LMFDB - Embedded newform 8004.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.66067$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.66067 q^{5} +4.50596 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.66067 q^{5} +4.50596 q^{7} +1.00000 q^{9} +6.15106 q^{11} +3.84454 q^{13} -2.66067 q^{15} +4.22506 q^{17} -0.858306 q^{19} +4.50596 q^{21} -1.00000 q^{23} +2.07919 q^{25} +1.00000 q^{27} +1.00000 q^{29} +2.78961 q^{31} +6.15106 q^{33} -11.9889 q^{35} +9.37710 q^{37} +3.84454 q^{39} +3.70763 q^{41} +9.59676 q^{43} -2.66067 q^{45} +1.42386 q^{47} +13.3037 q^{49} +4.22506 q^{51} -10.3493 q^{53} -16.3660 q^{55} -0.858306 q^{57} -10.1044 q^{59} -9.18530 q^{61} +4.50596 q^{63} -10.2291 q^{65} +9.98020 q^{67} -1.00000 q^{69} -6.89722 q^{71} +9.25073 q^{73} +2.07919 q^{75} +27.7164 q^{77} -15.7331 q^{79} +1.00000 q^{81} -3.42091 q^{83} -11.2415 q^{85} +1.00000 q^{87} +10.7016 q^{89} +17.3233 q^{91} +2.78961 q^{93} +2.28367 q^{95} -16.1955 q^{97} +6.15106 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.66067	−1.18989	−0.594945	−	0.803767i	$-0.702826\pi$
$5$	−2.66067	−1.18989	−0.594945	+	0.803767i	$0.702826\pi$
$6$	0	0
$7$	4.50596	1.70309	0.851546	−	0.524279i	$-0.175665\pi$
$7$	4.50596	1.70309	0.851546	+	0.524279i	$0.175665\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.15106	1.85461	0.927307	−	0.374301i	$-0.122117\pi$
$11$	6.15106	1.85461	0.927307	+	0.374301i	$0.122117\pi$
$12$	0	0
$13$	3.84454	1.06628	0.533141	−	0.846026i	$-0.321011\pi$
$13$	3.84454	1.06628	0.533141	+	0.846026i	$0.321011\pi$
$14$	0	0
$15$	−2.66067	−0.686983
$16$	0	0
$17$	4.22506	1.02473	0.512363	−	0.858769i	$-0.328770\pi$
$17$	4.22506	1.02473	0.512363	+	0.858769i	$0.328770\pi$
$18$	0	0
$19$	−0.858306	−0.196909	−0.0984545	−	0.995142i	$-0.531390\pi$
$19$	−0.858306	−0.196909	−0.0984545	+	0.995142i	$0.531390\pi$
$20$	0	0
$21$	4.50596	0.983281
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.07919	0.415837
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	2.78961	0.501029	0.250514	−	0.968113i	$-0.419400\pi$
$31$	2.78961	0.501029	0.250514	+	0.968113i	$0.419400\pi$
$32$	0	0
$33$	6.15106	1.07076
$34$	0	0
$35$	−11.9889	−2.02649
$36$	0	0
$37$	9.37710	1.54159	0.770793	−	0.637085i	$-0.219860\pi$
$37$	9.37710	1.54159	0.770793	+	0.637085i	$0.219860\pi$
$38$	0	0
$39$	3.84454	0.615618
$40$	0	0
$41$	3.70763	0.579034	0.289517	−	0.957173i	$-0.406505\pi$
$41$	3.70763	0.579034	0.289517	+	0.957173i	$0.406505\pi$
$42$	0	0
$43$	9.59676	1.46349	0.731746	−	0.681578i	$-0.238706\pi$
$43$	9.59676	1.46349	0.731746	+	0.681578i	$0.238706\pi$
$44$	0	0
$45$	−2.66067	−0.396630
$46$	0	0
$47$	1.42386	0.207691	0.103846	−	0.994593i	$-0.466885\pi$
$47$	1.42386	0.207691	0.103846	+	0.994593i	$0.466885\pi$
$48$	0	0
$49$	13.3037	1.90052
$50$	0	0
$51$	4.22506	0.591626
$52$	0	0
$53$	−10.3493	−1.42158	−0.710790	−	0.703405i	$-0.751662\pi$
$53$	−10.3493	−1.42158	−0.710790	+	0.703405i	$0.751662\pi$
$54$	0	0
$55$	−16.3660	−2.20679
$56$	0	0
$57$	−0.858306	−0.113685
$58$	0	0
$59$	−10.1044	−1.31547	−0.657737	−	0.753247i	$-0.728486\pi$
$59$	−10.1044	−1.31547	−0.657737	+	0.753247i	$0.728486\pi$
$60$	0	0
$61$	−9.18530	−1.17606	−0.588028	−	0.808840i	$-0.700096\pi$
$61$	−9.18530	−1.17606	−0.588028	+	0.808840i	$0.700096\pi$
$62$	0	0
$63$	4.50596	0.567698
$64$	0	0
$65$	−10.2291	−1.26876
$66$	0	0
$67$	9.98020	1.21928	0.609638	−	0.792680i	$-0.291315\pi$
$67$	9.98020	1.21928	0.609638	+	0.792680i	$0.291315\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.89722	−0.818549	−0.409275	−	0.912411i	$-0.634218\pi$
$71$	−6.89722	−0.818549	−0.409275	+	0.912411i	$0.634218\pi$
$72$	0	0
$73$	9.25073	1.08272	0.541358	−	0.840792i	$-0.317910\pi$
$73$	9.25073	1.08272	0.541358	+	0.840792i	$0.317910\pi$
$74$	0	0
$75$	2.07919	0.240084
$76$	0	0
$77$	27.7164	3.15858
$78$	0	0
$79$	−15.7331	−1.77011	−0.885054	−	0.465488i	$-0.845879\pi$
$79$	−15.7331	−1.77011	−0.885054	+	0.465488i	$0.845879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.42091	−0.375493	−0.187747	−	0.982217i	$-0.560118\pi$
$83$	−3.42091	−0.375493	−0.187747	+	0.982217i	$0.560118\pi$
$84$	0	0
$85$	−11.2415	−1.21931
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	10.7016	1.13437	0.567184	−	0.823591i	$-0.308033\pi$
$89$	10.7016	1.13437	0.567184	+	0.823591i	$0.308033\pi$
$90$	0	0
$91$	17.3233	1.81598
$92$	0	0
$93$	2.78961	0.289269
$94$	0	0
$95$	2.28367	0.234300
