LMFDB - Embedded newform 3344.2.a.bb.1.8

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 $ x^9 - x^8 - 22x^7 + 22x^6 + 152x^5 - 136x^4 - 341x^3 + 169x^2 + 196*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.93149$ 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)

$f(q)$	$=$	$q+2.93149 q^{3} +3.77651 q^{5} +2.25533 q^{7} +5.59362 q^{9} +O(q^{10})$	$q+2.93149 q^{3} +3.77651 q^{5} +2.25533 q^{7} +5.59362 q^{9} -1.00000 q^{11} -1.48217 q^{13} +11.0708 q^{15} +3.18952 q^{17} +1.00000 q^{19} +6.61148 q^{21} -8.69000 q^{23} +9.26206 q^{25} +7.60317 q^{27} -9.28316 q^{29} +3.52958 q^{31} -2.93149 q^{33} +8.51729 q^{35} +8.84895 q^{37} -4.34495 q^{39} -10.9014 q^{41} -5.47380 q^{43} +21.1244 q^{45} +3.55303 q^{47} -1.91348 q^{49} +9.35004 q^{51} -0.536113 q^{53} -3.77651 q^{55} +2.93149 q^{57} -13.7709 q^{59} +8.03558 q^{61} +12.6155 q^{63} -5.59742 q^{65} +8.22858 q^{67} -25.4746 q^{69} -6.42523 q^{71} +1.66168 q^{73} +27.1516 q^{75} -2.25533 q^{77} -12.0452 q^{79} +5.50775 q^{81} +5.79306 q^{83} +12.0453 q^{85} -27.2135 q^{87} +3.88846 q^{89} -3.34277 q^{91} +10.3469 q^{93} +3.77651 q^{95} +15.5922 q^{97} -5.59362 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * 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
$3$	2.93149	1.69250	0.846248	−	0.532790i	$-0.178856\pi$
$3$	2.93149	1.69250	0.846248	+	0.532790i	$0.178856\pi$
$4$	0	0
$5$	3.77651	1.68891	0.844454	−	0.535628i	$-0.179925\pi$
$5$	3.77651	1.68891	0.844454	+	0.535628i	$0.179925\pi$
$6$	0	0
$7$	2.25533	0.852435	0.426217	−	0.904621i	$-0.359846\pi$
$7$	2.25533	0.852435	0.426217	+	0.904621i	$0.359846\pi$
$8$	0	0
$9$	5.59362	1.86454
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.48217	−0.411079	−0.205539	−	0.978649i	$-0.565895\pi$
$13$	−1.48217	−0.411079	−0.205539	+	0.978649i	$0.565895\pi$
$14$	0	0
$15$	11.0708	2.85847
$16$	0	0
$17$	3.18952	0.773572	0.386786	−	0.922169i	$-0.373585\pi$
$17$	3.18952	0.773572	0.386786	+	0.922169i	$0.373585\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.61148	1.44274
$22$	0	0
$23$	−8.69000	−1.81199	−0.905995	−	0.423289i	$-0.860876\pi$
$23$	−8.69000	−1.81199	−0.905995	+	0.423289i	$0.860876\pi$
$24$	0	0
$25$	9.26206	1.85241
$26$	0	0
$27$	7.60317	1.46323
$28$	0	0
$29$	−9.28316	−1.72384	−0.861920	−	0.507044i	$-0.830738\pi$
$29$	−9.28316	−1.72384	−0.861920	+	0.507044i	$0.830738\pi$
$30$	0	0
$31$	3.52958	0.633932	0.316966	−	0.948437i	$-0.397336\pi$
$31$	3.52958	0.633932	0.316966	+	0.948437i	$0.397336\pi$
$32$	0	0
$33$	−2.93149	−0.510307
$34$	0	0
$35$	8.51729	1.43968
$36$	0	0
$37$	8.84895	1.45476	0.727379	−	0.686235i	$-0.240738\pi$
$37$	8.84895	1.45476	0.727379	+	0.686235i	$0.240738\pi$
$38$	0	0
$39$	−4.34495	−0.695749
$40$	0	0
$41$	−10.9014	−1.70251	−0.851253	−	0.524755i	$-0.824157\pi$
$41$	−10.9014	−1.70251	−0.851253	+	0.524755i	$0.824157\pi$
$42$	0	0
$43$	−5.47380	−0.834747	−0.417374	−	0.908735i	$-0.637049\pi$
$43$	−5.47380	−0.834747	−0.417374	+	0.908735i	$0.637049\pi$
$44$	0	0
$45$	21.1244	3.14904
$46$	0	0
$47$	3.55303	0.518263	0.259131	−	0.965842i	$-0.416564\pi$
$47$	3.55303	0.518263	0.259131	+	0.965842i	$0.416564\pi$
$48$	0	0
$49$	−1.91348	−0.273355
$50$	0	0
$51$	9.35004	1.30927
$52$	0	0
$53$	−0.536113	−0.0736408	−0.0368204	−	0.999322i	$-0.511723\pi$
$53$	−0.536113	−0.0736408	−0.0368204	+	0.999322i	$0.511723\pi$
$54$	0	0
$55$	−3.77651	−0.509225
$56$	0	0
$57$	2.93149	0.388285
$58$	0	0
$59$	−13.7709	−1.79281	−0.896407	−	0.443231i	$-0.853832\pi$
$59$	−13.7709	−1.79281	−0.896407	+	0.443231i	$0.853832\pi$
$60$	0	0
$61$	8.03558	1.02885	0.514425	−	0.857535i	$-0.328005\pi$
$61$	8.03558	1.02885	0.514425	+	0.857535i	$0.328005\pi$
$62$	0	0
$63$	12.6155	1.58940
$64$	0	0
$65$	−5.59742	−0.694275
$66$	0	0
$67$	8.22858	1.00528	0.502641	−	0.864495i	$-0.332362\pi$
$67$	8.22858	1.00528	0.502641	+	0.864495i	$0.332362\pi$
$68$	0	0
$69$	−25.4746	−3.06678
$70$	0	0
$71$	−6.42523	−0.762535	−0.381268	−	0.924465i	$-0.624512\pi$
$71$	−6.42523	−0.762535	−0.381268	+	0.924465i	$0.624512\pi$
$72$	0	0
$73$	1.66168	0.194485	0.0972423	−	0.995261i	$-0.468998\pi$
$73$	1.66168	0.194485	0.0972423	+	0.995261i	$0.468998\pi$
$74$	0	0
$75$	27.1516	3.13520
$76$	0	0
$77$	−2.25533	−0.257019
$78$	0	0
$79$	−12.0452	−1.35519	−0.677596	−	0.735435i	$-0.736978\pi$
$79$	−12.0452	−1.35519	−0.677596	+	0.735435i	$0.736978\pi$
$80$	0	0
$81$	5.50775	0.611972
$82$	0	0
$83$	5.79306	0.635870	0.317935	−	0.948112i	$-0.397011\pi$
