LMFDB - Embedded newform 3192.2.a.r.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.82843 q^{11} +5.65685 q^{13} -3.41421 q^{15} +2.24264 q^{17} +1.00000 q^{19} -1.00000 q^{21} -8.48528 q^{23} +6.65685 q^{25} +1.00000 q^{27} +10.2426 q^{29} +8.82843 q^{31} -2.82843 q^{33} +3.41421 q^{35} -6.00000 q^{37} +5.65685 q^{39} -8.82843 q^{41} -8.82843 q^{43} -3.41421 q^{45} -10.5858 q^{47} +1.00000 q^{49} +2.24264 q^{51} -5.07107 q^{53} +9.65685 q^{55} +1.00000 q^{57} -2.34315 q^{59} +2.00000 q^{61} -1.00000 q^{63} -19.3137 q^{65} -9.17157 q^{67} -8.48528 q^{69} +12.7279 q^{71} -8.82843 q^{73} +6.65685 q^{75} +2.82843 q^{77} +2.34315 q^{79} +1.00000 q^{81} -12.2426 q^{83} -7.65685 q^{85} +10.2426 q^{87} -16.8284 q^{89} -5.65685 q^{91} +8.82843 q^{93} -3.41421 q^{95} -0.343146 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 12 q^{31} + 4 q^{35} - 12 q^{37} - 12 q^{41} - 12 q^{43} - 4 q^{45} - 24 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{55} + 2 q^{57} - 16 q^{59} + 4 q^{61} - 2 q^{63} - 16 q^{65} - 24 q^{67} - 12 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} - 16 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} + 12 q^{93} - 4 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 12 * q^31 + 4 * q^35 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 4 * q^45 - 24 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 + 2 * q^57 - 16 * q^59 + 4 * q^61 - 2 * q^63 - 16 * q^65 - 24 * q^67 - 12 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 16 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 + 12 * q^93 - 4 * q^95 - 12 * q^97

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
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	10.2426	1.90201	0.951005	−	0.309175i	$-0.100053\pi$
$29$	10.2426	1.90201	0.951005	+	0.309175i	$0.100053\pi$
$30$	0	0
$31$	8.82843	1.58563	0.792816	−	0.609461i	$-0.208614\pi$
$31$	8.82843	1.58563	0.792816	+	0.609461i	$0.208614\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	3.41421	0.577107
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	5.65685	0.905822
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	−8.82843	−1.34632	−0.673161	−	0.739496i	$-0.735064\pi$
$43$	−8.82843	−1.34632	−0.673161	+	0.739496i	$0.735064\pi$
$44$	0	0
$45$	−3.41421	−0.508961
$46$	0	0
$47$	−10.5858	−1.54410	−0.772048	−	0.635564i	$-0.780767\pi$
$47$	−10.5858	−1.54410	−0.772048	+	0.635564i	$0.780767\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.24264	0.314033
$52$	0	0
$53$	−5.07107	−0.696565	−0.348282	−	0.937390i	$-0.613235\pi$
$53$	−5.07107	−0.696565	−0.348282	+	0.937390i	$0.613235\pi$
$54$	0	0
$55$	9.65685	1.30213
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−19.3137	−2.39557
$66$	0	0
$67$	−9.17157	−1.12049	−0.560243	−	0.828328i	$-0.689292\pi$
$67$	−9.17157	−1.12049	−0.560243	+	0.828328i	$0.689292\pi$
$68$	0	0
$69$	−8.48528	−1.02151
$70$	0	0
$71$	12.7279	1.51053	0.755263	−	0.655422i	$-0.227509\pi$
$71$	12.7279	1.51053	0.755263	+	0.655422i	$0.227509\pi$
$72$	0	0
$73$	−8.82843	−1.03329	−0.516645	−	0.856200i	$-0.672819\pi$
$73$	−8.82843	−1.03329	−0.516645	+	0.856200i	$0.672819\pi$
$74$	0	0
$75$	6.65685	0.768667
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.2426	−1.34380	−0.671902	−	0.740640i	$-0.734522\pi$
$83$	−12.2426	−1.34380	−0.671902	+	0.740640i	$0.734522\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	0	0
$87$	10.2426	1.09813
$88$	0	0
$89$	−16.8284	−1.78381	−0.891905	−	0.452223i	$-0.850631\pi$
$89$	−16.8284	−1.78381	−0.891905	+	0.452223i	$0.850631\pi$
$90$	0	0
$91$	−5.65685	−0.592999
$92$	0	0
$93$	8.82843	0.915465
$94$	0	0
$95$	−3.41421	−0.350291
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	9.07107	0.902605	0.451302	−	0.892371i	$-0.350960\pi$
$101$	9.07107	0.902605	0.451302	+	0.892371i	$0.350960\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	3.41421	0.333193
$106$	0	0
$107$	−16.7279	−1.61715	−0.808575	−	0.588394i	$-0.799761\pi$
$107$	−16.7279	−1.61715	−0.808575	+	0.588394i	$0.799761\pi$
$108$	0	0
$109$	−16.1421	−1.54614	−0.773068	−	0.634323i	$-0.781279\pi$
