LMFDB - Embedded newform 8024.2.a.bc.1.4

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			1.4
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)

$f(q)$	$=$	$q-2.83790 q^{3} +2.04357 q^{5} +3.25946 q^{7} +5.05368 q^{9} +O(q^{10})$	$q-2.83790 q^{3} +2.04357 q^{5} +3.25946 q^{7} +5.05368 q^{9} +5.18832 q^{11} +5.12728 q^{13} -5.79944 q^{15} -1.00000 q^{17} +0.485963 q^{19} -9.25002 q^{21} -3.79068 q^{23} -0.823831 q^{25} -5.82813 q^{27} +8.37542 q^{29} +6.95051 q^{31} -14.7239 q^{33} +6.66093 q^{35} +8.02893 q^{37} -14.5507 q^{39} +11.3313 q^{41} +6.57404 q^{43} +10.3275 q^{45} -6.70873 q^{47} +3.62409 q^{49} +2.83790 q^{51} -6.98854 q^{53} +10.6027 q^{55} -1.37911 q^{57} -1.00000 q^{59} +12.4674 q^{61} +16.4723 q^{63} +10.4779 q^{65} -12.1470 q^{67} +10.7576 q^{69} +1.61652 q^{71} +11.9773 q^{73} +2.33795 q^{75} +16.9111 q^{77} +16.5361 q^{79} +1.37862 q^{81} +5.89570 q^{83} -2.04357 q^{85} -23.7686 q^{87} -3.57802 q^{89} +16.7122 q^{91} -19.7248 q^{93} +0.993098 q^{95} -16.9318 q^{97} +26.2201 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.83790	−1.63846	−0.819231	−	0.573463i	$-0.805599\pi$
$3$	−2.83790	−1.63846	−0.819231	+	0.573463i	$0.805599\pi$
$4$	0	0
$5$	2.04357	0.913911	0.456956	−	0.889489i	$-0.348940\pi$
$5$	2.04357	0.913911	0.456956	+	0.889489i	$0.348940\pi$
$6$	0	0
$7$	3.25946	1.23196	0.615980	−	0.787762i	$-0.288760\pi$
$7$	3.25946	1.23196	0.615980	+	0.787762i	$0.288760\pi$
$8$	0	0
$9$	5.05368	1.68456
$10$	0	0
$11$	5.18832	1.56434	0.782169	−	0.623066i	$-0.214113\pi$
$11$	5.18832	1.56434	0.782169	+	0.623066i	$0.214113\pi$
$12$	0	0
$13$	5.12728	1.42205	0.711025	−	0.703166i	$-0.248231\pi$
$13$	5.12728	1.42205	0.711025	+	0.703166i	$0.248231\pi$
$14$	0	0
$15$	−5.79944	−1.49741
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.485963	0.111488	0.0557438	−	0.998445i	$-0.482247\pi$
$19$	0.485963	0.111488	0.0557438	+	0.998445i	$0.482247\pi$
$20$	0	0
$21$	−9.25002	−2.01852
$22$	0	0
$23$	−3.79068	−0.790411	−0.395206	−	0.918593i	$-0.629327\pi$
$23$	−3.79068	−0.790411	−0.395206	+	0.918593i	$0.629327\pi$
$24$	0	0
$25$	−0.823831	−0.164766
$26$	0	0
$27$	−5.82813	−1.12162
$28$	0	0
$29$	8.37542	1.55528	0.777638	−	0.628713i	$-0.216418\pi$
$29$	8.37542	1.55528	0.777638	+	0.628713i	$0.216418\pi$
$30$	0	0
$31$	6.95051	1.24835	0.624174	−	0.781285i	$-0.285436\pi$
$31$	6.95051	1.24835	0.624174	+	0.781285i	$0.285436\pi$
$32$	0	0
$33$	−14.7239	−2.56311
