LMFDB - Embedded newform 3344.2.a.q.1.1

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:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66908$ 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)

$f(q)$	$=$	$q-2.66908 q^{3} -4.12398 q^{5} +4.21417 q^{7} +4.12398 q^{9} +O(q^{10})$	$q-2.66908 q^{3} -4.12398 q^{5} +4.21417 q^{7} +4.12398 q^{9} +1.00000 q^{11} -2.21417 q^{13} +11.0072 q^{15} -3.45490 q^{17} +1.00000 q^{19} -11.2480 q^{21} +5.45490 q^{23} +12.0072 q^{25} -3.00000 q^{27} +5.57889 q^{29} -7.00724 q^{31} -2.66908 q^{33} -17.3792 q^{35} -2.90981 q^{37} +5.90981 q^{39} -11.9170 q^{41} -1.46214 q^{43} -17.0072 q^{45} -7.58612 q^{47} +10.7593 q^{49} +9.22141 q^{51} -13.2214 q^{53} -4.12398 q^{55} -2.66908 q^{57} -4.79306 q^{59} +8.90981 q^{61} +17.3792 q^{63} +9.13122 q^{65} -1.30437 q^{67} -14.5596 q^{69} +6.80030 q^{71} -1.45490 q^{73} -32.0483 q^{75} +4.21417 q^{77} +9.15777 q^{79} -4.36471 q^{81} +13.2142 q^{83} +14.2480 q^{85} -14.8905 q^{87} +8.24797 q^{89} -9.33092 q^{91} +18.7029 q^{93} -4.12398 q^{95} +2.18038 q^{97} +4.12398 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9	$ 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45} + 12 q^{47} + 27 q^{49} - 3 q^{51} - 9 q^{53} - 3 q^{55} + 3 q^{59} + 24 q^{61} - 6 q^{65} + 3 q^{69} - 21 q^{71} - 3 q^{73} - 36 q^{75} + 6 q^{77} + 6 q^{79} - 9 q^{81} + 33 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} - 36 q^{91} + 36 q^{93} - 3 q^{95} + 12 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * 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
$3$	−2.66908	−1.54099	−0.770497	−	0.637444i	$-0.779992\pi$
$3$	−2.66908	−1.54099	−0.770497	+	0.637444i	$0.779992\pi$
$4$	0	0
$5$	−4.12398	−1.84430	−0.922151	−	0.386831i	$-0.873570\pi$
$5$	−4.12398	−1.84430	−0.922151	+	0.386831i	$0.873570\pi$
$6$	0	0
$7$	4.21417	1.59281	0.796404	−	0.604765i	$-0.206733\pi$
$7$	4.21417	1.59281	0.796404	+	0.604765i	$0.206733\pi$
$8$	0	0
$9$	4.12398	1.37466
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.21417	−0.614102	−0.307051	−	0.951693i	$-0.599342\pi$
$13$	−2.21417	−0.614102	−0.307051	+	0.951693i	$0.599342\pi$
$14$	0	0
$15$	11.0072	2.84206
$16$	0	0
$17$	−3.45490	−0.837937	−0.418969	−	0.908001i	$-0.637608\pi$
$17$	−3.45490	−0.837937	−0.418969	+	0.908001i	$0.637608\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−11.2480	−2.45451
$22$	0	0
$23$	5.45490	1.13743	0.568713	−	0.822536i	$-0.307441\pi$
$23$	5.45490	1.13743	0.568713	+	0.822536i	$0.307441\pi$
$24$	0	0
$25$	12.0072	2.40145
$26$	0	0
$27$	−3.00000	−0.577350
$28$	0	0
$29$	5.57889	1.03597	0.517987	−	0.855389i	$-0.326682\pi$
$29$	5.57889	1.03597	0.517987	+	0.855389i	$0.326682\pi$
$30$	0	0
$31$	−7.00724	−1.25854	−0.629268	−	0.777188i	$-0.716645\pi$
$31$	−7.00724	−1.25854	−0.629268	+	0.777188i	$0.716645\pi$
$32$	0	0
$33$	−2.66908	−0.464627
$34$	0	0
$35$	−17.3792	−2.93762
$36$	0	0
$37$	−2.90981	−0.478370	−0.239185	−	0.970974i	$-0.576880\pi$
$37$	−2.90981	−0.478370	−0.239185	+	0.970974i	$0.576880\pi$
$38$	0	0
$39$	5.90981	0.946327
$40$	0	0
$41$	−11.9170	−1.86113	−0.930565	−	0.366127i	$-0.880684\pi$
$41$	−11.9170	−1.86113	−0.930565	+	0.366127i	$0.880684\pi$
$42$	0	0
$43$	−1.46214	−0.222974	−0.111487	−	0.993766i	$-0.535561\pi$
$43$	−1.46214	−0.222974	−0.111487	+	0.993766i	$0.535561\pi$
$44$	0	0
$45$	−17.0072	−2.53529
$46$	0	0
$47$	−7.58612	−1.10655	−0.553275	−	0.832999i	$-0.686622\pi$
$47$	−7.58612	−1.10655	−0.553275	+	0.832999i	$0.686622\pi$
$48$	0	0
$49$	10.7593	1.53704
$50$	0	0
$51$	9.22141	1.29126
$52$	0	0
$53$	−13.2214	−1.81610	−0.908050	−	0.418861i	$-0.862429\pi$
$53$	−13.2214	−1.81610	−0.908050	+	0.418861i	$0.862429\pi$
$54$	0	0
$55$	−4.12398	−0.556078
$56$	0	0
$57$	−2.66908	−0.353528
$58$	0	0
$59$	−4.79306	−0.624004	−0.312002	−	0.950082i	$-0.600999\pi$
$59$	−4.79306	−0.624004	−0.312002	+	0.950082i	$0.600999\pi$
$60$	0	0
$61$	8.90981	1.14078	0.570392	−	0.821373i	$-0.306791\pi$
$61$	8.90981	1.14078	0.570392	+	0.821373i	$0.306791\pi$
$62$	0	0
$63$	17.3792	2.18957
$64$	0	0
$65$	9.13122	1.13259
$66$	0	0
$67$	−1.30437	−0.159354	−0.0796769	−	0.996821i	$-0.525389\pi$
$67$	−1.30437	−0.159354	−0.0796769	+	0.996821i	$0.525389\pi$
$68$	0	0
$69$	−14.5596	−1.75277
$70$	0	0
$71$	6.80030	0.807047	0.403524	−	0.914969i	$-0.367785\pi$
$71$	6.80030	0.807047	0.403524	+	0.914969i	$0.367785\pi$
$72$	0	0
$73$	−1.45490	−0.170284	−0.0851418	−	0.996369i	$-0.527134\pi$
$73$	−1.45490	−0.170284	−0.0851418	+	0.996369i	$0.527134\pi$
$74$	0	0
$75$	−32.0483	−3.70061
$76$	0	0
