LMFDB - Embedded newform 3344.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 $ x^6 - 2x^5 - 12x^4 + 28x^3 + 16x^2 - 60*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.24102$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24102 q^{3} +2.83235 q^{5} +4.46564 q^{7} -1.45986 q^{9} +O(q^{10})$	$q+1.24102 q^{3} +2.83235 q^{5} +4.46564 q^{7} -1.45986 q^{9} -1.00000 q^{11} +2.14252 q^{13} +3.51500 q^{15} +7.81594 q^{17} -1.00000 q^{19} +5.54196 q^{21} -3.81016 q^{23} +3.02218 q^{25} -5.53479 q^{27} +1.86854 q^{29} +1.22996 q^{31} -1.24102 q^{33} +12.6482 q^{35} -8.10815 q^{37} +2.65892 q^{39} +10.9225 q^{41} -2.97782 q^{43} -4.13484 q^{45} -6.28504 q^{47} +12.9419 q^{49} +9.69975 q^{51} +6.43768 q^{53} -2.83235 q^{55} -1.24102 q^{57} -7.40754 q^{59} -9.83806 q^{61} -6.51922 q^{63} +6.06836 q^{65} +13.7013 q^{67} -4.72850 q^{69} +2.27693 q^{71} +2.77299 q^{73} +3.75060 q^{75} -4.46564 q^{77} -15.9183 q^{79} -2.48921 q^{81} -6.37820 q^{83} +22.1374 q^{85} +2.31890 q^{87} +1.65892 q^{89} +9.56772 q^{91} +1.52641 q^{93} -2.83235 q^{95} -5.99423 q^{97} +1.45986 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * 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
$3$	1.24102	0.716505	0.358252	−	0.933625i	$-0.383373\pi$
$3$	1.24102	0.716505	0.358252	+	0.933625i	$0.383373\pi$
$4$	0	0
$5$	2.83235	1.26666	0.633332	−	0.773880i	$-0.281687\pi$
$5$	2.83235	1.26666	0.633332	+	0.773880i	$0.281687\pi$
$6$	0	0
$7$	4.46564	1.68785	0.843926	−	0.536460i	$-0.180239\pi$
$7$	4.46564	1.68785	0.843926	+	0.536460i	$0.180239\pi$
$8$	0	0
$9$	−1.45986	−0.486621
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.14252	0.594229	0.297114	−	0.954842i	$-0.403976\pi$
$13$	2.14252	0.594229	0.297114	+	0.954842i	$0.403976\pi$
$14$	0	0
$15$	3.51500	0.907570
$16$	0	0
$17$	7.81594	1.89564	0.947822	−	0.318801i	$-0.103280\pi$
$17$	7.81594	1.89564	0.947822	+	0.318801i	$0.103280\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.54196	1.20935
$22$	0	0
$23$	−3.81016	−0.794474	−0.397237	−	0.917716i	$-0.630031\pi$
$23$	−3.81016	−0.794474	−0.397237	+	0.917716i	$0.630031\pi$
$24$	0	0
$25$	3.02218	0.604436
$26$	0	0
$27$	−5.53479	−1.06517
$28$	0	0
$29$	1.86854	0.346979	0.173490	−	0.984836i	$-0.444496\pi$
$29$	1.86854	0.346979	0.173490	+	0.984836i	$0.444496\pi$
$30$	0	0
$31$	1.22996	0.220907	0.110454	−	0.993881i	$-0.464770\pi$
$31$	1.22996	0.220907	0.110454	+	0.993881i	$0.464770\pi$
$32$	0	0
$33$	−1.24102	−0.216034
$34$	0	0
$35$	12.6482	2.13794
$36$	0	0
$37$	−8.10815	−1.33297	−0.666485	−	0.745518i	$-0.732202\pi$
$37$	−8.10815	−1.33297	−0.666485	+	0.745518i	$0.732202\pi$
$38$	0	0
$39$	2.65892	0.425768
$40$	0	0
$41$	10.9225	1.70582	0.852908	−	0.522062i	$-0.174837\pi$
$41$	10.9225	1.70582	0.852908	+	0.522062i	$0.174837\pi$
$42$	0	0
$43$	−2.97782	−0.454113	−0.227057	−	0.973882i	$-0.572910\pi$
$43$	−2.97782	−0.454113	−0.227057	+	0.973882i	$0.572910\pi$
$44$	0	0
$45$	−4.13484	−0.616385
$46$	0	0
$47$	−6.28504	−0.916768	−0.458384	−	0.888754i	$-0.651572\pi$
$47$	−6.28504	−0.916768	−0.458384	+	0.888754i	$0.651572\pi$
$48$	0	0
$49$	12.9419	1.84884
$50$	0	0
$51$	9.69975	1.35824
$52$	0	0
$53$	6.43768	0.884284	0.442142	−	0.896945i	$-0.354219\pi$
$53$	6.43768	0.884284	0.442142	+	0.896945i	$0.354219\pi$
$54$	0	0
$55$	−2.83235	−0.381913
$56$	0	0
$57$	−1.24102	−0.164377
$58$	0	0
$59$	−7.40754	−0.964380	−0.482190	−	0.876067i	$-0.660159\pi$
$59$	−7.40754	−0.964380	−0.482190	+	0.876067i	$0.660159\pi$
$60$	0	0
$61$	−9.83806	−1.25963	−0.629817	−	0.776743i	$-0.716870\pi$
$61$	−9.83806	−1.25963	−0.629817	+	0.776743i	$0.716870\pi$
$62$	0	0
$63$	−6.51922	−0.821344
$64$	0	0
$65$	6.06836	0.752688
$66$	0	0
$67$	13.7013	1.67388	0.836941	−	0.547293i	$-0.184342\pi$
$67$	13.7013	1.67388	0.836941	+	0.547293i	$0.184342\pi$
$68$	0	0
$69$	−4.72850	−0.569244
$70$	0	0
$71$	2.27693	0.270222	0.135111	−	0.990830i	$-0.456861\pi$
$71$	2.27693	0.270222	0.135111	+	0.990830i	$0.456861\pi$
$72$	0	0
$73$	2.77299	0.324554	0.162277	−	0.986745i	$-0.448116\pi$
$73$	2.77299	0.324554	0.162277	+	0.986745i	$0.448116\pi$
$74$	0	0
$75$	3.75060	0.433081
$76$	0	0
$77$	−4.46564	−0.508906
$78$	0	0
$79$	−15.9183	−1.79095	−0.895475	−	0.445112i	$-0.853164\pi$
$79$	−15.9183	−1.79095	−0.895475	+	0.445112i	$0.853164\pi$
$80$	0	0
$81$	−2.48921	−0.276579
$82$	0	0
$83$	−6.37820	−0.700098	−0.350049	−	0.936731i	$-0.613835\pi$
$83$	−6.37820	−0.700098	−0.350049	+	0.936731i	$0.613835\pi$
$84$	0	0
$85$	22.1374	2.40114
