LMFDB - Embedded newform 1071.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1071,2,Mod(1,1071)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1071, 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("1071.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1071 = 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1071.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:	$8.55197805648$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.453749.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 $ x^5 - 2x^4 - 5x^3 + 10*x^2 + x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.32183$ of defining polynomial
Character	$\chi$	$=$	1071.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.877834 q^{2} -1.22941 q^{4} -3.03818 q^{5} -1.00000 q^{7} +2.83488 q^{8} +O(q^{10})$	\(q-0.877834 q^{2} -1.22941 q^{4} -3.03818 q^{5} -1.00000 q^{7} +2.83488 q^{8} +2.66702 q^{10} -4.78178 q^{11} -4.39933 q^{13} +0.877834 q^{14} -0.0297440 q^{16} -1.00000 q^{17} -2.64366 q^{19} +3.73516 q^{20} +4.19761 q^{22} +8.45881 q^{23} +4.23054 q^{25} +3.86188 q^{26} +1.22941 q^{28} -7.04298 q^{29} -3.42063 q^{31} -5.64366 q^{32} +0.877834 q^{34} +3.03818 q^{35} +9.66977 q^{37} +2.32069 q^{38} -8.61289 q^{40} -1.33675 q^{41} +2.52513 q^{43} +5.87875 q^{44} -7.42544 q^{46} +5.57082 q^{47} +1.00000 q^{49} -3.71372 q^{50} +5.40856 q^{52} -5.17630 q^{53} +14.5279 q^{55} -2.83488 q^{56} +6.18257 q^{58} -9.28732 q^{59} +6.66325 q^{61} +3.00275 q^{62} +5.01368 q^{64} +13.3659 q^{65} +5.28251 q^{67} +1.22941 q^{68} -2.66702 q^{70} +11.9310 q^{71} +12.7155 q^{73} -8.48845 q^{74} +3.25013 q^{76} +4.78178 q^{77} +1.94051 q^{79} +0.0903677 q^{80} +1.17345 q^{82} +4.58417 q^{83} +3.03818 q^{85} -2.21664 q^{86} -13.5558 q^{88} +13.3956 q^{89} +4.39933 q^{91} -10.3993 q^{92} -4.89026 q^{94} +8.03191 q^{95} -15.1668 q^{97} -0.877834 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 10 q^{4} - 5 q^{7} - 6 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 10 * q^4 - 5 * q^7 - 6 * q^8	$ 5 q - 2 q^{2} + 10 q^{4} - 5 q^{7} - 6 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{14} + 4 q^{16} - 5 q^{17} + 6 q^{19} + 19 q^{20} + 6 q^{22} + 10 q^{23} + 21 q^{25} + 26 q^{26} - 10 q^{28} + 8 q^{29} - 9 q^{32} + 2 q^{34} + 8 q^{37} - 14 q^{38} + 5 q^{40} - 18 q^{41} + 8 q^{43} + 14 q^{44} + 8 q^{46} + 10 q^{47} + 5 q^{49} - 27 q^{50} + 4 q^{52} - 4 q^{53} - 24 q^{55} + 6 q^{56} + 12 q^{58} - 8 q^{59} + 22 q^{61} - 16 q^{62} - 16 q^{64} + 30 q^{65} + 16 q^{67} - 10 q^{68} - 4 q^{70} + 2 q^{71} + 10 q^{73} + 40 q^{74} + 24 q^{76} - 2 q^{77} + 18 q^{79} + 4 q^{80} - 31 q^{82} + 12 q^{83} - 23 q^{86} - 46 q^{88} - 20 q^{89} - 2 q^{91} - 28 q^{92} - 42 q^{94} + 22 q^{95} + 12 q^{97} - 2 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 10 * q^4 - 5 * q^7 - 6 * q^8 + 4 * q^10 + 2 * q^11 + 2 * q^13 + 2 * q^14 + 4 * q^16 - 5 * q^17 + 6 * q^19 + 19 * q^20 + 6 * q^22 + 10 * q^23 + 21 * q^25 + 26 * q^26 - 10 * q^28 + 8 * q^29 - 9 * q^32 + 2 * q^34 + 8 * q^37 - 14 * q^38 + 5 * q^40 - 18 * q^41 + 8 * q^43 + 14 * q^44 + 8 * q^46 + 10 * q^47 + 5 * q^49 - 27 * q^50 + 4 * q^52 - 4 * q^53 - 24 * q^55 + 6 * q^56 + 12 * q^58 - 8 * q^59 + 22 * q^61 - 16 * q^62 - 16 * q^64 + 30 * q^65 + 16 * q^67 - 10 * q^68 - 4 * q^70 + 2 * q^71 + 10 * q^73 + 40 * q^74 + 24 * q^76 - 2 * q^77 + 18 * q^79 + 4 * q^80 - 31 * q^82 + 12 * q^83 - 23 * q^86 - 46 * q^88 - 20 * q^89 - 2 * q^91 - 28 * q^92 - 42 * q^94 + 22 * q^95 + 12 * q^97 - 2 * 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 $

$2$	−0.877834	−0.620723	−0.310361	−	0.950619i	$-0.600450\pi$
$2$	−0.877834	−0.620723	−0.310361	+	0.950619i	$0.600450\pi$
$3$	0	0
$3$	0	0
$4$	−1.22941	−0.614704
$5$	−3.03818	−1.35872	−0.679358	−	0.733807i	$-0.737742\pi$
$5$	−3.03818	−1.35872	−0.679358	+	0.733807i	$0.737742\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.83488	1.00228
$9$	0	0
$10$	2.66702	0.843385
$11$	−4.78178	−1.44176	−0.720880	−	0.693060i	$-0.756262\pi$
$11$	−4.78178	−1.44176	−0.720880	+	0.693060i	$0.756262\pi$
$12$	0	0
$13$	−4.39933	−1.22015	−0.610077	−	0.792342i	$-0.708861\pi$
$13$	−4.39933	−1.22015	−0.610077	+	0.792342i	$0.708861\pi$
$14$	0.877834	0.234611
$15$	0	0
$16$	−0.0297440	−0.00743600
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.64366	−0.606497	−0.303248	−	0.952912i	$-0.598071\pi$
$19$	−2.64366	−0.606497	−0.303248	+	0.952912i	$0.598071\pi$
$20$	3.73516	0.835207
$21$	0	0
$22$	4.19761	0.894933
$23$	8.45881	1.76378	0.881892	−	0.471451i	$-0.156270\pi$
$23$	8.45881	1.76378	0.881892	+	0.471451i	$0.156270\pi$
$24$	0	0
$25$	4.23054	0.846109
$26$	3.86188	0.757377
$27$	0	0
$28$	1.22941	0.232336
$29$	−7.04298	−1.30785	−0.653925	−	0.756560i	$-0.726879\pi$
$29$	−7.04298	−1.30785	−0.653925	+	0.756560i	$0.726879\pi$
$30$	0	0
$31$	−3.42063	−0.614364	−0.307182	−	0.951651i	$-0.599386\pi$
$31$	−3.42063	−0.614364	−0.307182	+	0.951651i	$0.599386\pi$
$32$	−5.64366	−0.997667
$33$	0	0
$34$	0.877834	0.150547
$35$	3.03818	0.513546
$36$	0	0
$37$	9.66977	1.58970	0.794850	−	0.606806i	$-0.207549\pi$
$37$	9.66977	1.58970	0.794850	+	0.606806i	$0.207549\pi$
$38$	2.32069	0.376466
$39$	0	0
$40$	−8.61289	−1.36182
$41$	−1.33675	−0.208766	−0.104383	−	0.994537i	$-0.533287\pi$
$41$	−1.33675	−0.208766	−0.104383	+	0.994537i	$0.533287\pi$
$42$	0	0
$43$	2.52513	0.385078	0.192539	−	0.981289i	$-0.438328\pi$
