LMFDB - Embedded newform 7728.2.a.cd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.509552$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.41546 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.67510 q^{11} -4.66537 q^{13} +4.41546 q^{15} +6.24991 q^{17} +0.694209 q^{19} -1.00000 q^{21} +1.00000 q^{23} +14.4963 q^{25} +1.00000 q^{27} +5.60012 q^{29} -4.24991 q^{31} +1.67510 q^{33} -4.41546 q^{35} +9.26901 q^{37} -4.66537 q^{39} -5.15582 q^{41} -4.20376 q^{43} +4.41546 q^{45} +1.92501 q^{47} +1.00000 q^{49} +6.24991 q^{51} -1.84066 q^{53} +7.39636 q^{55} +0.694209 q^{57} -9.39779 q^{59} +7.72125 q^{61} -1.00000 q^{63} -20.5998 q^{65} +8.22728 q^{67} +1.00000 q^{69} +10.6654 q^{71} -11.9117 q^{73} +14.4963 q^{75} -1.67510 q^{77} -0.581012 q^{79} +1.00000 q^{81} -6.95032 q^{83} +27.5962 q^{85} +5.60012 q^{87} -13.4963 q^{89} +4.66537 q^{91} -4.24991 q^{93} +3.06525 q^{95} +10.0999 q^{97} +1.67510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9 + 5 * q^11 + 7 * q^13 + 5 * q^15 + 12 * q^17 - 3 * q^19 - 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 + 5 * q^33 - 5 * q^35 + 20 * q^37 + 7 * q^39 + 3 * q^41 - 9 * q^43 + 5 * q^45 - 7 * q^47 + 4 * q^49 + 12 * q^51 - 6 * q^53 + 21 * q^55 - 3 * q^57 + 2 * q^59 + 24 * q^61 - 4 * q^63 - 14 * q^65 - q^67 + 4 * q^69 + 17 * q^71 + 16 * q^73 + 7 * q^75 - 5 * q^77 + 10 * q^79 + 4 * q^81 - 8 * q^83 + 17 * q^85 + 6 * q^87 - 3 * q^89 - 7 * q^91 - 4 * q^93 + 3 * q^95 - 2 * q^97 + 5 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.41546	1.97465	0.987327	−	0.158699i	$-0.0507301\pi$
$5$	4.41546	1.97465	0.987327	+	0.158699i	$0.0507301\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.67510	0.505063	0.252531	−	0.967589i	$-0.418737\pi$
$11$	1.67510	0.505063	0.252531	+	0.967589i	$0.418737\pi$
$12$	0	0
$13$	−4.66537	−1.29394	−0.646970	−	0.762515i	$-0.723964\pi$
$13$	−4.66537	−1.29394	−0.646970	+	0.762515i	$0.723964\pi$
$14$	0	0
$15$	4.41546	1.14007
$16$	0	0
$17$	6.24991	1.51583	0.757913	−	0.652356i	$-0.226219\pi$
$17$	6.24991	1.51583	0.757913	+	0.652356i	$0.226219\pi$
$18$	0	0
$19$	0.694209	0.159262	0.0796312	−	0.996824i	$-0.474626\pi$
$19$	0.694209	0.159262	0.0796312	+	0.996824i	$0.474626\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	14.4963	2.89926
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.60012	1.03992	0.519958	−	0.854192i	$-0.325948\pi$
$29$	5.60012	1.03992	0.519958	+	0.854192i	$0.325948\pi$
$30$	0	0
$31$	−4.24991	−0.763306	−0.381653	−	0.924306i	$-0.624645\pi$
$31$	−4.24991	−0.763306	−0.381653	+	0.924306i	$0.624645\pi$
$32$	0	0
$33$	1.67510	0.291598
$34$	0	0
$35$	−4.41546	−0.746349
$36$	0	0
$37$	9.26901	1.52382	0.761908	−	0.647685i	$-0.224263\pi$
$37$	9.26901	1.52382	0.761908	+	0.647685i	$0.224263\pi$
$38$	0	0
$39$	−4.66537	−0.747057
$40$	0	0
$41$	−5.15582	−0.805203	−0.402602	−	0.915375i	$-0.631894\pi$
$41$	−5.15582	−0.805203	−0.402602	+	0.915375i	$0.631894\pi$
$42$	0	0
$43$	−4.20376	−0.641068	−0.320534	−	0.947237i	$-0.603862\pi$
$43$	−4.20376	−0.641068	−0.320534	+	0.947237i	$0.603862\pi$
$44$	0	0
$45$	4.41546	0.658218
$46$	0	0
$47$	1.92501	0.280792	0.140396	−	0.990095i	$-0.455162\pi$
$47$	1.92501	0.280792	0.140396	+	0.990095i	$0.455162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.24991	0.875162
$52$	0	0
$53$	−1.84066	−0.252833	−0.126417	−	0.991977i	$-0.540348\pi$
$53$	−1.84066	−0.252833	−0.126417	+	0.991977i	$0.540348\pi$
$54$	0	0
$55$	7.39636	0.997324
$56$	0	0
$57$	0.694209	0.0919502
$58$	0	0
$59$	−9.39779	−1.22349	−0.611744	−	0.791056i	$-0.709532\pi$
$59$	−9.39779	−1.22349	−0.611744	+	0.791056i	$0.709532\pi$
$60$	0	0
$61$	7.72125	0.988605	0.494302	−	0.869290i	$-0.335424\pi$
$61$	7.72125	0.988605	0.494302	+	0.869290i	$0.335424\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−20.5998	−2.55508
$66$	0	0
$67$	8.22728	1.00512	0.502561	−	0.864542i	$-0.332391\pi$
$67$	8.22728	1.00512	0.502561	+	0.864542i	$0.332391\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.6654	1.26575	0.632873	−	0.774255i	$-0.281875\pi$
$71$	10.6654	1.26575	0.632873	+	0.774255i	$0.281875\pi$
$72$	0	0
$73$	−11.9117	−1.39416	−0.697082	−	0.716991i	$-0.745519\pi$
$73$	−11.9117	−1.39416	−0.697082	+	0.716991i	$0.745519\pi$
$74$	0	0
$75$	14.4963	1.67389
$76$	0	0
$77$	−1.67510	−0.190896
$78$	0	0
$79$	−0.581012	−0.0653689	−0.0326845	−	0.999466i	$-0.510406\pi$
$79$	−0.581012	−0.0653689	−0.0326845	+	0.999466i	$0.510406\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.95032	−0.762897	−0.381449	−	0.924390i	$-0.624575\pi$
