LMFDB - Embedded newform 2352.3.f.j.97.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.7
Root			$0.662827 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.j.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +0.929003i q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +0.929003i q^{5} -3.00000 q^{9} -9.68980 q^{11} +15.9753i q^{13} -1.60908 q^{15} +10.5557i q^{17} +7.22049i q^{19} +11.3001 q^{23} +24.1370 q^{25} -5.19615i q^{27} +46.3148 q^{29} +0.483049i q^{31} -16.7832i q^{33} +2.48131 q^{37} -27.6700 q^{39} +55.8520i q^{41} -60.6786 q^{43} -2.78701i q^{45} +36.5867i q^{47} -18.2831 q^{51} -28.5229 q^{53} -9.00185i q^{55} -12.5063 q^{57} -94.0715i q^{59} -110.193i q^{61} -14.8411 q^{65} -82.0309 q^{67} +19.5723i q^{69} -127.349 q^{71} -46.2546i q^{73} +41.8064i q^{75} -18.7003 q^{79} +9.00000 q^{81} +59.6357i q^{83} -9.80632 q^{85} +80.2197i q^{87} +71.1281i q^{89} -0.836665 q^{93} -6.70786 q^{95} +102.239i q^{97} +29.0694 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100}) $ 8 * q - 24 * q^9 + 16 * q^23 + 72 * q^25 + 80 * q^29 + 128 * q^37 + 112 * q^43 + 96 * q^51 - 144 * q^53 - 192 * q^57 + 240 * q^65 + 64 * q^67 - 224 * q^71 + 432 * q^79 + 72 * q^81 - 96 * q^85 - 96 * q^93 + 272 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	0.929003i	0.185801i	0.995675	+	0.0929003i	$0.0296138\pi$
$5$	0.929003i	0.185801i	−0.995675	+	0.0929003i	$0.970386\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−9.68980	−0.880891	−0.440445	−	0.897779i	$-0.645179\pi$
$11$	−9.68980	−0.880891	−0.440445	+	0.897779i	$0.645179\pi$
$12$	0	0
$13$	15.9753i	1.22887i	0.788968	+	0.614434i	$0.210616\pi$
$13$	15.9753i	1.22887i	−0.788968	+	0.614434i	$0.789384\pi$
$14$	0	0
$15$	−1.60908	−0.107272
$16$	0	0
$17$	10.5557i	0.620926i	0.950585	+	0.310463i	$0.100484\pi$
$17$	10.5557i	0.620926i	−0.950585	+	0.310463i	$0.899516\pi$
$18$	0	0
$19$	7.22049i	0.380026i	0.981782	+	0.190013i	$0.0608530\pi$
$19$	7.22049i	0.380026i	−0.981782	+	0.190013i	$0.939147\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	11.3001	0.491307	0.245654	−	0.969358i	$-0.420997\pi$
$23$	11.3001	0.491307	0.245654	+	0.969358i	$0.420997\pi$
$24$	0	0
$25$	24.1370	0.965478
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	46.3148	1.59706	0.798532	−	0.601953i	$-0.205610\pi$
$29$	46.3148	1.59706	0.798532	+	0.601953i	$0.205610\pi$
$30$	0	0
$31$	0.483049i	0.0155822i	0.999970	+	0.00779111i	$0.00248001\pi$
$31$	0.483049i	0.0155822i	−0.999970	+	0.00779111i	$0.997520\pi$
$32$	0	0
$33$	− 16.7832i	− 0.508582i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.48131	0.0670623	0.0335312	−	0.999438i	$-0.489325\pi$
$37$	2.48131	0.0670623	0.0335312	+	0.999438i	$0.489325\pi$
$38$	0	0
$39$	−27.6700	−0.709487
$40$	0	0
$41$	55.8520i	1.36224i	0.732170	+	0.681122i	$0.238508\pi$
$41$	55.8520i	1.36224i	−0.732170	+	0.681122i	$0.761492\pi$
$42$	0	0
$43$	−60.6786	−1.41113	−0.705566	−	0.708645i	$-0.749307\pi$
$43$	−60.6786	−1.41113	−0.705566	+	0.708645i	$0.749307\pi$
$44$	0	0
$45$	− 2.78701i	− 0.0619335i
$46$	0	0
$47$	36.5867i	0.778440i	0.921145	+	0.389220i	$0.127255\pi$
$47$	36.5867i	0.778440i	−0.921145	+	0.389220i	$0.872745\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−18.2831	−0.358492
$52$	0	0
$53$	−28.5229	−0.538169	−0.269084	−	0.963117i	$-0.586721\pi$
$53$	−28.5229	−0.538169	−0.269084	+	0.963117i	$0.586721\pi$
$54$	0	0
$55$	− 9.00185i	− 0.163670i
$56$	0	0
$57$	−12.5063	−0.219408
$58$	0	0
$59$	− 94.0715i	− 1.59443i	−0.603694	−	0.797216i	$-0.706305\pi$
$59$	− 94.0715i	− 1.59443i	0.603694	−	0.797216i	$-0.293695\pi$
$60$	0	0
$61$	− 110.193i	− 1.80645i	−0.429171	−	0.903223i	$-0.641194\pi$
$61$	− 110.193i	− 1.80645i	0.429171	−	0.903223i	$-0.358806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.8411	−0.228324
$66$	0	0
$67$	−82.0309	−1.22434	−0.612171	−	0.790725i	$-0.709704\pi$
$67$	−82.0309	−1.22434	−0.612171	+	0.790725i	$0.709704\pi$
$68$	0	0
$69$	19.5723i	0.283656i
$70$	0	0
$71$	−127.349	−1.79365	−0.896827	−	0.442382i	$-0.854133\pi$
$71$	−127.349	−1.79365	−0.896827	+	0.442382i	$0.854133\pi$
$72$	0	0
$73$	− 46.2546i	− 0.633625i	−0.948488	−	0.316812i	$-0.897387\pi$
$73$	− 46.2546i	− 0.633625i	0.948488	−	0.316812i	$-0.102613\pi$
$74$	0	0
$75$	41.8064i	0.557419i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−18.7003	−0.236713	−0.118356	−	0.992971i	$-0.537763\pi$
$79$	−18.7003	−0.236713	−0.118356	+	0.992971i	$0.537763\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	59.6357i	0.718502i	0.933241	+	0.359251i	$0.116968\pi$
$83$	59.6357i	0.718502i	−0.933241	+	0.359251i	$0.883032\pi$
$84$	0	0
$85$	−9.80632	−0.115368
$86$	0	0
$87$	80.2197i	0.922065i
$88$	0	0
$89$	71.1281i	0.799192i	0.916691	+	0.399596i	$0.130850\pi$
