LMFDB - Embedded newform 3344.2.a.z.1.4

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:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.870457$ 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)

$f(q)$	$=$	$q+1.87046 q^{3} +2.83301 q^{5} +3.83301 q^{7} +0.498611 q^{9} +O(q^{10})$	$q+1.87046 q^{3} +2.83301 q^{5} +3.83301 q^{7} +0.498611 q^{9} +1.00000 q^{11} +6.12876 q^{13} +5.29902 q^{15} -3.70874 q^{17} -1.00000 q^{19} +7.16948 q^{21} -1.01807 q^{23} +3.02594 q^{25} -4.67874 q^{27} -3.56805 q^{29} +5.29902 q^{31} +1.87046 q^{33} +10.8590 q^{35} +1.68789 q^{37} +11.4636 q^{39} -6.71110 q^{41} +3.59858 q^{43} +1.41257 q^{45} +6.48461 q^{47} +7.69196 q^{49} -6.93704 q^{51} -6.34969 q^{53} +2.83301 q^{55} -1.87046 q^{57} -10.8133 q^{59} -0.743693 q^{61} +1.91118 q^{63} +17.3628 q^{65} -2.07574 q^{67} -1.90426 q^{69} +2.65687 q^{71} -11.0837 q^{73} +5.65990 q^{75} +3.83301 q^{77} +1.81581 q^{79} -10.2472 q^{81} -12.1416 q^{83} -10.5069 q^{85} -6.67389 q^{87} -1.51341 q^{89} +23.4916 q^{91} +9.91160 q^{93} -2.83301 q^{95} -13.3901 q^{97} +0.498611 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * 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
$3$	1.87046	1.07991	0.539955	−	0.841694i	$-0.318441\pi$
$3$	1.87046	1.07991	0.539955	+	0.841694i	$0.318441\pi$
$4$	0	0
$5$	2.83301	1.26696	0.633480	−	0.773759i	$-0.281626\pi$
$5$	2.83301	1.26696	0.633480	+	0.773759i	$0.281626\pi$
$6$	0	0
$7$	3.83301	1.44874	0.724371	−	0.689411i	$-0.242130\pi$
$7$	3.83301	1.44874	0.724371	+	0.689411i	$0.242130\pi$
$8$	0	0
$9$	0.498611	0.166204
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.12876	1.69981	0.849906	−	0.526934i	$-0.176659\pi$
$13$	6.12876	1.69981	0.849906	+	0.526934i	$0.176659\pi$
$14$	0	0
$15$	5.29902	1.36820
$16$	0	0
$17$	−3.70874	−0.899502	−0.449751	−	0.893154i	$-0.648487\pi$
$17$	−3.70874	−0.899502	−0.449751	+	0.893154i	$0.648487\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.16948	1.56451
$22$	0	0
$23$	−1.01807	−0.212282	−0.106141	−	0.994351i	$-0.533850\pi$
$23$	−1.01807	−0.212282	−0.106141	+	0.994351i	$0.533850\pi$
$24$	0	0
$25$	3.02594	0.605188
$26$	0	0
$27$	−4.67874	−0.900424
$28$	0	0
$29$	−3.56805	−0.662571	−0.331285	−	0.943531i	$-0.607482\pi$
$29$	−3.56805	−0.662571	−0.331285	+	0.943531i	$0.607482\pi$
$30$	0	0
$31$	5.29902	0.951733	0.475866	−	0.879518i	$-0.342135\pi$
$31$	5.29902	0.951733	0.475866	+	0.879518i	$0.342135\pi$
$32$	0	0
$33$	1.87046	0.325605
$34$	0	0
$35$	10.8590	1.83550
$36$	0	0
$37$	1.68789	0.277488	0.138744	−	0.990328i	$-0.455693\pi$
$37$	1.68789	0.277488	0.138744	+	0.990328i	$0.455693\pi$
$38$	0	0
$39$	11.4636	1.83564
$40$	0	0
$41$	−6.71110	−1.04810	−0.524049	−	0.851688i	$-0.675579\pi$
$41$	−6.71110	−1.04810	−0.524049	+	0.851688i	$0.675579\pi$
$42$	0	0
$43$	3.59858	0.548778	0.274389	−	0.961619i	$-0.411524\pi$
$43$	3.59858	0.548778	0.274389	+	0.961619i	$0.411524\pi$
$44$	0	0
$45$	1.41257	0.210573
$46$	0	0
$47$	6.48461	0.945877	0.472939	−	0.881095i	$-0.343193\pi$
$47$	6.48461	0.945877	0.472939	+	0.881095i	$0.343193\pi$
$48$	0	0
$49$	7.69196	1.09885
$50$	0	0
$51$	−6.93704	−0.971380
$52$	0	0
$53$	−6.34969	−0.872197	−0.436098	−	0.899899i	$-0.643640\pi$
$53$	−6.34969	−0.872197	−0.436098	+	0.899899i	$0.643640\pi$
$54$	0	0
$55$	2.83301	0.382003
$56$	0	0
$57$	−1.87046	−0.247748
$58$	0	0
$59$	−10.8133	−1.40777	−0.703884	−	0.710315i	$-0.748553\pi$
$59$	−10.8133	−1.40777	−0.703884	+	0.710315i	$0.748553\pi$
$60$	0	0
$61$	−0.743693	−0.0952202	−0.0476101	−	0.998866i	$-0.515160\pi$
$61$	−0.743693	−0.0952202	−0.0476101	+	0.998866i	$0.515160\pi$
$62$	0	0
$63$	1.91118	0.240786
$64$	0	0
$65$	17.3628	2.15359
$66$	0	0
$67$	−2.07574	−0.253592	−0.126796	−	0.991929i	$-0.540469\pi$
$67$	−2.07574	−0.253592	−0.126796	+	0.991929i	$0.540469\pi$
$68$	0	0
$69$	−1.90426	−0.229245
$70$	0	0
$71$	2.65687	0.315313	0.157656	−	0.987494i	$-0.449606\pi$
$71$	2.65687	0.315313	0.157656	+	0.987494i	$0.449606\pi$
$72$	0	0
$73$	−11.0837	−1.29725	−0.648623	−	0.761110i	$-0.724655\pi$
$73$	−11.0837	−1.29725	−0.648623	+	0.761110i	$0.724655\pi$
$74$	0	0
$75$	5.65990	0.653548
$76$	0	0
$77$	3.83301	0.436812
$78$	0	0
$79$	1.81581	0.204295	0.102147	−	0.994769i	$-0.467429\pi$
$79$	1.81581	0.204295	0.102147	+	0.994769i	$0.467429\pi$
$80$	0	0
$81$	−10.2472	−1.13858
$82$	0	0
$83$	−12.1416	−1.33272	−0.666358	−	0.745632i	$-0.732148\pi$
$83$	−12.1416	−1.33272	−0.666358	+	0.745632i	$0.732148\pi$
$84$	0	0
