LMFDB - Embedded newform 4005.2.a.t.1.8

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} - 19x^{8} - x^{7} + 128x^{6} + 14x^{5} - 358x^{4} - 59x^{3} + 344x^{2} + 71x - 21 $ x^10 - 19x^8 - x^7 + 128x^6 + 14x^5 - 358x^4 - 59x^3 + 344x^2 + 71*x - 21
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.8
Root			$-2.11901$ 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)

$f(q)$	$=$	$q+2.11901 q^{2} +2.49018 q^{4} +1.00000 q^{5} +1.69488 q^{7} +1.03870 q^{8} +O(q^{10})$	\(q+2.11901 q^{2} +2.49018 q^{4} +1.00000 q^{5} +1.69488 q^{7} +1.03870 q^{8} +2.11901 q^{10} +1.48120 q^{11} -0.319402 q^{13} +3.59146 q^{14} -2.77935 q^{16} +2.82926 q^{17} +8.40501 q^{19} +2.49018 q^{20} +3.13868 q^{22} -5.00664 q^{23} +1.00000 q^{25} -0.676814 q^{26} +4.22056 q^{28} +6.70584 q^{29} +1.54807 q^{31} -7.96687 q^{32} +5.99523 q^{34} +1.69488 q^{35} -2.62298 q^{37} +17.8103 q^{38} +1.03870 q^{40} -0.927190 q^{41} +5.55533 q^{43} +3.68847 q^{44} -10.6091 q^{46} -12.6174 q^{47} -4.12739 q^{49} +2.11901 q^{50} -0.795370 q^{52} +10.1370 q^{53} +1.48120 q^{55} +1.76048 q^{56} +14.2097 q^{58} -13.3357 q^{59} -4.50581 q^{61} +3.28037 q^{62} -11.3231 q^{64} -0.319402 q^{65} +7.52765 q^{67} +7.04539 q^{68} +3.59146 q^{70} +13.1461 q^{71} +16.2029 q^{73} -5.55811 q^{74} +20.9300 q^{76} +2.51046 q^{77} +3.44702 q^{79} -2.77935 q^{80} -1.96472 q^{82} +13.3448 q^{83} +2.82926 q^{85} +11.7718 q^{86} +1.53853 q^{88} +1.00000 q^{89} -0.541348 q^{91} -12.4675 q^{92} -26.7364 q^{94} +8.40501 q^{95} +13.6540 q^{97} -8.74595 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * 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 $

$2$	2.11901	1.49836	0.749182	−	0.662365i	$-0.230447\pi$
$2$	2.11901	1.49836	0.749182	+	0.662365i	$0.230447\pi$
$3$	0	0
$3$	0	0
$4$	2.49018	1.24509
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.69488	0.640604	0.320302	−	0.947315i	$-0.396216\pi$
$7$	1.69488	0.640604	0.320302	+	0.947315i	$0.396216\pi$
$8$	1.03870	0.367237
$9$	0	0
$10$	2.11901	0.670088
$11$	1.48120	0.446600	0.223300	−	0.974750i	$-0.428317\pi$
$11$	1.48120	0.446600	0.223300	+	0.974750i	$0.428317\pi$
$12$	0	0
$13$	−0.319402	−0.0885861	−0.0442931	−	0.999019i	$-0.514104\pi$
$13$	−0.319402	−0.0885861	−0.0442931	+	0.999019i	$0.514104\pi$
$14$	3.59146	0.959857
$15$	0	0
$16$	−2.77935	−0.694837
$17$	2.82926	0.686197	0.343099	−	0.939299i	$-0.388523\pi$
$17$	2.82926	0.686197	0.343099	+	0.939299i	$0.388523\pi$
$18$	0	0
$19$	8.40501	1.92824	0.964121	−	0.265464i	$-0.0855251\pi$
$19$	8.40501	1.92824	0.964121	+	0.265464i	$0.0855251\pi$
$20$	2.49018	0.556822
$21$	0	0
$22$	3.13868	0.669169
$23$	−5.00664	−1.04396	−0.521978	−	0.852959i	$-0.674806\pi$
$23$	−5.00664	−1.04396	−0.521978	+	0.852959i	$0.674806\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.676814	−0.132734
$27$	0	0
$28$	4.22056	0.797611
$29$	6.70584	1.24524	0.622621	−	0.782523i	$-0.286068\pi$
$29$	6.70584	1.24524	0.622621	+	0.782523i	$0.286068\pi$
$30$	0	0
$31$	1.54807	0.278041	0.139021	−	0.990289i	$-0.455605\pi$
$31$	1.54807	0.278041	0.139021	+	0.990289i	$0.455605\pi$
$32$	−7.96687	−1.40836
$33$	0	0
$34$	5.99523	1.02817
$35$	1.69488	0.286487
$36$	0	0
$37$	−2.62298	−0.431215	−0.215608	−	0.976480i	$-0.569173\pi$
$37$	−2.62298	−0.431215	−0.215608	+	0.976480i	$0.569173\pi$
$38$	17.8103	2.88921
$39$	0	0
$40$	1.03870	0.164234
$41$	−0.927190	−0.144803	−0.0724013	−	0.997376i	$-0.523066\pi$
$41$	−0.927190	−0.144803	−0.0724013	+	0.997376i	$0.523066\pi$
$42$	0	0
$43$	5.55533	0.847180	0.423590	−	0.905854i	$-0.360770\pi$
$43$	5.55533	0.847180	0.423590	+	0.905854i	$0.360770\pi$
$44$	3.68847	0.556058
$45$	0	0
$46$	−10.6091	−1.56423
$47$	−12.6174	−1.84044	−0.920222	−	0.391397i	$-0.871992\pi$
$47$	−12.6174	−1.84044	−0.920222	+	0.391397i	$0.871992\pi$
$48$	0	0
$49$	−4.12739	−0.589627
$50$	2.11901	0.299673
$51$	0	0
$52$	−0.795370	−0.110298
$53$	10.1370	1.39243	0.696215	−	0.717833i	$-0.254866\pi$
$53$	10.1370	1.39243	0.696215	+	0.717833i	$0.254866\pi$
$54$	0	0
$55$	1.48120	0.199726
$56$	1.76048	0.235254
$57$	0	0
$58$	14.2097	1.86583
$59$	−13.3357	−1.73616	−0.868081	−	0.496423i	$-0.834646\pi$
$59$	−13.3357	−1.73616	−0.868081	+	0.496423i	$0.834646\pi$
$60$	0	0
$61$	−4.50581	−0.576909	−0.288455	−	0.957494i	$-0.593141\pi$
$61$	−4.50581	−0.576909	−0.288455	+	0.957494i	$0.593141\pi$
$62$	3.28037	0.416607
$63$	0	0
$64$	−11.3231	−1.41539
$65$	−0.319402	−0.0396169
$66$	0	0
$67$	7.52765	0.919649	0.459824	−	0.888010i	$-0.347912\pi$
$67$	7.52765	0.919649	0.459824	+	0.888010i	$0.347912\pi$
$68$	7.04539	0.854379
$69$	0	0
$70$	3.59146	0.429261
$71$	13.1461	1.56016	0.780079	−	0.625681i	$-0.215179\pi$
$71$	13.1461	1.56016	0.780079	+	0.625681i	$0.215179\pi$
$72$	0	0
$73$	16.2029	1.89641	0.948204	−	0.317663i	$-0.102898\pi$
$73$	16.2029	1.89641	0.948204	+	0.317663i	$0.102898\pi$
$74$	−5.55811	−0.646117