$83$	5.79306	0.635870	0.317935	+	0.948112i	$0.397011\pi$
$84$	0	0
$85$	12.0453	1.30649
$86$	0	0
$87$	−27.2135	−2.91759
$88$	0	0
$89$	3.88846	0.412176	0.206088	−	0.978533i	$-0.433927\pi$
$89$	3.88846	0.412176	0.206088	+	0.978533i	$0.433927\pi$
$90$	0	0
$91$	−3.34277	−0.350418
$92$	0	0
$93$	10.3469	1.07293
$94$	0	0
$95$	3.77651	0.387462
$96$	0	0
$97$	15.5922	1.58315	0.791575	−	0.611072i	$-0.209262\pi$
$97$	15.5922	1.58315	0.791575	+	0.611072i	$0.209262\pi$
$98$	0	0
$99$	−5.59362	−0.562180
$100$	0	0
$101$	3.50272	0.348534	0.174267	−	0.984698i	$-0.444244\pi$
$101$	3.50272	0.348534	0.174267	+	0.984698i	$0.444244\pi$
$102$	0	0
$103$	−12.9261	−1.27364	−0.636821	−	0.771012i	$-0.719751\pi$
$103$	−12.9261	−1.27364	−0.636821	+	0.771012i	$0.719751\pi$
$104$	0	0
$105$	24.9683	2.43666
$106$	0	0
$107$	−3.30266	−0.319280	−0.159640	−	0.987175i	$-0.551033\pi$
$107$	−3.30266	−0.319280	−0.159640	+	0.987175i	$0.551033\pi$
$108$	0	0
$109$	2.61931	0.250884	0.125442	−	0.992101i	$-0.459965\pi$
$109$	2.61931	0.250884	0.125442	+	0.992101i	$0.459965\pi$
$110$	0	0
$111$	25.9406	2.46217
$112$	0	0
$113$	8.84603	0.832164	0.416082	−	0.909327i	$-0.363403\pi$
$113$	8.84603	0.832164	0.416082	+	0.909327i	$0.363403\pi$
$114$	0	0
$115$	−32.8179	−3.06028
$116$	0	0
$117$	−8.29068	−0.766473
$118$	0	0
$119$	7.19342	0.659420
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−31.9572	−2.88148
$124$	0	0
$125$	16.0957	1.43965
$126$	0	0
$127$	−14.7088	−1.30519	−0.652596	−	0.757706i	$-0.726320\pi$
$127$	−14.7088	−1.30519	−0.652596	+	0.757706i	$0.726320\pi$
$128$	0	0
$129$	−16.0464	−1.41281
$130$	0	0
$131$	−4.13239	−0.361048	−0.180524	−	0.983571i	$-0.557779\pi$
$131$	−4.13239	−0.361048	−0.180524	+	0.983571i	$0.557779\pi$
$132$	0	0
$133$	2.25533	0.195562
$134$	0	0
$135$	28.7135	2.47126
$136$	0	0
$137$	22.0386	1.88288	0.941442	−	0.337175i	$-0.109471\pi$
$137$	22.0386	1.88288	0.941442	+	0.337175i	$0.109471\pi$
$138$	0	0
$139$	−5.00835	−0.424802	−0.212401	−	0.977183i	$-0.568128\pi$
$139$	−5.00835	−0.424802	−0.212401	+	0.977183i	$0.568128\pi$
$140$	0	0
$141$	10.4157	0.877157
$142$	0	0
$143$	1.48217	0.123945
$144$	0	0
$145$	−35.0580	−2.91141
$146$	0	0
$147$	−5.60935	−0.462652
$148$	0	0
$149$	−11.5291	−0.944497	−0.472249	−	0.881465i	$-0.656558\pi$
$149$	−11.5291	−0.944497	−0.472249	+	0.881465i	$0.656558\pi$
$150$	0	0
$151$	−16.0994	−1.31015	−0.655073	−	0.755565i	$-0.727362\pi$
$151$	−16.0994	−1.31015	−0.655073	+	0.755565i	$0.727362\pi$
$152$	0	0
$153$	17.8410	1.44236
$154$	0	0
$155$	13.3295	1.07065
$156$	0	0
$157$	−7.66845	−0.612009	−0.306005	−	0.952030i	$-0.598992\pi$
$157$	−7.66845	−0.612009	−0.306005	+	0.952030i	$0.598992\pi$
$158$	0	0
$159$	−1.57161	−0.124637
$160$	0	0
$161$	−19.5988	−1.54460
$162$	0	0
$163$	13.6832	1.07175	0.535875	−	0.844298i	$-0.319982\pi$
$163$	13.6832	1.07175	0.535875	+	0.844298i	$0.319982\pi$
$164$	0	0
$165$	−11.0708	−0.861861
$166$	0	0
$167$	11.3695	0.879801	0.439901	−	0.898046i	$-0.355014\pi$
$167$	11.3695	0.879801	0.439901	+	0.898046i	$0.355014\pi$
$168$	0	0
$169$	−10.8032	−0.831014
$170$	0	0
$171$	5.59362	0.427755
$172$	0	0
$173$	−17.4564	−1.32719	−0.663593	−	0.748093i	$-0.730969\pi$
$173$	−17.4564	−1.32719	−0.663593	+	0.748093i	$0.730969\pi$
$174$	0	0
$175$	20.8890	1.57906
$176$	0	0
$177$	−40.3691	−3.03433
$178$	0	0
$179$	−15.8625	−1.18562	−0.592808	−	0.805344i	$-0.701981\pi$
$179$	−15.8625	−1.18562	−0.592808	+	0.805344i	$0.701981\pi$
$180$	0	0
$181$	5.02875	0.373784	0.186892	−	0.982380i	$-0.440159\pi$
$181$	5.02875	0.373784	0.186892	+	0.982380i	$0.440159\pi$
$182$	0	0
$183$	23.5562	1.74133
$184$	0	0
$185$	33.4182	2.45696
$186$	0	0
$187$	−3.18952	−0.233241
$188$	0	0
$189$	17.1477	1.24731
$190$	0	0
$191$	−7.86963	−0.569426	−0.284713	−	0.958613i	$-0.591898\pi$
$191$	−7.86963	−0.569426	−0.284713	+	0.958613i	$0.591898\pi$
$192$	0	0
$193$	26.8540	1.93299	0.966497	−	0.256680i	$-0.0826285\pi$
$193$	26.8540	1.93299	0.966497	+	0.256680i	$0.0826285\pi$
$194$	0	0
$195$	−16.4088	−1.17506
$196$	0	0
$197$	3.07581	0.219142	0.109571	−	0.993979i	$-0.465052\pi$
$197$	3.07581	0.219142	0.109571	+	0.993979i	$0.465052\pi$
$198$	0	0
$199$	9.75433	0.691466	0.345733	−	0.938333i	$-0.387630\pi$
$199$	9.75433	0.691466	0.345733	+	0.938333i	$0.387630\pi$
$200$	0	0
$201$	24.1220	1.70143
$202$	0	0
$203$	−20.9366	−1.46946
$204$	0	0
$205$	−41.1692	−2.87538
$206$	0	0
$207$	−48.6086	−3.37853
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	18.9943	1.30762	0.653810	−	0.756659i	$-0.273169\pi$
