.56345	0.781723	−	0.623625i	$-0.214341\pi$
$257$	25.0640	1.56345	0.781723	+	0.623625i	$0.214341\pi$
$258$	−2.29865	−0.143108
$259$	0.101235	0.00629043
$260$	0	0
$261$	7.91488	0.489919
$262$	−23.0221	−1.42231
$263$	17.8501	1.10069	0.550343	−	0.834939i	$-0.314497\pi$
$263$	17.8501	1.10069	0.550343	+	0.834939i	$0.314497\pi$
$264$	−1.40728	−0.0866123
$265$	−2.27369	−0.139672
$266$	16.5513	1.01483
$267$	7.17174	0.438903
$268$	−26.2487	−1.60339
$269$	−6.75838	−0.412066	−0.206033	−	0.978545i	$-0.566055\pi$
$269$	−6.75838	−0.412066	−0.206033	+	0.978545i	$0.566055\pi$
$270$	4.43048	0.269630
$271$	−7.27080	−0.441670	−0.220835	−	0.975311i	$-0.570878\pi$
$271$	−7.27080	−0.441670	−0.220835	+	0.975311i	$0.570878\pi$
$272$	−124.238	−7.53306
$273$	0	0
$274$	−27.9864	−1.69072
$275$	−0.332564	−0.0200544
$276$	−45.2824	−2.72568
$277$	−13.1166	−0.788102	−0.394051	−	0.919089i	$-0.628927\pi$
$277$	−13.1166	−0.788102	−0.394051	+	0.919089i	$0.628927\pi$
$278$	35.8371	2.14937
$279$	2.66844	0.159755
$280$	16.6158	0.992983
$281$	−18.2095	−1.08629	−0.543143	−	0.839640i	$-0.682766\pi$
$281$	−18.2095	−1.08629	−0.543143	+	0.839640i	$0.682766\pi$
$282$	7.19100	0.428218
$283$	−22.5038	−1.33771	−0.668857	−	0.743391i	$-0.733216\pi$
$283$	−22.5038	−1.33771	−0.668857	+	0.743391i	$0.733216\pi$
$284$	61.4777	3.64803
$285$	9.46158	0.560455
$286$	0	0
$287$	3.72584	0.219929
$288$	−28.0204	−1.65112
$289$	33.0243	1.94261
$290$	35.0667	2.05919
$291$	13.9093	0.815376
$292$	−8.59977	−0.503263
$293$	10.6254	0.620744	0.310372	−	0.950615i	$-0.399546\pi$
$293$	10.6254	0.620744	0.310372	+	0.950615i	$0.399546\pi$
$294$	−2.78394	−0.162363
$295$	−12.6280	−0.735231
$296$	−1.05696	−0.0614348
$297$	0.134788	0.00782119
$298$	63.2500	3.66397
$299$	0	0
$300$	−14.1879	−0.819138
$301$	0.825682	0.0475915
$302$	−7.44756	−0.428559
$303$	12.2952	0.706341
$304$	−104.433	−5.98965
$305$	6.01468	0.344400
$306$	19.6902	1.12562
$307$	−25.4720	−1.45377	−0.726883	−	0.686761i	$-0.759032\pi$
$307$	−25.4720	−1.45377	−0.726883	+	0.686761i	$0.759032\pi$
$308$	0.775076	0.0441641
$309$	8.10004	0.460795
$310$	11.8225	0.671470
$311$	−1.16454	−0.0660348	−0.0330174	−	0.999455i	$-0.510512\pi$
$311$	−1.16454	−0.0660348	−0.0330174	+	0.999455i	$0.510512\pi$
$312$	0	0
$313$	1.26134	0.0712953	0.0356476	−	0.999364i	$-0.488651\pi$
$313$	1.26134	0.0712953	0.0356476	+	0.999364i	$0.488651\pi$
$314$	−54.8480	−3.09525
$315$	−1.59144	−0.0896675
$316$	75.2671	4.23410
$317$	−4.95320	−0.278200	−0.139100	−	0.990278i	$-0.544421\pi$
$317$	−4.95320	−0.278200	−0.139100	+	0.990278i	$0.544421\pi$
$318$	−3.97742	−0.223043
$319$	1.06683	0.0597310
$320$	−68.2345	−3.81442
$321$	−5.28792	−0.295143
$322$	21.9228	1.22171
$323$	42.0498	2.33971
$324$	5.75034	0.319463
$325$	0	0
$326$	16.6966	0.924741
$327$	−0.0300910	−0.00166403
$328$	−38.9005	−2.14792
$329$	−2.58303	−0.142407
$330$	0.597175	0.0328734
$331$	12.8190	0.704597	0.352298	−	0.935888i	$-0.385400\pi$
$331$	12.8190	0.704597	0.352298	+	0.935888i	$0.385400\pi$
$332$	89.8728	4.93241
$333$	0.101235	0.00554764
$334$	−16.4126	−0.898059
$335$	7.26449	0.396901
$336$	17.5657	0.958287
$337$	−4.17281	−0.227307	−0.113654	−	0.993520i	$-0.536255\pi$
$337$	−4.17281	−0.227307	−0.113654	+	0.993520i	$0.536255\pi$
$338$	0	0
$339$	−5.81240	−0.315686
$340$	64.7253	3.51022
$341$	0.359673	0.0194774
$342$	16.5513	0.894994
$343$	1.00000	0.0539949
$344$	−8.62072	−0.464798
$345$	12.5322	0.674711
$346$	−12.5544	−0.674931
$347$	−2.68888	−0.144347	−0.0721733	−	0.997392i	$-0.522993\pi$
$347$	−2.68888	−0.144347	−0.0721733	+	0.997392i	$0.522993\pi$
$348$	45.5132	2.43976
$349$	1.62817	0.0871537	0.0435768	−	0.999050i	$-0.486125\pi$
$349$	1.62817	0.0871537	0.0435768	+	0.999050i	$0.486125\pi$
$350$	6.86886	0.367156
$351$	0	0
$352$	−3.77682	−0.201305
$353$	21.5506	1.14702	0.573511	−	0.819198i	$-0.305581\pi$
$353$	21.5506	1.14702	0.573511	+	0.819198i	$0.305581\pi$
$354$	−22.0905	−1.17409
$355$	−17.0143	−0.903027
$356$	41.2399	2.18571
$357$	−7.07278	−0.374331
$358$	−18.1106	−0.957173
$359$	−13.6959	−0.722841	−0.361421	−	0.932403i	$-0.617708\pi$
$359$	−13.6959	−0.722841	−0.361421	+	0.932403i	$0.617708\pi$
