LMFDB - Embedded newform 2368.2.a.bf.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75660$ 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)

$f(q)$	$=$	$q-2.75660 q^{3} -0.520530 q^{5} +3.84224 q^{7} +4.59883 q^{9} +O(q^{10})$	$q-2.75660 q^{3} -0.520530 q^{5} +3.84224 q^{7} +4.59883 q^{9} +1.15043 q^{11} -5.55777 q^{13} +1.43489 q^{15} -2.00000 q^{17} +1.68447 q^{19} -10.5915 q^{21} -9.24224 q^{23} -4.72905 q^{25} -4.40734 q^{27} +8.28331 q^{29} +8.99267 q^{31} -3.17127 q^{33} -2.00000 q^{35} -1.00000 q^{37} +15.3205 q^{39} -7.82139 q^{41} +8.06745 q^{43} -2.39383 q^{45} +4.62990 q^{47} +7.76278 q^{49} +5.51320 q^{51} -2.60617 q^{53} -0.598834 q^{55} -4.64341 q^{57} -3.04106 q^{59} -3.82202 q^{61} +17.6698 q^{63} +2.89299 q^{65} -5.78033 q^{67} +25.4772 q^{69} +1.19882 q^{71} -5.42756 q^{73} +13.0361 q^{75} +4.42023 q^{77} +15.4135 q^{79} -1.64723 q^{81} -12.7244 q^{83} +1.04106 q^{85} -22.8337 q^{87} -4.89681 q^{89} -21.3543 q^{91} -24.7892 q^{93} -0.876818 q^{95} -12.4721 q^{97} +5.29064 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + q^9	$ 4 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + q^{9} - 3 q^{11} - 3 q^{13} + 8 q^{15} - 8 q^{17} - 12 q^{19} - 8 q^{21} + q^{23} + 3 q^{25} - 6 q^{27} - 3 q^{29} + 19 q^{31} - 10 q^{33} - 8 q^{35} - 4 q^{37} + 5 q^{39} - 17 q^{41} + 2 q^{43} - 10 q^{45} + 10 q^{47} - 6 q^{49} + 6 q^{51} - 10 q^{53} + 15 q^{55} + 2 q^{57} - 14 q^{59} - 9 q^{61} + 18 q^{63} - 30 q^{65} - 7 q^{67} + 9 q^{69} + 16 q^{71} - 7 q^{73} - 14 q^{75} - 14 q^{77} + 21 q^{79} - 8 q^{81} + 12 q^{83} + 6 q^{85} - 19 q^{87} - 14 q^{91} - 30 q^{93} + 2 q^{95} - 32 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + q^9 - 3 * q^11 - 3 * q^13 + 8 * q^15 - 8 * q^17 - 12 * q^19 - 8 * q^21 + q^23 + 3 * q^25 - 6 * q^27 - 3 * q^29 + 19 * q^31 - 10 * q^33 - 8 * q^35 - 4 * q^37 + 5 * q^39 - 17 * q^41 + 2 * q^43 - 10 * q^45 + 10 * q^47 - 6 * q^49 + 6 * q^51 - 10 * q^53 + 15 * q^55 + 2 * q^57 - 14 * q^59 - 9 * q^61 + 18 * q^63 - 30 * q^65 - 7 * q^67 + 9 * q^69 + 16 * q^71 - 7 * q^73 - 14 * q^75 - 14 * q^77 + 21 * q^79 - 8 * q^81 + 12 * q^83 + 6 * q^85 - 19 * q^87 - 14 * q^91 - 30 * q^93 + 2 * q^95 - 32 * q^97 + 2 * 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
$3$	−2.75660	−1.59152	−0.795761	−	0.605610i	$-0.792929\pi$
$3$	−2.75660	−1.59152	−0.795761	+	0.605610i	$0.792929\pi$
$4$	0	0
$5$	−0.520530	−0.232788	−0.116394	−	0.993203i	$-0.537134\pi$
$5$	−0.520530	−0.232788	−0.116394	+	0.993203i	$0.537134\pi$
$6$	0	0
$7$	3.84224	1.45223	0.726114	−	0.687574i	$-0.241324\pi$
$7$	3.84224	1.45223	0.726114	+	0.687574i	$0.241324\pi$
$8$	0	0
$9$	4.59883	1.53294
$10$	0	0
$11$	1.15043	0.346868	0.173434	−	0.984846i	$-0.444514\pi$
$11$	1.15043	0.346868	0.173434	+	0.984846i	$0.444514\pi$
$12$	0	0
$13$	−5.55777	−1.54145	−0.770724	−	0.637169i	$-0.780106\pi$
$13$	−5.55777	−1.54145	−0.770724	+	0.637169i	$0.780106\pi$
$14$	0	0
$15$	1.43489	0.370488
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	1.68447	0.386444	0.193222	−	0.981155i	$-0.438106\pi$
$19$	1.68447	0.386444	0.193222	+	0.981155i	$0.438106\pi$
$20$	0	0
$21$	−10.5915	−2.31125
$22$	0	0
$23$	−9.24224	−1.92714	−0.963571	−	0.267454i	$-0.913818\pi$
$23$	−9.24224	−1.92714	−0.963571	+	0.267454i	$0.913818\pi$
$24$	0	0
$25$	−4.72905	−0.945810
$26$	0	0
$27$	−4.40734	−0.848194
$28$	0	0
$29$	8.28331	1.53817	0.769086	−	0.639146i	$-0.220712\pi$
$29$	8.28331	1.53817	0.769086	+	0.639146i	$0.220712\pi$
$30$	0	0
$31$	8.99267	1.61513	0.807565	−	0.589778i	$-0.200785\pi$
$31$	8.99267	1.61513	0.807565	+	0.589778i	$0.200785\pi$
$32$	0	0
$33$	−3.17127	−0.552048
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	15.3205	2.45325
$40$	0	0
$41$	−7.82139	−1.22150	−0.610748	−	0.791825i	$-0.709131\pi$
$41$	−7.82139	−1.22150	−0.610748	+	0.791825i	$0.709131\pi$
$42$	0	0
$43$	8.06745	1.23028	0.615138	−	0.788420i	$-0.289100\pi$
$43$	8.06745	1.23028	0.615138	+	0.788420i	$0.289100\pi$
$44$	0	0
$45$	−2.39383	−0.356851
$46$	0	0
$47$	4.62990	0.675340	0.337670	−	0.941264i	$-0.390361\pi$
$47$	4.62990	0.675340	0.337670	+	0.941264i	$0.390361\pi$
$48$	0	0
$49$	7.76278	1.10897
$50$	0	0
$51$	5.51320	0.772002
$52$	0	0
$53$	−2.60617	−0.357985	−0.178992	−	0.983850i	$-0.557284\pi$
$53$	−2.60617	−0.357985	−0.178992	+	0.983850i	$0.557284\pi$
$54$	0	0
$55$	−0.598834	−0.0807468
$56$	0	0
$57$	−4.64341	−0.615035
$58$	0	0
$59$	−3.04106	−0.395912	−0.197956	−	0.980211i	$-0.563430\pi$
$59$	−3.04106	−0.395912	−0.197956	+	0.980211i	$0.563430\pi$
$60$	0	0
$61$	−3.82202	−0.489359	−0.244680	−	0.969604i	$-0.578683\pi$
$61$	−3.82202	−0.489359	−0.244680	+	0.969604i	$0.578683\pi$
$62$	0	0
$63$	17.6698	2.22619
$64$	0	0
$65$	2.89299	0.358831
$66$	0	0
