LMFDB - Embedded newform 3332.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} +1.79129 q^{5} +4.79129 q^{9} +O(q^{10})$	$q-2.79129 q^{3} +1.79129 q^{5} +4.79129 q^{9} -3.58258 q^{11} +4.00000 q^{13} -5.00000 q^{15} +1.00000 q^{17} +2.00000 q^{19} +1.58258 q^{23} -1.79129 q^{25} -5.00000 q^{27} +0.208712 q^{31} +10.0000 q^{33} +5.58258 q^{37} -11.1652 q^{39} +1.20871 q^{41} -2.79129 q^{43} +8.58258 q^{45} -1.58258 q^{47} -2.79129 q^{51} -9.79129 q^{53} -6.41742 q^{55} -5.58258 q^{57} +11.5826 q^{59} +6.79129 q^{61} +7.16515 q^{65} -10.9564 q^{67} -4.41742 q^{69} +4.41742 q^{71} +14.7913 q^{73} +5.00000 q^{75} -10.0000 q^{79} -0.417424 q^{81} -9.58258 q^{83} +1.79129 q^{85} -11.1652 q^{89} -0.582576 q^{93} +3.58258 q^{95} +9.79129 q^{97} -17.1652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 5 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 + 5 * q^9	$ 2 q - q^{3} - q^{5} + 5 q^{9} + 2 q^{11} + 8 q^{13} - 10 q^{15} + 2 q^{17} + 4 q^{19} - 6 q^{23} + q^{25} - 10 q^{27} + 5 q^{31} + 20 q^{33} + 2 q^{37} - 4 q^{39} + 7 q^{41} - q^{43} + 8 q^{45} + 6 q^{47} - q^{51} - 15 q^{53} - 22 q^{55} - 2 q^{57} + 14 q^{59} + 9 q^{61} - 4 q^{65} + q^{67} - 18 q^{69} + 18 q^{71} + 25 q^{73} + 10 q^{75} - 20 q^{79} - 10 q^{81} - 10 q^{83} - q^{85} - 4 q^{89} + 8 q^{93} - 2 q^{95} + 15 q^{97} - 16 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 5 * q^9 + 2 * q^11 + 8 * q^13 - 10 * q^15 + 2 * q^17 + 4 * q^19 - 6 * q^23 + q^25 - 10 * q^27 + 5 * q^31 + 20 * q^33 + 2 * q^37 - 4 * q^39 + 7 * q^41 - q^43 + 8 * q^45 + 6 * q^47 - q^51 - 15 * q^53 - 22 * q^55 - 2 * q^57 + 14 * q^59 + 9 * q^61 - 4 * q^65 + q^67 - 18 * q^69 + 18 * q^71 + 25 * q^73 + 10 * q^75 - 20 * q^79 - 10 * q^81 - 10 * q^83 - q^85 - 4 * q^89 + 8 * q^93 - 2 * q^95 + 15 * q^97 - 16 * 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
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	0	0
$5$	1.79129	0.801088	0.400544	−	0.916277i	$-0.368821\pi$
$5$	1.79129	0.801088	0.400544	+	0.916277i	$0.368821\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	−3.58258	−1.08019	−0.540094	−	0.841605i	$-0.681611\pi$
$11$	−3.58258	−1.08019	−0.540094	+	0.841605i	$0.681611\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−5.00000	−1.29099
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.58258	0.329990	0.164995	−	0.986294i	$-0.447239\pi$
$23$	1.58258	0.329990	0.164995	+	0.986294i	$0.447239\pi$
$24$	0	0
$25$	−1.79129	−0.358258
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0.208712	0.0374858	0.0187429	−	0.999824i	$-0.494034\pi$
$31$	0.208712	0.0374858	0.0187429	+	0.999824i	$0.494034\pi$
$32$	0	0
$33$	10.0000	1.74078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.58258	0.917770	0.458885	−	0.888496i	$-0.348249\pi$
$37$	5.58258	0.917770	0.458885	+	0.888496i	$0.348249\pi$
$38$	0	0
$39$	−11.1652	−1.78786
$40$	0	0
$41$	1.20871	0.188769	0.0943846	−	0.995536i	$-0.469912\pi$
$41$	1.20871	0.188769	0.0943846	+	0.995536i	$0.469912\pi$
$42$	0	0
$43$	−2.79129	−0.425667	−0.212834	−	0.977088i	$-0.568269\pi$
$43$	−2.79129	−0.425667	−0.212834	+	0.977088i	$0.568269\pi$
$44$	0	0
$45$	8.58258	1.27941
$46$	0	0
$47$	−1.58258	−0.230842	−0.115421	−	0.993317i	$-0.536822\pi$
$47$	−1.58258	−0.230842	−0.115421	+	0.993317i	$0.536822\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.79129	−0.390858
$52$	0	0
$53$	−9.79129	−1.34494	−0.672468	−	0.740126i	$-0.734766\pi$
$53$	−9.79129	−1.34494	−0.672468	+	0.740126i	$0.734766\pi$
$54$	0	0
$55$	−6.41742	−0.865325
$56$	0	0
$57$	−5.58258	−0.739430
$58$	0	0
$59$	11.5826	1.50792	0.753961	−	0.656919i	$-0.228141\pi$
$59$	11.5826	1.50792	0.753961	+	0.656919i	$0.228141\pi$
$60$	0	0
$61$	6.79129	0.869535	0.434768	−	0.900543i	$-0.356830\pi$
$61$	6.79129	0.869535	0.434768	+	0.900543i	$0.356830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.16515	0.888728
$66$	0	0
$67$	−10.9564	−1.33854	−0.669271	−	0.743018i	$-0.733393\pi$
$67$	−10.9564	−1.33854	−0.669271	+	0.743018i	$0.733393\pi$
$68$	0	0
$69$	−4.41742	−0.531795
$70$	0	0
$71$	4.41742	0.524252	0.262126	−	0.965034i	$-0.415576\pi$
$71$	4.41742	0.524252	0.262126	+	0.965034i	$0.415576\pi$
$72$	0	0
$73$	14.7913	1.73119	0.865595	−	0.500745i	$-0.166941\pi$
$73$	14.7913	1.73119	0.865595	+	0.500745i	$0.166941\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	−9.58258	−1.05182	−0.525912	−	0.850539i	$-0.676276\pi$