$77$	4.21417	0.480250
$78$	0	0
$79$	9.15777	1.03033	0.515165	−	0.857091i	$-0.327731\pi$
$79$	9.15777	1.03033	0.515165	+	0.857091i	$0.327731\pi$
$80$	0	0
$81$	−4.36471	−0.484968
$82$	0	0
$83$	13.2142	1.45044	0.725222	−	0.688515i	$-0.241737\pi$
$83$	13.2142	1.45044	0.725222	+	0.688515i	$0.241737\pi$
$84$	0	0
$85$	14.2480	1.54541
$86$	0	0
$87$	−14.8905	−1.59643
$88$	0	0
$89$	8.24797	0.874283	0.437141	−	0.899393i	$-0.355991\pi$
$89$	8.24797	0.874283	0.437141	+	0.899393i	$0.355991\pi$
$90$	0	0
$91$	−9.33092	−0.978146
$92$	0	0
$93$	18.7029	1.93940
$94$	0	0
$95$	−4.12398	−0.423112
$96$	0	0
$97$	2.18038	0.221384	0.110692	−	0.993855i	$-0.464693\pi$
$97$	2.18038	0.221384	0.110692	+	0.993855i	$0.464693\pi$
$98$	0	0
$99$	4.12398	0.414476
$100$	0	0
$101$	−6.24797	−0.621696	−0.310848	−	0.950460i	$-0.600613\pi$
$101$	−6.24797	−0.621696	−0.310848	+	0.950460i	$0.600613\pi$
$102$	0	0
$103$	−0.605441	−0.0596559	−0.0298280	−	0.999555i	$-0.509496\pi$
$103$	−0.605441	−0.0596559	−0.0298280	+	0.999555i	$0.509496\pi$
$104$	0	0
$105$	46.3864	4.52685
$106$	0	0
$107$	−3.88325	−0.375408	−0.187704	−	0.982226i	$-0.560105\pi$
$107$	−3.88325	−0.375408	−0.187704	+	0.982226i	$0.560105\pi$
$108$	0	0
$109$	−4.97345	−0.476370	−0.238185	−	0.971220i	$-0.576552\pi$
$109$	−4.97345	−0.476370	−0.238185	+	0.971220i	$0.576552\pi$
$110$	0	0
$111$	7.76651	0.737164
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	−22.4959	−2.09776
$116$	0	0
$117$	−9.13122	−0.844182
$118$	0	0
$119$	−14.5596	−1.33467
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	31.8075	2.86799
$124$	0	0
$125$	−28.8977	−2.58469
$126$	0	0
$127$	2.48146	0.220194	0.110097	−	0.993921i	$-0.464884\pi$
$127$	2.48146	0.220194	0.110097	+	0.993921i	$0.464884\pi$
$128$	0	0
$129$	3.90257	0.343602
$130$	0	0
$131$	−11.0072	−0.961707	−0.480853	−	0.876801i	$-0.659673\pi$
$131$	−11.0072	−0.961707	−0.480853	+	0.876801i	$0.659673\pi$
$132$	0	0
$133$	4.21417	0.365415
$134$	0	0
$135$	12.3719	1.06481
$136$	0	0
$137$	6.94360	0.593232	0.296616	−	0.954997i	$-0.404142\pi$
$137$	6.94360	0.593232	0.296616	+	0.954997i	$0.404142\pi$
$138$	0	0
$139$	4.37195	0.370824	0.185412	−	0.982661i	$-0.440638\pi$
$139$	4.37195	0.370824	0.185412	+	0.982661i	$0.440638\pi$
$140$	0	0
$141$	20.2480	1.70519
$142$	0	0
$143$	−2.21417	−0.185159
$144$	0	0
$145$	−23.0072	−1.91065
$146$	0	0
$147$	−28.7173	−2.36857
$148$	0	0
$149$	11.7665	0.963950	0.481975	−	0.876185i	$-0.339920\pi$
$149$	11.7665	0.963950	0.481975	+	0.876185i	$0.339920\pi$
$150$	0	0
$151$	19.5861	1.59390	0.796948	−	0.604048i	$-0.206446\pi$
$151$	19.5861	1.59390	0.796948	+	0.604048i	$0.206446\pi$
$152$	0	0
$153$	−14.2480	−1.15188
$154$	0	0
$155$	28.8977	2.32112
$156$	0	0
$157$	20.6199	1.64565	0.822824	−	0.568296i	$-0.192397\pi$
$157$	20.6199	1.64565	0.822824	+	0.568296i	$0.192397\pi$
$158$	0	0
$159$	35.2890	2.79860
$160$	0	0
$161$	22.9879	1.81170
$162$	0	0
$163$	17.8341	1.39687	0.698437	−	0.715672i	$-0.253879\pi$
$163$	17.8341	1.39687	0.698437	+	0.715672i	$0.253879\pi$
$164$	0	0
$165$	11.0072	0.856912
$166$	0	0
$167$	−4.18038	−0.323488	−0.161744	−	0.986833i	$-0.551712\pi$
$167$	−4.18038	−0.323488	−0.161744	+	0.986833i	$0.551712\pi$
$168$	0	0
$169$	−8.09743	−0.622879
$170$	0	0
$171$	4.12398	0.315369
$172$	0	0
$173$	23.7101	1.80265	0.901323	−	0.433147i	$-0.142597\pi$
$173$	23.7101	1.80265	0.901323	+	0.433147i	$0.142597\pi$
$174$	0	0
$175$	50.6006	3.82505
$176$	0	0
$177$	12.7931	0.961585
$178$	0	0
$179$	10.8269	0.809237	0.404619	−	0.914486i	$-0.367404\pi$
$179$	10.8269	0.809237	0.404619	+	0.914486i	$0.367404\pi$
$180$	0	0
$181$	−17.8341	−1.32560	−0.662799	−	0.748798i	$-0.730632\pi$
$181$	−17.8341	−1.32560	−0.662799	+	0.748798i	$0.730632\pi$
$182$	0	0
$183$	−23.7810	−1.75794
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	−3.45490	−0.252648
$188$	0	0
$189$	−12.6425	−0.919608
$190$	0	0
$191$	−0.612679	−0.0443319	−0.0221659	−	0.999754i	$-0.507056\pi$
$191$	−0.612679	−0.0443319	−0.0221659	+	0.999754i	$0.507056\pi$
$192$	0	0
$193$	11.6425	0.838047	0.419024	−	0.907975i	$-0.362372\pi$
$193$	11.6425	0.838047	0.419024	+	0.907975i	$0.362372\pi$
$194$	0	0
$195$	−24.3719	−1.74531
$196$	0	0
$197$	−12.9098	−0.919786	−0.459893	−	0.887974i	$-0.652112\pi$
$197$	−12.9098	−0.919786	−0.459893	+	0.887974i	$0.652112\pi$
$198$	0	0
$199$	18.3647	1.30184	0.650920	−	0.759146i	$-0.274383\pi$
$199$	18.3647	1.30184	0.650920	+	0.759146i	$0.274383\pi$
$200$	0	0
$201$	3.48146	0.245563
$202$	0	0
$203$	23.5104	1.65011
$204$	0	0
$205$	49.1457	3.43248
$206$	0	0
$207$	22.4959	1.56358