$96$	0	0
$97$	−16.1955	−1.64440	−0.822202	−	0.569196i	$-0.807254\pi$
$97$	−16.1955	−1.64440	−0.822202	+	0.569196i	$0.807254\pi$
$98$	0	0
$99$	6.15106	0.618205
$100$	0	0
$101$	−11.8040	−1.17454	−0.587272	−	0.809389i	$-0.699798\pi$
$101$	−11.8040	−1.17454	−0.587272	+	0.809389i	$0.699798\pi$
$102$	0	0
$103$	1.92445	0.189622	0.0948108	−	0.995495i	$-0.469775\pi$
$103$	1.92445	0.189622	0.0948108	+	0.995495i	$0.469775\pi$
$104$	0	0
$105$	−11.9889	−1.17000
$106$	0	0
$107$	6.80742	0.658098	0.329049	−	0.944313i	$-0.393272\pi$
$107$	6.80742	0.658098	0.329049	+	0.944313i	$0.393272\pi$
$108$	0	0
$109$	−17.3800	−1.66470	−0.832352	−	0.554247i	$-0.813006\pi$
$109$	−17.3800	−1.66470	−0.832352	+	0.554247i	$0.813006\pi$
$110$	0	0
$111$	9.37710	0.890035
$112$	0	0
$113$	−9.53054	−0.896557	−0.448279	−	0.893894i	$-0.647963\pi$
$113$	−9.53054	−0.896557	−0.448279	+	0.893894i	$0.647963\pi$
$114$	0	0
$115$	2.66067	0.248109
$116$	0	0
$117$	3.84454	0.355427
$118$	0	0
$119$	19.0379	1.74520
$120$	0	0
$121$	26.8355	2.43959
$122$	0	0
$123$	3.70763	0.334306
$124$	0	0
$125$	7.77133	0.695089
$126$	0	0
$127$	−2.64090	−0.234342	−0.117171	−	0.993112i	$-0.537383\pi$
$127$	−2.64090	−0.234342	−0.117171	+	0.993112i	$0.537383\pi$
$128$	0	0
$129$	9.59676	0.844947
$130$	0	0
$131$	−6.94350	−0.606656	−0.303328	−	0.952886i	$-0.598098\pi$
$131$	−6.94350	−0.606656	−0.303328	+	0.952886i	$0.598098\pi$
$132$	0	0
$133$	−3.86749	−0.335354
$134$	0	0
$135$	−2.66067	−0.228994
$136$	0	0
$137$	−20.4921	−1.75076	−0.875379	−	0.483437i	$-0.839388\pi$
$137$	−20.4921	−1.75076	−0.875379	+	0.483437i	$0.839388\pi$
$138$	0	0
$139$	−14.9941	−1.27178	−0.635891	−	0.771779i	$-0.719367\pi$
$139$	−14.9941	−1.27178	−0.635891	+	0.771779i	$0.719367\pi$
$140$	0	0
$141$	1.42386	0.119911
$142$	0	0
$143$	23.6480	1.97754
$144$	0	0
$145$	−2.66067	−0.220957
$146$	0	0
$147$	13.3037	1.09727
$148$	0	0
$149$	−19.1400	−1.56801	−0.784005	−	0.620755i	$-0.786826\pi$
$149$	−19.1400	−1.56801	−0.784005	+	0.620755i	$0.786826\pi$
$150$	0	0
$151$	−10.3673	−0.843677	−0.421838	−	0.906671i	$-0.638615\pi$
$151$	−10.3673	−0.843677	−0.421838	+	0.906671i	$0.638615\pi$
$152$	0	0
$153$	4.22506	0.341575
$154$	0	0
$155$	−7.42224	−0.596169
$156$	0	0
$157$	10.3237	0.823920	0.411960	−	0.911202i	$-0.364844\pi$
$157$	10.3237	0.823920	0.411960	+	0.911202i	$0.364844\pi$
$158$	0	0
$159$	−10.3493	−0.820749
$160$	0	0
$161$	−4.50596	−0.355119
$162$	0	0
$163$	7.71033	0.603920	0.301960	−	0.953321i	$-0.402359\pi$
$163$	7.71033	0.603920	0.301960	+	0.953321i	$0.402359\pi$
$164$	0	0
$165$	−16.3660	−1.27409
$166$	0	0
$167$	0.0962495	0.00744801	0.00372401	−	0.999993i	$-0.498815\pi$
$167$	0.0962495	0.00744801	0.00372401	+	0.999993i	$0.498815\pi$
$168$	0	0
$169$	1.78046	0.136958
$170$	0	0
$171$	−0.858306	−0.0656363
$172$	0	0
$173$	21.4817	1.63322	0.816611	−	0.577188i	$-0.195850\pi$
$173$	21.4817	1.63322	0.816611	+	0.577188i	$0.195850\pi$
$174$	0	0
$175$	9.36873	0.708210
$176$	0	0
$177$	−10.1044	−0.759490
$178$	0	0
$179$	7.57395	0.566103	0.283052	−	0.959105i	$-0.408653\pi$
$179$	7.57395	0.566103	0.283052	+	0.959105i	$0.408653\pi$
$180$	0	0
$181$	−14.7920	−1.09948	−0.549740	−	0.835336i	$-0.685273\pi$
$181$	−14.7920	−1.09948	−0.549740	+	0.835336i	$0.685273\pi$
$182$	0	0
$183$	−9.18530	−0.678997
$184$	0	0
$185$	−24.9494	−1.83432
$186$	0	0
$187$	25.9886	1.90047
$188$	0	0
$189$	4.50596	0.327760
$190$	0	0
$191$	−3.32289	−0.240436	−0.120218	−	0.992748i	$-0.538359\pi$
$191$	−3.32289	−0.240436	−0.120218	+	0.992748i	$0.538359\pi$
$192$	0	0
$193$	−25.3966	−1.82809	−0.914045	−	0.405612i	$-0.867058\pi$
$193$	−25.3966	−1.82809	−0.914045	+	0.405612i	$0.867058\pi$
$194$	0	0
$195$	−10.2291	−0.732518
$196$	0	0
$197$	−0.253487	−0.0180602	−0.00903011	−	0.999959i	$-0.502874\pi$
$197$	−0.253487	−0.0180602	−0.00903011	+	0.999959i	$0.502874\pi$
$198$	0	0
$199$	1.12906	0.0800369	0.0400185	−	0.999199i	$-0.487258\pi$
$199$	1.12906	0.0800369	0.0400185	+	0.999199i	$0.487258\pi$
$200$	0	0
$201$	9.98020	0.703949
$202$	0	0
$203$	4.50596	0.316256
$204$	0	0
$205$	−9.86479	−0.688987
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−5.27949	−0.365190
$210$	0	0
$211$	15.6886	1.08005	0.540023	−	0.841650i	$-0.318415\pi$
$211$	15.6886	1.08005	0.540023	+	0.841650i	$0.318415\pi$
$212$	0	0
$213$	−6.89722	−0.472590
$214$	0	0
$215$	−25.5338	−1.74139
$216$	0	0
$217$	12.5699	0.853298
$218$	0	0
$219$	9.25073	0.625106
