LMFDB - Embedded newform 4016.2.a.k.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-0.932399$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04458 q^{3} -3.52221 q^{5} +4.32039 q^{7} +6.26949 q^{9} +O(q^{10})$	$q+3.04458 q^{3} -3.52221 q^{5} +4.32039 q^{7} +6.26949 q^{9} -0.664032 q^{11} +1.58654 q^{13} -10.7237 q^{15} -7.29240 q^{17} +3.05837 q^{19} +13.1538 q^{21} +4.67674 q^{23} +7.40597 q^{25} +9.95423 q^{27} +4.75304 q^{29} +1.10951 q^{31} -2.02170 q^{33} -15.2173 q^{35} +10.9031 q^{37} +4.83037 q^{39} -8.90418 q^{41} -0.765356 q^{43} -22.0825 q^{45} +0.788675 q^{47} +11.6657 q^{49} -22.2023 q^{51} -5.96054 q^{53} +2.33886 q^{55} +9.31147 q^{57} +0.649805 q^{59} +3.81583 q^{61} +27.0866 q^{63} -5.58814 q^{65} +4.61492 q^{67} +14.2387 q^{69} +5.78274 q^{71} +10.7341 q^{73} +22.5481 q^{75} -2.86888 q^{77} +1.26490 q^{79} +11.4980 q^{81} -13.6710 q^{83} +25.6854 q^{85} +14.4710 q^{87} +4.47002 q^{89} +6.85448 q^{91} +3.37799 q^{93} -10.7722 q^{95} -9.75452 q^{97} -4.16314 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	3.04458	1.75779	0.878896	−	0.477014i	$-0.158281\pi$
$3$	3.04458	1.75779	0.878896	+	0.477014i	$0.158281\pi$
$4$	0	0
$5$	−3.52221	−1.57518	−0.787590	−	0.616199i	$-0.788672\pi$
$5$	−3.52221	−1.57518	−0.787590	+	0.616199i	$0.788672\pi$
$6$	0	0
$7$	4.32039	1.63295	0.816476	−	0.577379i	$-0.195924\pi$
$7$	4.32039	1.63295	0.816476	+	0.577379i	$0.195924\pi$
$8$	0	0
$9$	6.26949	2.08983
$10$	0	0
$11$	−0.664032	−0.200213	−0.100107	−	0.994977i	$-0.531918\pi$
$11$	−0.664032	−0.200213	−0.100107	+	0.994977i	$0.531918\pi$
$12$	0	0
$13$	1.58654	0.440028	0.220014	−	0.975497i	$-0.429390\pi$
$13$	1.58654	0.440028	0.220014	+	0.975497i	$0.429390\pi$
$14$	0	0
$15$	−10.7237	−2.76884
$16$	0	0
$17$	−7.29240	−1.76867	−0.884333	−	0.466856i	$-0.845387\pi$
$17$	−7.29240	−1.76867	−0.884333	+	0.466856i	$0.845387\pi$
$18$	0	0
$19$	3.05837	0.701639	0.350819	−	0.936443i	$-0.385903\pi$
$19$	3.05837	0.701639	0.350819	+	0.936443i	$0.385903\pi$
$20$	0	0
$21$	13.1538	2.87039
$22$	0	0
$23$	4.67674	0.975169	0.487584	−	0.873076i	$-0.337878\pi$
$23$	4.67674	0.975169	0.487584	+	0.873076i	$0.337878\pi$
$24$	0	0
$25$	7.40597	1.48119
$26$	0	0
$27$	9.95423	1.91569
$28$	0	0
$29$	4.75304	0.882617	0.441308	−	0.897356i	$-0.354515\pi$
$29$	4.75304	0.882617	0.441308	+	0.897356i	$0.354515\pi$
$30$	0	0
$31$	1.10951	0.199274	0.0996369	−	0.995024i	$-0.468232\pi$
$31$	1.10951	0.199274	0.0996369	+	0.995024i	$0.468232\pi$
$32$	0	0
$33$	−2.02170	−0.351933
$34$	0	0
$35$	−15.2173	−2.57220
$36$	0	0
$37$	10.9031	1.79247	0.896233	−	0.443584i	$-0.146293\pi$
$37$	10.9031	1.79247	0.896233	+	0.443584i	$0.146293\pi$
$38$	0	0
$39$	4.83037	0.773477
$40$	0	0
$41$	−8.90418	−1.39060	−0.695300	−	0.718720i	$-0.744728\pi$
$41$	−8.90418	−1.39060	−0.695300	+	0.718720i	$0.744728\pi$
$42$	0	0
$43$	−0.765356	−0.116716	−0.0583578	−	0.998296i	$-0.518586\pi$
$43$	−0.765356	−0.116716	−0.0583578	+	0.998296i	$0.518586\pi$
$44$	0	0
$45$	−22.0825	−3.29186
$46$	0	0
$47$	0.788675	0.115040	0.0575200	−	0.998344i	$-0.481681\pi$
$47$	0.788675	0.115040	0.0575200	+	0.998344i	$0.481681\pi$
$48$	0	0
$49$	11.6657	1.66653
$50$	0	0
$51$	−22.2023	−3.10895
$52$	0	0
$53$	−5.96054	−0.818743	−0.409372	−	0.912368i	$-0.634252\pi$
$53$	−5.96054	−0.818743	−0.409372	+	0.912368i	$0.634252\pi$
$54$	0	0
$55$	2.33886	0.315372
$56$	0	0
$57$	9.31147	1.23333
$58$	0	0
$59$	0.649805	0.0845974	0.0422987	−	0.999105i	$-0.486532\pi$
$59$	0.649805	0.0845974	0.0422987	+	0.999105i	$0.486532\pi$
$60$	0	0
$61$	3.81583	0.488567	0.244284	−	0.969704i	$-0.421447\pi$
$61$	3.81583	0.488567	0.244284	+	0.969704i	$0.421447\pi$
$62$	0	0
$63$	27.0866	3.41259
$64$	0	0
$65$	−5.58814	−0.693124
$66$	0	0
$67$	4.61492	0.563802	0.281901	−	0.959444i	$-0.409035\pi$
$67$	4.61492	0.563802	0.281901	+	0.959444i	$0.409035\pi$
$68$	0	0
$69$	14.2387	1.71414
$70$	0	0
$71$	5.78274	0.686285	0.343142	−	0.939283i	$-0.388509\pi$
$71$	5.78274	0.686285	0.343142	+	0.939283i	$0.388509\pi$
$72$	0	0
$73$	10.7341	1.25633	0.628163	−	0.778082i	$-0.283807\pi$
$73$	10.7341	1.25633	0.628163	+	0.778082i	$0.283807\pi$
$74$	0	0
$75$	22.5481	2.60363
$76$	0	0
$77$	−2.86888	−0.326939
$78$	0	0
$79$	1.26490	0.142313	0.0711563	−	0.997465i	$-0.477331\pi$
$79$	1.26490	0.142313	0.0711563	+	0.997465i	$0.477331\pi$
$80$	0	0
$81$	11.4980	1.27756
$82$	0	0
$83$	−13.6710	−1.50059	−0.750296	−	0.661103i	$-0.770089\pi$
$83$	−13.6710	−1.50059	−0.750296	+	0.661103i	$0.770089\pi$