$43$	2.52513	0.385078	0.192539	+	0.981289i	$0.438328\pi$
$44$	5.87875	0.886255
$45$	0	0
$46$	−7.42544	−1.09482
$47$	5.57082	0.812588	0.406294	−	0.913742i	$-0.366821\pi$
$47$	5.57082	0.812588	0.406294	+	0.913742i	$0.366821\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.71372	−0.525199
$51$	0	0
$52$	5.40856	0.750033
$53$	−5.17630	−0.711020	−0.355510	−	0.934673i	$-0.615693\pi$
$53$	−5.17630	−0.711020	−0.355510	+	0.934673i	$0.615693\pi$
$54$	0	0
$55$	14.5279	1.95894
$56$	−2.83488	−0.378827
$57$	0	0
$58$	6.18257	0.811812
$59$	−9.28732	−1.20911	−0.604553	−	0.796565i	$-0.706648\pi$
$59$	−9.28732	−1.20911	−0.604553	+	0.796565i	$0.706648\pi$
$60$	0	0
$61$	6.66325	0.853141	0.426571	−	0.904454i	$-0.359722\pi$
$61$	6.66325	0.853141	0.426571	+	0.904454i	$0.359722\pi$
$62$	3.00275	0.381350
$63$	0	0
$64$	5.01368	0.626710
$65$	13.3659	1.65784
$66$	0	0
$67$	5.28251	0.645362	0.322681	−	0.946508i	$-0.395416\pi$
$67$	5.28251	0.645362	0.322681	+	0.946508i	$0.395416\pi$
$68$	1.22941	0.149088
$69$	0	0
$70$	−2.66702	−0.318770
$71$	11.9310	1.41595	0.707973	−	0.706239i	$-0.249610\pi$
$71$	11.9310	1.41595	0.707973	+	0.706239i	$0.249610\pi$
$72$	0	0
$73$	12.7155	1.48823	0.744116	−	0.668050i	$-0.232871\pi$
$73$	12.7155	1.48823	0.744116	+	0.668050i	$0.232871\pi$
$74$	−8.48845	−0.986763
$75$	0	0
$76$	3.25013	0.372816
$77$	4.78178	0.544934
$78$	0	0
$79$	1.94051	0.218325	0.109162	−	0.994024i	$-0.465183\pi$
$79$	1.94051	0.218325	0.109162	+	0.994024i	$0.465183\pi$
$80$	0.0903677	0.0101034
$81$	0	0
$82$	1.17345	0.129586
$83$	4.58417	0.503178	0.251589	−	0.967834i	$-0.419047\pi$
$83$	4.58417	0.503178	0.251589	+	0.967834i	$0.419047\pi$
$84$	0	0
$85$	3.03818	0.329537
$86$	−2.21664	−0.239027
$87$	0	0
$88$	−13.5558	−1.44505
$89$	13.3956	1.41993	0.709965	−	0.704237i	$-0.248711\pi$
$89$	13.3956	1.41993	0.709965	+	0.704237i	$0.248711\pi$
$90$	0	0
$91$	4.39933	0.461175
$92$	−10.3993	−1.08420
$93$	0	0
$94$	−4.89026	−0.504392
$95$	8.03191	0.824057
$96$	0	0
$97$	−15.1668	−1.53995	−0.769976	−	0.638073i	$-0.779732\pi$
$97$	−15.1668	−1.53995	−0.769976	+	0.638073i	$0.779732\pi$
$98$	−0.877834	−0.0886746
$99$	0	0
$100$	−5.20106	−0.520106
$101$	−3.24786	−0.323174	−0.161587	−	0.986858i	$-0.551661\pi$
$101$	−3.24786	−0.323174	−0.161587	+	0.986858i	$0.551661\pi$
$102$	0	0
$103$	−12.7031	−1.25168	−0.625839	−	0.779952i	$-0.715243\pi$
$103$	−12.7031	−1.25168	−0.625839	+	0.779952i	$0.715243\pi$
$104$	−12.4716	−1.22294
$105$	0	0
$106$	4.54393	0.441346
$107$	−3.91410	−0.378390	−0.189195	−	0.981940i	$-0.560588\pi$
$107$	−3.91410	−0.378390	−0.189195	+	0.981940i	$0.560588\pi$
$108$	0	0
$109$	−3.56356	−0.341327	−0.170663	−	0.985329i	$-0.554591\pi$
$109$	−3.56356	−0.341327	−0.170663	+	0.985329i	$0.554591\pi$
$110$	−12.7531	−1.21596
$111$	0	0
$112$	0.0297440	0.00281054
$113$	−0.966622	−0.0909322	−0.0454661	−	0.998966i	$-0.514477\pi$
$113$	−0.966622	−0.0909322	−0.0454661	+	0.998966i	$0.514477\pi$
$114$	0	0
$115$	−25.6994	−2.39648
$116$	8.65869	0.803940
$117$	0	0
$118$	8.15272	0.750519
$119$	1.00000	0.0916698
$120$	0	0
$121$	11.8654	1.07867
$122$	−5.84923	−0.529564
$123$	0	0
$124$	4.20535	0.377652
$125$	2.33775	0.209095
$126$	0	0
$127$	−5.55875	−0.493260	−0.246630	−	0.969110i	$-0.579323\pi$
$127$	−5.55875	−0.493260	−0.246630	+	0.969110i	$0.579323\pi$
$128$	6.88613	0.608654
$129$	0	0
$130$	−11.7331	−1.02906
$131$	3.51134	0.306787	0.153393	−	0.988165i	$-0.450980\pi$
$131$	3.51134	0.306787	0.153393	+	0.988165i	$0.450980\pi$
$132$	0	0
$133$	2.64366	0.229234
$134$	−4.63717	−0.400590
$135$	0	0
$136$	−2.83488	−0.243089
$137$	1.93780	0.165557	0.0827786	−	0.996568i	$-0.473621\pi$
$137$	1.93780	0.165557	0.0827786	+	0.996568i	$0.473621\pi$
$138$	0	0
$139$	4.69834	0.398508	0.199254	−	0.979948i	$-0.436148\pi$
$139$	4.69834	0.398508	0.199254	+	0.979948i	$0.436148\pi$
$140$	−3.73516	−0.315679
$141$	0	0
$142$	−10.4734	−0.878910
$143$	21.0366	1.75917
$144$	0	0
$145$	21.3979	1.77700
$146$	−11.1621	−0.923780
$147$	0	0
$148$	−11.8881	−0.977194
$149$	−4.24162	−0.347487	−0.173743	−	0.984791i	$-0.555586\pi$
$149$	−4.24162	−0.347487	−0.173743	+	0.984791i	$0.555586\pi$
$150$	0	0
$151$	12.4584	1.01385	0.506924	−	0.861991i	$-0.330783\pi$
$151$	12.4584	1.01385	0.506924	+	0.861991i	$0.330783\pi$
$152$	−7.49446	−0.607881
$153$	0	0
$154$	−4.19761	−0.338253
$155$	10.3925	0.834746
$156$	0	0
$157$	−17.6472	−1.40840	−0.704199	−	0.710002i	$-0.748694\pi$
$157$	−17.6472	−1.40840	−0.704199	+	0.710002i	$0.748694\pi$
$158$	−1.70345	−0.135519
$159$	0	0
$160$	17.1465	1.35555
$161$	−8.45881	−0.666648
$162$	0	0
$163$	−2.67557	−0.209567	−0.104783	−	0.994495i	$-0.533415\pi$
$163$	−2.67557	−0.209567	−0.104783	+	0.994495i	$0.533415\pi$
$164$	1.64341	0.128329
$165$	0	0
$166$	−4.02414	−0.312334
$167$	1.95783	0.151501	0.0757507	−	0.997127i	$-0.475865\pi$
$167$	1.95783	0.151501	0.0757507	+	0.997127i	$0.475865\pi$
$168$	0	0
$169$	6.35407	0.488775
$170$	−2.66702	−0.204551
$171$	0	0
$172$	−3.10441	−0.236709
$173$	−6.01732	−0.457488	−0.228744	−	0.973487i	$-0.573462\pi$
$173$	−6.01732	−0.457488	−0.228744	+	0.973487i	$0.573462\pi$
$174$	0	0
$175$	−4.23054	−0.319799
$176$	0.142229	0.0107209
$177$	0	0
$178$	−11.7591	−0.881382
$179$	−1.75112	−0.130885	−0.0654423	−	0.997856i	$-0.520846\pi$
$179$	−1.75112	−0.130885	−0.0654423	+	0.997856i	$0.520846\pi$
$180$	0	0
$181$	2.76491	0.205514	0.102757	−	0.994707i	$-0.467234\pi$
