LMFDB - Embedded newform 668.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(1,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, 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("668.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.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:	$5.33400685502$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
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			$1.47217$ of defining polynomial
Character	$\chi$	$=$	668.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04732 q^{3} +1.35854 q^{5} -3.73540 q^{7} +6.28618 q^{9} +O(q^{10})$	$q-3.04732 q^{3} +1.35854 q^{5} -3.73540 q^{7} +6.28618 q^{9} +3.46328 q^{11} +5.65282 q^{13} -4.13990 q^{15} -5.24666 q^{17} -5.91175 q^{19} +11.3830 q^{21} -3.74509 q^{23} -3.15438 q^{25} -10.0141 q^{27} -2.15273 q^{29} +10.5485 q^{31} -10.5537 q^{33} -5.07467 q^{35} -6.12685 q^{37} -17.2260 q^{39} -9.96026 q^{41} -6.73611 q^{43} +8.54001 q^{45} +3.69405 q^{47} +6.95320 q^{49} +15.9883 q^{51} -10.3613 q^{53} +4.70499 q^{55} +18.0150 q^{57} -2.38711 q^{59} -7.77996 q^{61} -23.4814 q^{63} +7.67956 q^{65} -10.1595 q^{67} +11.4125 q^{69} -0.525143 q^{71} +4.96794 q^{73} +9.61241 q^{75} -12.9367 q^{77} -3.47337 q^{79} +11.6575 q^{81} +16.2088 q^{83} -7.12778 q^{85} +6.56007 q^{87} +1.12957 q^{89} -21.1155 q^{91} -32.1447 q^{93} -8.03133 q^{95} +3.37062 q^{97} +21.7708 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.04732	−1.75937	−0.879687	−	0.475554i	$-0.842248\pi$
$3$	−3.04732	−1.75937	−0.879687	+	0.475554i	$0.842248\pi$
$4$	0	0
$5$	1.35854	0.607556	0.303778	−	0.952743i	$-0.401752\pi$
$5$	1.35854	0.607556	0.303778	+	0.952743i	$0.401752\pi$
$6$	0	0
$7$	−3.73540	−1.41185	−0.705924	−	0.708288i	$-0.749468\pi$
$7$	−3.73540	−1.41185	−0.705924	+	0.708288i	$0.749468\pi$
$8$	0	0
$9$	6.28618	2.09539
$10$	0	0
$11$	3.46328	1.04422	0.522109	−	0.852879i	$-0.325145\pi$
$11$	3.46328	1.04422	0.522109	+	0.852879i	$0.325145\pi$
$12$	0	0
$13$	5.65282	1.56781	0.783904	−	0.620881i	$-0.213225\pi$
$13$	5.65282	1.56781	0.783904	+	0.620881i	$0.213225\pi$
$14$	0	0
$15$	−4.13990	−1.06892
$16$	0	0
$17$	−5.24666	−1.27250	−0.636252	−	0.771482i	$-0.719516\pi$
$17$	−5.24666	−1.27250	−0.636252	+	0.771482i	$0.719516\pi$
$18$	0	0
$19$	−5.91175	−1.35625	−0.678124	−	0.734947i	$-0.737207\pi$
$19$	−5.91175	−1.35625	−0.678124	+	0.734947i	$0.737207\pi$
$20$	0	0
$21$	11.3830	2.48397
$22$	0	0
$23$	−3.74509	−0.780905	−0.390452	−	0.920623i	$-0.627681\pi$
$23$	−3.74509	−0.780905	−0.390452	+	0.920623i	$0.627681\pi$
$24$	0	0
$25$	−3.15438	−0.630876
$26$	0	0
$27$	−10.0141	−1.92721
$28$	0	0
$29$	−2.15273	−0.399752	−0.199876	−	0.979821i	$-0.564054\pi$
$29$	−2.15273	−0.399752	−0.199876	+	0.979821i	$0.564054\pi$
$30$	0	0
$31$	10.5485	1.89456	0.947282	−	0.320401i	$-0.103817\pi$
$31$	10.5485	1.89456	0.947282	+	0.320401i	$0.103817\pi$
$32$	0	0
$33$	−10.5537	−1.83717
$34$	0	0
$35$	−5.07467	−0.857776
$36$	0	0
$37$	−6.12685	−1.00725	−0.503624	−	0.863923i	$-0.668000\pi$
$37$	−6.12685	−1.00725	−0.503624	+	0.863923i	$0.668000\pi$
$38$	0	0
$39$	−17.2260	−2.75836
$40$	0	0
$41$	−9.96026	−1.55553	−0.777765	−	0.628555i	$-0.783647\pi$
$41$	−9.96026	−1.55553	−0.777765	+	0.628555i	$0.783647\pi$
$42$	0	0
$43$	−6.73611	−1.02725	−0.513624	−	0.858016i	$-0.671697\pi$
$43$	−6.73611	−1.02725	−0.513624	+	0.858016i	$0.671697\pi$
$44$	0	0
$45$	8.54001	1.27307
$46$	0	0
$47$	3.69405	0.538832	0.269416	−	0.963024i	$-0.413169\pi$
$47$	3.69405	0.538832	0.269416	+	0.963024i	$0.413169\pi$
$48$	0	0
$49$	6.95320	0.993314
$50$	0	0
$51$	15.9883	2.23881
$52$	0	0
$53$	−10.3613	−1.42323	−0.711616	−	0.702569i	$-0.752036\pi$
$53$	−10.3613	−1.42323	−0.711616	+	0.702569i	$0.752036\pi$
$54$	0	0
$55$	4.70499	0.634421
$56$	0	0
$57$	18.0150	2.38615
$58$	0	0
$59$	−2.38711	−0.310776	−0.155388	−	0.987854i	$-0.549663\pi$
$59$	−2.38711	−0.310776	−0.155388	+	0.987854i	$0.549663\pi$
$60$	0	0
$61$	−7.77996	−0.996121	−0.498061	−	0.867142i	$-0.665954\pi$
$61$	−7.77996	−0.996121	−0.498061	+	0.867142i	$0.665954\pi$
$62$	0	0
$63$	−23.4814	−2.95838
$64$	0	0
$65$	7.67956	0.952532
$66$	0	0
$67$	−10.1595	−1.24118	−0.620590	−	0.784135i	$-0.713107\pi$
$67$	−10.1595	−1.24118	−0.620590	+	0.784135i	$0.713107\pi$
$68$	0	0
$69$	11.4125	1.37390
$70$	0	0
$71$	−0.525143	−0.0623229	−0.0311615	−	0.999514i	$-0.509921\pi$
$71$	−0.525143	−0.0623229	−0.0311615	+	0.999514i	$0.509921\pi$
$72$	0	0
$73$	4.96794	0.581454	0.290727	−	0.956806i	$-0.406103\pi$
