LMFDB - Embedded newform 1449.2.a.r.1.1

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			1.1
Root			$2.69017$ 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)

$f(q)$	$=$	$q-2.69017 q^{2} +5.23702 q^{4} +3.51109 q^{5} +1.00000 q^{7} -8.70812 q^{8} +O(q^{10})$	\(q-2.69017 q^{2} +5.23702 q^{4} +3.51109 q^{5} +1.00000 q^{7} -8.70812 q^{8} -9.44544 q^{10} +3.78810 q^{11} +1.24125 q^{13} -2.69017 q^{14} +12.9523 q^{16} -5.98512 q^{17} -3.38034 q^{19} +18.3877 q^{20} -10.1906 q^{22} +1.00000 q^{23} +7.32778 q^{25} -3.33918 q^{26} +5.23702 q^{28} +7.02088 q^{29} +6.26984 q^{31} -17.4276 q^{32} +16.1010 q^{34} +3.51109 q^{35} +4.84066 q^{37} +9.09369 q^{38} -30.5750 q^{40} +1.78094 q^{41} +3.02219 q^{43} +19.8383 q^{44} -2.69017 q^{46} -3.90322 q^{47} +1.00000 q^{49} -19.7130 q^{50} +6.50046 q^{52} +2.47403 q^{53} +13.3004 q^{55} -8.70812 q^{56} -18.8873 q^{58} +2.89143 q^{59} -10.3518 q^{61} -16.8669 q^{62} +20.9787 q^{64} +4.35815 q^{65} +11.4466 q^{67} -31.3442 q^{68} -9.44544 q^{70} -2.70157 q^{71} -14.1893 q^{73} -13.0222 q^{74} -17.7029 q^{76} +3.78810 q^{77} +3.31407 q^{79} +45.4767 q^{80} -4.79102 q^{82} -7.02219 q^{83} -21.0143 q^{85} -8.13020 q^{86} -32.9872 q^{88} +1.59107 q^{89} +1.24125 q^{91} +5.23702 q^{92} +10.5003 q^{94} -11.8687 q^{95} -11.6270 q^{97} -2.69017 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.69017	−1.90224	−0.951119	−	0.308825i	$-0.900064\pi$
$2$	−2.69017	−1.90224	−0.951119	+	0.308825i	$0.900064\pi$
$3$	0	0
$3$	0	0
$4$	5.23702	2.61851
$5$	3.51109	1.57021	0.785104	−	0.619363i	$-0.212609\pi$
$5$	3.51109	1.57021	0.785104	+	0.619363i	$0.212609\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−8.70812	−3.07879
$9$	0	0
$10$	−9.44544	−2.98691
$11$	3.78810	1.14215	0.571077	−	0.820896i	$-0.306526\pi$
$11$	3.78810	1.14215	0.571077	+	0.820896i	$0.306526\pi$
$12$	0	0
$13$	1.24125	0.344261	0.172131	−	0.985074i	$-0.444935\pi$
$13$	1.24125	0.344261	0.172131	+	0.985074i	$0.444935\pi$
$14$	−2.69017	−0.718978
$15$	0	0
$16$	12.9523	3.23807
$17$	−5.98512	−1.45161	−0.725803	−	0.687903i	$-0.758532\pi$
$17$	−5.98512	−1.45161	−0.725803	+	0.687903i	$0.758532\pi$
$18$	0	0
$19$	−3.38034	−0.775503	−0.387752	−	0.921764i	$-0.626748\pi$
$19$	−3.38034	−0.775503	−0.387752	+	0.921764i	$0.626748\pi$
$20$	18.3877	4.11160
$21$	0	0
$22$	−10.1906	−2.17265
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.32778	1.46556
$26$	−3.33918	−0.654867
$27$	0	0
$28$	5.23702	0.989703
$29$	7.02088	1.30374	0.651872	−	0.758329i	$-0.273984\pi$
$29$	7.02088	1.30374	0.651872	+	0.758329i	$0.273984\pi$
$30$	0	0
$31$	6.26984	1.12610	0.563048	−	0.826424i	$-0.309628\pi$
$31$	6.26984	1.12610	0.563048	+	0.826424i	$0.309628\pi$
$32$	−17.4276	−3.08080
$33$	0	0
$34$	16.1010	2.76130
$35$	3.51109	0.593483
$36$	0	0
$37$	4.84066	0.795799	0.397899	−	0.917429i	$-0.369739\pi$
$37$	4.84066	0.795799	0.397899	+	0.917429i	$0.369739\pi$
$38$	9.09369	1.47519
$39$	0	0
$40$	−30.5750	−4.83434
$41$	1.78094	0.278135	0.139068	−	0.990283i	$-0.455589\pi$
$41$	1.78094	0.278135	0.139068	+	0.990283i	$0.455589\pi$
$42$	0	0
$43$	3.02219	0.460879	0.230440	−	0.973087i	$-0.425984\pi$
$43$	3.02219	0.460879	0.230440	+	0.973087i	$0.425984\pi$
$44$	19.8383	2.99074
$45$	0	0
$46$	−2.69017	−0.396644
$47$	−3.90322	−0.569343	−0.284671	−	0.958625i	$-0.591884\pi$
$47$	−3.90322	−0.569343	−0.284671	+	0.958625i	$0.591884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−19.7130	−2.78784
$51$	0	0
$52$	6.50046	0.901451
$53$	2.47403	0.339834	0.169917	−	0.985458i	$-0.445650\pi$
$53$	2.47403	0.339834	0.169917	+	0.985458i	$0.445650\pi$
$54$	0	0
$55$	13.3004	1.79342
$56$	−8.70812	−1.16367
$57$	0	0
$58$	−18.8873	−2.48003
$59$	2.89143	0.376433	0.188216	−	0.982128i	$-0.439729\pi$
$59$	2.89143	0.376433	0.188216	+	0.982128i	$0.439729\pi$
$60$	0	0
$61$	−10.3518	−1.32541	−0.662703	−	0.748882i	$-0.730591\pi$
$61$	−10.3518	−1.32541	−0.662703	+	0.748882i	$0.730591\pi$
$62$	−16.8669	−2.14210
$63$	0	0
$64$	20.9787	2.62234
$65$	4.35815	0.540562
$66$	0	0
$67$	11.4466	1.39843	0.699213	−	0.714913i	$-0.253534\pi$
$67$	11.4466	1.39843	0.699213	+	0.714913i	$0.253534\pi$
$68$	−31.3442	−3.80104
$69$	0	0
$70$	−9.44544	−1.12895
$71$	−2.70157	−0.320617	−0.160309	−	0.987067i	$-0.551249\pi$
$71$	−2.70157	−0.320617	−0.160309	+	0.987067i	$0.551249\pi$
$72$	0	0
$73$	−14.1893	−1.66073	−0.830367	−	0.557217i	$-0.811869\pi$
$73$	−14.1893	−1.66073	−0.830367	+	0.557217i	$0.811869\pi$
$74$	−13.0222	−1.51380
$75$	0	0
$76$	−17.7029	−2.03066
$77$	3.78810	0.431694
$78$	0	0
$79$	3.31407	0.372862	0.186431	−	0.982468i	$-0.440308\pi$
$79$	3.31407	0.372862	0.186431	+	0.982468i	$0.440308\pi$
$80$	45.4767	5.08445
$81$	0	0
$82$	−4.79102	−0.529080
$83$	−7.02219	−0.770785	−0.385393	−	0.922753i	$-0.625934\pi$
$83$	−7.02219	−0.770785	−0.385393	+	0.922753i	$0.625934\pi$
