LMFDB - Embedded newform 4012.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.24628$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34833 q^{3} +3.93023 q^{5} -4.81217 q^{7} +2.51465 q^{9} +O(q^{10})$	$q-2.34833 q^{3} +3.93023 q^{5} -4.81217 q^{7} +2.51465 q^{9} -2.50895 q^{11} -4.25199 q^{13} -9.22948 q^{15} +1.00000 q^{17} +7.47278 q^{19} +11.3006 q^{21} -2.47615 q^{23} +10.4467 q^{25} +1.13976 q^{27} +6.29468 q^{29} +5.98076 q^{31} +5.89184 q^{33} -18.9130 q^{35} +3.40521 q^{37} +9.98507 q^{39} +6.76903 q^{41} -3.05061 q^{43} +9.88316 q^{45} -13.5025 q^{47} +16.1570 q^{49} -2.34833 q^{51} -11.4629 q^{53} -9.86075 q^{55} -17.5486 q^{57} -1.00000 q^{59} -4.74419 q^{61} -12.1009 q^{63} -16.7113 q^{65} -7.93289 q^{67} +5.81482 q^{69} +2.82210 q^{71} -5.82442 q^{73} -24.5324 q^{75} +12.0735 q^{77} -8.11556 q^{79} -10.2205 q^{81} +16.8760 q^{83} +3.93023 q^{85} -14.7820 q^{87} -12.0533 q^{89} +20.4613 q^{91} -14.0448 q^{93} +29.3698 q^{95} +9.46244 q^{97} -6.30913 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.34833	−1.35581	−0.677904	−	0.735150i	$-0.737112\pi$
$3$	−2.34833	−1.35581	−0.677904	+	0.735150i	$0.737112\pi$
$4$	0	0
$5$	3.93023	1.75765	0.878827	−	0.477141i	$-0.158327\pi$
$5$	3.93023	1.75765	0.878827	+	0.477141i	$0.158327\pi$
$6$	0	0
$7$	−4.81217	−1.81883	−0.909416	−	0.415889i	$-0.863471\pi$
$7$	−4.81217	−1.81883	−0.909416	+	0.415889i	$0.863471\pi$
$8$	0	0
$9$	2.51465	0.838217
$10$	0	0
$11$	−2.50895	−0.756476	−0.378238	−	0.925708i	$-0.623470\pi$
$11$	−2.50895	−0.756476	−0.378238	+	0.925708i	$0.623470\pi$
$12$	0	0
$13$	−4.25199	−1.17929	−0.589645	−	0.807663i	$-0.700732\pi$
$13$	−4.25199	−1.17929	−0.589645	+	0.807663i	$0.700732\pi$
$14$	0	0
$15$	−9.22948	−2.38304
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	7.47278	1.71437	0.857187	−	0.515005i	$-0.172210\pi$
$19$	7.47278	1.71437	0.857187	+	0.515005i	$0.172210\pi$
$20$	0	0
$21$	11.3006	2.46599
$22$	0	0
$23$	−2.47615	−0.516314	−0.258157	−	0.966103i	$-0.583115\pi$
$23$	−2.47615	−0.516314	−0.258157	+	0.966103i	$0.583115\pi$
$24$	0	0
$25$	10.4467	2.08935
$26$	0	0
$27$	1.13976	0.219347
$28$	0	0
$29$	6.29468	1.16889	0.584446	−	0.811432i	$-0.301312\pi$
$29$	6.29468	1.16889	0.584446	+	0.811432i	$0.301312\pi$
$30$	0	0
$31$	5.98076	1.07418	0.537088	−	0.843526i	$-0.319524\pi$
$31$	5.98076	1.07418	0.537088	+	0.843526i	$0.319524\pi$
$32$	0	0
$33$	5.89184	1.02564
$34$	0	0
$35$	−18.9130	−3.19687
$36$	0	0
$37$	3.40521	0.559813	0.279907	−	0.960027i	$-0.409696\pi$
$37$	3.40521	0.559813	0.279907	+	0.960027i	$0.409696\pi$
$38$	0	0
$39$	9.98507	1.59889
$40$	0	0
$41$	6.76903	1.05714	0.528572	−	0.848888i	$-0.322728\pi$
$41$	6.76903	1.05714	0.528572	+	0.848888i	$0.322728\pi$
$42$	0	0
$43$	−3.05061	−0.465213	−0.232607	−	0.972571i	$-0.574725\pi$
$43$	−3.05061	−0.465213	−0.232607	+	0.972571i	$0.574725\pi$
$44$	0	0
$45$	9.88316	1.47329
$46$	0	0
$47$	−13.5025	−1.96955	−0.984774	−	0.173838i	$-0.944383\pi$
$47$	−13.5025	−1.96955	−0.984774	+	0.173838i	$0.944383\pi$
$48$	0	0
$49$	16.1570	2.30815
$50$	0	0
$51$	−2.34833	−0.328832
$52$	0	0
$53$	−11.4629	−1.57454	−0.787272	−	0.616606i	$-0.788507\pi$
$53$	−11.4629	−1.57454	−0.787272	+	0.616606i	$0.788507\pi$
$54$	0	0
$55$	−9.86075	−1.32962
$56$	0	0
$57$	−17.5486	−2.32436
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−4.74419	−0.607431	−0.303716	−	0.952763i	$-0.598227\pi$
$61$	−4.74419	−0.607431	−0.303716	+	0.952763i	$0.598227\pi$
$62$	0	0
$63$	−12.1009	−1.52457
$64$	0	0
$65$	−16.7113	−2.07278
$66$	0	0
$67$	−7.93289	−0.969157	−0.484578	−	0.874748i	$-0.661027\pi$
$67$	−7.93289	−0.969157	−0.484578	+	0.874748i	$0.661027\pi$
$68$	0	0
$69$	5.81482	0.700022
$70$	0	0
$71$	2.82210	0.334921	0.167461	−	0.985879i	$-0.446443\pi$
$71$	2.82210	0.334921	0.167461	+	0.985879i	$0.446443\pi$
$72$	0	0
$73$	−5.82442	−0.681697	−0.340849	−	0.940118i	$-0.610714\pi$
$73$	−5.82442	−0.681697	−0.340849	+	0.940118i	$0.610714\pi$
$74$	0	0
$75$	−24.5324	−2.83275
$76$	0	0
$77$	12.0735	1.37590
$78$	0	0
$79$	−8.11556	−0.913072	−0.456536	−	0.889705i	$-0.650910\pi$
$79$	−8.11556	−0.913072	−0.456536	+	0.889705i	$0.650910\pi$
$80$	0	0
$81$	−10.2205	−1.13561
$82$	0	0
$83$	16.8760	1.85238	0.926191	−	0.377054i	$-0.123063\pi$
$83$	16.8760	1.85238	0.926191	+	0.377054i	$0.123063\pi$
$84$	0	0
$85$	3.93023	0.426294
$86$	0	0
$87$	−14.7820	−1.58479
$88$	0	0
$89$	−12.0533	−1.27764	−0.638822	−	0.769355i	$-0.720578\pi$
