LMFDB - Embedded newform 3344.2.a.v.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.576096652.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 $ x^6 - 2x^5 - 8x^4 + 11x^3 + 16x^2 - 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$-1.13326$ 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)

$f(q)$	$=$	$q-2.13326 q^{3} +0.368439 q^{5} -4.29504 q^{7} +1.55080 q^{9} +O(q^{10})$	$q-2.13326 q^{3} +0.368439 q^{5} -4.29504 q^{7} +1.55080 q^{9} -1.00000 q^{11} +4.29504 q^{13} -0.785977 q^{15} +1.76482 q^{17} -1.00000 q^{19} +9.16244 q^{21} +2.34414 q^{23} -4.86425 q^{25} +3.09153 q^{27} -1.93765 q^{29} +6.63918 q^{31} +2.13326 q^{33} -1.58246 q^{35} -2.57932 q^{37} -9.16244 q^{39} +3.79334 q^{41} +11.7240 q^{43} +0.571375 q^{45} -8.32356 q^{47} +11.4474 q^{49} -3.76482 q^{51} +7.31345 q^{53} -0.368439 q^{55} +2.13326 q^{57} +0.635940 q^{59} -11.2466 q^{61} -6.66074 q^{63} +1.58246 q^{65} -5.63536 q^{67} -5.00066 q^{69} +2.65279 q^{71} +0.519535 q^{73} +10.3767 q^{75} +4.29504 q^{77} +3.02123 q^{79} -11.2474 q^{81} -12.0641 q^{83} +0.650229 q^{85} +4.13351 q^{87} +4.05192 q^{89} -18.4474 q^{91} -14.1631 q^{93} -0.368439 q^{95} -18.4338 q^{97} -1.55080 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + q^{5} + 4 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 + q^5 + 4 * q^9	$ 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43} + 11 q^{45} - 16 q^{47} + 2 q^{49} - 15 q^{51} + 17 q^{53} - q^{55} + 4 q^{57} - 19 q^{59} - 6 q^{61} - 2 q^{63} + 6 q^{65} - 14 q^{67} + q^{69} - q^{71} - 5 q^{73} - 18 q^{75} + 2 q^{81} - 11 q^{83} - 26 q^{85} - 14 q^{87} + 12 q^{89} - 44 q^{91} - 6 q^{93} - q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * 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
$3$	−2.13326	−1.23164	−0.615819	−	0.787888i	$-0.711175\pi$
$3$	−2.13326	−1.23164	−0.615819	+	0.787888i	$0.711175\pi$
$4$	0	0
$5$	0.368439	0.164771	0.0823856	−	0.996601i	$-0.473746\pi$
$5$	0.368439	0.164771	0.0823856	+	0.996601i	$0.473746\pi$
$6$	0	0
$7$	−4.29504	−1.62337	−0.811686	−	0.584094i	$-0.801450\pi$
$7$	−4.29504	−1.62337	−0.811686	+	0.584094i	$0.801450\pi$
$8$	0	0
$9$	1.55080	0.516932
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.29504	1.19123	0.595615	−	0.803270i	$-0.296908\pi$
$13$	4.29504	1.19123	0.595615	+	0.803270i	$0.296908\pi$
$14$	0	0
$15$	−0.785977	−0.202938
$16$	0	0
$17$	1.76482	0.428032	0.214016	−	0.976830i	$-0.431346\pi$
$17$	1.76482	0.428032	0.214016	+	0.976830i	$0.431346\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	9.16244	1.99941
$22$	0	0
$23$	2.34414	0.488787	0.244393	−	0.969676i	$-0.421411\pi$
$23$	2.34414	0.488787	0.244393	+	0.969676i	$0.421411\pi$
$24$	0	0
$25$	−4.86425	−0.972850
$26$	0	0
$27$	3.09153	0.594964
$28$	0	0
$29$	−1.93765	−0.359813	−0.179906	−	0.983684i	$-0.557579\pi$
$29$	−1.93765	−0.359813	−0.179906	+	0.983684i	$0.557579\pi$
$30$	0	0
$31$	6.63918	1.19243	0.596216	−	0.802824i	$-0.296670\pi$
$31$	6.63918	1.19243	0.596216	+	0.802824i	$0.296670\pi$
$32$	0	0
$33$	2.13326	0.371353
$34$	0	0
$35$	−1.58246	−0.267485
$36$	0	0
$37$	−2.57932	−0.424037	−0.212019	−	0.977266i	$-0.568004\pi$
$37$	−2.57932	−0.424037	−0.212019	+	0.977266i	$0.568004\pi$
$38$	0	0
$39$	−9.16244	−1.46716
$40$	0	0
$41$	3.79334	0.592420	0.296210	−	0.955123i	$-0.404277\pi$
$41$	3.79334	0.592420	0.296210	+	0.955123i	$0.404277\pi$
$42$	0	0
$43$	11.7240	1.78789	0.893946	−	0.448174i	$-0.147925\pi$
$43$	11.7240	1.78789	0.893946	+	0.448174i	$0.147925\pi$
$44$	0	0
$45$	0.571375	0.0851756
$46$	0	0
$47$	−8.32356	−1.21412	−0.607058	−	0.794657i	$-0.707650\pi$
$47$	−8.32356	−1.21412	−0.607058	+	0.794657i	$0.707650\pi$
$48$	0	0
$49$	11.4474	1.63534
$50$	0	0
$51$	−3.76482	−0.527180
$52$	0	0
$53$	7.31345	1.00458	0.502290	−	0.864699i	$-0.332491\pi$
$53$	7.31345	1.00458	0.502290	+	0.864699i	$0.332491\pi$
$54$	0	0
$55$	−0.368439	−0.0496804
$56$	0	0
$57$	2.13326	0.282557
$58$	0	0
$59$	0.635940	0.0827924	0.0413962	−	0.999143i	$-0.486819\pi$
$59$	0.635940	0.0827924	0.0413962	+	0.999143i	$0.486819\pi$
$60$	0	0
$61$	−11.2466	−1.43998	−0.719990	−	0.693985i	$-0.755853\pi$
$61$	−11.2466	−1.43998	−0.719990	+	0.693985i	$0.755853\pi$
$62$	0	0
$63$	−6.66074	−0.839174
$64$	0	0
$65$	1.58246	0.196280
$66$	0	0
$67$	−5.63536	−0.688469	−0.344234	−	0.938884i	$-0.611862\pi$
$67$	−5.63536	−0.688469	−0.344234	+	0.938884i	$0.611862\pi$
$68$	0	0
$69$	−5.00066	−0.602008
$70$	0	0
$71$	2.65279	0.314829	0.157414	−	0.987533i	$-0.449684\pi$
$71$	2.65279	0.314829	0.157414	+	0.987533i	$0.449684\pi$
$72$	0	0
$73$	0.519535	0.0608069	0.0304035	−	0.999538i	$-0.490321\pi$