$85$	−10.5069	−1.13963
$86$	0	0
$87$	−6.67389	−0.715516
$88$	0	0
$89$	−1.51341	−0.160421	−0.0802104	−	0.996778i	$-0.525559\pi$
$89$	−1.51341	−0.160421	−0.0802104	+	0.996778i	$0.525559\pi$
$90$	0	0
$91$	23.4916	2.46259
$92$	0	0
$93$	9.91160	1.02778
$94$	0	0
$95$	−2.83301	−0.290661
$96$	0	0
$97$	−13.3901	−1.35955	−0.679777	−	0.733419i	$-0.737923\pi$
$97$	−13.3901	−1.35955	−0.679777	+	0.733419i	$0.737923\pi$
$98$	0	0
$99$	0.498611	0.0501123
$100$	0	0
$101$	−18.4417	−1.83502	−0.917509	−	0.397715i	$-0.869803\pi$
$101$	−18.4417	−1.83502	−0.917509	+	0.397715i	$0.869803\pi$
$102$	0	0
$103$	13.4662	1.32686	0.663431	−	0.748237i	$-0.269100\pi$
$103$	13.4662	1.32686	0.663431	+	0.748237i	$0.269100\pi$
$104$	0	0
$105$	20.3112	1.98217
$106$	0	0
$107$	16.3720	1.58274	0.791370	−	0.611338i	$-0.209368\pi$
$107$	16.3720	1.58274	0.791370	+	0.611338i	$0.209368\pi$
$108$	0	0
$109$	7.60741	0.728658	0.364329	−	0.931270i	$-0.381298\pi$
$109$	7.60741	0.728658	0.364329	+	0.931270i	$0.381298\pi$
$110$	0	0
$111$	3.15713	0.299662
$112$	0	0
$113$	−10.0046	−0.941150	−0.470575	−	0.882360i	$-0.655953\pi$
$113$	−10.0046	−0.941150	−0.470575	+	0.882360i	$0.655953\pi$
$114$	0	0
$115$	−2.88420	−0.268953
$116$	0	0
$117$	3.05586	0.282515
$118$	0	0
$119$	−14.2156	−1.30315
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.5528	−1.13185
$124$	0	0
$125$	−5.59253	−0.500211
$126$	0	0
$127$	20.6361	1.83116	0.915578	−	0.402140i	$-0.131734\pi$
$127$	20.6361	1.83116	0.915578	+	0.402140i	$0.131734\pi$
$128$	0	0
$129$	6.73099	0.592630
$130$	0	0
$131$	−6.06512	−0.529912	−0.264956	−	0.964261i	$-0.585357\pi$
$131$	−6.06512	−0.529912	−0.264956	+	0.964261i	$0.585357\pi$
$132$	0	0
$133$	−3.83301	−0.332364
$134$	0	0
$135$	−13.2549	−1.14080
$136$	0	0
$137$	10.2035	0.871748	0.435874	−	0.900008i	$-0.356439\pi$
$137$	10.2035	0.871748	0.435874	+	0.900008i	$0.356439\pi$
$138$	0	0
$139$	21.7492	1.84474	0.922372	−	0.386304i	$-0.126248\pi$
$139$	21.7492	1.84474	0.922372	+	0.386304i	$0.126248\pi$
$140$	0	0
$141$	12.1292	1.02146
$142$	0	0
$143$	6.12876	0.512513
$144$	0	0
$145$	−10.1083	−0.839451
$146$	0	0
$147$	14.3875	1.18666
$148$	0	0
$149$	11.6216	0.952075	0.476038	−	0.879425i	$-0.342073\pi$
$149$	11.6216	0.952075	0.476038	+	0.879425i	$0.342073\pi$
$150$	0	0
$151$	−19.2317	−1.56506	−0.782528	−	0.622616i	$-0.786070\pi$
$151$	−19.2317	−1.56506	−0.782528	+	0.622616i	$0.786070\pi$
$152$	0	0
$153$	−1.84922	−0.149500
$154$	0	0
$155$	15.0122	1.20581
$156$	0	0
$157$	−5.03393	−0.401752	−0.200876	−	0.979617i	$-0.564379\pi$
$157$	−5.03393	−0.401752	−0.200876	+	0.979617i	$0.564379\pi$
$158$	0	0
$159$	−11.8768	−0.941893
$160$	0	0
$161$	−3.90227	−0.307542
$162$	0	0
$163$	−13.3585	−1.04632	−0.523158	−	0.852236i	$-0.675246\pi$
$163$	−13.3585	−1.04632	−0.523158	+	0.852236i	$0.675246\pi$
$164$	0	0
$165$	5.29902	0.412528
$166$	0	0
$167$	6.88619	0.532869	0.266435	−	0.963853i	$-0.414154\pi$
$167$	6.88619	0.532869	0.266435	+	0.963853i	$0.414154\pi$
$168$	0	0
$169$	24.5617	1.88936
$170$	0	0
$171$	−0.498611	−0.0381297
$172$	0	0
$173$	−14.3759	−1.09298	−0.546490	−	0.837465i	$-0.684036\pi$
$173$	−14.3759	−1.09298	−0.546490	+	0.837465i	$0.684036\pi$
$174$	0	0
$175$	11.5985	0.876761
$176$	0	0
$177$	−20.2258	−1.52026
$178$	0	0
$179$	2.40313	0.179619	0.0898094	−	0.995959i	$-0.471374\pi$
$179$	2.40313	0.179619	0.0898094	+	0.995959i	$0.471374\pi$
$180$	0	0
$181$	−6.94819	−0.516455	−0.258227	−	0.966084i	$-0.583138\pi$
$181$	−6.94819	−0.516455	−0.258227	+	0.966084i	$0.583138\pi$
$182$	0	0
$183$	−1.39105	−0.102829
$184$	0	0
$185$	4.78182	0.351566
$186$	0	0
$187$	−3.70874	−0.271210
$188$	0	0
$189$	−17.9337	−1.30448
$190$	0	0
$191$	−15.1886	−1.09901	−0.549504	−	0.835491i	$-0.685183\pi$
$191$	−15.1886	−1.09901	−0.549504	+	0.835491i	$0.685183\pi$
$192$	0	0
$193$	−10.9331	−0.786983	−0.393491	−	0.919328i	$-0.628733\pi$
$193$	−10.9331	−0.786983	−0.393491	+	0.919328i	$0.628733\pi$
$194$	0	0
$195$	32.4764	2.32569
$196$	0	0
$197$	−12.3440	−0.879470	−0.439735	−	0.898128i	$-0.644928\pi$
$197$	−12.3440	−0.879470	−0.439735	+	0.898128i	$0.644928\pi$
$198$	0	0
$199$	−12.9921	−0.920982	−0.460491	−	0.887664i	$-0.652327\pi$
$199$	−12.9921	−0.920982	−0.460491	+	0.887664i	$0.652327\pi$
$200$	0	0
$201$	−3.88258	−0.273856
$202$	0	0
$203$	−13.6764	−0.959894
$204$	0	0
$205$	−19.0126	−1.32790
$206$	0	0
$207$	−0.507620	−0.0352820
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	9.42821	0.649065	0.324532	−	0.945875i	$-0.394793\pi$
