LMFDB - Embedded newform 4005.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 8x^{8} + 35x^{7} + 29x^{6} - 103x^{5} - 57x^{4} + 106x^{3} + 29x^{2} - 39x + 2 $ x^10 - 4x^9 - 8x^8 + 35x^7 + 29x^6 - 103x^5 - 57x^4 + 106x^3 + 29x^2 - 39*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.0538492$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.946151 q^{2} -1.10480 q^{4} -1.00000 q^{5} +2.86767 q^{7} +2.93761 q^{8} +O(q^{10})$	\(q-0.946151 q^{2} -1.10480 q^{4} -1.00000 q^{5} +2.86767 q^{7} +2.93761 q^{8} +0.946151 q^{10} -0.403071 q^{11} -4.88811 q^{13} -2.71325 q^{14} -0.569823 q^{16} -6.87104 q^{17} +4.96794 q^{19} +1.10480 q^{20} +0.381366 q^{22} -4.24601 q^{23} +1.00000 q^{25} +4.62489 q^{26} -3.16820 q^{28} +7.26323 q^{29} +7.51228 q^{31} -5.33608 q^{32} +6.50104 q^{34} -2.86767 q^{35} -1.37435 q^{37} -4.70042 q^{38} -2.93761 q^{40} -8.13284 q^{41} +3.35592 q^{43} +0.445313 q^{44} +4.01736 q^{46} +4.51219 q^{47} +1.22353 q^{49} -0.946151 q^{50} +5.40038 q^{52} +11.5380 q^{53} +0.403071 q^{55} +8.42409 q^{56} -6.87211 q^{58} -8.09315 q^{59} -4.94084 q^{61} -7.10775 q^{62} +6.18838 q^{64} +4.88811 q^{65} -9.11713 q^{67} +7.59111 q^{68} +2.71325 q^{70} +9.34335 q^{71} -0.0512243 q^{73} +1.30034 q^{74} -5.48857 q^{76} -1.15588 q^{77} +0.244588 q^{79} +0.569823 q^{80} +7.69490 q^{82} -13.1293 q^{83} +6.87104 q^{85} -3.17521 q^{86} -1.18407 q^{88} -1.00000 q^{89} -14.0175 q^{91} +4.69098 q^{92} -4.26921 q^{94} -4.96794 q^{95} +14.2848 q^{97} -1.15764 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) $ 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8	$ 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) $ 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8 + 6 * q^10 + 10 * q^11 + 7 * q^13 + 7 * q^14 + 22 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^20 + 8 * q^22 - 6 * q^23 + 10 * q^25 + 14 * q^26 + 16 * q^28 + 3 * q^29 + 12 * q^31 - 21 * q^32 - 6 * q^34 - 7 * q^35 - 19 * q^37 - 6 * q^38 + 15 * q^40 - 13 * q^41 + 9 * q^43 + 26 * q^44 + 8 * q^46 - 21 * q^47 + 53 * q^49 - 6 * q^50 - 43 * q^52 - 7 * q^53 - 10 * q^55 + 53 * q^56 - 42 * q^58 + 19 * q^59 + 4 * q^61 + 28 * q^62 + 5 * q^64 - 7 * q^65 - 6 * q^67 - 2 * q^68 - 7 * q^70 + 6 * q^71 + 6 * q^73 - 2 * q^76 + 40 * q^77 + 25 * q^79 - 22 * q^80 + q^82 + 22 * q^83 + 11 * q^85 + 2 * q^86 + 20 * q^88 - 10 * q^89 - 10 * q^91 - 10 * q^92 + 25 * q^94 - 10 * q^95 + 40 * q^97 + 56 * 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 $

$2$	−0.946151	−0.669030	−0.334515	−	0.942390i	$-0.608572\pi$
$2$	−0.946151	−0.669030	−0.334515	+	0.942390i	$0.608572\pi$
$3$	0	0
$3$	0	0
$4$	−1.10480	−0.552399
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.86767	1.08388	0.541939	−	0.840418i	$-0.317691\pi$
$7$	2.86767	1.08388	0.541939	+	0.840418i	$0.317691\pi$
$8$	2.93761	1.03860
$9$	0	0
$10$	0.946151	0.299199
$11$	−0.403071	−0.121531	−0.0607653	−	0.998152i	$-0.519354\pi$
$11$	−0.403071	−0.121531	−0.0607653	+	0.998152i	$0.519354\pi$
$12$	0	0
$13$	−4.88811	−1.35572	−0.677859	−	0.735192i	$-0.737092\pi$
$13$	−4.88811	−1.35572	−0.677859	+	0.735192i	$0.737092\pi$
$14$	−2.71325	−0.725146
$15$	0	0
$16$	−0.569823	−0.142456
$17$	−6.87104	−1.66647	−0.833236	−	0.552918i	$-0.813514\pi$
$17$	−6.87104	−1.66647	−0.833236	+	0.552918i	$0.813514\pi$
$18$	0	0
$19$	4.96794	1.13972	0.569861	−	0.821741i	$-0.306997\pi$
$19$	4.96794	1.13972	0.569861	+	0.821741i	$0.306997\pi$
$20$	1.10480	0.247040
$21$	0	0
$22$	0.381366	0.0813075
$23$	−4.24601	−0.885354	−0.442677	−	0.896681i	$-0.645971\pi$
$23$	−4.24601	−0.885354	−0.442677	+	0.896681i	$0.645971\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.62489	0.907015
$27$	0	0
$28$	−3.16820	−0.598733
$29$	7.26323	1.34875	0.674374	−	0.738390i	$-0.264414\pi$
$29$	7.26323	1.34875	0.674374	+	0.738390i	$0.264414\pi$
$30$	0	0
$31$	7.51228	1.34924	0.674622	−	0.738163i	$-0.264306\pi$
$31$	7.51228	1.34924	0.674622	+	0.738163i	$0.264306\pi$
$32$	−5.33608	−0.943294
$33$	0	0
$34$	6.50104	1.11492
$35$	−2.86767	−0.484725
$36$	0	0
$37$	−1.37435	−0.225942	−0.112971	−	0.993598i	$-0.536037\pi$
$37$	−1.37435	−0.225942	−0.112971	+	0.993598i	$0.536037\pi$
$38$	−4.70042	−0.762508
$39$	0	0
$40$	−2.93761	−0.464477
$41$	−8.13284	−1.27014	−0.635068	−	0.772456i	$-0.719028\pi$
$41$	−8.13284	−1.27014	−0.635068	+	0.772456i	$0.719028\pi$
$42$	0	0
$43$	3.35592	0.511773	0.255887	−	0.966707i	$-0.417633\pi$
$43$	3.35592	0.511773	0.255887	+	0.966707i	$0.417633\pi$
$44$	0.445313	0.0671334
$45$	0	0
$46$	4.01736	0.592328
$47$	4.51219	0.658171	0.329085	−	0.944300i	$-0.393260\pi$
$47$	4.51219	0.658171	0.329085	+	0.944300i	$0.393260\pi$
$48$	0	0
$49$	1.22353	0.174790
$50$	−0.946151	−0.133806
$51$	0	0
$52$	5.40038	0.748897
$53$	11.5380	1.58487	0.792436	−	0.609955i	$-0.208813\pi$
$53$	11.5380	1.58487	0.792436	+	0.609955i	$0.208813\pi$
$54$	0	0
$55$	0.403071	0.0543501
$56$	8.42409	1.12572
$57$	0	0
$58$	−6.87211	−0.902352
$59$	−8.09315	−1.05364	−0.526819	−	0.849978i	$-0.676615\pi$
$59$	−8.09315	−1.05364	−0.526819	+	0.849978i	$0.676615\pi$
$60$	0	0
