LMFDB - Embedded newform 3344.2.a.y.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.57500224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 18*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.94641$ 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)

$f(q)$	$=$	$q+2.94641 q^{3} -3.26823 q^{5} +4.50474 q^{7} +5.68132 q^{9} +O(q^{10})$	$q+2.94641 q^{3} -3.26823 q^{5} +4.50474 q^{7} +5.68132 q^{9} -1.00000 q^{11} +5.39613 q^{13} -9.62953 q^{15} +3.19596 q^{17} +1.00000 q^{19} +13.2728 q^{21} -1.01938 q^{23} +5.68132 q^{25} +7.90025 q^{27} +4.22036 q^{29} -3.59076 q^{31} -2.94641 q^{33} -14.7225 q^{35} -9.26094 q^{37} +15.8992 q^{39} -9.49030 q^{41} +9.64397 q^{43} -18.5678 q^{45} -10.3221 q^{47} +13.2927 q^{49} +9.41661 q^{51} -4.95945 q^{53} +3.26823 q^{55} +2.94641 q^{57} -2.15410 q^{59} +6.38751 q^{61} +25.5928 q^{63} -17.6358 q^{65} +5.37994 q^{67} -3.00352 q^{69} +6.48606 q^{71} +14.4644 q^{73} +16.7395 q^{75} -4.50474 q^{77} -16.5586 q^{79} +6.23341 q^{81} +11.7606 q^{83} -10.4451 q^{85} +12.4349 q^{87} +7.92706 q^{89} +24.3081 q^{91} -10.5799 q^{93} -3.26823 q^{95} -2.72448 q^{97} -5.68132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.94641	1.70111	0.850555	−	0.525887i	$-0.176266\pi$
$3$	2.94641	1.70111	0.850555	+	0.525887i	$0.176266\pi$
$4$	0	0
$5$	−3.26823	−1.46160	−0.730798	−	0.682594i	$-0.760852\pi$
$5$	−3.26823	−1.46160	−0.730798	+	0.682594i	$0.760852\pi$
$6$	0	0
$7$	4.50474	1.70263	0.851315	−	0.524654i	$-0.175805\pi$
$7$	4.50474	1.70263	0.851315	+	0.524654i	$0.175805\pi$
$8$	0	0
$9$	5.68132	1.89377
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.39613	1.49662	0.748308	−	0.663351i	$-0.230867\pi$
$13$	5.39613	1.49662	0.748308	+	0.663351i	$0.230867\pi$
$14$	0	0
$15$	−9.62953	−2.48633
$16$	0	0
$17$	3.19596	0.775135	0.387567	−	0.921841i	$-0.373315\pi$
$17$	3.19596	0.775135	0.387567	+	0.921841i	$0.373315\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	13.2728	2.89636
$22$	0	0
$23$	−1.01938	−0.212556	−0.106278	−	0.994336i	$-0.533893\pi$
$23$	−1.01938	−0.212556	−0.106278	+	0.994336i	$0.533893\pi$
$24$	0	0
$25$	5.68132	1.13626
$26$	0	0
$27$	7.90025	1.52040
$28$	0	0
$29$	4.22036	0.783701	0.391851	−	0.920029i	$-0.371835\pi$
$29$	4.22036	0.783701	0.391851	+	0.920029i	$0.371835\pi$
$30$	0	0
$31$	−3.59076	−0.644920	−0.322460	−	0.946583i	$-0.604510\pi$
$31$	−3.59076	−0.644920	−0.322460	+	0.946583i	$0.604510\pi$
$32$	0	0
$33$	−2.94641	−0.512904
$34$	0	0
$35$	−14.7225	−2.48856
$36$	0	0
$37$	−9.26094	−1.52249	−0.761245	−	0.648465i	$-0.775411\pi$
$37$	−9.26094	−1.52249	−0.761245	+	0.648465i	$0.775411\pi$
$38$	0	0
$39$	15.8992	2.54591
$40$	0	0
$41$	−9.49030	−1.48214	−0.741068	−	0.671430i	$-0.765680\pi$
$41$	−9.49030	−1.48214	−0.741068	+	0.671430i	$0.765680\pi$
$42$	0	0
$43$	9.64397	1.47069	0.735346	−	0.677692i	$-0.237020\pi$
$43$	9.64397	1.47069	0.735346	+	0.677692i	$0.237020\pi$
$44$	0	0
$45$	−18.5678	−2.76793
$46$	0	0
$47$	−10.3221	−1.50563	−0.752815	−	0.658232i	$-0.771305\pi$
$47$	−10.3221	−1.50563	−0.752815	+	0.658232i	$0.771305\pi$
$48$	0	0
$49$	13.2927	1.89895
$50$	0	0
$51$	9.41661	1.31859
$52$	0	0
$53$	−4.95945	−0.681233	−0.340617	−	0.940202i	$-0.610636\pi$
$53$	−4.95945	−0.681233	−0.340617	+	0.940202i	$0.610636\pi$
$54$	0	0
$55$	3.26823	0.440688
$56$	0	0
$57$	2.94641	0.390261
$58$	0	0
$59$	−2.15410	−0.280440	−0.140220	−	0.990120i	$-0.544781\pi$
$59$	−2.15410	−0.280440	−0.140220	+	0.990120i	$0.544781\pi$
$60$	0	0
$61$	6.38751	0.817837	0.408918	−	0.912571i	$-0.365906\pi$
$61$	6.38751	0.817837	0.408918	+	0.912571i	$0.365906\pi$
$62$	0	0
$63$	25.5928	3.22439
$64$	0	0
$65$	−17.6358	−2.18745
$66$	0	0
$67$	5.37994	0.657265	0.328632	−	0.944458i	$-0.393412\pi$
$67$	5.37994	0.657265	0.328632	+	0.944458i	$0.393412\pi$
$68$	0	0
$69$	−3.00352	−0.361581
$70$	0	0
$71$	6.48606	0.769754	0.384877	−	0.922968i	$-0.374244\pi$
$71$	6.48606	0.769754	0.384877	+	0.922968i	$0.374244\pi$
$72$	0	0
$73$	14.4644	1.69293	0.846466	−	0.532442i	$-0.178726\pi$
$73$	14.4644	1.69293	0.846466	+	0.532442i	$0.178726\pi$
$74$	0	0
$75$	16.7395	1.93291
$76$	0	0
$77$	−4.50474	−0.513362
$78$	0	0
$79$	−16.5586	−1.86299	−0.931494	−	0.363757i	$-0.881494\pi$
$79$	−16.5586	−1.86299	−0.931494	+	0.363757i	$0.881494\pi$
$80$	0	0
$81$	6.23341	0.692601
$82$	0	0
$83$	11.7606	1.29090	0.645448	−	0.763804i	$-0.276671\pi$
$83$	11.7606	1.29090	0.645448	+	0.763804i	$0.276671\pi$
$84$	0	0
$85$	−10.4451	−1.13293
$86$	0	0
