LMFDB - Embedded newform 3344.2.a.w.1.2

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:	6.6.744786576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 $ x^6 - x^5 - 17x^4 + 13x^3 + 69x^2 - 21x - 30
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			1.2
Root			$2.82559$ 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)

$f(q)$	$=$	$q-2.82559 q^{3} -0.343563 q^{5} -4.77460 q^{7} +4.98394 q^{9} +O(q^{10})$	$q-2.82559 q^{3} -0.343563 q^{5} -4.77460 q^{7} +4.98394 q^{9} +1.00000 q^{11} +1.86340 q^{13} +0.970768 q^{15} -5.69469 q^{17} +1.00000 q^{19} +13.4910 q^{21} +1.06385 q^{23} -4.88196 q^{25} -5.60580 q^{27} -9.31093 q^{29} -2.97077 q^{31} -2.82559 q^{33} +1.64038 q^{35} -3.54764 q^{37} -5.26521 q^{39} +2.82372 q^{41} -8.09454 q^{43} -1.71230 q^{45} -3.00756 q^{47} +15.7968 q^{49} +16.0908 q^{51} -4.33830 q^{53} -0.343563 q^{55} -2.82559 q^{57} -14.6566 q^{59} +1.03595 q^{61} -23.7963 q^{63} -0.640197 q^{65} -7.93109 q^{67} -3.00601 q^{69} +11.4107 q^{71} -16.3062 q^{73} +13.7944 q^{75} -4.77460 q^{77} +13.3459 q^{79} +0.887845 q^{81} +3.40742 q^{83} +1.95649 q^{85} +26.3088 q^{87} +0.0562918 q^{89} -8.89700 q^{91} +8.39416 q^{93} -0.343563 q^{95} +11.4778 q^{97} +4.98394 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * 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
$3$	−2.82559	−1.63135	−0.815677	−	0.578508i	$-0.803635\pi$
$3$	−2.82559	−1.63135	−0.815677	+	0.578508i	$0.803635\pi$
$4$	0	0
$5$	−0.343563	−0.153646	−0.0768231	−	0.997045i	$-0.524478\pi$
$5$	−0.343563	−0.153646	−0.0768231	+	0.997045i	$0.524478\pi$
$6$	0	0
$7$	−4.77460	−1.80463	−0.902314	−	0.431079i	$-0.858133\pi$
$7$	−4.77460	−1.80463	−0.902314	+	0.431079i	$0.858133\pi$
$8$	0	0
$9$	4.98394	1.66131
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.86340	0.516815	0.258407	−	0.966036i	$-0.416802\pi$
$13$	1.86340	0.516815	0.258407	+	0.966036i	$0.416802\pi$
$14$	0	0
$15$	0.970768	0.250651
$16$	0	0
$17$	−5.69469	−1.38116	−0.690582	−	0.723254i	$-0.742646\pi$
$17$	−5.69469	−1.38116	−0.690582	+	0.723254i	$0.742646\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	13.4910	2.94399
$22$	0	0
$23$	1.06385	0.221829	0.110914	−	0.993830i	$-0.464622\pi$
$23$	1.06385	0.221829	0.110914	+	0.993830i	$0.464622\pi$
$24$	0	0
$25$	−4.88196	−0.976393
$26$	0	0
$27$	−5.60580	−1.07884
$28$	0	0
$29$	−9.31093	−1.72900	−0.864498	−	0.502636i	$-0.832364\pi$
$29$	−9.31093	−1.72900	−0.864498	+	0.502636i	$0.832364\pi$
$30$	0	0
$31$	−2.97077	−0.533566	−0.266783	−	0.963757i	$-0.585961\pi$
$31$	−2.97077	−0.533566	−0.266783	+	0.963757i	$0.585961\pi$
$32$	0	0
$33$	−2.82559	−0.491872
$34$	0	0
$35$	1.64038	0.277274
$36$	0	0
$37$	−3.54764	−0.583229	−0.291614	−	0.956536i	$-0.594192\pi$
$37$	−3.54764	−0.583229	−0.291614	+	0.956536i	$0.594192\pi$
$38$	0	0
$39$	−5.26521	−0.843108
$40$	0	0
$41$	2.82372	0.440992	0.220496	−	0.975388i	$-0.429232\pi$
$41$	2.82372	0.440992	0.220496	+	0.975388i	$0.429232\pi$
$42$	0	0
$43$	−8.09454	−1.23441	−0.617203	−	0.786804i	$-0.711734\pi$
$43$	−8.09454	−1.23441	−0.617203	+	0.786804i	$0.711734\pi$
$44$	0	0
$45$	−1.71230	−0.255255
$46$	0	0
$47$	−3.00756	−0.438698	−0.219349	−	0.975646i	$-0.570393\pi$
$47$	−3.00756	−0.438698	−0.219349	+	0.975646i	$0.570393\pi$
$48$	0	0
$49$	15.7968	2.25668
$50$	0	0
$51$	16.0908	2.25317
$52$	0	0
$53$	−4.33830	−0.595911	−0.297956	−	0.954580i	$-0.596305\pi$
$53$	−4.33830	−0.595911	−0.297956	+	0.954580i	$0.596305\pi$
$54$	0	0
$55$	−0.343563	−0.0463261
$56$	0	0
$57$	−2.82559	−0.374258
$58$	0	0
$59$	−14.6566	−1.90812	−0.954061	−	0.299613i	$-0.903143\pi$
$59$	−14.6566	−1.90812	−0.954061	+	0.299613i	$0.903143\pi$
$60$	0	0
$61$	1.03595	0.132640	0.0663201	−	0.997798i	$-0.478874\pi$
$61$	1.03595	0.132640	0.0663201	+	0.997798i	$0.478874\pi$
$62$	0	0
$63$	−23.7963	−2.99805
$64$	0	0
$65$	−0.640197	−0.0794067
$66$	0	0
$67$	−7.93109	−0.968937	−0.484468	−	0.874809i	$-0.660987\pi$
$67$	−7.93109	−0.968937	−0.484468	+	0.874809i	$0.660987\pi$
$68$	0	0
$69$	−3.00601	−0.361881
$70$	0	0
$71$	11.4107	1.35421	0.677103	−	0.735889i	$-0.263235\pi$
$71$	11.4107	1.35421	0.677103	+	0.735889i	$0.263235\pi$
$72$	0	0
$73$	−16.3062	−1.90849	−0.954247	−	0.299019i	$-0.903341\pi$
$73$	−16.3062	−1.90849	−0.954247	+	0.299019i	$0.903341\pi$
$74$	0	0
$75$	13.7944	1.59284
$76$	0	0
$77$	−4.77460	−0.544116
$78$	0	0
$79$	13.3459	1.50153	0.750763	−	0.660571i	$-0.229686\pi$
$79$	13.3459	1.50153	0.750763	+	0.660571i	$0.229686\pi$
$80$	0	0
$81$	0.887845	0.0986494
$82$	0	0