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−2.13122	−0.146719	−0.0733596	−	0.997306i	$-0.523372\pi$
$211$	−2.13122	−0.146719	−0.0733596	+	0.997306i	$0.523372\pi$
$212$	0	0
$213$	−18.1505	−1.24365
$214$	0	0
$215$	6.02985	0.411232
$216$	0	0
$217$	−29.5297	−2.00461
$218$	0	0
$219$	3.88325	0.262406
$220$	0	0
$221$	7.64976	0.514579
$222$	0	0
$223$	−6.90981	−0.462715	−0.231357	−	0.972869i	$-0.574317\pi$
$223$	−6.90981	−0.462715	−0.231357	+	0.972869i	$0.574317\pi$
$224$	0	0
$225$	49.5176	3.30118
$226$	0	0
$227$	2.72548	0.180896	0.0904482	−	0.995901i	$-0.471170\pi$
$227$	2.72548	0.180896	0.0904482	+	0.995901i	$0.471170\pi$
$228$	0	0
$229$	3.48870	0.230539	0.115270	−	0.993334i	$-0.463227\pi$
$229$	3.48870	0.230539	0.115270	+	0.993334i	$0.463227\pi$
$230$	0	0
$231$	−11.2480	−0.740062
$232$	0	0
$233$	−17.0902	−1.11962	−0.559808	−	0.828622i	$-0.689125\pi$
$233$	−17.0902	−1.11962	−0.559808	+	0.828622i	$0.689125\pi$
$234$	0	0
$235$	31.2850	2.04081
$236$	0	0
$237$	−24.4428	−1.58773
$238$	0	0
$239$	−6.06035	−0.392011	−0.196006	−	0.980603i	$-0.562797\pi$
$239$	−6.06035	−0.392011	−0.196006	+	0.980603i	$0.562797\pi$
$240$	0	0
$241$	4.85341	0.312635	0.156318	−	0.987707i	$-0.450038\pi$
$241$	4.85341	0.312635	0.156318	+	0.987707i	$0.450038\pi$
$242$	0	0
$243$	20.6498	1.32468
$244$	0	0
$245$	−44.3711	−2.83476
$246$	0	0
$247$	−2.21417	−0.140885
$248$	0	0
$249$	−35.2697	−2.23513
$250$	0	0
$251$	1.58612	0.100115	0.0500576	−	0.998746i	$-0.484059\pi$
$251$	1.58612	0.100115	0.0500576	+	0.998746i	$0.484059\pi$
$252$	0	0
$253$	5.45490	0.342947
$254$	0	0
$255$	−38.0289	−2.38147
$256$	0	0
$257$	19.5185	1.21753	0.608767	−	0.793349i	$-0.291665\pi$
$257$	19.5185	1.21753	0.608767	+	0.793349i	$0.291665\pi$
$258$	0	0
$259$	−12.2624	−0.761951
$260$	0	0
$261$	23.0072	1.42411
$262$	0	0
$263$	13.0483	0.804591	0.402295	−	0.915510i	$-0.368213\pi$
$263$	13.0483	0.804591	0.402295	+	0.915510i	$0.368213\pi$
$264$	0	0
$265$	54.5249	3.34944
$266$	0	0
$267$	−22.0145	−1.34726
$268$	0	0
$269$	23.6006	1.43895	0.719477	−	0.694516i	$-0.244382\pi$
$269$	23.6006	1.43895	0.719477	+	0.694516i	$0.244382\pi$
$270$	0	0
$271$	16.4090	0.996778	0.498389	−	0.866954i	$-0.333925\pi$
$271$	16.4090	0.996778	0.498389	+	0.866954i	$0.333925\pi$
$272$	0	0
$273$	24.9050	1.50732
$274$	0	0
$275$	12.0072	0.724064
$276$	0	0
$277$	−11.5861	−0.696143	−0.348071	−	0.937468i	$-0.613163\pi$
$277$	−11.5861	−0.696143	−0.348071	+	0.937468i	$0.613163\pi$
$278$	0	0
$279$	−28.8977	−1.73006
$280$	0	0
$281$	7.03379	0.419601	0.209800	−	0.977744i	$-0.432719\pi$
$281$	7.03379	0.419601	0.209800	+	0.977744i	$0.432719\pi$
$282$	0	0
$283$	−8.33092	−0.495222	−0.247611	−	0.968860i	$-0.579645\pi$
$283$	−8.33092	−0.495222	−0.247611	+	0.968860i	$0.579645\pi$
$284$	0	0
$285$	11.0072	0.652012
$286$	0	0
$287$	−50.2205	−2.96442
$288$	0	0
$289$	−5.06364	−0.297861
$290$	0	0
$291$	−5.81962	−0.341152
$292$	0	0
$293$	10.2142	0.596718	0.298359	−	0.954454i	$-0.403561\pi$
$293$	10.2142	0.596718	0.298359	+	0.954454i	$0.403561\pi$
$294$	0	0
$295$	19.7665	1.15085
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−12.0781	−0.698495
$300$	0	0
$301$	−6.16172	−0.355156
$302$	0	0
$303$	16.6763	0.958029
$304$	0	0
$305$	−36.7439	−2.10395
$306$	0	0
$307$	1.93242	0.110289	0.0551444	−	0.998478i	$-0.482438\pi$
$307$	1.93242	0.110289	0.0551444	+	0.998478i	$0.482438\pi$
$308$	0	0
$309$	1.61597	0.0919294
$310$	0	0
$311$	27.7173	1.57171	0.785853	−	0.618413i	$-0.212224\pi$
$311$	27.7173	1.57171	0.785853	+	0.618413i	$0.212224\pi$
$312$	0	0
$313$	11.5258	0.651476	0.325738	−	0.945460i	$-0.394387\pi$
$313$	11.5258	0.651476	0.325738	+	0.945460i	$0.394387\pi$
$314$	0	0
$315$	−71.6715	−4.03823
$316$	0	0
$317$	−33.8977	−1.90389	−0.951943	−	0.306275i	$-0.900917\pi$
$317$	−33.8977	−1.90389	−0.951943	+	0.306275i	$0.900917\pi$
$318$	0	0
$319$	5.57889	0.312358
$320$	0	0
$321$	10.3647	0.578502
$322$	0	0
$323$	−3.45490	−0.192236
$324$	0	0
$325$	−26.5861	−1.47473
$326$	0	0
$327$	13.2745	0.734083
$328$	0	0
$329$	−31.9693	−1.76252
$330$	0	0
$331$	16.2697	0.894262	0.447131	−	0.894468i	$-0.352446\pi$
$331$	16.2697	0.894262	0.447131	+	0.894468i	$0.352446\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	0	0
$335$	5.37919	0.293896
$336$	0	0
$337$	−6.82685	−0.371882	−0.185941	−	0.982561i	$-0.559533\pi$
$337$	−6.82685	−0.371882	−0.185941	+	0.982561i	$0.559533\pi$
$338$	0	0
$339$	21.3526	1.15972
$340$	0	0
$341$	−7.00724	−0.379463
$342$	0	0
$343$	15.8422	0.855400
$344$	0	0
$345$	60.0434	3.23263
$346$	0	0
