LMFDB - Embedded newform 1449.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11948$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11948 q^{2} +2.49221 q^{4} -2.40920 q^{5} +1.00000 q^{7} -1.04322 q^{8} +O(q^{10})$	\(q-2.11948 q^{2} +2.49221 q^{4} -2.40920 q^{5} +1.00000 q^{7} -1.04322 q^{8} +5.10626 q^{10} -5.87722 q^{11} -6.24994 q^{13} -2.11948 q^{14} -2.77332 q^{16} +5.42479 q^{17} -2.23897 q^{19} -6.00423 q^{20} +12.4567 q^{22} +1.00000 q^{23} +0.804258 q^{25} +13.2466 q^{26} +2.49221 q^{28} -0.642864 q^{29} +7.84074 q^{31} +7.96445 q^{32} -11.4977 q^{34} -2.40920 q^{35} +0.557492 q^{37} +4.74545 q^{38} +2.51334 q^{40} -2.56847 q^{41} -8.81841 q^{43} -14.6472 q^{44} -2.11948 q^{46} -4.26766 q^{47} +1.00000 q^{49} -1.70461 q^{50} -15.5761 q^{52} -3.01559 q^{53} +14.1594 q^{55} -1.04322 q^{56} +1.36254 q^{58} -4.17024 q^{59} -0.148289 q^{61} -16.6183 q^{62} -11.3339 q^{64} +15.0574 q^{65} +13.3396 q^{67} +13.5197 q^{68} +5.10626 q^{70} +7.93141 q^{71} +4.28111 q^{73} -1.18159 q^{74} -5.57996 q^{76} -5.87722 q^{77} -0.861628 q^{79} +6.68149 q^{80} +5.44382 q^{82} +4.81841 q^{83} -13.0694 q^{85} +18.6905 q^{86} +6.13125 q^{88} -6.32964 q^{89} -6.24994 q^{91} +2.49221 q^{92} +9.04523 q^{94} +5.39412 q^{95} +10.4822 q^{97} -2.11948 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8	$ 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8 - 8 * q^10 + 4 * q^11 - 6 * q^13 - 2 * q^14 + 10 * q^16 + 12 * q^17 + 6 * q^19 + 14 * q^22 + 5 * q^23 + 19 * q^25 - q^26 + 12 * q^28 + 4 * q^29 + 30 * q^31 - 8 * q^32 + 6 * q^34 + 4 * q^35 + 4 * q^37 + 40 * q^38 - 50 * q^40 - 6 * q^41 - 12 * q^43 + 26 * q^44 - 2 * q^46 - 10 * q^47 + 5 * q^49 + 2 * q^50 - 21 * q^52 - 16 * q^53 + 18 * q^55 - 3 * q^56 + 13 * q^58 - 22 * q^59 - 18 * q^61 - 15 * q^62 + 25 * q^64 + 26 * q^65 - 2 * q^67 - 12 * q^68 - 8 * q^70 - 4 * q^71 - 2 * q^73 - 38 * q^74 + 10 * q^76 + 4 * q^77 + 30 * q^79 + 10 * q^80 - 7 * q^82 - 8 * q^83 - 12 * q^85 - 8 * q^86 + 4 * q^88 + 20 * q^89 - 6 * q^91 + 12 * q^92 - 25 * q^94 - 8 * q^95 - 12 * q^97 - 2 * 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 $

$2$	−2.11948	−1.49870	−0.749350	−	0.662174i	$-0.769634\pi$
$2$	−2.11948	−1.49870	−0.749350	+	0.662174i	$0.769634\pi$
$3$	0	0
$3$	0	0
$4$	2.49221	1.24610
$5$	−2.40920	−1.07743	−0.538714	−	0.842489i	$-0.681090\pi$
$5$	−2.40920	−1.07743	−0.538714	+	0.842489i	$0.681090\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.04322	−0.368835
$9$	0	0
$10$	5.10626	1.61474
$11$	−5.87722	−1.77205	−0.886024	−	0.463640i	$-0.846543\pi$
$11$	−5.87722	−1.77205	−0.886024	+	0.463640i	$0.846543\pi$
$12$	0	0
$13$	−6.24994	−1.73342	−0.866711	−	0.498811i	$-0.833770\pi$
$13$	−6.24994	−1.73342	−0.866711	+	0.498811i	$0.833770\pi$
$14$	−2.11948	−0.566456
$15$	0	0
$16$	−2.77332	−0.693330
$17$	5.42479	1.31570	0.657852	−	0.753147i	$-0.271465\pi$
$17$	5.42479	1.31570	0.657852	+	0.753147i	$0.271465\pi$
$18$	0	0
$19$	−2.23897	−0.513654	−0.256827	−	0.966457i	$-0.582677\pi$
$19$	−2.23897	−0.513654	−0.256827	+	0.966457i	$0.582677\pi$
$20$	−6.00423	−1.34259
$21$	0	0
$22$	12.4567	2.65577
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.804258	0.160852
$26$	13.2466	2.59788
$27$	0	0
$28$	2.49221	0.470983
$29$	−0.642864	−0.119377	−0.0596884	−	0.998217i	$-0.519011\pi$
$29$	−0.642864	−0.119377	−0.0596884	+	0.998217i	$0.519011\pi$
$30$	0	0
$31$	7.84074	1.40824	0.704119	−	0.710082i	$-0.251342\pi$
$31$	7.84074	1.40824	0.704119	+	0.710082i	$0.251342\pi$
$32$	7.96445	1.40793
$33$	0	0
$34$	−11.4977	−1.97185
$35$	−2.40920	−0.407230
$36$	0	0
$37$	0.557492	0.0916511	0.0458256	−	0.998949i	$-0.485408\pi$
$37$	0.557492	0.0916511	0.0458256	+	0.998949i	$0.485408\pi$
$38$	4.74545	0.769813
$39$	0	0
$40$	2.51334	0.397393
$41$	−2.56847	−0.401127	−0.200564	−	0.979681i	$-0.564277\pi$
$41$	−2.56847	−0.401127	−0.200564	+	0.979681i	$0.564277\pi$
$42$	0	0
$43$	−8.81841	−1.34479	−0.672397	−	0.740191i	$-0.734735\pi$
$43$	−8.81841	−1.34479	−0.672397	+	0.740191i	$0.734735\pi$
$44$	−14.6472	−2.20815
$45$	0	0
$46$	−2.11948	−0.312501
$47$	−4.26766	−0.622502	−0.311251	−	0.950328i	$-0.600748\pi$
$47$	−4.26766	−0.622502	−0.311251	+	0.950328i	$0.600748\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.70461	−0.241068
$51$	0	0
$52$	−15.5761	−2.16002
$53$	−3.01559	−0.414223	−0.207111	−	0.978317i	$-0.566406\pi$
$53$	−3.01559	−0.414223	−0.207111	+	0.978317i	$0.566406\pi$
$54$	0	0
$55$	14.1594	1.90925
$56$	−1.04322	−0.139407
$57$	0	0
$58$	1.36254	0.178910
$59$	−4.17024	−0.542919	−0.271459	−	0.962450i	$-0.587506\pi$
$59$	−4.17024	−0.542919	−0.271459	+	0.962450i	$0.587506\pi$
$60$	0	0
$61$	−0.148289	−0.0189865	−0.00949325	−	0.999955i	$-0.503022\pi$
$61$	−0.148289	−0.0189865	−0.00949325	+	0.999955i	$0.503022\pi$
$62$	−16.6183	−2.11053
$63$	0	0
$64$	−11.3339	−1.41673
$65$	15.0574	1.86764
$66$	0	0
$67$	13.3396	1.62969	0.814843	−	0.579681i	$-0.196823\pi$
$67$	13.3396	1.62969	0.814843	+	0.579681i	$0.196823\pi$
$68$	13.5197	1.63950
$69$	0	0
