LMFDB - Embedded newform 7248.2.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 $ x^10 - 2x^9 - 27x^8 + 45x^7 + 258x^6 - 289x^5 - 1133x^4 + 510x^3 + 2070x^2 + 341*x - 500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3624)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.48327$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.48327 q^{5} -0.175714 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.48327 q^{5} -0.175714 q^{7} +1.00000 q^{9} +4.12168 q^{11} -4.08864 q^{13} +1.48327 q^{15} +5.45907 q^{17} -2.11616 q^{19} +0.175714 q^{21} -0.716411 q^{23} -2.79990 q^{25} -1.00000 q^{27} -1.63297 q^{29} -8.89959 q^{31} -4.12168 q^{33} +0.260632 q^{35} +3.96963 q^{37} +4.08864 q^{39} +8.82946 q^{41} +10.4769 q^{43} -1.48327 q^{45} -9.26291 q^{47} -6.96912 q^{49} -5.45907 q^{51} +6.62829 q^{53} -6.11358 q^{55} +2.11616 q^{57} +6.61618 q^{59} -0.398176 q^{61} -0.175714 q^{63} +6.06457 q^{65} -6.69175 q^{67} +0.716411 q^{69} -10.5331 q^{71} +7.22891 q^{73} +2.79990 q^{75} -0.724237 q^{77} -5.29435 q^{79} +1.00000 q^{81} -12.0708 q^{83} -8.09729 q^{85} +1.63297 q^{87} -1.20472 q^{89} +0.718432 q^{91} +8.89959 q^{93} +3.13884 q^{95} +15.4148 q^{97} +4.12168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * 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
$5$	−1.48327	−0.663340	−0.331670	−	0.943395i	$-0.607612\pi$
$5$	−1.48327	−0.663340	−0.331670	+	0.943395i	$0.607612\pi$
$6$	0	0
$7$	−0.175714	−0.0664137	−0.0332068	−	0.999449i	$-0.510572\pi$
$7$	−0.175714	−0.0664137	−0.0332068	+	0.999449i	$0.510572\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.12168	1.24273	0.621366	−	0.783520i	$-0.286578\pi$
$11$	4.12168	1.24273	0.621366	+	0.783520i	$0.286578\pi$
$12$	0	0
$13$	−4.08864	−1.13398	−0.566992	−	0.823723i	$-0.691893\pi$
$13$	−4.08864	−1.13398	−0.566992	+	0.823723i	$0.691893\pi$
$14$	0	0
$15$	1.48327	0.382980
$16$	0	0
$17$	5.45907	1.32402	0.662009	−	0.749495i	$-0.269704\pi$
$17$	5.45907	1.32402	0.662009	+	0.749495i	$0.269704\pi$
$18$	0	0
$19$	−2.11616	−0.485480	−0.242740	−	0.970091i	$-0.578046\pi$
$19$	−2.11616	−0.485480	−0.242740	+	0.970091i	$0.578046\pi$
$20$	0	0
$21$	0.175714	0.0383440
$22$	0	0
$23$	−0.716411	−0.149382	−0.0746910	−	0.997207i	$-0.523797\pi$
$23$	−0.716411	−0.149382	−0.0746910	+	0.997207i	$0.523797\pi$
$24$	0	0
$25$	−2.79990	−0.559980
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.63297	−0.303236	−0.151618	−	0.988439i	$-0.548448\pi$
$29$	−1.63297	−0.303236	−0.151618	+	0.988439i	$0.548448\pi$
$30$	0	0
$31$	−8.89959	−1.59841	−0.799206	−	0.601057i	$-0.794747\pi$
$31$	−8.89959	−1.59841	−0.799206	+	0.601057i	$0.794747\pi$
$32$	0	0
$33$	−4.12168	−0.717492
$34$	0	0
$35$	0.260632	0.0440548
$36$	0	0
$37$	3.96963	0.652602	0.326301	−	0.945266i	$-0.394198\pi$
$37$	3.96963	0.652602	0.326301	+	0.945266i	$0.394198\pi$
$38$	0	0
$39$	4.08864	0.654706
$40$	0	0
$41$	8.82946	1.37893	0.689465	−	0.724319i	$-0.257846\pi$
$41$	8.82946	1.37893	0.689465	+	0.724319i	$0.257846\pi$
$42$	0	0
$43$	10.4769	1.59772	0.798859	−	0.601518i	$-0.205437\pi$
$43$	10.4769	1.59772	0.798859	+	0.601518i	$0.205437\pi$
$44$	0	0
$45$	−1.48327	−0.221113
$46$	0	0
$47$	−9.26291	−1.35113	−0.675567	−	0.737298i	$-0.736101\pi$
$47$	−9.26291	−1.35113	−0.675567	+	0.737298i	$0.736101\pi$
$48$	0	0
$49$	−6.96912	−0.995589
$50$	0	0
$51$	−5.45907	−0.764423
$52$	0	0
$53$	6.62829	0.910466	0.455233	−	0.890372i	$-0.349556\pi$
$53$	6.62829	0.910466	0.455233	+	0.890372i	$0.349556\pi$
$54$	0	0
$55$	−6.11358	−0.824354
$56$	0	0
$57$	2.11616	0.280292
$58$	0	0
$59$	6.61618	0.861354	0.430677	−	0.902506i	$-0.358275\pi$
$59$	6.61618	0.861354	0.430677	+	0.902506i	$0.358275\pi$
$60$	0	0
$61$	−0.398176	−0.0509812	−0.0254906	−	0.999675i	$-0.508115\pi$
$61$	−0.398176	−0.0509812	−0.0254906	+	0.999675i	$0.508115\pi$
$62$	0	0
$63$	−0.175714	−0.0221379
$64$	0	0
$65$	6.06457	0.752218
$66$	0	0
$67$	−6.69175	−0.817527	−0.408764	−	0.912640i	$-0.634040\pi$
$67$	−6.69175	−0.817527	−0.408764	+	0.912640i	$0.634040\pi$
$68$	0	0
$69$	0.716411	0.0862458
$70$	0	0
$71$	−10.5331	−1.25005	−0.625026	−	0.780604i	$-0.714912\pi$
$71$	−10.5331	−1.25005	−0.625026	+	0.780604i	$0.714912\pi$
$72$	0	0
$73$	7.22891	0.846080	0.423040	−	0.906111i	$-0.360963\pi$
$73$	7.22891	0.846080	0.423040	+	0.906111i	$0.360963\pi$
$74$	0	0
$75$	2.79990	0.323305
$76$	0	0
$77$	−0.724237	−0.0825345
$78$	0	0
$79$	−5.29435	−0.595661	−0.297831	−	0.954619i	$-0.596263\pi$
