LMFDB - Embedded newform 3192.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.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:	$25.4882483252$
Analytic rank:	$1$
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:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -1.05863 q^{11} +3.55691 q^{13} -4.49828 q^{17} -1.00000 q^{19} -1.00000 q^{21} -7.43965 q^{23} -5.00000 q^{25} +1.00000 q^{27} -9.55691 q^{29} +6.61555 q^{31} -1.05863 q^{33} -8.61555 q^{37} +3.55691 q^{39} -0.117266 q^{41} -1.88273 q^{43} -0.941367 q^{47} +1.00000 q^{49} -4.49828 q^{51} -2.44309 q^{53} -1.00000 q^{57} +12.9966 q^{59} +6.99656 q^{61} -1.00000 q^{63} +0.824101 q^{67} -7.43965 q^{69} +1.43965 q^{71} +0.117266 q^{73} -5.00000 q^{75} +1.05863 q^{77} -3.67418 q^{79} +1.00000 q^{81} -9.67418 q^{83} -9.55691 q^{87} +5.11383 q^{89} -3.55691 q^{91} +6.61555 q^{93} -12.0552 q^{97} -1.05863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 3 q^{19} - 3 q^{21} - 4 q^{23} - 15 q^{25} + 3 q^{27} - 12 q^{29} + 4 q^{31} - 4 q^{33} - 10 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{47} + 3 q^{49} + 4 q^{51} - 24 q^{53} - 3 q^{57} + 4 q^{59} - 14 q^{61} - 3 q^{63} - 4 q^{69} - 14 q^{71} + 2 q^{73} - 15 q^{75} + 4 q^{77} + 4 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 3 * q^19 - 3 * q^21 - 4 * q^23 - 15 * q^25 + 3 * q^27 - 12 * q^29 + 4 * q^31 - 4 * q^33 - 10 * q^37 - 6 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^47 + 3 * q^49 + 4 * q^51 - 24 * q^53 - 3 * q^57 + 4 * q^59 - 14 * q^61 - 3 * q^63 - 4 * q^69 - 14 * q^71 + 2 * q^73 - 15 * q^75 + 4 * q^77 + 4 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^87 - 18 * q^89 + 6 * q^91 + 4 * q^93 - 2 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.05863	−0.319190	−0.159595	−	0.987183i	$-0.551019\pi$
$11$	−1.05863	−0.319190	−0.159595	+	0.987183i	$0.551019\pi$
$12$	0	0
$13$	3.55691	0.986511	0.493255	−	0.869885i	$-0.335807\pi$
$13$	3.55691	0.986511	0.493255	+	0.869885i	$0.335807\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.49828	−1.09099	−0.545497	−	0.838113i	$-0.683659\pi$
$17$	−4.49828	−1.09099	−0.545497	+	0.838113i	$0.683659\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−7.43965	−1.55127	−0.775637	−	0.631179i	$-0.782571\pi$
$23$	−7.43965	−1.55127	−0.775637	+	0.631179i	$0.782571\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.55691	−1.77467	−0.887337	−	0.461121i	$-0.847447\pi$
$29$	−9.55691	−1.77467	−0.887337	+	0.461121i	$0.847447\pi$
$30$	0	0
$31$	6.61555	1.18819	0.594094	−	0.804396i	$-0.297511\pi$
$31$	6.61555	1.18819	0.594094	+	0.804396i	$0.297511\pi$
$32$	0	0
$33$	−1.05863	−0.184284
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.61555	−1.41639	−0.708194	−	0.706018i	$-0.750490\pi$
$37$	−8.61555	−1.41639	−0.708194	+	0.706018i	$0.750490\pi$
$38$	0	0
$39$	3.55691	0.569562
$40$	0	0
$41$	−0.117266	−0.0183139	−0.00915696	−	0.999958i	$-0.502915\pi$
$41$	−0.117266	−0.0183139	−0.00915696	+	0.999958i	$0.502915\pi$
$42$	0	0
$43$	−1.88273	−0.287114	−0.143557	−	0.989642i	$-0.545854\pi$
$43$	−1.88273	−0.287114	−0.143557	+	0.989642i	$0.545854\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.941367	−0.137312	−0.0686562	−	0.997640i	$-0.521871\pi$
$47$	−0.941367	−0.137312	−0.0686562	+	0.997640i	$0.521871\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.49828	−0.629885
$52$	0	0
$53$	−2.44309	−0.335584	−0.167792	−	0.985822i	$-0.553664\pi$
$53$	−2.44309	−0.335584	−0.167792	+	0.985822i	$0.553664\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	12.9966	1.69201	0.846004	−	0.533176i	$-0.179002\pi$
$59$	12.9966	1.69201	0.846004	+	0.533176i	$0.179002\pi$
$60$	0	0
$61$	6.99656	0.895818	0.447909	−	0.894079i	$-0.352169\pi$
$61$	6.99656	0.895818	0.447909	+	0.894079i	$0.352169\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.824101	0.100680	0.0503400	−	0.998732i	$-0.483970\pi$
$67$	0.824101	0.100680	0.0503400	+	0.998732i	$0.483970\pi$
$68$	0	0
$69$	−7.43965	−0.895628
$70$	0	0
$71$	1.43965	0.170855	0.0854274	−	0.996344i	$-0.472774\pi$
$71$	1.43965	0.170855	0.0854274	+	0.996344i	$0.472774\pi$
$72$	0	0
$73$	0.117266	0.0137250	0.00686249	−	0.999976i	$-0.497816\pi$
$73$	0.117266	0.0137250	0.00686249	+	0.999976i	$0.497816\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	1.05863	0.120642
$78$	0	0
$79$	−3.67418	−0.413378	−0.206689	−	0.978407i	$-0.566269\pi$
$79$	−3.67418	−0.413378	−0.206689	+	0.978407i	$0.566269\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.67418	−1.06188	−0.530940	−	0.847410i	$-0.678161\pi$