$89$	71.1281i	0.799192i	−0.916691	+	0.399596i	$0.869150\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.836665	−0.00899640
$94$	0	0
$95$	−6.70786	−0.0706090
$96$	0	0
$97$	102.239i	1.05401i	0.849861	+	0.527007i	$0.176686\pi$
$97$	102.239i	1.05401i	−0.849861	+	0.527007i	$0.823314\pi$
$98$	0	0
$99$	29.0694	0.293630
$100$	0	0
$101$	108.929i	1.07851i	0.842143	+	0.539255i	$0.181294\pi$
$101$	108.929i	1.07851i	−0.842143	+	0.539255i	$0.818706\pi$
$102$	0	0
$103$	− 0.747187i	− 0.00725424i	−0.999993	−	0.00362712i	$-0.998845\pi$
$103$	− 0.747187i	− 0.00725424i	0.999993	−	0.00362712i	$-0.00115455\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.2926	−0.142922	−0.0714608	−	0.997443i	$-0.522766\pi$
$107$	−15.2926	−0.142922	−0.0714608	+	0.997443i	$0.522766\pi$
$108$	0	0
$109$	−54.2232	−0.497461	−0.248730	−	0.968573i	$-0.580013\pi$
$109$	−54.2232	−0.497461	−0.248730	+	0.968573i	$0.580013\pi$
$110$	0	0
$111$	4.29775i	0.0387185i
$112$	0	0
$113$	65.2511	0.577444	0.288722	−	0.957413i	$-0.406770\pi$
$113$	65.2511	0.577444	0.288722	+	0.957413i	$0.406770\pi$
$114$	0	0
$115$	10.4978i	0.0912852i
$116$	0	0
$117$	− 47.9259i	− 0.409623i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−27.1078	−0.224032
$122$	0	0
$123$	−96.7385	−0.786492
$124$	0	0
$125$	45.6484i	0.365187i
$126$	0	0
$127$	235.761	1.85639	0.928193	−	0.372098i	$-0.121361\pi$
$127$	235.761	1.85639	0.928193	+	0.372098i	$0.121361\pi$
$128$	0	0
$129$	− 105.098i	− 0.814717i
$130$	0	0
$131$	− 226.529i	− 1.72923i	−0.502433	−	0.864616i	$-0.667561\pi$
$131$	− 226.529i	− 1.72923i	0.502433	−	0.864616i	$-0.332439\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.82724	0.0357573
$136$	0	0
$137$	−253.743	−1.85214	−0.926070	−	0.377353i	$-0.876834\pi$
$137$	−253.743	−1.85214	−0.926070	+	0.377353i	$0.876834\pi$
$138$	0	0
$139$	148.040i	1.06503i	0.846419	+	0.532517i	$0.178754\pi$
$139$	148.040i	1.06503i	−0.846419	+	0.532517i	$0.821246\pi$
$140$	0	0
$141$	−63.3700	−0.449432
$142$	0	0
$143$	− 154.797i	− 1.08250i
$144$	0	0
$145$	43.0266i	0.296735i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−245.465	−1.64742	−0.823708	−	0.567014i	$-0.808098\pi$
$149$	−245.465	−1.64742	−0.823708	+	0.567014i	$0.808098\pi$
$150$	0	0
$151$	−176.254	−1.16725	−0.583623	−	0.812025i	$-0.698365\pi$
$151$	−176.254	−1.16725	−0.583623	+	0.812025i	$0.698365\pi$
$152$	0	0
$153$	− 31.6672i	− 0.206975i
$154$	0	0
$155$	−0.448754	−0.00289519
$156$	0	0
$157$	− 207.656i	− 1.32265i	−0.750099	−	0.661326i	$-0.769994\pi$
$157$	− 207.656i	− 1.32265i	0.750099	−	0.661326i	$-0.230006\pi$
$158$	0	0
$159$	− 49.4032i	− 0.310712i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−95.0241	−0.582970	−0.291485	−	0.956575i	$-0.594149\pi$
$163$	−95.0241	−0.582970	−0.291485	+	0.956575i	$0.594149\pi$
$164$	0	0
$165$	15.5917	0.0944949
$166$	0	0
$167$	158.478i	0.948972i	0.880263	+	0.474486i	$0.157366\pi$
$167$	158.478i	0.948972i	−0.880263	+	0.474486i	$0.842634\pi$
$168$	0	0
$169$	−86.2098	−0.510117
$170$	0	0
$171$	− 21.6615i	− 0.126675i
$172$	0	0
$173$	215.991i	1.24850i	0.781224	+	0.624251i	$0.214596\pi$
$173$	215.991i	1.24850i	−0.781224	+	0.624251i	$0.785404\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	162.937	0.920546
$178$	0	0
$179$	262.059	1.46402	0.732008	−	0.681296i	$-0.238584\pi$
$179$	262.059	1.46402	0.732008	+	0.681296i	$0.238584\pi$
$180$	0	0
$181$	− 83.8554i	− 0.463290i	−0.972800	−	0.231645i	$-0.925589\pi$
$181$	− 83.8554i	− 0.463290i	0.972800	−	0.231645i	$-0.0744107\pi$
$182$	0	0
$183$	190.860	1.04295
$184$	0	0
$185$	2.30514i	0.0124602i
$186$	0	0
$187$	− 102.283i	− 0.546968i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−233.692	−1.22352	−0.611760	−	0.791043i	$-0.709538\pi$
$191$	−233.692	−1.22352	−0.611760	+	0.791043i	$0.709538\pi$
$192$	0	0
$193$	−223.639	−1.15875	−0.579375	−	0.815061i	$-0.696703\pi$
$193$	−223.639	−1.15875	−0.579375	+	0.815061i	$0.696703\pi$
$194$	0	0
$195$	− 25.7055i	− 0.131823i
$196$	0	0
$197$	−133.006	−0.675158	−0.337579	−	0.941297i	$-0.609608\pi$
$197$	−133.006	−0.675158	−0.337579	+	0.941297i	$0.609608\pi$
$198$	0	0
$199$	60.6538i	0.304793i	0.988319	+	0.152397i	$0.0486991\pi$
$199$	60.6538i	0.304793i	−0.988319	+	0.152397i	$0.951301\pi$
$200$	0	0
$201$	− 142.082i	− 0.706874i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−51.8867	−0.253106
$206$	0	0
$207$	−33.9002	−0.163769
$208$	0	0
$209$	− 69.9651i	− 0.334761i
$210$	0	0
$211$	169.145	0.801637	0.400819	−	0.916157i	$-0.368726\pi$
$211$	169.145	0.801637	0.400819	+	0.916157i	$0.368726\pi$
$212$	0	0
$213$	− 220.576i	− 1.03557i
$214$	0	0
$215$	− 56.3706i	− 0.262189i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	80.1153	0.365823
$220$	0	0