$75$	0	0
$76$	20.9300	2.40084
$77$	2.51046	0.286094
$78$	0	0
$79$	3.44702	0.387820	0.193910	−	0.981019i	$-0.437883\pi$
$79$	3.44702	0.387820	0.193910	+	0.981019i	$0.437883\pi$
$80$	−2.77935	−0.310741
$81$	0	0
$82$	−1.96472	−0.216967
$83$	13.3448	1.46478	0.732390	−	0.680885i	$-0.238405\pi$
$83$	13.3448	1.46478	0.732390	+	0.680885i	$0.238405\pi$
$84$	0	0
$85$	2.82926	0.306877
$86$	11.7718	1.26938
$87$	0	0
$88$	1.53853	0.164008
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−0.541348	−0.0567486
$92$	−12.4675	−1.29982
$93$	0	0
$94$	−26.7364	−2.75765
$95$	8.40501	0.862336
$96$	0	0
$97$	13.6540	1.38636	0.693179	−	0.720766i	$-0.256210\pi$
$97$	13.6540	1.38636	0.693179	+	0.720766i	$0.256210\pi$
$98$	−8.74595	−0.883475
$99$	0	0
$100$	2.49018	0.249018
$101$	−8.19473	−0.815407	−0.407703	−	0.913114i	$-0.633670\pi$
$101$	−8.19473	−0.815407	−0.407703	+	0.913114i	$0.633670\pi$
$102$	0	0
$103$	7.01702	0.691408	0.345704	−	0.938344i	$-0.387640\pi$
$103$	7.01702	0.691408	0.345704	+	0.938344i	$0.387640\pi$
$104$	−0.331764	−0.0325322
$105$	0	0
$106$	21.4805	2.08637
$107$	0.269147	0.0260195	0.0130097	−	0.999915i	$-0.495859\pi$
$107$	0.269147	0.0260195	0.0130097	+	0.999915i	$0.495859\pi$
$108$	0	0
$109$	−14.4318	−1.38231	−0.691157	−	0.722704i	$-0.742899\pi$
$109$	−14.4318	−1.38231	−0.691157	+	0.722704i	$0.742899\pi$
$110$	3.13868	0.299262
$111$	0	0
$112$	−4.71066	−0.445116
$113$	−12.7048	−1.19516	−0.597582	−	0.801808i	$-0.703872\pi$
$113$	−12.7048	−1.19516	−0.597582	+	0.801808i	$0.703872\pi$
$114$	0	0
$115$	−5.00664	−0.466872
$116$	16.6988	1.55044
$117$	0	0
$118$	−28.2584	−2.60140
$119$	4.79526	0.439581
$120$	0	0
$121$	−8.80603	−0.800548
$122$	−9.54783	−0.864420
$123$	0	0
$124$	3.85498	0.346187
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.0049	1.59767	0.798837	−	0.601548i	$-0.205449\pi$
$127$	18.0049	1.59767	0.798837	+	0.601548i	$0.205449\pi$
$128$	−8.06005	−0.712415
$129$	0	0
$130$	−0.676814	−0.0593606
$131$	−18.0542	−1.57741	−0.788703	−	0.614774i	$-0.789247\pi$
$131$	−18.0542	−1.57741	−0.788703	+	0.614774i	$0.789247\pi$
$132$	0	0
$133$	14.2455	1.23524
$134$	15.9511	1.37797
$135$	0	0
$136$	2.93877	0.251997
$137$	−13.8051	−1.17945	−0.589724	−	0.807605i	$-0.700764\pi$
$137$	−13.8051	−1.17945	−0.589724	+	0.807605i	$0.700764\pi$
$138$	0	0
$139$	6.59996	0.559801	0.279900	−	0.960029i	$-0.409699\pi$
$139$	6.59996	0.559801	0.279900	+	0.960029i	$0.409699\pi$
$140$	4.22056	0.356703
$141$	0	0
$142$	27.8567	2.33768
$143$	−0.473100	−0.0395626
$144$	0	0
$145$	6.70584	0.556889
$146$	34.3341	2.84151
$147$	0	0
$148$	−6.53171	−0.536903
$149$	4.56999	0.374388	0.187194	−	0.982323i	$-0.440061\pi$
$149$	4.56999	0.374388	0.187194	+	0.982323i	$0.440061\pi$
$150$	0	0
$151$	−1.43780	−0.117006	−0.0585031	−	0.998287i	$-0.518633\pi$
$151$	−1.43780	−0.117006	−0.0585031	+	0.998287i	$0.518633\pi$
$152$	8.73032	0.708123
$153$	0	0
$154$	5.31968	0.428672
$155$	1.54807	0.124344
$156$	0	0
$157$	2.11732	0.168981	0.0844904	−	0.996424i	$-0.473074\pi$
$157$	2.11732	0.168981	0.0844904	+	0.996424i	$0.473074\pi$
$158$	7.30425	0.581095
$159$	0	0
$160$	−7.96687	−0.629836
$161$	−8.48565	−0.668763
$162$	0	0
$163$	−16.8921	−1.32309	−0.661547	−	0.749903i	$-0.730100\pi$
$163$	−16.8921	−1.32309	−0.661547	+	0.749903i	$0.730100\pi$
$164$	−2.30887	−0.180293
$165$	0	0
$166$	28.2777	2.19477
$167$	−22.7346	−1.75925	−0.879627	−	0.475664i	$-0.842208\pi$
$167$	−22.7346	−1.75925	−0.879627	+	0.475664i	$0.842208\pi$
$168$	0	0
$169$	−12.8980	−0.992152
$170$	5.99523	0.459813
$171$	0	0
$172$	13.8338	1.05482
$173$	−11.7197	−0.891034	−0.445517	−	0.895274i	$-0.646980\pi$
$173$	−11.7197	−0.891034	−0.445517	+	0.895274i	$0.646980\pi$
$174$	0	0
$175$	1.69488	0.128121
$176$	−4.11679	−0.310314
$177$	0	0
$178$	2.11901	0.158826
$179$	−7.64183	−0.571178	−0.285589	−	0.958352i	$-0.592189\pi$
$179$	−7.64183	−0.571178	−0.285589	+	0.958352i	$0.592189\pi$
$180$	0	0
$181$	20.8213	1.54763	0.773817	−	0.633409i	$-0.218345\pi$
$181$	20.8213	1.54763	0.773817	+	0.633409i	$0.218345\pi$
$182$	−1.14712	−0.0850301
$183$	0	0
$184$	−5.20042	−0.383380
$185$	−2.62298	−0.192845
$186$	0	0
$187$	4.19072	0.306456
$188$	−31.4198	−2.29152
$189$	0	0
$190$	17.8103	1.29209
$191$	−18.1818	−1.31559	−0.657793	−	0.753199i	$-0.728510\pi$
$191$	−18.1818	−1.31559	−0.657793	+	0.753199i	$0.728510\pi$
$192$	0	0
$193$	6.53672	0.470523	0.235262	−	0.971932i	$-0.424405\pi$
$193$	6.53672	0.470523	0.235262	+	0.971932i	$0.424405\pi$
$194$	28.9330	2.07727
$195$	0	0
$196$	−10.2780	−0.734140
$197$	−20.8489	−1.48542	−0.742710	−	0.669613i	$-0.766460\pi$
$197$	−20.8489	−1.48542	−0.742710	+	0.669613i	$0.766460\pi$
$198$	0	0
$199$	3.88956	0.275724	0.137862	−	0.990451i	$-0.455977\pi$
$199$	3.88956	0.275724	0.137862	+	0.990451i	$0.455977\pi$
$200$	1.03870	0.0734475
$201$	0	0
$202$	−17.3647	−1.22178
$203$	11.3656	0.797707