$83$	−6.95032	−0.762897	−0.381449	+	0.924390i	$0.624575\pi$
$84$	0	0
$85$	27.5962	2.99323
$86$	0	0
$87$	5.60012	0.600396
$88$	0	0
$89$	−13.4963	−1.43060	−0.715302	−	0.698816i	$-0.753711\pi$
$89$	−13.4963	−1.43060	−0.715302	+	0.698816i	$0.753711\pi$
$90$	0	0
$91$	4.66537	0.489064
$92$	0	0
$93$	−4.24991	−0.440695
$94$	0	0
$95$	3.06525	0.314488
$96$	0	0
$97$	10.0999	1.02549	0.512746	−	0.858540i	$-0.328628\pi$
$97$	10.0999	1.02549	0.512746	+	0.858540i	$0.328628\pi$
$98$	0	0
$99$	1.67510	0.168354
$100$	0	0
$101$	−13.7830	−1.37146	−0.685729	−	0.727857i	$-0.740516\pi$
$101$	−13.7830	−1.37146	−0.685729	+	0.727857i	$0.740516\pi$
$102$	0	0
$103$	−16.5553	−1.63125	−0.815623	−	0.578584i	$-0.803605\pi$
$103$	−16.5553	−1.63125	−0.815623	+	0.578584i	$0.803605\pi$
$104$	0	0
$105$	−4.41546	−0.430905
$106$	0	0
$107$	1.33463	0.129024	0.0645118	−	0.997917i	$-0.479451\pi$
$107$	1.33463	0.129024	0.0645118	+	0.997917i	$0.479451\pi$
$108$	0	0
$109$	4.00794	0.383891	0.191945	−	0.981406i	$-0.438520\pi$
$109$	4.00794	0.383891	0.191945	+	0.981406i	$0.438520\pi$
$110$	0	0
$111$	9.26901	0.879776
$112$	0	0
$113$	−9.06525	−0.852787	−0.426394	−	0.904538i	$-0.640216\pi$
$113$	−9.06525	−0.852787	−0.426394	+	0.904538i	$0.640216\pi$
$114$	0	0
$115$	4.41546	0.411744
$116$	0	0
$117$	−4.66537	−0.431314
$118$	0	0
$119$	−6.24991	−0.572928
$120$	0	0
$121$	−8.19403	−0.744911
$122$	0	0
$123$	−5.15582	−0.464884
$124$	0	0
$125$	41.9305	3.75038
$126$	0	0
$127$	21.4977	1.90761	0.953807	−	0.300419i	$-0.0971267\pi$
$127$	21.4977	1.90761	0.953807	+	0.300419i	$0.0971267\pi$
$128$	0	0
$129$	−4.20376	−0.370121
$130$	0	0
$131$	−1.17529	−0.102685	−0.0513426	−	0.998681i	$-0.516350\pi$
$131$	−1.17529	−0.102685	−0.0513426	+	0.998681i	$0.516350\pi$
$132$	0	0
$133$	−0.694209	−0.0601956
$134$	0	0
$135$	4.41546	0.380022
$136$	0	0
$137$	15.8562	1.35469	0.677345	−	0.735666i	$-0.263131\pi$
$137$	15.8562	1.35469	0.677345	+	0.735666i	$0.263131\pi$
$138$	0	0
$139$	1.93438	0.164072	0.0820361	−	0.996629i	$-0.473858\pi$
$139$	1.93438	0.164072	0.0820361	+	0.996629i	$0.473858\pi$
$140$	0	0
$141$	1.92501	0.162115
$142$	0	0
$143$	−7.81498	−0.653521
$144$	0	0
$145$	24.7271	2.05347
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	2.61301	0.214067	0.107033	−	0.994255i	$-0.465865\pi$
$149$	2.61301	0.214067	0.107033	+	0.994255i	$0.465865\pi$
$150$	0	0
$151$	9.26281	0.753797	0.376898	−	0.926255i	$-0.376991\pi$
$151$	9.26281	0.753797	0.376898	+	0.926255i	$0.376991\pi$
$152$	0	0
$153$	6.24991	0.505275
$154$	0	0
$155$	−18.7653	−1.50727
$156$	0	0
$157$	−7.77504	−0.620516	−0.310258	−	0.950652i	$-0.600415\pi$
$157$	−7.77504	−0.620516	−0.310258	+	0.950652i	$0.600415\pi$
$158$	0	0
$159$	−1.84066	−0.145973
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	7.12077	0.557742	0.278871	−	0.960329i	$-0.410040\pi$
$163$	7.12077	0.557742	0.278871	+	0.960329i	$0.410040\pi$
$164$	0	0
$165$	7.39636	0.575805
$166$	0	0
$167$	11.9867	0.927562	0.463781	−	0.885950i	$-0.346493\pi$
$167$	11.9867	0.927562	0.463781	+	0.885950i	$0.346493\pi$
$168$	0	0
$169$	8.76567	0.674282
$170$	0	0
$171$	0.694209	0.0530875
$172$	0	0
$173$	−2.09993	−0.159655	−0.0798275	−	0.996809i	$-0.525437\pi$
$173$	−2.09993	−0.159655	−0.0798275	+	0.996809i	$0.525437\pi$
$174$	0	0
$175$	−14.4963	−1.09582
$176$	0	0
$177$	−9.39779	−0.706381
$178$	0	0
$179$	15.9726	1.19385	0.596924	−	0.802298i	$-0.296389\pi$
$179$	15.9726	1.19385	0.596924	+	0.802298i	$0.296389\pi$
$180$	0	0
$181$	−2.89970	−0.215533	−0.107767	−	0.994176i	$-0.534370\pi$
$181$	−2.89970	−0.215533	−0.107767	+	0.994176i	$0.534370\pi$
$182$	0	0
$183$	7.72125	0.570771
$184$	0	0
$185$	40.9270	3.00901
$186$	0	0
$187$	10.4692	0.765587
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	21.5486	1.55921	0.779603	−	0.626275i	$-0.215421\pi$
$191$	21.5486	1.55921	0.779603	+	0.626275i	$0.215421\pi$
$192$	0	0
$193$	4.58722	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$193$	4.58722	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$194$	0	0
$195$	−20.5998	−1.47518
$196$	0	0
$197$	−11.8656	−0.845389	−0.422695	−	0.906272i	$-0.638916\pi$
$197$	−11.8656	−0.845389	−0.422695	+	0.906272i	$0.638916\pi$
$198$	0	0
$199$	2.52866	0.179252	0.0896259	−	0.995976i	$-0.471433\pi$
$199$	2.52866	0.179252	0.0896259	+	0.995976i	$0.471433\pi$
$200$	0	0
$201$	8.22728	0.580307
$202$	0	0
$203$	−5.60012	−0.393051
$204$	0	0
$205$	−22.7653	−1.59000
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	1.16287	0.0804376
$210$	0	0
$211$	21.9797	1.51314	0.756572	−	0.653911i	$-0.226873\pi$
