LMFDB - Embedded newform 2898.2.a.bh.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:	$4$
Coefficient field:	4.4.271296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 18x^{2} - 8x + 24 $ x^4 - 18x^2 - 8x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.76613$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.76613 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.76613 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.76613 q^{10} -5.76613 q^{11} -4.26484 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.26484 q^{17} -8.03097 q^{19} -3.76613 q^{20} +5.76613 q^{22} -1.00000 q^{23} +9.18372 q^{25} +4.26484 q^{26} +1.00000 q^{28} -2.00000 q^{29} +0.0811192 q^{31} -1.00000 q^{32} +4.26484 q^{34} -3.76613 q^{35} +0.579832 q^{37} +8.03097 q^{38} +3.76613 q^{40} +2.00000 q^{41} +6.76355 q^{43} -5.76613 q^{44} +1.00000 q^{46} +3.91888 q^{47} +1.00000 q^{49} -9.18372 q^{50} -4.26484 q^{52} -3.42017 q^{53} +21.7160 q^{55} -1.00000 q^{56} +2.00000 q^{58} +11.5323 q^{59} -11.2984 q^{61} -0.0811192 q^{62} +1.00000 q^{64} +16.0619 q^{65} -13.8281 q^{67} -4.26484 q^{68} +3.76613 q^{70} +5.18630 q^{71} -0.345960 q^{73} -0.579832 q^{74} -8.03097 q^{76} -5.76613 q^{77} +3.65404 q^{79} -3.76613 q^{80} -2.00000 q^{82} -11.8687 q^{83} +16.0619 q^{85} -6.76355 q^{86} +5.76613 q^{88} -7.10518 q^{89} -4.26484 q^{91} -1.00000 q^{92} -3.91888 q^{94} +30.2457 q^{95} +0.427080 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 8 q^{11} - 4 q^{14} + 4 q^{16} + 8 q^{22} - 4 q^{23} + 16 q^{25} + 4 q^{28} - 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{37} + 8 q^{41} + 8 q^{43} - 8 q^{44} + 4 q^{46} + 12 q^{47} + 4 q^{49} - 16 q^{50} - 12 q^{53} + 36 q^{55} - 4 q^{56} + 8 q^{58} + 16 q^{59} - 4 q^{62} + 4 q^{64} + 24 q^{67} + 4 q^{71} + 12 q^{73} - 4 q^{74} - 8 q^{77} + 28 q^{79} - 8 q^{82} - 8 q^{83} - 8 q^{86} + 8 q^{88} - 8 q^{89} - 4 q^{92} - 12 q^{94} + 36 q^{95} - 8 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 8 * q^11 - 4 * q^14 + 4 * q^16 + 8 * q^22 - 4 * q^23 + 16 * q^25 + 4 * q^28 - 8 * q^29 + 4 * q^31 - 4 * q^32 + 4 * q^37 + 8 * q^41 + 8 * q^43 - 8 * q^44 + 4 * q^46 + 12 * q^47 + 4 * q^49 - 16 * q^50 - 12 * q^53 + 36 * q^55 - 4 * q^56 + 8 * q^58 + 16 * q^59 - 4 * q^62 + 4 * q^64 + 24 * q^67 + 4 * q^71 + 12 * q^73 - 4 * q^74 - 8 * q^77 + 28 * q^79 - 8 * q^82 - 8 * q^83 - 8 * q^86 + 8 * q^88 - 8 * q^89 - 4 * q^92 - 12 * q^94 + 36 * q^95 - 8 * q^97 - 4 * 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
$5$	−3.76613	−1.68426	−0.842132	−	0.539272i	$-0.818700\pi$
$5$	−3.76613	−1.68426	−0.842132	+	0.539272i	$0.818700\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.76613	1.19095
$11$	−5.76613	−1.73855	−0.869277	−	0.494326i	$-0.835415\pi$
$11$	−5.76613	−1.73855	−0.869277	+	0.494326i	$0.835415\pi$
$12$	0	0
$13$	−4.26484	−1.18285	−0.591427	−	0.806359i	$-0.701435\pi$
$13$	−4.26484	−1.18285	−0.591427	+	0.806359i	$0.701435\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.26484	−1.03438	−0.517188	−	0.855872i	$-0.673021\pi$
$17$	−4.26484	−1.03438	−0.517188	+	0.855872i	$0.673021\pi$
$18$	0	0
$19$	−8.03097	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$19$	−8.03097	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$20$	−3.76613	−0.842132
$21$	0	0
$22$	5.76613	1.22934
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	9.18372	1.83674
$26$	4.26484	0.836404
$27$	0	0
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0.0811192	0.0145694	0.00728472	−	0.999973i	$-0.497681\pi$
$31$	0.0811192	0.0145694	0.00728472	+	0.999973i	$0.497681\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.26484	0.731414
$35$	−3.76613	−0.636592
$36$	0	0
$37$	0.579832	0.0953238	0.0476619	−	0.998864i	$-0.484823\pi$
$37$	0.579832	0.0953238	0.0476619	+	0.998864i	$0.484823\pi$
$38$	8.03097	1.30280
$39$	0	0
$40$	3.76613	0.595477
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	6.76355	1.03143	0.515716	−	0.856759i	$-0.327526\pi$
$43$	6.76355	1.03143	0.515716	+	0.856759i	$0.327526\pi$
$44$	−5.76613	−0.869277
$45$	0	0
$46$	1.00000	0.147442
$47$	3.91888	0.571628	0.285814	−	0.958285i	$-0.407736\pi$
$47$	3.91888	0.571628	0.285814	+	0.958285i	$0.407736\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.18372	−1.29877
$51$	0	0
$52$	−4.26484	−0.591427
$53$	−3.42017	−0.469796	−0.234898	−	0.972020i	$-0.575476\pi$
$53$	−3.42017	−0.469796	−0.234898	+	0.972020i	$0.575476\pi$
$54$	0	0
$55$	21.7160	2.92818
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	11.5323	1.50137	0.750686	−	0.660659i	$-0.229723\pi$
$59$	11.5323	1.50137	0.750686	+	0.660659i	$0.229723\pi$
$60$	0	0
$61$	−11.2984	−1.44661	−0.723305	−	0.690529i	$-0.757378\pi$
$61$	−11.2984	−1.44661	−0.723305	+	0.690529i	$0.757378\pi$
$62$	−0.0811192	−0.0103021
$63$	0	0
$64$	1.00000	0.125000
$65$	16.0619	1.99224
$66$	0	0
$67$	−13.8281	−1.68937	−0.844684	−	0.535266i	$-0.820211\pi$
$67$	−13.8281	−1.68937	−0.844684	+	0.535266i	$0.820211\pi$
$68$	−4.26484	−0.517188
$69$	0	0
$70$	3.76613	0.450138
