LMFDB - Embedded newform 5445.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
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:	no (minimal twist has level 1815)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.13353$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.13353 q^{2} +2.55193 q^{4} -1.00000 q^{5} +4.82155 q^{7} -1.17756 q^{8} +O(q^{10})$	$q-2.13353 q^{2} +2.55193 q^{4} -1.00000 q^{5} +4.82155 q^{7} -1.17756 q^{8} +2.13353 q^{10} +4.26705 q^{13} -10.2869 q^{14} -2.59152 q^{16} -4.64166 q^{17} -6.37371 q^{19} -2.55193 q^{20} +5.14344 q^{23} +1.00000 q^{25} -9.10386 q^{26} +12.3042 q^{28} -4.26705 q^{29} -6.39075 q^{31} +7.88417 q^{32} +9.90309 q^{34} -4.82155 q^{35} +6.14344 q^{37} +13.5985 q^{38} +1.17756 q^{40} -3.46410 q^{41} -1.55216 q^{43} -10.9737 q^{46} -5.35116 q^{47} +16.2473 q^{49} -2.13353 q^{50} +10.8892 q^{52} -2.24730 q^{53} -5.67764 q^{56} +9.10386 q^{58} +13.5985 q^{59} -6.80205 q^{61} +13.6348 q^{62} -11.6381 q^{64} -4.26705 q^{65} +6.14344 q^{67} -11.8452 q^{68} +10.2869 q^{70} +10.4946 q^{71} +5.62449 q^{73} -13.1072 q^{74} -16.2653 q^{76} +16.3914 q^{79} +2.59152 q^{80} +7.39075 q^{82} -6.17899 q^{83} +4.64166 q^{85} +3.31158 q^{86} +1.39075 q^{89} +20.5738 q^{91} +13.1257 q^{92} +11.4168 q^{94} +6.37371 q^{95} +3.93573 q^{97} -34.6640 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 14 * q^4 - 6 * q^5	$ 6 q + 14 q^{4} - 6 q^{5} - 8 q^{14} + 10 q^{16} - 14 q^{20} + 4 q^{23} + 6 q^{25} - 52 q^{26} + 18 q^{31} + 26 q^{34} + 10 q^{37} + 20 q^{38} + 68 q^{49} + 16 q^{53} + 76 q^{56} + 52 q^{58} + 20 q^{59} + 16 q^{64} + 10 q^{67} + 8 q^{70} + 4 q^{71} - 10 q^{80} - 12 q^{82} + 12 q^{86} - 48 q^{89} + 16 q^{91} - 30 q^{92} + 2 q^{97}+O(q^{100}) $ 6 * q + 14 * q^4 - 6 * q^5 - 8 * q^14 + 10 * q^16 - 14 * q^20 + 4 * q^23 + 6 * q^25 - 52 * q^26 + 18 * q^31 + 26 * q^34 + 10 * q^37 + 20 * q^38 + 68 * q^49 + 16 * q^53 + 76 * q^56 + 52 * q^58 + 20 * q^59 + 16 * q^64 + 10 * q^67 + 8 * q^70 + 4 * q^71 - 10 * q^80 - 12 * q^82 + 12 * q^86 - 48 * q^89 + 16 * q^91 - 30 * q^92 + 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	$-0.772031\pi$
$2$	−2.13353	−1.50863	−0.754315	+	0.656513i	$0.772031\pi$
$3$	0	0
$3$	0	0
$4$	2.55193	1.27596
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	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
$9$	0	0
$10$	2.13353	0.674680
$11$	0	0
$11$	0	0
$12$	0	0
$13$	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
$15$	0	0
$16$	−2.59152	−0.647879
$17$	−4.64166	−1.12577	−0.562884	−	0.826536i	$-0.690308\pi$
$17$	−4.64166	−1.12577	−0.562884	+	0.826536i	$0.690308\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	0	0
$23$	5.14344	1.07248	0.536241	−	0.844065i	$-0.319844\pi$
$23$	5.14344	1.07248	0.536241	+	0.844065i	$0.319844\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−9.10386	−1.78541
$27$	0	0
$28$	12.3042	2.32528
$29$	−4.26705	−0.792371	−0.396186	−	0.918170i	$-0.629666\pi$
$29$	−4.26705	−0.792371	−0.396186	+	0.918170i	$0.629666\pi$
$30$	0	0
$31$	−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
$36$	0	0
$37$	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
$39$	0	0
$40$	1.17756	0.186188
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	−10.9737	−1.61798
$47$	−5.35116	−0.780547	−0.390274	−	0.920699i	$-0.627620\pi$
$47$	−5.35116	−0.780547	−0.390274	+	0.920699i	$0.627620\pi$
$48$	0	0
$49$	16.2473	2.32104
$50$	−2.13353	−0.301726
$51$	0	0
$52$	10.8892	1.51006
$53$	−2.24730	−0.308691	−0.154345	−	0.988017i	$-0.549327\pi$
$53$	−2.24730	−0.308691	−0.154345	+	0.988017i	$0.549327\pi$