$211$	18.9943	1.30762	0.653810	+	0.756659i	$0.273169\pi$
$212$	0	0
$213$	−18.8355	−1.29059
$214$	0	0
$215$	−20.6719	−1.40981
$216$	0	0
$217$	7.96038	0.540386
$218$	0	0
$219$	4.87118	0.329164
$220$	0	0
$221$	−4.72740	−0.317999
$222$	0	0
$223$	4.84219	0.324257	0.162129	−	0.986770i	$-0.448164\pi$
$223$	4.84219	0.324257	0.162129	+	0.986770i	$0.448164\pi$
$224$	0	0
$225$	51.8085	3.45390
$226$	0	0
$227$	3.61321	0.239817	0.119909	−	0.992785i	$-0.461740\pi$
$227$	3.61321	0.239817	0.119909	+	0.992785i	$0.461740\pi$
$228$	0	0
$229$	4.24313	0.280394	0.140197	−	0.990124i	$-0.455226\pi$
$229$	4.24313	0.280394	0.140197	+	0.990124i	$0.455226\pi$
$230$	0	0
$231$	−6.61148	−0.435003
$232$	0	0
$233$	17.3473	1.13646	0.568229	−	0.822871i	$-0.307629\pi$
$233$	17.3473	1.13646	0.568229	+	0.822871i	$0.307629\pi$
$234$	0	0
$235$	13.4181	0.875298
$236$	0	0
$237$	−35.3104	−2.29366
$238$	0	0
$239$	7.29783	0.472058	0.236029	−	0.971746i	$-0.424154\pi$
$239$	7.29783	0.472058	0.236029	+	0.971746i	$0.424154\pi$
$240$	0	0
$241$	−8.13057	−0.523736	−0.261868	−	0.965104i	$-0.584339\pi$
$241$	−8.13057	−0.523736	−0.261868	+	0.965104i	$0.584339\pi$
$242$	0	0
$243$	−6.66363	−0.427472
$244$	0	0
$245$	−7.22629	−0.461671
$246$	0	0
$247$	−1.48217	−0.0943080
$248$	0	0
$249$	16.9823	1.07621
$250$	0	0
$251$	10.2146	0.644743	0.322371	−	0.946613i	$-0.395520\pi$
$251$	10.2146	0.644743	0.322371	+	0.946613i	$0.395520\pi$
$252$	0	0
$253$	8.69000	0.546335
$254$	0	0
$255$	35.3106	2.21123
$256$	0	0
$257$	25.2332	1.57400	0.787001	−	0.616952i	$-0.211633\pi$
$257$	25.2332	1.57400	0.787001	+	0.616952i	$0.211633\pi$
$258$	0	0
$259$	19.9573	1.24009
$260$	0	0
$261$	−51.9265	−3.21417
$262$	0	0
$263$	−12.7499	−0.786190	−0.393095	−	0.919498i	$-0.628596\pi$
$263$	−12.7499	−0.786190	−0.393095	+	0.919498i	$0.628596\pi$
$264$	0	0
$265$	−2.02464	−0.124373
$266$	0	0
$267$	11.3990	0.697606
$268$	0	0
$269$	−13.2182	−0.805929	−0.402964	−	0.915216i	$-0.632020\pi$
$269$	−13.2182	−0.805929	−0.402964	+	0.915216i	$0.632020\pi$
$270$	0	0
$271$	13.6104	0.826774	0.413387	−	0.910555i	$-0.364346\pi$
$271$	13.6104	0.826774	0.413387	+	0.910555i	$0.364346\pi$
$272$	0	0
$273$	−9.79930	−0.593081
$274$	0	0
$275$	−9.26206	−0.558523
$276$	0	0
$277$	−21.5893	−1.29717	−0.648587	−	0.761140i	$-0.724640\pi$
$277$	−21.5893	−1.29717	−0.648587	+	0.761140i	$0.724640\pi$
$278$	0	0
$279$	19.7432	1.18199
$280$	0	0
$281$	−9.87413	−0.589041	−0.294521	−	0.955645i	$-0.595160\pi$
$281$	−9.87413	−0.589041	−0.294521	+	0.955645i	$0.595160\pi$
$282$	0	0
$283$	−25.0226	−1.48744	−0.743719	−	0.668492i	$-0.766940\pi$
$283$	−25.0226	−1.48744	−0.743719	+	0.668492i	$0.766940\pi$
$284$	0	0
$285$	11.0708	0.655778
$286$	0	0
$287$	−24.5862	−1.45128
$288$	0	0
$289$	−6.82696	−0.401586
$290$	0	0
$291$	45.7084	2.67947
$292$	0	0
$293$	17.0395	0.995459	0.497729	−	0.867332i	$-0.334167\pi$
$293$	17.0395	0.995459	0.497729	+	0.867332i	$0.334167\pi$
$294$	0	0
$295$	−52.0059	−3.02790
$296$	0	0
$297$	−7.60317	−0.441181
$298$	0	0
$299$	12.8800	0.744871
$300$	0	0
$301$	−12.3452	−0.711568
$302$	0	0
$303$	10.2682	0.589892
$304$	0	0
$305$	30.3465	1.73763
$306$	0	0
$307$	31.5905	1.80297	0.901483	−	0.432815i	$-0.142480\pi$
$307$	31.5905	1.80297	0.901483	+	0.432815i	$0.142480\pi$
$308$	0	0
$309$	−37.8926	−2.15563
$310$	0	0
$311$	32.7463	1.85687	0.928437	−	0.371489i	$-0.121153\pi$
$311$	32.7463	1.85687	0.928437	+	0.371489i	$0.121153\pi$
$312$	0	0
$313$	−6.39364	−0.361390	−0.180695	−	0.983539i	$-0.557835\pi$
$313$	−6.39364	−0.361390	−0.180695	+	0.983539i	$0.557835\pi$
$314$	0	0
$315$	47.6425	2.68435
$316$	0	0
$317$	−18.5664	−1.04279	−0.521395	−	0.853315i	$-0.674588\pi$
$317$	−18.5664	−1.04279	−0.521395	+	0.853315i	$0.674588\pi$
$318$	0	0
$319$	9.28316	0.519757
$320$	0	0
$321$	−9.68170	−0.540380
$322$	0	0
$323$	3.18952	0.177470
$324$	0	0
$325$	−13.7279	−0.761488
$326$	0	0
$327$	7.67847	0.424621
$328$	0	0
$329$	8.01326	0.441785
$330$	0	0
$331$	24.4201	1.34225	0.671124	−	0.741345i	$-0.265812\pi$
$331$	24.4201	1.34225	0.671124	+	0.741345i	$0.265812\pi$
$332$	0	0
$333$	49.4977	2.71246
$334$	0	0
$335$	31.0754	1.69783
$336$	0	0
$337$	−6.07143	−0.330732	−0.165366	−	0.986232i	$-0.552881\pi$
$337$	−6.07143	−0.330732	−0.165366	+	0.986232i	$0.552881\pi$
$338$	0	0
$339$	25.9320	1.40843
$340$	0	0
$341$	−3.52958	−0.191138
$342$	0	0
$343$	−20.1029	−1.08545
$344$	0	0
$345$	−96.2053	−5.17952
$346$	0	0
