LMFDB - Embedded newform 8925.2.a.cu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8925,2,Mod(1,8925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8925 = 3 \cdot 5^{2} \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8925.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:	$71.2664838040$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{14} - 3 x^{13} - 18 x^{12} + 54 x^{11} + 124 x^{10} - 366 x^{9} - 416 x^{8} + 1164 x^{7} + 727 x^{6} + \cdots - 40$ x^14 - 3x^13 - 18x^12 + 54x^11 + 124x^10 - 366x^9 - 416x^8 + 1164x^7 + 727x^6 - 1783x^5 - 662x^4 + 1242x^3 + 284x^2 - 300*x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 1785)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.80343$ of defining polynomial
Character	$\chi$	$=$	8925.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80343 q^{2} -1.00000 q^{3} +5.85923 q^{4} +2.80343 q^{6} -1.00000 q^{7} -10.8191 q^{8} +1.00000 q^{9} +3.37847 q^{11} -5.85923 q^{12} +2.98612 q^{13} +2.80343 q^{14} +18.6122 q^{16} +1.00000 q^{17} -2.80343 q^{18} -8.20136 q^{19} +1.00000 q^{21} -9.47132 q^{22} +5.84024 q^{23} +10.8191 q^{24} -8.37138 q^{26} -1.00000 q^{27} -5.85923 q^{28} +6.80762 q^{29} -0.491153 q^{31} -30.5397 q^{32} -3.37847 q^{33} -2.80343 q^{34} +5.85923 q^{36} -2.87265 q^{37} +22.9920 q^{38} -2.98612 q^{39} +8.72402 q^{41} -2.80343 q^{42} -7.93970 q^{43} +19.7953 q^{44} -16.3727 q^{46} -5.43126 q^{47} -18.6122 q^{48} +1.00000 q^{49} -1.00000 q^{51} +17.4964 q^{52} +0.419890 q^{53} +2.80343 q^{54} +10.8191 q^{56} +8.20136 q^{57} -19.0847 q^{58} -4.91201 q^{59} -2.97619 q^{61} +1.37692 q^{62} -1.00000 q^{63} +48.3918 q^{64} +9.47132 q^{66} -16.0362 q^{67} +5.85923 q^{68} -5.84024 q^{69} -9.40560 q^{71} -10.8191 q^{72} -8.53815 q^{73} +8.05329 q^{74} -48.0537 q^{76} -3.37847 q^{77} +8.37138 q^{78} +7.24024 q^{79} +1.00000 q^{81} -24.4572 q^{82} -0.00138848 q^{83} +5.85923 q^{84} +22.2584 q^{86} -6.80762 q^{87} -36.5521 q^{88} +7.58835 q^{89} -2.98612 q^{91} +34.2194 q^{92} +0.491153 q^{93} +15.2262 q^{94} +30.5397 q^{96} +10.1653 q^{97} -2.80343 q^{98} +3.37847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$14 q - 3 q^{2} - 14 q^{3} + 17 q^{4} + 3 q^{6} - 14 q^{7} - 15 q^{8} + 14 q^{9} + 4 q^{11} - 17 q^{12} + q^{13} + 3 q^{14} + 19 q^{16} + 14 q^{17} - 3 q^{18} - 6 q^{19} + 14 q^{21} - 12 q^{22} - 7 q^{23}+ \cdots + 4 q^{99}+O(q^{100})$ 14 * q - 3 * q^2 - 14 * q^3 + 17 * q^4 + 3 * q^6 - 14 * q^7 - 15 * q^8 + 14 * q^9 + 4 * q^11 - 17 * q^12 + q^13 + 3 * q^14 + 19 * q^16 + 14 * q^17 - 3 * q^18 - 6 * q^19 + 14 * q^21 - 12 * q^22 - 7 * q^23 + 15 * q^24 - 2 * q^26 - 14 * q^27 - 17 * q^28 + 20 * q^29 - 7 * q^31 - 33 * q^32 - 4 * q^33 - 3 * q^34 + 17 * q^36 - 9 * q^37 - 18 * q^38 - q^39 + 25 * q^41 - 3 * q^42 - 28 * q^43 + 24 * q^44 - 10 * q^46 - 29 * q^47 - 19 * q^48 + 14 * q^49 - 14 * q^51 - 4 * q^52 - 26 * q^53 + 3 * q^54 + 15 * q^56 + 6 * q^57 - 2 * q^58 + 14 * q^59 - 9 * q^61 - 28 * q^62 - 14 * q^63 + 27 * q^64 + 12 * q^66 - 32 * q^67 + 17 * q^68 + 7 * q^69 + 2 * q^71 - 15 * q^72 - 18 * q^73 + 64 * q^74 - 42 * q^76 - 4 * q^77 + 2 * q^78 - 4 * q^79 + 14 * q^81 - 42 * q^82 - 51 * q^83 + 17 * q^84 + 44 * q^86 - 20 * q^87 - 32 * q^88 + 38 * q^89 - q^91 + 16 * q^92 + 7 * q^93 + 8 * q^94 + 33 * q^96 - 14 * q^97 - 3 * q^98 + 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.80343	−1.98233	−0.991163	−	0.132649i	$-0.957652\pi$
$2$	−2.80343	−1.98233	−0.991163	+	0.132649i	$0.957652\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.85923	2.92962
$5$	0	0
$5$	0	0
$6$	2.80343	1.14450
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−10.8191	−3.82513
$9$	1.00000	0.333333
$10$	0	0
$11$	3.37847	1.01865	0.509324	−	0.860575i	$-0.329895\pi$
$11$	3.37847	1.01865	0.509324	+	0.860575i	$0.329895\pi$
$12$	−5.85923	−1.69142
$13$	2.98612	0.828200	0.414100	−	0.910231i	$-0.364096\pi$
$13$	2.98612	0.828200	0.414100	+	0.910231i	$0.364096\pi$
$14$	2.80343	0.749249
$15$	0	0
$16$	18.6122	4.65304
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.80343	−0.660775
$19$	−8.20136	−1.88152	−0.940761	−	0.339072i	$-0.889887\pi$
$19$	−8.20136	−1.88152	−0.940761	+	0.339072i	$0.889887\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−9.47132	−2.01929
$23$	5.84024	1.21778	0.608888	−	0.793257i	$-0.291616\pi$
$23$	5.84024	1.21778	0.608888	+	0.793257i	$0.291616\pi$
$24$	10.8191	2.20844
$25$	0	0
$26$	−8.37138	−1.64176
$27$	−1.00000	−0.192450
$28$	−5.85923	−1.10729
$29$	6.80762	1.26414	0.632072	−	0.774910i	$-0.282205\pi$
$29$	6.80762	1.26414	0.632072	+	0.774910i	$0.282205\pi$
$30$	0	0
$31$	−0.491153	−0.0882137	−0.0441069	−	0.999027i	$-0.514044\pi$
$31$	−0.491153	−0.0882137	−0.0441069	+	0.999027i	$0.514044\pi$
$32$	−30.5397	−5.39871
$33$	−3.37847	−0.588117
$34$	−2.80343	−0.480785
$35$	0	0
$36$	5.85923	0.976539
$37$	−2.87265	−0.472261	−0.236131	−	0.971721i	$-0.575879\pi$
$37$	−2.87265	−0.472261	−0.236131	+	0.971721i	$0.575879\pi$
$38$	22.9920	3.72979
$39$	−2.98612	−0.478161
$40$	0	0
$41$	8.72402	1.36246	0.681232	−	0.732068i	$-0.261445\pi$
$41$	8.72402	1.36246	0.681232	+	0.732068i	$0.261445\pi$
$42$	−2.80343	−0.432579