$83$	−9.67418	−1.06188	−0.530940	+	0.847410i	$0.678161\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.55691	−1.02461
$88$	0	0
$89$	5.11383	0.542065	0.271032	−	0.962570i	$-0.412635\pi$
$89$	5.11383	0.542065	0.271032	+	0.962570i	$0.412635\pi$
$90$	0	0
$91$	−3.55691	−0.372866
$92$	0	0
$93$	6.61555	0.686000
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0552	−1.22402	−0.612010	−	0.790850i	$-0.709639\pi$
$97$	−12.0552	−1.22402	−0.612010	+	0.790850i	$0.709639\pi$
$98$	0	0
$99$	−1.05863	−0.106397
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−10.3810	−1.02287	−0.511436	−	0.859321i	$-0.670886\pi$
$103$	−10.3810	−1.02287	−0.511436	+	0.859321i	$0.670886\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.5535	1.60028	0.800142	−	0.599810i	$-0.204757\pi$
$107$	16.5535	1.60028	0.800142	+	0.599810i	$0.204757\pi$
$108$	0	0
$109$	15.4948	1.48414	0.742068	−	0.670324i	$-0.233845\pi$
$109$	15.4948	1.48414	0.742068	+	0.670324i	$0.233845\pi$
$110$	0	0
$111$	−8.61555	−0.817752
$112$	0	0
$113$	−7.67418	−0.721926	−0.360963	−	0.932580i	$-0.617552\pi$
$113$	−7.67418	−0.721926	−0.360963	+	0.932580i	$0.617552\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.55691	0.328837
$118$	0	0
$119$	4.49828	0.412357
$120$	0	0
$121$	−9.87930	−0.898118
$122$	0	0
$123$	−0.117266	−0.0105735
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.43965	−0.305219	−0.152610	−	0.988287i	$-0.548768\pi$
$127$	−3.43965	−0.305219	−0.152610	+	0.988287i	$0.548768\pi$
$128$	0	0
$129$	−1.88273	−0.165765
$130$	0	0
$131$	−18.4362	−1.61078	−0.805390	−	0.592746i	$-0.798044\pi$
$131$	−18.4362	−1.61078	−0.805390	+	0.592746i	$0.798044\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.8793	−1.78384	−0.891919	−	0.452195i	$-0.850641\pi$
$137$	−20.8793	−1.78384	−0.891919	+	0.452195i	$0.850641\pi$
$138$	0	0
$139$	−5.88273	−0.498967	−0.249483	−	0.968379i	$-0.580261\pi$
$139$	−5.88273	−0.498967	−0.249483	+	0.968379i	$0.580261\pi$
$140$	0	0
$141$	−0.941367	−0.0792774
$142$	0	0
$143$	−3.76547	−0.314884
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	4.38101	0.358907	0.179453	−	0.983766i	$-0.442567\pi$
$149$	4.38101	0.358907	0.179453	+	0.983766i	$0.442567\pi$
$150$	0	0
$151$	−14.5535	−1.18435	−0.592173	−	0.805811i	$-0.701730\pi$
$151$	−14.5535	−1.18435	−0.592173	+	0.805811i	$0.701730\pi$
$152$	0	0
$153$	−4.49828	−0.363664
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.11383	−0.408128	−0.204064	−	0.978958i	$-0.565415\pi$
$157$	−5.11383	−0.408128	−0.204064	+	0.978958i	$0.565415\pi$
$158$	0	0
$159$	−2.44309	−0.193749
$160$	0	0
$161$	7.43965	0.586326
$162$	0	0
$163$	6.11727	0.479141	0.239571	−	0.970879i	$-0.422993\pi$
$163$	6.11727	0.479141	0.239571	+	0.970879i	$0.422993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.88273	0.145690	0.0728452	−	0.997343i	$-0.476792\pi$
$167$	1.88273	0.145690	0.0728452	+	0.997343i	$0.476792\pi$
$168$	0	0
$169$	−0.348361	−0.0267970
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−24.8793	−1.89154	−0.945769	−	0.324840i	$-0.894689\pi$
$173$	−24.8793	−1.89154	−0.945769	+	0.324840i	$0.894689\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	12.9966	0.976881
$178$	0	0
$179$	13.6742	1.02206	0.511028	−	0.859564i	$-0.329265\pi$
$179$	13.6742	1.02206	0.511028	+	0.859564i	$0.329265\pi$
$180$	0	0
$181$	−0.443086	−0.0329343	−0.0164672	−	0.999864i	$-0.505242\pi$
$181$	−0.443086	−0.0329343	−0.0164672	+	0.999864i	$0.505242\pi$
$182$	0	0
$183$	6.99656	0.517201
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.76203	0.348234
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.67418	−0.265854	−0.132927	−	0.991126i	$-0.542438\pi$
$191$	−3.67418	−0.265854	−0.132927	+	0.991126i	$0.542438\pi$
$192$	0	0
$193$	15.2311	1.09636	0.548179	−	0.836361i	$-0.315321\pi$
$193$	15.2311	1.09636	0.548179	+	0.836361i	$0.315321\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.2637	−1.30124	−0.650619	−	0.759404i	$-0.725491\pi$
$197$	−18.2637	−1.30124	−0.650619	+	0.759404i	$0.725491\pi$
$198$	0	0
$199$	10.8793	0.771213	0.385606	−	0.922663i	$-0.373992\pi$
$199$	10.8793	0.771213	0.385606	+	0.922663i	$0.373992\pi$
$200$	0	0
$201$	0.824101	0.0581276
$202$	0	0
$203$	9.55691	0.670764
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.43965	−0.517091
$208$	0	0
$209$	1.05863	0.0732272
$210$	0	0
$211$	14.9414	1.02861	0.514303	−	0.857609i	$-0.328051\pi$
$211$	14.9414	1.02861	0.514303	+	0.857609i	$0.328051\pi$