$181$	2.76491	0.205514	0.102757	+	0.994707i	$0.467234\pi$
$182$	−3.86188	−0.286262
$183$	0	0
$184$	23.9798	1.76781
$185$	−29.3785	−2.15995
$186$	0	0
$187$	4.78178	0.349678
$188$	−6.84881	−0.499501
$189$	0	0
$190$	−7.05069	−0.511511
$191$	−18.8461	−1.36365	−0.681827	−	0.731514i	$-0.738814\pi$
$191$	−18.8461	−1.36365	−0.681827	+	0.731514i	$0.738814\pi$
$192$	0	0
$193$	11.9107	0.857348	0.428674	−	0.903459i	$-0.358981\pi$
$193$	11.9107	0.857348	0.428674	+	0.903459i	$0.358981\pi$
$194$	13.3139	0.955883
$195$	0	0
$196$	−1.22941	−0.0878148
$197$	17.2682	1.23031	0.615153	−	0.788408i	$-0.289094\pi$
$197$	17.2682	1.23031	0.615153	+	0.788408i	$0.289094\pi$
$198$	0	0
$199$	−15.2451	−1.08070	−0.540350	−	0.841440i	$-0.681708\pi$
$199$	−15.2451	−1.08070	−0.540350	+	0.841440i	$0.681708\pi$
$200$	11.9931	0.848040
$201$	0	0
$202$	2.85108	0.200601
$203$	7.04298	0.494321
$204$	0	0
$205$	4.06130	0.283653
$206$	11.1513	0.776945
$207$	0	0
$208$	0.130854	0.00907306
$209$	12.6414	0.874423
$210$	0	0
$211$	8.51331	0.586080	0.293040	−	0.956100i	$-0.405333\pi$
$211$	8.51331	0.586080	0.293040	+	0.956100i	$0.405333\pi$
$212$	6.36378	0.437066
$213$	0	0
$214$	3.43593	0.234875
$215$	−7.67179	−0.523212
$216$	0	0
$217$	3.42063	0.232208
$218$	3.12821	0.211869
$219$	0	0
$220$	−17.8607	−1.20417
$221$	4.39933	0.295931
$222$	0	0
$223$	−2.49093	−0.166805	−0.0834027	−	0.996516i	$-0.526579\pi$
$223$	−2.49093	−0.166805	−0.0834027	+	0.996516i	$0.526579\pi$
$224$	5.64366	0.377083
$225$	0	0
$226$	0.848534	0.0564437
$227$	16.9842	1.12728	0.563640	−	0.826020i	$-0.309400\pi$
$227$	16.9842	1.12728	0.563640	+	0.826020i	$0.309400\pi$
$228$	0	0
$229$	−16.3303	−1.07914	−0.539568	−	0.841942i	$-0.681413\pi$
$229$	−16.3303	−1.07914	−0.539568	+	0.841942i	$0.681413\pi$
$230$	22.5598	1.48755
$231$	0	0
$232$	−19.9660	−1.31083
$233$	5.77825	0.378546	0.189273	−	0.981925i	$-0.439387\pi$
$233$	5.77825	0.378546	0.189273	+	0.981925i	$0.439387\pi$
$234$	0	0
$235$	−16.9252	−1.10408
$236$	11.4179	0.743241
$237$	0	0
$238$	−0.877834	−0.0569015
$239$	−7.44275	−0.481432	−0.240716	−	0.970596i	$-0.577382\pi$
$239$	−7.44275	−0.481432	−0.240716	+	0.970596i	$0.577382\pi$
$240$	0	0
$241$	21.9036	1.41093	0.705467	−	0.708743i	$-0.250737\pi$
$241$	21.9036	1.41093	0.705467	+	0.708743i	$0.250737\pi$
$242$	−10.4159	−0.669557
$243$	0	0
$244$	−8.19184	−0.524429
$245$	−3.03818	−0.194102
$246$	0	0
$247$	11.6303	0.740019
$248$	−9.69710	−0.615766
$249$	0	0
$250$	−2.05216	−0.129790
$251$	−16.5769	−1.04632	−0.523162	−	0.852233i	$-0.675248\pi$
$251$	−16.5769	−1.04632	−0.523162	+	0.852233i	$0.675248\pi$
$252$	0	0
$253$	−40.4482	−2.54296
$254$	4.87966	0.306177
$255$	0	0
$256$	−16.0723	−1.00452
$257$	−12.1158	−0.755764	−0.377882	−	0.925854i	$-0.623347\pi$
$257$	−12.1158	−0.755764	−0.377882	+	0.925854i	$0.623347\pi$
$258$	0	0
$259$	−9.66977	−0.600850
$260$	−16.4322	−1.01908
$261$	0	0
$262$	−3.08237	−0.190430
$263$	−3.20135	−0.197404	−0.0987018	−	0.995117i	$-0.531469\pi$
$263$	−3.20135	−0.197404	−0.0987018	+	0.995117i	$0.531469\pi$
$264$	0	0
$265$	15.7265	0.966074
$266$	−2.32069	−0.142291
$267$	0	0
$268$	−6.49436	−0.396706
$269$	−15.4400	−0.941396	−0.470698	−	0.882294i	$-0.655998\pi$
$269$	−15.4400	−0.941396	−0.470698	+	0.882294i	$0.655998\pi$
$270$	0	0
$271$	−24.8878	−1.51182	−0.755912	−	0.654673i	$-0.772806\pi$
$271$	−24.8878	−1.51182	−0.755912	+	0.654673i	$0.772806\pi$
$272$	0.0297440	0.00180349
$273$	0	0
$274$	−1.70106	−0.102765
$275$	−20.2295	−1.21989
$276$	0	0
$277$	−11.2870	−0.678172	−0.339086	−	0.940755i	$-0.610118\pi$
$277$	−11.2870	−0.678172	−0.339086	+	0.940755i	$0.610118\pi$
$278$	−4.12437	−0.247363
$279$	0	0
$280$	8.61289	0.514719
$281$	−13.0332	−0.777495	−0.388747	−	0.921344i	$-0.627092\pi$
$281$	−13.0332	−0.777495	−0.388747	+	0.921344i	$0.627092\pi$
$282$	0	0
$283$	21.7190	1.29106	0.645531	−	0.763734i	$-0.276636\pi$
$283$	21.7190	1.29106	0.645531	+	0.763734i	$0.276636\pi$
$284$	−14.6680	−0.870387
$285$	0	0
$286$	−18.4667	−1.09196
$287$	1.33675	0.0789061
$288$	0	0
$289$	1.00000	0.0588235
$290$	−18.7838	−1.10302
$291$	0	0
$292$	−15.6325	−0.914822
$293$	28.9400	1.69069	0.845346	−	0.534219i	$-0.179394\pi$
$293$	28.9400	1.69069	0.845346	+	0.534219i	$0.179394\pi$
$294$	0	0
$295$	28.2165	1.64283
$296$	27.4127	1.59333
$297$	0	0
$298$	3.72344	0.215693
$299$	−37.2131	−2.15209
$300$	0	0
$301$	−2.52513	−0.145546
$302$	−10.9364	−0.629318
$303$	0	0
$304$	0.0786330	0.00450991
$305$	−20.2441	−1.15918
$306$	0	0
$307$	−11.5836	−0.661110	−0.330555	−	0.943787i	$-0.607236\pi$
$307$	−11.5836	−0.661110	−0.330555	+	0.943787i	$0.607236\pi$
$308$	−5.87875	−0.334973
$309$	0	0
$310$	−9.12289	−0.518146
$311$	15.7999	0.895932	0.447966	−	0.894051i	$-0.352149\pi$
$311$	15.7999	0.895932	0.447966	+	0.894051i	$0.352149\pi$
$312$	0	0
$313$	20.2658	1.14549	0.572745	−	0.819733i	$-0.305878\pi$
$313$	20.2658	1.14549	0.572745	+	0.819733i	$0.305878\pi$
$314$	15.4913	0.874225
$315$	0	0
$316$	−2.38568	−0.134205
$317$	−6.36537	−0.357515	−0.178757	−	0.983893i	$-0.557208\pi$
$317$	−6.36537	−0.357515	−0.178757	+	0.983893i	$0.557208\pi$
$318$	0	0
$319$	33.6780	1.88561
$320$	−15.2325	−0.851521
$321$	0	0
$322$	7.42544	0.413803
$323$	2.64366	0.147097
$324$	0	0
$325$	−18.6115	−1.03238
$326$	2.34871	0.130083
$327$	0	0
$328$	−3.78954	−0.209242
$329$	−5.57082	−0.307130
$330$	0	0
$331$	24.5910	1.35165	0.675823	−	0.737064i	$-0.263788\pi$