$109$	−16.1421	−1.54614	−0.773068	+	0.634323i	$0.781279\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	8.58579	0.807683	0.403841	−	0.914829i	$-0.367675\pi$
$113$	8.58579	0.807683	0.403841	+	0.914829i	$0.367675\pi$
$114$	0	0
$115$	28.9706	2.70152
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	−2.24264	−0.205583
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−8.82843	−0.796032
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−8.48528	−0.752947	−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	−8.82843	−0.777300
$130$	0	0
$131$	9.89949	0.864923	0.432461	−	0.901652i	$-0.357645\pi$
$131$	9.89949	0.864923	0.432461	+	0.901652i	$0.357645\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	−11.6569	−0.995912	−0.497956	−	0.867202i	$-0.665916\pi$
$137$	−11.6569	−0.995912	−0.497956	+	0.867202i	$0.665916\pi$
$138$	0	0
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	0	0
$141$	−10.5858	−0.891484
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	−34.9706	−2.90415
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	18.4853	1.51437	0.757187	−	0.653199i	$-0.226573\pi$
$149$	18.4853	1.51437	0.757187	+	0.653199i	$0.226573\pi$
$150$	0	0
$151$	8.48528	0.690522	0.345261	−	0.938507i	$-0.387790\pi$
$151$	8.48528	0.690522	0.345261	+	0.938507i	$0.387790\pi$
$152$	0	0
$153$	2.24264	0.181307
$154$	0	0
$155$	−30.1421	−2.42107
$156$	0	0
$157$	−7.65685	−0.611083	−0.305542	−	0.952179i	$-0.598838\pi$
$157$	−7.65685	−0.611083	−0.305542	+	0.952179i	$0.598838\pi$
$158$	0	0
$159$	−5.07107	−0.402162
$160$	0	0
$161$	8.48528	0.668734
$162$	0	0
$163$	−1.51472	−0.118642	−0.0593210	−	0.998239i	$-0.518894\pi$
$163$	−1.51472	−0.118642	−0.0593210	+	0.998239i	$0.518894\pi$
$164$	0	0
$165$	9.65685	0.751785
$166$	0	0
$167$	−22.8284	−1.76652	−0.883258	−	0.468887i	$-0.844655\pi$
$167$	−22.8284	−1.76652	−0.883258	+	0.468887i	$0.844655\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−3.17157	−0.241130	−0.120565	−	0.992705i	$-0.538471\pi$
$173$	−3.17157	−0.241130	−0.120565	+	0.992705i	$0.538471\pi$
$174$	0	0
$175$	−6.65685	−0.503211
$176$	0	0
$177$	−2.34315	−0.176122
$178$	0	0
$179$	3.07107	0.229542	0.114771	−	0.993392i	$-0.463387\pi$
$179$	3.07107	0.229542	0.114771	+	0.993392i	$0.463387\pi$
$180$	0	0
$181$	−9.31371	−0.692283	−0.346141	−	0.938182i	$-0.612508\pi$
$181$	−9.31371	−0.692283	−0.346141	+	0.938182i	$0.612508\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	20.4853	1.50611
$186$	0	0
$187$	−6.34315	−0.463857
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−14.8284	−1.07295	−0.536474	−	0.843917i	$-0.680244\pi$
$191$	−14.8284	−1.07295	−0.536474	+	0.843917i	$0.680244\pi$
$192$	0	0
$193$	−0.343146	−0.0247002	−0.0123501	−	0.999924i	$-0.503931\pi$
$193$	−0.343146	−0.0247002	−0.0123501	+	0.999924i	$0.503931\pi$
$194$	0	0
$195$	−19.3137	−1.38308
$196$	0	0
$197$	14.9706	1.06661	0.533304	−	0.845924i	$-0.320950\pi$
$197$	14.9706	1.06661	0.533304	+	0.845924i	$0.320950\pi$
$198$	0	0
$199$	−14.1421	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$199$	−14.1421	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$200$	0	0
$201$	−9.17157	−0.646913
$202$	0	0
$203$	−10.2426	−0.718892
$204$	0	0
$205$	30.1421	2.10522
$206$	0	0
$207$	−8.48528	−0.589768
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	28.9706	1.99442	0.997208	−	0.0746754i	$-0.0237921\pi$
$211$	28.9706	1.99442	0.997208	+	0.0746754i	$0.0237921\pi$
$212$	0	0
$213$	12.7279	0.872103
$214$	0	0
$215$	30.1421	2.05568
$216$	0	0
$217$	−8.82843	−0.599313
$218$	0	0
$219$	−8.82843	−0.596570
$220$	0	0
$221$	12.6863	0.853372
$222$	0	0
$223$	−0.828427	−0.0554756	−0.0277378	−	0.999615i	$-0.508830\pi$
$223$	−0.828427	−0.0554756	−0.0277378	+	0.999615i	$0.508830\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	11.5147	0.764259	0.382129	−	0.924109i	$-0.375191\pi$
$227$	11.5147	0.764259	0.382129	+	0.924109i	$0.375191\pi$
$228$	0	0
$229$	10.4853	0.692887	0.346443	−	0.938071i	$-0.387389\pi$
$229$	10.4853	0.692887	0.346443	+	0.938071i	$0.387389\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	0	0
$233$	10.4853	0.686914	0.343457	−	0.939168i	$-0.388402\pi$
$233$	10.4853	0.686914	0.343457	+	0.939168i	$0.388402\pi$
$234$	0	0
$235$	36.1421	2.35765
$236$	0	0
$237$	2.34315	0.152204
