LMFDB - Embedded newform 668.2.a.c.1.7

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			1.7
Root			$-0.656969$ 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)

$f(q)$	$=$	$q+2.62809 q^{3} -3.04428 q^{5} -5.08545 q^{7} +3.90685 q^{9} +O(q^{10})$	$q+2.62809 q^{3} -3.04428 q^{5} -5.08545 q^{7} +3.90685 q^{9} -5.20467 q^{11} -2.31129 q^{13} -8.00064 q^{15} +4.27784 q^{17} +1.59965 q^{19} -13.3650 q^{21} -2.43992 q^{23} +4.26765 q^{25} +2.38328 q^{27} +4.31901 q^{29} -4.46082 q^{31} -13.6783 q^{33} +15.4815 q^{35} -4.67425 q^{37} -6.07428 q^{39} -6.47545 q^{41} +0.211895 q^{43} -11.8935 q^{45} +11.7940 q^{47} +18.8618 q^{49} +11.2425 q^{51} +1.30865 q^{53} +15.8445 q^{55} +4.20403 q^{57} -2.08988 q^{59} -10.0019 q^{61} -19.8681 q^{63} +7.03623 q^{65} -12.4830 q^{67} -6.41231 q^{69} +10.6069 q^{71} +15.7077 q^{73} +11.2158 q^{75} +26.4681 q^{77} -1.89111 q^{79} -5.45708 q^{81} +0.0733009 q^{83} -13.0229 q^{85} +11.3507 q^{87} -13.6047 q^{89} +11.7540 q^{91} -11.7234 q^{93} -4.86979 q^{95} -13.9375 q^{97} -20.3339 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.62809	1.51733	0.758664	−	0.651482i	$-0.225853\pi$
$3$	2.62809	1.51733	0.758664	+	0.651482i	$0.225853\pi$
$4$	0	0
$5$	−3.04428	−1.36144	−0.680722	−	0.732542i	$-0.738334\pi$
$5$	−3.04428	−1.36144	−0.680722	+	0.732542i	$0.738334\pi$
$6$	0	0
$7$	−5.08545	−1.92212	−0.961060	−	0.276341i	$-0.910878\pi$
$7$	−5.08545	−1.92212	−0.961060	+	0.276341i	$0.910878\pi$
$8$	0	0
$9$	3.90685	1.30228
$10$	0	0
$11$	−5.20467	−1.56927	−0.784634	−	0.619959i	$-0.787149\pi$
$11$	−5.20467	−1.56927	−0.784634	+	0.619959i	$0.787149\pi$
$12$	0	0
$13$	−2.31129	−0.641037	−0.320519	−	0.947242i	$-0.603857\pi$
$13$	−2.31129	−0.641037	−0.320519	+	0.947242i	$0.603857\pi$
$14$	0	0
$15$	−8.00064	−2.06576
$16$	0	0
$17$	4.27784	1.03753	0.518764	−	0.854917i	$-0.326392\pi$
$17$	4.27784	1.03753	0.518764	+	0.854917i	$0.326392\pi$
$18$	0	0
$19$	1.59965	0.366985	0.183493	−	0.983021i	$-0.441260\pi$
$19$	1.59965	0.366985	0.183493	+	0.983021i	$0.441260\pi$
$20$	0	0
$21$	−13.3650	−2.91648
$22$	0	0
$23$	−2.43992	−0.508758	−0.254379	−	0.967105i	$-0.581871\pi$
$23$	−2.43992	−0.508758	−0.254379	+	0.967105i	$0.581871\pi$
$24$	0	0
$25$	4.26765	0.853531
$26$	0	0
$27$	2.38328	0.458662
$28$	0	0
$29$	4.31901	0.802019	0.401010	−	0.916074i	$-0.368659\pi$
$29$	4.31901	0.802019	0.401010	+	0.916074i	$0.368659\pi$
$30$	0	0
$31$	−4.46082	−0.801188	−0.400594	−	0.916256i	$-0.631196\pi$
$31$	−4.46082	−0.801188	−0.400594	+	0.916256i	$0.631196\pi$
$32$	0	0
$33$	−13.6783	−2.38109
$34$	0	0
$35$	15.4815	2.61686
$36$	0	0
$37$	−4.67425	−0.768441	−0.384221	−	0.923241i	$-0.625530\pi$
$37$	−4.67425	−0.768441	−0.384221	+	0.923241i	$0.625530\pi$
$38$	0	0
$39$	−6.07428	−0.972663
$40$	0	0
$41$	−6.47545	−1.01130	−0.505648	−	0.862740i	$-0.668746\pi$
$41$	−6.47545	−1.01130	−0.505648	+	0.862740i	$0.668746\pi$
$42$	0	0
$43$	0.211895	0.0323136	0.0161568	−	0.999869i	$-0.494857\pi$
$43$	0.211895	0.0323136	0.0161568	+	0.999869i	$0.494857\pi$
$44$	0	0
$45$	−11.8935	−1.77299
$46$	0	0
$47$	11.7940	1.72033	0.860164	−	0.510018i	$-0.170361\pi$
$47$	11.7940	1.72033	0.860164	+	0.510018i	$0.170361\pi$
$48$	0	0
$49$	18.8618	2.69454
$50$	0	0
$51$	11.2425	1.57427
$52$	0	0
$53$	1.30865	0.179756	0.0898782	−	0.995953i	$-0.471352\pi$
$53$	1.30865	0.179756	0.0898782	+	0.995953i	$0.471352\pi$
$54$	0	0
$55$	15.8445	2.13647
$56$	0	0
$57$	4.20403	0.556837
$58$	0	0
$59$	−2.08988	−0.272079	−0.136039	−	0.990703i	$-0.543437\pi$
$59$	−2.08988	−0.272079	−0.136039	+	0.990703i	$0.543437\pi$
$60$	0	0
$61$	−10.0019	−1.28061	−0.640304	−	0.768122i	$-0.721192\pi$
$61$	−10.0019	−1.28061	−0.640304	+	0.768122i	$0.721192\pi$
$62$	0	0
$63$	−19.8681	−2.50314
$64$	0	0
$65$	7.03623	0.872737
$66$	0	0
$67$	−12.4830	−1.52504	−0.762520	−	0.646965i	$-0.776038\pi$
$67$	−12.4830	−1.52504	−0.762520	+	0.646965i	$0.776038\pi$
$68$	0	0
$69$	−6.41231	−0.771952
$70$	0	0
$71$	10.6069	1.25880	0.629402	−	0.777079i	$-0.283300\pi$
$71$	10.6069	1.25880	0.629402	+	0.777079i	$0.283300\pi$
$72$	0	0
$73$	15.7077	1.83845	0.919223	−	0.393737i	$-0.128818\pi$
$73$	15.7077	1.83845	0.919223	+	0.393737i	$0.128818\pi$
$74$	0	0
$75$	11.2158	1.29509
$76$	0	0
$77$	26.4681	3.01632
$78$	0	0
$79$	−1.89111	−0.212766	−0.106383	−	0.994325i	$-0.533927\pi$
$79$	−1.89111	−0.212766	−0.106383	+	0.994325i	$0.533927\pi$
$80$	0	0
$81$	−5.45708	−0.606342
$82$	0	0
$83$	0.0733009	0.00804582	0.00402291	−	0.999992i	$-0.498719\pi$