$220$	0	0
$221$	16.2434	1.09265
$222$	0	0
$223$	20.0873	1.34515	0.672573	−	0.740031i	$-0.265189\pi$
$223$	20.0873	1.34515	0.672573	+	0.740031i	$0.265189\pi$
$224$	0	0
$225$	2.07919	0.138612
$226$	0	0
$227$	−25.5827	−1.69798	−0.848992	−	0.528406i	$-0.822790\pi$
$227$	−25.5827	−1.69798	−0.848992	+	0.528406i	$0.822790\pi$
$228$	0	0
$229$	6.25529	0.413361	0.206680	−	0.978408i	$-0.433734\pi$
$229$	6.25529	0.413361	0.206680	+	0.978408i	$0.433734\pi$
$230$	0	0
$231$	27.7164	1.82361
$232$	0	0
$233$	−9.58808	−0.628136	−0.314068	−	0.949400i	$-0.601692\pi$
$233$	−9.58808	−0.628136	−0.314068	+	0.949400i	$0.601692\pi$
$234$	0	0
$235$	−3.78843	−0.247130
$236$	0	0
$237$	−15.7331	−1.02197
$238$	0	0
$239$	−21.1993	−1.37127	−0.685636	−	0.727945i	$-0.740476\pi$
$239$	−21.1993	−1.37127	−0.685636	+	0.727945i	$0.740476\pi$
$240$	0	0
$241$	−1.82868	−0.117796	−0.0588979	−	0.998264i	$-0.518759\pi$
$241$	−1.82868	−0.117796	−0.0588979	+	0.998264i	$0.518759\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−35.3967	−2.26141
$246$	0	0
$247$	−3.29979	−0.209961
$248$	0	0
$249$	−3.42091	−0.216791
$250$	0	0
$251$	11.0564	0.697877	0.348938	−	0.937146i	$-0.386542\pi$
$251$	11.0564	0.697877	0.348938	+	0.937146i	$0.386542\pi$
$252$	0	0
$253$	−6.15106	−0.386714
$254$	0	0
$255$	−11.2415	−0.703970
$256$	0	0
$257$	2.59156	0.161657	0.0808286	−	0.996728i	$-0.474243\pi$
$257$	2.59156	0.161657	0.0808286	+	0.996728i	$0.474243\pi$
$258$	0	0
$259$	42.2529	2.62546
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	24.1075	1.48653	0.743265	−	0.668997i	$-0.233276\pi$
$263$	24.1075	1.48653	0.743265	+	0.668997i	$0.233276\pi$
$264$	0	0
$265$	27.5360	1.69152
$266$	0	0
$267$	10.7016	0.654927
$268$	0	0
$269$	−13.4333	−0.819044	−0.409522	−	0.912300i	$-0.634304\pi$
$269$	−13.4333	−0.819044	−0.409522	+	0.912300i	$0.634304\pi$
$270$	0	0
$271$	13.8591	0.841878	0.420939	−	0.907089i	$-0.361701\pi$
$271$	13.8591	0.841878	0.420939	+	0.907089i	$0.361701\pi$
$272$	0	0
$273$	17.3233	1.04846
$274$	0	0
$275$	12.7892	0.771218
$276$	0	0
$277$	4.68681	0.281603	0.140802	−	0.990038i	$-0.455032\pi$
$277$	4.68681	0.281603	0.140802	+	0.990038i	$0.455032\pi$
$278$	0	0
$279$	2.78961	0.167010
$280$	0	0
$281$	−15.3565	−0.916091	−0.458045	−	0.888929i	$-0.651450\pi$
$281$	−15.3565	−0.916091	−0.458045	+	0.888929i	$0.651450\pi$
$282$	0	0
$283$	−1.78577	−0.106153	−0.0530764	−	0.998590i	$-0.516903\pi$
$283$	−1.78577	−0.106153	−0.0530764	+	0.998590i	$0.516903\pi$
$284$	0	0
$285$	2.28367	0.135273
$286$	0	0
$287$	16.7064	0.986149
$288$	0	0
$289$	0.851090	0.0500641
$290$	0	0
$291$	−16.1955	−0.949397
$292$	0	0
$293$	6.96689	0.407010	0.203505	−	0.979074i	$-0.434767\pi$
$293$	6.96689	0.407010	0.203505	+	0.979074i	$0.434767\pi$
$294$	0	0
$295$	26.8844	1.56527
$296$	0	0
$297$	6.15106	0.356921
$298$	0	0
$299$	−3.84454	−0.222335
$300$	0	0
$301$	43.2426	2.49246
$302$	0	0
$303$	−11.8040	−0.678124
$304$	0	0
$305$	24.4391	1.39938
$306$	0	0
$307$	24.1568	1.37870	0.689352	−	0.724427i	$-0.257895\pi$
$307$	24.1568	1.37870	0.689352	+	0.724427i	$0.257895\pi$
$308$	0	0
$309$	1.92445	0.109478
$310$	0	0
$311$	0.454262	0.0257589	0.0128794	−	0.999917i	$-0.495900\pi$
$311$	0.454262	0.0257589	0.0128794	+	0.999917i	$0.495900\pi$
$312$	0	0
$313$	5.73746	0.324300	0.162150	−	0.986766i	$-0.448157\pi$
$313$	5.73746	0.324300	0.162150	+	0.986766i	$0.448157\pi$
$314$	0	0
$315$	−11.9889	−0.675497
$316$	0	0
$317$	21.5454	1.21011	0.605056	−	0.796183i	$-0.293151\pi$
$317$	21.5454	1.21011	0.605056	+	0.796183i	$0.293151\pi$
$318$	0	0
$319$	6.15106	0.344393
$320$	0	0
$321$	6.80742	0.379953
$322$	0	0
$323$	−3.62639	−0.201778
$324$	0	0
$325$	7.99351	0.443400
$326$	0	0
$327$	−17.3800	−0.961118
$328$	0	0
$329$	6.41586	0.353718
$330$	0	0
$331$	17.9555	0.986922	0.493461	−	0.869768i	$-0.335732\pi$
$331$	17.9555	0.986922	0.493461	+	0.869768i	$0.335732\pi$
$332$	0	0
$333$	9.37710	0.513862
$334$	0	0
$335$	−26.5541	−1.45080
$336$	0	0
$337$	17.3613	0.945732	0.472866	−	0.881134i	$-0.343219\pi$
$337$	17.3613	0.945732	0.472866	+	0.881134i	$0.343219\pi$
$338$	0	0
$339$	−9.53054	−0.517628
$340$	0	0
$341$	17.1591	0.929215
$342$	0	0
$343$	28.4041	1.53368
$344$	0	0
$345$	2.66067	0.143246
$346$	0	0
$347$	1.59595	0.0856751	0.0428375	−	0.999082i	$-0.486360\pi$
$347$	1.59595	0.0856751	0.0428375	+	0.999082i	$0.486360\pi$
$348$	0	0
$349$	−20.5408	−1.09952	−0.549761	−	0.835322i	$-0.685281\pi$