$221$	−168.631	−0.763036
$222$	0	0
$223$	162.093i	0.726874i	0.931619	+	0.363437i	$0.118397\pi$
$223$	162.093i	0.726874i	−0.931619	+	0.363437i	$0.881603\pi$
$224$	0	0
$225$	−72.4109	−0.321826
$226$	0	0
$227$	− 354.210i	− 1.56040i	−0.625533	−	0.780198i	$-0.715118\pi$
$227$	− 354.210i	− 1.56040i	0.625533	−	0.780198i	$-0.284882\pi$
$228$	0	0
$229$	− 131.456i	− 0.574043i	−0.957924	−	0.287021i	$-0.907335\pi$
$229$	− 131.456i	− 0.574043i	0.957924	−	0.287021i	$-0.0926651\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−142.949	−0.613514	−0.306757	−	0.951788i	$-0.599244\pi$
$233$	−142.949	−0.613514	−0.306757	+	0.951788i	$0.599244\pi$
$234$	0	0
$235$	−33.9891	−0.144635
$236$	0	0
$237$	− 32.3899i	− 0.136666i
$238$	0	0
$239$	−47.5259	−0.198853	−0.0994266	−	0.995045i	$-0.531701\pi$
$239$	−47.5259	−0.198853	−0.0994266	+	0.995045i	$0.531701\pi$
$240$	0	0
$241$	− 237.152i	− 0.984033i	−0.870586	−	0.492017i	$-0.836260\pi$
$241$	− 237.152i	− 0.984033i	0.870586	−	0.492017i	$-0.163740\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−115.349	−0.467002
$248$	0	0
$249$	−103.292	−0.414827
$250$	0	0
$251$	309.248i	1.23206i	0.787722	+	0.616031i	$0.211261\pi$
$251$	309.248i	1.23206i	−0.787722	+	0.616031i	$0.788739\pi$
$252$	0	0
$253$	−109.495	−0.432788
$254$	0	0
$255$	− 16.9850i	− 0.0666080i
$256$	0	0
$257$	179.345i	0.697840i	0.937153	+	0.348920i	$0.113451\pi$
$257$	179.345i	0.697840i	−0.937153	+	0.348920i	$0.886549\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−138.945	−0.532355
$262$	0	0
$263$	−74.7695	−0.284295	−0.142147	−	0.989846i	$-0.545401\pi$
$263$	−74.7695	−0.284295	−0.142147	+	0.989846i	$0.545401\pi$
$264$	0	0
$265$	− 26.4979i	− 0.0999921i
$266$	0	0
$267$	−123.198	−0.461414
$268$	0	0
$269$	− 56.8302i	− 0.211265i	−0.994405	−	0.105632i	$-0.966313\pi$
$269$	− 56.8302i	− 0.211265i	0.994405	−	0.105632i	$-0.0336867\pi$
$270$	0	0
$271$	− 13.5044i	− 0.0498316i	−0.999690	−	0.0249158i	$-0.992068\pi$
$271$	− 13.5044i	− 0.0498316i	0.999690	−	0.0249158i	$-0.00793176\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−233.882	−0.850481
$276$	0	0
$277$	−249.452	−0.900548	−0.450274	−	0.892890i	$-0.648674\pi$
$277$	−249.452	−0.900548	−0.450274	+	0.892890i	$0.648674\pi$
$278$	0	0
$279$	− 1.44915i	− 0.00519408i
$280$	0	0
$281$	200.268	0.712696	0.356348	−	0.934353i	$-0.384022\pi$
$281$	200.268	0.712696	0.356348	+	0.934353i	$0.384022\pi$
$282$	0	0
$283$	− 68.7680i	− 0.242997i	−0.992592	−	0.121498i	$-0.961230\pi$
$283$	− 68.7680i	− 0.242997i	0.992592	−	0.121498i	$-0.0387699\pi$
$284$	0	0
$285$	− 11.6183i	− 0.0407661i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	177.576	0.614451
$290$	0	0
$291$	−177.084	−0.608536
$292$	0	0
$293$	253.164i	0.864040i	0.901864	+	0.432020i	$0.142199\pi$
$293$	253.164i	0.864040i	−0.901864	+	0.432020i	$0.857801\pi$
$294$	0	0
$295$	87.3927	0.296247
$296$	0	0
$297$	50.3497i	0.169527i
$298$	0	0
$299$	180.522i	0.603752i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−188.671	−0.622678
$304$	0	0
$305$	102.370	0.335639
$306$	0	0
$307$	− 529.913i	− 1.72610i	−0.505116	−	0.863051i	$-0.668550\pi$
$307$	− 529.913i	− 1.72610i	0.505116	−	0.863051i	$-0.331450\pi$
$308$	0	0
$309$	1.29417	0.00418824
$310$	0	0
$311$	− 30.5056i	− 0.0980887i	−0.998797	−	0.0490444i	$-0.984382\pi$
$311$	− 30.5056i	− 0.0980887i	0.998797	−	0.0490444i	$-0.0156176\pi$
$312$	0	0
$313$	8.60202i	0.0274825i	0.999906	+	0.0137412i	$0.00437411\pi$
$313$	8.60202i	0.0274825i	−0.999906	+	0.0137412i	$0.995626\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−480.152	−1.51467	−0.757337	−	0.653024i	$-0.773500\pi$
$317$	−480.152	−1.51467	−0.757337	+	0.653024i	$0.773500\pi$
$318$	0	0
$319$	−448.781	−1.40684
$320$	0	0
$321$	− 26.4876i	− 0.0825158i
$322$	0	0
$323$	−76.2176	−0.235968
$324$	0	0
$325$	385.595i	1.18645i
$326$	0	0
$327$	− 93.9174i	− 0.287209i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	89.0195	0.268941	0.134471	−	0.990918i	$-0.457067\pi$
$331$	89.0195	0.268941	0.134471	+	0.990918i	$0.457067\pi$
$332$	0	0
$333$	−7.44392	−0.0223541
$334$	0	0
$335$	− 76.2070i	− 0.227484i
$336$	0	0
$337$	495.701	1.47092	0.735461	−	0.677567i	$-0.236966\pi$
$337$	495.701	1.47092	0.735461	+	0.677567i	$0.236966\pi$
$338$	0	0
$339$	113.018i	0.333387i
$340$	0	0
$341$	− 4.68065i	− 0.0137262i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−18.1827	−0.0527035
$346$	0	0
$347$	60.3534	0.173929	0.0869645	−	0.996211i	$-0.472283\pi$
$347$	60.3534	0.173929	0.0869645	+	0.996211i	$0.472283\pi$
$348$	0	0
$349$	− 72.2171i	− 0.206926i	−0.994633	−	0.103463i	$-0.967008\pi$
$349$	− 72.2171i	− 0.206926i	0.994633	−	0.103463i	$-0.0329923\pi$
$350$	0	0
$351$	83.0100	0.236496
$352$	0	0
