LMFDB - Embedded newform 8024.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33387 q^{3} -3.73347 q^{5} -3.52781 q^{7} +8.11466 q^{9} +O(q^{10})$	$q-3.33387 q^{3} -3.73347 q^{5} -3.52781 q^{7} +8.11466 q^{9} -3.60041 q^{11} +3.84913 q^{13} +12.4469 q^{15} -1.00000 q^{17} +8.37428 q^{19} +11.7612 q^{21} -3.05864 q^{23} +8.93882 q^{25} -17.0516 q^{27} +4.19543 q^{29} +7.55444 q^{31} +12.0033 q^{33} +13.1710 q^{35} +5.73849 q^{37} -12.8325 q^{39} -3.13130 q^{41} -4.16094 q^{43} -30.2959 q^{45} -8.52182 q^{47} +5.44544 q^{49} +3.33387 q^{51} -5.85266 q^{53} +13.4420 q^{55} -27.9187 q^{57} -1.00000 q^{59} -9.21081 q^{61} -28.6270 q^{63} -14.3706 q^{65} -13.4893 q^{67} +10.1971 q^{69} +7.66607 q^{71} +5.45921 q^{73} -29.8008 q^{75} +12.7015 q^{77} +7.10601 q^{79} +32.5037 q^{81} -14.7698 q^{83} +3.73347 q^{85} -13.9870 q^{87} +13.2884 q^{89} -13.5790 q^{91} -25.1855 q^{93} -31.2651 q^{95} +16.8573 q^{97} -29.2161 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−3.33387	−1.92481	−0.962404	−	0.271622i	$-0.912440\pi$
$3$	−3.33387	−1.92481	−0.962404	+	0.271622i	$0.912440\pi$
$4$	0	0
$5$	−3.73347	−1.66966	−0.834830	−	0.550508i	$-0.814434\pi$
$5$	−3.73347	−1.66966	−0.834830	+	0.550508i	$0.814434\pi$
$6$	0	0
$7$	−3.52781	−1.33339	−0.666693	−	0.745332i	$-0.732291\pi$
$7$	−3.52781	−1.33339	−0.666693	+	0.745332i	$0.732291\pi$
$8$	0	0
$9$	8.11466	2.70489
$10$	0	0
$11$	−3.60041	−1.08556	−0.542782	−	0.839874i	$-0.682629\pi$
$11$	−3.60041	−1.08556	−0.542782	+	0.839874i	$0.682629\pi$
$12$	0	0
$13$	3.84913	1.06756	0.533779	−	0.845624i	$-0.320772\pi$
$13$	3.84913	1.06756	0.533779	+	0.845624i	$0.320772\pi$
$14$	0	0
$15$	12.4469	3.21377
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	8.37428	1.92119	0.960595	−	0.277951i	$-0.0896552\pi$
$19$	8.37428	1.92119	0.960595	+	0.277951i	$0.0896552\pi$
$20$	0	0
$21$	11.7612	2.56651
$22$	0	0
$23$	−3.05864	−0.637770	−0.318885	−	0.947793i	$-0.603308\pi$
$23$	−3.05864	−0.637770	−0.318885	+	0.947793i	$0.603308\pi$
$24$	0	0
$25$	8.93882	1.78776
$26$	0	0
$27$	−17.0516	−3.28158
$28$	0	0
$29$	4.19543	0.779072	0.389536	−	0.921011i	$-0.372635\pi$
$29$	4.19543	0.779072	0.389536	+	0.921011i	$0.372635\pi$
$30$	0	0
$31$	7.55444	1.35682	0.678409	−	0.734685i	$-0.262670\pi$
$31$	7.55444	1.35682	0.678409	+	0.734685i	$0.262670\pi$
$32$	0	0
$33$	12.0033	2.08950
$34$	0	0
$35$	13.1710	2.22630
$36$	0	0
$37$	5.73849	0.943402	0.471701	−	0.881759i	$-0.343640\pi$
$37$	5.73849	0.943402	0.471701	+	0.881759i	$0.343640\pi$
$38$	0	0
$39$	−12.8325	−2.05484
$40$	0	0
$41$	−3.13130	−0.489026	−0.244513	−	0.969646i	$-0.578628\pi$
$41$	−3.13130	−0.489026	−0.244513	+	0.969646i	$0.578628\pi$
$42$	0	0
$43$	−4.16094	−0.634537	−0.317268	−	0.948336i	$-0.602766\pi$
$43$	−4.16094	−0.634537	−0.317268	+	0.948336i	$0.602766\pi$
$44$	0	0
$45$	−30.2959	−4.51624
$46$	0	0
$47$	−8.52182	−1.24304	−0.621518	−	0.783400i	$-0.713484\pi$
$47$	−8.52182	−1.24304	−0.621518	+	0.783400i	$0.713484\pi$
$48$	0	0
$49$	5.44544	0.777920
$50$	0	0
$51$	3.33387	0.466835
$52$	0	0
$53$	−5.85266	−0.803924	−0.401962	−	0.915656i	$-0.631672\pi$
$53$	−5.85266	−0.803924	−0.401962	+	0.915656i	$0.631672\pi$
$54$	0	0
$55$	13.4420	1.81252
$56$	0	0
$57$	−27.9187	−3.69792
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−9.21081	−1.17932	−0.589661	−	0.807650i	$-0.700739\pi$
$61$	−9.21081	−1.17932	−0.589661	+	0.807650i	$0.700739\pi$
$62$	0	0
$63$	−28.6270	−3.60666
$64$	0	0
$65$	−14.3706	−1.78246
$66$	0	0
$67$	−13.4893	−1.64798	−0.823989	−	0.566606i	$-0.808256\pi$
$67$	−13.4893	−1.64798	−0.823989	+	0.566606i	$0.808256\pi$
$68$	0	0
$69$	10.1971	1.22759
$70$	0	0
$71$	7.66607	0.909795	0.454897	−	0.890544i	$-0.349676\pi$
$71$	7.66607	0.909795	0.454897	+	0.890544i	$0.349676\pi$
$72$	0	0
$73$	5.45921	0.638952	0.319476	−	0.947594i	$-0.396493\pi$
$73$	5.45921	0.638952	0.319476	+	0.947594i	$0.396493\pi$
$74$	0	0
$75$	−29.8008	−3.44110
$76$	0	0
$77$	12.7015	1.44748
$78$	0	0
$79$	7.10601	0.799489	0.399744	−	0.916627i	$-0.369099\pi$
$79$	7.10601	0.799489	0.399744	+	0.916627i	$0.369099\pi$
$80$	0	0
$81$	32.5037	3.61152
$82$	0	0
$83$	−14.7698	−1.62119	−0.810596	−	0.585606i	$-0.800857\pi$
$83$	−14.7698	−1.62119	−0.810596	+	0.585606i	$0.800857\pi$
$84$	0	0
$85$	3.73347	0.404952
$86$	0	0
$87$	−13.9870	−1.49956
$88$	0	0
$89$	13.2884	1.40857	0.704284	−	0.709918i	$-0.251268\pi$
$89$	13.2884	1.40857	0.704284	+	0.709918i	$0.251268\pi$
