LMFDB - Embedded newform 3192.2.a.x.1.3

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:	$4$
Coefficient field:	4.4.9248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.662153$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.69614 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.32431 q^{11} -1.69614 q^{15} +0.102658 q^{17} -1.00000 q^{19} +1.00000 q^{21} -9.16400 q^{23} -2.12311 q^{25} -1.00000 q^{27} -4.81925 q^{29} -0.675693 q^{31} -1.32431 q^{33} -1.69614 q^{35} +0.406517 q^{37} +4.04090 q^{41} -4.91779 q^{43} +1.69614 q^{45} +3.66906 q^{47} +1.00000 q^{49} -0.102658 q^{51} +5.42696 q^{53} +2.24621 q^{55} +1.00000 q^{57} -9.43318 q^{59} -0.648614 q^{61} -1.00000 q^{63} -8.92190 q^{67} +9.16400 q^{69} +5.46786 q^{71} +4.04090 q^{73} +2.12311 q^{75} -1.32431 q^{77} +8.55628 q^{79} +1.00000 q^{81} -13.4679 q^{83} +0.174122 q^{85} +4.81925 q^{87} -4.85393 q^{89} +0.675693 q^{93} -1.69614 q^{95} +2.71659 q^{97} +1.32431 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{7} + 4 q^{9} - 4 q^{17} - 4 q^{19} + 4 q^{21} + 4 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{37} - 8 q^{41} - 12 q^{43} - 8 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{53} - 24 q^{55}+ \cdots - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^7 + 4 * q^9 - 4 * q^17 - 4 * q^19 + 4 * q^21 + 4 * q^23 + 8 * q^25 - 4 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^37 - 8 * q^41 - 12 * q^43 - 8 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^53 - 24 * q^55 + 4 * q^57 + 8 * q^61 - 4 * q^63 - 8 * q^67 - 4 * q^69 - 12 * q^71 - 8 * q^73 - 8 * q^75 - 20 * q^79 + 4 * q^81 - 20 * q^83 - 4 * q^85 - 4 * q^87 + 8 * q^93 - 8 * q^97

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
$5$	1.69614	0.758537	0.379269	−	0.925287i	$-0.376176\pi$
$5$	1.69614	0.758537	0.379269	+	0.925287i	$0.376176\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.32431	0.399294	0.199647	−	0.979868i	$-0.436021\pi$
$11$	1.32431	0.399294	0.199647	+	0.979868i	$0.436021\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−1.69614	−0.437942
$16$	0	0
$17$	0.102658	0.0248982	0.0124491	−	0.999923i	$-0.496037\pi$
$17$	0.102658	0.0248982	0.0124491	+	0.999923i	$0.496037\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−9.16400	−1.91083	−0.955413	−	0.295272i	$-0.904590\pi$
$23$	−9.16400	−1.91083	−0.955413	+	0.295272i	$0.904590\pi$
$24$	0	0
$25$	−2.12311	−0.424621
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.81925	−0.894912	−0.447456	−	0.894306i	$-0.647670\pi$
$29$	−4.81925	−0.894912	−0.447456	+	0.894306i	$0.647670\pi$
$30$	0	0
$31$	−0.675693	−0.121358	−0.0606790	−	0.998157i	$-0.519327\pi$
$31$	−0.675693	−0.121358	−0.0606790	+	0.998157i	$0.519327\pi$
$32$	0	0
$33$	−1.32431	−0.230532
$34$	0	0
$35$	−1.69614	−0.286700
$36$	0	0
$37$	0.406517	0.0668309	0.0334155	−	0.999442i	$-0.489362\pi$
$37$	0.406517	0.0668309	0.0334155	+	0.999442i	$0.489362\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.04090	0.631082	0.315541	−	0.948912i	$-0.397814\pi$
$41$	4.04090	0.631082	0.315541	+	0.948912i	$0.397814\pi$
$42$	0	0
$43$	−4.91779	−0.749956	−0.374978	−	0.927034i	$-0.622350\pi$
$43$	−4.91779	−0.749956	−0.374978	+	0.927034i	$0.622350\pi$
$44$	0	0
$45$	1.69614	0.252846
$46$	0	0
$47$	3.66906	0.535188	0.267594	−	0.963532i	$-0.413771\pi$
$47$	3.66906	0.535188	0.267594	+	0.963532i	$0.413771\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.102658	−0.0143750
$52$	0	0
$53$	5.42696	0.745451	0.372725	−	0.927942i	$-0.378423\pi$
$53$	5.42696	0.745451	0.372725	+	0.927942i	$0.378423\pi$
$54$	0	0
$55$	2.24621	0.302879
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−9.43318	−1.22810	−0.614048	−	0.789269i	$-0.710460\pi$
$59$	−9.43318	−1.22810	−0.614048	+	0.789269i	$0.710460\pi$
$60$	0	0
$61$	−0.648614	−0.0830465	−0.0415232	−	0.999138i	$-0.513221\pi$
$61$	−0.648614	−0.0830465	−0.0415232	+	0.999138i	$0.513221\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.92190	−1.08998	−0.544992	−	0.838441i	$-0.683467\pi$
$67$	−8.92190	−1.08998	−0.544992	+	0.838441i	$0.683467\pi$
$68$	0	0
$69$	9.16400	1.10322
$70$	0	0
$71$	5.46786	0.648916	0.324458	−	0.945900i	$-0.394818\pi$
$71$	5.46786	0.648916	0.324458	+	0.945900i	$0.394818\pi$
$72$	0	0
$73$	4.04090	0.472951	0.236476	−	0.971637i	$-0.424008\pi$
$73$	4.04090	0.472951	0.236476	+	0.971637i	$0.424008\pi$
$74$	0	0
$75$	2.12311	0.245155
$76$	0	0
$77$	−1.32431	−0.150919
$78$	0	0
$79$	8.55628	0.962657	0.481329	−	0.876540i	$-0.340154\pi$
