LMFDB - Embedded newform 2704.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(1,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, 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("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.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:	$21.5915487066$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 676)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35690 q^{3} -4.24698 q^{5} -2.13706 q^{7} -1.15883 q^{9} +3.35690 q^{11} +5.76271 q^{15} -0.0609989 q^{17} +6.18598 q^{19} +2.89977 q^{21} +5.09783 q^{23} +13.0368 q^{25} +5.64310 q^{27} -2.13706 q^{29} -2.85086 q^{31} -4.55496 q^{33} +9.07606 q^{35} -3.47219 q^{37} -5.39612 q^{41} +8.61356 q^{43} +4.92154 q^{45} -6.96077 q^{47} -2.43296 q^{49} +0.0827692 q^{51} +0.0217703 q^{53} -14.2567 q^{55} -8.39373 q^{57} -5.81163 q^{59} +10.6799 q^{61} +2.47650 q^{63} -7.40581 q^{67} -6.91723 q^{69} +0.0489173 q^{71} -4.51573 q^{73} -17.6896 q^{75} -7.17390 q^{77} -12.8170 q^{79} -4.18060 q^{81} +3.55496 q^{83} +0.259061 q^{85} +2.89977 q^{87} -2.34721 q^{89} +3.86831 q^{93} -26.2717 q^{95} +3.42327 q^{97} -3.89008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{5} - q^{7} + 5 q^{9} + 6 q^{11} - 10 q^{17} + 4 q^{19} - 14 q^{21} - 3 q^{23} + 11 q^{25} + 21 q^{27} - q^{29} + 5 q^{31} - 14 q^{33} + 12 q^{35} - 4 q^{37} - 25 q^{41} - 5 q^{43} - 11 q^{45}+ \cdots - 11 q^{99}+O(q^{100}) $ 3 * q - 8 * q^5 - q^7 + 5 * q^9 + 6 * q^11 - 10 * q^17 + 4 * q^19 - 14 * q^21 - 3 * q^23 + 11 * q^25 + 21 * q^27 - q^29 + 5 * q^31 - 14 * q^33 + 12 * q^35 - 4 * q^37 - 25 * q^41 - 5 * q^43 - 11 * q^45 - 8 * q^47 + 12 * q^49 + 7 * q^51 - 3 * q^53 - 16 * q^55 + 7 * q^57 + 9 * q^59 + 8 * q^61 - 18 * q^63 - 9 * q^67 - 14 * q^69 - 9 * q^71 - q^73 - 7 * q^75 + 12 * q^77 - 9 * q^79 - q^81 + 11 * q^83 + 15 * q^85 - 14 * q^87 - 25 * q^89 + 14 * q^93 - 27 * q^95 + 13 * q^97 - 11 * q^99

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
$3$	−1.35690	−0.783404	−0.391702	−	0.920092i	$-0.628114\pi$
$3$	−1.35690	−0.783404	−0.391702	+	0.920092i	$0.628114\pi$
$4$	0	0
$5$	−4.24698	−1.89931	−0.949654	−	0.313302i	$-0.898565\pi$
$5$	−4.24698	−1.89931	−0.949654	+	0.313302i	$0.898565\pi$
$6$	0	0
$7$	−2.13706	−0.807734	−0.403867	−	0.914818i	$-0.632334\pi$
$7$	−2.13706	−0.807734	−0.403867	+	0.914818i	$0.632334\pi$
$8$	0	0
$9$	−1.15883	−0.386278
$10$	0	0
$11$	3.35690	1.01214	0.506071	−	0.862492i	$-0.331097\pi$
$11$	3.35690	1.01214	0.506071	+	0.862492i	$0.331097\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.76271	1.48793
$16$	0	0
$17$	−0.0609989	−0.0147944	−0.00739721	−	0.999973i	$-0.502355\pi$
$17$	−0.0609989	−0.0147944	−0.00739721	+	0.999973i	$0.502355\pi$
$18$	0	0
$19$	6.18598	1.41916	0.709581	−	0.704624i	$-0.248884\pi$
$19$	6.18598	1.41916	0.709581	+	0.704624i	$0.248884\pi$
$20$	0	0
$21$	2.89977	0.632782
$22$	0	0
$23$	5.09783	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$23$	5.09783	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$24$	0	0
$25$	13.0368	2.60737
$26$	0	0
$27$	5.64310	1.08602
$28$	0	0
$29$	−2.13706	−0.396843	−0.198421	−	0.980117i	$-0.563581\pi$
$29$	−2.13706	−0.396843	−0.198421	+	0.980117i	$0.563581\pi$
$30$	0	0
$31$	−2.85086	−0.512029	−0.256014	−	0.966673i	$-0.582409\pi$
$31$	−2.85086	−0.512029	−0.256014	+	0.966673i	$0.582409\pi$
$32$	0	0
$33$	−4.55496	−0.792916
$34$	0	0
$35$	9.07606	1.53413
$36$	0	0
$37$	−3.47219	−0.570824	−0.285412	−	0.958405i	$-0.592130\pi$
$37$	−3.47219	−0.570824	−0.285412	+	0.958405i	$0.592130\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.39612	−0.842733	−0.421367	−	0.906890i	$-0.638449\pi$
$41$	−5.39612	−0.842733	−0.421367	+	0.906890i	$0.638449\pi$
$42$	0	0
$43$	8.61356	1.31356	0.656778	−	0.754084i	$-0.271919\pi$
$43$	8.61356	1.31356	0.656778	+	0.754084i	$0.271919\pi$
$44$	0	0
$45$	4.92154	0.733660
$46$	0	0
$47$	−6.96077	−1.01533	−0.507666	−	0.861554i	$-0.669492\pi$
$47$	−6.96077	−1.01533	−0.507666	+	0.861554i	$0.669492\pi$
$48$	0	0
$49$	−2.43296	−0.347566
$50$	0	0
$51$	0.0827692	0.0115900
$52$	0	0
$53$	0.0217703	0.00299038	0.00149519	−	0.999999i	$-0.499524\pi$
$53$	0.0217703	0.00299038	0.00149519	+	0.999999i	$0.499524\pi$
$54$	0	0
$55$	−14.2567	−1.92237
$56$	0	0
$57$	−8.39373	−1.11178
$58$	0	0
$59$	−5.81163	−0.756609	−0.378305	−	0.925681i	$-0.623493\pi$
$59$	−5.81163	−0.756609	−0.378305	+	0.925681i	$0.623493\pi$
$60$	0	0
$61$	10.6799	1.36743	0.683713	−	0.729751i	$-0.260364\pi$
$61$	10.6799	1.36743	0.683713	+	0.729751i	$0.260364\pi$
$62$	0	0
$63$	2.47650	0.312010
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.40581	−0.904764	−0.452382	−	0.891824i	$-0.649426\pi$
$67$	−7.40581	−0.904764	−0.452382	+	0.891824i	$0.649426\pi$
$68$	0	0
$69$	−6.91723	−0.832737
$70$	0	0