$347$	14.0282	0.753075	0.376538	−	0.926401i	$-0.377115\pi$
$347$	14.0282	0.753075	0.376538	+	0.926401i	$0.377115\pi$
$348$	0	0
$349$	33.8097	1.80979	0.904897	−	0.425631i	$-0.139948\pi$
$349$	33.8097	1.80979	0.904897	+	0.425631i	$0.139948\pi$
$350$	0	0
$351$	−11.2692	−0.601504
$352$	0	0
$353$	−3.61192	−0.192243	−0.0961217	−	0.995370i	$-0.530644\pi$
$353$	−3.61192	−0.192243	−0.0961217	+	0.995370i	$0.530644\pi$
$354$	0	0
$355$	−24.2650	−1.28785
$356$	0	0
$357$	21.0874	1.11607
$358$	0	0
$359$	6.47982	0.341992	0.170996	−	0.985272i	$-0.445301\pi$
$359$	6.47982	0.341992	0.170996	+	0.985272i	$0.445301\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.93149	0.153863
$364$	0	0
$365$	6.27534	0.328467
$366$	0	0
$367$	−15.1209	−0.789305	−0.394653	−	0.918830i	$-0.629135\pi$
$367$	−15.1209	−0.789305	−0.394653	+	0.918830i	$0.629135\pi$
$368$	0	0
$369$	−60.9781	−3.17439
$370$	0	0
$371$	−1.20911	−0.0627740
$372$	0	0
$373$	28.8109	1.49177	0.745886	−	0.666074i	$-0.232026\pi$
$373$	28.8109	1.49177	0.745886	+	0.666074i	$0.232026\pi$
$374$	0	0
$375$	47.1845	2.43660
$376$	0	0
$377$	13.7592	0.708634
$378$	0	0
$379$	−5.07551	−0.260712	−0.130356	−	0.991467i	$-0.541612\pi$
$379$	−5.07551	−0.260712	−0.130356	+	0.991467i	$0.541612\pi$
$380$	0	0
$381$	−43.1186	−2.20903
$382$	0	0
$383$	−7.44022	−0.380178	−0.190089	−	0.981767i	$-0.560878\pi$
$383$	−7.44022	−0.380178	−0.190089	+	0.981767i	$0.560878\pi$
$384$	0	0
$385$	−8.51729	−0.434081
$386$	0	0
$387$	−30.6184	−1.55642
$388$	0	0
$389$	−0.0442624	−0.00224419	−0.00112210	−	0.999999i	$-0.500357\pi$
$389$	−0.0442624	−0.00224419	−0.00112210	+	0.999999i	$0.500357\pi$
$390$	0	0
$391$	−27.7169	−1.40170
$392$	0	0
$393$	−12.1140	−0.611072
$394$	0	0
$395$	−45.4889	−2.28879
$396$	0	0
$397$	3.87687	0.194574	0.0972872	−	0.995256i	$-0.468983\pi$
$397$	3.87687	0.194574	0.0972872	+	0.995256i	$0.468983\pi$
$398$	0	0
$399$	6.61148	0.330988
$400$	0	0
$401$	11.1686	0.557733	0.278866	−	0.960330i	$-0.410041\pi$
$401$	11.1686	0.557733	0.278866	+	0.960330i	$0.410041\pi$
$402$	0	0
$403$	−5.23143	−0.260596
$404$	0	0
$405$	20.8001	1.03356
$406$	0	0
$407$	−8.84895	−0.438626
$408$	0	0
$409$	5.10522	0.252437	0.126218	−	0.992002i	$-0.459716\pi$
$409$	5.10522	0.252437	0.126218	+	0.992002i	$0.459716\pi$
$410$	0	0
$411$	64.6059	3.18677
$412$	0	0
$413$	−31.0579	−1.52826
$414$	0	0
$415$	21.8776	1.07393
$416$	0	0
$417$	−14.6819	−0.718976
$418$	0	0
$419$	15.7366	0.768785	0.384392	−	0.923170i	$-0.374411\pi$
$419$	15.7366	0.768785	0.384392	+	0.923170i	$0.374411\pi$
$420$	0	0
$421$	−34.1587	−1.66479	−0.832397	−	0.554180i	$-0.813032\pi$
$421$	−34.1587	−1.66479	−0.832397	+	0.554180i	$0.813032\pi$
$422$	0	0
$423$	19.8743	0.966322
$424$	0	0
$425$	29.5415	1.43297
$426$	0	0
$427$	18.1229	0.877028
$428$	0	0
$429$	4.34495	0.209776
$430$	0	0
$431$	−14.9079	−0.718089	−0.359044	−	0.933321i	$-0.616897\pi$
$431$	−14.9079	−0.718089	−0.359044	+	0.933321i	$0.616897\pi$
$432$	0	0
$433$	19.8317	0.953053	0.476526	−	0.879160i	$-0.341896\pi$
$433$	19.8317	0.953053	0.476526	+	0.879160i	$0.341896\pi$
$434$	0	0
$435$	−102.772	−4.92754
$436$	0	0
$437$	−8.69000	−0.415699
$438$	0	0
$439$	10.2401	0.488735	0.244367	−	0.969683i	$-0.421420\pi$
$439$	10.2401	0.488735	0.244367	+	0.969683i	$0.421420\pi$
$440$	0	0
$441$	−10.7033	−0.509681
$442$	0	0
$443$	−8.30971	−0.394806	−0.197403	−	0.980322i	$-0.563251\pi$
$443$	−8.30971	−0.394806	−0.197403	+	0.980322i	$0.563251\pi$
$444$	0	0
$445$	14.6848	0.696127
$446$	0	0
$447$	−33.7973	−1.59856
$448$	0	0
$449$	8.56504	0.404209	0.202105	−	0.979364i	$-0.435222\pi$
$449$	8.56504	0.404209	0.202105	+	0.979364i	$0.435222\pi$
$450$	0	0
$451$	10.9014	0.513325
$452$	0	0
$453$	−47.1951	−2.21742
$454$	0	0
$455$	−12.6240	−0.591824
$456$	0	0
$457$	−7.14025	−0.334007	−0.167003	−	0.985956i	$-0.553409\pi$
$457$	−7.14025	−0.334007	−0.167003	+	0.985956i	$0.553409\pi$
$458$	0	0
$459$	24.2505	1.13192
$460$	0	0
$461$	−4.34504	−0.202369	−0.101184	−	0.994868i	$-0.532263\pi$
$461$	−4.34504	−0.202369	−0.101184	+	0.994868i	$0.532263\pi$
$462$	0	0
$463$	34.0051	1.58035	0.790177	−	0.612879i	$-0.209989\pi$
$463$	34.0051	1.58035	0.790177	+	0.612879i	$0.209989\pi$
$464$	0	0
$465$	39.0753	1.81207
$466$	0	0
$467$	27.7609	1.28462	0.642311	−	0.766444i	$-0.277976\pi$
$467$	27.7609	1.28462	0.642311	+	0.766444i	$0.277976\pi$
$468$	0	0
$469$	18.5582	0.856937
$470$	0	0
$471$	−22.4800	−1.03582
$472$	0	0
$473$	5.47380	0.251686
$474$	0	0
