LMFDB - Embedded newform 4005.2.a.u.1.1

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.75145$ 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)

$f(q)$	$=$	$q-2.75145 q^{2} +5.57047 q^{4} +1.00000 q^{5} -4.05689 q^{7} -9.82397 q^{8} +O(q^{10})$	\(q-2.75145 q^{2} +5.57047 q^{4} +1.00000 q^{5} -4.05689 q^{7} -9.82397 q^{8} -2.75145 q^{10} +5.98172 q^{11} -1.19173 q^{13} +11.1623 q^{14} +15.8892 q^{16} +4.77273 q^{17} -4.76512 q^{19} +5.57047 q^{20} -16.4584 q^{22} -0.291033 q^{23} +1.00000 q^{25} +3.27898 q^{26} -22.5988 q^{28} -4.91815 q^{29} -2.46506 q^{31} -24.0704 q^{32} -13.1319 q^{34} -4.05689 q^{35} +4.09424 q^{37} +13.1110 q^{38} -9.82397 q^{40} -5.80645 q^{41} -8.92265 q^{43} +33.3210 q^{44} +0.800761 q^{46} -3.80957 q^{47} +9.45836 q^{49} -2.75145 q^{50} -6.63850 q^{52} +9.51147 q^{53} +5.98172 q^{55} +39.8548 q^{56} +13.5320 q^{58} +9.57946 q^{59} -5.87185 q^{61} +6.78249 q^{62} +34.4501 q^{64} -1.19173 q^{65} -4.40203 q^{67} +26.5864 q^{68} +11.1623 q^{70} +12.9413 q^{71} -5.72908 q^{73} -11.2651 q^{74} -26.5440 q^{76} -24.2672 q^{77} -6.02402 q^{79} +15.8892 q^{80} +15.9761 q^{82} -13.3496 q^{83} +4.77273 q^{85} +24.5502 q^{86} -58.7642 q^{88} +1.00000 q^{89} +4.83472 q^{91} -1.62119 q^{92} +10.4818 q^{94} -4.76512 q^{95} +12.0413 q^{97} -26.0242 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * 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 $

$2$	−2.75145	−1.94557	−0.972784	−	0.231713i	$-0.925567\pi$
$2$	−2.75145	−1.94557	−0.972784	+	0.231713i	$0.925567\pi$
$3$	0	0
$3$	0	0
$4$	5.57047	2.78524
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.05689	−1.53336	−0.766680	−	0.642029i	$-0.778093\pi$
$7$	−4.05689	−1.53336	−0.766680	+	0.642029i	$0.778093\pi$
$8$	−9.82397	−3.47330
$9$	0	0
$10$	−2.75145	−0.870085
$11$	5.98172	1.80356	0.901778	−	0.432199i	$-0.142262\pi$
$11$	5.98172	1.80356	0.901778	+	0.432199i	$0.142262\pi$
$12$	0	0
$13$	−1.19173	−0.330526	−0.165263	−	0.986250i	$-0.552847\pi$
$13$	−1.19173	−0.330526	−0.165263	+	0.986250i	$0.552847\pi$
$14$	11.1623	2.98326
$15$	0	0
$16$	15.8892	3.97230
$17$	4.77273	1.15756	0.578778	−	0.815485i	$-0.303530\pi$
$17$	4.77273	1.15756	0.578778	+	0.815485i	$0.303530\pi$
$18$	0	0
$19$	−4.76512	−1.09319	−0.546597	−	0.837396i	$-0.684077\pi$
$19$	−4.76512	−1.09319	−0.546597	+	0.837396i	$0.684077\pi$
$20$	5.57047	1.24560
$21$	0	0
$22$	−16.4584	−3.50894
$23$	−0.291033	−0.0606845	−0.0303422	−	0.999540i	$-0.509660\pi$
$23$	−0.291033	−0.0606845	−0.0303422	+	0.999540i	$0.509660\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.27898	0.643062
$27$	0	0
$28$	−22.5988	−4.27077
$29$	−4.91815	−0.913277	−0.456639	−	0.889652i	$-0.650947\pi$
$29$	−4.91815	−0.913277	−0.456639	+	0.889652i	$0.650947\pi$
$30$	0	0
$31$	−2.46506	−0.442738	−0.221369	−	0.975190i	$-0.571052\pi$
$31$	−2.46506	−0.442738	−0.221369	+	0.975190i	$0.571052\pi$
$32$	−24.0704	−4.25509
$33$	0	0
$34$	−13.1319	−2.25211
$35$	−4.05689	−0.685740
$36$	0	0
$37$	4.09424	0.673088	0.336544	−	0.941668i	$-0.390742\pi$
$37$	4.09424	0.673088	0.336544	+	0.941668i	$0.390742\pi$
$38$	13.1110	2.12688
$39$	0	0
$40$	−9.82397	−1.55331
$41$	−5.80645	−0.906815	−0.453407	−	0.891303i	$-0.649792\pi$
$41$	−5.80645	−0.906815	−0.453407	+	0.891303i	$0.649792\pi$
$42$	0	0
$43$	−8.92265	−1.36069	−0.680345	−	0.732892i	$-0.738170\pi$
$43$	−8.92265	−1.36069	−0.680345	+	0.732892i	$0.738170\pi$
$44$	33.3210	5.02333
$45$	0	0
$46$	0.800761	0.118066
$47$	−3.80957	−0.555683	−0.277842	−	0.960627i	$-0.589619\pi$
$47$	−3.80957	−0.555683	−0.277842	+	0.960627i	$0.589619\pi$
$48$	0	0
$49$	9.45836	1.35119
$50$	−2.75145	−0.389114
$51$	0	0
$52$	−6.63850	−0.920594
$53$	9.51147	1.30650	0.653250	−	0.757142i	$-0.273405\pi$
$53$	9.51147	1.30650	0.653250	+	0.757142i	$0.273405\pi$
$54$	0	0
$55$	5.98172	0.806575
$56$	39.8548	5.32582
$57$	0	0
$58$	13.5320	1.77684
$59$	9.57946	1.24714	0.623570	−	0.781768i	$-0.285682\pi$
$59$	9.57946	1.24714	0.623570	+	0.781768i	$0.285682\pi$
$60$	0	0
$61$	−5.87185	−0.751814	−0.375907	−	0.926657i	$-0.622669\pi$
$61$	−5.87185	−0.751814	−0.375907	+	0.926657i	$0.622669\pi$
$62$	6.78249	0.861376
$63$	0	0
$64$	34.4501	4.30626
$65$	−1.19173	−0.147816
$66$	0	0
$67$	−4.40203	−0.537794	−0.268897	−	0.963169i	$-0.586659\pi$
$67$	−4.40203	−0.537794	−0.268897	+	0.963169i	$0.586659\pi$
$68$	26.5864	3.22407
$69$	0	0
$70$	11.1623	1.33415
$71$	12.9413	1.53585	0.767926	−	0.640539i	$-0.221289\pi$
$71$	12.9413	1.53585	0.767926	+	0.640539i	$0.221289\pi$
$72$	0	0
$73$	−5.72908	−0.670538	−0.335269	−	0.942122i	$-0.608827\pi$
$73$	−5.72908	−0.670538	−0.335269	+	0.942122i	$0.608827\pi$
$74$	−11.2651	−1.30954
$75$	0	0
$76$	−26.5440	−3.04480
$77$	−24.2672	−2.76550
$78$	0	0
$79$	−6.02402	−0.677755	−0.338878	−	0.940830i	$-0.610047\pi$
