LMFDB - Embedded newform 3822.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.80560$ of defining polynomial
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.80560 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.80560 q^{10} +5.28822 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.80560 q^{15} +1.00000 q^{16} -6.15962 q^{17} -1.00000 q^{18} -5.48261 q^{19} -3.80560 q^{20} -5.28822 q^{22} +6.28822 q^{23} -1.00000 q^{24} +9.48261 q^{25} +1.00000 q^{26} +1.00000 q^{27} -3.00000 q^{29} +3.80560 q^{30} +8.48261 q^{31} -1.00000 q^{32} +5.28822 q^{33} +6.15962 q^{34} +1.00000 q^{36} +1.32299 q^{37} +5.48261 q^{38} -1.00000 q^{39} +3.80560 q^{40} -5.32299 q^{41} +1.61121 q^{43} +5.28822 q^{44} -3.80560 q^{45} -6.28822 q^{46} -7.09382 q^{47} +1.00000 q^{48} -9.48261 q^{50} -6.15962 q^{51} -1.00000 q^{52} -0.354020 q^{53} -1.00000 q^{54} -20.1248 q^{55} -5.48261 q^{57} +3.00000 q^{58} +1.67701 q^{59} -3.80560 q^{60} +9.77083 q^{61} -8.48261 q^{62} +1.00000 q^{64} +3.80560 q^{65} -5.28822 q^{66} -8.44784 q^{67} -6.15962 q^{68} +6.28822 q^{69} -15.4826 q^{71} -1.00000 q^{72} -1.71178 q^{73} -1.32299 q^{74} +9.48261 q^{75} -5.48261 q^{76} +1.00000 q^{78} +2.87141 q^{79} -3.80560 q^{80} +1.00000 q^{81} +5.32299 q^{82} +7.44784 q^{83} +23.4411 q^{85} -1.61121 q^{86} -3.00000 q^{87} -5.28822 q^{88} -10.6770 q^{89} +3.80560 q^{90} +6.28822 q^{92} +8.48261 q^{93} +7.09382 q^{94} +20.8646 q^{95} -1.00000 q^{96} +3.15962 q^{97} +5.28822 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 3 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} - 6 q^{19} - 3 q^{20} + 3 q^{22} - 3 q^{24}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 + 3 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 6 * q^17 - 3 * q^18 - 6 * q^19 - 3 * q^20 + 3 * q^22 - 3 * q^24 + 18 * q^25 + 3 * q^26 + 3 * q^27 - 9 * q^29 + 3 * q^30 + 15 * q^31 - 3 * q^32 - 3 * q^33 + 6 * q^34 + 3 * q^36 + 6 * q^37 + 6 * q^38 - 3 * q^39 + 3 * q^40 - 18 * q^41 - 12 * q^43 - 3 * q^44 - 3 * q^45 + 6 * q^47 + 3 * q^48 - 18 * q^50 - 6 * q^51 - 3 * q^52 + 3 * q^53 - 3 * q^54 - 27 * q^55 - 6 * q^57 + 9 * q^58 + 3 * q^59 - 3 * q^60 - 15 * q^62 + 3 * q^64 + 3 * q^65 + 3 * q^66 + 6 * q^67 - 6 * q^68 - 36 * q^71 - 3 * q^72 - 24 * q^73 - 6 * q^74 + 18 * q^75 - 6 * q^76 + 3 * q^78 + 15 * q^79 - 3 * q^80 + 3 * q^81 + 18 * q^82 - 9 * q^83 - 24 * q^85 + 12 * q^86 - 9 * q^87 + 3 * q^88 - 30 * q^89 + 3 * q^90 + 15 * q^93 - 6 * q^94 + 6 * q^95 - 3 * q^96 - 3 * q^97 - 3 * q^99

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.80560	−1.70192	−0.850959	−	0.525233i	$-0.823978\pi$
$5$	−3.80560	−1.70192	−0.850959	+	0.525233i	$0.823978\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.80560	1.20344
$11$	5.28822	1.59446	0.797229	−	0.603678i	$-0.206299\pi$
$11$	5.28822	1.59446	0.797229	+	0.603678i	$0.206299\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.80560	−0.982602
$16$	1.00000	0.250000
$17$	−6.15962	−1.49393	−0.746964	−	0.664864i	$-0.768489\pi$
$17$	−6.15962	−1.49393	−0.746964	+	0.664864i	$0.768489\pi$
$18$	−1.00000	−0.235702
$19$	−5.48261	−1.25780	−0.628899	−	0.777487i	$-0.716494\pi$
$19$	−5.48261	−1.25780	−0.628899	+	0.777487i	$0.716494\pi$
$20$	−3.80560	−0.850959
$21$	0	0
$22$	−5.28822	−1.12745
$23$	6.28822	1.31118	0.655592	−	0.755115i	$-0.272419\pi$
$23$	6.28822	1.31118	0.655592	+	0.755115i	$0.272419\pi$
$24$	−1.00000	−0.204124
$25$	9.48261	1.89652
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	3.80560	0.694805
$31$	8.48261	1.52352	0.761761	−	0.647858i	$-0.224335\pi$
$31$	8.48261	1.52352	0.761761	+	0.647858i	$0.224335\pi$
$32$	−1.00000	−0.176777
$33$	5.28822	0.920560
$34$	6.15962	1.05637
$35$	0	0
$36$	1.00000	0.166667
$37$	1.32299	0.217498	0.108749	−	0.994069i	$-0.465315\pi$
$37$	1.32299	0.217498	0.108749	+	0.994069i	$0.465315\pi$
$38$	5.48261	0.889397
$39$	−1.00000	−0.160128
$40$	3.80560	0.601719
$41$	−5.32299	−0.831311	−0.415656	−	0.909522i	$-0.636448\pi$
$41$	−5.32299	−0.831311	−0.415656	+	0.909522i	$0.636448\pi$
$42$	0	0
$43$	1.61121	0.245707	0.122853	−	0.992425i	$-0.460796\pi$
$43$	1.61121	0.245707	0.122853	+	0.992425i	$0.460796\pi$
$44$	5.28822	0.797229
$45$	−3.80560	−0.567306
$46$	−6.28822	−0.927147
$47$	−7.09382	−1.03474	−0.517370	−	0.855762i	$-0.673089\pi$
$47$	−7.09382	−1.03474	−0.517370	+	0.855762i	$0.673089\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−9.48261	−1.34104
$51$	−6.15962	−0.862520
$52$	−1.00000	−0.138675
$53$	−0.354020	−0.0486284	−0.0243142	−	0.999704i	$-0.507740\pi$
$53$	−0.354020	−0.0486284	−0.0243142	+	0.999704i	$0.507740\pi$
$54$	−1.00000	−0.136083
$55$	−20.1248	−2.71363
$56$	0	0
$57$	−5.48261	−0.726190
$58$	3.00000	0.393919
$59$	1.67701	0.218328	0.109164	−	0.994024i	$-0.465183\pi$
$59$	1.67701	0.218328	0.109164	+	0.994024i	$0.465183\pi$
$60$	−3.80560	−0.491301
$61$	9.77083	1.25103	0.625513	−	0.780214i	$-0.284890\pi$
$61$	9.77083	1.25103	0.625513	+	0.780214i	$0.284890\pi$
$62$	−8.48261	−1.07729
$63$	0	0
$64$	1.00000	0.125000
$65$	3.80560	0.472027
$66$	−5.28822	−0.650934
$67$	−8.44784	−1.03207	−0.516034	−	0.856568i	$-0.672592\pi$
$67$	−8.44784	−1.03207	−0.516034	+	0.856568i	$0.672592\pi$
$68$	−6.15962	−0.746964
$69$	6.28822	0.757012
$70$	0	0
$71$	−15.4826	−1.83745	−0.918724	−	0.394900i	$-0.870779\pi$
$71$	−15.4826	−1.83745	−0.918724	+	0.394900i	$0.870779\pi$
$72$	−1.00000	−0.117851
$73$	−1.71178	−0.200349	−0.100175	−	0.994970i	$-0.531940\pi$