$79$	−5.29435	−0.595661	−0.297831	+	0.954619i	$0.596263\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0708	−1.32494	−0.662470	−	0.749089i	$-0.730492\pi$
$83$	−12.0708	−1.32494	−0.662470	+	0.749089i	$0.730492\pi$
$84$	0	0
$85$	−8.09729	−0.878275
$86$	0	0
$87$	1.63297	0.175073
$88$	0	0
$89$	−1.20472	−0.127700	−0.0638498	−	0.997960i	$-0.520338\pi$
$89$	−1.20472	−0.127700	−0.0638498	+	0.997960i	$0.520338\pi$
$90$	0	0
$91$	0.718432	0.0753121
$92$	0	0
$93$	8.89959	0.922844
$94$	0	0
$95$	3.13884	0.322039
$96$	0	0
$97$	15.4148	1.56514	0.782568	−	0.622565i	$-0.213910\pi$
$97$	15.4148	1.56514	0.782568	+	0.622565i	$0.213910\pi$
$98$	0	0
$99$	4.12168	0.414244
$100$	0	0
$101$	15.3295	1.52534	0.762672	−	0.646785i	$-0.223887\pi$
$101$	15.3295	1.52534	0.762672	+	0.646785i	$0.223887\pi$
$102$	0	0
$103$	1.15129	0.113440	0.0567198	−	0.998390i	$-0.481936\pi$
$103$	1.15129	0.113440	0.0567198	+	0.998390i	$0.481936\pi$
$104$	0	0
$105$	−0.260632	−0.0254351
$106$	0	0
$107$	−2.71300	−0.262276	−0.131138	−	0.991364i	$-0.541863\pi$
$107$	−2.71300	−0.262276	−0.131138	+	0.991364i	$0.541863\pi$
$108$	0	0
$109$	−8.15699	−0.781298	−0.390649	−	0.920540i	$-0.627749\pi$
$109$	−8.15699	−0.781298	−0.390649	+	0.920540i	$0.627749\pi$
$110$	0	0
$111$	−3.96963	−0.376780
$112$	0	0
$113$	−4.82139	−0.453559	−0.226779	−	0.973946i	$-0.572820\pi$
$113$	−4.82139	−0.453559	−0.226779	+	0.973946i	$0.572820\pi$
$114$	0	0
$115$	1.06263	0.0990911
$116$	0	0
$117$	−4.08864	−0.377995
$118$	0	0
$119$	−0.959235	−0.0879330
$120$	0	0
$121$	5.98823	0.544385
$122$	0	0
$123$	−8.82946	−0.796125
$124$	0	0
$125$	11.5694	1.03480
$126$	0	0
$127$	−3.87564	−0.343907	−0.171954	−	0.985105i	$-0.555008\pi$
$127$	−3.87564	−0.343907	−0.171954	+	0.985105i	$0.555008\pi$
$128$	0	0
$129$	−10.4769	−0.922443
$130$	0	0
$131$	8.19391	0.715905	0.357952	−	0.933740i	$-0.383475\pi$
$131$	8.19391	0.715905	0.357952	+	0.933740i	$0.383475\pi$
$132$	0	0
$133$	0.371839	0.0322425
$134$	0	0
$135$	1.48327	0.127660
$136$	0	0
$137$	−14.8020	−1.26462	−0.632311	−	0.774715i	$-0.717893\pi$
$137$	−14.8020	−1.26462	−0.632311	+	0.774715i	$0.717893\pi$
$138$	0	0
$139$	12.9994	1.10259	0.551295	−	0.834310i	$-0.314134\pi$
$139$	12.9994	1.10259	0.551295	+	0.834310i	$0.314134\pi$
$140$	0	0
$141$	9.26291	0.780078
$142$	0	0
$143$	−16.8521	−1.40924
$144$	0	0
$145$	2.42215	0.201148
$146$	0	0
$147$	6.96912	0.574804
$148$	0	0
$149$	11.4229	0.935804	0.467902	−	0.883780i	$-0.345010\pi$
$149$	11.4229	0.935804	0.467902	+	0.883780i	$0.345010\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	5.45907	0.441340
$154$	0	0
$155$	13.2005	1.06029
$156$	0	0
$157$	−6.74558	−0.538356	−0.269178	−	0.963090i	$-0.586752\pi$
$157$	−6.74558	−0.538356	−0.269178	+	0.963090i	$0.586752\pi$
$158$	0	0
$159$	−6.62829	−0.525658
$160$	0	0
$161$	0.125883	0.00992101
$162$	0	0
$163$	17.0088	1.33223	0.666116	−	0.745848i	$-0.267955\pi$
$163$	17.0088	1.33223	0.666116	+	0.745848i	$0.267955\pi$
$164$	0	0
$165$	6.11358	0.475941
$166$	0	0
$167$	−7.93379	−0.613935	−0.306967	−	0.951720i	$-0.599314\pi$
$167$	−7.93379	−0.613935	−0.306967	+	0.951720i	$0.599314\pi$
$168$	0	0
$169$	3.71698	0.285922
$170$	0	0
$171$	−2.11616	−0.161827
$172$	0	0
$173$	12.6178	0.959317	0.479658	−	0.877455i	$-0.340761\pi$
$173$	12.6178	0.959317	0.479658	+	0.877455i	$0.340761\pi$
$174$	0	0
$175$	0.491982	0.0371903
$176$	0	0
$177$	−6.61618	−0.497303
$178$	0	0
$179$	−8.27629	−0.618599	−0.309299	−	0.950965i	$-0.600095\pi$
$179$	−8.27629	−0.618599	−0.309299	+	0.950965i	$0.600095\pi$
$180$	0	0
$181$	9.97420	0.741376	0.370688	−	0.928757i	$-0.379122\pi$
$181$	9.97420	0.741376	0.370688	+	0.928757i	$0.379122\pi$
$182$	0	0
$183$	0.398176	0.0294340
$184$	0	0
$185$	−5.88804	−0.432897
$186$	0	0
$187$	22.5005	1.64540
$188$	0	0
$189$	0.175714	0.0127813
$190$	0	0
$191$	−12.4929	−0.903951	−0.451976	−	0.892030i	$-0.649281\pi$
$191$	−12.4929	−0.903951	−0.451976	+	0.892030i	$0.649281\pi$
$192$	0	0
$193$	−15.4013	−1.10861	−0.554306	−	0.832313i	$-0.687016\pi$
$193$	−15.4013	−1.10861	−0.554306	+	0.832313i	$0.687016\pi$
$194$	0	0
$195$	−6.06457	−0.434293
$196$	0	0
$197$	−12.3978	−0.883304	−0.441652	−	0.897186i	$-0.645607\pi$
$197$	−12.3978	−0.883304	−0.441652	+	0.897186i	$0.645607\pi$
$198$	0	0
$199$	0.228835	0.0162217	0.00811084	−	0.999967i	$-0.497418\pi$
$199$	0.228835	0.0162217	0.00811084	+	0.999967i	$0.497418\pi$
$200$	0	0
$201$	6.69175	0.472000
$202$	0	0
$203$	0.286937	0.0201390
$204$	0	0
$205$	−13.0965	−0.914699
$206$	0	0
$207$	−0.716411	−0.0497940
$208$	0	0
$209$	−8.72213	−0.603322