$84$	0	0
$85$	25.6854	2.78597
$86$	0	0
$87$	14.4710	1.55146
$88$	0	0
$89$	4.47002	0.473821	0.236910	−	0.971531i	$-0.423865\pi$
$89$	4.47002	0.473821	0.236910	+	0.971531i	$0.423865\pi$
$90$	0	0
$91$	6.85448	0.718545
$92$	0	0
$93$	3.37799	0.350282
$94$	0	0
$95$	−10.7722	−1.10521
$96$	0	0
$97$	−9.75452	−0.990422	−0.495211	−	0.868773i	$-0.664909\pi$
$97$	−9.75452	−0.990422	−0.495211	+	0.868773i	$0.664909\pi$
$98$	0	0
$99$	−4.16314	−0.418411
$100$	0	0
$101$	10.0831	1.00330	0.501652	−	0.865070i	$-0.332726\pi$
$101$	10.0831	1.00330	0.501652	+	0.865070i	$0.332726\pi$
$102$	0	0
$103$	−2.34676	−0.231233	−0.115616	−	0.993294i	$-0.536884\pi$
$103$	−2.34676	−0.231233	−0.115616	+	0.993294i	$0.536884\pi$
$104$	0	0
$105$	−46.3304	−4.52138
$106$	0	0
$107$	19.7069	1.90514	0.952568	−	0.304326i	$-0.0984314\pi$
$107$	19.7069	1.90514	0.952568	+	0.304326i	$0.0984314\pi$
$108$	0	0
$109$	0.710465	0.0680502	0.0340251	−	0.999421i	$-0.489167\pi$
$109$	0.710465	0.0680502	0.0340251	+	0.999421i	$0.489167\pi$
$110$	0	0
$111$	33.1955	3.15078
$112$	0	0
$113$	0.267769	0.0251896	0.0125948	−	0.999921i	$-0.495991\pi$
$113$	0.267769	0.0251896	0.0125948	+	0.999921i	$0.495991\pi$
$114$	0	0
$115$	−16.4725	−1.53607
$116$	0	0
$117$	9.94682	0.919584
$118$	0	0
$119$	−31.5060	−2.88815
$120$	0	0
$121$	−10.5591	−0.959915
$122$	0	0
$123$	−27.1095	−2.44438
$124$	0	0
$125$	−8.47433	−0.757967
$126$	0	0
$127$	−9.24584	−0.820436	−0.410218	−	0.911988i	$-0.634547\pi$
$127$	−9.24584	−0.820436	−0.410218	+	0.911988i	$0.634547\pi$
$128$	0	0
$129$	−2.33019	−0.205162
$130$	0	0
$131$	15.0591	1.31572	0.657860	−	0.753140i	$-0.271462\pi$
$131$	15.0591	1.31572	0.657860	+	0.753140i	$0.271462\pi$
$132$	0	0
$133$	13.2134	1.14574
$134$	0	0
$135$	−35.0609	−3.01756
$136$	0	0
$137$	−2.30209	−0.196681	−0.0983406	−	0.995153i	$-0.531353\pi$
$137$	−2.30209	−0.196681	−0.0983406	+	0.995153i	$0.531353\pi$
$138$	0	0
$139$	−19.9900	−1.69553	−0.847765	−	0.530372i	$-0.822052\pi$
$139$	−19.9900	−1.69553	−0.847765	+	0.530372i	$0.822052\pi$
$140$	0	0
$141$	2.40119	0.202216
$142$	0	0
$143$	−1.05352	−0.0880994
$144$	0	0
$145$	−16.7412	−1.39028
$146$	0	0
$147$	35.5173	2.92942
$148$	0	0
$149$	12.0917	0.990593	0.495297	−	0.868724i	$-0.335059\pi$
$149$	12.0917	0.990593	0.495297	+	0.868724i	$0.335059\pi$
$150$	0	0
$151$	2.27473	0.185115	0.0925576	−	0.995707i	$-0.470496\pi$
$151$	2.27473	0.185115	0.0925576	+	0.995707i	$0.470496\pi$
$152$	0	0
$153$	−45.7196	−3.69621
$154$	0	0
$155$	−3.90792	−0.313892
$156$	0	0
$157$	−3.22306	−0.257228	−0.128614	−	0.991695i	$-0.541053\pi$
$157$	−3.22306	−0.257228	−0.128614	+	0.991695i	$0.541053\pi$
$158$	0	0
$159$	−18.1474	−1.43918
$160$	0	0
$161$	20.2053	1.59240
$162$	0	0
$163$	5.01391	0.392720	0.196360	−	0.980532i	$-0.437088\pi$
$163$	5.01391	0.392720	0.196360	+	0.980532i	$0.437088\pi$
$164$	0	0
$165$	7.12086	0.554358
$166$	0	0
$167$	−2.08792	−0.161568	−0.0807842	−	0.996732i	$-0.525742\pi$
$167$	−2.08792	−0.161568	−0.0807842	+	0.996732i	$0.525742\pi$
$168$	0	0
$169$	−10.4829	−0.806375
$170$	0	0
$171$	19.1744	1.46631
$172$	0	0
$173$	10.7830	0.819814	0.409907	−	0.912127i	$-0.365561\pi$
$173$	10.7830	0.819814	0.409907	+	0.912127i	$0.365561\pi$
$174$	0	0
$175$	31.9966	2.41872
$176$	0	0
$177$	1.97839	0.148705
$178$	0	0
$179$	−7.89735	−0.590276	−0.295138	−	0.955455i	$-0.595366\pi$
$179$	−7.89735	−0.590276	−0.295138	+	0.955455i	$0.595366\pi$
$180$	0	0
$181$	11.6313	0.864545	0.432272	−	0.901743i	$-0.357712\pi$
$181$	11.6313	0.864545	0.432272	+	0.901743i	$0.357712\pi$
$182$	0	0
$183$	11.6176	0.858799
$184$	0	0
$185$	−38.4032	−2.82346
$186$	0	0
$187$	4.84239	0.354110
$188$	0	0
$189$	43.0061	3.12824
$190$	0	0
$191$	−11.6811	−0.845218	−0.422609	−	0.906312i	$-0.638886\pi$
$191$	−11.6811	−0.845218	−0.422609	+	0.906312i	$0.638886\pi$
$192$	0	0
$193$	−3.66189	−0.263588	−0.131794	−	0.991277i	$-0.542074\pi$
$193$	−3.66189	−0.263588	−0.131794	+	0.991277i	$0.542074\pi$
$194$	0	0
$195$	−17.0136	−1.21837
$196$	0	0
$197$	−14.6018	−1.04034	−0.520168	−	0.854064i	$-0.674131\pi$
$197$	−14.6018	−1.04034	−0.520168	+	0.854064i	$0.674131\pi$
$198$	0	0
$199$	1.98657	0.140824	0.0704120	−	0.997518i	$-0.477569\pi$
$199$	1.98657	0.140824	0.0704120	+	0.997518i	$0.477569\pi$
$200$	0	0
$201$	14.0505	0.991046
$202$	0	0
$203$	20.5350	1.44127
$204$	0	0
$205$	31.3624	2.19045
$206$	0	0
$207$	29.3208	2.03794
$208$	0	0
$209$	−2.03086	−0.140477
$210$	0	0
