LMFDB - Embedded newform 1859.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,2,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, 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("1859.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1859.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:	$14.8441897358$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 18x^{5} + 7x^{4} - 22x^{3} - 3x^{2} + 6x + 1 $ x^8 - 2x^7 - 8x^6 + 18x^5 + 7x^4 - 22x^3 - 3x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.69931$ of defining polynomial
Character	$\chi$	$=$	1859.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69931 q^{2} -1.59679 q^{3} +5.28626 q^{4} -1.39555 q^{5} -4.31023 q^{6} -4.07855 q^{7} +8.87064 q^{8} -0.450256 q^{9} +O(q^{10})$	\(q+2.69931 q^{2} -1.59679 q^{3} +5.28626 q^{4} -1.39555 q^{5} -4.31023 q^{6} -4.07855 q^{7} +8.87064 q^{8} -0.450256 q^{9} -3.76701 q^{10} +1.00000 q^{11} -8.44106 q^{12} -11.0093 q^{14} +2.22840 q^{15} +13.3721 q^{16} -4.19112 q^{17} -1.21538 q^{18} -3.06871 q^{19} -7.37723 q^{20} +6.51259 q^{21} +2.69931 q^{22} -4.66037 q^{23} -14.1646 q^{24} -3.05245 q^{25} +5.50934 q^{27} -21.5603 q^{28} +1.18231 q^{29} +6.01513 q^{30} -4.60617 q^{31} +18.3540 q^{32} -1.59679 q^{33} -11.3131 q^{34} +5.69180 q^{35} -2.38017 q^{36} -2.26880 q^{37} -8.28340 q^{38} -12.3794 q^{40} -6.57863 q^{41} +17.5795 q^{42} +4.36704 q^{43} +5.28626 q^{44} +0.628353 q^{45} -12.5798 q^{46} -4.67269 q^{47} -21.3524 q^{48} +9.63454 q^{49} -8.23950 q^{50} +6.69235 q^{51} +3.57009 q^{53} +14.8714 q^{54} -1.39555 q^{55} -36.1793 q^{56} +4.90009 q^{57} +3.19141 q^{58} -2.70685 q^{59} +11.7799 q^{60} -8.11135 q^{61} -12.4335 q^{62} +1.83639 q^{63} +22.7991 q^{64} -4.31023 q^{66} +15.3727 q^{67} -22.1554 q^{68} +7.44165 q^{69} +15.3639 q^{70} -10.7433 q^{71} -3.99405 q^{72} -10.7336 q^{73} -6.12420 q^{74} +4.87413 q^{75} -16.2220 q^{76} -4.07855 q^{77} +4.42287 q^{79} -18.6613 q^{80} -7.44650 q^{81} -17.7577 q^{82} +11.5956 q^{83} +34.4273 q^{84} +5.84891 q^{85} +11.7880 q^{86} -1.88790 q^{87} +8.87064 q^{88} +8.50954 q^{89} +1.69612 q^{90} -24.6360 q^{92} +7.35509 q^{93} -12.6130 q^{94} +4.28253 q^{95} -29.3076 q^{96} +4.35531 q^{97} +26.0066 q^{98} -0.450256 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 4 q^{4} - 8 q^{5} - 4 q^{6} - 14 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 + 4 * q^4 - 8 * q^5 - 4 * q^6 - 14 * q^7 + 6 * q^8 + 2 * q^9	$ 8 q - 2 q^{2} + 4 q^{4} - 8 q^{5} - 4 q^{6} - 14 q^{7} + 6 q^{8} + 2 q^{9} + 10 q^{10} + 8 q^{11} - 8 q^{12} - 6 q^{14} - 2 q^{15} + 12 q^{16} + 6 q^{17} - 6 q^{18} - 4 q^{19} - 16 q^{20} - 2 q^{21} - 2 q^{22} - 14 q^{23} - 26 q^{24} + 12 q^{25} + 24 q^{27} - 20 q^{28} - 10 q^{29} + 14 q^{30} - 32 q^{31} + 40 q^{32} - 14 q^{34} + 10 q^{35} - 24 q^{37} + 12 q^{38} - 14 q^{40} - 2 q^{41} - 2 q^{42} + 14 q^{43} + 4 q^{44} + 4 q^{45} - 14 q^{46} - 18 q^{47} - 6 q^{48} + 2 q^{49} - 38 q^{50} + 12 q^{51} - 8 q^{53} - 2 q^{54} - 8 q^{55} - 28 q^{56} - 24 q^{57} + 14 q^{58} - 18 q^{59} + 14 q^{60} - 4 q^{61} - 8 q^{63} + 6 q^{64} - 4 q^{66} + 14 q^{67} - 34 q^{68} - 10 q^{69} - 18 q^{70} + 12 q^{71} - 8 q^{72} - 26 q^{73} + 2 q^{74} - 18 q^{75} - 54 q^{76} - 14 q^{77} + 26 q^{79} - 24 q^{80} - 16 q^{81} - 64 q^{82} - 16 q^{83} + 74 q^{84} - 56 q^{85} - 32 q^{86} - 18 q^{87} + 6 q^{88} + 8 q^{89} - 20 q^{90} - 30 q^{92} - 48 q^{93} - 4 q^{94} + 22 q^{95} - 16 q^{96} - 20 q^{97} + 46 q^{98} + 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 + 4 * q^4 - 8 * q^5 - 4 * q^6 - 14 * q^7 + 6 * q^8 + 2 * q^9 + 10 * q^10 + 8 * q^11 - 8 * q^12 - 6 * q^14 - 2 * q^15 + 12 * q^16 + 6 * q^17 - 6 * q^18 - 4 * q^19 - 16 * q^20 - 2 * q^21 - 2 * q^22 - 14 * q^23 - 26 * q^24 + 12 * q^25 + 24 * q^27 - 20 * q^28 - 10 * q^29 + 14 * q^30 - 32 * q^31 + 40 * q^32 - 14 * q^34 + 10 * q^35 - 24 * q^37 + 12 * q^38 - 14 * q^40 - 2 * q^41 - 2 * q^42 + 14 * q^43 + 4 * q^44 + 4 * q^45 - 14 * q^46 - 18 * q^47 - 6 * q^48 + 2 * q^49 - 38 * q^50 + 12 * q^51 - 8 * q^53 - 2 * q^54 - 8 * q^55 - 28 * q^56 - 24 * q^57 + 14 * q^58 - 18 * q^59 + 14 * q^60 - 4 * q^61 - 8 * q^63 + 6 * q^64 - 4 * q^66 + 14 * q^67 - 34 * q^68 - 10 * q^69 - 18 * q^70 + 12 * q^71 - 8 * q^72 - 26 * q^73 + 2 * q^74 - 18 * q^75 - 54 * q^76 - 14 * q^77 + 26 * q^79 - 24 * q^80 - 16 * q^81 - 64 * q^82 - 16 * q^83 + 74 * q^84 - 56 * q^85 - 32 * q^86 - 18 * q^87 + 6 * q^88 + 8 * q^89 - 20 * q^90 - 30 * q^92 - 48 * q^93 - 4 * q^94 + 22 * q^95 - 16 * q^96 - 20 * q^97 + 46 * q^98 + 2 * 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 $

$2$	2.69931	1.90870	0.954350	−	0.298692i	$-0.0965504\pi$
$2$	2.69931	1.90870	0.954350	+	0.298692i	$0.0965504\pi$
$3$	−1.59679	−0.921908	−0.460954	−	0.887424i	$-0.652493\pi$
$3$	−1.59679	−0.921908	−0.460954	+	0.887424i	$0.652493\pi$
$4$	5.28626	2.64313
$5$	−1.39555	−0.624107	−0.312054	−	0.950064i	$-0.601017\pi$
$5$	−1.39555	−0.624107	−0.312054	+	0.950064i	$0.601017\pi$
$6$	−4.31023	−1.75965
$7$	−4.07855	−1.54155	−0.770773	−	0.637110i	$-0.780130\pi$
$7$	−4.07855	−1.54155	−0.770773	+	0.637110i	$0.780130\pi$
$8$	8.87064	3.13624
$9$	−0.450256	−0.150085
$10$	−3.76701	−1.19123
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−8.44106	−2.43673
$13$	0	0
$13$	0	0
$14$	−11.0093	−2.94235
$15$	2.22840	0.575370
$16$	13.3721	3.34301
$17$	−4.19112	−1.01650	−0.508248	−	0.861211i	$-0.669707\pi$
$17$	−4.19112	−1.01650	−0.508248	+	0.861211i	$0.669707\pi$
$18$	−1.21538	−0.286467
$19$	−3.06871	−0.704011	−0.352005	−	0.935998i	$-0.614500\pi$
$19$	−3.06871	−0.704011	−0.352005	+	0.935998i	$0.614500\pi$
$20$	−7.37723	−1.64960
$21$	6.51259	1.42116
$22$	2.69931	0.575494
$23$	−4.66037	−0.971755	−0.485877	−	0.874027i	$-0.661500\pi$
$23$	−4.66037	−0.971755	−0.485877	+	0.874027i	$0.661500\pi$
$24$	−14.1646	−2.89133
$25$	−3.05245	−0.610490
$26$	0	0
$27$	5.50934	1.06027
$28$	−21.5603	−4.07451
$29$	1.18231	0.219549	0.109774	−	0.993957i	$-0.464987\pi$
$29$	1.18231	0.219549	0.109774	+	0.993957i	$0.464987\pi$
