LMFDB - Embedded newform 4005.2.a.u.1.4

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			1.4
Root			$1.62987$ 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)

$f(q)$	$=$	$q-1.62987 q^{2} +0.656467 q^{4} +1.00000 q^{5} -1.43414 q^{7} +2.18978 q^{8} +O(q^{10})$	\(q-1.62987 q^{2} +0.656467 q^{4} +1.00000 q^{5} -1.43414 q^{7} +2.18978 q^{8} -1.62987 q^{10} -6.20398 q^{11} +3.99234 q^{13} +2.33746 q^{14} -4.88199 q^{16} +5.83909 q^{17} -8.48123 q^{19} +0.656467 q^{20} +10.1117 q^{22} -1.19084 q^{23} +1.00000 q^{25} -6.50699 q^{26} -0.941465 q^{28} +4.83046 q^{29} +9.78262 q^{31} +3.57743 q^{32} -9.51694 q^{34} -1.43414 q^{35} +0.211343 q^{37} +13.8233 q^{38} +2.18978 q^{40} +5.06083 q^{41} -7.08633 q^{43} -4.07271 q^{44} +1.94091 q^{46} -3.37303 q^{47} -4.94325 q^{49} -1.62987 q^{50} +2.62084 q^{52} +7.97873 q^{53} -6.20398 q^{55} -3.14045 q^{56} -7.87301 q^{58} -8.77392 q^{59} -14.8347 q^{61} -15.9444 q^{62} +3.93324 q^{64} +3.99234 q^{65} +12.7266 q^{67} +3.83317 q^{68} +2.33746 q^{70} +9.96518 q^{71} -8.54823 q^{73} -0.344461 q^{74} -5.56765 q^{76} +8.89736 q^{77} -4.89907 q^{79} -4.88199 q^{80} -8.24849 q^{82} +2.27331 q^{83} +5.83909 q^{85} +11.5498 q^{86} -13.5853 q^{88} +1.00000 q^{89} -5.72557 q^{91} -0.781745 q^{92} +5.49759 q^{94} -8.48123 q^{95} +0.714336 q^{97} +8.05684 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.62987	−1.15249	−0.576245	−	0.817277i	$-0.695483\pi$
$2$	−1.62987	−1.15249	−0.576245	+	0.817277i	$0.695483\pi$
$3$	0	0
$3$	0	0
$4$	0.656467	0.328234
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.43414	−0.542053	−0.271027	−	0.962572i	$-0.587363\pi$
$7$	−1.43414	−0.542053	−0.271027	+	0.962572i	$0.587363\pi$
$8$	2.18978	0.774204
$9$	0	0
$10$	−1.62987	−0.515409
$11$	−6.20398	−1.87057	−0.935285	−	0.353896i	$-0.884857\pi$
$11$	−6.20398	−1.87057	−0.935285	+	0.353896i	$0.884857\pi$
$12$	0	0
$13$	3.99234	1.10728	0.553638	−	0.832757i	$-0.313239\pi$
$13$	3.99234	1.10728	0.553638	+	0.832757i	$0.313239\pi$
$14$	2.33746	0.624711
$15$	0	0
$16$	−4.88199	−1.22050
$17$	5.83909	1.41619	0.708093	−	0.706119i	$-0.249556\pi$
$17$	5.83909	1.41619	0.708093	+	0.706119i	$0.249556\pi$
$18$	0	0
$19$	−8.48123	−1.94573	−0.972864	−	0.231380i	$-0.925676\pi$
$19$	−8.48123	−1.94573	−0.972864	+	0.231380i	$0.925676\pi$
$20$	0.656467	0.146791
$21$	0	0
$22$	10.1117	2.15581
$23$	−1.19084	−0.248307	−0.124153	−	0.992263i	$-0.539621\pi$
$23$	−1.19084	−0.248307	−0.124153	+	0.992263i	$0.539621\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.50699	−1.27613
$27$	0	0
$28$	−0.941465	−0.177920
$29$	4.83046	0.896994	0.448497	−	0.893784i	$-0.351960\pi$
$29$	4.83046	0.896994	0.448497	+	0.893784i	$0.351960\pi$
$30$	0	0
$31$	9.78262	1.75701	0.878505	−	0.477733i	$-0.158541\pi$
$31$	9.78262	1.75701	0.878505	+	0.477733i	$0.158541\pi$
$32$	3.57743	0.632406
$33$	0	0
$34$	−9.51694	−1.63214
$35$	−1.43414	−0.242414
$36$	0	0
$37$	0.211343	0.0347445	0.0173723	−	0.999849i	$-0.494470\pi$
$37$	0.211343	0.0347445	0.0173723	+	0.999849i	$0.494470\pi$
$38$	13.8233	2.24243
$39$	0	0
$40$	2.18978	0.346235
$41$	5.06083	0.790370	0.395185	−	0.918602i	$-0.370681\pi$
$41$	5.06083	0.790370	0.395185	+	0.918602i	$0.370681\pi$
$42$	0	0
$43$	−7.08633	−1.08065	−0.540327	−	0.841455i	$-0.681700\pi$
$43$	−7.08633	−1.08065	−0.540327	+	0.841455i	$0.681700\pi$
$44$	−4.07271	−0.613984
$45$	0	0
$46$	1.94091	0.286171
$47$	−3.37303	−0.492007	−0.246003	−	0.969269i	$-0.579117\pi$
$47$	−3.37303	−0.492007	−0.246003	+	0.969269i	$0.579117\pi$
$48$	0	0
$49$	−4.94325	−0.706178
$50$	−1.62987	−0.230498
$51$	0	0
$52$	2.62084	0.363445
$53$	7.97873	1.09596	0.547982	−	0.836490i	$-0.315396\pi$
$53$	7.97873	1.09596	0.547982	+	0.836490i	$0.315396\pi$
$54$	0	0
$55$	−6.20398	−0.836544
$56$	−3.14045	−0.419660
$57$	0	0
$58$	−7.87301	−1.03378
$59$	−8.77392	−1.14227	−0.571134	−	0.820857i	$-0.693496\pi$
$59$	−8.77392	−1.14227	−0.571134	+	0.820857i	$0.693496\pi$
$60$	0	0
$61$	−14.8347	−1.89938	−0.949692	−	0.313184i	$-0.898604\pi$
$61$	−14.8347	−1.89938	−0.949692	+	0.313184i	$0.898604\pi$
$62$	−15.9444	−2.02494
$63$	0	0
$64$	3.93324	0.491655
$65$	3.99234	0.495189
$66$	0	0
$67$	12.7266	1.55480	0.777402	−	0.629005i	$-0.216537\pi$
$67$	12.7266	1.55480	0.777402	+	0.629005i	$0.216537\pi$
$68$	3.83317	0.464840
$69$	0	0
$70$	2.33746	0.279379
$71$	9.96518	1.18265	0.591325	−	0.806433i	$-0.298605\pi$
$71$	9.96518	1.18265	0.591325	+	0.806433i	$0.298605\pi$
$72$	0	0
$73$	−8.54823	−1.00049	−0.500247	−	0.865883i	$-0.666758\pi$
$73$	−8.54823	−1.00049	−0.500247	+	0.865883i	$0.666758\pi$
$74$	−0.344461	−0.0400428
$75$	0	0
$76$	−5.56765	−0.638653
$77$	8.89736	1.01395
$78$	0	0
$79$	−4.89907	−0.551188	−0.275594	−	0.961274i	$-0.588875\pi$
$79$	−4.89907	−0.551188	−0.275594	+	0.961274i	$0.588875\pi$
$80$	−4.88199	−0.545823
$81$	0	0
