LMFDB - Embedded newform 1449.2.a.j.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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ 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)

$f(q)$	$=$	$q-1.30278 q^{2} -0.302776 q^{4} +4.30278 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})$	\(q-1.30278 q^{2} -0.302776 q^{4} +4.30278 q^{5} -1.00000 q^{7} +3.00000 q^{8} -5.60555 q^{10} +5.00000 q^{11} -1.30278 q^{13} +1.30278 q^{14} -3.30278 q^{16} -1.60555 q^{17} +5.60555 q^{19} -1.30278 q^{20} -6.51388 q^{22} -1.00000 q^{23} +13.5139 q^{25} +1.69722 q^{26} +0.302776 q^{28} +8.21110 q^{29} +3.00000 q^{31} -1.69722 q^{32} +2.09167 q^{34} -4.30278 q^{35} -9.00000 q^{37} -7.30278 q^{38} +12.9083 q^{40} -2.21110 q^{41} -12.5139 q^{43} -1.51388 q^{44} +1.30278 q^{46} +1.39445 q^{47} +1.00000 q^{49} -17.6056 q^{50} +0.394449 q^{52} -5.51388 q^{53} +21.5139 q^{55} -3.00000 q^{56} -10.6972 q^{58} +6.90833 q^{59} -11.9083 q^{61} -3.90833 q^{62} +8.81665 q^{64} -5.60555 q^{65} -1.09167 q^{67} +0.486122 q^{68} +5.60555 q^{70} +9.90833 q^{71} +12.2111 q^{73} +11.7250 q^{74} -1.69722 q^{76} -5.00000 q^{77} -1.00000 q^{79} -14.2111 q^{80} +2.88057 q^{82} +1.60555 q^{83} -6.90833 q^{85} +16.3028 q^{86} +15.0000 q^{88} -0.0916731 q^{89} +1.30278 q^{91} +0.302776 q^{92} -1.81665 q^{94} +24.1194 q^{95} -10.3944 q^{97} -1.30278 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8	$ 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} + 10 q^{11} + q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} - 2 q^{23} + 9 q^{25} + 7 q^{26} - 3 q^{28} + 2 q^{29} + 6 q^{31} - 7 q^{32} + 15 q^{34} - 5 q^{35} - 18 q^{37} - 11 q^{38} + 15 q^{40} + 10 q^{41} - 7 q^{43} + 15 q^{44} - q^{46} + 10 q^{47} + 2 q^{49} - 28 q^{50} + 8 q^{52} + 7 q^{53} + 25 q^{55} - 6 q^{56} - 25 q^{58} + 3 q^{59} - 13 q^{61} + 3 q^{62} - 4 q^{64} - 4 q^{65} - 13 q^{67} + 19 q^{68} + 4 q^{70} + 9 q^{71} + 10 q^{73} - 9 q^{74} - 7 q^{76} - 10 q^{77} - 2 q^{79} - 14 q^{80} + 31 q^{82} - 4 q^{83} - 3 q^{85} + 29 q^{86} + 30 q^{88} - 11 q^{89} - q^{91} - 3 q^{92} + 18 q^{94} + 23 q^{95} - 28 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8 - 4 * q^10 + 10 * q^11 + q^13 - q^14 - 3 * q^16 + 4 * q^17 + 4 * q^19 + q^20 + 5 * q^22 - 2 * q^23 + 9 * q^25 + 7 * q^26 - 3 * q^28 + 2 * q^29 + 6 * q^31 - 7 * q^32 + 15 * q^34 - 5 * q^35 - 18 * q^37 - 11 * q^38 + 15 * q^40 + 10 * q^41 - 7 * q^43 + 15 * q^44 - q^46 + 10 * q^47 + 2 * q^49 - 28 * q^50 + 8 * q^52 + 7 * q^53 + 25 * q^55 - 6 * q^56 - 25 * q^58 + 3 * q^59 - 13 * q^61 + 3 * q^62 - 4 * q^64 - 4 * q^65 - 13 * q^67 + 19 * q^68 + 4 * q^70 + 9 * q^71 + 10 * q^73 - 9 * q^74 - 7 * q^76 - 10 * q^77 - 2 * q^79 - 14 * q^80 + 31 * q^82 - 4 * q^83 - 3 * q^85 + 29 * q^86 + 30 * q^88 - 11 * q^89 - q^91 - 3 * q^92 + 18 * q^94 + 23 * q^95 - 28 * q^97 + 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 $

$2$	−1.30278	−0.921201	−0.460601	−	0.887607i	$-0.652366\pi$
$2$	−1.30278	−0.921201	−0.460601	+	0.887607i	$0.652366\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	4.30278	1.92426	0.962130	−	0.272591i	$-0.0878807\pi$
$5$	4.30278	1.92426	0.962130	+	0.272591i	$0.0878807\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.00000	1.06066
$9$	0	0
$10$	−5.60555	−1.77263
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−1.30278	−0.361325	−0.180662	−	0.983545i	$-0.557824\pi$
$13$	−1.30278	−0.361325	−0.180662	+	0.983545i	$0.557824\pi$
$14$	1.30278	0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−1.60555	−0.389403	−0.194702	−	0.980863i	$-0.562374\pi$
$17$	−1.60555	−0.389403	−0.194702	+	0.980863i	$0.562374\pi$
$18$	0	0
$19$	5.60555	1.28600	0.643001	−	0.765865i	$-0.277689\pi$
$19$	5.60555	1.28600	0.643001	+	0.765865i	$0.277689\pi$
$20$	−1.30278	−0.291309
$21$	0	0
$22$	−6.51388	−1.38876
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	13.5139	2.70278
$26$	1.69722	0.332853
$27$	0	0
$28$	0.302776	0.0572192
$29$	8.21110	1.52476	0.762382	−	0.647128i	$-0.224030\pi$
$29$	8.21110	1.52476	0.762382	+	0.647128i	$0.224030\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.69722	−0.300030
$33$	0	0
$34$	2.09167	0.358719
$35$	−4.30278	−0.727302
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	−7.30278	−1.18467
$39$	0	0
$40$	12.9083	2.04099
$41$	−2.21110	−0.345316	−0.172658	−	0.984982i	$-0.555236\pi$
$41$	−2.21110	−0.345316	−0.172658	+	0.984982i	$0.555236\pi$
$42$	0	0
$43$	−12.5139	−1.90835	−0.954174	−	0.299252i	$-0.903263\pi$
$43$	−12.5139	−1.90835	−0.954174	+	0.299252i	$0.903263\pi$
$44$	−1.51388	−0.228226
$45$	0	0
$46$	1.30278	0.192084
$47$	1.39445	0.203401	0.101701	−	0.994815i	$-0.467572\pi$
$47$	1.39445	0.203401	0.101701	+	0.994815i	$0.467572\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−17.6056	−2.48980
$51$	0	0
$52$	0.394449	0.0547002
$53$	−5.51388	−0.757389	−0.378695	−	0.925522i	$-0.623627\pi$
$53$	−5.51388	−0.757389	−0.378695	+	0.925522i	$0.623627\pi$
$54$	0	0
$55$	21.5139	2.90093
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−10.6972	−1.40461
$59$	6.90833	0.899388	0.449694	−	0.893183i	$-0.351533\pi$
$59$	6.90833	0.899388	0.449694	+	0.893183i	$0.351533\pi$
