LMFDB - Embedded newform 8024.2.a.ba.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.797176 q^{3} -2.84462 q^{5} -0.0983465 q^{7} -2.36451 q^{9} +O(q^{10})$	$q-0.797176 q^{3} -2.84462 q^{5} -0.0983465 q^{7} -2.36451 q^{9} +1.03173 q^{11} -3.18174 q^{13} +2.26766 q^{15} -1.00000 q^{17} -6.45786 q^{19} +0.0783995 q^{21} +4.89456 q^{23} +3.09187 q^{25} +4.27646 q^{27} -1.16958 q^{29} -8.98516 q^{31} -0.822467 q^{33} +0.279758 q^{35} -6.37205 q^{37} +2.53641 q^{39} +4.73035 q^{41} -11.2246 q^{43} +6.72614 q^{45} -10.9596 q^{47} -6.99033 q^{49} +0.797176 q^{51} +4.31769 q^{53} -2.93487 q^{55} +5.14805 q^{57} +1.00000 q^{59} +14.5672 q^{61} +0.232541 q^{63} +9.05086 q^{65} +1.67437 q^{67} -3.90183 q^{69} -8.89605 q^{71} -2.54572 q^{73} -2.46477 q^{75} -0.101467 q^{77} -8.31819 q^{79} +3.68444 q^{81} -4.78412 q^{83} +2.84462 q^{85} +0.932365 q^{87} -18.8138 q^{89} +0.312913 q^{91} +7.16276 q^{93} +18.3702 q^{95} +9.43104 q^{97} -2.43953 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	−0.797176	−0.460250	−0.230125	−	0.973161i	$-0.573913\pi$
$3$	−0.797176	−0.460250	−0.230125	+	0.973161i	$0.573913\pi$
$4$	0	0
$5$	−2.84462	−1.27215	−0.636077	−	0.771626i	$-0.719444\pi$
$5$	−2.84462	−1.27215	−0.636077	+	0.771626i	$0.719444\pi$
$6$	0	0
$7$	−0.0983465	−0.0371715	−0.0185857	−	0.999827i	$-0.505916\pi$
$7$	−0.0983465	−0.0371715	−0.0185857	+	0.999827i	$0.505916\pi$
$8$	0	0
$9$	−2.36451	−0.788170
$10$	0	0
$11$	1.03173	0.311077	0.155539	−	0.987830i	$-0.450289\pi$
$11$	1.03173	0.311077	0.155539	+	0.987830i	$0.450289\pi$
$12$	0	0
$13$	−3.18174	−0.882457	−0.441229	−	0.897395i	$-0.645457\pi$
$13$	−3.18174	−0.882457	−0.441229	+	0.897395i	$0.645457\pi$
$14$	0	0
$15$	2.26766	0.585508
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.45786	−1.48154	−0.740768	−	0.671761i	$-0.765538\pi$
$19$	−6.45786	−1.48154	−0.740768	+	0.671761i	$0.765538\pi$
$20$	0	0
$21$	0.0783995	0.0171082
$22$	0	0
$23$	4.89456	1.02059	0.510293	−	0.860001i	$-0.329537\pi$
$23$	4.89456	1.02059	0.510293	+	0.860001i	$0.329537\pi$
$24$	0	0
$25$	3.09187	0.618374
$26$	0	0
$27$	4.27646	0.823005
$28$	0	0
$29$	−1.16958	−0.217186	−0.108593	−	0.994086i	$-0.534635\pi$
$29$	−1.16958	−0.217186	−0.108593	+	0.994086i	$0.534635\pi$
$30$	0	0
$31$	−8.98516	−1.61378	−0.806892	−	0.590700i	$-0.798852\pi$
$31$	−8.98516	−1.61378	−0.806892	+	0.590700i	$0.798852\pi$
$32$	0	0
$33$	−0.822467	−0.143173
$34$	0	0
$35$	0.279758	0.0472878
$36$	0	0
$37$	−6.37205	−1.04756	−0.523780	−	0.851854i	$-0.675478\pi$
$37$	−6.37205	−1.04756	−0.523780	+	0.851854i	$0.675478\pi$
$38$	0	0
$39$	2.53641	0.406151
$40$	0	0
$41$	4.73035	0.738756	0.369378	−	0.929279i	$-0.379571\pi$
$41$	4.73035	0.738756	0.369378	+	0.929279i	$0.379571\pi$
$42$	0	0
$43$	−11.2246	−1.71173	−0.855866	−	0.517198i	$-0.826975\pi$
$43$	−11.2246	−1.71173	−0.855866	+	0.517198i	$0.826975\pi$
$44$	0	0
$45$	6.72614	1.00267
$46$	0	0
$47$	−10.9596	−1.59862	−0.799309	−	0.600920i	$-0.794801\pi$
$47$	−10.9596	−1.59862	−0.799309	+	0.600920i	$0.794801\pi$
$48$	0	0
$49$	−6.99033	−0.998618
$50$	0	0
$51$	0.797176	0.111627
$52$	0	0
$53$	4.31769	0.593080	0.296540	−	0.955020i	$-0.404167\pi$
$53$	4.31769	0.593080	0.296540	+	0.955020i	$0.404167\pi$
$54$	0	0
$55$	−2.93487	−0.395738
$56$	0	0
$57$	5.14805	0.681876
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	14.5672	1.86513	0.932567	−	0.360998i	$-0.117564\pi$
$61$	14.5672	1.86513	0.932567	+	0.360998i	$0.117564\pi$
$62$	0	0
$63$	0.232541	0.0292974
$64$	0	0
$65$	9.05086	1.12262
$66$	0	0
$67$	1.67437	0.204557	0.102278	−	0.994756i	$-0.467387\pi$
$67$	1.67437	0.204557	0.102278	+	0.994756i	$0.467387\pi$
$68$	0	0
$69$	−3.90183	−0.469725
$70$	0	0
$71$	−8.89605	−1.05577	−0.527883	−	0.849317i	$-0.677014\pi$
$71$	−8.89605	−1.05577	−0.527883	+	0.849317i	$0.677014\pi$
$72$	0	0
$73$	−2.54572	−0.297954	−0.148977	−	0.988841i	$-0.547598\pi$
$73$	−2.54572	−0.297954	−0.148977	+	0.988841i	$0.547598\pi$
$74$	0	0
$75$	−2.46477	−0.284607
$76$	0	0
$77$	−0.101467	−0.0115632
$78$	0	0
$79$	−8.31819	−0.935869	−0.467934	−	0.883763i	$-0.655002\pi$
$79$	−8.31819	−0.935869	−0.467934	+	0.883763i	$0.655002\pi$
$80$	0	0
$81$	3.68444	0.409382
$82$	0	0
$83$	−4.78412	−0.525126	−0.262563	−	0.964915i	$-0.584568\pi$
$83$	−4.78412	−0.525126	−0.262563	+	0.964915i	$0.584568\pi$
$84$	0	0
$85$	2.84462	0.308543
$86$	0	0
$87$	0.932365	0.0999600
$88$	0	0
$89$	−18.8138	−1.99425	−0.997127	−	0.0757471i	$-0.975866\pi$