$82$	−8.24849	−0.910893
$83$	2.27331	0.249528	0.124764	−	0.992186i	$-0.460183\pi$
$83$	2.27331	0.249528	0.124764	+	0.992186i	$0.460183\pi$
$84$	0	0
$85$	5.83909	0.633338
$86$	11.5498	1.24544
$87$	0	0
$88$	−13.5853	−1.44820
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−5.72557	−0.600203
$92$	−0.781745	−0.0815026
$93$	0	0
$94$	5.49759	0.567033
$95$	−8.48123	−0.870156
$96$	0	0
$97$	0.714336	0.0725298	0.0362649	−	0.999342i	$-0.488454\pi$
$97$	0.714336	0.0725298	0.0362649	+	0.999342i	$0.488454\pi$
$98$	8.05684	0.813863
$99$	0	0
$100$	0.656467	0.0656467
$101$	2.53350	0.252092	0.126046	−	0.992024i	$-0.459771\pi$
$101$	2.53350	0.252092	0.126046	+	0.992024i	$0.459771\pi$
$102$	0	0
$103$	−9.01070	−0.887851	−0.443926	−	0.896064i	$-0.646415\pi$
$103$	−9.01070	−0.887851	−0.443926	+	0.896064i	$0.646415\pi$
$104$	8.74235	0.857258
$105$	0	0
$106$	−13.0043	−1.26309
$107$	9.02820	0.872789	0.436395	−	0.899755i	$-0.356255\pi$
$107$	9.02820	0.872789	0.436395	+	0.899755i	$0.356255\pi$
$108$	0	0
$109$	−15.9963	−1.53217	−0.766084	−	0.642740i	$-0.777797\pi$
$109$	−15.9963	−1.53217	−0.766084	+	0.642740i	$0.777797\pi$
$110$	10.1117	0.964109
$111$	0	0
$112$	7.00144	0.661574
$113$	−7.80412	−0.734150	−0.367075	−	0.930191i	$-0.619641\pi$
$113$	−7.80412	−0.734150	−0.367075	+	0.930191i	$0.619641\pi$
$114$	0	0
$115$	−1.19084	−0.111046
$116$	3.17104	0.294424
$117$	0	0
$118$	14.3003	1.31645
$119$	−8.37406	−0.767649
$120$	0	0
$121$	27.4893	2.49903
$122$	24.1785	2.18902
$123$	0	0
$124$	6.42197	0.576710
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.63591	−0.500106	−0.250053	−	0.968232i	$-0.580448\pi$
$127$	−5.63591	−0.500106	−0.250053	+	0.968232i	$0.580448\pi$
$128$	−13.5655	−1.19903
$129$	0	0
$130$	−6.50699	−0.570701
$131$	1.02956	0.0899529	0.0449764	−	0.998988i	$-0.485679\pi$
$131$	1.02956	0.0899529	0.0449764	+	0.998988i	$0.485679\pi$
$132$	0	0
$133$	12.1633	1.05469
$134$	−20.7427	−1.79190
$135$	0	0
$136$	12.7863	1.09642
$137$	4.56353	0.389888	0.194944	−	0.980814i	$-0.437547\pi$
$137$	4.56353	0.389888	0.194944	+	0.980814i	$0.437547\pi$
$138$	0	0
$139$	−15.1718	−1.28685	−0.643427	−	0.765507i	$-0.722488\pi$
$139$	−15.1718	−1.28685	−0.643427	+	0.765507i	$0.722488\pi$
$140$	−0.941465	−0.0795683
$141$	0	0
$142$	−16.2419	−1.36299
$143$	−24.7684	−2.07124
$144$	0	0
$145$	4.83046	0.401148
$146$	13.9325	1.15306
$147$	0	0
$148$	0.138740	0.0114043
$149$	23.9176	1.95941	0.979705	−	0.200445i	$-0.0642387\pi$
$149$	23.9176	1.95941	0.979705	+	0.200445i	$0.0642387\pi$
$150$	0	0
$151$	2.17844	0.177279	0.0886397	−	0.996064i	$-0.471748\pi$
$151$	2.17844	0.177279	0.0886397	+	0.996064i	$0.471748\pi$
$152$	−18.5720	−1.50639
$153$	0	0
$154$	−14.5015	−1.16857
$155$	9.78262	0.785759
$156$	0	0
$157$	−20.7908	−1.65928	−0.829642	−	0.558295i	$-0.811456\pi$
$157$	−20.7908	−1.65928	−0.829642	+	0.558295i	$0.811456\pi$
$158$	7.98483	0.635239
$159$	0	0
$160$	3.57743	0.282821
$161$	1.70782	0.134595
$162$	0	0
$163$	−10.5679	−0.827738	−0.413869	−	0.910336i	$-0.635823\pi$
$163$	−10.5679	−0.827738	−0.413869	+	0.910336i	$0.635823\pi$
$164$	3.32227	0.259426
$165$	0	0
$166$	−3.70519	−0.287578
$167$	−13.2920	−1.02856	−0.514282	−	0.857621i	$-0.671942\pi$
$167$	−13.2920	−1.02856	−0.514282	+	0.857621i	$0.671942\pi$
$168$	0	0
$169$	2.93879	0.226061
$170$	−9.51694	−0.729916
$171$	0	0
$172$	−4.65194	−0.354707
$173$	−8.17737	−0.621714	−0.310857	−	0.950457i	$-0.600616\pi$
$173$	−8.17737	−0.621714	−0.310857	+	0.950457i	$0.600616\pi$
$174$	0	0
$175$	−1.43414	−0.108411
$176$	30.2877	2.28302
$177$	0	0
$178$	−1.62987	−0.122164
$179$	−9.41628	−0.703806	−0.351903	−	0.936037i	$-0.614465\pi$
$179$	−9.41628	−0.703806	−0.351903	+	0.936037i	$0.614465\pi$
$180$	0	0
$181$	16.9919	1.26300	0.631499	−	0.775377i	$-0.282440\pi$
$181$	16.9919	1.26300	0.631499	+	0.775377i	$0.282440\pi$
$182$	9.33192	0.691728
$183$	0	0
$184$	−2.60767	−0.192240
$185$	0.211343	0.0155382
$186$	0	0
$187$	−36.2256	−2.64908
$188$	−2.21428	−0.161493
$189$	0	0
$190$	13.8233	1.00285
$191$	−2.11778	−0.153237	−0.0766185	−	0.997060i	$-0.524412\pi$
$191$	−2.11778	−0.153237	−0.0766185	+	0.997060i	$0.524412\pi$
$192$	0	0
$193$	23.9383	1.72312	0.861560	−	0.507657i	$-0.169488\pi$
$193$	23.9383	1.72312	0.861560	+	0.507657i	$0.169488\pi$
$194$	−1.16427	−0.0835899
$195$	0	0
$196$	−3.24508	−0.231791
$197$	−9.60786	−0.684532	−0.342266	−	0.939603i	$-0.611194\pi$
$197$	−9.60786	−0.684532	−0.342266	+	0.939603i	$0.611194\pi$
$198$	0	0
$199$	−1.23963	−0.0878747	−0.0439374	−	0.999034i	$-0.513990\pi$
$199$	−1.23963	−0.0878747	−0.0439374	+	0.999034i	$0.513990\pi$
$200$	2.18978	0.154841
$201$	0	0
$202$	−4.12926	−0.290534
$203$	−6.92755	−0.486218
$204$	0	0
$205$	5.06083	0.353464
$206$	14.6863	1.02324
$207$	0	0
$208$	−19.4906	−1.35143
$209$	52.6173	3.63962
$210$	0	0
$211$	5.54877	0.381993	0.190996	−	0.981591i	$-0.438828\pi$
