LMFDB - Embedded newform 8024.2.a.bb.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:	$32$
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)

$f(q)$	$=$	$q-1.11553 q^{3} -3.13935 q^{5} +0.425117 q^{7} -1.75560 q^{9} +O(q^{10})$	$q-1.11553 q^{3} -3.13935 q^{5} +0.425117 q^{7} -1.75560 q^{9} +5.06615 q^{11} +4.90156 q^{13} +3.50203 q^{15} +1.00000 q^{17} +0.636135 q^{19} -0.474229 q^{21} +1.07690 q^{23} +4.85552 q^{25} +5.30500 q^{27} +10.0578 q^{29} +4.65573 q^{31} -5.65142 q^{33} -1.33459 q^{35} +2.41557 q^{37} -5.46782 q^{39} -8.48041 q^{41} +5.24121 q^{43} +5.51144 q^{45} +3.53002 q^{47} -6.81928 q^{49} -1.11553 q^{51} -10.0259 q^{53} -15.9044 q^{55} -0.709625 q^{57} +1.00000 q^{59} -13.9340 q^{61} -0.746336 q^{63} -15.3877 q^{65} +11.6565 q^{67} -1.20131 q^{69} -15.1726 q^{71} +12.8755 q^{73} -5.41646 q^{75} +2.15371 q^{77} +0.414250 q^{79} -0.651066 q^{81} -3.12316 q^{83} -3.13935 q^{85} -11.2197 q^{87} +2.83442 q^{89} +2.08374 q^{91} -5.19359 q^{93} -1.99705 q^{95} -11.3560 q^{97} -8.89413 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * 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
$3$	−1.11553	−0.644050	−0.322025	−	0.946731i	$-0.604363\pi$
$3$	−1.11553	−0.644050	−0.322025	+	0.946731i	$0.604363\pi$
$4$	0	0
$5$	−3.13935	−1.40396	−0.701980	−	0.712197i	$-0.747700\pi$
$5$	−3.13935	−1.40396	−0.701980	+	0.712197i	$0.747700\pi$
$6$	0	0
$7$	0.425117	0.160679	0.0803396	−	0.996768i	$-0.474400\pi$
$7$	0.425117	0.160679	0.0803396	+	0.996768i	$0.474400\pi$
$8$	0	0
$9$	−1.75560	−0.585200
$10$	0	0
$11$	5.06615	1.52750	0.763751	−	0.645511i	$-0.223356\pi$
$11$	5.06615	1.52750	0.763751	+	0.645511i	$0.223356\pi$
$12$	0	0
$13$	4.90156	1.35945	0.679724	−	0.733468i	$-0.262099\pi$
$13$	4.90156	1.35945	0.679724	+	0.733468i	$0.262099\pi$
$14$	0	0
$15$	3.50203	0.904220
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.636135	0.145939	0.0729697	−	0.997334i	$-0.476752\pi$
$19$	0.636135	0.145939	0.0729697	+	0.997334i	$0.476752\pi$
$20$	0	0
$21$	−0.474229	−0.103485
$22$	0	0
$23$	1.07690	0.224549	0.112275	−	0.993677i	$-0.464186\pi$
$23$	1.07690	0.224549	0.112275	+	0.993677i	$0.464186\pi$
$24$	0	0
$25$	4.85552	0.971104
$26$	0	0
$27$	5.30500	1.02095
$28$	0	0
$29$	10.0578	1.86769	0.933844	−	0.357681i	$-0.116433\pi$
$29$	10.0578	1.86769	0.933844	+	0.357681i	$0.116433\pi$
$30$	0	0
$31$	4.65573	0.836193	0.418097	−	0.908403i	$-0.362697\pi$
$31$	4.65573	0.836193	0.418097	+	0.908403i	$0.362697\pi$
$32$	0	0
$33$	−5.65142	−0.983787
$34$	0	0
$35$	−1.33459	−0.225587
$36$	0	0
$37$	2.41557	0.397117	0.198558	−	0.980089i	$-0.436374\pi$
$37$	2.41557	0.397117	0.198558	+	0.980089i	$0.436374\pi$
$38$	0	0
$39$	−5.46782	−0.875552
$40$	0	0
$41$	−8.48041	−1.32442	−0.662209	−	0.749319i	$-0.730381\pi$
$41$	−8.48041	−1.32442	−0.662209	+	0.749319i	$0.730381\pi$
$42$	0	0
$43$	5.24121	0.799277	0.399638	−	0.916673i	$-0.369136\pi$
$43$	5.24121	0.799277	0.399638	+	0.916673i	$0.369136\pi$
$44$	0	0
$45$	5.51144	0.821598
$46$	0	0
$47$	3.53002	0.514906	0.257453	−	0.966291i	$-0.417117\pi$
$47$	3.53002	0.514906	0.257453	+	0.966291i	$0.417117\pi$
$48$	0	0
$49$	−6.81928	−0.974182
$50$	0	0
$51$	−1.11553	−0.156205
$52$	0	0
$53$	−10.0259	−1.37716	−0.688581	−	0.725159i	$-0.741766\pi$
$53$	−10.0259	−1.37716	−0.688581	+	0.725159i	$0.741766\pi$
$54$	0	0
$55$	−15.9044	−2.14455
$56$	0	0
$57$	−0.709625	−0.0939922
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−13.9340	−1.78407	−0.892035	−	0.451967i	$-0.850722\pi$
$61$	−13.9340	−1.78407	−0.892035	+	0.451967i	$0.850722\pi$
$62$	0	0
$63$	−0.746336	−0.0940295
$64$	0	0
$65$	−15.3877	−1.90861
$66$	0	0
$67$	11.6565	1.42407	0.712035	−	0.702144i	$-0.247774\pi$
$67$	11.6565	1.42407	0.712035	+	0.702144i	$0.247774\pi$
$68$	0	0
$69$	−1.20131	−0.144621
$70$	0	0
$71$	−15.1726	−1.80066	−0.900329	−	0.435209i	$-0.856674\pi$
$71$	−15.1726	−1.80066	−0.900329	+	0.435209i	$0.856674\pi$
$72$	0	0
$73$	12.8755	1.50696	0.753480	−	0.657471i	$-0.228374\pi$
$73$	12.8755	1.50696	0.753480	+	0.657471i	$0.228374\pi$
$74$	0	0
$75$	−5.41646	−0.625439
$76$	0	0
$77$	2.15371	0.245438
$78$	0	0
$79$	0.414250	0.0466068	0.0233034	−	0.999728i	$-0.492582\pi$
$79$	0.414250	0.0466068	0.0233034	+	0.999728i	$0.492582\pi$
$80$	0	0
$81$	−0.651066	−0.0723407
$82$	0	0
$83$	−3.12316	−0.342811	−0.171406	−	0.985201i	$-0.554831\pi$
$83$	−3.12316	−0.342811	−0.171406	+	0.985201i	$0.554831\pi$
$84$	0	0
$85$	−3.13935	−0.340510