$353$	− 383.361i	− 1.08601i	−0.839730	−	0.543004i	$-0.817287\pi$
$353$	− 383.361i	− 1.08601i	0.839730	−	0.543004i	$-0.182713\pi$
$354$	0	0
$355$	− 118.308i	− 0.333262i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	80.0448	0.222966	0.111483	−	0.993766i	$-0.464440\pi$
$359$	80.0448	0.222966	0.111483	+	0.993766i	$0.464440\pi$
$360$	0	0
$361$	308.865	0.855580
$362$	0	0
$363$	− 46.9521i	− 0.129345i
$364$	0	0
$365$	42.9707	0.117728
$366$	0	0
$367$	− 81.4316i	− 0.221884i	−0.993827	−	0.110942i	$-0.964613\pi$
$367$	− 81.4316i	− 0.221884i	0.993827	−	0.110942i	$-0.0353868\pi$
$368$	0	0
$369$	− 167.556i	− 0.454081i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−532.252	−1.42695	−0.713475	−	0.700681i	$-0.752880\pi$
$373$	−532.252	−1.42695	−0.713475	+	0.700681i	$0.752880\pi$
$374$	0	0
$375$	−79.0653	−0.210841
$376$	0	0
$377$	739.893i	1.96258i
$378$	0	0
$379$	−440.518	−1.16232	−0.581159	−	0.813790i	$-0.697400\pi$
$379$	−440.518	−1.16232	−0.581159	+	0.813790i	$0.697400\pi$
$380$	0	0
$381$	408.350i	1.07179i
$382$	0	0
$383$	699.290i	1.82582i	0.408159	+	0.912911i	$0.366171\pi$
$383$	699.290i	1.82582i	−0.408159	+	0.912911i	$0.633829\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	182.036	0.470377
$388$	0	0
$389$	624.462	1.60530	0.802650	−	0.596450i	$-0.203423\pi$
$389$	624.462	1.60530	0.802650	+	0.596450i	$0.203423\pi$
$390$	0	0
$391$	119.281i	0.305065i
$392$	0	0
$393$	392.361	0.998373
$394$	0	0
$395$	− 17.3726i	− 0.0439814i
$396$	0	0
$397$	− 632.547i	− 1.59332i	−0.604430	−	0.796658i	$-0.706599\pi$
$397$	− 632.547i	− 1.59332i	0.604430	−	0.796658i	$-0.293401\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	467.343	1.16544	0.582721	−	0.812672i	$-0.301988\pi$
$401$	467.343	1.16544	0.582721	+	0.812672i	$0.301988\pi$
$402$	0	0
$403$	−7.71685	−0.0191485
$404$	0	0
$405$	8.36103i	0.0206445i
$406$	0	0
$407$	−24.0434	−0.0590746
$408$	0	0
$409$	− 324.296i	− 0.792900i	−0.918056	−	0.396450i	$-0.870242\pi$
$409$	− 324.296i	− 0.792900i	0.918056	−	0.396450i	$-0.129758\pi$
$410$	0	0
$411$	− 439.496i	− 1.06933i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−55.4017	−0.133498
$416$	0	0
$417$	−256.413	−0.614898
$418$	0	0
$419$	28.8211i	0.0687854i	0.999408	+	0.0343927i	$0.0109497\pi$
$419$	28.8211i	0.0687854i	−0.999408	+	0.0343927i	$0.989050\pi$
$420$	0	0
$421$	−0.326830	−0.000776319 0	−0.000388160	−	1.00000i	$-0.500124\pi$
$421$	−0.326830	−0.000776319 0	−0.000388160		1.00000i	$0.500124\pi$
$422$	0	0
$423$	− 109.760i	− 0.259480i
$424$	0	0
$425$	254.784i	0.599491i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	268.117	0.624981
$430$	0	0
$431$	−391.337	−0.907975	−0.453988	−	0.891008i	$-0.649999\pi$
$431$	−391.337	−0.907975	−0.453988	+	0.891008i	$0.649999\pi$
$432$	0	0
$433$	470.579i	1.08679i	0.839478	+	0.543394i	$0.182861\pi$
$433$	470.579i	1.08679i	−0.839478	+	0.543394i	$0.817139\pi$
$434$	0	0
$435$	−74.5243	−0.171320
$436$	0	0
$437$	81.5920i	0.186709i
$438$	0	0
$439$	− 134.401i	− 0.306153i	−0.988214	−	0.153077i	$-0.951082\pi$
$439$	− 134.401i	− 0.306153i	0.988214	−	0.153077i	$-0.0489181\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−329.056	−0.742790	−0.371395	−	0.928475i	$-0.621120\pi$
$443$	−329.056	−0.742790	−0.371395	+	0.928475i	$0.621120\pi$
$444$	0	0
$445$	−66.0782	−0.148490
$446$	0	0
$447$	− 425.158i	− 0.951136i
$448$	0	0
$449$	130.592	0.290851	0.145426	−	0.989369i	$-0.453545\pi$
$449$	130.592	0.290851	0.145426	+	0.989369i	$0.453545\pi$
$450$	0	0
$451$	− 541.194i	− 1.19999i
$452$	0	0
$453$	− 305.281i	− 0.673909i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	316.282	0.692082	0.346041	−	0.938219i	$-0.387526\pi$
$457$	316.282	0.692082	0.346041	+	0.938219i	$0.387526\pi$
$458$	0	0
$459$	54.8493	0.119497
$460$	0	0
$461$	− 31.4853i	− 0.0682979i	−0.999417	−	0.0341490i	$-0.989128\pi$
$461$	− 31.4853i	− 0.0682979i	0.999417	−	0.0341490i	$-0.0108721\pi$
$462$	0	0
$463$	−667.424	−1.44152	−0.720761	−	0.693184i	$-0.756207\pi$
$463$	−667.424	−1.44152	−0.720761	+	0.693184i	$0.756207\pi$
$464$	0	0
$465$	− 0.777265i	− 0.00167154i
$466$	0	0
$467$	864.295i	1.85074i	0.379065	+	0.925370i	$0.376246\pi$
$467$	864.295i	1.85074i	−0.379065	+	0.925370i	$0.623754\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	359.671	0.763633
$472$	0	0
$473$	587.964	1.24305
$474$	0	0
$475$	174.281i	0.366907i
$476$	0	0
$477$	85.5688	0.179390
$478$	0	0
$479$	626.397i	1.30772i	0.756616	+	0.653859i	$0.226851\pi$
$479$	626.397i	1.30772i	−0.756616	+	0.653859i	$0.773149\pi$
$480$	0	0
$481$	39.6396i	0.0824108i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−94.9808	−0.195837
$486$	0	0
$487$	485.848	0.997634	0.498817	−	0.866707i	$-0.333768\pi$
$487$	485.848	0.997634	0.498817	+	0.866707i	$0.333768\pi$