$86$	0	0
$87$	2.31890	0.248612
$88$	0	0
$89$	1.65892	0.175845	0.0879225	−	0.996127i	$-0.471977\pi$
$89$	1.65892	0.175845	0.0879225	+	0.996127i	$0.471977\pi$
$90$	0	0
$91$	9.56772	1.00297
$92$	0	0
$93$	1.52641	0.158281
$94$	0	0
$95$	−2.83235	−0.290593
$96$	0	0
$97$	−5.99423	−0.608622	−0.304311	−	0.952573i	$-0.598426\pi$
$97$	−5.99423	−0.608622	−0.304311	+	0.952573i	$0.598426\pi$
$98$	0	0
$99$	1.45986	0.146722
$100$	0	0
$101$	11.9627	1.19033	0.595165	−	0.803603i	$-0.297087\pi$
$101$	11.9627	1.19033	0.595165	+	0.803603i	$0.297087\pi$
$102$	0	0
$103$	0.753204	0.0742154	0.0371077	−	0.999311i	$-0.488186\pi$
$103$	0.753204	0.0742154	0.0371077	+	0.999311i	$0.488186\pi$
$104$	0	0
$105$	15.6967	1.53184
$106$	0	0
$107$	1.05803	0.102284	0.0511420	−	0.998691i	$-0.483714\pi$
$107$	1.05803	0.102284	0.0511420	+	0.998691i	$0.483714\pi$
$108$	0	0
$109$	7.10240	0.680286	0.340143	−	0.940374i	$-0.389524\pi$
$109$	7.10240	0.680286	0.340143	+	0.940374i	$0.389524\pi$
$110$	0	0
$111$	−10.0624	−0.955080
$112$	0	0
$113$	−20.8752	−1.96378	−0.981888	−	0.189460i	$-0.939326\pi$
$113$	−20.8752	−1.96378	−0.981888	+	0.189460i	$0.939326\pi$
$114$	0	0
$115$	−10.7917	−1.00633
$116$	0	0
$117$	−3.12779	−0.289164
$118$	0	0
$119$	34.9031	3.19956
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	13.5551	1.22222
$124$	0	0
$125$	−5.60187	−0.501046
$126$	0	0
$127$	19.9848	1.77336	0.886682	−	0.462380i	$-0.153004\pi$
$127$	19.9848	1.77336	0.886682	+	0.462380i	$0.153004\pi$
$128$	0	0
$129$	−3.69554	−0.325374
$130$	0	0
$131$	−14.5666	−1.27269	−0.636346	−	0.771404i	$-0.719555\pi$
$131$	−14.5666	−1.27269	−0.636346	+	0.771404i	$0.719555\pi$
$132$	0	0
$133$	−4.46564	−0.387220
$134$	0	0
$135$	−15.6764	−1.34921
$136$	0	0
$137$	2.89754	0.247554	0.123777	−	0.992310i	$-0.460499\pi$
$137$	2.89754	0.247554	0.123777	+	0.992310i	$0.460499\pi$
$138$	0	0
$139$	−18.2399	−1.54709	−0.773544	−	0.633743i	$-0.781518\pi$
$139$	−18.2399	−1.54709	−0.773544	+	0.633743i	$0.781518\pi$
$140$	0	0
$141$	−7.79988	−0.656868
$142$	0	0
$143$	−2.14252	−0.179167
$144$	0	0
$145$	5.29235	0.439506
$146$	0	0
$147$	16.0612	1.32471
$148$	0	0
$149$	−18.3109	−1.50009	−0.750044	−	0.661387i	$-0.769968\pi$
$149$	−18.3109	−1.50009	−0.750044	+	0.661387i	$0.769968\pi$
$150$	0	0
$151$	15.3574	1.24977	0.624884	−	0.780718i	$-0.285146\pi$
$151$	15.3574	1.24977	0.624884	+	0.780718i	$0.285146\pi$
$152$	0	0
$153$	−11.4102	−0.922460
$154$	0	0
$155$	3.48367	0.279815
$156$	0	0
$157$	−2.81022	−0.224280	−0.112140	−	0.993692i	$-0.535771\pi$
$157$	−2.81022	−0.224280	−0.112140	+	0.993692i	$0.535771\pi$
$158$	0	0
$159$	7.98931	0.633593
$160$	0	0
$161$	−17.0148	−1.34095
$162$	0	0
$163$	6.73427	0.527469	0.263734	−	0.964595i	$-0.415046\pi$
$163$	6.73427	0.527469	0.263734	+	0.964595i	$0.415046\pi$
$164$	0	0
$165$	−3.51500	−0.273643
$166$	0	0
$167$	8.27586	0.640405	0.320203	−	0.947349i	$-0.396249\pi$
$167$	8.27586	0.640405	0.320203	+	0.947349i	$0.396249\pi$
$168$	0	0
$169$	−8.40960	−0.646892
$170$	0	0
$171$	1.45986	0.111639
$172$	0	0
$173$	−4.83263	−0.367418	−0.183709	−	0.982981i	$-0.558810\pi$
$173$	−4.83263	−0.367418	−0.183709	+	0.982981i	$0.558810\pi$
$174$	0	0
$175$	13.4960	1.02020
$176$	0	0
$177$	−9.19293	−0.690983
$178$	0	0
$179$	7.19714	0.537940	0.268970	−	0.963149i	$-0.413317\pi$
$179$	7.19714	0.537940	0.268970	+	0.963149i	$0.413317\pi$
$180$	0	0
$181$	2.75664	0.204899	0.102450	−	0.994738i	$-0.467332\pi$
$181$	2.75664	0.204899	0.102450	+	0.994738i	$0.467332\pi$
$182$	0	0
$183$	−12.2093	−0.902534
$184$	0	0
$185$	−22.9651	−1.68843
$186$	0	0
$187$	−7.81594	−0.571558
$188$	0	0
$189$	−24.7164	−1.79785
$190$	0	0
$191$	−13.9906	−1.01232	−0.506161	−	0.862439i	$-0.668936\pi$
$191$	−13.9906	−1.01232	−0.506161	+	0.862439i	$0.668936\pi$
$192$	0	0
$193$	−6.78688	−0.488530	−0.244265	−	0.969709i	$-0.578547\pi$
$193$	−6.78688	−0.488530	−0.244265	+	0.969709i	$0.578547\pi$
$194$	0	0
$195$	7.53098	0.539304
$196$	0	0
$197$	−1.99861	−0.142395	−0.0711975	−	0.997462i	$-0.522682\pi$
$197$	−1.99861	−0.142395	−0.0711975	+	0.997462i	$0.522682\pi$
$198$	0	0
$199$	−1.31094	−0.0929304	−0.0464652	−	0.998920i	$-0.514796\pi$
$199$	−1.31094	−0.0929304	−0.0464652	+	0.998920i	$0.514796\pi$
$200$	0	0
$201$	17.0036	1.19934
$202$	0	0
$203$	8.34422	0.585649
$204$	0	0
$205$	30.9364	2.16069
$206$	0	0
$207$	5.56232	0.386608
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−7.31644	−0.503685	−0.251842	−	0.967768i	$-0.581036\pi$