$30$	6.01513	1.09821
$31$	−4.60617	−0.827292	−0.413646	−	0.910438i	$-0.635745\pi$
$31$	−4.60617	−0.827292	−0.413646	+	0.910438i	$0.635745\pi$
$32$	18.3540	3.24456
$33$	−1.59679	−0.277966
$34$	−11.3131	−1.94019
$35$	5.69180	0.962090
$36$	−2.38017	−0.396695
$37$	−2.26880	−0.372989	−0.186494	−	0.982456i	$-0.559713\pi$
$37$	−2.26880	−0.372989	−0.186494	+	0.982456i	$0.559713\pi$
$38$	−8.28340	−1.34374
$39$	0	0
$40$	−12.3794	−1.95735
$41$	−6.57863	−1.02741	−0.513705	−	0.857967i	$-0.671727\pi$
$41$	−6.57863	−1.02741	−0.513705	+	0.857967i	$0.671727\pi$
$42$	17.5795	2.71257
$43$	4.36704	0.665968	0.332984	−	0.942932i	$-0.391944\pi$
$43$	4.36704	0.665968	0.332984	+	0.942932i	$0.391944\pi$
$44$	5.28626	0.796934
$45$	0.628353	0.0936693
$46$	−12.5798	−1.85479
$47$	−4.67269	−0.681582	−0.340791	−	0.940139i	$-0.610695\pi$
$47$	−4.67269	−0.681582	−0.340791	+	0.940139i	$0.610695\pi$
$48$	−21.3524	−3.08195
$49$	9.63454	1.37636
$50$	−8.23950	−1.16524
$51$	6.69235	0.937116
$52$	0	0
$53$	3.57009	0.490390	0.245195	−	0.969474i	$-0.421148\pi$
$53$	3.57009	0.490390	0.245195	+	0.969474i	$0.421148\pi$
$54$	14.8714	2.02374
$55$	−1.39555	−0.188175
$56$	−36.1793	−4.83466
$57$	4.90009	0.649033
$58$	3.19141	0.419053
$59$	−2.70685	−0.352401	−0.176201	−	0.984354i	$-0.556381\pi$
$59$	−2.70685	−0.352401	−0.176201	+	0.984354i	$0.556381\pi$
$60$	11.7799	1.52078
$61$	−8.11135	−1.03855	−0.519276	−	0.854607i	$-0.673798\pi$
$61$	−8.11135	−1.03855	−0.519276	+	0.854607i	$0.673798\pi$
$62$	−12.4335	−1.57905
$63$	1.83639	0.231363
$64$	22.7991	2.84988
$65$	0	0
$66$	−4.31023	−0.530553
$67$	15.3727	1.87807	0.939037	−	0.343817i	$-0.111720\pi$
$67$	15.3727	1.87807	0.939037	+	0.343817i	$0.111720\pi$
$68$	−22.1554	−2.68673
$69$	7.44165	0.895869
$70$	15.3639	1.83634
$71$	−10.7433	−1.27499	−0.637495	−	0.770454i	$-0.720029\pi$
$71$	−10.7433	−1.27499	−0.637495	+	0.770454i	$0.720029\pi$
$72$	−3.99405	−0.470704
$73$	−10.7336	−1.25627	−0.628136	−	0.778104i	$-0.716182\pi$
$73$	−10.7336	−1.25627	−0.628136	+	0.778104i	$0.716182\pi$
$74$	−6.12420	−0.711923
$75$	4.87413	0.562816
$76$	−16.2220	−1.86079
$77$	−4.07855	−0.464794
$78$	0	0
$79$	4.42287	0.497612	0.248806	−	0.968553i	$-0.419962\pi$
$79$	4.42287	0.497612	0.248806	+	0.968553i	$0.419962\pi$
$80$	−18.6613	−2.08640
$81$	−7.44650	−0.827389
$82$	−17.7577	−1.96101
$83$	11.5956	1.27278	0.636390	−	0.771367i	$-0.280427\pi$
$83$	11.5956	1.27278	0.636390	+	0.771367i	$0.280427\pi$
$84$	34.4273	3.75632
$85$	5.84891	0.634403
$86$	11.7880	1.27113
$87$	−1.88790	−0.202404
$88$	8.87064	0.945613
$89$	8.50954	0.902009	0.451005	−	0.892522i	$-0.351066\pi$
$89$	8.50954	0.902009	0.451005	+	0.892522i	$0.351066\pi$
$90$	1.69612	0.178786
$91$	0	0
$92$	−24.6360	−2.56848
$93$	7.35509	0.762688
$94$	−12.6130	−1.30093
$95$	4.28253	0.439378
$96$	−29.3076	−2.99119
$97$	4.35531	0.442215	0.221107	−	0.975249i	$-0.429033\pi$
$97$	4.35531	0.442215	0.221107	+	0.975249i	$0.429033\pi$
$98$	26.0066	2.62706
$99$	−0.450256	−0.0452524
$100$	−16.1361	−1.61361
$101$	10.7474	1.06941	0.534705	−	0.845039i	$-0.320423\pi$
$101$	10.7474	1.06941	0.534705	+	0.845039i	$0.320423\pi$
$102$	18.0647	1.78867
$103$	−9.39568	−0.925784	−0.462892	−	0.886415i	$-0.653188\pi$
$103$	−9.39568	−0.925784	−0.462892	+	0.886415i	$0.653188\pi$
$104$	0	0
$105$	−9.08862	−0.886959
$106$	9.63677	0.936006
$107$	13.5322	1.30821	0.654105	−	0.756404i	$-0.273046\pi$
$107$	13.5322	1.30821	0.654105	+	0.756404i	$0.273046\pi$
$108$	29.1238	2.80244
$109$	−9.84990	−0.943449	−0.471725	−	0.881746i	$-0.656368\pi$
$109$	−9.84990	−0.943449	−0.471725	+	0.881746i	$0.656368\pi$
$110$	−3.76701	−0.359170
$111$	3.62280	0.343861
$112$	−54.5386	−5.15341
$113$	0.869196	0.0817670	0.0408835	−	0.999164i	$-0.486983\pi$
$113$	0.869196	0.0817670	0.0408835	+	0.999164i	$0.486983\pi$
$114$	13.2269	1.23881
$115$	6.50377	0.606480
$116$	6.24998	0.580296
$117$	0	0
$118$	−7.30661	−0.672628
$119$	17.0937	1.56698
$120$	19.7673	1.80450
$121$	1.00000	0.0909091
$122$	−21.8950	−1.98228
$123$	10.5047	0.947177
$124$	−24.3494	−2.18664
$125$	11.2376	1.00512
$126$	4.95698	0.441603
$127$	6.60777	0.586344	0.293172	−	0.956060i	$-0.405289\pi$
$127$	6.60777	0.586344	0.293172	+	0.956060i	$0.405289\pi$
$128$	24.8336	2.19500
$129$	−6.97326	−0.613961
$130$	0	0
$131$	6.22070	0.543505	0.271752	−	0.962367i	$-0.412397\pi$
$131$	6.22070	0.543505	0.271752	+	0.962367i	$0.412397\pi$
$132$	−8.44106	−0.734700
$133$	12.5159	1.08526
$134$	41.4956	3.58468
$135$	−7.68854	−0.661724
$136$	−37.1779	−3.18798
$137$	−14.7490	−1.26009	−0.630045	−	0.776559i	$-0.716964\pi$
$137$	−14.7490	−1.26009	−0.630045	+	0.776559i	$0.716964\pi$
$138$	20.0873	1.70994
$139$	9.20266	0.780560	0.390280	−	0.920696i	$-0.372378\pi$
$139$	9.20266	0.780560	0.390280	+	0.920696i	$0.372378\pi$
$140$	30.0884	2.54293
$141$	7.46131	0.628356
$142$	−28.9994	−2.43357
$143$	0	0
$144$	−6.02084	−0.501737
$145$	−1.64996	−0.137022
$146$	−28.9733	−2.39784
$147$	−15.3844	−1.26888
$148$	−11.9935	−0.985858
$149$	−16.0582	−1.31554	−0.657768	−	0.753221i	$-0.728499\pi$
$149$	−16.0582	−1.31554	−0.657768	+	0.753221i	$0.728499\pi$
$150$	13.1568	1.07425
$151$	−10.4601	−0.851232	−0.425616	−	0.904904i	$-0.639943\pi$
$151$	−10.4601	−0.851232	−0.425616	+	0.904904i	$0.639943\pi$
$152$	−27.2214	−2.20795
$153$	1.88708	0.152561
$154$	−11.0093	−0.887151
$155$	6.42812	0.516319
$156$	0	0
$157$	3.19211	0.254758	0.127379	−	0.991854i	$-0.459344\pi$
$157$	3.19211	0.254758	0.127379	+	0.991854i	$0.459344\pi$
$158$	11.9387	0.949791
$159$	−5.70069	−0.452094
$160$	−25.6139	−2.02496
$161$	19.0075	1.49800
$162$	−20.1004	−1.57924
$163$	13.4920	1.05677	0.528387	−	0.849004i	$-0.322797\pi$
$163$	13.4920	1.05677	0.528387	+	0.849004i	$0.322797\pi$
$164$	−34.7764	−2.71558
$165$	2.22840	0.173481
$166$	31.3000	2.42935
$167$	−18.5662	−1.43669	−0.718346	−	0.695686i	$-0.755101\pi$
$167$	−18.5662	−1.43669	−0.718346	+	0.695686i	$0.755101\pi$
$168$	57.7708	4.45712
$169$	0	0
$170$	15.7880	1.21088
$171$	1.38170	0.105662