$73$	−1.71178	−0.200349	−0.100175	+	0.994970i	$0.531940\pi$
$74$	−1.32299	−0.153794
$75$	9.48261	1.09496
$76$	−5.48261	−0.628899
$77$	0	0
$78$	1.00000	0.113228
$79$	2.87141	0.323059	0.161529	−	0.986868i	$-0.448357\pi$
$79$	2.87141	0.323059	0.161529	+	0.986868i	$0.448357\pi$
$80$	−3.80560	−0.425479
$81$	1.00000	0.111111
$82$	5.32299	0.587826
$83$	7.44784	0.817507	0.408753	−	0.912645i	$-0.365964\pi$
$83$	7.44784	0.817507	0.408753	+	0.912645i	$0.365964\pi$
$84$	0	0
$85$	23.4411	2.54254
$86$	−1.61121	−0.173741
$87$	−3.00000	−0.321634
$88$	−5.28822	−0.563726
$89$	−10.6770	−1.13176	−0.565880	−	0.824487i	$-0.691464\pi$
$89$	−10.6770	−1.13176	−0.565880	+	0.824487i	$0.691464\pi$
$90$	3.80560	0.401146
$91$	0	0
$92$	6.28822	0.655592
$93$	8.48261	0.879606
$94$	7.09382	0.731672
$95$	20.8646	2.14067
$96$	−1.00000	−0.102062
$97$	3.15962	0.320811	0.160406	−	0.987051i	$-0.448720\pi$
$97$	3.15962	0.320811	0.160406	+	0.987051i	$0.448720\pi$
$98$	0	0
$99$	5.28822	0.531486
$100$	9.48261	0.948261
$101$	4.96523	0.494058	0.247029	−	0.969008i	$-0.420546\pi$
$101$	4.96523	0.494058	0.247029	+	0.969008i	$0.420546\pi$
$102$	6.15962	0.609894
$103$	5.61121	0.552889	0.276444	−	0.961030i	$-0.410844\pi$
$103$	5.61121	0.552889	0.276444	+	0.961030i	$0.410844\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0.354020	0.0343855
$107$	−17.0590	−1.64916	−0.824580	−	0.565745i	$-0.808589\pi$
$107$	−17.0590	−1.64916	−0.824580	+	0.565745i	$0.808589\pi$
$108$	1.00000	0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	20.1248	1.91883
$111$	1.32299	0.125573
$112$	0	0
$113$	−5.45158	−0.512842	−0.256421	−	0.966565i	$-0.582543\pi$
$113$	−5.45158	−0.512842	−0.256421	+	0.966565i	$0.582543\pi$
$114$	5.48261	0.513494
$115$	−23.9305	−2.23153
$116$	−3.00000	−0.278543
$117$	−1.00000	−0.0924500
$118$	−1.67701	−0.154381
$119$	0	0
$120$	3.80560	0.347402
$121$	16.9652	1.54229
$122$	−9.77083	−0.884609
$123$	−5.32299	−0.479958
$124$	8.48261	0.761761
$125$	−17.0590	−1.52581
$126$	0	0
$127$	−10.1248	−0.898435	−0.449218	−	0.893422i	$-0.648297\pi$
$127$	−10.1248	−0.898435	−0.449218	+	0.893422i	$0.648297\pi$
$128$	−1.00000	−0.0883883
$129$	1.61121	0.141859
$130$	−3.80560	−0.333773
$131$	−10.0628	−0.879190	−0.439595	−	0.898196i	$-0.644878\pi$
$131$	−10.0628	−0.879190	−0.439595	+	0.898196i	$0.644878\pi$
$132$	5.28822	0.460280
$133$	0	0
$134$	8.44784	0.729782
$135$	−3.80560	−0.327534
$136$	6.15962	0.528183
$137$	−6.67701	−0.570455	−0.285228	−	0.958460i	$-0.592069\pi$
$137$	−6.67701	−0.570455	−0.285228	+	0.958460i	$0.592069\pi$
$138$	−6.28822	−0.535288
$139$	12.2882	1.04227	0.521136	−	0.853473i	$-0.325508\pi$
$139$	12.2882	1.04227	0.521136	+	0.853473i	$0.325508\pi$
$140$	0	0
$141$	−7.09382	−0.597407
$142$	15.4826	1.29927
$143$	−5.28822	−0.442223
$144$	1.00000	0.0833333
$145$	11.4168	0.948114
$146$	1.71178	0.141668
$147$	0	0
$148$	1.32299	0.108749
$149$	−18.2882	−1.49823	−0.749115	−	0.662441i	$-0.769521\pi$
$149$	−18.2882	−1.49823	−0.749115	+	0.662441i	$0.769521\pi$
$150$	−9.48261	−0.774252
$151$	−15.2882	−1.24414	−0.622069	−	0.782963i	$-0.713708\pi$
$151$	−15.2882	−1.24414	−0.622069	+	0.782963i	$0.713708\pi$
$152$	5.48261	0.444699
$153$	−6.15962	−0.497976
$154$	0	0
$155$	−32.2815	−2.59291
$156$	−1.00000	−0.0800641
$157$	−10.4168	−0.831352	−0.415676	−	0.909513i	$-0.636455\pi$
$157$	−10.4168	−0.831352	−0.415676	+	0.909513i	$0.636455\pi$
$158$	−2.87141	−0.228437
$159$	−0.354020	−0.0280756
$160$	3.80560	0.300859
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−9.73980	−0.762880	−0.381440	−	0.924394i	$-0.624572\pi$
$163$	−9.73980	−0.762880	−0.381440	+	0.924394i	$0.624572\pi$
$164$	−5.32299	−0.415656
$165$	−20.1248	−1.56672
$166$	−7.44784	−0.578064
$167$	8.70502	0.673615	0.336808	−	0.941574i	$-0.390653\pi$
$167$	8.70502	0.673615	0.336808	+	0.941574i	$0.390653\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−23.4411	−1.79785
$171$	−5.48261	−0.419266
$172$	1.61121	0.122853
$173$	12.0590	0.916832	0.458416	−	0.888738i	$-0.348417\pi$
$173$	12.0590	0.916832	0.458416	+	0.888738i	$0.348417\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	5.28822	0.398614
$177$	1.67701	0.126052
$178$	10.6770	0.800276
$179$	−4.25719	−0.318197	−0.159098	−	0.987263i	$-0.550859\pi$
$179$	−4.25719	−0.318197	−0.159098	+	0.987263i	$0.550859\pi$
$180$	−3.80560	−0.283653
$181$	−12.1596	−0.903818	−0.451909	−	0.892064i	$-0.649257\pi$
$181$	−12.1596	−0.903818	−0.451909	+	0.892064i	$0.649257\pi$
$182$	0	0
$183$	9.77083	0.722280
$184$	−6.28822	−0.463573
$185$	−5.03477	−0.370164
$186$	−8.48261	−0.621975
$187$	−32.5734	−2.38200
$188$	−7.09382	−0.517370
$189$	0	0
$190$	−20.8646	−1.51368
$191$	−24.1876	−1.75016	−0.875078	−	0.483982i	$-0.839190\pi$
$191$	−24.1876	−1.75016	−0.875078	+	0.483982i	$0.839190\pi$
$192$	1.00000	0.0721688
$193$	1.54842	0.111458	0.0557288	−	0.998446i	$-0.482252\pi$
$193$	1.54842	0.111458	0.0557288	+	0.998446i	$0.482252\pi$
$194$	−3.15962	−0.226848
$195$	3.80560	0.272525
$196$	0	0
$197$	−18.2882	−1.30298	−0.651491	−	0.758657i	$-0.725856\pi$
$197$	−18.2882	−1.30298	−0.651491	+	0.758657i	$0.725856\pi$
$198$	−5.28822	−0.375817
$199$	26.2187	1.85859	0.929296	−	0.369336i	$-0.120415\pi$
$199$	26.2187	1.85859	0.929296	+	0.369336i	$0.120415\pi$
$200$	−9.48261	−0.670522
$201$	−8.44784	−0.595865
$202$	−4.96523	−0.349352
$203$	0	0
$204$	−6.15962	−0.431260
$205$	20.2572	1.41482
$206$	−5.61121	−0.390951
