LMFDB - Embedded newform 3330.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.68585 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.68585 q^{7} -1.00000 q^{8} -1.00000 q^{10} -2.14637 q^{11} -1.48929 q^{13} -3.68585 q^{14} +1.00000 q^{16} +0.707269 q^{17} -1.48929 q^{19} +1.00000 q^{20} +2.14637 q^{22} -5.88240 q^{23} +1.00000 q^{25} +1.48929 q^{26} +3.68585 q^{28} -1.80344 q^{29} +10.0288 q^{31} -1.00000 q^{32} -0.707269 q^{34} +3.68585 q^{35} -1.00000 q^{37} +1.48929 q^{38} -1.00000 q^{40} +9.00735 q^{41} +2.36435 q^{43} -2.14637 q^{44} +5.88240 q^{46} +11.2713 q^{47} +6.58546 q^{49} -1.00000 q^{50} -1.48929 q^{52} +5.51806 q^{53} -2.14637 q^{55} -3.68585 q^{56} +1.80344 q^{58} -13.6644 q^{59} -3.63565 q^{61} -10.0288 q^{62} +1.00000 q^{64} -1.48929 q^{65} +15.6644 q^{67} +0.707269 q^{68} -3.68585 q^{70} +7.66442 q^{71} +15.0533 q^{73} +1.00000 q^{74} -1.48929 q^{76} -7.91117 q^{77} +12.2927 q^{79} +1.00000 q^{80} -9.00735 q^{82} -5.15371 q^{83} +0.707269 q^{85} -2.36435 q^{86} +2.14637 q^{88} +4.46787 q^{89} -5.48929 q^{91} -5.88240 q^{92} -11.2713 q^{94} -1.48929 q^{95} -7.00735 q^{97} -6.58546 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 5 q^{11} + 3 q^{13} + q^{14} + 3 q^{16} + 5 q^{17} + 3 q^{19} + 3 q^{20} + 5 q^{22} - q^{23} + 3 q^{25} - 3 q^{26} - q^{28} - 10 q^{29} + 12 q^{31} - 3 q^{32} - 5 q^{34} - q^{35} - 3 q^{37} - 3 q^{38} - 3 q^{40} - 6 q^{41} + 16 q^{43} - 5 q^{44} + q^{46} + 16 q^{47} + 14 q^{49} - 3 q^{50} + 3 q^{52} - 9 q^{53} - 5 q^{55} + q^{56} + 10 q^{58} - 14 q^{59} - 2 q^{61} - 12 q^{62} + 3 q^{64} + 3 q^{65} + 20 q^{67} + 5 q^{68} + q^{70} - 4 q^{71} + 17 q^{73} + 3 q^{74} + 3 q^{76} + 11 q^{77} + 34 q^{79} + 3 q^{80} + 6 q^{82} + 19 q^{83} + 5 q^{85} - 16 q^{86} + 5 q^{88} - 9 q^{89} - 9 q^{91} - q^{92} - 16 q^{94} + 3 q^{95} + 12 q^{97} - 14 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8 - 3 * q^10 - 5 * q^11 + 3 * q^13 + q^14 + 3 * q^16 + 5 * q^17 + 3 * q^19 + 3 * q^20 + 5 * q^22 - q^23 + 3 * q^25 - 3 * q^26 - q^28 - 10 * q^29 + 12 * q^31 - 3 * q^32 - 5 * q^34 - q^35 - 3 * q^37 - 3 * q^38 - 3 * q^40 - 6 * q^41 + 16 * q^43 - 5 * q^44 + q^46 + 16 * q^47 + 14 * q^49 - 3 * q^50 + 3 * q^52 - 9 * q^53 - 5 * q^55 + q^56 + 10 * q^58 - 14 * q^59 - 2 * q^61 - 12 * q^62 + 3 * q^64 + 3 * q^65 + 20 * q^67 + 5 * q^68 + q^70 - 4 * q^71 + 17 * q^73 + 3 * q^74 + 3 * q^76 + 11 * q^77 + 34 * q^79 + 3 * q^80 + 6 * q^82 + 19 * q^83 + 5 * q^85 - 16 * q^86 + 5 * q^88 - 9 * q^89 - 9 * q^91 - q^92 - 16 * q^94 + 3 * q^95 + 12 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.68585	1.39312	0.696559	−	0.717499i	$-0.254713\pi$
$7$	3.68585	1.39312	0.696559	+	0.717499i	$0.254713\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−2.14637	−0.647154	−0.323577	−	0.946202i	$-0.604885\pi$
$11$	−2.14637	−0.647154	−0.323577	+	0.946202i	$0.604885\pi$
$12$	0	0
$13$	−1.48929	−0.413054	−0.206527	−	0.978441i	$-0.566216\pi$
$13$	−1.48929	−0.413054	−0.206527	+	0.978441i	$0.566216\pi$
$14$	−3.68585	−0.985084
$15$	0	0
$16$	1.00000	0.250000
$17$	0.707269	0.171538	0.0857690	−	0.996315i	$-0.472665\pi$
$17$	0.707269	0.171538	0.0857690	+	0.996315i	$0.472665\pi$
$18$	0	0
$19$	−1.48929	−0.341666	−0.170833	−	0.985300i	$-0.554646\pi$
$19$	−1.48929	−0.341666	−0.170833	+	0.985300i	$0.554646\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.14637	0.457607
$23$	−5.88240	−1.22657	−0.613283	−	0.789863i	$-0.710151\pi$
$23$	−5.88240	−1.22657	−0.613283	+	0.789863i	$0.710151\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.48929	0.292074
$27$	0	0
$28$	3.68585	0.696559
$29$	−1.80344	−0.334891	−0.167445	−	0.985881i	$-0.553552\pi$
$29$	−1.80344	−0.334891	−0.167445	+	0.985881i	$0.553552\pi$
$30$	0	0
$31$	10.0288	1.80122	0.900610	−	0.434628i	$-0.143120\pi$
$31$	10.0288	1.80122	0.900610	+	0.434628i	$0.143120\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.707269	−0.121296
$35$	3.68585	0.623022
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	1.48929	0.241595
$39$	0	0
$40$	−1.00000	−0.158114
$41$	9.00735	1.40671	0.703356	−	0.710838i	$-0.251684\pi$
$41$	9.00735	1.40671	0.703356	+	0.710838i	$0.251684\pi$
$42$	0	0
$43$	2.36435	0.360559	0.180280	−	0.983615i	$-0.442300\pi$
$43$	2.36435	0.360559	0.180280	+	0.983615i	$0.442300\pi$
$44$	−2.14637	−0.323577
$45$	0	0
$46$	5.88240	0.867313
$47$	11.2713	1.64409	0.822045	−	0.569423i	$-0.192833\pi$
$47$	11.2713	1.64409	0.822045	+	0.569423i	$0.192833\pi$
$48$	0	0
$49$	6.58546	0.940780
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.48929	−0.206527
$53$	5.51806	0.757964	0.378982	−	0.925404i	$-0.376274\pi$
$53$	5.51806	0.757964	0.378982	+	0.925404i	$0.376274\pi$
$54$	0	0
$55$	−2.14637	−0.289416
$56$	−3.68585	−0.492542
$57$	0	0
$58$	1.80344	0.236804
$59$	−13.6644	−1.77896	−0.889478	−	0.456978i	$-0.848932\pi$
$59$	−13.6644	−1.77896	−0.889478	+	0.456978i	$0.848932\pi$
$60$	0	0
$61$	−3.63565	−0.465498	−0.232749	−	0.972537i	$-0.574772\pi$
$61$	−3.63565	−0.465498	−0.232749	+	0.972537i	$0.574772\pi$
$62$	−10.0288	−1.27365
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.48929	−0.184724
$66$	0	0
