LMFDB - Embedded newform 2394.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.23607 q^{10} +5.23607 q^{11} -3.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.70820 q^{17} +1.00000 q^{19} -1.23607 q^{20} -5.23607 q^{22} +0.763932 q^{23} -3.47214 q^{25} +3.23607 q^{26} -1.00000 q^{28} -0.472136 q^{29} -2.00000 q^{31} -1.00000 q^{32} -5.70820 q^{34} +1.23607 q^{35} -1.52786 q^{37} -1.00000 q^{38} +1.23607 q^{40} +6.00000 q^{41} -8.94427 q^{43} +5.23607 q^{44} -0.763932 q^{46} -8.94427 q^{47} +1.00000 q^{49} +3.47214 q^{50} -3.23607 q^{52} +3.52786 q^{53} -6.47214 q^{55} +1.00000 q^{56} +0.472136 q^{58} +8.00000 q^{59} +8.47214 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +13.7082 q^{67} +5.70820 q^{68} -1.23607 q^{70} +1.52786 q^{71} -3.52786 q^{73} +1.52786 q^{74} +1.00000 q^{76} -5.23607 q^{77} +5.23607 q^{79} -1.23607 q^{80} -6.00000 q^{82} -10.0000 q^{83} -7.05573 q^{85} +8.94427 q^{86} -5.23607 q^{88} +6.00000 q^{89} +3.23607 q^{91} +0.763932 q^{92} +8.94427 q^{94} -1.23607 q^{95} +9.23607 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 12 q^{37} - 2 q^{38} - 2 q^{40} + 12 q^{41} + 6 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 16 q^{53} - 4 q^{55} + 2 q^{56} - 8 q^{58} + 16 q^{59} + 8 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{65} + 14 q^{67} - 2 q^{68} + 2 q^{70} + 12 q^{71} - 16 q^{73} + 12 q^{74} + 2 q^{76} - 6 q^{77} + 6 q^{79} + 2 q^{80} - 12 q^{82} - 20 q^{83} - 32 q^{85} - 6 q^{88} + 12 q^{89} + 2 q^{91} + 6 q^{92} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^19 + 2 * q^20 - 6 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 - 12 * q^37 - 2 * q^38 - 2 * q^40 + 12 * q^41 + 6 * q^44 - 6 * q^46 + 2 * q^49 - 2 * q^50 - 2 * q^52 + 16 * q^53 - 4 * q^55 + 2 * q^56 - 8 * q^58 + 16 * q^59 + 8 * q^61 + 4 * q^62 + 2 * q^64 + 8 * q^65 + 14 * q^67 - 2 * q^68 + 2 * q^70 + 12 * q^71 - 16 * q^73 + 12 * q^74 + 2 * q^76 - 6 * q^77 + 6 * q^79 + 2 * q^80 - 12 * q^82 - 20 * q^83 - 32 * q^85 - 6 * q^88 + 12 * q^89 + 2 * q^91 + 6 * q^92 + 2 * q^95 + 14 * 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
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.23607	0.390879
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.70820	1.38444	0.692221	−	0.721685i	$-0.256632\pi$
$17$	5.70820	1.38444	0.692221	+	0.721685i	$0.256632\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.23607	−0.276393
$21$	0	0
$22$	−5.23607	−1.11633
$23$	0.763932	0.159291	0.0796454	−	0.996823i	$-0.474621\pi$
$23$	0.763932	0.159291	0.0796454	+	0.996823i	$0.474621\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	3.23607	0.634645
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.70820	−0.978949
$35$	1.23607	0.208934
$36$	0	0
$37$	−1.52786	−0.251179	−0.125590	−	0.992082i	$-0.540082\pi$
$37$	−1.52786	−0.251179	−0.125590	+	0.992082i	$0.540082\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	1.23607	0.195440
$41$	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
$43$	−8.94427	−1.36399	−0.681994	−	0.731357i	$-0.738887\pi$
$43$	−8.94427	−1.36399	−0.681994	+	0.731357i	$0.738887\pi$
$44$	5.23607	0.789367
$45$	0	0
$46$	−0.763932	−0.112636
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.47214	0.491034
$51$	0	0
$52$	−3.23607	−0.448762
$53$	3.52786	0.484589	0.242295	−	0.970203i	$-0.422100\pi$
$53$	3.52786	0.484589	0.242295	+	0.970203i	$0.422100\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	1.00000	0.133631
$57$	0	0
$58$	0.472136	0.0619945
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	8.47214	1.08475	0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214	1.08475	0.542373	+	0.840138i	$0.317526\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	13.7082	1.67472	0.837362	−	0.546649i	$-0.184097\pi$
$67$	13.7082	1.67472	0.837362	+	0.546649i	$0.184097\pi$
$68$	5.70820	0.692221
$69$	0	0
$70$	−1.23607	−0.147738
$71$	1.52786	0.181324	0.0906621	−	0.995882i	$-0.471102\pi$
$71$	1.52786	0.181324	0.0906621	+	0.995882i	$0.471102\pi$
$72$	0	0
$73$	−3.52786	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$73$	−3.52786	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$74$	1.52786	0.177611
$75$	0	0
$76$	1.00000	0.114708
$77$	−5.23607	−0.596705
$78$	0	0
$79$	5.23607	0.589104	0.294552	−	0.955636i	$-0.404830\pi$
$79$	5.23607	0.589104	0.294552	+	0.955636i	$0.404830\pi$
$80$	−1.23607	−0.138197
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	−7.05573	−0.765301
$86$	8.94427	0.964486
$87$	0	0
$88$	−5.23607	−0.558167
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	0.763932	0.0796454
$93$	0	0
$94$	8.94427	0.922531
$95$	−1.23607	−0.126818
$96$	0	0
$97$	9.23607	0.937781	0.468890	−	0.883256i	$-0.344654\pi$