$207$	6.28822	0.437061
$208$	−1.00000	−0.0693375
$209$	−28.9932	−2.00550
$210$	0	0
$211$	−9.89942	−0.681504	−0.340752	−	0.940153i	$-0.610682\pi$
$211$	−9.89942	−0.681504	−0.340752	+	0.940153i	$0.610682\pi$
$212$	−0.354020	−0.0243142
$213$	−15.4826	−1.06085
$214$	17.0590	1.16613
$215$	−6.13161	−0.418172
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−4.00000	−0.270914
$219$	−1.71178	−0.115672
$220$	−20.1248	−1.35682
$221$	6.15962	0.414341
$222$	−1.32299	−0.0887933
$223$	−2.96897	−0.198817	−0.0994085	−	0.995047i	$-0.531695\pi$
$223$	−2.96897	−0.198817	−0.0994085	+	0.995047i	$0.531695\pi$
$224$	0	0
$225$	9.48261	0.632174
$226$	5.45158	0.362634
$227$	−18.4131	−1.22212	−0.611059	−	0.791585i	$-0.709256\pi$
$227$	−18.4131	−1.22212	−0.611059	+	0.791585i	$0.709256\pi$
$228$	−5.48261	−0.363095
$229$	1.42357	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$229$	1.42357	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$230$	23.9305	1.57793
$231$	0	0
$232$	3.00000	0.196960
$233$	10.7361	0.703342	0.351671	−	0.936124i	$-0.385614\pi$
$233$	10.7361	0.703342	0.351671	+	0.936124i	$0.385614\pi$
$234$	1.00000	0.0653720
$235$	26.9963	1.76104
$236$	1.67701	0.109164
$237$	2.87141	0.186518
$238$	0	0
$239$	2.12859	0.137687	0.0688436	−	0.997627i	$-0.478069\pi$
$239$	2.12859	0.137687	0.0688436	+	0.997627i	$0.478069\pi$
$240$	−3.80560	−0.245651
$241$	−18.7398	−1.20714	−0.603568	−	0.797311i	$-0.706255\pi$
$241$	−18.7398	−1.20714	−0.603568	+	0.797311i	$0.706255\pi$
$242$	−16.9652	−1.09057
$243$	1.00000	0.0641500
$244$	9.77083	0.625513
$245$	0	0
$246$	5.32299	0.339381
$247$	5.48261	0.348850
$248$	−8.48261	−0.538646
$249$	7.44784	0.471988
$250$	17.0590	1.07891
$251$	7.77457	0.490727	0.245363	−	0.969431i	$-0.421093\pi$
$251$	7.77457	0.490727	0.245363	+	0.969431i	$0.421093\pi$
$252$	0	0
$253$	33.2534	2.09063
$254$	10.1248	0.635290
$255$	23.4411	1.46794
$256$	1.00000	0.0625000
$257$	−6.57643	−0.410227	−0.205113	−	0.978738i	$-0.565756\pi$
$257$	−6.57643	−0.410227	−0.205113	+	0.978738i	$0.565756\pi$
$258$	−1.61121	−0.100309
$259$	0	0
$260$	3.80560	0.236013
$261$	−3.00000	−0.185695
$262$	10.0628	0.621681
$263$	−20.4759	−1.26260	−0.631298	−	0.775541i	$-0.717477\pi$
$263$	−20.4759	−1.26260	−0.631298	+	0.775541i	$0.717477\pi$
$264$	−5.28822	−0.325467
$265$	1.34726	0.0827615
$266$	0	0
$267$	−10.6770	−0.653422
$268$	−8.44784	−0.516034
$269$	24.1529	1.47263	0.736313	−	0.676641i	$-0.236565\pi$
$269$	24.1529	1.47263	0.736313	+	0.676641i	$0.236565\pi$
$270$	3.80560	0.231602
$271$	−30.4411	−1.84916	−0.924582	−	0.380983i	$-0.875586\pi$
$271$	−30.4411	−1.84916	−0.924582	+	0.380983i	$0.875586\pi$
$272$	−6.15962	−0.373482
$273$	0	0
$274$	6.67701	0.403373
$275$	50.1461	3.02392
$276$	6.28822	0.378506
$277$	−9.12485	−0.548259	−0.274130	−	0.961693i	$-0.588390\pi$
$277$	−9.12485	−0.548259	−0.274130	+	0.961693i	$0.588390\pi$
$278$	−12.2882	−0.736998
$279$	8.48261	0.507841
$280$	0	0
$281$	−6.70804	−0.400168	−0.200084	−	0.979779i	$-0.564122\pi$
$281$	−6.70804	−0.400168	−0.200084	+	0.979779i	$0.564122\pi$
$282$	7.09382	0.422431
$283$	−26.8646	−1.59694	−0.798469	−	0.602036i	$-0.794356\pi$
$283$	−26.8646	−1.59694	−0.798469	+	0.602036i	$0.794356\pi$
$284$	−15.4826	−0.918724
$285$	20.8646	1.23592
$286$	5.28822	0.312699
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	20.9410	1.23182
$290$	−11.4168	−0.670418
$291$	3.15962	0.185220
$292$	−1.71178	−0.100175
$293$	−9.80560	−0.572849	−0.286425	−	0.958103i	$-0.592467\pi$
$293$	−9.80560	−0.572849	−0.286425	+	0.958103i	$0.592467\pi$
$294$	0	0
$295$	−6.38203	−0.371576
$296$	−1.32299	−0.0768972
$297$	5.28822	0.306853
$298$	18.2882	1.05941
$299$	−6.28822	−0.363657
$300$	9.48261	0.547479
$301$	0	0
$302$	15.2882	0.879738
$303$	4.96523	0.285245
$304$	−5.48261	−0.314449
$305$	−37.1839	−2.12914
$306$	6.15962	0.352122
$307$	−20.5174	−1.17099	−0.585495	−	0.810676i	$-0.699100\pi$
$307$	−20.5174	−1.17099	−0.585495	+	0.810676i	$0.699100\pi$
$308$	0	0
$309$	5.61121	0.319210
$310$	32.2815	1.83346
$311$	22.5454	1.27843	0.639216	−	0.769027i	$-0.279259\pi$
$311$	22.5454	1.27843	0.639216	+	0.769027i	$0.279259\pi$
$312$	1.00000	0.0566139
$313$	−32.0938	−1.81405	−0.907025	−	0.421077i	$-0.861652\pi$
$313$	−32.0938	−1.81405	−0.907025	+	0.421077i	$0.861652\pi$
$314$	10.4168	0.587855
$315$	0	0
$316$	2.87141	0.161529
$317$	31.4478	1.76629	0.883143	−	0.469103i	$-0.155423\pi$
$317$	31.4478	1.76629	0.883143	+	0.469103i	$0.155423\pi$
$318$	0.354020	0.0198525
$319$	−15.8646	−0.888250
$320$	−3.80560	−0.212740
$321$	−17.0590	−0.952143
$322$	0	0
$323$	33.7708	1.87906
$324$	1.00000	0.0555556
$325$	−9.48261	−0.526001
$326$	9.73980	0.539438
$327$	4.00000	0.221201
$328$	5.32299	0.293913
$329$	0	0
$330$	20.1248	1.10784
$331$	16.8957	0.928670	0.464335	−	0.885660i	$-0.346293\pi$
$331$	16.8957	0.928670	0.464335	+	0.885660i	$0.346293\pi$
$332$	7.44784	0.408753
$333$	1.32299	0.0724994
$334$	−8.70502	−0.476318
$335$	32.1491	1.75649
$336$	0	0
$337$	−7.25719	−0.395324	−0.197662	−	0.980270i	$-0.563335\pi$
$337$	−7.25719	−0.395324	−0.197662	+	0.980270i	$0.563335\pi$
$338$	−1.00000	−0.0543928
$339$	−5.45158	−0.296089
$340$	23.4411	1.27127
$341$	44.8579	2.42919
$342$	5.48261	0.296466
$343$	0	0
$344$	−1.61121	−0.0868704
$345$	−23.9305	−1.28837
$346$	−12.0590	−0.648298
$347$	8.03103	0.431128	0.215564	−	0.976490i	$-0.430841\pi$
$347$	8.03103	0.431128	0.215564	+	0.976490i	$0.430841\pi$
