LMFDB - Embedded newform 2672.2.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2672,2,Mod(1,2672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2672 = 2^{4} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2672.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:	$21.3360274201$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 $ x^9 - x^8 - 13x^7 + 8x^6 + 56x^5 - 15x^4 - 81x^3 + 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1336)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.63574$ of defining polynomial
Character	$\chi$	$=$	2672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63574 q^{3} -0.530251 q^{5} +0.325283 q^{7} -0.324343 q^{9} +O(q^{10})$	$q+1.63574 q^{3} -0.530251 q^{5} +0.325283 q^{7} -0.324343 q^{9} -3.05624 q^{11} +1.58465 q^{13} -0.867355 q^{15} -2.75292 q^{17} +1.04080 q^{19} +0.532080 q^{21} -2.36742 q^{23} -4.71883 q^{25} -5.43777 q^{27} -6.32884 q^{29} -7.66901 q^{31} -4.99922 q^{33} -0.172482 q^{35} -0.416257 q^{37} +2.59208 q^{39} +5.37869 q^{41} +9.93267 q^{43} +0.171983 q^{45} +3.44772 q^{47} -6.89419 q^{49} -4.50307 q^{51} -1.87999 q^{53} +1.62057 q^{55} +1.70249 q^{57} +11.6115 q^{59} -15.3908 q^{61} -0.105503 q^{63} -0.840262 q^{65} -13.1715 q^{67} -3.87248 q^{69} +16.7791 q^{71} -7.62612 q^{73} -7.71880 q^{75} -0.994144 q^{77} -8.44815 q^{79} -7.92177 q^{81} +11.0056 q^{83} +1.45974 q^{85} -10.3524 q^{87} -5.95109 q^{89} +0.515460 q^{91} -12.5445 q^{93} -0.551886 q^{95} +4.93453 q^{97} +0.991269 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 8 q^{5} - 2 q^{7}+O(q^{10}) $ 9 * q + q^3 - 8 * q^5 - 2 * q^7	$ 9 q + q^{3} - 8 q^{5} - 2 q^{7} + 10 q^{11} - 13 q^{13} - 2 q^{15} - 8 q^{17} + q^{19} - 19 q^{21} + 3 q^{23} + 3 q^{25} + 10 q^{27} - 25 q^{29} + q^{31} - 12 q^{33} + 17 q^{35} - 35 q^{37} + 4 q^{39} - 16 q^{41} - 9 q^{43} - 24 q^{45} + q^{47} - q^{49} + 10 q^{51} - 29 q^{53} - 9 q^{55} - 17 q^{57} + 14 q^{59} - 28 q^{61} - 4 q^{63} - 31 q^{65} - 19 q^{67} - 19 q^{69} + 9 q^{71} + 7 q^{75} - 33 q^{77} + 18 q^{79} - 27 q^{81} + 13 q^{83} - 36 q^{85} - 18 q^{87} - 21 q^{89} - 20 q^{91} - 35 q^{93} + 12 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) $ 9 * q + q^3 - 8 * q^5 - 2 * q^7 + 10 * q^11 - 13 * q^13 - 2 * q^15 - 8 * q^17 + q^19 - 19 * q^21 + 3 * q^23 + 3 * q^25 + 10 * q^27 - 25 * q^29 + q^31 - 12 * q^33 + 17 * q^35 - 35 * q^37 + 4 * q^39 - 16 * q^41 - 9 * q^43 - 24 * q^45 + q^47 - q^49 + 10 * q^51 - 29 * q^53 - 9 * q^55 - 17 * q^57 + 14 * q^59 - 28 * q^61 - 4 * q^63 - 31 * q^65 - 19 * q^67 - 19 * q^69 + 9 * q^71 + 7 * q^75 - 33 * q^77 + 18 * q^79 - 27 * q^81 + 13 * q^83 - 36 * q^85 - 18 * q^87 - 21 * q^89 - 20 * q^91 - 35 * q^93 + 12 * q^95 + 2 * q^97 + 3 * 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
$3$	1.63574	0.944397	0.472199	−	0.881492i	$-0.343461\pi$
$3$	1.63574	0.944397	0.472199	+	0.881492i	$0.343461\pi$
$4$	0	0
$5$	−0.530251	−0.237135	−0.118568	−	0.992946i	$-0.537830\pi$
$5$	−0.530251	−0.237135	−0.118568	+	0.992946i	$0.537830\pi$
$6$	0	0
$7$	0.325283	0.122946	0.0614728	−	0.998109i	$-0.480420\pi$
$7$	0.325283	0.122946	0.0614728	+	0.998109i	$0.480420\pi$
$8$	0	0
$9$	−0.324343	−0.108114
$10$	0	0
$11$	−3.05624	−0.921490	−0.460745	−	0.887532i	$-0.652418\pi$
$11$	−3.05624	−0.921490	−0.460745	+	0.887532i	$0.652418\pi$
$12$	0	0
$13$	1.58465	0.439503	0.219751	−	0.975556i	$-0.429475\pi$
$13$	1.58465	0.439503	0.219751	+	0.975556i	$0.429475\pi$
$14$	0	0
$15$	−0.867355	−0.223950
$16$	0	0
$17$	−2.75292	−0.667681	−0.333840	−	0.942630i	$-0.608345\pi$
$17$	−2.75292	−0.667681	−0.333840	+	0.942630i	$0.608345\pi$
$18$	0	0
$19$	1.04080	0.238776	0.119388	−	0.992848i	$-0.461907\pi$
$19$	1.04080	0.238776	0.119388	+	0.992848i	$0.461907\pi$
$20$	0	0
$21$	0.532080	0.116109
$22$	0	0
$23$	−2.36742	−0.493640	−0.246820	−	0.969061i	$-0.579386\pi$
$23$	−2.36742	−0.493640	−0.246820	+	0.969061i	$0.579386\pi$
$24$	0	0
$25$	−4.71883	−0.943767
$26$	0	0
$27$	−5.43777	−1.04650
$28$	0	0
$29$	−6.32884	−1.17524	−0.587618	−	0.809139i	$-0.699934\pi$
$29$	−6.32884	−1.17524	−0.587618	+	0.809139i	$0.699934\pi$
$30$	0	0
$31$	−7.66901	−1.37740	−0.688698	−	0.725049i	$-0.741817\pi$
$31$	−7.66901	−1.37740	−0.688698	+	0.725049i	$0.741817\pi$
$32$	0	0
$33$	−4.99922	−0.870253
$34$	0	0
$35$	−0.172482	−0.0291548
$36$	0	0
$37$	−0.416257	−0.0684322	−0.0342161	−	0.999414i	$-0.510893\pi$
$37$	−0.416257	−0.0684322	−0.0342161	+	0.999414i	$0.510893\pi$
$38$	0	0
$39$	2.59208	0.415065
$40$	0	0
$41$	5.37869	0.840010	0.420005	−	0.907522i	$-0.362028\pi$
$41$	5.37869	0.840010	0.420005	+	0.907522i	$0.362028\pi$
$42$	0	0
$43$	9.93267	1.51472	0.757359	−	0.652999i	$-0.226489\pi$
$43$	9.93267	1.51472	0.757359	+	0.652999i	$0.226489\pi$
$44$	0	0
$45$	0.171983	0.0256377
$46$	0	0
$47$	3.44772	0.502902	0.251451	−	0.967870i	$-0.419092\pi$