$83$	0.0733009	0.00804582	0.00402291	+	0.999992i	$0.498719\pi$
$84$	0	0
$85$	−13.0229	−1.41254
$86$	0	0
$87$	11.3507	1.21693
$88$	0	0
$89$	−13.6047	−1.44210	−0.721050	−	0.692883i	$-0.756340\pi$
$89$	−13.6047	−1.44210	−0.721050	+	0.692883i	$0.756340\pi$
$90$	0	0
$91$	11.7540	1.23215
$92$	0	0
$93$	−11.7234	−1.21566
$94$	0	0
$95$	−4.86979	−0.499630
$96$	0	0
$97$	−13.9375	−1.41514	−0.707570	−	0.706643i	$-0.750209\pi$
$97$	−13.9375	−1.41514	−0.707570	+	0.706643i	$0.750209\pi$
$98$	0	0
$99$	−20.3339	−2.04363
$100$	0	0
$101$	0.0118148	0.00117562	0.000587809	−	1.00000i	$-0.499813\pi$
$101$	0.0118148	0.00117562	0.000587809		1.00000i	$0.499813\pi$
$102$	0	0
$103$	−17.9196	−1.76567	−0.882835	−	0.469684i	$-0.844368\pi$
$103$	−17.9196	−1.76567	−0.882835	+	0.469684i	$0.844368\pi$
$104$	0	0
$105$	40.6869	3.97063
$106$	0	0
$107$	−12.4182	−1.20052	−0.600258	−	0.799806i	$-0.704935\pi$
$107$	−12.4182	−1.20052	−0.600258	+	0.799806i	$0.704935\pi$
$108$	0	0
$109$	12.3019	1.17831	0.589154	−	0.808021i	$-0.299461\pi$
$109$	12.3019	1.17831	0.589154	+	0.808021i	$0.299461\pi$
$110$	0	0
$111$	−12.2843	−1.16598
$112$	0	0
$113$	−11.5977	−1.09102	−0.545510	−	0.838104i	$-0.683664\pi$
$113$	−11.5977	−1.09102	−0.545510	+	0.838104i	$0.683664\pi$
$114$	0	0
$115$	7.42779	0.692645
$116$	0	0
$117$	−9.02987	−0.834812
$118$	0	0
$119$	−21.7547	−1.99425
$120$	0	0
$121$	16.0886	1.46260
$122$	0	0
$123$	−17.0181	−1.53447
$124$	0	0
$125$	2.22947	0.199410
$126$	0	0
$127$	−10.3903	−0.921993	−0.460996	−	0.887402i	$-0.652508\pi$
$127$	−10.3903	−0.921993	−0.460996	+	0.887402i	$0.652508\pi$
$128$	0	0
$129$	0.556878	0.0490304
$130$	0	0
$131$	−6.34100	−0.554016	−0.277008	−	0.960868i	$-0.589343\pi$
$131$	−6.34100	−0.554016	−0.277008	+	0.960868i	$0.589343\pi$
$132$	0	0
$133$	−8.13495	−0.705390
$134$	0	0
$135$	−7.25537	−0.624443
$136$	0	0
$137$	17.9743	1.53565	0.767823	−	0.640662i	$-0.221340\pi$
$137$	17.9743	1.53565	0.767823	+	0.640662i	$0.221340\pi$
$138$	0	0
$139$	−15.7565	−1.33645	−0.668224	−	0.743960i	$-0.732945\pi$
$139$	−15.7565	−1.33645	−0.668224	+	0.743960i	$0.732945\pi$
$140$	0	0
$141$	30.9956	2.61030
$142$	0	0
$143$	12.0295	1.00596
$144$	0	0
$145$	−13.1483	−1.09190
$146$	0	0
$147$	49.5705	4.08850
$148$	0	0
$149$	20.5131	1.68050	0.840248	−	0.542202i	$-0.182409\pi$
$149$	20.5131	1.68050	0.840248	+	0.542202i	$0.182409\pi$
$150$	0	0
$151$	−3.50097	−0.284905	−0.142452	−	0.989802i	$-0.545499\pi$
$151$	−3.50097	−0.284905	−0.142452	+	0.989802i	$0.545499\pi$
$152$	0	0
$153$	16.7129	1.35115
$154$	0	0
$155$	13.5800	1.09077
$156$	0	0
$157$	−5.07915	−0.405360	−0.202680	−	0.979245i	$-0.564965\pi$
$157$	−5.07915	−0.405360	−0.202680	+	0.979245i	$0.564965\pi$
$158$	0	0
$159$	3.43924	0.272749
$160$	0	0
$161$	12.4081	0.977893
$162$	0	0
$163$	−6.95080	−0.544429	−0.272214	−	0.962237i	$-0.587756\pi$
$163$	−6.95080	−0.544429	−0.272214	+	0.962237i	$0.587756\pi$
$164$	0	0
$165$	41.6407	3.24173
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−7.65793	−0.589071
$170$	0	0
$171$	6.24960	0.477919
$172$	0	0
$173$	0.0212476	0.00161543	0.000807713	−	1.00000i	$-0.499743\pi$
$173$	0.0212476	0.00161543	0.000807713		1.00000i	$0.499743\pi$
$174$	0	0
$175$	−21.7029	−1.64059
$176$	0	0
$177$	−5.49238	−0.412832
$178$	0	0
$179$	5.82850	0.435642	0.217821	−	0.975989i	$-0.430105\pi$
$179$	5.82850	0.435642	0.217821	+	0.975989i	$0.430105\pi$
$180$	0	0
$181$	25.1573	1.86993	0.934964	−	0.354743i	$-0.115432\pi$
$181$	25.1573	1.86993	0.934964	+	0.354743i	$0.115432\pi$
$182$	0	0
$183$	−26.2858	−1.94310
$184$	0	0
$185$	14.2297	1.04619
$186$	0	0
$187$	−22.2648	−1.62816
$188$	0	0
$189$	−12.1200	−0.881603
$190$	0	0
$191$	19.2286	1.39134	0.695668	−	0.718364i	$-0.255109\pi$
$191$	19.2286	1.39134	0.695668	+	0.718364i	$0.255109\pi$
$192$	0	0
$193$	−0.794796	−0.0572106	−0.0286053	−	0.999591i	$-0.509107\pi$
$193$	−0.794796	−0.0572106	−0.0286053	+	0.999591i	$0.509107\pi$
$194$	0	0
$195$	18.4918	1.32423
$196$	0	0
$197$	−14.2473	−1.01508	−0.507541	−	0.861628i	$-0.669445\pi$
$197$	−14.2473	−1.01508	−0.507541	+	0.861628i	$0.669445\pi$
$198$	0	0
$199$	−8.74481	−0.619903	−0.309952	−	0.950752i	$-0.600313\pi$
$199$	−8.74481	−0.619903	−0.309952	+	0.950752i	$0.600313\pi$
$200$	0	0
$201$	−32.8064	−2.31398
$202$	0	0
$203$	−21.9641	−1.54158
$204$	0	0
$205$	19.7131	1.37682
$206$	0	0
$207$	−9.53238	−0.662546
$208$	0	0
$209$	−8.32567	−0.575899
$210$	0	0
$211$	12.6406	0.870214	0.435107	−	0.900379i	$-0.356711\pi$