$71$	0.0489173	0.00580542	0.00290271	−	0.999996i	$-0.499076\pi$
$71$	0.0489173	0.00580542	0.00290271	+	0.999996i	$0.499076\pi$
$72$	0	0
$73$	−4.51573	−0.528526	−0.264263	−	0.964451i	$-0.585129\pi$
$73$	−4.51573	−0.528526	−0.264263	+	0.964451i	$0.585129\pi$
$74$	0	0
$75$	−17.6896	−2.04262
$76$	0	0
$77$	−7.17390	−0.817542
$78$	0	0
$79$	−12.8170	−1.44203	−0.721013	−	0.692922i	$-0.756323\pi$
$79$	−12.8170	−1.44203	−0.721013	+	0.692922i	$0.756323\pi$
$80$	0	0
$81$	−4.18060	−0.464512
$82$	0	0
$83$	3.55496	0.390207	0.195104	−	0.980783i	$-0.437496\pi$
$83$	3.55496	0.390207	0.195104	+	0.980783i	$0.437496\pi$
$84$	0	0
$85$	0.259061	0.0280991
$86$	0	0
$87$	2.89977	0.310888
$88$	0	0
$89$	−2.34721	−0.248803	−0.124402	−	0.992232i	$-0.539701\pi$
$89$	−2.34721	−0.248803	−0.124402	+	0.992232i	$0.539701\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.86831	0.401125
$94$	0	0
$95$	−26.2717	−2.69542
$96$	0	0
$97$	3.42327	0.347581	0.173790	−	0.984783i	$-0.444399\pi$
$97$	3.42327	0.347581	0.173790	+	0.984783i	$0.444399\pi$
$98$	0	0
$99$	−3.89008	−0.390968
$100$	0	0
$101$	−11.3787	−1.13222	−0.566110	−	0.824330i	$-0.691552\pi$
$101$	−11.3787	−1.13222	−0.566110	+	0.824330i	$0.691552\pi$
$102$	0	0
$103$	6.07606	0.598692	0.299346	−	0.954145i	$-0.403231\pi$
$103$	6.07606	0.598692	0.299346	+	0.954145i	$0.403231\pi$
$104$	0	0
$105$	−12.3153	−1.20185
$106$	0	0
$107$	16.0368	1.55034	0.775170	−	0.631753i	$-0.217664\pi$
$107$	16.0368	1.55034	0.775170	+	0.631753i	$0.217664\pi$
$108$	0	0
$109$	14.9095	1.42807	0.714034	−	0.700111i	$-0.246866\pi$
$109$	14.9095	1.42807	0.714034	+	0.700111i	$0.246866\pi$
$110$	0	0
$111$	4.71140	0.447186
$112$	0	0
$113$	8.24459	0.775585	0.387793	−	0.921747i	$-0.373238\pi$
$113$	8.24459	0.775585	0.387793	+	0.921747i	$0.373238\pi$
$114$	0	0
$115$	−21.6504	−2.01891
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.130359	0.0119500
$120$	0	0
$121$	0.268750	0.0244318
$122$	0	0
$123$	7.32198	0.660201
$124$	0	0
$125$	−34.1323	−3.05288
$126$	0	0
$127$	6.64310	0.589480	0.294740	−	0.955577i	$-0.404767\pi$
$127$	6.64310	0.589480	0.294740	+	0.955577i	$0.404767\pi$
$128$	0	0
$129$	−11.6877	−1.02905
$130$	0	0
$131$	−7.86294	−0.686988	−0.343494	−	0.939155i	$-0.611610\pi$
$131$	−7.86294	−0.686988	−0.343494	+	0.939155i	$0.611610\pi$
$132$	0	0
$133$	−13.2198	−1.14630
$134$	0	0
$135$	−23.9661	−2.06268
$136$	0	0
$137$	0.987918	0.0844036	0.0422018	−	0.999109i	$-0.486563\pi$
$137$	0.987918	0.0844036	0.0422018	+	0.999109i	$0.486563\pi$
$138$	0	0
$139$	12.0368	1.02095	0.510476	−	0.859892i	$-0.329469\pi$
$139$	12.0368	1.02095	0.510476	+	0.859892i	$0.329469\pi$
$140$	0	0
$141$	9.44504	0.795416
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.07606	0.753726
$146$	0	0
$147$	3.30127	0.272284
$148$	0	0
$149$	−2.35929	−0.193280	−0.0966402	−	0.995319i	$-0.530810\pi$
$149$	−2.35929	−0.193280	−0.0966402	+	0.995319i	$0.530810\pi$
$150$	0	0
$151$	−7.43967	−0.605431	−0.302716	−	0.953081i	$-0.597893\pi$
$151$	−7.43967	−0.605431	−0.302716	+	0.953081i	$0.597893\pi$
$152$	0	0
$153$	0.0706876	0.00571475
$154$	0	0
$155$	12.1075	0.972500
$156$	0	0
$157$	−10.9487	−0.873801	−0.436900	−	0.899510i	$-0.643924\pi$
$157$	−10.9487	−0.873801	−0.436900	+	0.899510i	$0.643924\pi$
$158$	0	0
$159$	−0.0295400	−0.00234267
$160$	0	0
$161$	−10.8944	−0.858599
$162$	0	0
$163$	7.02475	0.550221	0.275111	−	0.961413i	$-0.411286\pi$
$163$	7.02475	0.550221	0.275111	+	0.961413i	$0.411286\pi$
$164$	0	0
$165$	19.3448	1.50599
$166$	0	0
$167$	−15.4819	−1.19802	−0.599012	−	0.800740i	$-0.704440\pi$
$167$	−15.4819	−1.19802	−0.599012	+	0.800740i	$0.704440\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−7.16852	−0.548191
$172$	0	0
$173$	13.7802	1.04769	0.523843	−	0.851815i	$-0.324498\pi$
$173$	13.7802	1.04769	0.523843	+	0.851815i	$0.324498\pi$
$174$	0	0
$175$	−27.8605	−2.10606
$176$	0	0
$177$	7.88577	0.592731
$178$	0	0
$179$	−22.1226	−1.65352	−0.826760	−	0.562555i	$-0.809819\pi$
$179$	−22.1226	−1.65352	−0.826760	+	0.562555i	$0.809819\pi$
$180$	0	0
$181$	−13.6407	−1.01391	−0.506953	−	0.861974i	$-0.669228\pi$
$181$	−13.6407	−1.01391	−0.506953	+	0.861974i	$0.669228\pi$
$182$	0	0
$183$	−14.4916	−1.07125
$184$	0	0
$185$	14.7463	1.08417
$186$	0	0
$187$	−0.204767	−0.0149740
$188$	0	0
$189$	−12.0597	−0.877212
$190$	0	0
$191$	−4.49157	−0.324998	−0.162499	−	0.986709i	$-0.551955\pi$
$191$	−4.49157	−0.324998	−0.162499	+	0.986709i	$0.551955\pi$
$192$	0	0
$193$	−25.7754	−1.85535	−0.927676	−	0.373385i	$-0.878197\pi$
$193$	−25.7754	−1.85535	−0.927676	+	0.373385i	$0.878197\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.6233	1.32685	0.663426	−	0.748242i	$-0.269102\pi$
$197$	18.6233	1.32685	0.663426	+	0.748242i	$0.269102\pi$
$198$	0	0
$199$	11.9661	0.848258	0.424129	−	0.905602i	$-0.360580\pi$
$199$	11.9661	0.848258	0.424129	+	0.905602i	$0.360580\pi$