$211$	21.9797	1.51314	0.756572	+	0.653911i	$0.226873\pi$
$212$	0	0
$213$	10.6654	0.730779
$214$	0	0
$215$	−18.5615	−1.26589
$216$	0	0
$217$	4.24991	0.288503
$218$	0	0
$219$	−11.9117	−0.804921
$220$	0	0
$221$	−29.1581	−1.96139
$222$	0	0
$223$	−0.0302726	−0.00202720	−0.00101360	−	0.999999i	$-0.500323\pi$
$223$	−0.0302726	−0.00202720	−0.00101360	+	0.999999i	$0.500323\pi$
$224$	0	0
$225$	14.4963	0.966419
$226$	0	0
$227$	−21.2393	−1.40970	−0.704852	−	0.709355i	$-0.748986\pi$
$227$	−21.2393	−1.40970	−0.704852	+	0.709355i	$0.748986\pi$
$228$	0	0
$229$	17.4360	1.15220	0.576102	−	0.817378i	$-0.304573\pi$
$229$	17.4360	1.15220	0.576102	+	0.817378i	$0.304573\pi$
$230$	0	0
$231$	−1.67510	−0.110214
$232$	0	0
$233$	−11.2082	−0.734272	−0.367136	−	0.930167i	$-0.619662\pi$
$233$	−11.2082	−0.734272	−0.367136	+	0.930167i	$0.619662\pi$
$234$	0	0
$235$	8.49982	0.554467
$236$	0	0
$237$	−0.581012	−0.0377408
$238$	0	0
$239$	18.5654	1.20090	0.600449	−	0.799663i	$-0.294989\pi$
$239$	18.5654	1.20090	0.600449	+	0.799663i	$0.294989\pi$
$240$	0	0
$241$	−13.6556	−0.879637	−0.439818	−	0.898087i	$-0.644957\pi$
$241$	−13.6556	−0.879637	−0.439818	+	0.898087i	$0.644957\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.41546	0.282093
$246$	0	0
$247$	−3.23874	−0.206076
$248$	0	0
$249$	−6.95032	−0.440459
$250$	0	0
$251$	−0.604526	−0.0381574	−0.0190787	−	0.999818i	$-0.506073\pi$
$251$	−0.604526	−0.0381574	−0.0190787	+	0.999818i	$0.506073\pi$
$252$	0	0
$253$	1.67510	0.105313
$254$	0	0
$255$	27.5962	1.72814
$256$	0	0
$257$	14.1602	0.883291	0.441645	−	0.897190i	$-0.354395\pi$
$257$	14.1602	0.883291	0.441645	+	0.897190i	$0.354395\pi$
$258$	0	0
$259$	−9.26901	−0.575948
$260$	0	0
$261$	5.60012	0.346639
$262$	0	0
$263$	−20.5788	−1.26895	−0.634473	−	0.772945i	$-0.718783\pi$
$263$	−20.5788	−1.26895	−0.634473	+	0.772945i	$0.718783\pi$
$264$	0	0
$265$	−8.12734	−0.499259
$266$	0	0
$267$	−13.4963	−0.825960
$268$	0	0
$269$	−0.496655	−0.0302816	−0.0151408	−	0.999885i	$-0.504820\pi$
$269$	−0.496655	−0.0302816	−0.0151408	+	0.999885i	$0.504820\pi$
$270$	0	0
$271$	−24.0547	−1.46122	−0.730609	−	0.682797i	$-0.760764\pi$
$271$	−24.0547	−1.46122	−0.730609	+	0.682797i	$0.760764\pi$
$272$	0	0
$273$	4.66537	0.282361
$274$	0	0
$275$	24.2828	1.46431
$276$	0	0
$277$	23.2402	1.39637	0.698183	−	0.715919i	$-0.253992\pi$
$277$	23.2402	1.39637	0.698183	+	0.715919i	$0.253992\pi$
$278$	0	0
$279$	−4.24991	−0.254435
$280$	0	0
$281$	−15.0697	−0.898985	−0.449492	−	0.893284i	$-0.648395\pi$
$281$	−15.0697	−0.898985	−0.449492	+	0.893284i	$0.648395\pi$
$282$	0	0
$283$	−9.22691	−0.548483	−0.274241	−	0.961661i	$-0.588427\pi$
$283$	−9.22691	−0.548483	−0.274241	+	0.961661i	$0.588427\pi$
$284$	0	0
$285$	3.06525	0.181570
$286$	0	0
$287$	5.15582	0.304338
$288$	0	0
$289$	22.0614	1.29773
$290$	0	0
$291$	10.0999	0.592069
$292$	0	0
$293$	−12.3881	−0.723718	−0.361859	−	0.932233i	$-0.617858\pi$
$293$	−12.3881	−0.723718	−0.361859	+	0.932233i	$0.617858\pi$
$294$	0	0
$295$	−41.4956	−2.41596
$296$	0	0
$297$	1.67510	0.0971994
$298$	0	0
$299$	−4.66537	−0.269805
$300$	0	0
$301$	4.20376	0.242301
$302$	0	0
$303$	−13.7830	−0.791811
$304$	0	0
$305$	34.0929	1.95215
$306$	0	0
$307$	−0.823281	−0.0469871	−0.0234936	−	0.999724i	$-0.507479\pi$
$307$	−0.823281	−0.0469871	−0.0234936	+	0.999724i	$0.507479\pi$
$308$	0	0
$309$	−16.5553	−0.941800
$310$	0	0
$311$	30.6632	1.73875	0.869375	−	0.494152i	$-0.164521\pi$
$311$	30.6632	1.73875	0.869375	+	0.494152i	$0.164521\pi$
$312$	0	0
$313$	13.1323	0.742282	0.371141	−	0.928577i	$-0.378967\pi$
$313$	13.1323	0.742282	0.371141	+	0.928577i	$0.378967\pi$
$314$	0	0
$315$	−4.41546	−0.248783
$316$	0	0
$317$	−15.2269	−0.855229	−0.427614	−	0.903961i	$-0.640646\pi$
$317$	−15.2269	−0.855229	−0.427614	+	0.903961i	$0.640646\pi$
$318$	0	0
$319$	9.38078	0.525223
$320$	0	0
$321$	1.33463	0.0744918
$322$	0	0
$323$	4.33874	0.241414
$324$	0	0
$325$	−67.6305	−3.75147
$326$	0	0
$327$	4.00794	0.221639
$328$	0	0
$329$	−1.92501	−0.106129
$330$	0	0
$331$	−25.5674	−1.40531	−0.702655	−	0.711530i	$-0.748002\pi$
$331$	−25.5674	−1.40531	−0.702655	+	0.711530i	$0.748002\pi$
$332$	0	0
$333$	9.26901	0.507939
$334$	0	0
$335$	36.3272	1.98477
$336$	0	0
$337$	16.2890	0.887318	0.443659	−	0.896196i	$-0.353680\pi$
$337$	16.2890	0.887318	0.443659	+	0.896196i	$0.353680\pi$
$338$	0	0
$339$	−9.06525	−0.492357
$340$	0	0
$341$	−7.11904	−0.385518
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.41546	0.237720
$346$	0	0
$347$	−21.6729	−1.16346	−0.581732	−	0.813380i	$-0.697625\pi$