$211$	12.6406	0.870214	0.435107	+	0.900379i	$0.356711\pi$
$212$	0	0
$213$	27.8758	1.91002
$214$	0	0
$215$	−0.645067	−0.0439932
$216$	0	0
$217$	22.6853	1.53998
$218$	0	0
$219$	41.2812	2.78953
$220$	0	0
$221$	−9.88734	−0.665094
$222$	0	0
$223$	−7.46048	−0.499590	−0.249795	−	0.968299i	$-0.580363\pi$
$223$	−7.46048	−0.499590	−0.249795	+	0.968299i	$0.580363\pi$
$224$	0	0
$225$	16.6731	1.11154
$226$	0	0
$227$	26.2068	1.73941	0.869705	−	0.493573i	$-0.164309\pi$
$227$	26.2068	1.73941	0.869705	+	0.493573i	$0.164309\pi$
$228$	0	0
$229$	−23.9910	−1.58537	−0.792685	−	0.609632i	$-0.791317\pi$
$229$	−23.9910	−1.58537	−0.792685	+	0.609632i	$0.791317\pi$
$230$	0	0
$231$	69.5605	4.57675
$232$	0	0
$233$	1.27525	0.0835441	0.0417720	−	0.999127i	$-0.486700\pi$
$233$	1.27525	0.0835441	0.0417720	+	0.999127i	$0.486700\pi$
$234$	0	0
$235$	−35.9042	−2.34213
$236$	0	0
$237$	−4.97000	−0.322836
$238$	0	0
$239$	−12.9028	−0.834613	−0.417306	−	0.908766i	$-0.637026\pi$
$239$	−12.9028	−0.834613	−0.417306	+	0.908766i	$0.637026\pi$
$240$	0	0
$241$	0.564182	0.0363421	0.0181711	−	0.999835i	$-0.494216\pi$
$241$	0.564182	0.0363421	0.0181711	+	0.999835i	$0.494216\pi$
$242$	0	0
$243$	−21.4915	−1.37868
$244$	0	0
$245$	−57.4207	−3.66847
$246$	0	0
$247$	−3.69727	−0.235251
$248$	0	0
$249$	0.192641	0.0122081
$250$	0	0
$251$	9.92977	0.626762	0.313381	−	0.949628i	$-0.398538\pi$
$251$	9.92977	0.626762	0.313381	+	0.949628i	$0.398538\pi$
$252$	0	0
$253$	12.6990	0.798377
$254$	0	0
$255$	−34.2254	−2.14328
$256$	0	0
$257$	18.1466	1.13196	0.565978	−	0.824420i	$-0.308499\pi$
$257$	18.1466	1.13196	0.565978	+	0.824420i	$0.308499\pi$
$258$	0	0
$259$	23.7706	1.47704
$260$	0	0
$261$	16.8737	1.04446
$262$	0	0
$263$	−10.0706	−0.620979	−0.310490	−	0.950577i	$-0.600493\pi$
$263$	−10.0706	−0.620979	−0.310490	+	0.950577i	$0.600493\pi$
$264$	0	0
$265$	−3.98389	−0.244728
$266$	0	0
$267$	−35.7545	−2.18814
$268$	0	0
$269$	−23.1164	−1.40943	−0.704717	−	0.709489i	$-0.748926\pi$
$269$	−23.1164	−1.40943	−0.704717	+	0.709489i	$0.748926\pi$
$270$	0	0
$271$	−2.46093	−0.149491	−0.0747454	−	0.997203i	$-0.523814\pi$
$271$	−2.46093	−0.149491	−0.0747454	+	0.997203i	$0.523814\pi$
$272$	0	0
$273$	30.8905	1.86958
$274$	0	0
$275$	−22.2117	−1.33942
$276$	0	0
$277$	−5.34328	−0.321046	−0.160523	−	0.987032i	$-0.551318\pi$
$277$	−5.34328	−0.321046	−0.160523	+	0.987032i	$0.551318\pi$
$278$	0	0
$279$	−17.4278	−1.04337
$280$	0	0
$281$	7.91721	0.472301	0.236151	−	0.971716i	$-0.424114\pi$
$281$	7.91721	0.472301	0.236151	+	0.971716i	$0.424114\pi$
$282$	0	0
$283$	1.66200	0.0987957	0.0493978	−	0.998779i	$-0.484270\pi$
$283$	1.66200	0.0987957	0.0493978	+	0.998779i	$0.484270\pi$
$284$	0	0
$285$	−12.7982	−0.758103
$286$	0	0
$287$	32.9306	1.94383
$288$	0	0
$289$	1.29990	0.0764646
$290$	0	0
$291$	−36.6290	−2.14723
$292$	0	0
$293$	−9.10191	−0.531739	−0.265870	−	0.964009i	$-0.585659\pi$
$293$	−9.10191	−0.531739	−0.265870	+	0.964009i	$0.585659\pi$
$294$	0	0
$295$	6.36217	0.370420
$296$	0	0
$297$	−12.4042	−0.719764
$298$	0	0
$299$	5.63936	0.326133
$300$	0	0
$301$	−1.07758	−0.0621107
$302$	0	0
$303$	0.0310504	0.00178380
$304$	0	0
$305$	30.4485	1.74348
$306$	0	0
$307$	−0.0993981	−0.00567295	−0.00283647	−	0.999996i	$-0.500903\pi$
$307$	−0.0993981	−0.00567295	−0.00283647	+	0.999996i	$0.500903\pi$
$308$	0	0
$309$	−47.0943	−2.67910
$310$	0	0
$311$	12.1343	0.688074	0.344037	−	0.938956i	$-0.388205\pi$
$311$	12.1343	0.688074	0.344037	+	0.938956i	$0.388205\pi$
$312$	0	0
$313$	30.9429	1.74899	0.874497	−	0.485031i	$-0.161192\pi$
$313$	30.9429	1.74899	0.874497	+	0.485031i	$0.161192\pi$
$314$	0	0
$315$	60.4840	3.40789
$316$	0	0
$317$	−26.9334	−1.51273	−0.756365	−	0.654150i	$-0.773027\pi$
$317$	−26.9334	−1.51273	−0.756365	+	0.654150i	$0.773027\pi$
$318$	0	0
$319$	−22.4790	−1.25858
$320$	0	0
$321$	−32.6362	−1.82158
$322$	0	0
$323$	6.84305	0.380758
$324$	0	0
$325$	−9.86379	−0.547145
$326$	0	0
$327$	32.3305	1.78788
$328$	0	0
$329$	−59.9776	−3.30667
$330$	0	0
$331$	−25.8294	−1.41971	−0.709855	−	0.704348i	$-0.751239\pi$
$331$	−25.8294	−1.41971	−0.709855	+	0.704348i	$0.751239\pi$
$332$	0	0
$333$	−18.2616	−1.00073
$334$	0	0
$335$	38.0017	2.07626
$336$	0	0
$337$	−3.84238	−0.209308	−0.104654	−	0.994509i	$-0.533373\pi$
$337$	−3.84238	−0.209308	−0.104654	+	0.994509i	$0.533373\pi$
$338$	0	0
$339$	−30.4798	−1.65544
$340$	0	0
$341$	23.2171	1.25728
$342$	0	0
$343$	−60.3226	−3.25712
$344$	0	0
$345$	19.5209	1.05097
$346$	0	0