$211$	−0.0799503	−0.00550400	−0.00275200	−	0.999996i	$-0.500876\pi$
$211$	−0.0799503	−0.00550400	−0.00275200	+	0.999996i	$0.500876\pi$
$212$	0	0
$213$	17.6060	1.20635
$214$	0	0
$215$	2.69574	0.183848
$216$	0	0
$217$	4.79351	0.325405
$218$	0	0
$219$	32.6807	2.20836
$220$	0	0
$221$	−11.5697	−0.778263
$222$	0	0
$223$	−14.5527	−0.974522	−0.487261	−	0.873256i	$-0.662004\pi$
$223$	−14.5527	−0.974522	−0.487261	+	0.873256i	$0.662004\pi$
$224$	0	0
$225$	46.4316	3.09544
$226$	0	0
$227$	24.4866	1.62523	0.812617	−	0.582798i	$-0.198042\pi$
$227$	24.4866	1.62523	0.812617	+	0.582798i	$0.198042\pi$
$228$	0	0
$229$	−24.8729	−1.64365	−0.821824	−	0.569742i	$-0.807043\pi$
$229$	−24.8729	−1.64365	−0.821824	+	0.569742i	$0.807043\pi$
$230$	0	0
$231$	−8.73453	−0.574690
$232$	0	0
$233$	−5.46960	−0.358325	−0.179163	−	0.983819i	$-0.557339\pi$
$233$	−5.46960	−0.358325	−0.179163	+	0.983819i	$0.557339\pi$
$234$	0	0
$235$	−2.77788	−0.181209
$236$	0	0
$237$	3.85110	0.250156
$238$	0	0
$239$	14.2892	0.924295	0.462147	−	0.886803i	$-0.347079\pi$
$239$	14.2892	0.924295	0.462147	+	0.886803i	$0.347079\pi$
$240$	0	0
$241$	−7.58300	−0.488464	−0.244232	−	0.969717i	$-0.578536\pi$
$241$	−7.58300	−0.488464	−0.244232	+	0.969717i	$0.578536\pi$
$242$	0	0
$243$	5.14400	0.329988
$244$	0	0
$245$	−41.0892	−2.62509
$246$	0	0
$247$	4.85224	0.308741
$248$	0	0
$249$	−41.6226	−2.63773
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−3.10551	−0.195242
$254$	0	0
$255$	78.2012	4.89715
$256$	0	0
$257$	22.7307	1.41790	0.708952	−	0.705257i	$-0.249168\pi$
$257$	22.7307	1.41790	0.708952	+	0.705257i	$0.249168\pi$
$258$	0	0
$259$	47.1058	2.92701
$260$	0	0
$261$	29.7991	1.84452
$262$	0	0
$263$	−14.5293	−0.895914	−0.447957	−	0.894055i	$-0.647848\pi$
$263$	−14.5293	−0.895914	−0.447957	+	0.894055i	$0.647848\pi$
$264$	0	0
$265$	20.9943	1.28967
$266$	0	0
$267$	13.6093	0.832878
$268$	0	0
$269$	−2.82898	−0.172486	−0.0862428	−	0.996274i	$-0.527486\pi$
$269$	−2.82898	−0.172486	−0.0862428	+	0.996274i	$0.527486\pi$
$270$	0	0
$271$	0.509052	0.0309227	0.0154614	−	0.999880i	$-0.495078\pi$
$271$	0.509052	0.0309227	0.0154614	+	0.999880i	$0.495078\pi$
$272$	0	0
$273$	20.8690	1.26305
$274$	0	0
$275$	−4.91780	−0.296554
$276$	0	0
$277$	27.2207	1.63553	0.817766	−	0.575550i	$-0.195212\pi$
$277$	27.2207	1.63553	0.817766	+	0.575550i	$0.195212\pi$
$278$	0	0
$279$	6.95605	0.416448
$280$	0	0
$281$	−0.00412650	−0.000246166 0	−0.000123083	−	1.00000i	$-0.500039\pi$
$281$	−0.00412650	−0.000246166 0	−0.000123083		1.00000i	$0.500039\pi$
$282$	0	0
$283$	−7.35293	−0.437086	−0.218543	−	0.975827i	$-0.570130\pi$
$283$	−7.35293	−0.437086	−0.218543	+	0.975827i	$0.570130\pi$
$284$	0	0
$285$	−32.7970	−1.94272
$286$	0	0
$287$	−38.4695	−2.27078
$288$	0	0
$289$	36.1791	2.12818
$290$	0	0
$291$	−29.6985	−1.74095
$292$	0	0
$293$	−11.0325	−0.644527	−0.322263	−	0.946650i	$-0.604444\pi$
$293$	−11.0325	−0.644527	−0.322263	+	0.946650i	$0.604444\pi$
$294$	0	0
$295$	−2.28875	−0.133256
$296$	0	0
$297$	−6.60993	−0.383547
$298$	0	0
$299$	7.41986	0.429102
$300$	0	0
$301$	−3.30663	−0.190591
$302$	0	0
$303$	30.6988	1.76360
$304$	0	0
$305$	−13.4402	−0.769582
$306$	0	0
$307$	−25.2446	−1.44078	−0.720391	−	0.693568i	$-0.756038\pi$
$307$	−25.2446	−1.44078	−0.720391	+	0.693568i	$0.756038\pi$
$308$	0	0
$309$	−7.14489	−0.406459
$310$	0	0
$311$	13.0405	0.739458	0.369729	−	0.929140i	$-0.379451\pi$
$311$	13.0405	0.739458	0.369729	+	0.929140i	$0.379451\pi$
$312$	0	0
$313$	−14.6191	−0.826319	−0.413159	−	0.910659i	$-0.635575\pi$
$313$	−14.6191	−0.826319	−0.413159	+	0.910659i	$0.635575\pi$
$314$	0	0
$315$	−95.4048	−5.37545
$316$	0	0
$317$	7.88988	0.443140	0.221570	−	0.975144i	$-0.428882\pi$
$317$	7.88988	0.443140	0.221570	+	0.975144i	$0.428882\pi$
$318$	0	0
$319$	−3.15617	−0.176712
$320$	0	0
$321$	59.9992	3.34883
$322$	0	0
$323$	−22.3029	−1.24096
$324$	0	0
$325$	11.7499	0.651767
$326$	0	0
$327$	2.16307	0.119618
$328$	0	0
$329$	3.40738	0.187855
$330$	0	0
$331$	−8.65287	−0.475605	−0.237802	−	0.971314i	$-0.576427\pi$
$331$	−8.65287	−0.475605	−0.237802	+	0.971314i	$0.576427\pi$
$332$	0	0
$333$	68.3571	3.74595
$334$	0	0
$335$	−16.2547	−0.888090
$336$	0	0
$337$	−6.27236	−0.341677	−0.170839	−	0.985299i	$-0.554648\pi$
$337$	−6.27236	−0.341677	−0.170839	+	0.985299i	$0.554648\pi$
$338$	0	0
$339$	0.815247	0.0442781
$340$	0	0
$341$	−0.736750	−0.0398972
$342$	0	0
$343$	20.1578	1.08842
$344$	0	0
$345$	−50.1518	−2.70008
$346$	0	0
$347$	−11.0589	−0.593675	−0.296837	−	0.954928i	$-0.595932\pi$
