LMFDB - Embedded newform 5445.2.a.y.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +O(q^{10})$	$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +2.41421 q^{10} +1.17157 q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843 q^{17} +3.82843 q^{20} +2.82843 q^{23} +1.00000 q^{25} +2.82843 q^{26} +7.65685 q^{28} -3.65685 q^{29} -1.58579 q^{32} +16.4853 q^{34} +2.00000 q^{35} -7.65685 q^{37} +4.41421 q^{40} +6.00000 q^{41} +6.00000 q^{43} +6.82843 q^{46} -2.82843 q^{47} -3.00000 q^{49} +2.41421 q^{50} +4.48528 q^{52} -11.6569 q^{53} +8.82843 q^{56} -8.82843 q^{58} -1.65685 q^{59} +9.31371 q^{61} -9.82843 q^{64} +1.17157 q^{65} +12.4853 q^{67} +26.1421 q^{68} +4.82843 q^{70} -11.3137 q^{71} +1.17157 q^{73} -18.4853 q^{74} -4.00000 q^{79} +3.00000 q^{80} +14.4853 q^{82} -6.00000 q^{83} +6.82843 q^{85} +14.4853 q^{86} +13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} -6.82843 q^{94} +3.65685 q^{97} -7.24264 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{20} + 2 q^{25} + 4 q^{28} + 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{35} - 4 q^{37} + 6 q^{40} + 12 q^{41} + 12 q^{43} + 8 q^{46} - 6 q^{49} + 2 q^{50} - 8 q^{52} - 12 q^{53} + 12 q^{56} - 12 q^{58} + 8 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} + 8 q^{67} + 24 q^{68} + 4 q^{70} + 8 q^{73} - 20 q^{74} - 8 q^{79} + 6 q^{80} + 12 q^{82} - 12 q^{83} + 8 q^{85} + 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{92} - 8 q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^8 + 2 * q^10 + 8 * q^13 + 4 * q^14 + 6 * q^16 + 8 * q^17 + 2 * q^20 + 2 * q^25 + 4 * q^28 + 4 * q^29 - 6 * q^32 + 16 * q^34 + 4 * q^35 - 4 * q^37 + 6 * q^40 + 12 * q^41 + 12 * q^43 + 8 * q^46 - 6 * q^49 + 2 * q^50 - 8 * q^52 - 12 * q^53 + 12 * q^56 - 12 * q^58 + 8 * q^59 - 4 * q^61 - 14 * q^64 + 8 * q^65 + 8 * q^67 + 24 * q^68 + 4 * q^70 + 8 * q^73 - 20 * q^74 - 8 * q^79 + 6 * q^80 + 12 * q^82 - 12 * q^83 + 8 * q^85 + 12 * q^86 + 4 * q^89 + 16 * q^91 + 16 * q^92 - 8 * q^94 - 4 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.17157	0.324936	0.162468	−	0.986714i	$-0.448055\pi$
$13$	1.17157	0.324936	0.162468	+	0.986714i	$0.448055\pi$
$14$	4.82843	1.29045
$15$	0	0
$16$	3.00000	0.750000
$17$	6.82843	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$17$	6.82843	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.82843	0.856062
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.82843	0.554700
$27$	0	0
$28$	7.65685	1.44701
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	16.4853	2.82720
$35$	2.00000	0.338062
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	0	0
$40$	4.41421	0.697948
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	6.82843	1.00680
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	2.41421	0.341421
$51$	0	0
$52$	4.48528	0.621997
$53$	−11.6569	−1.60119	−0.800596	−	0.599204i	$-0.795484\pi$
$53$	−11.6569	−1.60119	−0.800596	+	0.599204i	$0.795484\pi$
$54$	0	0
$55$	0	0
$56$	8.82843	1.17975
$57$	0	0
$58$	−8.82843	−1.15923
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	0	0
$63$	0	0
$64$	−9.82843	−1.22855
$65$	1.17157	0.145316
$66$	0	0
$67$	12.4853	1.52532	0.762660	−	0.646800i	$-0.223893\pi$
$67$	12.4853	1.52532	0.762660	+	0.646800i	$0.223893\pi$
$68$	26.1421	3.17020
$69$	0	0
$70$	4.82843	0.577107
$71$	−11.3137	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$72$	0	0
$73$	1.17157	0.137122	0.0685611	−	0.997647i	$-0.478159\pi$
$73$	1.17157	0.137122	0.0685611	+	0.997647i	$0.478159\pi$
$74$	−18.4853	−2.14887
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	14.4853	1.59963
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	6.82843	0.740647
$86$	14.4853	1.56199
$87$	0	0
$88$	0	0
$89$	13.3137	1.41125	0.705625	−	0.708585i	$-0.250666\pi$
$89$	13.3137	1.41125	0.705625	+	0.708585i	$0.250666\pi$
$90$	0	0
$91$	2.34315	0.245628
$92$	10.8284	1.12894
$93$	0	0
$94$	−6.82843	−0.704298
$95$	0	0
$96$	0	0
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	−7.24264	−0.731617
$99$	0	0
$100$	3.82843	0.382843
$101$	9.31371	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$102$	0	0
$103$	6.82843	0.672825	0.336412	−	0.941715i	$-0.390786\pi$
$103$	6.82843	0.672825	0.336412	+	0.941715i	$0.390786\pi$
$104$	5.17157	0.507114
$105$	0	0
$106$	−28.1421	−2.73341
$107$	7.65685	0.740216	0.370108	−	0.928989i	$-0.379321\pi$
$107$	7.65685	0.740216	0.370108	+	0.928989i	$0.379321\pi$
$108$	0	0