$67$	−5.78033	−0.706180	−0.353090	−	0.935589i	$-0.614869\pi$
$67$	−5.78033	−0.706180	−0.353090	+	0.935589i	$0.614869\pi$
$68$	0	0
$69$	25.4772	3.06709
$70$	0	0
$71$	1.19882	0.142274	0.0711372	−	0.997467i	$-0.477337\pi$
$71$	1.19882	0.142274	0.0711372	+	0.997467i	$0.477337\pi$
$72$	0	0
$73$	−5.42756	−0.635248	−0.317624	−	0.948217i	$-0.602885\pi$
$73$	−5.42756	−0.635248	−0.317624	+	0.948217i	$0.602885\pi$
$74$	0	0
$75$	13.0361	1.50528
$76$	0	0
$77$	4.42023	0.503731
$78$	0	0
$79$	15.4135	1.73416	0.867078	−	0.498172i	$-0.165995\pi$
$79$	15.4135	1.73416	0.867078	+	0.498172i	$0.165995\pi$
$80$	0	0
$81$	−1.64723	−0.183025
$82$	0	0
$83$	−12.7244	−1.39668	−0.698341	−	0.715765i	$-0.746078\pi$
$83$	−12.7244	−1.39668	−0.698341	+	0.715765i	$0.746078\pi$
$84$	0	0
$85$	1.04106	0.112919
$86$	0	0
$87$	−22.8337	−2.44803
$88$	0	0
$89$	−4.89681	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$89$	−4.89681	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$90$	0	0
$91$	−21.3543	−2.23854
$92$	0	0
$93$	−24.7892	−2.57052
$94$	0	0
$95$	−0.876818	−0.0899597
$96$	0	0
$97$	−12.4721	−1.26635	−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721	−1.26635	−0.633177	+	0.774007i	$0.718249\pi$
$98$	0	0
$99$	5.29064	0.531729
$100$	0	0
$101$	−3.81850	−0.379955	−0.189978	−	0.981788i	$-0.560842\pi$
$101$	−3.81850	−0.379955	−0.189978	+	0.981788i	$0.560842\pi$
$102$	0	0
$103$	−13.0601	−1.28685	−0.643426	−	0.765508i	$-0.722488\pi$
$103$	−13.0601	−1.28685	−0.643426	+	0.765508i	$0.722488\pi$
$104$	0	0
$105$	5.51320	0.538033
$106$	0	0
$107$	−8.51671	−0.823342	−0.411671	−	0.911333i	$-0.635055\pi$
$107$	−8.51671	−0.823342	−0.411671	+	0.911333i	$0.635055\pi$
$108$	0	0
$109$	2.02702	0.194153	0.0970767	−	0.995277i	$-0.469051\pi$
$109$	2.02702	0.194153	0.0970767	+	0.995277i	$0.469051\pi$
$110$	0	0
$111$	2.75660	0.261645
$112$	0	0
$113$	−7.76659	−0.730620	−0.365310	−	0.930886i	$-0.619037\pi$
$113$	−7.76659	−0.730620	−0.365310	+	0.930886i	$0.619037\pi$
$114$	0	0
$115$	4.81087	0.448616
$116$	0	0
$117$	−25.5593	−2.36296
$118$	0	0
$119$	−7.68447	−0.704434
$120$	0	0
$121$	−9.67651	−0.879683
$122$	0	0
$123$	21.5604	1.94404
$124$	0	0
$125$	5.06426	0.452962
$126$	0	0
$127$	1.54137	0.136775	0.0683874	−	0.997659i	$-0.478215\pi$
$127$	1.54137	0.136775	0.0683874	+	0.997659i	$0.478215\pi$
$128$	0	0
$129$	−22.2387	−1.95801
$130$	0	0
$131$	10.8000	0.943602	0.471801	−	0.881705i	$-0.343604\pi$
$131$	10.8000	0.943602	0.471801	+	0.881705i	$0.343604\pi$
$132$	0	0
$133$	6.47214	0.561205
$134$	0	0
$135$	2.29416	0.197449
$136$	0	0
$137$	18.1757	1.55285	0.776426	−	0.630208i	$-0.217031\pi$
$137$	18.1757	1.55285	0.776426	+	0.630208i	$0.217031\pi$
$138$	0	0
$139$	−10.3454	−0.877489	−0.438744	−	0.898612i	$-0.644577\pi$
$139$	−10.3454	−0.877489	−0.438744	+	0.898612i	$0.644577\pi$
$140$	0	0
$141$	−12.7628	−1.07482
$142$	0	0
$143$	−6.39383	−0.534679
$144$	0	0
$145$	−4.31171	−0.358068
$146$	0	0
$147$	−21.3989	−1.76495
$148$	0	0
$149$	−0.695321	−0.0569630	−0.0284815	−	0.999594i	$-0.509067\pi$
$149$	−0.695321	−0.0569630	−0.0284815	+	0.999594i	$0.509067\pi$
$150$	0	0
$151$	−10.3279	−0.840471	−0.420236	−	0.907415i	$-0.638053\pi$
$151$	−10.3279	−0.840471	−0.420236	+	0.907415i	$0.638053\pi$
$152$	0	0
$153$	−9.19767	−0.743587
$154$	0	0
$155$	−4.68096	−0.375983
$156$	0	0
$157$	3.20789	0.256017	0.128009	−	0.991773i	$-0.459141\pi$
$157$	3.20789	0.256017	0.128009	+	0.991773i	$0.459141\pi$
$158$	0	0
$159$	7.18416	0.569741
$160$	0	0
$161$	−35.5109	−2.79865
$162$	0	0
$163$	−12.3953	−0.970878	−0.485439	−	0.874271i	$-0.661340\pi$
$163$	−12.3953	−0.970878	−0.485439	+	0.874271i	$0.661340\pi$
$164$	0	0
$165$	1.65074	0.128510
$166$	0	0
$167$	−6.58798	−0.509793	−0.254897	−	0.966968i	$-0.582041\pi$
$167$	−6.58798	−0.509793	−0.254897	+	0.966968i	$0.582041\pi$
$168$	0	0
$169$	17.8888	1.37606
$170$	0	0
$171$	7.74660	0.592398
$172$	0	0
$173$	−2.26362	−0.172100	−0.0860498	−	0.996291i	$-0.527424\pi$
$173$	−2.26362	−0.172100	−0.0860498	+	0.996291i	$0.527424\pi$
$174$	0	0
$175$	−18.1701	−1.37353
$176$	0	0
$177$	8.38298	0.630104
$178$	0	0
$179$	−22.2511	−1.66312	−0.831562	−	0.555432i	$-0.812553\pi$
$179$	−22.2511	−1.66312	−0.831562	+	0.555432i	$0.812553\pi$
$180$	0	0
$181$	18.0572	1.34218	0.671092	−	0.741374i	$-0.265826\pi$
$181$	18.0572	1.34218	0.671092	+	0.741374i	$0.265826\pi$
$182$	0	0
$183$	10.5358	0.778827
$184$	0	0
$185$	0.520530	0.0382701
$186$	0	0
$187$	−2.30086	−0.168256
$188$	0	0
$189$	−16.9340	−1.23177
$190$	0	0
$191$	−13.6707	−0.989180	−0.494590	−	0.869127i	$-0.664682\pi$
$191$	−13.6707	−0.989180	−0.494590	+	0.869127i	$0.664682\pi$
$192$	0	0
$193$	23.6551	1.70273	0.851367	−	0.524571i	$-0.175774\pi$
$193$	23.6551	1.70273	0.851367	+	0.524571i	$0.175774\pi$
$194$	0	0