$347$	−11.0589	−0.593675	−0.296837	+	0.954928i	$0.595932\pi$
$348$	0	0
$349$	24.2097	1.29591	0.647957	−	0.761677i	$-0.275624\pi$
$349$	24.2097	1.29591	0.647957	+	0.761677i	$0.275624\pi$
$350$	0	0
$351$	15.7928	0.842959
$352$	0	0
$353$	14.2700	0.759518	0.379759	−	0.925086i	$-0.376007\pi$
$353$	14.2700	0.759518	0.379759	+	0.925086i	$0.376007\pi$
$354$	0	0
$355$	−20.3680	−1.08102
$356$	0	0
$357$	−95.9226	−5.07676
$358$	0	0
$359$	−5.89745	−0.311255	−0.155628	−	0.987816i	$-0.549740\pi$
$359$	−5.89745	−0.311255	−0.155628	+	0.987816i	$0.549740\pi$
$360$	0	0
$361$	−9.64636	−0.507703
$362$	0	0
$363$	−32.1479	−1.68733
$364$	0	0
$365$	−37.8076	−1.97894
$366$	0	0
$367$	−32.2132	−1.68151	−0.840756	−	0.541414i	$-0.817889\pi$
$367$	−32.2132	−1.68151	−0.840756	+	0.541414i	$0.817889\pi$
$368$	0	0
$369$	−55.8247	−2.90612
$370$	0	0
$371$	−25.7518	−1.33697
$372$	0	0
$373$	−3.64751	−0.188861	−0.0944304	−	0.995531i	$-0.530103\pi$
$373$	−3.64751	−0.188861	−0.0944304	+	0.995531i	$0.530103\pi$
$374$	0	0
$375$	−25.8008	−1.33235
$376$	0	0
$377$	7.54090	0.388376
$378$	0	0
$379$	17.8922	0.919061	0.459531	−	0.888162i	$-0.348018\pi$
$379$	17.8922	0.919061	0.459531	+	0.888162i	$0.348018\pi$
$380$	0	0
$381$	−28.1497	−1.44215
$382$	0	0
$383$	−27.5314	−1.40679	−0.703395	−	0.710799i	$-0.748333\pi$
$383$	−27.5314	−1.40679	−0.703395	+	0.710799i	$0.748333\pi$
$384$	0	0
$385$	10.1048	0.514987
$386$	0	0
$387$	−4.79839	−0.243916
$388$	0	0
$389$	35.8027	1.81527	0.907635	−	0.419761i	$-0.137886\pi$
$389$	35.8027	1.81527	0.907635	+	0.419761i	$0.137886\pi$
$390$	0	0
$391$	−34.1047	−1.72475
$392$	0	0
$393$	45.8487	2.31276
$394$	0	0
$395$	−4.45525	−0.224168
$396$	0	0
$397$	−35.4931	−1.78135	−0.890673	−	0.454645i	$-0.849766\pi$
$397$	−35.4931	−1.78135	−0.890673	+	0.454645i	$0.849766\pi$
$398$	0	0
$399$	40.2292	2.01398
$400$	0	0
$401$	17.4780	0.872811	0.436405	−	0.899750i	$-0.356251\pi$
$401$	17.4780	0.872811	0.436405	+	0.899750i	$0.356251\pi$
$402$	0	0
$403$	1.76028	0.0876860
$404$	0	0
$405$	−40.4985	−2.01238
$406$	0	0
$407$	−7.24004	−0.358875
$408$	0	0
$409$	−33.9525	−1.67884	−0.839421	−	0.543481i	$-0.817106\pi$
$409$	−33.9525	−1.67884	−0.839421	+	0.543481i	$0.817106\pi$
$410$	0	0
$411$	−7.00892	−0.345724
$412$	0	0
$413$	2.80741	0.138144
$414$	0	0
$415$	48.1523	2.36370
$416$	0	0
$417$	−60.8613	−2.98039
$418$	0	0
$419$	8.35911	0.408369	0.204185	−	0.978932i	$-0.434546\pi$
$419$	8.35911	0.408369	0.204185	+	0.978932i	$0.434546\pi$
$420$	0	0
$421$	−17.5702	−0.856320	−0.428160	−	0.903703i	$-0.640838\pi$
$421$	−17.5702	−0.856320	−0.428160	+	0.903703i	$0.640838\pi$
$422$	0	0
$423$	4.94459	0.240414
$424$	0	0
$425$	−54.0073	−2.61974
$426$	0	0
$427$	16.4859	0.797807
$428$	0	0
$429$	−3.20752	−0.154860
$430$	0	0
$431$	−35.4367	−1.70693	−0.853464	−	0.521152i	$-0.825502\pi$
$431$	−35.4367	−1.70693	−0.853464	+	0.521152i	$0.825502\pi$
$432$	0	0
$433$	−12.9322	−0.621480	−0.310740	−	0.950495i	$-0.600577\pi$
$433$	−12.9322	−0.621480	−0.310740	+	0.950495i	$0.600577\pi$
$434$	0	0
$435$	−50.9700	−2.44382
$436$	0	0
$437$	14.3032	0.684216
$438$	0	0
$439$	−27.6356	−1.31897	−0.659486	−	0.751716i	$-0.729226\pi$
$439$	−27.6356	−1.31897	−0.659486	+	0.751716i	$0.729226\pi$
$440$	0	0
$441$	73.1382	3.48277
$442$	0	0
$443$	−14.4931	−0.688589	−0.344294	−	0.938862i	$-0.611882\pi$
$443$	−14.4931	−0.688589	−0.344294	+	0.938862i	$0.611882\pi$
$444$	0	0
$445$	−15.7443	−0.746353
$446$	0	0
$447$	36.8143	1.74126
$448$	0	0
$449$	13.5151	0.637817	0.318908	−	0.947786i	$-0.396684\pi$
$449$	13.5151	0.637817	0.318908	+	0.947786i	$0.396684\pi$
$450$	0	0
$451$	5.91266	0.278416
$452$	0	0
$453$	6.92561	0.325394
$454$	0	0
$455$	−24.1429	−1.13184
$456$	0	0
$457$	6.92075	0.323739	0.161870	−	0.986812i	$-0.448248\pi$
$457$	6.92075	0.323739	0.161870	+	0.986812i	$0.448248\pi$
$458$	0	0
$459$	−72.5902	−3.38822
$460$	0	0
$461$	−6.81809	−0.317550	−0.158775	−	0.987315i	$-0.550754\pi$
$461$	−6.81809	−0.317550	−0.158775	+	0.987315i	$0.550754\pi$
$462$	0	0
$463$	−3.89445	−0.180991	−0.0904953	−	0.995897i	$-0.528845\pi$
$463$	−3.89445	−0.180991	−0.0904953	+	0.995897i	$0.528845\pi$
$464$	0	0
$465$	−11.8980	−0.551757
$466$	0	0
$467$	−32.4193	−1.50018	−0.750092	−	0.661333i	$-0.769991\pi$
$467$	−32.4193	−1.50018	−0.750092	+	0.661333i	$0.769991\pi$
$468$	0	0
$469$	19.9382	0.920662
$470$	0	0
$471$	−9.81288	−0.452153
$472$	0	0
$473$	0.508221	0.0233680
$474$	0	0
$475$	22.6502	1.03926
$476$	0	0