$475$	9.26206	0.424973
$476$	0	0
$477$	−2.99882	−0.137306
$478$	0	0
$479$	−31.5480	−1.44147	−0.720733	−	0.693213i	$-0.756195\pi$
$479$	−31.5480	−1.44147	−0.720733	+	0.693213i	$0.756195\pi$
$480$	0	0
$481$	−13.1156	−0.598021
$482$	0	0
$483$	−57.4537	−2.61423
$484$	0	0
$485$	58.8842	2.67379
$486$	0	0
$487$	8.66076	0.392456	0.196228	−	0.980558i	$-0.437131\pi$
$487$	8.66076	0.392456	0.196228	+	0.980558i	$0.437131\pi$
$488$	0	0
$489$	40.1121	1.81393
$490$	0	0
$491$	3.72129	0.167939	0.0839696	−	0.996468i	$-0.473240\pi$
$491$	3.72129	0.167939	0.0839696	+	0.996468i	$0.473240\pi$
$492$	0	0
$493$	−29.6088	−1.33351
$494$	0	0
$495$	−21.1244	−0.949471
$496$	0	0
$497$	−14.4910	−0.650012
$498$	0	0
$499$	1.54473	0.0691516	0.0345758	−	0.999402i	$-0.488992\pi$
$499$	1.54473	0.0691516	0.0345758	+	0.999402i	$0.488992\pi$
$500$	0	0
$501$	33.3297	1.48906
$502$	0	0
$503$	−9.92478	−0.442524	−0.221262	−	0.975214i	$-0.571018\pi$
$503$	−9.92478	−0.442524	−0.221262	+	0.975214i	$0.571018\pi$
$504$	0	0
$505$	13.2281	0.588642
$506$	0	0
$507$	−31.6694	−1.40649
$508$	0	0
$509$	3.73981	0.165764	0.0828820	−	0.996559i	$-0.473588\pi$
$509$	3.73981	0.165764	0.0828820	+	0.996559i	$0.473588\pi$
$510$	0	0
$511$	3.74763	0.165785
$512$	0	0
$513$	7.60317	0.335688
$514$	0	0
$515$	−48.8154	−2.15106
$516$	0	0
$517$	−3.55303	−0.156262
$518$	0	0
$519$	−51.1733	−2.24626
$520$	0	0
$521$	−32.2820	−1.41430	−0.707151	−	0.707063i	$-0.750020\pi$
$521$	−32.2820	−1.41430	−0.707151	+	0.707063i	$0.750020\pi$
$522$	0	0
$523$	3.77759	0.165183	0.0825913	−	0.996584i	$-0.473680\pi$
$523$	3.77759	0.165183	0.0825913	+	0.996584i	$0.473680\pi$
$524$	0	0
$525$	61.2359	2.67255
$526$	0	0
$527$	11.2577	0.490392
$528$	0	0
$529$	52.5160	2.28331
$530$	0	0
$531$	−77.0290	−3.34278
$532$	0	0
$533$	16.1576	0.699865
$534$	0	0
$535$	−12.4725	−0.539234
$536$	0	0
$537$	−46.5006	−2.00665
$538$	0	0
$539$	1.91348	0.0824195
$540$	0	0
$541$	27.5215	1.18324	0.591620	−	0.806217i	$-0.298489\pi$
$541$	27.5215	1.18324	0.591620	+	0.806217i	$0.298489\pi$
$542$	0	0
$543$	14.7417	0.632628
$544$	0	0
$545$	9.89186	0.423721
$546$	0	0
$547$	28.1999	1.20574	0.602870	−	0.797840i	$-0.294024\pi$
$547$	28.1999	1.20574	0.602870	+	0.797840i	$0.294024\pi$
$548$	0	0
$549$	44.9480	1.91833
$550$	0	0
$551$	−9.28316	−0.395476
$552$	0	0
$553$	−27.1659	−1.15521
$554$	0	0
$555$	97.9651	4.15839
$556$	0	0
$557$	−2.52660	−0.107056	−0.0535278	−	0.998566i	$-0.517047\pi$
$557$	−2.52660	−0.107056	−0.0535278	+	0.998566i	$0.517047\pi$
$558$	0	0
$559$	8.11308	0.343147
$560$	0	0
$561$	−9.35004	−0.394759
$562$	0	0
$563$	−24.0174	−1.01221	−0.506106	−	0.862471i	$-0.668916\pi$
$563$	−24.0174	−1.01221	−0.506106	+	0.862471i	$0.668916\pi$
$564$	0	0
$565$	33.4072	1.40545
$566$	0	0
$567$	12.4218	0.521666
$568$	0	0
$569$	−28.5879	−1.19847	−0.599233	−	0.800574i	$-0.704528\pi$
$569$	−28.5879	−1.19847	−0.599233	+	0.800574i	$0.704528\pi$
$570$	0	0
$571$	35.4625	1.48406	0.742029	−	0.670368i	$-0.233864\pi$
$571$	35.4625	1.48406	0.742029	+	0.670368i	$0.233864\pi$
$572$	0	0
$573$	−23.0697	−0.963751
$574$	0	0
$575$	−80.4873	−3.35655
$576$	0	0
$577$	15.7421	0.655353	0.327677	−	0.944790i	$-0.393734\pi$
$577$	15.7421	0.655353	0.327677	+	0.944790i	$0.393734\pi$
$578$	0	0
$579$	78.7222	3.27158
$580$	0	0
$581$	13.0653	0.542038
$582$	0	0
$583$	0.536113	0.0222035
$584$	0	0
$585$	−31.3099	−1.29450
$586$	0	0
$587$	8.14624	0.336231	0.168116	−	0.985767i	$-0.446232\pi$
$587$	8.14624	0.336231	0.168116	+	0.985767i	$0.446232\pi$
$588$	0	0
$589$	3.52958	0.145434
$590$	0	0
$591$	9.01670	0.370897
$592$	0	0
$593$	15.4004	0.632420	0.316210	−	0.948689i	$-0.397590\pi$
$593$	15.4004	0.632420	0.316210	+	0.948689i	$0.397590\pi$
$594$	0	0
$595$	27.1661	1.11370
$596$	0	0
$597$	28.5947	1.17030
$598$	0	0
$599$	5.61717	0.229511	0.114756	−	0.993394i	$-0.463391\pi$
$599$	5.61717	0.229511	0.114756	+	0.993394i	$0.463391\pi$
$600$	0	0
$601$	−17.7938	−0.725824	−0.362912	−	0.931823i	$-0.618218\pi$
$601$	−17.7938	−0.725824	−0.362912	+	0.931823i	$0.618218\pi$
$602$	0	0
$603$	46.0276	1.87439
$604$	0	0
$605$	3.77651	0.153537
$606$	0	0
$607$	−11.9015	−0.483065	−0.241533	−	0.970393i	$-0.577650\pi$
$607$	−11.9015	−0.483065	−0.241533	+	0.970393i	$0.577650\pi$
$608$	0	0
$609$	−61.3754	−2.48706
$610$	0	0
$611$	−5.26618	−0.213047
$612$	0	0
$613$	−17.7013	−0.714948	−0.357474	−	0.933923i	$-0.616362\pi$
$613$	−17.7013	−0.714948	−0.357474	+	0.933923i	$0.616362\pi$
$614$	0	0