$79$	8.55628	0.962657	0.481329	+	0.876540i	$0.340154\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.4679	−1.47829	−0.739145	−	0.673546i	$-0.764770\pi$
$83$	−13.4679	−1.47829	−0.739145	+	0.673546i	$0.764770\pi$
$84$	0	0
$85$	0.174122	0.0188862
$86$	0	0
$87$	4.81925	0.516677
$88$	0	0
$89$	−4.85393	−0.514515	−0.257258	−	0.966343i	$-0.582819\pi$
$89$	−4.85393	−0.514515	−0.257258	+	0.966343i	$0.582819\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.675693	0.0700661
$94$	0	0
$95$	−1.69614	−0.174020
$96$	0	0
$97$	2.71659	0.275828	0.137914	−	0.990444i	$-0.455960\pi$
$97$	2.71659	0.275828	0.137914	+	0.990444i	$0.455960\pi$
$98$	0	0
$99$	1.32431	0.133098
$100$	0	0
$101$	−13.7779	−1.37096	−0.685478	−	0.728094i	$-0.740407\pi$
$101$	−13.7779	−1.37096	−0.685478	+	0.728094i	$0.740407\pi$
$102$	0	0
$103$	−9.05513	−0.892229	−0.446114	−	0.894976i	$-0.647193\pi$
$103$	−9.05513	−0.892229	−0.446114	+	0.894976i	$0.647193\pi$
$104$	0	0
$105$	1.69614	0.165526
$106$	0	0
$107$	−3.22165	−0.311449	−0.155724	−	0.987801i	$-0.549771\pi$
$107$	−3.22165	−0.311449	−0.155724	+	0.987801i	$0.549771\pi$
$108$	0	0
$109$	12.2830	1.17650	0.588249	−	0.808680i	$-0.299818\pi$
$109$	12.2830	1.17650	0.588249	+	0.808680i	$0.299818\pi$
$110$	0	0
$111$	−0.406517	−0.0385849
$112$	0	0
$113$	−11.4679	−1.07881	−0.539403	−	0.842048i	$-0.681350\pi$
$113$	−11.4679	−1.07881	−0.539403	+	0.842048i	$0.681350\pi$
$114$	0	0
$115$	−15.5434	−1.44943
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.102658	−0.00941062
$120$	0	0
$121$	−9.24621	−0.840565
$122$	0	0
$123$	−4.04090	−0.364355
$124$	0	0
$125$	−12.0818	−1.08063
$126$	0	0
$127$	−10.9178	−0.968797	−0.484399	−	0.874847i	$-0.660962\pi$
$127$	−10.9178	−0.968797	−0.484399	+	0.874847i	$0.660962\pi$
$128$	0	0
$129$	4.91779	0.432987
$130$	0	0
$131$	2.81925	0.246319	0.123159	−	0.992387i	$-0.460697\pi$
$131$	2.81925	0.246319	0.123159	+	0.992387i	$0.460697\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−1.69614	−0.145981
$136$	0	0
$137$	20.3280	1.73674	0.868369	−	0.495918i	$-0.165168\pi$
$137$	20.3280	1.73674	0.868369	+	0.495918i	$0.165168\pi$
$138$	0	0
$139$	4.82546	0.409290	0.204645	−	0.978836i	$-0.434396\pi$
$139$	4.82546	0.409290	0.204645	+	0.978836i	$0.434396\pi$
$140$	0	0
$141$	−3.66906	−0.308991
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.17412	−0.678824
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	19.8806	1.62868	0.814341	−	0.580387i	$-0.197099\pi$
$149$	19.8806	1.62868	0.814341	+	0.580387i	$0.197099\pi$
$150$	0	0
$151$	−17.1640	−1.39679	−0.698393	−	0.715714i	$-0.746101\pi$
$151$	−17.1640	−1.39679	−0.698393	+	0.715714i	$0.746101\pi$
$152$	0	0
$153$	0.102658	0.00829938
$154$	0	0
$155$	−1.14607	−0.0920546
$156$	0	0
$157$	4.24621	0.338885	0.169442	−	0.985540i	$-0.445803\pi$
$157$	4.24621	0.338885	0.169442	+	0.985540i	$0.445803\pi$
$158$	0	0
$159$	−5.42696	−0.430386
$160$	0	0
$161$	9.16400	0.722224
$162$	0	0
$163$	3.70235	0.289991	0.144995	−	0.989432i	$-0.453683\pi$
$163$	3.70235	0.289991	0.144995	+	0.989432i	$0.453683\pi$
$164$	0	0
$165$	−2.24621	−0.174867
$166$	0	0
$167$	−20.2871	−1.56986	−0.784932	−	0.619582i	$-0.787302\pi$
$167$	−20.2871	−1.56986	−0.784932	+	0.619582i	$0.787302\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−5.39228	−0.409968	−0.204984	−	0.978765i	$-0.565714\pi$
$173$	−5.39228	−0.409968	−0.204984	+	0.978765i	$0.565714\pi$
$174$	0	0
$175$	2.12311	0.160492
$176$	0	0
$177$	9.43318	0.709041
$178$	0	0
$179$	3.63228	0.271489	0.135745	−	0.990744i	$-0.456657\pi$
$179$	3.63228	0.271489	0.135745	+	0.990744i	$0.456657\pi$
$180$	0	0
$181$	−17.7433	−1.31885	−0.659423	−	0.751772i	$-0.729199\pi$
$181$	−17.7433	−1.31885	−0.659423	+	0.751772i	$0.729199\pi$
$182$	0	0
$183$	0.648614	0.0479469
$184$	0	0
$185$	0.689510	0.0506938
$186$	0	0
$187$	0.135950	0.00994167
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	19.9486	1.44343	0.721714	−	0.692192i	$-0.243355\pi$
$191$	19.9486	1.44343	0.721714	+	0.692192i	$0.243355\pi$
$192$	0	0
$193$	−6.48419	−0.466743	−0.233371	−	0.972388i	$-0.574976\pi$
$193$	−6.48419	−0.466743	−0.233371	+	0.972388i	$0.574976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.48831	0.604767	0.302383	−	0.953186i	$-0.402218\pi$
$197$	8.48831	0.604767	0.302383	+	0.953186i	$0.402218\pi$
$198$	0	0
$199$	−13.2279	−0.937698	−0.468849	−	0.883278i	$-0.655331\pi$
$199$	−13.2279	−0.937698	−0.468849	+	0.883278i	$0.655331\pi$
$200$	0	0
$201$	8.92190	0.629303
$202$	0	0
$203$	4.81925	0.338245
$204$	0	0
$205$	6.85393	0.478699
$206$	0	0
$207$	−9.16400	−0.636942
$208$	0	0
$209$	−1.32431	−0.0916042