$109$	7.65685	0.733394	0.366697	−	0.930341i	$-0.380489\pi$
$109$	7.65685	0.733394	0.366697	+	0.930341i	$0.380489\pi$
$110$	0	0
$111$	0	0
$112$	6.00000	0.566947
$113$	−19.6569	−1.84916	−0.924581	−	0.380986i	$-0.875584\pi$
$113$	−19.6569	−1.84916	−0.924581	+	0.380986i	$0.875584\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	−14.0000	−1.29987
$117$	0	0
$118$	−4.00000	−0.368230
$119$	13.6569	1.25192
$120$	0	0
$121$	0	0
$122$	22.4853	2.03572
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	2.82843	0.248069
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	30.1421	2.60388
$135$	0	0
$136$	30.1421	2.58467
$137$	10.9706	0.937278	0.468639	−	0.883390i	$-0.344744\pi$
$137$	10.9706	0.937278	0.468639	+	0.883390i	$0.344744\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	7.65685	0.647122
$141$	0	0
$142$	−27.3137	−2.29212
$143$	0	0
$144$	0	0
$145$	−3.65685	−0.303685
$146$	2.82843	0.234082
$147$	0	0
$148$	−29.3137	−2.40957
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−9.65685	−0.768258
$159$	0	0
$160$	−1.58579	−0.125367
$161$	5.65685	0.445823
$162$	0	0
$163$	16.4853	1.29123	0.645613	−	0.763664i	$-0.276602\pi$
$163$	16.4853	1.29123	0.645613	+	0.763664i	$0.276602\pi$
$164$	22.9706	1.79370
$165$	0	0
$166$	−14.4853	−1.12428
$167$	−22.9706	−1.77752	−0.888758	−	0.458377i	$-0.848431\pi$
$167$	−22.9706	−1.77752	−0.888758	+	0.458377i	$0.848431\pi$
$168$	0	0
$169$	−11.6274	−0.894417
$170$	16.4853	1.26436
$171$	0	0
$172$	22.9706	1.75149
$173$	−22.1421	−1.68344	−0.841718	−	0.539918i	$-0.818455\pi$
$173$	−22.1421	−1.68344	−0.841718	+	0.539918i	$0.818455\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	32.1421	2.40915
$179$	−9.65685	−0.721787	−0.360894	−	0.932607i	$-0.617528\pi$
$179$	−9.65685	−0.721787	−0.360894	+	0.932607i	$0.617528\pi$
$180$	0	0
$181$	21.3137	1.58424	0.792118	−	0.610368i	$-0.208979\pi$
$181$	21.3137	1.58424	0.792118	+	0.610368i	$0.208979\pi$
$182$	5.65685	0.419314
$183$	0	0
$184$	12.4853	0.920427
$185$	−7.65685	−0.562943
$186$	0	0
$187$	0	0
$188$	−10.8284	−0.789744
$189$	0	0
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	1.17157	0.0843317	0.0421658	−	0.999111i	$-0.486574\pi$
$193$	1.17157	0.0843317	0.0421658	+	0.999111i	$0.486574\pi$
$194$	8.82843	0.633844
$195$	0	0
$196$	−11.4853	−0.820377
$197$	−10.8284	−0.771493	−0.385747	−	0.922605i	$-0.626056\pi$
$197$	−10.8284	−0.771493	−0.385747	+	0.922605i	$0.626056\pi$
$198$	0	0
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	4.41421	0.312132
$201$	0	0
$202$	22.4853	1.58206
$203$	−7.31371	−0.513322
$204$	0	0
$205$	6.00000	0.419058
$206$	16.4853	1.14858
$207$	0	0
$208$	3.51472	0.243702
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−44.6274	−3.06502
$213$	0	0
$214$	18.4853	1.26363
$215$	6.00000	0.409197
$216$	0	0
$217$	0	0
$218$	18.4853	1.25198
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−10.8284	−0.725125	−0.362563	−	0.931959i	$-0.618098\pi$
$223$	−10.8284	−0.725125	−0.362563	+	0.931959i	$0.618098\pi$
$224$	−3.17157	−0.211910
$225$	0	0
$226$	−47.4558	−3.15672
$227$	25.3137	1.68013	0.840065	−	0.542486i	$-0.182517\pi$
$227$	25.3137	1.68013	0.840065	+	0.542486i	$0.182517\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	6.82843	0.450253
$231$	0	0
$232$	−16.1421	−1.05978
$233$	−6.14214	−0.402385	−0.201192	−	0.979552i	$-0.564482\pi$
$233$	−6.14214	−0.402385	−0.201192	+	0.979552i	$0.564482\pi$
$234$	0	0
$235$	−2.82843	−0.184506
$236$	−6.34315	−0.412904
$237$	0	0
$238$	32.9706	2.13716
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	35.6569	2.28270
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	2.41421	0.152688
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−10.4853	−0.657905
$255$	0	0
$256$	−29.9706	−1.87316
$257$	9.31371	0.580973	0.290487	−	0.956879i	$-0.406183\pi$
$257$	9.31371	0.580973	0.290487	+	0.956879i	$0.406183\pi$
$258$	0	0
$259$	−15.3137	−0.951548
$260$	4.48528	0.278165
$261$	0	0
$262$	−27.3137	−1.68745
$263$	−10.9706	−0.676474	−0.338237	−	0.941061i	$-0.609831\pi$
$263$	−10.9706	−0.676474	−0.338237	+	0.941061i	$0.609831\pi$
$264$	0	0
$265$	−11.6569	−0.716075
$266$	0	0
$267$	0	0
$268$	47.7990	2.91979
$269$	−17.3137	−1.05564	−0.527818	−	0.849358i	$-0.676990\pi$
$269$	−17.3137	−1.05564	−0.527818	+	0.849358i	$0.676990\pi$
$270$	0	0
$271$	−7.31371	−0.444276	−0.222138	−	0.975015i	$-0.571304\pi$
$271$	−7.31371	−0.444276	−0.222138	+	0.975015i	$0.571304\pi$
