LMFDB - Embedded newform 1568.3.c.h.97.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(97,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1568.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (of order $2$, degree $1$, 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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 $ x^16 + 36x^14 + 522x^12 + 3644x^10 + 12219x^8 + 15156x^6 + 15478x^4 - 10992*x^2 + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.2
Root			$-0.707107 + 3.42121i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.h.97.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.83832i q^{3} +0.0515539i q^{5} -14.4094 q^{9} +O(q^{10})$	$q-4.83832i q^{3} +0.0515539i q^{5} -14.4094 q^{9} +1.78993 q^{11} +5.87602i q^{13} +0.249434 q^{15} +26.5867i q^{17} -26.3426i q^{19} -25.6772 q^{23} +24.9973 q^{25} +26.1723i q^{27} -27.1749 q^{29} +29.7046i q^{31} -8.66025i q^{33} -61.7258 q^{37} +28.4301 q^{39} +65.7376i q^{41} -9.52546 q^{43} -0.742859i q^{45} +70.7807i q^{47} +128.635 q^{51} +9.73110 q^{53} +0.0922778i q^{55} -127.454 q^{57} +62.7620i q^{59} +76.4215i q^{61} -0.302932 q^{65} +103.047 q^{67} +124.234i q^{69} +90.1681 q^{71} -33.2767i q^{73} -120.945i q^{75} -64.8645 q^{79} -3.05424 q^{81} +29.1364i q^{83} -1.37065 q^{85} +131.481i q^{87} -21.6725i q^{89} +143.720 q^{93} +1.35807 q^{95} -123.061i q^{97} -25.7918 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 4.83832i	− 1.61277i	−0.591388	−	0.806387i	$-0.701420\pi$
$3$	− 4.83832i	− 1.61277i	0.591388	−	0.806387i	$-0.298580\pi$
$4$	0	0
$5$	0.0515539i	0.0103108i	0.999987	+	0.00515539i	$0.00164102\pi$
$5$	0.0515539i	0.0103108i	−0.999987	+	0.00515539i	$0.998359\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−14.4094	−1.60104
$10$	0	0
$11$	1.78993	0.162721	0.0813604	−	0.996685i	$-0.474074\pi$
$11$	1.78993	0.162721	0.0813604	+	0.996685i	$0.474074\pi$
$12$	0	0
$13$	5.87602i	0.452002i	0.974127	+	0.226001i	$0.0725652\pi$
$13$	5.87602i	0.452002i	−0.974127	+	0.226001i	$0.927435\pi$
$14$	0	0
$15$	0.249434	0.0166290
$16$	0	0
$17$	26.5867i	1.56392i	0.623326	+	0.781962i	$0.285781\pi$
$17$	26.5867i	1.56392i	−0.623326	+	0.781962i	$0.714219\pi$
$18$	0	0
$19$	− 26.3426i	− 1.38645i	−0.720719	−	0.693227i	$-0.756188\pi$
$19$	− 26.3426i	− 1.38645i	0.720719	−	0.693227i	$-0.243812\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−25.6772	−1.11640	−0.558199	−	0.829707i	$-0.688508\pi$
$23$	−25.6772	−1.11640	−0.558199	+	0.829707i	$0.688508\pi$
$24$	0	0
$25$	24.9973	0.999894
$26$	0	0
$27$	26.1723i	0.969345i
$28$	0	0
$29$	−27.1749	−0.937066	−0.468533	−	0.883446i	$-0.655217\pi$
$29$	−27.1749	−0.937066	−0.468533	+	0.883446i	$0.655217\pi$
$30$	0	0
$31$	29.7046i	0.958213i	0.877757	+	0.479106i	$0.159039\pi$
$31$	29.7046i	0.958213i	−0.877757	+	0.479106i	$0.840961\pi$
$32$	0	0
$33$	− 8.66025i	− 0.262432i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−61.7258	−1.66826	−0.834132	−	0.551565i	$-0.814031\pi$
$37$	−61.7258	−1.66826	−0.834132	+	0.551565i	$0.814031\pi$
$38$	0	0
$39$	28.4301	0.728977
$40$	0	0
$41$	65.7376i	1.60336i	0.597755	+	0.801679i	$0.296059\pi$
$41$	65.7376i	1.60336i	−0.597755	+	0.801679i	$0.703941\pi$
$42$	0	0
$43$	−9.52546	−0.221522	−0.110761	−	0.993847i	$-0.535329\pi$
$43$	−9.52546	−0.221522	−0.110761	+	0.993847i	$0.535329\pi$
$44$	0	0
$45$	− 0.742859i	− 0.0165080i
$46$	0	0
$47$	70.7807i	1.50597i	0.658037	+	0.752986i	$0.271387\pi$
$47$	70.7807i	1.50597i	−0.658037	+	0.752986i	$0.728613\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	128.635	2.52226
$52$	0	0
$53$	9.73110	0.183606	0.0918028	−	0.995777i	$-0.470737\pi$
$53$	9.73110	0.183606	0.0918028	+	0.995777i	$0.470737\pi$
$54$	0	0
$55$	0.0922778i	0.00167778i
$56$	0	0
$57$	−127.454	−2.23604
$58$	0	0
$59$	62.7620i	1.06376i	0.846819	+	0.531881i	$0.178515\pi$
$59$	62.7620i	1.06376i	−0.846819	+	0.531881i	$0.821485\pi$
$60$	0	0
$61$	76.4215i	1.25281i	0.779497	+	0.626406i	$0.215475\pi$
$61$	76.4215i	1.25281i	−0.779497	+	0.626406i	$0.784525\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.302932	−0.00466049
$66$	0	0
$67$	103.047	1.53802	0.769009	−	0.639238i	$-0.220750\pi$
$67$	103.047	1.53802	0.769009	+	0.639238i	$0.220750\pi$
$68$	0	0
$69$	124.234i	1.80050i
$70$	0	0
$71$	90.1681	1.26997	0.634986	−	0.772523i	$-0.281006\pi$
$71$	90.1681	1.26997	0.634986	+	0.772523i	$0.281006\pi$
$72$	0	0
$73$	− 33.2767i	− 0.455845i	−0.973679	−	0.227922i	$-0.926807\pi$
$73$	− 33.2767i	− 0.455845i	0.973679	−	0.227922i	$-0.0731933\pi$
$74$	0	0
$75$	− 120.945i	− 1.61260i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−64.8645	−0.821070	−0.410535	−	0.911845i	$-0.634658\pi$
$79$	−64.8645	−0.821070	−0.410535	+	0.911845i	$0.634658\pi$