$211$	−7.31644	−0.503685	−0.251842	+	0.967768i	$0.581036\pi$
$212$	0	0
$213$	2.82572	0.193615
$214$	0	0
$215$	−8.43421	−0.575208
$216$	0	0
$217$	5.49255	0.372859
$218$	0	0
$219$	3.44134	0.232544
$220$	0	0
$221$	16.7458	1.12645
$222$	0	0
$223$	7.69849	0.515529	0.257765	−	0.966208i	$-0.417014\pi$
$223$	7.69849	0.515529	0.257765	+	0.966208i	$0.417014\pi$
$224$	0	0
$225$	−4.41197	−0.294131
$226$	0	0
$227$	−9.61894	−0.638431	−0.319216	−	0.947682i	$-0.603419\pi$
$227$	−9.61894	−0.638431	−0.319216	+	0.947682i	$0.603419\pi$
$228$	0	0
$229$	12.1361	0.801976	0.400988	−	0.916083i	$-0.368667\pi$
$229$	12.1361	0.801976	0.400988	+	0.916083i	$0.368667\pi$
$230$	0	0
$231$	−5.54196	−0.364634
$232$	0	0
$233$	−13.5356	−0.886745	−0.443372	−	0.896338i	$-0.646218\pi$
$233$	−13.5356	−0.886745	−0.443372	+	0.896338i	$0.646218\pi$
$234$	0	0
$235$	−17.8014	−1.16124
$236$	0	0
$237$	−19.7550	−1.28322
$238$	0	0
$239$	−18.0084	−1.16487	−0.582433	−	0.812879i	$-0.697899\pi$
$239$	−18.0084	−1.16487	−0.582433	+	0.812879i	$0.697899\pi$
$240$	0	0
$241$	24.4331	1.57387	0.786936	−	0.617034i	$-0.211666\pi$
$241$	24.4331	1.57387	0.786936	+	0.617034i	$0.211666\pi$
$242$	0	0
$243$	13.5152	0.867001
$244$	0	0
$245$	36.6560	2.34186
$246$	0	0
$247$	−2.14252	−0.136325
$248$	0	0
$249$	−7.91549	−0.501624
$250$	0	0
$251$	27.0408	1.70680	0.853400	−	0.521256i	$-0.174536\pi$
$251$	27.0408	1.70680	0.853400	+	0.521256i	$0.174536\pi$
$252$	0	0
$253$	3.81016	0.239543
$254$	0	0
$255$	27.4731	1.72043
$256$	0	0
$257$	−26.1689	−1.63237	−0.816184	−	0.577792i	$-0.803915\pi$
$257$	−26.1689	−1.63237	−0.816184	+	0.577792i	$0.803915\pi$
$258$	0	0
$259$	−36.2080	−2.24986
$260$	0	0
$261$	−2.72781	−0.168847
$262$	0	0
$263$	−12.1061	−0.746494	−0.373247	−	0.927732i	$-0.621756\pi$
$263$	−12.1061	−0.746494	−0.373247	+	0.927732i	$0.621756\pi$
$264$	0	0
$265$	18.2337	1.12009
$266$	0	0
$267$	2.05875	0.125994
$268$	0	0
$269$	27.5041	1.67696	0.838479	−	0.544934i	$-0.183445\pi$
$269$	27.5041	1.67696	0.838479	+	0.544934i	$0.183445\pi$
$270$	0	0
$271$	4.45270	0.270482	0.135241	−	0.990813i	$-0.456819\pi$
$271$	4.45270	0.270482	0.135241	+	0.990813i	$0.456819\pi$
$272$	0	0
$273$	11.8738	0.718633
$274$	0	0
$275$	−3.02218	−0.182244
$276$	0	0
$277$	14.0559	0.844538	0.422269	−	0.906471i	$-0.361234\pi$
$277$	14.0559	0.844538	0.422269	+	0.906471i	$0.361234\pi$
$278$	0	0
$279$	−1.79557	−0.107498
$280$	0	0
$281$	−12.0282	−0.717543	−0.358772	−	0.933425i	$-0.616804\pi$
$281$	−12.0282	−0.717543	−0.358772	+	0.933425i	$0.616804\pi$
$282$	0	0
$283$	19.1496	1.13833	0.569164	−	0.822224i	$-0.307267\pi$
$283$	19.1496	1.13833	0.569164	+	0.822224i	$0.307267\pi$
$284$	0	0
$285$	−3.51500	−0.208211
$286$	0	0
$287$	48.7761	2.87916
$288$	0	0
$289$	44.0889	2.59346
$290$	0	0
$291$	−7.43897	−0.436080
$292$	0	0
$293$	17.3732	1.01495	0.507476	−	0.861666i	$-0.330579\pi$
$293$	17.3732	1.01495	0.507476	+	0.861666i	$0.330579\pi$
$294$	0	0
$295$	−20.9807	−1.22155
$296$	0	0
$297$	5.53479	0.321161
$298$	0	0
$299$	−8.16336	−0.472099
$300$	0	0
$301$	−13.2979	−0.766476
$302$	0	0
$303$	14.8459	0.852877
$304$	0	0
$305$	−27.8648	−1.59553
$306$	0	0
$307$	25.4419	1.45205	0.726023	−	0.687670i	$-0.241366\pi$
$307$	25.4419	1.45205	0.726023	+	0.687670i	$0.241366\pi$
$308$	0	0
$309$	0.934744	0.0531757
$310$	0	0
$311$	6.22913	0.353222	0.176611	−	0.984281i	$-0.443487\pi$
$311$	6.22913	0.353222	0.176611	+	0.984281i	$0.443487\pi$
$312$	0	0
$313$	23.4024	1.32278	0.661392	−	0.750041i	$-0.269966\pi$
$313$	23.4024	1.32278	0.661392	+	0.750041i	$0.269966\pi$
$314$	0	0
$315$	−18.4647	−1.04037
$316$	0	0
$317$	−13.9440	−0.783171	−0.391585	−	0.920142i	$-0.628073\pi$
$317$	−13.9440	−0.783171	−0.391585	+	0.920142i	$0.628073\pi$
$318$	0	0
$319$	−1.86854	−0.104618
$320$	0	0
$321$	1.31304	0.0732870
$322$	0	0
$323$	−7.81594	−0.434890
$324$	0	0
$325$	6.47509	0.359173
$326$	0	0
$327$	8.81424	0.487428
$328$	0	0
$329$	−28.0667	−1.54737
$330$	0	0
$331$	−8.96372	−0.492691	−0.246345	−	0.969182i	$-0.579230\pi$
$331$	−8.96372	−0.492691	−0.246345	+	0.969182i	$0.579230\pi$
$332$	0	0
$333$	11.8368	0.648652
$334$	0	0
$335$	38.8069	2.12024
$336$	0	0
$337$	7.52217	0.409759	0.204879	−	0.978787i	$-0.434320\pi$
$337$	7.52217	0.409759	0.204879	+	0.978787i	$0.434320\pi$
$338$	0	0
$339$	−25.9066	−1.40706
$340$	0	0
$341$	−1.22996	−0.0666061
$342$	0	0
$343$	26.5344	1.43272
$344$	0	0
$345$	−13.3927	−0.721041
$346$	0	0
$347$	7.09789	0.381035	0.190517	−	0.981684i	$-0.438983\pi$
