LMFDB - Embedded newform 1672.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1672,2,Mod(1,1672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1672 = 2^{3} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1672.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.3509872180$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14822$ of defining polynomial
Character	$\chi$	$=$	1672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14822 q^{3} +0.677107 q^{5} -1.67711 q^{7} +6.91132 q^{9} +O(q^{10})$	$q-3.14822 q^{3} +0.677107 q^{5} -1.67711 q^{7} +6.91132 q^{9} -1.00000 q^{11} -4.16289 q^{13} -2.13169 q^{15} +6.86889 q^{17} +1.00000 q^{19} +5.27991 q^{21} +2.14077 q^{23} -4.54153 q^{25} -12.3137 q^{27} +2.01005 q^{29} -2.13169 q^{31} +3.14822 q^{33} -1.13558 q^{35} +2.81297 q^{37} +13.1057 q^{39} +10.9829 q^{41} +3.39032 q^{43} +4.67970 q^{45} +1.22974 q^{47} -4.18731 q^{49} -21.6248 q^{51} -11.7292 q^{53} -0.677107 q^{55} -3.14822 q^{57} -8.37650 q^{59} +9.52619 q^{61} -11.5910 q^{63} -2.81872 q^{65} -9.64636 q^{67} -6.73963 q^{69} +12.6011 q^{71} -14.1743 q^{73} +14.2977 q^{75} +1.67711 q^{77} -11.2387 q^{79} +18.0324 q^{81} -7.75976 q^{83} +4.65098 q^{85} -6.32808 q^{87} -4.08041 q^{89} +6.98160 q^{91} +6.71102 q^{93} +0.677107 q^{95} -12.4487 q^{97} -6.91132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	$ 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 6 * q^11 - 3 * q^13 - q^15 - q^17 + 6 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - 16 * q^27 - 7 * q^29 - q^31 + 4 * q^33 - 32 * q^35 + 5 * q^37 - 13 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^47 - 7 * q^49 - 35 * q^51 - 11 * q^53 + 3 * q^55 - 4 * q^57 - 18 * q^59 + 16 * q^61 - 6 * q^63 + 10 * q^65 - 18 * q^67 - 2 * q^69 - 7 * q^71 - 21 * q^73 - 23 * q^75 + 3 * q^77 - 22 * q^79 - 10 * q^81 - 16 * q^83 - 3 * q^87 - 13 * q^89 - 7 * q^91 - 9 * q^93 - 3 * q^95 - 15 * q^97 - 6 * 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
$3$	−3.14822	−1.81763	−0.908814	−	0.417201i	$-0.863011\pi$
$3$	−3.14822	−1.81763	−0.908814	+	0.417201i	$0.863011\pi$
$4$	0	0
$5$	0.677107	0.302812	0.151406	−	0.988472i	$-0.451620\pi$
$5$	0.677107	0.302812	0.151406	+	0.988472i	$0.451620\pi$
$6$	0	0
$7$	−1.67711	−0.633887	−0.316943	−	0.948444i	$-0.602657\pi$
$7$	−1.67711	−0.633887	−0.316943	+	0.948444i	$0.602657\pi$
$8$	0	0
$9$	6.91132	2.30377
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.16289	−1.15458	−0.577288	−	0.816540i	$-0.695889\pi$
$13$	−4.16289	−1.15458	−0.577288	+	0.816540i	$0.695889\pi$
$14$	0	0
$15$	−2.13169	−0.550399
$16$	0	0
$17$	6.86889	1.66595	0.832976	−	0.553310i	$-0.186635\pi$
$17$	6.86889	1.66595	0.832976	+	0.553310i	$0.186635\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	5.27991	1.15217
$22$	0	0
$23$	2.14077	0.446382	0.223191	−	0.974775i	$-0.428353\pi$
$23$	2.14077	0.446382	0.223191	+	0.974775i	$0.428353\pi$
$24$	0	0
$25$	−4.54153	−0.908305
$26$	0	0
$27$	−12.3137	−2.36977
$28$	0	0
$29$	2.01005	0.373257	0.186628	−	0.982431i	$-0.440244\pi$
$29$	2.01005	0.373257	0.186628	+	0.982431i	$0.440244\pi$
$30$	0	0
$31$	−2.13169	−0.382862	−0.191431	−	0.981506i	$-0.561313\pi$
$31$	−2.13169	−0.382862	−0.191431	+	0.981506i	$0.561313\pi$
$32$	0	0
$33$	3.14822	0.548036
$34$	0	0
$35$	−1.13558	−0.191948
$36$	0	0
$37$	2.81297	0.462450	0.231225	−	0.972900i	$-0.425727\pi$
$37$	2.81297	0.462450	0.231225	+	0.972900i	$0.425727\pi$
$38$	0	0
$39$	13.1057	2.09859
$40$	0	0
$41$	10.9829	1.71524	0.857622	−	0.514280i	$-0.171941\pi$
$41$	10.9829	1.71524	0.857622	+	0.514280i	$0.171941\pi$
$42$	0	0
$43$	3.39032	0.517019	0.258510	−	0.966009i	$-0.416769\pi$
$43$	3.39032	0.517019	0.258510	+	0.966009i	$0.416769\pi$
$44$	0	0
$45$	4.67970	0.697609
$46$	0	0
$47$	1.22974	0.179376	0.0896878	−	0.995970i	$-0.471413\pi$
$47$	1.22974	0.179376	0.0896878	+	0.995970i	$0.471413\pi$
$48$	0	0
$49$	−4.18731	−0.598187
$50$	0	0
$51$	−21.6248	−3.02808
$52$	0	0
$53$	−11.7292	−1.61113	−0.805564	−	0.592508i	$-0.798138\pi$
$53$	−11.7292	−1.61113	−0.805564	+	0.592508i	$0.798138\pi$
$54$	0	0
$55$	−0.677107	−0.0913011
$56$	0	0
$57$	−3.14822	−0.416993
$58$	0	0
$59$	−8.37650	−1.09053	−0.545264	−	0.838264i	$-0.683571\pi$
$59$	−8.37650	−1.09053	−0.545264	+	0.838264i	$0.683571\pi$
$60$	0	0
$61$	9.52619	1.21970	0.609852	−	0.792516i	$-0.291229\pi$
$61$	9.52619	1.21970	0.609852	+	0.792516i	$0.291229\pi$
$62$	0	0
$63$	−11.5910	−1.46033
$64$	0	0
$65$	−2.81872	−0.349619
$66$	0	0
$67$	−9.64636	−1.17849	−0.589245	−	0.807954i	$-0.700575\pi$
$67$	−9.64636	−1.17849	−0.589245	+	0.807954i	$0.700575\pi$
$68$	0	0
$69$	−6.73963	−0.811356
$70$	0	0