$80$	0	0
$81$	−3.05424	−0.0377067
$82$	0	0
$83$	29.1364i	0.351041i	0.984476	+	0.175520i	$0.0561608\pi$
$83$	29.1364i	0.351041i	−0.984476	+	0.175520i	$0.943839\pi$
$84$	0	0
$85$	−1.37065	−0.0161253
$86$	0	0
$87$	131.481i	1.51128i
$88$	0	0
$89$	− 21.6725i	− 0.243511i	−0.992560	−	0.121755i	$-0.961148\pi$
$89$	− 21.6725i	− 0.243511i	0.992560	−	0.121755i	$-0.0388523\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	143.720	1.54538
$94$	0	0
$95$	1.35807	0.0142954
$96$	0	0
$97$	− 123.061i	− 1.26867i	−0.773056	−	0.634337i	$-0.781273\pi$
$97$	− 123.061i	− 1.26867i	0.773056	−	0.634337i	$-0.218727\pi$
$98$	0	0
$99$	−25.7918	−0.260523
$100$	0	0
$101$	− 56.5276i	− 0.559679i	−0.960047	−	0.279840i	$-0.909719\pi$
$101$	− 56.5276i	− 0.559679i	0.960047	−	0.279840i	$-0.0902813\pi$
$102$	0	0
$103$	− 29.5675i	− 0.287063i	−0.989646	−	0.143532i	$-0.954154\pi$
$103$	− 29.5675i	− 0.287063i	0.989646	−	0.143532i	$-0.0458459\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−29.8108	−0.278605	−0.139303	−	0.990250i	$-0.544486\pi$
$107$	−29.8108	−0.278605	−0.139303	+	0.990250i	$0.544486\pi$
$108$	0	0
$109$	81.5503	0.748167	0.374084	−	0.927395i	$-0.377957\pi$
$109$	81.5503	0.748167	0.374084	+	0.927395i	$0.377957\pi$
$110$	0	0
$111$	298.649i	2.69053i
$112$	0	0
$113$	43.3994	0.384065	0.192033	−	0.981389i	$-0.438492\pi$
$113$	43.3994	0.384065	0.192033	+	0.981389i	$0.438492\pi$
$114$	0	0
$115$	− 1.32376i	− 0.0115109i
$116$	0	0
$117$	− 84.6698i	− 0.723674i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−117.796	−0.973522
$122$	0	0
$123$	318.060	2.58585
$124$	0	0
$125$	2.57756i	0.0206205i
$126$	0	0
$127$	−94.0304	−0.740397	−0.370199	−	0.928953i	$-0.620710\pi$
$127$	−94.0304	−0.740397	−0.370199	+	0.928953i	$0.620710\pi$
$128$	0	0
$129$	46.0873i	0.357266i
$130$	0	0
$131$	− 125.784i	− 0.960180i	−0.877219	−	0.480090i	$-0.840604\pi$
$131$	− 125.784i	− 0.960180i	0.877219	−	0.480090i	$-0.159396\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.34928	−0.00999470
$136$	0	0
$137$	−249.856	−1.82377	−0.911884	−	0.410448i	$-0.865372\pi$
$137$	−249.856	−1.82377	−0.911884	+	0.410448i	$0.865372\pi$
$138$	0	0
$139$	2.08301i	0.0149857i	0.999972	+	0.00749284i	$0.00238507\pi$
$139$	2.08301i	0.0149857i	−0.999972	+	0.00749284i	$0.997615\pi$
$140$	0	0
$141$	342.460	2.42879
$142$	0	0
$143$	10.5177i	0.0735501i
$144$	0	0
$145$	− 1.40097i	− 0.00966187i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.6244	−0.145130	−0.0725652	−	0.997364i	$-0.523119\pi$
$149$	−21.6244	−0.145130	−0.0725652	+	0.997364i	$0.523119\pi$
$150$	0	0
$151$	−88.0180	−0.582900	−0.291450	−	0.956586i	$-0.594138\pi$
$151$	−88.0180	−0.582900	−0.291450	+	0.956586i	$0.594138\pi$
$152$	0	0
$153$	− 383.098i	− 2.50391i
$154$	0	0
$155$	−1.53139	−0.00987992
$156$	0	0
$157$	164.623i	1.04855i	0.851548	+	0.524277i	$0.175664\pi$
$157$	164.623i	1.04855i	−0.851548	+	0.524277i	$0.824336\pi$
$158$	0	0
$159$	− 47.0822i	− 0.296114i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	164.884	1.01156	0.505779	−	0.862663i	$-0.331205\pi$
$163$	164.884	1.01156	0.505779	+	0.862663i	$0.331205\pi$
$164$	0	0
$165$	0.446470	0.00270588
$166$	0	0
$167$	− 18.8929i	− 0.113131i	−0.998399	−	0.0565655i	$-0.981985\pi$
$167$	− 18.8929i	− 0.113131i	0.998399	−	0.0565655i	$-0.0180150\pi$
$168$	0	0
$169$	134.472	0.795695
$170$	0	0
$171$	379.581i	2.21977i
$172$	0	0
$173$	98.4856i	0.569281i	0.958634	+	0.284641i	$0.0918742\pi$
$173$	98.4856i	0.569281i	−0.958634	+	0.284641i	$0.908126\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	303.663	1.71561
$178$	0	0
$179$	70.2963	0.392717	0.196358	−	0.980532i	$-0.437088\pi$
$179$	70.2963	0.392717	0.196358	+	0.980532i	$0.437088\pi$
$180$	0	0
$181$	− 204.167i	− 1.12800i	−0.825776	−	0.563999i	$-0.809262\pi$
$181$	− 204.167i	− 1.12800i	0.825776	−	0.563999i	$-0.190738\pi$
$182$	0	0
$183$	369.752	2.02050
$184$	0	0
$185$	− 3.18220i	− 0.0172011i
$186$	0	0
$187$	47.5883i	0.254483i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	190.703	0.998447	0.499224	−	0.866473i	$-0.333619\pi$
$191$	190.703	0.998447	0.499224	+	0.866473i	$0.333619\pi$
$192$	0	0
$193$	85.5481	0.443254	0.221627	−	0.975131i	$-0.428863\pi$
$193$	85.5481	0.443254	0.221627	+	0.975131i	$0.428863\pi$
$194$	0	0
$195$	1.46568i	0.00751631i
$196$	0	0
$197$	275.164	1.39677	0.698386	−	0.715721i	$-0.253902\pi$
$197$	275.164	1.39677	0.698386	+	0.715721i	$0.253902\pi$
$198$	0	0
$199$	206.878i	1.03959i	0.854292	+	0.519793i	$0.173991\pi$
$199$	206.878i	1.03959i	−0.854292	+	0.519793i	$0.826009\pi$
$200$	0	0
$201$	− 498.576i	− 2.48048i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.38903	−0.0165319
$206$	0	0
$207$	369.992	1.78740
