LMFDB - Embedded newform 5445.2.a.bn.1.4

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:	$4$
Coefficient field:	4.4.27648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 3 $ x^4 - 6*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.33441$ 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)

$f(q)$	$=$	$q+2.33441 q^{2} +3.44949 q^{4} +1.00000 q^{5} -1.04930 q^{7} +3.38371 q^{8} +O(q^{10})$	$q+2.33441 q^{2} +3.44949 q^{4} +1.00000 q^{5} -1.04930 q^{7} +3.38371 q^{8} +2.33441 q^{10} +5.71812 q^{13} -2.44949 q^{14} +1.00000 q^{16} -1.04930 q^{17} +4.66883 q^{19} +3.44949 q^{20} +4.89898 q^{23} +1.00000 q^{25} +13.3485 q^{26} -3.61953 q^{28} +2.57024 q^{29} -2.00000 q^{31} -4.43300 q^{32} -2.44949 q^{34} -1.04930 q^{35} -6.89898 q^{37} +10.8990 q^{38} +3.38371 q^{40} +6.76742 q^{41} +1.04930 q^{43} +11.4362 q^{46} -9.79796 q^{47} -5.89898 q^{49} +2.33441 q^{50} +19.7246 q^{52} +10.8990 q^{53} -3.55051 q^{56} +6.00000 q^{58} +10.8990 q^{59} -7.23907 q^{61} -4.66883 q^{62} -12.3485 q^{64} +5.71812 q^{65} +8.89898 q^{67} -3.61953 q^{68} -2.44949 q^{70} +1.10102 q^{71} +1.52094 q^{73} -16.1051 q^{74} +16.1051 q^{76} +9.80930 q^{79} +1.00000 q^{80} +15.7980 q^{82} -3.61953 q^{83} -1.04930 q^{85} +2.44949 q^{86} +3.79796 q^{89} -6.00000 q^{91} +16.8990 q^{92} -22.8725 q^{94} +4.66883 q^{95} +16.6969 q^{97} -13.7707 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^4 + 4 * q^5	$ 4 q + 4 q^{4} + 4 q^{5} + 4 q^{16} + 4 q^{20} + 4 q^{25} + 24 q^{26} - 8 q^{31} - 8 q^{37} + 24 q^{38} - 4 q^{49} + 24 q^{53} - 24 q^{56} + 24 q^{58} + 24 q^{59} - 20 q^{64} + 16 q^{67} + 24 q^{71} + 4 q^{80} + 24 q^{82} - 24 q^{89} - 24 q^{91} + 48 q^{92} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 4 * q^5 + 4 * q^16 + 4 * q^20 + 4 * q^25 + 24 * q^26 - 8 * q^31 - 8 * q^37 + 24 * q^38 - 4 * q^49 + 24 * q^53 - 24 * q^56 + 24 * q^58 + 24 * q^59 - 20 * q^64 + 16 * q^67 + 24 * q^71 + 4 * q^80 + 24 * q^82 - 24 * q^89 - 24 * q^91 + 48 * q^92 + 8 * 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 $

$2$	2.33441	1.65068	0.825340	−	0.564636i	$-0.190983\pi$
$2$	2.33441	1.65068	0.825340	+	0.564636i	$0.190983\pi$
$3$	0	0
$3$	0	0
$4$	3.44949	1.72474
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.04930	−0.396596	−0.198298	−	0.980142i	$-0.563541\pi$
$7$	−1.04930	−0.396596	−0.198298	+	0.980142i	$0.563541\pi$
$8$	3.38371	1.19632
$9$	0	0
$10$	2.33441	0.738207
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.71812	1.58592	0.792961	−	0.609272i	$-0.208538\pi$
$13$	5.71812	1.58592	0.792961	+	0.609272i	$0.208538\pi$
$14$	−2.44949	−0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.04930	−0.254491	−0.127246	−	0.991871i	$-0.540614\pi$
$17$	−1.04930	−0.254491	−0.127246	+	0.991871i	$0.540614\pi$
$18$	0	0
$19$	4.66883	1.07110	0.535551	−	0.844503i	$-0.320104\pi$
$19$	4.66883	1.07110	0.535551	+	0.844503i	$0.320104\pi$
$20$	3.44949	0.771329
$21$	0	0
$22$	0	0
$23$	4.89898	1.02151	0.510754	−	0.859727i	$-0.329366\pi$
$23$	4.89898	1.02151	0.510754	+	0.859727i	$0.329366\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	13.3485	2.61785
$27$	0	0
$28$	−3.61953	−0.684027
$29$	2.57024	0.477281	0.238641	−	0.971108i	$-0.423298\pi$
$29$	2.57024	0.477281	0.238641	+	0.971108i	$0.423298\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−4.43300	−0.783652
$33$	0	0
$34$	−2.44949	−0.420084
$35$	−1.04930	−0.177363
$36$	0	0
$37$	−6.89898	−1.13419	−0.567093	−	0.823654i	$-0.691932\pi$
$37$	−6.89898	−1.13419	−0.567093	+	0.823654i	$0.691932\pi$
$38$	10.8990	1.76805
$39$	0	0
$40$	3.38371	0.535011
$41$	6.76742	1.05689	0.528447	−	0.848967i	$-0.322775\pi$
$41$	6.76742	1.05689	0.528447	+	0.848967i	$0.322775\pi$
$42$	0	0
$43$	1.04930	0.160016	0.0800080	−	0.996794i	$-0.474505\pi$
$43$	1.04930	0.160016	0.0800080	+	0.996794i	$0.474505\pi$
$44$	0	0
$45$	0	0
$46$	11.4362	1.68618
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	−5.89898	−0.842711
$50$	2.33441	0.330136
$51$	0	0
$52$	19.7246	2.73531
$53$	10.8990	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$54$	0	0
$55$	0	0
$56$	−3.55051	−0.474457
$57$	0	0
$58$	6.00000	0.787839
$59$	10.8990	1.41893	0.709463	−	0.704743i	$-0.248937\pi$
$59$	10.8990	1.41893	0.709463	+	0.704743i	$0.248937\pi$
$60$	0	0
$61$	−7.23907	−0.926867	−0.463434	−	0.886132i	$-0.653383\pi$
$61$	−7.23907	−0.926867	−0.463434	+	0.886132i	$0.653383\pi$
$62$	−4.66883	−0.592942
$63$	0	0
$64$	−12.3485	−1.54356
$65$	5.71812	0.709246
$66$	0	0
$67$	8.89898	1.08718	0.543592	−	0.839350i	$-0.317064\pi$
$67$	8.89898	1.08718	0.543592	+	0.839350i	$0.317064\pi$
$68$	−3.61953	−0.438933
$69$	0	0
$70$	−2.44949	−0.292770
$71$	1.10102	0.130667	0.0653335	−	0.997863i	$-0.479189\pi$