$84$	0	0
$85$	−21.0143	−2.27932
$86$	−8.13020	−0.876702
$87$	0	0
$88$	−32.9872	−3.51645
$89$	1.59107	0.168653	0.0843265	−	0.996438i	$-0.473126\pi$
$89$	1.59107	0.168653	0.0843265	+	0.996438i	$0.473126\pi$
$90$	0	0
$91$	1.24125	0.130119
$92$	5.23702	0.545997
$93$	0	0
$94$	10.5003	1.08302
$95$	−11.8687	−1.21770
$96$	0	0
$97$	−11.6270	−1.18054	−0.590270	−	0.807206i	$-0.700979\pi$
$97$	−11.6270	−1.18054	−0.590270	+	0.807206i	$0.700979\pi$
$98$	−2.69017	−0.271748
$99$	0	0
$100$	38.3757	3.83757
$101$	0.366626	0.0364806	0.0182403	−	0.999834i	$-0.494194\pi$
$101$	0.366626	0.0364806	0.0182403	+	0.999834i	$0.494194\pi$
$102$	0	0
$103$	−0.474030	−0.0467076	−0.0233538	−	0.999727i	$-0.507434\pi$
$103$	−0.474030	−0.0467076	−0.0233538	+	0.999727i	$0.507434\pi$
$104$	−10.8090	−1.05991
$105$	0	0
$106$	−6.65556	−0.646445
$107$	5.74697	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$107$	5.74697	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$108$	0	0
$109$	15.4877	1.48346	0.741728	−	0.670700i	$-0.234006\pi$
$109$	15.4877	1.48346	0.741728	+	0.670700i	$0.234006\pi$
$110$	−35.7802	−3.41151
$111$	0	0
$112$	12.9523	1.22388
$113$	4.48250	0.421678	0.210839	−	0.977521i	$-0.432380\pi$
$113$	4.48250	0.421678	0.210839	+	0.977521i	$0.432380\pi$
$114$	0	0
$115$	3.51109	0.327411
$116$	36.7684	3.41386
$117$	0	0
$118$	−7.77845	−0.716064
$119$	−5.98512	−0.548655
$120$	0	0
$121$	3.34968	0.304516
$122$	27.8480	2.52124
$123$	0	0
$124$	32.8353	2.94869
$125$	8.17306	0.731021
$126$	0	0
$127$	−2.13061	−0.189061	−0.0945307	−	0.995522i	$-0.530135\pi$
$127$	−2.13061	−0.189061	−0.0945307	+	0.995522i	$0.530135\pi$
$128$	−21.5811	−1.90751
$129$	0	0
$130$	−11.7242	−1.02828
$131$	18.1242	1.58352	0.791760	−	0.610832i	$-0.209165\pi$
$131$	18.1242	1.58352	0.791760	+	0.610832i	$0.209165\pi$
$132$	0	0
$133$	−3.38034	−0.293113
$134$	−30.7933	−2.66014
$135$	0	0
$136$	52.1192	4.46918
$137$	15.3062	1.30770	0.653849	−	0.756625i	$-0.273153\pi$
$137$	15.3062	1.30770	0.653849	+	0.756625i	$0.273153\pi$
$138$	0	0
$139$	13.5273	1.14737	0.573687	−	0.819074i	$-0.305512\pi$
$139$	13.5273	1.14737	0.573687	+	0.819074i	$0.305512\pi$
$140$	18.3877	1.55404
$141$	0	0
$142$	7.26768	0.609890
$143$	4.70198	0.393200
$144$	0	0
$145$	24.6510	2.04715
$146$	38.1717	3.15911
$147$	0	0
$148$	25.3506	2.08381
$149$	−5.75773	−0.471691	−0.235846	−	0.971791i	$-0.575786\pi$
$149$	−5.75773	−0.471691	−0.235846	+	0.971791i	$0.575786\pi$
$150$	0	0
$151$	−9.57813	−0.779457	−0.389728	−	0.920930i	$-0.627431\pi$
$151$	−9.57813	−0.779457	−0.389728	+	0.920930i	$0.627431\pi$
$152$	29.4364	2.38761
$153$	0	0
$154$	−10.1906	−0.821184
$155$	22.0140	1.76821
$156$	0	0
$157$	−5.03706	−0.402001	−0.201001	−	0.979591i	$-0.564419\pi$
$157$	−5.03706	−0.402001	−0.201001	+	0.979591i	$0.564419\pi$
$158$	−8.91540	−0.709271
$159$	0	0
$160$	−61.1901	−4.83750
$161$	1.00000	0.0788110
$162$	0	0
$163$	−7.35713	−0.576255	−0.288127	−	0.957592i	$-0.593033\pi$
$163$	−7.35713	−0.576255	−0.288127	+	0.957592i	$0.593033\pi$
$164$	9.32679	0.728300
$165$	0	0
$166$	18.8909	1.46622
$167$	−20.9904	−1.62428	−0.812142	−	0.583460i	$-0.801698\pi$
$167$	−20.9904	−1.62428	−0.812142	+	0.583460i	$0.801698\pi$
$168$	0	0
$169$	−11.4593	−0.881484
$170$	56.5321	4.33582
$171$	0	0
$172$	15.8272	1.20682
$173$	12.3369	0.937955	0.468978	−	0.883210i	$-0.344622\pi$
$173$	12.3369	0.937955	0.468978	+	0.883210i	$0.344622\pi$
$174$	0	0
$175$	7.32778	0.553928
$176$	49.0646	3.69838
$177$	0	0
$178$	−4.28025	−0.320818
$179$	12.9082	0.964807	0.482404	−	0.875949i	$-0.339764\pi$
$179$	12.9082	0.964807	0.482404	+	0.875949i	$0.339764\pi$
$180$	0	0
$181$	−5.86925	−0.436258	−0.218129	−	0.975920i	$-0.569995\pi$
$181$	−5.86925	−0.436258	−0.218129	+	0.975920i	$0.569995\pi$
$182$	−3.33918	−0.247516
$183$	0	0
$184$	−8.70812	−0.641971
$185$	16.9960	1.24957
$186$	0	0
$187$	−22.6722	−1.65796
$188$	−20.4412	−1.49083
$189$	0	0
$190$	31.9288	2.31636
$191$	−11.9343	−0.863539	−0.431769	−	0.901984i	$-0.642111\pi$
$191$	−11.9343	−0.863539	−0.431769	+	0.901984i	$0.642111\pi$
$192$	0	0
$193$	0.752900	0.0541949	0.0270975	−	0.999633i	$-0.491374\pi$
$193$	0.752900	0.0541949	0.0270975	+	0.999633i	$0.491374\pi$
$194$	31.2785	2.24567
$195$	0	0
$196$	5.23702	0.374073
$197$	−25.4074	−1.81020	−0.905100	−	0.425200i	$-0.860204\pi$
$197$	−25.4074	−1.81020	−0.905100	+	0.425200i	$0.860204\pi$
$198$	0	0
$199$	15.1988	1.07741	0.538707	−	0.842493i	$-0.318913\pi$
$199$	15.1988	1.07741	0.538707	+	0.842493i	$0.318913\pi$
$200$	−63.8112	−4.51213
$201$	0	0
$202$	−0.986286	−0.0693948
$203$	7.02088	0.492769
$204$	0	0
$205$	6.25303	0.436731
$206$	1.27522	0.0888489
$207$	0	0
$208$	16.0771	1.11474
$209$	−12.8051	−0.885744
$210$	0	0
$211$	−24.2797	−1.67148	−0.835742	−	0.549123i	$-0.814962\pi$
$211$	−24.2797	−1.67148	−0.835742	+	0.549123i	$0.814962\pi$