$86$	0	0
$87$	−11.2197	−1.20288
$88$	0	0
$89$	2.83442	0.300448	0.150224	−	0.988652i	$-0.452001\pi$
$89$	2.83442	0.300448	0.150224	+	0.988652i	$0.452001\pi$
$90$	0	0
$91$	2.08374	0.218435
$92$	0	0
$93$	−5.19359	−0.538550
$94$	0	0
$95$	−1.99705	−0.204893
$96$	0	0
$97$	−11.3560	−1.15303	−0.576514	−	0.817087i	$-0.695587\pi$
$97$	−11.3560	−1.15303	−0.576514	+	0.817087i	$0.695587\pi$
$98$	0	0
$99$	−8.89413	−0.893894
$100$	0	0
$101$	12.9990	1.29345	0.646724	−	0.762724i	$-0.276138\pi$
$101$	12.9990	1.29345	0.646724	+	0.762724i	$0.276138\pi$
$102$	0	0
$103$	−2.15253	−0.212095	−0.106048	−	0.994361i	$-0.533820\pi$
$103$	−2.15253	−0.212095	−0.106048	+	0.994361i	$0.533820\pi$
$104$	0	0
$105$	1.48877	0.145289
$106$	0	0
$107$	−10.6249	−1.02715	−0.513576	−	0.858044i	$-0.671680\pi$
$107$	−10.6249	−1.02715	−0.513576	+	0.858044i	$0.671680\pi$
$108$	0	0
$109$	−0.872525	−0.0835727	−0.0417864	−	0.999127i	$-0.513305\pi$
$109$	−0.872525	−0.0835727	−0.0417864	+	0.999127i	$0.513305\pi$
$110$	0	0
$111$	−2.69463	−0.255763
$112$	0	0
$113$	5.15810	0.485233	0.242617	−	0.970122i	$-0.421994\pi$
$113$	5.15810	0.485233	0.242617	+	0.970122i	$0.421994\pi$
$114$	0	0
$115$	−3.38077	−0.315258
$116$	0	0
$117$	−8.60518	−0.795549
$118$	0	0
$119$	0.425117	0.0389704
$120$	0	0
$121$	14.6659	1.33326
$122$	0	0
$123$	9.46013	0.852991
$124$	0	0
$125$	0.453571	0.0405686
$126$	0	0
$127$	19.1000	1.69485	0.847424	−	0.530917i	$-0.178152\pi$
$127$	19.1000	1.69485	0.847424	+	0.530917i	$0.178152\pi$
$128$	0	0
$129$	−5.84671	−0.514774
$130$	0	0
$131$	11.4524	1.00060	0.500299	−	0.865852i	$-0.333223\pi$
$131$	11.4524	1.00060	0.500299	+	0.865852i	$0.333223\pi$
$132$	0	0
$133$	0.270432	0.0234494
$134$	0	0
$135$	−16.6542	−1.43337
$136$	0	0
$137$	3.55557	0.303773	0.151887	−	0.988398i	$-0.451465\pi$
$137$	3.55557	0.303773	0.151887	+	0.988398i	$0.451465\pi$
$138$	0	0
$139$	14.2879	1.21189	0.605943	−	0.795508i	$-0.292796\pi$
$139$	14.2879	1.21189	0.605943	+	0.795508i	$0.292796\pi$
$140$	0	0
$141$	−3.93783	−0.331625
$142$	0	0
$143$	24.8320	2.07656
$144$	0	0
$145$	−31.5750	−2.62216
$146$	0	0
$147$	7.60708	0.627422
$148$	0	0
$149$	−21.4361	−1.75611	−0.878056	−	0.478558i	$-0.841160\pi$
$149$	−21.4361	−1.75611	−0.878056	+	0.478558i	$0.841160\pi$
$150$	0	0
$151$	7.92526	0.644949	0.322474	−	0.946578i	$-0.395485\pi$
$151$	7.92526	0.644949	0.322474	+	0.946578i	$0.395485\pi$
$152$	0	0
$153$	−1.75560	−0.141932
$154$	0	0
$155$	−14.6160	−1.17398
$156$	0	0
$157$	11.1103	0.886698	0.443349	−	0.896349i	$-0.353790\pi$
$157$	11.1103	0.886698	0.443349	+	0.896349i	$0.353790\pi$
$158$	0	0
$159$	11.1841	0.886961
$160$	0	0
$161$	0.457809	0.0360804
$162$	0	0
$163$	−23.1324	−1.81187	−0.905936	−	0.423414i	$-0.860831\pi$
$163$	−23.1324	−1.81187	−0.905936	+	0.423414i	$0.860831\pi$
$164$	0	0
$165$	17.7418	1.38120
$166$	0	0
$167$	−3.08313	−0.238580	−0.119290	−	0.992859i	$-0.538062\pi$
$167$	−3.08313	−0.238580	−0.119290	+	0.992859i	$0.538062\pi$
$168$	0	0
$169$	11.0253	0.848100
$170$	0	0
$171$	−1.11680	−0.0854037
$172$	0	0
$173$	19.7767	1.50359	0.751796	−	0.659396i	$-0.229188\pi$
$173$	19.7767	1.50359	0.751796	+	0.659396i	$0.229188\pi$
$174$	0	0
$175$	2.06416	0.156036
$176$	0	0
$177$	−1.11553	−0.0838481
$178$	0	0
$179$	−12.8205	−0.958251	−0.479125	−	0.877746i	$-0.659046\pi$
$179$	−12.8205	−0.958251	−0.479125	+	0.877746i	$0.659046\pi$
$180$	0	0
$181$	1.84090	0.136833	0.0684166	−	0.997657i	$-0.478205\pi$
$181$	1.84090	0.136833	0.0684166	+	0.997657i	$0.478205\pi$
$182$	0	0
$183$	15.5438	1.14903
$184$	0	0
$185$	−7.58331	−0.557536
$186$	0	0
$187$	5.06615	0.370473
$188$	0	0
$189$	2.25525	0.164045
$190$	0	0
$191$	6.47921	0.468820	0.234410	−	0.972138i	$-0.424684\pi$
$191$	6.47921	0.468820	0.234410	+	0.972138i	$0.424684\pi$
$192$	0	0
$193$	−20.5212	−1.47715	−0.738574	−	0.674173i	$-0.764500\pi$
$193$	−20.5212	−1.47715	−0.738574	+	0.674173i	$0.764500\pi$
$194$	0	0
$195$	17.1654	1.22924
$196$	0	0
$197$	−7.87635	−0.561167	−0.280583	−	0.959830i	$-0.590528\pi$
$197$	−7.87635	−0.561167	−0.280583	+	0.959830i	$0.590528\pi$
$198$	0	0
$199$	20.9215	1.48309	0.741543	−	0.670905i	$-0.234094\pi$
$199$	20.9215	1.48309	0.741543	+	0.670905i	$0.234094\pi$
$200$	0	0
$201$	−13.0032	−0.917172
$202$	0	0
$203$	4.27574	0.300098
$204$	0	0
$205$	26.6230	1.85943
$206$	0	0
$207$	−1.89061	−0.131406
$208$	0	0
$209$	3.22275	0.222923
$210$	0	0
$211$	−3.70602	−0.255133	−0.127566	−	0.991830i	$-0.540717\pi$