$97$	9.23607	0.937781	0.468890	+	0.883256i	$0.344654\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.47214	−0.347214
$101$	−0.291796	−0.0290348	−0.0145174	−	0.999895i	$-0.504621\pi$
$101$	−0.291796	−0.0290348	−0.0145174	+	0.999895i	$0.504621\pi$
$102$	0	0
$103$	10.9443	1.07837	0.539186	−	0.842187i	$-0.318732\pi$
$103$	10.9443	1.07837	0.539186	+	0.842187i	$0.318732\pi$
$104$	3.23607	0.317323
$105$	0	0
$106$	−3.52786	−0.342656
$107$	14.4721	1.39907	0.699537	−	0.714596i	$-0.253390\pi$
$107$	14.4721	1.39907	0.699537	+	0.714596i	$0.253390\pi$
$108$	0	0
$109$	9.52786	0.912604	0.456302	−	0.889825i	$-0.349174\pi$
$109$	9.52786	0.912604	0.456302	+	0.889825i	$0.349174\pi$
$110$	6.47214	0.617094
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−0.944272	−0.0880538
$116$	−0.472136	−0.0438367
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−5.70820	−0.523270
$120$	0	0
$121$	16.4164	1.49240
$122$	−8.47214	−0.767031
$123$	0	0
$124$	−2.00000	−0.179605
$125$	10.4721	0.936656
$126$	0	0
$127$	3.70820	0.329050	0.164525	−	0.986373i	$-0.447391\pi$
$127$	3.70820	0.329050	0.164525	+	0.986373i	$0.447391\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	−7.52786	−0.657713	−0.328856	−	0.944380i	$-0.606663\pi$
$131$	−7.52786	−0.657713	−0.328856	+	0.944380i	$0.606663\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−13.7082	−1.18421
$135$	0	0
$136$	−5.70820	−0.489474
$137$	11.4164	0.975370	0.487685	−	0.873020i	$-0.337842\pi$
$137$	11.4164	0.975370	0.487685	+	0.873020i	$0.337842\pi$
$138$	0	0
$139$	−13.8885	−1.17801	−0.589005	−	0.808129i	$-0.700480\pi$
$139$	−13.8885	−1.17801	−0.589005	+	0.808129i	$0.700480\pi$
$140$	1.23607	0.104467
$141$	0	0
$142$	−1.52786	−0.128216
$143$	−16.9443	−1.41695
$144$	0	0
$145$	0.583592	0.0484647
$146$	3.52786	0.291968
$147$	0	0
$148$	−1.52786	−0.125590
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	5.23607	0.426105	0.213053	−	0.977041i	$-0.431659\pi$
$151$	5.23607	0.426105	0.213053	+	0.977041i	$0.431659\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	5.23607	0.421934
$155$	2.47214	0.198567
$156$	0	0
$157$	−13.4164	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$158$	−5.23607	−0.416559
$159$	0	0
$160$	1.23607	0.0977198
$161$	−0.763932	−0.0602063
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	10.0000	0.776151
$167$	−0.944272	−0.0730700	−0.0365350	−	0.999332i	$-0.511632\pi$
$167$	−0.944272	−0.0730700	−0.0365350	+	0.999332i	$0.511632\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	7.05573	0.541150
$171$	0	0
$172$	−8.94427	−0.681994
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	5.23607	0.394683
$177$	0	0
$178$	−6.00000	−0.449719
$179$	21.8885	1.63603	0.818013	−	0.575199i	$-0.195076\pi$
$179$	21.8885	1.63603	0.818013	+	0.575199i	$0.195076\pi$
$180$	0	0
$181$	−0.763932	−0.0567826	−0.0283913	−	0.999597i	$-0.509038\pi$
$181$	−0.763932	−0.0567826	−0.0283913	+	0.999597i	$0.509038\pi$
$182$	−3.23607	−0.239873
$183$	0	0
$184$	−0.763932	−0.0563178
$185$	1.88854	0.138849
$186$	0	0
$187$	29.8885	2.18567
$188$	−8.94427	−0.652328
$189$	0	0
$190$	1.23607	0.0896738
$191$	12.7639	0.923566	0.461783	−	0.886993i	$-0.347210\pi$
$191$	12.7639	0.923566	0.461783	+	0.886993i	$0.347210\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−9.23607	−0.663111
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.94427	0.494759	0.247379	−	0.968919i	$-0.420431\pi$
$197$	6.94427	0.494759	0.247379	+	0.968919i	$0.420431\pi$
$198$	0	0
$199$	−11.4164	−0.809288	−0.404644	−	0.914474i	$-0.632605\pi$
$199$	−11.4164	−0.809288	−0.404644	+	0.914474i	$0.632605\pi$
$200$	3.47214	0.245517
$201$	0	0
$202$	0.291796	0.0205307
$203$	0.472136	0.0331374
$204$	0	0
$205$	−7.41641	−0.517984
$206$	−10.9443	−0.762524
$207$	0	0
$208$	−3.23607	−0.224381
$209$	5.23607	0.362186
$210$	0	0
$211$	27.5967	1.89984	0.949919	−	0.312496i	$-0.101165\pi$
$211$	27.5967	1.89984	0.949919	+	0.312496i	$0.101165\pi$
$212$	3.52786	0.242295
$213$	0	0
$214$	−14.4721	−0.989295
$215$	11.0557	0.753994
$216$	0	0
$217$	2.00000	0.135769
$218$	−9.52786	−0.645308
$219$	0	0
$220$	−6.47214	−0.436351
$221$	−18.4721	−1.24257
$222$	0	0
$223$	19.5279	1.30768	0.653841	−	0.756632i	$-0.273156\pi$
$223$	19.5279	1.30768	0.653841	+	0.756632i	$0.273156\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−15.4164	−1.02322	−0.511611	−	0.859217i	$-0.670951\pi$
$227$	−15.4164	−1.02322	−0.511611	+	0.859217i	$0.670951\pi$
$228$	0	0
$229$	−18.9443	−1.25187	−0.625936	−	0.779874i	$-0.715283\pi$
$229$	−18.9443	−1.25187	−0.625936	+	0.779874i	$0.715283\pi$
$230$	0.944272	0.0622634
$231$	0	0
$232$	0.472136	0.0309972
$233$	14.4721	0.948101	0.474051	−	0.880498i	$-0.342791\pi$