$238$	0	0
$239$	−2.34315	−0.151565	−0.0757827	−	0.997124i	$-0.524146\pi$
$239$	−2.34315	−0.151565	−0.0757827	+	0.997124i	$0.524146\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.41421	−0.218126
$246$	0	0
$247$	5.65685	0.359937
$248$	0	0
$249$	−12.2426	−0.775846
$250$	0	0
$251$	−4.24264	−0.267793	−0.133897	−	0.990995i	$-0.542749\pi$
$251$	−4.24264	−0.267793	−0.133897	+	0.990995i	$0.542749\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	−7.65685	−0.479491
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	10.2426	0.634004
$262$	0	0
$263$	1.65685	0.102166	0.0510830	−	0.998694i	$-0.483733\pi$
$263$	1.65685	0.102166	0.0510830	+	0.998694i	$0.483733\pi$
$264$	0	0
$265$	17.3137	1.06357
$266$	0	0
$267$	−16.8284	−1.02988
$268$	0	0
$269$	−11.6569	−0.710731	−0.355365	−	0.934727i	$-0.615644\pi$
$269$	−11.6569	−0.710731	−0.355365	+	0.934727i	$0.615644\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	−5.65685	−0.342368
$274$	0	0
$275$	−18.8284	−1.13540
$276$	0	0
$277$	−11.3137	−0.679775	−0.339887	−	0.940466i	$-0.610389\pi$
$277$	−11.3137	−0.679775	−0.339887	+	0.940466i	$0.610389\pi$
$278$	0	0
$279$	8.82843	0.528544
$280$	0	0
$281$	1.75736	0.104835	0.0524176	−	0.998625i	$-0.483307\pi$
$281$	1.75736	0.104835	0.0524176	+	0.998625i	$0.483307\pi$
$282$	0	0
$283$	14.1421	0.840663	0.420331	−	0.907371i	$-0.361914\pi$
$283$	14.1421	0.840663	0.420331	+	0.907371i	$0.361914\pi$
$284$	0	0
$285$	−3.41421	−0.202241
$286$	0	0
$287$	8.82843	0.521126
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	−0.343146	−0.0201156
$292$	0	0
$293$	1.51472	0.0884908	0.0442454	−	0.999021i	$-0.485912\pi$
$293$	1.51472	0.0884908	0.0442454	+	0.999021i	$0.485912\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−2.82843	−0.164122
$298$	0	0
$299$	−48.0000	−2.77591
$300$	0	0
$301$	8.82843	0.508862
$302$	0	0
$303$	9.07107	0.521119
$304$	0	0
$305$	−6.82843	−0.390995
$306$	0	0
$307$	−16.1421	−0.921280	−0.460640	−	0.887587i	$-0.652380\pi$
$307$	−16.1421	−0.921280	−0.460640	+	0.887587i	$0.652380\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−28.0416	−1.59009	−0.795047	−	0.606547i	$-0.792554\pi$
$311$	−28.0416	−1.59009	−0.795047	+	0.606547i	$0.792554\pi$
$312$	0	0
$313$	2.97056	0.167906	0.0839531	−	0.996470i	$-0.473245\pi$
$313$	2.97056	0.167906	0.0839531	+	0.996470i	$0.473245\pi$
$314$	0	0
$315$	3.41421	0.192369
$316$	0	0
$317$	27.2132	1.52845	0.764223	−	0.644952i	$-0.223123\pi$
$317$	27.2132	1.52845	0.764223	+	0.644952i	$0.223123\pi$
$318$	0	0
$319$	−28.9706	−1.62204
$320$	0	0
$321$	−16.7279	−0.933662
$322$	0	0
$323$	2.24264	0.124784
$324$	0	0
$325$	37.6569	2.08883
$326$	0	0
$327$	−16.1421	−0.892662
$328$	0	0
$329$	10.5858	0.583613
$330$	0	0
$331$	−14.1421	−0.777322	−0.388661	−	0.921381i	$-0.627062\pi$
$331$	−14.1421	−0.777322	−0.388661	+	0.921381i	$0.627062\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	31.3137	1.71085
$336$	0	0
$337$	−8.82843	−0.480915	−0.240458	−	0.970660i	$-0.577297\pi$
$337$	−8.82843	−0.480915	−0.240458	+	0.970660i	$0.577297\pi$
$338$	0	0
$339$	8.58579	0.466316
$340$	0	0
$341$	−24.9706	−1.35223
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	28.9706	1.55972
$346$	0	0
$347$	20.9706	1.12576	0.562879	−	0.826539i	$-0.309694\pi$
$347$	20.9706	1.12576	0.562879	+	0.826539i	$0.309694\pi$
$348$	0	0
$349$	29.3137	1.56913	0.784563	−	0.620049i	$-0.212887\pi$
$349$	29.3137	1.56913	0.784563	+	0.620049i	$0.212887\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	0.100505	0.00534934	0.00267467	−	0.999996i	$-0.499149\pi$
$353$	0.100505	0.00534934	0.00267467	+	0.999996i	$0.499149\pi$
$354$	0	0
$355$	−43.4558	−2.30640
$356$	0	0
$357$	−2.24264	−0.118693
$358$	0	0
$359$	4.48528	0.236724	0.118362	−	0.992971i	$-0.462236\pi$
$359$	4.48528	0.236724	0.118362	+	0.992971i	$0.462236\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	30.1421	1.57771
$366$	0	0
$367$	5.85786	0.305778	0.152889	−	0.988243i	$-0.451142\pi$
$367$	5.85786	0.305778	0.152889	+	0.988243i	$0.451142\pi$
$368$	0	0
$369$	−8.82843	−0.459590
$370$	0	0
$371$	5.07107	0.263277
$372$	0	0
$373$	24.3431	1.26044	0.630220	−	0.776416i	$-0.282965\pi$