$347$	7.09789	0.381035	0.190517	+	0.981684i	$0.438983\pi$
$348$	0	0
$349$	−30.8496	−1.65134	−0.825670	−	0.564153i	$-0.809203\pi$
$349$	−30.8496	−1.65134	−0.825670	+	0.564153i	$0.809203\pi$
$350$	0	0
$351$	−11.8584	−0.632955
$352$	0	0
$353$	32.7894	1.74521	0.872603	−	0.488430i	$-0.162430\pi$
$353$	32.7894	1.74521	0.872603	+	0.488430i	$0.162430\pi$
$354$	0	0
$355$	6.44906	0.342281
$356$	0	0
$357$	43.3156	2.29250
$358$	0	0
$359$	−9.72206	−0.513111	−0.256555	−	0.966530i	$-0.582588\pi$
$359$	−9.72206	−0.513111	−0.256555	+	0.966530i	$0.582588\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.24102	0.0651368
$364$	0	0
$365$	7.85407	0.411101
$366$	0	0
$367$	−13.3115	−0.694856	−0.347428	−	0.937707i	$-0.612945\pi$
$367$	−13.3115	−0.694856	−0.347428	+	0.937707i	$0.612945\pi$
$368$	0	0
$369$	−15.9454	−0.830086
$370$	0	0
$371$	28.7483	1.49254
$372$	0	0
$373$	−6.88925	−0.356712	−0.178356	−	0.983966i	$-0.557078\pi$
$373$	−6.88925	−0.356712	−0.178356	+	0.983966i	$0.557078\pi$
$374$	0	0
$375$	−6.95204	−0.359002
$376$	0	0
$377$	4.00339	0.206185
$378$	0	0
$379$	−15.6685	−0.804837	−0.402418	−	0.915456i	$-0.631830\pi$
$379$	−15.6685	−0.804837	−0.402418	+	0.915456i	$0.631830\pi$
$380$	0	0
$381$	24.8016	1.27062
$382$	0	0
$383$	38.2914	1.95660	0.978299	−	0.207198i	$-0.0664346\pi$
$383$	38.2914	1.95660	0.978299	+	0.207198i	$0.0664346\pi$
$384$	0	0
$385$	−12.6482	−0.644613
$386$	0	0
$387$	4.34721	0.220981
$388$	0	0
$389$	13.4533	0.682108	0.341054	−	0.940044i	$-0.389216\pi$
$389$	13.4533	0.682108	0.341054	+	0.940044i	$0.389216\pi$
$390$	0	0
$391$	−29.7800	−1.50604
$392$	0	0
$393$	−18.0775	−0.911889
$394$	0	0
$395$	−45.0862	−2.26853
$396$	0	0
$397$	−30.5925	−1.53539	−0.767697	−	0.640813i	$-0.778597\pi$
$397$	−30.5925	−1.53539	−0.767697	+	0.640813i	$0.778597\pi$
$398$	0	0
$399$	−5.54196	−0.277445
$400$	0	0
$401$	−15.2750	−0.762798	−0.381399	−	0.924411i	$-0.624558\pi$
$401$	−15.2750	−0.762798	−0.381399	+	0.924411i	$0.624558\pi$
$402$	0	0
$403$	2.63522	0.131270
$404$	0	0
$405$	−7.05030	−0.350332
$406$	0	0
$407$	8.10815	0.401906
$408$	0	0
$409$	−15.3255	−0.757799	−0.378900	−	0.925438i	$-0.623697\pi$
$409$	−15.3255	−0.757799	−0.378900	+	0.925438i	$0.623697\pi$
$410$	0	0
$411$	3.59592	0.177374
$412$	0	0
$413$	−33.0794	−1.62773
$414$	0	0
$415$	−18.0653	−0.886789
$416$	0	0
$417$	−22.6361	−1.10850
$418$	0	0
$419$	−5.42923	−0.265235	−0.132618	−	0.991167i	$-0.542338\pi$
$419$	−5.42923	−0.265235	−0.132618	+	0.991167i	$0.542338\pi$
$420$	0	0
$421$	−13.4074	−0.653437	−0.326719	−	0.945122i	$-0.605943\pi$
$421$	−13.4074	−0.653437	−0.326719	+	0.945122i	$0.605943\pi$
$422$	0	0
$423$	9.17531	0.446119
$424$	0	0
$425$	23.6212	1.14580
$426$	0	0
$427$	−43.9332	−2.12608
$428$	0	0
$429$	−2.65892	−0.128374
$430$	0	0
$431$	9.96718	0.480102	0.240051	−	0.970760i	$-0.422836\pi$
$431$	9.96718	0.480102	0.240051	+	0.970760i	$0.422836\pi$
$432$	0	0
$433$	23.3082	1.12012	0.560061	−	0.828451i	$-0.310778\pi$
$433$	23.3082	1.12012	0.560061	+	0.828451i	$0.310778\pi$
$434$	0	0
$435$	6.56793	0.314908
$436$	0	0
$437$	3.81016	0.182265
$438$	0	0
$439$	11.5107	0.549374	0.274687	−	0.961534i	$-0.411426\pi$
$439$	11.5107	0.549374	0.274687	+	0.961534i	$0.411426\pi$
$440$	0	0
$441$	−18.8934	−0.899686
$442$	0	0
$443$	−31.2930	−1.48678	−0.743389	−	0.668860i	$-0.766783\pi$
$443$	−31.2930	−1.48678	−0.743389	+	0.668860i	$0.766783\pi$
$444$	0	0
$445$	4.69863	0.222736
$446$	0	0
$447$	−22.7243	−1.07482
$448$	0	0
$449$	−34.4750	−1.62697	−0.813487	−	0.581583i	$-0.802434\pi$
$449$	−34.4750	−1.62697	−0.813487	+	0.581583i	$0.802434\pi$
$450$	0	0
$451$	−10.9225	−0.514323
$452$	0	0
$453$	19.0589	0.895465
$454$	0	0
$455$	27.0991	1.27043
$456$	0	0
$457$	29.2398	1.36778	0.683891	−	0.729584i	$-0.260286\pi$
$457$	29.2398	1.36778	0.683891	+	0.729584i	$0.260286\pi$
$458$	0	0
$459$	−43.2596	−2.01918
$460$	0	0
$461$	9.66554	0.450169	0.225085	−	0.974339i	$-0.427734\pi$
$461$	9.66554	0.450169	0.225085	+	0.974339i	$0.427734\pi$
$462$	0	0
$463$	7.41991	0.344833	0.172416	−	0.985024i	$-0.444843\pi$
$463$	7.41991	0.344833	0.172416	+	0.985024i	$0.444843\pi$
$464$	0	0
$465$	4.32332	0.200489
$466$	0	0
$467$	3.09212	0.143086	0.0715430	−	0.997438i	$-0.477208\pi$
$467$	3.09212	0.143086	0.0715430	+	0.997438i	$0.477208\pi$
$468$	0	0
$469$	61.1851	2.82526
$470$	0	0
$471$	−3.48755	−0.160698
$472$	0	0
$473$	2.97782	0.136920
$474$	0	0
$475$	−3.02218	−0.138667
$476$	0	0
$477$	−9.39813	−0.430311
$478$	0	0
$479$	−30.3725	−1.38776	−0.693879	−	0.720092i	$-0.744100\pi$
