LMFDB - Embedded newform 2368.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 116x^{4} - 221x^{2} + 128 $ x^8 - 20x^6 + 116x^4 - 221*x^2 + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.36494$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36494 q^{3} +1.03927 q^{5} -5.11652 q^{7} +2.59295 q^{9} +O(q^{10})$	$q-2.36494 q^{3} +1.03927 q^{5} -5.11652 q^{7} +2.59295 q^{9} -4.82275 q^{11} -5.73403 q^{13} -2.45781 q^{15} +2.00000 q^{17} -6.54057 q^{19} +12.1003 q^{21} +0.940894 q^{23} -3.91992 q^{25} +0.962658 q^{27} -5.73403 q^{29} +1.51691 q^{31} +11.4055 q^{33} -5.31744 q^{35} -1.00000 q^{37} +13.5606 q^{39} -2.36624 q^{41} +8.42236 q^{43} +2.69476 q^{45} -0.200912 q^{47} +19.1788 q^{49} -4.72988 q^{51} -13.2862 q^{53} -5.01212 q^{55} +15.4681 q^{57} -6.72630 q^{59} -5.03927 q^{61} -13.2669 q^{63} -5.95919 q^{65} +8.64503 q^{67} -2.22516 q^{69} -4.93079 q^{71} +7.59295 q^{73} +9.27039 q^{75} +24.6757 q^{77} +0.940894 q^{79} -10.0555 q^{81} +9.84641 q^{83} +2.07853 q^{85} +13.5606 q^{87} -12.8110 q^{89} +29.3383 q^{91} -3.58741 q^{93} -6.79740 q^{95} +2.00000 q^{97} -12.5051 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 8 * q + 2 * q^5 + 16 * q^9	$ 8 q + 2 q^{5} + 16 q^{9} - 2 q^{13} + 16 q^{17} - 2 q^{21} + 22 q^{25} - 2 q^{29} + 30 q^{33} - 8 q^{37} + 36 q^{41} - 16 q^{45} + 42 q^{49} + 2 q^{53} + 36 q^{57} - 34 q^{61} + 12 q^{65} - 2 q^{69} + 56 q^{73} + 14 q^{77} + 48 q^{81} + 4 q^{85} + 20 q^{89} + 12 q^{93} + 16 q^{97}+O(q^{100}) $ 8 * q + 2 * q^5 + 16 * q^9 - 2 * q^13 + 16 * q^17 - 2 * q^21 + 22 * q^25 - 2 * q^29 + 30 * q^33 - 8 * q^37 + 36 * q^41 - 16 * q^45 + 42 * q^49 + 2 * q^53 + 36 * q^57 - 34 * q^61 + 12 * q^65 - 2 * q^69 + 56 * q^73 + 14 * q^77 + 48 * q^81 + 4 * q^85 + 20 * q^89 + 12 * q^93 + 16 * q^97

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
$3$	−2.36494	−1.36540	−0.682700	−	0.730699i	$-0.739194\pi$
$3$	−2.36494	−1.36540	−0.682700	+	0.730699i	$0.739194\pi$
$4$	0	0
$5$	1.03927	0.464774	0.232387	−	0.972623i	$-0.425346\pi$
$5$	1.03927	0.464774	0.232387	+	0.972623i	$0.425346\pi$
$6$	0	0
$7$	−5.11652	−1.93386	−0.966932	−	0.255034i	$-0.917913\pi$
$7$	−5.11652	−1.93386	−0.966932	+	0.255034i	$0.917913\pi$
$8$	0	0
$9$	2.59295	0.864315
$10$	0	0
$11$	−4.82275	−1.45411	−0.727056	−	0.686578i	$-0.759112\pi$
$11$	−4.82275	−1.45411	−0.727056	+	0.686578i	$0.759112\pi$
$12$	0	0
$13$	−5.73403	−1.59033	−0.795167	−	0.606390i	$-0.792617\pi$
$13$	−5.73403	−1.59033	−0.795167	+	0.606390i	$0.792617\pi$
$14$	0	0
$15$	−2.45781	−0.634603
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−6.54057	−1.50051	−0.750255	−	0.661149i	$-0.770069\pi$
$19$	−6.54057	−1.50051	−0.750255	+	0.661149i	$0.770069\pi$
$20$	0	0
$21$	12.1003	2.64050
$22$	0	0
$23$	0.940894	0.196190	0.0980950	−	0.995177i	$-0.468725\pi$
$23$	0.940894	0.196190	0.0980950	+	0.995177i	$0.468725\pi$
$24$	0	0
$25$	−3.91992	−0.783985
$26$	0	0
$27$	0.962658	0.185264
$28$	0	0
$29$	−5.73403	−1.06478	−0.532391	−	0.846498i	$-0.678707\pi$
$29$	−5.73403	−1.06478	−0.532391	+	0.846498i	$0.678707\pi$
$30$	0	0
$31$	1.51691	0.272445	0.136223	−	0.990678i	$-0.456504\pi$
$31$	1.51691	0.272445	0.136223	+	0.990678i	$0.456504\pi$
$32$	0	0
$33$	11.4055	1.98544
$34$	0	0
$35$	−5.31744	−0.898811
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	13.5606	2.17144
$40$	0	0
$41$	−2.36624	−0.369545	−0.184773	−	0.982781i	$-0.559155\pi$
$41$	−2.36624	−0.369545	−0.184773	+	0.982781i	$0.559155\pi$
$42$	0	0
$43$	8.42236	1.28440	0.642199	−	0.766538i	$-0.278022\pi$
$43$	8.42236	1.28440	0.642199	+	0.766538i	$0.278022\pi$
$44$	0	0
$45$	2.69476	0.401712
$46$	0	0
$47$	−0.200912	−0.0293060	−0.0146530	−	0.999893i	$-0.504664\pi$
$47$	−0.200912	−0.0293060	−0.0146530	+	0.999893i	$0.504664\pi$
$48$	0	0
$49$	19.1788	2.73983
$50$	0	0
$51$	−4.72988	−0.662316
$52$	0	0
$53$	−13.2862	−1.82500	−0.912498	−	0.409082i	$-0.865849\pi$
$53$	−13.2862	−1.82500	−0.912498	+	0.409082i	$0.865849\pi$
$54$	0	0
$55$	−5.01212	−0.675834
$56$	0	0
$57$	15.4681	2.04879
$58$	0	0
$59$	−6.72630	−0.875690	−0.437845	−	0.899051i	$-0.644258\pi$
$59$	−6.72630	−0.875690	−0.437845	+	0.899051i	$0.644258\pi$
$60$	0	0
$61$	−5.03927	−0.645212	−0.322606	−	0.946533i	$-0.604559\pi$
$61$	−5.03927	−0.645212	−0.322606	+	0.946533i	$0.604559\pi$
$62$	0	0
$63$	−13.2669	−1.67147
$64$	0	0
$65$	−5.95919	−0.739147
$66$	0	0
$67$	8.64503	1.05616	0.528080	−	0.849195i	$-0.322912\pi$
$67$	8.64503	1.05616	0.528080	+	0.849195i	$0.322912\pi$
$68$	0	0
$69$	−2.22516	−0.267878
$70$	0	0
$71$	−4.93079	−0.585178	−0.292589	−	0.956238i	$-0.594517\pi$
$71$	−4.93079	−0.585178	−0.292589	+	0.956238i	$0.594517\pi$