$195$	−7.97481	−0.571088
$196$	0	0
$197$	−21.3645	−1.52216	−0.761079	−	0.648660i	$-0.775330\pi$
$197$	−21.3645	−1.52216	−0.761079	+	0.648660i	$0.775330\pi$
$198$	0	0
$199$	−8.39001	−0.594752	−0.297376	−	0.954760i	$-0.596112\pi$
$199$	−8.39001	−0.594752	−0.297376	+	0.954760i	$0.596112\pi$
$200$	0	0
$201$	15.9340	1.12390
$202$	0	0
$203$	31.8264	2.23378
$204$	0	0
$205$	4.07127	0.284350
$206$	0	0
$207$	−42.5035	−2.95420
$208$	0	0
$209$	1.93787	0.134045
$210$	0	0
$211$	−0.864237	−0.0594965	−0.0297483	−	0.999557i	$-0.509471\pi$
$211$	−0.864237	−0.0594965	−0.0297483	+	0.999557i	$0.509471\pi$
$212$	0	0
$213$	−3.30468	−0.226433
$214$	0	0
$215$	−4.19935	−0.286394
$216$	0	0
$217$	34.5519	2.34554
$218$	0	0
$219$	14.9616	1.01101
$220$	0	0
$221$	11.1155	0.747713
$222$	0	0
$223$	1.76775	0.118377	0.0591886	−	0.998247i	$-0.481149\pi$
$223$	1.76775	0.118377	0.0591886	+	0.998247i	$0.481149\pi$
$224$	0	0
$225$	−21.7481	−1.44987
$226$	0	0
$227$	−9.64278	−0.640014	−0.320007	−	0.947415i	$-0.603685\pi$
$227$	−9.64278	−0.640014	−0.320007	+	0.947415i	$0.603685\pi$
$228$	0	0
$229$	−15.7000	−1.03749	−0.518743	−	0.854930i	$-0.673600\pi$
$229$	−15.7000	−1.03749	−0.518743	+	0.854930i	$0.673600\pi$
$230$	0	0
$231$	−12.1848	−0.801700
$232$	0	0
$233$	1.29734	0.0849919	0.0424959	−	0.999097i	$-0.486469\pi$
$233$	1.29734	0.0849919	0.0424959	+	0.999097i	$0.486469\pi$
$234$	0	0
$235$	−2.41000	−0.157211
$236$	0	0
$237$	−42.4889	−2.75995
$238$	0	0
$239$	−4.53520	−0.293358	−0.146679	−	0.989184i	$-0.546858\pi$
$239$	−4.53520	−0.293358	−0.146679	+	0.989184i	$0.546858\pi$
$240$	0	0
$241$	−30.1543	−1.94241	−0.971204	−	0.238248i	$-0.923427\pi$
$241$	−30.1543	−1.94241	−0.971204	+	0.238248i	$0.923427\pi$
$242$	0	0
$243$	17.7628	1.13948
$244$	0	0
$245$	−4.04076	−0.258155
$246$	0	0
$247$	−9.36191	−0.595684
$248$	0	0
$249$	35.0760	2.22285
$250$	0	0
$251$	−18.7583	−1.18402	−0.592008	−	0.805932i	$-0.701664\pi$
$251$	−18.7583	−1.18402	−0.592008	+	0.805932i	$0.701664\pi$
$252$	0	0
$253$	−10.6326	−0.668463
$254$	0	0
$255$	−2.86979	−0.179713
$256$	0	0
$257$	−13.1572	−0.820726	−0.410363	−	0.911922i	$-0.634598\pi$
$257$	−13.1572	−0.820726	−0.410363	+	0.911922i	$0.634598\pi$
$258$	0	0
$259$	−3.84224	−0.238745
$260$	0	0
$261$	38.0935	2.35793
$262$	0	0
$263$	22.7367	1.40201	0.701003	−	0.713158i	$-0.252736\pi$
$263$	22.7367	1.40201	0.701003	+	0.713158i	$0.252736\pi$
$264$	0	0
$265$	1.35659	0.0833346
$266$	0	0
$267$	13.4985	0.826097
$268$	0	0
$269$	12.8405	0.782896	0.391448	−	0.920200i	$-0.371974\pi$
$269$	12.8405	0.782896	0.391448	+	0.920200i	$0.371974\pi$
$270$	0	0
$271$	13.5337	0.822116	0.411058	−	0.911609i	$-0.365159\pi$
$271$	13.5337	0.822116	0.411058	+	0.911609i	$0.365159\pi$
$272$	0	0
$273$	58.8652	3.56268
$274$	0	0
$275$	−5.44044	−0.328071
$276$	0	0
$277$	32.9282	1.97846	0.989232	−	0.146353i	$-0.0467534\pi$
$277$	32.9282	1.97846	0.989232	+	0.146353i	$0.0467534\pi$
$278$	0	0
$279$	41.3558	2.47591
$280$	0	0
$281$	−3.02639	−0.180539	−0.0902697	−	0.995917i	$-0.528773\pi$
$281$	−3.02639	−0.180539	−0.0902697	+	0.995917i	$0.528773\pi$
$282$	0	0
$283$	2.59595	0.154313	0.0771565	−	0.997019i	$-0.475416\pi$
$283$	2.59595	0.154313	0.0771565	+	0.997019i	$0.475416\pi$
$284$	0	0
$285$	2.41704	0.143173
$286$	0	0
$287$	−30.0516	−1.77389
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	34.3807	2.01543
$292$	0	0
$293$	−28.4845	−1.66408	−0.832041	−	0.554714i	$-0.812828\pi$
$293$	−28.4845	−1.66408	−0.832041	+	0.554714i	$0.812828\pi$
$294$	0	0
$295$	1.58296	0.0921637
$296$	0	0
$297$	−5.07034	−0.294211
$298$	0	0
$299$	51.3663	2.97059
$300$	0	0
$301$	30.9971	1.78664
$302$	0	0
$303$	10.5261	0.604707
$304$	0	0
$305$	1.98948	0.113917
$306$	0	0
$307$	12.0434	0.687354	0.343677	−	0.939088i	$-0.388327\pi$
$307$	12.0434	0.687354	0.343677	+	0.939088i	$0.388327\pi$
$308$	0	0
$309$	36.0015	2.04805
$310$	0	0
$311$	−24.9742	−1.41616	−0.708078	−	0.706134i	$-0.750437\pi$
$311$	−24.9742	−1.41616	−0.708078	+	0.706134i	$0.750437\pi$
$312$	0	0
$313$	−6.93724	−0.392116	−0.196058	−	0.980592i	$-0.562814\pi$
$313$	−6.93724	−0.392116	−0.196058	+	0.980592i	$0.562814\pi$
$314$	0	0
$315$	−9.19767	−0.518230
$316$	0	0
$317$	−25.1830	−1.41442	−0.707209	−	0.707004i	$-0.750046\pi$
$317$	−25.1830	−1.41442	−0.707209	+	0.707004i	$0.750046\pi$
$318$	0	0
$319$	9.52937	0.533542
$320$	0	0
$321$	23.4772	1.31037
$322$	0	0
$323$	−3.36894	−0.187453
$324$	0	0
$325$	26.2830	1.45792
$326$	0	0
$327$	−5.58768	−0.309000
$328$	0	0
$329$	17.7892	0.980749
$330$	0	0
$331$	−8.36957	−0.460033	−0.230017	−	0.973187i	$-0.573878\pi$
$331$	−8.36957	−0.460033	−0.230017	+	0.973187i	$0.573878\pi$
$332$	0	0
$333$	−4.59883	−0.252015
$334$	0	0
$335$	3.00884	0.164390
$336$	0	0
$337$	−10.3718	−0.564990	−0.282495	−	0.959269i	$-0.591162\pi$