$204$	0	0
$205$	−0.927190	−0.0647577
$206$	14.8691	1.03598
$207$	0	0
$208$	0.887729	0.0615530
$209$	12.4495	0.861153
$210$	0	0
$211$	6.62600	0.456153	0.228076	−	0.973643i	$-0.426756\pi$
$211$	6.62600	0.456153	0.228076	+	0.973643i	$0.426756\pi$
$212$	25.2431	1.73370
$213$	0	0
$214$	0.570325	0.0389866
$215$	5.55533	0.378870
$216$	0	0
$217$	2.62379	0.178114
$218$	−30.5810	−2.07121
$219$	0	0
$220$	3.68847	0.248677
$221$	−0.903672	−0.0607876
$222$	0	0
$223$	−0.398579	−0.0266908	−0.0133454	−	0.999911i	$-0.504248\pi$
$223$	−0.398579	−0.0266908	−0.0133454	+	0.999911i	$0.504248\pi$
$224$	−13.5029	−0.902199
$225$	0	0
$226$	−26.9215	−1.79079
$227$	−14.7601	−0.979661	−0.489831	−	0.871818i	$-0.662941\pi$
$227$	−14.7601	−0.979661	−0.489831	+	0.871818i	$0.662941\pi$
$228$	0	0
$229$	−13.3853	−0.884525	−0.442262	−	0.896886i	$-0.645824\pi$
$229$	−13.3853	−0.884525	−0.442262	+	0.896886i	$0.645824\pi$
$230$	−10.6091	−0.699543
$231$	0	0
$232$	6.96538	0.457300
$233$	24.5999	1.61159	0.805795	−	0.592194i	$-0.201738\pi$
$233$	24.5999	1.61159	0.805795	+	0.592194i	$0.201738\pi$
$234$	0	0
$235$	−12.6174	−0.823071
$236$	−33.2084	−2.16168
$237$	0	0
$238$	10.1612	0.658652
$239$	16.0281	1.03677	0.518384	−	0.855148i	$-0.326534\pi$
$239$	16.0281	1.03677	0.518384	+	0.855148i	$0.326534\pi$
$240$	0	0
$241$	−28.0314	−1.80566	−0.902830	−	0.429998i	$-0.858514\pi$
$241$	−28.0314	−1.80566	−0.902830	+	0.429998i	$0.858514\pi$
$242$	−18.6600	−1.19951
$243$	0	0
$244$	−11.2203	−0.718305
$245$	−4.12739	−0.263689
$246$	0	0
$247$	−2.68458	−0.170816
$248$	1.60799	0.102107
$249$	0	0
$250$	2.11901	0.134018
$251$	10.4556	0.659953	0.329977	−	0.943989i	$-0.392959\pi$
$251$	10.4556	0.659953	0.329977	+	0.943989i	$0.392959\pi$
$252$	0	0
$253$	−7.41586	−0.466231
$254$	38.1524	2.39390
$255$	0	0
$256$	5.56697	0.347936
$257$	31.8885	1.98915	0.994574	−	0.104028i	$-0.0331731\pi$
$257$	31.8885	1.98915	0.994574	+	0.104028i	$0.0331731\pi$
$258$	0	0
$259$	−4.44563	−0.276238
$260$	−0.795370	−0.0493267
$261$	0	0
$262$	−38.2570	−2.36353
$263$	0.718791	0.0443225	0.0221613	−	0.999754i	$-0.492945\pi$
$263$	0.718791	0.0443225	0.0221613	+	0.999754i	$0.492945\pi$
$264$	0	0
$265$	10.1370	0.622714
$266$	30.1862	1.85084
$267$	0	0
$268$	18.7452	1.14505
$269$	−5.72343	−0.348964	−0.174482	−	0.984660i	$-0.555825\pi$
$269$	−5.72343	−0.348964	−0.174482	+	0.984660i	$0.555825\pi$
$270$	0	0
$271$	−0.502817	−0.0305440	−0.0152720	−	0.999883i	$-0.504861\pi$
$271$	−0.502817	−0.0305440	−0.0152720	+	0.999883i	$0.504861\pi$
$272$	−7.86351	−0.476796
$273$	0	0
$274$	−29.2531	−1.76724
$275$	1.48120	0.0893200
$276$	0	0
$277$	−6.45300	−0.387723	−0.193862	−	0.981029i	$-0.562101\pi$
$277$	−6.45300	−0.387723	−0.193862	+	0.981029i	$0.562101\pi$
$278$	13.9853	0.838785
$279$	0	0
$280$	1.76048	0.105209
$281$	14.9276	0.890504	0.445252	−	0.895405i	$-0.353114\pi$
$281$	14.9276	0.890504	0.445252	+	0.895405i	$0.353114\pi$
$282$	0	0
$283$	−7.84367	−0.466258	−0.233129	−	0.972446i	$-0.574896\pi$
$283$	−7.84367	−0.466258	−0.233129	+	0.972446i	$0.574896\pi$
$284$	32.7363	1.94254
$285$	0	0
$286$	−1.00250	−0.0592791
$287$	−1.57147	−0.0927612
$288$	0	0
$289$	−8.99526	−0.529133
$290$	14.2097	0.834423
$291$	0	0
$292$	40.3482	2.36120
$293$	−10.9256	−0.638282	−0.319141	−	0.947707i	$-0.603394\pi$
$293$	−10.9256	−0.638282	−0.319141	+	0.947707i	$0.603394\pi$
$294$	0	0
$295$	−13.3357	−0.776435
$296$	−2.72450	−0.158358
$297$	0	0
$298$	9.68384	0.560970
$299$	1.59913	0.0924801
$300$	0	0
$301$	9.41561	0.542707
$302$	−3.04670	−0.175318
$303$	0	0
$304$	−23.3605	−1.33981
$305$	−4.50581	−0.258002
$306$	0	0
$307$	−21.8705	−1.24821	−0.624107	−	0.781339i	$-0.714537\pi$
$307$	−21.8705	−1.24821	−0.624107	+	0.781339i	$0.714537\pi$
$308$	6.25152	0.356213
$309$	0	0
$310$	3.28037	0.186312
$311$	−11.3078	−0.641206	−0.320603	−	0.947214i	$-0.603886\pi$
$311$	−11.3078	−0.641206	−0.320603	+	0.947214i	$0.603886\pi$
$312$	0	0
$313$	−0.734611	−0.0415227	−0.0207613	−	0.999784i	$-0.506609\pi$
$313$	−0.734611	−0.0415227	−0.0207613	+	0.999784i	$0.506609\pi$
$314$	4.48662	0.253194
$315$	0	0
$316$	8.58371	0.482871
$317$	24.5672	1.37983	0.689917	−	0.723888i	$-0.257647\pi$
$317$	24.5672	1.37983	0.689917	+	0.723888i	$0.257647\pi$
$318$	0	0
$319$	9.93272	0.556125
$320$	−11.3231	−0.632982
$321$	0	0
$322$	−17.9811	−1.00205
$323$	23.7800	1.32315
$324$	0	0
$325$	−0.319402	−0.0177172
$326$	−35.7945	−1.98248
$327$	0	0
$328$	−0.963076	−0.0531770
$329$	−21.3850	−1.17900
$330$	0	0
$331$	−13.4396	−0.738708	−0.369354	−	0.929289i	$-0.620421\pi$
$331$	−13.4396	−0.738708	−0.369354	+	0.929289i	$0.620421\pi$
$332$	33.2310	1.82379
$333$	0	0
$334$	−48.1747	−2.63600
$335$	7.52765	0.411280
$336$	0	0
$337$	35.5108	1.93440	0.967199	−	0.254020i	$-0.0817529\pi$
$337$	35.5108	1.93440	0.967199	+	0.254020i	$0.0817529\pi$
$338$	−27.3309	−1.48660
$339$	0	0
$340$	7.04539	0.382090