$373$	24.3431	1.26044	0.630220	+	0.776416i	$0.282965\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	57.9411	2.98412
$378$	0	0
$379$	14.8284	0.761685	0.380843	−	0.924640i	$-0.375634\pi$
$379$	14.8284	0.761685	0.380843	+	0.924640i	$0.375634\pi$
$380$	0	0
$381$	−8.48528	−0.434714
$382$	0	0
$383$	−20.4853	−1.04675	−0.523374	−	0.852103i	$-0.675327\pi$
$383$	−20.4853	−1.04675	−0.523374	+	0.852103i	$0.675327\pi$
$384$	0	0
$385$	−9.65685	−0.492159
$386$	0	0
$387$	−8.82843	−0.448774
$388$	0	0
$389$	35.4558	1.79768	0.898841	−	0.438274i	$-0.144410\pi$
$389$	35.4558	1.79768	0.898841	+	0.438274i	$0.144410\pi$
$390$	0	0
$391$	−19.0294	−0.962360
$392$	0	0
$393$	9.89949	0.499363
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	1.31371	0.0659331	0.0329666	−	0.999456i	$-0.489505\pi$
$397$	1.31371	0.0659331	0.0329666	+	0.999456i	$0.489505\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−12.3848	−0.618466	−0.309233	−	0.950986i	$-0.600072\pi$
$401$	−12.3848	−0.618466	−0.309233	+	0.950986i	$0.600072\pi$
$402$	0	0
$403$	49.9411	2.48774
$404$	0	0
$405$	−3.41421	−0.169654
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	−18.6274	−0.921066	−0.460533	−	0.887642i	$-0.652342\pi$
$409$	−18.6274	−0.921066	−0.460533	+	0.887642i	$0.652342\pi$
$410$	0	0
$411$	−11.6569	−0.574990
$412$	0	0
$413$	2.34315	0.115299
$414$	0	0
$415$	41.7990	2.05183
$416$	0	0
$417$	−12.4853	−0.611407
$418$	0	0
$419$	−4.72792	−0.230974	−0.115487	−	0.993309i	$-0.536843\pi$
$419$	−4.72792	−0.230974	−0.115487	+	0.993309i	$0.536843\pi$
$420$	0	0
$421$	−1.51472	−0.0738229	−0.0369114	−	0.999319i	$-0.511752\pi$
$421$	−1.51472	−0.0738229	−0.0369114	+	0.999319i	$0.511752\pi$
$422$	0	0
$423$	−10.5858	−0.514699
$424$	0	0
$425$	14.9289	0.724160
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−16.0000	−0.772487
$430$	0	0
$431$	−33.4142	−1.60951	−0.804753	−	0.593610i	$-0.797702\pi$
$431$	−33.4142	−1.60951	−0.804753	+	0.593610i	$0.797702\pi$
$432$	0	0
$433$	4.62742	0.222379	0.111190	−	0.993799i	$-0.464534\pi$
$433$	4.62742	0.222379	0.111190	+	0.993799i	$0.464534\pi$
$434$	0	0
$435$	−34.9706	−1.67671
$436$	0	0
$437$	−8.48528	−0.405906
$438$	0	0
$439$	−41.7990	−1.99496	−0.997478	−	0.0709697i	$-0.977391\pi$
$439$	−41.7990	−1.99496	−0.997478	+	0.0709697i	$0.977391\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0.686292	0.0326067	0.0163033	−	0.999867i	$-0.494810\pi$
$443$	0.686292	0.0326067	0.0163033	+	0.999867i	$0.494810\pi$
$444$	0	0
$445$	57.4558	2.72367
$446$	0	0
$447$	18.4853	0.874324
$448$	0	0
$449$	16.8701	0.796147	0.398074	−	0.917353i	$-0.369679\pi$
$449$	16.8701	0.796147	0.398074	+	0.917353i	$0.369679\pi$
$450$	0	0
$451$	24.9706	1.17582
$452$	0	0
$453$	8.48528	0.398673
$454$	0	0
$455$	19.3137	0.905441
$456$	0	0
$457$	−27.3137	−1.27768	−0.638841	−	0.769339i	$-0.720586\pi$
$457$	−27.3137	−1.27768	−0.638841	+	0.769339i	$0.720586\pi$
$458$	0	0
$459$	2.24264	0.104678
$460$	0	0
$461$	−38.5269	−1.79438	−0.897189	−	0.441648i	$-0.854394\pi$
$461$	−38.5269	−1.79438	−0.897189	+	0.441648i	$0.854394\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	−30.1421	−1.39781
$466$	0	0
$467$	9.89949	0.458094	0.229047	−	0.973415i	$-0.426439\pi$
$467$	9.89949	0.458094	0.229047	+	0.973415i	$0.426439\pi$
$468$	0	0
$469$	9.17157	0.423504
$470$	0	0
$471$	−7.65685	−0.352809
$472$	0	0
$473$	24.9706	1.14815
$474$	0	0
$475$	6.65685	0.305437
$476$	0	0
$477$	−5.07107	−0.232188
$478$	0	0
$479$	−0.727922	−0.0332596	−0.0166298	−	0.999862i	$-0.505294\pi$
$479$	−0.727922	−0.0332596	−0.0166298	+	0.999862i	$0.505294\pi$
$480$	0	0
$481$	−33.9411	−1.54758
$482$	0	0
$483$	8.48528	0.386094
$484$	0	0
$485$	1.17157	0.0531984
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−1.51472	−0.0684979
$490$	0	0
$491$	−9.65685	−0.435808	−0.217904	−	0.975970i	$-0.569922\pi$
$491$	−9.65685	−0.435808	−0.217904	+	0.975970i	$0.569922\pi$
$492$	0	0
$493$	22.9706	1.03454
$494$	0	0
$495$	9.65685	0.434043
$496$	0	0
$497$	−12.7279	−0.570925
$498$	0	0
$499$	43.5980	1.95171	0.975857	−	0.218411i	$-0.0700874\pi$
$499$	43.5980	1.95171	0.975857	+	0.218411i	$0.0700874\pi$
$500$	0	0
$501$	−22.8284	−1.01990
$502$	0	0
$503$	42.1838	1.88088	0.940441	−	0.339958i	$-0.110413\pi$
