LMFDB - Embedded newform 7728.2.a.bl.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.19258$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.19258 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.19258 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} +3.19258 q^{13} -2.19258 q^{15} +1.00000 q^{17} +3.38516 q^{19} +1.00000 q^{21} -1.00000 q^{23} -0.192582 q^{25} +1.00000 q^{27} -5.00000 q^{29} +3.38516 q^{31} -5.00000 q^{33} -2.19258 q^{35} +5.38516 q^{37} +3.19258 q^{39} -1.00000 q^{41} -5.19258 q^{43} -2.19258 q^{45} -10.3852 q^{47} +1.00000 q^{49} +1.00000 q^{51} -10.5777 q^{53} +10.9629 q^{55} +3.38516 q^{57} +12.5777 q^{59} +9.19258 q^{61} +1.00000 q^{63} -7.00000 q^{65} -2.19258 q^{67} -1.00000 q^{69} -13.5777 q^{71} -9.00000 q^{73} -0.192582 q^{75} -5.00000 q^{77} +9.77033 q^{79} +1.00000 q^{81} +11.3852 q^{83} -2.19258 q^{85} -5.00000 q^{87} +3.57775 q^{89} +3.19258 q^{91} +3.38516 q^{93} -7.42225 q^{95} -17.7703 q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 10 q^{11} + q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 10 q^{33} + q^{35} + q^{39} - 2 q^{41} - 5 q^{43} + q^{45} - 10 q^{47} + 2 q^{49} + 2 q^{51} - 5 q^{53} - 5 q^{55} - 4 q^{57} + 9 q^{59} + 13 q^{61} + 2 q^{63} - 14 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} - 18 q^{73} + 5 q^{75} - 10 q^{77} - 2 q^{79} + 2 q^{81} + 12 q^{83} + q^{85} - 10 q^{87} - 9 q^{89} + q^{91} - 4 q^{93} - 31 q^{95} - 14 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 10 * q^11 + q^13 + q^15 + 2 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 10 * q^33 + q^35 + q^39 - 2 * q^41 - 5 * q^43 + q^45 - 10 * q^47 + 2 * q^49 + 2 * q^51 - 5 * q^53 - 5 * q^55 - 4 * q^57 + 9 * q^59 + 13 * q^61 + 2 * q^63 - 14 * q^65 + q^67 - 2 * q^69 - 11 * q^71 - 18 * q^73 + 5 * q^75 - 10 * q^77 - 2 * q^79 + 2 * q^81 + 12 * q^83 + q^85 - 10 * q^87 - 9 * q^89 + q^91 - 4 * q^93 - 31 * q^95 - 14 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.19258	−0.980553	−0.490276	−	0.871567i	$-0.663104\pi$
$5$	−2.19258	−0.980553	−0.490276	+	0.871567i	$0.663104\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	3.19258	0.885463	0.442732	−	0.896654i	$-0.354009\pi$
$13$	3.19258	0.885463	0.442732	+	0.896654i	$0.354009\pi$
$14$	0	0
$15$	−2.19258	−0.566122
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	3.38516	0.776610	0.388305	−	0.921531i	$-0.373061\pi$
$19$	3.38516	0.776610	0.388305	+	0.921531i	$0.373061\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.192582	−0.0385165
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	3.38516	0.607994	0.303997	−	0.952673i	$-0.401679\pi$
$31$	3.38516	0.607994	0.303997	+	0.952673i	$0.401679\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	−2.19258	−0.370614
$36$	0	0
$37$	5.38516	0.885316	0.442658	−	0.896691i	$-0.354036\pi$
$37$	5.38516	0.885316	0.442658	+	0.896691i	$0.354036\pi$
$38$	0	0
$39$	3.19258	0.511222
$40$	0	0
$41$	−1.00000	−0.156174	−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000	−0.156174	−0.0780869	+	0.996947i	$0.524881\pi$
$42$	0	0
$43$	−5.19258	−0.791861	−0.395931	−	0.918280i	$-0.629578\pi$
$43$	−5.19258	−0.791861	−0.395931	+	0.918280i	$0.629578\pi$
$44$	0	0
$45$	−2.19258	−0.326851
$46$	0	0
$47$	−10.3852	−1.51483	−0.757416	−	0.652933i	$-0.773538\pi$
$47$	−10.3852	−1.51483	−0.757416	+	0.652933i	$0.773538\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−10.5777	−1.45297	−0.726483	−	0.687185i	$-0.758846\pi$
$53$	−10.5777	−1.45297	−0.726483	+	0.687185i	$0.758846\pi$
$54$	0	0
$55$	10.9629	1.47824
$56$	0	0
$57$	3.38516	0.448376
$58$	0	0
$59$	12.5777	1.63748	0.818742	−	0.574162i	$-0.194672\pi$
$59$	12.5777	1.63748	0.818742	+	0.574162i	$0.194672\pi$
$60$	0	0
$61$	9.19258	1.17699	0.588495	−	0.808501i	$-0.299721\pi$
$61$	9.19258	1.17699	0.588495	+	0.808501i	$0.299721\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−7.00000	−0.868243
$66$	0	0
$67$	−2.19258	−0.267867	−0.133933	−	0.990990i	$-0.542761\pi$
$67$	−2.19258	−0.267867	−0.133933	+	0.990990i	$0.542761\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−13.5777	−1.61138	−0.805691	−	0.592336i	$-0.798206\pi$
$71$	−13.5777	−1.61138	−0.805691	+	0.592336i	$0.798206\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	0	0
$75$	−0.192582	−0.0222375
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	9.77033	1.09925	0.549624	−	0.835412i	$-0.314771\pi$
$79$	9.77033	1.09925	0.549624	+	0.835412i	$0.314771\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3852	1.24968	0.624842	−	0.780751i	$-0.285163\pi$