$54$	0	0
$55$	0	0
$56$	−5.67764	−0.758706
$57$	0	0
$58$	9.10386	1.19540
$59$	13.5985	1.77037	0.885185	−	0.465240i	$-0.154032\pi$
$59$	13.5985	1.77037	0.885185	+	0.465240i	$0.154032\pi$
$60$	0	0
$61$	−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
$63$	0	0
$64$	−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
$69$	0	0
$70$	10.2869	1.22952
$71$	10.4946	1.24548	0.622740	−	0.782429i	$-0.286019\pi$
$71$	10.4946	1.24548	0.622740	+	0.782429i	$0.286019\pi$
$72$	0	0
$73$	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
$75$	0	0
$76$	−16.2653	−1.86575
$77$	0	0
$78$	0	0
$79$	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
$81$	0	0
$82$	7.39075	0.816172
$83$	−6.17899	−0.678232	−0.339116	−	0.940745i	$-0.610128\pi$
$83$	−6.17899	−0.678232	−0.339116	+	0.940745i	$0.610128\pi$
$84$	0	0
$85$	4.64166	0.503458
$86$	3.31158	0.357096
$87$	0	0
$88$	0	0
$89$	1.39075	0.147419	0.0737095	−	0.997280i	$-0.476516\pi$
$89$	1.39075	0.147419	0.0737095	+	0.997280i	$0.476516\pi$
$90$	0	0
$91$	20.5738	2.15672
$92$	13.1257	1.36845
$93$	0	0
$94$	11.4168	1.17756
$95$	6.37371	0.653929
$96$	0	0
$97$	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	$-0.220616\pi$
$101$	15.4623	1.53856	0.769278	+	0.638914i	$0.220616\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	4.79468	0.465700
$107$	5.44461	0.526350	0.263175	−	0.964748i	$-0.415230\pi$
$107$	5.44461	0.526350	0.263175	+	0.964748i	$0.415230\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	−12.4951	−1.18068
$113$	2.45502	0.230949	0.115474	−	0.993310i	$-0.463161\pi$
$113$	2.45502	0.230949	0.115474	+	0.993310i	$0.463161\pi$
$114$	0	0
$115$	−5.14344	−0.479629
$116$	−10.8892	−1.01104
$117$	0	0
$118$	−29.0127	−2.67083
$119$	−22.3800	−2.05157
$120$	0	0
$121$	0	0
$122$	14.5123	1.31389
$123$	0	0
$124$	−16.3087	−1.46457
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−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
$129$	0	0
$130$	9.10386	0.798461
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	−30.7311	−2.66473
$134$	−13.1072	−1.13229
$135$	0	0
$136$	5.46581	0.468689
$137$	6.53419	0.558254	0.279127	−	0.960254i	$-0.409955\pi$
$137$	6.53419	0.558254	0.279127	+	0.960254i	$0.409955\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	−22.3905	−1.87897
$143$	0	0
$144$	0	0
$145$	4.26705	0.354359
$146$	−12.0000	−0.993127
$147$	0	0
$148$	15.6776	1.28869
$149$	12.3042	1.00800	0.504001	−	0.863703i	$-0.331861\pi$
$149$	12.3042	1.00800	0.504001	+	0.863703i	$0.331861\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	0	0
$155$	6.39075	0.513317
$156$	0	0
$157$	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
$159$	0	0
$160$	−7.88417	−0.623299
$161$	24.7994	1.95446
$162$	0	0
$163$	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	$-0.685607\pi$
$167$	−14.2310	−1.10123	−0.550614	+	0.834760i	$0.685607\pi$
$168$	0	0
$169$	5.20772	0.400594
$170$	−9.90309	−0.759532
$171$	0	0
$172$	−3.96101	−0.302024
$173$	18.1772	1.38199	0.690993	−	0.722861i	$-0.257173\pi$
$173$	18.1772	1.38199	0.690993	+	0.722861i	$0.257173\pi$
$174$	0	0
$175$	4.82155	0.364475
$176$	0	0
$177$	0	0
$178$	−2.96720	−0.222401
$179$	9.10386	0.680454	0.340227	−	0.940343i	$-0.389496\pi$
$179$	9.10386	0.680454	0.340227	+	0.940343i	$0.389496\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	−6.05669	−0.446505
$185$	−6.14344	−0.451675
$186$	0	0
$187$	0	0
$188$	−13.6558	−0.995951
$189$	0	0
$190$	−13.5985	−0.986536
$191$	18.0931	1.30917	0.654584	−	0.755989i	$-0.272844\pi$
$191$	18.0931	1.30917	0.654584	+	0.755989i	$0.272844\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	41.4620	2.96157
