LMFDB - Embedded newform 6008.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.502181 q^{3} -0.673128 q^{5} +2.13432 q^{7} -2.74781 q^{9} +O(q^{10})$	$q-0.502181 q^{3} -0.673128 q^{5} +2.13432 q^{7} -2.74781 q^{9} +4.57214 q^{11} +6.19886 q^{13} +0.338032 q^{15} +5.95643 q^{17} +6.84954 q^{19} -1.07181 q^{21} -8.41783 q^{23} -4.54690 q^{25} +2.88644 q^{27} -2.10012 q^{29} +5.72872 q^{31} -2.29604 q^{33} -1.43667 q^{35} +10.7385 q^{37} -3.11295 q^{39} +8.04092 q^{41} +0.663328 q^{43} +1.84963 q^{45} -12.2883 q^{47} -2.44469 q^{49} -2.99121 q^{51} +4.33138 q^{53} -3.07763 q^{55} -3.43971 q^{57} +11.8442 q^{59} -5.00040 q^{61} -5.86471 q^{63} -4.17263 q^{65} +8.75350 q^{67} +4.22728 q^{69} -6.26627 q^{71} +3.72925 q^{73} +2.28337 q^{75} +9.75839 q^{77} -15.1159 q^{79} +6.79393 q^{81} -1.89819 q^{83} -4.00944 q^{85} +1.05464 q^{87} -7.79885 q^{89} +13.2303 q^{91} -2.87685 q^{93} -4.61062 q^{95} -15.9700 q^{97} -12.5634 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−0.502181	−0.289934	−0.144967	−	0.989436i	$-0.546308\pi$
$3$	−0.502181	−0.289934	−0.144967	+	0.989436i	$0.546308\pi$
$4$	0	0
$5$	−0.673128	−0.301032	−0.150516	−	0.988608i	$-0.548094\pi$
$5$	−0.673128	−0.301032	−0.150516	+	0.988608i	$0.548094\pi$
$6$	0	0
$7$	2.13432	0.806696	0.403348	−	0.915047i	$-0.367846\pi$
$7$	2.13432	0.806696	0.403348	+	0.915047i	$0.367846\pi$
$8$	0	0
$9$	−2.74781	−0.915938
$10$	0	0
$11$	4.57214	1.37855	0.689275	−	0.724499i	$-0.257929\pi$
$11$	4.57214	1.37855	0.689275	+	0.724499i	$0.257929\pi$
$12$	0	0
$13$	6.19886	1.71925	0.859627	−	0.510922i	$-0.170696\pi$
$13$	6.19886	1.71925	0.859627	+	0.510922i	$0.170696\pi$
$14$	0	0
$15$	0.338032	0.0872796
$16$	0	0
$17$	5.95643	1.44465	0.722324	−	0.691555i	$-0.243074\pi$
$17$	5.95643	1.44465	0.722324	+	0.691555i	$0.243074\pi$
$18$	0	0
$19$	6.84954	1.57139	0.785696	−	0.618613i	$-0.212305\pi$
$19$	6.84954	1.57139	0.785696	+	0.618613i	$0.212305\pi$
$20$	0	0
$21$	−1.07181	−0.233889
$22$	0	0
$23$	−8.41783	−1.75524	−0.877620	−	0.479357i	$-0.840870\pi$
$23$	−8.41783	−1.75524	−0.877620	+	0.479357i	$0.840870\pi$
$24$	0	0
$25$	−4.54690	−0.909380
$26$	0	0
$27$	2.88644	0.555496
$28$	0	0
$29$	−2.10012	−0.389983	−0.194992	−	0.980805i	$-0.562468\pi$
$29$	−2.10012	−0.389983	−0.194992	+	0.980805i	$0.562468\pi$
$30$	0	0
$31$	5.72872	1.02891	0.514454	−	0.857518i	$-0.327995\pi$
$31$	5.72872	1.02891	0.514454	+	0.857518i	$0.327995\pi$
$32$	0	0
$33$	−2.29604	−0.399689
$34$	0	0
$35$	−1.43667	−0.242841
$36$	0	0
$37$	10.7385	1.76540	0.882700	−	0.469938i	$-0.155724\pi$
$37$	10.7385	1.76540	0.882700	+	0.469938i	$0.155724\pi$
$38$	0	0
$39$	−3.11295	−0.498471
$40$	0	0
$41$	8.04092	1.25578	0.627891	−	0.778302i	$-0.283919\pi$
$41$	8.04092	1.25578	0.627891	+	0.778302i	$0.283919\pi$
$42$	0	0
$43$	0.663328	0.101157	0.0505783	−	0.998720i	$-0.483894\pi$
$43$	0.663328	0.101157	0.0505783	+	0.998720i	$0.483894\pi$
$44$	0	0
$45$	1.84963	0.275727
$46$	0	0
$47$	−12.2883	−1.79243	−0.896217	−	0.443617i	$-0.853695\pi$
$47$	−12.2883	−1.79243	−0.896217	+	0.443617i	$0.853695\pi$
$48$	0	0
$49$	−2.44469	−0.349242
$50$	0	0
$51$	−2.99121	−0.418853
$52$	0	0
$53$	4.33138	0.594961	0.297480	−	0.954728i	$-0.403854\pi$
$53$	4.33138	0.594961	0.297480	+	0.954728i	$0.403854\pi$
$54$	0	0
$55$	−3.07763	−0.414988
$56$	0	0
$57$	−3.43971	−0.455600
$58$	0	0
$59$	11.8442	1.54199	0.770994	−	0.636843i	$-0.219760\pi$
$59$	11.8442	1.54199	0.770994	+	0.636843i	$0.219760\pi$
$60$	0	0
$61$	−5.00040	−0.640235	−0.320118	−	0.947378i	$-0.603722\pi$
$61$	−5.00040	−0.640235	−0.320118	+	0.947378i	$0.603722\pi$
$62$	0	0
$63$	−5.86471	−0.738884
$64$	0	0
$65$	−4.17263	−0.517551
$66$	0	0
$67$	8.75350	1.06941	0.534705	−	0.845039i	$-0.320423\pi$
$67$	8.75350	1.06941	0.534705	+	0.845039i	$0.320423\pi$
$68$	0	0
$69$	4.22728	0.508904
$70$	0	0
$71$	−6.26627	−0.743669	−0.371834	−	0.928299i	$-0.621271\pi$
$71$	−6.26627	−0.743669	−0.371834	+	0.928299i	$0.621271\pi$
$72$	0	0
$73$	3.72925	0.436476	0.218238	−	0.975896i	$-0.429969\pi$
$73$	3.72925	0.436476	0.218238	+	0.975896i	$0.429969\pi$
$74$	0	0
$75$	2.28337	0.263660
$76$	0	0
$77$	9.75839	1.11207
$78$	0	0
$79$	−15.1159	−1.70067	−0.850336	−	0.526241i	$-0.823601\pi$
$79$	−15.1159	−1.70067	−0.850336	+	0.526241i	$0.823601\pi$
$80$	0	0
$81$	6.79393	0.754881
$82$	0	0
$83$	−1.89819	−0.208353	−0.104177	−	0.994559i	$-0.533221\pi$
$83$	−1.89819	−0.208353	−0.104177	+	0.994559i	$0.533221\pi$
