LMFDB - Embedded newform 1568.3.d.e.1471.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(1471,1568)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("1568.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.d (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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1568.1471
Dual form			1568.3.d.e.1471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000i q^{3} +9.00000 q^{5} -16.0000 q^{9} +O(q^{10})$	$q+5.00000i q^{3} +9.00000 q^{5} -16.0000 q^{9} +3.00000i q^{11} +16.0000 q^{13} +45.0000i q^{15} -7.00000 q^{17} -11.0000i q^{19} +19.0000i q^{23} +56.0000 q^{25} -35.0000i q^{27} -32.0000 q^{29} +11.0000i q^{31} -15.0000 q^{33} -1.00000 q^{37} +80.0000i q^{39} -40.0000 q^{41} +40.0000i q^{43} -144.000 q^{45} +85.0000i q^{47} -35.0000i q^{51} +7.00000 q^{53} +27.0000i q^{55} +55.0000 q^{57} +53.0000i q^{59} +79.0000 q^{61} +144.000 q^{65} -11.0000i q^{67} -95.0000 q^{69} +48.0000i q^{71} +143.000 q^{73} +280.000i q^{75} +35.0000i q^{79} +31.0000 q^{81} +8.00000i q^{83} -63.0000 q^{85} -160.000i q^{87} -97.0000 q^{89} -55.0000 q^{93} -99.0000i q^{95} -88.0000 q^{97} -48.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{5} - 32 q^{9}+O(q^{10}) $ 2 * q + 18 * q^5 - 32 * q^9	$ 2 q + 18 q^{5} - 32 q^{9} + 32 q^{13} - 14 q^{17} + 112 q^{25} - 64 q^{29} - 30 q^{33} - 2 q^{37} - 80 q^{41} - 288 q^{45} + 14 q^{53} + 110 q^{57} + 158 q^{61} + 288 q^{65} - 190 q^{69} + 286 q^{73} + 62 q^{81} - 126 q^{85} - 194 q^{89} - 110 q^{93} - 176 q^{97}+O(q^{100}) $ 2 * q + 18 * q^5 - 32 * q^9 + 32 * q^13 - 14 * q^17 + 112 * q^25 - 64 * q^29 - 30 * q^33 - 2 * q^37 - 80 * q^41 - 288 * q^45 + 14 * q^53 + 110 * q^57 + 158 * q^61 + 288 * q^65 - 190 * q^69 + 286 * q^73 + 62 * q^81 - 126 * q^85 - 194 * q^89 - 110 * q^93 - 176 * q^97

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
$3$	5.00000i	1.66667i	0.552771	+	0.833333i	$0.313571\pi$
$3$	5.00000i	1.66667i	−0.552771	+	0.833333i	$0.686429\pi$
$4$	0	0
$5$	9.00000	1.80000	0.900000	−	0.435890i	$-0.143566\pi$
$5$	9.00000	1.80000	0.900000	+	0.435890i	$0.143566\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−16.0000	−1.77778
$10$	0	0
$11$	3.00000i	0.272727i	0.990659	+	0.136364i	$0.0435416\pi$
$11$	3.00000i	0.272727i	−0.990659	+	0.136364i	$0.956458\pi$
$12$	0	0
$13$	16.0000	1.23077	0.615385	−	0.788227i	$-0.289001\pi$
$13$	16.0000	1.23077	0.615385	+	0.788227i	$0.289001\pi$
$14$	0	0
$15$	45.0000i	3.00000i
$16$	0	0
$17$	−7.00000	−0.411765	−0.205882	−	0.978577i	$-0.566006\pi$
$17$	−7.00000	−0.411765	−0.205882	+	0.978577i	$0.566006\pi$
$18$	0	0
$19$	− 11.0000i	− 0.578947i	−0.957186	−	0.289474i	$-0.906520\pi$
$19$	− 11.0000i	− 0.578947i	0.957186	−	0.289474i	$-0.0934803\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	19.0000i	0.826087i	0.910711	+	0.413043i	$0.135534\pi$
$23$	19.0000i	0.826087i	−0.910711	+	0.413043i	$0.864466\pi$
$24$	0	0
$25$	56.0000	2.24000
$26$	0	0
$27$	− 35.0000i	− 1.29630i
$28$	0	0
$29$	−32.0000	−1.10345	−0.551724	−	0.834027i	$-0.686030\pi$
$29$	−32.0000	−1.10345	−0.551724	+	0.834027i	$0.686030\pi$
$30$	0	0
$31$	11.0000i	0.354839i	0.984135	+	0.177419i	$0.0567749\pi$
$31$	11.0000i	0.354839i	−0.984135	+	0.177419i	$0.943225\pi$
$32$	0	0
$33$	−15.0000	−0.454545
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.0270270	−0.0135135	−	0.999909i	$-0.504302\pi$
$37$	−1.00000	−0.0270270	−0.0135135	+	0.999909i	$0.504302\pi$
$38$	0	0
$39$	80.0000i	2.05128i
$40$	0	0
$41$	−40.0000	−0.975610	−0.487805	−	0.872953i	$-0.662202\pi$
$41$	−40.0000	−0.975610	−0.487805	+	0.872953i	$0.662202\pi$
$42$	0	0
$43$	40.0000i	0.930233i	0.885250	+	0.465116i	$0.153987\pi$
$43$	40.0000i	0.930233i	−0.885250	+	0.465116i	$0.846013\pi$
$44$	0	0
$45$	−144.000	−3.20000
$46$	0	0
$47$	85.0000i	1.80851i	0.426992	+	0.904255i	$0.359573\pi$
$47$	85.0000i	1.80851i	−0.426992	+	0.904255i	$0.640427\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 35.0000i	− 0.686275i
$52$	0	0
$53$	7.00000	0.132075	0.0660377	−	0.997817i	$-0.478964\pi$
$53$	7.00000	0.132075	0.0660377	+	0.997817i	$0.478964\pi$
$54$	0	0
$55$	27.0000i	0.490909i
$56$	0	0
$57$	55.0000	0.964912
$58$	0	0
$59$	53.0000i	0.898305i	0.893455	+	0.449153i	$0.148274\pi$
$59$	53.0000i	0.898305i	−0.893455	+	0.449153i	$0.851726\pi$
$60$	0	0
$61$	79.0000	1.29508	0.647541	−	0.762031i	$-0.275797\pi$
$61$	79.0000	1.29508	0.647541	+	0.762031i	$0.275797\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	144.000	2.21538
$66$	0	0
$67$	− 11.0000i	− 0.164179i	−0.996625	−	0.0820896i	$-0.973841\pi$
$67$	− 11.0000i	− 0.164179i	0.996625	−	0.0820896i	$-0.0261594\pi$
$68$	0	0
$69$	−95.0000	−1.37681
$70$	0	0
$71$	48.0000i	0.676056i	0.941136	+	0.338028i	$0.109760\pi$
$71$	48.0000i	0.676056i	−0.941136	+	0.338028i	$0.890240\pi$