$197$	15.4623	1.10164	0.550822	−	0.834623i	$-0.314314\pi$
$197$	15.4623	1.10164	0.550822	+	0.834623i	$0.314314\pi$
$198$	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
$201$	0	0
$202$	−32.9892	−2.32111
$203$	−20.5738	−1.44400
$204$	0	0
$205$	3.46410	0.241943
$206$	−15.4623	−1.07731
$207$	0	0
$208$	−11.0581	−0.766743
$209$	0	0
$210$	0	0
$211$	−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
$213$	0	0
$214$	−11.6162	−0.794067
$215$	1.55216	0.105857
$216$	0	0
$217$	−30.8133	−2.09174
$218$	6.20772	0.420440
$219$	0	0
$220$	0	0
$221$	−19.8062	−1.33231
$222$	0	0
$223$	−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
$225$	0	0
$226$	−5.23785	−0.348417
$227$	20.1577	1.33791	0.668957	−	0.743301i	$-0.266741\pi$
$227$	20.1577	1.33791	0.668957	+	0.743301i	$0.266741\pi$
$228$	0	0
$229$	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	$-0.495534\pi$
$233$	0.428342	0.0280616	0.0140308	+	0.999902i	$0.495534\pi$
$234$	0	0
$235$	5.35116	0.349071
$236$	34.7023	2.25893
$237$	0	0
$238$	47.7482	3.09506
$239$	18.1235	1.17231	0.586154	−	0.810199i	$-0.300641\pi$
$239$	18.1235	1.17231	0.586154	+	0.810199i	$0.300641\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−17.3584	−1.11125
$245$	−16.2473	−1.03800
$246$	0	0
$247$	−27.1969	−1.73050
$248$	7.52546	0.477867
$249$	0	0
$250$	2.13353	0.134936
$251$	−24.0931	−1.52074	−0.760371	−	0.649489i	$-0.774983\pi$
$251$	−24.0931	−1.52074	−0.760371	+	0.649489i	$0.774983\pi$
$252$	0	0
$253$	0	0
$254$	21.1039	1.32417
$255$	0	0
$256$	3.94268	0.246417
$257$	−14.6627	−0.914637	−0.457318	−	0.889303i	$-0.651190\pi$
$257$	−14.6627	−0.914637	−0.457318	+	0.889303i	$0.651190\pi$
$258$	0	0
$259$	29.6209	1.84055
$260$	−10.8892	−0.675320
$261$	0	0
$262$	7.39075	0.456602
$263$	−6.55360	−0.404112	−0.202056	−	0.979374i	$-0.564762\pi$
$263$	−6.55360	−0.404112	−0.202056	+	0.979374i	$0.564762\pi$
$264$	0	0
$265$	2.24730	0.138051
$266$	65.5656	4.02009
$267$	0	0
$268$	15.6776	0.957664
$269$	−26.7815	−1.63290	−0.816448	−	0.577419i	$-0.804060\pi$
$269$	−26.7815	−1.63290	−0.816448	+	0.577419i	$0.804060\pi$
$270$	0	0
$271$	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
$273$	0	0
$274$	−13.9409	−0.842198
$275$	0	0
$276$	0	0
$277$	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
$279$	0	0
$280$	5.67764	0.339304
$281$	−4.62683	−0.276013	−0.138007	−	0.990431i	$-0.544070\pi$
$281$	−4.62683	−0.276013	−0.138007	+	0.990431i	$0.544070\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	0	0
$287$	−16.7023	−0.985907
$288$	0	0
$289$	4.54498	0.267352
$290$	−9.10386	−0.534597
$291$	0	0
$292$	14.3533	0.839964
$293$	9.21475	0.538331	0.269166	−	0.963094i	$-0.413252\pi$
$293$	9.21475	0.538331	0.269166	+	0.963094i	$0.413252\pi$
$294$	0	0
$295$	−13.5985	−0.791733
$296$	−7.23425	−0.420482
$297$	0	0
$298$	−26.2514	−1.52070
$299$	21.9473	1.26925
$300$	0	0
$301$	−7.48382	−0.431360
$302$	12.3838	0.712607
$303$	0	0
$304$	16.5176	0.947347
$305$	6.80205	0.389484
$306$	0	0
$307$	−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
$309$	0	0
$310$	−13.6348	−0.774406
$311$	12.9753	0.735762	0.367881	−	0.929873i	$-0.380083\pi$
$311$	12.9753	0.735762	0.367881	+	0.929873i	$0.380083\pi$
$312$	0	0
$313$	−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
$315$	0	0
$316$	41.8297	2.35311
$317$	14.2473	0.800208	0.400104	−	0.916470i	$-0.368974\pi$
$317$	14.2473	0.800208	0.400104	+	0.916470i	$0.368974\pi$
$318$	0	0
$319$	0	0
$320$	11.6381	0.650587
$321$	0	0
$322$	−52.9100	−2.94856
$323$	29.5846	1.64613
$324$	0	0
$325$	4.26705	0.236693