$211$	−3.70602	−0.255133	−0.127566	+	0.991830i	$0.540717\pi$
$212$	0	0
$213$	16.9255	1.15971
$214$	0	0
$215$	−16.4540	−1.12215
$216$	0	0
$217$	1.97923	0.134359
$218$	0	0
$219$	−14.3629	−0.970557
$220$	0	0
$221$	4.90156	0.329715
$222$	0	0
$223$	−0.715401	−0.0479068	−0.0239534	−	0.999713i	$-0.507625\pi$
$223$	−0.715401	−0.0479068	−0.0239534	+	0.999713i	$0.507625\pi$
$224$	0	0
$225$	−8.52435	−0.568290
$226$	0	0
$227$	1.23805	0.0821724	0.0410862	−	0.999156i	$-0.486918\pi$
$227$	1.23805	0.0821724	0.0410862	+	0.999156i	$0.486918\pi$
$228$	0	0
$229$	−25.4567	−1.68222	−0.841112	−	0.540862i	$-0.818098\pi$
$229$	−25.4567	−1.68222	−0.841112	+	0.540862i	$0.818098\pi$
$230$	0	0
$231$	−2.40252	−0.158074
$232$	0	0
$233$	29.3468	1.92257	0.961286	−	0.275554i	$-0.0888612\pi$
$233$	29.3468	1.92257	0.961286	+	0.275554i	$0.0888612\pi$
$234$	0	0
$235$	−11.0820	−0.722908
$236$	0	0
$237$	−0.462107	−0.0300171
$238$	0	0
$239$	−12.8584	−0.831741	−0.415871	−	0.909424i	$-0.636523\pi$
$239$	−12.8584	−0.831741	−0.415871	+	0.909424i	$0.636523\pi$
$240$	0	0
$241$	22.6215	1.45718	0.728589	−	0.684951i	$-0.240176\pi$
$241$	22.6215	1.45718	0.728589	+	0.684951i	$0.240176\pi$
$242$	0	0
$243$	−15.1887	−0.974357
$244$	0	0
$245$	21.4081	1.36771
$246$	0	0
$247$	3.11805	0.198397
$248$	0	0
$249$	3.48397	0.220788
$250$	0	0
$251$	−25.3315	−1.59891	−0.799454	−	0.600728i	$-0.794878\pi$
$251$	−25.3315	−1.59891	−0.799454	+	0.600728i	$0.794878\pi$
$252$	0	0
$253$	5.45574	0.342999
$254$	0	0
$255$	3.50203	0.219306
$256$	0	0
$257$	3.72859	0.232583	0.116291	−	0.993215i	$-0.462899\pi$
$257$	3.72859	0.232583	0.116291	+	0.993215i	$0.462899\pi$
$258$	0	0
$259$	1.02690	0.0638084
$260$	0	0
$261$	−17.6575	−1.09297
$262$	0	0
$263$	8.97041	0.553139	0.276570	−	0.960994i	$-0.410802\pi$
$263$	8.97041	0.553139	0.276570	+	0.960994i	$0.410802\pi$
$264$	0	0
$265$	31.4748	1.93348
$266$	0	0
$267$	−3.16187	−0.193503
$268$	0	0
$269$	2.98487	0.181991	0.0909954	−	0.995851i	$-0.470995\pi$
$269$	2.98487	0.181991	0.0909954	+	0.995851i	$0.470995\pi$
$270$	0	0
$271$	−3.84212	−0.233392	−0.116696	−	0.993168i	$-0.537230\pi$
$271$	−3.84212	−0.233392	−0.116696	+	0.993168i	$0.537230\pi$
$272$	0	0
$273$	−2.32446	−0.140683
$274$	0	0
$275$	24.5988	1.48336
$276$	0	0
$277$	−7.68887	−0.461980	−0.230990	−	0.972956i	$-0.574196\pi$
$277$	−7.68887	−0.461980	−0.230990	+	0.972956i	$0.574196\pi$
$278$	0	0
$279$	−8.17359	−0.489340
$280$	0	0
$281$	23.5269	1.40350	0.701750	−	0.712424i	$-0.252403\pi$
$281$	23.5269	1.40350	0.701750	+	0.712424i	$0.252403\pi$
$282$	0	0
$283$	5.68512	0.337945	0.168973	−	0.985621i	$-0.445955\pi$
$283$	5.68512	0.337945	0.168973	+	0.985621i	$0.445955\pi$
$284$	0	0
$285$	2.22776	0.131961
$286$	0	0
$287$	−3.60517	−0.212806
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.6679	0.742608
$292$	0	0
$293$	−9.31022	−0.543909	−0.271954	−	0.962310i	$-0.587670\pi$
$293$	−9.31022	−0.543909	−0.271954	+	0.962310i	$0.587670\pi$
$294$	0	0
$295$	−3.13935	−0.182780
$296$	0	0
$297$	26.8759	1.55950
$298$	0	0
$299$	5.27850	0.305263
$300$	0	0
$301$	2.22813	0.128427
$302$	0	0
$303$	−14.5007	−0.833045
$304$	0	0
$305$	43.7438	2.50476
$306$	0	0
$307$	−25.7142	−1.46759	−0.733793	−	0.679373i	$-0.762252\pi$
$307$	−25.7142	−1.46759	−0.733793	+	0.679373i	$0.762252\pi$
$308$	0	0
$309$	2.40121	0.136600
$310$	0	0
$311$	20.7956	1.17921	0.589605	−	0.807691i	$-0.299283\pi$
$311$	20.7956	1.17921	0.589605	+	0.807691i	$0.299283\pi$
$312$	0	0
$313$	24.2703	1.37184	0.685920	−	0.727677i	$-0.259400\pi$
$313$	24.2703	1.37184	0.685920	+	0.727677i	$0.259400\pi$
$314$	0	0
$315$	2.34301	0.132014
$316$	0	0
$317$	−3.80304	−0.213600	−0.106800	−	0.994281i	$-0.534060\pi$
$317$	−3.80304	−0.213600	−0.106800	+	0.994281i	$0.534060\pi$
$318$	0	0
$319$	50.9543	2.85290
$320$	0	0
$321$	11.8524	0.661537
$322$	0	0
$323$	0.636135	0.0353955
$324$	0	0
$325$	23.7996	1.32017
$326$	0	0
$327$	0.973325	0.0538250
$328$	0	0
$329$	1.50067	0.0827347
$330$	0	0
$331$	27.4088	1.50652	0.753262	−	0.657721i	$-0.228479\pi$
$331$	27.4088	1.50652	0.753262	+	0.657721i	$0.228479\pi$
$332$	0	0
$333$	−4.24077	−0.232393
$334$	0	0
$335$	−36.5939	−1.99934
$336$	0	0
$337$	−8.24476	−0.449121	−0.224560	−	0.974460i	$-0.572095\pi$
$337$	−8.24476	−0.449121	−0.224560	+	0.974460i	$0.572095\pi$
$338$	0	0
$339$	−5.75400	−0.312514
$340$	0	0
$341$	23.5866	1.27729
$342$	0	0
$343$	−5.87481	−0.317210
$344$	0	0
$345$	3.77134	0.203042
$346$	0	0