$360$	16.6158	0.875729
$361$	16.3465	0.860340
$362$	32.0644	1.68527
$363$	−10.9818	−0.576397
$364$	0	0
$365$	2.38004	0.124577
$366$	10.5216	0.549973
$367$	−20.6517	−1.07801	−0.539005	−	0.842303i	$-0.681200\pi$
$367$	−20.6517	−1.07801	−0.539005	+	0.842303i	$0.681200\pi$
$368$	−138.325	−7.21071
$369$	3.72584	0.193959
$370$	0.448519	0.0233174
$371$	1.42870	0.0741744
$372$	15.3444	0.795570
$373$	−4.70300	−0.243512	−0.121756	−	0.992560i	$-0.538853\pi$
$373$	−4.70300	−0.243512	−0.121756	+	0.992560i	$0.538853\pi$
$374$	2.65401	0.137235
$375$	11.8838	0.613676
$376$	26.9687	1.39080
$377$	0	0
$378$	−2.78394	−0.143191
$379$	−6.02884	−0.309681	−0.154840	−	0.987939i	$-0.549486\pi$
$379$	−6.02884	−0.309681	−0.154840	+	0.987939i	$0.549486\pi$
$380$	54.4072	2.79103
$381$	20.9759	1.07463
$382$	63.0769	3.22729
$383$	0.635311	0.0324629	0.0162314	−	0.999868i	$-0.494833\pi$
$383$	0.635311	0.0324629	0.0162314	+	0.999868i	$0.494833\pi$
$384$	−63.3232	−3.23145
$385$	−0.214507	−0.0109323
$386$	35.7740	1.82085
$387$	0.825682	0.0419718
$388$	79.9830	4.06052
$389$	−14.3172	−0.725908	−0.362954	−	0.931807i	$-0.618232\pi$
$389$	−14.3172	−0.725908	−0.362954	+	0.931807i	$0.618232\pi$
$390$	0	0
$391$	55.6964	2.81669
$392$	−10.4407	−0.527336
$393$	8.26962	0.417147
$394$	17.1360	0.863298
$395$	−20.8306	−1.04810
$396$	0.775076	0.0389490
$397$	7.56097	0.379474	0.189737	−	0.981835i	$-0.439236\pi$
$397$	7.56097	0.379474	0.189737	+	0.981835i	$0.439236\pi$
$398$	16.9959	0.851930
$399$	−5.94529	−0.297637
$400$	−43.3401	−2.16701
$401$	9.40084	0.469455	0.234728	−	0.972061i	$-0.424580\pi$
$401$	9.40084	0.469455	0.234728	+	0.972061i	$0.424580\pi$
$402$	12.7079	0.633814
$403$	0	0
$404$	70.7015	3.51753
$405$	−1.59144	−0.0790793
$406$	−22.0346	−1.09356
$407$	0.0136452	0.000676369 0
$408$	73.8450	3.65587
$409$	1.55862	0.0770690	0.0385345	−	0.999257i	$-0.487731\pi$
$409$	1.55862	0.0770690	0.0385345	+	0.999257i	$0.487731\pi$
$410$	16.5073	0.815236
$411$	10.0528	0.495868
$412$	46.5779	2.29473
$413$	7.93495	0.390453
$414$	21.9228	1.07745
$415$	−24.8728	−1.22096
$416$	0	0
$417$	−12.8728	−0.630383
$418$	2.23092	0.109118
$419$	29.7054	1.45120	0.725602	−	0.688115i	$-0.241562\pi$
$419$	29.7054	1.45120	0.725602	+	0.688115i	$0.241562\pi$
$420$	−9.15132	−0.446539
$421$	38.8767	1.89474	0.947368	−	0.320146i	$-0.103732\pi$
$421$	38.8767	1.89474	0.947368	+	0.320146i	$0.103732\pi$
$422$	−9.32678	−0.454021
$423$	−2.58303	−0.125591
$424$	−14.9167	−0.724417
$425$	17.4508	0.846488
$426$	−29.7635	−1.44205
$427$	−3.77939	−0.182898
$428$	−30.4073	−1.46979
$429$	0	0
$430$	3.65817	0.176413
$431$	22.2975	1.07403	0.537015	−	0.843572i	$-0.319552\pi$
$431$	22.2975	1.07403	0.537015	+	0.843572i	$0.319552\pi$
$432$	17.5657	0.845130
$433$	34.2544	1.64616	0.823082	−	0.567923i	$-0.192253\pi$
$433$	34.2544	1.64616	0.823082	+	0.567923i	$0.192253\pi$
$434$	−7.42877	−0.356592
$435$	−12.5961	−0.603935
$436$	−0.173033	−0.00828679
$437$	46.8176	2.23959
$438$	4.16345	0.198937
$439$	32.4533	1.54891	0.774455	−	0.632630i	$-0.218024\pi$
$439$	32.4533	1.54891	0.774455	+	0.632630i	$0.218024\pi$
$440$	2.23961	0.106769
$441$	1.00000	0.0476190
$442$	0	0
$443$	−11.0332	−0.524205	−0.262103	−	0.965040i	$-0.584416\pi$
$443$	−11.0332	−0.524205	−0.262103	+	0.965040i	$0.584416\pi$
$444$	0.582134	0.0276269
$445$	−11.4134	−0.541047
$446$	−40.4237	−1.91412
$447$	−22.7196	−1.07460
$448$	42.8759	2.02570
$449$	−20.3805	−0.961816	−0.480908	−	0.876771i	$-0.659693\pi$
$449$	−20.3805	−0.961816	−0.480908	+	0.876771i	$0.659693\pi$
$450$	6.86886	0.323801
$451$	0.502198	0.0236476
$452$	−33.4232	−1.57210
$453$	2.67518	0.125691
$454$	41.3435	1.94035
$455$	0	0
$456$	62.0731	2.90684
$457$	−25.7914	−1.20647	−0.603234	−	0.797564i	$-0.706122\pi$
$457$	−25.7914	−1.20647	−0.603234	+	0.797564i	$0.706122\pi$
$458$	−60.2105	−2.81345
$459$	−7.07278	−0.330129
$460$	72.0643	3.36002
$461$	−1.21019	−0.0563640	−0.0281820	−	0.999603i	$-0.508972\pi$
$461$	−1.21019	−0.0563640	−0.0281820	+	0.999603i	$0.508972\pi$
$462$	−0.375242	−0.0174578
$463$	21.3847	0.993833	0.496917	−	0.867798i	$-0.334465\pi$
