LMFDB - Embedded newform 1815.2.a.y.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1815.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:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.437199552.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 49x^{2} - 48 $ x^6 - 13x^4 + 49x^2 - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.13353$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13353 q^{2} -1.00000 q^{3} +2.55193 q^{4} +1.00000 q^{5} -2.13353 q^{6} +4.82155 q^{7} +1.17756 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.13353 q^{2} -1.00000 q^{3} +2.55193 q^{4} +1.00000 q^{5} -2.13353 q^{6} +4.82155 q^{7} +1.17756 q^{8} +1.00000 q^{9} +2.13353 q^{10} -2.55193 q^{12} +4.26705 q^{13} +10.2869 q^{14} -1.00000 q^{15} -2.59152 q^{16} +4.64166 q^{17} +2.13353 q^{18} -6.37371 q^{19} +2.55193 q^{20} -4.82155 q^{21} -5.14344 q^{23} -1.17756 q^{24} +1.00000 q^{25} +9.10386 q^{26} -1.00000 q^{27} +12.3042 q^{28} +4.26705 q^{29} -2.13353 q^{30} -6.39075 q^{31} -7.88417 q^{32} +9.90309 q^{34} +4.82155 q^{35} +2.55193 q^{36} +6.14344 q^{37} -13.5985 q^{38} -4.26705 q^{39} +1.17756 q^{40} +3.46410 q^{41} -10.2869 q^{42} -1.55216 q^{43} +1.00000 q^{45} -10.9737 q^{46} +5.35116 q^{47} +2.59152 q^{48} +16.2473 q^{49} +2.13353 q^{50} -4.64166 q^{51} +10.8892 q^{52} +2.24730 q^{53} -2.13353 q^{54} +5.67764 q^{56} +6.37371 q^{57} +9.10386 q^{58} -13.5985 q^{59} -2.55193 q^{60} -6.80205 q^{61} -13.6348 q^{62} +4.82155 q^{63} -11.6381 q^{64} +4.26705 q^{65} +6.14344 q^{67} +11.8452 q^{68} +5.14344 q^{69} +10.2869 q^{70} -10.4946 q^{71} +1.17756 q^{72} +5.62449 q^{73} +13.1072 q^{74} -1.00000 q^{75} -16.2653 q^{76} -9.10386 q^{78} +16.3914 q^{79} -2.59152 q^{80} +1.00000 q^{81} +7.39075 q^{82} +6.17899 q^{83} -12.3042 q^{84} +4.64166 q^{85} -3.31158 q^{86} -4.26705 q^{87} -1.39075 q^{89} +2.13353 q^{90} +20.5738 q^{91} -13.1257 q^{92} +6.39075 q^{93} +11.4168 q^{94} -6.37371 q^{95} +7.88417 q^{96} +3.93573 q^{97} +34.6640 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 14 q^{4} + 6 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 14 * q^4 + 6 * q^5 + 6 * q^9	$ 6 q - 6 q^{3} + 14 q^{4} + 6 q^{5} + 6 q^{9} - 14 q^{12} + 8 q^{14} - 6 q^{15} + 10 q^{16} + 14 q^{20} - 4 q^{23} + 6 q^{25} + 52 q^{26} - 6 q^{27} + 18 q^{31} + 26 q^{34} + 14 q^{36} + 10 q^{37} - 20 q^{38} - 8 q^{42} + 6 q^{45} - 10 q^{48} + 68 q^{49} - 16 q^{53} - 76 q^{56} + 52 q^{58} - 20 q^{59} - 14 q^{60} + 16 q^{64} + 10 q^{67} + 4 q^{69} + 8 q^{70} - 4 q^{71} - 6 q^{75} - 52 q^{78} + 10 q^{80} + 6 q^{81} - 12 q^{82} - 12 q^{86} + 48 q^{89} + 16 q^{91} + 30 q^{92} - 18 q^{93} + 2 q^{97}+O(q^{100}) $ 6 * q - 6 * q^3 + 14 * q^4 + 6 * q^5 + 6 * q^9 - 14 * q^12 + 8 * q^14 - 6 * q^15 + 10 * q^16 + 14 * q^20 - 4 * q^23 + 6 * q^25 + 52 * q^26 - 6 * q^27 + 18 * q^31 + 26 * q^34 + 14 * q^36 + 10 * q^37 - 20 * q^38 - 8 * q^42 + 6 * q^45 - 10 * q^48 + 68 * q^49 - 16 * q^53 - 76 * q^56 + 52 * q^58 - 20 * q^59 - 14 * q^60 + 16 * q^64 + 10 * q^67 + 4 * q^69 + 8 * q^70 - 4 * q^71 - 6 * q^75 - 52 * q^78 + 10 * q^80 + 6 * q^81 - 12 * q^82 - 12 * q^86 + 48 * q^89 + 16 * q^91 + 30 * q^92 - 18 * q^93 + 2 * q^97

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 $

$2$	2.13353	1.50863	0.754315	−	0.656513i	$-0.227969\pi$
$2$	2.13353	1.50863	0.754315	+	0.656513i	$0.227969\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.55193	1.27596
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.13353	−0.871008
$7$	4.82155	1.82237	0.911186	−	0.411994i	$-0.135168\pi$
$7$	4.82155	1.82237	0.911186	+	0.411994i	$0.135168\pi$
$8$	1.17756	0.416329
$9$	1.00000	0.333333
$10$	2.13353	0.674680
$11$	0	0
$11$	0	0
$12$	−2.55193	−0.736679
$13$	4.26705	1.18347	0.591733	−	0.806134i	$-0.298444\pi$
$13$	4.26705	1.18347	0.591733	+	0.806134i	$0.298444\pi$
$14$	10.2869	2.74929
$15$	−1.00000	−0.258199
$16$	−2.59152	−0.647879
$17$	4.64166	1.12577	0.562884	−	0.826536i	$-0.309692\pi$
$17$	4.64166	1.12577	0.562884	+	0.826536i	$0.309692\pi$
$18$	2.13353	0.502877
$19$	−6.37371	−1.46223	−0.731114	−	0.682255i	$-0.760999\pi$
$19$	−6.37371	−1.46223	−0.731114	+	0.682255i	$0.760999\pi$
$20$	2.55193	0.570629
$21$	−4.82155	−1.05215
$22$	0	0
$23$	−5.14344	−1.07248	−0.536241	−	0.844065i	$-0.680156\pi$
$23$	−5.14344	−1.07248	−0.536241	+	0.844065i	$0.680156\pi$
$24$	−1.17756	−0.240367
$25$	1.00000	0.200000
$26$	9.10386	1.78541
$27$	−1.00000	−0.192450
$28$	12.3042	2.32528
$29$	4.26705	0.792371	0.396186	−	0.918170i	$-0.370334\pi$
$29$	4.26705	0.792371	0.396186	+	0.918170i	$0.370334\pi$
$30$	−2.13353	−0.389527
$31$	−6.39075	−1.14781	−0.573906	−	0.818921i	$-0.694573\pi$
$31$	−6.39075	−1.14781	−0.573906	+	0.818921i	$0.694573\pi$
$32$	−7.88417	−1.39374
$33$	0	0
$34$	9.90309	1.69837
$35$	4.82155	0.814990
$36$	2.55193	0.425322
$37$	6.14344	1.00998	0.504988	−	0.863126i	$-0.331497\pi$
$37$	6.14344	1.00998	0.504988	+	0.863126i	$0.331497\pi$
$38$	−13.5985	−2.20596
$39$	−4.26705	−0.683275
$40$	1.17756	0.186188
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	−10.2869	−1.58730
$43$	−1.55216	−0.236702	−0.118351	−	0.992972i	$-0.537761\pi$
$43$	−1.55216	−0.236702	−0.118351	+	0.992972i	$0.537761\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−10.9737	−1.61798
$47$	5.35116	0.780547	0.390274	−	0.920699i	$-0.372380\pi$
$47$	5.35116	0.780547	0.390274	+	0.920699i	$0.372380\pi$
$48$	2.59152	0.374053
$49$	16.2473	2.32104
$50$	2.13353	0.301726
$51$	−4.64166	−0.649962
$52$	10.8892	1.51006
$53$	2.24730	0.308691	0.154345	−	0.988017i	$-0.450673\pi$
$53$	2.24730	0.308691	0.154345	+	0.988017i	$0.450673\pi$
$54$	−2.13353	−0.290336
$55$	0	0