$233$	14.4721	0.948101	0.474051	+	0.880498i	$0.342791\pi$
$234$	0	0
$235$	11.0557	0.721196
$236$	8.00000	0.520756
$237$	0	0
$238$	5.70820	0.370008
$239$	3.81966	0.247073	0.123537	−	0.992340i	$-0.460576\pi$
$239$	3.81966	0.247073	0.123537	+	0.992340i	$0.460576\pi$
$240$	0	0
$241$	−5.23607	−0.337285	−0.168642	−	0.985677i	$-0.553938\pi$
$241$	−5.23607	−0.337285	−0.168642	+	0.985677i	$0.553938\pi$
$242$	−16.4164	−1.05529
$243$	0	0
$244$	8.47214	0.542373
$245$	−1.23607	−0.0789695
$246$	0	0
$247$	−3.23607	−0.205906
$248$	2.00000	0.127000
$249$	0	0
$250$	−10.4721	−0.662316
$251$	25.4164	1.60427	0.802135	−	0.597143i	$-0.203698\pi$
$251$	25.4164	1.60427	0.802135	+	0.597143i	$0.203698\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−3.70820	−0.232673
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.4721	−0.777990	−0.388995	−	0.921240i	$-0.627178\pi$
$257$	−12.4721	−0.777990	−0.388995	+	0.921240i	$0.627178\pi$
$258$	0	0
$259$	1.52786	0.0949369
$260$	4.00000	0.248069
$261$	0	0
$262$	7.52786	0.465073
$263$	−5.70820	−0.351983	−0.175991	−	0.984392i	$-0.556313\pi$
$263$	−5.70820	−0.351983	−0.175991	+	0.984392i	$0.556313\pi$
$264$	0	0
$265$	−4.36068	−0.267874
$266$	1.00000	0.0613139
$267$	0	0
$268$	13.7082	0.837362
$269$	10.9443	0.667284	0.333642	−	0.942700i	$-0.391722\pi$
$269$	10.9443	0.667284	0.333642	+	0.942700i	$0.391722\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	5.70820	0.346111
$273$	0	0
$274$	−11.4164	−0.689690
$275$	−18.1803	−1.09632
$276$	0	0
$277$	2.94427	0.176904	0.0884521	−	0.996080i	$-0.471808\pi$
$277$	2.94427	0.176904	0.0884521	+	0.996080i	$0.471808\pi$
$278$	13.8885	0.832980
$279$	0	0
$280$	−1.23607	−0.0738692
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	13.8885	0.825588	0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885	0.825588	0.412794	+	0.910824i	$0.364553\pi$
$284$	1.52786	0.0906621
$285$	0	0
$286$	16.9443	1.00194
$287$	−6.00000	−0.354169
$288$	0	0
$289$	15.5836	0.916682
$290$	−0.583592	−0.0342697
$291$	0	0
$292$	−3.52786	−0.206453
$293$	−19.8885	−1.16190	−0.580951	−	0.813939i	$-0.697319\pi$
$293$	−19.8885	−1.16190	−0.580951	+	0.813939i	$0.697319\pi$
$294$	0	0
$295$	−9.88854	−0.575733
$296$	1.52786	0.0888053
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−2.47214	−0.142967
$300$	0	0
$301$	8.94427	0.515539
$302$	−5.23607	−0.301302
$303$	0	0
$304$	1.00000	0.0573539
$305$	−10.4721	−0.599633
$306$	0	0
$307$	31.4164	1.79303	0.896515	−	0.443014i	$-0.146091\pi$
$307$	31.4164	1.79303	0.896515	+	0.443014i	$0.146091\pi$
$308$	−5.23607	−0.298353
$309$	0	0
$310$	−2.47214	−0.140408
$311$	−28.3607	−1.60819	−0.804093	−	0.594503i	$-0.797349\pi$
$311$	−28.3607	−1.60819	−0.804093	+	0.594503i	$0.797349\pi$
$312$	0	0
$313$	−10.9443	−0.618607	−0.309303	−	0.950963i	$-0.600096\pi$
$313$	−10.9443	−0.618607	−0.309303	+	0.950963i	$0.600096\pi$
$314$	13.4164	0.757132
$315$	0	0
$316$	5.23607	0.294552
$317$	7.88854	0.443065	0.221532	−	0.975153i	$-0.428894\pi$
$317$	7.88854	0.443065	0.221532	+	0.975153i	$0.428894\pi$
$318$	0	0
$319$	−2.47214	−0.138413
$320$	−1.23607	−0.0690983
$321$	0	0
$322$	0.763932	0.0425723
$323$	5.70820	0.317613
$324$	0	0
$325$	11.2361	0.623265
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	8.94427	0.493114
$330$	0	0
$331$	−19.5967	−1.07713	−0.538567	−	0.842582i	$-0.681034\pi$
$331$	−19.5967	−1.07713	−0.538567	+	0.842582i	$0.681034\pi$
$332$	−10.0000	−0.548821
$333$	0	0
$334$	0.944272	0.0516683
$335$	−16.9443	−0.925764
$336$	0	0
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	2.52786	0.137498
$339$	0	0
$340$	−7.05573	−0.382651
$341$	−10.4721	−0.567098
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.94427	0.482243
$345$	0	0
$346$	2.00000	0.107521
$347$	19.7082	1.05799	0.528996	−	0.848624i	$-0.322569\pi$
$347$	19.7082	1.05799	0.528996	+	0.848624i	$0.322569\pi$
$348$	0	0
$349$	28.8328	1.54339	0.771693	−	0.635996i	$-0.219410\pi$
$349$	28.8328	1.54339	0.771693	+	0.635996i	$0.219410\pi$
$350$	−3.47214	−0.185593
$351$	0	0
$352$	−5.23607	−0.279083
$353$	1.70820	0.0909185	0.0454593	−	0.998966i	$-0.485525\pi$
$353$	1.70820	0.0909185	0.0454593	+	0.998966i	$0.485525\pi$
$354$	0	0
$355$	−1.88854	−0.100233
$356$	6.00000	0.317999
$357$	0	0
$358$	−21.8885	−1.15685
$359$	29.1246	1.53714	0.768569	−	0.639767i	$-0.220969\pi$
$359$	29.1246	1.53714	0.768569	+	0.639767i	$0.220969\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.763932	0.0401514
$363$	0	0
$364$	3.23607	0.169616
$365$	4.36068	0.228248
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	0.763932	0.0398227
$369$	0	0
$370$	−1.88854	−0.0981807
$371$	−3.52786	−0.183158
$372$	0	0
$373$	12.9443	0.670229	0.335114	−	0.942177i	$-0.391225\pi$