$83$	11.3852	1.24968	0.624842	+	0.780751i	$0.285163\pi$
$84$	0	0
$85$	−2.19258	−0.237819
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	3.57775	0.379240	0.189620	−	0.981858i	$-0.439274\pi$
$89$	3.57775	0.379240	0.189620	+	0.981858i	$0.439274\pi$
$90$	0	0
$91$	3.19258	0.334674
$92$	0	0
$93$	3.38516	0.351025
$94$	0	0
$95$	−7.42225	−0.761507
$96$	0	0
$97$	−17.7703	−1.80430	−0.902152	−	0.431419i	$-0.858013\pi$
$97$	−17.7703	−1.80430	−0.902152	+	0.431419i	$0.858013\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	14.5777	1.45054	0.725270	−	0.688465i	$-0.241715\pi$
$101$	14.5777	1.45054	0.725270	+	0.688465i	$0.241715\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−2.19258	−0.213974
$106$	0	0
$107$	−17.5777	−1.69930	−0.849652	−	0.527343i	$-0.823188\pi$
$107$	−17.5777	−1.69930	−0.849652	+	0.527343i	$0.823188\pi$
$108$	0	0
$109$	3.42225	0.327792	0.163896	−	0.986478i	$-0.447594\pi$
$109$	3.42225	0.327792	0.163896	+	0.986478i	$0.447594\pi$
$110$	0	0
$111$	5.38516	0.511137
$112$	0	0
$113$	4.57775	0.430638	0.215319	−	0.976544i	$-0.430921\pi$
$113$	4.57775	0.430638	0.215319	+	0.976544i	$0.430921\pi$
$114$	0	0
$115$	2.19258	0.204459
$116$	0	0
$117$	3.19258	0.295154
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	11.3852	1.01832
$126$	0	0
$127$	−9.19258	−0.815710	−0.407855	−	0.913047i	$-0.633723\pi$
$127$	−9.19258	−0.815710	−0.407855	+	0.913047i	$0.633723\pi$
$128$	0	0
$129$	−5.19258	−0.457181
$130$	0	0
$131$	−9.77033	−0.853638	−0.426819	−	0.904337i	$-0.640366\pi$
$131$	−9.77033	−0.853638	−0.426819	+	0.904337i	$0.640366\pi$
$132$	0	0
$133$	3.38516	0.293531
$134$	0	0
$135$	−2.19258	−0.188707
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−0.577747	−0.0490039	−0.0245019	−	0.999700i	$-0.507800\pi$
$139$	−0.577747	−0.0490039	−0.0245019	+	0.999700i	$0.507800\pi$
$140$	0	0
$141$	−10.3852	−0.874589
$142$	0	0
$143$	−15.9629	−1.33489
$144$	0	0
$145$	10.9629	0.910420
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	10.3852	0.850786	0.425393	−	0.905009i	$-0.360136\pi$
$149$	10.3852	0.850786	0.425393	+	0.905009i	$0.360136\pi$
$150$	0	0
$151$	2.38516	0.194102	0.0970510	−	0.995279i	$-0.469059\pi$
$151$	2.38516	0.194102	0.0970510	+	0.995279i	$0.469059\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−7.42225	−0.596170
$156$	0	0
$157$	−7.61484	−0.607730	−0.303865	−	0.952715i	$-0.598277\pi$
$157$	−7.61484	−0.607730	−0.303865	+	0.952715i	$0.598277\pi$
$158$	0	0
$159$	−10.5777	−0.838870
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−19.3481	−1.51546	−0.757729	−	0.652569i	$-0.773691\pi$
$163$	−19.3481	−1.51546	−0.757729	+	0.652569i	$0.773691\pi$
$164$	0	0
$165$	10.9629	0.853462
$166$	0	0
$167$	−23.0000	−1.77979	−0.889897	−	0.456162i	$-0.849224\pi$
$167$	−23.0000	−1.77979	−0.889897	+	0.456162i	$0.849224\pi$
$168$	0	0
$169$	−2.80742	−0.215955
$170$	0	0
$171$	3.38516	0.258870
$172$	0	0
$173$	−15.7703	−1.19900	−0.599498	−	0.800376i	$-0.704633\pi$
$173$	−15.7703	−1.19900	−0.599498	+	0.800376i	$0.704633\pi$
$174$	0	0
$175$	−0.192582	−0.0145579
$176$	0	0
$177$	12.5777	0.945401
$178$	0	0
$179$	−20.5777	−1.53805	−0.769027	−	0.639217i	$-0.779259\pi$
$179$	−20.5777	−1.53805	−0.769027	+	0.639217i	$0.779259\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	9.19258	0.679535
$184$	0	0
$185$	−11.8074	−0.868099
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−10.3852	−0.751444	−0.375722	−	0.926732i	$-0.622605\pi$
$191$	−10.3852	−0.751444	−0.375722	+	0.926732i	$0.622605\pi$
$192$	0	0
$193$	−8.38516	−0.603577	−0.301789	−	0.953375i	$-0.597584\pi$
$193$	−8.38516	−0.603577	−0.301789	+	0.953375i	$0.597584\pi$
$194$	0	0
$195$	−7.00000	−0.501280
$196$	0	0
$197$	9.19258	0.654944	0.327472	−	0.944861i	$-0.393803\pi$
$197$	9.19258	0.654944	0.327472	+	0.944861i	$0.393803\pi$
$198$	0	0
$199$	13.3481	0.946220	0.473110	−	0.881003i	$-0.343131\pi$
$199$	13.3481	0.946220	0.473110	+	0.881003i	$0.343131\pi$
$200$	0	0
$201$	−2.19258	−0.154653
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	2.19258	0.153137
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−16.9258	−1.17078
$210$	0	0
$211$	−13.3852	−0.921473	−0.460736	−	0.887537i	$-0.652415\pi$
$211$	−13.3852	−0.921473	−0.460736	+	0.887537i	$0.652415\pi$
$212$	0	0
$213$	−13.5777	−0.930332
$214$	0	0
$215$	11.3852	0.776462
$216$	0	0
$217$	3.38516	0.229800
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	3.19258	0.214756
$222$	0	0