$71$	1.10102	0.130667	0.0653335	+	0.997863i	$0.479189\pi$
$72$	0	0
$73$	1.52094	0.178013	0.0890064	−	0.996031i	$-0.471631\pi$
$73$	1.52094	0.178013	0.0890064	+	0.996031i	$0.471631\pi$
$74$	−16.1051	−1.87218
$75$	0	0
$76$	16.1051	1.84738
$77$	0	0
$78$	0	0
$79$	9.80930	1.10363	0.551816	−	0.833966i	$-0.313935\pi$
$79$	9.80930	1.10363	0.551816	+	0.833966i	$0.313935\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	15.7980	1.74459
$83$	−3.61953	−0.397295	−0.198648	−	0.980071i	$-0.563655\pi$
$83$	−3.61953	−0.397295	−0.198648	+	0.980071i	$0.563655\pi$
$84$	0	0
$85$	−1.04930	−0.113812
$86$	2.44949	0.264135
$87$	0	0
$88$	0	0
$89$	3.79796	0.402583	0.201291	−	0.979531i	$-0.435486\pi$
$89$	3.79796	0.402583	0.201291	+	0.979531i	$0.435486\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	16.8990	1.76184
$93$	0	0
$94$	−22.8725	−2.35912
$95$	4.66883	0.479012
$96$	0	0
$97$	16.6969	1.69532	0.847659	−	0.530542i	$-0.178012\pi$
$97$	16.6969	1.69532	0.847659	+	0.530542i	$0.178012\pi$
$98$	−13.7707	−1.39105
$99$	0	0
$100$	3.44949	0.344949
$101$	−18.2037	−1.81133	−0.905666	−	0.423991i	$-0.860629\pi$
$101$	−18.2037	−1.81133	−0.905666	+	0.423991i	$0.860629\pi$
$102$	0	0
$103$	12.8990	1.27097	0.635487	−	0.772111i	$-0.280799\pi$
$103$	12.8990	1.27097	0.635487	+	0.772111i	$0.280799\pi$
$104$	19.3485	1.89727
$105$	0	0
$106$	25.4427	2.47122
$107$	−7.81671	−0.755670	−0.377835	−	0.925873i	$-0.623331\pi$
$107$	−7.81671	−0.755670	−0.377835	+	0.925873i	$0.623331\pi$
$108$	0	0
$109$	−13.5348	−1.29640	−0.648201	−	0.761469i	$-0.724478\pi$
$109$	−13.5348	−1.29640	−0.648201	+	0.761469i	$0.724478\pi$
$110$	0	0
$111$	0	0
$112$	−1.04930	−0.0991491
$113$	1.10102	0.103575	0.0517876	−	0.998658i	$-0.483508\pi$
$113$	1.10102	0.103575	0.0517876	+	0.998658i	$0.483508\pi$
$114$	0	0
$115$	4.89898	0.456832
$116$	8.86601	0.823188
$117$	0	0
$118$	25.4427	2.34219
$119$	1.10102	0.100930
$120$	0	0
$121$	0	0
$122$	−16.8990	−1.52996
$123$	0	0
$124$	−6.89898	−0.619547
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.7246	−1.75028	−0.875138	−	0.483873i	$-0.839230\pi$
$127$	−19.7246	−1.75028	−0.875138	+	0.483873i	$0.839230\pi$
$128$	−19.9604	−1.76427
$129$	0	0
$130$	13.3485	1.17074
$131$	15.6334	1.36590	0.682949	−	0.730466i	$-0.260697\pi$
$131$	15.6334	1.36590	0.682949	+	0.730466i	$0.260697\pi$
$132$	0	0
$133$	−4.89898	−0.424795
$134$	20.7739	1.79459
$135$	0	0
$136$	−3.55051	−0.304454
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	10.9646	0.930005	0.465003	−	0.885309i	$-0.346053\pi$
$139$	10.9646	0.930005	0.465003	+	0.885309i	$0.346053\pi$
$140$	−3.61953	−0.305906
$141$	0	0
$142$	2.57024	0.215690
$143$	0	0
$144$	0	0
$145$	2.57024	0.213447
$146$	3.55051	0.293842
$147$	0	0
$148$	−23.7980	−1.95618
$149$	0.471647	0.0386389	0.0193194	−	0.999813i	$-0.493850\pi$
$149$	0.471647	0.0386389	0.0193194	+	0.999813i	$0.493850\pi$
$150$	0	0
$151$	0.471647	0.0383821	0.0191911	−	0.999816i	$-0.493891\pi$
$151$	0.471647	0.0383821	0.0191911	+	0.999816i	$0.493891\pi$
$152$	15.7980	1.28138
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	7.79796	0.622345	0.311172	−	0.950353i	$-0.399278\pi$
$157$	7.79796	0.622345	0.311172	+	0.950353i	$0.399278\pi$
$158$	22.8990	1.82174
$159$	0	0
$160$	−4.43300	−0.350460
$161$	−5.14048	−0.405126
$162$	0	0
$163$	−6.69694	−0.524545	−0.262272	−	0.964994i	$-0.584472\pi$
$163$	−6.69694	−0.524545	−0.262272	+	0.964994i	$0.584472\pi$
$164$	23.3441	1.82287
$165$	0	0
$166$	−8.44949	−0.655808
$167$	12.0139	0.929663	0.464832	−	0.885399i	$-0.346115\pi$
$167$	12.0139	0.929663	0.464832	+	0.885399i	$0.346115\pi$
$168$	0	0
$169$	19.6969	1.51515
$170$	−2.44949	−0.187867
$171$	0	0
$172$	3.61953	0.275987
$173$	13.4288	1.02098	0.510488	−	0.859885i	$-0.329465\pi$
$173$	13.4288	1.02098	0.510488	+	0.859885i	$0.329465\pi$
$174$	0	0
$175$	−1.04930	−0.0793193
$176$	0	0
$177$	0	0
$178$	8.86601	0.664536
$179$	21.7980	1.62926	0.814628	−	0.579984i	$-0.196941\pi$
$179$	21.7980	1.62926	0.814628	+	0.579984i	$0.196941\pi$
$180$	0	0
$181$	3.10102	0.230497	0.115249	−	0.993337i	$-0.463234\pi$
$181$	3.10102	0.230497	0.115249	+	0.993337i	$0.463234\pi$
$182$	−14.0065	−1.03823
$183$	0	0
$184$	16.5767	1.22205
$185$	−6.89898	−0.507223
$186$	0	0
$187$	0	0
$188$	−33.7980	−2.46497
$189$	0	0
$190$	10.8990	0.790695
$191$	−19.5959	−1.41791	−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959	−1.41791	−0.708955	+	0.705253i	$0.750833\pi$
$192$	0	0
$193$	−20.1963	−1.45376	−0.726879	−	0.686765i	$-0.759030\pi$
$193$	−20.1963	−1.45376	−0.726879	+	0.686765i	$0.759030\pi$
$194$	38.9776	2.79843
$195$	0	0
$196$	−20.3485	−1.45346
$197$	−19.7246	−1.40532	−0.702660	−	0.711526i	$-0.748005\pi$