$212$	12.9565	0.889858
$213$	0	0
$214$	−15.4603	−1.05685
$215$	10.6112	0.723677
$216$	0	0
$217$	6.26984	0.425625
$218$	−41.6647	−2.82189
$219$	0	0
$220$	69.6542	4.69609
$221$	−7.42905	−0.499732
$222$	0	0
$223$	17.7344	1.18758	0.593791	−	0.804619i	$-0.297631\pi$
$223$	17.7344	1.18758	0.593791	+	0.804619i	$0.297631\pi$
$224$	−17.4276	−1.16443
$225$	0	0
$226$	−12.0587	−0.802133
$227$	15.7087	1.04263	0.521313	−	0.853366i	$-0.325442\pi$
$227$	15.7087	1.04263	0.521313	+	0.853366i	$0.325442\pi$
$228$	0	0
$229$	−4.51633	−0.298448	−0.149224	−	0.988803i	$-0.547678\pi$
$229$	−4.51633	−0.298448	−0.149224	+	0.988803i	$0.547678\pi$
$230$	−9.44544	−0.622814
$231$	0	0
$232$	−61.1386	−4.01395
$233$	16.2641	1.06549	0.532747	−	0.846275i	$-0.321160\pi$
$233$	16.2641	1.06549	0.532747	+	0.846275i	$0.321160\pi$
$234$	0	0
$235$	−13.7046	−0.893987
$236$	15.1425	0.985692
$237$	0	0
$238$	16.1010	1.04367
$239$	−2.14756	−0.138914	−0.0694571	−	0.997585i	$-0.522127\pi$
$239$	−2.14756	−0.138914	−0.0694571	+	0.997585i	$0.522127\pi$
$240$	0	0
$241$	12.4676	0.803111	0.401555	−	0.915835i	$-0.368470\pi$
$241$	12.4676	0.803111	0.401555	+	0.915835i	$0.368470\pi$
$242$	−9.01121	−0.579262
$243$	0	0
$244$	−54.2123	−3.47059
$245$	3.51109	0.224316
$246$	0	0
$247$	−4.19585	−0.266976
$248$	−54.5985	−3.46701
$249$	0	0
$250$	−21.9869	−1.39057
$251$	−4.91478	−0.310218	−0.155109	−	0.987897i	$-0.549573\pi$
$251$	−4.91478	−0.310218	−0.155109	+	0.987897i	$0.549573\pi$
$252$	0	0
$253$	3.78810	0.238156
$254$	5.73172	0.359640
$255$	0	0
$256$	16.0993	1.00620
$257$	−28.4782	−1.77642	−0.888212	−	0.459433i	$-0.848052\pi$
$257$	−28.4782	−1.77642	−0.888212	+	0.459433i	$0.848052\pi$
$258$	0	0
$259$	4.84066	0.300784
$260$	22.8237	1.41547
$261$	0	0
$262$	−48.7572	−3.01223
$263$	−1.89322	−0.116741	−0.0583703	−	0.998295i	$-0.518590\pi$
$263$	−1.89322	−0.116741	−0.0583703	+	0.998295i	$0.518590\pi$
$264$	0	0
$265$	8.68655	0.533611
$266$	9.09369	0.557570
$267$	0	0
$268$	59.9461	3.66179
$269$	2.51156	0.153133	0.0765663	−	0.997064i	$-0.475604\pi$
$269$	2.51156	0.153133	0.0765663	+	0.997064i	$0.475604\pi$
$270$	0	0
$271$	25.1098	1.52531	0.762656	−	0.646804i	$-0.223895\pi$
$271$	25.1098	1.52531	0.762656	+	0.646804i	$0.223895\pi$
$272$	−77.5211	−4.70041
$273$	0	0
$274$	−41.1763	−2.48755
$275$	27.7583	1.67389
$276$	0	0
$277$	9.48643	0.569984	0.284992	−	0.958530i	$-0.408009\pi$
$277$	9.48643	0.569984	0.284992	+	0.958530i	$0.408009\pi$
$278$	−36.3909	−2.18258
$279$	0	0
$280$	−30.5750	−1.82721
$281$	−8.71197	−0.519713	−0.259856	−	0.965647i	$-0.583675\pi$
$281$	−8.71197	−0.519713	−0.259856	+	0.965647i	$0.583675\pi$
$282$	0	0
$283$	30.5543	1.81627	0.908133	−	0.418683i	$-0.137508\pi$
$283$	30.5543	1.81627	0.908133	+	0.418683i	$0.137508\pi$
$284$	−14.1482	−0.839538
$285$	0	0
$286$	−12.6491	−0.747959
$287$	1.78094	0.105125
$288$	0	0
$289$	18.8217	1.10716
$290$	−66.3153	−3.89417
$291$	0	0
$292$	−74.3096	−4.34864
$293$	−12.7063	−0.742312	−0.371156	−	0.928570i	$-0.621039\pi$
$293$	−12.7063	−0.742312	−0.371156	+	0.928570i	$0.621039\pi$
$294$	0	0
$295$	10.1521	0.591078
$296$	−42.1530	−2.45009
$297$	0	0
$298$	15.4893	0.897269
$299$	1.24125	0.0717835
$300$	0	0
$301$	3.02219	0.174196
$302$	25.7668	1.48271
$303$	0	0
$304$	−43.7832	−2.51114
$305$	−36.3460	−2.08116
$306$	0	0
$307$	22.1153	1.26219	0.631094	−	0.775706i	$-0.282606\pi$
$307$	22.1153	1.26219	0.631094	+	0.775706i	$0.282606\pi$
$308$	19.8383	1.13039
$309$	0	0
$310$	−59.2214	−3.36355
$311$	−21.3945	−1.21317	−0.606586	−	0.795018i	$-0.707461\pi$
$311$	−21.3945	−1.21317	−0.606586	+	0.795018i	$0.707461\pi$
$312$	0	0
$313$	−29.9051	−1.69034	−0.845169	−	0.534498i	$-0.820501\pi$
$313$	−29.9051	−1.69034	−0.845169	+	0.534498i	$0.820501\pi$
$314$	13.5506	0.764702
$315$	0	0
$316$	17.3558	0.976341
$317$	−7.06103	−0.396587	−0.198294	−	0.980143i	$-0.563540\pi$
$317$	−7.06103	−0.396587	−0.198294	+	0.980143i	$0.563540\pi$
$318$	0	0
$319$	26.5958	1.48908
$320$	73.6583	4.11762
$321$	0	0
$322$	−2.69017	−0.149917
$323$	20.2318	1.12573
$324$	0	0
$325$	9.09562	0.504534
$326$	19.7919	1.09617
$327$	0	0
$328$	−15.5086	−0.856320
$329$	−3.90322	−0.215191
$330$	0	0
$331$	16.2063	0.890777	0.445388	−	0.895338i	$-0.353066\pi$
$331$	16.2063	0.890777	0.445388	+	0.895338i	$0.353066\pi$
$332$	−36.7753	−2.01831
$333$	0	0
$334$	56.4677	3.08977
$335$	40.1901	2.19582
$336$	0	0
$337$	5.21804	0.284245	0.142122	−	0.989849i	$-0.454607\pi$
$337$	5.21804	0.284245	0.142122	+	0.989849i	$0.454607\pi$
$338$	30.8274	1.67679
$339$	0	0
$340$	−110.052	−5.96843
$341$	23.7508	1.28618
$342$	0	0
$343$	1.00000	0.0539949
$344$	−26.3176	−1.41895
$345$	0	0
$346$	−33.1883	−1.78421
$347$	−4.19094	−0.224982	−0.112491	−	0.993653i	$-0.535883\pi$
$347$	−4.19094	−0.224982	−0.112491	+	0.993653i	$0.535883\pi$
$348$	0	0