$56$	5.67764	0.758706
$57$	6.37371	0.844218
$58$	9.10386	1.19540
$59$	−13.5985	−1.77037	−0.885185	−	0.465240i	$-0.845968\pi$
$59$	−13.5985	−1.77037	−0.885185	+	0.465240i	$0.845968\pi$
$60$	−2.55193	−0.329453
$61$	−6.80205	−0.870913	−0.435457	−	0.900210i	$-0.643413\pi$
$61$	−6.80205	−0.870913	−0.435457	+	0.900210i	$0.643413\pi$
$62$	−13.6348	−1.73162
$63$	4.82155	0.607458
$64$	−11.6381	−1.45476
$65$	4.26705	0.529262
$66$	0	0
$67$	6.14344	0.750541	0.375271	−	0.926915i	$-0.377550\pi$
$67$	6.14344	0.750541	0.375271	+	0.926915i	$0.377550\pi$
$68$	11.8452	1.43644
$69$	5.14344	0.619198
$70$	10.2869	1.22952
$71$	−10.4946	−1.24548	−0.622740	−	0.782429i	$-0.713981\pi$
$71$	−10.4946	−1.24548	−0.622740	+	0.782429i	$0.713981\pi$
$72$	1.17756	0.138776
$73$	5.62449	0.658297	0.329149	−	0.944278i	$-0.393238\pi$
$73$	5.62449	0.658297	0.329149	+	0.944278i	$0.393238\pi$
$74$	13.1072	1.52368
$75$	−1.00000	−0.115470
$76$	−16.2653	−1.86575
$77$	0	0
$78$	−9.10386	−1.03081
$79$	16.3914	1.84418	0.922089	−	0.386979i	$-0.126481\pi$
$79$	16.3914	1.84418	0.922089	+	0.386979i	$0.126481\pi$
$80$	−2.59152	−0.289740
$81$	1.00000	0.111111
$82$	7.39075	0.816172
$83$	6.17899	0.678232	0.339116	−	0.940745i	$-0.389872\pi$
$83$	6.17899	0.678232	0.339116	+	0.940745i	$0.389872\pi$
$84$	−12.3042	−1.34250
$85$	4.64166	0.503458
$86$	−3.31158	−0.357096
$87$	−4.26705	−0.457476
$88$	0	0
$89$	−1.39075	−0.147419	−0.0737095	−	0.997280i	$-0.523484\pi$
$89$	−1.39075	−0.147419	−0.0737095	+	0.997280i	$0.523484\pi$
$90$	2.13353	0.224893
$91$	20.5738	2.15672
$92$	−13.1257	−1.36845
$93$	6.39075	0.662690
$94$	11.4168	1.17756
$95$	−6.37371	−0.653929
$96$	7.88417	0.804675
$97$	3.93573	0.399613	0.199806	−	0.979835i	$-0.435969\pi$
$97$	3.93573	0.399613	0.199806	+	0.979835i	$0.435969\pi$
$98$	34.6640	3.50160
$99$	0	0
$100$	2.55193	0.255193
$101$	−15.4623	−1.53856	−0.769278	−	0.638914i	$-0.779384\pi$
$101$	−15.4623	−1.53856	−0.769278	+	0.638914i	$0.779384\pi$
$102$	−9.90309	−0.980552
$103$	7.24730	0.714098	0.357049	−	0.934086i	$-0.383783\pi$
$103$	7.24730	0.714098	0.357049	+	0.934086i	$0.383783\pi$
$104$	5.02469	0.492711
$105$	−4.82155	−0.470535
$106$	4.79468	0.465700
$107$	−5.44461	−0.526350	−0.263175	−	0.964748i	$-0.584770\pi$
$107$	−5.44461	−0.526350	−0.263175	+	0.964748i	$0.584770\pi$
$108$	−2.55193	−0.245560
$109$	−2.90961	−0.278690	−0.139345	−	0.990244i	$-0.544500\pi$
$109$	−2.90961	−0.278690	−0.139345	+	0.990244i	$0.544500\pi$
$110$	0	0
$111$	−6.14344	−0.583110
$112$	−12.4951	−1.18068
$113$	−2.45502	−0.230949	−0.115474	−	0.993310i	$-0.536839\pi$
$113$	−2.45502	−0.230949	−0.115474	+	0.993310i	$0.536839\pi$
$114$	13.5985	1.27361
$115$	−5.14344	−0.479629
$116$	10.8892	1.01104
$117$	4.26705	0.394489
$118$	−29.0127	−2.67083
$119$	22.3800	2.05157
$120$	−1.17756	−0.107496
$121$	0	0
$122$	−14.5123	−1.31389
$123$	−3.46410	−0.312348
$124$	−16.3087	−1.46457
$125$	1.00000	0.0894427
$126$	10.2869	0.916429
$127$	−9.89154	−0.877733	−0.438866	−	0.898552i	$-0.644620\pi$
$127$	−9.89154	−0.877733	−0.438866	+	0.898552i	$0.644620\pi$
$128$	−9.06173	−0.800951
$129$	1.55216	0.136660
$130$	9.10386	0.798461
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−30.7311	−2.66473
$134$	13.1072	1.13229
$135$	−1.00000	−0.0860663
$136$	5.46581	0.468689
$137$	−6.53419	−0.558254	−0.279127	−	0.960254i	$-0.590045\pi$
$137$	−6.53419	−0.558254	−0.279127	+	0.960254i	$0.590045\pi$
$138$	10.9737	0.934141
$139$	−19.6608	−1.66761	−0.833803	−	0.552062i	$-0.813841\pi$
$139$	−19.6608	−1.66761	−0.833803	+	0.552062i	$0.813841\pi$
$140$	12.3042	1.03990
$141$	−5.35116	−0.450649
$142$	−22.3905	−1.87897
$143$	0	0
$144$	−2.59152	−0.215960
$145$	4.26705	0.354359
$146$	12.0000	0.993127
$147$	−16.2473	−1.34005
$148$	15.6776	1.28869
$149$	−12.3042	−1.00800	−0.504001	−	0.863703i	$-0.668139\pi$
$149$	−12.3042	−1.00800	−0.504001	+	0.863703i	$0.668139\pi$
$150$	−2.13353	−0.174202
$151$	−5.80438	−0.472354	−0.236177	−	0.971710i	$-0.575895\pi$
$151$	−5.80438	−0.472354	−0.236177	+	0.971710i	$0.575895\pi$
$152$	−7.50539	−0.608768
$153$	4.64166	0.375256
$154$	0	0
$155$	−6.39075	−0.513317
$156$	−10.8892	−0.871835
$157$	0.351162	0.0280258	0.0140129	−	0.999902i	$-0.495539\pi$
$157$	0.351162	0.0280258	0.0140129	+	0.999902i	$0.495539\pi$
$158$	34.9715	2.78218
$159$	−2.24730	−0.178223
$160$	−7.88417	−0.623299
$161$	−24.7994	−1.95446
$162$	2.13353	0.167626
$163$	14.6381	1.14654	0.573270	−	0.819366i	$-0.305674\pi$
$163$	14.6381	1.14654	0.573270	+	0.819366i	$0.305674\pi$
$164$	8.84014	0.690299
$165$	0	0
$166$	13.1830	1.02320
$167$	14.2310	1.10123	0.550614	−	0.834760i	$-0.314393\pi$
$167$	14.2310	1.10123	0.550614	+	0.834760i	$0.314393\pi$
$168$	−5.67764	−0.438039
$169$	5.20772	0.400594
$170$	9.90309	0.759532
$171$	−6.37371	−0.487410
$172$	−3.96101	−0.302024
$173$	−18.1772	−1.38199	−0.690993	−	0.722861i	$-0.742827\pi$
$173$	−18.1772	−1.38199	−0.690993	+	0.722861i	$0.742827\pi$
$174$	−9.10386	−0.690162
$175$	4.82155	0.364475
$176$	0	0
$177$	13.5985	1.02212
$178$	−2.96720	−0.222401
$179$	−9.10386	−0.680454	−0.340227	−	0.940343i	$-0.610504\pi$
$179$	−9.10386	−0.680454	−0.340227	+	0.940343i	$0.610504\pi$
$180$	2.55193	0.190210
$181$	13.2473	0.984664	0.492332	−	0.870407i	$-0.336145\pi$
$181$	13.2473	0.984664	0.492332	+	0.870407i	$0.336145\pi$
$182$	43.8947	3.25369
$183$	6.80205	0.502822
$184$	−6.05669	−0.446505
$185$	6.14344	0.451675
$186$	13.6348	0.999754
$187$	0	0
$188$	13.6558	0.995951
$189$	−4.82155	−0.350716
$190$	−13.5985	−0.986536
$191$	−18.0931	−1.30917	−0.654584	−	0.755989i	$-0.727156\pi$
$191$	−18.0931	−1.30917	−0.654584	+	0.755989i	$0.727156\pi$
$192$	11.6381	0.839904