$223$	−12.5777	−0.842268	−0.421134	−	0.906998i	$-0.638368\pi$
$223$	−12.5777	−0.842268	−0.421134	+	0.906998i	$0.638368\pi$
$224$	0	0
$225$	−0.192582	−0.0128388
$226$	0	0
$227$	−0.577747	−0.0383464	−0.0191732	−	0.999816i	$-0.506103\pi$
$227$	−0.577747	−0.0383464	−0.0191732	+	0.999816i	$0.506103\pi$
$228$	0	0
$229$	4.19258	0.277054	0.138527	−	0.990359i	$-0.455763\pi$
$229$	4.19258	0.277054	0.138527	+	0.990359i	$0.455763\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	−2.96291	−0.194107	−0.0970534	−	0.995279i	$-0.530942\pi$
$233$	−2.96291	−0.194107	−0.0970534	+	0.995279i	$0.530942\pi$
$234$	0	0
$235$	22.7703	1.48537
$236$	0	0
$237$	9.77033	0.634651
$238$	0	0
$239$	−22.1926	−1.43552	−0.717759	−	0.696291i	$-0.754832\pi$
$239$	−22.1926	−1.43552	−0.717759	+	0.696291i	$0.754832\pi$
$240$	0	0
$241$	−19.7703	−1.27352	−0.636759	−	0.771063i	$-0.719726\pi$
$241$	−19.7703	−1.27352	−0.636759	+	0.771063i	$0.719726\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.19258	−0.140079
$246$	0	0
$247$	10.8074	0.687660
$248$	0	0
$249$	11.3852	0.721506
$250$	0	0
$251$	−11.6148	−0.733122	−0.366561	−	0.930394i	$-0.619465\pi$
$251$	−11.6148	−0.733122	−0.366561	+	0.930394i	$0.619465\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	−2.19258	−0.137305
$256$	0	0
$257$	−3.15549	−0.196834	−0.0984172	−	0.995145i	$-0.531378\pi$
$257$	−3.15549	−0.196834	−0.0984172	+	0.995145i	$0.531378\pi$
$258$	0	0
$259$	5.38516	0.334618
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	−24.1555	−1.48949	−0.744746	−	0.667348i	$-0.767429\pi$
$263$	−24.1555	−1.48949	−0.744746	+	0.667348i	$0.767429\pi$
$264$	0	0
$265$	23.1926	1.42471
$266$	0	0
$267$	3.57775	0.218955
$268$	0	0
$269$	29.9629	1.82687	0.913435	−	0.406984i	$-0.133419\pi$
$269$	29.9629	1.82687	0.913435	+	0.406984i	$0.133419\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	0	0
$273$	3.19258	0.193224
$274$	0	0
$275$	0.962912	0.0580658
$276$	0	0
$277$	−13.1926	−0.792665	−0.396333	−	0.918107i	$-0.629717\pi$
$277$	−13.1926	−0.792665	−0.396333	+	0.918107i	$0.629717\pi$
$278$	0	0
$279$	3.38516	0.202665
$280$	0	0
$281$	−16.3852	−0.977457	−0.488728	−	0.872436i	$-0.662539\pi$
$281$	−16.3852	−0.977457	−0.488728	+	0.872436i	$0.662539\pi$
$282$	0	0
$283$	5.03709	0.299424	0.149712	−	0.988730i	$-0.452165\pi$
$283$	5.03709	0.299424	0.149712	+	0.988730i	$0.452165\pi$
$284$	0	0
$285$	−7.42225	−0.439656
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−17.7703	−1.04172
$292$	0	0
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	−27.5777	−1.60564
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−3.19258	−0.184632
$300$	0	0
$301$	−5.19258	−0.299295
$302$	0	0
$303$	14.5777	0.837470
$304$	0	0
$305$	−20.1555	−1.15410
$306$	0	0
$307$	0.614835	0.0350905	0.0175452	−	0.999846i	$-0.494415\pi$
$307$	0.614835	0.0350905	0.0175452	+	0.999846i	$0.494415\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	3.96291	0.224716	0.112358	−	0.993668i	$-0.464160\pi$
$311$	3.96291	0.224716	0.112358	+	0.993668i	$0.464160\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	0	0
$315$	−2.19258	−0.123538
$316$	0	0
$317$	22.1926	1.24646	0.623230	−	0.782039i	$-0.285820\pi$
$317$	22.1926	1.24646	0.623230	+	0.782039i	$0.285820\pi$
$318$	0	0
$319$	25.0000	1.39973
$320$	0	0
$321$	−17.5777	−0.981094
$322$	0	0
$323$	3.38516	0.188356
$324$	0	0
$325$	−0.614835	−0.0341049
$326$	0	0
$327$	3.42225	0.189251
$328$	0	0
$329$	−10.3852	−0.572553
$330$	0	0
$331$	−13.2297	−0.727168	−0.363584	−	0.931561i	$-0.618447\pi$
$331$	−13.2297	−0.727168	−0.363584	+	0.931561i	$0.618447\pi$
$332$	0	0
$333$	5.38516	0.295105
$334$	0	0
$335$	4.80742	0.262657
$336$	0	0
$337$	−11.9629	−0.651661	−0.325831	−	0.945428i	$-0.605644\pi$
$337$	−11.9629	−0.651661	−0.325831	+	0.945428i	$0.605644\pi$
$338$	0	0
$339$	4.57775	0.248629
$340$	0	0
$341$	−16.9258	−0.916585
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.19258	0.118045
$346$	0	0
$347$	22.1555	1.18937	0.594685	−	0.803959i	$-0.297277\pi$
$347$	22.1555	1.18937	0.594685	+	0.803959i	$0.297277\pi$
$348$	0	0
$349$	−4.19258	−0.224424	−0.112212	−	0.993684i	$-0.535794\pi$
$349$	−4.19258	−0.224424	−0.112212	+	0.993684i	$0.535794\pi$
$350$	0	0
$351$	3.19258	0.170407
$352$	0	0
$353$	−29.7703	−1.58451	−0.792257	−	0.610187i	$-0.791094\pi$
$353$	−29.7703	−1.58451	−0.792257	+	0.610187i	$0.791094\pi$
$354$	0	0
$355$	29.7703	1.58005
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	1.42225	0.0750636	0.0375318	−	0.999295i	$-0.488050\pi$
$359$	1.42225	0.0750636	0.0375318	+	0.999295i	$0.488050\pi$
