LMFDB - Embedded newform 3344.2.a.y.1.5

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:	6.6.57500224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 18*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.67361$ 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)

$f(q)$	$=$	$q+2.67361 q^{3} +3.02460 q^{5} +0.969288 q^{7} +4.14820 q^{9} +O(q^{10})$	$q+2.67361 q^{3} +3.02460 q^{5} +0.969288 q^{7} +4.14820 q^{9} -1.00000 q^{11} -1.84959 q^{13} +8.08661 q^{15} -4.65727 q^{17} +1.00000 q^{19} +2.59150 q^{21} +8.83619 q^{23} +4.14820 q^{25} +3.06985 q^{27} +4.08776 q^{29} -5.58577 q^{31} -2.67361 q^{33} +2.93171 q^{35} +1.04199 q^{37} -4.94510 q^{39} +7.65296 q^{41} +5.53564 q^{43} +12.5467 q^{45} +3.35475 q^{47} -6.06048 q^{49} -12.4517 q^{51} +5.65116 q^{53} -3.02460 q^{55} +2.67361 q^{57} +6.98839 q^{59} -13.2254 q^{61} +4.02080 q^{63} -5.59428 q^{65} +16.0252 q^{67} +23.6245 q^{69} +1.66300 q^{71} +13.4425 q^{73} +11.0907 q^{75} -0.969288 q^{77} -5.63914 q^{79} -4.23701 q^{81} +8.89958 q^{83} -14.0864 q^{85} +10.9291 q^{87} -3.27111 q^{89} -1.79279 q^{91} -14.9342 q^{93} +3.02460 q^{95} -5.00721 q^{97} -4.14820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.67361	1.54361	0.771805	−	0.635859i	$-0.219354\pi$
$3$	2.67361	1.54361	0.771805	+	0.635859i	$0.219354\pi$
$4$	0	0
$5$	3.02460	1.35264	0.676321	−	0.736607i	$-0.263573\pi$
$5$	3.02460	1.35264	0.676321	+	0.736607i	$0.263573\pi$
$6$	0	0
$7$	0.969288	0.366356	0.183178	−	0.983080i	$-0.441361\pi$
$7$	0.969288	0.366356	0.183178	+	0.983080i	$0.441361\pi$
$8$	0	0
$9$	4.14820	1.38273
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.84959	−0.512985	−0.256493	−	0.966546i	$-0.582567\pi$
$13$	−1.84959	−0.512985	−0.256493	+	0.966546i	$0.582567\pi$
$14$	0	0
$15$	8.08661	2.08795
$16$	0	0
$17$	−4.65727	−1.12955	−0.564777	−	0.825244i	$-0.691038\pi$
$17$	−4.65727	−1.12955	−0.564777	+	0.825244i	$0.691038\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.59150	0.565512
$22$	0	0
$23$	8.83619	1.84247	0.921236	−	0.389004i	$-0.127181\pi$
$23$	8.83619	1.84247	0.921236	+	0.389004i	$0.127181\pi$
$24$	0	0
$25$	4.14820	0.829641
$26$	0	0
$27$	3.06985	0.590794
$28$	0	0
$29$	4.08776	0.759077	0.379539	−	0.925176i	$-0.376083\pi$
$29$	4.08776	0.759077	0.379539	+	0.925176i	$0.376083\pi$
$30$	0	0
$31$	−5.58577	−1.00323	−0.501617	−	0.865090i	$-0.667261\pi$
$31$	−5.58577	−1.00323	−0.501617	+	0.865090i	$0.667261\pi$
$32$	0	0
$33$	−2.67361	−0.465416
$34$	0	0
$35$	2.93171	0.495549
$36$	0	0
$37$	1.04199	0.171303	0.0856514	−	0.996325i	$-0.472703\pi$
$37$	1.04199	0.171303	0.0856514	+	0.996325i	$0.472703\pi$
$38$	0	0
$39$	−4.94510	−0.791850
$40$	0	0
$41$	7.65296	1.19519	0.597596	−	0.801798i	$-0.296123\pi$
$41$	7.65296	1.19519	0.597596	+	0.801798i	$0.296123\pi$
$42$	0	0
$43$	5.53564	0.844177	0.422088	−	0.906555i	$-0.361297\pi$
$43$	5.53564	0.844177	0.422088	+	0.906555i	$0.361297\pi$
$44$	0	0
$45$	12.5467	1.87035
$46$	0	0
$47$	3.35475	0.489340	0.244670	−	0.969606i	$-0.421320\pi$
$47$	3.35475	0.489340	0.244670	+	0.969606i	$0.421320\pi$
$48$	0	0
$49$	−6.06048	−0.865783
$50$	0	0
$51$	−12.4517	−1.74359
$52$	0	0
$53$	5.65116	0.776246	0.388123	−	0.921608i	$-0.373124\pi$
$53$	5.65116	0.776246	0.388123	+	0.921608i	$0.373124\pi$
$54$	0	0
$55$	−3.02460	−0.407837
$56$	0	0
$57$	2.67361	0.354129
$58$	0	0
$59$	6.98839	0.909811	0.454906	−	0.890540i	$-0.349673\pi$
$59$	6.98839	0.909811	0.454906	+	0.890540i	$0.349673\pi$
$60$	0	0
$61$	−13.2254	−1.69334	−0.846670	−	0.532119i	$-0.821396\pi$
$61$	−13.2254	−1.69334	−0.846670	+	0.532119i	$0.821396\pi$
$62$	0	0
$63$	4.02080	0.506574
$64$	0	0
$65$	−5.59428	−0.693886
$66$	0	0
$67$	16.0252	1.95779	0.978894	−	0.204369i	$-0.0655143\pi$
$67$	16.0252	1.95779	0.978894	+	0.204369i	$0.0655143\pi$
$68$	0	0
$69$	23.6245	2.84406
$70$	0	0
$71$	1.66300	0.197362	0.0986812	−	0.995119i	$-0.468538\pi$
$71$	1.66300	0.197362	0.0986812	+	0.995119i	$0.468538\pi$
$72$	0	0
$73$	13.4425	1.57333	0.786663	−	0.617383i	$-0.211807\pi$
$73$	13.4425	1.57333	0.786663	+	0.617383i	$0.211807\pi$
$74$	0	0
$75$	11.0907	1.28064
$76$	0	0
$77$	−0.969288	−0.110461
$78$	0	0
$79$	−5.63914	−0.634453	−0.317226	−	0.948350i	$-0.602751\pi$
$79$	−5.63914	−0.634453	−0.317226	+	0.948350i	$0.602751\pi$
$80$	0	0
$81$	−4.23701	−0.470779
$82$	0	0
$83$	8.89958	0.976856	0.488428	−	0.872604i	$-0.337570\pi$
$83$	8.89958	0.976856	0.488428	+	0.872604i	$0.337570\pi$
$84$	0	0
$85$	−14.0864	−1.52788