$193$	−16.0705	−1.15678	−0.578391	−	0.815760i	$-0.696319\pi$
$193$	−16.0705	−1.15678	−0.578391	+	0.815760i	$0.696319\pi$
$194$	8.39697	0.602867
$195$	−4.26705	−0.305570
$196$	41.4620	2.96157
$197$	−15.4623	−1.10164	−0.550822	−	0.834623i	$-0.685686\pi$
$197$	−15.4623	−1.10164	−0.550822	+	0.834623i	$0.685686\pi$
$198$	0	0
$199$	20.3907	1.44546	0.722731	−	0.691130i	$-0.242887\pi$
$199$	20.3907	1.44546	0.722731	+	0.691130i	$0.242887\pi$
$200$	1.17756	0.0832657
$201$	−6.14344	−0.433325
$202$	−32.9892	−2.32111
$203$	20.5738	1.44400
$204$	−11.8452	−0.829329
$205$	3.46410	0.241943
$206$	15.4623	1.07731
$207$	−5.14344	−0.357494
$208$	−11.0581	−0.766743
$209$	0	0
$210$	−10.2869	−0.709863
$211$	−4.14090	−0.285071	−0.142536	−	0.989790i	$-0.545526\pi$
$211$	−4.14090	−0.285071	−0.142536	+	0.989790i	$0.545526\pi$
$212$	5.73496	0.393879
$213$	10.4946	0.719079
$214$	−11.6162	−0.794067
$215$	−1.55216	−0.105857
$216$	−1.17756	−0.0801225
$217$	−30.8133	−2.09174
$218$	−6.20772	−0.420440
$219$	−5.62449	−0.380068
$220$	0	0
$221$	19.8062	1.33231
$222$	−13.1072	−0.879697
$223$	−25.5342	−1.70990	−0.854948	−	0.518714i	$-0.826411\pi$
$223$	−25.5342	−1.70990	−0.854948	+	0.518714i	$0.826411\pi$
$224$	−38.0139	−2.53991
$225$	1.00000	0.0666667
$226$	−5.23785	−0.348417
$227$	−20.1577	−1.33791	−0.668957	−	0.743301i	$-0.733259\pi$
$227$	−20.1577	−1.33791	−0.668957	+	0.743301i	$0.733259\pi$
$228$	16.2653	1.07719
$229$	7.96041	0.526039	0.263019	−	0.964791i	$-0.415282\pi$
$229$	7.96041	0.526039	0.263019	+	0.964791i	$0.415282\pi$
$230$	−10.9737	−0.723582
$231$	0	0
$232$	5.02469	0.329887
$233$	−0.428342	−0.0280616	−0.0140308	−	0.999902i	$-0.504466\pi$
$233$	−0.428342	−0.0280616	−0.0140308	+	0.999902i	$0.504466\pi$
$234$	9.10386	0.595138
$235$	5.35116	0.349071
$236$	−34.7023	−2.25893
$237$	−16.3914	−1.06474
$238$	47.7482	3.09506
$239$	−18.1235	−1.17231	−0.586154	−	0.810199i	$-0.699359\pi$
$239$	−18.1235	−1.17231	−0.586154	+	0.810199i	$0.699359\pi$
$240$	2.59152	0.167282
$241$	−20.4637	−1.31819	−0.659093	−	0.752062i	$-0.729059\pi$
$241$	−20.4637	−1.31819	−0.659093	+	0.752062i	$0.729059\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−17.3584	−1.11125
$245$	16.2473	1.03800
$246$	−7.39075	−0.471217
$247$	−27.1969	−1.73050
$248$	−7.52546	−0.477867
$249$	−6.17899	−0.391578
$250$	2.13353	0.134936
$251$	24.0931	1.52074	0.760371	−	0.649489i	$-0.225017\pi$
$251$	24.0931	1.52074	0.760371	+	0.649489i	$0.225017\pi$
$252$	12.3042	0.775095
$253$	0	0
$254$	−21.1039	−1.32417
$255$	−4.64166	−0.290672
$256$	3.94268	0.246417
$257$	14.6627	0.914637	0.457318	−	0.889303i	$-0.348810\pi$
$257$	14.6627	0.914637	0.457318	+	0.889303i	$0.348810\pi$
$258$	3.31158	0.206170
$259$	29.6209	1.84055
$260$	10.8892	0.675320
$261$	4.26705	0.264124
$262$	7.39075	0.456602
$263$	6.55360	0.404112	0.202056	−	0.979374i	$-0.435238\pi$
$263$	6.55360	0.404112	0.202056	+	0.979374i	$0.435238\pi$
$264$	0	0
$265$	2.24730	0.138051
$266$	−65.5656	−4.02009
$267$	1.39075	0.0851124
$268$	15.6776	0.957664
$269$	26.7815	1.63290	0.816448	−	0.577419i	$-0.195940\pi$
$269$	26.7815	1.63290	0.816448	+	0.577419i	$0.195940\pi$
$270$	−2.13353	−0.129842
$271$	5.08483	0.308881	0.154441	−	0.988002i	$-0.450642\pi$
$271$	5.08483	0.308881	0.154441	+	0.988002i	$0.450642\pi$
$272$	−12.0289	−0.729361
$273$	−20.5738	−1.24518
$274$	−13.9409	−0.842198
$275$	0	0
$276$	13.1257	0.790075
$277$	19.4809	1.17049	0.585247	−	0.810855i	$-0.300998\pi$
$277$	19.4809	1.17049	0.585247	+	0.810855i	$0.300998\pi$
$278$	−41.9468	−2.51580
$279$	−6.39075	−0.382604
$280$	5.67764	0.339304
$281$	4.62683	0.276013	0.138007	−	0.990431i	$-0.455930\pi$
$281$	4.62683	0.276013	0.138007	+	0.990431i	$0.455930\pi$
$282$	−11.4168	−0.679863
$283$	19.1211	1.13663	0.568316	−	0.822810i	$-0.307595\pi$
$283$	19.1211	1.13663	0.568316	+	0.822810i	$0.307595\pi$
$284$	−26.7815	−1.58919
$285$	6.37371	0.377546
$286$	0	0
$287$	16.7023	0.985907
$288$	−7.88417	−0.464579
$289$	4.54498	0.267352
$290$	9.10386	0.534597
$291$	−3.93573	−0.230716
$292$	14.3533	0.839964
$293$	−9.21475	−0.538331	−0.269166	−	0.963094i	$-0.586748\pi$
$293$	−9.21475	−0.538331	−0.269166	+	0.963094i	$0.586748\pi$
$294$	−34.6640	−2.02165
$295$	−13.5985	−0.791733
$296$	7.23425	0.420482
$297$	0	0
$298$	−26.2514	−1.52070
$299$	−21.9473	−1.26925
$300$	−2.55193	−0.147336
$301$	−7.48382	−0.431360
$302$	−12.3838	−0.712607
$303$	15.4623	0.888286
$304$	16.5176	0.947347
$305$	−6.80205	−0.389484
$306$	9.90309	0.566122
$307$	−3.65882	−0.208820	−0.104410	−	0.994534i	$-0.533295\pi$
$307$	−3.65882	−0.208820	−0.104410	+	0.994534i	$0.533295\pi$
$308$	0	0
$309$	−7.24730	−0.412285
$310$	−13.6348	−0.774406
$311$	−12.9753	−0.735762	−0.367881	−	0.929873i	$-0.619917\pi$
$311$	−12.9753	−0.735762	−0.367881	+	0.929873i	$0.619917\pi$
$312$	−5.02469	−0.284467
$313$	−4.48071	−0.253264	−0.126632	−	0.991950i	$-0.540417\pi$
$313$	−4.48071	−0.253264	−0.126632	+	0.991950i	$0.540417\pi$
$314$	0.749213	0.0422806
$315$	4.82155	0.271663
$316$	41.8297	2.35311
$317$	−14.2473	−0.800208	−0.400104	−	0.916470i	$-0.631026\pi$
$317$	−14.2473	−0.800208	−0.400104	+	0.916470i	$0.631026\pi$
$318$	−4.79468	−0.268872
$319$	0	0
$320$	−11.6381	−0.650587
$321$	5.44461	0.303888
$322$	−52.9100	−2.94856
$323$	−29.5846	−1.64613
$324$	2.55193	0.141774
$325$	4.26705	0.236693
$326$	31.2306	1.72971
$327$	2.90961	0.160902
$328$	4.07917	0.225235
$329$	25.8009	1.42245
$330$	0	0
$331$	1.68842	0.0928041	0.0464021	−	0.998923i	$-0.485224\pi$
$331$	1.68842	0.0928041	0.0464021	+	0.998923i	$0.485224\pi$
$332$	15.7683	0.865400
$333$	6.14344	0.336659
$334$	30.3622	1.66135
$335$	6.14344	0.335652
$336$	12.4951	0.681664