$71$	5.18630	0.615500	0.307750	−	0.951467i	$-0.400424\pi$
$71$	5.18630	0.615500	0.307750	+	0.951467i	$0.400424\pi$
$72$	0	0
$73$	−0.345960	−0.0404916	−0.0202458	−	0.999795i	$-0.506445\pi$
$73$	−0.345960	−0.0404916	−0.0202458	+	0.999795i	$0.506445\pi$
$74$	−0.579832	−0.0674041
$75$	0	0
$76$	−8.03097	−0.921215
$77$	−5.76613	−0.657111
$78$	0	0
$79$	3.65404	0.411112	0.205556	−	0.978645i	$-0.434100\pi$
$79$	3.65404	0.411112	0.205556	+	0.978645i	$0.434100\pi$
$80$	−3.76613	−0.421066
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−11.8687	−1.30276	−0.651381	−	0.758751i	$-0.725810\pi$
$83$	−11.8687	−1.30276	−0.651381	+	0.758751i	$0.725810\pi$
$84$	0	0
$85$	16.0619	1.74216
$86$	−6.76355	−0.729333
$87$	0	0
$88$	5.76613	0.614671
$89$	−7.10518	−0.753147	−0.376574	−	0.926387i	$-0.622898\pi$
$89$	−7.10518	−0.753147	−0.376574	+	0.926387i	$0.622898\pi$
$90$	0	0
$91$	−4.26484	−0.447077
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−3.91888	−0.404202
$95$	30.2457	3.10314
$96$	0	0
$97$	0.427080	0.0433634	0.0216817	−	0.999765i	$-0.493098\pi$
$97$	0.427080	0.0433634	0.0216817	+	0.999765i	$0.493098\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	9.18372	0.918372
$101$	−9.26227	−0.921630	−0.460815	−	0.887496i	$-0.652443\pi$
$101$	−9.26227	−0.921630	−0.460815	+	0.887496i	$0.652443\pi$
$102$	0	0
$103$	−20.0619	−1.97676	−0.988381	−	0.151998i	$-0.951429\pi$
$103$	−20.0619	−1.97676	−0.988381	+	0.151998i	$0.951429\pi$
$104$	4.26484	0.418202
$105$	0	0
$106$	3.42017	0.332196
$107$	−5.76613	−0.557433	−0.278716	−	0.960373i	$-0.589909\pi$
$107$	−5.76613	−0.557433	−0.278716	+	0.960373i	$0.589909\pi$
$108$	0	0
$109$	−3.60389	−0.345190	−0.172595	−	0.984993i	$-0.555215\pi$
$109$	−3.60389	−0.345190	−0.172595	+	0.984993i	$0.555215\pi$
$110$	−21.7160	−2.07054
$111$	0	0
$112$	1.00000	0.0944911
$113$	20.9023	1.96632	0.983160	−	0.182745i	$-0.0584984\pi$
$113$	20.9023	1.96632	0.983160	+	0.182745i	$0.0584984\pi$
$114$	0	0
$115$	3.76613	0.351193
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−11.5323	−1.06163
$119$	−4.26484	−0.390957
$120$	0	0
$121$	22.2482	2.02257
$122$	11.2984	1.02291
$123$	0	0
$124$	0.0811192	0.00728472
$125$	−15.7564	−1.40930
$126$	0	0
$127$	−4.52968	−0.401944	−0.200972	−	0.979597i	$-0.564410\pi$
$127$	−4.52968	−0.401944	−0.200972	+	0.979597i	$0.564410\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−16.0619	−1.40873
$131$	7.53226	0.658096	0.329048	−	0.944313i	$-0.393272\pi$
$131$	7.53226	0.658096	0.329048	+	0.944313i	$0.393272\pi$
$132$	0	0
$133$	−8.03097	−0.696373
$134$	13.8281	1.19456
$135$	0	0
$136$	4.26484	0.365707
$137$	5.53226	0.472653	0.236326	−	0.971674i	$-0.424057\pi$
$137$	5.53226	0.472653	0.236326	+	0.971674i	$0.424057\pi$
$138$	0	0
$139$	20.0619	1.70163	0.850816	−	0.525464i	$-0.176108\pi$
$139$	20.0619	1.70163	0.850816	+	0.525464i	$0.176108\pi$
$140$	−3.76613	−0.318296
$141$	0	0
$142$	−5.18630	−0.435224
$143$	24.5916	2.05645
$144$	0	0
$145$	7.53226	0.625520
$146$	0.345960	0.0286319
$147$	0	0
$148$	0.579832	0.0476619
$149$	4.76355	0.390246	0.195123	−	0.980779i	$-0.437489\pi$
$149$	4.76355	0.390246	0.195123	+	0.980779i	$0.437489\pi$
$150$	0	0
$151$	17.8782	1.45491	0.727454	−	0.686156i	$-0.240703\pi$
$151$	17.8782	1.45491	0.727454	+	0.686156i	$0.240703\pi$
$152$	8.03097	0.651398
$153$	0	0
$154$	5.76613	0.464648
$155$	−0.305505	−0.0245388
$156$	0	0
$157$	19.1361	1.52723	0.763615	−	0.645671i	$-0.223422\pi$
$157$	19.1361	1.52723	0.763615	+	0.645671i	$0.223422\pi$
$158$	−3.65404	−0.290700
$159$	0	0
$160$	3.76613	0.297739
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−7.18630	−0.562874	−0.281437	−	0.959580i	$-0.590811\pi$
$163$	−7.18630	−0.562874	−0.281437	+	0.959580i	$0.590811\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	11.8687	0.921192
$167$	−15.5131	−1.20044	−0.600219	−	0.799836i	$-0.704920\pi$
$167$	−15.5131	−1.20044	−0.600219	+	0.799836i	$0.704920\pi$
$168$	0	0
$169$	5.18887	0.399144
$170$	−16.0619	−1.23189
$171$	0	0
$172$	6.76355	0.515716
$173$	8.10260	0.616029	0.308015	−	0.951382i	$-0.400335\pi$
$173$	8.10260	0.616029	0.308015	+	0.951382i	$0.400335\pi$
$174$	0	0
$175$	9.18372	0.694224
$176$	−5.76613	−0.434638
$177$	0	0
$178$	7.10518	0.532556
$179$	−24.5916	−1.83806	−0.919032	−	0.394183i	$-0.871028\pi$
$179$	−24.5916	−1.83806	−0.919032	+	0.394183i	$0.871028\pi$
$180$	0	0
$181$	10.6065	0.788372	0.394186	−	0.919031i	$-0.371027\pi$
$181$	10.6065	0.788372	0.394186	+	0.919031i	$0.371027\pi$
$182$	4.26484	0.316131
$183$	0	0
$184$	1.00000	0.0737210
$185$	−2.18372	−0.160550
$186$	0	0
$187$	24.5916	1.79832
$188$	3.91888	0.285814
$189$	0	0
$190$	−30.2457	−2.19425
$191$	−0.813704	−0.0588775	−0.0294388	−	0.999567i	$-0.509372\pi$
$191$	−0.813704	−0.0588775	−0.0294388	+	0.999567i	$0.509372\pi$
$192$	0	0
$193$	5.81628	0.418665	0.209332	−	0.977845i	$-0.432871\pi$
$193$	5.81628	0.418665	0.209332	+	0.977845i	$0.432871\pi$
$194$	−0.427080	−0.0306625
$195$	0	0
$196$	1.00000	0.0714286
$197$	15.2216	1.08449	0.542247	−	0.840219i	$-0.317574\pi$
$197$	15.2216	1.08449	0.542247	+	0.840219i	$0.317574\pi$
