LMFDB - Embedded newform 2898.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, 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("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -4.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.47214 q^{19} -2.00000 q^{20} +1.00000 q^{23} -1.00000 q^{25} +4.47214 q^{26} +1.00000 q^{28} +2.00000 q^{29} -2.47214 q^{31} -1.00000 q^{32} +4.47214 q^{34} -2.00000 q^{35} +6.94427 q^{37} -2.47214 q^{38} +2.00000 q^{40} +6.00000 q^{41} -4.94427 q^{43} -1.00000 q^{46} +2.47214 q^{47} +1.00000 q^{49} +1.00000 q^{50} -4.47214 q^{52} +10.9443 q^{53} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +2.47214 q^{62} +1.00000 q^{64} +8.94427 q^{65} +4.94427 q^{67} -4.47214 q^{68} +2.00000 q^{70} -4.94427 q^{71} +14.9443 q^{73} -6.94427 q^{74} +2.47214 q^{76} -4.94427 q^{79} -2.00000 q^{80} -6.00000 q^{82} -2.47214 q^{83} +8.94427 q^{85} +4.94427 q^{86} -9.41641 q^{89} -4.47214 q^{91} +1.00000 q^{92} -2.47214 q^{94} -4.94427 q^{95} -0.472136 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{20} + 2 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{35} - 4 q^{37} + 4 q^{38} + 4 q^{40} + 12 q^{41} + 8 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{53} - 2 q^{56} - 4 q^{58} - 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 4 q^{70} + 8 q^{71} + 12 q^{73} + 4 q^{74} - 4 q^{76} + 8 q^{79} - 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{86} + 8 q^{89} + 2 q^{92} + 4 q^{94} + 8 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 - 2 * q^14 + 2 * q^16 - 4 * q^19 - 4 * q^20 + 2 * q^23 - 2 * q^25 + 2 * q^28 + 4 * q^29 + 4 * q^31 - 2 * q^32 - 4 * q^35 - 4 * q^37 + 4 * q^38 + 4 * q^40 + 12 * q^41 + 8 * q^43 - 2 * q^46 - 4 * q^47 + 2 * q^49 + 2 * q^50 + 4 * q^53 - 2 * q^56 - 4 * q^58 - 8 * q^59 - 12 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^67 + 4 * q^70 + 8 * q^71 + 12 * q^73 + 4 * q^74 - 4 * q^76 + 8 * q^79 - 4 * q^80 - 12 * q^82 + 4 * q^83 - 8 * q^86 + 8 * q^89 + 2 * q^92 + 4 * q^94 + 8 * q^95 + 8 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	4.47214	0.877058
$27$	0	0
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.47214	0.766965
$35$	−2.00000	−0.338062
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	−2.47214	−0.401033
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.94427	−0.753994	−0.376997	−	0.926214i	$-0.623043\pi$
$43$	−4.94427	−0.753994	−0.376997	+	0.926214i	$0.623043\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−4.47214	−0.620174
$53$	10.9443	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$53$	10.9443	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	2.47214	0.313962
$63$	0	0
$64$	1.00000	0.125000
$65$	8.94427	1.10940
$66$	0	0
$67$	4.94427	0.604039	0.302019	−	0.953302i	$-0.402339\pi$
$67$	4.94427	0.604039	0.302019	+	0.953302i	$0.402339\pi$
$68$	−4.47214	−0.542326
$69$	0	0
$70$	2.00000	0.239046
$71$	−4.94427	−0.586777	−0.293389	−	0.955993i	$-0.594783\pi$
$71$	−4.94427	−0.586777	−0.293389	+	0.955993i	$0.594783\pi$
$72$	0	0
$73$	14.9443	1.74909	0.874547	−	0.484940i	$-0.161159\pi$
$73$	14.9443	1.74909	0.874547	+	0.484940i	$0.161159\pi$
$74$	−6.94427	−0.807255
$75$	0	0
$76$	2.47214	0.283573
$77$	0	0
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−2.47214	−0.271352	−0.135676	−	0.990753i	$-0.543321\pi$
$83$	−2.47214	−0.271352	−0.135676	+	0.990753i	$0.543321\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	4.94427	0.533155
$87$	0	0
$88$	0	0
$89$	−9.41641	−0.998137	−0.499069	−	0.866562i	$-0.666324\pi$
$89$	−9.41641	−0.998137	−0.499069	+	0.866562i	$0.666324\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	1.00000	0.104257
$93$	0	0
$94$	−2.47214	−0.254981
$95$	−4.94427	−0.507272
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	9.41641	0.936968	0.468484	−	0.883472i	$-0.344800\pi$
$101$	9.41641	0.936968	0.468484	+	0.883472i	$0.344800\pi$
$102$	0	0
$103$	−4.94427	−0.487174	−0.243587	−	0.969879i	$-0.578324\pi$
$103$	−4.94427	−0.487174	−0.243587	+	0.969879i	$0.578324\pi$
$104$	4.47214	0.438529
$105$	0	0
$106$	−10.9443	−1.06300
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	18.9443	1.78213	0.891064	−	0.453878i	$-0.149960\pi$
