LMFDB - Embedded newform 2898.2.a.bi.1.3

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			1.3
Root			$-1.51727$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.51727 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.51727 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.51727 q^{10} +3.51727 q^{11} +6.28616 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.28616 q^{17} +4.76889 q^{19} +1.51727 q^{20} +3.51727 q^{22} +1.00000 q^{23} -2.69790 q^{25} +6.28616 q^{26} +1.00000 q^{28} +2.00000 q^{29} +1.41174 q^{31} +1.00000 q^{32} -6.28616 q^{34} +1.51727 q^{35} -6.39169 q^{37} +4.76889 q^{38} +1.51727 q^{40} -2.00000 q^{41} -12.0896 q^{43} +3.51727 q^{44} +1.00000 q^{46} -2.58826 q^{47} +1.00000 q^{49} -2.69790 q^{50} +6.28616 q^{52} +10.3917 q^{53} +5.33663 q^{55} +1.00000 q^{56} +2.00000 q^{58} -7.03453 q^{59} -4.55180 q^{61} +1.41174 q^{62} +1.00000 q^{64} +9.53779 q^{65} +14.0205 q^{67} -6.28616 q^{68} +1.51727 q^{70} -9.90896 q^{71} +8.87442 q^{73} -6.39169 q^{74} +4.76889 q^{76} +3.51727 q^{77} +12.8744 q^{79} +1.51727 q^{80} -2.00000 q^{82} -3.59237 q^{83} -9.53779 q^{85} -12.0896 q^{86} +3.51727 q^{88} +10.4972 q^{89} +6.28616 q^{91} +1.00000 q^{92} -2.58826 q^{94} +7.23569 q^{95} -7.46268 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.51727	0.678543	0.339271	−	0.940689i	$-0.389820\pi$
$5$	1.51727	0.678543	0.339271	+	0.940689i	$0.389820\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.51727	0.479802
$11$	3.51727	1.06050	0.530248	−	0.847843i	$-0.322099\pi$
$11$	3.51727	1.06050	0.530248	+	0.847843i	$0.322099\pi$
$12$	0	0
$13$	6.28616	1.74347	0.871734	−	0.489980i	$-0.162996\pi$
$13$	6.28616	1.74347	0.871734	+	0.489980i	$0.162996\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.28616	−1.52462	−0.762309	−	0.647213i	$-0.775934\pi$
$17$	−6.28616	−1.52462	−0.762309	+	0.647213i	$0.775934\pi$
$18$	0	0
$19$	4.76889	1.09406	0.547030	−	0.837113i	$-0.315759\pi$
$19$	4.76889	1.09406	0.547030	+	0.837113i	$0.315759\pi$
$20$	1.51727	0.339271
$21$	0	0
$22$	3.51727	0.749884
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.69790	−0.539580
$26$	6.28616	1.23282
$27$	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
$31$	1.41174	0.253556	0.126778	−	0.991931i	$-0.459536\pi$
$31$	1.41174	0.253556	0.126778	+	0.991931i	$0.459536\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.28616	−1.07807
$35$	1.51727	0.256465
$36$	0	0
$37$	−6.39169	−1.05079	−0.525394	−	0.850859i	$-0.676082\pi$
$37$	−6.39169	−1.05079	−0.525394	+	0.850859i	$0.676082\pi$
$38$	4.76889	0.773617
$39$	0	0
$40$	1.51727	0.239901
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−12.0896	−1.84364	−0.921822	−	0.387612i	$-0.873300\pi$
$43$	−12.0896	−1.84364	−0.921822	+	0.387612i	$0.873300\pi$
$44$	3.51727	0.530248
$45$	0	0
$46$	1.00000	0.147442
$47$	−2.58826	−0.377537	−0.188768	−	0.982022i	$-0.560450\pi$
$47$	−2.58826	−0.377537	−0.188768	+	0.982022i	$0.560450\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.69790	−0.381541
$51$	0	0
$52$	6.28616	0.871734
$53$	10.3917	1.42741	0.713704	−	0.700447i	$-0.247016\pi$
$53$	10.3917	1.42741	0.713704	+	0.700447i	$0.247016\pi$
$54$	0	0
$55$	5.33663	0.719592
$56$	1.00000	0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−7.03453	−0.915818	−0.457909	−	0.888999i	$-0.651402\pi$
$59$	−7.03453	−0.915818	−0.457909	+	0.888999i	$0.651402\pi$
$60$	0	0
$61$	−4.55180	−0.582799	−0.291399	−	0.956602i	$-0.594121\pi$
$61$	−4.55180	−0.582799	−0.291399	+	0.956602i	$0.594121\pi$
$62$	1.41174	0.179291
$63$	0	0
$64$	1.00000	0.125000
$65$	9.53779	1.18302
$66$	0	0
$67$	14.0205	1.71288	0.856439	−	0.516247i	$-0.172672\pi$
$67$	14.0205	1.71288	0.856439	+	0.516247i	$0.172672\pi$
$68$	−6.28616	−0.762309
$69$	0	0
$70$	1.51727	0.181348
$71$	−9.90896	−1.17598	−0.587988	−	0.808869i	$-0.700080\pi$
$71$	−9.90896	−1.17598	−0.587988	+	0.808869i	$0.700080\pi$
$72$	0	0
$73$	8.87442	1.03867	0.519336	−	0.854570i	$-0.326179\pi$
$73$	8.87442	1.03867	0.519336	+	0.854570i	$0.326179\pi$
$74$	−6.39169	−0.743019
$75$	0	0
$76$	4.76889	0.547030
$77$	3.51727	0.400830
$78$	0	0
$79$	12.8744	1.44849	0.724243	−	0.689545i	$-0.242189\pi$
$79$	12.8744	1.44849	0.724243	+	0.689545i	$0.242189\pi$
$80$	1.51727	0.169636
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−3.59237	−0.394314	−0.197157	−	0.980372i	$-0.563171\pi$
$83$	−3.59237	−0.394314	−0.197157	+	0.980372i	$0.563171\pi$
$84$	0	0
$85$	−9.53779	−1.03452
$86$	−12.0896	−1.30365
$87$	0	0
$88$	3.51727	0.374942
$89$	10.4972	1.11270	0.556351	−	0.830947i	$-0.312201\pi$
$89$	10.4972	1.11270	0.556351	+	0.830947i	$0.312201\pi$
$90$	0	0
$91$	6.28616	0.658969
$92$	1.00000	0.104257
$93$	0	0
$94$	−2.58826	−0.266959
$95$	7.23569	0.742366
$96$	0	0