$73$	0.519535	0.0608069	0.0304035	+	0.999538i	$0.490321\pi$
$74$	0	0
$75$	10.3767	1.19820
$76$	0	0
$77$	4.29504	0.489465
$78$	0	0
$79$	3.02123	0.339915	0.169958	−	0.985451i	$-0.445637\pi$
$79$	3.02123	0.339915	0.169958	+	0.985451i	$0.445637\pi$
$80$	0	0
$81$	−11.2474	−1.24971
$82$	0	0
$83$	−12.0641	−1.32420	−0.662102	−	0.749413i	$-0.730336\pi$
$83$	−12.0641	−1.32420	−0.662102	+	0.749413i	$0.730336\pi$
$84$	0	0
$85$	0.650229	0.0705273
$86$	0	0
$87$	4.13351	0.443159
$88$	0	0
$89$	4.05192	0.429502	0.214751	−	0.976669i	$-0.431106\pi$
$89$	4.05192	0.429502	0.214751	+	0.976669i	$0.431106\pi$
$90$	0	0
$91$	−18.4474	−1.93381
$92$	0	0
$93$	−14.1631	−1.46864
$94$	0	0
$95$	−0.368439	−0.0378011
$96$	0	0
$97$	−18.4338	−1.87167	−0.935836	−	0.352436i	$-0.885354\pi$
$97$	−18.4338	−1.87167	−0.935836	+	0.352436i	$0.885354\pi$
$98$	0	0
$99$	−1.55080	−0.155861
$100$	0	0
$101$	6.98008	0.694544	0.347272	−	0.937764i	$-0.387108\pi$
$101$	6.98008	0.694544	0.347272	+	0.937764i	$0.387108\pi$
$102$	0	0
$103$	−5.96899	−0.588142	−0.294071	−	0.955784i	$-0.595010\pi$
$103$	−5.96899	−0.588142	−0.294071	+	0.955784i	$0.595010\pi$
$104$	0	0
$105$	3.37580	0.329445
$106$	0	0
$107$	−6.57658	−0.635782	−0.317891	−	0.948127i	$-0.602975\pi$
$107$	−6.57658	−0.635782	−0.317891	+	0.948127i	$0.602975\pi$
$108$	0	0
$109$	−8.78605	−0.841551	−0.420776	−	0.907165i	$-0.638242\pi$
$109$	−8.78605	−0.841551	−0.420776	+	0.907165i	$0.638242\pi$
$110$	0	0
$111$	5.50236	0.522260
$112$	0	0
$113$	17.9008	1.68396	0.841982	−	0.539505i	$-0.181389\pi$
$113$	17.9008	1.68396	0.841982	+	0.539505i	$0.181389\pi$
$114$	0	0
$115$	0.863673	0.0805379
$116$	0	0
$117$	6.66074	0.615785
$118$	0	0
$119$	−7.57997	−0.694855
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.09218	−0.729647
$124$	0	0
$125$	−3.63438	−0.325069
$126$	0	0
$127$	10.5097	0.932587	0.466293	−	0.884630i	$-0.345589\pi$
$127$	10.5097	0.932587	0.466293	+	0.884630i	$0.345589\pi$
$128$	0	0
$129$	−25.0103	−2.20204
$130$	0	0
$131$	−15.0578	−1.31560	−0.657802	−	0.753191i	$-0.728514\pi$
$131$	−15.0578	−1.31560	−0.657802	+	0.753191i	$0.728514\pi$
$132$	0	0
$133$	4.29504	0.372427
$134$	0	0
$135$	1.13904	0.0980330
$136$	0	0
$137$	−11.1409	−0.951829	−0.475915	−	0.879491i	$-0.657883\pi$
$137$	−11.1409	−0.951829	−0.475915	+	0.879491i	$0.657883\pi$
$138$	0	0
$139$	−1.78283	−0.151218	−0.0756090	−	0.997138i	$-0.524090\pi$
$139$	−1.78283	−0.151218	−0.0756090	+	0.997138i	$0.524090\pi$
$140$	0	0
$141$	17.7563	1.49535
$142$	0	0
$143$	−4.29504	−0.359169
$144$	0	0
$145$	−0.713907	−0.0592868
$146$	0	0
$147$	−24.4202	−2.01415
$148$	0	0
$149$	−17.0249	−1.39474	−0.697368	−	0.716713i	$-0.745646\pi$
$149$	−17.0249	−1.39474	−0.697368	+	0.716713i	$0.745646\pi$
$150$	0	0
$151$	11.8398	0.963508	0.481754	−	0.876306i	$-0.340000\pi$
$151$	11.8398	0.963508	0.481754	+	0.876306i	$0.340000\pi$
$152$	0	0
$153$	2.73688	0.221264
$154$	0	0
$155$	2.44614	0.196478
$156$	0	0
$157$	15.7949	1.26057	0.630285	−	0.776363i	$-0.282938\pi$
$157$	15.7949	1.26057	0.630285	+	0.776363i	$0.282938\pi$
$158$	0	0
$159$	−15.6015	−1.23728
$160$	0	0
$161$	−10.0682	−0.793483
$162$	0	0
$163$	16.4299	1.28689	0.643443	−	0.765494i	$-0.277505\pi$
$163$	16.4299	1.28689	0.643443	+	0.765494i	$0.277505\pi$
$164$	0	0
$165$	0.785977	0.0611882
$166$	0	0
$167$	9.92808	0.768258	0.384129	−	0.923279i	$-0.374502\pi$
$167$	9.92808	0.768258	0.384129	+	0.923279i	$0.374502\pi$
$168$	0	0
$169$	5.44737	0.419029
$170$	0	0
$171$	−1.55080	−0.118592
$172$	0	0
$173$	−14.8140	−1.12629	−0.563145	−	0.826358i	$-0.690409\pi$
$173$	−14.8140	−1.12629	−0.563145	+	0.826358i	$0.690409\pi$
$174$	0	0
$175$	20.8922	1.57930
$176$	0	0
$177$	−1.35663	−0.101970
$178$	0	0
$179$	−11.6413	−0.870109	−0.435054	−	0.900404i	$-0.643271\pi$
$179$	−11.6413	−0.870109	−0.435054	+	0.900404i	$0.643271\pi$
$180$	0	0
$181$	−2.35395	−0.174968	−0.0874840	−	0.996166i	$-0.527883\pi$
$181$	−2.35395	−0.174968	−0.0874840	+	0.996166i	$0.527883\pi$
$182$	0	0
$183$	23.9919	1.77353
$184$	0	0
$185$	−0.950323	−0.0698691
$186$	0	0
$187$	−1.76482	−0.129056
$188$	0	0
$189$	−13.2782	−0.965849
$190$	0	0
$191$	−17.4735	−1.26434	−0.632171	−	0.774829i	$-0.717836\pi$
$191$	−17.4735	−1.26434	−0.632171	+	0.774829i	$0.717836\pi$
$192$	0	0
$193$	1.12828	0.0812152	0.0406076	−	0.999175i	$-0.487071\pi$
$193$	1.12828	0.0812152	0.0406076	+	0.999175i	$0.487071\pi$
$194$	0	0
$195$	−3.37580	−0.241746
$196$	0	0
$197$	−27.0948	−1.93043	−0.965213	−	0.261464i	$-0.915795\pi$
$197$	−27.0948	−1.93043	−0.965213	+	0.261464i	$0.915795\pi$
$198$	0	0