$84$	0	0
$85$	−4.00944	−0.434885
$86$	0	0
$87$	1.05464	0.113070
$88$	0	0
$89$	−7.79885	−0.826676	−0.413338	−	0.910578i	$-0.635637\pi$
$89$	−7.79885	−0.826676	−0.413338	+	0.910578i	$0.635637\pi$
$90$	0	0
$91$	13.2303	1.38692
$92$	0	0
$93$	−2.87685	−0.298316
$94$	0	0
$95$	−4.61062	−0.473039
$96$	0	0
$97$	−15.9700	−1.62150	−0.810752	−	0.585390i	$-0.800941\pi$
$97$	−15.9700	−1.62150	−0.810752	+	0.585390i	$0.800941\pi$
$98$	0	0
$99$	−12.5634	−1.26267
$100$	0	0
$101$	−19.8111	−1.97128	−0.985641	−	0.168857i	$-0.945993\pi$
$101$	−19.8111	−1.97128	−0.985641	+	0.168857i	$0.945993\pi$
$102$	0	0
$103$	3.97184	0.391357	0.195679	−	0.980668i	$-0.437309\pi$
$103$	3.97184	0.391357	0.195679	+	0.980668i	$0.437309\pi$
$104$	0	0
$105$	0.721468	0.0704081
$106$	0	0
$107$	−9.49639	−0.918051	−0.459025	−	0.888423i	$-0.651801\pi$
$107$	−9.49639	−0.918051	−0.459025	+	0.888423i	$0.651801\pi$
$108$	0	0
$109$	12.6924	1.21571	0.607857	−	0.794046i	$-0.292029\pi$
$109$	12.6924	1.21571	0.607857	+	0.794046i	$0.292029\pi$
$110$	0	0
$111$	−5.39267	−0.511850
$112$	0	0
$113$	−9.77418	−0.919478	−0.459739	−	0.888054i	$-0.652057\pi$
$113$	−9.77418	−0.919478	−0.459739	+	0.888054i	$0.652057\pi$
$114$	0	0
$115$	5.66628	0.528384
$116$	0	0
$117$	−17.0333	−1.57473
$118$	0	0
$119$	12.7129	1.16539
$120$	0	0
$121$	9.90443	0.900403
$122$	0	0
$123$	−4.03800	−0.364094
$124$	0	0
$125$	6.42629	0.574785
$126$	0	0
$127$	16.3731	1.45288	0.726441	−	0.687229i	$-0.241173\pi$
$127$	16.3731	1.45288	0.726441	+	0.687229i	$0.241173\pi$
$128$	0	0
$129$	−0.333111	−0.0293288
$130$	0	0
$131$	−1.45056	−0.126736	−0.0633682	−	0.997990i	$-0.520184\pi$
$131$	−1.45056	−0.126736	−0.0633682	+	0.997990i	$0.520184\pi$
$132$	0	0
$133$	14.6191	1.26764
$134$	0	0
$135$	−1.94295	−0.167222
$136$	0	0
$137$	20.6292	1.76248	0.881238	−	0.472674i	$-0.156711\pi$
$137$	20.6292	1.76248	0.881238	+	0.472674i	$0.156711\pi$
$138$	0	0
$139$	−5.81960	−0.493612	−0.246806	−	0.969065i	$-0.579381\pi$
$139$	−5.81960	−0.493612	−0.246806	+	0.969065i	$0.579381\pi$
$140$	0	0
$141$	6.17095	0.519688
$142$	0	0
$143$	28.3420	2.37008
$144$	0	0
$145$	1.41365	0.117398
$146$	0	0
$147$	1.22768	0.101257
$148$	0	0
$149$	−1.77199	−0.145167	−0.0725835	−	0.997362i	$-0.523124\pi$
$149$	−1.77199	−0.145167	−0.0725835	+	0.997362i	$0.523124\pi$
$150$	0	0
$151$	5.53123	0.450125	0.225062	−	0.974344i	$-0.427741\pi$
$151$	5.53123	0.450125	0.225062	+	0.974344i	$0.427741\pi$
$152$	0	0
$153$	−16.3672	−1.32321
$154$	0	0
$155$	−3.85616	−0.309734
$156$	0	0
$157$	−20.7615	−1.65694	−0.828472	−	0.560030i	$-0.810790\pi$
$157$	−20.7615	−1.65694	−0.828472	+	0.560030i	$0.810790\pi$
$158$	0	0
$159$	−2.17514	−0.172500
$160$	0	0
$161$	−17.9663	−1.41594
$162$	0	0
$163$	2.48070	0.194303	0.0971516	−	0.995270i	$-0.469027\pi$
$163$	2.48070	0.194303	0.0971516	+	0.995270i	$0.469027\pi$
$164$	0	0
$165$	1.54553	0.120319
$166$	0	0
$167$	−8.90364	−0.688984	−0.344492	−	0.938789i	$-0.611949\pi$
$167$	−8.90364	−0.688984	−0.344492	+	0.938789i	$0.611949\pi$
$168$	0	0
$169$	25.4258	1.95583
$170$	0	0
$171$	−18.8213	−1.43930
$172$	0	0
$173$	−9.73205	−0.739914	−0.369957	−	0.929049i	$-0.620628\pi$
$173$	−9.73205	−0.739914	−0.369957	+	0.929049i	$0.620628\pi$
$174$	0	0
$175$	−9.70452	−0.733593
$176$	0	0
$177$	−5.94795	−0.447075
$178$	0	0
$179$	12.9967	0.971418	0.485709	−	0.874121i	$-0.338562\pi$
$179$	12.9967	0.971418	0.485709	+	0.874121i	$0.338562\pi$
$180$	0	0
$181$	−13.5114	−1.00429	−0.502147	−	0.864782i	$-0.667456\pi$
$181$	−13.5114	−1.00429	−0.502147	+	0.864782i	$0.667456\pi$
$182$	0	0
$183$	2.51110	0.185626
$184$	0	0
$185$	−7.22839	−0.531442
$186$	0	0
$187$	27.2336	1.99152
$188$	0	0
$189$	6.16058	0.448117
$190$	0	0
$191$	−5.14399	−0.372206	−0.186103	−	0.982530i	$-0.559586\pi$
$191$	−5.14399	−0.372206	−0.186103	+	0.982530i	$0.559586\pi$
$192$	0	0
$193$	1.47560	0.106216	0.0531082	−	0.998589i	$-0.483087\pi$
$193$	1.47560	0.106216	0.0531082	+	0.998589i	$0.483087\pi$
$194$	0	0
$195$	2.09541	0.150056
$196$	0	0
$197$	22.2114	1.58250	0.791249	−	0.611494i	$-0.209431\pi$
$197$	22.2114	1.58250	0.791249	+	0.611494i	$0.209431\pi$
$198$	0	0
$199$	15.7309	1.11513	0.557566	−	0.830132i	$-0.311735\pi$
$199$	15.7309	1.11513	0.557566	+	0.830132i	$0.311735\pi$
$200$	0	0
$201$	−4.39584	−0.310059
$202$	0	0
$203$	−4.48233	−0.314598
$204$	0	0
$205$	−5.41257	−0.378030
$206$	0	0
$207$	23.1306	1.60769
$208$	0	0
$209$	31.3170	2.16624
$210$	0	0