$200$	0	0
$201$	10.0489	0.708796
$202$	0	0
$203$	4.56704	0.320543
$204$	0	0
$205$	22.9172	1.60061
$206$	0	0
$207$	−5.90754	−0.410603
$208$	0	0
$209$	20.7657	1.43639
$210$	0	0
$211$	−2.96077	−0.203828	−0.101914	−	0.994793i	$-0.532497\pi$
$211$	−2.96077	−0.203828	−0.101914	+	0.994793i	$0.532497\pi$
$212$	0	0
$213$	−0.0663757	−0.00454799
$214$	0	0
$215$	−36.5816	−2.49485
$216$	0	0
$217$	6.09246	0.413583
$218$	0	0
$219$	6.12737	0.414050
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.44265	−0.364467	−0.182233	−	0.983255i	$-0.558333\pi$
$223$	−5.44265	−0.364467	−0.182233	+	0.983255i	$0.558333\pi$
$224$	0	0
$225$	−15.1075	−1.00717
$226$	0	0
$227$	22.4480	1.48993	0.744964	−	0.667105i	$-0.232467\pi$
$227$	22.4480	1.48993	0.744964	+	0.667105i	$0.232467\pi$
$228$	0	0
$229$	16.6136	1.09786	0.548928	−	0.835870i	$-0.315036\pi$
$229$	16.6136	1.09786	0.548928	+	0.835870i	$0.315036\pi$
$230$	0	0
$231$	9.73423	0.640466
$232$	0	0
$233$	9.73125	0.637515	0.318758	−	0.947836i	$-0.396734\pi$
$233$	9.73125	0.637515	0.318758	+	0.947836i	$0.396734\pi$
$234$	0	0
$235$	29.5623	1.92843
$236$	0	0
$237$	17.3913	1.12969
$238$	0	0
$239$	−2.82908	−0.182998	−0.0914991	−	0.995805i	$-0.529166\pi$
$239$	−2.82908	−0.182998	−0.0914991	+	0.995805i	$0.529166\pi$
$240$	0	0
$241$	−18.3448	−1.18169	−0.590847	−	0.806784i	$-0.701206\pi$
$241$	−18.3448	−1.18169	−0.590847	+	0.806784i	$0.701206\pi$
$242$	0	0
$243$	−11.2567	−0.722116
$244$	0	0
$245$	10.3327	0.660134
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4.82371	−0.305690
$250$	0	0
$251$	2.53750	0.160166	0.0800828	−	0.996788i	$-0.474482\pi$
$251$	2.53750	0.160166	0.0800828	+	0.996788i	$0.474482\pi$
$252$	0	0
$253$	17.1129	1.07588
$254$	0	0
$255$	−0.351519	−0.0220130
$256$	0	0
$257$	−23.4862	−1.46503	−0.732514	−	0.680752i	$-0.761653\pi$
$257$	−23.4862	−1.46503	−0.732514	+	0.680752i	$0.761653\pi$
$258$	0	0
$259$	7.42029	0.461074
$260$	0	0
$261$	2.47650	0.153292
$262$	0	0
$263$	15.8092	0.974839	0.487420	−	0.873168i	$-0.337938\pi$
$263$	15.8092	0.974839	0.487420	+	0.873168i	$0.337938\pi$
$264$	0	0
$265$	−0.0924579	−0.00567964
$266$	0	0
$267$	3.18492	0.194914
$268$	0	0
$269$	−30.8582	−1.88145	−0.940727	−	0.339164i	$-0.889856\pi$
$269$	−30.8582	−1.88145	−0.940727	+	0.339164i	$0.889856\pi$
$270$	0	0
$271$	16.1933	0.983671	0.491836	−	0.870688i	$-0.336326\pi$
$271$	16.1933	0.983671	0.491836	+	0.870688i	$0.336326\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	43.7633	2.63903
$276$	0	0
$277$	13.1075	0.787555	0.393777	−	0.919206i	$-0.371168\pi$
$277$	13.1075	0.787555	0.393777	+	0.919206i	$0.371168\pi$
$278$	0	0
$279$	3.30367	0.197785
$280$	0	0
$281$	−14.2687	−0.851202	−0.425601	−	0.904911i	$-0.639937\pi$
$281$	−14.2687	−0.851202	−0.425601	+	0.904911i	$0.639937\pi$
$282$	0	0
$283$	7.95646	0.472962	0.236481	−	0.971636i	$-0.424006\pi$
$283$	7.95646	0.472962	0.236481	+	0.971636i	$0.424006\pi$
$284$	0	0
$285$	35.6480	2.11161
$286$	0	0
$287$	11.5319	0.680704
$288$	0	0
$289$	−16.9963	−0.999781
$290$	0	0
$291$	−4.64502	−0.272296
$292$	0	0
$293$	−19.6300	−1.14679	−0.573397	−	0.819278i	$-0.694375\pi$
$293$	−19.6300	−1.14679	−0.573397	+	0.819278i	$0.694375\pi$
$294$	0	0
$295$	24.6819	1.43703
$296$	0	0
$297$	18.9433	1.09920
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−18.4077	−1.06100
$302$	0	0
$303$	15.4397	0.886986
$304$	0	0
$305$	−45.3575	−2.59716
$306$	0	0
$307$	−15.6775	−0.894765	−0.447382	−	0.894343i	$-0.647644\pi$
$307$	−15.6775	−0.894765	−0.447382	+	0.894343i	$0.647644\pi$
$308$	0	0
$309$	−8.24459	−0.469018
$310$	0	0
$311$	−16.4983	−0.935531	−0.467766	−	0.883853i	$-0.654941\pi$
$311$	−16.4983	−0.935531	−0.467766	+	0.883853i	$0.654941\pi$
$312$	0	0
$313$	−11.3937	−0.644012	−0.322006	−	0.946738i	$-0.604357\pi$
$313$	−11.3937	−0.644012	−0.322006	+	0.946738i	$0.604357\pi$
$314$	0	0
$315$	−10.5176	−0.592602
$316$	0	0
$317$	−25.4470	−1.42924	−0.714622	−	0.699511i	$-0.753401\pi$
$317$	−25.4470	−1.42924	−0.714622	+	0.699511i	$0.753401\pi$
$318$	0	0
$319$	−7.17390	−0.401661
$320$	0	0
$321$	−21.7603	−1.21454
$322$	0	0
$323$	−0.377338	−0.0209957
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.2306	−1.11875
$328$	0	0
$329$	14.8756	0.820119
$330$	0	0
$331$	8.95539	0.492233	0.246117	−	0.969240i	$-0.420845\pi$
$331$	8.95539	0.492233	0.246117	+	0.969240i	$0.420845\pi$
$332$	0	0
$333$	4.02369	0.220497
$334$	0	0
$335$	31.4523	1.71842
$336$	0	0
$337$	19.3110	1.05194	0.525968	−	0.850505i	$-0.323703\pi$
$337$	19.3110	1.05194	0.525968	+	0.850505i	$0.323703\pi$
$338$	0	0
$339$	−11.1870	−0.607597
$340$	0	0
$341$	−9.57002	−0.518246
$342$	0	0
$343$	20.1588	1.08847
$344$	0	0
$345$	29.3773	1.58162
$346$	0	0
$347$	−36.0411	−1.93479	−0.967395	−	0.253272i	$-0.918493\pi$
$347$	−36.0411	−1.93479	−0.967395	+	0.253272i	$0.918493\pi$