$83$	−9.58258	−1.05182	−0.525912	+	0.850539i	$0.676276\pi$
$84$	0	0
$85$	1.79129	0.194292
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.1652	−1.18350	−0.591752	−	0.806120i	$-0.701563\pi$
$89$	−11.1652	−1.18350	−0.591752	+	0.806120i	$0.701563\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.582576	−0.0604103
$94$	0	0
$95$	3.58258	0.367565
$96$	0	0
$97$	9.79129	0.994155	0.497077	−	0.867706i	$-0.334407\pi$
$97$	9.79129	0.994155	0.497077	+	0.867706i	$0.334407\pi$
$98$	0	0
$99$	−17.1652	−1.72516
$100$	0	0
$101$	−0.834849	−0.0830705	−0.0415353	−	0.999137i	$-0.513225\pi$
$101$	−0.834849	−0.0830705	−0.0415353	+	0.999137i	$0.513225\pi$
$102$	0	0
$103$	−7.58258	−0.747133	−0.373567	−	0.927603i	$-0.621865\pi$
$103$	−7.58258	−0.747133	−0.373567	+	0.927603i	$0.621865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.41742	0.620396	0.310198	−	0.950672i	$-0.399605\pi$
$107$	6.41742	0.620396	0.310198	+	0.950672i	$0.399605\pi$
$108$	0	0
$109$	19.5826	1.87567	0.937835	−	0.347081i	$-0.112827\pi$
$109$	19.5826	1.87567	0.937835	+	0.347081i	$0.112827\pi$
$110$	0	0
$111$	−15.5826	−1.47903
$112$	0	0
$113$	−19.5826	−1.84217	−0.921087	−	0.389357i	$-0.872697\pi$
$113$	−19.5826	−1.84217	−0.921087	+	0.389357i	$0.872697\pi$
$114$	0	0
$115$	2.83485	0.264351
$116$	0	0
$117$	19.1652	1.77182
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.83485	0.166804
$122$	0	0
$123$	−3.37386	−0.304211
$124$	0	0
$125$	−12.1652	−1.08808
$126$	0	0
$127$	19.7913	1.75619	0.878096	−	0.478484i	$-0.158813\pi$
$127$	19.7913	1.75619	0.878096	+	0.478484i	$0.158813\pi$
$128$	0	0
$129$	7.79129	0.685985
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.95644	−0.770848
$136$	0	0
$137$	15.5390	1.32759	0.663794	−	0.747916i	$-0.268945\pi$
$137$	15.5390	1.32759	0.663794	+	0.747916i	$0.268945\pi$
$138$	0	0
$139$	8.20871	0.696254	0.348127	−	0.937447i	$-0.386818\pi$
$139$	8.20871	0.696254	0.348127	+	0.937447i	$0.386818\pi$
$140$	0	0
$141$	4.41742	0.372014
$142$	0	0
$143$	−14.3303	−1.19836
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.5390	1.27301	0.636503	−	0.771274i	$-0.280380\pi$
$149$	15.5390	1.27301	0.636503	+	0.771274i	$0.280380\pi$
$150$	0	0
$151$	−5.95644	−0.484728	−0.242364	−	0.970185i	$-0.577923\pi$
$151$	−5.95644	−0.484728	−0.242364	+	0.970185i	$0.577923\pi$
$152$	0	0
$153$	4.79129	0.387353
$154$	0	0
$155$	0.373864	0.0300294
$156$	0	0
$157$	−5.16515	−0.412224	−0.206112	−	0.978528i	$-0.566081\pi$
$157$	−5.16515	−0.412224	−0.206112	+	0.978528i	$0.566081\pi$
$158$	0	0
$159$	27.3303	2.16743
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.417424	0.0326952	0.0163476	−	0.999866i	$-0.494796\pi$
$163$	0.417424	0.0326952	0.0163476	+	0.999866i	$0.494796\pi$
$164$	0	0
$165$	17.9129	1.39452
$166$	0	0
$167$	5.20871	0.403062	0.201531	−	0.979482i	$-0.435408\pi$
$167$	5.20871	0.403062	0.201531	+	0.979482i	$0.435408\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	9.58258	0.732798
$172$	0	0
$173$	11.6261	0.883919	0.441959	−	0.897035i	$-0.354284\pi$
$173$	11.6261	0.883919	0.441959	+	0.897035i	$0.354284\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−32.3303	−2.43009
$178$	0	0
$179$	−13.2087	−0.987265	−0.493633	−	0.869670i	$-0.664331\pi$
$179$	−13.2087	−0.987265	−0.493633	+	0.869670i	$0.664331\pi$
$180$	0	0
$181$	1.16515	0.0866050	0.0433025	−	0.999062i	$-0.486212\pi$
$181$	1.16515	0.0866050	0.0433025	+	0.999062i	$0.486212\pi$
$182$	0	0
$183$	−18.9564	−1.40130
$184$	0	0
$185$	10.0000	0.735215
$186$	0	0
$187$	−3.58258	−0.261984
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.79129	−0.563758	−0.281879	−	0.959450i	$-0.590958\pi$
$191$	−7.79129	−0.563758	−0.281879	+	0.959450i	$0.590958\pi$
$192$	0	0
$193$	20.7477	1.49345	0.746727	−	0.665131i	$-0.231624\pi$
$193$	20.7477	1.49345	0.746727	+	0.665131i	$0.231624\pi$
$194$	0	0
$195$	−20.0000	−1.43223
$196$	0	0
$197$	23.1652	1.65045	0.825224	−	0.564805i	$-0.191049\pi$
$197$	23.1652	1.65045	0.825224	+	0.564805i	$0.191049\pi$
$198$	0	0
$199$	15.5390	1.10153	0.550766	−	0.834660i	$-0.314336\pi$
$199$	15.5390	1.10153	0.550766	+	0.834660i	$0.314336\pi$
$200$	0	0
$201$	30.5826	2.15713
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.16515	0.151221
$206$	0	0
$207$	7.58258	0.527025
$208$	0	0
$209$	−7.16515	−0.495624
$210$	0	0
$211$	21.9129	1.50854	0.754272	−	0.656562i	$-0.227990\pi$
$211$	21.9129	1.50854	0.754272	+	0.656562i	$0.227990\pi$
$212$	0	0