$79$	−6.02402	−0.677755	−0.338878	+	0.940830i	$0.610047\pi$
$80$	15.8892	1.77647
$81$	0	0
$82$	15.9761	1.76427
$83$	−13.3496	−1.46531	−0.732657	−	0.680598i	$-0.761720\pi$
$83$	−13.3496	−1.46531	−0.732657	+	0.680598i	$0.761720\pi$
$84$	0	0
$85$	4.77273	0.517675
$86$	24.5502	2.64732
$87$	0	0
$88$	−58.7642	−6.26429
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	4.83472	0.506816
$92$	−1.62119	−0.169021
$93$	0	0
$94$	10.4818	1.08112
$95$	−4.76512	−0.488891
$96$	0	0
$97$	12.0413	1.22261	0.611305	−	0.791395i	$-0.290645\pi$
$97$	12.0413	1.22261	0.611305	+	0.791395i	$0.290645\pi$
$98$	−26.0242	−2.62884
$99$	0	0
$100$	5.57047	0.557047
$101$	−4.48797	−0.446570	−0.223285	−	0.974753i	$-0.571678\pi$
$101$	−4.48797	−0.446570	−0.223285	+	0.974753i	$0.571678\pi$
$102$	0	0
$103$	−8.91374	−0.878297	−0.439148	−	0.898415i	$-0.644720\pi$
$103$	−8.91374	−0.878297	−0.439148	+	0.898415i	$0.644720\pi$
$104$	11.7075	1.14802
$105$	0	0
$106$	−26.1703	−2.54189
$107$	−5.27407	−0.509863	−0.254932	−	0.966959i	$-0.582053\pi$
$107$	−5.27407	−0.509863	−0.254932	+	0.966959i	$0.582053\pi$
$108$	0	0
$109$	15.0186	1.43852	0.719259	−	0.694742i	$-0.244481\pi$
$109$	15.0186	1.43852	0.719259	+	0.694742i	$0.244481\pi$
$110$	−16.4584	−1.56925
$111$	0	0
$112$	−64.4608	−6.09097
$113$	−13.0409	−1.22679	−0.613393	−	0.789778i	$-0.710196\pi$
$113$	−13.0409	−1.22679	−0.613393	+	0.789778i	$0.710196\pi$
$114$	0	0
$115$	−0.291033	−0.0271389
$116$	−27.3964	−2.54369
$117$	0	0
$118$	−26.3574	−2.42639
$119$	−19.3624	−1.77495
$120$	0	0
$121$	24.7810	2.25282
$122$	16.1561	1.46271
$123$	0	0
$124$	−13.7315	−1.23313
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.9697	1.50582	0.752910	−	0.658124i	$-0.228650\pi$
$127$	16.9697	1.50582	0.752910	+	0.658124i	$0.228650\pi$
$128$	−46.6469	−4.12304
$129$	0	0
$130$	3.27898	0.287586
$131$	−4.02829	−0.351953	−0.175977	−	0.984394i	$-0.556308\pi$
$131$	−4.02829	−0.351953	−0.175977	+	0.984394i	$0.556308\pi$
$132$	0	0
$133$	19.3316	1.67626
$134$	12.1120	1.04631
$135$	0	0
$136$	−46.8872	−4.02054
$137$	−18.3534	−1.56804	−0.784019	−	0.620737i	$-0.786833\pi$
$137$	−18.3534	−1.56804	−0.784019	+	0.620737i	$0.786833\pi$
$138$	0	0
$139$	−22.2592	−1.88800	−0.944001	−	0.329942i	$-0.892971\pi$
$139$	−22.2592	−1.88800	−0.944001	+	0.329942i	$0.892971\pi$
$140$	−22.5988	−1.90995
$141$	0	0
$142$	−35.6074	−2.98810
$143$	−7.12859	−0.596123
$144$	0	0
$145$	−4.91815	−0.408430
$146$	15.7633	1.30458
$147$	0	0
$148$	22.8068	1.87471
$149$	8.29467	0.679526	0.339763	−	0.940511i	$-0.389653\pi$
$149$	8.29467	0.679526	0.339763	+	0.940511i	$0.389653\pi$
$150$	0	0
$151$	14.1481	1.15135	0.575677	−	0.817677i	$-0.304739\pi$
$151$	14.1481	1.15135	0.575677	+	0.817677i	$0.304739\pi$
$152$	46.8124	3.79699
$153$	0	0
$154$	66.7699	5.38047
$155$	−2.46506	−0.197998
$156$	0	0
$157$	−6.73020	−0.537129	−0.268564	−	0.963262i	$-0.586549\pi$
$157$	−6.73020	−0.537129	−0.268564	+	0.963262i	$0.586549\pi$
$158$	16.5748	1.31862
$159$	0	0
$160$	−24.0704	−1.90293
$161$	1.18069	0.0930512
$162$	0	0
$163$	17.5933	1.37802	0.689008	−	0.724754i	$-0.258047\pi$
$163$	17.5933	1.37802	0.689008	+	0.724754i	$0.258047\pi$
$164$	−32.3447	−2.52569
$165$	0	0
$166$	36.7309	2.85087
$167$	14.6628	1.13464	0.567320	−	0.823497i	$-0.307980\pi$
$167$	14.6628	1.13464	0.567320	+	0.823497i	$0.307980\pi$
$168$	0	0
$169$	−11.5798	−0.890752
$170$	−13.1319	−1.00717
$171$	0	0
$172$	−49.7034	−3.78985
$173$	10.6695	0.811186	0.405593	−	0.914054i	$-0.367065\pi$
$173$	10.6695	0.811186	0.405593	+	0.914054i	$0.367065\pi$
$174$	0	0
$175$	−4.05689	−0.306672
$176$	95.0448	7.16427
$177$	0	0
$178$	−2.75145	−0.206230
$179$	−17.9844	−1.34422	−0.672108	−	0.740453i	$-0.734611\pi$
$179$	−17.9844	−1.34422	−0.672108	+	0.740453i	$0.734611\pi$
$180$	0	0
$181$	−22.1752	−1.64827	−0.824135	−	0.566394i	$-0.808338\pi$
$181$	−22.1752	−1.64827	−0.824135	+	0.566394i	$0.808338\pi$
$182$	−13.3025	−0.986045
$183$	0	0
$184$	2.85910	0.210775
$185$	4.09424	0.301014
$186$	0	0
$187$	28.5491	2.08772
$188$	−21.2211	−1.54771
$189$	0	0
$190$	13.1110	0.951171
$191$	−8.29493	−0.600200	−0.300100	−	0.953908i	$-0.597020\pi$
$191$	−8.29493	−0.600200	−0.300100	+	0.953908i	$0.597020\pi$
$192$	0	0
$193$	−14.0704	−1.01281	−0.506405	−	0.862296i	$-0.669026\pi$
$193$	−14.0704	−1.01281	−0.506405	+	0.862296i	$0.669026\pi$
$194$	−33.1311	−2.37867
$195$	0	0
$196$	52.6875	3.76339
$197$	23.5952	1.68109	0.840544	−	0.541743i	$-0.182235\pi$
$197$	23.5952	1.68109	0.840544	+	0.541743i	$0.182235\pi$
$198$	0	0
$199$	−23.3891	−1.65801	−0.829004	−	0.559243i	$-0.811092\pi$
$199$	−23.3891	−1.65801	−0.829004	+	0.559243i	$0.811092\pi$
$200$	−9.82397	−0.694660
$201$	0	0
$202$	12.3484	0.868832
$203$	19.9524	1.40038
$204$	0	0
$205$	−5.80645	−0.405540
$206$	24.5257	1.70879
$207$	0	0
$208$	−18.9356	−1.31295
$209$	−28.5036	−1.97164