$72$	0	0
$73$	7.59295	0.888687	0.444344	−	0.895856i	$-0.353437\pi$
$73$	7.59295	0.888687	0.444344	+	0.895856i	$0.353437\pi$
$74$	0	0
$75$	9.27039	1.07045
$76$	0	0
$77$	24.6757	2.81206
$78$	0	0
$79$	0.940894	0.105859	0.0529294	−	0.998598i	$-0.483144\pi$
$79$	0.940894	0.105859	0.0529294	+	0.998598i	$0.483144\pi$
$80$	0	0
$81$	−10.0555	−1.11727
$82$	0	0
$83$	9.84641	1.08078	0.540392	−	0.841414i	$-0.318276\pi$
$83$	9.84641	1.08078	0.540392	+	0.841414i	$0.318276\pi$
$84$	0	0
$85$	2.07853	0.225449
$86$	0	0
$87$	13.5606	1.45385
$88$	0	0
$89$	−12.8110	−1.35797	−0.678983	−	0.734154i	$-0.737579\pi$
$89$	−12.8110	−1.35797	−0.678983	+	0.734154i	$0.737579\pi$
$90$	0	0
$91$	29.3383	3.07549
$92$	0	0
$93$	−3.58741	−0.371997
$94$	0	0
$95$	−6.79740	−0.697399
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−12.5051	−1.25681
$100$	0	0
$101$	−4.02174	−0.400178	−0.200089	−	0.979778i	$-0.564123\pi$
$101$	−4.02174	−0.400178	−0.200089	+	0.979778i	$0.564123\pi$
$102$	0	0
$103$	−17.7741	−1.75133	−0.875666	−	0.482917i	$-0.839577\pi$
$103$	−17.7741	−1.75133	−0.875666	+	0.482917i	$0.839577\pi$
$104$	0	0
$105$	12.5754	1.22724
$106$	0	0
$107$	−8.25475	−0.798016	−0.399008	−	0.916947i	$-0.630646\pi$
$107$	−8.25475	−0.798016	−0.399008	+	0.916947i	$0.630646\pi$
$108$	0	0
$109$	−7.26443	−0.695806	−0.347903	−	0.937531i	$-0.613106\pi$
$109$	−7.26443	−0.695806	−0.347903	+	0.937531i	$0.613106\pi$
$110$	0	0
$111$	2.36494	0.224470
$112$	0	0
$113$	−0.892642	−0.0839727	−0.0419864	−	0.999118i	$-0.513369\pi$
$113$	−0.892642	−0.0839727	−0.0419864	+	0.999118i	$0.513369\pi$
$114$	0	0
$115$	0.977841	0.0911841
$116$	0	0
$117$	−14.8680	−1.37455
$118$	0	0
$119$	−10.2330	−0.938062
$120$	0	0
$121$	12.2589	1.11444
$122$	0	0
$123$	5.59603	0.504577
$124$	0	0
$125$	−9.27018	−0.829150
$126$	0	0
$127$	−5.30225	−0.470499	−0.235249	−	0.971935i	$-0.575591\pi$
$127$	−5.30225	−0.470499	−0.235249	+	0.971935i	$0.575591\pi$
$128$	0	0
$129$	−19.9184	−1.75372
$130$	0	0
$131$	16.1861	1.41418	0.707091	−	0.707122i	$-0.250007\pi$
$131$	16.1861	1.41418	0.707091	+	0.707122i	$0.250007\pi$
$132$	0	0
$133$	33.4650	2.90178
$134$	0	0
$135$	1.00046	0.0861058
$136$	0	0
$137$	12.3503	1.05515	0.527577	−	0.849507i	$-0.323101\pi$
$137$	12.3503	1.05515	0.527577	+	0.849507i	$0.323101\pi$
$138$	0	0
$139$	10.3296	0.876142	0.438071	−	0.898940i	$-0.355662\pi$
$139$	10.3296	0.876142	0.438071	+	0.898940i	$0.355662\pi$
$140$	0	0
$141$	0.475144	0.0400144
$142$	0	0
$143$	27.6538	2.31253
$144$	0	0
$145$	−5.95919	−0.494884
$146$	0	0
$147$	−45.3568	−3.74096
$148$	0	0
$149$	23.3647	1.91411	0.957056	−	0.289905i	$-0.0936236\pi$
$149$	23.3647	1.91411	0.957056	+	0.289905i	$0.0936236\pi$
$150$	0	0
$151$	10.4188	0.847868	0.423934	−	0.905693i	$-0.360649\pi$
$151$	10.4188	0.847868	0.423934	+	0.905693i	$0.360649\pi$
$152$	0	0
$153$	5.18589	0.419255
$154$	0	0
$155$	1.57648	0.126626
$156$	0	0
$157$	10.9144	0.871063	0.435531	−	0.900174i	$-0.356561\pi$
$157$	10.9144	0.871063	0.435531	+	0.900174i	$0.356561\pi$
$158$	0	0
$159$	31.4210	2.49185
$160$	0	0
$161$	−4.81411	−0.379405
$162$	0	0
$163$	0.0711007	0.00556904	0.00278452	−	0.999996i	$-0.499114\pi$
$163$	0.0711007	0.00556904	0.00278452	+	0.999996i	$0.499114\pi$
$164$	0	0
$165$	11.8534	0.922784
$166$	0	0
$167$	−11.9357	−0.923611	−0.461806	−	0.886981i	$-0.652798\pi$
$167$	−11.9357	−0.923611	−0.461806	+	0.886981i	$0.652798\pi$
$168$	0	0
$169$	19.8791	1.52916
$170$	0	0
$171$	−16.9593	−1.29691
$172$	0	0
$173$	11.8647	0.902054	0.451027	−	0.892510i	$-0.351058\pi$
$173$	11.8647	0.902054	0.451027	+	0.892510i	$0.351058\pi$
$174$	0	0
$175$	20.0564	1.51612
$176$	0	0
$177$	15.9073	1.19567
$178$	0	0
$179$	−17.8821	−1.33657	−0.668286	−	0.743904i	$-0.732972\pi$
$179$	−17.8821	−1.33657	−0.668286	+	0.743904i	$0.732972\pi$
$180$	0	0
$181$	−11.8966	−0.884270	−0.442135	−	0.896948i	$-0.645779\pi$
$181$	−11.8966	−0.884270	−0.442135	+	0.896948i	$0.645779\pi$
$182$	0	0
$183$	11.9176	0.880972
$184$	0	0
$185$	−1.03927	−0.0764084
$186$	0	0
$187$	−9.64549	−0.705348
$188$	0	0
$189$	−4.92546	−0.358275
$190$	0	0
$191$	−12.3375	−0.892711	−0.446355	−	0.894856i	$-0.647278\pi$
$191$	−12.3375	−0.892711	−0.446355	+	0.894856i	$0.647278\pi$
$192$	0	0
$193$	−15.3895	−1.10776	−0.553881	−	0.832596i	$-0.686854\pi$
$193$	−15.3895	−1.10776	−0.553881	+	0.832596i	$0.686854\pi$
$194$	0	0
$195$	14.0931	1.00923
$196$	0	0
$197$	25.2862	1.80156	0.900782	−	0.434271i	$-0.142994\pi$
$197$	25.2862	1.80156	0.900782	+	0.434271i	$0.142994\pi$
$198$	0	0
$199$	−1.99642	−0.141522	−0.0707611	−	0.997493i	$-0.522543\pi$
$199$	−1.99642	−0.141522	−0.0707611	+	0.997493i	$0.522543\pi$