$72$	0	0
$73$	143.000	1.95890	0.979452	−	0.201677i	$-0.0646392\pi$
$73$	143.000	1.95890	0.979452	+	0.201677i	$0.0646392\pi$
$74$	0	0
$75$	280.000i	3.73333i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	35.0000i	0.443038i	0.975156	+	0.221519i	$0.0711015\pi$
$79$	35.0000i	0.443038i	−0.975156	+	0.221519i	$0.928899\pi$
$80$	0	0
$81$	31.0000	0.382716
$82$	0	0
$83$	8.00000i	0.0963855i	0.998838	+	0.0481928i	$0.0153462\pi$
$83$	8.00000i	0.0963855i	−0.998838	+	0.0481928i	$0.984654\pi$
$84$	0	0
$85$	−63.0000	−0.741176
$86$	0	0
$87$	− 160.000i	− 1.83908i
$88$	0	0
$89$	−97.0000	−1.08989	−0.544944	−	0.838473i	$-0.683449\pi$
$89$	−97.0000	−1.08989	−0.544944	+	0.838473i	$0.683449\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−55.0000	−0.591398
$94$	0	0
$95$	− 99.0000i	− 1.04211i
$96$	0	0
$97$	−88.0000	−0.907216	−0.453608	−	0.891201i	$-0.649863\pi$
$97$	−88.0000	−0.907216	−0.453608	+	0.891201i	$0.649863\pi$
$98$	0	0
$99$	− 48.0000i	− 0.484848i
$100$	0	0
$101$	−15.0000	−0.148515	−0.0742574	−	0.997239i	$-0.523659\pi$
$101$	−15.0000	−0.148515	−0.0742574	+	0.997239i	$0.523659\pi$
$102$	0	0
$103$	− 93.0000i	− 0.902913i	−0.892293	−	0.451456i	$-0.850905\pi$
$103$	− 93.0000i	− 0.902913i	0.892293	−	0.451456i	$-0.149095\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.0000i	− 0.102804i	−0.998678	−	0.0514019i	$-0.983631\pi$
$107$	− 11.0000i	− 0.102804i	0.998678	−	0.0514019i	$-0.0163689\pi$
$108$	0	0
$109$	113.000	1.03670	0.518349	−	0.855169i	$-0.326547\pi$
$109$	113.000	1.03670	0.518349	+	0.855169i	$0.326547\pi$
$110$	0	0
$111$	− 5.00000i	− 0.0450450i
$112$	0	0
$113$	56.0000	0.495575	0.247788	−	0.968814i	$-0.420296\pi$
$113$	56.0000	0.495575	0.247788	+	0.968814i	$0.420296\pi$
$114$	0	0
$115$	171.000i	1.48696i
$116$	0	0
$117$	−256.000	−2.18803
$118$	0	0
$119$	0	0
$120$	0	0
$121$	112.000	0.925620
$122$	0	0
$123$	− 200.000i	− 1.62602i
$124$	0	0
$125$	279.000	2.23200
$126$	0	0
$127$	− 96.0000i	− 0.755906i	−0.925825	−	0.377953i	$-0.876628\pi$
$127$	− 96.0000i	− 0.755906i	0.925825	−	0.377953i	$-0.123372\pi$
$128$	0	0
$129$	−200.000	−1.55039
$130$	0	0
$131$	− 141.000i	− 1.07634i	−0.842838	−	0.538168i	$-0.819117\pi$
$131$	− 141.000i	− 1.07634i	0.842838	−	0.538168i	$-0.180883\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 315.000i	− 2.33333i
$136$	0	0
$137$	9.00000	0.0656934	0.0328467	−	0.999460i	$-0.489543\pi$
$137$	9.00000	0.0656934	0.0328467	+	0.999460i	$0.489543\pi$
$138$	0	0
$139$	136.000i	0.978417i	0.872167	+	0.489209i	$0.162714\pi$
$139$	136.000i	0.978417i	−0.872167	+	0.489209i	$0.837286\pi$
$140$	0	0
$141$	−425.000	−3.01418
$142$	0	0
$143$	48.0000i	0.335664i
$144$	0	0
$145$	−288.000	−1.98621
$146$	0	0
$147$	0	0
$148$	0	0
$149$	57.0000	0.382550	0.191275	−	0.981536i	$-0.438738\pi$
$149$	57.0000	0.382550	0.191275	+	0.981536i	$0.438738\pi$
$150$	0	0
$151$	27.0000i	0.178808i	0.995995	+	0.0894040i	$0.0284962\pi$
$151$	27.0000i	0.178808i	−0.995995	+	0.0894040i	$0.971504\pi$
$152$	0	0
$153$	112.000	0.732026
$154$	0	0
$155$	99.0000i	0.638710i
$156$	0	0
$157$	167.000	1.06369	0.531847	−	0.846840i	$-0.321498\pi$
$157$	167.000	1.06369	0.531847	+	0.846840i	$0.321498\pi$
$158$	0	0
$159$	35.0000i	0.220126i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.0000i	0.116564i	0.998300	+	0.0582822i	$0.0185623\pi$
$163$	19.0000i	0.116564i	−0.998300	+	0.0582822i	$0.981438\pi$
$164$	0	0
$165$	−135.000	−0.818182
$166$	0	0
$167$	− 190.000i	− 1.13772i	−0.822433	−	0.568862i	$-0.807384\pi$
$167$	− 190.000i	− 1.13772i	0.822433	−	0.568862i	$-0.192616\pi$
$168$	0	0
$169$	87.0000	0.514793
$170$	0	0
$171$	176.000i	1.02924i
$172$	0	0
$173$	−145.000	−0.838150	−0.419075	−	0.907952i	$-0.637646\pi$
$173$	−145.000	−0.838150	−0.419075	+	0.907952i	$0.637646\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−265.000	−1.49718
$178$	0	0
$179$	− 317.000i	− 1.77095i	−0.464687	−	0.885475i	$-0.653833\pi$
$179$	− 317.000i	− 1.77095i	0.464687	−	0.885475i	$-0.346167\pi$
$180$	0	0
$181$	−186.000	−1.02762	−0.513812	−	0.857903i	$-0.671767\pi$
$181$	−186.000	−1.02762	−0.513812	+	0.857903i	$0.671767\pi$
$182$	0	0
$183$	395.000i	2.15847i
$184$	0	0
$185$	−9.00000	−0.0486486
$186$	0	0
$187$	− 21.0000i	− 0.112299i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 283.000i	− 1.48168i	−0.671684	−	0.740838i	$-0.734429\pi$
$191$	− 283.000i	− 1.48168i	0.671684	−	0.740838i	$-0.265571\pi$
$192$	0	0
$193$	−295.000	−1.52850	−0.764249	−	0.644922i	$-0.776890\pi$
$193$	−295.000	−1.52850	−0.764249	+	0.644922i	$0.776890\pi$
$194$	0	0
$195$	720.000i	3.69231i
$196$	0	0
$197$	128.000	0.649746	0.324873	−	0.945758i	$-0.394678\pi$
$197$	128.000	0.649746	0.324873	+	0.945758i	$0.394678\pi$
$198$	0	0
$199$	283.000i	1.42211i	0.703136	+	0.711055i	$0.251782\pi$
$199$	283.000i	1.42211i	−0.703136	+	0.711055i	$0.748218\pi$