$97$	−7.46268	−0.757720	−0.378860	−	0.925454i	$-0.623684\pi$
$97$	−7.46268	−0.757720	−0.378860	+	0.925454i	$0.623684\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−2.69790	−0.269790
$101$	−17.8930	−1.78042	−0.890211	−	0.455549i	$-0.849443\pi$
$101$	−17.8930	−1.78042	−0.890211	+	0.455549i	$0.849443\pi$
$102$	0	0
$103$	5.53779	0.545654	0.272827	−	0.962063i	$-0.412041\pi$
$103$	5.53779	0.545654	0.272827	+	0.962063i	$0.412041\pi$
$104$	6.28616	0.616409
$105$	0	0
$106$	10.3917	1.00933
$107$	3.51727	0.340027	0.170014	−	0.985442i	$-0.445619\pi$
$107$	3.51727	0.340027	0.170014	+	0.985442i	$0.445619\pi$
$108$	0	0
$109$	1.30621	0.125112	0.0625562	−	0.998041i	$-0.480075\pi$
$109$	1.30621	0.125112	0.0625562	+	0.998041i	$0.480075\pi$
$110$	5.33663	0.508828
$111$	0	0
$112$	1.00000	0.0944911
$113$	−9.24559	−0.869752	−0.434876	−	0.900490i	$-0.643208\pi$
$113$	−9.24559	−0.869752	−0.434876	+	0.900490i	$0.643208\pi$
$114$	0	0
$115$	1.51727	0.141486
$116$	2.00000	0.185695
$117$	0	0
$118$	−7.03453	−0.647581
$119$	−6.28616	−0.576251
$120$	0	0
$121$	1.37117	0.124652
$122$	−4.55180	−0.412101
$123$	0	0
$124$	1.41174	0.126778
$125$	−11.6798	−1.04467
$126$	0	0
$127$	16.5723	1.47056	0.735278	−	0.677766i	$-0.237052\pi$
$127$	16.5723	1.47056	0.735278	+	0.677766i	$0.237052\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	9.53779	0.836519
$131$	−3.03453	−0.265129	−0.132564	−	0.991174i	$-0.542321\pi$
$131$	−3.03453	−0.265129	−0.132564	+	0.991174i	$0.542321\pi$
$132$	0	0
$133$	4.76889	0.413515
$134$	14.0205	1.21119
$135$	0	0
$136$	−6.28616	−0.539034
$137$	−1.03453	−0.0883863	−0.0441931	−	0.999023i	$-0.514072\pi$
$137$	−1.03453	−0.0883863	−0.0441931	+	0.999023i	$0.514072\pi$
$138$	0	0
$139$	−5.53779	−0.469709	−0.234854	−	0.972031i	$-0.575461\pi$
$139$	−5.53779	−0.469709	−0.234854	+	0.972031i	$0.575461\pi$
$140$	1.51727	0.128232
$141$	0	0
$142$	−9.90896	−0.831541
$143$	22.1101	1.84894
$144$	0	0
$145$	3.03453	0.252004
$146$	8.87442	0.734452
$147$	0	0
$148$	−6.39169	−0.525394
$149$	14.0896	1.15426	0.577132	−	0.816651i	$-0.304172\pi$
$149$	14.0896	1.15426	0.577132	+	0.816651i	$0.304172\pi$
$150$	0	0
$151$	4.16011	0.338545	0.169273	−	0.985569i	$-0.445858\pi$
$151$	4.16011	0.338545	0.169273	+	0.985569i	$0.445858\pi$
$152$	4.76889	0.386808
$153$	0	0
$154$	3.51727	0.283429
$155$	2.14199	0.172048
$156$	0	0
$157$	9.72832	0.776405	0.388202	−	0.921574i	$-0.373096\pi$
$157$	9.72832	0.776405	0.388202	+	0.921574i	$0.373096\pi$
$158$	12.8744	1.02423
$159$	0	0
$160$	1.51727	0.119951
$161$	1.00000	0.0788110
$162$	0	0
$163$	−11.9090	−0.932781	−0.466391	−	0.884579i	$-0.654446\pi$
$163$	−11.9090	−0.932781	−0.466391	+	0.884579i	$0.654446\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−3.59237	−0.278822
$167$	−15.9150	−1.23154	−0.615769	−	0.787926i	$-0.711155\pi$
$167$	−15.9150	−1.23154	−0.615769	+	0.787926i	$0.711155\pi$
$168$	0	0
$169$	26.5158	2.03968
$170$	−9.53779	−0.731515
$171$	0	0
$172$	−12.0896	−0.921822
$173$	5.10964	0.388479	0.194239	−	0.980954i	$-0.437776\pi$
$173$	5.10964	0.388479	0.194239	+	0.980954i	$0.437776\pi$
$174$	0	0
$175$	−2.69790	−0.203942
$176$	3.51727	0.265124
$177$	0	0
$178$	10.4972	0.786800
$179$	−22.1101	−1.65259	−0.826293	−	0.563240i	$-0.809555\pi$
$179$	−22.1101	−1.65259	−0.826293	+	0.563240i	$0.809555\pi$
$180$	0	0
$181$	22.3006	1.65759	0.828797	−	0.559550i	$-0.189026\pi$
$181$	22.3006	1.65759	0.828797	+	0.559550i	$0.189026\pi$
$182$	6.28616	0.465961
$183$	0	0
$184$	1.00000	0.0737210
$185$	−9.69790	−0.713004
$186$	0	0
$187$	−22.1101	−1.61685
$188$	−2.58826	−0.188768
$189$	0	0
$190$	7.23569	0.524932
$191$	−3.90896	−0.282842	−0.141421	−	0.989950i	$-0.545167\pi$
$191$	−3.90896	−0.282842	−0.141421	+	0.989950i	$0.545167\pi$
$192$	0	0
$193$	17.6979	1.27392	0.636961	−	0.770896i	$-0.280191\pi$
$193$	17.6979	1.27392	0.636961	+	0.770896i	$0.280191\pi$
$194$	−7.46268	−0.535789
$195$	0	0
$196$	1.00000	0.0714286
$197$	24.3212	1.73281	0.866406	−	0.499341i	$-0.166425\pi$
$197$	24.3212	1.73281	0.866406	+	0.499341i	$0.166425\pi$
$198$	0	0
$199$	−14.6414	−1.03790	−0.518950	−	0.854804i	$-0.673677\pi$
$199$	−14.6414	−1.03790	−0.518950	+	0.854804i	$0.673677\pi$
$200$	−2.69790	−0.190770
$201$	0	0
$202$	−17.8930	−1.25895
$203$	2.00000	0.140372
$204$	0	0
$205$	−3.03453	−0.211941
$206$	5.53779	0.385836
$207$	0	0
$208$	6.28616	0.435867
$209$	16.7735	1.16025
$210$	0	0
$211$	−14.9435	−1.02875	−0.514376	−	0.857565i	$-0.671976\pi$
$211$	−14.9435	−1.02875	−0.514376	+	0.857565i	$0.671976\pi$
$212$	10.3917	0.713704
$213$	0	0
$214$	3.51727	0.240435
$215$	−18.3431	−1.25099
$216$	0	0
$217$	1.41174	0.0958351
$218$	1.30621	0.0884678
$219$	0	0
$220$	5.33663	0.359796
$221$	−39.5158	−2.65812
$222$	0	0
$223$	18.5564	1.24263	0.621314	−	0.783562i	$-0.286599\pi$