$347$	−21.6729	−1.16346	−0.581732	+	0.813380i	$0.697625\pi$
$348$	0	0
$349$	−21.5198	−1.15193	−0.575964	−	0.817475i	$-0.695373\pi$
$349$	−21.5198	−1.15193	−0.575964	+	0.817475i	$0.695373\pi$
$350$	0	0
$351$	−4.66537	−0.249019
$352$	0	0
$353$	−20.1852	−1.07435	−0.537174	−	0.843471i	$-0.680508\pi$
$353$	−20.1852	−1.07435	−0.537174	+	0.843471i	$0.680508\pi$
$354$	0	0
$355$	47.0925	2.49941
$356$	0	0
$357$	−6.24991	−0.330780
$358$	0	0
$359$	−23.2492	−1.22705	−0.613524	−	0.789676i	$-0.710249\pi$
$359$	−23.2492	−1.22705	−0.613524	+	0.789676i	$0.710249\pi$
$360$	0	0
$361$	−18.5181	−0.974635
$362$	0	0
$363$	−8.19403	−0.430075
$364$	0	0
$365$	−52.5959	−2.75299
$366$	0	0
$367$	−27.7171	−1.44682	−0.723409	−	0.690419i	$-0.757426\pi$
$367$	−27.7171	−1.44682	−0.723409	+	0.690419i	$0.757426\pi$
$368$	0	0
$369$	−5.15582	−0.268401
$370$	0	0
$371$	1.84066	0.0955621
$372$	0	0
$373$	−24.7303	−1.28048	−0.640242	−	0.768173i	$-0.721166\pi$
$373$	−24.7303	−1.28048	−0.640242	+	0.768173i	$0.721166\pi$
$374$	0	0
$375$	41.9305	2.16528
$376$	0	0
$377$	−26.1266	−1.34559
$378$	0	0
$379$	3.11140	0.159822	0.0799109	−	0.996802i	$-0.474536\pi$
$379$	3.11140	0.159822	0.0799109	+	0.996802i	$0.474536\pi$
$380$	0	0
$381$	21.4977	1.10136
$382$	0	0
$383$	18.9503	0.968316	0.484158	−	0.874980i	$-0.339126\pi$
$383$	18.9503	0.968316	0.484158	+	0.874980i	$0.339126\pi$
$384$	0	0
$385$	−7.39636	−0.376953
$386$	0	0
$387$	−4.20376	−0.213689
$388$	0	0
$389$	−20.1693	−1.02262	−0.511312	−	0.859395i	$-0.670840\pi$
$389$	−20.1693	−1.02262	−0.511312	+	0.859395i	$0.670840\pi$
$390$	0	0
$391$	6.24991	0.316071
$392$	0	0
$393$	−1.17529	−0.0592854
$394$	0	0
$395$	−2.56543	−0.129081
$396$	0	0
$397$	8.41634	0.422404	0.211202	−	0.977442i	$-0.432262\pi$
$397$	8.41634	0.422404	0.211202	+	0.977442i	$0.432262\pi$
$398$	0	0
$399$	−0.694209	−0.0347539
$400$	0	0
$401$	−5.86329	−0.292799	−0.146399	−	0.989226i	$-0.546768\pi$
$401$	−5.86329	−0.292799	−0.146399	+	0.989226i	$0.546768\pi$
$402$	0	0
$403$	19.8274	0.987673
$404$	0	0
$405$	4.41546	0.219406
$406$	0	0
$407$	15.5266	0.769623
$408$	0	0
$409$	−26.2686	−1.29890	−0.649451	−	0.760404i	$-0.725001\pi$
$409$	−26.2686	−1.29890	−0.649451	+	0.760404i	$0.725001\pi$
$410$	0	0
$411$	15.8562	0.782130
$412$	0	0
$413$	9.39779	0.462435
$414$	0	0
$415$	−30.6889	−1.50646
$416$	0	0
$417$	1.93438	0.0947271
$418$	0	0
$419$	0.800648	0.0391142	0.0195571	−	0.999809i	$-0.493774\pi$
$419$	0.800648	0.0391142	0.0195571	+	0.999809i	$0.493774\pi$
$420$	0	0
$421$	23.3845	1.13969	0.569846	−	0.821751i	$-0.307003\pi$
$421$	23.3845	1.13969	0.569846	+	0.821751i	$0.307003\pi$
$422$	0	0
$423$	1.92501	0.0935973
$424$	0	0
$425$	90.6005	4.39477
$426$	0	0
$427$	−7.72125	−0.373658
$428$	0	0
$429$	−7.81498	−0.377311
$430$	0	0
$431$	33.4853	1.61293	0.806465	−	0.591281i	$-0.201378\pi$
$431$	33.4853	1.61293	0.806465	+	0.591281i	$0.201378\pi$
$432$	0	0
$433$	24.1373	1.15996	0.579982	−	0.814629i	$-0.303059\pi$
$433$	24.1373	1.15996	0.579982	+	0.814629i	$0.303059\pi$
$434$	0	0
$435$	24.7271	1.18557
$436$	0	0
$437$	0.694209	0.0332085
$438$	0	0
$439$	16.4469	0.784968	0.392484	−	0.919759i	$-0.371616\pi$
$439$	16.4469	0.784968	0.392484	+	0.919759i	$0.371616\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−9.58542	−0.455417	−0.227709	−	0.973729i	$-0.573123\pi$
$443$	−9.58542	−0.455417	−0.227709	+	0.973729i	$0.573123\pi$
$444$	0	0
$445$	−59.5923	−2.82495
$446$	0	0
$447$	2.61301	0.123591
$448$	0	0
$449$	4.17319	0.196945	0.0984725	−	0.995140i	$-0.468604\pi$
$449$	4.17319	0.196945	0.0984725	+	0.995140i	$0.468604\pi$
$450$	0	0
$451$	−8.63653	−0.406678
$452$	0	0
$453$	9.26281	0.435205
$454$	0	0
$455$	20.5998	0.965731
$456$	0	0
$457$	8.02668	0.375472	0.187736	−	0.982220i	$-0.439885\pi$
$457$	8.02668	0.375472	0.187736	+	0.982220i	$0.439885\pi$
$458$	0	0
$459$	6.24991	0.291721
$460$	0	0
$461$	11.6192	0.541158	0.270579	−	0.962698i	$-0.412785\pi$
$461$	11.6192	0.541158	0.270579	+	0.962698i	$0.412785\pi$
$462$	0	0
$463$	35.8714	1.66708	0.833542	−	0.552456i	$-0.186309\pi$
$463$	35.8714	1.66708	0.833542	+	0.552456i	$0.186309\pi$
$464$	0	0
$465$	−18.7653	−0.870220
$466$	0	0
$467$	13.1731	0.609579	0.304790	−	0.952420i	$-0.401414\pi$
$467$	13.1731	0.609579	0.304790	+	0.952420i	$0.401414\pi$
$468$	0	0
$469$	−8.22728	−0.379900
$470$	0	0
$471$	−7.77504	−0.358255
$472$	0	0
$473$	−7.04174	−0.323779
$474$	0	0
$475$	10.0635	0.461743
$476$	0	0
$477$	−1.84066	−0.0842778
$478$	0	0
$479$	−5.23080	−0.239002	−0.119501	−	0.992834i	$-0.538129\pi$
$479$	−5.23080	−0.239002	−0.119501	+	0.992834i	$0.538129\pi$