$349$	−18.6611	−0.998903	−0.499452	−	0.866342i	$-0.666465\pi$
$349$	−18.6611	−0.998903	−0.499452	+	0.866342i	$0.666465\pi$
$350$	−19.7130	−1.05370
$351$	0	0
$352$	−66.0176	−3.51875
$353$	11.5781	0.616241	0.308121	−	0.951347i	$-0.400300\pi$
$353$	11.5781	0.616241	0.308121	+	0.951347i	$0.400300\pi$
$354$	0	0
$355$	−9.48546	−0.503436
$356$	8.33246	0.441619
$357$	0	0
$358$	−34.7254	−1.83529
$359$	−29.4910	−1.55647	−0.778237	−	0.627970i	$-0.783886\pi$
$359$	−29.4910	−1.55647	−0.778237	+	0.627970i	$0.783886\pi$
$360$	0	0
$361$	−7.57330	−0.398595
$362$	15.7893	0.829866
$363$	0	0
$364$	6.50046	0.340716
$365$	−49.8200	−2.60770
$366$	0	0
$367$	−24.7858	−1.29381	−0.646903	−	0.762572i	$-0.723936\pi$
$367$	−24.7858	−1.29381	−0.646903	+	0.762572i	$0.723936\pi$
$368$	12.9523	0.675185
$369$	0	0
$370$	−45.7221	−2.37698
$371$	2.47403	0.128445
$372$	0	0
$373$	17.1524	0.888117	0.444058	−	0.895998i	$-0.353538\pi$
$373$	17.1524	0.888117	0.444058	+	0.895998i	$0.353538\pi$
$374$	60.9922	3.15383
$375$	0	0
$376$	33.9897	1.75288
$377$	8.71467	0.448829
$378$	0	0
$379$	32.8211	1.68591	0.842953	−	0.537986i	$-0.180815\pi$
$379$	32.8211	1.68591	0.842953	+	0.537986i	$0.180815\pi$
$380$	−62.1565	−3.18856
$381$	0	0
$382$	32.1054	1.64266
$383$	8.42609	0.430553	0.215277	−	0.976553i	$-0.430935\pi$
$383$	8.42609	0.430553	0.215277	+	0.976553i	$0.430935\pi$
$384$	0	0
$385$	13.3004	0.677849
$386$	−2.02543	−0.103092
$387$	0	0
$388$	−60.8906	−3.09125
$389$	25.3004	1.28278	0.641390	−	0.767215i	$-0.278358\pi$
$389$	25.3004	1.28278	0.641390	+	0.767215i	$0.278358\pi$
$390$	0	0
$391$	−5.98512	−0.302681
$392$	−8.70812	−0.439827
$393$	0	0
$394$	68.3501	3.44343
$395$	11.6360	0.585471
$396$	0	0
$397$	20.3650	1.02209	0.511045	−	0.859554i	$-0.329259\pi$
$397$	20.3650	1.02209	0.511045	+	0.859554i	$0.329259\pi$
$398$	−40.8874	−2.04950
$399$	0	0
$400$	94.9116	4.74558
$401$	−4.10277	−0.204883	−0.102441	−	0.994739i	$-0.532665\pi$
$401$	−4.10277	−0.204883	−0.102441	+	0.994739i	$0.532665\pi$
$402$	0	0
$403$	7.78245	0.387672
$404$	1.92002	0.0955248
$405$	0	0
$406$	−18.8873	−0.937363
$407$	18.3369	0.908925
$408$	0	0
$409$	−26.8449	−1.32739	−0.663697	−	0.748002i	$-0.731013\pi$
$409$	−26.8449	−1.32739	−0.663697	+	0.748002i	$0.731013\pi$
$410$	−16.8217	−0.830766
$411$	0	0
$412$	−2.48250	−0.122304
$413$	2.89143	0.142278
$414$	0	0
$415$	−24.6556	−1.21029
$416$	−21.6321	−1.06060
$417$	0	0
$418$	34.4478	1.68490
$419$	−0.149494	−0.00730327	−0.00365163	−	0.999993i	$-0.501162\pi$
$419$	−0.149494	−0.00730327	−0.00365163	+	0.999993i	$0.501162\pi$
$420$	0	0
$421$	−10.5269	−0.513049	−0.256524	−	0.966538i	$-0.582577\pi$
$421$	−10.5269	−0.513049	−0.256524	+	0.966538i	$0.582577\pi$
$422$	65.3165	3.17956
$423$	0	0
$424$	−21.5442	−1.04628
$425$	−43.8577	−2.12741
$426$	0	0
$427$	−10.3518	−0.500956
$428$	30.0969	1.45479
$429$	0	0
$430$	−28.5459	−1.37661
$431$	2.28665	0.110144	0.0550720	−	0.998482i	$-0.482461\pi$
$431$	2.28665	0.110144	0.0550720	+	0.998482i	$0.482461\pi$
$432$	0	0
$433$	−20.4569	−0.983094	−0.491547	−	0.870851i	$-0.663568\pi$
$433$	−20.4569	−0.983094	−0.491547	+	0.870851i	$0.663568\pi$
$434$	−16.8669	−0.809639
$435$	0	0
$436$	81.1096	3.88444
$437$	−3.38034	−0.161704
$438$	0	0
$439$	8.93821	0.426597	0.213299	−	0.976987i	$-0.431579\pi$
$439$	8.93821	0.426597	0.213299	+	0.976987i	$0.431579\pi$
$440$	−115.821	−5.52156
$441$	0	0
$442$	19.9854	0.950609
$443$	6.71589	0.319082	0.159541	−	0.987191i	$-0.448999\pi$
$443$	6.71589	0.319082	0.159541	+	0.987191i	$0.448999\pi$
$444$	0	0
$445$	5.58640	0.264821
$446$	−47.7085	−2.25906
$447$	0	0
$448$	20.9787	0.991151
$449$	−39.7178	−1.87440	−0.937200	−	0.348792i	$-0.886592\pi$
$449$	−39.7178	−1.87440	−0.937200	+	0.348792i	$0.886592\pi$
$450$	0	0
$451$	6.74636	0.317674
$452$	23.4749	1.10417
$453$	0	0
$454$	−42.2592	−1.98332
$455$	4.35815	0.204313
$456$	0	0
$457$	−35.7388	−1.67179	−0.835895	−	0.548889i	$-0.815051\pi$
$457$	−35.7388	−1.67179	−0.835895	+	0.548889i	$0.815051\pi$
$458$	12.1497	0.567719
$459$	0	0
$460$	18.3877	0.857329
$461$	18.3516	0.854720	0.427360	−	0.904082i	$-0.359444\pi$
$461$	18.3516	0.854720	0.427360	+	0.904082i	$0.359444\pi$
$462$	0	0
$463$	11.1821	0.519678	0.259839	−	0.965652i	$-0.416330\pi$
$463$	11.1821	0.519678	0.259839	+	0.965652i	$0.416330\pi$
$464$	90.9365	4.22162
$465$	0	0
$466$	−43.7531	−2.02682
$467$	−31.7231	−1.46797	−0.733984	−	0.679167i	$-0.762341\pi$
$467$	−31.7231	−1.46797	−0.733984	+	0.679167i	$0.762341\pi$
$468$	0	0
$469$	11.4466	0.528556
$470$	36.8676	1.70058
$471$	0	0
$472$	−25.1790	−1.15896
$473$	11.4483	0.526395
$474$	0	0
$475$	−24.7704	−1.13654
$476$	−31.3442	−1.43666
$477$	0	0
$478$	5.77731	0.264248
$479$	9.81725	0.448562	0.224281	−	0.974525i	$-0.427997\pi$
$479$	9.81725	0.448562	0.224281	+	0.974525i	$0.427997\pi$
$480$	0	0
$481$	6.00847	0.273963
$482$	−33.5400	−1.52771