$199$	−9.84021	−0.697554	−0.348777	−	0.937206i	$-0.613403\pi$
$199$	−9.84021	−0.697554	−0.348777	+	0.937206i	$0.613403\pi$
$200$	0	0
$201$	12.0217	0.847945
$202$	0	0
$203$	8.32229	0.584110
$204$	0	0
$205$	1.39762	0.0976138
$206$	0	0
$207$	3.63528	0.252670
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−23.1342	−1.59263	−0.796314	−	0.604884i	$-0.793219\pi$
$211$	−23.1342	−1.59263	−0.796314	+	0.604884i	$0.793219\pi$
$212$	0	0
$213$	−5.65910	−0.387755
$214$	0	0
$215$	4.31958	0.294593
$216$	0	0
$217$	−28.5155	−1.93576
$218$	0	0
$219$	−1.10830	−0.0748921
$220$	0	0
$221$	7.57997	0.509884
$222$	0	0
$223$	−10.2571	−0.686864	−0.343432	−	0.939178i	$-0.611590\pi$
$223$	−10.2571	−0.686864	−0.343432	+	0.939178i	$0.611590\pi$
$224$	0	0
$225$	−7.54347	−0.502898
$226$	0	0
$227$	−9.44257	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$227$	−9.44257	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$228$	0	0
$229$	−7.80092	−0.515499	−0.257750	−	0.966212i	$-0.582981\pi$
$229$	−7.80092	−0.515499	−0.257750	+	0.966212i	$0.582981\pi$
$230$	0	0
$231$	−9.16244	−0.602844
$232$	0	0
$233$	−0.284355	−0.0186287	−0.00931436	−	0.999957i	$-0.502965\pi$
$233$	−0.284355	−0.0186287	−0.00931436	+	0.999957i	$0.502965\pi$
$234$	0	0
$235$	−3.06673	−0.200051
$236$	0	0
$237$	−6.44508	−0.418653
$238$	0	0
$239$	13.3450	0.863214	0.431607	−	0.902062i	$-0.357947\pi$
$239$	13.3450	0.863214	0.431607	+	0.902062i	$0.357947\pi$
$240$	0	0
$241$	25.3523	1.63309	0.816544	−	0.577283i	$-0.195887\pi$
$241$	25.3523	1.63309	0.816544	+	0.577283i	$0.195887\pi$
$242$	0	0
$243$	14.7191	0.944230
$244$	0	0
$245$	4.21766	0.269457
$246$	0	0
$247$	−4.29504	−0.273287
$248$	0	0
$249$	25.7358	1.63094
$250$	0	0
$251$	−11.4410	−0.722149	−0.361074	−	0.932537i	$-0.617590\pi$
$251$	−11.4410	−0.722149	−0.361074	+	0.932537i	$0.617590\pi$
$252$	0	0
$253$	−2.34414	−0.147375
$254$	0	0
$255$	−1.38711	−0.0868641
$256$	0	0
$257$	6.77306	0.422492	0.211246	−	0.977433i	$-0.432248\pi$
$257$	6.77306	0.422492	0.211246	+	0.977433i	$0.432248\pi$
$258$	0	0
$259$	11.0783	0.688371
$260$	0	0
$261$	−3.00491	−0.185999
$262$	0	0
$263$	17.7941	1.09723	0.548616	−	0.836074i	$-0.315155\pi$
$263$	17.7941	1.09723	0.548616	+	0.836074i	$0.315155\pi$
$264$	0	0
$265$	2.69457	0.165526
$266$	0	0
$267$	−8.64379	−0.528992
$268$	0	0
$269$	−15.2225	−0.928131	−0.464065	−	0.885801i	$-0.653610\pi$
$269$	−15.2225	−0.928131	−0.464065	+	0.885801i	$0.653610\pi$
$270$	0	0
$271$	−13.8640	−0.842181	−0.421090	−	0.907019i	$-0.638352\pi$
$271$	−13.8640	−0.842181	−0.421090	+	0.907019i	$0.638352\pi$
$272$	0	0
$273$	39.3530	2.38175
$274$	0	0
$275$	4.86425	0.293325
$276$	0	0
$277$	−15.9664	−0.959331	−0.479665	−	0.877451i	$-0.659242\pi$
$277$	−15.9664	−0.959331	−0.479665	+	0.877451i	$0.659242\pi$
$278$	0	0
$279$	10.2960	0.616407
$280$	0	0
$281$	−6.50948	−0.388323	−0.194161	−	0.980970i	$-0.562199\pi$
$281$	−6.50948	−0.388323	−0.194161	+	0.980970i	$0.562199\pi$
$282$	0	0
$283$	21.6521	1.28709	0.643543	−	0.765410i	$-0.277464\pi$
$283$	21.6521	1.28709	0.643543	+	0.765410i	$0.277464\pi$
$284$	0	0
$285$	0.785977	0.0465573
$286$	0	0
$287$	−16.2926	−0.961719
$288$	0	0
$289$	−13.8854	−0.816789
$290$	0	0
$291$	39.3242	2.30522
$292$	0	0
$293$	−1.27143	−0.0742777	−0.0371389	−	0.999310i	$-0.511824\pi$
$293$	−1.27143	−0.0742777	−0.0371389	+	0.999310i	$0.511824\pi$
$294$	0	0
$295$	0.234306	0.0136418
$296$	0	0
$297$	−3.09153	−0.179389
$298$	0	0
$299$	10.0682	0.582257
$300$	0	0
$301$	−50.3550	−2.90242
$302$	0	0
$303$	−14.8903	−0.855427
$304$	0	0
$305$	−4.14369	−0.237267
$306$	0	0
$307$	−20.1122	−1.14786	−0.573932	−	0.818903i	$-0.694583\pi$
$307$	−20.1122	−1.14786	−0.573932	+	0.818903i	$0.694583\pi$
$308$	0	0
$309$	12.7334	0.724378
$310$	0	0
$311$	−12.2293	−0.693458	−0.346729	−	0.937965i	$-0.612708\pi$
$311$	−12.2293	−0.693458	−0.346729	+	0.937965i	$0.612708\pi$
$312$	0	0
$313$	−25.6832	−1.45170	−0.725850	−	0.687853i	$-0.758554\pi$
$313$	−25.6832	−1.45170	−0.725850	+	0.687853i	$0.758554\pi$
$314$	0	0
$315$	−2.45408	−0.138272
$316$	0	0
$317$	10.1568	0.570464	0.285232	−	0.958459i	$-0.407929\pi$
$317$	10.1568	0.570464	0.285232	+	0.958459i	$0.407929\pi$
$318$	0	0
$319$	1.93765	0.108488
$320$	0	0
$321$	14.0295	0.783053
$322$	0	0
$323$	−1.76482	−0.0981972
$324$	0	0
$325$	−20.8922	−1.15889
$326$	0	0
$327$	18.7429	1.03649
$328$	0	0
$329$	35.7500	1.97096
$330$	0	0
$331$	−22.9975	−1.26405	−0.632027	−	0.774946i	$-0.717777\pi$
$331$	−22.9975	−1.26405	−0.632027	+	0.774946i	$0.717777\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−2.07629	−0.113440
$336$	0	0