$210$	0	0
$211$	−13.6656	−0.940777	−0.470388	−	0.882459i	$-0.655886\pi$
$211$	−13.6656	−0.940777	−0.470388	+	0.882459i	$0.655886\pi$
$212$	0	0
$213$	−5.46786	−0.374652
$214$	0	0
$215$	−8.34127	−0.568870
$216$	0	0
$217$	0.675693	0.0458690
$218$	0	0
$219$	−4.04090	−0.273058
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−21.5705	−1.44447	−0.722234	−	0.691648i	$-0.756885\pi$
$223$	−21.5705	−1.44447	−0.722234	+	0.691648i	$0.756885\pi$
$224$	0	0
$225$	−2.12311	−0.141540
$226$	0	0
$227$	11.3381	0.752538	0.376269	−	0.926511i	$-0.377207\pi$
$227$	11.3381	0.752538	0.376269	+	0.926511i	$0.377207\pi$
$228$	0	0
$229$	11.9867	0.792106	0.396053	−	0.918228i	$-0.370380\pi$
$229$	11.9867	0.792106	0.396053	+	0.918228i	$0.370380\pi$
$230$	0	0
$231$	1.32431	0.0871330
$232$	0	0
$233$	−26.3689	−1.72748	−0.863742	−	0.503934i	$-0.831885\pi$
$233$	−26.3689	−1.72748	−0.863742	+	0.503934i	$0.831885\pi$
$234$	0	0
$235$	6.22325	0.405960
$236$	0	0
$237$	−8.55628	−0.555790
$238$	0	0
$239$	−11.6155	−0.751346	−0.375673	−	0.926752i	$-0.622588\pi$
$239$	−11.6155	−0.751346	−0.375673	+	0.926752i	$0.622588\pi$
$240$	0	0
$241$	−10.1782	−0.655638	−0.327819	−	0.944741i	$-0.606314\pi$
$241$	−10.1782	−0.655638	−0.327819	+	0.944741i	$0.606314\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.69614	0.108362
$246$	0	0
$247$	0	0
$248$	0	0
$249$	13.4679	0.853491
$250$	0	0
$251$	22.8317	1.44112	0.720561	−	0.693391i	$-0.243884\pi$
$251$	22.8317	1.44112	0.720561	+	0.693391i	$0.243884\pi$
$252$	0	0
$253$	−12.1360	−0.762981
$254$	0	0
$255$	−0.174122	−0.0109039
$256$	0	0
$257$	0.730405	0.0455614	0.0227807	−	0.999740i	$-0.492748\pi$
$257$	0.730405	0.0455614	0.0227807	+	0.999740i	$0.492748\pi$
$258$	0	0
$259$	−0.406517	−0.0252597
$260$	0	0
$261$	−4.81925	−0.298304
$262$	0	0
$263$	27.3408	1.68591	0.842954	−	0.537985i	$-0.180814\pi$
$263$	27.3408	1.68591	0.842954	+	0.537985i	$0.180814\pi$
$264$	0	0
$265$	9.20490	0.565452
$266$	0	0
$267$	4.85393	0.297056
$268$	0	0
$269$	−7.84381	−0.478245	−0.239123	−	0.970989i	$-0.576860\pi$
$269$	−7.84381	−0.478245	−0.239123	+	0.970989i	$0.576860\pi$
$270$	0	0
$271$	−2.70277	−0.164182	−0.0820909	−	0.996625i	$-0.526160\pi$
$271$	−2.70277	−0.164182	−0.0820909	+	0.996625i	$0.526160\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.81164	−0.169548
$276$	0	0
$277$	−0.958686	−0.0576019	−0.0288009	−	0.999585i	$-0.509169\pi$
$277$	−0.958686	−0.0576019	−0.0288009	+	0.999585i	$0.509169\pi$
$278$	0	0
$279$	−0.675693	−0.0404527
$280$	0	0
$281$	0.335052	0.0199876	0.00999378	−	0.999950i	$-0.496819\pi$
$281$	0.335052	0.0199876	0.00999378	+	0.999950i	$0.496819\pi$
$282$	0	0
$283$	−16.6977	−0.992577	−0.496289	−	0.868158i	$-0.665304\pi$
$283$	−16.6977	−0.992577	−0.496289	+	0.868158i	$0.665304\pi$
$284$	0	0
$285$	1.69614	0.100471
$286$	0	0
$287$	−4.04090	−0.238527
$288$	0	0
$289$	−16.9895	−0.999380
$290$	0	0
$291$	−2.71659	−0.159249
$292$	0	0
$293$	9.87648	0.576990	0.288495	−	0.957481i	$-0.406845\pi$
$293$	9.87648	0.576990	0.288495	+	0.957481i	$0.406845\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	−1.32431	−0.0768441
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.91779	0.283457
$302$	0	0
$303$	13.7779	0.791522
$304$	0	0
$305$	−1.10014	−0.0629939
$306$	0	0
$307$	−16.5154	−0.942583	−0.471291	−	0.881978i	$-0.656212\pi$
$307$	−16.5154	−0.942583	−0.471291	+	0.881978i	$0.656212\pi$
$308$	0	0
$309$	9.05513	0.515128
$310$	0	0
$311$	−10.9920	−0.623298	−0.311649	−	0.950197i	$-0.600881\pi$
$311$	−10.9920	−0.623298	−0.311649	+	0.950197i	$0.600881\pi$
$312$	0	0
$313$	−6.48419	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$313$	−6.48419	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$314$	0	0
$315$	−1.69614	−0.0955667
$316$	0	0
$317$	−7.95205	−0.446632	−0.223316	−	0.974746i	$-0.571688\pi$
$317$	−7.95205	−0.446632	−0.223316	+	0.974746i	$0.571688\pi$
$318$	0	0
$319$	−6.38216	−0.357332
$320$	0	0
$321$	3.22165	0.179815
$322$	0	0
$323$	−0.102658	−0.00571203
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.2830	−0.679251
$328$	0	0
$329$	−3.66906	−0.202282
$330$	0	0
$331$	24.3266	1.33711	0.668556	−	0.743662i	$-0.266913\pi$
$331$	24.3266	1.33711	0.668556	+	0.743662i	$0.266913\pi$
$332$	0	0
$333$	0.406517	0.0222770
$334$	0	0
$335$	−15.1328	−0.826794
$336$	0	0
$337$	−21.8765	−1.19169	−0.595844	−	0.803100i	$-0.703182\pi$
$337$	−21.8765	−1.19169	−0.595844	+	0.803100i	$0.703182\pi$
$338$	0	0
$339$	11.4679	0.622849
$340$	0	0
$341$	−0.894825	−0.0484575
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	15.5434	0.836831