$483$	0	0
$484$	17.5423	0.797378
$485$	−40.8234	−1.85369
$486$	0	0
$487$	−36.2778	−1.64390	−0.821952	−	0.569557i	$-0.807115\pi$
$487$	−36.2778	−1.64390	−0.821952	+	0.569557i	$0.807115\pi$
$488$	90.1443	4.08064
$489$	0	0
$490$	−9.44544	−0.426701
$491$	17.6053	0.794514	0.397257	−	0.917707i	$-0.369962\pi$
$491$	17.6053	0.794514	0.397257	+	0.917707i	$0.369962\pi$
$492$	0	0
$493$	−42.0208	−1.89252
$494$	11.2876	0.507851
$495$	0	0
$496$	81.2089	3.64639
$497$	−2.70157	−0.121182
$498$	0	0
$499$	12.1142	0.542308	0.271154	−	0.962536i	$-0.412595\pi$
$499$	12.1142	0.542308	0.271154	+	0.962536i	$0.412595\pi$
$500$	42.8024	1.91418
$501$	0	0
$502$	13.2216	0.590109
$503$	−10.2723	−0.458020	−0.229010	−	0.973424i	$-0.573549\pi$
$503$	−10.2723	−0.458020	−0.229010	+	0.973424i	$0.573549\pi$
$504$	0	0
$505$	1.28726	0.0572822
$506$	−10.1906	−0.453029
$507$	0	0
$508$	−11.1581	−0.495059
$509$	6.45082	0.285928	0.142964	−	0.989728i	$-0.454337\pi$
$509$	6.45082	0.285928	0.142964	+	0.989728i	$0.454337\pi$
$510$	0	0
$511$	−14.1893	−0.627698
$512$	−0.147625	−0.00652419
$513$	0	0
$514$	76.6113	3.37918
$515$	−1.66436	−0.0733407
$516$	0	0
$517$	−14.7858	−0.650277
$518$	−13.0222	−0.572162
$519$	0	0
$520$	−37.9513	−1.66428
$521$	−34.4858	−1.51085	−0.755425	−	0.655235i	$-0.772570\pi$
$521$	−34.4858	−1.51085	−0.755425	+	0.655235i	$0.772570\pi$
$522$	0	0
$523$	−25.7852	−1.12751	−0.563753	−	0.825943i	$-0.690643\pi$
$523$	−25.7852	−1.12751	−0.563753	+	0.825943i	$0.690643\pi$
$524$	94.9168	4.14646
$525$	0	0
$526$	5.09307	0.222068
$527$	−37.5258	−1.63465
$528$	0	0
$529$	1.00000	0.0434783
$530$	−23.3683	−1.01505
$531$	0	0
$532$	−17.7029	−0.767518
$533$	2.21059	0.0957513
$534$	0	0
$535$	20.1781	0.872377
$536$	−99.6785	−4.30546
$537$	0	0
$538$	−6.75653	−0.291295
$539$	3.78810	0.163165
$540$	0	0
$541$	−21.1475	−0.909203	−0.454601	−	0.890695i	$-0.650218\pi$
$541$	−21.1475	−0.909203	−0.454601	+	0.890695i	$0.650218\pi$
$542$	−67.5496	−2.90151
$543$	0	0
$544$	104.307	4.47211
$545$	54.3789	2.32934
$546$	0	0
$547$	−3.97188	−0.169825	−0.0849126	−	0.996388i	$-0.527061\pi$
$547$	−3.97188	−0.169825	−0.0849126	+	0.996388i	$0.527061\pi$
$548$	80.1589	3.42422
$549$	0	0
$550$	−74.6747	−3.18414
$551$	−23.7329	−1.01106
$552$	0	0
$553$	3.31407	0.140928
$554$	−25.5201	−1.08425
$555$	0	0
$556$	70.8429	3.00441
$557$	13.5585	0.574491	0.287246	−	0.957857i	$-0.407260\pi$
$557$	13.5585	0.574491	0.287246	+	0.957857i	$0.407260\pi$
$558$	0	0
$559$	3.75130	0.158663
$560$	45.4767	1.92174
$561$	0	0
$562$	23.4367	0.988617
$563$	32.4740	1.36862	0.684309	−	0.729193i	$-0.260104\pi$
$563$	32.4740	1.36862	0.684309	+	0.729193i	$0.260104\pi$
$564$	0	0
$565$	15.7385	0.662123
$566$	−82.1963	−3.45497
$567$	0	0
$568$	23.5256	0.987111
$569$	26.7707	1.12228	0.561142	−	0.827719i	$-0.310362\pi$
$569$	26.7707	1.12228	0.561142	+	0.827719i	$0.310362\pi$
$570$	0	0
$571$	5.30884	0.222168	0.111084	−	0.993811i	$-0.464568\pi$
$571$	5.30884	0.222168	0.111084	+	0.993811i	$0.464568\pi$
$572$	24.6244	1.02960
$573$	0	0
$574$	−4.79102	−0.199973
$575$	7.32778	0.305590
$576$	0	0
$577$	43.9737	1.83065	0.915325	−	0.402717i	$-0.131934\pi$
$577$	43.9737	1.83065	0.915325	+	0.402717i	$0.131934\pi$
$578$	−50.6336	−2.10608
$579$	0	0
$580$	129.097	5.36048
$581$	−7.02219	−0.291329
$582$	0	0
$583$	9.37187	0.388143
$584$	123.562	5.11304
$585$	0	0
$586$	34.1822	1.41205
$587$	34.5314	1.42526	0.712631	−	0.701539i	$-0.247503\pi$
$587$	34.5314	1.42526	0.712631	+	0.701539i	$0.247503\pi$
$588$	0	0
$589$	−21.1942	−0.873292
$590$	−27.3109	−1.12437
$591$	0	0
$592$	62.6976	2.57686
$593$	−2.06074	−0.0846245	−0.0423123	−	0.999104i	$-0.513472\pi$
$593$	−2.06074	−0.0846245	−0.0423123	+	0.999104i	$0.513472\pi$
$594$	0	0
$595$	−21.0143	−0.861504
$596$	−30.1533	−1.23513
$597$	0	0
$598$	−3.33918	−0.136549
$599$	28.2627	1.15478	0.577392	−	0.816467i	$-0.304070\pi$
$599$	28.2627	1.15478	0.577392	+	0.816467i	$0.304070\pi$
$600$	0	0
$601$	−25.0036	−1.01992	−0.509959	−	0.860199i	$-0.670340\pi$
$601$	−25.0036	−1.01992	−0.509959	+	0.860199i	$0.670340\pi$
$602$	−8.13020	−0.331362
$603$	0	0
$604$	−50.1608	−2.04101
$605$	11.7610	0.478154
$606$	0	0
$607$	−29.0255	−1.17811	−0.589054	−	0.808093i	$-0.700500\pi$
$607$	−29.0255	−1.17811	−0.589054	+	0.808093i	$0.700500\pi$
$608$	58.9114	2.38917
$609$	0	0
$610$	97.7768	3.95887
$611$	−4.84487	−0.196003
$612$	0	0
$613$	−25.2180	−1.01855	−0.509274	−	0.860605i	$-0.670086\pi$
$613$	−25.2180	−1.01855	−0.509274	+	0.860605i	$0.670086\pi$
$614$	−59.4940	−2.40098
$615$	0	0
$616$	−32.9872	−1.32909
$617$	−41.7462	−1.68064	−0.840319	−	0.542093i	$-0.817632\pi$
$617$	−41.7462	−1.68064	−0.840319	+	0.542093i	$0.817632\pi$
$618$	0	0
$619$	−0.503782	−0.0202487	−0.0101243	−	0.999949i	$-0.503223\pi$
$619$	−0.503782	−0.0202487	−0.0101243	+	0.999949i	$0.503223\pi$
$620$	115.288	4.63006