$200$	0	0
$201$	55.0000	0.273632
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−360.000	−1.75610
$206$	0	0
$207$	− 304.000i	− 1.46860i
$208$	0	0
$209$	33.0000	0.157895
$210$	0	0
$211$	40.0000i	0.189573i	0.995498	+	0.0947867i	$0.0302169\pi$
$211$	40.0000i	0.189573i	−0.995498	+	0.0947867i	$0.969783\pi$
$212$	0	0
$213$	−240.000	−1.12676
$214$	0	0
$215$	360.000i	1.67442i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	715.000i	3.26484i
$220$	0	0
$221$	−112.000	−0.506787
$222$	0	0
$223$	− 176.000i	− 0.789238i	−0.918845	−	0.394619i	$-0.870877\pi$
$223$	− 176.000i	− 0.789238i	0.918845	−	0.394619i	$-0.129123\pi$
$224$	0	0
$225$	−896.000	−3.98222
$226$	0	0
$227$	387.000i	1.70485i	0.522853	+	0.852423i	$0.324868\pi$
$227$	387.000i	1.70485i	−0.522853	+	0.852423i	$0.675132\pi$
$228$	0	0
$229$	−183.000	−0.799127	−0.399563	−	0.916706i	$-0.630838\pi$
$229$	−183.000	−0.799127	−0.399563	+	0.916706i	$0.630838\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	361.000	1.54936	0.774678	−	0.632356i	$-0.217912\pi$
$233$	361.000	1.54936	0.774678	+	0.632356i	$0.217912\pi$
$234$	0	0
$235$	765.000i	3.25532i
$236$	0	0
$237$	−175.000	−0.738397
$238$	0	0
$239$	− 50.0000i	− 0.209205i	−0.994514	−	0.104603i	$-0.966643\pi$
$239$	− 50.0000i	− 0.209205i	0.994514	−	0.104603i	$-0.0333570\pi$
$240$	0	0
$241$	−129.000	−0.535270	−0.267635	−	0.963520i	$-0.586242\pi$
$241$	−129.000	−0.535270	−0.267635	+	0.963520i	$0.586242\pi$
$242$	0	0
$243$	− 160.000i	− 0.658436i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 176.000i	− 0.712551i
$248$	0	0
$249$	−40.0000	−0.160643
$250$	0	0
$251$	− 394.000i	− 1.56972i	−0.619672	−	0.784861i	$-0.712735\pi$
$251$	− 394.000i	− 1.56972i	0.619672	−	0.784861i	$-0.287265\pi$
$252$	0	0
$253$	−57.0000	−0.225296
$254$	0	0
$255$	− 315.000i	− 1.23529i
$256$	0	0
$257$	89.0000	0.346304	0.173152	−	0.984895i	$-0.444605\pi$
$257$	89.0000	0.346304	0.173152	+	0.984895i	$0.444605\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	512.000	1.96169
$262$	0	0
$263$	45.0000i	0.171103i	0.996334	+	0.0855513i	$0.0272652\pi$
$263$	45.0000i	0.171103i	−0.996334	+	0.0855513i	$0.972735\pi$
$264$	0	0
$265$	63.0000	0.237736
$266$	0	0
$267$	− 485.000i	− 1.81648i
$268$	0	0
$269$	−457.000	−1.69888	−0.849442	−	0.527681i	$-0.823062\pi$
$269$	−457.000	−1.69888	−0.849442	+	0.527681i	$0.823062\pi$
$270$	0	0
$271$	− 397.000i	− 1.46494i	−0.680797	−	0.732472i	$-0.738366\pi$
$271$	− 397.000i	− 1.46494i	0.680797	−	0.732472i	$-0.261634\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	168.000i	0.610909i
$276$	0	0
$277$	231.000	0.833935	0.416968	−	0.908921i	$-0.363093\pi$
$277$	231.000	0.833935	0.416968	+	0.908921i	$0.363093\pi$
$278$	0	0
$279$	− 176.000i	− 0.630824i
$280$	0	0
$281$	104.000	0.370107	0.185053	−	0.982728i	$-0.440754\pi$
$281$	104.000	0.370107	0.185053	+	0.982728i	$0.440754\pi$
$282$	0	0
$283$	− 397.000i	− 1.40283i	−0.712755	−	0.701413i	$-0.752553\pi$
$283$	− 397.000i	− 1.40283i	0.712755	−	0.701413i	$-0.247447\pi$
$284$	0	0
$285$	495.000	1.73684
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−240.000	−0.830450
$290$	0	0
$291$	− 440.000i	− 1.51203i
$292$	0	0
$293$	154.000	0.525597	0.262799	−	0.964851i	$-0.415355\pi$
$293$	154.000	0.525597	0.262799	+	0.964851i	$0.415355\pi$
$294$	0	0
$295$	477.000i	1.61695i
$296$	0	0
$297$	105.000	0.353535
$298$	0	0
$299$	304.000i	1.01672i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 75.0000i	− 0.247525i
$304$	0	0
$305$	711.000	2.33115
$306$	0	0
$307$	− 376.000i	− 1.22476i	−0.790565	−	0.612378i	$-0.790213\pi$
$307$	− 376.000i	− 1.22476i	0.790565	−	0.612378i	$-0.209787\pi$
$308$	0	0
$309$	465.000	1.50485
$310$	0	0
$311$	− 323.000i	− 1.03859i	−0.854597	−	0.519293i	$-0.826195\pi$
$311$	− 323.000i	− 1.03859i	0.854597	−	0.519293i	$-0.173805\pi$
$312$	0	0
$313$	191.000	0.610224	0.305112	−	0.952317i	$-0.401306\pi$
$313$	191.000	0.610224	0.305112	+	0.952317i	$0.401306\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	335.000	1.05678	0.528391	−	0.849001i	$-0.322795\pi$
$317$	335.000	1.05678	0.528391	+	0.849001i	$0.322795\pi$
$318$	0	0
$319$	− 96.0000i	− 0.300940i
$320$	0	0
$321$	55.0000	0.171340
$322$	0	0
$323$	77.0000i	0.238390i
$324$	0	0
$325$	896.000	2.75692
$326$	0	0
$327$	565.000i	1.72783i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	35.0000i	0.105740i	0.998601	+	0.0528701i	$0.0168369\pi$
$331$	35.0000i	0.105740i	−0.998601	+	0.0528701i	$0.983163\pi$
$332$	0	0
$333$	16.0000	0.0480480
$334$	0	0
$335$	− 99.0000i	− 0.295522i
$336$	0	0
$337$	184.000	0.545994	0.272997	−	0.962015i	$-0.411985\pi$
$337$	184.000	0.545994	0.272997	+	0.962015i	$0.411985\pi$
$338$	0	0
$339$	280.000i	0.825959i
$340$	0	0
$341$	−33.0000	−0.0967742
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−855.000	−2.47826