$43$	−7.93970	−1.21079	−0.605396	−	0.795924i	$-0.706985\pi$
$43$	−7.93970	−1.21079	−0.605396	+	0.795924i	$0.706985\pi$
$44$	19.7953	2.98425
$45$	0	0
$46$	−16.3727	−2.41403
$47$	−5.43126	−0.792231	−0.396115	−	0.918201i	$-0.629642\pi$
$47$	−5.43126	−0.792231	−0.396115	+	0.918201i	$0.629642\pi$
$48$	−18.6122	−2.68643
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	17.4964	2.42631
$53$	0.419890	0.0576763	0.0288381	−	0.999584i	$-0.490819\pi$
$53$	0.419890	0.0576763	0.0288381	+	0.999584i	$0.490819\pi$
$54$	2.80343	0.381499
$55$	0	0
$56$	10.8191	1.44576
$57$	8.20136	1.08630
$58$	−19.0847	−2.50595
$59$	−4.91201	−0.639490	−0.319745	−	0.947504i	$-0.603597\pi$
$59$	−4.91201	−0.639490	−0.319745	+	0.947504i	$0.603597\pi$
$60$	0	0
$61$	−2.97619	−0.381062	−0.190531	−	0.981681i	$-0.561021\pi$
$61$	−2.97619	−0.381062	−0.190531	+	0.981681i	$0.561021\pi$
$62$	1.37692	0.174868
$63$	−1.00000	−0.125988
$64$	48.3918	6.04897
$65$	0	0
$66$	9.47132	1.16584
$67$	−16.0362	−1.95913	−0.979566	−	0.201124i	$-0.935541\pi$
$67$	−16.0362	−1.95913	−0.979566	+	0.201124i	$0.935541\pi$
$68$	5.85923	0.710537
$69$	−5.84024	−0.703083
$70$	0	0
$71$	−9.40560	−1.11624	−0.558120	−	0.829761i	$-0.688477\pi$
$71$	−9.40560	−1.11624	−0.558120	+	0.829761i	$0.688477\pi$
$72$	−10.8191	−1.27504
$73$	−8.53815	−0.999315	−0.499657	−	0.866223i	$-0.666541\pi$
$73$	−8.53815	−0.999315	−0.499657	+	0.866223i	$0.666541\pi$
$74$	8.05329	0.936176
$75$	0	0
$76$	−48.0537	−5.51214
$77$	−3.37847	−0.385013
$78$	8.37138	0.947872
$79$	7.24024	0.814590	0.407295	−	0.913297i	$-0.366472\pi$
$79$	7.24024	0.814590	0.407295	+	0.913297i	$0.366472\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−24.4572	−2.70085
$83$	−0.00138848	−0.000152405 0	−7.62027e−5	−	1.00000i	$-0.500024\pi$
$83$	−0.00138848	−0.000152405 0	−7.62027e−5		1.00000i	$0.500024\pi$
$84$	5.85923	0.639295
$85$	0	0
$86$	22.2584	2.40019
$87$	−6.80762	−0.729854
$88$	−36.5521	−3.89646
$89$	7.58835	0.804364	0.402182	−	0.915560i	$-0.368252\pi$
$89$	7.58835	0.804364	0.402182	+	0.915560i	$0.368252\pi$
$90$	0	0
$91$	−2.98612	−0.313030
$92$	34.2194	3.56761
$93$	0.491153	0.0509302
$94$	15.2262	1.57046
$95$	0	0
$96$	30.5397	3.11695
$97$	10.1653	1.03213	0.516064	−	0.856550i	$-0.327397\pi$
$97$	10.1653	1.03213	0.516064	+	0.856550i	$0.327397\pi$
$98$	−2.80343	−0.283189
$99$	3.37847	0.339549
$100$	0	0
$101$	−13.7816	−1.37132	−0.685659	−	0.727923i	$-0.740486\pi$
$101$	−13.7816	−1.37132	−0.685659	+	0.727923i	$0.740486\pi$
$102$	2.80343	0.277581
$103$	3.83305	0.377681	0.188841	−	0.982008i	$-0.439527\pi$
$103$	3.83305	0.377681	0.188841	+	0.982008i	$0.439527\pi$
$104$	−32.3071	−3.16797
$105$	0	0
$106$	−1.17713	−0.114333
$107$	3.76661	0.364132	0.182066	−	0.983286i	$-0.441722\pi$
$107$	3.76661	0.364132	0.182066	+	0.983286i	$0.441722\pi$
$108$	−5.85923	−0.563805
$109$	−13.1888	−1.26326	−0.631628	−	0.775272i	$-0.717613\pi$
$109$	−13.1888	−1.26326	−0.631628	+	0.775272i	$0.717613\pi$
$110$	0	0
$111$	2.87265	0.272660
$112$	−18.6122	−1.75868
$113$	15.3505	1.44405	0.722025	−	0.691867i	$-0.243212\pi$
$113$	15.3505	1.44405	0.722025	+	0.691867i	$0.243212\pi$
$114$	−22.9920	−2.15339
$115$	0	0
$116$	39.8875	3.70346
$117$	2.98612	0.276067
$118$	13.7705	1.26768
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	0.414084	0.0376440
$122$	8.34354	0.755389
$123$	−8.72402	−0.786618
$124$	−2.87778	−0.258432
$125$	0	0
$126$	2.80343	0.249750
$127$	11.2110	0.994815	0.497407	−	0.867517i	$-0.334285\pi$
$127$	11.2110	0.994815	0.497407	+	0.867517i	$0.334285\pi$
$128$	−74.5836	−6.59232
$129$	7.93970	0.699052
$130$	0	0
$131$	−9.55141	−0.834510	−0.417255	−	0.908789i	$-0.637008\pi$
$131$	−9.55141	−0.834510	−0.417255	+	0.908789i	$0.637008\pi$
$132$	−19.7953	−1.72296
$133$	8.20136	0.711148
$134$	44.9564	3.88364
$135$	0	0
$136$	−10.8191	−0.927731
$137$	12.6457	1.08040	0.540199	−	0.841537i	$-0.318349\pi$
$137$	12.6457	1.08040	0.540199	+	0.841537i	$0.318349\pi$
$138$	16.3727	1.39374
$139$	−8.08133	−0.685449	−0.342725	−	0.939436i	$-0.611350\pi$
$139$	−8.08133	−0.685449	−0.342725	+	0.939436i	$0.611350\pi$
$140$	0	0
$141$	5.43126	0.457395
$142$	26.3680	2.21275
$143$	10.0885	0.843644
$144$	18.6122	1.55101
$145$	0	0
$146$	23.9361	1.98097
$147$	−1.00000	−0.0824786
$148$	−16.8315	−1.38354
$149$	−14.1370	−1.15815	−0.579074	−	0.815275i	$-0.696586\pi$
$149$	−14.1370	−1.15815	−0.579074	+	0.815275i	$0.696586\pi$
$150$	0	0
$151$	−13.7500	−1.11896	−0.559479	−	0.828845i	$-0.688999\pi$
$151$	−13.7500	−1.11896	−0.559479	+	0.828845i	$0.688999\pi$
$152$	88.7314	7.19706
$153$	1.00000	0.0808452
$154$	9.47132	0.763221
$155$	0	0
$156$	−17.4964	−1.40083
$157$	−9.26551	−0.739468	−0.369734	−	0.929138i	$-0.620551\pi$
$157$	−9.26551	−0.739468	−0.369734	+	0.929138i	$0.620551\pi$
$158$	−20.2975	−1.61478
$159$	−0.419890	−0.0332994
$160$	0	0
$161$	−5.84024	−0.460276
$162$	−2.80343	−0.220258
$163$	6.61304	0.517973	0.258987	−	0.965881i	$-0.416611\pi$
$163$	6.61304	0.517973	0.258987	+	0.965881i	$0.416611\pi$
$164$	51.1161	3.99150
$165$	0	0
$166$	0.00389251	0.000302117 0
$167$	−5.97930	−0.462692	−0.231346	−	0.972872i	$-0.574313\pi$
$167$	−5.97930	−0.462692	−0.231346	+	0.972872i	$0.574313\pi$
$168$	−10.8191	−0.834712