$113$	18.9443	1.78213	0.891064	+	0.453878i	$0.149960\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	2.00000	0.185695
$117$	0	0
$118$	4.00000	0.368230
$119$	−4.47214	−0.409960
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	0	0
$124$	−2.47214	−0.222004
$125$	12.0000	1.07331
$126$	0	0
$127$	20.9443	1.85850	0.929252	−	0.369447i	$-0.120453\pi$
$127$	20.9443	1.85850	0.929252	+	0.369447i	$0.120453\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.94427	−0.784465
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	−4.94427	−0.427120
$135$	0	0
$136$	4.47214	0.383482
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	4.94427	0.414914
$143$	0	0
$144$	0	0
$145$	−4.00000	−0.332182
$146$	−14.9443	−1.23680
$147$	0	0
$148$	6.94427	0.570816
$149$	1.05573	0.0864886	0.0432443	−	0.999065i	$-0.486231\pi$
$149$	1.05573	0.0864886	0.0432443	+	0.999065i	$0.486231\pi$
$150$	0	0
$151$	4.94427	0.402359	0.201180	−	0.979554i	$-0.435523\pi$
$151$	4.94427	0.402359	0.201180	+	0.979554i	$0.435523\pi$
$152$	−2.47214	−0.200517
$153$	0	0
$154$	0	0
$155$	4.94427	0.397133
$156$	0	0
$157$	−2.94427	−0.234978	−0.117489	−	0.993074i	$-0.537485\pi$
$157$	−2.94427	−0.234978	−0.117489	+	0.993074i	$0.537485\pi$
$158$	4.94427	0.393345
$159$	0	0
$160$	2.00000	0.158114
$161$	1.00000	0.0788110
$162$	0	0
$163$	−8.94427	−0.700569	−0.350285	−	0.936643i	$-0.613915\pi$
$163$	−8.94427	−0.700569	−0.350285	+	0.936643i	$0.613915\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	2.47214	0.191875
$167$	15.4164	1.19296	0.596479	−	0.802629i	$-0.296566\pi$
$167$	15.4164	1.19296	0.596479	+	0.802629i	$0.296566\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	−8.94427	−0.685994
$171$	0	0
$172$	−4.94427	−0.376997
$173$	−0.472136	−0.0358958	−0.0179479	−	0.999839i	$-0.505713\pi$
$173$	−0.472136	−0.0358958	−0.0179479	+	0.999839i	$0.505713\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	9.41641	0.705790
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.47214	0.331497
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−13.8885	−1.02111
$186$	0	0
$187$	0	0
$188$	2.47214	0.180299
$189$	0	0
$190$	4.94427	0.358695
$191$	3.05573	0.221105	0.110552	−	0.993870i	$-0.464738\pi$
$191$	3.05573	0.221105	0.110552	+	0.993870i	$0.464738\pi$
$192$	0	0
$193$	11.8885	0.855756	0.427878	−	0.903836i	$-0.359261\pi$
$193$	11.8885	0.855756	0.427878	+	0.903836i	$0.359261\pi$
$194$	0.472136	0.0338974
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.9443	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$197$	14.9443	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$198$	0	0
$199$	3.05573	0.216615	0.108307	−	0.994117i	$-0.465457\pi$
$199$	3.05573	0.216615	0.108307	+	0.994117i	$0.465457\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−9.41641	−0.662536
$203$	2.00000	0.140372
$204$	0	0
$205$	−12.0000	−0.838116
$206$	4.94427	0.344484
$207$	0	0
$208$	−4.47214	−0.310087
$209$	0	0
$210$	0	0
$211$	−0.944272	−0.0650064	−0.0325032	−	0.999472i	$-0.510348\pi$
$211$	−0.944272	−0.0650064	−0.0325032	+	0.999472i	$0.510348\pi$
$212$	10.9443	0.751656
$213$	0	0
$214$	−8.00000	−0.546869
$215$	9.88854	0.674393
$216$	0	0
$217$	−2.47214	−0.167820
$218$	2.94427	0.199411
$219$	0	0
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	18.4721	1.23699	0.618493	−	0.785790i	$-0.287744\pi$
$223$	18.4721	1.23699	0.618493	+	0.785790i	$0.287744\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−18.9443	−1.26015
$227$	−5.52786	−0.366897	−0.183449	−	0.983029i	$-0.558726\pi$
$227$	−5.52786	−0.366897	−0.183449	+	0.983029i	$0.558726\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−2.00000	−0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	−4.00000	−0.260378
$237$	0	0
$238$	4.47214	0.289886
$239$	−20.9443	−1.35477	−0.677386	−	0.735628i	$-0.736887\pi$
$239$	−20.9443	−1.35477	−0.677386	+	0.735628i	$0.736887\pi$
$240$	0	0
$241$	20.4721	1.31873	0.659363	−	0.751825i	$-0.270826\pi$
$241$	20.4721	1.31873	0.659363	+	0.751825i	$0.270826\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−11.0557	−0.703459
$248$	2.47214	0.156981
$249$	0	0
$250$	−12.0000	−0.758947
$251$	5.52786	0.348916	0.174458	−	0.984665i	$-0.444183\pi$
$251$	5.52786	0.348916	0.174458	+	0.984665i	$0.444183\pi$
$252$	0	0
$253$	0	0