$337$	−5.26472	−0.286787	−0.143394	−	0.989666i	$-0.545802\pi$
$337$	−5.26472	−0.286787	−0.143394	+	0.989666i	$0.545802\pi$
$338$	11.1108	0.604348
$339$	2.45502	0.133338
$340$	11.8452	0.642395
$341$	0	0
$342$	−13.5985	−0.735321
$343$	44.5863	2.40743
$344$	−1.82776	−0.0985460
$345$	5.14344	0.276914
$346$	−38.7815	−2.08491
$347$	−30.5500	−1.64001	−0.820005	−	0.572356i	$-0.806029\pi$
$347$	−30.5500	−1.64001	−0.820005	+	0.572356i	$0.806029\pi$
$348$	−10.8892	−0.583723
$349$	34.6223	1.85329	0.926646	−	0.375936i	$-0.122679\pi$
$349$	34.6223	1.85329	0.926646	+	0.375936i	$0.122679\pi$
$350$	10.2869	0.549857
$351$	−4.26705	−0.227758
$352$	0	0
$353$	1.83187	0.0975005	0.0487502	−	0.998811i	$-0.484476\pi$
$353$	1.83187	0.0975005	0.0487502	+	0.998811i	$0.484476\pi$
$354$	29.0127	1.54201
$355$	−10.4946	−0.556996
$356$	−3.54909	−0.188101
$357$	−22.3800	−1.18447
$358$	−19.4233	−1.02655
$359$	16.5713	0.874599	0.437300	−	0.899316i	$-0.355935\pi$
$359$	16.5713	0.874599	0.437300	+	0.899316i	$0.355935\pi$
$360$	1.17756	0.0620626
$361$	21.6242	1.13811
$362$	28.2635	1.48549
$363$	0	0
$364$	52.5028	2.75190
$365$	5.62449	0.294399
$366$	14.5123	0.758572
$367$	−4.11465	−0.214783	−0.107391	−	0.994217i	$-0.534250\pi$
$367$	−4.11465	−0.214783	−0.107391	+	0.994217i	$0.534250\pi$
$368$	13.3293	0.694839
$369$	3.46410	0.180334
$370$	13.1072	0.681411
$371$	10.8355	0.562550
$372$	16.3087	0.845569
$373$	−21.0868	−1.09183	−0.545917	−	0.837840i	$-0.683818\pi$
$373$	−21.0868	−1.09183	−0.545917	+	0.837840i	$0.683818\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	6.30129	0.324964
$377$	18.2077	0.937745
$378$	−10.2869	−0.529100
$379$	−3.14344	−0.161468	−0.0807339	−	0.996736i	$-0.525726\pi$
$379$	−3.14344	−0.161468	−0.0807339	+	0.996736i	$0.525726\pi$
$380$	−16.2653	−0.834390
$381$	9.89154	0.506759
$382$	−38.6020	−1.97505
$383$	−10.9100	−0.557477	−0.278739	−	0.960367i	$-0.589916\pi$
$383$	−10.9100	−0.557477	−0.278739	+	0.960367i	$0.589916\pi$
$384$	9.06173	0.462429
$385$	0	0
$386$	−34.2869	−1.74516
$387$	−1.55216	−0.0789008
$388$	10.0437	0.509891
$389$	−29.4699	−1.49418	−0.747092	−	0.664721i	$-0.768551\pi$
$389$	−29.4699	−1.49418	−0.747092	+	0.664721i	$0.768551\pi$
$390$	−9.10386	−0.460992
$391$	−23.8741	−1.20737
$392$	19.1321	0.966317
$393$	−3.46410	−0.174741
$394$	−32.9892	−1.66197
$395$	16.3914	0.824741
$396$	0	0
$397$	7.03959	0.353307	0.176653	−	0.984273i	$-0.443473\pi$
$397$	7.03959	0.353307	0.176653	+	0.984273i	$0.443473\pi$
$398$	43.5042	2.18067
$399$	30.7311	1.53848
$400$	−2.59152	−0.129576
$401$	−1.50539	−0.0751758	−0.0375879	−	0.999293i	$-0.511967\pi$
$401$	−1.50539	−0.0751758	−0.0375879	+	0.999293i	$0.511967\pi$
$402$	−13.1072	−0.653727
$403$	−27.2696	−1.35840
$404$	−39.4587	−1.96314
$405$	1.00000	0.0496904
$406$	43.8947	2.17846
$407$	0	0
$408$	−5.46581	−0.270598
$409$	−37.2298	−1.84089	−0.920446	−	0.390869i	$-0.872175\pi$
$409$	−37.2298	−1.84089	−0.920446	+	0.390869i	$0.872175\pi$
$410$	7.39075	0.365003
$411$	6.53419	0.322308
$412$	18.4946	0.911164
$413$	−65.5656	−3.22627
$414$	−10.9737	−0.539326
$415$	6.17899	0.303315
$416$	−33.6422	−1.64944
$417$	19.6608	0.962793
$418$	0	0
$419$	−25.4838	−1.24497	−0.622483	−	0.782633i	$-0.713876\pi$
$419$	−25.4838	−1.24497	−0.622483	+	0.782633i	$0.713876\pi$
$420$	−12.3042	−0.600386
$421$	21.0288	1.02488	0.512440	−	0.858723i	$-0.328742\pi$
$421$	21.0288	1.02488	0.512440	+	0.858723i	$0.328742\pi$
$422$	−8.83471	−0.430067
$423$	5.35116	0.260182
$424$	2.64632	0.128517
$425$	4.64166	0.225153
$426$	22.3905	1.08482
$427$	−32.7964	−1.58713
$428$	−13.8942	−0.671604
$429$	0	0
$430$	−3.31158	−0.159698
$431$	31.5904	1.52166	0.760829	−	0.648953i	$-0.224793\pi$
$431$	31.5904	1.52166	0.760829	+	0.648953i	$0.224793\pi$
$432$	2.59152	0.124684
$433$	9.93573	0.477481	0.238740	−	0.971083i	$-0.423266\pi$
$433$	9.93573	0.477481	0.238740	+	0.971083i	$0.423266\pi$
$434$	−65.7409	−3.15566
$435$	−4.26705	−0.204589
$436$	−7.42511	−0.355598
$437$	32.7828	1.56821
$438$	−12.0000	−0.573382
$439$	29.2463	1.39585	0.697925	−	0.716171i	$-0.254107\pi$
$439$	29.2463	1.39585	0.697925	+	0.716171i	$0.254107\pi$
$440$	0	0
$441$	16.2473	0.773681
$442$	42.2570	2.00996
$443$	−28.7023	−1.36369	−0.681844	−	0.731497i	$-0.738822\pi$
$443$	−28.7023	−1.36369	−0.681844	+	0.731497i	$0.738822\pi$
$444$	−15.6776	−0.744028
$445$	−1.39075	−0.0659278
$446$	−54.4778	−2.57960
$447$	12.3042	0.581971
$448$	−56.1134	−2.65111
$449$	−12.2077	−0.576118	−0.288059	−	0.957613i	$-0.593010\pi$
$449$	−12.2077	−0.576118	−0.288059	+	0.957613i	$0.593010\pi$
$450$	2.13353	0.100575
$451$	0	0
$452$	−6.26504	−0.294683
$453$	5.80438	0.272714
$454$	−43.0070	−2.01842
$455$	20.5738	0.964514
$456$	7.50539	0.351472
$457$	13.1072	0.613129	0.306564	−	0.951850i	$-0.400821\pi$
$457$	13.1072	0.613129	0.306564	+	0.951850i	$0.400821\pi$
$458$	16.9837	0.793598
$459$	−4.64166	−0.216654
$460$	−13.1257	−0.611989
$461$	3.77014	0.175593	0.0877966	−	0.996138i	$-0.472017\pi$
$461$	3.77014	0.175593	0.0877966	+	0.996138i	$0.472017\pi$
$462$	0	0
$463$	−1.87145	−0.0869738	−0.0434869	−	0.999054i	$-0.513847\pi$
$463$	−1.87145	−0.0869738	−0.0434869	+	0.999054i	$0.513847\pi$
$464$	−11.0581	−0.513361
$465$	6.39075	0.296364
$466$	−0.913878	−0.0423346
$467$	27.8458	1.28855	0.644274	−	0.764795i	$-0.277160\pi$
$467$	27.8458	1.28855	0.644274	+	0.764795i	$0.277160\pi$
$468$	10.8892	0.503354
$469$	29.6209	1.36777
$470$	11.4168	0.526620
$471$	−0.351162	−0.0161807
$472$	−16.0129	−0.737056
$473$	0	0
$474$	−34.9715	−1.60629
$475$	−6.37371	−0.292446
$476$	57.1121	2.61773
$477$	2.24730	0.102897
$478$	−38.6669	−1.76858
$479$	2.66115	0.121591	0.0607956	−	0.998150i	$-0.480636\pi$