$210$	0	0
$211$	−11.1040	−0.764434	−0.382217	−	0.924073i	$-0.624839\pi$
$211$	−11.1040	−0.764434	−0.382217	+	0.924073i	$0.624839\pi$
$212$	52.9834	3.63891
$213$	0	0
$214$	14.5113	0.991974
$215$	−8.92265	−0.608519
$216$	0	0
$217$	10.0005	0.678876
$218$	−41.3228	−2.79874
$219$	0	0
$220$	33.3210	2.24650
$221$	−5.68780	−0.382603
$222$	0	0
$223$	3.95692	0.264975	0.132488	−	0.991185i	$-0.457704\pi$
$223$	3.95692	0.264975	0.132488	+	0.991185i	$0.457704\pi$
$224$	97.6511	6.52459
$225$	0	0
$226$	35.8814	2.38679
$227$	13.5191	0.897292	0.448646	−	0.893710i	$-0.351906\pi$
$227$	13.5191	0.897292	0.448646	+	0.893710i	$0.351906\pi$
$228$	0	0
$229$	4.42941	0.292704	0.146352	−	0.989233i	$-0.453247\pi$
$229$	4.42941	0.292704	0.146352	+	0.989233i	$0.453247\pi$
$230$	0.800761	0.0528006
$231$	0	0
$232$	48.3157	3.17208
$233$	16.5626	1.08505	0.542525	−	0.840040i	$-0.317469\pi$
$233$	16.5626	1.08505	0.542525	+	0.840040i	$0.317469\pi$
$234$	0	0
$235$	−3.80957	−0.248509
$236$	53.3621	3.47358
$237$	0	0
$238$	53.2748	3.45329
$239$	−0.614190	−0.0397286	−0.0198643	−	0.999803i	$-0.506323\pi$
$239$	−0.614190	−0.0397286	−0.0198643	+	0.999803i	$0.506323\pi$
$240$	0	0
$241$	18.1766	1.17086	0.585428	−	0.810725i	$-0.300927\pi$
$241$	18.1766	1.17086	0.585428	+	0.810725i	$0.300927\pi$
$242$	−68.1836	−4.38301
$243$	0	0
$244$	−32.7090	−2.09398
$245$	9.45836	0.604272
$246$	0	0
$247$	5.67874	0.361329
$248$	24.2167	1.53776
$249$	0	0
$250$	−2.75145	−0.174017
$251$	−21.8774	−1.38089	−0.690445	−	0.723385i	$-0.742585\pi$
$251$	−21.8774	−1.38089	−0.690445	+	0.723385i	$0.742585\pi$
$252$	0	0
$253$	−1.74088	−0.109448
$254$	−46.6913	−2.92967
$255$	0	0
$256$	59.4463	3.71539
$257$	5.00807	0.312395	0.156198	−	0.987726i	$-0.450076\pi$
$257$	5.00807	0.312395	0.156198	+	0.987726i	$0.450076\pi$
$258$	0	0
$259$	−16.6099	−1.03209
$260$	−6.63850	−0.411702
$261$	0	0
$262$	11.0836	0.684749
$263$	−22.7735	−1.40428	−0.702138	−	0.712041i	$-0.747771\pi$
$263$	−22.7735	−1.40428	−0.702138	+	0.712041i	$0.747771\pi$
$264$	0	0
$265$	9.51147	0.584285
$266$	−53.1899	−3.26128
$267$	0	0
$268$	−24.5214	−1.49788
$269$	−22.2032	−1.35376	−0.676878	−	0.736095i	$-0.736667\pi$
$269$	−22.2032	−1.35376	−0.676878	+	0.736095i	$0.736667\pi$
$270$	0	0
$271$	−0.952121	−0.0578373	−0.0289186	−	0.999582i	$-0.509206\pi$
$271$	−0.952121	−0.0578373	−0.0289186	+	0.999582i	$0.509206\pi$
$272$	75.8349	4.59817
$273$	0	0
$274$	50.4985	3.05072
$275$	5.98172	0.360711
$276$	0	0
$277$	−2.87171	−0.172544	−0.0862722	−	0.996272i	$-0.527495\pi$
$277$	−2.87171	−0.172544	−0.0862722	+	0.996272i	$0.527495\pi$
$278$	61.2451	3.67324
$279$	0	0
$280$	39.8548	2.38178
$281$	11.7144	0.698823	0.349411	−	0.936969i	$-0.386382\pi$
$281$	11.7144	0.698823	0.349411	+	0.936969i	$0.386382\pi$
$282$	0	0
$283$	−29.6040	−1.75978	−0.879888	−	0.475181i	$-0.842382\pi$
$283$	−29.6040	−1.75978	−0.879888	+	0.475181i	$0.842382\pi$
$284$	72.0892	4.27771
$285$	0	0
$286$	19.6140	1.15980
$287$	23.5561	1.39047
$288$	0	0
$289$	5.77894	0.339937
$290$	13.5320	0.794628
$291$	0	0
$292$	−31.9137	−1.86761
$293$	−25.0054	−1.46083	−0.730416	−	0.683003i	$-0.760674\pi$
$293$	−25.0054	−1.46083	−0.730416	+	0.683003i	$0.760674\pi$
$294$	0	0
$295$	9.57946	0.557738
$296$	−40.2217	−2.33784
$297$	0	0
$298$	−22.8224	−1.32206
$299$	0.346832	0.0200578
$300$	0	0
$301$	36.1982	2.08643
$302$	−38.9277	−2.24004
$303$	0	0
$304$	−75.7141	−4.34250
$305$	−5.87185	−0.336221
$306$	0	0
$307$	−13.8196	−0.788724	−0.394362	−	0.918955i	$-0.629034\pi$
$307$	−13.8196	−0.788724	−0.394362	+	0.918955i	$0.629034\pi$
$308$	−135.180	−7.70257
$309$	0	0
$310$	6.78249	0.385219
$311$	30.0444	1.70366	0.851831	−	0.523817i	$-0.175492\pi$
$311$	30.0444	1.70366	0.851831	+	0.523817i	$0.175492\pi$
$312$	0	0
$313$	16.8501	0.952424	0.476212	−	0.879331i	$-0.342010\pi$
$313$	16.8501	0.952424	0.476212	+	0.879331i	$0.342010\pi$
$314$	18.5178	1.04502
$315$	0	0
$316$	−33.5566	−1.88771
$317$	5.76327	0.323698	0.161849	−	0.986816i	$-0.448254\pi$
$317$	5.76327	0.323698	0.161849	+	0.986816i	$0.448254\pi$
$318$	0	0
$319$	−29.4190	−1.64715
$320$	34.4501	1.92582
$321$	0	0
$322$	−3.24860	−0.181037
$323$	−22.7426	−1.26543
$324$	0	0
$325$	−1.19173	−0.0661053
$326$	−48.4071	−2.68102
$327$	0	0
$328$	57.0424	3.14964
$329$	15.4550	0.852063
$330$	0	0
$331$	−15.7644	−0.866490	−0.433245	−	0.901276i	$-0.642631\pi$
$331$	−15.7644	−0.866490	−0.433245	+	0.901276i	$0.642631\pi$
$332$	−74.3638	−4.08124
$333$	0	0
$334$	−40.3439	−2.20752
$335$	−4.40203	−0.240509
$336$	0	0
$337$	3.64536	0.198576	0.0992878	−	0.995059i	$-0.468344\pi$
$337$	3.64536	0.198576	0.0992878	+	0.995059i	$0.468344\pi$
$338$	31.8612	1.73302
$339$	0	0
$340$	26.5864	1.44185
$341$	−14.7453	−0.798502
$342$	0	0
$343$	−9.97329	−0.538507
$344$	87.6558	4.72609
$345$	0	0
$346$	−29.3566	−1.57822