$463$	21.3847	0.993833	0.496917	+	0.867798i	$0.334465\pi$
$464$	139.030	6.45432
$465$	−4.24666	−0.196934
$466$	−21.4814	−0.995106
$467$	15.8716	0.734450	0.367225	−	0.930132i	$-0.380308\pi$
$467$	15.8716	0.734450	0.367225	+	0.930132i	$0.380308\pi$
$468$	0	0
$469$	−4.56472	−0.210779
$470$	−11.4440	−0.527875
$471$	19.7016	0.907800
$472$	−82.8466	−3.81332
$473$	0.111292	0.00511721
$474$	−36.4395	−1.67372
$475$	14.6689	0.673055
$476$	−40.6709	−1.86415
$477$	1.42870	0.0654157
$478$	10.6988	0.489350
$479$	−22.6704	−1.03584	−0.517919	−	0.855429i	$-0.673293\pi$
$479$	−22.6704	−1.03584	−0.517919	+	0.855429i	$0.673293\pi$
$480$	44.5929	2.03538
$481$	0	0
$482$	−13.5918	−0.619090
$483$	−7.87475	−0.358313
$484$	−63.1492	−2.87042
$485$	−22.1358	−1.00513
$486$	−2.78394	−0.126282
$487$	−34.9729	−1.58477	−0.792386	−	0.610019i	$-0.791162\pi$
$487$	−34.9729	−1.58477	−0.792386	+	0.610019i	$0.791162\pi$
$488$	39.4596	1.78625
$489$	−5.99748	−0.271215
$490$	4.43048	0.200149
$491$	−4.33486	−0.195629	−0.0978147	−	0.995205i	$-0.531185\pi$
$491$	−4.33486	−0.195629	−0.0978147	+	0.995205i	$0.531185\pi$
$492$	21.4248	0.965906
$493$	−55.9802	−2.52122
$494$	0	0
$495$	−0.214507	−0.00964138
$496$	46.8729	2.10466
$497$	10.6911	0.479563
$498$	−43.5106	−1.94976
$499$	−20.8879	−0.935073	−0.467536	−	0.883974i	$-0.654858\pi$
$499$	−20.8879	−0.935073	−0.467536	+	0.883974i	$0.654858\pi$
$500$	68.3358	3.05607
$501$	5.89546	0.263390
$502$	−31.6496	−1.41259
$503$	−35.1712	−1.56821	−0.784103	−	0.620631i	$-0.786877\pi$
$503$	−35.1712	−1.56821	−0.784103	+	0.620631i	$0.786877\pi$
$504$	−10.4407	−0.465067
$505$	−19.5671	−0.870724
$506$	2.95493	0.131363
$507$	0	0
$508$	120.618	5.35157
$509$	40.6851	1.80334	0.901668	−	0.432429i	$-0.142343\pi$
$509$	40.6851	1.80334	0.901668	+	0.432429i	$0.142343\pi$
$510$	−31.3358	−1.38757
$511$	−1.49552	−0.0661581
$512$	−125.401	−5.54201
$513$	−5.94529	−0.262491
$514$	−69.7766	−3.07771
$515$	−12.8907	−0.568034
$516$	4.74795	0.209017
$517$	−0.348161	−0.0153121
$518$	−0.281832	−0.0123830
$519$	4.50959	0.197949
$520$	0	0
$521$	−5.14269	−0.225305	−0.112653	−	0.993634i	$-0.535935\pi$
$521$	−5.14269	−0.225305	−0.112653	+	0.993634i	$0.535935\pi$
$522$	−22.0346	−0.964427
$523$	9.83526	0.430066	0.215033	−	0.976607i	$-0.431014\pi$
$523$	9.83526	0.430066	0.215033	+	0.976607i	$0.431014\pi$
$524$	47.5531	2.07737
$525$	−2.46732	−0.107682
$526$	−49.6937	−2.16675
$527$	−18.8733	−0.822133
$528$	2.36764	0.103038
$529$	39.0116	1.69616
$530$	6.32983	0.274950
$531$	7.93495	0.344348
$532$	−34.1874	−1.48221
$533$	0	0
$534$	−19.9657	−0.864001
$535$	8.41541	0.363830
$536$	47.6590	2.05855
$537$	6.50536	0.280727
$538$	18.8149	0.811170
$539$	0.134788	0.00580573
$540$	−9.15132	−0.393810
$541$	12.6834	0.545302	0.272651	−	0.962113i	$-0.412100\pi$
$541$	12.6834	0.545302	0.272651	+	0.962113i	$0.412100\pi$
$542$	20.2415	0.869447
$543$	−11.5176	−0.494269
$544$	198.183	8.49701
$545$	0.0478880	0.00205130
$546$	0	0
$547$	15.1082	0.645979	0.322990	−	0.946403i	$-0.395312\pi$
$547$	15.1082	0.645979	0.322990	+	0.946403i	$0.395312\pi$
$548$	57.8069	2.46939
$549$	−3.77939	−0.161301
$550$	0.925840	0.0394780
$551$	−47.0562	−2.00466
$552$	82.2180	3.49943
$553$	13.0892	0.556608
$554$	36.5160	1.55141
$555$	−0.161109	−0.00683871
$556$	−74.0229	−3.13927
$557$	−0.658544	−0.0279034	−0.0139517	−	0.999903i	$-0.504441\pi$
$557$	−0.658544	−0.0279034	−0.0139517	+	0.999903i	$0.504441\pi$
$558$	−7.42877	−0.314485
$559$	0	0
$560$	−27.9548	−1.18130
$561$	−0.953326	−0.0402495
$562$	50.6941	2.13840
$563$	4.87871	0.205613	0.102807	−	0.994701i	$-0.467218\pi$
$563$	4.87871	0.205613	0.102807	+	0.994701i	$0.467218\pi$
$564$	−14.8533	−0.625435
$565$	9.25009	0.389154
$566$	62.6494	2.63335
$567$	1.00000	0.0419961
$568$	−111.623	−4.68361
$569$	−0.747003	−0.0313160	−0.0156580	−	0.999877i	$-0.504984\pi$
$569$	−0.747003	−0.0313160	−0.0156580	+	0.999877i	$0.504984\pi$
$570$	−26.3405	−1.10328
$571$	26.2447	1.09831	0.549153	−	0.835722i	$-0.314950\pi$
$571$	26.2447	1.09831	0.549153	+	0.835722i	$0.314950\pi$
$572$	0	0
$573$	−22.6574	−0.946526
$574$	−10.3725	−0.432941