$373$	12.9443	0.670229	0.335114	+	0.942177i	$0.391225\pi$
$374$	−29.8885	−1.54550
$375$	0	0
$376$	8.94427	0.461266
$377$	1.52786	0.0786890
$378$	0	0
$379$	13.1246	0.674166	0.337083	−	0.941475i	$-0.390560\pi$
$379$	13.1246	0.674166	0.337083	+	0.941475i	$0.390560\pi$
$380$	−1.23607	−0.0634089
$381$	0	0
$382$	−12.7639	−0.653060
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	6.47214	0.329851
$386$	18.0000	0.916176
$387$	0	0
$388$	9.23607	0.468890
$389$	−29.4164	−1.49147	−0.745736	−	0.666242i	$-0.767902\pi$
$389$	−29.4164	−1.49147	−0.745736	+	0.666242i	$0.767902\pi$
$390$	0	0
$391$	4.36068	0.220529
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−6.94427	−0.349847
$395$	−6.47214	−0.325649
$396$	0	0
$397$	1.41641	0.0710875	0.0355437	−	0.999368i	$-0.488684\pi$
$397$	1.41641	0.0710875	0.0355437	+	0.999368i	$0.488684\pi$
$398$	11.4164	0.572253
$399$	0	0
$400$	−3.47214	−0.173607
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	6.47214	0.322400
$404$	−0.291796	−0.0145174
$405$	0	0
$406$	−0.472136	−0.0234317
$407$	−8.00000	−0.396545
$408$	0	0
$409$	3.12461	0.154502	0.0772511	−	0.997012i	$-0.475386\pi$
$409$	3.12461	0.154502	0.0772511	+	0.997012i	$0.475386\pi$
$410$	7.41641	0.366270
$411$	0	0
$412$	10.9443	0.539186
$413$	−8.00000	−0.393654
$414$	0	0
$415$	12.3607	0.606762
$416$	3.23607	0.158661
$417$	0	0
$418$	−5.23607	−0.256104
$419$	14.3607	0.701565	0.350783	−	0.936457i	$-0.385916\pi$
$419$	14.3607	0.701565	0.350783	+	0.936457i	$0.385916\pi$
$420$	0	0
$421$	−34.8328	−1.69765	−0.848824	−	0.528676i	$-0.822689\pi$
$421$	−34.8328	−1.69765	−0.848824	+	0.528676i	$0.822689\pi$
$422$	−27.5967	−1.34339
$423$	0	0
$424$	−3.52786	−0.171328
$425$	−19.8197	−0.961395
$426$	0	0
$427$	−8.47214	−0.409995
$428$	14.4721	0.699537
$429$	0	0
$430$	−11.0557	−0.533155
$431$	22.8328	1.09982	0.549909	−	0.835225i	$-0.314662\pi$
$431$	22.8328	1.09982	0.549909	+	0.835225i	$0.314662\pi$
$432$	0	0
$433$	−18.7639	−0.901737	−0.450869	−	0.892590i	$-0.648886\pi$
$433$	−18.7639	−0.901737	−0.450869	+	0.892590i	$0.648886\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	9.52786	0.456302
$437$	0.763932	0.0365438
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	6.47214	0.308547
$441$	0	0
$442$	18.4721	0.878630
$443$	−7.70820	−0.366228	−0.183114	−	0.983092i	$-0.558618\pi$
$443$	−7.70820	−0.366228	−0.183114	+	0.983092i	$0.558618\pi$
$444$	0	0
$445$	−7.41641	−0.351571
$446$	−19.5279	−0.924671
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−32.8328	−1.54948	−0.774738	−	0.632282i	$-0.782118\pi$
$449$	−32.8328	−1.54948	−0.774738	+	0.632282i	$0.782118\pi$
$450$	0	0
$451$	31.4164	1.47934
$452$	10.0000	0.470360
$453$	0	0
$454$	15.4164	0.723528
$455$	−4.00000	−0.187523
$456$	0	0
$457$	35.3050	1.65150	0.825748	−	0.564039i	$-0.190753\pi$
$457$	35.3050	1.65150	0.825748	+	0.564039i	$0.190753\pi$
$458$	18.9443	0.885208
$459$	0	0
$460$	−0.944272	−0.0440269
$461$	1.81966	0.0847500	0.0423750	−	0.999102i	$-0.486508\pi$
$461$	1.81966	0.0847500	0.0423750	+	0.999102i	$0.486508\pi$
$462$	0	0
$463$	−41.3050	−1.91960	−0.959802	−	0.280678i	$-0.909441\pi$
$463$	−41.3050	−1.91960	−0.959802	+	0.280678i	$0.909441\pi$
$464$	−0.472136	−0.0219184
$465$	0	0
$466$	−14.4721	−0.670409
$467$	−19.8885	−0.920332	−0.460166	−	0.887833i	$-0.652210\pi$
$467$	−19.8885	−0.920332	−0.460166	+	0.887833i	$0.652210\pi$
$468$	0	0
$469$	−13.7082	−0.632986
$470$	−11.0557	−0.509963
$471$	0	0
$472$	−8.00000	−0.368230
$473$	−46.8328	−2.15338
$474$	0	0
$475$	−3.47214	−0.159313
$476$	−5.70820	−0.261635
$477$	0	0
$478$	−3.81966	−0.174707
$479$	−34.4721	−1.57507	−0.787536	−	0.616269i	$-0.788644\pi$
$479$	−34.4721	−1.57507	−0.787536	+	0.616269i	$0.788644\pi$
$480$	0	0
$481$	4.94427	0.225439
$482$	5.23607	0.238496
$483$	0	0
$484$	16.4164	0.746200
$485$	−11.4164	−0.518392
$486$	0	0
$487$	−28.0689	−1.27192	−0.635961	−	0.771721i	$-0.719396\pi$
$487$	−28.0689	−1.27192	−0.635961	+	0.771721i	$0.719396\pi$
$488$	−8.47214	−0.383516
$489$	0	0
$490$	1.23607	0.0558399
$491$	−0.291796	−0.0131686	−0.00658429	−	0.999978i	$-0.502096\pi$
$491$	−0.291796	−0.0131686	−0.00658429	+	0.999978i	$0.502096\pi$
$492$	0	0
$493$	−2.69505	−0.121379
$494$	3.23607	0.145598
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−1.52786	−0.0685341
$498$	0	0
$499$	25.3050	1.13281	0.566403	−	0.824129i	$-0.308335\pi$
$499$	25.3050	1.13281	0.566403	+	0.824129i	$0.308335\pi$
$500$	10.4721	0.468328
$501$	0	0
$502$	−25.4164	−1.13439
$503$	−17.5279	−0.781529	−0.390764	−	0.920491i	$-0.627789\pi$
$503$	−17.5279	−0.781529	−0.390764	+	0.920491i	$0.627789\pi$
$504$	0	0
$505$	0.360680	0.0160500
$506$	−4.00000	−0.177822
$507$	0	0
$508$	3.70820	0.164525