$60$	0	0
$61$	−11.9083	−1.52471	−0.762353	−	0.647162i	$-0.775956\pi$
$61$	−11.9083	−1.52471	−0.762353	+	0.647162i	$0.775956\pi$
$62$	−3.90833	−0.496358
$63$	0	0
$64$	8.81665	1.10208
$65$	−5.60555	−0.695283
$66$	0	0
$67$	−1.09167	−0.133369	−0.0666845	−	0.997774i	$-0.521242\pi$
$67$	−1.09167	−0.133369	−0.0666845	+	0.997774i	$0.521242\pi$
$68$	0.486122	0.0589509
$69$	0	0
$70$	5.60555	0.669992
$71$	9.90833	1.17590	0.587951	−	0.808897i	$-0.299935\pi$
$71$	9.90833	1.17590	0.587951	+	0.808897i	$0.299935\pi$
$72$	0	0
$73$	12.2111	1.42920	0.714601	−	0.699533i	$-0.246608\pi$
$73$	12.2111	1.42920	0.714601	+	0.699533i	$0.246608\pi$
$74$	11.7250	1.36300
$75$	0	0
$76$	−1.69722	−0.194685
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−14.2111	−1.58885
$81$	0	0
$82$	2.88057	0.318106
$83$	1.60555	0.176232	0.0881161	−	0.996110i	$-0.471915\pi$
$83$	1.60555	0.176232	0.0881161	+	0.996110i	$0.471915\pi$
$84$	0	0
$85$	−6.90833	−0.749313
$86$	16.3028	1.75797
$87$	0	0
$88$	15.0000	1.59901
$89$	−0.0916731	−0.00971733	−0.00485866	−	0.999988i	$-0.501547\pi$
$89$	−0.0916731	−0.00971733	−0.00485866	+	0.999988i	$0.501547\pi$
$90$	0	0
$91$	1.30278	0.136568
$92$	0.302776	0.0315665
$93$	0	0
$94$	−1.81665	−0.187374
$95$	24.1194	2.47460
$96$	0	0
$97$	−10.3944	−1.05540	−0.527698	−	0.849432i	$-0.676945\pi$
$97$	−10.3944	−1.05540	−0.527698	+	0.849432i	$0.676945\pi$
$98$	−1.30278	−0.131600
$99$	0	0
$100$	−4.09167	−0.409167
$101$	−5.69722	−0.566895	−0.283448	−	0.958988i	$-0.591478\pi$
$101$	−5.69722	−0.566895	−0.283448	+	0.958988i	$0.591478\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−3.90833	−0.383243
$105$	0	0
$106$	7.18335	0.697708
$107$	10.5139	1.01641	0.508207	−	0.861235i	$-0.330308\pi$
$107$	10.5139	1.01641	0.508207	+	0.861235i	$0.330308\pi$
$108$	0	0
$109$	6.90833	0.661698	0.330849	−	0.943684i	$-0.392665\pi$
$109$	6.90833	0.661698	0.330849	+	0.943684i	$0.392665\pi$
$110$	−28.0278	−2.67234
$111$	0	0
$112$	3.30278	0.312083
$113$	−1.69722	−0.159661	−0.0798307	−	0.996808i	$-0.525438\pi$
$113$	−1.69722	−0.159661	−0.0798307	+	0.996808i	$0.525438\pi$
$114$	0	0
$115$	−4.30278	−0.401236
$116$	−2.48612	−0.230831
$117$	0	0
$118$	−9.00000	−0.828517
$119$	1.60555	0.147181
$120$	0	0
$121$	14.0000	1.27273
$122$	15.5139	1.40456
$123$	0	0
$124$	−0.908327	−0.0815702
$125$	36.6333	3.27658
$126$	0	0
$127$	5.30278	0.470545	0.235273	−	0.971929i	$-0.424402\pi$
$127$	5.30278	0.470545	0.235273	+	0.971929i	$0.424402\pi$
$128$	−8.09167	−0.715210
$129$	0	0
$130$	7.30278	0.640496
$131$	−17.6056	−1.53820	−0.769102	−	0.639126i	$-0.779296\pi$
$131$	−17.6056	−1.53820	−0.769102	+	0.639126i	$0.779296\pi$
$132$	0	0
$133$	−5.60555	−0.486063
$134$	1.42221	0.122860
$135$	0	0
$136$	−4.81665	−0.413025
$137$	4.81665	0.411515	0.205757	−	0.978603i	$-0.434034\pi$
$137$	4.81665	0.411515	0.205757	+	0.978603i	$0.434034\pi$
$138$	0	0
$139$	5.09167	0.431870	0.215935	−	0.976408i	$-0.430720\pi$
$139$	5.09167	0.431870	0.215935	+	0.976408i	$0.430720\pi$
$140$	1.30278	0.110105
$141$	0	0
$142$	−12.9083	−1.08324
$143$	−6.51388	−0.544718
$144$	0	0
$145$	35.3305	2.93404
$146$	−15.9083	−1.31658
$147$	0	0
$148$	2.72498	0.223992
$149$	−1.39445	−0.114238	−0.0571188	−	0.998367i	$-0.518191\pi$
$149$	−1.39445	−0.114238	−0.0571188	+	0.998367i	$0.518191\pi$
$150$	0	0
$151$	9.39445	0.764509	0.382255	−	0.924057i	$-0.375148\pi$
$151$	9.39445	0.764509	0.382255	+	0.924057i	$0.375148\pi$
$152$	16.8167	1.36401
$153$	0	0
$154$	6.51388	0.524903
$155$	12.9083	1.03682
$156$	0	0
$157$	17.8167	1.42192	0.710962	−	0.703231i	$-0.248260\pi$
$157$	17.8167	1.42192	0.710962	+	0.703231i	$0.248260\pi$
$158$	1.30278	0.103643
$159$	0	0
$160$	−7.30278	−0.577335
$161$	1.00000	0.0788110
$162$	0	0
$163$	18.7250	1.46665	0.733327	−	0.679876i	$-0.237967\pi$
$163$	18.7250	1.46665	0.733327	+	0.679876i	$0.237967\pi$
$164$	0.669468	0.0522767
$165$	0	0
$166$	−2.09167	−0.162345
$167$	18.8167	1.45608	0.728038	−	0.685537i	$-0.240432\pi$
$167$	18.8167	1.45608	0.728038	+	0.685537i	$0.240432\pi$
$168$	0	0
$169$	−11.3028	−0.869444
$170$	9.00000	0.690268
$171$	0	0
$172$	3.78890	0.288901
$173$	3.78890	0.288065	0.144032	−	0.989573i	$-0.453993\pi$
$173$	3.78890	0.288065	0.144032	+	0.989573i	$0.453993\pi$
$174$	0	0
$175$	−13.5139	−1.02155
$176$	−16.5139	−1.24478
$177$	0	0
$178$	0.119429	0.00895162
$179$	6.30278	0.471092	0.235546	−	0.971863i	$-0.424312\pi$
$179$	6.30278	0.471092	0.235546	+	0.971863i	$0.424312\pi$
$180$	0	0
$181$	14.8167	1.10131	0.550657	−	0.834732i	$-0.314377\pi$
$181$	14.8167	1.10131	0.550657	+	0.834732i	$0.314377\pi$
$182$	−1.69722	−0.125807
$183$	0	0
$184$	−3.00000	−0.221163
$185$	−38.7250	−2.84712
$186$	0	0
$187$	−8.02776	−0.587048
$188$	−0.422205	−0.0307925
$189$	0	0
$190$	−31.4222	−2.27961
$191$	−2.60555	−0.188531	−0.0942655	−	0.995547i	$-0.530050\pi$
$191$	−2.60555	−0.188531	−0.0942655	+	0.995547i	$0.530050\pi$
$192$	0	0
$193$	−27.0278	−1.94550	−0.972750	−	0.231856i	$-0.925520\pi$
$193$	−27.0278	−1.94550	−0.972750	+	0.231856i	$0.925520\pi$
$194$	13.5416	0.972233
$195$	0	0
$196$	−0.302776	−0.0216268
$197$	17.9083	1.27592	0.637958	−	0.770071i	$-0.279779\pi$