$254$	−20.9443	−1.31416
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.94427	0.183659	0.0918293	−	0.995775i	$-0.470729\pi$
$257$	2.94427	0.183659	0.0918293	+	0.995775i	$0.470729\pi$
$258$	0	0
$259$	6.94427	0.431496
$260$	8.94427	0.554700
$261$	0	0
$262$	12.0000	0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−21.8885	−1.34460
$266$	−2.47214	−0.151576
$267$	0	0
$268$	4.94427	0.302019
$269$	−16.4721	−1.00432	−0.502162	−	0.864774i	$-0.667462\pi$
$269$	−16.4721	−1.00432	−0.502162	+	0.864774i	$0.667462\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	−4.47214	−0.271163
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	0	0
$277$	−11.8885	−0.714313	−0.357157	−	0.934044i	$-0.616254\pi$
$277$	−11.8885	−0.714313	−0.357157	+	0.934044i	$0.616254\pi$
$278$	−8.94427	−0.536442
$279$	0	0
$280$	2.00000	0.119523
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	18.4721	1.09805	0.549027	−	0.835804i	$-0.314998\pi$
$283$	18.4721	1.09805	0.549027	+	0.835804i	$0.314998\pi$
$284$	−4.94427	−0.293389
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	3.00000	0.176471
$290$	4.00000	0.234888
$291$	0	0
$292$	14.9443	0.874547
$293$	−27.8885	−1.62927	−0.814633	−	0.579977i	$-0.803062\pi$
$293$	−27.8885	−1.62927	−0.814633	+	0.579977i	$0.803062\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−6.94427	−0.403628
$297$	0	0
$298$	−1.05573	−0.0611567
$299$	−4.47214	−0.258630
$300$	0	0
$301$	−4.94427	−0.284983
$302$	−4.94427	−0.284511
$303$	0	0
$304$	2.47214	0.141787
$305$	12.0000	0.687118
$306$	0	0
$307$	24.9443	1.42364	0.711822	−	0.702360i	$-0.247870\pi$
$307$	24.9443	1.42364	0.711822	+	0.702360i	$0.247870\pi$
$308$	0	0
$309$	0	0
$310$	−4.94427	−0.280816
$311$	2.47214	0.140182	0.0700910	−	0.997541i	$-0.477671\pi$
$311$	2.47214	0.140182	0.0700910	+	0.997541i	$0.477671\pi$
$312$	0	0
$313$	−13.4164	−0.758340	−0.379170	−	0.925327i	$-0.623790\pi$
$313$	−13.4164	−0.758340	−0.379170	+	0.925327i	$0.623790\pi$
$314$	2.94427	0.166155
$315$	0	0
$316$	−4.94427	−0.278137
$317$	13.0557	0.733283	0.366641	−	0.930362i	$-0.380508\pi$
$317$	13.0557	0.733283	0.366641	+	0.930362i	$0.380508\pi$
$318$	0	0
$319$	0	0
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	−11.0557	−0.615157
$324$	0	0
$325$	4.47214	0.248069
$326$	8.94427	0.495377
$327$	0	0
$328$	−6.00000	−0.331295
$329$	2.47214	0.136293
$330$	0	0
$331$	−13.8885	−0.763383	−0.381692	−	0.924290i	$-0.624658\pi$
$331$	−13.8885	−0.763383	−0.381692	+	0.924290i	$0.624658\pi$
$332$	−2.47214	−0.135676
$333$	0	0
$334$	−15.4164	−0.843548
$335$	−9.88854	−0.540269
$336$	0	0
$337$	−2.94427	−0.160385	−0.0801924	−	0.996779i	$-0.525553\pi$
$337$	−2.94427	−0.160385	−0.0801924	+	0.996779i	$0.525553\pi$
$338$	−7.00000	−0.380750
$339$	0	0
$340$	8.94427	0.485071
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	4.94427	0.266577
$345$	0	0
$346$	0.472136	0.0253822
$347$	0.944272	0.0506912	0.0253456	−	0.999679i	$-0.491931\pi$
$347$	0.944272	0.0506912	0.0253456	+	0.999679i	$0.491931\pi$
$348$	0	0
$349$	24.4721	1.30996	0.654982	−	0.755645i	$-0.272676\pi$
$349$	24.4721	1.30996	0.654982	+	0.755645i	$0.272676\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	9.88854	0.524829
$356$	−9.41641	−0.499069
$357$	0	0
$358$	−20.0000	−1.05703
$359$	30.8328	1.62729	0.813647	−	0.581359i	$-0.197479\pi$
$359$	30.8328	1.62729	0.813647	+	0.581359i	$0.197479\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	−2.00000	−0.105118
$363$	0	0
$364$	−4.47214	−0.234404
$365$	−29.8885	−1.56444
$366$	0	0
$367$	30.8328	1.60946	0.804730	−	0.593641i	$-0.202310\pi$
$367$	30.8328	1.60946	0.804730	+	0.593641i	$0.202310\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	13.8885	0.722031
$371$	10.9443	0.568198
$372$	0	0
$373$	6.94427	0.359561	0.179780	−	0.983707i	$-0.442461\pi$
$373$	6.94427	0.359561	0.179780	+	0.983707i	$0.442461\pi$
$374$	0	0
$375$	0	0
$376$	−2.47214	−0.127491
$377$	−8.94427	−0.460653
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	−4.94427	−0.253636
$381$	0	0
$382$	−3.05573	−0.156345
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	−11.8885	−0.605111
$387$	0	0
$388$	−0.472136	−0.0239691
$389$	18.9443	0.960513	0.480256	−	0.877128i	$-0.340544\pi$
$389$	18.9443	0.960513	0.480256	+	0.877128i	$0.340544\pi$