$87$	12.4349	1.33316
$88$	0	0
$89$	7.92706	0.840266	0.420133	−	0.907462i	$-0.361983\pi$
$89$	7.92706	0.840266	0.420133	+	0.907462i	$0.361983\pi$
$90$	0	0
$91$	24.3081	2.54818
$92$	0	0
$93$	−10.5799	−1.09708
$94$	0	0
$95$	−3.26823	−0.335313
$96$	0	0
$97$	−2.72448	−0.276629	−0.138315	−	0.990388i	$-0.544169\pi$
$97$	−2.72448	−0.276629	−0.138315	+	0.990388i	$0.544169\pi$
$98$	0	0
$99$	−5.68132	−0.570994
$100$	0	0
$101$	10.3275	1.02762	0.513812	−	0.857903i	$-0.328233\pi$
$101$	10.3275	1.02762	0.513812	+	0.857903i	$0.328233\pi$
$102$	0	0
$103$	−11.9528	−1.17774	−0.588871	−	0.808227i	$-0.700427\pi$
$103$	−11.9528	−1.17774	−0.588871	+	0.808227i	$0.700427\pi$
$104$	0	0
$105$	−43.3785	−4.23331
$106$	0	0
$107$	20.0324	1.93660	0.968301	−	0.249786i	$-0.0803604\pi$
$107$	20.0324	1.93660	0.968301	+	0.249786i	$0.0803604\pi$
$108$	0	0
$109$	3.28509	0.314654	0.157327	−	0.987547i	$-0.449712\pi$
$109$	3.28509	0.314654	0.157327	+	0.987547i	$0.449712\pi$
$110$	0	0
$111$	−27.2865	−2.58992
$112$	0	0
$113$	−5.73871	−0.539853	−0.269926	−	0.962881i	$-0.586999\pi$
$113$	−5.73871	−0.539853	−0.269926	+	0.962881i	$0.586999\pi$
$114$	0	0
$115$	3.33158	0.310671
$116$	0	0
$117$	30.6571	2.83425
$118$	0	0
$119$	14.3970	1.31977
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−27.9623	−2.52127
$124$	0	0
$125$	−2.22670	−0.199162
$126$	0	0
$127$	−8.64208	−0.766860	−0.383430	−	0.923570i	$-0.625257\pi$
$127$	−8.64208	−0.766860	−0.383430	+	0.923570i	$0.625257\pi$
$128$	0	0
$129$	28.4151	2.50181
$130$	0	0
$131$	3.40471	0.297471	0.148735	−	0.988877i	$-0.452480\pi$
$131$	3.40471	0.297471	0.148735	+	0.988877i	$0.452480\pi$
$132$	0	0
$133$	4.50474	0.390610
$134$	0	0
$135$	−25.8198	−2.22222
$136$	0	0
$137$	−18.4467	−1.57600	−0.788002	−	0.615672i	$-0.788885\pi$
$137$	−18.4467	−1.57600	−0.788002	+	0.615672i	$0.788885\pi$
$138$	0	0
$139$	18.9534	1.60761	0.803803	−	0.594896i	$-0.202807\pi$
$139$	18.9534	1.60761	0.803803	+	0.594896i	$0.202807\pi$
$140$	0	0
$141$	−30.4131	−2.56124
$142$	0	0
$143$	−5.39613	−0.451247
$144$	0	0
$145$	−13.7931	−1.14545
$146$	0	0
$147$	39.1656	3.23032
$148$	0	0
$149$	−3.63702	−0.297956	−0.148978	−	0.988840i	$-0.547598\pi$
$149$	−3.63702	−0.297956	−0.148978	+	0.988840i	$0.547598\pi$
$150$	0	0
$151$	−14.1036	−1.14774	−0.573869	−	0.818947i	$-0.694558\pi$
$151$	−14.1036	−1.14774	−0.573869	+	0.818947i	$0.694558\pi$
$152$	0	0
$153$	18.1573	1.46793
$154$	0	0
$155$	11.7354	0.942613
$156$	0	0
$157$	5.99943	0.478807	0.239403	−	0.970920i	$-0.423048\pi$
$157$	5.99943	0.478807	0.239403	+	0.970920i	$0.423048\pi$
$158$	0	0
$159$	−14.6126	−1.15885
$160$	0	0
$161$	−4.59206	−0.361905
$162$	0	0
$163$	0.872118	0.0683096	0.0341548	−	0.999417i	$-0.489126\pi$
$163$	0.872118	0.0683096	0.0341548	+	0.999417i	$0.489126\pi$
$164$	0	0
$165$	9.62953	0.749658
$166$	0	0
$167$	0.735620	0.0569239	0.0284620	−	0.999595i	$-0.490939\pi$
$167$	0.735620	0.0569239	0.0284620	+	0.999595i	$0.490939\pi$
$168$	0	0
$169$	16.1182	1.23986
$170$	0	0
$171$	5.68132	0.434461
$172$	0	0
$173$	−9.13245	−0.694327	−0.347164	−	0.937805i	$-0.612855\pi$
$173$	−9.13245	−0.694327	−0.347164	+	0.937805i	$0.612855\pi$
$174$	0	0
$175$	25.5928	1.93464
$176$	0	0
$177$	−6.34687	−0.477060
$178$	0	0
$179$	−12.4332	−0.929303	−0.464652	−	0.885494i	$-0.653820\pi$
$179$	−12.4332	−0.929303	−0.464652	+	0.885494i	$0.653820\pi$
$180$	0	0
$181$	0.710184	0.0527875	0.0263938	−	0.999652i	$-0.491598\pi$
$181$	0.710184	0.0527875	0.0263938	+	0.999652i	$0.491598\pi$
$182$	0	0
$183$	18.8202	1.39123
$184$	0	0
$185$	30.2669	2.22526
$186$	0	0
$187$	−3.19596	−0.233712
$188$	0	0
$189$	35.5886	2.58869
$190$	0	0
$191$	14.7000	1.06366	0.531828	−	0.846852i	$-0.321505\pi$
$191$	14.7000	1.06366	0.531828	+	0.846852i	$0.321505\pi$
$192$	0	0
$193$	−5.13566	−0.369673	−0.184836	−	0.982769i	$-0.559176\pi$
$193$	−5.13566	−0.369673	−0.184836	+	0.982769i	$0.559176\pi$
$194$	0	0
$195$	−51.9622	−3.72109
$196$	0	0
$197$	5.12832	0.365378	0.182689	−	0.983171i	$-0.441520\pi$
$197$	5.12832	0.365378	0.182689	+	0.983171i	$0.441520\pi$
$198$	0	0
$199$	9.98380	0.707733	0.353866	−	0.935296i	$-0.384867\pi$
$199$	9.98380	0.707733	0.353866	+	0.935296i	$0.384867\pi$
$200$	0	0
$201$	15.8515	1.11808
$202$	0	0
$203$	19.0116	1.33435
$204$	0	0
$205$	31.0165	2.16628
$206$	0	0
$207$	−5.79144	−0.402533
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	17.8722	1.23037	0.615186	−	0.788382i	$-0.289081\pi$
$211$	17.8722	1.23037	0.615186	+	0.788382i	$0.289081\pi$