$341$	2.29301	0.124173
$342$	0	0
$343$	−18.8596	−1.01832
$344$	5.77034	0.311116
$345$	0	0
$346$	−24.8342	−1.33509
$347$	3.12727	0.167881	0.0839404	−	0.996471i	$-0.473249\pi$
$347$	3.12727	0.167881	0.0839404	+	0.996471i	$0.473249\pi$
$348$	0	0
$349$	−4.60223	−0.246351	−0.123176	−	0.992385i	$-0.539308\pi$
$349$	−4.60223	−0.246351	−0.123176	+	0.992385i	$0.539308\pi$
$350$	3.59146	0.191971
$351$	0	0
$352$	−11.8006	−0.628972
$353$	8.01070	0.426366	0.213183	−	0.977012i	$-0.431617\pi$
$353$	8.01070	0.426366	0.213183	+	0.977012i	$0.431617\pi$
$354$	0	0
$355$	13.1461	0.697724
$356$	2.49018	0.131980
$357$	0	0
$358$	−16.1931	−0.855831
$359$	0.232206	0.0122554	0.00612769	−	0.999981i	$-0.498049\pi$
$359$	0.232206	0.0122554	0.00612769	+	0.999981i	$0.498049\pi$
$360$	0	0
$361$	51.6442	2.71812
$362$	44.1204	2.31892
$363$	0	0
$364$	−1.34806	−0.0706573
$365$	16.2029	0.848099
$366$	0	0
$367$	−17.7362	−0.925823	−0.462912	−	0.886404i	$-0.653195\pi$
$367$	−17.7362	−0.925823	−0.462912	+	0.886404i	$0.653195\pi$
$368$	13.9152	0.725380
$369$	0	0
$370$	−5.55811	−0.288952
$371$	17.1811	0.891996
$372$	0	0
$373$	−33.1601	−1.71696	−0.858482	−	0.512844i	$-0.828592\pi$
$373$	−33.1601	−1.71696	−0.858482	+	0.512844i	$0.828592\pi$
$374$	8.88016	0.459182
$375$	0	0
$376$	−13.1058	−0.675880
$377$	−2.14186	−0.110311
$378$	0	0
$379$	3.93616	0.202187	0.101093	−	0.994877i	$-0.467766\pi$
$379$	3.93616	0.202187	0.101093	+	0.994877i	$0.467766\pi$
$380$	20.9300	1.07369
$381$	0	0
$382$	−38.5272	−1.97123
$383$	−8.91118	−0.455340	−0.227670	−	0.973738i	$-0.573111\pi$
$383$	−8.91118	−0.455340	−0.227670	+	0.973738i	$0.573111\pi$
$384$	0	0
$385$	2.51046	0.127945
$386$	13.8513	0.705015
$387$	0	0
$388$	34.0011	1.72614
$389$	−19.3554	−0.981356	−0.490678	−	0.871341i	$-0.663251\pi$
$389$	−19.3554	−0.981356	−0.490678	+	0.871341i	$0.663251\pi$
$390$	0	0
$391$	−14.1651	−0.716360
$392$	−4.28713	−0.216533
$393$	0	0
$394$	−44.1789	−2.22570
$395$	3.44702	0.173438
$396$	0	0
$397$	−9.96167	−0.499962	−0.249981	−	0.968251i	$-0.580424\pi$
$397$	−9.96167	−0.499962	−0.249981	+	0.968251i	$0.580424\pi$
$398$	8.24201	0.413134
$399$	0	0
$400$	−2.77935	−0.138967
$401$	0.496910	0.0248145	0.0124072	−	0.999923i	$-0.496051\pi$
$401$	0.496910	0.0248145	0.0124072	+	0.999923i	$0.496051\pi$
$402$	0	0
$403$	−0.494456	−0.0246306
$404$	−20.4064	−1.01526
$405$	0	0
$406$	24.0837	1.19526
$407$	−3.88517	−0.192581
$408$	0	0
$409$	18.0517	0.892600	0.446300	−	0.894883i	$-0.352741\pi$
$409$	18.0517	0.892600	0.446300	+	0.894883i	$0.352741\pi$
$410$	−1.96472	−0.0970306
$411$	0	0
$412$	17.4737	0.860866
$413$	−22.6024	−1.11219
$414$	0	0
$415$	13.3448	0.655070
$416$	2.54463	0.124761
$417$	0	0
$418$	26.3806	1.29032
$419$	−19.0507	−0.930689	−0.465344	−	0.885130i	$-0.654070\pi$
$419$	−19.0507	−0.930689	−0.465344	+	0.885130i	$0.654070\pi$
$420$	0	0
$421$	−32.2617	−1.57234	−0.786169	−	0.618012i	$-0.787938\pi$
$421$	−32.2617	−1.57234	−0.786169	+	0.618012i	$0.787938\pi$
$422$	14.0405	0.683483
$423$	0	0
$424$	10.5294	0.511352
$425$	2.82926	0.137239
$426$	0	0
$427$	−7.63679	−0.369570
$428$	0.670227	0.0323966
$429$	0	0
$430$	11.7718	0.567685
$431$	11.7352	0.565267	0.282633	−	0.959228i	$-0.408792\pi$
$431$	11.7352	0.565267	0.282633	+	0.959228i	$0.408792\pi$
$432$	0	0
$433$	−6.87063	−0.330181	−0.165091	−	0.986278i	$-0.552792\pi$
$433$	−6.87063	−0.330181	−0.165091	+	0.986278i	$0.552792\pi$
$434$	5.55983	0.266880
$435$	0	0
$436$	−35.9378	−1.72111
$437$	−42.0809	−2.01300
$438$	0	0
$439$	−26.7783	−1.27806	−0.639029	−	0.769183i	$-0.720664\pi$
$439$	−26.7783	−1.27806	−0.639029	+	0.769183i	$0.720664\pi$
$440$	1.53853	0.0733467
$441$	0	0
$442$	−1.91489	−0.0910819
$443$	5.92477	0.281494	0.140747	−	0.990046i	$-0.455050\pi$
$443$	5.92477	0.281494	0.140747	+	0.990046i	$0.455050\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−0.844591	−0.0399925
$447$	0	0
$448$	−19.1913	−0.906706
$449$	0.691608	0.0326390	0.0163195	−	0.999867i	$-0.494805\pi$
$449$	0.691608	0.0326390	0.0163195	+	0.999867i	$0.494805\pi$
$450$	0	0
$451$	−1.37336	−0.0646689
$452$	−31.6372	−1.48809
$453$	0	0
$454$	−31.2767	−1.46789
$455$	−0.541348	−0.0253788
$456$	0	0
$457$	−29.6701	−1.38791	−0.693954	−	0.720020i	$-0.744133\pi$
$457$	−29.6701	−1.38791	−0.693954	+	0.720020i	$0.744133\pi$
$458$	−28.3635	−1.32534
$459$	0	0
$460$	−12.4675	−0.581298
$461$	10.5560	0.491640	0.245820	−	0.969315i	$-0.420943\pi$
$461$	10.5560	0.491640	0.245820	+	0.969315i	$0.420943\pi$
$462$	0	0
$463$	−21.4822	−0.998360	−0.499180	−	0.866498i	$-0.666365\pi$
$463$	−21.4822	−0.998360	−0.499180	+	0.866498i	$0.666365\pi$
$464$	−18.6379	−0.865241
$465$	0	0
$466$	52.1272	2.41475
$467$	0.833725	0.0385802	0.0192901	−	0.999814i	$-0.493859\pi$
$467$	0.833725	0.0385802	0.0192901	+	0.999814i	$0.493859\pi$
$468$	0	0
$469$	12.7585	0.589131
$470$	−26.7364	−1.23326
$471$	0	0