$83$	3.40742	0.374012	0.187006	−	0.982359i	$-0.440122\pi$
$83$	3.40742	0.374012	0.187006	+	0.982359i	$0.440122\pi$
$84$	0	0
$85$	1.95649	0.212211
$86$	0	0
$87$	26.3088	2.82060
$88$	0	0
$89$	0.0562918	0.00596692	0.00298346	−	0.999996i	$-0.499050\pi$
$89$	0.0562918	0.00596692	0.00298346	+	0.999996i	$0.499050\pi$
$90$	0	0
$91$	−8.89700	−0.932659
$92$	0	0
$93$	8.39416	0.870434
$94$	0	0
$95$	−0.343563	−0.0352489
$96$	0	0
$97$	11.4778	1.16539	0.582696	−	0.812690i	$-0.301998\pi$
$97$	11.4778	1.16539	0.582696	+	0.812690i	$0.301998\pi$
$98$	0	0
$99$	4.98394	0.500905
$100$	0	0
$101$	4.64361	0.462057	0.231028	−	0.972947i	$-0.425791\pi$
$101$	4.64361	0.462057	0.231028	+	0.972947i	$0.425791\pi$
$102$	0	0
$103$	−9.73334	−0.959055	−0.479527	−	0.877527i	$-0.659192\pi$
$103$	−9.73334	−0.959055	−0.479527	+	0.877527i	$0.659192\pi$
$104$	0	0
$105$	−4.63503	−0.452333
$106$	0	0
$107$	9.06541	0.876386	0.438193	−	0.898881i	$-0.355619\pi$
$107$	9.06541	0.876386	0.438193	+	0.898881i	$0.355619\pi$
$108$	0	0
$109$	5.74943	0.550695	0.275348	−	0.961345i	$-0.411207\pi$
$109$	5.74943	0.550695	0.275348	+	0.961345i	$0.411207\pi$
$110$	0	0
$111$	10.0242	0.951452
$112$	0	0
$113$	2.68930	0.252988	0.126494	−	0.991967i	$-0.459628\pi$
$113$	2.68930	0.252988	0.126494	+	0.991967i	$0.459628\pi$
$114$	0	0
$115$	−0.365501	−0.0340831
$116$	0	0
$117$	9.28709	0.858592
$118$	0	0
$119$	27.1898	2.49249
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.97868	−0.719413
$124$	0	0
$125$	3.39508	0.303665
$126$	0	0
$127$	13.1746	1.16906	0.584530	−	0.811372i	$-0.301279\pi$
$127$	13.1746	1.16906	0.584530	+	0.811372i	$0.301279\pi$
$128$	0	0
$129$	22.8718	2.01375
$130$	0	0
$131$	−19.6659	−1.71822	−0.859110	−	0.511791i	$-0.828982\pi$
$131$	−19.6659	−1.71822	−0.859110	+	0.511791i	$0.828982\pi$
$132$	0	0
$133$	−4.77460	−0.414010
$134$	0	0
$135$	1.92595	0.165759
$136$	0	0
$137$	9.00937	0.769722	0.384861	−	0.922974i	$-0.374249\pi$
$137$	9.00937	0.769722	0.384861	+	0.922974i	$0.374249\pi$
$138$	0	0
$139$	19.8943	1.68741	0.843707	−	0.536804i	$-0.180369\pi$
$139$	19.8943	1.68741	0.843707	+	0.536804i	$0.180369\pi$
$140$	0	0
$141$	8.49812	0.715671
$142$	0	0
$143$	1.86340	0.155826
$144$	0	0
$145$	3.19890	0.265654
$146$	0	0
$147$	−44.6352	−3.68145
$148$	0	0
$149$	−13.0292	−1.06739	−0.533695	−	0.845677i	$-0.679197\pi$
$149$	−13.0292	−1.06739	−0.533695	+	0.845677i	$0.679197\pi$
$150$	0	0
$151$	1.73820	0.141453	0.0707264	−	0.997496i	$-0.477468\pi$
$151$	1.73820	0.141453	0.0707264	+	0.997496i	$0.477468\pi$
$152$	0	0
$153$	−28.3820	−2.29455
$154$	0	0
$155$	1.02065	0.0819804
$156$	0	0
$157$	−6.95506	−0.555074	−0.277537	−	0.960715i	$-0.589518\pi$
$157$	−6.95506	−0.555074	−0.277537	+	0.960715i	$0.589518\pi$
$158$	0	0
$159$	12.2582	0.972142
$160$	0	0
$161$	−5.07947	−0.400318
$162$	0	0
$163$	−14.7201	−1.15297	−0.576484	−	0.817108i	$-0.695576\pi$
$163$	−14.7201	−1.15297	−0.576484	+	0.817108i	$0.695576\pi$
$164$	0	0
$165$	0.970768	0.0755742
$166$	0	0
$167$	5.95276	0.460638	0.230319	−	0.973115i	$-0.426023\pi$
$167$	5.95276	0.460638	0.230319	+	0.973115i	$0.426023\pi$
$168$	0	0
$169$	−9.52773	−0.732902
$170$	0	0
$171$	4.98394	0.381131
$172$	0	0
$173$	−0.925392	−0.0703563	−0.0351781	−	0.999381i	$-0.511200\pi$
$173$	−0.925392	−0.0703563	−0.0351781	+	0.999381i	$0.511200\pi$
$174$	0	0
$175$	23.3094	1.76203
$176$	0	0
$177$	41.4134	3.11282
$178$	0	0
$179$	−18.9602	−1.41716	−0.708578	−	0.705633i	$-0.750663\pi$
$179$	−18.9602	−1.41716	−0.708578	+	0.705633i	$0.750663\pi$
$180$	0	0
$181$	6.27828	0.466661	0.233331	−	0.972397i	$-0.425038\pi$
$181$	6.27828	0.466661	0.233331	+	0.972397i	$0.425038\pi$
$182$	0	0
$183$	−2.92718	−0.216383
$184$	0	0
$185$	1.21884	0.0896109
$186$	0	0
$187$	−5.69469	−0.416437
$188$	0	0
$189$	26.7654	1.94690
$190$	0	0
$191$	18.2273	1.31888	0.659440	−	0.751758i	$-0.270794\pi$
$191$	18.2273	1.31888	0.659440	+	0.751758i	$0.270794\pi$
$192$	0	0
$193$	−5.51799	−0.397194	−0.198597	−	0.980081i	$-0.563638\pi$
$193$	−5.51799	−0.397194	−0.198597	+	0.980081i	$0.563638\pi$
$194$	0	0
$195$	1.80893	0.129540
$196$	0	0
$197$	24.5897	1.75195	0.875973	−	0.482359i	$-0.160220\pi$
$197$	24.5897	1.75195	0.875973	+	0.482359i	$0.160220\pi$
$198$	0	0
$199$	17.1425	1.21520	0.607601	−	0.794243i	$-0.292132\pi$
$199$	17.1425	1.21520	0.607601	+	0.794243i	$0.292132\pi$
$200$	0	0
$201$	22.4100	1.58068
$202$	0	0
$203$	44.4560	3.12020
$204$	0	0
$205$	−0.970128	−0.0677567
$206$	0	0
$207$	5.30218	0.368527