$197$	17.9083	1.27592	0.637958	+	0.770071i	$0.279779\pi$
$198$	0	0
$199$	−5.51388	−0.390868	−0.195434	−	0.980717i	$-0.562612\pi$
$199$	−5.51388	−0.390868	−0.195434	+	0.980717i	$0.562612\pi$
$200$	40.5416	2.86673
$201$	0	0
$202$	7.42221	0.522225
$203$	−8.21110	−0.576306
$204$	0	0
$205$	−9.51388	−0.664478
$206$	5.21110	0.363075
$207$	0	0
$208$	4.30278	0.298344
$209$	28.0278	1.93872
$210$	0	0
$211$	1.42221	0.0979086	0.0489543	−	0.998801i	$-0.484411\pi$
$211$	1.42221	0.0979086	0.0489543	+	0.998801i	$0.484411\pi$
$212$	1.66947	0.114660
$213$	0	0
$214$	−13.6972	−0.936323
$215$	−53.8444	−3.67216
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−9.00000	−0.609557
$219$	0	0
$220$	−6.51388	−0.439166
$221$	2.09167	0.140701
$222$	0	0
$223$	9.09167	0.608823	0.304412	−	0.952541i	$-0.401540\pi$
$223$	9.09167	0.608823	0.304412	+	0.952541i	$0.401540\pi$
$224$	1.69722	0.113401
$225$	0	0
$226$	2.21110	0.147080
$227$	−19.3305	−1.28301	−0.641506	−	0.767118i	$-0.721690\pi$
$227$	−19.3305	−1.28301	−0.641506	+	0.767118i	$0.721690\pi$
$228$	0	0
$229$	−15.5139	−1.02519	−0.512593	−	0.858632i	$-0.671315\pi$
$229$	−15.5139	−1.02519	−0.512593	+	0.858632i	$0.671315\pi$
$230$	5.60555	0.369619
$231$	0	0
$232$	24.6333	1.61726
$233$	−25.3305	−1.65946	−0.829729	−	0.558166i	$-0.811505\pi$
$233$	−25.3305	−1.65946	−0.829729	+	0.558166i	$0.811505\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	−2.09167	−0.136156
$237$	0	0
$238$	−2.09167	−0.135583
$239$	24.9083	1.61119	0.805593	−	0.592470i	$-0.201847\pi$
$239$	24.9083	1.61119	0.805593	+	0.592470i	$0.201847\pi$
$240$	0	0
$241$	−24.0278	−1.54776	−0.773882	−	0.633330i	$-0.781688\pi$
$241$	−24.0278	−1.54776	−0.773882	+	0.633330i	$0.781688\pi$
$242$	−18.2389	−1.17244
$243$	0	0
$244$	3.60555	0.230822
$245$	4.30278	0.274894
$246$	0	0
$247$	−7.30278	−0.464664
$248$	9.00000	0.571501
$249$	0	0
$250$	−47.7250	−3.01839
$251$	−2.18335	−0.137812	−0.0689058	−	0.997623i	$-0.521951\pi$
$251$	−2.18335	−0.137812	−0.0689058	+	0.997623i	$0.521951\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	−6.90833	−0.433467
$255$	0	0
$256$	−7.09167	−0.443230
$257$	9.02776	0.563136	0.281568	−	0.959541i	$-0.409146\pi$
$257$	9.02776	0.563136	0.281568	+	0.959541i	$0.409146\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	1.69722	0.105257
$261$	0	0
$262$	22.9361	1.41700
$263$	−17.6056	−1.08560	−0.542802	−	0.839860i	$-0.682637\pi$
$263$	−17.6056	−1.08560	−0.542802	+	0.839860i	$0.682637\pi$
$264$	0	0
$265$	−23.7250	−1.45741
$266$	7.30278	0.447762
$267$	0	0
$268$	0.330532	0.0201905
$269$	9.90833	0.604121	0.302061	−	0.953289i	$-0.402325\pi$
$269$	9.90833	0.604121	0.302061	+	0.953289i	$0.402325\pi$
$270$	0	0
$271$	−3.60555	−0.219022	−0.109511	−	0.993986i	$-0.534928\pi$
$271$	−3.60555	−0.219022	−0.109511	+	0.993986i	$0.534928\pi$
$272$	5.30278	0.321528
$273$	0	0
$274$	−6.27502	−0.379088
$275$	67.5694	4.07459
$276$	0	0
$277$	−8.69722	−0.522566	−0.261283	−	0.965262i	$-0.584146\pi$
$277$	−8.69722	−0.522566	−0.261283	+	0.965262i	$0.584146\pi$
$278$	−6.63331	−0.397839
$279$	0	0
$280$	−12.9083	−0.771420
$281$	−6.18335	−0.368868	−0.184434	−	0.982845i	$-0.559045\pi$
$281$	−6.18335	−0.368868	−0.184434	+	0.982845i	$0.559045\pi$
$282$	0	0
$283$	13.6972	0.814215	0.407108	−	0.913380i	$-0.366537\pi$
$283$	13.6972	0.814215	0.407108	+	0.913380i	$0.366537\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	8.48612	0.501795
$287$	2.21110	0.130517
$288$	0	0
$289$	−14.4222	−0.848365
$290$	−46.0278	−2.70284
$291$	0	0
$292$	−3.69722	−0.216364
$293$	−32.8444	−1.91879	−0.959395	−	0.282064i	$-0.908981\pi$
$293$	−32.8444	−1.91879	−0.959395	+	0.282064i	$0.908981\pi$
$294$	0	0
$295$	29.7250	1.73066
$296$	−27.0000	−1.56934
$297$	0	0
$298$	1.81665	0.105236
$299$	1.30278	0.0753415
$300$	0	0
$301$	12.5139	0.721288
$302$	−12.2389	−0.704267
$303$	0	0
$304$	−18.5139	−1.06184
$305$	−51.2389	−2.93393
$306$	0	0
$307$	3.78890	0.216244	0.108122	−	0.994138i	$-0.465516\pi$
$307$	3.78890	0.216244	0.108122	+	0.994138i	$0.465516\pi$
$308$	1.51388	0.0862612
$309$	0	0
$310$	−16.8167	−0.955122
$311$	−8.51388	−0.482778	−0.241389	−	0.970428i	$-0.577603\pi$
$311$	−8.51388	−0.482778	−0.241389	+	0.970428i	$0.577603\pi$
$312$	0	0
$313$	−13.2111	−0.746736	−0.373368	−	0.927683i	$-0.621797\pi$
$313$	−13.2111	−0.746736	−0.373368	+	0.927683i	$0.621797\pi$
$314$	−23.2111	−1.30988
$315$	0	0
$316$	0.302776	0.0170325
$317$	12.4861	0.701290	0.350645	−	0.936508i	$-0.385962\pi$
$317$	12.4861	0.701290	0.350645	+	0.936508i	$0.385962\pi$
$318$	0	0
$319$	41.0555	2.29867
$320$	37.9361	2.12069
$321$	0	0
$322$	−1.30278	−0.0726008
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−17.6056	−0.976580
$326$	−24.3944	−1.35108
$327$	0	0
$328$	−6.63331	−0.366263
$329$	−1.39445	−0.0768784
$330$	0	0
$331$	−25.6333	−1.40893	−0.704467	−	0.709737i	$-0.748814\pi$
$331$	−25.6333	−1.40893	−0.704467	+	0.709737i	$0.748814\pi$
$332$	−0.486122	−0.0266794
$333$	0	0
$334$	−24.5139	−1.34134
$335$	−4.69722	−0.256637
$336$	0	0
$337$	−19.3028	−1.05149	−0.525745	−	0.850642i	$-0.676213\pi$
$337$	−19.3028	−1.05149	−0.525745	+	0.850642i	$0.676213\pi$