$61$	−4.94084	−0.632610	−0.316305	−	0.948658i	$-0.602442\pi$
$61$	−4.94084	−0.632610	−0.316305	+	0.948658i	$0.602442\pi$
$62$	−7.10775	−0.902685
$63$	0	0
$64$	6.18838	0.773547
$65$	4.88811	0.606295
$66$	0	0
$67$	−9.11713	−1.11383	−0.556917	−	0.830568i	$-0.688016\pi$
$67$	−9.11713	−1.11383	−0.556917	+	0.830568i	$0.688016\pi$
$68$	7.59111	0.920558
$69$	0	0
$70$	2.71325	0.324295
$71$	9.34335	1.10885	0.554426	−	0.832233i	$-0.312938\pi$
$71$	9.34335	1.10885	0.554426	+	0.832233i	$0.312938\pi$
$72$	0	0
$73$	−0.0512243	−0.00599534	−0.00299767	−	0.999996i	$-0.500954\pi$
$73$	−0.0512243	−0.00599534	−0.00299767	+	0.999996i	$0.500954\pi$
$74$	1.30034	0.151162
$75$	0	0
$76$	−5.48857	−0.629582
$77$	−1.15588	−0.131724
$78$	0	0
$79$	0.244588	0.0275183	0.0137592	−	0.999905i	$-0.495620\pi$
$79$	0.244588	0.0275183	0.0137592	+	0.999905i	$0.495620\pi$
$80$	0.569823	0.0637081
$81$	0	0
$82$	7.69490	0.849759
$83$	−13.1293	−1.44113	−0.720564	−	0.693389i	$-0.756117\pi$
$83$	−13.1293	−1.44113	−0.720564	+	0.693389i	$0.756117\pi$
$84$	0	0
$85$	6.87104	0.745269
$86$	−3.17521	−0.342392
$87$	0	0
$88$	−1.18407	−0.126222
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−14.0175	−1.46943
$92$	4.69098	0.489069
$93$	0	0
$94$	−4.26921	−0.440336
$95$	−4.96794	−0.509699
$96$	0	0
$97$	14.2848	1.45040	0.725200	−	0.688539i	$-0.241747\pi$
$97$	14.2848	1.45040	0.725200	+	0.688539i	$0.241747\pi$
$98$	−1.15764	−0.116940
$99$	0	0
$100$	−1.10480	−0.110480
$101$	15.3685	1.52923	0.764613	−	0.644490i	$-0.222930\pi$
$101$	15.3685	1.52923	0.764613	+	0.644490i	$0.222930\pi$
$102$	0	0
$103$	10.8757	1.07161	0.535805	−	0.844341i	$-0.320008\pi$
$103$	10.8757	1.07161	0.535805	+	0.844341i	$0.320008\pi$
$104$	−14.3593	−1.40805
$105$	0	0
$106$	−10.9167	−1.06033
$107$	−12.3748	−1.19631	−0.598157	−	0.801379i	$-0.704100\pi$
$107$	−12.3748	−1.19631	−0.598157	+	0.801379i	$0.704100\pi$
$108$	0	0
$109$	12.6258	1.20934	0.604668	−	0.796478i	$-0.293306\pi$
$109$	12.6258	1.20934	0.604668	+	0.796478i	$0.293306\pi$
$110$	−0.381366	−0.0363618
$111$	0	0
$112$	−1.63406	−0.154404
$113$	14.5337	1.36722	0.683609	−	0.729848i	$-0.260409\pi$
$113$	14.5337	1.36722	0.683609	+	0.729848i	$0.260409\pi$
$114$	0	0
$115$	4.24601	0.395942
$116$	−8.02440	−0.745047
$117$	0	0
$118$	7.65734	0.704915
$119$	−19.7039	−1.80625
$120$	0	0
$121$	−10.8375	−0.985230
$122$	4.67478	0.423235
$123$	0	0
$124$	−8.29955	−0.745322
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.62219	−0.232682	−0.116341	−	0.993209i	$-0.537117\pi$
$127$	−2.62219	−0.232682	−0.116341	+	0.993209i	$0.537117\pi$
$128$	4.81702	0.425768
$129$	0	0
$130$	−4.62489	−0.405629
$131$	9.03755	0.789614	0.394807	−	0.918764i	$-0.370811\pi$
$131$	9.03755	0.789614	0.394807	+	0.918764i	$0.370811\pi$
$132$	0	0
$133$	14.2464	1.23532
$134$	8.62618	0.745189
$135$	0	0
$136$	−20.1844	−1.73080
$137$	−20.6482	−1.76410	−0.882049	−	0.471157i	$-0.843836\pi$
$137$	−20.6482	−1.76410	−0.882049	+	0.471157i	$0.843836\pi$
$138$	0	0
$139$	−13.8417	−1.17404	−0.587020	−	0.809572i	$-0.699699\pi$
$139$	−13.8417	−1.17404	−0.587020	+	0.809572i	$0.699699\pi$
$140$	3.16820	0.267762
$141$	0	0
$142$	−8.84022	−0.741854
$143$	1.97026	0.164761
$144$	0	0
$145$	−7.26323	−0.603178
$146$	0.0484659	0.00401106
$147$	0	0
$148$	1.51838	0.124810
$149$	18.3887	1.50646	0.753229	−	0.657758i	$-0.228495\pi$
$149$	18.3887	1.50646	0.753229	+	0.657758i	$0.228495\pi$
$150$	0	0
$151$	−19.3290	−1.57297	−0.786485	−	0.617610i	$-0.788101\pi$
$151$	−19.3290	−1.57297	−0.786485	+	0.617610i	$0.788101\pi$
$152$	14.5938	1.18372
$153$	0	0
$154$	1.09363	0.0881274
$155$	−7.51228	−0.603401
$156$	0	0
$157$	−11.6925	−0.933164	−0.466582	−	0.884478i	$-0.654515\pi$
$157$	−11.6925	−0.933164	−0.466582	+	0.884478i	$0.654515\pi$
$158$	−0.231418	−0.0184106
$159$	0	0
$160$	5.33608	0.421854
$161$	−12.1762	−0.959615
$162$	0	0
$163$	−23.1674	−1.81461	−0.907304	−	0.420475i	$-0.861863\pi$
$163$	−23.1674	−1.81461	−0.907304	+	0.420475i	$0.861863\pi$
$164$	8.98516	0.701623
$165$	0	0
$166$	12.4223	0.964157
$167$	6.18653	0.478728	0.239364	−	0.970930i	$-0.423061\pi$
$167$	6.18653	0.478728	0.239364	+	0.970930i	$0.423061\pi$
$168$	0	0
$169$	10.8936	0.837970
$170$	−6.50104	−0.498607
$171$	0	0
$172$	−3.70762	−0.282703
$173$	21.2626	1.61657	0.808283	−	0.588794i	$-0.200397\pi$
$173$	21.2626	1.61657	0.808283	+	0.588794i	$0.200397\pi$
$174$	0	0
$175$	2.86767	0.216775
$176$	0.229679	0.0173127
$177$	0	0
$178$	0.946151	0.0709170
$179$	−0.934252	−0.0698293	−0.0349146	−	0.999390i	$-0.511116\pi$
$179$	−0.934252	−0.0698293	−0.0349146	+	0.999390i	$0.511116\pi$
$180$	0	0
$181$	11.4131	0.848328	0.424164	−	0.905585i	$-0.360568\pi$
$181$	11.4131	0.848328	0.424164	+	0.905585i	$0.360568\pi$
$182$	13.2627	0.983093
$183$	0	0
$184$	−12.4731	−0.919530
$185$	1.37435	0.101044
$186$	0	0
$187$	2.76952	0.202527
$188$	−4.98506	−0.363573
$189$	0	0
$190$	4.70042	0.341004
$191$	14.8971	1.07792	0.538959	−	0.842332i	$-0.318818\pi$
$191$	14.8971	1.07792	0.538959	+	0.842332i	$0.318818\pi$
$192$	0	0
$193$	5.71543	0.411406	0.205703	−	0.978614i	$-0.434052\pi$