$71$	12.6011	1.49547	0.747736	−	0.663996i	$-0.231141\pi$
$71$	12.6011	1.49547	0.747736	+	0.663996i	$0.231141\pi$
$72$	0	0
$73$	−14.1743	−1.65897	−0.829485	−	0.558528i	$-0.811366\pi$
$73$	−14.1743	−1.65897	−0.829485	+	0.558528i	$0.811366\pi$
$74$	0	0
$75$	14.2977	1.65096
$76$	0	0
$77$	1.67711	0.191124
$78$	0	0
$79$	−11.2387	−1.26445	−0.632225	−	0.774785i	$-0.717858\pi$
$79$	−11.2387	−1.26445	−0.632225	+	0.774785i	$0.717858\pi$
$80$	0	0
$81$	18.0324	2.00360
$82$	0	0
$83$	−7.75976	−0.851744	−0.425872	−	0.904783i	$-0.640033\pi$
$83$	−7.75976	−0.851744	−0.425872	+	0.904783i	$0.640033\pi$
$84$	0	0
$85$	4.65098	0.504469
$86$	0	0
$87$	−6.32808	−0.678442
$88$	0	0
$89$	−4.08041	−0.432523	−0.216261	−	0.976336i	$-0.569386\pi$
$89$	−4.08041	−0.432523	−0.216261	+	0.976336i	$0.569386\pi$
$90$	0	0
$91$	6.98160	0.731871
$92$	0	0
$93$	6.71102	0.695901
$94$	0	0
$95$	0.677107	0.0694697
$96$	0	0
$97$	−12.4487	−1.26398	−0.631988	−	0.774978i	$-0.717761\pi$
$97$	−12.4487	−1.26398	−0.631988	+	0.774978i	$0.717761\pi$
$98$	0	0
$99$	−6.91132	−0.694614
$100$	0	0
$101$	11.5645	1.15071	0.575353	−	0.817905i	$-0.304865\pi$
$101$	11.5645	1.15071	0.575353	+	0.817905i	$0.304865\pi$
$102$	0	0
$103$	10.0938	0.994569	0.497284	−	0.867588i	$-0.334331\pi$
$103$	10.0938	0.994569	0.497284	+	0.867588i	$0.334331\pi$
$104$	0	0
$105$	3.57506	0.348891
$106$	0	0
$107$	−14.3080	−1.38320	−0.691601	−	0.722280i	$-0.743094\pi$
$107$	−14.3080	−1.38320	−0.691601	+	0.722280i	$0.743094\pi$
$108$	0	0
$109$	−20.0639	−1.92177	−0.960887	−	0.276940i	$-0.910680\pi$
$109$	−20.0639	−1.92177	−0.960887	+	0.276940i	$0.910680\pi$
$110$	0	0
$111$	−8.85586	−0.840561
$112$	0	0
$113$	−14.7940	−1.39171	−0.695853	−	0.718185i	$-0.744973\pi$
$113$	−14.7940	−1.39171	−0.695853	+	0.718185i	$0.744973\pi$
$114$	0	0
$115$	1.44953	0.135170
$116$	0	0
$117$	−28.7710	−2.65988
$118$	0	0
$119$	−11.5199	−1.05602
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−34.5767	−3.11768
$124$	0	0
$125$	−6.46064	−0.577857
$126$	0	0
$127$	10.5225	0.933718	0.466859	−	0.884332i	$-0.345386\pi$
$127$	10.5225	0.933718	0.466859	+	0.884332i	$0.345386\pi$
$128$	0	0
$129$	−10.6735	−0.939749
$130$	0	0
$131$	−9.76083	−0.852808	−0.426404	−	0.904533i	$-0.640220\pi$
$131$	−9.76083	−0.852808	−0.426404	+	0.904533i	$0.640220\pi$
$132$	0	0
$133$	−1.67711	−0.145424
$134$	0	0
$135$	−8.33770	−0.717595
$136$	0	0
$137$	1.72444	0.147329	0.0736644	−	0.997283i	$-0.476531\pi$
$137$	1.72444	0.147329	0.0736644	+	0.997283i	$0.476531\pi$
$138$	0	0
$139$	−2.73416	−0.231908	−0.115954	−	0.993255i	$-0.536993\pi$
$139$	−2.73416	−0.231908	−0.115954	+	0.993255i	$0.536993\pi$
$140$	0	0
$141$	−3.87149	−0.326038
$142$	0	0
$143$	4.16289	0.348118
$144$	0	0
$145$	1.36102	0.113026
$146$	0	0
$147$	13.1826	1.08728
$148$	0	0
$149$	−18.6670	−1.52926	−0.764632	−	0.644468i	$-0.777079\pi$
$149$	−18.6670	−1.52926	−0.764632	+	0.644468i	$0.777079\pi$
$150$	0	0
$151$	−10.5284	−0.856791	−0.428395	−	0.903591i	$-0.640921\pi$
$151$	−10.5284	−0.856791	−0.428395	+	0.903591i	$0.640921\pi$
$152$	0	0
$153$	47.4731	3.83797
$154$	0	0
$155$	−1.44338	−0.115935
$156$	0	0
$157$	−11.6625	−0.930765	−0.465383	−	0.885110i	$-0.654083\pi$
$157$	−11.6625	−0.930765	−0.465383	+	0.885110i	$0.654083\pi$
$158$	0	0
$159$	36.9261	2.92843
$160$	0	0
$161$	−3.59030	−0.282956
$162$	0	0
$163$	14.9612	1.17185	0.585926	−	0.810364i	$-0.300731\pi$
$163$	14.9612	1.17185	0.585926	+	0.810364i	$0.300731\pi$
$164$	0	0
$165$	2.13169	0.165951
$166$	0	0
$167$	−10.5989	−0.820164	−0.410082	−	0.912049i	$-0.634500\pi$
$167$	−10.5989	−0.820164	−0.410082	+	0.912049i	$0.634500\pi$
$168$	0	0
$169$	4.32961	0.333047
$170$	0	0
$171$	6.91132	0.528522
$172$	0	0
$173$	3.76338	0.286124	0.143062	−	0.989714i	$-0.454305\pi$
$173$	3.76338	0.286124	0.143062	+	0.989714i	$0.454305\pi$
$174$	0	0
$175$	7.61663	0.575763
$176$	0	0
$177$	26.3711	1.98217
$178$	0	0
$179$	−3.50407	−0.261907	−0.130953	−	0.991389i	$-0.541804\pi$
$179$	−3.50407	−0.261907	−0.130953	+	0.991389i	$0.541804\pi$
$180$	0	0
$181$	25.3354	1.88317	0.941584	−	0.336778i	$-0.109337\pi$
$181$	25.3354	1.88317	0.941584	+	0.336778i	$0.109337\pi$
$182$	0	0
$183$	−29.9906	−2.21697
$184$	0	0
$185$	1.90468	0.140035
$186$	0	0
$187$	−6.86889	−0.502303
$188$	0	0
$189$	20.6514	1.50217
$190$	0	0
$191$	4.57468	0.331012	0.165506	−	0.986209i	$-0.447074\pi$
$191$	4.57468	0.331012	0.165506	+	0.986209i	$0.447074\pi$
$192$	0	0
$193$	1.55348	0.111822	0.0559111	−	0.998436i	$-0.482194\pi$
$193$	1.55348	0.111822	0.0559111	+	0.998436i	$0.482194\pi$
$194$	0	0
$195$	8.87396	0.635478
$196$	0	0
$197$	−14.8166	−1.05564	−0.527821	−	0.849356i	$-0.676991\pi$