$89$	−12.0533	−1.27764	−0.638822	+	0.769355i	$0.720578\pi$
$90$	0	0
$91$	20.4613	2.14493
$92$	0	0
$93$	−14.0448	−1.45638
$94$	0	0
$95$	29.3698	3.01327
$96$	0	0
$97$	9.46244	0.960765	0.480383	−	0.877059i	$-0.340498\pi$
$97$	9.46244	0.960765	0.480383	+	0.877059i	$0.340498\pi$
$98$	0	0
$99$	−6.30913	−0.634091
$100$	0	0
$101$	4.62365	0.460071	0.230035	−	0.973182i	$-0.426116\pi$
$101$	4.62365	0.460071	0.230035	+	0.973182i	$0.426116\pi$
$102$	0	0
$103$	7.79944	0.768502	0.384251	−	0.923229i	$-0.374460\pi$
$103$	7.79944	0.768502	0.384251	+	0.923229i	$0.374460\pi$
$104$	0	0
$105$	44.4139	4.33435
$106$	0	0
$107$	−15.8880	−1.53595	−0.767974	−	0.640481i	$-0.778735\pi$
$107$	−15.8880	−1.53595	−0.767974	+	0.640481i	$0.778735\pi$
$108$	0	0
$109$	−16.9174	−1.62040	−0.810199	−	0.586155i	$-0.800641\pi$
$109$	−16.9174	−1.62040	−0.810199	+	0.586155i	$0.800641\pi$
$110$	0	0
$111$	−7.99656	−0.759000
$112$	0	0
$113$	−17.2011	−1.61815	−0.809073	−	0.587709i	$-0.800030\pi$
$113$	−17.2011	−1.61815	−0.809073	+	0.587709i	$0.800030\pi$
$114$	0	0
$115$	−9.73186	−0.907500
$116$	0	0
$117$	−10.6923	−0.988500
$118$	0	0
$119$	−4.81217	−0.441131
$120$	0	0
$121$	−4.70518	−0.427743
$122$	0	0
$123$	−15.8959	−1.43329
$124$	0	0
$125$	21.4069	1.91469
$126$	0	0
$127$	0.528999	0.0469411	0.0234705	−	0.999725i	$-0.492528\pi$
$127$	0.528999	0.0469411	0.0234705	+	0.999725i	$0.492528\pi$
$128$	0	0
$129$	7.16383	0.630740
$130$	0	0
$131$	0.530059	0.0463114	0.0231557	−	0.999732i	$-0.492629\pi$
$131$	0.530059	0.0463114	0.0231557	+	0.999732i	$0.492629\pi$
$132$	0	0
$133$	−35.9603	−3.11816
$134$	0	0
$135$	4.47953	0.385536
$136$	0	0
$137$	8.32648	0.711379	0.355689	−	0.934604i	$-0.384246\pi$
$137$	8.32648	0.711379	0.355689	+	0.934604i	$0.384246\pi$
$138$	0	0
$139$	−16.6367	−1.41111	−0.705555	−	0.708655i	$-0.749302\pi$
$139$	−16.6367	−1.41111	−0.705555	+	0.708655i	$0.749302\pi$
$140$	0	0
$141$	31.7084	2.67033
$142$	0	0
$143$	10.6680	0.892105
$144$	0	0
$145$	24.7396	2.05451
$146$	0	0
$147$	−37.9420	−3.12941
$148$	0	0
$149$	−9.53456	−0.781101	−0.390551	−	0.920581i	$-0.627715\pi$
$149$	−9.53456	−0.781101	−0.390551	+	0.920581i	$0.627715\pi$
$150$	0	0
$151$	12.9156	1.05106	0.525528	−	0.850776i	$-0.323868\pi$
$151$	12.9156	1.05106	0.525528	+	0.850776i	$0.323868\pi$
$152$	0	0
$153$	2.51465	0.203297
$154$	0	0
$155$	23.5058	1.88803
$156$	0	0
$157$	−11.9074	−0.950317	−0.475159	−	0.879900i	$-0.657609\pi$
$157$	−11.9074	−0.950317	−0.475159	+	0.879900i	$0.657609\pi$
$158$	0	0
$159$	26.9186	2.13478
$160$	0	0
$161$	11.9157	0.939087
$162$	0	0
$163$	−6.73479	−0.527510	−0.263755	−	0.964590i	$-0.584961\pi$
$163$	−6.73479	−0.527510	−0.263755	+	0.964590i	$0.584961\pi$
$164$	0	0
$165$	23.1563	1.80271
$166$	0	0
$167$	−0.379383	−0.0293575	−0.0146788	−	0.999892i	$-0.504673\pi$
$167$	−0.379383	−0.0293575	−0.0146788	+	0.999892i	$0.504673\pi$
$168$	0	0
$169$	5.07941	0.390724
$170$	0	0
$171$	18.7914	1.43702
$172$	0	0
$173$	13.7353	1.04427	0.522136	−	0.852862i	$-0.325135\pi$
$173$	13.7353	1.04427	0.522136	+	0.852862i	$0.325135\pi$
$174$	0	0
$175$	−50.2715	−3.80017
$176$	0	0
$177$	2.34833	0.176511
$178$	0	0
$179$	1.75765	0.131373	0.0656863	−	0.997840i	$-0.479076\pi$
$179$	1.75765	0.131373	0.0656863	+	0.997840i	$0.479076\pi$
$180$	0	0
$181$	−0.383138	−0.0284784	−0.0142392	−	0.999899i	$-0.504533\pi$
$181$	−0.383138	−0.0284784	−0.0142392	+	0.999899i	$0.504533\pi$
$182$	0	0
$183$	11.1409	0.823560
$184$	0	0
$185$	13.3833	0.983958
$186$	0	0
$187$	−2.50895	−0.183472
$188$	0	0
$189$	−5.48473	−0.398955
$190$	0	0
$191$	7.60737	0.550450	0.275225	−	0.961380i	$-0.411248\pi$
$191$	7.60737	0.550450	0.275225	+	0.961380i	$0.411248\pi$
$192$	0	0
$193$	3.74911	0.269867	0.134934	−	0.990855i	$-0.456918\pi$
$193$	3.74911	0.269867	0.134934	+	0.990855i	$0.456918\pi$
$194$	0	0
$195$	39.2436	2.81030
$196$	0	0
$197$	−2.24800	−0.160163	−0.0800817	−	0.996788i	$-0.525518\pi$
$197$	−2.24800	−0.160163	−0.0800817	+	0.996788i	$0.525518\pi$
$198$	0	0
$199$	−23.2119	−1.64545	−0.822723	−	0.568443i	$-0.807546\pi$
$199$	−23.2119	−1.64545	−0.822723	+	0.568443i	$0.807546\pi$
$200$	0	0
$201$	18.6290	1.31399
$202$	0	0
$203$	−30.2911	−2.12602
$204$	0	0
$205$	26.6038	1.85809
$206$	0	0
$207$	−6.22666	−0.432783
$208$	0	0
$209$	−18.7488	−1.29688
$210$	0	0
$211$	−26.1811	−1.80238	−0.901192	−	0.433420i	$-0.857307\pi$
$211$	−26.1811	−1.80238	−0.901192	+	0.433420i	$0.857307\pi$