$503$	42.1838	1.88088	0.940441	+	0.339958i	$0.110413\pi$
$504$	0	0
$505$	−30.9706	−1.37817
$506$	0	0
$507$	19.0000	0.843820
$508$	0	0
$509$	−14.9706	−0.663559	−0.331779	−	0.943357i	$-0.607649\pi$
$509$	−14.9706	−0.663559	−0.331779	+	0.943357i	$0.607649\pi$
$510$	0	0
$511$	8.82843	0.390547
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−27.3137	−1.20359
$516$	0	0
$517$	29.9411	1.31681
$518$	0	0
$519$	−3.17157	−0.139217
$520$	0	0
$521$	−38.9706	−1.70733	−0.853666	−	0.520821i	$-0.825626\pi$
$521$	−38.9706	−1.70733	−0.853666	+	0.520821i	$0.825626\pi$
$522$	0	0
$523$	−0.142136	−0.00621516	−0.00310758	−	0.999995i	$-0.500989\pi$
$523$	−0.142136	−0.00621516	−0.00310758	+	0.999995i	$0.500989\pi$
$524$	0	0
$525$	−6.65685	−0.290529
$526$	0	0
$527$	19.7990	0.862458
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	−2.34315	−0.101684
$532$	0	0
$533$	−49.9411	−2.16319
$534$	0	0
$535$	57.1127	2.46920
$536$	0	0
$537$	3.07107	0.132526
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	−9.31371	−0.399689
$544$	0	0
$545$	55.1127	2.36077
$546$	0	0
$547$	−35.5980	−1.52206	−0.761030	−	0.648717i	$-0.775306\pi$
$547$	−35.5980	−1.52206	−0.761030	+	0.648717i	$0.775306\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	10.2426	0.436351
$552$	0	0
$553$	−2.34315	−0.0996407
$554$	0	0
$555$	20.4853	0.869552
$556$	0	0
$557$	3.17157	0.134384	0.0671919	−	0.997740i	$-0.478596\pi$
$557$	3.17157	0.134384	0.0671919	+	0.997740i	$0.478596\pi$
$558$	0	0
$559$	−49.9411	−2.11228
$560$	0	0
$561$	−6.34315	−0.267808
$562$	0	0
$563$	−24.4853	−1.03193	−0.515966	−	0.856609i	$-0.672567\pi$
$563$	−24.4853	−1.03193	−0.515966	+	0.856609i	$0.672567\pi$
$564$	0	0
$565$	−29.3137	−1.23324
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−18.5269	−0.776689	−0.388344	−	0.921514i	$-0.626953\pi$
$569$	−18.5269	−0.776689	−0.388344	+	0.921514i	$0.626953\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	0	0
$573$	−14.8284	−0.619466
$574$	0	0
$575$	−56.4853	−2.35560
$576$	0	0
$577$	4.34315	0.180808	0.0904038	−	0.995905i	$-0.471184\pi$
$577$	4.34315	0.180808	0.0904038	+	0.995905i	$0.471184\pi$
$578$	0	0
$579$	−0.343146	−0.0142607
$580$	0	0
$581$	12.2426	0.507910
$582$	0	0
$583$	14.3431	0.594032
$584$	0	0
$585$	−19.3137	−0.798524
$586$	0	0
$587$	−0.928932	−0.0383411	−0.0191706	−	0.999816i	$-0.506103\pi$
$587$	−0.928932	−0.0383411	−0.0191706	+	0.999816i	$0.506103\pi$
$588$	0	0
$589$	8.82843	0.363769
$590$	0	0
$591$	14.9706	0.615807
$592$	0	0
$593$	−48.3848	−1.98692	−0.993462	−	0.114161i	$-0.963582\pi$
$593$	−48.3848	−1.98692	−0.993462	+	0.114161i	$0.963582\pi$
$594$	0	0
$595$	7.65685	0.313900
$596$	0	0
$597$	−14.1421	−0.578799
$598$	0	0
$599$	22.1005	0.903002	0.451501	−	0.892271i	$-0.350889\pi$
$599$	22.1005	0.903002	0.451501	+	0.892271i	$0.350889\pi$
$600$	0	0
$601$	−9.31371	−0.379914	−0.189957	−	0.981792i	$-0.560835\pi$
$601$	−9.31371	−0.379914	−0.189957	+	0.981792i	$0.560835\pi$
$602$	0	0
$603$	−9.17157	−0.373495
$604$	0	0
$605$	10.2426	0.416423
$606$	0	0
$607$	25.9411	1.05292	0.526459	−	0.850201i	$-0.323519\pi$
$607$	25.9411	1.05292	0.526459	+	0.850201i	$0.323519\pi$
$608$	0	0
$609$	−10.2426	−0.415053
$610$	0	0
$611$	−59.8823	−2.42258
$612$	0	0
$613$	−12.2843	−0.496157	−0.248079	−	0.968740i	$-0.579799\pi$
$613$	−12.2843	−0.496157	−0.248079	+	0.968740i	$0.579799\pi$
$614$	0	0
$615$	30.1421	1.21545
$616$	0	0
$617$	−35.9411	−1.44694	−0.723468	−	0.690358i	$-0.757453\pi$
$617$	−35.9411	−1.44694	−0.723468	+	0.690358i	$0.757453\pi$
$618$	0	0
$619$	11.5147	0.462816	0.231408	−	0.972857i	$-0.425667\pi$
$619$	11.5147	0.462816	0.231408	+	0.972857i	$0.425667\pi$
$620$	0	0
$621$	−8.48528	−0.340503
$622$	0	0
$623$	16.8284	0.674217
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	−2.82843	−0.112956
$628$	0	0
$629$	−13.4558	−0.536520
$630$	0	0
$631$	15.4558	0.615287	0.307644	−	0.951502i	$-0.400460\pi$
$631$	15.4558	0.615287	0.307644	+	0.951502i	$0.400460\pi$
$632$	0	0
$633$	28.9706	1.15148
$634$	0	0
$635$	28.9706	1.14966
$636$	0	0
$637$	5.65685	0.224133
$638$	0	0
$639$	12.7279	0.503509
$640$	0	0
$641$	47.6985	1.88398	0.941988	−	0.335645i	$-0.108954\pi$
$641$	47.6985	1.88398	0.941988	+	0.335645i	$0.108954\pi$