$349$	−20.5408	−1.09952	−0.549761	+	0.835322i	$0.685281\pi$
$350$	0	0
$351$	3.84454	0.205206
$352$	0	0
$353$	−14.9359	−0.794959	−0.397479	−	0.917611i	$-0.630115\pi$
$353$	−14.9359	−0.794959	−0.397479	+	0.917611i	$0.630115\pi$
$354$	0	0
$355$	18.3513	0.973983
$356$	0	0
$357$	19.0379	1.00759
$358$	0	0
$359$	−5.69722	−0.300688	−0.150344	−	0.988634i	$-0.548038\pi$
$359$	−5.69722	−0.300688	−0.150344	+	0.988634i	$0.548038\pi$
$360$	0	0
$361$	−18.2633	−0.961227
$362$	0	0
$363$	26.8355	1.40850
$364$	0	0
$365$	−24.6132	−1.28831
$366$	0	0
$367$	12.7041	0.663149	0.331575	−	0.943429i	$-0.392420\pi$
$367$	12.7041	0.663149	0.331575	+	0.943429i	$0.392420\pi$
$368$	0	0
$369$	3.70763	0.193011
$370$	0	0
$371$	−46.6333	−2.42108
$372$	0	0
$373$	−9.52773	−0.493327	−0.246664	−	0.969101i	$-0.579334\pi$
$373$	−9.52773	−0.493327	−0.246664	+	0.969101i	$0.579334\pi$
$374$	0	0
$375$	7.77133	0.401310
$376$	0	0
$377$	3.84454	0.198004
$378$	0	0
$379$	33.3175	1.71140	0.855702	−	0.517469i	$-0.173126\pi$
$379$	33.3175	1.71140	0.855702	+	0.517469i	$0.173126\pi$
$380$	0	0
$381$	−2.64090	−0.135297
$382$	0	0
$383$	0.184440	0.00942444	0.00471222	−	0.999989i	$-0.498500\pi$
$383$	0.184440	0.00942444	0.00471222	+	0.999989i	$0.498500\pi$
$384$	0	0
$385$	−73.7444	−3.75836
$386$	0	0
$387$	9.59676	0.487831
$388$	0	0
$389$	−7.74842	−0.392860	−0.196430	−	0.980518i	$-0.562935\pi$
$389$	−7.74842	−0.392860	−0.196430	+	0.980518i	$0.562935\pi$
$390$	0	0
$391$	−4.22506	−0.213670
$392$	0	0
$393$	−6.94350	−0.350253
$394$	0	0
$395$	41.8606	2.10623
$396$	0	0
$397$	22.5519	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$397$	22.5519	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$398$	0	0
$399$	−3.86749	−0.193617
$400$	0	0
$401$	−16.3478	−0.816371	−0.408185	−	0.912899i	$-0.633838\pi$
$401$	−16.3478	−0.816371	−0.408185	+	0.912899i	$0.633838\pi$
$402$	0	0
$403$	10.7248	0.534238
$404$	0	0
$405$	−2.66067	−0.132210
$406$	0	0
$407$	57.6791	2.85905
$408$	0	0
$409$	−33.5469	−1.65878	−0.829392	−	0.558666i	$-0.811313\pi$
$409$	−33.5469	−1.65878	−0.829392	+	0.558666i	$0.811313\pi$
$410$	0	0
$411$	−20.4921	−1.01080
$412$	0	0
$413$	−45.5298	−2.24038
$414$	0	0
$415$	9.10192	0.446796
$416$	0	0
$417$	−14.9941	−0.734264
$418$	0	0
$419$	−15.4412	−0.754352	−0.377176	−	0.926142i	$-0.623105\pi$
$419$	−15.4412	−0.754352	−0.377176	+	0.926142i	$0.623105\pi$
$420$	0	0
$421$	17.3809	0.847095	0.423548	−	0.905874i	$-0.360785\pi$
$421$	17.3809	0.847095	0.423548	+	0.905874i	$0.360785\pi$
$422$	0	0
$423$	1.42386	0.0692305
$424$	0	0
$425$	8.78468	0.426119
$426$	0	0
$427$	−41.3886	−2.00293
$428$	0	0
$429$	23.6480	1.14173
$430$	0	0
$431$	33.6104	1.61896	0.809479	−	0.587149i	$-0.199750\pi$
$431$	33.6104	1.61896	0.809479	+	0.587149i	$0.199750\pi$
$432$	0	0
$433$	8.90181	0.427794	0.213897	−	0.976856i	$-0.431384\pi$
$433$	8.90181	0.427794	0.213897	+	0.976856i	$0.431384\pi$
$434$	0	0
$435$	−2.66067	−0.127570
$436$	0	0
$437$	0.858306	0.0410584
$438$	0	0
$439$	−31.4551	−1.50127	−0.750635	−	0.660717i	$-0.770253\pi$
$439$	−31.4551	−1.50127	−0.750635	+	0.660717i	$0.770253\pi$
$440$	0	0
$441$	13.3037	0.633508
$442$	0	0
$443$	23.8050	1.13101	0.565504	−	0.824746i	$-0.308682\pi$
$443$	23.8050	1.13101	0.565504	+	0.824746i	$0.308682\pi$
$444$	0	0
$445$	−28.4735	−1.34977
$446$	0	0
$447$	−19.1400	−0.905291
$448$	0	0
$449$	14.1310	0.666884	0.333442	−	0.942771i	$-0.391790\pi$
$449$	14.1310	0.666884	0.333442	+	0.942771i	$0.391790\pi$
$450$	0	0
$451$	22.8059	1.07389
$452$	0	0
$453$	−10.3673	−0.487097
$454$	0	0
$455$	−46.0917	−2.16081
$456$	0	0
$457$	−3.64371	−0.170446	−0.0852228	−	0.996362i	$-0.527160\pi$
$457$	−3.64371	−0.170446	−0.0852228	+	0.996362i	$0.527160\pi$
$458$	0	0
$459$	4.22506	0.197209
$460$	0	0
$461$	17.7980	0.828936	0.414468	−	0.910064i	$-0.363968\pi$
$461$	17.7980	0.828936	0.414468	+	0.910064i	$0.363968\pi$
$462$	0	0
$463$	14.0672	0.653759	0.326880	−	0.945066i	$-0.394003\pi$
$463$	14.0672	0.653759	0.326880	+	0.945066i	$0.394003\pi$
$464$	0	0
$465$	−7.42224	−0.344198
$466$	0	0
$467$	−19.4946	−0.902104	−0.451052	−	0.892498i	$-0.648951\pi$
$467$	−19.4946	−0.902104	−0.451052	+	0.892498i	$0.648951\pi$
$468$	0	0
$469$	44.9704	2.07654
$470$	0	0
$471$	10.3237	0.475691
$472$	0	0
$473$	59.0302	2.71421
$474$	0	0
$475$	−1.78458	−0.0818821
$476$	0	0
$477$	−10.3493	−0.473860
$478$	0	0
$479$	14.2515	0.651166	0.325583	−	0.945514i	$-0.394440\pi$