$211$	9.42821	0.649065	0.324532	+	0.945875i	$0.394793\pi$
$212$	0	0
$213$	4.96957	0.340509
$214$	0	0
$215$	10.1948	0.695280
$216$	0	0
$217$	20.3112	1.37881
$218$	0	0
$219$	−20.7316	−1.40091
$220$	0	0
$221$	−22.7300	−1.52898
$222$	0	0
$223$	−10.9595	−0.733900	−0.366950	−	0.930241i	$-0.619598\pi$
$223$	−10.9595	−0.733900	−0.366950	+	0.930241i	$0.619598\pi$
$224$	0	0
$225$	1.50877	0.100584
$226$	0	0
$227$	19.1738	1.27261	0.636306	−	0.771437i	$-0.280462\pi$
$227$	19.1738	1.27261	0.636306	+	0.771437i	$0.280462\pi$
$228$	0	0
$229$	4.44137	0.293494	0.146747	−	0.989174i	$-0.453120\pi$
$229$	4.44137	0.293494	0.146747	+	0.989174i	$0.453120\pi$
$230$	0	0
$231$	7.16948	0.471717
$232$	0	0
$233$	24.9949	1.63747	0.818736	−	0.574170i	$-0.194675\pi$
$233$	24.9949	1.63747	0.818736	+	0.574170i	$0.194675\pi$
$234$	0	0
$235$	18.3710	1.19839
$236$	0	0
$237$	3.39640	0.220620
$238$	0	0
$239$	8.50862	0.550377	0.275188	−	0.961390i	$-0.411260\pi$
$239$	8.50862	0.550377	0.275188	+	0.961390i	$0.411260\pi$
$240$	0	0
$241$	20.4857	1.31960	0.659800	−	0.751442i	$-0.270641\pi$
$241$	20.4857	1.31960	0.659800	+	0.751442i	$0.270641\pi$
$242$	0	0
$243$	−5.13076	−0.329138
$244$	0	0
$245$	21.7914	1.39220
$246$	0	0
$247$	−6.12876	−0.389964
$248$	0	0
$249$	−22.7104	−1.43921
$250$	0	0
$251$	−14.0342	−0.885828	−0.442914	−	0.896564i	$-0.646055\pi$
$251$	−14.0342	−0.885828	−0.442914	+	0.896564i	$0.646055\pi$
$252$	0	0
$253$	−1.01807	−0.0640055
$254$	0	0
$255$	−19.6527	−1.23070
$256$	0	0
$257$	16.4673	1.02720	0.513601	−	0.858029i	$-0.328311\pi$
$257$	16.4673	1.02720	0.513601	+	0.858029i	$0.328311\pi$
$258$	0	0
$259$	6.46971	0.402008
$260$	0	0
$261$	−1.77907	−0.110122
$262$	0	0
$263$	27.0368	1.66716	0.833580	−	0.552399i	$-0.186288\pi$
$263$	27.0368	1.66716	0.833580	+	0.552399i	$0.186288\pi$
$264$	0	0
$265$	−17.9887	−1.10504
$266$	0	0
$267$	−2.83076	−0.173240
$268$	0	0
$269$	2.69523	0.164331	0.0821657	−	0.996619i	$-0.473816\pi$
$269$	2.69523	0.164331	0.0821657	+	0.996619i	$0.473816\pi$
$270$	0	0
$271$	−0.753780	−0.0457889	−0.0228944	−	0.999738i	$-0.507288\pi$
$271$	−0.753780	−0.0457889	−0.0228944	+	0.999738i	$0.507288\pi$
$272$	0	0
$273$	43.9400	2.65937
$274$	0	0
$275$	3.02594	0.182471
$276$	0	0
$277$	12.0493	0.723972	0.361986	−	0.932184i	$-0.382099\pi$
$277$	12.0493	0.723972	0.361986	+	0.932184i	$0.382099\pi$
$278$	0	0
$279$	2.64215	0.158181
$280$	0	0
$281$	30.3906	1.81295	0.906475	−	0.422258i	$-0.138763\pi$
$281$	30.3906	1.81295	0.906475	+	0.422258i	$0.138763\pi$
$282$	0	0
$283$	−9.49645	−0.564505	−0.282253	−	0.959340i	$-0.591082\pi$
$283$	−9.49645	−0.564505	−0.282253	+	0.959340i	$0.591082\pi$
$284$	0	0
$285$	−5.29902	−0.313887
$286$	0	0
$287$	−25.7237	−1.51842
$288$	0	0
$289$	−3.24523	−0.190896
$290$	0	0
$291$	−25.0455	−1.46819
$292$	0	0
$293$	−17.9766	−1.05020	−0.525101	−	0.851040i	$-0.675972\pi$
$293$	−17.9766	−1.05020	−0.525101	+	0.851040i	$0.675972\pi$
$294$	0	0
$295$	−30.6341	−1.78359
$296$	0	0
$297$	−4.67874	−0.271488
$298$	0	0
$299$	−6.23950	−0.360840
$300$	0	0
$301$	13.7934	0.795037
$302$	0	0
$303$	−34.4944	−1.98165
$304$	0	0
$305$	−2.10689	−0.120640
$306$	0	0
$307$	−11.4216	−0.651867	−0.325933	−	0.945393i	$-0.605679\pi$
$307$	−11.4216	−0.651867	−0.325933	+	0.945393i	$0.605679\pi$
$308$	0	0
$309$	25.1879	1.43289
$310$	0	0
$311$	−25.6594	−1.45501	−0.727507	−	0.686101i	$-0.759321\pi$
$311$	−25.6594	−1.45501	−0.727507	+	0.686101i	$0.759321\pi$
$312$	0	0
$313$	−11.6229	−0.656962	−0.328481	−	0.944511i	$-0.606537\pi$
$313$	−11.6229	−0.656962	−0.328481	+	0.944511i	$0.606537\pi$
$314$	0	0
$315$	5.41439	0.305066
$316$	0	0
$317$	30.1542	1.69363	0.846813	−	0.531890i	$-0.178518\pi$
$317$	30.1542	1.69363	0.846813	+	0.531890i	$0.178518\pi$
$318$	0	0
$319$	−3.56805	−0.199773
$320$	0	0
$321$	30.6231	1.70921
$322$	0	0
$323$	3.70874	0.206360
$324$	0	0
$325$	18.5453	1.02871
$326$	0	0
$327$	14.2293	0.786884
$328$	0	0
$329$	24.8556	1.37033
$330$	0	0
$331$	25.4171	1.39705	0.698526	−	0.715585i	$-0.253840\pi$
$331$	25.4171	1.39705	0.698526	+	0.715585i	$0.253840\pi$
$332$	0	0
$333$	0.841602	0.0461195
$334$	0	0
$335$	−5.88058	−0.321291
$336$	0	0
$337$	−9.43868	−0.514158	−0.257079	−	0.966390i	$-0.582760\pi$
$337$	−9.43868	−0.514158	−0.257079	+	0.966390i	$0.582760\pi$
$338$	0	0
$339$	−18.7131	−1.01636
$340$	0	0
$341$	5.29902	0.286958
$342$	0	0
$343$	2.65229	0.143210
$344$	0	0
$345$	−5.39477	−0.290445
$346$	0	0