$326$	−31.2306	−1.72971
$327$	0	0
$328$	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
$333$	0	0
$334$	30.3622	1.66135
$335$	−6.14344	−0.335652
$336$	0	0
$337$	−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
$339$	0	0
$340$	11.8452	0.642395
$341$	0	0
$342$	0	0
$343$	44.5863	2.40743
$344$	1.82776	0.0985460
$345$	0	0
$346$	−38.7815	−2.08491
$347$	30.5500	1.64001	0.820005	−	0.572356i	$-0.193971\pi$
$347$	30.5500	1.64001	0.820005	+	0.572356i	$0.193971\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	0	0
$353$	−1.83187	−0.0975005	−0.0487502	−	0.998811i	$-0.515524\pi$
$353$	−1.83187	−0.0975005	−0.0487502	+	0.998811i	$0.515524\pi$
$354$	0	0
$355$	−10.4946	−0.556996
$356$	3.54909	0.188101
$357$	0	0
$358$	−19.4233	−1.02655
$359$	−16.5713	−0.874599	−0.437300	−	0.899316i	$-0.644065\pi$
$359$	−16.5713	−0.874599	−0.437300	+	0.899316i	$0.644065\pi$
$360$	0	0
$361$	21.6242	1.13811
$362$	−28.2635	−1.48549
$363$	0	0
$364$	52.5028	2.75190
$365$	−5.62449	−0.294399
$366$	0	0
$367$	−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
$369$	0	0
$370$	13.1072	0.681411
$371$	−10.8355	−0.562550
$372$	0	0
$373$	−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
$375$	0	0
$376$	6.30129	0.324964
$377$	−18.2077	−0.937745
$378$	0	0
$379$	−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
$381$	0	0
$382$	−38.6020	−1.97505
$383$	10.9100	0.557477	0.278739	−	0.960367i	$-0.410084\pi$
$383$	10.9100	0.557477	0.278739	+	0.960367i	$0.410084\pi$
$384$	0	0
$385$	0	0
$386$	34.2869	1.74516
$387$	0	0
$388$	10.0437	0.509891
$389$	29.4699	1.49418	0.747092	−	0.664721i	$-0.231449\pi$
$389$	29.4699	1.49418	0.747092	+	0.664721i	$0.231449\pi$
$390$	0	0
$391$	−23.8741	−1.20737
$392$	−19.1321	−0.966317
$393$	0	0
$394$	−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
$399$	0	0
$400$	−2.59152	−0.129576
$401$	1.50539	0.0751758	0.0375879	−	0.999293i	$-0.488033\pi$
$401$	1.50539	0.0751758	0.0375879	+	0.999293i	$0.488033\pi$
$402$	0	0
$403$	−27.2696	−1.35840
$404$	39.4587	1.96314
$405$	0	0
$406$	43.8947	2.17846
$407$	0	0
$408$	0	0
$409$	−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
$411$	0	0
$412$	18.4946	0.911164
$413$	65.5656	3.22627
$414$	0	0
$415$	6.17899	0.303315
$416$	33.6422	1.64944
$417$	0	0
$418$	0	0
$419$	25.4838	1.24497	0.622483	−	0.782633i	$-0.286124\pi$
$419$	25.4838	1.24497	0.622483	+	0.782633i	$0.286124\pi$
$420$	0	0
$421$	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
$423$	0	0
$424$	2.64632	0.128517
$425$	−4.64166	−0.225153
$426$	0	0
$427$	−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	$-0.775207\pi$
$431$	−31.5904	−1.52166	−0.760829	+	0.648953i	$0.775207\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	−7.42511	−0.355598
$437$	−32.7828	−1.56821
$438$	0	0
$439$	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
$441$	0	0
$442$	42.2570	2.00996
$443$	28.7023	1.36369	0.681844	−	0.731497i	$-0.261178\pi$
$443$	28.7023	1.36369	0.681844	+	0.731497i	$0.261178\pi$
$444$	0	0
$445$	−1.39075	−0.0659278
$446$	54.4778	2.57960
$447$	0	0
$448$	−56.1134	−2.65111
$449$	12.2077	0.576118	0.288059	−	0.957613i	$-0.406990\pi$
$449$	12.2077	0.576118	0.288059	+	0.957613i	$0.406990\pi$
$450$	0	0
$451$	0	0
$452$	6.26504	0.294683
$453$	0	0
$454$	−43.0070	−2.01842
$455$	−20.5738	−0.964514
$456$	0	0
$457$	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
$459$	0	0
$460$	−13.1257	−0.611989
$461$	−3.77014	−0.175593	−0.0877966	−	0.996138i	$-0.527983\pi$
$461$	−3.77014	−0.175593	−0.0877966	+	0.996138i	$0.527983\pi$
$462$	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