$210$	0	0
$211$	−17.6734	−1.21669	−0.608344	−	0.793673i	$-0.708166\pi$
$211$	−17.6734	−1.21669	−0.608344	+	0.793673i	$0.708166\pi$
$212$	0	0
$213$	10.5331	0.721718
$214$	0	0
$215$	−15.5402	−1.05983
$216$	0	0
$217$	1.56378	0.106156
$218$	0	0
$219$	−7.22891	−0.488485
$220$	0	0
$221$	−22.3202	−1.50142
$222$	0	0
$223$	−18.1746	−1.21706	−0.608530	−	0.793531i	$-0.708240\pi$
$223$	−18.1746	−1.21706	−0.608530	+	0.793531i	$0.708240\pi$
$224$	0	0
$225$	−2.79990	−0.186660
$226$	0	0
$227$	−15.6799	−1.04071	−0.520356	−	0.853949i	$-0.674201\pi$
$227$	−15.6799	−1.04071	−0.520356	+	0.853949i	$0.674201\pi$
$228$	0	0
$229$	−21.8796	−1.44584	−0.722922	−	0.690930i	$-0.757201\pi$
$229$	−21.8796	−1.44584	−0.722922	+	0.690930i	$0.757201\pi$
$230$	0	0
$231$	0.724237	0.0476513
$232$	0	0
$233$	−4.97208	−0.325732	−0.162866	−	0.986648i	$-0.552074\pi$
$233$	−4.97208	−0.325732	−0.162866	+	0.986648i	$0.552074\pi$
$234$	0	0
$235$	13.7394	0.896262
$236$	0	0
$237$	5.29435	0.343905
$238$	0	0
$239$	−12.7720	−0.826154	−0.413077	−	0.910696i	$-0.635546\pi$
$239$	−12.7720	−0.826154	−0.413077	+	0.910696i	$0.635546\pi$
$240$	0	0
$241$	1.21114	0.0780167	0.0390083	−	0.999239i	$-0.487580\pi$
$241$	1.21114	0.0780167	0.0390083	+	0.999239i	$0.487580\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	10.3371	0.660414
$246$	0	0
$247$	8.65222	0.550527
$248$	0	0
$249$	12.0708	0.764954
$250$	0	0
$251$	−16.2295	−1.02440	−0.512199	−	0.858867i	$-0.671169\pi$
$251$	−16.2295	−1.02440	−0.512199	+	0.858867i	$0.671169\pi$
$252$	0	0
$253$	−2.95282	−0.185642
$254$	0	0
$255$	8.09729	0.507072
$256$	0	0
$257$	2.57233	0.160457	0.0802287	−	0.996776i	$-0.474435\pi$
$257$	2.57233	0.160457	0.0802287	+	0.996776i	$0.474435\pi$
$258$	0	0
$259$	−0.697519	−0.0433417
$260$	0	0
$261$	−1.63297	−0.101079
$262$	0	0
$263$	−16.7211	−1.03107	−0.515534	−	0.856869i	$-0.672406\pi$
$263$	−16.7211	−1.03107	−0.515534	+	0.856869i	$0.672406\pi$
$264$	0	0
$265$	−9.83157	−0.603948
$266$	0	0
$267$	1.20472	0.0737274
$268$	0	0
$269$	−0.811246	−0.0494625	−0.0247313	−	0.999694i	$-0.507873\pi$
$269$	−0.811246	−0.0494625	−0.0247313	+	0.999694i	$0.507873\pi$
$270$	0	0
$271$	−7.63991	−0.464091	−0.232046	−	0.972705i	$-0.574542\pi$
$271$	−7.63991	−0.464091	−0.232046	+	0.972705i	$0.574542\pi$
$272$	0	0
$273$	−0.718432	−0.0434815
$274$	0	0
$275$	−11.5403	−0.695906
$276$	0	0
$277$	10.1519	0.609969	0.304984	−	0.952357i	$-0.401349\pi$
$277$	10.1519	0.609969	0.304984	+	0.952357i	$0.401349\pi$
$278$	0	0
$279$	−8.89959	−0.532804
$280$	0	0
$281$	−4.45901	−0.266002	−0.133001	−	0.991116i	$-0.542461\pi$
$281$	−4.45901	−0.266002	−0.133001	+	0.991116i	$0.542461\pi$
$282$	0	0
$283$	16.7681	0.996758	0.498379	−	0.866959i	$-0.333929\pi$
$283$	16.7681	0.996758	0.498379	+	0.866959i	$0.333929\pi$
$284$	0	0
$285$	−3.13884	−0.185929
$286$	0	0
$287$	−1.55146	−0.0915798
$288$	0	0
$289$	12.8014	0.753026
$290$	0	0
$291$	−15.4148	−0.903631
$292$	0	0
$293$	−27.4062	−1.60109	−0.800545	−	0.599273i	$-0.795456\pi$
$293$	−27.4062	−1.60109	−0.800545	+	0.599273i	$0.795456\pi$
$294$	0	0
$295$	−9.81361	−0.571370
$296$	0	0
$297$	−4.12168	−0.239164
$298$	0	0
$299$	2.92915	0.169397
$300$	0	0
$301$	−1.84095	−0.106110
$302$	0	0
$303$	−15.3295	−0.880658
$304$	0	0
$305$	0.590603	0.0338179
$306$	0	0
$307$	−19.4219	−1.10847	−0.554233	−	0.832361i	$-0.686988\pi$
$307$	−19.4219	−1.10847	−0.554233	+	0.832361i	$0.686988\pi$
$308$	0	0
$309$	−1.15129	−0.0654944
$310$	0	0
$311$	14.7683	0.837434	0.418717	−	0.908117i	$-0.362480\pi$
$311$	14.7683	0.837434	0.418717	+	0.908117i	$0.362480\pi$
$312$	0	0
$313$	34.2310	1.93485	0.967424	−	0.253160i	$-0.0814700\pi$
$313$	34.2310	1.93485	0.967424	+	0.253160i	$0.0814700\pi$
$314$	0	0
$315$	0.260632	0.0146849
$316$	0	0
$317$	−27.9167	−1.56796	−0.783978	−	0.620788i	$-0.786813\pi$
$317$	−27.9167	−1.56796	−0.783978	+	0.620788i	$0.786813\pi$
$318$	0	0
$319$	−6.73060	−0.376841
$320$	0	0
$321$	2.71300	0.151425
$322$	0	0
$323$	−11.5523	−0.642785
$324$	0	0
$325$	11.4478	0.635009
$326$	0	0
$327$	8.15699	0.451082
$328$	0	0
$329$	1.62762	0.0897338
$330$	0	0
$331$	−11.5972	−0.637438	−0.318719	−	0.947849i	$-0.603253\pi$
$331$	−11.5972	−0.637438	−0.318719	+	0.947849i	$0.603253\pi$
$332$	0	0
$333$	3.96963	0.217534
$334$	0	0
$335$	9.92569	0.542298
$336$	0	0
$337$	−31.9375	−1.73975	−0.869874	−	0.493274i	$-0.835800\pi$
$337$	−31.9375	−1.73975	−0.869874	+	0.493274i	$0.835800\pi$
$338$	0	0
$339$	4.82139	0.261862
$340$	0	0
$341$	−36.6812	−1.98640
$342$	0	0