$197$	−14.8166	−1.05564	−0.527821	+	0.849356i	$0.676991\pi$
$198$	0	0
$199$	−8.99201	−0.637427	−0.318713	−	0.947851i	$-0.603251\pi$
$199$	−8.99201	−0.637427	−0.318713	+	0.947851i	$0.603251\pi$
$200$	0	0
$201$	30.3689	2.14206
$202$	0	0
$203$	−3.37107	−0.236602
$204$	0	0
$205$	7.43662	0.519396
$206$	0	0
$207$	14.7955	1.02836
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	2.56841	0.176817	0.0884083	−	0.996084i	$-0.471822\pi$
$211$	2.56841	0.176817	0.0884083	+	0.996084i	$0.471822\pi$
$212$	0	0
$213$	−39.6710	−2.71821
$214$	0	0
$215$	2.29561	0.156559
$216$	0	0
$217$	3.57506	0.242691
$218$	0	0
$219$	44.6237	3.01539
$220$	0	0
$221$	−28.5944	−1.92347
$222$	0	0
$223$	−11.2774	−0.755188	−0.377594	−	0.925971i	$-0.623248\pi$
$223$	−11.2774	−0.755188	−0.377594	+	0.925971i	$0.623248\pi$
$224$	0	0
$225$	−31.3879	−2.09253
$226$	0	0
$227$	4.50002	0.298677	0.149339	−	0.988786i	$-0.452286\pi$
$227$	4.50002	0.298677	0.149339	+	0.988786i	$0.452286\pi$
$228$	0	0
$229$	17.9898	1.18880	0.594401	−	0.804169i	$-0.297389\pi$
$229$	17.9898	1.18880	0.594401	+	0.804169i	$0.297389\pi$
$230$	0	0
$231$	−5.27991	−0.347393
$232$	0	0
$233$	−9.65321	−0.632403	−0.316201	−	0.948692i	$-0.602408\pi$
$233$	−9.65321	−0.632403	−0.316201	+	0.948692i	$0.602408\pi$
$234$	0	0
$235$	0.832664	0.0543170
$236$	0	0
$237$	35.3819	2.29830
$238$	0	0
$239$	5.36583	0.347087	0.173543	−	0.984826i	$-0.444478\pi$
$239$	5.36583	0.347087	0.173543	+	0.984826i	$0.444478\pi$
$240$	0	0
$241$	2.30762	0.148647	0.0743236	−	0.997234i	$-0.476320\pi$
$241$	2.30762	0.148647	0.0743236	+	0.997234i	$0.476320\pi$
$242$	0	0
$243$	−19.8288	−1.27202
$244$	0	0
$245$	−2.83526	−0.181138
$246$	0	0
$247$	−4.16289	−0.264878
$248$	0	0
$249$	24.4295	1.54815
$250$	0	0
$251$	11.1322	0.702659	0.351330	−	0.936252i	$-0.385730\pi$
$251$	11.1322	0.702659	0.351330	+	0.936252i	$0.385730\pi$
$252$	0	0
$253$	−2.14077	−0.134589
$254$	0	0
$255$	−14.6423	−0.916937
$256$	0	0
$257$	8.10554	0.505609	0.252805	−	0.967517i	$-0.418647\pi$
$257$	8.10554	0.505609	0.252805	+	0.967517i	$0.418647\pi$
$258$	0	0
$259$	−4.71765	−0.293141
$260$	0	0
$261$	13.8921	0.859898
$262$	0	0
$263$	26.5787	1.63891	0.819455	−	0.573143i	$-0.194276\pi$
$263$	26.5787	1.63891	0.819455	+	0.573143i	$0.194276\pi$
$264$	0	0
$265$	−7.94192	−0.487868
$266$	0	0
$267$	12.8460	0.786165
$268$	0	0
$269$	−4.21563	−0.257032	−0.128516	−	0.991707i	$-0.541021\pi$
$269$	−4.21563	−0.257032	−0.128516	+	0.991707i	$0.541021\pi$
$270$	0	0
$271$	0.870973	0.0529079	0.0264539	−	0.999650i	$-0.491578\pi$
$271$	0.870973	0.0529079	0.0264539	+	0.999650i	$0.491578\pi$
$272$	0	0
$273$	−21.9797	−1.33027
$274$	0	0
$275$	4.54153	0.273864
$276$	0	0
$277$	−17.0705	−1.02567	−0.512835	−	0.858487i	$-0.671405\pi$
$277$	−17.0705	−1.02567	−0.512835	+	0.858487i	$0.671405\pi$
$278$	0	0
$279$	−14.7328	−0.882027
$280$	0	0
$281$	−3.82059	−0.227917	−0.113959	−	0.993486i	$-0.536353\pi$
$281$	−3.82059	−0.227917	−0.113959	+	0.993486i	$0.536353\pi$
$282$	0	0
$283$	−17.7958	−1.05785	−0.528926	−	0.848668i	$-0.677405\pi$
$283$	−17.7958	−1.05785	−0.528926	+	0.848668i	$0.677405\pi$
$284$	0	0
$285$	−2.13169	−0.126270
$286$	0	0
$287$	−18.4195	−1.08727
$288$	0	0
$289$	30.1817	1.77539
$290$	0	0
$291$	39.1914	2.29744
$292$	0	0
$293$	−1.68332	−0.0983403	−0.0491702	−	0.998790i	$-0.515658\pi$
$293$	−1.68332	−0.0983403	−0.0491702	+	0.998790i	$0.515658\pi$
$294$	0	0
$295$	−5.67179	−0.330224
$296$	0	0
$297$	12.3137	0.714514
$298$	0	0
$299$	−8.91179	−0.515382
$300$	0	0
$301$	−5.68593	−0.327732
$302$	0	0
$303$	−36.4075	−2.09156
$304$	0	0
$305$	6.45025	0.369340
$306$	0	0
$307$	7.93624	0.452945	0.226473	−	0.974018i	$-0.427281\pi$
$307$	7.93624	0.452945	0.226473	+	0.974018i	$0.427281\pi$
$308$	0	0
$309$	−31.7775	−1.80776
$310$	0	0
$311$	29.8589	1.69314	0.846570	−	0.532277i	$-0.178663\pi$
$311$	29.8589	1.69314	0.846570	+	0.532277i	$0.178663\pi$
$312$	0	0
$313$	18.8447	1.06516	0.532582	−	0.846378i	$-0.321222\pi$
$313$	18.8447	1.06516	0.532582	+	0.846378i	$0.321222\pi$
$314$	0	0
$315$	−7.84836	−0.442205
$316$	0	0
$317$	5.77789	0.324518	0.162259	−	0.986748i	$-0.448122\pi$
$317$	5.77789	0.324518	0.162259	+	0.986748i	$0.448122\pi$
$318$	0	0
$319$	−2.01005	−0.112541
$320$	0	0
$321$	45.0447	2.51415
$322$	0	0
$323$	6.86889	0.382195
$324$	0	0
$325$	18.9059	1.04871
$326$	0	0
$327$	63.1657	3.49307
$328$	0	0
$329$	−2.06240	−0.113704
$330$	0	0
$331$	10.4610	0.574991	0.287496	−	0.957782i	$-0.407177\pi$
$331$	10.4610	0.574991	0.287496	+	0.957782i	$0.407177\pi$
$332$	0	0
$333$	19.4413	1.06538
$334$	0	0
$335$	−6.53162	−0.356861
$336$	0	0
$337$	−26.5246	−1.44489	−0.722444	−	0.691430i	$-0.756981\pi$