$509$	13.4164	0.594672	0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164	0.594672	0.297336	+	0.954773i	$0.403902\pi$
$510$	0	0
$511$	3.52786	0.156064
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.4721	0.550122
$515$	−13.5279	−0.596109
$516$	0	0
$517$	−46.8328	−2.05970
$518$	−1.52786	−0.0671305
$519$	0	0
$520$	−4.00000	−0.175412
$521$	−18.3607	−0.804396	−0.402198	−	0.915553i	$-0.631754\pi$
$521$	−18.3607	−0.804396	−0.402198	+	0.915553i	$0.631754\pi$
$522$	0	0
$523$	14.4721	0.632822	0.316411	−	0.948622i	$-0.397522\pi$
$523$	14.4721	0.632822	0.316411	+	0.948622i	$0.397522\pi$
$524$	−7.52786	−0.328856
$525$	0	0
$526$	5.70820	0.248890
$527$	−11.4164	−0.497307
$528$	0	0
$529$	−22.4164	−0.974626
$530$	4.36068	0.189416
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−19.4164	−0.841018
$534$	0	0
$535$	−17.8885	−0.773389
$536$	−13.7082	−0.592104
$537$	0	0
$538$	−10.9443	−0.471841
$539$	5.23607	0.225533
$540$	0	0
$541$	−45.7771	−1.96811	−0.984055	−	0.177862i	$-0.943082\pi$
$541$	−45.7771	−1.96811	−0.984055	+	0.177862i	$0.943082\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	−5.70820	−0.244737
$545$	−11.7771	−0.504475
$546$	0	0
$547$	−29.1246	−1.24528	−0.622639	−	0.782509i	$-0.713940\pi$
$547$	−29.1246	−1.24528	−0.622639	+	0.782509i	$0.713940\pi$
$548$	11.4164	0.487685
$549$	0	0
$550$	18.1803	0.775212
$551$	−0.472136	−0.0201137
$552$	0	0
$553$	−5.23607	−0.222660
$554$	−2.94427	−0.125090
$555$	0	0
$556$	−13.8885	−0.589005
$557$	−35.8885	−1.52065	−0.760323	−	0.649545i	$-0.774959\pi$
$557$	−35.8885	−1.52065	−0.760323	+	0.649545i	$0.774959\pi$
$558$	0	0
$559$	28.9443	1.22421
$560$	1.23607	0.0522334
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	−12.3607	−0.520018
$566$	−13.8885	−0.583779
$567$	0	0
$568$	−1.52786	−0.0641078
$569$	14.9443	0.626496	0.313248	−	0.949671i	$-0.398583\pi$
$569$	14.9443	0.626496	0.313248	+	0.949671i	$0.398583\pi$
$570$	0	0
$571$	−2.47214	−0.103456	−0.0517278	−	0.998661i	$-0.516473\pi$
$571$	−2.47214	−0.103456	−0.0517278	+	0.998661i	$0.516473\pi$
$572$	−16.9443	−0.708476
$573$	0	0
$574$	6.00000	0.250435
$575$	−2.65248	−0.110616
$576$	0	0
$577$	−5.41641	−0.225488	−0.112744	−	0.993624i	$-0.535964\pi$
$577$	−5.41641	−0.225488	−0.112744	+	0.993624i	$0.535964\pi$
$578$	−15.5836	−0.648192
$579$	0	0
$580$	0.583592	0.0242323
$581$	10.0000	0.414870
$582$	0	0
$583$	18.4721	0.765038
$584$	3.52786	0.145984
$585$	0	0
$586$	19.8885	0.821588
$587$	8.83282	0.364569	0.182285	−	0.983246i	$-0.441651\pi$
$587$	8.83282	0.364569	0.182285	+	0.983246i	$0.441651\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	9.88854	0.407105
$591$	0	0
$592$	−1.52786	−0.0627948
$593$	−34.6525	−1.42301	−0.711503	−	0.702683i	$-0.751985\pi$
$593$	−34.6525	−1.42301	−0.711503	+	0.702683i	$0.751985\pi$
$594$	0	0
$595$	7.05573	0.289257
$596$	6.00000	0.245770
$597$	0	0
$598$	2.47214	0.101093
$599$	−12.5836	−0.514152	−0.257076	−	0.966391i	$-0.582759\pi$
$599$	−12.5836	−0.514152	−0.257076	+	0.966391i	$0.582759\pi$
$600$	0	0
$601$	−11.7082	−0.477588	−0.238794	−	0.971070i	$-0.576752\pi$
$601$	−11.7082	−0.477588	−0.238794	+	0.971070i	$0.576752\pi$
$602$	−8.94427	−0.364541
$603$	0	0
$604$	5.23607	0.213053
$605$	−20.2918	−0.824979
$606$	0	0
$607$	−1.41641	−0.0574902	−0.0287451	−	0.999587i	$-0.509151\pi$
$607$	−1.41641	−0.0574902	−0.0287451	+	0.999587i	$0.509151\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	10.4721	0.424004
$611$	28.9443	1.17096
$612$	0	0
$613$	43.8885	1.77264	0.886321	−	0.463072i	$-0.153253\pi$
$613$	43.8885	1.77264	0.886321	+	0.463072i	$0.153253\pi$
$614$	−31.4164	−1.26786
$615$	0	0
$616$	5.23607	0.210967
$617$	23.4164	0.942709	0.471355	−	0.881944i	$-0.343765\pi$
$617$	23.4164	0.942709	0.471355	+	0.881944i	$0.343765\pi$
$618$	0	0
$619$	−6.11146	−0.245640	−0.122820	−	0.992429i	$-0.539194\pi$
$619$	−6.11146	−0.245640	−0.122820	+	0.992429i	$0.539194\pi$
$620$	2.47214	0.0992834
$621$	0	0
$622$	28.3607	1.13716
$623$	−6.00000	−0.240385
$624$	0	0
$625$	4.41641	0.176656
$626$	10.9443	0.437421
$627$	0	0
$628$	−13.4164	−0.535373
$629$	−8.72136	−0.347743
$630$	0	0
$631$	2.47214	0.0984142	0.0492071	−	0.998789i	$-0.484331\pi$
$631$	2.47214	0.0984142	0.0492071	+	0.998789i	$0.484331\pi$
$632$	−5.23607	−0.208280
$633$	0	0
$634$	−7.88854	−0.313294
$635$	−4.58359	−0.181894
$636$	0	0
$637$	−3.23607	−0.128218
$638$	2.47214	0.0978728
$639$	0	0
$640$	1.23607	0.0488599
$641$	−12.1115	−0.478374	−0.239187	−	0.970974i	$-0.576881\pi$
$641$	−12.1115	−0.478374	−0.239187	+	0.970974i	$0.576881\pi$
$642$	0	0
$643$	−44.9443	−1.77243	−0.886215	−	0.463275i	$-0.846674\pi$
$643$	−44.9443	−1.77243	−0.886215	+	0.463275i	$0.846674\pi$
$644$	−0.763932	−0.0301031
$645$	0	0
$646$	−5.70820	−0.224586