$347$	35.1560	1.88727	0.943636	−	0.330985i	$-0.107381\pi$
$347$	35.1560	1.88727	0.943636	+	0.330985i	$0.107381\pi$
$348$	0	0
$349$	21.5526	1.15368	0.576841	−	0.816856i	$-0.304285\pi$
$349$	21.5526	1.15368	0.576841	+	0.816856i	$0.304285\pi$
$350$	0	0
$351$	−28.6749	−1.53055
$352$	0	0
$353$	8.48984	0.451869	0.225934	−	0.974143i	$-0.427457\pi$
$353$	8.48984	0.451869	0.225934	+	0.974143i	$0.427457\pi$
$354$	0	0
$355$	7.52695	0.399489
$356$	0	0
$357$	−26.5898	−1.40728
$358$	0	0
$359$	2.61783	0.138164	0.0690820	−	0.997611i	$-0.477993\pi$
$359$	2.61783	0.138164	0.0690820	+	0.997611i	$0.477993\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.87046	0.0981735
$364$	0	0
$365$	−31.4002	−1.64356
$366$	0	0
$367$	4.37102	0.228166	0.114083	−	0.993471i	$-0.463607\pi$
$367$	4.37102	0.228166	0.114083	+	0.993471i	$0.463607\pi$
$368$	0	0
$369$	−3.34622	−0.174197
$370$	0	0
$371$	−24.3384	−1.26359
$372$	0	0
$373$	−2.66161	−0.137813	−0.0689066	−	0.997623i	$-0.521951\pi$
$373$	−2.66161	−0.137813	−0.0689066	+	0.997623i	$0.521951\pi$
$374$	0	0
$375$	−10.4606	−0.540182
$376$	0	0
$377$	−21.8677	−1.12625
$378$	0	0
$379$	−17.7952	−0.914078	−0.457039	−	0.889447i	$-0.651090\pi$
$379$	−17.7952	−0.914078	−0.457039	+	0.889447i	$0.651090\pi$
$380$	0	0
$381$	38.5989	1.97748
$382$	0	0
$383$	32.3374	1.65236	0.826181	−	0.563405i	$-0.190509\pi$
$383$	32.3374	1.65236	0.826181	+	0.563405i	$0.190509\pi$
$384$	0	0
$385$	10.8590	0.553423
$386$	0	0
$387$	1.79429	0.0912088
$388$	0	0
$389$	13.1909	0.668804	0.334402	−	0.942431i	$-0.391466\pi$
$389$	13.1909	0.668804	0.334402	+	0.942431i	$0.391466\pi$
$390$	0	0
$391$	3.77576	0.190948
$392$	0	0
$393$	−11.3445	−0.572256
$394$	0	0
$395$	5.14421	0.258833
$396$	0	0
$397$	30.8885	1.55025	0.775124	−	0.631809i	$-0.217687\pi$
$397$	30.8885	1.55025	0.775124	+	0.631809i	$0.217687\pi$
$398$	0	0
$399$	−7.16948	−0.358923
$400$	0	0
$401$	−26.8786	−1.34225	−0.671126	−	0.741343i	$-0.734189\pi$
$401$	−26.8786	−1.34225	−0.671126	+	0.741343i	$0.734189\pi$
$402$	0	0
$403$	32.4764	1.61777
$404$	0	0
$405$	−29.0305	−1.44254
$406$	0	0
$407$	1.68789	0.0836658
$408$	0	0
$409$	28.5926	1.41381	0.706906	−	0.707307i	$-0.250090\pi$
$409$	28.5926	1.41381	0.706906	+	0.707307i	$0.250090\pi$
$410$	0	0
$411$	19.0853	0.941408
$412$	0	0
$413$	−41.4474	−2.03949
$414$	0	0
$415$	−34.3973	−1.68850
$416$	0	0
$417$	40.6810	1.99216
$418$	0	0
$419$	−17.3779	−0.848965	−0.424482	−	0.905436i	$-0.639544\pi$
$419$	−17.3779	−0.848965	−0.424482	+	0.905436i	$0.639544\pi$
$420$	0	0
$421$	8.93888	0.435654	0.217827	−	0.975987i	$-0.430103\pi$
$421$	8.93888	0.435654	0.217827	+	0.975987i	$0.430103\pi$
$422$	0	0
$423$	3.23329	0.157208
$424$	0	0
$425$	−11.2224	−0.544368
$426$	0	0
$427$	−2.85058	−0.137949
$428$	0	0
$429$	11.4636	0.553467
$430$	0	0
$431$	13.0145	0.626887	0.313444	−	0.949607i	$-0.398517\pi$
$431$	13.0145	0.626887	0.313444	+	0.949607i	$0.398517\pi$
$432$	0	0
$433$	−5.00494	−0.240522	−0.120261	−	0.992742i	$-0.538373\pi$
$433$	−5.00494	−0.240522	−0.120261	+	0.992742i	$0.538373\pi$
$434$	0	0
$435$	−18.9072	−0.906531
$436$	0	0
$437$	1.01807	0.0487009
$438$	0	0
$439$	−24.6582	−1.17687	−0.588437	−	0.808543i	$-0.700256\pi$
$439$	−24.6582	−1.17687	−0.588437	+	0.808543i	$0.700256\pi$
$440$	0	0
$441$	3.83529	0.182633
$442$	0	0
$443$	−9.81103	−0.466136	−0.233068	−	0.972460i	$-0.574877\pi$
$443$	−9.81103	−0.466136	−0.233068	+	0.972460i	$0.574877\pi$
$444$	0	0
$445$	−4.28749	−0.203247
$446$	0	0
$447$	21.7376	1.02815
$448$	0	0
$449$	−19.6064	−0.925285	−0.462642	−	0.886545i	$-0.653099\pi$
$449$	−19.6064	−0.925285	−0.462642	+	0.886545i	$0.653099\pi$
$450$	0	0
$451$	−6.71110	−0.316013
$452$	0	0
$453$	−35.9721	−1.69012
$454$	0	0
$455$	66.5519	3.12000
$456$	0	0
$457$	25.8767	1.21046	0.605230	−	0.796051i	$-0.293081\pi$
$457$	25.8767	1.21046	0.605230	+	0.796051i	$0.293081\pi$
$458$	0	0
$459$	17.3522	0.809934
$460$	0	0
$461$	−34.2353	−1.59450	−0.797249	−	0.603650i	$-0.793712\pi$
$461$	−34.2353	−1.59450	−0.797249	+	0.603650i	$0.793712\pi$
$462$	0	0
$463$	21.5544	1.00172	0.500859	−	0.865529i	$-0.333018\pi$
$463$	21.5544	1.00172	0.500859	+	0.865529i	$0.333018\pi$
$464$	0	0
$465$	28.0796	1.30216
$466$	0	0
$467$	−6.66020	−0.308198	−0.154099	−	0.988055i	$-0.549247\pi$
$467$	−6.66020	−0.308198	−0.154099	+	0.988055i	$0.549247\pi$
$468$	0	0
$469$	−7.95632	−0.367389
$470$	0	0
$471$	−9.41576	−0.433855
$472$	0	0
$473$	3.59858	0.165463
$474$	0	0
$475$	−3.02594	−0.138840
$476$	0	0