$615$	−120.687	−4.86656
$616$	0	0
$617$	34.6485	1.39490	0.697448	−	0.716636i	$-0.254319\pi$
$617$	34.6485	1.39490	0.697448	+	0.716636i	$0.254319\pi$
$618$	0	0
$619$	34.3351	1.38005	0.690023	−	0.723788i	$-0.257600\pi$
$619$	34.3351	1.38005	0.690023	+	0.723788i	$0.257600\pi$
$620$	0	0
$621$	−66.0716	−2.65136
$622$	0	0
$623$	8.76976	0.351353
$624$	0	0
$625$	14.4755	0.579019
$626$	0	0
$627$	−2.93149	−0.117072
$628$	0	0
$629$	28.2239	1.12536
$630$	0	0
$631$	−36.1359	−1.43855	−0.719274	−	0.694726i	$-0.755525\pi$
$631$	−36.1359	−1.43855	−0.719274	+	0.694726i	$0.755525\pi$
$632$	0	0
$633$	55.6815	2.21314
$634$	0	0
$635$	−55.5479	−2.20435
$636$	0	0
$637$	2.83610	0.112370
$638$	0	0
$639$	−35.9403	−1.42178
$640$	0	0
$641$	−29.3080	−1.15759	−0.578797	−	0.815471i	$-0.696478\pi$
$641$	−29.3080	−1.15759	−0.578797	+	0.815471i	$0.696478\pi$
$642$	0	0
$643$	−2.40830	−0.0949741	−0.0474870	−	0.998872i	$-0.515121\pi$
$643$	−2.40830	−0.0949741	−0.0474870	+	0.998872i	$0.515121\pi$
$644$	0	0
$645$	−60.5994	−2.38610
$646$	0	0
$647$	42.0325	1.65247	0.826234	−	0.563327i	$-0.190479\pi$
$647$	42.0325	1.65247	0.826234	+	0.563327i	$0.190479\pi$
$648$	0	0
$649$	13.7709	0.540554
$650$	0	0
$651$	23.3357	0.914600
$652$	0	0
$653$	42.0257	1.64459	0.822297	−	0.569059i	$-0.192692\pi$
$653$	42.0257	1.64459	0.822297	+	0.569059i	$0.192692\pi$
$654$	0	0
$655$	−15.6060	−0.609777
$656$	0	0
$657$	9.29479	0.362624
$658$	0	0
$659$	−22.4644	−0.875087	−0.437544	−	0.899197i	$-0.644151\pi$
$659$	−22.4644	−0.875087	−0.437544	+	0.899197i	$0.644151\pi$
$660$	0	0
$661$	−40.0391	−1.55734	−0.778671	−	0.627433i	$-0.784106\pi$
$661$	−40.0391	−1.55734	−0.778671	+	0.627433i	$0.784106\pi$
$662$	0	0
$663$	−13.8583	−0.538212
$664$	0	0
$665$	8.51729	0.330286
$666$	0	0
$667$	80.6706	3.12358
$668$	0	0
$669$	14.1948	0.548804
$670$	0	0
$671$	−8.03558	−0.310210
$672$	0	0
$673$	−26.5037	−1.02164	−0.510821	−	0.859687i	$-0.670659\pi$
$673$	−26.5037	−1.02164	−0.510821	+	0.859687i	$0.670659\pi$
$674$	0	0
$675$	70.4211	2.71051
$676$	0	0
$677$	39.9827	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$677$	39.9827	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$678$	0	0
$679$	35.1656	1.34953
$680$	0	0
$681$	10.5921	0.405889
$682$	0	0
$683$	34.2509	1.31057	0.655287	−	0.755380i	$-0.272548\pi$
$683$	34.2509	1.31057	0.655287	+	0.755380i	$0.272548\pi$
$684$	0	0
$685$	83.2291	3.18002
$686$	0	0
$687$	12.4387	0.474566
$688$	0	0
$689$	0.794609	0.0302722
$690$	0	0
$691$	−22.1464	−0.842489	−0.421244	−	0.906947i	$-0.638406\pi$
$691$	−22.1464	−0.842489	−0.421244	+	0.906947i	$0.638406\pi$
$692$	0	0
$693$	−12.6155	−0.479222
$694$	0	0
$695$	−18.9141	−0.717452
$696$	0	0
$697$	−34.7701	−1.31701
$698$	0	0
$699$	50.8533	1.92345
$700$	0	0
$701$	−10.0821	−0.380797	−0.190399	−	0.981707i	$-0.560978\pi$
$701$	−10.0821	−0.380797	−0.190399	+	0.981707i	$0.560978\pi$
$702$	0	0
$703$	8.84895	0.333745
$704$	0	0
$705$	39.3349	1.48144
$706$	0	0
$707$	7.89979	0.297102
$708$	0	0
$709$	−21.1417	−0.793994	−0.396997	−	0.917820i	$-0.629948\pi$
$709$	−21.1417	−0.793994	−0.396997	+	0.917820i	$0.629948\pi$
$710$	0	0
$711$	−67.3763	−2.52681
$712$	0	0
$713$	−30.6721	−1.14868
$714$	0	0
$715$	5.59742	0.209332
$716$	0	0
$717$	21.3935	0.798955
$718$	0	0
$719$	−4.38950	−0.163701	−0.0818504	−	0.996645i	$-0.526083\pi$
$719$	−4.38950	−0.163701	−0.0818504	+	0.996645i	$0.526083\pi$
$720$	0	0
$721$	−29.1525	−1.08570
$722$	0	0
$723$	−23.8347	−0.886421
$724$	0	0
$725$	−85.9812	−3.19326
$726$	0	0
$727$	16.6462	0.617373	0.308686	−	0.951164i	$-0.400111\pi$
$727$	16.6462	0.617373	0.308686	+	0.951164i	$0.400111\pi$
$728$	0	0
$729$	−36.0576	−1.33547
$730$	0	0
$731$	−17.4588	−0.645737
$732$	0	0
$733$	4.70441	0.173761	0.0868806	−	0.996219i	$-0.472310\pi$
$733$	4.70441	0.173761	0.0868806	+	0.996219i	$0.472310\pi$
$734$	0	0
$735$	−21.1838	−0.781376
$736$	0	0
$737$	−8.22858	−0.303104
$738$	0	0
$739$	0.937699	0.0344938	0.0172469	−	0.999851i	$-0.494510\pi$
$739$	0.937699	0.0344938	0.0172469	+	0.999851i	$0.494510\pi$
$740$	0	0
$741$	−4.34495	−0.159616
$742$	0	0
$743$	−19.2602	−0.706587	−0.353294	−	0.935512i	$-0.614938\pi$
$743$	−19.2602	−0.706587	−0.353294	+	0.935512i	$0.614938\pi$
$744$	0	0
$745$	−43.5397	−1.59517
$746$	0	0
$747$	32.4042	1.18561
$748$	0	0
$749$	−7.44858	−0.272165
$750$	0	0
$751$	−27.8101	−1.01480	−0.507402	−	0.861710i	$-0.669394\pi$
$751$	−27.8101	−1.01480	−0.507402	+	0.861710i	$0.669394\pi$
$752$	0	0
$753$	29.9441	1.09122