$479$	2.66115	0.121591	0.0607956	+	0.998150i	$0.480636\pi$
$480$	7.88417	0.359862
$481$	26.2144	1.19527
$482$	−43.6599	−1.98865
$483$	24.7994	1.12841
$484$	0	0
$485$	3.93573	0.178712
$486$	−2.13353	−0.0967787
$487$	11.8062	0.534989	0.267495	−	0.963559i	$-0.413804\pi$
$487$	11.8062	0.534989	0.267495	+	0.963559i	$0.413804\pi$
$488$	−8.00979	−0.362586
$489$	−14.6381	−0.661956
$490$	34.6640	1.56596
$491$	29.8156	1.34556	0.672780	−	0.739843i	$-0.265100\pi$
$491$	29.8156	1.34556	0.672780	+	0.739843i	$0.265100\pi$
$492$	−8.84014	−0.398544
$493$	19.8062	0.892026
$494$	−58.0253	−2.61068
$495$	0	0
$496$	16.5617	0.743643
$497$	−50.6002	−2.26973
$498$	−13.1830	−0.590746
$499$	−10.0288	−0.448951	−0.224475	−	0.974480i	$-0.572067\pi$
$499$	−10.0288	−0.448951	−0.224475	+	0.974480i	$0.572067\pi$
$500$	2.55193	0.114126
$501$	−14.2310	−0.635795
$502$	51.4032	2.29424
$503$	26.1996	1.16818	0.584090	−	0.811689i	$-0.301451\pi$
$503$	26.1996	1.16818	0.584090	+	0.811689i	$0.301451\pi$
$504$	5.67764	0.252902
$505$	−15.4623	−0.688063
$506$	0	0
$507$	−5.20772	−0.231283
$508$	−25.2425	−1.11996
$509$	−17.6776	−0.783547	−0.391774	−	0.920062i	$-0.628138\pi$
$509$	−17.6776	−0.783547	−0.391774	+	0.920062i	$0.628138\pi$
$510$	−9.90309	−0.438516
$511$	27.1188	1.19966
$512$	26.5353	1.17270
$513$	6.37371	0.281406
$514$	31.2833	1.37985
$515$	7.24730	0.319354
$516$	3.96101	0.174374
$517$	0	0
$518$	63.1969	2.77671
$519$	18.1772	0.797890
$520$	5.02469	0.220347
$521$	33.7568	1.47891	0.739456	−	0.673205i	$-0.235083\pi$
$521$	33.7568	1.47891	0.739456	+	0.673205i	$0.235083\pi$
$522$	9.10386	0.398465
$523$	−3.71255	−0.162339	−0.0811693	−	0.996700i	$-0.525865\pi$
$523$	−3.71255	−0.162339	−0.0811693	+	0.996700i	$0.525865\pi$
$524$	8.84014	0.386183
$525$	−4.82155	−0.210430
$526$	13.9823	0.609656
$527$	−29.6637	−1.29217
$528$	0	0
$529$	3.45502	0.150218
$530$	4.79468	0.208268
$531$	−13.5985	−0.590123
$532$	−78.4237	−3.40010
$533$	14.7815	0.640258
$534$	2.96720	0.128403
$535$	−5.44461	−0.235391
$536$	7.23425	0.312472
$537$	9.10386	0.392861
$538$	57.1390	2.46344
$539$	0	0
$540$	−2.55193	−0.109818
$541$	−22.3905	−0.962643	−0.481322	−	0.876544i	$-0.659843\pi$
$541$	−22.3905	−0.962643	−0.481322	+	0.876544i	$0.659843\pi$
$542$	10.8486	0.465988
$543$	−13.2473	−0.568496
$544$	−36.5956	−1.56902
$545$	−2.90961	−0.124634
$546$	−43.8947	−1.87852
$547$	−19.2324	−0.822320	−0.411160	−	0.911563i	$-0.634876\pi$
$547$	−19.2324	−0.822320	−0.411160	+	0.911563i	$0.634876\pi$
$548$	−16.6748	−0.712312
$549$	−6.80205	−0.290304
$550$	0	0
$551$	−27.1969	−1.15863
$552$	6.05669	0.257790
$553$	79.0319	3.36078
$554$	41.5630	1.76584
$555$	−6.14344	−0.260775
$556$	−50.1729	−2.12781
$557$	−37.9214	−1.60678	−0.803390	−	0.595453i	$-0.796973\pi$
$557$	−37.9214	−1.60678	−0.803390	+	0.595453i	$0.796973\pi$
$558$	−13.6348	−0.577208
$559$	−6.62315	−0.280130
$560$	−12.4951	−0.528015
$561$	0	0
$562$	9.87145	0.416402
$563$	−2.46258	−0.103785	−0.0518927	−	0.998653i	$-0.516525\pi$
$563$	−2.46258	−0.103785	−0.0518927	+	0.998653i	$0.516525\pi$
$564$	−13.6558	−0.575012
$565$	−2.45502	−0.103284
$566$	40.7954	1.71476
$567$	4.82155	0.202486
$568$	−12.3580	−0.518529
$569$	−6.48503	−0.271867	−0.135933	−	0.990718i	$-0.543403\pi$
$569$	−6.48503	−0.271867	−0.135933	+	0.990718i	$0.543403\pi$
$570$	13.5985	0.569577
$571$	−31.4643	−1.31674	−0.658369	−	0.752695i	$-0.728754\pi$
$571$	−31.4643	−1.31674	−0.658369	+	0.752695i	$0.728754\pi$
$572$	0	0
$573$	18.0931	0.755849
$574$	35.6348	1.48737
$575$	−5.14344	−0.214496
$576$	−11.6381	−0.484919
$577$	12.4658	0.518958	0.259479	−	0.965749i	$-0.416449\pi$
$577$	12.4658	0.518958	0.259479	+	0.965749i	$0.416449\pi$
$578$	9.69683	0.403335
$579$	16.0705	0.667869
$580$	10.8892	0.452150
$581$	29.7923	1.23599
$582$	−8.39697	−0.348066
$583$	0	0
$584$	6.62315	0.274068
$585$	4.26705	0.176421
$586$	−19.6599	−0.812143
$587$	30.4195	1.25555	0.627775	−	0.778395i	$-0.283966\pi$
$587$	30.4195	1.25555	0.627775	+	0.778395i	$0.283966\pi$
$588$	−41.4620	−1.70986
$589$	40.7328	1.67836
$590$	−29.0127	−1.19443
$591$	15.4623	0.636034
$592$	−15.9208	−0.654342
$593$	19.6756	0.807981	0.403990	−	0.914763i	$-0.367623\pi$
$593$	19.6756	0.807981	0.403990	+	0.914763i	$0.367623\pi$
$594$	0	0
$595$	22.3800	0.917489
$596$	−31.3996	−1.28618
$597$	−20.3907	−0.834538
$598$	−46.8252	−1.91482
$599$	8.01390	0.327439	0.163720	−	0.986507i	$-0.447651\pi$
$599$	8.01390	0.327439	0.163720	+	0.986507i	$0.447651\pi$
$600$	−1.17756	−0.0480735
$601$	30.7299	1.25350	0.626749	−	0.779221i	$-0.284385\pi$
$601$	30.7299	1.25350	0.626749	+	0.779221i	$0.284385\pi$
$602$	−15.9669	−0.650763
$603$	6.14344	0.250180
$604$	−14.8124	−0.602707
$605$	0	0
$606$	32.9892	1.34010
$607$	33.2260	1.34860	0.674301	−	0.738457i	$-0.264445\pi$
$607$	33.2260	1.34860	0.674301	+	0.738457i	$0.264445\pi$
$608$	50.2514	2.03796
$609$	−20.5738	−0.833692
$610$	−14.5123	−0.587588
$611$	22.8337	0.923752
$612$	11.8452	0.478813
$613$	20.9793	0.847347	0.423674	−	0.905815i	$-0.360740\pi$
$613$	20.9793	0.847347	0.423674	+	0.905815i	$0.360740\pi$
$614$	−7.80618	−0.315032
$615$	−3.46410	−0.139686
$616$	0	0
$617$	14.5738	0.586718	0.293359	−	0.956002i	$-0.405227\pi$
$617$	14.5738	0.586718	0.293359	+	0.956002i	$0.405227\pi$
$618$	−15.4623	−0.621985
$619$	5.21850	0.209749	0.104875	−	0.994485i	$-0.466556\pi$
$619$	5.21850	0.209749	0.104875	+	0.994485i	$0.466556\pi$
$620$	−16.3087	−0.654975
$621$	5.14344	0.206399
$622$	−27.6832	−1.10999
$623$	−6.70555	−0.268652
$624$	11.0581	0.442679
$625$	1.00000	0.0400000
$626$	−9.55970	−0.382082
$627$	0	0
$628$	0.896141	0.0357599
$629$	28.5158	1.13700
$630$	10.2869	0.409839
