LMFDB - Embedded newform 1666.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1666,2,Mod(1,1666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1666, 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("1666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1666 = 2 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1666.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:	$13.3030769767$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.74912$ of defining polynomial
Character	$\chi$	$=$	1666.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.334904 q^{3} +1.00000 q^{4} +0.808530 q^{5} +0.334904 q^{6} -1.00000 q^{8} -2.88784 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.334904 q^{3} +1.00000 q^{4} +0.808530 q^{5} +0.334904 q^{6} -1.00000 q^{8} -2.88784 q^{9} -0.808530 q^{10} +4.57754 q^{11} -0.334904 q^{12} -5.03127 q^{13} -0.270780 q^{15} +1.00000 q^{16} -1.00000 q^{17} +2.88784 q^{18} +1.53304 q^{19} +0.808530 q^{20} -4.57754 q^{22} +1.64520 q^{23} +0.334904 q^{24} -4.34628 q^{25} +5.03127 q^{26} +1.97186 q^{27} -4.02656 q^{29} +0.270780 q^{30} -0.856566 q^{31} -1.00000 q^{32} -1.53304 q^{33} +1.00000 q^{34} -2.88784 q^{36} -0.0547002 q^{37} -1.53304 q^{38} +1.68499 q^{39} -0.808530 q^{40} -10.0907 q^{41} -7.27391 q^{43} +4.57754 q^{44} -2.33490 q^{45} -1.64520 q^{46} +2.91245 q^{47} -0.334904 q^{48} +4.34628 q^{50} +0.334904 q^{51} -5.03127 q^{52} +13.5608 q^{53} -1.97186 q^{54} +3.70108 q^{55} -0.513421 q^{57} +4.02656 q^{58} -9.41421 q^{59} -0.270780 q^{60} -10.6334 q^{61} +0.856566 q^{62} +1.00000 q^{64} -4.06793 q^{65} +1.53304 q^{66} +13.4420 q^{67} -1.00000 q^{68} -0.550984 q^{69} -9.37941 q^{71} +2.88784 q^{72} +4.95668 q^{73} +0.0547002 q^{74} +1.45559 q^{75} +1.53304 q^{76} -1.68499 q^{78} +0.541560 q^{79} +0.808530 q^{80} +8.00313 q^{81} +10.0907 q^{82} -14.9124 q^{83} -0.808530 q^{85} +7.27391 q^{86} +1.34851 q^{87} -4.57754 q^{88} -1.63853 q^{89} +2.33490 q^{90} +1.64520 q^{92} +0.286867 q^{93} -2.91245 q^{94} +1.23951 q^{95} +0.334904 q^{96} +1.23951 q^{97} -13.2192 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{20} + 12 q^{23} - 12 q^{27} + 4 q^{30} - 12 q^{31} - 4 q^{32} + 4 q^{34} - 4 q^{37} + 4 q^{39} + 4 q^{40} - 20 q^{41} + 8 q^{43} - 8 q^{45} - 12 q^{46} - 8 q^{47} + 12 q^{54} - 8 q^{55} + 12 q^{57} - 32 q^{59} - 4 q^{60} + 4 q^{61} + 12 q^{62} + 4 q^{64} - 28 q^{65} - 4 q^{68} - 24 q^{71} + 4 q^{74} + 28 q^{75} - 4 q^{78} + 8 q^{79} - 4 q^{80} - 8 q^{81} + 20 q^{82} - 40 q^{83} + 4 q^{85} - 8 q^{86} - 32 q^{87} - 24 q^{89} + 8 q^{90} + 12 q^{92} - 16 q^{93} + 8 q^{94} + 28 q^{95} + 28 q^{97} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8 + 4 * q^10 - 4 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^20 + 12 * q^23 - 12 * q^27 + 4 * q^30 - 12 * q^31 - 4 * q^32 + 4 * q^34 - 4 * q^37 + 4 * q^39 + 4 * q^40 - 20 * q^41 + 8 * q^43 - 8 * q^45 - 12 * q^46 - 8 * q^47 + 12 * q^54 - 8 * q^55 + 12 * q^57 - 32 * q^59 - 4 * q^60 + 4 * q^61 + 12 * q^62 + 4 * q^64 - 28 * q^65 - 4 * q^68 - 24 * q^71 + 4 * q^74 + 28 * q^75 - 4 * q^78 + 8 * q^79 - 4 * q^80 - 8 * q^81 + 20 * q^82 - 40 * q^83 + 4 * q^85 - 8 * q^86 - 32 * q^87 - 24 * q^89 + 8 * q^90 + 12 * q^92 - 16 * q^93 + 8 * q^94 + 28 * q^95 + 28 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.334904	−0.193357	−0.0966785	−	0.995316i	$-0.530822\pi$
$3$	−0.334904	−0.193357	−0.0966785	+	0.995316i	$0.530822\pi$
$4$	1.00000	0.500000
$5$	0.808530	0.361585	0.180793	−	0.983521i	$-0.442134\pi$
$5$	0.808530	0.361585	0.180793	+	0.983521i	$0.442134\pi$
$6$	0.334904	0.136724
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.88784	−0.962613
$10$	−0.808530	−0.255680
$11$	4.57754	1.38018	0.690091	−	0.723723i	$-0.257571\pi$
$11$	4.57754	1.38018	0.690091	+	0.723723i	$0.257571\pi$
$12$	−0.334904	−0.0966785
$13$	−5.03127	−1.39542	−0.697712	−	0.716378i	$-0.745798\pi$
$13$	−5.03127	−1.39542	−0.697712	+	0.716378i	$0.745798\pi$
$14$	0	0
$15$	−0.270780	−0.0699151
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.88784	0.680670
$19$	1.53304	0.351703	0.175852	−	0.984417i	$-0.443732\pi$
$19$	1.53304	0.351703	0.175852	+	0.984417i	$0.443732\pi$
$20$	0.808530	0.180793
$21$	0	0
$22$	−4.57754	−0.975936
$23$	1.64520	0.343048	0.171524	−	0.985180i	$-0.445131\pi$
$23$	1.64520	0.343048	0.171524	+	0.985180i	$0.445131\pi$
$24$	0.334904	0.0683620
$25$	−4.34628	−0.869256
$26$	5.03127	0.986714
$27$	1.97186	0.379485
$28$	0	0
$29$	−4.02656	−0.747714	−0.373857	−	0.927486i	$-0.621965\pi$
$29$	−4.02656	−0.747714	−0.373857	+	0.927486i	$0.621965\pi$
$30$	0.270780	0.0494374
$31$	−0.856566	−0.153844	−0.0769219	−	0.997037i	$-0.524509\pi$
$31$	−0.856566	−0.153844	−0.0769219	+	0.997037i	$0.524509\pi$
$32$	−1.00000	−0.176777
$33$	−1.53304	−0.266868
$34$	1.00000	0.171499
$35$	0	0
$36$	−2.88784	−0.481307
$37$	−0.0547002	−0.00899266	−0.00449633	−	0.999990i	$-0.501431\pi$
$37$	−0.0547002	−0.00899266	−0.00449633	+	0.999990i	$0.501431\pi$
$38$	−1.53304	−0.248692
$39$	1.68499	0.269815
$40$	−0.808530	−0.127840
$41$	−10.0907	−1.57590	−0.787950	−	0.615739i	$-0.788858\pi$
$41$	−10.0907	−1.57590	−0.787950	+	0.615739i	$0.788858\pi$
$42$	0	0
$43$	−7.27391	−1.10926	−0.554631	−	0.832097i	$-0.687140\pi$
$43$	−7.27391	−1.10926	−0.554631	+	0.832097i	$0.687140\pi$
$44$	4.57754	0.690091
$45$	−2.33490	−0.348067
$46$	−1.64520	−0.242571
$47$	2.91245	0.424824	0.212412	−	0.977180i	$-0.431868\pi$
$47$	2.91245	0.424824	0.212412	+	0.977180i	$0.431868\pi$
$48$	−0.334904	−0.0483392
$49$	0	0
$50$	4.34628	0.614657
$51$	0.334904	0.0468959
$52$	−5.03127	−0.697712
$53$	13.5608	1.86272	0.931358	−	0.364104i	$-0.118625\pi$