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−4.10198	−0.282392	−0.141196	−	0.989982i	$-0.545095\pi$
$211$	−4.10198	−0.282392	−0.141196	+	0.989982i	$0.545095\pi$
$212$	0	0
$213$	−32.2420	−2.20919
$214$	0	0
$215$	2.78099	0.189662
$216$	0	0
$217$	14.1842	0.962888
$218$	0	0
$219$	46.0745	3.11343
$220$	0	0
$221$	−10.6115	−0.713806
$222$	0	0
$223$	−26.2530	−1.75803	−0.879015	−	0.476793i	$-0.841799\pi$
$223$	−26.2530	−1.75803	−0.879015	+	0.476793i	$0.841799\pi$
$224$	0	0
$225$	−24.3314	−1.62209
$226$	0	0
$227$	2.54225	0.168735	0.0843677	−	0.996435i	$-0.473113\pi$
$227$	2.54225	0.168735	0.0843677	+	0.996435i	$0.473113\pi$
$228$	0	0
$229$	−7.23905	−0.478370	−0.239185	−	0.970974i	$-0.576880\pi$
$229$	−7.23905	−0.478370	−0.239185	+	0.970974i	$0.576880\pi$
$230$	0	0
$231$	13.4910	0.887645
$232$	0	0
$233$	24.6807	1.61689	0.808444	−	0.588573i	$-0.200310\pi$
$233$	24.6807	1.61689	0.808444	+	0.588573i	$0.200310\pi$
$234$	0	0
$235$	1.03329	0.0674043
$236$	0	0
$237$	−37.7099	−2.44952
$238$	0	0
$239$	22.3130	1.44331	0.721654	−	0.692254i	$-0.243382\pi$
$239$	22.3130	1.44331	0.721654	+	0.692254i	$0.243382\pi$
$240$	0	0
$241$	9.08973	0.585521	0.292760	−	0.956186i	$-0.405426\pi$
$241$	9.08973	0.585521	0.292760	+	0.956186i	$0.405426\pi$
$242$	0	0
$243$	14.3087	0.917904
$244$	0	0
$245$	−5.42720	−0.346731
$246$	0	0
$247$	1.86340	0.118565
$248$	0	0
$249$	−9.62795	−0.610146
$250$	0	0
$251$	2.11865	0.133728	0.0668640	−	0.997762i	$-0.478701\pi$
$251$	2.11865	0.133728	0.0668640	+	0.997762i	$0.478701\pi$
$252$	0	0
$253$	1.06385	0.0668838
$254$	0	0
$255$	−5.52822	−0.346191
$256$	0	0
$257$	−4.83262	−0.301450	−0.150725	−	0.988576i	$-0.548161\pi$
$257$	−4.83262	−0.301450	−0.150725	+	0.988576i	$0.548161\pi$
$258$	0	0
$259$	16.9386	1.05251
$260$	0	0
$261$	−46.4051	−2.87241
$262$	0	0
$263$	−10.7931	−0.665529	−0.332764	−	0.943010i	$-0.607981\pi$
$263$	−10.7931	−0.665529	−0.332764	+	0.943010i	$0.607981\pi$
$264$	0	0
$265$	1.49048	0.0915596
$266$	0	0
$267$	−0.159057	−0.00973415
$268$	0	0
$269$	−16.0956	−0.981365	−0.490683	−	0.871338i	$-0.663253\pi$
$269$	−16.0956	−0.981365	−0.490683	+	0.871338i	$0.663253\pi$
$270$	0	0
$271$	22.2262	1.35014	0.675072	−	0.737752i	$-0.264113\pi$
$271$	22.2262	1.35014	0.675072	+	0.737752i	$0.264113\pi$
$272$	0	0
$273$	25.1392	1.52150
$274$	0	0
$275$	−4.88196	−0.294394
$276$	0	0
$277$	−1.12740	−0.0677392	−0.0338696	−	0.999426i	$-0.510783\pi$
$277$	−1.12740	−0.0677392	−0.0338696	+	0.999426i	$0.510783\pi$
$278$	0	0
$279$	−14.8061	−0.886420
$280$	0	0
$281$	1.51489	0.0903705	0.0451852	−	0.998979i	$-0.485612\pi$
$281$	1.51489	0.0903705	0.0451852	+	0.998979i	$0.485612\pi$
$282$	0	0
$283$	−5.74917	−0.341753	−0.170876	−	0.985292i	$-0.554660\pi$
$283$	−5.74917	−0.341753	−0.170876	+	0.985292i	$0.554660\pi$
$284$	0	0
$285$	0.970768	0.0575034
$286$	0	0
$287$	−13.4821	−0.795826
$288$	0	0
$289$	15.4295	0.907615
$290$	0	0
$291$	−32.4315	−1.90117
$292$	0	0
$293$	−0.00163269	−9.53826e−5 0	−4.76913e−5	−	1.00000i	$-0.500015\pi$
$293$	−0.00163269	−9.53826e−5 0	−4.76913e−5		1.00000i	$0.500015\pi$
$294$	0	0
$295$	5.03546	0.293176
$296$	0	0
$297$	−5.60580	−0.325281
$298$	0	0
$299$	1.98238	0.114644
$300$	0	0
$301$	38.6482	2.22764
$302$	0	0
$303$	−13.1209	−0.753778
$304$	0	0
$305$	−0.355916	−0.0203797
$306$	0	0
$307$	27.0663	1.54475	0.772377	−	0.635164i	$-0.219068\pi$
$307$	27.0663	1.54475	0.772377	+	0.635164i	$0.219068\pi$
$308$	0	0
$309$	27.5024	1.56456
$310$	0	0
$311$	−21.8440	−1.23866	−0.619329	−	0.785131i	$-0.712595\pi$
$311$	−21.8440	−1.23866	−0.619329	+	0.785131i	$0.712595\pi$
$312$	0	0
$313$	−4.54317	−0.256795	−0.128398	−	0.991723i	$-0.540983\pi$
$313$	−4.54317	−0.256795	−0.128398	+	0.991723i	$0.540983\pi$
$314$	0	0
$315$	8.17554	0.460640
$316$	0	0
$317$	−8.22001	−0.461682	−0.230841	−	0.972992i	$-0.574148\pi$
$317$	−8.22001	−0.461682	−0.230841	+	0.972992i	$0.574148\pi$
$318$	0	0
$319$	−9.31093	−0.521312
$320$	0	0
$321$	−25.6151	−1.42969
$322$	0	0
$323$	−5.69469	−0.316861
$324$	0	0
$325$	−9.09706	−0.504614
$326$	0	0
$327$	−16.2455	−0.898378
$328$	0	0
$329$	14.3599	0.791687
$330$	0	0
$331$	10.5060	0.577463	0.288732	−	0.957410i	$-0.406766\pi$
$331$	10.5060	0.577463	0.288732	+	0.957410i	$0.406766\pi$
$332$	0	0
$333$	−17.6812	−0.968926
$334$	0	0
$335$	2.72483	0.148874
$336$	0	0
$337$	21.5797	1.17552	0.587760	−	0.809035i	$-0.300010\pi$
$337$	21.5797	1.17552	0.587760	+	0.809035i	$0.300010\pi$
$338$	0	0
$339$	−7.59885	−0.412713
$340$	0	0
$341$	−2.97077	−0.160876
$342$	0	0
$343$	−42.0011	−2.26785
$344$	0	0