$169$	−4.08310	−0.314085
$170$	0	0
$171$	−8.20136	−0.627174
$172$	−46.5206	−3.54716
$173$	9.67349	0.735462	0.367731	−	0.929932i	$-0.380135\pi$
$173$	9.67349	0.735462	0.367731	+	0.929932i	$0.380135\pi$
$174$	19.0847	1.44681
$175$	0	0
$176$	62.8807	4.73981
$177$	4.91201	0.369210
$178$	−21.2734	−1.59451
$179$	−16.1835	−1.20961	−0.604807	−	0.796372i	$-0.706750\pi$
$179$	−16.1835	−1.20961	−0.604807	+	0.796372i	$0.706750\pi$
$180$	0	0
$181$	1.01069	0.0751243	0.0375621	−	0.999294i	$-0.488041\pi$
$181$	1.01069	0.0751243	0.0375621	+	0.999294i	$0.488041\pi$
$182$	8.37138	0.620528
$183$	2.97619	0.220006
$184$	−63.1862	−4.65815
$185$	0	0
$186$	−1.37692	−0.100960
$187$	3.37847	0.247058
$188$	−31.8230	−2.32093
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.2168	0.811616	0.405808	−	0.913958i	$-0.366990\pi$
$191$	11.2168	0.811616	0.405808	+	0.913958i	$0.366990\pi$
$192$	−48.3918	−3.49237
$193$	−13.0305	−0.937955	−0.468978	−	0.883210i	$-0.655378\pi$
$193$	−13.0305	−0.937955	−0.468978	+	0.883210i	$0.655378\pi$
$194$	−28.4977	−2.04601
$195$	0	0
$196$	5.85923	0.418517
$197$	−3.36117	−0.239473	−0.119737	−	0.992806i	$-0.538205\pi$
$197$	−3.36117	−0.239473	−0.119737	+	0.992806i	$0.538205\pi$
$198$	−9.47132	−0.673098
$199$	3.11797	0.221027	0.110513	−	0.993875i	$-0.464751\pi$
$199$	3.11797	0.221027	0.110513	+	0.993875i	$0.464751\pi$
$200$	0	0
$201$	16.0362	1.13111
$202$	38.6357	2.71840
$203$	−6.80762	−0.477802
$204$	−5.85923	−0.410228
$205$	0	0
$206$	−10.7457	−0.748688
$207$	5.84024	0.405925
$208$	55.5781	3.85365
$209$	−27.7081	−1.91661
$210$	0	0
$211$	1.18361	0.0814831	0.0407416	−	0.999170i	$-0.487028\pi$
$211$	1.18361	0.0814831	0.0407416	+	0.999170i	$0.487028\pi$
$212$	2.46023	0.168969
$213$	9.40560	0.644461
$214$	−10.5594	−0.721829
$215$	0	0
$216$	10.8191	0.736147
$217$	0.491153	0.0333417
$218$	36.9739	2.50419
$219$	8.53815	0.576955
$220$	0	0
$221$	2.98612	0.200868
$222$	−8.05329	−0.540501
$223$	−0.645242	−0.0432086	−0.0216043	−	0.999767i	$-0.506877\pi$
$223$	−0.645242	−0.0432086	−0.0216043	+	0.999767i	$0.506877\pi$
$224$	30.5397	2.04052
$225$	0	0
$226$	−43.0340	−2.86258
$227$	−20.6531	−1.37079	−0.685397	−	0.728170i	$-0.740371\pi$
$227$	−20.6531	−1.37079	−0.685397	+	0.728170i	$0.740371\pi$
$228$	48.0537	3.18243
$229$	8.46699	0.559515	0.279757	−	0.960071i	$-0.409746\pi$
$229$	8.46699	0.559515	0.279757	+	0.960071i	$0.409746\pi$
$230$	0	0
$231$	3.37847	0.222287
$232$	−73.6524	−4.83552
$233$	24.4882	1.60427	0.802136	−	0.597141i	$-0.203697\pi$
$233$	24.4882	1.60427	0.802136	+	0.597141i	$0.203697\pi$
$234$	−8.37138	−0.547254
$235$	0	0
$236$	−28.7806	−1.87346
$237$	−7.24024	−0.470304
$238$	2.80343	0.181720
$239$	−21.1704	−1.36940	−0.684700	−	0.728825i	$-0.740067\pi$
$239$	−21.1704	−1.36940	−0.684700	+	0.728825i	$0.740067\pi$
$240$	0	0
$241$	9.48083	0.610714	0.305357	−	0.952238i	$-0.401224\pi$
$241$	9.48083	0.610714	0.305357	+	0.952238i	$0.401224\pi$
$242$	−1.16086	−0.0746226
$243$	−1.00000	−0.0641500
$244$	−17.4382	−1.11637
$245$	0	0
$246$	24.4572	1.55933
$247$	−24.4902	−1.55828
$248$	5.31384	0.337429
$249$	0.00138848	8.79912e−5 0
$250$	0	0
$251$	−2.33881	−0.147624	−0.0738121	−	0.997272i	$-0.523517\pi$
$251$	−2.33881	−0.147624	−0.0738121	+	0.997272i	$0.523517\pi$
$252$	−5.85923	−0.369097
$253$	19.7311	1.24048
$254$	−31.4293	−1.97205
$255$	0	0
$256$	112.306	7.01916
$257$	−22.9292	−1.43028	−0.715142	−	0.698979i	$-0.753638\pi$
$257$	−22.9292	−1.43028	−0.715142	+	0.698979i	$0.753638\pi$
$258$	−22.2584	−1.38575
$259$	2.87265	0.178498
$260$	0	0
$261$	6.80762	0.421381
$262$	26.7767	1.65427
$263$	6.17182	0.380571	0.190285	−	0.981729i	$-0.439059\pi$
$263$	6.17182	0.380571	0.190285	+	0.981729i	$0.439059\pi$
$264$	36.5521	2.24962
$265$	0	0
$266$	−22.9920	−1.40973
$267$	−7.58835	−0.464400
$268$	−93.9598	−5.73951
$269$	−21.9597	−1.33891	−0.669453	−	0.742855i	$-0.733471\pi$
$269$	−21.9597	−1.33891	−0.669453	+	0.742855i	$0.733471\pi$
$270$	0	0
$271$	−29.7006	−1.80418	−0.902091	−	0.431546i	$-0.857968\pi$
$271$	−29.7006	−1.80418	−0.902091	+	0.431546i	$0.857968\pi$
$272$	18.6122	1.12853
$273$	2.98612	0.180728
$274$	−35.4515	−2.14170
$275$	0	0
$276$	−34.2194	−2.05976
$277$	0.330767	0.0198739	0.00993694	−	0.999951i	$-0.496837\pi$
$277$	0.330767	0.0198739	0.00993694	+	0.999951i	$0.496837\pi$
$278$	22.6555	1.35878
$279$	−0.491153	−0.0294046
$280$	0	0
$281$	−8.44777	−0.503952	−0.251976	−	0.967733i	$-0.581080\pi$
$281$	−8.44777	−0.503952	−0.251976	+	0.967733i	$0.581080\pi$
$282$	−15.2262	−0.906706
$283$	−20.5216	−1.21989	−0.609943	−	0.792446i	$-0.708807\pi$
$283$	−20.5216	−1.21989	−0.609943	+	0.792446i	$0.708807\pi$
$284$	−55.1096	−3.27015
$285$	0	0
$286$	−28.2825	−1.67238
$287$	−8.72402	−0.514963
$288$	−30.5397	−1.79957
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.1653	−0.595899
$292$	−50.0270	−2.92761
$293$	17.9089	1.04625	0.523124	−	0.852257i	$-0.324766\pi$
$293$	17.9089	1.04625	0.523124	+	0.852257i	$0.324766\pi$
$294$	2.80343	0.163500
$295$	0	0
$296$	31.0795	1.80646
$297$	−3.37847	−0.196039
$298$	39.6321	2.29583
$299$	17.4397	1.00856
$300$	0	0
$301$	7.93970	0.457637
$302$	38.5471	2.21814
$303$	13.7816	0.791731