$53$	13.5608	1.86272	0.931358	+	0.364104i	$0.118625\pi$
$54$	−1.97186	−0.268336
$55$	3.70108	0.499054
$56$	0	0
$57$	−0.513421	−0.0680042
$58$	4.02656	0.528713
$59$	−9.41421	−1.22563	−0.612813	−	0.790228i	$-0.709962\pi$
$59$	−9.41421	−1.22563	−0.612813	+	0.790228i	$0.709962\pi$
$60$	−0.270780	−0.0349575
$61$	−10.6334	−1.36147	−0.680735	−	0.732529i	$-0.738340\pi$
$61$	−10.6334	−1.36147	−0.680735	+	0.732529i	$0.738340\pi$
$62$	0.856566	0.108784
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.06793	−0.504565
$66$	1.53304	0.188704
$67$	13.4420	1.64220	0.821098	−	0.570787i	$-0.193362\pi$
$67$	13.4420	1.64220	0.821098	+	0.570787i	$0.193362\pi$
$68$	−1.00000	−0.121268
$69$	−0.550984	−0.0663306
$70$	0	0
$71$	−9.37941	−1.11313	−0.556566	−	0.830804i	$-0.687881\pi$
$71$	−9.37941	−1.11313	−0.556566	+	0.830804i	$0.687881\pi$
$72$	2.88784	0.340335
$73$	4.95668	0.580135	0.290067	−	0.957006i	$-0.406322\pi$
$73$	4.95668	0.580135	0.290067	+	0.957006i	$0.406322\pi$
$74$	0.0547002	0.00635877
$75$	1.45559	0.168077
$76$	1.53304	0.175852
$77$	0	0
$78$	−1.68499	−0.190788
$79$	0.541560	0.0609302	0.0304651	−	0.999536i	$-0.490301\pi$
$79$	0.541560	0.0609302	0.0304651	+	0.999536i	$0.490301\pi$
$80$	0.808530	0.0903964
$81$	8.00313	0.889237
$82$	10.0907	1.11433
$83$	−14.9124	−1.63685	−0.818427	−	0.574611i	$-0.805154\pi$
$83$	−14.9124	−1.63685	−0.818427	+	0.574611i	$0.805154\pi$
$84$	0	0
$85$	−0.808530	−0.0876974
$86$	7.27391	0.784366
$87$	1.34851	0.144576
$88$	−4.57754	−0.487968
$89$	−1.63853	−0.173684	−0.0868422	−	0.996222i	$-0.527678\pi$
$89$	−1.63853	−0.173684	−0.0868422	+	0.996222i	$0.527678\pi$
$90$	2.33490	0.246120
$91$	0	0
$92$	1.64520	0.171524
$93$	0.286867	0.0297468
$94$	−2.91245	−0.300396
$95$	1.23951	0.127171
$96$	0.334904	0.0341810
$97$	1.23951	0.125853	0.0629264	−	0.998018i	$-0.479957\pi$
$97$	1.23951	0.125853	0.0629264	+	0.998018i	$0.479957\pi$
$98$	0	0
$99$	−13.2192	−1.32858
$100$	−4.34628	−0.434628
$101$	−10.9030	−1.08489	−0.542446	−	0.840091i	$-0.682502\pi$
$101$	−10.9030	−1.08489	−0.542446	+	0.840091i	$0.682502\pi$
$102$	−0.334904	−0.0331604
$103$	−1.32077	−0.130139	−0.0650696	−	0.997881i	$-0.520727\pi$
$103$	−1.32077	−0.130139	−0.0650696	+	0.997881i	$0.520727\pi$
$104$	5.03127	0.493357
$105$	0	0
$106$	−13.5608	−1.31714
$107$	−4.53775	−0.438681	−0.219340	−	0.975648i	$-0.570391\pi$
$107$	−4.53775	−0.438681	−0.219340	+	0.975648i	$0.570391\pi$
$108$	1.97186	0.189742
$109$	−14.9949	−1.43625	−0.718125	−	0.695914i	$-0.754999\pi$
$109$	−14.9949	−1.43625	−0.718125	+	0.695914i	$0.754999\pi$
$110$	−3.70108	−0.352884
$111$	0.0183193	0.00173879
$112$	0	0
$113$	10.6948	1.00608	0.503041	−	0.864263i	$-0.332215\pi$
$113$	10.6948	1.00608	0.503041	+	0.864263i	$0.332215\pi$
$114$	0.513421	0.0480862
$115$	1.33019	0.124041
$116$	−4.02656	−0.373857
$117$	14.5295	1.34325
$118$	9.41421	0.866649
$119$	0	0
$120$	0.270780	0.0247187
$121$	9.95392	0.904901
$122$	10.6334	0.962705
$123$	3.37941	0.304711
$124$	−0.856566	−0.0769219
$125$	−7.55675	−0.675896
$126$	0	0
$127$	8.33333	0.739463	0.369732	−	0.929139i	$-0.379450\pi$
$127$	8.33333	0.739463	0.369732	+	0.929139i	$0.379450\pi$
$128$	−1.00000	−0.0883883
$129$	2.43606	0.214483
$130$	4.06793	0.356781
$131$	−15.8840	−1.38779	−0.693897	−	0.720074i	$-0.744108\pi$
$131$	−15.8840	−1.38779	−0.693897	+	0.720074i	$0.744108\pi$
$132$	−1.53304	−0.133434
$133$	0	0
$134$	−13.4420	−1.16121
$135$	1.59431	0.137216
$136$	1.00000	0.0857493
$137$	−5.45178	−0.465777	−0.232888	−	0.972503i	$-0.574818\pi$
$137$	−5.45178	−0.465777	−0.232888	+	0.972503i	$0.574818\pi$
$138$	0.550984	0.0469028
$139$	−10.9899	−0.932151	−0.466076	−	0.884745i	$-0.654333\pi$
$139$	−10.9899	−0.932151	−0.466076	+	0.884745i	$0.654333\pi$
$140$	0	0
$141$	−0.975391	−0.0821427
$142$	9.37941	0.787103
$143$	−23.0309	−1.92594
$144$	−2.88784	−0.240653
$145$	−3.25559	−0.270362
$146$	−4.95668	−0.410217
$147$	0	0
$148$	−0.0547002	−0.00449633
$149$	−1.28334	−0.105135	−0.0525676	−	0.998617i	$-0.516740\pi$
$149$	−1.28334	−0.105135	−0.0525676	+	0.998617i	$0.516740\pi$
$150$	−1.45559	−0.118848
$151$	−9.49157	−0.772413	−0.386207	−	0.922412i	$-0.626215\pi$
$151$	−9.49157	−0.772413	−0.386207	+	0.922412i	$0.626215\pi$
$152$	−1.53304	−0.124346
$153$	2.88784	0.233468
$154$	0	0
$155$	−0.692559	−0.0556277
$156$	1.68499	0.134907
$157$	5.13677	0.409959	0.204979	−	0.978766i	$-0.434287\pi$
$157$	5.13677	0.409959	0.204979	+	0.978766i	$0.434287\pi$
$158$	−0.541560	−0.0430842
$159$	−4.54156	−0.360169
$160$	−0.808530	−0.0639199
$161$	0	0
$162$	−8.00313	−0.628786
$163$	−3.25678	−0.255090	−0.127545	−	0.991833i	$-0.540710\pi$
$163$	−3.25678	−0.255090	−0.127545	+	0.991833i	$0.540710\pi$
$164$	−10.0907	−0.787950
$165$	−1.23951	−0.0964955
$166$	14.9124	1.15743
$167$	4.03441	0.312192	0.156096	−	0.987742i	$-0.450109\pi$
$167$	4.03441	0.312192	0.156096	+	0.987742i	$0.450109\pi$
$168$	0	0
$169$	12.3137	0.947208
$170$	0.808530	0.0620114
$171$	−4.42717	−0.338554
$172$	−7.27391	−0.554631
$173$	−23.4337	−1.78163	−0.890816	−	0.454364i	$-0.849867\pi$
$173$	−23.4337	−1.78163	−0.890816	+	0.454364i	$0.849867\pi$
$174$	−1.34851	−0.102230
$175$	0	0
$176$	4.57754	0.345045
$177$	3.15286	0.236983
$178$	1.63853	0.122813
$179$	−3.38883	−0.253293	−0.126647	−	0.991948i	$-0.540421\pi$
$179$	−3.38883	−0.253293	−0.126647	+	0.991948i	$0.540421\pi$
$180$	−2.33490	−0.174033
$181$	8.98990	0.668214	0.334107	−	0.942535i	$-0.391565\pi$
$181$	8.98990	0.668214	0.334107	+	0.942535i	$0.391565\pi$
$182$	0	0
$183$	3.56118	0.263250
$184$	−1.64520	−0.121286
$185$	−0.0442268	−0.00325162
$186$	−0.286867	−0.0210341
$187$	−4.57754	−0.334743