$347$	21.2099	1.13861	0.569304	−	0.822127i	$-0.307213\pi$
$347$	21.2099	1.13861	0.569304	+	0.822127i	$0.307213\pi$
$348$	0	0
$349$	−21.9673	−1.17588	−0.587942	−	0.808903i	$-0.700062\pi$
$349$	−21.9673	−1.17588	−0.587942	+	0.808903i	$0.700062\pi$
$350$	11.1623	0.596651
$351$	0	0
$352$	−143.983	−7.67429
$353$	−17.9952	−0.957790	−0.478895	−	0.877872i	$-0.658963\pi$
$353$	−17.9952	−0.957790	−0.478895	+	0.877872i	$0.658963\pi$
$354$	0	0
$355$	12.9413	0.686854
$356$	5.57047	0.295234
$357$	0	0
$358$	49.4831	2.61526
$359$	13.7314	0.724715	0.362357	−	0.932039i	$-0.381972\pi$
$359$	13.7314	0.724715	0.362357	+	0.932039i	$0.381972\pi$
$360$	0	0
$361$	3.70639	0.195073
$362$	61.0139	3.20682
$363$	0	0
$364$	26.9317	1.41160
$365$	−5.72908	−0.299874
$366$	0	0
$367$	−12.0839	−0.630772	−0.315386	−	0.948963i	$-0.602134\pi$
$367$	−12.0839	−0.630772	−0.315386	+	0.948963i	$0.602134\pi$
$368$	−4.62428	−0.241057
$369$	0	0
$370$	−11.2651	−0.585644
$371$	−38.5870	−2.00334
$372$	0	0
$373$	−9.27551	−0.480268	−0.240134	−	0.970740i	$-0.577191\pi$
$373$	−9.27551	−0.480268	−0.240134	+	0.970740i	$0.577191\pi$
$374$	−78.5515	−4.06180
$375$	0	0
$376$	37.4251	1.93005
$377$	5.86110	0.301862
$378$	0	0
$379$	−25.2400	−1.29649	−0.648246	−	0.761431i	$-0.724497\pi$
$379$	−25.2400	−1.29649	−0.648246	+	0.761431i	$0.724497\pi$
$380$	−26.5440	−1.36168
$381$	0	0
$382$	22.8231	1.16773
$383$	−20.9895	−1.07251	−0.536256	−	0.844055i	$-0.680162\pi$
$383$	−20.9895	−1.07251	−0.536256	+	0.844055i	$0.680162\pi$
$384$	0	0
$385$	−24.2672	−1.23677
$386$	38.7140	1.97049
$387$	0	0
$388$	67.0758	3.40526
$389$	−4.55474	−0.230935	−0.115467	−	0.993311i	$-0.536837\pi$
$389$	−4.55474	−0.230935	−0.115467	+	0.993311i	$0.536837\pi$
$390$	0	0
$391$	−1.38902	−0.0702457
$392$	−92.9187	−4.69310
$393$	0	0
$394$	−64.9210	−3.27067
$395$	−6.02402	−0.303101
$396$	0	0
$397$	−11.0580	−0.554984	−0.277492	−	0.960728i	$-0.589503\pi$
$397$	−11.0580	−0.554984	−0.277492	+	0.960728i	$0.589503\pi$
$398$	64.3539	3.22577
$399$	0	0
$400$	15.8892	0.794461
$401$	−14.4334	−0.720769	−0.360385	−	0.932804i	$-0.617355\pi$
$401$	−14.4334	−0.720769	−0.360385	+	0.932804i	$0.617355\pi$
$402$	0	0
$403$	2.93768	0.146336
$404$	−25.0001	−1.24380
$405$	0	0
$406$	−54.8980	−2.72454
$407$	24.4906	1.21395
$408$	0	0
$409$	−14.3240	−0.708274	−0.354137	−	0.935194i	$-0.615225\pi$
$409$	−14.3240	−0.708274	−0.354137	+	0.935194i	$0.615225\pi$
$410$	15.9761	0.789006
$411$	0	0
$412$	−49.6537	−2.44626
$413$	−38.8628	−1.91231
$414$	0	0
$415$	−13.3496	−0.655308
$416$	28.6854	1.40642
$417$	0	0
$418$	78.4263	3.83595
$419$	−7.54856	−0.368771	−0.184386	−	0.982854i	$-0.559029\pi$
$419$	−7.54856	−0.368771	−0.184386	+	0.982854i	$0.559029\pi$
$420$	0	0
$421$	14.8701	0.724725	0.362363	−	0.932037i	$-0.381970\pi$
$421$	14.8701	0.724725	0.362363	+	0.932037i	$0.381970\pi$
$422$	30.5522	1.48726
$423$	0	0
$424$	−93.4404	−4.53787
$425$	4.77273	0.231511
$426$	0	0
$427$	23.8215	1.15280
$428$	−29.3790	−1.42009
$429$	0	0
$430$	24.5502	1.18392
$431$	19.8615	0.956696	0.478348	−	0.878170i	$-0.341236\pi$
$431$	19.8615	0.956696	0.478348	+	0.878170i	$0.341236\pi$
$432$	0	0
$433$	−21.4644	−1.03151	−0.515757	−	0.856735i	$-0.672489\pi$
$433$	−21.4644	−1.03151	−0.515757	+	0.856735i	$0.672489\pi$
$434$	−27.5158	−1.32080
$435$	0	0
$436$	83.6605	4.00661
$437$	1.38681	0.0663399
$438$	0	0
$439$	−27.4103	−1.30822	−0.654111	−	0.756398i	$-0.726957\pi$
$439$	−27.4103	−1.30822	−0.654111	+	0.756398i	$0.726957\pi$
$440$	−58.7642	−2.80148
$441$	0	0
$442$	15.6497	0.744380
$443$	−28.9979	−1.37773	−0.688866	−	0.724889i	$-0.741891\pi$
$443$	−28.9979	−1.37773	−0.688866	+	0.724889i	$0.741891\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−10.8873	−0.515527
$447$	0	0
$448$	−139.760	−6.60305
$449$	−13.8076	−0.651622	−0.325811	−	0.945435i	$-0.605637\pi$
$449$	−13.8076	−0.651622	−0.325811	+	0.945435i	$0.605637\pi$
$450$	0	0
$451$	−34.7325	−1.63549
$452$	−72.6440	−3.41689
$453$	0	0
$454$	−37.1970	−1.74574
$455$	4.83472	0.226655
$456$	0	0
$457$	−6.56276	−0.306993	−0.153496	−	0.988149i	$-0.549053\pi$
$457$	−6.56276	−0.306993	−0.153496	+	0.988149i	$0.549053\pi$
$458$	−12.1873	−0.569475
$459$	0	0
$460$	−1.62119	−0.0755883
$461$	17.6279	0.821015	0.410508	−	0.911857i	$-0.365352\pi$
$461$	17.6279	0.821015	0.410508	+	0.911857i	$0.365352\pi$
$462$	0	0
$463$	10.9429	0.508559	0.254279	−	0.967131i	$-0.418162\pi$
$463$	10.9429	0.508559	0.254279	+	0.967131i	$0.418162\pi$
$464$	−78.1455	−3.62781
$465$	0	0
$466$	−45.5710	−2.11104
$467$	−39.2107	−1.81445	−0.907227	−	0.420641i	$-0.861805\pi$
$467$	−39.2107	−1.81445	−0.907227	+	0.420641i	$0.861805\pi$
$468$	0	0
$469$	17.8586	0.824632
$470$	10.4818	0.483492
$471$	0	0
$472$	−94.1083	−4.33169
$473$	−53.3728	−2.45408
$474$	0	0
$475$	−4.76512	−0.218639