$223$	18.5564	1.24263	0.621314	+	0.783562i	$0.286599\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.24559	−0.615008
$227$	−20.0145	−1.32841	−0.664204	−	0.747551i	$-0.731229\pi$
$227$	−20.0145	−1.32841	−0.664204	+	0.747551i	$0.731229\pi$
$228$	0	0
$229$	−3.44820	−0.227863	−0.113932	−	0.993489i	$-0.536344\pi$
$229$	−3.44820	−0.227863	−0.113932	+	0.993489i	$0.536344\pi$
$230$	1.51727	0.100046
$231$	0	0
$232$	2.00000	0.131306
$233$	−5.62883	−0.368757	−0.184378	−	0.982855i	$-0.559027\pi$
$233$	−5.62883	−0.368757	−0.184378	+	0.982855i	$0.559027\pi$
$234$	0	0
$235$	−3.92708	−0.256175
$236$	−7.03453	−0.457909
$237$	0	0
$238$	−6.28616	−0.407471
$239$	−6.82348	−0.441374	−0.220687	−	0.975345i	$-0.570830\pi$
$239$	−6.82348	−0.441374	−0.220687	+	0.975345i	$0.570830\pi$
$240$	0	0
$241$	−17.5318	−1.12932	−0.564660	−	0.825324i	$-0.690993\pi$
$241$	−17.5318	−1.12932	−0.564660	+	0.825324i	$0.690993\pi$
$242$	1.37117	0.0881421
$243$	0	0
$244$	−4.55180	−0.291399
$245$	1.51727	0.0969347
$246$	0	0
$247$	29.9780	1.90746
$248$	1.41174	0.0896455
$249$	0	0
$250$	−11.6798	−0.738694
$251$	−28.1647	−1.77774	−0.888870	−	0.458160i	$-0.848509\pi$
$251$	−28.1647	−1.77774	−0.888870	+	0.458160i	$0.848509\pi$
$252$	0	0
$253$	3.51727	0.221129
$254$	16.5723	1.03984
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.24559	0.576724	0.288362	−	0.957521i	$-0.406889\pi$
$257$	9.24559	0.576724	0.288362	+	0.957521i	$0.406889\pi$
$258$	0	0
$259$	−6.39169	−0.397160
$260$	9.53779	0.591508
$261$	0	0
$262$	−3.03453	−0.187474
$263$	5.74884	0.354489	0.177244	−	0.984167i	$-0.443282\pi$
$263$	5.74884	0.354489	0.177244	+	0.984167i	$0.443282\pi$
$264$	0	0
$265$	15.7670	0.968557
$266$	4.76889	0.292400
$267$	0	0
$268$	14.0205	0.856439
$269$	1.71384	0.104495	0.0522473	−	0.998634i	$-0.483362\pi$
$269$	1.71384	0.104495	0.0522473	+	0.998634i	$0.483362\pi$
$270$	0	0
$271$	−14.7484	−0.895900	−0.447950	−	0.894059i	$-0.647846\pi$
$271$	−14.7484	−0.895900	−0.447950	+	0.894059i	$0.647846\pi$
$272$	−6.28616	−0.381154
$273$	0	0
$274$	−1.03453	−0.0624985
$275$	−9.48923	−0.572222
$276$	0	0
$277$	1.63874	0.0984621	0.0492310	−	0.998787i	$-0.484323\pi$
$277$	1.63874	0.0984621	0.0492310	+	0.998787i	$0.484323\pi$
$278$	−5.53779	−0.332134
$279$	0	0
$280$	1.51727	0.0906741
$281$	−29.2137	−1.74274	−0.871372	−	0.490623i	$-0.836769\pi$
$281$	−29.2137	−1.74274	−0.871372	+	0.490623i	$0.836769\pi$
$282$	0	0
$283$	−2.83796	−0.168699	−0.0843497	−	0.996436i	$-0.526881\pi$
$283$	−2.83796	−0.168699	−0.0843497	+	0.996436i	$0.526881\pi$
$284$	−9.90896	−0.587988
$285$	0	0
$286$	22.1101	1.30740
$287$	−2.00000	−0.118056
$288$	0	0
$289$	22.5158	1.32446
$290$	3.03453	0.178194
$291$	0	0
$292$	8.87442	0.519336
$293$	−2.12147	−0.123937	−0.0619687	−	0.998078i	$-0.519738\pi$
$293$	−2.12147	−0.123937	−0.0619687	+	0.998078i	$0.519738\pi$
$294$	0	0
$295$	−10.6733	−0.621422
$296$	−6.39169	−0.371509
$297$	0	0
$298$	14.0896	0.816188
$299$	6.28616	0.363538
$300$	0	0
$301$	−12.0896	−0.696832
$302$	4.16011	0.239388
$303$	0	0
$304$	4.76889	0.273515
$305$	−6.90630	−0.395454
$306$	0	0
$307$	−2.50325	−0.142868	−0.0714340	−	0.997445i	$-0.522758\pi$
$307$	−2.50325	−0.142868	−0.0714340	+	0.997445i	$0.522758\pi$
$308$	3.51727	0.200415
$309$	0	0
$310$	2.14199	0.121657
$311$	25.6419	1.45402	0.727008	−	0.686629i	$-0.240910\pi$
$311$	25.6419	1.45402	0.727008	+	0.686629i	$0.240910\pi$
$312$	0	0
$313$	−16.3552	−0.924452	−0.462226	−	0.886762i	$-0.652949\pi$
$313$	−16.3552	−0.924452	−0.462226	+	0.886762i	$0.652949\pi$
$314$	9.72832	0.549001
$315$	0	0
$316$	12.8744	0.724243
$317$	5.42768	0.304849	0.152424	−	0.988315i	$-0.451292\pi$
$317$	5.42768	0.304849	0.152424	+	0.988315i	$0.451292\pi$
$318$	0	0
$319$	7.03453	0.393858
$320$	1.51727	0.0848178
$321$	0	0
$322$	1.00000	0.0557278
$323$	−29.9780	−1.66802
$324$	0	0
$325$	−16.9594	−0.940740
$326$	−11.9090	−0.659576
$327$	0	0
$328$	−2.00000	−0.110432
$329$	−2.58826	−0.142695
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−3.59237	−0.197157
$333$	0	0
$334$	−15.9150	−0.870830
$335$	21.2729	1.16226
$336$	0	0
$337$	−2.57232	−0.140123	−0.0700616	−	0.997543i	$-0.522320\pi$
$337$	−2.57232	−0.140123	−0.0700616	+	0.997543i	$0.522320\pi$
$338$	26.5158	1.44227
$339$	0	0
$340$	−9.53779	−0.517259
$341$	4.96547	0.268895
$342$	0	0
$343$	1.00000	0.0539949
$344$	−12.0896	−0.651827
$345$	0	0
$346$	5.10964	0.274696
$347$	−6.16011	−0.330692	−0.165346	−	0.986236i	$-0.552874\pi$
$347$	−6.16011	−0.330692	−0.165346	+	0.986236i	$0.552874\pi$
$348$	0	0
$349$	−8.14417	−0.435948	−0.217974	−	0.975955i	$-0.569945\pi$
$349$	−8.14417	−0.435948	−0.217974	+	0.975955i	$0.569945\pi$
$350$	−2.69790	−0.144209
$351$	0	0
$352$	3.51727	0.187471
$353$	10.3712	0.552002	0.276001	−	0.961157i	$-0.410991\pi$