$479$	−30.3725	−1.38776	−0.693879	+	0.720092i	$0.744100\pi$
$480$	0	0
$481$	−17.3719	−0.792090
$482$	0	0
$483$	−21.1158	−0.960800
$484$	0	0
$485$	−16.9777	−0.770919
$486$	0	0
$487$	36.6521	1.66086	0.830432	−	0.557120i	$-0.188094\pi$
$487$	36.6521	1.66086	0.830432	+	0.557120i	$0.188094\pi$
$488$	0	0
$489$	8.35738	0.377934
$490$	0	0
$491$	−14.8522	−0.670270	−0.335135	−	0.942170i	$-0.608782\pi$
$491$	−14.8522	−0.670270	−0.335135	+	0.942170i	$0.608782\pi$
$492$	0	0
$493$	14.6044	0.657749
$494$	0	0
$495$	4.13484	0.185847
$496$	0	0
$497$	10.1680	0.456095
$498$	0	0
$499$	−27.9369	−1.25063	−0.625314	−	0.780374i	$-0.715029\pi$
$499$	−27.9369	−1.25063	−0.625314	+	0.780374i	$0.715029\pi$
$500$	0	0
$501$	10.2705	0.458853
$502$	0	0
$503$	−8.65406	−0.385865	−0.192933	−	0.981212i	$-0.561800\pi$
$503$	−8.65406	−0.385865	−0.192933	+	0.981212i	$0.561800\pi$
$504$	0	0
$505$	33.8824	1.50775
$506$	0	0
$507$	−10.4365	−0.463501
$508$	0	0
$509$	−26.6016	−1.17910	−0.589548	−	0.807733i	$-0.700694\pi$
$509$	−26.6016	−1.17910	−0.589548	+	0.807733i	$0.700694\pi$
$510$	0	0
$511$	12.3832	0.547799
$512$	0	0
$513$	5.53479	0.244367
$514$	0	0
$515$	2.13334	0.0940060
$516$	0	0
$517$	6.28504	0.276416
$518$	0	0
$519$	−5.99740	−0.263257
$520$	0	0
$521$	4.17685	0.182991	0.0914955	−	0.995805i	$-0.470835\pi$
$521$	4.17685	0.182991	0.0914955	+	0.995805i	$0.470835\pi$
$522$	0	0
$523$	−3.07716	−0.134555	−0.0672773	−	0.997734i	$-0.521431\pi$
$523$	−3.07716	−0.134555	−0.0672773	+	0.997734i	$0.521431\pi$
$524$	0	0
$525$	16.7488	0.730977
$526$	0	0
$527$	9.61329	0.418762
$528$	0	0
$529$	−8.48265	−0.368811
$530$	0	0
$531$	10.8140	0.469288
$532$	0	0
$533$	23.4018	1.01364
$534$	0	0
$535$	2.99672	0.129560
$536$	0	0
$537$	8.93182	0.385436
$538$	0	0
$539$	−12.9419	−0.557447
$540$	0	0
$541$	−39.9766	−1.71873	−0.859365	−	0.511363i	$-0.829141\pi$
$541$	−39.9766	−1.71873	−0.859365	+	0.511363i	$0.829141\pi$
$542$	0	0
$543$	3.42105	0.146811
$544$	0	0
$545$	20.1164	0.861694
$546$	0	0
$547$	−13.7551	−0.588125	−0.294063	−	0.955786i	$-0.595007\pi$
$547$	−13.7551	−0.588125	−0.294063	+	0.955786i	$0.595007\pi$
$548$	0	0
$549$	14.3622	0.612965
$550$	0	0
$551$	−1.86854	−0.0796025
$552$	0	0
$553$	−71.0854	−3.02286
$554$	0	0
$555$	−28.5002	−1.20976
$556$	0	0
$557$	3.37375	0.142950	0.0714751	−	0.997442i	$-0.477229\pi$
$557$	3.37375	0.142950	0.0714751	+	0.997442i	$0.477229\pi$
$558$	0	0
$559$	−6.38004	−0.269847
$560$	0	0
$561$	−9.69975	−0.409524
$562$	0	0
$563$	−32.4119	−1.36600	−0.682999	−	0.730419i	$-0.739325\pi$
$563$	−32.4119	−1.36600	−0.682999	+	0.730419i	$0.739325\pi$
$564$	0	0
$565$	−59.1259	−2.48744
$566$	0	0
$567$	−11.1159	−0.466824
$568$	0	0
$569$	29.7569	1.24748	0.623738	−	0.781633i	$-0.285613\pi$
$569$	29.7569	1.24748	0.623738	+	0.781633i	$0.285613\pi$
$570$	0	0
$571$	−11.9446	−0.499865	−0.249933	−	0.968263i	$-0.580409\pi$
$571$	−11.9446	−0.499865	−0.249933	+	0.968263i	$0.580409\pi$
$572$	0	0
$573$	−17.3626	−0.725334
$574$	0	0
$575$	−11.5150	−0.480209
$576$	0	0
$577$	−4.05443	−0.168788	−0.0843941	−	0.996432i	$-0.526895\pi$
$577$	−4.05443	−0.168788	−0.0843941	+	0.996432i	$0.526895\pi$
$578$	0	0
$579$	−8.42266	−0.350034
$580$	0	0
$581$	−28.4827	−1.18166
$582$	0	0
$583$	−6.43768	−0.266622
$584$	0	0
$585$	−8.85898	−0.366274
$586$	0	0
$587$	−17.7020	−0.730640	−0.365320	−	0.930882i	$-0.619040\pi$
$587$	−17.7020	−0.730640	−0.365320	+	0.930882i	$0.619040\pi$
$588$	0	0
$589$	−1.22996	−0.0506796
$590$	0	0
$591$	−2.48032	−0.102027
$592$	0	0
$593$	9.42897	0.387201	0.193601	−	0.981080i	$-0.437983\pi$
$593$	9.42897	0.387201	0.193601	+	0.981080i	$0.437983\pi$
$594$	0	0
$595$	98.8577	4.05277
$596$	0	0
$597$	−1.62691	−0.0665851
$598$	0	0
$599$	−2.75320	−0.112493	−0.0562464	−	0.998417i	$-0.517913\pi$
$599$	−2.75320	−0.112493	−0.0562464	+	0.998417i	$0.517913\pi$
$600$	0	0
$601$	−45.2029	−1.84386	−0.921932	−	0.387351i	$-0.873390\pi$
$601$	−45.2029	−1.84386	−0.921932	+	0.387351i	$0.873390\pi$
$602$	0	0
$603$	−20.0020	−0.814546
$604$	0	0
$605$	2.83235	0.115151
$606$	0	0
$607$	38.2093	1.55087	0.775433	−	0.631430i	$-0.217532\pi$
$607$	38.2093	1.55087	0.775433	+	0.631430i	$0.217532\pi$
$608$	0	0
$609$	10.3554	0.419621
$610$	0	0
$611$	−13.4658	−0.544770
$612$	0	0
$613$	−2.78003	−0.112284	−0.0561421	−	0.998423i	$-0.517880\pi$
$613$	−2.78003	−0.112284	−0.0561421	+	0.998423i	$0.517880\pi$
$614$	0	0
$615$	38.3928	1.54815
$616$	0	0
$617$	21.4168	0.862207	0.431104	−	0.902302i	$-0.358124\pi$