$338$	14.7250	0.800933
$339$	0	0
$340$	2.09167	0.113437
$341$	15.0000	0.812296
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−37.5416	−2.02411
$345$	0	0
$346$	−4.93608	−0.265365
$347$	28.6333	1.53712	0.768558	−	0.639780i	$-0.220974\pi$
$347$	28.6333	1.53712	0.768558	+	0.639780i	$0.220974\pi$
$348$	0	0
$349$	−20.9083	−1.11920	−0.559599	−	0.828764i	$-0.689045\pi$
$349$	−20.9083	−1.11920	−0.559599	+	0.828764i	$0.689045\pi$
$350$	17.6056	0.941056
$351$	0	0
$352$	−8.48612	−0.452312
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	42.6333	2.26274
$356$	0.0277564	0.00147109
$357$	0	0
$358$	−8.21110	−0.433970
$359$	−18.5416	−0.978590	−0.489295	−	0.872118i	$-0.662746\pi$
$359$	−18.5416	−0.978590	−0.489295	+	0.872118i	$0.662746\pi$
$360$	0	0
$361$	12.4222	0.653800
$362$	−19.3028	−1.01453
$363$	0	0
$364$	−0.394449	−0.0206747
$365$	52.5416	2.75015
$366$	0	0
$367$	3.48612	0.181974	0.0909870	−	0.995852i	$-0.470998\pi$
$367$	3.48612	0.181974	0.0909870	+	0.995852i	$0.470998\pi$
$368$	3.30278	0.172169
$369$	0	0
$370$	50.4500	2.62277
$371$	5.51388	0.286266
$372$	0	0
$373$	1.78890	0.0926256	0.0463128	−	0.998927i	$-0.485253\pi$
$373$	1.78890	0.0926256	0.0463128	+	0.998927i	$0.485253\pi$
$374$	10.4584	0.540789
$375$	0	0
$376$	4.18335	0.215740
$377$	−10.6972	−0.550935
$378$	0	0
$379$	6.42221	0.329887	0.164943	−	0.986303i	$-0.447256\pi$
$379$	6.42221	0.329887	0.164943	+	0.986303i	$0.447256\pi$
$380$	−7.30278	−0.374624
$381$	0	0
$382$	3.39445	0.173675
$383$	16.8167	0.859291	0.429645	−	0.902998i	$-0.358639\pi$
$383$	16.8167	0.859291	0.429645	+	0.902998i	$0.358639\pi$
$384$	0	0
$385$	−21.5139	−1.09645
$386$	35.2111	1.79220
$387$	0	0
$388$	3.14719	0.159774
$389$	−6.63331	−0.336322	−0.168161	−	0.985760i	$-0.553783\pi$
$389$	−6.63331	−0.336322	−0.168161	+	0.985760i	$0.553783\pi$
$390$	0	0
$391$	1.60555	0.0811962
$392$	3.00000	0.151523
$393$	0	0
$394$	−23.3305	−1.17538
$395$	−4.30278	−0.216496
$396$	0	0
$397$	−34.6056	−1.73680	−0.868401	−	0.495862i	$-0.834852\pi$
$397$	−34.6056	−1.73680	−0.868401	+	0.495862i	$0.834852\pi$
$398$	7.18335	0.360069
$399$	0	0
$400$	−44.6333	−2.23167
$401$	−15.4222	−0.770148	−0.385074	−	0.922886i	$-0.625824\pi$
$401$	−15.4222	−0.770148	−0.385074	+	0.922886i	$0.625824\pi$
$402$	0	0
$403$	−3.90833	−0.194688
$404$	1.72498	0.0858210
$405$	0	0
$406$	10.6972	0.530894
$407$	−45.0000	−2.23057
$408$	0	0
$409$	−12.0278	−0.594734	−0.297367	−	0.954763i	$-0.596109\pi$
$409$	−12.0278	−0.594734	−0.297367	+	0.954763i	$0.596109\pi$
$410$	12.3944	0.612118
$411$	0	0
$412$	1.21110	0.0596667
$413$	−6.90833	−0.339937
$414$	0	0
$415$	6.90833	0.339116
$416$	2.21110	0.108408
$417$	0	0
$418$	−36.5139	−1.78595
$419$	10.1194	0.494366	0.247183	−	0.968969i	$-0.420495\pi$
$419$	10.1194	0.494366	0.247183	+	0.968969i	$0.420495\pi$
$420$	0	0
$421$	13.3028	0.648338	0.324169	−	0.945999i	$-0.394915\pi$
$421$	13.3028	0.648338	0.324169	+	0.945999i	$0.394915\pi$
$422$	−1.85281	−0.0901936
$423$	0	0
$424$	−16.5416	−0.803333
$425$	−21.6972	−1.05247
$426$	0	0
$427$	11.9083	0.576284
$428$	−3.18335	−0.153873
$429$	0	0
$430$	70.1472	3.38280
$431$	−23.7250	−1.14279	−0.571396	−	0.820674i	$-0.693598\pi$
$431$	−23.7250	−1.14279	−0.571396	+	0.820674i	$0.693598\pi$
$432$	0	0
$433$	15.0278	0.722188	0.361094	−	0.932529i	$-0.382403\pi$
$433$	15.0278	0.722188	0.361094	+	0.932529i	$0.382403\pi$
$434$	3.90833	0.187606
$435$	0	0
$436$	−2.09167	−0.100173
$437$	−5.60555	−0.268150
$438$	0	0
$439$	5.78890	0.276289	0.138145	−	0.990412i	$-0.455886\pi$
$439$	5.78890	0.276289	0.138145	+	0.990412i	$0.455886\pi$
$440$	64.5416	3.07690
$441$	0	0
$442$	−2.72498	−0.129614
$443$	−24.8444	−1.18039	−0.590197	−	0.807259i	$-0.700950\pi$
$443$	−24.8444	−1.18039	−0.590197	+	0.807259i	$0.700950\pi$
$444$	0	0
$445$	−0.394449	−0.0186987
$446$	−11.8444	−0.560849
$447$	0	0
$448$	−8.81665	−0.416548
$449$	3.27502	0.154558	0.0772789	−	0.997010i	$-0.475377\pi$
$449$	3.27502	0.154558	0.0772789	+	0.997010i	$0.475377\pi$
$450$	0	0
$451$	−11.0555	−0.520584
$452$	0.513878	0.0241708
$453$	0	0
$454$	25.1833	1.18191
$455$	5.60555	0.262792
$456$	0	0
$457$	−36.1194	−1.68960	−0.844798	−	0.535086i	$-0.820279\pi$
$457$	−36.1194	−1.68960	−0.844798	+	0.535086i	$0.820279\pi$
$458$	20.2111	0.944403
$459$	0	0
$460$	1.30278	0.0607422
$461$	24.7250	1.15156	0.575779	−	0.817606i	$-0.304699\pi$
$461$	24.7250	1.15156	0.575779	+	0.817606i	$0.304699\pi$
$462$	0	0
$463$	2.81665	0.130901	0.0654505	−	0.997856i	$-0.479152\pi$
$463$	2.81665	0.130901	0.0654505	+	0.997856i	$0.479152\pi$
$464$	−27.1194	−1.25899
$465$	0	0
$466$	33.0000	1.52870
$467$	−16.3944	−0.758645	−0.379322	−	0.925265i	$-0.623843\pi$
$467$	−16.3944	−0.758645	−0.379322	+	0.925265i	$0.623843\pi$
$468$	0	0
$469$	1.09167	0.0504088
$470$	−7.81665	−0.360555
$471$	0	0
$472$	20.7250	0.953945
$473$	−62.5694	−2.87694
$474$	0	0
$475$	75.7527	3.47577
$476$	−0.486122	−0.0222814
$477$	0	0
$478$	−32.4500	−1.48423
$479$	3.78890	0.173119	0.0865596	−	0.996247i	$-0.472413\pi$
$479$	3.78890	0.173119	0.0865596	+	0.996247i	$0.472413\pi$
$480$	0	0
$481$	11.7250	0.534613
$482$	31.3028	1.42580
$483$	0	0
$484$	−4.23886	−0.192675