$198$	0	0
$199$	−2.53483	−0.179689	−0.0898447	−	0.995956i	$-0.528637\pi$
$199$	−2.53483	−0.179689	−0.0898447	+	0.995956i	$0.528637\pi$
$200$	−9.18372	−0.649387
$201$	0	0
$202$	9.26227	0.651691
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−7.53226	−0.526076
$206$	20.0619	1.39778
$207$	0	0
$208$	−4.26484	−0.295714
$209$	46.3076	3.20316
$210$	0	0
$211$	−14.7186	−1.01327	−0.506633	−	0.862162i	$-0.669110\pi$
$211$	−14.7186	−1.01327	−0.506633	+	0.862162i	$0.669110\pi$
$212$	−3.42017	−0.234898
$213$	0	0
$214$	5.76613	0.394164
$215$	−25.4724	−1.73720
$216$	0	0
$217$	0.0811192	0.00550673
$218$	3.60389	0.244086
$219$	0	0
$220$	21.7160	1.46409
$221$	18.1889	1.22352
$222$	0	0
$223$	−24.9782	−1.67267	−0.836334	−	0.548221i	$-0.815305\pi$
$223$	−24.9782	−1.67267	−0.836334	+	0.548221i	$0.815305\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−20.9023	−1.39040
$227$	18.8713	1.25253	0.626266	−	0.779609i	$-0.284582\pi$
$227$	18.8713	1.25253	0.626266	+	0.779609i	$0.284582\pi$
$228$	0	0
$229$	3.29838	0.217963	0.108982	−	0.994044i	$-0.465241\pi$
$229$	3.29838	0.217963	0.108982	+	0.994044i	$0.465241\pi$
$230$	−3.76613	−0.248331
$231$	0	0
$232$	2.00000	0.131306
$233$	−15.2482	−0.998945	−0.499473	−	0.866330i	$-0.666473\pi$
$233$	−15.2482	−0.998945	−0.499473	+	0.866330i	$0.666473\pi$
$234$	0	0
$235$	−14.7590	−0.962772
$236$	11.5323	0.750686
$237$	0	0
$238$	4.26484	0.276449
$239$	4.16224	0.269233	0.134616	−	0.990898i	$-0.457020\pi$
$239$	4.16224	0.269233	0.134616	+	0.990898i	$0.457020\pi$
$240$	0	0
$241$	−18.6374	−1.20054	−0.600271	−	0.799796i	$-0.704941\pi$
$241$	−18.6374	−1.20054	−0.600271	+	0.799796i	$0.704941\pi$
$242$	−22.2482	−1.43017
$243$	0	0
$244$	−11.2984	−0.723305
$245$	−3.76613	−0.240609
$246$	0	0
$247$	34.2508	2.17933
$248$	−0.0811192	−0.00515107
$249$	0	0
$250$	15.7564	0.996524
$251$	−8.39841	−0.530103	−0.265052	−	0.964234i	$-0.585389\pi$
$251$	−8.39841	−0.530103	−0.265052	+	0.964234i	$0.585389\pi$
$252$	0	0
$253$	5.76613	0.362513
$254$	4.52968	0.284218
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.9023	−1.30385	−0.651924	−	0.758284i	$-0.726038\pi$
$257$	−20.9023	−1.30385	−0.651924	+	0.758284i	$0.726038\pi$
$258$	0	0
$259$	0.579832	0.0360290
$260$	16.0619	0.996119
$261$	0	0
$262$	−7.53226	−0.465344
$263$	12.6919	0.782617	0.391309	−	0.920259i	$-0.372022\pi$
$263$	12.6919	0.782617	0.391309	+	0.920259i	$0.372022\pi$
$264$	0	0
$265$	12.8808	0.791261
$266$	8.03097	0.492410
$267$	0	0
$268$	−13.8281	−0.844684
$269$	−12.2648	−0.747801	−0.373900	−	0.927469i	$-0.621980\pi$
$269$	−12.2648	−0.747801	−0.373900	+	0.927469i	$0.621980\pi$
$270$	0	0
$271$	−29.7971	−1.81004	−0.905022	−	0.425364i	$-0.860146\pi$
$271$	−29.7971	−1.81004	−0.905022	+	0.425364i	$0.860146\pi$
$272$	−4.26484	−0.258594
$273$	0	0
$274$	−5.53226	−0.334216
$275$	−52.9545	−3.19328
$276$	0	0
$277$	29.8997	1.79650	0.898249	−	0.439486i	$-0.144839\pi$
$277$	29.8997	1.79650	0.898249	+	0.439486i	$0.144839\pi$
$278$	−20.0619	−1.20324
$279$	0	0
$280$	3.76613	0.225069
$281$	−3.99485	−0.238313	−0.119156	−	0.992875i	$-0.538019\pi$
$281$	−3.99485	−0.238313	−0.119156	+	0.992875i	$0.538019\pi$
$282$	0	0
$283$	0.966456	0.0574499	0.0287249	−	0.999587i	$-0.490855\pi$
$283$	0.966456	0.0574499	0.0287249	+	0.999587i	$0.490855\pi$
$284$	5.18630	0.307750
$285$	0	0
$286$	−24.5916	−1.45413
$287$	2.00000	0.118056
$288$	0	0
$289$	1.18887	0.0699335
$290$	−7.53226	−0.442309
$291$	0	0
$292$	−0.345960	−0.0202458
$293$	28.1336	1.64358	0.821790	−	0.569790i	$-0.192975\pi$
$293$	28.1336	1.64358	0.821790	+	0.569790i	$0.192975\pi$
$294$	0	0
$295$	−43.4320	−2.52871
$296$	−0.579832	−0.0337021
$297$	0	0
$298$	−4.76355	−0.275945
$299$	4.26484	0.246642
$300$	0	0
$301$	6.76355	0.389845
$302$	−17.8782	−1.02878
$303$	0	0
$304$	−8.03097	−0.460608
$305$	42.5512	2.43647
$306$	0	0
$307$	27.5942	1.57488	0.787442	−	0.616389i	$-0.211405\pi$
$307$	27.5942	1.57488	0.787442	+	0.616389i	$0.211405\pi$
$308$	−5.76613	−0.328556
$309$	0	0
$310$	0.305505	0.0173515
$311$	19.9542	1.13150	0.565749	−	0.824577i	$-0.308587\pi$
$311$	19.9542	1.13150	0.565749	+	0.824577i	$0.308587\pi$
$312$	0	0
$313$	−14.7997	−0.836527	−0.418263	−	0.908326i	$-0.637361\pi$
$313$	−14.7997	−0.836527	−0.418263	+	0.908326i	$0.637361\pi$
$314$	−19.1361	−1.07992
$315$	0	0
$316$	3.65404	0.205556
$317$	−26.5297	−1.49006	−0.745028	−	0.667034i	$-0.767564\pi$
$317$	−26.5297	−1.49006	−0.745028	+	0.667034i	$0.767564\pi$
$318$	0	0
$319$	11.5323	0.645682
$320$	−3.76613	−0.210533
$321$	0	0
$322$	1.00000	0.0557278
$323$	34.2508	1.90577
$324$	0	0
$325$	−39.1671	−2.17260
$326$	7.18630	0.398012
$327$	0	0
$328$	−2.00000	−0.110432
$329$	3.91888	0.216055
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−11.8687	−0.651381
$333$	0	0
$334$	15.5131	0.848838
$335$	52.0783	2.84534
$336$	0	0
$337$	18.5297	1.00938	0.504688	−	0.863302i	$-0.331608\pi$
$337$	18.5297	1.00938	0.504688	+	0.863302i	$0.331608\pi$
$338$	−5.18887	−0.282237
$339$	0	0
$340$	16.0619	0.871081
$341$	−0.467743	−0.0253297