$193$	5.71543	0.411406	0.205703	+	0.978614i	$0.434052\pi$
$194$	−13.5156	−0.970360
$195$	0	0
$196$	−1.35175	−0.0965539
$197$	13.1229	0.934965	0.467483	−	0.884002i	$-0.345161\pi$
$197$	13.1229	0.934965	0.467483	+	0.884002i	$0.345161\pi$
$198$	0	0
$199$	11.2996	0.801010	0.400505	−	0.916294i	$-0.368835\pi$
$199$	11.2996	0.801010	0.400505	+	0.916294i	$0.368835\pi$
$200$	2.93761	0.207720
$201$	0	0
$202$	−14.5410	−1.02310
$203$	20.8285	1.46188
$204$	0	0
$205$	8.13284	0.568022
$206$	−10.2900	−0.716940
$207$	0	0
$208$	2.78535	0.193130
$209$	−2.00243	−0.138511
$210$	0	0
$211$	13.5310	0.931511	0.465755	−	0.884913i	$-0.345783\pi$
$211$	13.5310	0.931511	0.465755	+	0.884913i	$0.345783\pi$
$212$	−12.7472	−0.875482
$213$	0	0
$214$	11.7084	0.800370
$215$	−3.35592	−0.228872
$216$	0	0
$217$	21.5427	1.46242
$218$	−11.9459	−0.809081
$219$	0	0
$220$	−0.445313	−0.0300230
$221$	33.5864	2.25926
$222$	0	0
$223$	26.1408	1.75051	0.875257	−	0.483658i	$-0.160692\pi$
$223$	26.1408	1.75051	0.875257	+	0.483658i	$0.160692\pi$
$224$	−15.3021	−1.02242
$225$	0	0
$226$	−13.7511	−0.914709
$227$	18.8086	1.24837	0.624187	−	0.781275i	$-0.285431\pi$
$227$	18.8086	1.24837	0.624187	+	0.781275i	$0.285431\pi$
$228$	0	0
$229$	−9.43438	−0.623441	−0.311721	−	0.950174i	$-0.600905\pi$
$229$	−9.43438	−0.623441	−0.311721	+	0.950174i	$0.600905\pi$
$230$	−4.01736	−0.264897
$231$	0	0
$232$	21.3365	1.40081
$233$	20.7725	1.36085	0.680425	−	0.732818i	$-0.261795\pi$
$233$	20.7725	1.36085	0.680425	+	0.732818i	$0.261795\pi$
$234$	0	0
$235$	−4.51219	−0.294343
$236$	8.94130	0.582029
$237$	0	0
$238$	18.6428	1.20844
$239$	4.68057	0.302761	0.151380	−	0.988476i	$-0.451628\pi$
$239$	4.68057	0.302761	0.151380	+	0.988476i	$0.451628\pi$
$240$	0	0
$241$	25.0182	1.61156	0.805781	−	0.592214i	$-0.201746\pi$
$241$	25.0182	1.61156	0.805781	+	0.592214i	$0.201746\pi$
$242$	10.2539	0.659148
$243$	0	0
$244$	5.45864	0.349453
$245$	−1.22353	−0.0781685
$246$	0	0
$247$	−24.2838	−1.54514
$248$	22.0681	1.40133
$249$	0	0
$250$	0.946151	0.0598398
$251$	21.6414	1.36600	0.682998	−	0.730421i	$-0.260676\pi$
$251$	21.6414	1.36600	0.682998	+	0.730421i	$0.260676\pi$
$252$	0	0
$253$	1.71144	0.107598
$254$	2.48099	0.155671
$255$	0	0
$256$	−16.9344	−1.05840
$257$	19.1317	1.19340	0.596702	−	0.802463i	$-0.296477\pi$
$257$	19.1317	1.19340	0.596702	+	0.802463i	$0.296477\pi$
$258$	0	0
$259$	−3.94118	−0.244893
$260$	−5.40038	−0.334917
$261$	0	0
$262$	−8.55088	−0.528275
$263$	24.5633	1.51464	0.757320	−	0.653044i	$-0.226508\pi$
$263$	24.5633	1.51464	0.757320	+	0.653044i	$0.226508\pi$
$264$	0	0
$265$	−11.5380	−0.708776
$266$	−13.4792	−0.826465
$267$	0	0
$268$	10.0726	0.615282
$269$	11.9477	0.728466	0.364233	−	0.931308i	$-0.381331\pi$
$269$	11.9477	0.728466	0.364233	+	0.931308i	$0.381331\pi$
$270$	0	0
$271$	18.1681	1.10363	0.551816	−	0.833966i	$-0.313935\pi$
$271$	18.1681	1.10363	0.551816	+	0.833966i	$0.313935\pi$
$272$	3.91527	0.237398
$273$	0	0
$274$	19.5364	1.18023
$275$	−0.403071	−0.0243061
$276$	0	0
$277$	−12.5983	−0.756956	−0.378478	−	0.925610i	$-0.623552\pi$
$277$	−12.5983	−0.756956	−0.378478	+	0.925610i	$0.623552\pi$
$278$	13.0964	0.785468
$279$	0	0
$280$	−8.42409	−0.503436
$281$	−23.9903	−1.43114	−0.715570	−	0.698541i	$-0.753833\pi$
$281$	−23.9903	−1.43114	−0.715570	+	0.698541i	$0.753833\pi$
$282$	0	0
$283$	9.57047	0.568905	0.284453	−	0.958690i	$-0.408188\pi$
$283$	9.57047	0.568905	0.284453	+	0.958690i	$0.408188\pi$
$284$	−10.3225	−0.612529
$285$	0	0
$286$	−1.86416	−0.110230
$287$	−23.3223	−1.37667
$288$	0	0
$289$	30.2112	1.77713
$290$	6.87211	0.403544
$291$	0	0
$292$	0.0565925	0.00331182
$293$	3.73291	0.218079	0.109040	−	0.994037i	$-0.465222\pi$
$293$	3.73291	0.218079	0.109040	+	0.994037i	$0.465222\pi$
$294$	0	0
$295$	8.09315	0.471201
$296$	−4.03730	−0.234663
$297$	0	0
$298$	−17.3985	−1.00787
$299$	20.7550	1.20029
$300$	0	0
$301$	9.62368	0.554700
$302$	18.2881	1.05236
$303$	0	0
$304$	−2.83084	−0.162360
$305$	4.94084	0.282912
$306$	0	0
$307$	4.84228	0.276363	0.138182	−	0.990407i	$-0.455874\pi$
$307$	4.84228	0.276363	0.138182	+	0.990407i	$0.455874\pi$
$308$	1.27701	0.0727644
$309$	0	0
$310$	7.10775	0.403693
$311$	−22.1048	−1.25345	−0.626724	−	0.779241i	$-0.715605\pi$
$311$	−22.1048	−1.25345	−0.626724	+	0.779241i	$0.715605\pi$
$312$	0	0
$313$	−17.8746	−1.01033	−0.505166	−	0.863022i	$-0.668569\pi$
$313$	−17.8746	−1.01033	−0.505166	+	0.863022i	$0.668569\pi$
$314$	11.0629	0.624314
$315$	0	0
$316$	−0.270221	−0.0152011
$317$	−16.9094	−0.949725	−0.474862	−	0.880060i	$-0.657502\pi$
$317$	−16.9094	−0.949725	−0.474862	+	0.880060i	$0.657502\pi$
$318$	0	0
$319$	−2.92760	−0.163914
$320$	−6.18838	−0.345941
$321$	0	0
$322$	11.5205	0.642011
$323$	−34.1349	−1.89932
$324$	0	0
$325$	−4.88811	−0.271143
$326$	21.9198	1.21403
$327$	0	0
$328$	−23.8911	−1.31917
$329$	12.9395	0.713376
$330$	0	0
$331$	24.1452	1.32714	0.663571	−	0.748114i	$-0.269040\pi$
$331$	24.1452	1.32714	0.663571	+	0.748114i	$0.269040\pi$
$332$	14.5052	0.796078
$333$	0	0
$334$	−5.85339	−0.320283
$335$	9.11713	0.498122
$336$	0	0