$86$	0	0
$87$	10.9291	1.17172
$88$	0	0
$89$	−3.27111	−0.346737	−0.173368	−	0.984857i	$-0.555465\pi$
$89$	−3.27111	−0.346737	−0.173368	+	0.984857i	$0.555465\pi$
$90$	0	0
$91$	−1.79279	−0.187935
$92$	0	0
$93$	−14.9342	−1.54860
$94$	0	0
$95$	3.02460	0.310317
$96$	0	0
$97$	−5.00721	−0.508405	−0.254202	−	0.967151i	$-0.581813\pi$
$97$	−5.00721	−0.508405	−0.254202	+	0.967151i	$0.581813\pi$
$98$	0	0
$99$	−4.14820	−0.416910
$100$	0	0
$101$	−12.6207	−1.25580	−0.627901	−	0.778293i	$-0.716086\pi$
$101$	−12.6207	−1.25580	−0.627901	+	0.778293i	$0.716086\pi$
$102$	0	0
$103$	8.61871	0.849227	0.424614	−	0.905375i	$-0.360410\pi$
$103$	8.61871	0.849227	0.424614	+	0.905375i	$0.360410\pi$
$104$	0	0
$105$	7.83825	0.764935
$106$	0	0
$107$	−15.7496	−1.52257	−0.761284	−	0.648419i	$-0.775430\pi$
$107$	−15.7496	−1.52257	−0.761284	+	0.648419i	$0.775430\pi$
$108$	0	0
$109$	−11.2609	−1.07860	−0.539301	−	0.842113i	$-0.681312\pi$
$109$	−11.2609	−1.07860	−0.539301	+	0.842113i	$0.681312\pi$
$110$	0	0
$111$	2.78589	0.264425
$112$	0	0
$113$	−14.3356	−1.34858	−0.674291	−	0.738466i	$-0.735551\pi$
$113$	−14.3356	−1.34858	−0.674291	+	0.738466i	$0.735551\pi$
$114$	0	0
$115$	26.7259	2.49221
$116$	0	0
$117$	−7.67250	−0.709323
$118$	0	0
$119$	−4.51424	−0.413819
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.4610	1.84491
$124$	0	0
$125$	−2.57634	−0.230435
$126$	0	0
$127$	0.975074	0.0865239	0.0432619	−	0.999064i	$-0.486225\pi$
$127$	0.975074	0.0865239	0.0432619	+	0.999064i	$0.486225\pi$
$128$	0	0
$129$	14.8002	1.30308
$130$	0	0
$131$	−14.9828	−1.30905	−0.654526	−	0.756040i	$-0.727132\pi$
$131$	−14.9828	−1.30905	−0.654526	+	0.756040i	$0.727132\pi$
$132$	0	0
$133$	0.969288	0.0840479
$134$	0	0
$135$	9.28508	0.799132
$136$	0	0
$137$	4.85656	0.414924	0.207462	−	0.978243i	$-0.433480\pi$
$137$	4.85656	0.414924	0.207462	+	0.978243i	$0.433480\pi$
$138$	0	0
$139$	−7.56887	−0.641984	−0.320992	−	0.947082i	$-0.604016\pi$
$139$	−7.56887	−0.641984	−0.320992	+	0.947082i	$0.604016\pi$
$140$	0	0
$141$	8.96930	0.755351
$142$	0	0
$143$	1.84959	0.154671
$144$	0	0
$145$	12.3638	1.02676
$146$	0	0
$147$	−16.2034	−1.33643
$148$	0	0
$149$	−4.99722	−0.409388	−0.204694	−	0.978826i	$-0.565620\pi$
$149$	−4.99722	−0.409388	−0.204694	+	0.978826i	$0.565620\pi$
$150$	0	0
$151$	−4.90781	−0.399392	−0.199696	−	0.979858i	$-0.563995\pi$
$151$	−4.90781	−0.399392	−0.199696	+	0.979858i	$0.563995\pi$
$152$	0	0
$153$	−19.3193	−1.56187
$154$	0	0
$155$	−16.8947	−1.35702
$156$	0	0
$157$	−16.6033	−1.32509	−0.662546	−	0.749022i	$-0.730524\pi$
$157$	−16.6033	−1.32509	−0.662546	+	0.749022i	$0.730524\pi$
$158$	0	0
$159$	15.1090	1.19822
$160$	0	0
$161$	8.56481	0.675001
$162$	0	0
$163$	19.5034	1.52763	0.763814	−	0.645436i	$-0.223324\pi$
$163$	19.5034	1.52763	0.763814	+	0.645436i	$0.223324\pi$
$164$	0	0
$165$	−8.08661	−0.629542
$166$	0	0
$167$	−11.6679	−0.902888	−0.451444	−	0.892300i	$-0.649091\pi$
$167$	−11.6679	−0.902888	−0.451444	+	0.892300i	$0.649091\pi$
$168$	0	0
$169$	−9.57900	−0.736846
$170$	0	0
$171$	4.14820	0.317221
$172$	0	0
$173$	−11.5637	−0.879172	−0.439586	−	0.898201i	$-0.644875\pi$
$173$	−11.5637	−0.879172	−0.439586	+	0.898201i	$0.644875\pi$
$174$	0	0
$175$	4.02080	0.303944
$176$	0	0
$177$	18.6843	1.40439
$178$	0	0
$179$	6.90831	0.516351	0.258176	−	0.966098i	$-0.416879\pi$
$179$	6.90831	0.516351	0.258176	+	0.966098i	$0.416879\pi$
$180$	0	0
$181$	−16.2025	−1.20432	−0.602161	−	0.798375i	$-0.705694\pi$
$181$	−16.2025	−1.20432	−0.602161	+	0.798375i	$0.705694\pi$
$182$	0	0
$183$	−35.3596	−2.61386
$184$	0	0
$185$	3.15161	0.231711
$186$	0	0
$187$	4.65727	0.340573
$188$	0	0
$189$	2.97557	0.216441
$190$	0	0
$191$	−6.32642	−0.457764	−0.228882	−	0.973454i	$-0.573507\pi$
$191$	−6.32642	−0.457764	−0.228882	+	0.973454i	$0.573507\pi$
$192$	0	0
$193$	−7.82970	−0.563594	−0.281797	−	0.959474i	$-0.590930\pi$
$193$	−7.82970	−0.563594	−0.281797	+	0.959474i	$0.590930\pi$
$194$	0	0
$195$	−14.9569	−1.07109
$196$	0	0
$197$	−10.2294	−0.728814	−0.364407	−	0.931240i	$-0.618728\pi$
$197$	−10.2294	−0.728814	−0.364407	+	0.931240i	$0.618728\pi$
$198$	0	0
$199$	−16.2673	−1.15316	−0.576580	−	0.817041i	$-0.695613\pi$
$199$	−16.2673	−1.15316	−0.576580	+	0.817041i	$0.695613\pi$
$200$	0	0
$201$	42.8451	3.02206
$202$	0	0
$203$	3.96221	0.278093
$204$	0	0
$205$	23.1471	1.61667
$206$	0	0
$207$	36.6543	2.54765
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−12.8310	−0.883320	−0.441660	−	0.897182i	$-0.645610\pi$