$73$	4.96794	0.581454	0.290727	+	0.956806i	$0.406103\pi$
$74$	0	0
$75$	9.61241	1.10995
$76$	0	0
$77$	−12.9367	−1.47428
$78$	0	0
$79$	−3.47337	−0.390785	−0.195393	−	0.980725i	$-0.562598\pi$
$79$	−3.47337	−0.390785	−0.195393	+	0.980725i	$0.562598\pi$
$80$	0	0
$81$	11.6575	1.29528
$82$	0	0
$83$	16.2088	1.77915	0.889575	−	0.456789i	$-0.151000\pi$
$83$	16.2088	1.77915	0.889575	+	0.456789i	$0.151000\pi$
$84$	0	0
$85$	−7.12778	−0.773117
$86$	0	0
$87$	6.56007	0.703313
$88$	0	0
$89$	1.12957	0.119735	0.0598673	−	0.998206i	$-0.480932\pi$
$89$	1.12957	0.119735	0.0598673	+	0.998206i	$0.480932\pi$
$90$	0	0
$91$	−21.1155	−2.21351
$92$	0	0
$93$	−32.1447	−3.33325
$94$	0	0
$95$	−8.03133	−0.823997
$96$	0	0
$97$	3.37062	0.342235	0.171117	−	0.985251i	$-0.445262\pi$
$97$	3.37062	0.342235	0.171117	+	0.985251i	$0.445262\pi$
$98$	0	0
$99$	21.7708	2.18805
$100$	0	0
$101$	2.81750	0.280351	0.140176	−	0.990127i	$-0.455233\pi$
$101$	2.81750	0.280351	0.140176	+	0.990127i	$0.455233\pi$
$102$	0	0
$103$	−0.231833	−0.0228432	−0.0114216	−	0.999935i	$-0.503636\pi$
$103$	−0.231833	−0.0228432	−0.0114216	+	0.999935i	$0.503636\pi$
$104$	0	0
$105$	15.4642	1.50915
$106$	0	0
$107$	−18.7696	−1.81452	−0.907262	−	0.420566i	$-0.861832\pi$
$107$	−18.7696	−1.81452	−0.907262	+	0.420566i	$0.861832\pi$
$108$	0	0
$109$	1.02824	0.0984873	0.0492437	−	0.998787i	$-0.484319\pi$
$109$	1.02824	0.0984873	0.0492437	+	0.998787i	$0.484319\pi$
$110$	0	0
$111$	18.6705	1.77212
$112$	0	0
$113$	−1.36583	−0.128487	−0.0642433	−	0.997934i	$-0.520463\pi$
$113$	−1.36583	−0.128487	−0.0642433	+	0.997934i	$0.520463\pi$
$114$	0	0
$115$	−5.08784	−0.474443
$116$	0	0
$117$	35.5346	3.28518
$118$	0	0
$119$	19.5984	1.79658
$120$	0	0
$121$	0.994304	0.0903912
$122$	0	0
$123$	30.3521	2.73676
$124$	0	0
$125$	−11.0780	−0.990848
$126$	0	0
$127$	12.9283	1.14720	0.573601	−	0.819135i	$-0.305546\pi$
$127$	12.9283	1.14720	0.573601	+	0.819135i	$0.305546\pi$
$128$	0	0
$129$	20.5271	1.80731
$130$	0	0
$131$	18.5957	1.62471	0.812356	−	0.583162i	$-0.198185\pi$
$131$	18.5957	1.62471	0.812356	+	0.583162i	$0.198185\pi$
$132$	0	0
$133$	22.0827	1.91482
$134$	0	0
$135$	−13.6045	−1.17089
$136$	0	0
$137$	5.14323	0.439416	0.219708	−	0.975566i	$-0.429490\pi$
$137$	5.14323	0.439416	0.219708	+	0.975566i	$0.429490\pi$
$138$	0	0
$139$	−2.75646	−0.233800	−0.116900	−	0.993144i	$-0.537296\pi$
$139$	−2.75646	−0.233800	−0.116900	+	0.993144i	$0.537296\pi$
$140$	0	0
$141$	−11.2570	−0.948007
$142$	0	0
$143$	19.5773	1.63713
$144$	0	0
$145$	−2.92456	−0.242872
$146$	0	0
$147$	−21.1886	−1.74761
$148$	0	0
$149$	15.0712	1.23468	0.617339	−	0.786697i	$-0.288211\pi$
$149$	15.0712	1.23468	0.617339	+	0.786697i	$0.288211\pi$
$150$	0	0
$151$	−11.7438	−0.955694	−0.477847	−	0.878443i	$-0.658583\pi$
$151$	−11.7438	−0.955694	−0.477847	+	0.878443i	$0.658583\pi$
$152$	0	0
$153$	−32.9815	−2.66640
$154$	0	0
$155$	14.3305	1.15105
$156$	0	0
$157$	−19.5884	−1.56332	−0.781662	−	0.623702i	$-0.785628\pi$
$157$	−19.5884	−1.56332	−0.781662	+	0.623702i	$0.785628\pi$
$158$	0	0
$159$	31.5742	2.50400
$160$	0	0
$161$	13.9894	1.10252
$162$	0	0
$163$	−17.7961	−1.39390	−0.696950	−	0.717119i	$-0.745460\pi$
$163$	−17.7961	−1.39390	−0.696950	+	0.717119i	$0.745460\pi$
$164$	0	0
$165$	−14.3376	−1.11618
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	18.9543	1.45803
$170$	0	0
$171$	−37.1623	−2.84187
$172$	0	0
$173$	14.2960	1.08690	0.543451	−	0.839441i	$-0.317117\pi$
$173$	14.2960	1.08690	0.543451	+	0.839441i	$0.317117\pi$
$174$	0	0
$175$	11.7829	0.890701
$176$	0	0
$177$	7.27431	0.546770
$178$	0	0
$179$	−21.8108	−1.63022	−0.815108	−	0.579310i	$-0.803322\pi$
$179$	−21.8108	−1.63022	−0.815108	+	0.579310i	$0.803322\pi$
$180$	0	0
$181$	−10.6609	−0.792418	−0.396209	−	0.918160i	$-0.629674\pi$
$181$	−10.6609	−0.792418	−0.396209	+	0.918160i	$0.629674\pi$
$182$	0	0
$183$	23.7080	1.75255
$184$	0	0
$185$	−8.32355	−0.611959
$186$	0	0
$187$	−18.1707	−1.32877
$188$	0	0
$189$	37.4065	2.72092
$190$	0	0
$191$	8.42495	0.609608	0.304804	−	0.952415i	$-0.401409\pi$
$191$	8.42495	0.609608	0.304804	+	0.952415i	$0.401409\pi$
$192$	0	0
$193$	19.9264	1.43433	0.717167	−	0.696902i	$-0.245439\pi$
$193$	19.9264	1.43433	0.717167	+	0.696902i	$0.245439\pi$
$194$	0	0
$195$	−23.4021	−1.67586
$196$	0	0
$197$	−2.67355	−0.190483	−0.0952414	−	0.995454i	$-0.530362\pi$
$197$	−2.67355	−0.190483	−0.0952414	+	0.995454i	$0.530362\pi$
$198$	0	0
$199$	−22.9740	−1.62858	−0.814292	−	0.580456i	$-0.802875\pi$
$199$	−22.9740	−1.62858	−0.814292	+	0.580456i	$0.802875\pi$