$304$	−152.645	−8.75479
$305$	0	0
$306$	−2.80343	−0.160262
$307$	19.8061	1.13039	0.565196	−	0.824957i	$-0.308801\pi$
$307$	19.8061	1.13039	0.565196	+	0.824957i	$0.308801\pi$
$308$	−19.7953	−1.12794
$309$	−3.83305	−0.218054
$310$	0	0
$311$	−4.30912	−0.244348	−0.122174	−	0.992509i	$-0.538987\pi$
$311$	−4.30912	−0.244348	−0.122174	+	0.992509i	$0.538987\pi$
$312$	32.3071	1.82903
$313$	27.4270	1.55026	0.775131	−	0.631800i	$-0.217684\pi$
$313$	27.4270	1.55026	0.775131	+	0.631800i	$0.217684\pi$
$314$	25.9752	1.46587
$315$	0	0
$316$	42.4223	2.38644
$317$	22.6461	1.27193	0.635967	−	0.771716i	$-0.280601\pi$
$317$	22.6461	1.27193	0.635967	+	0.771716i	$0.280601\pi$
$318$	1.17713	0.0660103
$319$	22.9994	1.28772
$320$	0	0
$321$	−3.76661	−0.210232
$322$	16.3727	0.912417
$323$	−8.20136	−0.456336
$324$	5.85923	0.325513
$325$	0	0
$326$	−18.5392	−1.02679
$327$	13.1888	0.729341
$328$	−94.3861	−5.21160
$329$	5.43126	0.299435
$330$	0	0
$331$	18.3691	1.00966	0.504829	−	0.863219i	$-0.331555\pi$
$331$	18.3691	1.00966	0.504829	+	0.863219i	$0.331555\pi$
$332$	−0.00813542	−0.000446489 0
$333$	−2.87265	−0.157420
$334$	16.7626	0.917206
$335$	0	0
$336$	18.6122	1.01538
$337$	22.7393	1.23869	0.619345	−	0.785119i	$-0.287398\pi$
$337$	22.7393	1.23869	0.619345	+	0.785119i	$0.287398\pi$
$338$	11.4467	0.622618
$339$	−15.3505	−0.833722
$340$	0	0
$341$	−1.65935	−0.0898588
$342$	22.9920	1.24326
$343$	−1.00000	−0.0539949
$344$	85.9004	4.63144
$345$	0	0
$346$	−27.1190	−1.45793
$347$	5.38634	0.289154	0.144577	−	0.989494i	$-0.453818\pi$
$347$	5.38634	0.289154	0.144577	+	0.989494i	$0.453818\pi$
$348$	−39.8875	−2.13819
$349$	−4.94714	−0.264814	−0.132407	−	0.991195i	$-0.542271\pi$
$349$	−4.94714	−0.264814	−0.132407	+	0.991195i	$0.542271\pi$
$350$	0	0
$351$	−2.98612	−0.159387
$352$	−103.178	−5.49939
$353$	−24.9950	−1.33035	−0.665175	−	0.746687i	$-0.731643\pi$
$353$	−24.9950	−1.33035	−0.665175	+	0.746687i	$0.731643\pi$
$354$	−13.7705	−0.731894
$355$	0	0
$356$	44.4619	2.35648
$357$	1.00000	0.0529256
$358$	45.3694	2.39785
$359$	30.5447	1.61209	0.806045	−	0.591855i	$-0.201604\pi$
$359$	30.5447	1.61209	0.806045	+	0.591855i	$0.201604\pi$
$360$	0	0
$361$	48.2623	2.54012
$362$	−2.83341	−0.148921
$363$	−0.414084	−0.0217338
$364$	−17.4964	−0.917059
$365$	0	0
$366$	−8.34354	−0.436124
$367$	−1.26006	−0.0657744	−0.0328872	−	0.999459i	$-0.510470\pi$
$367$	−1.26006	−0.0657744	−0.0328872	+	0.999459i	$0.510470\pi$
$368$	108.700	5.66636
$369$	8.72402	0.454154
$370$	0	0
$371$	−0.419890	−0.0217996
$372$	2.87778	0.149206
$373$	−20.5582	−1.06447	−0.532233	−	0.846598i	$-0.678647\pi$
$373$	−20.5582	−1.06447	−0.532233	+	0.846598i	$0.678647\pi$
$374$	−9.47132	−0.489750
$375$	0	0
$376$	58.7614	3.03039
$377$	20.3284	1.04696
$378$	−2.80343	−0.144193
$379$	−3.33561	−0.171339	−0.0856693	−	0.996324i	$-0.527303\pi$
$379$	−3.33561	−0.171339	−0.0856693	+	0.996324i	$0.527303\pi$
$380$	0	0
$381$	−11.2110	−0.574357
$382$	−31.4454	−1.60889
$383$	1.72119	0.0879485	0.0439742	−	0.999033i	$-0.485998\pi$
$383$	1.72119	0.0879485	0.0439742	+	0.999033i	$0.485998\pi$
$384$	74.5836	3.80608
$385$	0	0
$386$	36.5301	1.85933
$387$	−7.93970	−0.403598
$388$	59.5608	3.02374
$389$	−20.4186	−1.03526	−0.517632	−	0.855603i	$-0.673186\pi$
$389$	−20.4186	−1.03526	−0.517632	+	0.855603i	$0.673186\pi$
$390$	0	0
$391$	5.84024	0.295354
$392$	−10.8191	−0.546447
$393$	9.55141	0.481805
$394$	9.42281	0.474714
$395$	0	0
$396$	19.7953	0.994750
$397$	−7.58930	−0.380896	−0.190448	−	0.981697i	$-0.560994\pi$
$397$	−7.58930	−0.380896	−0.190448	+	0.981697i	$0.560994\pi$
$398$	−8.74101	−0.438147
$399$	−8.20136	−0.410582
$400$	0	0
$401$	−1.65683	−0.0827380	−0.0413690	−	0.999144i	$-0.513172\pi$
$401$	−1.65683	−0.0827380	−0.0413690	+	0.999144i	$0.513172\pi$
$402$	−44.9564	−2.24222
$403$	−1.46664	−0.0730586
$404$	−80.7495	−4.01744
$405$	0	0
$406$	19.0847	0.947159
$407$	−9.70518	−0.481068
$408$	10.8191	0.535625
$409$	−6.04520	−0.298916	−0.149458	−	0.988768i	$-0.547753\pi$
$409$	−6.04520	−0.298916	−0.149458	+	0.988768i	$0.547753\pi$
$410$	0	0
$411$	−12.6457	−0.623769
$412$	22.4587	1.10646
$413$	4.91201	0.241704
$414$	−16.3727	−0.804676
$415$	0	0
$416$	−91.1952	−4.47121
$417$	8.08133	0.395744
$418$	77.6777	3.79934
$419$	−2.06017	−0.100646	−0.0503230	−	0.998733i	$-0.516025\pi$
$419$	−2.06017	−0.100646	−0.0503230	+	0.998733i	$0.516025\pi$
$420$	0	0
$421$	9.05757	0.441439	0.220719	−	0.975337i	$-0.429159\pi$
$421$	9.05757	0.441439	0.220719	+	0.975337i	$0.429159\pi$
$422$	−3.31817	−0.161526
$423$	−5.43126	−0.264077
$424$	−4.54283	−0.220619
$425$	0	0
$426$	−26.3680	−1.27753
$427$	2.97619	0.144028
$428$	22.0695	1.06677
$429$	−10.0885	−0.487078
$430$	0	0
$431$	13.5639	0.653348	0.326674	−	0.945137i	$-0.394072\pi$
$431$	13.5639	0.653348	0.326674	+	0.945137i	$0.394072\pi$
$432$	−18.6122	−0.895478
$433$	5.31344	0.255348	0.127674	−	0.991816i	$-0.459249\pi$
$433$	5.31344	0.255348	0.127674	+	0.991816i	$0.459249\pi$
$434$	−1.37692	−0.0660940
$435$	0	0
$436$	−77.2762	−3.70086
$437$	−47.8979	−2.29127
$438$	−23.9361	−1.14371
$439$	22.5460	1.07606	0.538030	−	0.842926i	$-0.319169\pi$
$439$	22.5460	1.07606	0.538030	+	0.842926i	$0.319169\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−8.37138	−0.398186