$477$	−3.16602	−0.144962
$478$	0	0
$479$	8.06349	0.368430	0.184215	−	0.982886i	$-0.441026\pi$
$479$	8.06349	0.368430	0.184215	+	0.982886i	$0.441026\pi$
$480$	0	0
$481$	10.3447	0.471677
$482$	0	0
$483$	−7.29903	−0.332117
$484$	0	0
$485$	−37.9341	−1.72250
$486$	0	0
$487$	−23.8768	−1.08196	−0.540981	−	0.841035i	$-0.681947\pi$
$487$	−23.8768	−1.08196	−0.540981	+	0.841035i	$0.681947\pi$
$488$	0	0
$489$	−24.9865	−1.12993
$490$	0	0
$491$	−2.42952	−0.109643	−0.0548214	−	0.998496i	$-0.517459\pi$
$491$	−2.42952	−0.109643	−0.0548214	+	0.998496i	$0.517459\pi$
$492$	0	0
$493$	13.2330	0.595984
$494$	0	0
$495$	1.41257	0.0634902
$496$	0	0
$497$	10.1838	0.456807
$498$	0	0
$499$	−30.9479	−1.38542	−0.692709	−	0.721217i	$-0.743583\pi$
$499$	−30.9479	−1.38542	−0.692709	+	0.721217i	$0.743583\pi$
$500$	0	0
$501$	12.8803	0.575450
$502$	0	0
$503$	10.1030	0.450472	0.225236	−	0.974304i	$-0.427685\pi$
$503$	10.1030	0.450472	0.225236	+	0.974304i	$0.427685\pi$
$504$	0	0
$505$	−52.2455	−2.32490
$506$	0	0
$507$	45.9416	2.04034
$508$	0	0
$509$	30.0363	1.33134	0.665668	−	0.746248i	$-0.268147\pi$
$509$	30.0363	1.33134	0.665668	+	0.746248i	$0.268147\pi$
$510$	0	0
$511$	−42.4839	−1.87937
$512$	0	0
$513$	4.67874	0.206572
$514$	0	0
$515$	38.1498	1.68108
$516$	0	0
$517$	6.48461	0.285193
$518$	0	0
$519$	−26.8895	−1.18032
$520$	0	0
$521$	−26.3804	−1.15574	−0.577872	−	0.816127i	$-0.696117\pi$
$521$	−26.3804	−1.15574	−0.577872	+	0.816127i	$0.696117\pi$
$522$	0	0
$523$	−22.2738	−0.973965	−0.486982	−	0.873412i	$-0.661902\pi$
$523$	−22.2738	−0.973965	−0.486982	+	0.873412i	$0.661902\pi$
$524$	0	0
$525$	21.6944	0.946823
$526$	0	0
$527$	−19.6527	−0.856085
$528$	0	0
$529$	−21.9635	−0.954936
$530$	0	0
$531$	−5.39161	−0.233976
$532$	0	0
$533$	−41.1307	−1.78157
$534$	0	0
$535$	46.3820	2.00527
$536$	0	0
$537$	4.49496	0.193972
$538$	0	0
$539$	7.69196	0.331316
$540$	0	0
$541$	25.9664	1.11638	0.558192	−	0.829712i	$-0.311495\pi$
$541$	25.9664	1.11638	0.558192	+	0.829712i	$0.311495\pi$
$542$	0	0
$543$	−12.9963	−0.557724
$544$	0	0
$545$	21.5519	0.923180
$546$	0	0
$547$	9.05138	0.387009	0.193505	−	0.981099i	$-0.438015\pi$
$547$	9.05138	0.387009	0.193505	+	0.981099i	$0.438015\pi$
$548$	0	0
$549$	−0.370813	−0.0158259
$550$	0	0
$551$	3.56805	0.152004
$552$	0	0
$553$	6.96002	0.295970
$554$	0	0
$555$	8.94419	0.379660
$556$	0	0
$557$	9.46486	0.401039	0.200519	−	0.979690i	$-0.435737\pi$
$557$	9.46486	0.401039	0.200519	+	0.979690i	$0.435737\pi$
$558$	0	0
$559$	22.0548	0.932819
$560$	0	0
$561$	−6.93704	−0.292882
$562$	0	0
$563$	14.9480	0.629982	0.314991	−	0.949095i	$-0.397999\pi$
$563$	14.9480	0.629982	0.314991	+	0.949095i	$0.397999\pi$
$564$	0	0
$565$	−28.3430	−1.19240
$566$	0	0
$567$	−39.2777	−1.64951
$568$	0	0
$569$	−37.8266	−1.58577	−0.792887	−	0.609369i	$-0.791423\pi$
$569$	−37.8266	−1.58577	−0.792887	+	0.609369i	$0.791423\pi$
$570$	0	0
$571$	−21.6033	−0.904071	−0.452035	−	0.892000i	$-0.649302\pi$
$571$	−21.6033	−0.904071	−0.452035	+	0.892000i	$0.649302\pi$
$572$	0	0
$573$	−28.4096	−1.18683
$574$	0	0
$575$	−3.08062	−0.128471
$576$	0	0
$577$	−27.5079	−1.14517	−0.572584	−	0.819846i	$-0.694059\pi$
$577$	−27.5079	−1.14517	−0.572584	+	0.819846i	$0.694059\pi$
$578$	0	0
$579$	−20.4499	−0.849870
$580$	0	0
$581$	−46.5389	−1.93076
$582$	0	0
$583$	−6.34969	−0.262977
$584$	0	0
$585$	8.65729	0.357935
$586$	0	0
$587$	16.4534	0.679106	0.339553	−	0.940587i	$-0.389724\pi$
$587$	16.4534	0.679106	0.339553	+	0.940587i	$0.389724\pi$
$588$	0	0
$589$	−5.29902	−0.218342
$590$	0	0
$591$	−23.0888	−0.949748
$592$	0	0
$593$	−0.937222	−0.0384871	−0.0192435	−	0.999815i	$-0.506126\pi$
$593$	−0.937222	−0.0384871	−0.0192435	+	0.999815i	$0.506126\pi$
$594$	0	0
$595$	−40.2730	−1.65103
$596$	0	0
$597$	−24.3011	−0.994577
$598$	0	0
$599$	−9.50619	−0.388412	−0.194206	−	0.980961i	$-0.562213\pi$
$599$	−9.50619	−0.388412	−0.194206	+	0.980961i	$0.562213\pi$
$600$	0	0
$601$	−33.0715	−1.34901	−0.674507	−	0.738268i	$-0.735644\pi$
$601$	−33.0715	−1.34901	−0.674507	+	0.738268i	$0.735644\pi$
$602$	0	0
$603$	−1.03498	−0.0421478
$604$	0	0
$605$	2.83301	0.115178
$606$	0	0
$607$	16.8867	0.685411	0.342705	−	0.939443i	$-0.388657\pi$
$607$	16.8867	0.685411	0.342705	+	0.939443i	$0.388657\pi$
$608$	0	0
$609$	−25.5811	−1.03660
$610$	0	0
$611$	39.7426	1.60781
$612$	0	0
$613$	−12.5772	−0.507987	−0.253993	−	0.967206i	$-0.581744\pi$
$613$	−12.5772	−0.507987	−0.253993	+	0.967206i	$0.581744\pi$
$614$	0	0
$615$	−35.5623	−1.43401