$647$	34.2492	1.34648	0.673238	−	0.739426i	$-0.264903\pi$
$647$	34.2492	1.34648	0.673238	+	0.739426i	$0.264903\pi$
$648$	0	0
$649$	41.8885	1.64427
$650$	−11.2361	−0.440715
$651$	0	0
$652$	4.00000	0.156652
$653$	17.4164	0.681557	0.340778	−	0.940144i	$-0.389309\pi$
$653$	17.4164	0.681557	0.340778	+	0.940144i	$0.389309\pi$
$654$	0	0
$655$	9.30495	0.363575
$656$	6.00000	0.234261
$657$	0	0
$658$	−8.94427	−0.348684
$659$	14.4721	0.563754	0.281877	−	0.959450i	$-0.409043\pi$
$659$	14.4721	0.563754	0.281877	+	0.959450i	$0.409043\pi$
$660$	0	0
$661$	−40.7639	−1.58553	−0.792767	−	0.609525i	$-0.791360\pi$
$661$	−40.7639	−1.58553	−0.792767	+	0.609525i	$0.791360\pi$
$662$	19.5967	0.761649
$663$	0	0
$664$	10.0000	0.388075
$665$	1.23607	0.0479327
$666$	0	0
$667$	−0.360680	−0.0139656
$668$	−0.944272	−0.0365350
$669$	0	0
$670$	16.9443	0.654614
$671$	44.3607	1.71253
$672$	0	0
$673$	−44.2492	−1.70568	−0.852841	−	0.522171i	$-0.825122\pi$
$673$	−44.2492	−1.70568	−0.852841	+	0.522171i	$0.825122\pi$
$674$	11.5279	0.444037
$675$	0	0
$676$	−2.52786	−0.0972255
$677$	−19.5279	−0.750517	−0.375258	−	0.926920i	$-0.622446\pi$
$677$	−19.5279	−0.750517	−0.375258	+	0.926920i	$0.622446\pi$
$678$	0	0
$679$	−9.23607	−0.354448
$680$	7.05573	0.270575
$681$	0	0
$682$	10.4721	0.400999
$683$	45.8885	1.75588	0.877938	−	0.478774i	$-0.158919\pi$
$683$	45.8885	1.75588	0.877938	+	0.478774i	$0.158919\pi$
$684$	0	0
$685$	−14.1115	−0.539171
$686$	1.00000	0.0381802
$687$	0	0
$688$	−8.94427	−0.340997
$689$	−11.4164	−0.434931
$690$	0	0
$691$	26.8328	1.02077	0.510384	−	0.859946i	$-0.329503\pi$
$691$	26.8328	1.02077	0.510384	+	0.859946i	$0.329503\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−19.7082	−0.748113
$695$	17.1672	0.651188
$696$	0	0
$697$	34.2492	1.29728
$698$	−28.8328	−1.09134
$699$	0	0
$700$	3.47214	0.131234
$701$	−36.4721	−1.37753	−0.688767	−	0.724983i	$-0.741848\pi$
$701$	−36.4721	−1.37753	−0.688767	+	0.724983i	$0.741848\pi$
$702$	0	0
$703$	−1.52786	−0.0576245
$704$	5.23607	0.197342
$705$	0	0
$706$	−1.70820	−0.0642891
$707$	0.291796	0.0109741
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	1.88854	0.0708758
$711$	0	0
$712$	−6.00000	−0.224860
$713$	−1.52786	−0.0572190
$714$	0	0
$715$	20.9443	0.783271
$716$	21.8885	0.818013
$717$	0	0
$718$	−29.1246	−1.08692
$719$	2.47214	0.0921951	0.0460976	−	0.998937i	$-0.485321\pi$
$719$	2.47214	0.0921951	0.0460976	+	0.998937i	$0.485321\pi$
$720$	0	0
$721$	−10.9443	−0.407586
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−0.763932	−0.0283913
$725$	1.63932	0.0608828
$726$	0	0
$727$	43.7771	1.62360	0.811801	−	0.583934i	$-0.198487\pi$
$727$	43.7771	1.62360	0.811801	+	0.583934i	$0.198487\pi$
$728$	−3.23607	−0.119937
$729$	0	0
$730$	−4.36068	−0.161396
$731$	−51.0557	−1.88836
$732$	0	0
$733$	−40.8328	−1.50819	−0.754097	−	0.656763i	$-0.771925\pi$
$733$	−40.8328	−1.50819	−0.754097	+	0.656763i	$0.771925\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−0.763932	−0.0281589
$737$	71.7771	2.64394
$738$	0	0
$739$	23.4164	0.861386	0.430693	−	0.902498i	$-0.358269\pi$
$739$	23.4164	0.861386	0.430693	+	0.902498i	$0.358269\pi$
$740$	1.88854	0.0694243
$741$	0	0
$742$	3.52786	0.129512
$743$	3.41641	0.125336	0.0626679	−	0.998034i	$-0.480039\pi$
$743$	3.41641	0.125336	0.0626679	+	0.998034i	$0.480039\pi$
$744$	0	0
$745$	−7.41641	−0.271716
$746$	−12.9443	−0.473923
$747$	0	0
$748$	29.8885	1.09283
$749$	−14.4721	−0.528800
$750$	0	0
$751$	−53.0132	−1.93448	−0.967239	−	0.253868i	$-0.918297\pi$
$751$	−53.0132	−1.93448	−0.967239	+	0.253868i	$0.918297\pi$
$752$	−8.94427	−0.326164
$753$	0	0
$754$	−1.52786	−0.0556415
$755$	−6.47214	−0.235545
$756$	0	0
$757$	0.111456	0.00405094	0.00202547	−	0.999998i	$-0.499355\pi$
$757$	0.111456	0.00405094	0.00202547	+	0.999998i	$0.499355\pi$
$758$	−13.1246	−0.476707
$759$	0	0
$760$	1.23607	0.0448369
$761$	−2.65248	−0.0961522	−0.0480761	−	0.998844i	$-0.515309\pi$
$761$	−2.65248	−0.0961522	−0.0480761	+	0.998844i	$0.515309\pi$
$762$	0	0
$763$	−9.52786	−0.344932
$764$	12.7639	0.461783
$765$	0	0
$766$	8.00000	0.289052
$767$	−25.8885	−0.934781
$768$	0	0
$769$	6.94427	0.250417	0.125208	−	0.992130i	$-0.460040\pi$
$769$	6.94427	0.250417	0.125208	+	0.992130i	$0.460040\pi$
$770$	−6.47214	−0.233240
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−18.5836	−0.668405	−0.334203	−	0.942501i	$-0.608467\pi$
$773$	−18.5836	−0.668405	−0.334203	+	0.942501i	$0.608467\pi$
$774$	0	0
$775$	6.94427	0.249446
$776$	−9.23607	−0.331556
$777$	0	0
$778$	29.4164	1.05463
$779$	6.00000	0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	−4.36068	−0.155938
$783$	0	0
$784$	1.00000	0.0357143
$785$	16.5836	0.591894
$786$	0	0