$337$	−15.9446	−0.868560	−0.434280	−	0.900778i	$-0.642997\pi$
$337$	−15.9446	−0.868560	−0.434280	+	0.900778i	$0.642997\pi$
$338$	0	0
$339$	−38.1870	−2.07404
$340$	0	0
$341$	−6.63918	−0.359532
$342$	0	0
$343$	−19.1016	−1.03139
$344$	0	0
$345$	−1.84244	−0.0991936
$346$	0	0
$347$	14.7552	0.792102	0.396051	−	0.918229i	$-0.370380\pi$
$347$	14.7552	0.792102	0.396051	+	0.918229i	$0.370380\pi$
$348$	0	0
$349$	35.0348	1.87537	0.937684	−	0.347489i	$-0.112966\pi$
$349$	35.0348	1.87537	0.937684	+	0.347489i	$0.112966\pi$
$350$	0	0
$351$	13.2782	0.708739
$352$	0	0
$353$	−3.53960	−0.188394	−0.0941971	−	0.995554i	$-0.530028\pi$
$353$	−3.53960	−0.188394	−0.0941971	+	0.995554i	$0.530028\pi$
$354$	0	0
$355$	0.977394	0.0518747
$356$	0	0
$357$	16.1701	0.855810
$358$	0	0
$359$	16.0685	0.848065	0.424032	−	0.905647i	$-0.360614\pi$
$359$	16.0685	0.848065	0.424032	+	0.905647i	$0.360614\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.13326	−0.111967
$364$	0	0
$365$	0.191417	0.0100192
$366$	0	0
$367$	3.31930	0.173266	0.0866331	−	0.996240i	$-0.472389\pi$
$367$	3.31930	0.173266	0.0866331	+	0.996240i	$0.472389\pi$
$368$	0	0
$369$	5.88270	0.306241
$370$	0	0
$371$	−31.4116	−1.63081
$372$	0	0
$373$	4.75685	0.246301	0.123150	−	0.992388i	$-0.460700\pi$
$373$	4.75685	0.246301	0.123150	+	0.992388i	$0.460700\pi$
$374$	0	0
$375$	7.75308	0.400367
$376$	0	0
$377$	−8.32229	−0.428620
$378$	0	0
$379$	−14.9302	−0.766912	−0.383456	−	0.923559i	$-0.625266\pi$
$379$	−14.9302	−0.766912	−0.383456	+	0.923559i	$0.625266\pi$
$380$	0	0
$381$	−22.4200	−1.14861
$382$	0	0
$383$	32.5424	1.66284	0.831421	−	0.555643i	$-0.187528\pi$
$383$	32.5424	1.66284	0.831421	+	0.555643i	$0.187528\pi$
$384$	0	0
$385$	1.58246	0.0806497
$386$	0	0
$387$	18.1815	0.924220
$388$	0	0
$389$	−26.4819	−1.34268	−0.671342	−	0.741148i	$-0.734282\pi$
$389$	−26.4819	−1.34268	−0.671342	+	0.741148i	$0.734282\pi$
$390$	0	0
$391$	4.13698	0.209216
$392$	0	0
$393$	32.1221	1.62035
$394$	0	0
$395$	1.11314	0.0560082
$396$	0	0
$397$	18.7685	0.941964	0.470982	−	0.882143i	$-0.343900\pi$
$397$	18.7685	0.941964	0.470982	+	0.882143i	$0.343900\pi$
$398$	0	0
$399$	−9.16244	−0.458696
$400$	0	0
$401$	5.76842	0.288061	0.144031	−	0.989573i	$-0.453994\pi$
$401$	5.76842	0.288061	0.144031	+	0.989573i	$0.453994\pi$
$402$	0	0
$403$	28.5155	1.42046
$404$	0	0
$405$	−4.14399	−0.205917
$406$	0	0
$407$	2.57932	0.127852
$408$	0	0
$409$	−11.9722	−0.591986	−0.295993	−	0.955190i	$-0.595650\pi$
$409$	−11.9722	−0.591986	−0.295993	+	0.955190i	$0.595650\pi$
$410$	0	0
$411$	23.7664	1.17231
$412$	0	0
$413$	−2.73139	−0.134403
$414$	0	0
$415$	−4.44488	−0.218191
$416$	0	0
$417$	3.80325	0.186246
$418$	0	0
$419$	−35.8202	−1.74993	−0.874965	−	0.484187i	$-0.839115\pi$
$419$	−35.8202	−1.74993	−0.874965	+	0.484187i	$0.839115\pi$
$420$	0	0
$421$	−17.5248	−0.854104	−0.427052	−	0.904227i	$-0.640448\pi$
$421$	−17.5248	−0.854104	−0.427052	+	0.904227i	$0.640448\pi$
$422$	0	0
$423$	−12.9082	−0.627616
$424$	0	0
$425$	−8.58453	−0.416411
$426$	0	0
$427$	48.3046	2.33762
$428$	0	0
$429$	9.16244	0.442367
$430$	0	0
$431$	10.2381	0.493153	0.246577	−	0.969123i	$-0.420694\pi$
$431$	10.2381	0.493153	0.246577	+	0.969123i	$0.420694\pi$
$432$	0	0
$433$	−27.0289	−1.29893	−0.649463	−	0.760393i	$-0.725006\pi$
$433$	−27.0289	−1.29893	−0.649463	+	0.760393i	$0.725006\pi$
$434$	0	0
$435$	1.52295	0.0730199
$436$	0	0
$437$	−2.34414	−0.112135
$438$	0	0
$439$	−35.5862	−1.69844	−0.849218	−	0.528042i	$-0.822926\pi$
$439$	−35.5862	−1.69844	−0.849218	+	0.528042i	$0.822926\pi$
$440$	0	0
$441$	17.7526	0.845360
$442$	0	0
$443$	−40.3060	−1.91500	−0.957499	−	0.288438i	$-0.906864\pi$
$443$	−40.3060	−1.91500	−0.957499	+	0.288438i	$0.906864\pi$
$444$	0	0
$445$	1.49289	0.0707696
$446$	0	0
$447$	36.3186	1.71781
$448$	0	0
$449$	−4.72627	−0.223047	−0.111523	−	0.993762i	$-0.535573\pi$
$449$	−4.72627	−0.223047	−0.111523	+	0.993762i	$0.535573\pi$
$450$	0	0
$451$	−3.79334	−0.178621
$452$	0	0
$453$	−25.2573	−1.18669
$454$	0	0
$455$	−6.79674	−0.318636
$456$	0	0
$457$	−0.557936	−0.0260992	−0.0130496	−	0.999915i	$-0.504154\pi$
$457$	−0.557936	−0.0260992	−0.0130496	+	0.999915i	$0.504154\pi$
$458$	0	0
$459$	5.45599	0.254664
$460$	0	0
$461$	3.88592	0.180985	0.0904926	−	0.995897i	$-0.471156\pi$
$461$	3.88592	0.180985	0.0904926	+	0.995897i	$0.471156\pi$
$462$	0	0
$463$	19.3618	0.899821	0.449911	−	0.893074i	$-0.351456\pi$
$463$	19.3618	0.899821	0.449911	+	0.893074i	$0.351456\pi$
$464$	0	0
$465$	−5.21824	−0.241990
$466$	0	0
$467$	3.62076	0.167549	0.0837745	−	0.996485i	$-0.473302\pi$
$467$	3.62076	0.167549	0.0837745	+	0.996485i	$0.473302\pi$