$212$	0	0
$213$	19.1106	1.30944
$214$	0	0
$215$	−31.5187	−2.14956
$216$	0	0
$217$	−16.1755	−1.09806
$218$	0	0
$219$	42.6181	2.87986
$220$	0	0
$221$	17.2458	1.16008
$222$	0	0
$223$	13.5866	0.909824	0.454912	−	0.890536i	$-0.349671\pi$
$223$	13.5866	0.909824	0.454912	+	0.890536i	$0.349671\pi$
$224$	0	0
$225$	32.2774	2.15182
$226$	0	0
$227$	−5.15739	−0.342308	−0.171154	−	0.985244i	$-0.554750\pi$
$227$	−5.15739	−0.342308	−0.171154	+	0.985244i	$0.554750\pi$
$228$	0	0
$229$	−17.6083	−1.16359	−0.581794	−	0.813336i	$-0.697649\pi$
$229$	−17.6083	−1.16359	−0.581794	+	0.813336i	$0.697649\pi$
$230$	0	0
$231$	−13.2728	−0.873286
$232$	0	0
$233$	−22.8393	−1.49625	−0.748125	−	0.663558i	$-0.769046\pi$
$233$	−22.8393	−1.49625	−0.748125	+	0.663558i	$0.769046\pi$
$234$	0	0
$235$	33.7349	2.20062
$236$	0	0
$237$	−48.7884	−3.16914
$238$	0	0
$239$	12.9219	0.835848	0.417924	−	0.908482i	$-0.362758\pi$
$239$	12.9219	0.835848	0.417924	+	0.908482i	$0.362758\pi$
$240$	0	0
$241$	−14.5380	−0.936477	−0.468238	−	0.883602i	$-0.655111\pi$
$241$	−14.5380	−0.936477	−0.468238	+	0.883602i	$0.655111\pi$
$242$	0	0
$243$	−5.33460	−0.342215
$244$	0	0
$245$	−43.4434	−2.77550
$246$	0	0
$247$	5.39613	0.343347
$248$	0	0
$249$	34.6516	2.19595
$250$	0	0
$251$	−22.5156	−1.42117	−0.710586	−	0.703611i	$-0.751570\pi$
$251$	−22.5156	−1.42117	−0.710586	+	0.703611i	$0.751570\pi$
$252$	0	0
$253$	1.01938	0.0640881
$254$	0	0
$255$	−30.7756	−1.92724
$256$	0	0
$257$	−8.75995	−0.546431	−0.273215	−	0.961953i	$-0.588087\pi$
$257$	−8.75995	−0.546431	−0.273215	+	0.961953i	$0.588087\pi$
$258$	0	0
$259$	−41.7181	−2.59224
$260$	0	0
$261$	23.9772	1.48415
$262$	0	0
$263$	5.97342	0.368337	0.184169	−	0.982895i	$-0.441041\pi$
$263$	5.97342	0.368337	0.184169	+	0.982895i	$0.441041\pi$
$264$	0	0
$265$	16.2086	0.995688
$266$	0	0
$267$	23.3563	1.42938
$268$	0	0
$269$	−13.5163	−0.824103	−0.412051	−	0.911161i	$-0.635188\pi$
$269$	−13.5163	−0.824103	−0.412051	+	0.911161i	$0.635188\pi$
$270$	0	0
$271$	7.13822	0.433616	0.216808	−	0.976214i	$-0.430435\pi$
$271$	7.13822	0.433616	0.216808	+	0.976214i	$0.430435\pi$
$272$	0	0
$273$	71.6217	4.33474
$274$	0	0
$275$	−5.68132	−0.342596
$276$	0	0
$277$	−5.01233	−0.301162	−0.150581	−	0.988598i	$-0.548114\pi$
$277$	−5.01233	−0.301162	−0.150581	+	0.988598i	$0.548114\pi$
$278$	0	0
$279$	−20.4003	−1.22133
$280$	0	0
$281$	−25.6480	−1.53003	−0.765016	−	0.644012i	$-0.777269\pi$
$281$	−25.6480	−1.53003	−0.765016	+	0.644012i	$0.777269\pi$
$282$	0	0
$283$	15.2620	0.907234	0.453617	−	0.891197i	$-0.350133\pi$
$283$	15.2620	0.907234	0.453617	+	0.891197i	$0.350133\pi$
$284$	0	0
$285$	−9.62953	−0.570404
$286$	0	0
$287$	−42.7513	−2.52353
$288$	0	0
$289$	−6.78582	−0.399166
$290$	0	0
$291$	−8.02744	−0.470577
$292$	0	0
$293$	−20.5905	−1.20291	−0.601456	−	0.798906i	$-0.705412\pi$
$293$	−20.5905	−1.20291	−0.601456	+	0.798906i	$0.705412\pi$
$294$	0	0
$295$	7.04010	0.409891
$296$	0	0
$297$	−7.90025	−0.458419
$298$	0	0
$299$	−5.50072	−0.318115
$300$	0	0
$301$	43.4436	2.50404
$302$	0	0
$303$	30.4290	1.74810
$304$	0	0
$305$	−20.8758	−1.19535
$306$	0	0
$307$	−32.1808	−1.83666	−0.918329	−	0.395818i	$-0.870461\pi$
$307$	−32.1808	−1.83666	−0.918329	+	0.395818i	$0.870461\pi$
$308$	0	0
$309$	−35.2178	−2.00347
$310$	0	0
$311$	−24.8839	−1.41103	−0.705517	−	0.708693i	$-0.749285\pi$
$311$	−24.8839	−1.41103	−0.705517	+	0.708693i	$0.749285\pi$
$312$	0	0
$313$	2.40539	0.135961	0.0679805	−	0.997687i	$-0.478344\pi$
$313$	2.40539	0.135961	0.0679805	+	0.997687i	$0.478344\pi$
$314$	0	0
$315$	−83.6432	−4.71276
$316$	0	0
$317$	−18.4338	−1.03535	−0.517673	−	0.855579i	$-0.673201\pi$
$317$	−18.4338	−1.03535	−0.517673	+	0.855579i	$0.673201\pi$
$318$	0	0
$319$	−4.22036	−0.236295
$320$	0	0
$321$	59.0235	3.29437
$322$	0	0
$323$	3.19596	0.177828
$324$	0	0
$325$	30.6571	1.70055
$326$	0	0
$327$	9.67921	0.535261
$328$	0	0
$329$	−46.4983	−2.56353
$330$	0	0
$331$	−30.0397	−1.65113	−0.825566	−	0.564306i	$-0.809144\pi$
$331$	−30.0397	−1.65113	−0.825566	+	0.564306i	$0.809144\pi$
$332$	0	0
$333$	−52.6143	−2.88325
$334$	0	0
$335$	−17.5829	−0.960655
$336$	0	0
$337$	1.02468	0.0558176	0.0279088	−	0.999610i	$-0.491115\pi$
$337$	1.02468	0.0558176	0.0279088	+	0.999610i	$0.491115\pi$
$338$	0	0
$339$	−16.9086	−0.918348
$340$	0	0
$341$	3.59076	0.194451
$342$	0	0
$343$	28.3468	1.53058
$344$	0	0
$345$	9.81619	0.528486
$346$	0	0
$347$	1.70510	0.0915345	0.0457673	−	0.998952i	$-0.485427\pi$