$337$	−10.3718	−0.564990	−0.282495	+	0.959269i	$0.591162\pi$
$338$	0	0
$339$	21.4094	1.16280
$340$	0	0
$341$	10.3454	0.560237
$342$	0	0
$343$	2.93076	0.158246
$344$	0	0
$345$	−13.2616	−0.713982
$346$	0	0
$347$	16.0341	0.860753	0.430377	−	0.902649i	$-0.358381\pi$
$347$	16.0341	0.860753	0.430377	+	0.902649i	$0.358381\pi$
$348$	0	0
$349$	−1.91085	−0.102285	−0.0511426	−	0.998691i	$-0.516286\pi$
$349$	−1.91085	−0.102285	−0.0511426	+	0.998691i	$0.516286\pi$
$350$	0	0
$351$	24.4950	1.30745
$352$	0	0
$353$	−25.7173	−1.36879	−0.684396	−	0.729110i	$-0.739934\pi$
$353$	−25.7173	−1.36879	−0.684396	+	0.729110i	$0.739934\pi$
$354$	0	0
$355$	−0.624025	−0.0331198
$356$	0	0
$357$	21.1830	1.12112
$358$	0	0
$359$	−26.4833	−1.39774	−0.698868	−	0.715250i	$-0.746313\pi$
$359$	−26.4833	−1.39774	−0.698868	+	0.715250i	$0.746313\pi$
$360$	0	0
$361$	−16.1626	−0.850661
$362$	0	0
$363$	26.6742	1.40004
$364$	0	0
$365$	2.82521	0.147878
$366$	0	0
$367$	35.5930	1.85794	0.928970	−	0.370155i	$-0.120696\pi$
$367$	35.5930	1.85794	0.928970	+	0.370155i	$0.120696\pi$
$368$	0	0
$369$	−35.9693	−1.87249
$370$	0	0
$371$	−10.0135	−0.519876
$372$	0	0
$373$	26.2490	1.35912	0.679560	−	0.733620i	$-0.262171\pi$
$373$	26.2490	1.35912	0.679560	+	0.733620i	$0.262171\pi$
$374$	0	0
$375$	−13.9601	−0.720899
$376$	0	0
$377$	−46.0367	−2.37101
$378$	0	0
$379$	5.61875	0.288616	0.144308	−	0.989533i	$-0.453904\pi$
$379$	5.61875	0.288616	0.144308	+	0.989533i	$0.453904\pi$
$380$	0	0
$381$	−4.24895	−0.217680
$382$	0	0
$383$	13.4217	0.685818	0.342909	−	0.939369i	$-0.388588\pi$
$383$	13.4217	0.685818	0.342909	+	0.939369i	$0.388588\pi$
$384$	0	0
$385$	−2.30086	−0.117263
$386$	0	0
$387$	37.1009	1.88594
$388$	0	0
$389$	−2.66629	−0.135186	−0.0675931	−	0.997713i	$-0.521532\pi$
$389$	−2.66629	−0.135186	−0.0675931	+	0.997713i	$0.521532\pi$
$390$	0	0
$391$	18.4845	0.934801
$392$	0	0
$393$	−29.7713	−1.50176
$394$	0	0
$395$	−8.02320	−0.403691
$396$	0	0
$397$	−24.9328	−1.25134	−0.625671	−	0.780087i	$-0.715175\pi$
$397$	−24.9328	−1.25134	−0.625671	+	0.780087i	$0.715175\pi$
$398$	0	0
$399$	−17.8411	−0.893171
$400$	0	0
$401$	24.2241	1.20969	0.604846	−	0.796343i	$-0.293235\pi$
$401$	24.2241	1.20969	0.604846	+	0.796343i	$0.293235\pi$
$402$	0	0
$403$	−49.9792	−2.48964
$404$	0	0
$405$	0.857432	0.0426061
$406$	0	0
$407$	−1.15043	−0.0570247
$408$	0	0
$409$	−1.83405	−0.0906877	−0.0453439	−	0.998971i	$-0.514438\pi$
$409$	−1.83405	−0.0906877	−0.0453439	+	0.998971i	$0.514438\pi$
$410$	0	0
$411$	−50.1030	−2.47140
$412$	0	0
$413$	−11.6845	−0.574955
$414$	0	0
$415$	6.62342	0.325131
$416$	0	0
$417$	28.5182	1.39654
$418$	0	0
$419$	−22.4117	−1.09488	−0.547442	−	0.836843i	$-0.684398\pi$
$419$	−22.4117	−1.09488	−0.547442	+	0.836843i	$0.684398\pi$
$420$	0	0
$421$	−10.6786	−0.520445	−0.260223	−	0.965549i	$-0.583796\pi$
$421$	−10.6786	−0.520445	−0.260223	+	0.965549i	$0.583796\pi$
$422$	0	0
$423$	21.2921	1.03526
$424$	0	0
$425$	9.45810	0.458785
$426$	0	0
$427$	−14.6851	−0.710662
$428$	0	0
$429$	17.6252	0.850954
$430$	0	0
$431$	−1.45106	−0.0698953	−0.0349476	−	0.999389i	$-0.511126\pi$
$431$	−1.45106	−0.0698953	−0.0349476	+	0.999389i	$0.511126\pi$
$432$	0	0
$433$	−16.6561	−0.800439	−0.400220	−	0.916419i	$-0.631066\pi$
$433$	−16.6561	−0.800439	−0.400220	+	0.916419i	$0.631066\pi$
$434$	0	0
$435$	11.8857	0.569874
$436$	0	0
$437$	−15.5683	−0.744733
$438$	0	0
$439$	37.2006	1.77549	0.887743	−	0.460340i	$-0.152272\pi$
$439$	37.2006	1.77549	0.887743	+	0.460340i	$0.152272\pi$
$440$	0	0
$441$	35.6997	1.69999
$442$	0	0
$443$	16.9024	0.803055	0.401528	−	0.915847i	$-0.368479\pi$
$443$	16.9024	0.803055	0.401528	+	0.915847i	$0.368479\pi$
$444$	0	0
$445$	2.54894	0.120831
$446$	0	0
$447$	1.91672	0.0906578
$448$	0	0
$449$	15.9156	0.751102	0.375551	−	0.926802i	$-0.377453\pi$
$449$	15.9156	0.751102	0.375551	+	0.926802i	$0.377453\pi$
$450$	0	0
$451$	−8.99797	−0.423698
$452$	0	0
$453$	28.4698	1.33763
$454$	0	0
$455$	11.1155	0.521105
$456$	0	0
$457$	−33.1954	−1.55281	−0.776407	−	0.630232i	$-0.782960\pi$
$457$	−33.1954	−1.55281	−0.776407	+	0.630232i	$0.782960\pi$
$458$	0	0
$459$	8.81469	0.411434
$460$	0	0
$461$	19.9490	0.929116	0.464558	−	0.885543i	$-0.346213\pi$
$461$	19.9490	0.929116	0.464558	+	0.885543i	$0.346213\pi$
$462$	0	0
$463$	5.88797	0.273637	0.136819	−	0.990596i	$-0.456312\pi$
$463$	5.88797	0.273637	0.136819	+	0.990596i	$0.456312\pi$
$464$	0	0
$465$	12.9035	0.598386
$466$	0	0
$467$	−12.2862	−0.568537	−0.284269	−	0.958745i	$-0.591751\pi$
$467$	−12.2862	−0.568537	−0.284269	+	0.958745i	$0.591751\pi$
$468$	0	0
$469$	−22.2094	−1.02553
$470$	0	0
$471$	−8.84286	−0.407458
$472$	0	0
$473$	9.28105	0.426743
$474$	0	0
$475$	−7.96595	−0.365503
$476$	0	0