$347$	−6.42446	−0.344883	−0.172442	−	0.985020i	$-0.555166\pi$
$347$	−6.42446	−0.344883	−0.172442	+	0.985020i	$0.555166\pi$
$348$	0	0
$349$	31.4502	1.68349	0.841745	−	0.539876i	$-0.181529\pi$
$349$	31.4502	1.68349	0.841745	+	0.539876i	$0.181529\pi$
$350$	0	0
$351$	−5.50845	−0.294019
$352$	0	0
$353$	9.31611	0.495847	0.247923	−	0.968780i	$-0.420252\pi$
$353$	9.31611	0.495847	0.247923	+	0.968780i	$0.420252\pi$
$354$	0	0
$355$	−32.2903	−1.71379
$356$	0	0
$357$	−57.1734	−3.02593
$358$	0	0
$359$	28.6321	1.51115	0.755573	−	0.655064i	$-0.227358\pi$
$359$	28.6321	1.51115	0.755573	+	0.655064i	$0.227358\pi$
$360$	0	0
$361$	−16.4411	−0.865322
$362$	0	0
$363$	42.2824	2.21925
$364$	0	0
$365$	−47.8187	−2.50294
$366$	0	0
$367$	1.73752	0.0906978	0.0453489	−	0.998971i	$-0.485560\pi$
$367$	1.73752	0.0906978	0.0453489	+	0.998971i	$0.485560\pi$
$368$	0	0
$369$	−25.2986	−1.31699
$370$	0	0
$371$	−6.65506	−0.345513
$372$	0	0
$373$	12.0346	0.623128	0.311564	−	0.950225i	$-0.399147\pi$
$373$	12.0346	0.623128	0.311564	+	0.950225i	$0.399147\pi$
$374$	0	0
$375$	5.85925	0.302570
$376$	0	0
$377$	−9.98249	−0.514124
$378$	0	0
$379$	−7.85505	−0.403487	−0.201743	−	0.979438i	$-0.564661\pi$
$379$	−7.85505	−0.403487	−0.201743	+	0.979438i	$0.564661\pi$
$380$	0	0
$381$	−27.3067	−1.39896
$382$	0	0
$383$	20.1447	1.02934	0.514672	−	0.857387i	$-0.327914\pi$
$383$	20.1447	1.02934	0.514672	+	0.857387i	$0.327914\pi$
$384$	0	0
$385$	−80.5764	−4.10655
$386$	0	0
$387$	0.827840	0.0420815
$388$	0	0
$389$	1.95858	0.0993042	0.0496521	−	0.998767i	$-0.484189\pi$
$389$	1.95858	0.0993042	0.0496521	+	0.998767i	$0.484189\pi$
$390$	0	0
$391$	−10.4376	−0.527850
$392$	0	0
$393$	−16.6647	−0.840624
$394$	0	0
$395$	5.75707	0.289670
$396$	0	0
$397$	2.81024	0.141042	0.0705210	−	0.997510i	$-0.477534\pi$
$397$	2.81024	0.141042	0.0705210	+	0.997510i	$0.477534\pi$
$398$	0	0
$399$	−21.3794	−1.07031
$400$	0	0
$401$	−14.0379	−0.701018	−0.350509	−	0.936559i	$-0.613991\pi$
$401$	−14.0379	−0.701018	−0.350509	+	0.936559i	$0.613991\pi$
$402$	0	0
$403$	10.3103	0.513591
$404$	0	0
$405$	16.6129	0.825501
$406$	0	0
$407$	24.3279	1.20589
$408$	0	0
$409$	17.9272	0.886443	0.443222	−	0.896412i	$-0.353835\pi$
$409$	17.9272	0.886443	0.443222	+	0.896412i	$0.353835\pi$
$410$	0	0
$411$	47.2380	2.33008
$412$	0	0
$413$	10.6280	0.522968
$414$	0	0
$415$	−0.223149	−0.0109539
$416$	0	0
$417$	−41.4095	−2.02783
$418$	0	0
$419$	−34.5292	−1.68686	−0.843431	−	0.537238i	$-0.819468\pi$
$419$	−34.5292	−1.68686	−0.843431	+	0.537238i	$0.819468\pi$
$420$	0	0
$421$	−12.9257	−0.629960	−0.314980	−	0.949098i	$-0.601998\pi$
$421$	−12.9257	−0.629960	−0.314980	+	0.949098i	$0.601998\pi$
$422$	0	0
$423$	46.0772	2.24035
$424$	0	0
$425$	18.2563	0.885562
$426$	0	0
$427$	50.8640	2.46148
$428$	0	0
$429$	31.6147	1.52637
$430$	0	0
$431$	−14.5498	−0.700840	−0.350420	−	0.936593i	$-0.613961\pi$
$431$	−14.5498	−0.700840	−0.350420	+	0.936593i	$0.613961\pi$
$432$	0	0
$433$	−26.2852	−1.26319	−0.631593	−	0.775300i	$-0.717599\pi$
$433$	−26.2852	−1.26319	−0.631593	+	0.775300i	$0.717599\pi$
$434$	0	0
$435$	−34.5548	−1.65678
$436$	0	0
$437$	−3.90302	−0.186707
$438$	0	0
$439$	12.6733	0.604863	0.302432	−	0.953171i	$-0.402202\pi$
$439$	12.6733	0.604863	0.302432	+	0.953171i	$0.402202\pi$
$440$	0	0
$441$	73.6902	3.50906
$442$	0	0
$443$	−9.61838	−0.456983	−0.228491	−	0.973546i	$-0.573379\pi$
$443$	−9.61838	−0.456983	−0.228491	+	0.973546i	$0.573379\pi$
$444$	0	0
$445$	41.4167	1.96334
$446$	0	0
$447$	53.9102	2.54986
$448$	0	0
$449$	19.1273	0.902673	0.451337	−	0.892354i	$-0.350947\pi$
$449$	19.1273	0.902673	0.451337	+	0.892354i	$0.350947\pi$
$450$	0	0
$451$	33.7026	1.58699
$452$	0	0
$453$	−9.20085	−0.432293
$454$	0	0
$455$	−35.7824	−1.67750
$456$	0	0
$457$	37.5083	1.75457	0.877283	−	0.479974i	$-0.159354\pi$
$457$	37.5083	1.75457	0.877283	+	0.479974i	$0.159354\pi$
$458$	0	0
$459$	10.1953	0.475875
$460$	0	0
$461$	17.1867	0.800466	0.400233	−	0.916413i	$-0.368929\pi$
$461$	17.1867	0.800466	0.400233	+	0.916413i	$0.368929\pi$
$462$	0	0
$463$	−22.8793	−1.06329	−0.531646	−	0.846967i	$-0.678426\pi$
$463$	−22.8793	−1.06329	−0.531646	+	0.846967i	$0.678426\pi$
$464$	0	0
$465$	35.6895	1.65506
$466$	0	0
$467$	−33.2033	−1.53647	−0.768234	−	0.640169i	$-0.778864\pi$
$467$	−33.2033	−1.53647	−0.768234	+	0.640169i	$0.778864\pi$
$468$	0	0
$469$	63.4816	2.93131
$470$	0	0
$471$	−13.3485	−0.615065
$472$	0	0
$473$	−1.10284	−0.0507088
$474$	0	0
$475$	6.82676	0.313233
$476$	0	0
$477$	5.11268	0.234094
$478$	0	0