$212$	0	0
$213$	1.43965	0.0986430
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.61555	−0.449093
$218$	0	0
$219$	0.117266	0.00792412
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	26.7259	1.78970	0.894851	−	0.446366i	$-0.147282\pi$
$223$	26.7259	1.78970	0.894851	+	0.446366i	$0.147282\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	10.1173	0.671507	0.335753	−	0.941950i	$-0.391009\pi$
$227$	10.1173	0.671507	0.335753	+	0.941950i	$0.391009\pi$
$228$	0	0
$229$	−9.11383	−0.602259	−0.301129	−	0.953583i	$-0.597364\pi$
$229$	−9.11383	−0.602259	−0.301129	+	0.953583i	$0.597364\pi$
$230$	0	0
$231$	1.05863	0.0696529
$232$	0	0
$233$	6.99656	0.458360	0.229180	−	0.973384i	$-0.426396\pi$
$233$	6.99656	0.458360	0.229180	+	0.973384i	$0.426396\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.67418	−0.238664
$238$	0	0
$239$	2.44309	0.158030	0.0790150	−	0.996873i	$-0.474822\pi$
$239$	2.44309	0.158030	0.0790150	+	0.996873i	$0.474822\pi$
$240$	0	0
$241$	24.0552	1.54953	0.774766	−	0.632248i	$-0.217868\pi$
$241$	24.0552	1.54953	0.774766	+	0.632248i	$0.217868\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.55691	−0.226321
$248$	0	0
$249$	−9.67418	−0.613076
$250$	0	0
$251$	−18.4362	−1.16368	−0.581842	−	0.813302i	$-0.697668\pi$
$251$	−18.4362	−1.16368	−0.581842	+	0.813302i	$0.697668\pi$
$252$	0	0
$253$	7.87586	0.495151
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.88273	−0.491711	−0.245856	−	0.969306i	$-0.579069\pi$
$257$	−7.88273	−0.491711	−0.245856	+	0.969306i	$0.579069\pi$
$258$	0	0
$259$	8.61555	0.535344
$260$	0	0
$261$	−9.55691	−0.591558
$262$	0	0
$263$	0.325819	0.0200909	0.0100454	−	0.999950i	$-0.496802\pi$
$263$	0.325819	0.0200909	0.0100454	+	0.999950i	$0.496802\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.11383	0.312961
$268$	0	0
$269$	2.99656	0.182704	0.0913518	−	0.995819i	$-0.470881\pi$
$269$	2.99656	0.182704	0.0913518	+	0.995819i	$0.470881\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−3.55691	−0.215274
$274$	0	0
$275$	5.29317	0.319190
$276$	0	0
$277$	−4.87930	−0.293168	−0.146584	−	0.989198i	$-0.546828\pi$
$277$	−4.87930	−0.293168	−0.146584	+	0.989198i	$0.546828\pi$
$278$	0	0
$279$	6.61555	0.396062
$280$	0	0
$281$	−1.44652	−0.0862924	−0.0431462	−	0.999069i	$-0.513738\pi$
$281$	−1.44652	−0.0862924	−0.0431462	+	0.999069i	$0.513738\pi$
$282$	0	0
$283$	−12.1104	−0.719888	−0.359944	−	0.932974i	$-0.617204\pi$
$283$	−12.1104	−0.719888	−0.359944	+	0.932974i	$0.617204\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.117266	0.00692201
$288$	0	0
$289$	3.23453	0.190267
$290$	0	0
$291$	−12.0552	−0.706688
$292$	0	0
$293$	30.3449	1.77277	0.886385	−	0.462949i	$-0.153209\pi$
$293$	30.3449	1.77277	0.886385	+	0.462949i	$0.153209\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.05863	−0.0614281
$298$	0	0
$299$	−26.4622	−1.53035
$300$	0	0
$301$	1.88273	0.108519
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.8793	0.620914	0.310457	−	0.950587i	$-0.399518\pi$
$307$	10.8793	0.620914	0.310457	+	0.950587i	$0.399518\pi$
$308$	0	0
$309$	−10.3810	−0.590555
$310$	0	0
$311$	0.706834	0.0400809	0.0200404	−	0.999799i	$-0.493621\pi$
$311$	0.706834	0.0400809	0.0200404	+	0.999799i	$0.493621\pi$
$312$	0	0
$313$	−3.23109	−0.182632	−0.0913161	−	0.995822i	$-0.529107\pi$
$313$	−3.23109	−0.182632	−0.0913161	+	0.995822i	$0.529107\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.9053	−0.949495	−0.474747	−	0.880122i	$-0.657461\pi$
$317$	−16.9053	−0.949495	−0.474747	+	0.880122i	$0.657461\pi$
$318$	0	0
$319$	10.1173	0.566458
$320$	0	0
$321$	16.5535	0.923925
$322$	0	0
$323$	4.49828	0.250291
$324$	0	0
$325$	−17.7846	−0.986511
$326$	0	0
$327$	15.4948	0.856867
$328$	0	0
$329$	0.941367	0.0518992
$330$	0	0
$331$	33.1690	1.82313	0.911567	−	0.411151i	$-0.134873\pi$
$331$	33.1690	1.82313	0.911567	+	0.411151i	$0.134873\pi$
$332$	0	0
$333$	−8.61555	−0.472129
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.2311	1.04758	0.523792	−	0.851846i	$-0.324517\pi$
$337$	19.2311	1.04758	0.523792	+	0.851846i	$0.324517\pi$
$338$	0	0
$339$	−7.67418	−0.416804
$340$	0	0
$341$	−7.00344	−0.379257
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.4070	1.09550	0.547752	−	0.836640i	$-0.315484\pi$
$347$	20.4070	1.09550	0.547752	+	0.836640i	$0.315484\pi$
$348$	0	0
$349$	−13.1138	−0.701967	−0.350984	−	0.936382i	$-0.614153\pi$
$349$	−13.1138	−0.701967	−0.350984	+	0.936382i	$0.614153\pi$
$350$	0	0
$351$	3.55691	0.189854
$352$	0	0