$89$	−18.8138	−1.99425	−0.997127	+	0.0757471i	$0.975866\pi$
$90$	0	0
$91$	0.312913	0.0328022
$92$	0	0
$93$	7.16276	0.742743
$94$	0	0
$95$	18.3702	1.88474
$96$	0	0
$97$	9.43104	0.957577	0.478789	−	0.877930i	$-0.341076\pi$
$97$	9.43104	0.957577	0.478789	+	0.877930i	$0.341076\pi$
$98$	0	0
$99$	−2.43953	−0.245182
$100$	0	0
$101$	2.56180	0.254909	0.127454	−	0.991844i	$-0.459319\pi$
$101$	2.56180	0.254909	0.127454	+	0.991844i	$0.459319\pi$
$102$	0	0
$103$	−10.9309	−1.07705	−0.538525	−	0.842610i	$-0.681018\pi$
$103$	−10.9309	−1.07705	−0.538525	+	0.842610i	$0.681018\pi$
$104$	0	0
$105$	−0.223017	−0.0217642
$106$	0	0
$107$	4.40760	0.426099	0.213049	−	0.977041i	$-0.431660\pi$
$107$	4.40760	0.426099	0.213049	+	0.977041i	$0.431660\pi$
$108$	0	0
$109$	−9.84506	−0.942986	−0.471493	−	0.881870i	$-0.656285\pi$
$109$	−9.84506	−0.942986	−0.471493	+	0.881870i	$0.656285\pi$
$110$	0	0
$111$	5.07965	0.482139
$112$	0	0
$113$	−21.1323	−1.98796	−0.993981	−	0.109550i	$-0.965059\pi$
$113$	−21.1323	−1.98796	−0.993981	+	0.109550i	$0.965059\pi$
$114$	0	0
$115$	−13.9232	−1.29834
$116$	0	0
$117$	7.52327	0.695526
$118$	0	0
$119$	0.0983465	0.00901541
$120$	0	0
$121$	−9.93554	−0.903231
$122$	0	0
$123$	−3.77092	−0.340012
$124$	0	0
$125$	5.42790	0.485486
$126$	0	0
$127$	1.43375	0.127225	0.0636124	−	0.997975i	$-0.479738\pi$
$127$	1.43375	0.127225	0.0636124	+	0.997975i	$0.479738\pi$
$128$	0	0
$129$	8.94796	0.787824
$130$	0	0
$131$	14.4895	1.26595	0.632977	−	0.774170i	$-0.281833\pi$
$131$	14.4895	1.26595	0.632977	+	0.774170i	$0.281833\pi$
$132$	0	0
$133$	0.635108	0.0550708
$134$	0	0
$135$	−12.1649	−1.04699
$136$	0	0
$137$	−11.5833	−0.989625	−0.494813	−	0.869000i	$-0.664763\pi$
$137$	−11.5833	−0.989625	−0.494813	+	0.869000i	$0.664763\pi$
$138$	0	0
$139$	19.8643	1.68487	0.842436	−	0.538797i	$-0.181121\pi$
$139$	19.8643	1.68487	0.842436	+	0.538797i	$0.181121\pi$
$140$	0	0
$141$	8.73671	0.735764
$142$	0	0
$143$	−3.28269	−0.274512
$144$	0	0
$145$	3.32703	0.276294
$146$	0	0
$147$	5.57252	0.459614
$148$	0	0
$149$	−16.8916	−1.38381	−0.691905	−	0.721988i	$-0.743228\pi$
$149$	−16.8916	−1.38381	−0.691905	+	0.721988i	$0.743228\pi$
$150$	0	0
$151$	13.1023	1.06625	0.533125	−	0.846037i	$-0.321018\pi$
$151$	13.1023	1.06625	0.533125	+	0.846037i	$0.321018\pi$
$152$	0	0
$153$	2.36451	0.191159
$154$	0	0
$155$	25.5594	2.05298
$156$	0	0
$157$	11.9667	0.955049	0.477525	−	0.878618i	$-0.341534\pi$
$157$	11.9667	0.955049	0.477525	+	0.878618i	$0.341534\pi$
$158$	0	0
$159$	−3.44196	−0.272965
$160$	0	0
$161$	−0.481363	−0.0379367
$162$	0	0
$163$	−12.2753	−0.961472	−0.480736	−	0.876865i	$-0.659630\pi$
$163$	−12.2753	−0.961472	−0.480736	+	0.876865i	$0.659630\pi$
$164$	0	0
$165$	2.33961	0.182138
$166$	0	0
$167$	−3.37065	−0.260829	−0.130415	−	0.991460i	$-0.541631\pi$
$167$	−3.37065	−0.260829	−0.130415	+	0.991460i	$0.541631\pi$
$168$	0	0
$169$	−2.87650	−0.221270
$170$	0	0
$171$	15.2697	1.16770
$172$	0	0
$173$	17.3377	1.31816	0.659079	−	0.752074i	$-0.270946\pi$
$173$	17.3377	1.31816	0.659079	+	0.752074i	$0.270946\pi$
$174$	0	0
$175$	−0.304075	−0.0229859
$176$	0	0
$177$	−0.797176	−0.0599194
$178$	0	0
$179$	23.3110	1.74235	0.871173	−	0.490975i	$-0.163360\pi$
$179$	23.3110	1.74235	0.871173	+	0.490975i	$0.163360\pi$
$180$	0	0
$181$	−15.2149	−1.13092	−0.565458	−	0.824777i	$-0.691300\pi$
$181$	−15.2149	−1.13092	−0.565458	+	0.824777i	$0.691300\pi$
$182$	0	0
$183$	−11.6126	−0.858427
$184$	0	0
$185$	18.1261	1.33266
$186$	0	0
$187$	−1.03173	−0.0754473
$188$	0	0
$189$	−0.420575	−0.0305923
$190$	0	0
$191$	10.9940	0.795497	0.397749	−	0.917494i	$-0.369792\pi$
$191$	10.9940	0.795497	0.397749	+	0.917494i	$0.369792\pi$
$192$	0	0
$193$	−1.82505	−0.131370	−0.0656851	−	0.997840i	$-0.520923\pi$
$193$	−1.82505	−0.131370	−0.0656851	+	0.997840i	$0.520923\pi$
$194$	0	0
$195$	−7.21513	−0.516686
$196$	0	0
$197$	18.3828	1.30972	0.654861	−	0.755749i	$-0.272727\pi$
$197$	18.3828	1.30972	0.654861	+	0.755749i	$0.272727\pi$
$198$	0	0
$199$	12.0886	0.856939	0.428470	−	0.903556i	$-0.359053\pi$
$199$	12.0886	0.856939	0.428470	+	0.903556i	$0.359053\pi$
$200$	0	0
$201$	−1.33477	−0.0941472
$202$	0	0
$203$	0.115025	0.00807314
$204$	0	0
$205$	−13.4560	−0.939811
$206$	0	0
$207$	−11.5732	−0.804396
$208$	0	0
$209$	−6.66274	−0.460872
$210$	0	0
$211$	−7.06966	−0.486695	−0.243348	−	0.969939i	$-0.578246\pi$
$211$	−7.06966	−0.486695	−0.243348	+	0.969939i	$0.578246\pi$
$212$	0	0
$213$	7.09172	0.485916