$353$	10.3712	0.552002	0.276001	+	0.961157i	$0.410991\pi$
$354$	0	0
$355$	−15.0345	−0.797950
$356$	10.4972	0.556351
$357$	0	0
$358$	−22.1101	−1.16856
$359$	−4.51316	−0.238195	−0.119098	−	0.992883i	$-0.538000\pi$
$359$	−4.51316	−0.238195	−0.119098	+	0.992883i	$0.538000\pi$
$360$	0	0
$361$	3.74234	0.196965
$362$	22.3006	1.17210
$363$	0	0
$364$	6.28616	0.329484
$365$	13.4649	0.704783
$366$	0	0
$367$	11.2456	0.587015	0.293508	−	0.955957i	$-0.405177\pi$
$367$	11.2456	0.587015	0.293508	+	0.955957i	$0.405177\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−9.69790	−0.504170
$371$	10.3917	0.539510
$372$	0	0
$373$	21.7875	1.12811	0.564057	−	0.825736i	$-0.309240\pi$
$373$	21.7875	1.12811	0.564057	+	0.825736i	$0.309240\pi$
$374$	−22.1101	−1.14329
$375$	0	0
$376$	−2.58826	−0.133479
$377$	12.5723	0.647507
$378$	0	0
$379$	−0.802961	−0.0412453	−0.0206227	−	0.999787i	$-0.506565\pi$
$379$	−0.802961	−0.0412453	−0.0206227	+	0.999787i	$0.506565\pi$
$380$	7.23569	0.371183
$381$	0	0
$382$	−3.90896	−0.200000
$383$	−2.25116	−0.115029	−0.0575144	−	0.998345i	$-0.518318\pi$
$383$	−2.25116	−0.115029	−0.0575144	+	0.998345i	$0.518318\pi$
$384$	0	0
$385$	5.33663	0.271980
$386$	17.6979	0.900799
$387$	0	0
$388$	−7.46268	−0.378860
$389$	−26.1488	−1.32579	−0.662897	−	0.748710i	$-0.730673\pi$
$389$	−26.1488	−1.32579	−0.662897	+	0.748710i	$0.730673\pi$
$390$	0	0
$391$	−6.28616	−0.317905
$392$	1.00000	0.0505076
$393$	0	0
$394$	24.3212	1.22528
$395$	19.5339	0.982859
$396$	0	0
$397$	−30.2862	−1.52002	−0.760009	−	0.649912i	$-0.774806\pi$
$397$	−30.2862	−1.52002	−0.760009	+	0.649912i	$0.774806\pi$
$398$	−14.6414	−0.733907
$399$	0	0
$400$	−2.69790	−0.134895
$401$	32.8524	1.64057	0.820286	−	0.571953i	$-0.193814\pi$
$401$	32.8524	1.64057	0.820286	+	0.571953i	$0.193814\pi$
$402$	0	0
$403$	8.87442	0.442066
$404$	−17.8930	−0.890211
$405$	0	0
$406$	2.00000	0.0992583
$407$	−22.4813	−1.11436
$408$	0	0
$409$	13.3958	0.662380	0.331190	−	0.943564i	$-0.392550\pi$
$409$	13.3958	0.662380	0.331190	+	0.943564i	$0.392550\pi$
$410$	−3.03453	−0.149865
$411$	0	0
$412$	5.53779	0.272827
$413$	−7.03453	−0.346147
$414$	0	0
$415$	−5.45059	−0.267559
$416$	6.28616	0.308204
$417$	0	0
$418$	16.7735	0.820417
$419$	−38.5177	−1.88171	−0.940857	−	0.338803i	$-0.889978\pi$
$419$	−38.5177	−1.88171	−0.940857	+	0.338803i	$0.889978\pi$
$420$	0	0
$421$	35.2342	1.71721	0.858606	−	0.512637i	$-0.171331\pi$
$421$	35.2342	1.71721	0.858606	+	0.512637i	$0.171331\pi$
$422$	−14.9435	−0.727438
$423$	0	0
$424$	10.3917	0.504665
$425$	16.9594	0.822653
$426$	0	0
$427$	−4.55180	−0.220277
$428$	3.51727	0.170014
$429$	0	0
$430$	−18.3431	−0.884585
$431$	10.4303	0.502412	0.251206	−	0.967934i	$-0.419173\pi$
$431$	10.4303	0.502412	0.251206	+	0.967934i	$0.419173\pi$
$432$	0	0
$433$	−0.217091	−0.0104327	−0.00521636	−	0.999986i	$-0.501660\pi$
$433$	−0.217091	−0.0104327	−0.00521636	+	0.999986i	$0.501660\pi$
$434$	1.41174	0.0677657
$435$	0	0
$436$	1.30621	0.0625562
$437$	4.76889	0.228127
$438$	0	0
$439$	−25.2516	−1.20519	−0.602597	−	0.798046i	$-0.705867\pi$
$439$	−25.2516	−1.20519	−0.602597	+	0.798046i	$0.705867\pi$
$440$	5.33663	0.254414
$441$	0	0
$442$	−39.5158	−1.87958
$443$	−4.22918	−0.200935	−0.100467	−	0.994940i	$-0.532034\pi$
$443$	−4.22918	−0.200935	−0.100467	+	0.994940i	$0.532034\pi$
$444$	0	0
$445$	15.9271	0.755016
$446$	18.5564	0.878670
$447$	0	0
$448$	1.00000	0.0472456
$449$	−7.32673	−0.345770	−0.172885	−	0.984942i	$-0.555309\pi$
$449$	−7.32673	−0.345770	−0.172885	+	0.984942i	$0.555309\pi$
$450$	0	0
$451$	−7.03453	−0.331243
$452$	−9.24559	−0.434876
$453$	0	0
$454$	−20.0145	−0.939326
$455$	9.53779	0.447138
$456$	0	0
$457$	18.3613	0.858904	0.429452	−	0.903090i	$-0.358707\pi$
$457$	18.3613	0.858904	0.429452	+	0.903090i	$0.358707\pi$
$458$	−3.44820	−0.161124
$459$	0	0
$460$	1.51727	0.0707430
$461$	−38.2543	−1.78168	−0.890840	−	0.454318i	$-0.849883\pi$
$461$	−38.2543	−1.78168	−0.890840	+	0.454318i	$0.849883\pi$
$462$	0	0
$463$	−23.3958	−1.08730	−0.543648	−	0.839314i	$-0.682957\pi$
$463$	−23.3958	−1.08730	−0.543648	+	0.839314i	$0.682957\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−5.62883	−0.260751
$467$	17.0891	0.790790	0.395395	−	0.918511i	$-0.370608\pi$
$467$	17.0891	0.790790	0.395395	+	0.918511i	$0.370608\pi$
$468$	0	0
$469$	14.0205	0.647407
$470$	−3.92708	−0.181143
$471$	0	0
$472$	−7.03453	−0.323791
$473$	−42.5223	−1.95518
$474$	0	0
$475$	−12.8660	−0.590332
$476$	−6.28616	−0.288126
$477$	0	0
$478$	−6.82348	−0.312099
$479$	24.5723	1.12274	0.561369	−	0.827566i	$-0.310275\pi$
$479$	24.5723	1.12274	0.561369	+	0.827566i	$0.310275\pi$
$480$	0	0
$481$	−40.1792	−1.83201
$482$	−17.5318	−0.798549
$483$	0	0
$484$	1.37117	0.0623259
$485$	−11.3229	−0.514146
$486$	0	0
$487$	11.2577	0.510133	0.255067	−	0.966923i	$-0.417903\pi$