$272$	20.4853	1.24210
$273$	0	0
$274$	26.4853	1.60003
$275$	0	0
$276$	0	0
$277$	−6.82843	−0.410280	−0.205140	−	0.978733i	$-0.565765\pi$
$277$	−6.82843	−0.410280	−0.205140	+	0.978733i	$0.565765\pi$
$278$	9.65685	0.579180
$279$	0	0
$280$	8.82843	0.527599
$281$	17.3137	1.03285	0.516425	−	0.856333i	$-0.327263\pi$
$281$	17.3137	1.03285	0.516425	+	0.856333i	$0.327263\pi$
$282$	0	0
$283$	−32.6274	−1.93950	−0.969749	−	0.244103i	$-0.921507\pi$
$283$	−32.6274	−1.93950	−0.969749	+	0.244103i	$0.921507\pi$
$284$	−43.3137	−2.57020
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	29.6274	1.74279
$290$	−8.82843	−0.518423
$291$	0	0
$292$	4.48528	0.262481
$293$	−9.17157	−0.535809	−0.267905	−	0.963445i	$-0.586331\pi$
$293$	−9.17157	−0.535809	−0.267905	+	0.963445i	$0.586331\pi$
$294$	0	0
$295$	−1.65685	−0.0964658
$296$	−33.7990	−1.96453
$297$	0	0
$298$	0.828427	0.0479895
$299$	3.31371	0.191637
$300$	0	0
$301$	12.0000	0.691669
$302$	28.9706	1.66707
$303$	0	0
$304$	0	0
$305$	9.31371	0.533301
$306$	0	0
$307$	16.3431	0.932753	0.466376	−	0.884586i	$-0.345559\pi$
$307$	16.3431	0.932753	0.466376	+	0.884586i	$0.345559\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.68629	−0.265735	−0.132868	−	0.991134i	$-0.542419\pi$
$311$	−4.68629	−0.265735	−0.132868	+	0.991134i	$0.542419\pi$
$312$	0	0
$313$	−1.31371	−0.0742552	−0.0371276	−	0.999311i	$-0.511821\pi$
$313$	−1.31371	−0.0742552	−0.0371276	+	0.999311i	$0.511821\pi$
$314$	−33.7990	−1.90739
$315$	0	0
$316$	−15.3137	−0.861463
$317$	1.31371	0.0737852	0.0368926	−	0.999319i	$-0.488254\pi$
$317$	1.31371	0.0737852	0.0368926	+	0.999319i	$0.488254\pi$
$318$	0	0
$319$	0	0
$320$	−9.82843	−0.549426
$321$	0	0
$322$	13.6569	0.761067
$323$	0	0
$324$	0	0
$325$	1.17157	0.0649872
$326$	39.7990	2.20426
$327$	0	0
$328$	26.4853	1.46241
$329$	−5.65685	−0.311872
$330$	0	0
$331$	−7.31371	−0.401998	−0.200999	−	0.979591i	$-0.564419\pi$
$331$	−7.31371	−0.401998	−0.200999	+	0.979591i	$0.564419\pi$
$332$	−22.9706	−1.26067
$333$	0	0
$334$	−55.4558	−3.03441
$335$	12.4853	0.682144
$336$	0	0
$337$	20.4853	1.11590	0.557952	−	0.829873i	$-0.311587\pi$
$337$	20.4853	1.11590	0.557952	+	0.829873i	$0.311587\pi$
$338$	−28.0711	−1.52686
$339$	0	0
$340$	26.1421	1.41776
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	26.4853	1.42799
$345$	0	0
$346$	−53.4558	−2.87380
$347$	10.9706	0.588931	0.294465	−	0.955662i	$-0.404858\pi$
$347$	10.9706	0.588931	0.294465	+	0.955662i	$0.404858\pi$
$348$	0	0
$349$	−26.9706	−1.44370	−0.721851	−	0.692049i	$-0.756708\pi$
$349$	−26.9706	−1.44370	−0.721851	+	0.692049i	$0.756708\pi$
$350$	4.82843	0.258090
$351$	0	0
$352$	0	0
$353$	−21.3137	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$354$	0	0
$355$	−11.3137	−0.600469
$356$	50.9706	2.70143
$357$	0	0
$358$	−23.3137	−1.23217
$359$	0.686292	0.0362211	0.0181105	−	0.999836i	$-0.494235\pi$
$359$	0.686292	0.0362211	0.0181105	+	0.999836i	$0.494235\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	51.4558	2.70446
$363$	0	0
$364$	8.97056	0.470185
$365$	1.17157	0.0613229
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	8.48528	0.442326
$369$	0	0
$370$	−18.4853	−0.961004
$371$	−23.3137	−1.21039
$372$	0	0
$373$	−35.7990	−1.85360	−0.926801	−	0.375554i	$-0.877453\pi$
$373$	−35.7990	−1.85360	−0.926801	+	0.375554i	$0.877453\pi$
$374$	0	0
$375$	0	0
$376$	−12.4853	−0.643879
$377$	−4.28427	−0.220651
$378$	0	0
$379$	33.6569	1.72884	0.864418	−	0.502773i	$-0.167687\pi$
$379$	33.6569	1.72884	0.864418	+	0.502773i	$0.167687\pi$
$380$	0	0
$381$	0	0
$382$	−8.00000	−0.409316
$383$	5.85786	0.299323	0.149661	−	0.988737i	$-0.452182\pi$
$383$	5.85786	0.299323	0.149661	+	0.988737i	$0.452182\pi$
$384$	0	0
$385$	0	0
$386$	2.82843	0.143963
$387$	0	0
$388$	14.0000	0.710742
$389$	−20.6274	−1.04585	−0.522926	−	0.852378i	$-0.675160\pi$
$389$	−20.6274	−1.04585	−0.522926	+	0.852378i	$0.675160\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	−13.2426	−0.668854
$393$	0	0
$394$	−26.1421	−1.31702
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−9.31371	−0.467442	−0.233721	−	0.972304i	$-0.575090\pi$
$397$	−9.31371	−0.467442	−0.233721	+	0.972304i	$0.575090\pi$
$398$	24.9706	1.25166
$399$	0	0
$400$	3.00000	0.150000
$401$	5.31371	0.265354	0.132677	−	0.991159i	$-0.457643\pi$
$401$	5.31371	0.265354	0.132677	+	0.991159i	$0.457643\pi$
$402$	0	0
$403$	0	0
$404$	35.6569	1.77399
$405$	0	0
$406$	−17.6569	−0.876295
$407$	0	0