$213$	−12.3303	−0.844858
$214$	0	0
$215$	−5.00000	−0.340997
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−41.2867	−2.78990
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	15.1652	1.01553	0.507767	−	0.861495i	$-0.330471\pi$
$223$	15.1652	1.01553	0.507767	+	0.861495i	$0.330471\pi$
$224$	0	0
$225$	−8.58258	−0.572172
$226$	0	0
$227$	27.3739	1.81687	0.908434	−	0.418029i	$-0.137279\pi$
$227$	27.3739	1.81687	0.908434	+	0.418029i	$0.137279\pi$
$228$	0	0
$229$	16.7477	1.10672	0.553360	−	0.832942i	$-0.313345\pi$
$229$	16.7477	1.10672	0.553360	+	0.832942i	$0.313345\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.7477	0.966156	0.483078	−	0.875577i	$-0.339519\pi$
$233$	14.7477	0.966156	0.483078	+	0.875577i	$0.339519\pi$
$234$	0	0
$235$	−2.83485	−0.184925
$236$	0	0
$237$	27.9129	1.81314
$238$	0	0
$239$	12.3739	0.800399	0.400199	−	0.916428i	$-0.368941\pi$
$239$	12.3739	0.800399	0.400199	+	0.916428i	$0.368941\pi$
$240$	0	0
$241$	−12.5390	−0.807709	−0.403854	−	0.914823i	$-0.632330\pi$
$241$	−12.5390	−0.807709	−0.403854	+	0.914823i	$0.632330\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	26.7477	1.69507
$250$	0	0
$251$	−10.4174	−0.657542	−0.328771	−	0.944410i	$-0.606634\pi$
$251$	−10.4174	−0.657542	−0.328771	+	0.944410i	$0.606634\pi$
$252$	0	0
$253$	−5.66970	−0.356451
$254$	0	0
$255$	−5.00000	−0.313112
$256$	0	0
$257$	−8.33030	−0.519630	−0.259815	−	0.965658i	$-0.583662\pi$
$257$	−8.33030	−0.519630	−0.259815	+	0.965658i	$0.583662\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.3303	1.87025	0.935123	−	0.354322i	$-0.115288\pi$
$263$	30.3303	1.87025	0.935123	+	0.354322i	$0.115288\pi$
$264$	0	0
$265$	−17.5390	−1.07741
$266$	0	0
$267$	31.1652	1.90728
$268$	0	0
$269$	−2.83485	−0.172844	−0.0864219	−	0.996259i	$-0.527543\pi$
$269$	−2.83485	−0.172844	−0.0864219	+	0.996259i	$0.527543\pi$
$270$	0	0
$271$	−3.16515	−0.192269	−0.0961346	−	0.995368i	$-0.530648\pi$
$271$	−3.16515	−0.192269	−0.0961346	+	0.995368i	$0.530648\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.41742	0.386985
$276$	0	0
$277$	−26.7477	−1.60712	−0.803558	−	0.595227i	$-0.797062\pi$
$277$	−26.7477	−1.60712	−0.803558	+	0.595227i	$0.797062\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	9.79129	0.584099	0.292050	−	0.956403i	$-0.405663\pi$
$281$	9.79129	0.584099	0.292050	+	0.956403i	$0.405663\pi$
$282$	0	0
$283$	26.1216	1.55277	0.776384	−	0.630261i	$-0.217052\pi$
$283$	26.1216	1.55277	0.776384	+	0.630261i	$0.217052\pi$
$284$	0	0
$285$	−10.0000	−0.592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−27.3303	−1.60213
$292$	0	0
$293$	12.7477	0.744730	0.372365	−	0.928086i	$-0.378547\pi$
$293$	12.7477	0.744730	0.372365	+	0.928086i	$0.378547\pi$
$294$	0	0
$295$	20.7477	1.20798
$296$	0	0
$297$	17.9129	1.03941
$298$	0	0
$299$	6.33030	0.366091
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.33030	0.133872
$304$	0	0
$305$	12.1652	0.696575
$306$	0	0
$307$	7.58258	0.432760	0.216380	−	0.976309i	$-0.430575\pi$
$307$	7.58258	0.432760	0.216380	+	0.976309i	$0.430575\pi$
$308$	0	0
$309$	21.1652	1.20404
$310$	0	0
$311$	−20.2087	−1.14593	−0.572965	−	0.819580i	$-0.694207\pi$
$311$	−20.2087	−1.14593	−0.572965	+	0.819580i	$0.694207\pi$
$312$	0	0
$313$	−31.7042	−1.79203	−0.896013	−	0.444028i	$-0.853549\pi$
$313$	−31.7042	−1.79203	−0.896013	+	0.444028i	$0.853549\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.3303	0.580208	0.290104	−	0.956995i	$-0.406310\pi$
$317$	10.3303	0.580208	0.290104	+	0.956995i	$0.406310\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−17.9129	−0.999799
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−7.16515	−0.397451
$326$	0	0
$327$	−54.6606	−3.02274
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.37386	−0.405304	−0.202652	−	0.979251i	$-0.564956\pi$
$331$	−7.37386	−0.405304	−0.202652	+	0.979251i	$0.564956\pi$
$332$	0	0
$333$	26.7477	1.46577
$334$	0	0
$335$	−19.6261	−1.07229
$336$	0	0
$337$	−17.1652	−0.935045	−0.467523	−	0.883981i	$-0.654853\pi$
$337$	−17.1652	−0.935045	−0.467523	+	0.883981i	$0.654853\pi$
$338$	0	0
$339$	54.6606	2.96876
$340$	0	0
$341$	−0.747727	−0.0404917
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.91288	−0.426015
$346$	0	0
$347$	0.834849	0.0448170	0.0224085	−	0.999749i	$-0.492867\pi$
$347$	0.834849	0.0448170	0.0224085	+	0.999749i	$0.492867\pi$
$348$	0	0
$349$	0.417424	0.0223442	0.0111721	−	0.999938i	$-0.496444\pi$