$348$	−3.00000	−0.160817
$349$	9.25344	0.495325	0.247663	−	0.968846i	$-0.420338\pi$
$349$	9.25344	0.495325	0.247663	+	0.968846i	$0.420338\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−5.28822	−0.281863
$353$	−12.7080	−0.676381	−0.338190	−	0.941078i	$-0.609815\pi$
$353$	−12.7080	−0.676381	−0.338190	+	0.941078i	$0.609815\pi$
$354$	−1.67701	−0.0891321
$355$	58.9207	3.12718
$356$	−10.6770	−0.565880
$357$	0	0
$358$	4.25719	0.224999
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	3.80560	0.200573
$361$	11.0590	0.582055
$362$	12.1596	0.639096
$363$	16.9652	0.890443
$364$	0	0
$365$	6.51437	0.340978
$366$	−9.77083	−0.510729
$367$	−27.0590	−1.41247	−0.706235	−	0.707977i	$-0.749608\pi$
$367$	−27.0590	−1.41247	−0.706235	+	0.707977i	$0.749608\pi$
$368$	6.28822	0.327796
$369$	−5.32299	−0.277104
$370$	5.03477	0.261745
$371$	0	0
$372$	8.48261	0.439803
$373$	3.70877	0.192033	0.0960164	−	0.995380i	$-0.469390\pi$
$373$	3.70877	0.192033	0.0960164	+	0.995380i	$0.469390\pi$
$374$	32.5734	1.68433
$375$	−17.0590	−0.880925
$376$	7.09382	0.365836
$377$	3.00000	0.154508
$378$	0	0
$379$	−19.2224	−0.987389	−0.493694	−	0.869635i	$-0.664354\pi$
$379$	−19.2224	−0.987389	−0.493694	+	0.869635i	$0.664354\pi$
$380$	20.8646	1.07033
$381$	−10.1248	−0.518712
$382$	24.1876	1.23755
$383$	−25.2224	−1.28881	−0.644403	−	0.764686i	$-0.722894\pi$
$383$	−25.2224	−1.28881	−0.644403	+	0.764686i	$0.722894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−1.54842	−0.0788124
$387$	1.61121	0.0819022
$388$	3.15962	0.160406
$389$	18.1286	0.919156	0.459578	−	0.888137i	$-0.348001\pi$
$389$	18.1286	0.919156	0.459578	+	0.888137i	$0.348001\pi$
$390$	−3.80560	−0.192704
$391$	−38.7330	−1.95881
$392$	0	0
$393$	−10.0628	−0.507601
$394$	18.2882	0.921347
$395$	−10.9274	−0.549819
$396$	5.28822	0.265743
$397$	14.6075	0.733127	0.366564	−	0.930393i	$-0.380534\pi$
$397$	14.6075	0.733127	0.366564	+	0.930393i	$0.380534\pi$
$398$	−26.2187	−1.31422
$399$	0	0
$400$	9.48261	0.474131
$401$	−21.2534	−1.06135	−0.530673	−	0.847577i	$-0.678061\pi$
$401$	−21.2534	−1.06135	−0.530673	+	0.847577i	$0.678061\pi$
$402$	8.44784	0.421340
$403$	−8.48261	−0.422549
$404$	4.96523	0.247029
$405$	−3.80560	−0.189102
$406$	0	0
$407$	6.99626	0.346792
$408$	6.15962	0.304947
$409$	15.9932	0.790815	0.395407	−	0.918506i	$-0.370603\pi$
$409$	15.9932	0.790815	0.395407	+	0.918506i	$0.370603\pi$
$410$	−20.2572	−1.00043
$411$	−6.67701	−0.329353
$412$	5.61121	0.276444
$413$	0	0
$414$	−6.28822	−0.309049
$415$	−28.3435	−1.39133
$416$	1.00000	0.0490290
$417$	12.2882	0.601757
$418$	28.9932	1.41811
$419$	5.61121	0.274125	0.137063	−	0.990562i	$-0.456234\pi$
$419$	5.61121	0.274125	0.137063	+	0.990562i	$0.456234\pi$
$420$	0	0
$421$	17.5802	0.856805	0.428403	−	0.903588i	$-0.359076\pi$
$421$	17.5802	0.856805	0.428403	+	0.903588i	$0.359076\pi$
$422$	9.89942	0.481896
$423$	−7.09382	−0.344913
$424$	0.354020	0.0171927
$425$	−58.4093	−2.83327
$426$	15.4826	0.750135
$427$	0	0
$428$	−17.0590	−0.824580
$429$	−5.28822	−0.255317
$430$	6.13161	0.295692
$431$	17.2224	0.829574	0.414787	−	0.909918i	$-0.363856\pi$
$431$	17.2224	0.829574	0.414787	+	0.909918i	$0.363856\pi$
$432$	1.00000	0.0481125
$433$	−35.0243	−1.68316	−0.841580	−	0.540133i	$-0.818374\pi$
$433$	−35.0243	−1.68316	−0.841580	+	0.540133i	$0.818374\pi$
$434$	0	0
$435$	11.4168	0.547394
$436$	4.00000	0.191565
$437$	−34.4759	−1.64920
$438$	1.71178	0.0817922
$439$	14.5136	0.692698	0.346349	−	0.938106i	$-0.387421\pi$
$439$	14.5136	0.692698	0.346349	+	0.938106i	$0.387421\pi$
$440$	20.1248	0.959415
$441$	0	0
$442$	−6.15962	−0.292983
$443$	10.3820	0.493265	0.246633	−	0.969109i	$-0.420676\pi$
$443$	10.3820	0.493265	0.246633	+	0.969109i	$0.420676\pi$
$444$	1.32299	0.0627863
$445$	40.6325	1.92616
$446$	2.96897	0.140585
$447$	−18.2882	−0.865003
$448$	0	0
$449$	31.0833	1.46691	0.733456	−	0.679737i	$-0.237906\pi$
$449$	31.0833	1.46691	0.733456	+	0.679737i	$0.237906\pi$
$450$	−9.48261	−0.447015
$451$	−28.1491	−1.32549
$452$	−5.45158	−0.256421
$453$	−15.2882	−0.718303
$454$	18.4131	0.864168
$455$	0	0
$456$	5.48261	0.256747
$457$	5.41681	0.253388	0.126694	−	0.991942i	$-0.459563\pi$
$457$	5.41681	0.253388	0.126694	+	0.991942i	$0.459563\pi$
$458$	−1.42357	−0.0665190
$459$	−6.15962	−0.287507
$460$	−23.9305	−1.11576
$461$	−9.54166	−0.444399	−0.222200	−	0.975001i	$-0.571324\pi$
$461$	−9.54166	−0.444399	−0.222200	+	0.975001i	$0.571324\pi$
$462$	0	0
$463$	−27.2224	−1.26513	−0.632566	−	0.774506i	$-0.717998\pi$
$463$	−27.2224	−1.26513	−0.632566	+	0.774506i	$0.717998\pi$
$464$	−3.00000	−0.139272
$465$	−32.2815	−1.49702
$466$	−10.7361	−0.497338
$467$	−13.0658	−0.604613	−0.302307	−	0.953211i	$-0.597757\pi$
$467$	−13.0658	−0.604613	−0.302307	+	0.953211i	$0.597757\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	−26.9963	−1.24524
$471$	−10.4168	−0.479981
$472$	−1.67701	−0.0771906
$473$	8.52040	0.391769
$474$	−2.87141	−0.131888
$475$	−51.9895	−2.38544
$476$	0	0
$477$	−0.354020	−0.0162095
$478$	−2.12859	−0.0973596
$479$	25.2119	1.15196	0.575981	−	0.817463i	$-0.304620\pi$
$479$	25.2119	1.15196	0.575981	+	0.817463i	$0.304620\pi$
$480$	3.80560	0.173701
$481$	−1.32299	−0.0603231
$482$	18.7398	0.853574
$483$	0	0
$484$	16.9652	0.771147
$485$	−12.0243	−0.545994
$486$	−1.00000	−0.0453609
$487$	13.9342	0.631419	0.315709	−	0.948856i	$-0.397758\pi$
$487$	13.9342	0.631419	0.315709	+	0.948856i	$0.397758\pi$