$348$	0	0
$349$	−23.2416	−1.24409	−0.622047	−	0.782980i	$-0.713699\pi$
$349$	−23.2416	−1.24409	−0.622047	+	0.782980i	$0.713699\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.2349	−1.28989	−0.644947	−	0.764228i	$-0.723120\pi$
$353$	−24.2349	−1.28989	−0.644947	+	0.764228i	$0.723120\pi$
$354$	0	0
$355$	−0.207751	−0.0110263
$356$	0	0
$357$	−0.176883	−0.00936164
$358$	0	0
$359$	17.7332	0.935921	0.467960	−	0.883749i	$-0.344989\pi$
$359$	17.7332	0.935921	0.467960	+	0.883749i	$0.344989\pi$
$360$	0	0
$361$	19.2664	1.01402
$362$	0	0
$363$	−0.364666	−0.0191400
$364$	0	0
$365$	19.1782	1.00383
$366$	0	0
$367$	−30.2083	−1.57686	−0.788431	−	0.615123i	$-0.789106\pi$
$367$	−30.2083	−1.57686	−0.788431	+	0.615123i	$0.789106\pi$
$368$	0	0
$369$	6.25321	0.325529
$370$	0	0
$371$	−0.0465244	−0.00241543
$372$	0	0
$373$	−0.681268	−0.0352747	−0.0176374	−	0.999844i	$-0.505614\pi$
$373$	−0.681268	−0.0352747	−0.0176374	+	0.999844i	$0.505614\pi$
$374$	0	0
$375$	46.3139	2.39164
$376$	0	0
$377$	0	0
$378$	0	0
$379$	36.4959	1.87467	0.937334	−	0.348433i	$-0.113286\pi$
$379$	36.4959	1.87467	0.937334	+	0.348433i	$0.113286\pi$
$380$	0	0
$381$	−9.01400	−0.461801
$382$	0	0
$383$	−5.20775	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$383$	−5.20775	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$384$	0	0
$385$	30.4674	1.55276
$386$	0	0
$387$	−9.98169	−0.507398
$388$	0	0
$389$	−17.5254	−0.888574	−0.444287	−	0.895885i	$-0.646543\pi$
$389$	−17.5254	−0.888574	−0.444287	+	0.895885i	$0.646543\pi$
$390$	0	0
$391$	−0.310962	−0.0157260
$392$	0	0
$393$	10.6692	0.538189
$394$	0	0
$395$	54.4336	2.73885
$396$	0	0
$397$	29.8049	1.49587	0.747933	−	0.663774i	$-0.231046\pi$
$397$	29.8049	1.49587	0.747933	+	0.663774i	$0.231046\pi$
$398$	0	0
$399$	17.9379	0.898020
$400$	0	0
$401$	−16.3341	−0.815684	−0.407842	−	0.913053i	$-0.633719\pi$
$401$	−16.3341	−0.815684	−0.407842	+	0.913053i	$0.633719\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	17.7549	0.882250
$406$	0	0
$407$	−11.6558	−0.577755
$408$	0	0
$409$	9.49827	0.469659	0.234830	−	0.972037i	$-0.424547\pi$
$409$	9.49827	0.469659	0.234830	+	0.972037i	$0.424547\pi$
$410$	0	0
$411$	−1.34050	−0.0661221
$412$	0	0
$413$	12.4198	0.611139
$414$	0	0
$415$	−15.0978	−0.741124
$416$	0	0
$417$	−16.3327	−0.799817
$418$	0	0
$419$	−12.4450	−0.607980	−0.303990	−	0.952675i	$-0.598319\pi$
$419$	−12.4450	−0.607980	−0.303990	+	0.952675i	$0.598319\pi$
$420$	0	0
$421$	−3.65578	−0.178172	−0.0890858	−	0.996024i	$-0.528395\pi$
$421$	−3.65578	−0.178172	−0.0890858	+	0.996024i	$0.528395\pi$
$422$	0	0
$423$	8.06638	0.392201
$424$	0	0
$425$	−0.795233	−0.0385745
$426$	0	0
$427$	−22.8237	−1.10452
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.6219	−1.71585	−0.857924	−	0.513777i	$-0.828246\pi$
$431$	−35.6219	−1.71585	−0.857924	+	0.513777i	$0.828246\pi$
$432$	0	0
$433$	−34.6926	−1.66722	−0.833610	−	0.552353i	$-0.813730\pi$
$433$	−34.6926	−1.66722	−0.833610	+	0.552353i	$0.813730\pi$
$434$	0	0
$435$	−12.3153	−0.590472
$436$	0	0
$437$	31.5351	1.50853
$438$	0	0
$439$	−9.48858	−0.452865	−0.226433	−	0.974027i	$-0.572706\pi$
$439$	−9.48858	−0.452865	−0.226433	+	0.974027i	$0.572706\pi$
$440$	0	0
$441$	2.81940	0.134257
$442$	0	0
$443$	−14.1849	−0.673946	−0.336973	−	0.941514i	$-0.609403\pi$
$443$	−14.1849	−0.673946	−0.336973	+	0.941514i	$0.609403\pi$
$444$	0	0
$445$	9.96854	0.472554
$446$	0	0
$447$	3.20131	0.151417
$448$	0	0
$449$	6.99894	0.330300	0.165150	−	0.986268i	$-0.447189\pi$
$449$	6.99894	0.330300	0.165150	+	0.986268i	$0.447189\pi$
$450$	0	0
$451$	−18.1142	−0.852966
$452$	0	0
$453$	10.0949	0.474297
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.21313	−0.150304	−0.0751519	−	0.997172i	$-0.523944\pi$
$457$	−3.21313	−0.150304	−0.0751519	+	0.997172i	$0.523944\pi$
$458$	0	0
$459$	−0.344223	−0.0160670
$460$	0	0
$461$	−15.5278	−0.723202	−0.361601	−	0.932333i	$-0.617770\pi$
$461$	−15.5278	−0.723202	−0.361601	+	0.932333i	$0.617770\pi$
$462$	0	0
$463$	32.0834	1.49104	0.745520	−	0.666483i	$-0.232201\pi$
$463$	32.0834	1.49104	0.745520	+	0.666483i	$0.232201\pi$
$464$	0	0
$465$	−16.4286	−0.761860
$466$	0	0
$467$	−7.24459	−0.335239	−0.167620	−	0.985852i	$-0.553608\pi$
$467$	−7.24459	−0.335239	−0.167620	+	0.985852i	$0.553608\pi$
$468$	0	0
$469$	15.8267	0.730809
$470$	0	0
$471$	14.8562	0.684539
$472$	0	0
$473$	28.9148	1.32951
$474$	0	0
$475$	80.6456	3.70027
$476$	0	0
$477$	−0.0252281	−0.00115512
$478$	0	0
$479$	5.78209	0.264190	0.132095	−	0.991237i	$-0.457830\pi$
$479$	5.78209	0.264190	0.132095	+	0.991237i	$0.457830\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.7826	0.672630
$484$	0	0
$485$	−14.5386	−0.660162
$486$	0	0
$487$	4.14782	0.187956	0.0939778	−	0.995574i	$-0.470042\pi$
$487$	4.14782	0.187956	0.0939778	+	0.995574i	$0.470042\pi$
$488$	0	0