$208$	0	0
$209$	− 47.1514i	− 0.225605i
$210$	0	0
$211$	78.7325	0.373140	0.186570	−	0.982442i	$-0.440263\pi$
$211$	78.7325	0.373140	0.186570	+	0.982442i	$0.440263\pi$
$212$	0	0
$213$	− 436.262i	− 2.04818i
$214$	0	0
$215$	− 0.491074i	− 0.00228407i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−161.003	−0.735175
$220$	0	0
$221$	−156.224	−0.706896
$222$	0	0
$223$	155.408i	0.696896i	0.937328	+	0.348448i	$0.113291\pi$
$223$	155.408i	0.696896i	−0.937328	+	0.348448i	$0.886709\pi$
$224$	0	0
$225$	−360.196	−1.60087
$226$	0	0
$227$	− 81.8668i	− 0.360647i	−0.983607	−	0.180323i	$-0.942286\pi$
$227$	− 81.8668i	− 0.360647i	0.983607	−	0.180323i	$-0.0577144\pi$
$228$	0	0
$229$	410.692i	1.79341i	0.442625	+	0.896707i	$0.354047\pi$
$229$	410.692i	1.79341i	−0.442625	+	0.896707i	$0.645953\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	37.8718	0.162540	0.0812699	−	0.996692i	$-0.474102\pi$
$233$	37.8718	0.162540	0.0812699	+	0.996692i	$0.474102\pi$
$234$	0	0
$235$	−3.64902	−0.0155277
$236$	0	0
$237$	313.836i	1.32420i
$238$	0	0
$239$	−428.133	−1.79135	−0.895676	−	0.444707i	$-0.853308\pi$
$239$	−428.133	−1.79135	−0.895676	+	0.444707i	$0.853308\pi$
$240$	0	0
$241$	208.025i	0.863174i	0.902071	+	0.431587i	$0.142046\pi$
$241$	208.025i	0.863174i	−0.902071	+	0.431587i	$0.857954\pi$
$242$	0	0
$243$	250.328i	1.03016i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	154.790	0.626680
$248$	0	0
$249$	140.971	0.566149
$250$	0	0
$251$	375.625i	1.49652i	0.663408	+	0.748258i	$0.269109\pi$
$251$	375.625i	1.49652i	−0.663408	+	0.748258i	$0.730891\pi$
$252$	0	0
$253$	−45.9603	−0.181661
$254$	0	0
$255$	6.63164i	0.0260064i
$256$	0	0
$257$	72.0964i	0.280531i	0.990114	+	0.140265i	$0.0447956\pi$
$257$	72.0964i	0.280531i	−0.990114	+	0.140265i	$0.955204\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	391.574	1.50028
$262$	0	0
$263$	209.100	0.795058	0.397529	−	0.917590i	$-0.369868\pi$
$263$	209.100	0.795058	0.397529	+	0.917590i	$0.369868\pi$
$264$	0	0
$265$	0.501676i	0.00189312i
$266$	0	0
$267$	−104.858	−0.392728
$268$	0	0
$269$	− 282.850i	− 1.05149i	−0.850643	−	0.525744i	$-0.823787\pi$
$269$	− 282.850i	− 1.05149i	0.850643	−	0.525744i	$-0.176213\pi$
$270$	0	0
$271$	− 197.927i	− 0.730359i	−0.930937	−	0.365179i	$-0.881008\pi$
$271$	− 197.927i	− 0.730359i	0.930937	−	0.365179i	$-0.118992\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	44.7435	0.162703
$276$	0	0
$277$	−237.318	−0.856745	−0.428372	−	0.903602i	$-0.640913\pi$
$277$	−237.318	−0.856745	−0.428372	+	0.903602i	$0.640913\pi$
$278$	0	0
$279$	− 428.025i	− 1.53414i
$280$	0	0
$281$	−239.870	−0.853628	−0.426814	−	0.904339i	$-0.640364\pi$
$281$	−239.870	−0.853628	−0.426814	+	0.904339i	$0.640364\pi$
$282$	0	0
$283$	− 138.174i	− 0.488249i	−0.969744	−	0.244124i	$-0.921499\pi$
$283$	− 138.174i	− 0.488249i	0.969744	−	0.244124i	$-0.0785005\pi$
$284$	0	0
$285$	− 6.57076i	− 0.0230553i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−417.853	−1.44586
$290$	0	0
$291$	−595.411	−2.04609
$292$	0	0
$293$	385.332i	1.31513i	0.753400	+	0.657563i	$0.228413\pi$
$293$	385.332i	1.31513i	−0.753400	+	0.657563i	$0.771587\pi$
$294$	0	0
$295$	−3.23562	−0.0109682
$296$	0	0
$297$	46.8466i	0.157733i
$298$	0	0
$299$	− 150.880i	− 0.504614i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−273.499	−0.902636
$304$	0	0
$305$	−3.93983	−0.0129175
$306$	0	0
$307$	− 222.533i	− 0.724864i	−0.932010	−	0.362432i	$-0.881946\pi$
$307$	− 222.533i	− 0.724864i	0.932010	−	0.362432i	$-0.118054\pi$
$308$	0	0
$309$	−143.057	−0.462968
$310$	0	0
$311$	− 198.425i	− 0.638022i	−0.947751	−	0.319011i	$-0.896649\pi$
$311$	− 198.425i	− 0.638022i	0.947751	−	0.319011i	$-0.103351\pi$
$312$	0	0
$313$	379.457i	1.21232i	0.795342	+	0.606161i	$0.207291\pi$
$313$	379.457i	1.21232i	−0.795342	+	0.606161i	$0.792709\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−259.626	−0.819010	−0.409505	−	0.912308i	$-0.634299\pi$
$317$	−259.626	−0.819010	−0.409505	+	0.912308i	$0.634299\pi$
$318$	0	0
$319$	−48.6411	−0.152480
$320$	0	0
$321$	144.234i	0.449327i
$322$	0	0
$323$	700.364	2.16831
$324$	0	0
$325$	146.885i	0.451954i
$326$	0	0
$327$	− 394.567i	− 1.20663i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−547.177	−1.65310	−0.826552	−	0.562861i	$-0.809701\pi$
$331$	−547.177	−1.65310	−0.826552	+	0.562861i	$0.809701\pi$
$332$	0	0
$333$	889.430	2.67096
$334$	0	0
$335$	5.31248i	0.0158582i
$336$	0	0
$337$	−408.705	−1.21277	−0.606387	−	0.795170i	$-0.707382\pi$
$337$	−408.705	−1.21277	−0.606387	+	0.795170i	$0.707382\pi$
$338$	0	0
$339$	− 209.980i	− 0.619411i
$340$	0	0
$341$	53.1691i	0.155921i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.40477	−0.0185645
$346$	0	0