$479$	2.08415	0.0952273	0.0476137	−	0.998866i	$-0.484838\pi$
$479$	2.08415	0.0952273	0.0476137	+	0.998866i	$0.484838\pi$
$480$	0	0
$481$	10.8036	0.492600
$482$	0	0
$483$	32.6095	1.48378
$484$	0	0
$485$	42.4297	1.92663
$486$	0	0
$487$	−4.35539	−0.197361	−0.0986807	−	0.995119i	$-0.531462\pi$
$487$	−4.35539	−0.197361	−0.0986807	+	0.995119i	$0.531462\pi$
$488$	0	0
$489$	−18.2673	−0.826077
$490$	0	0
$491$	−7.49495	−0.338242	−0.169121	−	0.985595i	$-0.554093\pi$
$491$	−7.49495	−0.338242	−0.169121	+	0.985595i	$0.554093\pi$
$492$	0	0
$493$	18.4760	0.832118
$494$	0	0
$495$	61.9020	2.78229
$496$	0	0
$497$	−53.9408	−2.41957
$498$	0	0
$499$	−18.1591	−0.812913	−0.406456	−	0.913670i	$-0.633236\pi$
$499$	−18.1591	−0.812913	−0.406456	+	0.913670i	$0.633236\pi$
$500$	0	0
$501$	2.62809	0.117414
$502$	0	0
$503$	−9.07749	−0.404745	−0.202373	−	0.979309i	$-0.564865\pi$
$503$	−9.07749	−0.404745	−0.202373	+	0.979309i	$0.564865\pi$
$504$	0	0
$505$	−0.0359676	−0.00160054
$506$	0	0
$507$	−20.1257	−0.893814
$508$	0	0
$509$	−2.55113	−0.113077	−0.0565384	−	0.998400i	$-0.518006\pi$
$509$	−2.55113	−0.113077	−0.0565384	+	0.998400i	$0.518006\pi$
$510$	0	0
$511$	−79.8807	−3.53371
$512$	0	0
$513$	3.81242	0.168322
$514$	0	0
$515$	54.5523	2.40386
$516$	0	0
$517$	−61.3838	−2.69966
$518$	0	0
$519$	0.0558406	0.00245113
$520$	0	0
$521$	4.95834	0.217229	0.108614	−	0.994084i	$-0.465359\pi$
$521$	4.95834	0.217229	0.108614	+	0.994084i	$0.465359\pi$
$522$	0	0
$523$	−27.3959	−1.19794	−0.598969	−	0.800772i	$-0.704423\pi$
$523$	−27.3959	−1.19794	−0.598969	+	0.800772i	$0.704423\pi$
$524$	0	0
$525$	−57.0372	−2.48931
$526$	0	0
$527$	−19.0827	−0.831255
$528$	0	0
$529$	−17.0468	−0.741166
$530$	0	0
$531$	−8.16483	−0.354323
$532$	0	0
$533$	14.9667	0.648278
$534$	0	0
$535$	37.8046	1.63444
$536$	0	0
$537$	15.3178	0.661012
$538$	0	0
$539$	−98.1696	−4.22846
$540$	0	0
$541$	5.78439	0.248690	0.124345	−	0.992239i	$-0.460317\pi$
$541$	5.78439	0.248690	0.124345	+	0.992239i	$0.460317\pi$
$542$	0	0
$543$	66.1156	2.83729
$544$	0	0
$545$	−37.4504	−1.60420
$546$	0	0
$547$	−35.0915	−1.50040	−0.750202	−	0.661209i	$-0.770044\pi$
$547$	−35.0915	−1.50040	−0.750202	+	0.661209i	$0.770044\pi$
$548$	0	0
$549$	−39.0758	−1.66771
$550$	0	0
$551$	6.90891	0.294329
$552$	0	0
$553$	9.61714	0.408962
$554$	0	0
$555$	37.3970	1.58741
$556$	0	0
$557$	30.1463	1.27734	0.638671	−	0.769480i	$-0.279485\pi$
$557$	30.1463	1.27734	0.638671	+	0.769480i	$0.279485\pi$
$558$	0	0
$559$	−0.489751	−0.0207142
$560$	0	0
$561$	−58.5137	−2.47045
$562$	0	0
$563$	11.3311	0.477549	0.238775	−	0.971075i	$-0.423254\pi$
$563$	11.3311	0.477549	0.238775	+	0.971075i	$0.423254\pi$
$564$	0	0
$565$	35.3067	1.48536
$566$	0	0
$567$	27.7517	1.16546
$568$	0	0
$569$	29.8478	1.25129	0.625643	−	0.780109i	$-0.284837\pi$
$569$	29.8478	1.25129	0.625643	+	0.780109i	$0.284837\pi$
$570$	0	0
$571$	30.5791	1.27970	0.639848	−	0.768501i	$-0.278997\pi$
$571$	30.5791	1.27970	0.639848	+	0.768501i	$0.278997\pi$
$572$	0	0
$573$	50.5346	2.11111
$574$	0	0
$575$	−10.4127	−0.434240
$576$	0	0
$577$	33.7066	1.40322	0.701611	−	0.712560i	$-0.252464\pi$
$577$	33.7066	1.40322	0.701611	+	0.712560i	$0.252464\pi$
$578$	0	0
$579$	−2.08879	−0.0868073
$580$	0	0
$581$	−0.372768	−0.0154650
$582$	0	0
$583$	−6.81108	−0.282086
$584$	0	0
$585$	27.4895	1.13655
$586$	0	0
$587$	−29.1916	−1.20487	−0.602434	−	0.798169i	$-0.705802\pi$
$587$	−29.1916	−1.20487	−0.602434	+	0.798169i	$0.705802\pi$
$588$	0	0
$589$	−7.13577	−0.294024
$590$	0	0
$591$	−37.4433	−1.54021
$592$	0	0
$593$	14.7720	0.606612	0.303306	−	0.952893i	$-0.401910\pi$
$593$	14.7720	0.606612	0.303306	+	0.952893i	$0.401910\pi$
$594$	0	0
$595$	66.2275	2.71506
$596$	0	0
$597$	−22.9821	−0.940596
$598$	0	0
$599$	5.68424	0.232252	0.116126	−	0.993234i	$-0.462952\pi$
$599$	5.68424	0.232252	0.116126	+	0.993234i	$0.462952\pi$
$600$	0	0
$601$	−27.7049	−1.13011	−0.565054	−	0.825054i	$-0.691145\pi$
$601$	−27.7049	−1.13011	−0.565054	+	0.825054i	$0.691145\pi$
$602$	0	0
$603$	−48.7691	−1.98603
$604$	0	0
$605$	−48.9784	−1.99125
$606$	0	0
$607$	31.6971	1.28655	0.643273	−	0.765637i	$-0.277576\pi$
$607$	31.6971	1.28655	0.643273	+	0.765637i	$0.277576\pi$
$608$	0	0
$609$	−57.7236	−2.33908
$610$	0	0
$611$	−27.2593	−1.10279
$612$	0	0
$613$	−27.9921	−1.13059	−0.565295	−	0.824889i	$-0.691237\pi$
$613$	−27.9921	−1.13059	−0.565295	+	0.824889i	$0.691237\pi$
$614$	0	0
$615$	51.8078	2.08909
$616$	0	0
$617$	2.46431	0.0992094	0.0496047	−	0.998769i	$-0.484204\pi$