$347$	5.09019	0.273256	0.136628	−	0.990622i	$-0.456374\pi$
$347$	5.09019	0.273256	0.136628	+	0.990622i	$0.456374\pi$
$348$	0	0
$349$	30.9919	1.65896	0.829478	−	0.558539i	$-0.188638\pi$
$349$	30.9919	1.65896	0.829478	+	0.558539i	$0.188638\pi$
$350$	0	0
$351$	6.64252	0.354552
$352$	0	0
$353$	−17.8301	−0.949003	−0.474501	−	0.880255i	$-0.657372\pi$
$353$	−17.8301	−0.949003	−0.474501	+	0.880255i	$0.657372\pi$
$354$	0	0
$355$	−28.0443	−1.48844
$356$	0	0
$357$	38.8606	2.05672
$358$	0	0
$359$	−24.4090	−1.28826	−0.644130	−	0.764916i	$-0.722780\pi$
$359$	−24.4090	−1.28826	−0.644130	+	0.764916i	$0.722780\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.66908	−0.140090
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	26.6232	1.38972	0.694860	−	0.719145i	$-0.255466\pi$
$367$	26.6232	1.38972	0.694860	+	0.719145i	$0.255466\pi$
$368$	0	0
$369$	−49.1457	−2.55842
$370$	0	0
$371$	−55.7173	−2.89270
$372$	0	0
$373$	−12.6160	−0.653230	−0.326615	−	0.945157i	$-0.605908\pi$
$373$	−12.6160	−0.653230	−0.326615	+	0.945157i	$0.605908\pi$
$374$	0	0
$375$	77.1303	3.98299
$376$	0	0
$377$	−12.3526	−0.636193
$378$	0	0
$379$	22.8905	1.17581	0.587903	−	0.808932i	$-0.299954\pi$
$379$	22.8905	1.17581	0.587903	+	0.808932i	$0.299954\pi$
$380$	0	0
$381$	−6.62321	−0.339317
$382$	0	0
$383$	0.0298464	0.00152508	0.000762539	−	1.00000i	$-0.499757\pi$
$383$	0.0298464	0.00152508	0.000762539		1.00000i	$0.499757\pi$
$384$	0	0
$385$	−17.3792	−0.885725
$386$	0	0
$387$	−6.02985	−0.306514
$388$	0	0
$389$	9.22865	0.467911	0.233956	−	0.972247i	$-0.424833\pi$
$389$	9.22865	0.467911	0.233956	+	0.972247i	$0.424833\pi$
$390$	0	0
$391$	−18.8462	−0.953092
$392$	0	0
$393$	29.3792	1.48198
$394$	0	0
$395$	−37.7665	−1.90024
$396$	0	0
$397$	17.2697	0.866740	0.433370	−	0.901216i	$-0.357324\pi$
$397$	17.2697	0.866740	0.433370	+	0.901216i	$0.357324\pi$
$398$	0	0
$399$	−11.2480	−0.563103
$400$	0	0
$401$	−13.2850	−0.663424	−0.331712	−	0.943381i	$-0.607626\pi$
$401$	−13.2850	−0.663424	−0.331712	+	0.943381i	$0.607626\pi$
$402$	0	0
$403$	15.5152	0.772870
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	−2.90981	−0.144234
$408$	0	0
$409$	1.94360	0.0961048	0.0480524	−	0.998845i	$-0.484699\pi$
$409$	1.94360	0.0961048	0.0480524	+	0.998845i	$0.484699\pi$
$410$	0	0
$411$	−18.5330	−0.914166
$412$	0	0
$413$	−20.1988	−0.993918
$414$	0	0
$415$	−54.4950	−2.67506
$416$	0	0
$417$	−11.6691	−0.571437
$418$	0	0
$419$	2.49593	0.121934	0.0609671	−	0.998140i	$-0.480582\pi$
$419$	2.49593	0.121934	0.0609671	+	0.998140i	$0.480582\pi$
$420$	0	0
$421$	15.6353	0.762017	0.381009	−	0.924571i	$-0.375577\pi$
$421$	15.6353	0.762017	0.381009	+	0.924571i	$0.375577\pi$
$422$	0	0
$423$	−31.2850	−1.52113
$424$	0	0
$425$	−41.4839	−2.01226
$426$	0	0
$427$	37.5475	1.81705
$428$	0	0
$429$	5.90981	0.285328
$430$	0	0
$431$	−0.262441	−0.0126413	−0.00632067	−	0.999980i	$-0.502012\pi$
$431$	−0.262441	−0.0126413	−0.00632067	+	0.999980i	$0.502012\pi$
$432$	0	0
$433$	−32.2769	−1.55113	−0.775565	−	0.631268i	$-0.782535\pi$
$433$	−32.2769	−1.55113	−0.775565	+	0.631268i	$0.782535\pi$
$434$	0	0
$435$	61.4081	2.94429
$436$	0	0
$437$	5.45490	0.260943
$438$	0	0
$439$	17.0902	0.815670	0.407835	−	0.913056i	$-0.366284\pi$
$439$	17.0902	0.815670	0.407835	+	0.913056i	$0.366284\pi$
$440$	0	0
$441$	44.3711	2.11291
$442$	0	0
$443$	−16.2624	−0.772652	−0.386326	−	0.922362i	$-0.626256\pi$
$443$	−16.2624	−0.772652	−0.386326	+	0.922362i	$0.626256\pi$
$444$	0	0
$445$	−34.0145	−1.61244
$446$	0	0
$447$	−31.4057	−1.48544
$448$	0	0
$449$	−27.3382	−1.29017	−0.645084	−	0.764112i	$-0.723178\pi$
$449$	−27.3382	−1.29017	−0.645084	+	0.764112i	$0.723178\pi$
$450$	0	0
$451$	−11.9170	−0.561152
$452$	0	0
$453$	−52.2769	−2.45618
$454$	0	0
$455$	38.4806	1.80400
$456$	0	0
$457$	10.4920	0.490794	0.245397	−	0.969423i	$-0.421082\pi$
$457$	10.4920	0.490794	0.245397	+	0.969423i	$0.421082\pi$
$458$	0	0
$459$	10.3647	0.483783
$460$	0	0
$461$	6.18038	0.287849	0.143925	−	0.989589i	$-0.454028\pi$
$461$	6.18038	0.287849	0.143925	+	0.989589i	$0.454028\pi$
$462$	0	0
$463$	11.1433	0.517873	0.258937	−	0.965894i	$-0.416628\pi$
$463$	11.1433	0.517873	0.258937	+	0.965894i	$0.416628\pi$
$464$	0	0
$465$	−77.1303	−3.57683
$466$	0	0
$467$	34.5104	1.59695	0.798476	−	0.602027i	$-0.205640\pi$
$467$	34.5104	1.59695	0.798476	+	0.602027i	$0.205640\pi$
$468$	0	0
$469$	−5.49683	−0.253820
$470$	0	0
$471$	−55.0362	−2.53593
$472$	0	0
$473$	−1.46214	−0.0672293
$474$	0	0
$475$	12.0072	0.550930
$476$	0	0
$477$	−54.5249	−2.49652
$478$	0	0
$479$	−41.6980	−1.90523	−0.952616	−	0.304176i	$-0.901619\pi$