$197$	−19.7246	−1.40532	−0.702660	+	0.711526i	$0.748005\pi$
$198$	0	0
$199$	−17.7980	−1.26166	−0.630832	−	0.775920i	$-0.717286\pi$
$199$	−17.7980	−1.26166	−0.630832	+	0.775920i	$0.717286\pi$
$200$	3.38371	0.239264
$201$	0	0
$202$	−42.4949	−2.98993
$203$	−2.69694	−0.189288
$204$	0	0
$205$	6.76742	0.472657
$206$	30.1116	2.09797
$207$	0	0
$208$	5.71812	0.396481
$209$	0	0
$210$	0	0
$211$	14.0065	0.964246	0.482123	−	0.876103i	$-0.339866\pi$
$211$	14.0065	0.964246	0.482123	+	0.876103i	$0.339866\pi$
$212$	37.5959	2.58210
$213$	0	0
$214$	−18.2474	−1.24737
$215$	1.04930	0.0715613
$216$	0	0
$217$	2.09859	0.142462
$218$	−31.5959	−2.13995
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−1.79796	−0.120400	−0.0602001	−	0.998186i	$-0.519174\pi$
$223$	−1.79796	−0.120400	−0.0602001	+	0.998186i	$0.519174\pi$
$224$	4.65153	0.310793
$225$	0	0
$226$	2.57024	0.170970
$227$	1.52094	0.100949	0.0504743	−	0.998725i	$-0.483927\pi$
$227$	1.52094	0.100949	0.0504743	+	0.998725i	$0.483927\pi$
$228$	0	0
$229$	−21.5959	−1.42710	−0.713549	−	0.700605i	$-0.752913\pi$
$229$	−21.5959	−1.42710	−0.713549	+	0.700605i	$0.752913\pi$
$230$	11.4362	0.754084
$231$	0	0
$232$	8.69694	0.570982
$233$	−10.3870	−0.680472	−0.340236	−	0.940340i	$-0.610507\pi$
$233$	−10.3870	−0.680472	−0.340236	+	0.940340i	$0.610507\pi$
$234$	0	0
$235$	−9.79796	−0.639148
$236$	37.5959	2.44729
$237$	0	0
$238$	2.57024	0.166604
$239$	−25.9144	−1.67626	−0.838131	−	0.545469i	$-0.816352\pi$
$239$	−25.9144	−1.67626	−0.838131	+	0.545469i	$0.816352\pi$
$240$	0	0
$241$	−16.5767	−1.06780	−0.533900	−	0.845547i	$-0.679274\pi$
$241$	−16.5767	−1.06780	−0.533900	+	0.845547i	$0.679274\pi$
$242$	0	0
$243$	0	0
$244$	−24.9711	−1.59861
$245$	−5.89898	−0.376872
$246$	0	0
$247$	26.6969	1.69869
$248$	−6.76742	−0.429732
$249$	0	0
$250$	2.33441	0.147641
$251$	13.1010	0.826929	0.413465	−	0.910520i	$-0.364319\pi$
$251$	13.1010	0.826929	0.413465	+	0.910520i	$0.364319\pi$
$252$	0	0
$253$	0	0
$254$	−46.0454	−2.88915
$255$	0	0
$256$	−21.8990	−1.36869
$257$	−25.5959	−1.59663	−0.798315	−	0.602240i	$-0.794275\pi$
$257$	−25.5959	−1.59663	−0.798315	+	0.602240i	$0.794275\pi$
$258$	0	0
$259$	7.23907	0.449814
$260$	19.7246	1.22327
$261$	0	0
$262$	36.4949	2.25466
$263$	−20.1963	−1.24535	−0.622677	−	0.782479i	$-0.713955\pi$
$263$	−20.1963	−1.24535	−0.622677	+	0.782479i	$0.713955\pi$
$264$	0	0
$265$	10.8990	0.669519
$266$	−11.4362	−0.701201
$267$	0	0
$268$	30.6969	1.87511
$269$	−7.59592	−0.463131	−0.231566	−	0.972819i	$-0.574385\pi$
$269$	−7.59592	−0.463131	−0.231566	+	0.972819i	$0.574385\pi$
$270$	0	0
$271$	−22.4008	−1.36075	−0.680377	−	0.732862i	$-0.738184\pi$
$271$	−22.4008	−1.36075	−0.680377	+	0.732862i	$0.738184\pi$
$272$	−1.04930	−0.0636229
$273$	0	0
$274$	42.0195	2.53849
$275$	0	0
$276$	0	0
$277$	26.4920	1.59175	0.795876	−	0.605460i	$-0.207011\pi$
$277$	26.4920	1.59175	0.795876	+	0.605460i	$0.207011\pi$
$278$	25.5959	1.53514
$279$	0	0
$280$	−3.55051	−0.212184
$281$	−6.76742	−0.403710	−0.201855	−	0.979415i	$-0.564697\pi$
$281$	−6.76742	−0.403710	−0.201855	+	0.979415i	$0.564697\pi$
$282$	0	0
$283$	−29.0623	−1.72757	−0.863786	−	0.503859i	$-0.831913\pi$
$283$	−29.0623	−1.72757	−0.863786	+	0.503859i	$0.831913\pi$
$284$	3.79796	0.225367
$285$	0	0
$286$	0	0
$287$	−7.10102	−0.419160
$288$	0	0
$289$	−15.8990	−0.935234
$290$	6.00000	0.352332
$291$	0	0
$292$	5.24648	0.307027
$293$	−29.0623	−1.69784	−0.848918	−	0.528525i	$-0.822745\pi$
$293$	−29.0623	−1.69784	−0.848918	+	0.528525i	$0.822745\pi$
$294$	0	0
$295$	10.8990	0.634563
$296$	−23.3441	−1.35685
$297$	0	0
$298$	1.10102	0.0637804
$299$	28.0130	1.62003
$300$	0	0
$301$	−1.10102	−0.0634618
$302$	1.10102	0.0633566
$303$	0	0
$304$	4.66883	0.267776
$305$	−7.23907	−0.414508
$306$	0	0
$307$	−15.5274	−0.886197	−0.443099	−	0.896473i	$-0.646121\pi$
$307$	−15.5274	−0.886197	−0.443099	+	0.896473i	$0.646121\pi$
$308$	0	0
$309$	0	0
$310$	−4.66883	−0.265172
$311$	22.8990	1.29848	0.649241	−	0.760583i	$-0.275087\pi$
$311$	22.8990	1.29848	0.649241	+	0.760583i	$0.275087\pi$
$312$	0	0
$313$	−21.1010	−1.19270	−0.596350	−	0.802724i	$-0.703383\pi$
$313$	−21.1010	−1.19270	−0.596350	+	0.802724i	$0.703383\pi$
$314$	18.2037	1.02729
$315$	0	0
$316$	33.8371	1.90349
$317$	−8.69694	−0.488469	−0.244234	−	0.969716i	$-0.578537\pi$
$317$	−8.69694	−0.488469	−0.244234	+	0.969716i	$0.578537\pi$
$318$	0	0
$319$	0	0
$320$	−12.3485	−0.690300
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−4.89898	−0.272587
$324$	0	0
$325$	5.71812	0.317184
$326$	−15.6334	−0.865856
$327$	0	0
$328$	22.8990	1.26438
$329$	10.2810	0.566807
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−12.4855	−0.685233
$333$	0	0
$334$	28.0454	1.53458