$70$	5.10626	0.610315
$71$	7.93141	0.941285	0.470643	−	0.882324i	$-0.344022\pi$
$71$	7.93141	0.941285	0.470643	+	0.882324i	$0.344022\pi$
$72$	0	0
$73$	4.28111	0.501066	0.250533	−	0.968108i	$-0.419394\pi$
$73$	4.28111	0.501066	0.250533	+	0.968108i	$0.419394\pi$
$74$	−1.18159	−0.137358
$75$	0	0
$76$	−5.57996	−0.640066
$77$	−5.87722	−0.669771
$78$	0	0
$79$	−0.861628	−0.0969407	−0.0484704	−	0.998825i	$-0.515435\pi$
$79$	−0.861628	−0.0969407	−0.0484704	+	0.998825i	$0.515435\pi$
$80$	6.68149	0.747013
$81$	0	0
$82$	5.44382	0.601169
$83$	4.81841	0.528889	0.264444	−	0.964401i	$-0.414811\pi$
$83$	4.81841	0.528889	0.264444	+	0.964401i	$0.414811\pi$
$84$	0	0
$85$	−13.0694	−1.41758
$86$	18.6905	2.01544
$87$	0	0
$88$	6.13125	0.653593
$89$	−6.32964	−0.670941	−0.335470	−	0.942051i	$-0.608895\pi$
$89$	−6.32964	−0.670941	−0.335470	+	0.942051i	$0.608895\pi$
$90$	0	0
$91$	−6.24994	−0.655172
$92$	2.49221	0.259830
$93$	0	0
$94$	9.04523	0.932944
$95$	5.39412	0.553425
$96$	0	0
$97$	10.4822	1.06430	0.532151	−	0.846649i	$-0.321384\pi$
$97$	10.4822	1.06430	0.532151	+	0.846649i	$0.321384\pi$
$98$	−2.11948	−0.214100
$99$	0	0
$100$	2.00438	0.200438
$101$	1.57308	0.156527	0.0782636	−	0.996933i	$-0.475062\pi$
$101$	1.57308	0.156527	0.0782636	+	0.996933i	$0.475062\pi$
$102$	0	0
$103$	5.01559	0.494201	0.247100	−	0.968990i	$-0.420522\pi$
$103$	5.01559	0.494201	0.247100	+	0.968990i	$0.420522\pi$
$104$	6.52008	0.639346
$105$	0	0
$106$	6.39148	0.620796
$107$	5.81204	0.561872	0.280936	−	0.959727i	$-0.409355\pi$
$107$	5.81204	0.561872	0.280936	+	0.959727i	$0.409355\pi$
$108$	0	0
$109$	7.65030	0.732766	0.366383	−	0.930464i	$-0.380596\pi$
$109$	7.65030	0.732766	0.366383	+	0.930464i	$0.380596\pi$
$110$	−30.0106	−2.86140
$111$	0	0
$112$	−2.77332	−0.262054
$113$	−10.4999	−0.987745	−0.493873	−	0.869534i	$-0.664419\pi$
$113$	−10.4999	−0.987745	−0.493873	+	0.869534i	$0.664419\pi$
$114$	0	0
$115$	−2.40920	−0.224659
$116$	−1.60215	−0.148756
$117$	0	0
$118$	8.83875	0.813673
$119$	5.42479	0.497290
$120$	0	0
$121$	23.5417	2.14015
$122$	0.314297	0.0284551
$123$	0	0
$124$	19.5407	1.75481
$125$	10.1084	0.904122
$126$	0	0
$127$	−17.9732	−1.59486	−0.797432	−	0.603409i	$-0.793809\pi$
$127$	−17.9732	−1.59486	−0.797432	+	0.603409i	$0.793809\pi$
$128$	8.09304	0.715331
$129$	0	0
$130$	−31.9138	−2.79903
$131$	13.0641	1.14142	0.570708	−	0.821153i	$-0.306669\pi$
$131$	13.0641	1.14142	0.570708	+	0.821153i	$0.306669\pi$
$132$	0	0
$133$	−2.23897	−0.194143
$134$	−28.2730	−2.44241
$135$	0	0
$136$	−5.65926	−0.485278
$137$	15.0262	1.28377	0.641887	−	0.766799i	$-0.278152\pi$
$137$	15.0262	1.28377	0.641887	+	0.766799i	$0.278152\pi$
$138$	0	0
$139$	−12.7987	−1.08557	−0.542786	−	0.839871i	$-0.682631\pi$
$139$	−12.7987	−1.08557	−0.542786	+	0.839871i	$0.682631\pi$
$140$	−6.00423	−0.507450
$141$	0	0
$142$	−16.8105	−1.41071
$143$	36.7322	3.07170
$144$	0	0
$145$	1.54879	0.128620
$146$	−9.07375	−0.750949
$147$	0	0
$148$	1.38939	0.114207
$149$	21.1303	1.73106	0.865532	−	0.500854i	$-0.166981\pi$
$149$	21.1303	1.73106	0.865532	+	0.500854i	$0.166981\pi$
$150$	0	0
$151$	19.5264	1.58904	0.794520	−	0.607239i	$-0.207723\pi$
$151$	19.5264	1.58904	0.794520	+	0.607239i	$0.207723\pi$
$152$	2.33574	0.189453
$153$	0	0
$154$	12.4567	1.00379
$155$	−18.8899	−1.51728
$156$	0	0
$157$	−4.60638	−0.367630	−0.183815	−	0.982961i	$-0.558845\pi$
$157$	−4.60638	−0.367630	−0.183815	+	0.982961i	$0.558845\pi$
$158$	1.82621	0.145285
$159$	0	0
$160$	−19.1880	−1.51694
$161$	1.00000	0.0788110
$162$	0	0
$163$	16.3229	1.27851	0.639254	−	0.768996i	$-0.279243\pi$
$163$	16.3229	1.27851	0.639254	+	0.768996i	$0.279243\pi$
$164$	−6.40115	−0.499846
$165$	0	0
$166$	−10.2125	−0.792646
$167$	−16.7254	−1.29425	−0.647125	−	0.762384i	$-0.724029\pi$
$167$	−16.7254	−1.29425	−0.647125	+	0.762384i	$0.724029\pi$
$168$	0	0
$169$	26.0617	2.00475
$170$	27.7004	2.12452
$171$	0	0
$172$	−21.9773	−1.67575
$173$	−9.27650	−0.705279	−0.352640	−	0.935759i	$-0.614716\pi$
$173$	−9.27650	−0.705279	−0.352640	+	0.935759i	$0.614716\pi$
$174$	0	0
$175$	0.804258	0.0607962
$176$	16.2994	1.22861
$177$	0	0
$178$	13.4156	1.00554
$179$	7.48254	0.559272	0.279636	−	0.960106i	$-0.409786\pi$
$179$	7.48254	0.559272	0.279636	+	0.960106i	$0.409786\pi$
$180$	0	0
$181$	−10.6482	−0.791472	−0.395736	−	0.918364i	$-0.629510\pi$
$181$	−10.6482	−0.791472	−0.395736	+	0.918364i	$0.629510\pi$
$182$	13.2466	0.981906
$183$	0	0
$184$	−1.04322	−0.0769074
$185$	−1.34311	−0.0987475
$186$	0	0
$187$	−31.8827	−2.33149
$188$	−10.6359	−0.775701
$189$	0	0
$190$	−11.4327	−0.829419
$191$	−3.30294	−0.238992	−0.119496	−	0.992835i	$-0.538128\pi$
$191$	−3.30294	−0.238992	−0.119496	+	0.992835i	$0.538128\pi$
$192$	0	0
$193$	9.38315	0.675414	0.337707	−	0.941251i	$-0.390349\pi$
$193$	9.38315	0.675414	0.337707	+	0.941251i	$0.390349\pi$
$194$	−22.2168	−1.59507
$195$	0	0
$196$	2.49221	0.178015
$197$	23.0929	1.64530	0.822651	−	0.568547i	$-0.192494\pi$
$197$	23.0929	1.64530	0.822651	+	0.568547i	$0.192494\pi$
$198$	0	0