$200$	0	0
$201$	30.9593	2.18370
$202$	0	0
$203$	8.04130	0.564389
$204$	0	0
$205$	−13.5314	−0.945072
$206$	0	0
$207$	−23.5423	−1.63630
$208$	0	0
$209$	−20.4740	−1.41622
$210$	0	0
$211$	−14.8171	−1.02005	−0.510025	−	0.860160i	$-0.670364\pi$
$211$	−14.8171	−1.02005	−0.510025	+	0.860160i	$0.670364\pi$
$212$	0	0
$213$	1.60028	0.109649
$214$	0	0
$215$	−9.15125	−0.624110
$216$	0	0
$217$	−39.4028	−2.67484
$218$	0	0
$219$	−15.1389	−1.02299
$220$	0	0
$221$	−29.6584	−1.99504
$222$	0	0
$223$	−4.33025	−0.289975	−0.144988	−	0.989433i	$-0.546314\pi$
$223$	−4.33025	−0.289975	−0.144988	+	0.989433i	$0.546314\pi$
$224$	0	0
$225$	−19.8290	−1.32193
$226$	0	0
$227$	2.81874	0.187086	0.0935432	−	0.995615i	$-0.470181\pi$
$227$	2.81874	0.187086	0.0935432	+	0.995615i	$0.470181\pi$
$228$	0	0
$229$	−4.12400	−0.272521	−0.136261	−	0.990673i	$-0.543508\pi$
$229$	−4.12400	−0.272521	−0.136261	+	0.990673i	$0.543508\pi$
$230$	0	0
$231$	39.4224	2.59380
$232$	0	0
$233$	3.19333	0.209202	0.104601	−	0.994514i	$-0.466643\pi$
$233$	3.19333	0.209202	0.104601	+	0.994514i	$0.466643\pi$
$234$	0	0
$235$	5.01850	0.327371
$236$	0	0
$237$	10.5845	0.687537
$238$	0	0
$239$	18.5384	1.19915	0.599574	−	0.800319i	$-0.295337\pi$
$239$	18.5384	1.19915	0.599574	+	0.800319i	$0.295337\pi$
$240$	0	0
$241$	−18.5958	−1.19786	−0.598931	−	0.800800i	$-0.704408\pi$
$241$	−18.5958	−1.19786	−0.598931	+	0.800800i	$0.704408\pi$
$242$	0	0
$243$	−5.48214	−0.351679
$244$	0	0
$245$	9.44617	0.603494
$246$	0	0
$247$	−33.4180	−2.12634
$248$	0	0
$249$	−49.3936	−3.13019
$250$	0	0
$251$	24.5175	1.54753	0.773767	−	0.633471i	$-0.218370\pi$
$251$	24.5175	1.54753	0.773767	+	0.633471i	$0.218370\pi$
$252$	0	0
$253$	−12.9703	−0.815435
$254$	0	0
$255$	21.7207	1.36020
$256$	0	0
$257$	3.01478	0.188057	0.0940284	−	0.995570i	$-0.470026\pi$
$257$	3.01478	0.188057	0.0940284	+	0.995570i	$0.470026\pi$
$258$	0	0
$259$	22.8862	1.42208
$260$	0	0
$261$	−13.5325	−0.837638
$262$	0	0
$263$	−29.3036	−1.80694	−0.903469	−	0.428653i	$-0.858988\pi$
$263$	−29.3036	−1.80694	−0.903469	+	0.428653i	$0.858988\pi$
$264$	0	0
$265$	−14.0762	−0.864693
$266$	0	0
$267$	−3.44218	−0.210658
$268$	0	0
$269$	−1.97312	−0.120303	−0.0601515	−	0.998189i	$-0.519158\pi$
$269$	−1.97312	−0.120303	−0.0601515	+	0.998189i	$0.519158\pi$
$270$	0	0
$271$	18.6146	1.13076	0.565380	−	0.824831i	$-0.308730\pi$
$271$	18.6146	1.13076	0.565380	+	0.824831i	$0.308730\pi$
$272$	0	0
$273$	64.3458	3.89439
$274$	0	0
$275$	−10.9245	−0.658772
$276$	0	0
$277$	−17.4286	−1.04718	−0.523592	−	0.851969i	$-0.675408\pi$
$277$	−17.4286	−1.04718	−0.523592	+	0.851969i	$0.675408\pi$
$278$	0	0
$279$	66.3097	3.96986
$280$	0	0
$281$	24.9852	1.49049	0.745246	−	0.666790i	$-0.232332\pi$
$281$	24.9852	1.49049	0.745246	+	0.666790i	$0.232332\pi$
$282$	0	0
$283$	1.77892	0.105746	0.0528728	−	0.998601i	$-0.483162\pi$
$283$	1.77892	0.105746	0.0528728	+	0.998601i	$0.483162\pi$
$284$	0	0
$285$	24.4741	1.44972
$286$	0	0
$287$	37.2055	2.19617
$288$	0	0
$289$	10.5275	0.619264
$290$	0	0
$291$	−10.2714	−0.602119
$292$	0	0
$293$	13.2815	0.775915	0.387957	−	0.921677i	$-0.373181\pi$
$293$	13.2815	0.775915	0.387957	+	0.921677i	$0.373181\pi$
$294$	0	0
$295$	−3.24298	−0.188814
$296$	0	0
$297$	−34.6815	−2.01242
$298$	0	0
$299$	−21.1703	−1.22431
$300$	0	0
$301$	25.1621	1.45032
$302$	0	0
$303$	−8.58582	−0.493243
$304$	0	0
$305$	−10.5694	−0.605199
$306$	0	0
$307$	0.916704	0.0523191	0.0261595	−	0.999658i	$-0.491672\pi$
$307$	0.916704	0.0523191	0.0261595	+	0.999658i	$0.491672\pi$
$308$	0	0
$309$	0.706469	0.0401896
$310$	0	0
$311$	−19.5632	−1.10933	−0.554664	−	0.832075i	$-0.687153\pi$
$311$	−19.5632	−1.10933	−0.554664	+	0.832075i	$0.687153\pi$
$312$	0	0
$313$	−19.0566	−1.07715	−0.538573	−	0.842579i	$-0.681036\pi$
$313$	−19.0566	−1.07715	−0.538573	+	0.842579i	$0.681036\pi$
$314$	0	0
$315$	−31.9003	−1.79738
$316$	0	0
$317$	6.33422	0.355765	0.177883	−	0.984052i	$-0.443075\pi$
$317$	6.33422	0.355765	0.177883	+	0.984052i	$0.443075\pi$
$318$	0	0
$319$	−7.45551	−0.417428
$320$	0	0
$321$	57.1970	3.19243
$322$	0	0
$323$	31.0170	1.72583
$324$	0	0
$325$	−17.8311	−0.989093
$326$	0	0
$327$	−3.13337	−0.173276
$328$	0	0
$329$	−13.7987	−0.760749
$330$	0	0
$331$	5.49519	0.302043	0.151022	−	0.988530i	$-0.451744\pi$
$331$	5.49519	0.302043	0.151022	+	0.988530i	$0.451744\pi$
$332$	0	0
$333$	−38.5145	−2.11058
$334$	0	0
$335$	−13.8021	−0.754087
$336$	0	0
$337$	9.69085	0.527894	0.263947	−	0.964537i	$-0.414976\pi$