$188$	2.91245	0.212412
$189$	0	0
$190$	−1.23951	−0.0899233
$191$	−7.38331	−0.534238	−0.267119	−	0.963664i	$-0.586072\pi$
$191$	−7.38331	−0.534238	−0.267119	+	0.963664i	$0.586072\pi$
$192$	−0.334904	−0.0241696
$193$	−2.41735	−0.174004	−0.0870022	−	0.996208i	$-0.527729\pi$
$193$	−2.41735	−0.174004	−0.0870022	+	0.996208i	$0.527729\pi$
$194$	−1.23951	−0.0889914
$195$	1.36237	0.0975611
$196$	0	0
$197$	4.61511	0.328813	0.164406	−	0.986393i	$-0.447429\pi$
$197$	4.61511	0.328813	0.164406	+	0.986393i	$0.447429\pi$
$198$	13.2192	0.939449
$199$	−23.8874	−1.69334	−0.846668	−	0.532122i	$-0.821395\pi$
$199$	−23.8874	−1.69334	−0.846668	+	0.532122i	$0.821395\pi$
$200$	4.34628	0.307328
$201$	−4.50176	−0.317530
$202$	10.9030	0.767134
$203$	0	0
$204$	0.334904	0.0234480
$205$	−8.15862	−0.569823
$206$	1.32077	0.0920223
$207$	−4.75107	−0.330222
$208$	−5.03127	−0.348856
$209$	7.01755	0.485414
$210$	0	0
$211$	11.6436	0.801580	0.400790	−	0.916170i	$-0.368736\pi$
$211$	11.6436	0.801580	0.400790	+	0.916170i	$0.368736\pi$
$212$	13.5608	0.931358
$213$	3.14120	0.215232
$214$	4.53775	0.310194
$215$	−5.88118	−0.401093
$216$	−1.97186	−0.134168
$217$	0	0
$218$	14.9949	1.01558
$219$	−1.66001	−0.112173
$220$	3.70108	0.249527
$221$	5.03127	0.338440
$222$	−0.0183193	−0.00122951
$223$	−23.1923	−1.55307	−0.776534	−	0.630075i	$-0.783024\pi$
$223$	−23.1923	−1.55307	−0.776534	+	0.630075i	$0.783024\pi$
$224$	0	0
$225$	12.5514	0.836757
$226$	−10.6948	−0.711407
$227$	25.8792	1.71766	0.858831	−	0.512258i	$-0.171191\pi$
$227$	25.8792	1.71766	0.858831	+	0.512258i	$0.171191\pi$
$228$	−0.513421	−0.0340021
$229$	17.5228	1.15794	0.578970	−	0.815349i	$-0.303455\pi$
$229$	17.5228	1.15794	0.578970	+	0.815349i	$0.303455\pi$
$230$	−1.33019	−0.0877103
$231$	0	0
$232$	4.02656	0.264357
$233$	−29.3857	−1.92512	−0.962560	−	0.271069i	$-0.912623\pi$
$233$	−29.3857	−1.92512	−0.962560	+	0.271069i	$0.912623\pi$
$234$	−14.5295	−0.949824
$235$	2.35480	0.153610
$236$	−9.41421	−0.612813
$237$	−0.181370	−0.0117813
$238$	0	0
$239$	−7.40922	−0.479263	−0.239631	−	0.970864i	$-0.577027\pi$
$239$	−7.40922	−0.479263	−0.239631	+	0.970864i	$0.577027\pi$
$240$	−0.270780	−0.0174788
$241$	11.8116	0.760850	0.380425	−	0.924812i	$-0.375778\pi$
$241$	11.8116	0.760850	0.380425	+	0.924812i	$0.375778\pi$
$242$	−9.95392	−0.639862
$243$	−8.59586	−0.551425
$244$	−10.6334	−0.680735
$245$	0	0
$246$	−3.37941	−0.215463
$247$	−7.71313	−0.490775
$248$	0.856566	0.0543920
$249$	4.99424	0.316497
$250$	7.55675	0.477931
$251$	−17.2883	−1.09123	−0.545615	−	0.838036i	$-0.683704\pi$
$251$	−17.2883	−1.09123	−0.545615	+	0.838036i	$0.683704\pi$
$252$	0	0
$253$	7.53097	0.473468
$254$	−8.33333	−0.522879
$255$	0.270780	0.0169569
$256$	1.00000	0.0625000
$257$	−3.89060	−0.242689	−0.121344	−	0.992610i	$-0.538721\pi$
$257$	−3.89060	−0.242689	−0.121344	+	0.992610i	$0.538721\pi$
$258$	−2.43606	−0.151663
$259$	0	0
$260$	−4.06793	−0.252283
$261$	11.6281	0.719759
$262$	15.8840	0.981319
$263$	−5.62372	−0.346774	−0.173387	−	0.984854i	$-0.555471\pi$
$263$	−5.62372	−0.346774	−0.173387	+	0.984854i	$0.555471\pi$
$264$	1.53304	0.0943520
$265$	10.9643	0.673531
$266$	0	0
$267$	0.548752	0.0335831
$268$	13.4420	0.821098
$269$	−7.71116	−0.470158	−0.235079	−	0.971976i	$-0.575535\pi$
$269$	−7.71116	−0.470158	−0.235079	+	0.971976i	$0.575535\pi$
$270$	−1.59431	−0.0970265
$271$	20.2091	1.22762	0.613808	−	0.789455i	$-0.289637\pi$
$271$	20.2091	1.22762	0.613808	+	0.789455i	$0.289637\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	5.45178	0.329354
$275$	−19.8953	−1.19973
$276$	−0.550984	−0.0331653
$277$	10.1924	0.612400	0.306200	−	0.951967i	$-0.400942\pi$
$277$	10.1924	0.612400	0.306200	+	0.951967i	$0.400942\pi$
$278$	10.9899	0.659130
$279$	2.47363	0.148092
$280$	0	0
$281$	22.4223	1.33760	0.668802	−	0.743440i	$-0.266807\pi$
$281$	22.4223	1.33760	0.668802	+	0.743440i	$0.266807\pi$
$282$	0.975391	0.0580837
$283$	28.0356	1.66654	0.833271	−	0.552864i	$-0.186465\pi$
$283$	28.0356	1.66654	0.833271	+	0.552864i	$0.186465\pi$
$284$	−9.37941	−0.556566
$285$	−0.415116	−0.0245893
$286$	23.0309	1.36184
$287$	0	0
$288$	2.88784	0.170168
$289$	1.00000	0.0588235
$290$	3.25559	0.191175
$291$	−0.415116	−0.0243345
$292$	4.95668	0.290067
$293$	−7.58226	−0.442960	−0.221480	−	0.975165i	$-0.571089\pi$
$293$	−7.58226	−0.442960	−0.221480	+	0.975165i	$0.571089\pi$
$294$	0	0
$295$	−7.61167	−0.443169
$296$	0.0547002	0.00317939
$297$	9.02628	0.523758
$298$	1.28334	0.0743418
$299$	−8.27744	−0.478697
$300$	1.45559	0.0840383
$301$	0	0
$302$	9.49157	0.546179
$303$	3.65147	0.209771
$304$	1.53304	0.0879258
$305$	−8.59744	−0.492288
$306$	−2.88784	−0.165087
$307$	13.6221	0.777452	0.388726	−	0.921354i	$-0.372915\pi$
$307$	13.6221	0.777452	0.388726	+	0.921354i	$0.372915\pi$
$308$	0	0
$309$	0.442331	0.0251633
$310$	0.692559	0.0393347
$311$	−7.38331	−0.418669	−0.209335	−	0.977844i	$-0.567130\pi$
$311$	−7.38331	−0.418669	−0.209335	+	0.977844i	$0.567130\pi$
$312$	−1.68499	−0.0953940
$313$	32.5103	1.83759	0.918794	−	0.394736i	$-0.129164\pi$
$313$	32.5103	1.83759	0.918794	+	0.394736i	$0.129164\pi$
$314$	−5.13677	−0.289885
$315$	0	0
$316$	0.541560	0.0304651
$317$	−1.16832	−0.0656196	−0.0328098	−	0.999462i	$-0.510446\pi$
$317$	−1.16832	−0.0656196	−0.0328098	+	0.999462i	$0.510446\pi$
$318$	4.54156	0.254678
$319$	−18.4318	−1.03198
$320$	0.808530	0.0451982
$321$	1.51971	0.0848220
$322$	0	0
$323$	−1.53304	−0.0853005
$324$	8.00313	0.444619
$325$	21.8673	1.21298
$326$	3.25678	0.180376
$327$	5.02185	0.277709
$328$	10.0907	0.557165
$329$	0	0
$330$	1.23951	0.0682326
$331$	4.19079	0.230347	0.115173	−	0.993345i	$-0.463258\pi$