$47$	3.44772	0.502902	0.251451	+	0.967870i	$0.419092\pi$
$48$	0	0
$49$	−6.89419	−0.984884
$50$	0	0
$51$	−4.50307	−0.630555
$52$	0	0
$53$	−1.87999	−0.258236	−0.129118	−	0.991629i	$-0.541215\pi$
$53$	−1.87999	−0.258236	−0.129118	+	0.991629i	$0.541215\pi$
$54$	0	0
$55$	1.62057	0.218518
$56$	0	0
$57$	1.70249	0.225500
$58$	0	0
$59$	11.6115	1.51169	0.755845	−	0.654751i	$-0.227226\pi$
$59$	11.6115	1.51169	0.755845	+	0.654751i	$0.227226\pi$
$60$	0	0
$61$	−15.3908	−1.97060	−0.985298	−	0.170846i	$-0.945350\pi$
$61$	−15.3908	−1.97060	−0.985298	+	0.170846i	$0.945350\pi$
$62$	0	0
$63$	−0.105503	−0.0132922
$64$	0	0
$65$	−0.840262	−0.104222
$66$	0	0
$67$	−13.1715	−1.60915	−0.804575	−	0.593851i	$-0.797607\pi$
$67$	−13.1715	−1.60915	−0.804575	+	0.593851i	$0.797607\pi$
$68$	0	0
$69$	−3.87248	−0.466192
$70$	0	0
$71$	16.7791	1.99132	0.995658	−	0.0930848i	$-0.0296728\pi$
$71$	16.7791	1.99132	0.995658	+	0.0930848i	$0.0296728\pi$
$72$	0	0
$73$	−7.62612	−0.892570	−0.446285	−	0.894891i	$-0.647253\pi$
$73$	−7.62612	−0.892570	−0.446285	+	0.894891i	$0.647253\pi$
$74$	0	0
$75$	−7.71880	−0.891291
$76$	0	0
$77$	−0.994144	−0.113293
$78$	0	0
$79$	−8.44815	−0.950491	−0.475245	−	0.879853i	$-0.657641\pi$
$79$	−8.44815	−0.950491	−0.475245	+	0.879853i	$0.657641\pi$
$80$	0	0
$81$	−7.92177	−0.880197
$82$	0	0
$83$	11.0056	1.20802	0.604008	−	0.796978i	$-0.293569\pi$
$83$	11.0056	1.20802	0.604008	+	0.796978i	$0.293569\pi$
$84$	0	0
$85$	1.45974	0.158331
$86$	0	0
$87$	−10.3524	−1.10989
$88$	0	0
$89$	−5.95109	−0.630815	−0.315407	−	0.948956i	$-0.602141\pi$
$89$	−5.95109	−0.630815	−0.315407	+	0.948956i	$0.602141\pi$
$90$	0	0
$91$	0.515460	0.0540349
$92$	0	0
$93$	−12.5445	−1.30081
$94$	0	0
$95$	−0.551886	−0.0566224
$96$	0	0
$97$	4.93453	0.501026	0.250513	−	0.968113i	$-0.419401\pi$
$97$	4.93453	0.501026	0.250513	+	0.968113i	$0.419401\pi$
$98$	0	0
$99$	0.991269	0.0996263
$100$	0	0
$101$	−7.75542	−0.771694	−0.385847	−	0.922563i	$-0.626091\pi$
$101$	−7.75542	−0.771694	−0.385847	+	0.922563i	$0.626091\pi$
$102$	0	0
$103$	−10.7502	−1.05925	−0.529623	−	0.848233i	$-0.677667\pi$
$103$	−10.7502	−1.05925	−0.529623	+	0.848233i	$0.677667\pi$
$104$	0	0
$105$	−0.282136	−0.0275337
$106$	0	0
$107$	−13.6838	−1.32286	−0.661430	−	0.750007i	$-0.730050\pi$
$107$	−13.6838	−1.32286	−0.661430	+	0.750007i	$0.730050\pi$
$108$	0	0
$109$	−11.6636	−1.11717	−0.558587	−	0.829446i	$-0.688656\pi$
$109$	−11.6636	−1.11717	−0.558587	+	0.829446i	$0.688656\pi$
$110$	0	0
$111$	−0.680889	−0.0646271
$112$	0	0
$113$	15.2895	1.43832	0.719159	−	0.694846i	$-0.244527\pi$
$113$	15.2895	1.43832	0.719159	+	0.694846i	$0.244527\pi$
$114$	0	0
$115$	1.25532	0.117060
$116$	0	0
$117$	−0.513970	−0.0475165
$118$	0	0
$119$	−0.895479	−0.0820884
$120$	0	0
$121$	−1.65941	−0.150855
$122$	0	0
$123$	8.79815	0.793303
$124$	0	0
$125$	5.15342	0.460936
$126$	0	0
$127$	19.9293	1.76844	0.884220	−	0.467070i	$-0.154690\pi$
$127$	19.9293	1.76844	0.884220	+	0.467070i	$0.154690\pi$
$128$	0	0
$129$	16.2473	1.43049
$130$	0	0
$131$	16.1071	1.40729	0.703643	−	0.710554i	$-0.251555\pi$
$131$	16.1071	1.40729	0.703643	+	0.710554i	$0.251555\pi$
$132$	0	0
$133$	0.338556	0.0293565
$134$	0	0
$135$	2.88338	0.248162
$136$	0	0
$137$	0.984127	0.0840797	0.0420398	−	0.999116i	$-0.486614\pi$
$137$	0.984127	0.0840797	0.0420398	+	0.999116i	$0.486614\pi$
$138$	0	0
$139$	4.24170	0.359776	0.179888	−	0.983687i	$-0.442426\pi$
$139$	4.24170	0.359776	0.179888	+	0.983687i	$0.442426\pi$
$140$	0	0
$141$	5.63959	0.474939
$142$	0	0
$143$	−4.84306	−0.404997
$144$	0	0
$145$	3.35587	0.278690
$146$	0	0
$147$	−11.2771	−0.930122
$148$	0	0
$149$	−4.83804	−0.396347	−0.198174	−	0.980167i	$-0.563501\pi$
$149$	−4.83804	−0.396347	−0.198174	+	0.980167i	$0.563501\pi$
$150$	0	0
$151$	−19.5402	−1.59016	−0.795079	−	0.606506i	$-0.792571\pi$
$151$	−19.5402	−1.59016	−0.795079	+	0.606506i	$0.792571\pi$
$152$	0	0
$153$	0.892889	0.0721858
$154$	0	0
$155$	4.06650	0.326629
$156$	0	0
$157$	−9.90764	−0.790716	−0.395358	−	0.918527i	$-0.629379\pi$
$157$	−9.90764	−0.790716	−0.395358	+	0.918527i	$0.629379\pi$
$158$	0	0
$159$	−3.07517	−0.243877
$160$	0	0
$161$	−0.770081	−0.0606909
$162$	0	0
$163$	19.1607	1.50078	0.750390	−	0.660995i	$-0.229866\pi$
$163$	19.1607	1.50078	0.750390	+	0.660995i	$0.229866\pi$
$164$	0	0
$165$	2.65084	0.206368
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−10.4889	−0.806837
$170$	0	0
$171$	−0.337577	−0.0258151
$172$	0	0
$173$	−13.0595	−0.992892	−0.496446	−	0.868068i	$-0.665362\pi$
$173$	−13.0595	−0.992892	−0.496446	+	0.868068i	$0.665362\pi$
$174$	0	0
$175$	−1.53496	−0.116032
$176$	0	0
$177$	18.9935	1.42764
$178$	0	0
$179$	−2.55029	−0.190618	−0.0953089	−	0.995448i	$-0.530384\pi$
$179$	−2.55029	−0.190618	−0.0953089	+	0.995448i	$0.530384\pi$
$180$	0	0
$181$	10.1911	0.757499	0.378750	−	0.925499i	$-0.376354\pi$