$616$	0	0
$617$	−21.2054	−0.853696	−0.426848	−	0.904323i	$-0.640376\pi$
$617$	−21.2054	−0.853696	−0.426848	+	0.904323i	$0.640376\pi$
$618$	0	0
$619$	−2.83787	−0.114064	−0.0570319	−	0.998372i	$-0.518164\pi$
$619$	−2.83787	−0.114064	−0.0570319	+	0.998372i	$0.518164\pi$
$620$	0	0
$621$	4.76328	0.191144
$622$	0	0
$623$	−5.80090	−0.232408
$624$	0	0
$625$	−30.9734	−1.23894
$626$	0	0
$627$	−1.87046	−0.0746989
$628$	0	0
$629$	−6.25996	−0.249601
$630$	0	0
$631$	−21.1498	−0.841962	−0.420981	−	0.907070i	$-0.638314\pi$
$631$	−21.1498	−0.841962	−0.420981	+	0.907070i	$0.638314\pi$
$632$	0	0
$633$	17.6351	0.700931
$634$	0	0
$635$	58.4622	2.32000
$636$	0	0
$637$	47.1422	1.86784
$638$	0	0
$639$	1.32475	0.0524061
$640$	0	0
$641$	35.8882	1.41750	0.708749	−	0.705461i	$-0.249260\pi$
$641$	35.8882	1.41750	0.708749	+	0.705461i	$0.249260\pi$
$642$	0	0
$643$	23.9922	0.946162	0.473081	−	0.881019i	$-0.343142\pi$
$643$	23.9922	0.946162	0.473081	+	0.881019i	$0.343142\pi$
$644$	0	0
$645$	19.0689	0.750839
$646$	0	0
$647$	−3.96962	−0.156062	−0.0780310	−	0.996951i	$-0.524863\pi$
$647$	−3.96962	−0.156062	−0.0780310	+	0.996951i	$0.524863\pi$
$648$	0	0
$649$	−10.8133	−0.424458
$650$	0	0
$651$	37.9912	1.48899
$652$	0	0
$653$	−9.43621	−0.369268	−0.184634	−	0.982807i	$-0.559110\pi$
$653$	−9.43621	−0.369268	−0.184634	+	0.982807i	$0.559110\pi$
$654$	0	0
$655$	−17.1825	−0.671377
$656$	0	0
$657$	−5.52644	−0.215607
$658$	0	0
$659$	31.8841	1.24203	0.621014	−	0.783799i	$-0.286721\pi$
$659$	31.8841	1.24203	0.621014	+	0.783799i	$0.286721\pi$
$660$	0	0
$661$	−27.9815	−1.08835	−0.544177	−	0.838970i	$-0.683158\pi$
$661$	−27.9815	−1.08835	−0.544177	+	0.838970i	$0.683158\pi$
$662$	0	0
$663$	−42.5155	−1.65116
$664$	0	0
$665$	−10.8590	−0.421092
$666$	0	0
$667$	3.63253	0.140652
$668$	0	0
$669$	−20.4992	−0.792545
$670$	0	0
$671$	−0.743693	−0.0287100
$672$	0	0
$673$	33.3736	1.28646	0.643228	−	0.765675i	$-0.277595\pi$
$673$	33.3736	1.28646	0.643228	+	0.765675i	$0.277595\pi$
$674$	0	0
$675$	−14.1576	−0.544926
$676$	0	0
$677$	18.6319	0.716081	0.358040	−	0.933706i	$-0.383445\pi$
$677$	18.6319	0.716081	0.358040	+	0.933706i	$0.383445\pi$
$678$	0	0
$679$	−51.3242	−1.96964
$680$	0	0
$681$	35.8638	1.37431
$682$	0	0
$683$	38.0254	1.45500	0.727501	−	0.686106i	$-0.240681\pi$
$683$	38.0254	1.45500	0.727501	+	0.686106i	$0.240681\pi$
$684$	0	0
$685$	28.9067	1.10447
$686$	0	0
$687$	8.30739	0.316947
$688$	0	0
$689$	−38.9157	−1.48257
$690$	0	0
$691$	−16.7840	−0.638495	−0.319247	−	0.947671i	$-0.603430\pi$
$691$	−16.7840	−0.638495	−0.319247	+	0.947671i	$0.603430\pi$
$692$	0	0
$693$	1.91118	0.0725997
$694$	0	0
$695$	61.6157	2.33722
$696$	0	0
$697$	24.8897	0.942766
$698$	0	0
$699$	46.7520	1.76832
$700$	0	0
$701$	−31.4816	−1.18904	−0.594522	−	0.804079i	$-0.702659\pi$
$701$	−31.4816	−1.18904	−0.594522	+	0.804079i	$0.702659\pi$
$702$	0	0
$703$	−1.68789	−0.0636601
$704$	0	0
$705$	34.3621	1.29415
$706$	0	0
$707$	−70.6872	−2.65847
$708$	0	0
$709$	38.9461	1.46265	0.731326	−	0.682028i	$-0.238902\pi$
$709$	38.9461	1.46265	0.731326	+	0.682028i	$0.238902\pi$
$710$	0	0
$711$	0.905382	0.0339545
$712$	0	0
$713$	−5.39477	−0.202036
$714$	0	0
$715$	17.3628	0.649333
$716$	0	0
$717$	15.9150	0.594357
$718$	0	0
$719$	−28.0860	−1.04743	−0.523715	−	0.851894i	$-0.675454\pi$
$719$	−28.0860	−1.04743	−0.523715	+	0.851894i	$0.675454\pi$
$720$	0	0
$721$	51.6160	1.92228
$722$	0	0
$723$	38.3176	1.42505
$724$	0	0
$725$	−10.7967	−0.400980
$726$	0	0
$727$	−44.4663	−1.64916	−0.824582	−	0.565742i	$-0.808590\pi$
$727$	−44.4663	−1.64916	−0.824582	+	0.565742i	$0.808590\pi$
$728$	0	0
$729$	21.1448	0.783140
$730$	0	0
$731$	−13.3462	−0.493627
$732$	0	0
$733$	27.3248	1.00926	0.504631	−	0.863335i	$-0.331628\pi$
$733$	27.3248	1.00926	0.504631	+	0.863335i	$0.331628\pi$
$734$	0	0
$735$	40.7599	1.50345
$736$	0	0
$737$	−2.07574	−0.0764608
$738$	0	0
$739$	32.4957	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$739$	32.4957	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$740$	0	0
$741$	−11.4636	−0.421125
$742$	0	0
$743$	−3.71370	−0.136242	−0.0681212	−	0.997677i	$-0.521700\pi$
$743$	−3.71370	−0.136242	−0.0681212	+	0.997677i	$0.521700\pi$
$744$	0	0
$745$	32.9240	1.20624
$746$	0	0
$747$	−6.05394	−0.221502
$748$	0	0
$749$	62.7540	2.29298
$750$	0	0
$751$	−10.8240	−0.394974	−0.197487	−	0.980305i	$-0.563278\pi$
$751$	−10.8240	−0.394974	−0.197487	+	0.980305i	$0.563278\pi$
$752$	0	0
$753$	−26.2503	−0.956614