$335$	8.89898	0.486203
$336$	0	0
$337$	−10.8586	−0.591506	−0.295753	−	0.955265i	$-0.595570\pi$
$337$	−10.8586	−0.591506	−0.295753	+	0.955265i	$0.595570\pi$
$338$	45.9808	2.50103
$339$	0	0
$340$	−3.61953	−0.196297
$341$	0	0
$342$	0	0
$343$	13.5348	0.730813
$344$	3.55051	0.191431
$345$	0	0
$346$	31.3485	1.68530
$347$	12.9572	0.695578	0.347789	−	0.937573i	$-0.386932\pi$
$347$	12.9572	0.695578	0.347789	+	0.937573i	$0.386932\pi$
$348$	0	0
$349$	−7.23907	−0.387498	−0.193749	−	0.981051i	$-0.562065\pi$
$349$	−7.23907	−0.387498	−0.193749	+	0.981051i	$0.562065\pi$
$350$	−2.44949	−0.130931
$351$	0	0
$352$	0	0
$353$	−10.8990	−0.580094	−0.290047	−	0.957012i	$-0.593671\pi$
$353$	−10.8990	−0.580094	−0.290047	+	0.957012i	$0.593671\pi$
$354$	0	0
$355$	1.10102	0.0584361
$356$	13.1010	0.694353
$357$	0	0
$358$	50.8855	2.68938
$359$	31.0549	1.63901	0.819506	−	0.573070i	$-0.194248\pi$
$359$	31.0549	1.63901	0.819506	+	0.573070i	$0.194248\pi$
$360$	0	0
$361$	2.79796	0.147261
$362$	7.23907	0.380477
$363$	0	0
$364$	−20.6969	−1.08481
$365$	1.52094	0.0796098
$366$	0	0
$367$	−23.5959	−1.23170	−0.615848	−	0.787865i	$-0.711187\pi$
$367$	−23.5959	−1.23170	−0.615848	+	0.787865i	$0.711187\pi$
$368$	4.89898	0.255377
$369$	0	0
$370$	−16.1051	−0.837263
$371$	−11.4362	−0.593740
$372$	0	0
$373$	9.91530	0.513395	0.256698	−	0.966492i	$-0.417366\pi$
$373$	9.91530	0.513395	0.256698	+	0.966492i	$0.417366\pi$
$374$	0	0
$375$	0	0
$376$	−33.1534	−1.70976
$377$	14.6969	0.756931
$378$	0	0
$379$	−23.5959	−1.21204	−0.606020	−	0.795449i	$-0.707235\pi$
$379$	−23.5959	−1.21204	−0.606020	+	0.795449i	$0.707235\pi$
$380$	16.1051	0.826173
$381$	0	0
$382$	−45.7450	−2.34052
$383$	16.8990	0.863498	0.431749	−	0.901994i	$-0.357897\pi$
$383$	16.8990	0.863498	0.431749	+	0.901994i	$0.357897\pi$
$384$	0	0
$385$	0	0
$386$	−47.1464	−2.39969
$387$	0	0
$388$	57.5959	2.92399
$389$	−31.5959	−1.60198	−0.800988	−	0.598680i	$-0.795692\pi$
$389$	−31.5959	−1.60198	−0.800988	+	0.598680i	$0.795692\pi$
$390$	0	0
$391$	−5.14048	−0.259965
$392$	−19.9604	−1.00815
$393$	0	0
$394$	−46.0454	−2.31973
$395$	9.80930	0.493560
$396$	0	0
$397$	28.6969	1.44026	0.720129	−	0.693840i	$-0.244083\pi$
$397$	28.6969	1.44026	0.720129	+	0.693840i	$0.244083\pi$
$398$	−41.5478	−2.08260
$399$	0	0
$400$	1.00000	0.0500000
$401$	8.20204	0.409590	0.204795	−	0.978805i	$-0.434347\pi$
$401$	8.20204	0.409590	0.204795	+	0.978805i	$0.434347\pi$
$402$	0	0
$403$	−11.4362	−0.569680
$404$	−62.7934	−3.12409
$405$	0	0
$406$	−6.29577	−0.312454
$407$	0	0
$408$	0	0
$409$	17.7320	0.876792	0.438396	−	0.898782i	$-0.355547\pi$
$409$	17.7320	0.876792	0.438396	+	0.898782i	$0.355547\pi$
$410$	15.7980	0.780206
$411$	0	0
$412$	44.4949	2.19211
$413$	−11.4362	−0.562741
$414$	0	0
$415$	−3.61953	−0.177676
$416$	−25.3485	−1.24281
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−5.30306	−0.258455	−0.129228	−	0.991615i	$-0.541250\pi$
$421$	−5.30306	−0.258455	−0.129228	+	0.991615i	$0.541250\pi$
$422$	32.6969	1.59166
$423$	0	0
$424$	36.8790	1.79100
$425$	−1.04930	−0.0508983
$426$	0	0
$427$	7.59592	0.367592
$428$	−26.9637	−1.30334
$429$	0	0
$430$	2.44949	0.118125
$431$	4.19718	0.202171	0.101086	−	0.994878i	$-0.467768\pi$
$431$	4.19718	0.202171	0.101086	+	0.994878i	$0.467768\pi$
$432$	0	0
$433$	−12.6969	−0.610176	−0.305088	−	0.952324i	$-0.598686\pi$
$433$	−12.6969	−0.610176	−0.305088	+	0.952324i	$0.598686\pi$
$434$	4.89898	0.235159
$435$	0	0
$436$	−46.6883	−2.23596
$437$	22.8725	1.09414
$438$	0	0
$439$	−39.9209	−1.90532	−0.952659	−	0.304039i	$-0.901665\pi$
$439$	−39.9209	−1.90532	−0.952659	+	0.304039i	$0.901665\pi$
$440$	0	0
$441$	0	0
$442$	−14.0065	−0.666221
$443$	−2.69694	−0.128135	−0.0640677	−	0.997946i	$-0.520407\pi$
$443$	−2.69694	−0.128135	−0.0640677	+	0.997946i	$0.520407\pi$
$444$	0	0
$445$	3.79796	0.180041
$446$	−4.19718	−0.198742
$447$	0	0
$448$	12.9572	0.612170
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	3.79796	0.178641
$453$	0	0
$454$	3.55051	0.166634
$455$	−6.00000	−0.281284
$456$	0	0
$457$	27.4353	1.28337	0.641685	−	0.766968i	$-0.278236\pi$
$457$	27.4353	1.28337	0.641685	+	0.766968i	$0.278236\pi$
$458$	−50.4138	−2.35568
$459$	0	0
$460$	16.8990	0.787919
$461$	−13.0632	−0.608413	−0.304207	−	0.952606i	$-0.598391\pi$
$461$	−13.0632	−0.608413	−0.304207	+	0.952606i	$0.598391\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	2.57024	0.119320
$465$	0	0
$466$	−24.2474	−1.12324
$467$	−4.89898	−0.226698	−0.113349	−	0.993555i	$-0.536158\pi$
$467$	−4.89898	−0.226698	−0.113349	+	0.993555i	$0.536158\pi$
$468$	0	0
$469$	−9.33766	−0.431173
$470$	−22.8725	−1.05503
$471$	0	0
$472$	36.8790	1.69749
$473$	0	0
$474$	0	0
$475$	4.66883	0.214221
$476$	3.79796	0.174079