$443$	−14.2922	−0.679041	−0.339521	−	0.940599i	$-0.610265\pi$
$443$	−14.2922	−0.679041	−0.339521	+	0.940599i	$0.610265\pi$
$444$	16.8315	0.798790
$445$	0	0
$446$	1.80889	0.0856536
$447$	14.1370	0.668657
$448$	−48.3918	−2.28630
$449$	2.03457	0.0960175	0.0480087	−	0.998847i	$-0.484712\pi$
$449$	2.03457	0.0960175	0.0480087	+	0.998847i	$0.484712\pi$
$450$	0	0
$451$	29.4739	1.38787
$452$	89.9419	4.23051
$453$	13.7500	0.646030
$454$	57.8995	2.71736
$455$	0	0
$456$	−88.7314	−4.15523
$457$	−6.67114	−0.312063	−0.156031	−	0.987752i	$-0.549870\pi$
$457$	−6.67114	−0.312063	−0.156031	+	0.987752i	$0.549870\pi$
$458$	−23.7366	−1.10914
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	33.1967	1.54612	0.773062	−	0.634330i	$-0.218724\pi$
$461$	33.1967	1.54612	0.773062	+	0.634330i	$0.218724\pi$
$462$	−9.47132	−0.440646
$463$	−1.31144	−0.0609477	−0.0304739	−	0.999536i	$-0.509702\pi$
$463$	−1.31144	−0.0609477	−0.0304739	+	0.999536i	$0.509702\pi$
$464$	126.705	5.88211
$465$	0	0
$466$	−68.6509	−3.18019
$467$	−35.3652	−1.63650	−0.818252	−	0.574859i	$-0.805057\pi$
$467$	−35.3652	−1.63650	−0.818252	+	0.574859i	$0.805057\pi$
$468$	17.4964	0.808770
$469$	16.0362	0.740482
$470$	0	0
$471$	9.26551	0.426932
$472$	53.1436	2.44613
$473$	−26.8241	−1.23337
$474$	20.2975	0.932296
$475$	0	0
$476$	−5.85923	−0.268558
$477$	0.419890	0.0192254
$478$	59.3498	2.71460
$479$	26.9709	1.23233	0.616166	−	0.787616i	$-0.288685\pi$
$479$	26.9709	1.23233	0.616166	+	0.787616i	$0.288685\pi$
$480$	0	0
$481$	−8.57808	−0.391127
$482$	−26.5789	−1.21063
$483$	5.84024	0.265740
$484$	2.42621	0.110282
$485$	0	0
$486$	2.80343	0.127166
$487$	33.3994	1.51347	0.756737	−	0.653720i	$-0.226792\pi$
$487$	33.3994	1.51347	0.756737	+	0.653720i	$0.226792\pi$
$488$	32.1997	1.45761
$489$	−6.61304	−0.299052
$490$	0	0
$491$	31.6594	1.42877	0.714385	−	0.699753i	$-0.246707\pi$
$491$	31.6594	1.42877	0.714385	+	0.699753i	$0.246707\pi$
$492$	−51.1161	−2.30449
$493$	6.80762	0.306600
$494$	68.6567	3.08901
$495$	0	0
$496$	−9.14142	−0.410462
$497$	9.40560	0.421899
$498$	−0.00389251	−0.000174427 0
$499$	−7.88306	−0.352894	−0.176447	−	0.984310i	$-0.556460\pi$
$499$	−7.88306	−0.352894	−0.176447	+	0.984310i	$0.556460\pi$
$500$	0	0
$501$	5.97930	0.267135
$502$	6.55669	0.292639
$503$	−14.1621	−0.631455	−0.315727	−	0.948850i	$-0.602249\pi$
$503$	−14.1621	−0.631455	−0.315727	+	0.948850i	$0.602249\pi$
$504$	10.8191	0.481921
$505$	0	0
$506$	−55.3148	−2.45904
$507$	4.08310	0.181337
$508$	65.6878	2.91443
$509$	−9.81696	−0.435129	−0.217565	−	0.976046i	$-0.569811\pi$
$509$	−9.81696	−0.435129	−0.217565	+	0.976046i	$0.569811\pi$
$510$	0	0
$511$	8.53815	0.377705
$512$	−165.677	−7.32194
$513$	8.20136	0.362099
$514$	64.2805	2.83529
$515$	0	0
$516$	46.5206	2.04795
$517$	−18.3494	−0.807005
$518$	−8.05329	−0.353841
$519$	−9.67349	−0.424619
$520$	0	0
$521$	26.3114	1.15272	0.576362	−	0.817195i	$-0.304472\pi$
$521$	26.3114	1.15272	0.576362	+	0.817195i	$0.304472\pi$
$522$	−19.0847	−0.835315
$523$	−34.5725	−1.51175	−0.755875	−	0.654716i	$-0.772788\pi$
$523$	−34.5725	−1.51175	−0.755875	+	0.654716i	$0.772788\pi$
$524$	−55.9639	−2.44480
$525$	0	0
$526$	−17.3023	−0.754416
$527$	−0.491153	−0.0213950
$528$	−62.8807	−2.73653
$529$	11.1085	0.482976
$530$	0	0
$531$	−4.91201	−0.213163
$532$	48.0537	2.08339
$533$	26.0510	1.12839
$534$	21.2734	0.920592
$535$	0	0
$536$	173.497	7.49393
$537$	16.1835	0.698371
$538$	61.5625	2.65415
$539$	3.37847	0.145521
$540$	0	0
$541$	2.91743	0.125430	0.0627151	−	0.998031i	$-0.480024\pi$
$541$	2.91743	0.125430	0.0627151	+	0.998031i	$0.480024\pi$
$542$	83.2636	3.57648
$543$	−1.01069	−0.0433730
$544$	−30.5397	−1.30938
$545$	0	0
$546$	−8.37138	−0.358262
$547$	17.4499	0.746105	0.373053	−	0.927810i	$-0.378311\pi$
$547$	17.4499	0.746105	0.373053	+	0.927810i	$0.378311\pi$
$548$	74.0944	3.16516
$549$	−2.97619	−0.127021
$550$	0	0
$551$	−55.8318	−2.37851
$552$	63.1862	2.68938
$553$	−7.24024	−0.307886
$554$	−0.927283	−0.0393965
$555$	0	0
$556$	−47.3504	−2.00810
$557$	−1.58407	−0.0671191	−0.0335596	−	0.999437i	$-0.510684\pi$
$557$	−1.58407	−0.0671191	−0.0335596	+	0.999437i	$0.510684\pi$
$558$	1.37692	0.0582895
$559$	−23.7089	−1.00278
$560$	0	0
$561$	−3.37847	−0.142639
$562$	23.6828	0.998997
$563$	−21.9428	−0.924780	−0.462390	−	0.886677i	$-0.653008\pi$
$563$	−21.9428	−0.924780	−0.462390	+	0.886677i	$0.653008\pi$
$564$	31.8230	1.33999
$565$	0	0
$566$	57.5310	2.41821
$567$	−1.00000	−0.0419961
$568$	101.760	4.26976
$569$	−25.0638	−1.05073	−0.525365	−	0.850877i	$-0.676071\pi$
$569$	−25.0638	−1.05073	−0.525365	+	0.850877i	$0.676071\pi$
$570$	0	0
$571$	4.84213	0.202637	0.101318	−	0.994854i	$-0.467694\pi$
$571$	4.84213	0.202637	0.101318	+	0.994854i	$0.467694\pi$
$572$	59.1110	2.47156
$573$	−11.2168	−0.468587
$574$	24.4572	1.02082
$575$	0	0
$576$	48.3918	2.01632
$577$	−21.1366	−0.879927	−0.439964	−	0.898016i	$-0.645009\pi$
$577$	−21.1366	−0.879927	−0.439964	+	0.898016i	$0.645009\pi$
$578$	−2.80343	−0.116607
$579$	13.0305	0.541529
$580$	0	0
$581$	0.00138848	5.76038e−5 0
$582$	28.4977	1.18127
$583$	1.41859	0.0587518
$584$	92.3751	3.82251
$585$	0	0