$754$	0	0
$755$	−60.7994	−2.21272
$756$	0	0
$757$	−40.1605	−1.45966	−0.729829	−	0.683629i	$-0.760401\pi$
$757$	−40.1605	−1.45966	−0.729829	+	0.683629i	$0.760401\pi$
$758$	0	0
$759$	25.4746	0.924670
$760$	0	0
$761$	−7.88418	−0.285801	−0.142901	−	0.989737i	$-0.545643\pi$
$761$	−7.88418	−0.285801	−0.142901	+	0.989737i	$0.545643\pi$
$762$	0	0
$763$	5.90741	0.213863
$764$	0	0
$765$	67.3767	2.43601
$766$	0	0
$767$	20.4107	0.736988
$768$	0	0
$769$	15.0393	0.542330	0.271165	−	0.962533i	$-0.412591\pi$
$769$	15.0393	0.542330	0.271165	+	0.962533i	$0.412591\pi$
$770$	0	0
$771$	73.9708	2.66399
$772$	0	0
$773$	6.24984	0.224791	0.112396	−	0.993664i	$-0.464148\pi$
$773$	6.24984	0.224791	0.112396	+	0.993664i	$0.464148\pi$
$774$	0	0
$775$	32.6912	1.17430
$776$	0	0
$777$	58.5046	2.09884
$778$	0	0
$779$	−10.9014	−0.390582
$780$	0	0
$781$	6.42523	0.229913
$782$	0	0
$783$	−70.5815	−2.52238
$784$	0	0
$785$	−28.9600	−1.03363
$786$	0	0
$787$	14.8053	0.527750	0.263875	−	0.964557i	$-0.414999\pi$
$787$	14.8053	0.527750	0.263875	+	0.964557i	$0.414999\pi$
$788$	0	0
$789$	−37.3760	−1.33062
$790$	0	0
$791$	19.9507	0.709366
$792$	0	0
$793$	−11.9101	−0.422939
$794$	0	0
$795$	−5.93521	−0.210500
$796$	0	0
$797$	−42.4773	−1.50462	−0.752312	−	0.658807i	$-0.771061\pi$
$797$	−42.4773	−1.50462	−0.752312	+	0.658807i	$0.771061\pi$
$798$	0	0
$799$	11.3325	0.400914
$800$	0	0
$801$	21.7506	0.768519
$802$	0	0
$803$	−1.66168	−0.0586393
$804$	0	0
$805$	−74.0152	−2.60869
$806$	0	0
$807$	−38.7490	−1.36403
$808$	0	0
$809$	−52.5493	−1.84753	−0.923767	−	0.382955i	$-0.874907\pi$
$809$	−52.5493	−1.84753	−0.923767	+	0.382955i	$0.874907\pi$
$810$	0	0
$811$	−41.5803	−1.46008	−0.730041	−	0.683403i	$-0.760499\pi$
$811$	−41.5803	−1.46008	−0.730041	+	0.683403i	$0.760499\pi$
$812$	0	0
$813$	39.8988	1.39931
$814$	0	0
$815$	51.6747	1.81009
$816$	0	0
$817$	−5.47380	−0.191504
$818$	0	0
$819$	−18.6982	−0.653369
$820$	0	0
$821$	4.26098	0.148709	0.0743546	−	0.997232i	$-0.476310\pi$
$821$	4.26098	0.148709	0.0743546	+	0.997232i	$0.476310\pi$
$822$	0	0
$823$	−50.8790	−1.77353	−0.886764	−	0.462222i	$-0.847052\pi$
$823$	−50.8790	−1.77353	−0.886764	+	0.462222i	$0.847052\pi$
$824$	0	0
$825$	−27.1516	−0.945298
$826$	0	0
$827$	−18.5242	−0.644150	−0.322075	−	0.946714i	$-0.604380\pi$
$827$	−18.5242	−0.644150	−0.322075	+	0.946714i	$0.604380\pi$
$828$	0	0
$829$	−27.0511	−0.939524	−0.469762	−	0.882793i	$-0.655660\pi$
$829$	−27.0511	−0.939524	−0.469762	+	0.882793i	$0.655660\pi$
$830$	0	0
$831$	−63.2887	−2.19546
$832$	0	0
$833$	−6.10309	−0.211460
$834$	0	0
$835$	42.9372	1.48590
$836$	0	0
$837$	26.8360	0.927589
$838$	0	0
$839$	6.47528	0.223551	0.111776	−	0.993733i	$-0.464346\pi$
$839$	6.47528	0.223551	0.111776	+	0.993733i	$0.464346\pi$
$840$	0	0
$841$	57.1771	1.97162
$842$	0	0
$843$	−28.9459	−0.996950
$844$	0	0
$845$	−40.7984	−1.40351
$846$	0	0
$847$	2.25533	0.0774941
$848$	0	0
$849$	−73.3534	−2.51748
$850$	0	0
$851$	−76.8974	−2.63601
$852$	0	0
$853$	−31.9260	−1.09312	−0.546562	−	0.837418i	$-0.684064\pi$
$853$	−31.9260	−1.09312	−0.546562	+	0.837418i	$0.684064\pi$
$854$	0	0
$855$	21.1244	0.722439
$856$	0	0
$857$	−44.2258	−1.51072	−0.755362	−	0.655308i	$-0.772539\pi$
$857$	−44.2258	−1.51072	−0.755362	+	0.655308i	$0.772539\pi$
$858$	0	0
$859$	−45.6383	−1.55716	−0.778580	−	0.627545i	$-0.784060\pi$
$859$	−45.6383	−1.55716	−0.778580	+	0.627545i	$0.784060\pi$
$860$	0	0
$861$	−72.0741	−2.45628
$862$	0	0
$863$	−15.7740	−0.536954	−0.268477	−	0.963286i	$-0.586520\pi$
$863$	−15.7740	−0.536954	−0.268477	+	0.963286i	$0.586520\pi$
$864$	0	0
$865$	−65.9244	−2.24150
$866$	0	0
$867$	−20.0132	−0.679683
$868$	0	0
$869$	12.0452	0.408606
$870$	0	0
$871$	−12.1961	−0.413250
$872$	0	0
$873$	87.2170	2.95185
$874$	0	0
$875$	36.3012	1.22721
$876$	0	0
$877$	9.77173	0.329968	0.164984	−	0.986296i	$-0.447243\pi$
$877$	9.77173	0.329968	0.164984	+	0.986296i	$0.447243\pi$
$878$	0	0
$879$	49.9511	1.68481
$880$	0	0
$881$	27.3955	0.922977	0.461489	−	0.887146i	$-0.347316\pi$
$881$	27.3955	0.922977	0.461489	+	0.887146i	$0.347316\pi$
$882$	0	0
$883$	−9.76363	−0.328572	−0.164286	−	0.986413i	$-0.552532\pi$
$883$	−9.76363	−0.328572	−0.164286	+	0.986413i	$0.552532\pi$
$884$	0	0
$885$	−152.455	−5.12471
$886$	0	0
$887$	−18.2333	−0.612215	−0.306108	−	0.951997i	$-0.599027\pi$
$887$	−18.2333	−0.612215	−0.306108	+	0.951997i	$0.599027\pi$
$888$	0	0
$889$	−33.1731	−1.11259
$890$	0	0
$891$	−5.50775	−0.184516
$892$	0	0