$212$	0	0
$213$	−6.62721	−0.454089
$214$	0	0
$215$	−11.9896	−0.817684
$216$	0	0
$217$	−28.7804	−1.95374
$218$	0	0
$219$	13.6777	0.924251
$220$	0	0
$221$	−4.25199	−0.286020
$222$	0	0
$223$	−2.29267	−0.153528	−0.0767642	−	0.997049i	$-0.524459\pi$
$223$	−2.29267	−0.153528	−0.0767642	+	0.997049i	$0.524459\pi$
$224$	0	0
$225$	26.2699	1.75132
$226$	0	0
$227$	−16.4607	−1.09253	−0.546267	−	0.837611i	$-0.683952\pi$
$227$	−16.4607	−1.09253	−0.546267	+	0.837611i	$0.683952\pi$
$228$	0	0
$229$	20.4313	1.35014	0.675069	−	0.737755i	$-0.264114\pi$
$229$	20.4313	1.35014	0.675069	+	0.737755i	$0.264114\pi$
$230$	0	0
$231$	−28.3526	−1.86546
$232$	0	0
$233$	−23.1877	−1.51908	−0.759540	−	0.650461i	$-0.774576\pi$
$233$	−23.1877	−1.51908	−0.759540	+	0.650461i	$0.774576\pi$
$234$	0	0
$235$	−53.0681	−3.46178
$236$	0	0
$237$	19.0580	1.23795
$238$	0	0
$239$	16.4275	1.06261	0.531303	−	0.847182i	$-0.321703\pi$
$239$	16.4275	1.06261	0.531303	+	0.847182i	$0.321703\pi$
$240$	0	0
$241$	−12.3496	−0.795507	−0.397754	−	0.917492i	$-0.630210\pi$
$241$	−12.3496	−0.795507	−0.397754	+	0.917492i	$0.630210\pi$
$242$	0	0
$243$	20.5818	1.32032
$244$	0	0
$245$	63.5009	4.05692
$246$	0	0
$247$	−31.7742	−2.02174
$248$	0	0
$249$	−39.6304	−2.51148
$250$	0	0
$251$	−11.0264	−0.695977	−0.347989	−	0.937499i	$-0.613135\pi$
$251$	−11.0264	−0.695977	−0.347989	+	0.937499i	$0.613135\pi$
$252$	0	0
$253$	6.21254	0.390579
$254$	0	0
$255$	−9.22948	−0.577972
$256$	0	0
$257$	−25.2167	−1.57298	−0.786489	−	0.617605i	$-0.788103\pi$
$257$	−25.2167	−1.57298	−0.786489	+	0.617605i	$0.788103\pi$
$258$	0	0
$259$	−16.3865	−1.01821
$260$	0	0
$261$	15.8289	0.979785
$262$	0	0
$263$	28.0874	1.73194	0.865970	−	0.500096i	$-0.166702\pi$
$263$	28.0874	1.73194	0.865970	+	0.500096i	$0.166702\pi$
$264$	0	0
$265$	−45.0517	−2.76750
$266$	0	0
$267$	28.3050	1.73224
$268$	0	0
$269$	−2.97514	−0.181398	−0.0906988	−	0.995878i	$-0.528910\pi$
$269$	−2.97514	−0.181398	−0.0906988	+	0.995878i	$0.528910\pi$
$270$	0	0
$271$	10.3539	0.628952	0.314476	−	0.949265i	$-0.398171\pi$
$271$	10.3539	0.628952	0.314476	+	0.949265i	$0.398171\pi$
$272$	0	0
$273$	−48.0499	−2.90811
$274$	0	0
$275$	−26.2103	−1.58054
$276$	0	0
$277$	6.31386	0.379363	0.189682	−	0.981846i	$-0.439254\pi$
$277$	6.31386	0.379363	0.189682	+	0.981846i	$0.439254\pi$
$278$	0	0
$279$	15.0395	0.900392
$280$	0	0
$281$	−6.93928	−0.413963	−0.206981	−	0.978345i	$-0.566364\pi$
$281$	−6.93928	−0.413963	−0.206981	+	0.978345i	$0.566364\pi$
$282$	0	0
$283$	−12.5429	−0.745597	−0.372799	−	0.927912i	$-0.621602\pi$
$283$	−12.5429	−0.745597	−0.372799	+	0.927912i	$0.621602\pi$
$284$	0	0
$285$	−68.9699	−4.08542
$286$	0	0
$287$	−32.5737	−1.92277
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−22.2209	−1.30261
$292$	0	0
$293$	−7.00779	−0.409400	−0.204700	−	0.978825i	$-0.565622\pi$
$293$	−7.00779	−0.409400	−0.204700	+	0.978825i	$0.565622\pi$
$294$	0	0
$295$	−3.93023	−0.228827
$296$	0	0
$297$	−2.85960	−0.165931
$298$	0	0
$299$	10.5286	0.608883
$300$	0	0
$301$	14.6801	0.846145
$302$	0	0
$303$	−10.8579	−0.623768
$304$	0	0
$305$	−18.6458	−1.06765
$306$	0	0
$307$	23.1426	1.32082	0.660408	−	0.750907i	$-0.270383\pi$
$307$	23.1426	1.32082	0.660408	+	0.750907i	$0.270383\pi$
$308$	0	0
$309$	−18.3157	−1.04194
$310$	0	0
$311$	19.2746	1.09296	0.546482	−	0.837471i	$-0.315967\pi$
$311$	19.2746	1.09296	0.546482	+	0.837471i	$0.315967\pi$
$312$	0	0
$313$	14.3610	0.811731	0.405865	−	0.913933i	$-0.366970\pi$
$313$	14.3610	0.811731	0.405865	+	0.913933i	$0.366970\pi$
$314$	0	0
$315$	−47.5595	−2.67967
$316$	0	0
$317$	28.7875	1.61686	0.808432	−	0.588589i	$-0.200316\pi$
$317$	28.7875	1.61686	0.808432	+	0.588589i	$0.200316\pi$
$318$	0	0
$319$	−15.7930	−0.884240
$320$	0	0
$321$	37.3102	2.08245
$322$	0	0
$323$	7.47278	0.415797
$324$	0	0
$325$	−44.4194	−2.46394
$326$	0	0
$327$	39.7277	2.19695
$328$	0	0
$329$	64.9766	3.58228
$330$	0	0
$331$	−22.9677	−1.26242	−0.631209	−	0.775613i	$-0.717441\pi$
$331$	−22.9677	−1.26242	−0.631209	+	0.775613i	$0.717441\pi$
$332$	0	0
$333$	8.56292	0.469245
$334$	0	0
$335$	−31.1781	−1.70344
$336$	0	0
$337$	−7.56081	−0.411863	−0.205932	−	0.978566i	$-0.566022\pi$
$337$	−7.56081	−0.411863	−0.205932	+	0.978566i	$0.566022\pi$
$338$	0	0
$339$	40.3939	2.19390
$340$	0	0
$341$	−15.0054	−0.812589
$342$	0	0
$343$	−44.0652	−2.37930
$344$	0	0
$345$	22.8536	1.23040
$346$	0	0
$347$	−13.5462	−0.727196	−0.363598	−	0.931556i	$-0.618452\pi$