$211$	−12.8310	−0.883320	−0.441660	+	0.897182i	$0.645610\pi$
$212$	0	0
$213$	4.44623	0.304651
$214$	0	0
$215$	16.7431	1.14187
$216$	0	0
$217$	−5.41421	−0.367541
$218$	0	0
$219$	35.9400	2.42860
$220$	0	0
$221$	8.61406	0.579445
$222$	0	0
$223$	−0.651776	−0.0436461	−0.0218231	−	0.999762i	$-0.506947\pi$
$223$	−0.651776	−0.0436461	−0.0218231	+	0.999762i	$0.506947\pi$
$224$	0	0
$225$	17.2076	1.14717
$226$	0	0
$227$	−6.63048	−0.440081	−0.220040	−	0.975491i	$-0.570619\pi$
$227$	−6.63048	−0.440081	−0.220040	+	0.975491i	$0.570619\pi$
$228$	0	0
$229$	−10.0694	−0.665405	−0.332702	−	0.943032i	$-0.607960\pi$
$229$	−10.0694	−0.665405	−0.332702	+	0.943032i	$0.607960\pi$
$230$	0	0
$231$	−2.59150	−0.170508
$232$	0	0
$233$	−1.23993	−0.0812304	−0.0406152	−	0.999175i	$-0.512932\pi$
$233$	−1.23993	−0.0812304	−0.0406152	+	0.999175i	$0.512932\pi$
$234$	0	0
$235$	10.1468	0.661903
$236$	0	0
$237$	−15.0769	−0.979348
$238$	0	0
$239$	16.3466	1.05737	0.528686	−	0.848817i	$-0.322685\pi$
$239$	16.3466	1.05737	0.528686	+	0.848817i	$0.322685\pi$
$240$	0	0
$241$	3.17786	0.204704	0.102352	−	0.994748i	$-0.467363\pi$
$241$	3.17786	0.204704	0.102352	+	0.994748i	$0.467363\pi$
$242$	0	0
$243$	−20.5377	−1.31749
$244$	0	0
$245$	−18.3305	−1.17109
$246$	0	0
$247$	−1.84959	−0.117687
$248$	0	0
$249$	23.7940	1.50789
$250$	0	0
$251$	27.1261	1.71218	0.856091	−	0.516825i	$-0.172886\pi$
$251$	27.1261	1.71218	0.856091	+	0.516825i	$0.172886\pi$
$252$	0	0
$253$	−8.83619	−0.555526
$254$	0	0
$255$	−37.6615	−2.35846
$256$	0	0
$257$	11.5115	0.718065	0.359032	−	0.933325i	$-0.383107\pi$
$257$	11.5115	0.718065	0.359032	+	0.933325i	$0.383107\pi$
$258$	0	0
$259$	1.00999	0.0627578
$260$	0	0
$261$	16.9569	1.04960
$262$	0	0
$263$	6.62648	0.408606	0.204303	−	0.978908i	$-0.434507\pi$
$263$	6.62648	0.408606	0.204303	+	0.978908i	$0.434507\pi$
$264$	0	0
$265$	17.0925	1.04998
$266$	0	0
$267$	−8.74568	−0.535227
$268$	0	0
$269$	−4.79395	−0.292292	−0.146146	−	0.989263i	$-0.546687\pi$
$269$	−4.79395	−0.292292	−0.146146	+	0.989263i	$0.546687\pi$
$270$	0	0
$271$	−12.5511	−0.762423	−0.381211	−	0.924488i	$-0.624493\pi$
$271$	−12.5511	−0.762423	−0.381211	+	0.924488i	$0.624493\pi$
$272$	0	0
$273$	−4.79323	−0.290099
$274$	0	0
$275$	−4.14820	−0.250146
$276$	0	0
$277$	−4.27079	−0.256607	−0.128303	−	0.991735i	$-0.540953\pi$
$277$	−4.27079	−0.256607	−0.128303	+	0.991735i	$0.540953\pi$
$278$	0	0
$279$	−23.1709	−1.38721
$280$	0	0
$281$	−1.92584	−0.114886	−0.0574430	−	0.998349i	$-0.518295\pi$
$281$	−1.92584	−0.114886	−0.0574430	+	0.998349i	$0.518295\pi$
$282$	0	0
$283$	−25.2590	−1.50149	−0.750746	−	0.660591i	$-0.770306\pi$
$283$	−25.2590	−1.50149	−0.750746	+	0.660591i	$0.770306\pi$
$284$	0	0
$285$	8.08661	0.479009
$286$	0	0
$287$	7.41792	0.437866
$288$	0	0
$289$	4.69017	0.275892
$290$	0	0
$291$	−13.3873	−0.784779
$292$	0	0
$293$	−7.91449	−0.462369	−0.231185	−	0.972910i	$-0.574260\pi$
$293$	−7.91449	−0.462369	−0.231185	+	0.972910i	$0.574260\pi$
$294$	0	0
$295$	21.1371	1.23065
$296$	0	0
$297$	−3.06985	−0.178131
$298$	0	0
$299$	−16.3434	−0.945161
$300$	0	0
$301$	5.36563	0.309270
$302$	0	0
$303$	−33.7428	−1.93847
$304$	0	0
$305$	−40.0016	−2.29048
$306$	0	0
$307$	−24.9862	−1.42604	−0.713018	−	0.701145i	$-0.752672\pi$
$307$	−24.9862	−1.42604	−0.713018	+	0.701145i	$0.752672\pi$
$308$	0	0
$309$	23.0431	1.31088
$310$	0	0
$311$	−16.0956	−0.912696	−0.456348	−	0.889801i	$-0.650843\pi$
$311$	−16.0956	−0.912696	−0.456348	+	0.889801i	$0.650843\pi$
$312$	0	0
$313$	35.2383	1.99178	0.995892	−	0.0905486i	$-0.0288621\pi$
$313$	35.2383	1.99178	0.995892	+	0.0905486i	$0.0288621\pi$
$314$	0	0
$315$	12.1613	0.685213
$316$	0	0
$317$	3.75601	0.210959	0.105479	−	0.994421i	$-0.466362\pi$
$317$	3.75601	0.210959	0.105479	+	0.994421i	$0.466362\pi$
$318$	0	0
$319$	−4.08776	−0.228870
$320$	0	0
$321$	−42.1082	−2.35025
$322$	0	0
$323$	−4.65727	−0.259137
$324$	0	0
$325$	−7.67250	−0.425594
$326$	0	0
$327$	−30.1074	−1.66494
$328$	0	0
$329$	3.25172	0.179273
$330$	0	0
$331$	−19.0695	−1.04816	−0.524078	−	0.851670i	$-0.675590\pi$
$331$	−19.0695	−1.04816	−0.524078	+	0.851670i	$0.675590\pi$
$332$	0	0
$333$	4.32240	0.236866
$334$	0	0
$335$	48.4698	2.64819
$336$	0	0
$337$	19.7795	1.07746	0.538730	−	0.842478i	$-0.318904\pi$
$337$	19.7795	1.07746	0.538730	+	0.842478i	$0.318904\pi$
$338$	0	0
$339$	−38.3279	−2.08169
$340$	0	0
$341$	5.58577	0.302486
$342$	0	0
$343$	−12.6594	−0.683541
$344$	0	0
$345$	71.4548	3.84700
$346$	0	0