$477$	−37.3695	−1.71103
$478$	0	0
$479$	36.5728	1.67105	0.835526	−	0.549451i	$-0.185163\pi$
$479$	36.5728	1.67105	0.835526	+	0.549451i	$0.185163\pi$
$480$	0	0
$481$	17.2983	0.788735
$482$	0	0
$483$	61.5169	2.79911
$484$	0	0
$485$	34.3575	1.56009
$486$	0	0
$487$	6.29767	0.285375	0.142687	−	0.989768i	$-0.454426\pi$
$487$	6.29767	0.285375	0.142687	+	0.989768i	$0.454426\pi$
$488$	0	0
$489$	15.2653	0.690319
$490$	0	0
$491$	32.1877	1.45261	0.726306	−	0.687372i	$-0.241236\pi$
$491$	32.1877	1.45261	0.726306	+	0.687372i	$0.241236\pi$
$492$	0	0
$493$	−34.6610	−1.56105
$494$	0	0
$495$	14.6635	0.659074
$496$	0	0
$497$	24.9837	1.12067
$498$	0	0
$499$	30.3176	1.35720	0.678601	−	0.734507i	$-0.262587\pi$
$499$	30.3176	1.35720	0.678601	+	0.734507i	$0.262587\pi$
$500$	0	0
$501$	−6.35686	−0.284003
$502$	0	0
$503$	18.2262	0.812667	0.406334	−	0.913725i	$-0.366807\pi$
$503$	18.2262	0.812667	0.406334	+	0.913725i	$0.366807\pi$
$504$	0	0
$505$	−35.5147	−1.58038
$506$	0	0
$507$	−31.9160	−1.41744
$508$	0	0
$509$	−16.8186	−0.745472	−0.372736	−	0.927937i	$-0.621580\pi$
$509$	−16.8186	−0.745472	−0.372736	+	0.927937i	$0.621580\pi$
$510$	0	0
$511$	46.3753	2.05152
$512$	0	0
$513$	30.4437	1.34412
$514$	0	0
$515$	8.26577	0.364233
$516$	0	0
$517$	−0.523705	−0.0230325
$518$	0	0
$519$	32.8296	1.44106
$520$	0	0
$521$	5.78518	0.253453	0.126727	−	0.991938i	$-0.459553\pi$
$521$	5.78518	0.253453	0.126727	+	0.991938i	$0.459553\pi$
$522$	0	0
$523$	20.2775	0.886671	0.443336	−	0.896356i	$-0.353795\pi$
$523$	20.2775	0.886671	0.443336	+	0.896356i	$0.353795\pi$
$524$	0	0
$525$	97.4165	4.25160
$526$	0	0
$527$	−8.09098	−0.352449
$528$	0	0
$529$	−1.12806	−0.0490462
$530$	0	0
$531$	4.07395	0.176794
$532$	0	0
$533$	−14.1269	−0.611903
$534$	0	0
$535$	−69.4118	−3.00093
$536$	0	0
$537$	−24.0441	−1.03758
$538$	0	0
$539$	−7.74643	−0.333662
$540$	0	0
$541$	8.05766	0.346426	0.173213	−	0.984884i	$-0.444585\pi$
$541$	8.05766	0.346426	0.173213	+	0.984884i	$0.444585\pi$
$542$	0	0
$543$	35.4123	1.51969
$544$	0	0
$545$	−2.50241	−0.107191
$546$	0	0
$547$	38.5607	1.64874	0.824369	−	0.566053i	$-0.191530\pi$
$547$	38.5607	1.64874	0.824369	+	0.566053i	$0.191530\pi$
$548$	0	0
$549$	23.9233	1.02102
$550$	0	0
$551$	14.5366	0.619278
$552$	0	0
$553$	5.46487	0.232390
$554$	0	0
$555$	−116.922	−4.96305
$556$	0	0
$557$	14.1932	0.601384	0.300692	−	0.953721i	$-0.402782\pi$
$557$	14.1932	0.601384	0.300692	+	0.953721i	$0.402782\pi$
$558$	0	0
$559$	−1.21427	−0.0513582
$560$	0	0
$561$	14.7430	0.622452
$562$	0	0
$563$	−42.8768	−1.80704	−0.903520	−	0.428545i	$-0.859026\pi$
$563$	−42.8768	−1.80704	−0.903520	+	0.428545i	$0.859026\pi$
$564$	0	0
$565$	−0.943141	−0.0396782
$566$	0	0
$567$	49.6759	2.08619
$568$	0	0
$569$	−19.5270	−0.818615	−0.409308	−	0.912396i	$-0.634230\pi$
$569$	−19.5270	−0.818615	−0.409308	+	0.912396i	$0.634230\pi$
$570$	0	0
$571$	−38.9931	−1.63181	−0.815906	−	0.578185i	$-0.803761\pi$
$571$	−38.9931	−1.63181	−0.815906	+	0.578185i	$0.803761\pi$
$572$	0	0
$573$	−35.5642	−1.48572
$574$	0	0
$575$	34.6358	1.44441
$576$	0	0
$577$	4.78742	0.199303	0.0996514	−	0.995022i	$-0.468227\pi$
$577$	4.78742	0.199303	0.0996514	+	0.995022i	$0.468227\pi$
$578$	0	0
$579$	−11.1489	−0.463333
$580$	0	0
$581$	−59.0642	−2.45039
$582$	0	0
$583$	3.95799	0.163923
$584$	0	0
$585$	−35.0348	−1.44851
$586$	0	0
$587$	−31.3281	−1.29305	−0.646525	−	0.762893i	$-0.723778\pi$
$587$	−31.3281	−1.29305	−0.646525	+	0.762893i	$0.723778\pi$
$588$	0	0
$589$	3.39329	0.139818
$590$	0	0
$591$	−44.4564	−1.82869
$592$	0	0
$593$	−23.2303	−0.953953	−0.476976	−	0.878916i	$-0.658267\pi$
$593$	−23.2303	−0.953953	−0.476976	+	0.878916i	$0.658267\pi$
$594$	0	0
$595$	110.971	4.54936
$596$	0	0
$597$	6.04827	0.247539
$598$	0	0
$599$	−12.7134	−0.519456	−0.259728	−	0.965682i	$-0.583633\pi$
$599$	−12.7134	−0.519456	−0.259728	+	0.965682i	$0.583633\pi$
$600$	0	0
$601$	34.4089	1.40357	0.701783	−	0.712391i	$-0.252387\pi$
$601$	34.4089	1.40357	0.701783	+	0.712391i	$0.252387\pi$
$602$	0	0
$603$	28.9332	1.17825
$604$	0	0
$605$	37.1912	1.51204
$606$	0	0
$607$	−16.5273	−0.670821	−0.335411	−	0.942072i	$-0.608875\pi$
$607$	−16.5273	−0.670821	−0.335411	+	0.942072i	$0.608875\pi$
$608$	0	0
$609$	62.5204	2.53345
$610$	0	0
$611$	1.25127	0.0506208
$612$	0	0
$613$	22.5836	0.912141	0.456071	−	0.889944i	$-0.349256\pi$
$613$	22.5836	0.912141	0.456071	+	0.889944i	$0.349256\pi$
$614$	0	0
$615$	95.4855	3.85035
$616$	0	0
$617$	−31.4822	−1.26743	−0.633713	−	0.773568i	$-0.718470\pi$