$347$	−13.5462	−0.727196	−0.363598	+	0.931556i	$0.618452\pi$
$348$	0	0
$349$	−5.29664	−0.283523	−0.141761	−	0.989901i	$-0.545277\pi$
$349$	−5.29664	−0.283523	−0.141761	+	0.989901i	$0.545277\pi$
$350$	0	0
$351$	−4.84625	−0.258674
$352$	0	0
$353$	3.94838	0.210151	0.105076	−	0.994464i	$-0.466492\pi$
$353$	3.94838	0.210151	0.105076	+	0.994464i	$0.466492\pi$
$354$	0	0
$355$	11.0915	0.588675
$356$	0	0
$357$	11.3006	0.598090
$358$	0	0
$359$	31.6859	1.67232	0.836160	−	0.548486i	$-0.184796\pi$
$359$	31.6859	1.67232	0.836160	+	0.548486i	$0.184796\pi$
$360$	0	0
$361$	36.8425	1.93908
$362$	0	0
$363$	11.0493	0.579938
$364$	0	0
$365$	−22.8913	−1.19819
$366$	0	0
$367$	12.4402	0.649375	0.324688	−	0.945821i	$-0.394741\pi$
$367$	12.4402	0.649375	0.324688	+	0.945821i	$0.394741\pi$
$368$	0	0
$369$	17.0217	0.886116
$370$	0	0
$371$	55.1613	2.86383
$372$	0	0
$373$	−0.972307	−0.0503442	−0.0251721	−	0.999683i	$-0.508013\pi$
$373$	−0.972307	−0.0503442	−0.0251721	+	0.999683i	$0.508013\pi$
$374$	0	0
$375$	−50.2704	−2.59595
$376$	0	0
$377$	−26.7649	−1.37846
$378$	0	0
$379$	−19.3777	−0.995364	−0.497682	−	0.867359i	$-0.665815\pi$
$379$	−19.3777	−0.995364	−0.497682	+	0.867359i	$0.665815\pi$
$380$	0	0
$381$	−1.24226	−0.0636431
$382$	0	0
$383$	15.2124	0.777319	0.388659	−	0.921382i	$-0.372938\pi$
$383$	15.2124	0.777319	0.388659	+	0.921382i	$0.372938\pi$
$384$	0	0
$385$	47.4517	2.41836
$386$	0	0
$387$	−7.67121	−0.389950
$388$	0	0
$389$	−0.472752	−0.0239695	−0.0119847	−	0.999928i	$-0.503815\pi$
$389$	−0.472752	−0.0239695	−0.0119847	+	0.999928i	$0.503815\pi$
$390$	0	0
$391$	−2.47615	−0.125224
$392$	0	0
$393$	−1.24475	−0.0627894
$394$	0	0
$395$	−31.8960	−1.60486
$396$	0	0
$397$	−14.9101	−0.748318	−0.374159	−	0.927365i	$-0.622069\pi$
$397$	−14.9101	−0.748318	−0.374159	+	0.927365i	$0.622069\pi$
$398$	0	0
$399$	84.4467	4.22762
$400$	0	0
$401$	−17.7197	−0.884881	−0.442440	−	0.896798i	$-0.645887\pi$
$401$	−17.7197	−0.884881	−0.442440	+	0.896798i	$0.645887\pi$
$402$	0	0
$403$	−25.4301	−1.26676
$404$	0	0
$405$	−40.1689	−1.99601
$406$	0	0
$407$	−8.54350	−0.423486
$408$	0	0
$409$	−4.58313	−0.226621	−0.113310	−	0.993560i	$-0.536145\pi$
$409$	−4.58313	−0.226621	−0.113310	+	0.993560i	$0.536145\pi$
$410$	0	0
$411$	−19.5533	−0.964493
$412$	0	0
$413$	4.81217	0.236792
$414$	0	0
$415$	66.3266	3.25585
$416$	0	0
$417$	39.0685	1.91319
$418$	0	0
$419$	−13.8412	−0.676185	−0.338093	−	0.941113i	$-0.609782\pi$
$419$	−13.8412	−0.676185	−0.338093	+	0.941113i	$0.609782\pi$
$420$	0	0
$421$	−21.6057	−1.05300	−0.526498	−	0.850176i	$-0.676495\pi$
$421$	−21.6057	−1.05300	−0.526498	+	0.850176i	$0.676495\pi$
$422$	0	0
$423$	−33.9542	−1.65091
$424$	0	0
$425$	10.4467	0.506741
$426$	0	0
$427$	22.8299	1.10481
$428$	0	0
$429$	−25.0520	−1.20952
$430$	0	0
$431$	−13.9866	−0.673710	−0.336855	−	0.941556i	$-0.609363\pi$
$431$	−13.9866	−0.673710	−0.336855	+	0.941556i	$0.609363\pi$
$432$	0	0
$433$	17.7117	0.851169	0.425584	−	0.904919i	$-0.360069\pi$
$433$	17.7117	0.851169	0.425584	+	0.904919i	$0.360069\pi$
$434$	0	0
$435$	−58.0966	−2.78552
$436$	0	0
$437$	−18.5037	−0.885154
$438$	0	0
$439$	−31.7236	−1.51408	−0.757042	−	0.653366i	$-0.773356\pi$
$439$	−31.7236	−1.51408	−0.757042	+	0.653366i	$0.773356\pi$
$440$	0	0
$441$	40.6293	1.93473
$442$	0	0
$443$	−14.7103	−0.698907	−0.349454	−	0.936954i	$-0.613633\pi$
$443$	−14.7103	−0.698907	−0.349454	+	0.936954i	$0.613633\pi$
$444$	0	0
$445$	−47.3721	−2.24565
$446$	0	0
$447$	22.3903	1.05902
$448$	0	0
$449$	−33.6222	−1.58673	−0.793365	−	0.608746i	$-0.791673\pi$
$449$	−33.6222	−1.58673	−0.793365	+	0.608746i	$0.791673\pi$
$450$	0	0
$451$	−16.9831	−0.799705
$452$	0	0
$453$	−30.3301	−1.42503
$454$	0	0
$455$	80.4177	3.77004
$456$	0	0
$457$	13.0576	0.610811	0.305405	−	0.952222i	$-0.401208\pi$
$457$	13.0576	0.610811	0.305405	+	0.952222i	$0.401208\pi$
$458$	0	0
$459$	1.13976	0.0531995
$460$	0	0
$461$	−1.17363	−0.0546615	−0.0273308	−	0.999626i	$-0.508701\pi$
$461$	−1.17363	−0.0546615	−0.0273308	+	0.999626i	$0.508701\pi$
$462$	0	0
$463$	29.5694	1.37420	0.687102	−	0.726561i	$-0.258882\pi$
$463$	29.5694	1.37420	0.687102	+	0.726561i	$0.258882\pi$
$464$	0	0
$465$	−55.1993	−2.55980
$466$	0	0
$467$	30.0834	1.39209	0.696046	−	0.717997i	$-0.254941\pi$
$467$	30.0834	1.39209	0.696046	+	0.717997i	$0.254941\pi$
$468$	0	0
$469$	38.1745	1.76273
$470$	0	0
$471$	27.9626	1.28845
$472$	0	0