$347$	−7.91764	−0.425041	−0.212521	−	0.977157i	$-0.568167\pi$
$347$	−7.91764	−0.425041	−0.212521	+	0.977157i	$0.568167\pi$
$348$	0	0
$349$	2.16334	0.115801	0.0579005	−	0.998322i	$-0.481559\pi$
$349$	2.16334	0.115801	0.0579005	+	0.998322i	$0.481559\pi$
$350$	0	0
$351$	26.0028	1.38793
$352$	0	0
$353$	23.6408	1.25827	0.629137	−	0.777295i	$-0.283409\pi$
$353$	23.6408	1.25827	0.629137	+	0.777295i	$0.283409\pi$
$354$	0	0
$355$	47.6322	2.52805
$356$	0	0
$357$	−0.474229	−0.0250989
$358$	0	0
$359$	3.28310	0.173275	0.0866377	−	0.996240i	$-0.472388\pi$
$359$	3.28310	0.173275	0.0866377	+	0.996240i	$0.472388\pi$
$360$	0	0
$361$	−18.5953	−0.978702
$362$	0	0
$363$	−16.3602	−0.858686
$364$	0	0
$365$	−40.4206	−2.11571
$366$	0	0
$367$	13.1685	0.687388	0.343694	−	0.939082i	$-0.388322\pi$
$367$	13.1685	0.687388	0.343694	+	0.939082i	$0.388322\pi$
$368$	0	0
$369$	14.8882	0.775050
$370$	0	0
$371$	−4.26218	−0.221281
$372$	0	0
$373$	22.0980	1.14419	0.572095	−	0.820187i	$-0.306131\pi$
$373$	22.0980	1.14419	0.572095	+	0.820187i	$0.306131\pi$
$374$	0	0
$375$	−0.505971	−0.0261282
$376$	0	0
$377$	49.2989	2.53903
$378$	0	0
$379$	−14.2907	−0.734066	−0.367033	−	0.930208i	$-0.619626\pi$
$379$	−14.2907	−0.734066	−0.367033	+	0.930208i	$0.619626\pi$
$380$	0	0
$381$	−21.3065	−1.09157
$382$	0	0
$383$	−24.3087	−1.24211	−0.621057	−	0.783765i	$-0.713297\pi$
$383$	−24.3087	−1.24211	−0.621057	+	0.783765i	$0.713297\pi$
$384$	0	0
$385$	−6.76124	−0.344585
$386$	0	0
$387$	−9.20146	−0.467737
$388$	0	0
$389$	−19.2015	−0.973553	−0.486776	−	0.873527i	$-0.661827\pi$
$389$	−19.2015	−0.973553	−0.486776	+	0.873527i	$0.661827\pi$
$390$	0	0
$391$	1.07690	0.0544612
$392$	0	0
$393$	−12.7754	−0.644435
$394$	0	0
$395$	−1.30048	−0.0654340
$396$	0	0
$397$	4.04559	0.203042	0.101521	−	0.994833i	$-0.467629\pi$
$397$	4.04559	0.203042	0.101521	+	0.994833i	$0.467629\pi$
$398$	0	0
$399$	−0.301674	−0.0151026
$400$	0	0
$401$	18.4028	0.918992	0.459496	−	0.888180i	$-0.348030\pi$
$401$	18.4028	0.918992	0.459496	+	0.888180i	$0.348030\pi$
$402$	0	0
$403$	22.8203	1.13676
$404$	0	0
$405$	2.04392	0.101563
$406$	0	0
$407$	12.2376	0.606596
$408$	0	0
$409$	6.17836	0.305500	0.152750	−	0.988265i	$-0.451187\pi$
$409$	6.17836	0.305500	0.152750	+	0.988265i	$0.451187\pi$
$410$	0	0
$411$	−3.96634	−0.195645
$412$	0	0
$413$	0.425117	0.0209186
$414$	0	0
$415$	9.80470	0.481294
$416$	0	0
$417$	−15.9386	−0.780515
$418$	0	0
$419$	27.4976	1.34335	0.671673	−	0.740848i	$-0.265576\pi$
$419$	27.4976	1.34335	0.671673	+	0.740848i	$0.265576\pi$
$420$	0	0
$421$	33.4208	1.62883	0.814414	−	0.580284i	$-0.197058\pi$
$421$	33.4208	1.62883	0.814414	+	0.580284i	$0.197058\pi$
$422$	0	0
$423$	−6.19730	−0.301323
$424$	0	0
$425$	4.85552	0.235527
$426$	0	0
$427$	−5.92359	−0.286663
$428$	0	0
$429$	−27.7008	−1.33741
$430$	0	0
$431$	−29.7830	−1.43460	−0.717299	−	0.696766i	$-0.754622\pi$
$431$	−29.7830	−1.43460	−0.717299	+	0.696766i	$0.754622\pi$
$432$	0	0
$433$	18.1622	0.872821	0.436411	−	0.899748i	$-0.356249\pi$
$433$	18.1622	0.872821	0.436411	+	0.899748i	$0.356249\pi$
$434$	0	0
$435$	35.2227	1.68880
$436$	0	0
$437$	0.685054	0.0327706
$438$	0	0
$439$	−10.9728	−0.523701	−0.261851	−	0.965108i	$-0.584333\pi$
$439$	−10.9728	−0.523701	−0.261851	+	0.965108i	$0.584333\pi$
$440$	0	0
$441$	11.9719	0.570092
$442$	0	0
$443$	21.7437	1.03307	0.516537	−	0.856265i	$-0.327221\pi$
$443$	21.7437	1.03307	0.516537	+	0.856265i	$0.327221\pi$
$444$	0	0
$445$	−8.89824	−0.421817
$446$	0	0
$447$	23.9125	1.13102
$448$	0	0
$449$	22.3117	1.05295	0.526477	−	0.850189i	$-0.323513\pi$
$449$	22.3117	1.05295	0.526477	+	0.850189i	$0.323513\pi$
$450$	0	0
$451$	−42.9630	−2.02305
$452$	0	0
$453$	−8.84084	−0.415379
$454$	0	0
$455$	−6.54158	−0.306674
$456$	0	0
$457$	15.9251	0.744943	0.372472	−	0.928044i	$-0.378510\pi$
$457$	15.9251	0.744943	0.372472	+	0.928044i	$0.378510\pi$
$458$	0	0
$459$	5.30500	0.247616
$460$	0	0
$461$	27.7015	1.29019	0.645094	−	0.764103i	$-0.276818\pi$
$461$	27.7015	1.29019	0.645094	+	0.764103i	$0.276818\pi$
$462$	0	0
$463$	−25.6341	−1.19132	−0.595659	−	0.803237i	$-0.703109\pi$
$463$	−25.6341	−1.19132	−0.595659	+	0.803237i	$0.703109\pi$
$464$	0	0
$465$	16.3045	0.756102
$466$	0	0
$467$	7.35356	0.340282	0.170141	−	0.985420i	$-0.445578\pi$
$467$	7.35356	0.340282	0.170141	+	0.985420i	$0.445578\pi$
$468$	0	0
$469$	4.95538	0.228818
$470$	0	0
$471$	−12.3938	−0.571078
$472$	0	0
$473$	26.5527	1.22090
$474$	0	0
$475$	3.08877	0.141722