$347$	13.8434	0.743152	0.371576	−	0.928402i	$-0.378817\pi$
$347$	13.8434	0.743152	0.371576	+	0.928402i	$0.378817\pi$
$348$	0	0
$349$	28.1062	1.50449	0.752245	−	0.658883i	$-0.228971\pi$
$349$	28.1062	1.50449	0.752245	+	0.658883i	$0.228971\pi$
$350$	0	0
$351$	−5.67799	−0.303068
$352$	0	0
$353$	7.43671	0.395816	0.197908	−	0.980221i	$-0.436585\pi$
$353$	7.43671	0.395816	0.197908	+	0.980221i	$0.436585\pi$
$354$	0	0
$355$	5.02992	0.266961
$356$	0	0
$357$	−12.0693	−0.638776
$358$	0	0
$359$	10.2344	0.540152	0.270076	−	0.962839i	$-0.412951\pi$
$359$	10.2344	0.540152	0.270076	+	0.962839i	$0.412951\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.67361	0.140328
$364$	0	0
$365$	40.6582	2.12815
$366$	0	0
$367$	−8.82204	−0.460507	−0.230253	−	0.973131i	$-0.573955\pi$
$367$	−8.82204	−0.460507	−0.230253	+	0.973131i	$0.573955\pi$
$368$	0	0
$369$	31.7460	1.65263
$370$	0	0
$371$	5.47760	0.284383
$372$	0	0
$373$	22.6493	1.17273	0.586367	−	0.810045i	$-0.300558\pi$
$373$	22.6493	1.17273	0.586367	+	0.810045i	$0.300558\pi$
$374$	0	0
$375$	−6.88814	−0.355702
$376$	0	0
$377$	−7.56070	−0.389396
$378$	0	0
$379$	19.1676	0.984575	0.492288	−	0.870433i	$-0.336161\pi$
$379$	19.1676	0.984575	0.492288	+	0.870433i	$0.336161\pi$
$380$	0	0
$381$	2.60697	0.133559
$382$	0	0
$383$	−30.4177	−1.55427	−0.777137	−	0.629332i	$-0.783329\pi$
$383$	−30.4177	−1.55427	−0.777137	+	0.629332i	$0.783329\pi$
$384$	0	0
$385$	−2.93171	−0.149414
$386$	0	0
$387$	22.9630	1.16727
$388$	0	0
$389$	−12.5570	−0.636664	−0.318332	−	0.947979i	$-0.603123\pi$
$389$	−12.5570	−0.636664	−0.318332	+	0.947979i	$0.603123\pi$
$390$	0	0
$391$	−41.1525	−2.08117
$392$	0	0
$393$	−40.0581	−2.02067
$394$	0	0
$395$	−17.0561	−0.858187
$396$	0	0
$397$	26.8655	1.34834	0.674170	−	0.738576i	$-0.264502\pi$
$397$	26.8655	1.34834	0.674170	+	0.738576i	$0.264502\pi$
$398$	0	0
$399$	2.59150	0.129737
$400$	0	0
$401$	18.2412	0.910920	0.455460	−	0.890256i	$-0.349475\pi$
$401$	18.2412	0.910920	0.455460	+	0.890256i	$0.349475\pi$
$402$	0	0
$403$	10.3314	0.514644
$404$	0	0
$405$	−12.8153	−0.636796
$406$	0	0
$407$	−1.04199	−0.0516497
$408$	0	0
$409$	−32.3976	−1.60196	−0.800980	−	0.598691i	$-0.795688\pi$
$409$	−32.3976	−1.60196	−0.800980	+	0.598691i	$0.795688\pi$
$410$	0	0
$411$	12.9846	0.640482
$412$	0	0
$413$	6.77376	0.333315
$414$	0	0
$415$	26.9177	1.32134
$416$	0	0
$417$	−20.2362	−0.990973
$418$	0	0
$419$	33.0152	1.61290	0.806449	−	0.591304i	$-0.201387\pi$
$419$	33.0152	1.61290	0.806449	+	0.591304i	$0.201387\pi$
$420$	0	0
$421$	13.0812	0.637539	0.318769	−	0.947832i	$-0.396730\pi$
$421$	13.0812	0.637539	0.318769	+	0.947832i	$0.396730\pi$
$422$	0	0
$423$	13.9162	0.676628
$424$	0	0
$425$	−19.3193	−0.937124
$426$	0	0
$427$	−12.8192	−0.620366
$428$	0	0
$429$	4.94510	0.238752
$430$	0	0
$431$	25.8903	1.24709	0.623546	−	0.781786i	$-0.285691\pi$
$431$	25.8903	1.24709	0.623546	+	0.781786i	$0.285691\pi$
$432$	0	0
$433$	−30.1103	−1.44701	−0.723505	−	0.690319i	$-0.757470\pi$
$433$	−30.1103	−1.44701	−0.723505	+	0.690319i	$0.757470\pi$
$434$	0	0
$435$	33.0561	1.58492
$436$	0	0
$437$	8.83619	0.422692
$438$	0	0
$439$	13.1810	0.629094	0.314547	−	0.949242i	$-0.398147\pi$
$439$	13.1810	0.629094	0.314547	+	0.949242i	$0.398147\pi$
$440$	0	0
$441$	−25.1401	−1.19715
$442$	0	0
$443$	1.24227	0.0590223	0.0295111	−	0.999564i	$-0.490605\pi$
$443$	1.24227	0.0590223	0.0295111	+	0.999564i	$0.490605\pi$
$444$	0	0
$445$	−9.89380	−0.469011
$446$	0	0
$447$	−13.3606	−0.631936
$448$	0	0
$449$	−11.2083	−0.528954	−0.264477	−	0.964392i	$-0.585199\pi$
$449$	−11.2083	−0.528954	−0.264477	+	0.964392i	$0.585199\pi$
$450$	0	0
$451$	−7.65296	−0.360364
$452$	0	0
$453$	−13.1216	−0.616505
$454$	0	0
$455$	−5.42247	−0.254209
$456$	0	0
$457$	24.6855	1.15474	0.577370	−	0.816483i	$-0.304079\pi$
$457$	24.6855	1.15474	0.577370	+	0.816483i	$0.304079\pi$
$458$	0	0
$459$	−14.2971	−0.667333
$460$	0	0
$461$	7.47546	0.348167	0.174084	−	0.984731i	$-0.444304\pi$
$461$	7.47546	0.348167	0.174084	+	0.984731i	$0.444304\pi$
$462$	0	0
$463$	26.9185	1.25101	0.625505	−	0.780220i	$-0.284893\pi$
$463$	26.9185	1.25101	0.625505	+	0.780220i	$0.284893\pi$
$464$	0	0
$465$	−45.1699	−2.09470
$466$	0	0
$467$	−23.1476	−1.07114	−0.535572	−	0.844489i	$-0.679904\pi$
$467$	−23.1476	−1.07114	−0.535572	+	0.844489i	$0.679904\pi$
$468$	0	0
$469$	15.5330	0.717248
$470$	0	0
$471$	−44.3909	−2.04543
$472$	0	0
$473$	−5.53564	−0.254529
$474$	0	0
$475$	4.14820	0.190333
$476$	0	0