$343$	2.45457	0.132534
$344$	0	0
$345$	−1.06263	−0.0572103
$346$	0	0
$347$	−28.7055	−1.54099	−0.770496	−	0.637445i	$-0.779991\pi$
$347$	−28.7055	−1.54099	−0.770496	+	0.637445i	$0.779991\pi$
$348$	0	0
$349$	31.7521	1.69965	0.849824	−	0.527066i	$-0.176708\pi$
$349$	31.7521	1.69965	0.849824	+	0.527066i	$0.176708\pi$
$350$	0	0
$351$	4.08864	0.218235
$352$	0	0
$353$	0.0639637	0.00340444	0.00170222	−	0.999999i	$-0.499458\pi$
$353$	0.0639637	0.00340444	0.00170222	+	0.999999i	$0.499458\pi$
$354$	0	0
$355$	15.6235	0.829209
$356$	0	0
$357$	0.959235	0.0507681
$358$	0	0
$359$	1.47636	0.0779195	0.0389597	−	0.999241i	$-0.487596\pi$
$359$	1.47636	0.0779195	0.0389597	+	0.999241i	$0.487596\pi$
$360$	0	0
$361$	−14.5219	−0.764309
$362$	0	0
$363$	−5.98823	−0.314301
$364$	0	0
$365$	−10.7225	−0.561239
$366$	0	0
$367$	22.6854	1.18417	0.592084	−	0.805876i	$-0.298305\pi$
$367$	22.6854	1.18417	0.592084	+	0.805876i	$0.298305\pi$
$368$	0	0
$369$	8.82946	0.459643
$370$	0	0
$371$	−1.16468	−0.0604674
$372$	0	0
$373$	9.88463	0.511807	0.255903	−	0.966702i	$-0.417627\pi$
$373$	9.88463	0.511807	0.255903	+	0.966702i	$0.417627\pi$
$374$	0	0
$375$	−11.5694	−0.597440
$376$	0	0
$377$	6.67665	0.343865
$378$	0	0
$379$	−22.2501	−1.14291	−0.571455	−	0.820633i	$-0.693621\pi$
$379$	−22.2501	−1.14291	−0.571455	+	0.820633i	$0.693621\pi$
$380$	0	0
$381$	3.87564	0.198555
$382$	0	0
$383$	−21.4644	−1.09678	−0.548389	−	0.836224i	$-0.684759\pi$
$383$	−21.4644	−1.09678	−0.548389	+	0.836224i	$0.684759\pi$
$384$	0	0
$385$	1.07424	0.0547484
$386$	0	0
$387$	10.4769	0.532573
$388$	0	0
$389$	−5.05650	−0.256375	−0.128187	−	0.991750i	$-0.540916\pi$
$389$	−5.05650	−0.256375	−0.128187	+	0.991750i	$0.540916\pi$
$390$	0	0
$391$	−3.91094	−0.197785
$392$	0	0
$393$	−8.19391	−0.413328
$394$	0	0
$395$	7.85297	0.395126
$396$	0	0
$397$	−5.18287	−0.260121	−0.130060	−	0.991506i	$-0.541517\pi$
$397$	−5.18287	−0.260121	−0.130060	+	0.991506i	$0.541517\pi$
$398$	0	0
$399$	−0.371839	−0.0186152
$400$	0	0
$401$	−32.7166	−1.63379	−0.816894	−	0.576788i	$-0.804306\pi$
$401$	−32.7166	−1.63379	−0.816894	+	0.576788i	$0.804306\pi$
$402$	0	0
$403$	36.3872	1.81258
$404$	0	0
$405$	−1.48327	−0.0737044
$406$	0	0
$407$	16.3615	0.811011
$408$	0	0
$409$	−11.3712	−0.562270	−0.281135	−	0.959668i	$-0.590711\pi$
$409$	−11.3712	−0.562270	−0.281135	+	0.959668i	$0.590711\pi$
$410$	0	0
$411$	14.8020	0.730130
$412$	0	0
$413$	−1.16256	−0.0572057
$414$	0	0
$415$	17.9043	0.878886
$416$	0	0
$417$	−12.9994	−0.636581
$418$	0	0
$419$	8.00365	0.391004	0.195502	−	0.980703i	$-0.437366\pi$
$419$	8.00365	0.391004	0.195502	+	0.980703i	$0.437366\pi$
$420$	0	0
$421$	−8.84464	−0.431061	−0.215531	−	0.976497i	$-0.569148\pi$
$421$	−8.84464	−0.431061	−0.215531	+	0.976497i	$0.569148\pi$
$422$	0	0
$423$	−9.26291	−0.450378
$424$	0	0
$425$	−15.2849	−0.741424
$426$	0	0
$427$	0.0699651	0.00338585
$428$	0	0
$429$	16.8521	0.813625
$430$	0	0
$431$	12.9901	0.625710	0.312855	−	0.949801i	$-0.398715\pi$
$431$	12.9901	0.625710	0.312855	+	0.949801i	$0.398715\pi$
$432$	0	0
$433$	18.1649	0.872947	0.436474	−	0.899717i	$-0.356227\pi$
$433$	18.1649	0.872947	0.436474	+	0.899717i	$0.356227\pi$
$434$	0	0
$435$	−2.42215	−0.116133
$436$	0	0
$437$	1.51604	0.0725220
$438$	0	0
$439$	7.64407	0.364832	0.182416	−	0.983221i	$-0.441608\pi$
$439$	7.64407	0.364832	0.182416	+	0.983221i	$0.441608\pi$
$440$	0	0
$441$	−6.96912	−0.331863
$442$	0	0
$443$	25.0084	1.18819	0.594093	−	0.804397i	$-0.297511\pi$
$443$	25.0084	1.18819	0.594093	+	0.804397i	$0.297511\pi$
$444$	0	0
$445$	1.78692	0.0847082
$446$	0	0
$447$	−11.4229	−0.540287
$448$	0	0
$449$	−24.3211	−1.14778	−0.573891	−	0.818932i	$-0.694567\pi$
$449$	−24.3211	−1.14778	−0.573891	+	0.818932i	$0.694567\pi$
$450$	0	0
$451$	36.3922	1.71364
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	−1.06563	−0.0499575
$456$	0	0
$457$	−34.0492	−1.59276	−0.796378	−	0.604800i	$-0.793253\pi$
$457$	−34.0492	−1.59276	−0.796378	+	0.604800i	$0.793253\pi$
$458$	0	0
$459$	−5.45907	−0.254808
$460$	0	0
$461$	12.8149	0.596851	0.298425	−	0.954433i	$-0.403539\pi$
$461$	12.8149	0.596851	0.298425	+	0.954433i	$0.403539\pi$
$462$	0	0
$463$	31.9201	1.48345	0.741726	−	0.670704i	$-0.234008\pi$
$463$	31.9201	1.48345	0.741726	+	0.670704i	$0.234008\pi$
$464$	0	0
$465$	−13.2005	−0.612159
$466$	0	0
$467$	−28.6662	−1.32651	−0.663257	−	0.748392i	$-0.730826\pi$
$467$	−28.6662	−1.32651	−0.663257	+	0.748392i	$0.730826\pi$
$468$	0	0
$469$	1.17583	0.0542950
$470$	0	0
$471$	6.74558	0.310820
$472$	0	0