$90$	0	0
$91$	−13.5790	−1.42347
$92$	0	0
$93$	−25.1855	−2.61161
$94$	0	0
$95$	−31.2651	−3.20773
$96$	0	0
$97$	16.8573	1.71160	0.855801	−	0.517304i	$-0.173065\pi$
$97$	16.8573	1.71160	0.855801	+	0.517304i	$0.173065\pi$
$98$	0	0
$99$	−29.2161	−2.93632
$100$	0	0
$101$	−13.1195	−1.30544	−0.652719	−	0.757600i	$-0.726372\pi$
$101$	−13.1195	−1.30544	−0.652719	+	0.757600i	$0.726372\pi$
$102$	0	0
$103$	−3.38002	−0.333043	−0.166521	−	0.986038i	$-0.553253\pi$
$103$	−3.38002	−0.333043	−0.166521	+	0.986038i	$0.553253\pi$
$104$	0	0
$105$	−43.9103	−4.28520
$106$	0	0
$107$	−0.913489	−0.0883103	−0.0441552	−	0.999025i	$-0.514060\pi$
$107$	−0.913489	−0.0883103	−0.0441552	+	0.999025i	$0.514060\pi$
$108$	0	0
$109$	−6.92364	−0.663164	−0.331582	−	0.943426i	$-0.607582\pi$
$109$	−6.92364	−0.663164	−0.331582	+	0.943426i	$0.607582\pi$
$110$	0	0
$111$	−19.1314	−1.81587
$112$	0	0
$113$	7.09172	0.667133	0.333566	−	0.942727i	$-0.391748\pi$
$113$	7.09172	0.667133	0.333566	+	0.942727i	$0.391748\pi$
$114$	0	0
$115$	11.4193	1.06486
$116$	0	0
$117$	31.2344	2.88762
$118$	0	0
$119$	3.52781	0.323394
$120$	0	0
$121$	1.96292	0.178447
$122$	0	0
$123$	10.4393	0.941282
$124$	0	0
$125$	−14.7055	−1.31530
$126$	0	0
$127$	18.4178	1.63431	0.817157	−	0.576416i	$-0.195549\pi$
$127$	18.4178	1.63431	0.817157	+	0.576416i	$0.195549\pi$
$128$	0	0
$129$	13.8720	1.22136
$130$	0	0
$131$	3.55743	0.310814	0.155407	−	0.987851i	$-0.450331\pi$
$131$	3.55743	0.310814	0.155407	+	0.987851i	$0.450331\pi$
$132$	0	0
$133$	−29.5428	−2.56169
$134$	0	0
$135$	63.6616	5.47912
$136$	0	0
$137$	−3.13022	−0.267433	−0.133716	−	0.991020i	$-0.542691\pi$
$137$	−3.13022	−0.267433	−0.133716	+	0.991020i	$0.542691\pi$
$138$	0	0
$139$	2.96976	0.251891	0.125946	−	0.992037i	$-0.459803\pi$
$139$	2.96976	0.251891	0.125946	+	0.992037i	$0.459803\pi$
$140$	0	0
$141$	28.4106	2.39260
$142$	0	0
$143$	−13.8584	−1.15890
$144$	0	0
$145$	−15.6635	−1.30079
$146$	0	0
$147$	−18.1544	−1.49735
$148$	0	0
$149$	−13.8099	−1.13135	−0.565677	−	0.824627i	$-0.691385\pi$
$149$	−13.8099	−1.13135	−0.565677	+	0.824627i	$0.691385\pi$
$150$	0	0
$151$	−13.2611	−1.07917	−0.539586	−	0.841931i	$-0.681419\pi$
$151$	−13.2611	−1.07917	−0.539586	+	0.841931i	$0.681419\pi$
$152$	0	0
$153$	−8.11466	−0.656031
$154$	0	0
$155$	−28.2043	−2.26542
$156$	0	0
$157$	8.34842	0.666276	0.333138	−	0.942878i	$-0.391892\pi$
$157$	8.34842	0.666276	0.333138	+	0.942878i	$0.391892\pi$
$158$	0	0
$159$	19.5120	1.54740
$160$	0	0
$161$	10.7903	0.850395
$162$	0	0
$163$	5.00726	0.392199	0.196099	−	0.980584i	$-0.437172\pi$
$163$	5.00726	0.392199	0.196099	+	0.980584i	$0.437172\pi$
$164$	0	0
$165$	−44.8139	−3.48876
$166$	0	0
$167$	−10.3027	−0.797250	−0.398625	−	0.917114i	$-0.630512\pi$
$167$	−10.3027	−0.797250	−0.398625	+	0.917114i	$0.630512\pi$
$168$	0	0
$169$	1.81582	0.139678
$170$	0	0
$171$	67.9544	5.19660
$172$	0	0
$173$	21.8205	1.65898	0.829490	−	0.558521i	$-0.188631\pi$
$173$	21.8205	1.65898	0.829490	+	0.558521i	$0.188631\pi$
$174$	0	0
$175$	−31.5345	−2.38378
$176$	0	0
$177$	3.33387	0.250589
$178$	0	0
$179$	2.38729	0.178435	0.0892173	−	0.996012i	$-0.471563\pi$
$179$	2.38729	0.178435	0.0892173	+	0.996012i	$0.471563\pi$
$180$	0	0
$181$	10.0514	0.747116	0.373558	−	0.927607i	$-0.378138\pi$
$181$	10.0514	0.747116	0.373558	+	0.927607i	$0.378138\pi$
$182$	0	0
$183$	30.7076	2.26997
$184$	0	0
$185$	−21.4245	−1.57516
$186$	0	0
$187$	3.60041	0.263288
$188$	0	0
$189$	60.1547	4.37561
$190$	0	0
$191$	−4.45834	−0.322594	−0.161297	−	0.986906i	$-0.551568\pi$
$191$	−4.45834	−0.322594	−0.161297	+	0.986906i	$0.551568\pi$
$192$	0	0
$193$	−14.0148	−1.00881	−0.504405	−	0.863467i	$-0.668288\pi$
$193$	−14.0148	−1.00881	−0.504405	+	0.863467i	$0.668288\pi$
$194$	0	0
$195$	47.9097	3.43089
$196$	0	0
$197$	−27.2784	−1.94351	−0.971754	−	0.235995i	$-0.924165\pi$
$197$	−27.2784	−1.94351	−0.971754	+	0.235995i	$0.924165\pi$
$198$	0	0
$199$	23.7257	1.68187	0.840937	−	0.541133i	$-0.182005\pi$
$199$	23.7257	1.68187	0.840937	+	0.541133i	$0.182005\pi$
$200$	0	0
$201$	44.9714	3.17204
$202$	0	0
$203$	−14.8007	−1.03880
$204$	0	0
$205$	11.6906	0.816508
$206$	0	0
$207$	−24.8198	−1.72510
$208$	0	0
$209$	−30.1508	−2.08557
$210$	0	0
$211$	2.59053	0.178339	0.0891696	−	0.996016i	$-0.471579\pi$
$211$	2.59053	0.178339	0.0891696	+	0.996016i	$0.471579\pi$
$212$	0	0
$213$	−25.5576	−1.75118
$214$	0	0
$215$	15.5347	1.05946
$216$	0	0