$575$	19.4295	0.810265
$576$	42.8759	1.78650
$577$	34.6422	1.44217	0.721087	−	0.692844i	$-0.243643\pi$
$577$	34.6422	1.44217	0.721087	+	0.692844i	$0.243643\pi$
$578$	−91.9377	−3.82410
$579$	−12.8501	−0.534033
$580$	−72.4316	−3.00756
$581$	15.6291	0.648406
$582$	−38.7226	−1.60510
$583$	0.192572	0.00797550
$584$	15.6143	0.646126
$585$	0	0
$586$	−29.5806	−1.22196
$587$	17.7493	0.732592	0.366296	−	0.930498i	$-0.380626\pi$
$587$	17.7493	0.732592	0.366296	+	0.930498i	$0.380626\pi$
$588$	5.75034	0.237140
$589$	−15.8646	−0.653690
$590$	35.1557	1.44734
$591$	−6.15529	−0.253195
$592$	1.77826	0.0730860
$593$	24.5573	1.00845	0.504223	−	0.863573i	$-0.331779\pi$
$593$	24.5573	1.00845	0.504223	+	0.863573i	$0.331779\pi$
$594$	−0.375242	−0.0153964
$595$	11.2559	0.461448
$596$	−130.645	−5.35143
$597$	−6.10499	−0.249861
$598$	0	0
$599$	28.0271	1.14516	0.572578	−	0.819850i	$-0.305943\pi$
$599$	28.0271	1.14516	0.572578	+	0.819850i	$0.305943\pi$
$600$	25.7605	1.05167
$601$	30.6035	1.24834	0.624171	−	0.781288i	$-0.285437\pi$
$601$	30.6035	1.24834	0.624171	+	0.781288i	$0.285437\pi$
$602$	−2.29865	−0.0936860
$603$	−4.56472	−0.185890
$604$	15.3832	0.625934
$605$	17.4769	0.710539
$606$	−34.2291	−1.39046
$607$	−21.5723	−0.875593	−0.437797	−	0.899074i	$-0.644241\pi$
$607$	−21.5723	−0.875593	−0.437797	+	0.899074i	$0.644241\pi$
$608$	166.590	6.75610
$609$	7.91488	0.320727
$610$	−16.7445	−0.677966
$611$	0	0
$612$	−40.6709	−1.64402
$613$	−22.6684	−0.915569	−0.457784	−	0.889063i	$-0.651357\pi$
$613$	−22.6684	−0.915569	−0.457784	+	0.889063i	$0.651357\pi$
$614$	70.9127	2.86180
$615$	−5.92946	−0.239099
$616$	−1.40728	−0.0567011
$617$	−11.7948	−0.474841	−0.237421	−	0.971407i	$-0.576302\pi$
$617$	−11.7948	−0.474841	−0.237421	+	0.971407i	$0.576302\pi$
$618$	−22.5500	−0.907096
$619$	5.95151	0.239211	0.119606	−	0.992821i	$-0.461837\pi$
$619$	5.95151	0.239211	0.119606	+	0.992821i	$0.461837\pi$
$620$	−24.4197	−0.980719
$621$	−7.87475	−0.316003
$622$	3.24200	0.129992
$623$	7.17174	0.287330
$624$	0	0
$625$	−6.57578	−0.263031
$626$	−3.51150	−0.140348
$627$	−0.801353	−0.0320030
$628$	113.291	4.52079
$629$	−0.716012	−0.0285493
$630$	4.43048	0.176515
$631$	−1.30868	−0.0520978	−0.0260489	−	0.999661i	$-0.508293\pi$
$631$	−1.30868	−0.0520978	−0.0260489	+	0.999661i	$0.508293\pi$
$632$	−136.660	−5.43605
$633$	3.35021	0.133159
$634$	13.7894	0.547648
$635$	−33.3819	−1.32472
$636$	8.21550	0.325766
$637$	0	0
$638$	−2.96999	−0.117583
$639$	10.6911	0.422935
$640$	100.775	3.98349
$641$	32.0757	1.26691	0.633457	−	0.773778i	$-0.281635\pi$
$641$	32.0757	1.26691	0.633457	+	0.773778i	$0.281635\pi$
$642$	14.7213	0.581002
$643$	−14.3540	−0.566067	−0.283033	−	0.959110i	$-0.591341\pi$
$643$	−14.3540	−0.566067	−0.283033	+	0.959110i	$0.591341\pi$
$644$	−45.2824	−1.78438
$645$	−1.31402	−0.0517397
$646$	−117.064	−4.60582
$647$	26.7087	1.05003	0.525014	−	0.851094i	$-0.324060\pi$
$647$	26.7087	1.05003	0.525014	+	0.851094i	$0.324060\pi$
$648$	−10.4407	−0.410150
$649$	1.06954	0.0419830
$650$	0	0
$651$	2.66844	0.104584
$652$	−34.4875	−1.35064
$653$	−18.9610	−0.742001	−0.371000	−	0.928633i	$-0.620985\pi$
$653$	−18.9610	−0.742001	−0.371000	+	0.928633i	$0.620985\pi$
$654$	0.0837716	0.00327573
$655$	−13.1606	−0.514227
$656$	65.4470	2.55527
$657$	−1.49552	−0.0583460
$658$	7.19100	0.280334
$659$	−1.32694	−0.0516901	−0.0258451	−	0.999666i	$-0.508228\pi$
$659$	−1.32694	−0.0516901	−0.0258451	+	0.999666i	$0.508228\pi$
$660$	−1.23349	−0.0480135
$661$	−33.0787	−1.28661	−0.643307	−	0.765608i	$-0.722438\pi$
$661$	−33.0787	−1.28661	−0.643307	+	0.765608i	$0.722438\pi$
$662$	−35.6874	−1.38703
$663$	0	0
$664$	−163.179	−6.33259
$665$	9.46158	0.366904
$666$	−0.281832	−0.0109208
$667$	−62.3276	−2.41334
$668$	33.9009	1.31166
$669$	14.5203	0.561387
$670$	−20.2239	−0.781318
$671$	−0.509417	−0.0196658
$672$	−28.0204	−1.08091
$673$	6.08084	0.234399	0.117200	−	0.993108i	$-0.462608\pi$
$673$	6.08084	0.234399	0.117200	+	0.993108i	$0.462608\pi$
$674$	11.6169	0.447465
$675$	−2.46732	−0.0949670
$676$	0	0
$677$	−14.2996	−0.549579	−0.274790	−	0.961504i	$-0.588608\pi$