$331$	24.5910	1.35165	0.675823	+	0.737064i	$0.263788\pi$
$332$	−5.63581	−0.309305
$333$	0	0
$334$	−1.71865	−0.0940403
$335$	−16.0492	−0.876863
$336$	0	0
$337$	7.11956	0.387827	0.193913	−	0.981019i	$-0.437882\pi$
$337$	7.11956	0.387827	0.193913	+	0.981019i	$0.437882\pi$
$338$	−5.57782	−0.303393
$339$	0	0
$340$	−3.73516	−0.202568
$341$	16.3567	0.885766
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.15844	0.385957
$345$	0	0
$346$	5.28221	0.283973
$347$	−1.51477	−0.0813173	−0.0406586	−	0.999173i	$-0.512946\pi$
$347$	−1.51477	−0.0813173	−0.0406586	+	0.999173i	$0.512946\pi$
$348$	0	0
$349$	24.7937	1.32717	0.663587	−	0.748099i	$-0.269033\pi$
$349$	24.7937	1.32717	0.663587	+	0.748099i	$0.269033\pi$
$350$	3.71372	0.198506
$351$	0	0
$352$	26.9867	1.43840
$353$	11.8416	0.630267	0.315133	−	0.949047i	$-0.397951\pi$
$353$	11.8416	0.630267	0.315133	+	0.949047i	$0.397951\pi$
$354$	0	0
$355$	−36.2485	−1.92387
$356$	−16.4686	−0.872836
$357$	0	0
$358$	1.53719	0.0812430
$359$	26.6743	1.40781	0.703907	−	0.710292i	$-0.251437\pi$
$359$	26.6743	1.40781	0.703907	+	0.710292i	$0.251437\pi$
$360$	0	0
$361$	−12.0111	−0.632162
$362$	−2.42713	−0.127567
$363$	0	0
$364$	−5.40856	−0.283486
$365$	−38.6319	−2.02209
$366$	0	0
$367$	−28.6897	−1.49759	−0.748796	−	0.662801i	$-0.769368\pi$
$367$	−28.6897	−1.49759	−0.748796	+	0.662801i	$0.769368\pi$
$368$	−0.251599	−0.0131155
$369$	0	0
$370$	25.7895	1.34073
$371$	5.17630	0.268740
$372$	0	0
$373$	−26.8946	−1.39255	−0.696275	−	0.717775i	$-0.745161\pi$
$373$	−26.8946	−1.39255	−0.696275	+	0.717775i	$0.745161\pi$
$374$	−4.19761	−0.217053
$375$	0	0
$376$	15.7926	0.814443
$377$	30.9844	1.59578
$378$	0	0
$379$	0.960453	0.0493351	0.0246676	−	0.999696i	$-0.492147\pi$
$379$	0.960453	0.0493351	0.0246676	+	0.999696i	$0.492147\pi$
$380$	−9.87449	−0.506551
$381$	0	0
$382$	16.5437	0.846451
$383$	10.9913	0.561627	0.280814	−	0.959762i	$-0.409396\pi$
$383$	10.9913	0.561627	0.280814	+	0.959762i	$0.409396\pi$
$384$	0	0
$385$	−14.5279	−0.740411
$386$	−10.4556	−0.532175
$387$	0	0
$388$	18.6461	0.946614
$389$	25.4239	1.28904	0.644520	−	0.764587i	$-0.277057\pi$
$389$	25.4239	1.28904	0.644520	+	0.764587i	$0.277057\pi$
$390$	0	0
$391$	−8.45881	−0.427781
$392$	2.83488	0.143183
$393$	0	0
$394$	−15.1586	−0.763679
$395$	−5.89563	−0.296641
$396$	0	0
$397$	−15.0140	−0.753533	−0.376767	−	0.926308i	$-0.622964\pi$
$397$	−15.0140	−0.753533	−0.376767	+	0.926308i	$0.622964\pi$
$398$	13.3827	0.670815
$399$	0	0
$400$	−0.125833	−0.00629166
$401$	11.0656	0.552588	0.276294	−	0.961073i	$-0.410894\pi$
$401$	11.0656	0.552588	0.276294	+	0.961073i	$0.410894\pi$
$402$	0	0
$403$	15.0485	0.749618
$404$	3.99294	0.198656
$405$	0	0
$406$	−6.18257	−0.306836
$407$	−46.2387	−2.29197
$408$	0	0
$409$	28.5650	1.41245	0.706225	−	0.707988i	$-0.250397\pi$
$409$	28.5650	1.41245	0.706225	+	0.707988i	$0.250397\pi$
$410$	−3.56515	−0.176070
$411$	0	0
$412$	15.6173	0.769411
$413$	9.28732	0.456999
$414$	0	0
$415$	−13.9275	−0.683676
$416$	24.8283	1.21731
$417$	0	0
$418$	−11.0970	−0.542774
$419$	1.98566	0.0970057	0.0485029	−	0.998823i	$-0.484555\pi$
$419$	1.98566	0.0970057	0.0485029	+	0.998823i	$0.484555\pi$
$420$	0	0
$421$	25.1984	1.22810	0.614048	−	0.789269i	$-0.289540\pi$
$421$	25.1984	1.22810	0.614048	+	0.789269i	$0.289540\pi$
$422$	−7.47327	−0.363793
$423$	0	0
$424$	−14.6742	−0.712643
$425$	−4.23054	−0.205211
$426$	0	0
$427$	−6.66325	−0.322457
$428$	4.81202	0.232598
$429$	0	0
$430$	6.73456	0.324769
$431$	27.4984	1.32455	0.662275	−	0.749261i	$-0.269591\pi$
$431$	27.4984	1.32455	0.662275	+	0.749261i	$0.269591\pi$
$432$	0	0
$433$	28.1977	1.35509	0.677547	−	0.735480i	$-0.263043\pi$
$433$	28.1977	1.35509	0.677547	+	0.735480i	$0.263043\pi$
$434$	−3.00275	−0.144137
$435$	0	0
$436$	4.38106	0.209815
$437$	−22.3622	−1.06973
$438$	0	0
$439$	6.31298	0.301302	0.150651	−	0.988587i	$-0.451863\pi$
$439$	6.31298	0.301302	0.150651	+	0.988587i	$0.451863\pi$
$440$	41.1849	1.96341
$441$	0	0
$442$	−3.86188	−0.183691
$443$	−26.4557	−1.25695	−0.628473	−	0.777831i	$-0.716320\pi$
$443$	−26.4557	−1.25695	−0.628473	+	0.777831i	$0.716320\pi$
$444$	0	0
$445$	−40.6982	−1.92928
$446$	2.18663	0.103540
$447$	0	0
$448$	−5.01368	−0.236874
$449$	34.5555	1.63077	0.815387	−	0.578916i	$-0.196524\pi$
$449$	34.5555	1.63077	0.815387	+	0.578916i	$0.196524\pi$
$450$	0	0
$451$	6.39206	0.300990
$452$	1.18837	0.0558963
$453$	0	0
$454$	−14.9093	−0.699728
$455$	−13.3659	−0.626605
$456$	0	0
$457$	26.5922	1.24393	0.621965	−	0.783045i	$-0.286334\pi$
$457$	26.5922	1.24393	0.621965	+	0.783045i	$0.286334\pi$
$458$	14.3353	0.669844
$459$	0	0
$460$	31.5950	1.47313
$461$	0.310199	0.0144474	0.00722371	−	0.999974i	$-0.497701\pi$
$461$	0.310199	0.0144474	0.00722371	+	0.999974i	$0.497701\pi$
$462$	0	0
$463$	22.4654	1.04406	0.522028	−	0.852928i	$-0.325175\pi$
$463$	22.4654	1.04406	0.522028	+	0.852928i	$0.325175\pi$
$464$	0.209487	0.00972517
$465$	0	0
$466$	−5.07235	−0.234972
$467$	11.6126	0.537365	0.268682	−	0.963229i	$-0.413412\pi$
$467$	11.6126	0.537365	0.268682	+	0.963229i	$0.413412\pi$
$468$	0	0
$469$	−5.28251	−0.243924
$470$	14.8575	0.685325
$471$	0	0
$472$	−26.3285	−1.21187
$473$	−12.0746	−0.555190
$474$	0	0
$475$	−11.1841	−0.513162
$476$	−1.22941	−0.0563498
$477$	0	0
$478$	6.53350	0.298836
$479$	−31.7976	−1.45287	−0.726435	−	0.687235i	$-0.758824\pi$
$479$	−31.7976	−1.45287	−0.726435	+	0.687235i	$0.758824\pi$