$642$	0	0
$643$	21.1716	0.834925	0.417463	−	0.908694i	$-0.362919\pi$
$643$	21.1716	0.834925	0.417463	+	0.908694i	$0.362919\pi$
$644$	0	0
$645$	30.1421	1.18685
$646$	0	0
$647$	0.727922	0.0286176	0.0143088	−	0.999898i	$-0.495445\pi$
$647$	0.727922	0.0286176	0.0143088	+	0.999898i	$0.495445\pi$
$648$	0	0
$649$	6.62742	0.260149
$650$	0	0
$651$	−8.82843	−0.346013
$652$	0	0
$653$	37.5980	1.47132	0.735661	−	0.677350i	$-0.236872\pi$
$653$	37.5980	1.47132	0.735661	+	0.677350i	$0.236872\pi$
$654$	0	0
$655$	−33.7990	−1.32064
$656$	0	0
$657$	−8.82843	−0.344430
$658$	0	0
$659$	16.2426	0.632723	0.316362	−	0.948639i	$-0.397539\pi$
$659$	16.2426	0.632723	0.316362	+	0.948639i	$0.397539\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	0	0
$663$	12.6863	0.492695
$664$	0	0
$665$	3.41421	0.132398
$666$	0	0
$667$	−86.9117	−3.36523
$668$	0	0
$669$	−0.828427	−0.0320288
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	21.3137	0.821583	0.410792	−	0.911729i	$-0.365252\pi$
$673$	21.3137	0.821583	0.410792	+	0.911729i	$0.365252\pi$
$674$	0	0
$675$	6.65685	0.256222
$676$	0	0
$677$	39.9411	1.53506	0.767531	−	0.641012i	$-0.221485\pi$
$677$	39.9411	1.53506	0.767531	+	0.641012i	$0.221485\pi$
$678$	0	0
$679$	0.343146	0.0131687
$680$	0	0
$681$	11.5147	0.441245
$682$	0	0
$683$	−21.8995	−0.837961	−0.418980	−	0.907995i	$-0.637612\pi$
$683$	−21.8995	−0.837961	−0.418980	+	0.907995i	$0.637612\pi$
$684$	0	0
$685$	39.7990	1.52064
$686$	0	0
$687$	10.4853	0.400038
$688$	0	0
$689$	−28.6863	−1.09286
$690$	0	0
$691$	28.9706	1.10209	0.551046	−	0.834475i	$-0.314229\pi$
$691$	28.9706	1.10209	0.551046	+	0.834475i	$0.314229\pi$
$692$	0	0
$693$	2.82843	0.107443
$694$	0	0
$695$	42.6274	1.61695
$696$	0	0
$697$	−19.7990	−0.749940
$698$	0	0
$699$	10.4853	0.396590
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	36.1421	1.36119
$706$	0	0
$707$	−9.07107	−0.341153
$708$	0	0
$709$	−1.65685	−0.0622245	−0.0311122	−	0.999516i	$-0.509905\pi$
$709$	−1.65685	−0.0622245	−0.0311122	+	0.999516i	$0.509905\pi$
$710$	0	0
$711$	2.34315	0.0878748
$712$	0	0
$713$	−74.9117	−2.80546
$714$	0	0
$715$	54.6274	2.04295
$716$	0	0
$717$	−2.34315	−0.0875064
$718$	0	0
$719$	6.38478	0.238112	0.119056	−	0.992888i	$-0.462013\pi$
$719$	6.38478	0.238112	0.119056	+	0.992888i	$0.462013\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	68.1838	2.53228
$726$	0	0
$727$	−26.6274	−0.987556	−0.493778	−	0.869588i	$-0.664385\pi$
$727$	−26.6274	−0.987556	−0.493778	+	0.869588i	$0.664385\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.7990	−0.732292
$732$	0	0
$733$	−20.8284	−0.769316	−0.384658	−	0.923059i	$-0.625681\pi$
$733$	−20.8284	−0.769316	−0.384658	+	0.923059i	$0.625681\pi$
$734$	0	0
$735$	−3.41421	−0.125935
$736$	0	0
$737$	25.9411	0.955554
$738$	0	0
$739$	−37.7990	−1.39046	−0.695229	−	0.718788i	$-0.744697\pi$
$739$	−37.7990	−1.39046	−0.695229	+	0.718788i	$0.744697\pi$
$740$	0	0
$741$	5.65685	0.207810
$742$	0	0
$743$	−39.0711	−1.43338	−0.716689	−	0.697393i	$-0.754343\pi$
$743$	−39.0711	−1.43338	−0.716689	+	0.697393i	$0.754343\pi$
$744$	0	0
$745$	−63.1127	−2.31227
$746$	0	0
$747$	−12.2426	−0.447935
$748$	0	0
$749$	16.7279	0.611225
$750$	0	0
$751$	12.2010	0.445221	0.222611	−	0.974907i	$-0.428542\pi$
$751$	12.2010	0.445221	0.222611	+	0.974907i	$0.428542\pi$
$752$	0	0
$753$	−4.24264	−0.154610
$754$	0	0
$755$	−28.9706	−1.05435
$756$	0	0
$757$	4.34315	0.157854	0.0789272	−	0.996880i	$-0.474851\pi$
$757$	4.34315	0.157854	0.0789272	+	0.996880i	$0.474851\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	47.2132	1.71148	0.855739	−	0.517408i	$-0.173103\pi$
$761$	47.2132	1.71148	0.855739	+	0.517408i	$0.173103\pi$
$762$	0	0
$763$	16.1421	0.584385
$764$	0	0
$765$	−7.65685	−0.276834
$766$	0	0
$767$	−13.2548	−0.478604
$768$	0	0
$769$	1.02944	0.0371225	0.0185612	−	0.999828i	$-0.494091\pi$
$769$	1.02944	0.0371225	0.0185612	+	0.999828i	$0.494091\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−13.5147	−0.486091	−0.243045	−	0.970015i	$-0.578146\pi$
$773$	−13.5147	−0.486091	−0.243045	+	0.970015i	$0.578146\pi$
$774$	0	0
$775$	58.7696	2.11106
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−8.82843	−0.316311