$488$	−9.77083	−0.442305
$489$	−9.73980	−0.440449
$490$	0	0
$491$	−12.3510	−0.557393	−0.278697	−	0.960379i	$-0.589902\pi$
$491$	−12.3510	−0.557393	−0.278697	+	0.960379i	$0.589902\pi$
$492$	−5.32299	−0.239979
$493$	18.4789	0.832246
$494$	−5.48261	−0.246674
$495$	−20.1248	−0.904545
$496$	8.48261	0.380881
$497$	0	0
$498$	−7.44784	−0.333746
$499$	−34.7641	−1.55625	−0.778127	−	0.628107i	$-0.783830\pi$
$499$	−34.7641	−1.55625	−0.778127	+	0.628107i	$0.783830\pi$
$500$	−17.0590	−0.762904
$501$	8.70502	0.388912
$502$	−7.77457	−0.346996
$503$	5.68075	0.253292	0.126646	−	0.991948i	$-0.459579\pi$
$503$	5.68075	0.253292	0.126646	+	0.991948i	$0.459579\pi$
$504$	0	0
$505$	−18.8957	−0.840847
$506$	−33.2534	−1.47830
$507$	1.00000	0.0444116
$508$	−10.1248	−0.449218
$509$	9.38578	0.416017	0.208009	−	0.978127i	$-0.433302\pi$
$509$	9.38578	0.416017	0.208009	+	0.978127i	$0.433302\pi$
$510$	−23.4411	−1.03799
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−5.48261	−0.242063
$514$	6.57643	0.290074
$515$	−21.3540	−0.940971
$516$	1.61121	0.0709294
$517$	−37.5136	−1.64985
$518$	0	0
$519$	12.0590	0.529333
$520$	−3.80560	−0.166887
$521$	42.0485	1.84218	0.921090	−	0.389350i	$-0.127300\pi$
$521$	42.0485	1.84218	0.921090	+	0.389350i	$0.127300\pi$
$522$	3.00000	0.131306
$523$	−20.8336	−0.910990	−0.455495	−	0.890238i	$-0.650538\pi$
$523$	−20.8336	−0.910990	−0.455495	+	0.890238i	$0.650538\pi$
$524$	−10.0628	−0.439595
$525$	0	0
$526$	20.4759	0.892790
$527$	−52.2497	−2.27603
$528$	5.28822	0.230140
$529$	16.5417	0.719202
$530$	−1.34726	−0.0585212
$531$	1.67701	0.0727760
$532$	0	0
$533$	5.32299	0.230564
$534$	10.6770	0.462039
$535$	64.9199	2.80673
$536$	8.44784	0.364891
$537$	−4.25719	−0.183711
$538$	−24.1529	−1.04130
$539$	0	0
$540$	−3.80560	−0.163767
$541$	16.0310	0.689228	0.344614	−	0.938745i	$-0.388010\pi$
$541$	16.0310	0.689228	0.344614	+	0.938745i	$0.388010\pi$
$542$	30.4411	1.30756
$543$	−12.1596	−0.521819
$544$	6.15962	0.264092
$545$	−15.2224	−0.652056
$546$	0	0
$547$	23.2534	0.994245	0.497123	−	0.867680i	$-0.334390\pi$
$547$	23.2534	0.994245	0.497123	+	0.867680i	$0.334390\pi$
$548$	−6.67701	−0.285228
$549$	9.77083	0.417009
$550$	−50.1461	−2.13824
$551$	16.4478	0.700701
$552$	−6.28822	−0.267644
$553$	0	0
$554$	9.12485	0.387678
$555$	−5.03477	−0.213714
$556$	12.2882	0.521136
$557$	35.0590	1.48550	0.742750	−	0.669569i	$-0.233521\pi$
$557$	35.0590	1.48550	0.742750	+	0.669569i	$0.233521\pi$
$558$	−8.48261	−0.359098
$559$	−1.61121	−0.0681467
$560$	0	0
$561$	−32.5734	−1.37525
$562$	6.70804	0.282962
$563$	27.5174	1.15972	0.579860	−	0.814716i	$-0.303107\pi$
$563$	27.5174	1.15972	0.579860	+	0.814716i	$0.303107\pi$
$564$	−7.09382	−0.298704
$565$	20.7466	0.872814
$566$	26.8646	1.12921
$567$	0	0
$568$	15.4826	0.649636
$569$	−18.0280	−0.755774	−0.377887	−	0.925852i	$-0.623349\pi$
$569$	−18.0280	−0.755774	−0.377887	+	0.925852i	$0.623349\pi$
$570$	−20.8646	−0.873924
$571$	5.29196	0.221462	0.110731	−	0.993850i	$-0.464681\pi$
$571$	5.29196	0.221462	0.110731	+	0.993850i	$0.464681\pi$
$572$	−5.28822	−0.221111
$573$	−24.1876	−1.01045
$574$	0	0
$575$	59.6287	2.48669
$576$	1.00000	0.0416667
$577$	8.16337	0.339845	0.169923	−	0.985457i	$-0.445648\pi$
$577$	8.16337	0.339845	0.169923	+	0.985457i	$0.445648\pi$
$578$	−20.9410	−0.871029
$579$	1.54842	0.0643500
$580$	11.4168	0.474057
$581$	0	0
$582$	−3.15962	−0.130971
$583$	−1.87214	−0.0775359
$584$	1.71178	0.0708341
$585$	3.80560	0.157342
$586$	9.80560	0.405066
$587$	−14.4546	−0.596605	−0.298303	−	0.954471i	$-0.596420\pi$
$587$	−14.4546	−0.596605	−0.298303	+	0.954471i	$0.596420\pi$
$588$	0	0
$589$	−46.5069	−1.91628
$590$	6.38203	0.262744
$591$	−18.2882	−0.752277
$592$	1.32299	0.0543746
$593$	−3.96897	−0.162986	−0.0814930	−	0.996674i	$-0.525969\pi$
$593$	−3.96897	−0.162986	−0.0814930	+	0.996674i	$0.525969\pi$
$594$	−5.28822	−0.216978
$595$	0	0
$596$	−18.2882	−0.749115
$597$	26.2187	1.07306
$598$	6.28822	0.257144
$599$	−21.2534	−0.868392	−0.434196	−	0.900818i	$-0.642967\pi$
$599$	−21.2534	−0.868392	−0.434196	+	0.900818i	$0.642967\pi$
$600$	−9.48261	−0.387126
$601$	27.2497	1.11154	0.555769	−	0.831337i	$-0.312424\pi$
$601$	27.2497	1.11154	0.555769	+	0.831337i	$0.312424\pi$
$602$	0	0
$603$	−8.44784	−0.344023
$604$	−15.2882	−0.622069
$605$	−64.5629	−2.62486
$606$	−4.96523	−0.201699
$607$	9.92369	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$607$	9.92369	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$608$	5.48261	0.222349
$609$	0	0
$610$	37.1839	1.50553
$611$	7.09382	0.286985
$612$	−6.15962	−0.248988
$613$	−9.86839	−0.398581	−0.199290	−	0.979940i	$-0.563864\pi$
$613$	−9.86839	−0.398581	−0.199290	+	0.979940i	$0.563864\pi$
$614$	20.5174	0.828014
$615$	20.2572	0.816849
$616$	0	0
$617$	−41.9865	−1.69031	−0.845156	−	0.534520i	$-0.820493\pi$
$617$	−41.9865	−1.69031	−0.845156	+	0.534520i	$0.820493\pi$
$618$	−5.61121	−0.225716
$619$	25.0348	1.00623	0.503116	−	0.864219i	$-0.332187\pi$
$619$	25.0348	1.00623	0.503116	+	0.864219i	$0.332187\pi$
$620$	−32.2815	−1.29645
$621$	6.28822	0.252337
$622$	−22.5454	−0.903988
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	17.5069	0.700275
$626$	32.0938	1.28273
$627$	−28.9932	−1.15788
$628$	−10.4168	−0.415676
$629$	−8.14912	−0.324927
$630$	0	0
$631$	−4.55216	−0.181219	−0.0906093	−	0.995887i	$-0.528881\pi$
$631$	−4.55216	−0.181219	−0.0906093	+	0.995887i	$0.528881\pi$
$632$	−2.87141	−0.114218