$480$	0	0
$481$	−42.5405	−1.93968
$482$	−19.2277	−0.875799
$483$	0	0
$484$	−14.5874	−0.663064
$485$	46.0794	2.09236
$486$	0	0
$487$	−23.0166	−1.04298	−0.521490	−	0.853257i	$-0.674624\pi$
$487$	−23.0166	−1.04298	−0.521490	+	0.853257i	$0.674624\pi$
$488$	18.8895	0.855089
$489$	0	0
$490$	2.66702	0.120484
$491$	33.3176	1.50360	0.751802	−	0.659389i	$-0.229185\pi$
$491$	33.3176	1.50360	0.751802	+	0.659389i	$0.229185\pi$
$492$	0	0
$493$	7.04298	0.317200
$494$	−10.2095	−0.459347
$495$	0	0
$496$	0.101743	0.00456841
$497$	−11.9310	−0.535177
$498$	0	0
$499$	37.0236	1.65741	0.828703	−	0.559689i	$-0.189079\pi$
$499$	37.0236	1.65741	0.828703	+	0.559689i	$0.189079\pi$
$500$	−2.87405	−0.128531
$501$	0	0
$502$	14.5518	0.649477
$503$	12.4172	0.553654	0.276827	−	0.960920i	$-0.410717\pi$
$503$	12.4172	0.553654	0.276827	+	0.960920i	$0.410717\pi$
$504$	0	0
$505$	9.86759	0.439102
$506$	35.5068	1.57847
$507$	0	0
$508$	6.83397	0.303208
$509$	−11.9257	−0.528597	−0.264299	−	0.964441i	$-0.585140\pi$
$509$	−11.9257	−0.528597	−0.264299	+	0.964441i	$0.585140\pi$
$510$	0	0
$511$	−12.7155	−0.562499
$512$	0.336507	0.0148716
$513$	0	0
$514$	10.6357	0.469120
$515$	38.5945	1.70067
$516$	0	0
$517$	−26.6385	−1.17156
$518$	8.48845	0.372961
$519$	0	0
$520$	37.8909	1.66163
$521$	−26.3980	−1.15652	−0.578259	−	0.815853i	$-0.696268\pi$
$521$	−26.3980	−1.15652	−0.578259	+	0.815853i	$0.696268\pi$
$522$	0	0
$523$	13.6272	0.595874	0.297937	−	0.954586i	$-0.403701\pi$
$523$	13.6272	0.595874	0.297937	+	0.954586i	$0.403701\pi$
$524$	−4.31686	−0.188583
$525$	0	0
$526$	2.81025	0.122533
$527$	3.42063	0.149005
$528$	0	0
$529$	48.5515	2.11094
$530$	−13.8053	−0.599664
$531$	0	0
$532$	−3.25013	−0.140911
$533$	5.88081	0.254726
$534$	0	0
$535$	11.8917	0.514125
$536$	14.9753	0.646835
$537$	0	0
$538$	13.5538	0.584346
$539$	−4.78178	−0.205966
$540$	0	0
$541$	26.3358	1.13226	0.566132	−	0.824314i	$-0.308439\pi$
$541$	26.3358	1.13226	0.566132	+	0.824314i	$0.308439\pi$
$542$	21.8473	0.938424
$543$	0	0
$544$	5.64366	0.241970
$545$	10.8267	0.463766
$546$	0	0
$547$	−2.25139	−0.0962624	−0.0481312	−	0.998841i	$-0.515327\pi$
$547$	−2.25139	−0.0962624	−0.0481312	+	0.998841i	$0.515327\pi$
$548$	−2.38234	−0.101769
$549$	0	0
$550$	17.7582	0.757211
$551$	18.6192	0.793206
$552$	0	0
$553$	−1.94051	−0.0825190
$554$	9.90815	0.420957
$555$	0	0
$556$	−5.77618	−0.244964
$557$	−30.3355	−1.28536	−0.642679	−	0.766136i	$-0.722177\pi$
$557$	−30.3355	−1.28536	−0.642679	+	0.766136i	$0.722177\pi$
$558$	0	0
$559$	−11.1089	−0.469854
$560$	−0.0903677	−0.00381873
$561$	0	0
$562$	11.4410	0.482608
$563$	−10.9867	−0.463035	−0.231518	−	0.972831i	$-0.574369\pi$
$563$	−10.9867	−0.463035	−0.231518	+	0.972831i	$0.574369\pi$
$564$	0	0
$565$	2.93677	0.123551
$566$	−19.0657	−0.801391
$567$	0	0
$568$	33.8229	1.41918
$569$	28.8479	1.20937	0.604684	−	0.796465i	$-0.293299\pi$
$569$	28.8479	1.20937	0.604684	+	0.796465i	$0.293299\pi$
$570$	0	0
$571$	−10.5165	−0.440100	−0.220050	−	0.975489i	$-0.570622\pi$
$571$	−10.5165	−0.440100	−0.220050	+	0.975489i	$0.570622\pi$
$572$	−25.8626	−1.08137
$573$	0	0
$574$	−1.17345	−0.0489788
$575$	35.7854	1.49235
$576$	0	0
$577$	−36.8288	−1.53320	−0.766601	−	0.642123i	$-0.778054\pi$
$577$	−36.8288	−1.53320	−0.766601	+	0.642123i	$0.778054\pi$
$578$	−0.877834	−0.0365131
$579$	0	0
$580$	−26.3067	−1.09233
$581$	−4.58417	−0.190183
$582$	0	0
$583$	24.7519	1.02512
$584$	36.0469	1.49163
$585$	0	0
$586$	−25.4045	−1.04945
$587$	−30.1420	−1.24409	−0.622047	−	0.782980i	$-0.713699\pi$
$587$	−30.1420	−1.24409	−0.622047	+	0.782980i	$0.713699\pi$
$588$	0	0
$589$	9.04298	0.372610
$590$	−24.7694	−1.01974
$591$	0	0
$592$	−0.287618	−0.0118210
$593$	7.88925	0.323973	0.161986	−	0.986793i	$-0.448210\pi$
$593$	7.88925	0.323973	0.161986	+	0.986793i	$0.448210\pi$
$594$	0	0
$595$	−3.03818	−0.124553
$596$	5.21467	0.213601
$597$	0	0
$598$	32.6669	1.33585
$599$	−29.8059	−1.21783	−0.608917	−	0.793234i	$-0.708396\pi$
$599$	−29.8059	−1.21783	−0.608917	+	0.793234i	$0.708396\pi$
$600$	0	0
$601$	11.9432	0.487175	0.243587	−	0.969879i	$-0.421676\pi$
$601$	11.9432	0.487175	0.243587	+	0.969879i	$0.421676\pi$
$602$	2.21664	0.0903436
$603$	0	0
$604$	−15.3164	−0.623216
$605$	−36.0493	−1.46561
$606$	0	0
$607$	−4.67281	−0.189664	−0.0948318	−	0.995493i	$-0.530231\pi$
$607$	−4.67281	−0.189664	−0.0948318	+	0.995493i	$0.530231\pi$
$608$	14.9199	0.605082
$609$	0	0
$610$	17.7710	0.719527
$611$	−24.5079	−0.991483
$612$	0	0
$613$	42.3894	1.71209	0.856046	−	0.516900i	$-0.172914\pi$
$613$	42.3894	1.71209	0.856046	+	0.516900i	$0.172914\pi$
$614$	10.1685	0.410366
$615$	0	0
$616$	13.5558	0.546178
$617$	23.2809	0.937255	0.468628	−	0.883396i	$-0.344749\pi$
$617$	23.2809	0.937255	0.468628	+	0.883396i	$0.344749\pi$
$618$	0	0
$619$	−43.2201	−1.73716	−0.868582	−	0.495545i	$-0.834968\pi$
$619$	−43.2201	−1.73716	−0.868582	+	0.495545i	$0.834968\pi$
$620$	−12.7766	−0.513121
$621$	0	0
$622$	−13.8697	−0.556125
$623$	−13.3956	−0.536683
$624$	0	0
$625$	−28.2552	−1.13021
$626$	−17.7900	−0.711032
$627$	0	0
$628$	21.6956	0.865748
$629$	−9.66977	−0.385559
$630$	0	0
$631$	30.6825	1.22145	0.610725	−	0.791843i	$-0.290878\pi$
$631$	30.6825	1.22145	0.610725	+	0.791843i	$0.290878\pi$
$632$	5.50113	0.218823
$633$	0	0
$634$	5.58774	0.221917
$635$	16.8885	0.670200
$636$	0	0
$637$	−4.39933	−0.174308
$638$	−29.5637	−1.17044
$639$	0	0