$617$	−31.4822	−1.26743	−0.633713	+	0.773568i	$0.718470\pi$
$618$	0	0
$619$	11.2329	0.451487	0.225744	−	0.974187i	$-0.427519\pi$
$619$	11.2329	0.451487	0.225744	+	0.974187i	$0.427519\pi$
$620$	0	0
$621$	46.5534	1.86812
$622$	0	0
$623$	19.3122	0.773727
$624$	0	0
$625$	−7.18148	−0.287259
$626$	0	0
$627$	−6.18311	−0.246930
$628$	0	0
$629$	−79.5101	−3.17027
$630$	0	0
$631$	−12.5546	−0.499791	−0.249895	−	0.968273i	$-0.580396\pi$
$631$	−12.5546	−0.499791	−0.249895	+	0.968273i	$0.580396\pi$
$632$	0	0
$633$	−0.243415	−0.00967489
$634$	0	0
$635$	32.5658	1.29233
$636$	0	0
$637$	18.5082	0.733322
$638$	0	0
$639$	36.2548	1.43422
$640$	0	0
$641$	−27.1054	−1.07060	−0.535300	−	0.844662i	$-0.679802\pi$
$641$	−27.1054	−1.07060	−0.535300	+	0.844662i	$0.679802\pi$
$642$	0	0
$643$	−44.0977	−1.73904	−0.869521	−	0.493896i	$-0.835572\pi$
$643$	−44.0977	−1.73904	−0.869521	+	0.493896i	$0.835572\pi$
$644$	0	0
$645$	8.20742	0.323167
$646$	0	0
$647$	4.94688	0.194482	0.0972409	−	0.995261i	$-0.468998\pi$
$647$	4.94688	0.194482	0.0972409	+	0.995261i	$0.468998\pi$
$648$	0	0
$649$	−0.431491	−0.0169375
$650$	0	0
$651$	14.5942	0.571993
$652$	0	0
$653$	22.9199	0.896926	0.448463	−	0.893801i	$-0.351972\pi$
$653$	22.9199	0.896926	0.448463	+	0.893801i	$0.351972\pi$
$654$	0	0
$655$	−53.0413	−2.07250
$656$	0	0
$657$	67.2970	2.62551
$658$	0	0
$659$	4.47370	0.174270	0.0871352	−	0.996196i	$-0.472229\pi$
$659$	4.47370	0.174270	0.0871352	+	0.996196i	$0.472229\pi$
$660$	0	0
$661$	1.37456	0.0534643	0.0267321	−	0.999643i	$-0.491490\pi$
$661$	1.37456	0.0534643	0.0267321	+	0.999643i	$0.491490\pi$
$662$	0	0
$663$	−35.2249	−1.36802
$664$	0	0
$665$	−46.5402	−1.80475
$666$	0	0
$667$	22.2287	0.860700
$668$	0	0
$669$	−44.3070	−1.71301
$670$	0	0
$671$	−2.53383	−0.0978176
$672$	0	0
$673$	28.7690	1.10897	0.554483	−	0.832195i	$-0.312916\pi$
$673$	28.7690	1.10897	0.554483	+	0.832195i	$0.312916\pi$
$674$	0	0
$675$	73.7207	2.83751
$676$	0	0
$677$	−33.7230	−1.29608	−0.648040	−	0.761606i	$-0.724411\pi$
$677$	−33.7230	−1.29608	−0.648040	+	0.761606i	$0.724411\pi$
$678$	0	0
$679$	−42.1433	−1.61731
$680$	0	0
$681$	74.5515	2.85682
$682$	0	0
$683$	3.53688	0.135335	0.0676675	−	0.997708i	$-0.478444\pi$
$683$	3.53688	0.135335	0.0676675	+	0.997708i	$0.478444\pi$
$684$	0	0
$685$	8.10846	0.309808
$686$	0	0
$687$	−75.7276	−2.88919
$688$	0	0
$689$	−9.45666	−0.360270
$690$	0	0
$691$	−33.1634	−1.26160	−0.630798	−	0.775947i	$-0.717273\pi$
$691$	−33.1634	−1.26160	−0.630798	+	0.775947i	$0.717273\pi$
$692$	0	0
$693$	−17.9864	−0.683246
$694$	0	0
$695$	70.4090	2.67077
$696$	0	0
$697$	64.9328	2.45951
$698$	0	0
$699$	−16.6526	−0.629861
$700$	0	0
$701$	9.48377	0.358197	0.179098	−	0.983831i	$-0.442682\pi$
$701$	9.48377	0.358197	0.179098	+	0.983831i	$0.442682\pi$
$702$	0	0
$703$	33.3459	1.25766
$704$	0	0
$705$	−8.45748	−0.318527
$706$	0	0
$707$	43.5628	1.63835
$708$	0	0
$709$	−51.5623	−1.93646	−0.968231	−	0.250059i	$-0.919550\pi$
$709$	−51.5623	−1.93646	−0.968231	+	0.250059i	$0.919550\pi$
$710$	0	0
$711$	7.93029	0.297409
$712$	0	0
$713$	5.18889	0.194325
$714$	0	0
$715$	3.71070	0.138772
$716$	0	0
$717$	43.5048	1.62472
$718$	0	0
$719$	−29.3492	−1.09454	−0.547269	−	0.836956i	$-0.684333\pi$
$719$	−29.3492	−1.09454	−0.547269	+	0.836956i	$0.684333\pi$
$720$	0	0
$721$	−10.1389	−0.377592
$722$	0	0
$723$	−23.0871	−0.858618
$724$	0	0
$725$	35.2008	1.30733
$726$	0	0
$727$	33.7517	1.25178	0.625890	−	0.779912i	$-0.284736\pi$
$727$	33.7517	1.25178	0.625890	+	0.779912i	$0.284736\pi$
$728$	0	0
$729$	−18.8327	−0.697509
$730$	0	0
$731$	5.58128	0.206431
$732$	0	0
$733$	−11.2827	−0.416735	−0.208368	−	0.978051i	$-0.566815\pi$
$733$	−11.2827	−0.416735	−0.208368	+	0.978051i	$0.566815\pi$
$734$	0	0
$735$	−125.100	−4.61437
$736$	0	0
$737$	−3.06445	−0.112881
$738$	0	0
$739$	−52.6457	−1.93660	−0.968301	−	0.249786i	$-0.919640\pi$
$739$	−52.6457	−1.93660	−0.968301	+	0.249786i	$0.919640\pi$
$740$	0	0
$741$	14.7731	0.542702
$742$	0	0
$743$	−8.43132	−0.309315	−0.154657	−	0.987968i	$-0.549427\pi$
$743$	−8.43132	−0.309315	−0.154657	+	0.987968i	$0.549427\pi$
$744$	0	0
$745$	−42.5896	−1.56036
$746$	0	0
$747$	−85.7104	−3.13598
$748$	0	0
$749$	85.1413	3.11100
$750$	0	0
$751$	−15.1033	−0.551127	−0.275564	−	0.961283i	$-0.588864\pi$
$751$	−15.1033	−0.551127	−0.275564	+	0.961283i	$0.588864\pi$
$752$	0	0
$753$	−3.04458	−0.110951
$754$	0	0
$755$	−8.01209	−0.291590
$756$	0	0
$757$	9.50795	0.345572	0.172786	−	0.984959i	$-0.444723\pi$