$479$	−41.6980	−1.90523	−0.952616	+	0.304176i	$0.901619\pi$
$480$	0	0
$481$	6.44282	0.293768
$482$	0	0
$483$	−61.3566	−2.79182
$484$	0	0
$485$	−8.99187	−0.408300
$486$	0	0
$487$	−9.10137	−0.412423	−0.206211	−	0.978507i	$-0.566113\pi$
$487$	−9.10137	−0.412423	−0.206211	+	0.978507i	$0.566113\pi$
$488$	0	0
$489$	−47.6006	−2.15257
$490$	0	0
$491$	30.8413	1.39185	0.695925	−	0.718115i	$-0.254995\pi$
$491$	30.8413	1.39185	0.695925	+	0.718115i	$0.254995\pi$
$492$	0	0
$493$	−19.2745	−0.868081
$494$	0	0
$495$	−17.0072	−0.764418
$496$	0	0
$497$	28.6577	1.28547
$498$	0	0
$499$	−21.7665	−0.974403	−0.487201	−	0.873290i	$-0.661982\pi$
$499$	−21.7665	−0.974403	−0.487201	+	0.873290i	$0.661982\pi$
$500$	0	0
$501$	11.1578	0.498493
$502$	0	0
$503$	−38.0893	−1.69832	−0.849159	−	0.528138i	$-0.822891\pi$
$503$	−38.0893	−1.69832	−0.849159	+	0.528138i	$0.822891\pi$
$504$	0	0
$505$	25.7665	1.14659
$506$	0	0
$507$	21.6127	0.959853
$508$	0	0
$509$	14.9388	0.662149	0.331074	−	0.943605i	$-0.392589\pi$
$509$	14.9388	0.662149	0.331074	+	0.943605i	$0.392589\pi$
$510$	0	0
$511$	−6.13122	−0.271229
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	2.49683	0.110023
$516$	0	0
$517$	−7.58612	−0.333637
$518$	0	0
$519$	−63.2842	−2.77787
$520$	0	0
$521$	7.22536	0.316549	0.158274	−	0.987395i	$-0.449407\pi$
$521$	7.22536	0.316549	0.158274	+	0.987395i	$0.449407\pi$
$522$	0	0
$523$	−4.90586	−0.214518	−0.107259	−	0.994231i	$-0.534207\pi$
$523$	−4.90586	−0.214518	−0.107259	+	0.994231i	$0.534207\pi$
$524$	0	0
$525$	−135.057	−5.89437
$526$	0	0
$527$	24.2093	1.05458
$528$	0	0
$529$	6.75598	0.293738
$530$	0	0
$531$	−19.7665	−0.857793
$532$	0	0
$533$	26.3864	1.14292
$534$	0	0
$535$	16.0145	0.692366
$536$	0	0
$537$	−28.8977	−1.24703
$538$	0	0
$539$	10.7593	0.463435
$540$	0	0
$541$	30.8712	1.32726	0.663628	−	0.748063i	$-0.269016\pi$
$541$	30.8712	1.32726	0.663628	+	0.748063i	$0.269016\pi$
$542$	0	0
$543$	47.6006	2.04274
$544$	0	0
$545$	20.5104	0.878569
$546$	0	0
$547$	−2.54115	−0.108652	−0.0543259	−	0.998523i	$-0.517301\pi$
$547$	−2.54115	−0.108652	−0.0543259	+	0.998523i	$0.517301\pi$
$548$	0	0
$549$	36.7439	1.56819
$550$	0	0
$551$	5.57889	0.237669
$552$	0	0
$553$	38.5925	1.64112
$554$	0	0
$555$	−32.0289	−1.35955
$556$	0	0
$557$	23.9017	1.01275	0.506373	−	0.862314i	$-0.330986\pi$
$557$	23.9017	1.01275	0.506373	+	0.862314i	$0.330986\pi$
$558$	0	0
$559$	3.23744	0.136929
$560$	0	0
$561$	9.22141	0.389328
$562$	0	0
$563$	10.6232	0.447715	0.223857	−	0.974622i	$-0.428135\pi$
$563$	10.6232	0.447715	0.223857	+	0.974622i	$0.428135\pi$
$564$	0	0
$565$	32.9919	1.38798
$566$	0	0
$567$	−18.3937	−0.772461
$568$	0	0
$569$	29.3373	1.22988	0.614941	−	0.788573i	$-0.289180\pi$
$569$	29.3373	1.22988	0.614941	+	0.788573i	$0.289180\pi$
$570$	0	0
$571$	−19.8639	−0.831280	−0.415640	−	0.909529i	$-0.636442\pi$
$571$	−19.8639	−0.831280	−0.415640	+	0.909529i	$0.636442\pi$
$572$	0	0
$573$	1.63529	0.0683151
$574$	0	0
$575$	65.4983	2.73147
$576$	0	0
$577$	−3.17709	−0.132264	−0.0661320	−	0.997811i	$-0.521066\pi$
$577$	−3.17709	−0.132264	−0.0661320	+	0.997811i	$0.521066\pi$
$578$	0	0
$579$	−31.0748	−1.29143
$580$	0	0
$581$	55.6868	2.31028
$582$	0	0
$583$	−13.2214	−0.547575
$584$	0	0
$585$	37.6570	1.55693
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	−7.00724	−0.288728
$590$	0	0
$591$	34.4573	1.41738
$592$	0	0
$593$	−3.45885	−0.142038	−0.0710190	−	0.997475i	$-0.522625\pi$
$593$	−3.45885	−0.142038	−0.0710190	+	0.997475i	$0.522625\pi$
$594$	0	0
$595$	60.0434	2.46154
$596$	0	0
$597$	−49.0169	−2.00613
$598$	0	0
$599$	−40.3864	−1.65014	−0.825072	−	0.565027i	$-0.808866\pi$
$599$	−40.3864	−1.65014	−0.825072	+	0.565027i	$0.808866\pi$
$600$	0	0
$601$	18.3309	0.747734	0.373867	−	0.927482i	$-0.378032\pi$
$601$	18.3309	0.747734	0.373867	+	0.927482i	$0.378032\pi$
$602$	0	0
$603$	−5.37919	−0.219057
$604$	0	0
$605$	−4.12398	−0.167664
$606$	0	0
$607$	−12.4283	−0.504451	−0.252226	−	0.967668i	$-0.581162\pi$
$607$	−12.4283	−0.504451	−0.252226	+	0.967668i	$0.581162\pi$
$608$	0	0
$609$	−62.7511	−2.54280
$610$	0	0
$611$	16.7970	0.679534
$612$	0	0
$613$	−0.0531082	−0.00214502	−0.00107251	−	0.999999i	$-0.500341\pi$
$613$	−0.0531082	−0.00214502	−0.00107251	+	0.999999i	$0.500341\pi$
$614$	0	0
$615$	−131.174	−5.28944
$616$	0	0
$617$	−17.4887	−0.704068	−0.352034	−	0.935987i	$-0.614510\pi$
$617$	−17.4887	−0.704068	−0.352034	+	0.935987i	$0.614510\pi$
$618$	0	0
$619$	−25.2995	−1.01687	−0.508437	−	0.861099i	$-0.669776\pi$
$619$	−25.2995	−1.01687	−0.508437	+	0.861099i	$0.669776\pi$