$477$	0	0
$478$	−60.4949	−2.76697
$479$	29.1683	1.33273	0.666366	−	0.745625i	$-0.267849\pi$
$479$	29.1683	1.33273	0.666366	+	0.745625i	$0.267849\pi$
$480$	0	0
$481$	−39.4492	−1.79873
$482$	−38.6969	−1.76260
$483$	0	0
$484$	0	0
$485$	16.6969	0.758169
$486$	0	0
$487$	8.89898	0.403251	0.201626	−	0.979463i	$-0.435378\pi$
$487$	8.89898	0.403251	0.201626	+	0.979463i	$0.435378\pi$
$488$	−24.4949	−1.10883
$489$	0	0
$490$	−13.7707	−0.622095
$491$	−37.3506	−1.68561	−0.842805	−	0.538219i	$-0.819097\pi$
$491$	−37.3506	−1.68561	−0.842805	+	0.538219i	$0.819097\pi$
$492$	0	0
$493$	−2.69694	−0.121464
$494$	62.3217	2.80399
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−1.15530	−0.0518221
$498$	0	0
$499$	13.7980	0.617681	0.308841	−	0.951114i	$-0.400059\pi$
$499$	13.7980	0.617681	0.308841	+	0.951114i	$0.400059\pi$
$500$	3.44949	0.154266
$501$	0	0
$502$	30.5832	1.36500
$503$	24.3934	1.08765	0.543825	−	0.839199i	$-0.316976\pi$
$503$	24.3934	1.08765	0.543825	+	0.839199i	$0.316976\pi$
$504$	0	0
$505$	−18.2037	−0.810053
$506$	0	0
$507$	0	0
$508$	−68.0398	−3.01878
$509$	−9.79796	−0.434287	−0.217143	−	0.976140i	$-0.569674\pi$
$509$	−9.79796	−0.434287	−0.217143	+	0.976140i	$0.569674\pi$
$510$	0	0
$511$	−1.59592	−0.0705993
$512$	−11.2004	−0.494993
$513$	0	0
$514$	−59.7515	−2.63552
$515$	12.8990	0.568397
$516$	0	0
$517$	0	0
$518$	16.8990	0.742499
$519$	0	0
$520$	19.3485	0.848487
$521$	23.3939	1.02490	0.512452	−	0.858716i	$-0.328737\pi$
$521$	23.3939	1.02490	0.512452	+	0.858716i	$0.328737\pi$
$522$	0	0
$523$	−11.5422	−0.504707	−0.252354	−	0.967635i	$-0.581205\pi$
$523$	−11.5422	−0.504707	−0.252354	+	0.967635i	$0.581205\pi$
$524$	53.9274	2.35583
$525$	0	0
$526$	−47.1464	−2.05568
$527$	2.09859	0.0914160
$528$	0	0
$529$	1.00000	0.0434783
$530$	25.4427	1.10516
$531$	0	0
$532$	−16.8990	−0.732664
$533$	38.6969	1.67615
$534$	0	0
$535$	−7.81671	−0.337946
$536$	30.1116	1.30062
$537$	0	0
$538$	−17.7320	−0.764482
$539$	0	0
$540$	0	0
$541$	−31.2669	−1.34427	−0.672134	−	0.740430i	$-0.734622\pi$
$541$	−31.2669	−1.34427	−0.672134	+	0.740430i	$0.734622\pi$
$542$	−52.2929	−2.24617
$543$	0	0
$544$	4.65153	0.199433
$545$	−13.5348	−0.579769
$546$	0	0
$547$	−32.1042	−1.37267	−0.686337	−	0.727283i	$-0.740783\pi$
$547$	−32.1042	−1.37267	−0.686337	+	0.727283i	$0.740783\pi$
$548$	62.0908	2.65239
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−10.2929	−0.437697
$554$	61.8434	2.62747
$555$	0	0
$556$	37.8223	1.60402
$557$	5.24648	0.222300	0.111150	−	0.993804i	$-0.464547\pi$
$557$	5.24648	0.222300	0.111150	+	0.993804i	$0.464547\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	−1.04930	−0.0443408
$561$	0	0
$562$	−15.7980	−0.666397
$563$	6.87342	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$563$	6.87342	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$564$	0	0
$565$	1.10102	0.0463203
$566$	−67.8434	−2.85167
$567$	0	0
$568$	3.72553	0.156320
$569$	−5.61212	−0.235272	−0.117636	−	0.993057i	$-0.537532\pi$
$569$	−5.61212	−0.235272	−0.117636	+	0.993057i	$0.537532\pi$
$570$	0	0
$571$	−10.9646	−0.458854	−0.229427	−	0.973326i	$-0.573685\pi$
$571$	−10.9646	−0.458854	−0.229427	+	0.973326i	$0.573685\pi$
$572$	0	0
$573$	0	0
$574$	−16.5767	−0.691899
$575$	4.89898	0.204302
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−37.1148	−1.54377
$579$	0	0
$580$	8.86601	0.368141
$581$	3.79796	0.157566
$582$	0	0
$583$	0	0
$584$	5.14643	0.212961
$585$	0	0
$586$	−67.8434	−2.80258
$587$	14.6969	0.606608	0.303304	−	0.952894i	$-0.401910\pi$
$587$	14.6969	0.606608	0.303304	+	0.952894i	$0.401910\pi$
$588$	0	0
$589$	−9.33766	−0.384751
$590$	25.4427	1.04746
$591$	0	0
$592$	−6.89898	−0.283546
$593$	−17.6260	−0.723814	−0.361907	−	0.932214i	$-0.617874\pi$
$593$	−17.6260	−0.723814	−0.361907	+	0.932214i	$0.617874\pi$
$594$	0	0
$595$	1.10102	0.0451374
$596$	1.62694	0.0666422
$597$	0	0
$598$	65.3939	2.67415
$599$	31.5959	1.29097	0.645487	−	0.763771i	$-0.276654\pi$
$599$	31.5959	1.29097	0.645487	+	0.763771i	$0.276654\pi$
$600$	0	0
$601$	−3.04189	−0.124081	−0.0620405	−	0.998074i	$-0.519761\pi$
$601$	−3.04189	−0.124081	−0.0620405	+	0.998074i	$0.519761\pi$
$602$	−2.57024	−0.104755
$603$	0	0
$604$	1.62694	0.0661994
$605$	0	0
$606$	0	0
$607$	−15.5274	−0.630239	−0.315119	−	0.949052i	$-0.602045\pi$
$607$	−15.5274	−0.630239	−0.315119	+	0.949052i	$0.602045\pi$
$608$	−20.6969	−0.839372
$609$	0	0
$610$	−16.8990	−0.684220
$611$	−56.0259	−2.26657
$612$	0	0
$613$	−29.5339	−1.19286	−0.596432	−	0.802664i	$-0.703415\pi$
$613$	−29.5339	−1.19286	−0.596432	+	0.802664i	$0.703415\pi$
$614$	−36.2474	−1.46283
$615$	0	0
$616$	0	0
$617$	13.1010	0.527427	0.263714	−	0.964601i	$-0.415053\pi$