$476$	−107.858	−4.94366
$477$	0	0
$478$	1.68991	0.0772948
$479$	3.46110	0.158142	0.0790709	−	0.996869i	$-0.474805\pi$
$479$	3.46110	0.158142	0.0790709	+	0.996869i	$0.474805\pi$
$480$	0	0
$481$	−4.87922	−0.222473
$482$	−50.0119	−2.27798
$483$	0	0
$484$	138.042	6.27462
$485$	12.0413	0.546768
$486$	0	0
$487$	26.1287	1.18400	0.592002	−	0.805937i	$-0.298338\pi$
$487$	26.1287	1.18400	0.592002	+	0.805937i	$0.298338\pi$
$488$	57.6849	2.61127
$489$	0	0
$490$	−26.0242	−1.17565
$491$	−30.6804	−1.38459	−0.692294	−	0.721615i	$-0.743400\pi$
$491$	−30.6804	−1.38459	−0.692294	+	0.721615i	$0.743400\pi$
$492$	0	0
$493$	−23.4730	−1.05717
$494$	−15.6248	−0.702991
$495$	0	0
$496$	−39.1679	−1.75869
$497$	−52.5015	−2.35501
$498$	0	0
$499$	26.5638	1.18916	0.594580	−	0.804037i	$-0.297318\pi$
$499$	26.5638	1.18916	0.594580	+	0.804037i	$0.297318\pi$
$500$	5.57047	0.249119
$501$	0	0
$502$	60.1946	2.68662
$503$	−12.1708	−0.542667	−0.271334	−	0.962485i	$-0.587465\pi$
$503$	−12.1708	−0.542667	−0.271334	+	0.962485i	$0.587465\pi$
$504$	0	0
$505$	−4.48797	−0.199712
$506$	4.78993	0.212938
$507$	0	0
$508$	94.5294	4.19406
$509$	3.08866	0.136902	0.0684512	−	0.997654i	$-0.478194\pi$
$509$	3.08866	0.136902	0.0684512	+	0.997654i	$0.478194\pi$
$510$	0	0
$511$	23.2423	1.02818
$512$	−70.2697	−3.10551
$513$	0	0
$514$	−13.7795	−0.607786
$515$	−8.91374	−0.392786
$516$	0	0
$517$	−22.7878	−1.00221
$518$	45.7012	2.00800
$519$	0	0
$520$	11.7075	0.513409
$521$	24.0249	1.05255	0.526275	−	0.850315i	$-0.323588\pi$
$521$	24.0249	1.05255	0.526275	+	0.850315i	$0.323588\pi$
$522$	0	0
$523$	30.5368	1.33528	0.667641	−	0.744483i	$-0.267304\pi$
$523$	30.5368	1.33528	0.667641	+	0.744483i	$0.267304\pi$
$524$	−22.4395	−0.980273
$525$	0	0
$526$	62.6602	2.73211
$527$	−11.7651	−0.512494
$528$	0	0
$529$	−22.9153	−0.996317
$530$	−26.1703	−1.13677
$531$	0	0
$532$	107.686	4.66878
$533$	6.91972	0.299726
$534$	0	0
$535$	−5.27407	−0.228018
$536$	43.2454	1.86792
$537$	0	0
$538$	61.0911	2.63382
$539$	56.5772	2.43695
$540$	0	0
$541$	16.3153	0.701451	0.350725	−	0.936478i	$-0.385935\pi$
$541$	16.3153	0.701451	0.350725	+	0.936478i	$0.385935\pi$
$542$	2.61971	0.112526
$543$	0	0
$544$	−114.882	−4.92551
$545$	15.0186	0.643325
$546$	0	0
$547$	−14.3516	−0.613632	−0.306816	−	0.951769i	$-0.599264\pi$
$547$	−14.3516	−0.613632	−0.306816	+	0.951769i	$0.599264\pi$
$548$	−102.237	−4.36736
$549$	0	0
$550$	−16.4584	−0.701788
$551$	23.4356	0.998389
$552$	0	0
$553$	24.4388	1.03924
$554$	7.90136	0.335697
$555$	0	0
$556$	−123.994	−5.25853
$557$	−11.8867	−0.503658	−0.251829	−	0.967772i	$-0.581032\pi$
$557$	−11.8867	−0.503658	−0.251829	+	0.967772i	$0.581032\pi$
$558$	0	0
$559$	10.6334	0.449744
$560$	−64.4608	−2.72397
$561$	0	0
$562$	−32.2316	−1.35961
$563$	39.9659	1.68436	0.842180	−	0.539196i	$-0.181272\pi$
$563$	39.9659	1.68436	0.842180	+	0.539196i	$0.181272\pi$
$564$	0	0
$565$	−13.0409	−0.548635
$566$	81.4539	3.42376
$567$	0	0
$568$	−127.135	−5.33447
$569$	−0.206225	−0.00864540	−0.00432270	−	0.999991i	$-0.501376\pi$
$569$	−0.206225	−0.00864540	−0.00432270	+	0.999991i	$0.501376\pi$
$570$	0	0
$571$	−30.6208	−1.28144	−0.640720	−	0.767774i	$-0.721364\pi$
$571$	−30.6208	−1.28144	−0.640720	+	0.767774i	$0.721364\pi$
$572$	−39.7096	−1.66034
$573$	0	0
$574$	−64.8135	−2.70526
$575$	−0.291033	−0.0121369
$576$	0	0
$577$	−28.9398	−1.20478	−0.602389	−	0.798203i	$-0.705784\pi$
$577$	−28.9398	−1.20478	−0.602389	+	0.798203i	$0.705784\pi$
$578$	−15.9004	−0.661371
$579$	0	0
$580$	−27.3964	−1.13757
$581$	54.1580	2.24685
$582$	0	0
$583$	56.8949	2.35635
$584$	56.2823	2.32898
$585$	0	0
$586$	68.8011	2.84215
$587$	28.5727	1.17932	0.589661	−	0.807651i	$-0.299261\pi$
$587$	28.5727	1.17932	0.589661	+	0.807651i	$0.299261\pi$
$588$	0	0
$589$	11.7463	0.483998
$590$	−26.3574	−1.08512
$591$	0	0
$592$	65.0542	2.67371
$593$	−4.97569	−0.204327	−0.102163	−	0.994768i	$-0.532576\pi$
$593$	−4.97569	−0.204327	−0.102163	+	0.994768i	$0.532576\pi$
$594$	0	0
$595$	−19.3624	−0.793782
$596$	46.2052	1.89264
$597$	0	0
$598$	−0.954291	−0.0390239
$599$	25.2750	1.03271	0.516355	−	0.856375i	$-0.327289\pi$
$599$	25.2750	1.03271	0.516355	+	0.856375i	$0.327289\pi$
$600$	0	0
$601$	−18.6729	−0.761683	−0.380841	−	0.924640i	$-0.624366\pi$
$601$	−18.6729	−0.761683	−0.380841	+	0.924640i	$0.624366\pi$
$602$	−99.5975	−4.05929
$603$	0	0
$604$	78.8114	3.20679
$605$	24.7810	1.00749
$606$	0	0
$607$	29.2329	1.18653	0.593263	−	0.805009i	$-0.297839\pi$
$607$	29.2329	1.18653	0.593263	+	0.805009i	$0.297839\pi$
$608$	114.699	4.65164
$609$	0	0
$610$	16.1561	0.654142
$611$	4.53998	0.183668
$612$	0	0
$613$	−21.1908	−0.855887	−0.427943	−	0.903806i	$-0.640762\pi$
$613$	−21.1908	−0.855887	−0.427943	+	0.903806i	$0.640762\pi$
$614$	38.0238	1.53452
$615$	0	0
$616$	238.400	9.60541