$620$	0	0
$621$	−16.3647	−0.656693
$622$	0	0
$623$	34.7584	1.39256
$624$	0	0
$625$	59.1376	2.36550
$626$	0	0
$627$	−2.66908	−0.106593
$628$	0	0
$629$	10.0531	0.400844
$630$	0	0
$631$	−9.02261	−0.359184	−0.179592	−	0.983741i	$-0.557478\pi$
$631$	−9.02261	−0.359184	−0.179592	+	0.983741i	$0.557478\pi$
$632$	0	0
$633$	5.68840	0.226093
$634$	0	0
$635$	−10.2335	−0.406104
$636$	0	0
$637$	−23.8229	−0.943898
$638$	0	0
$639$	28.0443	1.10942
$640$	0	0
$641$	30.4573	1.20299	0.601495	−	0.798876i	$-0.294572\pi$
$641$	30.4573	1.20299	0.601495	+	0.798876i	$0.294572\pi$
$642$	0	0
$643$	36.7584	1.44961	0.724804	−	0.688955i	$-0.241930\pi$
$643$	36.7584	1.44961	0.724804	+	0.688955i	$0.241930\pi$
$644$	0	0
$645$	−16.0941	−0.633706
$646$	0	0
$647$	41.0700	1.61463	0.807314	−	0.590122i	$-0.200921\pi$
$647$	41.0700	1.61463	0.807314	+	0.590122i	$0.200921\pi$
$648$	0	0
$649$	−4.79306	−0.188144
$650$	0	0
$651$	78.8172	3.08909
$652$	0	0
$653$	28.4806	1.11453	0.557265	−	0.830335i	$-0.311851\pi$
$653$	28.4806	1.11453	0.557265	+	0.830335i	$0.311851\pi$
$654$	0	0
$655$	45.3937	1.77368
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	10.5740	0.411906	0.205953	−	0.978562i	$-0.433971\pi$
$659$	10.5740	0.411906	0.205953	+	0.978562i	$0.433971\pi$
$660$	0	0
$661$	7.70287	0.299607	0.149803	−	0.988716i	$-0.452136\pi$
$661$	7.70287	0.299607	0.149803	+	0.988716i	$0.452136\pi$
$662$	0	0
$663$	−20.4178	−0.792962
$664$	0	0
$665$	−17.3792	−0.673936
$666$	0	0
$667$	30.4323	1.17834
$668$	0	0
$669$	18.4428	0.713041
$670$	0	0
$671$	8.90981	0.343959
$672$	0	0
$673$	−31.5176	−1.21492	−0.607458	−	0.794352i	$-0.707811\pi$
$673$	−31.5176	−1.21492	−0.607458	+	0.794352i	$0.707811\pi$
$674$	0	0
$675$	−36.0217	−1.38648
$676$	0	0
$677$	11.9928	0.460919	0.230460	−	0.973082i	$-0.425977\pi$
$677$	11.9928	0.460919	0.230460	+	0.973082i	$0.425977\pi$
$678$	0	0
$679$	9.18852	0.352623
$680$	0	0
$681$	−7.27452	−0.278760
$682$	0	0
$683$	−39.1722	−1.49888	−0.749442	−	0.662070i	$-0.769678\pi$
$683$	−39.1722	−1.49888	−0.749442	+	0.662070i	$0.769678\pi$
$684$	0	0
$685$	−28.6353	−1.09410
$686$	0	0
$687$	−9.31160	−0.355260
$688$	0	0
$689$	29.2745	1.11527
$690$	0	0
$691$	49.1722	1.87060	0.935300	−	0.353855i	$-0.115129\pi$
$691$	49.1722	1.87060	0.935300	+	0.353855i	$0.115129\pi$
$692$	0	0
$693$	17.3792	0.660181
$694$	0	0
$695$	−18.0298	−0.683911
$696$	0	0
$697$	41.1722	1.55951
$698$	0	0
$699$	45.6151	1.72532
$700$	0	0
$701$	37.8486	1.42952	0.714760	−	0.699370i	$-0.246536\pi$
$701$	37.8486	1.42952	0.714760	+	0.699370i	$0.246536\pi$
$702$	0	0
$703$	−2.90981	−0.109745
$704$	0	0
$705$	−83.5023	−3.14488
$706$	0	0
$707$	−26.3300	−0.990242
$708$	0	0
$709$	27.7511	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$709$	27.7511	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$710$	0	0
$711$	37.7665	1.41635
$712$	0	0
$713$	−38.2238	−1.43149
$714$	0	0
$715$	9.13122	0.341488
$716$	0	0
$717$	16.1755	0.604087
$718$	0	0
$719$	17.0410	0.635523	0.317762	−	0.948171i	$-0.397069\pi$
$719$	17.0410	0.635523	0.317762	+	0.948171i	$0.397069\pi$
$720$	0	0
$721$	−2.55144	−0.0950204
$722$	0	0
$723$	−12.9541	−0.481769
$724$	0	0
$725$	66.9870	2.48784
$726$	0	0
$727$	−13.5225	−0.501521	−0.250761	−	0.968049i	$-0.580681\pi$
$727$	−13.5225	−0.501521	−0.250761	+	0.968049i	$0.580681\pi$
$728$	0	0
$729$	−42.0217	−1.55636
$730$	0	0
$731$	5.05156	0.186839
$732$	0	0
$733$	−25.6682	−0.948076	−0.474038	−	0.880504i	$-0.657204\pi$
$733$	−25.6682	−0.948076	−0.474038	+	0.880504i	$0.657204\pi$
$734$	0	0
$735$	118.430	4.36835
$736$	0	0
$737$	−1.30437	−0.0480470
$738$	0	0
$739$	−21.3493	−0.785348	−0.392674	−	0.919678i	$-0.628450\pi$
$739$	−21.3493	−0.785348	−0.392674	+	0.919678i	$0.628450\pi$
$740$	0	0
$741$	5.90981	0.217102
$742$	0	0
$743$	7.55718	0.277246	0.138623	−	0.990345i	$-0.455732\pi$
$743$	7.55718	0.277246	0.138623	+	0.990345i	$0.455732\pi$
$744$	0	0
$745$	−48.5249	−1.77781
$746$	0	0
$747$	54.4950	1.99387
$748$	0	0
$749$	−16.3647	−0.597954
$750$	0	0
$751$	53.5861	1.95539	0.977693	−	0.210040i	$-0.0673595\pi$
$751$	53.5861	1.95539	0.977693	+	0.210040i	$0.0673595\pi$
$752$	0	0
$753$	−4.23349	−0.154277
$754$	0	0
$755$	−80.7728	−2.93962
$756$	0	0
$757$	−13.9436	−0.506789	−0.253394	−	0.967363i	$-0.581547\pi$
$757$	−13.9436	−0.506789	−0.253394	+	0.967363i	$0.581547\pi$
$758$	0	0
$759$	−14.5596	−0.528479
$760$	0	0
$761$	−45.7994	−1.66023	−0.830114	−	0.557594i	$-0.811724\pi$
$761$	−45.7994	−1.66023	−0.830114	+	0.557594i	$0.811724\pi$
$762$	0	0
$763$	−20.9590	−0.758766
$764$	0	0