$67$	15.6644	1.91371	0.956857	−	0.290559i	$-0.0938414\pi$
$67$	15.6644	1.91371	0.956857	+	0.290559i	$0.0938414\pi$
$68$	0.707269	0.0857690
$69$	0	0
$70$	−3.68585	−0.440543
$71$	7.66442	0.909600	0.454800	−	0.890594i	$-0.349711\pi$
$71$	7.66442	0.909600	0.454800	+	0.890594i	$0.349711\pi$
$72$	0	0
$73$	15.0533	1.76186	0.880929	−	0.473248i	$-0.156918\pi$
$73$	15.0533	1.76186	0.880929	+	0.473248i	$0.156918\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−1.48929	−0.170833
$77$	−7.91117	−0.901562
$78$	0	0
$79$	12.2927	1.38304	0.691520	−	0.722357i	$-0.256941\pi$
$79$	12.2927	1.38304	0.691520	+	0.722357i	$0.256941\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.00735	−0.994695
$83$	−5.15371	−0.565693	−0.282847	−	0.959165i	$-0.591279\pi$
$83$	−5.15371	−0.565693	−0.282847	+	0.959165i	$0.591279\pi$
$84$	0	0
$85$	0.707269	0.0767141
$86$	−2.36435	−0.254954
$87$	0	0
$88$	2.14637	0.228803
$89$	4.46787	0.473593	0.236796	−	0.971559i	$-0.423903\pi$
$89$	4.46787	0.473593	0.236796	+	0.971559i	$0.423903\pi$
$90$	0	0
$91$	−5.48929	−0.575434
$92$	−5.88240	−0.613283
$93$	0	0
$94$	−11.2713	−1.16255
$95$	−1.48929	−0.152798
$96$	0	0
$97$	−7.00735	−0.711488	−0.355744	−	0.934583i	$-0.615773\pi$
$97$	−7.00735	−0.711488	−0.355744	+	0.934583i	$0.615773\pi$
$98$	−6.58546	−0.665232
$99$	0	0
$100$	1.00000	0.100000
$101$	2.39312	0.238124	0.119062	−	0.992887i	$-0.462011\pi$
$101$	2.39312	0.238124	0.119062	+	0.992887i	$0.462011\pi$
$102$	0	0
$103$	−16.0147	−1.57797	−0.788987	−	0.614410i	$-0.789394\pi$
$103$	−16.0147	−1.57797	−0.788987	+	0.614410i	$0.789394\pi$
$104$	1.48929	0.146037
$105$	0	0
$106$	−5.51806	−0.535961
$107$	−13.0533	−1.26191	−0.630956	−	0.775818i	$-0.717337\pi$
$107$	−13.0533	−1.26191	−0.630956	+	0.775818i	$0.717337\pi$
$108$	0	0
$109$	−1.12494	−0.107750	−0.0538750	−	0.998548i	$-0.517157\pi$
$109$	−1.12494	−0.107750	−0.0538750	+	0.998548i	$0.517157\pi$
$110$	2.14637	0.204648
$111$	0	0
$112$	3.68585	0.348280
$113$	16.4464	1.54715	0.773576	−	0.633704i	$-0.218466\pi$
$113$	16.4464	1.54715	0.773576	+	0.633704i	$0.218466\pi$
$114$	0	0
$115$	−5.88240	−0.548537
$116$	−1.80344	−0.167445
$117$	0	0
$118$	13.6644	1.25791
$119$	2.60688	0.238973
$120$	0	0
$121$	−6.39312	−0.581192
$122$	3.63565	0.329157
$123$	0	0
$124$	10.0288	0.900610
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.1537	1.69962	0.849809	−	0.527091i	$-0.176717\pi$
$127$	19.1537	1.69962	0.849809	+	0.527091i	$0.176717\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.48929	0.130619
$131$	13.6644	1.19387	0.596933	−	0.802291i	$-0.296386\pi$
$131$	13.6644	1.19387	0.596933	+	0.802291i	$0.296386\pi$
$132$	0	0
$133$	−5.48929	−0.475982
$134$	−15.6644	−1.35320
$135$	0	0
$136$	−0.707269	−0.0606478
$137$	0.978577	0.0836055	0.0418027	−	0.999126i	$-0.486690\pi$
$137$	0.978577	0.0836055	0.0418027	+	0.999126i	$0.486690\pi$
$138$	0	0
$139$	10.7820	0.914519	0.457259	−	0.889333i	$-0.348831\pi$
$139$	10.7820	0.914519	0.457259	+	0.889333i	$0.348831\pi$
$140$	3.68585	0.311511
$141$	0	0
$142$	−7.66442	−0.643184
$143$	3.19656	0.267310
$144$	0	0
$145$	−1.80344	−0.149768
$146$	−15.0533	−1.24582
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−13.6644	−1.11943	−0.559717	−	0.828684i	$-0.689090\pi$
$149$	−13.6644	−1.11943	−0.559717	+	0.828684i	$0.689090\pi$
$150$	0	0
$151$	−0.568250	−0.0462435	−0.0231218	−	0.999733i	$-0.507361\pi$
$151$	−0.568250	−0.0462435	−0.0231218	+	0.999733i	$0.507361\pi$
$152$	1.48929	0.120797
$153$	0	0
$154$	7.91117	0.637500
$155$	10.0288	0.805530
$156$	0	0
$157$	−2.68164	−0.214018	−0.107009	−	0.994258i	$-0.534127\pi$
$157$	−2.68164	−0.214018	−0.107009	+	0.994258i	$0.534127\pi$
$158$	−12.2927	−0.977957
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−21.6816	−1.70875
$162$	0	0
$163$	−4.93260	−0.386351	−0.193175	−	0.981164i	$-0.561879\pi$
$163$	−4.93260	−0.386351	−0.193175	+	0.981164i	$0.561879\pi$
$164$	9.00735	0.703356
$165$	0	0
$166$	5.15371	0.400006
$167$	−14.8181	−1.14666	−0.573331	−	0.819324i	$-0.694349\pi$
$167$	−14.8181	−1.14666	−0.573331	+	0.819324i	$0.694349\pi$
$168$	0	0
$169$	−10.7820	−0.829386
$170$	−0.707269	−0.0542451
$171$	0	0
$172$	2.36435	0.180280
$173$	−3.56090	−0.270730	−0.135365	−	0.990796i	$-0.543221\pi$
$173$	−3.56090	−0.270730	−0.135365	+	0.990796i	$0.543221\pi$
$174$	0	0
$175$	3.68585	0.278624
$176$	−2.14637	−0.161788
$177$	0	0
$178$	−4.46787	−0.334881
$179$	−22.0575	−1.64866	−0.824329	−	0.566111i	$-0.808447\pi$
$179$	−22.0575	−1.64866	−0.824329	+	0.566111i	$0.808447\pi$
$180$	0	0
$181$	11.6216	0.863825	0.431913	−	0.901915i	$-0.357839\pi$
$181$	11.6216	0.863825	0.431913	+	0.901915i	$0.357839\pi$
$182$	5.48929	0.406893
$183$	0	0
$184$	5.88240	0.433657
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−1.51806	−0.111011
$188$	11.2713	0.822045
$189$	0	0
$190$	1.48929	0.108044
$191$	8.17092	0.591227	0.295614	−	0.955308i	$-0.404476\pi$
$191$	8.17092	0.591227	0.295614	+	0.955308i	$0.404476\pi$
$192$	0	0
$193$	3.41454	0.245784	0.122892	−	0.992420i	$-0.460783\pi$
$193$	3.41454	0.245784	0.122892	+	0.992420i	$0.460783\pi$
$194$	7.00735	0.503098
$195$	0	0
$196$	6.58546	0.470390