$345$	1.03275	0.0556016
$346$	0	0
$347$	21.5171	1.15510	0.577549	−	0.816356i	$-0.304009\pi$
$347$	21.5171	1.15510	0.577549	+	0.816356i	$0.304009\pi$
$348$	0	0
$349$	23.7204	1.26972	0.634862	−	0.772625i	$-0.281057\pi$
$349$	23.7204	1.26972	0.634862	+	0.772625i	$0.281057\pi$
$350$	0	0
$351$	−10.4459	−0.557559
$352$	0	0
$353$	13.4757	0.717241	0.358621	−	0.933483i	$-0.383247\pi$
$353$	13.4757	0.717241	0.358621	+	0.933483i	$0.383247\pi$
$354$	0	0
$355$	−3.92031	−0.208069
$356$	0	0
$357$	−76.8273	−4.06613
$358$	0	0
$359$	14.4579	0.763058	0.381529	−	0.924357i	$-0.375398\pi$
$359$	14.4579	0.763058	0.381529	+	0.924357i	$0.375398\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.82559	−0.148305
$364$	0	0
$365$	5.60221	0.293233
$366$	0	0
$367$	0.692406	0.0361433	0.0180716	−	0.999837i	$-0.494247\pi$
$367$	0.692406	0.0361433	0.0180716	+	0.999837i	$0.494247\pi$
$368$	0	0
$369$	14.0733	0.732625
$370$	0	0
$371$	20.7136	1.07540
$372$	0	0
$373$	15.2174	0.787929	0.393964	−	0.919126i	$-0.371103\pi$
$373$	15.2174	0.787929	0.393964	+	0.919126i	$0.371103\pi$
$374$	0	0
$375$	−9.59310	−0.495385
$376$	0	0
$377$	−17.3500	−0.893571
$378$	0	0
$379$	−30.7668	−1.58039	−0.790193	−	0.612859i	$-0.790020\pi$
$379$	−30.7668	−1.58039	−0.790193	+	0.612859i	$0.790020\pi$
$380$	0	0
$381$	−37.2261	−1.90715
$382$	0	0
$383$	−31.1031	−1.58929	−0.794646	−	0.607073i	$-0.792344\pi$
$383$	−31.1031	−1.58929	−0.794646	+	0.607073i	$0.792344\pi$
$384$	0	0
$385$	1.64038	0.0836014
$386$	0	0
$387$	−40.3427	−2.05074
$388$	0	0
$389$	16.8274	0.853181	0.426590	−	0.904445i	$-0.359715\pi$
$389$	16.8274	0.853181	0.426590	+	0.904445i	$0.359715\pi$
$390$	0	0
$391$	−6.05831	−0.306382
$392$	0	0
$393$	55.5678	2.80302
$394$	0	0
$395$	−4.58515	−0.230704
$396$	0	0
$397$	18.3895	0.922943	0.461472	−	0.887155i	$-0.347322\pi$
$397$	18.3895	0.922943	0.461472	+	0.887155i	$0.347322\pi$
$398$	0	0
$399$	13.4910	0.675397
$400$	0	0
$401$	−18.4034	−0.919022	−0.459511	−	0.888172i	$-0.651975\pi$
$401$	−18.4034	−0.919022	−0.459511	+	0.888172i	$0.651975\pi$
$402$	0	0
$403$	−5.53574	−0.275755
$404$	0	0
$405$	−0.305031	−0.0151571
$406$	0	0
$407$	−3.54764	−0.175850
$408$	0	0
$409$	−39.3705	−1.94675	−0.973373	−	0.229225i	$-0.926381\pi$
$409$	−39.3705	−1.94675	−0.973373	+	0.229225i	$0.926381\pi$
$410$	0	0
$411$	−25.4568	−1.25569
$412$	0	0
$413$	69.9792	3.44345
$414$	0	0
$415$	−1.17066	−0.0574656
$416$	0	0
$417$	−56.2131	−2.75277
$418$	0	0
$419$	−17.2402	−0.842237	−0.421118	−	0.907006i	$-0.638362\pi$
$419$	−17.2402	−0.842237	−0.421118	+	0.907006i	$0.638362\pi$
$420$	0	0
$421$	7.86946	0.383534	0.191767	−	0.981440i	$-0.438578\pi$
$421$	7.86946	0.383534	0.191767	+	0.981440i	$0.438578\pi$
$422$	0	0
$423$	−14.9895	−0.728815
$424$	0	0
$425$	27.8013	1.34856
$426$	0	0
$427$	−4.94626	−0.239366
$428$	0	0
$429$	−5.26521	−0.254207
$430$	0	0
$431$	21.9971	1.05957	0.529783	−	0.848133i	$-0.322274\pi$
$431$	21.9971	1.05957	0.529783	+	0.848133i	$0.322274\pi$
$432$	0	0
$433$	24.7111	1.18754	0.593771	−	0.804634i	$-0.297639\pi$
$433$	24.7111	1.18754	0.593771	+	0.804634i	$0.297639\pi$
$434$	0	0
$435$	−9.03876	−0.433375
$436$	0	0
$437$	1.06385	0.0508909
$438$	0	0
$439$	−27.3096	−1.30342	−0.651708	−	0.758470i	$-0.725947\pi$
$439$	−27.3096	−1.30342	−0.651708	+	0.758470i	$0.725947\pi$
$440$	0	0
$441$	78.7303	3.74906
$442$	0	0
$443$	38.0147	1.80613	0.903066	−	0.429501i	$-0.141311\pi$
$443$	38.0147	1.80613	0.903066	+	0.429501i	$0.141311\pi$
$444$	0	0
$445$	−0.0193398	−0.000916795 0
$446$	0	0
$447$	36.8150	1.74129
$448$	0	0
$449$	39.1470	1.84746	0.923730	−	0.383043i	$-0.125124\pi$
$449$	39.1470	1.84746	0.923730	+	0.383043i	$0.125124\pi$
$450$	0	0
$451$	2.82372	0.132964
$452$	0	0
$453$	−4.91144	−0.230759
$454$	0	0
$455$	3.05668	0.143300
$456$	0	0
$457$	4.72692	0.221116	0.110558	−	0.993870i	$-0.464736\pi$
$457$	4.72692	0.221116	0.110558	+	0.993870i	$0.464736\pi$
$458$	0	0
$459$	31.9233	1.49005
$460$	0	0
$461$	−11.1666	−0.520079	−0.260040	−	0.965598i	$-0.583736\pi$
$461$	−11.1666	−0.520079	−0.260040	+	0.965598i	$0.583736\pi$
$462$	0	0
$463$	−30.5320	−1.41894	−0.709470	−	0.704735i	$-0.751066\pi$
$463$	−30.5320	−1.41894	−0.709470	+	0.704735i	$0.751066\pi$
$464$	0	0
$465$	−2.88393	−0.133739
$466$	0	0
$467$	−11.2216	−0.519274	−0.259637	−	0.965706i	$-0.583603\pi$
$467$	−11.2216	−0.519274	−0.259637	+	0.965706i	$0.583603\pi$
$468$	0	0
$469$	37.8678	1.74857
$470$	0	0
$471$	19.6521	0.905522
$472$	0	0
$473$	−8.09454	−0.372187
$474$	0	0
$475$	−4.88196	−0.224000
$476$	0	0