$677$	−14.2996	−0.549579	−0.274790	+	0.961504i	$0.588608\pi$
$678$	16.1814	0.621442
$679$	13.9093	0.533789
$680$	−117.520	−4.50668
$681$	−14.8507	−0.569080
$682$	−1.00131	−0.0383421
$683$	40.8971	1.56488	0.782441	−	0.622724i	$-0.213974\pi$
$683$	40.8971	1.56488	0.782441	+	0.622724i	$0.213974\pi$
$684$	−34.1874	−1.30719
$685$	−15.9984	−0.611268
$686$	−2.78394	−0.106291
$687$	21.6278	0.825151
$688$	14.5037	0.552948
$689$	0	0
$690$	−34.8889	−1.32820
$691$	−37.7189	−1.43489	−0.717447	−	0.696613i	$-0.754689\pi$
$691$	−37.7189	−1.43489	−0.717447	+	0.696613i	$0.754689\pi$
$692$	25.9317	0.985774
$693$	0.134788	0.00512017
$694$	7.48568	0.284153
$695$	20.4863	0.777089
$696$	−82.6370	−3.13235
$697$	−26.3521	−0.998156
$698$	−4.53272	−0.171566
$699$	7.71618	0.291853
$700$	−14.1879	−0.536252
$701$	3.52274	0.133052	0.0665261	−	0.997785i	$-0.478808\pi$
$701$	3.52274	0.133052	0.0665261	+	0.997785i	$0.478808\pi$
$702$	0	0
$703$	−0.601870	−0.0227000
$704$	5.77916	0.217810
$705$	4.11073	0.154819
$706$	−59.9956	−2.25797
$707$	12.2952	0.462409
$708$	45.6286	1.71483
$709$	2.41078	0.0905388	0.0452694	−	0.998975i	$-0.485585\pi$
$709$	2.41078	0.0905388	0.0452694	+	0.998975i	$0.485585\pi$
$710$	47.3669	1.77765
$711$	13.0892	0.490882
$712$	−74.8781	−2.80618
$713$	−21.0133	−0.786953
$714$	19.6902	0.736888
$715$	0	0
$716$	37.4080	1.39800
$717$	−3.84302	−0.143520
$718$	38.1286	1.42295
$719$	42.2743	1.57656	0.788282	−	0.615314i	$-0.210971\pi$
$719$	42.2743	1.57656	0.788282	+	0.615314i	$0.210971\pi$
$720$	−27.9548	−1.04181
$721$	8.10004	0.301661
$722$	−45.5076	−1.69362
$723$	4.88222	0.181572
$724$	−66.2303	−2.46143
$725$	−19.5285	−0.725270
$726$	30.5728	1.13466
$727$	18.8872	0.700488	0.350244	−	0.936659i	$-0.386099\pi$
$727$	18.8872	0.700488	0.350244	+	0.936659i	$0.386099\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.62589	−0.245235
$731$	−5.83987	−0.215995
$732$	−21.7328	−0.803266
$733$	35.3689	1.30638	0.653191	−	0.757193i	$-0.273430\pi$
$733$	35.3689	1.30638	0.653191	+	0.757193i	$0.273430\pi$
$734$	57.4931	2.12211
$735$	−1.59144	−0.0587012
$736$	220.654	8.13341
$737$	−0.615270	−0.0226637
$738$	−10.3725	−0.381818
$739$	−18.9800	−0.698191	−0.349095	−	0.937087i	$-0.613511\pi$
$739$	−18.9800	−0.698191	−0.349095	+	0.937087i	$0.613511\pi$
$740$	−0.926433	−0.0340563
$741$	0	0
$742$	−3.97742	−0.146016
$743$	41.4651	1.52121	0.760603	−	0.649217i	$-0.224903\pi$
$743$	41.4651	1.52121	0.760603	+	0.649217i	$0.224903\pi$
$744$	−27.8604	−1.02141
$745$	36.1569	1.32468
$746$	13.0929	0.479365
$747$	15.6291	0.571840
$748$	−5.48195	−0.200440
$749$	−5.28792	−0.193216
$750$	−33.0838	−1.20805
$751$	7.35910	0.268537	0.134269	−	0.990945i	$-0.457132\pi$
$751$	7.35910	0.268537	0.134269	+	0.990945i	$0.457132\pi$
$752$	−45.3726	−1.65457
$753$	11.3686	0.414296
$754$	0	0
$755$	−4.25740	−0.154943
$756$	5.75034	0.209138
$757$	−19.5518	−0.710623	−0.355311	−	0.934748i	$-0.615625\pi$
$757$	−19.5518	−0.710623	−0.355311	+	0.934748i	$0.615625\pi$
$758$	16.7840	0.609621
$759$	−1.06142	−0.0385271
$760$	−98.7857	−3.58333
$761$	37.1023	1.34496	0.672478	−	0.740117i	$-0.265230\pi$
$761$	37.1023	1.34496	0.672478	+	0.740117i	$0.265230\pi$
$762$	−58.3956	−2.11545
$763$	−0.0300910	−0.00108937
$764$	−130.288	−4.71364
$765$	11.2559	0.406959
$766$	−1.76867	−0.0639046
$767$	0	0
$768$	90.5363	3.26695
$769$	−30.7408	−1.10854	−0.554271	−	0.832337i	$-0.687003\pi$
$769$	−30.7408	−1.10854	−0.554271	+	0.832337i	$0.687003\pi$
$770$	0.597175	0.0215207
$771$	25.0640	0.902656
$772$	−73.8925	−2.65945
$773$	−1.97846	−0.0711603	−0.0355802	−	0.999367i	$-0.511328\pi$
$773$	−1.97846	−0.0711603	−0.0355802	+	0.999367i	$0.511328\pi$
$774$	−2.29865	−0.0826233
$775$	−6.58387	−0.236500
$776$	−145.223	−5.21320
$777$	0.101235	0.00363178
$778$	39.8581	1.42898
$779$	−22.1512	−0.793649
$780$	0	0
$781$	1.44104	0.0515644
$782$	−155.056	−5.54477
$783$	7.91488	0.282855
$784$	17.5657	0.627346
$785$	−31.3539	−1.11907
$786$	−23.0221	−0.821172
$787$	9.71005	0.346126	0.173063	−	0.984911i	$-0.444634\pi$
$787$	9.71005	0.346126	0.173063	+	0.984911i	$0.444634\pi$
$788$	−35.3950	−1.26089
$789$	17.8501	0.635481