$754$	0	0
$755$	−54.4837	−1.98286
$756$	0	0
$757$	16.0604	0.583724	0.291862	−	0.956461i	$-0.405725\pi$
$757$	16.0604	0.583724	0.291862	+	0.956461i	$0.405725\pi$
$758$	0	0
$759$	−1.90426	−0.0691201
$760$	0	0
$761$	−24.4280	−0.885515	−0.442757	−	0.896641i	$-0.646000\pi$
$761$	−24.4280	−0.885515	−0.442757	+	0.896641i	$0.646000\pi$
$762$	0	0
$763$	29.1593	1.05564
$764$	0	0
$765$	−5.23885	−0.189411
$766$	0	0
$767$	−66.2719	−2.39294
$768$	0	0
$769$	−29.2657	−1.05535	−0.527674	−	0.849447i	$-0.676936\pi$
$769$	−29.2657	−1.05535	−0.527674	+	0.849447i	$0.676936\pi$
$770$	0	0
$771$	30.8014	1.10928
$772$	0	0
$773$	17.0485	0.613194	0.306597	−	0.951839i	$-0.400810\pi$
$773$	17.0485	0.613194	0.306597	+	0.951839i	$0.400810\pi$
$774$	0	0
$775$	16.0345	0.575978
$776$	0	0
$777$	12.1013	0.434132
$778$	0	0
$779$	6.71110	0.240450
$780$	0	0
$781$	2.65687	0.0950704
$782$	0	0
$783$	16.6940	0.596595
$784$	0	0
$785$	−14.2612	−0.509003
$786$	0	0
$787$	−37.5808	−1.33961	−0.669806	−	0.742536i	$-0.733623\pi$
$787$	−37.5808	−1.33961	−0.669806	+	0.742536i	$0.733623\pi$
$788$	0	0
$789$	50.5712	1.80038
$790$	0	0
$791$	−38.3476	−1.36348
$792$	0	0
$793$	−4.55792	−0.161856
$794$	0	0
$795$	−33.6472	−1.19334
$796$	0	0
$797$	17.4740	0.618959	0.309480	−	0.950906i	$-0.399845\pi$
$797$	17.4740	0.618959	0.309480	+	0.950906i	$0.399845\pi$
$798$	0	0
$799$	−24.0497	−0.850819
$800$	0	0
$801$	−0.754600	−0.0266625
$802$	0	0
$803$	−11.0837	−0.391135
$804$	0	0
$805$	−11.0552	−0.389643
$806$	0	0
$807$	5.04132	0.177463
$808$	0	0
$809$	−36.0380	−1.26703	−0.633514	−	0.773731i	$-0.718388\pi$
$809$	−36.0380	−1.26703	−0.633514	+	0.773731i	$0.718388\pi$
$810$	0	0
$811$	51.5131	1.80887	0.904435	−	0.426611i	$-0.140293\pi$
$811$	51.5131	1.80887	0.904435	+	0.426611i	$0.140293\pi$
$812$	0	0
$813$	−1.40991	−0.0494478
$814$	0	0
$815$	−37.8447	−1.32564
$816$	0	0
$817$	−3.59858	−0.125898
$818$	0	0
$819$	11.7132	0.409291
$820$	0	0
$821$	39.3130	1.37203	0.686017	−	0.727586i	$-0.259358\pi$
$821$	39.3130	1.37203	0.686017	+	0.727586i	$0.259358\pi$
$822$	0	0
$823$	24.3573	0.849040	0.424520	−	0.905418i	$-0.360443\pi$
$823$	24.3573	0.849040	0.424520	+	0.905418i	$0.360443\pi$
$824$	0	0
$825$	5.65990	0.197052
$826$	0	0
$827$	0.290693	0.0101084	0.00505420	−	0.999987i	$-0.498391\pi$
$827$	0.290693	0.0101084	0.00505420	+	0.999987i	$0.498391\pi$
$828$	0	0
$829$	−44.2644	−1.53737	−0.768684	−	0.639629i	$-0.779088\pi$
$829$	−44.2644	−1.53737	−0.768684	+	0.639629i	$0.779088\pi$
$830$	0	0
$831$	22.5377	0.781824
$832$	0	0
$833$	−28.5275	−0.988419
$834$	0	0
$835$	19.5086	0.675124
$836$	0	0
$837$	−24.7928	−0.856963
$838$	0	0
$839$	−14.9485	−0.516080	−0.258040	−	0.966134i	$-0.583077\pi$
$839$	−14.9485	−0.516080	−0.258040	+	0.966134i	$0.583077\pi$
$840$	0	0
$841$	−16.2690	−0.561000
$842$	0	0
$843$	56.8443	1.95782
$844$	0	0
$845$	69.5835	2.39374
$846$	0	0
$847$	3.83301	0.131704
$848$	0	0
$849$	−17.7627	−0.609614
$850$	0	0
$851$	−1.71839	−0.0589058
$852$	0	0
$853$	5.52071	0.189026	0.0945128	−	0.995524i	$-0.469871\pi$
$853$	5.52071	0.189026	0.0945128	+	0.995524i	$0.469871\pi$
$854$	0	0
$855$	−1.41257	−0.0483088
$856$	0	0
$857$	−30.5165	−1.04242	−0.521212	−	0.853427i	$-0.674520\pi$
$857$	−30.5165	−1.04242	−0.521212	+	0.853427i	$0.674520\pi$
$858$	0	0
$859$	−55.2698	−1.88578	−0.942891	−	0.333101i	$-0.891905\pi$
$859$	−55.2698	−1.88578	−0.942891	+	0.333101i	$0.891905\pi$
$860$	0	0
$861$	−48.1151	−1.63976
$862$	0	0
$863$	21.7474	0.740291	0.370145	−	0.928974i	$-0.379308\pi$
$863$	21.7474	0.740291	0.370145	+	0.928974i	$0.379308\pi$
$864$	0	0
$865$	−40.7271	−1.38476
$866$	0	0
$867$	−6.07007	−0.206150
$868$	0	0
$869$	1.81581	0.0615972
$870$	0	0
$871$	−12.7217	−0.431058
$872$	0	0
$873$	−6.67642	−0.225963
$874$	0	0
$875$	−21.4362	−0.724676
$876$	0	0
$877$	−38.7805	−1.30952	−0.654762	−	0.755835i	$-0.727231\pi$
$877$	−38.7805	−1.30952	−0.654762	+	0.755835i	$0.727231\pi$
$878$	0	0
$879$	−33.6244	−1.13412
$880$	0	0
$881$	−51.5855	−1.73796	−0.868979	−	0.494849i	$-0.835223\pi$
$881$	−51.5855	−1.73796	−0.868979	+	0.494849i	$0.835223\pi$
$882$	0	0
$883$	7.82960	0.263487	0.131743	−	0.991284i	$-0.457942\pi$
$883$	7.82960	0.263487	0.131743	+	0.991284i	$0.457942\pi$
$884$	0	0
$885$	−57.2998	−1.92611
$886$	0	0
$887$	−26.0841	−0.875820	−0.437910	−	0.899019i	$-0.644281\pi$
$887$	−26.0841	−0.875820	−0.437910	+	0.899019i	$0.644281\pi$
$888$	0	0
$889$	79.0983	2.65287
$890$	0	0
$891$	−10.2472	−0.343295