$211$	−3.02533	−0.208272	−0.104136	−	0.994563i	$-0.533208\pi$
$211$	−3.02533	−0.208272	−0.104136	+	0.994563i	$0.533208\pi$
$212$	0	0
$213$	3.14680	0.215615
$214$	0	0
$215$	−0.446505	−0.0304514
$216$	0	0
$217$	12.2269	0.830016
$218$	0	0
$219$	−1.87276	−0.126549
$220$	0	0
$221$	36.9231	2.48372
$222$	0	0
$223$	24.1643	1.61816	0.809080	−	0.587698i	$-0.199966\pi$
$223$	24.1643	1.61816	0.809080	+	0.587698i	$0.199966\pi$
$224$	0	0
$225$	12.4940	0.832935
$226$	0	0
$227$	9.89048	0.656454	0.328227	−	0.944599i	$-0.393549\pi$
$227$	9.89048	0.656454	0.328227	+	0.944599i	$0.393549\pi$
$228$	0	0
$229$	8.60616	0.568711	0.284356	−	0.958719i	$-0.408220\pi$
$229$	8.60616	0.568711	0.284356	+	0.958719i	$0.408220\pi$
$230$	0	0
$231$	−4.90048	−0.322428
$232$	0	0
$233$	−4.25146	−0.278523	−0.139261	−	0.990256i	$-0.544473\pi$
$233$	−4.25146	−0.278523	−0.139261	+	0.990256i	$0.544473\pi$
$234$	0	0
$235$	8.27161	0.539580
$236$	0	0
$237$	7.59092	0.493083
$238$	0	0
$239$	16.0177	1.03610	0.518050	−	0.855350i	$-0.326658\pi$
$239$	16.0177	1.03610	0.518050	+	0.855350i	$0.326658\pi$
$240$	0	0
$241$	6.48671	0.417846	0.208923	−	0.977932i	$-0.433004\pi$
$241$	6.48671	0.417846	0.208923	+	0.977932i	$0.433004\pi$
$242$	0	0
$243$	−12.0711	−0.774362
$244$	0	0
$245$	1.64559	0.105133
$246$	0	0
$247$	42.4593	2.70162
$248$	0	0
$249$	0.953234	0.0604087
$250$	0	0
$251$	20.0292	1.26423	0.632115	−	0.774874i	$-0.282187\pi$
$251$	20.0292	1.26423	0.632115	+	0.774874i	$0.282187\pi$
$252$	0	0
$253$	−38.4875	−2.41969
$254$	0	0
$255$	2.01347	0.126088
$256$	0	0
$257$	−9.17288	−0.572189	−0.286094	−	0.958201i	$-0.592357\pi$
$257$	−9.17288	−0.572189	−0.286094	+	0.958201i	$0.592357\pi$
$258$	0	0
$259$	22.9194	1.42414
$260$	0	0
$261$	5.77075	0.357201
$262$	0	0
$263$	−28.7061	−1.77009	−0.885047	−	0.465502i	$-0.845874\pi$
$263$	−28.7061	−1.77009	−0.885047	+	0.465502i	$0.845874\pi$
$264$	0	0
$265$	−2.91557	−0.179102
$266$	0	0
$267$	3.91643	0.239682
$268$	0	0
$269$	8.91228	0.543391	0.271696	−	0.962383i	$-0.412416\pi$
$269$	8.91228	0.543391	0.271696	+	0.962383i	$0.412416\pi$
$270$	0	0
$271$	8.09794	0.491915	0.245957	−	0.969281i	$-0.420898\pi$
$271$	8.09794	0.491915	0.245957	+	0.969281i	$0.420898\pi$
$272$	0	0
$273$	−6.64402	−0.402114
$274$	0	0
$275$	−20.7890	−1.25363
$276$	0	0
$277$	10.5348	0.632973	0.316487	−	0.948597i	$-0.397497\pi$
$277$	10.5348	0.632973	0.316487	+	0.948597i	$0.397497\pi$
$278$	0	0
$279$	−15.7415	−0.942416
$280$	0	0
$281$	−9.37494	−0.559262	−0.279631	−	0.960108i	$-0.590212\pi$
$281$	−9.37494	−0.559262	−0.279631	+	0.960108i	$0.590212\pi$
$282$	0	0
$283$	19.2728	1.14565	0.572825	−	0.819678i	$-0.305848\pi$
$283$	19.2728	1.14565	0.572825	+	0.819678i	$0.305848\pi$
$284$	0	0
$285$	2.31537	0.137150
$286$	0	0
$287$	17.1619	1.01303
$288$	0	0
$289$	18.4791	1.08701
$290$	0	0
$291$	8.01981	0.470129
$292$	0	0
$293$	−31.2574	−1.82608	−0.913039	−	0.407873i	$-0.866271\pi$
$293$	−31.2574	−1.82608	−0.913039	+	0.407873i	$0.866271\pi$
$294$	0	0
$295$	−7.97269	−0.464188
$296$	0	0
$297$	13.1972	0.765780
$298$	0	0
$299$	−52.1810	−3.01770
$300$	0	0
$301$	1.41575	0.0816026
$302$	0	0
$303$	9.94877	0.571542
$304$	0	0
$305$	3.36591	0.192731
$306$	0	0
$307$	14.6670	0.837089	0.418544	−	0.908196i	$-0.362540\pi$
$307$	14.6670	0.837089	0.418544	+	0.908196i	$0.362540\pi$
$308$	0	0
$309$	−1.99458	−0.113468
$310$	0	0
$311$	11.8901	0.674227	0.337114	−	0.941464i	$-0.390549\pi$
$311$	11.8901	0.674227	0.337114	+	0.941464i	$0.390549\pi$
$312$	0	0
$313$	−16.5167	−0.933576	−0.466788	−	0.884369i	$-0.654589\pi$
$313$	−16.5167	−0.933576	−0.466788	+	0.884369i	$0.654589\pi$
$314$	0	0
$315$	3.94770	0.222428
$316$	0	0
$317$	10.1504	0.570102	0.285051	−	0.958512i	$-0.407989\pi$
$317$	10.1504	0.570102	0.285051	+	0.958512i	$0.407989\pi$
$318$	0	0
$319$	−9.60205	−0.537612
$320$	0	0
$321$	4.76891	0.266175
$322$	0	0
$323$	40.7988	2.27011
$324$	0	0
$325$	−28.1856	−1.56345
$326$	0	0
$327$	−6.37390	−0.352477
$328$	0	0
$329$	−26.2271	−1.44595
$330$	0	0
$331$	−7.03197	−0.386512	−0.193256	−	0.981148i	$-0.561905\pi$
$331$	−7.03197	−0.386512	−0.193256	+	0.981148i	$0.561905\pi$
$332$	0	0
$333$	−29.5074	−1.61700
$334$	0	0
$335$	−5.89223	−0.321927
$336$	0	0
$337$	8.63549	0.470405	0.235202	−	0.971946i	$-0.424425\pi$
$337$	8.63549	0.470405	0.235202	+	0.971946i	$0.424425\pi$
$338$	0	0
$339$	4.90841	0.266588
$340$	0	0
$341$	26.1925	1.41840
$342$	0	0
$343$	−20.1580	−1.08843
$344$	0	0
$345$	−2.84550	−0.153197