$200$	0	0
$201$	−20.4450	−1.44208
$202$	0	0
$203$	29.3383	2.05915
$204$	0	0
$205$	−2.45916	−0.171755
$206$	0	0
$207$	2.43969	0.169570
$208$	0	0
$209$	31.5435	2.18191
$210$	0	0
$211$	−18.4914	−1.27300	−0.636502	−	0.771275i	$-0.719619\pi$
$211$	−18.4914	−1.27300	−0.636502	+	0.771275i	$0.719619\pi$
$212$	0	0
$213$	11.6610	0.799001
$214$	0	0
$215$	8.75308	0.596955
$216$	0	0
$217$	−7.76131	−0.526872
$218$	0	0
$219$	−17.9569	−1.21341
$220$	0	0
$221$	−11.4681	−0.771425
$222$	0	0
$223$	−0.200912	−0.0134540	−0.00672702	−	0.999977i	$-0.502141\pi$
$223$	−0.200912	−0.0134540	−0.00672702	+	0.999977i	$0.502141\pi$
$224$	0	0
$225$	−10.1642	−0.677610
$226$	0	0
$227$	17.8821	1.18688	0.593439	−	0.804879i	$-0.297770\pi$
$227$	17.8821	1.18688	0.593439	+	0.804879i	$0.297770\pi$
$228$	0	0
$229$	−8.71075	−0.575622	−0.287811	−	0.957687i	$-0.592928\pi$
$229$	−8.71075	−0.575622	−0.287811	+	0.957687i	$0.592928\pi$
$230$	0	0
$231$	−58.3566	−3.83958
$232$	0	0
$233$	16.5451	1.08390	0.541951	−	0.840410i	$-0.317686\pi$
$233$	16.5451	1.08390	0.541951	+	0.840410i	$0.317686\pi$
$234$	0	0
$235$	−0.208801	−0.0136207
$236$	0	0
$237$	−2.22516	−0.144540
$238$	0	0
$239$	−0.929359	−0.0601152	−0.0300576	−	0.999548i	$-0.509569\pi$
$239$	−0.929359	−0.0601152	−0.0300576	+	0.999548i	$0.509569\pi$
$240$	0	0
$241$	−8.65395	−0.557450	−0.278725	−	0.960371i	$-0.589912\pi$
$241$	−8.65395	−0.557450	−0.278725	+	0.960371i	$0.589912\pi$
$242$	0	0
$243$	20.8926	1.34026
$244$	0	0
$245$	19.9319	1.27340
$246$	0	0
$247$	37.5038	2.38631
$248$	0	0
$249$	−23.2862	−1.47570
$250$	0	0
$251$	0.851676	0.0537573	0.0268787	−	0.999639i	$-0.491443\pi$
$251$	0.851676	0.0537573	0.0268787	+	0.999639i	$0.491443\pi$
$252$	0	0
$253$	−4.53770	−0.285282
$254$	0	0
$255$	−4.91561	−0.307828
$256$	0	0
$257$	−1.84293	−0.114959	−0.0574794	−	0.998347i	$-0.518306\pi$
$257$	−1.84293	−0.114959	−0.0574794	+	0.998347i	$0.518306\pi$
$258$	0	0
$259$	5.11652	0.317925
$260$	0	0
$261$	−14.8680	−0.920308
$262$	0	0
$263$	−17.0456	−1.05108	−0.525539	−	0.850769i	$-0.676136\pi$
$263$	−17.0456	−1.05108	−0.525539	+	0.850769i	$0.676136\pi$
$264$	0	0
$265$	−13.8079	−0.848211
$266$	0	0
$267$	30.2973	1.85417
$268$	0	0
$269$	−22.0755	−1.34596	−0.672982	−	0.739659i	$-0.734987\pi$
$269$	−22.0755	−1.34596	−0.672982	+	0.739659i	$0.734987\pi$
$270$	0	0
$271$	20.0795	1.21974	0.609870	−	0.792502i	$-0.291222\pi$
$271$	20.0795	1.21974	0.609870	+	0.792502i	$0.291222\pi$
$272$	0	0
$273$	−69.3834	−4.19927
$274$	0	0
$275$	18.9048	1.14000
$276$	0	0
$277$	−23.1613	−1.39163	−0.695813	−	0.718223i	$-0.744956\pi$
$277$	−23.1613	−1.39163	−0.695813	+	0.718223i	$0.744956\pi$
$278$	0	0
$279$	3.93327	0.235479
$280$	0	0
$281$	2.12510	0.126773	0.0633865	−	0.997989i	$-0.479810\pi$
$281$	2.12510	0.126773	0.0633865	+	0.997989i	$0.479810\pi$
$282$	0	0
$283$	5.20281	0.309275	0.154637	−	0.987971i	$-0.450579\pi$
$283$	5.20281	0.309275	0.154637	+	0.987971i	$0.450579\pi$
$284$	0	0
$285$	16.0755	0.952228
$286$	0	0
$287$	12.1069	0.714650
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−4.72988	−0.277271
$292$	0	0
$293$	19.0963	1.11562	0.557808	−	0.829970i	$-0.311642\pi$
$293$	19.0963	1.11562	0.557808	+	0.829970i	$0.311642\pi$
$294$	0	0
$295$	−6.99042	−0.406998
$296$	0	0
$297$	−4.64266	−0.269394
$298$	0	0
$299$	−5.39512	−0.312008
$300$	0	0
$301$	−43.0932	−2.48385
$302$	0	0
$303$	9.51118	0.546403
$304$	0	0
$305$	−5.23715	−0.299878
$306$	0	0
$307$	−23.7973	−1.35819	−0.679093	−	0.734052i	$-0.737627\pi$
$307$	−23.7973	−1.35819	−0.679093	+	0.734052i	$0.737627\pi$
$308$	0	0
$309$	42.0346	2.39127
$310$	0	0
$311$	−22.3733	−1.26867	−0.634337	−	0.773057i	$-0.718727\pi$
$311$	−22.3733	−1.26867	−0.634337	+	0.773057i	$0.718727\pi$
$312$	0	0
$313$	26.4938	1.49752	0.748759	−	0.662842i	$-0.230650\pi$
$313$	26.4938	1.49752	0.748759	+	0.662842i	$0.230650\pi$
$314$	0	0
$315$	−13.7878	−0.776856
$316$	0	0
$317$	23.5466	1.32251	0.661254	−	0.750162i	$-0.270024\pi$
$317$	23.5466	1.32251	0.661254	+	0.750162i	$0.270024\pi$
$318$	0	0
$319$	27.6538	1.54831
$320$	0	0
$321$	19.5220	1.08961
$322$	0	0
$323$	−13.0811	−0.727854
$324$	0	0
$325$	22.4770	1.24680
$326$	0	0
$327$	17.1799	0.950053
$328$	0	0
$329$	1.02797	0.0566738
$330$	0	0
$331$	−2.39824	−0.131819	−0.0659096	−	0.997826i	$-0.520995\pi$
$331$	−2.39824	−0.131819	−0.0659096	+	0.997826i	$0.520995\pi$
$332$	0	0
$333$	−2.59295	−0.142093
$334$	0	0
$335$	8.98450	0.490876
$336$	0	0
$337$	3.43588	0.187164	0.0935821	−	0.995612i	$-0.470168\pi$
$337$	3.43588	0.187164	0.0935821	+	0.995612i	$0.470168\pi$
$338$	0	0
$339$	2.11105	0.114656
$340$	0	0
$341$	−7.31568	−0.396166