$476$	0	0
$477$	17.6015	0.805915
$478$	0	0
$479$	−18.9850	−0.867445	−0.433723	−	0.901046i	$-0.642800\pi$
$479$	−18.9850	−0.867445	−0.433723	+	0.901046i	$0.642800\pi$
$480$	0	0
$481$	11.8400	0.539860
$482$	0	0
$483$	−0.510698	−0.0232376
$484$	0	0
$485$	35.6505	1.61881
$486$	0	0
$487$	10.1543	0.460137	0.230069	−	0.973174i	$-0.426105\pi$
$487$	10.1543	0.460137	0.230069	+	0.973174i	$0.426105\pi$
$488$	0	0
$489$	25.8048	1.16694
$490$	0	0
$491$	−28.4757	−1.28509	−0.642546	−	0.766247i	$-0.722122\pi$
$491$	−28.4757	−1.28509	−0.642546	+	0.766247i	$0.722122\pi$
$492$	0	0
$493$	10.0578	0.452981
$494$	0	0
$495$	27.9218	1.25499
$496$	0	0
$497$	−6.45014	−0.289328
$498$	0	0
$499$	−8.48323	−0.379761	−0.189881	−	0.981807i	$-0.560810\pi$
$499$	−8.48323	−0.379761	−0.189881	+	0.981807i	$0.560810\pi$
$500$	0	0
$501$	3.43931	0.153657
$502$	0	0
$503$	−24.0217	−1.07108	−0.535538	−	0.844511i	$-0.679891\pi$
$503$	−24.0217	−1.07108	−0.535538	+	0.844511i	$0.679891\pi$
$504$	0	0
$505$	−40.8084	−1.81595
$506$	0	0
$507$	−12.2990	−0.546219
$508$	0	0
$509$	28.6821	1.27131	0.635656	−	0.771973i	$-0.280730\pi$
$509$	28.6821	1.27131	0.635656	+	0.771973i	$0.280730\pi$
$510$	0	0
$511$	5.47358	0.242137
$512$	0	0
$513$	3.37470	0.148996
$514$	0	0
$515$	6.75755	0.297773
$516$	0	0
$517$	17.8836	0.786520
$518$	0	0
$519$	−22.0614	−0.968387
$520$	0	0
$521$	−1.04350	−0.0457165	−0.0228582	−	0.999739i	$-0.507277\pi$
$521$	−1.04350	−0.0457165	−0.0228582	+	0.999739i	$0.507277\pi$
$522$	0	0
$523$	18.6488	0.815455	0.407727	−	0.913104i	$-0.366321\pi$
$523$	18.6488	0.815455	0.407727	+	0.913104i	$0.366321\pi$
$524$	0	0
$525$	−2.30263	−0.100495
$526$	0	0
$527$	4.65573	0.202807
$528$	0	0
$529$	−21.8403	−0.949578
$530$	0	0
$531$	−1.75560	−0.0761866
$532$	0	0
$533$	−41.5673	−1.80048
$534$	0	0
$535$	33.3554	1.44208
$536$	0	0
$537$	14.3016	0.617161
$538$	0	0
$539$	−34.5475	−1.48806
$540$	0	0
$541$	10.5867	0.455158	0.227579	−	0.973760i	$-0.426919\pi$
$541$	10.5867	0.455158	0.227579	+	0.973760i	$0.426919\pi$
$542$	0	0
$543$	−2.05357	−0.0881273
$544$	0	0
$545$	2.73916	0.117333
$546$	0	0
$547$	26.0323	1.11306	0.556529	−	0.830828i	$-0.312133\pi$
$547$	26.0323	1.11306	0.556529	+	0.830828i	$0.312133\pi$
$548$	0	0
$549$	24.4626	1.04404
$550$	0	0
$551$	6.39812	0.272569
$552$	0	0
$553$	0.176105	0.00748873
$554$	0	0
$555$	8.45938	0.359081
$556$	0	0
$557$	−29.8512	−1.26484	−0.632419	−	0.774627i	$-0.717938\pi$
$557$	−29.8512	−1.26484	−0.632419	+	0.774627i	$0.717938\pi$
$558$	0	0
$559$	25.6901	1.08658
$560$	0	0
$561$	−5.65142	−0.238603
$562$	0	0
$563$	−14.2449	−0.600352	−0.300176	−	0.953884i	$-0.597045\pi$
$563$	−14.2449	−0.600352	−0.300176	+	0.953884i	$0.597045\pi$
$564$	0	0
$565$	−16.1931	−0.681248
$566$	0	0
$567$	−0.276779	−0.0116236
$568$	0	0
$569$	−46.0889	−1.93215	−0.966073	−	0.258271i	$-0.916847\pi$
$569$	−46.0889	−1.93215	−0.966073	+	0.258271i	$0.916847\pi$
$570$	0	0
$571$	1.35118	0.0565450	0.0282725	−	0.999600i	$-0.490999\pi$
$571$	1.35118	0.0565450	0.0282725	+	0.999600i	$0.490999\pi$
$572$	0	0
$573$	−7.22774	−0.301943
$574$	0	0
$575$	5.22891	0.218061
$576$	0	0
$577$	31.0805	1.29390	0.646948	−	0.762534i	$-0.276045\pi$
$577$	31.0805	1.29390	0.646948	+	0.762534i	$0.276045\pi$
$578$	0	0
$579$	22.8919	0.951356
$580$	0	0
$581$	−1.32771	−0.0550827
$582$	0	0
$583$	−50.7927	−2.10362
$584$	0	0
$585$	27.0147	1.11692
$586$	0	0
$587$	−47.8733	−1.97594	−0.987971	−	0.154642i	$-0.950578\pi$
$587$	−47.8733	−1.97594	−0.987971	+	0.154642i	$0.950578\pi$
$588$	0	0
$589$	2.96167	0.122033
$590$	0	0
$591$	8.78628	0.361419
$592$	0	0
$593$	−17.0427	−0.699859	−0.349929	−	0.936776i	$-0.613794\pi$
$593$	−17.0427	−0.699859	−0.349929	+	0.936776i	$0.613794\pi$
$594$	0	0
$595$	−1.33459	−0.0547129
$596$	0	0
$597$	−23.3385	−0.955181
$598$	0	0
$599$	−17.0441	−0.696403	−0.348201	−	0.937420i	$-0.613207\pi$
$599$	−17.0441	−0.696403	−0.348201	+	0.937420i	$0.613207\pi$
$600$	0	0
$601$	24.2337	0.988514	0.494257	−	0.869316i	$-0.335440\pi$
$601$	24.2337	0.988514	0.494257	+	0.869316i	$0.335440\pi$
$602$	0	0
$603$	−20.4642	−0.833366
$604$	0	0
$605$	−46.0413	−1.87184
$606$	0	0
$607$	8.15079	0.330830	0.165415	−	0.986224i	$-0.447104\pi$
$607$	8.15079	0.330830	0.165415	+	0.986224i	$0.447104\pi$
$608$	0	0
$609$	−4.76971	−0.193278
$610$	0	0
$611$	17.3026	0.699988
$612$	0	0
$613$	5.83164	0.235538	0.117769	−	0.993041i	$-0.462426\pi$
$613$	5.83164	0.235538	0.117769	+	0.993041i	$0.462426\pi$