$473$	43.1826	1.98554
$474$	0	0
$475$	5.92504	0.271859
$476$	0	0
$477$	6.62829	0.303489
$478$	0	0
$479$	28.1031	1.28406	0.642032	−	0.766678i	$-0.278092\pi$
$479$	28.1031	1.28406	0.642032	+	0.766678i	$0.278092\pi$
$480$	0	0
$481$	−16.2304	−0.740041
$482$	0	0
$483$	−0.125883	−0.00572790
$484$	0	0
$485$	−22.8644	−1.03822
$486$	0	0
$487$	−14.8577	−0.673265	−0.336633	−	0.941636i	$-0.609288\pi$
$487$	−14.8577	−0.673265	−0.336633	+	0.941636i	$0.609288\pi$
$488$	0	0
$489$	−17.0088	−0.769165
$490$	0	0
$491$	−18.5227	−0.835916	−0.417958	−	0.908466i	$-0.637254\pi$
$491$	−18.5227	−0.835916	−0.417958	+	0.908466i	$0.637254\pi$
$492$	0	0
$493$	−8.91452	−0.401490
$494$	0	0
$495$	−6.11358	−0.274785
$496$	0	0
$497$	1.85082	0.0830205
$498$	0	0
$499$	−2.63646	−0.118024	−0.0590120	−	0.998257i	$-0.518795\pi$
$499$	−2.63646	−0.118024	−0.0590120	+	0.998257i	$0.518795\pi$
$500$	0	0
$501$	7.93379	0.354456
$502$	0	0
$503$	−33.3884	−1.48871	−0.744357	−	0.667782i	$-0.767244\pi$
$503$	−33.3884	−1.48871	−0.744357	+	0.667782i	$0.767244\pi$
$504$	0	0
$505$	−22.7379	−1.01182
$506$	0	0
$507$	−3.71698	−0.165077
$508$	0	0
$509$	0.305025	0.0135200	0.00676000	−	0.999977i	$-0.497848\pi$
$509$	0.305025	0.0135200	0.00676000	+	0.999977i	$0.497848\pi$
$510$	0	0
$511$	−1.27022	−0.0561913
$512$	0	0
$513$	2.11616	0.0934307
$514$	0	0
$515$	−1.70767	−0.0752490
$516$	0	0
$517$	−38.1787	−1.67910
$518$	0	0
$519$	−12.6178	−0.553862
$520$	0	0
$521$	39.3562	1.72423	0.862114	−	0.506714i	$-0.169140\pi$
$521$	39.3562	1.72423	0.862114	+	0.506714i	$0.169140\pi$
$522$	0	0
$523$	−22.4963	−0.983696	−0.491848	−	0.870681i	$-0.663678\pi$
$523$	−22.4963	−0.983696	−0.491848	+	0.870681i	$0.663678\pi$
$524$	0	0
$525$	−0.491982	−0.0214718
$526$	0	0
$527$	−48.5835	−2.11633
$528$	0	0
$529$	−22.4868	−0.977685
$530$	0	0
$531$	6.61618	0.287118
$532$	0	0
$533$	−36.1005	−1.56368
$534$	0	0
$535$	4.02412	0.173978
$536$	0	0
$537$	8.27629	0.357148
$538$	0	0
$539$	−28.7245	−1.23725
$540$	0	0
$541$	1.33729	0.0574946	0.0287473	−	0.999587i	$-0.490848\pi$
$541$	1.33729	0.0574946	0.0287473	+	0.999587i	$0.490848\pi$
$542$	0	0
$543$	−9.97420	−0.428034
$544$	0	0
$545$	12.0990	0.518266
$546$	0	0
$547$	−7.23893	−0.309514	−0.154757	−	0.987953i	$-0.549459\pi$
$547$	−7.23893	−0.309514	−0.154757	+	0.987953i	$0.549459\pi$
$548$	0	0
$549$	−0.398176	−0.0169937
$550$	0	0
$551$	3.45564	0.147215
$552$	0	0
$553$	0.930292	0.0395601
$554$	0	0
$555$	5.88804	0.249933
$556$	0	0
$557$	14.3590	0.608411	0.304205	−	0.952607i	$-0.401609\pi$
$557$	14.3590	0.608411	0.304205	+	0.952607i	$0.401609\pi$
$558$	0	0
$559$	−42.8364	−1.81179
$560$	0	0
$561$	−22.5005	−0.949973
$562$	0	0
$563$	−42.1882	−1.77802	−0.889010	−	0.457888i	$-0.848606\pi$
$563$	−42.1882	−1.77802	−0.889010	+	0.457888i	$0.848606\pi$
$564$	0	0
$565$	7.15144	0.300864
$566$	0	0
$567$	−0.175714	−0.00737930
$568$	0	0
$569$	−20.1010	−0.842676	−0.421338	−	0.906904i	$-0.638439\pi$
$569$	−20.1010	−0.842676	−0.421338	+	0.906904i	$0.638439\pi$
$570$	0	0
$571$	−0.0956906	−0.00400453	−0.00200226	−	0.999998i	$-0.500637\pi$
$571$	−0.0956906	−0.00400453	−0.00200226	+	0.999998i	$0.500637\pi$
$572$	0	0
$573$	12.4929	0.521897
$574$	0	0
$575$	2.00588	0.0836510
$576$	0	0
$577$	−26.4358	−1.10054	−0.550268	−	0.834988i	$-0.685474\pi$
$577$	−26.4358	−1.10054	−0.550268	+	0.834988i	$0.685474\pi$
$578$	0	0
$579$	15.4013	0.640057
$580$	0	0
$581$	2.12101	0.0879941
$582$	0	0
$583$	27.3197	1.13147
$584$	0	0
$585$	6.06457	0.250739
$586$	0	0
$587$	−22.3699	−0.923304	−0.461652	−	0.887061i	$-0.652743\pi$
$587$	−22.3699	−0.923304	−0.461652	+	0.887061i	$0.652743\pi$
$588$	0	0
$589$	18.8329	0.775998
$590$	0	0
$591$	12.3978	0.509976
$592$	0	0
$593$	30.0566	1.23428	0.617138	−	0.786855i	$-0.288292\pi$
$593$	30.0566	1.23428	0.617138	+	0.786855i	$0.288292\pi$
$594$	0	0
$595$	1.42281	0.0583294
$596$	0	0
$597$	−0.228835	−0.00936559
$598$	0	0
$599$	−2.11898	−0.0865792	−0.0432896	−	0.999063i	$-0.513784\pi$
$599$	−2.11898	−0.0865792	−0.0432896	+	0.999063i	$0.513784\pi$
$600$	0	0
$601$	−21.9456	−0.895181	−0.447591	−	0.894239i	$-0.647718\pi$
$601$	−21.9456	−0.895181	−0.447591	+	0.894239i	$0.647718\pi$
$602$	0	0
$603$	−6.69175	−0.272509
$604$	0	0
$605$	−8.88219	−0.361112
$606$	0	0
$607$	−27.6210	−1.12110	−0.560552	−	0.828119i	$-0.689411\pi$
$607$	−27.6210	−1.12110	−0.560552	+	0.828119i	$0.689411\pi$
$608$	0	0
$609$	−0.286937	−0.0116273
$610$	0	0
$611$	37.8727	1.53217
$612$	0	0
$613$	−32.0177	−1.29318	−0.646591	−	0.762837i	$-0.723806\pi$