$473$	7.65382	0.351923
$474$	0	0
$475$	78.0661	3.58192
$476$	0	0
$477$	−28.8251	−1.31981
$478$	0	0
$479$	−9.63007	−0.440009	−0.220005	−	0.975499i	$-0.570607\pi$
$479$	−9.63007	−0.440009	−0.220005	+	0.975499i	$0.570607\pi$
$480$	0	0
$481$	−14.4789	−0.660182
$482$	0	0
$483$	−27.9819	−1.27322
$484$	0	0
$485$	37.1896	1.68869
$486$	0	0
$487$	−34.6848	−1.57172	−0.785859	−	0.618406i	$-0.787779\pi$
$487$	−34.6848	−1.57172	−0.785859	+	0.618406i	$0.787779\pi$
$488$	0	0
$489$	15.8155	0.715202
$490$	0	0
$491$	−18.2672	−0.824387	−0.412193	−	0.911096i	$-0.635237\pi$
$491$	−18.2672	−0.824387	−0.412193	+	0.911096i	$0.635237\pi$
$492$	0	0
$493$	6.29468	0.283498
$494$	0	0
$495$	−24.7963	−1.11451
$496$	0	0
$497$	−13.5804	−0.609165
$498$	0	0
$499$	21.2768	0.952480	0.476240	−	0.879315i	$-0.341999\pi$
$499$	21.2768	0.952480	0.476240	+	0.879315i	$0.341999\pi$
$500$	0	0
$501$	0.890916	0.0398032
$502$	0	0
$503$	−27.2597	−1.21545	−0.607725	−	0.794148i	$-0.707918\pi$
$503$	−27.2597	−1.21545	−0.607725	+	0.794148i	$0.707918\pi$
$504$	0	0
$505$	18.1720	0.808644
$506$	0	0
$507$	−11.9281	−0.529747
$508$	0	0
$509$	26.0733	1.15568	0.577838	−	0.816151i	$-0.303896\pi$
$509$	26.0733	1.15568	0.577838	+	0.816151i	$0.303896\pi$
$510$	0	0
$511$	28.0281	1.23989
$512$	0	0
$513$	8.51719	0.376043
$514$	0	0
$515$	30.6536	1.35076
$516$	0	0
$517$	33.8772	1.48992
$518$	0	0
$519$	−32.2549	−1.41583
$520$	0	0
$521$	−38.0758	−1.66813	−0.834066	−	0.551665i	$-0.813993\pi$
$521$	−38.0758	−1.66813	−0.834066	+	0.551665i	$0.813993\pi$
$522$	0	0
$523$	23.1028	1.01022	0.505108	−	0.863056i	$-0.331453\pi$
$523$	23.1028	1.01022	0.505108	+	0.863056i	$0.331453\pi$
$524$	0	0
$525$	118.054	5.15230
$526$	0	0
$527$	5.98076	0.260526
$528$	0	0
$529$	−16.8687	−0.733420
$530$	0	0
$531$	−2.51465	−0.109127
$532$	0	0
$533$	−28.7818	−1.24668
$534$	0	0
$535$	−62.4434	−2.69966
$536$	0	0
$537$	−4.12753	−0.178116
$538$	0	0
$539$	−40.5372	−1.74606
$540$	0	0
$541$	2.25111	0.0967829	0.0483914	−	0.998828i	$-0.484591\pi$
$541$	2.25111	0.0967829	0.0483914	+	0.998828i	$0.484591\pi$
$542$	0	0
$543$	0.899733	0.0386112
$544$	0	0
$545$	−66.4895	−2.84810
$546$	0	0
$547$	26.7187	1.14241	0.571204	−	0.820808i	$-0.306476\pi$
$547$	26.7187	1.14241	0.571204	+	0.820808i	$0.306476\pi$
$548$	0	0
$549$	−11.9300	−0.509159
$550$	0	0
$551$	47.0388	2.00392
$552$	0	0
$553$	39.0535	1.66072
$554$	0	0
$555$	−31.4283	−1.33406
$556$	0	0
$557$	37.1573	1.57440	0.787202	−	0.616695i	$-0.211529\pi$
$557$	37.1573	1.57440	0.787202	+	0.616695i	$0.211529\pi$
$558$	0	0
$559$	12.9712	0.548621
$560$	0	0
$561$	5.89184	0.248754
$562$	0	0
$563$	16.2655	0.685509	0.342755	−	0.939425i	$-0.388640\pi$
$563$	16.2655	0.685509	0.342755	+	0.939425i	$0.388640\pi$
$564$	0	0
$565$	−67.6044	−2.84414
$566$	0	0
$567$	49.1828	2.06548
$568$	0	0
$569$	−44.7192	−1.87473	−0.937364	−	0.348352i	$-0.886741\pi$
$569$	−44.7192	−1.87473	−0.937364	+	0.348352i	$0.886741\pi$
$570$	0	0
$571$	−44.8176	−1.87556	−0.937779	−	0.347232i	$-0.887122\pi$
$571$	−44.8176	−1.87556	−0.937779	+	0.347232i	$0.887122\pi$
$572$	0	0
$573$	−17.8646	−0.746305
$574$	0	0
$575$	−25.8677	−1.07876
$576$	0	0
$577$	−10.2620	−0.427214	−0.213607	−	0.976920i	$-0.568521\pi$
$577$	−10.2620	−0.427214	−0.213607	+	0.976920i	$0.568521\pi$
$578$	0	0
$579$	−8.80415	−0.365888
$580$	0	0
$581$	−81.2103	−3.36917
$582$	0	0
$583$	28.7597	1.19111
$584$	0	0
$585$	−42.0231	−1.73744
$586$	0	0
$587$	−2.38416	−0.0984047	−0.0492024	−	0.998789i	$-0.515668\pi$
$587$	−2.38416	−0.0984047	−0.0492024	+	0.998789i	$0.515668\pi$
$588$	0	0
$589$	44.6929	1.84154
$590$	0	0
$591$	5.27904	0.217151
$592$	0	0
$593$	6.48951	0.266492	0.133246	−	0.991083i	$-0.457460\pi$
$593$	6.48951	0.266492	0.133246	+	0.991083i	$0.457460\pi$
$594$	0	0
$595$	−18.9130	−0.775356
$596$	0	0
$597$	54.5091	2.23091
$598$	0	0
$599$	−6.00519	−0.245365	−0.122683	−	0.992446i	$-0.539150\pi$
$599$	−6.00519	−0.245365	−0.122683	+	0.992446i	$0.539150\pi$
$600$	0	0
$601$	3.83208	0.156314	0.0781569	−	0.996941i	$-0.475096\pi$
$601$	3.83208	0.156314	0.0781569	+	0.996941i	$0.475096\pi$
$602$	0	0
$603$	−19.9484	−0.812363
$604$	0	0
$605$	−18.4924	−0.751825
$606$	0	0
$607$	40.0159	1.62419	0.812097	−	0.583522i	$-0.198326\pi$
$607$	40.0159	1.62419	0.812097	+	0.583522i	$0.198326\pi$
$608$	0	0
$609$	71.1335	2.88247
$610$	0	0
$611$	57.4127	2.32267
$612$	0	0
$613$	−16.9947	−0.686410	−0.343205	−	0.939261i	$-0.611513\pi$