$390$	0	0
$391$	−4.47214	−0.226166
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−14.9443	−0.752882
$395$	9.88854	0.497547
$396$	0	0
$397$	−36.4721	−1.83048	−0.915242	−	0.402905i	$-0.868001\pi$
$397$	−36.4721	−1.83048	−0.915242	+	0.402905i	$0.868001\pi$
$398$	−3.05573	−0.153170
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	11.0557	0.550725
$404$	9.41641	0.468484
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	0	0
$409$	−15.8885	−0.785638	−0.392819	−	0.919616i	$-0.628500\pi$
$409$	−15.8885	−0.785638	−0.392819	+	0.919616i	$0.628500\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	−4.94427	−0.243587
$413$	−4.00000	−0.196827
$414$	0	0
$415$	4.94427	0.242705
$416$	4.47214	0.219265
$417$	0	0
$418$	0	0
$419$	7.41641	0.362315	0.181158	−	0.983454i	$-0.442016\pi$
$419$	7.41641	0.362315	0.181158	+	0.983454i	$0.442016\pi$
$420$	0	0
$421$	32.8328	1.60017	0.800087	−	0.599884i	$-0.204787\pi$
$421$	32.8328	1.60017	0.800087	+	0.599884i	$0.204787\pi$
$422$	0.944272	0.0459664
$423$	0	0
$424$	−10.9443	−0.531501
$425$	4.47214	0.216930
$426$	0	0
$427$	−6.00000	−0.290360
$428$	8.00000	0.386695
$429$	0	0
$430$	−9.88854	−0.476868
$431$	−17.8885	−0.861661	−0.430830	−	0.902433i	$-0.641779\pi$
$431$	−17.8885	−0.861661	−0.430830	+	0.902433i	$0.641779\pi$
$432$	0	0
$433$	25.4164	1.22143	0.610717	−	0.791849i	$-0.290881\pi$
$433$	25.4164	1.22143	0.610717	+	0.791849i	$0.290881\pi$
$434$	2.47214	0.118666
$435$	0	0
$436$	−2.94427	−0.141005
$437$	2.47214	0.118258
$438$	0	0
$439$	28.3607	1.35358	0.676791	−	0.736175i	$-0.263370\pi$
$439$	28.3607	1.35358	0.676791	+	0.736175i	$0.263370\pi$
$440$	0	0
$441$	0	0
$442$	−20.0000	−0.951303
$443$	−34.8328	−1.65496	−0.827479	−	0.561497i	$-0.810225\pi$
$443$	−34.8328	−1.65496	−0.827479	+	0.561497i	$0.810225\pi$
$444$	0	0
$445$	18.8328	0.892761
$446$	−18.4721	−0.874681
$447$	0	0
$448$	1.00000	0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	18.9443	0.891064
$453$	0	0
$454$	5.52786	0.259436
$455$	8.94427	0.419314
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−6.58359	−0.306628	−0.153314	−	0.988177i	$-0.548995\pi$
$461$	−6.58359	−0.306628	−0.153314	+	0.988177i	$0.548995\pi$
$462$	0	0
$463$	6.11146	0.284023	0.142012	−	0.989865i	$-0.454643\pi$
$463$	6.11146	0.284023	0.142012	+	0.989865i	$0.454643\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−33.3050	−1.54117	−0.770585	−	0.637338i	$-0.780036\pi$
$467$	−33.3050	−1.54117	−0.770585	+	0.637338i	$0.780036\pi$
$468$	0	0
$469$	4.94427	0.228305
$470$	4.94427	0.228062
$471$	0	0
$472$	4.00000	0.184115
$473$	0	0
$474$	0	0
$475$	−2.47214	−0.113429
$476$	−4.47214	−0.204980
$477$	0	0
$478$	20.9443	0.957969
$479$	4.94427	0.225910	0.112955	−	0.993600i	$-0.463968\pi$
$479$	4.94427	0.225910	0.112955	+	0.993600i	$0.463968\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	−20.4721	−0.932480
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0.944272	0.0428772
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	2.00000	0.0903508
$491$	−40.9443	−1.84779	−0.923895	−	0.382647i	$-0.875013\pi$
$491$	−40.9443	−1.84779	−0.923895	+	0.382647i	$0.875013\pi$
$492$	0	0
$493$	−8.94427	−0.402830
$494$	11.0557	0.497421
$495$	0	0
$496$	−2.47214	−0.111002
$497$	−4.94427	−0.221781
$498$	0	0
$499$	32.9443	1.47479	0.737394	−	0.675463i	$-0.236056\pi$
$499$	32.9443	1.47479	0.737394	+	0.675463i	$0.236056\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−5.52786	−0.246721
$503$	9.88854	0.440908	0.220454	−	0.975397i	$-0.429246\pi$
$503$	9.88854	0.440908	0.220454	+	0.975397i	$0.429246\pi$
$504$	0	0
$505$	−18.8328	−0.838049
$506$	0	0
$507$	0	0
$508$	20.9443	0.929252
$509$	−6.58359	−0.291813	−0.145906	−	0.989298i	$-0.546610\pi$
$509$	−6.58359	−0.291813	−0.145906	+	0.989298i	$0.546610\pi$
$510$	0	0
$511$	14.9443	0.661096
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.94427	−0.129866
$515$	9.88854	0.435741
$516$	0	0
$517$	0	0
$518$	−6.94427	−0.305114
$519$	0	0
$520$	−8.94427	−0.392232
$521$	−33.4164	−1.46400	−0.732000	−	0.681305i	$-0.761413\pi$
$521$	−33.4164	−1.46400	−0.732000	+	0.681305i	$0.761413\pi$
$522$	0	0
$523$	−25.3050	−1.10651	−0.553254	−	0.833013i	$-0.686614\pi$