$342$	0	0
$343$	−62.3132	−3.36459
$344$	0	0
$345$	−2.31254	−0.124503
$346$	0	0
$347$	−23.5783	−1.26575	−0.632875	−	0.774254i	$-0.718125\pi$
$347$	−23.5783	−1.26575	−0.632875	+	0.774254i	$0.718125\pi$
$348$	0	0
$349$	3.99692	0.213950	0.106975	−	0.994262i	$-0.465884\pi$
$349$	3.99692	0.213950	0.106975	+	0.994262i	$0.465884\pi$
$350$	0	0
$351$	−5.51991	−0.294631
$352$	0	0
$353$	−25.6251	−1.36389	−0.681944	−	0.731404i	$-0.738865\pi$
$353$	−25.6251	−1.36389	−0.681944	+	0.731404i	$0.738865\pi$
$354$	0	0
$355$	−5.12441	−0.271976
$356$	0	0
$357$	24.2006	1.28083
$358$	0	0
$359$	−8.69437	−0.458871	−0.229436	−	0.973324i	$-0.573688\pi$
$359$	−8.69437	−0.458871	−0.229436	+	0.973324i	$0.573688\pi$
$360$	0	0
$361$	23.7791	1.25153
$362$	0	0
$363$	−28.9915	−1.52166
$364$	0	0
$365$	7.89110	0.413039
$366$	0	0
$367$	−7.94943	−0.414957	−0.207479	−	0.978240i	$-0.566526\pi$
$367$	−7.94943	−0.414957	−0.207479	+	0.978240i	$0.566526\pi$
$368$	0	0
$369$	−6.13555	−0.319404
$370$	0	0
$371$	67.9790	3.52929
$372$	0	0
$373$	−29.0684	−1.50510	−0.752552	−	0.658533i	$-0.771177\pi$
$373$	−29.0684	−1.50510	−0.752552	+	0.658533i	$0.771177\pi$
$374$	0	0
$375$	21.9234	1.13212
$376$	0	0
$377$	32.8791	1.69336
$378$	0	0
$379$	−10.7162	−0.550454	−0.275227	−	0.961379i	$-0.588753\pi$
$379$	−10.7162	−0.550454	−0.275227	+	0.961379i	$0.588753\pi$
$380$	0	0
$381$	12.5395	0.642419
$382$	0	0
$383$	−5.24633	−0.268075	−0.134038	−	0.990976i	$-0.542794\pi$
$383$	−5.24633	−0.268075	−0.134038	+	0.990976i	$0.542794\pi$
$384$	0	0
$385$	25.6446	1.30697
$386$	0	0
$387$	21.8387	1.11012
$388$	0	0
$389$	−9.89110	−0.501499	−0.250749	−	0.968052i	$-0.580677\pi$
$389$	−9.89110	−0.501499	−0.250749	+	0.968052i	$0.580677\pi$
$390$	0	0
$391$	1.88179	0.0951661
$392$	0	0
$393$	−38.2791	−1.93092
$394$	0	0
$395$	0.977841	0.0492005
$396$	0	0
$397$	−6.18189	−0.310260	−0.155130	−	0.987894i	$-0.549580\pi$
$397$	−6.18189	−0.310260	−0.155130	+	0.987894i	$0.549580\pi$
$398$	0	0
$399$	−79.1427	−3.96209
$400$	0	0
$401$	−20.2791	−1.01269	−0.506345	−	0.862331i	$-0.669004\pi$
$401$	−20.2791	−1.01269	−0.506345	+	0.862331i	$0.669004\pi$
$402$	0	0
$403$	−8.69802	−0.433279
$404$	0	0
$405$	−10.4503	−0.519281
$406$	0	0
$407$	4.82275	0.239055
$408$	0	0
$409$	−17.6251	−0.871507	−0.435753	−	0.900066i	$-0.643518\pi$
$409$	−17.6251	−0.871507	−0.435753	+	0.900066i	$0.643518\pi$
$410$	0	0
$411$	−29.2076	−1.44071
$412$	0	0
$413$	34.4153	1.69346
$414$	0	0
$415$	10.2330	0.502320
$416$	0	0
$417$	−24.4288	−1.19628
$418$	0	0
$419$	−5.21303	−0.254673	−0.127337	−	0.991860i	$-0.540643\pi$
$419$	−5.21303	−0.254673	−0.127337	+	0.991860i	$0.540643\pi$
$420$	0	0
$421$	−2.10890	−0.102781	−0.0513907	−	0.998679i	$-0.516365\pi$
$421$	−2.10890	−0.102781	−0.0513907	+	0.998679i	$0.516365\pi$
$422$	0	0
$423$	−0.520953	−0.0253296
$424$	0	0
$425$	−7.83985	−0.380288
$426$	0	0
$427$	25.7835	1.24775
$428$	0	0
$429$	−65.3996	−3.15752
$430$	0	0
$431$	−14.0447	−0.676507	−0.338254	−	0.941055i	$-0.609836\pi$
$431$	−14.0447	−0.676507	−0.338254	+	0.941055i	$0.609836\pi$
$432$	0	0
$433$	18.2092	0.875077	0.437539	−	0.899200i	$-0.355850\pi$
$433$	18.2092	0.875077	0.437539	+	0.899200i	$0.355850\pi$
$434$	0	0
$435$	14.0931	0.675714
$436$	0	0
$437$	−6.15399	−0.294385
$438$	0	0
$439$	33.3130	1.58994	0.794972	−	0.606646i	$-0.207486\pi$
$439$	33.3130	1.58994	0.794972	+	0.606646i	$0.207486\pi$
$440$	0	0
$441$	49.7296	2.36808
$442$	0	0
$443$	−38.3265	−1.82094	−0.910472	−	0.413570i	$-0.864282\pi$
$443$	−38.3265	−1.82094	−0.910472	+	0.413570i	$0.864282\pi$
$444$	0	0
$445$	−13.3141	−0.631148
$446$	0	0
$447$	−55.2561	−2.61353
$448$	0	0
$449$	−24.7294	−1.16705	−0.583526	−	0.812094i	$-0.698328\pi$
$449$	−24.7294	−1.16705	−0.583526	+	0.812094i	$0.698328\pi$
$450$	0	0
$451$	11.4118	0.537361
$452$	0	0
$453$	−24.6398	−1.15768
$454$	0	0
$455$	30.4903	1.42941
$456$	0	0
$457$	−22.8576	−1.06923	−0.534616	−	0.845095i	$-0.679544\pi$
$457$	−22.8576	−1.06923	−0.534616	+	0.845095i	$0.679544\pi$
$458$	0	0
$459$	1.92532	0.0898661
$460$	0	0
$461$	2.95029	0.137409	0.0687043	−	0.997637i	$-0.478113\pi$
$461$	2.95029	0.137409	0.0687043	+	0.997637i	$0.478113\pi$
$462$	0	0
$463$	30.4649	1.41583	0.707913	−	0.706300i	$-0.249637\pi$
$463$	30.4649	1.41583	0.707913	+	0.706300i	$0.249637\pi$
$464$	0	0
$465$	−3.72827	−0.172895
$466$	0	0
$467$	−11.8580	−0.548723	−0.274362	−	0.961627i	$-0.588466\pi$
$467$	−11.8580	−0.548723	−0.274362	+	0.961627i	$0.588466\pi$
$468$	0	0
$469$	−44.2325	−2.04247
$470$	0	0
$471$	−25.8119	−1.18935
$472$	0	0
$473$	−40.6189	−1.86766
$474$	0	0
$475$	25.6385	1.17638
$476$	0	0
$477$	−34.4503	−1.57737