$346$	0	0
$347$	17.2878	0.928057	0.464028	−	0.885820i	$-0.346404\pi$
$347$	17.2878	0.928057	0.464028	+	0.885820i	$0.346404\pi$
$348$	0	0
$349$	9.65188	0.516653	0.258327	−	0.966058i	$-0.416829\pi$
$349$	9.65188	0.516653	0.258327	+	0.966058i	$0.416829\pi$
$350$	0	0
$351$	17.8927	0.955039
$352$	0	0
$353$	−20.0584	−1.06760	−0.533802	−	0.845610i	$-0.679237\pi$
$353$	−20.0584	−1.06760	−0.533802	+	0.845610i	$0.679237\pi$
$354$	0	0
$355$	4.21800	0.223868
$356$	0	0
$357$	−6.38418	−0.337887
$358$	0	0
$359$	−2.39653	−0.126484	−0.0632419	−	0.997998i	$-0.520144\pi$
$359$	−2.39653	−0.126484	−0.0632419	+	0.997998i	$0.520144\pi$
$360$	0	0
$361$	27.9162	1.46927
$362$	0	0
$363$	−4.97382	−0.261058
$364$	0	0
$365$	−2.51026	−0.131393
$366$	0	0
$367$	−21.3049	−1.11210	−0.556052	−	0.831147i	$-0.687685\pi$
$367$	−21.3049	−1.11210	−0.556052	+	0.831147i	$0.687685\pi$
$368$	0	0
$369$	−22.0950	−1.15022
$370$	0	0
$371$	9.24454	0.479952
$372$	0	0
$373$	−4.69075	−0.242878	−0.121439	−	0.992599i	$-0.538751\pi$
$373$	−4.69075	−0.242878	−0.121439	+	0.992599i	$0.538751\pi$
$374$	0	0
$375$	−3.22716	−0.166650
$376$	0	0
$377$	−13.0184	−0.670480
$378$	0	0
$379$	−28.9108	−1.48505	−0.742525	−	0.669818i	$-0.766372\pi$
$379$	−28.9108	−1.48505	−0.742525	+	0.669818i	$0.766372\pi$
$380$	0	0
$381$	−8.22228	−0.421240
$382$	0	0
$383$	28.7211	1.46758	0.733790	−	0.679377i	$-0.237750\pi$
$383$	28.7211	1.46758	0.733790	+	0.679377i	$0.237750\pi$
$384$	0	0
$385$	−6.56865	−0.334769
$386$	0	0
$387$	−1.82270	−0.0926531
$388$	0	0
$389$	−31.7983	−1.61224	−0.806118	−	0.591755i	$-0.798435\pi$
$389$	−31.7983	−1.61224	−0.806118	+	0.591755i	$0.798435\pi$
$390$	0	0
$391$	−50.1403	−2.53570
$392$	0	0
$393$	0.728445	0.0367452
$394$	0	0
$395$	10.1749	0.511957
$396$	0	0
$397$	32.1263	1.61237	0.806185	−	0.591663i	$-0.201529\pi$
$397$	32.1263	1.61237	0.806185	+	0.591663i	$0.201529\pi$
$398$	0	0
$399$	−7.34143	−0.367531
$400$	0	0
$401$	−18.3102	−0.914370	−0.457185	−	0.889372i	$-0.651142\pi$
$401$	−18.3102	−0.914370	−0.457185	+	0.889372i	$0.651142\pi$
$402$	0	0
$403$	35.5115	1.76895
$404$	0	0
$405$	−4.57318	−0.227243
$406$	0	0
$407$	49.0979	2.43369
$408$	0	0
$409$	−32.7745	−1.62059	−0.810297	−	0.586020i	$-0.800694\pi$
$409$	−32.7745	−1.62059	−0.810297	+	0.586020i	$0.800694\pi$
$410$	0	0
$411$	−10.3596	−0.511002
$412$	0	0
$413$	25.2793	1.24392
$414$	0	0
$415$	1.27772	0.0627210
$416$	0	0
$417$	2.92249	0.143115
$418$	0	0
$419$	24.0615	1.17548	0.587740	−	0.809050i	$-0.300018\pi$
$419$	24.0615	1.17548	0.587740	+	0.809050i	$0.300018\pi$
$420$	0	0
$421$	−35.5690	−1.73353	−0.866763	−	0.498719i	$-0.833804\pi$
$421$	−35.5690	−1.73353	−0.866763	+	0.498719i	$0.833804\pi$
$422$	0	0
$423$	33.7660	1.64176
$424$	0	0
$425$	−27.0833	−1.31373
$426$	0	0
$427$	−10.6724	−0.516475
$428$	0	0
$429$	−14.2328	−0.687167
$430$	0	0
$431$	9.24482	0.445307	0.222654	−	0.974898i	$-0.428528\pi$
$431$	9.24482	0.445307	0.222654	+	0.974898i	$0.428528\pi$
$432$	0	0
$433$	−38.6764	−1.85867	−0.929335	−	0.369239i	$-0.879619\pi$
$433$	−38.6764	−1.85867	−0.929335	+	0.369239i	$0.879619\pi$
$434$	0	0
$435$	−0.709910	−0.0340376
$436$	0	0
$437$	−57.6583	−2.75817
$438$	0	0
$439$	31.8165	1.51852	0.759259	−	0.650789i	$-0.225562\pi$
$439$	31.8165	1.51852	0.759259	+	0.650789i	$0.225562\pi$
$440$	0	0
$441$	6.71756	0.319884
$442$	0	0
$443$	29.9844	1.42460	0.712302	−	0.701874i	$-0.247653\pi$
$443$	29.9844	1.42460	0.712302	+	0.701874i	$0.247653\pi$
$444$	0	0
$445$	5.24963	0.248856
$446$	0	0
$447$	0.889859	0.0420889
$448$	0	0
$449$	24.5905	1.16050	0.580250	−	0.814439i	$-0.302955\pi$
$449$	24.5905	1.16050	0.580250	+	0.814439i	$0.302955\pi$
$450$	0	0
$451$	36.7642	1.73116
$452$	0	0
$453$	−2.77768	−0.130507
$454$	0	0
$455$	−8.90571	−0.417506
$456$	0	0
$457$	1.56222	0.0730777	0.0365388	−	0.999332i	$-0.488367\pi$
$457$	1.56222	0.0730777	0.0365388	+	0.999332i	$0.488367\pi$
$458$	0	0
$459$	17.1929	0.802496
$460$	0	0
$461$	0.545580	0.0254102	0.0127051	−	0.999919i	$-0.495956\pi$
$461$	0.545580	0.0254102	0.0127051	+	0.999919i	$0.495956\pi$
$462$	0	0
$463$	1.99141	0.0925486	0.0462743	−	0.998929i	$-0.485265\pi$
$463$	1.99141	0.0925486	0.0462743	+	0.998929i	$0.485265\pi$
$464$	0	0
$465$	1.93649	0.0898027
$466$	0	0
$467$	4.99855	0.231306	0.115653	−	0.993290i	$-0.463104\pi$
$467$	4.99855	0.231306	0.115653	+	0.993290i	$0.463104\pi$
$468$	0	0
$469$	18.6827	0.862689
$470$	0	0
$471$	10.4260	0.480405
$472$	0	0