$790$	57.9913	2.06324
$791$	−5.81240	−0.206665
$792$	−1.40728	−0.0500056
$793$	0	0
$794$	−21.0493	−0.747012
$795$	−2.27369	−0.0806395
$796$	−35.1058	−1.24429
$797$	−35.5792	−1.26028	−0.630140	−	0.776481i	$-0.717003\pi$
$797$	−35.5792	−1.26028	−0.630140	+	0.776481i	$0.717003\pi$
$798$	16.5513	0.585911
$799$	18.2692	0.646317
$800$	69.1353	2.44430
$801$	7.17174	0.253401
$802$	−26.1714	−0.924144
$803$	−0.201579	−0.00711356
$804$	−26.2487	−0.925720
$805$	12.5322	0.441702
$806$	0	0
$807$	−6.75838	−0.237906
$808$	−128.371	−4.51607
$809$	−22.4774	−0.790265	−0.395133	−	0.918624i	$-0.629301\pi$
$809$	−22.4774	−0.790265	−0.395133	+	0.918624i	$0.629301\pi$
$810$	4.43048	0.155671
$811$	−3.81370	−0.133917	−0.0669586	−	0.997756i	$-0.521330\pi$
$811$	−3.81370	−0.133917	−0.0669586	+	0.997756i	$0.521330\pi$
$812$	45.5132	1.59720
$813$	−7.27080	−0.254998
$814$	−0.0379876	−0.00133146
$815$	9.54463	0.334334
$816$	−124.238	−4.34921
$817$	−4.90892	−0.171741
$818$	−4.33912	−0.151714
$819$	0	0
$820$	−34.0964	−1.19070
$821$	−13.8299	−0.482668	−0.241334	−	0.970442i	$-0.577585\pi$
$821$	−13.8299	−0.482668	−0.241334	+	0.970442i	$0.577585\pi$
$822$	−27.9864	−0.976137
$823$	37.3025	1.30028	0.650142	−	0.759813i	$-0.274709\pi$
$823$	37.3025	1.30028	0.650142	+	0.759813i	$0.274709\pi$
$824$	−84.5702	−2.94614
$825$	−0.332564	−0.0115784
$826$	−22.0905	−0.768625
$827$	30.2891	1.05326	0.526628	−	0.850096i	$-0.323456\pi$
$827$	30.2891	1.05326	0.526628	+	0.850096i	$0.323456\pi$
$828$	−45.2824	−1.57367
$829$	16.4068	0.569831	0.284916	−	0.958553i	$-0.408034\pi$
$829$	16.4068	0.569831	0.284916	+	0.958553i	$0.408034\pi$
$830$	69.2446	2.40351
$831$	−13.1166	−0.455011
$832$	0	0
$833$	−7.07278	−0.245057
$834$	35.8371	1.24094
$835$	−9.38228	−0.324687
$836$	−4.60805	−0.159373
$837$	2.66844	0.0922346
$838$	−82.6981	−2.85676
$839$	−6.43862	−0.222286	−0.111143	−	0.993804i	$-0.535451\pi$
$839$	−6.43862	−0.222286	−0.111143	+	0.993804i	$0.535451\pi$
$840$	16.6158	0.573299
$841$	33.6453	1.16018
$842$	−108.231	−3.72987
$843$	−18.2095	−0.627167
$844$	19.2648	0.663122
$845$	0	0
$846$	7.19100	0.247232
$847$	−10.9818	−0.377340
$848$	25.0961	0.861804
$849$	−22.5038	−0.772330
$850$	−48.5820	−1.66635
$851$	−0.797199	−0.0273276
$852$	61.4777	2.10619
$853$	−2.02935	−0.0694836	−0.0347418	−	0.999396i	$-0.511061\pi$
$853$	−2.02935	−0.0694836	−0.0347418	+	0.999396i	$0.511061\pi$
$854$	10.5216	0.360042
$855$	9.46158	0.323579
$856$	55.2097	1.88703
$857$	−49.1347	−1.67841	−0.839205	−	0.543815i	$-0.816979\pi$
$857$	−49.1347	−1.67841	−0.839205	+	0.543815i	$0.816979\pi$
$858$	0	0
$859$	18.4424	0.629246	0.314623	−	0.949217i	$-0.398122\pi$
$859$	18.4424	0.629246	0.314623	+	0.949217i	$0.398122\pi$
$860$	−7.55608	−0.257660
$861$	3.72584	0.126976
$862$	−62.0748	−2.11428
$863$	−44.9156	−1.52894	−0.764472	−	0.644657i	$-0.777000\pi$
$863$	−44.9156	−1.52894	−0.764472	+	0.644657i	$0.777000\pi$
$864$	−28.0204	−0.953275
$865$	−7.17675	−0.244017
$866$	−95.3624	−3.24055
$867$	33.0243	1.12156
$868$	15.3444	0.520823
$869$	1.76426	0.0598485
$870$	35.0667	1.18887
$871$	0	0
$872$	0.314171	0.0106392
$873$	13.9093	0.470758
$874$	−130.338	−4.40874
$875$	11.8838	0.401746
$876$	−8.59977	−0.290559
$877$	5.10048	0.172231	0.0861154	−	0.996285i	$-0.472555\pi$
$877$	5.10048	0.172231	0.0861154	+	0.996285i	$0.472555\pi$
$878$	−90.3480	−3.04910
$879$	10.6254	0.358387
$880$	−3.76797	−0.127018
$881$	−20.3571	−0.685849	−0.342925	−	0.939363i	$-0.611418\pi$
$881$	−20.3571	−0.685849	−0.342925	+	0.939363i	$0.611418\pi$
$882$	−2.78394	−0.0937402
$883$	−58.6403	−1.97340	−0.986702	−	0.162538i	$-0.948032\pi$
$883$	−58.6403	−1.97340	−0.986702	+	0.162538i	$0.948032\pi$
$884$	0	0
$885$	−12.6280	−0.424486
$886$	30.7159	1.03192
$887$	−24.1256	−0.810058	−0.405029	−	0.914304i	$-0.632739\pi$
$887$	−24.1256	−0.810058	−0.405029	+	0.914304i	$0.632739\pi$
$888$	−1.05696	−0.0354694
$889$	20.9759	0.703508
$890$	31.7742	1.06508
$891$	0.134788	0.00451557
$892$	83.4965	2.79567
$893$	15.3568	0.513897
$894$	63.2500	2.11540
$895$	−10.3529	−0.346059
$896$	−63.3232	−2.11548
$897$	0	0
$898$	56.7382	1.89338
$899$	21.1203	0.704403