$346$	0	0
$347$	−2.01382	−0.108107	−0.0540537	−	0.998538i	$-0.517214\pi$
$347$	−2.01382	−0.108107	−0.0540537	+	0.998538i	$0.517214\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.58999	−0.510424	−0.255212	−	0.966885i	$-0.582145\pi$
$353$	−9.58999	−0.510424	−0.255212	+	0.966885i	$0.582145\pi$
$354$	0	0
$355$	9.27426	0.492227
$356$	0	0
$357$	0.102658	0.00543322
$358$	0	0
$359$	18.9260	0.998877	0.499439	−	0.866349i	$-0.333540\pi$
$359$	18.9260	0.998877	0.499439	+	0.866349i	$0.333540\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.24621	0.485300
$364$	0	0
$365$	6.85393	0.358751
$366$	0	0
$367$	32.0483	1.67291	0.836454	−	0.548038i	$-0.184625\pi$
$367$	32.0483	1.67291	0.836454	+	0.548038i	$0.184625\pi$
$368$	0	0
$369$	4.04090	0.210361
$370$	0	0
$371$	−5.42696	−0.281754
$372$	0	0
$373$	4.89071	0.253231	0.126616	−	0.991952i	$-0.459588\pi$
$373$	4.89071	0.253231	0.126616	+	0.991952i	$0.459588\pi$
$374$	0	0
$375$	12.0818	0.623901
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.07810	−0.363577	−0.181789	−	0.983338i	$-0.558189\pi$
$379$	−7.07810	−0.363577	−0.181789	+	0.983338i	$0.558189\pi$
$380$	0	0
$381$	10.9178	0.559335
$382$	0	0
$383$	−21.9867	−1.12347	−0.561735	−	0.827317i	$-0.689866\pi$
$383$	−21.9867	−1.12347	−0.561735	+	0.827317i	$0.689866\pi$
$384$	0	0
$385$	−2.24621	−0.114478
$386$	0	0
$387$	−4.91779	−0.249985
$388$	0	0
$389$	10.7680	0.545960	0.272980	−	0.962020i	$-0.411991\pi$
$389$	10.7680	0.545960	0.272980	+	0.962020i	$0.411991\pi$
$390$	0	0
$391$	−0.940755	−0.0475760
$392$	0	0
$393$	−2.81925	−0.142212
$394$	0	0
$395$	14.5127	0.730211
$396$	0	0
$397$	−5.86405	−0.294308	−0.147154	−	0.989114i	$-0.547011\pi$
$397$	−5.86405	−0.294308	−0.147154	+	0.989114i	$0.547011\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	30.4036	1.51828	0.759141	−	0.650926i	$-0.225619\pi$
$401$	30.4036	1.51828	0.759141	+	0.650926i	$0.225619\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.69614	0.0842819
$406$	0	0
$407$	0.538353	0.0266852
$408$	0	0
$409$	−8.78045	−0.434165	−0.217083	−	0.976153i	$-0.569654\pi$
$409$	−8.78045	−0.434165	−0.217083	+	0.976153i	$0.569654\pi$
$410$	0	0
$411$	−20.3280	−1.00271
$412$	0	0
$413$	9.43318	0.464176
$414$	0	0
$415$	−22.8434	−1.12134
$416$	0	0
$417$	−4.82546	−0.236304
$418$	0	0
$419$	0.708986	0.0346362	0.0173181	−	0.999850i	$-0.494487\pi$
$419$	0.708986	0.0346362	0.0173181	+	0.999850i	$0.494487\pi$
$420$	0	0
$421$	11.7446	0.572399	0.286199	−	0.958170i	$-0.407608\pi$
$421$	11.7446	0.572399	0.286199	+	0.958170i	$0.407608\pi$
$422$	0	0
$423$	3.66906	0.178396
$424$	0	0
$425$	−0.217953	−0.0105723
$426$	0	0
$427$	0.648614	0.0313886
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.3985	1.03073	0.515365	−	0.856971i	$-0.327656\pi$
$431$	21.3985	1.03073	0.515365	+	0.856971i	$0.327656\pi$
$432$	0	0
$433$	17.8494	0.857787	0.428894	−	0.903355i	$-0.358904\pi$
$433$	17.8494	0.857787	0.428894	+	0.903355i	$0.358904\pi$
$434$	0	0
$435$	8.17412	0.391919
$436$	0	0
$437$	9.16400	0.438374
$438$	0	0
$439$	−29.7883	−1.42172	−0.710858	−	0.703336i	$-0.751693\pi$
$439$	−29.7883	−1.42172	−0.710858	+	0.703336i	$0.751693\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−13.0731	−0.621120	−0.310560	−	0.950554i	$-0.600517\pi$
$443$	−13.0731	−0.621120	−0.310560	+	0.950554i	$0.600517\pi$
$444$	0	0
$445$	−8.23295	−0.390279
$446$	0	0
$447$	−19.8806	−0.940320
$448$	0	0
$449$	−14.9295	−0.704567	−0.352284	−	0.935893i	$-0.614595\pi$
$449$	−14.9295	−0.704567	−0.352284	+	0.935893i	$0.614595\pi$
$450$	0	0
$451$	5.35139	0.251987
$452$	0	0
$453$	17.1640	0.806435
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.33854	0.202948	0.101474	−	0.994838i	$-0.467644\pi$
$457$	4.33854	0.202948	0.101474	+	0.994838i	$0.467644\pi$
$458$	0	0
$459$	−0.102658	−0.00479165
$460$	0	0
$461$	13.9832	0.651265	0.325632	−	0.945496i	$-0.394423\pi$
$461$	13.9832	0.651265	0.325632	+	0.945496i	$0.394423\pi$
$462$	0	0
$463$	−29.2972	−1.36156	−0.680779	−	0.732489i	$-0.738359\pi$
$463$	−29.2972	−1.36156	−0.680779	+	0.732489i	$0.738359\pi$
$464$	0	0
$465$	1.14607	0.0531478
$466$	0	0
$467$	−3.91116	−0.180987	−0.0904934	−	0.995897i	$-0.528844\pi$
$467$	−3.91116	−0.180987	−0.0904934	+	0.995897i	$0.528844\pi$
$468$	0	0
$469$	8.92190	0.411975
$470$	0	0
$471$	−4.24621	−0.195655
$472$	0	0
$473$	−6.51266	−0.299453
$474$	0	0
$475$	2.12311	0.0974148
$476$	0	0
$477$	5.42696	0.248484
$478$	0	0
$479$	35.0613	1.60199	0.800997	−	0.598669i	$-0.204303\pi$
$479$	35.0613	1.60199	0.800997	+	0.598669i	$0.204303\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−9.16400	−0.416977