$217$	−26.6506	−1.80916
$218$	0	0
$219$	−18.2003	−1.22986
$220$	0	0
$221$	−3.84913	−0.258921
$222$	0	0
$223$	−18.2876	−1.22463	−0.612313	−	0.790615i	$-0.709761\pi$
$223$	−18.2876	−1.22463	−0.612313	+	0.790615i	$0.709761\pi$
$224$	0	0
$225$	72.5355	4.83570
$226$	0	0
$227$	−9.59589	−0.636901	−0.318451	−	0.947939i	$-0.603163\pi$
$227$	−9.59589	−0.636901	−0.318451	+	0.947939i	$0.603163\pi$
$228$	0	0
$229$	−7.65066	−0.505570	−0.252785	−	0.967523i	$-0.581346\pi$
$229$	−7.65066	−0.505570	−0.252785	+	0.967523i	$0.581346\pi$
$230$	0	0
$231$	−42.3452	−2.78611
$232$	0	0
$233$	−6.31643	−0.413803	−0.206902	−	0.978362i	$-0.566338\pi$
$233$	−6.31643	−0.413803	−0.206902	+	0.978362i	$0.566338\pi$
$234$	0	0
$235$	31.8160	2.07545
$236$	0	0
$237$	−23.6905	−1.53886
$238$	0	0
$239$	16.4039	1.06108	0.530540	−	0.847660i	$-0.321989\pi$
$239$	16.4039	1.06108	0.530540	+	0.847660i	$0.321989\pi$
$240$	0	0
$241$	−28.4081	−1.82993	−0.914964	−	0.403536i	$-0.867781\pi$
$241$	−28.4081	−1.82993	−0.914964	+	0.403536i	$0.867781\pi$
$242$	0	0
$243$	−57.2082	−3.66991
$244$	0	0
$245$	−20.3304	−1.29886
$246$	0	0
$247$	32.2337	2.05098
$248$	0	0
$249$	49.2404	3.12048
$250$	0	0
$251$	−23.7712	−1.50042	−0.750211	−	0.661198i	$-0.770048\pi$
$251$	−23.7712	−1.50042	−0.750211	+	0.661198i	$0.770048\pi$
$252$	0	0
$253$	11.0123	0.692340
$254$	0	0
$255$	−12.4469	−0.779455
$256$	0	0
$257$	18.2512	1.13848	0.569238	−	0.822173i	$-0.307238\pi$
$257$	18.2512	1.13848	0.569238	+	0.822173i	$0.307238\pi$
$258$	0	0
$259$	−20.2443	−1.25792
$260$	0	0
$261$	34.0445	2.10730
$262$	0	0
$263$	−11.0692	−0.682556	−0.341278	−	0.939962i	$-0.610860\pi$
$263$	−11.0692	−0.682556	−0.341278	+	0.939962i	$0.610860\pi$
$264$	0	0
$265$	21.8507	1.34228
$266$	0	0
$267$	−44.3018	−2.71122
$268$	0	0
$269$	10.9954	0.670401	0.335200	−	0.942147i	$-0.391196\pi$
$269$	10.9954	0.670401	0.335200	+	0.942147i	$0.391196\pi$
$270$	0	0
$271$	−28.1934	−1.71262	−0.856312	−	0.516459i	$-0.827250\pi$
$271$	−28.1934	−1.71262	−0.856312	+	0.516459i	$0.827250\pi$
$272$	0	0
$273$	45.2706	2.73990
$274$	0	0
$275$	−32.1834	−1.94073
$276$	0	0
$277$	−31.2650	−1.87853	−0.939266	−	0.343190i	$-0.888493\pi$
$277$	−31.2650	−1.87853	−0.939266	+	0.343190i	$0.888493\pi$
$278$	0	0
$279$	61.3017	3.67004
$280$	0	0
$281$	−9.19068	−0.548270	−0.274135	−	0.961691i	$-0.588392\pi$
$281$	−9.19068	−0.548270	−0.274135	+	0.961691i	$0.588392\pi$
$282$	0	0
$283$	22.8751	1.35978	0.679892	−	0.733312i	$-0.262027\pi$
$283$	22.8751	1.35978	0.679892	+	0.733312i	$0.262027\pi$
$284$	0	0
$285$	104.234	6.17427
$286$	0	0
$287$	11.0466	0.652061
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−56.2001	−3.29451
$292$	0	0
$293$	7.82706	0.457262	0.228631	−	0.973513i	$-0.426575\pi$
$293$	7.82706	0.457262	0.228631	+	0.973513i	$0.426575\pi$
$294$	0	0
$295$	3.73347	0.217371
$296$	0	0
$297$	61.3926	3.56236
$298$	0	0
$299$	−11.7731	−0.680856
$300$	0	0
$301$	14.6790	0.846083
$302$	0	0
$303$	43.7386	2.51272
$304$	0	0
$305$	34.3883	1.96907
$306$	0	0
$307$	23.5661	1.34499	0.672494	−	0.740103i	$-0.265223\pi$
$307$	23.5661	1.34499	0.672494	+	0.740103i	$0.265223\pi$
$308$	0	0
$309$	11.2685	0.641044
$310$	0	0
$311$	−8.43202	−0.478136	−0.239068	−	0.971003i	$-0.576842\pi$
$311$	−8.43202	−0.478136	−0.239068	+	0.971003i	$0.576842\pi$
$312$	0	0
$313$	−0.190088	−0.0107444	−0.00537219	−	0.999986i	$-0.501710\pi$
$313$	−0.190088	−0.0107444	−0.00537219	+	0.999986i	$0.501710\pi$
$314$	0	0
$315$	106.878	6.02189
$316$	0	0
$317$	16.0860	0.903477	0.451739	−	0.892150i	$-0.350804\pi$
$317$	16.0860	0.903477	0.451739	+	0.892150i	$0.350804\pi$
$318$	0	0
$319$	−15.1053	−0.845732
$320$	0	0
$321$	3.04545	0.169980
$322$	0	0
$323$	−8.37428	−0.465957
$324$	0	0
$325$	34.4067	1.90854
$326$	0	0
$327$	23.0825	1.27646
$328$	0	0
$329$	30.0634	1.65745
$330$	0	0
$331$	−3.56954	−0.196200	−0.0980999	−	0.995177i	$-0.531276\pi$
$331$	−3.56954	−0.196200	−0.0980999	+	0.995177i	$0.531276\pi$
$332$	0	0
$333$	46.5659	2.55180
$334$	0	0
$335$	50.3618	2.75156
$336$	0	0
$337$	−1.65351	−0.0900722	−0.0450361	−	0.998985i	$-0.514340\pi$
$337$	−1.65351	−0.0900722	−0.0450361	+	0.998985i	$0.514340\pi$
$338$	0	0
$339$	−23.6428	−1.28410
$340$	0	0
$341$	−27.1991	−1.47291
$342$	0	0
$343$	5.48419	0.296119
$344$	0	0
$345$	−38.0706	−2.04965
$346$	0	0
$347$	−4.76475	−0.255785	−0.127892	−	0.991788i	$-0.540821\pi$
$347$	−4.76475	−0.255785	−0.127892	+	0.991788i	$0.540821\pi$