$488$	0	0
$489$	− 164.587i	− 0.336578i
$490$	0	0
$491$	6.00051	0.0122210	0.00611049	−	0.999981i	$-0.498055\pi$
$491$	6.00051	0.0122210	0.00611049	+	0.999981i	$0.498055\pi$
$492$	0	0
$493$	488.888i	0.991658i
$494$	0	0
$495$	27.0056i	0.0545567i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	687.551	1.37786	0.688929	−	0.724829i	$-0.258081\pi$
$499$	687.551	1.37786	0.688929	+	0.724829i	$0.258081\pi$
$500$	0	0
$501$	−274.492	−0.547889
$502$	0	0
$503$	435.270i	0.865348i	0.901550	+	0.432674i	$0.142430\pi$
$503$	435.270i	0.865348i	−0.901550	+	0.432674i	$0.857570\pi$
$504$	0	0
$505$	−101.196	−0.200388
$506$	0	0
$507$	− 149.320i	− 0.294516i
$508$	0	0
$509$	− 483.105i	− 0.949126i	−0.880222	−	0.474563i	$-0.842606\pi$
$509$	− 483.105i	− 0.949126i	0.880222	−	0.474563i	$-0.157394\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	37.5188	0.0731360
$514$	0	0
$515$	0.694139	0.00134784
$516$	0	0
$517$	− 354.517i	− 0.685720i
$518$	0	0
$519$	−374.107	−0.720823
$520$	0	0
$521$	− 314.569i	− 0.603779i	−0.953343	−	0.301890i	$-0.902383\pi$
$521$	− 314.569i	− 0.603779i	0.953343	−	0.301890i	$-0.0976174\pi$
$522$	0	0
$523$	− 156.567i	− 0.299363i	−0.988734	−	0.149682i	$-0.952175\pi$
$523$	− 156.567i	− 0.299363i	0.988734	−	0.149682i	$-0.0478249\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.09894	−0.00967541
$528$	0	0
$529$	−401.309	−0.758617
$530$	0	0
$531$	282.215i	0.531478i
$532$	0	0
$533$	−892.251	−1.67402
$534$	0	0
$535$	− 14.2069i	− 0.0265549i
$536$	0	0
$537$	453.899i	0.845250i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	772.066	1.42711	0.713554	−	0.700600i	$-0.247084\pi$
$541$	772.066	1.42711	0.713554	+	0.700600i	$0.247084\pi$
$542$	0	0
$543$	145.242	0.267480
$544$	0	0
$545$	− 50.3736i	− 0.0924285i
$546$	0	0
$547$	−48.0113	−0.0877721	−0.0438860	−	0.999037i	$-0.513974\pi$
$547$	−48.0113	−0.0877721	−0.0438860	+	0.999037i	$0.513974\pi$
$548$	0	0
$549$	330.580i	0.602149i
$550$	0	0
$551$	334.416i	0.606925i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.99262	−0.00719391
$556$	0	0
$557$	151.405	0.271823	0.135911	−	0.990721i	$-0.456604\pi$
$557$	151.405	0.271823	0.135911	+	0.990721i	$0.456604\pi$
$558$	0	0
$559$	− 969.359i	− 1.73409i
$560$	0	0
$561$	177.159	0.315792
$562$	0	0
$563$	846.301i	1.50320i	0.659620	+	0.751599i	$0.270717\pi$
$563$	846.301i	1.50320i	−0.659620	+	0.751599i	$0.729283\pi$
$564$	0	0
$565$	60.6185i	0.107289i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.5774	−0.0379215	−0.0189608	−	0.999820i	$-0.506036\pi$
$569$	−21.5774	−0.0379215	−0.0189608	+	0.999820i	$0.506036\pi$
$570$	0	0
$571$	−283.245	−0.496051	−0.248026	−	0.968753i	$-0.579782\pi$
$571$	−283.245	−0.496051	−0.248026	+	0.968753i	$0.579782\pi$
$572$	0	0
$573$	− 404.767i	− 0.706400i
$574$	0	0
$575$	272.749	0.474346
$576$	0	0
$577$	469.306i	0.813355i	0.913572	+	0.406678i	$0.133313\pi$
$577$	469.306i	0.813355i	−0.913572	+	0.406678i	$0.866687\pi$
$578$	0	0
$579$	− 387.354i	− 0.669004i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	276.381	0.474068
$584$	0	0
$585$	44.5233	0.0761082
$586$	0	0
$587$	− 554.500i	− 0.944634i	−0.881429	−	0.472317i	$-0.843418\pi$
$587$	− 554.500i	− 0.944634i	0.881429	−	0.472317i	$-0.156582\pi$
$588$	0	0
$589$	−3.48785	−0.00592165
$590$	0	0
$591$	− 230.373i	− 0.389803i
$592$	0	0
$593$	816.481i	1.37687i	0.725300	+	0.688433i	$0.241701\pi$
$593$	816.481i	1.37687i	−0.725300	+	0.688433i	$0.758299\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−105.055	−0.175972
$598$	0	0
$599$	868.683	1.45022	0.725111	−	0.688632i	$-0.241789\pi$
$599$	868.683	1.45022	0.725111	+	0.688632i	$0.241789\pi$
$600$	0	0
$601$	− 705.861i	− 1.17448i	−0.809413	−	0.587239i	$-0.800215\pi$
$601$	− 705.861i	− 1.17448i	0.809413	−	0.587239i	$-0.199785\pi$
$602$	0	0
$603$	246.093	0.408114
$604$	0	0
$605$	− 25.1833i	− 0.0416252i
$606$	0	0
$607$	− 535.811i	− 0.882720i	−0.897330	−	0.441360i	$-0.854496\pi$
$607$	− 535.811i	− 0.882720i	0.897330	−	0.441360i	$-0.145504\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−584.483	−0.956600
$612$	0	0
$613$	−284.494	−0.464101	−0.232050	−	0.972704i	$-0.574543\pi$
$613$	−284.494	−0.464101	−0.232050	+	0.972704i	$0.574543\pi$
$614$	0	0
$615$	− 89.8703i	− 0.146131i
$616$	0	0
$617$	−766.565	−1.24241	−0.621203	−	0.783650i	$-0.713356\pi$
$617$	−766.565	−1.24241	−0.621203	+	0.783650i	$0.713356\pi$
$618$	0	0
$619$	543.006i	0.877232i	0.898675	+	0.438616i	$0.144531\pi$
$619$	543.006i	0.877232i	−0.898675	+	0.438616i	$0.855469\pi$
$620$	0	0
$621$	− 58.7169i	− 0.0945521i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	561.016	0.897626
$626$	0	0
$627$	121.183	0.193274
$628$	0	0
$629$	26.1920i	0.0416408i
$630$	0	0