$484$	0	0
$485$	4.60772	0.209226
$486$	0	0
$487$	−16.6922	−0.756397	−0.378199	−	0.925724i	$-0.623456\pi$
$487$	−16.6922	−0.756397	−0.378199	+	0.925724i	$0.623456\pi$
$488$	0	0
$489$	−3.70235	−0.167426
$490$	0	0
$491$	−6.82685	−0.308091	−0.154046	−	0.988064i	$-0.549230\pi$
$491$	−6.82685	−0.308091	−0.154046	+	0.988064i	$0.549230\pi$
$492$	0	0
$493$	−0.494733	−0.0222816
$494$	0	0
$495$	2.24621	0.100960
$496$	0	0
$497$	−5.46786	−0.245267
$498$	0	0
$499$	35.1410	1.57313	0.786564	−	0.617508i	$-0.211858\pi$
$499$	35.1410	1.57313	0.786564	+	0.617508i	$0.211858\pi$
$500$	0	0
$501$	20.2871	0.906361
$502$	0	0
$503$	32.4486	1.44681	0.723406	−	0.690423i	$-0.242576\pi$
$503$	32.4486	1.44681	0.723406	+	0.690423i	$0.242576\pi$
$504$	0	0
$505$	−23.3693	−1.03992
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	5.05924	0.224247	0.112124	−	0.993694i	$-0.464235\pi$
$509$	5.05924	0.224247	0.112124	+	0.993694i	$0.464235\pi$
$510$	0	0
$511$	−4.04090	−0.178759
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−15.3588	−0.676789
$516$	0	0
$517$	4.85896	0.213697
$518$	0	0
$519$	5.39228	0.236695
$520$	0	0
$521$	2.41063	0.105612	0.0528058	−	0.998605i	$-0.483184\pi$
$521$	2.41063	0.105612	0.0528058	+	0.998605i	$0.483184\pi$
$522$	0	0
$523$	−16.5154	−0.722167	−0.361084	−	0.932533i	$-0.617593\pi$
$523$	−16.5154	−0.722167	−0.361084	+	0.932533i	$0.617593\pi$
$524$	0	0
$525$	−2.12311	−0.0926599
$526$	0	0
$527$	−0.0693651	−0.00302159
$528$	0	0
$529$	60.9789	2.65126
$530$	0	0
$531$	−9.43318	−0.409365
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−5.46437	−0.236245
$536$	0	0
$537$	−3.63228	−0.156744
$538$	0	0
$539$	1.32431	0.0570419
$540$	0	0
$541$	−2.49242	−0.107158	−0.0535788	−	0.998564i	$-0.517063\pi$
$541$	−2.49242	−0.107158	−0.0535788	+	0.998564i	$0.517063\pi$
$542$	0	0
$543$	17.7433	0.761436
$544$	0	0
$545$	20.8337	0.892417
$546$	0	0
$547$	−19.8576	−0.849051	−0.424525	−	0.905416i	$-0.639559\pi$
$547$	−19.8576	−0.849051	−0.424525	+	0.905416i	$0.639559\pi$
$548$	0	0
$549$	−0.648614	−0.0276822
$550$	0	0
$551$	4.81925	0.205307
$552$	0	0
$553$	−8.55628	−0.363850
$554$	0	0
$555$	−0.689510	−0.0292681
$556$	0	0
$557$	37.7162	1.59809	0.799043	−	0.601274i	$-0.205340\pi$
$557$	37.7162	1.59809	0.799043	+	0.601274i	$0.205340\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.135950	−0.00573983
$562$	0	0
$563$	17.5025	0.737644	0.368822	−	0.929500i	$-0.379761\pi$
$563$	17.5025	0.737644	0.368822	+	0.929500i	$0.379761\pi$
$564$	0	0
$565$	−19.4511	−0.818314
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−1.02456	−0.0429519	−0.0214759	−	0.999769i	$-0.506837\pi$
$569$	−1.02456	−0.0429519	−0.0214759	+	0.999769i	$0.506837\pi$
$570$	0	0
$571$	31.2770	1.30890	0.654451	−	0.756105i	$-0.272900\pi$
$571$	31.2770	1.30890	0.654451	+	0.756105i	$0.272900\pi$
$572$	0	0
$573$	−19.9486	−0.833363
$574$	0	0
$575$	19.4561	0.811377
$576$	0	0
$577$	10.1636	0.423116	0.211558	−	0.977365i	$-0.432146\pi$
$577$	10.1636	0.423116	0.211558	+	0.977365i	$0.432146\pi$
$578$	0	0
$579$	6.48419	0.269474
$580$	0	0
$581$	13.4679	0.558741
$582$	0	0
$583$	7.18697	0.297654
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.1799	1.12184	0.560918	−	0.827872i	$-0.310448\pi$
$587$	27.1799	1.12184	0.560918	+	0.827872i	$0.310448\pi$
$588$	0	0
$589$	0.675693	0.0278414
$590$	0	0
$591$	−8.48831	−0.349162
$592$	0	0
$593$	20.2662	0.832235	0.416117	−	0.909311i	$-0.363391\pi$
$593$	20.2662	0.832235	0.416117	+	0.909311i	$0.363391\pi$
$594$	0	0
$595$	−0.174122	−0.00713830
$596$	0	0
$597$	13.2279	0.541380
$598$	0	0
$599$	32.3342	1.32114	0.660570	−	0.750764i	$-0.270315\pi$
$599$	32.3342	1.32114	0.660570	+	0.750764i	$0.270315\pi$
$600$	0	0
$601$	−11.4755	−0.468094	−0.234047	−	0.972225i	$-0.575197\pi$
$601$	−11.4755	−0.468094	−0.234047	+	0.972225i	$0.575197\pi$
$602$	0	0
$603$	−8.92190	−0.363328
$604$	0	0
$605$	−15.6829	−0.637600
$606$	0	0
$607$	−2.32473	−0.0943577	−0.0471788	−	0.998886i	$-0.515023\pi$
$607$	−2.32473	−0.0943577	−0.0471788	+	0.998886i	$0.515023\pi$
$608$	0	0
$609$	−4.81925	−0.195286
$610$	0	0
$611$	0	0
$612$	0	0
$613$	21.2334	0.857608	0.428804	−	0.903398i	$-0.358935\pi$
$613$	21.2334	0.857608	0.428804	+	0.903398i	$0.358935\pi$
$614$	0	0
$615$	−6.85393	−0.276377
$616$	0	0
$617$	−4.11026	−0.165473	−0.0827364	−	0.996571i	$-0.526366\pi$
$617$	−4.11026	−0.165473	−0.0827364	+	0.996571i	$0.526366\pi$
$618$	0	0
$619$	14.4515	0.580856	0.290428	−	0.956897i	$-0.406202\pi$
$619$	14.4515	0.580856	0.290428	+	0.956897i	$0.406202\pi$