$893$	3.55303	0.118898
$894$	0	0
$895$	−59.9048	−2.00240
$896$	0	0
$897$	37.7576	1.26069
$898$	0	0
$899$	−32.7657	−1.09280
$900$	0	0
$901$	−1.70994	−0.0569665
$902$	0	0
$903$	−36.1899	−1.20433
$904$	0	0
$905$	18.9912	0.631287
$906$	0	0
$907$	54.6088	1.81326	0.906628	−	0.421931i	$-0.138648\pi$
$907$	54.6088	1.81326	0.906628	+	0.421931i	$0.138648\pi$
$908$	0	0
$909$	19.5929	0.649855
$910$	0	0
$911$	−34.4568	−1.14161	−0.570803	−	0.821087i	$-0.693368\pi$
$911$	−34.4568	−1.14161	−0.570803	+	0.821087i	$0.693368\pi$
$912$	0	0
$913$	−5.79306	−0.191722
$914$	0	0
$915$	88.9604	2.94094
$916$	0	0
$917$	−9.31990	−0.307770
$918$	0	0
$919$	13.7052	0.452093	0.226047	−	0.974116i	$-0.427420\pi$
$919$	13.7052	0.452093	0.226047	+	0.974116i	$0.427420\pi$
$920$	0	0
$921$	92.6072	3.05151
$922$	0	0
$923$	9.52326	0.313462
$924$	0	0
$925$	81.9596	2.69481
$926$	0	0
$927$	−72.3035	−2.37476
$928$	0	0
$929$	−23.5489	−0.772616	−0.386308	−	0.922370i	$-0.626250\pi$
$929$	−23.5489	−0.772616	−0.386308	+	0.922370i	$0.626250\pi$
$930$	0	0
$931$	−1.91348	−0.0627119
$932$	0	0
$933$	95.9955	3.14275
$934$	0	0
$935$	−12.0453	−0.393922
$936$	0	0
$937$	−39.4420	−1.28851	−0.644257	−	0.764809i	$-0.722833\pi$
$937$	−39.4420	−1.28851	−0.644257	+	0.764809i	$0.722833\pi$
$938$	0	0
$939$	−18.7429	−0.611651
$940$	0	0
$941$	27.7454	0.904475	0.452238	−	0.891897i	$-0.350626\pi$
$941$	27.7454	0.904475	0.452238	+	0.891897i	$0.350626\pi$
$942$	0	0
$943$	94.7328	3.08492
$944$	0	0
$945$	64.7584	2.10659
$946$	0	0
$947$	−30.1366	−0.979306	−0.489653	−	0.871917i	$-0.662877\pi$
$947$	−30.1366	−0.979306	−0.489653	+	0.871917i	$0.662877\pi$
$948$	0	0
$949$	−2.46288	−0.0799485
$950$	0	0
$951$	−54.4270	−1.76492
$952$	0	0
$953$	−8.66883	−0.280811	−0.140405	−	0.990094i	$-0.544841\pi$
$953$	−8.66883	−0.280811	−0.140405	+	0.990094i	$0.544841\pi$
$954$	0	0
$955$	−29.7198	−0.961709
$956$	0	0
$957$	27.2135	0.879687
$958$	0	0
$959$	49.7043	1.60504
$960$	0	0
$961$	−18.5420	−0.598131
$962$	0	0
$963$	−18.4738	−0.595310
$964$	0	0
$965$	101.415	3.26465
$966$	0	0
$967$	28.8252	0.926956	0.463478	−	0.886108i	$-0.346601\pi$
$967$	28.8252	0.926956	0.463478	+	0.886108i	$0.346601\pi$
$968$	0	0
$969$	9.35004	0.300367
$970$	0	0
$971$	−9.36801	−0.300634	−0.150317	−	0.988638i	$-0.548029\pi$
$971$	−9.36801	−0.300634	−0.150317	+	0.988638i	$0.548029\pi$
$972$	0	0
$973$	−11.2955	−0.362116
$974$	0	0
$975$	−40.2432	−1.28881
$976$	0	0
$977$	−25.4895	−0.815481	−0.407741	−	0.913098i	$-0.633683\pi$
$977$	−25.4895	−0.815481	−0.407741	+	0.913098i	$0.633683\pi$
$978$	0	0
$979$	−3.88846	−0.124276
$980$	0	0
$981$	14.6514	0.467784
$982$	0	0
$983$	9.15810	0.292098	0.146049	−	0.989277i	$-0.453344\pi$
$983$	9.15810	0.292098	0.146049	+	0.989277i	$0.453344\pi$
$984$	0	0
$985$	11.6158	0.370111
$986$	0	0
$987$	23.4908	0.747719
$988$	0	0
$989$	47.5673	1.51255
$990$	0	0
$991$	−6.25667	−0.198750	−0.0993749	−	0.995050i	$-0.531684\pi$
$991$	−6.25667	−0.198750	−0.0993749	+	0.995050i	$0.531684\pi$
$992$	0	0
$993$	71.5871	2.27175
$994$	0	0
$995$	36.8374	1.16782
$996$	0	0
$997$	−22.9411	−0.726552	−0.363276	−	0.931682i	$-0.618342\pi$
$997$	−22.9411	−0.726552	−0.363276	+	0.931682i	$0.618342\pi$
$998$	0	0
$999$	67.2801	2.12865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.bb.1.8		9
4.3	odd	2		1672.2.a.k.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.k.1.2	✓	9	4.3	odd	2
3344.2.a.bb.1.8		9	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)

\(f(q)\)	\(=\)	\(q+2.93149 q^{3} +3.77651 q^{5} +2.25533 q^{7} +5.59362 q^{9} +O(q^{10})\)	\(q+2.93149 q^{3} +3.77651 q^{5} +2.25533 q^{7} +5.59362 q^{9} -1.00000 q^{11} -1.48217 q^{13} +11.0708 q^{15} +3.18952 q^{17} +1.00000 q^{19} +6.61148 q^{21} -8.69000 q^{23} +9.26206 q^{25} +7.60317 q^{27} -9.28316 q^{29} +3.52958 q^{31} -2.93149 q^{33} +8.51729 q^{35} +8.84895 q^{37} -4.34495 q^{39} -10.9014 q^{41} -5.47380 q^{43} +21.1244 q^{45} +3.55303 q^{47} -1.91348 q^{49} +9.35004 q^{51} -0.536113 q^{53} -3.77651 q^{55} +2.93149 q^{57} -13.7709 q^{59} +8.03558 q^{61} +12.6155 q^{63} -5.59742 q^{65} +8.22858 q^{67} -25.4746 q^{69} -6.42523 q^{71} +1.66168 q^{73} +27.1516 q^{75} -2.25533 q^{77} -12.0452 q^{79} +5.50775 q^{81} +5.79306 q^{83} +12.0453 q^{85} -27.2135 q^{87} +3.88846 q^{89} -3.34277 q^{91} +10.3469 q^{93} +3.77651 q^{95} +15.5922 q^{97} -5.59362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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