$617$	2.46431	0.0992094	0.0496047	+	0.998769i	$0.484204\pi$
$618$	0	0
$619$	11.3778	0.457311	0.228656	−	0.973507i	$-0.426567\pi$
$619$	11.3778	0.457311	0.228656	+	0.973507i	$0.426567\pi$
$620$	0	0
$621$	−5.81499	−0.233348
$622$	0	0
$623$	69.1863	2.77189
$624$	0	0
$625$	−28.1254	−1.12502
$626$	0	0
$627$	−21.8806	−0.873827
$628$	0	0
$629$	−19.9957	−0.797280
$630$	0	0
$631$	0.661698	0.0263418	0.0131709	−	0.999913i	$-0.495807\pi$
$631$	0.661698	0.0263418	0.0131709	+	0.999913i	$0.495807\pi$
$632$	0	0
$633$	33.2206	1.32040
$634$	0	0
$635$	31.6311	1.25524
$636$	0	0
$637$	−43.5952	−1.72730
$638$	0	0
$639$	41.4395	1.63932
$640$	0	0
$641$	32.0433	1.26564	0.632818	−	0.774301i	$-0.281898\pi$
$641$	32.0433	1.26564	0.632818	+	0.774301i	$0.281898\pi$
$642$	0	0
$643$	−43.1882	−1.70317	−0.851587	−	0.524213i	$-0.824360\pi$
$643$	−43.1882	−1.70317	−0.851587	+	0.524213i	$0.824360\pi$
$644$	0	0
$645$	−1.69529	−0.0667521
$646$	0	0
$647$	−1.66679	−0.0655283	−0.0327642	−	0.999463i	$-0.510431\pi$
$647$	−1.66679	−0.0655283	−0.0327642	+	0.999463i	$0.510431\pi$
$648$	0	0
$649$	10.8771	0.426965
$650$	0	0
$651$	59.6190	2.33665
$652$	0	0
$653$	−7.25965	−0.284092	−0.142046	−	0.989860i	$-0.545368\pi$
$653$	−7.25965	−0.284092	−0.142046	+	0.989860i	$0.545368\pi$
$654$	0	0
$655$	19.3038	0.754262
$656$	0	0
$657$	61.3676	2.39418
$658$	0	0
$659$	13.2040	0.514353	0.257177	−	0.966364i	$-0.417208\pi$
$659$	13.2040	0.514353	0.257177	+	0.966364i	$0.417208\pi$
$660$	0	0
$661$	−0.0792928	−0.00308413	−0.00154207	−	0.999999i	$-0.500491\pi$
$661$	−0.0792928	−0.00308413	−0.00154207	+	0.999999i	$0.500491\pi$
$662$	0	0
$663$	−25.9848	−1.00917
$664$	0	0
$665$	24.7651	0.960349
$666$	0	0
$667$	−10.5380	−0.408033
$668$	0	0
$669$	−19.6068	−0.758042
$670$	0	0
$671$	52.0565	2.00962
$672$	0	0
$673$	−8.33681	−0.321360	−0.160680	−	0.987007i	$-0.551369\pi$
$673$	−8.33681	−0.321360	−0.160680	+	0.987007i	$0.551369\pi$
$674$	0	0
$675$	10.1710	0.391482
$676$	0	0
$677$	11.8503	0.455444	0.227722	−	0.973726i	$-0.426872\pi$
$677$	11.8503	0.455444	0.227722	+	0.973726i	$0.426872\pi$
$678$	0	0
$679$	70.8785	2.72007
$680$	0	0
$681$	68.8739	2.63925
$682$	0	0
$683$	−12.5601	−0.480597	−0.240298	−	0.970699i	$-0.577245\pi$
$683$	−12.5601	−0.480597	−0.240298	+	0.970699i	$0.577245\pi$
$684$	0	0
$685$	−54.7188	−2.09070
$686$	0	0
$687$	−63.0505	−2.40553
$688$	0	0
$689$	−3.02467	−0.115231
$690$	0	0
$691$	27.0610	1.02945	0.514724	−	0.857356i	$-0.327894\pi$
$691$	27.0610	1.02945	0.514724	+	0.857356i	$0.327894\pi$
$692$	0	0
$693$	103.407	3.92810
$694$	0	0
$695$	47.9672	1.81950
$696$	0	0
$697$	−27.7009	−1.04925
$698$	0	0
$699$	3.35146	0.126764
$700$	0	0
$701$	−45.9613	−1.73594	−0.867968	−	0.496620i	$-0.834574\pi$
$701$	−45.9613	−1.73594	−0.867968	+	0.496620i	$0.834574\pi$
$702$	0	0
$703$	−7.47717	−0.282007
$704$	0	0
$705$	−94.3593	−3.55378
$706$	0	0
$707$	−0.0600837	−0.00225968
$708$	0	0
$709$	−8.25749	−0.310116	−0.155058	−	0.987905i	$-0.549557\pi$
$709$	−8.25749	−0.310116	−0.155058	+	0.987905i	$0.549557\pi$
$710$	0	0
$711$	−7.38827	−0.277082
$712$	0	0
$713$	10.8840	0.407610
$714$	0	0
$715$	−36.6213	−1.36956
$716$	0	0
$717$	−33.9097	−1.26638
$718$	0	0
$719$	14.6033	0.544611	0.272306	−	0.962211i	$-0.412214\pi$
$719$	14.6033	0.544611	0.272306	+	0.962211i	$0.412214\pi$
$720$	0	0
$721$	91.1292	3.39383
$722$	0	0
$723$	1.48272	0.0551429
$724$	0	0
$725$	18.4320	0.684548
$726$	0	0
$727$	−43.0804	−1.59776	−0.798881	−	0.601489i	$-0.794574\pi$
$727$	−43.0804	−1.59776	−0.798881	+	0.601489i	$0.794574\pi$
$728$	0	0
$729$	−40.1104	−1.48557
$730$	0	0
$731$	0.906451	0.0335263
$732$	0	0
$733$	24.5740	0.907663	0.453831	−	0.891088i	$-0.350057\pi$
$733$	24.5740	0.907663	0.453831	+	0.891088i	$0.350057\pi$
$734$	0	0
$735$	−150.907	−5.56627
$736$	0	0
$737$	64.9699	2.39320
$738$	0	0
$739$	−2.87480	−0.105751	−0.0528757	−	0.998601i	$-0.516839\pi$
$739$	−2.87480	−0.105751	−0.0528757	+	0.998601i	$0.516839\pi$
$740$	0	0
$741$	−9.71674	−0.356953
$742$	0	0
$743$	−34.3105	−1.25873	−0.629365	−	0.777110i	$-0.716685\pi$
$743$	−34.3105	−1.25873	−0.629365	+	0.777110i	$0.716685\pi$
$744$	0	0
$745$	−62.4476	−2.28790
$746$	0	0
$747$	0.286375	0.0104779
$748$	0	0
$749$	63.1523	2.30754
$750$	0	0
$751$	19.8641	0.724851	0.362426	−	0.932013i	$-0.381949\pi$
$751$	19.8641	0.724851	0.362426	+	0.932013i	$0.381949\pi$
$752$	0	0
$753$	26.0963	0.951003
$754$	0	0
$755$	10.6579	0.387882
$756$	0	0
$757$	−9.00475	−0.327283	−0.163642	−	0.986520i	$-0.552324\pi$