$480$	0	0
$481$	−43.2434	−1.97173
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	44.5959	2.02499
$486$	0	0
$487$	−28.9779	−1.31311	−0.656557	−	0.754277i	$-0.727988\pi$
$487$	−28.9779	−1.31311	−0.656557	+	0.754277i	$0.727988\pi$
$488$	0	0
$489$	7.12077	0.322012
$490$	0	0
$491$	16.6912	0.753262	0.376631	−	0.926363i	$-0.377083\pi$
$491$	16.6912	0.753262	0.376631	+	0.926363i	$0.377083\pi$
$492$	0	0
$493$	35.0002	1.57633
$494$	0	0
$495$	7.39636	0.332441
$496$	0	0
$497$	−10.6654	−0.478407
$498$	0	0
$499$	−9.57213	−0.428507	−0.214254	−	0.976778i	$-0.568732\pi$
$499$	−9.57213	−0.428507	−0.214254	+	0.976778i	$0.568732\pi$
$500$	0	0
$501$	11.9867	0.535528
$502$	0	0
$503$	20.6348	0.920060	0.460030	−	0.887903i	$-0.347839\pi$
$503$	20.6348	0.920060	0.460030	+	0.887903i	$0.347839\pi$
$504$	0	0
$505$	−60.8582	−2.70815
$506$	0	0
$507$	8.76567	0.389297
$508$	0	0
$509$	6.23665	0.276434	0.138217	−	0.990402i	$-0.455863\pi$
$509$	6.23665	0.276434	0.138217	+	0.990402i	$0.455863\pi$
$510$	0	0
$511$	11.9117	0.526945
$512$	0	0
$513$	0.694209	0.0306501
$514$	0	0
$515$	−73.0994	−3.22115
$516$	0	0
$517$	3.22460	0.141818
$518$	0	0
$519$	−2.09993	−0.0921769
$520$	0	0
$521$	12.6615	0.554709	0.277355	−	0.960768i	$-0.410542\pi$
$521$	12.6615	0.554709	0.277355	+	0.960768i	$0.410542\pi$
$522$	0	0
$523$	−43.3446	−1.89533	−0.947663	−	0.319272i	$-0.896562\pi$
$523$	−43.3446	−1.89533	−0.947663	+	0.319272i	$0.896562\pi$
$524$	0	0
$525$	−14.4963	−0.632670
$526$	0	0
$527$	−26.5615	−1.15704
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.39779	−0.407829
$532$	0	0
$533$	24.0538	1.04189
$534$	0	0
$535$	5.89301	0.254777
$536$	0	0
$537$	15.9726	0.689268
$538$	0	0
$539$	1.67510	0.0721518
$540$	0	0
$541$	5.87439	0.252560	0.126280	−	0.991995i	$-0.459696\pi$
$541$	5.87439	0.252560	0.126280	+	0.991995i	$0.459696\pi$
$542$	0	0
$543$	−2.89970	−0.124438
$544$	0	0
$545$	17.6969	0.758051
$546$	0	0
$547$	−13.8580	−0.592524	−0.296262	−	0.955107i	$-0.595740\pi$
$547$	−13.8580	−0.592524	−0.296262	+	0.955107i	$0.595740\pi$
$548$	0	0
$549$	7.72125	0.329535
$550$	0	0
$551$	3.88765	0.165620
$552$	0	0
$553$	0.581012	0.0247071
$554$	0	0
$555$	40.9270	1.73725
$556$	0	0
$557$	−18.5921	−0.787773	−0.393887	−	0.919159i	$-0.628870\pi$
$557$	−18.5921	−0.787773	−0.393887	+	0.919159i	$0.628870\pi$
$558$	0	0
$559$	19.6121	0.829503
$560$	0	0
$561$	10.4692	0.442012
$562$	0	0
$563$	−5.09582	−0.214763	−0.107382	−	0.994218i	$-0.534247\pi$
$563$	−5.09582	−0.214763	−0.107382	+	0.994218i	$0.534247\pi$
$564$	0	0
$565$	−40.0273	−1.68396
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−29.3028	−1.22844	−0.614218	−	0.789136i	$-0.710529\pi$
$569$	−29.3028	−1.22844	−0.614218	+	0.789136i	$0.710529\pi$
$570$	0	0
$571$	5.10736	0.213736	0.106868	−	0.994273i	$-0.465918\pi$
$571$	5.10736	0.213736	0.106868	+	0.994273i	$0.465918\pi$
$572$	0	0
$573$	21.5486	0.900208
$574$	0	0
$575$	14.4963	0.604537
$576$	0	0
$577$	24.7379	1.02985	0.514926	−	0.857235i	$-0.327819\pi$
$577$	24.7379	1.02985	0.514926	+	0.857235i	$0.327819\pi$
$578$	0	0
$579$	4.58722	0.190638
$580$	0	0
$581$	6.95032	0.288348
$582$	0	0
$583$	−3.08329	−0.127697
$584$	0	0
$585$	−20.5998	−0.851695
$586$	0	0
$587$	−38.8319	−1.60276	−0.801382	−	0.598152i	$-0.795902\pi$
$587$	−38.8319	−1.60276	−0.801382	+	0.598152i	$0.795902\pi$
$588$	0	0
$589$	−2.95032	−0.121566
$590$	0	0
$591$	−11.8656	−0.488086
$592$	0	0
$593$	36.9162	1.51596	0.757982	−	0.652275i	$-0.226185\pi$
$593$	36.9162	1.51596	0.757982	+	0.652275i	$0.226185\pi$
$594$	0	0
$595$	−27.5962	−1.13133
$596$	0	0
$597$	2.52866	0.103491
$598$	0	0
$599$	−40.4224	−1.65161	−0.825807	−	0.563953i	$-0.809280\pi$
$599$	−40.4224	−1.65161	−0.825807	+	0.563953i	$0.809280\pi$
$600$	0	0
$601$	2.98464	0.121746	0.0608730	−	0.998146i	$-0.480612\pi$
$601$	2.98464	0.121746	0.0608730	+	0.998146i	$0.480612\pi$
$602$	0	0
$603$	8.22728	0.335041
$604$	0	0
$605$	−36.1804	−1.47094
$606$	0	0
$607$	29.1893	1.18476	0.592378	−	0.805660i	$-0.298189\pi$
$607$	29.1893	1.18476	0.592378	+	0.805660i	$0.298189\pi$
$608$	0	0
$609$	−5.60012	−0.226928
$610$	0	0
$611$	−8.98090	−0.363328
$612$	0	0
$613$	−5.88595	−0.237731	−0.118866	−	0.992910i	$-0.537926\pi$
$613$	−5.88595	−0.237731	−0.118866	+	0.992910i	$0.537926\pi$
$614$	0	0
$615$	−22.7653	−0.917986
$616$	0	0
$617$	−8.07326	−0.325017	−0.162509	−	0.986707i	$-0.551959\pi$
$617$	−8.07326	−0.325017	−0.162509	+	0.986707i	$0.551959\pi$
$618$	0	0
$619$	−10.8589	−0.436457	−0.218228	−	0.975898i	$-0.570028\pi$