$523$	−25.3050	−1.10651	−0.553254	+	0.833013i	$0.686614\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	11.0557	0.481595
$528$	0	0
$529$	1.00000	0.0434783
$530$	21.8885	0.950778
$531$	0	0
$532$	2.47214	0.107181
$533$	−26.8328	−1.16226
$534$	0	0
$535$	−16.0000	−0.691740
$536$	−4.94427	−0.213560
$537$	0	0
$538$	16.4721	0.710164
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−15.4164	−0.662191
$543$	0	0
$544$	4.47214	0.191741
$545$	5.88854	0.252238
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	14.0000	0.598050
$549$	0	0
$550$	0	0
$551$	4.94427	0.210633
$552$	0	0
$553$	−4.94427	−0.210252
$554$	11.8885	0.505096
$555$	0	0
$556$	8.94427	0.379322
$557$	17.0557	0.722674	0.361337	−	0.932435i	$-0.382320\pi$
$557$	17.0557	0.722674	0.361337	+	0.932435i	$0.382320\pi$
$558$	0	0
$559$	22.1115	0.935215
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	10.0000	0.421825
$563$	−26.4721	−1.11567	−0.557834	−	0.829953i	$-0.688367\pi$
$563$	−26.4721	−1.11567	−0.557834	+	0.829953i	$0.688367\pi$
$564$	0	0
$565$	−37.8885	−1.59398
$566$	−18.4721	−0.776442
$567$	0	0
$568$	4.94427	0.207457
$569$	7.88854	0.330705	0.165352	−	0.986235i	$-0.447124\pi$
$569$	7.88854	0.330705	0.165352	+	0.986235i	$0.447124\pi$
$570$	0	0
$571$	−12.9443	−0.541701	−0.270850	−	0.962621i	$-0.587305\pi$
$571$	−12.9443	−0.541701	−0.270850	+	0.962621i	$0.587305\pi$
$572$	0	0
$573$	0	0
$574$	−6.00000	−0.250435
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−2.94427	−0.122572	−0.0612858	−	0.998120i	$-0.519520\pi$
$577$	−2.94427	−0.122572	−0.0612858	+	0.998120i	$0.519520\pi$
$578$	−3.00000	−0.124784
$579$	0	0
$580$	−4.00000	−0.166091
$581$	−2.47214	−0.102561
$582$	0	0
$583$	0	0
$584$	−14.9443	−0.618398
$585$	0	0
$586$	27.8885	1.15207
$587$	0.944272	0.0389743	0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272	0.0389743	0.0194871	+	0.999810i	$0.493797\pi$
$588$	0	0
$589$	−6.11146	−0.251818
$590$	−8.00000	−0.329355
$591$	0	0
$592$	6.94427	0.285408
$593$	28.8328	1.18402	0.592011	−	0.805930i	$-0.298334\pi$
$593$	28.8328	1.18402	0.592011	+	0.805930i	$0.298334\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	1.05573	0.0432443
$597$	0	0
$598$	4.47214	0.182879
$599$	−33.8885	−1.38465	−0.692324	−	0.721587i	$-0.743413\pi$
$599$	−33.8885	−1.38465	−0.692324	+	0.721587i	$0.743413\pi$
$600$	0	0
$601$	40.8328	1.66561	0.832803	−	0.553570i	$-0.186735\pi$
$601$	40.8328	1.66561	0.832803	+	0.553570i	$0.186735\pi$
$602$	4.94427	0.201513
$603$	0	0
$604$	4.94427	0.201180
$605$	22.0000	0.894427
$606$	0	0
$607$	−28.3607	−1.15112	−0.575562	−	0.817758i	$-0.695217\pi$
$607$	−28.3607	−1.15112	−0.575562	+	0.817758i	$0.695217\pi$
$608$	−2.47214	−0.100258
$609$	0	0
$610$	−12.0000	−0.485866
$611$	−11.0557	−0.447267
$612$	0	0
$613$	40.8328	1.64922	0.824611	−	0.565700i	$-0.191394\pi$
$613$	40.8328	1.64922	0.824611	+	0.565700i	$0.191394\pi$
$614$	−24.9443	−1.00667
$615$	0	0
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	−34.4721	−1.38555	−0.692776	−	0.721153i	$-0.743613\pi$
$619$	−34.4721	−1.38555	−0.692776	+	0.721153i	$0.743613\pi$
$620$	4.94427	0.198567
$621$	0	0
$622$	−2.47214	−0.0991236
$623$	−9.41641	−0.377260
$624$	0	0
$625$	−19.0000	−0.760000
$626$	13.4164	0.536228
$627$	0	0
$628$	−2.94427	−0.117489
$629$	−31.0557	−1.23827
$630$	0	0
$631$	−11.0557	−0.440122	−0.220061	−	0.975486i	$-0.570626\pi$
$631$	−11.0557	−0.440122	−0.220061	+	0.975486i	$0.570626\pi$
$632$	4.94427	0.196673
$633$	0	0
$634$	−13.0557	−0.518509
$635$	−41.8885	−1.66230
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	0	0
$640$	2.00000	0.0790569
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−10.4721	−0.412981	−0.206490	−	0.978449i	$-0.566204\pi$
$643$	−10.4721	−0.412981	−0.206490	+	0.978449i	$0.566204\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	11.0557	0.434982
$647$	−12.3607	−0.485948	−0.242974	−	0.970033i	$-0.578123\pi$
$647$	−12.3607	−0.485948	−0.242974	+	0.970033i	$0.578123\pi$
$648$	0	0
$649$	0	0
$650$	−4.47214	−0.175412
$651$	0	0
$652$	−8.94427	−0.350285
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	24.0000	0.937758
$656$	6.00000	0.234261
$657$	0	0
$658$	−2.47214	−0.0963739
$659$	−46.8328	−1.82435	−0.912174	−	0.409804i	$-0.865597\pi$