$337$	9.69085	0.527894	0.263947	+	0.964537i	$0.414976\pi$
$338$	0	0
$339$	4.16213	0.226056
$340$	0	0
$341$	36.5324	1.97834
$342$	0	0
$343$	0.174827	0.00943975
$344$	0	0
$345$	15.5043	0.834723
$346$	0	0
$347$	−5.49206	−0.294829	−0.147415	−	0.989075i	$-0.547095\pi$
$347$	−5.49206	−0.294829	−0.147415	+	0.989075i	$0.547095\pi$
$348$	0	0
$349$	25.6773	1.37447	0.687236	−	0.726434i	$-0.258824\pi$
$349$	25.6773	1.37447	0.687236	+	0.726434i	$0.258824\pi$
$350$	0	0
$351$	−56.6077	−3.02149
$352$	0	0
$353$	6.17231	0.328519	0.164260	−	0.986417i	$-0.447477\pi$
$353$	6.17231	0.328519	0.164260	+	0.986417i	$0.447477\pi$
$354$	0	0
$355$	−0.713425	−0.0378647
$356$	0	0
$357$	−59.7226	−3.16086
$358$	0	0
$359$	−0.966490	−0.0510094	−0.0255047	−	0.999675i	$-0.508119\pi$
$359$	−0.966490	−0.0510094	−0.0255047	+	0.999675i	$0.508119\pi$
$360$	0	0
$361$	15.9488	0.839409
$362$	0	0
$363$	−3.02997	−0.159032
$364$	0	0
$365$	6.74913	0.353266
$366$	0	0
$367$	−4.35207	−0.227176	−0.113588	−	0.993528i	$-0.536234\pi$
$367$	−4.35207	−0.227176	−0.113588	+	0.993528i	$0.536234\pi$
$368$	0	0
$369$	−62.6120	−3.25945
$370$	0	0
$371$	38.7035	2.00939
$372$	0	0
$373$	18.9655	0.981998	0.490999	−	0.871160i	$-0.336632\pi$
$373$	18.9655	0.981998	0.490999	+	0.871160i	$0.336632\pi$
$374$	0	0
$375$	33.7583	1.74327
$376$	0	0
$377$	−12.1690	−0.626735
$378$	0	0
$379$	14.2692	0.732962	0.366481	−	0.930426i	$-0.380562\pi$
$379$	14.2692	0.732962	0.366481	+	0.930426i	$0.380562\pi$
$380$	0	0
$381$	−39.3967	−2.01835
$382$	0	0
$383$	−4.89756	−0.250254	−0.125127	−	0.992141i	$-0.539934\pi$
$383$	−4.89756	−0.250254	−0.125127	+	0.992141i	$0.539934\pi$
$384$	0	0
$385$	−17.5750	−0.895706
$386$	0	0
$387$	−42.3444	−2.15249
$388$	0	0
$389$	0.524174	0.0265767	0.0132883	−	0.999912i	$-0.495770\pi$
$389$	0.524174	0.0265767	0.0132883	+	0.999912i	$0.495770\pi$
$390$	0	0
$391$	19.6492	0.993703
$392$	0	0
$393$	−56.6671	−2.85847
$394$	0	0
$395$	−4.71871	−0.237424
$396$	0	0
$397$	−20.8607	−1.04697	−0.523484	−	0.852035i	$-0.675368\pi$
$397$	−20.8607	−1.04697	−0.523484	+	0.852035i	$0.675368\pi$
$398$	0	0
$399$	−67.2932	−3.36888
$400$	0	0
$401$	17.1324	0.855553	0.427776	−	0.903885i	$-0.359297\pi$
$401$	17.1324	0.855553	0.427776	+	0.903885i	$0.359297\pi$
$402$	0	0
$403$	59.6287	2.97031
$404$	0	0
$405$	15.8372	0.786957
$406$	0	0
$407$	−21.2190	−1.05179
$408$	0	0
$409$	7.68942	0.380217	0.190109	−	0.981763i	$-0.439116\pi$
$409$	7.68942	0.380217	0.190109	+	0.981763i	$0.439116\pi$
$410$	0	0
$411$	−15.6731	−0.773096
$412$	0	0
$413$	8.91682	0.438768
$414$	0	0
$415$	22.0203	1.08093
$416$	0	0
$417$	8.39982	0.411341
$418$	0	0
$419$	−32.1640	−1.57131	−0.785656	−	0.618663i	$-0.787675\pi$
$419$	−32.1640	−1.57131	−0.785656	+	0.618663i	$0.787675\pi$
$420$	0	0
$421$	−4.11781	−0.200690	−0.100345	−	0.994953i	$-0.531995\pi$
$421$	−4.11781	−0.200690	−0.100345	+	0.994953i	$0.531995\pi$
$422$	0	0
$423$	23.2215	1.12907
$424$	0	0
$425$	16.5500	0.802791
$426$	0	0
$427$	29.0612	1.40637
$428$	0	0
$429$	−59.6583	−2.88033
$430$	0	0
$431$	−19.0788	−0.918994	−0.459497	−	0.888179i	$-0.651970\pi$
$431$	−19.0788	−0.918994	−0.459497	+	0.888179i	$0.651970\pi$
$432$	0	0
$433$	20.0805	0.965006	0.482503	−	0.875894i	$-0.339728\pi$
$433$	20.0805	0.965006	0.482503	+	0.875894i	$0.339728\pi$
$434$	0	0
$435$	8.91209	0.427302
$436$	0	0
$437$	22.1400	1.05910
$438$	0	0
$439$	−17.6120	−0.840577	−0.420288	−	0.907391i	$-0.638071\pi$
$439$	−17.6120	−0.840577	−0.420288	+	0.907391i	$0.638071\pi$
$440$	0	0
$441$	43.7091	2.08138
$442$	0	0
$443$	−16.6825	−0.792610	−0.396305	−	0.918119i	$-0.629708\pi$
$443$	−16.6825	−0.792610	−0.396305	+	0.918119i	$0.629708\pi$
$444$	0	0
$445$	1.53457	0.0727454
$446$	0	0
$447$	−45.9268	−2.17226
$448$	0	0
$449$	18.7438	0.884572	0.442286	−	0.896874i	$-0.354168\pi$
$449$	18.7438	0.884572	0.442286	+	0.896874i	$0.354168\pi$
$450$	0	0
$451$	−34.4952	−1.62431
$452$	0	0
$453$	35.7870	1.68142
$454$	0	0
$455$	−28.6862	−1.34483
$456$	0	0
$457$	0.235876	0.0110338	0.00551691	−	0.999985i	$-0.498244\pi$
$457$	0.235876	0.0110338	0.00551691	+	0.999985i	$0.498244\pi$
$458$	0	0
$459$	52.5404	2.45238
$460$	0	0
$461$	−4.84630	−0.225714	−0.112857	−	0.993611i	$-0.536000\pi$
$461$	−4.84630	−0.225714	−0.112857	+	0.993611i	$0.536000\pi$
$462$	0	0
$463$	18.1375	0.842922	0.421461	−	0.906847i	$-0.361517\pi$
$463$	18.1375	0.842922	0.421461	+	0.906847i	$0.361517\pi$
$464$	0	0
$465$	−43.6697	−2.02513
$466$	0	0
$467$	14.1549	0.655009	0.327504	−	0.944850i	$-0.393792\pi$