$757$	−9.00475	−0.327283	−0.163642	+	0.986520i	$0.552324\pi$
$758$	0	0
$759$	33.3740	1.21140
$760$	0	0
$761$	16.5583	0.600237	0.300119	−	0.953902i	$-0.402974\pi$
$761$	16.5583	0.600237	0.300119	+	0.953902i	$0.402974\pi$
$762$	0	0
$763$	−62.5607	−2.26485
$764$	0	0
$765$	−50.8787	−1.83952
$766$	0	0
$767$	4.83032	0.174413
$768$	0	0
$769$	34.2931	1.23664	0.618320	−	0.785927i	$-0.287814\pi$
$769$	34.2931	1.23664	0.618320	+	0.785927i	$0.287814\pi$
$770$	0	0
$771$	47.6910	1.71755
$772$	0	0
$773$	−28.3661	−1.02026	−0.510128	−	0.860099i	$-0.670402\pi$
$773$	−28.3661	−1.02026	−0.510128	+	0.860099i	$0.670402\pi$
$774$	0	0
$775$	−19.0372	−0.683838
$776$	0	0
$777$	62.4714	2.24115
$778$	0	0
$779$	−10.3585	−0.371131
$780$	0	0
$781$	−55.2054	−1.97540
$782$	0	0
$783$	10.2934	0.367856
$784$	0	0
$785$	15.4624	0.551876
$786$	0	0
$787$	26.9779	0.961658	0.480829	−	0.876814i	$-0.340336\pi$
$787$	26.9779	0.961658	0.480829	+	0.876814i	$0.340336\pi$
$788$	0	0
$789$	−26.4664	−0.942229
$790$	0	0
$791$	58.9796	2.09707
$792$	0	0
$793$	23.1172	0.820918
$794$	0	0
$795$	−10.4700	−0.371333
$796$	0	0
$797$	−6.97928	−0.247219	−0.123609	−	0.992331i	$-0.539447\pi$
$797$	−6.97928	−0.247219	−0.123609	+	0.992331i	$0.539447\pi$
$798$	0	0
$799$	50.4527	1.78489
$800$	0	0
$801$	−53.1517	−1.87802
$802$	0	0
$803$	−81.7534	−2.88502
$804$	0	0
$805$	−37.7737	−1.33135
$806$	0	0
$807$	−60.7520	−2.13857
$808$	0	0
$809$	31.7088	1.11482	0.557410	−	0.830237i	$-0.311795\pi$
$809$	31.7088	1.11482	0.557410	+	0.830237i	$0.311795\pi$
$810$	0	0
$811$	22.6829	0.796503	0.398252	−	0.917276i	$-0.369617\pi$
$811$	22.6829	0.796503	0.398252	+	0.917276i	$0.369617\pi$
$812$	0	0
$813$	−6.46754	−0.226827
$814$	0	0
$815$	21.1602	0.741210
$816$	0	0
$817$	0.338958	0.0118586
$818$	0	0
$819$	45.9210	1.60461
$820$	0	0
$821$	−53.9761	−1.88378	−0.941889	−	0.335925i	$-0.890951\pi$
$821$	−53.9761	−1.88378	−0.941889	+	0.335925i	$0.890951\pi$
$822$	0	0
$823$	45.9530	1.60182	0.800911	−	0.598784i	$-0.204349\pi$
$823$	45.9530	1.60182	0.800911	+	0.598784i	$0.204349\pi$
$824$	0	0
$825$	−58.3744	−2.03234
$826$	0	0
$827$	2.83746	0.0986681	0.0493340	−	0.998782i	$-0.484290\pi$
$827$	2.83746	0.0986681	0.0493340	+	0.998782i	$0.484290\pi$
$828$	0	0
$829$	37.6293	1.30692	0.653460	−	0.756961i	$-0.273317\pi$
$829$	37.6293	1.30692	0.653460	+	0.756961i	$0.273317\pi$
$830$	0	0
$831$	−14.0426	−0.487132
$832$	0	0
$833$	80.6877	2.79566
$834$	0	0
$835$	−3.04428	−0.105352
$836$	0	0
$837$	−10.6314	−0.367474
$838$	0	0
$839$	−9.62932	−0.332441	−0.166221	−	0.986089i	$-0.553156\pi$
$839$	−9.62932	−0.332441	−0.166221	+	0.986089i	$0.553156\pi$
$840$	0	0
$841$	−10.3462	−0.356765
$842$	0	0
$843$	20.8071	0.716636
$844$	0	0
$845$	23.3129	0.801988
$846$	0	0
$847$	−81.8180	−2.81130
$848$	0	0
$849$	4.36788	0.149905
$850$	0	0
$851$	11.4048	0.390950
$852$	0	0
$853$	−44.1581	−1.51194	−0.755972	−	0.654604i	$-0.772836\pi$
$853$	−44.1581	−1.51194	−0.755972	+	0.654604i	$0.772836\pi$
$854$	0	0
$855$	−19.0255	−0.650660
$856$	0	0
$857$	−46.8099	−1.59899	−0.799497	−	0.600669i	$-0.794901\pi$
$857$	−46.8099	−1.59899	−0.799497	+	0.600669i	$0.794901\pi$
$858$	0	0
$859$	−5.86869	−0.200237	−0.100119	−	0.994976i	$-0.531922\pi$
$859$	−5.86869	−0.200237	−0.100119	+	0.994976i	$0.531922\pi$
$860$	0	0
$861$	86.5445	2.94943
$862$	0	0
$863$	−36.0240	−1.22627	−0.613135	−	0.789978i	$-0.710092\pi$
$863$	−36.0240	−1.22627	−0.613135	+	0.789978i	$0.710092\pi$
$864$	0	0
$865$	−0.0646838	−0.00219931
$866$	0	0
$867$	3.41625	0.116022
$868$	0	0
$869$	9.84261	0.333888
$870$	0	0
$871$	28.8518	0.977607
$872$	0	0
$873$	−54.4517	−1.84291
$874$	0	0
$875$	−11.3379	−0.383290
$876$	0	0
$877$	−1.82003	−0.0614579	−0.0307290	−	0.999528i	$-0.509783\pi$
$877$	−1.82003	−0.0614579	−0.0307290	+	0.999528i	$0.509783\pi$
$878$	0	0
$879$	−23.9206	−0.806823
$880$	0	0
$881$	−6.46872	−0.217937	−0.108968	−	0.994045i	$-0.534755\pi$
$881$	−6.46872	−0.217937	−0.108968	+	0.994045i	$0.534755\pi$
$882$	0	0
$883$	−0.603541	−0.0203108	−0.0101554	−	0.999948i	$-0.503233\pi$
$883$	−0.603541	−0.0203108	−0.0101554	+	0.999948i	$0.503233\pi$
$884$	0	0
$885$	16.7203	0.562048
$886$	0	0
$887$	25.1649	0.844956	0.422478	−	0.906373i	$-0.361160\pi$
$887$	25.1649	0.844956	0.422478	+	0.906373i	$0.361160\pi$
$888$	0	0
$889$	52.8395	1.77218
$890$	0	0
$891$	28.4023	0.951514
$892$	0	0
$893$	18.8663	0.631335
$894$	0	0
$895$	−17.7436	−0.593103
$896$	0	0
$897$	14.8207	0.494850
$898$	0	0