$479$	14.2515	0.651166	0.325583	+	0.945514i	$0.394440\pi$
$480$	0	0
$481$	36.0506	1.64377
$482$	0	0
$483$	−4.50596	−0.205028
$484$	0	0
$485$	43.0909	1.95666
$486$	0	0
$487$	36.9603	1.67483	0.837415	−	0.546567i	$-0.184066\pi$
$487$	36.9603	1.67483	0.837415	+	0.546567i	$0.184066\pi$
$488$	0	0
$489$	7.71033	0.348673
$490$	0	0
$491$	−3.79439	−0.171238	−0.0856192	−	0.996328i	$-0.527287\pi$
$491$	−3.79439	−0.171238	−0.0856192	+	0.996328i	$0.527287\pi$
$492$	0	0
$493$	4.22506	0.190287
$494$	0	0
$495$	−16.3660	−0.735595
$496$	0	0
$497$	−31.0786	−1.39407
$498$	0	0
$499$	26.9320	1.20564	0.602821	−	0.797877i	$-0.294043\pi$
$499$	26.9320	1.20564	0.602821	+	0.797877i	$0.294043\pi$
$500$	0	0
$501$	0.0962495	0.00430011
$502$	0	0
$503$	−11.0828	−0.494160	−0.247080	−	0.968995i	$-0.579471\pi$
$503$	−11.0828	−0.494160	−0.247080	+	0.968995i	$0.579471\pi$
$504$	0	0
$505$	31.4067	1.39758
$506$	0	0
$507$	1.78046	0.0790729
$508$	0	0
$509$	−27.8188	−1.23305	−0.616524	−	0.787336i	$-0.711460\pi$
$509$	−27.8188	−1.23305	−0.616524	+	0.787336i	$0.711460\pi$
$510$	0	0
$511$	41.6834	1.84397
$512$	0	0
$513$	−0.858306	−0.0378951
$514$	0	0
$515$	−5.12033	−0.225629
$516$	0	0
$517$	8.75825	0.385187
$518$	0	0
$519$	21.4817	0.942941
$520$	0	0
$521$	−15.3356	−0.671864	−0.335932	−	0.941886i	$-0.609051\pi$
$521$	−15.3356	−0.671864	−0.335932	+	0.941886i	$0.609051\pi$
$522$	0	0
$523$	−5.70567	−0.249492	−0.124746	−	0.992189i	$-0.539812\pi$
$523$	−5.70567	−0.249492	−0.124746	+	0.992189i	$0.539812\pi$
$524$	0	0
$525$	9.36873	0.408885
$526$	0	0
$527$	11.7863	0.513417
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.1044	−0.438492
$532$	0	0
$533$	14.2541	0.617414
$534$	0	0
$535$	−18.1123	−0.783064
$536$	0	0
$537$	7.57395	0.326840
$538$	0	0
$539$	81.8317	3.52474
$540$	0	0
$541$	22.4060	0.963309	0.481654	−	0.876361i	$-0.340036\pi$
$541$	22.4060	0.963309	0.481654	+	0.876361i	$0.340036\pi$
$542$	0	0
$543$	−14.7920	−0.634785
$544$	0	0
$545$	46.2426	1.98081
$546$	0	0
$547$	−18.9413	−0.809869	−0.404935	−	0.914346i	$-0.632706\pi$
$547$	−18.9413	−0.809869	−0.404935	+	0.914346i	$0.632706\pi$
$548$	0	0
$549$	−9.18530	−0.392019
$550$	0	0
$551$	−0.858306	−0.0365651
$552$	0	0
$553$	−70.8926	−3.01466
$554$	0	0
$555$	−24.9494	−1.05904
$556$	0	0
$557$	13.6798	0.579631	0.289816	−	0.957083i	$-0.406406\pi$
$557$	13.6798	0.579631	0.289816	+	0.957083i	$0.406406\pi$
$558$	0	0
$559$	36.8951	1.56050
$560$	0	0
$561$	25.9886	1.09724
$562$	0	0
$563$	−43.1118	−1.81695	−0.908473	−	0.417944i	$-0.862751\pi$
$563$	−43.1118	−1.81695	−0.908473	+	0.417944i	$0.862751\pi$
$564$	0	0
$565$	25.3576	1.06680
$566$	0	0
$567$	4.50596	0.189233
$568$	0	0
$569$	22.6148	0.948060	0.474030	−	0.880509i	$-0.342799\pi$
$569$	22.6148	0.948060	0.474030	+	0.880509i	$0.342799\pi$
$570$	0	0
$571$	8.64251	0.361678	0.180839	−	0.983513i	$-0.442119\pi$
$571$	8.64251	0.361678	0.180839	+	0.983513i	$0.442119\pi$
$572$	0	0
$573$	−3.32289	−0.138816
$574$	0	0
$575$	−2.07919	−0.0867081
$576$	0	0
$577$	37.8604	1.57615	0.788075	−	0.615579i	$-0.211078\pi$
$577$	37.8604	1.57615	0.788075	+	0.615579i	$0.211078\pi$
$578$	0	0
$579$	−25.3966	−1.05545
$580$	0	0
$581$	−15.4145	−0.639500
$582$	0	0
$583$	−63.6589	−2.63648
$584$	0	0
$585$	−10.2291	−0.422919
$586$	0	0
$587$	0.224299	0.00925780	0.00462890	−	0.999989i	$-0.498527\pi$
$587$	0.224299	0.00925780	0.00462890	+	0.999989i	$0.498527\pi$
$588$	0	0
$589$	−2.39434	−0.0986570
$590$	0	0
$591$	−0.253487	−0.0104271
$592$	0	0
$593$	−13.6493	−0.560509	−0.280254	−	0.959926i	$-0.590419\pi$
$593$	−13.6493	−0.560509	−0.280254	+	0.959926i	$0.590419\pi$
$594$	0	0
$595$	−50.6537	−2.07660
$596$	0	0
$597$	1.12906	0.0462093
$598$	0	0
$599$	−37.0226	−1.51270	−0.756351	−	0.654166i	$-0.773020\pi$
$599$	−37.0226	−1.51270	−0.756351	+	0.654166i	$0.773020\pi$
$600$	0	0
$601$	2.52733	0.103092	0.0515460	−	0.998671i	$-0.483585\pi$
$601$	2.52733	0.103092	0.0515460	+	0.998671i	$0.483585\pi$
$602$	0	0
$603$	9.98020	0.406425
$604$	0	0
$605$	−71.4006	−2.90285
$606$	0	0
$607$	2.20912	0.0896654	0.0448327	−	0.998995i	$-0.485725\pi$
$607$	2.20912	0.0896654	0.0448327	+	0.998995i	$0.485725\pi$
$608$	0	0
$609$	4.50596	0.182591
$610$	0	0
$611$	5.47408	0.221458
$612$	0	0
$613$	−41.1140	−1.66058	−0.830289	−	0.557332i	$-0.811825\pi$
$613$	−41.1140	−1.66058	−0.830289	+	0.557332i	$0.811825\pi$
$614$	0	0
$615$	−9.86479	−0.397787
$616$	0	0