$199$	21.6149	1.53224	0.766118	−	0.642699i	$-0.222186\pi$
$199$	21.6149	1.53224	0.766118	+	0.642699i	$0.222186\pi$
$200$	−0.839020	−0.0593277
$201$	0	0
$202$	−3.33411	−0.234587
$203$	−0.642864	−0.0451202
$204$	0	0
$205$	6.18796	0.432186
$206$	−10.6304	−0.740659
$207$	0	0
$208$	17.3331	1.20183
$209$	13.1589	0.910219
$210$	0	0
$211$	15.4579	1.06416	0.532081	−	0.846693i	$-0.321410\pi$
$211$	15.4579	1.06416	0.532081	+	0.846693i	$0.321410\pi$
$212$	−7.51547	−0.516164
$213$	0	0
$214$	−12.3185	−0.842077
$215$	21.2453	1.44892
$216$	0	0
$217$	7.84074	0.532264
$218$	−16.2147	−1.09820
$219$	0	0
$220$	35.2882	2.37913
$221$	−33.9046	−2.28067
$222$	0	0
$223$	−11.0708	−0.741357	−0.370679	−	0.928761i	$-0.620875\pi$
$223$	−11.0708	−0.741357	−0.370679	+	0.928761i	$0.620875\pi$
$224$	7.96445	0.532147
$225$	0	0
$226$	22.2543	1.48033
$227$	2.44676	0.162397	0.0811984	−	0.996698i	$-0.474125\pi$
$227$	2.44676	0.162397	0.0811984	+	0.996698i	$0.474125\pi$
$228$	0	0
$229$	−5.74097	−0.379374	−0.189687	−	0.981845i	$-0.560747\pi$
$229$	−5.74097	−0.379374	−0.189687	+	0.981845i	$0.560747\pi$
$230$	5.10626	0.336697
$231$	0	0
$232$	0.670650	0.0440303
$233$	−8.66481	−0.567651	−0.283825	−	0.958876i	$-0.591604\pi$
$233$	−8.66481	−0.567651	−0.283825	+	0.958876i	$0.591604\pi$
$234$	0	0
$235$	10.2817	0.670701
$236$	−10.3931	−0.676533
$237$	0	0
$238$	−11.4977	−0.745288
$239$	0.995387	0.0643862	0.0321931	−	0.999482i	$-0.489751\pi$
$239$	0.995387	0.0643862	0.0321931	+	0.999482i	$0.489751\pi$
$240$	0	0
$241$	−13.9247	−0.896967	−0.448483	−	0.893791i	$-0.648036\pi$
$241$	−13.9247	−0.896967	−0.448483	+	0.893791i	$0.648036\pi$
$242$	−49.8961	−3.20745
$243$	0	0
$244$	−0.369568	−0.0236591
$245$	−2.40920	−0.153918
$246$	0	0
$247$	13.9934	0.890378
$248$	−8.17963	−0.519407
$249$	0	0
$250$	−21.4246	−1.35501
$251$	0.229739	0.0145010	0.00725049	−	0.999974i	$-0.497692\pi$
$251$	0.229739	0.0145010	0.00725049	+	0.999974i	$0.497692\pi$
$252$	0	0
$253$	−5.87722	−0.369497
$254$	38.0939	2.39022
$255$	0	0
$256$	5.51468	0.344667
$257$	4.24568	0.264838	0.132419	−	0.991194i	$-0.457726\pi$
$257$	4.24568	0.264838	0.132419	+	0.991194i	$0.457726\pi$
$258$	0	0
$259$	0.557492	0.0346409
$260$	37.5261	2.32727
$261$	0	0
$262$	−27.6892	−1.71064
$263$	−2.99220	−0.184507	−0.0922535	−	0.995736i	$-0.529407\pi$
$263$	−2.99220	−0.184507	−0.0922535	+	0.995736i	$0.529407\pi$
$264$	0	0
$265$	7.26516	0.446295
$266$	4.74545	0.290962
$267$	0	0
$268$	33.2449	2.03076
$269$	8.92878	0.544397	0.272199	−	0.962241i	$-0.412249\pi$
$269$	8.92878	0.544397	0.272199	+	0.962241i	$0.412249\pi$
$270$	0	0
$271$	20.9773	1.27428	0.637140	−	0.770748i	$-0.280117\pi$
$271$	20.9773	1.27428	0.637140	+	0.770748i	$0.280117\pi$
$272$	−15.0447	−0.912218
$273$	0	0
$274$	−31.8478	−1.92399
$275$	−4.72680	−0.285036
$276$	0	0
$277$	5.82584	0.350041	0.175020	−	0.984565i	$-0.444001\pi$
$277$	5.82584	0.350041	0.175020	+	0.984565i	$0.444001\pi$
$278$	27.1266	1.62695
$279$	0	0
$280$	2.51334	0.150201
$281$	21.1877	1.26395	0.631976	−	0.774988i	$-0.282244\pi$
$281$	21.1877	1.26395	0.631976	+	0.774988i	$0.282244\pi$
$282$	0	0
$283$	−12.8049	−0.761173	−0.380587	−	0.924745i	$-0.624278\pi$
$283$	−12.8049	−0.761173	−0.380587	+	0.924745i	$0.624278\pi$
$284$	19.7667	1.17294
$285$	0	0
$286$	−77.8533	−4.60356
$287$	−2.56847	−0.151612
$288$	0	0
$289$	12.4283	0.731079
$290$	−3.28263	−0.192763
$291$	0	0
$292$	10.6694	0.624381
$293$	3.11921	0.182226	0.0911132	−	0.995841i	$-0.470957\pi$
$293$	3.11921	0.182226	0.0911132	+	0.995841i	$0.470957\pi$
$294$	0	0
$295$	10.0469	0.584956
$296$	−0.581588	−0.0338041
$297$	0	0
$298$	−44.7854	−2.59435
$299$	−6.24994	−0.361443
$300$	0	0
$301$	−8.81841	−0.508284
$302$	−41.3859	−2.38149
$303$	0	0
$304$	6.20937	0.356132
$305$	0.357259	0.0204566
$306$	0	0
$307$	−16.1152	−0.919746	−0.459873	−	0.887985i	$-0.652105\pi$
$307$	−16.1152	−0.919746	−0.459873	+	0.887985i	$0.652105\pi$
$308$	−14.6472	−0.834604
$309$	0	0
$310$	40.0369	2.27394
$311$	−30.2428	−1.71491	−0.857457	−	0.514556i	$-0.827957\pi$
$311$	−30.2428	−1.71491	−0.857457	+	0.514556i	$0.827957\pi$
$312$	0	0
$313$	−20.4956	−1.15848	−0.579241	−	0.815156i	$-0.696651\pi$
$313$	−20.4956	−1.15848	−0.579241	+	0.815156i	$0.696651\pi$
$314$	9.76315	0.550967
$315$	0	0
$316$	−2.14736	−0.120798
$317$	−2.95042	−0.165712	−0.0828559	−	0.996562i	$-0.526404\pi$
$317$	−2.95042	−0.165712	−0.0828559	+	0.996562i	$0.526404\pi$
$318$	0	0
$319$	3.77825	0.211541
$320$	27.3056	1.52643
$321$	0	0
$322$	−2.11948	−0.118114
$323$	−12.1459	−0.675817
$324$	0	0
$325$	−5.02656	−0.278823
$326$	−34.5961	−1.91610
$327$	0	0
$328$	2.67948	0.147950
$329$	−4.26766	−0.235284
$330$	0	0
$331$	−21.2497	−1.16799	−0.583994	−	0.811758i	$-0.698511\pi$
$331$	−21.2497	−1.16799	−0.583994	+	0.811758i	$0.698511\pi$
$332$	12.0085	0.659050
$333$	0	0
$334$	35.4492	1.93969
$335$	−32.1377	−1.75587
$336$	0	0
$337$	−24.8118	−1.35158	−0.675792	−	0.737092i	$-0.736198\pi$
$337$	−24.8118	−1.35158	−0.675792	+	0.737092i	$0.736198\pi$
$338$	−55.2374	−3.00452
$339$	0	0