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	10.2426	0.366042
$784$	0	0
$785$	26.1421	0.933053
$786$	0	0
$787$	16.1421	0.575405	0.287702	−	0.957720i	$-0.407109\pi$
$787$	16.1421	0.575405	0.287702	+	0.957720i	$0.407109\pi$
$788$	0	0
$789$	1.65685	0.0589856
$790$	0	0
$791$	−8.58579	−0.305275
$792$	0	0
$793$	11.3137	0.401762
$794$	0	0
$795$	17.3137	0.614054
$796$	0	0
$797$	6.48528	0.229720	0.114860	−	0.993382i	$-0.463358\pi$
$797$	6.48528	0.229720	0.114860	+	0.993382i	$0.463358\pi$
$798$	0	0
$799$	−23.7401	−0.839865
$800$	0	0
$801$	−16.8284	−0.594603
$802$	0	0
$803$	24.9706	0.881192
$804$	0	0
$805$	−28.9706	−1.02108
$806$	0	0
$807$	−11.6569	−0.410341
$808$	0	0
$809$	−26.9706	−0.948234	−0.474117	−	0.880462i	$-0.657233\pi$
$809$	−26.9706	−0.948234	−0.474117	+	0.880462i	$0.657233\pi$
$810$	0	0
$811$	−44.9706	−1.57913	−0.789565	−	0.613667i	$-0.789694\pi$
$811$	−44.9706	−1.57913	−0.789565	+	0.613667i	$0.789694\pi$
$812$	0	0
$813$	−16.9706	−0.595184
$814$	0	0
$815$	5.17157	0.181152
$816$	0	0
$817$	−8.82843	−0.308868
$818$	0	0
$819$	−5.65685	−0.197666
$820$	0	0
$821$	−16.8284	−0.587316	−0.293658	−	0.955911i	$-0.594873\pi$
$821$	−16.8284	−0.587316	−0.293658	+	0.955911i	$0.594873\pi$
$822$	0	0
$823$	44.1421	1.53870	0.769349	−	0.638829i	$-0.220581\pi$
$823$	44.1421	1.53870	0.769349	+	0.638829i	$0.220581\pi$
$824$	0	0
$825$	−18.8284	−0.655522
$826$	0	0
$827$	−21.8995	−0.761520	−0.380760	−	0.924674i	$-0.624338\pi$
$827$	−21.8995	−0.761520	−0.380760	+	0.924674i	$0.624338\pi$
$828$	0	0
$829$	50.2843	1.74644	0.873222	−	0.487322i	$-0.162026\pi$
$829$	50.2843	1.74644	0.873222	+	0.487322i	$0.162026\pi$
$830$	0	0
$831$	−11.3137	−0.392468
$832$	0	0
$833$	2.24264	0.0777029
$834$	0	0
$835$	77.9411	2.69726
$836$	0	0
$837$	8.82843	0.305155
$838$	0	0
$839$	41.2548	1.42428	0.712138	−	0.702040i	$-0.247727\pi$
$839$	41.2548	1.42428	0.712138	+	0.702040i	$0.247727\pi$
$840$	0	0
$841$	75.9117	2.61764
$842$	0	0
$843$	1.75736	0.0605267
$844$	0	0
$845$	−64.8701	−2.23160
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	14.1421	0.485357
$850$	0	0
$851$	50.9117	1.74523
$852$	0	0
$853$	1.02944	0.0352473	0.0176236	−	0.999845i	$-0.494390\pi$
$853$	1.02944	0.0352473	0.0176236	+	0.999845i	$0.494390\pi$
$854$	0	0
$855$	−3.41421	−0.116764
$856$	0	0
$857$	−26.4853	−0.904720	−0.452360	−	0.891835i	$-0.649418\pi$
$857$	−26.4853	−0.904720	−0.452360	+	0.891835i	$0.649418\pi$
$858$	0	0
$859$	53.6569	1.83075	0.915374	−	0.402604i	$-0.131895\pi$
$859$	53.6569	1.83075	0.915374	+	0.402604i	$0.131895\pi$
$860$	0	0
$861$	8.82843	0.300872
$862$	0	0
$863$	27.7574	0.944871	0.472436	−	0.881365i	$-0.343375\pi$
$863$	27.7574	0.944871	0.472436	+	0.881365i	$0.343375\pi$
$864$	0	0
$865$	10.8284	0.368178
$866$	0	0
$867$	−11.9706	−0.406542
$868$	0	0
$869$	−6.62742	−0.224820
$870$	0	0
$871$	−51.8823	−1.75796
$872$	0	0
$873$	−0.343146	−0.0116137
$874$	0	0
$875$	5.65685	0.191237
$876$	0	0
$877$	18.6863	0.630991	0.315496	−	0.948927i	$-0.397829\pi$
$877$	18.6863	0.630991	0.315496	+	0.948927i	$0.397829\pi$
$878$	0	0
$879$	1.51472	0.0510902
$880$	0	0
$881$	20.3848	0.686781	0.343390	−	0.939193i	$-0.388425\pi$
$881$	20.3848	0.686781	0.343390	+	0.939193i	$0.388425\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	−42.4264	−1.42454	−0.712270	−	0.701906i	$-0.752333\pi$
$887$	−42.4264	−1.42454	−0.712270	+	0.701906i	$0.752333\pi$
$888$	0	0
$889$	8.48528	0.284587
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	−10.5858	−0.354240
$894$	0	0
$895$	−10.4853	−0.350484
$896$	0	0
$897$	−48.0000	−1.60267
$898$	0	0
$899$	90.4264	3.01589
$900$	0	0
$901$	−11.3726	−0.378876
$902$	0	0
$903$	8.82843	0.293792
$904$	0	0
$905$	31.7990	1.05703
$906$	0	0
$907$	−31.5980	−1.04919	−0.524597	−	0.851351i	$-0.675784\pi$
$907$	−31.5980	−1.04919	−0.524597	+	0.851351i	$0.675784\pi$
$908$	0	0
$909$	9.07107	0.300868
$910$	0	0
$911$	−16.4437	−0.544802	−0.272401	−	0.962184i	$-0.587818\pi$
$911$	−16.4437	−0.544802	−0.272401	+	0.962184i	$0.587818\pi$
$912$	0	0
$913$	34.6274	1.14600
$914$	0	0
$915$	−6.82843	−0.225741
$916$	0	0
$917$	−9.89949	−0.326910
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−16.1421	−0.531901
$922$	0	0