$477$	−11.9853	−0.548771
$478$	0	0
$479$	−24.7901	−1.13269	−0.566344	−	0.824169i	$-0.691643\pi$
$479$	−24.7901	−1.13269	−0.566344	+	0.824169i	$0.691643\pi$
$480$	0	0
$481$	5.55777	0.253413
$482$	0	0
$483$	97.8892	4.45411
$484$	0	0
$485$	6.49212	0.294792
$486$	0	0
$487$	34.6815	1.57157	0.785785	−	0.618500i	$-0.212259\pi$
$487$	34.6815	1.57157	0.785785	+	0.618500i	$0.212259\pi$
$488$	0	0
$489$	34.1690	1.54517
$490$	0	0
$491$	23.6780	1.06857	0.534287	−	0.845303i	$-0.320580\pi$
$491$	23.6780	1.06857	0.534287	+	0.845303i	$0.320580\pi$
$492$	0	0
$493$	−16.5666	−0.746123
$494$	0	0
$495$	−2.75394	−0.123780
$496$	0	0
$497$	4.60617	0.206615
$498$	0	0
$499$	1.01936	0.0456328	0.0228164	−	0.999740i	$-0.492737\pi$
$499$	1.01936	0.0456328	0.0228164	+	0.999740i	$0.492737\pi$
$500$	0	0
$501$	18.1604	0.811348
$502$	0	0
$503$	7.73990	0.345105	0.172553	−	0.985000i	$-0.444799\pi$
$503$	7.73990	0.345105	0.172553	+	0.985000i	$0.444799\pi$
$504$	0	0
$505$	1.98765	0.0884491
$506$	0	0
$507$	−49.3124	−2.19004
$508$	0	0
$509$	−36.4672	−1.61638	−0.808191	−	0.588921i	$-0.799553\pi$
$509$	−36.4672	−1.61638	−0.808191	+	0.588921i	$0.799553\pi$
$510$	0	0
$511$	−20.8540	−0.922525
$512$	0	0
$513$	−7.42404	−0.327779
$514$	0	0
$515$	6.79819	0.299564
$516$	0	0
$517$	5.32638	0.234254
$518$	0	0
$519$	6.23989	0.273901
$520$	0	0
$521$	−26.3985	−1.15654	−0.578270	−	0.815845i	$-0.696272\pi$
$521$	−26.3985	−1.15654	−0.578270	+	0.815845i	$0.696272\pi$
$522$	0	0
$523$	−3.67212	−0.160571	−0.0802853	−	0.996772i	$-0.525583\pi$
$523$	−3.67212	−0.160571	−0.0802853	+	0.996772i	$0.525583\pi$
$524$	0	0
$525$	50.0877	2.18601
$526$	0	0
$527$	−17.9853	−0.783453
$528$	0	0
$529$	62.4191	2.71387
$530$	0	0
$531$	−13.9853	−0.606912
$532$	0	0
$533$	43.4695	1.88287
$534$	0	0
$535$	4.43321	0.191664
$536$	0	0
$537$	61.3373	2.64690
$538$	0	0
$539$	8.93053	0.384665
$540$	0	0
$541$	13.2033	0.567655	0.283827	−	0.958875i	$-0.408396\pi$
$541$	13.2033	0.567655	0.283827	+	0.958875i	$0.408396\pi$
$542$	0	0
$543$	−49.7765	−2.13612
$544$	0	0
$545$	−1.05513	−0.0451966
$546$	0	0
$547$	−11.4657	−0.490239	−0.245120	−	0.969493i	$-0.578827\pi$
$547$	−11.4657	−0.490239	−0.245120	+	0.969493i	$0.578827\pi$
$548$	0	0
$549$	−17.5768	−0.750161
$550$	0	0
$551$	13.9530	0.594417
$552$	0	0
$553$	59.2224	2.51839
$554$	0	0
$555$	−1.43489	−0.0609078
$556$	0	0
$557$	−13.9082	−0.589310	−0.294655	−	0.955604i	$-0.595205\pi$
$557$	−13.9082	−0.589310	−0.294655	+	0.955604i	$0.595205\pi$
$558$	0	0
$559$	−44.8371	−1.89641
$560$	0	0
$561$	6.34255	0.267783
$562$	0	0
$563$	18.2387	0.768671	0.384335	−	0.923194i	$-0.374431\pi$
$563$	18.2387	0.768671	0.384335	+	0.923194i	$0.374431\pi$
$564$	0	0
$565$	4.04275	0.170080
$566$	0	0
$567$	−6.32904	−0.265795
$568$	0	0
$569$	40.4211	1.69454	0.847270	−	0.531162i	$-0.178244\pi$
$569$	40.4211	1.69454	0.847270	+	0.531162i	$0.178244\pi$
$570$	0	0
$571$	−28.6181	−1.19763	−0.598816	−	0.800887i	$-0.704362\pi$
$571$	−28.6181	−1.19763	−0.598816	+	0.800887i	$0.704362\pi$
$572$	0	0
$573$	37.6847	1.57430
$574$	0	0
$575$	43.7070	1.82271
$576$	0	0
$577$	38.0358	1.58345	0.791726	−	0.610877i	$-0.209183\pi$
$577$	38.0358	1.58345	0.791726	+	0.610877i	$0.209183\pi$
$578$	0	0
$579$	−65.2077	−2.70994
$580$	0	0
$581$	−48.8900	−2.02830
$582$	0	0
$583$	−2.99821	−0.124173
$584$	0	0
$585$	13.3044	0.550068
$586$	0	0
$587$	−25.3126	−1.04476	−0.522381	−	0.852712i	$-0.674956\pi$
$587$	−25.3126	−1.04476	−0.522381	+	0.852712i	$0.674956\pi$
$588$	0	0
$589$	15.1479	0.624158
$590$	0	0
$591$	58.8933	2.42255
$592$	0	0
$593$	−40.8212	−1.67633	−0.838163	−	0.545420i	$-0.816370\pi$
$593$	−40.8212	−1.67633	−0.838163	+	0.545420i	$0.816370\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	23.1279	0.946562
$598$	0	0
$599$	−2.93608	−0.119965	−0.0599825	−	0.998199i	$-0.519105\pi$
$599$	−2.93608	−0.119965	−0.0599825	+	0.998199i	$0.519105\pi$
$600$	0	0
$601$	19.7803	0.806857	0.403428	−	0.915011i	$-0.367818\pi$
$601$	19.7803	0.806857	0.403428	+	0.915011i	$0.367818\pi$
$602$	0	0
$603$	−26.5828	−1.08253
$604$	0	0
$605$	5.03692	0.204780
$606$	0	0
$607$	28.0840	1.13989	0.569946	−	0.821682i	$-0.306964\pi$
$607$	28.0840	1.13989	0.569946	+	0.821682i	$0.306964\pi$
$608$	0	0
$609$	−87.7326	−3.55511
$610$	0	0
$611$	−25.7319	−1.04100
$612$	0	0
$613$	−43.3822	−1.75219	−0.876095	−	0.482139i	$-0.839860\pi$
$613$	−43.3822	−1.75219	−0.876095	+	0.482139i	$0.839860\pi$
$614$	0	0
$615$	−11.2229	−0.452549
$616$	0	0
$617$	3.73224	0.150254	0.0751271	−	0.997174i	$-0.476064\pi$
$617$	3.73224	0.150254	0.0751271	+	0.997174i	$0.476064\pi$
$618$	0	0
$619$	34.2437	1.37637	0.688185	−	0.725535i	$-0.258408\pi$
$619$	34.2437	1.37637	0.688185	+	0.725535i	$0.258408\pi$
$620$	0	0
$621$	40.7337	1.63459
$622$	0	0
$623$	−18.8147	−0.753794