$614$	0	0
$615$	−29.6987	−1.19757
$616$	0	0
$617$	−11.9579	−0.481406	−0.240703	−	0.970599i	$-0.577378\pi$
$617$	−11.9579	−0.481406	−0.240703	+	0.970599i	$0.577378\pi$
$618$	0	0
$619$	−14.1823	−0.570034	−0.285017	−	0.958522i	$-0.591999\pi$
$619$	−14.1823	−0.570034	−0.285017	+	0.958522i	$0.591999\pi$
$620$	0	0
$621$	5.71296	0.229253
$622$	0	0
$623$	1.20496	0.0482757
$624$	0	0
$625$	−25.7015	−1.02806
$626$	0	0
$627$	−3.59507	−0.143573
$628$	0	0
$629$	2.41557	0.0963149
$630$	0	0
$631$	9.32799	0.371341	0.185671	−	0.982612i	$-0.440554\pi$
$631$	9.32799	0.371341	0.185671	+	0.982612i	$0.440554\pi$
$632$	0	0
$633$	4.13416	0.164318
$634$	0	0
$635$	−59.9615	−2.37950
$636$	0	0
$637$	−33.4251	−1.32435
$638$	0	0
$639$	26.6371	1.05375
$640$	0	0
$641$	33.4419	1.32088	0.660438	−	0.750881i	$-0.270371\pi$
$641$	33.4419	1.32088	0.660438	+	0.750881i	$0.270371\pi$
$642$	0	0
$643$	25.6913	1.01316	0.506582	−	0.862192i	$-0.330909\pi$
$643$	25.6913	1.01316	0.506582	+	0.862192i	$0.330909\pi$
$644$	0	0
$645$	18.3549	0.722722
$646$	0	0
$647$	19.2653	0.757396	0.378698	−	0.925520i	$-0.376372\pi$
$647$	19.2653	0.757396	0.378698	+	0.925520i	$0.376372\pi$
$648$	0	0
$649$	5.06615	0.198864
$650$	0	0
$651$	−2.20788	−0.0865337
$652$	0	0
$653$	−16.2853	−0.637291	−0.318646	−	0.947874i	$-0.603228\pi$
$653$	−16.2853	−0.637291	−0.318646	+	0.947874i	$0.603228\pi$
$654$	0	0
$655$	−35.9530	−1.40480
$656$	0	0
$657$	−22.6042	−0.881873
$658$	0	0
$659$	17.6709	0.688362	0.344181	−	0.938903i	$-0.388157\pi$
$659$	17.6709	0.688362	0.344181	+	0.938903i	$0.388157\pi$
$660$	0	0
$661$	−4.55417	−0.177137	−0.0885684	−	0.996070i	$-0.528229\pi$
$661$	−4.55417	−0.177137	−0.0885684	+	0.996070i	$0.528229\pi$
$662$	0	0
$663$	−5.46782	−0.212353
$664$	0	0
$665$	−0.848980	−0.0329220
$666$	0	0
$667$	10.8313	0.419388
$668$	0	0
$669$	0.798049	0.0308544
$670$	0	0
$671$	−70.5918	−2.72517
$672$	0	0
$673$	37.7130	1.45373	0.726864	−	0.686781i	$-0.240977\pi$
$673$	37.7130	1.45373	0.726864	+	0.686781i	$0.240977\pi$
$674$	0	0
$675$	25.7585	0.991446
$676$	0	0
$677$	−11.2833	−0.433651	−0.216826	−	0.976210i	$-0.569570\pi$
$677$	−11.2833	−0.433651	−0.216826	+	0.976210i	$0.569570\pi$
$678$	0	0
$679$	−4.82764	−0.185268
$680$	0	0
$681$	−1.38108	−0.0529231
$682$	0	0
$683$	−1.49232	−0.0571022	−0.0285511	−	0.999592i	$-0.509089\pi$
$683$	−1.49232	−0.0571022	−0.0285511	+	0.999592i	$0.509089\pi$
$684$	0	0
$685$	−11.1622	−0.426485
$686$	0	0
$687$	28.3976	1.08344
$688$	0	0
$689$	−49.1425	−1.87218
$690$	0	0
$691$	−5.31723	−0.202277	−0.101138	−	0.994872i	$-0.532249\pi$
$691$	−5.31723	−0.202277	−0.101138	+	0.994872i	$0.532249\pi$
$692$	0	0
$693$	−3.78105	−0.143630
$694$	0	0
$695$	−44.8548	−1.70144
$696$	0	0
$697$	−8.48041	−0.321219
$698$	0	0
$699$	−32.7371	−1.23823
$700$	0	0
$701$	37.5592	1.41859	0.709297	−	0.704910i	$-0.249013\pi$
$701$	37.5592	1.41859	0.709297	+	0.704910i	$0.249013\pi$
$702$	0	0
$703$	1.53663	0.0579549
$704$	0	0
$705$	12.3622	0.465588
$706$	0	0
$707$	5.52610	0.207830
$708$	0	0
$709$	11.8393	0.444632	0.222316	−	0.974975i	$-0.428638\pi$
$709$	11.8393	0.444632	0.222316	+	0.974975i	$0.428638\pi$
$710$	0	0
$711$	−0.727257	−0.0272743
$712$	0	0
$713$	5.01376	0.187767
$714$	0	0
$715$	−77.9565	−2.91541
$716$	0	0
$717$	14.3439	0.535683
$718$	0	0
$719$	1.82392	0.0680209	0.0340104	−	0.999421i	$-0.489172\pi$
$719$	1.82392	0.0680209	0.0340104	+	0.999421i	$0.489172\pi$
$720$	0	0
$721$	−0.915077	−0.0340793
$722$	0	0
$723$	−25.2349	−0.938495
$724$	0	0
$725$	48.8359	1.81372
$726$	0	0
$727$	−28.8361	−1.06947	−0.534736	−	0.845019i	$-0.679589\pi$
$727$	−28.8361	−1.06947	−0.534736	+	0.845019i	$0.679589\pi$
$728$	0	0
$729$	18.8966	0.699875
$730$	0	0
$731$	5.24121	0.193853
$732$	0	0
$733$	−37.9768	−1.40270	−0.701352	−	0.712815i	$-0.747420\pi$
$733$	−37.9768	−1.40270	−0.701352	+	0.712815i	$0.747420\pi$
$734$	0	0
$735$	−23.8813	−0.880875
$736$	0	0
$737$	59.0536	2.17527
$738$	0	0
$739$	5.52655	0.203298	0.101649	−	0.994820i	$-0.467588\pi$
$739$	5.52655	0.203298	0.101649	+	0.994820i	$0.467588\pi$
$740$	0	0
$741$	−3.47827	−0.127778
$742$	0	0
$743$	−21.1147	−0.774625	−0.387312	−	0.921949i	$-0.626597\pi$
$743$	−21.1147	−0.774625	−0.387312	+	0.921949i	$0.626597\pi$
$744$	0	0
$745$	67.2953	2.46551
$746$	0	0
$747$	5.48302	0.200613
$748$	0	0
$749$	−4.51684	−0.165042
$750$	0	0
$751$	−2.49272	−0.0909608	−0.0454804	−	0.998965i	$-0.514482\pi$
$751$	−2.49272	−0.0909608	−0.0454804	+	0.998965i	$0.514482\pi$