$337$	−31.0851	−1.69331	−0.846656	−	0.532140i	$-0.821388\pi$
$337$	−31.0851	−1.69331	−0.846656	+	0.532140i	$0.821388\pi$
$338$	−10.3070	−0.560626
$339$	0	0
$340$	−7.59111	−0.411686
$341$	−3.02798	−0.163974
$342$	0	0
$343$	−16.5650	−0.894426
$344$	9.85838	0.531528
$345$	0	0
$346$	−20.1176	−1.08153
$347$	1.73542	0.0931624	0.0465812	−	0.998915i	$-0.485167\pi$
$347$	1.73542	0.0931624	0.0465812	+	0.998915i	$0.485167\pi$
$348$	0	0
$349$	15.1429	0.810583	0.405292	−	0.914187i	$-0.367170\pi$
$349$	15.1429	0.810583	0.405292	+	0.914187i	$0.367170\pi$
$350$	−2.71325	−0.145029
$351$	0	0
$352$	2.15082	0.114639
$353$	−4.82900	−0.257022	−0.128511	−	0.991708i	$-0.541020\pi$
$353$	−4.82900	−0.257022	−0.128511	+	0.991708i	$0.541020\pi$
$354$	0	0
$355$	−9.34335	−0.495893
$356$	1.10480	0.0585542
$357$	0	0
$358$	0.883943	0.0467179
$359$	−2.89861	−0.152983	−0.0764915	−	0.997070i	$-0.524372\pi$
$359$	−2.89861	−0.152983	−0.0764915	+	0.997070i	$0.524372\pi$
$360$	0	0
$361$	5.68038	0.298968
$362$	−10.7985	−0.567557
$363$	0	0
$364$	15.4865	0.811713
$365$	0.0512243	0.00268120
$366$	0	0
$367$	3.52414	0.183959	0.0919793	−	0.995761i	$-0.470681\pi$
$367$	3.52414	0.183959	0.0919793	+	0.995761i	$0.470681\pi$
$368$	2.41947	0.126124
$369$	0	0
$370$	−1.30034	−0.0676016
$371$	33.0873	1.71781
$372$	0	0
$373$	21.3475	1.10533	0.552666	−	0.833403i	$-0.313611\pi$
$373$	21.3475	1.10533	0.552666	+	0.833403i	$0.313611\pi$
$374$	−2.62038	−0.135497
$375$	0	0
$376$	13.2550	0.683577
$377$	−35.5034	−1.82852
$378$	0	0
$379$	14.7869	0.759551	0.379775	−	0.925079i	$-0.376001\pi$
$379$	14.7869	0.759551	0.379775	+	0.925079i	$0.376001\pi$
$380$	5.48857	0.281558
$381$	0	0
$382$	−14.0949	−0.721159
$383$	17.9044	0.914872	0.457436	−	0.889242i	$-0.348768\pi$
$383$	17.9044	0.914872	0.457436	+	0.889242i	$0.348768\pi$
$384$	0	0
$385$	1.15588	0.0589089
$386$	−5.40766	−0.275243
$387$	0	0
$388$	−15.7818	−0.801200
$389$	−1.62374	−0.0823270	−0.0411635	−	0.999152i	$-0.513106\pi$
$389$	−1.62374	−0.0823270	−0.0411635	+	0.999152i	$0.513106\pi$
$390$	0	0
$391$	29.1745	1.47542
$392$	3.59425	0.181537
$393$	0	0
$394$	−12.4162	−0.625519
$395$	−0.244588	−0.0123066
$396$	0	0
$397$	18.8547	0.946293	0.473146	−	0.880984i	$-0.343118\pi$
$397$	18.8547	0.946293	0.473146	+	0.880984i	$0.343118\pi$
$398$	−10.6912	−0.535900
$399$	0	0
$400$	−0.569823	−0.0284911
$401$	28.0476	1.40063	0.700316	−	0.713833i	$-0.253042\pi$
$401$	28.0476	1.40063	0.700316	+	0.713833i	$0.253042\pi$
$402$	0	0
$403$	−36.7208	−1.82919
$404$	−16.9791	−0.844744
$405$	0	0
$406$	−19.7069	−0.978039
$407$	0.553961	0.0274588
$408$	0	0
$409$	26.9099	1.33061	0.665305	−	0.746572i	$-0.268302\pi$
$409$	26.9099	1.33061	0.665305	+	0.746572i	$0.268302\pi$
$410$	−7.69490	−0.380024
$411$	0	0
$412$	−12.0154	−0.591957
$413$	−23.2085	−1.14201
$414$	0	0
$415$	13.1293	0.644492
$416$	26.0833	1.27884
$417$	0	0
$418$	1.89460	0.0926680
$419$	17.1506	0.837862	0.418931	−	0.908018i	$-0.362405\pi$
$419$	17.1506	0.837862	0.418931	+	0.908018i	$0.362405\pi$
$420$	0	0
$421$	31.6248	1.54130	0.770648	−	0.637261i	$-0.219933\pi$
$421$	31.6248	1.54130	0.770648	+	0.637261i	$0.219933\pi$
$422$	−12.8023	−0.623208
$423$	0	0
$424$	33.8942	1.64605
$425$	−6.87104	−0.333294
$426$	0	0
$427$	−14.1687	−0.685672
$428$	13.6716	0.660843
$429$	0	0
$430$	3.17521	0.153122
$431$	−13.9644	−0.672641	−0.336320	−	0.941748i	$-0.609182\pi$
$431$	−13.9644	−0.672641	−0.336320	+	0.941748i	$0.609182\pi$
$432$	0	0
$433$	17.3295	0.832802	0.416401	−	0.909181i	$-0.363291\pi$
$433$	17.3295	0.832802	0.416401	+	0.909181i	$0.363291\pi$
$434$	−20.3827	−0.978399
$435$	0	0
$436$	−13.9490	−0.668036
$437$	−21.0939	−1.00906
$438$	0	0
$439$	−2.30828	−0.110168	−0.0550842	−	0.998482i	$-0.517543\pi$
$439$	−2.30828	−0.110168	−0.0550842	+	0.998482i	$0.517543\pi$
$440$	1.18407	0.0564481
$441$	0	0
$442$	−31.7778	−1.51152
$443$	−1.91195	−0.0908393	−0.0454197	−	0.998968i	$-0.514462\pi$
$443$	−1.91195	−0.0908393	−0.0454197	+	0.998968i	$0.514462\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−24.7331	−1.17115
$447$	0	0
$448$	17.7462	0.838430
$449$	−3.23368	−0.152607	−0.0763033	−	0.997085i	$-0.524312\pi$
$449$	−3.23368	−0.152607	−0.0763033	+	0.997085i	$0.524312\pi$
$450$	0	0
$451$	3.27812	0.154360
$452$	−16.0568	−0.755250
$453$	0	0
$454$	−17.7958	−0.835199
$455$	14.0175	0.657150
$456$	0	0
$457$	−24.2883	−1.13616	−0.568080	−	0.822973i	$-0.692314\pi$
$457$	−24.2883	−1.13616	−0.568080	+	0.822973i	$0.692314\pi$
$458$	8.92635	0.417101
$459$	0	0
$460$	−4.69098	−0.218718
$461$	−13.2919	−0.619065	−0.309533	−	0.950889i	$-0.600173\pi$
$461$	−13.2919	−0.619065	−0.309533	+	0.950889i	$0.600173\pi$
$462$	0	0
$463$	−15.5892	−0.724493	−0.362247	−	0.932082i	$-0.617990\pi$
$463$	−15.5892	−0.724493	−0.362247	+	0.932082i	$0.617990\pi$
$464$	−4.13875	−0.192137
$465$	0	0
$466$	−19.6539	−0.910449
$467$	−25.8755	−1.19737	−0.598687	−	0.800983i	$-0.704311\pi$
$467$	−25.8755	−1.19737	−0.598687	+	0.800983i	$0.704311\pi$
$468$	0	0
$469$	−26.1449	−1.20726
$470$	4.26921	0.196924
$471$	0	0
$472$	−23.7745	−1.09431
$473$	−1.35268	−0.0621961
$474$	0	0