$624$	0	0
$625$	21.0091	0.840366
$626$	0	0
$627$	−5.34192	−0.213336
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	17.6803	0.703843	0.351921	−	0.936030i	$-0.385528\pi$
$631$	17.6803	0.703843	0.351921	+	0.936030i	$0.385528\pi$
$632$	0	0
$633$	2.38235	0.0946901
$634$	0	0
$635$	−0.802332	−0.0318396
$636$	0	0
$637$	−43.1437	−1.70942
$638$	0	0
$639$	5.51320	0.218099
$640$	0	0
$641$	0.437781	0.0172913	0.00864566	−	0.999963i	$-0.497248\pi$
$641$	0.437781	0.0172913	0.00864566	+	0.999963i	$0.497248\pi$
$642$	0	0
$643$	−4.35490	−0.171741	−0.0858703	−	0.996306i	$-0.527367\pi$
$643$	−4.35490	−0.171741	−0.0858703	+	0.996306i	$0.527367\pi$
$644$	0	0
$645$	11.5759	0.455802
$646$	0	0
$647$	25.4065	0.998832	0.499416	−	0.866362i	$-0.333548\pi$
$647$	25.4065	0.998832	0.499416	+	0.866362i	$0.333548\pi$
$648$	0	0
$649$	−3.49853	−0.137329
$650$	0	0
$651$	−95.2458	−3.73298
$652$	0	0
$653$	−44.9923	−1.76069	−0.880343	−	0.474338i	$-0.842687\pi$
$653$	−44.9923	−1.76069	−0.880343	+	0.474338i	$0.842687\pi$
$654$	0	0
$655$	−5.62174	−0.219659
$656$	0	0
$657$	−24.9604	−0.973800
$658$	0	0
$659$	35.7279	1.39176	0.695881	−	0.718158i	$-0.255014\pi$
$659$	35.7279	1.39176	0.695881	+	0.718158i	$0.255014\pi$
$660$	0	0
$661$	17.5103	0.681072	0.340536	−	0.940231i	$-0.389391\pi$
$661$	17.5103	0.681072	0.340536	+	0.940231i	$0.389391\pi$
$662$	0	0
$663$	−30.6411	−1.19000
$664$	0	0
$665$	−3.36894	−0.130642
$666$	0	0
$667$	−76.5563	−2.96427
$668$	0	0
$669$	−4.87298	−0.188400
$670$	0	0
$671$	−4.39697	−0.169743
$672$	0	0
$673$	24.2033	0.932969	0.466485	−	0.884529i	$-0.345520\pi$
$673$	24.2033	0.932969	0.466485	+	0.884529i	$0.345520\pi$
$674$	0	0
$675$	20.8425	0.802230
$676$	0	0
$677$	−13.8243	−0.531310	−0.265655	−	0.964068i	$-0.585588\pi$
$677$	−13.8243	−0.531310	−0.265655	+	0.964068i	$0.585588\pi$
$678$	0	0
$679$	−47.9209	−1.83903
$680$	0	0
$681$	26.5813	1.01860
$682$	0	0
$683$	48.9983	1.87487	0.937434	−	0.348163i	$-0.113194\pi$
$683$	48.9983	1.87487	0.937434	+	0.348163i	$0.113194\pi$
$684$	0	0
$685$	−9.46098	−0.361486
$686$	0	0
$687$	43.2786	1.65118
$688$	0	0
$689$	14.4845	0.551815
$690$	0	0
$691$	−37.3677	−1.42153	−0.710767	−	0.703428i	$-0.751652\pi$
$691$	−37.3677	−1.42153	−0.710767	+	0.703428i	$0.751652\pi$
$692$	0	0
$693$	20.3279	0.772192
$694$	0	0
$695$	5.38511	0.204269
$696$	0	0
$697$	15.6428	0.592513
$698$	0	0
$699$	−3.57626	−0.135267
$700$	0	0
$701$	30.8484	1.16513	0.582564	−	0.812785i	$-0.302049\pi$
$701$	30.8484	1.16513	0.582564	+	0.812785i	$0.302049\pi$
$702$	0	0
$703$	−1.68447	−0.0635310
$704$	0	0
$705$	6.64341	0.250205
$706$	0	0
$707$	−14.6716	−0.551782
$708$	0	0
$709$	25.7846	0.968361	0.484180	−	0.874968i	$-0.339118\pi$
$709$	25.7846	0.968361	0.484180	+	0.874968i	$0.339118\pi$
$710$	0	0
$711$	70.8842	2.65837
$712$	0	0
$713$	−83.1124	−3.11258
$714$	0	0
$715$	3.32818	0.124467
$716$	0	0
$717$	12.5017	0.466885
$718$	0	0
$719$	−9.31203	−0.347280	−0.173640	−	0.984809i	$-0.555553\pi$
$719$	−9.31203	−0.347280	−0.173640	+	0.984809i	$0.555553\pi$
$720$	0	0
$721$	−50.1801	−1.86880
$722$	0	0
$723$	83.1233	3.09139
$724$	0	0
$725$	−39.1722	−1.45482
$726$	0	0
$727$	−20.3053	−0.753080	−0.376540	−	0.926400i	$-0.622886\pi$
$727$	−20.3053	−0.753080	−0.376540	+	0.926400i	$0.622886\pi$
$728$	0	0
$729$	−44.0231	−1.63049
$730$	0	0
$731$	−16.1349	−0.596771
$732$	0	0
$733$	22.4126	0.827828	0.413914	−	0.910316i	$-0.364161\pi$
$733$	22.4126	0.827828	0.413914	+	0.910316i	$0.364161\pi$
$734$	0	0
$735$	11.1388	0.410859
$736$	0	0
$737$	−6.64987	−0.244951
$738$	0	0
$739$	36.1722	1.33062	0.665308	−	0.746569i	$-0.268300\pi$
$739$	36.1722	1.33062	0.665308	+	0.746569i	$0.268300\pi$
$740$	0	0
$741$	25.8070	0.948045
$742$	0	0
$743$	23.0229	0.844629	0.422315	−	0.906449i	$-0.361218\pi$
$743$	23.0229	0.844629	0.422315	+	0.906449i	$0.361218\pi$
$744$	0	0
$745$	0.361936	0.0132603
$746$	0	0
$747$	−58.5173	−2.14104
$748$	0	0
$749$	−32.7232	−1.19568
$750$	0	0
$751$	19.9367	0.727501	0.363750	−	0.931496i	$-0.381496\pi$
$751$	19.9367	0.727501	0.363750	+	0.931496i	$0.381496\pi$
$752$	0	0
$753$	51.7092	1.88439
$754$	0	0
$755$	5.37598	0.195652
$756$	0	0
$757$	−8.04734	−0.292485	−0.146243	−	0.989249i	$-0.546718\pi$
$757$	−8.04734	−0.292485	−0.146243	+	0.989249i	$0.546718\pi$
$758$	0	0
$759$	29.3097	1.06387
$760$	0	0
$761$	4.84904	0.175778	0.0878888	−	0.996130i	$-0.471988\pi$
$761$	4.84904	0.175778	0.0878888	+	0.996130i	$0.471988\pi$
$762$	0	0
$763$	7.78829	0.281955
$764$	0	0
$765$	4.78766	0.173098
$766$	0	0
$767$	16.9015	0.610279
$768$	0	0
$769$	−37.6258	−1.35682	−0.678411	−	0.734683i	$-0.737331\pi$
$769$	−37.6258	−1.35682	−0.678411	+	0.734683i	$0.737331\pi$
$770$	0	0
$771$	36.2692	1.30620
$772$	0	0
$773$	35.0467	1.26054	0.630270	−	0.776376i	$-0.282944\pi$