$181$	10.1911	0.757499	0.378750	+	0.925499i	$0.376354\pi$
$182$	0	0
$183$	−25.1755	−1.86102
$184$	0	0
$185$	0.220721	0.0162277
$186$	0	0
$187$	8.41357	0.615261
$188$	0	0
$189$	−1.76882	−0.128663
$190$	0	0
$191$	5.81981	0.421106	0.210553	−	0.977582i	$-0.432473\pi$
$191$	5.81981	0.421106	0.210553	+	0.977582i	$0.432473\pi$
$192$	0	0
$193$	19.5068	1.40413	0.702065	−	0.712113i	$-0.252262\pi$
$193$	19.5068	1.40413	0.702065	+	0.712113i	$0.252262\pi$
$194$	0	0
$195$	−1.37445	−0.0984266
$196$	0	0
$197$	−19.7778	−1.40911	−0.704554	−	0.709650i	$-0.748853\pi$
$197$	−19.7778	−1.40911	−0.704554	+	0.709650i	$0.748853\pi$
$198$	0	0
$199$	−12.6469	−0.896516	−0.448258	−	0.893904i	$-0.647955\pi$
$199$	−12.6469	−0.896516	−0.448258	+	0.893904i	$0.647955\pi$
$200$	0	0
$201$	−21.5451	−1.51968
$202$	0	0
$203$	−2.05867	−0.144490
$204$	0	0
$205$	−2.85205	−0.199196
$206$	0	0
$207$	0.767854	0.0533695
$208$	0	0
$209$	−3.18094	−0.220030
$210$	0	0
$211$	−19.2357	−1.32424	−0.662121	−	0.749397i	$-0.730344\pi$
$211$	−19.2357	−1.32424	−0.662121	+	0.749397i	$0.730344\pi$
$212$	0	0
$213$	27.4464	1.88059
$214$	0	0
$215$	−5.26681	−0.359193
$216$	0	0
$217$	−2.49460	−0.169345
$218$	0	0
$219$	−12.4744	−0.842940
$220$	0	0
$221$	−4.36241	−0.293447
$222$	0	0
$223$	−18.6026	−1.24572	−0.622860	−	0.782333i	$-0.714030\pi$
$223$	−18.6026	−1.24572	−0.622860	+	0.782333i	$0.714030\pi$
$224$	0	0
$225$	1.53052	0.102035
$226$	0	0
$227$	−1.87046	−0.124147	−0.0620733	−	0.998072i	$-0.519771\pi$
$227$	−1.87046	−0.124147	−0.0620733	+	0.998072i	$0.519771\pi$
$228$	0	0
$229$	26.4416	1.74731	0.873655	−	0.486547i	$-0.161744\pi$
$229$	26.4416	1.74731	0.873655	+	0.486547i	$0.161744\pi$
$230$	0	0
$231$	−1.62616	−0.106994
$232$	0	0
$233$	4.24009	0.277778	0.138889	−	0.990308i	$-0.455647\pi$
$233$	4.24009	0.277778	0.138889	+	0.990308i	$0.455647\pi$
$234$	0	0
$235$	−1.82816	−0.119256
$236$	0	0
$237$	−13.8190	−0.897640
$238$	0	0
$239$	13.5115	0.873984	0.436992	−	0.899465i	$-0.356044\pi$
$239$	13.5115	0.873984	0.436992	+	0.899465i	$0.356044\pi$
$240$	0	0
$241$	22.0486	1.42028	0.710139	−	0.704062i	$-0.248632\pi$
$241$	22.0486	1.42028	0.710139	+	0.704062i	$0.248632\pi$
$242$	0	0
$243$	3.35533	0.215244
$244$	0	0
$245$	3.65565	0.233551
$246$	0	0
$247$	1.64931	0.104943
$248$	0	0
$249$	18.0023	1.14085
$250$	0	0
$251$	−2.01263	−0.127036	−0.0635181	−	0.997981i	$-0.520232\pi$
$251$	−2.01263	−0.127036	−0.0635181	+	0.997981i	$0.520232\pi$
$252$	0	0
$253$	7.23538	0.454885
$254$	0	0
$255$	2.38776	0.149527
$256$	0	0
$257$	19.3459	1.20677	0.603383	−	0.797452i	$-0.293819\pi$
$257$	19.3459	1.20677	0.603383	+	0.797452i	$0.293819\pi$
$258$	0	0
$259$	−0.135401	−0.00841343
$260$	0	0
$261$	2.05271	0.127060
$262$	0	0
$263$	11.4288	0.704731	0.352366	−	0.935862i	$-0.385377\pi$
$263$	11.4288	0.704731	0.352366	+	0.935862i	$0.385377\pi$
$264$	0	0
$265$	0.996864	0.0612369
$266$	0	0
$267$	−9.73446	−0.595739
$268$	0	0
$269$	−1.57052	−0.0957561	−0.0478780	−	0.998853i	$-0.515246\pi$
$269$	−1.57052	−0.0957561	−0.0478780	+	0.998853i	$0.515246\pi$
$270$	0	0
$271$	−8.08899	−0.491371	−0.245686	−	0.969350i	$-0.579013\pi$
$271$	−8.08899	−0.491371	−0.245686	+	0.969350i	$0.579013\pi$
$272$	0	0
$273$	0.843161	0.0510304
$274$	0	0
$275$	14.4219	0.869672
$276$	0	0
$277$	3.30062	0.198315	0.0991577	−	0.995072i	$-0.468385\pi$
$277$	3.30062	0.198315	0.0991577	+	0.995072i	$0.468385\pi$
$278$	0	0
$279$	2.48739	0.148916
$280$	0	0
$281$	9.00975	0.537477	0.268738	−	0.963213i	$-0.413393\pi$
$281$	9.00975	0.537477	0.268738	+	0.963213i	$0.413393\pi$
$282$	0	0
$283$	−5.39215	−0.320530	−0.160265	−	0.987074i	$-0.551235\pi$
$283$	−5.39215	−0.320530	−0.160265	+	0.987074i	$0.551235\pi$
$284$	0	0
$285$	−0.902745	−0.0534740
$286$	0	0
$287$	1.74960	0.103276
$288$	0	0
$289$	−9.42145	−0.554203
$290$	0	0
$291$	8.07162	0.473167
$292$	0	0
$293$	−16.5144	−0.964779	−0.482389	−	0.875957i	$-0.660231\pi$
$293$	−16.5144	−0.964779	−0.482389	+	0.875957i	$0.660231\pi$
$294$	0	0
$295$	−6.15702	−0.358475
$296$	0	0
$297$	16.6191	0.964340
$298$	0	0
$299$	−3.75152	−0.216956
$300$	0	0
$301$	3.23093	0.186228
$302$	0	0
$303$	−12.6859	−0.728785
$304$	0	0
$305$	8.16101	0.467298
$306$	0	0
$307$	−1.32046	−0.0753625	−0.0376812	−	0.999290i	$-0.511997\pi$
$307$	−1.32046	−0.0753625	−0.0376812	+	0.999290i	$0.511997\pi$
$308$	0	0
$309$	−17.5845	−1.00035
$310$	0	0
$311$	−23.5785	−1.33701	−0.668506	−	0.743707i	$-0.733066\pi$
$311$	−23.5785	−1.33701	−0.668506	+	0.743707i	$0.733066\pi$
$312$	0	0
$313$	20.8467	1.17833	0.589163	−	0.808014i	$-0.299458\pi$
$313$	20.8467	1.17833	0.589163	+	0.808014i	$0.299458\pi$
$314$	0	0
$315$	0.0559433	0.00315205
$316$	0	0
$317$	8.83476	0.496210	0.248105	−	0.968733i	$-0.420192\pi$
$317$	8.83476	0.496210	0.248105	+	0.968733i	$0.420192\pi$
$318$	0	0
$319$	19.3424	1.08297
$320$	0	0
$321$	−22.3831	−1.24930
$322$	0	0
$323$	−2.86524	−0.159426