$620$	0	0
$621$	9.16400	0.367739
$622$	0	0
$623$	4.85393	0.194469
$624$	0	0
$625$	−9.87689	−0.395076
$626$	0	0
$627$	1.32431	0.0528877
$628$	0	0
$629$	0.0417321	0.00166397
$630$	0	0
$631$	2.67981	0.106681	0.0533407	−	0.998576i	$-0.483013\pi$
$631$	2.67981	0.106681	0.0533407	+	0.998576i	$0.483013\pi$
$632$	0	0
$633$	13.6656	0.543158
$634$	0	0
$635$	−18.5181	−0.734869
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.46786	0.216305
$640$	0	0
$641$	−20.7775	−0.820662	−0.410331	−	0.911937i	$-0.634587\pi$
$641$	−20.7775	−0.820662	−0.410331	+	0.911937i	$0.634587\pi$
$642$	0	0
$643$	1.01835	0.0401598	0.0200799	−	0.999798i	$-0.493608\pi$
$643$	1.01835	0.0401598	0.0200799	+	0.999798i	$0.493608\pi$
$644$	0	0
$645$	8.34127	0.328437
$646$	0	0
$647$	16.1256	0.633964	0.316982	−	0.948432i	$-0.397331\pi$
$647$	16.1256	0.633964	0.316982	+	0.948432i	$0.397331\pi$
$648$	0	0
$649$	−12.4924	−0.490370
$650$	0	0
$651$	−0.675693	−0.0264825
$652$	0	0
$653$	−21.4373	−0.838906	−0.419453	−	0.907777i	$-0.637778\pi$
$653$	−21.4373	−0.838906	−0.419453	+	0.907777i	$0.637778\pi$
$654$	0	0
$655$	4.78184	0.186842
$656$	0	0
$657$	4.04090	0.157650
$658$	0	0
$659$	13.1890	0.513770	0.256885	−	0.966442i	$-0.417304\pi$
$659$	13.1890	0.513770	0.256885	+	0.966442i	$0.417304\pi$
$660$	0	0
$661$	−10.1921	−0.396425	−0.198212	−	0.980159i	$-0.563514\pi$
$661$	−10.1921	−0.396425	−0.198212	+	0.980159i	$0.563514\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.69614	0.0657735
$666$	0	0
$667$	44.1636	1.71002
$668$	0	0
$669$	21.5705	0.833964
$670$	0	0
$671$	−0.858964	−0.0331599
$672$	0	0
$673$	43.6868	1.68400	0.842001	−	0.539476i	$-0.181378\pi$
$673$	43.6868	1.68400	0.842001	+	0.539476i	$0.181378\pi$
$674$	0	0
$675$	2.12311	0.0817184
$676$	0	0
$677$	−5.37986	−0.206765	−0.103382	−	0.994642i	$-0.532967\pi$
$677$	−5.37986	−0.206765	−0.103382	+	0.994642i	$0.532967\pi$
$678$	0	0
$679$	−2.71659	−0.104253
$680$	0	0
$681$	−11.3381	−0.434478
$682$	0	0
$683$	15.3576	0.587642	0.293821	−	0.955860i	$-0.405073\pi$
$683$	15.3576	0.587642	0.293821	+	0.955860i	$0.405073\pi$
$684$	0	0
$685$	34.4792	1.31738
$686$	0	0
$687$	−11.9867	−0.457323
$688$	0	0
$689$	0	0
$690$	0	0
$691$	19.3588	0.736443	0.368221	−	0.929738i	$-0.379967\pi$
$691$	19.3588	0.736443	0.368221	+	0.929738i	$0.379967\pi$
$692$	0	0
$693$	−1.32431	−0.0503063
$694$	0	0
$695$	8.18466	0.310462
$696$	0	0
$697$	0.414829	0.0157128
$698$	0	0
$699$	26.3689	0.997363
$700$	0	0
$701$	24.8420	0.938269	0.469135	−	0.883127i	$-0.344566\pi$
$701$	24.8420	0.938269	0.469135	+	0.883127i	$0.344566\pi$
$702$	0	0
$703$	−0.406517	−0.0153321
$704$	0	0
$705$	−6.22325	−0.234381
$706$	0	0
$707$	13.7779	0.518172
$708$	0	0
$709$	33.5053	1.25832	0.629158	−	0.777277i	$-0.283400\pi$
$709$	33.5053	1.25832	0.629158	+	0.777277i	$0.283400\pi$
$710$	0	0
$711$	8.55628	0.320886
$712$	0	0
$713$	6.19205	0.231894
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.6155	0.433790
$718$	0	0
$719$	−9.84591	−0.367190	−0.183595	−	0.983002i	$-0.558774\pi$
$719$	−9.84591	−0.367190	−0.183595	+	0.983002i	$0.558774\pi$
$720$	0	0
$721$	9.05513	0.337231
$722$	0	0
$723$	10.1782	0.378533
$724$	0	0
$725$	10.2318	0.379998
$726$	0	0
$727$	−40.8737	−1.51592	−0.757962	−	0.652299i	$-0.773805\pi$
$727$	−40.8737	−1.51592	−0.757962	+	0.652299i	$0.773805\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.504849	−0.0186725
$732$	0	0
$733$	6.77130	0.250104	0.125052	−	0.992150i	$-0.460090\pi$
$733$	6.77130	0.250104	0.125052	+	0.992150i	$0.460090\pi$
$734$	0	0
$735$	−1.69614	−0.0625631
$736$	0	0
$737$	−11.8153	−0.435224
$738$	0	0
$739$	0.781840	0.0287604	0.0143802	−	0.999897i	$-0.495422\pi$
$739$	0.781840	0.0287604	0.0143802	+	0.999897i	$0.495422\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.1422	1.87623	0.938113	−	0.346329i	$-0.112572\pi$
$743$	51.1422	1.87623	0.938113	+	0.346329i	$0.112572\pi$
$744$	0	0
$745$	33.7203	1.23542
$746$	0	0
$747$	−13.4679	−0.492763
$748$	0	0
$749$	3.22165	0.117716
$750$	0	0
$751$	5.97703	0.218105	0.109053	−	0.994036i	$-0.465218\pi$
$751$	5.97703	0.218105	0.109053	+	0.994036i	$0.465218\pi$
$752$	0	0
$753$	−22.8317	−0.832032
$754$	0	0
$755$	−29.1126	−1.05951
$756$	0	0
$757$	−37.3046	−1.35586	−0.677930	−	0.735127i	$-0.737123\pi$
$757$	−37.3046	−1.35586	−0.677930	+	0.735127i	$0.737123\pi$
$758$	0	0
$759$	12.1360	0.440507
$760$	0	0
$761$	−46.5818	−1.68859	−0.844295	−	0.535879i	$-0.819980\pi$
$761$	−46.5818	−1.68859	−0.844295	+	0.535879i	$0.819980\pi$
$762$	0	0
$763$	−12.2830	−0.444674
$764$	0	0
$765$	0.174122	0.00629539