$331$	4.19079	0.230347	0.115173	+	0.993345i	$0.463258\pi$
$332$	−14.9124	−0.818427
$333$	0.157966	0.00865646
$334$	−4.03441	−0.220753
$335$	10.8682	0.593794
$336$	0	0
$337$	−7.53580	−0.410501	−0.205251	−	0.978709i	$-0.565801\pi$
$337$	−7.53580	−0.410501	−0.205251	+	0.978709i	$0.565801\pi$
$338$	−12.3137	−0.669777
$339$	−3.58173	−0.194533
$340$	−0.808530	−0.0438487
$341$	−3.92097	−0.212332
$342$	4.42717	0.239394
$343$	0	0
$344$	7.27391	0.392183
$345$	−0.445487	−0.0239842
$346$	23.4337	1.25980
$347$	20.5329	1.10227	0.551133	−	0.834418i	$-0.314196\pi$
$347$	20.5329	1.10227	0.551133	+	0.834418i	$0.314196\pi$
$348$	1.34851	0.0722878
$349$	32.7279	1.75189	0.875943	−	0.482415i	$-0.160240\pi$
$349$	32.7279	1.75189	0.875943	+	0.482415i	$0.160240\pi$
$350$	0	0
$351$	−9.92097	−0.529542
$352$	−4.57754	−0.243984
$353$	19.0363	1.01320	0.506599	−	0.862182i	$-0.330902\pi$
$353$	19.0363	1.01320	0.506599	+	0.862182i	$0.330902\pi$
$354$	−3.15286	−0.167572
$355$	−7.58353	−0.402492
$356$	−1.63853	−0.0868422
$357$	0	0
$358$	3.38883	0.179105
$359$	31.8874	1.68295	0.841477	−	0.540292i	$-0.181686\pi$
$359$	31.8874	1.68295	0.841477	+	0.540292i	$0.181686\pi$
$360$	2.33490	0.123060
$361$	−16.6498	−0.876305
$362$	−8.98990	−0.472499
$363$	−3.33361	−0.174969
$364$	0	0
$365$	4.00762	0.209768
$366$	−3.56118	−0.186146
$367$	−1.65462	−0.0863706	−0.0431853	−	0.999067i	$-0.513751\pi$
$367$	−1.65462	−0.0863706	−0.0431853	+	0.999067i	$0.513751\pi$
$368$	1.64520	0.0857619
$369$	29.1403	1.51698
$370$	0.0442268	0.00229924
$371$	0	0
$372$	0.286867	0.0148734
$373$	−7.07550	−0.366355	−0.183178	−	0.983080i	$-0.558638\pi$
$373$	−7.07550	−0.366355	−0.183178	+	0.983080i	$0.558638\pi$
$374$	4.57754	0.236699
$375$	2.53078	0.130689
$376$	−2.91245	−0.150198
$377$	20.2587	1.04338
$378$	0	0
$379$	27.1668	1.39547	0.697733	−	0.716358i	$-0.254192\pi$
$379$	27.1668	1.39547	0.697733	+	0.716358i	$0.254192\pi$
$380$	1.23951	0.0635854
$381$	−2.79086	−0.142980
$382$	7.38331	0.377763
$383$	6.03979	0.308619	0.154310	−	0.988023i	$-0.450685\pi$
$383$	6.03979	0.308619	0.154310	+	0.988023i	$0.450685\pi$
$384$	0.334904	0.0170905
$385$	0	0
$386$	2.41735	0.123040
$387$	21.0059	1.06779
$388$	1.23951	0.0629264
$389$	−17.5773	−0.891203	−0.445601	−	0.895231i	$-0.647010\pi$
$389$	−17.5773	−0.891203	−0.445601	+	0.895231i	$0.647010\pi$
$390$	−1.36237	−0.0689861
$391$	−1.64520	−0.0832013
$392$	0	0
$393$	5.31962	0.268340
$394$	−4.61511	−0.232506
$395$	0.437867	0.0220315
$396$	−13.2192	−0.664290
$397$	−6.35046	−0.318721	−0.159360	−	0.987220i	$-0.550943\pi$
$397$	−6.35046	−0.318721	−0.159360	+	0.987220i	$0.550943\pi$
$398$	23.8874	1.19737
$399$	0	0
$400$	−4.34628	−0.217314
$401$	−9.21360	−0.460105	−0.230053	−	0.973178i	$-0.573890\pi$
$401$	−9.21360	−0.460105	−0.230053	+	0.973178i	$0.573890\pi$
$402$	4.50176	0.224528
$403$	4.30962	0.214677
$404$	−10.9030	−0.542446
$405$	6.47077	0.321535
$406$	0	0
$407$	−0.250393	−0.0124115
$408$	−0.334904	−0.0165802
$409$	30.1157	1.48912	0.744562	−	0.667553i	$-0.232658\pi$
$409$	30.1157	1.48912	0.744562	+	0.667553i	$0.232658\pi$
$410$	8.15862	0.402925
$411$	1.82582	0.0900611
$412$	−1.32077	−0.0650696
$413$	0	0
$414$	4.75107	0.233502
$415$	−12.0572	−0.591863
$416$	5.03127	0.246678
$417$	3.68056	0.180238
$418$	−7.01755	−0.343240
$419$	−25.5564	−1.24851	−0.624257	−	0.781219i	$-0.714598\pi$
$419$	−25.5564	−1.24851	−0.624257	+	0.781219i	$0.714598\pi$
$420$	0	0
$421$	27.8969	1.35961	0.679805	−	0.733393i	$-0.262064\pi$
$421$	27.8969	1.35961	0.679805	+	0.733393i	$0.262064\pi$
$422$	−11.6436	−0.566803
$423$	−8.41068	−0.408941
$424$	−13.5608	−0.658570
$425$	4.34628	0.210826
$426$	−3.14120	−0.152192
$427$	0	0
$428$	−4.53775	−0.219340
$429$	7.71313	0.372394
$430$	5.88118	0.283615
$431$	32.9987	1.58949	0.794746	−	0.606942i	$-0.207604\pi$
$431$	32.9987	1.58949	0.794746	+	0.606942i	$0.207604\pi$
$432$	1.97186	0.0948712
$433$	9.96464	0.478870	0.239435	−	0.970912i	$-0.423038\pi$
$433$	9.96464	0.478870	0.239435	+	0.970912i	$0.423038\pi$
$434$	0	0
$435$	1.09031	0.0522764
$436$	−14.9949	−0.718125
$437$	2.52215	0.120651
$438$	1.66001	0.0793184
$439$	−30.6252	−1.46166	−0.730830	−	0.682559i	$-0.760867\pi$
$439$	−30.6252	−1.46166	−0.730830	+	0.682559i	$0.760867\pi$
$440$	−3.70108	−0.176442
$441$	0	0
$442$	−5.03127	−0.239313
$443$	−35.5045	−1.68687	−0.843435	−	0.537231i	$-0.819470\pi$
$443$	−35.5045	−1.68687	−0.843435	+	0.537231i	$0.819470\pi$
$444$	0.0183193	0.000869397 0
$445$	−1.32480	−0.0628017
$446$	23.1923	1.09819
$447$	0.429795	0.0203286
$448$	0	0
$449$	11.0402	0.521018	0.260509	−	0.965471i	$-0.416110\pi$
$449$	11.0402	0.521018	0.260509	+	0.965471i	$0.416110\pi$
$450$	−12.5514	−0.591677
$451$	−46.1906	−2.17503
$452$	10.6948	0.503041
$453$	3.17877	0.149351
$454$	−25.8792	−1.21457
$455$	0	0
$456$	0.513421	0.0240431
$457$	−24.9169	−1.16556	−0.582781	−	0.812629i	$-0.698036\pi$
$457$	−24.9169	−1.16556	−0.582781	+	0.812629i	$0.698036\pi$
$458$	−17.5228	−0.818788
$459$	−1.97186	−0.0920386
$460$	1.33019	0.0620205
$461$	−18.8991	−0.880220	−0.440110	−	0.897944i	$-0.645061\pi$
$461$	−18.8991	−0.880220	−0.440110	+	0.897944i	$0.645061\pi$
$462$	0	0
$463$	−4.64982	−0.216095	−0.108048	−	0.994146i	$-0.534460\pi$
$463$	−4.64982	−0.216095	−0.108048	+	0.994146i	$0.534460\pi$
$464$	−4.02656	−0.186928
$465$	0.231941	0.0107560
$466$	29.3857	1.36127
$467$	35.0313	1.62105	0.810527	−	0.585701i	$-0.199181\pi$
$467$	35.0313	1.62105	0.810527	+	0.585701i	$0.199181\pi$
$468$	14.5295	0.671627
$469$	0	0
$470$	−2.35480	−0.108619
$471$	−1.72032	−0.0792684
$472$	9.41421	0.433324
$473$	−33.2967	−1.53098
$474$	0.181370	0.00833062