$347$	1.70510	0.0915345	0.0457673	+	0.998952i	$0.485427\pi$
$348$	0	0
$349$	−25.7500	−1.37837	−0.689183	−	0.724587i	$-0.742031\pi$
$349$	−25.7500	−1.37837	−0.689183	+	0.724587i	$0.742031\pi$
$350$	0	0
$351$	42.6307	2.27546
$352$	0	0
$353$	−6.23845	−0.332039	−0.166020	−	0.986122i	$-0.553092\pi$
$353$	−6.23845	−0.332039	−0.166020	+	0.986122i	$0.553092\pi$
$354$	0	0
$355$	−21.1979	−1.12507
$356$	0	0
$357$	42.4193	2.24507
$358$	0	0
$359$	−16.3958	−0.865335	−0.432668	−	0.901554i	$-0.642428\pi$
$359$	−16.3958	−0.865335	−0.432668	+	0.901554i	$0.642428\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.94641	0.154646
$364$	0	0
$365$	−47.2730	−2.47438
$366$	0	0
$367$	−21.9944	−1.14810	−0.574048	−	0.818821i	$-0.694628\pi$
$367$	−21.9944	−1.14810	−0.574048	+	0.818821i	$0.694628\pi$
$368$	0	0
$369$	−53.9174	−2.80683
$370$	0	0
$371$	−22.3410	−1.15989
$372$	0	0
$373$	31.0023	1.60524	0.802618	−	0.596493i	$-0.203440\pi$
$373$	31.0023	1.60524	0.802618	+	0.596493i	$0.203440\pi$
$374$	0	0
$375$	−6.56076	−0.338796
$376$	0	0
$377$	22.7736	1.17290
$378$	0	0
$379$	−24.7850	−1.27312	−0.636559	−	0.771228i	$-0.719643\pi$
$379$	−24.7850	−1.27312	−0.636559	+	0.771228i	$0.719643\pi$
$380$	0	0
$381$	−25.4631	−1.30451
$382$	0	0
$383$	26.4303	1.35052	0.675261	−	0.737578i	$-0.264031\pi$
$383$	26.4303	1.35052	0.675261	+	0.737578i	$0.264031\pi$
$384$	0	0
$385$	14.7225	0.750329
$386$	0	0
$387$	54.7904	2.78515
$388$	0	0
$389$	31.4198	1.59304	0.796522	−	0.604609i	$-0.206671\pi$
$389$	31.4198	1.59304	0.796522	+	0.604609i	$0.206671\pi$
$390$	0	0
$391$	−3.25791	−0.164760
$392$	0	0
$393$	10.0317	0.506030
$394$	0	0
$395$	54.1173	2.72294
$396$	0	0
$397$	−7.38831	−0.370809	−0.185404	−	0.982662i	$-0.559359\pi$
$397$	−7.38831	−0.370809	−0.185404	+	0.982662i	$0.559359\pi$
$398$	0	0
$399$	13.2728	0.664471
$400$	0	0
$401$	18.5903	0.928355	0.464178	−	0.885742i	$-0.346350\pi$
$401$	18.5903	0.928355	0.464178	+	0.885742i	$0.346350\pi$
$402$	0	0
$403$	−19.3762	−0.965198
$404$	0	0
$405$	−20.3722	−1.01230
$406$	0	0
$407$	9.26094	0.459048
$408$	0	0
$409$	9.10092	0.450011	0.225006	−	0.974357i	$-0.427760\pi$
$409$	9.10092	0.450011	0.225006	+	0.974357i	$0.427760\pi$
$410$	0	0
$411$	−54.3514	−2.68096
$412$	0	0
$413$	−9.70367	−0.477486
$414$	0	0
$415$	−38.4364	−1.88677
$416$	0	0
$417$	55.8444	2.73471
$418$	0	0
$419$	2.42443	0.118441	0.0592205	−	0.998245i	$-0.481138\pi$
$419$	2.42443	0.118441	0.0592205	+	0.998245i	$0.481138\pi$
$420$	0	0
$421$	−8.88486	−0.433022	−0.216511	−	0.976280i	$-0.569468\pi$
$421$	−8.88486	−0.433022	−0.216511	+	0.976280i	$0.569468\pi$
$422$	0	0
$423$	−58.6430	−2.85132
$424$	0	0
$425$	18.1573	0.880757
$426$	0	0
$427$	28.7740	1.39247
$428$	0	0
$429$	−15.8992	−0.767620
$430$	0	0
$431$	−11.7567	−0.566299	−0.283150	−	0.959076i	$-0.591379\pi$
$431$	−11.7567	−0.566299	−0.283150	+	0.959076i	$0.591379\pi$
$432$	0	0
$433$	24.5262	1.17865	0.589326	−	0.807895i	$-0.299393\pi$
$433$	24.5262	1.17865	0.589326	+	0.807895i	$0.299393\pi$
$434$	0	0
$435$	−40.6401	−1.94854
$436$	0	0
$437$	−1.01938	−0.0487637
$438$	0	0
$439$	−15.6164	−0.745330	−0.372665	−	0.927966i	$-0.621556\pi$
$439$	−15.6164	−0.745330	−0.372665	+	0.927966i	$0.621556\pi$
$440$	0	0
$441$	75.5198	3.59618
$442$	0	0
$443$	32.0999	1.52511	0.762557	−	0.646921i	$-0.223944\pi$
$443$	32.0999	1.52511	0.762557	+	0.646921i	$0.223944\pi$
$444$	0	0
$445$	−25.9074	−1.22813
$446$	0	0
$447$	−10.7161	−0.506856
$448$	0	0
$449$	−11.5155	−0.543450	−0.271725	−	0.962375i	$-0.587594\pi$
$449$	−11.5155	−0.543450	−0.271725	+	0.962375i	$0.587594\pi$
$450$	0	0
$451$	9.49030	0.446881
$452$	0	0
$453$	−41.5551	−1.95243
$454$	0	0
$455$	−79.4445	−3.72442
$456$	0	0
$457$	10.0593	0.470554	0.235277	−	0.971928i	$-0.424400\pi$
$457$	10.0593	0.470554	0.235277	+	0.971928i	$0.424400\pi$
$458$	0	0
$459$	25.2489	1.17852
$460$	0	0
$461$	−7.67074	−0.357262	−0.178631	−	0.983916i	$-0.557167\pi$
$461$	−7.67074	−0.357262	−0.178631	+	0.983916i	$0.557167\pi$
$462$	0	0
$463$	21.7119	1.00904	0.504518	−	0.863401i	$-0.331670\pi$
$463$	21.7119	1.00904	0.504518	+	0.863401i	$0.331670\pi$
$464$	0	0
$465$	34.5774	1.60349
$466$	0	0
$467$	−17.0457	−0.788783	−0.394391	−	0.918943i	$-0.629045\pi$
$467$	−17.0457	−0.788783	−0.394391	+	0.918943i	$0.629045\pi$
$468$	0	0
$469$	24.2352	1.11908
$470$	0	0
$471$	17.6768	0.814503
$472$	0	0
$473$	−9.64397	−0.443430
$474$	0	0
$475$	5.68132	0.260677
$476$	0	0
$477$	−28.1762	−1.29010
$478$	0	0