$360$	0	0
$361$	−7.54066	−0.396877
$362$	0	0
$363$	14.0000	0.734809
$364$	0	0
$365$	19.7332	1.03289
$366$	0	0
$367$	−17.9629	−0.937656	−0.468828	−	0.883289i	$-0.655324\pi$
$367$	−17.9629	−0.937656	−0.468828	+	0.883289i	$0.655324\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	−10.5777	−0.549169
$372$	0	0
$373$	−8.15549	−0.422275	−0.211138	−	0.977456i	$-0.567717\pi$
$373$	−8.15549	−0.422275	−0.211138	+	0.977456i	$0.567717\pi$
$374$	0	0
$375$	11.3852	0.587927
$376$	0	0
$377$	−15.9629	−0.822132
$378$	0	0
$379$	−29.5407	−1.51740	−0.758701	−	0.651439i	$-0.774166\pi$
$379$	−29.5407	−1.51740	−0.758701	+	0.651439i	$0.774166\pi$
$380$	0	0
$381$	−9.19258	−0.470950
$382$	0	0
$383$	37.3852	1.91029	0.955146	−	0.296134i	$-0.0956976\pi$
$383$	37.3852	1.91029	0.955146	+	0.296134i	$0.0956976\pi$
$384$	0	0
$385$	10.9629	0.558722
$386$	0	0
$387$	−5.19258	−0.263954
$388$	0	0
$389$	28.9258	1.46660	0.733299	−	0.679907i	$-0.237980\pi$
$389$	28.9258	1.46660	0.733299	+	0.679907i	$0.237980\pi$
$390$	0	0
$391$	−1.00000	−0.0505722
$392$	0	0
$393$	−9.77033	−0.492848
$394$	0	0
$395$	−21.4223	−1.07787
$396$	0	0
$397$	−19.9258	−1.00005	−0.500024	−	0.866011i	$-0.666676\pi$
$397$	−19.9258	−1.00005	−0.500024	+	0.866011i	$0.666676\pi$
$398$	0	0
$399$	3.38516	0.169470
$400$	0	0
$401$	−6.61484	−0.330329	−0.165165	−	0.986266i	$-0.552816\pi$
$401$	−6.61484	−0.330329	−0.165165	+	0.986266i	$0.552816\pi$
$402$	0	0
$403$	10.8074	0.538356
$404$	0	0
$405$	−2.19258	−0.108950
$406$	0	0
$407$	−26.9258	−1.33466
$408$	0	0
$409$	20.9258	1.03472	0.517358	−	0.855769i	$-0.326916\pi$
$409$	20.9258	1.03472	0.517358	+	0.855769i	$0.326916\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	12.5777	0.618910
$414$	0	0
$415$	−24.9629	−1.22538
$416$	0	0
$417$	−0.577747	−0.0282924
$418$	0	0
$419$	−20.1926	−0.986472	−0.493236	−	0.869895i	$-0.664186\pi$
$419$	−20.1926	−0.986472	−0.493236	+	0.869895i	$0.664186\pi$
$420$	0	0
$421$	8.42225	0.410475	0.205238	−	0.978712i	$-0.434203\pi$
$421$	8.42225	0.410475	0.205238	+	0.978712i	$0.434203\pi$
$422$	0	0
$423$	−10.3852	−0.504944
$424$	0	0
$425$	−0.192582	−0.00934162
$426$	0	0
$427$	9.19258	0.444860
$428$	0	0
$429$	−15.9629	−0.770697
$430$	0	0
$431$	−27.9629	−1.34693	−0.673463	−	0.739221i	$-0.735194\pi$
$431$	−27.9629	−1.34693	−0.673463	+	0.739221i	$0.735194\pi$
$432$	0	0
$433$	−11.6148	−0.558173	−0.279087	−	0.960266i	$-0.590032\pi$
$433$	−11.6148	−0.558173	−0.279087	+	0.960266i	$0.590032\pi$
$434$	0	0
$435$	10.9629	0.525631
$436$	0	0
$437$	−3.38516	−0.161934
$438$	0	0
$439$	−34.9258	−1.66692	−0.833459	−	0.552581i	$-0.813643\pi$
$439$	−34.9258	−1.66692	−0.833459	+	0.552581i	$0.813643\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	30.3110	1.44012	0.720059	−	0.693913i	$-0.244115\pi$
$443$	30.3110	1.44012	0.720059	+	0.693913i	$0.244115\pi$
$444$	0	0
$445$	−7.84451	−0.371865
$446$	0	0
$447$	10.3852	0.491201
$448$	0	0
$449$	6.19258	0.292246	0.146123	−	0.989266i	$-0.453320\pi$
$449$	6.19258	0.292246	0.146123	+	0.989266i	$0.453320\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	0	0
$453$	2.38516	0.112065
$454$	0	0
$455$	−7.00000	−0.328165
$456$	0	0
$457$	29.3481	1.37285	0.686423	−	0.727203i	$-0.259180\pi$
$457$	29.3481	1.37285	0.686423	+	0.727203i	$0.259180\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−22.9629	−1.06949	−0.534745	−	0.845014i	$-0.679592\pi$
$461$	−22.9629	−1.06949	−0.534745	+	0.845014i	$0.679592\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	0	0
$465$	−7.42225	−0.344199
$466$	0	0
$467$	−24.9258	−1.15343	−0.576715	−	0.816946i	$-0.695666\pi$
$467$	−24.9258	−1.15343	−0.576715	+	0.816946i	$0.695666\pi$
$468$	0	0
$469$	−2.19258	−0.101244
$470$	0	0
$471$	−7.61484	−0.350873
$472$	0	0
$473$	25.9629	1.19378
$474$	0	0
$475$	−0.651923	−0.0299123
$476$	0	0
$477$	−10.5777	−0.484322
$478$	0	0
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	0	0
$481$	17.1926	0.783914
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	38.9629	1.76921
$486$	0	0
$487$	27.7703	1.25839	0.629197	−	0.777246i	$-0.283384\pi$
$487$	27.7703	1.25839	0.629197	+	0.777246i	$0.283384\pi$
$488$	0	0
$489$	−19.3481	−0.874950
$490$	0	0
$491$	−6.03709	−0.272450	−0.136225	−	0.990678i	$-0.543497\pi$
$491$	−6.03709	−0.272450	−0.136225	+	0.990678i	$0.543497\pi$
$492$	0	0
$493$	−5.00000	−0.225189
$494$	0	0
$495$	10.9629	0.492746
$496$	0	0
$497$	−13.5777	−0.609045
$498$	0	0
$499$	0.192582	0.00862117	0.00431059	−	0.999991i	$-0.498628\pi$