$477$	−21.6218	−0.989996
$478$	0	0
$479$	−15.0838	−0.689197	−0.344598	−	0.938750i	$-0.611985\pi$
$479$	−15.0838	−0.689197	−0.344598	+	0.938750i	$0.611985\pi$
$480$	0	0
$481$	−6.61069	−0.301421
$482$	0	0
$483$	14.3525	0.653060
$484$	0	0
$485$	−3.94335	−0.179058
$486$	0	0
$487$	24.3598	1.10385	0.551925	−	0.833894i	$-0.313894\pi$
$487$	24.3598	1.10385	0.551925	+	0.833894i	$0.313894\pi$
$488$	0	0
$489$	41.5930	1.88090
$490$	0	0
$491$	−15.4105	−0.695467	−0.347734	−	0.937593i	$-0.613049\pi$
$491$	−15.4105	−0.695467	−0.347734	+	0.937593i	$0.613049\pi$
$492$	0	0
$493$	53.0228	2.38803
$494$	0	0
$495$	−1.71230	−0.0769622
$496$	0	0
$497$	−54.4817	−2.44384
$498$	0	0
$499$	−11.7733	−0.527043	−0.263522	−	0.964653i	$-0.584884\pi$
$499$	−11.7733	−0.527043	−0.263522	+	0.964653i	$0.584884\pi$
$500$	0	0
$501$	−16.8200	−0.751464
$502$	0	0
$503$	21.6073	0.963421	0.481710	−	0.876330i	$-0.340016\pi$
$503$	21.6073	0.963421	0.481710	+	0.876330i	$0.340016\pi$
$504$	0	0
$505$	−1.59538	−0.0709933
$506$	0	0
$507$	26.9214	1.19562
$508$	0	0
$509$	−30.4087	−1.34784	−0.673921	−	0.738804i	$-0.735391\pi$
$509$	−30.4087	−1.34784	−0.673921	+	0.738804i	$0.735391\pi$
$510$	0	0
$511$	77.8555	3.44412
$512$	0	0
$513$	−5.60580	−0.247502
$514$	0	0
$515$	3.34402	0.147355
$516$	0	0
$517$	−3.00756	−0.132272
$518$	0	0
$519$	2.61478	0.114776
$520$	0	0
$521$	25.2167	1.10476	0.552381	−	0.833591i	$-0.313719\pi$
$521$	25.2167	1.10476	0.552381	+	0.833591i	$0.313719\pi$
$522$	0	0
$523$	−27.9677	−1.22294	−0.611472	−	0.791266i	$-0.709422\pi$
$523$	−27.9677	−1.22294	−0.611472	+	0.791266i	$0.709422\pi$
$524$	0	0
$525$	−65.8628	−2.87449
$526$	0	0
$527$	16.9176	0.736942
$528$	0	0
$529$	−21.8682	−0.950792
$530$	0	0
$531$	−73.0474	−3.16999
$532$	0	0
$533$	5.26174	0.227911
$534$	0	0
$535$	−3.11454	−0.134653
$536$	0	0
$537$	53.5738	2.31188
$538$	0	0
$539$	15.7968	0.680416
$540$	0	0
$541$	28.8814	1.24171	0.620855	−	0.783926i	$-0.286786\pi$
$541$	28.8814	1.24171	0.620855	+	0.783926i	$0.286786\pi$
$542$	0	0
$543$	−17.7398	−0.761289
$544$	0	0
$545$	−1.97529	−0.0846122
$546$	0	0
$547$	−16.4332	−0.702632	−0.351316	−	0.936257i	$-0.614266\pi$
$547$	−16.4332	−0.702632	−0.351316	+	0.936257i	$0.614266\pi$
$548$	0	0
$549$	5.16313	0.220357
$550$	0	0
$551$	−9.31093	−0.396659
$552$	0	0
$553$	−63.7211	−2.70970
$554$	0	0
$555$	−3.44394	−0.146187
$556$	0	0
$557$	0.814831	0.0345255	0.0172628	−	0.999851i	$-0.494505\pi$
$557$	0.814831	0.0345255	0.0172628	+	0.999851i	$0.494505\pi$
$558$	0	0
$559$	−15.0834	−0.637960
$560$	0	0
$561$	16.0908	0.679356
$562$	0	0
$563$	−24.0662	−1.01427	−0.507134	−	0.861867i	$-0.669295\pi$
$563$	−24.0662	−1.01427	−0.507134	+	0.861867i	$0.669295\pi$
$564$	0	0
$565$	−0.923945	−0.0388706
$566$	0	0
$567$	−4.23910	−0.178026
$568$	0	0
$569$	−21.5082	−0.901670	−0.450835	−	0.892607i	$-0.648874\pi$
$569$	−21.5082	−0.901670	−0.450835	+	0.892607i	$0.648874\pi$
$570$	0	0
$571$	−14.6812	−0.614388	−0.307194	−	0.951647i	$-0.599390\pi$
$571$	−14.6812	−0.614388	−0.307194	+	0.951647i	$0.599390\pi$
$572$	0	0
$573$	−51.5027	−2.15156
$574$	0	0
$575$	−5.19369	−0.216592
$576$	0	0
$577$	−22.0271	−0.917001	−0.458501	−	0.888694i	$-0.651613\pi$
$577$	−22.0271	−0.917001	−0.458501	+	0.888694i	$0.651613\pi$
$578$	0	0
$579$	15.5916	0.647963
$580$	0	0
$581$	−16.2690	−0.674954
$582$	0	0
$583$	−4.33830	−0.179674
$584$	0	0
$585$	−3.19070	−0.131919
$586$	0	0
$587$	−27.9648	−1.15423	−0.577116	−	0.816663i	$-0.695822\pi$
$587$	−27.9648	−1.15423	−0.577116	+	0.816663i	$0.695822\pi$
$588$	0	0
$589$	−2.97077	−0.122408
$590$	0	0
$591$	−69.4805	−2.85804
$592$	0	0
$593$	−14.4726	−0.594317	−0.297159	−	0.954828i	$-0.596039\pi$
$593$	−14.4726	−0.594317	−0.297159	+	0.954828i	$0.596039\pi$
$594$	0	0
$595$	−9.34144	−0.382962
$596$	0	0
$597$	−48.4377	−1.98242
$598$	0	0
$599$	−12.7403	−0.520554	−0.260277	−	0.965534i	$-0.583814\pi$
$599$	−12.7403	−0.520554	−0.260277	+	0.965534i	$0.583814\pi$
$600$	0	0
$601$	−0.00889305	−0.000362755 0	−0.000181378	−	1.00000i	$-0.500058\pi$
$601$	−0.00889305	−0.000362755 0	−0.000181378		1.00000i	$0.500058\pi$
$602$	0	0
$603$	−39.5281	−1.60971
$604$	0	0
$605$	−0.343563	−0.0139678
$606$	0	0
$607$	−10.0651	−0.408530	−0.204265	−	0.978916i	$-0.565480\pi$
$607$	−10.0651	−0.408530	−0.204265	+	0.978916i	$0.565480\pi$
$608$	0	0
$609$	−125.614	−5.09014
$610$	0	0
$611$	−5.60430	−0.226726
$612$	0	0
$613$	−17.0238	−0.687584	−0.343792	−	0.939046i	$-0.611712\pi$
$613$	−17.0238	−0.687584	−0.343792	+	0.939046i	$0.611712\pi$
$614$	0	0