$617$	21.4168	0.862207	0.431104	+	0.902302i	$0.358124\pi$
$618$	0	0
$619$	−5.63033	−0.226302	−0.113151	−	0.993578i	$-0.536094\pi$
$619$	−5.63033	−0.226302	−0.113151	+	0.993578i	$0.536094\pi$
$620$	0	0
$621$	21.0885	0.846251
$622$	0	0
$623$	7.40813	0.296800
$624$	0	0
$625$	−30.9773	−1.23909
$626$	0	0
$627$	1.24102	0.0495617
$628$	0	0
$629$	−63.3728	−2.52684
$630$	0	0
$631$	47.0442	1.87280	0.936400	−	0.350935i	$-0.114136\pi$
$631$	47.0442	1.87280	0.936400	+	0.350935i	$0.114136\pi$
$632$	0	0
$633$	−9.07987	−0.360893
$634$	0	0
$635$	56.6039	2.24626
$636$	0	0
$637$	27.7283	1.09864
$638$	0	0
$639$	−3.32401	−0.131496
$640$	0	0
$641$	−4.26926	−0.168626	−0.0843128	−	0.996439i	$-0.526870\pi$
$641$	−4.26926	−0.168626	−0.0843128	+	0.996439i	$0.526870\pi$
$642$	0	0
$643$	−9.72241	−0.383414	−0.191707	−	0.981452i	$-0.561402\pi$
$643$	−9.72241	−0.383414	−0.191707	+	0.981452i	$0.561402\pi$
$644$	0	0
$645$	−10.4670	−0.412139
$646$	0	0
$647$	−30.8738	−1.21377	−0.606887	−	0.794788i	$-0.707582\pi$
$647$	−30.8738	−1.21377	−0.606887	+	0.794788i	$0.707582\pi$
$648$	0	0
$649$	7.40754	0.290772
$650$	0	0
$651$	6.81638	0.267155
$652$	0	0
$653$	−2.21765	−0.0867832	−0.0433916	−	0.999058i	$-0.513816\pi$
$653$	−2.21765	−0.0867832	−0.0433916	+	0.999058i	$0.513816\pi$
$654$	0	0
$655$	−41.2577	−1.61207
$656$	0	0
$657$	−4.04819	−0.157935
$658$	0	0
$659$	−8.82521	−0.343781	−0.171891	−	0.985116i	$-0.554988\pi$
$659$	−8.82521	−0.343781	−0.171891	+	0.985116i	$0.554988\pi$
$660$	0	0
$661$	47.0328	1.82937	0.914683	−	0.404173i	$-0.132441\pi$
$661$	47.0328	1.82937	0.914683	+	0.404173i	$0.132441\pi$
$662$	0	0
$663$	20.7819	0.807104
$664$	0	0
$665$	−12.6482	−0.490477
$666$	0	0
$667$	−7.11944	−0.275666
$668$	0	0
$669$	9.55400	0.369379
$670$	0	0
$671$	9.83806	0.379794
$672$	0	0
$673$	23.0771	0.889556	0.444778	−	0.895641i	$-0.353283\pi$
$673$	23.0771	0.889556	0.444778	+	0.895641i	$0.353283\pi$
$674$	0	0
$675$	−16.7271	−0.643828
$676$	0	0
$677$	6.34225	0.243753	0.121876	−	0.992545i	$-0.461109\pi$
$677$	6.34225	0.243753	0.121876	+	0.992545i	$0.461109\pi$
$678$	0	0
$679$	−26.7680	−1.02726
$680$	0	0
$681$	−11.9373	−0.457439
$682$	0	0
$683$	−3.29657	−0.126140	−0.0630698	−	0.998009i	$-0.520089\pi$
$683$	−3.29657	−0.126140	−0.0630698	+	0.998009i	$0.520089\pi$
$684$	0	0
$685$	8.20685	0.313568
$686$	0	0
$687$	15.0612	0.574620
$688$	0	0
$689$	13.7929	0.525467
$690$	0	0
$691$	15.7705	0.599937	0.299969	−	0.953949i	$-0.403024\pi$
$691$	15.7705	0.599937	0.299969	+	0.953949i	$0.403024\pi$
$692$	0	0
$693$	6.51922	0.247645
$694$	0	0
$695$	−51.6617	−1.95964
$696$	0	0
$697$	85.3700	3.23362
$698$	0	0
$699$	−16.7979	−0.635357
$700$	0	0
$701$	−18.1060	−0.683855	−0.341927	−	0.939726i	$-0.611080\pi$
$701$	−18.1060	−0.683855	−0.341927	+	0.939726i	$0.611080\pi$
$702$	0	0
$703$	8.10815	0.305805
$704$	0	0
$705$	−22.0920	−0.832031
$706$	0	0
$707$	53.4209	2.00910
$708$	0	0
$709$	−25.6798	−0.964426	−0.482213	−	0.876054i	$-0.660167\pi$
$709$	−25.6798	−0.964426	−0.482213	+	0.876054i	$0.660167\pi$
$710$	0	0
$711$	23.2386	0.871514
$712$	0	0
$713$	−4.68635	−0.175505
$714$	0	0
$715$	−6.06836	−0.226944
$716$	0	0
$717$	−22.3488	−0.834632
$718$	0	0
$719$	27.8775	1.03966	0.519828	−	0.854271i	$-0.325996\pi$
$719$	27.8775	1.03966	0.519828	+	0.854271i	$0.325996\pi$
$720$	0	0
$721$	3.36354	0.125265
$722$	0	0
$723$	30.3220	1.12769
$724$	0	0
$725$	5.64707	0.209727
$726$	0	0
$727$	41.7941	1.55006	0.775029	−	0.631925i	$-0.217735\pi$
$727$	41.7941	1.55006	0.775029	+	0.631925i	$0.217735\pi$
$728$	0	0
$729$	24.2403	0.897789
$730$	0	0
$731$	−23.2744	−0.860836
$732$	0	0
$733$	9.36881	0.346045	0.173022	−	0.984918i	$-0.444647\pi$
$733$	9.36881	0.346045	0.173022	+	0.984918i	$0.444647\pi$
$734$	0	0
$735$	45.4909	1.67796
$736$	0	0
$737$	−13.7013	−0.504694
$738$	0	0
$739$	−34.2879	−1.26130	−0.630650	−	0.776067i	$-0.717212\pi$
$739$	−34.2879	−1.26130	−0.630650	+	0.776067i	$0.717212\pi$
$740$	0	0
$741$	−2.65892	−0.0976778
$742$	0	0
$743$	−25.3459	−0.929851	−0.464925	−	0.885350i	$-0.653919\pi$
$743$	−25.3459	−0.929851	−0.464925	+	0.885350i	$0.653919\pi$
$744$	0	0
$745$	−51.8629	−1.90011
$746$	0	0
$747$	9.31130	0.340683
$748$	0	0
$749$	4.72480	0.172640
$750$	0	0
$751$	−36.2198	−1.32168	−0.660840	−	0.750527i	$-0.729800\pi$
$751$	−36.2198	−1.32168	−0.660840	+	0.750527i	$0.729800\pi$
$752$	0	0
$753$	33.5583	1.22293
$754$	0	0
$755$	43.4975	1.58304
$756$	0	0
$757$	−33.7822	−1.22784	−0.613918	−	0.789370i	$-0.710407\pi$