$340$	−32.5717	−1.76645
$341$	−46.0817	−2.49546
$342$	0	0
$343$	1.00000	0.0539949
$344$	9.19956	0.496007
$345$	0	0
$346$	19.6614	1.05700
$347$	−11.8158	−0.634304	−0.317152	−	0.948375i	$-0.602726\pi$
$347$	−11.8158	−0.634304	−0.317152	+	0.948375i	$0.602726\pi$
$348$	0	0
$349$	−12.1614	−0.650984	−0.325492	−	0.945545i	$-0.605530\pi$
$349$	−12.1614	−0.650984	−0.325492	+	0.945545i	$0.605530\pi$
$350$	−1.70461	−0.0911152
$351$	0	0
$352$	−46.8088	−2.49492
$353$	−17.5264	−0.932838	−0.466419	−	0.884564i	$-0.654456\pi$
$353$	−17.5264	−0.932838	−0.466419	+	0.884564i	$0.654456\pi$
$354$	0	0
$355$	−19.1084	−1.01417
$356$	−15.7748	−0.836061
$357$	0	0
$358$	−15.8591	−0.838181
$359$	−7.70275	−0.406535	−0.203268	−	0.979123i	$-0.565156\pi$
$359$	−7.70275	−0.406535	−0.203268	+	0.979123i	$0.565156\pi$
$360$	0	0
$361$	−13.9870	−0.736160
$362$	22.5686	1.18618
$363$	0	0
$364$	−15.5761	−0.816411
$365$	−10.3141	−0.539863
$366$	0	0
$367$	15.0819	0.787271	0.393635	−	0.919267i	$-0.371217\pi$
$367$	15.0819	0.787271	0.393635	+	0.919267i	$0.371217\pi$
$368$	−2.77332	−0.144569
$369$	0	0
$370$	2.84670	0.147993
$371$	−3.01559	−0.156561
$372$	0	0
$373$	−21.5089	−1.11369	−0.556843	−	0.830618i	$-0.687988\pi$
$373$	−21.5089	−1.11369	−0.556843	+	0.830618i	$0.687988\pi$
$374$	67.5747	3.49421
$375$	0	0
$376$	4.45212	0.229600
$377$	4.01786	0.206930
$378$	0	0
$379$	34.7117	1.78302	0.891511	−	0.452999i	$-0.149646\pi$
$379$	34.7117	1.78302	0.891511	+	0.452999i	$0.149646\pi$
$380$	13.4433	0.689625
$381$	0	0
$382$	7.00052	0.358178
$383$	10.2963	0.526119	0.263059	−	0.964780i	$-0.415268\pi$
$383$	10.2963	0.526119	0.263059	+	0.964780i	$0.415268\pi$
$384$	0	0
$385$	14.1594	0.721630
$386$	−19.8874	−1.01224
$387$	0	0
$388$	26.1237	1.32623
$389$	26.1594	1.32633	0.663167	−	0.748471i	$-0.269212\pi$
$389$	26.1594	1.32633	0.663167	+	0.748471i	$0.269212\pi$
$390$	0	0
$391$	5.42479	0.274343
$392$	−1.04322	−0.0526907
$393$	0	0
$394$	−48.9450	−2.46582
$395$	2.07584	0.104447
$396$	0	0
$397$	−4.52380	−0.227043	−0.113522	−	0.993536i	$-0.536213\pi$
$397$	−4.52380	−0.227043	−0.113522	+	0.993536i	$0.536213\pi$
$398$	−45.8123	−2.29636
$399$	0	0
$400$	−2.23046	−0.111523
$401$	18.0222	0.899987	0.449994	−	0.893032i	$-0.351426\pi$
$401$	18.0222	0.899987	0.449994	+	0.893032i	$0.351426\pi$
$402$	0	0
$403$	−49.0041	−2.44107
$404$	3.92044	0.195049
$405$	0	0
$406$	1.36254	0.0676216
$407$	−3.27650	−0.162410
$408$	0	0
$409$	4.67260	0.231045	0.115523	−	0.993305i	$-0.463146\pi$
$409$	4.67260	0.231045	0.115523	+	0.993305i	$0.463146\pi$
$410$	−13.1153	−0.647717
$411$	0	0
$412$	12.4999	0.615825
$413$	−4.17024	−0.205204
$414$	0	0
$415$	−11.6085	−0.569840
$416$	−49.7773	−2.44053
$417$	0	0
$418$	−27.8900	−1.36415
$419$	12.7674	0.623728	0.311864	−	0.950127i	$-0.399047\pi$
$419$	12.7674	0.623728	0.311864	+	0.950127i	$0.399047\pi$
$420$	0	0
$421$	28.1367	1.37130	0.685649	−	0.727932i	$-0.259518\pi$
$421$	28.1367	1.37130	0.685649	+	0.727932i	$0.259518\pi$
$422$	−32.7626	−1.59486
$423$	0	0
$424$	3.14593	0.152780
$425$	4.36293	0.211633
$426$	0	0
$427$	−0.148289	−0.00717622
$428$	14.4848	0.700150
$429$	0	0
$430$	−45.0291	−2.17150
$431$	5.49352	0.264613	0.132307	−	0.991209i	$-0.457762\pi$
$431$	5.49352	0.264613	0.132307	+	0.991209i	$0.457762\pi$
$432$	0	0
$433$	−21.0177	−1.01005	−0.505023	−	0.863106i	$-0.668516\pi$
$433$	−21.0177	−1.01005	−0.505023	+	0.863106i	$0.668516\pi$
$434$	−16.6183	−0.797704
$435$	0	0
$436$	19.0661	0.913102
$437$	−2.23897	−0.107104
$438$	0	0
$439$	39.2674	1.87413	0.937066	−	0.349153i	$-0.113531\pi$
$439$	39.2674	1.87413	0.937066	+	0.349153i	$0.113531\pi$
$440$	−14.7714	−0.704199
$441$	0	0
$442$	71.8602	3.41804
$443$	−14.5489	−0.691240	−0.345620	−	0.938375i	$-0.612331\pi$
$443$	−14.5489	−0.691240	−0.345620	+	0.938375i	$0.612331\pi$
$444$	0	0
$445$	15.2494	0.722890
$446$	23.4644	1.11107
$447$	0	0
$448$	−11.3339	−0.535475
$449$	−11.3660	−0.536395	−0.268197	−	0.963364i	$-0.586428\pi$
$449$	−11.3660	−0.536395	−0.268197	+	0.963364i	$0.586428\pi$
$450$	0	0
$451$	15.0954	0.710816
$452$	−26.1679	−1.23083
$453$	0	0
$454$	−5.18586	−0.243384
$455$	15.0574	0.705900
$456$	0	0
$457$	−9.42744	−0.440997	−0.220499	−	0.975387i	$-0.570768\pi$
$457$	−9.42744	−0.440997	−0.220499	+	0.975387i	$0.570768\pi$
$458$	12.1679	0.568568
$459$	0	0
$460$	−6.00423	−0.279949
$461$	28.7697	1.33994	0.669968	−	0.742390i	$-0.266308\pi$
$461$	28.7697	1.33994	0.669968	+	0.742390i	$0.266308\pi$
$462$	0	0
$463$	−4.65928	−0.216535	−0.108268	−	0.994122i	$-0.534530\pi$
$463$	−4.65928	−0.216535	−0.108268	+	0.994122i	$0.534530\pi$
$464$	1.78287	0.0827675
$465$	0	0
$466$	18.3649	0.850738
$467$	−7.82926	−0.362295	−0.181147	−	0.983456i	$-0.557981\pi$
$467$	−7.82926	−0.362295	−0.181147	+	0.983456i	$0.557981\pi$
$468$	0	0
$469$	13.3396	0.615964
$470$	−21.7918	−1.00518
$471$	0	0
$472$	4.35049	0.200247
$473$	51.8277	2.38304
$474$	0	0
$475$	−1.80070	−0.0826220
$476$	13.5197	0.619674
$477$	0	0
$478$	−2.10971	−0.0964957
$479$	33.9427	1.55088	0.775440	−	0.631421i	$-0.217528\pi$