$477$	23.4422	1.07334
$478$	0	0
$479$	−4.76571	−0.217751	−0.108875	−	0.994055i	$-0.534725\pi$
$479$	−4.76571	−0.217751	−0.108875	+	0.994055i	$0.534725\pi$
$480$	0	0
$481$	−1.92727	−0.0878758
$482$	0	0
$483$	22.8990	1.04194
$484$	0	0
$485$	−15.1448	−0.687690
$486$	0	0
$487$	−13.1889	−0.597645	−0.298822	−	0.954309i	$-0.596594\pi$
$487$	−13.1889	−0.597645	−0.298822	+	0.954309i	$0.596594\pi$
$488$	0	0
$489$	52.1447	2.35806
$490$	0	0
$491$	18.9640	0.855831	0.427916	−	0.903819i	$-0.359248\pi$
$491$	18.9640	0.855831	0.427916	+	0.903819i	$0.359248\pi$
$492$	0	0
$493$	−19.0378	−0.857419
$494$	0	0
$495$	−12.5467	−0.563930
$496$	0	0
$497$	1.61193	0.0723049
$498$	0	0
$499$	−24.8350	−1.11177	−0.555883	−	0.831260i	$-0.687620\pi$
$499$	−24.8350	−1.11177	−0.555883	+	0.831260i	$0.687620\pi$
$500$	0	0
$501$	−31.1954	−1.39371
$502$	0	0
$503$	38.7702	1.72868	0.864338	−	0.502911i	$-0.167738\pi$
$503$	38.7702	1.72868	0.864338	+	0.502911i	$0.167738\pi$
$504$	0	0
$505$	−38.1725	−1.69865
$506$	0	0
$507$	−25.6105	−1.13740
$508$	0	0
$509$	−26.0143	−1.15306	−0.576531	−	0.817075i	$-0.695594\pi$
$509$	−26.0143	−1.15306	−0.576531	+	0.817075i	$0.695594\pi$
$510$	0	0
$511$	13.0297	0.576398
$512$	0	0
$513$	3.06985	0.135537
$514$	0	0
$515$	26.0682	1.14870
$516$	0	0
$517$	−3.35475	−0.147542
$518$	0	0
$519$	−30.9169	−1.35710
$520$	0	0
$521$	−9.59103	−0.420191	−0.210095	−	0.977681i	$-0.567377\pi$
$521$	−9.59103	−0.420191	−0.210095	+	0.977681i	$0.567377\pi$
$522$	0	0
$523$	40.4599	1.76919	0.884594	−	0.466362i	$-0.154436\pi$
$523$	40.4599	1.76919	0.884594	+	0.466362i	$0.154436\pi$
$524$	0	0
$525$	10.7501	0.469172
$526$	0	0
$527$	26.0144	1.13321
$528$	0	0
$529$	55.0782	2.39470
$530$	0	0
$531$	28.9893	1.25803
$532$	0	0
$533$	−14.1549	−0.613116
$534$	0	0
$535$	−47.6361	−2.05949
$536$	0	0
$537$	18.4701	0.797045
$538$	0	0
$539$	6.06048	0.261043
$540$	0	0
$541$	7.04034	0.302688	0.151344	−	0.988481i	$-0.451640\pi$
$541$	7.04034	0.302688	0.151344	+	0.988481i	$0.451640\pi$
$542$	0	0
$543$	−43.3192	−1.85901
$544$	0	0
$545$	−34.0598	−1.45896
$546$	0	0
$547$	40.8725	1.74758	0.873790	−	0.486303i	$-0.161655\pi$
$547$	40.8725	1.74758	0.873790	+	0.486303i	$0.161655\pi$
$548$	0	0
$549$	−54.8617	−2.34144
$550$	0	0
$551$	4.08776	0.174144
$552$	0	0
$553$	−5.46595	−0.232436
$554$	0	0
$555$	8.42620	0.357672
$556$	0	0
$557$	−19.2633	−0.816211	−0.408106	−	0.912935i	$-0.633810\pi$
$557$	−19.2633	−0.816211	−0.408106	+	0.912935i	$0.633810\pi$
$558$	0	0
$559$	−10.2387	−0.433050
$560$	0	0
$561$	12.4517	0.525713
$562$	0	0
$563$	−4.67190	−0.196897	−0.0984485	−	0.995142i	$-0.531388\pi$
$563$	−4.67190	−0.196897	−0.0984485	+	0.995142i	$0.531388\pi$
$564$	0	0
$565$	−43.3595	−1.82415
$566$	0	0
$567$	−4.10689	−0.172473
$568$	0	0
$569$	26.9661	1.13048	0.565239	−	0.824927i	$-0.308784\pi$
$569$	26.9661	1.13048	0.565239	+	0.824927i	$0.308784\pi$
$570$	0	0
$571$	33.5199	1.40276	0.701382	−	0.712785i	$-0.252567\pi$
$571$	33.5199	1.40276	0.701382	+	0.712785i	$0.252567\pi$
$572$	0	0
$573$	−16.9144	−0.706609
$574$	0	0
$575$	36.6543	1.52859
$576$	0	0
$577$	16.6163	0.691747	0.345873	−	0.938281i	$-0.387583\pi$
$577$	16.6163	0.691747	0.345873	+	0.938281i	$0.387583\pi$
$578$	0	0
$579$	−20.9336	−0.869970
$580$	0	0
$581$	8.62626	0.357877
$582$	0	0
$583$	−5.65116	−0.234047
$584$	0	0
$585$	−23.2062	−0.959460
$586$	0	0
$587$	−16.4649	−0.679580	−0.339790	−	0.940501i	$-0.610356\pi$
$587$	−16.4649	−0.679580	−0.339790	+	0.940501i	$0.610356\pi$
$588$	0	0
$589$	−5.58577	−0.230157
$590$	0	0
$591$	−27.3494	−1.12500
$592$	0	0
$593$	14.0948	0.578802	0.289401	−	0.957208i	$-0.406544\pi$
$593$	14.0948	0.578802	0.289401	+	0.957208i	$0.406544\pi$
$594$	0	0
$595$	−13.6538	−0.559749
$596$	0	0
$597$	−43.4925	−1.78003
$598$	0	0
$599$	12.5656	0.513416	0.256708	−	0.966489i	$-0.417362\pi$
$599$	12.5656	0.513416	0.256708	+	0.966489i	$0.417362\pi$
$600$	0	0
$601$	27.5612	1.12425	0.562123	−	0.827053i	$-0.309985\pi$
$601$	27.5612	1.12425	0.562123	+	0.827053i	$0.309985\pi$
$602$	0	0
$603$	66.4757	2.70710
$604$	0	0
$605$	3.02460	0.122967
$606$	0	0
$607$	−29.9679	−1.21636	−0.608180	−	0.793800i	$-0.708100\pi$
$607$	−29.9679	−1.21636	−0.608180	+	0.793800i	$0.708100\pi$
$608$	0	0
$609$	10.5934	0.429267
$610$	0	0
$611$	−6.20493	−0.251024
$612$	0	0
$613$	34.1855	1.38074	0.690370	−	0.723456i	$-0.257448\pi$
$613$	34.1855	1.38074	0.690370	+	0.723456i	$0.257448\pi$
$614$	0	0
$615$	61.8865	2.49550
$616$	0	0