$211$	5.54877	0.381993	0.190996	+	0.981591i	$0.438828\pi$
$212$	5.23778	0.359732
$213$	0	0
$214$	−14.7148	−1.00588
$215$	−7.08633	−0.483283
$216$	0	0
$217$	−14.0296	−0.952393
$218$	26.0719	1.76581
$219$	0	0
$220$	−4.07271	−0.274582
$221$	23.3116	1.56811
$222$	0	0
$223$	4.63787	0.310575	0.155287	−	0.987869i	$-0.450370\pi$
$223$	4.63787	0.310575	0.155287	+	0.987869i	$0.450370\pi$
$224$	−5.13053	−0.342798
$225$	0	0
$226$	12.7197	0.846100
$227$	9.21308	0.611494	0.305747	−	0.952113i	$-0.401094\pi$
$227$	9.21308	0.611494	0.305747	+	0.952113i	$0.401094\pi$
$228$	0	0
$229$	−19.8735	−1.31328	−0.656639	−	0.754205i	$-0.728023\pi$
$229$	−19.8735	−1.31328	−0.656639	+	0.754205i	$0.728023\pi$
$230$	1.94091	0.127979
$231$	0	0
$232$	10.5776	0.694456
$233$	−2.27158	−0.148816	−0.0744081	−	0.997228i	$-0.523707\pi$
$233$	−2.27158	−0.148816	−0.0744081	+	0.997228i	$0.523707\pi$
$234$	0	0
$235$	−3.37303	−0.220032
$236$	−5.75979	−0.374931
$237$	0	0
$238$	13.6486	0.884708
$239$	12.2618	0.793147	0.396573	−	0.918003i	$-0.370199\pi$
$239$	12.2618	0.793147	0.396573	+	0.918003i	$0.370199\pi$
$240$	0	0
$241$	−20.8625	−1.34387	−0.671937	−	0.740608i	$-0.734538\pi$
$241$	−20.8625	−1.34387	−0.671937	+	0.740608i	$0.734538\pi$
$242$	−44.8040	−2.88011
$243$	0	0
$244$	−9.73848	−0.623442
$245$	−4.94325	−0.315812
$246$	0	0
$247$	−33.8600	−2.15446
$248$	21.4218	1.36028
$249$	0	0
$250$	−1.62987	−0.103082
$251$	2.85507	0.180210	0.0901052	−	0.995932i	$-0.471280\pi$
$251$	2.85507	0.180210	0.0901052	+	0.995932i	$0.471280\pi$
$252$	0	0
$253$	7.38792	0.464475
$254$	9.18578	0.576367
$255$	0	0
$256$	14.2435	0.890219
$257$	−14.8291	−0.925012	−0.462506	−	0.886616i	$-0.653050\pi$
$257$	−14.8291	−0.925012	−0.462506	+	0.886616i	$0.653050\pi$
$258$	0	0
$259$	−0.303095	−0.0188334
$260$	2.62084	0.162538
$261$	0	0
$262$	−1.67804	−0.103670
$263$	−15.5155	−0.956724	−0.478362	−	0.878163i	$-0.658769\pi$
$263$	−15.5155	−0.956724	−0.478362	+	0.878163i	$0.658769\pi$
$264$	0	0
$265$	7.97873	0.490130
$266$	−19.8245	−1.21552
$267$	0	0
$268$	8.35461	0.510339
$269$	−9.63256	−0.587308	−0.293654	−	0.955912i	$-0.594871\pi$
$269$	−9.63256	−0.587308	−0.293654	+	0.955912i	$0.594871\pi$
$270$	0	0
$271$	−24.5282	−1.48998	−0.744991	−	0.667074i	$-0.767546\pi$
$271$	−24.5282	−1.48998	−0.744991	+	0.667074i	$0.767546\pi$
$272$	−28.5063	−1.72845
$273$	0	0
$274$	−7.43794	−0.449343
$275$	−6.20398	−0.374114
$276$	0	0
$277$	27.5073	1.65275	0.826376	−	0.563119i	$-0.190399\pi$
$277$	27.5073	1.65275	0.826376	+	0.563119i	$0.190399\pi$
$278$	24.7280	1.48309
$279$	0	0
$280$	−3.14045	−0.187678
$281$	−6.53961	−0.390120	−0.195060	−	0.980791i	$-0.562490\pi$
$281$	−6.53961	−0.390120	−0.195060	+	0.980791i	$0.562490\pi$
$282$	0	0
$283$	−3.15182	−0.187356	−0.0936780	−	0.995603i	$-0.529862\pi$
$283$	−3.15182	−0.187356	−0.0936780	+	0.995603i	$0.529862\pi$
$284$	6.54182	0.388185
$285$	0	0
$286$	40.3692	2.38708
$287$	−7.25794	−0.428422
$288$	0	0
$289$	17.0949	1.00558
$290$	−7.87301	−0.462319
$291$	0	0
$292$	−5.61163	−0.328396
$293$	−27.6634	−1.61611	−0.808055	−	0.589107i	$-0.799480\pi$
$293$	−27.6634	−1.61611	−0.808055	+	0.589107i	$0.799480\pi$
$294$	0	0
$295$	−8.77392	−0.510837
$296$	0.462794	0.0268994
$297$	0	0
$298$	−38.9826	−2.25820
$299$	−4.75423	−0.274944
$300$	0	0
$301$	10.1628	0.585772
$302$	−3.55058	−0.204313
$303$	0	0
$304$	41.4052	2.37475
$305$	−14.8347	−0.849431
$306$	0	0
$307$	−6.65296	−0.379705	−0.189852	−	0.981813i	$-0.560801\pi$
$307$	−6.65296	−0.379705	−0.189852	+	0.981813i	$0.560801\pi$
$308$	5.84083	0.332812
$309$	0	0
$310$	−15.9444	−0.905580
$311$	8.82743	0.500558	0.250279	−	0.968174i	$-0.419478\pi$
$311$	8.82743	0.500558	0.250279	+	0.968174i	$0.419478\pi$
$312$	0	0
$313$	16.8560	0.952757	0.476379	−	0.879240i	$-0.341949\pi$
$313$	16.8560	0.952757	0.476379	+	0.879240i	$0.341949\pi$
$314$	33.8862	1.91231
$315$	0	0
$316$	−3.21608	−0.180919
$317$	−34.8045	−1.95482	−0.977408	−	0.211363i	$-0.932210\pi$
$317$	−34.8045	−1.95482	−0.977408	+	0.211363i	$0.932210\pi$
$318$	0	0
$319$	−29.9681	−1.67789
$320$	3.93324	0.219875
$321$	0	0
$322$	−2.78353	−0.155120
$323$	−49.5226	−2.75551
$324$	0	0
$325$	3.99234	0.221455
$326$	17.2242	0.953960
$327$	0	0
$328$	11.0821	0.611907
$329$	4.83739	0.266694
$330$	0	0
$331$	3.66619	0.201512	0.100756	−	0.994911i	$-0.467874\pi$
$331$	3.66619	0.201512	0.100756	+	0.994911i	$0.467874\pi$
$332$	1.49235	0.0819035
$333$	0	0
$334$	21.6642	1.18541
$335$	12.7266	0.695329
$336$	0	0
$337$	−8.23420	−0.448545	−0.224273	−	0.974526i	$-0.572001\pi$
$337$	−8.23420	−0.448545	−0.224273	+	0.974526i	$0.572001\pi$
$338$	−4.78984	−0.260533
$339$	0	0
$340$	3.83317	0.207883
$341$	−60.6912	−3.28661
$342$	0	0
$343$	17.1283	0.924840
$344$	−15.5175	−0.836647
$345$	0	0
$346$	13.3280	0.716519
$347$	−30.1964	−1.62103	−0.810513	−	0.585720i	$-0.800812\pi$