$324$	0	0
$325$	−7.47770	−0.414788
$326$	0	0
$327$	−19.0787	−1.05506
$328$	0	0
$329$	1.12149	0.0618296
$330$	0	0
$331$	6.46457	0.355325	0.177662	−	0.984091i	$-0.443146\pi$
$331$	6.46457	0.355325	0.177662	+	0.984091i	$0.443146\pi$
$332$	0	0
$333$	0.135010	0.00739849
$334$	0	0
$335$	6.98418	0.381587
$336$	0	0
$337$	1.34979	0.0735279	0.0367640	−	0.999324i	$-0.488295\pi$
$337$	1.34979	0.0735279	0.0367640	+	0.999324i	$0.488295\pi$
$338$	0	0
$339$	25.0097	1.35834
$340$	0	0
$341$	23.4383	1.26926
$342$	0	0
$343$	−4.51955	−0.244033
$344$	0	0
$345$	2.05339	0.110551
$346$	0	0
$347$	6.95932	0.373596	0.186798	−	0.982398i	$-0.440189\pi$
$347$	6.95932	0.373596	0.186798	+	0.982398i	$0.440189\pi$
$348$	0	0
$349$	−26.3758	−1.41187	−0.705933	−	0.708279i	$-0.749472\pi$
$349$	−26.3758	−1.41187	−0.705933	+	0.708279i	$0.749472\pi$
$350$	0	0
$351$	−8.61696	−0.459939
$352$	0	0
$353$	−11.1894	−0.595554	−0.297777	−	0.954635i	$-0.596245\pi$
$353$	−11.1894	−0.595554	−0.297777	+	0.954635i	$0.596245\pi$
$354$	0	0
$355$	−8.89715	−0.472212
$356$	0	0
$357$	−1.46477	−0.0775240
$358$	0	0
$359$	8.14201	0.429719	0.214859	−	0.976645i	$-0.431071\pi$
$359$	8.14201	0.429719	0.214859	+	0.976645i	$0.431071\pi$
$360$	0	0
$361$	−17.9167	−0.942986
$362$	0	0
$363$	−2.71437	−0.142467
$364$	0	0
$365$	4.04376	0.211660
$366$	0	0
$367$	−27.9685	−1.45994	−0.729972	−	0.683477i	$-0.760467\pi$
$367$	−27.9685	−1.45994	−0.729972	+	0.683477i	$0.760467\pi$
$368$	0	0
$369$	−1.74454	−0.0908170
$370$	0	0
$371$	−0.611528	−0.0317490
$372$	0	0
$373$	−14.5966	−0.755781	−0.377891	−	0.925850i	$-0.623351\pi$
$373$	−14.5966	−0.755781	−0.377891	+	0.925850i	$0.623351\pi$
$374$	0	0
$375$	8.42968	0.435307
$376$	0	0
$377$	−10.0290	−0.516519
$378$	0	0
$379$	3.18010	0.163351	0.0816755	−	0.996659i	$-0.473973\pi$
$379$	3.18010	0.163351	0.0816755	+	0.996659i	$0.473973\pi$
$380$	0	0
$381$	32.5993	1.67011
$382$	0	0
$383$	16.5125	0.843751	0.421876	−	0.906654i	$-0.361372\pi$
$383$	16.5125	0.843751	0.421876	+	0.906654i	$0.361372\pi$
$384$	0	0
$385$	0.527146	0.0268658
$386$	0	0
$387$	−3.22159	−0.163763
$388$	0	0
$389$	18.0059	0.912933	0.456467	−	0.889741i	$-0.349115\pi$
$389$	18.0059	0.912933	0.456467	+	0.889741i	$0.349115\pi$
$390$	0	0
$391$	6.51730	0.329594
$392$	0	0
$393$	26.3471	1.32904
$394$	0	0
$395$	4.47964	0.225395
$396$	0	0
$397$	18.2942	0.918161	0.459080	−	0.888395i	$-0.348179\pi$
$397$	18.2942	0.918161	0.459080	+	0.888395i	$0.348179\pi$
$398$	0	0
$399$	0.553790	0.0277242
$400$	0	0
$401$	−10.3601	−0.517357	−0.258678	−	0.965964i	$-0.583287\pi$
$401$	−10.3601	−0.517357	−0.258678	+	0.965964i	$0.583287\pi$
$402$	0	0
$403$	−12.1527	−0.605369
$404$	0	0
$405$	4.20053	0.208726
$406$	0	0
$407$	1.27218	0.0630596
$408$	0	0
$409$	−9.17438	−0.453644	−0.226822	−	0.973936i	$-0.572834\pi$
$409$	−9.17438	−0.453644	−0.226822	+	0.973936i	$0.572834\pi$
$410$	0	0
$411$	1.60978	0.0794046
$412$	0	0
$413$	3.77703	0.185856
$414$	0	0
$415$	−5.83571	−0.286464
$416$	0	0
$417$	6.93834	0.339772
$418$	0	0
$419$	−3.20065	−0.156362	−0.0781811	−	0.996939i	$-0.524911\pi$
$419$	−3.20065	−0.156362	−0.0781811	+	0.996939i	$0.524911\pi$
$420$	0	0
$421$	−22.7508	−1.10881	−0.554404	−	0.832248i	$-0.687054\pi$
$421$	−22.7508	−1.10881	−0.554404	+	0.832248i	$0.687054\pi$
$422$	0	0
$423$	−1.11824	−0.0543709
$424$	0	0
$425$	12.9906	0.630135
$426$	0	0
$427$	−5.00639	−0.242276
$428$	0	0
$429$	−7.92201	−0.382478
$430$	0	0
$431$	35.7887	1.72388	0.861940	−	0.507010i	$-0.169249\pi$
$431$	35.7887	1.72388	0.861940	+	0.507010i	$0.169249\pi$
$432$	0	0
$433$	40.1659	1.93025	0.965125	−	0.261790i	$-0.0843129\pi$
$433$	40.1659	1.93025	0.965125	+	0.261790i	$0.0843129\pi$
$434$	0	0
$435$	5.48935	0.263194
$436$	0	0
$437$	−2.46401	−0.117870
$438$	0	0
$439$	27.8598	1.32968	0.664838	−	0.746988i	$-0.268501\pi$
$439$	27.8598	1.32968	0.664838	+	0.746988i	$0.268501\pi$
$440$	0	0
$441$	2.23608	0.106480
$442$	0	0
$443$	18.5283	0.880307	0.440154	−	0.897923i	$-0.354924\pi$
$443$	18.5283	0.880307	0.440154	+	0.897923i	$0.354924\pi$
$444$	0	0
$445$	3.15557	0.149589
$446$	0	0
$447$	−7.91379	−0.374309
$448$	0	0
$449$	−30.0799	−1.41956	−0.709778	−	0.704425i	$-0.751205\pi$
$449$	−30.0799	−1.41956	−0.709778	+	0.704425i	$0.751205\pi$
$450$	0	0
$451$	−16.4385	−0.774061
$452$	0	0
$453$	−31.9627	−1.50174
$454$	0	0
$455$	−0.273323	−0.0128136
$456$	0	0
$457$	13.5770	0.635104	0.317552	−	0.948241i	$-0.397139\pi$
$457$	13.5770	0.635104	0.317552	+	0.948241i	$0.397139\pi$
$458$	0	0
$459$	14.9697	0.698728
$460$	0	0
$461$	−36.1476	−1.68356	−0.841781	−	0.539819i	$-0.818493\pi$
$461$	−36.1476	−1.68356	−0.841781	+	0.539819i	$0.818493\pi$
$462$	0	0
$463$	32.3370	1.50283	0.751415	−	0.659830i	$-0.229372\pi$
$463$	32.3370	1.50283	0.751415	+	0.659830i	$0.229372\pi$
$464$	0	0
$465$	6.65175	0.308468
$466$	0	0
$467$	−40.7798	−1.88706	−0.943531	−	0.331283i	$-0.892519\pi$