$613$	−32.0177	−1.29318	−0.646591	+	0.762837i	$0.723806\pi$
$614$	0	0
$615$	13.0965	0.528102
$616$	0	0
$617$	−42.0327	−1.69217	−0.846087	−	0.533045i	$-0.821048\pi$
$617$	−42.0327	−1.69217	−0.846087	+	0.533045i	$0.821048\pi$
$618$	0	0
$619$	31.9713	1.28503	0.642517	−	0.766272i	$-0.277890\pi$
$619$	31.9713	1.28503	0.642517	+	0.766272i	$0.277890\pi$
$620$	0	0
$621$	0.716411	0.0287486
$622$	0	0
$623$	0.211685	0.00848100
$624$	0	0
$625$	−3.16106	−0.126442
$626$	0	0
$627$	8.72213	0.348328
$628$	0	0
$629$	21.6705	0.864058
$630$	0	0
$631$	−12.1979	−0.485592	−0.242796	−	0.970077i	$-0.578065\pi$
$631$	−12.1979	−0.485592	−0.242796	+	0.970077i	$0.578065\pi$
$632$	0	0
$633$	17.6734	0.702456
$634$	0	0
$635$	5.74863	0.228128
$636$	0	0
$637$	28.4942	1.12898
$638$	0	0
$639$	−10.5331	−0.416684
$640$	0	0
$641$	38.6091	1.52497	0.762483	−	0.647008i	$-0.223980\pi$
$641$	38.6091	1.52497	0.762483	+	0.647008i	$0.223980\pi$
$642$	0	0
$643$	−16.2721	−0.641707	−0.320854	−	0.947129i	$-0.603970\pi$
$643$	−16.2721	−0.641707	−0.320854	+	0.947129i	$0.603970\pi$
$644$	0	0
$645$	15.5402	0.611893
$646$	0	0
$647$	−4.02301	−0.158161	−0.0790804	−	0.996868i	$-0.525198\pi$
$647$	−4.02301	−0.158161	−0.0790804	+	0.996868i	$0.525198\pi$
$648$	0	0
$649$	27.2698	1.07043
$650$	0	0
$651$	−1.56378	−0.0612895
$652$	0	0
$653$	17.8777	0.699609	0.349805	−	0.936823i	$-0.386248\pi$
$653$	17.8777	0.699609	0.349805	+	0.936823i	$0.386248\pi$
$654$	0	0
$655$	−12.1538	−0.474888
$656$	0	0
$657$	7.22891	0.282027
$658$	0	0
$659$	−8.32644	−0.324352	−0.162176	−	0.986762i	$-0.551851\pi$
$659$	−8.32644	−0.324352	−0.162176	+	0.986762i	$0.551851\pi$
$660$	0	0
$661$	11.7893	0.458551	0.229276	−	0.973362i	$-0.426364\pi$
$661$	11.7893	0.458551	0.229276	+	0.973362i	$0.426364\pi$
$662$	0	0
$663$	22.3202	0.866844
$664$	0	0
$665$	−0.551539	−0.0213878
$666$	0	0
$667$	1.16988	0.0452980
$668$	0	0
$669$	18.1746	0.702670
$670$	0	0
$671$	−1.64115	−0.0633560
$672$	0	0
$673$	−29.2724	−1.12837	−0.564184	−	0.825649i	$-0.690809\pi$
$673$	−29.2724	−1.12837	−0.564184	+	0.825649i	$0.690809\pi$
$674$	0	0
$675$	2.79990	0.107768
$676$	0	0
$677$	−7.00406	−0.269188	−0.134594	−	0.990901i	$-0.542973\pi$
$677$	−7.00406	−0.269188	−0.134594	+	0.990901i	$0.542973\pi$
$678$	0	0
$679$	−2.70860	−0.103946
$680$	0	0
$681$	15.6799	0.600856
$682$	0	0
$683$	3.34363	0.127940	0.0639702	−	0.997952i	$-0.479624\pi$
$683$	3.34363	0.127940	0.0639702	+	0.997952i	$0.479624\pi$
$684$	0	0
$685$	21.9554	0.838874
$686$	0	0
$687$	21.8796	0.834758
$688$	0	0
$689$	−27.1007	−1.03245
$690$	0	0
$691$	28.2967	1.07646	0.538228	−	0.842799i	$-0.319094\pi$
$691$	28.2967	1.07646	0.538228	+	0.842799i	$0.319094\pi$
$692$	0	0
$693$	−0.724237	−0.0275115
$694$	0	0
$695$	−19.2816	−0.731393
$696$	0	0
$697$	48.2006	1.82573
$698$	0	0
$699$	4.97208	0.188061
$700$	0	0
$701$	30.2356	1.14198	0.570992	−	0.820956i	$-0.306559\pi$
$701$	30.2356	1.14198	0.570992	+	0.820956i	$0.306559\pi$
$702$	0	0
$703$	−8.40036	−0.316826
$704$	0	0
$705$	−13.7394	−0.517457
$706$	0	0
$707$	−2.69361	−0.101304
$708$	0	0
$709$	24.6904	0.927269	0.463635	−	0.886026i	$-0.346545\pi$
$709$	24.6904	0.927269	0.463635	+	0.886026i	$0.346545\pi$
$710$	0	0
$711$	−5.29435	−0.198554
$712$	0	0
$713$	6.37576	0.238774
$714$	0	0
$715$	24.9962	0.934805
$716$	0	0
$717$	12.7720	0.476980
$718$	0	0
$719$	35.7878	1.33466	0.667329	−	0.744763i	$-0.267437\pi$
$719$	35.7878	1.33466	0.667329	+	0.744763i	$0.267437\pi$
$720$	0	0
$721$	−0.202297	−0.00753394
$722$	0	0
$723$	−1.21114	−0.0450430
$724$	0	0
$725$	4.57217	0.169806
$726$	0	0
$727$	32.8476	1.21825	0.609124	−	0.793075i	$-0.291521\pi$
$727$	32.8476	1.21825	0.609124	+	0.793075i	$0.291521\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	57.1943	2.11541
$732$	0	0
$733$	−2.89665	−0.106990	−0.0534951	−	0.998568i	$-0.517036\pi$
$733$	−2.89665	−0.106990	−0.0534951	+	0.998568i	$0.517036\pi$
$734$	0	0
$735$	−10.3371	−0.381290
$736$	0	0
$737$	−27.5812	−1.01597
$738$	0	0
$739$	6.80285	0.250247	0.125123	−	0.992141i	$-0.460067\pi$
$739$	6.80285	0.250247	0.125123	+	0.992141i	$0.460067\pi$
$740$	0	0
$741$	−8.65222	−0.317847
$742$	0	0
$743$	44.8039	1.64370	0.821848	−	0.569707i	$-0.192943\pi$
$743$	44.8039	1.64370	0.821848	+	0.569707i	$0.192943\pi$
$744$	0	0
$745$	−16.9433	−0.620756
$746$	0	0
$747$	−12.0708	−0.441647
$748$	0	0
$749$	0.476713	0.0174187
$750$	0	0
$751$	−18.2890	−0.667373	−0.333687	−	0.942684i	$-0.608293\pi$
$751$	−18.2890	−0.667373	−0.333687	+	0.942684i	$0.608293\pi$
$752$	0	0