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.76355	−0.364666
$345$	0	0
$346$	−8.10260	−0.435599
$347$	19.8782	1.06712	0.533559	−	0.845763i	$-0.320854\pi$
$347$	19.8782	1.06712	0.533559	+	0.845763i	$0.320854\pi$
$348$	0	0
$349$	0.570346	0.0305299	0.0152650	−	0.999883i	$-0.495141\pi$
$349$	0.570346	0.0305299	0.0152650	+	0.999883i	$0.495141\pi$
$350$	−9.18372	−0.490891
$351$	0	0
$352$	5.76613	0.307336
$353$	−31.2482	−1.66318	−0.831588	−	0.555393i	$-0.812568\pi$
$353$	−31.2482	−1.66318	−0.831588	+	0.555393i	$0.812568\pi$
$354$	0	0
$355$	−19.5323	−1.03666
$356$	−7.10518	−0.376574
$357$	0	0
$358$	24.5916	1.29971
$359$	23.5537	1.24312	0.621559	−	0.783367i	$-0.286499\pi$
$359$	23.5537	1.24312	0.621559	+	0.783367i	$0.286499\pi$
$360$	0	0
$361$	45.4965	2.39455
$362$	−10.6065	−0.557463
$363$	0	0
$364$	−4.26484	−0.223538
$365$	1.30293	0.0681985
$366$	0	0
$367$	22.9023	1.19549	0.597745	−	0.801687i	$-0.296064\pi$
$367$	22.9023	1.19549	0.597745	+	0.801687i	$0.296064\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	2.18372	0.113526
$371$	−3.42017	−0.177566
$372$	0	0
$373$	−8.94728	−0.463272	−0.231636	−	0.972802i	$-0.574408\pi$
$373$	−8.94728	−0.463272	−0.231636	+	0.972802i	$0.574408\pi$
$374$	−24.5916	−1.27160
$375$	0	0
$376$	−3.91888	−0.202101
$377$	8.52968	0.439301
$378$	0	0
$379$	−25.9903	−1.33503	−0.667516	−	0.744595i	$-0.732642\pi$
$379$	−25.9903	−1.33503	−0.667516	+	0.744595i	$0.732642\pi$
$380$	30.2457	1.55157
$381$	0	0
$382$	0.813704	0.0416327
$383$	20.6919	1.05731	0.528654	−	0.848837i	$-0.322697\pi$
$383$	20.6919	1.05731	0.528654	+	0.848837i	$0.322697\pi$
$384$	0	0
$385$	21.7160	1.10675
$386$	−5.81628	−0.296041
$387$	0	0
$388$	0.427080	0.0216817
$389$	−32.8470	−1.66541	−0.832704	−	0.553719i	$-0.813208\pi$
$389$	−32.8470	−1.66541	−0.832704	+	0.553719i	$0.813208\pi$
$390$	0	0
$391$	4.26484	0.215682
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−15.2216	−0.766853
$395$	−13.7616	−0.692420
$396$	0	0
$397$	−19.7352	−0.990479	−0.495240	−	0.868756i	$-0.664920\pi$
$397$	−19.7352	−0.990479	−0.495240	+	0.868756i	$0.664920\pi$
$398$	2.53483	0.127060
$399$	0	0
$400$	9.18372	0.459186
$401$	−27.9048	−1.39350	−0.696751	−	0.717313i	$-0.745372\pi$
$401$	−27.9048	−1.39350	−0.696751	+	0.717313i	$0.745372\pi$
$402$	0	0
$403$	−0.345960	−0.0172335
$404$	−9.26227	−0.460815
$405$	0	0
$406$	2.00000	0.0992583
$407$	−3.34339	−0.165726
$408$	0	0
$409$	−10.3674	−0.512637	−0.256319	−	0.966592i	$-0.582510\pi$
$409$	−10.3674	−0.512637	−0.256319	+	0.966592i	$0.582510\pi$
$410$	7.53226	0.371992
$411$	0	0
$412$	−20.0619	−0.988381
$413$	11.5323	0.567465
$414$	0	0
$415$	44.6992	2.19420
$416$	4.26484	0.209101
$417$	0	0
$418$	−46.3076	−2.26498
$419$	7.27711	0.355510	0.177755	−	0.984075i	$-0.443117\pi$
$419$	7.27711	0.355510	0.177755	+	0.984075i	$0.443117\pi$
$420$	0	0
$421$	−25.8229	−1.25853	−0.629266	−	0.777190i	$-0.716644\pi$
$421$	−25.8229	−1.25853	−0.629266	+	0.777190i	$0.716644\pi$
$422$	14.7186	0.716488
$423$	0	0
$424$	3.42017	0.166098
$425$	−39.1671	−1.89988
$426$	0	0
$427$	−11.2984	−0.546767
$428$	−5.76613	−0.278716
$429$	0	0
$430$	25.4724	1.22839
$431$	8.83519	0.425576	0.212788	−	0.977098i	$-0.431746\pi$
$431$	8.83519	0.425576	0.212788	+	0.977098i	$0.431746\pi$
$432$	0	0
$433$	19.3294	0.928909	0.464455	−	0.885597i	$-0.346250\pi$
$433$	19.3294	0.928909	0.464455	+	0.885597i	$0.346250\pi$
$434$	−0.0811192	−0.00389384
$435$	0	0
$436$	−3.60389	−0.172595
$437$	8.03097	0.384173
$438$	0	0
$439$	−10.2029	−0.486958	−0.243479	−	0.969906i	$-0.578289\pi$
$439$	−10.2029	−0.486958	−0.243479	+	0.969906i	$0.578289\pi$
$440$	−21.7160	−1.03527
$441$	0	0
$442$	−18.1889	−0.865156
$443$	26.9427	1.28009	0.640044	−	0.768338i	$-0.278916\pi$
$443$	26.9427	1.28009	0.640044	+	0.768338i	$0.278916\pi$
$444$	0	0
$445$	26.7590	1.26850
$446$	24.9782	1.18275
$447$	0	0
$448$	1.00000	0.0472456
$449$	−25.4320	−1.20021	−0.600104	−	0.799922i	$-0.704874\pi$
$449$	−25.4320	−1.20021	−0.600104	+	0.799922i	$0.704874\pi$
$450$	0	0
$451$	−11.5323	−0.543033
$452$	20.9023	0.983160
$453$	0	0
$454$	−18.8713	−0.885674
$455$	16.0619	0.752995
$456$	0	0
$457$	−9.89970	−0.463088	−0.231544	−	0.972824i	$-0.574378\pi$
$457$	−9.89970	−0.463088	−0.231544	+	0.972824i	$0.574378\pi$
$458$	−3.29838	−0.154123
$459$	0	0
$460$	3.76613	0.175597
$461$	−17.1620	−0.799313	−0.399656	−	0.916665i	$-0.630871\pi$
$461$	−17.1620	−0.799313	−0.399656	+	0.916665i	$0.630871\pi$
$462$	0	0
$463$	0.367444	0.0170766	0.00853829	−	0.999964i	$-0.497282\pi$
$463$	0.367444	0.0170766	0.00853829	+	0.999964i	$0.497282\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	15.2482	0.706361
$467$	−31.7255	−1.46808	−0.734040	−	0.679107i	$-0.762367\pi$
$467$	−31.7255	−1.46808	−0.734040	+	0.679107i	$0.762367\pi$
$468$	0	0
$469$	−13.8281	−0.638521
$470$	14.7590	0.680782
$471$	0	0
$472$	−11.5323	−0.530815
$473$	−38.9995	−1.79320
$474$	0	0
$475$	−73.7542	−3.38407
$476$	−4.26484	−0.195479
$477$	0	0
$478$	−4.16224	−0.190376
$479$	−3.47032	−0.158563	−0.0792814	−	0.996852i	$-0.525263\pi$