$353$	−1.72938	−0.0920454	−0.0460227	−	0.998940i	$-0.514655\pi$
$353$	−1.72938	−0.0920454	−0.0460227	+	0.998940i	$0.514655\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.49828	0.238074
$358$	0	0
$359$	−13.7914	−0.727885	−0.363942	−	0.931421i	$-0.618569\pi$
$359$	−13.7914	−0.727885	−0.363942	+	0.931421i	$0.618569\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.87930	−0.518529
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.12070	−0.267299	−0.133649	−	0.991029i	$-0.542670\pi$
$367$	−5.12070	−0.267299	−0.133649	+	0.991029i	$0.542670\pi$
$368$	0	0
$369$	−0.117266	−0.00610464
$370$	0	0
$371$	2.44309	0.126839
$372$	0	0
$373$	−10.2637	−0.531437	−0.265718	−	0.964051i	$-0.585609\pi$
$373$	−10.2637	−0.531437	−0.265718	+	0.964051i	$0.585609\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−33.9931	−1.75073
$378$	0	0
$379$	23.1759	1.19047	0.595233	−	0.803553i	$-0.297060\pi$
$379$	23.1759	1.19047	0.595233	+	0.803553i	$0.297060\pi$
$380$	0	0
$381$	−3.43965	−0.176219
$382$	0	0
$383$	−2.87930	−0.147125	−0.0735626	−	0.997291i	$-0.523437\pi$
$383$	−2.87930	−0.147125	−0.0735626	+	0.997291i	$0.523437\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.88273	−0.0957047
$388$	0	0
$389$	−24.7259	−1.25365	−0.626827	−	0.779158i	$-0.715647\pi$
$389$	−24.7259	−1.25365	−0.626827	+	0.779158i	$0.715647\pi$
$390$	0	0
$391$	33.4656	1.69243
$392$	0	0
$393$	−18.4362	−0.929984
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−25.8759	−1.29867	−0.649336	−	0.760502i	$-0.724953\pi$
$397$	−25.8759	−1.29867	−0.649336	+	0.760502i	$0.724953\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	14.3189	0.715054	0.357527	−	0.933903i	$-0.383620\pi$
$401$	14.3189	0.715054	0.357527	+	0.933903i	$0.383620\pi$
$402$	0	0
$403$	23.5309	1.17216
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.12070	0.452097
$408$	0	0
$409$	33.2863	1.64590	0.822951	−	0.568113i	$-0.192326\pi$
$409$	33.2863	1.64590	0.822951	+	0.568113i	$0.192326\pi$
$410$	0	0
$411$	−20.8793	−1.02990
$412$	0	0
$413$	−12.9966	−0.639519
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.88273	−0.288079
$418$	0	0
$419$	13.2051	0.645112	0.322556	−	0.946550i	$-0.395458\pi$
$419$	13.2051	0.645112	0.322556	+	0.946550i	$0.395458\pi$
$420$	0	0
$421$	−33.3776	−1.62672	−0.813362	−	0.581758i	$-0.802365\pi$
$421$	−33.3776	−1.62672	−0.813362	+	0.581758i	$0.802365\pi$
$422$	0	0
$423$	−0.941367	−0.0457708
$424$	0	0
$425$	22.4914	1.09099
$426$	0	0
$427$	−6.99656	−0.338587
$428$	0	0
$429$	−3.76547	−0.181798
$430$	0	0
$431$	−4.78801	−0.230630	−0.115315	−	0.993329i	$-0.536788\pi$
$431$	−4.78801	−0.230630	−0.115315	+	0.993329i	$0.536788\pi$
$432$	0	0
$433$	32.4001	1.55705	0.778525	−	0.627613i	$-0.215968\pi$
$433$	32.4001	1.55705	0.778525	+	0.627613i	$0.215968\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.43965	0.355887
$438$	0	0
$439$	−5.61899	−0.268180	−0.134090	−	0.990969i	$-0.542811\pi$
$439$	−5.61899	−0.268180	−0.134090	+	0.990969i	$0.542811\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	32.0483	1.52266	0.761331	−	0.648364i	$-0.224546\pi$
$443$	32.0483	1.52266	0.761331	+	0.648364i	$0.224546\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.38101	0.207215
$448$	0	0
$449$	−24.5604	−1.15907	−0.579537	−	0.814946i	$-0.696767\pi$
$449$	−24.5604	−1.15907	−0.579537	+	0.814946i	$0.696767\pi$
$450$	0	0
$451$	0.124142	0.00584562
$452$	0	0
$453$	−14.5535	−0.683782
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.7586	1.67272	0.836358	−	0.548183i	$-0.184680\pi$
$457$	35.7586	1.67272	0.836358	+	0.548183i	$0.184680\pi$
$458$	0	0
$459$	−4.49828	−0.209962
$460$	0	0
$461$	−16.9966	−0.791609	−0.395804	−	0.918335i	$-0.629534\pi$
$461$	−16.9966	−0.791609	−0.395804	+	0.918335i	$0.629534\pi$
$462$	0	0
$463$	19.7655	0.918579	0.459290	−	0.888287i	$-0.348104\pi$
$463$	19.7655	0.918579	0.459290	+	0.888287i	$0.348104\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.67418	0.0774718	0.0387359	−	0.999249i	$-0.487667\pi$
$467$	1.67418	0.0774718	0.0387359	+	0.999249i	$0.487667\pi$
$468$	0	0
$469$	−0.824101	−0.0380534
$470$	0	0
$471$	−5.11383	−0.235633
$472$	0	0
$473$	1.99312	0.0916440
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	−2.44309	−0.111861
$478$	0	0
$479$	33.2863	1.52089	0.760445	−	0.649403i	$-0.224981\pi$
$479$	33.2863	1.52089	0.760445	+	0.649403i	$0.224981\pi$
$480$	0	0
$481$	−30.6448	−1.39728
$482$	0	0
$483$	7.43965	0.338516
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.3224	−0.966209	−0.483105	−	0.875563i	$-0.660491\pi$