$465$	0	0
$466$	−0.913878	−0.0423346
$467$	−27.8458	−1.28855	−0.644274	−	0.764795i	$-0.722840\pi$
$467$	−27.8458	−1.28855	−0.644274	+	0.764795i	$0.722840\pi$
$468$	0	0
$469$	29.6209	1.36777
$470$	−11.4168	−0.526620
$471$	0	0
$472$	−16.0129	−0.737056
$473$	0	0
$474$	0	0
$475$	−6.37371	−0.292446
$476$	−57.1121	−2.61773
$477$	0	0
$478$	−38.6669	−1.76858
$479$	−2.66115	−0.121591	−0.0607956	−	0.998150i	$-0.519364\pi$
$479$	−2.66115	−0.121591	−0.0607956	+	0.998150i	$0.519364\pi$
$480$	0	0
$481$	26.2144	1.19527
$482$	43.6599	1.98865
$483$	0	0
$484$	0	0
$485$	−3.93573	−0.178712
$486$	0	0
$487$	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
$489$	0	0
$490$	34.6640	1.56596
$491$	−29.8156	−1.34556	−0.672780	−	0.739843i	$-0.734900\pi$
$491$	−29.8156	−1.34556	−0.672780	+	0.739843i	$0.734900\pi$
$492$	0	0
$493$	19.8062	0.892026
$494$	58.0253	2.61068
$495$	0	0
$496$	16.5617	0.743643
$497$	50.6002	2.26973
$498$	0	0
$499$	−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
$501$	0	0
$502$	51.4032	2.29424
$503$	−26.1996	−1.16818	−0.584090	−	0.811689i	$-0.698549\pi$
$503$	−26.1996	−1.16818	−0.584090	+	0.811689i	$0.698549\pi$
$504$	0	0
$505$	−15.4623	−0.688063
$506$	0	0
$507$	0	0
$508$	−25.2425	−1.11996
$509$	17.6776	0.783547	0.391774	−	0.920062i	$-0.371862\pi$
$509$	17.6776	0.783547	0.391774	+	0.920062i	$0.371862\pi$
$510$	0	0
$511$	27.1188	1.19966
$512$	−26.5353	−1.17270
$513$	0	0
$514$	31.2833	1.37985
$515$	−7.24730	−0.319354
$516$	0	0
$517$	0	0
$518$	−63.1969	−2.77671
$519$	0	0
$520$	5.02469	0.220347
$521$	−33.7568	−1.47891	−0.739456	−	0.673205i	$-0.764917\pi$
$521$	−33.7568	−1.47891	−0.739456	+	0.673205i	$0.764917\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	13.9823	0.609656
$527$	29.6637	1.29217
$528$	0	0
$529$	3.45502	0.150218
$530$	−4.79468	−0.208268
$531$	0	0
$532$	−78.4237	−3.40010
$533$	−14.7815	−0.640258
$534$	0	0
$535$	−5.44461	−0.235391
$536$	−7.23425	−0.312472
$537$	0	0
$538$	57.1390	2.46344
$539$	0	0
$540$	0	0
$541$	−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
$543$	0	0
$544$	−36.5956	−1.56902
$545$	2.90961	0.124634
$546$	0	0
$547$	−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
$549$	0	0
$550$	0	0
$551$	27.1969	1.15863
$552$	0	0
$553$	79.0319	3.36078
$554$	−41.5630	−1.76584
$555$	0	0
$556$	−50.1729	−2.12781
$557$	37.9214	1.60678	0.803390	−	0.595453i	$-0.203027\pi$
$557$	37.9214	1.60678	0.803390	+	0.595453i	$0.203027\pi$
$558$	0	0
$559$	−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	$-0.483475\pi$
$563$	2.46258	0.103785	0.0518927	+	0.998653i	$0.483475\pi$
$564$	0	0
$565$	−2.45502	−0.103284
$566$	−40.7954	−1.71476
$567$	0	0
$568$	−12.3580	−0.518529
$569$	6.48503	0.271867	0.135933	−	0.990718i	$-0.456597\pi$
$569$	6.48503	0.271867	0.135933	+	0.990718i	$0.456597\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	35.6348	1.48737
$575$	5.14344	0.214496
$576$	0	0
$577$	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
$579$	0	0
$580$	10.8892	0.452150
$581$	−29.7923	−1.23599
$582$	0	0
$583$	0	0
$584$	−6.62315	−0.274068
$585$	0	0
$586$	−19.6599	−0.812143
$587$	−30.4195	−1.25555	−0.627775	−	0.778395i	$-0.716034\pi$
$587$	−30.4195	−1.25555	−0.627775	+	0.778395i	$0.716034\pi$
$588$	0	0
$589$	40.7328	1.67836
$590$	29.0127	1.19443
$591$	0	0
$592$	−15.9208	−0.654342
$593$	−19.6756	−0.807981	−0.403990	−	0.914763i	$-0.632377\pi$
$593$	−19.6756	−0.807981	−0.403990	+	0.914763i	$0.632377\pi$
$594$	0	0
$595$	22.3800	0.917489
$596$	31.3996	1.28618
$597$	0	0
$598$	−46.8252	−1.91482
$599$	−8.01390	−0.327439	−0.163720	−	0.986507i	$-0.552349\pi$