$640$	−20.9213	−0.826988
$641$	21.3561	0.843517	0.421758	−	0.906708i	$-0.361413\pi$
$641$	21.3561	0.843517	0.421758	+	0.906708i	$0.361413\pi$
$642$	0	0
$643$	22.7013	0.895253	0.447626	−	0.894221i	$-0.352269\pi$
$643$	22.7013	0.895253	0.447626	+	0.894221i	$0.352269\pi$
$644$	10.3993	0.409791
$645$	0	0
$646$	−2.32069	−0.0913065
$647$	−35.2630	−1.38633	−0.693165	−	0.720779i	$-0.743784\pi$
$647$	−35.2630	−1.38633	−0.693165	+	0.720779i	$0.743784\pi$
$648$	0	0
$649$	44.4099	1.74324
$650$	16.3378	0.640823
$651$	0	0
$652$	3.28936	0.128821
$653$	30.0593	1.17631	0.588155	−	0.808748i	$-0.299854\pi$
$653$	30.0593	1.17631	0.588155	+	0.808748i	$0.299854\pi$
$654$	0	0
$655$	−10.6681	−0.416836
$656$	0.0397604	0.00155238
$657$	0	0
$658$	4.89026	0.190642
$659$	18.9747	0.739148	0.369574	−	0.929201i	$-0.379504\pi$
$659$	18.9747	0.739148	0.369574	+	0.929201i	$0.379504\pi$
$660$	0	0
$661$	25.8038	1.00365	0.501825	−	0.864969i	$-0.332662\pi$
$661$	25.8038	1.00365	0.501825	+	0.864969i	$0.332662\pi$
$662$	−21.5869	−0.838997
$663$	0	0
$664$	12.9956	0.504327
$665$	−8.03191	−0.311464
$666$	0	0
$667$	−59.5753	−2.30676
$668$	−2.40697	−0.0931284
$669$	0	0
$670$	14.0886	0.544289
$671$	−31.8622	−1.23003
$672$	0	0
$673$	4.38401	0.168991	0.0844956	−	0.996424i	$-0.473072\pi$
$673$	4.38401	0.168991	0.0844956	+	0.996424i	$0.473072\pi$
$674$	−6.24979	−0.240733
$675$	0	0
$676$	−7.81174	−0.300452
$677$	32.1115	1.23415	0.617073	−	0.786906i	$-0.288318\pi$
$677$	32.1115	1.23415	0.617073	+	0.786906i	$0.288318\pi$
$678$	0	0
$679$	15.1668	0.582047
$680$	8.61289	0.330289
$681$	0	0
$682$	−14.3585	−0.549815
$683$	18.7093	0.715892	0.357946	−	0.933742i	$-0.383477\pi$
$683$	18.7093	0.715892	0.357946	+	0.933742i	$0.383477\pi$
$684$	0	0
$685$	−5.88738	−0.224945
$686$	0.877834	0.0335159
$687$	0	0
$688$	−0.0751073	−0.00286344
$689$	22.7722	0.867553
$690$	0	0
$691$	−40.9131	−1.55641	−0.778205	−	0.628011i	$-0.783869\pi$
$691$	−40.9131	−1.55641	−0.778205	+	0.628011i	$0.783869\pi$
$692$	7.39773	0.281220
$693$	0	0
$694$	1.32972	0.0504755
$695$	−14.2744	−0.541459
$696$	0	0
$697$	1.33675	0.0506331
$698$	−21.7647	−0.823807
$699$	0	0
$700$	5.20106	0.196582
$701$	17.6240	0.665649	0.332825	−	0.942989i	$-0.391998\pi$
$701$	17.6240	0.665649	0.332825	+	0.942989i	$0.391998\pi$
$702$	0	0
$703$	−25.5636	−0.964148
$704$	−23.9743	−0.903566
$705$	0	0
$706$	−10.3950	−0.391221
$707$	3.24786	0.122148
$708$	0	0
$709$	−8.18690	−0.307466	−0.153733	−	0.988112i	$-0.549130\pi$
$709$	−8.18690	−0.307466	−0.153733	+	0.988112i	$0.549130\pi$
$710$	31.8201	1.19419
$711$	0	0
$712$	37.9749	1.42317
$713$	−28.9345	−1.08361
$714$	0	0
$715$	−63.9130	−2.39021
$716$	2.15283	0.0804552
$717$	0	0
$718$	−23.4156	−0.873862
$719$	−42.6093	−1.58906	−0.794530	−	0.607225i	$-0.792283\pi$
$719$	−42.6093	−1.58906	−0.794530	+	0.607225i	$0.792283\pi$
$720$	0	0
$721$	12.7031	0.473090
$722$	10.5437	0.392397
$723$	0	0
$724$	−3.39919	−0.126330
$725$	−29.7956	−1.10658
$726$	0	0
$727$	−10.5470	−0.391166	−0.195583	−	0.980687i	$-0.562660\pi$
$727$	−10.5470	−0.391166	−0.195583	+	0.980687i	$0.562660\pi$
$728$	12.4716	0.462227
$729$	0	0
$730$	33.9124	1.25515
$731$	−2.52513	−0.0933951
$732$	0	0
$733$	6.61491	0.244327	0.122164	−	0.992510i	$-0.461017\pi$
$733$	6.61491	0.244327	0.122164	+	0.992510i	$0.461017\pi$
$734$	25.1848	0.929589
$735$	0	0
$736$	−47.7387	−1.75967
$737$	−25.2598	−0.930457
$738$	0	0
$739$	−0.834376	−0.0306930	−0.0153465	−	0.999882i	$-0.504885\pi$
$739$	−0.834376	−0.0306930	−0.0153465	+	0.999882i	$0.504885\pi$
$740$	36.1181	1.32773
$741$	0	0
$742$	−4.54393	−0.166813
$743$	−52.9082	−1.94101	−0.970506	−	0.241077i	$-0.922499\pi$
$743$	−52.9082	−1.94101	−0.970506	+	0.241077i	$0.922499\pi$
$744$	0	0
$745$	12.8868	0.472136
$746$	23.6090	0.864387
$747$	0	0
$748$	−5.87875	−0.214948
$749$	3.91410	0.143018
$750$	0	0
$751$	35.3681	1.29060	0.645300	−	0.763929i	$-0.276732\pi$
$751$	35.3681	1.29060	0.645300	+	0.763929i	$0.276732\pi$
$752$	−0.165699	−0.00604241
$753$	0	0
$754$	−27.1992	−0.990535
$755$	−37.8508	−1.37753
$756$	0	0
$757$	−19.6550	−0.714372	−0.357186	−	0.934033i	$-0.616264\pi$
$757$	−19.6550	−0.714372	−0.357186	+	0.934033i	$0.616264\pi$
$758$	−0.843118	−0.0306234
$759$	0	0
$760$	22.7695	0.825938
$761$	6.80776	0.246781	0.123390	−	0.992358i	$-0.460623\pi$
$761$	6.80776	0.246781	0.123390	+	0.992358i	$0.460623\pi$
$762$	0	0
$763$	3.56356	0.129009
$764$	23.1695	0.838243
$765$	0	0
$766$	−9.64851	−0.348615
$767$	40.8579	1.47529
$768$	0	0
$769$	44.3645	1.59983	0.799913	−	0.600116i	$-0.204879\pi$
$769$	44.3645	1.59983	0.799913	+	0.600116i	$0.204879\pi$
$770$	12.7531	0.459590
$771$	0	0
$772$	−14.6431	−0.527015
$773$	25.6974	0.924270	0.462135	−	0.886810i	$-0.347084\pi$
$773$	25.6974	0.924270	0.462135	+	0.886810i	$0.347084\pi$
$774$	0	0
$775$	−14.4711	−0.519819
$776$	−42.9960	−1.54347
$777$	0	0
$778$	−22.3179	−0.800137
$779$	3.53392	0.126616
$780$	0	0
$781$	−57.0513	−2.04146
$782$	7.42544	0.265533
$783$	0	0
$784$	−0.0297440	−0.00106229
$785$	53.6153	1.91361
$786$	0	0
$787$	25.6566	0.914558	0.457279	−	0.889323i	$-0.348824\pi$
$787$	25.6566	0.914558	0.457279	+	0.889323i	$0.348824\pi$
$788$	−21.2296	−0.756274
$789$	0	0
$790$	5.17538	0.184132
$791$	0.966622	0.0343691
$792$	0	0
$793$	−29.3138	−1.04096
$794$	13.1798	0.467735
$795$	0	0
$796$	18.7425	0.664310
$797$	−20.3683	−0.721483	−0.360741	−	0.932666i	$-0.617476\pi$