$478$	0	0
$479$	−3.78899	−0.173123	−0.0865616	−	0.996246i	$-0.527588\pi$
$479$	−3.78899	−0.173123	−0.0865616	+	0.996246i	$0.527588\pi$
$480$	0	0
$481$	5.73403	0.261449
$482$	0	0
$483$	11.3851	0.518039
$484$	0	0
$485$	2.07853	0.0943814
$486$	0	0
$487$	10.6829	0.484088	0.242044	−	0.970265i	$-0.422182\pi$
$487$	10.6829	0.484088	0.242044	+	0.970265i	$0.422182\pi$
$488$	0	0
$489$	−0.168149	−0.00760396
$490$	0	0
$491$	23.4107	1.05651	0.528255	−	0.849086i	$-0.322847\pi$
$491$	23.4107	1.05651	0.528255	+	0.849086i	$0.322847\pi$
$492$	0	0
$493$	−11.4681	−0.516496
$494$	0	0
$495$	−12.9962	−0.584134
$496$	0	0
$497$	25.2285	1.13165
$498$	0	0
$499$	−10.7495	−0.481214	−0.240607	−	0.970623i	$-0.577346\pi$
$499$	−10.7495	−0.481214	−0.240607	+	0.970623i	$0.577346\pi$
$500$	0	0
$501$	28.2272	1.26110
$502$	0	0
$503$	4.17928	0.186345	0.0931723	−	0.995650i	$-0.470299\pi$
$503$	4.17928	0.186345	0.0931723	+	0.995650i	$0.470299\pi$
$504$	0	0
$505$	−4.17966	−0.185993
$506$	0	0
$507$	−47.0129	−2.08792
$508$	0	0
$509$	−20.9929	−0.930495	−0.465247	−	0.885181i	$-0.654035\pi$
$509$	−20.9929	−0.930495	−0.465247	+	0.885181i	$0.654035\pi$
$510$	0	0
$511$	−38.8495	−1.71860
$512$	0	0
$513$	−6.29633	−0.277990
$514$	0	0
$515$	−18.4720	−0.813974
$516$	0	0
$517$	0.968947	0.0426142
$518$	0	0
$519$	−28.0593	−1.23166
$520$	0	0
$521$	33.4867	1.46708	0.733540	−	0.679646i	$-0.237867\pi$
$521$	33.4867	1.46708	0.733540	+	0.679646i	$0.237867\pi$
$522$	0	0
$523$	19.4360	0.849876	0.424938	−	0.905222i	$-0.360296\pi$
$523$	19.4360	0.849876	0.424938	+	0.905222i	$0.360296\pi$
$524$	0	0
$525$	−47.4322	−2.07011
$526$	0	0
$527$	3.03382	0.132155
$528$	0	0
$529$	−22.1147	−0.961509
$530$	0	0
$531$	−17.4409	−0.756872
$532$	0	0
$533$	13.5681	0.587701
$534$	0	0
$535$	−8.57889	−0.370898
$536$	0	0
$537$	42.2902	1.82496
$538$	0	0
$539$	−92.4945	−3.98402
$540$	0	0
$541$	27.4754	1.18126	0.590630	−	0.806942i	$-0.298879\pi$
$541$	27.4754	1.18126	0.590630	+	0.806942i	$0.298879\pi$
$542$	0	0
$543$	28.1349	1.20738
$544$	0	0
$545$	−7.54968	−0.323393
$546$	0	0
$547$	0.665947	0.0284738	0.0142369	−	0.999899i	$-0.495468\pi$
$547$	0.665947	0.0284738	0.0142369	+	0.999899i	$0.495468\pi$
$548$	0	0
$549$	−13.0665	−0.557667
$550$	0	0
$551$	37.5038	1.59772
$552$	0	0
$553$	−4.81411	−0.204717
$554$	0	0
$555$	2.45781	0.104328
$556$	0	0
$557$	13.7398	0.582174	0.291087	−	0.956697i	$-0.405983\pi$
$557$	13.7398	0.582174	0.291087	+	0.956697i	$0.405983\pi$
$558$	0	0
$559$	−48.2941	−2.04262
$560$	0	0
$561$	22.8110	0.963082
$562$	0	0
$563$	−5.36546	−0.226127	−0.113064	−	0.993588i	$-0.536066\pi$
$563$	−5.36546	−0.226127	−0.113064	+	0.993588i	$0.536066\pi$
$564$	0	0
$565$	−0.927694	−0.0390284
$566$	0	0
$567$	51.4490	2.16066
$568$	0	0
$569$	−4.61047	−0.193281	−0.0966405	−	0.995319i	$-0.530810\pi$
$569$	−4.61047	−0.193281	−0.0966405	+	0.995319i	$0.530810\pi$
$570$	0	0
$571$	10.6858	0.447188	0.223594	−	0.974682i	$-0.428221\pi$
$571$	10.6858	0.447188	0.223594	+	0.974682i	$0.428221\pi$
$572$	0	0
$573$	29.1775	1.21891
$574$	0	0
$575$	−3.68823	−0.153810
$576$	0	0
$577$	−32.4362	−1.35033	−0.675167	−	0.737665i	$-0.735929\pi$
$577$	−32.4362	−1.35033	−0.675167	+	0.737665i	$0.735929\pi$
$578$	0	0
$579$	36.3953	1.51254
$580$	0	0
$581$	−50.3794	−2.09009
$582$	0	0
$583$	64.0758	2.65375
$584$	0	0
$585$	−15.4519	−0.638856
$586$	0	0
$587$	46.2977	1.91091	0.955454	−	0.295139i	$-0.0953660\pi$
$587$	46.2977	1.91091	0.955454	+	0.295139i	$0.0953660\pi$
$588$	0	0
$589$	−9.92147	−0.408807
$590$	0	0
$591$	−59.8003	−2.45986
$592$	0	0
$593$	19.7500	0.811036	0.405518	−	0.914087i	$-0.367091\pi$
$593$	19.7500	0.811036	0.405518	+	0.914087i	$0.367091\pi$
$594$	0	0
$595$	−10.6349	−0.435987
$596$	0	0
$597$	4.72141	0.193234
$598$	0	0
$599$	−2.71378	−0.110882	−0.0554410	−	0.998462i	$-0.517656\pi$
$599$	−2.71378	−0.110882	−0.0554410	+	0.998462i	$0.517656\pi$
$600$	0	0
$601$	−31.8938	−1.30097	−0.650487	−	0.759517i	$-0.725435\pi$
$601$	−31.8938	−1.30097	−0.650487	+	0.759517i	$0.725435\pi$
$602$	0	0
$603$	22.4161	0.912855
$604$	0	0
$605$	12.7403	0.517965
$606$	0	0
$607$	−2.08564	−0.0846533	−0.0423267	−	0.999104i	$-0.513477\pi$
$607$	−2.08564	−0.0846533	−0.0423267	+	0.999104i	$0.513477\pi$
$608$	0	0
$609$	−69.3834	−2.81156
$610$	0	0
$611$	1.15203	0.0466063
$612$	0	0
$613$	−19.4898	−0.787186	−0.393593	−	0.919285i	$-0.628768\pi$
$613$	−19.4898	−0.787186	−0.393593	+	0.919285i	$0.628768\pi$
$614$	0	0
$615$	5.81577	0.234514
$616$	0	0
$617$	2.74111	0.110353	0.0551765	−	0.998477i	$-0.482428\pi$
$617$	2.74111	0.110353	0.0551765	+	0.998477i	$0.482428\pi$
$618$	0	0
$619$	−21.1538	−0.850243	−0.425122	−	0.905136i	$-0.639769\pi$