$347$	−30.1964	−1.62103	−0.810513	+	0.585720i	$0.800812\pi$
$348$	0	0
$349$	7.92169	0.424038	0.212019	−	0.977265i	$-0.431996\pi$
$349$	7.92169	0.424038	0.212019	+	0.977265i	$0.431996\pi$
$350$	2.33746	0.124942
$351$	0	0
$352$	−22.1943	−1.18296
$353$	−19.1412	−1.01878	−0.509391	−	0.860535i	$-0.670129\pi$
$353$	−19.1412	−1.01878	−0.509391	+	0.860535i	$0.670129\pi$
$354$	0	0
$355$	9.96518	0.528897
$356$	0.656467	0.0347927
$357$	0	0
$358$	15.3473	0.811129
$359$	−10.9539	−0.578126	−0.289063	−	0.957310i	$-0.593344\pi$
$359$	−10.9539	−0.578126	−0.289063	+	0.957310i	$0.593344\pi$
$360$	0	0
$361$	52.9312	2.78585
$362$	−27.6946	−1.45559
$363$	0	0
$364$	−3.75865	−0.197007
$365$	−8.54823	−0.447435
$366$	0	0
$367$	−1.55941	−0.0814008	−0.0407004	−	0.999171i	$-0.512959\pi$
$367$	−1.55941	−0.0814008	−0.0407004	+	0.999171i	$0.512959\pi$
$368$	5.81365	0.303057
$369$	0	0
$370$	−0.344461	−0.0179077
$371$	−11.4426	−0.594071
$372$	0	0
$373$	−28.0452	−1.45213	−0.726063	−	0.687628i	$-0.758652\pi$
$373$	−28.0452	−1.45213	−0.726063	+	0.687628i	$0.758652\pi$
$374$	59.0429	3.05303
$375$	0	0
$376$	−7.38619	−0.380914
$377$	19.2848	0.993220
$378$	0	0
$379$	−1.36801	−0.0702697	−0.0351349	−	0.999383i	$-0.511186\pi$
$379$	−1.36801	−0.0702697	−0.0351349	+	0.999383i	$0.511186\pi$
$380$	−5.56765	−0.285614
$381$	0	0
$382$	3.45169	0.176604
$383$	−8.72531	−0.445843	−0.222921	−	0.974836i	$-0.571559\pi$
$383$	−8.72531	−0.445843	−0.222921	+	0.974836i	$0.571559\pi$
$384$	0	0
$385$	8.89736	0.453452
$386$	−39.0163	−1.98588
$387$	0	0
$388$	0.468938	0.0238067
$389$	−35.7119	−1.81067	−0.905333	−	0.424702i	$-0.860379\pi$
$389$	−35.7119	−1.81067	−0.905333	+	0.424702i	$0.860379\pi$
$390$	0	0
$391$	−6.95340	−0.351648
$392$	−10.8246	−0.546726
$393$	0	0
$394$	15.6595	0.788916
$395$	−4.89907	−0.246499
$396$	0	0
$397$	−23.7219	−1.19057	−0.595284	−	0.803516i	$-0.702960\pi$
$397$	−23.7219	−1.19057	−0.595284	+	0.803516i	$0.702960\pi$
$398$	2.02042	0.101275
$399$	0	0
$400$	−4.88199	−0.244099
$401$	−7.94273	−0.396641	−0.198321	−	0.980137i	$-0.563549\pi$
$401$	−7.94273	−0.396641	−0.198321	+	0.980137i	$0.563549\pi$
$402$	0	0
$403$	39.0556	1.94550
$404$	1.66316	0.0827452
$405$	0	0
$406$	11.2910	0.560362
$407$	−1.31117	−0.0649921
$408$	0	0
$409$	−3.37471	−0.166869	−0.0834345	−	0.996513i	$-0.526589\pi$
$409$	−3.37471	−0.166869	−0.0834345	+	0.996513i	$0.526589\pi$
$410$	−8.24849	−0.407364
$411$	0	0
$412$	−5.91523	−0.291423
$413$	12.5830	0.619170
$414$	0	0
$415$	2.27331	0.111592
$416$	14.2823	0.700248
$417$	0	0
$418$	−85.7593	−4.19462
$419$	−32.6506	−1.59509	−0.797543	−	0.603261i	$-0.793868\pi$
$419$	−32.6506	−1.59509	−0.797543	+	0.603261i	$0.793868\pi$
$420$	0	0
$421$	−8.11883	−0.395687	−0.197844	−	0.980234i	$-0.563394\pi$
$421$	−8.11883	−0.395687	−0.197844	+	0.980234i	$0.563394\pi$
$422$	−9.04375	−0.440243
$423$	0	0
$424$	17.4717	0.848499
$425$	5.83909	0.283237
$426$	0	0
$427$	21.2750	1.02957
$428$	5.92672	0.286479
$429$	0	0
$430$	11.5498	0.556979
$431$	3.18260	0.153301	0.0766503	−	0.997058i	$-0.475578\pi$
$431$	3.18260	0.153301	0.0766503	+	0.997058i	$0.475578\pi$
$432$	0	0
$433$	−33.0180	−1.58674	−0.793372	−	0.608737i	$-0.791676\pi$
$433$	−33.0180	−1.58674	−0.793372	+	0.608737i	$0.791676\pi$
$434$	22.8664	1.09762
$435$	0	0
$436$	−10.5011	−0.502909
$437$	10.0998	0.483137
$438$	0	0
$439$	−0.958541	−0.0457486	−0.0228743	−	0.999738i	$-0.507282\pi$
$439$	−0.958541	−0.0457486	−0.0228743	+	0.999738i	$0.507282\pi$
$440$	−13.5853	−0.647656
$441$	0	0
$442$	−37.9949	−1.80723
$443$	17.8054	0.845960	0.422980	−	0.906139i	$-0.360984\pi$
$443$	17.8054	0.845960	0.422980	+	0.906139i	$0.360984\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−7.55911	−0.357934
$447$	0	0
$448$	−5.64081	−0.266503
$449$	29.4887	1.39166	0.695830	−	0.718207i	$-0.255037\pi$
$449$	29.4887	1.39166	0.695830	+	0.718207i	$0.255037\pi$
$450$	0	0
$451$	−31.3973	−1.47844
$452$	−5.12315	−0.240973
$453$	0	0
$454$	−15.0161	−0.704740
$455$	−5.72557	−0.268419
$456$	0	0
$457$	−18.8232	−0.880514	−0.440257	−	0.897872i	$-0.645113\pi$
$457$	−18.8232	−0.880514	−0.440257	+	0.897872i	$0.645113\pi$
$458$	32.3912	1.51354
$459$	0	0
$460$	−0.781745	−0.0364491
$461$	−6.92860	−0.322697	−0.161349	−	0.986897i	$-0.551584\pi$
$461$	−6.92860	−0.322697	−0.161349	+	0.986897i	$0.551584\pi$
$462$	0	0
$463$	14.6499	0.680839	0.340420	−	0.940274i	$-0.389431\pi$
$463$	14.6499	0.680839	0.340420	+	0.940274i	$0.389431\pi$
$464$	−23.5822	−1.09478
$465$	0	0
$466$	3.70237	0.171509
$467$	12.8528	0.594754	0.297377	−	0.954760i	$-0.403888\pi$
$467$	12.8528	0.594754	0.297377	+	0.954760i	$0.403888\pi$
$468$	0	0
$469$	−18.2517	−0.842786
$470$	5.49759	0.253585
$471$	0	0
$472$	−19.2130	−0.884348
$473$	43.9634	2.02144
$474$	0	0
$475$	−8.48123	−0.389145
$476$	−5.49730	−0.251968
$477$	0	0