$467$	14.1549	0.655009	0.327504	+	0.944850i	$0.393792\pi$
$468$	0	0
$469$	37.9498	1.75236
$470$	0	0
$471$	59.6922	2.75047
$472$	0	0
$473$	−23.3290	−1.07267
$474$	0	0
$475$	18.6479	0.855624
$476$	0	0
$477$	−65.1329	−2.98223
$478$	0	0
$479$	26.5238	1.21190	0.605952	−	0.795501i	$-0.292792\pi$
$479$	26.5238	1.21190	0.605952	+	0.795501i	$0.292792\pi$
$480$	0	0
$481$	−34.6339	−1.57917
$482$	0	0
$483$	−42.6302	−1.93974
$484$	0	0
$485$	4.57911	0.207927
$486$	0	0
$487$	7.12246	0.322750	0.161375	−	0.986893i	$-0.448407\pi$
$487$	7.12246	0.322750	0.161375	+	0.986893i	$0.448407\pi$
$488$	0	0
$489$	54.2306	2.45239
$490$	0	0
$491$	−4.06416	−0.183413	−0.0917066	−	0.995786i	$-0.529232\pi$
$491$	−4.06416	−0.183413	−0.0917066	+	0.995786i	$0.529232\pi$
$492$	0	0
$493$	11.2947	0.508686
$494$	0	0
$495$	29.5764	1.32936
$496$	0	0
$497$	1.96162	0.0879905
$498$	0	0
$499$	−17.1820	−0.769172	−0.384586	−	0.923089i	$-0.625656\pi$
$499$	−17.1820	−0.769172	−0.384586	+	0.923089i	$0.625656\pi$
$500$	0	0
$501$	−3.04732	−0.136144
$502$	0	0
$503$	34.1952	1.52469	0.762345	−	0.647171i	$-0.224048\pi$
$503$	34.1952	1.52469	0.762345	+	0.647171i	$0.224048\pi$
$504$	0	0
$505$	3.82767	0.170329
$506$	0	0
$507$	−57.7600	−2.56521
$508$	0	0
$509$	−2.31236	−0.102493	−0.0512467	−	0.998686i	$-0.516319\pi$
$509$	−2.31236	−0.102493	−0.0512467	+	0.998686i	$0.516319\pi$
$510$	0	0
$511$	−18.5572	−0.820924
$512$	0	0
$513$	59.2006	2.61377
$514$	0	0
$515$	−0.314953	−0.0138785
$516$	0	0
$517$	12.7935	0.562658
$518$	0	0
$519$	−43.5645	−1.91227
$520$	0	0
$521$	9.99538	0.437905	0.218953	−	0.975735i	$-0.429736\pi$
$521$	9.99538	0.437905	0.218953	+	0.975735i	$0.429736\pi$
$522$	0	0
$523$	31.1314	1.36128	0.680642	−	0.732617i	$-0.261701\pi$
$523$	31.1314	1.36128	0.680642	+	0.732617i	$0.261701\pi$
$524$	0	0
$525$	−35.9062	−1.56707
$526$	0	0
$527$	−55.3444	−2.41084
$528$	0	0
$529$	−8.97433	−0.390188
$530$	0	0
$531$	−15.0058	−0.651197
$532$	0	0
$533$	−56.3035	−2.43878
$534$	0	0
$535$	−25.4992	−1.10242
$536$	0	0
$537$	66.4645	2.86816
$538$	0	0
$539$	24.0809	1.03724
$540$	0	0
$541$	−9.66706	−0.415620	−0.207810	−	0.978169i	$-0.566633\pi$
$541$	−9.66706	−0.415620	−0.207810	+	0.978169i	$0.566633\pi$
$542$	0	0
$543$	32.4872	1.39416
$544$	0	0
$545$	1.39690	0.0598365
$546$	0	0
$547$	41.2694	1.76455	0.882276	−	0.470733i	$-0.156010\pi$
$547$	41.2694	1.76455	0.882276	+	0.470733i	$0.156010\pi$
$548$	0	0
$549$	−48.9062	−2.08727
$550$	0	0
$551$	12.7264	0.542163
$552$	0	0
$553$	12.9744	0.551729
$554$	0	0
$555$	25.3645	1.07666
$556$	0	0
$557$	8.04280	0.340785	0.170392	−	0.985376i	$-0.445497\pi$
$557$	8.04280	0.340785	0.170392	+	0.985376i	$0.445497\pi$
$558$	0	0
$559$	−38.0780	−1.61053
$560$	0	0
$561$	55.3719	2.33780
$562$	0	0
$563$	34.5717	1.45703	0.728513	−	0.685032i	$-0.240212\pi$
$563$	34.5717	1.45703	0.728513	+	0.685032i	$0.240212\pi$
$564$	0	0
$565$	−1.85553	−0.0780628
$566$	0	0
$567$	−43.5456	−1.82874
$568$	0	0
$569$	17.7585	0.744476	0.372238	−	0.928137i	$-0.378590\pi$
$569$	17.7585	0.744476	0.372238	+	0.928137i	$0.378590\pi$
$570$	0	0
$571$	−7.35563	−0.307824	−0.153912	−	0.988085i	$-0.549187\pi$
$571$	−7.35563	−0.307824	−0.153912	+	0.988085i	$0.549187\pi$
$572$	0	0
$573$	−25.6735	−1.07253
$574$	0	0
$575$	11.8134	0.492654
$576$	0	0
$577$	−32.3499	−1.34675	−0.673373	−	0.739303i	$-0.735155\pi$
$577$	−32.3499	−1.34675	−0.673373	+	0.739303i	$0.735155\pi$
$578$	0	0
$579$	−60.7222	−2.52353
$580$	0	0
$581$	−60.5464	−2.51189
$582$	0	0
$583$	−35.8840	−1.48616
$584$	0	0
$585$	48.2751	1.99593
$586$	0	0
$587$	30.2634	1.24910	0.624552	−	0.780984i	$-0.285282\pi$
$587$	30.2634	1.24910	0.624552	+	0.780984i	$0.285282\pi$
$588$	0	0
$589$	−62.3600	−2.56950
$590$	0	0
$591$	8.14718	0.335130
$592$	0	0
$593$	35.3338	1.45098	0.725492	−	0.688231i	$-0.241612\pi$
$593$	35.3338	1.45098	0.725492	+	0.688231i	$0.241612\pi$
$594$	0	0
$595$	26.6251	1.09152
$596$	0	0
$597$	70.0092	2.86529
$598$	0	0
$599$	23.7744	0.971394	0.485697	−	0.874127i	$-0.338566\pi$
$599$	23.7744	0.971394	0.485697	+	0.874127i	$0.338566\pi$
$600$	0	0
$601$	−3.75741	−0.153268	−0.0766340	−	0.997059i	$-0.524417\pi$
$601$	−3.75741	−0.153268	−0.0766340	+	0.997059i	$0.524417\pi$
$602$	0	0
$603$	−63.8645	−2.60076
$604$	0	0
$605$	1.35080	0.0549177
$606$	0	0
$607$	9.76801	0.396471	0.198236	−	0.980154i	$-0.436479\pi$
$607$	9.76801	0.396471	0.198236	+	0.980154i	$0.436479\pi$
$608$	0	0
$609$	−24.5045	−0.992971
$610$	0	0
$611$	20.8818	0.844786
$612$	0	0
$613$	−46.3303	−1.87126	−0.935632	−	0.352977i	$-0.885170\pi$