$619$	−10.8589	−0.436457	−0.218228	+	0.975898i	$0.570028\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	13.4963	0.540717
$624$	0	0
$625$	112.661	4.50644
$626$	0	0
$627$	1.16287	0.0464406
$628$	0	0
$629$	57.9305	2.30984
$630$	0	0
$631$	−23.1891	−0.923145	−0.461572	−	0.887103i	$-0.652715\pi$
$631$	−23.1891	−0.923145	−0.461572	+	0.887103i	$0.652715\pi$
$632$	0	0
$633$	21.9797	0.873614
$634$	0	0
$635$	94.9223	3.76688
$636$	0	0
$637$	−4.66537	−0.184849
$638$	0	0
$639$	10.6654	0.421915
$640$	0	0
$641$	−46.7138	−1.84509	−0.922543	−	0.385895i	$-0.873893\pi$
$641$	−46.7138	−1.84509	−0.922543	+	0.385895i	$0.873893\pi$
$642$	0	0
$643$	−33.7524	−1.33106	−0.665532	−	0.746369i	$-0.731795\pi$
$643$	−33.7524	−1.33106	−0.665532	+	0.746369i	$0.731795\pi$
$644$	0	0
$645$	−18.5615	−0.730860
$646$	0	0
$647$	−18.9277	−0.744124	−0.372062	−	0.928208i	$-0.621349\pi$
$647$	−18.9277	−0.744124	−0.372062	+	0.928208i	$0.621349\pi$
$648$	0	0
$649$	−15.7423	−0.617938
$650$	0	0
$651$	4.24991	0.166567
$652$	0	0
$653$	−10.1696	−0.397967	−0.198984	−	0.980003i	$-0.563764\pi$
$653$	−10.1696	−0.397967	−0.198984	+	0.980003i	$0.563764\pi$
$654$	0	0
$655$	−5.18943	−0.202768
$656$	0	0
$657$	−11.9117	−0.464722
$658$	0	0
$659$	4.60754	0.179484	0.0897421	−	0.995965i	$-0.471396\pi$
$659$	4.60754	0.179484	0.0897421	+	0.995965i	$0.471396\pi$
$660$	0	0
$661$	−31.1205	−1.21045	−0.605223	−	0.796056i	$-0.706916\pi$
$661$	−31.1205	−1.21045	−0.605223	+	0.796056i	$0.706916\pi$
$662$	0	0
$663$	−29.1581	−1.13241
$664$	0	0
$665$	−3.06525	−0.118865
$666$	0	0
$667$	5.60012	0.216837
$668$	0	0
$669$	−0.0302726	−0.00117041
$670$	0	0
$671$	12.9339	0.499308
$672$	0	0
$673$	−36.9447	−1.42411	−0.712057	−	0.702122i	$-0.752236\pi$
$673$	−36.9447	−1.42411	−0.712057	+	0.702122i	$0.752236\pi$
$674$	0	0
$675$	14.4963	0.557962
$676$	0	0
$677$	0.331401	0.0127368	0.00636838	−	0.999980i	$-0.497973\pi$
$677$	0.331401	0.0127368	0.00636838	+	0.999980i	$0.497973\pi$
$678$	0	0
$679$	−10.0999	−0.387600
$680$	0	0
$681$	−21.2393	−0.813893
$682$	0	0
$683$	16.8856	0.646109	0.323055	−	0.946380i	$-0.395290\pi$
$683$	16.8856	0.646109	0.323055	+	0.946380i	$0.395290\pi$
$684$	0	0
$685$	70.0126	2.67504
$686$	0	0
$687$	17.4360	0.665225
$688$	0	0
$689$	8.58734	0.327152
$690$	0	0
$691$	11.1963	0.425929	0.212964	−	0.977060i	$-0.431688\pi$
$691$	11.1963	0.425929	0.212964	+	0.977060i	$0.431688\pi$
$692$	0	0
$693$	−1.67510	−0.0636319
$694$	0	0
$695$	8.54119	0.323986
$696$	0	0
$697$	−32.2234	−1.22055
$698$	0	0
$699$	−11.2082	−0.423932
$700$	0	0
$701$	9.50609	0.359040	0.179520	−	0.983754i	$-0.442546\pi$
$701$	9.50609	0.359040	0.179520	+	0.983754i	$0.442546\pi$
$702$	0	0
$703$	6.43463	0.242687
$704$	0	0
$705$	8.49982	0.320122
$706$	0	0
$707$	13.7830	0.518362
$708$	0	0
$709$	36.2890	1.36286	0.681431	−	0.731882i	$-0.261358\pi$
$709$	36.2890	1.36286	0.681431	+	0.731882i	$0.261358\pi$
$710$	0	0
$711$	−0.581012	−0.0217896
$712$	0	0
$713$	−4.24991	−0.159160
$714$	0	0
$715$	−34.5067	−1.29048
$716$	0	0
$717$	18.5654	0.693339
$718$	0	0
$719$	33.5772	1.25222	0.626109	−	0.779736i	$-0.284647\pi$
$719$	33.5772	1.25222	0.626109	+	0.779736i	$0.284647\pi$
$720$	0	0
$721$	16.5553	0.616553
$722$	0	0
$723$	−13.6556	−0.507858
$724$	0	0
$725$	81.1809	3.01498
$726$	0	0
$727$	−37.5410	−1.39232	−0.696159	−	0.717887i	$-0.745109\pi$
$727$	−37.5410	−1.39232	−0.696159	+	0.717887i	$0.745109\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.2731	−0.971747
$732$	0	0
$733$	24.2222	0.894668	0.447334	−	0.894367i	$-0.352374\pi$
$733$	24.2222	0.894668	0.447334	+	0.894367i	$0.352374\pi$
$734$	0	0
$735$	4.41546	0.162867
$736$	0	0
$737$	13.7815	0.507650
$738$	0	0
$739$	19.4377	0.715028	0.357514	−	0.933908i	$-0.383624\pi$
$739$	19.4377	0.715028	0.357514	+	0.933908i	$0.383624\pi$
$740$	0	0
$741$	−3.23874	−0.118978
$742$	0	0
$743$	−35.7282	−1.31074	−0.655370	−	0.755308i	$-0.727487\pi$
$743$	−35.7282	−1.31074	−0.655370	+	0.755308i	$0.727487\pi$
$744$	0	0
$745$	11.5377	0.422707
$746$	0	0
$747$	−6.95032	−0.254299
$748$	0	0
$749$	−1.33463	−0.0487664
$750$	0	0
$751$	−14.4795	−0.528363	−0.264182	−	0.964473i	$-0.585102\pi$
$751$	−14.4795	−0.528363	−0.264182	+	0.964473i	$0.585102\pi$
$752$	0	0
$753$	−0.604526	−0.0220302
$754$	0	0
$755$	40.8996	1.48849
$756$	0	0
$757$	21.8034	0.792460	0.396230	−	0.918151i	$-0.370318\pi$
$757$	21.8034	0.792460	0.396230	+	0.918151i	$0.370318\pi$
$758$	0	0
$759$	1.67510	0.0608024
$760$	0	0
$761$	−5.98196	−0.216846	−0.108423	−	0.994105i	$-0.534580\pi$
$761$	−5.98196	−0.216846	−0.108423	+	0.994105i	$0.534580\pi$
$762$	0	0
$763$	−4.00794	−0.145097
$764$	0	0