$479$	22.6568	1.03521	0.517607	−	0.855618i	$-0.326823\pi$
$479$	22.6568	1.03521	0.517607	+	0.855618i	$0.326823\pi$
$480$	0	0
$481$	−49.9732	−2.27858
$482$	0	0
$483$	−13.5301	−0.615639
$484$	0	0
$485$	8.90423	0.404320
$486$	0	0
$487$	6.82492	0.309267	0.154633	−	0.987972i	$-0.450580\pi$
$487$	6.82492	0.309267	0.154633	+	0.987972i	$0.450580\pi$
$488$	0	0
$489$	2.56962	0.116202
$490$	0	0
$491$	26.6416	1.20232	0.601160	−	0.799129i	$-0.294706\pi$
$491$	26.6416	1.20232	0.601160	+	0.799129i	$0.294706\pi$
$492$	0	0
$493$	13.4881	0.607474
$494$	0	0
$495$	18.5678	0.834562
$496$	0	0
$497$	29.2180	1.31061
$498$	0	0
$499$	25.7582	1.15309	0.576547	−	0.817064i	$-0.304400\pi$
$499$	25.7582	1.15309	0.576547	+	0.817064i	$0.304400\pi$
$500$	0	0
$501$	2.16743	0.0968338
$502$	0	0
$503$	34.2458	1.52694	0.763472	−	0.645841i	$-0.223493\pi$
$503$	34.2458	1.52694	0.763472	+	0.645841i	$0.223493\pi$
$504$	0	0
$505$	−33.7526	−1.50197
$506$	0	0
$507$	47.4907	2.10914
$508$	0	0
$509$	9.26071	0.410474	0.205237	−	0.978712i	$-0.434203\pi$
$509$	9.26071	0.410474	0.205237	+	0.978712i	$0.434203\pi$
$510$	0	0
$511$	65.1584	2.88244
$512$	0	0
$513$	7.90025	0.348805
$514$	0	0
$515$	39.0644	1.72138
$516$	0	0
$517$	10.3221	0.453965
$518$	0	0
$519$	−26.9079	−1.18113
$520$	0	0
$521$	−28.0560	−1.22916	−0.614579	−	0.788855i	$-0.710674\pi$
$521$	−28.0560	−1.22916	−0.614579	+	0.788855i	$0.710674\pi$
$522$	0	0
$523$	−43.2887	−1.89288	−0.946441	−	0.322877i	$-0.895350\pi$
$523$	−43.2887	−1.89288	−0.946441	+	0.322877i	$0.895350\pi$
$524$	0	0
$525$	75.4069	3.29103
$526$	0	0
$527$	−11.4759	−0.499900
$528$	0	0
$529$	−21.9609	−0.954820
$530$	0	0
$531$	−12.2381	−0.531090
$532$	0	0
$533$	−51.2109	−2.21819
$534$	0	0
$535$	−65.4703	−2.83053
$536$	0	0
$537$	−36.6334	−1.58085
$538$	0	0
$539$	−13.2927	−0.572555
$540$	0	0
$541$	−44.9827	−1.93396	−0.966979	−	0.254858i	$-0.917971\pi$
$541$	−44.9827	−1.93396	−0.966979	+	0.254858i	$0.917971\pi$
$542$	0	0
$543$	2.09249	0.0897974
$544$	0	0
$545$	−10.7364	−0.459898
$546$	0	0
$547$	7.53568	0.322202	0.161101	−	0.986938i	$-0.448495\pi$
$547$	7.53568	0.322202	0.161101	+	0.986938i	$0.448495\pi$
$548$	0	0
$549$	36.2895	1.54880
$550$	0	0
$551$	4.22036	0.179793
$552$	0	0
$553$	−74.5921	−3.17198
$554$	0	0
$555$	89.1785	3.78542
$556$	0	0
$557$	12.6924	0.537794	0.268897	−	0.963169i	$-0.413341\pi$
$557$	12.6924	0.537794	0.268897	+	0.963169i	$0.413341\pi$
$558$	0	0
$559$	52.0401	2.20106
$560$	0	0
$561$	−9.41661	−0.397569
$562$	0	0
$563$	13.6269	0.574305	0.287152	−	0.957885i	$-0.407291\pi$
$563$	13.6269	0.574305	0.287152	+	0.957885i	$0.407291\pi$
$564$	0	0
$565$	18.7554	0.789046
$566$	0	0
$567$	28.0799	1.17924
$568$	0	0
$569$	4.90870	0.205783	0.102892	−	0.994693i	$-0.467190\pi$
$569$	4.90870	0.205783	0.102892	+	0.994693i	$0.467190\pi$
$570$	0	0
$571$	−15.3597	−0.642782	−0.321391	−	0.946947i	$-0.604150\pi$
$571$	−15.3597	−0.642782	−0.321391	+	0.946947i	$0.604150\pi$
$572$	0	0
$573$	43.3123	1.80940
$574$	0	0
$575$	−5.79144	−0.241520
$576$	0	0
$577$	21.4960	0.894892	0.447446	−	0.894311i	$-0.352334\pi$
$577$	21.4960	0.894892	0.447446	+	0.894311i	$0.352334\pi$
$578$	0	0
$579$	−15.1318	−0.628854
$580$	0	0
$581$	52.9785	2.19792
$582$	0	0
$583$	4.95945	0.205400
$584$	0	0
$585$	−100.194	−4.14253
$586$	0	0
$587$	35.6383	1.47095	0.735475	−	0.677552i	$-0.236959\pi$
$587$	35.6383	1.47095	0.735475	+	0.677552i	$0.236959\pi$
$588$	0	0
$589$	−3.59076	−0.147955
$590$	0	0
$591$	15.1101	0.621547
$592$	0	0
$593$	−42.3396	−1.73868	−0.869340	−	0.494215i	$-0.835455\pi$
$593$	−42.3396	−1.73868	−0.869340	+	0.494215i	$0.835455\pi$
$594$	0	0
$595$	−47.0526	−1.92897
$596$	0	0
$597$	29.4163	1.20393
$598$	0	0
$599$	5.73689	0.234403	0.117201	−	0.993108i	$-0.462608\pi$
$599$	5.73689	0.234403	0.117201	+	0.993108i	$0.462608\pi$
$600$	0	0
$601$	18.2015	0.742453	0.371226	−	0.928542i	$-0.378937\pi$
$601$	18.2015	0.742453	0.371226	+	0.928542i	$0.378937\pi$
$602$	0	0
$603$	30.5652	1.24471
$604$	0	0
$605$	−3.26823	−0.132872
$606$	0	0
$607$	−7.56532	−0.307067	−0.153533	−	0.988143i	$-0.549065\pi$
$607$	−7.56532	−0.307067	−0.153533	+	0.988143i	$0.549065\pi$
$608$	0	0
$609$	56.0160	2.26988
$610$	0	0
$611$	−55.6993	−2.25335
$612$	0	0
$613$	−14.4648	−0.584230	−0.292115	−	0.956383i	$-0.594359\pi$
$613$	−14.4648	−0.584230	−0.292115	+	0.956383i	$0.594359\pi$
$614$	0	0
$615$	91.3871	3.68509
$616$	0	0
$617$	13.4446	0.541259	0.270630	−	0.962684i	$-0.412768\pi$
$617$	13.4446	0.541259	0.270630	+	0.962684i	$0.412768\pi$