$659$	−46.8328	−1.82435	−0.912174	+	0.409804i	$0.865597\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	13.8885	0.539794
$663$	0	0
$664$	2.47214	0.0959375
$665$	−4.94427	−0.191731
$666$	0	0
$667$	2.00000	0.0774403
$668$	15.4164	0.596479
$669$	0	0
$670$	9.88854	0.382028
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	2.94427	0.113409
$675$	0	0
$676$	7.00000	0.269231
$677$	4.11146	0.158016	0.0790080	−	0.996874i	$-0.474825\pi$
$677$	4.11146	0.158016	0.0790080	+	0.996874i	$0.474825\pi$
$678$	0	0
$679$	−0.472136	−0.0181189
$680$	−8.94427	−0.342997
$681$	0	0
$682$	0	0
$683$	−16.9443	−0.648355	−0.324177	−	0.945996i	$-0.605087\pi$
$683$	−16.9443	−0.648355	−0.324177	+	0.945996i	$0.605087\pi$
$684$	0	0
$685$	−28.0000	−1.06983
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−4.94427	−0.188499
$689$	−48.9443	−1.86463
$690$	0	0
$691$	−15.0557	−0.572747	−0.286373	−	0.958118i	$-0.592450\pi$
$691$	−15.0557	−0.572747	−0.286373	+	0.958118i	$0.592450\pi$
$692$	−0.472136	−0.0179479
$693$	0	0
$694$	−0.944272	−0.0358441
$695$	−17.8885	−0.678551
$696$	0	0
$697$	−26.8328	−1.01637
$698$	−24.4721	−0.926284
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	20.8328	0.786845	0.393422	−	0.919358i	$-0.371291\pi$
$701$	20.8328	0.786845	0.393422	+	0.919358i	$0.371291\pi$
$702$	0	0
$703$	17.1672	0.647473
$704$	0	0
$705$	0	0
$706$	2.00000	0.0752710
$707$	9.41641	0.354140
$708$	0	0
$709$	6.94427	0.260798	0.130399	−	0.991462i	$-0.458374\pi$
$709$	6.94427	0.260798	0.130399	+	0.991462i	$0.458374\pi$
$710$	−9.88854	−0.371110
$711$	0	0
$712$	9.41641	0.352895
$713$	−2.47214	−0.0925822
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	−30.8328	−1.15067
$719$	5.52786	0.206155	0.103077	−	0.994673i	$-0.467131\pi$
$719$	5.52786	0.206155	0.103077	+	0.994673i	$0.467131\pi$
$720$	0	0
$721$	−4.94427	−0.184134
$722$	12.8885	0.479662
$723$	0	0
$724$	2.00000	0.0743294
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	3.05573	0.113331	0.0566653	−	0.998393i	$-0.481953\pi$
$727$	3.05573	0.113331	0.0566653	+	0.998393i	$0.481953\pi$
$728$	4.47214	0.165748
$729$	0	0
$730$	29.8885	1.10622
$731$	22.1115	0.817822
$732$	0	0
$733$	−50.9443	−1.88167	−0.940835	−	0.338866i	$-0.889957\pi$
$733$	−50.9443	−1.88167	−0.940835	+	0.338866i	$0.889957\pi$
$734$	−30.8328	−1.13806
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	−8.94427	−0.329020	−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427	−0.329020	−0.164510	+	0.986375i	$0.552604\pi$
$740$	−13.8885	−0.510553
$741$	0	0
$742$	−10.9443	−0.401777
$743$	22.8328	0.837655	0.418827	−	0.908066i	$-0.362441\pi$
$743$	22.8328	0.837655	0.418827	+	0.908066i	$0.362441\pi$
$744$	0	0
$745$	−2.11146	−0.0773578
$746$	−6.94427	−0.254248
$747$	0	0
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	17.8885	0.652762	0.326381	−	0.945238i	$-0.394171\pi$
$751$	17.8885	0.652762	0.326381	+	0.945238i	$0.394171\pi$
$752$	2.47214	0.0901495
$753$	0	0
$754$	8.94427	0.325731
$755$	−9.88854	−0.359881
$756$	0	0
$757$	−42.9443	−1.56084	−0.780418	−	0.625258i	$-0.784994\pi$
$757$	−42.9443	−1.56084	−0.780418	+	0.625258i	$0.784994\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	4.94427	0.179348
$761$	−21.0557	−0.763270	−0.381635	−	0.924313i	$-0.624639\pi$
$761$	−21.0557	−0.763270	−0.381635	+	0.924313i	$0.624639\pi$
$762$	0	0
$763$	−2.94427	−0.106590
$764$	3.05573	0.110552
$765$	0	0
$766$	16.0000	0.578103
$767$	17.8885	0.645918
$768$	0	0
$769$	30.3607	1.09483	0.547417	−	0.836860i	$-0.315611\pi$
$769$	30.3607	1.09483	0.547417	+	0.836860i	$0.315611\pi$
$770$	0	0
$771$	0	0
$772$	11.8885	0.427878
$773$	−5.05573	−0.181842	−0.0909210	−	0.995858i	$-0.528981\pi$
$773$	−5.05573	−0.181842	−0.0909210	+	0.995858i	$0.528981\pi$
$774$	0	0
$775$	2.47214	0.0888017
$776$	0.472136	0.0169487
$777$	0	0
$778$	−18.9443	−0.679185
$779$	14.8328	0.531441
$780$	0	0
$781$	0	0
$782$	4.47214	0.159923
$783$	0	0
$784$	1.00000	0.0357143
$785$	5.88854	0.210171
$786$	0	0
$787$	−0.583592	−0.0208028	−0.0104014	−	0.999946i	$-0.503311\pi$
$787$	−0.583592	−0.0208028	−0.0104014	+	0.999946i	$0.503311\pi$
$788$	14.9443	0.532368
$789$	0	0
$790$	−9.88854	−0.351819
$791$	18.9443	0.673581
$792$	0	0
$793$	26.8328	0.952861
$794$	36.4721	1.29435
$795$	0	0
$796$	3.05573	0.108307