$797$	−20.3683	−0.721483	−0.360741	+	0.932666i	$0.617476\pi$
$798$	0	0
$799$	−5.57082	−0.197082
$800$	−23.8757	−0.844135
$801$	0	0
$802$	−9.71373	−0.343004
$803$	−60.8026	−2.14568
$804$	0	0
$805$	25.6994	0.905785
$806$	−13.2101	−0.465305
$807$	0	0
$808$	−9.20731	−0.323912
$809$	36.5480	1.28496	0.642480	−	0.766303i	$-0.277906\pi$
$809$	36.5480	1.28496	0.642480	+	0.766303i	$0.277906\pi$
$810$	0	0
$811$	7.33037	0.257404	0.128702	−	0.991683i	$-0.458919\pi$
$811$	7.33037	0.257404	0.128702	+	0.991683i	$0.458919\pi$
$812$	−8.65869	−0.303861
$813$	0	0
$814$	40.5899	1.42268
$815$	8.12886	0.284742
$816$	0	0
$817$	−6.67557	−0.233549
$818$	−25.0754	−0.876739
$819$	0	0
$820$	−4.99299	−0.174363
$821$	−1.76321	−0.0615366	−0.0307683	−	0.999527i	$-0.509795\pi$
$821$	−1.76321	−0.0615366	−0.0307683	+	0.999527i	$0.509795\pi$
$822$	0	0
$823$	0.636323	0.0221808	0.0110904	−	0.999938i	$-0.496470\pi$
$823$	0.636323	0.0221808	0.0110904	+	0.999938i	$0.496470\pi$
$824$	−36.0119	−1.25454
$825$	0	0
$826$	−8.15272	−0.283670
$827$	6.81869	0.237109	0.118554	−	0.992948i	$-0.462174\pi$
$827$	6.81869	0.237109	0.118554	+	0.992948i	$0.462174\pi$
$828$	0	0
$829$	50.9480	1.76950	0.884749	−	0.466068i	$-0.154330\pi$
$829$	50.9480	1.76950	0.884749	+	0.466068i	$0.154330\pi$
$830$	12.2261	0.424373
$831$	0	0
$832$	−22.0568	−0.764683
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−5.94824	−0.205847
$836$	−15.5414	−0.537511
$837$	0	0
$838$	−1.74308	−0.0602137
$839$	−2.95064	−0.101867	−0.0509336	−	0.998702i	$-0.516220\pi$
$839$	−2.95064	−0.101867	−0.0509336	+	0.998702i	$0.516220\pi$
$840$	0	0
$841$	20.6036	0.710470
$842$	−22.1200	−0.762307
$843$	0	0
$844$	−10.4663	−0.360266
$845$	−19.3048	−0.664106
$846$	0	0
$847$	−11.8654	−0.407700
$848$	0.153964	0.00528714
$849$	0	0
$850$	3.71372	0.127379
$851$	81.7948	2.80389
$852$	0	0
$853$	−28.8764	−0.988709	−0.494355	−	0.869260i	$-0.664596\pi$
$853$	−28.8764	−0.988709	−0.494355	+	0.869260i	$0.664596\pi$
$854$	5.84923	0.200156
$855$	0	0
$856$	−11.0960	−0.379254
$857$	9.83977	0.336120	0.168060	−	0.985777i	$-0.446250\pi$
$857$	9.83977	0.336120	0.168060	+	0.985777i	$0.446250\pi$
$858$	0	0
$859$	−11.1344	−0.379900	−0.189950	−	0.981794i	$-0.560833\pi$
$859$	−11.1344	−0.379900	−0.189950	+	0.981794i	$0.560833\pi$
$860$	9.43175	0.321620
$861$	0	0
$862$	−24.1390	−0.822178
$863$	−22.7030	−0.772818	−0.386409	−	0.922328i	$-0.626285\pi$
$863$	−22.7030	−0.772818	−0.386409	+	0.922328i	$0.626285\pi$
$864$	0	0
$865$	18.2817	0.621596
$866$	−24.7529	−0.841137
$867$	0	0
$868$	−4.20535	−0.142739
$869$	−9.27910	−0.314772
$870$	0	0
$871$	−23.2395	−0.787440
$872$	−10.1023	−0.342106
$873$	0	0
$874$	19.6303	0.664005
$875$	−2.33775	−0.0790304
$876$	0	0
$877$	1.55717	0.0525819	0.0262910	−	0.999654i	$-0.491630\pi$
$877$	1.55717	0.0525819	0.0262910	+	0.999654i	$0.491630\pi$
$878$	−5.54175	−0.187025
$879$	0	0
$880$	−0.432118	−0.0145667
$881$	10.3326	0.348113	0.174056	−	0.984736i	$-0.444313\pi$
$881$	10.3326	0.348113	0.174056	+	0.984736i	$0.444313\pi$
$882$	0	0
$883$	23.6160	0.794743	0.397371	−	0.917658i	$-0.369922\pi$
$883$	23.6160	0.794743	0.397371	+	0.917658i	$0.369922\pi$
$884$	−5.40856	−0.181910
$885$	0	0
$886$	23.2237	0.780215
$887$	−13.9260	−0.467588	−0.233794	−	0.972286i	$-0.575114\pi$
$887$	−13.9260	−0.467588	−0.233794	+	0.972286i	$0.575114\pi$
$888$	0	0
$889$	5.55875	0.186435
$890$	35.7263	1.19755
$891$	0	0
$892$	3.06237	0.102536
$893$	−14.7274	−0.492832
$894$	0	0
$895$	5.32021	0.177835
$896$	−6.88613	−0.230050
$897$	0	0
$898$	−30.3340	−1.01226
$899$	24.0915	0.803495
$900$	0	0
$901$	5.17630	0.172448
$902$	−5.61117	−0.186831
$903$	0	0
$904$	−2.74026	−0.0911398
$905$	−8.40028	−0.279235
$906$	0	0
$907$	11.3628	0.377295	0.188648	−	0.982045i	$-0.439590\pi$
$907$	11.3628	0.377295	0.188648	+	0.982045i	$0.439590\pi$
$908$	−20.8805	−0.692943
$909$	0	0
$910$	11.7331	0.388948
$911$	−4.75507	−0.157543	−0.0787713	−	0.996893i	$-0.525100\pi$
$911$	−4.75507	−0.157543	−0.0787713	+	0.996893i	$0.525100\pi$
$912$	0	0
$913$	−21.9205	−0.725462
$914$	−23.3435	−0.772136
$915$	0	0
$916$	20.0766	0.663349
$917$	−3.51134	−0.115955
$918$	0	0
$919$	−32.6303	−1.07637	−0.538187	−	0.842825i	$-0.680891\pi$
$919$	−32.6303	−1.07637	−0.538187	+	0.842825i	$0.680891\pi$
$920$	−72.8548	−2.40195
$921$	0	0
$922$	−0.272303	−0.00896784
$923$	−52.4882	−1.72767
$924$	0	0
$925$	40.9084	1.34506
$926$	−19.7209	−0.648070
$927$	0	0
$928$	39.7482	1.30480
$929$	47.3546	1.55365	0.776827	−	0.629714i	$-0.216828\pi$
$929$	47.3546	1.55365	0.776827	+	0.629714i	$0.216828\pi$
$930$	0	0
$931$	−2.64366	−0.0866424
$932$	−7.10382	−0.232693
$933$	0	0
$934$	−10.1939	−0.333554
$935$	−14.5279	−0.475113
$936$	0	0
$937$	−51.6525	−1.68741	−0.843707	−	0.536803i	$-0.819632\pi$
$937$	−51.6525	−1.68741	−0.843707	+	0.536803i	$0.819632\pi$
$938$	4.63717	0.151409
$939$	0	0
$940$	20.8079	0.678680
$941$	54.7240	1.78395	0.891975	−	0.452084i	$-0.149319\pi$
$941$	54.7240	1.78395	0.891975	+	0.452084i	$0.149319\pi$
$942$	0	0
$943$	−11.3073	−0.368218
$944$	0.276242	0.00899091
$945$	0	0
$946$	10.5995	0.344619
$947$	36.9728	1.20145	0.600727	−	0.799454i	$-0.294878\pi$
$947$	36.9728	1.20145	0.600727	+	0.799454i	$0.294878\pi$
$948$	0	0
$949$	−55.9395	−1.81587
$950$	9.81779	0.318531
$951$	0	0
$952$	2.83488	0.0918791
$953$	7.91313	0.256331	0.128166	−	0.991753i	$-0.459091\pi$
$953$	7.91313	0.256331	0.128166	+	0.991753i	$0.459091\pi$