$472$	−13.8519	−0.637583
$473$	8.22858	0.378350
$474$	0	0
$475$	8.40501	0.385648
$476$	11.9411	0.547319
$477$	0	0
$478$	33.9635	1.55346
$479$	−34.2640	−1.56556	−0.782782	−	0.622296i	$-0.786200\pi$
$479$	−34.2640	−1.56556	−0.782782	+	0.622296i	$0.786200\pi$
$480$	0	0
$481$	0.837785	0.0381997
$482$	−59.3986	−2.70553
$483$	0	0
$484$	−21.9286	−0.996757
$485$	13.6540	0.619998
$486$	0	0
$487$	−9.39656	−0.425799	−0.212899	−	0.977074i	$-0.568291\pi$
$487$	−9.39656	−0.425799	−0.212899	+	0.977074i	$0.568291\pi$
$488$	−4.68020	−0.211863
$489$	0	0
$490$	−8.74595	−0.395102
$491$	21.3117	0.961783	0.480892	−	0.876780i	$-0.340313\pi$
$491$	21.3117	0.961783	0.480892	+	0.876780i	$0.340313\pi$
$492$	0	0
$493$	18.9726	0.854482
$494$	−5.68863	−0.255944
$495$	0	0
$496$	−4.30263	−0.193194
$497$	22.2811	0.999444
$498$	0	0
$499$	−4.20669	−0.188317	−0.0941586	−	0.995557i	$-0.530016\pi$
$499$	−4.20669	−0.188317	−0.0941586	+	0.995557i	$0.530016\pi$
$500$	2.49018	0.111364
$501$	0	0
$502$	22.1555	0.988850
$503$	17.3976	0.775722	0.387861	−	0.921718i	$-0.373214\pi$
$503$	17.3976	0.775722	0.387861	+	0.921718i	$0.373214\pi$
$504$	0	0
$505$	−8.19473	−0.364661
$506$	−15.7142	−0.698584
$507$	0	0
$508$	44.8355	1.98925
$509$	−0.571998	−0.0253534	−0.0126767	−	0.999920i	$-0.504035\pi$
$509$	−0.571998	−0.0253534	−0.0126767	+	0.999920i	$0.504035\pi$
$510$	0	0
$511$	27.4620	1.21485
$512$	27.9165	1.23375
$513$	0	0
$514$	67.5719	2.98047
$515$	7.01702	0.309207
$516$	0	0
$517$	−18.6890	−0.821942
$518$	−9.42032	−0.413905
$519$	0	0
$520$	−0.331764	−0.0145488
$521$	19.3719	0.848697	0.424348	−	0.905499i	$-0.360503\pi$
$521$	19.3719	0.848697	0.424348	+	0.905499i	$0.360503\pi$
$522$	0	0
$523$	11.9604	0.522992	0.261496	−	0.965204i	$-0.415784\pi$
$523$	11.9604	0.522992	0.261496	+	0.965204i	$0.415784\pi$
$524$	−44.9584	−1.96402
$525$	0	0
$526$	1.52312	0.0664113
$527$	4.37990	0.190791
$528$	0	0
$529$	2.06645	0.0898455
$530$	21.4805	0.933051
$531$	0	0
$532$	35.4739	1.53799
$533$	0.296146	0.0128275
$534$	0	0
$535$	0.269147	0.0116363
$536$	7.81900	0.337730
$537$	0	0
$538$	−12.1280	−0.522874
$539$	−6.11350	−0.263327
$540$	0	0
$541$	−40.3424	−1.73445	−0.867227	−	0.497914i	$-0.834100\pi$
$541$	−40.3424	−1.73445	−0.867227	+	0.497914i	$0.834100\pi$
$542$	−1.06547	−0.0457660
$543$	0	0
$544$	−22.5404	−0.966410
$545$	−14.4318	−0.618190
$546$	0	0
$547$	29.9772	1.28173	0.640866	−	0.767653i	$-0.278575\pi$
$547$	29.9772	1.28173	0.640866	+	0.767653i	$0.278575\pi$
$548$	−34.3772	−1.46852
$549$	0	0
$550$	3.13868	0.133834
$551$	56.3626	2.40113
$552$	0	0
$553$	5.84227	0.248439
$554$	−13.6740	−0.580951
$555$	0	0
$556$	16.4351	0.697004
$557$	−17.3802	−0.736421	−0.368211	−	0.929742i	$-0.620030\pi$
$557$	−17.3802	−0.736421	−0.368211	+	0.929742i	$0.620030\pi$
$558$	0	0
$559$	−1.77438	−0.0750484
$560$	−4.71066	−0.199062
$561$	0	0
$562$	31.6316	1.33430
$563$	−22.2885	−0.939350	−0.469675	−	0.882840i	$-0.655629\pi$
$563$	−22.2885	−0.939350	−0.469675	+	0.882840i	$0.655629\pi$
$564$	0	0
$565$	−12.7048	−0.534493
$566$	−16.6208	−0.698624
$567$	0	0
$568$	13.6549	0.572949
$569$	−33.8734	−1.42005	−0.710024	−	0.704178i	$-0.751316\pi$
$569$	−33.8734	−1.42005	−0.710024	+	0.704178i	$0.751316\pi$
$570$	0	0
$571$	−4.72783	−0.197853	−0.0989267	−	0.995095i	$-0.531541\pi$
$571$	−4.72783	−0.197853	−0.0989267	+	0.995095i	$0.531541\pi$
$572$	−1.17811	−0.0492591
$573$	0	0
$574$	−3.32996	−0.138990
$575$	−5.00664	−0.208791
$576$	0	0
$577$	−2.28568	−0.0951540	−0.0475770	−	0.998868i	$-0.515150\pi$
$577$	−2.28568	−0.0951540	−0.0475770	+	0.998868i	$0.515150\pi$
$578$	−19.0610	−0.792834
$579$	0	0
$580$	16.6988	0.693379
$581$	22.6178	0.938344
$582$	0	0
$583$	15.0150	0.621859
$584$	16.8300	0.696432
$585$	0	0
$586$	−23.1515	−0.956378
$587$	−23.5234	−0.970914	−0.485457	−	0.874261i	$-0.661347\pi$
$587$	−23.5234	−0.970914	−0.485457	+	0.874261i	$0.661347\pi$
$588$	0	0
$589$	13.0115	0.536131
$590$	−28.2584	−1.16338
$591$	0	0
$592$	7.29018	0.299624
$593$	42.3038	1.73721	0.868605	−	0.495506i	$-0.165017\pi$
$593$	42.3038	1.73721	0.868605	+	0.495506i	$0.165017\pi$
$594$	0	0
$595$	4.79526	0.196586
$596$	11.3801	0.466148
$597$	0	0
$598$	3.38857	0.138569
$599$	27.3481	1.11741	0.558707	−	0.829365i	$-0.311298\pi$
$599$	27.3481	1.11741	0.558707	+	0.829365i	$0.311298\pi$
$600$	0	0
$601$	−9.28359	−0.378686	−0.189343	−	0.981911i	$-0.560636\pi$
$601$	−9.28359	−0.378686	−0.189343	+	0.981911i	$0.560636\pi$
$602$	19.9517	0.813172
$603$	0	0
$604$	−3.58038	−0.145684
$605$	−8.80603	−0.358016
$606$	0	0
$607$	−8.10856	−0.329116	−0.164558	−	0.986367i	$-0.552620\pi$
$607$	−8.10856	−0.329116	−0.164558	+	0.986367i	$0.552620\pi$
$608$	−66.9616	−2.71565
$609$	0	0
$610$	−9.54783	−0.386580
$611$	4.03004	0.163038
$612$	0	0
$613$	−16.0542	−0.648425	−0.324212	−	0.945984i	$-0.605099\pi$
$613$	−16.0542	−0.648425	−0.324212	+	0.945984i	$0.605099\pi$