$473$	3.03282	0.139449
$474$	0	0
$475$	−31.1442	−1.42899
$476$	0	0
$477$	−11.9018	−0.544947
$478$	0	0
$479$	29.1561	1.33218	0.666089	−	0.745872i	$-0.267967\pi$
$479$	29.1561	1.33218	0.666089	+	0.745872i	$0.267967\pi$
$480$	0	0
$481$	66.5665	3.03517
$482$	0	0
$483$	9.02235	0.410531
$484$	0	0
$485$	10.7498	0.488125
$486$	0	0
$487$	−14.3712	−0.651223	−0.325612	−	0.945504i	$-0.605570\pi$
$487$	−14.3712	−0.651223	−0.325612	+	0.945504i	$0.605570\pi$
$488$	0	0
$489$	−1.24576	−0.0563352
$490$	0	0
$491$	−6.63850	−0.299591	−0.149796	−	0.988717i	$-0.547862\pi$
$491$	−6.63850	−0.299591	−0.149796	+	0.988717i	$0.547862\pi$
$492$	0	0
$493$	−12.5092	−0.563388
$494$	0	0
$495$	8.45677	0.380103
$496$	0	0
$497$	−13.3742	−0.599915
$498$	0	0
$499$	10.4479	0.467713	0.233857	−	0.972271i	$-0.424865\pi$
$499$	10.4479	0.467713	0.233857	+	0.972271i	$0.424865\pi$
$500$	0	0
$501$	4.47124	0.199760
$502$	0	0
$503$	12.3225	0.549435	0.274717	−	0.961525i	$-0.411416\pi$
$503$	12.3225	0.549435	0.274717	+	0.961525i	$0.411416\pi$
$504$	0	0
$505$	13.3354	0.593419
$506$	0	0
$507$	−12.7684	−0.567063
$508$	0	0
$509$	−23.2950	−1.03253	−0.516266	−	0.856429i	$-0.672678\pi$
$509$	−23.2950	−1.03253	−0.516266	+	0.856429i	$0.672678\pi$
$510$	0	0
$511$	7.95940	0.352103
$512$	0	0
$513$	19.7708	0.872902
$514$	0	0
$515$	−2.67356	−0.117811
$516$	0	0
$517$	−56.1838	−2.47096
$518$	0	0
$519$	4.88725	0.214527
$520$	0	0
$521$	22.9534	1.00561	0.502803	−	0.864401i	$-0.332302\pi$
$521$	22.9534	1.00561	0.502803	+	0.864401i	$0.332302\pi$
$522$	0	0
$523$	−30.1615	−1.31887	−0.659435	−	0.751762i	$-0.729204\pi$
$523$	−30.1615	−1.31887	−0.659435	+	0.751762i	$0.729204\pi$
$524$	0	0
$525$	4.87343	0.212694
$526$	0	0
$527$	34.1227	1.48641
$528$	0	0
$529$	47.8599	2.08087
$530$	0	0
$531$	−32.5457	−1.41236
$532$	0	0
$533$	49.8445	2.15901
$534$	0	0
$535$	6.39229	0.276363
$536$	0	0
$537$	−6.52669	−0.281647
$538$	0	0
$539$	−11.1775	−0.481447
$540$	0	0
$541$	0.722434	0.0310599	0.0155299	−	0.999879i	$-0.495056\pi$
$541$	0.722434	0.0310599	0.0155299	+	0.999879i	$0.495056\pi$
$542$	0	0
$543$	6.78517	0.291179
$544$	0	0
$545$	−8.54363	−0.365969
$546$	0	0
$547$	−6.77689	−0.289759	−0.144879	−	0.989449i	$-0.546279\pi$
$547$	−6.77689	−0.289759	−0.144879	+	0.989449i	$0.546279\pi$
$548$	0	0
$549$	13.7402	0.586416
$550$	0	0
$551$	−14.3849	−0.612817
$552$	0	0
$553$	−32.2621	−1.37192
$554$	0	0
$555$	3.62996	0.154083
$556$	0	0
$557$	39.7408	1.68387	0.841935	−	0.539579i	$-0.181416\pi$
$557$	39.7408	1.68387	0.841935	+	0.539579i	$0.181416\pi$
$558$	0	0
$559$	4.11187	0.173914
$560$	0	0
$561$	−13.6762	−0.577410
$562$	0	0
$563$	−20.6931	−0.872111	−0.436056	−	0.899920i	$-0.643625\pi$
$563$	−20.6931	−0.872111	−0.436056	+	0.899920i	$0.643625\pi$
$564$	0	0
$565$	6.57928	0.276792
$566$	0	0
$567$	14.5004	0.608959
$568$	0	0
$569$	−13.5588	−0.568415	−0.284208	−	0.958763i	$-0.591731\pi$
$569$	−13.5588	−0.568415	−0.284208	+	0.958763i	$0.591731\pi$
$570$	0	0
$571$	34.4577	1.44201	0.721005	−	0.692929i	$-0.243680\pi$
$571$	34.4577	1.44201	0.721005	+	0.692929i	$0.243680\pi$
$572$	0	0
$573$	2.58322	0.107915
$574$	0	0
$575$	38.2750	1.59618
$576$	0	0
$577$	−1.78011	−0.0741069	−0.0370534	−	0.999313i	$-0.511797\pi$
$577$	−1.78011	−0.0741069	−0.0370534	+	0.999313i	$0.511797\pi$
$578$	0	0
$579$	−0.741021	−0.0307958
$580$	0	0
$581$	−4.05133	−0.168078
$582$	0	0
$583$	19.8037	0.820184
$584$	0	0
$585$	11.4656	0.474044
$586$	0	0
$587$	1.85995	0.0767683	0.0383841	−	0.999263i	$-0.487779\pi$
$587$	1.85995	0.0767683	0.0383841	+	0.999263i	$0.487779\pi$
$588$	0	0
$589$	39.2391	1.61682
$590$	0	0
$591$	−11.1542	−0.458821
$592$	0	0
$593$	−11.7086	−0.480813	−0.240406	−	0.970672i	$-0.577281\pi$
$593$	−11.7086	−0.480813	−0.240406	+	0.970672i	$0.577281\pi$
$594$	0	0
$595$	−8.55742	−0.350820
$596$	0	0
$597$	−7.89975	−0.323315
$598$	0	0
$599$	−14.6502	−0.598589	−0.299295	−	0.954161i	$-0.596751\pi$
$599$	−14.6502	−0.598589	−0.299295	+	0.954161i	$0.596751\pi$
$600$	0	0
$601$	41.8384	1.70662	0.853311	−	0.521402i	$-0.174591\pi$
$601$	41.8384	1.70662	0.853311	+	0.521402i	$0.174591\pi$
$602$	0	0
$603$	−24.0530	−0.979513
$604$	0	0
$605$	−6.66695	−0.271050
$606$	0	0
$607$	17.8849	0.725927	0.362963	−	0.931803i	$-0.381765\pi$
$607$	17.8849	0.725927	0.362963	+	0.931803i	$0.381765\pi$
$608$	0	0
$609$	2.25094	0.0912128
$610$	0	0
$611$	−76.1734	−3.08165
$612$	0	0
$613$	18.7192	0.756060	0.378030	−	0.925793i	$-0.376602\pi$