$617$	−5.98419	−0.240914	−0.120457	−	0.992719i	$-0.538436\pi$
$617$	−5.98419	−0.240914	−0.120457	+	0.992719i	$0.538436\pi$
$618$	0	0
$619$	−25.0074	−1.00513	−0.502566	−	0.864539i	$-0.667610\pi$
$619$	−25.0074	−1.00513	−0.502566	+	0.864539i	$0.667610\pi$
$620$	−13.7315	−0.551472
$621$	0	0
$622$	−82.6657	−3.31459
$623$	−4.05689	−0.162536
$624$	0	0
$625$	1.00000	0.0400000
$626$	−46.3622	−1.85301
$627$	0	0
$628$	−37.4904	−1.49603
$629$	19.5407	0.779138
$630$	0	0
$631$	−35.9757	−1.43217	−0.716085	−	0.698013i	$-0.754068\pi$
$631$	−35.9757	−1.43217	−0.716085	+	0.698013i	$0.754068\pi$
$632$	59.1798	2.35405
$633$	0	0
$634$	−15.8573	−0.629776
$635$	16.9697	0.673423
$636$	0	0
$637$	−11.2718	−0.446605
$638$	80.9448	3.20464
$639$	0	0
$640$	−46.6469	−1.84388
$641$	−44.8886	−1.77299	−0.886497	−	0.462734i	$-0.846869\pi$
$641$	−44.8886	−1.77299	−0.886497	+	0.462734i	$0.846869\pi$
$642$	0	0
$643$	−36.6755	−1.44634	−0.723171	−	0.690669i	$-0.757316\pi$
$643$	−36.6755	−1.44634	−0.723171	+	0.690669i	$0.757316\pi$
$644$	6.57698	0.259170
$645$	0	0
$646$	62.5752	2.46199
$647$	−35.9147	−1.41195	−0.705977	−	0.708235i	$-0.749492\pi$
$647$	−35.9147	−1.41195	−0.705977	+	0.708235i	$0.749492\pi$
$648$	0	0
$649$	57.3016	2.24929
$650$	3.27898	0.128612
$651$	0	0
$652$	98.0031	3.83810
$653$	31.0033	1.21325	0.606626	−	0.794987i	$-0.292522\pi$
$653$	31.0033	1.21325	0.606626	+	0.794987i	$0.292522\pi$
$654$	0	0
$655$	−4.02829	−0.157398
$656$	−92.2599	−3.60214
$657$	0	0
$658$	−42.5237	−1.65775
$659$	−9.68874	−0.377420	−0.188710	−	0.982033i	$-0.560431\pi$
$659$	−9.68874	−0.377420	−0.188710	+	0.982033i	$0.560431\pi$
$660$	0	0
$661$	19.2868	0.750168	0.375084	−	0.926991i	$-0.377614\pi$
$661$	19.2868	0.750168	0.375084	+	0.926991i	$0.377614\pi$
$662$	43.3749	1.68582
$663$	0	0
$664$	131.147	5.08947
$665$	19.3316	0.749646
$666$	0	0
$667$	1.43134	0.0554217
$668$	81.6787	3.16024
$669$	0	0
$670$	12.1120	0.467926
$671$	−35.1238	−1.35594
$672$	0	0
$673$	12.5115	0.482282	0.241141	−	0.970490i	$-0.422478\pi$
$673$	12.5115	0.482282	0.241141	+	0.970490i	$0.422478\pi$
$674$	−10.0300	−0.386343
$675$	0	0
$676$	−64.5048	−2.48096
$677$	19.4714	0.748347	0.374173	−	0.927359i	$-0.377927\pi$
$677$	19.4714	0.748347	0.374173	+	0.927359i	$0.377927\pi$
$678$	0	0
$679$	−48.8503	−1.87470
$680$	−46.8872	−1.79804
$681$	0	0
$682$	40.5709	1.55354
$683$	31.5879	1.20868	0.604339	−	0.796728i	$-0.293437\pi$
$683$	31.5879	1.20868	0.604339	+	0.796728i	$0.293437\pi$
$684$	0	0
$685$	−18.3534	−0.701248
$686$	27.4410	1.04770
$687$	0	0
$688$	−141.774	−5.40508
$689$	−11.3351	−0.431833
$690$	0	0
$691$	−16.2113	−0.616706	−0.308353	−	0.951272i	$-0.599778\pi$
$691$	−16.2113	−0.616706	−0.308353	+	0.951272i	$0.599778\pi$
$692$	59.4341	2.25935
$693$	0	0
$694$	−58.3580	−2.21524
$695$	−22.2592	−0.844340
$696$	0	0
$697$	−27.7126	−1.04969
$698$	60.4419	2.28776
$699$	0	0
$700$	−22.5988	−0.854154
$701$	−0.685605	−0.0258949	−0.0129475	−	0.999916i	$-0.504121\pi$
$701$	−0.685605	−0.0258949	−0.0129475	+	0.999916i	$0.504121\pi$
$702$	0	0
$703$	−19.5095	−0.735816
$704$	206.071	7.76659
$705$	0	0
$706$	49.5130	1.86345
$707$	18.2072	0.684753
$708$	0	0
$709$	−12.4484	−0.467510	−0.233755	−	0.972295i	$-0.575101\pi$
$709$	−12.4484	−0.467510	−0.233755	+	0.972295i	$0.575101\pi$
$710$	−35.6074	−1.33632
$711$	0	0
$712$	−9.82397	−0.368169
$713$	0.717413	0.0268673
$714$	0	0
$715$	−7.12859	−0.266594
$716$	−100.181	−3.74396
$717$	0	0
$718$	−37.7812	−1.40998
$719$	−20.4765	−0.763644	−0.381822	−	0.924236i	$-0.624703\pi$
$719$	−20.4765	−0.763644	−0.381822	+	0.924236i	$0.624703\pi$
$720$	0	0
$721$	36.1621	1.34675
$722$	−10.1979	−0.379528
$723$	0	0
$724$	−123.526	−4.59082
$725$	−4.91815	−0.182655
$726$	0	0
$727$	−31.7422	−1.17725	−0.588626	−	0.808406i	$-0.700331\pi$
$727$	−31.7422	−1.17725	−0.588626	+	0.808406i	$0.700331\pi$
$728$	−47.4961	−1.76032
$729$	0	0
$730$	15.7633	0.583425
$731$	−42.5854	−1.57508
$732$	0	0
$733$	14.2914	0.527864	0.263932	−	0.964541i	$-0.414981\pi$
$733$	14.2914	0.527864	0.263932	+	0.964541i	$0.414981\pi$
$734$	33.2481	1.22721
$735$	0	0
$736$	7.00528	0.258218
$737$	−26.3317	−0.969942
$738$	0	0
$739$	16.0030	0.588680	0.294340	−	0.955701i	$-0.404900\pi$
$739$	16.0030	0.588680	0.294340	+	0.955701i	$0.404900\pi$
$740$	22.8068	0.838395
$741$	0	0
$742$	106.170	3.89763
$743$	−39.1411	−1.43595	−0.717974	−	0.696070i	$-0.754930\pi$
$743$	−39.1411	−1.43595	−0.717974	+	0.696070i	$0.754930\pi$
$744$	0	0
$745$	8.29467	0.303893
$746$	25.5211	0.934394
$747$	0	0
$748$	159.032	5.81479
$749$	21.3963	0.781804
$750$	0	0
$751$	−6.28646	−0.229396	−0.114698	−	0.993400i	$-0.536590\pi$
$751$	−6.28646	−0.229396	−0.114698	+	0.993400i	$0.536590\pi$
$752$	−60.5311	−2.20734
$753$	0	0
$754$	−16.1265	−0.587293
$755$	14.1481	0.514901
$756$	0	0
$757$	23.3143	0.847373	0.423686	−	0.905809i	$-0.360736\pi$