$475$	4.96794	0.227945
$476$	21.7688	0.997772
$477$	0	0
$478$	−4.42852	−0.202556
$479$	−12.9529	−0.591833	−0.295916	−	0.955214i	$-0.595625\pi$
$479$	−12.9529	−0.591833	−0.295916	+	0.955214i	$0.595625\pi$
$480$	0	0
$481$	6.71797	0.306313
$482$	−23.6709	−1.07818
$483$	0	0
$484$	11.9733	0.544241
$485$	−14.2848	−0.648638
$486$	0	0
$487$	−3.99951	−0.181235	−0.0906176	−	0.995886i	$-0.528884\pi$
$487$	−3.99951	−0.181235	−0.0906176	+	0.995886i	$0.528884\pi$
$488$	−14.5143	−0.657030
$489$	0	0
$490$	1.15764	0.0522970
$491$	29.8079	1.34521	0.672606	−	0.740001i	$-0.265175\pi$
$491$	29.8079	1.34521	0.672606	+	0.740001i	$0.265175\pi$
$492$	0	0
$493$	−49.9059	−2.24765
$494$	22.9761	1.03375
$495$	0	0
$496$	−4.28066	−0.192208
$497$	26.7936	1.20186
$498$	0	0
$499$	−43.4586	−1.94547	−0.972737	−	0.231911i	$-0.925502\pi$
$499$	−43.4586	−1.94547	−0.972737	+	0.231911i	$0.925502\pi$
$500$	1.10480	0.0494081
$501$	0	0
$502$	−20.4761	−0.913891
$503$	−17.1011	−0.762499	−0.381249	−	0.924472i	$-0.624506\pi$
$503$	−17.1011	−0.762499	−0.381249	+	0.924472i	$0.624506\pi$
$504$	0	0
$505$	−15.3685	−0.683891
$506$	−1.61928	−0.0719860
$507$	0	0
$508$	2.89699	0.128533
$509$	−12.4986	−0.553990	−0.276995	−	0.960871i	$-0.589338\pi$
$509$	−12.4986	−0.553990	−0.276995	+	0.960871i	$0.589338\pi$
$510$	0	0
$511$	−0.146894	−0.00649822
$512$	6.38845	0.282332
$513$	0	0
$514$	−18.1015	−0.798423
$515$	−10.8757	−0.479239
$516$	0	0
$517$	−1.81873	−0.0799878
$518$	3.72895	0.163841
$519$	0	0
$520$	14.3593	0.629699
$521$	−4.54637	−0.199180	−0.0995901	−	0.995029i	$-0.531753\pi$
$521$	−4.54637	−0.199180	−0.0995901	+	0.995029i	$0.531753\pi$
$522$	0	0
$523$	28.1405	1.23050	0.615249	−	0.788333i	$-0.289056\pi$
$523$	28.1405	1.23050	0.615249	+	0.788333i	$0.289056\pi$
$524$	−9.98467	−0.436182
$525$	0	0
$526$	−23.2406	−1.01334
$527$	−51.6171	−2.24848
$528$	0	0
$529$	−4.97141	−0.216148
$530$	10.9167	0.474192
$531$	0	0
$532$	−15.7394	−0.682390
$533$	39.7542	1.72195
$534$	0	0
$535$	12.3748	0.535008
$536$	−26.7826	−1.15683
$537$	0	0
$538$	−11.3044	−0.487366
$539$	−0.493170	−0.0212423
$540$	0	0
$541$	15.2392	0.655183	0.327592	−	0.944819i	$-0.393763\pi$
$541$	15.2392	0.655183	0.327592	+	0.944819i	$0.393763\pi$
$542$	−17.1897	−0.738362
$543$	0	0
$544$	36.6644	1.57197
$545$	−12.6258	−0.540831
$546$	0	0
$547$	−28.4760	−1.21755	−0.608774	−	0.793344i	$-0.708338\pi$
$547$	−28.4760	−1.21755	−0.608774	+	0.793344i	$0.708338\pi$
$548$	22.8122	0.974487
$549$	0	0
$550$	0.381366	0.0162615
$551$	36.0832	1.53720
$552$	0	0
$553$	0.701399	0.0298265
$554$	11.9199	0.506426
$555$	0	0
$556$	15.2923	0.648539
$557$	−12.7912	−0.541981	−0.270991	−	0.962582i	$-0.587351\pi$
$557$	−12.7912	−0.541981	−0.270991	+	0.962582i	$0.587351\pi$
$558$	0	0
$559$	−16.4041	−0.693820
$560$	1.63406	0.0690518
$561$	0	0
$562$	22.6984	0.957476
$563$	28.1865	1.18792	0.593959	−	0.804495i	$-0.297564\pi$
$563$	28.1865	1.18792	0.593959	+	0.804495i	$0.297564\pi$
$564$	0	0
$565$	−14.5337	−0.611438
$566$	−9.05511	−0.380615
$567$	0	0
$568$	27.4471	1.15165
$569$	−17.3573	−0.727657	−0.363828	−	0.931466i	$-0.618531\pi$
$569$	−17.3573	−0.727657	−0.363828	+	0.931466i	$0.618531\pi$
$570$	0	0
$571$	30.4265	1.27331	0.636654	−	0.771149i	$-0.280318\pi$
$571$	30.4265	1.27331	0.636654	+	0.771149i	$0.280318\pi$
$572$	−2.17674	−0.0910139
$573$	0	0
$574$	22.0664	0.921035
$575$	−4.24601	−0.177071
$576$	0	0
$577$	13.4095	0.558244	0.279122	−	0.960256i	$-0.409957\pi$
$577$	13.4095	0.558244	0.279122	+	0.960256i	$0.409957\pi$
$578$	−28.5843	−1.18895
$579$	0	0
$580$	8.02440	0.333195
$581$	−37.6505	−1.56200
$582$	0	0
$583$	−4.65065	−0.192610
$584$	−0.150477	−0.00622677
$585$	0	0
$586$	−3.53190	−0.145901
$587$	17.2768	0.713091	0.356546	−	0.934278i	$-0.383954\pi$
$587$	17.2768	0.713091	0.356546	+	0.934278i	$0.383954\pi$
$588$	0	0
$589$	37.3205	1.53776
$590$	−7.65734	−0.315248
$591$	0	0
$592$	0.783136	0.0321867
$593$	29.5353	1.21287	0.606436	−	0.795133i	$-0.292599\pi$
$593$	29.5353	1.21287	0.606436	+	0.795133i	$0.292599\pi$
$594$	0	0
$595$	19.7039	0.807780
$596$	−20.3158	−0.832167
$597$	0	0
$598$	−19.6373	−0.803029
$599$	−11.0013	−0.449503	−0.224751	−	0.974416i	$-0.572157\pi$
$599$	−11.0013	−0.449503	−0.224751	+	0.974416i	$0.572157\pi$
$600$	0	0
$601$	44.4792	1.81434	0.907172	−	0.420761i	$-0.138237\pi$
$601$	44.4792	1.81434	0.907172	+	0.420761i	$0.138237\pi$
$602$	−9.10545	−0.371110
$603$	0	0
$604$	21.3546	0.868907
$605$	10.8375	0.440608
$606$	0	0
$607$	21.5711	0.875542	0.437771	−	0.899086i	$-0.355768\pi$
$607$	21.5711	0.875542	0.437771	+	0.899086i	$0.355768\pi$
$608$	−26.5093	−1.07509
$609$	0	0
$610$	−4.67478	−0.189276
$611$	−22.0561	−0.892293
$612$	0	0
$613$	−15.2248	−0.614924	−0.307462	−	0.951560i	$-0.599480\pi$
$613$	−15.2248	−0.614924	−0.307462	+	0.951560i	$0.599480\pi$
$614$	−4.58152	−0.184895
$615$	0	0
$616$	−3.39551	−0.136809
$617$	33.4861	1.34810	0.674049	−	0.738686i	$-0.264553\pi$
$617$	33.4861	1.34810	0.674049	+	0.738686i	$0.264553\pi$
$618$	0	0
$619$	−9.55487	−0.384043	−0.192021	−	0.981391i	$-0.561504\pi$