$346$	0	0
$347$	− 187.000i	− 0.538905i	−0.963014	−	0.269452i	$-0.913157\pi$
$347$	− 187.000i	− 0.538905i	0.963014	−	0.269452i	$-0.0868427\pi$
$348$	0	0
$349$	−208.000	−0.595989	−0.297994	−	0.954568i	$-0.596318\pi$
$349$	−208.000	−0.595989	−0.297994	+	0.954568i	$0.596318\pi$
$350$	0	0
$351$	− 560.000i	− 1.59544i
$352$	0	0
$353$	−231.000	−0.654391	−0.327195	−	0.944957i	$-0.606104\pi$
$353$	−231.000	−0.654391	−0.327195	+	0.944957i	$0.606104\pi$
$354$	0	0
$355$	432.000i	1.21690i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	37.0000i	0.103064i	0.998671	+	0.0515320i	$0.0164104\pi$
$359$	37.0000i	0.103064i	−0.998671	+	0.0515320i	$0.983590\pi$
$360$	0	0
$361$	240.000	0.664820
$362$	0	0
$363$	560.000i	1.54270i
$364$	0	0
$365$	1287.00	3.52603
$366$	0	0
$367$	− 483.000i	− 1.31608i	−0.752985	−	0.658038i	$-0.771387\pi$
$367$	− 483.000i	− 1.31608i	0.752985	−	0.658038i	$-0.228613\pi$
$368$	0	0
$369$	640.000	1.73442
$370$	0	0
$371$	0	0
$372$	0	0
$373$	431.000	1.15550	0.577748	−	0.816215i	$-0.303932\pi$
$373$	431.000	1.15550	0.577748	+	0.816215i	$0.303932\pi$
$374$	0	0
$375$	1395.00i	3.72000i
$376$	0	0
$377$	−512.000	−1.35809
$378$	0	0
$379$	714.000i	1.88391i	0.335746	+	0.941953i	$0.391012\pi$
$379$	714.000i	1.88391i	−0.335746	+	0.941953i	$0.608988\pi$
$380$	0	0
$381$	480.000	1.25984
$382$	0	0
$383$	− 651.000i	− 1.69974i	−0.526993	−	0.849869i	$-0.676681\pi$
$383$	− 651.000i	− 1.69974i	0.526993	−	0.849869i	$-0.323319\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 640.000i	− 1.65375i
$388$	0	0
$389$	465.000	1.19537	0.597686	−	0.801730i	$-0.296087\pi$
$389$	465.000	1.19537	0.597686	+	0.801730i	$0.296087\pi$
$390$	0	0
$391$	− 133.000i	− 0.340153i
$392$	0	0
$393$	705.000	1.79389
$394$	0	0
$395$	315.000i	0.797468i
$396$	0	0
$397$	185.000	0.465995	0.232997	−	0.972477i	$-0.425147\pi$
$397$	185.000	0.465995	0.232997	+	0.972477i	$0.425147\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	105.000	0.261845	0.130923	−	0.991393i	$-0.458206\pi$
$401$	105.000	0.261845	0.130923	+	0.991393i	$0.458206\pi$
$402$	0	0
$403$	176.000i	0.436725i
$404$	0	0
$405$	279.000	0.688889
$406$	0	0
$407$	− 3.00000i	− 0.00737101i
$408$	0	0
$409$	−7.00000	−0.0171149	−0.00855746	−	0.999963i	$-0.502724\pi$
$409$	−7.00000	−0.0171149	−0.00855746	+	0.999963i	$0.502724\pi$
$410$	0	0
$411$	45.0000i	0.109489i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	72.0000i	0.173494i
$416$	0	0
$417$	−680.000	−1.63070
$418$	0	0
$419$	392.000i	0.935561i	0.883845	+	0.467780i	$0.154946\pi$
$419$	392.000i	0.935561i	−0.883845	+	0.467780i	$0.845054\pi$
$420$	0	0
$421$	64.0000	0.152019	0.0760095	−	0.997107i	$-0.475782\pi$
$421$	64.0000	0.152019	0.0760095	+	0.997107i	$0.475782\pi$
$422$	0	0
$423$	− 1360.00i	− 3.21513i
$424$	0	0
$425$	−392.000	−0.922353
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−240.000	−0.559441
$430$	0	0
$431$	731.000i	1.69606i	0.529952	+	0.848028i	$0.322210\pi$
$431$	731.000i	1.69606i	−0.529952	+	0.848028i	$0.677790\pi$
$432$	0	0
$433$	376.000	0.868360	0.434180	−	0.900826i	$-0.357038\pi$
$433$	376.000	0.868360	0.434180	+	0.900826i	$0.357038\pi$
$434$	0	0
$435$	− 1440.00i	− 3.31034i
$436$	0	0
$437$	209.000	0.478261
$438$	0	0
$439$	− 635.000i	− 1.44647i	−0.690602	−	0.723235i	$-0.742655\pi$
$439$	− 635.000i	− 1.44647i	0.690602	−	0.723235i	$-0.257345\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 491.000i	− 1.10835i	−0.832399	−	0.554176i	$-0.813033\pi$
$443$	− 491.000i	− 1.10835i	0.832399	−	0.554176i	$-0.186967\pi$
$444$	0	0
$445$	−873.000	−1.96180
$446$	0	0
$447$	285.000i	0.637584i
$448$	0	0
$449$	−344.000	−0.766147	−0.383073	−	0.923718i	$-0.625134\pi$
$449$	−344.000	−0.766147	−0.383073	+	0.923718i	$0.625134\pi$
$450$	0	0
$451$	− 120.000i	− 0.266075i
$452$	0	0
$453$	−135.000	−0.298013
$454$	0	0
$455$	0	0
$456$	0	0
$457$	527.000	1.15317	0.576586	−	0.817036i	$-0.304384\pi$
$457$	527.000	1.15317	0.576586	+	0.817036i	$0.304384\pi$
$458$	0	0
$459$	245.000i	0.533769i
$460$	0	0
$461$	−704.000	−1.52711	−0.763557	−	0.645740i	$-0.776549\pi$
$461$	−704.000	−1.52711	−0.763557	+	0.645740i	$0.776549\pi$
$462$	0	0
$463$	160.000i	0.345572i	0.984959	+	0.172786i	$0.0552770\pi$
$463$	160.000i	0.345572i	−0.984959	+	0.172786i	$0.944723\pi$
$464$	0	0
$465$	−495.000	−1.06452
$466$	0	0
$467$	− 541.000i	− 1.15846i	−0.815165	−	0.579229i	$-0.803354\pi$
$467$	− 541.000i	− 1.15846i	0.815165	−	0.579229i	$-0.196646\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	835.000i	1.77282i
$472$	0	0
$473$	−120.000	−0.253700
$474$	0	0
$475$	− 616.000i	− 1.29684i
$476$	0	0
$477$	−112.000	−0.234801
$478$	0	0
$479$	477.000i	0.995825i	0.867227	+	0.497912i	$0.165900\pi$
$479$	477.000i	0.995825i	−0.867227	+	0.497912i	$0.834100\pi$
$480$	0	0
$481$	−16.0000	−0.0332640