$478$	−19.9850	−0.914094
$479$	10.1027	0.461604	0.230802	−	0.973001i	$-0.425865\pi$
$479$	10.1027	0.461604	0.230802	+	0.973001i	$0.425865\pi$
$480$	0	0
$481$	0.843753	0.0384718
$482$	34.0032	1.54880
$483$	0	0
$484$	18.0459	0.820266
$485$	0.714336	0.0324363
$486$	0	0
$487$	1.56850	0.0710754	0.0355377	−	0.999368i	$-0.488686\pi$
$487$	1.56850	0.0710754	0.0355377	+	0.999368i	$0.488686\pi$
$488$	−32.4847	−1.47051
$489$	0	0
$490$	8.05684	0.363971
$491$	−29.1241	−1.31435	−0.657176	−	0.753737i	$-0.728249\pi$
$491$	−29.1241	−1.31435	−0.657176	+	0.753737i	$0.728249\pi$
$492$	0	0
$493$	28.2055	1.27031
$494$	55.1872	2.48299
$495$	0	0
$496$	−47.7586	−2.14442
$497$	−14.2915	−0.641059
$498$	0	0
$499$	34.3783	1.53898	0.769492	−	0.638657i	$-0.220510\pi$
$499$	34.3783	1.53898	0.769492	+	0.638657i	$0.220510\pi$
$500$	0.656467	0.0293581
$501$	0	0
$502$	−4.65338	−0.207691
$503$	−25.9663	−1.15778	−0.578890	−	0.815406i	$-0.696514\pi$
$503$	−25.9663	−1.15778	−0.578890	+	0.815406i	$0.696514\pi$
$504$	0	0
$505$	2.53350	0.112739
$506$	−12.0413	−0.535302
$507$	0	0
$508$	−3.69979	−0.164152
$509$	−11.5305	−0.511081	−0.255541	−	0.966798i	$-0.582253\pi$
$509$	−11.5305	−0.511081	−0.255541	+	0.966798i	$0.582253\pi$
$510$	0	0
$511$	12.2593	0.542321
$512$	3.91599	0.173064
$513$	0	0
$514$	24.1694	1.06607
$515$	−9.01070	−0.397059
$516$	0	0
$517$	20.9262	0.920333
$518$	0.494004	0.0217053
$519$	0	0
$520$	8.74235	0.383377
$521$	3.76687	0.165030	0.0825148	−	0.996590i	$-0.473705\pi$
$521$	3.76687	0.165030	0.0825148	+	0.996590i	$0.473705\pi$
$522$	0	0
$523$	−24.6646	−1.07851	−0.539254	−	0.842143i	$-0.681294\pi$
$523$	−24.6646	−1.07851	−0.539254	+	0.842143i	$0.681294\pi$
$524$	0.675871	0.0295256
$525$	0	0
$526$	25.2881	1.10262
$527$	57.1216	2.48825
$528$	0	0
$529$	−21.5819	−0.938344
$530$	−13.0043	−0.564870
$531$	0	0
$532$	7.98478	0.346184
$533$	20.2046	0.875158
$534$	0	0
$535$	9.02820	0.390323
$536$	27.8685	1.20373
$537$	0	0
$538$	15.6998	0.676866
$539$	30.6678	1.32096
$540$	0	0
$541$	−19.7697	−0.849967	−0.424983	−	0.905201i	$-0.639720\pi$
$541$	−19.7697	−0.849967	−0.424983	+	0.905201i	$0.639720\pi$
$542$	39.9777	1.71719
$543$	0	0
$544$	20.8889	0.895605
$545$	−15.9963	−0.685207
$546$	0	0
$547$	−13.4518	−0.575159	−0.287579	−	0.957757i	$-0.592851\pi$
$547$	−13.4518	−0.575159	−0.287579	+	0.957757i	$0.592851\pi$
$548$	2.99581	0.127975
$549$	0	0
$550$	10.1117	0.431163
$551$	−40.9682	−1.74531
$552$	0	0
$553$	7.02594	0.298773
$554$	−44.8332	−1.90478
$555$	0	0
$556$	−9.95979	−0.422389
$557$	−29.2755	−1.24044	−0.620222	−	0.784426i	$-0.712957\pi$
$557$	−29.2755	−1.24044	−0.620222	+	0.784426i	$0.712957\pi$
$558$	0	0
$559$	−28.2910	−1.19658
$560$	7.00144	0.295865
$561$	0	0
$562$	10.6587	0.449610
$563$	12.3184	0.519157	0.259578	−	0.965722i	$-0.416416\pi$
$563$	12.3184	0.519157	0.259578	+	0.965722i	$0.416416\pi$
$564$	0	0
$565$	−7.80412	−0.328322
$566$	5.13704	0.215926
$567$	0	0
$568$	21.8216	0.915612
$569$	2.99391	0.125511	0.0627557	−	0.998029i	$-0.480011\pi$
$569$	2.99391	0.125511	0.0627557	+	0.998029i	$0.480011\pi$
$570$	0	0
$571$	2.12517	0.0889357	0.0444678	−	0.999011i	$-0.485841\pi$
$571$	2.12517	0.0889357	0.0444678	+	0.999011i	$0.485841\pi$
$572$	−16.2596	−0.679850
$573$	0	0
$574$	11.8295	0.493753
$575$	−1.19084	−0.0496613
$576$	0	0
$577$	−0.618151	−0.0257340	−0.0128670	−	0.999917i	$-0.504096\pi$
$577$	−0.618151	−0.0257340	−0.0128670	+	0.999917i	$0.504096\pi$
$578$	−27.8625	−1.15893
$579$	0	0
$580$	3.17104	0.131670
$581$	−3.26024	−0.135257
$582$	0	0
$583$	−49.4999	−2.05008
$584$	−18.7187	−0.774587
$585$	0	0
$586$	45.0876	1.86255
$587$	14.7784	0.609970	0.304985	−	0.952357i	$-0.401349\pi$
$587$	14.7784	0.609970	0.304985	+	0.952357i	$0.401349\pi$
$588$	0	0
$589$	−82.9686	−3.41866
$590$	14.3003	0.588735
$591$	0	0
$592$	−1.03177	−0.0424056
$593$	44.0916	1.81062	0.905312	−	0.424748i	$-0.139637\pi$
$593$	44.0916	1.81062	0.905312	+	0.424748i	$0.139637\pi$
$594$	0	0
$595$	−8.37406	−0.343303
$596$	15.7012	0.643144
$597$	0	0
$598$	7.74876	0.316870
$599$	−29.1740	−1.19202	−0.596008	−	0.802978i	$-0.703247\pi$
$599$	−29.1740	−1.19202	−0.596008	+	0.802978i	$0.703247\pi$
$600$	0	0
$601$	11.4828	0.468393	0.234196	−	0.972189i	$-0.424754\pi$
$601$	11.4828	0.468393	0.234196	+	0.972189i	$0.424754\pi$
$602$	−16.5640	−0.675097
$603$	0	0
$604$	1.43008	0.0581891
$605$	27.4893	1.11760
$606$	0	0
$607$	−23.5713	−0.956729	−0.478365	−	0.878161i	$-0.658770\pi$
$607$	−23.5713	−0.956729	−0.478365	+	0.878161i	$0.658770\pi$
$608$	−30.3410	−1.23049
$609$	0	0
$610$	24.1785	0.978961
$611$	−13.4663	−0.544788
$612$	0	0
$613$	21.9040	0.884695	0.442348	−	0.896844i	$-0.354146\pi$
$613$	21.9040	0.884695	0.442348	+	0.896844i	$0.354146\pi$
$614$	10.8434	0.437606
$615$	0	0
$616$	19.4833	0.785003
$617$	−13.5730	−0.546429	−0.273215	−	0.961953i	$-0.588087\pi$