$347$	−267.149	−0.769883	−0.384941	−	0.922941i	$-0.625778\pi$
$347$	−267.149	−0.769883	−0.384941	+	0.922941i	$0.625778\pi$
$348$	0	0
$349$	466.080i	1.33547i	0.744398	+	0.667736i	$0.232737\pi$
$349$	466.080i	1.33547i	−0.744398	+	0.667736i	$0.767263\pi$
$350$	0	0
$351$	−153.789	−0.438146
$352$	0	0
$353$	265.818i	0.753026i	0.926411	+	0.376513i	$0.122877\pi$
$353$	265.818i	0.753026i	−0.926411	+	0.376513i	$0.877123\pi$
$354$	0	0
$355$	4.64851i	0.0130944i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	289.805	0.807257	0.403628	−	0.914923i	$-0.367749\pi$
$359$	289.805	0.807257	0.403628	+	0.914923i	$0.367749\pi$
$360$	0	0
$361$	−332.935	−0.922257
$362$	0	0
$363$	569.936i	1.57007i
$364$	0	0
$365$	1.71554	0.00470011
$366$	0	0
$367$	7.01434i	0.0191126i	0.999954	+	0.00955632i	$0.00304192\pi$
$367$	7.01434i	0.0191126i	−0.999954	+	0.00955632i	$0.996958\pi$
$368$	0	0
$369$	− 947.239i	− 2.56704i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−573.997	−1.53887	−0.769433	−	0.638728i	$-0.779461\pi$
$373$	−573.997	−1.53887	−0.769433	+	0.638728i	$0.779461\pi$
$374$	0	0
$375$	12.4711	0.0332561
$376$	0	0
$377$	− 159.680i	− 0.423555i
$378$	0	0
$379$	−678.807	−1.79105	−0.895524	−	0.445014i	$-0.853199\pi$
$379$	−678.807	−1.79105	−0.895524	+	0.445014i	$0.853199\pi$
$380$	0	0
$381$	454.950i	1.19409i
$382$	0	0
$383$	413.896i	1.08067i	0.841450	+	0.540334i	$0.181702\pi$
$383$	413.896i	1.08067i	−0.841450	+	0.540334i	$0.818298\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	137.256	0.354667
$388$	0	0
$389$	138.488	0.356009	0.178005	−	0.984030i	$-0.443036\pi$
$389$	138.488	0.356009	0.178005	+	0.984030i	$0.443036\pi$
$390$	0	0
$391$	− 682.672i	− 1.74596i
$392$	0	0
$393$	−608.582	−1.54855
$394$	0	0
$395$	− 3.34402i	− 0.00846587i
$396$	0	0
$397$	− 4.18005i	− 0.0105291i	−0.999986	−	0.00526455i	$-0.998324\pi$
$397$	− 4.18005i	− 0.0105291i	0.999986	−	0.00526455i	$-0.00167576\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−602.054	−1.50138	−0.750690	−	0.660654i	$-0.770279\pi$
$401$	−602.054	−1.50138	−0.750690	+	0.660654i	$0.770279\pi$
$402$	0	0
$403$	−174.545	−0.433114
$404$	0	0
$405$	− 0.157458i	0 0.000388785i
$406$	0	0
$407$	−110.485	−0.271461
$408$	0	0
$409$	− 179.293i	− 0.438368i	−0.975684	−	0.219184i	$-0.929660\pi$
$409$	− 179.293i	− 0.438368i	0.975684	−	0.219184i	$-0.0703395\pi$
$410$	0	0
$411$	1208.89i	2.94133i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.50209	−0.00361950
$416$	0	0
$417$	10.0783	0.0241685
$418$	0	0
$419$	− 605.541i	− 1.44521i	−0.691263	−	0.722603i	$-0.742946\pi$
$419$	− 605.541i	− 1.44521i	0.691263	−	0.722603i	$-0.257054\pi$
$420$	0	0
$421$	582.852	1.38445	0.692224	−	0.721683i	$-0.256631\pi$
$421$	582.852	1.38445	0.692224	+	0.721683i	$0.256631\pi$
$422$	0	0
$423$	− 1019.91i	− 2.41112i
$424$	0	0
$425$	664.597i	1.56376i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	50.8878	0.118620
$430$	0	0
$431$	275.405	0.638990	0.319495	−	0.947588i	$-0.396487\pi$
$431$	275.405	0.638990	0.319495	+	0.947588i	$0.396487\pi$
$432$	0	0
$433$	− 27.7972i	− 0.0641967i	−0.999485	−	0.0320984i	$-0.989781\pi$
$433$	− 27.7972i	− 0.0641967i	0.999485	−	0.0320984i	$-0.0102190\pi$
$434$	0	0
$435$	−6.77836	−0.0155824
$436$	0	0
$437$	676.404i	1.54784i
$438$	0	0
$439$	− 294.160i	− 0.670067i	−0.942206	−	0.335034i	$-0.891252\pi$
$439$	− 294.160i	− 0.670067i	0.942206	−	0.335034i	$-0.108748\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−290.890	−0.656637	−0.328319	−	0.944567i	$-0.606482\pi$
$443$	−290.890	−0.656637	−0.328319	+	0.944567i	$0.606482\pi$
$444$	0	0
$445$	1.11730	0.00251079
$446$	0	0
$447$	104.626i	0.234063i
$448$	0	0
$449$	433.407	0.965271	0.482635	−	0.875821i	$-0.339680\pi$
$449$	433.407	0.965271	0.482635	+	0.875821i	$0.339680\pi$
$450$	0	0
$451$	117.666i	0.260900i
$452$	0	0
$453$	425.859i	0.940087i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−57.3656	−0.125526	−0.0627632	−	0.998028i	$-0.519991\pi$
$457$	−57.3656	−0.125526	−0.0627632	+	0.998028i	$0.519991\pi$
$458$	0	0
$459$	−695.836	−1.51598
$460$	0	0
$461$	567.060i	1.23007i	0.788501	+	0.615033i	$0.210857\pi$
$461$	567.060i	1.23007i	−0.788501	+	0.615033i	$0.789143\pi$
$462$	0	0
$463$	−0.548177	−0.00118397	−0.000591984	−	1.00000i	$-0.500188\pi$
$463$	−0.548177	−0.00118397	−0.000591984		1.00000i	$0.500188\pi$
$464$	0	0
$465$	7.40935i	0.0159341i
$466$	0	0
$467$	102.209i	0.218862i	0.993994	+	0.109431i	$0.0349029\pi$
$467$	102.209i	0.218862i	−0.993994	+	0.109431i	$0.965097\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	796.499	1.69108
$472$	0	0
$473$	−17.0499	−0.0360463
$474$	0	0
$475$	− 658.496i	− 1.38631i
$476$	0	0
$477$	−140.219	−0.293960
$478$	0	0