$617$	−32.3758	−1.30340	−0.651700	−	0.758477i	$-0.725944\pi$
$617$	−32.3758	−1.30340	−0.651700	+	0.758477i	$0.725944\pi$
$618$	0	0
$619$	16.6473	0.669113	0.334557	−	0.942376i	$-0.391413\pi$
$619$	16.6473	0.669113	0.334557	+	0.942376i	$0.391413\pi$
$620$	0	0
$621$	27.1258	1.08852
$622$	0	0
$623$	−3.17065	−0.127029
$624$	0	0
$625$	−28.5334	−1.14134
$626$	0	0
$627$	−2.67361	−0.106774
$628$	0	0
$629$	−4.85285	−0.193496
$630$	0	0
$631$	−28.8558	−1.14873	−0.574366	−	0.818599i	$-0.694751\pi$
$631$	−28.8558	−1.14873	−0.574366	+	0.818599i	$0.694751\pi$
$632$	0	0
$633$	−34.3050	−1.36350
$634$	0	0
$635$	2.94921	0.117036
$636$	0	0
$637$	11.2094	0.444134
$638$	0	0
$639$	6.89848	0.272900
$640$	0	0
$641$	−1.96322	−0.0775427	−0.0387713	−	0.999248i	$-0.512344\pi$
$641$	−1.96322	−0.0775427	−0.0387713	+	0.999248i	$0.512344\pi$
$642$	0	0
$643$	38.0709	1.50137	0.750685	−	0.660660i	$-0.229724\pi$
$643$	38.0709	1.50137	0.750685	+	0.660660i	$0.229724\pi$
$644$	0	0
$645$	44.7645	1.76260
$646$	0	0
$647$	−37.4128	−1.47085	−0.735425	−	0.677606i	$-0.763018\pi$
$647$	−37.4128	−1.47085	−0.735425	+	0.677606i	$0.763018\pi$
$648$	0	0
$649$	−6.98839	−0.274318
$650$	0	0
$651$	−14.4755	−0.567340
$652$	0	0
$653$	44.0792	1.72495	0.862476	−	0.506098i	$-0.168913\pi$
$653$	44.0792	1.72495	0.862476	+	0.506098i	$0.168913\pi$
$654$	0	0
$655$	−45.3169	−1.77068
$656$	0	0
$657$	55.7622	2.17549
$658$	0	0
$659$	2.82050	0.109871	0.0549356	−	0.998490i	$-0.482505\pi$
$659$	2.82050	0.109871	0.0549356	+	0.998490i	$0.482505\pi$
$660$	0	0
$661$	−4.28117	−0.166518	−0.0832591	−	0.996528i	$-0.526533\pi$
$661$	−4.28117	−0.166518	−0.0832591	+	0.996528i	$0.526533\pi$
$662$	0	0
$663$	23.0307	0.894437
$664$	0	0
$665$	2.93171	0.113687
$666$	0	0
$667$	36.1202	1.39858
$668$	0	0
$669$	−1.74260	−0.0673726
$670$	0	0
$671$	13.2254	0.510561
$672$	0	0
$673$	−40.6368	−1.56643	−0.783216	−	0.621750i	$-0.786422\pi$
$673$	−40.6368	−1.56643	−0.783216	+	0.621750i	$0.786422\pi$
$674$	0	0
$675$	12.7344	0.490147
$676$	0	0
$677$	4.97819	0.191327	0.0956637	−	0.995414i	$-0.469503\pi$
$677$	4.97819	0.191327	0.0956637	+	0.995414i	$0.469503\pi$
$678$	0	0
$679$	−4.85342	−0.186257
$680$	0	0
$681$	−17.7273	−0.679313
$682$	0	0
$683$	−0.348085	−0.0133191	−0.00665956	−	0.999978i	$-0.502120\pi$
$683$	−0.348085	−0.0133191	−0.00665956	+	0.999978i	$0.502120\pi$
$684$	0	0
$685$	14.6892	0.561244
$686$	0	0
$687$	−26.9217	−1.02713
$688$	0	0
$689$	−10.4524	−0.398203
$690$	0	0
$691$	5.02411	0.191126	0.0955630	−	0.995423i	$-0.469535\pi$
$691$	5.02411	0.191126	0.0955630	+	0.995423i	$0.469535\pi$
$692$	0	0
$693$	−4.02080	−0.152738
$694$	0	0
$695$	−22.8928	−0.868374
$696$	0	0
$697$	−35.6419	−1.35003
$698$	0	0
$699$	−3.31509	−0.125388
$700$	0	0
$701$	−36.6906	−1.38578	−0.692892	−	0.721041i	$-0.743664\pi$
$701$	−36.6906	−1.38578	−0.692892	+	0.721041i	$0.743664\pi$
$702$	0	0
$703$	1.04199	0.0392995
$704$	0	0
$705$	27.1285	1.02172
$706$	0	0
$707$	−12.2331	−0.460071
$708$	0	0
$709$	52.9102	1.98708	0.993542	−	0.113461i	$-0.0361936\pi$
$709$	52.9102	1.98708	0.993542	+	0.113461i	$0.0361936\pi$
$710$	0	0
$711$	−23.3923	−0.877280
$712$	0	0
$713$	−49.3569	−1.84843
$714$	0	0
$715$	5.59428	0.209214
$716$	0	0
$717$	43.7044	1.63217
$718$	0	0
$719$	−34.3648	−1.28159	−0.640795	−	0.767712i	$-0.721395\pi$
$719$	−34.3648	−1.28159	−0.640795	+	0.767712i	$0.721395\pi$
$720$	0	0
$721$	8.35401	0.311120
$722$	0	0
$723$	8.49636	0.315983
$724$	0	0
$725$	16.9569	0.629762
$726$	0	0
$727$	9.01822	0.334467	0.167234	−	0.985917i	$-0.446517\pi$
$727$	9.01822	0.334467	0.167234	+	0.985917i	$0.446517\pi$
$728$	0	0
$729$	−42.1988	−1.56292
$730$	0	0
$731$	−25.7810	−0.953543
$732$	0	0
$733$	11.4550	0.423100	0.211550	−	0.977367i	$-0.432149\pi$
$733$	11.4550	0.423100	0.211550	+	0.977367i	$0.432149\pi$
$734$	0	0
$735$	−49.0087	−1.80771
$736$	0	0
$737$	−16.0252	−0.590295
$738$	0	0
$739$	2.02492	0.0744878	0.0372439	−	0.999306i	$-0.488142\pi$
$739$	2.02492	0.0744878	0.0372439	+	0.999306i	$0.488142\pi$
$740$	0	0
$741$	−4.94510	−0.181663
$742$	0	0
$743$	−15.4042	−0.565126	−0.282563	−	0.959249i	$-0.591185\pi$
$743$	−15.4042	−0.565126	−0.282563	+	0.959249i	$0.591185\pi$
$744$	0	0
$745$	−15.1146	−0.553755
$746$	0	0
$747$	36.9173	1.35073
$748$	0	0
$749$	−15.2659	−0.557802
$750$	0	0
$751$	−21.2986	−0.777196	−0.388598	−	0.921407i	$-0.627040\pi$
$751$	−21.2986	−0.777196	−0.388598	+	0.921407i	$0.627040\pi$
$752$	0	0
$753$	72.5246	2.64294
$754$	0	0
$755$	−14.8442	−0.540234