$923$	72.0000	2.36991
$924$	0	0
$925$	−39.9411	−1.31326
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	43.2132	1.41778	0.708890	−	0.705319i	$-0.249196\pi$
$929$	43.2132	1.41778	0.708890	+	0.705319i	$0.249196\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−28.0416	−0.918042
$934$	0	0
$935$	21.6569	0.708255
$936$	0	0
$937$	25.0294	0.817676	0.408838	−	0.912607i	$-0.365934\pi$
$937$	25.0294	0.817676	0.408838	+	0.912607i	$0.365934\pi$
$938$	0	0
$939$	2.97056	0.0969407
$940$	0	0
$941$	−30.2843	−0.987239	−0.493620	−	0.869678i	$-0.664326\pi$
$941$	−30.2843	−0.987239	−0.493620	+	0.869678i	$0.664326\pi$
$942$	0	0
$943$	74.9117	2.43946
$944$	0	0
$945$	3.41421	0.111064
$946$	0	0
$947$	4.48528	0.145752	0.0728760	−	0.997341i	$-0.476782\pi$
$947$	4.48528	0.145752	0.0728760	+	0.997341i	$0.476782\pi$
$948$	0	0
$949$	−49.9411	−1.62116
$950$	0	0
$951$	27.2132	0.882449
$952$	0	0
$953$	−50.0416	−1.62101	−0.810504	−	0.585734i	$-0.800807\pi$
$953$	−50.0416	−1.62101	−0.810504	+	0.585734i	$0.800807\pi$
$954$	0	0
$955$	50.6274	1.63826
$956$	0	0
$957$	−28.9706	−0.936485
$958$	0	0
$959$	11.6569	0.376419
$960$	0	0
$961$	46.9411	1.51423
$962$	0	0
$963$	−16.7279	−0.539050
$964$	0	0
$965$	1.17157	0.0377143
$966$	0	0
$967$	−24.8284	−0.798428	−0.399214	−	0.916858i	$-0.630717\pi$
$967$	−24.8284	−0.798428	−0.399214	+	0.916858i	$0.630717\pi$
$968$	0	0
$969$	2.24264	0.0720440
$970$	0	0
$971$	−12.6863	−0.407122	−0.203561	−	0.979062i	$-0.565252\pi$
$971$	−12.6863	−0.407122	−0.203561	+	0.979062i	$0.565252\pi$
$972$	0	0
$973$	12.4853	0.400260
$974$	0	0
$975$	37.6569	1.20598
$976$	0	0
$977$	61.1543	1.95650	0.978250	−	0.207429i	$-0.0665095\pi$
$977$	61.1543	1.95650	0.978250	+	0.207429i	$0.0665095\pi$
$978$	0	0
$979$	47.5980	1.52124
$980$	0	0
$981$	−16.1421	−0.515379
$982$	0	0
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	0	0
$985$	−51.1127	−1.62859
$986$	0	0
$987$	10.5858	0.336949
$988$	0	0
$989$	74.9117	2.38205
$990$	0	0
$991$	24.2843	0.771415	0.385708	−	0.922621i	$-0.373957\pi$
$991$	24.2843	0.771415	0.385708	+	0.922621i	$0.373957\pi$
$992$	0	0
$993$	−14.1421	−0.448787
$994$	0	0
$995$	48.2843	1.53071
$996$	0	0
$997$	19.6569	0.622539	0.311269	−	0.950322i	$-0.399246\pi$
$997$	19.6569	0.622539	0.311269	+	0.950322i	$0.399246\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.r.1.1	✓	2
3.2	odd	2		9576.2.a.by.1.2		2
4.3	odd	2		6384.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.r.1.1	✓	2	1.1	even	1	trivial
6384.2.a.bh.1.1		2	4.3	odd	2
9576.2.a.by.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.82843 q^{11} +5.65685 q^{13} -3.41421 q^{15} +2.24264 q^{17} +1.00000 q^{19} -1.00000 q^{21} -8.48528 q^{23} +6.65685 q^{25} +1.00000 q^{27} +10.2426 q^{29} +8.82843 q^{31} -2.82843 q^{33} +3.41421 q^{35} -6.00000 q^{37} +5.65685 q^{39} -8.82843 q^{41} -8.82843 q^{43} -3.41421 q^{45} -10.5858 q^{47} +1.00000 q^{49} +2.24264 q^{51} -5.07107 q^{53} +9.65685 q^{55} +1.00000 q^{57} -2.34315 q^{59} +2.00000 q^{61} -1.00000 q^{63} -19.3137 q^{65} -9.17157 q^{67} -8.48528 q^{69} +12.7279 q^{71} -8.82843 q^{73} +6.65685 q^{75} +2.82843 q^{77} +2.34315 q^{79} +1.00000 q^{81} -12.2426 q^{83} -7.65685 q^{85} +10.2426 q^{87} -16.8284 q^{89} -5.65685 q^{91} +8.82843 q^{93} -3.41421 q^{95} -0.343146 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 12 q^{31} + 4 q^{35} - 12 q^{37} - 12 q^{41} - 12 q^{43} - 4 q^{45} - 24 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{55} + 2 q^{57} - 16 q^{59} + 4 q^{61} - 2 q^{63} - 16 q^{65} - 24 q^{67} - 12 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} - 16 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} + 12 q^{93} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 12 * q^31 + 4 * q^35 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 4 * q^45 - 24 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 + 2 * q^57 - 16 * q^59 + 4 * q^61 - 2 * q^63 - 16 * q^65 - 24 * q^67 - 12 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 16 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 + 12 * q^93 - 4 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

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