$479$	33.9427	1.55088	0.775440	+	0.631421i	$0.217528\pi$
$480$	0	0
$481$	−3.48429	−0.158870
$482$	29.5131	1.34428
$483$	0	0
$484$	58.6707	2.66685
$485$	−25.2536	−1.14671
$486$	0	0
$487$	−11.6880	−0.529633	−0.264817	−	0.964299i	$-0.585311\pi$
$487$	−11.6880	−0.529633	−0.264817	+	0.964299i	$0.585311\pi$
$488$	0.154699	0.00700289
$489$	0	0
$490$	5.10626	0.230677
$491$	19.6742	0.887885	0.443943	−	0.896055i	$-0.353579\pi$
$491$	19.6742	0.887885	0.443943	+	0.896055i	$0.353579\pi$
$492$	0	0
$493$	−3.48740	−0.157065
$494$	−29.6588	−1.33441
$495$	0	0
$496$	−21.7449	−0.976374
$497$	7.93141	0.355772
$498$	0	0
$499$	−30.1698	−1.35059	−0.675294	−	0.737549i	$-0.735983\pi$
$499$	−30.1698	−1.35059	−0.675294	+	0.737549i	$0.735983\pi$
$500$	25.1922	1.12663
$501$	0	0
$502$	−0.486927	−0.0217326
$503$	−24.1110	−1.07506	−0.537529	−	0.843246i	$-0.680642\pi$
$503$	−24.1110	−1.07506	−0.537529	+	0.843246i	$0.680642\pi$
$504$	0	0
$505$	−3.78987	−0.168647
$506$	12.4567	0.553766
$507$	0	0
$508$	−44.7929	−1.98736
$509$	−21.5774	−0.956404	−0.478202	−	0.878250i	$-0.658711\pi$
$509$	−21.5774	−0.956404	−0.478202	+	0.878250i	$0.658711\pi$
$510$	0	0
$511$	4.28111	0.189385
$512$	−27.8744	−1.23188
$513$	0	0
$514$	−8.99864	−0.396913
$515$	−12.0836	−0.532466
$516$	0	0
$517$	25.0819	1.10310
$518$	−1.18159	−0.0519163
$519$	0	0
$520$	−15.7082	−0.688850
$521$	22.0862	0.967613	0.483807	−	0.875175i	$-0.339254\pi$
$521$	22.0862	0.967613	0.483807	+	0.875175i	$0.339254\pi$
$522$	0	0
$523$	5.79856	0.253553	0.126777	−	0.991931i	$-0.459537\pi$
$523$	5.79856	0.253553	0.126777	+	0.991931i	$0.459537\pi$
$524$	32.5585	1.42232
$525$	0	0
$526$	6.34191	0.276521
$527$	42.5343	1.85283
$528$	0	0
$529$	1.00000	0.0434783
$530$	−15.3984	−0.668863
$531$	0	0
$532$	−5.57996	−0.241922
$533$	16.0528	0.695322
$534$	0	0
$535$	−14.0024	−0.605376
$536$	−13.9161	−0.601085
$537$	0	0
$538$	−18.9244	−0.815888
$539$	−5.87722	−0.253150
$540$	0	0
$541$	29.6582	1.27511	0.637553	−	0.770407i	$-0.279947\pi$
$541$	29.6582	1.27511	0.637553	+	0.770407i	$0.279947\pi$
$542$	−44.4610	−1.90976
$543$	0	0
$544$	43.2055	1.85242
$545$	−18.4311	−0.789502
$546$	0	0
$547$	−7.24730	−0.309872	−0.154936	−	0.987924i	$-0.549517\pi$
$547$	−7.24730	−0.309872	−0.154936	+	0.987924i	$0.549517\pi$
$548$	37.4484	1.59972
$549$	0	0
$550$	10.0184	0.427184
$551$	1.43935	0.0613183
$552$	0	0
$553$	−0.861628	−0.0366402
$554$	−12.3478	−0.524606
$555$	0	0
$556$	−31.8970	−1.35274
$557$	−21.7634	−0.922146	−0.461073	−	0.887362i	$-0.652535\pi$
$557$	−21.7634	−0.922146	−0.461073	+	0.887362i	$0.652535\pi$
$558$	0	0
$559$	55.1145	2.33109
$560$	6.68149	0.282345
$561$	0	0
$562$	−44.9070	−1.89429
$563$	26.9844	1.13726	0.568629	−	0.822594i	$-0.307474\pi$
$563$	26.9844	1.13726	0.568629	+	0.822594i	$0.307474\pi$
$564$	0	0
$565$	25.2963	1.06422
$566$	27.1398	1.14077
$567$	0	0
$568$	−8.27423	−0.347179
$569$	−37.4420	−1.56965	−0.784826	−	0.619717i	$-0.787248\pi$
$569$	−37.4420	−1.56965	−0.784826	+	0.619717i	$0.787248\pi$
$570$	0	0
$571$	−3.32489	−0.139142	−0.0695711	−	0.997577i	$-0.522163\pi$
$571$	−3.32489	−0.139142	−0.0695711	+	0.997577i	$0.522163\pi$
$572$	91.5443	3.82766
$573$	0	0
$574$	5.44382	0.227221
$575$	0.804258	0.0335399
$576$	0	0
$577$	−41.0573	−1.70924	−0.854618	−	0.519257i	$-0.826209\pi$
$577$	−41.0573	−1.70924	−0.854618	+	0.519257i	$0.826209\pi$
$578$	−26.3417	−1.09567
$579$	0	0
$580$	3.85990	0.160274
$581$	4.81841	0.199901
$582$	0	0
$583$	17.7233	0.734022
$584$	−4.46616	−0.184811
$585$	0	0
$586$	−6.61112	−0.273103
$587$	31.9182	1.31741	0.658703	−	0.752403i	$-0.271105\pi$
$587$	31.9182	1.31741	0.658703	+	0.752403i	$0.271105\pi$
$588$	0	0
$589$	−17.5551	−0.723347
$590$	−21.2943	−0.876674
$591$	0	0
$592$	−1.54610	−0.0635445
$593$	−36.5062	−1.49913	−0.749566	−	0.661930i	$-0.769738\pi$
$593$	−36.5062	−1.49913	−0.749566	+	0.661930i	$0.769738\pi$
$594$	0	0
$595$	−13.0694	−0.535794
$596$	52.6611	2.15708
$597$	0	0
$598$	13.2466	0.541695
$599$	7.51073	0.306880	0.153440	−	0.988158i	$-0.450965\pi$
$599$	7.51073	0.306880	0.153440	+	0.988158i	$0.450965\pi$
$600$	0	0
$601$	−6.05329	−0.246919	−0.123460	−	0.992350i	$-0.539399\pi$
$601$	−6.05329	−0.246919	−0.123460	+	0.992350i	$0.539399\pi$
$602$	18.6905	0.761766
$603$	0	0
$604$	48.6639	1.98011
$605$	−56.7166	−2.30586
$606$	0	0
$607$	24.4047	0.990557	0.495278	−	0.868734i	$-0.335066\pi$
$607$	24.4047	0.990557	0.495278	+	0.868734i	$0.335066\pi$
$608$	−17.8321	−0.723188
$609$	0	0
$610$	−0.757204	−0.0306583
$611$	26.6726	1.07906
$612$	0	0
$613$	4.81180	0.194347	0.0971734	−	0.995267i	$-0.469020\pi$
$613$	4.81180	0.194347	0.0971734	+	0.995267i	$0.469020\pi$
$614$	34.1560	1.37842
$615$	0	0
$616$	6.13125	0.247035
$617$	38.1414	1.53552	0.767758	−	0.640740i	$-0.221372\pi$
$617$	38.1414	1.53552	0.767758	+	0.640740i	$0.221372\pi$
$618$	0	0
$619$	−17.8340	−0.716809	−0.358404	−	0.933566i	$-0.616679\pi$
$619$	−17.8340	−0.716809	−0.358404	+	0.933566i	$0.616679\pi$
$620$	−47.0776	−1.89068
$621$	0	0
$622$	64.0992	2.57014