$631$	13.8458	0.551191	0.275596	−	0.961274i	$-0.411125\pi$
$631$	13.8458	0.551191	0.275596	+	0.961274i	$0.411125\pi$
$632$	19.3018	0.767784
$633$	4.14090	0.164586
$634$	−30.3970	−1.20722
$635$	−9.89154	−0.392534
$636$	−5.73496	−0.227406
$637$	69.3281	2.74688
$638$	0	0
$639$	−10.4946	−0.415160
$640$	−9.06173	−0.358196
$641$	−49.5769	−1.95817	−0.979085	−	0.203453i	$-0.934784\pi$
$641$	−49.5769	−1.95817	−0.979085	+	0.203453i	$0.934784\pi$
$642$	11.6162	0.458455
$643$	4.25809	0.167923	0.0839613	−	0.996469i	$-0.473243\pi$
$643$	4.25809	0.167923	0.0839613	+	0.996469i	$0.473243\pi$
$644$	−63.2862	−2.49383
$645$	1.55216	0.0611163
$646$	−63.1194	−2.48340
$647$	−24.6273	−0.968198	−0.484099	−	0.875013i	$-0.660852\pi$
$647$	−24.6273	−0.968198	−0.484099	+	0.875013i	$0.660852\pi$
$648$	1.17756	0.0462587
$649$	0	0
$650$	9.10386	0.357083
$651$	30.8133	1.20767
$652$	37.3553	1.46295
$653$	−2.98921	−0.116977	−0.0584885	−	0.998288i	$-0.518628\pi$
$653$	−2.98921	−0.116977	−0.0584885	+	0.998288i	$0.518628\pi$
$654$	6.20772	0.242741
$655$	3.46410	0.135354
$656$	−8.97727	−0.350504
$657$	5.62449	0.219432
$658$	55.0468	2.14595
$659$	−28.5158	−1.11082	−0.555408	−	0.831578i	$-0.687438\pi$
$659$	−28.5158	−1.11082	−0.555408	+	0.831578i	$0.687438\pi$
$660$	0	0
$661$	36.0288	1.40136	0.700679	−	0.713477i	$-0.252881\pi$
$661$	36.0288	1.40136	0.700679	+	0.713477i	$0.252881\pi$
$662$	3.60229	0.140007
$663$	−19.8062	−0.769208
$664$	7.27610	0.282368
$665$	−30.7311	−1.19170
$666$	13.1072	0.507893
$667$	−21.9473	−0.849804
$668$	36.3165	1.40513
$669$	25.5342	0.987209
$670$	13.1072	0.506375
$671$	0	0
$672$	38.0139	1.46642
$673$	6.92435	0.266914	0.133457	−	0.991055i	$-0.457392\pi$
$673$	6.92435	0.266914	0.133457	+	0.991055i	$0.457392\pi$
$674$	−11.2324	−0.432656
$675$	−1.00000	−0.0384900
$676$	13.2897	0.511143
$677$	20.1725	0.775293	0.387647	−	0.921808i	$-0.373288\pi$
$677$	20.1725	0.775293	0.387647	+	0.921808i	$0.373288\pi$
$678$	5.23785	0.201158
$679$	18.9763	0.728243
$680$	5.46581	0.209604
$681$	20.1577	0.772445
$682$	0	0
$683$	43.4838	1.66386	0.831931	−	0.554879i	$-0.187235\pi$
$683$	43.4838	1.66386	0.831931	+	0.554879i	$0.187235\pi$
$684$	−16.2653	−0.621917
$685$	−6.53419	−0.249659
$686$	95.1260	3.63193
$687$	−7.96041	−0.303709
$688$	4.02245	0.153355
$689$	9.58936	0.365325
$690$	10.9737	0.417760
$691$	16.6776	0.634447	0.317224	−	0.948351i	$-0.397249\pi$
$691$	16.6776	0.634447	0.317224	+	0.948351i	$0.397249\pi$
$692$	−46.3869	−1.76337
$693$	0	0
$694$	−65.1792	−2.47417
$695$	−19.6608	−0.745776
$696$	−5.02469	−0.190460
$697$	16.0792	0.609042
$698$	73.8676	2.79593
$699$	0.428342	0.0162014
$700$	12.3042	0.465057
$701$	−3.90727	−0.147576	−0.0737878	−	0.997274i	$-0.523509\pi$
$701$	−3.90727	−0.147576	−0.0737878	+	0.997274i	$0.523509\pi$
$702$	−9.10386	−0.343603
$703$	−39.1565	−1.47682
$704$	0	0
$705$	−5.35116	−0.201536
$706$	3.90834	0.147092
$707$	−74.5522	−2.80382
$708$	34.7023	1.30419
$709$	41.6026	1.56242	0.781209	−	0.624270i	$-0.214603\pi$
$709$	41.6026	1.56242	0.781209	+	0.624270i	$0.214603\pi$
$710$	−22.3905	−0.840301
$711$	16.3914	0.614726
$712$	−1.63768	−0.0613747
$713$	32.8705	1.23101
$714$	−47.7482	−1.78693
$715$	0	0
$716$	−23.2324	−0.868236
$717$	18.1235	0.676833
$718$	35.3553	1.31945
$719$	4.91004	0.183114	0.0915568	−	0.995800i	$-0.470816\pi$
$719$	4.91004	0.183114	0.0915568	+	0.995800i	$0.470816\pi$
$720$	−2.59152	−0.0965801
$721$	34.9432	1.30135
$722$	46.1357	1.71699
$723$	20.4637	0.761055
$724$	33.8062	1.25640
$725$	4.26705	0.158474
$726$	0	0
$727$	17.7131	0.656943	0.328471	−	0.944514i	$-0.393467\pi$
$727$	17.7131	0.656943	0.328471	+	0.944514i	$0.393467\pi$
$728$	24.2268	0.897904
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	−7.20460	−0.266472
$732$	17.3584	0.641583
$733$	−14.7131	−0.543440	−0.271720	−	0.962376i	$-0.587593\pi$
$733$	−14.7131	−0.543440	−0.271720	+	0.962376i	$0.587593\pi$
$734$	−8.77870	−0.324028
$735$	−16.2473	−0.599291
$736$	40.5518	1.49476
$737$	0	0
$738$	7.39075	0.272057
$739$	14.9226	0.548938	0.274469	−	0.961596i	$-0.411498\pi$
$739$	14.9226	0.548938	0.274469	+	0.961596i	$0.411498\pi$
$740$	15.6776	0.576321
$741$	27.1969	0.999104
$742$	23.1178	0.848680
$743$	52.6882	1.93294	0.966471	−	0.256775i	$-0.0826598\pi$
$743$	52.6882	1.93294	0.966471	+	0.256775i	$0.0826598\pi$
$744$	7.52546	0.275897
$745$	−12.3042	−0.450793
$746$	−44.9892	−1.64717
$747$	6.17899	0.226077
$748$	0	0
$749$	−26.2514	−0.959206
$750$	−2.13353	−0.0779053
$751$	14.5985	0.532706	0.266353	−	0.963876i	$-0.414181\pi$
$751$	14.5985	0.532706	0.266353	+	0.963876i	$0.414181\pi$
$752$	−13.8676	−0.505700
$753$	−24.0931	−0.878000
$754$	38.8466	1.41471
$755$	−5.80438	−0.211243
$756$	−12.3042	−0.447501
$757$	10.0782	0.366297	0.183149	−	0.983085i	$-0.441371\pi$
$757$	10.0782	0.366297	0.183149	+	0.983085i	$0.441371\pi$
$758$	−6.70662	−0.243595
$759$	0	0
$760$	−7.50539	−0.272249
$761$	−18.5129	−0.671092	−0.335546	−	0.942024i	$-0.608921\pi$
$761$	−18.5129	−0.671092	−0.335546	+	0.942024i	$0.608921\pi$
$762$	21.1039	0.764512
$763$	−14.0288	−0.507877
$764$	−46.1722	−1.67045
$765$	4.64166	0.167819
$766$	−23.2768	−0.841027
$767$	−58.0253	−2.09517
$768$	−3.94268	−0.142269
$769$	21.7674	0.784954	0.392477	−	0.919762i	$-0.371618\pi$
$769$	21.7674	0.784954	0.392477	+	0.919762i	$0.371618\pi$
$770$	0	0
$771$	−14.6627	−0.528066
$772$	−41.0109	−1.47601
$773$	−44.9003	−1.61495	−0.807475	−	0.589902i	$-0.799166\pi$
$773$	−44.9003	−1.61495	−0.807475	+	0.589902i	$0.799166\pi$
$774$	−3.31158	−0.119032
$775$	−6.39075	−0.229562
$776$	4.63454	0.166370
$777$	−29.6209	−1.06264
$778$	−62.8748	−2.25417
$779$	−22.0792	−0.791068
$780$	−10.8892	−0.389896