$757$	−33.7822	−1.22784	−0.613918	+	0.789370i	$0.710407\pi$
$758$	0	0
$759$	4.72850	0.171634
$760$	0	0
$761$	−24.9369	−0.903963	−0.451981	−	0.892027i	$-0.649283\pi$
$761$	−24.9369	−0.903963	−0.451981	+	0.892027i	$0.649283\pi$
$762$	0	0
$763$	31.7167	1.14822
$764$	0	0
$765$	−32.3176	−1.16845
$766$	0	0
$767$	−15.8708	−0.573062
$768$	0	0
$769$	13.4936	0.486592	0.243296	−	0.969952i	$-0.421771\pi$
$769$	13.4936	0.486592	0.243296	+	0.969952i	$0.421771\pi$
$770$	0	0
$771$	−32.4761	−1.16960
$772$	0	0
$773$	−11.4133	−0.410509	−0.205254	−	0.978709i	$-0.565802\pi$
$773$	−11.4133	−0.410509	−0.205254	+	0.978709i	$0.565802\pi$
$774$	0	0
$775$	3.71716	0.133524
$776$	0	0
$777$	−44.9350	−1.61203
$778$	0	0
$779$	−10.9225	−0.391341
$780$	0	0
$781$	−2.27693	−0.0814751
$782$	0	0
$783$	−10.3420	−0.369592
$784$	0	0
$785$	−7.95952	−0.284087
$786$	0	0
$787$	−20.7206	−0.738609	−0.369304	−	0.929308i	$-0.620404\pi$
$787$	−20.7206	−0.738609	−0.369304	+	0.929308i	$0.620404\pi$
$788$	0	0
$789$	−15.0239	−0.534866
$790$	0	0
$791$	−93.2212	−3.31456
$792$	0	0
$793$	−21.0783	−0.748511
$794$	0	0
$795$	22.6285	0.802549
$796$	0	0
$797$	22.4629	0.795675	0.397838	−	0.917456i	$-0.369761\pi$
$797$	22.4629	0.795675	0.397838	+	0.917456i	$0.369761\pi$
$798$	0	0
$799$	−49.1235	−1.73786
$800$	0	0
$801$	−2.42179	−0.0855699
$802$	0	0
$803$	−2.77299	−0.0978567
$804$	0	0
$805$	−48.1918	−1.69854
$806$	0	0
$807$	34.1333	1.20155
$808$	0	0
$809$	1.49572	0.0525866	0.0262933	−	0.999654i	$-0.491630\pi$
$809$	1.49572	0.0525866	0.0262933	+	0.999654i	$0.491630\pi$
$810$	0	0
$811$	47.8558	1.68044	0.840222	−	0.542243i	$-0.182425\pi$
$811$	47.8558	1.68044	0.840222	+	0.542243i	$0.182425\pi$
$812$	0	0
$813$	5.52590	0.193802
$814$	0	0
$815$	19.0738	0.668126
$816$	0	0
$817$	2.97782	0.104181
$818$	0	0
$819$	−13.9676	−0.488066
$820$	0	0
$821$	−43.4124	−1.51510	−0.757552	−	0.652774i	$-0.773605\pi$
$821$	−43.4124	−1.51510	−0.757552	+	0.652774i	$0.773605\pi$
$822$	0	0
$823$	−11.7858	−0.410826	−0.205413	−	0.978675i	$-0.565854\pi$
$823$	−11.7858	−0.410826	−0.205413	+	0.978675i	$0.565854\pi$
$824$	0	0
$825$	−3.75060	−0.130579
$826$	0	0
$827$	−42.6416	−1.48279	−0.741397	−	0.671067i	$-0.765836\pi$
$827$	−42.6416	−1.48279	−0.741397	+	0.671067i	$0.765836\pi$
$828$	0	0
$829$	−56.4118	−1.95926	−0.979631	−	0.200808i	$-0.935643\pi$
$829$	−56.4118	−1.95926	−0.979631	+	0.200808i	$0.935643\pi$
$830$	0	0
$831$	17.4437	0.605115
$832$	0	0
$833$	101.153	3.50475
$834$	0	0
$835$	23.4401	0.811178
$836$	0	0
$837$	−6.80757	−0.235304
$838$	0	0
$839$	27.6179	0.953474	0.476737	−	0.879046i	$-0.341819\pi$
$839$	27.6179	0.953474	0.476737	+	0.879046i	$0.341819\pi$
$840$	0	0
$841$	−25.5086	−0.879605
$842$	0	0
$843$	−14.9273	−0.514123
$844$	0	0
$845$	−23.8189	−0.819395
$846$	0	0
$847$	4.46564	0.153441
$848$	0	0
$849$	23.7651	0.815617
$850$	0	0
$851$	30.8934	1.05901
$852$	0	0
$853$	22.2378	0.761409	0.380705	−	0.924697i	$-0.375681\pi$
$853$	22.2378	0.761409	0.380705	+	0.924697i	$0.375681\pi$
$854$	0	0
$855$	4.13484	0.141408
$856$	0	0
$857$	−15.2312	−0.520289	−0.260144	−	0.965570i	$-0.583770\pi$
$857$	−15.2312	−0.520289	−0.260144	+	0.965570i	$0.583770\pi$
$858$	0	0
$859$	−40.9212	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$859$	−40.9212	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$860$	0	0
$861$	60.5323	2.06293
$862$	0	0
$863$	13.2896	0.452384	0.226192	−	0.974083i	$-0.427372\pi$
$863$	13.2896	0.452384	0.226192	+	0.974083i	$0.427372\pi$
$864$	0	0
$865$	−13.6877	−0.465395
$866$	0	0
$867$	54.7153	1.85823
$868$	0	0
$869$	15.9183	0.539992
$870$	0	0
$871$	29.3554	0.994669
$872$	0	0
$873$	8.75075	0.296168
$874$	0	0
$875$	−25.0159	−0.845692
$876$	0	0
$877$	17.3501	0.585872	0.292936	−	0.956132i	$-0.405368\pi$
$877$	17.3501	0.585872	0.292936	+	0.956132i	$0.405368\pi$
$878$	0	0
$879$	21.5605	0.727217
$880$	0	0
$881$	−24.6924	−0.831907	−0.415954	−	0.909386i	$-0.636552\pi$
$881$	−24.6924	−0.831907	−0.415954	+	0.909386i	$0.636552\pi$
$882$	0	0
$883$	−14.7443	−0.496185	−0.248092	−	0.968736i	$-0.579804\pi$
$883$	−14.7443	−0.496185	−0.248092	+	0.968736i	$0.579804\pi$
$884$	0	0
$885$	−26.0376	−0.875243
$886$	0	0
$887$	22.0908	0.741737	0.370869	−	0.928685i	$-0.379060\pi$
$887$	22.0908	0.741737	0.370869	+	0.928685i	$0.379060\pi$
$888$	0	0
$889$	89.2448	2.99318
$890$	0	0
$891$	2.48921	0.0833917
$892$	0	0
$893$	6.28504	0.210321
$894$	0	0
$895$	20.3848	0.681389
$896$	0	0
$897$	−10.1309	−0.338261
$898$	0	0