$753$	16.2295	0.591436
$754$	0	0
$755$	1.48327	0.0539818
$756$	0	0
$757$	−34.5561	−1.25596	−0.627982	−	0.778228i	$-0.716119\pi$
$757$	−34.5561	−1.25596	−0.627982	+	0.778228i	$0.716119\pi$
$758$	0	0
$759$	2.95282	0.107180
$760$	0	0
$761$	−39.0673	−1.41619	−0.708095	−	0.706118i	$-0.750445\pi$
$761$	−39.0673	−1.41619	−0.708095	+	0.706118i	$0.750445\pi$
$762$	0	0
$763$	1.43330	0.0518888
$764$	0	0
$765$	−8.09729	−0.292758
$766$	0	0
$767$	−27.0512	−0.976762
$768$	0	0
$769$	50.3528	1.81577	0.907885	−	0.419219i	$-0.137696\pi$
$769$	50.3528	1.81577	0.907885	+	0.419219i	$0.137696\pi$
$770$	0	0
$771$	−2.57233	−0.0926401
$772$	0	0
$773$	10.4686	0.376530	0.188265	−	0.982118i	$-0.439714\pi$
$773$	10.4686	0.376530	0.188265	+	0.982118i	$0.439714\pi$
$774$	0	0
$775$	24.9180	0.895079
$776$	0	0
$777$	0.697519	0.0250234
$778$	0	0
$779$	−18.6845	−0.669443
$780$	0	0
$781$	−43.4142	−1.55348
$782$	0	0
$783$	1.63297	0.0583578
$784$	0	0
$785$	10.0055	0.357113
$786$	0	0
$787$	−2.16429	−0.0771487	−0.0385743	−	0.999256i	$-0.512282\pi$
$787$	−2.16429	−0.0771487	−0.0385743	+	0.999256i	$0.512282\pi$
$788$	0	0
$789$	16.7211	0.595288
$790$	0	0
$791$	0.847187	0.0301225
$792$	0	0
$793$	1.62800	0.0578119
$794$	0	0
$795$	9.83157	0.348690
$796$	0	0
$797$	−18.3430	−0.649744	−0.324872	−	0.945758i	$-0.605321\pi$
$797$	−18.3430	−0.649744	−0.324872	+	0.945758i	$0.605321\pi$
$798$	0	0
$799$	−50.5669	−1.78893
$800$	0	0
$801$	−1.20472	−0.0425665
$802$	0	0
$803$	29.7953	1.05145
$804$	0	0
$805$	−0.186720	−0.00658100
$806$	0	0
$807$	0.811246	0.0285572
$808$	0	0
$809$	−12.0274	−0.422859	−0.211430	−	0.977393i	$-0.567812\pi$
$809$	−12.0274	−0.422859	−0.211430	+	0.977393i	$0.567812\pi$
$810$	0	0
$811$	19.7894	0.694901	0.347450	−	0.937698i	$-0.387048\pi$
$811$	19.7894	0.694901	0.347450	+	0.937698i	$0.387048\pi$
$812$	0	0
$813$	7.63991	0.267943
$814$	0	0
$815$	−25.2287	−0.883723
$816$	0	0
$817$	−22.1709	−0.775661
$818$	0	0
$819$	0.718432	0.0251040
$820$	0	0
$821$	−3.73284	−0.130277	−0.0651386	−	0.997876i	$-0.520749\pi$
$821$	−3.73284	−0.130277	−0.0651386	+	0.997876i	$0.520749\pi$
$822$	0	0
$823$	44.4754	1.55031	0.775157	−	0.631768i	$-0.217671\pi$
$823$	44.4754	1.55031	0.775157	+	0.631768i	$0.217671\pi$
$824$	0	0
$825$	11.5403	0.401781
$826$	0	0
$827$	−16.3027	−0.566899	−0.283450	−	0.958987i	$-0.591479\pi$
$827$	−16.3027	−0.566899	−0.283450	+	0.958987i	$0.591479\pi$
$828$	0	0
$829$	34.9801	1.21491	0.607455	−	0.794354i	$-0.292190\pi$
$829$	34.9801	1.21491	0.607455	+	0.794354i	$0.292190\pi$
$830$	0	0
$831$	−10.1519	−0.352166
$832$	0	0
$833$	−38.0449	−1.31818
$834$	0	0
$835$	11.7680	0.407248
$836$	0	0
$837$	8.89959	0.307615
$838$	0	0
$839$	47.3110	1.63336	0.816679	−	0.577093i	$-0.195813\pi$
$839$	47.3110	1.63336	0.816679	+	0.577093i	$0.195813\pi$
$840$	0	0
$841$	−26.3334	−0.908048
$842$	0	0
$843$	4.45901	0.153577
$844$	0	0
$845$	−5.51330	−0.189663
$846$	0	0
$847$	−1.05222	−0.0361546
$848$	0	0
$849$	−16.7681	−0.575478
$850$	0	0
$851$	−2.84388	−0.0974871
$852$	0	0
$853$	−20.3314	−0.696134	−0.348067	−	0.937470i	$-0.613162\pi$
$853$	−20.3314	−0.696134	−0.348067	+	0.937470i	$0.613162\pi$
$854$	0	0
$855$	3.13884	0.107346
$856$	0	0
$857$	−39.4397	−1.34724	−0.673618	−	0.739080i	$-0.735261\pi$
$857$	−39.4397	−1.34724	−0.673618	+	0.739080i	$0.735261\pi$
$858$	0	0
$859$	42.3831	1.44609	0.723047	−	0.690799i	$-0.242741\pi$
$859$	42.3831	1.44609	0.723047	+	0.690799i	$0.242741\pi$
$860$	0	0
$861$	1.55146	0.0528736
$862$	0	0
$863$	−33.2233	−1.13093	−0.565466	−	0.824771i	$-0.691304\pi$
$863$	−33.2233	−1.13093	−0.565466	+	0.824771i	$0.691304\pi$
$864$	0	0
$865$	−18.7157	−0.636353
$866$	0	0
$867$	−12.8014	−0.434760
$868$	0	0
$869$	−21.8216	−0.740248
$870$	0	0
$871$	27.3602	0.927063
$872$	0	0
$873$	15.4148	0.521712
$874$	0	0
$875$	−2.03290	−0.0687247
$876$	0	0
$877$	−37.4467	−1.26449	−0.632243	−	0.774770i	$-0.717866\pi$
$877$	−37.4467	−1.26449	−0.632243	+	0.774770i	$0.717866\pi$
$878$	0	0
$879$	27.4062	0.924389
$880$	0	0
$881$	43.8990	1.47899	0.739497	−	0.673160i	$-0.235064\pi$
$881$	43.8990	1.47899	0.739497	+	0.673160i	$0.235064\pi$
$882$	0	0
$883$	43.0321	1.44815	0.724073	−	0.689723i	$-0.242268\pi$
$883$	43.0321	1.44815	0.724073	+	0.689723i	$0.242268\pi$
$884$	0	0
$885$	9.81361	0.329881
$886$	0	0
$887$	−25.9706	−0.872009	−0.436004	−	0.899945i	$-0.643607\pi$
$887$	−25.9706	−0.872009	−0.436004	+	0.899945i	$0.643607\pi$
$888$	0	0
$889$	0.681005	0.0228402
$890$	0	0
$891$	4.12168	0.138081
$892$	0	0
$893$	19.6018	0.655949