$599$	−8.01390	−0.327439	−0.163720	+	0.986507i	$0.552349\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	−14.8124	−0.602707
$605$	0	0
$606$	0	0
$607$	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
$609$	0	0
$610$	−14.5123	−0.587588
$611$	−22.8337	−0.923752
$612$	0	0
$613$	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
$615$	0	0
$616$	0	0
$617$	−14.5738	−0.586718	−0.293359	−	0.956002i	$-0.594773\pi$
$617$	−14.5738	−0.586718	−0.293359	+	0.956002i	$0.594773\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	−27.6832	−1.10999
$623$	6.70555	0.268652
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.55970	0.382082
$627$	0	0
$628$	0.896141	0.0357599
$629$	−28.5158	−1.13700
$630$	0	0
$631$	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
$633$	0	0
$634$	−30.3970	−1.20722
$635$	9.89154	0.392534
$636$	0	0
$637$	69.3281	2.74688
$638$	0	0
$639$	0	0
$640$	−9.06173	−0.358196
$641$	49.5769	1.95817	0.979085	−	0.203453i	$-0.0652163\pi$
$641$	49.5769	1.95817	0.979085	+	0.203453i	$0.0652163\pi$
$642$	0	0
$643$	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
$645$	0	0
$646$	−63.1194	−2.48340
$647$	24.6273	0.968198	0.484099	−	0.875013i	$-0.339148\pi$
$647$	24.6273	0.968198	0.484099	+	0.875013i	$0.339148\pi$
$648$	0	0
$649$	0	0
$650$	−9.10386	−0.357083
$651$	0	0
$652$	37.3553	1.46295
$653$	2.98921	0.116977	0.0584885	−	0.998288i	$-0.481372\pi$
$653$	2.98921	0.116977	0.0584885	+	0.998288i	$0.481372\pi$
$654$	0	0
$655$	3.46410	0.135354
$656$	8.97727	0.350504
$657$	0	0
$658$	55.0468	2.14595
$659$	28.5158	1.11082	0.555408	−	0.831578i	$-0.312562\pi$
$659$	28.5158	1.11082	0.555408	+	0.831578i	$0.312562\pi$
$660$	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
$663$	0	0
$664$	7.27610	0.282368
$665$	30.7311	1.19170
$666$	0	0
$667$	−21.9473	−0.849804
$668$	−36.3165	−1.40513
$669$	0	0
$670$	13.1072	0.506375
$671$	0	0
$672$	0	0
$673$	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
$675$	0	0
$676$	13.2897	0.511143
$677$	−20.1725	−0.775293	−0.387647	−	0.921808i	$-0.626712\pi$
$677$	−20.1725	−0.775293	−0.387647	+	0.921808i	$0.626712\pi$
$678$	0	0
$679$	18.9763	0.728243
$680$	−5.46581	−0.209604
$681$	0	0
$682$	0	0
$683$	−43.4838	−1.66386	−0.831931	−	0.554879i	$-0.812765\pi$
$683$	−43.4838	−1.66386	−0.831931	+	0.554879i	$0.812765\pi$
$684$	0	0
$685$	−6.53419	−0.249659
$686$	−95.1260	−3.63193
$687$	0	0
$688$	4.02245	0.153355
$689$	−9.58936	−0.365325
$690$	0	0
$691$	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
$696$	0	0
$697$	16.0792	0.609042
$698$	−73.8676	−2.79593
$699$	0	0
$700$	12.3042	0.465057
$701$	3.90727	0.147576	0.0737878	−	0.997274i	$-0.476491\pi$
$701$	3.90727	0.147576	0.0737878	+	0.997274i	$0.476491\pi$
$702$	0	0
$703$	−39.1565	−1.47682
$704$	0	0
$705$	0	0
$706$	3.90834	0.147092
$707$	74.5522	2.80382
$708$	0	0
$709$	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
$711$	0	0
$712$	−1.63768	−0.0613747
$713$	−32.8705	−1.23101
$714$	0	0
$715$	0	0
$716$	23.2324	0.868236
$717$	0	0
$718$	35.3553	1.31945
$719$	−4.91004	−0.183114	−0.0915568	−	0.995800i	$-0.529184\pi$
$719$	−4.91004	−0.183114	−0.0915568	+	0.995800i	$0.529184\pi$
$720$	0	0
$721$	34.9432	1.30135
$722$	−46.1357	−1.71699
$723$	0	0
$724$	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
$729$	0	0
$730$	12.0000	0.444140
$731$	7.20460	0.266472
$732$	0	0
$733$	−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
$735$	0	0
$736$	40.5518	1.49476
$737$	0	0
$738$	0	0
$739$	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
$741$	0	0
$742$	23.1178	0.848680