$408$	0	0
$409$	−1.02944	−0.0509024	−0.0254512	−	0.999676i	$-0.508102\pi$
$409$	−1.02944	−0.0509024	−0.0254512	+	0.999676i	$0.508102\pi$
$410$	14.4853	0.715377
$411$	0	0
$412$	26.1421	1.28793
$413$	−3.31371	−0.163057
$414$	0	0
$415$	−6.00000	−0.294528
$416$	−1.85786	−0.0910893
$417$	0	0
$418$	0	0
$419$	25.6569	1.25342	0.626710	−	0.779253i	$-0.284401\pi$
$419$	25.6569	1.25342	0.626710	+	0.779253i	$0.284401\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	38.6274	1.88035
$423$	0	0
$424$	−51.4558	−2.49892
$425$	6.82843	0.331227
$426$	0	0
$427$	18.6274	0.901444
$428$	29.3137	1.41693
$429$	0	0
$430$	14.4853	0.698542
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	0	0
$433$	−7.65685	−0.367965	−0.183982	−	0.982930i	$-0.558899\pi$
$433$	−7.65685	−0.367965	−0.183982	+	0.982930i	$0.558899\pi$
$434$	0	0
$435$	0	0
$436$	29.3137	1.40387
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	19.3137	0.918659
$443$	26.8284	1.27466	0.637329	−	0.770592i	$-0.280039\pi$
$443$	26.8284	1.27466	0.637329	+	0.770592i	$0.280039\pi$
$444$	0	0
$445$	13.3137	0.631130
$446$	−26.1421	−1.23787
$447$	0	0
$448$	−19.6569	−0.928699
$449$	−28.6274	−1.35101	−0.675506	−	0.737355i	$-0.736075\pi$
$449$	−28.6274	−1.35101	−0.675506	+	0.737355i	$0.736075\pi$
$450$	0	0
$451$	0	0
$452$	−75.2548	−3.53969
$453$	0	0
$454$	61.1127	2.86816
$455$	2.34315	0.109848
$456$	0	0
$457$	−0.485281	−0.0227005	−0.0113503	−	0.999936i	$-0.503613\pi$
$457$	−0.485281	−0.0227005	−0.0113503	+	0.999936i	$0.503613\pi$
$458$	3.17157	0.148198
$459$	0	0
$460$	10.8284	0.504878
$461$	12.6274	0.588117	0.294059	−	0.955787i	$-0.404994\pi$
$461$	12.6274	0.588117	0.294059	+	0.955787i	$0.404994\pi$
$462$	0	0
$463$	−6.14214	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$463$	−6.14214	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$464$	−10.9706	−0.509296
$465$	0	0
$466$	−14.8284	−0.686914
$467$	14.8284	0.686178	0.343089	−	0.939303i	$-0.388527\pi$
$467$	14.8284	0.686178	0.343089	+	0.939303i	$0.388527\pi$
$468$	0	0
$469$	24.9706	1.15303
$470$	−6.82843	−0.314972
$471$	0	0
$472$	−7.31371	−0.336641
$473$	0	0
$474$	0	0
$475$	0	0
$476$	52.2843	2.39645
$477$	0	0
$478$	−56.2843	−2.57438
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−8.97056	−0.409022
$482$	−14.4853	−0.659786
$483$	0	0
$484$	0	0
$485$	3.65685	0.166049
$486$	0	0
$487$	−24.4853	−1.10953	−0.554767	−	0.832006i	$-0.687193\pi$
$487$	−24.4853	−1.10953	−0.554767	+	0.832006i	$0.687193\pi$
$488$	41.1127	1.86108
$489$	0	0
$490$	−7.24264	−0.327189
$491$	−0.686292	−0.0309719	−0.0154860	−	0.999880i	$-0.504930\pi$
$491$	−0.686292	−0.0309719	−0.0154860	+	0.999880i	$0.504930\pi$
$492$	0	0
$493$	−24.9706	−1.12462
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.6274	−1.01498
$498$	0	0
$499$	9.65685	0.432300	0.216150	−	0.976360i	$-0.430650\pi$
$499$	9.65685	0.432300	0.216150	+	0.976360i	$0.430650\pi$
$500$	3.82843	0.171212
$501$	0	0
$502$	−28.9706	−1.29302
$503$	16.6274	0.741380	0.370690	−	0.928757i	$-0.379121\pi$
$503$	16.6274	0.741380	0.370690	+	0.928757i	$0.379121\pi$
$504$	0	0
$505$	9.31371	0.414455
$506$	0	0
$507$	0	0
$508$	−16.6274	−0.737722
$509$	13.3137	0.590120	0.295060	−	0.955479i	$-0.404660\pi$
$509$	13.3137	0.590120	0.295060	+	0.955479i	$0.404660\pi$
$510$	0	0
$511$	2.34315	0.103655
$512$	−31.2426	−1.38074
$513$	0	0
$514$	22.4853	0.991783
$515$	6.82843	0.300896
$516$	0	0
$517$	0	0
$518$	−36.9706	−1.62439
$519$	0	0
$520$	5.17157	0.226788
$521$	−25.3137	−1.10901	−0.554507	−	0.832179i	$-0.687093\pi$
$521$	−25.3137	−1.10901	−0.554507	+	0.832179i	$0.687093\pi$
$522$	0	0
$523$	41.5980	1.81895	0.909476	−	0.415756i	$-0.136483\pi$
$523$	41.5980	1.81895	0.909476	+	0.415756i	$0.136483\pi$
$524$	−43.3137	−1.89217
$525$	0	0
$526$	−26.4853	−1.15481
$527$	0	0
$528$	0	0
$529$	−15.0000	−0.652174
$530$	−28.1421	−1.22242
$531$	0	0
$532$	0	0
$533$	7.02944	0.304479
$534$	0	0
$535$	7.65685	0.331035
$536$	55.1127	2.38051
$537$	0	0
$538$	−41.7990	−1.80208
$539$	0	0
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	−17.6569	−0.758427
$543$	0	0
$544$	−10.8284	−0.464265
$545$	7.65685	0.327984
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	42.0000	1.79415
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−16.4853	−0.700392
$555$	0	0
$556$	15.3137	0.649446
$557$	9.85786	0.417691	0.208846	−	0.977949i	$-0.433029\pi$
$557$	9.85786	0.417691	0.208846	+	0.977949i	$0.433029\pi$
$558$	0	0