$781$	0	0
$782$	−50.9360	−1.82147
$783$	−4.26705	−0.152492
$784$	−42.1051	−1.50375
$785$	0.351162	0.0125335
$786$	−7.39075	−0.263619
$787$	−25.1054	−0.894911	−0.447455	−	0.894306i	$-0.647670\pi$
$787$	−25.1054	−0.894911	−0.447455	+	0.894306i	$0.647670\pi$
$788$	−39.4587	−1.40566
$789$	−6.55360	−0.233314
$790$	34.9715	1.24423
$791$	−11.8370	−0.420875
$792$	0	0
$793$	−29.0247	−1.03070
$794$	15.0191	0.533009
$795$	−2.24730	−0.0797036
$796$	52.0357	1.84436
$797$	2.98921	0.105883	0.0529417	−	0.998598i	$-0.483140\pi$
$797$	2.98921	0.105883	0.0529417	+	0.998598i	$0.483140\pi$
$798$	65.5656	2.32100
$799$	24.8383	0.878714
$800$	−7.88417	−0.278748
$801$	−1.39075	−0.0491397
$802$	−3.21179	−0.113412
$803$	0	0
$804$	−15.6776	−0.552908
$805$	−24.7994	−0.874062
$806$	−58.1805	−2.04932
$807$	−26.7815	−0.942753
$808$	−18.2077	−0.640545
$809$	−17.2909	−0.607914	−0.303957	−	0.952686i	$-0.598308\pi$
$809$	−17.2909	−0.607914	−0.303957	+	0.952686i	$0.598308\pi$
$810$	2.13353	0.0749644
$811$	−43.4625	−1.52617	−0.763087	−	0.646296i	$-0.776317\pi$
$811$	−43.4625	−1.52617	−0.763087	+	0.646296i	$0.776317\pi$
$812$	52.5028	1.84249
$813$	−5.08483	−0.178333
$814$	0	0
$815$	14.6381	0.512749
$816$	12.0289	0.421097
$817$	9.89303	0.346113
$818$	−79.4306	−2.77723
$819$	20.5738	0.718906
$820$	8.84014	0.308711
$821$	25.1351	0.877219	0.438610	−	0.898678i	$-0.355471\pi$
$821$	25.1351	0.877219	0.438610	+	0.898678i	$0.355471\pi$
$822$	13.9409	0.486243
$823$	−37.7419	−1.31560	−0.657800	−	0.753193i	$-0.728513\pi$
$823$	−37.7419	−1.31560	−0.657800	+	0.753193i	$0.728513\pi$
$824$	8.53410	0.297299
$825$	0	0
$826$	−139.886	−4.86725
$827$	−16.3190	−0.567467	−0.283733	−	0.958903i	$-0.591573\pi$
$827$	−16.3190	−0.567467	−0.283733	+	0.958903i	$0.591573\pi$
$828$	−13.1257	−0.456150
$829$	8.66374	0.300904	0.150452	−	0.988617i	$-0.451927\pi$
$829$	8.66374	0.300904	0.150452	+	0.988617i	$0.451927\pi$
$830$	13.1830	0.457590
$831$	−19.4809	−0.675785
$832$	−49.6601	−1.72166
$833$	75.4144	2.61295
$834$	41.9468	1.45250
$835$	14.2310	0.492485
$836$	0	0
$837$	6.39075	0.220897
$838$	−54.3704	−1.87819
$839$	11.5846	0.399944	0.199972	−	0.979802i	$-0.435915\pi$
$839$	11.5846	0.399944	0.199972	+	0.979802i	$0.435915\pi$
$840$	−5.67764	−0.195897
$841$	−10.7923	−0.372148
$842$	44.8655	1.54617
$843$	−4.62683	−0.159356
$844$	−10.5673	−0.363741
$845$	5.20772	0.179151
$846$	11.4168	0.392519
$847$	0	0
$848$	−5.82392	−0.199994
$849$	−19.1211	−0.656235
$850$	9.90309	0.339673
$851$	−31.5985	−1.08318
$852$	26.7815	0.917519
$853$	18.0121	0.616724	0.308362	−	0.951269i	$-0.400219\pi$
$853$	18.0121	0.616724	0.308362	+	0.951269i	$0.400219\pi$
$854$	−69.9719	−2.39439
$855$	−6.37371	−0.217976
$856$	−6.41132	−0.219135
$857$	13.0386	0.445391	0.222696	−	0.974888i	$-0.428514\pi$
$857$	13.0386	0.445391	0.222696	+	0.974888i	$0.428514\pi$
$858$	0	0
$859$	22.4442	0.765787	0.382894	−	0.923792i	$-0.374928\pi$
$859$	22.4442	0.765787	0.382894	+	0.923792i	$0.374928\pi$
$860$	−3.96101	−0.135069
$861$	−16.7023	−0.569214
$862$	67.3990	2.29562
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	7.88417	0.268225
$865$	−18.1772	−0.618043
$866$	21.1981	0.720342
$867$	−4.54498	−0.154356
$868$	−78.6333	−2.66899
$869$	0	0
$870$	−9.10386	−0.308650
$871$	26.2144	0.888241
$872$	−3.42622	−0.116027
$873$	3.93573	0.133204
$874$	69.9430	2.36586
$875$	4.82155	0.162998
$876$	−14.3533	−0.484953
$877$	41.0147	1.38497	0.692484	−	0.721433i	$-0.256516\pi$
$877$	41.0147	1.38497	0.692484	+	0.721433i	$0.256516\pi$
$878$	62.3977	2.10582
$879$	9.21475	0.310806
$880$	0	0
$881$	18.6232	0.627430	0.313715	−	0.949517i	$-0.398426\pi$
$881$	18.6232	0.627430	0.313715	+	0.949517i	$0.398426\pi$
$882$	34.6640	1.16720
$883$	−32.8458	−1.10535	−0.552674	−	0.833397i	$-0.686393\pi$
$883$	−32.8458	−1.10535	−0.552674	+	0.833397i	$0.686393\pi$
$884$	50.5440	1.69998
$885$	13.5985	0.457107
$886$	−61.2371	−2.05730
$887$	29.5710	0.992898	0.496449	−	0.868066i	$-0.334637\pi$
$887$	29.5710	0.992898	0.496449	+	0.868066i	$0.334637\pi$
$888$	−7.23425	−0.242765
$889$	−47.6925	−1.59956
$890$	−2.96720	−0.0994606
$891$	0	0
$892$	−65.1615	−2.18177
$893$	−34.1067	−1.14134
$894$	26.2514	0.877979
$895$	−9.10386	−0.304308
$896$	−43.6915	−1.45963
$897$	21.9473	0.732800
$898$	−26.0455	−0.869149
$899$	−27.2696	−0.909493
$900$	2.55193	0.0850643
$901$	10.4312	0.347514
$902$	0	0
$903$	7.48382	0.249046
$904$	−2.89092	−0.0961507
$905$	13.2473	0.440355
$906$	12.3838	0.411424
$907$	15.1681	0.503650	0.251825	−	0.967773i	$-0.418969\pi$
$907$	15.1681	0.503650	0.251825	+	0.967773i	$0.418969\pi$
$908$	−51.4410	−1.70713
$909$	−15.4623	−0.512852
$910$	43.8947	1.45509
$911$	−26.8961	−0.891109	−0.445554	−	0.895255i	$-0.646993\pi$
$911$	−26.8961	−0.891109	−0.445554	+	0.895255i	$0.646993\pi$
$912$	−16.5176	−0.546951
$913$	0	0
$914$	27.9645	0.924984
$915$	6.80205	0.224869
$916$	20.3144	0.671207
$917$	16.7023	0.551559
$918$	−9.90309	−0.326851
$919$	37.6878	1.24320	0.621602	−	0.783333i	$-0.286482\pi$
$919$	37.6878	1.24320	0.621602	+	0.783333i	$0.286482\pi$
$920$	−6.05669	−0.199683
$921$	3.65882	0.120562
$922$	8.04370	0.264905
$923$	−44.7810	−1.47399
$924$	0	0
$925$	6.14344	0.201995
$926$	−3.99279	−0.131211
$927$	7.24730	0.238033
$928$	−33.6422	−1.10436
$929$	46.7321	1.53323	0.766616	−	0.642106i	$-0.221939\pi$
$929$	46.7321	1.53323	0.766616	+	0.642106i	$0.221939\pi$
$930$	13.6348	0.447103
$931$	−103.556	−3.39390
$932$	−1.09310	−0.0358056
$933$	12.9753	0.424793
$934$	59.4096	1.94394
$935$	0	0
$936$	5.02469	0.164237
$937$	4.20946	0.137517	0.0687585	−	0.997633i	$-0.478096\pi$
$937$	4.20946	0.137517	0.0687585	+	0.997633i	$0.478096\pi$