$468$	0	0
$469$	24.2041	1.11764
$470$	0	0
$471$	−33.6946	−1.55257
$472$	0	0
$473$	−11.7240	−0.539070
$474$	0	0
$475$	4.86425	0.223187
$476$	0	0
$477$	11.3417	0.519300
$478$	0	0
$479$	34.9363	1.59628	0.798139	−	0.602473i	$-0.205818\pi$
$479$	34.9363	1.59628	0.798139	+	0.602473i	$0.205818\pi$
$480$	0	0
$481$	−11.0783	−0.505126
$482$	0	0
$483$	21.4780	0.977284
$484$	0	0
$485$	−6.79175	−0.308398
$486$	0	0
$487$	24.9504	1.13061	0.565306	−	0.824881i	$-0.308758\pi$
$487$	24.9504	1.13061	0.565306	+	0.824881i	$0.308758\pi$
$488$	0	0
$489$	−35.0492	−1.58498
$490$	0	0
$491$	32.3947	1.46195	0.730976	−	0.682403i	$-0.239065\pi$
$491$	32.3947	1.46195	0.730976	+	0.682403i	$0.239065\pi$
$492$	0	0
$493$	−3.41961	−0.154011
$494$	0	0
$495$	−0.571375	−0.0256814
$496$	0	0
$497$	−11.3939	−0.511084
$498$	0	0
$499$	5.67702	0.254138	0.127069	−	0.991894i	$-0.459443\pi$
$499$	5.67702	0.254138	0.127069	+	0.991894i	$0.459443\pi$
$500$	0	0
$501$	−21.1792	−0.946216
$502$	0	0
$503$	17.9667	0.801095	0.400547	−	0.916276i	$-0.368820\pi$
$503$	17.9667	0.801095	0.400547	+	0.916276i	$0.368820\pi$
$504$	0	0
$505$	2.57174	0.114441
$506$	0	0
$507$	−11.6207	−0.516092
$508$	0	0
$509$	−19.9900	−0.886040	−0.443020	−	0.896512i	$-0.646093\pi$
$509$	−19.9900	−0.886040	−0.443020	+	0.896512i	$0.646093\pi$
$510$	0	0
$511$	−2.23142	−0.0987123
$512$	0	0
$513$	−3.09153	−0.136494
$514$	0	0
$515$	−2.19921	−0.0969089
$516$	0	0
$517$	8.32356	0.366070
$518$	0	0
$519$	31.6022	1.38718
$520$	0	0
$521$	39.2081	1.71774	0.858870	−	0.512194i	$-0.171167\pi$
$521$	39.2081	1.71774	0.858870	+	0.512194i	$0.171167\pi$
$522$	0	0
$523$	15.6782	0.685559	0.342779	−	0.939416i	$-0.388632\pi$
$523$	15.6782	0.685559	0.342779	+	0.939416i	$0.388632\pi$
$524$	0	0
$525$	−44.5684	−1.94512
$526$	0	0
$527$	11.7170	0.510399
$528$	0	0
$529$	−17.5050	−0.761088
$530$	0	0
$531$	0.986215	0.0427981
$532$	0	0
$533$	16.2926	0.705709
$534$	0	0
$535$	−2.42307	−0.104758
$536$	0	0
$537$	24.8338	1.07166
$538$	0	0
$539$	−11.4474	−0.493073
$540$	0	0
$541$	−30.6920	−1.31955	−0.659775	−	0.751463i	$-0.729349\pi$
$541$	−30.6920	−1.31955	−0.659775	+	0.751463i	$0.729349\pi$
$542$	0	0
$543$	5.02159	0.215497
$544$	0	0
$545$	−3.23713	−0.138663
$546$	0	0
$547$	−28.0271	−1.19835	−0.599176	−	0.800617i	$-0.704505\pi$
$547$	−28.0271	−1.19835	−0.599176	+	0.800617i	$0.704505\pi$
$548$	0	0
$549$	−17.4412	−0.744372
$550$	0	0
$551$	1.93765	0.0825467
$552$	0	0
$553$	−12.9763	−0.551809
$554$	0	0
$555$	2.02729	0.0860535
$556$	0	0
$557$	41.1122	1.74198	0.870989	−	0.491302i	$-0.163479\pi$
$557$	41.1122	1.74198	0.870989	+	0.491302i	$0.163479\pi$
$558$	0	0
$559$	50.3550	2.12979
$560$	0	0
$561$	3.76482	0.158951
$562$	0	0
$563$	−16.1179	−0.679287	−0.339643	−	0.940554i	$-0.610306\pi$
$563$	−16.1179	−0.679287	−0.339643	+	0.940554i	$0.610306\pi$
$564$	0	0
$565$	6.59536	0.277469
$566$	0	0
$567$	48.3081	2.02875
$568$	0	0
$569$	−9.59832	−0.402383	−0.201191	−	0.979552i	$-0.564481\pi$
$569$	−9.59832	−0.402383	−0.201191	+	0.979552i	$0.564481\pi$
$570$	0	0
$571$	−20.1814	−0.844564	−0.422282	−	0.906464i	$-0.638771\pi$
$571$	−20.1814	−0.844564	−0.422282	+	0.906464i	$0.638771\pi$
$572$	0	0
$573$	37.2756	1.55721
$574$	0	0
$575$	−11.4025	−0.475516
$576$	0	0
$577$	32.6942	1.36108	0.680539	−	0.732712i	$-0.261746\pi$
$577$	32.6942	1.36108	0.680539	+	0.732712i	$0.261746\pi$
$578$	0	0
$579$	−2.40691	−0.100028
$580$	0	0
$581$	51.8157	2.14968
$582$	0	0
$583$	−7.31345	−0.302892
$584$	0	0
$585$	2.45408	0.101464
$586$	0	0
$587$	43.4049	1.79151	0.895755	−	0.444547i	$-0.146635\pi$
$587$	43.4049	1.79151	0.895755	+	0.444547i	$0.146635\pi$
$588$	0	0
$589$	−6.63918	−0.273563
$590$	0	0
$591$	57.8003	2.37759
$592$	0	0
$593$	−0.471672	−0.0193693	−0.00968463	−	0.999953i	$-0.503083\pi$
$593$	−0.471672	−0.0193693	−0.00968463	+	0.999953i	$0.503083\pi$
$594$	0	0
$595$	−2.79276	−0.114492
$596$	0	0
$597$	20.9917	0.859134
$598$	0	0
$599$	18.2715	0.746555	0.373277	−	0.927720i	$-0.378234\pi$
$599$	18.2715	0.746555	0.373277	+	0.927720i	$0.378234\pi$
$600$	0	0
$601$	−8.50378	−0.346876	−0.173438	−	0.984845i	$-0.555488\pi$
$601$	−8.50378	−0.346876	−0.173438	+	0.984845i	$0.555488\pi$
$602$	0	0
$603$	−8.73930	−0.355892
$604$	0	0
$605$	0.368439	0.0149792
$606$	0	0
$607$	−30.4070	−1.23418	−0.617092	−	0.786891i	$-0.711689\pi$
$607$	−30.4070	−1.23418	−0.617092	+	0.786891i	$0.711689\pi$
$608$	0	0
$609$	−17.7536	−0.719413
$610$	0	0
$611$	−35.7500	−1.44629
$612$	0	0
$613$	11.8580	0.478942	0.239471	−	0.970904i	$-0.423026\pi$
$613$	11.8580	0.478942	0.239471	+	0.970904i	$0.423026\pi$
$614$	0	0
$615$	−2.98148	−0.120225