$485$	−44.7250	−2.03086
$486$	0	0
$487$	20.8167	0.943293	0.471646	−	0.881788i	$-0.343660\pi$
$487$	20.8167	0.943293	0.471646	+	0.881788i	$0.343660\pi$
$488$	−35.7250	−1.61719
$489$	0	0
$490$	−5.60555	−0.253233
$491$	−28.9361	−1.30587	−0.652934	−	0.757415i	$-0.726462\pi$
$491$	−28.9361	−1.30587	−0.652934	+	0.757415i	$0.726462\pi$
$492$	0	0
$493$	−13.1833	−0.593748
$494$	9.51388	0.428050
$495$	0	0
$496$	−9.90833	−0.444897
$497$	−9.90833	−0.444449
$498$	0	0
$499$	21.0917	0.944193	0.472096	−	0.881547i	$-0.343497\pi$
$499$	21.0917	0.944193	0.472096	+	0.881547i	$0.343497\pi$
$500$	−11.0917	−0.496035
$501$	0	0
$502$	2.84441	0.126952
$503$	14.7250	0.656554	0.328277	−	0.944581i	$-0.393532\pi$
$503$	14.7250	0.656554	0.328277	+	0.944581i	$0.393532\pi$
$504$	0	0
$505$	−24.5139	−1.09085
$506$	6.51388	0.289577
$507$	0	0
$508$	−1.60555	−0.0712348
$509$	−31.4500	−1.39400	−0.696998	−	0.717074i	$-0.745481\pi$
$509$	−31.4500	−1.39400	−0.696998	+	0.717074i	$0.745481\pi$
$510$	0	0
$511$	−12.2111	−0.540187
$512$	25.4222	1.12351
$513$	0	0
$514$	−11.7611	−0.518762
$515$	−17.2111	−0.758412
$516$	0	0
$517$	6.97224	0.306639
$518$	−11.7250	−0.515166
$519$	0	0
$520$	−16.8167	−0.737459
$521$	21.6333	0.947772	0.473886	−	0.880586i	$-0.342851\pi$
$521$	21.6333	0.947772	0.473886	+	0.880586i	$0.342851\pi$
$522$	0	0
$523$	28.4222	1.24282	0.621408	−	0.783487i	$-0.286561\pi$
$523$	28.4222	1.24282	0.621408	+	0.783487i	$0.286561\pi$
$524$	5.33053	0.232865
$525$	0	0
$526$	22.9361	1.00006
$527$	−4.81665	−0.209817
$528$	0	0
$529$	1.00000	0.0434783
$530$	30.9083	1.34257
$531$	0	0
$532$	1.69722	0.0735840
$533$	2.88057	0.124771
$534$	0	0
$535$	45.2389	1.95585
$536$	−3.27502	−0.141459
$537$	0	0
$538$	−12.9083	−0.556517
$539$	5.00000	0.215365
$540$	0	0
$541$	15.8167	0.680011	0.340006	−	0.940423i	$-0.389571\pi$
$541$	15.8167	0.680011	0.340006	+	0.940423i	$0.389571\pi$
$542$	4.69722	0.201763
$543$	0	0
$544$	2.72498	0.116833
$545$	29.7250	1.27328
$546$	0	0
$547$	24.7250	1.05716	0.528582	−	0.848882i	$-0.322724\pi$
$547$	24.7250	1.05716	0.528582	+	0.848882i	$0.322724\pi$
$548$	−1.45837	−0.0622983
$549$	0	0
$550$	−88.0278	−3.75352
$551$	46.0278	1.96085
$552$	0	0
$553$	1.00000	0.0425243
$554$	11.3305	0.481388
$555$	0	0
$556$	−1.54163	−0.0653799
$557$	31.0278	1.31469	0.657344	−	0.753591i	$-0.271680\pi$
$557$	31.0278	1.31469	0.657344	+	0.753591i	$0.271680\pi$
$558$	0	0
$559$	16.3028	0.689534
$560$	14.2111	0.600529
$561$	0	0
$562$	8.05551	0.339801
$563$	3.66947	0.154650	0.0773248	−	0.997006i	$-0.475362\pi$
$563$	3.66947	0.154650	0.0773248	+	0.997006i	$0.475362\pi$
$564$	0	0
$565$	−7.30278	−0.307230
$566$	−17.8444	−0.750057
$567$	0	0
$568$	29.7250	1.24723
$569$	−25.2111	−1.05690	−0.528452	−	0.848963i	$-0.677227\pi$
$569$	−25.2111	−1.05690	−0.528452	+	0.848963i	$0.677227\pi$
$570$	0	0
$571$	−4.21110	−0.176229	−0.0881146	−	0.996110i	$-0.528084\pi$
$571$	−4.21110	−0.176229	−0.0881146	+	0.996110i	$0.528084\pi$
$572$	1.97224	0.0824636
$573$	0	0
$574$	−2.88057	−0.120233
$575$	−13.5139	−0.563568
$576$	0	0
$577$	−7.57779	−0.315468	−0.157734	−	0.987482i	$-0.550419\pi$
$577$	−7.57779	−0.315468	−0.157734	+	0.987482i	$0.550419\pi$
$578$	18.7889	0.781515
$579$	0	0
$580$	−10.6972	−0.444178
$581$	−1.60555	−0.0666095
$582$	0	0
$583$	−27.5694	−1.14181
$584$	36.6333	1.51590
$585$	0	0
$586$	42.7889	1.76759
$587$	−22.1472	−0.914112	−0.457056	−	0.889438i	$-0.651096\pi$
$587$	−22.1472	−0.914112	−0.457056	+	0.889438i	$0.651096\pi$
$588$	0	0
$589$	16.8167	0.692918
$590$	−38.7250	−1.59428
$591$	0	0
$592$	29.7250	1.22169
$593$	−0.972244	−0.0399253	−0.0199626	−	0.999801i	$-0.506355\pi$
$593$	−0.972244	−0.0399253	−0.0199626	+	0.999801i	$0.506355\pi$
$594$	0	0
$595$	6.90833	0.283214
$596$	0.422205	0.0172942
$597$	0	0
$598$	−1.69722	−0.0694047
$599$	21.4861	0.877899	0.438950	−	0.898512i	$-0.355351\pi$
$599$	21.4861	0.877899	0.438950	+	0.898512i	$0.355351\pi$
$600$	0	0
$601$	−5.93608	−0.242138	−0.121069	−	0.992644i	$-0.538632\pi$
$601$	−5.93608	−0.242138	−0.121069	+	0.992644i	$0.538632\pi$
$602$	−16.3028	−0.664452
$603$	0	0
$604$	−2.84441	−0.115737
$605$	60.2389	2.44906
$606$	0	0
$607$	−42.5139	−1.72559	−0.862793	−	0.505558i	$-0.831287\pi$
$607$	−42.5139	−1.72559	−0.862793	+	0.505558i	$0.831287\pi$
$608$	−9.51388	−0.385839
$609$	0	0
$610$	66.7527	2.70274
$611$	−1.81665	−0.0734939
$612$	0	0
$613$	−18.8167	−0.759997	−0.379999	−	0.924987i	$-0.624076\pi$
$613$	−18.8167	−0.759997	−0.379999	+	0.924987i	$0.624076\pi$
$614$	−4.93608	−0.199204
$615$	0	0
$616$	−15.0000	−0.604367
$617$	17.7250	0.713581	0.356790	−	0.934184i	$-0.383871\pi$
$617$	17.7250	0.713581	0.356790	+	0.934184i	$0.383871\pi$
$618$	0	0
$619$	35.1194	1.41157	0.705785	−	0.708427i	$-0.250595\pi$
$619$	35.1194	1.41157	0.705785	+	0.708427i	$0.250595\pi$
$620$	−3.90833	−0.156962
$621$	0	0
$622$	11.0917	0.444736
$623$	0.0916731	0.00367280
$624$	0	0
$625$	90.0555	3.60222
$626$	17.2111	0.687894
$627$	0	0
$628$	−5.39445	−0.215262
$629$	14.4500	0.576158
$630$	0	0
$631$	−48.0278	−1.91195	−0.955977	−	0.293440i	$-0.905200\pi$
$631$	−48.0278	−1.91195	−0.955977	+	0.293440i	$0.905200\pi$
$632$	−3.00000	−0.119334