$489$	−9.53186	−0.431046
$490$	0	0
$491$	−7.19375	−0.324649	−0.162325	−	0.986737i	$-0.551899\pi$
$491$	−7.19375	−0.324649	−0.162325	+	0.986737i	$0.551899\pi$
$492$	0	0
$493$	0.130359	0.00587105
$494$	0	0
$495$	16.5211	0.742569
$496$	0	0
$497$	−0.104539	−0.00468924
$498$	0	0
$499$	−41.5066	−1.85809	−0.929046	−	0.369964i	$-0.879370\pi$
$499$	−41.5066	−1.85809	−0.929046	+	0.369964i	$0.879370\pi$
$500$	0	0
$501$	21.0073	0.938537
$502$	0	0
$503$	25.8605	1.15306	0.576532	−	0.817074i	$-0.304405\pi$
$503$	25.8605	1.15306	0.576532	+	0.817074i	$0.304405\pi$
$504$	0	0
$505$	48.3250	2.15043
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.3924	−1.56874	−0.784370	−	0.620293i	$-0.787014\pi$
$509$	−35.3924	−1.56874	−0.784370	+	0.620293i	$0.787014\pi$
$510$	0	0
$511$	9.65040	0.426909
$512$	0	0
$513$	34.9081	1.54123
$514$	0	0
$515$	−25.8049	−1.13710
$516$	0	0
$517$	−23.3666	−1.02766
$518$	0	0
$519$	−18.6983	−0.820762
$520$	0	0
$521$	−34.2620	−1.50105	−0.750524	−	0.660844i	$-0.770199\pi$
$521$	−34.2620	−1.50105	−0.750524	+	0.660844i	$0.770199\pi$
$522$	0	0
$523$	14.6028	0.638536	0.319268	−	0.947664i	$-0.396563\pi$
$523$	14.6028	0.638536	0.319268	+	0.947664i	$0.396563\pi$
$524$	0	0
$525$	37.8039	1.64990
$526$	0	0
$527$	0.173899	0.00757516
$528$	0	0
$529$	2.98792	0.129909
$530$	0	0
$531$	6.73471	0.292261
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−68.1081	−2.94457
$536$	0	0
$537$	30.0180	1.29537
$538$	0	0
$539$	−8.16719	−0.351786
$540$	0	0
$541$	−18.8866	−0.811999	−0.406000	−	0.913873i	$-0.633077\pi$
$541$	−18.8866	−0.811999	−0.406000	+	0.913873i	$0.633077\pi$
$542$	0	0
$543$	18.5090	0.794298
$544$	0	0
$545$	−63.3202	−2.71234
$546$	0	0
$547$	34.6752	1.48260	0.741301	−	0.671172i	$-0.234209\pi$
$547$	34.6752	1.48260	0.741301	+	0.671172i	$0.234209\pi$
$548$	0	0
$549$	−12.3763	−0.528206
$550$	0	0
$551$	−13.2198	−0.563184
$552$	0	0
$553$	27.3907	1.16477
$554$	0	0
$555$	−20.0092	−0.849344
$556$	0	0
$557$	4.25667	0.180361	0.0901804	−	0.995925i	$-0.471256\pi$
$557$	4.25667	0.180361	0.0901804	+	0.995925i	$0.471256\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.277848	0.0117307
$562$	0	0
$563$	−21.1371	−0.890821	−0.445411	−	0.895326i	$-0.646942\pi$
$563$	−21.1371	−0.890821	−0.445411	+	0.895326i	$0.646942\pi$
$564$	0	0
$565$	−35.0146	−1.47307
$566$	0	0
$567$	8.93422	0.375202
$568$	0	0
$569$	35.4795	1.48738	0.743689	−	0.668526i	$-0.233074\pi$
$569$	35.4795	1.48738	0.743689	+	0.668526i	$0.233074\pi$
$570$	0	0
$571$	−3.94033	−0.164898	−0.0824488	−	0.996595i	$-0.526274\pi$
$571$	−3.94033	−0.164898	−0.0824488	+	0.996595i	$0.526274\pi$
$572$	0	0
$573$	6.09459	0.254605
$574$	0	0
$575$	66.4596	2.77156
$576$	0	0
$577$	16.6256	0.692135	0.346067	−	0.938210i	$-0.387517\pi$
$577$	16.6256	0.692135	0.346067	+	0.938210i	$0.387517\pi$
$578$	0	0
$579$	34.9745	1.45349
$580$	0	0
$581$	−7.59717	−0.315184
$582$	0	0
$583$	0.0730805	0.00302669
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.00298	−0.330318	−0.165159	−	0.986267i	$-0.552814\pi$
$587$	−8.00298	−0.330318	−0.165159	+	0.986267i	$0.552814\pi$
$588$	0	0
$589$	−17.6353	−0.726651
$590$	0	0
$591$	−25.2698	−1.03946
$592$	0	0
$593$	−15.8345	−0.650243	−0.325122	−	0.945672i	$-0.605405\pi$
$593$	−15.8345	−0.650243	−0.325122	+	0.945672i	$0.605405\pi$
$594$	0	0
$595$	−0.553630	−0.0226966
$596$	0	0
$597$	−16.2368	−0.664529
$598$	0	0
$599$	−25.9226	−1.05917	−0.529585	−	0.848257i	$-0.677652\pi$
$599$	−25.9226	−1.05917	−0.529585	+	0.848257i	$0.677652\pi$
$600$	0	0
$601$	20.0616	0.818329	0.409165	−	0.912461i	$-0.365820\pi$
$601$	20.0616	0.818329	0.409165	+	0.912461i	$0.365820\pi$
$602$	0	0
$603$	8.58211	0.349490
$604$	0	0
$605$	−1.14138	−0.0464035
$606$	0	0
$607$	−41.7614	−1.69504	−0.847521	−	0.530762i	$-0.821906\pi$
$607$	−41.7614	−1.69504	−0.847521	+	0.530762i	$0.821906\pi$
$608$	0	0
$609$	−6.19700	−0.251115
$610$	0	0
$611$	0	0
$612$	0	0
$613$	12.4179	0.501554	0.250777	−	0.968045i	$-0.419314\pi$
$613$	12.4179	0.501554	0.250777	+	0.968045i	$0.419314\pi$
$614$	0	0
$615$	−31.0963	−1.25392
$616$	0	0
$617$	17.2198	0.693244	0.346622	−	0.938005i	$-0.387329\pi$
$617$	17.2198	0.693244	0.346622	+	0.938005i	$0.387329\pi$
$618$	0	0
$619$	42.6136	1.71278	0.856392	−	0.516326i	$-0.172701\pi$
$619$	42.6136	1.71278	0.856392	+	0.516326i	$0.172701\pi$
$620$	0	0
$621$	28.7676	1.15440
$622$	0	0
$623$	5.01613	0.200967
$624$	0	0
$625$	79.7749	3.19100
$626$	0	0
$627$	−28.1769	−1.12528
$628$	0	0
$629$	0.211800	0.00844501
$630$	0	0
$631$	−26.8605	−1.06930	−0.534651	−	0.845073i	$-0.679557\pi$
$631$	−26.8605	−1.06930	−0.534651	+	0.845073i	$0.679557\pi$
$632$	0	0
$633$	4.01746	0.159680
$634$	0	0
$635$	−28.2131	−1.11960
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.0566871	−0.00224251
$640$	0	0
$641$	33.3045	1.31545	0.657725	−	0.753258i	$-0.271519\pi$