$614$	−46.3436	−1.87028
$615$	0	0
$616$	2.60763	0.105064
$617$	−38.7961	−1.56187	−0.780936	−	0.624610i	$-0.785258\pi$
$617$	−38.7961	−1.56187	−0.780936	+	0.624610i	$0.785258\pi$
$618$	0	0
$619$	−3.40366	−0.136805	−0.0684024	−	0.997658i	$-0.521790\pi$
$619$	−3.40366	−0.136805	−0.0684024	+	0.997658i	$0.521790\pi$
$620$	3.85498	0.154820
$621$	0	0
$622$	−23.9613	−0.960759
$623$	1.69488	0.0679039
$624$	0	0
$625$	1.00000	0.0400000
$626$	−1.55665	−0.0622161
$627$	0	0
$628$	5.27252	0.210397
$629$	−7.42110	−0.295899
$630$	0	0
$631$	43.8475	1.74554	0.872771	−	0.488130i	$-0.162321\pi$
$631$	43.8475	1.74554	0.872771	+	0.488130i	$0.162321\pi$
$632$	3.58043	0.142422
$633$	0	0
$634$	52.0581	2.06749
$635$	18.0049	0.714501
$636$	0	0
$637$	1.31829	0.0522327
$638$	21.0475	0.833278
$639$	0	0
$640$	−8.06005	−0.318602
$641$	−37.7328	−1.49036	−0.745178	−	0.666865i	$-0.767636\pi$
$641$	−37.7328	−1.49036	−0.745178	+	0.666865i	$0.767636\pi$
$642$	0	0
$643$	5.22355	0.205997	0.102998	−	0.994682i	$-0.467156\pi$
$643$	5.22355	0.205997	0.102998	+	0.994682i	$0.467156\pi$
$644$	−21.1308	−0.832671
$645$	0	0
$646$	50.3899	1.98257
$647$	25.1312	0.988009	0.494005	−	0.869459i	$-0.335533\pi$
$647$	25.1312	0.988009	0.494005	+	0.869459i	$0.335533\pi$
$648$	0	0
$649$	−19.7529	−0.775370
$650$	−0.676814	−0.0265468
$651$	0	0
$652$	−42.0645	−1.64737
$653$	14.3246	0.560565	0.280283	−	0.959918i	$-0.409572\pi$
$653$	14.3246	0.560565	0.280283	+	0.959918i	$0.409572\pi$
$654$	0	0
$655$	−18.0542	−0.705438
$656$	2.57698	0.100614
$657$	0	0
$658$	−45.3150	−1.76656
$659$	−28.4078	−1.10661	−0.553306	−	0.832978i	$-0.686634\pi$
$659$	−28.4078	−1.10661	−0.553306	+	0.832978i	$0.686634\pi$
$660$	0	0
$661$	−22.4607	−0.873620	−0.436810	−	0.899554i	$-0.643892\pi$
$661$	−22.4607	−0.873620	−0.436810	+	0.899554i	$0.643892\pi$
$662$	−28.4786	−1.10685
$663$	0	0
$664$	13.8613	0.537922
$665$	14.2455	0.552416
$666$	0	0
$667$	−33.5737	−1.29998
$668$	−56.6133	−2.19043
$669$	0	0
$670$	15.9511	0.616246
$671$	−6.67402	−0.257648
$672$	0	0
$673$	36.9634	1.42484	0.712418	−	0.701756i	$-0.247600\pi$
$673$	36.9634	1.42484	0.712418	+	0.701756i	$0.247600\pi$
$674$	75.2476	2.89843
$675$	0	0
$676$	−32.1184	−1.23532
$677$	9.24040	0.355138	0.177569	−	0.984108i	$-0.443177\pi$
$677$	9.24040	0.355138	0.177569	+	0.984108i	$0.443177\pi$
$678$	0	0
$679$	23.1419	0.888106
$680$	2.93877	0.112697
$681$	0	0
$682$	4.85890	0.186057
$683$	27.4336	1.04972	0.524859	−	0.851189i	$-0.324118\pi$
$683$	27.4336	1.04972	0.524859	+	0.851189i	$0.324118\pi$
$684$	0	0
$685$	−13.8051	−0.527465
$686$	−39.9635	−1.52581
$687$	0	0
$688$	−15.4402	−0.588652
$689$	−3.23779	−0.123350
$690$	0	0
$691$	7.97770	0.303486	0.151743	−	0.988420i	$-0.451511\pi$
$691$	7.97770	0.303486	0.151743	+	0.988420i	$0.451511\pi$
$692$	−29.1843	−1.10942
$693$	0	0
$694$	6.62671	0.251546
$695$	6.59996	0.250351
$696$	0	0
$697$	−2.62326	−0.0993632
$698$	−9.75214	−0.369124
$699$	0	0
$700$	4.22056	0.159522
$701$	−30.3161	−1.14502	−0.572511	−	0.819897i	$-0.694031\pi$
$701$	−30.3161	−1.14502	−0.572511	+	0.819897i	$0.694031\pi$
$702$	0	0
$703$	−22.0462	−0.831487
$704$	−16.7719	−0.632114
$705$	0	0
$706$	16.9747	0.638852
$707$	−13.8891	−0.522353
$708$	0	0
$709$	−39.8478	−1.49651	−0.748257	−	0.663409i	$-0.769109\pi$
$709$	−39.8478	−1.49651	−0.748257	+	0.663409i	$0.769109\pi$
$710$	27.8567	1.04544
$711$	0	0
$712$	1.03870	0.0389271
$713$	−7.75063	−0.290263
$714$	0	0
$715$	−0.473100	−0.0176929
$716$	−19.0296	−0.711169
$717$	0	0
$718$	0.492046	0.0183630
$719$	4.33634	0.161718	0.0808592	−	0.996726i	$-0.474234\pi$
$719$	4.33634	0.161718	0.0808592	+	0.996726i	$0.474234\pi$
$720$	0	0
$721$	11.8930	0.442918
$722$	109.434	4.07272
$723$	0	0
$724$	51.8489	1.92695
$725$	6.70584	0.249048
$726$	0	0
$727$	−3.88666	−0.144148	−0.0720740	−	0.997399i	$-0.522962\pi$
$727$	−3.88666	−0.144148	−0.0720740	+	0.997399i	$0.522962\pi$
$728$	−0.562300	−0.0208402
$729$	0	0
$730$	34.3341	1.27076
$731$	15.7175	0.581332
$732$	0	0
$733$	38.8407	1.43461	0.717307	−	0.696758i	$-0.245375\pi$
$733$	38.8407	1.43461	0.717307	+	0.696758i	$0.245375\pi$
$734$	−37.5832	−1.38722
$735$	0	0
$736$	39.8872	1.47026
$737$	11.1500	0.410715
$738$	0	0
$739$	21.4174	0.787851	0.393926	−	0.919142i	$-0.371117\pi$
$739$	21.4174	0.787851	0.393926	+	0.919142i	$0.371117\pi$
$740$	−6.53171	−0.240110
$741$	0	0
$742$	36.4068	1.33653
$743$	19.7668	0.725173	0.362586	−	0.931950i	$-0.381894\pi$
$743$	19.7668	0.725173	0.362586	+	0.931950i	$0.381894\pi$
$744$	0	0
$745$	4.56999	0.167431
$746$	−70.2664	−2.57264
$747$	0	0
$748$	10.4357	0.381566
$749$	0.456172	0.0166682
$750$	0	0
$751$	−35.6231	−1.29991	−0.649953	−	0.759974i	$-0.725212\pi$
$751$	−35.6231	−1.29991	−0.649953	+	0.759974i	$0.725212\pi$
$752$	35.0683	1.27881
$753$	0	0
$754$	−4.53861	−0.165286
$755$	−1.43780	−0.0523268
$756$	0	0
$757$	−16.7878	−0.610162	−0.305081	−	0.952326i	$-0.598684\pi$