$559$	7.02944	0.297314
$560$	6.00000	0.253546
$561$	0	0
$562$	41.7990	1.76318
$563$	0.343146	0.0144619	0.00723093	−	0.999974i	$-0.497698\pi$
$563$	0.343146	0.0144619	0.00723093	+	0.999974i	$0.497698\pi$
$564$	0	0
$565$	−19.6569	−0.826970
$566$	−78.7696	−3.31093
$567$	0	0
$568$	−49.9411	−2.09548
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	21.9411	0.918208	0.459104	−	0.888383i	$-0.348171\pi$
$571$	21.9411	0.918208	0.459104	+	0.888383i	$0.348171\pi$
$572$	0	0
$573$	0	0
$574$	28.9706	1.20921
$575$	2.82843	0.117954
$576$	0	0
$577$	−26.9706	−1.12280	−0.561400	−	0.827545i	$-0.689737\pi$
$577$	−26.9706	−1.12280	−0.561400	+	0.827545i	$0.689737\pi$
$578$	71.5269	2.97513
$579$	0	0
$580$	−14.0000	−0.581318
$581$	−12.0000	−0.497844
$582$	0	0
$583$	0	0
$584$	5.17157	0.214001
$585$	0	0
$586$	−22.1421	−0.914683
$587$	2.14214	0.0884154	0.0442077	−	0.999022i	$-0.485924\pi$
$587$	2.14214	0.0884154	0.0442077	+	0.999022i	$0.485924\pi$
$588$	0	0
$589$	0	0
$590$	−4.00000	−0.164677
$591$	0	0
$592$	−22.9706	−0.944084
$593$	3.51472	0.144332	0.0721661	−	0.997393i	$-0.477009\pi$
$593$	3.51472	0.144332	0.0721661	+	0.997393i	$0.477009\pi$
$594$	0	0
$595$	13.6569	0.559876
$596$	1.31371	0.0538116
$597$	0	0
$598$	8.00000	0.327144
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	−23.9411	−0.976579	−0.488289	−	0.872682i	$-0.662379\pi$
$601$	−23.9411	−0.976579	−0.488289	+	0.872682i	$0.662379\pi$
$602$	28.9706	1.18075
$603$	0	0
$604$	45.9411	1.86932
$605$	0	0
$606$	0	0
$607$	−38.2843	−1.55391	−0.776955	−	0.629556i	$-0.783237\pi$
$607$	−38.2843	−1.55391	−0.776955	+	0.629556i	$0.783237\pi$
$608$	0	0
$609$	0	0
$610$	22.4853	0.910402
$611$	−3.31371	−0.134058
$612$	0	0
$613$	25.4558	1.02815	0.514076	−	0.857745i	$-0.328135\pi$
$613$	25.4558	1.02815	0.514076	+	0.857745i	$0.328135\pi$
$614$	39.4558	1.59231
$615$	0	0
$616$	0	0
$617$	−0.343146	−0.0138145	−0.00690726	−	0.999976i	$-0.502199\pi$
$617$	−0.343146	−0.0138145	−0.00690726	+	0.999976i	$0.502199\pi$
$618$	0	0
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	0	0
$622$	−11.3137	−0.453638
$623$	26.6274	1.06680
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.17157	−0.126762
$627$	0	0
$628$	−53.5980	−2.13879
$629$	−52.2843	−2.08471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−17.6569	−0.702352
$633$	0	0
$634$	3.17157	0.125959
$635$	−4.34315	−0.172352
$636$	0	0
$637$	−3.51472	−0.139258
$638$	0	0
$639$	0	0
$640$	−20.5563	−0.812561
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−1.45584	−0.0574129	−0.0287064	−	0.999588i	$-0.509139\pi$
$643$	−1.45584	−0.0574129	−0.0287064	+	0.999588i	$0.509139\pi$
$644$	21.6569	0.853400
$645$	0	0
$646$	0	0
$647$	−27.1127	−1.06591	−0.532955	−	0.846144i	$-0.678919\pi$
$647$	−27.1127	−1.06591	−0.532955	+	0.846144i	$0.678919\pi$
$648$	0	0
$649$	0	0
$650$	2.82843	0.110940
$651$	0	0
$652$	63.1127	2.47168
$653$	−11.6569	−0.456168	−0.228084	−	0.973641i	$-0.573246\pi$
$653$	−11.6569	−0.456168	−0.228084	+	0.973641i	$0.573246\pi$
$654$	0	0
$655$	−11.3137	−0.442063
$656$	18.0000	0.702782
$657$	0	0
$658$	−13.6569	−0.532400
$659$	45.9411	1.78961	0.894806	−	0.446455i	$-0.147314\pi$
$659$	45.9411	1.78961	0.894806	+	0.446455i	$0.147314\pi$
$660$	0	0
$661$	44.6274	1.73581	0.867903	−	0.496734i	$-0.165468\pi$
$661$	44.6274	1.73581	0.867903	+	0.496734i	$0.165468\pi$
$662$	−17.6569	−0.686253
$663$	0	0
$664$	−26.4853	−1.02783
$665$	0	0
$666$	0	0
$667$	−10.3431	−0.400488
$668$	−87.9411	−3.40254
$669$	0	0
$670$	30.1421	1.16449
$671$	0	0
$672$	0	0
$673$	12.4853	0.481272	0.240636	−	0.970615i	$-0.422644\pi$
$673$	12.4853	0.481272	0.240636	+	0.970615i	$0.422644\pi$
$674$	49.4558	1.90497
$675$	0	0
$676$	−44.5147	−1.71210
$677$	22.8284	0.877368	0.438684	−	0.898641i	$-0.355445\pi$
$677$	22.8284	0.877368	0.438684	+	0.898641i	$0.355445\pi$
$678$	0	0
$679$	7.31371	0.280674
$680$	30.1421	1.15590
$681$	0	0
$682$	0	0
$683$	7.79899	0.298420	0.149210	−	0.988806i	$-0.452327\pi$
$683$	7.79899	0.298420	0.149210	+	0.988806i	$0.452327\pi$
$684$	0	0
$685$	10.9706	0.419164
$686$	−48.2843	−1.84350
$687$	0	0
$688$	18.0000	0.686244
$689$	−13.6569	−0.520285
$690$	0	0
$691$	−39.3137	−1.49556	−0.747782	−	0.663944i	$-0.768881\pi$
$691$	−39.3137	−1.49556	−0.747782	+	0.663944i	$0.768881\pi$
$692$	−84.7696	−3.22245
$693$	0	0
$694$	26.4853	1.00537
$695$	4.00000	0.151729
$696$	0	0
$697$	40.9706	1.55187
$698$	−65.1127	−2.46455
$699$	0	0
$700$	7.65685	0.289402