$773$	35.0467	1.26054	0.630270	+	0.776376i	$0.282944\pi$
$774$	0	0
$775$	−42.5268	−1.52761
$776$	0	0
$777$	10.5915	0.379968
$778$	0	0
$779$	−13.1749	−0.472040
$780$	0	0
$781$	1.37916	0.0493504
$782$	0	0
$783$	−36.5074	−1.30467
$784$	0	0
$785$	−1.66980	−0.0595979
$786$	0	0
$787$	17.2452	0.614726	0.307363	−	0.951592i	$-0.400553\pi$
$787$	17.2452	0.614726	0.307363	+	0.951592i	$0.400553\pi$
$788$	0	0
$789$	−62.6760	−2.23133
$790$	0	0
$791$	−29.8411	−1.06103
$792$	0	0
$793$	21.2419	0.754323
$794$	0	0
$795$	−3.73957	−0.132629
$796$	0	0
$797$	38.1431	1.35110	0.675549	−	0.737315i	$-0.263907\pi$
$797$	38.1431	1.35110	0.675549	+	0.737315i	$0.263907\pi$
$798$	0	0
$799$	−9.25980	−0.327588
$800$	0	0
$801$	−22.5196	−0.795691
$802$	0	0
$803$	−6.24403	−0.220347
$804$	0	0
$805$	18.4845	0.651493
$806$	0	0
$807$	−35.3960	−1.24600
$808$	0	0
$809$	−20.8487	−0.733003	−0.366501	−	0.930418i	$-0.619444\pi$
$809$	−20.8487	−0.733003	−0.366501	+	0.930418i	$0.619444\pi$
$810$	0	0
$811$	30.6262	1.07543	0.537716	−	0.843126i	$-0.319287\pi$
$811$	30.6262	1.07543	0.537716	+	0.843126i	$0.319287\pi$
$812$	0	0
$813$	−37.3071	−1.30842
$814$	0	0
$815$	6.45215	0.226009
$816$	0	0
$817$	13.5894	0.475433
$818$	0	0
$819$	−98.2048	−3.43155
$820$	0	0
$821$	−27.7687	−0.969135	−0.484568	−	0.874754i	$-0.661023\pi$
$821$	−27.7687	−0.969135	−0.484568	+	0.874754i	$0.661023\pi$
$822$	0	0
$823$	−8.02934	−0.279885	−0.139943	−	0.990160i	$-0.544692\pi$
$823$	−8.02934	−0.279885	−0.139943	+	0.990160i	$0.544692\pi$
$824$	0	0
$825$	14.9971	0.522132
$826$	0	0
$827$	−17.0170	−0.591741	−0.295870	−	0.955228i	$-0.595610\pi$
$827$	−17.0170	−0.591741	−0.295870	+	0.955228i	$0.595610\pi$
$828$	0	0
$829$	1.53075	0.0531652	0.0265826	−	0.999647i	$-0.491537\pi$
$829$	1.53075	0.0531652	0.0265826	+	0.999647i	$0.491537\pi$
$830$	0	0
$831$	−90.7699	−3.14877
$832$	0	0
$833$	−15.5256	−0.537928
$834$	0	0
$835$	3.42925	0.118674
$836$	0	0
$837$	−39.6338	−1.36994
$838$	0	0
$839$	34.1202	1.17796	0.588981	−	0.808147i	$-0.299529\pi$
$839$	34.1202	1.17796	0.588981	+	0.808147i	$0.299529\pi$
$840$	0	0
$841$	39.6131	1.36597
$842$	0	0
$843$	8.34255	0.287333
$844$	0	0
$845$	−9.31169	−0.320332
$846$	0	0
$847$	−37.1794	−1.27750
$848$	0	0
$849$	−7.15598	−0.245593
$850$	0	0
$851$	9.24224	0.316820
$852$	0	0
$853$	−11.3352	−0.388110	−0.194055	−	0.980991i	$-0.562164\pi$
$853$	−11.3352	−0.388110	−0.194055	+	0.980991i	$0.562164\pi$
$854$	0	0
$855$	−4.03234	−0.137903
$856$	0	0
$857$	45.7877	1.56408	0.782040	−	0.623229i	$-0.214179\pi$
$857$	45.7877	1.56408	0.782040	+	0.623229i	$0.214179\pi$
$858$	0	0
$859$	−26.5683	−0.906499	−0.453250	−	0.891384i	$-0.649735\pi$
$859$	−26.5683	−0.906499	−0.453250	+	0.891384i	$0.649735\pi$
$860$	0	0
$861$	82.8403	2.82319
$862$	0	0
$863$	−45.0932	−1.53499	−0.767496	−	0.641054i	$-0.778498\pi$
$863$	−45.0932	−1.53499	−0.767496	+	0.641054i	$0.778498\pi$
$864$	0	0
$865$	1.17828	0.0400628
$866$	0	0
$867$	35.8358	1.21705
$868$	0	0
$869$	17.7322	0.601523
$870$	0	0
$871$	32.1258	1.08854
$872$	0	0
$873$	−57.3573	−1.94125
$874$	0	0
$875$	19.4581	0.657804
$876$	0	0
$877$	−28.3004	−0.955637	−0.477818	−	0.878459i	$-0.658572\pi$
$877$	−28.3004	−0.955637	−0.477818	+	0.878459i	$0.658572\pi$
$878$	0	0
$879$	78.5203	2.64842
$880$	0	0
$881$	39.3657	1.32626	0.663132	−	0.748502i	$-0.269227\pi$
$881$	39.3657	1.32626	0.663132	+	0.748502i	$0.269227\pi$
$882$	0	0
$883$	−25.0728	−0.843767	−0.421883	−	0.906650i	$-0.638631\pi$
$883$	−25.0728	−0.843767	−0.421883	+	0.906650i	$0.638631\pi$
$884$	0	0
$885$	−4.36360	−0.146681
$886$	0	0
$887$	10.4316	0.350259	0.175129	−	0.984545i	$-0.443966\pi$
$887$	10.4316	0.350259	0.175129	+	0.984545i	$0.443966\pi$
$888$	0	0
$889$	5.92232	0.198628
$890$	0	0
$891$	−1.89502	−0.0634856
$892$	0	0
$893$	7.79893	0.260981
$894$	0	0
$895$	11.5824	0.387156
$896$	0	0
$897$	−141.596	−4.72776
$898$	0	0
$899$	74.4890	2.48435
$900$	0	0
$901$	5.21234	0.173648
$902$	0	0
$903$	−85.4464	−2.84348
$904$	0	0
$905$	−9.39934	−0.312444
$906$	0	0
$907$	22.2839	0.739924	0.369962	−	0.929047i	$-0.379371\pi$
$907$	22.2839	0.739924	0.369962	+	0.929047i	$0.379371\pi$
$908$	0	0
$909$	−17.5607	−0.582450
$910$	0	0
$911$	−40.3132	−1.33564	−0.667818	−	0.744324i	$-0.732772\pi$
$911$	−40.3132	−1.33564	−0.667818	+	0.744324i	$0.732772\pi$
$912$	0	0
$913$	−14.6385	−0.484464
$914$	0	0
$915$	−5.48419	−0.181302
$916$	0	0
$917$	41.4962	1.37033
$918$	0	0
$919$	21.2664	0.701513	0.350757	−	0.936467i	$-0.385924\pi$
$919$	21.2664	0.701513	0.350757	+	0.936467i	$0.385924\pi$
$920$	0	0
$921$	−33.1989	−1.09394
$922$	0	0
$923$	−6.66280	−0.219309
$924$	0	0
$925$	4.72905	0.155490
$926$	0	0
$927$	−60.0613	−1.97267
$928$	0	0
$929$	−1.29006	−0.0423257	−0.0211628	−	0.999776i	$-0.506737\pi$