$197$	−2.56825	−0.182980	−0.0914901	−	0.995806i	$-0.529163\pi$
$197$	−2.56825	−0.182980	−0.0914901	+	0.995806i	$0.529163\pi$
$198$	0	0
$199$	12.0575	0.854736	0.427368	−	0.904078i	$-0.359441\pi$
$199$	12.0575	0.854736	0.427368	+	0.904078i	$0.359441\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−2.39312	−0.168379
$203$	−6.64721	−0.466543
$204$	0	0
$205$	9.00735	0.629100
$206$	16.0147	1.11580
$207$	0	0
$208$	−1.48929	−0.103264
$209$	3.19656	0.221111
$210$	0	0
$211$	3.56825	0.245648	0.122824	−	0.992428i	$-0.460805\pi$
$211$	3.56825	0.245648	0.122824	+	0.992428i	$0.460805\pi$
$212$	5.51806	0.378982
$213$	0	0
$214$	13.0533	0.892307
$215$	2.36435	0.161247
$216$	0	0
$217$	36.9645	2.50931
$218$	1.12494	0.0761907
$219$	0	0
$220$	−2.14637	−0.144708
$221$	−1.05333	−0.0708545
$222$	0	0
$223$	8.88240	0.594810	0.297405	−	0.954751i	$-0.403879\pi$
$223$	8.88240	0.594810	0.297405	+	0.954751i	$0.403879\pi$
$224$	−3.68585	−0.246271
$225$	0	0
$226$	−16.4464	−1.09400
$227$	23.8610	1.58371	0.791854	−	0.610710i	$-0.209116\pi$
$227$	23.8610	1.58371	0.791854	+	0.610710i	$0.209116\pi$
$228$	0	0
$229$	−9.03612	−0.597123	−0.298562	−	0.954390i	$-0.596507\pi$
$229$	−9.03612	−0.597123	−0.298562	+	0.954390i	$0.596507\pi$
$230$	5.88240	0.387874
$231$	0	0
$232$	1.80344	0.118402
$233$	19.9143	1.30463	0.652315	−	0.757948i	$-0.273798\pi$
$233$	19.9143	1.30463	0.652315	+	0.757948i	$0.273798\pi$
$234$	0	0
$235$	11.2713	0.735259
$236$	−13.6644	−0.889478
$237$	0	0
$238$	−2.60688	−0.168979
$239$	5.61110	0.362952	0.181476	−	0.983395i	$-0.441913\pi$
$239$	5.61110	0.362952	0.181476	+	0.983395i	$0.441913\pi$
$240$	0	0
$241$	−16.7434	−1.07854	−0.539268	−	0.842134i	$-0.681299\pi$
$241$	−16.7434	−1.07854	−0.539268	+	0.842134i	$0.681299\pi$
$242$	6.39312	0.410965
$243$	0	0
$244$	−3.63565	−0.232749
$245$	6.58546	0.420730
$246$	0	0
$247$	2.21798	0.141127
$248$	−10.0288	−0.636827
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−7.17935	−0.453156	−0.226578	−	0.973993i	$-0.572754\pi$
$251$	−7.17935	−0.453156	−0.226578	+	0.973993i	$0.572754\pi$
$252$	0	0
$253$	12.6258	0.793777
$254$	−19.1537	−1.20181
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.5468	1.96784	0.983919	−	0.178618i	$-0.0571626\pi$
$257$	31.5468	1.96784	0.983919	+	0.178618i	$0.0571626\pi$
$258$	0	0
$259$	−3.68585	−0.229027
$260$	−1.48929	−0.0923618
$261$	0	0
$262$	−13.6644	−0.844191
$263$	2.56511	0.158172	0.0790859	−	0.996868i	$-0.474800\pi$
$263$	2.56511	0.158172	0.0790859	+	0.996868i	$0.474800\pi$
$264$	0	0
$265$	5.51806	0.338972
$266$	5.48929	0.336570
$267$	0	0
$268$	15.6644	0.956857
$269$	−9.19656	−0.560724	−0.280362	−	0.959894i	$-0.590455\pi$
$269$	−9.19656	−0.560724	−0.280362	+	0.959894i	$0.590455\pi$
$270$	0	0
$271$	−31.3288	−1.90309	−0.951546	−	0.307507i	$-0.900505\pi$
$271$	−31.3288	−1.90309	−0.951546	+	0.307507i	$0.900505\pi$
$272$	0.707269	0.0428845
$273$	0	0
$274$	−0.978577	−0.0591180
$275$	−2.14637	−0.129431
$276$	0	0
$277$	23.5040	1.41222	0.706109	−	0.708103i	$-0.250449\pi$
$277$	23.5040	1.41222	0.706109	+	0.708103i	$0.250449\pi$
$278$	−10.7820	−0.646663
$279$	0	0
$280$	−3.68585	−0.220271
$281$	−5.15371	−0.307445	−0.153722	−	0.988114i	$-0.549126\pi$
$281$	−5.15371	−0.307445	−0.153722	+	0.988114i	$0.549126\pi$
$282$	0	0
$283$	22.7753	1.35385	0.676925	−	0.736052i	$-0.263312\pi$
$283$	22.7753	1.35385	0.676925	+	0.736052i	$0.263312\pi$
$284$	7.66442	0.454800
$285$	0	0
$286$	−3.19656	−0.189016
$287$	33.1997	1.95972
$288$	0	0
$289$	−16.4998	−0.970575
$290$	1.80344	0.105902
$291$	0	0
$292$	15.0533	0.880929
$293$	−14.4538	−0.844399	−0.422200	−	0.906503i	$-0.638742\pi$
$293$	−14.4538	−0.844399	−0.422200	+	0.906503i	$0.638742\pi$
$294$	0	0
$295$	−13.6644	−0.795573
$296$	1.00000	0.0581238
$297$	0	0
$298$	13.6644	0.791559
$299$	8.76060	0.506638
$300$	0	0
$301$	8.71462	0.502302
$302$	0.568250	0.0326991
$303$	0	0
$304$	−1.48929	−0.0854166
$305$	−3.63565	−0.208177
$306$	0	0
$307$	−5.46365	−0.311827	−0.155914	−	0.987771i	$-0.549832\pi$
$307$	−5.46365	−0.311827	−0.155914	+	0.987771i	$0.549832\pi$
$308$	−7.91117	−0.450781
$309$	0	0
$310$	−10.0288	−0.569596
$311$	−1.13229	−0.0642062	−0.0321031	−	0.999485i	$-0.510220\pi$
$311$	−1.13229	−0.0642062	−0.0321031	+	0.999485i	$0.510220\pi$
$312$	0	0
$313$	3.41454	0.193001	0.0965006	−	0.995333i	$-0.469235\pi$
$313$	3.41454	0.193001	0.0965006	+	0.995333i	$0.469235\pi$
$314$	2.68164	0.151333
$315$	0	0
$316$	12.2927	0.691520
$317$	17.3429	0.974076	0.487038	−	0.873381i	$-0.338077\pi$
$317$	17.3429	0.974076	0.487038	+	0.873381i	$0.338077\pi$
$318$	0	0
$319$	3.87085	0.216726
$320$	1.00000	0.0559017
$321$	0	0
$322$	21.6816	1.20827
$323$	−1.05333	−0.0586087
$324$	0	0
$325$	−1.48929	−0.0826109
$326$	4.93260	0.273191
$327$	0	0
$328$	−9.00735	−0.497348
$329$	41.5443	2.29041
$330$	0	0
$331$	−1.12181	−0.0616601	−0.0308300	−	0.999525i	$-0.509815\pi$
$331$	−1.12181	−0.0616601	−0.0308300	+	0.999525i	$0.509815\pi$
$332$	−5.15371	−0.282847
$333$	0	0
$334$	14.8181	0.810812
$335$	15.6644	0.855839
$336$	0	0
$337$	−8.66863	−0.472211	−0.236105	−	0.971727i	$-0.575871\pi$
$337$	−8.66863	−0.472211	−0.236105	+	0.971727i	$0.575871\pi$
$338$	10.7820	0.586465