$938$	63.1969	2.06345
$939$	4.48071	0.146222
$940$	13.6558	0.445403
$941$	36.4081	1.18687	0.593435	−	0.804882i	$-0.297771\pi$
$941$	36.4081	1.18687	0.593435	+	0.804882i	$0.297771\pi$
$942$	−0.749213	−0.0244107
$943$	−17.8174	−0.580215
$944$	35.2406	1.14698
$945$	−4.82155	−0.156845
$946$	0	0
$947$	−47.5589	−1.54546	−0.772728	−	0.634737i	$-0.781108\pi$
$947$	−47.5589	−1.54546	−0.772728	+	0.634737i	$0.781108\pi$
$948$	−41.8297	−1.35857
$949$	24.0000	0.779073
$950$	−13.5985	−0.441192
$951$	14.2473	0.462000
$952$	26.3536	0.854126
$953$	−14.9654	−0.484777	−0.242388	−	0.970179i	$-0.577931\pi$
$953$	−14.9654	−0.484777	−0.242388	+	0.970179i	$0.577931\pi$
$954$	4.79468	0.155233
$955$	−18.0931	−0.585478
$956$	−46.2498	−1.49582
$957$	0	0
$958$	5.67764	0.183436
$959$	−31.5049	−1.01735
$960$	11.6381	0.375616
$961$	9.84166	0.317473
$962$	55.9291	1.80322
$963$	−5.44461	−0.175450
$964$	−52.2220	−1.68196
$965$	−16.0705	−0.517329
$966$	52.9100	1.70235
$967$	−28.6271	−0.920585	−0.460293	−	0.887767i	$-0.652255\pi$
$967$	−28.6271	−0.920585	−0.460293	+	0.887767i	$0.652255\pi$
$968$	0	0
$969$	29.5846	0.950393
$970$	8.39697	0.269611
$971$	8.89614	0.285491	0.142745	−	0.989759i	$-0.454407\pi$
$971$	8.89614	0.285491	0.142745	+	0.989759i	$0.454407\pi$
$972$	−2.55193	−0.0818532
$973$	−94.7954	−3.03900
$974$	25.1888	0.807101
$975$	−4.26705	−0.136655
$976$	17.6276	0.564246
$977$	34.8211	1.11403	0.557013	−	0.830504i	$-0.311948\pi$
$977$	34.8211	1.11403	0.557013	+	0.830504i	$0.311948\pi$
$978$	−31.2306	−0.998646
$979$	0	0
$980$	41.4620	1.32445
$981$	−2.90961	−0.0928966
$982$	63.6124	2.02995
$983$	14.3619	0.458075	0.229038	−	0.973418i	$-0.426442\pi$
$983$	14.3619	0.458075	0.229038	+	0.973418i	$0.426442\pi$
$984$	−4.07917	−0.130039
$985$	−15.4623	−0.492670
$986$	42.2570	1.34574
$987$	−25.8009	−0.821251
$988$	−69.4046	−2.20806
$989$	7.98346	0.253859
$990$	0	0
$991$	16.2118	0.514986	0.257493	−	0.966280i	$-0.417104\pi$
$991$	16.2118	0.514986	0.257493	+	0.966280i	$0.417104\pi$
$992$	50.3858	1.59975
$993$	−1.68842	−0.0535805
$994$	−107.957	−3.42418
$995$	20.3907	0.646430
$996$	−15.7683	−0.499639
$997$	−25.9659	−0.822349	−0.411175	−	0.911557i	$-0.634881\pi$
$997$	−25.9659	−0.822349	−0.411175	+	0.911557i	$0.634881\pi$
$998$	−21.3967	−0.677301
$999$	−6.14344	−0.194370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.2.a.y.1.5	yes	6
3.2	odd	2		5445.2.a.bz.1.2		6
5.4	even	2		9075.2.a.dq.1.2		6
11.10	odd	2	inner	1815.2.a.y.1.2	✓	6
33.32	even	2		5445.2.a.bz.1.5		6
55.54	odd	2		9075.2.a.dq.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.y.1.2	✓	6	11.10	odd	2	inner
1815.2.a.y.1.5	yes	6	1.1	even	1	trivial
5445.2.a.bz.1.2		6	3.2	odd	2
5445.2.a.bz.1.5		6	33.32	even	2
9075.2.a.dq.1.2		6	5.4	even	2
9075.2.a.dq.1.5		6	55.54	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13353 q^{2} -1.00000 q^{3} +2.55193 q^{4} +1.00000 q^{5} -2.13353 q^{6} +4.82155 q^{7} +1.17756 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.13353 q^{2} -1.00000 q^{3} +2.55193 q^{4} +1.00000 q^{5} -2.13353 q^{6} +4.82155 q^{7} +1.17756 q^{8} +1.00000 q^{9} +2.13353 q^{10} -2.55193 q^{12} +4.26705 q^{13} +10.2869 q^{14} -1.00000 q^{15} -2.59152 q^{16} +4.64166 q^{17} +2.13353 q^{18} -6.37371 q^{19} +2.55193 q^{20} -4.82155 q^{21} -5.14344 q^{23} -1.17756 q^{24} +1.00000 q^{25} +9.10386 q^{26} -1.00000 q^{27} +12.3042 q^{28} +4.26705 q^{29} -2.13353 q^{30} -6.39075 q^{31} -7.88417 q^{32} +9.90309 q^{34} +4.82155 q^{35} +2.55193 q^{36} +6.14344 q^{37} -13.5985 q^{38} -4.26705 q^{39} +1.17756 q^{40} +3.46410 q^{41} -10.2869 q^{42} -1.55216 q^{43} +1.00000 q^{45} -10.9737 q^{46} +5.35116 q^{47} +2.59152 q^{48} +16.2473 q^{49} +2.13353 q^{50} -4.64166 q^{51} +10.8892 q^{52} +2.24730 q^{53} -2.13353 q^{54} +5.67764 q^{56} +6.37371 q^{57} +9.10386 q^{58} -13.5985 q^{59} -2.55193 q^{60} -6.80205 q^{61} -13.6348 q^{62} +4.82155 q^{63} -11.6381 q^{64} +4.26705 q^{65} +6.14344 q^{67} +11.8452 q^{68} +5.14344 q^{69} +10.2869 q^{70} -10.4946 q^{71} +1.17756 q^{72} +5.62449 q^{73} +13.1072 q^{74} -1.00000 q^{75} -16.2653 q^{76} -9.10386 q^{78} +16.3914 q^{79} -2.59152 q^{80} +1.00000 q^{81} +7.39075 q^{82} +6.17899 q^{83} -12.3042 q^{84} +4.64166 q^{85} -3.31158 q^{86} -4.26705 q^{87} -1.39075 q^{89} +2.13353 q^{90} +20.5738 q^{91} -13.1257 q^{92} +6.39075 q^{93} +11.4168 q^{94} -6.37371 q^{95} +7.88417 q^{96} +3.93573 q^{97} +34.6640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 14 q^{4} + 6 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 14 * q^4 + 6 * q^5 + 6 * q^9	\( 6 q - 6 q^{3} + 14 q^{4} + 6 q^{5} + 6 q^{9} - 14 q^{12} + 8 q^{14} - 6 q^{15} + 10 q^{16} + 14 q^{20} - 4 q^{23} + 6 q^{25} + 52 q^{26} - 6 q^{27} + 18 q^{31} + 26 q^{34} + 14 q^{36} + 10 q^{37} - 20 q^{38} - 8 q^{42} + 6 q^{45} - 10 q^{48} + 68 q^{49} - 16 q^{53} - 76 q^{56} + 52 q^{58} - 20 q^{59} - 14 q^{60} + 16 q^{64} + 10 q^{67} + 4 q^{69} + 8 q^{70} - 4 q^{71} - 6 q^{75} - 52 q^{78} + 10 q^{80} + 6 q^{81} - 12 q^{82} - 12 q^{86} + 48 q^{89} + 16 q^{91} + 30 q^{92} - 18 q^{93} + 2 q^{97}+O(q^{100}) \) 6 * q - 6 * q^3 + 14 * q^4 + 6 * q^5 + 6 * q^9 - 14 * q^12 + 8 * q^14 - 6 * q^15 + 10 * q^16 + 14 * q^20 - 4 * q^23 + 6 * q^25 + 52 * q^26 - 6 * q^27 + 18 * q^31 + 26 * q^34 + 14 * q^36 + 10 * q^37 - 20 * q^38 - 8 * q^42 + 6 * q^45 - 10 * q^48 + 68 * q^49 - 16 * q^53 - 76 * q^56 + 52 * q^58 - 20 * q^59 - 14 * q^60 + 16 * q^64 + 10 * q^67 + 4 * q^69 + 8 * q^70 - 4 * q^71 - 6 * q^75 - 52 * q^78 + 10 * q^80 + 6 * q^81 - 12 * q^82 - 12 * q^86 + 48 * q^89 + 16 * q^91 + 30 * q^92 - 18 * q^93 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more