$743$	−52.6882	−1.93294	−0.966471	−	0.256775i	$-0.917340\pi$
$743$	−52.6882	−1.93294	−0.966471	+	0.256775i	$0.917340\pi$
$744$	0	0
$745$	−12.3042	−0.450793
$746$	44.9892	1.64717
$747$	0	0
$748$	0	0
$749$	26.2514	0.959206
$750$	0	0
$751$	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
$753$	0	0
$754$	38.8466	1.41471
$755$	5.80438	0.211243
$756$	0	0
$757$	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	$-0.391079\pi$
$761$	18.5129	0.671092	0.335546	+	0.942024i	$0.391079\pi$
$762$	0	0
$763$	−14.0288	−0.507877
$764$	46.1722	1.67045
$765$	0	0
$766$	−23.2768	−0.841027
$767$	58.0253	2.09517
$768$	0	0
$769$	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
$771$	0	0
$772$	−41.0109	−1.47601
$773$	44.9003	1.61495	0.807475	−	0.589902i	$-0.200834\pi$
$773$	44.9003	1.61495	0.807475	+	0.589902i	$0.200834\pi$
$774$	0	0
$775$	−6.39075	−0.229562
$776$	−4.63454	−0.166370
$777$	0	0
$778$	−62.8748	−2.25417
$779$	22.0792	0.791068
$780$	0	0
$781$	0	0
$782$	50.9360	1.82147
$783$	0	0
$784$	−42.1051	−1.50375
$785$	−0.351162	−0.0125335
$786$	0	0
$787$	−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
$789$	0	0
$790$	34.9715	1.24423
$791$	11.8370	0.420875
$792$	0	0
$793$	−29.0247	−1.03070
$794$	−15.0191	−0.533009
$795$	0	0
$796$	52.0357	1.84436
$797$	−2.98921	−0.105883	−0.0529417	−	0.998598i	$-0.516860\pi$
$797$	−2.98921	−0.105883	−0.0529417	+	0.998598i	$0.516860\pi$
$798$	0	0
$799$	24.8383	0.878714
$800$	7.88417	0.278748
$801$	0	0
$802$	−3.21179	−0.113412
$803$	0	0
$804$	0	0
$805$	−24.7994	−0.874062
$806$	58.1805	2.04932
$807$	0	0
$808$	−18.2077	−0.640545
$809$	17.2909	0.607914	0.303957	−	0.952686i	$-0.401692\pi$
$809$	17.2909	0.607914	0.303957	+	0.952686i	$0.401692\pi$
$810$	0	0
$811$	−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
$813$	0	0
$814$	0	0
$815$	−14.6381	−0.512749
$816$	0	0
$817$	9.89303	0.346113
$818$	79.4306	2.77723
$819$	0	0
$820$	8.84014	0.308711
$821$	−25.1351	−0.877219	−0.438610	−	0.898678i	$-0.644529\pi$
$821$	−25.1351	−0.877219	−0.438610	+	0.898678i	$0.644529\pi$
$822$	0	0
$823$	−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	$-0.408427\pi$
$827$	16.3190	0.567467	0.283733	+	0.958903i	$0.408427\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	−49.6601	−1.72166
$833$	−75.4144	−2.61295
$834$	0	0
$835$	14.2310	0.492485
$836$	0	0
$837$	0	0
$838$	−54.3704	−1.87819
$839$	−11.5846	−0.399944	−0.199972	−	0.979802i	$-0.564085\pi$
$839$	−11.5846	−0.399944	−0.199972	+	0.979802i	$0.564085\pi$
$840$	0	0
$841$	−10.7923	−0.372148
$842$	−44.8655	−1.54617
$843$	0	0
$844$	−10.5673	−0.363741
$845$	−5.20772	−0.179151
$846$	0	0
$847$	0	0
$848$	5.82392	0.199994
$849$	0	0
$850$	9.90309	0.339673
$851$	31.5985	1.08318
$852$	0	0
$853$	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
$855$	0	0
$856$	−6.41132	−0.219135
$857$	−13.0386	−0.445391	−0.222696	−	0.974888i	$-0.571486\pi$
$857$	−13.0386	−0.445391	−0.222696	+	0.974888i	$0.571486\pi$
$858$	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
$861$	0	0
$862$	67.3990	2.29562
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−18.1772	−0.618043
$866$	−21.1981	−0.720342
$867$	0	0
$868$	−78.6333	−2.66899
$869$	0	0
$870$	0	0
$871$	26.2144	0.888241
$872$	3.42622	0.116027
$873$	0	0
$874$	69.9430	2.36586
$875$	−4.82155	−0.162998
$876$	0	0
$877$	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
$879$	0	0
$880$	0	0
$881$	−18.6232	−0.627430	−0.313715	−	0.949517i	$-0.601574\pi$
$881$	−18.6232	−0.627430	−0.313715	+	0.949517i	$0.601574\pi$
$882$	0	0
$883$	−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
$885$	0	0