$214$	0	0
$215$	31.9297	2.17759
$216$	0	0
$217$	0.883659	0.0599867
$218$	0	0
$219$	2.02939	0.137133
$220$	0	0
$221$	3.18174	0.214027
$222$	0	0
$223$	9.69408	0.649164	0.324582	−	0.945858i	$-0.394776\pi$
$223$	9.69408	0.649164	0.324582	+	0.945858i	$0.394776\pi$
$224$	0	0
$225$	−7.31076	−0.487384
$226$	0	0
$227$	−4.42230	−0.293518	−0.146759	−	0.989172i	$-0.546884\pi$
$227$	−4.42230	−0.293518	−0.146759	+	0.989172i	$0.546884\pi$
$228$	0	0
$229$	5.15009	0.340328	0.170164	−	0.985416i	$-0.445570\pi$
$229$	5.15009	0.340328	0.170164	+	0.985416i	$0.445570\pi$
$230$	0	0
$231$	0.0808868	0.00532196
$232$	0	0
$233$	−15.3394	−1.00492	−0.502458	−	0.864602i	$-0.667571\pi$
$233$	−15.3394	−1.00492	−0.502458	+	0.864602i	$0.667571\pi$
$234$	0	0
$235$	31.1759	2.03369
$236$	0	0
$237$	6.63106	0.430734
$238$	0	0
$239$	−5.50491	−0.356083	−0.178042	−	0.984023i	$-0.556976\pi$
$239$	−5.50491	−0.356083	−0.178042	+	0.984023i	$0.556976\pi$
$240$	0	0
$241$	3.12878	0.201542	0.100771	−	0.994910i	$-0.467869\pi$
$241$	3.12878	0.201542	0.100771	+	0.994910i	$0.467869\pi$
$242$	0	0
$243$	−15.7665	−1.01142
$244$	0	0
$245$	19.8848	1.27040
$246$	0	0
$247$	20.5473	1.30739
$248$	0	0
$249$	3.81379	0.241689
$250$	0	0
$251$	−21.4842	−1.35607	−0.678035	−	0.735030i	$-0.737168\pi$
$251$	−21.4842	−1.35607	−0.678035	+	0.735030i	$0.737168\pi$
$252$	0	0
$253$	5.04984	0.317481
$254$	0	0
$255$	−2.26766	−0.142007
$256$	0	0
$257$	16.6315	1.03745	0.518723	−	0.854942i	$-0.326407\pi$
$257$	16.6315	1.03745	0.518723	+	0.854942i	$0.326407\pi$
$258$	0	0
$259$	0.626669	0.0389393
$260$	0	0
$261$	2.76550	0.171180
$262$	0	0
$263$	14.2047	0.875898	0.437949	−	0.899000i	$-0.355705\pi$
$263$	14.2047	0.875898	0.437949	+	0.899000i	$0.355705\pi$
$264$	0	0
$265$	−12.2822	−0.754488
$266$	0	0
$267$	14.9979	0.917855
$268$	0	0
$269$	−17.3686	−1.05898	−0.529491	−	0.848315i	$-0.677617\pi$
$269$	−17.3686	−1.05898	−0.529491	+	0.848315i	$0.677617\pi$
$270$	0	0
$271$	−22.7293	−1.38071	−0.690353	−	0.723472i	$-0.742545\pi$
$271$	−22.7293	−1.38071	−0.690353	+	0.723472i	$0.742545\pi$
$272$	0	0
$273$	−0.249447	−0.0150972
$274$	0	0
$275$	3.18996	0.192362
$276$	0	0
$277$	6.53167	0.392450	0.196225	−	0.980559i	$-0.437132\pi$
$277$	6.53167	0.392450	0.196225	+	0.980559i	$0.437132\pi$
$278$	0	0
$279$	21.2455	1.27194
$280$	0	0
$281$	−0.735247	−0.0438611	−0.0219306	−	0.999759i	$-0.506981\pi$
$281$	−0.735247	−0.0438611	−0.0219306	+	0.999759i	$0.506981\pi$
$282$	0	0
$283$	−7.80292	−0.463835	−0.231918	−	0.972735i	$-0.574500\pi$
$283$	−7.80292	−0.463835	−0.231918	+	0.972735i	$0.574500\pi$
$284$	0	0
$285$	−14.6443	−0.867451
$286$	0	0
$287$	−0.465213	−0.0274606
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.51820	−0.440725
$292$	0	0
$293$	2.40464	0.140481	0.0702403	−	0.997530i	$-0.477623\pi$
$293$	2.40464	0.140481	0.0702403	+	0.997530i	$0.477623\pi$
$294$	0	0
$295$	−2.84462	−0.165620
$296$	0	0
$297$	4.41213	0.256018
$298$	0	0
$299$	−15.5732	−0.900624
$300$	0	0
$301$	1.10390	0.0636276
$302$	0	0
$303$	−2.04221	−0.117322
$304$	0	0
$305$	−41.4380	−2.37274
$306$	0	0
$307$	31.9490	1.82342	0.911712	−	0.410829i	$-0.134761\pi$
$307$	31.9490	1.82342	0.911712	+	0.410829i	$0.134761\pi$
$308$	0	0
$309$	8.71382	0.495712
$310$	0	0
$311$	−9.95602	−0.564554	−0.282277	−	0.959333i	$-0.591090\pi$
$311$	−9.95602	−0.564554	−0.282277	+	0.959333i	$0.591090\pi$
$312$	0	0
$313$	17.1505	0.969402	0.484701	−	0.874680i	$-0.338928\pi$
$313$	17.1505	0.969402	0.484701	+	0.874680i	$0.338928\pi$
$314$	0	0
$315$	−0.661492	−0.0372708
$316$	0	0
$317$	−27.9753	−1.57125	−0.785624	−	0.618704i	$-0.787658\pi$
$317$	−27.9753	−1.57125	−0.785624	+	0.618704i	$0.787658\pi$
$318$	0	0
$319$	−1.20669	−0.0675617
$320$	0	0
$321$	−3.51363	−0.196112
$322$	0	0
$323$	6.45786	0.359325
$324$	0	0
$325$	−9.83754	−0.545689
$326$	0	0
$327$	7.84825	0.434009
$328$	0	0
$329$	1.07784	0.0594230
$330$	0	0
$331$	19.6388	1.07945	0.539724	−	0.841842i	$-0.318529\pi$
$331$	19.6388	1.07945	0.539724	+	0.841842i	$0.318529\pi$
$332$	0	0
$333$	15.0668	0.825655
$334$	0	0
$335$	−4.76295	−0.260228
$336$	0	0
$337$	−15.3258	−0.834847	−0.417424	−	0.908712i	$-0.637067\pi$
$337$	−15.3258	−0.834847	−0.417424	+	0.908712i	$0.637067\pi$
$338$	0	0
$339$	16.8462	0.914959
$340$	0	0
$341$	−9.27023	−0.502011
$342$	0	0
$343$	1.37590	0.0742916
$344$	0	0
$345$	11.0992	0.597562
$346$	0	0
$347$	7.01598	0.376638	0.188319	−	0.982108i	$-0.439696\pi$