$172$	23.0853	1.76024
$173$	−9.69490	−0.737090	−0.368545	−	0.929610i	$-0.620144\pi$
$173$	−9.69490	−0.737090	−0.368545	+	0.929610i	$0.620144\pi$
$174$	−5.09602	−0.386328
$175$	12.4496	0.941098
$176$	13.3721	1.00796
$177$	4.32227	0.324882
$178$	22.9699	1.72166
$179$	−21.8603	−1.63391	−0.816956	−	0.576699i	$-0.804340\pi$
$179$	−21.8603	−1.63391	−0.816956	+	0.576699i	$0.804340\pi$
$180$	3.32164	0.247580
$181$	11.8666	0.882036	0.441018	−	0.897498i	$-0.354618\pi$
$181$	11.8666	0.882036	0.441018	+	0.897498i	$0.354618\pi$
$182$	0	0
$183$	12.9521	0.957450
$184$	−41.3405	−3.04766
$185$	3.16622	0.232785
$186$	19.8537	1.45574
$187$	−4.19112	−0.306485
$188$	−24.7011	−1.80151
$189$	−22.4701	−1.63446
$190$	11.5599	0.838641
$191$	13.3643	0.967010	0.483505	−	0.875342i	$-0.339364\pi$
$191$	13.3643	0.967010	0.483505	+	0.875342i	$0.339364\pi$
$192$	−36.4053	−2.62733
$193$	8.98604	0.646829	0.323415	−	0.946257i	$-0.395169\pi$
$193$	8.98604	0.646829	0.323415	+	0.946257i	$0.395169\pi$
$194$	11.7563	0.844055
$195$	0	0
$196$	50.9307	3.63791
$197$	4.90459	0.349437	0.174719	−	0.984618i	$-0.444098\pi$
$197$	4.90459	0.349437	0.174719	+	0.984618i	$0.444098\pi$
$198$	−1.21538	−0.0863732
$199$	−7.51152	−0.532477	−0.266239	−	0.963907i	$-0.585781\pi$
$199$	−7.51152	−0.532477	−0.266239	+	0.963907i	$0.585781\pi$
$200$	−27.0772	−1.91465
$201$	−24.5470	−1.73141
$202$	29.0106	2.04118
$203$	−4.82209	−0.338445
$204$	35.3775	2.47692
$205$	9.18078	0.641214
$206$	−25.3618	−1.76704
$207$	2.09836	0.145846
$208$	0	0
$209$	−3.06871	−0.212267
$210$	−24.5330	−1.69294
$211$	17.5677	1.20941	0.604707	−	0.796448i	$-0.293290\pi$
$211$	17.5677	1.20941	0.604707	+	0.796448i	$0.293290\pi$
$212$	18.8724	1.29616
$213$	17.1548	1.17542
$214$	36.5276	2.49698
$215$	−6.09441	−0.415636
$216$	48.8714	3.32528
$217$	18.7865	1.27531
$218$	−26.5879	−1.80076
$219$	17.1393	1.15817
$220$	−7.37723	−0.497373
$221$	0	0
$222$	9.77907	0.656328
$223$	12.3275	0.825510	0.412755	−	0.910842i	$-0.364567\pi$
$223$	12.3275	0.825510	0.412755	+	0.910842i	$0.364567\pi$
$224$	−74.8578	−5.00164
$225$	1.37438	0.0916255
$226$	2.34623	0.156069
$227$	6.17877	0.410100	0.205050	−	0.978752i	$-0.434264\pi$
$227$	6.17877	0.410100	0.205050	+	0.978752i	$0.434264\pi$
$228$	25.9032	1.71548
$229$	−15.4004	−1.01769	−0.508844	−	0.860859i	$-0.669927\pi$
$229$	−15.4004	−1.01769	−0.508844	+	0.860859i	$0.669927\pi$
$230$	17.5557	1.15759
$231$	6.51259	0.428497
$232$	10.4878	0.688559
$233$	3.78478	0.247949	0.123975	−	0.992285i	$-0.460436\pi$
$233$	3.78478	0.247949	0.123975	+	0.992285i	$0.460436\pi$
$234$	0	0
$235$	6.52096	0.425380
$236$	−14.3091	−0.931443
$237$	−7.06240	−0.458752
$238$	46.1411	2.99088
$239$	14.4468	0.934484	0.467242	−	0.884129i	$-0.345248\pi$
$239$	14.4468	0.934484	0.467242	+	0.884129i	$0.345248\pi$
$240$	29.7983	1.92347
$241$	−16.6637	−1.07340	−0.536702	−	0.843772i	$-0.680330\pi$
$241$	−16.6637	−1.07340	−0.536702	+	0.843772i	$0.680330\pi$
$242$	2.69931	0.173518
$243$	−4.63750	−0.297496
$244$	−42.8787	−2.74503
$245$	−13.4455	−0.858999
$246$	28.3554	1.80788
$247$	0	0
$248$	−40.8597	−2.59459
$249$	−18.5157	−1.17339
$250$	30.3337	1.91847
$251$	−10.9799	−0.693048	−0.346524	−	0.938041i	$-0.612638\pi$
$251$	−10.9799	−0.693048	−0.346524	+	0.938041i	$0.612638\pi$
$252$	9.70763	0.611523
$253$	−4.66037	−0.292995
$254$	17.8364	1.11916
$255$	−9.33949	−0.584861
$256$	21.4355	1.33972
$257$	7.78157	0.485401	0.242700	−	0.970101i	$-0.421967\pi$
$257$	7.78157	0.485401	0.242700	+	0.970101i	$0.421967\pi$
$258$	−18.8230	−1.17187
$259$	9.25341	0.574979
$260$	0	0
$261$	−0.532340	−0.0329510
$262$	16.7916	1.03739
$263$	−3.84830	−0.237296	−0.118648	−	0.992936i	$-0.537856\pi$
$263$	−3.84830	−0.237296	−0.118648	+	0.992936i	$0.537856\pi$
$264$	−14.1646	−0.871769
$265$	−4.98223	−0.306056
$266$	33.7842	2.07144
$267$	−13.5880	−0.831570
$268$	81.2641	4.96399
$269$	9.92107	0.604899	0.302449	−	0.953165i	$-0.402196\pi$
$269$	9.92107	0.604899	0.302449	+	0.953165i	$0.402196\pi$
$270$	−20.7537	−1.26303
$271$	−9.02331	−0.548127	−0.274064	−	0.961712i	$-0.588368\pi$
$271$	−9.02331	−0.548127	−0.274064	+	0.961712i	$0.588368\pi$
$272$	−56.0439	−3.39816
$273$	0	0
$274$	−39.8120	−2.40513
$275$	−3.05245	−0.184070
$276$	39.3385	2.36790
$277$	−22.3851	−1.34499	−0.672494	−	0.740102i	$-0.734777\pi$
$277$	−22.3851	−1.34499	−0.672494	+	0.740102i	$0.734777\pi$
$278$	24.8408	1.48985
$279$	2.07395	0.124164
$280$	50.4899	3.01735
$281$	−22.4799	−1.34104	−0.670519	−	0.741893i	$-0.733928\pi$
$281$	−22.4799	−1.34104	−0.670519	+	0.741893i	$0.733928\pi$
$282$	20.1404	1.19934
$283$	−4.07879	−0.242459	−0.121229	−	0.992625i	$-0.538684\pi$
$283$	−4.07879	−0.242459	−0.121229	+	0.992625i	$0.538684\pi$
$284$	−56.7917	−3.36997
$285$	−6.83831	−0.405066
$286$	0	0
$287$	26.8312	1.58380
$288$	−8.26400	−0.486961
$289$	0.565503	0.0332649
$290$	−4.45376	−0.261534
$291$	−6.95452	−0.407681
$292$	−56.7406	−3.32049
$293$	−11.6160	−0.678616	−0.339308	−	0.940675i	$-0.610193\pi$
$293$	−11.6160	−0.678616	−0.339308	+	0.940675i	$0.610193\pi$
$294$	−41.5271	−2.42191
$295$	3.77753	0.219936
$296$	−20.1257	−1.16978
$297$	5.50934	0.319684
$298$	−43.3459	−2.51096
$299$	0	0
$300$	25.7659	1.48760
$301$	−17.8112	−1.02662
$302$	−28.2351	−1.62475
$303$	−17.1614	−0.985897
$304$	−41.0350	−2.35352
$305$	11.3198	0.648168
$306$	5.09380	0.291193
$307$	1.45109	0.0828179	0.0414090	−	0.999142i	$-0.486815\pi$
$307$	1.45109	0.0828179	0.0414090	+	0.999142i	$0.486815\pi$
$308$	−21.5603	−1.22851
$309$	15.0030	0.853488
$310$	17.3515	0.985498
$311$	−19.5008	−1.10579	−0.552896	−	0.833250i	$-0.686477\pi$
$311$	−19.5008	−1.10579	−0.552896	+	0.833250i	$0.686477\pi$
$312$	0	0
$313$	−19.6876	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$313$	−19.6876	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$314$	8.61649	0.486257
$315$	−2.56277	−0.144395
$316$	23.3804	1.31525
$317$	−23.9947	−1.34768	−0.673838	−	0.738880i	$-0.735355\pi$
$317$	−23.9947	−1.34768	−0.673838	+	0.738880i	$0.735355\pi$
$318$	−15.3879	−0.862912