$479$	450.134i	0.939738i	0.882736	+	0.469869i	$0.155699\pi$
$479$	450.134i	0.939738i	−0.882736	+	0.469869i	$0.844301\pi$
$480$	0	0
$481$	− 362.702i	− 0.754058i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.34429	0.0130810
$486$	0	0
$487$	−940.772	−1.93177	−0.965885	−	0.258972i	$-0.916616\pi$
$487$	−940.772	−1.93177	−0.965885	+	0.258972i	$0.916616\pi$
$488$	0	0
$489$	− 797.762i	− 1.63142i
$490$	0	0
$491$	514.670	1.04821	0.524103	−	0.851655i	$-0.324401\pi$
$491$	514.670	1.04821	0.524103	+	0.851655i	$0.324401\pi$
$492$	0	0
$493$	− 722.492i	− 1.46550i
$494$	0	0
$495$	− 1.32966i	− 0.00268619i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−248.007	−0.497008	−0.248504	−	0.968631i	$-0.579939\pi$
$499$	−248.007	−0.497008	−0.248504	+	0.968631i	$0.579939\pi$
$500$	0	0
$501$	−91.4099	−0.182455
$502$	0	0
$503$	164.798i	0.327630i	0.986491	+	0.163815i	$0.0523800\pi$
$503$	164.798i	0.327630i	−0.986491	+	0.163815i	$0.947620\pi$
$504$	0	0
$505$	2.91422	0.00577073
$506$	0	0
$507$	− 650.621i	− 1.28328i
$508$	0	0
$509$	− 126.488i	− 0.248502i	−0.992251	−	0.124251i	$-0.960347\pi$
$509$	− 126.488i	− 0.248502i	0.992251	−	0.124251i	$-0.0396529\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	689.448	1.34395
$514$	0	0
$515$	1.52432	0.00295984
$516$	0	0
$517$	126.692i	0.245053i
$518$	0	0
$519$	476.505	0.918122
$520$	0	0
$521$	− 51.6797i	− 0.0991932i	−0.998769	−	0.0495966i	$-0.984206\pi$
$521$	− 51.6797i	− 0.0991932i	0.998769	−	0.0495966i	$-0.0157936\pi$
$522$	0	0
$523$	− 343.615i	− 0.657008i	−0.944503	−	0.328504i	$-0.893456\pi$
$523$	− 343.615i	− 0.657008i	0.944503	−	0.328504i	$-0.106544\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−789.748	−1.49857
$528$	0	0
$529$	130.317	0.246346
$530$	0	0
$531$	− 904.361i	− 1.70313i
$532$	0	0
$533$	−386.276	−0.724720
$534$	0	0
$535$	− 1.53686i	− 0.00287264i
$536$	0	0
$537$	− 340.116i	− 0.633363i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	764.013	1.41222	0.706111	−	0.708101i	$-0.250448\pi$
$541$	764.013	1.41222	0.706111	+	0.708101i	$0.250448\pi$
$542$	0	0
$543$	−987.828	−1.81921
$544$	0	0
$545$	4.20423i	0.00771419i
$546$	0	0
$547$	−59.3354	−0.108474	−0.0542371	−	0.998528i	$-0.517273\pi$
$547$	−59.3354	−0.108474	−0.0542371	+	0.998528i	$0.517273\pi$
$548$	0	0
$549$	− 1101.19i	− 2.00580i
$550$	0	0
$551$	715.859i	1.29920i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−15.3965	−0.0277415
$556$	0	0
$557$	−730.096	−1.31076	−0.655382	−	0.755298i	$-0.727492\pi$
$557$	−730.096	−1.31076	−0.655382	+	0.755298i	$0.727492\pi$
$558$	0	0
$559$	− 55.9718i	− 0.100128i
$560$	0	0
$561$	230.248	0.410424
$562$	0	0
$563$	− 476.215i	− 0.845853i	−0.906164	−	0.422927i	$-0.861003\pi$
$563$	− 476.215i	− 0.845853i	0.906164	−	0.422927i	$-0.138997\pi$
$564$	0	0
$565$	2.23741i	0.00396001i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	417.402	0.733570	0.366785	−	0.930306i	$-0.380458\pi$
$569$	417.402	0.733570	0.366785	+	0.930306i	$0.380458\pi$
$570$	0	0
$571$	984.002	1.72330	0.861648	−	0.507506i	$-0.169433\pi$
$571$	984.002	1.72330	0.861648	+	0.507506i	$0.169433\pi$
$572$	0	0
$573$	− 922.685i	− 1.61027i
$574$	0	0
$575$	−641.861	−1.11628
$576$	0	0
$577$	92.7876i	0.160810i	0.996762	+	0.0804052i	$0.0256214\pi$
$577$	92.7876i	0.160810i	−0.996762	+	0.0804052i	$0.974379\pi$
$578$	0	0
$579$	− 413.909i	− 0.714869i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.4180	0.0298764
$584$	0	0
$585$	4.36506	0.00746163
$586$	0	0
$587$	648.667i	1.10506i	0.833495	+	0.552528i	$0.186337\pi$
$587$	648.667i	1.10506i	−0.833495	+	0.552528i	$0.813663\pi$
$588$	0	0
$589$	782.498	1.32852
$590$	0	0
$591$	− 1331.33i	− 2.25268i
$592$	0	0
$593$	523.532i	0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$593$	523.532i	0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1000.94	1.67662
$598$	0	0
$599$	−891.045	−1.48755	−0.743777	−	0.668428i	$-0.766968\pi$
$599$	−891.045	−1.48755	−0.743777	+	0.668428i	$0.766968\pi$
$600$	0	0
$601$	502.034i	0.835331i	0.908601	+	0.417666i	$0.137152\pi$
$601$	502.034i	0.835331i	−0.908601	+	0.417666i	$0.862848\pi$
$602$	0	0
$603$	−1484.85	−2.46243
$604$	0	0
$605$	− 6.07285i	− 0.0100378i
$606$	0	0
$607$	− 304.423i	− 0.501521i	−0.968049	−	0.250760i	$-0.919319\pi$
$607$	− 304.423i	− 0.501521i	0.968049	−	0.250760i	$-0.0806806\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−415.909	−0.680701
$612$	0	0
$613$	−498.106	−0.812572	−0.406286	−	0.913746i	$-0.633176\pi$
$613$	−498.106	−0.812572	−0.406286	+	0.913746i	$0.633176\pi$
$614$	0	0
$615$	16.3972i	0.0266622i
$616$	0	0
$617$	242.250	0.392626	0.196313	−	0.980541i	$-0.437103\pi$
$617$	242.250	0.392626	0.196313	+	0.980541i	$0.437103\pi$
$618$	0	0
$619$	− 540.597i	− 0.873339i	−0.899622	−	0.436669i	$-0.856158\pi$