$631$	967.080	1.53261	0.766307	−	0.642474i	$-0.222092\pi$
$631$	967.080	1.53261	0.766307	+	0.642474i	$0.222092\pi$
$632$	0	0
$633$	292.969i	0.462825i
$634$	0	0
$635$	219.023i	0.344918i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	382.048	0.597884
$640$	0	0
$641$	979.024	1.52734	0.763669	−	0.645608i	$-0.223396\pi$
$641$	979.024	1.52734	0.763669	+	0.645608i	$0.223396\pi$
$642$	0	0
$643$	991.244i	1.54159i	0.637081	+	0.770797i	$0.280142\pi$
$643$	991.244i	1.54159i	−0.637081	+	0.770797i	$0.719858\pi$
$644$	0	0
$645$	97.6368	0.151375
$646$	0	0
$647$	1232.42i	1.90482i	0.304822	+	0.952409i	$0.401403\pi$
$647$	1232.42i	1.90482i	−0.304822	+	0.952409i	$0.598597\pi$
$648$	0	0
$649$	911.534i	1.40452i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−781.069	−1.19612	−0.598062	−	0.801450i	$-0.704062\pi$
$653$	−781.069	−1.19612	−0.598062	+	0.801450i	$0.704062\pi$
$654$	0	0
$655$	210.447	0.321293
$656$	0	0
$657$	138.764i	0.211208i
$658$	0	0
$659$	549.328	0.833578	0.416789	−	0.909003i	$-0.363155\pi$
$659$	549.328	0.833578	0.416789	+	0.909003i	$0.363155\pi$
$660$	0	0
$661$	413.630i	0.625764i	0.949792	+	0.312882i	$0.101294\pi$
$661$	413.630i	0.625764i	−0.949792	+	0.312882i	$0.898706\pi$
$662$	0	0
$663$	− 292.078i	− 0.440539i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	523.361	0.784649
$668$	0	0
$669$	−280.753	−0.419661
$670$	0	0
$671$	1067.75i	1.59128i
$672$	0	0
$673$	553.924	0.823067	0.411533	−	0.911395i	$-0.364993\pi$
$673$	553.924	0.823067	0.411533	+	0.911395i	$0.364993\pi$
$674$	0	0
$675$	− 125.419i	− 0.185806i
$676$	0	0
$677$	− 110.386i	− 0.163052i	−0.996671	−	0.0815259i	$-0.974021\pi$
$677$	− 110.386i	− 0.163052i	0.996671	−	0.0815259i	$-0.0259793\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	613.509	0.900895
$682$	0	0
$683$	195.631	0.286429	0.143215	−	0.989692i	$-0.454256\pi$
$683$	195.631	0.286429	0.143215	+	0.989692i	$0.454256\pi$
$684$	0	0
$685$	− 235.728i	− 0.344129i
$686$	0	0
$687$	227.688	0.331424
$688$	0	0
$689$	− 455.662i	− 0.661338i
$690$	0	0
$691$	− 662.950i	− 0.959407i	−0.877431	−	0.479703i	$-0.840744\pi$
$691$	− 662.950i	− 0.959407i	0.877431	−	0.479703i	$-0.159256\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−137.529	−0.197884
$696$	0	0
$697$	−589.559	−0.845852
$698$	0	0
$699$	− 247.595i	− 0.354213i
$700$	0	0
$701$	−1236.29	−1.76361	−0.881807	−	0.471610i	$-0.843673\pi$
$701$	−1236.29	−1.76361	−0.881807	+	0.471610i	$0.843673\pi$
$702$	0	0
$703$	17.9162i	0.0254854i
$704$	0	0
$705$	− 58.8709i	− 0.0835048i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−900.501	−1.27010	−0.635050	−	0.772471i	$-0.719020\pi$
$709$	−900.501	−1.27010	−0.635050	+	0.772471i	$0.719020\pi$
$710$	0	0
$711$	56.1009	0.0789043
$712$	0	0
$713$	5.45848i	0.00765566i
$714$	0	0
$715$	143.807	0.201129
$716$	0	0
$717$	− 82.3173i	− 0.114808i
$718$	0	0
$719$	− 119.102i	− 0.165649i	−0.996564	−	0.0828246i	$-0.973606\pi$
$719$	− 119.102i	− 0.165649i	0.996564	−	0.0828246i	$-0.0263941\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	410.759	0.568132
$724$	0	0
$725$	1117.90	1.54193
$726$	0	0
$727$	− 815.672i	− 1.12197i	−0.827826	−	0.560985i	$-0.810423\pi$
$727$	− 815.672i	− 1.12197i	0.827826	−	0.560985i	$-0.189577\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 640.508i	− 0.876208i
$732$	0	0
$733$	223.741i	0.305241i	0.988285	+	0.152620i	$0.0487712\pi$
$733$	223.741i	0.305241i	−0.988285	+	0.152620i	$0.951229\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	794.863	1.07851
$738$	0	0
$739$	1273.79	1.72367	0.861836	−	0.507187i	$-0.169315\pi$
$739$	1273.79	1.72367	0.861836	+	0.507187i	$0.169315\pi$
$740$	0	0
$741$	− 199.791i	− 0.269624i
$742$	0	0
$743$	204.339	0.275019	0.137510	−	0.990500i	$-0.456090\pi$
$743$	204.339	0.275019	0.137510	+	0.990500i	$0.456090\pi$
$744$	0	0
$745$	− 228.038i	− 0.306091i
$746$	0	0
$747$	− 178.907i	− 0.239501i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	386.559	0.514726	0.257363	−	0.966315i	$-0.417146\pi$
$751$	386.559	0.514726	0.257363	+	0.966315i	$0.417146\pi$
$752$	0	0
$753$	−535.633	−0.711332
$754$	0	0
$755$	− 163.741i	− 0.216875i
$756$	0	0
$757$	−1254.03	−1.65658	−0.828290	−	0.560300i	$-0.810686\pi$
$757$	−1254.03	−1.65658	−0.828290	+	0.560300i	$0.810686\pi$
$758$	0	0
$759$	− 189.651i	− 0.249870i
$760$	0	0
$761$	810.611i	1.06519i	0.846370	+	0.532596i	$0.178783\pi$
$761$	810.611i	1.06519i	−0.846370	+	0.532596i	$0.821217\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	29.4190	0.0384562
$766$	0	0
$767$	1502.82	1.95935
$768$	0	0
$769$	− 1338.07i	− 1.74002i	−0.493036	−	0.870009i	$-0.664113\pi$
$769$	− 1338.07i	− 1.74002i	0.493036	−	0.870009i	$-0.335887\pi$
$770$	0	0
$771$	−310.634	−0.402898
$772$	0	0
$773$	516.124i	0.667690i	0.942628	+	0.333845i	$0.108346\pi$