$641$	33.3045	1.31545	0.657725	+	0.753258i	$0.271519\pi$
$642$	0	0
$643$	30.6746	1.20969	0.604843	−	0.796344i	$-0.293236\pi$
$643$	30.6746	1.20969	0.604843	+	0.796344i	$0.293236\pi$
$644$	0	0
$645$	49.6375	1.95447
$646$	0	0
$647$	3.25608	0.128010	0.0640048	−	0.997950i	$-0.479613\pi$
$647$	3.25608	0.128010	0.0640048	+	0.997950i	$0.479613\pi$
$648$	0	0
$649$	−19.5090	−0.765796
$650$	0	0
$651$	−8.26683	−0.324003
$652$	0	0
$653$	14.8183	0.579886	0.289943	−	0.957044i	$-0.406364\pi$
$653$	14.8183	0.579886	0.289943	+	0.957044i	$0.406364\pi$
$654$	0	0
$655$	33.3937	1.30480
$656$	0	0
$657$	5.23298	0.204158
$658$	0	0
$659$	−30.5392	−1.18964	−0.594818	−	0.803860i	$-0.702776\pi$
$659$	−30.5392	−1.18964	−0.594818	+	0.803860i	$0.702776\pi$
$660$	0	0
$661$	19.3676	0.753314	0.376657	−	0.926353i	$-0.377073\pi$
$661$	19.3676	0.753314	0.376657	+	0.926353i	$0.377073\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	56.1444	2.17718
$666$	0	0
$667$	−10.8944	−0.421833
$668$	0	0
$669$	7.38511	0.285525
$670$	0	0
$671$	35.8514	1.38403
$672$	0	0
$673$	41.5459	1.60148	0.800738	−	0.599015i	$-0.204441\pi$
$673$	41.5459	1.60148	0.800738	+	0.599015i	$0.204441\pi$
$674$	0	0
$675$	73.5682	2.83164
$676$	0	0
$677$	−40.6064	−1.56063	−0.780315	−	0.625387i	$-0.784941\pi$
$677$	−40.6064	−1.56063	−0.780315	+	0.625387i	$0.784941\pi$
$678$	0	0
$679$	−7.31575	−0.280753
$680$	0	0
$681$	−30.4596	−1.16722
$682$	0	0
$683$	−2.40581	−0.0920559	−0.0460279	−	0.998940i	$-0.514656\pi$
$683$	−2.40581	−0.0920559	−0.0460279	+	0.998940i	$0.514656\pi$
$684$	0	0
$685$	−4.19567	−0.160308
$686$	0	0
$687$	−22.5429	−0.860064
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.81269	0.259167	0.129583	−	0.991569i	$-0.458636\pi$
$691$	6.81269	0.259167	0.129583	+	0.991569i	$0.458636\pi$
$692$	0	0
$693$	8.31336	0.315798
$694$	0	0
$695$	−51.1202	−1.93910
$696$	0	0
$697$	0.329158	0.0124677
$698$	0	0
$699$	−13.2043	−0.499432
$700$	0	0
$701$	−12.7802	−0.482700	−0.241350	−	0.970438i	$-0.577590\pi$
$701$	−12.7802	−0.482700	−0.241350	+	0.970438i	$0.577590\pi$
$702$	0	0
$703$	−21.4789	−0.810092
$704$	0	0
$705$	−40.1129	−1.51074
$706$	0	0
$707$	24.3169	0.914532
$708$	0	0
$709$	4.94139	0.185578	0.0927890	−	0.995686i	$-0.470422\pi$
$709$	4.94139	0.185578	0.0927890	+	0.995686i	$0.470422\pi$
$710$	0	0
$711$	14.8528	0.557023
$712$	0	0
$713$	−14.5332	−0.544272
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.83877	0.143362
$718$	0	0
$719$	37.6155	1.40282	0.701410	−	0.712758i	$-0.252554\pi$
$719$	37.6155	1.40282	0.701410	+	0.712758i	$0.252554\pi$
$720$	0	0
$721$	−12.9849	−0.483584
$722$	0	0
$723$	24.8920	0.925744
$724$	0	0
$725$	−27.8605	−1.03471
$726$	0	0
$727$	17.1879	0.637464	0.318732	−	0.947845i	$-0.396743\pi$
$727$	17.1879	0.637464	0.318732	+	0.947845i	$0.396743\pi$
$728$	0	0
$729$	27.8159	1.03022
$730$	0	0
$731$	−0.525418	−0.0194333
$732$	0	0
$733$	19.2597	0.711371	0.355686	−	0.934606i	$-0.384247\pi$
$733$	19.2597	0.711371	0.355686	+	0.934606i	$0.384247\pi$
$734$	0	0
$735$	−14.0204	−0.517152
$736$	0	0
$737$	−24.8605	−0.915750
$738$	0	0
$739$	18.0785	0.665027	0.332513	−	0.943099i	$-0.392103\pi$
$739$	18.0785	0.665027	0.332513	+	0.943099i	$0.392103\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.8219	1.20412	0.602059	−	0.798451i	$-0.294347\pi$
$743$	32.8219	1.20412	0.602059	+	0.798451i	$0.294347\pi$
$744$	0	0
$745$	10.0199	0.367099
$746$	0	0
$747$	−4.11960	−0.150728
$748$	0	0
$749$	−34.2717	−1.25226
$750$	0	0
$751$	33.1702	1.21040	0.605198	−	0.796075i	$-0.293094\pi$
$751$	33.1702	1.21040	0.605198	+	0.796075i	$0.293094\pi$
$752$	0	0
$753$	−3.44312	−0.125474
$754$	0	0
$755$	31.5961	1.14990
$756$	0	0
$757$	−0.736627	−0.0267732	−0.0133866	−	0.999910i	$-0.504261\pi$
$757$	−0.736627	−0.0267732	−0.0133866	+	0.999910i	$0.504261\pi$
$758$	0	0
$759$	−23.2204	−0.842848
$760$	0	0
$761$	−36.3817	−1.31883	−0.659417	−	0.751777i	$-0.729197\pi$
$761$	−36.3817	−1.31883	−0.659417	+	0.751777i	$0.729197\pi$
$762$	0	0
$763$	−31.8625	−1.15350
$764$	0	0
$765$	−0.300209	−0.0108541
$766$	0	0
$767$	0	0
$768$	0	0
$769$	11.4523	0.412982	0.206491	−	0.978449i	$-0.433796\pi$
$769$	11.4523	0.412982	0.206491	+	0.978449i	$0.433796\pi$
$770$	0	0
$771$	31.8683	1.14771
$772$	0	0
$773$	−7.57732	−0.272537	−0.136269	−	0.990672i	$-0.543511\pi$
$773$	−7.57732	−0.272537	−0.136269	+	0.990672i	$0.543511\pi$
$774$	0	0
$775$	−37.1661	−1.33505
$776$	0	0
$777$	−10.0686	−0.361207
$778$	0	0
$779$	−33.3803	−1.19597
$780$	0	0
$781$	0.164210	0.00587591
$782$	0	0
$783$	−12.0597	−0.430977
$784$	0	0
$785$	46.4989	1.65962
$786$	0	0
$787$	−30.5066	−1.08744	−0.543722	−	0.839265i	$-0.682985\pi$
$787$	−30.5066	−1.08744	−0.543722	+	0.839265i	$0.682985\pi$
$788$	0	0
$789$	−21.4515	−0.763693
$790$	0	0
$791$	−17.6192	−0.626467