$797$	28.8328	1.02131	0.510655	−	0.859785i	$-0.329403\pi$
$797$	28.8328	1.02131	0.510655	+	0.859785i	$0.329403\pi$
$798$	0	0
$799$	−11.0557	−0.391124
$800$	1.00000	0.0353553
$801$	0	0
$802$	−22.0000	−0.776847
$803$	0	0
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	−11.0557	−0.389421
$807$	0	0
$808$	−9.41641	−0.331268
$809$	−35.8885	−1.26177	−0.630887	−	0.775875i	$-0.717309\pi$
$809$	−35.8885	−1.26177	−0.630887	+	0.775875i	$0.717309\pi$
$810$	0	0
$811$	−37.8885	−1.33045	−0.665223	−	0.746644i	$-0.731664\pi$
$811$	−37.8885	−1.33045	−0.665223	+	0.746644i	$0.731664\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	0	0
$815$	17.8885	0.626608
$816$	0	0
$817$	−12.2229	−0.427626
$818$	15.8885	0.555530
$819$	0	0
$820$	−12.0000	−0.419058
$821$	16.8328	0.587469	0.293735	−	0.955887i	$-0.405102\pi$
$821$	16.8328	0.587469	0.293735	+	0.955887i	$0.405102\pi$
$822$	0	0
$823$	20.9443	0.730071	0.365036	−	0.930994i	$-0.381057\pi$
$823$	20.9443	0.730071	0.365036	+	0.930994i	$0.381057\pi$
$824$	4.94427	0.172242
$825$	0	0
$826$	4.00000	0.139178
$827$	−33.8885	−1.17842	−0.589210	−	0.807980i	$-0.700561\pi$
$827$	−33.8885	−1.17842	−0.589210	+	0.807980i	$0.700561\pi$
$828$	0	0
$829$	13.4164	0.465971	0.232986	−	0.972480i	$-0.425151\pi$
$829$	13.4164	0.465971	0.232986	+	0.972480i	$0.425151\pi$
$830$	−4.94427	−0.171618
$831$	0	0
$832$	−4.47214	−0.155043
$833$	−4.47214	−0.154950
$834$	0	0
$835$	−30.8328	−1.06701
$836$	0	0
$837$	0	0
$838$	−7.41641	−0.256196
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−32.8328	−1.13149
$843$	0	0
$844$	−0.944272	−0.0325032
$845$	−14.0000	−0.481615
$846$	0	0
$847$	−11.0000	−0.377964
$848$	10.9443	0.375828
$849$	0	0
$850$	−4.47214	−0.153393
$851$	6.94427	0.238047
$852$	0	0
$853$	−6.36068	−0.217786	−0.108893	−	0.994054i	$-0.534731\pi$
$853$	−6.36068	−0.217786	−0.108893	+	0.994054i	$0.534731\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	−8.00000	−0.273434
$857$	47.8885	1.63584	0.817921	−	0.575331i	$-0.195127\pi$
$857$	47.8885	1.63584	0.817921	+	0.575331i	$0.195127\pi$
$858$	0	0
$859$	32.9443	1.12404	0.562022	−	0.827122i	$-0.310024\pi$
$859$	32.9443	1.12404	0.562022	+	0.827122i	$0.310024\pi$
$860$	9.88854	0.337197
$861$	0	0
$862$	17.8885	0.609286
$863$	−20.9443	−0.712951	−0.356476	−	0.934305i	$-0.616022\pi$
$863$	−20.9443	−0.712951	−0.356476	+	0.934305i	$0.616022\pi$
$864$	0	0
$865$	0.944272	0.0321062
$866$	−25.4164	−0.863685
$867$	0	0
$868$	−2.47214	−0.0839098
$869$	0	0
$870$	0	0
$871$	−22.1115	−0.749218
$872$	2.94427	0.0997056
$873$	0	0
$874$	−2.47214	−0.0836212
$875$	12.0000	0.405674
$876$	0	0
$877$	−34.7214	−1.17246	−0.586229	−	0.810146i	$-0.699388\pi$
$877$	−34.7214	−1.17246	−0.586229	+	0.810146i	$0.699388\pi$
$878$	−28.3607	−0.957127
$879$	0	0
$880$	0	0
$881$	−22.3607	−0.753350	−0.376675	−	0.926345i	$-0.622933\pi$
$881$	−22.3607	−0.753350	−0.376675	+	0.926345i	$0.622933\pi$
$882$	0	0
$883$	2.83282	0.0953318	0.0476659	−	0.998863i	$-0.484822\pi$
$883$	2.83282	0.0953318	0.0476659	+	0.998863i	$0.484822\pi$
$884$	20.0000	0.672673
$885$	0	0
$886$	34.8328	1.17023
$887$	5.52786	0.185608	0.0928038	−	0.995684i	$-0.470417\pi$
$887$	5.52786	0.185608	0.0928038	+	0.995684i	$0.470417\pi$
$888$	0	0
$889$	20.9443	0.702448
$890$	−18.8328	−0.631277
$891$	0	0
$892$	18.4721	0.618493
$893$	6.11146	0.204512
$894$	0	0
$895$	−40.0000	−1.33705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	−4.94427	−0.164901
$900$	0	0
$901$	−48.9443	−1.63057
$902$	0	0
$903$	0	0
$904$	−18.9443	−0.630077
$905$	−4.00000	−0.132964
$906$	0	0
$907$	−41.8885	−1.39089	−0.695443	−	0.718581i	$-0.744792\pi$
$907$	−41.8885	−1.39089	−0.695443	+	0.718581i	$0.744792\pi$
$908$	−5.52786	−0.183449
$909$	0	0
$910$	−8.94427	−0.296500
$911$	48.7214	1.61421	0.807105	−	0.590407i	$-0.201033\pi$
$911$	48.7214	1.61421	0.807105	+	0.590407i	$0.201033\pi$
$912$	0	0
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−14.0000	−0.462573
$917$	−12.0000	−0.396275
$918$	0	0
$919$	35.0557	1.15638	0.578191	−	0.815902i	$-0.303759\pi$
$919$	35.0557	1.15638	0.578191	+	0.815902i	$0.303759\pi$
$920$	2.00000	0.0659380
$921$	0	0
$922$	6.58359	0.216819
$923$	22.1115	0.727807
$924$	0	0
$925$	−6.94427	−0.228326
$926$	−6.11146	−0.200835