$319$	1.18231	0.0661964
$320$	−31.8171	−1.77863
$321$	−21.6081	−1.20605
$322$	51.3072	2.85924
$323$	12.8613	0.715624
$324$	−39.3642	−2.18690
$325$	0	0
$326$	36.4190	2.01706
$327$	15.7282	0.869774
$328$	−58.3566	−3.22221
$329$	19.0578	1.05069
$330$	6.01513	0.331122
$331$	−33.0216	−1.81503	−0.907515	−	0.420020i	$-0.862023\pi$
$331$	−33.0216	−1.81503	−0.907515	+	0.420020i	$0.862023\pi$
$332$	61.2973	3.36413
$333$	1.02154	0.0559801
$334$	−50.1158	−2.74221
$335$	−21.4533	−1.17212
$336$	87.0867	4.75097
$337$	29.7171	1.61880	0.809398	−	0.587261i	$-0.199794\pi$
$337$	29.7171	1.61880	0.809398	+	0.587261i	$0.199794\pi$
$338$	0	0
$339$	−1.38792	−0.0753817
$340$	30.9189	1.67681
$341$	−4.60617	−0.249438
$342$	3.72965	0.201676
$343$	−10.7451	−0.580181
$344$	38.7385	2.08864
$345$	−10.3852	−0.559118
$346$	−26.1695	−1.40688
$347$	−6.55705	−0.352001	−0.176000	−	0.984390i	$-0.556316\pi$
$347$	−6.55705	−0.352001	−0.176000	+	0.984390i	$0.556316\pi$
$348$	−9.97992	−0.534980
$349$	−10.0311	−0.536954	−0.268477	−	0.963286i	$-0.586520\pi$
$349$	−10.0311	−0.536954	−0.268477	+	0.963286i	$0.586520\pi$
$350$	33.6052	1.79627
$351$	0	0
$352$	18.3540	0.978273
$353$	−19.7033	−1.04870	−0.524350	−	0.851503i	$-0.675692\pi$
$353$	−19.7033	−1.04870	−0.524350	+	0.851503i	$0.675692\pi$
$354$	11.6671	0.620101
$355$	14.9927	0.795731
$356$	44.9837	2.38413
$357$	−27.2951	−1.44461
$358$	−59.0076	−3.11865
$359$	5.37147	0.283495	0.141748	−	0.989903i	$-0.454728\pi$
$359$	5.37147	0.283495	0.141748	+	0.989903i	$0.454728\pi$
$360$	5.57389	0.293770
$361$	−9.58301	−0.504369
$362$	32.0315	1.68354
$363$	−1.59679	−0.0838098
$364$	0	0
$365$	14.9792	0.784048
$366$	34.9618	1.82748
$367$	11.8702	0.619618	0.309809	−	0.950799i	$-0.399735\pi$
$367$	11.8702	0.619618	0.309809	+	0.950799i	$0.399735\pi$
$368$	−62.3188	−3.24859
$369$	2.96206	0.154199
$370$	8.54660	0.444317
$371$	−14.5608	−0.755958
$372$	38.8810	2.01588
$373$	36.0843	1.86838	0.934188	−	0.356781i	$-0.116126\pi$
$373$	36.0843	1.86838	0.934188	+	0.356781i	$0.116126\pi$
$374$	−11.3131	−0.584988
$375$	−17.9441	−0.926627
$376$	−41.4497	−2.13761
$377$	0	0
$378$	−60.6537	−3.11969
$379$	−24.9461	−1.28139	−0.640697	−	0.767794i	$-0.721355\pi$
$379$	−24.9461	−1.28139	−0.640697	+	0.767794i	$0.721355\pi$
$380$	22.6386	1.16133
$381$	−10.5512	−0.540556
$382$	36.0745	1.84573
$383$	28.2054	1.44123	0.720615	−	0.693336i	$-0.243860\pi$
$383$	28.2054	1.44123	0.720615	+	0.693336i	$0.243860\pi$
$384$	−39.6541	−2.02359
$385$	5.69180	0.290081
$386$	24.2561	1.23460
$387$	−1.96629	−0.0999519
$388$	23.0233	1.16883
$389$	9.87266	0.500564	0.250282	−	0.968173i	$-0.419477\pi$
$389$	9.87266	0.500564	0.250282	+	0.968173i	$0.419477\pi$
$390$	0	0
$391$	19.5322	0.987785
$392$	85.4646	4.31661
$393$	−9.93316	−0.501062
$394$	13.2390	0.666971
$395$	−6.17232	−0.310563
$396$	−2.38017	−0.119608
$397$	13.7825	0.691725	0.345862	−	0.938285i	$-0.387586\pi$
$397$	13.7825	0.691725	0.345862	+	0.938285i	$0.387586\pi$
$398$	−20.2759	−1.01634
$399$	−19.9853	−1.00051
$400$	−40.8175	−2.04088
$401$	−30.1251	−1.50437	−0.752187	−	0.658950i	$-0.771001\pi$
$401$	−30.1251	−1.50437	−0.752187	+	0.658950i	$0.771001\pi$
$402$	−66.2599	−3.30474
$403$	0	0
$404$	56.8137	2.82659
$405$	10.3919	0.516380
$406$	−13.0163	−0.645989
$407$	−2.26880	−0.112460
$408$	59.3654	2.93903
$409$	−6.80038	−0.336257	−0.168129	−	0.985765i	$-0.553772\pi$
$409$	−6.80038	−0.336257	−0.168129	+	0.985765i	$0.553772\pi$
$410$	24.7818	1.22388
$411$	23.5510	1.16169
$412$	−49.6681	−2.44697
$413$	11.0400	0.543243
$414$	5.66412	0.278376
$415$	−16.1822	−0.794352
$416$	0	0
$417$	−14.6947	−0.719604
$418$	−8.28340	−0.405154
$419$	−12.9279	−0.631568	−0.315784	−	0.948831i	$-0.602268\pi$
$419$	−12.9279	−0.631568	−0.315784	+	0.948831i	$0.602268\pi$
$420$	−48.0449	−2.34435
$421$	−19.0846	−0.930125	−0.465063	−	0.885278i	$-0.653968\pi$
$421$	−19.0846	−0.930125	−0.465063	+	0.885278i	$0.653968\pi$
$422$	47.4207	2.30841
$423$	2.10390	0.102295
$424$	31.6690	1.53798
$425$	12.7932	0.620561
$426$	46.3060	2.24353
$427$	33.0825	1.60098
$428$	71.5349	3.45777
$429$	0	0
$430$	−16.4507	−0.793323
$431$	15.4660	0.744972	0.372486	−	0.928038i	$-0.378506\pi$
$431$	15.4660	0.744972	0.372486	+	0.928038i	$0.378506\pi$
$432$	73.6712	3.54451
$433$	−11.6331	−0.559049	−0.279525	−	0.960138i	$-0.590177\pi$
$433$	−11.6331	−0.559049	−0.279525	+	0.960138i	$0.590177\pi$
$434$	50.7105	2.43418
$435$	2.63465	0.126322
$436$	−52.0692	−2.49366
$437$	14.3013	0.684126
$438$	46.2643	2.21059
$439$	3.70259	0.176715	0.0883575	−	0.996089i	$-0.471838\pi$
$439$	3.70259	0.176715	0.0883575	+	0.996089i	$0.471838\pi$
$440$	−12.3794	−0.590164
$441$	−4.33801	−0.206572
$442$	0	0
$443$	−9.12016	−0.433312	−0.216656	−	0.976248i	$-0.569515\pi$
$443$	−9.12016	−0.433312	−0.216656	+	0.976248i	$0.569515\pi$
$444$	19.1511	0.908871
$445$	−11.8755	−0.562951
$446$	33.2757	1.57565
$447$	25.6415	1.21280
$448$	−92.9870	−4.39322
$449$	−20.2928	−0.957676	−0.478838	−	0.877903i	$-0.658942\pi$
$449$	−20.2928	−0.957676	−0.478838	+	0.877903i	$0.658942\pi$
$450$	3.70988	0.174885
$451$	−6.57863	−0.309776
$452$	4.59480	0.216121
$453$	16.7026	0.784758
$454$	16.6784	0.782757
$455$	0	0
$456$	43.4669	2.03553
$457$	10.0312	0.469242	0.234621	−	0.972087i	$-0.424615\pi$
$457$	10.0312	0.469242	0.234621	+	0.972087i	$0.424615\pi$
$458$	−41.5705	−1.94246
$459$	−23.0903	−1.07776
$460$	34.3806	1.60301
$461$	27.5165	1.28157	0.640786	−	0.767719i	$-0.278608\pi$
$461$	27.5165	1.28157	0.640786	+	0.767719i	$0.278608\pi$
$462$	17.5795	0.817872
$463$	−11.2141	−0.521162	−0.260581	−	0.965452i	$-0.583914\pi$
$463$	−11.2141	−0.521162	−0.260581	+	0.965452i	$0.583914\pi$
$464$	15.8099	0.733955
$465$	−10.2644	−0.475999
$466$	10.2163	0.473261
$467$	21.6441	1.00157	0.500786	−	0.865571i	$-0.333044\pi$
$467$	21.6441	1.00157	0.500786	+	0.865571i	$0.333044\pi$
$468$	0	0
$469$	−62.6982	−2.89514
$470$	17.6021	0.811923
$471$	−5.09714	−0.234864
$472$	−24.0114	−1.10522
$473$	4.36704	0.200797
$474$	−19.0636	−0.875620