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.125456	0.00444946
$796$	0	0
$797$	36.2435	1.28381	0.641906	−	0.766784i	$-0.278144\pi$
$797$	36.2435	1.28381	0.641906	+	0.766784i	$0.278144\pi$
$798$	0	0
$799$	0.424600	0.0150213
$800$	0	0
$801$	2.72002	0.0961073
$802$	0	0
$803$	−15.1588	−0.534944
$804$	0	0
$805$	46.2683	1.63074
$806$	0	0
$807$	41.8713	1.47394
$808$	0	0
$809$	22.3236	0.784857	0.392429	−	0.919782i	$-0.371635\pi$
$809$	22.3236	0.784857	0.392429	+	0.919782i	$0.371635\pi$
$810$	0	0
$811$	−39.4922	−1.38676	−0.693379	−	0.720573i	$-0.743879\pi$
$811$	−39.4922	−1.38676	−0.693379	+	0.720573i	$0.743879\pi$
$812$	0	0
$813$	−21.9726	−0.770612
$814$	0	0
$815$	−29.8340	−1.04504
$816$	0	0
$817$	53.2833	1.86415
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.3830	−0.397269	−0.198634	−	0.980074i	$-0.563651\pi$
$821$	−11.3830	−0.397269	−0.198634	+	0.980074i	$0.563651\pi$
$822$	0	0
$823$	−25.6848	−0.895317	−0.447659	−	0.894205i	$-0.647742\pi$
$823$	−25.6848	−0.895317	−0.447659	+	0.894205i	$0.647742\pi$
$824$	0	0
$825$	−59.3822	−2.06742
$826$	0	0
$827$	−36.7251	−1.27706	−0.638529	−	0.769598i	$-0.720457\pi$
$827$	−36.7251	−1.27706	−0.638529	+	0.769598i	$0.720457\pi$
$828$	0	0
$829$	−11.5808	−0.402217	−0.201109	−	0.979569i	$-0.564454\pi$
$829$	−11.5808	−0.402217	−0.201109	+	0.979569i	$0.564454\pi$
$830$	0	0
$831$	−17.7855	−0.616974
$832$	0	0
$833$	0.148408	0.00514203
$834$	0	0
$835$	65.7512	2.27541
$836$	0	0
$837$	−16.0877	−0.556071
$838$	0	0
$839$	13.4577	0.464612	0.232306	−	0.972643i	$-0.425373\pi$
$839$	13.4577	0.464612	0.232306	+	0.972643i	$0.425373\pi$
$840$	0	0
$841$	−24.4330	−0.842516
$842$	0	0
$843$	19.3612	0.666835
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.574335	−0.0197344
$848$	0	0
$849$	−10.7961	−0.370521
$850$	0	0
$851$	−17.7006	−0.606770
$852$	0	0
$853$	5.34050	0.182855	0.0914277	−	0.995812i	$-0.470857\pi$
$853$	5.34050	0.182855	0.0914277	+	0.995812i	$0.470857\pi$
$854$	0	0
$855$	30.4446	1.04118
$856$	0	0
$857$	49.3381	1.68536	0.842679	−	0.538417i	$-0.180977\pi$
$857$	49.3381	1.68536	0.842679	+	0.538417i	$0.180977\pi$
$858$	0	0
$859$	4.92500	0.168039	0.0840194	−	0.996464i	$-0.473224\pi$
$859$	4.92500	0.168039	0.0840194	+	0.996464i	$0.473224\pi$
$860$	0	0
$861$	−15.6475	−0.533266
$862$	0	0
$863$	−40.0315	−1.36269	−0.681343	−	0.731964i	$-0.738604\pi$
$863$	−40.0315	−1.36269	−0.681343	+	0.731964i	$0.738604\pi$
$864$	0	0
$865$	−58.5241	−1.98988
$866$	0	0
$867$	23.0622	0.783233
$868$	0	0
$869$	−43.0253	−1.45953
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.96700	−0.134263
$874$	0	0
$875$	72.9428	2.46592
$876$	0	0
$877$	28.9379	0.977165	0.488582	−	0.872518i	$-0.337514\pi$
$877$	28.9379	0.977165	0.488582	+	0.872518i	$0.337514\pi$
$878$	0	0
$879$	26.6358	0.898404
$880$	0	0
$881$	21.6692	0.730053	0.365027	−	0.930997i	$-0.381060\pi$
$881$	21.6692	0.730053	0.365027	+	0.930997i	$0.381060\pi$
$882$	0	0
$883$	−11.7614	−0.395802	−0.197901	−	0.980222i	$-0.563412\pi$
$883$	−11.7614	−0.395802	−0.197901	+	0.980222i	$0.563412\pi$
$884$	0	0
$885$	−33.4907	−1.12578
$886$	0	0
$887$	−26.8485	−0.901483	−0.450742	−	0.892654i	$-0.648840\pi$
$887$	−26.8485	−0.901483	−0.450742	+	0.892654i	$0.648840\pi$
$888$	0	0
$889$	−14.1967	−0.476143
$890$	0	0
$891$	−14.0339	−0.470152
$892$	0	0
$893$	−43.0592	−1.44092
$894$	0	0
$895$	93.9542	3.14054
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.09246	0.203195
$900$	0	0
$901$	−0.00132796	−4.42409e−5 0
$902$	0	0
$903$	24.9774	0.831195
$904$	0	0
$905$	57.9318	1.92572
$906$	0	0
$907$	−16.4306	−0.545568	−0.272784	−	0.962075i	$-0.587944\pi$
$907$	−16.4306	−0.545568	−0.272784	+	0.962075i	$0.587944\pi$
$908$	0	0
$909$	13.1860	0.437351
$910$	0	0
$911$	−26.0519	−0.863138	−0.431569	−	0.902080i	$-0.642040\pi$
$911$	−26.0519	−0.863138	−0.431569	+	0.902080i	$0.642040\pi$
$912$	0	0
$913$	11.9336	0.394945
$914$	0	0
$915$	61.5454	2.03463
$916$	0	0
$917$	16.8036	0.554904
$918$	0	0
$919$	−4.36957	−0.144139	−0.0720694	−	0.997400i	$-0.522960\pi$
$919$	−4.36957	−0.144139	−0.0720694	+	0.997400i	$0.522960\pi$
$920$	0	0
$921$	21.2728	0.700963
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−45.2664	−1.48835
$926$	0	0
$927$	−7.04115	−0.231262
$928$	0	0
$929$	32.2868	1.05930	0.529648	−	0.848218i	$-0.322324\pi$
$929$	32.2868	1.05930	0.529648	+	0.848218i	$0.322324\pi$
$930$	0	0
$931$	−15.0502	−0.493252
$932$	0	0
$933$	22.3864	0.732899
$934$	0	0
$935$	0.869641	0.0284403
$936$	0	0
$937$	−48.5730	−1.58681	−0.793405	−	0.608693i	$-0.791694\pi$
$937$	−48.5730	−1.58681	−0.793405	+	0.608693i	$0.791694\pi$
$938$	0	0
$939$	15.4601	0.504522
$940$	0	0
$941$	−4.82849	−0.157404	−0.0787022	−	0.996898i	$-0.525078\pi$
$941$	−4.82849	−0.157404	−0.0787022	+	0.996898i	$0.525078\pi$
$942$	0	0
$943$	−27.5086	−0.895802
$944$	0	0
$945$	51.2172	1.66609