$616$	0	0
$617$	43.1793	1.73833	0.869167	−	0.494519i	$-0.164656\pi$
$617$	43.1793	1.73833	0.869167	+	0.494519i	$0.164656\pi$
$618$	0	0
$619$	−34.3194	−1.37941	−0.689706	−	0.724090i	$-0.742260\pi$
$619$	−34.3194	−1.37941	−0.689706	+	0.724090i	$0.742260\pi$
$620$	0	0
$621$	7.24696	0.290811
$622$	0	0
$623$	−17.4031	−0.697242
$624$	0	0
$625$	22.9822	0.919289
$626$	0	0
$627$	−2.13326	−0.0851942
$628$	0	0
$629$	−4.55203	−0.181501
$630$	0	0
$631$	2.90779	0.115757	0.0578787	−	0.998324i	$-0.481566\pi$
$631$	2.90779	0.115757	0.0578787	+	0.998324i	$0.481566\pi$
$632$	0	0
$633$	49.3514	1.96154
$634$	0	0
$635$	3.87220	0.153663
$636$	0	0
$637$	49.1669	1.94806
$638$	0	0
$639$	4.11395	0.162745
$640$	0	0
$641$	−21.5767	−0.852227	−0.426113	−	0.904670i	$-0.640118\pi$
$641$	−21.5767	−0.852227	−0.426113	+	0.904670i	$0.640118\pi$
$642$	0	0
$643$	18.3190	0.722432	0.361216	−	0.932482i	$-0.382362\pi$
$643$	18.3190	0.722432	0.361216	+	0.932482i	$0.382362\pi$
$644$	0	0
$645$	−9.21479	−0.362832
$646$	0	0
$647$	6.74136	0.265030	0.132515	−	0.991181i	$-0.457695\pi$
$647$	6.74136	0.265030	0.132515	+	0.991181i	$0.457695\pi$
$648$	0	0
$649$	−0.635940	−0.0249628
$650$	0	0
$651$	60.8311	2.38416
$652$	0	0
$653$	−19.1161	−0.748071	−0.374036	−	0.927414i	$-0.622026\pi$
$653$	−19.1161	−0.748071	−0.374036	+	0.927414i	$0.622026\pi$
$654$	0	0
$655$	−5.54788	−0.216774
$656$	0	0
$657$	0.805693	0.0314331
$658$	0	0
$659$	−8.91788	−0.347391	−0.173696	−	0.984799i	$-0.555571\pi$
$659$	−8.91788	−0.347391	−0.173696	+	0.984799i	$0.555571\pi$
$660$	0	0
$661$	10.2681	0.399383	0.199691	−	0.979859i	$-0.436006\pi$
$661$	10.2681	0.399383	0.199691	+	0.979859i	$0.436006\pi$
$662$	0	0
$663$	−16.1701	−0.627993
$664$	0	0
$665$	1.58246	0.0613653
$666$	0	0
$667$	−4.54212	−0.175872
$668$	0	0
$669$	21.8810	0.845968
$670$	0	0
$671$	11.2466	0.434170
$672$	0	0
$673$	−16.2347	−0.625800	−0.312900	−	0.949786i	$-0.601301\pi$
$673$	−16.2347	−0.625800	−0.312900	+	0.949786i	$0.601301\pi$
$674$	0	0
$675$	−15.0380	−0.578811
$676$	0	0
$677$	31.1769	1.19822	0.599112	−	0.800665i	$-0.295520\pi$
$677$	31.1769	1.19822	0.599112	+	0.800665i	$0.295520\pi$
$678$	0	0
$679$	79.1741	3.03842
$680$	0	0
$681$	20.1435	0.771899
$682$	0	0
$683$	28.9358	1.10720	0.553599	−	0.832783i	$-0.313254\pi$
$683$	28.9358	1.10720	0.553599	+	0.832783i	$0.313254\pi$
$684$	0	0
$685$	−4.10474	−0.156834
$686$	0	0
$687$	16.6414	0.634909
$688$	0	0
$689$	31.4116	1.19669
$690$	0	0
$691$	−6.96767	−0.265063	−0.132531	−	0.991179i	$-0.542311\pi$
$691$	−6.96767	−0.265063	−0.132531	+	0.991179i	$0.542311\pi$
$692$	0	0
$693$	6.66074	0.253020
$694$	0	0
$695$	−0.656866	−0.0249163
$696$	0	0
$697$	6.69457	0.253575
$698$	0	0
$699$	0.606603	0.0229438
$700$	0	0
$701$	−29.6325	−1.11920	−0.559601	−	0.828762i	$-0.689046\pi$
$701$	−29.6325	−1.11920	−0.559601	+	0.828762i	$0.689046\pi$
$702$	0	0
$703$	2.57932	0.0972808
$704$	0	0
$705$	6.54213	0.246391
$706$	0	0
$707$	−29.9797	−1.12750
$708$	0	0
$709$	35.9959	1.35185	0.675927	−	0.736968i	$-0.263743\pi$
$709$	35.9959	1.35185	0.675927	+	0.736968i	$0.263743\pi$
$710$	0	0
$711$	4.68532	0.175713
$712$	0	0
$713$	15.5632	0.582845
$714$	0	0
$715$	−1.58246	−0.0591807
$716$	0	0
$717$	−28.4683	−1.06317
$718$	0	0
$719$	−39.9296	−1.48912	−0.744562	−	0.667553i	$-0.767342\pi$
$719$	−39.9296	−1.48912	−0.744562	+	0.667553i	$0.767342\pi$
$720$	0	0
$721$	25.6371	0.954774
$722$	0	0
$723$	−54.0831	−2.01137
$724$	0	0
$725$	9.42523	0.350044
$726$	0	0
$727$	30.3904	1.12712	0.563559	−	0.826076i	$-0.309432\pi$
$727$	30.3904	1.12712	0.563559	+	0.826076i	$0.309432\pi$
$728$	0	0
$729$	2.34261	0.0867634
$730$	0	0
$731$	20.6907	0.765275
$732$	0	0
$733$	−5.38711	−0.198977	−0.0994887	−	0.995039i	$-0.531721\pi$
$733$	−5.38711	−0.198977	−0.0994887	+	0.995039i	$0.531721\pi$
$734$	0	0
$735$	−8.99737	−0.331873
$736$	0	0
$737$	5.63536	0.207581
$738$	0	0
$739$	23.1257	0.850692	0.425346	−	0.905031i	$-0.360152\pi$
$739$	23.1257	0.850692	0.425346	+	0.905031i	$0.360152\pi$
$740$	0	0
$741$	9.16244	0.336591
$742$	0	0
$743$	−0.424027	−0.0155560	−0.00777802	−	0.999970i	$-0.502476\pi$
$743$	−0.424027	−0.0155560	−0.00777802	+	0.999970i	$0.502476\pi$
$744$	0	0
$745$	−6.27265	−0.229812
$746$	0	0
$747$	−18.7089	−0.684525
$748$	0	0
$749$	28.2467	1.03211
$750$	0	0
$751$	−13.6179	−0.496923	−0.248462	−	0.968642i	$-0.579925\pi$
$751$	−13.6179	−0.496923	−0.248462	+	0.968642i	$0.579925\pi$
$752$	0	0
$753$	24.4066	0.889426
$754$	0	0
$755$	4.36224	0.158758
$756$	0	0
$757$	16.3559	0.594465	0.297232	−	0.954805i	$-0.403936\pi$
$757$	16.3559	0.594465	0.297232	+	0.954805i	$0.403936\pi$