$475$	9.36708	0.429791
$476$	90.3617	4.14172
$477$	−1.60745	−0.0736002
$478$	38.9963	1.78365
$479$	−11.9921	−0.547935	−0.273967	−	0.961739i	$-0.588336\pi$
$479$	−11.9921	−0.547935	−0.273967	+	0.961739i	$0.588336\pi$
$480$	40.9001	1.86682
$481$	0	0
$482$	−44.9805	−2.04881
$483$	−30.3511	−1.38102
$484$	5.28626	0.240285
$485$	−6.07804	−0.275989
$486$	−12.5181	−0.567830
$487$	14.3784	0.651549	0.325774	−	0.945448i	$-0.394375\pi$
$487$	14.3784	0.651549	0.325774	+	0.945448i	$0.394375\pi$
$488$	−71.9529	−3.25715
$489$	−21.5439	−0.974249
$490$	−36.2934	−1.63957
$491$	28.2400	1.27445	0.637227	−	0.770676i	$-0.280081\pi$
$491$	28.2400	1.27445	0.637227	+	0.770676i	$0.280081\pi$
$492$	55.5306	2.50351
$493$	−4.95519	−0.223171
$494$	0	0
$495$	0.628353	0.0282424
$496$	−61.5940	−2.76565
$497$	43.8169	1.96546
$498$	−49.9797	−2.23964
$499$	6.90992	0.309330	0.154665	−	0.987967i	$-0.450570\pi$
$499$	6.90992	0.309330	0.154665	+	0.987967i	$0.450570\pi$
$500$	59.4048	2.65666
$501$	29.6463	1.32450
$502$	−29.6383	−1.32282
$503$	−11.1543	−0.497346	−0.248673	−	0.968588i	$-0.579994\pi$
$503$	−11.1543	−0.497346	−0.248673	+	0.968588i	$0.579994\pi$
$504$	16.2899	0.725612
$505$	−14.9985	−0.667426
$506$	−12.5798	−0.559240
$507$	0	0
$508$	34.9304	1.54979
$509$	7.68209	0.340503	0.170251	−	0.985401i	$-0.445542\pi$
$509$	7.68209	0.340503	0.170251	+	0.985401i	$0.445542\pi$
$510$	−25.2101	−1.11632
$511$	43.7774	1.93660
$512$	8.19372	0.362115
$513$	−16.9066	−0.746443
$514$	21.0049	0.926484
$515$	13.1121	0.577789
$516$	−36.8625	−1.62278
$517$	−4.67269	−0.205505
$518$	24.9778	1.09746
$519$	15.4807	0.679529
$520$	0	0
$521$	16.6993	0.731608	0.365804	−	0.930692i	$-0.380794\pi$
$521$	16.6993	0.731608	0.365804	+	0.930692i	$0.380794\pi$
$522$	−1.43695	−0.0628936
$523$	14.2089	0.621313	0.310657	−	0.950522i	$-0.399451\pi$
$523$	14.2089	0.621313	0.310657	+	0.950522i	$0.399451\pi$
$524$	32.8843	1.43656
$525$	−19.8794	−0.867606
$526$	−10.3877	−0.452927
$527$	19.3050	0.840940
$528$	−21.3524	−0.929244
$529$	−1.28092	−0.0556923
$530$	−13.4486	−0.584168
$531$	1.21877	0.0528902
$532$	66.1622	2.86850
$533$	0	0
$534$	−36.6781	−1.58722
$535$	−18.8849	−0.816463
$536$	136.366	5.89010
$537$	34.9063	1.50632
$538$	26.7800	1.15457
$539$	9.63454	0.414989
$540$	−40.6437	−1.74902
$541$	13.5499	0.582556	0.291278	−	0.956638i	$-0.405920\pi$
$541$	13.5499	0.582556	0.291278	+	0.956638i	$0.405920\pi$
$542$	−24.3567	−1.04621
$543$	−18.9485	−0.813156
$544$	−76.9240	−3.29809
$545$	13.7460	0.588814
$546$	0	0
$547$	28.4599	1.21686	0.608428	−	0.793609i	$-0.291800\pi$
$547$	28.4599	1.21686	0.608428	+	0.793609i	$0.291800\pi$
$548$	−77.9670	−3.33058
$549$	3.65218	0.155871
$550$	−8.23950	−0.351334
$551$	−3.62816	−0.154565
$552$	66.0122	2.80966
$553$	−18.0389	−0.767091
$554$	−60.4242	−2.56718
$555$	−5.05579	−0.214606
$556$	48.6477	2.06312
$557$	−7.19853	−0.305012	−0.152506	−	0.988303i	$-0.548734\pi$
$557$	−7.19853	−0.305012	−0.152506	+	0.988303i	$0.548734\pi$
$558$	5.59824	0.236992
$559$	0	0
$560$	76.1111	3.21628
$561$	6.69235	0.282551
$562$	−60.6801	−2.55964
$563$	−5.21240	−0.219676	−0.109838	−	0.993949i	$-0.535033\pi$
$563$	−5.21240	−0.219676	−0.109838	+	0.993949i	$0.535033\pi$
$564$	39.4425	1.66083
$565$	−1.21300	−0.0510314
$566$	−11.0099	−0.462781
$567$	30.3709	1.27546
$568$	−95.2996	−3.99868
$569$	−12.9466	−0.542752	−0.271376	−	0.962473i	$-0.587479\pi$
$569$	−12.9466	−0.542752	−0.271376	+	0.962473i	$0.587479\pi$
$570$	−18.4587	−0.773150
$571$	−32.1928	−1.34723	−0.673613	−	0.739084i	$-0.735259\pi$
$571$	−32.1928	−1.34723	−0.673613	+	0.739084i	$0.735259\pi$
$572$	0	0
$573$	−21.3401	−0.891494
$574$	72.4258	3.02299
$575$	14.2256	0.593247
$576$	−10.2654	−0.427725
$577$	−3.52083	−0.146574	−0.0732871	−	0.997311i	$-0.523349\pi$
$577$	−3.52083	−0.146574	−0.0732871	+	0.997311i	$0.523349\pi$
$578$	1.52647	0.0634926
$579$	−14.3488	−0.596317
$580$	−8.72214	−0.362167
$581$	−47.2931	−1.96205
$582$	−18.7724	−0.778141
$583$	3.57009	0.147858
$584$	−95.2138	−3.93997
$585$	0	0
$586$	−31.3553	−1.29527
$587$	−3.29522	−0.136008	−0.0680041	−	0.997685i	$-0.521663\pi$
$587$	−3.29522	−0.136008	−0.0680041	+	0.997685i	$0.521663\pi$
$588$	−81.3258	−3.35382
$589$	14.1350	0.582423
$590$	10.1967	0.419792
$591$	−7.83161	−0.322149
$592$	−30.3385	−1.24691
$593$	−25.3977	−1.04296	−0.521479	−	0.853264i	$-0.674619\pi$
$593$	−25.3977	−1.04296	−0.521479	+	0.853264i	$0.674619\pi$
$594$	14.8714	0.610181
$595$	−23.8550	−0.977961
$596$	−84.8876	−3.47713
$597$	11.9943	0.490895
$598$	0	0
$599$	18.2140	0.744204	0.372102	−	0.928192i	$-0.378637\pi$
$599$	18.2140	0.744204	0.372102	+	0.928192i	$0.378637\pi$
$600$	43.2366	1.76513
$601$	−38.6690	−1.57734	−0.788670	−	0.614817i	$-0.789230\pi$
$601$	−38.6690	−1.57734	−0.788670	+	0.614817i	$0.789230\pi$
$602$	−48.0779	−1.95951
$603$	−6.92164	−0.281871
$604$	−55.2949	−2.24992
$605$	−1.39555	−0.0567370
$606$	−46.3239	−1.88178
$607$	−31.9320	−1.29608	−0.648040	−	0.761606i	$-0.724411\pi$
$607$	−31.9320	−1.29608	−0.648040	+	0.761606i	$0.724411\pi$
$608$	−56.3232	−2.28421
$609$	7.69988	0.312015
$610$	30.5555	1.23716
$611$	0	0
$612$	9.97558	0.403239
$613$	−37.7252	−1.52371	−0.761854	−	0.647749i	$-0.775711\pi$
$613$	−37.7252	−1.52371	−0.761854	+	0.647749i	$0.775711\pi$
$614$	3.91693	0.158074
$615$	−14.6598	−0.591140
$616$	−36.1793	−1.45771
$617$	7.12962	0.287028	0.143514	−	0.989648i	$-0.454160\pi$
$617$	7.12962	0.287028	0.143514	+	0.989648i	$0.454160\pi$
$618$	40.4976	1.62905
$619$	15.9319	0.640358	0.320179	−	0.947357i	$-0.396257\pi$
$619$	15.9319	0.640358	0.320179	+	0.947357i	$0.396257\pi$
$620$	33.9808	1.36470
$621$	−25.6756	−1.03033
$622$	−52.6388	−2.11062
$623$	−34.7065	−1.39049
$624$	0	0
$625$	−0.420305	−0.0168122
$626$	−53.1429	−2.12402
$627$	4.90009	0.195691
$628$	16.8743	0.673360
$629$	9.50883	0.379142
$630$	−6.91769	−0.275608
$631$	40.2488	1.60228	0.801140	−	0.598477i	$-0.204227\pi$
$631$	40.2488	1.60228	0.801140	+	0.598477i	$0.204227\pi$
$632$	39.2337	1.56063
$633$	−28.0520	−1.11497