$899$	−19.2663	−0.642568
$900$	0	0
$901$	5.59818	0.186502
$902$	0	0
$903$	−2.83197	−0.0942422
$904$	0	0
$905$	−76.5859	−2.54580
$906$	0	0
$907$	−0.167564	−0.00556388	−0.00278194	−	0.999996i	$-0.500886\pi$
$907$	−0.167564	−0.00556388	−0.00278194	+	0.999996i	$0.500886\pi$
$908$	0	0
$909$	0.0461587	0.00153099
$910$	0	0
$911$	−29.5214	−0.978087	−0.489043	−	0.872259i	$-0.662654\pi$
$911$	−29.5214	−0.978087	−0.489043	+	0.872259i	$0.662654\pi$
$912$	0	0
$913$	−0.381507	−0.0126260
$914$	0	0
$915$	80.0214	2.64542
$916$	0	0
$917$	32.2469	1.06488
$918$	0	0
$919$	−17.2206	−0.568055	−0.284028	−	0.958816i	$-0.591671\pi$
$919$	−17.2206	−0.568055	−0.284028	+	0.958816i	$0.591671\pi$
$920$	0	0
$921$	−0.261227	−0.00860772
$922$	0	0
$923$	−24.5156	−0.806941
$924$	0	0
$925$	−19.9481	−0.655888
$926$	0	0
$927$	−70.0091	−2.29940
$928$	0	0
$929$	6.90753	0.226629	0.113314	−	0.993559i	$-0.463853\pi$
$929$	6.90753	0.226629	0.113314	+	0.993559i	$0.463853\pi$
$930$	0	0
$931$	30.1723	0.988858
$932$	0	0
$933$	31.8901	1.04403
$934$	0	0
$935$	67.7802	2.21665
$936$	0	0
$937$	−7.54846	−0.246598	−0.123299	−	0.992370i	$-0.539347\pi$
$937$	−7.54846	−0.246598	−0.123299	+	0.992370i	$0.539347\pi$
$938$	0	0
$939$	81.3206	2.65380
$940$	0	0
$941$	10.7147	0.349291	0.174645	−	0.984631i	$-0.444122\pi$
$941$	10.7147	0.349291	0.174645	+	0.984631i	$0.444122\pi$
$942$	0	0
$943$	15.7996	0.514504
$944$	0	0
$945$	36.8968	1.20025
$946$	0	0
$947$	26.7102	0.867963	0.433982	−	0.900922i	$-0.357108\pi$
$947$	26.7102	0.867963	0.433982	+	0.900922i	$0.357108\pi$
$948$	0	0
$949$	−36.3051	−1.17851
$950$	0	0
$951$	−70.7833	−2.29531
$952$	0	0
$953$	19.5038	0.631790	0.315895	−	0.948794i	$-0.397695\pi$
$953$	19.5038	0.631790	0.315895	+	0.948794i	$0.397695\pi$
$954$	0	0
$955$	−58.5374	−1.89423
$956$	0	0
$957$	−59.0769	−1.90968
$958$	0	0
$959$	−91.4073	−2.95169
$960$	0	0
$961$	−11.1010	−0.358098
$962$	0	0
$963$	−48.5162	−1.56341
$964$	0	0
$965$	2.41958	0.0778891
$966$	0	0
$967$	34.5578	1.11130	0.555652	−	0.831415i	$-0.312469\pi$
$967$	34.5578	1.11130	0.555652	+	0.831415i	$0.312469\pi$
$968$	0	0
$969$	17.9842	0.577734
$970$	0	0
$971$	60.5841	1.94424	0.972118	−	0.234492i	$-0.0753426\pi$
$971$	60.5841	1.94424	0.972118	+	0.234492i	$0.0753426\pi$
$972$	0	0
$973$	80.1289	2.56881
$974$	0	0
$975$	−25.9229	−0.830198
$976$	0	0
$977$	8.77067	0.280599	0.140299	−	0.990109i	$-0.455194\pi$
$977$	8.77067	0.280599	0.140299	+	0.990109i	$0.455194\pi$
$978$	0	0
$979$	70.8083	2.26304
$980$	0	0
$981$	48.0616	1.53449
$982$	0	0
$983$	−24.5528	−0.783114	−0.391557	−	0.920154i	$-0.628063\pi$
$983$	−24.5528	−0.783114	−0.391557	+	0.920154i	$0.628063\pi$
$984$	0	0
$985$	43.3729	1.38198
$986$	0	0
$987$	−157.627	−5.01731
$988$	0	0
$989$	−0.517005	−0.0164398
$990$	0	0
$991$	−7.90644	−0.251156	−0.125578	−	0.992084i	$-0.540079\pi$
$991$	−7.90644	−0.251156	−0.125578	+	0.992084i	$0.540079\pi$
$992$	0	0
$993$	−67.8818	−2.15416
$994$	0	0
$995$	26.6217	0.843963
$996$	0	0
$997$	26.5355	0.840387	0.420193	−	0.907435i	$-0.361962\pi$
$997$	26.5355	0.840387	0.420193	+	0.907435i	$0.361962\pi$
$998$	0	0
$999$	−11.1400	−0.352455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.c.1.7	✓	7
3.2	odd	2		6012.2.a.g.1.7		7
4.3	odd	2		2672.2.a.k.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.7	✓	7	1.1	even	1	trivial
2672.2.a.k.1.1		7	4.3	odd	2
6012.2.a.g.1.7		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)

\(f(q)\)	\(=\)	\(q+2.62809 q^{3} -3.04428 q^{5} -5.08545 q^{7} +3.90685 q^{9} +O(q^{10})\)	\(q+2.62809 q^{3} -3.04428 q^{5} -5.08545 q^{7} +3.90685 q^{9} -5.20467 q^{11} -2.31129 q^{13} -8.00064 q^{15} +4.27784 q^{17} +1.59965 q^{19} -13.3650 q^{21} -2.43992 q^{23} +4.26765 q^{25} +2.38328 q^{27} +4.31901 q^{29} -4.46082 q^{31} -13.6783 q^{33} +15.4815 q^{35} -4.67425 q^{37} -6.07428 q^{39} -6.47545 q^{41} +0.211895 q^{43} -11.8935 q^{45} +11.7940 q^{47} +18.8618 q^{49} +11.2425 q^{51} +1.30865 q^{53} +15.8445 q^{55} +4.20403 q^{57} -2.08988 q^{59} -10.0019 q^{61} -19.8681 q^{63} +7.03623 q^{65} -12.4830 q^{67} -6.41231 q^{69} +10.6069 q^{71} +15.7077 q^{73} +11.2158 q^{75} +26.4681 q^{77} -1.89111 q^{79} -5.45708 q^{81} +0.0733009 q^{83} -13.0229 q^{85} +11.3507 q^{87} -13.6047 q^{89} +11.7540 q^{91} -11.7234 q^{93} -4.86979 q^{95} -13.9375 q^{97} -20.3339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) \) 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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