$617$	11.8305	0.476279	0.238139	−	0.971231i	$-0.423462\pi$
$617$	11.8305	0.476279	0.238139	+	0.971231i	$0.423462\pi$
$618$	0	0
$619$	−14.3248	−0.575764	−0.287882	−	0.957666i	$-0.592951\pi$
$619$	−14.3248	−0.575764	−0.287882	+	0.957666i	$0.592951\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	48.2210	1.93193
$624$	0	0
$625$	−31.0729	−1.24292
$626$	0	0
$627$	−5.27949	−0.210843
$628$	0	0
$629$	39.6188	1.57970
$630$	0	0
$631$	−6.43918	−0.256340	−0.128170	−	0.991752i	$-0.540910\pi$
$631$	−6.43918	−0.256340	−0.128170	+	0.991752i	$0.540910\pi$
$632$	0	0
$633$	15.6886	0.623565
$634$	0	0
$635$	7.02656	0.278841
$636$	0	0
$637$	51.1465	2.02650
$638$	0	0
$639$	−6.89722	−0.272850
$640$	0	0
$641$	35.5146	1.40274	0.701372	−	0.712796i	$-0.252571\pi$
$641$	35.5146	1.40274	0.701372	+	0.712796i	$0.252571\pi$
$642$	0	0
$643$	39.7108	1.56604	0.783020	−	0.621996i	$-0.213678\pi$
$643$	39.7108	1.56604	0.783020	+	0.621996i	$0.213678\pi$
$644$	0	0
$645$	−25.5338	−1.00539
$646$	0	0
$647$	−31.0848	−1.22207	−0.611035	−	0.791603i	$-0.709247\pi$
$647$	−31.0848	−1.22207	−0.611035	+	0.791603i	$0.709247\pi$
$648$	0	0
$649$	−62.1525	−2.43970
$650$	0	0
$651$	12.5699	0.492652
$652$	0	0
$653$	−1.90615	−0.0745933	−0.0372966	−	0.999304i	$-0.511875\pi$
$653$	−1.90615	−0.0745933	−0.0372966	+	0.999304i	$0.511875\pi$
$654$	0	0
$655$	18.4744	0.721854
$656$	0	0
$657$	9.25073	0.360905
$658$	0	0
$659$	21.7477	0.847171	0.423586	−	0.905856i	$-0.360771\pi$
$659$	21.7477	0.847171	0.423586	+	0.905856i	$0.360771\pi$
$660$	0	0
$661$	6.06779	0.236010	0.118005	−	0.993013i	$-0.462350\pi$
$661$	6.06779	0.236010	0.118005	+	0.993013i	$0.462350\pi$
$662$	0	0
$663$	16.2434	0.630840
$664$	0	0
$665$	10.2901	0.399035
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	20.0873	0.776620
$670$	0	0
$671$	−56.4993	−2.18113
$672$	0	0
$673$	14.2706	0.550091	0.275046	−	0.961431i	$-0.411307\pi$
$673$	14.2706	0.550091	0.275046	+	0.961431i	$0.411307\pi$
$674$	0	0
$675$	2.07919	0.0800279
$676$	0	0
$677$	16.5598	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$677$	16.5598	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$678$	0	0
$679$	−72.9762	−2.80057
$680$	0	0
$681$	−25.5827	−0.980331
$682$	0	0
$683$	−27.7531	−1.06194	−0.530971	−	0.847390i	$-0.678173\pi$
$683$	−27.7531	−1.06194	−0.530971	+	0.847390i	$0.678173\pi$
$684$	0	0
$685$	54.5228	2.08321
$686$	0	0
$687$	6.25529	0.238654
$688$	0	0
$689$	−39.7881	−1.51580
$690$	0	0
$691$	−3.64935	−0.138828	−0.0694138	−	0.997588i	$-0.522113\pi$
$691$	−3.64935	−0.138828	−0.0694138	+	0.997588i	$0.522113\pi$
$692$	0	0
$693$	27.7164	1.05286
$694$	0	0
$695$	39.8944	1.51328
$696$	0	0
$697$	15.6649	0.593352
$698$	0	0
$699$	−9.58808	−0.362655
$700$	0	0
$701$	−34.8711	−1.31706	−0.658531	−	0.752553i	$-0.728822\pi$
$701$	−34.8711	−1.31706	−0.658531	+	0.752553i	$0.728822\pi$
$702$	0	0
$703$	−8.04843	−0.303552
$704$	0	0
$705$	−3.78843	−0.142680
$706$	0	0
$707$	−53.1885	−2.00036
$708$	0	0
$709$	35.9470	1.35002	0.675009	−	0.737810i	$-0.264140\pi$
$709$	35.9470	1.35002	0.675009	+	0.737810i	$0.264140\pi$
$710$	0	0
$711$	−15.7331	−0.590036
$712$	0	0
$713$	−2.78961	−0.104472
$714$	0	0
$715$	−62.9195	−2.35306
$716$	0	0
$717$	−21.1993	−0.791704
$718$	0	0
$719$	−44.7319	−1.66822	−0.834110	−	0.551599i	$-0.814018\pi$
$719$	−44.7319	−1.66822	−0.834110	+	0.551599i	$0.814018\pi$
$720$	0	0
$721$	8.67149	0.322943
$722$	0	0
$723$	−1.82868	−0.0680095
$724$	0	0
$725$	2.07919	0.0772191
$726$	0	0
$727$	−49.9953	−1.85422	−0.927112	−	0.374785i	$-0.877717\pi$
$727$	−49.9953	−1.85422	−0.927112	+	0.374785i	$0.877717\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.5468	1.49968
$732$	0	0
$733$	−43.9035	−1.62161	−0.810807	−	0.585314i	$-0.800971\pi$
$733$	−43.9035	−1.62161	−0.810807	+	0.585314i	$0.800971\pi$
$734$	0	0
$735$	−35.3967	−1.30563
$736$	0	0
$737$	61.3888	2.26129
$738$	0	0
$739$	6.57470	0.241854	0.120927	−	0.992661i	$-0.461413\pi$
$739$	6.57470	0.241854	0.120927	+	0.992661i	$0.461413\pi$
$740$	0	0
$741$	−3.29979	−0.121221
$742$	0	0
$743$	−41.3832	−1.51820	−0.759101	−	0.650973i	$-0.774361\pi$
$743$	−41.3832	−1.51820	−0.759101	+	0.650973i	$0.774361\pi$
$744$	0	0
$745$	50.9253	1.86576
$746$	0	0
$747$	−3.42091	−0.125164
$748$	0	0
$749$	30.6740	1.12080
$750$	0	0
$751$	−34.6375	−1.26394	−0.631971	−	0.774992i	$-0.717754\pi$
$751$	−34.6375	−1.26394	−0.631971	+	0.774992i	$0.717754\pi$
$752$	0	0