$617$	−13.5730	−0.546429	−0.273215	+	0.961953i	$0.588087\pi$
$618$	0	0
$619$	−10.5154	−0.422648	−0.211324	−	0.977416i	$-0.567777\pi$
$619$	−10.5154	−0.422648	−0.211324	+	0.977416i	$0.567777\pi$
$620$	6.42197	0.257913
$621$	0	0
$622$	−14.3875	−0.576888
$623$	−1.43414	−0.0574575
$624$	0	0
$625$	1.00000	0.0400000
$626$	−27.4730	−1.09804
$627$	0	0
$628$	−13.6485	−0.544633
$629$	1.23405	0.0492048
$630$	0	0
$631$	−1.14571	−0.0456101	−0.0228051	−	0.999740i	$-0.507260\pi$
$631$	−1.14571	−0.0456101	−0.0228051	+	0.999740i	$0.507260\pi$
$632$	−10.7279	−0.426732
$633$	0	0
$634$	56.7267	2.25291
$635$	−5.63591	−0.223654
$636$	0	0
$637$	−19.7351	−0.781934
$638$	48.8440	1.93375
$639$	0	0
$640$	−13.5655	−0.536224
$641$	−13.2969	−0.525195	−0.262597	−	0.964905i	$-0.584579\pi$
$641$	−13.2969	−0.525195	−0.262597	+	0.964905i	$0.584579\pi$
$642$	0	0
$643$	45.6904	1.80185	0.900926	−	0.433972i	$-0.142888\pi$
$643$	45.6904	1.80185	0.900926	+	0.433972i	$0.142888\pi$
$644$	1.12113	0.0441787
$645$	0	0
$646$	80.7153	3.17570
$647$	22.9913	0.903881	0.451941	−	0.892048i	$-0.350732\pi$
$647$	22.9913	0.903881	0.451941	+	0.892048i	$0.350732\pi$
$648$	0	0
$649$	54.4332	2.13669
$650$	−6.50699	−0.255225
$651$	0	0
$652$	−6.93745	−0.271692
$653$	−9.73673	−0.381028	−0.190514	−	0.981685i	$-0.561015\pi$
$653$	−9.73673	−0.381028	−0.190514	+	0.981685i	$0.561015\pi$
$654$	0	0
$655$	1.02956	0.0402281
$656$	−24.7069	−0.964643
$657$	0	0
$658$	−7.88430	−0.307362
$659$	11.1806	0.435535	0.217767	−	0.976001i	$-0.430123\pi$
$659$	11.1806	0.435535	0.217767	+	0.976001i	$0.430123\pi$
$660$	0	0
$661$	−14.6673	−0.570490	−0.285245	−	0.958455i	$-0.592075\pi$
$661$	−14.6673	−0.570490	−0.285245	+	0.958455i	$0.592075\pi$
$662$	−5.97540	−0.232241
$663$	0	0
$664$	4.97804	0.193186
$665$	12.1633	0.471671
$666$	0	0
$667$	−5.75229	−0.222729
$668$	−8.72575	−0.337609
$669$	0	0
$670$	−20.7427	−0.801360
$671$	92.0340	3.55293
$672$	0	0
$673$	3.10851	0.119824	0.0599121	−	0.998204i	$-0.480918\pi$
$673$	3.10851	0.119824	0.0599121	+	0.998204i	$0.480918\pi$
$674$	13.4206	0.516944
$675$	0	0
$676$	1.92922	0.0742008
$677$	16.8848	0.648934	0.324467	−	0.945897i	$-0.394815\pi$
$677$	16.8848	0.648934	0.324467	+	0.945897i	$0.394815\pi$
$678$	0	0
$679$	−1.02446	−0.0393150
$680$	12.7863	0.490333
$681$	0	0
$682$	98.9185	3.78779
$683$	3.00712	0.115064	0.0575321	−	0.998344i	$-0.481677\pi$
$683$	3.00712	0.115064	0.0575321	+	0.998344i	$0.481677\pi$
$684$	0	0
$685$	4.56353	0.174363
$686$	−27.9168	−1.06587
$687$	0	0
$688$	34.5953	1.31894
$689$	31.8538	1.21353
$690$	0	0
$691$	−18.0413	−0.686322	−0.343161	−	0.939277i	$-0.611498\pi$
$691$	−18.0413	−0.686322	−0.343161	+	0.939277i	$0.611498\pi$
$692$	−5.36817	−0.204067
$693$	0	0
$694$	49.2161	1.86822
$695$	−15.1718	−0.575499
$696$	0	0
$697$	29.5507	1.11931
$698$	−12.9113	−0.488700
$699$	0	0
$700$	−0.941465	−0.0355840
$701$	8.18744	0.309235	0.154618	−	0.987974i	$-0.450585\pi$
$701$	8.18744	0.309235	0.154618	+	0.987974i	$0.450585\pi$
$702$	0	0
$703$	−1.79245	−0.0676034
$704$	−24.4017	−0.919674
$705$	0	0
$706$	31.1976	1.17414
$707$	−3.63338	−0.136647
$708$	0	0
$709$	−11.2972	−0.424274	−0.212137	−	0.977240i	$-0.568042\pi$
$709$	−11.2972	−0.424274	−0.212137	+	0.977240i	$0.568042\pi$
$710$	−16.2419	−0.609549
$711$	0	0
$712$	2.18978	0.0820655
$713$	−11.6495	−0.436277
$714$	0	0
$715$	−24.7684	−0.926286
$716$	−6.18148	−0.231013
$717$	0	0
$718$	17.8535	0.666285
$719$	−5.84785	−0.218088	−0.109044	−	0.994037i	$-0.534779\pi$
$719$	−5.84785	−0.218088	−0.109044	+	0.994037i	$0.534779\pi$
$720$	0	0
$721$	12.9226	0.481263
$722$	−86.2709	−3.21067
$723$	0	0
$724$	11.1546	0.414559
$725$	4.83046	0.179399
$726$	0	0
$727$	−0.772939	−0.0286667	−0.0143334	−	0.999897i	$-0.504563\pi$
$727$	−0.772939	−0.0286667	−0.0143334	+	0.999897i	$0.504563\pi$
$728$	−12.5377	−0.464680
$729$	0	0
$730$	13.9325	0.515664
$731$	−41.3777	−1.53041
$732$	0	0
$733$	1.08337	0.0400153	0.0200077	−	0.999800i	$-0.493631\pi$
$733$	1.08337	0.0400153	0.0200077	+	0.999800i	$0.493631\pi$
$734$	2.54164	0.0938136
$735$	0	0
$736$	−4.26013	−0.157031
$737$	−78.9556	−2.90837
$738$	0	0
$739$	44.0919	1.62195	0.810973	−	0.585083i	$-0.198938\pi$
$739$	44.0919	1.62195	0.810973	+	0.585083i	$0.198938\pi$
$740$	0.138740	0.00510017
$741$	0	0
$742$	18.6499	0.684661
$743$	33.9694	1.24622	0.623109	−	0.782135i	$-0.285869\pi$
$743$	33.9694	1.24622	0.623109	+	0.782135i	$0.285869\pi$
$744$	0	0
$745$	23.9176	0.876275
$746$	45.7100	1.67356
$747$	0	0
$748$	−23.7809	−0.869516
$749$	−12.9477	−0.473098
$750$	0	0
$751$	−5.89488	−0.215107	−0.107554	−	0.994199i	$-0.534302\pi$
$751$	−5.89488	−0.215107	−0.107554	+	0.994199i	$0.534302\pi$
$752$	16.4671	0.600492
$753$	0	0
$754$	−31.4317	−1.14468
$755$	2.17844	0.0792817
$756$	0	0
$757$	−26.3758	−0.958643	−0.479322	−	0.877639i	$-0.659117\pi$