$621$	0	0
$622$	57.5549	2.30774
$623$	1.59107	0.0637449
$624$	0	0
$625$	−7.94253	−0.317701
$626$	80.4499	3.21543
$627$	0	0
$628$	−26.3792	−1.05264
$629$	−28.9719	−1.15519
$630$	0	0
$631$	−17.6868	−0.704102	−0.352051	−	0.935981i	$-0.614516\pi$
$631$	−17.6868	−0.704102	−0.352051	+	0.935981i	$0.614516\pi$
$632$	−28.8593	−1.14796
$633$	0	0
$634$	18.9954	0.754403
$635$	−7.48079	−0.296866
$636$	0	0
$637$	1.24125	0.0491802
$638$	−71.5471	−2.83258
$639$	0	0
$640$	−75.7731	−2.99519
$641$	37.4714	1.48003	0.740015	−	0.672590i	$-0.234818\pi$
$641$	37.4714	1.48003	0.740015	+	0.672590i	$0.234818\pi$
$642$	0	0
$643$	0.178907	0.00705539	0.00352770	−	0.999994i	$-0.498877\pi$
$643$	0.178907	0.00705539	0.00352770	+	0.999994i	$0.498877\pi$
$644$	5.23702	0.206367
$645$	0	0
$646$	−54.4269	−2.14140
$647$	−2.94235	−0.115676	−0.0578379	−	0.998326i	$-0.518421\pi$
$647$	−2.94235	−0.115676	−0.0578379	+	0.998326i	$0.518421\pi$
$648$	0	0
$649$	10.9530	0.429944
$650$	−24.4688	−0.959744
$651$	0	0
$652$	−38.5294	−1.50893
$653$	−44.1719	−1.72858	−0.864290	−	0.502994i	$-0.832232\pi$
$653$	−44.1719	−1.72858	−0.864290	+	0.502994i	$0.832232\pi$
$654$	0	0
$655$	63.6358	2.48646
$656$	23.0672	0.900623
$657$	0	0
$658$	10.5003	0.409345
$659$	−13.9282	−0.542564	−0.271282	−	0.962500i	$-0.587448\pi$
$659$	−13.9282	−0.542564	−0.271282	+	0.962500i	$0.587448\pi$
$660$	0	0
$661$	−2.15327	−0.0837525	−0.0418763	−	0.999123i	$-0.513334\pi$
$661$	−2.15327	−0.0837525	−0.0418763	+	0.999123i	$0.513334\pi$
$662$	−43.5976	−1.69447
$663$	0	0
$664$	61.1501	2.37308
$665$	−11.8687	−0.460248
$666$	0	0
$667$	7.02088	0.271849
$668$	−109.927	−4.25320
$669$	0	0
$670$	−108.118	−4.17697
$671$	−39.2134	−1.51382
$672$	0	0
$673$	−2.96276	−0.114206	−0.0571030	−	0.998368i	$-0.518186\pi$
$673$	−2.96276	−0.114206	−0.0571030	+	0.998368i	$0.518186\pi$
$674$	−14.0374	−0.540701
$675$	0	0
$676$	−60.0125	−2.30817
$677$	−9.59927	−0.368930	−0.184465	−	0.982839i	$-0.559055\pi$
$677$	−9.59927	−0.368930	−0.184465	+	0.982839i	$0.559055\pi$
$678$	0	0
$679$	−11.6270	−0.446202
$680$	182.995	7.01755
$681$	0	0
$682$	−63.8936	−2.44661
$683$	24.1908	0.925636	0.462818	−	0.886453i	$-0.346838\pi$
$683$	24.1908	0.925636	0.462818	+	0.886453i	$0.346838\pi$
$684$	0	0
$685$	53.7416	2.05336
$686$	−2.69017	−0.102711
$687$	0	0
$688$	39.1443	1.49236
$689$	3.07089	0.116992
$690$	0	0
$691$	30.4784	1.15945	0.579726	−	0.814811i	$-0.303159\pi$
$691$	30.4784	1.15945	0.579726	+	0.814811i	$0.303159\pi$
$692$	64.6084	2.45604
$693$	0	0
$694$	11.2743	0.427968
$695$	47.4958	1.80162
$696$	0	0
$697$	−10.6591	−0.403743
$698$	50.2014	1.90015
$699$	0	0
$700$	38.3757	1.45047
$701$	−17.4101	−0.657570	−0.328785	−	0.944405i	$-0.606639\pi$
$701$	−17.4101	−0.657570	−0.328785	+	0.944405i	$0.606639\pi$
$702$	0	0
$703$	−16.3631	−0.617145
$704$	79.4694	2.99512
$705$	0	0
$706$	−31.1471	−1.17224
$707$	0.366626	0.0137884
$708$	0	0
$709$	13.6465	0.512504	0.256252	−	0.966610i	$-0.417512\pi$
$709$	13.6465	0.512504	0.256252	+	0.966610i	$0.417512\pi$
$710$	25.5175	0.957655
$711$	0	0
$712$	−13.8552	−0.519247
$713$	6.26984	0.234807
$714$	0	0
$715$	16.5091	0.617405
$716$	67.6007	2.52636
$717$	0	0
$718$	79.3358	2.96078
$719$	29.8264	1.11234	0.556169	−	0.831069i	$-0.312271\pi$
$719$	29.8264	1.11234	0.556169	+	0.831069i	$0.312271\pi$
$720$	0	0
$721$	−0.474030	−0.0176538
$722$	20.3735	0.758222
$723$	0	0
$724$	−30.7373	−1.14234
$725$	51.4474	1.91071
$726$	0	0
$727$	6.92789	0.256941	0.128471	−	0.991713i	$-0.458993\pi$
$727$	6.92789	0.256941	0.128471	+	0.991713i	$0.458993\pi$
$728$	−10.8090	−0.400607
$729$	0	0
$730$	134.024	4.96046
$731$	−18.0882	−0.669015
$732$	0	0
$733$	−10.8092	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$733$	−10.8092	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$734$	66.6779	2.46113
$735$	0	0
$736$	−17.4276	−0.642391
$737$	43.3609	1.59722
$738$	0	0
$739$	34.5455	1.27078	0.635388	−	0.772193i	$-0.280840\pi$
$739$	34.5455	1.27078	0.635388	+	0.772193i	$0.280840\pi$
$740$	89.0083	3.27201
$741$	0	0
$742$	−6.65556	−0.244333
$743$	−19.7750	−0.725475	−0.362737	−	0.931891i	$-0.618158\pi$
$743$	−19.7750	−0.725475	−0.362737	+	0.931891i	$0.618158\pi$
$744$	0	0
$745$	−20.2159	−0.740654
$746$	−46.1428	−1.68941
$747$	0	0
$748$	−118.735	−4.34137
$749$	5.74697	0.209990
$750$	0	0
$751$	−31.0704	−1.13378	−0.566888	−	0.823795i	$-0.691853\pi$
$751$	−31.0704	−1.13378	−0.566888	+	0.823795i	$0.691853\pi$
$752$	−50.5556	−1.84357
$753$	0	0
$754$	−23.4440	−0.853779
$755$	−33.6297	−1.22391
$756$	0	0
$757$	8.32104	0.302433	0.151217	−	0.988501i	$-0.451681\pi$
$757$	8.32104	0.302433	0.151217	+	0.988501i	$0.451681\pi$
$758$	−88.2944	−3.20700
$759$	0	0
$760$	103.354	3.74904
$761$	3.82034	0.138487	0.0692436	−	0.997600i	$-0.477941\pi$
$761$	3.82034	0.138487	0.0692436	+	0.997600i	$0.477941\pi$