$337$	−26.5246	−1.44489	−0.722444	+	0.691430i	$0.756981\pi$
$338$	0	0
$339$	46.5749	2.52960
$340$	0	0
$341$	2.13169	0.115437
$342$	0	0
$343$	18.7623	1.01307
$344$	0	0
$345$	−4.56345	−0.245688
$346$	0	0
$347$	−9.94655	−0.533959	−0.266979	−	0.963702i	$-0.586026\pi$
$347$	−9.94655	−0.533959	−0.266979	+	0.963702i	$0.586026\pi$
$348$	0	0
$349$	1.51832	0.0812738	0.0406369	−	0.999174i	$-0.487061\pi$
$349$	1.51832	0.0812738	0.0406369	+	0.999174i	$0.487061\pi$
$350$	0	0
$351$	51.2605	2.73609
$352$	0	0
$353$	−34.4107	−1.83150	−0.915749	−	0.401750i	$-0.868402\pi$
$353$	−34.4107	−1.83150	−0.915749	+	0.401750i	$0.868402\pi$
$354$	0	0
$355$	8.53227	0.452846
$356$	0	0
$357$	36.2671	1.91946
$358$	0	0
$359$	1.31071	0.0691769	0.0345884	−	0.999402i	$-0.488988\pi$
$359$	1.31071	0.0691769	0.0345884	+	0.999402i	$0.488988\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.14822	−0.165239
$364$	0	0
$365$	−9.59749	−0.502355
$366$	0	0
$367$	−14.2993	−0.746415	−0.373207	−	0.927748i	$-0.621742\pi$
$367$	−14.2993	−0.746415	−0.373207	+	0.927748i	$0.621742\pi$
$368$	0	0
$369$	75.9065	3.95153
$370$	0	0
$371$	19.6711	1.02127
$372$	0	0
$373$	−20.7508	−1.07443	−0.537217	−	0.843444i	$-0.680525\pi$
$373$	−20.7508	−1.07443	−0.537217	+	0.843444i	$0.680525\pi$
$374$	0	0
$375$	20.3395	1.05033
$376$	0	0
$377$	−8.36760	−0.430953
$378$	0	0
$379$	−9.68867	−0.497673	−0.248837	−	0.968545i	$-0.580048\pi$
$379$	−9.68867	−0.497673	−0.248837	+	0.968545i	$0.580048\pi$
$380$	0	0
$381$	−33.1271	−1.69715
$382$	0	0
$383$	−36.1499	−1.84717	−0.923587	−	0.383390i	$-0.874757\pi$
$383$	−36.1499	−1.84717	−0.923587	+	0.383390i	$0.874757\pi$
$384$	0	0
$385$	1.13558	0.0578746
$386$	0	0
$387$	23.4316	1.19109
$388$	0	0
$389$	−28.0423	−1.42180	−0.710900	−	0.703293i	$-0.751712\pi$
$389$	−28.0423	−1.42180	−0.710900	+	0.703293i	$0.751712\pi$
$390$	0	0
$391$	14.7047	0.743650
$392$	0	0
$393$	30.7293	1.55009
$394$	0	0
$395$	−7.60979	−0.382890
$396$	0	0
$397$	7.80222	0.391582	0.195791	−	0.980646i	$-0.437273\pi$
$397$	7.80222	0.391582	0.195791	+	0.980646i	$0.437273\pi$
$398$	0	0
$399$	5.27991	0.264326
$400$	0	0
$401$	−21.9629	−1.09677	−0.548387	−	0.836225i	$-0.684758\pi$
$401$	−21.9629	−1.09677	−0.548387	+	0.836225i	$0.684758\pi$
$402$	0	0
$403$	8.87396	0.442043
$404$	0	0
$405$	12.2098	0.606712
$406$	0	0
$407$	−2.81297	−0.139434
$408$	0	0
$409$	1.85738	0.0918414	0.0459207	−	0.998945i	$-0.485378\pi$
$409$	1.85738	0.0918414	0.0459207	+	0.998945i	$0.485378\pi$
$410$	0	0
$411$	−5.42892	−0.267789
$412$	0	0
$413$	14.0483	0.691271
$414$	0	0
$415$	−5.25419	−0.257918
$416$	0	0
$417$	8.60774	0.421523
$418$	0	0
$419$	3.01873	0.147474	0.0737372	−	0.997278i	$-0.476507\pi$
$419$	3.01873	0.147474	0.0737372	+	0.997278i	$0.476507\pi$
$420$	0	0
$421$	4.54661	0.221588	0.110794	−	0.993843i	$-0.464661\pi$
$421$	4.54661	0.221588	0.110794	+	0.993843i	$0.464661\pi$
$422$	0	0
$423$	8.49910	0.413240
$424$	0	0
$425$	−31.1953	−1.51319
$426$	0	0
$427$	−15.9764	−0.773154
$428$	0	0
$429$	−13.1057	−0.632749
$430$	0	0
$431$	−12.1446	−0.584984	−0.292492	−	0.956268i	$-0.594484\pi$
$431$	−12.1446	−0.584984	−0.292492	+	0.956268i	$0.594484\pi$
$432$	0	0
$433$	−4.70207	−0.225967	−0.112983	−	0.993597i	$-0.536041\pi$
$433$	−4.70207	−0.225967	−0.112983	+	0.993597i	$0.536041\pi$
$434$	0	0
$435$	−4.28479	−0.205440
$436$	0	0
$437$	2.14077	0.102407
$438$	0	0
$439$	26.9200	1.28482	0.642411	−	0.766360i	$-0.277934\pi$
$439$	26.9200	1.28482	0.642411	+	0.766360i	$0.277934\pi$
$440$	0	0
$441$	−28.9398	−1.37809
$442$	0	0
$443$	4.33483	0.205954	0.102977	−	0.994684i	$-0.467163\pi$
$443$	4.33483	0.205954	0.102977	+	0.994684i	$0.467163\pi$
$444$	0	0
$445$	−2.76288	−0.130973
$446$	0	0
$447$	58.7680	2.77963
$448$	0	0
$449$	10.4154	0.491533	0.245767	−	0.969329i	$-0.420960\pi$
$449$	10.4154	0.491533	0.245767	+	0.969329i	$0.420960\pi$
$450$	0	0
$451$	−10.9829	−0.517166
$452$	0	0
$453$	33.1458	1.55733
$454$	0	0
$455$	4.72729	0.221619
$456$	0	0
$457$	−10.4225	−0.487543	−0.243771	−	0.969833i	$-0.578385\pi$
$457$	−10.4225	−0.487543	−0.243771	+	0.969833i	$0.578385\pi$
$458$	0	0
$459$	−84.5815	−3.94793
$460$	0	0
$461$	34.1110	1.58871	0.794355	−	0.607454i	$-0.207809\pi$
$461$	34.1110	1.58871	0.794355	+	0.607454i	$0.207809\pi$
$462$	0	0
$463$	28.7298	1.33519	0.667594	−	0.744525i	$-0.267324\pi$
$463$	28.7298	1.33519	0.667594	+	0.744525i	$0.267324\pi$
$464$	0	0
$465$	4.54408	0.210727
$466$	0	0
$467$	25.6777	1.18822	0.594111	−	0.804383i	$-0.297504\pi$
$467$	25.6777	1.18822	0.594111	+	0.804383i	$0.297504\pi$
$468$	0	0
$469$	16.1780	0.747030
$470$	0	0
$471$	36.7160	1.69178
$472$	0	0
$473$	−3.39032	−0.155887