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	2.47214	0.0810210
$932$	6.00000	0.196537
$933$	0	0
$934$	33.3050	1.08977
$935$	0	0
$936$	0	0
$937$	40.2492	1.31488	0.657442	−	0.753505i	$-0.271638\pi$
$937$	40.2492	1.31488	0.657442	+	0.753505i	$0.271638\pi$
$938$	−4.94427	−0.161436
$939$	0	0
$940$	−4.94427	−0.161264
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	−0.944272	−0.0306847	−0.0153424	−	0.999882i	$-0.504884\pi$
$947$	−0.944272	−0.0306847	−0.0153424	+	0.999882i	$0.504884\pi$
$948$	0	0
$949$	−66.8328	−2.16949
$950$	2.47214	0.0802067
$951$	0	0
$952$	4.47214	0.144943
$953$	9.05573	0.293344	0.146672	−	0.989185i	$-0.453144\pi$
$953$	9.05573	0.293344	0.146672	+	0.989185i	$0.453144\pi$
$954$	0	0
$955$	−6.11146	−0.197762
$956$	−20.9443	−0.677386
$957$	0	0
$958$	−4.94427	−0.159742
$959$	14.0000	0.452084
$960$	0	0
$961$	−24.8885	−0.802856
$962$	31.0557	1.00128
$963$	0	0
$964$	20.4721	0.659363
$965$	−23.7771	−0.765412
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−0.944272	−0.0303187
$971$	23.4164	0.751468	0.375734	−	0.926727i	$-0.377391\pi$
$971$	23.4164	0.751468	0.375734	+	0.926727i	$0.377391\pi$
$972$	0	0
$973$	8.94427	0.286740
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−6.00000	−0.192055
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	40.9443	1.30658
$983$	22.1115	0.705246	0.352623	−	0.935765i	$-0.385290\pi$
$983$	22.1115	0.705246	0.352623	+	0.935765i	$0.385290\pi$
$984$	0	0
$985$	−29.8885	−0.952328
$986$	8.94427	0.284844
$987$	0	0
$988$	−11.0557	−0.351730
$989$	−4.94427	−0.157219
$990$	0	0
$991$	60.9443	1.93596	0.967979	−	0.251030i	$-0.0807693\pi$
$991$	60.9443	1.93596	0.967979	+	0.251030i	$0.0807693\pi$
$992$	2.47214	0.0784904
$993$	0	0
$994$	4.94427	0.156823
$995$	−6.11146	−0.193746
$996$	0	0
$997$	−46.3607	−1.46826	−0.734129	−	0.679010i	$-0.762409\pi$
$997$	−46.3607	−1.46826	−0.734129	+	0.679010i	$0.762409\pi$
$998$	−32.9443	−1.04283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.v.1.1		2
3.2	odd	2		966.2.a.p.1.1	✓	2
12.11	even	2		7728.2.a.bd.1.1		2
21.20	even	2		6762.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.p.1.1	✓	2	3.2	odd	2
2898.2.a.v.1.1		2	1.1	even	1	trivial
6762.2.a.bz.1.2		2	21.20	even	2
7728.2.a.bd.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -4.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.47214 q^{19} -2.00000 q^{20} +1.00000 q^{23} -1.00000 q^{25} +4.47214 q^{26} +1.00000 q^{28} +2.00000 q^{29} -2.47214 q^{31} -1.00000 q^{32} +4.47214 q^{34} -2.00000 q^{35} +6.94427 q^{37} -2.47214 q^{38} +2.00000 q^{40} +6.00000 q^{41} -4.94427 q^{43} -1.00000 q^{46} +2.47214 q^{47} +1.00000 q^{49} +1.00000 q^{50} -4.47214 q^{52} +10.9443 q^{53} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +2.47214 q^{62} +1.00000 q^{64} +8.94427 q^{65} +4.94427 q^{67} -4.47214 q^{68} +2.00000 q^{70} -4.94427 q^{71} +14.9443 q^{73} -6.94427 q^{74} +2.47214 q^{76} -4.94427 q^{79} -2.00000 q^{80} -6.00000 q^{82} -2.47214 q^{83} +8.94427 q^{85} +4.94427 q^{86} -9.41641 q^{89} -4.47214 q^{91} +1.00000 q^{92} -2.47214 q^{94} -4.94427 q^{95} -0.472136 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{20} + 2 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{35} - 4 q^{37} + 4 q^{38} + 4 q^{40} + 12 q^{41} + 8 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{53} - 2 q^{56} - 4 q^{58} - 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 4 q^{70} + 8 q^{71} + 12 q^{73} + 4 q^{74} - 4 q^{76} + 8 q^{79} - 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{86} + 8 q^{89} + 2 q^{92} + 4 q^{94} + 8 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 - 2 * q^14 + 2 * q^16 - 4 * q^19 - 4 * q^20 + 2 * q^23 - 2 * q^25 + 2 * q^28 + 4 * q^29 + 4 * q^31 - 2 * q^32 - 4 * q^35 - 4 * q^37 + 4 * q^38 + 4 * q^40 + 12 * q^41 + 8 * q^43 - 2 * q^46 - 4 * q^47 + 2 * q^49 + 2 * q^50 + 4 * q^53 - 2 * q^56 - 4 * q^58 - 8 * q^59 - 12 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^67 + 4 * q^70 + 8 * q^71 + 12 * q^73 + 4 * q^74 - 4 * q^76 + 8 * q^79 - 4 * q^80 - 12 * q^82 + 4 * q^83 - 8 * q^86 + 8 * q^89 + 2 * q^92 + 4 * q^94 + 8 * q^95 + 8 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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