$634$	−64.7690	−2.57231
$635$	−9.22145	−0.365942
$636$	−30.1354	−1.19494
$637$	0	0
$638$	3.19141	0.126349
$639$	4.83721	0.191357
$640$	−34.6565	−1.36992
$641$	−5.98952	−0.236572	−0.118286	−	0.992980i	$-0.537740\pi$
$641$	−5.98952	−0.236572	−0.118286	+	0.992980i	$0.537740\pi$
$642$	−58.3271	−2.30199
$643$	−46.7103	−1.84208	−0.921038	−	0.389473i	$-0.872657\pi$
$643$	−46.7103	−1.84208	−0.921038	+	0.389473i	$0.872657\pi$
$644$	100.479	3.95942
$645$	9.73151	0.383178
$646$	34.7167	1.36591
$647$	−12.1235	−0.476625	−0.238312	−	0.971189i	$-0.576594\pi$
$647$	−12.1235	−0.476625	−0.238312	+	0.971189i	$0.576594\pi$
$648$	−66.0552	−2.59489
$649$	−2.70685	−0.106253
$650$	0	0
$651$	−29.9981	−1.17572
$652$	71.3222	2.79319
$653$	36.4414	1.42606	0.713031	−	0.701132i	$-0.247322\pi$
$653$	36.4414	1.42606	0.713031	+	0.701132i	$0.247322\pi$
$654$	42.4554	1.66014
$655$	−8.68128	−0.339205
$656$	−87.9698	−3.43464
$657$	4.83286	0.188548
$658$	51.4428	2.00545
$659$	44.8117	1.74562	0.872809	−	0.488062i	$-0.162296\pi$
$659$	44.8117	1.74562	0.872809	+	0.488062i	$0.162296\pi$
$660$	11.7799	0.458532
$661$	−13.3733	−0.520161	−0.260080	−	0.965587i	$-0.583749\pi$
$661$	−13.3733	−0.520161	−0.260080	+	0.965587i	$0.583749\pi$
$662$	−89.1354	−3.46435
$663$	0	0
$664$	102.860	3.99175
$665$	−17.4665	−0.677322
$666$	2.75745	0.106849
$667$	−5.50999	−0.213348
$668$	−98.1456	−3.79737
$669$	−19.6844	−0.761044
$670$	−57.9091	−2.23722
$671$	−8.11135	−0.313135
$672$	119.532	4.61106
$673$	−40.1842	−1.54899	−0.774494	−	0.632581i	$-0.781996\pi$
$673$	−40.1842	−1.54899	−0.774494	+	0.632581i	$0.781996\pi$
$674$	80.2157	3.08979
$675$	−16.8170	−0.647286
$676$	0	0
$677$	29.2752	1.12514	0.562568	−	0.826751i	$-0.309813\pi$
$677$	29.2752	1.12514	0.562568	+	0.826751i	$0.309813\pi$
$678$	−3.74644	−0.143881
$679$	−17.7633	−0.681694
$680$	51.8835	1.98964
$681$	−9.86622	−0.378074
$682$	−12.4335	−0.476102
$683$	4.41459	0.168920	0.0844598	−	0.996427i	$-0.473084\pi$
$683$	4.41459	0.168920	0.0844598	+	0.996427i	$0.473084\pi$
$684$	7.30405	0.279277
$685$	20.5829	0.786432
$686$	−29.0044	−1.10739
$687$	24.5913	0.938215
$688$	58.3964	2.22634
$689$	0	0
$690$	−28.0328	−1.06719
$691$	30.9029	1.17560	0.587801	−	0.809006i	$-0.299994\pi$
$691$	30.9029	1.17560	0.587801	+	0.809006i	$0.299994\pi$
$692$	−51.2498	−1.94823
$693$	1.83639	0.0697586
$694$	−17.6995	−0.671864
$695$	−12.8427	−0.487153
$696$	−16.7469	−0.634788
$697$	27.5718	1.04436
$698$	−27.0771	−1.02488
$699$	−6.04351	−0.228587
$700$	65.8116	2.48745
$701$	−20.8403	−0.787127	−0.393563	−	0.919298i	$-0.628758\pi$
$701$	−20.8403	−0.787127	−0.393563	+	0.919298i	$0.628758\pi$
$702$	0	0
$703$	6.96230	0.262588
$704$	22.7991	0.859272
$705$	−10.4126	−0.392162
$706$	−53.1852	−2.00165
$707$	−43.8339	−1.64854
$708$	22.8487	0.858705
$709$	−15.0814	−0.566395	−0.283198	−	0.959062i	$-0.591395\pi$
$709$	−15.0814	−0.566395	−0.283198	+	0.959062i	$0.591395\pi$
$710$	40.4700	1.51881
$711$	−1.99142	−0.0746841
$712$	75.4850	2.82892
$713$	21.4665	0.803926
$714$	−73.6778	−2.75732
$715$	0	0
$716$	−115.559	−4.31865
$717$	−23.0685	−0.861508
$718$	14.4993	0.541107
$719$	−9.31036	−0.347218	−0.173609	−	0.984815i	$-0.555543\pi$
$719$	−9.31036	−0.347218	−0.173609	+	0.984815i	$0.555543\pi$
$720$	8.40237	0.313138
$721$	38.3207	1.42714
$722$	−25.8675	−0.962689
$723$	26.6085	0.989581
$724$	62.7299	2.33134
$725$	−3.60893	−0.134032
$726$	−4.31023	−0.159968
$727$	21.8723	0.811200	0.405600	−	0.914051i	$-0.367063\pi$
$727$	21.8723	0.811200	0.405600	+	0.914051i	$0.367063\pi$
$728$	0	0
$729$	29.7446	1.10165
$730$	40.4335	1.49651
$731$	−18.3028	−0.676954
$732$	68.4684	2.53067
$733$	−50.1177	−1.85114	−0.925570	−	0.378576i	$-0.876414\pi$
$733$	−50.1177	−1.85114	−0.925570	+	0.378576i	$0.876414\pi$
$734$	32.0412	1.18266
$735$	21.4696	0.791918
$736$	−85.5366	−3.15292
$737$	15.3727	0.566260
$738$	7.99552	0.294319
$739$	13.7331	0.505180	0.252590	−	0.967573i	$-0.418718\pi$
$739$	13.7331	0.505180	0.252590	+	0.967573i	$0.418718\pi$
$740$	16.7375	0.615282
$741$	0	0
$742$	−39.3040	−1.44290
$743$	−4.94343	−0.181357	−0.0906785	−	0.995880i	$-0.528904\pi$
$743$	−4.94343	−0.181357	−0.0906785	+	0.995880i	$0.528904\pi$
$744$	65.2444	2.39197
$745$	22.4099	0.821035
$746$	97.4027	3.56617
$747$	−5.22098	−0.191025
$748$	−22.1554	−0.810081
$749$	−55.1918	−2.01667
$750$	−48.4365	−1.76865
$751$	−47.7397	−1.74205	−0.871023	−	0.491242i	$-0.836543\pi$
$751$	−47.7397	−1.74205	−0.871023	+	0.491242i	$0.836543\pi$
$752$	−62.4835	−2.27854
$753$	17.5327	0.638927
$754$	0	0
$755$	14.5976	0.531260
$756$	−118.783	−4.32009
$757$	−3.89742	−0.141654	−0.0708270	−	0.997489i	$-0.522564\pi$
$757$	−3.89742	−0.141654	−0.0708270	+	0.997489i	$0.522564\pi$
$758$	−67.3372	−2.44580
$759$	7.44165	0.270115
$760$	37.9888	1.37800
$761$	20.7359	0.751677	0.375838	−	0.926685i	$-0.377355\pi$
$761$	20.7359	0.751677	0.375838	+	0.926685i	$0.377355\pi$
$762$	−28.4810	−1.03176
$763$	40.1733	1.45437
$764$	70.6474	2.55593
$765$	−2.63350	−0.0952145
$766$	76.1351	2.75087
$767$	0	0
$768$	−34.2280	−1.23510
$769$	−43.6737	−1.57491	−0.787457	−	0.616370i	$-0.788603\pi$
$769$	−43.6737	−1.57491	−0.787457	+	0.616370i	$0.788603\pi$
$770$	15.3639	0.553678
$771$	−12.4255	−0.447495
$772$	47.5026	1.70966
$773$	21.8081	0.784383	0.392191	−	0.919884i	$-0.371717\pi$
$773$	21.8081	0.784383	0.392191	+	0.919884i	$0.371717\pi$
$774$	−5.30761	−0.190778
$775$	14.0601	0.505054
$776$	38.6344	1.38689
$777$	−14.7758	−0.530078
$778$	26.6494	0.955426
$779$	20.1879	0.723307
$780$	0	0
$781$	−10.7433	−0.384424
$782$	52.7234	1.88539
$783$	6.51373	0.232782
$784$	128.834	4.60120
$785$	−4.45474	−0.158996
$786$	−26.8127	−0.956376
$787$	−33.9897	−1.21160	−0.605801	−	0.795616i	$-0.707147\pi$
$787$	−33.9897	−1.21160	−0.605801	+	0.795616i	$0.707147\pi$
$788$	25.9269	0.923609
$789$	6.14493	0.218765
$790$	−16.6610	−0.592771
$791$	−3.54505	−0.126048
$792$	−3.99405	−0.141923
$793$	0	0
$794$	37.2033	1.32029
$795$	7.95558	0.282155
$796$	−39.7079	−1.40741