$487$	11.2577	0.510133	0.255067	+	0.966923i	$0.417903\pi$
$488$	−4.55180	−0.206050
$489$	0	0
$490$	1.51727	0.0685432
$491$	26.9845	1.21779	0.608897	−	0.793249i	$-0.291612\pi$
$491$	26.9845	1.21779	0.608897	+	0.793249i	$0.291612\pi$
$492$	0	0
$493$	−12.5723	−0.566229
$494$	29.9780	1.34878
$495$	0	0
$496$	1.41174	0.0633890
$497$	−9.90896	−0.444477
$498$	0	0
$499$	9.23569	0.413446	0.206723	−	0.978400i	$-0.433720\pi$
$499$	9.23569	0.413446	0.206723	+	0.978400i	$0.433720\pi$
$500$	−11.6798	−0.522335
$501$	0	0
$502$	−28.1647	−1.25705
$503$	−38.4303	−1.71352	−0.856762	−	0.515712i	$-0.827527\pi$
$503$	−38.4303	−1.71352	−0.856762	+	0.515712i	$0.827527\pi$
$504$	0	0
$505$	−27.1485	−1.20809
$506$	3.51727	0.156362
$507$	0	0
$508$	16.5723	0.735278
$509$	−11.6008	−0.514197	−0.257099	−	0.966385i	$-0.582767\pi$
$509$	−11.6008	−0.514197	−0.257099	+	0.966385i	$0.582767\pi$
$510$	0	0
$511$	8.87442	0.392581
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.24559	0.407806
$515$	8.40230	0.370250
$516$	0	0
$517$	−9.10360	−0.400376
$518$	−6.39169	−0.280835
$519$	0	0
$520$	9.53779	0.418260
$521$	−28.1731	−1.23429	−0.617144	−	0.786850i	$-0.711710\pi$
$521$	−28.1731	−1.23429	−0.617144	+	0.786850i	$0.711710\pi$
$522$	0	0
$523$	−5.73436	−0.250746	−0.125373	−	0.992110i	$-0.540013\pi$
$523$	−5.73436	−0.250746	−0.125373	+	0.992110i	$0.540013\pi$
$524$	−3.03453	−0.132564
$525$	0	0
$526$	5.74884	0.250661
$527$	−8.87442	−0.386576
$528$	0	0
$529$	1.00000	0.0434783
$530$	15.7670	0.684873
$531$	0	0
$532$	4.76889	0.206758
$533$	−12.5723	−0.544568
$534$	0	0
$535$	5.33663	0.230723
$536$	14.0205	0.605594
$537$	0	0
$538$	1.71384	0.0738889
$539$	3.51727	0.151499
$540$	0	0
$541$	−22.3613	−0.961386	−0.480693	−	0.876889i	$-0.659615\pi$
$541$	−22.3613	−0.961386	−0.480693	+	0.876889i	$0.659615\pi$
$542$	−14.7484	−0.633497
$543$	0	0
$544$	−6.28616	−0.269517
$545$	1.98187	0.0848941
$546$	0	0
$547$	−40.5404	−1.73338	−0.866692	−	0.498844i	$-0.833758\pi$
$547$	−40.5404	−1.73338	−0.866692	+	0.498844i	$0.833758\pi$
$548$	−1.03453	−0.0441931
$549$	0	0
$550$	−9.48923	−0.404622
$551$	9.53779	0.406323
$552$	0	0
$553$	12.8744	0.547476
$554$	1.63874	0.0696232
$555$	0	0
$556$	−5.53779	−0.234854
$557$	41.9485	1.77742	0.888708	−	0.458473i	$-0.151603\pi$
$557$	41.9485	1.77742	0.888708	+	0.458473i	$0.151603\pi$
$558$	0	0
$559$	−75.9971	−3.21433
$560$	1.51727	0.0641162
$561$	0	0
$562$	−29.2137	−1.23231
$563$	−44.1937	−1.86254	−0.931270	−	0.364329i	$-0.881298\pi$
$563$	−44.1937	−1.86254	−0.931270	+	0.364329i	$0.881298\pi$
$564$	0	0
$565$	−14.0280	−0.590164
$566$	−2.83796	−0.119288
$567$	0	0
$568$	−9.90896	−0.415771
$569$	38.0410	1.59476	0.797382	−	0.603475i	$-0.206218\pi$
$569$	38.0410	1.59476	0.797382	+	0.603475i	$0.206218\pi$
$570$	0	0
$571$	27.1241	1.13511	0.567555	−	0.823336i	$-0.307890\pi$
$571$	27.1241	1.13511	0.567555	+	0.823336i	$0.307890\pi$
$572$	22.1101	0.924470
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	−2.69790	−0.112510
$576$	0	0
$577$	43.7679	1.82208	0.911041	−	0.412316i	$-0.135280\pi$
$577$	43.7679	1.82208	0.911041	+	0.412316i	$0.135280\pi$
$578$	22.5158	0.936534
$579$	0	0
$580$	3.03453	0.126002
$581$	−3.59237	−0.149037
$582$	0	0
$583$	36.5503	1.51376
$584$	8.87442	0.367226
$585$	0	0
$586$	−2.12147	−0.0876370
$587$	−31.4248	−1.29704	−0.648519	−	0.761198i	$-0.724611\pi$
$587$	−31.4248	−1.29704	−0.648519	+	0.761198i	$0.724611\pi$
$588$	0	0
$589$	6.73243	0.277405
$590$	−10.6733	−0.439412
$591$	0	0
$592$	−6.39169	−0.262697
$593$	−5.38373	−0.221083	−0.110542	−	0.993871i	$-0.535259\pi$
$593$	−5.38373	−0.221083	−0.110542	+	0.993871i	$0.535259\pi$
$594$	0	0
$595$	−9.53779	−0.391011
$596$	14.0896	0.577132
$597$	0	0
$598$	6.28616	0.257060
$599$	−13.6470	−0.557600	−0.278800	−	0.960349i	$-0.589937\pi$
$599$	−13.6470	−0.557600	−0.278800	+	0.960349i	$0.589937\pi$
$600$	0	0
$601$	−15.1256	−0.616985	−0.308493	−	0.951227i	$-0.599825\pi$
$601$	−15.1256	−0.616985	−0.308493	+	0.951227i	$0.599825\pi$
$602$	−12.0896	−0.492735
$603$	0	0
$604$	4.16011	0.169273
$605$	2.08043	0.0845815
$606$	0	0
$607$	13.2297	0.536975	0.268487	−	0.963283i	$-0.413476\pi$
$607$	13.2297	0.536975	0.268487	+	0.963283i	$0.413476\pi$
$608$	4.76889	0.193404
$609$	0	0
$610$	−6.90630	−0.279628
$611$	−16.2702	−0.658223
$612$	0	0
$613$	−42.2688	−1.70722	−0.853610	−	0.520913i	$-0.825592\pi$
$613$	−42.2688	−1.70722	−0.853610	+	0.520913i	$0.825592\pi$
$614$	−2.50325	−0.101023
$615$	0	0
$616$	3.51727	0.141715
$617$	6.32023	0.254443	0.127221	−	0.991874i	$-0.459394\pi$
$617$	6.32023	0.254443	0.127221	+	0.991874i	$0.459394\pi$
$618$	0	0
$619$	15.2721	0.613839	0.306920	−	0.951735i	$-0.400702\pi$
$619$	15.2721	0.613839	0.306920	+	0.951735i	$0.400702\pi$
$620$	2.14199	0.0860242
$621$	0	0