$633$	−9.89942	−0.393467
$634$	−31.4478	−1.24895
$635$	38.5312	1.52906
$636$	−0.354020	−0.0140378
$637$	0	0
$638$	15.8646	0.628087
$639$	−15.4826	−0.612483
$640$	3.80560	0.150430
$641$	33.1529	1.30946	0.654730	−	0.755863i	$-0.272782\pi$
$641$	33.1529	1.30946	0.654730	+	0.755863i	$0.272782\pi$
$642$	17.0590	0.673267
$643$	42.0590	1.65865	0.829323	−	0.558769i	$-0.188726\pi$
$643$	42.0590	1.65865	0.829323	+	0.558769i	$0.188726\pi$
$644$	0	0
$645$	−6.13161	−0.241432
$646$	−33.7708	−1.32870
$647$	16.2572	0.639136	0.319568	−	0.947563i	$-0.396462\pi$
$647$	16.2572	0.639136	0.319568	+	0.947563i	$0.396462\pi$
$648$	−1.00000	−0.0392837
$649$	8.86839	0.348115
$650$	9.48261	0.371939
$651$	0	0
$652$	−9.73980	−0.381440
$653$	30.9895	1.21271	0.606356	−	0.795193i	$-0.292631\pi$
$653$	30.9895	1.21271	0.606356	+	0.795193i	$0.292631\pi$
$654$	−4.00000	−0.156412
$655$	38.2950	1.49631
$656$	−5.32299	−0.207828
$657$	−1.71178	−0.0667831
$658$	0	0
$659$	3.25344	0.126736	0.0633680	−	0.997990i	$-0.479816\pi$
$659$	3.25344	0.126736	0.0633680	+	0.997990i	$0.479816\pi$
$660$	−20.1248	−0.783359
$661$	4.87214	0.189504	0.0947520	−	0.995501i	$-0.469794\pi$
$661$	4.87214	0.189504	0.0947520	+	0.995501i	$0.469794\pi$
$662$	−16.8957	−0.656669
$663$	6.15962	0.239220
$664$	−7.44784	−0.289032
$665$	0	0
$666$	−1.32299	−0.0512648
$667$	−18.8646	−0.730442
$668$	8.70502	0.336808
$669$	−2.96897	−0.114787
$670$	−32.1491	−1.24203
$671$	51.6703	1.99471
$672$	0	0
$673$	16.8714	0.650345	0.325172	−	0.945655i	$-0.394578\pi$
$673$	16.8714	0.650345	0.325172	+	0.945655i	$0.394578\pi$
$674$	7.25719	0.279536
$675$	9.48261	0.364986
$676$	1.00000	0.0384615
$677$	−30.7988	−1.18370	−0.591848	−	0.806050i	$-0.701601\pi$
$677$	−30.7988	−1.18370	−0.591848	+	0.806050i	$0.701601\pi$
$678$	5.45158	0.209367
$679$	0	0
$680$	−23.4411	−0.898924
$681$	−18.4131	−0.705590
$682$	−44.8579	−1.71770
$683$	36.3435	1.39065	0.695323	−	0.718697i	$-0.255261\pi$
$683$	36.3435	1.39065	0.695323	+	0.718697i	$0.255261\pi$
$684$	−5.48261	−0.209633
$685$	25.4100	0.970868
$686$	0	0
$687$	1.42357	0.0543125
$688$	1.61121	0.0614266
$689$	0.354020	0.0134871
$690$	23.9305	0.911017
$691$	−38.9547	−1.48191	−0.740954	−	0.671556i	$-0.765626\pi$
$691$	−38.9547	−1.48191	−0.740954	+	0.671556i	$0.765626\pi$
$692$	12.0590	0.458416
$693$	0	0
$694$	−8.03103	−0.304854
$695$	−46.7641	−1.77386
$696$	3.00000	0.113715
$697$	32.7876	1.24192
$698$	−9.25344	−0.350248
$699$	10.7361	0.406075
$700$	0	0
$701$	40.7323	1.53844	0.769219	−	0.638985i	$-0.220645\pi$
$701$	40.7323	1.53844	0.769219	+	0.638985i	$0.220645\pi$
$702$	1.00000	0.0377426
$703$	−7.25344	−0.273569
$704$	5.28822	0.199307
$705$	26.9963	1.01674
$706$	12.7080	0.478273
$707$	0	0
$708$	1.67701	0.0630259
$709$	−23.1218	−0.868359	−0.434179	−	0.900826i	$-0.642962\pi$
$709$	−23.1218	−0.868359	−0.434179	+	0.900826i	$0.642962\pi$
$710$	−58.9207	−2.21125
$711$	2.87141	0.107686
$712$	10.6770	0.400138
$713$	53.3405	1.99762
$714$	0	0
$715$	20.1248	0.752627
$716$	−4.25719	−0.159098
$717$	2.12859	0.0794938
$718$	18.0000	0.671754
$719$	26.3753	0.983632	0.491816	−	0.870699i	$-0.336333\pi$
$719$	26.3753	0.983632	0.491816	+	0.870699i	$0.336333\pi$
$720$	−3.80560	−0.141826
$721$	0	0
$722$	−11.0590	−0.411575
$723$	−18.7398	−0.696941
$724$	−12.1596	−0.451909
$725$	−28.4478	−1.05653
$726$	−16.9652	−0.629639
$727$	6.73980	0.249965	0.124983	−	0.992159i	$-0.460112\pi$
$727$	6.73980	0.249965	0.124983	+	0.992159i	$0.460112\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.51437	−0.241108
$731$	−9.92442	−0.367068
$732$	9.77083	0.361140
$733$	−23.9615	−0.885038	−0.442519	−	0.896759i	$-0.645915\pi$
$733$	−23.9615	−0.885038	−0.442519	+	0.896759i	$0.645915\pi$
$734$	27.0590	0.998768
$735$	0	0
$736$	−6.28822	−0.231787
$737$	−44.6740	−1.64559
$738$	5.32299	0.195942
$739$	5.93794	0.218431	0.109215	−	0.994018i	$-0.465166\pi$
$739$	5.93794	0.218431	0.109215	+	0.994018i	$0.465166\pi$
$740$	−5.03477	−0.185082
$741$	5.48261	0.201409
$742$	0	0
$743$	−22.3783	−0.820980	−0.410490	−	0.911865i	$-0.634642\pi$
$743$	−22.3783	−0.820980	−0.410490	+	0.911865i	$0.634642\pi$
$744$	−8.48261	−0.310988
$745$	69.5977	2.54986
$746$	−3.70877	−0.135788
$747$	7.44784	0.272502
$748$	−32.5734	−1.19100
$749$	0	0
$750$	17.0590	0.622908
$751$	41.2467	1.50511	0.752556	−	0.658528i	$-0.228821\pi$
$751$	41.2467	1.50511	0.752556	+	0.658528i	$0.228821\pi$
$752$	−7.09382	−0.258685
$753$	7.77457	0.283321
$754$	−3.00000	−0.109254
$755$	58.1809	2.11742
$756$	0	0
$757$	−41.9585	−1.52501	−0.762503	−	0.646984i	$-0.776030\pi$
$757$	−41.9585	−1.52501	−0.762503	+	0.646984i	$0.776030\pi$
$758$	19.2224	0.698189
$759$	33.2534	1.20702
$760$	−20.8646	−0.756840
$761$	−19.1604	−0.694562	−0.347281	−	0.937761i	$-0.612895\pi$
$761$	−19.1604	−0.694562	−0.347281	+	0.937761i	$0.612895\pi$
$762$	10.1248	0.366785
$763$	0	0
$764$	−24.1876	−0.875078
$765$	23.4411	0.847514
$766$	25.2224	0.911323
$767$	−1.67701	−0.0605533
$768$	1.00000	0.0360844
$769$	−31.7981	−1.14667	−0.573335	−	0.819321i	$-0.694351\pi$
$769$	−31.7981	−1.14667	−0.573335	+	0.819321i	$0.694351\pi$
$770$	0	0
$771$	−6.57643	−0.236844
$772$	1.54842	0.0557288
$773$	50.3753	1.81187	0.905936	−	0.423414i	$-0.139168\pi$
$773$	50.3753	1.81187	0.905936	+	0.423414i	$0.139168\pi$
$774$	−1.61121	−0.0579136
$775$	80.4373	2.88939
$776$	−3.15962	−0.113424
$777$	0	0
$778$	−18.1286	−0.649942
$779$	29.1839	1.04562
$780$	3.80560	0.136262