$475$	−6.66301	−0.305720
$476$	0	0
$477$	−39.1614	−1.79308
$478$	7.40922	0.338890
$479$	31.4223	1.43572	0.717861	−	0.696186i	$-0.245121\pi$
$479$	31.4223	1.43572	0.717861	+	0.696186i	$0.245121\pi$
$480$	0.270780	0.0123594
$481$	0.275212	0.0125486
$482$	−11.8116	−0.538002
$483$	0	0
$484$	9.95392	0.452451
$485$	1.00218	0.0455066
$486$	8.59586	0.389916
$487$	−27.2426	−1.23448	−0.617240	−	0.786775i	$-0.711749\pi$
$487$	−27.2426	−1.23448	−0.617240	+	0.786775i	$0.711749\pi$
$488$	10.6334	0.481353
$489$	1.09071	0.0493235
$490$	0	0
$491$	−31.2215	−1.40901	−0.704504	−	0.709700i	$-0.748831\pi$
$491$	−31.2215	−1.40901	−0.704504	+	0.709700i	$0.748831\pi$
$492$	3.37941	0.152356
$493$	4.02656	0.181347
$494$	7.71313	0.347030
$495$	−10.6881	−0.480396
$496$	−0.856566	−0.0384610
$497$	0	0
$498$	−4.99424	−0.223797
$499$	40.3384	1.80580	0.902898	−	0.429856i	$-0.141436\pi$
$499$	40.3384	1.80580	0.902898	+	0.429856i	$0.141436\pi$
$500$	−7.55675	−0.337948
$501$	−1.35114	−0.0603644
$502$	17.2883	0.771616
$503$	8.44772	0.376665	0.188333	−	0.982105i	$-0.439692\pi$
$503$	8.44772	0.376665	0.188333	+	0.982105i	$0.439692\pi$
$504$	0	0
$505$	−8.81542	−0.392281
$506$	−7.53097	−0.334792
$507$	−4.12391	−0.183149
$508$	8.33333	0.369732
$509$	8.99204	0.398565	0.199283	−	0.979942i	$-0.436139\pi$
$509$	8.99204	0.398565	0.199283	+	0.979942i	$0.436139\pi$
$510$	−0.270780	−0.0119903
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	3.02294	0.133466
$514$	3.89060	0.171607
$515$	−1.06788	−0.0470564
$516$	2.43606	0.107242
$517$	13.3319	0.586335
$518$	0	0
$519$	7.84804	0.344491
$520$	4.06793	0.178391
$521$	26.7860	1.17352	0.586759	−	0.809762i	$-0.300404\pi$
$521$	26.7860	1.17352	0.586759	+	0.809762i	$0.300404\pi$
$522$	−11.6281	−0.508946
$523$	4.32223	0.188998	0.0944990	−	0.995525i	$-0.469875\pi$
$523$	4.32223	0.188998	0.0944990	+	0.995525i	$0.469875\pi$
$524$	−15.8840	−0.693897
$525$	0	0
$526$	5.62372	0.245206
$527$	0.856566	0.0373126
$528$	−1.53304	−0.0667169
$529$	−20.2933	−0.882318
$530$	−10.9643	−0.476259
$531$	27.1867	1.17980
$532$	0	0
$533$	50.7690	2.19905
$534$	−0.548752	−0.0237468
$535$	−3.66891	−0.158621
$536$	−13.4420	−0.580604
$537$	1.13493	0.0489760
$538$	7.71116	0.332452
$539$	0	0
$540$	1.59431	0.0686081
$541$	−29.4779	−1.26736	−0.633678	−	0.773597i	$-0.718456\pi$
$541$	−29.4779	−1.26736	−0.633678	+	0.773597i	$0.718456\pi$
$542$	−20.2091	−0.868056
$543$	−3.01075	−0.129204
$544$	1.00000	0.0428746
$545$	−12.1238	−0.519327
$546$	0	0
$547$	−10.7510	−0.459679	−0.229839	−	0.973229i	$-0.573820\pi$
$547$	−10.7510	−0.459679	−0.229839	+	0.973229i	$0.573820\pi$
$548$	−5.45178	−0.232888
$549$	30.7076	1.31057
$550$	19.8953	0.848338
$551$	−6.17287	−0.262973
$552$	0.550984	0.0234514
$553$	0	0
$554$	−10.1924	−0.433032
$555$	0.0148117	0.000628723 0
$556$	−10.9899	−0.466076
$557$	29.2669	1.24008	0.620038	−	0.784572i	$-0.287117\pi$
$557$	29.2669	1.24008	0.620038	+	0.784572i	$0.287117\pi$
$558$	−2.47363	−0.104717
$559$	36.5970	1.54789
$560$	0	0
$561$	1.53304	0.0647249
$562$	−22.4223	−0.945829
$563$	7.54693	0.318065	0.159032	−	0.987273i	$-0.449163\pi$
$563$	7.54693	0.318065	0.159032	+	0.987273i	$0.449163\pi$
$564$	−0.975391	−0.0410714
$565$	8.64706	0.363784
$566$	−28.0356	−1.17842
$567$	0	0
$568$	9.37941	0.393551
$569$	−16.8347	−0.705747	−0.352874	−	0.935671i	$-0.614795\pi$
$569$	−16.8347	−0.705747	−0.352874	+	0.935671i	$0.614795\pi$
$570$	0.415116	0.0173873
$571$	38.4994	1.61115	0.805575	−	0.592494i	$-0.201857\pi$
$571$	38.4994	1.61115	0.805575	+	0.592494i	$0.201857\pi$
$572$	−23.0309	−0.962969
$573$	2.47270	0.103299
$574$	0	0
$575$	−7.15049	−0.298196
$576$	−2.88784	−0.120327
$577$	−2.57429	−0.107169	−0.0535846	−	0.998563i	$-0.517065\pi$
$577$	−2.57429	−0.107169	−0.0535846	+	0.998563i	$0.517065\pi$
$578$	−1.00000	−0.0415945
$579$	0.809579	0.0336450
$580$	−3.25559	−0.135181
$581$	0	0
$582$	0.415116	0.0172071
$583$	62.0751	2.57089
$584$	−4.95668	−0.205109
$585$	11.7475	0.485701
$586$	7.58226	0.313220
$587$	15.7775	0.651208	0.325604	−	0.945506i	$-0.394432\pi$
$587$	15.7775	0.651208	0.325604	+	0.945506i	$0.394432\pi$
$588$	0	0
$589$	−1.31315	−0.0541073
$590$	7.61167	0.313368
$591$	−1.54562	−0.0635782
$592$	−0.0547002	−0.00224817
$593$	−15.3087	−0.628654	−0.314327	−	0.949315i	$-0.601779\pi$
$593$	−15.3087	−0.628654	−0.314327	+	0.949315i	$0.601779\pi$
$594$	−9.02628	−0.370353
$595$	0	0
$596$	−1.28334	−0.0525676
$597$	8.00000	0.327418
$598$	8.27744	0.338490
$599$	19.0751	0.779388	0.389694	−	0.920944i	$-0.372581\pi$
$599$	19.0751	0.779388	0.389694	+	0.920944i	$0.372581\pi$
$600$	−1.45559	−0.0594241
$601$	12.9544	0.528423	0.264211	−	0.964465i	$-0.414888\pi$
$601$	12.9544	0.528423	0.264211	+	0.964465i	$0.414888\pi$
$602$	0	0
$603$	−38.8182	−1.58080
$604$	−9.49157	−0.386207
$605$	8.04804	0.327199
$606$	−3.65147	−0.148331
$607$	29.5310	1.19863	0.599313	−	0.800515i	$-0.295441\pi$
$607$	29.5310	1.19863	0.599313	+	0.800515i	$0.295441\pi$
$608$	−1.53304	−0.0621729
$609$	0	0
$610$	8.59744	0.348100
$611$	−14.6533	−0.592810
$612$	2.88784	0.116734
$613$	−37.6239	−1.51961	−0.759807	−	0.650148i	$-0.774707\pi$
$613$	−37.6239	−1.51961	−0.759807	+	0.650148i	$0.774707\pi$
$614$	−13.6221	−0.549741
$615$	2.73235	0.110179
$616$	0	0
$617$	−5.28873	−0.212916	−0.106458	−	0.994317i	$-0.533951\pi$
$617$	−5.28873	−0.212916	−0.106458	+	0.994317i	$0.533951\pi$
$618$	−0.442331	−0.0177931
$619$	1.50332	0.0604236	0.0302118	−	0.999544i	$-0.490382\pi$
$619$	1.50332	0.0604236	0.0302118	+	0.999544i	$0.490382\pi$
$620$	−0.692559	−0.0278139
$621$	3.24410	0.130181
$622$	7.38331	0.296044
$623$	0	0
$624$	1.68499	0.0674537
$625$	15.6215	0.624862