$339$	0	0
$340$	0.707269	0.0383570
$341$	−21.5254	−1.16567
$342$	0	0
$343$	−1.52792	−0.0825001
$344$	−2.36435	−0.127477
$345$	0	0
$346$	3.56090	0.191435
$347$	5.02142	0.269564	0.134782	−	0.990875i	$-0.456967\pi$
$347$	5.02142	0.269564	0.134782	+	0.990875i	$0.456967\pi$
$348$	0	0
$349$	16.1004	0.861834	0.430917	−	0.902392i	$-0.358190\pi$
$349$	16.1004	0.861834	0.430917	+	0.902392i	$0.358190\pi$
$350$	−3.68585	−0.197017
$351$	0	0
$352$	2.14637	0.114402
$353$	15.6686	0.833957	0.416979	−	0.908916i	$-0.363089\pi$
$353$	15.6686	0.833957	0.416979	+	0.908916i	$0.363089\pi$
$354$	0	0
$355$	7.66442	0.406785
$356$	4.46787	0.236796
$357$	0	0
$358$	22.0575	1.16578
$359$	−33.7648	−1.78204	−0.891019	−	0.453966i	$-0.850009\pi$
$359$	−33.7648	−1.78204	−0.891019	+	0.453966i	$0.850009\pi$
$360$	0	0
$361$	−16.7820	−0.883264
$362$	−11.6216	−0.610817
$363$	0	0
$364$	−5.48929	−0.287717
$365$	15.0533	0.787927
$366$	0	0
$367$	26.0361	1.35907	0.679537	−	0.733641i	$-0.262181\pi$
$367$	26.0361	1.35907	0.679537	+	0.733641i	$0.262181\pi$
$368$	−5.88240	−0.306641
$369$	0	0
$370$	1.00000	0.0519875
$371$	20.3387	1.05593
$372$	0	0
$373$	−26.3503	−1.36437	−0.682183	−	0.731182i	$-0.738969\pi$
$373$	−26.3503	−1.36437	−0.682183	+	0.731182i	$0.738969\pi$
$374$	1.51806	0.0784969
$375$	0	0
$376$	−11.2713	−0.581273
$377$	2.68585	0.138328
$378$	0	0
$379$	3.21377	0.165080	0.0825401	−	0.996588i	$-0.473697\pi$
$379$	3.21377	0.165080	0.0825401	+	0.996588i	$0.473697\pi$
$380$	−1.48929	−0.0763989
$381$	0	0
$382$	−8.17092	−0.418061
$383$	28.4250	1.45245	0.726225	−	0.687457i	$-0.241273\pi$
$383$	28.4250	1.45245	0.726225	+	0.687457i	$0.241273\pi$
$384$	0	0
$385$	−7.91117	−0.403191
$386$	−3.41454	−0.173795
$387$	0	0
$388$	−7.00735	−0.355744
$389$	−8.78202	−0.445266	−0.222633	−	0.974902i	$-0.571465\pi$
$389$	−8.78202	−0.445266	−0.222633	+	0.974902i	$0.571465\pi$
$390$	0	0
$391$	−4.16044	−0.210403
$392$	−6.58546	−0.332616
$393$	0	0
$394$	2.56825	0.129387
$395$	12.2927	0.618514
$396$	0	0
$397$	13.2138	0.663180	0.331590	−	0.943424i	$-0.392415\pi$
$397$	13.2138	0.663180	0.331590	+	0.943424i	$0.392415\pi$
$398$	−12.0575	−0.604390
$399$	0	0
$400$	1.00000	0.0500000
$401$	1.97437	0.0985951	0.0492976	−	0.998784i	$-0.484302\pi$
$401$	1.97437	0.0985951	0.0492976	+	0.998784i	$0.484302\pi$
$402$	0	0
$403$	−14.9357	−0.744002
$404$	2.39312	0.119062
$405$	0	0
$406$	6.64721	0.329896
$407$	2.14637	0.106391
$408$	0	0
$409$	−35.4868	−1.75471	−0.877354	−	0.479844i	$-0.840693\pi$
$409$	−35.4868	−1.75471	−0.877354	+	0.479844i	$0.840693\pi$
$410$	−9.00735	−0.444841
$411$	0	0
$412$	−16.0147	−0.788987
$413$	−50.3650	−2.47830
$414$	0	0
$415$	−5.15371	−0.252986
$416$	1.48929	0.0730184
$417$	0	0
$418$	−3.19656	−0.156349
$419$	27.4464	1.34085	0.670423	−	0.741979i	$-0.266113\pi$
$419$	27.4464	1.34085	0.670423	+	0.741979i	$0.266113\pi$
$420$	0	0
$421$	−16.9357	−0.825397	−0.412699	−	0.910868i	$-0.635414\pi$
$421$	−16.9357	−0.825397	−0.412699	+	0.910868i	$0.635414\pi$
$422$	−3.56825	−0.173700
$423$	0	0
$424$	−5.51806	−0.267981
$425$	0.707269	0.0343076
$426$	0	0
$427$	−13.4005	−0.648494
$428$	−13.0533	−0.630956
$429$	0	0
$430$	−2.36435	−0.114019
$431$	−12.9572	−0.624124	−0.312062	−	0.950062i	$-0.601020\pi$
$431$	−12.9572	−0.624124	−0.312062	+	0.950062i	$0.601020\pi$
$432$	0	0
$433$	−13.8824	−0.667146	−0.333573	−	0.942724i	$-0.608254\pi$
$433$	−13.8824	−0.667146	−0.333573	+	0.942724i	$0.608254\pi$
$434$	−36.9645	−1.77435
$435$	0	0
$436$	−1.12494	−0.0538750
$437$	8.76060	0.419076
$438$	0	0
$439$	−11.0502	−0.527397	−0.263698	−	0.964605i	$-0.584942\pi$
$439$	−11.0502	−0.527397	−0.263698	+	0.964605i	$0.584942\pi$
$440$	2.14637	0.102324
$441$	0	0
$442$	1.05333	0.0501017
$443$	23.7220	1.12706	0.563532	−	0.826094i	$-0.309442\pi$
$443$	23.7220	1.12706	0.563532	+	0.826094i	$0.309442\pi$
$444$	0	0
$445$	4.46787	0.211797
$446$	−8.88240	−0.420594
$447$	0	0
$448$	3.68585	0.174140
$449$	−10.8438	−0.511749	−0.255875	−	0.966710i	$-0.582363\pi$
$449$	−10.8438	−0.511749	−0.255875	+	0.966710i	$0.582363\pi$
$450$	0	0
$451$	−19.3331	−0.910358
$452$	16.4464	0.773576
$453$	0	0
$454$	−23.8610	−1.11985
$455$	−5.48929	−0.257342
$456$	0	0
$457$	1.43489	0.0671211	0.0335606	−	0.999437i	$-0.489315\pi$
$457$	1.43489	0.0671211	0.0335606	+	0.999437i	$0.489315\pi$
$458$	9.03612	0.422230
$459$	0	0
$460$	−5.88240	−0.274268
$461$	22.1537	1.03180	0.515901	−	0.856648i	$-0.327457\pi$
$461$	22.1537	1.03180	0.515901	+	0.856648i	$0.327457\pi$
$462$	0	0
$463$	−42.4653	−1.97353	−0.986766	−	0.162151i	$-0.948157\pi$
$463$	−42.4653	−1.97353	−0.986766	+	0.162151i	$0.948157\pi$
$464$	−1.80344	−0.0837227
$465$	0	0
$466$	−19.9143	−0.922513
$467$	28.6472	1.32563	0.662817	−	0.748781i	$-0.269361\pi$
$467$	28.6472	1.32563	0.662817	+	0.748781i	$0.269361\pi$
$468$	0	0
$469$	57.7367	2.66603
$470$	−11.2713	−0.519907
$471$	0	0
$472$	13.6644	0.628956
$473$	−5.07475	−0.233337
$474$	0	0
$475$	−1.48929	−0.0683332
$476$	2.60688	0.119486
$477$	0	0
$478$	−5.61110	−0.256646
$479$	6.41875	0.293280	0.146640	−	0.989190i	$-0.453154\pi$
$479$	6.41875	0.293280	0.146640	+	0.989190i	$0.453154\pi$
$480$	0	0
$481$	1.48929	0.0679057