$765$	58.7584	2.12441
$766$	0	0
$767$	10.6127	0.383202
$768$	0	0
$769$	−40.8751	−1.47399	−0.736997	−	0.675896i	$-0.763757\pi$
$769$	−40.8751	−1.47399	−0.736997	+	0.675896i	$0.763757\pi$
$770$	0	0
$771$	−52.0965	−1.87621
$772$	0	0
$773$	37.4983	1.34872	0.674361	−	0.738402i	$-0.264419\pi$
$773$	37.4983	1.34872	0.674361	+	0.738402i	$0.264419\pi$
$774$	0	0
$775$	−84.1376	−3.02231
$776$	0	0
$777$	32.7294	1.17416
$778$	0	0
$779$	−11.9170	−0.426972
$780$	0	0
$781$	6.80030	0.243334
$782$	0	0
$783$	−16.7367	−0.598119
$784$	0	0
$785$	−85.0362	−3.03507
$786$	0	0
$787$	−8.80754	−0.313955	−0.156977	−	0.987602i	$-0.550175\pi$
$787$	−8.80754	−0.313955	−0.156977	+	0.987602i	$0.550175\pi$
$788$	0	0
$789$	−34.8269	−1.23987
$790$	0	0
$791$	−33.7134	−1.19871
$792$	0	0
$793$	−19.7279	−0.700557
$794$	0	0
$795$	−145.531	−5.16146
$796$	0	0
$797$	−26.1312	−0.925615	−0.462808	−	0.886459i	$-0.653158\pi$
$797$	−26.1312	−0.925615	−0.462808	+	0.886459i	$0.653158\pi$
$798$	0	0
$799$	26.2093	0.927220
$800$	0	0
$801$	34.0145	1.20184
$802$	0	0
$803$	−1.45490	−0.0513425
$804$	0	0
$805$	−94.8018	−3.34132
$806$	0	0
$807$	−62.9919	−2.21742
$808$	0	0
$809$	−6.01053	−0.211319	−0.105659	−	0.994402i	$-0.533695\pi$
$809$	−6.01053	−0.211319	−0.105659	+	0.994402i	$0.533695\pi$
$810$	0	0
$811$	40.2809	1.41445	0.707226	−	0.706987i	$-0.249946\pi$
$811$	40.2809	1.41445	0.707226	+	0.706987i	$0.249946\pi$
$812$	0	0
$813$	−43.7970	−1.53603
$814$	0	0
$815$	−73.5475	−2.57626
$816$	0	0
$817$	−1.46214	−0.0511539
$818$	0	0
$819$	−38.4806	−1.34462
$820$	0	0
$821$	50.1496	1.75023	0.875117	−	0.483911i	$-0.160784\pi$
$821$	50.1496	1.75023	0.875117	+	0.483911i	$0.160784\pi$
$822$	0	0
$823$	42.7400	1.48982	0.744911	−	0.667164i	$-0.232492\pi$
$823$	42.7400	1.48982	0.744911	+	0.667164i	$0.232492\pi$
$824$	0	0
$825$	−32.0483	−1.11578
$826$	0	0
$827$	−49.0555	−1.70583	−0.852913	−	0.522052i	$-0.825167\pi$
$827$	−49.0555	−1.70583	−0.852913	+	0.522052i	$0.825167\pi$
$828$	0	0
$829$	49.9653	1.73537	0.867683	−	0.497117i	$-0.165608\pi$
$829$	49.9653	1.73537	0.867683	+	0.497117i	$0.165608\pi$
$830$	0	0
$831$	30.9243	1.07275
$832$	0	0
$833$	−37.1722	−1.28794
$834$	0	0
$835$	17.2398	0.596609
$836$	0	0
$837$	21.0217	0.726617
$838$	0	0
$839$	52.0257	1.79613	0.898063	−	0.439868i	$-0.144975\pi$
$839$	52.0257	1.79613	0.898063	+	0.439868i	$0.144975\pi$
$840$	0	0
$841$	2.12398	0.0732408
$842$	0	0
$843$	−18.7737	−0.646602
$844$	0	0
$845$	33.3937	1.14878
$846$	0	0
$847$	4.21417	0.144801
$848$	0	0
$849$	22.2359	0.763134
$850$	0	0
$851$	−15.8727	−0.544110
$852$	0	0
$853$	16.2335	0.555824	0.277912	−	0.960607i	$-0.410358\pi$
$853$	16.2335	0.555824	0.277912	+	0.960607i	$0.410358\pi$
$854$	0	0
$855$	−17.0072	−0.581635
$856$	0	0
$857$	9.60150	0.327981	0.163990	−	0.986462i	$-0.447563\pi$
$857$	9.60150	0.327981	0.163990	+	0.986462i	$0.447563\pi$
$858$	0	0
$859$	−19.4057	−0.662115	−0.331058	−	0.943611i	$-0.607405\pi$
$859$	−19.4057	−0.662115	−0.331058	+	0.943611i	$0.607405\pi$
$860$	0	0
$861$	134.043	4.56816
$862$	0	0
$863$	−1.54839	−0.0527077	−0.0263539	−	0.999653i	$-0.508390\pi$
$863$	−1.54839	−0.0527077	−0.0263539	+	0.999653i	$0.508390\pi$
$864$	0	0
$865$	−97.7801	−3.32462
$866$	0	0
$867$	13.5152	0.459002
$868$	0	0
$869$	9.15777	0.310656
$870$	0	0
$871$	2.88810	0.0978594
$872$	0	0
$873$	8.99187	0.304329
$874$	0	0
$875$	−121.780	−4.11692
$876$	0	0
$877$	−12.1200	−0.409265	−0.204632	−	0.978839i	$-0.565600\pi$
$877$	−12.1200	−0.409265	−0.204632	+	0.978839i	$0.565600\pi$
$878$	0	0
$879$	−27.2624	−0.919539
$880$	0	0
$881$	53.6836	1.80864	0.904322	−	0.426850i	$-0.140377\pi$
$881$	53.6836	1.80864	0.904322	+	0.426850i	$0.140377\pi$
$882$	0	0
$883$	−34.4283	−1.15861	−0.579303	−	0.815112i	$-0.696675\pi$
$883$	−34.4283	−1.15861	−0.579303	+	0.815112i	$0.696675\pi$
$884$	0	0
$885$	−52.7584	−1.77345
$886$	0	0
$887$	27.5330	0.924468	0.462234	−	0.886758i	$-0.347048\pi$
$887$	27.5330	0.924468	0.462234	+	0.886758i	$0.347048\pi$
$888$	0	0
$889$	10.4573	0.350727
$890$	0	0
$891$	−4.36471	−0.146223
$892$	0	0
$893$	−7.58612	−0.253860
$894$	0	0
$895$	−44.6498	−1.49248
$896$	0	0
$897$	32.2374	1.07638
$898$	0	0
$899$	−39.0926	−1.30381
$900$	0	0
$901$	45.6787	1.52178
$902$	0	0
$903$	16.4461	0.547292
$904$	0	0
$905$	73.5475	2.44480
$906$	0	0
$907$	−15.5677	−0.516917	−0.258459	−	0.966022i	$-0.583215\pi$
$907$	−15.5677	−0.516917	−0.258459	+	0.966022i	$0.583215\pi$
$908$	0	0
$909$	−25.7665	−0.854621
$910$	0	0
$911$	0.623208	0.0206478	0.0103239	−	0.999947i	$-0.496714\pi$