$487$	−21.3224	−0.966209	−0.483105	+	0.875563i	$0.660491\pi$
$488$	0	0
$489$	6.11727	0.276632
$490$	0	0
$491$	−8.82410	−0.398226	−0.199113	−	0.979977i	$-0.563806\pi$
$491$	−8.82410	−0.398226	−0.199113	+	0.979977i	$0.563806\pi$
$492$	0	0
$493$	42.9897	1.93616
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.43965	−0.0645770
$498$	0	0
$499$	−20.3449	−0.910764	−0.455382	−	0.890296i	$-0.650497\pi$
$499$	−20.3449	−0.910764	−0.455382	+	0.890296i	$0.650497\pi$
$500$	0	0
$501$	1.88273	0.0841143
$502$	0	0
$503$	0.706834	0.0315162	0.0157581	−	0.999876i	$-0.494984\pi$
$503$	0.706834	0.0315162	0.0157581	+	0.999876i	$0.494984\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.348361	−0.0154713
$508$	0	0
$509$	−23.4656	−1.04010	−0.520048	−	0.854137i	$-0.674086\pi$
$509$	−23.4656	−1.04010	−0.520048	+	0.854137i	$0.674086\pi$
$510$	0	0
$511$	−0.117266	−0.00518756
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.996562	0.0438288
$518$	0	0
$519$	−24.8793	−1.09208
$520$	0	0
$521$	−9.53093	−0.417558	−0.208779	−	0.977963i	$-0.566949\pi$
$521$	−9.53093	−0.417558	−0.208779	+	0.977963i	$0.566949\pi$
$522$	0	0
$523$	17.1070	0.748035	0.374017	−	0.927422i	$-0.377980\pi$
$523$	17.1070	0.748035	0.374017	+	0.927422i	$0.377980\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	−29.7586	−1.29630
$528$	0	0
$529$	32.3484	1.40645
$530$	0	0
$531$	12.9966	0.564003
$532$	0	0
$533$	−0.417106	−0.0180669
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.6742	0.590084
$538$	0	0
$539$	−1.05863	−0.0455986
$540$	0	0
$541$	28.8793	1.24162	0.620809	−	0.783962i	$-0.286804\pi$
$541$	28.8793	1.24162	0.620809	+	0.783962i	$0.286804\pi$
$542$	0	0
$543$	−0.443086	−0.0190146
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.8138	−1.36026	−0.680130	−	0.733092i	$-0.738077\pi$
$547$	−31.8138	−1.36026	−0.680130	+	0.733092i	$0.738077\pi$
$548$	0	0
$549$	6.99656	0.298606
$550$	0	0
$551$	9.55691	0.407138
$552$	0	0
$553$	3.67418	0.156242
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.0812	−1.27458	−0.637290	−	0.770624i	$-0.719945\pi$
$557$	−30.0812	−1.27458	−0.637290	+	0.770624i	$0.719945\pi$
$558$	0	0
$559$	−6.69672	−0.283241
$560$	0	0
$561$	4.76203	0.201053
$562$	0	0
$563$	−12.1104	−0.510392	−0.255196	−	0.966889i	$-0.582140\pi$
$563$	−12.1104	−0.510392	−0.255196	+	0.966889i	$0.582140\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−30.5535	−1.28087	−0.640434	−	0.768013i	$-0.721246\pi$
$569$	−30.5535	−1.28087	−0.640434	+	0.768013i	$0.721246\pi$
$570$	0	0
$571$	34.2277	1.43238	0.716191	−	0.697904i	$-0.245884\pi$
$571$	34.2277	1.43238	0.716191	+	0.697904i	$0.245884\pi$
$572$	0	0
$573$	−3.67418	−0.153491
$574$	0	0
$575$	37.1982	1.55127
$576$	0	0
$577$	13.7655	0.573064	0.286532	−	0.958071i	$-0.407498\pi$
$577$	13.7655	0.573064	0.286532	+	0.958071i	$0.407498\pi$
$578$	0	0
$579$	15.2311	0.632983
$580$	0	0
$581$	9.67418	0.401353
$582$	0	0
$583$	2.58633	0.107115
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.6639	0.852889	0.426445	−	0.904514i	$-0.359766\pi$
$587$	20.6639	0.852889	0.426445	+	0.904514i	$0.359766\pi$
$588$	0	0
$589$	−6.61555	−0.272589
$590$	0	0
$591$	−18.2637	−0.751270
$592$	0	0
$593$	−12.2637	−0.503612	−0.251806	−	0.967778i	$-0.581024\pi$
$593$	−12.2637	−0.503612	−0.251806	+	0.967778i	$0.581024\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.8793	0.445260
$598$	0	0
$599$	36.7880	1.50312	0.751559	−	0.659666i	$-0.229302\pi$
$599$	36.7880	1.50312	0.751559	+	0.659666i	$0.229302\pi$
$600$	0	0
$601$	−21.5208	−0.877853	−0.438926	−	0.898523i	$-0.644641\pi$
$601$	−21.5208	−0.877853	−0.438926	+	0.898523i	$0.644641\pi$
$602$	0	0
$603$	0.824101	0.0335600
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.4983	−1.48142	−0.740710	−	0.671825i	$-0.765511\pi$
$607$	−36.4983	−1.48142	−0.740710	+	0.671825i	$0.765511\pi$
$608$	0	0
$609$	9.55691	0.387266
$610$	0	0
$611$	−3.34836	−0.135460
$612$	0	0
$613$	12.1173	0.489412	0.244706	−	0.969597i	$-0.421309\pi$
$613$	12.1173	0.489412	0.244706	+	0.969597i	$0.421309\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.3449	0.738539	0.369269	−	0.929322i	$-0.379608\pi$
$617$	18.3449	0.738539	0.369269	+	0.929322i	$0.379608\pi$
$618$	0	0
$619$	17.6482	0.709341	0.354671	−	0.934991i	$-0.384593\pi$
$619$	17.6482	0.709341	0.354671	+	0.934991i	$0.384593\pi$
$620$	0	0
$621$	−7.43965	−0.298543
$622$	0	0
$623$	−5.11383	−0.204881
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	1.05863	0.0422777