$619$	− 540.597i	− 0.873339i	0.899622	−	0.436669i	$-0.143842\pi$
$620$	0	0
$621$	− 672.031i	− 1.08218i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	624.801	0.999681
$626$	0	0
$627$	−228.134	−0.363850
$628$	0	0
$629$	− 1641.09i	− 2.60904i
$630$	0	0
$631$	−808.138	−1.28073	−0.640363	−	0.768073i	$-0.721216\pi$
$631$	−808.138	−1.28073	−0.640363	+	0.768073i	$0.721216\pi$
$632$	0	0
$633$	− 380.933i	− 0.601790i
$634$	0	0
$635$	− 4.84763i	− 0.00763407i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1299.27	−2.03328
$640$	0	0
$641$	310.553	0.484482	0.242241	−	0.970216i	$-0.422117\pi$
$641$	310.553	0.484482	0.242241	+	0.970216i	$0.422117\pi$
$642$	0	0
$643$	− 342.761i	− 0.533065i	−0.963826	−	0.266533i	$-0.914122\pi$
$643$	− 342.761i	− 0.533065i	0.963826	−	0.266533i	$-0.0858780\pi$
$644$	0	0
$645$	−2.37598	−0.00368369
$646$	0	0
$647$	− 516.908i	− 0.798931i	−0.916748	−	0.399466i	$-0.869196\pi$
$647$	− 516.908i	− 0.798931i	0.916748	−	0.399466i	$-0.130804\pi$
$648$	0	0
$649$	112.340i	0.173096i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−763.553	−1.16930	−0.584650	−	0.811286i	$-0.698768\pi$
$653$	−763.553	−1.16930	−0.584650	+	0.811286i	$0.698768\pi$
$654$	0	0
$655$	6.48463	0.00990020
$656$	0	0
$657$	479.496i	0.729827i
$658$	0	0
$659$	593.617	0.900785	0.450392	−	0.892831i	$-0.351284\pi$
$659$	593.617	0.900785	0.450392	+	0.892831i	$0.351284\pi$
$660$	0	0
$661$	− 317.730i	− 0.480680i	−0.970689	−	0.240340i	$-0.922741\pi$
$661$	− 317.730i	− 0.480680i	0.970689	−	0.240340i	$-0.0772590\pi$
$662$	0	0
$663$	755.863i	1.14006i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	697.775	1.04614
$668$	0	0
$669$	751.914	1.12394
$670$	0	0
$671$	136.789i	0.203859i
$672$	0	0
$673$	567.441	0.843152	0.421576	−	0.906793i	$-0.361477\pi$
$673$	567.441	0.843152	0.421576	+	0.906793i	$0.361477\pi$
$674$	0	0
$675$	654.238i	0.969242i
$676$	0	0
$677$	624.802i	0.922898i	0.887167	+	0.461449i	$0.152670\pi$
$677$	624.802i	0.922898i	−0.887167	+	0.461449i	$0.847330\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−396.098	−0.581642
$682$	0	0
$683$	−92.8849	−0.135995	−0.0679977	−	0.997685i	$-0.521661\pi$
$683$	−92.8849	−0.135995	−0.0679977	+	0.997685i	$0.521661\pi$
$684$	0	0
$685$	− 12.8811i	− 0.0188045i
$686$	0	0
$687$	1987.06	2.89237
$688$	0	0
$689$	57.1801i	0.0829900i
$690$	0	0
$691$	− 487.730i	− 0.705832i	−0.935655	−	0.352916i	$-0.885190\pi$
$691$	− 487.730i	− 0.705832i	0.935655	−	0.352916i	$-0.114810\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.107387	−0.000154514 0
$696$	0	0
$697$	−1747.75	−2.50753
$698$	0	0
$699$	− 183.236i	− 0.262140i
$700$	0	0
$701$	548.723	0.782772	0.391386	−	0.920227i	$-0.371996\pi$
$701$	548.723	0.782772	0.391386	+	0.920227i	$0.371996\pi$
$702$	0	0
$703$	1626.02i	2.31297i
$704$	0	0
$705$	17.6551i	0.0250427i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−251.772	−0.355109	−0.177555	−	0.984111i	$-0.556819\pi$
$709$	−251.772	−0.355109	−0.177555	+	0.984111i	$0.556819\pi$
$710$	0	0
$711$	934.658	1.31457
$712$	0	0
$713$	− 762.730i	− 1.06975i
$714$	0	0
$715$	−0.542226	−0.000758358 0
$716$	0	0
$717$	2071.45i	2.88905i
$718$	0	0
$719$	402.578i	0.559914i	0.960013	+	0.279957i	$0.0903202\pi$
$719$	402.578i	0.559914i	−0.960013	+	0.279957i	$0.909680\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1006.49	1.39210
$724$	0	0
$725$	−679.300	−0.936966
$726$	0	0
$727$	− 419.973i	− 0.577680i	−0.957377	−	0.288840i	$-0.906730\pi$
$727$	− 419.973i	− 0.577680i	0.957377	−	0.288840i	$-0.0932695\pi$
$728$	0	0
$729$	1183.68	1.62371
$730$	0	0
$731$	− 253.251i	− 0.346444i
$732$	0	0
$733$	− 441.727i	− 0.602628i	−0.953525	−	0.301314i	$-0.902575\pi$
$733$	− 441.727i	− 0.602628i	0.953525	−	0.301314i	$-0.0974253\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	184.447	0.250268
$738$	0	0
$739$	−439.053	−0.594117	−0.297059	−	0.954859i	$-0.596006\pi$
$739$	−439.053	−0.594117	−0.297059	+	0.954859i	$0.596006\pi$
$740$	0	0
$741$	− 748.924i	− 1.01069i
$742$	0	0
$743$	882.165	1.18730	0.593651	−	0.804723i	$-0.297686\pi$
$743$	882.165	1.18730	0.593651	+	0.804723i	$0.297686\pi$
$744$	0	0
$745$	− 1.11482i	− 0.00149641i
$746$	0	0
$747$	− 419.837i	− 0.562031i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	115.636	0.153975	0.0769877	−	0.997032i	$-0.475470\pi$
$751$	115.636	0.153975	0.0769877	+	0.997032i	$0.475470\pi$
$752$	0	0
$753$	1817.40	2.41354
$754$	0	0
$755$	− 4.53767i	− 0.00601015i
$756$	0	0
$757$	−885.222	−1.16938	−0.584691	−	0.811256i	$-0.698784\pi$
$757$	−885.222	−1.16938	−0.584691	+	0.811256i	$0.698784\pi$
$758$	0	0
$759$	222.371i	0.292979i
$760$	0	0
$761$	− 554.033i	− 0.728033i	−0.931392	−	0.364017i	$-0.881405\pi$