$900$	−14.1879	−0.472930
$901$	−10.1049	−0.336643
$902$	−1.39809	−0.0465514
$903$	0.825682	0.0274770
$904$	60.6856	2.01837
$905$	18.3296	0.609298
$906$	−7.44756	−0.247429
$907$	18.8848	0.627060	0.313530	−	0.949578i	$-0.398488\pi$
$907$	18.8848	0.627060	0.313530	+	0.949578i	$0.398488\pi$
$908$	−85.3966	−2.83398
$909$	12.2952	0.407806
$910$	0	0
$911$	−15.0686	−0.499245	−0.249623	−	0.968343i	$-0.580307\pi$
$911$	−15.0686	−0.499245	−0.249623	+	0.968343i	$0.580307\pi$
$912$	−104.433	−3.45813
$913$	2.10662	0.0697189
$914$	71.8017	2.37499
$915$	6.01468	0.198839
$916$	124.367	4.10920
$917$	8.26962	0.273087
$918$	19.6902	0.649874
$919$	13.6354	0.449790	0.224895	−	0.974383i	$-0.427796\pi$
$919$	13.6354	0.449790	0.224895	+	0.974383i	$0.427796\pi$
$920$	−130.845	−4.31384
$921$	−25.4720	−0.839332
$922$	3.36909	0.110955
$923$	0	0
$924$	0.775076	0.0254981
$925$	−0.249778	−0.00821266
$926$	−59.5339	−1.95641
$927$	8.10004	0.266040
$928$	−221.778	−7.28023
$929$	−26.0326	−0.854103	−0.427051	−	0.904227i	$-0.640448\pi$
$929$	−26.0326	−0.854103	−0.427051	+	0.904227i	$0.640448\pi$
$930$	11.8225	0.387674
$931$	−5.94529	−0.194849
$932$	44.3706	1.45341
$933$	−1.16454	−0.0381252
$934$	−44.1856	−1.44580
$935$	1.51716	0.0496165
$936$	0	0
$937$	−25.5354	−0.834206	−0.417103	−	0.908859i	$-0.636954\pi$
$937$	−25.5354	−0.834206	−0.417103	+	0.908859i	$0.636954\pi$
$938$	12.7079	0.414928
$939$	1.26134	0.0411623
$940$	23.6381	0.770990
$941$	27.5605	0.898446	0.449223	−	0.893420i	$-0.351701\pi$
$941$	27.5605	0.898446	0.449223	+	0.893420i	$0.351701\pi$
$942$	−54.8480	−1.78704
$943$	−29.3400	−0.955443
$944$	139.383	4.53653
$945$	−1.59144	−0.0517696
$946$	−0.309831	−0.0100735
$947$	25.3044	0.822281	0.411141	−	0.911572i	$-0.365131\pi$
$947$	25.3044	0.822281	0.411141	+	0.911572i	$0.365131\pi$
$948$	75.2671	2.44456
$949$	0	0
$950$	−40.8374	−1.32494
$951$	−4.95320	−0.160619
$952$	73.8450	2.39333
$953$	−40.8004	−1.32166	−0.660828	−	0.750537i	$-0.729795\pi$
$953$	−40.8004	−1.32166	−0.660828	+	0.750537i	$0.729795\pi$
$954$	−3.97742	−0.128774
$955$	36.0579	1.16681
$956$	−22.0987	−0.714722
$957$	1.06683	0.0344857
$958$	63.1132	2.03910
$959$	10.0528	0.324622
$960$	−68.2345	−2.20226
$961$	−23.8794	−0.770305
$962$	0	0
$963$	−5.28792	−0.170401
$964$	28.0744	0.904215
$965$	20.4502	0.658315
$966$	21.9228	0.705356
$967$	0.377530	0.0121406	0.00607028	−	0.999982i	$-0.498068\pi$
$967$	0.377530	0.0121406	0.00607028	+	0.999982i	$0.498068\pi$
$968$	114.658	3.68526
$969$	42.0498	1.35083
$970$	61.6248	1.97865
$971$	28.1109	0.902120	0.451060	−	0.892494i	$-0.351046\pi$
$971$	28.1109	0.902120	0.451060	+	0.892494i	$0.351046\pi$
$972$	5.75034	0.184442
$973$	−12.8728	−0.412683
$974$	97.3625	3.11970
$975$	0	0
$976$	−66.3876	−2.12502
$977$	6.53388	0.209037	0.104519	−	0.994523i	$-0.466670\pi$
$977$	6.53388	0.209037	0.104519	+	0.994523i	$0.466670\pi$
$978$	16.6966	0.533899
$979$	0.966664	0.0308947
$980$	−9.15132	−0.292328
$981$	−0.0300910	−0.000960731 0
$982$	12.0680	0.385105
$983$	4.29547	0.137004	0.0685021	−	0.997651i	$-0.478178\pi$
$983$	4.29547	0.137004	0.0685021	+	0.997651i	$0.478178\pi$
$984$	−38.9005	−1.24010
$985$	9.79578	0.312119
$986$	155.846	4.96314
$987$	−2.58303	−0.0822186
$988$	0	0
$989$	−6.50204	−0.206753
$990$	0.597175	0.0189795
$991$	−7.50668	−0.238458	−0.119229	−	0.992867i	$-0.538042\pi$
$991$	−7.50668	−0.238458	−0.119229	+	0.992867i	$0.538042\pi$
$992$	−74.7707	−2.37397
$993$	12.8190	0.406799
$994$	−29.7635	−0.944042
$995$	9.71573	0.308009
$996$	89.8728	2.84773
$997$	−14.3663	−0.454985	−0.227493	−	0.973780i	$-0.573053\pi$
$997$	−14.3663	−0.454985	−0.227493	+	0.973780i	$0.573053\pi$
$998$	58.1508	1.84073
$999$	0.101235	0.00320293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.2.a.bg.1.2	✓	15
13.12	even	2		3549.2.a.bh.1.14	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3549.2.a.bg.1.2	✓	15	1.1	even	1	trivial