$34$	0	0
$35$	6.66093	1.12590
$36$	0	0
$37$	8.02893	1.31995	0.659974	−	0.751288i	$-0.270567\pi$
$37$	8.02893	1.31995	0.659974	+	0.751288i	$0.270567\pi$
$38$	0	0
$39$	−14.5507	−2.32998
$40$	0	0
$41$	11.3313	1.76965	0.884826	−	0.465921i	$-0.154277\pi$
$41$	11.3313	1.76965	0.884826	+	0.465921i	$0.154277\pi$
$42$	0	0
$43$	6.57404	1.00253	0.501266	−	0.865293i	$-0.332868\pi$
$43$	6.57404	1.00253	0.501266	+	0.865293i	$0.332868\pi$
$44$	0	0
$45$	10.3275	1.53954
$46$	0	0
$47$	−6.70873	−0.978568	−0.489284	−	0.872124i	$-0.662742\pi$
$47$	−6.70873	−0.978568	−0.489284	+	0.872124i	$0.662742\pi$
$48$	0	0
$49$	3.62409	0.517727
$50$	0	0
$51$	2.83790	0.397385
$52$	0	0
$53$	−6.98854	−0.959949	−0.479975	−	0.877282i	$-0.659354\pi$
$53$	−6.98854	−0.959949	−0.479975	+	0.877282i	$0.659354\pi$
$54$	0	0
$55$	10.6027	1.42967
$56$	0	0
$57$	−1.37911	−0.182668
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	12.4674	1.59628	0.798141	−	0.602471i	$-0.205817\pi$
$61$	12.4674	1.59628	0.798141	+	0.602471i	$0.205817\pi$
$62$	0	0
$63$	16.4723	2.07531
$64$	0	0
$65$	10.4779	1.29963
$66$	0	0
$67$	−12.1470	−1.48399	−0.741995	−	0.670406i	$-0.766120\pi$
$67$	−12.1470	−1.48399	−0.741995	+	0.670406i	$0.766120\pi$
$68$	0	0
$69$	10.7576	1.29506
$70$	0	0
$71$	1.61652	0.191846	0.0959229	−	0.995389i	$-0.469420\pi$
$71$	1.61652	0.191846	0.0959229	+	0.995389i	$0.469420\pi$
$72$	0	0
$73$	11.9773	1.40184	0.700918	−	0.713242i	$-0.252774\pi$
$73$	11.9773	1.40184	0.700918	+	0.713242i	$0.252774\pi$
$74$	0	0
$75$	2.33795	0.269963
$76$	0	0
$77$	16.9111	1.92720
$78$	0	0
$79$	16.5361	1.86046	0.930229	−	0.366979i	$-0.119608\pi$
$79$	16.5361	1.86046	0.930229	+	0.366979i	$0.119608\pi$
$80$	0	0
$81$	1.37862	0.153180
$82$	0	0
$83$	5.89570	0.647137	0.323569	−	0.946205i	$-0.395117\pi$
$83$	5.89570	0.647137	0.323569	+	0.946205i	$0.395117\pi$
$84$	0	0
$85$	−2.04357	−0.221656
$86$	0	0
$87$	−23.7686	−2.54826
$88$	0	0
$89$	−3.57802	−0.379269	−0.189635	−	0.981855i	$-0.560730\pi$
$89$	−3.57802	−0.379269	−0.189635	+	0.981855i	$0.560730\pi$
$90$	0	0
$91$	16.7122	1.75191
$92$	0	0
$93$	−19.7248	−2.04537
$94$	0	0
$95$	0.993098	0.101890
$96$	0	0
$97$	−16.9318	−1.71916	−0.859581	−	0.510999i	$-0.829276\pi$
$97$	−16.9318	−1.71916	−0.859581	+	0.510999i	$0.829276\pi$
$98$	0	0
$99$	26.2201	2.63522
$100$	0	0
$101$	−4.13324	−0.411273	−0.205636	−	0.978628i	$-0.565926\pi$
$101$	−4.13324	−0.411273	−0.205636	+	0.978628i	$0.565926\pi$