$482$	16.7434	0.762640
$483$	0	0
$484$	−6.39312	−0.290596
$485$	−7.00735	−0.318187
$486$	0	0
$487$	−39.9227	−1.80907	−0.904536	−	0.426398i	$-0.859782\pi$
$487$	−39.9227	−1.80907	−0.904536	+	0.426398i	$0.859782\pi$
$488$	3.63565	0.164578
$489$	0	0
$490$	−6.58546	−0.297501
$491$	−16.9038	−0.762859	−0.381430	−	0.924398i	$-0.624568\pi$
$491$	−16.9038	−0.762859	−0.381430	+	0.924398i	$0.624568\pi$
$492$	0	0
$493$	−1.27552	−0.0574465
$494$	−2.21798	−0.0997917
$495$	0	0
$496$	10.0288	0.450305
$497$	28.2499	1.26718
$498$	0	0
$499$	37.9118	1.69716	0.848582	−	0.529063i	$-0.177457\pi$
$499$	37.9118	1.69716	0.848582	+	0.529063i	$0.177457\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	7.17935	0.320430
$503$	36.0294	1.60647	0.803235	−	0.595662i	$-0.203110\pi$
$503$	36.0294	1.60647	0.803235	+	0.595662i	$0.203110\pi$
$504$	0	0
$505$	2.39312	0.106492
$506$	−12.6258	−0.561285
$507$	0	0
$508$	19.1537	0.849809
$509$	2.56825	0.113836	0.0569178	−	0.998379i	$-0.481873\pi$
$509$	2.56825	0.113836	0.0569178	+	0.998379i	$0.481873\pi$
$510$	0	0
$511$	55.4843	2.45448
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−31.5468	−1.39147
$515$	−16.0147	−0.705692
$516$	0	0
$517$	−24.1923	−1.06398
$518$	3.68585	0.161947
$519$	0	0
$520$	1.48929	0.0653096
$521$	10.2295	0.448164	0.224082	−	0.974570i	$-0.428062\pi$
$521$	10.2295	0.448164	0.224082	+	0.974570i	$0.428062\pi$
$522$	0	0
$523$	−16.2583	−0.710926	−0.355463	−	0.934690i	$-0.615677\pi$
$523$	−16.2583	−0.710926	−0.355463	+	0.934690i	$0.615677\pi$
$524$	13.6644	0.596933
$525$	0	0
$526$	−2.56511	−0.111844
$527$	7.09304	0.308978
$528$	0	0
$529$	11.6027	0.504464
$530$	−5.51806	−0.239689
$531$	0	0
$532$	−5.48929	−0.237991
$533$	−13.4145	−0.581048
$534$	0	0
$535$	−13.0533	−0.564345
$536$	−15.6644	−0.676600
$537$	0	0
$538$	9.19656	0.396492
$539$	−14.1348	−0.608829
$540$	0	0
$541$	−40.8757	−1.75738	−0.878691	−	0.477391i	$-0.841583\pi$
$541$	−40.8757	−1.75738	−0.878691	+	0.477391i	$0.841583\pi$
$542$	31.3288	1.34569
$543$	0	0
$544$	−0.707269	−0.0303239
$545$	−1.12494	−0.0481872
$546$	0	0
$547$	−22.5458	−0.963987	−0.481993	−	0.876175i	$-0.660087\pi$
$547$	−22.5458	−0.963987	−0.481993	+	0.876175i	$0.660087\pi$
$548$	0.978577	0.0418027
$549$	0	0
$550$	2.14637	0.0915213
$551$	2.68585	0.114421
$552$	0	0
$553$	45.3091	1.92674
$554$	−23.5040	−0.998588
$555$	0	0
$556$	10.7820	0.457259
$557$	25.7795	1.09231	0.546156	−	0.837683i	$-0.316090\pi$
$557$	25.7795	1.09231	0.546156	+	0.837683i	$0.316090\pi$
$558$	0	0
$559$	−3.52119	−0.148931
$560$	3.68585	0.155755
$561$	0	0
$562$	5.15371	0.217396
$563$	−36.1684	−1.52432	−0.762158	−	0.647391i	$-0.775860\pi$
$563$	−36.1684	−1.52432	−0.762158	+	0.647391i	$0.775860\pi$
$564$	0	0
$565$	16.4464	0.691907
$566$	−22.7753	−0.957317
$567$	0	0
$568$	−7.66442	−0.321592
$569$	−15.8824	−0.665825	−0.332913	−	0.942958i	$-0.608031\pi$
$569$	−15.8824	−0.665825	−0.332913	+	0.942958i	$0.608031\pi$
$570$	0	0
$571$	−11.9529	−0.500215	−0.250108	−	0.968218i	$-0.580466\pi$
$571$	−11.9529	−0.500215	−0.250108	+	0.968218i	$0.580466\pi$
$572$	3.19656	0.133655
$573$	0	0
$574$	−33.1997	−1.38573
$575$	−5.88240	−0.245313
$576$	0	0
$577$	8.20077	0.341402	0.170701	−	0.985323i	$-0.445397\pi$
$577$	8.20077	0.341402	0.170701	+	0.985323i	$0.445397\pi$
$578$	16.4998	0.686300
$579$	0	0
$580$	−1.80344	−0.0748839
$581$	−18.9958	−0.788078
$582$	0	0
$583$	−11.8438	−0.490519
$584$	−15.0533	−0.622911
$585$	0	0
$586$	14.4538	0.597081
$587$	26.4549	1.09191	0.545955	−	0.837815i	$-0.316167\pi$
$587$	26.4549	1.09191	0.545955	+	0.837815i	$0.316167\pi$
$588$	0	0
$589$	−14.9357	−0.615416
$590$	13.6644	0.562555
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−1.21377	−0.0498435	−0.0249218	−	0.999689i	$-0.507934\pi$
$593$	−1.21377	−0.0498435	−0.0249218	+	0.999689i	$0.507934\pi$
$594$	0	0
$595$	2.60688	0.106872
$596$	−13.6644	−0.559717
$597$	0	0
$598$	−8.76060	−0.358247
$599$	5.70727	0.233193	0.116596	−	0.993179i	$-0.462802\pi$
$599$	5.70727	0.233193	0.116596	+	0.993179i	$0.462802\pi$
$600$	0	0
$601$	−2.06427	−0.0842033	−0.0421016	−	0.999113i	$-0.513405\pi$
$601$	−2.06427	−0.0842033	−0.0421016	+	0.999113i	$0.513405\pi$
$602$	−8.71462	−0.355181
$603$	0	0
$604$	−0.568250	−0.0231218
$605$	−6.39312	−0.259917
$606$	0	0
$607$	29.6791	1.20464	0.602319	−	0.798255i	$-0.294243\pi$
$607$	29.6791	1.20464	0.602319	+	0.798255i	$0.294243\pi$
$608$	1.48929	0.0603986
$609$	0	0
$610$	3.63565	0.147203
$611$	−16.7862	−0.679098
$612$	0	0
$613$	−34.2969	−1.38524	−0.692620	−	0.721302i	$-0.743544\pi$
$613$	−34.2969	−1.38524	−0.692620	+	0.721302i	$0.743544\pi$
$614$	5.46365	0.220495
$615$	0	0
$616$	7.91117	0.318750
$617$	−26.7005	−1.07492	−0.537462	−	0.843288i	$-0.680617\pi$
$617$	−26.7005	−1.07492	−0.537462	+	0.843288i	$0.680617\pi$
$618$	0	0
$619$	5.02563	0.201997	0.100999	−	0.994887i	$-0.467796\pi$
$619$	5.02563	0.201997	0.100999	+	0.994887i	$0.467796\pi$
$620$	10.0288	0.402765
$621$	0	0
$622$	1.13229	0.0454007
$623$	16.4679	0.659771
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.41454	−0.136472
$627$	0	0
$628$	−2.68164	−0.107009
$629$	−0.707269	−0.0282007
$630$	0	0