$613$	−46.3303	−1.87126	−0.935632	+	0.352977i	$0.885170\pi$
$614$	0	0
$615$	41.2345	1.66273
$616$	0	0
$617$	8.47963	0.341377	0.170688	−	0.985325i	$-0.445401\pi$
$617$	8.47963	0.341377	0.170688	+	0.985325i	$0.445401\pi$
$618$	0	0
$619$	4.67093	0.187740	0.0938702	−	0.995584i	$-0.470076\pi$
$619$	4.67093	0.187740	0.0938702	+	0.995584i	$0.470076\pi$
$620$	0	0
$621$	37.5035	1.50497
$622$	0	0
$623$	−4.21941	−0.169047
$624$	0	0
$625$	0.722001	0.0288800
$626$	0	0
$627$	62.3910	2.49166
$628$	0	0
$629$	32.1455	1.28173
$630$	0	0
$631$	−8.82339	−0.351253	−0.175627	−	0.984457i	$-0.556195\pi$
$631$	−8.82339	−0.351253	−0.175627	+	0.984457i	$0.556195\pi$
$632$	0	0
$633$	45.1525	1.79465
$634$	0	0
$635$	17.5636	0.696989
$636$	0	0
$637$	39.3051	1.55733
$638$	0	0
$639$	−3.30114	−0.130591
$640$	0	0
$641$	−17.8502	−0.705040	−0.352520	−	0.935804i	$-0.614675\pi$
$641$	−17.8502	−0.705040	−0.352520	+	0.935804i	$0.614675\pi$
$642$	0	0
$643$	−12.2272	−0.482193	−0.241096	−	0.970501i	$-0.577507\pi$
$643$	−12.2272	−0.482193	−0.241096	+	0.970501i	$0.577507\pi$
$644$	0	0
$645$	27.8868	1.09804
$646$	0	0
$647$	20.9410	0.823274	0.411637	−	0.911348i	$-0.364957\pi$
$647$	20.9410	0.823274	0.411637	+	0.911348i	$0.364957\pi$
$648$	0	0
$649$	−8.26724	−0.324517
$650$	0	0
$651$	120.073	4.70604
$652$	0	0
$653$	−8.83643	−0.345796	−0.172898	−	0.984940i	$-0.555313\pi$
$653$	−8.83643	−0.345796	−0.172898	+	0.984940i	$0.555313\pi$
$654$	0	0
$655$	25.2629	0.987103
$656$	0	0
$657$	31.2294	1.21838
$658$	0	0
$659$	−17.9043	−0.697454	−0.348727	−	0.937224i	$-0.613386\pi$
$659$	−17.9043	−0.697454	−0.348727	+	0.937224i	$0.613386\pi$
$660$	0	0
$661$	−11.0640	−0.430341	−0.215171	−	0.976576i	$-0.569031\pi$
$661$	−11.0640	−0.430341	−0.215171	+	0.976576i	$0.569031\pi$
$662$	0	0
$663$	90.3788	3.51002
$664$	0	0
$665$	30.0002	1.16336
$666$	0	0
$667$	8.06216	0.312168
$668$	0	0
$669$	13.1957	0.510174
$670$	0	0
$671$	−26.9442	−1.04017
$672$	0	0
$673$	38.1470	1.47046	0.735229	−	0.677819i	$-0.237075\pi$
$673$	38.1470	1.47046	0.735229	+	0.677819i	$0.237075\pi$
$674$	0	0
$675$	31.5882	1.21583
$676$	0	0
$677$	−16.6262	−0.638997	−0.319499	−	0.947587i	$-0.603515\pi$
$677$	−16.6262	−0.638997	−0.319499	+	0.947587i	$0.603515\pi$
$678$	0	0
$679$	−12.5906	−0.483183
$680$	0	0
$681$	−8.58961	−0.329155
$682$	0	0
$683$	−34.2239	−1.30954	−0.654771	−	0.755827i	$-0.727235\pi$
$683$	−34.2239	−1.30954	−0.654771	+	0.755827i	$0.727235\pi$
$684$	0	0
$685$	6.98726	0.266970
$686$	0	0
$687$	12.5672	0.479467
$688$	0	0
$689$	−58.5704	−2.23136
$690$	0	0
$691$	−36.1393	−1.37481	−0.687403	−	0.726277i	$-0.741249\pi$
$691$	−36.1393	−1.37481	−0.687403	+	0.726277i	$0.741249\pi$
$692$	0	0
$693$	−81.3226	−3.08919
$694$	0	0
$695$	−3.74475	−0.142046
$696$	0	0
$697$	52.2581	1.97942
$698$	0	0
$699$	−9.73111	−0.368064
$700$	0	0
$701$	−19.6295	−0.741395	−0.370697	−	0.928754i	$-0.620881\pi$
$701$	−19.6295	−0.741395	−0.370697	+	0.928754i	$0.620881\pi$
$702$	0	0
$703$	36.2204	1.36608
$704$	0	0
$705$	−15.2930	−0.575967
$706$	0	0
$707$	−10.5245	−0.395813
$708$	0	0
$709$	20.7410	0.778946	0.389473	−	0.921038i	$-0.372657\pi$
$709$	20.7410	0.778946	0.389473	+	0.921038i	$0.372657\pi$
$710$	0	0
$711$	−21.8343	−0.818849
$712$	0	0
$713$	−39.5050	−1.47947
$714$	0	0
$715$	26.5964	0.994651
$716$	0	0
$717$	−56.4925	−2.10975
$718$	0	0
$719$	−43.2159	−1.61168	−0.805841	−	0.592132i	$-0.798287\pi$
$719$	−43.2159	−1.61168	−0.805841	+	0.592132i	$0.798287\pi$
$720$	0	0
$721$	0.865987	0.0322510
$722$	0	0
$723$	56.6675	2.10749
$724$	0	0
$725$	6.79053	0.252194
$726$	0	0
$727$	−33.6198	−1.24689	−0.623445	−	0.781867i	$-0.714267\pi$
$727$	−33.6198	−1.24689	−0.623445	+	0.781867i	$0.714267\pi$
$728$	0	0
$729$	−18.2668	−0.676548
$730$	0	0
$731$	35.3421	1.30718
$732$	0	0
$733$	−10.8105	−0.399296	−0.199648	−	0.979868i	$-0.563980\pi$
$733$	−10.8105	−0.399296	−0.199648	+	0.979868i	$0.563980\pi$
$734$	0	0
$735$	−28.7855	−1.06177
$736$	0	0
$737$	−35.1852	−1.29606
$738$	0	0
$739$	19.6414	0.722521	0.361260	−	0.932465i	$-0.382347\pi$
$739$	19.6414	0.722521	0.361260	+	0.932465i	$0.382347\pi$
$740$	0	0
$741$	101.836	3.74102
$742$	0	0
$743$	46.2216	1.69571	0.847853	−	0.530231i	$-0.177895\pi$
$743$	46.2216	1.69571	0.847853	+	0.530231i	$0.177895\pi$
$744$	0	0
$745$	20.4747	0.750136
$746$	0	0
$747$	101.892	3.72802
$748$	0	0
$749$	70.1119	2.56183
$750$	0	0
$751$	−40.5429	−1.47943	−0.739715	−	0.672920i	$-0.765040\pi$
$751$	−40.5429	−1.47943	−0.739715	+	0.672920i	$0.765040\pi$