$787$	−7.41641	−0.264366	−0.132183	−	0.991225i	$-0.542199\pi$
$787$	−7.41641	−0.264366	−0.132183	+	0.991225i	$0.542199\pi$
$788$	6.94427	0.247379
$789$	0	0
$790$	6.47214	0.230268
$791$	−10.0000	−0.355559
$792$	0	0
$793$	−27.4164	−0.973585
$794$	−1.41641	−0.0502664
$795$	0	0
$796$	−11.4164	−0.404644
$797$	45.4164	1.60873	0.804366	−	0.594134i	$-0.202505\pi$
$797$	45.4164	1.60873	0.804366	+	0.594134i	$0.202505\pi$
$798$	0	0
$799$	−51.0557	−1.80622
$800$	3.47214	0.122759
$801$	0	0
$802$	−30.0000	−1.05934
$803$	−18.4721	−0.651868
$804$	0	0
$805$	0.944272	0.0332812
$806$	−6.47214	−0.227971
$807$	0	0
$808$	0.291796	0.0102653
$809$	8.94427	0.314464	0.157232	−	0.987562i	$-0.449743\pi$
$809$	8.94427	0.314464	0.157232	+	0.987562i	$0.449743\pi$
$810$	0	0
$811$	−23.4164	−0.822261	−0.411131	−	0.911576i	$-0.634866\pi$
$811$	−23.4164	−0.822261	−0.411131	+	0.911576i	$0.634866\pi$
$812$	0.472136	0.0165687
$813$	0	0
$814$	8.00000	0.280400
$815$	−4.94427	−0.173190
$816$	0	0
$817$	−8.94427	−0.312920
$818$	−3.12461	−0.109249
$819$	0	0
$820$	−7.41641	−0.258992
$821$	6.94427	0.242357	0.121178	−	0.992631i	$-0.461333\pi$
$821$	6.94427	0.242357	0.121178	+	0.992631i	$0.461333\pi$
$822$	0	0
$823$	29.5279	1.02928	0.514638	−	0.857407i	$-0.327926\pi$
$823$	29.5279	1.02928	0.514638	+	0.857407i	$0.327926\pi$
$824$	−10.9443	−0.381262
$825$	0	0
$826$	8.00000	0.278356
$827$	−42.8328	−1.48944	−0.744721	−	0.667375i	$-0.767418\pi$
$827$	−42.8328	−1.48944	−0.744721	+	0.667375i	$0.767418\pi$
$828$	0	0
$829$	−23.2361	−0.807022	−0.403511	−	0.914975i	$-0.632210\pi$
$829$	−23.2361	−0.807022	−0.403511	+	0.914975i	$0.632210\pi$
$830$	−12.3607	−0.429045
$831$	0	0
$832$	−3.23607	−0.112190
$833$	5.70820	0.197778
$834$	0	0
$835$	1.16718	0.0403921
$836$	5.23607	0.181093
$837$	0	0
$838$	−14.3607	−0.496081
$839$	53.8885	1.86044	0.930220	−	0.367003i	$-0.119616\pi$
$839$	53.8885	1.86044	0.930220	+	0.367003i	$0.119616\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	34.8328	1.20042
$843$	0	0
$844$	27.5967	0.949919
$845$	3.12461	0.107490
$846$	0	0
$847$	−16.4164	−0.564074
$848$	3.52786	0.121147
$849$	0	0
$850$	19.8197	0.679809
$851$	−1.16718	−0.0400106
$852$	0	0
$853$	6.94427	0.237767	0.118884	−	0.992908i	$-0.462068\pi$
$853$	6.94427	0.237767	0.118884	+	0.992908i	$0.462068\pi$
$854$	8.47214	0.289911
$855$	0	0
$856$	−14.4721	−0.494647
$857$	31.8885	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$857$	31.8885	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$858$	0	0
$859$	22.1115	0.754433	0.377217	−	0.926125i	$-0.376881\pi$
$859$	22.1115	0.754433	0.377217	+	0.926125i	$0.376881\pi$
$860$	11.0557	0.376997
$861$	0	0
$862$	−22.8328	−0.777689
$863$	48.3607	1.64622	0.823108	−	0.567884i	$-0.192238\pi$
$863$	48.3607	1.64622	0.823108	+	0.567884i	$0.192238\pi$
$864$	0	0
$865$	2.47214	0.0840551
$866$	18.7639	0.637624
$867$	0	0
$868$	2.00000	0.0678844
$869$	27.4164	0.930038
$870$	0	0
$871$	−44.3607	−1.50310
$872$	−9.52786	−0.322654
$873$	0	0
$874$	−0.763932	−0.0258404
$875$	−10.4721	−0.354023
$876$	0	0
$877$	−19.4164	−0.655646	−0.327823	−	0.944739i	$-0.606315\pi$
$877$	−19.4164	−0.655646	−0.327823	+	0.944739i	$0.606315\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	−6.47214	−0.218176
$881$	43.5967	1.46881	0.734406	−	0.678711i	$-0.237461\pi$
$881$	43.5967	1.46881	0.734406	+	0.678711i	$0.237461\pi$
$882$	0	0
$883$	−5.52786	−0.186027	−0.0930137	−	0.995665i	$-0.529650\pi$
$883$	−5.52786	−0.186027	−0.0930137	+	0.995665i	$0.529650\pi$
$884$	−18.4721	−0.621285
$885$	0	0
$886$	7.70820	0.258962
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	−3.70820	−0.124369
$890$	7.41641	0.248599
$891$	0	0
$892$	19.5279	0.653841
$893$	−8.94427	−0.299309
$894$	0	0
$895$	−27.0557	−0.904373
$896$	1.00000	0.0334077
$897$	0	0
$898$	32.8328	1.09565
$899$	0.944272	0.0314932
$900$	0	0
$901$	20.1378	0.670886
$902$	−31.4164	−1.04605
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0.944272	0.0313887
$906$	0	0
$907$	23.2361	0.771541	0.385770	−	0.922595i	$-0.373936\pi$
$907$	23.2361	0.771541	0.385770	+	0.922595i	$0.373936\pi$
$908$	−15.4164	−0.511611
$909$	0	0
$910$	4.00000	0.132599
$911$	−42.2492	−1.39978	−0.699890	−	0.714251i	$-0.746767\pi$
$911$	−42.2492	−1.39978	−0.699890	+	0.714251i	$0.746767\pi$
$912$	0	0
$913$	−52.3607	−1.73289
$914$	−35.3050	−1.16778
$915$	0	0
$916$	−18.9443	−0.625936
$917$	7.52786	0.248592
$918$	0	0
$919$	−12.3607	−0.407741	−0.203871	−	0.978998i	$-0.565352\pi$
$919$	−12.3607	−0.407741	−0.203871	+	0.978998i	$0.565352\pi$
$920$	0.944272	0.0311317
$921$	0	0
$922$	−1.81966	−0.0599273
$923$	−4.94427	−0.162743
$924$	0	0
$925$	5.30495	0.174426
$926$	41.3050	1.35736
$927$	0	0
$928$	0.472136	0.0154986