$633$	0	0
$634$	−16.2666	−0.646030
$635$	22.8167	0.905451
$636$	0	0
$637$	−1.30278	−0.0516179
$638$	−53.4861	−2.11754
$639$	0	0
$640$	−34.8167	−1.37625
$641$	−43.5416	−1.71979	−0.859896	−	0.510470i	$-0.829471\pi$
$641$	−43.5416	−1.71979	−0.859896	+	0.510470i	$0.829471\pi$
$642$	0	0
$643$	−46.5694	−1.83652	−0.918259	−	0.395981i	$-0.870405\pi$
$643$	−46.5694	−1.83652	−0.918259	+	0.395981i	$0.870405\pi$
$644$	−0.302776	−0.0119310
$645$	0	0
$646$	11.7250	0.461313
$647$	23.3305	0.917218	0.458609	−	0.888638i	$-0.348348\pi$
$647$	23.3305	0.917218	0.458609	+	0.888638i	$0.348348\pi$
$648$	0	0
$649$	34.5416	1.35588
$650$	22.9361	0.899627
$651$	0	0
$652$	−5.66947	−0.222034
$653$	−12.4861	−0.488620	−0.244310	−	0.969697i	$-0.578561\pi$
$653$	−12.4861	−0.488620	−0.244310	+	0.969697i	$0.578561\pi$
$654$	0	0
$655$	−75.7527	−2.95990
$656$	7.30278	0.285125
$657$	0	0
$658$	1.81665	0.0708205
$659$	0.633308	0.0246702	0.0123351	−	0.999924i	$-0.496074\pi$
$659$	0.633308	0.0246702	0.0123351	+	0.999924i	$0.496074\pi$
$660$	0	0
$661$	−0.816654	−0.0317642	−0.0158821	−	0.999874i	$-0.505056\pi$
$661$	−0.816654	−0.0317642	−0.0158821	+	0.999874i	$0.505056\pi$
$662$	33.3944	1.29791
$663$	0	0
$664$	4.81665	0.186922
$665$	−24.1194	−0.935311
$666$	0	0
$667$	−8.21110	−0.317935
$668$	−5.69722	−0.220432
$669$	0	0
$670$	6.11943	0.236414
$671$	−59.5416	−2.29858
$672$	0	0
$673$	16.6333	0.641167	0.320583	−	0.947220i	$-0.396121\pi$
$673$	16.6333	0.641167	0.320583	+	0.947220i	$0.396121\pi$
$674$	25.1472	0.968633
$675$	0	0
$676$	3.42221	0.131623
$677$	24.1472	0.928052	0.464026	−	0.885822i	$-0.346404\pi$
$677$	24.1472	0.928052	0.464026	+	0.885822i	$0.346404\pi$
$678$	0	0
$679$	10.3944	0.398902
$680$	−20.7250	−0.794767
$681$	0	0
$682$	−19.5416	−0.748288
$683$	−27.4222	−1.04928	−0.524641	−	0.851324i	$-0.675800\pi$
$683$	−27.4222	−1.04928	−0.524641	+	0.851324i	$0.675800\pi$
$684$	0	0
$685$	20.7250	0.791861
$686$	1.30278	0.0497402
$687$	0	0
$688$	41.3305	1.57571
$689$	7.18335	0.273664
$690$	0	0
$691$	−13.4861	−0.513036	−0.256518	−	0.966539i	$-0.582575\pi$
$691$	−13.4861	−0.513036	−0.256518	+	0.966539i	$0.582575\pi$
$692$	−1.14719	−0.0436095
$693$	0	0
$694$	−37.3028	−1.41599
$695$	21.9083	0.831030
$696$	0	0
$697$	3.55004	0.134467
$698$	27.2389	1.03101
$699$	0	0
$700$	4.09167	0.154651
$701$	−28.5416	−1.07800	−0.539001	−	0.842305i	$-0.681198\pi$
$701$	−28.5416	−1.07800	−0.539001	+	0.842305i	$0.681198\pi$
$702$	0	0
$703$	−50.4500	−1.90276
$704$	44.0833	1.66145
$705$	0	0
$706$	−14.3305	−0.539337
$707$	5.69722	0.214266
$708$	0	0
$709$	−7.66947	−0.288033	−0.144016	−	0.989575i	$-0.546002\pi$
$709$	−7.66947	−0.288033	−0.144016	+	0.989575i	$0.546002\pi$
$710$	−55.5416	−2.08444
$711$	0	0
$712$	−0.275019	−0.0103068
$713$	−3.00000	−0.112351
$714$	0	0
$715$	−28.0278	−1.04818
$716$	−1.90833	−0.0713175
$717$	0	0
$718$	24.1556	0.901479
$719$	0.211103	0.00787280	0.00393640	−	0.999992i	$-0.498747\pi$
$719$	0.211103	0.00787280	0.00393640	+	0.999992i	$0.498747\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−16.1833	−0.602282
$723$	0	0
$724$	−4.48612	−0.166725
$725$	110.964	4.12109
$726$	0	0
$727$	46.8722	1.73839	0.869196	−	0.494467i	$-0.164637\pi$
$727$	46.8722	1.73839	0.869196	+	0.494467i	$0.164637\pi$
$728$	3.90833	0.144852
$729$	0	0
$730$	−68.4500	−2.53345
$731$	20.0917	0.743117
$732$	0	0
$733$	−19.3944	−0.716350	−0.358175	−	0.933654i	$-0.616601\pi$
$733$	−19.3944	−0.716350	−0.358175	+	0.933654i	$0.616601\pi$
$734$	−4.54163	−0.167635
$735$	0	0
$736$	1.69722	0.0625605
$737$	−5.45837	−0.201061
$738$	0	0
$739$	−14.5778	−0.536253	−0.268126	−	0.963384i	$-0.586404\pi$
$739$	−14.5778	−0.536253	−0.268126	+	0.963384i	$0.586404\pi$
$740$	11.7250	0.431019
$741$	0	0
$742$	−7.18335	−0.263709
$743$	−12.4861	−0.458071	−0.229036	−	0.973418i	$-0.573557\pi$
$743$	−12.4861	−0.458071	−0.229036	+	0.973418i	$0.573557\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−2.33053	−0.0853268
$747$	0	0
$748$	2.43061	0.0888719
$749$	−10.5139	−0.384169
$750$	0	0
$751$	12.3028	0.448935	0.224467	−	0.974482i	$-0.427936\pi$
$751$	12.3028	0.448935	0.224467	+	0.974482i	$0.427936\pi$
$752$	−4.60555	−0.167947
$753$	0	0
$754$	13.9361	0.507522
$755$	40.4222	1.47111
$756$	0	0
$757$	−46.6333	−1.69492	−0.847458	−	0.530862i	$-0.821868\pi$
$757$	−46.6333	−1.69492	−0.847458	+	0.530862i	$0.821868\pi$
$758$	−8.36669	−0.303892
$759$	0	0
$760$	72.3583	2.62471
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	−6.90833	−0.250098
$764$	0.788897	0.0285413
$765$	0	0
$766$	−21.9083	−0.791580
$767$	−9.00000	−0.324971
$768$	0	0
$769$	37.4500	1.35048	0.675240	−	0.737598i	$-0.264040\pi$
$769$	37.4500	1.35048	0.675240	+	0.737598i	$0.264040\pi$
$770$	28.0278	1.01005
$771$	0	0
$772$	8.18335	0.294525
$773$	25.8444	0.929559	0.464779	−	0.885427i	$-0.346134\pi$
$773$	25.8444	0.929559	0.464779	+	0.885427i	$0.346134\pi$
$774$	0	0
$775$	40.5416	1.45630
$776$	−31.1833	−1.11942
$777$	0	0
$778$	8.64171	0.309820
$779$	−12.3944	−0.444077
$780$	0	0
$781$	49.5416	1.77274
$782$	−2.09167	−0.0747981
$783$	0	0
$784$	−3.30278	−0.117956
$785$	76.6611	2.73615
$786$	0	0