$474$	0	0
$475$	−4.54153	−0.208380
$476$	0	0
$477$	−81.0642	−3.71167
$478$	0	0
$479$	−25.4486	−1.16278	−0.581389	−	0.813626i	$-0.697491\pi$
$479$	−25.4486	−1.16278	−0.581389	+	0.813626i	$0.697491\pi$
$480$	0	0
$481$	−11.7101	−0.533934
$482$	0	0
$483$	11.3031	0.514308
$484$	0	0
$485$	−8.42912	−0.382747
$486$	0	0
$487$	6.53376	0.296073	0.148036	−	0.988982i	$-0.452705\pi$
$487$	6.53376	0.296073	0.148036	+	0.988982i	$0.452705\pi$
$488$	0	0
$489$	−47.1013	−2.12999
$490$	0	0
$491$	−42.5368	−1.91966	−0.959831	−	0.280580i	$-0.909473\pi$
$491$	−42.5368	−1.91966	−0.959831	+	0.280580i	$0.909473\pi$
$492$	0	0
$493$	13.8068	0.621827
$494$	0	0
$495$	−4.67970	−0.210337
$496$	0	0
$497$	−21.1333	−0.947960
$498$	0	0
$499$	−43.6620	−1.95458	−0.977289	−	0.211912i	$-0.932031\pi$
$499$	−43.6620	−1.95458	−0.977289	+	0.211912i	$0.932031\pi$
$500$	0	0
$501$	33.3676	1.49075
$502$	0	0
$503$	−14.1374	−0.630354	−0.315177	−	0.949033i	$-0.602064\pi$
$503$	−14.1374	−0.630354	−0.315177	+	0.949033i	$0.602064\pi$
$504$	0	0
$505$	7.83037	0.348447
$506$	0	0
$507$	−13.6306	−0.605356
$508$	0	0
$509$	6.20694	0.275118	0.137559	−	0.990494i	$-0.456074\pi$
$509$	6.20694	0.275118	0.137559	+	0.990494i	$0.456074\pi$
$510$	0	0
$511$	23.7717	1.05160
$512$	0	0
$513$	−12.3137	−0.543663
$514$	0	0
$515$	6.83456	0.301167
$516$	0	0
$517$	−1.22974	−0.0540838
$518$	0	0
$519$	−11.8480	−0.520067
$520$	0	0
$521$	28.7876	1.26121	0.630605	−	0.776104i	$-0.282807\pi$
$521$	28.7876	1.26121	0.630605	+	0.776104i	$0.282807\pi$
$522$	0	0
$523$	−29.1445	−1.27440	−0.637199	−	0.770699i	$-0.719907\pi$
$523$	−29.1445	−1.27440	−0.637199	+	0.770699i	$0.719907\pi$
$524$	0	0
$525$	−23.9788	−1.04652
$526$	0	0
$527$	−14.6423	−0.637829
$528$	0	0
$529$	−18.4171	−0.800743
$530$	0	0
$531$	−57.8927	−2.51233
$532$	0	0
$533$	−45.7207	−1.98038
$534$	0	0
$535$	−9.68802	−0.418850
$536$	0	0
$537$	11.0316	0.476049
$538$	0	0
$539$	4.18731	0.180360
$540$	0	0
$541$	23.3632	1.00446	0.502230	−	0.864734i	$-0.332513\pi$
$541$	23.3632	1.00446	0.502230	+	0.864734i	$0.332513\pi$
$542$	0	0
$543$	−79.7616	−3.42290
$544$	0	0
$545$	−13.5854	−0.581935
$546$	0	0
$547$	−39.8726	−1.70483	−0.852416	−	0.522865i	$-0.824863\pi$
$547$	−39.8726	−1.70483	−0.852416	+	0.522865i	$0.824863\pi$
$548$	0	0
$549$	65.8385	2.80992
$550$	0	0
$551$	2.01005	0.0856309
$552$	0	0
$553$	18.8485	0.801519
$554$	0	0
$555$	−5.99637	−0.254532
$556$	0	0
$557$	−16.8257	−0.712929	−0.356465	−	0.934309i	$-0.616018\pi$
$557$	−16.8257	−0.712929	−0.356465	+	0.934309i	$0.616018\pi$
$558$	0	0
$559$	−14.1135	−0.596938
$560$	0	0
$561$	21.6248	0.913000
$562$	0	0
$563$	8.68841	0.366173	0.183086	−	0.983097i	$-0.441391\pi$
$563$	8.68841	0.366173	0.183086	+	0.983097i	$0.441391\pi$
$564$	0	0
$565$	−10.0171	−0.421424
$566$	0	0
$567$	−30.2422	−1.27005
$568$	0	0
$569$	20.7590	0.870262	0.435131	−	0.900367i	$-0.356702\pi$
$569$	20.7590	0.870262	0.435131	+	0.900367i	$0.356702\pi$
$570$	0	0
$571$	−28.5889	−1.19641	−0.598203	−	0.801344i	$-0.704119\pi$
$571$	−28.5889	−1.19641	−0.598203	+	0.801344i	$0.704119\pi$
$572$	0	0
$573$	−14.4021	−0.601657
$574$	0	0
$575$	−9.72237	−0.405451
$576$	0	0
$577$	8.97270	0.373538	0.186769	−	0.982404i	$-0.440198\pi$
$577$	8.97270	0.373538	0.186769	+	0.982404i	$0.440198\pi$
$578$	0	0
$579$	−4.89071	−0.203251
$580$	0	0
$581$	13.0139	0.539909
$582$	0	0
$583$	11.7292	0.485774
$584$	0	0
$585$	−19.4811	−0.805443
$586$	0	0
$587$	1.59725	0.0659254	0.0329627	−	0.999457i	$-0.489506\pi$
$587$	1.59725	0.0659254	0.0329627	+	0.999457i	$0.489506\pi$
$588$	0	0
$589$	−2.13169	−0.0878346
$590$	0	0
$591$	46.6461	1.91876
$592$	0	0
$593$	−0.933009	−0.0383141	−0.0191571	−	0.999816i	$-0.506098\pi$
$593$	−0.933009	−0.0383141	−0.0191571	+	0.999816i	$0.506098\pi$
$594$	0	0
$595$	−7.80018	−0.319776
$596$	0	0
$597$	28.3089	1.15860
$598$	0	0
$599$	−0.280302	−0.0114528	−0.00572641	−	0.999984i	$-0.501823\pi$
$599$	−0.280302	−0.0114528	−0.00572641	+	0.999984i	$0.501823\pi$
$600$	0	0
$601$	−15.5729	−0.635230	−0.317615	−	0.948220i	$-0.602882\pi$
$601$	−15.5729	−0.635230	−0.317615	+	0.948220i	$0.602882\pi$
$602$	0	0
$603$	−66.6691	−2.71497
$604$	0	0
$605$	0.677107	0.0275283
$606$	0	0
$607$	−0.305109	−0.0123840	−0.00619199	−	0.999981i	$-0.501971\pi$
$607$	−0.305109	−0.0123840	−0.00619199	+	0.999981i	$0.501971\pi$
$608$	0	0
$609$	10.6129	0.430055
$610$	0	0
$611$	−5.11925	−0.207103
$612$	0	0
$613$	29.6347	1.19693	0.598467	−	0.801148i	$-0.295777\pi$
$613$	29.6347	1.19693	0.598467	+	0.801148i	$0.295777\pi$
$614$	0	0
$615$	−23.4121	−0.944069
$616$	0	0
$617$	−11.8931	−0.478798	−0.239399	−	0.970921i	$-0.576950\pi$