$618$	0	0
$619$	−31.7907	−1.27777	−0.638887	−	0.769300i	$-0.720605\pi$
$619$	−31.7907	−1.27777	−0.638887	+	0.769300i	$0.720605\pi$
$620$	0	0
$621$	−8.05339	−0.323171
$622$	0	0
$623$	35.7093	1.43066
$624$	0	0
$625$	−21.1292	−0.845169
$626$	0	0
$627$	−2.94641	−0.117668
$628$	0	0
$629$	−29.5976	−1.18013
$630$	0	0
$631$	6.35280	0.252901	0.126451	−	0.991973i	$-0.459641\pi$
$631$	6.35280	0.252901	0.126451	+	0.991973i	$0.459641\pi$
$632$	0	0
$633$	52.6588	2.09300
$634$	0	0
$635$	28.2443	1.12084
$636$	0	0
$637$	71.7289	2.84200
$638$	0	0
$639$	36.8494	1.45774
$640$	0	0
$641$	−9.26589	−0.365981	−0.182990	−	0.983115i	$-0.558578\pi$
$641$	−9.26589	−0.365981	−0.182990	+	0.983115i	$0.558578\pi$
$642$	0	0
$643$	23.8530	0.940670	0.470335	−	0.882488i	$-0.344133\pi$
$643$	23.8530	0.940670	0.470335	+	0.882488i	$0.344133\pi$
$644$	0	0
$645$	−92.8669	−3.65663
$646$	0	0
$647$	11.4001	0.448182	0.224091	−	0.974568i	$-0.428059\pi$
$647$	11.4001	0.448182	0.224091	+	0.974568i	$0.428059\pi$
$648$	0	0
$649$	2.15410	0.0845559
$650$	0	0
$651$	−47.6595	−1.86792
$652$	0	0
$653$	−14.1919	−0.555371	−0.277685	−	0.960672i	$-0.589567\pi$
$653$	−14.1919	−0.555371	−0.277685	+	0.960672i	$0.589567\pi$
$654$	0	0
$655$	−11.1274	−0.434782
$656$	0	0
$657$	82.1770	3.20603
$658$	0	0
$659$	−1.69991	−0.0662191	−0.0331095	−	0.999452i	$-0.510541\pi$
$659$	−1.69991	−0.0662191	−0.0331095	+	0.999452i	$0.510541\pi$
$660$	0	0
$661$	−17.4577	−0.679025	−0.339513	−	0.940601i	$-0.610262\pi$
$661$	−17.4577	−0.679025	−0.339513	+	0.940601i	$0.610262\pi$
$662$	0	0
$663$	50.8132	1.97342
$664$	0	0
$665$	−14.7225	−0.570914
$666$	0	0
$667$	−4.30217	−0.166581
$668$	0	0
$669$	40.0316	1.54771
$670$	0	0
$671$	−6.38751	−0.246587
$672$	0	0
$673$	−13.7235	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$673$	−13.7235	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$674$	0	0
$675$	44.8838	1.72758
$676$	0	0
$677$	−15.7204	−0.604186	−0.302093	−	0.953278i	$-0.597685\pi$
$677$	−15.7204	−0.604186	−0.302093	+	0.953278i	$0.597685\pi$
$678$	0	0
$679$	−12.2731	−0.470998
$680$	0	0
$681$	−15.1958	−0.582303
$682$	0	0
$683$	−30.7965	−1.17840	−0.589198	−	0.807989i	$-0.700556\pi$
$683$	−30.7965	−1.17840	−0.589198	+	0.807989i	$0.700556\pi$
$684$	0	0
$685$	60.2879	2.30348
$686$	0	0
$687$	−51.8812	−1.97939
$688$	0	0
$689$	−26.7618	−1.01954
$690$	0	0
$691$	5.21888	0.198536	0.0992678	−	0.995061i	$-0.468350\pi$
$691$	5.21888	0.198536	0.0992678	+	0.995061i	$0.468350\pi$
$692$	0	0
$693$	−25.5928	−0.972192
$694$	0	0
$695$	−61.9440	−2.34967
$696$	0	0
$697$	−30.3306	−1.14885
$698$	0	0
$699$	−67.2938	−2.54528
$700$	0	0
$701$	2.56839	0.0970069	0.0485034	−	0.998823i	$-0.484555\pi$
$701$	2.56839	0.0970069	0.0485034	+	0.998823i	$0.484555\pi$
$702$	0	0
$703$	−9.26094	−0.349283
$704$	0	0
$705$	99.3969	3.74350
$706$	0	0
$707$	46.5226	1.74966
$708$	0	0
$709$	31.0388	1.16569	0.582843	−	0.812585i	$-0.301940\pi$
$709$	31.0388	1.16569	0.582843	+	0.812585i	$0.301940\pi$
$710$	0	0
$711$	−94.0746	−3.52807
$712$	0	0
$713$	3.66037	0.137082
$714$	0	0
$715$	17.6358	0.659540
$716$	0	0
$717$	38.0732	1.42187
$718$	0	0
$719$	−36.3524	−1.35571	−0.677857	−	0.735193i	$-0.737091\pi$
$719$	−36.3524	−1.35571	−0.677857	+	0.735193i	$0.737091\pi$
$720$	0	0
$721$	−53.8441	−2.00526
$722$	0	0
$723$	−42.8350	−1.59305
$724$	0	0
$725$	23.9772	0.890491
$726$	0	0
$727$	42.2802	1.56808	0.784042	−	0.620707i	$-0.213154\pi$
$727$	42.2802	1.56808	0.784042	+	0.620707i	$0.213154\pi$
$728$	0	0
$729$	−34.4181	−1.27474
$730$	0	0
$731$	30.8218	1.13998
$732$	0	0
$733$	3.52920	0.130354	0.0651770	−	0.997874i	$-0.479239\pi$
$733$	3.52920	0.130354	0.0651770	+	0.997874i	$0.479239\pi$
$734$	0	0
$735$	−128.002	−4.72143
$736$	0	0
$737$	−5.37994	−0.198173
$738$	0	0
$739$	−16.2076	−0.596207	−0.298103	−	0.954534i	$-0.596354\pi$
$739$	−16.2076	−0.596207	−0.298103	+	0.954534i	$0.596354\pi$
$740$	0	0
$741$	15.8992	0.584071
$742$	0	0
$743$	25.3207	0.928928	0.464464	−	0.885592i	$-0.346247\pi$
$743$	25.3207	0.928928	0.464464	+	0.885592i	$0.346247\pi$
$744$	0	0
$745$	11.8866	0.435492
$746$	0	0
$747$	66.8158	2.44466
$748$	0	0
$749$	90.2406	3.29732
$750$	0	0
$751$	21.1631	0.772253	0.386127	−	0.922446i	$-0.373813\pi$
$751$	21.1631	0.772253	0.386127	+	0.922446i	$0.373813\pi$
$752$	0	0
$753$	−66.3401	−2.41757
$754$	0	0
$755$	46.0939	1.67753
$756$	0	0
$757$	−48.5117	−1.76319	−0.881595	−	0.472007i	$-0.843530\pi$
$757$	−48.5117	−1.76319	−0.881595	+	0.472007i	$0.843530\pi$