$586$	−50.2063	−2.07400
$587$	−9.36669	−0.386605	−0.193302	−	0.981139i	$-0.561920\pi$
$587$	−9.36669	−0.386605	−0.193302	+	0.981139i	$0.561920\pi$
$588$	−5.85923	−0.241631
$589$	4.02813	0.165976
$590$	0	0
$591$	3.36117	0.138260
$592$	−53.4663	−2.19745
$593$	−12.1261	−0.497961	−0.248980	−	0.968509i	$-0.580095\pi$
$593$	−12.1261	−0.497961	−0.248980	+	0.968509i	$0.580095\pi$
$594$	9.47132	0.388613
$595$	0	0
$596$	−82.8319	−3.39293
$597$	−3.11797	−0.127610
$598$	−48.8909	−1.99930
$599$	19.9848	0.816558	0.408279	−	0.912857i	$-0.366129\pi$
$599$	19.9848	0.816558	0.408279	+	0.912857i	$0.366129\pi$
$600$	0	0
$601$	−29.9290	−1.22083	−0.610415	−	0.792081i	$-0.708998\pi$
$601$	−29.9290	−1.22083	−0.610415	+	0.792081i	$0.708998\pi$
$602$	−22.2584	−0.907185
$603$	−16.0362	−0.653044
$604$	−80.5643	−3.27812
$605$	0	0
$606$	−38.6357	−1.56947
$607$	22.5262	0.914309	0.457154	−	0.889387i	$-0.348869\pi$
$607$	22.5262	0.914309	0.457154	+	0.889387i	$0.348869\pi$
$608$	250.467	10.1578
$609$	6.80762	0.275859
$610$	0	0
$611$	−16.2184	−0.656126
$612$	5.85923	0.236846
$613$	−28.3171	−1.14372	−0.571858	−	0.820352i	$-0.693777\pi$
$613$	−28.3171	−1.14372	−0.571858	+	0.820352i	$0.693777\pi$
$614$	−55.5249	−2.24080
$615$	0	0
$616$	36.5521	1.47272
$617$	25.0388	1.00802	0.504012	−	0.863697i	$-0.331857\pi$
$617$	25.0388	1.00802	0.504012	+	0.863697i	$0.331857\pi$
$618$	10.7457	0.432255
$619$	−20.2234	−0.812847	−0.406424	−	0.913685i	$-0.633224\pi$
$619$	−20.2234	−0.812847	−0.406424	+	0.913685i	$0.633224\pi$
$620$	0	0
$621$	−5.84024	−0.234361
$622$	12.0803	0.484377
$623$	−7.58835	−0.304021
$624$	−55.5781	−2.22490
$625$	0	0
$626$	−76.8896	−3.07313
$627$	27.7081	1.10655
$628$	−54.2888	−2.16636
$629$	−2.87265	−0.114540
$630$	0	0
$631$	19.9504	0.794212	0.397106	−	0.917773i	$-0.370015\pi$
$631$	19.9504	0.794212	0.397106	+	0.917773i	$0.370015\pi$
$632$	−78.3329	−3.11591
$633$	−1.18361	−0.0470443
$634$	−63.4869	−2.52139
$635$	0	0
$636$	−2.46023	−0.0975545
$637$	2.98612	0.118314
$638$	−64.4772	−2.55268
$639$	−9.40560	−0.372080
$640$	0	0
$641$	24.0841	0.951264	0.475632	−	0.879644i	$-0.342219\pi$
$641$	24.0841	0.951264	0.475632	+	0.879644i	$0.342219\pi$
$642$	10.5594	0.416748
$643$	28.7066	1.13208	0.566039	−	0.824379i	$-0.308475\pi$
$643$	28.7066	1.13208	0.566039	+	0.824379i	$0.308475\pi$
$644$	−34.2194	−1.34843
$645$	0	0
$646$	22.9920	0.904607
$647$	0.387706	0.0152423	0.00762116	−	0.999971i	$-0.497574\pi$
$647$	0.387706	0.0152423	0.00762116	+	0.999971i	$0.497574\pi$
$648$	−10.8191	−0.425015
$649$	−16.5951	−0.651415
$650$	0	0
$651$	−0.491153	−0.0192498
$652$	38.7473	1.51746
$653$	18.8767	0.738704	0.369352	−	0.929290i	$-0.379580\pi$
$653$	18.8767	0.738704	0.369352	+	0.929290i	$0.379580\pi$
$654$	−36.9739	−1.44579
$655$	0	0
$656$	162.373	6.33960
$657$	−8.53815	−0.333105
$658$	−15.2262	−0.593578
$659$	−1.85135	−0.0721185	−0.0360592	−	0.999350i	$-0.511480\pi$
$659$	−1.85135	−0.0721185	−0.0360592	+	0.999350i	$0.511480\pi$
$660$	0	0
$661$	−40.4450	−1.57313	−0.786564	−	0.617509i	$-0.788142\pi$
$661$	−40.4450	−1.57313	−0.786564	+	0.617509i	$0.788142\pi$
$662$	−51.4966	−2.00147
$663$	−2.98612	−0.115971
$664$	0.0150221	0.000582970 0
$665$	0	0
$666$	8.05329	0.312059
$667$	39.7582	1.53944
$668$	−35.0341	−1.35551
$669$	0.645242	0.0249465
$670$	0	0
$671$	−10.0550	−0.388168
$672$	−30.5397	−1.17810
$673$	−27.7127	−1.06825	−0.534123	−	0.845407i	$-0.679358\pi$
$673$	−27.7127	−1.06825	−0.534123	+	0.845407i	$0.679358\pi$
$674$	−63.7482	−2.45549
$675$	0	0
$676$	−23.9239	−0.920148
$677$	22.7354	0.873792	0.436896	−	0.899512i	$-0.356078\pi$
$677$	22.7354	0.873792	0.436896	+	0.899512i	$0.356078\pi$
$678$	43.0340	1.65271
$679$	−10.1653	−0.390108
$680$	0	0
$681$	20.6531	0.791428
$682$	4.65187	0.178129
$683$	21.6837	0.829704	0.414852	−	0.909889i	$-0.363833\pi$
$683$	21.6837	0.829704	0.414852	+	0.909889i	$0.363833\pi$
$684$	−48.0537	−1.83738
$685$	0	0
$686$	2.80343	0.107036
$687$	−8.46699	−0.323036
$688$	−147.775	−5.63387
$689$	1.25384	0.0477675
$690$	0	0
$691$	7.63112	0.290301	0.145151	−	0.989410i	$-0.453633\pi$
$691$	7.63112	0.290301	0.145151	+	0.989410i	$0.453633\pi$
$692$	56.6792	2.15462
$693$	−3.37847	−0.128338
$694$	−15.1002	−0.573197
$695$	0	0
$696$	73.6524	2.79179
$697$	8.72402	0.330446
$698$	13.8690	0.524948
$699$	−24.4882	−0.926227
$700$	0	0
$701$	17.9592	0.678308	0.339154	−	0.940731i	$-0.389859\pi$
$701$	17.9592	0.678308	0.339154	+	0.940731i	$0.389859\pi$
$702$	8.37138	0.315957
$703$	23.5597	0.888569
$704$	163.490	6.16177
$705$	0	0
$706$	70.0718	2.63719
$707$	13.7816	0.518309
$708$	28.7806	1.08164
$709$	−48.8046	−1.83290	−0.916448	−	0.400155i	$-0.868956\pi$
$709$	−48.8046	−1.83290	−0.916448	+	0.400155i	$0.868956\pi$
$710$	0	0
$711$	7.24024	0.271530
$712$	−82.0992	−3.07680
$713$	−2.86846	−0.107424
$714$	−2.80343	−0.104916
$715$	0	0
$716$	−94.8231	−3.54370
$717$	21.1704	0.790624
$718$	−85.6301	−3.19569
$719$	21.4308	0.799234	0.399617	−	0.916682i	$-0.369143\pi$
$719$	21.4308	0.799234	0.399617	+	0.916682i	$0.369143\pi$
$720$	0	0
$721$	−3.83305	−0.142750
$722$	−135.300	−5.03535
$723$	−9.48083	−0.352596
$724$	5.92189	0.220085
$725$	0	0
$726$	1.16086	0.0430834