$348$	0	0
$349$	35.8292	1.91789	0.958947	−	0.283587i	$-0.0915243\pi$
$349$	35.8292	1.91789	0.958947	+	0.283587i	$0.0915243\pi$
$350$	0	0
$351$	−65.6338	−3.50327
$352$	0	0
$353$	8.46428	0.450508	0.225254	−	0.974300i	$-0.427679\pi$
$353$	8.46428	0.450508	0.225254	+	0.974300i	$0.427679\pi$
$354$	0	0
$355$	−28.6210	−1.51905
$356$	0	0
$357$	−11.7612	−0.622471
$358$	0	0
$359$	29.7315	1.56917	0.784584	−	0.620023i	$-0.212877\pi$
$359$	29.7315	1.56917	0.784584	+	0.620023i	$0.212877\pi$
$360$	0	0
$361$	51.1285	2.69097
$362$	0	0
$363$	−6.54411	−0.343476
$364$	0	0
$365$	−20.3818	−1.06683
$366$	0	0
$367$	−10.3475	−0.540137	−0.270069	−	0.962841i	$-0.587046\pi$
$367$	−10.3475	−0.540137	−0.270069	+	0.962841i	$0.587046\pi$
$368$	0	0
$369$	−25.4094	−1.32276
$370$	0	0
$371$	20.6471	1.07194
$372$	0	0
$373$	−16.7802	−0.868846	−0.434423	−	0.900709i	$-0.643048\pi$
$373$	−16.7802	−0.868846	−0.434423	+	0.900709i	$0.643048\pi$
$374$	0	0
$375$	49.0261	2.53170
$376$	0	0
$377$	16.1488	0.831704
$378$	0	0
$379$	15.3226	0.787069	0.393534	−	0.919310i	$-0.371252\pi$
$379$	15.3226	0.787069	0.393534	+	0.919310i	$0.371252\pi$
$380$	0	0
$381$	−61.4024	−3.14574
$382$	0	0
$383$	−11.3599	−0.580464	−0.290232	−	0.956956i	$-0.593733\pi$
$383$	−11.3599	−0.580464	−0.290232	+	0.956956i	$0.593733\pi$
$384$	0	0
$385$	−47.4209	−2.41679
$386$	0	0
$387$	−33.7646	−1.71635
$388$	0	0
$389$	5.90126	0.299206	0.149603	−	0.988746i	$-0.452200\pi$
$389$	5.90126	0.299206	0.149603	+	0.988746i	$0.452200\pi$
$390$	0	0
$391$	3.05864	0.154682
$392$	0	0
$393$	−11.8600	−0.598257
$394$	0	0
$395$	−26.5301	−1.33487
$396$	0	0
$397$	−28.9463	−1.45277	−0.726386	−	0.687287i	$-0.758802\pi$
$397$	−28.9463	−1.45277	−0.726386	+	0.687287i	$0.758802\pi$
$398$	0	0
$399$	98.4919	4.93076
$400$	0	0
$401$	17.8868	0.893224	0.446612	−	0.894728i	$-0.352630\pi$
$401$	17.8868	0.893224	0.446612	+	0.894728i	$0.352630\pi$
$402$	0	0
$403$	29.0780	1.44848
$404$	0	0
$405$	−121.352	−6.03001
$406$	0	0
$407$	−20.6609	−1.02412
$408$	0	0
$409$	10.0748	0.498165	0.249082	−	0.968482i	$-0.419871\pi$
$409$	10.0748	0.498165	0.249082	+	0.968482i	$0.419871\pi$
$410$	0	0
$411$	10.4357	0.514757
$412$	0	0
$413$	3.52781	0.173592
$414$	0	0
$415$	55.1425	2.70684
$416$	0	0
$417$	−9.90077	−0.484843
$418$	0	0
$419$	−30.4739	−1.48875	−0.744373	−	0.667764i	$-0.767252\pi$
$419$	−30.4739	−1.48875	−0.744373	+	0.667764i	$0.767252\pi$
$420$	0	0
$421$	−24.2931	−1.18397	−0.591986	−	0.805948i	$-0.701656\pi$
$421$	−24.2931	−1.18397	−0.591986	+	0.805948i	$0.701656\pi$
$422$	0	0
$423$	−69.1517	−3.36227
$424$	0	0
$425$	−8.93882	−0.433596
$426$	0	0
$427$	32.4940	1.57249
$428$	0	0
$429$	46.2022	2.23066
$430$	0	0
$431$	−22.0203	−1.06068	−0.530341	−	0.847784i	$-0.677936\pi$
$431$	−22.0203	−1.06068	−0.530341	+	0.847784i	$0.677936\pi$
$432$	0	0
$433$	−21.4822	−1.03237	−0.516184	−	0.856478i	$-0.672648\pi$
$433$	−21.4822	−1.03237	−0.516184	+	0.856478i	$0.672648\pi$
$434$	0	0
$435$	52.2201	2.50376
$436$	0	0
$437$	−25.6139	−1.22528
$438$	0	0
$439$	−2.06270	−0.0984473	−0.0492236	−	0.998788i	$-0.515675\pi$
$439$	−2.06270	−0.0984473	−0.0492236	+	0.998788i	$0.515675\pi$
$440$	0	0
$441$	44.1879	2.10418
$442$	0	0
$443$	−2.71837	−0.129154	−0.0645768	−	0.997913i	$-0.520570\pi$
$443$	−2.71837	−0.129154	−0.0645768	+	0.997913i	$0.520570\pi$
$444$	0	0
$445$	−49.6119	−2.35183
$446$	0	0
$447$	46.0405	2.17764
$448$	0	0
$449$	28.3028	1.33569	0.667845	−	0.744300i	$-0.267217\pi$
$449$	28.3028	1.33569	0.667845	+	0.744300i	$0.267217\pi$
$450$	0	0
$451$	11.2739	0.530869
$452$	0	0
$453$	44.2106	2.07720
$454$	0	0
$455$	50.6968	2.37670
$456$	0	0
$457$	11.4279	0.534573	0.267286	−	0.963617i	$-0.413873\pi$
$457$	11.4279	0.534573	0.267286	+	0.963617i	$0.413873\pi$
$458$	0	0
$459$	17.0516	0.795900
$460$	0	0
$461$	9.22732	0.429759	0.214879	−	0.976641i	$-0.431064\pi$
$461$	9.22732	0.429759	0.214879	+	0.976641i	$0.431064\pi$
$462$	0	0
$463$	−8.24136	−0.383008	−0.191504	−	0.981492i	$-0.561337\pi$
$463$	−8.24136	−0.383008	−0.191504	+	0.981492i	$0.561337\pi$
$464$	0	0
$465$	94.0293	4.36051
$466$	0	0
$467$	9.21328	0.426340	0.213170	−	0.977015i	$-0.431621\pi$
$467$	9.21328	0.426340	0.213170	+	0.977015i	$0.431621\pi$
$468$	0	0
$469$	47.5876	2.19739
$470$	0	0
$471$	−27.8325	−1.28245
$472$	0	0
$473$	14.9811	0.688830
$474$	0	0
$475$	74.8561	3.43464
$476$	0	0
$477$	−47.4923	−2.17452
$478$	0	0