$499$	0.192582	0.00862117	0.00431059	+	0.999991i	$0.498628\pi$
$500$	0	0
$501$	−23.0000	−1.02756
$502$	0	0
$503$	7.42225	0.330942	0.165471	−	0.986215i	$-0.447086\pi$
$503$	7.42225	0.330942	0.165471	+	0.986215i	$0.447086\pi$
$504$	0	0
$505$	−31.9629	−1.42233
$506$	0	0
$507$	−2.80742	−0.124682
$508$	0	0
$509$	10.3852	0.460314	0.230157	−	0.973153i	$-0.426076\pi$
$509$	10.3852	0.460314	0.230157	+	0.973153i	$0.426076\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	0	0
$513$	3.38516	0.149459
$514$	0	0
$515$	17.5407	0.772934
$516$	0	0
$517$	51.9258	2.28370
$518$	0	0
$519$	−15.7703	−0.692241
$520$	0	0
$521$	−1.22967	−0.0538728	−0.0269364	−	0.999637i	$-0.508575\pi$
$521$	−1.22967	−0.0538728	−0.0269364	+	0.999637i	$0.508575\pi$
$522$	0	0
$523$	2.77033	0.121138	0.0605690	−	0.998164i	$-0.480708\pi$
$523$	2.77033	0.121138	0.0605690	+	0.998164i	$0.480708\pi$
$524$	0	0
$525$	−0.192582	−0.00840499
$526$	0	0
$527$	3.38516	0.147460
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	12.5777	0.545828
$532$	0	0
$533$	−3.19258	−0.138286
$534$	0	0
$535$	38.5407	1.66626
$536$	0	0
$537$	−20.5777	−0.887995
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−18.3852	−0.790440	−0.395220	−	0.918587i	$-0.629332\pi$
$541$	−18.3852	−0.790440	−0.395220	+	0.918587i	$0.629332\pi$
$542$	0	0
$543$	7.00000	0.300399
$544$	0	0
$545$	−7.50357	−0.321418
$546$	0	0
$547$	−20.5777	−0.879841	−0.439920	−	0.898037i	$-0.644993\pi$
$547$	−20.5777	−0.879841	−0.439920	+	0.898037i	$0.644993\pi$
$548$	0	0
$549$	9.19258	0.392330
$550$	0	0
$551$	−16.9258	−0.721064
$552$	0	0
$553$	9.77033	0.415477
$554$	0	0
$555$	−11.8074	−0.501197
$556$	0	0
$557$	9.15549	0.387931	0.193965	−	0.981008i	$-0.437865\pi$
$557$	9.15549	0.387931	0.193965	+	0.981008i	$0.437865\pi$
$558$	0	0
$559$	−16.5777	−0.701164
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	9.96291	0.419887	0.209943	−	0.977714i	$-0.432672\pi$
$563$	9.96291	0.419887	0.209943	+	0.977714i	$0.432672\pi$
$564$	0	0
$565$	−10.0371	−0.422263
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	37.5407	1.57379	0.786893	−	0.617089i	$-0.211688\pi$
$569$	37.5407	1.57379	0.786893	+	0.617089i	$0.211688\pi$
$570$	0	0
$571$	7.77033	0.325178	0.162589	−	0.986694i	$-0.448016\pi$
$571$	7.77033	0.325178	0.162589	+	0.986694i	$0.448016\pi$
$572$	0	0
$573$	−10.3852	−0.433846
$574$	0	0
$575$	0.192582	0.00803124
$576$	0	0
$577$	−17.2297	−0.717281	−0.358640	−	0.933476i	$-0.616760\pi$
$577$	−17.2297	−0.717281	−0.358640	+	0.933476i	$0.616760\pi$
$578$	0	0
$579$	−8.38516	−0.348476
$580$	0	0
$581$	11.3852	0.472336
$582$	0	0
$583$	52.8887	2.19043
$584$	0	0
$585$	−7.00000	−0.289414
$586$	0	0
$587$	−20.3481	−0.839855	−0.419928	−	0.907558i	$-0.637945\pi$
$587$	−20.3481	−0.839855	−0.419928	+	0.907558i	$0.637945\pi$
$588$	0	0
$589$	11.4593	0.472174
$590$	0	0
$591$	9.19258	0.378132
$592$	0	0
$593$	−35.6148	−1.46253	−0.731263	−	0.682096i	$-0.761069\pi$
$593$	−35.6148	−1.46253	−0.731263	+	0.682096i	$0.761069\pi$
$594$	0	0
$595$	−2.19258	−0.0898871
$596$	0	0
$597$	13.3481	0.546300
$598$	0	0
$599$	30.7332	1.25573	0.627863	−	0.778324i	$-0.283930\pi$
$599$	30.7332	1.25573	0.627863	+	0.778324i	$0.283930\pi$
$600$	0	0
$601$	26.9629	1.09984	0.549920	−	0.835217i	$-0.314658\pi$
$601$	26.9629	1.09984	0.549920	+	0.835217i	$0.314658\pi$
$602$	0	0
$603$	−2.19258	−0.0892889
$604$	0	0
$605$	−30.6962	−1.24798
$606$	0	0
$607$	−0.348077	−0.0141280	−0.00706400	−	0.999975i	$-0.502249\pi$
$607$	−0.348077	−0.0141280	−0.00706400	+	0.999975i	$0.502249\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−33.1555	−1.34133
$612$	0	0
$613$	−8.54066	−0.344954	−0.172477	−	0.985014i	$-0.555177\pi$
$613$	−8.54066	−0.344954	−0.172477	+	0.985014i	$0.555177\pi$
$614$	0	0
$615$	2.19258	0.0884135
$616$	0	0
$617$	−29.5777	−1.19076	−0.595378	−	0.803446i	$-0.702998\pi$
$617$	−29.5777	−1.19076	−0.595378	+	0.803446i	$0.702998\pi$
$618$	0	0
$619$	−22.7332	−0.913726	−0.456863	−	0.889537i	$-0.651027\pi$
$619$	−22.7332	−0.913726	−0.456863	+	0.889537i	$0.651027\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	3.57775	0.143339
$624$	0	0
$625$	−24.0000	−0.960000
$626$	0	0
$627$	−16.9258	−0.675952
$628$	0	0
$629$	5.38516	0.214721
$630$	0	0
$631$	3.84451	0.153047	0.0765237	−	0.997068i	$-0.475618\pi$
$631$	3.84451	0.153047	0.0765237	+	0.997068i	$0.475618\pi$
$632$	0	0
$633$	−13.3852	−0.532013
$634$	0	0
$635$	20.1555	0.799846
$636$	0	0
$637$	3.19258	0.126495
$638$	0	0
$639$	−13.5777	−0.537127
$640$	0	0
$641$	22.7332	0.897909	0.448955	−	0.893555i	$-0.351797\pi$