$623$	−6.32964	−0.253592
$624$	0	0
$625$	−28.3745	−1.13498
$626$	43.4402	1.73622
$627$	0	0
$628$	−11.4801	−0.458104
$629$	3.02428	0.120586
$630$	0	0
$631$	22.2906	0.887377	0.443688	−	0.896181i	$-0.353670\pi$
$631$	22.2906	0.887377	0.443688	+	0.896181i	$0.353670\pi$
$632$	0.898870	0.0357551
$633$	0	0
$634$	6.25336	0.248352
$635$	43.3011	1.71835
$636$	0	0
$637$	−6.24994	−0.247632
$638$	−8.00793	−0.317037
$639$	0	0
$640$	−19.4978	−0.770718
$641$	−28.4927	−1.12540	−0.562698	−	0.826663i	$-0.690237\pi$
$641$	−28.4927	−1.12540	−0.562698	+	0.826663i	$0.690237\pi$
$642$	0	0
$643$	0.975186	0.0384576	0.0192288	−	0.999815i	$-0.493879\pi$
$643$	0.975186	0.0384576	0.0192288	+	0.999815i	$0.493879\pi$
$644$	2.49221	0.0982067
$645$	0	0
$646$	25.7431	1.01285
$647$	27.5193	1.08190	0.540948	−	0.841056i	$-0.318065\pi$
$647$	27.5193	1.08190	0.540948	+	0.841056i	$0.318065\pi$
$648$	0	0
$649$	24.5094	0.962078
$650$	10.6537	0.417873
$651$	0	0
$652$	40.6800	1.59315
$653$	26.1590	1.02368	0.511840	−	0.859081i	$-0.328964\pi$
$653$	26.1590	1.02368	0.511840	+	0.859081i	$0.328964\pi$
$654$	0	0
$655$	−31.4741	−1.22979
$656$	7.12318	0.278113
$657$	0	0
$658$	9.04523	0.352620
$659$	2.67085	0.104042	0.0520208	−	0.998646i	$-0.483434\pi$
$659$	2.67085	0.104042	0.0520208	+	0.998646i	$0.483434\pi$
$660$	0	0
$661$	−18.4928	−0.719285	−0.359643	−	0.933090i	$-0.617101\pi$
$661$	−18.4928	−0.719285	−0.359643	+	0.933090i	$0.617101\pi$
$662$	45.0384	1.75047
$663$	0	0
$664$	−5.02667	−0.195073
$665$	5.39412	0.209175
$666$	0	0
$667$	−0.642864	−0.0248918
$668$	−41.6831	−1.61277
$669$	0	0
$670$	68.1153	2.63152
$671$	0.871528	0.0336450
$672$	0	0
$673$	47.5002	1.83100	0.915499	−	0.402321i	$-0.131797\pi$
$673$	47.5002	1.83100	0.915499	+	0.402321i	$0.131797\pi$
$674$	52.5882	2.02562
$675$	0	0
$676$	64.9512	2.49812
$677$	−33.4288	−1.28477	−0.642386	−	0.766381i	$-0.722056\pi$
$677$	−33.4288	−1.28477	−0.642386	+	0.766381i	$0.722056\pi$
$678$	0	0
$679$	10.4822	0.402268
$680$	13.6343	0.522852
$681$	0	0
$682$	97.6694	3.73995
$683$	−46.7168	−1.78757	−0.893784	−	0.448498i	$-0.851959\pi$
$683$	−46.7168	−1.78757	−0.893784	+	0.448498i	$0.851959\pi$
$684$	0	0
$685$	−36.2012	−1.38317
$686$	−2.11948	−0.0809222
$687$	0	0
$688$	24.4563	0.932386
$689$	18.8472	0.718023
$690$	0	0
$691$	−22.8670	−0.869903	−0.434952	−	0.900454i	$-0.643235\pi$
$691$	−22.8670	−0.869903	−0.434952	+	0.900454i	$0.643235\pi$
$692$	−23.1190	−0.878851
$693$	0	0
$694$	25.0433	0.950631
$695$	30.8347	1.16963
$696$	0	0
$697$	−13.9334	−0.527765
$698$	25.7758	0.975630
$699$	0	0
$700$	2.00438	0.0757583
$701$	−39.0885	−1.47635	−0.738177	−	0.674607i	$-0.764313\pi$
$701$	−39.0885	−1.47635	−0.738177	+	0.674607i	$0.764313\pi$
$702$	0	0
$703$	−1.24821	−0.0470769
$704$	66.6116	2.51052
$705$	0	0
$706$	37.1470	1.39805
$707$	1.57308	0.0591617
$708$	0	0
$709$	18.3762	0.690132	0.345066	−	0.938578i	$-0.387856\pi$
$709$	18.3762	0.690132	0.345066	+	0.938578i	$0.387856\pi$
$710$	40.4999	1.51993
$711$	0	0
$712$	6.60323	0.247466
$713$	7.84074	0.293638
$714$	0	0
$715$	−88.4954	−3.30954
$716$	18.6480	0.696910
$717$	0	0
$718$	16.3258	0.609275
$719$	11.2232	0.418553	0.209276	−	0.977857i	$-0.432889\pi$
$719$	11.2232	0.418553	0.209276	+	0.977857i	$0.432889\pi$
$720$	0	0
$721$	5.01559	0.186790
$722$	29.6453	1.10328
$723$	0	0
$724$	−26.5374	−0.986256
$725$	−0.517028	−0.0192019
$726$	0	0
$727$	7.71953	0.286302	0.143151	−	0.989701i	$-0.454277\pi$
$727$	7.71953	0.286302	0.143151	+	0.989701i	$0.454277\pi$
$728$	6.52008	0.241650
$729$	0	0
$730$	21.8605	0.809093
$731$	−47.8380	−1.76935
$732$	0	0
$733$	−23.2083	−0.857217	−0.428609	−	0.903490i	$-0.640996\pi$
$733$	−23.2083	−0.857217	−0.428609	+	0.903490i	$0.640996\pi$
$734$	−31.9659	−1.17988
$735$	0	0
$736$	7.96445	0.293573
$737$	−78.3995	−2.88788
$738$	0	0
$739$	8.36545	0.307728	0.153864	−	0.988092i	$-0.450828\pi$
$739$	8.36545	0.307728	0.153864	+	0.988092i	$0.450828\pi$
$740$	−3.34731	−0.123050
$741$	0	0
$742$	6.39148	0.234639
$743$	−9.54735	−0.350258	−0.175129	−	0.984545i	$-0.556034\pi$
$743$	−9.54735	−0.350258	−0.175129	+	0.984545i	$0.556034\pi$
$744$	0	0
$745$	−50.9072	−1.86510
$746$	45.5877	1.66908
$747$	0	0
$748$	−79.4582	−2.90528
$749$	5.81204	0.212367
$750$	0	0
$751$	2.83381	0.103407	0.0517035	−	0.998662i	$-0.483535\pi$
$751$	2.83381	0.103407	0.0517035	+	0.998662i	$0.483535\pi$
$752$	11.8356	0.431599
$753$	0	0
$754$	−8.51578	−0.310126
$755$	−47.0432	−1.71208
$756$	0	0
$757$	49.7767	1.80916	0.904582	−	0.426300i	$-0.140183\pi$
$757$	49.7767	1.80916	0.904582	+	0.426300i	$0.140183\pi$
$758$	−73.5709	−2.67222
$759$	0	0
$760$	−5.62727	−0.204123
$761$	−48.7458	−1.76704	−0.883518	−	0.468398i	$-0.844831\pi$
$761$	−48.7458	−1.76704	−0.883518	+	0.468398i	$0.844831\pi$
$762$	0	0
$763$	7.65030	0.276959
$764$	−8.23161	−0.297809
$765$	0	0
$766$	−21.8229	−0.788495
$767$	26.0637	0.941107
$768$	0	0
$769$	−13.3190	−0.480296	−0.240148	−	0.970736i	$-0.577196\pi$
$769$	−13.3190	−0.480296	−0.240148	+	0.970736i	$0.577196\pi$
$770$	−30.0106	−1.08151