$929$	−42.0689	−1.38024	−0.690118	−	0.723697i	$-0.742441\pi$
$929$	−42.0689	−1.38024	−0.690118	+	0.723697i	$0.742441\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	14.4721	0.474051
$933$	0	0
$934$	19.8885	0.650773
$935$	−36.9443	−1.20821
$936$	0	0
$937$	−46.9443	−1.53360	−0.766801	−	0.641885i	$-0.778153\pi$
$937$	−46.9443	−1.53360	−0.766801	+	0.641885i	$0.778153\pi$
$938$	13.7082	0.447589
$939$	0	0
$940$	11.0557	0.360598
$941$	35.5279	1.15818	0.579088	−	0.815265i	$-0.303409\pi$
$941$	35.5279	1.15818	0.579088	+	0.815265i	$0.303409\pi$
$942$	0	0
$943$	4.58359	0.149262
$944$	8.00000	0.260378
$945$	0	0
$946$	46.8328	1.52267
$947$	−17.0132	−0.552853	−0.276427	−	0.961035i	$-0.589150\pi$
$947$	−17.0132	−0.552853	−0.276427	+	0.961035i	$0.589150\pi$
$948$	0	0
$949$	11.4164	0.370592
$950$	3.47214	0.112651
$951$	0	0
$952$	5.70820	0.185004
$953$	42.9443	1.39110	0.695551	−	0.718477i	$-0.255160\pi$
$953$	42.9443	1.39110	0.695551	+	0.718477i	$0.255160\pi$
$954$	0	0
$955$	−15.7771	−0.510535
$956$	3.81966	0.123537
$957$	0	0
$958$	34.4721	1.11374
$959$	−11.4164	−0.368655
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−4.94427	−0.159410
$963$	0	0
$964$	−5.23607	−0.168642
$965$	22.2492	0.716228
$966$	0	0
$967$	30.8328	0.991517	0.495758	−	0.868461i	$-0.334890\pi$
$967$	30.8328	0.991517	0.495758	+	0.868461i	$0.334890\pi$
$968$	−16.4164	−0.527643
$969$	0	0
$970$	11.4164	0.366559
$971$	34.4721	1.10626	0.553132	−	0.833094i	$-0.313433\pi$
$971$	34.4721	1.10626	0.553132	+	0.833094i	$0.313433\pi$
$972$	0	0
$973$	13.8885	0.445246
$974$	28.0689	0.899385
$975$	0	0
$976$	8.47214	0.271186
$977$	−27.8885	−0.892234	−0.446117	−	0.894975i	$-0.647193\pi$
$977$	−27.8885	−0.892234	−0.446117	+	0.894975i	$0.647193\pi$
$978$	0	0
$979$	31.4164	1.00407
$980$	−1.23607	−0.0394847
$981$	0	0
$982$	0.291796	0.00931159
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	−8.58359	−0.273496
$986$	2.69505	0.0858278
$987$	0	0
$988$	−3.23607	−0.102953
$989$	−6.83282	−0.217271
$990$	0	0
$991$	29.2361	0.928714	0.464357	−	0.885648i	$-0.346285\pi$
$991$	29.2361	0.928714	0.464357	+	0.885648i	$0.346285\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	1.52786	0.0484609
$995$	14.1115	0.447363
$996$	0	0
$997$	37.1935	1.17793	0.588965	−	0.808159i	$-0.299536\pi$
$997$	37.1935	1.17793	0.588965	+	0.808159i	$0.299536\pi$
$998$	−25.3050	−0.801014
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.s.1.1	✓	2
3.2	odd	2		2394.2.a.v.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.s.1.1	✓	2	1.1	even	1	trivial
2394.2.a.v.1.2	yes	2	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.23607 q^{10} +5.23607 q^{11} -3.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.70820 q^{17} +1.00000 q^{19} -1.23607 q^{20} -5.23607 q^{22} +0.763932 q^{23} -3.47214 q^{25} +3.23607 q^{26} -1.00000 q^{28} -0.472136 q^{29} -2.00000 q^{31} -1.00000 q^{32} -5.70820 q^{34} +1.23607 q^{35} -1.52786 q^{37} -1.00000 q^{38} +1.23607 q^{40} +6.00000 q^{41} -8.94427 q^{43} +5.23607 q^{44} -0.763932 q^{46} -8.94427 q^{47} +1.00000 q^{49} +3.47214 q^{50} -3.23607 q^{52} +3.52786 q^{53} -6.47214 q^{55} +1.00000 q^{56} +0.472136 q^{58} +8.00000 q^{59} +8.47214 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +13.7082 q^{67} +5.70820 q^{68} -1.23607 q^{70} +1.52786 q^{71} -3.52786 q^{73} +1.52786 q^{74} +1.00000 q^{76} -5.23607 q^{77} +5.23607 q^{79} -1.23607 q^{80} -6.00000 q^{82} -10.0000 q^{83} -7.05573 q^{85} +8.94427 q^{86} -5.23607 q^{88} +6.00000 q^{89} +3.23607 q^{91} +0.763932 q^{92} +8.94427 q^{94} -1.23607 q^{95} +9.23607 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 12 q^{37} - 2 q^{38} - 2 q^{40} + 12 q^{41} + 6 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 16 q^{53} - 4 q^{55} + 2 q^{56} - 8 q^{58} + 16 q^{59} + 8 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{65} + 14 q^{67} - 2 q^{68} + 2 q^{70} + 12 q^{71} - 16 q^{73} + 12 q^{74} + 2 q^{76} - 6 q^{77} + 6 q^{79} + 2 q^{80} - 12 q^{82} - 20 q^{83} - 32 q^{85} - 6 q^{88} + 12 q^{89} + 2 q^{91} + 6 q^{92} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^19 + 2 * q^20 - 6 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 - 12 * q^37 - 2 * q^38 - 2 * q^40 + 12 * q^41 + 6 * q^44 - 6 * q^46 + 2 * q^49 - 2 * q^50 - 2 * q^52 + 16 * q^53 - 4 * q^55 + 2 * q^56 - 8 * q^58 + 16 * q^59 + 8 * q^61 + 4 * q^62 + 2 * q^64 + 8 * q^65 + 14 * q^67 - 2 * q^68 + 2 * q^70 + 12 * q^71 - 16 * q^73 + 12 * q^74 + 2 * q^76 - 6 * q^77 + 6 * q^79 + 2 * q^80 - 12 * q^82 - 20 * q^83 - 32 * q^85 - 6 * q^88 + 12 * q^89 + 2 * q^91 + 6 * q^92 + 2 * q^95 + 14 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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