$467$	−40.7798	−1.88706	−0.943531	+	0.331283i	$0.892519\pi$
$468$	0	0
$469$	−4.28446	−0.197838
$470$	0	0
$471$	−16.2064	−0.746750
$472$	0	0
$473$	−30.3566	−1.39580
$474$	0	0
$475$	−4.91137	−0.225349
$476$	0	0
$477$	0.609760	0.0279190
$478$	0	0
$479$	−5.72483	−0.261574	−0.130787	−	0.991410i	$-0.541750\pi$
$479$	−5.72483	−0.261574	−0.130787	+	0.991410i	$0.541750\pi$
$480$	0	0
$481$	−0.659621	−0.0300761
$482$	0	0
$483$	−1.25966	−0.0573163
$484$	0	0
$485$	−2.61654	−0.118811
$486$	0	0
$487$	26.0427	1.18011	0.590053	−	0.807364i	$-0.299107\pi$
$487$	26.0427	1.18011	0.590053	+	0.807364i	$0.299107\pi$
$488$	0	0
$489$	31.3420	1.41733
$490$	0	0
$491$	−0.851757	−0.0384392	−0.0192196	−	0.999815i	$-0.506118\pi$
$491$	−0.851757	−0.0384392	−0.0192196	+	0.999815i	$0.506118\pi$
$492$	0	0
$493$	17.4228	0.784682
$494$	0	0
$495$	−0.525621	−0.0236249
$496$	0	0
$497$	5.45797	0.244824
$498$	0	0
$499$	18.0391	0.807542	0.403771	−	0.914860i	$-0.367699\pi$
$499$	18.0391	0.807542	0.403771	+	0.914860i	$0.367699\pi$
$500$	0	0
$501$	−1.63574	−0.0730796
$502$	0	0
$503$	2.08681	0.0930462	0.0465231	−	0.998917i	$-0.485186\pi$
$503$	2.08681	0.0930462	0.0465231	+	0.998917i	$0.485186\pi$
$504$	0	0
$505$	4.11232	0.182996
$506$	0	0
$507$	−17.1571	−0.761975
$508$	0	0
$509$	−22.6357	−1.00331	−0.501655	−	0.865068i	$-0.667275\pi$
$509$	−22.6357	−1.00331	−0.501655	+	0.865068i	$0.667275\pi$
$510$	0	0
$511$	−2.48065	−0.109738
$512$	0	0
$513$	−5.65965	−0.249879
$514$	0	0
$515$	5.70029	0.251185
$516$	0	0
$517$	−10.5371	−0.463419
$518$	0	0
$519$	−21.3619	−0.937684
$520$	0	0
$521$	38.9298	1.70554	0.852772	−	0.522283i	$-0.174920\pi$
$521$	38.9298	1.70554	0.852772	+	0.522283i	$0.174920\pi$
$522$	0	0
$523$	−38.6063	−1.68814	−0.844069	−	0.536235i	$-0.819846\pi$
$523$	−38.6063	−1.68814	−0.844069	+	0.536235i	$0.819846\pi$
$524$	0	0
$525$	−2.51080	−0.109580
$526$	0	0
$527$	21.1122	0.919660
$528$	0	0
$529$	−17.3953	−0.756319
$530$	0	0
$531$	−3.76611	−0.163435
$532$	0	0
$533$	8.52333	0.369186
$534$	0	0
$535$	7.25583	0.313697
$536$	0	0
$537$	−4.17163	−0.180019
$538$	0	0
$539$	21.0703	0.907562
$540$	0	0
$541$	−5.17279	−0.222396	−0.111198	−	0.993798i	$-0.535469\pi$
$541$	−5.17279	−0.222396	−0.111198	+	0.993798i	$0.535469\pi$
$542$	0	0
$543$	16.6700	0.715380
$544$	0	0
$545$	6.18466	0.264922
$546$	0	0
$547$	21.4615	0.917626	0.458813	−	0.888533i	$-0.348275\pi$
$547$	21.4615	0.917626	0.458813	+	0.888533i	$0.348275\pi$
$548$	0	0
$549$	4.99191	0.213049
$550$	0	0
$551$	−6.58707	−0.280619
$552$	0	0
$553$	−2.74804	−0.116859
$554$	0	0
$555$	0.361042	0.0153254
$556$	0	0
$557$	−27.6171	−1.17017	−0.585086	−	0.810971i	$-0.698939\pi$
$557$	−27.6171	−1.17017	−0.585086	+	0.810971i	$0.698939\pi$
$558$	0	0
$559$	15.7398	0.665722
$560$	0	0
$561$	13.7624	0.581051
$562$	0	0
$563$	11.2365	0.473560	0.236780	−	0.971563i	$-0.423908\pi$
$563$	11.2365	0.473560	0.236780	+	0.971563i	$0.423908\pi$
$564$	0	0
$565$	−8.10729	−0.341076
$566$	0	0
$567$	−2.57682	−0.108216
$568$	0	0
$569$	−33.4889	−1.40393	−0.701964	−	0.712212i	$-0.747693\pi$
$569$	−33.4889	−1.40393	−0.701964	+	0.712212i	$0.747693\pi$
$570$	0	0
$571$	−24.2608	−1.01528	−0.507641	−	0.861569i	$-0.669482\pi$
$571$	−24.2608	−1.01528	−0.507641	+	0.861569i	$0.669482\pi$
$572$	0	0
$573$	9.51971	0.397692
$574$	0	0
$575$	11.1714	0.465881
$576$	0	0
$577$	45.2939	1.88561	0.942805	−	0.333344i	$-0.108177\pi$
$577$	45.2939	1.88561	0.942805	+	0.333344i	$0.108177\pi$
$578$	0	0
$579$	31.9081	1.32606
$580$	0	0
$581$	3.57993	0.148520
$582$	0	0
$583$	5.74568	0.237962
$584$	0	0
$585$	0.272533	0.0112678
$586$	0	0
$587$	4.45465	0.183863	0.0919315	−	0.995765i	$-0.470696\pi$
$587$	4.45465	0.183863	0.0919315	+	0.995765i	$0.470696\pi$
$588$	0	0
$589$	−7.98192	−0.328889
$590$	0	0
$591$	−32.3514	−1.33076
$592$	0	0
$593$	1.59543	0.0655166	0.0327583	−	0.999463i	$-0.489571\pi$
$593$	1.59543	0.0655166	0.0327583	+	0.999463i	$0.489571\pi$
$594$	0	0
$595$	0.474828	0.0194661
$596$	0	0
$597$	−20.6871	−0.846667
$598$	0	0
$599$	33.1831	1.35583	0.677913	−	0.735142i	$-0.262885\pi$
$599$	33.1831	1.35583	0.677913	+	0.735142i	$0.262885\pi$
$600$	0	0
$601$	6.32486	0.257996	0.128998	−	0.991645i	$-0.458824\pi$
$601$	6.32486	0.257996	0.128998	+	0.991645i	$0.458824\pi$
$602$	0	0
$603$	4.27207	0.173972
$604$	0	0
$605$	0.879903	0.0357732
$606$	0	0
$607$	−4.48850	−0.182183	−0.0910913	−	0.995843i	$-0.529035\pi$
$607$	−4.48850	−0.182183	−0.0910913	+	0.995843i	$0.529035\pi$
$608$	0	0
$609$	−3.36745	−0.136456
$610$	0	0
$611$	5.46343	0.221027
$612$	0	0
$613$	22.1900	0.896247	0.448124	−	0.893972i	$-0.352092\pi$
$613$	22.1900	0.896247	0.448124	+	0.893972i	$0.352092\pi$
$614$	0	0
$615$	−4.66523	−0.188120
$616$	0	0
$617$	5.44902	0.219369	0.109685	−	0.993966i	$-0.465016\pi$
$617$	5.44902	0.219369	0.109685	+	0.993966i	$0.465016\pi$
$618$	0	0