$757$	9.50795	0.345572	0.172786	+	0.984959i	$0.444723\pi$
$758$	0	0
$759$	−9.45498	−0.343194
$760$	0	0
$761$	−42.0667	−1.52492	−0.762458	−	0.647037i	$-0.776008\pi$
$761$	−42.0667	−1.52492	−0.762458	+	0.647037i	$0.776008\pi$
$762$	0	0
$763$	3.06948	0.111123
$764$	0	0
$765$	161.034	5.82220
$766$	0	0
$767$	1.03094	0.0372252
$768$	0	0
$769$	−46.9307	−1.69236	−0.846182	−	0.532894i	$-0.821104\pi$
$769$	−46.9307	−1.69236	−0.846182	+	0.532894i	$0.821104\pi$
$770$	0	0
$771$	69.2056	2.49238
$772$	0	0
$773$	25.2021	0.906456	0.453228	−	0.891395i	$-0.350272\pi$
$773$	25.2021	0.906456	0.453228	+	0.891395i	$0.350272\pi$
$774$	0	0
$775$	8.21699	0.295163
$776$	0	0
$777$	143.418	5.14507
$778$	0	0
$779$	−27.2323	−0.975698
$780$	0	0
$781$	−3.83992	−0.137403
$782$	0	0
$783$	47.3128	1.69082
$784$	0	0
$785$	11.3523	0.405181
$786$	0	0
$787$	−29.2020	−1.04094	−0.520470	−	0.853880i	$-0.674243\pi$
$787$	−29.2020	−1.04094	−0.520470	+	0.853880i	$0.674243\pi$
$788$	0	0
$789$	−44.2356	−1.57483
$790$	0	0
$791$	1.15687	0.0411335
$792$	0	0
$793$	6.05399	0.214983
$794$	0	0
$795$	63.9188	2.26697
$796$	0	0
$797$	−35.8761	−1.27080	−0.635399	−	0.772184i	$-0.719164\pi$
$797$	−35.8761	−1.27080	−0.635399	+	0.772184i	$0.719164\pi$
$798$	0	0
$799$	−5.75133	−0.203467
$800$	0	0
$801$	28.0247	0.990205
$802$	0	0
$803$	−7.12776	−0.251533
$804$	0	0
$805$	−71.1675	−2.50832
$806$	0	0
$807$	−8.61305	−0.303194
$808$	0	0
$809$	13.0513	0.458860	0.229430	−	0.973325i	$-0.426314\pi$
$809$	13.0513	0.458860	0.229430	+	0.973325i	$0.426314\pi$
$810$	0	0
$811$	41.7964	1.46767	0.733835	−	0.679328i	$-0.237728\pi$
$811$	41.7964	1.46767	0.733835	+	0.679328i	$0.237728\pi$
$812$	0	0
$813$	1.54985	0.0543557
$814$	0	0
$815$	−17.6601	−0.618605
$816$	0	0
$817$	−2.34074	−0.0818922
$818$	0	0
$819$	42.9741	1.50164
$820$	0	0
$821$	−2.83830	−0.0990572	−0.0495286	−	0.998773i	$-0.515772\pi$
$821$	−2.83830	−0.0990572	−0.0495286	+	0.998773i	$0.515772\pi$
$822$	0	0
$823$	−4.66219	−0.162514	−0.0812568	−	0.996693i	$-0.525893\pi$
$823$	−4.66219	−0.162514	−0.0812568	+	0.996693i	$0.525893\pi$
$824$	0	0
$825$	−14.9727	−0.521281
$826$	0	0
$827$	−40.4435	−1.40636	−0.703179	−	0.711013i	$-0.748237\pi$
$827$	−40.4435	−1.40636	−0.703179	+	0.711013i	$0.748237\pi$
$828$	0	0
$829$	−1.17452	−0.0407929	−0.0203964	−	0.999792i	$-0.506493\pi$
$829$	−1.17452	−0.0407929	−0.0203964	+	0.999792i	$0.506493\pi$
$830$	0	0
$831$	82.8757	2.87492
$832$	0	0
$833$	−85.0712	−2.94754
$834$	0	0
$835$	7.35410	0.254499
$836$	0	0
$837$	11.0443	0.381747
$838$	0	0
$839$	23.4803	0.810630	0.405315	−	0.914177i	$-0.367162\pi$
$839$	23.4803	0.810630	0.405315	+	0.914177i	$0.367162\pi$
$840$	0	0
$841$	−6.40864	−0.220988
$842$	0	0
$843$	−0.0125635	−0.000432709 0
$844$	0	0
$845$	36.9229	1.27019
$846$	0	0
$847$	−45.6192	−1.56750
$848$	0	0
$849$	−22.3866	−0.768306
$850$	0	0
$851$	50.9912	1.74796
$852$	0	0
$853$	36.0921	1.23577	0.617885	−	0.786269i	$-0.287990\pi$
$853$	36.0921	1.23577	0.617885	+	0.786269i	$0.287990\pi$
$854$	0	0
$855$	−67.5364	−2.30970
$856$	0	0
$857$	54.4526	1.86006	0.930032	−	0.367477i	$-0.119779\pi$
$857$	54.4526	1.86006	0.930032	+	0.367477i	$0.119779\pi$
$858$	0	0
$859$	−36.2835	−1.23798	−0.618989	−	0.785400i	$-0.712457\pi$
$859$	−36.2835	−1.23798	−0.618989	+	0.785400i	$0.712457\pi$
$860$	0	0
$861$	−117.124	−3.99156
$862$	0	0
$863$	−25.8098	−0.878576	−0.439288	−	0.898346i	$-0.644769\pi$
$863$	−25.8098	−0.878576	−0.439288	+	0.898346i	$0.644769\pi$
$864$	0	0
$865$	−37.9799	−1.29135
$866$	0	0
$867$	110.150	3.74090
$868$	0	0
$869$	−0.839936	−0.0284929
$870$	0	0
$871$	7.32177	0.248089
$872$	0	0
$873$	−61.1559	−2.06981
$874$	0	0
$875$	−36.6124	−1.23772
$876$	0	0
$877$	−28.2213	−0.952967	−0.476484	−	0.879183i	$-0.658089\pi$
$877$	−28.2213	−0.952967	−0.476484	+	0.879183i	$0.658089\pi$
$878$	0	0
$879$	−33.5894	−1.13294
$880$	0	0
$881$	41.7912	1.40798	0.703991	−	0.710209i	$-0.251399\pi$
$881$	41.7912	1.40798	0.703991	+	0.710209i	$0.251399\pi$
$882$	0	0
$883$	40.3429	1.35765	0.678823	−	0.734302i	$-0.262490\pi$
$883$	40.3429	1.35765	0.678823	+	0.734302i	$0.262490\pi$
$884$	0	0
$885$	−6.96829	−0.234237
$886$	0	0
$887$	7.91141	0.265639	0.132820	−	0.991140i	$-0.457597\pi$
$887$	7.91141	0.265639	0.132820	+	0.991140i	$0.457597\pi$
$888$	0	0
$889$	−39.9456	−1.33973
$890$	0	0
$891$	−7.63506	−0.255784
$892$	0	0
$893$	2.41206	0.0807165
$894$	0	0
$895$	27.8161	0.929791
$896$	0	0
$897$	22.5904	0.754271
$898$	0	0