$347$	7.01598	0.376638	0.188319	+	0.982108i	$0.439696\pi$
$348$	0	0
$349$	13.0256	0.697244	0.348622	−	0.937263i	$-0.386650\pi$
$349$	13.0256	0.697244	0.348622	+	0.937263i	$0.386650\pi$
$350$	0	0
$351$	−13.6066	−0.726267
$352$	0	0
$353$	6.62257	0.352484	0.176242	−	0.984347i	$-0.443606\pi$
$353$	6.62257	0.352484	0.176242	+	0.984347i	$0.443606\pi$
$354$	0	0
$355$	25.3059	1.34310
$356$	0	0
$357$	−0.0783995	−0.00414934
$358$	0	0
$359$	−18.0634	−0.953347	−0.476674	−	0.879080i	$-0.658158\pi$
$359$	−18.0634	−0.953347	−0.476674	+	0.879080i	$0.658158\pi$
$360$	0	0
$361$	22.7040	1.19495
$362$	0	0
$363$	7.92038	0.415712
$364$	0	0
$365$	7.24162	0.379043
$366$	0	0
$367$	−0.421766	−0.0220160	−0.0110080	−	0.999939i	$-0.503504\pi$
$367$	−0.421766	−0.0220160	−0.0110080	+	0.999939i	$0.503504\pi$
$368$	0	0
$369$	−11.1850	−0.582265
$370$	0	0
$371$	−0.424629	−0.0220456
$372$	0	0
$373$	−23.7846	−1.23152	−0.615761	−	0.787933i	$-0.711151\pi$
$373$	−23.7846	−1.23152	−0.615761	+	0.787933i	$0.711151\pi$
$374$	0	0
$375$	−4.32699	−0.223445
$376$	0	0
$377$	3.72132	0.191658
$378$	0	0
$379$	−4.67848	−0.240317	−0.120159	−	0.992755i	$-0.538340\pi$
$379$	−4.67848	−0.240317	−0.120159	+	0.992755i	$0.538340\pi$
$380$	0	0
$381$	−1.14295	−0.0585552
$382$	0	0
$383$	26.0884	1.33306	0.666528	−	0.745480i	$-0.267780\pi$
$383$	26.0884	1.33306	0.666528	+	0.745480i	$0.267780\pi$
$384$	0	0
$385$	0.288634	0.0147102
$386$	0	0
$387$	26.5406	1.34914
$388$	0	0
$389$	15.7340	0.797743	0.398872	−	0.917007i	$-0.369402\pi$
$389$	15.7340	0.797743	0.398872	+	0.917007i	$0.369402\pi$
$390$	0	0
$391$	−4.89456	−0.247529
$392$	0	0
$393$	−11.5507	−0.582655
$394$	0	0
$395$	23.6621	1.19057
$396$	0	0
$397$	24.0671	1.20789	0.603946	−	0.797025i	$-0.293594\pi$
$397$	24.0671	1.20789	0.603946	+	0.797025i	$0.293594\pi$
$398$	0	0
$399$	−0.506293	−0.0253463
$400$	0	0
$401$	5.41025	0.270175	0.135087	−	0.990834i	$-0.456868\pi$
$401$	5.41025	0.270175	0.135087	+	0.990834i	$0.456868\pi$
$402$	0	0
$403$	28.5885	1.42409
$404$	0	0
$405$	−10.4808	−0.520797
$406$	0	0
$407$	−6.57421	−0.325872
$408$	0	0
$409$	21.0569	1.04120	0.520599	−	0.853801i	$-0.325709\pi$
$409$	21.0569	1.04120	0.520599	+	0.853801i	$0.325709\pi$
$410$	0	0
$411$	9.23391	0.455475
$412$	0	0
$413$	−0.0983465	−0.00483931
$414$	0	0
$415$	13.6090	0.668041
$416$	0	0
$417$	−15.8354	−0.775462
$418$	0	0
$419$	21.9341	1.07155	0.535775	−	0.844361i	$-0.320019\pi$
$419$	21.9341	1.07155	0.535775	+	0.844361i	$0.320019\pi$
$420$	0	0
$421$	13.9579	0.680264	0.340132	−	0.940378i	$-0.389528\pi$
$421$	13.9579	0.680264	0.340132	+	0.940378i	$0.389528\pi$
$422$	0	0
$423$	25.9140	1.25998
$424$	0	0
$425$	−3.09187	−0.149978
$426$	0	0
$427$	−1.43263	−0.0693297
$428$	0	0
$429$	2.61688	0.126344
$430$	0	0
$431$	−9.24105	−0.445126	−0.222563	−	0.974918i	$-0.571442\pi$
$431$	−9.24105	−0.445126	−0.222563	+	0.974918i	$0.571442\pi$
$432$	0	0
$433$	12.5434	0.602797	0.301398	−	0.953498i	$-0.402547\pi$
$433$	12.5434	0.602797	0.301398	+	0.953498i	$0.402547\pi$
$434$	0	0
$435$	−2.65223	−0.127165
$436$	0	0
$437$	−31.6084	−1.51203
$438$	0	0
$439$	32.8598	1.56831	0.784155	−	0.620565i	$-0.213097\pi$
$439$	32.8598	1.56831	0.784155	+	0.620565i	$0.213097\pi$
$440$	0	0
$441$	16.5287	0.787081
$442$	0	0
$443$	24.7834	1.17749	0.588746	−	0.808318i	$-0.299622\pi$
$443$	24.7834	1.17749	0.588746	+	0.808318i	$0.299622\pi$
$444$	0	0
$445$	53.5180	2.53700
$446$	0	0
$447$	13.4656	0.636899
$448$	0	0
$449$	−30.4269	−1.43593	−0.717967	−	0.696077i	$-0.754927\pi$
$449$	−30.4269	−1.43593	−0.717967	+	0.696077i	$0.754927\pi$
$450$	0	0
$451$	4.88042	0.229810
$452$	0	0
$453$	−10.4448	−0.490741
$454$	0	0
$455$	−0.890120	−0.0417295
$456$	0	0
$457$	−13.2572	−0.620148	−0.310074	−	0.950712i	$-0.600354\pi$
$457$	−13.2572	−0.620148	−0.310074	+	0.950712i	$0.600354\pi$
$458$	0	0
$459$	−4.27646	−0.199608
$460$	0	0
$461$	−1.40146	−0.0652723	−0.0326362	−	0.999467i	$-0.510390\pi$
$461$	−1.40146	−0.0652723	−0.0326362	+	0.999467i	$0.510390\pi$
$462$	0	0
$463$	28.9989	1.34769	0.673846	−	0.738872i	$-0.264641\pi$
$463$	28.9989	1.34769	0.673846	+	0.738872i	$0.264641\pi$
$464$	0	0
$465$	−20.3753	−0.944884
$466$	0	0
$467$	21.4748	0.993736	0.496868	−	0.867826i	$-0.334483\pi$
$467$	21.4748	0.993736	0.496868	+	0.867826i	$0.334483\pi$
$468$	0	0
$469$	−0.164668	−0.00760368
$470$	0	0
$471$	−9.53959	−0.439561
$472$	0	0
$473$	−11.5807	−0.532480
$474$	0	0
$475$	−19.9669	−0.916143
$476$	0	0
$477$	−10.2092	−0.467448