$631$	−3.57812	−0.142443	−0.0712213	−	0.997461i	$-0.522690\pi$
$631$	−3.57812	−0.142443	−0.0712213	+	0.997461i	$0.522690\pi$
$632$	−12.2927	−0.488979
$633$	0	0
$634$	−17.3429	−0.688775
$635$	19.1537	0.760092
$636$	0	0
$637$	−9.80765	−0.388593
$638$	−3.87085	−0.153248
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−11.8364	−0.467511	−0.233755	−	0.972295i	$-0.575101\pi$
$641$	−11.8364	−0.467511	−0.233755	+	0.972295i	$0.575101\pi$
$642$	0	0
$643$	−33.9540	−1.33902	−0.669508	−	0.742805i	$-0.733495\pi$
$643$	−33.9540	−1.33902	−0.669508	+	0.742805i	$0.733495\pi$
$644$	−21.6816	−0.854376
$645$	0	0
$646$	1.05333	0.0414426
$647$	40.5745	1.59515	0.797575	−	0.603220i	$-0.206116\pi$
$647$	40.5745	1.59515	0.797575	+	0.603220i	$0.206116\pi$
$648$	0	0
$649$	29.3288	1.15126
$650$	1.48929	0.0584147
$651$	0	0
$652$	−4.93260	−0.193175
$653$	−23.5149	−0.920210	−0.460105	−	0.887865i	$-0.652188\pi$
$653$	−23.5149	−0.920210	−0.460105	+	0.887865i	$0.652188\pi$
$654$	0	0
$655$	13.6644	0.533913
$656$	9.00735	0.351678
$657$	0	0
$658$	−41.5443	−1.61957
$659$	23.9718	0.933811	0.466905	−	0.884307i	$-0.345369\pi$
$659$	23.9718	0.933811	0.466905	+	0.884307i	$0.345369\pi$
$660$	0	0
$661$	−40.9473	−1.59266	−0.796332	−	0.604859i	$-0.793229\pi$
$661$	−40.9473	−1.59266	−0.796332	+	0.604859i	$0.793229\pi$
$662$	1.12181	0.0436003
$663$	0	0
$664$	5.15371	0.200003
$665$	−5.48929	−0.212865
$666$	0	0
$667$	10.6086	0.410766
$668$	−14.8181	−0.573331
$669$	0	0
$670$	−15.6644	−0.605169
$671$	7.80344	0.301249
$672$	0	0
$673$	13.2113	0.509256	0.254628	−	0.967039i	$-0.418047\pi$
$673$	13.2113	0.509256	0.254628	+	0.967039i	$0.418047\pi$
$674$	8.66863	0.333903
$675$	0	0
$676$	−10.7820	−0.414693
$677$	21.4318	0.823689	0.411845	−	0.911254i	$-0.364885\pi$
$677$	21.4318	0.823689	0.411845	+	0.911254i	$0.364885\pi$
$678$	0	0
$679$	−25.8280	−0.991188
$680$	−0.707269	−0.0271225
$681$	0	0
$682$	21.5254	0.824250
$683$	9.03190	0.345596	0.172798	−	0.984957i	$-0.444719\pi$
$683$	9.03190	0.345596	0.172798	+	0.984957i	$0.444719\pi$
$684$	0	0
$685$	0.978577	0.0373895
$686$	1.52792	0.0583364
$687$	0	0
$688$	2.36435	0.0901398
$689$	−8.21798	−0.313080
$690$	0	0
$691$	46.5468	1.77072	0.885362	−	0.464902i	$-0.153910\pi$
$691$	46.5468	1.77072	0.885362	+	0.464902i	$0.153910\pi$
$692$	−3.56090	−0.135365
$693$	0	0
$694$	−5.02142	−0.190611
$695$	10.7820	0.408985
$696$	0	0
$697$	6.37062	0.241304
$698$	−16.1004	−0.609409
$699$	0	0
$700$	3.68585	0.139312
$701$	−5.64973	−0.213387	−0.106694	−	0.994292i	$-0.534026\pi$
$701$	−5.64973	−0.213387	−0.106694	+	0.994292i	$0.534026\pi$
$702$	0	0
$703$	1.48929	0.0561696
$704$	−2.14637	−0.0808942
$705$	0	0
$706$	−15.6686	−0.589697
$707$	8.82065	0.331735
$708$	0	0
$709$	34.2959	1.28801	0.644004	−	0.765022i	$-0.277272\pi$
$709$	34.2959	1.28801	0.644004	+	0.765022i	$0.277272\pi$
$710$	−7.66442	−0.287641
$711$	0	0
$712$	−4.46787	−0.167440
$713$	−58.9933	−2.20932
$714$	0	0
$715$	3.19656	0.119544
$716$	−22.0575	−0.824329
$717$	0	0
$718$	33.7648	1.26009
$719$	−40.8585	−1.52376	−0.761882	−	0.647716i	$-0.775724\pi$
$719$	−40.8585	−1.52376	−0.761882	+	0.647716i	$0.775724\pi$
$720$	0	0
$721$	−59.0277	−2.19831
$722$	16.7820	0.624562
$723$	0	0
$724$	11.6216	0.431913
$725$	−1.80344	−0.0669782
$726$	0	0
$727$	−7.85677	−0.291391	−0.145696	−	0.989329i	$-0.546542\pi$
$727$	−7.85677	−0.291391	−0.145696	+	0.989329i	$0.546542\pi$
$728$	5.48929	0.203447
$729$	0	0
$730$	−15.0533	−0.557149
$731$	1.67223	0.0618496
$732$	0	0
$733$	−13.6686	−0.504863	−0.252431	−	0.967615i	$-0.581230\pi$
$733$	−13.6686	−0.504863	−0.252431	+	0.967615i	$0.581230\pi$
$734$	−26.0361	−0.961011
$735$	0	0
$736$	5.88240	0.216828
$737$	−33.6216	−1.23847
$738$	0	0
$739$	45.0894	1.65864	0.829321	−	0.558772i	$-0.188727\pi$
$739$	45.0894	1.65864	0.829321	+	0.558772i	$0.188727\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−20.3387	−0.746658
$743$	−5.63565	−0.206752	−0.103376	−	0.994642i	$-0.532965\pi$
$743$	−5.63565	−0.206752	−0.103376	+	0.994642i	$0.532965\pi$
$744$	0	0
$745$	−13.6644	−0.500626
$746$	26.3503	0.964752
$747$	0	0
$748$	−1.51806	−0.0555057
$749$	−48.1126	−1.75799
$750$	0	0
$751$	−9.70727	−0.354223	−0.177112	−	0.984191i	$-0.556675\pi$
$751$	−9.70727	−0.354223	−0.177112	+	0.984191i	$0.556675\pi$
$752$	11.2713	0.411022
$753$	0	0
$754$	−2.68585	−0.0978127
$755$	−0.568250	−0.0206807
$756$	0	0
$757$	−9.50398	−0.345428	−0.172714	−	0.984972i	$-0.555254\pi$
$757$	−9.50398	−0.345428	−0.172714	+	0.984972i	$0.555254\pi$
$758$	−3.21377	−0.116729
$759$	0	0
$760$	1.48929	0.0540222
$761$	−16.6227	−0.602571	−0.301285	−	0.953534i	$-0.597416\pi$
$761$	−16.6227	−0.602571	−0.301285	+	0.953534i	$0.597416\pi$
$762$	0	0
$763$	−4.14637	−0.150109
$764$	8.17092	0.295614
$765$	0	0
$766$	−28.4250	−1.02704
$767$	20.3503	0.734806
$768$	0	0
$769$	−1.06427	−0.0383785	−0.0191893	−	0.999816i	$-0.506109\pi$
$769$	−1.06427	−0.0383785	−0.0191893	+	0.999816i	$0.506109\pi$
$770$	7.91117	0.285099
$771$	0	0
$772$	3.41454	0.122892
$773$	−50.3618	−1.81139	−0.905695	−	0.423931i	$-0.860650\pi$
$773$	−50.3618	−1.81139	−0.905695	+	0.423931i	$0.860650\pi$
$774$	0	0
$775$	10.0288	0.360244