$628$	0	0
$629$	38.7552	1.54527
$630$	0	0
$631$	12.8862	0.512990	0.256495	−	0.966546i	$-0.417432\pi$
$631$	12.8862	0.512990	0.256495	+	0.966546i	$0.417432\pi$
$632$	0	0
$633$	14.9414	0.593866
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.55691	0.140930
$638$	0	0
$639$	1.43965	0.0569516
$640$	0	0
$641$	35.2051	1.39052	0.695259	−	0.718759i	$-0.255290\pi$
$641$	35.2051	1.39052	0.695259	+	0.718759i	$0.255290\pi$
$642$	0	0
$643$	−40.8724	−1.61185	−0.805925	−	0.592017i	$-0.798332\pi$
$643$	−40.8724	−1.61185	−0.805925	+	0.592017i	$0.798332\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.5174	−1.51427	−0.757137	−	0.653256i	$-0.773403\pi$
$647$	−38.5174	−1.51427	−0.757137	+	0.653256i	$0.773403\pi$
$648$	0	0
$649$	−13.7586	−0.540072
$650$	0	0
$651$	−6.61555	−0.259284
$652$	0	0
$653$	30.0812	1.17717	0.588584	−	0.808436i	$-0.299686\pi$
$653$	30.0812	1.17717	0.588584	+	0.808436i	$0.299686\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.117266	0.00457499
$658$	0	0
$659$	9.43965	0.367716	0.183858	−	0.982953i	$-0.441141\pi$
$659$	9.43965	0.367716	0.183858	+	0.982953i	$0.441141\pi$
$660$	0	0
$661$	5.20512	0.202456	0.101228	−	0.994863i	$-0.467723\pi$
$661$	5.20512	0.202456	0.101228	+	0.994863i	$0.467723\pi$
$662$	0	0
$663$	−16.0000	−0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	71.1001	2.75301
$668$	0	0
$669$	26.7259	1.03328
$670$	0	0
$671$	−7.40679	−0.285936
$672$	0	0
$673$	−23.3415	−0.899748	−0.449874	−	0.893092i	$-0.648531\pi$
$673$	−23.3415	−0.899748	−0.449874	+	0.893092i	$0.648531\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−2.88617	−0.110925	−0.0554623	−	0.998461i	$-0.517663\pi$
$677$	−2.88617	−0.110925	−0.0554623	+	0.998461i	$0.517663\pi$
$678$	0	0
$679$	12.0552	0.462636
$680$	0	0
$681$	10.1173	0.387694
$682$	0	0
$683$	7.79145	0.298131	0.149066	−	0.988827i	$-0.452373\pi$
$683$	7.79145	0.298131	0.149066	+	0.988827i	$0.452373\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.11383	−0.347714
$688$	0	0
$689$	−8.68985	−0.331057
$690$	0	0
$691$	−16.2345	−0.617591	−0.308795	−	0.951128i	$-0.599926\pi$
$691$	−16.2345	−0.617591	−0.308795	+	0.951128i	$0.599926\pi$
$692$	0	0
$693$	1.05863	0.0402141
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.527497	0.0199804
$698$	0	0
$699$	6.99656	0.264634
$700$	0	0
$701$	−23.6190	−0.892077	−0.446038	−	0.895014i	$-0.647166\pi$
$701$	−23.6190	−0.892077	−0.446038	+	0.895014i	$0.647166\pi$
$702$	0	0
$703$	8.61555	0.324942
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	−32.9897	−1.23895	−0.619477	−	0.785015i	$-0.712655\pi$
$709$	−32.9897	−1.23895	−0.619477	+	0.785015i	$0.712655\pi$
$710$	0	0
$711$	−3.67418	−0.137793
$712$	0	0
$713$	−49.2173	−1.84320
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.44309	0.0912387
$718$	0	0
$719$	52.2760	1.94956	0.974782	−	0.223157i	$-0.0716363\pi$
$719$	52.2760	1.94956	0.974782	+	0.223157i	$0.0716363\pi$
$720$	0	0
$721$	10.3810	0.386609
$722$	0	0
$723$	24.0552	0.894622
$724$	0	0
$725$	47.7846	1.77467
$726$	0	0
$727$	−39.2242	−1.45475	−0.727373	−	0.686242i	$-0.759259\pi$
$727$	−39.2242	−1.45475	−0.727373	+	0.686242i	$0.759259\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.46907	0.313240
$732$	0	0
$733$	15.7586	0.582057	0.291028	−	0.956714i	$-0.406003\pi$
$733$	15.7586	0.582057	0.291028	+	0.956714i	$0.406003\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.872420	−0.0321360
$738$	0	0
$739$	8.23453	0.302912	0.151456	−	0.988464i	$-0.451604\pi$
$739$	8.23453	0.302912	0.151456	+	0.988464i	$0.451604\pi$
$740$	0	0
$741$	−3.55691	−0.130667
$742$	0	0
$743$	−23.9018	−0.876873	−0.438437	−	0.898762i	$-0.644468\pi$
$743$	−23.9018	−0.876873	−0.438437	+	0.898762i	$0.644468\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.67418	−0.353960
$748$	0	0
$749$	−16.5535	−0.604851
$750$	0	0
$751$	−43.1329	−1.57394	−0.786972	−	0.616989i	$-0.788352\pi$
$751$	−43.1329	−1.57394	−0.786972	+	0.616989i	$0.788352\pi$
$752$	0	0
$753$	−18.4362	−0.671853
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19.4656	−0.707490	−0.353745	−	0.935342i	$-0.615092\pi$
$757$	−19.4656	−0.707490	−0.353745	+	0.935342i	$0.615092\pi$
$758$	0	0
$759$	7.87586	0.285876
$760$	0	0
$761$	−9.96391	−0.361191	−0.180596	−	0.983557i	$-0.557803\pi$
$761$	−9.96391	−0.361191	−0.180596	+	0.983557i	$0.557803\pi$
$762$	0	0
$763$	−15.4948	−0.560951
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46.2277	1.66918
$768$	0	0
$769$	−5.22422	−0.188390	−0.0941951	−	0.995554i	$-0.530028\pi$