$617$	13.1010	0.527427	0.263714	+	0.964601i	$0.415053\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−6.89898	−0.277070
$621$	0	0
$622$	53.4557	2.14338
$623$	−3.98518	−0.159663
$624$	0	0
$625$	1.00000	0.0400000
$626$	−49.2585	−1.96877
$627$	0	0
$628$	26.8990	1.07339
$629$	7.23907	0.288640
$630$	0	0
$631$	21.3939	0.851677	0.425838	−	0.904799i	$-0.359979\pi$
$631$	21.3939	0.851677	0.425838	+	0.904799i	$0.359979\pi$
$632$	33.1918	1.32030
$633$	0	0
$634$	−20.3023	−0.806306
$635$	−19.7246	−0.782747
$636$	0	0
$637$	−33.7311	−1.33647
$638$	0	0
$639$	0	0
$640$	−19.9604	−0.789005
$641$	−37.5959	−1.48495	−0.742475	−	0.669874i	$-0.766348\pi$
$641$	−37.5959	−1.48495	−0.742475	+	0.669874i	$0.766348\pi$
$642$	0	0
$643$	8.49490	0.335006	0.167503	−	0.985872i	$-0.446430\pi$
$643$	8.49490	0.335006	0.167503	+	0.985872i	$0.446430\pi$
$644$	−17.7320	−0.698739
$645$	0	0
$646$	−11.4362	−0.449953
$647$	40.8990	1.60790	0.803952	−	0.594694i	$-0.202727\pi$
$647$	40.8990	1.60790	0.803952	+	0.594694i	$0.202727\pi$
$648$	0	0
$649$	0	0
$650$	13.3485	0.523570
$651$	0	0
$652$	−23.1010	−0.904706
$653$	1.59592	0.0624531	0.0312265	−	0.999512i	$-0.490059\pi$
$653$	1.59592	0.0624531	0.0312265	+	0.999512i	$0.490059\pi$
$654$	0	0
$655$	15.6334	0.610849
$656$	6.76742	0.264223
$657$	0	0
$658$	24.0000	0.935617
$659$	36.4073	1.41823	0.709114	−	0.705094i	$-0.249095\pi$
$659$	36.4073	1.41823	0.709114	+	0.705094i	$0.249095\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	−23.3441	−0.907296
$663$	0	0
$664$	−12.2474	−0.475293
$665$	−4.89898	−0.189974
$666$	0	0
$667$	12.5915	0.487546
$668$	41.4418	1.60343
$669$	0	0
$670$	20.7739	0.802566
$671$	0	0
$672$	0	0
$673$	2.46424	0.0949894	0.0474947	−	0.998871i	$-0.484876\pi$
$673$	2.46424	0.0949894	0.0474947	+	0.998871i	$0.484876\pi$
$674$	−25.3485	−0.976387
$675$	0	0
$676$	67.9444	2.61325
$677$	30.2176	1.16136	0.580678	−	0.814134i	$-0.302788\pi$
$677$	30.2176	1.16136	0.580678	+	0.814134i	$0.302788\pi$
$678$	0	0
$679$	−17.5200	−0.672357
$680$	−3.55051	−0.136156
$681$	0	0
$682$	0	0
$683$	31.5959	1.20898	0.604492	−	0.796611i	$-0.293376\pi$
$683$	31.5959	1.20898	0.604492	+	0.796611i	$0.293376\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	31.5959	1.20634
$687$	0	0
$688$	1.04930	0.0400040
$689$	62.3217	2.37427
$690$	0	0
$691$	−18.2020	−0.692438	−0.346219	−	0.938154i	$-0.612535\pi$
$691$	−18.2020	−0.692438	−0.346219	+	0.938154i	$0.612535\pi$
$692$	46.3226	1.76092
$693$	0	0
$694$	30.2474	1.14818
$695$	10.9646	0.415911
$696$	0	0
$697$	−7.10102	−0.268970
$698$	−16.8990	−0.639636
$699$	0	0
$700$	−3.61953	−0.136805
$701$	−10.0213	−0.378499	−0.189250	−	0.981929i	$-0.560606\pi$
$701$	−10.0213	−0.378499	−0.189250	+	0.981929i	$0.560606\pi$
$702$	0	0
$703$	−32.2102	−1.21483
$704$	0	0
$705$	0	0
$706$	−25.4427	−0.957550
$707$	19.1010	0.718368
$708$	0	0
$709$	6.69694	0.251509	0.125754	−	0.992061i	$-0.459865\pi$
$709$	6.69694	0.251509	0.125754	+	0.992061i	$0.459865\pi$
$710$	2.57024	0.0964593
$711$	0	0
$712$	12.8512	0.481619
$713$	−9.79796	−0.366936
$714$	0	0
$715$	0	0
$716$	75.1918	2.81005
$717$	0	0
$718$	72.4949	2.70549
$719$	−14.2020	−0.529647	−0.264823	−	0.964297i	$-0.585314\pi$
$719$	−14.2020	−0.529647	−0.264823	+	0.964297i	$0.585314\pi$
$720$	0	0
$721$	−13.5348	−0.504064
$722$	6.53160	0.243081
$723$	0	0
$724$	10.6969	0.397549
$725$	2.57024	0.0954562
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−20.3023	−0.752452
$729$	0	0
$730$	3.55051	0.131410
$731$	−1.10102	−0.0407227
$732$	0	0
$733$	−32.5758	−1.20321	−0.601607	−	0.798792i	$-0.705473\pi$
$733$	−32.5758	−1.20321	−0.601607	+	0.798792i	$0.705473\pi$
$734$	−55.0826	−2.03314
$735$	0	0
$736$	−21.7172	−0.800507
$737$	0	0
$738$	0	0
$739$	−7.71071	−0.283643	−0.141822	−	0.989892i	$-0.545296\pi$
$739$	−7.71071	−0.283643	−0.141822	+	0.989892i	$0.545296\pi$
$740$	−23.7980	−0.874830
$741$	0	0
$742$	−26.6969	−0.980075
$743$	34.8864	1.27986	0.639929	−	0.768434i	$-0.278964\pi$
$743$	34.8864	1.27986	0.639929	+	0.768434i	$0.278964\pi$
$744$	0	0
$745$	0.471647	0.0172798
$746$	23.1464	0.847451
$747$	0	0
$748$	0	0
$749$	8.20204	0.299696
$750$	0	0
$751$	31.1918	1.13821	0.569103	−	0.822266i	$-0.307291\pi$
$751$	31.1918	1.13821	0.569103	+	0.822266i	$0.307291\pi$
$752$	−9.79796	−0.357295
$753$	0	0
$754$	34.3087	1.24945
$755$	0.471647	0.0171650
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−55.0826	−2.00069
$759$	0	0
$760$	15.7980	0.573052
$761$	22.4008	0.812030	0.406015	−	0.913866i	$-0.366918\pi$
$761$	22.4008	0.812030	0.406015	+	0.913866i	$0.366918\pi$
$762$	0	0
$763$	14.2020	0.514148
$764$	−67.5959	−2.44553
$765$	0	0
$766$	39.4492	1.42536
$767$	62.3217	2.25031
$768$	0	0
$769$	11.4362	0.412402	0.206201	−	0.978510i	$-0.433890\pi$