$756$	0	0
$757$	44.6493	1.62281	0.811403	−	0.584487i	$-0.198704\pi$
$757$	44.6493	1.62281	0.811403	+	0.584487i	$0.198704\pi$
$758$	0	0
$759$	−23.6245	−0.857517
$760$	0	0
$761$	33.8034	1.22537	0.612687	−	0.790326i	$-0.290089\pi$
$761$	33.8034	1.22537	0.612687	+	0.790326i	$0.290089\pi$
$762$	0	0
$763$	−10.9151	−0.395153
$764$	0	0
$765$	−58.4332	−2.11266
$766$	0	0
$767$	−12.9257	−0.466720
$768$	0	0
$769$	20.3344	0.733279	0.366639	−	0.930363i	$-0.380508\pi$
$769$	20.3344	0.733279	0.366639	+	0.930363i	$0.380508\pi$
$770$	0	0
$771$	30.7772	1.10841
$772$	0	0
$773$	−37.8998	−1.36316	−0.681581	−	0.731743i	$-0.738707\pi$
$773$	−37.8998	−1.36316	−0.681581	+	0.731743i	$0.738707\pi$
$774$	0	0
$775$	−23.1709	−0.832323
$776$	0	0
$777$	2.70033	0.0968737
$778$	0	0
$779$	7.65296	0.274196
$780$	0	0
$781$	−1.66300	−0.0595070
$782$	0	0
$783$	12.5488	0.448458
$784$	0	0
$785$	−50.2185	−1.79237
$786$	0	0
$787$	−24.1646	−0.861375	−0.430687	−	0.902501i	$-0.641729\pi$
$787$	−24.1646	−0.861375	−0.430687	+	0.902501i	$0.641729\pi$
$788$	0	0
$789$	17.7167	0.630729
$790$	0	0
$791$	−13.8953	−0.494061
$792$	0	0
$793$	24.4616	0.868658
$794$	0	0
$795$	45.6987	1.62077
$796$	0	0
$797$	−30.3914	−1.07652	−0.538260	−	0.842779i	$-0.680918\pi$
$797$	−30.3914	−1.07652	−0.538260	+	0.842779i	$0.680918\pi$
$798$	0	0
$799$	−15.6240	−0.552736
$800$	0	0
$801$	−13.5692	−0.479445
$802$	0	0
$803$	−13.4425	−0.474376
$804$	0	0
$805$	25.9051	0.913035
$806$	0	0
$807$	−12.8172	−0.451186
$808$	0	0
$809$	−3.39736	−0.119445	−0.0597224	−	0.998215i	$-0.519022\pi$
$809$	−3.39736	−0.119445	−0.0597224	+	0.998215i	$0.519022\pi$
$810$	0	0
$811$	−41.5382	−1.45860	−0.729301	−	0.684193i	$-0.760155\pi$
$811$	−41.5382	−1.45860	−0.729301	+	0.684193i	$0.760155\pi$
$812$	0	0
$813$	−33.5567	−1.17688
$814$	0	0
$815$	58.9901	2.06633
$816$	0	0
$817$	5.53564	0.193667
$818$	0	0
$819$	−7.43686	−0.259865
$820$	0	0
$821$	7.87958	0.274999	0.137500	−	0.990502i	$-0.456093\pi$
$821$	7.87958	0.274999	0.137500	+	0.990502i	$0.456093\pi$
$822$	0	0
$823$	20.9346	0.729734	0.364867	−	0.931060i	$-0.381114\pi$
$823$	20.9346	0.729734	0.364867	+	0.931060i	$0.381114\pi$
$824$	0	0
$825$	−11.0907	−0.386128
$826$	0	0
$827$	−10.6684	−0.370975	−0.185488	−	0.982647i	$-0.559386\pi$
$827$	−10.6684	−0.370975	−0.185488	+	0.982647i	$0.559386\pi$
$828$	0	0
$829$	46.4722	1.61405	0.807024	−	0.590519i	$-0.201077\pi$
$829$	46.4722	1.61405	0.807024	+	0.590519i	$0.201077\pi$
$830$	0	0
$831$	−11.4184	−0.396101
$832$	0	0
$833$	28.2253	0.977949
$834$	0	0
$835$	−35.2907	−1.22128
$836$	0	0
$837$	−17.1475	−0.592704
$838$	0	0
$839$	−37.8909	−1.30814	−0.654070	−	0.756434i	$-0.726940\pi$
$839$	−37.8909	−1.30814	−0.654070	+	0.756434i	$0.726940\pi$
$840$	0	0
$841$	−12.2902	−0.423801
$842$	0	0
$843$	−5.14895	−0.177339
$844$	0	0
$845$	−28.9726	−0.996689
$846$	0	0
$847$	0.969288	0.0333051
$848$	0	0
$849$	−67.5328	−2.31772
$850$	0	0
$851$	9.20725	0.315621
$852$	0	0
$853$	−32.9291	−1.12747	−0.563735	−	0.825956i	$-0.690636\pi$
$853$	−32.9291	−1.12747	−0.563735	+	0.825956i	$0.690636\pi$
$854$	0	0
$855$	12.5467	0.429087
$856$	0	0
$857$	5.84255	0.199578	0.0997888	−	0.995009i	$-0.468183\pi$
$857$	5.84255	0.199578	0.0997888	+	0.995009i	$0.468183\pi$
$858$	0	0
$859$	32.6377	1.11358	0.556792	−	0.830652i	$-0.312032\pi$
$859$	32.6377	1.11358	0.556792	+	0.830652i	$0.312032\pi$
$860$	0	0
$861$	19.8326	0.675895
$862$	0	0
$863$	1.74557	0.0594199	0.0297099	−	0.999559i	$-0.490542\pi$
$863$	1.74557	0.0594199	0.0297099	+	0.999559i	$0.490542\pi$
$864$	0	0
$865$	−34.9756	−1.18920
$866$	0	0
$867$	12.5397	0.425870
$868$	0	0
$869$	5.63914	0.191295
$870$	0	0
$871$	−29.6401	−1.00432
$872$	0	0
$873$	−20.7709	−0.702989
$874$	0	0
$875$	−2.49722	−0.0844213
$876$	0	0
$877$	23.3270	0.787696	0.393848	−	0.919176i	$-0.371144\pi$
$877$	23.3270	0.787696	0.393848	+	0.919176i	$0.371144\pi$
$878$	0	0
$879$	−21.1603	−0.713718
$880$	0	0
$881$	10.0551	0.338764	0.169382	−	0.985551i	$-0.445823\pi$
$881$	10.0551	0.338764	0.169382	+	0.985551i	$0.445823\pi$
$882$	0	0
$883$	−11.8721	−0.399528	−0.199764	−	0.979844i	$-0.564018\pi$
$883$	−11.8721	−0.399528	−0.199764	+	0.979844i	$0.564018\pi$
$884$	0	0
$885$	56.5124	1.89964
$886$	0	0
$887$	20.5554	0.690184	0.345092	−	0.938569i	$-0.387848\pi$
$887$	20.5554	0.690184	0.345092	+	0.938569i	$0.387848\pi$
$888$	0	0
$889$	0.945128	0.0316986
$890$	0	0
$891$	4.23701	0.141945
$892$	0	0
$893$	3.35475	0.112262
$894$	0	0
$895$	20.8949	0.698439