$479$	−3.47032	−0.158563	−0.0792814	+	0.996852i	$0.525263\pi$
$480$	0	0
$481$	−2.47289	−0.112754
$482$	18.6374	0.848912
$483$	0	0
$484$	22.2482	1.01128
$485$	−1.60844	−0.0730353
$486$	0	0
$487$	−30.4965	−1.38193	−0.690963	−	0.722890i	$-0.742813\pi$
$487$	−30.4965	−1.38193	−0.690963	+	0.722890i	$0.742813\pi$
$488$	11.2984	0.511454
$489$	0	0
$490$	3.76613	0.170136
$491$	28.9376	1.30593	0.652967	−	0.757386i	$-0.273524\pi$
$491$	28.9376	1.30593	0.652967	+	0.757386i	$0.273524\pi$
$492$	0	0
$493$	8.52968	0.384158
$494$	−34.2508	−1.54102
$495$	0	0
$496$	0.0811192	0.00364236
$497$	5.18630	0.232637
$498$	0	0
$499$	−28.2457	−1.26445	−0.632225	−	0.774785i	$-0.717858\pi$
$499$	−28.2457	−1.26445	−0.632225	+	0.774785i	$0.717858\pi$
$500$	−15.7564	−0.704649
$501$	0	0
$502$	8.39841	0.374840
$503$	19.1648	0.854517	0.427258	−	0.904130i	$-0.359479\pi$
$503$	19.1648	0.854517	0.427258	+	0.904130i	$0.359479\pi$
$504$	0	0
$505$	34.8829	1.55227
$506$	−5.76613	−0.256336
$507$	0	0
$508$	−4.52968	−0.200972
$509$	21.7019	0.961922	0.480961	−	0.876742i	$-0.340288\pi$
$509$	21.7019	0.961922	0.480961	+	0.876742i	$0.340288\pi$
$510$	0	0
$511$	−0.345960	−0.0153044
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	20.9023	0.921960
$515$	75.5558	3.32939
$516$	0	0
$517$	−22.5968	−0.993805
$518$	−0.579832	−0.0254764
$519$	0	0
$520$	−16.0619	−0.704363
$521$	17.1723	0.752331	0.376165	−	0.926553i	$-0.377242\pi$
$521$	17.1723	0.752331	0.376165	+	0.926553i	$0.377242\pi$
$522$	0	0
$523$	11.5632	0.505625	0.252812	−	0.967515i	$-0.418644\pi$
$523$	11.5632	0.505625	0.252812	+	0.967515i	$0.418644\pi$
$524$	7.53226	0.329048
$525$	0	0
$526$	−12.6919	−0.553394
$527$	−0.345960	−0.0150703
$528$	0	0
$529$	1.00000	0.0434783
$530$	−12.8808	−0.559506
$531$	0	0
$532$	−8.03097	−0.348187
$533$	−8.52968	−0.369462
$534$	0	0
$535$	21.7160	0.938864
$536$	13.8281	0.597282
$537$	0	0
$538$	12.2648	0.528775
$539$	−5.76613	−0.248365
$540$	0	0
$541$	5.89970	0.253648	0.126824	−	0.991925i	$-0.459522\pi$
$541$	5.89970	0.253648	0.126824	+	0.991925i	$0.459522\pi$
$542$	29.7971	1.27989
$543$	0	0
$544$	4.26484	0.182854
$545$	13.5727	0.581391
$546$	0	0
$547$	25.4268	1.08717	0.543586	−	0.839354i	$-0.317066\pi$
$547$	25.4268	1.08717	0.543586	+	0.839354i	$0.317066\pi$
$548$	5.53226	0.236326
$549$	0	0
$550$	52.9545	2.25799
$551$	16.0619	0.684262
$552$	0	0
$553$	3.65404	0.155386
$554$	−29.8997	−1.27032
$555$	0	0
$556$	20.0619	0.850816
$557$	42.0471	1.78159	0.890796	−	0.454403i	$-0.150147\pi$
$557$	42.0471	1.78159	0.890796	+	0.454403i	$0.150147\pi$
$558$	0	0
$559$	−28.8455	−1.22003
$560$	−3.76613	−0.159148
$561$	0	0
$562$	3.99485	0.168513
$563$	5.34420	0.225231	0.112616	−	0.993639i	$-0.464077\pi$
$563$	5.34420	0.225231	0.112616	+	0.993639i	$0.464077\pi$
$564$	0	0
$565$	−78.7206	−3.31180
$566$	−0.966456	−0.0406232
$567$	0	0
$568$	−5.18630	−0.217612
$569$	17.6561	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$569$	17.6561	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$570$	0	0
$571$	12.7687	0.534354	0.267177	−	0.963648i	$-0.413909\pi$
$571$	12.7687	0.534354	0.267177	+	0.963648i	$0.413909\pi$
$572$	24.5916	1.02823
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	−9.18372	−0.382988
$576$	0	0
$577$	−26.0972	−1.08644	−0.543221	−	0.839590i	$-0.682796\pi$
$577$	−26.0972	−1.08644	−0.543221	+	0.839590i	$0.682796\pi$
$578$	−1.18887	−0.0494505
$579$	0	0
$580$	7.53226	0.312760
$581$	−11.8687	−0.492398
$582$	0	0
$583$	19.7211	0.816766
$584$	0.345960	0.0143159
$585$	0	0
$586$	−28.1336	−1.16219
$587$	5.37517	0.221857	0.110928	−	0.993828i	$-0.464618\pi$
$587$	5.37517	0.221857	0.110928	+	0.993828i	$0.464618\pi$
$588$	0	0
$589$	−0.651466	−0.0268432
$590$	43.4320	1.78807
$591$	0	0
$592$	0.579832	0.0238310
$593$	35.0313	1.43856	0.719282	−	0.694719i	$-0.244471\pi$
$593$	35.0313	1.43856	0.719282	+	0.694719i	$0.244471\pi$
$594$	0	0
$595$	16.0619	0.658475
$596$	4.76355	0.195123
$597$	0	0
$598$	−4.26484	−0.174402
$599$	8.32448	0.340129	0.170064	−	0.985433i	$-0.445602\pi$
$599$	8.32448	0.340129	0.170064	+	0.985433i	$0.445602\pi$
$600$	0	0
$601$	−24.3460	−0.993092	−0.496546	−	0.868010i	$-0.665399\pi$
$601$	−24.3460	−0.993092	−0.496546	+	0.868010i	$0.665399\pi$
$602$	−6.76355	−0.275662
$603$	0	0
$604$	17.8782	0.727454
$605$	−83.7897	−3.40654
$606$	0	0
$607$	2.45371	0.0995931	0.0497965	−	0.998759i	$-0.484143\pi$
$607$	2.45371	0.0995931	0.0497965	+	0.998759i	$0.484143\pi$
$608$	8.03097	0.325699
$609$	0	0
$610$	−42.5512	−1.72285
$611$	−16.7134	−0.676152
$612$	0	0
$613$	14.2907	0.577194	0.288597	−	0.957451i	$-0.406811\pi$
$613$	14.2907	0.577194	0.288597	+	0.957451i	$0.406811\pi$
$614$	−27.5942	−1.11361
$615$	0	0
$616$	5.76613	0.232324
$617$	−33.7564	−1.35898	−0.679491	−	0.733683i	$-0.737800\pi$
$617$	−33.7564	−1.35898	−0.679491	+	0.733683i	$0.737800\pi$
$618$	0	0
$619$	−27.6252	−1.11035	−0.555175	−	0.831734i	$-0.687349\pi$
$619$	−27.6252	−1.11035	−0.555175	+	0.831734i	$0.687349\pi$
$620$	−0.305505	−0.0122694
$621$	0	0
$622$	−19.9542	−0.800090
$623$	−7.10518	−0.284663
$624$	0	0