$761$	− 554.033i	− 0.728033i	0.931392	−	0.364017i	$-0.118595\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	19.7502	0.0258172
$766$	0	0
$767$	−368.791	−0.480823
$768$	0	0
$769$	502.045i	0.652854i	0.945222	+	0.326427i	$0.105845\pi$
$769$	502.045i	0.652854i	−0.945222	+	0.326427i	$0.894155\pi$
$770$	0	0
$771$	348.826	0.452433
$772$	0	0
$773$	− 1188.92i	− 1.53806i	−0.639211	−	0.769032i	$-0.720739\pi$
$773$	− 1188.92i	− 1.53806i	0.639211	−	0.769032i	$-0.279261\pi$
$774$	0	0
$775$	742.536i	0.958111i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1731.70	2.22298
$780$	0	0
$781$	161.394	0.206651
$782$	0	0
$783$	− 711.230i	− 0.908340i
$784$	0	0
$785$	−8.48695	−0.0108114
$786$	0	0
$787$	− 283.004i	− 0.359598i	−0.983703	−	0.179799i	$-0.942455\pi$
$787$	− 283.004i	− 0.359598i	0.983703	−	0.179799i	$-0.0575448\pi$
$788$	0	0
$789$	− 1011.70i	− 1.28225i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−449.054	−0.566273
$794$	0	0
$795$	2.42727	0.00305317
$796$	0	0
$797$	999.015i	1.25347i	0.779233	+	0.626735i	$0.215609\pi$
$797$	999.015i	1.25347i	−0.779233	+	0.626735i	$0.784391\pi$
$798$	0	0
$799$	−1881.83	−2.35523
$800$	0	0
$801$	312.287i	0.389871i
$802$	0	0
$803$	− 59.5629i	− 0.0741754i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1368.52	−1.69581
$808$	0	0
$809$	706.226	0.872962	0.436481	−	0.899713i	$-0.356225\pi$
$809$	706.226	0.872962	0.436481	+	0.899713i	$0.356225\pi$
$810$	0	0
$811$	− 357.615i	− 0.440955i	−0.975392	−	0.220478i	$-0.929238\pi$
$811$	− 357.615i	− 0.440955i	0.975392	−	0.220478i	$-0.0707616\pi$
$812$	0	0
$813$	−957.636	−1.17790
$814$	0	0
$815$	8.50041i	0.0104299i
$816$	0	0
$817$	250.926i	0.307131i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1337.56	−1.62919	−0.814594	−	0.580031i	$-0.803040\pi$
$821$	−1337.56	−1.62919	−0.814594	+	0.580031i	$0.803040\pi$
$822$	0	0
$823$	−478.600	−0.581531	−0.290766	−	0.956794i	$-0.593910\pi$
$823$	−478.600	−0.581531	−0.290766	+	0.956794i	$0.593910\pi$
$824$	0	0
$825$	− 216.483i	− 0.262404i
$826$	0	0
$827$	−572.317	−0.692040	−0.346020	−	0.938227i	$-0.612467\pi$
$827$	−572.317	−0.692040	−0.346020	+	0.938227i	$0.612467\pi$
$828$	0	0
$829$	1186.46i	1.43119i	0.698515	+	0.715596i	$0.253845\pi$
$829$	1186.46i	1.43119i	−0.698515	+	0.715596i	$0.746155\pi$
$830$	0	0
$831$	1148.22i	1.38174i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.974001	0.00116647
$836$	0	0
$837$	−777.438	−0.928839
$838$	0	0
$839$	979.116i	1.16700i	0.812112	+	0.583502i	$0.198318\pi$
$839$	979.116i	1.16700i	−0.812112	+	0.583502i	$0.801682\pi$
$840$	0	0
$841$	−102.524	−0.121908
$842$	0	0
$843$	1160.57i	1.37671i
$844$	0	0
$845$	6.93257i	0.00820423i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−668.533	−0.787435
$850$	0	0
$851$	1584.94	1.86245
$852$	0	0
$853$	683.331i	0.801091i	0.916277	+	0.400546i	$0.131179\pi$
$853$	683.331i	0.801091i	−0.916277	+	0.400546i	$0.868821\pi$
$854$	0	0
$855$	−19.5689	−0.0228876
$856$	0	0
$857$	− 8.13871i	− 0.00949675i	−0.999989	−	0.00474837i	$-0.998489\pi$
$857$	− 8.13871i	− 0.00949675i	0.999989	−	0.00474837i	$-0.00151146\pi$
$858$	0	0
$859$	− 1052.97i	− 1.22581i	−0.790156	−	0.612906i	$-0.790000\pi$
$859$	− 1052.97i	− 1.22581i	0.790156	−	0.612906i	$-0.210000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1043.99	1.20972	0.604860	−	0.796332i	$-0.293229\pi$
$863$	1043.99	1.20972	0.604860	+	0.796332i	$0.293229\pi$
$864$	0	0
$865$	−5.07732	−0.00586973
$866$	0	0
$867$	2021.71i	2.33185i
$868$	0	0
$869$	−116.103	−0.133605
$870$	0	0
$871$	605.508i	0.695187i
$872$	0	0
$873$	1773.24i	2.03120i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	807.552	0.920812	0.460406	−	0.887708i	$-0.347704\pi$
$877$	807.552	0.920812	0.460406	+	0.887708i	$0.347704\pi$
$878$	0	0
$879$	1864.36	2.12100
$880$	0	0
$881$	681.043i	0.773034i	0.922282	+	0.386517i	$0.126322\pi$
$881$	681.043i	0.773034i	−0.922282	+	0.386517i	$0.873678\pi$
$882$	0	0
$883$	−996.177	−1.12817	−0.564086	−	0.825716i	$-0.690772\pi$
$883$	−996.177	−1.12817	−0.564086	+	0.825716i	$0.690772\pi$
$884$	0	0
$885$	15.6550i	0.0176893i
$886$	0	0
$887$	− 273.427i	− 0.308260i	−0.988051	−	0.154130i	$-0.950743\pi$
$887$	− 273.427i	− 0.308260i	0.988051	−	0.154130i	$-0.0492575\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.46688	−0.00613567
$892$	0	0
$893$	1864.55	2.08796
$894$	0	0
$895$	3.62405i	0.00404921i
$896$	0	0
$897$	−730.004	−0.813829
$898$	0	0
$899$	− 807.220i	− 0.897909i
$900$	0	0
$901$	258.718i	0.287145i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.5256	0.0116305
$906$	0	0
$907$	−31.9359	−0.0352105	−0.0176052	−	0.999845i	$-0.505604\pi$
$907$	−31.9359	−0.0352105	−0.0176052	+	0.999845i	$0.505604\pi$
$908$	0	0