$781$	−81.8754	−2.92973
$782$	38.7330	1.38509
$783$	−3.00000	−0.107211
$784$	0	0
$785$	39.6422	1.41489
$786$	10.0628	0.358928
$787$	22.0590	0.786320	0.393160	−	0.919470i	$-0.371382\pi$
$787$	22.0590	0.786320	0.393160	+	0.919470i	$0.371382\pi$
$788$	−18.2882	−0.651491
$789$	−20.4759	−0.728960
$790$	10.9274	0.388781
$791$	0	0
$792$	−5.28822	−0.187909
$793$	−9.77083	−0.346972
$794$	−14.6075	−0.518399
$795$	1.34726	0.0477824
$796$	26.2187	0.929296
$797$	52.5312	1.86075	0.930374	−	0.366611i	$-0.119482\pi$
$797$	52.5312	1.86075	0.930374	+	0.366611i	$0.119482\pi$
$798$	0	0
$799$	43.6952	1.54583
$800$	−9.48261	−0.335261
$801$	−10.6770	−0.377254
$802$	21.2534	0.750485
$803$	−9.05228	−0.319448
$804$	−8.44784	−0.297932
$805$	0	0
$806$	8.48261	0.298787
$807$	24.1529	0.850221
$808$	−4.96523	−0.174676
$809$	−46.9858	−1.65193	−0.825966	−	0.563721i	$-0.809369\pi$
$809$	−46.9858	−1.65193	−0.825966	+	0.563721i	$0.809369\pi$
$810$	3.80560	0.133715
$811$	−1.81236	−0.0636407	−0.0318203	−	0.999494i	$-0.510130\pi$
$811$	−1.81236	−0.0636407	−0.0318203	+	0.999494i	$0.510130\pi$
$812$	0	0
$813$	−30.4411	−1.06762
$814$	−6.99626	−0.245219
$815$	37.0658	1.29836
$816$	−6.15962	−0.215630
$817$	−8.83362	−0.309049
$818$	−15.9932	−0.559191
$819$	0	0
$820$	20.2572	0.707412
$821$	−35.1521	−1.22682	−0.613409	−	0.789765i	$-0.710202\pi$
$821$	−35.1521	−1.22682	−0.613409	+	0.789765i	$0.710202\pi$
$822$	6.67701	0.232887
$823$	34.6770	1.20876	0.604382	−	0.796694i	$-0.293420\pi$
$823$	34.6770	1.20876	0.604382	+	0.796694i	$0.293420\pi$
$824$	−5.61121	−0.195476
$825$	50.1461	1.74586
$826$	0	0
$827$	19.4138	0.675084	0.337542	−	0.941311i	$-0.390405\pi$
$827$	19.4138	0.675084	0.337542	+	0.941311i	$0.390405\pi$
$828$	6.28822	0.218531
$829$	−41.8964	−1.45512	−0.727561	−	0.686043i	$-0.759346\pi$
$829$	−41.8964	−1.45512	−0.727561	+	0.686043i	$0.759346\pi$
$830$	28.3435	0.983818
$831$	−9.12485	−0.316537
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	−12.2882	−0.425506
$835$	−33.1279	−1.14644
$836$	−28.9932	−1.00275
$837$	8.48261	0.293202
$838$	−5.61121	−0.193836
$839$	−15.9895	−0.552019	−0.276009	−	0.961155i	$-0.589012\pi$
$839$	−15.9895	−0.552019	−0.276009	+	0.961155i	$0.589012\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−17.5802	−0.605853
$843$	−6.70804	−0.231037
$844$	−9.89942	−0.340752
$845$	−3.80560	−0.130917
$846$	7.09382	0.243891
$847$	0	0
$848$	−0.354020	−0.0121571
$849$	−26.8646	−0.921992
$850$	58.4093	2.00342
$851$	8.31925	0.285180
$852$	−15.4826	−0.530426
$853$	44.2747	1.51594	0.757968	−	0.652291i	$-0.226192\pi$
$853$	44.2747	1.51594	0.757968	+	0.652291i	$0.226192\pi$
$854$	0	0
$855$	20.8646	0.713556
$856$	17.0590	0.583066
$857$	−23.8329	−0.814116	−0.407058	−	0.913402i	$-0.633445\pi$
$857$	−23.8329	−0.814116	−0.407058	+	0.913402i	$0.633445\pi$
$858$	5.28822	0.180537
$859$	14.6460	0.499714	0.249857	−	0.968283i	$-0.419616\pi$
$859$	14.6460	0.499714	0.249857	+	0.968283i	$0.419616\pi$
$860$	−6.13161	−0.209086
$861$	0	0
$862$	−17.2224	−0.586598
$863$	−21.1529	−0.720052	−0.360026	−	0.932942i	$-0.617232\pi$
$863$	−21.1529	−0.720052	−0.360026	+	0.932942i	$0.617232\pi$
$864$	−1.00000	−0.0340207
$865$	−45.8919	−1.56037
$866$	35.0243	1.19017
$867$	20.9410	0.711192
$868$	0	0
$869$	15.1846	0.515103
$870$	−11.4168	−0.387066
$871$	8.44784	0.286244
$872$	−4.00000	−0.135457
$873$	3.15962	0.106937
$874$	34.4759	1.16616
$875$	0	0
$876$	−1.71178	−0.0578358
$877$	6.22616	0.210242	0.105121	−	0.994459i	$-0.466477\pi$
$877$	6.22616	0.210242	0.105121	+	0.994459i	$0.466477\pi$
$878$	−14.5136	−0.489812
$879$	−9.80560	−0.330735
$880$	−20.1248	−0.678409
$881$	−46.9517	−1.58184	−0.790922	−	0.611918i	$-0.790398\pi$
$881$	−46.9517	−1.58184	−0.790922	+	0.611918i	$0.790398\pi$
$882$	0	0
$883$	−29.1143	−0.979776	−0.489888	−	0.871785i	$-0.662962\pi$
$883$	−29.1143	−0.979776	−0.489888	+	0.871785i	$0.662962\pi$
$884$	6.15962	0.207171
$885$	−6.38203	−0.214530
$886$	−10.3820	−0.348791
$887$	−43.6037	−1.46407	−0.732035	−	0.681267i	$-0.761429\pi$
$887$	−43.6037	−1.46407	−0.732035	+	0.681267i	$0.761429\pi$
$888$	−1.32299	−0.0443966
$889$	0	0
$890$	−40.6325	−1.36200
$891$	5.28822	0.177162
$892$	−2.96897	−0.0994085
$893$	38.8927	1.30149
$894$	18.2882	0.611649
$895$	16.2012	0.541545
$896$	0	0
$897$	−6.28822	−0.209957
$898$	−31.0833	−1.03726
$899$	−25.4478	−0.848733
$900$	9.48261	0.316087
$901$	2.18063	0.0726473
$902$	28.1491	0.937263
$903$	0	0
$904$	5.45158	0.181317
$905$	46.2747	1.53822
$906$	15.2882	0.507917
$907$	2.47585	0.0822093	0.0411047	−	0.999155i	$-0.486912\pi$
$907$	2.47585	0.0822093	0.0411047	+	0.999155i	$0.486912\pi$
$908$	−18.4131	−0.611059
$909$	4.96523	0.164686
$910$	0	0
$911$	46.0000	1.52405	0.762024	−	0.647549i	$-0.224206\pi$
$911$	46.0000	1.52405	0.762024	+	0.647549i	$0.224206\pi$
$912$	−5.48261	−0.181547
$913$	39.3858	1.30348
$914$	−5.41681	−0.179172
$915$	−37.1839	−1.22926
$916$	1.42357	0.0470360
$917$	0	0
$918$	6.15962	0.203298
$919$	35.5417	1.17241	0.586206	−	0.810162i	$-0.300621\pi$
$919$	35.5417	1.17241	0.586206	+	0.810162i	$0.300621\pi$
$920$	23.9305	0.788964
$921$	−20.5174	−0.676071
$922$	9.54166	0.314238
$923$	15.4826	0.509616
$924$	0	0
$925$	12.5454	0.412490
$926$	27.2224	0.894584
$927$	5.61121	0.184296
$928$	3.00000	0.0984798
$929$	−14.1876	−0.465481	−0.232741	−	0.972539i	$-0.574769\pi$
$929$	−14.1876	−0.465481	−0.232741	+	0.972539i	$0.574769\pi$
$930$	32.2815	1.05855
$931$	0	0