$773$	516.124i	0.667690i	−0.942628	+	0.333845i	$0.891654\pi$
$774$	0	0
$775$	11.6593i	0.0150443i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−403.279	−0.517688
$780$	0	0
$781$	1233.99	1.58001
$782$	0	0
$783$	− 240.659i	− 0.307355i
$784$	0	0
$785$	192.913	0.245749
$786$	0	0
$787$	1504.21i	1.91133i	0.294463	+	0.955663i	$0.404859\pi$
$787$	1504.21i	1.91133i	−0.294463	+	0.955663i	$0.595141\pi$
$788$	0	0
$789$	− 129.505i	− 0.164138i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1760.37	2.21988
$794$	0	0
$795$	45.8957	0.0577304
$796$	0	0
$797$	− 147.647i	− 0.185254i	−0.995701	−	0.0926268i	$-0.970474\pi$
$797$	− 147.647i	− 0.185254i	0.995701	−	0.0926268i	$-0.0295263\pi$
$798$	0	0
$799$	−386.200	−0.483354
$800$	0	0
$801$	− 213.384i	− 0.266397i
$802$	0	0
$803$	448.198i	0.558154i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	98.4328	0.121974
$808$	0	0
$809$	171.184	0.211600	0.105800	−	0.994387i	$-0.466260\pi$
$809$	171.184	0.211600	0.105800	+	0.994387i	$0.466260\pi$
$810$	0	0
$811$	− 668.261i	− 0.823997i	−0.911185	−	0.411998i	$-0.864831\pi$
$811$	− 668.261i	− 0.823997i	0.911185	−	0.411998i	$-0.135169\pi$
$812$	0	0
$813$	23.3902	0.0287703
$814$	0	0
$815$	− 88.2777i	− 0.108316i
$816$	0	0
$817$	− 438.130i	− 0.536266i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.6411	−0.0202693	−0.0101346	−	0.999949i	$-0.503226\pi$
$821$	−16.6411	−0.0202693	−0.0101346	+	0.999949i	$0.503226\pi$
$822$	0	0
$823$	310.695	0.377515	0.188758	−	0.982024i	$-0.439554\pi$
$823$	310.695	0.377515	0.188758	+	0.982024i	$0.439554\pi$
$824$	0	0
$825$	− 405.096i	− 0.491025i
$826$	0	0
$827$	−139.891	−0.169155	−0.0845777	−	0.996417i	$-0.526954\pi$
$827$	−139.891	−0.169155	−0.0845777	+	0.996417i	$0.526954\pi$
$828$	0	0
$829$	1442.37i	1.73989i	0.493150	+	0.869944i	$0.335845\pi$
$829$	1442.37i	1.73989i	−0.493150	+	0.869944i	$0.664155\pi$
$830$	0	0
$831$	− 432.063i	− 0.519932i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−147.227	−0.176320
$836$	0	0
$837$	2.51000	0.00299880
$838$	0	0
$839$	657.634i	0.783831i	0.920001	+	0.391915i	$0.128187\pi$
$839$	657.634i	0.783831i	−0.920001	+	0.391915i	$0.871813\pi$
$840$	0	0
$841$	1304.06	1.55061
$842$	0	0
$843$	346.874i	0.411475i
$844$	0	0
$845$	− 80.0892i	− 0.0947801i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	119.110	0.140294
$850$	0	0
$851$	28.0389	0.0329482
$852$	0	0
$853$	− 121.230i	− 0.142122i	−0.997472	−	0.0710609i	$-0.977362\pi$
$853$	− 121.230i	− 0.142122i	0.997472	−	0.0710609i	$-0.0226385\pi$
$854$	0	0
$855$	20.1236	0.0235363
$856$	0	0
$857$	− 1245.58i	− 1.45342i	−0.686945	−	0.726710i	$-0.741049\pi$
$857$	− 1245.58i	− 1.45342i	0.686945	−	0.726710i	$-0.258951\pi$
$858$	0	0
$859$	779.573i	0.907536i	0.891120	+	0.453768i	$0.149920\pi$
$859$	779.573i	0.907536i	−0.891120	+	0.453768i	$0.850080\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	840.750	0.974218	0.487109	−	0.873341i	$-0.338051\pi$
$863$	840.750	0.974218	0.487109	+	0.873341i	$0.338051\pi$
$864$	0	0
$865$	−200.656	−0.231972
$866$	0	0
$867$	307.571i	0.354753i
$868$	0	0
$869$	181.202	0.208518
$870$	0	0
$871$	− 1310.47i	− 1.50456i
$872$	0	0
$873$	− 306.718i	− 0.351338i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1474.24	−1.68101	−0.840504	−	0.541806i	$-0.817741\pi$
$877$	−1474.24	−1.68101	−0.840504	+	0.541806i	$0.817741\pi$
$878$	0	0
$879$	−438.493	−0.498854
$880$	0	0
$881$	− 583.982i	− 0.662863i	−0.943479	−	0.331431i	$-0.892468\pi$
$881$	− 583.982i	− 0.662863i	0.943479	−	0.331431i	$-0.107532\pi$
$882$	0	0
$883$	1226.32	1.38882	0.694408	−	0.719581i	$-0.255666\pi$
$883$	1226.32	1.38882	0.694408	+	0.719581i	$0.255666\pi$
$884$	0	0
$885$	151.369i	0.171038i
$886$	0	0
$887$	498.376i	0.561867i	0.959727	+	0.280934i	$0.0906441\pi$
$887$	498.376i	0.561867i	−0.959727	+	0.280934i	$0.909356\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−87.2082	−0.0978767
$892$	0	0
$893$	−264.174	−0.295827
$894$	0	0
$895$	243.453i	0.272015i
$896$	0	0
$897$	−312.673	−0.348576
$898$	0	0
$899$	22.3723i	0.0248858i
$900$	0	0
$901$	− 301.081i	− 0.334163i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	77.9020	0.0860795
$906$	0	0
$907$	−73.8246	−0.0813942	−0.0406971	−	0.999172i	$-0.512958\pi$
$907$	−73.8246	−0.0813942	−0.0406971	+	0.999172i	$0.512958\pi$
$908$	0	0
$909$	− 326.788i	− 0.359503i
$910$	0	0
$911$	−1102.46	−1.21017	−0.605084	−	0.796162i	$-0.706860\pi$
$911$	−1102.46	−1.21017	−0.605084	+	0.796162i	$0.706860\pi$
$912$	0	0
$913$	− 577.857i	− 0.632922i
$914$	0	0
$915$	177.310i	0.193781i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−982.172	−1.06874	−0.534370	−	0.845251i	$-0.679451\pi$
$919$	−982.172	−1.06874	−0.534370	+	0.845251i	$0.679451\pi$