$641$	22.7332	0.897909	0.448955	+	0.893555i	$0.351797\pi$
$642$	0	0
$643$	−39.5036	−1.55787	−0.778934	−	0.627105i	$-0.784240\pi$
$643$	−39.5036	−1.55787	−0.778934	+	0.627105i	$0.784240\pi$
$644$	0	0
$645$	11.3852	0.448290
$646$	0	0
$647$	36.8887	1.45025	0.725123	−	0.688619i	$-0.241783\pi$
$647$	36.8887	1.45025	0.725123	+	0.688619i	$0.241783\pi$
$648$	0	0
$649$	−62.8887	−2.46860
$650$	0	0
$651$	3.38516	0.132675
$652$	0	0
$653$	−30.9629	−1.21167	−0.605836	−	0.795589i	$-0.707161\pi$
$653$	−30.9629	−1.21167	−0.605836	+	0.795589i	$0.707161\pi$
$654$	0	0
$655$	21.4223	0.837037
$656$	0	0
$657$	−9.00000	−0.351123
$658$	0	0
$659$	−29.0000	−1.12968	−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000	−1.12968	−0.564840	+	0.825201i	$0.691062\pi$
$660$	0	0
$661$	28.5407	1.11010	0.555051	−	0.831816i	$-0.312699\pi$
$661$	28.5407	1.11010	0.555051	+	0.831816i	$0.312699\pi$
$662$	0	0
$663$	3.19258	0.123990
$664$	0	0
$665$	−7.42225	−0.287823
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	−12.5777	−0.486284
$670$	0	0
$671$	−45.9629	−1.77438
$672$	0	0
$673$	−15.7703	−0.607902	−0.303951	−	0.952688i	$-0.598306\pi$
$673$	−15.7703	−0.607902	−0.303951	+	0.952688i	$0.598306\pi$
$674$	0	0
$675$	−0.192582	−0.00741250
$676$	0	0
$677$	−27.1184	−1.04225	−0.521123	−	0.853482i	$-0.674487\pi$
$677$	−27.1184	−1.04225	−0.521123	+	0.853482i	$0.674487\pi$
$678$	0	0
$679$	−17.7703	−0.681963
$680$	0	0
$681$	−0.577747	−0.0221393
$682$	0	0
$683$	19.3852	0.741753	0.370876	−	0.928682i	$-0.379057\pi$
$683$	19.3852	0.741753	0.370876	+	0.928682i	$0.379057\pi$
$684$	0	0
$685$	−19.7332	−0.753968
$686$	0	0
$687$	4.19258	0.159957
$688$	0	0
$689$	−33.7703	−1.28655
$690$	0	0
$691$	−8.42225	−0.320398	−0.160199	−	0.987085i	$-0.551214\pi$
$691$	−8.42225	−0.320398	−0.160199	+	0.987085i	$0.551214\pi$
$692$	0	0
$693$	−5.00000	−0.189934
$694$	0	0
$695$	1.26676	0.0480509
$696$	0	0
$697$	−1.00000	−0.0378777
$698$	0	0
$699$	−2.96291	−0.112068
$700$	0	0
$701$	2.57775	0.0973602	0.0486801	−	0.998814i	$-0.484499\pi$
$701$	2.57775	0.0973602	0.0486801	+	0.998814i	$0.484499\pi$
$702$	0	0
$703$	18.2297	0.687545
$704$	0	0
$705$	22.7703	0.857580
$706$	0	0
$707$	14.5777	0.548253
$708$	0	0
$709$	15.1184	0.567784	0.283892	−	0.958856i	$-0.408374\pi$
$709$	15.1184	0.567784	0.283892	+	0.958856i	$0.408374\pi$
$710$	0	0
$711$	9.77033	0.366416
$712$	0	0
$713$	−3.38516	−0.126775
$714$	0	0
$715$	35.0000	1.30893
$716$	0	0
$717$	−22.1926	−0.828797
$718$	0	0
$719$	−22.9258	−0.854989	−0.427494	−	0.904018i	$-0.640604\pi$
$719$	−22.9258	−0.854989	−0.427494	+	0.904018i	$0.640604\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−19.7703	−0.735266
$724$	0	0
$725$	0.962912	0.0357617
$726$	0	0
$727$	52.4665	1.94587	0.972937	−	0.231070i	$-0.0742227\pi$
$727$	52.4665	1.94587	0.972937	+	0.231070i	$0.0742227\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.19258	−0.192055
$732$	0	0
$733$	−27.6148	−1.01998	−0.509989	−	0.860181i	$-0.670350\pi$
$733$	−27.6148	−1.01998	−0.509989	+	0.860181i	$0.670350\pi$
$734$	0	0
$735$	−2.19258	−0.0808746
$736$	0	0
$737$	10.9629	0.403824
$738$	0	0
$739$	28.9258	1.06405	0.532027	−	0.846728i	$-0.321431\pi$
$739$	28.9258	1.06405	0.532027	+	0.846728i	$0.321431\pi$
$740$	0	0
$741$	10.8074	0.397020
$742$	0	0
$743$	34.9629	1.28266	0.641332	−	0.767263i	$-0.278382\pi$
$743$	34.9629	1.28266	0.641332	+	0.767263i	$0.278382\pi$
$744$	0	0
$745$	−22.7703	−0.834240
$746$	0	0
$747$	11.3852	0.416561
$748$	0	0
$749$	−17.5777	−0.642277
$750$	0	0
$751$	−18.1184	−0.661150	−0.330575	−	0.943780i	$-0.607243\pi$
$751$	−18.1184	−0.661150	−0.330575	+	0.943780i	$0.607243\pi$
$752$	0	0
$753$	−11.6148	−0.423268
$754$	0	0
$755$	−5.22967	−0.190327
$756$	0	0
$757$	46.4665	1.68885	0.844427	−	0.535671i	$-0.179941\pi$
$757$	46.4665	1.68885	0.844427	+	0.535671i	$0.179941\pi$
$758$	0	0
$759$	5.00000	0.181489
$760$	0	0
$761$	16.7703	0.607924	0.303962	−	0.952684i	$-0.401690\pi$
$761$	16.7703	0.607924	0.303962	+	0.952684i	$0.401690\pi$
$762$	0	0
$763$	3.42225	0.123894
$764$	0	0
$765$	−2.19258	−0.0792730
$766$	0	0
$767$	40.1555	1.44993
$768$	0	0
$769$	−2.84451	−0.102575	−0.0512877	−	0.998684i	$-0.516333\pi$
$769$	−2.84451	−0.102575	−0.0512877	+	0.998684i	$0.516333\pi$
$770$	0	0
$771$	−3.15549	−0.113642
$772$	0	0
$773$	40.1555	1.44429	0.722146	−	0.691740i	$-0.243156\pi$
$773$	40.1555	1.44429	0.722146	+	0.691740i	$0.243156\pi$
$774$	0	0
$775$	−0.651923	−0.0234178
$776$	0	0
$777$	5.38516	0.193192
$778$	0	0