$619$	−21.1538	−0.850243	−0.425122	+	0.905136i	$0.639769\pi$
$620$	0	0
$621$	0.905760	0.0363469
$622$	0	0
$623$	65.5479	2.62612
$624$	0	0
$625$	9.96542	0.398617
$626$	0	0
$627$	−74.5985	−2.97918
$628$	0	0
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−8.25925	−0.328795	−0.164398	−	0.986394i	$-0.552568\pi$
$631$	−8.25925	−0.328795	−0.164398	+	0.986394i	$0.552568\pi$
$632$	0	0
$633$	43.7312	1.73816
$634$	0	0
$635$	−5.51046	−0.218676
$636$	0	0
$637$	−109.972	−4.35725
$638$	0	0
$639$	−12.7853	−0.505778
$640$	0	0
$641$	−18.6164	−0.735305	−0.367653	−	0.929963i	$-0.619838\pi$
$641$	−18.6164	−0.735305	−0.367653	+	0.929963i	$0.619838\pi$
$642$	0	0
$643$	−25.3862	−1.00113	−0.500567	−	0.865698i	$-0.666875\pi$
$643$	−25.3862	−1.00113	−0.500567	+	0.865698i	$0.666875\pi$
$644$	0	0
$645$	−20.7005	−0.815082
$646$	0	0
$647$	3.99779	0.157169	0.0785846	−	0.996907i	$-0.474960\pi$
$647$	3.99779	0.157169	0.0785846	+	0.996907i	$0.474960\pi$
$648$	0	0
$649$	32.4392	1.27335
$650$	0	0
$651$	18.3550	0.719391
$652$	0	0
$653$	0.0512530	0.00200569	0.00100284	−	0.999999i	$-0.499681\pi$
$653$	0.0512530	0.00200569	0.00100284	+	0.999999i	$0.499681\pi$
$654$	0	0
$655$	16.8216	0.657276
$656$	0	0
$657$	19.6881	0.768106
$658$	0	0
$659$	14.1446	0.550994	0.275497	−	0.961302i	$-0.411158\pi$
$659$	14.1446	0.550994	0.275497	+	0.961302i	$0.411158\pi$
$660$	0	0
$661$	−41.3592	−1.60868	−0.804342	−	0.594166i	$-0.797482\pi$
$661$	−41.3592	−1.60868	−0.804342	+	0.594166i	$0.797482\pi$
$662$	0	0
$663$	27.1213	1.05330
$664$	0	0
$665$	34.7791	1.34867
$666$	0	0
$667$	−5.39512	−0.208900
$668$	0	0
$669$	0.475144	0.0183701
$670$	0	0
$671$	24.3031	0.938211
$672$	0	0
$673$	4.25574	0.164047	0.0820234	−	0.996630i	$-0.473862\pi$
$673$	4.25574	0.164047	0.0820234	+	0.996630i	$0.473862\pi$
$674$	0	0
$675$	−3.77355	−0.145244
$676$	0	0
$677$	−4.67878	−0.179820	−0.0899101	−	0.995950i	$-0.528658\pi$
$677$	−4.67878	−0.179820	−0.0899101	+	0.995950i	$0.528658\pi$
$678$	0	0
$679$	−10.2330	−0.392708
$680$	0	0
$681$	−42.2902	−1.62056
$682$	0	0
$683$	16.9231	0.647545	0.323772	−	0.946135i	$-0.395049\pi$
$683$	16.9231	0.647545	0.323772	+	0.946135i	$0.395049\pi$
$684$	0	0
$685$	12.8352	0.490409
$686$	0	0
$687$	20.6004	0.785954
$688$	0	0
$689$	76.1833	2.90235
$690$	0	0
$691$	2.58847	0.0984701	0.0492350	−	0.998787i	$-0.484322\pi$
$691$	2.58847	0.0984701	0.0492350	+	0.998787i	$0.484322\pi$
$692$	0	0
$693$	63.9828	2.43050
$694$	0	0
$695$	10.7352	0.407208
$696$	0	0
$697$	−4.73249	−0.179256
$698$	0	0
$699$	−39.1281	−1.47996
$700$	0	0
$701$	−6.37423	−0.240751	−0.120376	−	0.992728i	$-0.538410\pi$
$701$	−6.37423	−0.240751	−0.120376	+	0.992728i	$0.538410\pi$
$702$	0	0
$703$	6.54057	0.246682
$704$	0	0
$705$	0.493802	0.0185977
$706$	0	0
$707$	20.5773	0.773890
$708$	0	0
$709$	35.8392	1.34597	0.672985	−	0.739656i	$-0.265012\pi$
$709$	35.8392	1.34597	0.672985	+	0.739656i	$0.265012\pi$
$710$	0	0
$711$	2.43969	0.0914955
$712$	0	0
$713$	1.42725	0.0534511
$714$	0	0
$715$	28.7397	1.07480
$716$	0	0
$717$	2.19788	0.0820813
$718$	0	0
$719$	23.0697	0.860356	0.430178	−	0.902744i	$-0.358451\pi$
$719$	23.0697	0.860356	0.430178	+	0.902744i	$0.358451\pi$
$720$	0	0
$721$	90.9415	3.38684
$722$	0	0
$723$	20.4661	0.761142
$724$	0	0
$725$	22.4770	0.834774
$726$	0	0
$727$	30.8856	1.14548	0.572741	−	0.819736i	$-0.305880\pi$
$727$	30.8856	1.14548	0.572741	+	0.819736i	$0.305880\pi$
$728$	0	0
$729$	−19.2434	−0.712718
$730$	0	0
$731$	16.8447	0.623024
$732$	0	0
$733$	−10.7077	−0.395497	−0.197748	−	0.980253i	$-0.563363\pi$
$733$	−10.7077	−0.395497	−0.197748	+	0.980253i	$0.563363\pi$
$734$	0	0
$735$	−47.1378	−1.73870
$736$	0	0
$737$	−41.6928	−1.53577
$738$	0	0
$739$	27.0211	0.993988	0.496994	−	0.867754i	$-0.334437\pi$
$739$	27.0211	0.993988	0.496994	+	0.867754i	$0.334437\pi$
$740$	0	0
$741$	−88.6944	−3.25827
$742$	0	0
$743$	−37.4959	−1.37559	−0.687796	−	0.725904i	$-0.741422\pi$
$743$	−37.4959	−1.37559	−0.687796	+	0.725904i	$0.741422\pi$
$744$	0	0
$745$	24.2822	0.889630
$746$	0	0
$747$	25.5312	0.934138
$748$	0	0
$749$	42.2356	1.54326
$750$	0	0
$751$	11.5425	0.421191	0.210595	−	0.977573i	$-0.432460\pi$
$751$	11.5425	0.421191	0.210595	+	0.977573i	$0.432460\pi$
$752$	0	0
$753$	−2.01416	−0.0734002
$754$	0	0
$755$	10.8279	0.394067
$756$	0	0
$757$	50.8680	1.84883	0.924415	−	0.381388i	$-0.124554\pi$
$757$	50.8680	1.84883	0.924415	+	0.381388i	$0.124554\pi$
$758$	0	0
$759$	10.7314	0.389524
$760$	0	0
$761$	−6.94167	−0.251635	−0.125818	−	0.992053i	$-0.540155\pi$
$761$	−6.94167	−0.251635	−0.125818	+	0.992053i	$0.540155\pi$
$762$	0	0
$763$	37.1686	1.34559