$757$	−26.3758	−0.958643	−0.479322	+	0.877639i	$0.659117\pi$
$758$	2.22967	0.0809852
$759$	0	0
$760$	−18.5720	−0.673678
$761$	41.1836	1.49290	0.746452	−	0.665439i	$-0.231756\pi$
$761$	41.1836	1.49290	0.746452	+	0.665439i	$0.231756\pi$
$762$	0	0
$763$	22.9409	0.830517
$764$	−1.39025	−0.0502975
$765$	0	0
$766$	14.2211	0.513829
$767$	−35.0285	−1.26481
$768$	0	0
$769$	−26.2489	−0.946560	−0.473280	−	0.880912i	$-0.656930\pi$
$769$	−26.2489	−0.946560	−0.473280	+	0.880912i	$0.656930\pi$
$770$	−14.5015	−0.522598
$771$	0	0
$772$	15.7147	0.565586
$773$	46.0843	1.65754	0.828768	−	0.559592i	$-0.189042\pi$
$773$	46.0843	1.65754	0.828768	+	0.559592i	$0.189042\pi$
$774$	0	0
$775$	9.78262	0.351402
$776$	1.56424	0.0561529
$777$	0	0
$778$	58.2057	2.08677
$779$	−42.9221	−1.53784
$780$	0	0
$781$	−61.8238	−2.21223
$782$	11.3331	0.405271
$783$	0	0
$784$	24.1329	0.861888
$785$	−20.7908	−0.742055
$786$	0	0
$787$	−26.3708	−0.940018	−0.470009	−	0.882662i	$-0.655749\pi$
$787$	−26.3708	−0.940018	−0.470009	+	0.882662i	$0.655749\pi$
$788$	−6.30725	−0.224686
$789$	0	0
$790$	7.98483	0.284088
$791$	11.1922	0.397948
$792$	0	0
$793$	−59.2251	−2.10314
$794$	38.6635	1.37212
$795$	0	0
$796$	−0.813774	−0.0288434
$797$	−18.4603	−0.653896	−0.326948	−	0.945042i	$-0.606020\pi$
$797$	−18.4603	−0.653896	−0.326948	+	0.945042i	$0.606020\pi$
$798$	0	0
$799$	−19.6954	−0.696773
$800$	3.57743	0.126481
$801$	0	0
$802$	12.9456	0.457125
$803$	53.0330	1.87149
$804$	0	0
$805$	1.70782	0.0601929
$806$	−63.6554	−2.24217
$807$	0	0
$808$	5.54780	0.195171
$809$	1.67385	0.0588494	0.0294247	−	0.999567i	$-0.490632\pi$
$809$	1.67385	0.0588494	0.0294247	+	0.999567i	$0.490632\pi$
$810$	0	0
$811$	−8.49022	−0.298132	−0.149066	−	0.988827i	$-0.547627\pi$
$811$	−8.49022	−0.298132	−0.149066	+	0.988827i	$0.547627\pi$
$812$	−4.54771	−0.159593
$813$	0	0
$814$	2.13703	0.0749028
$815$	−10.5679	−0.370176
$816$	0	0
$817$	60.1007	2.10266
$818$	5.50034	0.192315
$819$	0	0
$820$	3.32227	0.116019
$821$	21.5245	0.751209	0.375605	−	0.926780i	$-0.377435\pi$
$821$	21.5245	0.751209	0.375605	+	0.926780i	$0.377435\pi$
$822$	0	0
$823$	26.3340	0.917946	0.458973	−	0.888450i	$-0.348217\pi$
$823$	26.3340	0.917946	0.458973	+	0.888450i	$0.348217\pi$
$824$	−19.7315	−0.687378
$825$	0	0
$826$	−20.5086	−0.713587
$827$	−1.65103	−0.0574118	−0.0287059	−	0.999588i	$-0.509139\pi$
$827$	−1.65103	−0.0574118	−0.0287059	+	0.999588i	$0.509139\pi$
$828$	0	0
$829$	−25.7305	−0.893658	−0.446829	−	0.894619i	$-0.647447\pi$
$829$	−25.7305	−0.893658	−0.446829	+	0.894619i	$0.647447\pi$
$830$	−3.70519	−0.128609
$831$	0	0
$832$	15.7028	0.544398
$833$	−28.8640	−1.00008
$834$	0	0
$835$	−13.2920	−0.459988
$836$	34.5416	1.19465
$837$	0	0
$838$	53.2162	1.83832
$839$	−17.0596	−0.588964	−0.294482	−	0.955657i	$-0.595147\pi$
$839$	−17.0596	−0.588964	−0.294482	+	0.955657i	$0.595147\pi$
$840$	0	0
$841$	−5.66666	−0.195402
$842$	13.2326	0.456026
$843$	0	0
$844$	3.64258	0.125383
$845$	2.93879	0.101098
$846$	0	0
$847$	−39.4235	−1.35461
$848$	−38.9521	−1.33762
$849$	0	0
$850$	−9.51694	−0.326428
$851$	−0.251675	−0.00862730
$852$	0	0
$853$	−23.2687	−0.796705	−0.398352	−	0.917232i	$-0.630418\pi$
$853$	−23.2687	−0.796705	−0.398352	+	0.917232i	$0.630418\pi$
$854$	−34.6754	−1.18657
$855$	0	0
$856$	19.7698	0.675717
$857$	−19.1404	−0.653822	−0.326911	−	0.945055i	$-0.606008\pi$
$857$	−19.1404	−0.653822	−0.326911	+	0.945055i	$0.606008\pi$
$858$	0	0
$859$	6.32979	0.215970	0.107985	−	0.994153i	$-0.465560\pi$
$859$	6.32979	0.215970	0.107985	+	0.994153i	$0.465560\pi$
$860$	−4.65194	−0.158630
$861$	0	0
$862$	−5.18722	−0.176677
$863$	33.0443	1.12484	0.562421	−	0.826851i	$-0.309870\pi$
$863$	33.0443	1.12484	0.562421	+	0.826851i	$0.309870\pi$
$864$	0	0
$865$	−8.17737	−0.278039
$866$	53.8150	1.82871
$867$	0	0
$868$	−9.21000	−0.312608
$869$	30.3937	1.03104
$870$	0	0
$871$	50.8090	1.72160
$872$	−35.0284	−1.18621
$873$	0	0
$874$	−16.4613	−0.556810
$875$	−1.43414	−0.0484827
$876$	0	0
$877$	−16.4255	−0.554649	−0.277324	−	0.960776i	$-0.589448\pi$
$877$	−16.4255	−0.554649	−0.277324	+	0.960776i	$0.589448\pi$
$878$	1.56229	0.0527249
$879$	0	0
$880$	30.2877	1.02100
$881$	−8.73665	−0.294345	−0.147173	−	0.989111i	$-0.547017\pi$
$881$	−8.73665	−0.294345	−0.147173	+	0.989111i	$0.547017\pi$
$882$	0	0
$883$	−23.3023	−0.784184	−0.392092	−	0.919926i	$-0.628248\pi$
$883$	−23.3023	−0.784184	−0.392092	+	0.919926i	$0.628248\pi$
$884$	15.3033	0.514707
$885$	0	0
$886$	−29.0204	−0.974960
$887$	21.1191	0.709108	0.354554	−	0.935035i	$-0.384633\pi$
$887$	21.1191	0.709108	0.354554	+	0.935035i	$0.384633\pi$
$888$	0	0
$889$	8.08267	0.271084
$890$	−1.62987	−0.0546333
$891$	0	0
$892$	3.04461	0.101941
$893$	28.6074	0.957311
$894$	0	0
$895$	−9.41628	−0.314751
$896$	19.4548	0.649940
$897$	0	0
$898$	−48.0627	−1.60387
$899$	47.2546	1.57603
$900$	0	0
$901$	46.5885	1.55209