$753$	11.0564	0.402919
$754$	0	0
$755$	27.5839	1.00388
$756$	0	0
$757$	−21.7995	−0.792316	−0.396158	−	0.918182i	$-0.629657\pi$
$757$	−21.7995	−0.792316	−0.396158	+	0.918182i	$0.629657\pi$
$758$	0	0
$759$	−6.15106	−0.223269
$760$	0	0
$761$	−15.3830	−0.557632	−0.278816	−	0.960345i	$-0.589942\pi$
$761$	−15.3830	−0.557632	−0.278816	+	0.960345i	$0.589942\pi$
$762$	0	0
$763$	−78.3137	−2.83515
$764$	0	0
$765$	−11.2415	−0.406437
$766$	0	0
$767$	−38.8465	−1.40267
$768$	0	0
$769$	39.2174	1.41421	0.707107	−	0.707106i	$-0.250000\pi$
$769$	39.2174	1.41421	0.707107	+	0.707106i	$0.250000\pi$
$770$	0	0
$771$	2.59156	0.0933329
$772$	0	0
$773$	14.8887	0.535511	0.267756	−	0.963487i	$-0.413718\pi$
$773$	14.8887	0.535511	0.267756	+	0.963487i	$0.413718\pi$
$774$	0	0
$775$	5.80012	0.208346
$776$	0	0
$777$	42.2529	1.51581
$778$	0	0
$779$	−3.18228	−0.114017
$780$	0	0
$781$	−42.4252	−1.51809
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−27.4680	−0.980374
$786$	0	0
$787$	−3.41018	−0.121560	−0.0607799	−	0.998151i	$-0.519359\pi$
$787$	−3.41018	−0.121560	−0.0607799	+	0.998151i	$0.519359\pi$
$788$	0	0
$789$	24.1075	0.858249
$790$	0	0
$791$	−42.9442	−1.52692
$792$	0	0
$793$	−35.3132	−1.25401
$794$	0	0
$795$	27.5360	0.976601
$796$	0	0
$797$	−49.7997	−1.76400	−0.881998	−	0.471253i	$-0.843802\pi$
$797$	−49.7997	−1.76400	−0.881998	+	0.471253i	$0.843802\pi$
$798$	0	0
$799$	6.01589	0.212827
$800$	0	0
$801$	10.7016	0.378122
$802$	0	0
$803$	56.9018	2.00802
$804$	0	0
$805$	11.9889	0.422553
$806$	0	0
$807$	−13.4333	−0.472875
$808$	0	0
$809$	−28.5521	−1.00384	−0.501920	−	0.864914i	$-0.667373\pi$
$809$	−28.5521	−1.00384	−0.501920	+	0.864914i	$0.667373\pi$
$810$	0	0
$811$	−38.2013	−1.34143	−0.670715	−	0.741715i	$-0.734013\pi$
$811$	−38.2013	−1.34143	−0.670715	+	0.741715i	$0.734013\pi$
$812$	0	0
$813$	13.8591	0.486059
$814$	0	0
$815$	−20.5147	−0.718598
$816$	0	0
$817$	−8.23696	−0.288175
$818$	0	0
$819$	17.3233	0.605326
$820$	0	0
$821$	3.29638	0.115045	0.0575223	−	0.998344i	$-0.481680\pi$
$821$	3.29638	0.115045	0.0575223	+	0.998344i	$0.481680\pi$
$822$	0	0
$823$	30.0480	1.04741	0.523703	−	0.851901i	$-0.324550\pi$
$823$	30.0480	1.04741	0.523703	+	0.851901i	$0.324550\pi$
$824$	0	0
$825$	12.7892	0.445263
$826$	0	0
$827$	46.7872	1.62695	0.813476	−	0.581598i	$-0.197572\pi$
$827$	46.7872	1.62695	0.813476	+	0.581598i	$0.197572\pi$
$828$	0	0
$829$	−54.9898	−1.90987	−0.954937	−	0.296810i	$-0.904077\pi$
$829$	−54.9898	−1.90987	−0.954937	+	0.296810i	$0.904077\pi$
$830$	0	0
$831$	4.68681	0.162584
$832$	0	0
$833$	56.2088	1.94752
$834$	0	0
$835$	−0.256089	−0.00886231
$836$	0	0
$837$	2.78961	0.0964230
$838$	0	0
$839$	−11.8255	−0.408261	−0.204130	−	0.978944i	$-0.565437\pi$
$839$	−11.8255	−0.408261	−0.204130	+	0.978944i	$0.565437\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−15.3565	−0.528905
$844$	0	0
$845$	−4.73722	−0.162965
$846$	0	0
$847$	120.920	4.15486
$848$	0	0
$849$	−1.78577	−0.0612874
$850$	0	0
$851$	−9.37710	−0.321443
$852$	0	0
$853$	−18.6275	−0.637794	−0.318897	−	0.947789i	$-0.603312\pi$
$853$	−18.6275	−0.637794	−0.318897	+	0.947789i	$0.603312\pi$
$854$	0	0
$855$	2.28367	0.0781000
$856$	0	0
$857$	33.7115	1.15156	0.575782	−	0.817604i	$-0.304698\pi$
$857$	33.7115	1.15156	0.575782	+	0.817604i	$0.304698\pi$
$858$	0	0
$859$	16.3799	0.558874	0.279437	−	0.960164i	$-0.409852\pi$
$859$	16.3799	0.558874	0.279437	+	0.960164i	$0.409852\pi$
$860$	0	0
$861$	16.7064	0.569354
$862$	0	0
$863$	37.7378	1.28461	0.642305	−	0.766449i	$-0.277978\pi$
$863$	37.7378	1.28461	0.642305	+	0.766449i	$0.277978\pi$
$864$	0	0
$865$	−57.1558	−1.94335
$866$	0	0
$867$	0.851090	0.0289045
$868$	0	0
$869$	−96.7750	−3.28287
$870$	0	0
$871$	38.3693	1.30009
$872$	0	0
$873$	−16.1955	−0.548134
$874$	0	0
$875$	35.0173	1.18380
$876$	0	0
$877$	6.65458	0.224709	0.112355	−	0.993668i	$-0.464161\pi$
$877$	6.65458	0.224709	0.112355	+	0.993668i	$0.464161\pi$
$878$	0	0
$879$	6.96689	0.234987
$880$	0	0
$881$	−2.03894	−0.0686938	−0.0343469	−	0.999410i	$-0.510935\pi$
$881$	−2.03894	−0.0686938	−0.0343469	+	0.999410i	$0.510935\pi$
$882$	0	0
$883$	−23.8463	−0.802493	−0.401246	−	0.915970i	$-0.631423\pi$
$883$	−23.8463	−0.802493	−0.401246	+	0.915970i	$0.631423\pi$
$884$	0	0
$885$	26.8844	0.903709
$886$	0	0
$887$	8.98155	0.301571	0.150785	−	0.988567i	$-0.451820\pi$
$887$	8.98155	0.301571	0.150785	+	0.988567i	$0.451820\pi$
$888$	0	0
$889$	−11.8998	−0.399106