$349$	0.417424	0.0223442	0.0111721	+	0.999938i	$0.496444\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	−27.1652	−1.44586	−0.722928	−	0.690924i	$-0.757204\pi$
$353$	−27.1652	−1.44586	−0.722928	+	0.690924i	$0.757204\pi$
$354$	0	0
$355$	7.91288	0.419972
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.9564	1.00048	0.500241	−	0.865886i	$-0.333245\pi$
$359$	18.9564	1.00048	0.500241	+	0.865886i	$0.333245\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−5.12159	−0.268814
$364$	0	0
$365$	26.4955	1.38684
$366$	0	0
$367$	28.3739	1.48110	0.740552	−	0.671999i	$-0.234564\pi$
$367$	28.3739	1.48110	0.740552	+	0.671999i	$0.234564\pi$
$368$	0	0
$369$	5.79129	0.301482
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.1216	0.886522	0.443261	−	0.896393i	$-0.353821\pi$
$373$	17.1216	0.886522	0.443261	+	0.896393i	$0.353821\pi$
$374$	0	0
$375$	33.9564	1.75350
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.3303	0.530632	0.265316	−	0.964162i	$-0.414524\pi$
$379$	10.3303	0.530632	0.265316	+	0.964162i	$0.414524\pi$
$380$	0	0
$381$	−55.2432	−2.83019
$382$	0	0
$383$	−8.41742	−0.430110	−0.215055	−	0.976602i	$-0.568993\pi$
$383$	−8.41742	−0.430110	−0.215055	+	0.976602i	$0.568993\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.3739	−0.679832
$388$	0	0
$389$	−20.1216	−1.02021	−0.510103	−	0.860114i	$-0.670393\pi$
$389$	−20.1216	−1.02021	−0.510103	+	0.860114i	$0.670393\pi$
$390$	0	0
$391$	1.58258	0.0800343
$392$	0	0
$393$	−33.4955	−1.68962
$394$	0	0
$395$	−17.9129	−0.901295
$396$	0	0
$397$	31.2867	1.57024	0.785118	−	0.619346i	$-0.212602\pi$
$397$	31.2867	1.57024	0.785118	+	0.619346i	$0.212602\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.83485	−0.141566	−0.0707828	−	0.997492i	$-0.522550\pi$
$401$	−2.83485	−0.141566	−0.0707828	+	0.997492i	$0.522550\pi$
$402$	0	0
$403$	0.834849	0.0415868
$404$	0	0
$405$	−0.747727	−0.0371549
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−43.3739	−2.13947
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−17.1652	−0.842604
$416$	0	0
$417$	−22.9129	−1.12205
$418$	0	0
$419$	−24.2087	−1.18267	−0.591336	−	0.806425i	$-0.701399\pi$
$419$	−24.2087	−1.18267	−0.591336	+	0.806425i	$0.701399\pi$
$420$	0	0
$421$	−9.20871	−0.448805	−0.224403	−	0.974497i	$-0.572043\pi$
$421$	−9.20871	−0.448805	−0.224403	+	0.974497i	$0.572043\pi$
$422$	0	0
$423$	−7.58258	−0.368677
$424$	0	0
$425$	−1.79129	−0.0868902
$426$	0	0
$427$	0	0
$428$	0	0
$429$	40.0000	1.93122
$430$	0	0
$431$	−0.417424	−0.0201066	−0.0100533	−	0.999949i	$-0.503200\pi$
$431$	−0.417424	−0.0201066	−0.0100533	+	0.999949i	$0.503200\pi$
$432$	0	0
$433$	33.5826	1.61388	0.806938	−	0.590636i	$-0.201123\pi$
$433$	33.5826	1.61388	0.806938	+	0.590636i	$0.201123\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.16515	0.151410
$438$	0	0
$439$	1.95644	0.0933758	0.0466879	−	0.998910i	$-0.485133\pi$
$439$	1.95644	0.0933758	0.0466879	+	0.998910i	$0.485133\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.66970	−0.0793297	−0.0396649	−	0.999213i	$-0.512629\pi$
$443$	−1.66970	−0.0793297	−0.0396649	+	0.999213i	$0.512629\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	−43.3739	−2.05151
$448$	0	0
$449$	−1.58258	−0.0746864	−0.0373432	−	0.999303i	$-0.511889\pi$
$449$	−1.58258	−0.0746864	−0.0373432	+	0.999303i	$0.511889\pi$
$450$	0	0
$451$	−4.33030	−0.203906
$452$	0	0
$453$	16.6261	0.781164
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.79129	0.458017	0.229009	−	0.973424i	$-0.426452\pi$
$457$	9.79129	0.458017	0.229009	+	0.973424i	$0.426452\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	30.7477	1.43206	0.716032	−	0.698067i	$-0.245956\pi$
$461$	30.7477	1.43206	0.716032	+	0.698067i	$0.245956\pi$
$462$	0	0
$463$	32.3739	1.50454	0.752271	−	0.658854i	$-0.228959\pi$
$463$	32.3739	1.50454	0.752271	+	0.658854i	$0.228959\pi$
$464$	0	0
$465$	−1.04356	−0.0483940
$466$	0	0
$467$	31.1652	1.44215	0.721076	−	0.692856i	$-0.243648\pi$
$467$	31.1652	1.44215	0.721076	+	0.692856i	$0.243648\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.4174	0.664320
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	−3.58258	−0.164380
$476$	0	0
$477$	−46.9129	−2.14799
$478$	0	0
$479$	−33.8693	−1.54753	−0.773764	−	0.633474i	$-0.781629\pi$
$479$	−33.8693	−1.54753	−0.773764	+	0.633474i	$0.781629\pi$
$480$	0	0
$481$	22.3303	1.01817
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.5390	0.796406