$619$	−9.55487	−0.384043	−0.192021	+	0.981391i	$0.561504\pi$
$620$	8.29955	0.333318
$621$	0	0
$622$	20.9145	0.838594
$623$	−2.86767	−0.114891
$624$	0	0
$625$	1.00000	0.0400000
$626$	16.9121	0.675942
$627$	0	0
$628$	12.9179	0.515479
$629$	9.44321	0.376526
$630$	0	0
$631$	−33.4661	−1.33227	−0.666133	−	0.745833i	$-0.732052\pi$
$631$	−33.4661	−1.33227	−0.666133	+	0.745833i	$0.732052\pi$
$632$	0.718505	0.0285806
$633$	0	0
$634$	15.9988	0.635394
$635$	2.62219	0.104058
$636$	0	0
$637$	−5.98075	−0.236966
$638$	2.76995	0.109663
$639$	0	0
$640$	−4.81702	−0.190409
$641$	−8.13291	−0.321231	−0.160615	−	0.987017i	$-0.551348\pi$
$641$	−8.13291	−0.321231	−0.160615	+	0.987017i	$0.551348\pi$
$642$	0	0
$643$	9.71398	0.383082	0.191541	−	0.981485i	$-0.438651\pi$
$643$	9.71398	0.383082	0.191541	+	0.981485i	$0.438651\pi$
$644$	13.4522	0.530091
$645$	0	0
$646$	32.2967	1.27070
$647$	−11.0070	−0.432728	−0.216364	−	0.976313i	$-0.569420\pi$
$647$	−11.0070	−0.432728	−0.216364	+	0.976313i	$0.569420\pi$
$648$	0	0
$649$	3.26211	0.128049
$650$	4.62489	0.181403
$651$	0	0
$652$	25.5953	1.00239
$653$	1.19801	0.0468818	0.0234409	−	0.999725i	$-0.492538\pi$
$653$	1.19801	0.0468818	0.0234409	+	0.999725i	$0.492538\pi$
$654$	0	0
$655$	−9.03755	−0.353126
$656$	4.63428	0.180938
$657$	0	0
$658$	−12.2427	−0.477270
$659$	32.7232	1.27471	0.637357	−	0.770569i	$-0.280028\pi$
$659$	32.7232	1.27471	0.637357	+	0.770569i	$0.280028\pi$
$660$	0	0
$661$	−1.75597	−0.0682993	−0.0341497	−	0.999417i	$-0.510872\pi$
$661$	−1.75597	−0.0682993	−0.0341497	+	0.999417i	$0.510872\pi$
$662$	−22.8450	−0.887897
$663$	0	0
$664$	−38.5687	−1.49676
$665$	−14.2464	−0.552452
$666$	0	0
$667$	−30.8397	−1.19412
$668$	−6.83487	−0.264449
$669$	0	0
$670$	−8.62618	−0.333258
$671$	1.99151	0.0768815
$672$	0	0
$673$	2.45282	0.0945494	0.0472747	−	0.998882i	$-0.484946\pi$
$673$	2.45282	0.0945494	0.0472747	+	0.998882i	$0.484946\pi$
$674$	29.4112	1.13288
$675$	0	0
$676$	−12.0352	−0.462894
$677$	−39.2203	−1.50736	−0.753679	−	0.657243i	$-0.771723\pi$
$677$	−39.2203	−1.50736	−0.753679	+	0.657243i	$0.771723\pi$
$678$	0	0
$679$	40.9640	1.57206
$680$	20.1844	0.774037
$681$	0	0
$682$	2.86493	0.109704
$683$	−20.7220	−0.792903	−0.396452	−	0.918056i	$-0.629759\pi$
$683$	−20.7220	−0.792903	−0.396452	+	0.918056i	$0.629759\pi$
$684$	0	0
$685$	20.6482	0.788929
$686$	15.6730	0.598398
$687$	0	0
$688$	−1.91228	−0.0729050
$689$	−56.3992	−2.14864
$690$	0	0
$691$	−36.1025	−1.37341	−0.686703	−	0.726938i	$-0.740943\pi$
$691$	−36.1025	−1.37341	−0.686703	+	0.726938i	$0.740943\pi$
$692$	−23.4909	−0.892990
$693$	0	0
$694$	−1.64197	−0.0623284
$695$	13.8417	0.525047
$696$	0	0
$697$	55.8811	2.11665
$698$	−14.3275	−0.542304
$699$	0	0
$700$	−3.16820	−0.119747
$701$	−14.8612	−0.561298	−0.280649	−	0.959810i	$-0.590550\pi$
$701$	−14.8612	−0.561298	−0.280649	+	0.959810i	$0.590550\pi$
$702$	0	0
$703$	−6.82768	−0.257511
$704$	−2.49436	−0.0940096
$705$	0	0
$706$	4.56896	0.171955
$707$	44.0719	1.65749
$708$	0	0
$709$	22.2550	0.835805	0.417902	−	0.908492i	$-0.362765\pi$
$709$	22.2550	0.835805	0.417902	+	0.908492i	$0.362765\pi$
$710$	8.84022	0.331767
$711$	0	0
$712$	−2.93761	−0.110092
$713$	−31.8972	−1.19456
$714$	0	0
$715$	−1.97026	−0.0736834
$716$	1.03216	0.0385737
$717$	0	0
$718$	2.74253	0.102350
$719$	−25.2104	−0.940190	−0.470095	−	0.882616i	$-0.655780\pi$
$719$	−25.2104	−0.940190	−0.470095	+	0.882616i	$0.655780\pi$
$720$	0	0
$721$	31.1878	1.16149
$722$	−5.37450	−0.200018
$723$	0	0
$724$	−12.6092	−0.468616
$725$	7.26323	0.269749
$726$	0	0
$727$	−5.77942	−0.214347	−0.107173	−	0.994240i	$-0.534180\pi$
$727$	−5.77942	−0.214347	−0.107173	+	0.994240i	$0.534180\pi$
$728$	−41.1779	−1.52615
$729$	0	0
$730$	−0.0484659	−0.00179380
$731$	−23.0587	−0.852856
$732$	0	0
$733$	6.86477	0.253556	0.126778	−	0.991931i	$-0.459536\pi$
$733$	6.86477	0.253556	0.126778	+	0.991931i	$0.459536\pi$
$734$	−3.33437	−0.123074
$735$	0	0
$736$	22.6570	0.835149
$737$	3.67485	0.135365
$738$	0	0
$739$	49.7241	1.82913	0.914565	−	0.404440i	$-0.132533\pi$
$739$	49.7241	1.82913	0.914565	+	0.404440i	$0.132533\pi$
$740$	−1.51838	−0.0558168
$741$	0	0
$742$	−31.3056	−1.14926
$743$	29.9243	1.09782	0.548908	−	0.835883i	$-0.315044\pi$
$743$	29.9243	1.09782	0.548908	+	0.835883i	$0.315044\pi$
$744$	0	0
$745$	−18.3887	−0.673709
$746$	−20.1979	−0.739499
$747$	0	0
$748$	−3.05976	−0.111876
$749$	−35.4868	−1.29666
$750$	0	0
$751$	−3.99781	−0.145882	−0.0729411	−	0.997336i	$-0.523239\pi$
$751$	−3.99781	−0.145882	−0.0729411	+	0.997336i	$0.523239\pi$
$752$	−2.57115	−0.0937601
$753$	0	0
$754$	33.5916	1.22333
$755$	19.3290	0.703453
$756$	0	0
$757$	−42.7417	−1.55347	−0.776736	−	0.629826i	$-0.783126\pi$
$757$	−42.7417	−1.55347	−0.776736	+	0.629826i	$0.783126\pi$
$758$	−13.9906	−0.508162
$759$	0	0
$760$	−14.5938	−0.529374
$761$	12.5683	0.455599	0.227800	−	0.973708i	$-0.426847\pi$
$761$	12.5683	0.455599	0.227800	+	0.973708i	$0.426847\pi$
$762$	0	0
$763$	36.2067	1.31077
$764$	−16.4583	−0.595441
$765$	0	0
$766$	−16.9403	−0.612077
$767$	39.5602	1.42844
$768$	0	0
$769$	−16.4162	−0.591985	−0.295993	−	0.955190i	$-0.595650\pi$