$894$	0	0
$895$	12.2760	0.410341
$896$	0	0
$897$	−2.92915	−0.0978014
$898$	0	0
$899$	14.5328	0.484696
$900$	0	0
$901$	36.1843	1.20547
$902$	0	0
$903$	1.84095	0.0612628
$904$	0	0
$905$	−14.7945	−0.491784
$906$	0	0
$907$	−42.4775	−1.41044	−0.705222	−	0.708987i	$-0.749152\pi$
$907$	−42.4775	−1.41044	−0.705222	+	0.708987i	$0.749152\pi$
$908$	0	0
$909$	15.3295	0.508448
$910$	0	0
$911$	11.8152	0.391455	0.195728	−	0.980658i	$-0.437293\pi$
$911$	11.8152	0.391455	0.195728	+	0.980658i	$0.437293\pi$
$912$	0	0
$913$	−49.7519	−1.64655
$914$	0	0
$915$	−0.590603	−0.0195247
$916$	0	0
$917$	−1.43978	−0.0475459
$918$	0	0
$919$	6.89827	0.227553	0.113776	−	0.993506i	$-0.463705\pi$
$919$	6.89827	0.227553	0.113776	+	0.993506i	$0.463705\pi$
$920$	0	0
$921$	19.4219	0.639973
$922$	0	0
$923$	43.0662	1.41754
$924$	0	0
$925$	−11.1146	−0.365444
$926$	0	0
$927$	1.15129	0.0378132
$928$	0	0
$929$	−3.35926	−0.110214	−0.0551069	−	0.998480i	$-0.517550\pi$
$929$	−3.35926	−0.110214	−0.0551069	+	0.998480i	$0.517550\pi$
$930$	0	0
$931$	14.7478	0.483339
$932$	0	0
$933$	−14.7683	−0.483492
$934$	0	0
$935$	−33.3744	−1.09146
$936$	0	0
$937$	−51.5893	−1.68535	−0.842675	−	0.538423i	$-0.819020\pi$
$937$	−51.5893	−1.68535	−0.842675	+	0.538423i	$0.819020\pi$
$938$	0	0
$939$	−34.2310	−1.11709
$940$	0	0
$941$	−13.4148	−0.437309	−0.218655	−	0.975802i	$-0.570167\pi$
$941$	−13.4148	−0.437309	−0.218655	+	0.975802i	$0.570167\pi$
$942$	0	0
$943$	−6.32552	−0.205987
$944$	0	0
$945$	−0.260632	−0.00847836
$946$	0	0
$947$	−6.97008	−0.226497	−0.113249	−	0.993567i	$-0.536126\pi$
$947$	−6.97008	−0.226497	−0.113249	+	0.993567i	$0.536126\pi$
$948$	0	0
$949$	−29.5564	−0.959442
$950$	0	0
$951$	27.9167	0.905260
$952$	0	0
$953$	26.4107	0.855525	0.427763	−	0.903891i	$-0.359302\pi$
$953$	26.4107	0.855525	0.427763	+	0.903891i	$0.359302\pi$
$954$	0	0
$955$	18.5303	0.599627
$956$	0	0
$957$	6.73060	0.217569
$958$	0	0
$959$	2.60092	0.0839882
$960$	0	0
$961$	48.2026	1.55492
$962$	0	0
$963$	−2.71300	−0.0874253
$964$	0	0
$965$	22.8444	0.735387
$966$	0	0
$967$	−49.4558	−1.59039	−0.795196	−	0.606352i	$-0.792632\pi$
$967$	−49.4558	−1.59039	−0.795196	+	0.606352i	$0.792632\pi$
$968$	0	0
$969$	11.5523	0.371112
$970$	0	0
$971$	24.0342	0.771294	0.385647	−	0.922646i	$-0.373978\pi$
$971$	24.0342	0.771294	0.385647	+	0.922646i	$0.373978\pi$
$972$	0	0
$973$	−2.28417	−0.0732271
$974$	0	0
$975$	−11.4478	−0.366623
$976$	0	0
$977$	58.3481	1.86672	0.933360	−	0.358942i	$-0.116862\pi$
$977$	58.3481	1.86672	0.933360	+	0.358942i	$0.116862\pi$
$978$	0	0
$979$	−4.96545	−0.158696
$980$	0	0
$981$	−8.15699	−0.260433
$982$	0	0
$983$	−36.9906	−1.17982	−0.589909	−	0.807470i	$-0.700836\pi$
$983$	−36.9906	−1.17982	−0.589909	+	0.807470i	$0.700836\pi$
$984$	0	0
$985$	18.3893	0.585931
$986$	0	0
$987$	−1.62762	−0.0518078
$988$	0	0
$989$	−7.50579	−0.238670
$990$	0	0
$991$	−57.6512	−1.83135	−0.915675	−	0.401920i	$-0.868343\pi$
$991$	−57.6512	−1.83135	−0.915675	+	0.401920i	$0.868343\pi$
$992$	0	0
$993$	11.5972	0.368025
$994$	0	0
$995$	−0.339425	−0.0107605
$996$	0	0
$997$	−5.78845	−0.183322	−0.0916611	−	0.995790i	$-0.529218\pi$
$997$	−5.78845	−0.183322	−0.0916611	+	0.995790i	$0.529218\pi$
$998$	0	0
$999$	−3.96963	−0.125593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.bm.1.4		10
4.3	odd	2		3624.2.a.l.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3624.2.a.l.1.4	✓	10	4.3	odd	2
7248.2.a.bm.1.4		10	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:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.48327 q^{5} -0.175714 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.48327 q^{5} -0.175714 q^{7} +1.00000 q^{9} +4.12168 q^{11} -4.08864 q^{13} +1.48327 q^{15} +5.45907 q^{17} -2.11616 q^{19} +0.175714 q^{21} -0.716411 q^{23} -2.79990 q^{25} -1.00000 q^{27} -1.63297 q^{29} -8.89959 q^{31} -4.12168 q^{33} +0.260632 q^{35} +3.96963 q^{37} +4.08864 q^{39} +8.82946 q^{41} +10.4769 q^{43} -1.48327 q^{45} -9.26291 q^{47} -6.96912 q^{49} -5.45907 q^{51} +6.62829 q^{53} -6.11358 q^{55} +2.11616 q^{57} +6.61618 q^{59} -0.398176 q^{61} -0.175714 q^{63} +6.06457 q^{65} -6.69175 q^{67} +0.716411 q^{69} -10.5331 q^{71} +7.22891 q^{73} +2.79990 q^{75} -0.724237 q^{77} -5.29435 q^{79} +1.00000 q^{81} -12.0708 q^{83} -8.09729 q^{85} +1.63297 q^{87} -1.20472 q^{89} +0.718432 q^{91} +8.89959 q^{93} +3.13884 q^{95} +15.4148 q^{97} +4.12168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

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