$613$	18.7192	0.756060	0.378030	+	0.925793i	$0.376602\pi$
$614$	0	0
$615$	2.71809	0.109604
$616$	0	0
$617$	−14.4619	−0.582215	−0.291107	−	0.956690i	$-0.594024\pi$
$617$	−14.4619	−0.582215	−0.291107	+	0.956690i	$0.594024\pi$
$618$	0	0
$619$	−46.8001	−1.88105	−0.940527	−	0.339718i	$-0.889668\pi$
$619$	−46.8001	−1.88105	−0.940527	+	0.339718i	$0.889668\pi$
$620$	0	0
$621$	−24.2976	−0.975029
$622$	0	0
$623$	−16.6452	−0.666876
$624$	0	0
$625$	18.4088	0.736351
$626$	0	0
$627$	−15.7268	−0.628068
$628$	0	0
$629$	63.9632	2.55038
$630$	0	0
$631$	0.441258	0.0175662	0.00878310	−	0.999961i	$-0.497204\pi$
$631$	0.441258	0.0175662	0.00878310	+	0.999961i	$0.497204\pi$
$632$	0	0
$633$	1.51926	0.0603853
$634$	0	0
$635$	−11.0212	−0.437364
$636$	0	0
$637$	−15.1543	−0.600435
$638$	0	0
$639$	17.2185	0.681155
$640$	0	0
$641$	−16.8043	−0.663729	−0.331864	−	0.943327i	$-0.607678\pi$
$641$	−16.8043	−0.663729	−0.331864	+	0.943327i	$0.607678\pi$
$642$	0	0
$643$	−15.2811	−0.602626	−0.301313	−	0.953525i	$-0.597425\pi$
$643$	−15.2811	−0.602626	−0.301313	+	0.953525i	$0.597425\pi$
$644$	0	0
$645$	0.224226	0.00882890
$646$	0	0
$647$	−23.3205	−0.916825	−0.458413	−	0.888739i	$-0.651582\pi$
$647$	−23.3205	−0.916825	−0.458413	+	0.888739i	$0.651582\pi$
$648$	0	0
$649$	54.1534	2.12571
$650$	0	0
$651$	−6.14012	−0.240650
$652$	0	0
$653$	12.4865	0.488633	0.244317	−	0.969695i	$-0.421436\pi$
$653$	12.4865	0.488633	0.244317	+	0.969695i	$0.421436\pi$
$654$	0	0
$655$	0.976415	0.0381517
$656$	0	0
$657$	−10.2473	−0.399785
$658$	0	0
$659$	23.5357	0.916821	0.458411	−	0.888741i	$-0.348419\pi$
$659$	23.5357	0.916821	0.458411	+	0.888741i	$0.348419\pi$
$660$	0	0
$661$	0.342781	0.0133326	0.00666631	−	0.999978i	$-0.497878\pi$
$661$	0.342781	0.0133326	0.00666631	+	0.999978i	$0.497878\pi$
$662$	0	0
$663$	−18.5421	−0.720114
$664$	0	0
$665$	−9.84052	−0.381599
$666$	0	0
$667$	17.6785	0.684514
$668$	0	0
$669$	−12.1348	−0.469160
$670$	0	0
$671$	−22.8625	−0.882597
$672$	0	0
$673$	−4.48891	−0.173035	−0.0865174	−	0.996250i	$-0.527574\pi$
$673$	−4.48891	−0.173035	−0.0865174	+	0.996250i	$0.527574\pi$
$674$	0	0
$675$	−13.1244	−0.505157
$676$	0	0
$677$	19.3283	0.742846	0.371423	−	0.928464i	$-0.378870\pi$
$677$	19.3283	0.742846	0.371423	+	0.928464i	$0.378870\pi$
$678$	0	0
$679$	−34.0849	−1.30806
$680$	0	0
$681$	−4.96681	−0.190329
$682$	0	0
$683$	−6.60062	−0.252566	−0.126283	−	0.991994i	$-0.540305\pi$
$683$	−6.60062	−0.252566	−0.126283	+	0.991994i	$0.540305\pi$
$684$	0	0
$685$	−13.8861	−0.530562
$686$	0	0
$687$	−4.32185	−0.164889
$688$	0	0
$689$	26.8496	1.02289
$690$	0	0
$691$	−14.4772	−0.550737	−0.275369	−	0.961339i	$-0.588800\pi$
$691$	−14.4772	−0.550737	−0.275369	+	0.961339i	$0.588800\pi$
$692$	0	0
$693$	−26.8142	−1.01859
$694$	0	0
$695$	3.91734	0.148593
$696$	0	0
$697$	47.8952	1.81416
$698$	0	0
$699$	2.13500	0.0807533
$700$	0	0
$701$	−17.4031	−0.657307	−0.328653	−	0.944451i	$-0.606595\pi$
$701$	−17.4031	−0.657307	−0.328653	+	0.944451i	$0.606595\pi$
$702$	0	0
$703$	73.5538	2.77413
$704$	0	0
$705$	−4.15384	−0.156443
$706$	0	0
$707$	−42.2832	−1.59022
$708$	0	0
$709$	−27.4108	−1.02943	−0.514716	−	0.857361i	$-0.672103\pi$
$709$	−27.4108	−1.02943	−0.514716	+	0.857361i	$0.672103\pi$
$710$	0	0
$711$	41.5357	1.55771
$712$	0	0
$713$	−48.2234	−1.80598
$714$	0	0
$715$	−19.0778	−0.713470
$716$	0	0
$717$	−8.04379	−0.300401
$718$	0	0
$719$	20.9143	0.779971	0.389985	−	0.920821i	$-0.372480\pi$
$719$	20.9143	0.779971	0.389985	+	0.920821i	$0.372480\pi$
$720$	0	0
$721$	8.47717	0.315706
$722$	0	0
$723$	−3.25750	−0.121148
$724$	0	0
$725$	9.54905	0.354643
$726$	0	0
$727$	−21.8243	−0.809420	−0.404710	−	0.914445i	$-0.632627\pi$
$727$	−21.8243	−0.809420	−0.404710	+	0.914445i	$0.632627\pi$
$728$	0	0
$729$	−14.3199	−0.530366
$730$	0	0
$731$	3.95107	0.146135
$732$	0	0
$733$	−32.9107	−1.21558	−0.607792	−	0.794097i	$-0.707944\pi$
$733$	−32.9107	−1.21558	−0.607792	+	0.794097i	$0.707944\pi$
$734$	0	0
$735$	−0.826385	−0.0304817
$736$	0	0
$737$	40.0222	1.47424
$738$	0	0
$739$	24.3154	0.894455	0.447227	−	0.894420i	$-0.352411\pi$
$739$	24.3154	0.894455	0.447227	+	0.894420i	$0.352411\pi$
$740$	0	0
$741$	−21.3223	−0.783293
$742$	0	0
$743$	−31.5581	−1.15775	−0.578877	−	0.815415i	$-0.696509\pi$
$743$	−31.5581	−1.15775	−0.578877	+	0.815415i	$0.696509\pi$
$744$	0	0
$745$	1.19278	0.0436999
$746$	0	0
$747$	5.21587	0.190839
$748$	0	0
$749$	−20.2683	−0.740588