$765$	27.5962	0.997744
$766$	0	0
$767$	43.8441	1.58312
$768$	0	0
$769$	−16.0694	−0.579476	−0.289738	−	0.957106i	$-0.593568\pi$
$769$	−16.0694	−0.579476	−0.289738	+	0.957106i	$0.593568\pi$
$770$	0	0
$771$	14.1602	0.509968
$772$	0	0
$773$	36.5039	1.31295	0.656476	−	0.754347i	$-0.272046\pi$
$773$	36.5039	1.31295	0.656476	+	0.754347i	$0.272046\pi$
$774$	0	0
$775$	−61.6079	−2.21302
$776$	0	0
$777$	−9.26901	−0.332524
$778$	0	0
$779$	−3.57921	−0.128239
$780$	0	0
$781$	17.8656	0.639282
$782$	0	0
$783$	5.60012	0.200132
$784$	0	0
$785$	−34.3304	−1.22530
$786$	0	0
$787$	44.7402	1.59482	0.797408	−	0.603440i	$-0.206204\pi$
$787$	44.7402	1.59482	0.797408	+	0.603440i	$0.206204\pi$
$788$	0	0
$789$	−20.5788	−0.732626
$790$	0	0
$791$	9.06525	0.322323
$792$	0	0
$793$	−36.0225	−1.27920
$794$	0	0
$795$	−8.12734	−0.288247
$796$	0	0
$797$	−11.3919	−0.403521	−0.201761	−	0.979435i	$-0.564666\pi$
$797$	−11.3919	−0.403521	−0.201761	+	0.979435i	$0.564666\pi$
$798$	0	0
$799$	12.0312	0.425632
$800$	0	0
$801$	−13.4963	−0.476868
$802$	0	0
$803$	−19.9534	−0.704141
$804$	0	0
$805$	−4.41546	−0.155625
$806$	0	0
$807$	−0.496655	−0.0174831
$808$	0	0
$809$	−23.4803	−0.825523	−0.412761	−	0.910839i	$-0.635436\pi$
$809$	−23.4803	−0.825523	−0.412761	+	0.910839i	$0.635436\pi$
$810$	0	0
$811$	0.320003	0.0112368	0.00561841	−	0.999984i	$-0.498212\pi$
$811$	0.320003	0.0112368	0.00561841	+	0.999984i	$0.498212\pi$
$812$	0	0
$813$	−24.0547	−0.843634
$814$	0	0
$815$	31.4415	1.10135
$816$	0	0
$817$	−2.91829	−0.102098
$818$	0	0
$819$	4.66537	0.163021
$820$	0	0
$821$	50.4892	1.76208	0.881042	−	0.473038i	$-0.156843\pi$
$821$	50.4892	1.76208	0.881042	+	0.473038i	$0.156843\pi$
$822$	0	0
$823$	29.6259	1.03270	0.516348	−	0.856379i	$-0.327291\pi$
$823$	29.6259	1.03270	0.516348	+	0.856379i	$0.327291\pi$
$824$	0	0
$825$	24.2828	0.845418
$826$	0	0
$827$	3.13318	0.108951	0.0544757	−	0.998515i	$-0.482651\pi$
$827$	3.13318	0.108951	0.0544757	+	0.998515i	$0.482651\pi$
$828$	0	0
$829$	−20.1070	−0.698345	−0.349172	−	0.937059i	$-0.613537\pi$
$829$	−20.1070	−0.698345	−0.349172	+	0.937059i	$0.613537\pi$
$830$	0	0
$831$	23.2402	0.806193
$832$	0	0
$833$	6.24991	0.216547
$834$	0	0
$835$	52.9270	1.83161
$836$	0	0
$837$	−4.24991	−0.146898
$838$	0	0
$839$	6.63075	0.228919	0.114460	−	0.993428i	$-0.463486\pi$
$839$	6.63075	0.228919	0.114460	+	0.993428i	$0.463486\pi$
$840$	0	0
$841$	2.36131	0.0814244
$842$	0	0
$843$	−15.0697	−0.519029
$844$	0	0
$845$	38.7045	1.33147
$846$	0	0
$847$	8.19403	0.281550
$848$	0	0
$849$	−9.22691	−0.316667
$850$	0	0
$851$	9.26901	0.317738
$852$	0	0
$853$	−36.1804	−1.23879	−0.619397	−	0.785078i	$-0.712623\pi$
$853$	−36.1804	−1.23879	−0.619397	+	0.785078i	$0.712623\pi$
$854$	0	0
$855$	3.06525	0.104829
$856$	0	0
$857$	12.6983	0.433764	0.216882	−	0.976198i	$-0.430411\pi$
$857$	12.6983	0.433764	0.216882	+	0.976198i	$0.430411\pi$
$858$	0	0
$859$	−49.2759	−1.68127	−0.840636	−	0.541600i	$-0.817819\pi$
$859$	−49.2759	−1.68127	−0.840636	+	0.541600i	$0.817819\pi$
$860$	0	0
$861$	5.15582	0.175710
$862$	0	0
$863$	3.28011	0.111656	0.0558282	−	0.998440i	$-0.482220\pi$
$863$	3.28011	0.111656	0.0558282	+	0.998440i	$0.482220\pi$
$864$	0	0
$865$	−9.27218	−0.315263
$866$	0	0
$867$	22.0614	0.749243
$868$	0	0
$869$	−0.973255	−0.0330154
$870$	0	0
$871$	−38.3833	−1.30057
$872$	0	0
$873$	10.0999	0.341831
$874$	0	0
$875$	−41.9305	−1.41751
$876$	0	0
$877$	7.64380	0.258113	0.129056	−	0.991637i	$-0.458805\pi$
$877$	7.64380	0.258113	0.129056	+	0.991637i	$0.458805\pi$
$878$	0	0
$879$	−12.3881	−0.417839
$880$	0	0
$881$	5.21611	0.175735	0.0878676	−	0.996132i	$-0.471995\pi$
$881$	5.21611	0.175735	0.0878676	+	0.996132i	$0.471995\pi$
$882$	0	0
$883$	45.2052	1.52127	0.760637	−	0.649177i	$-0.224887\pi$
$883$	45.2052	1.52127	0.760637	+	0.649177i	$0.224887\pi$
$884$	0	0
$885$	−41.4956	−1.39486
$886$	0	0
$887$	8.59600	0.288626	0.144313	−	0.989532i	$-0.453903\pi$
$887$	8.59600	0.288626	0.144313	+	0.989532i	$0.453903\pi$
$888$	0	0
$889$	−21.4977	−0.721010
$890$	0	0
$891$	1.67510	0.0561181
$892$	0	0
$893$	1.33636	0.0447196
$894$	0	0
$895$	70.5263	2.35744
$896$	0	0
$897$	−4.66537	−0.155772
$898$	0	0
$899$	−23.8000	−0.793774
$900$	0	0
$901$	−11.5039	−0.383251
$902$	0	0
$903$	4.20376	0.139892
$904$	0	0
$905$	−12.8035	−0.425603
$906$	0	0
$907$	−2.50429	−0.0831537	−0.0415769	−	0.999135i	$-0.513238\pi$
$907$	−2.50429	−0.0831537	−0.0415769	+	0.999135i	$0.513238\pi$
$908$	0	0
$909$	−13.7830	−0.457152
$910$	0	0
$911$	−29.5776	−0.979951	−0.489975	−	0.871736i	$-0.662994\pi$