$909$	814.527i	0.896070i
$910$	0	0
$911$	1257.16	1.37998	0.689991	−	0.723818i	$-0.257615\pi$
$911$	1257.16	1.37998	0.689991	+	0.723818i	$0.257615\pi$
$912$	0	0
$913$	52.1520i	0.0571216i
$914$	0	0
$915$	19.0621i	0.0208330i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1382.98	−1.50488	−0.752438	−	0.658663i	$-0.771122\pi$
$919$	−1382.98	−1.50488	−0.752438	+	0.658663i	$0.771122\pi$
$920$	0	0
$921$	−1076.69	−1.16904
$922$	0	0
$923$	529.829i	0.574030i
$924$	0	0
$925$	−1542.98	−1.66809
$926$	0	0
$927$	426.049i	0.459600i
$928$	0	0
$929$	490.078i	0.527533i	0.964587	+	0.263766i	$0.0849648\pi$
$929$	490.078i	0.527533i	−0.964587	+	0.263766i	$0.915035\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−960.043	−1.02899
$934$	0	0
$935$	−2.45336	−0.00262392
$936$	0	0
$937$	1136.61i	1.21303i	0.795072	+	0.606515i	$0.207433\pi$
$937$	1136.61i	1.21303i	−0.795072	+	0.606515i	$0.792567\pi$
$938$	0	0
$939$	1835.94	1.95520
$940$	0	0
$941$	529.140i	0.562316i	0.959661	+	0.281158i	$0.0907185\pi$
$941$	529.140i	0.562316i	−0.959661	+	0.281158i	$0.909281\pi$
$942$	0	0
$943$	− 1687.96i	− 1.78999i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1164.16	−1.22931	−0.614657	−	0.788795i	$-0.710705\pi$
$947$	−1164.16	−1.22931	−0.614657	+	0.788795i	$0.710705\pi$
$948$	0	0
$949$	195.534	0.206043
$950$	0	0
$951$	1256.16i	1.32088i
$952$	0	0
$953$	−616.861	−0.647283	−0.323642	−	0.946180i	$-0.604907\pi$
$953$	−616.861	−0.647283	−0.323642	+	0.946180i	$0.604907\pi$
$954$	0	0
$955$	9.83150i	0.0102948i
$956$	0	0
$957$	235.342i	0.245916i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	78.6368	0.0818281
$962$	0	0
$963$	429.554	0.446059
$964$	0	0
$965$	4.41034i	0.00457030i
$966$	0	0
$967$	−624.887	−0.646212	−0.323106	−	0.946363i	$-0.604727\pi$
$967$	−624.887	−0.646212	−0.323106	+	0.946363i	$0.604727\pi$
$968$	0	0
$969$	− 3388.59i	− 3.49700i
$970$	0	0
$971$	− 744.764i	− 0.767007i	−0.923539	−	0.383504i	$-0.874717\pi$
$971$	− 744.764i	− 0.767007i	0.923539	−	0.383504i	$-0.125283\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	710.677	0.728899
$976$	0	0
$977$	−684.393	−0.700504	−0.350252	−	0.936655i	$-0.613904\pi$
$977$	−684.393	−0.700504	−0.350252	+	0.936655i	$0.613904\pi$
$978$	0	0
$979$	− 38.7922i	− 0.0396243i
$980$	0	0
$981$	−1175.09	−1.19785
$982$	0	0
$983$	909.341i	0.925067i	0.886602	+	0.462533i	$0.153059\pi$
$983$	909.341i	0.925067i	−0.886602	+	0.462533i	$0.846941\pi$
$984$	0	0
$985$	14.1858i	0.0144018i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	244.587	0.247307
$990$	0	0
$991$	−1325.02	−1.33705	−0.668526	−	0.743689i	$-0.733074\pi$
$991$	−1325.02	−1.33705	−0.668526	+	0.743689i	$0.733074\pi$
$992$	0	0
$993$	2647.42i	2.66608i
$994$	0	0
$995$	−10.6653	−0.0107189
$996$	0	0
$997$	937.109i	0.939928i	0.882685	+	0.469964i	$0.155733\pi$
$997$	937.109i	0.939928i	−0.882685	+	0.469964i	$0.844267\pi$
$998$	0	0
$999$	− 1615.51i	− 1.61712i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.h.97.2		16
4.3	odd	2	inner	1568.3.c.h.97.16		16
7.4	even	3		224.3.s.a.33.1	✓	16
7.5	odd	6		224.3.s.a.129.1	yes	16
7.6	odd	2	inner	1568.3.c.h.97.15		16
28.11	odd	6		224.3.s.a.33.8	yes	16
28.19	even	6		224.3.s.a.129.8	yes	16
28.27	even	2	inner	1568.3.c.h.97.1		16
56.5	odd	6		448.3.s.g.129.8		16
56.11	odd	6		448.3.s.g.257.1		16
56.19	even	6		448.3.s.g.129.1		16
56.53	even	6		448.3.s.g.257.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.a.33.1	✓	16	7.4	even	3
224.3.s.a.33.8	yes	16	28.11	odd	6
224.3.s.a.129.1	yes	16	7.5	odd	6
224.3.s.a.129.8	yes	16	28.19	even	6
448.3.s.g.129.1		16	56.19	even	6
448.3.s.g.129.8		16	56.5	odd	6
448.3.s.g.257.1		16	56.11	odd	6
448.3.s.g.257.8		16	56.53	even	6
1568.3.c.h.97.1		16	28.27	even	2	inner
1568.3.c.h.97.2		16	1.1	even	1	trivial
1568.3.c.h.97.15		16	7.6	odd	2	inner
1568.3.c.h.97.16		16	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.83832i q^{3} +0.0515539i q^{5} -14.4094 q^{9} +O(q^{10})\)	\(q-4.83832i q^{3} +0.0515539i q^{5} -14.4094 q^{9} +1.78993 q^{11} +5.87602i q^{13} +0.249434 q^{15} +26.5867i q^{17} -26.3426i q^{19} -25.6772 q^{23} +24.9973 q^{25} +26.1723i q^{27} -27.1749 q^{29} +29.7046i q^{31} -8.66025i q^{33} -61.7258 q^{37} +28.4301 q^{39} +65.7376i q^{41} -9.52546 q^{43} -0.742859i q^{45} +70.7807i q^{47} +128.635 q^{51} +9.73110 q^{53} +0.0922778i q^{55} -127.454 q^{57} +62.7620i q^{59} +76.4215i q^{61} -0.302932 q^{65} +103.047 q^{67} +124.234i q^{69} +90.1681 q^{71} -33.2767i q^{73} -120.945i q^{75} -64.8645 q^{79} -3.05424 q^{81} +29.1364i q^{83} -1.37065 q^{85} +131.481i q^{87} -21.6725i q^{89} +143.720 q^{93} +1.35807 q^{95} -123.061i q^{97} -25.7918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more