$622$	25.6419	1.02814
$623$	10.4972	0.420562
$624$	0	0
$625$	−4.23184	−0.169274
$626$	−16.3552	−0.653686
$627$	0	0
$628$	9.72832	0.388202
$629$	40.1792	1.60205
$630$	0	0
$631$	−45.3738	−1.80630	−0.903152	−	0.429322i	$-0.858753\pi$
$631$	−45.3738	−1.80630	−0.903152	+	0.429322i	$0.858753\pi$
$632$	12.8744	0.512117
$633$	0	0
$634$	5.42768	0.215561
$635$	25.1446	0.997835
$636$	0	0
$637$	6.28616	0.249067
$638$	7.03453	0.278500
$639$	0	0
$640$	1.51727	0.0599753
$641$	18.4994	0.730683	0.365341	−	0.930874i	$-0.380952\pi$
$641$	18.4994	0.730683	0.365341	+	0.930874i	$0.380952\pi$
$642$	0	0
$643$	3.41028	0.134488	0.0672442	−	0.997737i	$-0.478579\pi$
$643$	3.41028	0.134488	0.0672442	+	0.997737i	$0.478579\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	−29.9780	−1.17947
$647$	−16.6474	−0.654478	−0.327239	−	0.944942i	$-0.606118\pi$
$647$	−16.6474	−0.654478	−0.327239	+	0.944942i	$0.606118\pi$
$648$	0	0
$649$	−24.7423	−0.971222
$650$	−16.9594	−0.665204
$651$	0	0
$652$	−11.9090	−0.466391
$653$	−18.4631	−0.722519	−0.361259	−	0.932465i	$-0.617653\pi$
$653$	−18.4631	−0.722519	−0.361259	+	0.932465i	$0.617653\pi$
$654$	0	0
$655$	−4.60420	−0.179901
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−2.58826	−0.100901
$659$	27.8467	1.08475	0.542376	−	0.840136i	$-0.317525\pi$
$659$	27.8467	1.08475	0.542376	+	0.840136i	$0.317525\pi$
$660$	0	0
$661$	−34.4798	−1.34111	−0.670555	−	0.741860i	$-0.733944\pi$
$661$	−34.4798	−1.34111	−0.670555	+	0.741860i	$0.733944\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−3.59237	−0.139411
$665$	7.23569	0.280588
$666$	0	0
$667$	2.00000	0.0774403
$668$	−15.9150	−0.615769
$669$	0	0
$670$	21.2729	0.821843
$671$	−16.0099	−0.618056
$672$	0	0
$673$	−21.0065	−0.809741	−0.404871	−	0.914374i	$-0.632683\pi$
$673$	−21.0065	−0.809741	−0.404871	+	0.914374i	$0.632683\pi$
$674$	−2.57232	−0.0990821
$675$	0	0
$676$	26.5158	1.01984
$677$	4.73389	0.181938	0.0909691	−	0.995854i	$-0.471004\pi$
$677$	4.73389	0.181938	0.0909691	+	0.995854i	$0.471004\pi$
$678$	0	0
$679$	−7.46268	−0.286391
$680$	−9.53779	−0.365757
$681$	0	0
$682$	4.96547	0.190137
$683$	−10.9435	−0.418741	−0.209371	−	0.977836i	$-0.567141\pi$
$683$	−10.9435	−0.418741	−0.209371	+	0.977836i	$0.567141\pi$
$684$	0	0
$685$	−1.56967	−0.0599738
$686$	1.00000	0.0381802
$687$	0	0
$688$	−12.0896	−0.460911
$689$	65.3238	2.48864
$690$	0	0
$691$	28.7515	1.09376	0.546879	−	0.837212i	$-0.315816\pi$
$691$	28.7515	1.09376	0.546879	+	0.837212i	$0.315816\pi$
$692$	5.10964	0.194239
$693$	0	0
$694$	−6.16011	−0.233835
$695$	−8.40230	−0.318717
$696$	0	0
$697$	12.5723	0.476211
$698$	−8.14417	−0.308262
$699$	0	0
$700$	−2.69790	−0.101971
$701$	38.4098	1.45072	0.725359	−	0.688370i	$-0.241674\pi$
$701$	38.4098	1.45072	0.725359	+	0.688370i	$0.241674\pi$
$702$	0	0
$703$	−30.4813	−1.14962
$704$	3.51727	0.132562
$705$	0	0
$706$	10.3712	0.390324
$707$	−17.8930	−0.672936
$708$	0	0
$709$	−41.9667	−1.57609	−0.788046	−	0.615617i	$-0.788907\pi$
$709$	−41.9667	−1.57609	−0.788046	+	0.615617i	$0.788907\pi$
$710$	−15.0345	−0.564236
$711$	0	0
$712$	10.4972	0.393400
$713$	1.41174	0.0528701
$714$	0	0
$715$	33.5469	1.25458
$716$	−22.1101	−0.826293
$717$	0	0
$718$	−4.51316	−0.168429
$719$	−30.1442	−1.12419	−0.562094	−	0.827073i	$-0.690004\pi$
$719$	−30.1442	−1.12419	−0.562094	+	0.827073i	$0.690004\pi$
$720$	0	0
$721$	5.53779	0.206238
$722$	3.74234	0.139275
$723$	0	0
$724$	22.3006	0.828797
$725$	−5.39580	−0.200395
$726$	0	0
$727$	20.7105	0.768108	0.384054	−	0.923311i	$-0.374528\pi$
$727$	20.7105	0.768108	0.384054	+	0.923311i	$0.374528\pi$
$728$	6.28616	0.232981
$729$	0	0
$730$	13.4649	0.498357
$731$	75.9971	2.81085
$732$	0	0
$733$	12.7629	0.471407	0.235703	−	0.971825i	$-0.424261\pi$
$733$	12.7629	0.471407	0.235703	+	0.971825i	$0.424261\pi$
$734$	11.2456	0.415082
$735$	0	0
$736$	1.00000	0.0368605
$737$	49.3139	1.81650
$738$	0	0
$739$	−29.9780	−1.10276	−0.551380	−	0.834254i	$-0.685898\pi$
$739$	−29.9780	−1.10276	−0.551380	+	0.834254i	$0.685898\pi$
$740$	−9.69790	−0.356502
$741$	0	0
$742$	10.3917	0.381491
$743$	30.7614	1.12853	0.564263	−	0.825595i	$-0.309160\pi$
$743$	30.7614	1.12853	0.564263	+	0.825595i	$0.309160\pi$
$744$	0	0
$745$	21.3777	0.783217
$746$	21.7875	0.797697
$747$	0	0
$748$	−22.1101	−0.808426
$749$	3.51727	0.128518
$750$	0	0
$751$	15.0345	0.548618	0.274309	−	0.961642i	$-0.411551\pi$
$751$	15.0345	0.548618	0.274309	+	0.961642i	$0.411551\pi$
$752$	−2.58826	−0.0943841
$753$	0	0
$754$	12.5723	0.457857
$755$	6.31201	0.229717
$756$	0	0
$757$	26.2688	0.954754	0.477377	−	0.878698i	$-0.341588\pi$
$757$	26.2688	0.954754	0.477377	+	0.878698i	$0.341588\pi$
$758$	−0.802961	−0.0291648
$759$	0	0
$760$	7.23569	0.262466
$761$	−49.2328	−1.78469	−0.892343	−	0.451357i	$-0.850940\pi$