$625$	13.4221	0.536886
$626$	14.7997	0.591514
$627$	0	0
$628$	19.1361	0.763615
$629$	−2.47289	−0.0986007
$630$	0	0
$631$	−25.8834	−1.03040	−0.515200	−	0.857070i	$-0.672282\pi$
$631$	−25.8834	−1.03040	−0.515200	+	0.857070i	$0.672282\pi$
$632$	−3.65404	−0.145350
$633$	0	0
$634$	26.5297	1.05363
$635$	17.0594	0.676980
$636$	0	0
$637$	−4.26484	−0.168979
$638$	−11.5323	−0.456566
$639$	0	0
$640$	3.76613	0.148869
$641$	−8.22933	−0.325039	−0.162519	−	0.986705i	$-0.551962\pi$
$641$	−8.22933	−0.325039	−0.162519	+	0.986705i	$0.551962\pi$
$642$	0	0
$643$	−21.4961	−0.847725	−0.423862	−	0.905727i	$-0.639326\pi$
$643$	−21.4961	−0.847725	−0.423862	+	0.905727i	$0.639326\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−34.2508	−1.34758
$647$	−22.1645	−0.871378	−0.435689	−	0.900097i	$-0.643495\pi$
$647$	−22.1645	−0.871378	−0.435689	+	0.900097i	$0.643495\pi$
$648$	0	0
$649$	−66.4965	−2.61021
$650$	39.1671	1.53626
$651$	0	0
$652$	−7.18630	−0.281437
$653$	−22.9161	−0.896776	−0.448388	−	0.893839i	$-0.648002\pi$
$653$	−22.9161	−0.896776	−0.448388	+	0.893839i	$0.648002\pi$
$654$	0	0
$655$	−28.3674	−1.10841
$656$	2.00000	0.0780869
$657$	0	0
$658$	−3.91888	−0.152774
$659$	43.0307	1.67624	0.838119	−	0.545487i	$-0.183655\pi$
$659$	43.0307	1.67624	0.838119	+	0.545487i	$0.183655\pi$
$660$	0	0
$661$	14.9206	0.580346	0.290173	−	0.956974i	$-0.406287\pi$
$661$	14.9206	0.580346	0.290173	+	0.956974i	$0.406287\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	11.8687	0.460596
$665$	30.2457	1.17288
$666$	0	0
$667$	2.00000	0.0774403
$668$	−15.5131	−0.600219
$669$	0	0
$670$	−52.0783	−2.01196
$671$	65.1479	2.51501
$672$	0	0
$673$	39.1884	1.51060	0.755301	−	0.655378i	$-0.227491\pi$
$673$	39.1884	1.51060	0.755301	+	0.655378i	$0.227491\pi$
$674$	−18.5297	−0.713737
$675$	0	0
$676$	5.18887	0.199572
$677$	−20.9258	−0.804244	−0.402122	−	0.915586i	$-0.631727\pi$
$677$	−20.9258	−0.804244	−0.402122	+	0.915586i	$0.631727\pi$
$678$	0	0
$679$	0.427080	0.0163898
$680$	−16.0619	−0.615947
$681$	0	0
$682$	0.467743	0.0179108
$683$	10.7186	0.410134	0.205067	−	0.978748i	$-0.434259\pi$
$683$	10.7186	0.410134	0.205067	+	0.978748i	$0.434259\pi$
$684$	0	0
$685$	−20.8352	−0.796072
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	6.76355	0.257858
$689$	14.5865	0.555700
$690$	0	0
$691$	−30.0568	−1.14341	−0.571707	−	0.820458i	$-0.693719\pi$
$691$	−30.0568	−1.14341	−0.571707	+	0.820458i	$0.693719\pi$
$692$	8.10260	0.308015
$693$	0	0
$694$	−19.8782	−0.754567
$695$	−75.5558	−2.86600
$696$	0	0
$697$	−8.52968	−0.323085
$698$	−0.570346	−0.0215879
$699$	0	0
$700$	9.18372	0.347112
$701$	−46.9929	−1.77490	−0.887448	−	0.460907i	$-0.847524\pi$
$701$	−46.9929	−1.77490	−0.887448	+	0.460907i	$0.847524\pi$
$702$	0	0
$703$	−4.65661	−0.175628
$704$	−5.76613	−0.217319
$705$	0	0
$706$	31.2482	1.17604
$707$	−9.26227	−0.348343
$708$	0	0
$709$	26.4744	0.994266	0.497133	−	0.867674i	$-0.334386\pi$
$709$	26.4744	0.994266	0.497133	+	0.867674i	$0.334386\pi$
$710$	19.5323	0.733033
$711$	0	0
$712$	7.10518	0.266278
$713$	−0.0811192	−0.00303794
$714$	0	0
$715$	−92.6152	−3.46361
$716$	−24.5916	−0.919032
$717$	0	0
$718$	−23.5537	−0.879018
$719$	21.4297	0.799191	0.399596	−	0.916692i	$-0.369151\pi$
$719$	21.4297	0.799191	0.399596	+	0.916692i	$0.369151\pi$
$720$	0	0
$721$	−20.0619	−0.747146
$722$	−45.4965	−1.69320
$723$	0	0
$724$	10.6065	0.394186
$725$	−18.3674	−0.682150
$726$	0	0
$727$	17.5993	0.652724	0.326362	−	0.945245i	$-0.394177\pi$
$727$	17.5993	0.652724	0.326362	+	0.945245i	$0.394177\pi$
$728$	4.26484	0.158066
$729$	0	0
$730$	−1.30293	−0.0482236
$731$	−28.8455	−1.06689
$732$	0	0
$733$	26.6684	0.985020	0.492510	−	0.870307i	$-0.336080\pi$
$733$	26.6684	0.985020	0.492510	+	0.870307i	$0.336080\pi$
$734$	−22.9023	−0.845338
$735$	0	0
$736$	1.00000	0.0368605
$737$	79.7344	2.93705
$738$	0	0
$739$	−34.2508	−1.25994	−0.629968	−	0.776621i	$-0.716932\pi$
$739$	−34.2508	−1.25994	−0.629968	+	0.776621i	$0.716932\pi$
$740$	−2.18372	−0.0802752
$741$	0	0
$742$	3.42017	0.125558
$743$	−21.0911	−0.773759	−0.386880	−	0.922130i	$-0.626447\pi$
$743$	−21.0911	−0.773759	−0.386880	+	0.922130i	$0.626447\pi$
$744$	0	0
$745$	−17.9402	−0.657276
$746$	8.94728	0.327583
$747$	0	0
$748$	24.5916	0.899159
$749$	−5.76613	−0.210690
$750$	0	0
$751$	19.5323	0.712742	0.356371	−	0.934345i	$-0.384014\pi$
$751$	19.5323	0.712742	0.356371	+	0.934345i	$0.384014\pi$
$752$	3.91888	0.142907
$753$	0	0
$754$	−8.52968	−0.310633
$755$	−67.3317	−2.45045
$756$	0	0
$757$	−30.2907	−1.10093	−0.550466	−	0.834857i	$-0.685550\pi$
$757$	−30.2907	−1.10093	−0.550466	+	0.834857i	$0.685550\pi$
$758$	25.9903	0.944010
$759$	0	0
$760$	−30.2457	−1.09713
$761$	−35.4002	−1.28326	−0.641628	−	0.767016i	$-0.721741\pi$
$761$	−35.4002	−1.28326	−0.641628	+	0.767016i	$0.721741\pi$
$762$	0	0
$763$	−3.60389	−0.130470
$764$	−0.813704	−0.0294388
$765$	0	0
$766$	−20.6919	−0.747629
$767$	−49.1832	−1.77590
$768$	0	0
$769$	−25.4967	−0.919436	−0.459718	−	0.888065i	$-0.652050\pi$
$769$	−25.4967	−0.919436	−0.459718	+	0.888065i	$0.652050\pi$