$929$	−1.29006	−0.0423257	−0.0211628	+	0.999776i	$0.506737\pi$
$930$	0	0
$931$	13.0762	0.428554
$932$	0	0
$933$	68.8438	2.25384
$934$	0	0
$935$	1.19767	0.0391679
$936$	0	0
$937$	−19.3891	−0.633414	−0.316707	−	0.948523i	$-0.602577\pi$
$937$	−19.3891	−0.633414	−0.316707	+	0.948523i	$0.602577\pi$
$938$	0	0
$939$	19.1232	0.624061
$940$	0	0
$941$	30.9092	1.00761	0.503805	−	0.863817i	$-0.331933\pi$
$941$	30.9092	1.00761	0.503805	+	0.863817i	$0.331933\pi$
$942$	0	0
$943$	72.2872	2.35400
$944$	0	0
$945$	8.81469	0.286742
$946$	0	0
$947$	−7.92320	−0.257469	−0.128735	−	0.991679i	$-0.541092\pi$
$947$	−7.92320	−0.257469	−0.128735	+	0.991679i	$0.541092\pi$
$948$	0	0
$949$	30.1651	0.979202
$950$	0	0
$951$	69.4194	2.25108
$952$	0	0
$953$	0.440541	0.0142705	0.00713526	−	0.999975i	$-0.497729\pi$
$953$	0.440541	0.0142705	0.00713526	+	0.999975i	$0.497729\pi$
$954$	0	0
$955$	7.11603	0.230269
$956$	0	0
$957$	−26.2686	−0.849144
$958$	0	0
$959$	69.8352	2.25510
$960$	0	0
$961$	49.8680	1.60865
$962$	0	0
$963$	−39.1669	−1.26214
$964$	0	0
$965$	−12.3132	−0.396376
$966$	0	0
$967$	−11.7799	−0.378815	−0.189408	−	0.981899i	$-0.560657\pi$
$967$	−11.7799	−0.378815	−0.189408	+	0.981899i	$0.560657\pi$
$968$	0	0
$969$	9.28682	0.298336
$970$	0	0
$971$	−41.5593	−1.33370	−0.666850	−	0.745192i	$-0.732358\pi$
$971$	−41.5593	−1.33370	−0.666850	+	0.745192i	$0.732358\pi$
$972$	0	0
$973$	−39.7496	−1.27431
$974$	0	0
$975$	−72.4516	−2.32031
$976$	0	0
$977$	46.3877	1.48407	0.742037	−	0.670359i	$-0.233860\pi$
$977$	46.3877	1.48407	0.742037	+	0.670359i	$0.233860\pi$
$978$	0	0
$979$	−5.63344	−0.180045
$980$	0	0
$981$	9.32193	0.297626
$982$	0	0
$983$	12.0099	0.383058	0.191529	−	0.981487i	$-0.438655\pi$
$983$	12.0099	0.383058	0.191529	+	0.981487i	$0.438655\pi$
$984$	0	0
$985$	11.1209	0.354340
$986$	0	0
$987$	−49.0376	−1.56088
$988$	0	0
$989$	−74.5614	−2.37091
$990$	0	0
$991$	−5.03054	−0.159800	−0.0799001	−	0.996803i	$-0.525460\pi$
$991$	−5.03054	−0.159800	−0.0799001	+	0.996803i	$0.525460\pi$
$992$	0	0
$993$	23.0715	0.732153
$994$	0	0
$995$	4.36726	0.138451
$996$	0	0
$997$	−22.8223	−0.722790	−0.361395	−	0.932413i	$-0.617699\pi$
$997$	−22.8223	−0.722790	−0.361395	+	0.932413i	$0.617699\pi$
$998$	0	0
$999$	4.40734	0.139442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bf.1.1		4
4.3	odd	2		2368.2.a.bi.1.4		4
8.3	odd	2		1184.2.a.n.1.1	✓	4
8.5	even	2		1184.2.a.o.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.n.1.1	✓	4	8.3	odd	2
1184.2.a.o.1.4	yes	4	8.5	even	2
2368.2.a.bf.1.1		4	1.1	even	1	trivial
2368.2.a.bi.1.4		4	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.75660 q^{3} -0.520530 q^{5} +3.84224 q^{7} +4.59883 q^{9} +O(q^{10})\)	\(q-2.75660 q^{3} -0.520530 q^{5} +3.84224 q^{7} +4.59883 q^{9} +1.15043 q^{11} -5.55777 q^{13} +1.43489 q^{15} -2.00000 q^{17} +1.68447 q^{19} -10.5915 q^{21} -9.24224 q^{23} -4.72905 q^{25} -4.40734 q^{27} +8.28331 q^{29} +8.99267 q^{31} -3.17127 q^{33} -2.00000 q^{35} -1.00000 q^{37} +15.3205 q^{39} -7.82139 q^{41} +8.06745 q^{43} -2.39383 q^{45} +4.62990 q^{47} +7.76278 q^{49} +5.51320 q^{51} -2.60617 q^{53} -0.598834 q^{55} -4.64341 q^{57} -3.04106 q^{59} -3.82202 q^{61} +17.6698 q^{63} +2.89299 q^{65} -5.78033 q^{67} +25.4772 q^{69} +1.19882 q^{71} -5.42756 q^{73} +13.0361 q^{75} +4.42023 q^{77} +15.4135 q^{79} -1.64723 q^{81} -12.7244 q^{83} +1.04106 q^{85} -22.8337 q^{87} -4.89681 q^{89} -21.3543 q^{91} -24.7892 q^{93} -0.876818 q^{95} -12.4721 q^{97} +5.29064 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + q^9	\( 4 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + q^{9} - 3 q^{11} - 3 q^{13} + 8 q^{15} - 8 q^{17} - 12 q^{19} - 8 q^{21} + q^{23} + 3 q^{25} - 6 q^{27} - 3 q^{29} + 19 q^{31} - 10 q^{33} - 8 q^{35} - 4 q^{37} + 5 q^{39} - 17 q^{41} + 2 q^{43} - 10 q^{45} + 10 q^{47} - 6 q^{49} + 6 q^{51} - 10 q^{53} + 15 q^{55} + 2 q^{57} - 14 q^{59} - 9 q^{61} + 18 q^{63} - 30 q^{65} - 7 q^{67} + 9 q^{69} + 16 q^{71} - 7 q^{73} - 14 q^{75} - 14 q^{77} + 21 q^{79} - 8 q^{81} + 12 q^{83} + 6 q^{85} - 19 q^{87} - 14 q^{91} - 30 q^{93} + 2 q^{95} - 32 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + q^9 - 3 * q^11 - 3 * q^13 + 8 * q^15 - 8 * q^17 - 12 * q^19 - 8 * q^21 + q^23 + 3 * q^25 - 6 * q^27 - 3 * q^29 + 19 * q^31 - 10 * q^33 - 8 * q^35 - 4 * q^37 + 5 * q^39 - 17 * q^41 + 2 * q^43 - 10 * q^45 + 10 * q^47 - 6 * q^49 + 6 * q^51 - 10 * q^53 + 15 * q^55 + 2 * q^57 - 14 * q^59 - 9 * q^61 + 18 * q^63 - 30 * q^65 - 7 * q^67 + 9 * q^69 + 16 * q^71 - 7 * q^73 - 14 * q^75 - 14 * q^77 + 21 * q^79 - 8 * q^81 + 12 * q^83 + 6 * q^85 - 19 * q^87 - 14 * q^91 - 30 * q^93 + 2 * q^95 - 32 * q^97 + 2 * q^99

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