$479$	28.1451	1.28598	0.642990	−	0.765874i	$-0.277694\pi$
$479$	28.1451	1.28598	0.642990	+	0.765874i	$0.277694\pi$
$480$	0	0
$481$	22.0882	1.00714
$482$	0	0
$483$	−35.9734	−1.63685
$484$	0	0
$485$	−62.9364	−2.85779
$486$	0	0
$487$	−28.3233	−1.28345	−0.641725	−	0.766935i	$-0.721781\pi$
$487$	−28.3233	−1.28345	−0.641725	+	0.766935i	$0.721781\pi$
$488$	0	0
$489$	−16.6935	−0.754907
$490$	0	0
$491$	−1.42428	−0.0642768	−0.0321384	−	0.999483i	$-0.510232\pi$
$491$	−1.42428	−0.0642768	−0.0321384	+	0.999483i	$0.510232\pi$
$492$	0	0
$493$	−4.19543	−0.188953
$494$	0	0
$495$	109.077	4.90266
$496$	0	0
$497$	−27.0444	−1.21311
$498$	0	0
$499$	−1.69368	−0.0758194	−0.0379097	−	0.999281i	$-0.512070\pi$
$499$	−1.69368	−0.0758194	−0.0379097	+	0.999281i	$0.512070\pi$
$500$	0	0
$501$	34.3479	1.53455
$502$	0	0
$503$	17.2682	0.769950	0.384975	−	0.922927i	$-0.374210\pi$
$503$	17.2682	0.769950	0.384975	+	0.922927i	$0.374210\pi$
$504$	0	0
$505$	48.9813	2.17964
$506$	0	0
$507$	−6.05369	−0.268854
$508$	0	0
$509$	−6.07287	−0.269175	−0.134588	−	0.990902i	$-0.542971\pi$
$509$	−6.07287	−0.269175	−0.134588	+	0.990902i	$0.542971\pi$
$510$	0	0
$511$	−19.2591	−0.851970
$512$	0	0
$513$	−142.795	−6.30454
$514$	0	0
$515$	12.6192	0.556068
$516$	0	0
$517$	30.6820	1.34939
$518$	0	0
$519$	−72.7466	−3.19322
$520$	0	0
$521$	−26.4512	−1.15885	−0.579425	−	0.815026i	$-0.696723\pi$
$521$	−26.4512	−1.15885	−0.579425	+	0.815026i	$0.696723\pi$
$522$	0	0
$523$	−11.5054	−0.503097	−0.251549	−	0.967845i	$-0.580940\pi$
$523$	−11.5054	−0.503097	−0.251549	+	0.967845i	$0.580940\pi$
$524$	0	0
$525$	105.132	4.58832
$526$	0	0
$527$	−7.55444	−0.329077
$528$	0	0
$529$	−13.6447	−0.593249
$530$	0	0
$531$	−8.11466	−0.352146
$532$	0	0
$533$	−12.0528	−0.522064
$534$	0	0
$535$	3.41049	0.147448
$536$	0	0
$537$	−7.95891	−0.343452
$538$	0	0
$539$	−19.6058	−0.844481
$540$	0	0
$541$	−22.6748	−0.974865	−0.487433	−	0.873161i	$-0.662067\pi$
$541$	−22.6748	−0.974865	−0.487433	+	0.873161i	$0.662067\pi$
$542$	0	0
$543$	−33.5101	−1.43806
$544$	0	0
$545$	25.8492	1.10726
$546$	0	0
$547$	−3.90625	−0.167019	−0.0835096	−	0.996507i	$-0.526613\pi$
$547$	−3.90625	−0.167019	−0.0835096	+	0.996507i	$0.526613\pi$
$548$	0	0
$549$	−74.7425	−3.18993
$550$	0	0
$551$	35.1337	1.49675
$552$	0	0
$553$	−25.0686	−1.06603
$554$	0	0
$555$	71.4264	3.03188
$556$	0	0
$557$	−15.6673	−0.663844	−0.331922	−	0.943307i	$-0.607697\pi$
$557$	−15.6673	−0.663844	−0.331922	+	0.943307i	$0.607697\pi$
$558$	0	0
$559$	−16.0160	−0.677404
$560$	0	0
$561$	−12.0033	−0.506778
$562$	0	0
$563$	46.9410	1.97833	0.989163	−	0.146819i	$-0.0469036\pi$
$563$	46.9410	1.97833	0.989163	+	0.146819i	$0.0469036\pi$
$564$	0	0
$565$	−26.4767	−1.11388
$566$	0	0
$567$	−114.667	−4.81555
$568$	0	0
$569$	40.6335	1.70345	0.851723	−	0.523992i	$-0.175558\pi$
$569$	40.6335	1.70345	0.851723	+	0.523992i	$0.175558\pi$
$570$	0	0
$571$	−21.0849	−0.882377	−0.441188	−	0.897414i	$-0.645443\pi$
$571$	−21.0849	−0.882377	−0.441188	+	0.897414i	$0.645443\pi$
$572$	0	0
$573$	14.8635	0.620931
$574$	0	0
$575$	−27.3406	−1.14018
$576$	0	0
$577$	46.7978	1.94822	0.974109	−	0.226081i	$-0.0725913\pi$
$577$	46.7978	1.94822	0.974109	+	0.226081i	$0.0725913\pi$
$578$	0	0
$579$	46.7236	1.94177
$580$	0	0
$581$	52.1049	2.16168
$582$	0	0
$583$	21.0719	0.872710
$584$	0	0
$585$	−116.613	−4.82134
$586$	0	0
$587$	13.0847	0.540062	0.270031	−	0.962852i	$-0.412966\pi$
$587$	13.0847	0.540062	0.270031	+	0.962852i	$0.412966\pi$
$588$	0	0
$589$	63.2630	2.60671
$590$	0	0
$591$	90.9427	3.74088
$592$	0	0
$593$	40.3911	1.65866	0.829332	−	0.558756i	$-0.188721\pi$
$593$	40.3911	1.65866	0.829332	+	0.558756i	$0.188721\pi$
$594$	0	0
$595$	−13.1710	−0.539958
$596$	0	0
$597$	−79.0984	−3.23728
$598$	0	0
$599$	−20.3813	−0.832759	−0.416379	−	0.909191i	$-0.636701\pi$
$599$	−20.3813	−0.832759	−0.416379	+	0.909191i	$0.636701\pi$
$600$	0	0
$601$	−18.1328	−0.739654	−0.369827	−	0.929101i	$-0.620583\pi$
$601$	−18.1328	−0.739654	−0.369827	+	0.929101i	$0.620583\pi$
$602$	0	0
$603$	−109.461	−4.45759
$604$	0	0
$605$	−7.32850	−0.297946
$606$	0	0
$607$	−5.79445	−0.235190	−0.117595	−	0.993062i	$-0.537518\pi$
$607$	−5.79445	−0.235190	−0.117595	+	0.993062i	$0.537518\pi$
$608$	0	0
$609$	49.3435	1.99950
$610$	0	0
$611$	−32.8016	−1.32701
$612$	0	0
$613$	−37.9833	−1.53413	−0.767066	−	0.641568i	$-0.778284\pi$
$613$	−37.9833	−1.53413	−0.767066	+	0.641568i	$0.778284\pi$
$614$	0	0
$615$	−38.9749	−1.57162