$779$	−3.38516	−0.121286
$780$	0	0
$781$	67.8887	2.42925
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	16.6962	0.595911
$786$	0	0
$787$	−39.2739	−1.39996	−0.699982	−	0.714161i	$-0.746809\pi$
$787$	−39.2739	−1.39996	−0.699982	+	0.714161i	$0.746809\pi$
$788$	0	0
$789$	−24.1555	−0.859958
$790$	0	0
$791$	4.57775	0.162766
$792$	0	0
$793$	29.3481	1.04218
$794$	0	0
$795$	23.1926	0.822556
$796$	0	0
$797$	23.9258	0.847496	0.423748	−	0.905780i	$-0.360714\pi$
$797$	23.9258	0.847496	0.423748	+	0.905780i	$0.360714\pi$
$798$	0	0
$799$	−10.3852	−0.367401
$800$	0	0
$801$	3.57775	0.126413
$802$	0	0
$803$	45.0000	1.58802
$804$	0	0
$805$	2.19258	0.0772784
$806$	0	0
$807$	29.9629	1.05474
$808$	0	0
$809$	43.1184	1.51596	0.757981	−	0.652276i	$-0.226186\pi$
$809$	43.1184	1.51596	0.757981	+	0.652276i	$0.226186\pi$
$810$	0	0
$811$	−35.3110	−1.23994	−0.619968	−	0.784627i	$-0.712855\pi$
$811$	−35.3110	−1.23994	−0.619968	+	0.784627i	$0.712855\pi$
$812$	0	0
$813$	5.00000	0.175358
$814$	0	0
$815$	42.4223	1.48599
$816$	0	0
$817$	−17.5777	−0.614968
$818$	0	0
$819$	3.19258	0.111558
$820$	0	0
$821$	−5.61484	−0.195959	−0.0979795	−	0.995188i	$-0.531238\pi$
$821$	−5.61484	−0.195959	−0.0979795	+	0.995188i	$0.531238\pi$
$822$	0	0
$823$	17.3481	0.604716	0.302358	−	0.953194i	$-0.402226\pi$
$823$	17.3481	0.604716	0.302358	+	0.953194i	$0.402226\pi$
$824$	0	0
$825$	0.962912	0.0335243
$826$	0	0
$827$	43.2739	1.50478	0.752390	−	0.658717i	$-0.228901\pi$
$827$	43.2739	1.50478	0.752390	+	0.658717i	$0.228901\pi$
$828$	0	0
$829$	10.9258	0.379470	0.189735	−	0.981835i	$-0.439237\pi$
$829$	10.9258	0.379470	0.189735	+	0.981835i	$0.439237\pi$
$830$	0	0
$831$	−13.1926	−0.457646
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	50.4294	1.74518
$836$	0	0
$837$	3.38516	0.117008
$838$	0	0
$839$	25.5036	0.880481	0.440241	−	0.897880i	$-0.354893\pi$
$839$	25.5036	0.880481	0.440241	+	0.897880i	$0.354893\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−16.3852	−0.564335
$844$	0	0
$845$	6.15549	0.211755
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	5.03709	0.172872
$850$	0	0
$851$	−5.38516	−0.184601
$852$	0	0
$853$	29.9258	1.02464	0.512320	−	0.858794i	$-0.328786\pi$
$853$	29.9258	1.02464	0.512320	+	0.858794i	$0.328786\pi$
$854$	0	0
$855$	−7.42225	−0.253836
$856$	0	0
$857$	37.1555	1.26921	0.634604	−	0.772838i	$-0.281163\pi$
$857$	37.1555	1.26921	0.634604	+	0.772838i	$0.281163\pi$
$858$	0	0
$859$	50.7703	1.73226	0.866131	−	0.499818i	$-0.166600\pi$
$859$	50.7703	1.73226	0.866131	+	0.499818i	$0.166600\pi$
$860$	0	0
$861$	−1.00000	−0.0340799
$862$	0	0
$863$	−41.9258	−1.42717	−0.713586	−	0.700568i	$-0.752930\pi$
$863$	−41.9258	−1.42717	−0.713586	+	0.700568i	$0.752930\pi$
$864$	0	0
$865$	34.5777	1.17568
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−48.8516	−1.65718
$870$	0	0
$871$	−7.00000	−0.237186
$872$	0	0
$873$	−17.7703	−0.601435
$874$	0	0
$875$	11.3852	0.384889
$876$	0	0
$877$	48.0000	1.62084	0.810422	−	0.585846i	$-0.199238\pi$
$877$	48.0000	1.62084	0.810422	+	0.585846i	$0.199238\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	46.6962	1.57323	0.786617	−	0.617442i	$-0.211831\pi$
$881$	46.6962	1.57323	0.786617	+	0.617442i	$0.211831\pi$
$882$	0	0
$883$	−31.8887	−1.07314	−0.536571	−	0.843855i	$-0.680281\pi$
$883$	−31.8887	−1.07314	−0.536571	+	0.843855i	$0.680281\pi$
$884$	0	0
$885$	−27.5777	−0.927016
$886$	0	0
$887$	−39.8887	−1.33933	−0.669666	−	0.742662i	$-0.733563\pi$
$887$	−39.8887	−1.33933	−0.669666	+	0.742662i	$0.733563\pi$
$888$	0	0
$889$	−9.19258	−0.308309
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	−35.1555	−1.17643
$894$	0	0
$895$	45.1184	1.50814
$896$	0	0
$897$	−3.19258	−0.106597
$898$	0	0
$899$	−16.9258	−0.564508
$900$	0	0
$901$	−10.5777	−0.352396
$902$	0	0
$903$	−5.19258	−0.172798
$904$	0	0
$905$	−15.3481	−0.510187
$906$	0	0
$907$	52.1926	1.73303	0.866513	−	0.499154i	$-0.166356\pi$
$907$	52.1926	1.73303	0.866513	+	0.499154i	$0.166356\pi$
$908$	0	0
$909$	14.5777	0.483513
$910$	0	0
$911$	31.1555	1.03223	0.516114	−	0.856520i	$-0.327378\pi$
$911$	31.1555	1.03223	0.516114	+	0.856520i	$0.327378\pi$
$912$	0	0
$913$	−56.9258	−1.88397
$914$	0	0
$915$	−20.1555	−0.666320
$916$	0	0
$917$	−9.77033	−0.322645
$918$	0	0
$919$	−22.6148	−0.745995	−0.372997	−	0.927832i	$-0.621670\pi$
$919$	−22.6148	−0.745995	−0.372997	+	0.927832i	$0.621670\pi$
$920$	0	0
$921$	0.614835	0.0202595
$922$	0	0
$923$	−43.3481	−1.42682
$924$	0	0
$925$	−1.03709	−0.0340992