$932$	10.7361	0.351671
$933$	22.5454	0.738103
$934$	13.0658	0.427526
$935$	123.961	4.05397
$936$	1.00000	0.0326860
$937$	2.09683	0.0685006	0.0342503	−	0.999413i	$-0.489096\pi$
$937$	2.09683	0.0685006	0.0342503	+	0.999413i	$0.489096\pi$
$938$	0	0
$939$	−32.0938	−1.04734
$940$	26.9963	0.880521
$941$	7.90618	0.257734	0.128867	−	0.991662i	$-0.458866\pi$
$941$	7.90618	0.257734	0.128867	+	0.991662i	$0.458866\pi$
$942$	10.4168	0.339398
$943$	−33.4721	−1.09000
$944$	1.67701	0.0545820
$945$	0	0
$946$	−8.52040	−0.277022
$947$	−36.8541	−1.19760	−0.598799	−	0.800899i	$-0.704355\pi$
$947$	−36.8541	−1.19760	−0.598799	+	0.800899i	$0.704355\pi$
$948$	2.87141	0.0932590
$949$	1.71178	0.0555669
$950$	51.9895	1.68676
$951$	31.4478	1.01977
$952$	0	0
$953$	21.6527	0.701401	0.350701	−	0.936488i	$-0.385944\pi$
$953$	21.6527	0.701401	0.350701	+	0.936488i	$0.385944\pi$
$954$	0.354020	0.0114618
$955$	92.0485	2.97862
$956$	2.12859	0.0688436
$957$	−15.8646	−0.512831
$958$	−25.2119	−0.814560
$959$	0	0
$960$	−3.80560	−0.122825
$961$	40.9547	1.32112
$962$	1.32299	0.0426549
$963$	−17.0590	−0.549720
$964$	−18.7398	−0.603568
$965$	−5.89266	−0.189691
$966$	0	0
$967$	7.21867	0.232137	0.116068	−	0.993241i	$-0.462971\pi$
$967$	7.21867	0.232137	0.116068	+	0.993241i	$0.462971\pi$
$968$	−16.9652	−0.545283
$969$	33.7708	1.08488
$970$	12.0243	0.386076
$971$	27.7981	0.892084	0.446042	−	0.895012i	$-0.352833\pi$
$971$	27.7981	0.892084	0.446042	+	0.895012i	$0.352833\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−13.9342	−0.446480
$975$	−9.48261	−0.303687
$976$	9.77083	0.312757
$977$	−6.57643	−0.210399	−0.105199	−	0.994451i	$-0.533548\pi$
$977$	−6.57643	−0.210399	−0.105199	+	0.994451i	$0.533548\pi$
$978$	9.73980	0.311444
$979$	−56.4623	−1.80454
$980$	0	0
$981$	4.00000	0.127710
$982$	12.3510	0.394136
$983$	−10.4478	−0.333234	−0.166617	−	0.986022i	$-0.553284\pi$
$983$	−10.4478	−0.333234	−0.166617	+	0.986022i	$0.553284\pi$
$984$	5.32299	0.169691
$985$	69.5977	2.21757
$986$	−18.4789	−0.588487
$987$	0	0
$988$	5.48261	0.174425
$989$	10.1316	0.322166
$990$	20.1248	0.639610
$991$	10.9895	0.349093	0.174546	−	0.984649i	$-0.444154\pi$
$991$	10.9895	0.349093	0.174546	+	0.984649i	$0.444154\pi$
$992$	−8.48261	−0.269323
$993$	16.8957	0.536168
$994$	0	0
$995$	−99.7778	−3.16317
$996$	7.44784	0.235994
$997$	−44.2157	−1.40032	−0.700162	−	0.713984i	$-0.746889\pi$
$997$	−44.2157	−1.40032	−0.700162	+	0.713984i	$0.746889\pi$
$998$	34.7641	1.10044
$999$	1.32299	0.0418575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.bw.1.1		3
7.3	odd	6		546.2.i.k.79.1	✓	6
7.5	odd	6		546.2.i.k.235.1	yes	6
7.6	odd	2		3822.2.a.bv.1.3		3
21.5	even	6		1638.2.j.q.235.3		6
21.17	even	6		1638.2.j.q.1171.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.k.79.1	✓	6	7.3	odd	6
546.2.i.k.235.1	yes	6	7.5	odd	6
1638.2.j.q.235.3		6	21.5	even	6
1638.2.j.q.1171.3		6	21.17	even	6
3822.2.a.bv.1.3		3	7.6	odd	2
3822.2.a.bw.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.80560 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.80560 q^{10} +5.28822 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.80560 q^{15} +1.00000 q^{16} -6.15962 q^{17} -1.00000 q^{18} -5.48261 q^{19} -3.80560 q^{20} -5.28822 q^{22} +6.28822 q^{23} -1.00000 q^{24} +9.48261 q^{25} +1.00000 q^{26} +1.00000 q^{27} -3.00000 q^{29} +3.80560 q^{30} +8.48261 q^{31} -1.00000 q^{32} +5.28822 q^{33} +6.15962 q^{34} +1.00000 q^{36} +1.32299 q^{37} +5.48261 q^{38} -1.00000 q^{39} +3.80560 q^{40} -5.32299 q^{41} +1.61121 q^{43} +5.28822 q^{44} -3.80560 q^{45} -6.28822 q^{46} -7.09382 q^{47} +1.00000 q^{48} -9.48261 q^{50} -6.15962 q^{51} -1.00000 q^{52} -0.354020 q^{53} -1.00000 q^{54} -20.1248 q^{55} -5.48261 q^{57} +3.00000 q^{58} +1.67701 q^{59} -3.80560 q^{60} +9.77083 q^{61} -8.48261 q^{62} +1.00000 q^{64} +3.80560 q^{65} -5.28822 q^{66} -8.44784 q^{67} -6.15962 q^{68} +6.28822 q^{69} -15.4826 q^{71} -1.00000 q^{72} -1.71178 q^{73} -1.32299 q^{74} +9.48261 q^{75} -5.48261 q^{76} +1.00000 q^{78} +2.87141 q^{79} -3.80560 q^{80} +1.00000 q^{81} +5.32299 q^{82} +7.44784 q^{83} +23.4411 q^{85} -1.61121 q^{86} -3.00000 q^{87} -5.28822 q^{88} -10.6770 q^{89} +3.80560 q^{90} +6.28822 q^{92} +8.48261 q^{93} +7.09382 q^{94} +20.8646 q^{95} -1.00000 q^{96} +3.15962 q^{97} +5.28822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 3 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} - 6 q^{19} - 3 q^{20} + 3 q^{22} - 3 q^{24}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 + 3 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 6 * q^17 - 3 * q^18 - 6 * q^19 - 3 * q^20 + 3 * q^22 - 3 * q^24 + 18 * q^25 + 3 * q^26 + 3 * q^27 - 9 * q^29 + 3 * q^30 + 15 * q^31 - 3 * q^32 - 3 * q^33 + 6 * q^34 + 3 * q^36 + 6 * q^37 + 6 * q^38 - 3 * q^39 + 3 * q^40 - 18 * q^41 - 12 * q^43 - 3 * q^44 - 3 * q^45 + 6 * q^47 + 3 * q^48 - 18 * q^50 - 6 * q^51 - 3 * q^52 + 3 * q^53 - 3 * q^54 - 27 * q^55 - 6 * q^57 + 9 * q^58 + 3 * q^59 - 3 * q^60 - 15 * q^62 + 3 * q^64 + 3 * q^65 + 3 * q^66 + 6 * q^67 - 6 * q^68 - 36 * q^71 - 3 * q^72 - 24 * q^73 - 6 * q^74 + 18 * q^75 - 6 * q^76 + 3 * q^78 + 15 * q^79 - 3 * q^80 + 3 * q^81 + 18 * q^82 - 9 * q^83 - 24 * q^85 + 12 * q^86 - 9 * q^87 + 3 * q^88 - 30 * q^89 + 3 * q^90 + 15 * q^93 - 6 * q^94 + 6 * q^95 - 3 * q^96 - 3 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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