$771$	0	0
$772$	23.3847	0.841635
$773$	13.9252	0.500855	0.250427	−	0.968135i	$-0.419429\pi$
$773$	13.9252	0.500855	0.250427	+	0.968135i	$0.419429\pi$
$774$	0	0
$775$	6.30597	0.226517
$776$	−10.9352	−0.392552
$777$	0	0
$778$	−55.4444	−1.98778
$779$	5.75071	0.206040
$780$	0	0
$781$	−46.6146	−1.66800
$782$	−11.4977	−0.411159
$783$	0	0
$784$	−2.77332	−0.0990472
$785$	11.0977	0.396094
$786$	0	0
$787$	45.7185	1.62969	0.814844	−	0.579680i	$-0.196822\pi$
$787$	45.7185	1.62969	0.814844	+	0.579680i	$0.196822\pi$
$788$	57.5523	2.05022
$789$	0	0
$790$	−4.39970	−0.156534
$791$	−10.4999	−0.373333
$792$	0	0
$793$	0.926799	0.0329116
$794$	9.58812	0.340270
$795$	0	0
$796$	53.8687	1.90933
$797$	11.4211	0.404555	0.202277	−	0.979328i	$-0.435166\pi$
$797$	11.4211	0.404555	0.202277	+	0.979328i	$0.435166\pi$
$798$	0	0
$799$	−23.1511	−0.819029
$800$	6.40547	0.226468
$801$	0	0
$802$	−38.1978	−1.34881
$803$	−25.1610	−0.887913
$804$	0	0
$805$	−2.40920	−0.0849132
$806$	103.863	3.65843
$807$	0	0
$808$	−1.64107	−0.0577327
$809$	−17.6307	−0.619864	−0.309932	−	0.950759i	$-0.600306\pi$
$809$	−17.6307	−0.619864	−0.309932	+	0.950759i	$0.600306\pi$
$810$	0	0
$811$	30.9666	1.08738	0.543692	−	0.839285i	$-0.317026\pi$
$811$	30.9666	1.08738	0.543692	+	0.839285i	$0.317026\pi$
$812$	−1.60215	−0.0562244
$813$	0	0
$814$	6.94449	0.243404
$815$	−39.3252	−1.37750
$816$	0	0
$817$	19.7441	0.690759
$818$	−9.90349	−0.346267
$819$	0	0
$820$	15.4217	0.538548
$821$	−25.3060	−0.883187	−0.441593	−	0.897215i	$-0.645587\pi$
$821$	−25.3060	−0.883187	−0.441593	+	0.897215i	$0.645587\pi$
$822$	0	0
$823$	45.6057	1.58972	0.794858	−	0.606795i	$-0.207545\pi$
$823$	45.6057	1.58972	0.794858	+	0.606795i	$0.207545\pi$
$824$	−5.23237	−0.182278
$825$	0	0
$826$	8.83875	0.307539
$827$	25.4338	0.884420	0.442210	−	0.896912i	$-0.354195\pi$
$827$	25.4338	0.884420	0.442210	+	0.896912i	$0.354195\pi$
$828$	0	0
$829$	−5.12771	−0.178093	−0.0890463	−	0.996027i	$-0.528382\pi$
$829$	−5.12771	−0.178093	−0.0890463	+	0.996027i	$0.528382\pi$
$830$	24.6040	0.854019
$831$	0	0
$832$	70.8360	2.45580
$833$	5.42479	0.187958
$834$	0	0
$835$	40.2948	1.39446
$836$	32.7946	1.13423
$837$	0	0
$838$	−27.0603	−0.934782
$839$	7.81839	0.269921	0.134960	−	0.990851i	$-0.456909\pi$
$839$	7.81839	0.269921	0.134960	+	0.990851i	$0.456909\pi$
$840$	0	0
$841$	−28.5867	−0.985749
$842$	−59.6352	−2.05517
$843$	0	0
$844$	38.5242	1.32606
$845$	−62.7880	−2.15997
$846$	0	0
$847$	23.5417	0.808901
$848$	8.36319	0.287193
$849$	0	0
$850$	−9.24715	−0.317175
$851$	0.557492	0.0191106
$852$	0	0
$853$	−36.6540	−1.25501	−0.627505	−	0.778613i	$-0.715924\pi$
$853$	−36.6540	−1.25501	−0.627505	+	0.778613i	$0.715924\pi$
$854$	0.314297	0.0107550
$855$	0	0
$856$	−6.06326	−0.207238
$857$	−44.0599	−1.50506	−0.752529	−	0.658559i	$-0.771166\pi$
$857$	−44.0599	−1.50506	−0.752529	+	0.658559i	$0.771166\pi$
$858$	0	0
$859$	−6.04906	−0.206391	−0.103196	−	0.994661i	$-0.532907\pi$
$859$	−6.04906	−0.206391	−0.103196	+	0.994661i	$0.532907\pi$
$860$	52.9477	1.80550
$861$	0	0
$862$	−11.6434	−0.396576
$863$	25.7890	0.877867	0.438933	−	0.898520i	$-0.355356\pi$
$863$	25.7890	0.877867	0.438933	+	0.898520i	$0.355356\pi$
$864$	0	0
$865$	22.3490	0.759888
$866$	44.5467	1.51376
$867$	0	0
$868$	19.5407	0.663256
$869$	5.06397	0.171784
$870$	0	0
$871$	−83.3714	−2.82493
$872$	−7.98097	−0.270270
$873$	0	0
$874$	4.74545	0.160517
$875$	10.1084	0.341726
$876$	0	0
$877$	9.59839	0.324114	0.162057	−	0.986781i	$-0.448187\pi$
$877$	9.59839	0.324114	0.162057	+	0.986781i	$0.448187\pi$
$878$	−83.2266	−2.80876
$879$	0	0
$880$	−39.2686	−1.32374
$881$	9.38385	0.316150	0.158075	−	0.987427i	$-0.449471\pi$
$881$	9.38385	0.316150	0.158075	+	0.987427i	$0.449471\pi$
$882$	0	0
$883$	−4.98842	−0.167874	−0.0839368	−	0.996471i	$-0.526749\pi$
$883$	−4.98842	−0.167874	−0.0839368	+	0.996471i	$0.526749\pi$
$884$	−84.4973	−2.84195
$885$	0	0
$886$	30.8362	1.03596
$887$	−36.6169	−1.22948	−0.614738	−	0.788732i	$-0.710738\pi$
$887$	−36.6169	−1.22948	−0.614738	+	0.788732i	$0.710738\pi$
$888$	0	0
$889$	−17.9732	−0.602802
$890$	−32.3208	−1.08340
$891$	0	0
$892$	−27.5908	−0.923808
$893$	9.55514	0.319750
$894$	0	0
$895$	−18.0270	−0.602575
$896$	8.09304	0.270370
$897$	0	0
$898$	24.0900	0.803895
$899$	−5.04052	−0.168111
$900$	0	0
$901$	−16.3589	−0.544995
$902$	−31.9945	−1.06530
$903$	0	0
$904$	10.9537	0.364315
$905$	25.6536	0.852754
$906$	0	0
$907$	34.2361	1.13679	0.568395	−	0.822756i	$-0.307564\pi$
$907$	34.2361	1.13679	0.568395	+	0.822756i	$0.307564\pi$
$908$	6.09782	0.202363
$909$	0	0
$910$	−31.9138	−1.05793
$911$	−2.10958	−0.0698934	−0.0349467	−	0.999389i	$-0.511126\pi$
$911$	−2.10958	−0.0698934	−0.0349467	+	0.999389i	$0.511126\pi$
$912$	0	0
$913$	−28.3188	−0.937216
$914$	19.9813	0.660923
$915$	0	0
$916$	−14.3077	−0.472739
$917$	13.0641	0.431415
$918$	0	0
$919$	49.7639	1.64156	0.820779	−	0.571245i	$-0.193539\pi$
$919$	49.7639	1.64156	0.820779	+	0.571245i	$0.193539\pi$
$920$	2.51334	0.0828622
$921$	0	0
$922$	−60.9768	−2.00816