$617$	−11.8931	−0.478798	−0.239399	+	0.970921i	$0.576950\pi$
$618$	0	0
$619$	−9.11269	−0.366270	−0.183135	−	0.983088i	$-0.558625\pi$
$619$	−9.11269	−0.366270	−0.183135	+	0.983088i	$0.558625\pi$
$620$	0	0
$621$	−26.3608	−1.05782
$622$	0	0
$623$	6.84329	0.274170
$624$	0	0
$625$	18.3331	0.733324
$626$	0	0
$627$	3.14822	0.125728
$628$	0	0
$629$	19.3220	0.770418
$630$	0	0
$631$	27.0964	1.07869	0.539346	−	0.842084i	$-0.318671\pi$
$631$	27.0964	1.07869	0.539346	+	0.842084i	$0.318671\pi$
$632$	0	0
$633$	−8.08593	−0.321387
$634$	0	0
$635$	7.12483	0.282740
$636$	0	0
$637$	17.4313	0.690653
$638$	0	0
$639$	87.0900	3.44523
$640$	0	0
$641$	−31.8022	−1.25611	−0.628056	−	0.778168i	$-0.716149\pi$
$641$	−31.8022	−1.25611	−0.628056	+	0.778168i	$0.716149\pi$
$642$	0	0
$643$	−34.3861	−1.35606	−0.678028	−	0.735036i	$-0.737165\pi$
$643$	−34.3861	−1.35606	−0.678028	+	0.735036i	$0.737165\pi$
$644$	0	0
$645$	−7.22710	−0.284567
$646$	0	0
$647$	31.0285	1.21986	0.609928	−	0.792457i	$-0.291198\pi$
$647$	31.0285	1.21986	0.609928	+	0.792457i	$0.291198\pi$
$648$	0	0
$649$	8.37650	0.328806
$650$	0	0
$651$	−11.2551	−0.441122
$652$	0	0
$653$	−31.9348	−1.24971	−0.624853	−	0.780743i	$-0.714841\pi$
$653$	−31.9348	−1.24971	−0.624853	+	0.780743i	$0.714841\pi$
$654$	0	0
$655$	−6.60913	−0.258240
$656$	0	0
$657$	−97.9628	−3.82189
$658$	0	0
$659$	45.5268	1.77347	0.886736	−	0.462277i	$-0.152967\pi$
$659$	45.5268	1.77347	0.886736	+	0.462277i	$0.152967\pi$
$660$	0	0
$661$	−29.2167	−1.13640	−0.568200	−	0.822891i	$-0.692360\pi$
$661$	−29.2167	−1.13640	−0.568200	+	0.822891i	$0.692360\pi$
$662$	0	0
$663$	90.0216	3.49615
$664$	0	0
$665$	−1.13558	−0.0440359
$666$	0	0
$667$	4.30305	0.166615
$668$	0	0
$669$	35.5036	1.37265
$670$	0	0
$671$	−9.52619	−0.367754
$672$	0	0
$673$	45.9862	1.77264	0.886319	−	0.463076i	$-0.153254\pi$
$673$	45.9862	1.77264	0.886319	+	0.463076i	$0.153254\pi$
$674$	0	0
$675$	55.9230	2.15248
$676$	0	0
$677$	−16.9360	−0.650904	−0.325452	−	0.945559i	$-0.605516\pi$
$677$	−16.9360	−0.650904	−0.325452	+	0.945559i	$0.605516\pi$
$678$	0	0
$679$	20.8779	0.801218
$680$	0	0
$681$	−14.1671	−0.542884
$682$	0	0
$683$	1.67863	0.0642308	0.0321154	−	0.999484i	$-0.489776\pi$
$683$	1.67863	0.0642308	0.0321154	+	0.999484i	$0.489776\pi$
$684$	0	0
$685$	1.16763	0.0446129
$686$	0	0
$687$	−56.6361	−2.16080
$688$	0	0
$689$	48.8273	1.86017
$690$	0	0
$691$	41.2185	1.56803	0.784013	−	0.620744i	$-0.213170\pi$
$691$	41.2185	1.56803	0.784013	+	0.620744i	$0.213170\pi$
$692$	0	0
$693$	11.5910	0.440306
$694$	0	0
$695$	−1.85132	−0.0702244
$696$	0	0
$697$	75.4405	2.85751
$698$	0	0
$699$	30.3905	1.14947
$700$	0	0
$701$	−26.4831	−1.00025	−0.500126	−	0.865953i	$-0.666713\pi$
$701$	−26.4831	−1.00025	−0.500126	+	0.865953i	$0.666713\pi$
$702$	0	0
$703$	2.81297	0.106093
$704$	0	0
$705$	−2.62141	−0.0987281
$706$	0	0
$707$	−19.3948	−0.729418
$708$	0	0
$709$	36.8689	1.38464	0.692319	−	0.721591i	$-0.256589\pi$
$709$	36.8689	1.38464	0.692319	+	0.721591i	$0.256589\pi$
$710$	0	0
$711$	−77.6741	−2.91301
$712$	0	0
$713$	−4.56345	−0.170903
$714$	0	0
$715$	2.81872	0.105414
$716$	0	0
$717$	−16.8928	−0.630875
$718$	0	0
$719$	−19.7419	−0.736250	−0.368125	−	0.929776i	$-0.620000\pi$
$719$	−19.7419	−0.736250	−0.368125	+	0.929776i	$0.620000\pi$
$720$	0	0
$721$	−16.9283	−0.630444
$722$	0	0
$723$	−7.26492	−0.270185
$724$	0	0
$725$	−9.12869	−0.339031
$726$	0	0
$727$	33.7540	1.25187	0.625934	−	0.779876i	$-0.284718\pi$
$727$	33.7540	1.25187	0.625934	+	0.779876i	$0.284718\pi$
$728$	0	0
$729$	8.32841	0.308460
$730$	0	0
$731$	23.2878	0.861329
$732$	0	0
$733$	−35.0242	−1.29365	−0.646825	−	0.762639i	$-0.723904\pi$
$733$	−35.0242	−1.29365	−0.646825	+	0.762639i	$0.723904\pi$
$734$	0	0
$735$	8.92603	0.329242
$736$	0	0
$737$	9.64636	0.355328
$738$	0	0
$739$	−8.99477	−0.330878	−0.165439	−	0.986220i	$-0.552904\pi$
$739$	−8.99477	−0.330878	−0.165439	+	0.986220i	$0.552904\pi$
$740$	0	0
$741$	13.1057	0.481450
$742$	0	0
$743$	17.8359	0.654337	0.327168	−	0.944966i	$-0.393906\pi$
$743$	17.8359	0.654337	0.327168	+	0.944966i	$0.393906\pi$
$744$	0	0
$745$	−12.6396	−0.463079
$746$	0	0
$747$	−53.6301	−1.96222
$748$	0	0
$749$	23.9960	0.876794
$750$	0	0
$751$	−10.8155	−0.394662	−0.197331	−	0.980337i	$-0.563227\pi$
$751$	−10.8155	−0.394662	−0.197331	+	0.980337i	$0.563227\pi$
$752$	0	0
$753$	−35.0467	−1.27717
$754$	0	0
$755$	−7.12887	−0.259446
$756$	0	0
$757$	29.0949	1.05747	0.528737	−	0.848786i	$-0.322666\pi$
$757$	29.0949	1.05747	0.528737	+	0.848786i	$0.322666\pi$
$758$	0	0
$759$	6.73963	0.244633
$760$	0	0
$761$	−24.8989	−0.902586	−0.451293	−	0.892376i	$-0.649037\pi$
$761$	−24.8989	−0.902586	−0.451293	+	0.892376i	$0.649037\pi$