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−792.000	−1.63299
$486$	0	0
$487$	− 307.000i	− 0.630390i	−0.949027	−	0.315195i	$-0.897930\pi$
$487$	− 307.000i	− 0.630390i	0.949027	−	0.315195i	$-0.102070\pi$
$488$	0	0
$489$	−95.0000	−0.194274
$490$	0	0
$491$	− 22.0000i	− 0.0448065i	−0.999749	−	0.0224033i	$-0.992868\pi$
$491$	− 22.0000i	− 0.0448065i	0.999749	−	0.0224033i	$-0.00713178\pi$
$492$	0	0
$493$	224.000	0.454361
$494$	0	0
$495$	− 432.000i	− 0.872727i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	915.000i	1.83367i	0.399269	+	0.916834i	$0.369264\pi$
$499$	915.000i	1.83367i	−0.399269	+	0.916834i	$0.630736\pi$
$500$	0	0
$501$	950.000	1.89621
$502$	0	0
$503$	192.000i	0.381710i	0.981618	+	0.190855i	$0.0611260\pi$
$503$	192.000i	0.381710i	−0.981618	+	0.190855i	$0.938874\pi$
$504$	0	0
$505$	−135.000	−0.267327
$506$	0	0
$507$	435.000i	0.857988i
$508$	0	0
$509$	−71.0000	−0.139489	−0.0697446	−	0.997565i	$-0.522218\pi$
$509$	−71.0000	−0.139489	−0.0697446	+	0.997565i	$0.522218\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−385.000	−0.750487
$514$	0	0
$515$	− 837.000i	− 1.62524i
$516$	0	0
$517$	−255.000	−0.493230
$518$	0	0
$519$	− 725.000i	− 1.39692i
$520$	0	0
$521$	703.000	1.34933	0.674664	−	0.738125i	$-0.264288\pi$
$521$	703.000	1.34933	0.674664	+	0.738125i	$0.264288\pi$
$522$	0	0
$523$	325.000i	0.621415i	0.950506	+	0.310707i	$0.100566\pi$
$523$	325.000i	0.621415i	−0.950506	+	0.310707i	$0.899434\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 77.0000i	− 0.146110i
$528$	0	0
$529$	168.000	0.317580
$530$	0	0
$531$	− 848.000i	− 1.59699i
$532$	0	0
$533$	−640.000	−1.20075
$534$	0	0
$535$	− 99.0000i	− 0.185047i
$536$	0	0
$537$	1585.00	2.95158
$538$	0	0
$539$	0	0
$540$	0	0
$541$	265.000	0.489834	0.244917	−	0.969544i	$-0.421239\pi$
$541$	265.000	0.489834	0.244917	+	0.969544i	$0.421239\pi$
$542$	0	0
$543$	− 930.000i	− 1.71271i
$544$	0	0
$545$	1017.00	1.86606
$546$	0	0
$547$	134.000i	0.244973i	0.992470	+	0.122486i	$0.0390868\pi$
$547$	134.000i	0.244973i	−0.992470	+	0.122486i	$0.960913\pi$
$548$	0	0
$549$	−1264.00	−2.30237
$550$	0	0
$551$	352.000i	0.638838i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 45.0000i	− 0.0810811i
$556$	0	0
$557$	−575.000	−1.03232	−0.516158	−	0.856493i	$-0.672638\pi$
$557$	−575.000	−1.03232	−0.516158	+	0.856493i	$0.672638\pi$
$558$	0	0
$559$	640.000i	1.14490i
$560$	0	0
$561$	105.000	0.187166
$562$	0	0
$563$	181.000i	0.321492i	0.986996	+	0.160746i	$0.0513900\pi$
$563$	181.000i	0.321492i	−0.986996	+	0.160746i	$0.948610\pi$
$564$	0	0
$565$	504.000	0.892035
$566$	0	0
$567$	0	0
$568$	0	0
$569$	319.000	0.560633	0.280316	−	0.959908i	$-0.409561\pi$
$569$	319.000	0.560633	0.280316	+	0.959908i	$0.409561\pi$
$570$	0	0
$571$	163.000i	0.285464i	0.989761	+	0.142732i	$0.0455887\pi$
$571$	163.000i	0.285464i	−0.989761	+	0.142732i	$0.954411\pi$
$572$	0	0
$573$	1415.00	2.46946
$574$	0	0
$575$	1064.00i	1.85043i
$576$	0	0
$577$	719.000	1.24610	0.623050	−	0.782182i	$-0.285893\pi$
$577$	719.000	1.24610	0.623050	+	0.782182i	$0.285893\pi$
$578$	0	0
$579$	− 1475.00i	− 2.54750i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	21.0000i	0.0360206i
$584$	0	0
$585$	−2304.00	−3.93846
$586$	0	0
$587$	− 296.000i	− 0.504259i	−0.967694	−	0.252129i	$-0.918869\pi$
$587$	− 296.000i	− 0.504259i	0.967694	−	0.252129i	$-0.0811309\pi$
$588$	0	0
$589$	121.000	0.205433
$590$	0	0
$591$	640.000i	1.08291i
$592$	0	0
$593$	361.000	0.608769	0.304384	−	0.952549i	$-0.401549\pi$
$593$	361.000	0.608769	0.304384	+	0.952549i	$0.401549\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1415.00	−2.37018
$598$	0	0
$599$	125.000i	0.208681i	0.994542	+	0.104341i	$0.0332732\pi$
$599$	125.000i	0.208681i	−0.994542	+	0.104341i	$0.966727\pi$
$600$	0	0
$601$	1000.00	1.66389	0.831947	−	0.554855i	$-0.187226\pi$
$601$	1000.00	1.66389	0.831947	+	0.554855i	$0.187226\pi$
$602$	0	0
$603$	176.000i	0.291874i
$604$	0	0
$605$	1008.00	1.66612
$606$	0	0
$607$	− 411.000i	− 0.677100i	−0.940948	−	0.338550i	$-0.890063\pi$
$607$	− 411.000i	− 0.677100i	0.940948	−	0.338550i	$-0.109937\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1360.00i	2.22586i
$612$	0	0
$613$	−1039.00	−1.69494	−0.847471	−	0.530841i	$-0.821876\pi$
$613$	−1039.00	−1.69494	−0.847471	+	0.530841i	$0.821876\pi$
$614$	0	0
$615$	− 1800.00i	− 2.92683i
$616$	0	0
$617$	−248.000	−0.401945	−0.200972	−	0.979597i	$-0.564410\pi$
$617$	−248.000	−0.401945	−0.200972	+	0.979597i	$0.564410\pi$
$618$	0	0
$619$	− 603.000i	− 0.974152i	−0.873360	−	0.487076i	$-0.838063\pi$
$619$	− 603.000i	− 0.974152i	0.873360	−	0.487076i	$-0.161937\pi$
$620$	0	0
$621$	665.000	1.07085
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1111.00	1.77760
$626$	0	0
$627$	165.000i	0.263158i
$628$	0	0
$629$	7.00000	0.0111288