$766$	0	0
$767$	0	0
$768$	0	0
$769$	20.3822	0.735000	0.367500	−	0.930024i	$-0.380214\pi$
$769$	20.3822	0.735000	0.367500	+	0.930024i	$0.380214\pi$
$770$	0	0
$771$	−0.730405	−0.0263049
$772$	0	0
$773$	27.9380	1.00486	0.502430	−	0.864618i	$-0.332439\pi$
$773$	27.9380	1.00486	0.502430	+	0.864618i	$0.332439\pi$
$774$	0	0
$775$	1.43457	0.0515312
$776$	0	0
$777$	0.406517	0.0145837
$778$	0	0
$779$	−4.04090	−0.144780
$780$	0	0
$781$	7.24113	0.259108
$782$	0	0
$783$	4.81925	0.172226
$784$	0	0
$785$	7.20217	0.257057
$786$	0	0
$787$	−35.8200	−1.27685	−0.638423	−	0.769686i	$-0.720413\pi$
$787$	−35.8200	−1.27685	−0.638423	+	0.769686i	$0.720413\pi$
$788$	0	0
$789$	−27.3408	−0.973360
$790$	0	0
$791$	11.4679	0.407750
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−9.20490	−0.326464
$796$	0	0
$797$	−15.6385	−0.553944	−0.276972	−	0.960878i	$-0.589331\pi$
$797$	−15.6385	−0.553944	−0.276972	+	0.960878i	$0.589331\pi$
$798$	0	0
$799$	0.376657	0.0133252
$800$	0	0
$801$	−4.85393	−0.171505
$802$	0	0
$803$	5.35139	0.188846
$804$	0	0
$805$	15.5434	0.547834
$806$	0	0
$807$	7.84381	0.276115
$808$	0	0
$809$	−23.9256	−0.841179	−0.420590	−	0.907251i	$-0.638177\pi$
$809$	−23.9256	−0.841179	−0.420590	+	0.907251i	$0.638177\pi$
$810$	0	0
$811$	−37.6868	−1.32336	−0.661681	−	0.749786i	$-0.730157\pi$
$811$	−37.6868	−1.32336	−0.661681	+	0.749786i	$0.730157\pi$
$812$	0	0
$813$	2.70277	0.0947904
$814$	0	0
$815$	6.27972	0.219969
$816$	0	0
$817$	4.91779	0.172052
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.4166	0.747446	0.373723	−	0.927540i	$-0.378081\pi$
$821$	21.4166	0.747446	0.373723	+	0.927540i	$0.378081\pi$
$822$	0	0
$823$	−33.0272	−1.15126	−0.575628	−	0.817712i	$-0.695242\pi$
$823$	−33.0272	−1.15126	−0.575628	+	0.817712i	$0.695242\pi$
$824$	0	0
$825$	2.81164	0.0978889
$826$	0	0
$827$	7.71407	0.268245	0.134122	−	0.990965i	$-0.457179\pi$
$827$	7.71407	0.268245	0.134122	+	0.990965i	$0.457179\pi$
$828$	0	0
$829$	1.79741	0.0624266	0.0312133	−	0.999513i	$-0.490063\pi$
$829$	1.79741	0.0624266	0.0312133	+	0.999513i	$0.490063\pi$
$830$	0	0
$831$	0.958686	0.0332565
$832$	0	0
$833$	0.102658	0.00355688
$834$	0	0
$835$	−34.4098	−1.19080
$836$	0	0
$837$	0.675693	0.0233554
$838$	0	0
$839$	−1.49012	−0.0514445	−0.0257223	−	0.999669i	$-0.508189\pi$
$839$	−1.49012	−0.0514445	−0.0257223	+	0.999669i	$0.508189\pi$
$840$	0	0
$841$	−5.77486	−0.199133
$842$	0	0
$843$	−0.335052	−0.0115398
$844$	0	0
$845$	−22.0498	−0.758537
$846$	0	0
$847$	9.24621	0.317704
$848$	0	0
$849$	16.6977	0.573065
$850$	0	0
$851$	−3.72532	−0.127702
$852$	0	0
$853$	−0.648614	−0.0222081	−0.0111041	−	0.999938i	$-0.503535\pi$
$853$	−0.648614	−0.0222081	−0.0111041	+	0.999938i	$0.503535\pi$
$854$	0	0
$855$	−1.69614	−0.0580068
$856$	0	0
$857$	−16.5875	−0.566617	−0.283309	−	0.959029i	$-0.591432\pi$
$857$	−16.5875	−0.566617	−0.283309	+	0.959029i	$0.591432\pi$
$858$	0	0
$859$	26.9205	0.918516	0.459258	−	0.888303i	$-0.348115\pi$
$859$	26.9205	0.918516	0.459258	+	0.888303i	$0.348115\pi$
$860$	0	0
$861$	4.04090	0.137713
$862$	0	0
$863$	18.2267	0.620445	0.310223	−	0.950664i	$-0.399596\pi$
$863$	18.2267	0.620445	0.310223	+	0.950664i	$0.399596\pi$
$864$	0	0
$865$	−9.14607	−0.310976
$866$	0	0
$867$	16.9895	0.576992
$868$	0	0
$869$	11.3311	0.384383
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.71659	0.0919426
$874$	0	0
$875$	12.0818	0.408439
$876$	0	0
$877$	19.0634	0.643724	0.321862	−	0.946787i	$-0.395691\pi$
$877$	19.0634	0.643724	0.321862	+	0.946787i	$0.395691\pi$
$878$	0	0
$879$	−9.87648	−0.333125
$880$	0	0
$881$	−45.8354	−1.54423	−0.772116	−	0.635481i	$-0.780802\pi$
$881$	−45.8354	−1.54423	−0.772116	+	0.635481i	$0.780802\pi$
$882$	0	0
$883$	−23.0592	−0.776005	−0.388003	−	0.921658i	$-0.626835\pi$
$883$	−23.0592	−0.776005	−0.388003	+	0.921658i	$0.626835\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−26.3128	−0.883497	−0.441749	−	0.897139i	$-0.645642\pi$
$887$	−26.3128	−0.883497	−0.441749	+	0.897139i	$0.645642\pi$
$888$	0	0
$889$	10.9178	0.366171
$890$	0	0
$891$	1.32431	0.0443660
$892$	0	0
$893$	−3.66906	−0.122780
$894$	0	0
$895$	6.16086	0.205935
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.25633	0.108605
$900$	0	0
$901$	0.557120	0.0185603
$902$	0	0
$903$	−4.91779	−0.163654
$904$	0	0
$905$	−30.0951	−1.00039
$906$	0	0
$907$	34.9904	1.16184	0.580919	−	0.813961i	$-0.302693\pi$
$907$	34.9904	1.16184	0.580919	+	0.813961i	$0.302693\pi$
$908$	0	0
$909$	−13.7779	−0.456985
$910$	0	0
$911$	6.88583	0.228138	0.114069	−	0.993473i	$-0.463612\pi$