$761$	−49.2328	−1.78469	−0.892343	+	0.451357i	$0.850940\pi$
$762$	0	0
$763$	1.30621	0.0472880
$764$	−3.90896	−0.141421
$765$	0	0
$766$	−2.25116	−0.0813377
$767$	−44.2202	−1.59670
$768$	0	0
$769$	−41.8201	−1.50807	−0.754036	−	0.656833i	$-0.771895\pi$
$769$	−41.8201	−1.50807	−0.754036	+	0.656833i	$0.771895\pi$
$770$	5.33663	0.192319
$771$	0	0
$772$	17.6979	0.636961
$773$	32.9420	1.18484	0.592421	−	0.805628i	$-0.298172\pi$
$773$	32.9420	1.18484	0.592421	+	0.805628i	$0.298172\pi$
$774$	0	0
$775$	−3.80873	−0.136814
$776$	−7.46268	−0.267895
$777$	0	0
$778$	−26.1488	−0.937478
$779$	−9.53779	−0.341727
$780$	0	0
$781$	−34.8524	−1.24712
$782$	−6.28616	−0.224793
$783$	0	0
$784$	1.00000	0.0357143
$785$	14.7605	0.526824
$786$	0	0
$787$	−9.47935	−0.337902	−0.168951	−	0.985624i	$-0.554038\pi$
$787$	−9.47935	−0.337902	−0.168951	+	0.985624i	$0.554038\pi$
$788$	24.3212	0.866406
$789$	0	0
$790$	19.5339	0.694986
$791$	−9.24559	−0.328735
$792$	0	0
$793$	−28.6134	−1.01609
$794$	−30.2862	−1.07482
$795$	0	0
$796$	−14.6414	−0.518950
$797$	16.4426	0.582428	0.291214	−	0.956658i	$-0.405941\pi$
$797$	16.4426	0.582428	0.291214	+	0.956658i	$0.405941\pi$
$798$	0	0
$799$	16.2702	0.575599
$800$	−2.69790	−0.0953852
$801$	0	0
$802$	32.8524	1.16006
$803$	31.2137	1.10151
$804$	0	0
$805$	1.51727	0.0534766
$806$	8.87442	0.312588
$807$	0	0
$808$	−17.8930	−0.629474
$809$	−7.98187	−0.280628	−0.140314	−	0.990107i	$-0.544811\pi$
$809$	−7.98187	−0.280628	−0.140314	+	0.990107i	$0.544811\pi$
$810$	0	0
$811$	−24.9655	−0.876656	−0.438328	−	0.898815i	$-0.644429\pi$
$811$	−24.9655	−0.876656	−0.438328	+	0.898815i	$0.644429\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	−22.4813	−0.787968
$815$	−18.0691	−0.632932
$816$	0	0
$817$	−57.6539	−2.01706
$818$	13.3958	0.468373
$819$	0	0
$820$	−3.03453	−0.105971
$821$	40.8645	1.42618	0.713091	−	0.701072i	$-0.247295\pi$
$821$	40.8645	1.42618	0.713091	+	0.701072i	$0.247295\pi$
$822$	0	0
$823$	46.7334	1.62902	0.814511	−	0.580148i	$-0.197005\pi$
$823$	46.7334	1.62902	0.814511	+	0.580148i	$0.197005\pi$
$824$	5.53779	0.192918
$825$	0	0
$826$	−7.03453	−0.244763
$827$	1.58634	0.0551623	0.0275812	−	0.999620i	$-0.491220\pi$
$827$	1.58634	0.0551623	0.0275812	+	0.999620i	$0.491220\pi$
$828$	0	0
$829$	−23.5719	−0.818684	−0.409342	−	0.912381i	$-0.634242\pi$
$829$	−23.5719	−0.818684	−0.409342	+	0.912381i	$0.634242\pi$
$830$	−5.45059	−0.189193
$831$	0	0
$832$	6.28616	0.217933
$833$	−6.28616	−0.217803
$834$	0	0
$835$	−24.1473	−0.835652
$836$	16.7735	0.580123
$837$	0	0
$838$	−38.5177	−1.33057
$839$	−0.138139	−0.00476908	−0.00238454	−	0.999997i	$-0.500759\pi$
$839$	−0.138139	−0.00476908	−0.00238454	+	0.999997i	$0.500759\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	35.2342	1.21425
$843$	0	0
$844$	−14.9435	−0.514376
$845$	40.2316	1.38401
$846$	0	0
$847$	1.37117	0.0471139
$848$	10.3917	0.356852
$849$	0	0
$850$	16.9594	0.581704
$851$	−6.39169	−0.219104
$852$	0	0
$853$	−40.1731	−1.37550	−0.687751	−	0.725947i	$-0.741402\pi$
$853$	−40.1731	−1.37550	−0.687751	+	0.725947i	$0.741402\pi$
$854$	−4.55180	−0.155759
$855$	0	0
$856$	3.51727	0.120218
$857$	−15.3777	−0.525291	−0.262646	−	0.964892i	$-0.584595\pi$
$857$	−15.3777	−0.525291	−0.262646	+	0.964892i	$0.584595\pi$
$858$	0	0
$859$	−15.8300	−0.540112	−0.270056	−	0.962845i	$-0.587042\pi$
$859$	−15.8300	−0.540112	−0.270056	+	0.962845i	$0.587042\pi$
$860$	−18.3431	−0.625496
$861$	0	0
$862$	10.4303	0.355259
$863$	−36.7105	−1.24964	−0.624819	−	0.780769i	$-0.714827\pi$
$863$	−36.7105	−1.24964	−0.624819	+	0.780769i	$0.714827\pi$
$864$	0	0
$865$	7.75269	0.263599
$866$	−0.217091	−0.00737705
$867$	0	0
$868$	1.41174	0.0479176
$869$	45.2828	1.53611
$870$	0	0
$871$	88.1352	2.98635
$872$	1.30621	0.0442339
$873$	0	0
$874$	4.76889	0.161310
$875$	−11.6798	−0.394848
$876$	0	0
$877$	31.9962	1.08043	0.540217	−	0.841526i	$-0.318342\pi$
$877$	31.9962	1.08043	0.540217	+	0.841526i	$0.318342\pi$
$878$	−25.2516	−0.852201
$879$	0	0
$880$	5.33663	0.179898
$881$	12.1761	0.410222	0.205111	−	0.978739i	$-0.434245\pi$
$881$	12.1761	0.410222	0.205111	+	0.978739i	$0.434245\pi$
$882$	0	0
$883$	−51.9060	−1.74678	−0.873389	−	0.487024i	$-0.838082\pi$
$883$	−51.9060	−1.74678	−0.873389	+	0.487024i	$0.838082\pi$
$884$	−39.5158	−1.32906
$885$	0	0
$886$	−4.22918	−0.142082
$887$	−50.5245	−1.69645	−0.848223	−	0.529639i	$-0.822328\pi$
$887$	−50.5245	−1.69645	−0.848223	+	0.529639i	$0.822328\pi$
$888$	0	0
$889$	16.5723	0.555818
$890$	15.9271	0.533877
$891$	0	0
$892$	18.5564	0.621314
$893$	−12.3431	−0.413047
$894$	0	0
$895$	−33.5469	−1.12135
$896$	1.00000	0.0334077
$897$	0	0
$898$	−7.32673	−0.244496
$899$	2.82348	0.0941683
$900$	0	0
$901$	−65.3238	−2.17625
$902$	−7.03453	−0.234224
$903$	0	0
$904$	−9.24559	−0.307504
$905$	33.8360	1.12475
$906$	0	0