$911$	−29.5776	−0.979951	−0.489975	+	0.871736i	$0.662994\pi$
$912$	0	0
$913$	−11.6425	−0.385311
$914$	0	0
$915$	34.0929	1.12708
$916$	0	0
$917$	1.17529	0.0388114
$918$	0	0
$919$	−23.5583	−0.777117	−0.388558	−	0.921424i	$-0.627027\pi$
$919$	−23.5583	−0.777117	−0.388558	+	0.921424i	$0.627027\pi$
$920$	0	0
$921$	−0.823281	−0.0271280
$922$	0	0
$923$	−49.7579	−1.63780
$924$	0	0
$925$	134.366	4.41794
$926$	0	0
$927$	−16.5553	−0.543749
$928$	0	0
$929$	−37.9337	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$929$	−37.9337	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$930$	0	0
$931$	0.694209	0.0227518
$932$	0	0
$933$	30.6632	1.00387
$934$	0	0
$935$	46.2265	1.51177
$936$	0	0
$937$	−42.3490	−1.38348	−0.691741	−	0.722146i	$-0.743156\pi$
$937$	−42.3490	−1.38348	−0.691741	+	0.722146i	$0.743156\pi$
$938$	0	0
$939$	13.1323	0.428556
$940$	0	0
$941$	−33.9735	−1.10750	−0.553752	−	0.832682i	$-0.686804\pi$
$941$	−33.9735	−1.10750	−0.553752	+	0.832682i	$0.686804\pi$
$942$	0	0
$943$	−5.15582	−0.167896
$944$	0	0
$945$	−4.41546	−0.143635
$946$	0	0
$947$	27.7471	0.901659	0.450829	−	0.892610i	$-0.351128\pi$
$947$	27.7471	0.901659	0.450829	+	0.892610i	$0.351128\pi$
$948$	0	0
$949$	55.5727	1.80397
$950$	0	0
$951$	−15.2269	−0.493766
$952$	0	0
$953$	−6.25582	−0.202646	−0.101323	−	0.994854i	$-0.532308\pi$
$953$	−6.25582	−0.202646	−0.101323	+	0.994854i	$0.532308\pi$
$954$	0	0
$955$	95.1472	3.07889
$956$	0	0
$957$	9.38078	0.303237
$958$	0	0
$959$	−15.8562	−0.512024
$960$	0	0
$961$	−12.9383	−0.417364
$962$	0	0
$963$	1.33463	0.0430079
$964$	0	0
$965$	20.2547	0.652021
$966$	0	0
$967$	−43.1613	−1.38797	−0.693987	−	0.719988i	$-0.744147\pi$
$967$	−43.1613	−1.38797	−0.693987	+	0.719988i	$0.744147\pi$
$968$	0	0
$969$	4.33874	0.139381
$970$	0	0
$971$	−6.69189	−0.214753	−0.107377	−	0.994218i	$-0.534245\pi$
$971$	−6.69189	−0.214753	−0.107377	+	0.994218i	$0.534245\pi$
$972$	0	0
$973$	−1.93438	−0.0620135
$974$	0	0
$975$	−67.6305	−2.16591
$976$	0	0
$977$	−47.1871	−1.50965	−0.754825	−	0.655926i	$-0.772278\pi$
$977$	−47.1871	−1.50965	−0.754825	+	0.655926i	$0.772278\pi$
$978$	0	0
$979$	−22.6077	−0.722545
$980$	0	0
$981$	4.00794	0.127964
$982$	0	0
$983$	−18.4505	−0.588480	−0.294240	−	0.955732i	$-0.595066\pi$
$983$	−18.4505	−0.588480	−0.294240	+	0.955732i	$0.595066\pi$
$984$	0	0
$985$	−52.3921	−1.66935
$986$	0	0
$987$	−1.92501	−0.0612738
$988$	0	0
$989$	−4.20376	−0.133672
$990$	0	0
$991$	24.3764	0.774342	0.387171	−	0.922008i	$-0.373452\pi$
$991$	24.3764	0.774342	0.387171	+	0.922008i	$0.373452\pi$
$992$	0	0
$993$	−25.5674	−0.811356
$994$	0	0
$995$	11.1652	0.353960
$996$	0	0
$997$	59.5570	1.88619	0.943094	−	0.332525i	$-0.107901\pi$
$997$	59.5570	1.88619	0.943094	+	0.332525i	$0.107901\pi$
$998$	0	0
$999$	9.26901	0.293259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cd.1.4		4
4.3	odd	2		483.2.a.i.1.2	✓	4
12.11	even	2		1449.2.a.p.1.3		4
28.27	even	2		3381.2.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.2	✓	4	4.3	odd	2
1449.2.a.p.1.3		4	12.11	even	2
3381.2.a.w.1.2		4	28.27	even	2
7728.2.a.cd.1.4		4	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.41546 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.67510 q^{11} -4.66537 q^{13} +4.41546 q^{15} +6.24991 q^{17} +0.694209 q^{19} -1.00000 q^{21} +1.00000 q^{23} +14.4963 q^{25} +1.00000 q^{27} +5.60012 q^{29} -4.24991 q^{31} +1.67510 q^{33} -4.41546 q^{35} +9.26901 q^{37} -4.66537 q^{39} -5.15582 q^{41} -4.20376 q^{43} +4.41546 q^{45} +1.92501 q^{47} +1.00000 q^{49} +6.24991 q^{51} -1.84066 q^{53} +7.39636 q^{55} +0.694209 q^{57} -9.39779 q^{59} +7.72125 q^{61} -1.00000 q^{63} -20.5998 q^{65} +8.22728 q^{67} +1.00000 q^{69} +10.6654 q^{71} -11.9117 q^{73} +14.4963 q^{75} -1.67510 q^{77} -0.581012 q^{79} +1.00000 q^{81} -6.95032 q^{83} +27.5962 q^{85} +5.60012 q^{87} -13.4963 q^{89} +4.66537 q^{91} -4.24991 q^{93} +3.06525 q^{95} +10.0999 q^{97} +1.67510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9 + 5 * q^11 + 7 * q^13 + 5 * q^15 + 12 * q^17 - 3 * q^19 - 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 + 5 * q^33 - 5 * q^35 + 20 * q^37 + 7 * q^39 + 3 * q^41 - 9 * q^43 + 5 * q^45 - 7 * q^47 + 4 * q^49 + 12 * q^51 - 6 * q^53 + 21 * q^55 - 3 * q^57 + 2 * q^59 + 24 * q^61 - 4 * q^63 - 14 * q^65 - q^67 + 4 * q^69 + 17 * q^71 + 16 * q^73 + 7 * q^75 - 5 * q^77 + 10 * q^79 + 4 * q^81 - 8 * q^83 + 17 * q^85 + 6 * q^87 - 3 * q^89 - 7 * q^91 - 4 * q^93 + 3 * q^95 - 2 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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