$787$	−29.1472	−1.03898	−0.519492	−	0.854475i	$-0.673879\pi$
$787$	−29.1472	−1.03898	−0.519492	+	0.854475i	$0.673879\pi$
$788$	−5.42221	−0.193158
$789$	0	0
$790$	5.60555	0.199437
$791$	1.69722	0.0603464
$792$	0	0
$793$	15.5139	0.550914
$794$	45.0833	1.59995
$795$	0	0
$796$	1.66947	0.0591727
$797$	2.97224	0.105282	0.0526411	−	0.998613i	$-0.483236\pi$
$797$	2.97224	0.105282	0.0526411	+	0.998613i	$0.483236\pi$
$798$	0	0
$799$	−2.23886	−0.0792051
$800$	−22.9361	−0.810913
$801$	0	0
$802$	20.0917	0.709462
$803$	61.0555	2.15460
$804$	0	0
$805$	4.30278	0.151653
$806$	5.09167	0.179347
$807$	0	0
$808$	−17.0917	−0.601283
$809$	20.9361	0.736073	0.368037	−	0.929811i	$-0.380030\pi$
$809$	20.9361	0.736073	0.368037	+	0.929811i	$0.380030\pi$
$810$	0	0
$811$	−17.6056	−0.618215	−0.309107	−	0.951027i	$-0.600030\pi$
$811$	−17.6056	−0.618215	−0.309107	+	0.951027i	$0.600030\pi$
$812$	2.48612	0.0872458
$813$	0	0
$814$	58.6249	2.05480
$815$	80.5694	2.82222
$816$	0	0
$817$	−70.1472	−2.45414
$818$	15.6695	0.547870
$819$	0	0
$820$	2.88057	0.100594
$821$	26.6056	0.928540	0.464270	−	0.885694i	$-0.346317\pi$
$821$	26.6056	0.928540	0.464270	+	0.885694i	$0.346317\pi$
$822$	0	0
$823$	−11.5139	−0.401349	−0.200674	−	0.979658i	$-0.564313\pi$
$823$	−11.5139	−0.401349	−0.200674	+	0.979658i	$0.564313\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	9.00000	0.313150
$827$	−0.908327	−0.0315856	−0.0157928	−	0.999875i	$-0.505027\pi$
$827$	−0.908327	−0.0315856	−0.0157928	+	0.999875i	$0.505027\pi$
$828$	0	0
$829$	34.4500	1.19650	0.598248	−	0.801311i	$-0.295864\pi$
$829$	34.4500	1.19650	0.598248	+	0.801311i	$0.295864\pi$
$830$	−9.00000	−0.312395
$831$	0	0
$832$	−11.4861	−0.398210
$833$	−1.60555	−0.0556291
$834$	0	0
$835$	80.9638	2.80187
$836$	−8.48612	−0.293499
$837$	0	0
$838$	−13.1833	−0.455411
$839$	−34.6972	−1.19788	−0.598941	−	0.800793i	$-0.704411\pi$
$839$	−34.6972	−1.19788	−0.598941	+	0.800793i	$0.704411\pi$
$840$	0	0
$841$	38.4222	1.32490
$842$	−17.3305	−0.597250
$843$	0	0
$844$	−0.430609	−0.0148222
$845$	−48.6333	−1.67304
$846$	0	0
$847$	−14.0000	−0.481046
$848$	18.2111	0.625372
$849$	0	0
$850$	28.2666	0.969537
$851$	9.00000	0.308516
$852$	0	0
$853$	−7.76114	−0.265736	−0.132868	−	0.991134i	$-0.542419\pi$
$853$	−7.76114	−0.265736	−0.132868	+	0.991134i	$0.542419\pi$
$854$	−15.5139	−0.530874
$855$	0	0
$856$	31.5416	1.07807
$857$	−0.238859	−0.00815927	−0.00407963	−	0.999992i	$-0.501299\pi$
$857$	−0.238859	−0.00815927	−0.00407963	+	0.999992i	$0.501299\pi$
$858$	0	0
$859$	28.7889	0.982265	0.491132	−	0.871085i	$-0.336583\pi$
$859$	28.7889	0.982265	0.491132	+	0.871085i	$0.336583\pi$
$860$	16.3028	0.555920
$861$	0	0
$862$	30.9083	1.05274
$863$	56.2389	1.91439	0.957197	−	0.289439i	$-0.0934687\pi$
$863$	56.2389	1.91439	0.957197	+	0.289439i	$0.0934687\pi$
$864$	0	0
$865$	16.3028	0.554311
$866$	−19.5778	−0.665281
$867$	0	0
$868$	0.908327	0.0308306
$869$	−5.00000	−0.169613
$870$	0	0
$871$	1.42221	0.0481896
$872$	20.7250	0.701836
$873$	0	0
$874$	7.30278	0.247020
$875$	−36.6333	−1.23843
$876$	0	0
$877$	−29.2111	−0.986389	−0.493194	−	0.869919i	$-0.664171\pi$
$877$	−29.2111	−0.986389	−0.493194	+	0.869919i	$0.664171\pi$
$878$	−7.54163	−0.254518
$879$	0	0
$880$	−71.0555	−2.39528
$881$	22.1833	0.747376	0.373688	−	0.927554i	$-0.378093\pi$
$881$	22.1833	0.747376	0.373688	+	0.927554i	$0.378093\pi$
$882$	0	0
$883$	−36.5139	−1.22879	−0.614395	−	0.788999i	$-0.710600\pi$
$883$	−36.5139	−1.22879	−0.614395	+	0.788999i	$0.710600\pi$
$884$	−0.633308	−0.0213004
$885$	0	0
$886$	32.3667	1.08738
$887$	−34.9361	−1.17304	−0.586519	−	0.809935i	$-0.699502\pi$
$887$	−34.9361	−1.17304	−0.586519	+	0.809935i	$0.699502\pi$
$888$	0	0
$889$	−5.30278	−0.177849
$890$	0.513878	0.0172252
$891$	0	0
$892$	−2.75274	−0.0921685
$893$	7.81665	0.261574
$894$	0	0
$895$	27.1194	0.906503
$896$	8.09167	0.270324
$897$	0	0
$898$	−4.26662	−0.142379
$899$	24.6333	0.821567
$900$	0	0
$901$	8.85281	0.294930
$902$	14.4029	0.479563
$903$	0	0
$904$	−5.09167	−0.169347
$905$	63.7527	2.11921
$906$	0	0
$907$	53.1472	1.76472	0.882362	−	0.470572i	$-0.155952\pi$
$907$	53.1472	1.76472	0.882362	+	0.470572i	$0.155952\pi$
$908$	5.85281	0.194232
$909$	0	0
$910$	−7.30278	−0.242085
$911$	−26.2389	−0.869332	−0.434666	−	0.900592i	$-0.643134\pi$
$911$	−26.2389	−0.869332	−0.434666	+	0.900592i	$0.643134\pi$
$912$	0	0
$913$	8.02776	0.265680
$914$	47.0555	1.55646
$915$	0	0
$916$	4.69722	0.155201
$917$	17.6056	0.581387
$918$	0	0
$919$	−17.6056	−0.580754	−0.290377	−	0.956912i	$-0.593781\pi$
$919$	−17.6056	−0.580754	−0.290377	+	0.956912i	$0.593781\pi$
$920$	−12.9083	−0.425575
$921$	0	0
$922$	−32.2111	−1.06082
$923$	−12.9083	−0.424883
$924$	0	0
$925$	−121.625	−3.99900
$926$	−3.66947	−0.120586
$927$	0	0
$928$	−13.9361	−0.457474
$929$	−0.669468	−0.0219645	−0.0109823	−	0.999940i	$-0.503496\pi$
$929$	−0.669468	−0.0219645	−0.0109823	+	0.999940i	$0.503496\pi$
$930$	0	0
$931$	5.60555	0.183715
$932$	7.66947	0.251222
$933$	0	0
$934$	21.3583	0.698865
$935$	−34.5416	−1.12963
$936$	0	0
$937$	−33.4222	−1.09186	−0.545928	−	0.837832i	$-0.683823\pi$
$937$	−33.4222	−1.09186	−0.545928	+	0.837832i	$0.683823\pi$
$938$	−1.42221	−0.0464366