$626$	−32.5103	−1.29937
$627$	−2.35021	−0.0938582
$628$	5.13677	0.204979
$629$	0.0547002	0.00218104
$630$	0	0
$631$	3.37179	0.134229	0.0671144	−	0.997745i	$-0.478621\pi$
$631$	3.37179	0.134229	0.0671144	+	0.997745i	$0.478621\pi$
$632$	−0.541560	−0.0215421
$633$	−3.89949	−0.154991
$634$	1.16832	0.0464000
$635$	6.73774	0.267379
$636$	−4.54156	−0.180085
$637$	0	0
$638$	18.4318	0.729720
$639$	27.0862	1.07151
$640$	−0.808530	−0.0319599
$641$	−6.60187	−0.260758	−0.130379	−	0.991464i	$-0.541619\pi$
$641$	−6.60187	−0.260758	−0.130379	+	0.991464i	$0.541619\pi$
$642$	−1.51971	−0.0599782
$643$	38.7093	1.52655	0.763273	−	0.646076i	$-0.223591\pi$
$643$	38.7093	1.52655	0.763273	+	0.646076i	$0.223591\pi$
$644$	0	0
$645$	1.96963	0.0775541
$646$	1.53304	0.0603166
$647$	−9.96520	−0.391772	−0.195886	−	0.980627i	$-0.562758\pi$
$647$	−9.96520	−0.391772	−0.195886	+	0.980627i	$0.562758\pi$
$648$	−8.00313	−0.314393
$649$	−43.0940	−1.69159
$650$	−21.8673	−0.857707
$651$	0	0
$652$	−3.25678	−0.127545
$653$	−27.7874	−1.08741	−0.543703	−	0.839278i	$-0.682978\pi$
$653$	−27.7874	−1.08741	−0.543703	+	0.839278i	$0.682978\pi$
$654$	−5.02185	−0.196370
$655$	−12.8427	−0.501806
$656$	−10.0907	−0.393975
$657$	−14.3141	−0.558446
$658$	0	0
$659$	−13.3200	−0.518873	−0.259436	−	0.965760i	$-0.583537\pi$
$659$	−13.3200	−0.518873	−0.259436	+	0.965760i	$0.583537\pi$
$660$	−1.23951	−0.0482477
$661$	−19.9620	−0.776433	−0.388217	−	0.921568i	$-0.626909\pi$
$661$	−19.9620	−0.776433	−0.388217	+	0.921568i	$0.626909\pi$
$662$	−4.19079	−0.162880
$663$	−1.68499	−0.0654397
$664$	14.9124	0.578715
$665$	0	0
$666$	−0.157966	−0.00612104
$667$	−6.62449	−0.256501
$668$	4.03441	0.156096
$669$	7.76718	0.300296
$670$	−10.8682	−0.419876
$671$	−48.6750	−1.87908
$672$	0	0
$673$	−1.96021	−0.0755604	−0.0377802	−	0.999286i	$-0.512029\pi$
$673$	−1.96021	−0.0755604	−0.0377802	+	0.999286i	$0.512029\pi$
$674$	7.53580	0.290268
$675$	−8.57026	−0.329869
$676$	12.3137	0.473604
$677$	40.1222	1.54202	0.771011	−	0.636821i	$-0.219751\pi$
$677$	40.1222	1.54202	0.771011	+	0.636821i	$0.219751\pi$
$678$	3.58173	0.137555
$679$	0	0
$680$	0.808530	0.0310057
$681$	−8.66705	−0.332122
$682$	3.92097	0.150142
$683$	26.9457	1.03105	0.515524	−	0.856875i	$-0.327597\pi$
$683$	26.9457	1.03105	0.515524	+	0.856875i	$0.327597\pi$
$684$	−4.42717	−0.169277
$685$	−4.40792	−0.168418
$686$	0	0
$687$	−5.86846	−0.223896
$688$	−7.27391	−0.277315
$689$	−68.2280	−2.59928
$690$	0.445487	0.0169594
$691$	−37.1774	−1.41429	−0.707147	−	0.707067i	$-0.750018\pi$
$691$	−37.1774	−1.41429	−0.707147	+	0.707067i	$0.750018\pi$
$692$	−23.4337	−0.890816
$693$	0	0
$694$	−20.5329	−0.779419
$695$	−8.88566	−0.337052
$696$	−1.34851	−0.0511152
$697$	10.0907	0.382212
$698$	−32.7279	−1.23877
$699$	9.84138	0.372235
$700$	0	0
$701$	30.2970	1.14430	0.572152	−	0.820148i	$-0.306109\pi$
$701$	30.2970	1.14430	0.572152	+	0.820148i	$0.306109\pi$
$702$	9.92097	0.374443
$703$	−0.0838575	−0.00316275
$704$	4.57754	0.172523
$705$	−0.788632	−0.0297016
$706$	−19.0363	−0.716439
$707$	0	0
$708$	3.15286	0.118492
$709$	30.5414	1.14701	0.573504	−	0.819203i	$-0.305584\pi$
$709$	30.5414	1.14701	0.573504	+	0.819203i	$0.305584\pi$
$710$	7.58353	0.284605
$711$	−1.56394	−0.0586522
$712$	1.63853	0.0614067
$713$	−1.40922	−0.0527758
$714$	0	0
$715$	−18.6211	−0.696391
$716$	−3.38883	−0.126647
$717$	2.48138	0.0926688
$718$	−31.8874	−1.19003
$719$	−37.8913	−1.41311	−0.706554	−	0.707659i	$-0.749751\pi$
$719$	−37.8913	−1.41311	−0.706554	+	0.707659i	$0.749751\pi$
$720$	−2.33490	−0.0870167
$721$	0	0
$722$	16.6498	0.619641
$723$	−3.95574	−0.147116
$724$	8.98990	0.334107
$725$	17.5006	0.649955
$726$	3.33361	0.123722
$727$	−46.8044	−1.73588	−0.867939	−	0.496671i	$-0.834555\pi$
$727$	−46.8044	−1.73588	−0.867939	+	0.496671i	$0.834555\pi$
$728$	0	0
$729$	−21.1306	−0.782615
$730$	−4.00762	−0.148329
$731$	7.27391	0.269035
$732$	3.56118	0.131625
$733$	−30.1036	−1.11190	−0.555951	−	0.831215i	$-0.687646\pi$
$733$	−30.1036	−1.11190	−0.555951	+	0.831215i	$0.687646\pi$
$734$	1.65462	0.0610732
$735$	0	0
$736$	−1.64520	−0.0606428
$737$	61.5312	2.26653
$738$	−29.1403	−1.07267
$739$	−32.0233	−1.17800	−0.588998	−	0.808134i	$-0.700478\pi$
$739$	−32.0233	−1.17800	−0.588998	+	0.808134i	$0.700478\pi$
$740$	−0.0442268	−0.00162581
$741$	2.58316	0.0948947
$742$	0	0
$743$	15.4999	0.568636	0.284318	−	0.958730i	$-0.408233\pi$
$743$	15.4999	0.568636	0.284318	+	0.958730i	$0.408233\pi$
$744$	−0.286867	−0.0105171
$745$	−1.03762	−0.0380153
$746$	7.07550	0.259052
$747$	43.0648	1.57566
$748$	−4.57754	−0.167372
$749$	0	0
$750$	−2.53078	−0.0924112
$751$	45.9242	1.67580	0.837899	−	0.545825i	$-0.183784\pi$
$751$	45.9242	1.67580	0.837899	+	0.545825i	$0.183784\pi$
$752$	2.91245	0.106206
$753$	5.78993	0.210997
$754$	−20.2587	−0.737779
$755$	−7.67422	−0.279293
$756$	0	0
$757$	−13.3137	−0.483895	−0.241947	−	0.970289i	$-0.577786\pi$
$757$	−13.3137	−0.483895	−0.241947	+	0.970289i	$0.577786\pi$
$758$	−27.1668	−0.986744
$759$	−2.52215	−0.0915483
$760$	−1.23951	−0.0449616
$761$	−12.6604	−0.458939	−0.229469	−	0.973316i	$-0.573699\pi$
$761$	−12.6604	−0.458939	−0.229469	+	0.973316i	$0.573699\pi$
$762$	2.79086	0.101102
$763$	0	0
$764$	−7.38331	−0.267119
$765$	2.33490	0.0844186
$766$	−6.03979	−0.218227
$767$	47.3655	1.71027
$768$	−0.334904	−0.0120848
$769$	−26.4379	−0.953374	−0.476687	−	0.879073i	$-0.658162\pi$
$769$	−26.4379	−0.953374	−0.476687	+	0.879073i	$0.658162\pi$
$770$	0	0
$771$	1.30298	0.0469256
$772$	−2.41735	−0.0870022
$773$	−18.5164	−0.665987	−0.332994	−	0.942929i	$-0.608059\pi$
$773$	−18.5164	−0.665987	−0.332994	+	0.942929i	$0.608059\pi$