$757$	−16.7878	−0.610162	−0.305081	+	0.952326i	$0.598684\pi$
$758$	8.34074	0.302949
$759$	0	0
$760$	8.73032	0.316682
$761$	−37.3301	−1.35321	−0.676607	−	0.736345i	$-0.736550\pi$
$761$	−37.3301	−1.35321	−0.676607	+	0.736345i	$0.736550\pi$
$762$	0	0
$763$	−24.4601	−0.885516
$764$	−45.2759	−1.63803
$765$	0	0
$766$	−18.8828	−0.682265
$767$	4.25945	0.153800
$768$	0	0
$769$	15.0089	0.541234	0.270617	−	0.962687i	$-0.412772\pi$
$769$	15.0089	0.541234	0.270617	+	0.962687i	$0.412772\pi$
$770$	5.31968	0.191708
$771$	0	0
$772$	16.2776	0.585845
$773$	44.6758	1.60688	0.803439	−	0.595387i	$-0.203001\pi$
$773$	44.6758	1.60688	0.803439	+	0.595387i	$0.203001\pi$
$774$	0	0
$775$	1.54807	0.0556083
$776$	14.1825	0.509122
$777$	0	0
$778$	−41.0141	−1.47043
$779$	−7.79304	−0.279215
$780$	0	0
$781$	19.4721	0.696767
$782$	−30.0159	−1.07337
$783$	0	0
$784$	11.4714	0.409695
$785$	2.11732	0.0755705
$786$	0	0
$787$	2.49787	0.0890396	0.0445198	−	0.999009i	$-0.485824\pi$
$787$	2.49787	0.0890396	0.0445198	+	0.999009i	$0.485824\pi$
$788$	−51.9175	−1.84949
$789$	0	0
$790$	7.30425	0.259873
$791$	−21.5330	−0.765626
$792$	0	0
$793$	1.43916	0.0511062
$794$	−21.1088	−0.749125
$795$	0	0
$796$	9.68573	0.343302
$797$	−9.64189	−0.341533	−0.170767	−	0.985311i	$-0.554624\pi$
$797$	−9.64189	−0.341533	−0.170767	+	0.985311i	$0.554624\pi$
$798$	0	0
$799$	−35.6981	−1.26291
$800$	−7.96687	−0.281671
$801$	0	0
$802$	1.05295	0.0371811
$803$	23.9998	0.846936
$804$	0	0
$805$	−8.48565	−0.299080
$806$	−1.04776	−0.0369056
$807$	0	0
$808$	−8.51191	−0.299448
$809$	−36.3025	−1.27633	−0.638164	−	0.769900i	$-0.720306\pi$
$809$	−36.3025	−1.27633	−0.638164	+	0.769900i	$0.720306\pi$
$810$	0	0
$811$	−28.9585	−1.01687	−0.508436	−	0.861100i	$-0.669776\pi$
$811$	−28.9585	−1.01687	−0.508436	+	0.861100i	$0.669776\pi$
$812$	28.3024	0.993219
$813$	0	0
$814$	−8.23270	−0.288556
$815$	−16.8921	−0.591706
$816$	0	0
$817$	46.6926	1.63357
$818$	38.2517	1.33744
$819$	0	0
$820$	−2.30887	−0.0806294
$821$	29.6355	1.03429	0.517143	−	0.855899i	$-0.326996\pi$
$821$	29.6355	1.03429	0.517143	+	0.855899i	$0.326996\pi$
$822$	0	0
$823$	14.5222	0.506213	0.253107	−	0.967438i	$-0.418548\pi$
$823$	14.5222	0.506213	0.253107	+	0.967438i	$0.418548\pi$
$824$	7.28861	0.253911
$825$	0	0
$826$	−47.8946	−1.66647
$827$	−3.68209	−0.128039	−0.0640195	−	0.997949i	$-0.520392\pi$
$827$	−3.68209	−0.128039	−0.0640195	+	0.997949i	$0.520392\pi$
$828$	0	0
$829$	−27.2182	−0.945328	−0.472664	−	0.881243i	$-0.656708\pi$
$829$	−27.2182	−0.945328	−0.472664	+	0.881243i	$0.656708\pi$
$830$	28.2777	0.981533
$831$	0	0
$832$	3.61663	0.125384
$833$	−11.6775	−0.404600
$834$	0	0
$835$	−22.7346	−0.786762
$836$	31.0017	1.07221
$837$	0	0
$838$	−40.3686	−1.39451
$839$	57.2445	1.97630	0.988149	−	0.153497i	$-0.0490536\pi$
$839$	57.2445	1.97630	0.988149	+	0.153497i	$0.0490536\pi$
$840$	0	0
$841$	15.9682	0.550629
$842$	−68.3627	−2.35593
$843$	0	0
$844$	16.5000	0.567952
$845$	−12.8980	−0.443704
$846$	0	0
$847$	−14.9252	−0.512834
$848$	−28.1744	−0.967512
$849$	0	0
$850$	5.99523	0.205635
$851$	13.1323	0.450170
$852$	0	0
$853$	−12.7251	−0.435699	−0.217849	−	0.975982i	$-0.569904\pi$
$853$	−12.7251	−0.435699	−0.217849	+	0.975982i	$0.569904\pi$
$854$	−16.1824	−0.553751
$855$	0	0
$856$	0.279565	0.00955532
$857$	−26.8788	−0.918163	−0.459081	−	0.888394i	$-0.651821\pi$
$857$	−26.8788	−0.918163	−0.459081	+	0.888394i	$0.651821\pi$
$858$	0	0
$859$	20.8053	0.709868	0.354934	−	0.934891i	$-0.384503\pi$
$859$	20.8053	0.709868	0.354934	+	0.934891i	$0.384503\pi$
$860$	13.8338	0.471728
$861$	0	0
$862$	24.8670	0.846975
$863$	−13.2449	−0.450861	−0.225430	−	0.974259i	$-0.572379\pi$
$863$	−13.2449	−0.450861	−0.225430	+	0.974259i	$0.572379\pi$
$864$	0	0
$865$	−11.7197	−0.398482
$866$	−14.5589	−0.494732
$867$	0	0
$868$	6.53372	0.221769
$869$	5.10574	0.173200
$870$	0	0
$871$	−2.40435	−0.0814682
$872$	−14.9904	−0.507638
$873$	0	0
$874$	−89.1696	−3.01621
$875$	1.69488	0.0572974
$876$	0	0
$877$	17.2714	0.583215	0.291607	−	0.956538i	$-0.405810\pi$
$877$	17.2714	0.583215	0.291607	+	0.956538i	$0.405810\pi$
$878$	−56.7433	−1.91499
$879$	0	0
$880$	−4.11679	−0.138777
$881$	39.6846	1.33701	0.668504	−	0.743709i	$-0.266935\pi$
$881$	39.6846	1.33701	0.668504	+	0.743709i	$0.266935\pi$
$882$	0	0
$883$	−20.5626	−0.691988	−0.345994	−	0.938237i	$-0.612458\pi$
$883$	−20.5626	−0.691988	−0.345994	+	0.938237i	$0.612458\pi$
$884$	−2.25031	−0.0756862
$885$	0	0
$886$	12.5546	0.421781
$887$	7.24097	0.243128	0.121564	−	0.992584i	$-0.461209\pi$
$887$	7.24097	0.243128	0.121564	+	0.992584i	$0.461209\pi$
$888$	0	0
$889$	30.5161	1.02348
$890$	2.11901	0.0710292
$891$	0	0
$892$	−0.992535	−0.0332325
$893$	−106.050	−3.54882
$894$	0	0
$895$	−7.64183	−0.255438
$896$	−13.6608	−0.456376
$897$	0	0
$898$	1.46552	0.0489051
$899$	10.3811	0.346229
$900$	0	0