$757$	23.3143	0.847373	0.423686	+	0.905809i	$0.360736\pi$
$758$	69.4466	2.52241
$759$	0	0
$760$	46.8124	1.69807
$761$	−16.9437	−0.614210	−0.307105	−	0.951676i	$-0.599360\pi$
$761$	−16.9437	−0.614210	−0.307105	+	0.951676i	$0.599360\pi$
$762$	0	0
$763$	−60.9287	−2.20577
$764$	−46.2067	−1.67170
$765$	0	0
$766$	57.7515	2.08665
$767$	−11.4161	−0.412212
$768$	0	0
$769$	−43.1592	−1.55636	−0.778180	−	0.628041i	$-0.783857\pi$
$769$	−43.1592	−1.55636	−0.778180	+	0.628041i	$0.783857\pi$
$770$	66.7699	2.40622
$771$	0	0
$772$	−78.3788	−2.82091
$773$	−29.8265	−1.07278	−0.536392	−	0.843969i	$-0.680213\pi$
$773$	−29.8265	−1.07278	−0.536392	+	0.843969i	$0.680213\pi$
$774$	0	0
$775$	−2.46506	−0.0885475
$776$	−118.294	−4.24649
$777$	0	0
$778$	12.5321	0.449299
$779$	27.6684	0.991325
$780$	0	0
$781$	77.4113	2.76999
$782$	3.82182	0.136668
$783$	0	0
$784$	150.286	5.36735
$785$	−6.73020	−0.240211
$786$	0	0
$787$	28.1230	1.00248	0.501239	−	0.865309i	$-0.332878\pi$
$787$	28.1230	1.00248	0.501239	+	0.865309i	$0.332878\pi$
$788$	131.436	4.68223
$789$	0	0
$790$	16.5748	0.589704
$791$	52.9055	1.88110
$792$	0	0
$793$	6.99766	0.248494
$794$	30.4255	1.07976
$795$	0	0
$796$	−130.288	−4.61794
$797$	36.5266	1.29384	0.646920	−	0.762558i	$-0.276057\pi$
$797$	36.5266	1.29384	0.646920	+	0.762558i	$0.276057\pi$
$798$	0	0
$799$	−18.1821	−0.643235
$800$	−24.0704	−0.851018
$801$	0	0
$802$	39.7127	1.40231
$803$	−34.2698	−1.20935
$804$	0	0
$805$	1.18069	0.0416138
$806$	−8.08289	−0.284708
$807$	0	0
$808$	44.0897	1.55107
$809$	48.4047	1.70182	0.850910	−	0.525311i	$-0.176051\pi$
$809$	48.4047	1.70182	0.850910	+	0.525311i	$0.176051\pi$
$810$	0	0
$811$	10.9956	0.386109	0.193054	−	0.981188i	$-0.438161\pi$
$811$	10.9956	0.386109	0.193054	+	0.981188i	$0.438161\pi$
$812$	111.144	3.90040
$813$	0	0
$814$	−67.3845	−2.36183
$815$	17.5933	0.616267
$816$	0	0
$817$	42.5175	1.48750
$818$	39.4116	1.37800
$819$	0	0
$820$	−32.3447	−1.12952
$821$	−24.1333	−0.842257	−0.421129	−	0.907001i	$-0.638366\pi$
$821$	−24.1333	−0.842257	−0.421129	+	0.907001i	$0.638366\pi$
$822$	0	0
$823$	−0.366980	−0.0127921	−0.00639606	−	0.999980i	$-0.502036\pi$
$823$	−0.366980	−0.0127921	−0.00639606	+	0.999980i	$0.502036\pi$
$824$	87.5683	3.05059
$825$	0	0
$826$	106.929	3.72054
$827$	−10.8520	−0.377362	−0.188681	−	0.982038i	$-0.560421\pi$
$827$	−10.8520	−0.377362	−0.188681	+	0.982038i	$0.560421\pi$
$828$	0	0
$829$	2.09254	0.0726770	0.0363385	−	0.999340i	$-0.488431\pi$
$829$	2.09254	0.0726770	0.0363385	+	0.999340i	$0.488431\pi$
$830$	36.7309	1.27495
$831$	0	0
$832$	−41.0552	−1.42333
$833$	45.1422	1.56408
$834$	0	0
$835$	14.6628	0.507427
$836$	−158.779	−5.49147
$837$	0	0
$838$	20.7695	0.717469
$839$	−46.9409	−1.62058	−0.810291	−	0.586028i	$-0.800691\pi$
$839$	−46.9409	−1.62058	−0.810291	+	0.586028i	$0.800691\pi$
$840$	0	0
$841$	−4.81182	−0.165925
$842$	−40.9144	−1.41000
$843$	0	0
$844$	−61.8547	−2.12913
$845$	−11.5798	−0.398357
$846$	0	0
$847$	−100.534	−3.45438
$848$	151.130	5.18982
$849$	0	0
$850$	−13.1319	−0.450421
$851$	−1.19156	−0.0408460
$852$	0	0
$853$	29.8408	1.02173	0.510865	−	0.859661i	$-0.329325\pi$
$853$	29.8408	1.02173	0.510865	+	0.859661i	$0.329325\pi$
$854$	−65.5436	−2.24285
$855$	0	0
$856$	51.8123	1.77091
$857$	10.7346	0.366687	0.183343	−	0.983049i	$-0.441308\pi$
$857$	10.7346	0.366687	0.183343	+	0.983049i	$0.441308\pi$
$858$	0	0
$859$	1.90879	0.0651269	0.0325635	−	0.999470i	$-0.489633\pi$
$859$	1.90879	0.0651269	0.0325635	+	0.999470i	$0.489633\pi$
$860$	−49.7034	−1.69487
$861$	0	0
$862$	−54.6480	−1.86132
$863$	−40.7868	−1.38840	−0.694199	−	0.719783i	$-0.744241\pi$
$863$	−40.7868	−1.38840	−0.694199	+	0.719783i	$0.744241\pi$
$864$	0	0
$865$	10.6695	0.362774
$866$	59.0583	2.00688
$867$	0	0
$868$	55.7074	1.89083
$869$	−36.0340	−1.22237
$870$	0	0
$871$	5.24603	0.177755
$872$	−147.542	−4.99640
$873$	0	0
$874$	−3.81573	−0.129069
$875$	−4.05689	−0.137148
$876$	0	0
$877$	−55.4865	−1.87365	−0.936823	−	0.349804i	$-0.886248\pi$
$877$	−55.4865	−1.87365	−0.936823	+	0.349804i	$0.886248\pi$
$878$	75.4181	2.54524
$879$	0	0
$880$	95.0448	3.20396
$881$	13.2235	0.445510	0.222755	−	0.974874i	$-0.428495\pi$
$881$	13.2235	0.445510	0.222755	+	0.974874i	$0.428495\pi$
$882$	0	0
$883$	−16.2163	−0.545720	−0.272860	−	0.962054i	$-0.587970\pi$
$883$	−16.2163	−0.545720	−0.272860	+	0.962054i	$0.587970\pi$
$884$	−31.6837	−1.06564
$885$	0	0
$886$	79.7863	2.68047
$887$	12.5532	0.421494	0.210747	−	0.977541i	$-0.432410\pi$
$887$	12.5532	0.421494	0.210747	+	0.977541i	$0.432410\pi$
$888$	0	0
$889$	−68.8443	−2.30896
$890$	−2.75145	−0.0922288
$891$	0	0
$892$	22.0419	0.738018
$893$	18.1531	0.607470
$894$	0	0
$895$	−17.9844	−0.601151
$896$	189.241	6.32211
$897$	0	0
$898$	37.9910	1.26777
$899$	12.1235	0.404342
$900$	0	0