$613$	−16.9947	−0.686410	−0.343205	+	0.939261i	$0.611513\pi$
$614$	0	0
$615$	−62.4746	−2.51922
$616$	0	0
$617$	19.0244	0.765892	0.382946	−	0.923771i	$-0.374909\pi$
$617$	19.0244	0.765892	0.382946	+	0.923771i	$0.374909\pi$
$618$	0	0
$619$	−25.3873	−1.02040	−0.510201	−	0.860055i	$-0.670429\pi$
$619$	−25.3873	−1.02040	−0.510201	+	0.860055i	$0.670429\pi$
$620$	0	0
$621$	−2.82222	−0.113252
$622$	0	0
$623$	58.0024	2.32382
$624$	0	0
$625$	31.9004	1.27602
$626$	0	0
$627$	44.0284	1.75833
$628$	0	0
$629$	3.40521	0.135775
$630$	0	0
$631$	−17.5803	−0.699861	−0.349931	−	0.936776i	$-0.613795\pi$
$631$	−17.5803	−0.699861	−0.349931	+	0.936776i	$0.613795\pi$
$632$	0	0
$633$	61.4819	2.44369
$634$	0	0
$635$	2.07909	0.0825061
$636$	0	0
$637$	−68.6995	−2.72197
$638$	0	0
$639$	7.09658	0.280737
$640$	0	0
$641$	8.51880	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$641$	8.51880	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$642$	0	0
$643$	0.432933	0.0170732	0.00853660	−	0.999964i	$-0.497283\pi$
$643$	0.432933	0.0170732	0.00853660	+	0.999964i	$0.497283\pi$
$644$	0	0
$645$	28.1555	1.10862
$646$	0	0
$647$	−28.1218	−1.10558	−0.552790	−	0.833321i	$-0.686437\pi$
$647$	−28.1218	−1.10558	−0.552790	+	0.833321i	$0.686437\pi$
$648$	0	0
$649$	2.50895	0.0984848
$650$	0	0
$651$	67.5860	2.64890
$652$	0	0
$653$	−23.1192	−0.904724	−0.452362	−	0.891834i	$-0.649419\pi$
$653$	−23.1192	−0.904724	−0.452362	+	0.891834i	$0.649419\pi$
$654$	0	0
$655$	2.08325	0.0813994
$656$	0	0
$657$	−14.6464	−0.571410
$658$	0	0
$659$	−28.8708	−1.12465	−0.562323	−	0.826917i	$-0.690092\pi$
$659$	−28.8708	−1.12465	−0.562323	+	0.826917i	$0.690092\pi$
$660$	0	0
$661$	−14.5532	−0.566053	−0.283026	−	0.959112i	$-0.591338\pi$
$661$	−14.5532	−0.566053	−0.283026	+	0.959112i	$0.591338\pi$
$662$	0	0
$663$	9.98507	0.387788
$664$	0	0
$665$	−141.332	−5.48064
$666$	0	0
$667$	−15.5866	−0.603515
$668$	0	0
$669$	5.38394	0.208155
$670$	0	0
$671$	11.9029	0.459507
$672$	0	0
$673$	−23.9089	−0.921621	−0.460810	−	0.887499i	$-0.652441\pi$
$673$	−23.9089	−0.921621	−0.460810	+	0.887499i	$0.652441\pi$
$674$	0	0
$675$	11.9068	0.458292
$676$	0	0
$677$	34.9081	1.34163	0.670813	−	0.741627i	$-0.265945\pi$
$677$	34.9081	1.34163	0.670813	+	0.741627i	$0.265945\pi$
$678$	0	0
$679$	−45.5349	−1.74747
$680$	0	0
$681$	38.6551	1.48127
$682$	0	0
$683$	−15.7398	−0.602268	−0.301134	−	0.953582i	$-0.597365\pi$
$683$	−15.7398	−0.602268	−0.301134	+	0.953582i	$0.597365\pi$
$684$	0	0
$685$	32.7250	1.25036
$686$	0	0
$687$	−47.9794	−1.83053
$688$	0	0
$689$	48.7399	1.85684
$690$	0	0
$691$	21.6726	0.824465	0.412233	−	0.911079i	$-0.364749\pi$
$691$	21.6726	0.824465	0.412233	+	0.911079i	$0.364749\pi$
$692$	0	0
$693$	30.3606	1.15330
$694$	0	0
$695$	−65.3863	−2.48024
$696$	0	0
$697$	6.76903	0.256395
$698$	0	0
$699$	54.4524	2.05958
$700$	0	0
$701$	−31.4286	−1.18704	−0.593521	−	0.804819i	$-0.702263\pi$
$701$	−31.4286	−1.18704	−0.593521	+	0.804819i	$0.702263\pi$
$702$	0	0
$703$	25.4464	0.959729
$704$	0	0
$705$	124.621	4.69352
$706$	0	0
$707$	−22.2498	−0.836791
$708$	0	0
$709$	10.4246	0.391503	0.195751	−	0.980654i	$-0.437285\pi$
$709$	10.4246	0.391503	0.195751	+	0.980654i	$0.437285\pi$
$710$	0	0
$711$	−20.4078	−0.765352
$712$	0	0
$713$	−14.8093	−0.554611
$714$	0	0
$715$	41.9278	1.56801
$716$	0	0
$717$	−38.5771	−1.44069
$718$	0	0
$719$	17.3111	0.645597	0.322798	−	0.946468i	$-0.395376\pi$
$719$	17.3111	0.645597	0.322798	+	0.946468i	$0.395376\pi$
$720$	0	0
$721$	−37.5323	−1.39778
$722$	0	0
$723$	29.0009	1.07856
$724$	0	0
$725$	65.7588	2.44222
$726$	0	0
$727$	−10.1194	−0.375308	−0.187654	−	0.982235i	$-0.560088\pi$
$727$	−10.1194	−0.375308	−0.187654	+	0.982235i	$0.560088\pi$
$728$	0	0
$729$	−17.6713	−0.654494
$730$	0	0
$731$	−3.05061	−0.112831
$732$	0	0
$733$	17.8784	0.660355	0.330177	−	0.943919i	$-0.392891\pi$
$733$	17.8784	0.660355	0.330177	+	0.943919i	$0.392891\pi$
$734$	0	0
$735$	−149.121	−5.50041
$736$	0	0
$737$	19.9032	0.733144
$738$	0	0
$739$	29.6156	1.08943	0.544713	−	0.838622i	$-0.316639\pi$
$739$	29.6156	1.08943	0.544713	+	0.838622i	$0.316639\pi$
$740$	0	0
$741$	74.6163	2.74110
$742$	0	0
$743$	31.2480	1.14638	0.573189	−	0.819424i	$-0.305706\pi$
$743$	31.2480	1.14638	0.573189	+	0.819424i	$0.305706\pi$
$744$	0	0
$745$	−37.4730	−1.37291
$746$	0	0
$747$	42.4373	1.55270
$748$	0	0
$749$	76.4557	2.79363
$750$	0	0
$751$	5.43113	0.198185	0.0990923	−	0.995078i	$-0.468406\pi$