$797$	0.605869	0.0214610	0.0107305	−	0.999942i	$-0.496584\pi$
$797$	0.605869	0.0214610	0.0107305	+	0.999942i	$0.496584\pi$
$798$	−53.9464	−1.90968
$799$	19.5838	0.692825
$800$	−56.0247	−1.98077
$801$	−3.83147	−0.135378
$802$	−81.3168	−2.87140
$803$	−10.7336	−0.378780
$804$	−129.762	−4.57635
$805$	−26.5259	−0.934916
$806$	0	0
$807$	−15.8419	−0.557661
$808$	95.3366	3.35393
$809$	−52.6004	−1.84933	−0.924665	−	0.380781i	$-0.875655\pi$
$809$	−52.6004	−1.84933	−0.924665	+	0.380781i	$0.875655\pi$
$810$	28.0511	0.985614
$811$	−52.1220	−1.83025	−0.915125	−	0.403171i	$-0.867908\pi$
$811$	−52.1220	−1.83025	−0.915125	+	0.403171i	$0.867908\pi$
$812$	−25.4909	−0.894553
$813$	14.4083	0.505323
$814$	−6.12420	−0.214653
$815$	−18.8287	−0.659541
$816$	89.4905	3.13279
$817$	−13.4012	−0.468848
$818$	−18.3563	−0.641814
$819$	0	0
$820$	48.5320	1.69481
$821$	−4.42068	−0.154283	−0.0771414	−	0.997020i	$-0.524579\pi$
$821$	−4.42068	−0.154283	−0.0771414	+	0.997020i	$0.524579\pi$
$822$	63.5715	2.21731
$823$	−53.0850	−1.85043	−0.925214	−	0.379447i	$-0.876114\pi$
$823$	−53.0850	−1.85043	−0.925214	+	0.379447i	$0.876114\pi$
$824$	−83.3457	−2.90349
$825$	4.87413	0.169695
$826$	29.8003	1.03689
$827$	2.10269	0.0731178	0.0365589	−	0.999332i	$-0.488360\pi$
$827$	2.10269	0.0731178	0.0365589	+	0.999332i	$0.488360\pi$
$828$	11.0925	0.385490
$829$	−34.2497	−1.18954	−0.594771	−	0.803895i	$-0.702757\pi$
$829$	−34.2497	−1.18954	−0.594771	+	0.803895i	$0.702757\pi$
$830$	−43.6807	−1.51618
$831$	35.7443	1.23996
$832$	0	0
$833$	−40.3795	−1.39907
$834$	−39.6656	−1.37351
$835$	25.9099	0.896651
$836$	−16.2220	−0.561050
$837$	−25.3770	−0.877156
$838$	−34.8963	−1.20547
$839$	5.62944	0.194350	0.0971750	−	0.995267i	$-0.469019\pi$
$839$	5.62944	0.194350	0.0971750	+	0.995267i	$0.469019\pi$
$840$	−80.6219	−2.78172
$841$	−27.6022	−0.951798
$842$	−51.5151	−1.77533
$843$	35.8957	1.23631
$844$	92.8677	3.19664
$845$	0	0
$846$	5.67909	0.195251
$847$	−4.07855	−0.140141
$848$	47.7395	1.63938
$849$	6.51297	0.223525
$850$	34.5328	1.18446
$851$	10.5735	0.362454
$852$	90.6846	3.10680
$853$	−17.4844	−0.598655	−0.299328	−	0.954150i	$-0.596762\pi$
$853$	−17.4844	−0.598655	−0.299328	+	0.954150i	$0.596762\pi$
$854$	89.2999	3.05578
$855$	−1.92823	−0.0659442
$856$	120.039	4.10287
$857$	53.4328	1.82523	0.912615	−	0.408820i	$-0.134060\pi$
$857$	53.4328	1.82523	0.912615	+	0.408820i	$0.134060\pi$
$858$	0	0
$859$	47.1736	1.60954	0.804771	−	0.593585i	$-0.202288\pi$
$859$	47.1736	1.60954	0.804771	+	0.593585i	$0.202288\pi$
$860$	−32.2167	−1.09858
$861$	−42.8439	−1.46012
$862$	41.7475	1.42193
$863$	23.2983	0.793083	0.396541	−	0.918017i	$-0.370210\pi$
$863$	23.2983	0.793083	0.396541	+	0.918017i	$0.370210\pi$
$864$	101.119	3.44012
$865$	13.5297	0.460023
$866$	−31.4012	−1.06706
$867$	−0.902990	−0.0306672
$868$	99.3103	3.37081
$869$	4.42287	0.150036
$870$	7.11173	0.241110
$871$	0	0
$872$	−87.3749	−2.95889
$873$	−1.96100	−0.0663699
$874$	38.6037	1.30579
$875$	−45.8330	−1.54944
$876$	90.6029	3.06119
$877$	24.2122	0.817587	0.408794	−	0.912627i	$-0.365950\pi$
$877$	24.2122	0.817587	0.408794	+	0.912627i	$0.365950\pi$
$878$	9.99443	0.337296
$879$	18.5484	0.625622
$880$	−18.6613	−0.629073
$881$	46.0485	1.55141	0.775707	−	0.631093i	$-0.217393\pi$
$881$	46.0485	1.55141	0.775707	+	0.631093i	$0.217393\pi$
$882$	−11.7096	−0.394283
$883$	23.7759	0.800121	0.400061	−	0.916489i	$-0.368989\pi$
$883$	23.7759	0.800121	0.400061	+	0.916489i	$0.368989\pi$
$884$	0	0
$885$	−6.03193	−0.202761
$886$	−24.6181	−0.827062
$887$	−36.5576	−1.22749	−0.613743	−	0.789506i	$-0.710337\pi$
$887$	−36.5576	−1.22749	−0.613743	+	0.789506i	$0.710337\pi$
$888$	32.1366	1.07843
$889$	−26.9501	−0.903877
$890$	−32.0555	−1.07450
$891$	−7.44650	−0.249467
$892$	65.1664	2.18193
$893$	14.3391	0.479841
$894$	69.2144	2.31488
$895$	30.5070	1.01974
$896$	−101.285	−3.38370
$897$	0	0
$898$	−54.7765	−1.82791
$899$	−5.44590	−0.181631
$900$	7.26535	0.242178
$901$	−14.9627	−0.498479
$902$	−17.7577	−0.591268
$903$	28.4408	0.946450
$904$	7.71032	0.256441
$905$	−16.5604	−0.550485
$906$	45.0855	1.49787
$907$	39.2212	1.30232	0.651158	−	0.758942i	$-0.274283\pi$
$907$	39.2212	1.30232	0.651158	+	0.758942i	$0.274283\pi$
$908$	32.6626	1.08395
$909$	−4.83909	−0.160502
$910$	0	0
$911$	−26.4511	−0.876364	−0.438182	−	0.898886i	$-0.644377\pi$
$911$	−26.4511	−0.876364	−0.438182	+	0.898886i	$0.644377\pi$
$912$	65.5243	2.16973
$913$	11.5956	0.383758
$914$	27.0774	0.895641
$915$	−18.0753	−0.597551
$916$	−81.4107	−2.68988
$917$	−25.3714	−0.837838
$918$	−62.3279	−2.05713
$919$	−1.75563	−0.0579128	−0.0289564	−	0.999581i	$-0.509218\pi$
$919$	−1.75563	−0.0579128	−0.0289564	+	0.999581i	$0.509218\pi$
$920$	57.6926	1.90207
$921$	−2.31708	−0.0763505
$922$	74.2756	2.44614
$923$	0	0
$924$	34.4273	1.13257
$925$	6.92540	0.227706
$926$	−30.2702	−0.994741
$927$	4.23046	0.138947
$928$	21.7001	0.712340
$929$	−15.6803	−0.514454	−0.257227	−	0.966351i	$-0.582809\pi$
$929$	−15.6803	−0.514454	−0.257227	+	0.966351i	$0.582809\pi$
$930$	−27.7067	−0.908539
$931$	−29.5656	−0.968974
$932$	20.0074	0.655363
$933$	31.1388	1.01944
$934$	58.4242	1.91170
$935$	5.84891	0.191280
$936$	0	0
$937$	2.81020	0.0918053	0.0459027	−	0.998946i	$-0.485384\pi$
$937$	2.81020	0.0918053	0.0459027	+	0.998946i	$0.485384\pi$
$938$	−169.242	−5.52594
$939$	31.4370	1.02591
$940$	34.4715	1.12434
$941$	34.8295	1.13541	0.567704	−	0.823233i	$-0.307832\pi$
$941$	34.8295	1.13541	0.567704	+	0.823233i	$0.307832\pi$
$942$	−13.7587	−0.448284
$943$	30.6589	0.998390
$944$	−36.1961	−1.17808
$945$	31.3581	1.02008
$946$	11.7880	0.383261
$947$	15.4082	0.500701	0.250350	−	0.968155i	$-0.419454\pi$
$947$	15.4082	0.500701	0.250350	+	0.968155i	$0.419454\pi$
$948$	−37.3337	−1.21254
$949$	0	0
$950$	25.2846	0.820342
$951$	38.3145	1.24243
$952$	151.632	4.91442
$953$	−3.63200	−0.117652	−0.0588260	−	0.998268i	$-0.518736\pi$
$953$	−3.63200	−0.117652	−0.0588260	+	0.998268i	$0.518736\pi$
$954$	−4.33901	−0.140481
$955$	−18.6506	−0.603518
$956$	76.3694	2.46996