$701$	−12.6274	−0.476931	−0.238465	−	0.971151i	$-0.576644\pi$
$701$	−12.6274	−0.476931	−0.238465	+	0.971151i	$0.576644\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−51.4558	−1.93657
$707$	18.6274	0.700556
$708$	0	0
$709$	24.6274	0.924902	0.462451	−	0.886645i	$-0.346970\pi$
$709$	24.6274	0.924902	0.462451	+	0.886645i	$0.346970\pi$
$710$	−27.3137	−1.02507
$711$	0	0
$712$	58.7696	2.20248
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−36.9706	−1.38165
$717$	0	0
$718$	1.65685	0.0618333
$719$	−18.3431	−0.684084	−0.342042	−	0.939685i	$-0.611118\pi$
$719$	−18.3431	−0.684084	−0.342042	+	0.939685i	$0.611118\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	−45.8701	−1.70711
$723$	0	0
$724$	81.5980	3.03257
$725$	−3.65685	−0.135812
$726$	0	0
$727$	−19.5147	−0.723761	−0.361880	−	0.932225i	$-0.617865\pi$
$727$	−19.5147	−0.723761	−0.361880	+	0.932225i	$0.617865\pi$
$728$	10.3431	0.383342
$729$	0	0
$730$	2.82843	0.104685
$731$	40.9706	1.51535
$732$	0	0
$733$	17.4558	0.644746	0.322373	−	0.946613i	$-0.395519\pi$
$733$	17.4558	0.644746	0.322373	+	0.946613i	$0.395519\pi$
$734$	−20.4853	−0.756126
$735$	0	0
$736$	−4.48528	−0.165330
$737$	0	0
$738$	0	0
$739$	−29.9411	−1.10140	−0.550701	−	0.834703i	$-0.685640\pi$
$739$	−29.9411	−1.10140	−0.550701	+	0.834703i	$0.685640\pi$
$740$	−29.3137	−1.07759
$741$	0	0
$742$	−56.2843	−2.06626
$743$	−49.5980	−1.81957	−0.909787	−	0.415076i	$-0.863755\pi$
$743$	−49.5980	−1.81957	−0.909787	+	0.415076i	$0.863755\pi$
$744$	0	0
$745$	0.343146	0.0125719
$746$	−86.4264	−3.16430
$747$	0	0
$748$	0	0
$749$	15.3137	0.559551
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−8.48528	−0.309426
$753$	0	0
$754$	−10.3431	−0.376675
$755$	12.0000	0.436725
$756$	0	0
$757$	13.3137	0.483895	0.241947	−	0.970289i	$-0.422214\pi$
$757$	13.3137	0.483895	0.241947	+	0.970289i	$0.422214\pi$
$758$	81.2548	2.95131
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	15.3137	0.554393
$764$	−12.6863	−0.458974
$765$	0	0
$766$	14.1421	0.510976
$767$	−1.94113	−0.0700900
$768$	0	0
$769$	18.9706	0.684096	0.342048	−	0.939682i	$-0.388879\pi$
$769$	18.9706	0.684096	0.342048	+	0.939682i	$0.388879\pi$
$770$	0	0
$771$	0	0
$772$	4.48528	0.161429
$773$	−26.2843	−0.945380	−0.472690	−	0.881229i	$-0.656717\pi$
$773$	−26.2843	−0.945380	−0.472690	+	0.881229i	$0.656717\pi$
$774$	0	0
$775$	0	0
$776$	16.1421	0.579469
$777$	0	0
$778$	−49.7990	−1.78538
$779$	0	0
$780$	0	0
$781$	0	0
$782$	46.6274	1.66739
$783$	0	0
$784$	−9.00000	−0.321429
$785$	−14.0000	−0.499681
$786$	0	0
$787$	14.9706	0.533643	0.266821	−	0.963746i	$-0.414027\pi$
$787$	14.9706	0.533643	0.266821	+	0.963746i	$0.414027\pi$
$788$	−41.4558	−1.47680
$789$	0	0
$790$	−9.65685	−0.343575
$791$	−39.3137	−1.39783
$792$	0	0
$793$	10.9117	0.387485
$794$	−22.4853	−0.797973
$795$	0	0
$796$	39.5980	1.40351
$797$	−32.6274	−1.15572	−0.577861	−	0.816135i	$-0.696113\pi$
$797$	−32.6274	−1.15572	−0.577861	+	0.816135i	$0.696113\pi$
$798$	0	0
$799$	−19.3137	−0.683270
$800$	−1.58579	−0.0560660
$801$	0	0
$802$	12.8284	0.452988
$803$	0	0
$804$	0	0
$805$	5.65685	0.199378
$806$	0	0
$807$	0	0
$808$	41.1127	1.44634
$809$	−10.9706	−0.385704	−0.192852	−	0.981228i	$-0.561774\pi$
$809$	−10.9706	−0.385704	−0.192852	+	0.981228i	$0.561774\pi$
$810$	0	0
$811$	−53.9411	−1.89413	−0.947065	−	0.321043i	$-0.895967\pi$
$811$	−53.9411	−1.89413	−0.947065	+	0.321043i	$0.895967\pi$
$812$	−28.0000	−0.982607
$813$	0	0
$814$	0	0
$815$	16.4853	0.577454
$816$	0	0
$817$	0	0
$818$	−2.48528	−0.0868958
$819$	0	0
$820$	22.9706	0.802167
$821$	−41.3137	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$821$	−41.3137	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$822$	0	0
$823$	19.5147	0.680240	0.340120	−	0.940382i	$-0.389532\pi$
$823$	19.5147	0.680240	0.340120	+	0.940382i	$0.389532\pi$
$824$	30.1421	1.05005
$825$	0	0
$826$	−8.00000	−0.278356
$827$	22.2843	0.774900	0.387450	−	0.921891i	$-0.373356\pi$
$827$	22.2843	0.774900	0.387450	+	0.921891i	$0.373356\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	−14.4853	−0.502791
$831$	0	0
$832$	−11.5147	−0.399201
$833$	−20.4853	−0.709773
$834$	0	0
$835$	−22.9706	−0.794929
$836$	0	0
$837$	0	0
$838$	61.9411	2.13972
$839$	−26.3431	−0.909466	−0.454733	−	0.890628i	$-0.650265\pi$
$839$	−26.3431	−0.909466	−0.454733	+	0.890628i	$0.650265\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	−14.4853	−0.499196
$843$	0	0