$762$	0	0
$763$	15.4877	0.560694
$764$	−62.5004	−2.26118
$765$	0	0
$766$	−22.6676	−0.819014
$767$	3.58900	0.129591
$768$	0	0
$769$	8.05891	0.290612	0.145306	−	0.989387i	$-0.453583\pi$
$769$	8.05891	0.290612	0.145306	+	0.989387i	$0.453583\pi$
$770$	−35.7802	−1.28943
$771$	0	0
$772$	3.94295	0.141910
$773$	20.3254	0.731054	0.365527	−	0.930801i	$-0.380889\pi$
$773$	20.3254	0.731054	0.365527	+	0.930801i	$0.380889\pi$
$774$	0	0
$775$	45.9440	1.65036
$776$	101.249	3.63463
$777$	0	0
$778$	−68.0623	−2.44015
$779$	−6.02017	−0.215695
$780$	0	0
$781$	−10.2338	−0.366194
$782$	16.1010	0.575771
$783$	0	0
$784$	12.9523	0.462582
$785$	−17.6856	−0.631226
$786$	0	0
$787$	−47.7951	−1.70371	−0.851856	−	0.523775i	$-0.824523\pi$
$787$	−47.7951	−1.70371	−0.851856	+	0.523775i	$0.824523\pi$
$788$	−133.059	−4.74002
$789$	0	0
$790$	−31.3028	−1.11370
$791$	4.48250	0.159379
$792$	0	0
$793$	−12.8491	−0.456286
$794$	−54.7853	−1.94426
$795$	0	0
$796$	79.5964	2.82122
$797$	7.57090	0.268175	0.134088	−	0.990969i	$-0.457190\pi$
$797$	7.57090	0.268175	0.134088	+	0.990969i	$0.457190\pi$
$798$	0	0
$799$	23.3612	0.826461
$800$	−127.706	−4.51509
$801$	0	0
$802$	11.0372	0.389735
$803$	−53.7505	−1.89681
$804$	0	0
$805$	3.51109	0.123750
$806$	−20.9361	−0.737444
$807$	0	0
$808$	−3.19262	−0.112316
$809$	−8.55279	−0.300700	−0.150350	−	0.988633i	$-0.548040\pi$
$809$	−8.55279	−0.300700	−0.150350	+	0.988633i	$0.548040\pi$
$810$	0	0
$811$	−45.0023	−1.58024	−0.790122	−	0.612950i	$-0.789983\pi$
$811$	−45.0023	−1.58024	−0.790122	+	0.612950i	$0.789983\pi$
$812$	36.7684	1.29032
$813$	0	0
$814$	−49.3293	−1.72899
$815$	−25.8316	−0.904841
$816$	0	0
$817$	−10.2160	−0.357413
$818$	72.2173	2.52502
$819$	0	0
$820$	32.7472	1.14358
$821$	−41.4358	−1.44612	−0.723060	−	0.690785i	$-0.757265\pi$
$821$	−41.4358	−1.44612	−0.723060	+	0.690785i	$0.757265\pi$
$822$	0	0
$823$	−19.1508	−0.667553	−0.333777	−	0.942652i	$-0.608323\pi$
$823$	−19.1508	−0.667553	−0.333777	+	0.942652i	$0.608323\pi$
$824$	4.12791	0.143803
$825$	0	0
$826$	−7.77845	−0.270647
$827$	32.2820	1.12256	0.561278	−	0.827627i	$-0.310310\pi$
$827$	32.2820	1.12256	0.561278	+	0.827627i	$0.310310\pi$
$828$	0	0
$829$	9.85700	0.342348	0.171174	−	0.985241i	$-0.445244\pi$
$829$	9.85700	0.342348	0.171174	+	0.985241i	$0.445244\pi$
$830$	66.3277	2.30227
$831$	0	0
$832$	26.0399	0.902770
$833$	−5.98512	−0.207372
$834$	0	0
$835$	−73.6991	−2.55046
$836$	−67.0603	−2.31933
$837$	0	0
$838$	0.402165	0.0138925
$839$	−11.7098	−0.404267	−0.202133	−	0.979358i	$-0.564787\pi$
$839$	−11.7098	−0.404267	−0.202133	+	0.979358i	$0.564787\pi$
$840$	0	0
$841$	20.2927	0.699748
$842$	28.3191	0.975941
$843$	0	0
$844$	−127.153	−4.37679
$845$	−40.2347	−1.38411
$846$	0	0
$847$	3.34968	0.115096
$848$	32.0444	1.10041
$849$	0	0
$850$	117.985	4.04684
$851$	4.84066	0.165936
$852$	0	0
$853$	−29.2441	−1.00130	−0.500650	−	0.865650i	$-0.666906\pi$
$853$	−29.2441	−1.00130	−0.500650	+	0.865650i	$0.666906\pi$
$854$	27.8480	0.952938
$855$	0	0
$856$	−50.0453	−1.71051
$857$	−38.6873	−1.32153	−0.660766	−	0.750592i	$-0.729768\pi$
$857$	−38.6873	−1.32153	−0.660766	+	0.750592i	$0.729768\pi$
$858$	0	0
$859$	−49.3913	−1.68521	−0.842604	−	0.538534i	$-0.818979\pi$
$859$	−49.3913	−1.68521	−0.842604	+	0.538534i	$0.818979\pi$
$860$	55.5709	1.89495
$861$	0	0
$862$	−6.15148	−0.209520
$863$	2.32156	0.0790267	0.0395134	−	0.999219i	$-0.487419\pi$
$863$	2.32156	0.0790267	0.0395134	+	0.999219i	$0.487419\pi$
$864$	0	0
$865$	43.3159	1.47279
$866$	55.0325	1.87008
$867$	0	0
$868$	32.8353	1.11450
$869$	12.5540	0.425865
$870$	0	0
$871$	14.2081	0.481424
$872$	−134.869	−4.56725
$873$	0	0
$874$	9.09369	0.307599
$875$	8.17306	0.276300
$876$	0	0
$877$	36.2775	1.22500	0.612502	−	0.790469i	$-0.290163\pi$
$877$	36.2775	1.22500	0.612502	+	0.790469i	$0.290163\pi$
$878$	−24.0453	−0.811490
$879$	0	0
$880$	172.270	5.80723
$881$	−30.1252	−1.01494	−0.507472	−	0.861668i	$-0.669420\pi$
$881$	−30.1252	−1.01494	−0.507472	+	0.861668i	$0.669420\pi$
$882$	0	0
$883$	1.94158	0.0653394	0.0326697	−	0.999466i	$-0.489599\pi$
$883$	1.94158	0.0653394	0.0326697	+	0.999466i	$0.489599\pi$
$884$	−38.9060	−1.30855
$885$	0	0
$886$	−18.0669	−0.606969
$887$	−52.3596	−1.75806	−0.879031	−	0.476765i	$-0.841809\pi$
$887$	−52.3596	−1.75806	−0.879031	+	0.476765i	$0.841809\pi$
$888$	0	0
$889$	−2.13061	−0.0714585
$890$	−15.0284	−0.503752
$891$	0	0
$892$	92.8752	3.10969
$893$	13.1942	0.441527
$894$	0	0
$895$	45.3221	1.51495
$896$	−21.5811	−0.720972
$897$	0	0
$898$	106.848	3.56555
$899$	44.0198	1.46814
$900$	0	0
$901$	−14.8074	−0.493305
$902$	−18.1489	−0.604291
$903$	0	0
$904$	−39.0342	−1.29826
$905$	−20.6075	−0.685016
$906$	0	0
$907$	9.62846	0.319708	0.159854	−	0.987141i	$-0.448898\pi$
$907$	9.62846	0.319708	0.159854	+	0.987141i	$0.448898\pi$
$908$	82.2669	2.73012
$909$	0	0
$910$	−11.7242	−0.388652