$764$	0	0
$765$	5.38953	0.194859
$766$	0	0
$767$	38.5688	1.39264
$768$	0	0
$769$	−23.5146	−0.847959	−0.423980	−	0.905672i	$-0.639367\pi$
$769$	−23.5146	−0.847959	−0.423980	+	0.905672i	$0.639367\pi$
$770$	0	0
$771$	4.35842	0.156965
$772$	0	0
$773$	14.2684	0.513200	0.256600	−	0.966518i	$-0.417398\pi$
$773$	14.2684	0.513200	0.256600	+	0.966518i	$0.417398\pi$
$774$	0	0
$775$	−5.94618	−0.213593
$776$	0	0
$777$	−12.1003	−0.434095
$778$	0	0
$779$	15.4766	0.554506
$780$	0	0
$781$	23.7800	0.850914
$782$	0	0
$783$	−5.51991	−0.197266
$784$	0	0
$785$	11.3430	0.404848
$786$	0	0
$787$	54.8978	1.95690	0.978448	−	0.206491i	$-0.0662044\pi$
$787$	54.8978	1.95690	0.978448	+	0.206491i	$0.0662044\pi$
$788$	0	0
$789$	40.3119	1.43514
$790$	0	0
$791$	4.56722	0.162392
$792$	0	0
$793$	28.8953	1.02610
$794$	0	0
$795$	32.6548	1.15815
$796$	0	0
$797$	−3.62353	−0.128352	−0.0641760	−	0.997939i	$-0.520442\pi$
$797$	−3.62353	−0.128352	−0.0641760	+	0.997939i	$0.520442\pi$
$798$	0	0
$799$	−0.401824	−0.0142155
$800$	0	0
$801$	−33.2183	−1.17371
$802$	0	0
$803$	−36.6189	−1.29225
$804$	0	0
$805$	−5.00314	−0.176338
$806$	0	0
$807$	52.2071	1.83778
$808$	0	0
$809$	−9.86382	−0.346793	−0.173397	−	0.984852i	$-0.555474\pi$
$809$	−9.86382	−0.346793	−0.173397	+	0.984852i	$0.555474\pi$
$810$	0	0
$811$	−21.6675	−0.760848	−0.380424	−	0.924812i	$-0.624222\pi$
$811$	−21.6675	−0.760848	−0.380424	+	0.924812i	$0.624222\pi$
$812$	0	0
$813$	−47.4867	−1.66543
$814$	0	0
$815$	0.0738926	0.00258835
$816$	0	0
$817$	−55.0870	−1.92725
$818$	0	0
$819$	76.0726	2.65819
$820$	0	0
$821$	16.9224	0.590595	0.295298	−	0.955405i	$-0.404581\pi$
$821$	16.9224	0.590595	0.295298	+	0.955405i	$0.404581\pi$
$822$	0	0
$823$	−36.1357	−1.25961	−0.629806	−	0.776753i	$-0.716866\pi$
$823$	−36.1357	−1.25961	−0.629806	+	0.776753i	$0.716866\pi$
$824$	0	0
$825$	−44.7087	−1.55656
$826$	0	0
$827$	39.9021	1.38753	0.693766	−	0.720201i	$-0.255950\pi$
$827$	39.9021	1.38753	0.693766	+	0.720201i	$0.255950\pi$
$828$	0	0
$829$	40.5566	1.40859	0.704294	−	0.709908i	$-0.251264\pi$
$829$	40.5566	1.40859	0.704294	+	0.709908i	$0.251264\pi$
$830$	0	0
$831$	54.7751	1.90013
$832$	0	0
$833$	38.3576	1.32901
$834$	0	0
$835$	−12.4044	−0.429271
$836$	0	0
$837$	1.46027	0.0504742
$838$	0	0
$839$	5.10863	0.176370	0.0881848	−	0.996104i	$-0.471893\pi$
$839$	5.10863	0.176370	0.0881848	+	0.996104i	$0.471893\pi$
$840$	0	0
$841$	3.87911	0.133763
$842$	0	0
$843$	−5.02574	−0.173096
$844$	0	0
$845$	20.6597	0.710716
$846$	0	0
$847$	−62.7229	−2.15518
$848$	0	0
$849$	−12.3043	−0.422284
$850$	0	0
$851$	−0.940894	−0.0322534
$852$	0	0
$853$	23.4111	0.801579	0.400790	−	0.916170i	$-0.368736\pi$
$853$	23.4111	0.801579	0.400790	+	0.916170i	$0.368736\pi$
$854$	0	0
$855$	−17.6253	−0.602772
$856$	0	0
$857$	49.8368	1.70239	0.851196	−	0.524848i	$-0.175878\pi$
$857$	49.8368	1.70239	0.851196	+	0.524848i	$0.175878\pi$
$858$	0	0
$859$	−25.0583	−0.854977	−0.427489	−	0.904021i	$-0.640602\pi$
$859$	−25.0583	−0.854977	−0.427489	+	0.904021i	$0.640602\pi$
$860$	0	0
$861$	−28.6322	−0.975783
$862$	0	0
$863$	18.7773	0.639188	0.319594	−	0.947555i	$-0.396454\pi$
$863$	18.7773	0.639188	0.319594	+	0.947555i	$0.396454\pi$
$864$	0	0
$865$	12.3306	0.419252
$866$	0	0
$867$	30.7442	1.04413
$868$	0	0
$869$	−4.53770	−0.153931
$870$	0	0
$871$	−49.5709	−1.67965
$872$	0	0
$873$	5.18589	0.175516
$874$	0	0
$875$	47.4311	1.60346
$876$	0	0
$877$	6.40727	0.216358	0.108179	−	0.994131i	$-0.465498\pi$
$877$	6.40727	0.216358	0.108179	+	0.994131i	$0.465498\pi$
$878$	0	0
$879$	−45.1616	−1.52326
$880$	0	0
$881$	−37.9723	−1.27932	−0.639660	−	0.768658i	$-0.720925\pi$
$881$	−37.9723	−1.27932	−0.639660	+	0.768658i	$0.720925\pi$
$882$	0	0
$883$	−34.7704	−1.17012	−0.585058	−	0.810991i	$-0.698928\pi$
$883$	−34.7704	−1.17012	−0.585058	+	0.810991i	$0.698928\pi$
$884$	0	0
$885$	16.5319	0.555715
$886$	0	0
$887$	−5.07300	−0.170335	−0.0851673	−	0.996367i	$-0.527142\pi$
$887$	−5.07300	−0.170335	−0.0851673	+	0.996367i	$0.527142\pi$
$888$	0	0
$889$	27.1291	0.909881
$890$	0	0
$891$	48.4950	1.62464
$892$	0	0
$893$	1.31408	0.0439739
$894$	0	0
$895$	−18.5843	−0.621205
$896$	0	0
$897$	12.7591	0.426015
$898$	0	0
$899$	−8.69802	−0.290095
$900$	0	0
$901$	−26.5723	−0.885253
$902$	0	0
$903$	101.913	3.39145
$904$	0	0
$905$	−12.3638	−0.410986
$906$	0	0
$907$	−13.0100	−0.431991	−0.215996	−	0.976394i	$-0.569300\pi$
$907$	−13.0100	−0.431991	−0.215996	+	0.976394i	$0.569300\pi$
$908$	0	0
$909$	−10.4282	−0.345880
$910$	0	0
$911$	−21.8618	−0.724314	−0.362157	−	0.932117i	$-0.617960\pi$
$911$	−21.8618	−0.724314	−0.362157	+	0.932117i	$0.617960\pi$
$912$	0	0
$913$	−47.4867	−1.57158