$923$	−49.5708	−1.63164
$924$	0	0
$925$	0.448367	0.0147422
$926$	9.87527	0.324521
$927$	0	0
$928$	−5.12005	−0.168074
$929$	45.0863	1.47923	0.739617	−	0.673028i	$-0.235007\pi$
$929$	45.0863	1.47923	0.739617	+	0.673028i	$0.235007\pi$
$930$	0	0
$931$	−2.23897	−0.0733791
$932$	−21.5945	−0.707351
$933$	0	0
$934$	16.5940	0.542971
$935$	76.8118	2.51201
$936$	0	0
$937$	14.8779	0.486040	0.243020	−	0.970021i	$-0.421862\pi$
$937$	14.8779	0.486040	0.243020	+	0.970021i	$0.421862\pi$
$938$	−28.2730	−0.923145
$939$	0	0
$940$	25.6240	0.835763
$941$	−30.3917	−0.990742	−0.495371	−	0.868681i	$-0.664968\pi$
$941$	−30.3917	−0.990742	−0.495371	+	0.868681i	$0.664968\pi$
$942$	0	0
$943$	−2.56847	−0.0836408
$944$	11.5654	0.376422
$945$	0	0
$946$	−109.848	−3.57146
$947$	26.5595	0.863068	0.431534	−	0.902097i	$-0.357972\pi$
$947$	26.5595	0.863068	0.431534	+	0.902097i	$0.357972\pi$
$948$	0	0
$949$	−26.7567	−0.868559
$950$	3.81656	0.123826
$951$	0	0
$952$	−5.65926	−0.183418
$953$	−8.75467	−0.283592	−0.141796	−	0.989896i	$-0.545288\pi$
$953$	−8.75467	−0.283592	−0.141796	+	0.989896i	$0.545288\pi$
$954$	0	0
$955$	7.95745	0.257497
$956$	2.48071	0.0802319
$957$	0	0
$958$	−71.9409	−2.32430
$959$	15.0262	0.485221
$960$	0	0
$961$	30.4771	0.983134
$962$	7.38489	0.238099
$963$	0	0
$964$	−34.7031	−1.11771
$965$	−22.6059	−0.727710
$966$	0	0
$967$	−19.3848	−0.623372	−0.311686	−	0.950185i	$-0.600894\pi$
$967$	−19.3848	−0.623372	−0.311686	+	0.950185i	$0.600894\pi$
$968$	−24.5592	−0.789363
$969$	0	0
$970$	53.5247	1.71857
$971$	17.9424	0.575799	0.287899	−	0.957661i	$-0.407043\pi$
$971$	17.9424	0.575799	0.287899	+	0.957661i	$0.407043\pi$
$972$	0	0
$973$	−12.7987	−0.410308
$974$	24.7725	0.793762
$975$	0	0
$976$	0.411254	0.0131639
$977$	−5.12609	−0.163998	−0.0819991	−	0.996632i	$-0.526130\pi$
$977$	−5.12609	−0.163998	−0.0819991	+	0.996632i	$0.526130\pi$
$978$	0	0
$979$	37.2007	1.18894
$980$	−6.00423	−0.191798
$981$	0	0
$982$	−41.6992	−1.33067
$983$	48.1332	1.53521	0.767606	−	0.640922i	$-0.221448\pi$
$983$	48.1332	1.53521	0.767606	+	0.640922i	$0.221448\pi$
$984$	0	0
$985$	−55.6355	−1.77269
$986$	7.39148	0.235393
$987$	0	0
$988$	34.8744	1.10950
$989$	−8.81841	−0.280409
$990$	0	0
$991$	−6.50910	−0.206769	−0.103384	−	0.994641i	$-0.532967\pi$
$991$	−6.50910	−0.206769	−0.103384	+	0.994641i	$0.532967\pi$
$992$	62.4471	1.98270
$993$	0	0
$994$	−16.8105	−0.533196
$995$	−52.0746	−1.65088
$996$	0	0
$997$	43.2321	1.36917	0.684587	−	0.728931i	$-0.259982\pi$
$997$	43.2321	1.36917	0.684587	+	0.728931i	$0.259982\pi$
$998$	63.9445	2.02413
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.r.1.2		5
3.2	odd	2		161.2.a.d.1.4	✓	5
12.11	even	2		2576.2.a.bd.1.4		5
15.14	odd	2		4025.2.a.p.1.2		5
21.20	even	2		1127.2.a.h.1.4		5
69.68	even	2		3703.2.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.4	✓	5	3.2	odd	2
1127.2.a.h.1.4		5	21.20	even	2
1449.2.a.r.1.2		5	1.1	even	1	trivial
2576.2.a.bd.1.4		5	12.11	even	2
3703.2.a.j.1.4		5	69.68	even	2
4025.2.a.p.1.2		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11948 q^{2} +2.49221 q^{4} -2.40920 q^{5} +1.00000 q^{7} -1.04322 q^{8} +O(q^{10})\)	\(q-2.11948 q^{2} +2.49221 q^{4} -2.40920 q^{5} +1.00000 q^{7} -1.04322 q^{8} +5.10626 q^{10} -5.87722 q^{11} -6.24994 q^{13} -2.11948 q^{14} -2.77332 q^{16} +5.42479 q^{17} -2.23897 q^{19} -6.00423 q^{20} +12.4567 q^{22} +1.00000 q^{23} +0.804258 q^{25} +13.2466 q^{26} +2.49221 q^{28} -0.642864 q^{29} +7.84074 q^{31} +7.96445 q^{32} -11.4977 q^{34} -2.40920 q^{35} +0.557492 q^{37} +4.74545 q^{38} +2.51334 q^{40} -2.56847 q^{41} -8.81841 q^{43} -14.6472 q^{44} -2.11948 q^{46} -4.26766 q^{47} +1.00000 q^{49} -1.70461 q^{50} -15.5761 q^{52} -3.01559 q^{53} +14.1594 q^{55} -1.04322 q^{56} +1.36254 q^{58} -4.17024 q^{59} -0.148289 q^{61} -16.6183 q^{62} -11.3339 q^{64} +15.0574 q^{65} +13.3396 q^{67} +13.5197 q^{68} +5.10626 q^{70} +7.93141 q^{71} +4.28111 q^{73} -1.18159 q^{74} -5.57996 q^{76} -5.87722 q^{77} -0.861628 q^{79} +6.68149 q^{80} +5.44382 q^{82} +4.81841 q^{83} -13.0694 q^{85} +18.6905 q^{86} +6.13125 q^{88} -6.32964 q^{89} -6.24994 q^{91} +2.49221 q^{92} +9.04523 q^{94} +5.39412 q^{95} +10.4822 q^{97} -2.11948 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8 - 8 * q^10 + 4 * q^11 - 6 * q^13 - 2 * q^14 + 10 * q^16 + 12 * q^17 + 6 * q^19 + 14 * q^22 + 5 * q^23 + 19 * q^25 - q^26 + 12 * q^28 + 4 * q^29 + 30 * q^31 - 8 * q^32 + 6 * q^34 + 4 * q^35 + 4 * q^37 + 40 * q^38 - 50 * q^40 - 6 * q^41 - 12 * q^43 + 26 * q^44 - 2 * q^46 - 10 * q^47 + 5 * q^49 + 2 * q^50 - 21 * q^52 - 16 * q^53 + 18 * q^55 - 3 * q^56 + 13 * q^58 - 22 * q^59 - 18 * q^61 - 15 * q^62 + 25 * q^64 + 26 * q^65 - 2 * q^67 - 12 * q^68 - 8 * q^70 - 4 * q^71 - 2 * q^73 - 38 * q^74 + 10 * q^76 + 4 * q^77 + 30 * q^79 + 10 * q^80 - 7 * q^82 - 8 * q^83 - 12 * q^85 - 8 * q^86 + 4 * q^88 + 20 * q^89 - 6 * q^91 + 12 * q^92 - 25 * q^94 - 8 * q^95 - 12 * q^97 - 2 * q^98

Introduction

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