$619$	−5.11122	−0.205437	−0.102719	−	0.994710i	$-0.532754\pi$
$619$	−5.11122	−0.205437	−0.102719	+	0.994710i	$0.532754\pi$
$620$	0	0
$621$	12.8735	0.516594
$622$	0	0
$623$	−1.93579	−0.0775559
$624$	0	0
$625$	20.8616	0.834462
$626$	0	0
$627$	−5.20320	−0.207796
$628$	0	0
$629$	1.14592	0.0456908
$630$	0	0
$631$	8.87913	0.353472	0.176736	−	0.984258i	$-0.443446\pi$
$631$	8.87913	0.353472	0.176736	+	0.984258i	$0.443446\pi$
$632$	0	0
$633$	−31.4647	−1.25061
$634$	0	0
$635$	−10.5675	−0.419360
$636$	0	0
$637$	−10.9249	−0.432859
$638$	0	0
$639$	−5.44219	−0.215290
$640$	0	0
$641$	−26.0788	−1.03005	−0.515025	−	0.857175i	$-0.672217\pi$
$641$	−26.0788	−1.03005	−0.515025	+	0.857175i	$0.672217\pi$
$642$	0	0
$643$	−21.0090	−0.828515	−0.414257	−	0.910160i	$-0.635959\pi$
$643$	−21.0090	−0.828515	−0.414257	+	0.910160i	$0.635959\pi$
$644$	0	0
$645$	−8.61515	−0.339221
$646$	0	0
$647$	−0.953157	−0.0374725	−0.0187362	−	0.999824i	$-0.505964\pi$
$647$	−0.953157	−0.0374725	−0.0187362	+	0.999824i	$0.505964\pi$
$648$	0	0
$649$	−35.4875	−1.39301
$650$	0	0
$651$	−4.08053	−0.159929
$652$	0	0
$653$	16.3972	0.641672	0.320836	−	0.947135i	$-0.396036\pi$
$653$	16.3972	0.641672	0.320836	+	0.947135i	$0.396036\pi$
$654$	0	0
$655$	−8.54082	−0.333717
$656$	0	0
$657$	2.47348	0.0964995
$658$	0	0
$659$	13.3973	0.521883	0.260942	−	0.965355i	$-0.415967\pi$
$659$	13.3973	0.521883	0.260942	+	0.965355i	$0.415967\pi$
$660$	0	0
$661$	−12.4625	−0.484734	−0.242367	−	0.970185i	$-0.577924\pi$
$661$	−12.4625	−0.484734	−0.242367	+	0.970185i	$0.577924\pi$
$662$	0	0
$663$	−7.13578	−0.277131
$664$	0	0
$665$	−0.179520	−0.00696147
$666$	0	0
$667$	14.9830	0.580144
$668$	0	0
$669$	−30.4290	−1.17645
$670$	0	0
$671$	47.0381	1.81588
$672$	0	0
$673$	24.5484	0.946272	0.473136	−	0.880990i	$-0.343122\pi$
$673$	24.5484	0.946272	0.473136	+	0.880990i	$0.343122\pi$
$674$	0	0
$675$	25.6599	0.987652
$676$	0	0
$677$	18.1335	0.696927	0.348464	−	0.937322i	$-0.386704\pi$
$677$	18.1335	0.696927	0.348464	+	0.937322i	$0.386704\pi$
$678$	0	0
$679$	1.60512	0.0615989
$680$	0	0
$681$	−3.05959	−0.117244
$682$	0	0
$683$	−26.0161	−0.995478	−0.497739	−	0.867327i	$-0.665836\pi$
$683$	−26.0161	−0.995478	−0.497739	+	0.867327i	$0.665836\pi$
$684$	0	0
$685$	−0.521835	−0.0199383
$686$	0	0
$687$	43.2517	1.65015
$688$	0	0
$689$	−2.97912	−0.113495
$690$	0	0
$691$	−14.6266	−0.556423	−0.278211	−	0.960520i	$-0.589742\pi$
$691$	−14.6266	−0.556423	−0.278211	+	0.960520i	$0.589742\pi$
$692$	0	0
$693$	0.322443	0.0122486
$694$	0	0
$695$	−2.24917	−0.0853158
$696$	0	0
$697$	−14.8071	−0.560858
$698$	0	0
$699$	6.93570	0.262332
$700$	0	0
$701$	29.6438	1.11963	0.559816	−	0.828617i	$-0.310872\pi$
$701$	29.6438	1.11963	0.559816	+	0.828617i	$0.310872\pi$
$702$	0	0
$703$	−0.433241	−0.0163400
$704$	0	0
$705$	−2.99040	−0.112625
$706$	0	0
$707$	−2.52271	−0.0948763
$708$	0	0
$709$	29.7647	1.11784	0.558918	−	0.829223i	$-0.311217\pi$
$709$	29.7647	1.11784	0.558918	+	0.829223i	$0.311217\pi$
$710$	0	0
$711$	2.74010	0.102762
$712$	0	0
$713$	18.1557	0.679938
$714$	0	0
$715$	2.56804	0.0960393
$716$	0	0
$717$	22.1013	0.825388
$718$	0	0
$719$	18.5515	0.691853	0.345927	−	0.938262i	$-0.387565\pi$
$719$	18.5515	0.691853	0.345927	+	0.938262i	$0.387565\pi$
$720$	0	0
$721$	−3.49686	−0.130230
$722$	0	0
$723$	36.0659	1.34131
$724$	0	0
$725$	29.8647	1.10915
$726$	0	0
$727$	−0.275879	−0.0102318	−0.00511590	−	0.999987i	$-0.501628\pi$
$727$	−0.275879	−0.0102318	−0.00511590	+	0.999987i	$0.501628\pi$
$728$	0	0
$729$	29.2538	1.08347
$730$	0	0
$731$	−27.3438	−1.01135
$732$	0	0
$733$	−25.3558	−0.936538	−0.468269	−	0.883586i	$-0.655122\pi$
$733$	−25.3558	−0.936538	−0.468269	+	0.883586i	$0.655122\pi$
$734$	0	0
$735$	5.97971	0.220565
$736$	0	0
$737$	40.2551	1.48282
$738$	0	0
$739$	−52.8383	−1.94369	−0.971843	−	0.235628i	$-0.924285\pi$
$739$	−52.8383	−1.94369	−0.971843	+	0.235628i	$0.924285\pi$
$740$	0	0
$741$	2.69784	0.0991077
$742$	0	0
$743$	17.4756	0.641119	0.320560	−	0.947228i	$-0.396129\pi$
$743$	17.4756	0.641119	0.320560	+	0.947228i	$0.396129\pi$
$744$	0	0
$745$	2.56537	0.0939880
$746$	0	0
$747$	−3.56957	−0.130604
$748$	0	0
$749$	−4.45110	−0.162640
$750$	0	0
$751$	6.19664	0.226119	0.113059	−	0.993588i	$-0.463935\pi$
$751$	6.19664	0.226119	0.113059	+	0.993588i	$0.463935\pi$
$752$	0	0
$753$	−3.29215	−0.119973
$754$	0	0
$755$	10.3612	0.377083
$756$	0	0
$757$	−35.7428	−1.29910	−0.649548	−	0.760321i	$-0.725042\pi$
$757$	−35.7428	−1.29910	−0.649548	+	0.760321i	$0.725042\pi$
$758$	0	0
$759$	11.8352	0.429592
$760$	0	0
$761$	15.6157	0.566069	0.283034	−	0.959110i	$-0.408659\pi$
$761$	15.6157	0.566069	0.283034	+	0.959110i	$0.408659\pi$
$762$	0	0
$763$	−3.79399	−0.137352
$764$	0	0
$765$	−0.473455	−0.0171178
$766$	0	0
$767$	18.4002	0.664392
$768$	0	0
$769$	38.7389	1.39696	0.698479	−	0.715630i	$-0.253860\pi$