$774$	−21.0059	−0.755041
$775$	3.72288	0.133730
$776$	−1.23951	−0.0444957
$777$	0	0
$778$	17.5773	0.630175
$779$	−15.4694	−0.554249
$780$	1.36237	0.0487806
$781$	−42.9347	−1.53632
$782$	1.64520	0.0588322
$783$	−7.93982	−0.283746
$784$	0	0
$785$	4.15323	0.148235
$786$	−5.31962	−0.189745
$787$	−36.9939	−1.31869	−0.659346	−	0.751840i	$-0.729167\pi$
$787$	−36.9939	−1.31869	−0.659346	+	0.751840i	$0.729167\pi$
$788$	4.61511	0.164406
$789$	1.88341	0.0670511
$790$	−0.437867	−0.0155786
$791$	0	0
$792$	13.2192	0.469724
$793$	53.4997	1.89983
$794$	6.35046	0.225370
$795$	−3.67199	−0.130232
$796$	−23.8874	−0.846668
$797$	−5.28833	−0.187322	−0.0936611	−	0.995604i	$-0.529857\pi$
$797$	−5.28833	−0.187322	−0.0936611	+	0.995604i	$0.529857\pi$
$798$	0	0
$799$	−2.91245	−0.103035
$800$	4.34628	0.153664
$801$	4.73183	0.167191
$802$	9.21360	0.325344
$803$	22.6894	0.800692
$804$	−4.50176	−0.158765
$805$	0	0
$806$	−4.30962	−0.151800
$807$	2.58250	0.0909082
$808$	10.9030	0.383567
$809$	−10.9644	−0.385489	−0.192744	−	0.981249i	$-0.561739\pi$
$809$	−10.9644	−0.385489	−0.192744	+	0.981249i	$0.561739\pi$
$810$	−6.47077	−0.227360
$811$	−10.9103	−0.383113	−0.191556	−	0.981482i	$-0.561353\pi$
$811$	−10.9103	−0.383113	−0.191556	+	0.981482i	$0.561353\pi$
$812$	0	0
$813$	−6.76811	−0.237368
$814$	0.250393	0.00877626
$815$	−2.63320	−0.0922370
$816$	0.334904	0.0117240
$817$	−11.1512	−0.390131
$818$	−30.1157	−1.05297
$819$	0	0
$820$	−8.15862	−0.284911
$821$	−33.7670	−1.17848	−0.589239	−	0.807959i	$-0.700572\pi$
$821$	−33.7670	−1.17848	−0.589239	+	0.807959i	$0.700572\pi$
$822$	−1.82582	−0.0636828
$823$	−52.6425	−1.83500	−0.917502	−	0.397731i	$-0.869798\pi$
$823$	−52.6425	−1.83500	−0.917502	+	0.397731i	$0.869798\pi$
$824$	1.32077	0.0460111
$825$	6.66301	0.231976
$826$	0	0
$827$	−56.2640	−1.95649	−0.978245	−	0.207452i	$-0.933483\pi$
$827$	−56.2640	−1.95649	−0.978245	+	0.207452i	$0.933483\pi$
$828$	−4.75107	−0.165111
$829$	20.1471	0.699739	0.349869	−	0.936799i	$-0.386226\pi$
$829$	20.1471	0.699739	0.349869	+	0.936799i	$0.386226\pi$
$830$	12.0572	0.418510
$831$	−3.41347	−0.118412
$832$	−5.03127	−0.174428
$833$	0	0
$834$	−3.68056	−0.127447
$835$	3.26194	0.112884
$836$	7.01755	0.242707
$837$	−1.68903	−0.0583814
$838$	25.5564	0.882833
$839$	−40.4901	−1.39787	−0.698936	−	0.715184i	$-0.746343\pi$
$839$	−40.4901	−1.39787	−0.698936	+	0.715184i	$0.746343\pi$
$840$	0	0
$841$	−12.7868	−0.440924
$842$	−27.8969	−0.961390
$843$	−7.50933	−0.258635
$844$	11.6436	0.400790
$845$	9.95600	0.342497
$846$	8.41068	0.289165
$847$	0	0
$848$	13.5608	0.465679
$849$	−9.38923	−0.322238
$850$	−4.34628	−0.149076
$851$	−0.0899928	−0.00308491
$852$	3.14120	0.107616
$853$	45.0101	1.54111	0.770557	−	0.637371i	$-0.219978\pi$
$853$	45.0101	1.54111	0.770557	+	0.637371i	$0.219978\pi$
$854$	0	0
$855$	−3.57950	−0.122416
$856$	4.53775	0.155097
$857$	1.15230	0.0393617	0.0196809	−	0.999806i	$-0.493735\pi$
$857$	1.15230	0.0393617	0.0196809	+	0.999806i	$0.493735\pi$
$858$	−7.71313	−0.263322
$859$	47.8869	1.63388	0.816940	−	0.576722i	$-0.195669\pi$
$859$	47.8869	1.63388	0.816940	+	0.576722i	$0.195669\pi$
$860$	−5.88118	−0.200546
$861$	0	0
$862$	−32.9987	−1.12394
$863$	50.4773	1.71827	0.859133	−	0.511752i	$-0.171003\pi$
$863$	50.4773	1.71827	0.859133	+	0.511752i	$0.171003\pi$
$864$	−1.97186	−0.0670841
$865$	−18.9469	−0.644212
$866$	−9.96464	−0.338612
$867$	−0.334904	−0.0113739
$868$	0	0
$869$	2.47901	0.0840948
$870$	−1.09031	−0.0369650
$871$	−67.6302	−2.29156
$872$	14.9949	0.507791
$873$	−3.57950	−0.121148
$874$	−2.52215	−0.0853131
$875$	0	0
$876$	−1.66001	−0.0560866
$877$	36.8877	1.24561	0.622805	−	0.782377i	$-0.285993\pi$
$877$	36.8877	1.24561	0.622805	+	0.782377i	$0.285993\pi$
$878$	30.6252	1.03355
$879$	2.53933	0.0856494
$880$	3.70108	0.124763
$881$	−12.9489	−0.436260	−0.218130	−	0.975920i	$-0.569996\pi$
$881$	−12.9489	−0.436260	−0.218130	+	0.975920i	$0.569996\pi$
$882$	0	0
$883$	−16.7816	−0.564745	−0.282372	−	0.959305i	$-0.591121\pi$
$883$	−16.7816	−0.564745	−0.282372	+	0.959305i	$0.591121\pi$
$884$	5.03127	0.169220
$885$	2.54918	0.0856897
$886$	35.5045	1.19280
$887$	−27.7294	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$887$	−27.7294	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$888$	−0.0183193	−0.000614756 0
$889$	0	0
$890$	1.32480	0.0444075
$891$	36.6347	1.22731
$892$	−23.1923	−0.776534
$893$	4.46489	0.149412
$894$	−0.429795	−0.0143745
$895$	−2.73997	−0.0915872
$896$	0	0
$897$	2.77215	0.0925594
$898$	−11.0402	−0.368415
$899$	3.44902	0.115031
$900$	12.5514	0.418379
$901$	−13.5608	−0.451775
$902$	46.1906	1.53798
$903$	0	0
$904$	−10.6948	−0.355703
$905$	7.26860	0.241616
$906$	−3.17877	−0.105607
$907$	34.4587	1.14418	0.572092	−	0.820190i	$-0.306132\pi$
$907$	34.4587	1.14418	0.572092	+	0.820190i	$0.306132\pi$
$908$	25.8792	0.858831
$909$	31.4862	1.04433
$910$	0	0
$911$	−41.1560	−1.36356	−0.681779	−	0.731558i	$-0.738793\pi$
$911$	−41.1560	−1.36356	−0.681779	+	0.731558i	$0.738793\pi$
$912$	−0.513421	−0.0170011
$913$	−68.2624	−2.25916
$914$	24.9169	0.824177
$915$	2.87932	0.0951873
$916$	17.5228	0.578970
$917$	0	0
$918$	1.97186	0.0650811
$919$	16.8884	0.557096	0.278548	−	0.960422i	$-0.410147\pi$
$919$	16.8884	0.557096	0.278548	+	0.960422i	$0.410147\pi$
$920$	−1.33019	−0.0438551
$921$	−4.56208	−0.150326
$922$	18.8991	0.622409
$923$	47.1904	1.55329
$924$	0	0
$925$	0.237743	0.00781693
$926$	4.64982	0.152802
$927$	3.81417	0.125274
$928$	4.02656	0.132178
$929$	14.7949	0.485405	0.242702	−	0.970101i	$-0.421966\pi$
$929$	14.7949	0.485405	0.242702	+	0.970101i	$0.421966\pi$
$930$	−0.231941	−0.00760564
$931$	0	0