$844$	61.2548	2.10848
$845$	−11.6274	−0.399995
$846$	0	0
$847$	0	0
$848$	−34.9706	−1.20089
$849$	0	0
$850$	16.4853	0.565440
$851$	−21.6569	−0.742387
$852$	0	0
$853$	15.5147	0.531214	0.265607	−	0.964081i	$-0.414428\pi$
$853$	15.5147	0.531214	0.265607	+	0.964081i	$0.414428\pi$
$854$	44.9706	1.53886
$855$	0	0
$856$	33.7990	1.15523
$857$	24.7696	0.846112	0.423056	−	0.906104i	$-0.360957\pi$
$857$	24.7696	0.846112	0.423056	+	0.906104i	$0.360957\pi$
$858$	0	0
$859$	24.2843	0.828569	0.414284	−	0.910148i	$-0.364032\pi$
$859$	24.2843	0.828569	0.414284	+	0.910148i	$0.364032\pi$
$860$	22.9706	0.783290
$861$	0	0
$862$	−27.3137	−0.930309
$863$	9.17157	0.312204	0.156102	−	0.987741i	$-0.450107\pi$
$863$	9.17157	0.312204	0.156102	+	0.987741i	$0.450107\pi$
$864$	0	0
$865$	−22.1421	−0.752855
$866$	−18.4853	−0.628155
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	14.6274	0.495631
$872$	33.7990	1.14458
$873$	0	0
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	49.4558	1.67001	0.835003	−	0.550246i	$-0.185466\pi$
$877$	49.4558	1.67001	0.835003	+	0.550246i	$0.185466\pi$
$878$	38.6274	1.30361
$879$	0	0
$880$	0	0
$881$	7.37258	0.248389	0.124194	−	0.992258i	$-0.460365\pi$
$881$	7.37258	0.248389	0.124194	+	0.992258i	$0.460365\pi$
$882$	0	0
$883$	37.1716	1.25092	0.625462	−	0.780255i	$-0.284911\pi$
$883$	37.1716	1.25092	0.625462	+	0.780255i	$0.284911\pi$
$884$	30.6274	1.03011
$885$	0	0
$886$	64.7696	2.17598
$887$	38.2843	1.28546	0.642730	−	0.766093i	$-0.277802\pi$
$887$	38.2843	1.28546	0.642730	+	0.766093i	$0.277802\pi$
$888$	0	0
$889$	−8.68629	−0.291329
$890$	32.1421	1.07741
$891$	0	0
$892$	−41.4558	−1.38804
$893$	0	0
$894$	0	0
$895$	−9.65685	−0.322793
$896$	−41.1127	−1.37348
$897$	0	0
$898$	−69.1127	−2.30632
$899$	0	0
$900$	0	0
$901$	−79.5980	−2.65179
$902$	0	0
$903$	0	0
$904$	−86.7696	−2.88591
$905$	21.3137	0.708492
$906$	0	0
$907$	−27.5147	−0.913611	−0.456806	−	0.889567i	$-0.651007\pi$
$907$	−27.5147	−0.913611	−0.456806	+	0.889567i	$0.651007\pi$
$908$	96.9117	3.21613
$909$	0	0
$910$	5.65685	0.187523
$911$	9.94113	0.329364	0.164682	−	0.986347i	$-0.447340\pi$
$911$	9.94113	0.329364	0.164682	+	0.986347i	$0.447340\pi$
$912$	0	0
$913$	0	0
$914$	−1.17157	−0.0387522
$915$	0	0
$916$	5.02944	0.166177
$917$	−22.6274	−0.747223
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	12.4853	0.411628
$921$	0	0
$922$	30.4853	1.00398
$923$	−13.2548	−0.436288
$924$	0	0
$925$	−7.65685	−0.251756
$926$	−14.8284	−0.487292
$927$	0	0
$928$	5.79899	0.190361
$929$	−5.31371	−0.174337	−0.0871686	−	0.996194i	$-0.527782\pi$
$929$	−5.31371	−0.174337	−0.0871686	+	0.996194i	$0.527782\pi$
$930$	0	0
$931$	0	0
$932$	−23.5147	−0.770250
$933$	0	0
$934$	35.7990	1.17138
$935$	0	0
$936$	0	0
$937$	1.45584	0.0475604	0.0237802	−	0.999717i	$-0.492430\pi$
$937$	1.45584	0.0475604	0.0237802	+	0.999717i	$0.492430\pi$
$938$	60.2843	1.96835
$939$	0	0
$940$	−10.8284	−0.353184
$941$	−6.68629	−0.217967	−0.108983	−	0.994044i	$-0.534760\pi$
$941$	−6.68629	−0.217967	−0.108983	+	0.994044i	$0.534760\pi$
$942$	0	0
$943$	16.9706	0.552638
$944$	−4.97056	−0.161778
$945$	0	0
$946$	0	0
$947$	−41.1716	−1.33790	−0.668948	−	0.743309i	$-0.733255\pi$
$947$	−41.1716	−1.33790	−0.668948	+	0.743309i	$0.733255\pi$
$948$	0	0
$949$	1.37258	0.0445559
$950$	0	0
$951$	0	0
$952$	60.2843	1.95382
$953$	53.1716	1.72240	0.861198	−	0.508269i	$-0.169715\pi$
$953$	53.1716	1.72240	0.861198	+	0.508269i	$0.169715\pi$
$954$	0	0
$955$	−3.31371	−0.107229
$956$	−89.2548	−2.88671
$957$	0	0
$958$	−86.9117	−2.80799
$959$	21.9411	0.708516
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−21.6569	−0.698245
$963$	0	0
$964$	−22.9706	−0.739832
$965$	1.17157	0.0377143
$966$	0	0
$967$	14.9706	0.481421	0.240710	−	0.970597i	$-0.422620\pi$
$967$	14.9706	0.481421	0.240710	+	0.970597i	$0.422620\pi$
$968$	0	0
$969$	0	0
$970$	8.82843	0.283464
$971$	8.68629	0.278756	0.139378	−	0.990239i	$-0.455490\pi$
$971$	8.68629	0.278756	0.139378	+	0.990239i	$0.455490\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	−59.1127	−1.89409
$975$	0	0
$976$	27.9411	0.894374
$977$	−32.3431	−1.03475	−0.517374	−	0.855759i	$-0.673091\pi$
$977$	−32.3431	−1.03475	−0.517374	+	0.855759i	$0.673091\pi$
$978$	0	0
$979$	0	0
$980$	−11.4853	−0.366884
$981$	0	0
$982$	−1.65685	−0.0528723
$983$	21.8579	0.697158	0.348579	−	0.937279i	$-0.386664\pi$
$983$	21.8579	0.697158	0.348579	+	0.937279i	$0.386664\pi$
$984$	0	0
$985$	−10.8284	−0.345022