$751$	5.43113	0.198185	0.0990923	+	0.995078i	$0.468406\pi$
$752$	0	0
$753$	25.8935	0.943612
$754$	0	0
$755$	50.7613	1.84739
$756$	0	0
$757$	−35.0991	−1.27570	−0.637849	−	0.770161i	$-0.720176\pi$
$757$	−35.0991	−1.27570	−0.637849	+	0.770161i	$0.720176\pi$
$758$	0	0
$759$	−14.5891	−0.529550
$760$	0	0
$761$	−14.3400	−0.519826	−0.259913	−	0.965632i	$-0.583694\pi$
$761$	−14.3400	−0.519826	−0.259913	+	0.965632i	$0.583694\pi$
$762$	0	0
$763$	81.4097	2.94723
$764$	0	0
$765$	9.88316	0.357326
$766$	0	0
$767$	4.25199	0.153530
$768$	0	0
$769$	−24.6823	−0.890066	−0.445033	−	0.895514i	$-0.646808\pi$
$769$	−24.6823	−0.890066	−0.445033	+	0.895514i	$0.646808\pi$
$770$	0	0
$771$	59.2172	2.13266
$772$	0	0
$773$	−13.6175	−0.489786	−0.244893	−	0.969550i	$-0.578753\pi$
$773$	−13.6175	−0.489786	−0.244893	+	0.969550i	$0.578753\pi$
$774$	0	0
$775$	62.4793	2.24432
$776$	0	0
$777$	38.4808	1.38049
$778$	0	0
$779$	50.5835	1.81234
$780$	0	0
$781$	−7.08049	−0.253360
$782$	0	0
$783$	7.17443	0.256393
$784$	0	0
$785$	−46.7990	−1.67033
$786$	0	0
$787$	13.5829	0.484179	0.242089	−	0.970254i	$-0.422167\pi$
$787$	13.5829	0.484179	0.242089	+	0.970254i	$0.422167\pi$
$788$	0	0
$789$	−65.9583	−2.34818
$790$	0	0
$791$	82.7748	2.94313
$792$	0	0
$793$	20.1722	0.716337
$794$	0	0
$795$	105.796	3.75220
$796$	0	0
$797$	−7.10010	−0.251499	−0.125749	−	0.992062i	$-0.540133\pi$
$797$	−7.10010	−0.251499	−0.125749	+	0.992062i	$0.540133\pi$
$798$	0	0
$799$	−13.5025	−0.477686
$800$	0	0
$801$	−30.3097	−1.07094
$802$	0	0
$803$	14.6132	0.515688
$804$	0	0
$805$	46.8314	1.65059
$806$	0	0
$807$	6.98661	0.245940
$808$	0	0
$809$	−0.759267	−0.0266944	−0.0133472	−	0.999911i	$-0.504249\pi$
$809$	−0.759267	−0.0266944	−0.0133472	+	0.999911i	$0.504249\pi$
$810$	0	0
$811$	29.3945	1.03218	0.516090	−	0.856535i	$-0.327387\pi$
$811$	29.3945	1.03218	0.516090	+	0.856535i	$0.327387\pi$
$812$	0	0
$813$	−24.3143	−0.852739
$814$	0	0
$815$	−26.4693	−0.927179
$816$	0	0
$817$	−22.7965	−0.797550
$818$	0	0
$819$	51.4530	1.79792
$820$	0	0
$821$	20.7579	0.724454	0.362227	−	0.932090i	$-0.382016\pi$
$821$	20.7579	0.724454	0.362227	+	0.932090i	$0.382016\pi$
$822$	0	0
$823$	−50.0934	−1.74615	−0.873073	−	0.487590i	$-0.837876\pi$
$823$	−50.0934	−1.74615	−0.873073	+	0.487590i	$0.837876\pi$
$824$	0	0
$825$	61.5504	2.14291
$826$	0	0
$827$	16.0191	0.557039	0.278519	−	0.960431i	$-0.410156\pi$
$827$	16.0191	0.557039	0.278519	+	0.960431i	$0.410156\pi$
$828$	0	0
$829$	−4.17713	−0.145078	−0.0725388	−	0.997366i	$-0.523110\pi$
$829$	−4.17713	−0.145078	−0.0725388	+	0.997366i	$0.523110\pi$
$830$	0	0
$831$	−14.8270	−0.514344
$832$	0	0
$833$	16.1570	0.559808
$834$	0	0
$835$	−1.49106	−0.0516004
$836$	0	0
$837$	6.81664	0.235617
$838$	0	0
$839$	−8.40615	−0.290213	−0.145106	−	0.989416i	$-0.546352\pi$
$839$	−8.40615	−0.290213	−0.145106	+	0.989416i	$0.546352\pi$
$840$	0	0
$841$	10.6230	0.366310
$842$	0	0
$843$	16.2957	0.561254
$844$	0	0
$845$	19.9633	0.686757
$846$	0	0
$847$	22.6421	0.777993
$848$	0	0
$849$	29.4548	1.01089
$850$	0	0
$851$	−8.43182	−0.289039
$852$	0	0
$853$	50.3085	1.72253	0.861265	−	0.508156i	$-0.169673\pi$
$853$	50.3085	1.72253	0.861265	+	0.508156i	$0.169673\pi$
$854$	0	0
$855$	73.8547	2.52578
$856$	0	0
$857$	−29.7650	−1.01675	−0.508376	−	0.861135i	$-0.669754\pi$
$857$	−29.7650	−1.01675	−0.508376	+	0.861135i	$0.669754\pi$
$858$	0	0
$859$	−21.7620	−0.742511	−0.371256	−	0.928531i	$-0.621073\pi$
$859$	−21.7620	−0.742511	−0.371256	+	0.928531i	$0.621073\pi$
$860$	0	0
$861$	76.4939	2.60690
$862$	0	0
$863$	−45.0925	−1.53497	−0.767483	−	0.641069i	$-0.778491\pi$
$863$	−45.0925	−1.53497	−0.767483	+	0.641069i	$0.778491\pi$
$864$	0	0
$865$	53.9827	1.83547
$866$	0	0
$867$	−2.34833	−0.0797534
$868$	0	0
$869$	20.3615	0.690717
$870$	0	0
$871$	33.7306	1.14292
$872$	0	0
$873$	23.7947	0.805329
$874$	0	0
$875$	−103.014	−3.48250
$876$	0	0
$877$	26.8913	0.908053	0.454027	−	0.890988i	$-0.349987\pi$
$877$	26.8913	0.908053	0.454027	+	0.890988i	$0.349987\pi$
$878$	0	0
$879$	16.4566	0.555068
$880$	0	0
$881$	35.4559	1.19454	0.597270	−	0.802040i	$-0.296252\pi$
$881$	35.4559	1.19454	0.597270	+	0.802040i	$0.296252\pi$
$882$	0	0
$883$	23.7649	0.799753	0.399877	−	0.916569i	$-0.369053\pi$
$883$	23.7649	0.799753	0.399877	+	0.916569i	$0.369053\pi$
$884$	0	0
$885$	9.22948	0.310246
$886$	0	0
$887$	22.1456	0.743577	0.371788	−	0.928318i	$-0.378745\pi$