$769$	−5.22422	−0.188390	−0.0941951	+	0.995554i	$0.530028\pi$
$770$	0	0
$771$	−7.88273	−0.283890
$772$	0	0
$773$	22.3449	0.803691	0.401846	−	0.915707i	$-0.368369\pi$
$773$	22.3449	0.803691	0.401846	+	0.915707i	$0.368369\pi$
$774$	0	0
$775$	−33.0777	−1.18819
$776$	0	0
$777$	8.61555	0.309081
$778$	0	0
$779$	0.117266	0.00420150
$780$	0	0
$781$	−1.52406	−0.0545351
$782$	0	0
$783$	−9.55691	−0.341536
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.3449	1.29556	0.647778	−	0.761829i	$-0.275698\pi$
$787$	36.3449	1.29556	0.647778	+	0.761829i	$0.275698\pi$
$788$	0	0
$789$	0.325819	0.0115995
$790$	0	0
$791$	7.67418	0.272862
$792$	0	0
$793$	24.8862	0.883734
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.8793	−1.16464	−0.582322	−	0.812958i	$-0.697856\pi$
$797$	−32.8793	−1.16464	−0.582322	+	0.812958i	$0.697856\pi$
$798$	0	0
$799$	4.23453	0.149807
$800$	0	0
$801$	5.11383	0.180688
$802$	0	0
$803$	−0.124142	−0.00438088
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.99656	0.105484
$808$	0	0
$809$	1.00344	0.0352790	0.0176395	−	0.999844i	$-0.494385\pi$
$809$	1.00344	0.0352790	0.0176395	+	0.999844i	$0.494385\pi$
$810$	0	0
$811$	23.1138	0.811636	0.405818	−	0.913954i	$-0.366987\pi$
$811$	23.1138	0.811636	0.405818	+	0.913954i	$0.366987\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.88273	0.0658685
$818$	0	0
$819$	−3.55691	−0.124289
$820$	0	0
$821$	−10.3741	−0.362060	−0.181030	−	0.983478i	$-0.557943\pi$
$821$	−10.3741	−0.362060	−0.181030	+	0.983478i	$0.557943\pi$
$822$	0	0
$823$	33.9931	1.18493	0.592463	−	0.805598i	$-0.298156\pi$
$823$	33.9931	1.18493	0.592463	+	0.805598i	$0.298156\pi$
$824$	0	0
$825$	5.29317	0.184284
$826$	0	0
$827$	−53.8950	−1.87411	−0.937056	−	0.349180i	$-0.886460\pi$
$827$	−53.8950	−1.87411	−0.937056	+	0.349180i	$0.886460\pi$
$828$	0	0
$829$	−34.7811	−1.20800	−0.603999	−	0.796985i	$-0.706427\pi$
$829$	−34.7811	−1.20800	−0.603999	+	0.796985i	$0.706427\pi$
$830$	0	0
$831$	−4.87930	−0.169261
$832$	0	0
$833$	−4.49828	−0.155856
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.61555	0.228667
$838$	0	0
$839$	25.4656	0.879171	0.439586	−	0.898201i	$-0.355125\pi$
$839$	25.4656	0.879171	0.439586	+	0.898201i	$0.355125\pi$
$840$	0	0
$841$	62.3346	2.14947
$842$	0	0
$843$	−1.44652	−0.0498209
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.87930	0.339457
$848$	0	0
$849$	−12.1104	−0.415628
$850$	0	0
$851$	64.0966	2.19720
$852$	0	0
$853$	45.2242	1.54845	0.774224	−	0.632912i	$-0.218140\pi$
$853$	45.2242	1.54845	0.774224	+	0.632912i	$0.218140\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	55.5760	1.89844	0.949220	−	0.314612i	$-0.101874\pi$
$857$	55.5760	1.89844	0.949220	+	0.314612i	$0.101874\pi$
$858$	0	0
$859$	−42.7552	−1.45879	−0.729394	−	0.684094i	$-0.760198\pi$
$859$	−42.7552	−1.45879	−0.729394	+	0.684094i	$0.760198\pi$
$860$	0	0
$861$	0.117266	0.00399643
$862$	0	0
$863$	51.1982	1.74281	0.871404	−	0.490566i	$-0.163210\pi$
$863$	51.1982	1.74281	0.871404	+	0.490566i	$0.163210\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.23453	0.109850
$868$	0	0
$869$	3.88961	0.131946
$870$	0	0
$871$	2.93125	0.0993218
$872$	0	0
$873$	−12.0552	−0.408007
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.6052	1.06723	0.533616	−	0.845727i	$-0.320833\pi$
$877$	31.6052	1.06723	0.533616	+	0.845727i	$0.320833\pi$
$878$	0	0
$879$	30.3449	1.02351
$880$	0	0
$881$	−15.2672	−0.514365	−0.257182	−	0.966363i	$-0.582794\pi$
$881$	−15.2672	−0.514365	−0.257182	+	0.966363i	$0.582794\pi$
$882$	0	0
$883$	32.1104	1.08060	0.540300	−	0.841472i	$-0.318311\pi$
$883$	32.1104	1.08060	0.540300	+	0.841472i	$0.318311\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.11727	0.205398	0.102699	−	0.994712i	$-0.467252\pi$
$887$	6.11727	0.205398	0.102699	+	0.994712i	$0.467252\pi$
$888$	0	0
$889$	3.43965	0.115362
$890$	0	0
$891$	−1.05863	−0.0354655
$892$	0	0
$893$	0.941367	0.0315016
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−26.4622	−0.883547
$898$	0	0
$899$	−63.2242	−2.10865
$900$	0	0
$901$	10.9897	0.366120
$902$	0	0
$903$	1.88273	0.0626535
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25.9311	−0.861026	−0.430513	−	0.902584i	$-0.641667\pi$
$907$	−25.9311	−0.861026	−0.430513	+	0.902584i	$0.641667\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	17.0225	0.563982	0.281991	−	0.959417i	$-0.409005\pi$
$911$	17.0225	0.563982	0.281991	+	0.959417i	$0.409005\pi$
$912$	0	0
$913$	10.2414	0.338941
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.4362	0.608817