$770$	−21.7160	−0.782590
$771$	0	0
$772$	5.81628	0.209332
$773$	−9.14129	−0.328790	−0.164395	−	0.986395i	$-0.552567\pi$
$773$	−9.14129	−0.328790	−0.164395	+	0.986395i	$0.552567\pi$
$774$	0	0
$775$	0.744976	0.0267603
$776$	−0.427080	−0.0153313
$777$	0	0
$778$	32.8470	1.17762
$779$	−16.0619	−0.575479
$780$	0	0
$781$	−29.9048	−1.07008
$782$	−4.26484	−0.152510
$783$	0	0
$784$	1.00000	0.0357143
$785$	−72.0692	−2.57226
$786$	0	0
$787$	6.43163	0.229263	0.114631	−	0.993408i	$-0.463431\pi$
$787$	6.43163	0.229263	0.114631	+	0.993408i	$0.463431\pi$
$788$	15.2216	0.542247
$789$	0	0
$790$	13.7616	0.489615
$791$	20.9023	0.743199
$792$	0	0
$793$	48.1858	1.71113
$794$	19.7352	0.700375
$795$	0	0
$796$	−2.53483	−0.0898447
$797$	−2.91197	−0.103147	−0.0515736	−	0.998669i	$-0.516424\pi$
$797$	−2.91197	−0.103147	−0.0515736	+	0.998669i	$0.516424\pi$
$798$	0	0
$799$	−16.7134	−0.591278
$800$	−9.18372	−0.324694
$801$	0	0
$802$	27.9048	0.985354
$803$	1.99485	0.0703968
$804$	0	0
$805$	3.76613	0.132739
$806$	0.345960	0.0121859
$807$	0	0
$808$	9.26227	0.325845
$809$	−7.57271	−0.266242	−0.133121	−	0.991100i	$-0.542500\pi$
$809$	−7.57271	−0.266242	−0.133121	+	0.991100i	$0.542500\pi$
$810$	0	0
$811$	−20.4677	−0.718720	−0.359360	−	0.933199i	$-0.617005\pi$
$811$	−20.4677	−0.718720	−0.359360	+	0.933199i	$0.617005\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	3.34339	0.117186
$815$	27.0645	0.948029
$816$	0	0
$817$	−54.3179	−1.90034
$818$	10.3674	0.362489
$819$	0	0
$820$	−7.53226	−0.263038
$821$	17.4939	0.610541	0.305271	−	0.952266i	$-0.401253\pi$
$821$	17.4939	0.610541	0.305271	+	0.952266i	$0.401253\pi$
$822$	0	0
$823$	−27.6295	−0.963104	−0.481552	−	0.876418i	$-0.659927\pi$
$823$	−27.6295	−0.963104	−0.481552	+	0.876418i	$0.659927\pi$
$824$	20.0619	0.698891
$825$	0	0
$826$	−11.5323	−0.401259
$827$	−12.8306	−0.446165	−0.223083	−	0.974800i	$-0.571612\pi$
$827$	−12.8306	−0.446165	−0.223083	+	0.974800i	$0.571612\pi$
$828$	0	0
$829$	−35.9593	−1.24892	−0.624460	−	0.781057i	$-0.714681\pi$
$829$	−35.9593	−1.24892	−0.624460	+	0.781057i	$0.714681\pi$
$830$	−44.6992	−1.55153
$831$	0	0
$832$	−4.26484	−0.147857
$833$	−4.26484	−0.147768
$834$	0	0
$835$	58.4242	2.02185
$836$	46.3076	1.60158
$837$	0	0
$838$	−7.27711	−0.251384
$839$	18.1290	0.625883	0.312942	−	0.949772i	$-0.398686\pi$
$839$	18.1290	0.625883	0.312942	+	0.949772i	$0.398686\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	25.8229	0.889917
$843$	0	0
$844$	−14.7186	−0.506633
$845$	−19.5420	−0.672264
$846$	0	0
$847$	22.2482	0.764458
$848$	−3.42017	−0.117449
$849$	0	0
$850$	39.1671	1.34342
$851$	−0.579832	−0.0198764
$852$	0	0
$853$	−29.1723	−0.998839	−0.499420	−	0.866360i	$-0.666453\pi$
$853$	−29.1723	−0.998839	−0.499420	+	0.866360i	$0.666453\pi$
$854$	11.2984	0.386623
$855$	0	0
$856$	5.76613	0.197082
$857$	−23.9402	−0.817780	−0.408890	−	0.912584i	$-0.634084\pi$
$857$	−23.9402	−0.817780	−0.408890	+	0.912584i	$0.634084\pi$
$858$	0	0
$859$	47.0262	1.60451	0.802256	−	0.596980i	$-0.203633\pi$
$859$	47.0262	1.60451	0.802256	+	0.596980i	$0.203633\pi$
$860$	−25.4724	−0.868602
$861$	0	0
$862$	−8.83519	−0.300928
$863$	33.5993	1.14373	0.571867	−	0.820346i	$-0.306219\pi$
$863$	33.5993	1.14373	0.571867	+	0.820346i	$0.306219\pi$
$864$	0	0
$865$	−30.5154	−1.03756
$866$	−19.3294	−0.656838
$867$	0	0
$868$	0.0811192	0.00275336
$869$	−21.0697	−0.714739
$870$	0	0
$871$	58.9745	1.99828
$872$	3.60389	0.122043
$873$	0	0
$874$	−8.03097	−0.271652
$875$	−15.7564	−0.532665
$876$	0	0
$877$	51.8235	1.74996	0.874978	−	0.484163i	$-0.160876\pi$
$877$	51.8235	1.74996	0.874978	+	0.484163i	$0.160876\pi$
$878$	10.2029	0.344331
$879$	0	0
$880$	21.7160	0.732045
$881$	−48.3268	−1.62817	−0.814085	−	0.580745i	$-0.802761\pi$
$881$	−48.3268	−1.62817	−0.814085	+	0.580745i	$0.802761\pi$
$882$	0	0
$883$	−0.0317803	−0.00106949	−0.000534746	−	1.00000i	$-0.500170\pi$
$883$	−0.0317803	−0.00106949	−0.000534746		1.00000i	$0.500170\pi$
$884$	18.1889	0.611758
$885$	0	0
$886$	−26.9427	−0.905159
$887$	−37.8754	−1.27173	−0.635865	−	0.771800i	$-0.719357\pi$
$887$	−37.8754	−1.27173	−0.635865	+	0.771800i	$0.719357\pi$
$888$	0	0
$889$	−4.52968	−0.151921
$890$	−26.7590	−0.896964
$891$	0	0
$892$	−24.9782	−0.836334
$893$	−31.4724	−1.05318
$894$	0	0
$895$	92.6152	3.09578
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	25.4320	0.848675
$899$	−0.162238	−0.00541095
$900$	0	0
$901$	14.5865	0.485946
$902$	11.5323	0.383982
$903$	0	0
$904$	−20.9023	−0.695199
$905$	−39.9453	−1.32783
$906$	0	0
$907$	7.14997	0.237411	0.118705	−	0.992930i	$-0.462126\pi$
$907$	7.14997	0.237411	0.118705	+	0.992930i	$0.462126\pi$
$908$	18.8713	0.626266
$909$	0	0
$910$	−16.0619	−0.532448
$911$	−17.3153	−0.573682	−0.286841	−	0.957978i	$-0.592605\pi$
$911$	−17.3153	−0.573682	−0.286841	+	0.957978i	$0.592605\pi$
$912$	0	0
$913$	68.4366	2.26492
$914$	9.89970	0.327453
$915$	0	0
$916$	3.29838	0.108982
$917$	7.53226	0.248737
$918$	0	0
$919$	50.5180	1.66643	0.833217	−	0.552946i	$-0.186497\pi$
$919$	50.5180	1.66643	0.833217	+	0.552946i	$0.186497\pi$