$920$	0	0
$921$	917.837	0.996566
$922$	0	0
$923$	− 2034.44i	− 2.20416i
$924$	0	0
$925$	59.8912	0.0647472
$926$	0	0
$927$	2.24156i	0.00241808i
$928$	0	0
$929$	648.814i	0.698400i	0.937048	+	0.349200i	$0.113547\pi$
$929$	648.814i	0.698400i	−0.937048	+	0.349200i	$0.886453\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	52.8372	0.0566316
$934$	0	0
$935$	95.0212	0.101627
$936$	0	0
$937$	1348.17i	1.43881i	0.694590	+	0.719406i	$0.255586\pi$
$937$	1348.17i	1.43881i	−0.694590	+	0.719406i	$0.744414\pi$
$938$	0	0
$939$	−14.8991	−0.0158670
$940$	0	0
$941$	1187.26i	1.26170i	0.775904	+	0.630851i	$0.217294\pi$
$941$	1187.26i	1.26170i	−0.775904	+	0.630851i	$0.782706\pi$
$942$	0	0
$943$	631.131i	0.669280i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	132.564	0.139983	0.0699915	−	0.997548i	$-0.477703\pi$
$947$	132.564	0.139983	0.0699915	+	0.997548i	$0.477703\pi$
$948$	0	0
$949$	738.930	0.778641
$950$	0	0
$951$	− 831.648i	− 0.874498i
$952$	0	0
$953$	1521.99	1.59705	0.798524	−	0.601963i	$-0.205615\pi$
$953$	1521.99	1.59705	0.798524	+	0.601963i	$0.205615\pi$
$954$	0	0
$955$	− 217.101i	− 0.227331i
$956$	0	0
$957$	− 777.312i	− 0.812238i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	960.767	0.999757
$962$	0	0
$963$	45.8778	0.0476405
$964$	0	0
$965$	− 207.761i	− 0.215296i
$966$	0	0
$967$	−645.014	−0.667026	−0.333513	−	0.942746i	$-0.608234\pi$
$967$	−645.014	−0.667026	−0.333513	+	0.942746i	$0.608234\pi$
$968$	0	0
$969$	− 132.013i	− 0.136236i
$970$	0	0
$971$	1609.85i	1.65793i	0.559301	+	0.828964i	$0.311069\pi$
$971$	1609.85i	1.65793i	−0.559301	+	0.828964i	$0.688931\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−667.870	−0.684995
$976$	0	0
$977$	−255.143	−0.261150	−0.130575	−	0.991438i	$-0.541682\pi$
$977$	−255.143	−0.261150	−0.130575	+	0.991438i	$0.541682\pi$
$978$	0	0
$979$	− 689.217i	− 0.704001i
$980$	0	0
$981$	162.670	0.165820
$982$	0	0
$983$	− 481.938i	− 0.490272i	−0.969489	−	0.245136i	$-0.921167\pi$
$983$	− 481.938i	− 0.490272i	0.969489	−	0.245136i	$-0.0788327\pi$
$984$	0	0
$985$	− 123.563i	− 0.125445i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−685.673	−0.693299
$990$	0	0
$991$	−1126.22	−1.13644	−0.568222	−	0.822875i	$-0.692368\pi$
$991$	−1126.22	−1.13644	−0.568222	+	0.822875i	$0.692368\pi$
$992$	0	0
$993$	154.186i	0.155273i
$994$	0	0
$995$	−56.3476	−0.0566307
$996$	0	0
$997$	− 1060.71i	− 1.06391i	−0.846774	−	0.531953i	$-0.821458\pi$
$997$	− 1060.71i	− 1.06391i	0.846774	−	0.531953i	$-0.178542\pi$
$998$	0	0
$999$	− 12.8932i	− 0.0129062i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.j.97.7		8
4.3	odd	2		588.3.d.c.97.3	✓	8
7.6	odd	2	inner	2352.3.f.j.97.2		8
12.11	even	2		1764.3.d.h.685.3		8
28.3	even	6		588.3.m.f.313.3		8
28.11	odd	6		588.3.m.e.313.2		8
28.19	even	6		588.3.m.e.325.2		8
28.23	odd	6		588.3.m.f.325.3		8
28.27	even	2		588.3.d.c.97.6	yes	8
84.11	even	6		1764.3.z.m.901.3		8
84.23	even	6		1764.3.z.l.325.2		8
84.47	odd	6		1764.3.z.m.325.3		8
84.59	odd	6		1764.3.z.l.901.2		8
84.83	odd	2		1764.3.d.h.685.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.3	✓	8	4.3	odd	2
588.3.d.c.97.6	yes	8	28.27	even	2
588.3.m.e.313.2		8	28.11	odd	6
588.3.m.e.325.2		8	28.19	even	6
588.3.m.f.313.3		8	28.3	even	6
588.3.m.f.325.3		8	28.23	odd	6
1764.3.d.h.685.3		8	12.11	even	2
1764.3.d.h.685.6		8	84.83	odd	2
1764.3.z.l.325.2		8	84.23	even	6
1764.3.z.l.901.2		8	84.59	odd	6
1764.3.z.m.325.3		8	84.47	odd	6
1764.3.z.m.901.3		8	84.11	even	6
2352.3.f.j.97.2		8	7.6	odd	2	inner
2352.3.f.j.97.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +0.929003i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +0.929003i q^{5} -3.00000 q^{9} -9.68980 q^{11} +15.9753i q^{13} -1.60908 q^{15} +10.5557i q^{17} +7.22049i q^{19} +11.3001 q^{23} +24.1370 q^{25} -5.19615i q^{27} +46.3148 q^{29} +0.483049i q^{31} -16.7832i q^{33} +2.48131 q^{37} -27.6700 q^{39} +55.8520i q^{41} -60.6786 q^{43} -2.78701i q^{45} +36.5867i q^{47} -18.2831 q^{51} -28.5229 q^{53} -9.00185i q^{55} -12.5063 q^{57} -94.0715i q^{59} -110.193i q^{61} -14.8411 q^{65} -82.0309 q^{67} +19.5723i q^{69} -127.349 q^{71} -46.2546i q^{73} +41.8064i q^{75} -18.7003 q^{79} +9.00000 q^{81} +59.6357i q^{83} -9.80632 q^{85} +80.2197i q^{87} +71.1281i q^{89} -0.836665 q^{93} -6.70786 q^{95} +102.239i q^{97} +29.0694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100}) \) 8 * q - 24 * q^9 + 16 * q^23 + 72 * q^25 + 80 * q^29 + 128 * q^37 + 112 * q^43 + 96 * q^51 - 144 * q^53 - 192 * q^57 + 240 * q^65 + 64 * q^67 - 224 * q^71 + 432 * q^79 + 72 * q^81 - 96 * q^85 - 96 * q^93 + 272 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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