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−51.8074	−1.69975	−0.849873	−	0.526987i	$-0.823322\pi$
$929$	−51.8074	−1.69975	−0.849873	+	0.526987i	$0.823322\pi$
$930$	0	0
$931$	3.38516	0.110944
$932$	0	0
$933$	3.96291	0.129740
$934$	0	0
$935$	10.9629	0.358526
$936$	0	0
$937$	−42.1555	−1.37716	−0.688580	−	0.725160i	$-0.741766\pi$
$937$	−42.1555	−1.37716	−0.688580	+	0.725160i	$0.741766\pi$
$938$	0	0
$939$	4.00000	0.130535
$940$	0	0
$941$	−8.77033	−0.285905	−0.142952	−	0.989730i	$-0.545660\pi$
$941$	−8.77033	−0.285905	−0.142952	+	0.989730i	$0.545660\pi$
$942$	0	0
$943$	1.00000	0.0325645
$944$	0	0
$945$	−2.19258	−0.0713247
$946$	0	0
$947$	−37.2297	−1.20980	−0.604901	−	0.796301i	$-0.706787\pi$
$947$	−37.2297	−1.20980	−0.604901	+	0.796301i	$0.706787\pi$
$948$	0	0
$949$	−28.7332	−0.932720
$950$	0	0
$951$	22.1926	0.719644
$952$	0	0
$953$	−32.8074	−1.06274	−0.531368	−	0.847141i	$-0.678322\pi$
$953$	−32.8074	−1.06274	−0.531368	+	0.847141i	$0.678322\pi$
$954$	0	0
$955$	22.7703	0.736831
$956$	0	0
$957$	25.0000	0.808135
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	−19.5407	−0.630344
$962$	0	0
$963$	−17.5777	−0.566435
$964$	0	0
$965$	18.3852	0.591839
$966$	0	0
$967$	37.0813	1.19245	0.596227	−	0.802816i	$-0.296666\pi$
$967$	37.0813	1.19245	0.596227	+	0.802816i	$0.296666\pi$
$968$	0	0
$969$	3.38516	0.108747
$970$	0	0
$971$	−16.4223	−0.527015	−0.263508	−	0.964657i	$-0.584879\pi$
$971$	−16.4223	−0.527015	−0.263508	+	0.964657i	$0.584879\pi$
$972$	0	0
$973$	−0.577747	−0.0185217
$974$	0	0
$975$	−0.614835	−0.0196905
$976$	0	0
$977$	5.96291	0.190770	0.0953852	−	0.995440i	$-0.469592\pi$
$977$	5.96291	0.190770	0.0953852	+	0.995440i	$0.469592\pi$
$978$	0	0
$979$	−17.8887	−0.571726
$980$	0	0
$981$	3.42225	0.109264
$982$	0	0
$983$	−6.61484	−0.210980	−0.105490	−	0.994420i	$-0.533641\pi$
$983$	−6.61484	−0.210980	−0.105490	+	0.994420i	$0.533641\pi$
$984$	0	0
$985$	−20.1555	−0.642207
$986$	0	0
$987$	−10.3852	−0.330563
$988$	0	0
$989$	5.19258	0.165115
$990$	0	0
$991$	51.5036	1.63606	0.818032	−	0.575172i	$-0.195065\pi$
$991$	51.5036	1.63606	0.818032	+	0.575172i	$0.195065\pi$
$992$	0	0
$993$	−13.2297	−0.419831
$994$	0	0
$995$	−29.2668	−0.927819
$996$	0	0
$997$	−25.6962	−0.813805	−0.406903	−	0.913472i	$-0.633391\pi$
$997$	−25.6962	−0.813805	−0.406903	+	0.913472i	$0.633391\pi$
$998$	0	0
$999$	5.38516	0.170379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bl.1.1		2
4.3	odd	2		3864.2.a.h.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.h.1.1	✓	2	4.3	odd	2
7728.2.a.bl.1.1		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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.19258 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.19258 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} +3.19258 q^{13} -2.19258 q^{15} +1.00000 q^{17} +3.38516 q^{19} +1.00000 q^{21} -1.00000 q^{23} -0.192582 q^{25} +1.00000 q^{27} -5.00000 q^{29} +3.38516 q^{31} -5.00000 q^{33} -2.19258 q^{35} +5.38516 q^{37} +3.19258 q^{39} -1.00000 q^{41} -5.19258 q^{43} -2.19258 q^{45} -10.3852 q^{47} +1.00000 q^{49} +1.00000 q^{51} -10.5777 q^{53} +10.9629 q^{55} +3.38516 q^{57} +12.5777 q^{59} +9.19258 q^{61} +1.00000 q^{63} -7.00000 q^{65} -2.19258 q^{67} -1.00000 q^{69} -13.5777 q^{71} -9.00000 q^{73} -0.192582 q^{75} -5.00000 q^{77} +9.77033 q^{79} +1.00000 q^{81} +11.3852 q^{83} -2.19258 q^{85} -5.00000 q^{87} +3.57775 q^{89} +3.19258 q^{91} +3.38516 q^{93} -7.42225 q^{95} -17.7703 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 10 q^{11} + q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 10 q^{33} + q^{35} + q^{39} - 2 q^{41} - 5 q^{43} + q^{45} - 10 q^{47} + 2 q^{49} + 2 q^{51} - 5 q^{53} - 5 q^{55} - 4 q^{57} + 9 q^{59} + 13 q^{61} + 2 q^{63} - 14 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} - 18 q^{73} + 5 q^{75} - 10 q^{77} - 2 q^{79} + 2 q^{81} + 12 q^{83} + q^{85} - 10 q^{87} - 9 q^{89} + q^{91} - 4 q^{93} - 31 q^{95} - 14 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 10 * q^11 + q^13 + q^15 + 2 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 10 * q^33 + q^35 + q^39 - 2 * q^41 - 5 * q^43 + q^45 - 10 * q^47 + 2 * q^49 + 2 * q^51 - 5 * q^53 - 5 * q^55 - 4 * q^57 + 9 * q^59 + 13 * q^61 + 2 * q^63 - 14 * q^65 + q^67 - 2 * q^69 - 11 * q^71 - 18 * q^73 + 5 * q^75 - 10 * q^77 - 2 * q^79 + 2 * q^81 + 12 * q^83 + q^85 - 10 * q^87 - 9 * q^89 + q^91 - 4 * q^93 - 31 * q^95 - 14 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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