$752$	0	0
$753$	−74.7129	−2.72269
$754$	0	0
$755$	−15.9543	−0.580637
$756$	0	0
$757$	−33.4877	−1.21713	−0.608566	−	0.793504i	$-0.708255\pi$
$757$	−33.4877	−1.21713	−0.608566	+	0.793504i	$0.708255\pi$
$758$	0	0
$759$	39.5246	1.43465
$760$	0	0
$761$	−16.6129	−0.602216	−0.301108	−	0.953590i	$-0.597356\pi$
$761$	−16.6129	−0.602216	−0.301108	+	0.953590i	$0.597356\pi$
$762$	0	0
$763$	−3.84088	−0.139049
$764$	0	0
$765$	−44.8066	−1.61998
$766$	0	0
$767$	−13.4939	−0.487237
$768$	0	0
$769$	21.1342	0.762119	0.381059	−	0.924551i	$-0.375559\pi$
$769$	21.1342	0.762119	0.381059	+	0.924551i	$0.375559\pi$
$770$	0	0
$771$	−9.18701	−0.330862
$772$	0	0
$773$	−19.6281	−0.705975	−0.352988	−	0.935628i	$-0.614834\pi$
$773$	−19.6281	−0.705975	−0.352988	+	0.935628i	$0.614834\pi$
$774$	0	0
$775$	−33.2739	−1.19523
$776$	0	0
$777$	−69.7417	−2.50197
$778$	0	0
$779$	58.8825	2.10969
$780$	0	0
$781$	−1.81872	−0.0650787
$782$	0	0
$783$	21.5576	0.770405
$784$	0	0
$785$	−26.6116	−0.949807
$786$	0	0
$787$	−7.14146	−0.254566	−0.127283	−	0.991866i	$-0.540626\pi$
$787$	−7.14146	−0.254566	−0.127283	+	0.991866i	$0.540626\pi$
$788$	0	0
$789$	89.2976	3.17908
$790$	0	0
$791$	5.10193	0.181404
$792$	0	0
$793$	−43.9787	−1.56173
$794$	0	0
$795$	42.8947	1.52132
$796$	0	0
$797$	−32.3622	−1.14633	−0.573164	−	0.819441i	$-0.694284\pi$
$797$	−32.3622	−1.14633	−0.573164	+	0.819441i	$0.694284\pi$
$798$	0	0
$799$	−19.3814	−0.685666
$800$	0	0
$801$	7.10071	0.250891
$802$	0	0
$803$	17.2054	0.607165
$804$	0	0
$805$	19.0051	0.669842
$806$	0	0
$807$	6.01272	0.211658
$808$	0	0
$809$	17.6100	0.619134	0.309567	−	0.950878i	$-0.399816\pi$
$809$	17.6100	0.619134	0.309567	+	0.950878i	$0.399816\pi$
$810$	0	0
$811$	20.6634	0.725588	0.362794	−	0.931869i	$-0.381823\pi$
$811$	20.6634	0.725588	0.362794	+	0.931869i	$0.381823\pi$
$812$	0	0
$813$	−56.7249	−1.98943
$814$	0	0
$815$	−24.1767	−0.846873
$816$	0	0
$817$	39.8222	1.39320
$818$	0	0
$819$	−132.736	−4.63817
$820$	0	0
$821$	−14.5424	−0.507534	−0.253767	−	0.967265i	$-0.581670\pi$
$821$	−14.5424	−0.507534	−0.253767	+	0.967265i	$0.581670\pi$
$822$	0	0
$823$	−18.4446	−0.642939	−0.321469	−	0.946920i	$-0.604177\pi$
$823$	−18.4446	−0.642939	−0.321469	+	0.946920i	$0.604177\pi$
$824$	0	0
$825$	33.2905	1.15903
$826$	0	0
$827$	36.6524	1.27453	0.637265	−	0.770645i	$-0.280066\pi$
$827$	36.6524	1.27453	0.637265	+	0.770645i	$0.280066\pi$
$828$	0	0
$829$	52.5477	1.82506	0.912528	−	0.409015i	$-0.134128\pi$
$829$	52.5477	1.82506	0.912528	+	0.409015i	$0.134128\pi$
$830$	0	0
$831$	53.1107	1.84239
$832$	0	0
$833$	−36.4811	−1.26399
$834$	0	0
$835$	1.35854	0.0470141
$836$	0	0
$837$	−105.633	−3.65122
$838$	0	0
$839$	44.7378	1.54452	0.772260	−	0.635307i	$-0.219126\pi$
$839$	44.7378	1.54452	0.772260	+	0.635307i	$0.219126\pi$
$840$	0	0
$841$	−24.3658	−0.840198
$842$	0	0
$843$	−76.1380	−2.62233
$844$	0	0
$845$	25.7501	0.885832
$846$	0	0
$847$	−3.71412	−0.127619
$848$	0	0
$849$	−5.42094	−0.186046
$850$	0	0
$851$	22.9456	0.786564
$852$	0	0
$853$	−23.9929	−0.821501	−0.410751	−	0.911748i	$-0.634733\pi$
$853$	−23.9929	−0.821501	−0.410751	+	0.911748i	$0.634733\pi$
$854$	0	0
$855$	−50.4864	−1.72660
$856$	0	0
$857$	−47.8544	−1.63467	−0.817337	−	0.576160i	$-0.804551\pi$
$857$	−47.8544	−1.63467	−0.817337	+	0.576160i	$0.804551\pi$
$858$	0	0
$859$	4.54528	0.155083	0.0775415	−	0.996989i	$-0.475293\pi$
$859$	4.54528	0.155083	0.0775415	+	0.996989i	$0.475293\pi$
$860$	0	0
$861$	−113.377	−3.86389
$862$	0	0
$863$	3.00469	0.102281	0.0511405	−	0.998691i	$-0.483714\pi$
$863$	3.00469	0.102281	0.0511405	+	0.998691i	$0.483714\pi$
$864$	0	0
$865$	19.4216	0.660354
$866$	0	0
$867$	−32.0807	−1.08952
$868$	0	0
$869$	−12.0293	−0.408065
$870$	0	0
$871$	−57.4298	−1.94593
$872$	0	0
$873$	21.1883	0.717117
$874$	0	0
$875$	41.3808	1.39893
$876$	0	0
$877$	−39.3309	−1.32811	−0.664055	−	0.747683i	$-0.731166\pi$
$877$	−39.3309	−1.32811	−0.664055	+	0.747683i	$0.731166\pi$
$878$	0	0
$879$	−40.4731	−1.36512
$880$	0	0
$881$	−24.6172	−0.829373	−0.414687	−	0.909964i	$-0.636109\pi$
$881$	−24.6172	−0.829373	−0.414687	+	0.909964i	$0.636109\pi$
$882$	0	0
$883$	24.2688	0.816711	0.408355	−	0.912823i	$-0.366102\pi$
$883$	24.2688	0.816711	0.408355	+	0.912823i	$0.366102\pi$
$884$	0	0
$885$	9.88241	0.332194
$886$	0	0
$887$	14.7424	0.495001	0.247501	−	0.968888i	$-0.420391\pi$
$887$	14.7424	0.495001	0.247501	+	0.968888i	$0.420391\pi$
$888$	0	0
$889$	−48.2923	−1.61967
$890$	0	0
$891$	40.3733	1.35256
$892$	0	0
$893$	−21.8383	−0.730790
$894$	0	0