$762$	0	0
$763$	33.6493	1.21819
$764$	0	0
$765$	32.1444	1.16218
$766$	0	0
$767$	34.8704	1.25910
$768$	0	0
$769$	−22.4656	−0.810129	−0.405064	−	0.914288i	$-0.632751\pi$
$769$	−22.4656	−0.810129	−0.405064	+	0.914288i	$0.632751\pi$
$770$	0	0
$771$	−25.5180	−0.919010
$772$	0	0
$773$	−0.491907	−0.0176927	−0.00884633	−	0.999961i	$-0.502816\pi$
$773$	−0.491907	−0.0176927	−0.00884633	+	0.999961i	$0.502816\pi$
$774$	0	0
$775$	9.68110	0.347756
$776$	0	0
$777$	14.8522	0.532821
$778$	0	0
$779$	10.9829	0.393504
$780$	0	0
$781$	−12.6011	−0.450902
$782$	0	0
$783$	−24.7511	−0.884534
$784$	0	0
$785$	−7.89673	−0.281846
$786$	0	0
$787$	30.7601	1.09648	0.548239	−	0.836322i	$-0.315298\pi$
$787$	30.7601	1.09648	0.548239	+	0.836322i	$0.315298\pi$
$788$	0	0
$789$	−83.6756	−2.97893
$790$	0	0
$791$	24.8112	0.882184
$792$	0	0
$793$	−39.6564	−1.40824
$794$	0	0
$795$	25.0030	0.886763
$796$	0	0
$797$	−46.8176	−1.65836	−0.829182	−	0.558978i	$-0.811193\pi$
$797$	−46.8176	−1.65836	−0.829182	+	0.558978i	$0.811193\pi$
$798$	0	0
$799$	8.44693	0.298831
$800$	0	0
$801$	−28.2010	−0.996434
$802$	0	0
$803$	14.1743	0.500199
$804$	0	0
$805$	−2.43102	−0.0856822
$806$	0	0
$807$	13.2718	0.467188
$808$	0	0
$809$	−15.3194	−0.538601	−0.269300	−	0.963056i	$-0.586792\pi$
$809$	−15.3194	−0.538601	−0.269300	+	0.963056i	$0.586792\pi$
$810$	0	0
$811$	21.0369	0.738706	0.369353	−	0.929289i	$-0.379579\pi$
$811$	21.0369	0.738706	0.369353	+	0.929289i	$0.379579\pi$
$812$	0	0
$813$	−2.74202	−0.0961668
$814$	0	0
$815$	10.1303	0.354851
$816$	0	0
$817$	3.39032	0.118612
$818$	0	0
$819$	48.2521	1.68606
$820$	0	0
$821$	−49.3342	−1.72178	−0.860888	−	0.508794i	$-0.830092\pi$
$821$	−49.3342	−1.72178	−0.860888	+	0.508794i	$0.830092\pi$
$822$	0	0
$823$	44.7400	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$823$	44.7400	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$824$	0	0
$825$	−14.2977	−0.497784
$826$	0	0
$827$	−42.7710	−1.48729	−0.743646	−	0.668573i	$-0.766905\pi$
$827$	−42.7710	−1.48729	−0.743646	+	0.668573i	$0.766905\pi$
$828$	0	0
$829$	48.6840	1.69086	0.845432	−	0.534083i	$-0.179343\pi$
$829$	48.6840	1.69086	0.845432	+	0.534083i	$0.179343\pi$
$830$	0	0
$831$	53.7419	1.86429
$832$	0	0
$833$	−28.7622	−0.996551
$834$	0	0
$835$	−7.17656	−0.248355
$836$	0	0
$837$	26.2489	0.907296
$838$	0	0
$839$	−50.1921	−1.73282	−0.866412	−	0.499330i	$-0.833579\pi$
$839$	−50.1921	−1.73282	−0.866412	+	0.499330i	$0.833579\pi$
$840$	0	0
$841$	−24.9597	−0.860680
$842$	0	0
$843$	12.0281	0.414268
$844$	0	0
$845$	2.93161	0.100851
$846$	0	0
$847$	−1.67711	−0.0576261
$848$	0	0
$849$	56.0253	1.92278
$850$	0	0
$851$	6.02193	0.206429
$852$	0	0
$853$	−25.6101	−0.876874	−0.438437	−	0.898762i	$-0.644468\pi$
$853$	−25.6101	−0.876874	−0.438437	+	0.898762i	$0.644468\pi$
$854$	0	0
$855$	4.67970	0.160042
$856$	0	0
$857$	−0.780477	−0.0266606	−0.0133303	−	0.999911i	$-0.504243\pi$
$857$	−0.780477	−0.0266606	−0.0133303	+	0.999911i	$0.504243\pi$
$858$	0	0
$859$	18.5597	0.633247	0.316624	−	0.948551i	$-0.397451\pi$
$859$	18.5597	0.633247	0.316624	+	0.948551i	$0.397451\pi$
$860$	0	0
$861$	57.9889	1.97625
$862$	0	0
$863$	20.5115	0.698220	0.349110	−	0.937082i	$-0.386484\pi$
$863$	20.5115	0.698220	0.349110	+	0.937082i	$0.386484\pi$
$864$	0	0
$865$	2.54821	0.0866417
$866$	0	0
$867$	−95.0187	−3.22700
$868$	0	0
$869$	11.2387	0.381246
$870$	0	0
$871$	40.1567	1.36066
$872$	0	0
$873$	−86.0371	−2.91191
$874$	0	0
$875$	10.8352	0.366296
$876$	0	0
$877$	14.4072	0.486497	0.243248	−	0.969964i	$-0.421787\pi$
$877$	14.4072	0.486497	0.243248	+	0.969964i	$0.421787\pi$
$878$	0	0
$879$	5.29945	0.178746
$880$	0	0
$881$	33.3794	1.12458	0.562291	−	0.826940i	$-0.309920\pi$
$881$	33.3794	1.12458	0.562291	+	0.826940i	$0.309920\pi$
$882$	0	0
$883$	35.6404	1.19940	0.599698	−	0.800226i	$-0.295287\pi$
$883$	35.6404	1.19940	0.599698	+	0.800226i	$0.295287\pi$
$884$	0	0
$885$	17.8561	0.600225
$886$	0	0
$887$	−12.6762	−0.425626	−0.212813	−	0.977093i	$-0.568263\pi$
$887$	−12.6762	−0.425626	−0.212813	+	0.977093i	$0.568263\pi$
$888$	0	0
$889$	−17.6473	−0.591871
$890$	0	0
$891$	−18.0324	−0.604107
$892$	0	0
$893$	1.22974	0.0411516
$894$	0	0
$895$	−2.37263	−0.0793083
$896$	0	0
$897$	28.0563	0.936773
$898$	0	0
$899$	−4.28479	−0.142906
$900$	0	0
$901$	−80.5666	−2.68406
$902$	0	0
$903$	17.9006	0.595694
$904$	0	0
$905$	17.1548	0.570245
$906$	0	0
$907$	36.6608	1.21730	0.608651	−	0.793438i	$-0.291711\pi$
$907$	36.6608	1.21730	0.608651	+	0.793438i	$0.291711\pi$
$908$	0	0
$909$	79.9256	2.65097
$910$	0	0
$911$	−47.6952	−1.58021	−0.790106	−	0.612970i	$-0.789974\pi$
$911$	−47.6952	−1.58021	−0.790106	+	0.612970i	$0.789974\pi$