$727$	−10.3818	−0.385038	−0.192519	−	0.981293i	$-0.561666\pi$
$727$	−10.3818	−0.385038	−0.192519	+	0.981293i	$0.561666\pi$
$728$	32.3071	1.19738
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.93970	−0.293660
$732$	17.4382	0.644534
$733$	−44.0321	−1.62636	−0.813181	−	0.582011i	$-0.802266\pi$
$733$	−44.0321	−1.62636	−0.813181	+	0.582011i	$0.802266\pi$
$734$	3.53249	0.130386
$735$	0	0
$736$	−178.359	−6.57442
$737$	−54.1778	−1.99567
$738$	−24.4572	−0.900282
$739$	−18.5124	−0.680989	−0.340494	−	0.940247i	$-0.610594\pi$
$739$	−18.5124	−0.680989	−0.340494	+	0.940247i	$0.610594\pi$
$740$	0	0
$741$	24.4902	0.899671
$742$	1.17713	0.0432139
$743$	−16.9028	−0.620103	−0.310051	−	0.950720i	$-0.600346\pi$
$743$	−16.9028	−0.620103	−0.310051	+	0.950720i	$0.600346\pi$
$744$	−5.31384	−0.194815
$745$	0	0
$746$	57.6336	2.11012
$747$	−0.00138848	−5.08018e−5 0
$748$	19.7953	0.723787
$749$	−3.76661	−0.137629
$750$	0	0
$751$	−2.02812	−0.0740073	−0.0370037	−	0.999315i	$-0.511781\pi$
$751$	−2.02812	−0.0740073	−0.0370037	+	0.999315i	$0.511781\pi$
$752$	−101.088	−3.68628
$753$	2.33881	0.0852309
$754$	−56.9892	−2.07542
$755$	0	0
$756$	5.85923	0.213098
$757$	13.1802	0.479042	0.239521	−	0.970891i	$-0.423010\pi$
$757$	13.1802	0.479042	0.239521	+	0.970891i	$0.423010\pi$
$758$	9.35115	0.339649
$759$	−19.7311	−0.716194
$760$	0	0
$761$	7.26906	0.263503	0.131752	−	0.991283i	$-0.457940\pi$
$761$	7.26906	0.263503	0.131752	+	0.991283i	$0.457940\pi$
$762$	31.4293	1.13856
$763$	13.1888	0.477466
$764$	65.7216	2.37773
$765$	0	0
$766$	−4.82523	−0.174343
$767$	−14.6679	−0.529625
$768$	−112.306	−4.05251
$769$	20.8807	0.752977	0.376488	−	0.926421i	$-0.377132\pi$
$769$	20.8807	0.752977	0.376488	+	0.926421i	$0.377132\pi$
$770$	0	0
$771$	22.9292	0.825775
$772$	−76.3487	−2.74785
$773$	−41.1634	−1.48054	−0.740272	−	0.672307i	$-0.765303\pi$
$773$	−41.1634	−1.48054	−0.740272	+	0.672307i	$0.765303\pi$
$774$	22.2584	0.800062
$775$	0	0
$776$	−109.979	−3.94802
$777$	−2.87265	−0.103056
$778$	57.2422	2.05223
$779$	−71.5488	−2.56350
$780$	0	0
$781$	−31.7766	−1.13705
$782$	−16.3727	−0.585488
$783$	−6.80762	−0.243285
$784$	18.6122	0.664720
$785$	0	0
$786$	−26.7767	−0.955094
$787$	37.7603	1.34601	0.673004	−	0.739639i	$-0.265004\pi$
$787$	37.7603	1.34601	0.673004	+	0.739639i	$0.265004\pi$
$788$	−19.6939	−0.701565
$789$	−6.17182	−0.219723
$790$	0	0
$791$	−15.3505	−0.545799
$792$	−36.5521	−1.29882
$793$	−8.88725	−0.315595
$794$	21.2761	0.755061
$795$	0	0
$796$	18.2689	0.647524
$797$	−45.9127	−1.62631	−0.813155	−	0.582047i	$-0.802252\pi$
$797$	−45.9127	−1.62631	−0.813155	+	0.582047i	$0.802252\pi$
$798$	22.9920	0.813907
$799$	−5.43126	−0.192144
$800$	0	0
$801$	7.58835	0.268121
$802$	4.64480	0.164014
$803$	−28.8459	−1.01795
$804$	93.9598	3.31371
$805$	0	0
$806$	4.11163	0.144826
$807$	21.9597	0.773018
$808$	149.104	5.24547
$809$	28.9703	1.01854	0.509270	−	0.860607i	$-0.329915\pi$
$809$	28.9703	1.01854	0.509270	+	0.860607i	$0.329915\pi$
$810$	0	0
$811$	−28.6073	−1.00454	−0.502270	−	0.864711i	$-0.667502\pi$
$811$	−28.6073	−1.00454	−0.502270	+	0.864711i	$0.667502\pi$
$812$	−39.8875	−1.39978
$813$	29.7006	1.04164
$814$	27.2078	0.953633
$815$	0	0
$816$	−18.6122	−0.651556
$817$	65.1163	2.27813
$818$	16.9473	0.592549
$819$	−2.98612	−0.104343
$820$	0	0
$821$	−21.6958	−0.757188	−0.378594	−	0.925563i	$-0.623592\pi$
$821$	−21.6958	−0.757188	−0.378594	+	0.925563i	$0.623592\pi$
$822$	35.4515	1.23651
$823$	−33.1277	−1.15476	−0.577379	−	0.816476i	$-0.695925\pi$
$823$	−33.1277	−1.15476	−0.577379	+	0.816476i	$0.695925\pi$
$824$	−41.4701	−1.44468
$825$	0	0
$826$	−13.7705	−0.479137
$827$	−31.8796	−1.10856	−0.554280	−	0.832330i	$-0.687006\pi$
$827$	−31.8796	−1.10856	−0.554280	+	0.832330i	$0.687006\pi$
$828$	34.2194	1.18920
$829$	12.0622	0.418937	0.209469	−	0.977815i	$-0.432827\pi$
$829$	12.0622	0.418937	0.209469	+	0.977815i	$0.432827\pi$
$830$	0	0
$831$	−0.330767	−0.0114742
$832$	144.503	5.00976
$833$	1.00000	0.0346479
$834$	−22.6555	−0.784495
$835$	0	0
$836$	−162.348	−5.61493
$837$	0.491153	0.0169767
$838$	5.77555	0.199513
$839$	−31.0708	−1.07268	−0.536342	−	0.844001i	$-0.680194\pi$
$839$	−31.0708	−1.07268	−0.536342	+	0.844001i	$0.680194\pi$
$840$	0	0
$841$	17.3438	0.598060
$842$	−25.3923	−0.875076
$843$	8.44777	0.290957
$844$	6.93505	0.238714
$845$	0	0
$846$	15.2262	0.523487
$847$	−0.414084	−0.0142281
$848$	7.81505	0.268370
$849$	20.5216	0.704301
$850$	0	0
$851$	−16.7770	−0.575108
$852$	55.1096	1.88802
$853$	−0.590902	−0.0202321	−0.0101161	−	0.999949i	$-0.503220\pi$
$853$	−0.590902	−0.0202321	−0.0101161	+	0.999949i	$0.503220\pi$
$854$	−8.34354	−0.285510
$855$	0	0
$856$	−40.7514	−1.39285
$857$	27.9253	0.953909	0.476955	−	0.878928i	$-0.341741\pi$
$857$	27.9253	0.953909	0.476955	+	0.878928i	$0.341741\pi$
$858$	28.2825	0.965548
$859$	10.1726	0.347084	0.173542	−	0.984826i	$-0.444479\pi$
$859$	10.1726	0.347084	0.173542	+	0.984826i	$0.444479\pi$
$860$	0	0
$861$	8.72402	0.297314
$862$	−38.0254	−1.29515
$863$	29.5369	1.00545	0.502723	−	0.864447i	$-0.332331\pi$
$863$	29.5369	1.00545	0.502723	+	0.864447i	$0.332331\pi$
$864$	30.5397	1.03898
$865$	0	0
$866$	−14.8959	−0.506182
$867$	−1.00000	−0.0339618