$896$	0	0
$897$	−43.6958	−1.45896
$898$	0	0
$899$	−22.8333	−0.761532
$900$	0	0
$901$	−26.3190	−0.876812
$902$	0	0
$903$	14.3456	0.477392
$904$	0	0
$905$	−49.0061	−1.62902
$906$	0	0
$907$	48.2269	1.60135	0.800674	−	0.599101i	$-0.204475\pi$
$907$	48.2269	1.60135	0.800674	+	0.599101i	$0.204475\pi$
$908$	0	0
$909$	−52.3531	−1.73644
$910$	0	0
$911$	−57.5598	−1.90704	−0.953521	−	0.301327i	$-0.902571\pi$
$911$	−57.5598	−1.90704	−0.953521	+	0.301327i	$0.902571\pi$
$912$	0	0
$913$	−8.89958	−0.294533
$914$	0	0
$915$	−106.949	−3.53561
$916$	0	0
$917$	−14.5226	−0.479579
$918$	0	0
$919$	−32.8946	−1.08509	−0.542546	−	0.840026i	$-0.682540\pi$
$919$	−32.8946	−1.08509	−0.542546	+	0.840026i	$0.682540\pi$
$920$	0	0
$921$	−66.8034	−2.20125
$922$	0	0
$923$	−3.07588	−0.101244
$924$	0	0
$925$	4.32240	0.142120
$926$	0	0
$927$	35.7522	1.17426
$928$	0	0
$929$	37.1170	1.21777	0.608884	−	0.793259i	$-0.291617\pi$
$929$	37.1170	1.21777	0.608884	+	0.793259i	$0.291617\pi$
$930$	0	0
$931$	−6.06048	−0.198624
$932$	0	0
$933$	−43.0333	−1.40885
$934$	0	0
$935$	14.0864	0.460674
$936$	0	0
$937$	28.0894	0.917641	0.458820	−	0.888529i	$-0.348272\pi$
$937$	28.0894	0.917641	0.458820	+	0.888529i	$0.348272\pi$
$938$	0	0
$939$	94.2134	3.07454
$940$	0	0
$941$	−19.7448	−0.643663	−0.321831	−	0.946797i	$-0.604298\pi$
$941$	−19.7448	−0.643663	−0.321831	+	0.946797i	$0.604298\pi$
$942$	0	0
$943$	67.6230	2.20211
$944$	0	0
$945$	8.99991	0.292767
$946$	0	0
$947$	9.93635	0.322888	0.161444	−	0.986882i	$-0.448385\pi$
$947$	9.93635	0.322888	0.161444	+	0.986882i	$0.448385\pi$
$948$	0	0
$949$	−24.8632	−0.807093
$950$	0	0
$951$	10.0421	0.325638
$952$	0	0
$953$	−40.0494	−1.29733	−0.648663	−	0.761076i	$-0.724672\pi$
$953$	−40.0494	−1.29733	−0.648663	+	0.761076i	$0.724672\pi$
$954$	0	0
$955$	−19.1349	−0.619191
$956$	0	0
$957$	−10.9291	−0.353287
$958$	0	0
$959$	4.70741	0.152010
$960$	0	0
$961$	0.200781	0.00647680
$962$	0	0
$963$	−65.3324	−2.10531
$964$	0	0
$965$	−23.6817	−0.762341
$966$	0	0
$967$	−49.2715	−1.58446	−0.792232	−	0.610220i	$-0.791081\pi$
$967$	−49.2715	−1.58446	−0.792232	+	0.610220i	$0.791081\pi$
$968$	0	0
$969$	−12.4517	−0.400007
$970$	0	0
$971$	55.6709	1.78656	0.893281	−	0.449498i	$-0.148397\pi$
$971$	55.6709	1.78656	0.893281	+	0.449498i	$0.148397\pi$
$972$	0	0
$973$	−7.33642	−0.235195
$974$	0	0
$975$	−20.5133	−0.656951
$976$	0	0
$977$	−17.1309	−0.548067	−0.274034	−	0.961720i	$-0.588358\pi$
$977$	−17.1309	−0.548067	−0.274034	+	0.961720i	$0.588358\pi$
$978$	0	0
$979$	3.27111	0.104545
$980$	0	0
$981$	−46.7127	−1.49142
$982$	0	0
$983$	35.1356	1.12065	0.560326	−	0.828272i	$-0.310676\pi$
$983$	35.1356	1.12065	0.560326	+	0.828272i	$0.310676\pi$
$984$	0	0
$985$	−30.9398	−0.985824
$986$	0	0
$987$	8.69383	0.276728
$988$	0	0
$989$	48.9139	1.55537
$990$	0	0
$991$	2.91682	0.0926559	0.0463280	−	0.998926i	$-0.485248\pi$
$991$	2.91682	0.0926559	0.0463280	+	0.998926i	$0.485248\pi$
$992$	0	0
$993$	−50.9845	−1.61794
$994$	0	0
$995$	−49.2022	−1.55981
$996$	0	0
$997$	−6.94047	−0.219807	−0.109903	−	0.993942i	$-0.535054\pi$
$997$	−6.94047	−0.219807	−0.109903	+	0.993942i	$0.535054\pi$
$998$	0	0
$999$	3.19877	0.101205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.y.1.5		6
4.3	odd	2		1672.2.a.h.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.h.1.2	✓	6	4.3	odd	2
3344.2.a.y.1.5		6	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67361 q^{3} +3.02460 q^{5} +0.969288 q^{7} +4.14820 q^{9} +O(q^{10})\)	\(q+2.67361 q^{3} +3.02460 q^{5} +0.969288 q^{7} +4.14820 q^{9} -1.00000 q^{11} -1.84959 q^{13} +8.08661 q^{15} -4.65727 q^{17} +1.00000 q^{19} +2.59150 q^{21} +8.83619 q^{23} +4.14820 q^{25} +3.06985 q^{27} +4.08776 q^{29} -5.58577 q^{31} -2.67361 q^{33} +2.93171 q^{35} +1.04199 q^{37} -4.94510 q^{39} +7.65296 q^{41} +5.53564 q^{43} +12.5467 q^{45} +3.35475 q^{47} -6.06048 q^{49} -12.4517 q^{51} +5.65116 q^{53} -3.02460 q^{55} +2.67361 q^{57} +6.98839 q^{59} -13.2254 q^{61} +4.02080 q^{63} -5.59428 q^{65} +16.0252 q^{67} +23.6245 q^{69} +1.66300 q^{71} +13.4425 q^{73} +11.0907 q^{75} -0.969288 q^{77} -5.63914 q^{79} -4.23701 q^{81} +8.89958 q^{83} -14.0864 q^{85} +10.9291 q^{87} -3.27111 q^{89} -1.79279 q^{91} -14.9342 q^{93} +3.02460 q^{95} -5.00721 q^{97} -4.14820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Introduction

L-functions

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Database

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