$630$	0	0
$631$	− 816.000i	− 1.29319i	−0.762835	−	0.646593i	$-0.776193\pi$
$631$	− 816.000i	− 1.29319i	0.762835	−	0.646593i	$-0.223807\pi$
$632$	0	0
$633$	−200.000	−0.315956
$634$	0	0
$635$	− 864.000i	− 1.36063i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 768.000i	− 1.20188i
$640$	0	0
$641$	639.000	0.996880	0.498440	−	0.866924i	$-0.333906\pi$
$641$	639.000	0.996880	0.498440	+	0.866924i	$0.333906\pi$
$642$	0	0
$643$	664.000i	1.03266i	0.856390	+	0.516330i	$0.172702\pi$
$643$	664.000i	1.03266i	−0.856390	+	0.516330i	$0.827298\pi$
$644$	0	0
$645$	−1800.00	−2.79070
$646$	0	0
$647$	1261.00i	1.94900i	0.224398	+	0.974498i	$0.427959\pi$
$647$	1261.00i	1.94900i	−0.224398	+	0.974498i	$0.572041\pi$
$648$	0	0
$649$	−159.000	−0.244992
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−593.000	−0.908116	−0.454058	−	0.890972i	$-0.650024\pi$
$653$	−593.000	−0.908116	−0.454058	+	0.890972i	$0.650024\pi$
$654$	0	0
$655$	− 1269.00i	− 1.93740i
$656$	0	0
$657$	−2288.00	−3.48250
$658$	0	0
$659$	− 1192.00i	− 1.80880i	−0.426684	−	0.904401i	$-0.640318\pi$
$659$	− 1192.00i	− 1.80880i	0.426684	−	0.904401i	$-0.359682\pi$
$660$	0	0
$661$	−473.000	−0.715582	−0.357791	−	0.933802i	$-0.616470\pi$
$661$	−473.000	−0.715582	−0.357791	+	0.933802i	$0.616470\pi$
$662$	0	0
$663$	− 560.000i	− 0.844646i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 608.000i	− 0.911544i
$668$	0	0
$669$	880.000	1.31540
$670$	0	0
$671$	237.000i	0.353204i
$672$	0	0
$673$	824.000	1.22437	0.612184	−	0.790715i	$-0.290291\pi$
$673$	824.000	1.22437	0.612184	+	0.790715i	$0.290291\pi$
$674$	0	0
$675$	− 1960.00i	− 2.90370i
$676$	0	0
$677$	−263.000	−0.388479	−0.194239	−	0.980954i	$-0.562224\pi$
$677$	−263.000	−0.388479	−0.194239	+	0.980954i	$0.562224\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1935.00	−2.84141
$682$	0	0
$683$	643.000i	0.941435i	0.882284	+	0.470717i	$0.156005\pi$
$683$	643.000i	0.941435i	−0.882284	+	0.470717i	$0.843995\pi$
$684$	0	0
$685$	81.0000	0.118248
$686$	0	0
$687$	− 915.000i	− 1.33188i
$688$	0	0
$689$	112.000	0.162554
$690$	0	0
$691$	− 125.000i	− 0.180897i	−0.995901	−	0.0904486i	$-0.971170\pi$
$691$	− 125.000i	− 0.180897i	0.995901	−	0.0904486i	$-0.0288301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1224.00i	1.76115i
$696$	0	0
$697$	280.000	0.401722
$698$	0	0
$699$	1805.00i	2.58226i
$700$	0	0
$701$	−54.0000	−0.0770328	−0.0385164	−	0.999258i	$-0.512263\pi$
$701$	−54.0000	−0.0770328	−0.0385164	+	0.999258i	$0.512263\pi$
$702$	0	0
$703$	11.0000i	0.0156472i
$704$	0	0
$705$	−3825.00	−5.42553
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1151.00	1.62341	0.811707	−	0.584065i	$-0.198539\pi$
$709$	1151.00	1.62341	0.811707	+	0.584065i	$0.198539\pi$
$710$	0	0
$711$	− 560.000i	− 0.787623i
$712$	0	0
$713$	−209.000	−0.293128
$714$	0	0
$715$	432.000i	0.604196i
$716$	0	0
$717$	250.000	0.348675
$718$	0	0
$719$	403.000i	0.560501i	0.959927	+	0.280250i	$0.0904175\pi$
$719$	403.000i	0.560501i	−0.959927	+	0.280250i	$0.909583\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 645.000i	− 0.892116i
$724$	0	0
$725$	−1792.00	−2.47172
$726$	0	0
$727$	496.000i	0.682256i	0.940017	+	0.341128i	$0.110809\pi$
$727$	496.000i	0.682256i	−0.940017	+	0.341128i	$0.889191\pi$
$728$	0	0
$729$	1079.00	1.48011
$730$	0	0
$731$	− 280.000i	− 0.383037i
$732$	0	0
$733$	−161.000	−0.219645	−0.109823	−	0.993951i	$-0.535028\pi$
$733$	−161.000	−0.219645	−0.109823	+	0.993951i	$0.535028\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.0000	0.0447761
$738$	0	0
$739$	547.000i	0.740189i	0.928994	+	0.370095i	$0.120675\pi$
$739$	547.000i	0.740189i	−0.928994	+	0.370095i	$0.879325\pi$
$740$	0	0
$741$	880.000	1.18758
$742$	0	0
$743$	− 256.000i	− 0.344549i	−0.985049	−	0.172275i	$-0.944888\pi$
$743$	− 256.000i	− 0.344549i	0.985049	−	0.172275i	$-0.0551116\pi$
$744$	0	0
$745$	513.000	0.688591
$746$	0	0
$747$	− 128.000i	− 0.171352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 1131.00i	− 1.50599i	−0.658025	−	0.752996i	$-0.728608\pi$
$751$	− 1131.00i	− 1.50599i	0.658025	−	0.752996i	$-0.271392\pi$
$752$	0	0
$753$	1970.00	2.61620
$754$	0	0
$755$	243.000i	0.321854i
$756$	0	0
$757$	−336.000	−0.443857	−0.221929	−	0.975063i	$-0.571235\pi$
$757$	−336.000	−0.443857	−0.221929	+	0.975063i	$0.571235\pi$
$758$	0	0
$759$	− 285.000i	− 0.375494i
$760$	0	0
$761$	345.000	0.453351	0.226675	−	0.973970i	$-0.427214\pi$
$761$	345.000	0.453351	0.226675	+	0.973970i	$0.427214\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1008.00	1.31765
$766$	0	0
$767$	848.000i	1.10561i
$768$	0	0
$769$	568.000	0.738622	0.369311	−	0.929306i	$-0.379594\pi$
$769$	568.000	0.738622	0.369311	+	0.929306i	$0.379594\pi$
$770$	0	0
$771$	445.000i	0.577173i
$772$	0	0
$773$	−559.000	−0.723157	−0.361578	−	0.932342i	$-0.617762\pi$