3549.2.a.bh.1.14	yes	15	13.12	even	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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3549.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78394 q^{2} +1.00000 q^{3} +5.75034 q^{4} -1.59144 q^{5} -2.78394 q^{6} +1.00000 q^{7} -10.4407 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.78394 q^{2} +1.00000 q^{3} +5.75034 q^{4} -1.59144 q^{5} -2.78394 q^{6} +1.00000 q^{7} -10.4407 q^{8} +1.00000 q^{9} +4.43048 q^{10} +0.134788 q^{11} +5.75034 q^{12} -2.78394 q^{14} -1.59144 q^{15} +17.5657 q^{16} -7.07278 q^{17} -2.78394 q^{18} -5.94529 q^{19} -9.15132 q^{20} +1.00000 q^{21} -0.375242 q^{22} -7.87475 q^{23} -10.4407 q^{24} -2.46732 q^{25} +1.00000 q^{27} +5.75034 q^{28} +7.91488 q^{29} +4.43048 q^{30} +2.66844 q^{31} -28.0204 q^{32} +0.134788 q^{33} +19.6902 q^{34} -1.59144 q^{35} +5.75034 q^{36} +0.101235 q^{37} +16.5513 q^{38} +16.6158 q^{40} +3.72584 q^{41} -2.78394 q^{42} +0.825682 q^{43} +0.775076 q^{44} -1.59144 q^{45} +21.9228 q^{46} -2.58303 q^{47} +17.5657 q^{48} +1.00000 q^{49} +6.86886 q^{50} -7.07278 q^{51} +1.42870 q^{53} -2.78394 q^{54} -0.214507 q^{55} -10.4407 q^{56} -5.94529 q^{57} -22.0346 q^{58} +7.93495 q^{59} -9.15132 q^{60} -3.77939 q^{61} -7.42877 q^{62} +1.00000 q^{63} +42.8759 q^{64} -0.375242 q^{66} -4.56472 q^{67} -40.6709 q^{68} -7.87475 q^{69} +4.43048 q^{70} +10.6911 q^{71} -10.4407 q^{72} -1.49552 q^{73} -0.281832 q^{74} -2.46732 q^{75} -34.1874 q^{76} +0.134788 q^{77} +13.0892 q^{79} -27.9548 q^{80} +1.00000 q^{81} -10.3725 q^{82} +15.6291 q^{83} +5.75034 q^{84} +11.2559 q^{85} -2.29865 q^{86} +7.91488 q^{87} -1.40728 q^{88} +7.17174 q^{89} +4.43048 q^{90} -45.2824 q^{92} +2.66844 q^{93} +7.19100 q^{94} +9.46158 q^{95} -28.0204 q^{96} +13.9093 q^{97} -2.78394 q^{98} +0.134788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 2 q^{2} + 15 q^{3} + 28 q^{4} + 9 q^{5} - 2 q^{6} + 15 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q - 2 * q^2 + 15 * q^3 + 28 * q^4 + 9 * q^5 - 2 * q^6 + 15 * q^7 - 9 * q^8 + 15 * q^9	\( 15 q - 2 q^{2} + 15 q^{3} + 28 q^{4} + 9 q^{5} - 2 q^{6} + 15 q^{7} - 9 q^{8} + 15 q^{9} + 21 q^{10} - 5 q^{11} + 28 q^{12} - 2 q^{14} + 9 q^{15} + 50 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} + 23 q^{20} + 15 q^{21} + 21 q^{22} + 4 q^{23} - 9 q^{24} + 50 q^{25} + 15 q^{27} + 28 q^{28} + 9 q^{29} + 21 q^{30} - 7 q^{31} - 35 q^{32} - 5 q^{33} + 2 q^{34} + 9 q^{35} + 28 q^{36} - 17 q^{37} - 12 q^{38} + 46 q^{40} + 22 q^{41} - 2 q^{42} + 36 q^{43} - 29 q^{44} + 9 q^{45} + q^{46} + 12 q^{47} + 50 q^{48} + 15 q^{49} - 53 q^{50} - q^{51} - 5 q^{53} - 2 q^{54} + 43 q^{55} - 9 q^{56} - 3 q^{57} - 29 q^{58} + 29 q^{59} + 23 q^{60} + 12 q^{61} + 14 q^{62} + 15 q^{63} + 95 q^{64} + 21 q^{66} + 12 q^{67} - 16 q^{68} + 4 q^{69} + 21 q^{70} - 36 q^{71} - 9 q^{72} + 29 q^{73} - 5 q^{74} + 50 q^{75} - 25 q^{76} - 5 q^{77} + 35 q^{79} + 89 q^{80} + 15 q^{81} + 51 q^{82} + 10 q^{83} + 28 q^{84} - 23 q^{85} - 19 q^{86} + 9 q^{87} + 73 q^{88} - 25 q^{89} + 21 q^{90} - 31 q^{92} - 7 q^{93} - 19 q^{94} - 7 q^{95} - 35 q^{96} + 26 q^{97} - 2 q^{98} - 5 q^{99}+O(q^{100}) \) 15 * q - 2 * q^2 + 15 * q^3 + 28 * q^4 + 9 * q^5 - 2 * q^6 + 15 * q^7 - 9 * q^8 + 15 * q^9 + 21 * q^10 - 5 * q^11 + 28 * q^12 - 2 * q^14 + 9 * q^15 + 50 * q^16 - q^17 - 2 * q^18 - 3 * q^19 + 23 * q^20 + 15 * q^21 + 21 * q^22 + 4 * q^23 - 9 * q^24 + 50 * q^25 + 15 * q^27 + 28 * q^28 + 9 * q^29 + 21 * q^30 - 7 * q^31 - 35 * q^32 - 5 * q^33 + 2 * q^34 + 9 * q^35 + 28 * q^36 - 17 * q^37 - 12 * q^38 + 46 * q^40 + 22 * q^41 - 2 * q^42 + 36 * q^43 - 29 * q^44 + 9 * q^45 + q^46 + 12 * q^47 + 50 * q^48 + 15 * q^49 - 53 * q^50 - q^51 - 5 * q^53 - 2 * q^54 + 43 * q^55 - 9 * q^56 - 3 * q^57 - 29 * q^58 + 29 * q^59 + 23 * q^60 + 12 * q^61 + 14 * q^62 + 15 * q^63 + 95 * q^64 + 21 * q^66 + 12 * q^67 - 16 * q^68 + 4 * q^69 + 21 * q^70 - 36 * q^71 - 9 * q^72 + 29 * q^73 - 5 * q^74 + 50 * q^75 - 25 * q^76 - 5 * q^77 + 35 * q^79 + 89 * q^80 + 15 * q^81 + 51 * q^82 + 10 * q^83 + 28 * q^84 - 23 * q^85 - 19 * q^86 + 9 * q^87 + 73 * q^88 - 25 * q^89 + 21 * q^90 - 31 * q^92 - 7 * q^93 - 19 * q^94 - 7 * q^95 - 35 * q^96 + 26 * q^97 - 2 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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