$102$	0	0
$103$	−8.32394	−0.820183	−0.410091	−	0.912044i	$-0.634503\pi$
$103$	−8.32394	−0.820183	−0.410091	+	0.912044i	$0.634503\pi$
$104$	0	0
$105$	−18.9031	−1.84475
$106$	0	0
$107$	−11.9869	−1.15882	−0.579409	−	0.815037i	$-0.696717\pi$
$107$	−11.9869	−1.15882	−0.579409	+	0.815037i	$0.696717\pi$
$108$	0	0
$109$	8.22198	0.787523	0.393762	−	0.919213i	$-0.371174\pi$
$109$	8.22198	0.787523	0.393762	+	0.919213i	$0.371174\pi$
$110$	0	0
$111$	−22.7853	−2.16269
$112$	0	0
$113$	0.616465	0.0579921	0.0289961	−	0.999580i	$-0.490769\pi$
$113$	0.616465	0.0579921	0.0289961	+	0.999580i	$0.490769\pi$
$114$	0	0
$115$	−7.74651	−0.722366
$116$	0	0
$117$	25.9116	2.39553
$118$	0	0
$119$	−3.25946	−0.298794
$120$	0	0
$121$	15.9187	1.44715
$122$	0	0
$123$	−32.1571	−2.89951
$124$	0	0
$125$	−11.9014	−1.06449
$126$	0	0
$127$	−11.8831	−1.05445	−0.527225	−	0.849725i	$-0.676768\pi$
$127$	−11.8831	−1.05445	−0.527225	+	0.849725i	$0.676768\pi$
$128$	0	0
$129$	−18.6565	−1.64261
$130$	0	0
$131$	−13.8682	−1.21167	−0.605834	−	0.795591i	$-0.707160\pi$
$131$	−13.8682	−1.21167	−0.605834	+	0.795591i	$0.707160\pi$
$132$	0	0
$133$	1.58398	0.137348
$134$	0	0
$135$	−11.9102	−1.02506
$136$	0	0
$137$	9.44284	0.806756	0.403378	−	0.915033i	$-0.367836\pi$
$137$	9.44284	0.806756	0.403378	+	0.915033i	$0.367836\pi$
$138$	0	0
$139$	−6.26466	−0.531362	−0.265681	−	0.964061i	$-0.585597\pi$
$139$	−6.26466	−0.531362	−0.265681	+	0.964061i	$0.585597\pi$
$140$	0	0
$141$	19.0387	1.60335
$142$	0	0
$143$	26.6020	2.22457
$144$	0	0
$145$	17.1157	1.42138
$146$	0	0
$147$	−10.2848	−0.848276
$148$	0	0
$149$	−4.72685	−0.387239	−0.193619	−	0.981077i	$-0.562023\pi$
$149$	−4.72685	−0.387239	−0.193619	+	0.981077i	$0.562023\pi$
$150$	0	0
$151$	−16.2265	−1.32050	−0.660248	−	0.751048i	$-0.729549\pi$
$151$	−16.2265	−1.32050	−0.660248	+	0.751048i	$0.729549\pi$
$152$	0	0
$153$	−5.05368	−0.408566
$154$	0	0
$155$	14.2038	1.14088
$156$	0	0
$157$	10.8468	0.865672	0.432836	−	0.901473i	$-0.357513\pi$
$157$	10.8468	0.865672	0.432836	+	0.901473i	$0.357513\pi$
$158$	0	0
$159$	19.8328	1.57284
$160$	0	0
$161$	−12.3556	−0.973756
$162$	0	0
$163$	−5.40341	−0.423228	−0.211614	−	0.977353i	$-0.567872\pi$
$163$	−5.40341	−0.423228	−0.211614	+	0.977353i	$0.567872\pi$
$164$	0	0
$165$	−30.0894	−2.34245
$166$	0	0
$167$	1.51177	0.116985	0.0584923	−	0.998288i	$-0.481371\pi$
$167$	1.51177	0.116985	0.0584923	+	0.998288i	$0.481371\pi$
$168$	0	0