$899$	5.27354	0.175882
$900$	0	0
$901$	43.4666	1.44808
$902$	0	0
$903$	−10.0673	−0.335019
$904$	0	0
$905$	−40.9678	−1.36181
$906$	0	0
$907$	8.57698	0.284794	0.142397	−	0.989810i	$-0.454519\pi$
$907$	8.57698	0.284794	0.142397	+	0.989810i	$0.454519\pi$
$908$	0	0
$909$	63.2157	2.09673
$910$	0	0
$911$	−24.9038	−0.825099	−0.412549	−	0.910935i	$-0.635362\pi$
$911$	−24.9038	−0.825099	−0.412549	+	0.910935i	$0.635362\pi$
$912$	0	0
$913$	9.07801	0.300438
$914$	0	0
$915$	−40.9197	−1.35276
$916$	0	0
$917$	65.0612	2.14851
$918$	0	0
$919$	−44.8569	−1.47969	−0.739846	−	0.672777i	$-0.765101\pi$
$919$	−44.8569	−1.47969	−0.739846	+	0.672777i	$0.765101\pi$
$920$	0	0
$921$	−76.8592	−2.53260
$922$	0	0
$923$	9.17457	0.301985
$924$	0	0
$925$	80.7483	2.65499
$926$	0	0
$927$	−14.7130	−0.483237
$928$	0	0
$929$	−43.8396	−1.43833	−0.719165	−	0.694840i	$-0.755475\pi$
$929$	−43.8396	−1.43833	−0.719165	+	0.694840i	$0.755475\pi$
$930$	0	0
$931$	35.6782	1.16931
$932$	0	0
$933$	39.7028	1.29981
$934$	0	0
$935$	−17.0559	−0.557788
$936$	0	0
$937$	−10.0111	−0.327049	−0.163524	−	0.986539i	$-0.552286\pi$
$937$	−10.0111	−0.327049	−0.163524	+	0.986539i	$0.552286\pi$
$938$	0	0
$939$	−44.5090	−1.45250
$940$	0	0
$941$	3.29436	0.107393	0.0536966	−	0.998557i	$-0.482900\pi$
$941$	3.29436	0.107393	0.0536966	+	0.998557i	$0.482900\pi$
$942$	0	0
$943$	−41.6426	−1.35607
$944$	0	0
$945$	−151.477	−4.92754
$946$	0	0
$947$	32.3254	1.05043	0.525217	−	0.850969i	$-0.323984\pi$
$947$	32.3254	1.05043	0.525217	+	0.850969i	$0.323984\pi$
$948$	0	0
$949$	17.0301	0.552819
$950$	0	0
$951$	24.0214	0.778947
$952$	0	0
$953$	−17.8393	−0.577873	−0.288936	−	0.957348i	$-0.593302\pi$
$953$	−17.8393	−0.577873	−0.288936	+	0.957348i	$0.593302\pi$
$954$	0	0
$955$	41.1435	1.33137
$956$	0	0
$957$	−9.60922	−0.310622
$958$	0	0
$959$	−9.94594	−0.321171
$960$	0	0
$961$	−29.7690	−0.960290
$962$	0	0
$963$	123.552	3.98141
$964$	0	0
$965$	12.8979	0.415199
$966$	0	0
$967$	−8.39357	−0.269919	−0.134959	−	0.990851i	$-0.543090\pi$
$967$	−8.39357	−0.269919	−0.134959	+	0.990851i	$0.543090\pi$
$968$	0	0
$969$	−67.9029	−2.18136
$970$	0	0
$971$	48.5852	1.55917	0.779587	−	0.626294i	$-0.215429\pi$
$971$	48.5852	1.55917	0.779587	+	0.626294i	$0.215429\pi$
$972$	0	0
$973$	−86.3646	−2.76872
$974$	0	0
$975$	35.7735	1.14567
$976$	0	0
$977$	−37.9594	−1.21443	−0.607215	−	0.794538i	$-0.707713\pi$
$977$	−37.9594	−1.21443	−0.607215	+	0.794538i	$0.707713\pi$
$978$	0	0
$979$	−2.96823	−0.0948652
$980$	0	0
$981$	4.45425	0.142213
$982$	0	0
$983$	32.8128	1.04657	0.523283	−	0.852159i	$-0.324707\pi$
$983$	32.8128	1.04657	0.523283	+	0.852159i	$0.324707\pi$
$984$	0	0
$985$	51.4307	1.63872
$986$	0	0
$987$	10.3741	0.330210
$988$	0	0
$989$	−3.57937	−0.113817
$990$	0	0
$991$	−6.73259	−0.213868	−0.106934	−	0.994266i	$-0.534103\pi$
$991$	−6.73259	−0.213868	−0.106934	+	0.994266i	$0.534103\pi$
$992$	0	0
$993$	−26.3444	−0.836014
$994$	0	0
$995$	−6.99711	−0.221823
$996$	0	0
$997$	36.4158	1.15330	0.576650	−	0.816992i	$-0.304360\pi$
$997$	36.4158	1.15330	0.576650	+	0.816992i	$0.304360\pi$
$998$	0	0
$999$	108.532	3.43381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.16		17
4.3	odd	2		251.2.a.b.1.6	✓	17
12.11	even	2		2259.2.a.k.1.12		17
20.19	odd	2		6275.2.a.e.1.12		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.6	✓	17	4.3	odd	2
2259.2.a.k.1.12		17	12.11	even	2
4016.2.a.k.1.16		17	1.1	even	1	trivial
6275.2.a.e.1.12		17	20.19	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04458 q^{3} -3.52221 q^{5} +4.32039 q^{7} +6.26949 q^{9} +O(q^{10})\)	\(q+3.04458 q^{3} -3.52221 q^{5} +4.32039 q^{7} +6.26949 q^{9} -0.664032 q^{11} +1.58654 q^{13} -10.7237 q^{15} -7.29240 q^{17} +3.05837 q^{19} +13.1538 q^{21} +4.67674 q^{23} +7.40597 q^{25} +9.95423 q^{27} +4.75304 q^{29} +1.10951 q^{31} -2.02170 q^{33} -15.2173 q^{35} +10.9031 q^{37} +4.83037 q^{39} -8.90418 q^{41} -0.765356 q^{43} -22.0825 q^{45} +0.788675 q^{47} +11.6657 q^{49} -22.2023 q^{51} -5.96054 q^{53} +2.33886 q^{55} +9.31147 q^{57} +0.649805 q^{59} +3.81583 q^{61} +27.0866 q^{63} -5.58814 q^{65} +4.61492 q^{67} +14.2387 q^{69} +5.78274 q^{71} +10.7341 q^{73} +22.5481 q^{75} -2.86888 q^{77} +1.26490 q^{79} +11.4980 q^{81} -13.6710 q^{83} +25.6854 q^{85} +14.4710 q^{87} +4.47002 q^{89} +6.85448 q^{91} +3.37799 q^{93} -10.7722 q^{95} -9.75452 q^{97} -4.16314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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