$758$	0	0
$759$	3.00352	0.109021
$760$	0	0
$761$	−6.42681	−0.232972	−0.116486	−	0.993192i	$-0.537163\pi$
$761$	−6.42681	−0.232972	−0.116486	+	0.993192i	$0.537163\pi$
$762$	0	0
$763$	14.7985	0.535740
$764$	0	0
$765$	−59.3421	−2.14552
$766$	0	0
$767$	−11.6238	−0.419712
$768$	0	0
$769$	−6.22562	−0.224501	−0.112251	−	0.993680i	$-0.535806\pi$
$769$	−6.22562	−0.224501	−0.112251	+	0.993680i	$0.535806\pi$
$770$	0	0
$771$	−25.8104	−0.929539
$772$	0	0
$773$	−42.5482	−1.53035	−0.765177	−	0.643820i	$-0.777349\pi$
$773$	−42.5482	−1.53035	−0.765177	+	0.643820i	$0.777349\pi$
$774$	0	0
$775$	−20.4003	−0.732799
$776$	0	0
$777$	−122.919	−4.40968
$778$	0	0
$779$	−9.49030	−0.340025
$780$	0	0
$781$	−6.48606	−0.232090
$782$	0	0
$783$	33.3419	1.19154
$784$	0	0
$785$	−19.6075	−0.699822
$786$	0	0
$787$	29.8477	1.06396	0.531978	−	0.846758i	$-0.321449\pi$
$787$	29.8477	1.06396	0.531978	+	0.846758i	$0.321449\pi$
$788$	0	0
$789$	17.6001	0.626582
$790$	0	0
$791$	−25.8514	−0.919170
$792$	0	0
$793$	34.4678	1.22399
$794$	0	0
$795$	47.7572	1.69377
$796$	0	0
$797$	−0.403408	−0.0142894	−0.00714472	−	0.999974i	$-0.502274\pi$
$797$	−0.403408	−0.0142894	−0.00714472	+	0.999974i	$0.502274\pi$
$798$	0	0
$799$	−32.9890	−1.16707
$800$	0	0
$801$	45.0361	1.59127
$802$	0	0
$803$	−14.4644	−0.510438
$804$	0	0
$805$	15.0079	0.528959
$806$	0	0
$807$	−39.8245	−1.40189
$808$	0	0
$809$	24.2101	0.851183	0.425592	−	0.904915i	$-0.360066\pi$
$809$	24.2101	0.851183	0.425592	+	0.904915i	$0.360066\pi$
$810$	0	0
$811$	−3.48323	−0.122313	−0.0611563	−	0.998128i	$-0.519479\pi$
$811$	−3.48323	−0.122313	−0.0611563	+	0.998128i	$0.519479\pi$
$812$	0	0
$813$	21.0321	0.737628
$814$	0	0
$815$	−2.85028	−0.0998410
$816$	0	0
$817$	9.64397	0.337400
$818$	0	0
$819$	138.102	4.82568
$820$	0	0
$821$	28.8355	1.00637	0.503183	−	0.864180i	$-0.332162\pi$
$821$	28.8355	1.00637	0.503183	+	0.864180i	$0.332162\pi$
$822$	0	0
$823$	−14.0923	−0.491227	−0.245613	−	0.969368i	$-0.578989\pi$
$823$	−14.0923	−0.491227	−0.245613	+	0.969368i	$0.578989\pi$
$824$	0	0
$825$	−16.7395	−0.582794
$826$	0	0
$827$	36.0724	1.25436	0.627180	−	0.778874i	$-0.284209\pi$
$827$	36.0724	1.25436	0.627180	+	0.778874i	$0.284209\pi$
$828$	0	0
$829$	−12.6519	−0.439418	−0.219709	−	0.975565i	$-0.570511\pi$
$829$	−12.6519	−0.439418	−0.219709	+	0.975565i	$0.570511\pi$
$830$	0	0
$831$	−14.7684	−0.512309
$832$	0	0
$833$	42.4828	1.47194
$834$	0	0
$835$	−2.40417	−0.0831998
$836$	0	0
$837$	−28.3679	−0.980539
$838$	0	0
$839$	46.9119	1.61958	0.809789	−	0.586721i	$-0.199582\pi$
$839$	46.9119	1.61958	0.809789	+	0.586721i	$0.199582\pi$
$840$	0	0
$841$	−11.1886	−0.385813
$842$	0	0
$843$	−75.5694	−2.60275
$844$	0	0
$845$	−52.6779	−1.81217
$846$	0	0
$847$	4.50474	0.154785
$848$	0	0
$849$	44.9682	1.54330
$850$	0	0
$851$	9.44045	0.323614
$852$	0	0
$853$	8.48291	0.290449	0.145225	−	0.989399i	$-0.453610\pi$
$853$	8.48291	0.290449	0.145225	+	0.989399i	$0.453610\pi$
$854$	0	0
$855$	−18.5678	−0.635007
$856$	0	0
$857$	25.4016	0.867702	0.433851	−	0.900985i	$-0.357154\pi$
$857$	25.4016	0.867702	0.433851	+	0.900985i	$0.357154\pi$
$858$	0	0
$859$	22.6944	0.774323	0.387162	−	0.922012i	$-0.373456\pi$
$859$	22.6944	0.774323	0.387162	+	0.922012i	$0.373456\pi$
$860$	0	0
$861$	−125.963	−4.29280
$862$	0	0
$863$	−17.4235	−0.593103	−0.296552	−	0.955017i	$-0.595837\pi$
$863$	−17.4235	−0.593103	−0.296552	+	0.955017i	$0.595837\pi$
$864$	0	0
$865$	29.8469	1.01483
$866$	0	0
$867$	−19.9938	−0.679025
$868$	0	0
$869$	16.5586	0.561712
$870$	0	0
$871$	29.0309	0.983673
$872$	0	0
$873$	−15.4787	−0.523873
$874$	0	0
$875$	−10.0307	−0.339099
$876$	0	0
$877$	54.7462	1.84865	0.924325	−	0.381607i	$-0.124629\pi$
$877$	54.7462	1.84865	0.924325	+	0.381607i	$0.124629\pi$
$878$	0	0
$879$	−60.6681	−2.04628
$880$	0	0
$881$	−19.3479	−0.651848	−0.325924	−	0.945396i	$-0.605675\pi$
$881$	−19.3479	−0.651848	−0.325924	+	0.945396i	$0.605675\pi$
$882$	0	0
$883$	17.2313	0.579879	0.289939	−	0.957045i	$-0.406365\pi$
$883$	17.2313	0.579879	0.289939	+	0.957045i	$0.406365\pi$
$884$	0	0
$885$	20.7430	0.697268
$886$	0	0
$887$	−9.30136	−0.312309	−0.156155	−	0.987733i	$-0.549910\pi$
$887$	−9.30136	−0.312309	−0.156155	+	0.987733i	$0.549910\pi$
$888$	0	0
$889$	−38.9303	−1.30568
$890$	0	0
$891$	−6.23341	−0.208827
$892$	0	0
$893$	−10.3221	−0.345415
$894$	0	0
$895$	40.6346	1.35827
$896$	0	0
$897$	−16.2074	−0.541148
$898$	0	0
$899$	−15.1543	−0.505425
$900$	0	0