$912$	0	0
$913$	7.75976	0.256810
$914$	0	0
$915$	−20.3068	−0.671323
$916$	0	0
$917$	16.3700	0.540584
$918$	0	0
$919$	−29.7694	−0.982001	−0.491001	−	0.871159i	$-0.663369\pi$
$919$	−29.7694	−0.982001	−0.491001	+	0.871159i	$0.663369\pi$
$920$	0	0
$921$	−24.9851	−0.823286
$922$	0	0
$923$	−52.4568	−1.72664
$924$	0	0
$925$	−12.7752	−0.420045
$926$	0	0
$927$	69.7613	2.29126
$928$	0	0
$929$	−50.6180	−1.66072	−0.830362	−	0.557225i	$-0.811866\pi$
$929$	−50.6180	−1.66072	−0.830362	+	0.557225i	$0.811866\pi$
$930$	0	0
$931$	−4.18731	−0.137234
$932$	0	0
$933$	−94.0024	−3.07750
$934$	0	0
$935$	−4.65098	−0.152103
$936$	0	0
$937$	16.1084	0.526238	0.263119	−	0.964763i	$-0.415249\pi$
$937$	16.1084	0.526238	0.263119	+	0.964763i	$0.415249\pi$
$938$	0	0
$939$	−59.3273	−1.93607
$940$	0	0
$941$	16.8109	0.548018	0.274009	−	0.961727i	$-0.411650\pi$
$941$	16.8109	0.548018	0.274009	+	0.961727i	$0.411650\pi$
$942$	0	0
$943$	23.5119	0.765654
$944$	0	0
$945$	13.9832	0.454874
$946$	0	0
$947$	36.1350	1.17423	0.587114	−	0.809504i	$-0.300264\pi$
$947$	36.1350	1.17423	0.587114	+	0.809504i	$0.300264\pi$
$948$	0	0
$949$	59.0058	1.91541
$950$	0	0
$951$	−18.1901	−0.589854
$952$	0	0
$953$	−29.1171	−0.943197	−0.471598	−	0.881813i	$-0.656323\pi$
$953$	−29.1171	−0.943197	−0.471598	+	0.881813i	$0.656323\pi$
$954$	0	0
$955$	3.09755	0.100234
$956$	0	0
$957$	6.32808	0.204558
$958$	0	0
$959$	−2.89207	−0.0933898
$960$	0	0
$961$	−26.4559	−0.853417
$962$	0	0
$963$	−98.8868	−3.18658
$964$	0	0
$965$	1.05187	0.0338610
$966$	0	0
$967$	10.1650	0.326883	0.163442	−	0.986553i	$-0.447740\pi$
$967$	10.1650	0.326883	0.163442	+	0.986553i	$0.447740\pi$
$968$	0	0
$969$	−21.6248	−0.694689
$970$	0	0
$971$	15.9729	0.512594	0.256297	−	0.966598i	$-0.417497\pi$
$971$	15.9729	0.512594	0.256297	+	0.966598i	$0.417497\pi$
$972$	0	0
$973$	4.58547	0.147003
$974$	0	0
$975$	−59.5199	−1.90616
$976$	0	0
$977$	42.7941	1.36910	0.684552	−	0.728964i	$-0.259998\pi$
$977$	42.7941	1.36910	0.684552	+	0.728964i	$0.259998\pi$
$978$	0	0
$979$	4.08041	0.130410
$980$	0	0
$981$	−138.668	−4.42733
$982$	0	0
$983$	13.8981	0.443280	0.221640	−	0.975129i	$-0.428859\pi$
$983$	13.8981	0.443280	0.221640	+	0.975129i	$0.428859\pi$
$984$	0	0
$985$	−10.0325	−0.319661
$986$	0	0
$987$	6.49290	0.206671
$988$	0	0
$989$	7.25790	0.230788
$990$	0	0
$991$	−49.5286	−1.57333	−0.786664	−	0.617381i	$-0.788194\pi$
$991$	−49.5286	−1.57333	−0.786664	+	0.617381i	$0.788194\pi$
$992$	0	0
$993$	−32.9337	−1.04512
$994$	0	0
$995$	−6.08855	−0.193020
$996$	0	0
$997$	51.6810	1.63675	0.818377	−	0.574682i	$-0.194874\pi$
$997$	51.6810	1.63675	0.818377	+	0.574682i	$0.194874\pi$
$998$	0	0
$999$	−34.6381	−1.09590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1672.2.a.i.1.1	✓	6
4.3	odd	2		3344.2.a.z.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.1	✓	6	1.1	even	1	trivial
3344.2.a.z.1.6		6	4.3	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:	\( N \)	\(=\)	\( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1672.a (trivial)

\(f(q)\)	\(=\)	\(q-3.14822 q^{3} +0.677107 q^{5} -1.67711 q^{7} +6.91132 q^{9} +O(q^{10})\)	\(q-3.14822 q^{3} +0.677107 q^{5} -1.67711 q^{7} +6.91132 q^{9} -1.00000 q^{11} -4.16289 q^{13} -2.13169 q^{15} +6.86889 q^{17} +1.00000 q^{19} +5.27991 q^{21} +2.14077 q^{23} -4.54153 q^{25} -12.3137 q^{27} +2.01005 q^{29} -2.13169 q^{31} +3.14822 q^{33} -1.13558 q^{35} +2.81297 q^{37} +13.1057 q^{39} +10.9829 q^{41} +3.39032 q^{43} +4.67970 q^{45} +1.22974 q^{47} -4.18731 q^{49} -21.6248 q^{51} -11.7292 q^{53} -0.677107 q^{55} -3.14822 q^{57} -8.37650 q^{59} +9.52619 q^{61} -11.5910 q^{63} -2.81872 q^{65} -9.64636 q^{67} -6.73963 q^{69} +12.6011 q^{71} -14.1743 q^{73} +14.2977 q^{75} +1.67711 q^{77} -11.2387 q^{79} +18.0324 q^{81} -7.75976 q^{83} +4.65098 q^{85} -6.32808 q^{87} -4.08041 q^{89} +6.98160 q^{91} +6.71102 q^{93} +0.677107 q^{95} -12.4487 q^{97} -6.91132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	\( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 6 * q^11 - 3 * q^13 - q^15 - q^17 + 6 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - 16 * q^27 - 7 * q^29 - q^31 + 4 * q^33 - 32 * q^35 + 5 * q^37 - 13 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^47 - 7 * q^49 - 35 * q^51 - 11 * q^53 + 3 * q^55 - 4 * q^57 - 18 * q^59 + 16 * q^61 - 6 * q^63 + 10 * q^65 - 18 * q^67 - 2 * q^69 - 7 * q^71 - 21 * q^73 - 23 * q^75 + 3 * q^77 - 22 * q^79 - 10 * q^81 - 16 * q^83 - 3 * q^87 - 13 * q^89 - 7 * q^91 - 9 * q^93 - 3 * q^95 - 15 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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