$776$	7.00735	0.251549
$777$	0	0
$778$	8.78202	0.314851
$779$	−13.4145	−0.480626
$780$	0	0
$781$	−16.4507	−0.588651
$782$	4.16044	0.148777
$783$	0	0
$784$	6.58546	0.235195
$785$	−2.68164	−0.0957117
$786$	0	0
$787$	14.2070	0.506426	0.253213	−	0.967411i	$-0.418513\pi$
$787$	14.2070	0.506426	0.253213	+	0.967411i	$0.418513\pi$
$788$	−2.56825	−0.0914901
$789$	0	0
$790$	−12.2927	−0.437356
$791$	60.6191	2.15537
$792$	0	0
$793$	5.41454	0.192276
$794$	−13.2138	−0.468939
$795$	0	0
$796$	12.0575	0.427368
$797$	−26.4225	−0.935933	−0.467966	−	0.883746i	$-0.655013\pi$
$797$	−26.4225	−0.935933	−0.467966	+	0.883746i	$0.655013\pi$
$798$	0	0
$799$	7.97185	0.282024
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−1.97437	−0.0697173
$803$	−32.3099	−1.14019
$804$	0	0
$805$	−21.6816	−0.764177
$806$	14.9357	0.526089
$807$	0	0
$808$	−2.39312	−0.0841895
$809$	−29.8824	−1.05061	−0.525305	−	0.850914i	$-0.676049\pi$
$809$	−29.8824	−1.05061	−0.525305	+	0.850914i	$0.676049\pi$
$810$	0	0
$811$	9.13650	0.320826	0.160413	−	0.987050i	$-0.448717\pi$
$811$	9.13650	0.320826	0.160413	+	0.987050i	$0.448717\pi$
$812$	−6.64721	−0.233271
$813$	0	0
$814$	−2.14637	−0.0752301
$815$	−4.93260	−0.172781
$816$	0	0
$817$	−3.52119	−0.123191
$818$	35.4868	1.24077
$819$	0	0
$820$	9.00735	0.314550
$821$	−33.7392	−1.17751	−0.588753	−	0.808313i	$-0.700381\pi$
$821$	−33.7392	−1.17751	−0.588753	+	0.808313i	$0.700381\pi$
$822$	0	0
$823$	−11.4746	−0.399979	−0.199990	−	0.979798i	$-0.564091\pi$
$823$	−11.4746	−0.399979	−0.199990	+	0.979798i	$0.564091\pi$
$824$	16.0147	0.557898
$825$	0	0
$826$	50.3650	1.75242
$827$	−3.89540	−0.135457	−0.0677283	−	0.997704i	$-0.521575\pi$
$827$	−3.89540	−0.135457	−0.0677283	+	0.997704i	$0.521575\pi$
$828$	0	0
$829$	−10.9389	−0.379923	−0.189961	−	0.981792i	$-0.560836\pi$
$829$	−10.9389	−0.379923	−0.189961	+	0.981792i	$0.560836\pi$
$830$	5.15371	0.178888
$831$	0	0
$832$	−1.48929	−0.0516318
$833$	4.65769	0.161380
$834$	0	0
$835$	−14.8181	−0.512803
$836$	3.19656	0.110555
$837$	0	0
$838$	−27.4464	−0.948122
$839$	35.1512	1.21355	0.606777	−	0.794872i	$-0.292462\pi$
$839$	35.1512	1.21355	0.606777	+	0.794872i	$0.292462\pi$
$840$	0	0
$841$	−25.7476	−0.887848
$842$	16.9357	0.583644
$843$	0	0
$844$	3.56825	0.122824
$845$	−10.7820	−0.370913
$846$	0	0
$847$	−23.5640	−0.809670
$848$	5.51806	0.189491
$849$	0	0
$850$	−0.707269	−0.0242591
$851$	5.88240	0.201646
$852$	0	0
$853$	−38.8610	−1.33057	−0.665287	−	0.746587i	$-0.731691\pi$
$853$	−38.8610	−1.33057	−0.665287	+	0.746587i	$0.731691\pi$
$854$	13.4005	0.458554
$855$	0	0
$856$	13.0533	0.446154
$857$	27.8866	0.952589	0.476294	−	0.879286i	$-0.341980\pi$
$857$	27.8866	0.952589	0.476294	+	0.879286i	$0.341980\pi$
$858$	0	0
$859$	−44.3822	−1.51430	−0.757150	−	0.653241i	$-0.773409\pi$
$859$	−44.3822	−1.51430	−0.757150	+	0.653241i	$0.773409\pi$
$860$	2.36435	0.0806235
$861$	0	0
$862$	12.9572	0.441322
$863$	13.3429	0.454198	0.227099	−	0.973872i	$-0.427076\pi$
$863$	13.3429	0.454198	0.227099	+	0.973872i	$0.427076\pi$
$864$	0	0
$865$	−3.56090	−0.121074
$866$	13.8824	0.471743
$867$	0	0
$868$	36.9645	1.25466
$869$	−26.3847	−0.895039
$870$	0	0
$871$	−23.3288	−0.790468
$872$	1.12494	0.0380954
$873$	0	0
$874$	−8.76060	−0.296332
$875$	3.68585	0.124604
$876$	0	0
$877$	−21.7324	−0.733852	−0.366926	−	0.930250i	$-0.619590\pi$
$877$	−21.7324	−0.733852	−0.366926	+	0.930250i	$0.619590\pi$
$878$	11.0502	0.372926
$879$	0	0
$880$	−2.14637	−0.0723540
$881$	−41.2144	−1.38855	−0.694274	−	0.719711i	$-0.744274\pi$
$881$	−41.2144	−1.38855	−0.694274	+	0.719711i	$0.744274\pi$
$882$	0	0
$883$	3.65287	0.122929	0.0614644	−	0.998109i	$-0.480423\pi$
$883$	3.65287	0.122929	0.0614644	+	0.998109i	$0.480423\pi$
$884$	−1.05333	−0.0354272
$885$	0	0
$886$	−23.7220	−0.796955
$887$	50.8297	1.70669	0.853347	−	0.521343i	$-0.174569\pi$
$887$	50.8297	1.70669	0.853347	+	0.521343i	$0.174569\pi$
$888$	0	0
$889$	70.5976	2.36777
$890$	−4.46787	−0.149763
$891$	0	0
$892$	8.88240	0.297405
$893$	−16.7862	−0.561730
$894$	0	0
$895$	−22.0575	−0.737302
$896$	−3.68585	−0.123135
$897$	0	0
$898$	10.8438	0.361861
$899$	−18.0863	−0.603212
$900$	0	0
$901$	3.90275	0.130019
$902$	19.3331	0.643720
$903$	0	0
$904$	−16.4464	−0.547001
$905$	11.6216	0.386314
$906$	0	0
$907$	−33.4549	−1.11085	−0.555425	−	0.831566i	$-0.687445\pi$
$907$	−33.4549	−1.11085	−0.555425	+	0.831566i	$0.687445\pi$
$908$	23.8610	0.791854
$909$	0	0
$910$	5.48929	0.181968
$911$	−19.2432	−0.637554	−0.318777	−	0.947830i	$-0.603272\pi$
$911$	−19.2432	−0.637554	−0.318777	+	0.947830i	$0.603272\pi$
$912$	0	0
$913$	11.0617	0.366090
$914$	−1.43489	−0.0474618
$915$	0	0
$916$	−9.03612	−0.298562
$917$	50.3650	1.66320
$918$	0	0
$919$	−38.1642	−1.25892	−0.629460	−	0.777033i	$-0.716724\pi$
$919$	−38.1642	−1.25892	−0.629460	+	0.777033i	$0.716724\pi$
$920$	5.88240	0.193937
$921$	0	0
$922$	−22.1537	−0.729594
$923$	−11.4145	−0.375714
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	42.4653	1.39550
$927$	0	0
$928$	1.80344	0.0592009
$929$	−31.9284	−1.04754	−0.523768	−	0.851861i	$-0.675474\pi$
$929$	−31.9284	−1.04754	−0.523768	+	0.851861i	$0.675474\pi$
$930$	0	0
$931$	−9.80765	−0.321433