$478$	0	0
$479$	−19.6291	−0.896878	−0.448439	−	0.893813i	$-0.648020\pi$
$479$	−19.6291	−0.896878	−0.448439	+	0.893813i	$0.648020\pi$
$480$	0	0
$481$	20.2742	0.924426
$482$	0	0
$483$	0.383731	0.0174604
$484$	0	0
$485$	−26.8277	−1.21819
$486$	0	0
$487$	15.8700	0.719140	0.359570	−	0.933118i	$-0.382923\pi$
$487$	15.8700	0.719140	0.359570	+	0.933118i	$0.382923\pi$
$488$	0	0
$489$	9.78554	0.442517
$490$	0	0
$491$	−20.6193	−0.930535	−0.465267	−	0.885170i	$-0.654042\pi$
$491$	−20.6193	−0.930535	−0.465267	+	0.885170i	$0.654042\pi$
$492$	0	0
$493$	1.16958	0.0526755
$494$	0	0
$495$	6.93953	0.311909
$496$	0	0
$497$	0.874895	0.0392444
$498$	0	0
$499$	−30.7022	−1.37442	−0.687209	−	0.726460i	$-0.741164\pi$
$499$	−30.7022	−1.37442	−0.687209	+	0.726460i	$0.741164\pi$
$500$	0	0
$501$	2.68701	0.120047
$502$	0	0
$503$	5.30109	0.236364	0.118182	−	0.992992i	$-0.462293\pi$
$503$	5.30109	0.236364	0.118182	+	0.992992i	$0.462293\pi$
$504$	0	0
$505$	−7.28736	−0.324283
$506$	0	0
$507$	2.29308	0.101839
$508$	0	0
$509$	27.6511	1.22561	0.612807	−	0.790233i	$-0.290040\pi$
$509$	27.6511	1.22561	0.612807	+	0.790233i	$0.290040\pi$
$510$	0	0
$511$	0.250363	0.0110754
$512$	0	0
$513$	−27.6168	−1.21931
$514$	0	0
$515$	31.0942	1.37017
$516$	0	0
$517$	−11.3073	−0.497294
$518$	0	0
$519$	−13.8212	−0.606682
$520$	0	0
$521$	14.4066	0.631163	0.315582	−	0.948898i	$-0.397800\pi$
$521$	14.4066	0.631163	0.315582	+	0.948898i	$0.397800\pi$
$522$	0	0
$523$	−10.9776	−0.480018	−0.240009	−	0.970771i	$-0.577150\pi$
$523$	−10.9776	−0.480018	−0.240009	+	0.970771i	$0.577150\pi$
$524$	0	0
$525$	0.242401	0.0105793
$526$	0	0
$527$	8.98516	0.391400
$528$	0	0
$529$	0.956718	0.0415965
$530$	0	0
$531$	−2.36451	−0.102611
$532$	0	0
$533$	−15.0507	−0.651920
$534$	0	0
$535$	−12.5380	−0.542063
$536$	0	0
$537$	−18.5830	−0.801915
$538$	0	0
$539$	−7.21210	−0.310647
$540$	0	0
$541$	8.56267	0.368138	0.184069	−	0.982913i	$-0.441073\pi$
$541$	8.56267	0.368138	0.184069	+	0.982913i	$0.441073\pi$
$542$	0	0
$543$	12.1290	0.520504
$544$	0	0
$545$	28.0055	1.19962
$546$	0	0
$547$	−15.7788	−0.674655	−0.337327	−	0.941387i	$-0.609523\pi$
$547$	−15.7788	−0.674655	−0.337327	+	0.941387i	$0.609523\pi$
$548$	0	0
$549$	−34.4442	−1.47004
$550$	0	0
$551$	7.55302	0.321769
$552$	0	0
$553$	0.818064	0.0347876
$554$	0	0
$555$	−14.4497	−0.613355
$556$	0	0
$557$	−29.7462	−1.26039	−0.630193	−	0.776439i	$-0.717024\pi$
$557$	−29.7462	−1.26039	−0.630193	+	0.776439i	$0.717024\pi$
$558$	0	0
$559$	35.7137	1.51053
$560$	0	0
$561$	0.822467	0.0347246
$562$	0	0
$563$	−32.7480	−1.38016	−0.690082	−	0.723731i	$-0.742425\pi$
$563$	−32.7480	−1.38016	−0.690082	+	0.723731i	$0.742425\pi$
$564$	0	0
$565$	60.1135	2.52899
$566$	0	0
$567$	−0.362352	−0.0152173
$568$	0	0
$569$	−3.48984	−0.146302	−0.0731509	−	0.997321i	$-0.523305\pi$
$569$	−3.48984	−0.146302	−0.0731509	+	0.997321i	$0.523305\pi$
$570$	0	0
$571$	−7.20695	−0.301601	−0.150801	−	0.988564i	$-0.548185\pi$
$571$	−7.20695	−0.301601	−0.150801	+	0.988564i	$0.548185\pi$
$572$	0	0
$573$	−8.76415	−0.366127
$574$	0	0
$575$	15.1334	0.631104
$576$	0	0
$577$	−2.29667	−0.0956118	−0.0478059	−	0.998857i	$-0.515223\pi$
$577$	−2.29667	−0.0956118	−0.0478059	+	0.998857i	$0.515223\pi$
$578$	0	0
$579$	1.45489	0.0604631
$580$	0	0
$581$	0.470502	0.0195197
$582$	0	0
$583$	4.45467	0.184493
$584$	0	0
$585$	−21.4008	−0.884816
$586$	0	0
$587$	15.1911	0.627004	0.313502	−	0.949588i	$-0.398498\pi$
$587$	15.1911	0.627004	0.313502	+	0.949588i	$0.398498\pi$
$588$	0	0
$589$	58.0249	2.39088
$590$	0	0
$591$	−14.6543	−0.602799
$592$	0	0
$593$	−14.8494	−0.609791	−0.304896	−	0.952386i	$-0.598622\pi$
$593$	−14.8494	−0.609791	−0.304896	+	0.952386i	$0.598622\pi$
$594$	0	0
$595$	−0.279758	−0.0114690
$596$	0	0
$597$	−9.63676	−0.394406
$598$	0	0
$599$	26.5387	1.08434	0.542172	−	0.840268i	$-0.317602\pi$
$599$	26.5387	1.08434	0.542172	+	0.840268i	$0.317602\pi$
$600$	0	0
$601$	6.61641	0.269889	0.134945	−	0.990853i	$-0.456914\pi$
$601$	6.61641	0.269889	0.134945	+	0.990853i	$0.456914\pi$
$602$	0	0
$603$	−3.95906	−0.161226
$604$	0	0
$605$	28.2629	1.14905
$606$	0	0
$607$	21.6135	0.877267	0.438633	−	0.898666i	$-0.355463\pi$
$607$	21.6135	0.877267	0.438633	+	0.898666i	$0.355463\pi$
$608$	0	0
$609$	−0.0916948	−0.00371566
$610$	0	0
$611$	34.8706	1.41071
$612$	0	0
$613$	−28.9405	−1.16889	−0.584447	−	0.811432i	$-0.698689\pi$
$613$	−28.9405	−1.16889	−0.584447	+	0.811432i	$0.698689\pi$
$614$	0	0