$868$	2.87778	0.0976783
$869$	24.4610	0.829781
$870$	0	0
$871$	−47.8859	−1.62255
$872$	142.691	4.83212
$873$	10.1653	0.344043
$874$	134.279	4.54204
$875$	0	0
$876$	50.0270	1.69026
$877$	−23.9038	−0.807173	−0.403587	−	0.914941i	$-0.632237\pi$
$877$	−23.9038	−0.807173	−0.403587	+	0.914941i	$0.632237\pi$
$878$	−63.2061	−2.13310
$879$	−17.9089	−0.604052
$880$	0	0
$881$	38.0782	1.28289	0.641443	−	0.767170i	$-0.278336\pi$
$881$	38.0782	1.28289	0.641443	+	0.767170i	$0.278336\pi$
$882$	−2.80343	−0.0943965
$883$	−17.5620	−0.591007	−0.295503	−	0.955342i	$-0.595487\pi$
$883$	−17.5620	−0.591007	−0.295503	+	0.955342i	$0.595487\pi$
$884$	17.4964	0.588466
$885$	0	0
$886$	40.0671	1.34608
$887$	−35.3014	−1.18530	−0.592652	−	0.805459i	$-0.701919\pi$
$887$	−35.3014	−1.18530	−0.592652	+	0.805459i	$0.701919\pi$
$888$	−31.0795	−1.04296
$889$	−11.2110	−0.376005
$890$	0	0
$891$	3.37847	0.113183
$892$	−3.78062	−0.126585
$893$	44.5437	1.49060
$894$	−39.6321	−1.32550
$895$	0	0
$896$	74.5836	2.49166
$897$	−17.4397	−0.582293
$898$	−5.70379	−0.190338
$899$	−3.34359	−0.111515
$900$	0	0
$901$	0.419890	0.0139886
$902$	−82.6280	−2.75121
$903$	−7.93970	−0.264217
$904$	−166.078	−5.52368
$905$	0	0
$906$	−38.5471	−1.28064
$907$	33.4468	1.11058	0.555292	−	0.831656i	$-0.312607\pi$
$907$	33.4468	1.11058	0.555292	+	0.831656i	$0.312607\pi$
$908$	−121.011	−4.01590
$909$	−13.7816	−0.457106
$910$	0	0
$911$	−16.4216	−0.544071	−0.272036	−	0.962287i	$-0.587697\pi$
$911$	−16.4216	−0.544071	−0.272036	+	0.962287i	$0.587697\pi$
$912$	152.645	5.05458
$913$	−0.00469094	−0.000155247 0
$914$	18.7021	0.618610
$915$	0	0
$916$	49.6101	1.63916
$917$	9.55141	0.315415
$918$	2.80343	0.0925271
$919$	−44.7007	−1.47454	−0.737271	−	0.675598i	$-0.763886\pi$
$919$	−44.7007	−1.47454	−0.737271	+	0.675598i	$0.763886\pi$
$920$	0	0
$921$	−19.8061	−0.652632
$922$	−93.0647	−3.06492
$923$	−28.0862	−0.924469
$924$	19.7953	0.651217
$925$	0	0
$926$	3.67653	0.120818
$927$	3.83305	0.125894
$928$	−207.903	−6.82475
$929$	−45.7602	−1.50134	−0.750672	−	0.660676i	$-0.770270\pi$
$929$	−45.7602	−1.50134	−0.750672	+	0.660676i	$0.770270\pi$
$930$	0	0
$931$	−8.20136	−0.268789
$932$	143.482	4.69991
$933$	4.30912	0.141074
$934$	99.1438	3.24409
$935$	0	0
$936$	−32.3071	−1.05599
$937$	−21.8797	−0.714779	−0.357390	−	0.933955i	$-0.616333\pi$
$937$	−21.8797	−0.714779	−0.357390	+	0.933955i	$0.616333\pi$
$938$	−44.9564	−1.46788
$939$	−27.4270	−0.895045
$940$	0	0
$941$	21.9504	0.715563	0.357782	−	0.933805i	$-0.383533\pi$
$941$	21.9504	0.715563	0.357782	+	0.933805i	$0.383533\pi$
$942$	−25.9752	−0.846318
$943$	50.9504	1.65917
$944$	−91.4232	−2.97557
$945$	0	0
$946$	75.1995	2.44495
$947$	6.14205	0.199590	0.0997949	−	0.995008i	$-0.468181\pi$
$947$	6.14205	0.199590	0.0997949	+	0.995008i	$0.468181\pi$
$948$	−42.4223	−1.37781
$949$	−25.4959	−0.827632
$950$	0	0
$951$	−22.6461	−0.734352
$952$	10.8191	0.350649
$953$	−51.4151	−1.66550	−0.832750	−	0.553650i	$-0.813235\pi$
$953$	−51.4151	−1.66550	−0.832750	+	0.553650i	$0.813235\pi$
$954$	−1.17713	−0.0381111
$955$	0	0
$956$	−124.042	−4.01182
$957$	−22.9994	−0.743464
$958$	−75.6112	−2.44289
$959$	−12.6457	−0.408352
$960$	0	0
$961$	−30.7588	−0.992218
$962$	24.0481	0.775341
$963$	3.76661	0.121377
$964$	55.5504	1.78916
$965$	0	0
$966$	−16.3727	−0.526784
$967$	−19.1624	−0.616221	−0.308110	−	0.951351i	$-0.599697\pi$
$967$	−19.1624	−0.616221	−0.308110	+	0.951351i	$0.599697\pi$
$968$	−4.48001	−0.143993
$969$	8.20136	0.263466
$970$	0	0
$971$	30.0374	0.963946	0.481973	−	0.876186i	$-0.339920\pi$
$971$	30.0374	0.963946	0.481973	+	0.876186i	$0.339920\pi$
$972$	−5.85923	−0.187935
$973$	8.08133	0.259076
$974$	−93.6331	−3.00020
$975$	0	0
$976$	−55.3933	−1.77310
$977$	−55.6869	−1.78158	−0.890792	−	0.454412i	$-0.849849\pi$
$977$	−55.6869	−1.78158	−0.890792	+	0.454412i	$0.849849\pi$
$978$	18.5392	0.592819
$979$	25.6371	0.819364
$980$	0	0
$981$	−13.1888	−0.421085
$982$	−88.7550	−2.83229
$983$	33.8445	1.07947	0.539736	−	0.841835i	$-0.318524\pi$
$983$	33.8445	1.07947	0.539736	+	0.841835i	$0.318524\pi$
$984$	94.3861	3.00892
$985$	0	0
$986$	−19.0847	−0.607781
$987$	−5.43126	−0.172879
$988$	−143.494	−4.56515
$989$	−46.3698	−1.47447
$990$	0	0
$991$	−32.8709	−1.04418	−0.522089	−	0.852891i	$-0.674847\pi$
$991$	−32.8709	−1.04418	−0.522089	+	0.852891i	$0.674847\pi$
$992$	14.9997	0.476241
$993$	−18.3691	−0.582927
$994$	−26.3680	−0.836341
$995$	0	0
$996$	0.00813542	0.000257781 0
$997$	17.3006	0.547915	0.273958	−	0.961742i	$-0.411667\pi$
$997$	17.3006	0.547915	0.273958	+	0.961742i	$0.411667\pi$
$998$	22.0996	0.699551
$999$	2.87265	0.0908867

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8925.2.a.cu.1.1		14
5.2	odd	4		1785.2.g.g.1429.1	✓	28
5.3	odd	4		1785.2.g.g.1429.28	yes	28
5.4	even	2		8925.2.a.cx.1.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.g.g.1429.1	✓	28	5.2	odd	4
1785.2.g.g.1429.28	yes	28	5.3	odd	4
8925.2.a.cu.1.1		14	1.1	even	1	trivial
8925.2.a.cx.1.14		14	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8925.2.a.cu.1.1

Level	$8925$
Weight	$2$
Character	8925.1
Self dual	yes
Analytic conductor	$71.266$
Analytic rank	$1$
Dimension	$14$
CM	no
Inner twists	$1$