$773$	−559.000	−0.723157	−0.361578	+	0.932342i	$0.617762\pi$
$774$	0	0
$775$	616.000i	0.794839i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	440.000i	0.564827i
$780$	0	0
$781$	−144.000	−0.184379
$782$	0	0
$783$	1120.00i	1.43040i
$784$	0	0
$785$	1503.00	1.91465
$786$	0	0
$787$	661.000i	0.839898i	0.907548	+	0.419949i	$0.137952\pi$
$787$	661.000i	0.839898i	−0.907548	+	0.419949i	$0.862048\pi$
$788$	0	0
$789$	−225.000	−0.285171
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1264.00	1.59395
$794$	0	0
$795$	315.000i	0.396226i
$796$	0	0
$797$	−736.000	−0.923463	−0.461731	−	0.887020i	$-0.652772\pi$
$797$	−736.000	−0.923463	−0.461731	+	0.887020i	$0.652772\pi$
$798$	0	0
$799$	− 595.000i	− 0.744681i
$800$	0	0
$801$	1552.00	1.93758
$802$	0	0
$803$	429.000i	0.534247i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2285.00i	− 2.83147i
$808$	0	0
$809$	287.000	0.354759	0.177379	−	0.984143i	$-0.443238\pi$
$809$	287.000	0.354759	0.177379	+	0.984143i	$0.443238\pi$
$810$	0	0
$811$	− 1272.00i	− 1.56843i	−0.620487	−	0.784217i	$-0.713065\pi$
$811$	− 1272.00i	− 1.56843i	0.620487	−	0.784217i	$-0.286935\pi$
$812$	0	0
$813$	1985.00	2.44157
$814$	0	0
$815$	171.000i	0.209816i
$816$	0	0
$817$	440.000	0.538556
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−743.000	−0.904994	−0.452497	−	0.891766i	$-0.649467\pi$
$821$	−743.000	−0.904994	−0.452497	+	0.891766i	$0.649467\pi$
$822$	0	0
$823$	− 227.000i	− 0.275820i	−0.990445	−	0.137910i	$-0.955962\pi$
$823$	− 227.000i	− 0.275820i	0.990445	−	0.137910i	$-0.0440385\pi$
$824$	0	0
$825$	−840.000	−1.01818
$826$	0	0
$827$	− 120.000i	− 0.145103i	−0.997365	−	0.0725514i	$-0.976886\pi$
$827$	− 120.000i	− 0.145103i	0.997365	−	0.0725514i	$-0.0231141\pi$
$828$	0	0
$829$	241.000	0.290712	0.145356	−	0.989379i	$-0.453567\pi$
$829$	241.000	0.290712	0.145356	+	0.989379i	$0.453567\pi$
$830$	0	0
$831$	1155.00i	1.38989i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 1710.00i	− 2.04790i
$836$	0	0
$837$	385.000	0.459976
$838$	0	0
$839$	880.000i	1.04887i	0.851451	+	0.524434i	$0.175723\pi$
$839$	880.000i	1.04887i	−0.851451	+	0.524434i	$0.824277\pi$
$840$	0	0
$841$	183.000	0.217598
$842$	0	0
$843$	520.000i	0.616845i
$844$	0	0
$845$	783.000	0.926627
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1985.00	2.33804
$850$	0	0
$851$	− 19.0000i	− 0.0223267i
$852$	0	0
$853$	176.000	0.206331	0.103165	−	0.994664i	$-0.467103\pi$
$853$	176.000	0.206331	0.103165	+	0.994664i	$0.467103\pi$
$854$	0	0
$855$	1584.00i	1.85263i
$856$	0	0
$857$	255.000	0.297550	0.148775	−	0.988871i	$-0.452467\pi$
$857$	255.000	0.297550	0.148775	+	0.988871i	$0.452467\pi$
$858$	0	0
$859$	645.000i	0.750873i	0.926848	+	0.375437i	$0.122507\pi$
$859$	645.000i	0.750873i	−0.926848	+	0.375437i	$0.877493\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	835.000i	0.967555i	0.875191	+	0.483778i	$0.160736\pi$
$863$	835.000i	0.967555i	−0.875191	+	0.483778i	$0.839264\pi$
$864$	0	0
$865$	−1305.00	−1.50867
$866$	0	0
$867$	− 1200.00i	− 1.38408i
$868$	0	0
$869$	−105.000	−0.120829
$870$	0	0
$871$	− 176.000i	− 0.202067i
$872$	0	0
$873$	1408.00	1.61283
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1143.00	−1.30331	−0.651653	−	0.758517i	$-0.725924\pi$
$877$	−1143.00	−1.30331	−0.651653	+	0.758517i	$0.725924\pi$
$878$	0	0
$879$	770.000i	0.875995i
$880$	0	0
$881$	−930.000	−1.05562	−0.527809	−	0.849363i	$-0.676986\pi$
$881$	−930.000	−1.05562	−0.527809	+	0.849363i	$0.676986\pi$
$882$	0	0
$883$	1320.00i	1.49490i	0.664316	+	0.747452i	$0.268723\pi$
$883$	1320.00i	1.49490i	−0.664316	+	0.747452i	$0.731277\pi$
$884$	0	0
$885$	−2385.00	−2.69492
$886$	0	0
$887$	1061.00i	1.19617i	0.801434	+	0.598083i	$0.204071\pi$
$887$	1061.00i	1.19617i	−0.801434	+	0.598083i	$0.795929\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	93.0000i	0.104377i
$892$	0	0
$893$	935.000	1.04703
$894$	0	0
$895$	− 2853.00i	− 3.18771i
$896$	0	0
$897$	−1520.00	−1.69454
$898$	0	0
$899$	− 352.000i	− 0.391546i
$900$	0	0
$901$	−49.0000	−0.0543840
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1674.00	−1.84972
$906$	0	0
$907$	− 891.000i	− 0.982359i	−0.871058	−	0.491180i	$-0.836566\pi$
$907$	− 891.000i	− 0.982359i	0.871058	−	0.491180i	$-0.163434\pi$
$908$	0	0
$909$	240.000	0.264026
$910$	0	0
$911$	− 464.000i	− 0.509330i	−0.967029	−	0.254665i	$-0.918035\pi$
$911$	− 464.000i	− 0.509330i	0.967029	−	0.254665i	$-0.0819653\pi$
$912$	0	0
$913$	−24.0000	−0.0262870
$914$	0	0
$915$	3555.00i	3.88525i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1445.00i	1.57236i	0.617997	+	0.786181i	$0.287944\pi$
$919$	1445.00i	1.57236i	−0.617997	+	0.786181i	$0.712056\pi$
$920$	0	0
$921$	1880.00	2.04126
$922$	0	0
$923$	768.000i	0.832069i
$924$	0	0
$925$	−56.0000	−0.0605405
$926$	0	0
$927$	1488.00i	1.60518i
$928$	0	0