$946$	0	0
$947$	−35.1065	−1.14081	−0.570403	−	0.821365i	$-0.693213\pi$
$947$	−35.1065	−1.14081	−0.570403	+	0.821365i	$0.693213\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	34.5289	1.11968
$952$	0	0
$953$	−28.1866	−0.913053	−0.456526	−	0.889710i	$-0.650907\pi$
$953$	−28.1866	−0.913053	−0.456526	+	0.889710i	$0.650907\pi$
$954$	0	0
$955$	19.0756	0.617272
$956$	0	0
$957$	9.73423	0.314663
$958$	0	0
$959$	−2.11124	−0.0681756
$960$	0	0
$961$	−22.8726	−0.737827
$962$	0	0
$963$	−18.5840	−0.598862
$964$	0	0
$965$	109.468	3.52388
$966$	0	0
$967$	44.6674	1.43641	0.718203	−	0.695834i	$-0.244965\pi$
$967$	44.6674	1.43641	0.718203	+	0.695834i	$0.244965\pi$
$968$	0	0
$969$	0.512009	0.0164481
$970$	0	0
$971$	14.9108	0.478510	0.239255	−	0.970957i	$-0.423097\pi$
$971$	14.9108	0.478510	0.239255	+	0.970957i	$0.423097\pi$
$972$	0	0
$973$	−25.7235	−0.824657
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.2127	1.12655	0.563276	−	0.826269i	$-0.309541\pi$
$977$	35.2127	1.12655	0.563276	+	0.826269i	$0.309541\pi$
$978$	0	0
$979$	−7.87933	−0.251824
$980$	0	0
$981$	−17.2776	−0.551631
$982$	0	0
$983$	−30.5646	−0.974861	−0.487430	−	0.873162i	$-0.662066\pi$
$983$	−30.5646	−0.974861	−0.487430	+	0.873162i	$0.662066\pi$
$984$	0	0
$985$	−79.0926	−2.52010
$986$	0	0
$987$	−20.1847	−0.642485
$988$	0	0
$989$	43.9105	1.39627
$990$	0	0
$991$	−62.5739	−1.98772	−0.993862	−	0.110626i	$-0.964714\pi$
$991$	−62.5739	−1.98772	−0.993862	+	0.110626i	$0.964714\pi$
$992$	0	0
$993$	−12.1515	−0.385617
$994$	0	0
$995$	−50.8200	−1.61110
$996$	0	0
$997$	−1.88444	−0.0596809	−0.0298405	−	0.999555i	$-0.509500\pi$
$997$	−1.88444	−0.0596809	−0.0298405	+	0.999555i	$0.509500\pi$
$998$	0	0
$999$	−19.5939	−0.619924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.x.1.2		3
4.3	odd	2		676.2.a.g.1.2	✓	3
12.11	even	2		6084.2.a.bc.1.3		3
13.5	odd	4		2704.2.f.n.337.4		6
13.8	odd	4		2704.2.f.n.337.3		6
13.12	even	2		2704.2.a.y.1.2		3
52.3	odd	6		676.2.e.f.529.2		6
52.7	even	12		676.2.h.e.361.3		12
52.11	even	12		676.2.h.e.485.3		12
52.15	even	12		676.2.h.e.485.4		12
52.19	even	12		676.2.h.e.361.4		12
52.23	odd	6		676.2.e.g.529.2		6
52.31	even	4		676.2.d.e.337.4		6
52.35	odd	6		676.2.e.f.653.2		6
52.43	odd	6		676.2.e.g.653.2		6
52.47	even	4		676.2.d.e.337.3		6
52.51	odd	2		676.2.a.h.1.2	yes	3
156.47	odd	4		6084.2.b.p.4393.6		6
156.83	odd	4		6084.2.b.p.4393.1		6
156.155	even	2		6084.2.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
676.2.a.g.1.2	✓	3	4.3	odd	2
676.2.a.h.1.2	yes	3	52.51	odd	2
676.2.d.e.337.3		6	52.47	even	4
676.2.d.e.337.4		6	52.31	even	4
676.2.e.f.529.2		6	52.3	odd	6
676.2.e.f.653.2		6	52.35	odd	6
676.2.e.g.529.2		6	52.23	odd	6
676.2.e.g.653.2		6	52.43	odd	6
676.2.h.e.361.3		12	52.7	even	12
676.2.h.e.361.4		12	52.19	even	12
676.2.h.e.485.3		12	52.11	even	12
676.2.h.e.485.4		12	52.15	even	12
2704.2.a.x.1.2		3	1.1	even	1	trivial
2704.2.a.y.1.2		3	13.12	even	2
2704.2.f.n.337.3		6	13.8	odd	4
2704.2.f.n.337.4		6	13.5	odd	4
6084.2.a.x.1.1		3	156.155	even	2
6084.2.a.bc.1.3		3	12.11	even	2
6084.2.b.p.4393.1		6	156.83	odd	4
6084.2.b.p.4393.6		6	156.47	odd	4

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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

\(f(q)\)	\(=\)	\(q-1.35690 q^{3} -4.24698 q^{5} -2.13706 q^{7} -1.15883 q^{9} +3.35690 q^{11} +5.76271 q^{15} -0.0609989 q^{17} +6.18598 q^{19} +2.89977 q^{21} +5.09783 q^{23} +13.0368 q^{25} +5.64310 q^{27} -2.13706 q^{29} -2.85086 q^{31} -4.55496 q^{33} +9.07606 q^{35} -3.47219 q^{37} -5.39612 q^{41} +8.61356 q^{43} +4.92154 q^{45} -6.96077 q^{47} -2.43296 q^{49} +0.0827692 q^{51} +0.0217703 q^{53} -14.2567 q^{55} -8.39373 q^{57} -5.81163 q^{59} +10.6799 q^{61} +2.47650 q^{63} -7.40581 q^{67} -6.91723 q^{69} +0.0489173 q^{71} -4.51573 q^{73} -17.6896 q^{75} -7.17390 q^{77} -12.8170 q^{79} -4.18060 q^{81} +3.55496 q^{83} +0.259061 q^{85} +2.89977 q^{87} -2.34721 q^{89} +3.86831 q^{93} -26.2717 q^{95} +3.42327 q^{97} -3.89008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} - q^{7} + 5 q^{9} + 6 q^{11} - 10 q^{17} + 4 q^{19} - 14 q^{21} - 3 q^{23} + 11 q^{25} + 21 q^{27} - q^{29} + 5 q^{31} - 14 q^{33} + 12 q^{35} - 4 q^{37} - 25 q^{41} - 5 q^{43} - 11 q^{45}+ \cdots - 11 q^{99}+O(q^{100}) \) 3 * q - 8 * q^5 - q^7 + 5 * q^9 + 6 * q^11 - 10 * q^17 + 4 * q^19 - 14 * q^21 - 3 * q^23 + 11 * q^25 + 21 * q^27 - q^29 + 5 * q^31 - 14 * q^33 + 12 * q^35 - 4 * q^37 - 25 * q^41 - 5 * q^43 - 11 * q^45 - 8 * q^47 + 12 * q^49 + 7 * q^51 - 3 * q^53 - 16 * q^55 + 7 * q^57 + 9 * q^59 + 8 * q^61 - 18 * q^63 - 9 * q^67 - 14 * q^69 - 9 * q^71 - q^73 - 7 * q^75 + 12 * q^77 - 9 * q^79 - q^81 + 11 * q^83 + 15 * q^85 - 14 * q^87 - 25 * q^89 + 14 * q^93 - 27 * q^95 + 13 * q^97 - 11 * q^99

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