$907$	−31.9804	−1.06189	−0.530946	−	0.847406i	$-0.678163\pi$
$907$	−31.9804	−1.06189	−0.530946	+	0.847406i	$0.678163\pi$
$908$	−20.0145	−0.664204
$909$	0	0
$910$	9.53779	0.316175
$911$	4.04709	0.134086	0.0670431	−	0.997750i	$-0.478643\pi$
$911$	4.04709	0.134086	0.0670431	+	0.997750i	$0.478643\pi$
$912$	0	0
$913$	−12.6353	−0.418168
$914$	18.3613	0.607337
$915$	0	0
$916$	−3.44820	−0.113932
$917$	−3.03453	−0.100209
$918$	0	0
$919$	−5.77904	−0.190633	−0.0953165	−	0.995447i	$-0.530386\pi$
$919$	−5.77904	−0.190633	−0.0953165	+	0.995447i	$0.530386\pi$
$920$	1.51727	0.0500228
$921$	0	0
$922$	−38.2543	−1.25984
$923$	−62.2893	−2.05028
$924$	0	0
$925$	17.2441	0.566984
$926$	−23.3958	−0.768834
$927$	0	0
$928$	2.00000	0.0656532
$929$	29.4977	0.967788	0.483894	−	0.875127i	$-0.339222\pi$
$929$	29.4977	0.967788	0.483894	+	0.875127i	$0.339222\pi$
$930$	0	0
$931$	4.76889	0.156294
$932$	−5.62883	−0.184378
$933$	0	0
$934$	17.0891	0.559173
$935$	−33.5469	−1.09710
$936$	0	0
$937$	18.4972	0.604278	0.302139	−	0.953264i	$-0.402299\pi$
$937$	18.4972	0.604278	0.302139	+	0.953264i	$0.402299\pi$
$938$	14.0205	0.457786
$939$	0	0
$940$	−3.92708	−0.128087
$941$	42.8850	1.39801	0.699006	−	0.715116i	$-0.253626\pi$
$941$	42.8850	1.39801	0.699006	+	0.715116i	$0.253626\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	−7.03453	−0.228955
$945$	0	0
$946$	−42.5223	−1.38252
$947$	4.19293	0.136252	0.0681259	−	0.997677i	$-0.478298\pi$
$947$	4.19293	0.136252	0.0681259	+	0.997677i	$0.478298\pi$
$948$	0	0
$949$	55.7860	1.81089
$950$	−12.8660	−0.417428
$951$	0	0
$952$	−6.28616	−0.203736
$953$	−7.25381	−0.234974	−0.117487	−	0.993074i	$-0.537484\pi$
$953$	−7.25381	−0.234974	−0.117487	+	0.993074i	$0.537484\pi$
$954$	0	0
$955$	−5.93093	−0.191920
$956$	−6.82348	−0.220687
$957$	0	0
$958$	24.5723	0.793896
$959$	−1.03453	−0.0334069
$960$	0	0
$961$	−29.0070	−0.935709
$962$	−40.1792	−1.29543
$963$	0	0
$964$	−17.5318	−0.564660
$965$	26.8524	0.864411
$966$	0	0
$967$	3.72593	0.119818	0.0599090	−	0.998204i	$-0.480919\pi$
$967$	3.72593	0.119818	0.0599090	+	0.998204i	$0.480919\pi$
$968$	1.37117	0.0440711
$969$	0	0
$970$	−11.3229	−0.363556
$971$	54.6969	1.75531	0.877654	−	0.479295i	$-0.159108\pi$
$971$	54.6969	1.75531	0.877654	+	0.479295i	$0.159108\pi$
$972$	0	0
$973$	−5.53779	−0.177533
$974$	11.2577	0.360719
$975$	0	0
$976$	−4.55180	−0.145700
$977$	38.0410	1.21704	0.608520	−	0.793538i	$-0.291763\pi$
$977$	38.0410	1.21704	0.608520	+	0.793538i	$0.291763\pi$
$978$	0	0
$979$	36.9215	1.18002
$980$	1.51727	0.0484673
$981$	0	0
$982$	26.9845	0.861111
$983$	18.8964	0.602701	0.301351	−	0.953513i	$-0.402563\pi$
$983$	18.8964	0.602701	0.301351	+	0.953513i	$0.402563\pi$
$984$	0	0
$985$	36.9017	1.17579
$986$	−12.5723	−0.400384
$987$	0	0
$988$	29.9780	0.953728
$989$	−12.0896	−0.384427
$990$	0	0
$991$	2.22918	0.0708123	0.0354062	−	0.999373i	$-0.488728\pi$
$991$	2.22918	0.0708123	0.0354062	+	0.999373i	$0.488728\pi$
$992$	1.41174	0.0448228
$993$	0	0
$994$	−9.90896	−0.314293
$995$	−22.2149	−0.704260
$996$	0	0
$997$	25.5028	0.807681	0.403841	−	0.914829i	$-0.367675\pi$
$997$	25.5028	0.807681	0.403841	+	0.914829i	$0.367675\pi$
$998$	9.23569	0.292351
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bi.1.3	yes	4
3.2	odd	2		2898.2.a.bh.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.bh.1.2	✓	4	3.2	odd	2
2898.2.a.bi.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.51727 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.51727 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.51727 q^{10} +3.51727 q^{11} +6.28616 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.28616 q^{17} +4.76889 q^{19} +1.51727 q^{20} +3.51727 q^{22} +1.00000 q^{23} -2.69790 q^{25} +6.28616 q^{26} +1.00000 q^{28} +2.00000 q^{29} +1.41174 q^{31} +1.00000 q^{32} -6.28616 q^{34} +1.51727 q^{35} -6.39169 q^{37} +4.76889 q^{38} +1.51727 q^{40} -2.00000 q^{41} -12.0896 q^{43} +3.51727 q^{44} +1.00000 q^{46} -2.58826 q^{47} +1.00000 q^{49} -2.69790 q^{50} +6.28616 q^{52} +10.3917 q^{53} +5.33663 q^{55} +1.00000 q^{56} +2.00000 q^{58} -7.03453 q^{59} -4.55180 q^{61} +1.41174 q^{62} +1.00000 q^{64} +9.53779 q^{65} +14.0205 q^{67} -6.28616 q^{68} +1.51727 q^{70} -9.90896 q^{71} +8.87442 q^{73} -6.39169 q^{74} +4.76889 q^{76} +3.51727 q^{77} +12.8744 q^{79} +1.51727 q^{80} -2.00000 q^{82} -3.59237 q^{83} -9.53779 q^{85} -12.0896 q^{86} +3.51727 q^{88} +10.4972 q^{89} +6.28616 q^{91} +1.00000 q^{92} -2.58826 q^{94} +7.23569 q^{95} -7.46268 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8}+O(q^{10}) \) 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

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Groups

Database

Properties

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