$615$	2.74118	0.110535
$616$	0	0
$617$	−39.2503	−1.58016	−0.790078	−	0.613006i	$-0.789960\pi$
$617$	−39.2503	−1.58016	−0.790078	+	0.613006i	$0.789960\pi$
$618$	0	0
$619$	49.2905	1.98115	0.990577	−	0.136960i	$-0.0437331\pi$
$619$	49.2905	1.98115	0.990577	+	0.136960i	$0.0437331\pi$
$620$	0	0
$621$	−5.96374	−0.239317
$622$	0	0
$623$	−0.268771	−0.0107681
$624$	0	0
$625$	23.2434	0.929736
$626$	0	0
$627$	−2.82559	−0.112843
$628$	0	0
$629$	20.2027	0.805535
$630$	0	0
$631$	32.9741	1.31268	0.656340	−	0.754465i	$-0.272104\pi$
$631$	32.9741	1.31268	0.656340	+	0.754465i	$0.272104\pi$
$632$	0	0
$633$	11.5905	0.460681
$634$	0	0
$635$	−4.52633	−0.179622
$636$	0	0
$637$	29.4358	1.16629
$638$	0	0
$639$	56.8704	2.24976
$640$	0	0
$641$	−5.23839	−0.206904	−0.103452	−	0.994634i	$-0.532989\pi$
$641$	−5.23839	−0.206904	−0.103452	+	0.994634i	$0.532989\pi$
$642$	0	0
$643$	−12.3440	−0.486799	−0.243400	−	0.969926i	$-0.578263\pi$
$643$	−12.3440	−0.486799	−0.243400	+	0.969926i	$0.578263\pi$
$644$	0	0
$645$	−7.85793	−0.309406
$646$	0	0
$647$	31.0492	1.22067	0.610334	−	0.792144i	$-0.291035\pi$
$647$	31.0492	1.22067	0.610334	+	0.792144i	$0.291035\pi$
$648$	0	0
$649$	−14.6566	−0.575320
$650$	0	0
$651$	−40.0788	−1.57081
$652$	0	0
$653$	−24.2141	−0.947570	−0.473785	−	0.880640i	$-0.657113\pi$
$653$	−24.2141	−0.947570	−0.473785	+	0.880640i	$0.657113\pi$
$654$	0	0
$655$	6.75650	0.263998
$656$	0	0
$657$	−81.2691	−3.17061
$658$	0	0
$659$	−36.4592	−1.42025	−0.710124	−	0.704076i	$-0.751361\pi$
$659$	−36.4592	−1.42025	−0.710124	+	0.704076i	$0.751361\pi$
$660$	0	0
$661$	47.0828	1.83131	0.915655	−	0.401965i	$-0.131673\pi$
$661$	47.0828	1.83131	0.915655	+	0.401965i	$0.131673\pi$
$662$	0	0
$663$	29.9837	1.16447
$664$	0	0
$665$	1.64038	0.0636111
$666$	0	0
$667$	−9.90545	−0.383541
$668$	0	0
$669$	74.1801	2.86797
$670$	0	0
$671$	1.03595	0.0399925
$672$	0	0
$673$	33.4857	1.29078	0.645390	−	0.763853i	$-0.276695\pi$
$673$	33.4857	1.29078	0.645390	+	0.763853i	$0.276695\pi$
$674$	0	0
$675$	27.3673	1.05337
$676$	0	0
$677$	−39.8223	−1.53050	−0.765249	−	0.643735i	$-0.777384\pi$
$677$	−39.8223	−1.53050	−0.765249	+	0.643735i	$0.777384\pi$
$678$	0	0
$679$	−54.8018	−2.10310
$680$	0	0
$681$	−7.18336	−0.275267
$682$	0	0
$683$	−1.36966	−0.0524085	−0.0262043	−	0.999657i	$-0.508342\pi$
$683$	−1.36966	−0.0524085	−0.0262043	+	0.999657i	$0.508342\pi$
$684$	0	0
$685$	−3.09529	−0.118265
$686$	0	0
$687$	20.4546	0.780390
$688$	0	0
$689$	−8.08400	−0.307976
$690$	0	0
$691$	−29.5333	−1.12350	−0.561751	−	0.827307i	$-0.689872\pi$
$691$	−29.5333	−1.12350	−0.561751	+	0.827307i	$0.689872\pi$
$692$	0	0
$693$	−23.7963	−0.903947
$694$	0	0
$695$	−6.83496	−0.259265
$696$	0	0
$697$	−16.0802	−0.609082
$698$	0	0
$699$	−69.7376	−2.63772
$700$	0	0
$701$	40.5434	1.53130	0.765651	−	0.643256i	$-0.222417\pi$
$701$	40.5434	1.53130	0.765651	+	0.643256i	$0.222417\pi$
$702$	0	0
$703$	−3.54764	−0.133802
$704$	0	0
$705$	−2.91964	−0.109960
$706$	0	0
$707$	−22.1714	−0.833841
$708$	0	0
$709$	10.1180	0.379991	0.189995	−	0.981785i	$-0.439153\pi$
$709$	10.1180	0.379991	0.189995	+	0.981785i	$0.439153\pi$
$710$	0	0
$711$	66.5150	2.49451
$712$	0	0
$713$	−3.16046	−0.118360
$714$	0	0
$715$	−0.640197	−0.0239420
$716$	0	0
$717$	−63.0473	−2.35455
$718$	0	0
$719$	10.5326	0.392799	0.196400	−	0.980524i	$-0.437075\pi$
$719$	10.5326	0.392799	0.196400	+	0.980524i	$0.437075\pi$
$720$	0	0
$721$	46.4728	1.73074
$722$	0	0
$723$	−25.6838	−0.955191
$724$	0	0
$725$	45.4556	1.68818
$726$	0	0
$727$	12.8599	0.476947	0.238474	−	0.971149i	$-0.423353\pi$
$727$	12.8599	0.476947	0.238474	+	0.971149i	$0.423353\pi$
$728$	0	0
$729$	−43.0940	−1.59608
$730$	0	0
$731$	46.0959	1.70492
$732$	0	0
$733$	−13.3272	−0.492252	−0.246126	−	0.969238i	$-0.579158\pi$
$733$	−13.3272	−0.492252	−0.246126	+	0.969238i	$0.579158\pi$
$734$	0	0
$735$	15.3350	0.565641
$736$	0	0
$737$	−7.93109	−0.292145
$738$	0	0
$739$	19.7362	0.726009	0.363004	−	0.931787i	$-0.381751\pi$
$739$	19.7362	0.726009	0.363004	+	0.931787i	$0.381751\pi$
$740$	0	0
$741$	−5.26521	−0.193422
$742$	0	0
$743$	−10.9311	−0.401022	−0.200511	−	0.979691i	$-0.564260\pi$
$743$	−10.9311	−0.401022	−0.200511	+	0.979691i	$0.564260\pi$
$744$	0	0
$745$	4.47634	0.164000
$746$	0	0
$747$	16.9824	0.621352
$748$	0	0
$749$	−43.2837	−1.58155
$750$	0	0
$751$	0.00491790	0.000179457 0	8.97284e−5	−	1.00000i	$-0.499971\pi$
$751$	0.00491790	0.000179457 0	8.97284e−5		1.00000i	$0.499971\pi$
$752$	0	0
$753$	−5.98643	−0.218158
$754$	0	0
$755$	−0.597182	−0.0217337
$756$	0	0