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−10.0583	−0.366544
$754$	0	0
$755$	−3.72322	−0.135502
$756$	0	0
$757$	−23.4160	−0.851071	−0.425535	−	0.904942i	$-0.639914\pi$
$757$	−23.4160	−0.851071	−0.425535	+	0.904942i	$0.639914\pi$
$758$	0	0
$759$	19.3277	0.701550
$760$	0	0
$761$	−1.84012	−0.0667042	−0.0333521	−	0.999444i	$-0.510618\pi$
$761$	−1.84012	−0.0667042	−0.0333521	+	0.999444i	$0.510618\pi$
$762$	0	0
$763$	27.0897	0.980712
$764$	0	0
$765$	11.0172	0.398328
$766$	0	0
$767$	73.4207	2.65107
$768$	0	0
$769$	28.9317	1.04330	0.521651	−	0.853159i	$-0.325316\pi$
$769$	28.9317	1.04330	0.521651	+	0.853159i	$0.325316\pi$
$770$	0	0
$771$	4.60645	0.165897
$772$	0	0
$773$	−24.2384	−0.871796	−0.435898	−	0.899996i	$-0.643569\pi$
$773$	−24.2384	−0.871796	−0.435898	+	0.899996i	$0.643569\pi$
$774$	0	0
$775$	−26.0479	−0.935668
$776$	0	0
$777$	−11.5097	−0.412907
$778$	0	0
$779$	55.0766	1.97332
$780$	0	0
$781$	−28.6502	−1.02519
$782$	0	0
$783$	−6.06189	−0.216634
$784$	0	0
$785$	13.9751	0.498794
$786$	0	0
$787$	11.2889	0.402405	0.201202	−	0.979550i	$-0.435515\pi$
$787$	11.2889	0.402405	0.201202	+	0.979550i	$0.435515\pi$
$788$	0	0
$789$	14.4157	0.513211
$790$	0	0
$791$	−20.8612	−0.741739
$792$	0	0
$793$	−30.9968	−1.10073
$794$	0	0
$795$	1.46415	0.0519279
$796$	0	0
$797$	9.44020	0.334389	0.167195	−	0.985924i	$-0.446529\pi$
$797$	9.44020	0.334389	0.167195	+	0.985924i	$0.446529\pi$
$798$	0	0
$799$	−73.1945	−2.58943
$800$	0	0
$801$	21.4298	0.757184
$802$	0	0
$803$	17.0506	0.601704
$804$	0	0
$805$	12.0936	0.426245
$806$	0	0
$807$	−4.47558	−0.157548
$808$	0	0
$809$	−17.1121	−0.601628	−0.300814	−	0.953683i	$-0.597258\pi$
$809$	−17.1121	−0.601628	−0.300814	+	0.953683i	$0.597258\pi$
$810$	0	0
$811$	25.8300	0.907015	0.453508	−	0.891252i	$-0.350172\pi$
$811$	25.8300	0.907015	0.453508	+	0.891252i	$0.350172\pi$
$812$	0	0
$813$	−4.06663	−0.142623
$814$	0	0
$815$	−1.66983	−0.0584915
$816$	0	0
$817$	4.54349	0.158957
$818$	0	0
$819$	−36.3545	−1.27033
$820$	0	0
$821$	−48.4600	−1.69127	−0.845633	−	0.533765i	$-0.820777\pi$
$821$	−48.4600	−1.69127	−0.845633	+	0.533765i	$0.820777\pi$
$822$	0	0
$823$	−43.1922	−1.50559	−0.752793	−	0.658257i	$-0.771294\pi$
$823$	−43.1922	−1.50559	−0.752793	+	0.658257i	$0.771294\pi$
$824$	0	0
$825$	10.4399	0.363469
$826$	0	0
$827$	−39.6208	−1.37775	−0.688874	−	0.724881i	$-0.741895\pi$
$827$	−39.6208	−1.37775	−0.688874	+	0.724881i	$0.741895\pi$
$828$	0	0
$829$	20.8106	0.722783	0.361391	−	0.932414i	$-0.382302\pi$
$829$	20.8106	0.722783	0.361391	+	0.932414i	$0.382302\pi$
$830$	0	0
$831$	−5.29036	−0.183521
$832$	0	0
$833$	−14.5616	−0.504531
$834$	0	0
$835$	5.99329	0.207406
$836$	0	0
$837$	16.5356	0.571555
$838$	0	0
$839$	−30.1973	−1.04252	−0.521262	−	0.853396i	$-0.674539\pi$
$839$	−30.1973	−1.04252	−0.521262	+	0.853396i	$0.674539\pi$
$840$	0	0
$841$	−24.5895	−0.847913
$842$	0	0
$843$	4.70792	0.162149
$844$	0	0
$845$	−17.1149	−0.588769
$846$	0	0
$847$	21.1392	0.726351
$848$	0	0
$849$	−9.67844	−0.332163
$850$	0	0
$851$	−90.3949	−3.09870
$852$	0	0
$853$	44.2562	1.51530	0.757652	−	0.652659i	$-0.226346\pi$
$853$	44.2562	1.51530	0.757652	+	0.652659i	$0.226346\pi$
$854$	0	0
$855$	12.6691	0.433275
$856$	0	0
$857$	−31.5582	−1.07801	−0.539003	−	0.842304i	$-0.681199\pi$
$857$	−31.5582	−1.07801	−0.539003	+	0.842304i	$0.681199\pi$
$858$	0	0
$859$	0.625261	0.0213336	0.0106668	−	0.999943i	$-0.496605\pi$
$859$	0.625261	0.0213336	0.0106668	+	0.999943i	$0.496605\pi$
$860$	0	0
$861$	−8.61837	−0.293713
$862$	0	0
$863$	41.5699	1.41506	0.707529	−	0.706685i	$-0.249810\pi$
$863$	41.5699	1.41506	0.707529	+	0.706685i	$0.249810\pi$
$864$	0	0
$865$	6.55092	0.222738
$866$	0	0
$867$	−9.27985	−0.315160
$868$	0	0
$869$	−69.1119	−2.34446
$870$	0	0
$871$	54.2617	1.83859
$872$	0	0
$873$	43.8825	1.48520
$874$	0	0
$875$	13.7157	0.463676
$876$	0	0
$877$	38.8143	1.31067	0.655333	−	0.755340i	$-0.272528\pi$
$877$	38.8143	1.31067	0.655333	+	0.755340i	$0.272528\pi$
$878$	0	0
$879$	15.6969	0.529443
$880$	0	0
$881$	−50.7974	−1.71141	−0.855704	−	0.517466i	$-0.826875\pi$
$881$	−50.7974	−1.71141	−0.855704	+	0.517466i	$0.826875\pi$
$882$	0	0
$883$	37.0376	1.24641	0.623207	−	0.782057i	$-0.285829\pi$
$883$	37.0376	1.24641	0.623207	+	0.782057i	$0.285829\pi$
$884$	0	0
$885$	4.00373	0.134584
$886$	0	0
$887$	−47.9332	−1.60944	−0.804720	−	0.593654i	$-0.797685\pi$
$887$	−47.9332	−1.60944	−0.804720	+	0.593654i	$0.797685\pi$
$888$	0	0
$889$	34.9455	1.17203