$911$	24.6036	0.815154	0.407577	−	0.913171i	$-0.366374\pi$
$911$	24.6036	0.815154	0.407577	+	0.913171i	$0.366374\pi$
$912$	0	0
$913$	−26.6007	−0.880356
$914$	96.1434	3.18014
$915$	0	0
$916$	−23.6521	−0.781488
$917$	18.1242	0.598514
$918$	0	0
$919$	18.4685	0.609220	0.304610	−	0.952477i	$-0.401474\pi$
$919$	18.4685	0.609220	0.304610	+	0.952477i	$0.401474\pi$
$920$	−30.5750	−1.00803
$921$	0	0
$922$	−49.3690	−1.62588
$923$	−3.35333	−0.110376
$924$	0	0
$925$	35.4713	1.16629
$926$	−30.0819	−0.988551
$927$	0	0
$928$	−122.357	−4.01657
$929$	−21.6032	−0.708778	−0.354389	−	0.935098i	$-0.615311\pi$
$929$	−21.6032	−0.708778	−0.354389	+	0.935098i	$0.615311\pi$
$930$	0	0
$931$	−3.38034	−0.110786
$932$	85.1751	2.79000
$933$	0	0
$934$	85.3404	2.79242
$935$	−79.6043	−2.60334
$936$	0	0
$937$	4.95446	0.161855	0.0809276	−	0.996720i	$-0.474212\pi$
$937$	4.95446	0.161855	0.0809276	+	0.996720i	$0.474212\pi$
$938$	−30.7933	−1.00544
$939$	0	0
$940$	−71.7710	−2.34091
$941$	37.1398	1.21072	0.605361	−	0.795951i	$-0.293029\pi$
$941$	37.1398	1.21072	0.605361	+	0.795951i	$0.293029\pi$
$942$	0	0
$943$	1.78094	0.0579953
$944$	37.4507	1.21892
$945$	0	0
$946$	−30.7980	−1.00133
$947$	11.0644	0.359543	0.179772	−	0.983708i	$-0.442464\pi$
$947$	11.0644	0.359543	0.179772	+	0.983708i	$0.442464\pi$
$948$	0	0
$949$	−17.6125	−0.571726
$950$	66.6366	2.16198
$951$	0	0
$952$	52.1192	1.68919
$953$	−19.3888	−0.628065	−0.314033	−	0.949412i	$-0.601680\pi$
$953$	−19.3888	−0.628065	−0.314033	+	0.949412i	$0.601680\pi$
$954$	0	0
$955$	−41.9026	−1.35594
$956$	−11.2468	−0.363748
$957$	0	0
$958$	−26.4101	−0.853271
$959$	15.3062	0.494263
$960$	0	0
$961$	8.31092	0.268094
$962$	−16.1638	−0.521142
$963$	0	0
$964$	65.2932	2.10295
$965$	2.64350	0.0850974
$966$	0	0
$967$	−32.5154	−1.04563	−0.522813	−	0.852448i	$-0.675117\pi$
$967$	−32.5154	−1.04563	−0.522813	+	0.852448i	$0.675117\pi$
$968$	−29.1694	−0.937540
$969$	0	0
$970$	109.822	3.52617
$971$	−1.32316	−0.0424622	−0.0212311	−	0.999775i	$-0.506759\pi$
$971$	−1.32316	−0.0424622	−0.0212311	+	0.999775i	$0.506759\pi$
$972$	0	0
$973$	13.5273	0.433667
$974$	97.5934	3.12709
$975$	0	0
$976$	−134.079	−4.29176
$977$	31.6195	1.01160	0.505799	−	0.862651i	$-0.331198\pi$
$977$	31.6195	1.01160	0.505799	+	0.862651i	$0.331198\pi$
$978$	0	0
$979$	6.02713	0.192628
$980$	18.3877	0.587372
$981$	0	0
$982$	−47.3611	−1.51135
$983$	12.1696	0.388149	0.194074	−	0.980987i	$-0.437830\pi$
$983$	12.1696	0.388149	0.194074	+	0.980987i	$0.437830\pi$
$984$	0	0
$985$	−89.2076	−2.84239
$986$	113.043	3.60003
$987$	0	0
$988$	−21.9737	−0.699078
$989$	3.02219	0.0961000
$990$	0	0
$991$	2.18738	0.0694844	0.0347422	−	0.999396i	$-0.488939\pi$
$991$	2.18738	0.0694844	0.0347422	+	0.999396i	$0.488939\pi$
$992$	−109.269	−3.46928
$993$	0	0
$994$	7.26768	0.230517
$995$	53.3644	1.69177
$996$	0	0
$997$	−3.78052	−0.119730	−0.0598652	−	0.998206i	$-0.519067\pi$
$997$	−3.78052	−0.119730	−0.0598652	+	0.998206i	$0.519067\pi$
$998$	−32.5894	−1.03160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.r.1.1		5
3.2	odd	2		161.2.a.d.1.5	✓	5
12.11	even	2		2576.2.a.bd.1.3		5
15.14	odd	2		4025.2.a.p.1.1		5
21.20	even	2		1127.2.a.h.1.5		5
69.68	even	2		3703.2.a.j.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.5	✓	5	3.2	odd	2
1127.2.a.h.1.5		5	21.20	even	2
1449.2.a.r.1.1		5	1.1	even	1	trivial
2576.2.a.bd.1.3		5	12.11	even	2
3703.2.a.j.1.5		5	69.68	even	2
4025.2.a.p.1.1		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)

\(f(q)\)	\(=\)	\(q-2.69017 q^{2} +5.23702 q^{4} +3.51109 q^{5} +1.00000 q^{7} -8.70812 q^{8} +O(q^{10})\)	\(q-2.69017 q^{2} +5.23702 q^{4} +3.51109 q^{5} +1.00000 q^{7} -8.70812 q^{8} -9.44544 q^{10} +3.78810 q^{11} +1.24125 q^{13} -2.69017 q^{14} +12.9523 q^{16} -5.98512 q^{17} -3.38034 q^{19} +18.3877 q^{20} -10.1906 q^{22} +1.00000 q^{23} +7.32778 q^{25} -3.33918 q^{26} +5.23702 q^{28} +7.02088 q^{29} +6.26984 q^{31} -17.4276 q^{32} +16.1010 q^{34} +3.51109 q^{35} +4.84066 q^{37} +9.09369 q^{38} -30.5750 q^{40} +1.78094 q^{41} +3.02219 q^{43} +19.8383 q^{44} -2.69017 q^{46} -3.90322 q^{47} +1.00000 q^{49} -19.7130 q^{50} +6.50046 q^{52} +2.47403 q^{53} +13.3004 q^{55} -8.70812 q^{56} -18.8873 q^{58} +2.89143 q^{59} -10.3518 q^{61} -16.8669 q^{62} +20.9787 q^{64} +4.35815 q^{65} +11.4466 q^{67} -31.3442 q^{68} -9.44544 q^{70} -2.70157 q^{71} -14.1893 q^{73} -13.0222 q^{74} -17.7029 q^{76} +3.78810 q^{77} +3.31407 q^{79} +45.4767 q^{80} -4.79102 q^{82} -7.02219 q^{83} -21.0143 q^{85} -8.13020 q^{86} -32.9872 q^{88} +1.59107 q^{89} +1.24125 q^{91} +5.23702 q^{92} +10.5003 q^{94} -11.8687 q^{95} -11.6270 q^{97} -2.69017 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 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

L-functions

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