$902$	51.1734	1.70389
$903$	0	0
$904$	−17.0893	−0.568382
$905$	16.9919	0.564830
$906$	0	0
$907$	−28.3435	−0.941130	−0.470565	−	0.882365i	$-0.655950\pi$
$907$	−28.3435	−0.941130	−0.470565	+	0.882365i	$0.655950\pi$
$908$	6.04809	0.200713
$909$	0	0
$910$	9.33192	0.309350
$911$	49.0501	1.62510	0.812551	−	0.582890i	$-0.198078\pi$
$911$	49.0501	1.62510	0.812551	+	0.582890i	$0.198078\pi$
$912$	0	0
$913$	−14.1035	−0.466759
$914$	30.6794	1.01478
$915$	0	0
$916$	−13.0463	−0.431062
$917$	−1.47653	−0.0487592
$918$	0	0
$919$	−33.4429	−1.10318	−0.551590	−	0.834115i	$-0.685979\pi$
$919$	−33.4429	−1.10318	−0.551590	+	0.834115i	$0.685979\pi$
$920$	−2.60767	−0.0859723
$921$	0	0
$922$	11.2927	0.371905
$923$	39.7844	1.30952
$924$	0	0
$925$	0.211343	0.00694891
$926$	−23.8774	−0.784661
$927$	0	0
$928$	17.2806	0.567264
$929$	37.7330	1.23798	0.618990	−	0.785399i	$-0.287542\pi$
$929$	37.7330	1.23798	0.618990	+	0.785399i	$0.287542\pi$
$930$	0	0
$931$	41.9248	1.37403
$932$	−1.49122	−0.0488465
$933$	0	0
$934$	−20.9483	−0.685449
$935$	−36.2256	−1.18470
$936$	0	0
$937$	46.4064	1.51603	0.758015	−	0.652237i	$-0.226169\pi$
$937$	46.4064	1.51603	0.758015	+	0.652237i	$0.226169\pi$
$938$	29.7479	0.971303
$939$	0	0
$940$	−2.21428	−0.0722220
$941$	−33.7507	−1.10024	−0.550120	−	0.835085i	$-0.685418\pi$
$941$	−33.7507	−1.10024	−0.550120	+	0.835085i	$0.685418\pi$
$942$	0	0
$943$	−6.02663	−0.196254
$944$	42.8342	1.39413
$945$	0	0
$946$	−71.6545	−2.32969
$947$	28.4902	0.925806	0.462903	−	0.886409i	$-0.346808\pi$
$947$	28.4902	0.925806	0.462903	+	0.886409i	$0.346808\pi$
$948$	0	0
$949$	−34.1274	−1.10782
$950$	13.8233	0.448486
$951$	0	0
$952$	−18.3373	−0.594317
$953$	−59.8000	−1.93711	−0.968556	−	0.248796i	$-0.919965\pi$
$953$	−59.8000	−1.93711	−0.968556	+	0.248796i	$0.919965\pi$
$954$	0	0
$955$	−2.11778	−0.0685296
$956$	8.04944	0.260338
$957$	0	0
$958$	−16.4660	−0.531994
$959$	−6.54473	−0.211340
$960$	0	0
$961$	64.6997	2.08709
$962$	−1.37521	−0.0443384
$963$	0	0
$964$	−13.6956	−0.441105
$965$	23.9383	0.770602
$966$	0	0
$967$	51.7784	1.66508	0.832541	−	0.553963i	$-0.186885\pi$
$967$	51.7784	1.66508	0.832541	+	0.553963i	$0.186885\pi$
$968$	60.1956	1.93476
$969$	0	0
$970$	−1.16427	−0.0373825
$971$	15.8665	0.509180	0.254590	−	0.967049i	$-0.418059\pi$
$971$	15.8665	0.509180	0.254590	+	0.967049i	$0.418059\pi$
$972$	0	0
$973$	21.7585	0.697544
$974$	−2.55644	−0.0819137
$975$	0	0
$976$	72.4226	2.31819
$977$	21.9655	0.702738	0.351369	−	0.936237i	$-0.385716\pi$
$977$	21.9655	0.702738	0.351369	+	0.936237i	$0.385716\pi$
$978$	0	0
$979$	−6.20398	−0.198280
$980$	−3.24508	−0.103660
$981$	0	0
$982$	47.4684	1.51478
$983$	38.4532	1.22647	0.613234	−	0.789901i	$-0.289868\pi$
$983$	38.4532	1.22647	0.613234	+	0.789901i	$0.289868\pi$
$984$	0	0
$985$	−9.60786	−0.306132
$986$	−45.9712	−1.46402
$987$	0	0
$988$	−22.2280	−0.707166
$989$	8.43865	0.268334
$990$	0	0
$991$	43.9109	1.39488	0.697438	−	0.716645i	$-0.254323\pi$
$991$	43.9109	1.39488	0.697438	+	0.716645i	$0.254323\pi$
$992$	34.9966	1.11114
$993$	0	0
$994$	23.2932	0.738814
$995$	−1.23963	−0.0392988
$996$	0	0
$997$	31.9880	1.01307	0.506535	−	0.862220i	$-0.330926\pi$
$997$	31.9880	1.01307	0.506535	+	0.862220i	$0.330926\pi$
$998$	−56.0320	−1.77366
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.u.1.4	✓	12
3.2	odd	2		4005.2.a.v.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.4	✓	12	1.1	even	1	trivial
4005.2.a.v.1.9	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)

\(f(q)\)	\(=\)	\(q-1.62987 q^{2} +0.656467 q^{4} +1.00000 q^{5} -1.43414 q^{7} +2.18978 q^{8} +O(q^{10})\)	\(q-1.62987 q^{2} +0.656467 q^{4} +1.00000 q^{5} -1.43414 q^{7} +2.18978 q^{8} -1.62987 q^{10} -6.20398 q^{11} +3.99234 q^{13} +2.33746 q^{14} -4.88199 q^{16} +5.83909 q^{17} -8.48123 q^{19} +0.656467 q^{20} +10.1117 q^{22} -1.19084 q^{23} +1.00000 q^{25} -6.50699 q^{26} -0.941465 q^{28} +4.83046 q^{29} +9.78262 q^{31} +3.57743 q^{32} -9.51694 q^{34} -1.43414 q^{35} +0.211343 q^{37} +13.8233 q^{38} +2.18978 q^{40} +5.06083 q^{41} -7.08633 q^{43} -4.07271 q^{44} +1.94091 q^{46} -3.37303 q^{47} -4.94325 q^{49} -1.62987 q^{50} +2.62084 q^{52} +7.97873 q^{53} -6.20398 q^{55} -3.14045 q^{56} -7.87301 q^{58} -8.77392 q^{59} -14.8347 q^{61} -15.9444 q^{62} +3.93324 q^{64} +3.99234 q^{65} +12.7266 q^{67} +3.83317 q^{68} +2.33746 q^{70} +9.96518 q^{71} -8.54823 q^{73} -0.344461 q^{74} -5.56765 q^{76} +8.89736 q^{77} -4.89907 q^{79} -4.88199 q^{80} -8.24849 q^{82} +2.27331 q^{83} +5.83909 q^{85} +11.5498 q^{86} -13.5853 q^{88} +1.00000 q^{89} -5.72557 q^{91} -0.781745 q^{92} +5.49759 q^{94} -8.48123 q^{95} +0.714336 q^{97} +8.05684 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) \) 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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