$757$	3.46494	0.125935	0.0629677	−	0.998016i	$-0.479943\pi$
$757$	3.46494	0.125935	0.0629677	+	0.998016i	$0.479943\pi$
$758$	0	0
$759$	−3.00601	−0.109111
$760$	0	0
$761$	10.8298	0.392580	0.196290	−	0.980546i	$-0.437111\pi$
$761$	10.8298	0.392580	0.196290	+	0.980546i	$0.437111\pi$
$762$	0	0
$763$	−27.4512	−0.993800
$764$	0	0
$765$	9.75101	0.352549
$766$	0	0
$767$	−27.3111	−0.986146
$768$	0	0
$769$	−5.55885	−0.200457	−0.100229	−	0.994964i	$-0.531957\pi$
$769$	−5.55885	−0.200457	−0.100229	+	0.994964i	$0.531957\pi$
$770$	0	0
$771$	13.6550	0.491772
$772$	0	0
$773$	−20.6807	−0.743834	−0.371917	−	0.928266i	$-0.621299\pi$
$773$	−20.6807	−0.743834	−0.371917	+	0.928266i	$0.621299\pi$
$774$	0	0
$775$	14.5032	0.520970
$776$	0	0
$777$	−47.8614	−1.71702
$778$	0	0
$779$	2.82372	0.101170
$780$	0	0
$781$	11.4107	0.408308
$782$	0	0
$783$	52.1952	1.86530
$784$	0	0
$785$	2.38950	0.0852851
$786$	0	0
$787$	−7.09548	−0.252926	−0.126463	−	0.991971i	$-0.540363\pi$
$787$	−7.09548	−0.252926	−0.126463	+	0.991971i	$0.540363\pi$
$788$	0	0
$789$	30.4967	1.08571
$790$	0	0
$791$	−12.8403	−0.456549
$792$	0	0
$793$	1.93040	0.0685504
$794$	0	0
$795$	−4.21149	−0.149366
$796$	0	0
$797$	−21.9429	−0.777258	−0.388629	−	0.921394i	$-0.627051\pi$
$797$	−21.9429	−0.777258	−0.388629	+	0.921394i	$0.627051\pi$
$798$	0	0
$799$	17.1271	0.605914
$800$	0	0
$801$	0.280555	0.00991293
$802$	0	0
$803$	−16.3062	−0.575433
$804$	0	0
$805$	1.74512	0.0615074
$806$	0	0
$807$	45.4795	1.60095
$808$	0	0
$809$	3.89093	0.136798	0.0683990	−	0.997658i	$-0.478211\pi$
$809$	3.89093	0.136798	0.0683990	+	0.997658i	$0.478211\pi$
$810$	0	0
$811$	8.09828	0.284369	0.142185	−	0.989840i	$-0.454587\pi$
$811$	8.09828	0.284369	0.142185	+	0.989840i	$0.454587\pi$
$812$	0	0
$813$	−62.8019	−2.20256
$814$	0	0
$815$	5.05729	0.177149
$816$	0	0
$817$	−8.09454	−0.283192
$818$	0	0
$819$	−44.3421	−1.54944
$820$	0	0
$821$	−29.0408	−1.01353	−0.506765	−	0.862084i	$-0.669159\pi$
$821$	−29.0408	−1.01353	−0.506765	+	0.862084i	$0.669159\pi$
$822$	0	0
$823$	46.6098	1.62471	0.812357	−	0.583160i	$-0.198184\pi$
$823$	46.6098	1.62471	0.812357	+	0.583160i	$0.198184\pi$
$824$	0	0
$825$	13.7944	0.480260
$826$	0	0
$827$	−55.5230	−1.93072	−0.965361	−	0.260917i	$-0.915975\pi$
$827$	−55.5230	−1.93072	−0.965361	+	0.260917i	$0.915975\pi$
$828$	0	0
$829$	16.7515	0.581802	0.290901	−	0.956753i	$-0.406045\pi$
$829$	16.7515	0.581802	0.290901	+	0.956753i	$0.406045\pi$
$830$	0	0
$831$	3.18558	0.110507
$832$	0	0
$833$	−89.9578	−3.11685
$834$	0	0
$835$	−2.04515	−0.0707754
$836$	0	0
$837$	16.6535	0.575630
$838$	0	0
$839$	−40.4855	−1.39771	−0.698857	−	0.715261i	$-0.746308\pi$
$839$	−40.4855	−1.39771	−0.698857	+	0.715261i	$0.746308\pi$
$840$	0	0
$841$	57.6934	1.98943
$842$	0	0
$843$	−4.28044	−0.147426
$844$	0	0
$845$	3.27338	0.112608
$846$	0	0
$847$	−4.77460	−0.164057
$848$	0	0
$849$	16.2448	0.557519
$850$	0	0
$851$	−3.77417	−0.129377
$852$	0	0
$853$	51.7573	1.77214	0.886068	−	0.463555i	$-0.153426\pi$
$853$	51.7573	1.77214	0.886068	+	0.463555i	$0.153426\pi$
$854$	0	0
$855$	−1.71230	−0.0585594
$856$	0	0
$857$	−27.8604	−0.951693	−0.475846	−	0.879528i	$-0.657858\pi$
$857$	−27.8604	−0.951693	−0.475846	+	0.879528i	$0.657858\pi$
$858$	0	0
$859$	51.6139	1.76104	0.880521	−	0.474007i	$-0.157193\pi$
$859$	51.6139	1.76104	0.880521	+	0.474007i	$0.157193\pi$
$860$	0	0
$861$	38.0950	1.29827
$862$	0	0
$863$	−1.36733	−0.0465444	−0.0232722	−	0.999729i	$-0.507408\pi$
$863$	−1.36733	−0.0465444	−0.0232722	+	0.999729i	$0.507408\pi$
$864$	0	0
$865$	0.317931	0.0108100
$866$	0	0
$867$	−43.5973	−1.48064
$868$	0	0
$869$	13.3459	0.452727
$870$	0	0
$871$	−14.7788	−0.500761
$872$	0	0
$873$	57.2046	1.93608
$874$	0	0
$875$	−16.2102	−0.548003
$876$	0	0
$877$	−3.25271	−0.109836	−0.0549181	−	0.998491i	$-0.517490\pi$
$877$	−3.25271	−0.109836	−0.0549181	+	0.998491i	$0.517490\pi$
$878$	0	0
$879$	0.00461330	0.000155603 0
$880$	0	0
$881$	9.48028	0.319399	0.159699	−	0.987166i	$-0.448948\pi$
$881$	9.48028	0.319399	0.159699	+	0.987166i	$0.448948\pi$
$882$	0	0
$883$	−33.3488	−1.12228	−0.561138	−	0.827722i	$-0.689636\pi$
$883$	−33.3488	−1.12228	−0.561138	+	0.827722i	$0.689636\pi$
$884$	0	0
$885$	−14.2281	−0.478273
$886$	0	0
$887$	−38.2336	−1.28376	−0.641880	−	0.766805i	$-0.721845\pi$
$887$	−38.2336	−1.28376	−0.641880	+	0.766805i	$0.721845\pi$
$888$	0	0
$889$	−62.9036	−2.10972
$890$	0	0
$891$	0.887845	0.0297439
$892$	0	0
$893$	−3.00756	−0.100644
$894$	0	0
$895$	6.51405	0.217741
$896$	0	0
$897$	−5.60140	−0.187025