$899$	2.29823	0.0766503
$900$	0	0
$901$	50.3165	1.67629
$902$	0	0
$903$	−16.5029	−0.549183
$904$	0	0
$905$	7.80775	0.259539
$906$	0	0
$907$	37.1803	1.23455	0.617275	−	0.786747i	$-0.288236\pi$
$907$	37.1803	1.23455	0.617275	+	0.786747i	$0.288236\pi$
$908$	0	0
$909$	−17.4639	−0.579240
$910$	0	0
$911$	−6.79211	−0.225033	−0.112516	−	0.993650i	$-0.535891\pi$
$911$	−6.79211	−0.225033	−0.112516	+	0.993650i	$0.535891\pi$
$912$	0	0
$913$	6.37820	0.211088
$914$	0	0
$915$	−34.5808	−1.14321
$916$	0	0
$917$	−65.0492	−2.14811
$918$	0	0
$919$	33.7646	1.11379	0.556896	−	0.830582i	$-0.311992\pi$
$919$	33.7646	1.11379	0.556896	+	0.830582i	$0.311992\pi$
$920$	0	0
$921$	31.5740	1.04040
$922$	0	0
$923$	4.87838	0.160574
$924$	0	0
$925$	−24.5043	−0.805696
$926$	0	0
$927$	−1.09958	−0.0361148
$928$	0	0
$929$	4.77431	0.156640	0.0783200	−	0.996928i	$-0.475044\pi$
$929$	4.77431	0.156640	0.0783200	+	0.996928i	$0.475044\pi$
$930$	0	0
$931$	−12.9419	−0.424154
$932$	0	0
$933$	7.73050	0.253085
$934$	0	0
$935$	−22.1374	−0.723972
$936$	0	0
$937$	−52.6610	−1.72036	−0.860180	−	0.509990i	$-0.829649\pi$
$937$	−52.6610	−1.72036	−0.860180	+	0.509990i	$0.829649\pi$
$938$	0	0
$939$	29.0429	0.947781
$940$	0	0
$941$	−26.6960	−0.870264	−0.435132	−	0.900367i	$-0.643298\pi$
$941$	−26.6960	−0.870264	−0.435132	+	0.900367i	$0.643298\pi$
$942$	0	0
$943$	−41.6167	−1.35523
$944$	0	0
$945$	−70.0053	−2.27727
$946$	0	0
$947$	−49.0764	−1.59477	−0.797384	−	0.603472i	$-0.793783\pi$
$947$	−49.0764	−1.59477	−0.797384	+	0.603472i	$0.793783\pi$
$948$	0	0
$949$	5.94119	0.192859
$950$	0	0
$951$	−17.3048	−0.561146
$952$	0	0
$953$	−47.8969	−1.55153	−0.775767	−	0.631020i	$-0.782637\pi$
$953$	−47.8969	−1.55153	−0.775767	+	0.631020i	$0.782637\pi$
$954$	0	0
$955$	−39.6261	−1.28227
$956$	0	0
$957$	−2.31890	−0.0749594
$958$	0	0
$959$	12.9394	0.417834
$960$	0	0
$961$	−29.4872	−0.951200
$962$	0	0
$963$	−1.54459	−0.0497736
$964$	0	0
$965$	−19.2228	−0.618803
$966$	0	0
$967$	33.0737	1.06358	0.531789	−	0.846877i	$-0.321520\pi$
$967$	33.0737	1.06358	0.531789	+	0.846877i	$0.321520\pi$
$968$	0	0
$969$	−9.69975	−0.311601
$970$	0	0
$971$	2.21427	0.0710593	0.0355296	−	0.999369i	$-0.488688\pi$
$971$	2.21427	0.0710593	0.0355296	+	0.999369i	$0.488688\pi$
$972$	0	0
$973$	−81.4527	−2.61125
$974$	0	0
$975$	8.03573	0.257349
$976$	0	0
$977$	−32.6454	−1.04442	−0.522210	−	0.852817i	$-0.674892\pi$
$977$	−32.6454	−1.04442	−0.522210	+	0.852817i	$0.674892\pi$
$978$	0	0
$979$	−1.65892	−0.0530193
$980$	0	0
$981$	−10.3685	−0.331042
$982$	0	0
$983$	47.4633	1.51385	0.756923	−	0.653505i	$-0.226702\pi$
$983$	47.4633	1.51385	0.756923	+	0.653505i	$0.226702\pi$
$984$	0	0
$985$	−5.66075	−0.180366
$986$	0	0
$987$	−34.8314	−1.10870
$988$	0	0
$989$	11.3460	0.360781
$990$	0	0
$991$	−44.0157	−1.39821	−0.699103	−	0.715021i	$-0.746417\pi$
$991$	−44.0157	−1.39821	−0.699103	+	0.715021i	$0.746417\pi$
$992$	0	0
$993$	−11.1242	−0.353015
$994$	0	0
$995$	−3.71305	−0.117712
$996$	0	0
$997$	24.1273	0.764120	0.382060	−	0.924138i	$-0.375215\pi$
$997$	24.1273	0.764120	0.382060	+	0.924138i	$0.375215\pi$
$998$	0	0
$999$	44.8769	1.41984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.x.1.4		6
4.3	odd	2		836.2.a.d.1.3	✓	6
12.11	even	2		7524.2.a.r.1.2		6
44.43	even	2		9196.2.a.k.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.d.1.3	✓	6	4.3	odd	2
3344.2.a.x.1.4		6	1.1	even	1	trivial
7524.2.a.r.1.2		6	12.11	even	2
9196.2.a.k.1.3		6	44.43	even	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+1.24102 q^{3} +2.83235 q^{5} +4.46564 q^{7} -1.45986 q^{9} +O(q^{10})\)	\(q+1.24102 q^{3} +2.83235 q^{5} +4.46564 q^{7} -1.45986 q^{9} -1.00000 q^{11} +2.14252 q^{13} +3.51500 q^{15} +7.81594 q^{17} -1.00000 q^{19} +5.54196 q^{21} -3.81016 q^{23} +3.02218 q^{25} -5.53479 q^{27} +1.86854 q^{29} +1.22996 q^{31} -1.24102 q^{33} +12.6482 q^{35} -8.10815 q^{37} +2.65892 q^{39} +10.9225 q^{41} -2.97782 q^{43} -4.13484 q^{45} -6.28504 q^{47} +12.9419 q^{49} +9.69975 q^{51} +6.43768 q^{53} -2.83235 q^{55} -1.24102 q^{57} -7.40754 q^{59} -9.83806 q^{61} -6.51922 q^{63} +6.06836 q^{65} +13.7013 q^{67} -4.72850 q^{69} +2.27693 q^{71} +2.77299 q^{73} +3.75060 q^{75} -4.46564 q^{77} -15.9183 q^{79} -2.48921 q^{81} -6.37820 q^{83} +22.1374 q^{85} +2.31890 q^{87} +1.65892 q^{89} +9.56772 q^{91} +1.52641 q^{93} -2.83235 q^{95} -5.99423 q^{97} +1.45986 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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