$911$	0.623208	0.0206478	0.0103239	+	0.999947i	$0.496714\pi$
$912$	0	0
$913$	13.2142	0.437325
$914$	0	0
$915$	98.0724	3.24217
$916$	0	0
$917$	−46.3864	−1.53181
$918$	0	0
$919$	44.4887	1.46755	0.733773	−	0.679394i	$-0.237757\pi$
$919$	44.4887	1.46755	0.733773	+	0.679394i	$0.237757\pi$
$920$	0	0
$921$	−5.15777	−0.169954
$922$	0	0
$923$	−15.0571	−0.495609
$924$	0	0
$925$	−34.9388	−1.14878
$926$	0	0
$927$	−2.49683	−0.0820067
$928$	0	0
$929$	44.1224	1.44761	0.723805	−	0.690005i	$-0.242391\pi$
$929$	44.1224	1.44761	0.723805	+	0.690005i	$0.242391\pi$
$930$	0	0
$931$	10.7593	0.352621
$932$	0	0
$933$	−73.9798	−2.42199
$934$	0	0
$935$	14.2480	0.465958
$936$	0	0
$937$	24.8672	0.812377	0.406188	−	0.913789i	$-0.366858\pi$
$937$	24.8672	0.812377	0.406188	+	0.913789i	$0.366858\pi$
$938$	0	0
$939$	−30.7632	−1.00392
$940$	0	0
$941$	29.8301	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$941$	29.8301	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$942$	0	0
$943$	−65.0063	−2.11690
$944$	0	0
$945$	52.1376	1.69603
$946$	0	0
$947$	−9.07572	−0.294921	−0.147461	−	0.989068i	$-0.547110\pi$
$947$	−9.07572	−0.294921	−0.147461	+	0.989068i	$0.547110\pi$
$948$	0	0
$949$	3.22141	0.104571
$950$	0	0
$951$	90.4757	2.93388
$952$	0	0
$953$	1.27058	0.0411580	0.0205790	−	0.999788i	$-0.493449\pi$
$953$	1.27058	0.0411580	0.0205790	+	0.999788i	$0.493449\pi$
$954$	0	0
$955$	2.52668	0.0817613
$956$	0	0
$957$	−14.8905	−0.481341
$958$	0	0
$959$	29.2615	0.944905
$960$	0	0
$961$	18.1014	0.583915
$962$	0	0
$963$	−16.0145	−0.516059
$964$	0	0
$965$	−48.0136	−1.54561
$966$	0	0
$967$	−6.67632	−0.214696	−0.107348	−	0.994222i	$-0.534236\pi$
$967$	−6.67632	−0.214696	−0.107348	+	0.994222i	$0.534236\pi$
$968$	0	0
$969$	9.22141	0.296234
$970$	0	0
$971$	0.191566	0.00614764	0.00307382	−	0.999995i	$-0.499022\pi$
$971$	0.191566	0.00614764	0.00307382	+	0.999995i	$0.499022\pi$
$972$	0	0
$973$	18.4242	0.590651
$974$	0	0
$975$	70.9605	2.27255
$976$	0	0
$977$	16.4959	0.527752	0.263876	−	0.964557i	$-0.414999\pi$
$977$	16.4959	0.527752	0.263876	+	0.964557i	$0.414999\pi$
$978$	0	0
$979$	8.24797	0.263606
$980$	0	0
$981$	−20.5104	−0.654847
$982$	0	0
$983$	35.0072	1.11656	0.558279	−	0.829653i	$-0.311462\pi$
$983$	35.0072	1.11656	0.558279	+	0.829653i	$0.311462\pi$
$984$	0	0
$985$	53.2398	1.69636
$986$	0	0
$987$	85.3285	2.71604
$988$	0	0
$989$	−7.97584	−0.253617
$990$	0	0
$991$	11.3759	0.361367	0.180684	−	0.983541i	$-0.442169\pi$
$991$	11.3759	0.361367	0.180684	+	0.983541i	$0.442169\pi$
$992$	0	0
$993$	−43.4251	−1.37805
$994$	0	0
$995$	−75.7358	−2.40099
$996$	0	0
$997$	8.89533	0.281718	0.140859	−	0.990030i	$-0.455014\pi$
$997$	8.89533	0.281718	0.140859	+	0.990030i	$0.455014\pi$
$998$	0	0
$999$	8.72942	0.276187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.q.1.1		3
4.3	odd	2		418.2.a.g.1.3	✓	3
12.11	even	2		3762.2.a.bg.1.3		3
44.43	even	2		4598.2.a.bo.1.3		3
76.75	even	2		7942.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.3	✓	3	4.3	odd	2
3344.2.a.q.1.1		3	1.1	even	1	trivial
3762.2.a.bg.1.3		3	12.11	even	2
4598.2.a.bo.1.3		3	44.43	even	2
7942.2.a.bi.1.1		3	76.75	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)

\(f(q)\)	\(=\)	\(q-2.66908 q^{3} -4.12398 q^{5} +4.21417 q^{7} +4.12398 q^{9} +O(q^{10})\)	\(q-2.66908 q^{3} -4.12398 q^{5} +4.21417 q^{7} +4.12398 q^{9} +1.00000 q^{11} -2.21417 q^{13} +11.0072 q^{15} -3.45490 q^{17} +1.00000 q^{19} -11.2480 q^{21} +5.45490 q^{23} +12.0072 q^{25} -3.00000 q^{27} +5.57889 q^{29} -7.00724 q^{31} -2.66908 q^{33} -17.3792 q^{35} -2.90981 q^{37} +5.90981 q^{39} -11.9170 q^{41} -1.46214 q^{43} -17.0072 q^{45} -7.58612 q^{47} +10.7593 q^{49} +9.22141 q^{51} -13.2214 q^{53} -4.12398 q^{55} -2.66908 q^{57} -4.79306 q^{59} +8.90981 q^{61} +17.3792 q^{63} +9.13122 q^{65} -1.30437 q^{67} -14.5596 q^{69} +6.80030 q^{71} -1.45490 q^{73} -32.0483 q^{75} +4.21417 q^{77} +9.15777 q^{79} -4.36471 q^{81} +13.2142 q^{83} +14.2480 q^{85} -14.8905 q^{87} +8.24797 q^{89} -9.33092 q^{91} +18.7029 q^{93} -4.12398 q^{95} +2.18038 q^{97} +4.12398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9	\( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45} + 12 q^{47} + 27 q^{49} - 3 q^{51} - 9 q^{53} - 3 q^{55} + 3 q^{59} + 24 q^{61} - 6 q^{65} + 3 q^{69} - 21 q^{71} - 3 q^{73} - 36 q^{75} + 6 q^{77} + 6 q^{79} - 9 q^{81} + 33 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} - 36 q^{91} + 36 q^{93} - 3 q^{95} + 12 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * q^99

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