$898$	0	0
$899$	27.6606	0.922533
$900$	0	0
$901$	24.7053	0.823052
$902$	0	0
$903$	−109.204	−3.63408
$904$	0	0
$905$	−2.15699	−0.0717007
$906$	0	0
$907$	52.9937	1.75963	0.879814	−	0.475318i	$-0.157667\pi$
$907$	52.9937	1.75963	0.879814	+	0.475318i	$0.157667\pi$
$908$	0	0
$909$	23.1435	0.767621
$910$	0	0
$911$	−11.4793	−0.380325	−0.190162	−	0.981753i	$-0.560901\pi$
$911$	−11.4793	−0.380325	−0.190162	+	0.981753i	$0.560901\pi$
$912$	0	0
$913$	3.40742	0.112769
$914$	0	0
$915$	1.00567	0.0332464
$916$	0	0
$917$	93.8969	3.10075
$918$	0	0
$919$	50.7962	1.67561	0.837806	−	0.545968i	$-0.183838\pi$
$919$	50.7962	1.67561	0.837806	+	0.545968i	$0.183838\pi$
$920$	0	0
$921$	−76.4781	−2.52004
$922$	0	0
$923$	21.2628	0.699874
$924$	0	0
$925$	17.3195	0.569461
$926$	0	0
$927$	−48.5104	−1.59329
$928$	0	0
$929$	25.5521	0.838339	0.419169	−	0.907908i	$-0.362321\pi$
$929$	25.5521	0.838339	0.419169	+	0.907908i	$0.362321\pi$
$930$	0	0
$931$	15.7968	0.517719
$932$	0	0
$933$	61.7221	2.02069
$934$	0	0
$935$	1.95649	0.0639839
$936$	0	0
$937$	−33.0081	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$937$	−33.0081	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$938$	0	0
$939$	12.8371	0.418924
$940$	0	0
$941$	−10.2690	−0.334761	−0.167380	−	0.985892i	$-0.553531\pi$
$941$	−10.2690	−0.334761	−0.167380	+	0.985892i	$0.553531\pi$
$942$	0	0
$943$	3.00403	0.0978245
$944$	0	0
$945$	−9.19562	−0.299134
$946$	0	0
$947$	23.4610	0.762379	0.381190	−	0.924497i	$-0.375514\pi$
$947$	23.4610	0.762379	0.381190	+	0.924497i	$0.375514\pi$
$948$	0	0
$949$	−30.3850	−0.986338
$950$	0	0
$951$	23.2264	0.753166
$952$	0	0
$953$	38.9755	1.26254	0.631271	−	0.775563i	$-0.282534\pi$
$953$	38.9755	1.26254	0.631271	+	0.775563i	$0.282534\pi$
$954$	0	0
$955$	−6.26222	−0.202641
$956$	0	0
$957$	26.3088	0.850444
$958$	0	0
$959$	−43.0161	−1.38906
$960$	0	0
$961$	−22.1745	−0.715308
$962$	0	0
$963$	45.1814	1.45595
$964$	0	0
$965$	1.89578	0.0610273
$966$	0	0
$967$	51.3942	1.65273	0.826363	−	0.563137i	$-0.190406\pi$
$967$	51.3942	1.65273	0.826363	+	0.563137i	$0.190406\pi$
$968$	0	0
$969$	16.0908	0.516912
$970$	0	0
$971$	−4.66223	−0.149618	−0.0748090	−	0.997198i	$-0.523835\pi$
$971$	−4.66223	−0.149618	−0.0748090	+	0.997198i	$0.523835\pi$
$972$	0	0
$973$	−94.9874	−3.04515
$974$	0	0
$975$	25.7045	0.823204
$976$	0	0
$977$	−51.7991	−1.65720	−0.828600	−	0.559841i	$-0.810862\pi$
$977$	−51.7991	−1.65720	−0.828600	+	0.559841i	$0.810862\pi$
$978$	0	0
$979$	0.0562918	0.00179909
$980$	0	0
$981$	28.6548	0.914877
$982$	0	0
$983$	−51.7747	−1.65136	−0.825678	−	0.564142i	$-0.809207\pi$
$983$	−51.7747	−1.65136	−0.825678	+	0.564142i	$0.809207\pi$
$984$	0	0
$985$	−8.44814	−0.269180
$986$	0	0
$987$	−40.5751	−1.29152
$988$	0	0
$989$	−8.61140	−0.273826
$990$	0	0
$991$	21.2223	0.674148	0.337074	−	0.941478i	$-0.390563\pi$
$991$	21.2223	0.674148	0.337074	+	0.941478i	$0.390563\pi$
$992$	0	0
$993$	−29.6857	−0.942047
$994$	0	0
$995$	−5.88955	−0.186711
$996$	0	0
$997$	51.7325	1.63839	0.819193	−	0.573518i	$-0.194422\pi$
$997$	51.7325	1.63839	0.819193	+	0.573518i	$0.194422\pi$
$998$	0	0
$999$	19.8874	0.629209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.w.1.2		6
4.3	odd	2		836.2.a.e.1.5	✓	6
12.11	even	2		7524.2.a.q.1.5		6
44.43	even	2		9196.2.a.n.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.5	✓	6	4.3	odd	2
3344.2.a.w.1.2		6	1.1	even	1	trivial
7524.2.a.q.1.5		6	12.11	even	2
9196.2.a.n.1.5		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)

\(f(q)\)	\(=\)	\(q-2.82559 q^{3} -0.343563 q^{5} -4.77460 q^{7} +4.98394 q^{9} +O(q^{10})\)	\(q-2.82559 q^{3} -0.343563 q^{5} -4.77460 q^{7} +4.98394 q^{9} +1.00000 q^{11} +1.86340 q^{13} +0.970768 q^{15} -5.69469 q^{17} +1.00000 q^{19} +13.4910 q^{21} +1.06385 q^{23} -4.88196 q^{25} -5.60580 q^{27} -9.31093 q^{29} -2.97077 q^{31} -2.82559 q^{33} +1.64038 q^{35} -3.54764 q^{37} -5.26521 q^{39} +2.82372 q^{41} -8.09454 q^{43} -1.71230 q^{45} -3.00756 q^{47} +15.7968 q^{49} +16.0908 q^{51} -4.33830 q^{53} -0.343563 q^{55} -2.82559 q^{57} -14.6566 q^{59} +1.03595 q^{61} -23.7963 q^{63} -0.640197 q^{65} -7.93109 q^{67} -3.00601 q^{69} +11.4107 q^{71} -16.3062 q^{73} +13.7944 q^{75} -4.77460 q^{77} +13.3459 q^{79} +0.887845 q^{81} +3.40742 q^{83} +1.95649 q^{85} +26.3088 q^{87} +0.0562918 q^{89} -8.89700 q^{91} +8.39416 q^{93} -0.343563 q^{95} +11.4778 q^{97} +4.98394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * q^99

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