$901$	−15.8502	−0.528048
$902$	0	0
$903$	128.002	4.25965
$904$	0	0
$905$	−2.32104	−0.0771541
$906$	0	0
$907$	−29.8731	−0.991921	−0.495961	−	0.868345i	$-0.665184\pi$
$907$	−29.8731	−0.991921	−0.495961	+	0.868345i	$0.665184\pi$
$908$	0	0
$909$	58.6738	1.94609
$910$	0	0
$911$	11.3876	0.377288	0.188644	−	0.982046i	$-0.439591\pi$
$911$	11.3876	0.377288	0.188644	+	0.982046i	$0.439591\pi$
$912$	0	0
$913$	−11.7606	−0.389220
$914$	0	0
$915$	−61.5087	−2.03342
$916$	0	0
$917$	15.3373	0.506483
$918$	0	0
$919$	−36.8388	−1.21520	−0.607599	−	0.794244i	$-0.707867\pi$
$919$	−36.8388	−1.21520	−0.607599	+	0.794244i	$0.707867\pi$
$920$	0	0
$921$	−94.8179	−3.12436
$922$	0	0
$923$	34.9996	1.15203
$924$	0	0
$925$	−52.6143	−1.72995
$926$	0	0
$927$	−67.9075	−2.23038
$928$	0	0
$929$	−15.8374	−0.519608	−0.259804	−	0.965661i	$-0.583658\pi$
$929$	−15.8374	−0.519608	−0.259804	+	0.965661i	$0.583658\pi$
$930$	0	0
$931$	13.2927	0.435649
$932$	0	0
$933$	−73.3180	−2.40032
$934$	0	0
$935$	10.4451	0.341592
$936$	0	0
$937$	28.7080	0.937849	0.468924	−	0.883238i	$-0.344642\pi$
$937$	28.7080	0.937849	0.468924	+	0.883238i	$0.344642\pi$
$938$	0	0
$939$	7.08727	0.231284
$940$	0	0
$941$	−28.5074	−0.929316	−0.464658	−	0.885490i	$-0.653823\pi$
$941$	−28.5074	−0.929316	−0.464658	+	0.885490i	$0.653823\pi$
$942$	0	0
$943$	9.67426	0.315037
$944$	0	0
$945$	−116.312	−3.78361
$946$	0	0
$947$	−22.3550	−0.726439	−0.363219	−	0.931704i	$-0.618322\pi$
$947$	−22.3550	−0.726439	−0.363219	+	0.931704i	$0.618322\pi$
$948$	0	0
$949$	78.0518	2.53367
$950$	0	0
$951$	−54.3135	−1.76124
$952$	0	0
$953$	32.7765	1.06173	0.530867	−	0.847455i	$-0.321866\pi$
$953$	32.7765	1.06173	0.530867	+	0.847455i	$0.321866\pi$
$954$	0	0
$955$	−48.0430	−1.55464
$956$	0	0
$957$	−12.4349	−0.401963
$958$	0	0
$959$	−83.0974	−2.68335
$960$	0	0
$961$	−18.1064	−0.584078
$962$	0	0
$963$	113.810	3.66748
$964$	0	0
$965$	16.7845	0.540313
$966$	0	0
$967$	−0.984453	−0.0316579	−0.0158289	−	0.999875i	$-0.505039\pi$
$967$	−0.984453	−0.0316579	−0.0158289	+	0.999875i	$0.505039\pi$
$968$	0	0
$969$	9.41661	0.302505
$970$	0	0
$971$	−31.2522	−1.00293	−0.501465	−	0.865178i	$-0.667205\pi$
$971$	−31.2522	−1.00293	−0.501465	+	0.865178i	$0.667205\pi$
$972$	0	0
$973$	85.3801	2.73716
$974$	0	0
$975$	90.3283	2.89282
$976$	0	0
$977$	−24.4218	−0.781322	−0.390661	−	0.920535i	$-0.627754\pi$
$977$	−24.4218	−0.781322	−0.390661	+	0.920535i	$0.627754\pi$
$978$	0	0
$979$	−7.92706	−0.253350
$980$	0	0
$981$	18.6636	0.595884
$982$	0	0
$983$	35.8832	1.14450	0.572248	−	0.820080i	$-0.306071\pi$
$983$	35.8832	1.14450	0.572248	+	0.820080i	$0.306071\pi$
$984$	0	0
$985$	−16.7605	−0.534034
$986$	0	0
$987$	−137.003	−4.36085
$988$	0	0
$989$	−9.83091	−0.312605
$990$	0	0
$991$	12.1779	0.386845	0.193422	−	0.981116i	$-0.438041\pi$
$991$	12.1779	0.386845	0.193422	+	0.981116i	$0.438041\pi$
$992$	0	0
$993$	−88.5092	−2.80875
$994$	0	0
$995$	−32.6293	−1.03442
$996$	0	0
$997$	−9.61391	−0.304476	−0.152238	−	0.988344i	$-0.548648\pi$
$997$	−9.61391	−0.304476	−0.152238	+	0.988344i	$0.548648\pi$
$998$	0	0
$999$	−73.1637	−2.31480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.y.1.6		6
4.3	odd	2		1672.2.a.h.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.h.1.1	✓	6	4.3	odd	2
3344.2.a.y.1.6		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)

\(f(q)\)	\(=\)	\(q+2.94641 q^{3} -3.26823 q^{5} +4.50474 q^{7} +5.68132 q^{9} +O(q^{10})\)	\(q+2.94641 q^{3} -3.26823 q^{5} +4.50474 q^{7} +5.68132 q^{9} -1.00000 q^{11} +5.39613 q^{13} -9.62953 q^{15} +3.19596 q^{17} +1.00000 q^{19} +13.2728 q^{21} -1.01938 q^{23} +5.68132 q^{25} +7.90025 q^{27} +4.22036 q^{29} -3.59076 q^{31} -2.94641 q^{33} -14.7225 q^{35} -9.26094 q^{37} +15.8992 q^{39} -9.49030 q^{41} +9.64397 q^{43} -18.5678 q^{45} -10.3221 q^{47} +13.2927 q^{49} +9.41661 q^{51} -4.95945 q^{53} +3.26823 q^{55} +2.94641 q^{57} -2.15410 q^{59} +6.38751 q^{61} +25.5928 q^{63} -17.6358 q^{65} +5.37994 q^{67} -3.00352 q^{69} +6.48606 q^{71} +14.4644 q^{73} +16.7395 q^{75} -4.50474 q^{77} -16.5586 q^{79} +6.23341 q^{81} +11.7606 q^{83} -10.4451 q^{85} +12.4349 q^{87} +7.92706 q^{89} +24.3081 q^{91} -10.5799 q^{93} -3.26823 q^{95} -2.72448 q^{97} -5.68132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Introduction

L-functions

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Varieties

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Database

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