$616$	0	0
$617$	17.4170	0.701184	0.350592	−	0.936528i	$-0.385980\pi$
$617$	17.4170	0.701184	0.350592	+	0.936528i	$0.385980\pi$
$618$	0	0
$619$	−17.9537	−0.721619	−0.360809	−	0.932640i	$-0.617499\pi$
$619$	−17.9537	−0.721619	−0.360809	+	0.932640i	$0.617499\pi$
$620$	0	0
$621$	52.1546	2.09289
$622$	0	0
$623$	−46.8790	−1.87817
$624$	0	0
$625$	10.2084	0.408336
$626$	0	0
$627$	100.519	4.01433
$628$	0	0
$629$	−5.73849	−0.228809
$630$	0	0
$631$	−20.8511	−0.830069	−0.415035	−	0.909806i	$-0.636231\pi$
$631$	−20.8511	−0.830069	−0.415035	+	0.909806i	$0.636231\pi$
$632$	0	0
$633$	−8.63647	−0.343269
$634$	0	0
$635$	−68.7623	−2.72875
$636$	0	0
$637$	20.9602	0.830474
$638$	0	0
$639$	62.2075	2.46089
$640$	0	0
$641$	−29.1414	−1.15102	−0.575508	−	0.817796i	$-0.695196\pi$
$641$	−29.1414	−1.15102	−0.575508	+	0.817796i	$0.695196\pi$
$642$	0	0
$643$	−15.2342	−0.600777	−0.300389	−	0.953817i	$-0.597116\pi$
$643$	−15.2342	−0.600777	−0.300389	+	0.953817i	$0.597116\pi$
$644$	0	0
$645$	−51.7907	−2.03926
$646$	0	0
$647$	39.9957	1.57240	0.786198	−	0.617975i	$-0.212047\pi$
$647$	39.9957	1.57240	0.786198	+	0.617975i	$0.212047\pi$
$648$	0	0
$649$	3.60041	0.141328
$650$	0	0
$651$	88.8496	3.48229
$652$	0	0
$653$	13.0758	0.511694	0.255847	−	0.966717i	$-0.417646\pi$
$653$	13.0758	0.511694	0.255847	+	0.966717i	$0.417646\pi$
$654$	0	0
$655$	−13.2816	−0.518954
$656$	0	0
$657$	44.2996	1.72829
$658$	0	0
$659$	16.3558	0.637130	0.318565	−	0.947901i	$-0.396799\pi$
$659$	16.3558	0.637130	0.318565	+	0.947901i	$0.396799\pi$
$660$	0	0
$661$	11.2483	0.437506	0.218753	−	0.975780i	$-0.429801\pi$
$661$	11.2483	0.437506	0.218753	+	0.975780i	$0.429801\pi$
$662$	0	0
$663$	12.8325	0.498373
$664$	0	0
$665$	110.297	4.27715
$666$	0	0
$667$	−12.8323	−0.496869
$668$	0	0
$669$	60.9683	2.35717
$670$	0	0
$671$	33.1626	1.28023
$672$	0	0
$673$	32.7718	1.26326	0.631630	−	0.775270i	$-0.282386\pi$
$673$	32.7718	1.26326	0.631630	+	0.775270i	$0.282386\pi$
$674$	0	0
$675$	−152.421	−5.86669
$676$	0	0
$677$	−3.63241	−0.139605	−0.0698023	−	0.997561i	$-0.522237\pi$
$677$	−3.63241	−0.139605	−0.0698023	+	0.997561i	$0.522237\pi$
$678$	0	0
$679$	−59.4695	−2.28223
$680$	0	0
$681$	31.9914	1.22591
$682$	0	0
$683$	6.48202	0.248028	0.124014	−	0.992280i	$-0.460423\pi$
$683$	6.48202	0.248028	0.124014	+	0.992280i	$0.460423\pi$
$684$	0	0
$685$	11.6866	0.446522
$686$	0	0
$687$	25.5063	0.973125
$688$	0	0
$689$	−22.5276	−0.858235
$690$	0	0
$691$	−10.8561	−0.412986	−0.206493	−	0.978448i	$-0.566205\pi$
$691$	−10.8561	−0.412986	−0.206493	+	0.978448i	$0.566205\pi$
$692$	0	0
$693$	103.069	3.91526
$694$	0	0
$695$	−11.0875	−0.420573
$696$	0	0
$697$	3.13130	0.118606
$698$	0	0
$699$	21.0581	0.796492
$700$	0	0
$701$	−18.9181	−0.714528	−0.357264	−	0.934004i	$-0.616290\pi$
$701$	−18.9181	−0.714528	−0.357264	+	0.934004i	$0.616290\pi$
$702$	0	0
$703$	48.0557	1.81246
$704$	0	0
$705$	−106.070	−3.99484
$706$	0	0
$707$	46.2831	1.74065
$708$	0	0
$709$	−22.6952	−0.852337	−0.426168	−	0.904644i	$-0.640137\pi$
$709$	−22.6952	−0.852337	−0.426168	+	0.904644i	$0.640137\pi$
$710$	0	0
$711$	57.6628	2.16253
$712$	0	0
$713$	−23.1063	−0.865338
$714$	0	0
$715$	51.7401	1.93497
$716$	0	0
$717$	−54.6884	−2.04237
$718$	0	0
$719$	10.4413	0.389394	0.194697	−	0.980863i	$-0.437628\pi$
$719$	10.4413	0.389394	0.194697	+	0.980863i	$0.437628\pi$
$720$	0	0
$721$	11.9241	0.444075
$722$	0	0
$723$	94.7088	3.52226
$724$	0	0
$725$	37.5022	1.39280
$726$	0	0
$727$	23.0075	0.853302	0.426651	−	0.904416i	$-0.359693\pi$
$727$	23.0075	0.853302	0.426651	+	0.904416i	$0.359693\pi$
$728$	0	0
$729$	93.2134	3.45235
$730$	0	0
$731$	4.16094	0.153898
$732$	0	0
$733$	−1.74990	−0.0646341	−0.0323170	−	0.999478i	$-0.510289\pi$
$733$	−1.74990	−0.0646341	−0.0323170	+	0.999478i	$0.510289\pi$
$734$	0	0
$735$	67.7788	2.50006
$736$	0	0
$737$	48.5669	1.78898
$738$	0	0
$739$	30.6311	1.12678	0.563391	−	0.826191i	$-0.309497\pi$
$739$	30.6311	1.12678	0.563391	+	0.826191i	$0.309497\pi$
$740$	0	0
$741$	−107.463	−3.94774
$742$	0	0
$743$	−43.7138	−1.60371	−0.801853	−	0.597522i	$-0.796152\pi$
$743$	−43.7138	−1.60371	−0.801853	+	0.597522i	$0.796152\pi$
$744$	0	0
$745$	51.5591	1.88898
$746$	0	0
$747$	−119.852	−4.38514
$748$	0	0
$749$	3.22262	0.117752
$750$	0	0
$751$	−42.3814	−1.54652	−0.773260	−	0.634089i	$-0.781375\pi$
$751$	−42.3814	−1.54652	−0.773260	+	0.634089i	$0.781375\pi$
$752$	0	0
$753$	79.2499	2.88803
$754$	0	0