$486$	0	0
$487$	−32.7477	−1.48394	−0.741971	−	0.670432i	$-0.766109\pi$
$487$	−32.7477	−1.48394	−0.741971	+	0.670432i	$0.766109\pi$
$488$	0	0
$489$	−1.16515	−0.0526900
$490$	0	0
$491$	1.04356	0.0470952	0.0235476	−	0.999723i	$-0.492504\pi$
$491$	1.04356	0.0470952	0.0235476	+	0.999723i	$0.492504\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−30.7477	−1.38201
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.417424	−0.0186865	−0.00934324	−	0.999956i	$-0.502974\pi$
$499$	−0.417424	−0.0186865	−0.00934324	+	0.999956i	$0.502974\pi$
$500$	0	0
$501$	−14.5390	−0.649555
$502$	0	0
$503$	−3.79129	−0.169045	−0.0845226	−	0.996422i	$-0.526937\pi$
$503$	−3.79129	−0.169045	−0.0845226	+	0.996422i	$0.526937\pi$
$504$	0	0
$505$	−1.49545	−0.0665468
$506$	0	0
$507$	−8.37386	−0.371896
$508$	0	0
$509$	8.33030	0.369234	0.184617	−	0.982811i	$-0.440896\pi$
$509$	8.33030	0.369234	0.184617	+	0.982811i	$0.440896\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	−13.5826	−0.598520
$516$	0	0
$517$	5.66970	0.249353
$518$	0	0
$519$	−32.4519	−1.42448
$520$	0	0
$521$	−6.04356	−0.264773	−0.132387	−	0.991198i	$-0.542264\pi$
$521$	−6.04356	−0.264773	−0.132387	+	0.991198i	$0.542264\pi$
$522$	0	0
$523$	−33.0780	−1.44640	−0.723201	−	0.690638i	$-0.757330\pi$
$523$	−33.0780	−1.44640	−0.723201	+	0.690638i	$0.757330\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.208712	0.00909164
$528$	0	0
$529$	−20.4955	−0.891107
$530$	0	0
$531$	55.4955	2.40830
$532$	0	0
$533$	4.83485	0.209421
$534$	0	0
$535$	11.4955	0.496992
$536$	0	0
$537$	36.8693	1.59103
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.8348	0.551813	0.275907	−	0.961184i	$-0.411022\pi$
$541$	12.8348	0.551813	0.275907	+	0.961184i	$0.411022\pi$
$542$	0	0
$543$	−3.25227	−0.139568
$544$	0	0
$545$	35.0780	1.50258
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	32.5390	1.38873
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−27.9129	−1.18484
$556$	0	0
$557$	−24.3303	−1.03091	−0.515454	−	0.856917i	$-0.672377\pi$
$557$	−24.3303	−1.03091	−0.515454	+	0.856917i	$0.672377\pi$
$558$	0	0
$559$	−11.1652	−0.472236
$560$	0	0
$561$	10.0000	0.422200
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−35.0780	−1.47574
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.9564	−1.25584	−0.627920	−	0.778278i	$-0.716093\pi$
$569$	−29.9564	−1.25584	−0.627920	+	0.778278i	$0.716093\pi$
$570$	0	0
$571$	−15.5826	−0.652110	−0.326055	−	0.945351i	$-0.605720\pi$
$571$	−15.5826	−0.652110	−0.326055	+	0.945351i	$0.605720\pi$
$572$	0	0
$573$	21.7477	0.908524
$574$	0	0
$575$	−2.83485	−0.118221
$576$	0	0
$577$	9.91288	0.412679	0.206339	−	0.978481i	$-0.433845\pi$
$577$	9.91288	0.412679	0.206339	+	0.978481i	$0.433845\pi$
$578$	0	0
$579$	−57.9129	−2.40678
$580$	0	0
$581$	0	0
$582$	0	0
$583$	35.0780	1.45278
$584$	0	0
$585$	34.3303	1.41938
$586$	0	0
$587$	26.4174	1.09036	0.545182	−	0.838318i	$-0.316461\pi$
$587$	26.4174	1.09036	0.545182	+	0.838318i	$0.316461\pi$
$588$	0	0
$589$	0.417424	0.0171997
$590$	0	0
$591$	−64.6606	−2.65978
$592$	0	0
$593$	25.0780	1.02983	0.514916	−	0.857241i	$-0.327823\pi$
$593$	25.0780	1.02983	0.514916	+	0.857241i	$0.327823\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−43.3739	−1.77517
$598$	0	0
$599$	−13.2087	−0.539693	−0.269847	−	0.962903i	$-0.586973\pi$
$599$	−13.2087	−0.539693	−0.269847	+	0.962903i	$0.586973\pi$
$600$	0	0
$601$	−23.4955	−0.958400	−0.479200	−	0.877706i	$-0.659073\pi$
$601$	−23.4955	−0.958400	−0.479200	+	0.877706i	$0.659073\pi$
$602$	0	0
$603$	−52.4955	−2.13778
$604$	0	0
$605$	3.28674	0.133625
$606$	0	0
$607$	−39.3739	−1.59814	−0.799068	−	0.601241i	$-0.794673\pi$
$607$	−39.3739	−1.59814	−0.799068	+	0.601241i	$0.794673\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.33030	−0.256097
$612$	0	0
$613$	−23.2867	−0.940543	−0.470271	−	0.882522i	$-0.655844\pi$
$613$	−23.2867	−0.940543	−0.470271	+	0.882522i	$0.655844\pi$
$614$	0	0
$615$	−6.04356	−0.243700
$616$	0	0
$617$	−15.4955	−0.623823	−0.311912	−	0.950111i	$-0.600969\pi$
$617$	−15.4955	−0.623823	−0.311912	+	0.950111i	$0.600969\pi$
$618$	0	0
$619$	−28.6606	−1.15197	−0.575983	−	0.817461i	$-0.695381\pi$
$619$	−28.6606	−1.15197	−0.575983	+	0.817461i	$0.695381\pi$
$620$	0	0
$621$	−7.91288	−0.317533
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−12.8348	−0.513394
$626$	0	0
$627$	20.0000	0.798723
$628$	0	0
$629$	5.58258	0.222592