$914$	0	0
$915$	12.3855	0.409453
$916$	0	0
$917$	−82.8164	−2.73484
$918$	0	0
$919$	−13.4046	−0.442176	−0.221088	−	0.975254i	$-0.570961\pi$
$919$	−13.4046	−0.442176	−0.221088	+	0.975254i	$0.570961\pi$
$920$	0	0
$921$	56.2793	1.85447
$922$	0	0
$923$	28.2733	0.930628
$924$	0	0
$925$	3.91992	0.128886
$926$	0	0
$927$	−46.0872	−1.51370
$928$	0	0
$929$	49.3649	1.61961	0.809805	−	0.586699i	$-0.199573\pi$
$929$	49.3649	1.61961	0.809805	+	0.586699i	$0.199573\pi$
$930$	0	0
$931$	−125.440	−4.11114
$932$	0	0
$933$	52.9115	1.73225
$934$	0	0
$935$	−10.0242	−0.327828
$936$	0	0
$937$	21.8343	0.713296	0.356648	−	0.934239i	$-0.383920\pi$
$937$	21.8343	0.713296	0.356648	+	0.934239i	$0.383920\pi$
$938$	0	0
$939$	−62.6563	−2.04471
$940$	0	0
$941$	54.7760	1.78565	0.892823	−	0.450408i	$-0.148721\pi$
$941$	54.7760	1.78565	0.892823	+	0.450408i	$0.148721\pi$
$942$	0	0
$943$	−2.22639	−0.0725011
$944$	0	0
$945$	−5.11887	−0.166517
$946$	0	0
$947$	27.1258	0.881470	0.440735	−	0.897637i	$-0.354718\pi$
$947$	27.1258	0.881470	0.440735	+	0.897637i	$0.354718\pi$
$948$	0	0
$949$	−43.5382	−1.41331
$950$	0	0
$951$	−55.6863	−1.80575
$952$	0	0
$953$	−16.9217	−0.548149	−0.274075	−	0.961708i	$-0.588371\pi$
$953$	−16.9217	−0.548149	−0.274075	+	0.961708i	$0.588371\pi$
$954$	0	0
$955$	−12.8220	−0.414909
$956$	0	0
$957$	−65.3996	−2.11407
$958$	0	0
$959$	−63.1904	−2.04052
$960$	0	0
$961$	−28.6990	−0.925774
$962$	0	0
$963$	−21.4041	−0.689738
$964$	0	0
$965$	−15.9938	−0.514860
$966$	0	0
$967$	−18.0033	−0.578948	−0.289474	−	0.957186i	$-0.593480\pi$
$967$	−18.0033	−0.578948	−0.289474	+	0.957186i	$0.593480\pi$
$968$	0	0
$969$	30.9361	0.993812
$970$	0	0
$971$	11.6789	0.374792	0.187396	−	0.982284i	$-0.439995\pi$
$971$	11.6789	0.374792	0.187396	+	0.982284i	$0.439995\pi$
$972$	0	0
$973$	−52.8514	−1.69434
$974$	0	0
$975$	−53.1567	−1.70238
$976$	0	0
$977$	−42.6718	−1.36519	−0.682595	−	0.730797i	$-0.739149\pi$
$977$	−42.6718	−1.36519	−0.682595	+	0.730797i	$0.739149\pi$
$978$	0	0
$979$	61.7843	1.97464
$980$	0	0
$981$	−18.8363	−0.601396
$982$	0	0
$983$	−2.08999	−0.0666604	−0.0333302	−	0.999444i	$-0.510611\pi$
$983$	−2.08999	−0.0666604	−0.0333302	+	0.999444i	$0.510611\pi$
$984$	0	0
$985$	26.2791	0.837321
$986$	0	0
$987$	−2.43109	−0.0773824
$988$	0	0
$989$	7.92455	0.251986
$990$	0	0
$991$	38.2517	1.21511	0.607553	−	0.794279i	$-0.292151\pi$
$991$	38.2517	1.21511	0.607553	+	0.794279i	$0.292151\pi$
$992$	0	0
$993$	5.67170	0.179986
$994$	0	0
$995$	−2.07481	−0.0657759
$996$	0	0
$997$	27.8288	0.881346	0.440673	−	0.897668i	$-0.354740\pi$
$997$	27.8288	0.881346	0.440673	+	0.897668i	$0.354740\pi$
$998$	0	0
$999$	−0.962658	−0.0304572

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bj.1.2		8
4.3	odd	2	inner	2368.2.a.bj.1.7		8
8.3	odd	2		1184.2.a.p.1.2	✓	8
8.5	even	2		1184.2.a.p.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.p.1.2	✓	8	8.3	odd	2
1184.2.a.p.1.7	yes	8	8.5	even	2
2368.2.a.bj.1.2		8	1.1	even	1	trivial
2368.2.a.bj.1.7		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36494 q^{3} +1.03927 q^{5} -5.11652 q^{7} +2.59295 q^{9} +O(q^{10})\)	\(q-2.36494 q^{3} +1.03927 q^{5} -5.11652 q^{7} +2.59295 q^{9} -4.82275 q^{11} -5.73403 q^{13} -2.45781 q^{15} +2.00000 q^{17} -6.54057 q^{19} +12.1003 q^{21} +0.940894 q^{23} -3.91992 q^{25} +0.962658 q^{27} -5.73403 q^{29} +1.51691 q^{31} +11.4055 q^{33} -5.31744 q^{35} -1.00000 q^{37} +13.5606 q^{39} -2.36624 q^{41} +8.42236 q^{43} +2.69476 q^{45} -0.200912 q^{47} +19.1788 q^{49} -4.72988 q^{51} -13.2862 q^{53} -5.01212 q^{55} +15.4681 q^{57} -6.72630 q^{59} -5.03927 q^{61} -13.2669 q^{63} -5.95919 q^{65} +8.64503 q^{67} -2.22516 q^{69} -4.93079 q^{71} +7.59295 q^{73} +9.27039 q^{75} +24.6757 q^{77} +0.940894 q^{79} -10.0555 q^{81} +9.84641 q^{83} +2.07853 q^{85} +13.5606 q^{87} -12.8110 q^{89} +29.3383 q^{91} -3.58741 q^{93} -6.79740 q^{95} +2.00000 q^{97} -12.5051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 8 * q + 2 * q^5 + 16 * q^9	\( 8 q + 2 q^{5} + 16 q^{9} - 2 q^{13} + 16 q^{17} - 2 q^{21} + 22 q^{25} - 2 q^{29} + 30 q^{33} - 8 q^{37} + 36 q^{41} - 16 q^{45} + 42 q^{49} + 2 q^{53} + 36 q^{57} - 34 q^{61} + 12 q^{65} - 2 q^{69} + 56 q^{73} + 14 q^{77} + 48 q^{81} + 4 q^{85} + 20 q^{89} + 12 q^{93} + 16 q^{97}+O(q^{100}) \) 8 * q + 2 * q^5 + 16 * q^9 - 2 * q^13 + 16 * q^17 - 2 * q^21 + 22 * q^25 - 2 * q^29 + 30 * q^33 - 8 * q^37 + 36 * q^41 - 16 * q^45 + 42 * q^49 + 2 * q^53 + 36 * q^57 - 34 * q^61 + 12 * q^65 - 2 * q^69 + 56 * q^73 + 14 * q^77 + 48 * q^81 + 4 * q^85 + 20 * q^89 + 12 * q^93 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more