$615$	10.7268	0.432548
$616$	0	0
$617$	−22.1462	−0.891573	−0.445787	−	0.895139i	$-0.647076\pi$
$617$	−22.1462	−0.891573	−0.445787	+	0.895139i	$0.647076\pi$
$618$	0	0
$619$	−12.8883	−0.518023	−0.259012	−	0.965874i	$-0.583397\pi$
$619$	−12.8883	−0.518023	−0.259012	+	0.965874i	$0.583397\pi$
$620$	0	0
$621$	20.9314	0.839948
$622$	0	0
$623$	1.85027	0.0741294
$624$	0	0
$625$	−30.8997	−1.23599
$626$	0	0
$627$	5.31138	0.212116
$628$	0	0
$629$	6.37205	0.254070
$630$	0	0
$631$	26.2936	1.04673	0.523366	−	0.852108i	$-0.324676\pi$
$631$	26.2936	1.04673	0.523366	+	0.852108i	$0.324676\pi$
$632$	0	0
$633$	5.63576	0.224001
$634$	0	0
$635$	−4.07848	−0.161849
$636$	0	0
$637$	22.2414	0.881238
$638$	0	0
$639$	21.0348	0.832123
$640$	0	0
$641$	−12.5166	−0.494376	−0.247188	−	0.968968i	$-0.579507\pi$
$641$	−12.5166	−0.494376	−0.247188	+	0.968968i	$0.579507\pi$
$642$	0	0
$643$	4.44378	0.175246	0.0876228	−	0.996154i	$-0.472073\pi$
$643$	4.44378	0.175246	0.0876228	+	0.996154i	$0.472073\pi$
$644$	0	0
$645$	−25.4536	−1.00223
$646$	0	0
$647$	48.1536	1.89311	0.946557	−	0.322538i	$-0.104536\pi$
$647$	48.1536	1.89311	0.946557	+	0.322538i	$0.104536\pi$
$648$	0	0
$649$	1.03173	0.0404988
$650$	0	0
$651$	−0.704432	−0.0276089
$652$	0	0
$653$	9.30426	0.364104	0.182052	−	0.983289i	$-0.441726\pi$
$653$	9.30426	0.364104	0.182052	+	0.983289i	$0.441726\pi$
$654$	0	0
$655$	−41.2172	−1.61049
$656$	0	0
$657$	6.01939	0.234839
$658$	0	0
$659$	−11.5076	−0.448271	−0.224135	−	0.974558i	$-0.571956\pi$
$659$	−11.5076	−0.448271	−0.224135	+	0.974558i	$0.571956\pi$
$660$	0	0
$661$	13.7153	0.533465	0.266733	−	0.963771i	$-0.414056\pi$
$661$	13.7153	0.533465	0.266733	+	0.963771i	$0.414056\pi$
$662$	0	0
$663$	−2.53641	−0.0985060
$664$	0	0
$665$	−1.80664	−0.0700586
$666$	0	0
$667$	−5.72460	−0.221658
$668$	0	0
$669$	−7.72789	−0.298778
$670$	0	0
$671$	15.0293	0.580200
$672$	0	0
$673$	30.0006	1.15644	0.578218	−	0.815882i	$-0.303748\pi$
$673$	30.0006	1.15644	0.578218	+	0.815882i	$0.303748\pi$
$674$	0	0
$675$	13.2223	0.508925
$676$	0	0
$677$	−14.7692	−0.567626	−0.283813	−	0.958880i	$-0.591599\pi$
$677$	−14.7692	−0.567626	−0.283813	+	0.958880i	$0.591599\pi$
$678$	0	0
$679$	−0.927510	−0.0355946
$680$	0	0
$681$	3.52535	0.135092
$682$	0	0
$683$	−26.0593	−0.997131	−0.498566	−	0.866852i	$-0.666140\pi$
$683$	−26.0593	−0.997131	−0.498566	+	0.866852i	$0.666140\pi$
$684$	0	0
$685$	32.9500	1.25896
$686$	0	0
$687$	−4.10553	−0.156636
$688$	0	0
$689$	−13.7378	−0.523367
$690$	0	0
$691$	−25.9278	−0.986339	−0.493169	−	0.869933i	$-0.664162\pi$
$691$	−25.9278	−0.986339	−0.493169	+	0.869933i	$0.664162\pi$
$692$	0	0
$693$	0.239919	0.00911376
$694$	0	0
$695$	−56.5065	−2.14341
$696$	0	0
$697$	−4.73035	−0.179175
$698$	0	0
$699$	12.2282	0.462512
$700$	0	0
$701$	−39.2772	−1.48348	−0.741739	−	0.670688i	$-0.765999\pi$
$701$	−39.2772	−1.48348	−0.741739	+	0.670688i	$0.765999\pi$
$702$	0	0
$703$	41.1498	1.55200
$704$	0	0
$705$	−24.8526	−0.936005
$706$	0	0
$707$	−0.251944	−0.00947534
$708$	0	0
$709$	−26.4157	−0.992063	−0.496031	−	0.868305i	$-0.665210\pi$
$709$	−26.4157	−0.992063	−0.496031	+	0.868305i	$0.665210\pi$
$710$	0	0
$711$	19.6684	0.737624
$712$	0	0
$713$	−43.9784	−1.64700
$714$	0	0
$715$	9.33800	0.349222
$716$	0	0
$717$	4.38839	0.163887
$718$	0	0
$719$	−27.6232	−1.03017	−0.515086	−	0.857138i	$-0.672240\pi$
$719$	−27.6232	−1.03017	−0.515086	+	0.857138i	$0.672240\pi$
$720$	0	0
$721$	1.07501	0.0400355
$722$	0	0
$723$	−2.49419	−0.0927597
$724$	0	0
$725$	−3.61621	−0.134303
$726$	0	0
$727$	−21.1853	−0.785721	−0.392861	−	0.919598i	$-0.628514\pi$
$727$	−21.1853	−0.785721	−0.392861	+	0.919598i	$0.628514\pi$
$728$	0	0
$729$	1.51538	0.0561252
$730$	0	0
$731$	11.2246	0.415156
$732$	0	0
$733$	−29.1836	−1.07792	−0.538960	−	0.842331i	$-0.681183\pi$
$733$	−29.1836	−1.07792	−0.538960	+	0.842331i	$0.681183\pi$
$734$	0	0
$735$	−15.8517	−0.584699
$736$	0	0
$737$	1.72749	0.0636329
$738$	0	0
$739$	12.7964	0.470725	0.235362	−	0.971908i	$-0.424372\pi$
$739$	12.7964	0.470725	0.235362	+	0.971908i	$0.424372\pi$
$740$	0	0
$741$	−16.3798	−0.601727
$742$	0	0
$743$	−16.9044	−0.620161	−0.310080	−	0.950710i	$-0.600356\pi$
$743$	−16.9044	−0.620161	−0.310080	+	0.950710i	$0.600356\pi$
$744$	0	0
$745$	48.0501	1.76042
$746$	0	0
$747$	11.3121	0.413888
$748$	0	0
$749$	−0.433472	−0.0158387
$750$	0	0
$751$	−34.8125	−1.27032	−0.635162	−	0.772379i	$-0.719067\pi$
$751$	−34.8125	−1.27032	−0.635162	+	0.772379i	$0.719067\pi$