$929$	−1601.00	−1.72336	−0.861679	−	0.507453i	$-0.830587\pi$
$929$	−1601.00	−1.72336	−0.861679	+	0.507453i	$0.830587\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1615.00	1.73098
$934$	0	0
$935$	− 189.000i	− 0.202139i
$936$	0	0
$937$	1554.00	1.65848	0.829242	−	0.558889i	$-0.188772\pi$
$937$	1554.00	1.65848	0.829242	+	0.558889i	$0.188772\pi$
$938$	0	0
$939$	955.000i	1.01704i
$940$	0	0
$941$	−985.000	−1.04676	−0.523379	−	0.852100i	$-0.675329\pi$
$941$	−985.000	−1.04676	−0.523379	+	0.852100i	$0.675329\pi$
$942$	0	0
$943$	− 760.000i	− 0.805938i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	323.000i	0.341077i	0.985351	+	0.170539i	$0.0545507\pi$
$947$	323.000i	0.341077i	−0.985351	+	0.170539i	$0.945449\pi$
$948$	0	0
$949$	2288.00	2.41096
$950$	0	0
$951$	1675.00i	1.76130i
$952$	0	0
$953$	−168.000	−0.176285	−0.0881427	−	0.996108i	$-0.528093\pi$
$953$	−168.000	−0.176285	−0.0881427	+	0.996108i	$0.528093\pi$
$954$	0	0
$955$	− 2547.00i	− 2.66702i
$956$	0	0
$957$	480.000	0.501567
$958$	0	0
$959$	0	0
$960$	0	0
$961$	840.000	0.874089
$962$	0	0
$963$	176.000i	0.182762i
$964$	0	0
$965$	−2655.00	−2.75130
$966$	0	0
$967$	− 112.000i	− 0.115822i	−0.998322	−	0.0579111i	$-0.981556\pi$
$967$	− 112.000i	− 0.115822i	0.998322	−	0.0579111i	$-0.0184440\pi$
$968$	0	0
$969$	−385.000	−0.397317
$970$	0	0
$971$	− 1885.00i	− 1.94130i	−0.240501	−	0.970649i	$-0.577312\pi$
$971$	− 1885.00i	− 1.94130i	0.240501	−	0.970649i	$-0.422688\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4480.00i	4.59487i
$976$	0	0
$977$	25.0000	0.0255885	0.0127943	−	0.999918i	$-0.495927\pi$
$977$	25.0000	0.0255885	0.0127943	+	0.999918i	$0.495927\pi$
$978$	0	0
$979$	− 291.000i	− 0.297242i
$980$	0	0
$981$	−1808.00	−1.84302
$982$	0	0
$983$	− 85.0000i	− 0.0864700i	−0.999065	−	0.0432350i	$-0.986234\pi$
$983$	− 85.0000i	− 0.0864700i	0.999065	−	0.0432350i	$-0.0137664\pi$
$984$	0	0
$985$	1152.00	1.16954
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−760.000	−0.768453
$990$	0	0
$991$	157.000i	0.158426i	0.996858	+	0.0792129i	$0.0252407\pi$
$991$	157.000i	0.158426i	−0.996858	+	0.0792129i	$0.974759\pi$
$992$	0	0
$993$	−175.000	−0.176234
$994$	0	0
$995$	2547.00i	2.55980i
$996$	0	0
$997$	−297.000	−0.297894	−0.148947	−	0.988845i	$-0.547588\pi$
$997$	−297.000	−0.297894	−0.148947	+	0.988845i	$0.547588\pi$
$998$	0	0
$999$	35.0000i	0.0350350i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.d.e.1471.2		2
4.3	odd	2	inner	1568.3.d.e.1471.1		2
7.2	even	3		224.3.r.a.95.1	✓	4
7.4	even	3		224.3.r.a.191.2	yes	4
7.6	odd	2		1568.3.d.a.1471.1		2
28.11	odd	6		224.3.r.a.191.1	yes	4
28.23	odd	6		224.3.r.a.95.2	yes	4
28.27	even	2		1568.3.d.a.1471.2		2
56.11	odd	6		448.3.r.c.191.2		4
56.37	even	6		448.3.r.c.319.2		4
56.51	odd	6		448.3.r.c.319.1		4
56.53	even	6		448.3.r.c.191.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.r.a.95.1	✓	4	7.2	even	3
224.3.r.a.95.2	yes	4	28.23	odd	6
224.3.r.a.191.1	yes	4	28.11	odd	6
224.3.r.a.191.2	yes	4	7.4	even	3
448.3.r.c.191.1		4	56.53	even	6
448.3.r.c.191.2		4	56.11	odd	6
448.3.r.c.319.1		4	56.51	odd	6
448.3.r.c.319.2		4	56.37	even	6
1568.3.d.a.1471.1		2	7.6	odd	2
1568.3.d.a.1471.2		2	28.27	even	2
1568.3.d.e.1471.1		2	4.3	odd	2	inner
1568.3.d.e.1471.2		2	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{3} +9.00000 q^{5} -16.0000 q^{9} +O(q^{10})\)	\(q+5.00000i q^{3} +9.00000 q^{5} -16.0000 q^{9} +3.00000i q^{11} +16.0000 q^{13} +45.0000i q^{15} -7.00000 q^{17} -11.0000i q^{19} +19.0000i q^{23} +56.0000 q^{25} -35.0000i q^{27} -32.0000 q^{29} +11.0000i q^{31} -15.0000 q^{33} -1.00000 q^{37} +80.0000i q^{39} -40.0000 q^{41} +40.0000i q^{43} -144.000 q^{45} +85.0000i q^{47} -35.0000i q^{51} +7.00000 q^{53} +27.0000i q^{55} +55.0000 q^{57} +53.0000i q^{59} +79.0000 q^{61} +144.000 q^{65} -11.0000i q^{67} -95.0000 q^{69} +48.0000i q^{71} +143.000 q^{73} +280.000i q^{75} +35.0000i q^{79} +31.0000 q^{81} +8.00000i q^{83} -63.0000 q^{85} -160.000i q^{87} -97.0000 q^{89} -55.0000 q^{93} -99.0000i q^{95} -88.0000 q^{97} -48.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{5} - 32 q^{9}+O(q^{10}) \) 2 * q + 18 * q^5 - 32 * q^9	\( 2 q + 18 q^{5} - 32 q^{9} + 32 q^{13} - 14 q^{17} + 112 q^{25} - 64 q^{29} - 30 q^{33} - 2 q^{37} - 80 q^{41} - 288 q^{45} + 14 q^{53} + 110 q^{57} + 158 q^{61} + 288 q^{65} - 190 q^{69} + 286 q^{73} + 62 q^{81} - 126 q^{85} - 194 q^{89} - 110 q^{93} - 176 q^{97}+O(q^{100}) \) 2 * q + 18 * q^5 - 32 * q^9 + 32 * q^13 - 14 * q^17 + 112 * q^25 - 64 * q^29 - 30 * q^33 - 2 * q^37 - 80 * q^41 - 288 * q^45 + 14 * q^53 + 110 * q^57 + 158 * q^61 + 288 * q^65 - 190 * q^69 + 286 * q^73 + 62 * q^81 - 126 * q^85 - 194 * q^89 - 110 * q^93 - 176 * q^97

Introduction

L-functions

Modular forms

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