$901$	28.6804	0.955482
$902$	−2.91015	−0.0968975
$903$	0	0
$904$	−13.1965	−0.438909
$905$	20.8213	0.692123
$906$	0	0
$907$	28.2581	0.938296	0.469148	−	0.883120i	$-0.344561\pi$
$907$	28.2581	0.938296	0.469148	+	0.883120i	$0.344561\pi$
$908$	−36.7553	−1.21977
$909$	0	0
$910$	−1.14712	−0.0380266
$911$	−20.4140	−0.676346	−0.338173	−	0.941084i	$-0.609809\pi$
$911$	−20.4140	−0.676346	−0.338173	+	0.941084i	$0.609809\pi$
$912$	0	0
$913$	19.7664	0.654171
$914$	−62.8710	−2.07959
$915$	0	0
$916$	−33.3318	−1.10132
$917$	−30.5998	−1.01049
$918$	0	0
$919$	4.98893	0.164570	0.0822849	−	0.996609i	$-0.473778\pi$
$919$	4.98893	0.164570	0.0822849	+	0.996609i	$0.473778\pi$
$920$	−5.20042	−0.171453
$921$	0	0
$922$	22.3682	0.736656
$923$	−4.19890	−0.138208
$924$	0	0
$925$	−2.62298	−0.0862431
$926$	−45.5208	−1.49591
$927$	0	0
$928$	−53.4245	−1.75374
$929$	−7.08398	−0.232418	−0.116209	−	0.993225i	$-0.537074\pi$
$929$	−7.08398	−0.232418	−0.116209	+	0.993225i	$0.537074\pi$
$930$	0	0
$931$	−34.6907	−1.13694
$932$	61.2582	2.00658
$933$	0	0
$934$	1.76667	0.0578072
$935$	4.19072	0.137051
$936$	0	0
$937$	−17.0622	−0.557398	−0.278699	−	0.960379i	$-0.589903\pi$
$937$	−17.0622	−0.557398	−0.278699	+	0.960379i	$0.589903\pi$
$938$	27.0352	0.882732
$939$	0	0
$940$	−31.4198	−1.02480
$941$	42.3077	1.37919	0.689595	−	0.724195i	$-0.257789\pi$
$941$	42.3077	1.37919	0.689595	+	0.724195i	$0.257789\pi$
$942$	0	0
$943$	4.64211	0.151168
$944$	37.0646	1.20635
$945$	0	0
$946$	17.4364	0.566906
$947$	43.4876	1.41316	0.706579	−	0.707634i	$-0.250237\pi$
$947$	43.4876	1.41316	0.706579	+	0.707634i	$0.250237\pi$
$948$	0	0
$949$	−5.17524	−0.167995
$950$	17.8103	0.577841
$951$	0	0
$952$	4.98086	0.161431
$953$	5.76627	0.186788	0.0933939	−	0.995629i	$-0.470228\pi$
$953$	5.76627	0.186788	0.0933939	+	0.995629i	$0.470228\pi$
$954$	0	0
$955$	−18.1818	−0.588348
$956$	39.9128	1.29087
$957$	0	0
$958$	−72.6057	−2.34578
$959$	−23.3980	−0.755559
$960$	0	0
$961$	−28.6035	−0.922693
$962$	1.77527	0.0572370
$963$	0	0
$964$	−69.8033	−2.24821
$965$	6.53672	0.210424
$966$	0	0
$967$	36.4037	1.17066	0.585332	−	0.810794i	$-0.300964\pi$
$967$	36.4037	1.17066	0.585332	+	0.810794i	$0.300964\pi$
$968$	−9.14686	−0.293991
$969$	0	0
$970$	28.9330	0.928982
$971$	56.2151	1.80403	0.902014	−	0.431706i	$-0.142088\pi$
$971$	56.2151	1.80403	0.902014	+	0.431706i	$0.142088\pi$
$972$	0	0
$973$	11.1861	0.358611
$974$	−19.9114	−0.638001
$975$	0	0
$976$	12.5232	0.400858
$977$	−21.0645	−0.673913	−0.336957	−	0.941520i	$-0.609398\pi$
$977$	−21.0645	−0.673913	−0.336957	+	0.941520i	$0.609398\pi$
$978$	0	0
$979$	1.48120	0.0473395
$980$	−10.2780	−0.328317
$981$	0	0
$982$	45.1596	1.44110
$983$	17.3301	0.552743	0.276371	−	0.961051i	$-0.410868\pi$
$983$	17.3301	0.552743	0.276371	+	0.961051i	$0.410868\pi$
$984$	0	0
$985$	−20.8489	−0.664300
$986$	40.2030	1.28032
$987$	0	0
$988$	−6.68509	−0.212681
$989$	−27.8135	−0.884419
$990$	0	0
$991$	−11.0194	−0.350042	−0.175021	−	0.984565i	$-0.555999\pi$
$991$	−11.0194	−0.350042	−0.175021	+	0.984565i	$0.555999\pi$
$992$	−12.3333	−0.391581
$993$	0	0
$994$	47.2138	1.49753
$995$	3.88956	0.123307
$996$	0	0
$997$	22.3031	0.706347	0.353173	−	0.935558i	$-0.385103\pi$
$997$	22.3031	0.706347	0.353173	+	0.935558i	$0.385103\pi$
$998$	−8.91400	−0.282168
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.t.1.8		10
3.2	odd	2		1335.2.a.i.1.3	✓	10
15.14	odd	2		6675.2.a.ba.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.i.1.3	✓	10	3.2	odd	2
4005.2.a.t.1.8		10	1.1	even	1	trivial
6675.2.a.ba.1.8		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)

\(f(q)\)	\(=\)	\(q+2.11901 q^{2} +2.49018 q^{4} +1.00000 q^{5} +1.69488 q^{7} +1.03870 q^{8} +O(q^{10})\)	\(q+2.11901 q^{2} +2.49018 q^{4} +1.00000 q^{5} +1.69488 q^{7} +1.03870 q^{8} +2.11901 q^{10} +1.48120 q^{11} -0.319402 q^{13} +3.59146 q^{14} -2.77935 q^{16} +2.82926 q^{17} +8.40501 q^{19} +2.49018 q^{20} +3.13868 q^{22} -5.00664 q^{23} +1.00000 q^{25} -0.676814 q^{26} +4.22056 q^{28} +6.70584 q^{29} +1.54807 q^{31} -7.96687 q^{32} +5.99523 q^{34} +1.69488 q^{35} -2.62298 q^{37} +17.8103 q^{38} +1.03870 q^{40} -0.927190 q^{41} +5.55533 q^{43} +3.68847 q^{44} -10.6091 q^{46} -12.6174 q^{47} -4.12739 q^{49} +2.11901 q^{50} -0.795370 q^{52} +10.1370 q^{53} +1.48120 q^{55} +1.76048 q^{56} +14.2097 q^{58} -13.3357 q^{59} -4.50581 q^{61} +3.28037 q^{62} -11.3231 q^{64} -0.319402 q^{65} +7.52765 q^{67} +7.04539 q^{68} +3.59146 q^{70} +13.1461 q^{71} +16.2029 q^{73} -5.55811 q^{74} +20.9300 q^{76} +2.51046 q^{77} +3.44702 q^{79} -2.77935 q^{80} -1.96472 q^{82} +13.3448 q^{83} +2.82926 q^{85} +11.7718 q^{86} +1.53853 q^{88} +1.00000 q^{89} -0.541348 q^{91} -12.4675 q^{92} -26.7364 q^{94} +8.40501 q^{95} +13.6540 q^{97} -8.74595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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