$769$	11.4362	0.412402	0.206201	+	0.978510i	$0.433890\pi$
$770$	0	0
$771$	0	0
$772$	−69.6668	−2.50736
$773$	−20.2020	−0.726617	−0.363308	−	0.931669i	$-0.618353\pi$
$773$	−20.2020	−0.726617	−0.363308	+	0.931669i	$0.618353\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	56.4976	2.02815
$777$	0	0
$778$	−73.7580	−2.64435
$779$	31.5959	1.13204
$780$	0	0
$781$	0	0
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−5.89898	−0.210678
$785$	7.79796	0.278321
$786$	0	0
$787$	−44.6957	−1.59323	−0.796615	−	0.604487i	$-0.793378\pi$
$787$	−44.6957	−1.59323	−0.796615	+	0.604487i	$0.793378\pi$
$788$	−68.0398	−2.42382
$789$	0	0
$790$	22.8990	0.814709
$791$	−1.15530	−0.0410776
$792$	0	0
$793$	−41.3939	−1.46994
$794$	66.9905	2.37741
$795$	0	0
$796$	−61.3939	−2.17605
$797$	1.10102	0.0390001	0.0195001	−	0.999810i	$-0.493793\pi$
$797$	1.10102	0.0390001	0.0195001	+	0.999810i	$0.493793\pi$
$798$	0	0
$799$	10.2810	0.363714
$800$	−4.43300	−0.156730
$801$	0	0
$802$	19.1470	0.676103
$803$	0	0
$804$	0	0
$805$	−5.14048	−0.181178
$806$	−26.6969	−0.940360
$807$	0	0
$808$	−61.5959	−2.16694
$809$	31.5265	1.10841	0.554206	−	0.832379i	$-0.313022\pi$
$809$	31.5265	1.10841	0.554206	+	0.832379i	$0.313022\pi$
$810$	0	0
$811$	−16.1051	−0.565526	−0.282763	−	0.959190i	$-0.591251\pi$
$811$	−16.1051	−0.565526	−0.282763	+	0.959190i	$0.591251\pi$
$812$	−9.30306	−0.326473
$813$	0	0
$814$	0	0
$815$	−6.69694	−0.234584
$816$	0	0
$817$	4.89898	0.171394
$818$	41.3939	1.44730
$819$	0	0
$820$	23.3441	0.815213
$821$	−39.9209	−1.39325	−0.696624	−	0.717437i	$-0.745315\pi$
$821$	−39.9209	−1.39325	−0.696624	+	0.717437i	$0.745315\pi$
$822$	0	0
$823$	39.1010	1.36298	0.681488	−	0.731829i	$-0.261333\pi$
$823$	39.1010	1.36298	0.681488	+	0.731829i	$0.261333\pi$
$824$	43.6464	1.52049
$825$	0	0
$826$	−26.6969	−0.928905
$827$	−43.0688	−1.49765	−0.748824	−	0.662769i	$-0.769381\pi$
$827$	−43.0688	−1.49765	−0.748824	+	0.662769i	$0.769381\pi$
$828$	0	0
$829$	−42.6969	−1.48293	−0.741463	−	0.670994i	$-0.765868\pi$
$829$	−42.6969	−1.48293	−0.741463	+	0.670994i	$0.765868\pi$
$830$	−8.44949	−0.293286
$831$	0	0
$832$	−70.6101	−2.44796
$833$	6.18977	0.214463
$834$	0	0
$835$	12.0139	0.415758
$836$	0	0
$837$	0	0
$838$	28.0130	0.967692
$839$	−30.4949	−1.05280	−0.526400	−	0.850237i	$-0.676459\pi$
$839$	−30.4949	−1.05280	−0.526400	+	0.850237i	$0.676459\pi$
$840$	0	0
$841$	−22.3939	−0.772203
$842$	−12.3795	−0.426627
$843$	0	0
$844$	48.3152	1.66308
$845$	19.6969	0.677595
$846$	0	0
$847$	0	0
$848$	10.8990	0.374272
$849$	0	0
$850$	−2.44949	−0.0840168
$851$	−33.7980	−1.15858
$852$	0	0
$853$	15.9991	0.547798	0.273899	−	0.961758i	$-0.411687\pi$
$853$	15.9991	0.547798	0.273899	+	0.961758i	$0.411687\pi$
$854$	17.7320	0.606777
$855$	0	0
$856$	−26.4495	−0.904025
$857$	−18.5693	−0.634316	−0.317158	−	0.948373i	$-0.602728\pi$
$857$	−18.5693	−0.634316	−0.317158	+	0.948373i	$0.602728\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	3.61953	0.123425
$861$	0	0
$862$	9.79796	0.333720
$863$	−21.3031	−0.725165	−0.362582	−	0.931952i	$-0.618105\pi$
$863$	−21.3031	−0.725165	−0.362582	+	0.931952i	$0.618105\pi$
$864$	0	0
$865$	13.4288	0.456594
$866$	−29.6399	−1.00721
$867$	0	0
$868$	7.23907	0.245710
$869$	0	0
$870$	0	0
$871$	50.8855	1.72419
$872$	−45.7980	−1.55091
$873$	0	0
$874$	53.3939	1.80607
$875$	−1.04930	−0.0354727
$876$	0	0
$877$	16.2111	0.547409	0.273705	−	0.961814i	$-0.411751\pi$
$877$	16.2111	0.547409	0.273705	+	0.961814i	$0.411751\pi$
$878$	−93.1918	−3.14507
$879$	0	0
$880$	0	0
$881$	21.7980	0.734392	0.367196	−	0.930144i	$-0.380318\pi$
$881$	21.7980	0.734392	0.367196	+	0.930144i	$0.380318\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−20.6969	−0.696113
$885$	0	0
$886$	−6.29577	−0.211511
$887$	29.5339	0.991652	0.495826	−	0.868422i	$-0.334865\pi$
$887$	29.5339	0.991652	0.495826	+	0.868422i	$0.334865\pi$
$888$	0	0
$889$	20.6969	0.694153
$890$	8.86601	0.297189
$891$	0	0
$892$	−6.20204	−0.207660
$893$	−45.7450	−1.53080
$894$	0	0
$895$	21.7980	0.728625
$896$	20.9444	0.699703
$897$	0	0
$898$	42.0195	1.40221
$899$	−5.14048	−0.171444
$900$	0	0
$901$	−11.4362	−0.380997
$902$	0	0
$903$	0	0
$904$	3.72553	0.123909
$905$	3.10102	0.103081
$906$	0	0
$907$	−9.39388	−0.311919	−0.155959	−	0.987763i	$-0.549847\pi$
$907$	−9.39388	−0.311919	−0.155959	+	0.987763i	$0.549847\pi$
$908$	5.24648	0.174110
$909$	0	0
$910$	−14.0065	−0.464310
$911$	14.2020	0.470535	0.235267	−	0.971931i	$-0.424403\pi$
$911$	14.2020	0.470535	0.235267	+	0.971931i	$0.424403\pi$
$912$	0	0
$913$	0	0
$914$	64.0454	2.11843
$915$	0	0
$916$	−74.4949	−2.46138
$917$	−16.4041	−0.541711
$918$	0	0
$919$	19.3590	0.638593	0.319297	−	0.947655i	$-0.396553\pi$