$887$	22.1456	0.743577	0.371788	+	0.928318i	$0.378745\pi$
$888$	0	0
$889$	−2.54564	−0.0853779
$890$	0	0
$891$	25.6427	0.859062
$892$	0	0
$893$	−100.902	−3.37654
$894$	0	0
$895$	6.90796	0.230907
$896$	0	0
$897$	−24.7246	−0.825529
$898$	0	0
$899$	37.6469	1.25560
$900$	0	0
$901$	−11.4629	−0.381883
$902$	0	0
$903$	−34.4736	−1.14721
$904$	0	0
$905$	−1.50582	−0.0500551
$906$	0	0
$907$	−36.7485	−1.22022	−0.610108	−	0.792319i	$-0.708874\pi$
$907$	−36.7485	−1.22022	−0.610108	+	0.792319i	$0.708874\pi$
$908$	0	0
$909$	11.6269	0.385639
$910$	0	0
$911$	−1.45632	−0.0482501	−0.0241250	−	0.999709i	$-0.507680\pi$
$911$	−1.45632	−0.0482501	−0.0241250	+	0.999709i	$0.507680\pi$
$912$	0	0
$913$	−42.3410	−1.40128
$914$	0	0
$915$	43.7864	1.44753
$916$	0	0
$917$	−2.55073	−0.0842327
$918$	0	0
$919$	42.0590	1.38740	0.693699	−	0.720265i	$-0.255980\pi$
$919$	42.0590	1.38740	0.693699	+	0.720265i	$0.255980\pi$
$920$	0	0
$921$	−54.3464	−1.79077
$922$	0	0
$923$	−11.9995	−0.394969
$924$	0	0
$925$	35.5733	1.16964
$926$	0	0
$927$	19.6129	0.644171
$928$	0	0
$929$	18.4889	0.606602	0.303301	−	0.952895i	$-0.401911\pi$
$929$	18.4889	0.606602	0.303301	+	0.952895i	$0.401911\pi$
$930$	0	0
$931$	120.738	3.95703
$932$	0	0
$933$	−45.2632	−1.48185
$934$	0	0
$935$	−9.86075	−0.322481
$936$	0	0
$937$	29.2612	0.955921	0.477960	−	0.878381i	$-0.341376\pi$
$937$	29.2612	0.955921	0.477960	+	0.878381i	$0.341376\pi$
$938$	0	0
$939$	−33.7243	−1.10055
$940$	0	0
$941$	−29.8135	−0.971893	−0.485947	−	0.873988i	$-0.661525\pi$
$941$	−29.8135	−0.971893	−0.485947	+	0.873988i	$0.661525\pi$
$942$	0	0
$943$	−16.7611	−0.545818
$944$	0	0
$945$	−21.5563	−0.701225
$946$	0	0
$947$	−28.2876	−0.919223	−0.459611	−	0.888120i	$-0.652011\pi$
$947$	−28.2876	−0.919223	−0.459611	+	0.888120i	$0.652011\pi$
$948$	0	0
$949$	24.7654	0.803918
$950$	0	0
$951$	−67.6024	−2.19216
$952$	0	0
$953$	33.4156	1.08244	0.541218	−	0.840882i	$-0.317963\pi$
$953$	33.4156	1.08244	0.541218	+	0.840882i	$0.317963\pi$
$954$	0	0
$955$	29.8987	0.967500
$956$	0	0
$957$	37.0872	1.19886
$958$	0	0
$959$	−40.0685	−1.29388
$960$	0	0
$961$	4.76945	0.153853
$962$	0	0
$963$	−39.9527	−1.28746
$964$	0	0
$965$	14.7349	0.474333
$966$	0	0
$967$	32.9681	1.06018	0.530092	−	0.847940i	$-0.322158\pi$
$967$	32.9681	1.06018	0.530092	+	0.847940i	$0.322158\pi$
$968$	0	0
$969$	−17.5486	−0.563741
$970$	0	0
$971$	−15.0104	−0.481707	−0.240853	−	0.970561i	$-0.577427\pi$
$971$	−15.0104	−0.481707	−0.240853	+	0.970561i	$0.577427\pi$
$972$	0	0
$973$	80.0589	2.56657
$974$	0	0
$975$	104.311	3.34063
$976$	0	0
$977$	4.05009	0.129574	0.0647869	−	0.997899i	$-0.479363\pi$
$977$	4.05009	0.129574	0.0647869	+	0.997899i	$0.479363\pi$
$978$	0	0
$979$	30.2410	0.966507
$980$	0	0
$981$	−42.5415	−1.35824
$982$	0	0
$983$	−7.10830	−0.226719	−0.113360	−	0.993554i	$-0.536161\pi$
$983$	−7.10830	−0.226719	−0.113360	+	0.993554i	$0.536161\pi$
$984$	0	0
$985$	−8.83516	−0.281512
$986$	0	0
$987$	−152.586	−4.85688
$988$	0	0
$989$	7.55377	0.240196
$990$	0	0
$991$	20.6035	0.654492	0.327246	−	0.944939i	$-0.393879\pi$
$991$	20.6035	0.654492	0.327246	+	0.944939i	$0.393879\pi$
$992$	0	0
$993$	53.9357	1.71160
$994$	0	0
$995$	−91.2280	−2.89212
$996$	0	0
$997$	13.8242	0.437816	0.218908	−	0.975746i	$-0.429751\pi$
$997$	13.8242	0.437816	0.218908	+	0.975746i	$0.429751\pi$
$998$	0	0
$999$	3.88113	0.122793

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.3	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34833 q^{3} +3.93023 q^{5} -4.81217 q^{7} +2.51465 q^{9} +O(q^{10})\)	\(q-2.34833 q^{3} +3.93023 q^{5} -4.81217 q^{7} +2.51465 q^{9} -2.50895 q^{11} -4.25199 q^{13} -9.22948 q^{15} +1.00000 q^{17} +7.47278 q^{19} +11.3006 q^{21} -2.47615 q^{23} +10.4467 q^{25} +1.13976 q^{27} +6.29468 q^{29} +5.98076 q^{31} +5.89184 q^{33} -18.9130 q^{35} +3.40521 q^{37} +9.98507 q^{39} +6.76903 q^{41} -3.05061 q^{43} +9.88316 q^{45} -13.5025 q^{47} +16.1570 q^{49} -2.34833 q^{51} -11.4629 q^{53} -9.86075 q^{55} -17.5486 q^{57} -1.00000 q^{59} -4.74419 q^{61} -12.1009 q^{63} -16.7113 q^{65} -7.93289 q^{67} +5.81482 q^{69} +2.82210 q^{71} -5.82442 q^{73} -24.5324 q^{75} +12.0735 q^{77} -8.11556 q^{79} -10.2205 q^{81} +16.8760 q^{83} +3.93023 q^{85} -14.7820 q^{87} -12.0533 q^{89} +20.4613 q^{91} -14.0448 q^{93} +29.3698 q^{95} +9.46244 q^{97} -6.30913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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