$769$	38.7389	1.39696	0.698479	+	0.715630i	$0.253860\pi$
$770$	0	0
$771$	31.6450	1.13967
$772$	0	0
$773$	47.9765	1.72559	0.862797	−	0.505550i	$-0.168710\pi$
$773$	47.9765	1.72559	0.862797	+	0.505550i	$0.168710\pi$
$774$	0	0
$775$	36.1888	1.29994
$776$	0	0
$777$	−0.221482	−0.00794562
$778$	0	0
$779$	5.59815	0.200575
$780$	0	0
$781$	−51.2810	−1.83498
$782$	0	0
$783$	34.4148	1.22988
$784$	0	0
$785$	5.25354	0.187507
$786$	0	0
$787$	24.7022	0.880537	0.440268	−	0.897866i	$-0.354883\pi$
$787$	24.7022	0.880537	0.440268	+	0.897866i	$0.354883\pi$
$788$	0	0
$789$	18.6946	0.665546
$790$	0	0
$791$	4.97343	0.176835
$792$	0	0
$793$	−24.3891	−0.866082
$794$	0	0
$795$	1.63061	0.0578319
$796$	0	0
$797$	−49.3491	−1.74803	−0.874017	−	0.485896i	$-0.838493\pi$
$797$	−49.3491	−1.74803	−0.874017	+	0.485896i	$0.838493\pi$
$798$	0	0
$799$	−9.49129	−0.335778
$800$	0	0
$801$	1.93019	0.0682001
$802$	0	0
$803$	23.3072	0.822495
$804$	0	0
$805$	0.408336	0.0143920
$806$	0	0
$807$	−2.56896	−0.0904317
$808$	0	0
$809$	40.8832	1.43738	0.718689	−	0.695332i	$-0.244743\pi$
$809$	40.8832	1.43738	0.718689	+	0.695332i	$0.244743\pi$
$810$	0	0
$811$	−3.36442	−0.118141	−0.0590705	−	0.998254i	$-0.518814\pi$
$811$	−3.36442	−0.118141	−0.0590705	+	0.998254i	$0.518814\pi$
$812$	0	0
$813$	−13.2315	−0.464049
$814$	0	0
$815$	−10.1600	−0.355888
$816$	0	0
$817$	10.3379	0.361679
$818$	0	0
$819$	−0.167186	−0.00584194
$820$	0	0
$821$	−2.80181	−0.0977839	−0.0488919	−	0.998804i	$-0.515569\pi$
$821$	−2.80181	−0.0977839	−0.0488919	+	0.998804i	$0.515569\pi$
$822$	0	0
$823$	−15.6511	−0.545562	−0.272781	−	0.962076i	$-0.587943\pi$
$823$	−15.6511	−0.545562	−0.272781	+	0.962076i	$0.587943\pi$
$824$	0	0
$825$	23.5905	0.821316
$826$	0	0
$827$	−9.42820	−0.327851	−0.163925	−	0.986473i	$-0.552416\pi$
$827$	−9.42820	−0.327851	−0.163925	+	0.986473i	$0.552416\pi$
$828$	0	0
$829$	−21.7337	−0.754844	−0.377422	−	0.926041i	$-0.623189\pi$
$829$	−21.7337	−0.754844	−0.377422	+	0.926041i	$0.623189\pi$
$830$	0	0
$831$	5.39898	0.187288
$832$	0	0
$833$	18.9791	0.657588
$834$	0	0
$835$	0.530251	0.0183501
$836$	0	0
$837$	41.7023	1.44144
$838$	0	0
$839$	−29.9380	−1.03358	−0.516788	−	0.856113i	$-0.672872\pi$
$839$	−29.9380	−1.03358	−0.516788	+	0.856113i	$0.672872\pi$
$840$	0	0
$841$	11.0542	0.381179
$842$	0	0
$843$	14.7376	0.507592
$844$	0	0
$845$	5.56174	0.191330
$846$	0	0
$847$	−0.539778	−0.0185470
$848$	0	0
$849$	−8.82018	−0.302708
$850$	0	0
$851$	0.985452	0.0337809
$852$	0	0
$853$	−40.8801	−1.39971	−0.699853	−	0.714287i	$-0.746751\pi$
$853$	−40.8801	−1.39971	−0.699853	+	0.714287i	$0.746751\pi$
$854$	0	0
$855$	0.179000	0.00612169
$856$	0	0
$857$	−46.6092	−1.59214	−0.796069	−	0.605205i	$-0.793091\pi$
$857$	−46.6092	−1.59214	−0.796069	+	0.605205i	$0.793091\pi$
$858$	0	0
$859$	−12.2003	−0.416269	−0.208134	−	0.978100i	$-0.566739\pi$
$859$	−12.2003	−0.416269	−0.208134	+	0.978100i	$0.566739\pi$
$860$	0	0
$861$	2.86189	0.0975331
$862$	0	0
$863$	−1.98470	−0.0675600	−0.0337800	−	0.999429i	$-0.510755\pi$
$863$	−1.98470	−0.0675600	−0.0337800	+	0.999429i	$0.510755\pi$
$864$	0	0
$865$	6.92479	0.235450
$866$	0	0
$867$	−15.4111	−0.523387
$868$	0	0
$869$	25.8195	0.875868
$870$	0	0
$871$	−20.8721	−0.707225
$872$	0	0
$873$	−1.60048	−0.0541680
$874$	0	0
$875$	1.67632	0.0566701
$876$	0	0
$877$	0.512526	0.0173068	0.00865339	−	0.999963i	$-0.497246\pi$
$877$	0.512526	0.0173068	0.00865339	+	0.999963i	$0.497246\pi$
$878$	0	0
$879$	−27.0133	−0.911134
$880$	0	0
$881$	−52.6344	−1.77330	−0.886649	−	0.462443i	$-0.846973\pi$
$881$	−52.6344	−1.77330	−0.886649	+	0.462443i	$0.846973\pi$
$882$	0	0
$883$	42.2301	1.42115	0.710577	−	0.703619i	$-0.248434\pi$
$883$	42.2301	1.42115	0.710577	+	0.703619i	$0.248434\pi$
$884$	0	0
$885$	−10.0713	−0.338543
$886$	0	0
$887$	20.9975	0.705026	0.352513	−	0.935807i	$-0.385327\pi$
$887$	20.9975	0.705026	0.352513	+	0.935807i	$0.385327\pi$
$888$	0	0
$889$	6.48268	0.217422
$890$	0	0
$891$	24.2108	0.811093
$892$	0	0
$893$	3.58840	0.120081
$894$	0	0
$895$	1.35230	0.0452023
$896$	0	0
$897$	−6.13653	−0.204893
$898$	0	0
$899$	48.5359	1.61876
$900$	0	0
$901$	5.17544	0.172419
$902$	0	0
$903$	5.28498	0.175873
$904$	0	0
$905$	−5.40385	−0.179630
$906$	0	0
$907$	−44.0561	−1.46286	−0.731429	−	0.681917i	$-0.761146\pi$
$907$	−44.0561	−1.46286	−0.731429	+	0.681917i	$0.761146\pi$
$908$	0	0
$909$	2.51542	0.0834311
$910$	0	0
$911$	15.2150	0.504096	0.252048	−	0.967715i	$-0.418896\pi$
$911$	15.2150	0.504096	0.252048	+	0.967715i	$0.418896\pi$
$912$	0	0
$913$	−33.6356	−1.11318
$914$	0	0
$915$	13.3493	0.441315
$916$	0	0
$917$	5.23938	0.173020
$918$	0	0
$919$	11.9563	0.394401	0.197200	−	0.980363i	$-0.436815\pi$
$919$	11.9563	0.394401	0.197200	+	0.980363i	$0.436815\pi$
$920$	0	0
$921$	−2.15993	−0.0711721
$922$	0	0
$923$	26.5890	0.875189
$924$	0	0
$925$	1.96425	0.0645840
$926$	0	0
$927$	3.48674	0.114520
$928$	0	0