$630$	0	0
$631$	−18.9564	−0.754644	−0.377322	−	0.926082i	$-0.623155\pi$
$631$	−18.9564	−0.754644	−0.377322	+	0.926082i	$0.623155\pi$
$632$	0	0
$633$	−61.1652	−2.43110
$634$	0	0
$635$	35.4519	1.40687
$636$	0	0
$637$	0	0
$638$	0	0
$639$	21.1652	0.837280
$640$	0	0
$641$	−41.9129	−1.65546	−0.827730	−	0.561127i	$-0.810368\pi$
$641$	−41.9129	−1.65546	−0.827730	+	0.561127i	$0.810368\pi$
$642$	0	0
$643$	1.95644	0.0771544	0.0385772	−	0.999256i	$-0.487717\pi$
$643$	1.95644	0.0771544	0.0385772	+	0.999256i	$0.487717\pi$
$644$	0	0
$645$	13.9564	0.549534
$646$	0	0
$647$	−4.83485	−0.190078	−0.0950388	−	0.995474i	$-0.530298\pi$
$647$	−4.83485	−0.190078	−0.0950388	+	0.995474i	$0.530298\pi$
$648$	0	0
$649$	−41.4955	−1.62884
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.9129	0.779251	0.389626	−	0.920973i	$-0.372604\pi$
$653$	19.9129	0.779251	0.389626	+	0.920973i	$0.372604\pi$
$654$	0	0
$655$	21.4955	0.839897
$656$	0	0
$657$	70.8693	2.76488
$658$	0	0
$659$	−26.9564	−1.05007	−0.525037	−	0.851079i	$-0.675948\pi$
$659$	−26.9564	−1.05007	−0.525037	+	0.851079i	$0.675948\pi$
$660$	0	0
$661$	43.4955	1.69178	0.845889	−	0.533360i	$-0.179071\pi$
$661$	43.4955	1.69178	0.845889	+	0.533360i	$0.179071\pi$
$662$	0	0
$663$	−11.1652	−0.433619
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−42.3303	−1.63658
$670$	0	0
$671$	−24.3303	−0.939261
$672$	0	0
$673$	−13.0780	−0.504121	−0.252061	−	0.967711i	$-0.581108\pi$
$673$	−13.0780	−0.504121	−0.252061	+	0.967711i	$0.581108\pi$
$674$	0	0
$675$	8.95644	0.344734
$676$	0	0
$677$	−26.8348	−1.03135	−0.515674	−	0.856785i	$-0.672458\pi$
$677$	−26.8348	−1.03135	−0.515674	+	0.856785i	$0.672458\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−76.4083	−2.92797
$682$	0	0
$683$	32.7477	1.25306	0.626528	−	0.779399i	$-0.284475\pi$
$683$	32.7477	1.25306	0.626528	+	0.779399i	$0.284475\pi$
$684$	0	0
$685$	27.8348	1.06351
$686$	0	0
$687$	−46.7477	−1.78354
$688$	0	0
$689$	−39.1652	−1.49207
$690$	0	0
$691$	−19.6261	−0.746613	−0.373307	−	0.927708i	$-0.621776\pi$
$691$	−19.6261	−0.746613	−0.373307	+	0.927708i	$0.621776\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.7042	0.557761
$696$	0	0
$697$	1.20871	0.0457832
$698$	0	0
$699$	−41.1652	−1.55701
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	11.1652	0.421102
$704$	0	0
$705$	7.91288	0.298016
$706$	0	0
$707$	0	0
$708$	0	0
$709$	30.8348	1.15803	0.579014	−	0.815318i	$-0.303438\pi$
$709$	30.8348	1.15803	0.579014	+	0.815318i	$0.303438\pi$
$710$	0	0
$711$	−47.9129	−1.79687
$712$	0	0
$713$	0.330303	0.0123699
$714$	0	0
$715$	−25.6697	−0.959992
$716$	0	0
$717$	−34.5390	−1.28988
$718$	0	0
$719$	17.8693	0.666413	0.333207	−	0.942854i	$-0.391869\pi$
$719$	17.8693	0.666413	0.333207	+	0.942854i	$0.391869\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	35.0000	1.30166
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.3303	1.19906	0.599532	−	0.800351i	$-0.295353\pi$
$727$	32.3303	1.19906	0.599532	+	0.800351i	$0.295353\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	−2.79129	−0.103240
$732$	0	0
$733$	−14.7477	−0.544720	−0.272360	−	0.962195i	$-0.587804\pi$
$733$	−14.7477	−0.544720	−0.272360	+	0.962195i	$0.587804\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.2523	1.44588
$738$	0	0
$739$	29.9564	1.10197	0.550983	−	0.834517i	$-0.314253\pi$
$739$	29.9564	1.10197	0.550983	+	0.834517i	$0.314253\pi$
$740$	0	0
$741$	−22.3303	−0.820324
$742$	0	0
$743$	45.1652	1.65695	0.828474	−	0.560027i	$-0.189209\pi$
$743$	45.1652	1.65695	0.828474	+	0.560027i	$0.189209\pi$
$744$	0	0
$745$	27.8348	1.01979
$746$	0	0
$747$	−45.9129	−1.67986
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.83485	0.249407	0.124704	−	0.992194i	$-0.460202\pi$
$751$	6.83485	0.249407	0.124704	+	0.992194i	$0.460202\pi$
$752$	0	0
$753$	29.0780	1.05966
$754$	0	0
$755$	−10.6697	−0.388310
$756$	0	0
$757$	54.6170	1.98509	0.992545	−	0.121878i	$-0.0388916\pi$
$757$	54.6170	1.98509	0.992545	+	0.121878i	$0.0388916\pi$
$758$	0	0
$759$	15.8258	0.574439
$760$	0	0
$761$	−31.1652	−1.12974	−0.564868	−	0.825181i	$-0.691073\pi$
$761$	−31.1652	−1.12974	−0.564868	+	0.825181i	$0.691073\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.58258	0.310304
$766$	0	0
$767$	46.3303	1.67289
$768$	0	0
$769$	37.4955	1.35212	0.676060	−	0.736846i	$-0.263686\pi$
$769$	37.4955	1.35212	0.676060	+	0.736846i	$0.263686\pi$
$770$	0	0
$771$	23.2523	0.837410