$755$	49.5099	1.80185
$756$	0	0
$757$	−49.1668	−1.78700	−0.893499	−	0.449064i	$-0.851757\pi$
$757$	−49.1668	−1.78700	−0.893499	+	0.449064i	$0.851757\pi$
$758$	0	0
$759$	−36.7137	−1.33262
$760$	0	0
$761$	26.5756	0.963363	0.481682	−	0.876346i	$-0.340026\pi$
$761$	26.5756	0.963363	0.481682	+	0.876346i	$0.340026\pi$
$762$	0	0
$763$	24.4253	0.884255
$764$	0	0
$765$	30.2959	1.09535
$766$	0	0
$767$	−3.84913	−0.138984
$768$	0	0
$769$	46.0163	1.65939	0.829695	−	0.558217i	$-0.188514\pi$
$769$	46.0163	1.65939	0.829695	+	0.558217i	$0.188514\pi$
$770$	0	0
$771$	−60.8469	−2.19135
$772$	0	0
$773$	−29.7752	−1.07094	−0.535469	−	0.844555i	$-0.679865\pi$
$773$	−29.7752	−1.07094	−0.535469	+	0.844555i	$0.679865\pi$
$774$	0	0
$775$	67.5278	2.42567
$776$	0	0
$777$	67.4918	2.42125
$778$	0	0
$779$	−26.2223	−0.939513
$780$	0	0
$781$	−27.6009	−0.987639
$782$	0	0
$783$	−71.5387	−2.55659
$784$	0	0
$785$	−31.1686	−1.11246
$786$	0	0
$787$	−9.30567	−0.331711	−0.165856	−	0.986150i	$-0.553039\pi$
$787$	−9.30567	−0.331711	−0.165856	+	0.986150i	$0.553039\pi$
$788$	0	0
$789$	36.9032	1.31379
$790$	0	0
$791$	−25.0182	−0.889546
$792$	0	0
$793$	−35.4536	−1.25899
$794$	0	0
$795$	−72.8474	−2.58363
$796$	0	0
$797$	−10.4028	−0.368488	−0.184244	−	0.982881i	$-0.558984\pi$
$797$	−10.4028	−0.368488	−0.184244	+	0.982881i	$0.558984\pi$
$798$	0	0
$799$	8.52182	0.301480
$800$	0	0
$801$	107.831	3.81002
$802$	0	0
$803$	−19.6554	−0.693623
$804$	0	0
$805$	−40.2853	−1.41987
$806$	0	0
$807$	−36.6571	−1.29039
$808$	0	0
$809$	13.8395	0.486571	0.243285	−	0.969955i	$-0.421775\pi$
$809$	13.8395	0.486571	0.243285	+	0.969955i	$0.421775\pi$
$810$	0	0
$811$	5.45049	0.191393	0.0956963	−	0.995411i	$-0.469492\pi$
$811$	5.45049	0.191393	0.0956963	+	0.995411i	$0.469492\pi$
$812$	0	0
$813$	93.9928	3.29647
$814$	0	0
$815$	−18.6945	−0.654839
$816$	0	0
$817$	−34.8448	−1.21907
$818$	0	0
$819$	−110.189	−3.85031
$820$	0	0
$821$	21.6283	0.754833	0.377417	−	0.926044i	$-0.376812\pi$
$821$	21.6283	0.754833	0.377417	+	0.926044i	$0.376812\pi$
$822$	0	0
$823$	2.01185	0.0701286	0.0350643	−	0.999385i	$-0.488836\pi$
$823$	2.01185	0.0701286	0.0350643	+	0.999385i	$0.488836\pi$
$824$	0	0
$825$	107.295	3.73553
$826$	0	0
$827$	46.3318	1.61112	0.805558	−	0.592517i	$-0.201866\pi$
$827$	46.3318	1.61112	0.805558	+	0.592517i	$0.201866\pi$
$828$	0	0
$829$	22.6764	0.787584	0.393792	−	0.919200i	$-0.371163\pi$
$829$	22.6764	0.787584	0.393792	+	0.919200i	$0.371163\pi$
$830$	0	0
$831$	104.233	3.61581
$832$	0	0
$833$	−5.44544	−0.188673
$834$	0	0
$835$	38.4650	1.33114
$836$	0	0
$837$	−128.815	−4.45250
$838$	0	0
$839$	−54.2688	−1.87357	−0.936783	−	0.349910i	$-0.886212\pi$
$839$	−54.2688	−1.87357	−0.936783	+	0.349910i	$0.886212\pi$
$840$	0	0
$841$	−11.3984	−0.393047
$842$	0	0
$843$	30.6405	1.05531
$844$	0	0
$845$	−6.77931	−0.233215
$846$	0	0
$847$	−6.92480	−0.237939
$848$	0	0
$849$	−76.2626	−2.61732
$850$	0	0
$851$	−17.5520	−0.601674
$852$	0	0
$853$	−42.5838	−1.45804	−0.729021	−	0.684491i	$-0.760024\pi$
$853$	−42.5838	−1.45804	−0.729021	+	0.684491i	$0.760024\pi$
$854$	0	0
$855$	−253.706	−8.67656
$856$	0	0
$857$	−52.3786	−1.78922	−0.894610	−	0.446848i	$-0.852546\pi$
$857$	−52.3786	−1.78922	−0.894610	+	0.446848i	$0.852546\pi$
$858$	0	0
$859$	15.3587	0.524031	0.262016	−	0.965064i	$-0.415613\pi$
$859$	15.3587	0.524031	0.262016	+	0.965064i	$0.415613\pi$
$860$	0	0
$861$	−36.8279	−1.25509
$862$	0	0
$863$	22.8784	0.778791	0.389395	−	0.921071i	$-0.372684\pi$
$863$	22.8784	0.778791	0.389395	+	0.921071i	$0.372684\pi$
$864$	0	0
$865$	−81.4662	−2.76993
$866$	0	0
$867$	−3.33387	−0.113224
$868$	0	0
$869$	−25.5845	−0.867895
$870$	0	0
$871$	−51.9220	−1.75931
$872$	0	0
$873$	136.791	4.62969
$874$	0	0
$875$	51.8781	1.75380
$876$	0	0
$877$	20.0545	0.677191	0.338596	−	0.940932i	$-0.390048\pi$
$877$	20.0545	0.677191	0.338596	+	0.940932i	$0.390048\pi$
$878$	0	0
$879$	−26.0944	−0.880141
$880$	0	0
$881$	34.0776	1.14810	0.574052	−	0.818819i	$-0.305371\pi$
$881$	34.0776	1.14810	0.574052	+	0.818819i	$0.305371\pi$
$882$	0	0
$883$	−17.8147	−0.599512	−0.299756	−	0.954016i	$-0.596905\pi$
$883$	−17.8147	−0.599512	−0.299756	+	0.954016i	$0.596905\pi$
$884$	0	0
$885$	−12.4469	−0.418398
$886$	0	0
$887$	23.5518	0.790791	0.395395	−	0.918511i	$-0.370608\pi$
$887$	23.5518	0.790791	0.395395	+	0.918511i	$0.370608\pi$
$888$	0	0
$889$	−64.9744	−2.17917
$890$	0	0
$891$	−117.026	−3.92053
$892$	0	0