$986$	−60.2843	−1.91984
$987$	0	0
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	−57.9411	−1.84056	−0.920280	−	0.391260i	$-0.872039\pi$
$991$	−57.9411	−1.84056	−0.920280	+	0.391260i	$0.872039\pi$
$992$	0	0
$993$	0	0
$994$	−54.6274	−1.73268
$995$	10.3431	0.327900
$996$	0	0
$997$	41.4558	1.31292	0.656460	−	0.754361i	$-0.272053\pi$
$997$	41.4558	1.31292	0.656460	+	0.754361i	$0.272053\pi$
$998$	23.3137	0.737983
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.y.1.2		2
3.2	odd	2		605.2.a.d.1.1		2
11.10	odd	2		495.2.a.b.1.1		2
12.11	even	2		9680.2.a.bn.1.2		2
15.14	odd	2		3025.2.a.o.1.2		2
33.2	even	10		605.2.g.f.81.1		8
33.5	odd	10		605.2.g.l.366.2		8
33.8	even	10		605.2.g.f.251.2		8
33.14	odd	10		605.2.g.l.251.1		8
33.17	even	10		605.2.g.f.366.1		8
33.20	odd	10		605.2.g.l.81.2		8
33.26	odd	10		605.2.g.l.511.1		8
33.29	even	10		605.2.g.f.511.2		8
33.32	even	2		55.2.a.b.1.2	✓	2
44.43	even	2		7920.2.a.ch.1.2		2
55.32	even	4		2475.2.c.l.199.1		4
55.43	even	4		2475.2.c.l.199.4		4
55.54	odd	2		2475.2.a.x.1.2		2
132.131	odd	2		880.2.a.m.1.2		2
165.32	odd	4		275.2.b.d.199.4		4
165.98	odd	4		275.2.b.d.199.1		4
165.164	even	2		275.2.a.c.1.1		2
231.230	odd	2		2695.2.a.f.1.2		2
264.131	odd	2		3520.2.a.bo.1.1		2
264.197	even	2		3520.2.a.bn.1.2		2
429.428	even	2		9295.2.a.g.1.1		2
660.263	even	4		4400.2.b.q.4049.3		4
660.527	even	4		4400.2.b.q.4049.2		4
660.659	odd	2		4400.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.2	✓	2	33.32	even	2
275.2.a.c.1.1		2	165.164	even	2
275.2.b.d.199.1		4	165.98	odd	4
275.2.b.d.199.4		4	165.32	odd	4
495.2.a.b.1.1		2	11.10	odd	2
605.2.a.d.1.1		2	3.2	odd	2
605.2.g.f.81.1		8	33.2	even	10
605.2.g.f.251.2		8	33.8	even	10
605.2.g.f.366.1		8	33.17	even	10
605.2.g.f.511.2		8	33.29	even	10
605.2.g.l.81.2		8	33.20	odd	10
605.2.g.l.251.1		8	33.14	odd	10
605.2.g.l.366.2		8	33.5	odd	10
605.2.g.l.511.1		8	33.26	odd	10
880.2.a.m.1.2		2	132.131	odd	2
2475.2.a.x.1.2		2	55.54	odd	2
2475.2.c.l.199.1		4	55.32	even	4
2475.2.c.l.199.4		4	55.43	even	4
2695.2.a.f.1.2		2	231.230	odd	2
3025.2.a.o.1.2		2	15.14	odd	2
3520.2.a.bn.1.2		2	264.197	even	2
3520.2.a.bo.1.1		2	264.131	odd	2
4400.2.a.bn.1.1		2	660.659	odd	2
4400.2.b.q.4049.2		4	660.527	even	4
4400.2.b.q.4049.3		4	660.263	even	4
5445.2.a.y.1.2		2	1.1	even	1	trivial
7920.2.a.ch.1.2		2	44.43	even	2
9295.2.a.g.1.1		2	429.428	even	2
9680.2.a.bn.1.2		2	12.11	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)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +2.41421 q^{10} +1.17157 q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843 q^{17} +3.82843 q^{20} +2.82843 q^{23} +1.00000 q^{25} +2.82843 q^{26} +7.65685 q^{28} -3.65685 q^{29} -1.58579 q^{32} +16.4853 q^{34} +2.00000 q^{35} -7.65685 q^{37} +4.41421 q^{40} +6.00000 q^{41} +6.00000 q^{43} +6.82843 q^{46} -2.82843 q^{47} -3.00000 q^{49} +2.41421 q^{50} +4.48528 q^{52} -11.6569 q^{53} +8.82843 q^{56} -8.82843 q^{58} -1.65685 q^{59} +9.31371 q^{61} -9.82843 q^{64} +1.17157 q^{65} +12.4853 q^{67} +26.1421 q^{68} +4.82843 q^{70} -11.3137 q^{71} +1.17157 q^{73} -18.4853 q^{74} -4.00000 q^{79} +3.00000 q^{80} +14.4853 q^{82} -6.00000 q^{83} +6.82843 q^{85} +14.4853 q^{86} +13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} -6.82843 q^{94} +3.65685 q^{97} -7.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{20} + 2 q^{25} + 4 q^{28} + 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{35} - 4 q^{37} + 6 q^{40} + 12 q^{41} + 12 q^{43} + 8 q^{46} - 6 q^{49} + 2 q^{50} - 8 q^{52} - 12 q^{53} + 12 q^{56} - 12 q^{58} + 8 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} + 8 q^{67} + 24 q^{68} + 4 q^{70} + 8 q^{73} - 20 q^{74} - 8 q^{79} + 6 q^{80} + 12 q^{82} - 12 q^{83} + 8 q^{85} + 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{92} - 8 q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^8 + 2 * q^10 + 8 * q^13 + 4 * q^14 + 6 * q^16 + 8 * q^17 + 2 * q^20 + 2 * q^25 + 4 * q^28 + 4 * q^29 - 6 * q^32 + 16 * q^34 + 4 * q^35 - 4 * q^37 + 6 * q^40 + 12 * q^41 + 12 * q^43 + 8 * q^46 - 6 * q^49 + 2 * q^50 - 8 * q^52 - 12 * q^53 + 12 * q^56 - 12 * q^58 + 8 * q^59 - 4 * q^61 - 14 * q^64 + 8 * q^65 + 8 * q^67 + 24 * q^68 + 4 * q^70 + 8 * q^73 - 20 * q^74 - 8 * q^79 + 6 * q^80 + 12 * q^82 - 12 * q^83 + 8 * q^85 + 12 * q^86 + 4 * q^89 + 16 * q^91 + 16 * q^92 - 8 * q^94 - 4 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more