$954$	0	0
$955$	57.2578	1.85282
$956$	9.15017	0.295938
$957$	0	0
$958$	27.9130	0.901830
$959$	−1.93780	−0.0625747
$960$	0	0
$961$	−19.2993	−0.622557
$962$	37.3435	1.20400
$963$	0	0
$964$	−26.9284	−0.867306
$965$	−36.1867	−1.16489
$966$	0	0
$967$	8.05906	0.259162	0.129581	−	0.991569i	$-0.458637\pi$
$967$	8.05906	0.259162	0.129581	+	0.991569i	$0.458637\pi$
$968$	33.6371	1.08114
$969$	0	0
$970$	−40.4500	−1.29877
$971$	1.81193	0.0581476	0.0290738	−	0.999577i	$-0.490744\pi$
$971$	1.81193	0.0581476	0.0290738	+	0.999577i	$0.490744\pi$
$972$	0	0
$973$	−4.69834	−0.150622
$974$	20.2047	0.647401
$975$	0	0
$976$	−0.198192	−0.00634396
$977$	−15.1934	−0.486079	−0.243040	−	0.970016i	$-0.578145\pi$
$977$	−15.1934	−0.486079	−0.243040	+	0.970016i	$0.578145\pi$
$978$	0	0
$979$	−64.0547	−2.04720
$980$	3.73516	0.119315
$981$	0	0
$982$	−29.2474	−0.933321
$983$	−11.6472	−0.371488	−0.185744	−	0.982598i	$-0.559470\pi$
$983$	−11.6472	−0.371488	−0.185744	+	0.982598i	$0.559470\pi$
$984$	0	0
$985$	−52.4638	−1.67164
$986$	−6.18257	−0.196893
$987$	0	0
$988$	−14.2984	−0.454892
$989$	21.3596	0.679195
$990$	0	0
$991$	−47.1759	−1.49859	−0.749296	−	0.662235i	$-0.769608\pi$
$991$	−47.1759	−1.49859	−0.749296	+	0.662235i	$0.769608\pi$
$992$	19.3049	0.612931
$993$	0	0
$994$	10.4734	0.332197
$995$	46.3175	1.46836
$996$	0	0
$997$	39.7574	1.25913	0.629564	−	0.776949i	$-0.283234\pi$
$997$	39.7574	1.25913	0.629564	+	0.776949i	$0.283234\pi$
$998$	−32.5006	−1.02879
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1071.2.a.m.1.3		5
3.2	odd	2		119.2.a.b.1.3	✓	5
7.6	odd	2		7497.2.a.br.1.3		5
12.11	even	2		1904.2.a.t.1.2		5
15.14	odd	2		2975.2.a.m.1.3		5
21.2	odd	6		833.2.e.i.18.3		10
21.5	even	6		833.2.e.h.18.3		10
21.11	odd	6		833.2.e.i.324.3		10
21.17	even	6		833.2.e.h.324.3		10
21.20	even	2		833.2.a.g.1.3		5
24.5	odd	2		7616.2.a.bt.1.2		5
24.11	even	2		7616.2.a.bq.1.4		5
51.50	odd	2		2023.2.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.2.a.b.1.3	✓	5	3.2	odd	2
833.2.a.g.1.3		5	21.20	even	2
833.2.e.h.18.3		10	21.5	even	6
833.2.e.h.324.3		10	21.17	even	6
833.2.e.i.18.3		10	21.2	odd	6
833.2.e.i.324.3		10	21.11	odd	6
1071.2.a.m.1.3		5	1.1	even	1	trivial
1904.2.a.t.1.2		5	12.11	even	2
2023.2.a.j.1.3		5	51.50	odd	2
2975.2.a.m.1.3		5	15.14	odd	2
7497.2.a.br.1.3		5	7.6	odd	2
7616.2.a.bq.1.4		5	24.11	even	2
7616.2.a.bt.1.2		5	24.5	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:	\( N \)	\(=\)	\( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1071.a (trivial)

\(f(q)\)	\(=\)	\(q-0.877834 q^{2} -1.22941 q^{4} -3.03818 q^{5} -1.00000 q^{7} +2.83488 q^{8} +O(q^{10})\)	\(q-0.877834 q^{2} -1.22941 q^{4} -3.03818 q^{5} -1.00000 q^{7} +2.83488 q^{8} +2.66702 q^{10} -4.78178 q^{11} -4.39933 q^{13} +0.877834 q^{14} -0.0297440 q^{16} -1.00000 q^{17} -2.64366 q^{19} +3.73516 q^{20} +4.19761 q^{22} +8.45881 q^{23} +4.23054 q^{25} +3.86188 q^{26} +1.22941 q^{28} -7.04298 q^{29} -3.42063 q^{31} -5.64366 q^{32} +0.877834 q^{34} +3.03818 q^{35} +9.66977 q^{37} +2.32069 q^{38} -8.61289 q^{40} -1.33675 q^{41} +2.52513 q^{43} +5.87875 q^{44} -7.42544 q^{46} +5.57082 q^{47} +1.00000 q^{49} -3.71372 q^{50} +5.40856 q^{52} -5.17630 q^{53} +14.5279 q^{55} -2.83488 q^{56} +6.18257 q^{58} -9.28732 q^{59} +6.66325 q^{61} +3.00275 q^{62} +5.01368 q^{64} +13.3659 q^{65} +5.28251 q^{67} +1.22941 q^{68} -2.66702 q^{70} +11.9310 q^{71} +12.7155 q^{73} -8.48845 q^{74} +3.25013 q^{76} +4.78178 q^{77} +1.94051 q^{79} +0.0903677 q^{80} +1.17345 q^{82} +4.58417 q^{83} +3.03818 q^{85} -2.21664 q^{86} -13.5558 q^{88} +13.3956 q^{89} +4.39933 q^{91} -10.3993 q^{92} -4.89026 q^{94} +8.03191 q^{95} -15.1668 q^{97} -0.877834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 10 q^{4} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 10 * q^4 - 5 * q^7 - 6 * q^8	\( 5 q - 2 q^{2} + 10 q^{4} - 5 q^{7} - 6 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{14} + 4 q^{16} - 5 q^{17} + 6 q^{19} + 19 q^{20} + 6 q^{22} + 10 q^{23} + 21 q^{25} + 26 q^{26} - 10 q^{28} + 8 q^{29} - 9 q^{32} + 2 q^{34} + 8 q^{37} - 14 q^{38} + 5 q^{40} - 18 q^{41} + 8 q^{43} + 14 q^{44} + 8 q^{46} + 10 q^{47} + 5 q^{49} - 27 q^{50} + 4 q^{52} - 4 q^{53} - 24 q^{55} + 6 q^{56} + 12 q^{58} - 8 q^{59} + 22 q^{61} - 16 q^{62} - 16 q^{64} + 30 q^{65} + 16 q^{67} - 10 q^{68} - 4 q^{70} + 2 q^{71} + 10 q^{73} + 40 q^{74} + 24 q^{76} - 2 q^{77} + 18 q^{79} + 4 q^{80} - 31 q^{82} + 12 q^{83} - 23 q^{86} - 46 q^{88} - 20 q^{89} - 2 q^{91} - 28 q^{92} - 42 q^{94} + 22 q^{95} + 12 q^{97} - 2 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 10 * q^4 - 5 * q^7 - 6 * q^8 + 4 * q^10 + 2 * q^11 + 2 * q^13 + 2 * q^14 + 4 * q^16 - 5 * q^17 + 6 * q^19 + 19 * q^20 + 6 * q^22 + 10 * q^23 + 21 * q^25 + 26 * q^26 - 10 * q^28 + 8 * q^29 - 9 * q^32 + 2 * q^34 + 8 * q^37 - 14 * q^38 + 5 * q^40 - 18 * q^41 + 8 * q^43 + 14 * q^44 + 8 * q^46 + 10 * q^47 + 5 * q^49 - 27 * q^50 + 4 * q^52 - 4 * q^53 - 24 * q^55 + 6 * q^56 + 12 * q^58 - 8 * q^59 + 22 * q^61 - 16 * q^62 - 16 * q^64 + 30 * q^65 + 16 * q^67 - 10 * q^68 - 4 * q^70 + 2 * q^71 + 10 * q^73 + 40 * q^74 + 24 * q^76 - 2 * q^77 + 18 * q^79 + 4 * q^80 - 31 * q^82 + 12 * q^83 - 23 * q^86 - 46 * q^88 - 20 * q^89 - 2 * q^91 - 28 * q^92 - 42 * q^94 + 22 * q^95 + 12 * q^97 - 2 * q^98

Introduction

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