$893$	−71.3641	−2.38811
$894$	0	0
$895$	−8.91289	−0.297925
$896$	0	0
$897$	39.2500	1.31052
$898$	0	0
$899$	31.6941	1.05706
$900$	0	0
$901$	5.85266	0.194980
$902$	0	0
$903$	−48.9378	−1.62855
$904$	0	0
$905$	−37.5267	−1.24743
$906$	0	0
$907$	9.03728	0.300078	0.150039	−	0.988680i	$-0.452060\pi$
$907$	9.03728	0.300078	0.150039	+	0.988680i	$0.452060\pi$
$908$	0	0
$909$	−106.460	−3.53106
$910$	0	0
$911$	−16.5956	−0.549837	−0.274919	−	0.961468i	$-0.588651\pi$
$911$	−16.5956	−0.549837	−0.274919	+	0.961468i	$0.588651\pi$
$912$	0	0
$913$	53.1771	1.75991
$914$	0	0
$915$	−114.646	−3.79008
$916$	0	0
$917$	−12.5499	−0.414435
$918$	0	0
$919$	1.29845	0.0428321	0.0214160	−	0.999771i	$-0.493183\pi$
$919$	1.29845	0.0428321	0.0214160	+	0.999771i	$0.493183\pi$
$920$	0	0
$921$	−78.5662	−2.58884
$922$	0	0
$923$	29.5077	0.971258
$924$	0	0
$925$	51.2953	1.68658
$926$	0	0
$927$	−27.4277	−0.900843
$928$	0	0
$929$	17.4430	0.572286	0.286143	−	0.958187i	$-0.407627\pi$
$929$	17.4430	0.572286	0.286143	+	0.958187i	$0.407627\pi$
$930$	0	0
$931$	45.6016	1.49453
$932$	0	0
$933$	28.1112	0.920320
$934$	0	0
$935$	−13.4420	−0.439601
$936$	0	0
$937$	−17.4632	−0.570499	−0.285249	−	0.958453i	$-0.592076\pi$
$937$	−17.4632	−0.570499	−0.285249	+	0.958453i	$0.592076\pi$
$938$	0	0
$939$	0.633727	0.0206809
$940$	0	0
$941$	40.4156	1.31751	0.658755	−	0.752358i	$-0.271083\pi$
$941$	40.4156	1.31751	0.658755	+	0.752358i	$0.271083\pi$
$942$	0	0
$943$	9.57751	0.311887
$944$	0	0
$945$	−224.586	−7.30578
$946$	0	0
$947$	25.6589	0.833801	0.416901	−	0.908952i	$-0.363116\pi$
$947$	25.6589	0.833801	0.416901	+	0.908952i	$0.363116\pi$
$948$	0	0
$949$	21.0132	0.682118
$950$	0	0
$951$	−53.6284	−1.73902
$952$	0	0
$953$	12.3449	0.399891	0.199945	−	0.979807i	$-0.435924\pi$
$953$	12.3449	0.399891	0.199945	+	0.979807i	$0.435924\pi$
$954$	0	0
$955$	16.6451	0.538622
$956$	0	0
$957$	50.3589	1.62787
$958$	0	0
$959$	11.0428	0.356592
$960$	0	0
$961$	26.0696	0.840954
$962$	0	0
$963$	−7.41265	−0.238869
$964$	0	0
$965$	52.3240	1.68437
$966$	0	0
$967$	17.3525	0.558018	0.279009	−	0.960288i	$-0.409994\pi$
$967$	17.3525	0.558018	0.279009	+	0.960288i	$0.409994\pi$
$968$	0	0
$969$	27.9187	0.896878
$970$	0	0
$971$	−6.21073	−0.199312	−0.0996558	−	0.995022i	$-0.531774\pi$
$971$	−6.21073	−0.199312	−0.0996558	+	0.995022i	$0.531774\pi$
$972$	0	0
$973$	−10.4767	−0.335869
$974$	0	0
$975$	−114.707	−3.67357
$976$	0	0
$977$	4.52575	0.144792	0.0723958	−	0.997376i	$-0.476936\pi$
$977$	4.52575	0.144792	0.0723958	+	0.997376i	$0.476936\pi$
$978$	0	0
$979$	−47.8437	−1.52909
$980$	0	0
$981$	−56.1830	−1.79378
$982$	0	0
$983$	4.06315	0.129594	0.0647971	−	0.997898i	$-0.479360\pi$
$983$	4.06315	0.129594	0.0647971	+	0.997898i	$0.479360\pi$
$984$	0	0
$985$	101.843	3.24500
$986$	0	0
$987$	−100.227	−3.19027
$988$	0	0
$989$	12.7268	0.404689
$990$	0	0
$991$	−48.2150	−1.53160	−0.765800	−	0.643079i	$-0.777657\pi$
$991$	−48.2150	−1.53160	−0.765800	+	0.643079i	$0.777657\pi$
$992$	0	0
$993$	11.9004	0.377647
$994$	0	0
$995$	−88.5794	−2.80816
$996$	0	0
$997$	30.9967	0.981675	0.490837	−	0.871251i	$-0.336691\pi$
$997$	30.9967	0.981675	0.490837	+	0.871251i	$0.336691\pi$
$998$	0	0
$999$	−97.8503	−3.09585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.3	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.3	✓	33	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.33387 q^{3} -3.73347 q^{5} -3.52781 q^{7} +8.11466 q^{9} +O(q^{10})\)	\(q-3.33387 q^{3} -3.73347 q^{5} -3.52781 q^{7} +8.11466 q^{9} -3.60041 q^{11} +3.84913 q^{13} +12.4469 q^{15} -1.00000 q^{17} +8.37428 q^{19} +11.7612 q^{21} -3.05864 q^{23} +8.93882 q^{25} -17.0516 q^{27} +4.19543 q^{29} +7.55444 q^{31} +12.0033 q^{33} +13.1710 q^{35} +5.73849 q^{37} -12.8325 q^{39} -3.13130 q^{41} -4.16094 q^{43} -30.2959 q^{45} -8.52182 q^{47} +5.44544 q^{49} +3.33387 q^{51} -5.85266 q^{53} +13.4420 q^{55} -27.9187 q^{57} -1.00000 q^{59} -9.21081 q^{61} -28.6270 q^{63} -14.3706 q^{65} -13.4893 q^{67} +10.1971 q^{69} +7.66607 q^{71} +5.45921 q^{73} -29.8008 q^{75} +12.7015 q^{77} +7.10601 q^{79} +32.5037 q^{81} -14.7698 q^{83} +3.73347 q^{85} -13.9870 q^{87} +13.2884 q^{89} -13.5790 q^{91} -25.1855 q^{93} -31.2651 q^{95} +16.8573 q^{97} -29.2161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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Database

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