$892$	0	0
$893$	−6.48461	−0.216999
$894$	0	0
$895$	6.80810	0.227570
$896$	0	0
$897$	−11.6707	−0.389674
$898$	0	0
$899$	−18.9072	−0.630590
$900$	0	0
$901$	23.5494	0.784543
$902$	0	0
$903$	25.7999	0.858568
$904$	0	0
$905$	−19.6843	−0.654328
$906$	0	0
$907$	52.5096	1.74355	0.871777	−	0.489903i	$-0.162968\pi$
$907$	52.5096	1.74355	0.871777	+	0.489903i	$0.162968\pi$
$908$	0	0
$909$	−9.19523	−0.304987
$910$	0	0
$911$	10.2679	0.340190	0.170095	−	0.985428i	$-0.445592\pi$
$911$	10.2679	0.340190	0.170095	+	0.985428i	$0.445592\pi$
$912$	0	0
$913$	−12.1416	−0.401829
$914$	0	0
$915$	−3.94085	−0.130280
$916$	0	0
$917$	−23.2476	−0.767705
$918$	0	0
$919$	18.6639	0.615665	0.307833	−	0.951441i	$-0.400396\pi$
$919$	18.6639	0.615665	0.307833	+	0.951441i	$0.400396\pi$
$920$	0	0
$921$	−21.3637	−0.703957
$922$	0	0
$923$	16.2833	0.535973
$924$	0	0
$925$	5.10747	0.167933
$926$	0	0
$927$	6.71438	0.220529
$928$	0	0
$929$	−24.0742	−0.789849	−0.394925	−	0.918713i	$-0.629229\pi$
$929$	−24.0742	−0.789849	−0.394925	+	0.918713i	$0.629229\pi$
$930$	0	0
$931$	−7.69196	−0.252094
$932$	0	0
$933$	−47.9949	−1.57128
$934$	0	0
$935$	−10.5069	−0.343612
$936$	0	0
$937$	9.95255	0.325136	0.162568	−	0.986697i	$-0.448022\pi$
$937$	9.95255	0.325136	0.162568	+	0.986697i	$0.448022\pi$
$938$	0	0
$939$	−21.7400	−0.709460
$940$	0	0
$941$	−28.7364	−0.936779	−0.468390	−	0.883522i	$-0.655166\pi$
$941$	−28.7364	−0.936779	−0.468390	+	0.883522i	$0.655166\pi$
$942$	0	0
$943$	6.83236	0.222492
$944$	0	0
$945$	−50.8062	−1.65273
$946$	0	0
$947$	−1.35752	−0.0441135	−0.0220567	−	0.999757i	$-0.507021\pi$
$947$	−1.35752	−0.0441135	−0.0220567	+	0.999757i	$0.507021\pi$
$948$	0	0
$949$	−67.9292	−2.20508
$950$	0	0
$951$	56.4021	1.82896
$952$	0	0
$953$	−57.1898	−1.85256	−0.926280	−	0.376835i	$-0.877012\pi$
$953$	−57.1898	−1.85256	−0.926280	+	0.376835i	$0.877012\pi$
$954$	0	0
$955$	−43.0294	−1.39240
$956$	0	0
$957$	−6.67389	−0.215736
$958$	0	0
$959$	39.1103	1.26294
$960$	0	0
$961$	−2.92035	−0.0942050
$962$	0	0
$963$	8.16324	0.263057
$964$	0	0
$965$	−30.9736	−0.997076
$966$	0	0
$967$	−31.9125	−1.02624	−0.513118	−	0.858318i	$-0.671510\pi$
$967$	−31.9125	−1.02624	−0.513118	+	0.858318i	$0.671510\pi$
$968$	0	0
$969$	6.93704	0.222850
$970$	0	0
$971$	−22.4554	−0.720627	−0.360314	−	0.932831i	$-0.617330\pi$
$971$	−22.4554	−0.720627	−0.360314	+	0.932831i	$0.617330\pi$
$972$	0	0
$973$	83.3649	2.67256
$974$	0	0
$975$	34.6881	1.11091
$976$	0	0
$977$	39.6048	1.26707	0.633535	−	0.773714i	$-0.281603\pi$
$977$	39.6048	1.26707	0.633535	+	0.773714i	$0.281603\pi$
$978$	0	0
$979$	−1.51341	−0.0483687
$980$	0	0
$981$	3.79313	0.121105
$982$	0	0
$983$	50.2521	1.60279	0.801396	−	0.598134i	$-0.204091\pi$
$983$	50.2521	1.60279	0.801396	+	0.598134i	$0.204091\pi$
$984$	0	0
$985$	−34.9705	−1.11425
$986$	0	0
$987$	46.4913	1.47983
$988$	0	0
$989$	−3.66360	−0.116496
$990$	0	0
$991$	−35.6965	−1.13394	−0.566968	−	0.823740i	$-0.691884\pi$
$991$	−35.6965	−1.13394	−0.566968	+	0.823740i	$0.691884\pi$
$992$	0	0
$993$	47.5416	1.50869
$994$	0	0
$995$	−36.8066	−1.16685
$996$	0	0
$997$	42.6983	1.35227	0.676134	−	0.736778i	$-0.263654\pi$
$997$	42.6983	1.35227	0.676134	+	0.736778i	$0.263654\pi$
$998$	0	0
$999$	−7.89722	−0.249857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.z.1.4		6
4.3	odd	2		1672.2.a.i.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.3	✓	6	4.3	odd	2
3344.2.a.z.1.4		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)

\(f(q)\)	\(=\)	\(q+1.87046 q^{3} +2.83301 q^{5} +3.83301 q^{7} +0.498611 q^{9} +O(q^{10})\)	\(q+1.87046 q^{3} +2.83301 q^{5} +3.83301 q^{7} +0.498611 q^{9} +1.00000 q^{11} +6.12876 q^{13} +5.29902 q^{15} -3.70874 q^{17} -1.00000 q^{19} +7.16948 q^{21} -1.01807 q^{23} +3.02594 q^{25} -4.67874 q^{27} -3.56805 q^{29} +5.29902 q^{31} +1.87046 q^{33} +10.8590 q^{35} +1.68789 q^{37} +11.4636 q^{39} -6.71110 q^{41} +3.59858 q^{43} +1.41257 q^{45} +6.48461 q^{47} +7.69196 q^{49} -6.93704 q^{51} -6.34969 q^{53} +2.83301 q^{55} -1.87046 q^{57} -10.8133 q^{59} -0.743693 q^{61} +1.91118 q^{63} +17.3628 q^{65} -2.07574 q^{67} -1.90426 q^{69} +2.65687 q^{71} -11.0837 q^{73} +5.65990 q^{75} +3.83301 q^{77} +1.81581 q^{79} -10.2472 q^{81} -12.1416 q^{83} -10.5069 q^{85} -6.67389 q^{87} -1.51341 q^{89} +23.4916 q^{91} +9.91160 q^{93} -2.83301 q^{95} -13.3901 q^{97} +0.498611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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