$895$	−29.6308	−0.990447
$896$	0	0
$897$	64.5127	2.15402
$898$	0	0
$899$	−22.7080	−0.757356
$900$	0	0
$901$	54.3622	1.81107
$902$	0	0
$903$	−76.6769	−2.55165
$904$	0	0
$905$	−14.4832	−0.481438
$906$	0	0
$907$	4.04344	0.134260	0.0671300	−	0.997744i	$-0.478616\pi$
$907$	4.04344	0.134260	0.0671300	+	0.997744i	$0.478616\pi$
$908$	0	0
$909$	17.7113	0.587447
$910$	0	0
$911$	26.1517	0.866445	0.433223	−	0.901287i	$-0.357376\pi$
$911$	26.1517	0.866445	0.433223	+	0.901287i	$0.357376\pi$
$912$	0	0
$913$	56.1357	1.85782
$914$	0	0
$915$	32.2082	1.06477
$916$	0	0
$917$	−69.4623	−2.29385
$918$	0	0
$919$	−11.7942	−0.389053	−0.194527	−	0.980897i	$-0.562317\pi$
$919$	−11.7942	−0.389053	−0.194527	+	0.980897i	$0.562317\pi$
$920$	0	0
$921$	−2.79349	−0.0920488
$922$	0	0
$923$	−2.96853	−0.0977105
$924$	0	0
$925$	19.3264	0.635448
$926$	0	0
$927$	−1.45734	−0.0478654
$928$	0	0
$929$	−35.4933	−1.16450	−0.582248	−	0.813011i	$-0.697827\pi$
$929$	−35.4933	−1.16450	−0.582248	+	0.813011i	$0.697827\pi$
$930$	0	0
$931$	−41.1056	−1.34718
$932$	0	0
$933$	59.6154	1.95172
$934$	0	0
$935$	−24.6855	−0.807302
$936$	0	0
$937$	19.6733	0.642700	0.321350	−	0.946961i	$-0.395863\pi$
$937$	19.6733	0.642700	0.321350	+	0.946961i	$0.395863\pi$
$938$	0	0
$939$	58.0718	1.89510
$940$	0	0
$941$	3.50813	0.114362	0.0571809	−	0.998364i	$-0.481789\pi$
$941$	3.50813	0.114362	0.0571809	+	0.998364i	$0.481789\pi$
$942$	0	0
$943$	37.3020	1.21472
$944$	0	0
$945$	50.8181	1.65311
$946$	0	0
$947$	−29.4558	−0.957183	−0.478592	−	0.878038i	$-0.658853\pi$
$947$	−29.4558	−0.957183	−0.478592	+	0.878038i	$0.658853\pi$
$948$	0	0
$949$	28.0829	0.911609
$950$	0	0
$951$	−19.3024	−0.625924
$952$	0	0
$953$	−6.45573	−0.209121	−0.104561	−	0.994519i	$-0.533344\pi$
$953$	−6.45573	−0.209121	−0.104561	+	0.994519i	$0.533344\pi$
$954$	0	0
$955$	11.4456	0.370371
$956$	0	0
$957$	22.7193	0.734412
$958$	0	0
$959$	−19.2120	−0.620388
$960$	0	0
$961$	80.2706	2.58937
$962$	0	0
$963$	−117.989	−3.80214
$964$	0	0
$965$	27.0707	0.871438
$966$	0	0
$967$	−46.5083	−1.49560	−0.747802	−	0.663921i	$-0.768891\pi$
$967$	−46.5083	−1.49560	−0.747802	+	0.663921i	$0.768891\pi$
$968$	0	0
$969$	−94.5187	−3.03638
$970$	0	0
$971$	−22.5794	−0.724607	−0.362304	−	0.932060i	$-0.618010\pi$
$971$	−22.5794	−0.724607	−0.362304	+	0.932060i	$0.618010\pi$
$972$	0	0
$973$	10.2965	0.330090
$974$	0	0
$975$	54.3372	1.74018
$976$	0	0
$977$	32.6174	1.04352	0.521762	−	0.853091i	$-0.325275\pi$
$977$	32.6174	1.04352	0.521762	+	0.853091i	$0.325275\pi$
$978$	0	0
$979$	3.91203	0.125029
$980$	0	0
$981$	6.46369	0.206370
$982$	0	0
$983$	−38.6062	−1.23135	−0.615674	−	0.788001i	$-0.711116\pi$
$983$	−38.6062	−1.23135	−0.615674	+	0.788001i	$0.711116\pi$
$984$	0	0
$985$	−3.63212	−0.115729
$986$	0	0
$987$	42.0492	1.33844
$988$	0	0
$989$	25.2273	0.802182
$990$	0	0
$991$	4.63294	0.147170	0.0735851	−	0.997289i	$-0.476556\pi$
$991$	4.63294	0.147170	0.0735851	+	0.997289i	$0.476556\pi$
$992$	0	0
$993$	−16.7456	−0.531406
$994$	0	0
$995$	−31.2110	−0.989456
$996$	0	0
$997$	−34.0242	−1.07756	−0.538779	−	0.842447i	$-0.681114\pi$
$997$	−34.0242	−1.07756	−0.538779	+	0.842447i	$0.681114\pi$
$998$	0	0
$999$	61.3547	1.94118

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.c.1.2	✓	7
3.2	odd	2		6012.2.a.g.1.2		7
4.3	odd	2		2672.2.a.k.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.2	✓	7	1.1	even	1	trivial
2672.2.a.k.1.6		7	4.3	odd	2
6012.2.a.g.1.2		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.a (trivial)

\(f(q)\)	\(=\)	\(q-3.04732 q^{3} +1.35854 q^{5} -3.73540 q^{7} +6.28618 q^{9} +O(q^{10})\)	\(q-3.04732 q^{3} +1.35854 q^{5} -3.73540 q^{7} +6.28618 q^{9} +3.46328 q^{11} +5.65282 q^{13} -4.13990 q^{15} -5.24666 q^{17} -5.91175 q^{19} +11.3830 q^{21} -3.74509 q^{23} -3.15438 q^{25} -10.0141 q^{27} -2.15273 q^{29} +10.5485 q^{31} -10.5537 q^{33} -5.07467 q^{35} -6.12685 q^{37} -17.2260 q^{39} -9.96026 q^{41} -6.73611 q^{43} +8.54001 q^{45} +3.69405 q^{47} +6.95320 q^{49} +15.9883 q^{51} -10.3613 q^{53} +4.70499 q^{55} +18.0150 q^{57} -2.38711 q^{59} -7.77996 q^{61} -23.4814 q^{63} +7.67956 q^{65} -10.1595 q^{67} +11.4125 q^{69} -0.525143 q^{71} +4.96794 q^{73} +9.61241 q^{75} -12.9367 q^{77} -3.47337 q^{79} +11.6575 q^{81} +16.2088 q^{83} -7.12778 q^{85} +6.56007 q^{87} +1.12957 q^{89} -21.1155 q^{91} -32.1447 q^{93} -8.03133 q^{95} +3.37062 q^{97} +21.7708 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * q^99

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