$752$	0	0
$753$	17.1267	0.624131
$754$	0	0
$755$	−37.2711	−1.35643
$756$	0	0
$757$	36.9850	1.34424	0.672121	−	0.740441i	$-0.265384\pi$
$757$	36.9850	1.34424	0.672121	+	0.740441i	$0.265384\pi$
$758$	0	0
$759$	−4.02562	−0.146121
$760$	0	0
$761$	15.3352	0.555901	0.277950	−	0.960595i	$-0.410345\pi$
$761$	15.3352	0.555901	0.277950	+	0.960595i	$0.410345\pi$
$762$	0	0
$763$	0.968227	0.0350522
$764$	0	0
$765$	−6.72614	−0.243184
$766$	0	0
$767$	−3.18174	−0.114886
$768$	0	0
$769$	−12.4619	−0.449389	−0.224694	−	0.974429i	$-0.572138\pi$
$769$	−12.4619	−0.449389	−0.224694	+	0.974429i	$0.572138\pi$
$770$	0	0
$771$	−13.2583	−0.477485
$772$	0	0
$773$	−6.36710	−0.229009	−0.114504	−	0.993423i	$-0.536528\pi$
$773$	−6.36710	−0.229009	−0.114504	+	0.993423i	$0.536528\pi$
$774$	0	0
$775$	−27.7810	−0.997922
$776$	0	0
$777$	−0.499566	−0.0179218
$778$	0	0
$779$	−30.5479	−1.09449
$780$	0	0
$781$	−9.17828	−0.328425
$782$	0	0
$783$	−5.00168	−0.178746
$784$	0	0
$785$	−34.0408	−1.21497
$786$	0	0
$787$	−37.9370	−1.35231	−0.676153	−	0.736761i	$-0.736354\pi$
$787$	−37.9370	−1.35231	−0.676153	+	0.736761i	$0.736354\pi$
$788$	0	0
$789$	−11.3236	−0.403132
$790$	0	0
$791$	2.07829	0.0738955
$792$	0	0
$793$	−46.3490	−1.64590
$794$	0	0
$795$	9.79106	0.347253
$796$	0	0
$797$	−51.2073	−1.81386	−0.906928	−	0.421285i	$-0.861579\pi$
$797$	−51.2073	−1.81386	−0.906928	+	0.421285i	$0.861579\pi$
$798$	0	0
$799$	10.9596	0.387722
$800$	0	0
$801$	44.4853	1.57181
$802$	0	0
$803$	−2.62649	−0.0926867
$804$	0	0
$805$	1.36929	0.0482613
$806$	0	0
$807$	13.8458	0.487397
$808$	0	0
$809$	−33.6124	−1.18175	−0.590874	−	0.806764i	$-0.701217\pi$
$809$	−33.6124	−1.18175	−0.590874	+	0.806764i	$0.701217\pi$
$810$	0	0
$811$	9.02485	0.316905	0.158453	−	0.987367i	$-0.449349\pi$
$811$	9.02485	0.316905	0.158453	+	0.987367i	$0.449349\pi$
$812$	0	0
$813$	18.1193	0.635470
$814$	0	0
$815$	34.9185	1.22314
$816$	0	0
$817$	72.4867	2.53599
$818$	0	0
$819$	−0.739887	−0.0258537
$820$	0	0
$821$	7.35785	0.256791	0.128395	−	0.991723i	$-0.459017\pi$
$821$	7.35785	0.256791	0.128395	+	0.991723i	$0.459017\pi$
$822$	0	0
$823$	28.4499	0.991702	0.495851	−	0.868408i	$-0.334856\pi$
$823$	28.4499	0.991702	0.495851	+	0.868408i	$0.334856\pi$
$824$	0	0
$825$	−2.54296	−0.0885346
$826$	0	0
$827$	20.8701	0.725724	0.362862	−	0.931843i	$-0.381800\pi$
$827$	20.8701	0.725724	0.362862	+	0.931843i	$0.381800\pi$
$828$	0	0
$829$	34.1702	1.18678	0.593390	−	0.804915i	$-0.297789\pi$
$829$	34.1702	1.18678	0.593390	+	0.804915i	$0.297789\pi$
$830$	0	0
$831$	−5.20689	−0.180625
$832$	0	0
$833$	6.99033	0.242201
$834$	0	0
$835$	9.58824	0.331815
$836$	0	0
$837$	−38.4247	−1.32815
$838$	0	0
$839$	−11.4269	−0.394499	−0.197250	−	0.980353i	$-0.563201\pi$
$839$	−11.4269	−0.394499	−0.197250	+	0.980353i	$0.563201\pi$
$840$	0	0
$841$	−27.6321	−0.952830
$842$	0	0
$843$	0.586121	0.0201871
$844$	0	0
$845$	8.18257	0.281489
$846$	0	0
$847$	0.977125	0.0335744
$848$	0	0
$849$	6.22030	0.213480
$850$	0	0
$851$	−31.1884	−1.06912
$852$	0	0
$853$	18.1363	0.620977	0.310488	−	0.950577i	$-0.399507\pi$
$853$	18.1363	0.620977	0.310488	+	0.950577i	$0.399507\pi$
$854$	0	0
$855$	−43.4365	−1.48550
$856$	0	0
$857$	−27.3291	−0.933544	−0.466772	−	0.884378i	$-0.654583\pi$
$857$	−27.3291	−0.933544	−0.466772	+	0.884378i	$0.654583\pi$
$858$	0	0
$859$	37.9652	1.29535	0.647677	−	0.761915i	$-0.275741\pi$
$859$	37.9652	1.29535	0.647677	+	0.761915i	$0.275741\pi$
$860$	0	0
$861$	0.370857	0.0126388
$862$	0	0
$863$	−24.4305	−0.831623	−0.415812	−	0.909451i	$-0.636502\pi$
$863$	−24.4305	−0.831623	−0.415812	+	0.909451i	$0.636502\pi$
$864$	0	0
$865$	−49.3191	−1.67690
$866$	0	0
$867$	−0.797176	−0.0270735
$868$	0	0
$869$	−8.58209	−0.291127
$870$	0	0
$871$	−5.32742	−0.180513
$872$	0	0
$873$	−22.2998	−0.754734
$874$	0	0
$875$	−0.533815	−0.0180462
$876$	0	0
$877$	25.1678	0.849855	0.424928	−	0.905227i	$-0.360300\pi$
$877$	25.1678	0.849855	0.424928	+	0.905227i	$0.360300\pi$
$878$	0	0
$879$	−1.91692	−0.0646562
$880$	0	0
$881$	41.7810	1.40764	0.703819	−	0.710380i	$-0.251477\pi$
$881$	41.7810	1.40764	0.703819	+	0.710380i	$0.251477\pi$
$882$	0	0
$883$	−3.46563	−0.116628	−0.0583139	−	0.998298i	$-0.518572\pi$
$883$	−3.46563	−0.116628	−0.0583139	+	0.998298i	$0.518572\pi$
$884$	0	0
$885$	2.26766	0.0762267
$886$	0	0
$887$	13.0270	0.437405	0.218703	−	0.975792i	$-0.429817\pi$
$887$	13.0270	0.437405	0.218703	+	0.975792i	$0.429817\pi$
$888$	0	0