$932$	19.9143	0.652315
$933$	0	0
$934$	−28.6472	−0.937365
$935$	−1.51806	−0.0496458
$936$	0	0
$937$	−14.5939	−0.476761	−0.238381	−	0.971172i	$-0.576617\pi$
$937$	−14.5939	−0.476761	−0.238381	+	0.971172i	$0.576617\pi$
$938$	−57.7367	−1.88517
$939$	0	0
$940$	11.2713	0.367630
$941$	−16.1579	−0.526733	−0.263367	−	0.964696i	$-0.584833\pi$
$941$	−16.1579	−0.526733	−0.263367	+	0.964696i	$0.584833\pi$
$942$	0	0
$943$	−52.9848	−1.72542
$944$	−13.6644	−0.444739
$945$	0	0
$946$	5.07475	0.164994
$947$	−30.7967	−1.00076	−0.500379	−	0.865806i	$-0.666806\pi$
$947$	−30.7967	−1.00076	−0.500379	+	0.865806i	$0.666806\pi$
$948$	0	0
$949$	−22.4187	−0.727743
$950$	1.48929	0.0483189
$951$	0	0
$952$	−2.60688	−0.0844896
$953$	44.7152	1.44847	0.724234	−	0.689554i	$-0.242193\pi$
$953$	44.7152	1.44847	0.724234	+	0.689554i	$0.242193\pi$
$954$	0	0
$955$	8.17092	0.264405
$956$	5.61110	0.181476
$957$	0	0
$958$	−6.41875	−0.207380
$959$	3.60688	0.116472
$960$	0	0
$961$	69.5762	2.24439
$962$	−1.48929	−0.0480166
$963$	0	0
$964$	−16.7434	−0.539268
$965$	3.41454	0.109918
$966$	0	0
$967$	−44.5657	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$967$	−44.5657	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$968$	6.39312	0.205483
$969$	0	0
$970$	7.00735	0.224992
$971$	4.65708	0.149453	0.0747264	−	0.997204i	$-0.476192\pi$
$971$	4.65708	0.149453	0.0747264	+	0.997204i	$0.476192\pi$
$972$	0	0
$973$	39.7409	1.27403
$974$	39.9227	1.27921
$975$	0	0
$976$	−3.63565	−0.116374
$977$	−31.2155	−0.998671	−0.499336	−	0.866409i	$-0.666422\pi$
$977$	−31.2155	−0.998671	−0.499336	+	0.866409i	$0.666422\pi$
$978$	0	0
$979$	−9.58967	−0.306487
$980$	6.58546	0.210365
$981$	0	0
$982$	16.9038	0.539423
$983$	16.5567	0.528076	0.264038	−	0.964512i	$-0.414945\pi$
$983$	16.5567	0.528076	0.264038	+	0.964512i	$0.414945\pi$
$984$	0	0
$985$	−2.56825	−0.0818312
$986$	1.27552	0.0406208
$987$	0	0
$988$	2.21798	0.0705634
$989$	−13.9080	−0.442250
$990$	0	0
$991$	30.9645	0.983620	0.491810	−	0.870703i	$-0.336336\pi$
$991$	30.9645	0.983620	0.491810	+	0.870703i	$0.336336\pi$
$992$	−10.0288	−0.318414
$993$	0	0
$994$	−28.2499	−0.896032
$995$	12.0575	0.382250
$996$	0	0
$997$	−26.5598	−0.841158	−0.420579	−	0.907256i	$-0.638173\pi$
$997$	−26.5598	−0.841158	−0.420579	+	0.907256i	$0.638173\pi$
$998$	−37.9118	−1.20008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bh.1.3	✓	3
3.2	odd	2		3330.2.a.bi.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bh.1.3	✓	3	1.1	even	1	trivial
3330.2.a.bi.1.3	yes	3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.68585 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.68585 q^{7} -1.00000 q^{8} -1.00000 q^{10} -2.14637 q^{11} -1.48929 q^{13} -3.68585 q^{14} +1.00000 q^{16} +0.707269 q^{17} -1.48929 q^{19} +1.00000 q^{20} +2.14637 q^{22} -5.88240 q^{23} +1.00000 q^{25} +1.48929 q^{26} +3.68585 q^{28} -1.80344 q^{29} +10.0288 q^{31} -1.00000 q^{32} -0.707269 q^{34} +3.68585 q^{35} -1.00000 q^{37} +1.48929 q^{38} -1.00000 q^{40} +9.00735 q^{41} +2.36435 q^{43} -2.14637 q^{44} +5.88240 q^{46} +11.2713 q^{47} +6.58546 q^{49} -1.00000 q^{50} -1.48929 q^{52} +5.51806 q^{53} -2.14637 q^{55} -3.68585 q^{56} +1.80344 q^{58} -13.6644 q^{59} -3.63565 q^{61} -10.0288 q^{62} +1.00000 q^{64} -1.48929 q^{65} +15.6644 q^{67} +0.707269 q^{68} -3.68585 q^{70} +7.66442 q^{71} +15.0533 q^{73} +1.00000 q^{74} -1.48929 q^{76} -7.91117 q^{77} +12.2927 q^{79} +1.00000 q^{80} -9.00735 q^{82} -5.15371 q^{83} +0.707269 q^{85} -2.36435 q^{86} +2.14637 q^{88} +4.46787 q^{89} -5.48929 q^{91} -5.88240 q^{92} -11.2713 q^{94} -1.48929 q^{95} -7.00735 q^{97} -6.58546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 5 q^{11} + 3 q^{13} + q^{14} + 3 q^{16} + 5 q^{17} + 3 q^{19} + 3 q^{20} + 5 q^{22} - q^{23} + 3 q^{25} - 3 q^{26} - q^{28} - 10 q^{29} + 12 q^{31} - 3 q^{32} - 5 q^{34} - q^{35} - 3 q^{37} - 3 q^{38} - 3 q^{40} - 6 q^{41} + 16 q^{43} - 5 q^{44} + q^{46} + 16 q^{47} + 14 q^{49} - 3 q^{50} + 3 q^{52} - 9 q^{53} - 5 q^{55} + q^{56} + 10 q^{58} - 14 q^{59} - 2 q^{61} - 12 q^{62} + 3 q^{64} + 3 q^{65} + 20 q^{67} + 5 q^{68} + q^{70} - 4 q^{71} + 17 q^{73} + 3 q^{74} + 3 q^{76} + 11 q^{77} + 34 q^{79} + 3 q^{80} + 6 q^{82} + 19 q^{83} + 5 q^{85} - 16 q^{86} + 5 q^{88} - 9 q^{89} - 9 q^{91} - q^{92} - 16 q^{94} + 3 q^{95} + 12 q^{97} - 14 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8 - 3 * q^10 - 5 * q^11 + 3 * q^13 + q^14 + 3 * q^16 + 5 * q^17 + 3 * q^19 + 3 * q^20 + 5 * q^22 - q^23 + 3 * q^25 - 3 * q^26 - q^28 - 10 * q^29 + 12 * q^31 - 3 * q^32 - 5 * q^34 - q^35 - 3 * q^37 - 3 * q^38 - 3 * q^40 - 6 * q^41 + 16 * q^43 - 5 * q^44 + q^46 + 16 * q^47 + 14 * q^49 - 3 * q^50 + 3 * q^52 - 9 * q^53 - 5 * q^55 + q^56 + 10 * q^58 - 14 * q^59 - 2 * q^61 - 12 * q^62 + 3 * q^64 + 3 * q^65 + 20 * q^67 + 5 * q^68 + q^70 - 4 * q^71 + 17 * q^73 + 3 * q^74 + 3 * q^76 + 11 * q^77 + 34 * q^79 + 3 * q^80 + 6 * q^82 + 19 * q^83 + 5 * q^85 - 16 * q^86 + 5 * q^88 - 9 * q^89 - 9 * q^91 - q^92 - 16 * q^94 + 3 * q^95 + 12 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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