$758$	0	0
$759$	5.00066	0.181512
$760$	0	0
$761$	45.0288	1.63229	0.816146	−	0.577845i	$-0.196106\pi$
$761$	45.0288	1.63229	0.816146	+	0.577845i	$0.196106\pi$
$762$	0	0
$763$	37.7365	1.36615
$764$	0	0
$765$	1.00837	0.0364578
$766$	0	0
$767$	2.73139	0.0986248
$768$	0	0
$769$	−3.44528	−0.124240	−0.0621200	−	0.998069i	$-0.519786\pi$
$769$	−3.44528	−0.124240	−0.0621200	+	0.998069i	$0.519786\pi$
$770$	0	0
$771$	−14.4487	−0.520357
$772$	0	0
$773$	0.0468702	0.00168580	0.000842901	−	1.00000i	$-0.499732\pi$
$773$	0.0468702	0.00168580	0.000842901		1.00000i	$0.499732\pi$
$774$	0	0
$775$	−32.2946	−1.16006
$776$	0	0
$777$	−23.6328	−0.847823
$778$	0	0
$779$	−3.79334	−0.135911
$780$	0	0
$781$	−2.65279	−0.0949244
$782$	0	0
$783$	−5.99030	−0.214076
$784$	0	0
$785$	5.81947	0.207706
$786$	0	0
$787$	−19.8161	−0.706368	−0.353184	−	0.935554i	$-0.614901\pi$
$787$	−19.8161	−0.706368	−0.353184	+	0.935554i	$0.614901\pi$
$788$	0	0
$789$	−37.9595	−1.35139
$790$	0	0
$791$	−76.8846	−2.73370
$792$	0	0
$793$	−48.3046	−1.71535
$794$	0	0
$795$	−5.74821	−0.203868
$796$	0	0
$797$	−6.71688	−0.237924	−0.118962	−	0.992899i	$-0.537957\pi$
$797$	−6.71688	−0.237924	−0.118962	+	0.992899i	$0.537957\pi$
$798$	0	0
$799$	−14.6896	−0.519680
$800$	0	0
$801$	6.28370	0.222024
$802$	0	0
$803$	−0.519535	−0.0183340
$804$	0	0
$805$	−3.70951	−0.130743
$806$	0	0
$807$	32.4735	1.14312
$808$	0	0
$809$	−9.18834	−0.323045	−0.161522	−	0.986869i	$-0.551640\pi$
$809$	−9.18834	−0.323045	−0.161522	+	0.986869i	$0.551640\pi$
$810$	0	0
$811$	−35.2974	−1.23946	−0.619730	−	0.784815i	$-0.712758\pi$
$811$	−35.2974	−1.23946	−0.619730	+	0.784815i	$0.712758\pi$
$812$	0	0
$813$	29.5756	1.03726
$814$	0	0
$815$	6.05341	0.212042
$816$	0	0
$817$	−11.7240	−0.410171
$818$	0	0
$819$	−28.6081	−0.999649
$820$	0	0
$821$	49.4473	1.72572	0.862861	−	0.505441i	$-0.168670\pi$
$821$	49.4473	1.72572	0.862861	+	0.505441i	$0.168670\pi$
$822$	0	0
$823$	−39.8595	−1.38941	−0.694707	−	0.719292i	$-0.744466\pi$
$823$	−39.8595	−1.38941	−0.694707	+	0.719292i	$0.744466\pi$
$824$	0	0
$825$	−10.3767	−0.361271
$826$	0	0
$827$	−4.30898	−0.149838	−0.0749190	−	0.997190i	$-0.523870\pi$
$827$	−4.30898	−0.149838	−0.0749190	+	0.997190i	$0.523870\pi$
$828$	0	0
$829$	−2.92252	−0.101503	−0.0507516	−	0.998711i	$-0.516162\pi$
$829$	−2.92252	−0.101503	−0.0507516	+	0.998711i	$0.516162\pi$
$830$	0	0
$831$	34.0606	1.18155
$832$	0	0
$833$	20.2026	0.699977
$834$	0	0
$835$	3.65790	0.126587
$836$	0	0
$837$	20.5252	0.709454
$838$	0	0
$839$	−46.3066	−1.59868	−0.799341	−	0.600878i	$-0.794818\pi$
$839$	−46.3066	−1.59868	−0.799341	+	0.600878i	$0.794818\pi$
$840$	0	0
$841$	−25.2455	−0.870535
$842$	0	0
$843$	13.8864	0.478273
$844$	0	0
$845$	2.00703	0.0690438
$846$	0	0
$847$	−4.29504	−0.147579
$848$	0	0
$849$	−46.1896	−1.58522
$850$	0	0
$851$	−6.04628	−0.207264
$852$	0	0
$853$	55.6663	1.90598	0.952988	−	0.303007i	$-0.0979906\pi$
$853$	55.6663	1.90598	0.952988	+	0.303007i	$0.0979906\pi$
$854$	0	0
$855$	−0.571375	−0.0195406
$856$	0	0
$857$	37.4470	1.27916	0.639582	−	0.768723i	$-0.279107\pi$
$857$	37.4470	1.27916	0.639582	+	0.768723i	$0.279107\pi$
$858$	0	0
$859$	−9.88100	−0.337135	−0.168568	−	0.985690i	$-0.553914\pi$
$859$	−9.88100	−0.337135	−0.168568	+	0.985690i	$0.553914\pi$
$860$	0	0
$861$	34.7562	1.18449
$862$	0	0
$863$	−16.5374	−0.562941	−0.281470	−	0.959570i	$-0.590822\pi$
$863$	−16.5374	−0.562941	−0.281470	+	0.959570i	$0.590822\pi$
$864$	0	0
$865$	−5.45808	−0.185580
$866$	0	0
$867$	29.6212	1.00599
$868$	0	0
$869$	−3.02123	−0.102488
$870$	0	0
$871$	−24.2041	−0.820125
$872$	0	0
$873$	−28.5871	−0.967528
$874$	0	0
$875$	15.6098	0.527708
$876$	0	0
$877$	11.6237	0.392506	0.196253	−	0.980553i	$-0.437123\pi$
$877$	11.6237	0.392506	0.196253	+	0.980553i	$0.437123\pi$
$878$	0	0
$879$	2.71229	0.0914833
$880$	0	0
$881$	−10.3316	−0.348079	−0.174040	−	0.984739i	$-0.555682\pi$
$881$	−10.3316	−0.348079	−0.174040	+	0.984739i	$0.555682\pi$
$882$	0	0
$883$	−18.4388	−0.620515	−0.310258	−	0.950653i	$-0.600415\pi$
$883$	−18.4388	−0.620515	−0.310258	+	0.950653i	$0.600415\pi$
$884$	0	0
$885$	−0.499835	−0.0168018
$886$	0	0
$887$	39.4700	1.32527	0.662636	−	0.748941i	$-0.269437\pi$
$887$	39.4700	1.32527	0.662636	+	0.748941i	$0.269437\pi$
$888$	0	0
$889$	−45.1397	−1.51394
$890$	0	0
$891$	11.2474	0.376803
$892$	0	0
$893$	8.32356	0.278537
$894$	0	0
$895$	−4.28910	−0.143369
$896$	0	0
$897$	−21.4780	−0.717130
$898$	0	0
$899$	−12.8644	−0.429052
$900$	0	0
$901$	12.9069	0.429992
$902$	0	0
$903$	107.420	3.57473
$904$	0	0
$905$	−0.867289	−0.0288297
$906$	0	0