$889$	−0.141004	−0.00472913
$890$	0	0
$891$	3.80133	0.127349
$892$	0	0
$893$	70.7754	2.36841
$894$	0	0
$895$	−66.3110	−2.21653
$896$	0	0
$897$	12.4146	0.414512
$898$	0	0
$899$	10.5089	0.350492
$900$	0	0
$901$	−4.31769	−0.143843
$902$	0	0
$903$	−0.880001	−0.0292846
$904$	0	0
$905$	43.2807	1.43870
$906$	0	0
$907$	53.4168	1.77367	0.886837	−	0.462082i	$-0.152897\pi$
$907$	53.4168	1.77367	0.886837	+	0.462082i	$0.152897\pi$
$908$	0	0
$909$	−6.05741	−0.200912
$910$	0	0
$911$	30.6332	1.01492	0.507461	−	0.861675i	$-0.330584\pi$
$911$	30.6332	1.01492	0.507461	+	0.861675i	$0.330584\pi$
$912$	0	0
$913$	−4.93590	−0.163355
$914$	0	0
$915$	33.0334	1.09205
$916$	0	0
$917$	−1.42499	−0.0470574
$918$	0	0
$919$	19.7010	0.649877	0.324938	−	0.945735i	$-0.394656\pi$
$919$	19.7010	0.649877	0.324938	+	0.945735i	$0.394656\pi$
$920$	0	0
$921$	−25.4690	−0.839231
$922$	0	0
$923$	28.3049	0.931668
$924$	0	0
$925$	−19.7016	−0.647784
$926$	0	0
$927$	25.8461	0.848898
$928$	0	0
$929$	−28.4201	−0.932434	−0.466217	−	0.884670i	$-0.654383\pi$
$929$	−28.4201	−0.932434	−0.466217	+	0.884670i	$0.654383\pi$
$930$	0	0
$931$	45.1426	1.47949
$932$	0	0
$933$	7.93670	0.259836
$934$	0	0
$935$	2.93487	0.0959805
$936$	0	0
$937$	−41.1822	−1.34536	−0.672682	−	0.739931i	$-0.734858\pi$
$937$	−41.1822	−1.34536	−0.672682	+	0.739931i	$0.734858\pi$
$938$	0	0
$939$	−13.6719	−0.446167
$940$	0	0
$941$	38.5433	1.25647	0.628237	−	0.778022i	$-0.283777\pi$
$941$	38.5433	1.25647	0.628237	+	0.778022i	$0.283777\pi$
$942$	0	0
$943$	23.1530	0.753964
$944$	0	0
$945$	1.19638	0.0389181
$946$	0	0
$947$	−28.1272	−0.914010	−0.457005	−	0.889464i	$-0.651078\pi$
$947$	−28.1272	−0.914010	−0.457005	+	0.889464i	$0.651078\pi$
$948$	0	0
$949$	8.09984	0.262932
$950$	0	0
$951$	22.3012	0.723166
$952$	0	0
$953$	−57.2744	−1.85530	−0.927650	−	0.373451i	$-0.878174\pi$
$953$	−57.2744	−1.85530	−0.927650	+	0.373451i	$0.878174\pi$
$954$	0	0
$955$	−31.2737	−1.01199
$956$	0	0
$957$	0.961945	0.0310953
$958$	0	0
$959$	1.13917	0.0367858
$960$	0	0
$961$	49.7332	1.60430
$962$	0	0
$963$	−10.4218	−0.335838
$964$	0	0
$965$	5.19158	0.167123
$966$	0	0
$967$	−8.70261	−0.279857	−0.139929	−	0.990162i	$-0.544687\pi$
$967$	−8.70261	−0.279857	−0.139929	+	0.990162i	$0.544687\pi$
$968$	0	0
$969$	−5.14805	−0.165379
$970$	0	0
$971$	39.5981	1.27076	0.635382	−	0.772198i	$-0.280843\pi$
$971$	39.5981	1.27076	0.635382	+	0.772198i	$0.280843\pi$
$972$	0	0
$973$	−1.95359	−0.0626291
$974$	0	0
$975$	7.84226	0.251153
$976$	0	0
$977$	19.2543	0.615998	0.307999	−	0.951387i	$-0.400341\pi$
$977$	19.2543	0.615998	0.307999	+	0.951387i	$0.400341\pi$
$978$	0	0
$979$	−19.4106	−0.620367
$980$	0	0
$981$	23.2788	0.743233
$982$	0	0
$983$	15.5804	0.496937	0.248469	−	0.968640i	$-0.420073\pi$
$983$	15.5804	0.496937	0.248469	+	0.968640i	$0.420073\pi$
$984$	0	0
$985$	−52.2922	−1.66617
$986$	0	0
$987$	−0.859225	−0.0273494
$988$	0	0
$989$	−54.9394	−1.74697
$990$	0	0
$991$	11.6110	0.368835	0.184417	−	0.982848i	$-0.440960\pi$
$991$	11.6110	0.368835	0.184417	+	0.982848i	$0.440960\pi$
$992$	0	0
$993$	−15.6556	−0.496815
$994$	0	0
$995$	−34.3875	−1.09016
$996$	0	0
$997$	31.8425	1.00846	0.504232	−	0.863568i	$-0.331776\pi$
$997$	31.8425	1.00846	0.504232	+	0.863568i	$0.331776\pi$
$998$	0	0
$999$	−27.2498	−0.862146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.12	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.12	✓	30	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.797176 q^{3} -2.84462 q^{5} -0.0983465 q^{7} -2.36451 q^{9} +O(q^{10})\)	\(q-0.797176 q^{3} -2.84462 q^{5} -0.0983465 q^{7} -2.36451 q^{9} +1.03173 q^{11} -3.18174 q^{13} +2.26766 q^{15} -1.00000 q^{17} -6.45786 q^{19} +0.0783995 q^{21} +4.89456 q^{23} +3.09187 q^{25} +4.27646 q^{27} -1.16958 q^{29} -8.98516 q^{31} -0.822467 q^{33} +0.279758 q^{35} -6.37205 q^{37} +2.53641 q^{39} +4.73035 q^{41} -11.2246 q^{43} +6.72614 q^{45} -10.9596 q^{47} -6.99033 q^{49} +0.797176 q^{51} +4.31769 q^{53} -2.93487 q^{55} +5.14805 q^{57} +1.00000 q^{59} +14.5672 q^{61} +0.232541 q^{63} +9.05086 q^{65} +1.67437 q^{67} -3.90183 q^{69} -8.89605 q^{71} -2.54572 q^{73} -2.46477 q^{75} -0.101467 q^{77} -8.31819 q^{79} +3.68444 q^{81} -4.78412 q^{83} +2.84462 q^{85} +0.932365 q^{87} -18.8138 q^{89} +0.312913 q^{91} +7.16276 q^{93} +18.3702 q^{95} +9.43104 q^{97} -2.43953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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