$918$	0	0
$919$	0.469065	0.0154730	0.00773652	−	0.999970i	$-0.497537\pi$
$919$	0.469065	0.0154730	0.00773652	+	0.999970i	$0.497537\pi$
$920$	0	0
$921$	10.8793	0.358485
$922$	0	0
$923$	5.12070	0.168550
$924$	0	0
$925$	43.0777	1.41639
$926$	0	0
$927$	−10.3810	−0.340957
$928$	0	0
$929$	39.9570	1.31095	0.655474	−	0.755218i	$-0.272469\pi$
$929$	39.9570	1.31095	0.655474	+	0.755218i	$0.272469\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0.706834	0.0231407
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.87930	0.290074	0.145037	−	0.989426i	$-0.453670\pi$
$937$	8.87930	0.290074	0.145037	+	0.989426i	$0.453670\pi$
$938$	0	0
$939$	−3.23109	−0.105443
$940$	0	0
$941$	−24.7689	−0.807443	−0.403722	−	0.914882i	$-0.632284\pi$
$941$	−24.7689	−0.807443	−0.403722	+	0.914882i	$0.632284\pi$
$942$	0	0
$943$	0.872420	0.0284099
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.7034	−1.16020	−0.580102	−	0.814544i	$-0.696987\pi$
$947$	−35.7034	−1.16020	−0.580102	+	0.814544i	$0.696987\pi$
$948$	0	0
$949$	0.417106	0.0135398
$950$	0	0
$951$	−16.9053	−0.548191
$952$	0	0
$953$	−35.0225	−1.13449	−0.567246	−	0.823549i	$-0.691991\pi$
$953$	−35.0225	−1.13449	−0.567246	+	0.823549i	$0.691991\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.1173	0.327045
$958$	0	0
$959$	20.8793	0.674228
$960$	0	0
$961$	12.7655	0.411789
$962$	0	0
$963$	16.5535	0.533428
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.8724	1.05711	0.528553	−	0.848900i	$-0.322735\pi$
$967$	32.8724	1.05711	0.528553	+	0.848900i	$0.322735\pi$
$968$	0	0
$969$	4.49828	0.144506
$970$	0	0
$971$	−51.2242	−1.64386	−0.821932	−	0.569586i	$-0.807104\pi$
$971$	−51.2242	−1.64386	−0.821932	+	0.569586i	$0.807104\pi$
$972$	0	0
$973$	5.88273	0.188592
$974$	0	0
$975$	−17.7846	−0.569562
$976$	0	0
$977$	−51.8950	−1.66027	−0.830133	−	0.557565i	$-0.811736\pi$
$977$	−51.8950	−1.66027	−0.830133	+	0.557565i	$0.811736\pi$
$978$	0	0
$979$	−5.41367	−0.173022
$980$	0	0
$981$	15.4948	0.494712
$982$	0	0
$983$	25.9931	0.829052	0.414526	−	0.910037i	$-0.363947\pi$
$983$	25.9931	0.829052	0.414526	+	0.910037i	$0.363947\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.941367	0.0299640
$988$	0	0
$989$	14.0069	0.445393
$990$	0	0
$991$	21.3224	0.677328	0.338664	−	0.940907i	$-0.390025\pi$
$991$	21.3224	0.677328	0.338664	+	0.940907i	$0.390025\pi$
$992$	0	0
$993$	33.1690	1.05259
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.5829	0.810218	0.405109	−	0.914268i	$-0.367234\pi$
$997$	25.5829	0.810218	0.405109	+	0.914268i	$0.367234\pi$
$998$	0	0
$999$	−8.61555	−0.272584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.v.1.2	✓	3
3.2	odd	2		9576.2.a.cc.1.2		3
4.3	odd	2		6384.2.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.v.1.2	✓	3	1.1	even	1	trivial
6384.2.a.bv.1.2		3	4.3	odd	2
9576.2.a.cc.1.2		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:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -1.05863 q^{11} +3.55691 q^{13} -4.49828 q^{17} -1.00000 q^{19} -1.00000 q^{21} -7.43965 q^{23} -5.00000 q^{25} +1.00000 q^{27} -9.55691 q^{29} +6.61555 q^{31} -1.05863 q^{33} -8.61555 q^{37} +3.55691 q^{39} -0.117266 q^{41} -1.88273 q^{43} -0.941367 q^{47} +1.00000 q^{49} -4.49828 q^{51} -2.44309 q^{53} -1.00000 q^{57} +12.9966 q^{59} +6.99656 q^{61} -1.00000 q^{63} +0.824101 q^{67} -7.43965 q^{69} +1.43965 q^{71} +0.117266 q^{73} -5.00000 q^{75} +1.05863 q^{77} -3.67418 q^{79} +1.00000 q^{81} -9.67418 q^{83} -9.55691 q^{87} +5.11383 q^{89} -3.55691 q^{91} +6.61555 q^{93} -12.0552 q^{97} -1.05863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 3 q^{19} - 3 q^{21} - 4 q^{23} - 15 q^{25} + 3 q^{27} - 12 q^{29} + 4 q^{31} - 4 q^{33} - 10 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{47} + 3 q^{49} + 4 q^{51} - 24 q^{53} - 3 q^{57} + 4 q^{59} - 14 q^{61} - 3 q^{63} - 4 q^{69} - 14 q^{71} + 2 q^{73} - 15 q^{75} + 4 q^{77} + 4 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 3 * q^19 - 3 * q^21 - 4 * q^23 - 15 * q^25 + 3 * q^27 - 12 * q^29 + 4 * q^31 - 4 * q^33 - 10 * q^37 - 6 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^47 + 3 * q^49 + 4 * q^51 - 24 * q^53 - 3 * q^57 + 4 * q^59 - 14 * q^61 - 3 * q^63 - 4 * q^69 - 14 * q^71 + 2 * q^73 - 15 * q^75 + 4 * q^77 + 4 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^87 - 18 * q^89 + 6 * q^91 + 4 * q^93 - 2 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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