$932$	−29.3857	−0.962560
$933$	2.47270	0.0809526
$934$	−35.0313	−1.14626
$935$	−3.70108	−0.121038
$936$	−14.5295	−0.474912
$937$	6.29002	0.205486	0.102743	−	0.994708i	$-0.467238\pi$
$937$	6.29002	0.205486	0.102743	+	0.994708i	$0.467238\pi$
$938$	0	0
$939$	−10.8878	−0.355310
$940$	2.35480	0.0768052
$941$	24.9436	0.813138	0.406569	−	0.913620i	$-0.366725\pi$
$941$	24.9436	0.813138	0.406569	+	0.913620i	$0.366725\pi$
$942$	1.72032	0.0560512
$943$	−16.6012	−0.540609
$944$	−9.41421	−0.306407
$945$	0	0
$946$	33.2967	1.08257
$947$	−23.2729	−0.756267	−0.378134	−	0.925751i	$-0.623434\pi$
$947$	−23.2729	−0.756267	−0.378134	+	0.925751i	$0.623434\pi$
$948$	−0.181370	−0.00589064
$949$	−24.9384	−0.809534
$950$	6.66301	0.216177
$951$	0.391276	0.0126880
$952$	0	0
$953$	−33.9078	−1.09838	−0.549191	−	0.835697i	$-0.685064\pi$
$953$	−33.9078	−1.09838	−0.549191	+	0.835697i	$0.685064\pi$
$954$	39.1614	1.26790
$955$	−5.96963	−0.193173
$956$	−7.40922	−0.239631
$957$	6.17287	0.199541
$958$	−31.4223	−1.01521
$959$	0	0
$960$	−0.270780	−0.00873938
$961$	−30.2663	−0.976332
$962$	−0.275212	−0.00887319
$963$	13.1043	0.422280
$964$	11.8116	0.380425
$965$	−1.95450	−0.0629175
$966$	0	0
$967$	−51.3699	−1.65195	−0.825973	−	0.563710i	$-0.809374\pi$
$967$	−51.3699	−1.65195	−0.825973	+	0.563710i	$0.809374\pi$
$968$	−9.95392	−0.319931
$969$	0.513421	0.0164934
$970$	−1.00218	−0.0321780
$971$	28.1295	0.902719	0.451360	−	0.892342i	$-0.350939\pi$
$971$	28.1295	0.902719	0.451360	+	0.892342i	$0.350939\pi$
$972$	−8.59586	−0.275712
$973$	0	0
$974$	27.2426	0.872910
$975$	−7.32345	−0.234538
$976$	−10.6334	−0.340368
$977$	51.7694	1.65625	0.828124	−	0.560544i	$-0.189408\pi$
$977$	51.7694	1.65625	0.828124	+	0.560544i	$0.189408\pi$
$978$	−1.09071	−0.0348770
$979$	−7.50047	−0.239716
$980$	0	0
$981$	43.3028	1.38255
$982$	31.2215	0.996319
$983$	−30.1453	−0.961485	−0.480743	−	0.876862i	$-0.659633\pi$
$983$	−30.1453	−0.961485	−0.480743	+	0.876862i	$0.659633\pi$
$984$	−3.37941	−0.107732
$985$	3.73145	0.118894
$986$	−4.02656	−0.128232
$987$	0	0
$988$	−7.71313	−0.245387
$989$	−11.9670	−0.380530
$990$	10.6881	0.339691
$991$	26.9755	0.856906	0.428453	−	0.903564i	$-0.359059\pi$
$991$	26.9755	0.856906	0.428453	+	0.903564i	$0.359059\pi$
$992$	0.856566	0.0271960
$993$	−1.40351	−0.0445392
$994$	0	0
$995$	−19.3137	−0.612286
$996$	4.99424	0.158249
$997$	37.1968	1.17803	0.589017	−	0.808120i	$-0.299515\pi$
$997$	37.1968	1.17803	0.589017	+	0.808120i	$0.299515\pi$
$998$	−40.3384	−1.27689
$999$	−0.107861	−0.00341258

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1666.2.a.v.1.2	✓	4
7.6	odd	2		1666.2.a.w.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1666.2.a.v.1.2	✓	4	1.1	even	1	trivial
1666.2.a.w.1.3	yes	4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1666.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.334904 q^{3} +1.00000 q^{4} +0.808530 q^{5} +0.334904 q^{6} -1.00000 q^{8} -2.88784 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.334904 q^{3} +1.00000 q^{4} +0.808530 q^{5} +0.334904 q^{6} -1.00000 q^{8} -2.88784 q^{9} -0.808530 q^{10} +4.57754 q^{11} -0.334904 q^{12} -5.03127 q^{13} -0.270780 q^{15} +1.00000 q^{16} -1.00000 q^{17} +2.88784 q^{18} +1.53304 q^{19} +0.808530 q^{20} -4.57754 q^{22} +1.64520 q^{23} +0.334904 q^{24} -4.34628 q^{25} +5.03127 q^{26} +1.97186 q^{27} -4.02656 q^{29} +0.270780 q^{30} -0.856566 q^{31} -1.00000 q^{32} -1.53304 q^{33} +1.00000 q^{34} -2.88784 q^{36} -0.0547002 q^{37} -1.53304 q^{38} +1.68499 q^{39} -0.808530 q^{40} -10.0907 q^{41} -7.27391 q^{43} +4.57754 q^{44} -2.33490 q^{45} -1.64520 q^{46} +2.91245 q^{47} -0.334904 q^{48} +4.34628 q^{50} +0.334904 q^{51} -5.03127 q^{52} +13.5608 q^{53} -1.97186 q^{54} +3.70108 q^{55} -0.513421 q^{57} +4.02656 q^{58} -9.41421 q^{59} -0.270780 q^{60} -10.6334 q^{61} +0.856566 q^{62} +1.00000 q^{64} -4.06793 q^{65} +1.53304 q^{66} +13.4420 q^{67} -1.00000 q^{68} -0.550984 q^{69} -9.37941 q^{71} +2.88784 q^{72} +4.95668 q^{73} +0.0547002 q^{74} +1.45559 q^{75} +1.53304 q^{76} -1.68499 q^{78} +0.541560 q^{79} +0.808530 q^{80} +8.00313 q^{81} +10.0907 q^{82} -14.9124 q^{83} -0.808530 q^{85} +7.27391 q^{86} +1.34851 q^{87} -4.57754 q^{88} -1.63853 q^{89} +2.33490 q^{90} +1.64520 q^{92} +0.286867 q^{93} -2.91245 q^{94} +1.23951 q^{95} +0.334904 q^{96} +1.23951 q^{97} -13.2192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{20} + 12 q^{23} - 12 q^{27} + 4 q^{30} - 12 q^{31} - 4 q^{32} + 4 q^{34} - 4 q^{37} + 4 q^{39} + 4 q^{40} - 20 q^{41} + 8 q^{43} - 8 q^{45} - 12 q^{46} - 8 q^{47} + 12 q^{54} - 8 q^{55} + 12 q^{57} - 32 q^{59} - 4 q^{60} + 4 q^{61} + 12 q^{62} + 4 q^{64} - 28 q^{65} - 4 q^{68} - 24 q^{71} + 4 q^{74} + 28 q^{75} - 4 q^{78} + 8 q^{79} - 4 q^{80} - 8 q^{81} + 20 q^{82} - 40 q^{83} + 4 q^{85} - 8 q^{86} - 32 q^{87} - 24 q^{89} + 8 q^{90} + 12 q^{92} - 16 q^{93} + 8 q^{94} + 28 q^{95} + 28 q^{97} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8 + 4 * q^10 - 4 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^20 + 12 * q^23 - 12 * q^27 + 4 * q^30 - 12 * q^31 - 4 * q^32 + 4 * q^34 - 4 * q^37 + 4 * q^39 + 4 * q^40 - 20 * q^41 + 8 * q^43 - 8 * q^45 - 12 * q^46 - 8 * q^47 + 12 * q^54 - 8 * q^55 + 12 * q^57 - 32 * q^59 - 4 * q^60 + 4 * q^61 + 12 * q^62 + 4 * q^64 - 28 * q^65 - 4 * q^68 - 24 * q^71 + 4 * q^74 + 28 * q^75 - 4 * q^78 + 8 * q^79 - 4 * q^80 - 8 * q^81 + 20 * q^82 - 40 * q^83 + 4 * q^85 - 8 * q^86 - 32 * q^87 - 24 * q^89 + 8 * q^90 + 12 * q^92 - 16 * q^93 + 8 * q^94 + 28 * q^95 + 28 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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