$957$	−1.88790	−0.0610271
$958$	−32.3705	−1.04584
$959$	60.1544	1.94249
$960$	50.8054	1.63974
$961$	−9.78320	−0.315587
$962$	0	0
$963$	−6.09296	−0.196343
$964$	−88.0888	−2.83715
$965$	−12.5404	−0.403691
$966$	−81.9270	−2.63596
$967$	54.3074	1.74641	0.873205	−	0.487354i	$-0.162038\pi$
$967$	54.3074	1.74641	0.873205	+	0.487354i	$0.162038\pi$
$968$	8.87064	0.285113
$969$	−20.5369	−0.659740
$970$	−16.4065	−0.526781
$971$	−42.4467	−1.36218	−0.681090	−	0.732200i	$-0.738494\pi$
$971$	−42.4467	−1.36218	−0.681090	+	0.732200i	$0.738494\pi$
$972$	−24.5151	−0.786321
$973$	−37.5335	−1.20327
$974$	38.8118	1.24361
$975$	0	0
$976$	−108.465	−3.47189
$977$	−24.0093	−0.768127	−0.384063	−	0.923307i	$-0.625476\pi$
$977$	−24.0093	−0.768127	−0.384063	+	0.923307i	$0.625476\pi$
$978$	−58.1536	−1.85955
$979$	8.50954	0.271966
$980$	−71.0762	−2.27045
$981$	4.43497	0.141598
$982$	76.2285	2.43255
$983$	15.7310	0.501742	0.250871	−	0.968021i	$-0.419283\pi$
$983$	15.7310	0.501742	0.250871	+	0.968021i	$0.419283\pi$
$984$	93.1834	2.97058
$985$	−6.84458	−0.218086
$986$	−13.3756	−0.425965
$987$	−30.4313	−0.968639
$988$	0	0
$989$	−20.3521	−0.647158
$990$	1.69612	0.0539062
$991$	15.9513	0.506711	0.253355	−	0.967373i	$-0.418466\pi$
$991$	15.9513	0.506711	0.253355	+	0.967373i	$0.418466\pi$
$992$	−84.5418	−2.68420
$993$	52.7286	1.67329
$994$	118.275	3.75146
$995$	10.4827	0.332323
$996$	−97.8790	−3.10142
$997$	−19.3300	−0.612186	−0.306093	−	0.952002i	$-0.599022\pi$
$997$	−19.3300	−0.612186	−0.306093	+	0.952002i	$0.599022\pi$
$998$	18.6520	0.590419
$999$	−12.4996	−0.395470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.2.a.o.1.8		8
13.6	odd	12		143.2.j.b.23.1	✓	16
13.11	odd	12		143.2.j.b.56.1	yes	16
13.12	even	2		1859.2.a.p.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.j.b.23.1	✓	16	13.6	odd	12
143.2.j.b.56.1	yes	16	13.11	odd	12
1859.2.a.o.1.8		8	1.1	even	1	trivial
1859.2.a.p.1.1		8	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:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69931 q^{2} -1.59679 q^{3} +5.28626 q^{4} -1.39555 q^{5} -4.31023 q^{6} -4.07855 q^{7} +8.87064 q^{8} -0.450256 q^{9} +O(q^{10})\)	\(q+2.69931 q^{2} -1.59679 q^{3} +5.28626 q^{4} -1.39555 q^{5} -4.31023 q^{6} -4.07855 q^{7} +8.87064 q^{8} -0.450256 q^{9} -3.76701 q^{10} +1.00000 q^{11} -8.44106 q^{12} -11.0093 q^{14} +2.22840 q^{15} +13.3721 q^{16} -4.19112 q^{17} -1.21538 q^{18} -3.06871 q^{19} -7.37723 q^{20} +6.51259 q^{21} +2.69931 q^{22} -4.66037 q^{23} -14.1646 q^{24} -3.05245 q^{25} +5.50934 q^{27} -21.5603 q^{28} +1.18231 q^{29} +6.01513 q^{30} -4.60617 q^{31} +18.3540 q^{32} -1.59679 q^{33} -11.3131 q^{34} +5.69180 q^{35} -2.38017 q^{36} -2.26880 q^{37} -8.28340 q^{38} -12.3794 q^{40} -6.57863 q^{41} +17.5795 q^{42} +4.36704 q^{43} +5.28626 q^{44} +0.628353 q^{45} -12.5798 q^{46} -4.67269 q^{47} -21.3524 q^{48} +9.63454 q^{49} -8.23950 q^{50} +6.69235 q^{51} +3.57009 q^{53} +14.8714 q^{54} -1.39555 q^{55} -36.1793 q^{56} +4.90009 q^{57} +3.19141 q^{58} -2.70685 q^{59} +11.7799 q^{60} -8.11135 q^{61} -12.4335 q^{62} +1.83639 q^{63} +22.7991 q^{64} -4.31023 q^{66} +15.3727 q^{67} -22.1554 q^{68} +7.44165 q^{69} +15.3639 q^{70} -10.7433 q^{71} -3.99405 q^{72} -10.7336 q^{73} -6.12420 q^{74} +4.87413 q^{75} -16.2220 q^{76} -4.07855 q^{77} +4.42287 q^{79} -18.6613 q^{80} -7.44650 q^{81} -17.7577 q^{82} +11.5956 q^{83} +34.4273 q^{84} +5.84891 q^{85} +11.7880 q^{86} -1.88790 q^{87} +8.87064 q^{88} +8.50954 q^{89} +1.69612 q^{90} -24.6360 q^{92} +7.35509 q^{93} -12.6130 q^{94} +4.28253 q^{95} -29.3076 q^{96} +4.35531 q^{97} +26.0066 q^{98} -0.450256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 4 q^{4} - 8 q^{5} - 4 q^{6} - 14 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 4 * q^4 - 8 * q^5 - 4 * q^6 - 14 * q^7 + 6 * q^8 + 2 * q^9	\( 8 q - 2 q^{2} + 4 q^{4} - 8 q^{5} - 4 q^{6} - 14 q^{7} + 6 q^{8} + 2 q^{9} + 10 q^{10} + 8 q^{11} - 8 q^{12} - 6 q^{14} - 2 q^{15} + 12 q^{16} + 6 q^{17} - 6 q^{18} - 4 q^{19} - 16 q^{20} - 2 q^{21} - 2 q^{22} - 14 q^{23} - 26 q^{24} + 12 q^{25} + 24 q^{27} - 20 q^{28} - 10 q^{29} + 14 q^{30} - 32 q^{31} + 40 q^{32} - 14 q^{34} + 10 q^{35} - 24 q^{37} + 12 q^{38} - 14 q^{40} - 2 q^{41} - 2 q^{42} + 14 q^{43} + 4 q^{44} + 4 q^{45} - 14 q^{46} - 18 q^{47} - 6 q^{48} + 2 q^{49} - 38 q^{50} + 12 q^{51} - 8 q^{53} - 2 q^{54} - 8 q^{55} - 28 q^{56} - 24 q^{57} + 14 q^{58} - 18 q^{59} + 14 q^{60} - 4 q^{61} - 8 q^{63} + 6 q^{64} - 4 q^{66} + 14 q^{67} - 34 q^{68} - 10 q^{69} - 18 q^{70} + 12 q^{71} - 8 q^{72} - 26 q^{73} + 2 q^{74} - 18 q^{75} - 54 q^{76} - 14 q^{77} + 26 q^{79} - 24 q^{80} - 16 q^{81} - 64 q^{82} - 16 q^{83} + 74 q^{84} - 56 q^{85} - 32 q^{86} - 18 q^{87} + 6 q^{88} + 8 q^{89} - 20 q^{90} - 30 q^{92} - 48 q^{93} - 4 q^{94} + 22 q^{95} - 16 q^{96} - 20 q^{97} + 46 q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 4 * q^4 - 8 * q^5 - 4 * q^6 - 14 * q^7 + 6 * q^8 + 2 * q^9 + 10 * q^10 + 8 * q^11 - 8 * q^12 - 6 * q^14 - 2 * q^15 + 12 * q^16 + 6 * q^17 - 6 * q^18 - 4 * q^19 - 16 * q^20 - 2 * q^21 - 2 * q^22 - 14 * q^23 - 26 * q^24 + 12 * q^25 + 24 * q^27 - 20 * q^28 - 10 * q^29 + 14 * q^30 - 32 * q^31 + 40 * q^32 - 14 * q^34 + 10 * q^35 - 24 * q^37 + 12 * q^38 - 14 * q^40 - 2 * q^41 - 2 * q^42 + 14 * q^43 + 4 * q^44 + 4 * q^45 - 14 * q^46 - 18 * q^47 - 6 * q^48 + 2 * q^49 - 38 * q^50 + 12 * q^51 - 8 * q^53 - 2 * q^54 - 8 * q^55 - 28 * q^56 - 24 * q^57 + 14 * q^58 - 18 * q^59 + 14 * q^60 - 4 * q^61 - 8 * q^63 + 6 * q^64 - 4 * q^66 + 14 * q^67 - 34 * q^68 - 10 * q^69 - 18 * q^70 + 12 * q^71 - 8 * q^72 - 26 * q^73 + 2 * q^74 - 18 * q^75 - 54 * q^76 - 14 * q^77 + 26 * q^79 - 24 * q^80 - 16 * q^81 - 64 * q^82 - 16 * q^83 + 74 * q^84 - 56 * q^85 - 32 * q^86 - 18 * q^87 + 6 * q^88 + 8 * q^89 - 20 * q^90 - 30 * q^92 - 48 * q^93 - 4 * q^94 + 22 * q^95 - 16 * q^96 - 20 * q^97 + 46 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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