$911$	6.88583	0.228138	0.114069	+	0.993473i	$0.463612\pi$
$912$	0	0
$913$	−17.8356	−0.590272
$914$	0	0
$915$	1.10014	0.0363695
$916$	0	0
$917$	−2.81925	−0.0930997
$918$	0	0
$919$	−41.1484	−1.35736	−0.678681	−	0.734433i	$-0.737448\pi$
$919$	−41.1484	−1.35736	−0.678681	+	0.734433i	$0.737448\pi$
$920$	0	0
$921$	16.5154	0.544201
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−0.863078	−0.0283778
$926$	0	0
$927$	−9.05513	−0.297410
$928$	0	0
$929$	−37.6052	−1.23379	−0.616893	−	0.787047i	$-0.711609\pi$
$929$	−37.6052	−1.23379	−0.616893	+	0.787047i	$0.711609\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	10.9920	0.359861
$934$	0	0
$935$	0.230591	0.00754113
$936$	0	0
$937$	−19.9992	−0.653344	−0.326672	−	0.945138i	$-0.605927\pi$
$937$	−19.9992	−0.653344	−0.326672	+	0.945138i	$0.605927\pi$
$938$	0	0
$939$	6.48419	0.211604
$940$	0	0
$941$	−54.1718	−1.76595	−0.882975	−	0.469419i	$-0.844463\pi$
$941$	−54.1718	−1.76595	−0.882975	+	0.469419i	$0.844463\pi$
$942$	0	0
$943$	−37.0308	−1.20589
$944$	0	0
$945$	1.69614	0.0551755
$946$	0	0
$947$	40.7649	1.32468	0.662340	−	0.749203i	$-0.269563\pi$
$947$	40.7649	1.32468	0.662340	+	0.749203i	$0.269563\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	7.95205	0.257863
$952$	0	0
$953$	22.9419	0.743162	0.371581	−	0.928401i	$-0.378816\pi$
$953$	22.9419	0.743162	0.371581	+	0.928401i	$0.378816\pi$
$954$	0	0
$955$	33.8356	1.09489
$956$	0	0
$957$	6.38216	0.206306
$958$	0	0
$959$	−20.3280	−0.656425
$960$	0	0
$961$	−30.5434	−0.985272
$962$	0	0
$963$	−3.22165	−0.103816
$964$	0	0
$965$	−10.9981	−0.354042
$966$	0	0
$967$	17.9762	0.578076	0.289038	−	0.957318i	$-0.406665\pi$
$967$	17.9762	0.578076	0.289038	+	0.957318i	$0.406665\pi$
$968$	0	0
$969$	0.102658	0.00329784
$970$	0	0
$971$	13.3514	0.428466	0.214233	−	0.976783i	$-0.431275\pi$
$971$	13.3514	0.428466	0.214233	+	0.976783i	$0.431275\pi$
$972$	0	0
$973$	−4.82546	−0.154697
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.8734	−0.411857	−0.205928	−	0.978567i	$-0.566021\pi$
$977$	−12.8734	−0.411857	−0.205928	+	0.978567i	$0.566021\pi$
$978$	0	0
$979$	−6.42809	−0.205443
$980$	0	0
$981$	12.2830	0.392166
$982$	0	0
$983$	12.6007	0.401901	0.200951	−	0.979601i	$-0.435597\pi$
$983$	12.6007	0.401901	0.200951	+	0.979601i	$0.435597\pi$
$984$	0	0
$985$	14.3974	0.458738
$986$	0	0
$987$	3.66906	0.116788
$988$	0	0
$989$	45.0666	1.43304
$990$	0	0
$991$	0.00970237	0.000308206 0	0.000154103	−	1.00000i	$-0.499951\pi$
$991$	0.00970237	0.000308206 0	0.000154103		1.00000i	$0.499951\pi$
$992$	0	0
$993$	−24.3266	−0.771982
$994$	0	0
$995$	−22.4363	−0.711279
$996$	0	0
$997$	25.8153	0.817580	0.408790	−	0.912628i	$-0.365951\pi$
$997$	25.8153	0.817580	0.408790	+	0.912628i	$0.365951\pi$
$998$	0	0
$999$	−0.406517	−0.0128616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.x.1.3	✓	4
3.2	odd	2		9576.2.a.cj.1.2		4
4.3	odd	2		6384.2.a.cb.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.x.1.3	✓	4	1.1	even	1	trivial
6384.2.a.cb.1.3		4	4.3	odd	2
9576.2.a.cj.1.2		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.69614 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.32431 q^{11} -1.69614 q^{15} +0.102658 q^{17} -1.00000 q^{19} +1.00000 q^{21} -9.16400 q^{23} -2.12311 q^{25} -1.00000 q^{27} -4.81925 q^{29} -0.675693 q^{31} -1.32431 q^{33} -1.69614 q^{35} +0.406517 q^{37} +4.04090 q^{41} -4.91779 q^{43} +1.69614 q^{45} +3.66906 q^{47} +1.00000 q^{49} -0.102658 q^{51} +5.42696 q^{53} +2.24621 q^{55} +1.00000 q^{57} -9.43318 q^{59} -0.648614 q^{61} -1.00000 q^{63} -8.92190 q^{67} +9.16400 q^{69} +5.46786 q^{71} +4.04090 q^{73} +2.12311 q^{75} -1.32431 q^{77} +8.55628 q^{79} +1.00000 q^{81} -13.4679 q^{83} +0.174122 q^{85} +4.81925 q^{87} -4.85393 q^{89} +0.675693 q^{93} -1.69614 q^{95} +2.71659 q^{97} +1.32431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{7} + 4 q^{9} - 4 q^{17} - 4 q^{19} + 4 q^{21} + 4 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{37} - 8 q^{41} - 12 q^{43} - 8 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{53} - 24 q^{55}+ \cdots - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^7 + 4 * q^9 - 4 * q^17 - 4 * q^19 + 4 * q^21 + 4 * q^23 + 8 * q^25 - 4 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^37 - 8 * q^41 - 12 * q^43 - 8 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^53 - 24 * q^55 + 4 * q^57 + 8 * q^61 - 4 * q^63 - 8 * q^67 - 4 * q^69 - 12 * q^71 - 8 * q^73 - 8 * q^75 - 20 * q^79 + 4 * q^81 - 20 * q^83 - 4 * q^85 - 4 * q^87 + 8 * q^93 - 8 * q^97

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