$920$	−3.76613	−0.124166
$921$	0	0
$922$	17.1620	0.565199
$923$	−22.1187	−0.728047
$924$	0	0
$925$	5.32502	0.175085
$926$	−0.367444	−0.0120750
$927$	0	0
$928$	2.00000	0.0656532
$929$	7.38384	0.242256	0.121128	−	0.992637i	$-0.461349\pi$
$929$	7.38384	0.242256	0.121128	+	0.992637i	$0.461349\pi$
$930$	0	0
$931$	−8.03097	−0.263204
$932$	−15.2482	−0.499473
$933$	0	0
$934$	31.7255	1.03809
$935$	−92.6152	−3.02884
$936$	0	0
$937$	15.1052	0.493465	0.246732	−	0.969084i	$-0.420643\pi$
$937$	15.1052	0.493465	0.246732	+	0.969084i	$0.420643\pi$
$938$	13.8281	0.451502
$939$	0	0
$940$	−14.7590	−0.481386
$941$	43.3220	1.41226	0.706128	−	0.708084i	$-0.250440\pi$
$941$	43.3220	1.41226	0.706128	+	0.708084i	$0.250440\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	11.5323	0.375343
$945$	0	0
$946$	38.9995	1.26798
$947$	4.20269	0.136569	0.0682846	−	0.997666i	$-0.478247\pi$
$947$	4.20269	0.136569	0.0682846	+	0.997666i	$0.478247\pi$
$948$	0	0
$949$	1.47547	0.0478957
$950$	73.7542	2.39290
$951$	0	0
$952$	4.26484	0.138224
$953$	−14.6729	−0.475303	−0.237652	−	0.971350i	$-0.576378\pi$
$953$	−14.6729	−0.475303	−0.237652	+	0.971350i	$0.576378\pi$
$954$	0	0
$955$	3.06451	0.0991653
$956$	4.16224	0.134616
$957$	0	0
$958$	3.47032	0.112121
$959$	5.53226	0.178646
$960$	0	0
$961$	−30.9934	−0.999788
$962$	2.47289	0.0797292
$963$	0	0
$964$	−18.6374	−0.600271
$965$	−21.9048	−0.705142
$966$	0	0
$967$	56.5369	1.81810	0.909052	−	0.416682i	$-0.136807\pi$
$967$	56.5369	1.81810	0.909052	+	0.416682i	$0.136807\pi$
$968$	−22.2482	−0.715085
$969$	0	0
$970$	1.60844	0.0516438
$971$	14.2500	0.457304	0.228652	−	0.973508i	$-0.426568\pi$
$971$	14.2500	0.457304	0.228652	+	0.973508i	$0.426568\pi$
$972$	0	0
$973$	20.0619	0.643156
$974$	30.4965	0.977170
$975$	0	0
$976$	−11.2984	−0.361652
$977$	17.6561	0.564870	0.282435	−	0.959286i	$-0.408858\pi$
$977$	17.6561	0.564870	0.282435	+	0.959286i	$0.408858\pi$
$978$	0	0
$979$	40.9694	1.30939
$980$	−3.76613	−0.120305
$981$	0	0
$982$	−28.9376	−0.923435
$983$	−5.40323	−0.172336	−0.0861681	−	0.996281i	$-0.527462\pi$
$983$	−5.40323	−0.172336	−0.0861681	+	0.996281i	$0.527462\pi$
$984$	0	0
$985$	−57.3265	−1.82657
$986$	−8.52968	−0.271640
$987$	0	0
$988$	34.2508	1.08966
$989$	−6.76355	−0.215069
$990$	0	0
$991$	24.9427	0.792332	0.396166	−	0.918179i	$-0.370340\pi$
$991$	24.9427	0.792332	0.396166	+	0.918179i	$0.370340\pi$
$992$	−0.0811192	−0.00257554
$993$	0	0
$994$	−5.18630	−0.164499
$995$	9.54650	0.302644
$996$	0	0
$997$	28.8948	0.915108	0.457554	−	0.889182i	$-0.348726\pi$
$997$	28.8948	0.915108	0.457554	+	0.889182i	$0.348726\pi$
$998$	28.2457	0.894101
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bh.1.1	✓	4
3.2	odd	2		2898.2.a.bi.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.bh.1.1	✓	4	1.1	even	1	trivial
2898.2.a.bi.1.4	yes	4	3.2	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.76613 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.76613 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.76613 q^{10} -5.76613 q^{11} -4.26484 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.26484 q^{17} -8.03097 q^{19} -3.76613 q^{20} +5.76613 q^{22} -1.00000 q^{23} +9.18372 q^{25} +4.26484 q^{26} +1.00000 q^{28} -2.00000 q^{29} +0.0811192 q^{31} -1.00000 q^{32} +4.26484 q^{34} -3.76613 q^{35} +0.579832 q^{37} +8.03097 q^{38} +3.76613 q^{40} +2.00000 q^{41} +6.76355 q^{43} -5.76613 q^{44} +1.00000 q^{46} +3.91888 q^{47} +1.00000 q^{49} -9.18372 q^{50} -4.26484 q^{52} -3.42017 q^{53} +21.7160 q^{55} -1.00000 q^{56} +2.00000 q^{58} +11.5323 q^{59} -11.2984 q^{61} -0.0811192 q^{62} +1.00000 q^{64} +16.0619 q^{65} -13.8281 q^{67} -4.26484 q^{68} +3.76613 q^{70} +5.18630 q^{71} -0.345960 q^{73} -0.579832 q^{74} -8.03097 q^{76} -5.76613 q^{77} +3.65404 q^{79} -3.76613 q^{80} -2.00000 q^{82} -11.8687 q^{83} +16.0619 q^{85} -6.76355 q^{86} +5.76613 q^{88} -7.10518 q^{89} -4.26484 q^{91} -1.00000 q^{92} -3.91888 q^{94} +30.2457 q^{95} +0.427080 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 8 q^{11} - 4 q^{14} + 4 q^{16} + 8 q^{22} - 4 q^{23} + 16 q^{25} + 4 q^{28} - 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{37} + 8 q^{41} + 8 q^{43} - 8 q^{44} + 4 q^{46} + 12 q^{47} + 4 q^{49} - 16 q^{50} - 12 q^{53} + 36 q^{55} - 4 q^{56} + 8 q^{58} + 16 q^{59} - 4 q^{62} + 4 q^{64} + 24 q^{67} + 4 q^{71} + 12 q^{73} - 4 q^{74} - 8 q^{77} + 28 q^{79} - 8 q^{82} - 8 q^{83} - 8 q^{86} + 8 q^{88} - 8 q^{89} - 4 q^{92} - 12 q^{94} + 36 q^{95} - 8 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 8 * q^11 - 4 * q^14 + 4 * q^16 + 8 * q^22 - 4 * q^23 + 16 * q^25 + 4 * q^28 - 8 * q^29 + 4 * q^31 - 4 * q^32 + 4 * q^37 + 8 * q^41 + 8 * q^43 - 8 * q^44 + 4 * q^46 + 12 * q^47 + 4 * q^49 - 16 * q^50 - 12 * q^53 + 36 * q^55 - 4 * q^56 + 8 * q^58 + 16 * q^59 - 4 * q^62 + 4 * q^64 + 24 * q^67 + 4 * q^71 + 12 * q^73 - 4 * q^74 - 8 * q^77 + 28 * q^79 - 8 * q^82 - 8 * q^83 - 8 * q^86 + 8 * q^88 - 8 * q^89 - 4 * q^92 - 12 * q^94 + 36 * q^95 - 8 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more