$890$	0	0
$891$	31.0628	1.04064
$892$	0	0
$893$	−84.1692	−2.81662
$894$	0	0
$895$	−8.74844	−0.292428
$896$	0	0
$897$	26.2043	0.874936
$898$	0	0
$899$	−12.0310	−0.401257
$900$	0	0
$901$	25.7996	0.859508
$902$	0	0
$903$	−0.710964	−0.0236594
$904$	0	0
$905$	9.09491	0.302325
$906$	0	0
$907$	−13.1100	−0.435310	−0.217655	−	0.976026i	$-0.569841\pi$
$907$	−13.1100	−0.435310	−0.217655	+	0.976026i	$0.569841\pi$
$908$	0	0
$909$	54.4373	1.80557
$910$	0	0
$911$	−3.32267	−0.110085	−0.0550424	−	0.998484i	$-0.517529\pi$
$911$	−3.32267	−0.110085	−0.0550424	+	0.998484i	$0.517529\pi$
$912$	0	0
$913$	−8.67877	−0.287225
$914$	0	0
$915$	−1.69030	−0.0558795
$916$	0	0
$917$	−3.09596	−0.102238
$918$	0	0
$919$	−50.6933	−1.67222	−0.836108	−	0.548565i	$-0.815174\pi$
$919$	−50.6933	−1.67222	−0.836108	+	0.548565i	$0.815174\pi$
$920$	0	0
$921$	−7.36548	−0.242701
$922$	0	0
$923$	−38.8437	−1.27856
$924$	0	0
$925$	−48.8269	−1.60542
$926$	0	0
$927$	−10.9139	−0.358459
$928$	0	0
$929$	−12.4152	−0.407331	−0.203665	−	0.979041i	$-0.565285\pi$
$929$	−12.4152	−0.407331	−0.203665	+	0.979041i	$0.565285\pi$
$930$	0	0
$931$	−16.7450	−0.548795
$932$	0	0
$933$	−5.97100	−0.195482
$934$	0	0
$935$	−18.3317	−0.599511
$936$	0	0
$937$	44.2248	1.44476	0.722380	−	0.691496i	$-0.243048\pi$
$937$	44.2248	1.44476	0.722380	+	0.691496i	$0.243048\pi$
$938$	0	0
$939$	8.29435	0.270676
$940$	0	0
$941$	−21.2130	−0.691526	−0.345763	−	0.938322i	$-0.612380\pi$
$941$	−21.2130	−0.691526	−0.345763	+	0.938322i	$0.612380\pi$
$942$	0	0
$943$	−67.6872	−2.20420
$944$	0	0
$945$	−4.14686	−0.134897
$946$	0	0
$947$	47.7873	1.55288	0.776440	−	0.630192i	$-0.217024\pi$
$947$	47.7873	1.55288	0.776440	+	0.630192i	$0.217024\pi$
$948$	0	0
$949$	23.1171	0.750412
$950$	0	0
$951$	−5.09733	−0.165292
$952$	0	0
$953$	19.7314	0.639164	0.319582	−	0.947559i	$-0.396458\pi$
$953$	19.7314	0.639164	0.319582	+	0.947559i	$0.396458\pi$
$954$	0	0
$955$	3.46257	0.112046
$956$	0	0
$957$	4.82197	0.155872
$958$	0	0
$959$	44.0293	1.42178
$960$	0	0
$961$	1.81823	0.0586524
$962$	0	0
$963$	26.0943	0.840878
$964$	0	0
$965$	−0.993271	−0.0319745
$966$	0	0
$967$	5.16997	0.166255	0.0831276	−	0.996539i	$-0.473509\pi$
$967$	5.16997	0.166255	0.0831276	+	0.996539i	$0.473509\pi$
$968$	0	0
$969$	−20.4884	−0.658182
$970$	0	0
$971$	16.4414	0.527631	0.263815	−	0.964573i	$-0.415019\pi$
$971$	16.4414	0.527631	0.263815	+	0.964573i	$0.415019\pi$
$972$	0	0
$973$	−12.4209	−0.398195
$974$	0	0
$975$	14.1543	0.453299
$976$	0	0
$977$	−17.9718	−0.574969	−0.287485	−	0.957785i	$-0.592819\pi$
$977$	−17.9718	−0.574969	−0.287485	+	0.957785i	$0.592819\pi$
$978$	0	0
$979$	−35.6574	−1.13962
$980$	0	0
$981$	−34.8764	−1.11352
$982$	0	0
$983$	15.8261	0.504775	0.252388	−	0.967626i	$-0.418784\pi$
$983$	15.8261	0.504775	0.252388	+	0.967626i	$0.418784\pi$
$984$	0	0
$985$	−14.9511	−0.476383
$986$	0	0
$987$	13.1708	0.419230
$988$	0	0
$989$	−5.58378	−0.177554
$990$	0	0
$991$	5.73303	0.182116	0.0910578	−	0.995846i	$-0.470975\pi$
$991$	5.73303	0.182116	0.0910578	+	0.995846i	$0.470975\pi$
$992$	0	0
$993$	3.53132	0.112063
$994$	0	0
$995$	−10.5889	−0.335691
$996$	0	0
$997$	−45.1093	−1.42863	−0.714313	−	0.699826i	$-0.753261\pi$
$997$	−45.1093	−1.42863	−0.714313	+	0.699826i	$0.753261\pi$
$998$	0	0
$999$	30.9961	0.980673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.19	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.19	✓	49	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.502181 q^{3} -0.673128 q^{5} +2.13432 q^{7} -2.74781 q^{9} +O(q^{10})\)	\(q-0.502181 q^{3} -0.673128 q^{5} +2.13432 q^{7} -2.74781 q^{9} +4.57214 q^{11} +6.19886 q^{13} +0.338032 q^{15} +5.95643 q^{17} +6.84954 q^{19} -1.07181 q^{21} -8.41783 q^{23} -4.54690 q^{25} +2.88644 q^{27} -2.10012 q^{29} +5.72872 q^{31} -2.29604 q^{33} -1.43667 q^{35} +10.7385 q^{37} -3.11295 q^{39} +8.04092 q^{41} +0.663328 q^{43} +1.84963 q^{45} -12.2883 q^{47} -2.44469 q^{49} -2.99121 q^{51} +4.33138 q^{53} -3.07763 q^{55} -3.43971 q^{57} +11.8442 q^{59} -5.00040 q^{61} -5.86471 q^{63} -4.17263 q^{65} +8.75350 q^{67} +4.22728 q^{69} -6.26627 q^{71} +3.72925 q^{73} +2.28337 q^{75} +9.75839 q^{77} -15.1159 q^{79} +6.79393 q^{81} -1.89819 q^{83} -4.00944 q^{85} +1.05464 q^{87} -7.79885 q^{89} +13.2303 q^{91} -2.87685 q^{93} -4.61062 q^{95} -15.9700 q^{97} -12.5634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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