$890$	0	0
$891$	6.15106	0.206068
$892$	0	0
$893$	−1.22211	−0.0408963
$894$	0	0
$895$	−20.1518	−0.673601
$896$	0	0
$897$	−3.84454	−0.128365
$898$	0	0
$899$	2.78961	0.0930387
$900$	0	0
$901$	−43.7262	−1.45673
$902$	0	0
$903$	43.2426	1.43902
$904$	0	0
$905$	39.3567	1.30826
$906$	0	0
$907$	−4.44155	−0.147479	−0.0737396	−	0.997278i	$-0.523493\pi$
$907$	−4.44155	−0.147479	−0.0737396	+	0.997278i	$0.523493\pi$
$908$	0	0
$909$	−11.8040	−0.391515
$910$	0	0
$911$	47.9687	1.58927	0.794636	−	0.607086i	$-0.207662\pi$
$911$	47.9687	1.58927	0.794636	+	0.607086i	$0.207662\pi$
$912$	0	0
$913$	−21.0422	−0.696395
$914$	0	0
$915$	24.4391	0.807931
$916$	0	0
$917$	−31.2871	−1.03319
$918$	0	0
$919$	−34.5039	−1.13818	−0.569090	−	0.822275i	$-0.692704\pi$
$919$	−34.5039	−1.13818	−0.569090	+	0.822275i	$0.692704\pi$
$920$	0	0
$921$	24.1568	0.795995
$922$	0	0
$923$	−26.5166	−0.872805
$924$	0	0
$925$	19.4967	0.641049
$926$	0	0
$927$	1.92445	0.0632072
$928$	0	0
$929$	−40.1585	−1.31756	−0.658779	−	0.752337i	$-0.728927\pi$
$929$	−40.1585	−1.31756	−0.658779	+	0.752337i	$0.728927\pi$
$930$	0	0
$931$	−11.4186	−0.374230
$932$	0	0
$933$	0.454262	0.0148719
$934$	0	0
$935$	−69.1471	−2.26135
$936$	0	0
$937$	41.9037	1.36894	0.684468	−	0.729043i	$-0.260035\pi$
$937$	41.9037	1.36894	0.684468	+	0.729043i	$0.260035\pi$
$938$	0	0
$939$	5.73746	0.187235
$940$	0	0
$941$	24.5083	0.798946	0.399473	−	0.916745i	$-0.369193\pi$
$941$	24.5083	0.798946	0.399473	+	0.916745i	$0.369193\pi$
$942$	0	0
$943$	−3.70763	−0.120737
$944$	0	0
$945$	−11.9889	−0.389999
$946$	0	0
$947$	−28.9664	−0.941280	−0.470640	−	0.882325i	$-0.655977\pi$
$947$	−28.9664	−0.941280	−0.470640	+	0.882325i	$0.655977\pi$
$948$	0	0
$949$	35.5648	1.15448
$950$	0	0
$951$	21.5454	0.698659
$952$	0	0
$953$	6.90221	0.223584	0.111792	−	0.993732i	$-0.464341\pi$
$953$	6.90221	0.223584	0.111792	+	0.993732i	$0.464341\pi$
$954$	0	0
$955$	8.84113	0.286092
$956$	0	0
$957$	6.15106	0.198836
$958$	0	0
$959$	−92.3366	−2.98170
$960$	0	0
$961$	−23.2181	−0.748970
$962$	0	0
$963$	6.80742	0.219366
$964$	0	0
$965$	67.5722	2.17523
$966$	0	0
$967$	−4.82704	−0.155227	−0.0776136	−	0.996984i	$-0.524730\pi$
$967$	−4.82704	−0.155227	−0.0776136	+	0.996984i	$0.524730\pi$
$968$	0	0
$969$	−3.62639	−0.116496
$970$	0	0
$971$	−24.6570	−0.791281	−0.395641	−	0.918405i	$-0.629477\pi$
$971$	−24.6570	−0.791281	−0.395641	+	0.918405i	$0.629477\pi$
$972$	0	0
$973$	−67.5628	−2.16596
$974$	0	0
$975$	7.99351	0.255997
$976$	0	0
$977$	60.5259	1.93639	0.968197	−	0.250189i	$-0.0804928\pi$
$977$	60.5259	1.93639	0.968197	+	0.250189i	$0.0804928\pi$
$978$	0	0
$979$	65.8262	2.10381
$980$	0	0
$981$	−17.3800	−0.554901
$982$	0	0
$983$	−50.6356	−1.61502	−0.807512	−	0.589851i	$-0.799187\pi$
$983$	−50.6356	−1.61502	−0.807512	+	0.589851i	$0.799187\pi$
$984$	0	0
$985$	0.674447	0.0214897
$986$	0	0
$987$	6.41586	0.204219
$988$	0	0
$989$	−9.59676	−0.305159
$990$	0	0
$991$	17.6079	0.559333	0.279666	−	0.960097i	$-0.409776\pi$
$991$	17.6079	0.559333	0.279666	+	0.960097i	$0.409776\pi$
$992$	0	0
$993$	17.9555	0.569800
$994$	0	0
$995$	−3.00406	−0.0952351
$996$	0	0
$997$	−36.8699	−1.16768	−0.583840	−	0.811868i	$-0.698451\pi$
$997$	−36.8699	−1.16768	−0.583840	+	0.811868i	$0.698451\pi$
$998$	0	0
$999$	9.37710	0.296678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.5	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.5	✓	18	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.66067 q^{5} +4.50596 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.66067 q^{5} +4.50596 q^{7} +1.00000 q^{9} +6.15106 q^{11} +3.84454 q^{13} -2.66067 q^{15} +4.22506 q^{17} -0.858306 q^{19} +4.50596 q^{21} -1.00000 q^{23} +2.07919 q^{25} +1.00000 q^{27} +1.00000 q^{29} +2.78961 q^{31} +6.15106 q^{33} -11.9889 q^{35} +9.37710 q^{37} +3.84454 q^{39} +3.70763 q^{41} +9.59676 q^{43} -2.66067 q^{45} +1.42386 q^{47} +13.3037 q^{49} +4.22506 q^{51} -10.3493 q^{53} -16.3660 q^{55} -0.858306 q^{57} -10.1044 q^{59} -9.18530 q^{61} +4.50596 q^{63} -10.2291 q^{65} +9.98020 q^{67} -1.00000 q^{69} -6.89722 q^{71} +9.25073 q^{73} +2.07919 q^{75} +27.7164 q^{77} -15.7331 q^{79} +1.00000 q^{81} -3.42091 q^{83} -11.2415 q^{85} +1.00000 q^{87} +10.7016 q^{89} +17.3233 q^{91} +2.78961 q^{93} +2.28367 q^{95} -16.1955 q^{97} +6.15106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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