$769$	−16.4162	−0.591985	−0.295993	+	0.955190i	$0.595650\pi$
$770$	−1.09363	−0.0394118
$771$	0	0
$772$	−6.31440	−0.227260
$773$	−11.1866	−0.402353	−0.201176	−	0.979555i	$-0.564476\pi$
$773$	−11.1866	−0.402353	−0.201176	+	0.979555i	$0.564476\pi$
$774$	0	0
$775$	7.51228	0.269849
$776$	41.9631	1.50639
$777$	0	0
$778$	1.53630	0.0550792
$779$	−40.4034	−1.44760
$780$	0	0
$781$	−3.76603	−0.134759
$782$	−27.6035	−0.987098
$783$	0	0
$784$	−0.697195	−0.0248998
$785$	11.6925	0.417324
$786$	0	0
$787$	−48.5681	−1.73127	−0.865633	−	0.500679i	$-0.833084\pi$
$787$	−48.5681	−1.73127	−0.865633	+	0.500679i	$0.833084\pi$
$788$	−14.4981	−0.516474
$789$	0	0
$790$	0.231418	0.00823347
$791$	41.6779	1.48190
$792$	0	0
$793$	24.1514	0.857641
$794$	−17.8394	−0.633098
$795$	0	0
$796$	−12.4838	−0.442478
$797$	44.1242	1.56296	0.781480	−	0.623930i	$-0.214465\pi$
$797$	44.1242	1.56296	0.781480	+	0.623930i	$0.214465\pi$
$798$	0	0
$799$	−31.0034	−1.09682
$800$	−5.33608	−0.188659
$801$	0	0
$802$	−26.5373	−0.937065
$803$	0.0206470	0.000728618 0
$804$	0	0
$805$	12.1762	0.429153
$806$	34.7434	1.22379
$807$	0	0
$808$	45.1467	1.58826
$809$	−20.4601	−0.719340	−0.359670	−	0.933080i	$-0.617111\pi$
$809$	−20.4601	−0.719340	−0.359670	+	0.933080i	$0.617111\pi$
$810$	0	0
$811$	19.2587	0.676266	0.338133	−	0.941098i	$-0.390205\pi$
$811$	19.2587	0.676266	0.338133	+	0.941098i	$0.390205\pi$
$812$	−23.0113	−0.807540
$813$	0	0
$814$	−0.524131	−0.0183708
$815$	23.1674	0.811517
$816$	0	0
$817$	16.6720	0.583280
$818$	−25.4608	−0.890217
$819$	0	0
$820$	−8.98516	−0.313775
$821$	−38.0578	−1.32823	−0.664113	−	0.747632i	$-0.731191\pi$
$821$	−38.0578	−1.32823	−0.664113	+	0.747632i	$0.731191\pi$
$822$	0	0
$823$	−28.8754	−1.00653	−0.503267	−	0.864131i	$-0.667869\pi$
$823$	−28.8754	−1.00653	−0.503267	+	0.864131i	$0.667869\pi$
$824$	31.9484	1.11298
$825$	0	0
$826$	21.9587	0.764041
$827$	−19.4701	−0.677043	−0.338521	−	0.940959i	$-0.609927\pi$
$827$	−19.4701	−0.677043	−0.338521	+	0.940959i	$0.609927\pi$
$828$	0	0
$829$	2.72567	0.0946665	0.0473333	−	0.998879i	$-0.484928\pi$
$829$	2.72567	0.0946665	0.0473333	+	0.998879i	$0.484928\pi$
$830$	−12.4223	−0.431184
$831$	0	0
$832$	−30.2495	−1.04871
$833$	−8.40692	−0.291283
$834$	0	0
$835$	−6.18653	−0.214094
$836$	2.21228	0.0765134
$837$	0	0
$838$	−16.2271	−0.560554
$839$	32.5256	1.12291	0.561454	−	0.827508i	$-0.310242\pi$
$839$	32.5256	1.12291	0.561454	+	0.827508i	$0.310242\pi$
$840$	0	0
$841$	23.7544	0.819119
$842$	−29.9218	−1.03117
$843$	0	0
$844$	−14.9490	−0.514566
$845$	−10.8936	−0.374751
$846$	0	0
$847$	−31.0785	−1.06787
$848$	−6.57464	−0.225774
$849$	0	0
$850$	6.50104	0.222984
$851$	5.83550	0.200038
$852$	0	0
$853$	38.8589	1.33050	0.665252	−	0.746619i	$-0.268324\pi$
$853$	38.8589	1.33050	0.665252	+	0.746619i	$0.268324\pi$
$854$	13.4057	0.458735
$855$	0	0
$856$	−36.3522	−1.24249
$857$	−23.6584	−0.808156	−0.404078	−	0.914725i	$-0.632408\pi$
$857$	−23.6584	−0.808156	−0.404078	+	0.914725i	$0.632408\pi$
$858$	0	0
$859$	19.9239	0.679796	0.339898	−	0.940462i	$-0.389607\pi$
$859$	19.9239	0.679796	0.339898	+	0.940462i	$0.389607\pi$
$860$	3.70762	0.126429
$861$	0	0
$862$	13.2124	0.450016
$863$	10.9560	0.372947	0.186474	−	0.982460i	$-0.440294\pi$
$863$	10.9560	0.372947	0.186474	+	0.982460i	$0.440294\pi$
$864$	0	0
$865$	−21.2626	−0.722951
$866$	−16.3963	−0.557169
$867$	0	0
$868$	−23.8004	−0.807837
$869$	−0.0985866	−0.00334432
$870$	0	0
$871$	44.5655	1.51005
$872$	37.0897	1.25602
$873$	0	0
$874$	19.9580	0.675090
$875$	−2.86767	−0.0969449
$876$	0	0
$877$	−16.1523	−0.545423	−0.272711	−	0.962096i	$-0.587920\pi$
$877$	−16.1523	−0.545423	−0.272711	+	0.962096i	$0.587920\pi$
$878$	2.18398	0.0737059
$879$	0	0
$880$	−0.229679	−0.00774248
$881$	−5.00512	−0.168627	−0.0843134	−	0.996439i	$-0.526870\pi$
$881$	−5.00512	−0.168627	−0.0843134	+	0.996439i	$0.526870\pi$
$882$	0	0
$883$	−11.7342	−0.394889	−0.197444	−	0.980314i	$-0.563264\pi$
$883$	−11.7342	−0.394889	−0.197444	+	0.980314i	$0.563264\pi$
$884$	−37.1062	−1.24802
$885$	0	0
$886$	1.80899	0.0607742
$887$	0.100369	0.00337008	0.00168504	−	0.999999i	$-0.499464\pi$
$887$	0.100369	0.00337008	0.00168504	+	0.999999i	$0.499464\pi$
$888$	0	0
$889$	−7.51958	−0.252199
$890$	−0.946151	−0.0317150
$891$	0	0
$892$	−28.8803	−0.966983
$893$	22.4163	0.750132
$894$	0	0
$895$	0.934252	0.0312286
$896$	13.8136	0.461480
$897$	0	0
$898$	3.05955	0.102098
$899$	54.5634	1.81979
$900$	0	0
$901$	−79.2783	−2.64114
$902$	−3.10159	−0.103272
$903$	0	0
$904$	42.6944	1.41999
$905$	−11.4131	−0.379384
$906$	0	0
$907$	5.23673	0.173883	0.0869414	−	0.996213i	$-0.472291\pi$
$907$	5.23673	0.173883	0.0869414	+	0.996213i	$0.472291\pi$
$908$	−20.7798	−0.689600
$909$	0	0
$910$	−13.2627	−0.439653
$911$	−31.6428	−1.04837	−0.524186	−	0.851604i	$-0.675630\pi$
$911$	−31.6428	−1.04837	−0.524186	+	0.851604i	$0.675630\pi$
$912$	0	0
$913$	5.29204	0.175141
$914$	22.9804	0.760125
$915$	0	0
$916$	10.4231	0.344389
$917$	25.9167	0.855845
$918$	0	0
$919$	9.55119	0.315065	0.157532	−	0.987514i	$-0.449646\pi$