$886$	−61.2371	−2.05730
$887$	−29.5710	−0.992898	−0.496449	−	0.868066i	$-0.665363\pi$
$887$	−29.5710	−0.992898	−0.496449	+	0.868066i	$0.665363\pi$
$888$	0	0
$889$	−47.6925	−1.59956
$890$	2.96720	0.0994606
$891$	0	0
$892$	−65.1615	−2.18177
$893$	34.1067	1.14134
$894$	0	0
$895$	−9.10386	−0.304308
$896$	43.6915	1.45963
$897$	0	0
$898$	−26.0455	−0.869149
$899$	27.2696	0.909493
$900$	0	0
$901$	10.4312	0.347514
$902$	0	0
$903$	0	0
$904$	−2.89092	−0.0961507
$905$	−13.2473	−0.440355
$906$	0	0
$907$	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
$909$	0	0
$910$	43.8947	1.45509
$911$	26.8961	0.891109	0.445554	−	0.895255i	$-0.353007\pi$
$911$	26.8961	0.891109	0.445554	+	0.895255i	$0.353007\pi$
$912$	0	0
$913$	0	0
$914$	−27.9645	−0.924984
$915$	0	0
$916$	20.3144	0.671207
$917$	−16.7023	−0.551559
$918$	0	0
$919$	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
$921$	0	0
$922$	8.04370	0.264905
$923$	44.7810	1.47399
$924$	0	0
$925$	6.14344	0.201995
$926$	3.99279	0.131211
$927$	0	0
$928$	−33.6422	−1.10436
$929$	−46.7321	−1.53323	−0.766616	−	0.642106i	$-0.778061\pi$
$929$	−46.7321	−1.53323	−0.766616	+	0.642106i	$0.778061\pi$
$930$	0	0
$931$	−103.556	−3.39390
$932$	1.09310	0.0358056
$933$	0	0
$934$	59.4096	1.94394
$935$	0	0
$936$	0	0
$937$	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
$939$	0	0
$940$	13.6558	0.445403
$941$	−36.4081	−1.18687	−0.593435	−	0.804882i	$-0.702229\pi$
$941$	−36.4081	−1.18687	−0.593435	+	0.804882i	$0.702229\pi$
$942$	0	0
$943$	−17.8174	−0.580215
$944$	−35.2406	−1.14698
$945$	0	0
$946$	0	0
$947$	47.5589	1.54546	0.772728	−	0.634737i	$-0.218892\pi$
$947$	47.5589	1.54546	0.772728	+	0.634737i	$0.218892\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	13.5985	0.441192
$951$	0	0
$952$	26.3536	0.854126
$953$	14.9654	0.484777	0.242388	−	0.970179i	$-0.422069\pi$
$953$	14.9654	0.484777	0.242388	+	0.970179i	$0.422069\pi$
$954$	0	0
$955$	−18.0931	−0.585478
$956$	46.2498	1.49582
$957$	0	0
$958$	5.67764	0.183436
$959$	31.5049	1.01735
$960$	0	0
$961$	9.84166	0.317473
$962$	−55.9291	−1.80322
$963$	0	0
$964$	−52.2220	−1.68196
$965$	16.0705	0.517329
$966$	0	0
$967$	−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
$969$	0	0
$970$	8.39697	0.269611
$971$	−8.89614	−0.285491	−0.142745	−	0.989759i	$-0.545593\pi$
$971$	−8.89614	−0.285491	−0.142745	+	0.989759i	$0.545593\pi$
$972$	0	0
$973$	−94.7954	−3.03900
$974$	−25.1888	−0.807101
$975$	0	0
$976$	17.6276	0.564246
$977$	−34.8211	−1.11403	−0.557013	−	0.830504i	$-0.688052\pi$
$977$	−34.8211	−1.11403	−0.557013	+	0.830504i	$0.688052\pi$
$978$	0	0
$979$	0	0
$980$	−41.4620	−1.32445
$981$	0	0
$982$	63.6124	2.02995
$983$	−14.3619	−0.458075	−0.229038	−	0.973418i	$-0.573558\pi$
$983$	−14.3619	−0.458075	−0.229038	+	0.973418i	$0.573558\pi$
$984$	0	0
$985$	−15.4623	−0.492670
$986$	−42.2570	−1.34574
$987$	0	0
$988$	−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
$993$	0	0
$994$	−107.957	−3.42418
$995$	−20.3907	−0.646430
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bz.1.2		6
3.2	odd	2		1815.2.a.y.1.5	yes	6
11.10	odd	2	inner	5445.2.a.bz.1.5		6
15.14	odd	2		9075.2.a.dq.1.2		6
33.32	even	2		1815.2.a.y.1.2	✓	6
165.164	even	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	33.32	even	2
1815.2.a.y.1.5	yes	6	3.2	odd	2
5445.2.a.bz.1.2		6	1.1	even	1	trivial
5445.2.a.bz.1.5		6	11.10	odd	2	inner
9075.2.a.dq.1.2		6	15.14	odd	2
9075.2.a.dq.1.5		6	165.164	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

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

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