$907$	−43.8936	−1.45746	−0.728731	−	0.684800i	$-0.759890\pi$
$907$	−43.8936	−1.45746	−0.728731	+	0.684800i	$0.759890\pi$
$908$	0	0
$909$	10.8247	0.359032
$910$	0	0
$911$	−34.7253	−1.15050	−0.575251	−	0.817977i	$-0.695096\pi$
$911$	−34.7253	−1.15050	−0.575251	+	0.817977i	$0.695096\pi$
$912$	0	0
$913$	12.0641	0.399263
$914$	0	0
$915$	8.83957	0.292227
$916$	0	0
$917$	64.6737	2.13572
$918$	0	0
$919$	9.18965	0.303139	0.151569	−	0.988447i	$-0.451567\pi$
$919$	9.18965	0.303139	0.151569	+	0.988447i	$0.451567\pi$
$920$	0	0
$921$	42.9046	1.41375
$922$	0	0
$923$	11.3939	0.375033
$924$	0	0
$925$	12.5465	0.412525
$926$	0	0
$927$	−9.25670	−0.304030
$928$	0	0
$929$	−24.5639	−0.805917	−0.402958	−	0.915218i	$-0.632018\pi$
$929$	−24.5639	−0.805917	−0.402958	+	0.915218i	$0.632018\pi$
$930$	0	0
$931$	−11.4474	−0.375172
$932$	0	0
$933$	26.0882	0.854090
$934$	0	0
$935$	−0.650229	−0.0212648
$936$	0	0
$937$	−33.1667	−1.08351	−0.541754	−	0.840537i	$-0.682240\pi$
$937$	−33.1667	−1.08351	−0.541754	+	0.840537i	$0.682240\pi$
$938$	0	0
$939$	54.7890	1.78797
$940$	0	0
$941$	28.0332	0.913855	0.456927	−	0.889504i	$-0.348950\pi$
$941$	28.0332	0.913855	0.456927	+	0.889504i	$0.348950\pi$
$942$	0	0
$943$	8.89212	0.289567
$944$	0	0
$945$	−4.89222	−0.159144
$946$	0	0
$947$	−26.1601	−0.850089	−0.425044	−	0.905173i	$-0.639742\pi$
$947$	−26.1601	−0.850089	−0.425044	+	0.905173i	$0.639742\pi$
$948$	0	0
$949$	2.23142	0.0724350
$950$	0	0
$951$	−21.6671	−0.702605
$952$	0	0
$953$	−22.6523	−0.733779	−0.366890	−	0.930264i	$-0.619577\pi$
$953$	−22.6523	−0.733779	−0.366890	+	0.930264i	$0.619577\pi$
$954$	0	0
$955$	−6.43794	−0.208327
$956$	0	0
$957$	−4.13351	−0.133618
$958$	0	0
$959$	47.8505	1.54517
$960$	0	0
$961$	13.0787	0.421893
$962$	0	0
$963$	−10.1989	−0.328656
$964$	0	0
$965$	0.415702	0.0133819
$966$	0	0
$967$	−22.0751	−0.709888	−0.354944	−	0.934888i	$-0.615500\pi$
$967$	−22.0751	−0.709888	−0.354944	+	0.934888i	$0.615500\pi$
$968$	0	0
$969$	3.76482	0.120943
$970$	0	0
$971$	−50.8293	−1.63119	−0.815595	−	0.578623i	$-0.803590\pi$
$971$	−50.8293	−1.63119	−0.815595	+	0.578623i	$0.803590\pi$
$972$	0	0
$973$	7.65734	0.245483
$974$	0	0
$975$	44.5684	1.42733
$976$	0	0
$977$	−28.5689	−0.913999	−0.456999	−	0.889467i	$-0.651076\pi$
$977$	−28.5689	−0.913999	−0.456999	+	0.889467i	$0.651076\pi$
$978$	0	0
$979$	−4.05192	−0.129500
$980$	0	0
$981$	−13.6254	−0.435025
$982$	0	0
$983$	−11.1620	−0.356012	−0.178006	−	0.984029i	$-0.556965\pi$
$983$	−11.1620	−0.356012	−0.178006	+	0.984029i	$0.556965\pi$
$984$	0	0
$985$	−9.98280	−0.318079
$986$	0	0
$987$	−76.2641	−2.42751
$988$	0	0
$989$	27.4827	0.873898
$990$	0	0
$991$	7.53878	0.239477	0.119739	−	0.992805i	$-0.461794\pi$
$991$	7.53878	0.239477	0.119739	+	0.992805i	$0.461794\pi$
$992$	0	0
$993$	49.0596	1.55686
$994$	0	0
$995$	−3.62552	−0.114937
$996$	0	0
$997$	−20.6859	−0.655129	−0.327565	−	0.944829i	$-0.606228\pi$
$997$	−20.6859	−0.655129	−0.327565	+	0.944829i	$0.606228\pi$
$998$	0	0
$999$	−7.97403	−0.252287

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.j.1.5	✓	6	4.3	odd	2
3344.2.a.v.1.2		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)

\(f(q)\)	\(=\)	\(q-2.13326 q^{3} +0.368439 q^{5} -4.29504 q^{7} +1.55080 q^{9} +O(q^{10})\)	\(q-2.13326 q^{3} +0.368439 q^{5} -4.29504 q^{7} +1.55080 q^{9} -1.00000 q^{11} +4.29504 q^{13} -0.785977 q^{15} +1.76482 q^{17} -1.00000 q^{19} +9.16244 q^{21} +2.34414 q^{23} -4.86425 q^{25} +3.09153 q^{27} -1.93765 q^{29} +6.63918 q^{31} +2.13326 q^{33} -1.58246 q^{35} -2.57932 q^{37} -9.16244 q^{39} +3.79334 q^{41} +11.7240 q^{43} +0.571375 q^{45} -8.32356 q^{47} +11.4474 q^{49} -3.76482 q^{51} +7.31345 q^{53} -0.368439 q^{55} +2.13326 q^{57} +0.635940 q^{59} -11.2466 q^{61} -6.66074 q^{63} +1.58246 q^{65} -5.63536 q^{67} -5.00066 q^{69} +2.65279 q^{71} +0.519535 q^{73} +10.3767 q^{75} +4.29504 q^{77} +3.02123 q^{79} -11.2474 q^{81} -12.0641 q^{83} +0.650229 q^{85} +4.13351 q^{87} +4.05192 q^{89} -18.4474 q^{91} -14.1631 q^{93} -0.368439 q^{95} -18.4338 q^{97} -1.55080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + q^{5} + 4 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 + q^5 + 4 * q^9	\( 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43} + 11 q^{45} - 16 q^{47} + 2 q^{49} - 15 q^{51} + 17 q^{53} - q^{55} + 4 q^{57} - 19 q^{59} - 6 q^{61} - 2 q^{63} + 6 q^{65} - 14 q^{67} + q^{69} - q^{71} - 5 q^{73} - 18 q^{75} + 2 q^{81} - 11 q^{83} - 26 q^{85} - 14 q^{87} + 12 q^{89} - 44 q^{91} - 6 q^{93} - q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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