$929$	1.27850	0.0419464	0.0209732	−	0.999780i	$-0.493324\pi$
$929$	1.27850	0.0419464	0.0209732	+	0.999780i	$0.493324\pi$
$930$	0	0
$931$	−7.17549	−0.235167
$932$	0	0
$933$	−38.5683	−1.26267
$934$	0	0
$935$	−4.46131	−0.145900
$936$	0	0
$937$	−9.14783	−0.298846	−0.149423	−	0.988773i	$-0.547742\pi$
$937$	−9.14783	−0.298846	−0.149423	+	0.988773i	$0.547742\pi$
$938$	0	0
$939$	34.0999	1.11281
$940$	0	0
$941$	−34.0274	−1.10926	−0.554631	−	0.832096i	$-0.687141\pi$
$941$	−34.0274	−1.10926	−0.554631	+	0.832096i	$0.687141\pi$
$942$	0	0
$943$	−12.7336	−0.414663
$944$	0	0
$945$	0.937917	0.0305105
$946$	0	0
$947$	24.4147	0.793370	0.396685	−	0.917955i	$-0.370161\pi$
$947$	24.4147	0.793370	0.396685	+	0.917955i	$0.370161\pi$
$948$	0	0
$949$	−12.0847	−0.392287
$950$	0	0
$951$	14.4514	0.468619
$952$	0	0
$953$	46.1182	1.49391	0.746957	−	0.664872i	$-0.231514\pi$
$953$	46.1182	1.49391	0.746957	+	0.664872i	$0.231514\pi$
$954$	0	0
$955$	−3.08596	−0.0998593
$956$	0	0
$957$	31.6393	1.02275
$958$	0	0
$959$	0.320120	0.0103372
$960$	0	0
$961$	27.8137	0.897217
$962$	0	0
$963$	4.43823	0.143020
$964$	0	0
$965$	−10.3435	−0.332969
$966$	0	0
$967$	−12.6187	−0.405789	−0.202894	−	0.979201i	$-0.565035\pi$
$967$	−12.6187	−0.405789	−0.202894	+	0.979201i	$0.565035\pi$
$968$	0	0
$969$	−4.68680	−0.150562
$970$	0	0
$971$	−58.3997	−1.87414	−0.937069	−	0.349146i	$-0.886472\pi$
$971$	−58.3997	−1.87414	−0.937069	+	0.349146i	$0.886472\pi$
$972$	0	0
$973$	1.37976	0.0442329
$974$	0	0
$975$	−12.2316	−0.391724
$976$	0	0
$977$	21.2512	0.679885	0.339942	−	0.940446i	$-0.389592\pi$
$977$	21.2512	0.679885	0.339942	+	0.940446i	$0.389592\pi$
$978$	0	0
$979$	18.1880	0.581290
$980$	0	0
$981$	3.78302	0.120783
$982$	0	0
$983$	−57.2580	−1.82625	−0.913124	−	0.407682i	$-0.866337\pi$
$983$	−57.2580	−1.82625	−0.913124	+	0.407682i	$0.866337\pi$
$984$	0	0
$985$	10.4872	0.334150
$986$	0	0
$987$	1.83447	0.0583917
$988$	0	0
$989$	−23.5147	−0.747726
$990$	0	0
$991$	−24.0372	−0.763568	−0.381784	−	0.924252i	$-0.624690\pi$
$991$	−24.0372	−0.763568	−0.381784	+	0.924252i	$0.624690\pi$
$992$	0	0
$993$	10.5744	0.335568
$994$	0	0
$995$	6.70604	0.212596
$996$	0	0
$997$	44.0011	1.39353	0.696764	−	0.717301i	$-0.254623\pi$
$997$	44.0011	1.39353	0.696764	+	0.717301i	$0.254623\pi$
$998$	0	0
$999$	2.26351	0.0716142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.m.1.7		9
4.3	odd	2		1336.2.a.c.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.c.1.3	✓	9	4.3	odd	2
2672.2.a.m.1.7		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2672 = 2^{4} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2672.a (trivial)

\(f(q)\)	\(=\)	\(q+1.63574 q^{3} -0.530251 q^{5} +0.325283 q^{7} -0.324343 q^{9} +O(q^{10})\)	\(q+1.63574 q^{3} -0.530251 q^{5} +0.325283 q^{7} -0.324343 q^{9} -3.05624 q^{11} +1.58465 q^{13} -0.867355 q^{15} -2.75292 q^{17} +1.04080 q^{19} +0.532080 q^{21} -2.36742 q^{23} -4.71883 q^{25} -5.43777 q^{27} -6.32884 q^{29} -7.66901 q^{31} -4.99922 q^{33} -0.172482 q^{35} -0.416257 q^{37} +2.59208 q^{39} +5.37869 q^{41} +9.93267 q^{43} +0.171983 q^{45} +3.44772 q^{47} -6.89419 q^{49} -4.50307 q^{51} -1.87999 q^{53} +1.62057 q^{55} +1.70249 q^{57} +11.6115 q^{59} -15.3908 q^{61} -0.105503 q^{63} -0.840262 q^{65} -13.1715 q^{67} -3.87248 q^{69} +16.7791 q^{71} -7.62612 q^{73} -7.71880 q^{75} -0.994144 q^{77} -8.44815 q^{79} -7.92177 q^{81} +11.0056 q^{83} +1.45974 q^{85} -10.3524 q^{87} -5.95109 q^{89} +0.515460 q^{91} -12.5445 q^{93} -0.551886 q^{95} +4.93453 q^{97} +0.991269 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 8 q^{5} - 2 q^{7}+O(q^{10}) \) 9 * q + q^3 - 8 * q^5 - 2 * q^7	\( 9 q + q^{3} - 8 q^{5} - 2 q^{7} + 10 q^{11} - 13 q^{13} - 2 q^{15} - 8 q^{17} + q^{19} - 19 q^{21} + 3 q^{23} + 3 q^{25} + 10 q^{27} - 25 q^{29} + q^{31} - 12 q^{33} + 17 q^{35} - 35 q^{37} + 4 q^{39} - 16 q^{41} - 9 q^{43} - 24 q^{45} + q^{47} - q^{49} + 10 q^{51} - 29 q^{53} - 9 q^{55} - 17 q^{57} + 14 q^{59} - 28 q^{61} - 4 q^{63} - 31 q^{65} - 19 q^{67} - 19 q^{69} + 9 q^{71} + 7 q^{75} - 33 q^{77} + 18 q^{79} - 27 q^{81} + 13 q^{83} - 36 q^{85} - 18 q^{87} - 21 q^{89} - 20 q^{91} - 35 q^{93} + 12 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) 9 * q + q^3 - 8 * q^5 - 2 * q^7 + 10 * q^11 - 13 * q^13 - 2 * q^15 - 8 * q^17 + q^19 - 19 * q^21 + 3 * q^23 + 3 * q^25 + 10 * q^27 - 25 * q^29 + q^31 - 12 * q^33 + 17 * q^35 - 35 * q^37 + 4 * q^39 - 16 * q^41 - 9 * q^43 - 24 * q^45 + q^47 - q^49 + 10 * q^51 - 29 * q^53 - 9 * q^55 - 17 * q^57 + 14 * q^59 - 28 * q^61 - 4 * q^63 - 31 * q^65 - 19 * q^67 - 19 * q^69 + 9 * q^71 + 7 * q^75 - 33 * q^77 + 18 * q^79 - 27 * q^81 + 13 * q^83 - 36 * q^85 - 18 * q^87 - 21 * q^89 - 20 * q^91 - 35 * q^93 + 12 * q^95 + 2 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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