$939$	0	0
$940$	−1.81665	−0.0592527
$941$	−56.8444	−1.85307	−0.926537	−	0.376203i	$-0.877230\pi$
$941$	−56.8444	−1.85307	−0.926537	+	0.376203i	$0.877230\pi$
$942$	0	0
$943$	2.21110	0.0720034
$944$	−22.8167	−0.742619
$945$	0	0
$946$	81.5139	2.65024
$947$	−29.6333	−0.962953	−0.481477	−	0.876459i	$-0.659899\pi$
$947$	−29.6333	−0.962953	−0.481477	+	0.876459i	$0.659899\pi$
$948$	0	0
$949$	−15.9083	−0.516406
$950$	−98.6888	−3.20189
$951$	0	0
$952$	4.81665	0.156109
$953$	11.6695	0.378011	0.189006	−	0.981976i	$-0.439474\pi$
$953$	11.6695	0.378011	0.189006	+	0.981976i	$0.439474\pi$
$954$	0	0
$955$	−11.2111	−0.362783
$956$	−7.54163	−0.243914
$957$	0	0
$958$	−4.93608	−0.159478
$959$	−4.81665	−0.155538
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−15.2750	−0.492486
$963$	0	0
$964$	7.27502	0.234313
$965$	−116.294	−3.74365
$966$	0	0
$967$	−45.2666	−1.45568	−0.727838	−	0.685749i	$-0.759475\pi$
$967$	−45.2666	−1.45568	−0.727838	+	0.685749i	$0.759475\pi$
$968$	42.0000	1.34993
$969$	0	0
$970$	58.2666	1.87083
$971$	45.9083	1.47327	0.736634	−	0.676291i	$-0.236414\pi$
$971$	45.9083	1.47327	0.736634	+	0.676291i	$0.236414\pi$
$972$	0	0
$973$	−5.09167	−0.163232
$974$	−27.1194	−0.868963
$975$	0	0
$976$	39.3305	1.25894
$977$	30.1472	0.964494	0.482247	−	0.876035i	$-0.339821\pi$
$977$	30.1472	0.964494	0.482247	+	0.876035i	$0.339821\pi$
$978$	0	0
$979$	−0.458365	−0.0146494
$980$	−1.30278	−0.0416156
$981$	0	0
$982$	37.6972	1.20297
$983$	1.18335	0.0377429	0.0188714	−	0.999822i	$-0.493993\pi$
$983$	1.18335	0.0377429	0.0188714	+	0.999822i	$0.493993\pi$
$984$	0	0
$985$	77.0555	2.45519
$986$	17.1749	0.546962
$987$	0	0
$988$	2.21110	0.0703445
$989$	12.5139	0.397918
$990$	0	0
$991$	17.4861	0.555465	0.277732	−	0.960658i	$-0.410417\pi$
$991$	17.4861	0.555465	0.277732	+	0.960658i	$0.410417\pi$
$992$	−5.09167	−0.161661
$993$	0	0
$994$	12.9083	0.409427
$995$	−23.7250	−0.752132
$996$	0	0
$997$	36.0278	1.14101	0.570505	−	0.821294i	$-0.306747\pi$
$997$	36.0278	1.14101	0.570505	+	0.821294i	$0.306747\pi$
$998$	−27.4777	−0.869792
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.j.1.1		2
3.2	odd	2		483.2.a.f.1.2	✓	2
12.11	even	2		7728.2.a.x.1.1		2
21.20	even	2		3381.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.f.1.2	✓	2	3.2	odd	2
1449.2.a.j.1.1		2	1.1	even	1	trivial
3381.2.a.p.1.2		2	21.20	even	2
7728.2.a.x.1.1		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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{2} -0.302776 q^{4} +4.30278 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-1.30278 q^{2} -0.302776 q^{4} +4.30278 q^{5} -1.00000 q^{7} +3.00000 q^{8} -5.60555 q^{10} +5.00000 q^{11} -1.30278 q^{13} +1.30278 q^{14} -3.30278 q^{16} -1.60555 q^{17} +5.60555 q^{19} -1.30278 q^{20} -6.51388 q^{22} -1.00000 q^{23} +13.5139 q^{25} +1.69722 q^{26} +0.302776 q^{28} +8.21110 q^{29} +3.00000 q^{31} -1.69722 q^{32} +2.09167 q^{34} -4.30278 q^{35} -9.00000 q^{37} -7.30278 q^{38} +12.9083 q^{40} -2.21110 q^{41} -12.5139 q^{43} -1.51388 q^{44} +1.30278 q^{46} +1.39445 q^{47} +1.00000 q^{49} -17.6056 q^{50} +0.394449 q^{52} -5.51388 q^{53} +21.5139 q^{55} -3.00000 q^{56} -10.6972 q^{58} +6.90833 q^{59} -11.9083 q^{61} -3.90833 q^{62} +8.81665 q^{64} -5.60555 q^{65} -1.09167 q^{67} +0.486122 q^{68} +5.60555 q^{70} +9.90833 q^{71} +12.2111 q^{73} +11.7250 q^{74} -1.69722 q^{76} -5.00000 q^{77} -1.00000 q^{79} -14.2111 q^{80} +2.88057 q^{82} +1.60555 q^{83} -6.90833 q^{85} +16.3028 q^{86} +15.0000 q^{88} -0.0916731 q^{89} +1.30278 q^{91} +0.302776 q^{92} -1.81665 q^{94} +24.1194 q^{95} -10.3944 q^{97} -1.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8	\( 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} + 10 q^{11} + q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} - 2 q^{23} + 9 q^{25} + 7 q^{26} - 3 q^{28} + 2 q^{29} + 6 q^{31} - 7 q^{32} + 15 q^{34} - 5 q^{35} - 18 q^{37} - 11 q^{38} + 15 q^{40} + 10 q^{41} - 7 q^{43} + 15 q^{44} - q^{46} + 10 q^{47} + 2 q^{49} - 28 q^{50} + 8 q^{52} + 7 q^{53} + 25 q^{55} - 6 q^{56} - 25 q^{58} + 3 q^{59} - 13 q^{61} + 3 q^{62} - 4 q^{64} - 4 q^{65} - 13 q^{67} + 19 q^{68} + 4 q^{70} + 9 q^{71} + 10 q^{73} - 9 q^{74} - 7 q^{76} - 10 q^{77} - 2 q^{79} - 14 q^{80} + 31 q^{82} - 4 q^{83} - 3 q^{85} + 29 q^{86} + 30 q^{88} - 11 q^{89} - q^{91} - 3 q^{92} + 18 q^{94} + 23 q^{95} - 28 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8 - 4 * q^10 + 10 * q^11 + q^13 - q^14 - 3 * q^16 + 4 * q^17 + 4 * q^19 + q^20 + 5 * q^22 - 2 * q^23 + 9 * q^25 + 7 * q^26 - 3 * q^28 + 2 * q^29 + 6 * q^31 - 7 * q^32 + 15 * q^34 - 5 * q^35 - 18 * q^37 - 11 * q^38 + 15 * q^40 + 10 * q^41 - 7 * q^43 + 15 * q^44 - q^46 + 10 * q^47 + 2 * q^49 - 28 * q^50 + 8 * q^52 + 7 * q^53 + 25 * q^55 - 6 * q^56 - 25 * q^58 + 3 * q^59 - 13 * q^61 + 3 * q^62 - 4 * q^64 - 4 * q^65 - 13 * q^67 + 19 * q^68 + 4 * q^70 + 9 * q^71 + 10 * q^73 - 9 * q^74 - 7 * q^76 - 10 * q^77 - 2 * q^79 - 14 * q^80 + 31 * q^82 - 4 * q^83 - 3 * q^85 + 29 * q^86 + 30 * q^88 - 11 * q^89 - q^91 - 3 * q^92 + 18 * q^94 + 23 * q^95 - 28 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more