$919$	9.55119	0.315065	0.157532	+	0.987514i	$0.449646\pi$
$920$	12.4731	0.411226
$921$	0	0
$922$	12.5761	0.414173
$923$	−45.6713	−1.50329
$924$	0	0
$925$	−1.37435	−0.0451883
$926$	14.7498	0.484707
$927$	0	0
$928$	−38.7571	−1.27227
$929$	3.50757	0.115079	0.0575397	−	0.998343i	$-0.481674\pi$
$929$	3.50757	0.115079	0.0575397	+	0.998343i	$0.481674\pi$
$930$	0	0
$931$	6.07842	0.199212
$932$	−22.9494	−0.751732
$933$	0	0
$934$	24.4821	0.801079
$935$	−2.76952	−0.0905729
$936$	0	0
$937$	−38.5990	−1.26097	−0.630486	−	0.776200i	$-0.717145\pi$
$937$	−38.5990	−1.26097	−0.630486	+	0.776200i	$0.717145\pi$
$938$	24.7370	0.807693
$939$	0	0
$940$	4.98506	0.162595
$941$	−21.7370	−0.708606	−0.354303	−	0.935131i	$-0.615282\pi$
$941$	−21.7370	−0.708606	−0.354303	+	0.935131i	$0.615282\pi$
$942$	0	0
$943$	34.5321	1.12452
$944$	4.61166	0.150097
$945$	0	0
$946$	1.27984	0.0416110
$947$	−53.5427	−1.73990	−0.869951	−	0.493138i	$-0.835850\pi$
$947$	−53.5427	−1.73990	−0.869951	+	0.493138i	$0.835850\pi$
$948$	0	0
$949$	0.250390	0.00812799
$950$	−4.70042	−0.152502
$951$	0	0
$952$	−57.8822	−1.87597
$953$	−24.1863	−0.783473	−0.391736	−	0.920078i	$-0.628125\pi$
$953$	−24.1863	−0.783473	−0.391736	+	0.920078i	$0.628125\pi$
$954$	0	0
$955$	−14.8971	−0.482060
$956$	−5.17109	−0.167245
$957$	0	0
$958$	12.2554	0.395954
$959$	−59.2123	−1.91207
$960$	0	0
$961$	25.4343	0.820461
$962$	−6.35621	−0.204933
$963$	0	0
$964$	−27.6400	−0.890225
$965$	−5.71543	−0.183986
$966$	0	0
$967$	22.2263	0.714751	0.357375	−	0.933961i	$-0.383672\pi$
$967$	22.2263	0.714751	0.357375	+	0.933961i	$0.383672\pi$
$968$	−31.8364	−1.02326
$969$	0	0
$970$	13.5156	0.433958
$971$	40.1597	1.28879	0.644393	−	0.764695i	$-0.277110\pi$
$971$	40.1597	1.28879	0.644393	+	0.764695i	$0.277110\pi$
$972$	0	0
$973$	−39.6935	−1.27252
$974$	3.78414	0.121252
$975$	0	0
$976$	2.81540	0.0901189
$977$	39.3218	1.25802	0.629008	−	0.777399i	$-0.283461\pi$
$977$	39.3218	1.25802	0.629008	+	0.777399i	$0.283461\pi$
$978$	0	0
$979$	0.403071	0.0128822
$980$	1.35175	0.0431802
$981$	0	0
$982$	−28.2028	−0.899986
$983$	−35.1754	−1.12192	−0.560961	−	0.827842i	$-0.689568\pi$
$983$	−35.1754	−1.12192	−0.560961	+	0.827842i	$0.689568\pi$
$984$	0	0
$985$	−13.1229	−0.418129
$986$	47.2185	1.50374
$987$	0	0
$988$	26.8287	0.853535
$989$	−14.2493	−0.453101
$990$	0	0
$991$	28.4860	0.904888	0.452444	−	0.891793i	$-0.350552\pi$
$991$	28.4860	0.904888	0.452444	+	0.891793i	$0.350552\pi$
$992$	−40.0861	−1.27273
$993$	0	0
$994$	−25.3508	−0.804079
$995$	−11.2996	−0.358223
$996$	0	0
$997$	19.1798	0.607432	0.303716	−	0.952763i	$-0.401773\pi$
$997$	19.1798	0.607432	0.303716	+	0.952763i	$0.401773\pi$
$998$	41.1184	1.30158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.r.1.5		10
3.2	odd	2		1335.2.a.k.1.6	✓	10
15.14	odd	2		6675.2.a.y.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.k.1.6	✓	10	3.2	odd	2
4005.2.a.r.1.5		10	1.1	even	1	trivial
6675.2.a.y.1.5		10	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-0.946151 q^{2} -1.10480 q^{4} -1.00000 q^{5} +2.86767 q^{7} +2.93761 q^{8} +O(q^{10})\)	\(q-0.946151 q^{2} -1.10480 q^{4} -1.00000 q^{5} +2.86767 q^{7} +2.93761 q^{8} +0.946151 q^{10} -0.403071 q^{11} -4.88811 q^{13} -2.71325 q^{14} -0.569823 q^{16} -6.87104 q^{17} +4.96794 q^{19} +1.10480 q^{20} +0.381366 q^{22} -4.24601 q^{23} +1.00000 q^{25} +4.62489 q^{26} -3.16820 q^{28} +7.26323 q^{29} +7.51228 q^{31} -5.33608 q^{32} +6.50104 q^{34} -2.86767 q^{35} -1.37435 q^{37} -4.70042 q^{38} -2.93761 q^{40} -8.13284 q^{41} +3.35592 q^{43} +0.445313 q^{44} +4.01736 q^{46} +4.51219 q^{47} +1.22353 q^{49} -0.946151 q^{50} +5.40038 q^{52} +11.5380 q^{53} +0.403071 q^{55} +8.42409 q^{56} -6.87211 q^{58} -8.09315 q^{59} -4.94084 q^{61} -7.10775 q^{62} +6.18838 q^{64} +4.88811 q^{65} -9.11713 q^{67} +7.59111 q^{68} +2.71325 q^{70} +9.34335 q^{71} -0.0512243 q^{73} +1.30034 q^{74} -5.48857 q^{76} -1.15588 q^{77} +0.244588 q^{79} +0.569823 q^{80} +7.69490 q^{82} -13.1293 q^{83} +6.87104 q^{85} -3.17521 q^{86} -1.18407 q^{88} -1.00000 q^{89} -14.0175 q^{91} +4.69098 q^{92} -4.26921 q^{94} -4.96794 q^{95} +14.2848 q^{97} -1.15764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) \) 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) \) 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8 + 6 * q^10 + 10 * q^11 + 7 * q^13 + 7 * q^14 + 22 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^20 + 8 * q^22 - 6 * q^23 + 10 * q^25 + 14 * q^26 + 16 * q^28 + 3 * q^29 + 12 * q^31 - 21 * q^32 - 6 * q^34 - 7 * q^35 - 19 * q^37 - 6 * q^38 + 15 * q^40 - 13 * q^41 + 9 * q^43 + 26 * q^44 + 8 * q^46 - 21 * q^47 + 53 * q^49 - 6 * q^50 - 43 * q^52 - 7 * q^53 - 10 * q^55 + 53 * q^56 - 42 * q^58 + 19 * q^59 + 4 * q^61 + 28 * q^62 + 5 * q^64 - 7 * q^65 - 6 * q^67 - 2 * q^68 - 7 * q^70 + 6 * q^71 + 6 * q^73 - 2 * q^76 + 40 * q^77 + 25 * q^79 - 22 * q^80 + q^82 + 22 * q^83 + 11 * q^85 + 2 * q^86 + 20 * q^88 - 10 * q^89 - 10 * q^91 - 10 * q^92 + 25 * q^94 - 10 * q^95 + 40 * q^97 + 56 * q^98

Introduction

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