$752$	0	0
$753$	28.2579	1.02978
$754$	0	0
$755$	−24.8802	−0.905482
$756$	0	0
$757$	−13.6761	−0.497067	−0.248533	−	0.968623i	$-0.579949\pi$
$757$	−13.6761	−0.497067	−0.248533	+	0.968623i	$0.579949\pi$
$758$	0	0
$759$	−6.08602	−0.220909
$760$	0	0
$761$	24.6132	0.892226	0.446113	−	0.894977i	$-0.352808\pi$
$761$	24.6132	0.892226	0.446113	+	0.894977i	$0.352808\pi$
$762$	0	0
$763$	−0.370925	−0.0134284
$764$	0	0
$765$	5.51144	0.199267
$766$	0	0
$767$	4.90156	0.176985
$768$	0	0
$769$	16.2758	0.586921	0.293460	−	0.955971i	$-0.405193\pi$
$769$	16.2758	0.586921	0.293460	+	0.955971i	$0.405193\pi$
$770$	0	0
$771$	−4.15934	−0.149795
$772$	0	0
$773$	−40.2626	−1.44815	−0.724073	−	0.689724i	$-0.757732\pi$
$773$	−40.2626	−1.44815	−0.724073	+	0.689724i	$0.757732\pi$
$774$	0	0
$775$	22.6060	0.812031
$776$	0	0
$777$	−1.14553	−0.0410957
$778$	0	0
$779$	−5.39469	−0.193285
$780$	0	0
$781$	−76.8667	−2.75051
$782$	0	0
$783$	53.3566	1.90681
$784$	0	0
$785$	−34.8791	−1.24489
$786$	0	0
$787$	48.9973	1.74656	0.873282	−	0.487216i	$-0.161987\pi$
$787$	48.9973	1.74656	0.873282	+	0.487216i	$0.161987\pi$
$788$	0	0
$789$	−10.0067	−0.356249
$790$	0	0
$791$	2.19280	0.0779668
$792$	0	0
$793$	−68.2985	−2.42535
$794$	0	0
$795$	−35.1110	−1.24526
$796$	0	0
$797$	−20.6650	−0.731993	−0.365996	−	0.930616i	$-0.619272\pi$
$797$	−20.6650	−0.731993	−0.365996	+	0.930616i	$0.619272\pi$
$798$	0	0
$799$	3.53002	0.124883
$800$	0	0
$801$	−4.97611	−0.175822
$802$	0	0
$803$	65.2291	2.30188
$804$	0	0
$805$	−1.43722	−0.0506554
$806$	0	0
$807$	−3.32970	−0.117211
$808$	0	0
$809$	−7.69632	−0.270588	−0.135294	−	0.990805i	$-0.543198\pi$
$809$	−7.69632	−0.270588	−0.135294	+	0.990805i	$0.543198\pi$
$810$	0	0
$811$	39.2933	1.37977	0.689887	−	0.723917i	$-0.257660\pi$
$811$	39.2933	1.37977	0.689887	+	0.723917i	$0.257660\pi$
$812$	0	0
$813$	4.28599	0.150316
$814$	0	0
$815$	72.6208	2.54380
$816$	0	0
$817$	3.33411	0.116646
$818$	0	0
$819$	−3.65821	−0.127828
$820$	0	0
$821$	44.3697	1.54851	0.774257	−	0.632872i	$-0.218124\pi$
$821$	44.3697	1.54851	0.774257	+	0.632872i	$0.218124\pi$
$822$	0	0
$823$	−18.2308	−0.635487	−0.317743	−	0.948177i	$-0.602925\pi$
$823$	−18.2308	−0.635487	−0.317743	+	0.948177i	$0.602925\pi$
$824$	0	0
$825$	−27.4406	−0.955359
$826$	0	0
$827$	37.5575	1.30600	0.653001	−	0.757357i	$-0.273510\pi$
$827$	37.5575	1.30600	0.653001	+	0.757357i	$0.273510\pi$
$828$	0	0
$829$	30.4760	1.05848	0.529238	−	0.848474i	$-0.322478\pi$
$829$	30.4760	1.05848	0.529238	+	0.848474i	$0.322478\pi$
$830$	0	0
$831$	8.57714	0.297538
$832$	0	0
$833$	−6.81928	−0.236274
$834$	0	0
$835$	9.67902	0.334956
$836$	0	0
$837$	24.6986	0.853709
$838$	0	0
$839$	29.6649	1.02415	0.512073	−	0.858942i	$-0.328878\pi$
$839$	29.6649	1.02415	0.512073	+	0.858942i	$0.328878\pi$
$840$	0	0
$841$	72.1594	2.48826
$842$	0	0
$843$	−26.2449	−0.903923
$844$	0	0
$845$	−34.6123	−1.19070
$846$	0	0
$847$	6.23471	0.214227
$848$	0	0
$849$	−6.34190	−0.217653
$850$	0	0
$851$	2.60133	0.0891723
$852$	0	0
$853$	18.5277	0.634376	0.317188	−	0.948363i	$-0.397261\pi$
$853$	18.5277	0.634376	0.317188	+	0.948363i	$0.397261\pi$
$854$	0	0
$855$	3.50602	0.119903
$856$	0	0
$857$	40.2965	1.37650	0.688252	−	0.725472i	$-0.258378\pi$
$857$	40.2965	1.37650	0.688252	+	0.725472i	$0.258378\pi$
$858$	0	0
$859$	47.8778	1.63357	0.816785	−	0.576942i	$-0.195754\pi$
$859$	47.8778	1.63357	0.816785	+	0.576942i	$0.195754\pi$
$860$	0	0
$861$	4.02166	0.137058
$862$	0	0
$863$	−36.4964	−1.24235	−0.621176	−	0.783671i	$-0.713345\pi$
$863$	−36.4964	−1.24235	−0.621176	+	0.783671i	$0.713345\pi$
$864$	0	0
$865$	−62.0859	−2.11098
$866$	0	0
$867$	−1.11553	−0.0378853
$868$	0	0
$869$	2.09865	0.0711919
$870$	0	0
$871$	57.1351	1.93595
$872$	0	0
$873$	19.9366	0.674752
$874$	0	0
$875$	0.192821	0.00651853
$876$	0	0
$877$	29.3330	0.990504	0.495252	−	0.868749i	$-0.335076\pi$
$877$	29.3330	0.990504	0.495252	+	0.868749i	$0.335076\pi$
$878$	0	0
$879$	10.3858	0.350304
$880$	0	0
$881$	4.91080	0.165449	0.0827245	−	0.996572i	$-0.473638\pi$
$881$	4.91080	0.165449	0.0827245	+	0.996572i	$0.473638\pi$
$882$	0	0
$883$	−46.3875	−1.56106	−0.780531	−	0.625117i	$-0.785051\pi$
$883$	−46.3875	−1.56106	−0.780531	+	0.625117i	$0.785051\pi$
$884$	0	0
$885$	3.50203	0.117719
$886$	0	0
$887$	51.5111	1.72957	0.864787	−	0.502140i	$-0.167454\pi$
$887$	51.5111	1.72957	0.864787	+	0.502140i	$0.167454\pi$
$888$	0	0
$889$	8.11972	0.272327
$890$	0	0