$772$	0	0
$773$	31.9129	1.14783	0.573913	−	0.818916i	$-0.305425\pi$
$773$	31.9129	1.14783	0.573913	+	0.818916i	$0.305425\pi$
$774$	0	0
$775$	−0.373864	−0.0134296
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.41742	0.0866132
$780$	0	0
$781$	−15.8258	−0.566290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9.25227	−0.330228
$786$	0	0
$787$	22.3303	0.795989	0.397995	−	0.917388i	$-0.369706\pi$
$787$	22.3303	0.795989	0.397995	+	0.917388i	$0.369706\pi$
$788$	0	0
$789$	−84.6606	−3.01400
$790$	0	0
$791$	0	0
$792$	0	0
$793$	27.1652	0.964663
$794$	0	0
$795$	48.9564	1.73631
$796$	0	0
$797$	−23.0780	−0.817466	−0.408733	−	0.912654i	$-0.634029\pi$
$797$	−23.0780	−0.817466	−0.408733	+	0.912654i	$0.634029\pi$
$798$	0	0
$799$	−1.58258	−0.0559875
$800$	0	0
$801$	−53.4955	−1.89017
$802$	0	0
$803$	−52.9909	−1.87001
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.91288	0.278547
$808$	0	0
$809$	−27.4955	−0.966689	−0.483344	−	0.875430i	$-0.660578\pi$
$809$	−27.4955	−0.966689	−0.483344	+	0.875430i	$0.660578\pi$
$810$	0	0
$811$	1.29583	0.0455029	0.0227514	−	0.999741i	$-0.492757\pi$
$811$	1.29583	0.0455029	0.0227514	+	0.999741i	$0.492757\pi$
$812$	0	0
$813$	8.83485	0.309852
$814$	0	0
$815$	0.747727	0.0261917
$816$	0	0
$817$	−5.58258	−0.195310
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	11.0780	0.386156	0.193078	−	0.981183i	$-0.438153\pi$
$823$	11.0780	0.386156	0.193078	+	0.981183i	$0.438153\pi$
$824$	0	0
$825$	−17.9129	−0.623646
$826$	0	0
$827$	40.3303	1.40242	0.701211	−	0.712954i	$-0.252643\pi$
$827$	40.3303	1.40242	0.701211	+	0.712954i	$0.252643\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	74.6606	2.58995
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9.33030	0.322888
$836$	0	0
$837$	−1.04356	−0.0360707
$838$	0	0
$839$	−51.1652	−1.76642	−0.883209	−	0.468980i	$-0.844622\pi$
$839$	−51.1652	−1.76642	−0.883209	+	0.468980i	$0.844622\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−27.3303	−0.941306
$844$	0	0
$845$	5.37386	0.184867
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−72.9129	−2.50236
$850$	0	0
$851$	8.83485	0.302855
$852$	0	0
$853$	−11.4955	−0.393597	−0.196798	−	0.980444i	$-0.563054\pi$
$853$	−11.4955	−0.393597	−0.196798	+	0.980444i	$0.563054\pi$
$854$	0	0
$855$	17.1652	0.587036
$856$	0	0
$857$	−3.28674	−0.112273	−0.0561365	−	0.998423i	$-0.517878\pi$
$857$	−3.28674	−0.112273	−0.0561365	+	0.998423i	$0.517878\pi$
$858$	0	0
$859$	7.91288	0.269984	0.134992	−	0.990847i	$-0.456899\pi$
$859$	7.91288	0.269984	0.134992	+	0.990847i	$0.456899\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3739	−0.387171	−0.193585	−	0.981083i	$-0.562012\pi$
$863$	−11.3739	−0.387171	−0.193585	+	0.981083i	$0.562012\pi$
$864$	0	0
$865$	20.8258	0.708097
$866$	0	0
$867$	−2.79129	−0.0947971
$868$	0	0
$869$	35.8258	1.21531
$870$	0	0
$871$	−43.8258	−1.48498
$872$	0	0
$873$	46.9129	1.58776
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.9129	−0.807481	−0.403740	−	0.914874i	$-0.632290\pi$
$877$	−23.9129	−0.807481	−0.403740	+	0.914874i	$0.632290\pi$
$878$	0	0
$879$	−35.5826	−1.20017
$880$	0	0
$881$	−40.1216	−1.35173	−0.675865	−	0.737025i	$-0.736230\pi$
$881$	−40.1216	−1.35173	−0.675865	+	0.737025i	$0.736230\pi$
$882$	0	0
$883$	−34.5390	−1.16233	−0.581165	−	0.813786i	$-0.697403\pi$
$883$	−34.5390	−1.16233	−0.581165	+	0.813786i	$0.697403\pi$
$884$	0	0
$885$	−57.9129	−1.94672
$886$	0	0
$887$	20.2087	0.678542	0.339271	−	0.940689i	$-0.389820\pi$
$887$	20.2087	0.678542	0.339271	+	0.940689i	$0.389820\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.49545	0.0500996
$892$	0	0
$893$	−3.16515	−0.105918
$894$	0	0
$895$	−23.6606	−0.790887
$896$	0	0
$897$	−17.6697	−0.589974
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−9.79129	−0.326195
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.08712	0.0693783
$906$	0	0
$907$	−13.5826	−0.451002	−0.225501	−	0.974243i	$-0.572402\pi$
$907$	−13.5826	−0.451002	−0.225501	+	0.974243i	$0.572402\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	−9.66970	−0.320371	−0.160186	−	0.987087i	$-0.551209\pi$
$911$	−9.66970	−0.320371	−0.160186	+	0.987087i	$0.551209\pi$
$912$	0	0
$913$	34.3303	1.13617
$914$	0	0
$915$	−33.9564	−1.12257
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.8784	−0.490793	−0.245397	−	0.969423i	$-0.578918\pi$
$919$	−14.8784	−0.490793	−0.245397	+	0.969423i	$0.578918\pi$
$920$	0	0