$919$	19.3590	0.638593	0.319297	+	0.947655i	$0.396553\pi$
$920$	16.5767	0.546518
$921$	0	0
$922$	−30.4949	−1.00430
$923$	6.29577	0.207228
$924$	0	0
$925$	−6.89898	−0.226837
$926$	18.6753	0.613709
$927$	0	0
$928$	−11.3939	−0.374022
$929$	53.3939	1.75180	0.875898	−	0.482496i	$-0.160270\pi$
$929$	53.3939	1.75180	0.875898	+	0.482496i	$0.160270\pi$
$930$	0	0
$931$	−27.5413	−0.902630
$932$	−35.8297	−1.17364
$933$	0	0
$934$	−11.4362	−0.374205
$935$	0	0
$936$	0	0
$937$	32.7878	1.07113	0.535565	−	0.844494i	$-0.320099\pi$
$937$	32.7878	1.07113	0.535565	+	0.844494i	$0.320099\pi$
$938$	−21.7980	−0.711729
$939$	0	0
$940$	−33.7980	−1.10237
$941$	−42.9628	−1.40055	−0.700273	−	0.713875i	$-0.746938\pi$
$941$	−42.9628	−1.40055	−0.700273	+	0.713875i	$0.746938\pi$
$942$	0	0
$943$	33.1534	1.07962
$944$	10.8990	0.354732
$945$	0	0
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	8.69694	0.282315
$950$	10.8990	0.353610
$951$	0	0
$952$	3.72553	0.120745
$953$	0.106000	0.00343369	0.00171684	−	0.999999i	$-0.499454\pi$
$953$	0.106000	0.00343369	0.00171684	+	0.999999i	$0.499454\pi$
$954$	0	0
$955$	−19.5959	−0.634109
$956$	−89.3914	−2.89112
$957$	0	0
$958$	68.0908	2.19991
$959$	−18.8873	−0.609903
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−92.0908	−2.96913
$963$	0	0
$964$	−57.1812	−1.84168
$965$	−20.1963	−0.650140
$966$	0	0
$967$	32.1042	1.03240	0.516200	−	0.856468i	$-0.327346\pi$
$967$	32.1042	1.03240	0.516200	+	0.856468i	$0.327346\pi$
$968$	0	0
$969$	0	0
$970$	38.9776	1.25149
$971$	29.3939	0.943294	0.471647	−	0.881787i	$-0.343660\pi$
$971$	29.3939	0.943294	0.471647	+	0.881787i	$0.343660\pi$
$972$	0	0
$973$	−11.5051	−0.368837
$974$	20.7739	0.665639
$975$	0	0
$976$	−7.23907	−0.231717
$977$	33.1918	1.06190	0.530950	−	0.847403i	$-0.321835\pi$
$977$	33.1918	1.06190	0.530950	+	0.847403i	$0.321835\pi$
$978$	0	0
$979$	0	0
$980$	−20.3485	−0.650008
$981$	0	0
$982$	−87.1918	−2.78240
$983$	2.69694	0.0860190	0.0430095	−	0.999075i	$-0.486305\pi$
$983$	2.69694	0.0860190	0.0430095	+	0.999075i	$0.486305\pi$
$984$	0	0
$985$	−19.7246	−0.628478
$986$	−6.29577	−0.200498
$987$	0	0
$988$	92.0908	2.92980
$989$	5.14048	0.163458
$990$	0	0
$991$	25.1918	0.800245	0.400123	−	0.916462i	$-0.368968\pi$
$991$	25.1918	0.800245	0.400123	+	0.916462i	$0.368968\pi$
$992$	8.86601	0.281496
$993$	0	0
$994$	−2.69694	−0.0855417
$995$	−17.7980	−0.564233
$996$	0	0
$997$	−13.1692	−0.417072	−0.208536	−	0.978015i	$-0.566870\pi$
$997$	−13.1692	−0.417072	−0.208536	+	0.978015i	$0.566870\pi$
$998$	32.2102	1.01959
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bn.1.4	yes	4
3.2	odd	2		5445.2.a.bm.1.1	✓	4
11.10	odd	2	inner	5445.2.a.bn.1.1	yes	4
33.32	even	2		5445.2.a.bm.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.bm.1.1	✓	4	3.2	odd	2
5445.2.a.bm.1.4	yes	4	33.32	even	2
5445.2.a.bn.1.1	yes	4	11.10	odd	2	inner
5445.2.a.bn.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.33441 q^{2} +3.44949 q^{4} +1.00000 q^{5} -1.04930 q^{7} +3.38371 q^{8} +O(q^{10})\)	\(q+2.33441 q^{2} +3.44949 q^{4} +1.00000 q^{5} -1.04930 q^{7} +3.38371 q^{8} +2.33441 q^{10} +5.71812 q^{13} -2.44949 q^{14} +1.00000 q^{16} -1.04930 q^{17} +4.66883 q^{19} +3.44949 q^{20} +4.89898 q^{23} +1.00000 q^{25} +13.3485 q^{26} -3.61953 q^{28} +2.57024 q^{29} -2.00000 q^{31} -4.43300 q^{32} -2.44949 q^{34} -1.04930 q^{35} -6.89898 q^{37} +10.8990 q^{38} +3.38371 q^{40} +6.76742 q^{41} +1.04930 q^{43} +11.4362 q^{46} -9.79796 q^{47} -5.89898 q^{49} +2.33441 q^{50} +19.7246 q^{52} +10.8990 q^{53} -3.55051 q^{56} +6.00000 q^{58} +10.8990 q^{59} -7.23907 q^{61} -4.66883 q^{62} -12.3485 q^{64} +5.71812 q^{65} +8.89898 q^{67} -3.61953 q^{68} -2.44949 q^{70} +1.10102 q^{71} +1.52094 q^{73} -16.1051 q^{74} +16.1051 q^{76} +9.80930 q^{79} +1.00000 q^{80} +15.7980 q^{82} -3.61953 q^{83} -1.04930 q^{85} +2.44949 q^{86} +3.79796 q^{89} -6.00000 q^{91} +16.8990 q^{92} -22.8725 q^{94} +4.66883 q^{95} +16.6969 q^{97} -13.7707 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^4 + 4 * q^5	\( 4 q + 4 q^{4} + 4 q^{5} + 4 q^{16} + 4 q^{20} + 4 q^{25} + 24 q^{26} - 8 q^{31} - 8 q^{37} + 24 q^{38} - 4 q^{49} + 24 q^{53} - 24 q^{56} + 24 q^{58} + 24 q^{59} - 20 q^{64} + 16 q^{67} + 24 q^{71} + 4 q^{80} + 24 q^{82} - 24 q^{89} - 24 q^{91} + 48 q^{92} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 4 * q^5 + 4 * q^16 + 4 * q^20 + 4 * q^25 + 24 * q^26 - 8 * q^31 - 8 * q^37 + 24 * q^38 - 4 * q^49 + 24 * q^53 - 24 * q^56 + 24 * q^58 + 24 * q^59 - 20 * q^64 + 16 * q^67 + 24 * q^71 + 4 * q^80 + 24 * q^82 - 24 * q^89 - 24 * q^91 + 48 * q^92 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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