$901$	45.3956	1.51235
$902$	95.5648	3.18196
$903$	0	0
$904$	128.114	4.26099
$905$	−22.1752	−0.737128
$906$	0	0
$907$	−13.6863	−0.454446	−0.227223	−	0.973843i	$-0.572965\pi$
$907$	−13.6863	−0.454446	−0.227223	+	0.973843i	$0.572965\pi$
$908$	75.3076	2.49917
$909$	0	0
$910$	−13.3025	−0.440973
$911$	−20.4430	−0.677306	−0.338653	−	0.940911i	$-0.609971\pi$
$911$	−20.4430	−0.677306	−0.338653	+	0.940911i	$0.609971\pi$
$912$	0	0
$913$	−79.8538	−2.64278
$914$	18.0571	0.597276
$915$	0	0
$916$	24.6739	0.815249
$917$	16.3423	0.539671
$918$	0	0
$919$	−2.95574	−0.0975008	−0.0487504	−	0.998811i	$-0.515524\pi$
$919$	−2.95574	−0.0975008	−0.0487504	+	0.998811i	$0.515524\pi$
$920$	2.85910	0.0942616
$921$	0	0
$922$	−48.5024	−1.59734
$923$	−15.4225	−0.507639
$924$	0	0
$925$	4.09424	0.134618
$926$	−30.1088	−0.989435
$927$	0	0
$928$	118.382	3.88608
$929$	−50.8173	−1.66726	−0.833630	−	0.552323i	$-0.813742\pi$
$929$	−50.8173	−1.66726	−0.833630	+	0.552323i	$0.813742\pi$
$930$	0	0
$931$	−45.0702	−1.47712
$932$	92.2613	3.02212
$933$	0	0
$934$	107.886	3.53015
$935$	28.5491	0.933656
$936$	0	0
$937$	32.2062	1.05213	0.526065	−	0.850444i	$-0.323667\pi$
$937$	32.2062	1.05213	0.526065	+	0.850444i	$0.323667\pi$
$938$	−49.1369	−1.60438
$939$	0	0
$940$	−21.2211	−0.692157
$941$	25.4012	0.828055	0.414028	−	0.910264i	$-0.364122\pi$
$941$	25.4012	0.828055	0.414028	+	0.910264i	$0.364122\pi$
$942$	0	0
$943$	1.68987	0.0550296
$944$	152.210	4.95402
$945$	0	0
$946$	146.852	4.77459
$947$	−59.8832	−1.94594	−0.972970	−	0.230930i	$-0.925823\pi$
$947$	−59.8832	−1.94594	−0.972970	+	0.230930i	$0.925823\pi$
$948$	0	0
$949$	6.82752	0.221631
$950$	13.1110	0.425377
$951$	0	0
$952$	190.216	6.16494
$953$	23.5567	0.763075	0.381538	−	0.924353i	$-0.375395\pi$
$953$	23.5567	0.763075	0.381538	+	0.924353i	$0.375395\pi$
$954$	0	0
$955$	−8.29493	−0.268418
$956$	−3.42133	−0.110654
$957$	0	0
$958$	−9.52305	−0.307676
$959$	74.4578	2.40437
$960$	0	0
$961$	−24.9235	−0.803983
$962$	13.4249	0.432837
$963$	0	0
$964$	101.252	3.26111
$965$	−14.0704	−0.452942
$966$	0	0
$967$	4.35690	0.140108	0.0700542	−	0.997543i	$-0.477683\pi$
$967$	4.35690	0.140108	0.0700542	+	0.997543i	$0.477683\pi$
$968$	−243.448	−7.82470
$969$	0	0
$970$	−33.1311	−1.06377
$971$	44.9232	1.44165	0.720827	−	0.693115i	$-0.243762\pi$
$971$	44.9232	1.44165	0.720827	+	0.693115i	$0.243762\pi$
$972$	0	0
$973$	90.3032	2.89499
$974$	−71.8917	−2.30356
$975$	0	0
$976$	−93.2991	−2.98643
$977$	3.90081	0.124798	0.0623990	−	0.998051i	$-0.480125\pi$
$977$	3.90081	0.124798	0.0623990	+	0.998051i	$0.480125\pi$
$978$	0	0
$979$	5.98172	0.191177
$980$	52.6875	1.68304
$981$	0	0
$982$	84.4156	2.69381
$983$	−0.843117	−0.0268913	−0.0134456	−	0.999910i	$-0.504280\pi$
$983$	−0.843117	−0.0268913	−0.0134456	+	0.999910i	$0.504280\pi$
$984$	0	0
$985$	23.5952	0.751806
$986$	64.5847	2.05680
$987$	0	0
$988$	31.6332	1.00639
$989$	2.59678	0.0825728
$990$	0	0
$991$	−1.74204	−0.0553376	−0.0276688	−	0.999617i	$-0.508808\pi$
$991$	−1.74204	−0.0553376	−0.0276688	+	0.999617i	$0.508808\pi$
$992$	59.3350	1.88389
$993$	0	0
$994$	144.455	4.58184
$995$	−23.3891	−0.741484
$996$	0	0
$997$	21.6420	0.685409	0.342704	−	0.939443i	$-0.388657\pi$
$997$	21.6420	0.685409	0.342704	+	0.939443i	$0.388657\pi$
$998$	−73.0890	−2.31359
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.u.1.1	✓	12
3.2	odd	2		4005.2.a.v.1.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.1	✓	12	1.1	even	1	trivial
4005.2.a.v.1.12	yes	12	3.2	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)

\(f(q)\)	\(=\)	\(q-2.75145 q^{2} +5.57047 q^{4} +1.00000 q^{5} -4.05689 q^{7} -9.82397 q^{8} +O(q^{10})\)	\(q-2.75145 q^{2} +5.57047 q^{4} +1.00000 q^{5} -4.05689 q^{7} -9.82397 q^{8} -2.75145 q^{10} +5.98172 q^{11} -1.19173 q^{13} +11.1623 q^{14} +15.8892 q^{16} +4.77273 q^{17} -4.76512 q^{19} +5.57047 q^{20} -16.4584 q^{22} -0.291033 q^{23} +1.00000 q^{25} +3.27898 q^{26} -22.5988 q^{28} -4.91815 q^{29} -2.46506 q^{31} -24.0704 q^{32} -13.1319 q^{34} -4.05689 q^{35} +4.09424 q^{37} +13.1110 q^{38} -9.82397 q^{40} -5.80645 q^{41} -8.92265 q^{43} +33.3210 q^{44} +0.800761 q^{46} -3.80957 q^{47} +9.45836 q^{49} -2.75145 q^{50} -6.63850 q^{52} +9.51147 q^{53} +5.98172 q^{55} +39.8548 q^{56} +13.5320 q^{58} +9.57946 q^{59} -5.87185 q^{61} +6.78249 q^{62} +34.4501 q^{64} -1.19173 q^{65} -4.40203 q^{67} +26.5864 q^{68} +11.1623 q^{70} +12.9413 q^{71} -5.72908 q^{73} -11.2651 q^{74} -26.5440 q^{76} -24.2672 q^{77} -6.02402 q^{79} +15.8892 q^{80} +15.9761 q^{82} -13.3496 q^{83} +4.77273 q^{85} +24.5502 q^{86} -58.7642 q^{88} +1.00000 q^{89} +4.83472 q^{91} -1.62119 q^{92} +10.4818 q^{94} -4.76512 q^{95} +12.0413 q^{97} -26.0242 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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