$921$	−21.1652	−0.697415
$922$	0	0
$923$	17.6697	0.581605
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	−36.3303	−1.19324
$928$	0	0
$929$	5.37386	0.176311	0.0881554	−	0.996107i	$-0.471903\pi$
$929$	5.37386	0.176311	0.0881554	+	0.996107i	$0.471903\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	56.4083	1.84673
$934$	0	0
$935$	−6.41742	−0.209872
$936$	0	0
$937$	16.6606	0.544278	0.272139	−	0.962258i	$-0.412269\pi$
$937$	16.6606	0.544278	0.272139	+	0.962258i	$0.412269\pi$
$938$	0	0
$939$	88.4955	2.88794
$940$	0	0
$941$	4.28674	0.139744	0.0698719	−	0.997556i	$-0.477741\pi$
$941$	4.28674	0.139744	0.0698719	+	0.997556i	$0.477741\pi$
$942$	0	0
$943$	1.91288	0.0622919
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.0871215	0.00283107	0.00141553	−	0.999999i	$-0.499549\pi$
$947$	0.0871215	0.00283107	0.00141553	+	0.999999i	$0.499549\pi$
$948$	0	0
$949$	59.1652	1.92058
$950$	0	0
$951$	−28.8348	−0.935034
$952$	0	0
$953$	−8.29583	−0.268728	−0.134364	−	0.990932i	$-0.542899\pi$
$953$	−8.29583	−0.268728	−0.134364	+	0.990932i	$0.542899\pi$
$954$	0	0
$955$	−13.9564	−0.451620
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9564	−0.998595
$962$	0	0
$963$	30.7477	0.990832
$964$	0	0
$965$	37.1652	1.19639
$966$	0	0
$967$	39.5390	1.27149	0.635745	−	0.771900i	$-0.280693\pi$
$967$	39.5390	1.27149	0.635745	+	0.771900i	$0.280693\pi$
$968$	0	0
$969$	−5.58258	−0.179338
$970$	0	0
$971$	−57.1652	−1.83452	−0.917259	−	0.398292i	$-0.869603\pi$
$971$	−57.1652	−1.83452	−0.917259	+	0.398292i	$0.869603\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	−48.5390	−1.55290	−0.776450	−	0.630178i	$-0.782982\pi$
$977$	−48.5390	−1.55290	−0.776450	+	0.630178i	$0.782982\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	93.8258	2.99563
$982$	0	0
$983$	−8.62614	−0.275131	−0.137566	−	0.990493i	$-0.543928\pi$
$983$	−8.62614	−0.275131	−0.137566	+	0.990493i	$0.543928\pi$
$984$	0	0
$985$	41.4955	1.32216
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.41742	−0.140466
$990$	0	0
$991$	45.0780	1.43195	0.715975	−	0.698126i	$-0.245982\pi$
$991$	45.0780	1.43195	0.715975	+	0.698126i	$0.245982\pi$
$992$	0	0
$993$	20.5826	0.653168
$994$	0	0
$995$	27.8348	0.882424
$996$	0	0
$997$	1.29583	0.0410395	0.0205197	−	0.999789i	$-0.493468\pi$
$997$	1.29583	0.0410395	0.0205197	+	0.999789i	$0.493468\pi$
$998$	0	0
$999$	−27.9129	−0.883124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.i.1.1	✓	2
7.6	odd	2		3332.2.a.m.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.i.1.1	✓	2	1.1	even	1	trivial
3332.2.a.m.1.2	yes	2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} +1.79129 q^{5} +4.79129 q^{9} +O(q^{10})\)	\(q-2.79129 q^{3} +1.79129 q^{5} +4.79129 q^{9} -3.58258 q^{11} +4.00000 q^{13} -5.00000 q^{15} +1.00000 q^{17} +2.00000 q^{19} +1.58258 q^{23} -1.79129 q^{25} -5.00000 q^{27} +0.208712 q^{31} +10.0000 q^{33} +5.58258 q^{37} -11.1652 q^{39} +1.20871 q^{41} -2.79129 q^{43} +8.58258 q^{45} -1.58258 q^{47} -2.79129 q^{51} -9.79129 q^{53} -6.41742 q^{55} -5.58258 q^{57} +11.5826 q^{59} +6.79129 q^{61} +7.16515 q^{65} -10.9564 q^{67} -4.41742 q^{69} +4.41742 q^{71} +14.7913 q^{73} +5.00000 q^{75} -10.0000 q^{79} -0.417424 q^{81} -9.58258 q^{83} +1.79129 q^{85} -11.1652 q^{89} -0.582576 q^{93} +3.58258 q^{95} +9.79129 q^{97} -17.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 5 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 + 5 * q^9	\( 2 q - q^{3} - q^{5} + 5 q^{9} + 2 q^{11} + 8 q^{13} - 10 q^{15} + 2 q^{17} + 4 q^{19} - 6 q^{23} + q^{25} - 10 q^{27} + 5 q^{31} + 20 q^{33} + 2 q^{37} - 4 q^{39} + 7 q^{41} - q^{43} + 8 q^{45} + 6 q^{47} - q^{51} - 15 q^{53} - 22 q^{55} - 2 q^{57} + 14 q^{59} + 9 q^{61} - 4 q^{65} + q^{67} - 18 q^{69} + 18 q^{71} + 25 q^{73} + 10 q^{75} - 20 q^{79} - 10 q^{81} - 10 q^{83} - q^{85} - 4 q^{89} + 8 q^{93} - 2 q^{95} + 15 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 5 * q^9 + 2 * q^11 + 8 * q^13 - 10 * q^15 + 2 * q^17 + 4 * q^19 - 6 * q^23 + q^25 - 10 * q^27 + 5 * q^31 + 20 * q^33 + 2 * q^37 - 4 * q^39 + 7 * q^41 - q^43 + 8 * q^45 + 6 * q^47 - q^51 - 15 * q^53 - 22 * q^55 - 2 * q^57 + 14 * q^59 + 9 * q^61 - 4 * q^65 + q^67 - 18 * q^69 + 18 * q^71 + 25 * q^73 + 10 * q^75 - 20 * q^79 - 10 * q^81 - 10 * q^83 - q^85 - 4 * q^89 + 8 * q^93 - 2 * q^95 + 15 * q^97 - 16 * q^99

Introduction

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