$891$	−3.29840	−0.110500
$892$	0	0
$893$	2.24557	0.0751451
$894$	0	0
$895$	40.2481	1.34535
$896$	0	0
$897$	−5.88830	−0.196605
$898$	0	0
$899$	46.8264	1.56175
$900$	0	0
$901$	−10.0259	−0.334011
$902$	0	0
$903$	−2.48553	−0.0827134
$904$	0	0
$905$	−5.77923	−0.192108
$906$	0	0
$907$	48.3761	1.60630	0.803151	−	0.595776i	$-0.203155\pi$
$907$	48.3761	1.60630	0.803151	+	0.595776i	$0.203155\pi$
$908$	0	0
$909$	−22.8211	−0.756926
$910$	0	0
$911$	−18.0203	−0.597040	−0.298520	−	0.954403i	$-0.596493\pi$
$911$	−18.0203	−0.597040	−0.298520	+	0.954403i	$0.596493\pi$
$912$	0	0
$913$	−15.8224	−0.523645
$914$	0	0
$915$	−48.7974	−1.61319
$916$	0	0
$917$	4.86860	0.160775
$918$	0	0
$919$	8.72495	0.287810	0.143905	−	0.989592i	$-0.454034\pi$
$919$	8.72495	0.287810	0.143905	+	0.989592i	$0.454034\pi$
$920$	0	0
$921$	28.6849	0.945198
$922$	0	0
$923$	−74.3695	−2.44790
$924$	0	0
$925$	11.7288	0.385642
$926$	0	0
$927$	3.77898	0.124118
$928$	0	0
$929$	36.3117	1.19135	0.595674	−	0.803226i	$-0.296885\pi$
$929$	36.3117	1.19135	0.595674	+	0.803226i	$0.296885\pi$
$930$	0	0
$931$	−4.33798	−0.142172
$932$	0	0
$933$	−23.1981	−0.759470
$934$	0	0
$935$	−15.9044	−0.520130
$936$	0	0
$937$	−20.0613	−0.655376	−0.327688	−	0.944786i	$-0.606269\pi$
$937$	−20.0613	−0.655376	−0.327688	+	0.944786i	$0.606269\pi$
$938$	0	0
$939$	−27.0742	−0.883534
$940$	0	0
$941$	55.7135	1.81621	0.908104	−	0.418745i	$-0.137530\pi$
$941$	55.7135	1.81621	0.908104	+	0.418745i	$0.137530\pi$
$942$	0	0
$943$	−9.13257	−0.297397
$944$	0	0
$945$	−7.08001	−0.230313
$946$	0	0
$947$	−33.1926	−1.07861	−0.539307	−	0.842110i	$-0.681314\pi$
$947$	−33.1926	−1.07861	−0.539307	+	0.842110i	$0.681314\pi$
$948$	0	0
$949$	63.1099	2.04864
$950$	0	0
$951$	4.24239	0.137569
$952$	0	0
$953$	−50.9438	−1.65023	−0.825116	−	0.564963i	$-0.808890\pi$
$953$	−50.9438	−1.65023	−0.825116	+	0.564963i	$0.808890\pi$
$954$	0	0
$955$	−20.3405	−0.658204
$956$	0	0
$957$	−56.8409	−1.83741
$958$	0	0
$959$	1.51153	0.0488100
$960$	0	0
$961$	−9.32422	−0.300781
$962$	0	0
$963$	18.6532	0.601089
$964$	0	0
$965$	64.4232	2.07386
$966$	0	0
$967$	1.03306	0.0332209	0.0166105	−	0.999862i	$-0.494712\pi$
$967$	1.03306	0.0332209	0.0166105	+	0.999862i	$0.494712\pi$
$968$	0	0
$969$	−0.709625	−0.0227965
$970$	0	0
$971$	−48.7764	−1.56531	−0.782654	−	0.622457i	$-0.786135\pi$
$971$	−48.7764	−1.56531	−0.782654	+	0.622457i	$0.786135\pi$
$972$	0	0
$973$	6.07404	0.194725
$974$	0	0
$975$	−26.5491	−0.850252
$976$	0	0
$977$	−18.4778	−0.591156	−0.295578	−	0.955319i	$-0.595512\pi$
$977$	−18.4778	−0.591156	−0.295578	+	0.955319i	$0.595512\pi$
$978$	0	0
$979$	14.3596	0.458935
$980$	0	0
$981$	1.53180	0.0489068
$982$	0	0
$983$	−54.1404	−1.72681	−0.863405	−	0.504511i	$-0.831673\pi$
$983$	−54.1404	−1.72681	−0.863405	+	0.504511i	$0.831673\pi$
$984$	0	0
$985$	24.7266	0.787856
$986$	0	0
$987$	−1.67404	−0.0532852
$988$	0	0
$989$	5.64426	0.179477
$990$	0	0
$991$	−16.5106	−0.524477	−0.262239	−	0.965003i	$-0.584461\pi$
$991$	−16.5106	−0.524477	−0.262239	+	0.965003i	$0.584461\pi$
$992$	0	0
$993$	−30.5752	−0.970276
$994$	0	0
$995$	−65.6799	−2.08219
$996$	0	0
$997$	22.4667	0.711527	0.355764	−	0.934576i	$-0.384221\pi$
$997$	22.4667	0.711527	0.355764	+	0.934576i	$0.384221\pi$
$998$	0	0
$999$	12.8146	0.405435

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.12	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.12	✓	32	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)

\(f(q)\)	\(=\)	\(q-1.11553 q^{3} -3.13935 q^{5} +0.425117 q^{7} -1.75560 q^{9} +O(q^{10})\)	\(q-1.11553 q^{3} -3.13935 q^{5} +0.425117 q^{7} -1.75560 q^{9} +5.06615 q^{11} +4.90156 q^{13} +3.50203 q^{15} +1.00000 q^{17} +0.636135 q^{19} -0.474229 q^{21} +1.07690 q^{23} +4.85552 q^{25} +5.30500 q^{27} +10.0578 q^{29} +4.65573 q^{31} -5.65142 q^{33} -1.33459 q^{35} +2.41557 q^{37} -5.46782 q^{39} -8.48041 q^{41} +5.24121 q^{43} +5.51144 q^{45} +3.53002 q^{47} -6.81928 q^{49} -1.11553 q^{51} -10.0259 q^{53} -15.9044 q^{55} -0.709625 q^{57} +1.00000 q^{59} -13.9340 q^{61} -0.746336 q^{63} -15.3877 q^{65} +11.6565 q^{67} -1.20131 q^{69} -15.1726 q^{71} +12.8755 q^{73} -5.41646 q^{75} +2.15371 q^{77} +0.414250 q^{79} -0.651066 q^{81} -3.12316 q^{83} -3.13935 q^{85} -11.2197 q^{87} +2.83442 q^{89} +2.08374 q^{91} -5.19359 q^{93} -1.99705 q^{95} -11.3560 q^{97} -8.89413 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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