$169$	13.2890	1.02223
$170$	0	0
$171$	2.45590	0.187807
$172$	0	0
$173$	−20.1007	−1.52823	−0.764115	−	0.645080i	$-0.776824\pi$
$173$	−20.1007	−1.52823	−0.764115	+	0.645080i	$0.776824\pi$
$174$	0	0
$175$	−2.68525	−0.202985
$176$	0	0
$177$	2.83790	0.213310
$178$	0	0
$179$	−14.2152	−1.06249	−0.531247	−	0.847217i	$-0.678276\pi$
$179$	−14.2152	−1.06249	−0.531247	+	0.847217i	$0.678276\pi$
$180$	0	0
$181$	−26.0461	−1.93599	−0.967996	−	0.250965i	$-0.919252\pi$
$181$	−26.0461	−1.93599	−0.967996	+	0.250965i	$0.919252\pi$
$182$	0	0
$183$	−35.3811	−2.61545
$184$	0	0
$185$	16.4077	1.20632
$186$	0	0
$187$	−5.18832	−0.379408
$188$	0	0
$189$	−18.9966	−1.38180
$190$	0	0
$191$	2.67635	0.193654	0.0968271	−	0.995301i	$-0.469131\pi$
$191$	2.67635	0.193654	0.0968271	+	0.995301i	$0.469131\pi$
$192$	0	0
$193$	6.29105	0.452839	0.226420	−	0.974030i	$-0.427298\pi$
$193$	6.29105	0.452839	0.226420	+	0.974030i	$0.427298\pi$
$194$	0	0
$195$	−29.7353	−2.12939
$196$	0	0
$197$	6.06203	0.431902	0.215951	−	0.976404i	$-0.430715\pi$
$197$	6.06203	0.431902	0.215951	+	0.976404i	$0.430715\pi$
$198$	0	0
$199$	−14.4158	−1.02191	−0.510955	−	0.859607i	$-0.670708\pi$
$199$	−14.4158	−1.02191	−0.510955	+	0.859607i	$0.670708\pi$
$200$	0	0
$201$	34.4719	2.43146
$202$	0	0
$203$	27.2993	1.91604
$204$	0	0
$205$	23.1563	1.61731
$206$	0	0
$207$	−19.1569	−1.33149
$208$	0	0
$209$	2.52133	0.174404
$210$	0	0
$211$	−10.4748	−0.721118	−0.360559	−	0.932736i	$-0.617414\pi$
$211$	−10.4748	−0.721118	−0.360559	+	0.932736i	$0.617414\pi$
$212$	0	0
$213$	−4.58752	−0.314332
$214$	0	0
$215$	13.4345	0.916225
$216$	0	0
$217$	22.6549	1.53792
$218$	0	0
$219$	−33.9904	−2.29686
$220$	0	0
$221$	−5.12728	−0.344898
$222$	0	0
$223$	−17.0272	−1.14023	−0.570113	−	0.821566i	$-0.693101\pi$
$223$	−17.0272	−1.14023	−0.570113	+	0.821566i	$0.693101\pi$
$224$	0	0
$225$	−4.16338	−0.277558
$226$	0	0
$227$	−10.1510	−0.673745	−0.336873	−	0.941550i	$-0.609369\pi$
$227$	−10.1510	−0.673745	−0.336873	+	0.941550i	$0.609369\pi$
$228$	0	0
$229$	−16.5978	−1.09681	−0.548406	−	0.836212i	$-0.684765\pi$
$229$	−16.5978	−1.09681	−0.548406	+	0.836212i	$0.684765\pi$
$230$	0	0
$231$	−47.9921	−3.15765
$232$	0	0
$233$	−19.4929	−1.27702	−0.638511	−	0.769613i	$-0.720449\pi$
$233$	−19.4929	−1.27702	−0.638511	+	0.769613i	$0.720449\pi$
$234$	0	0
$235$	−13.7097	−0.894325
$236$	0	0
$237$	−46.9278	−3.04829
$238$	0	0
$239$	9.16880	0.593080	0.296540	−	0.955020i	$-0.404167\pi$
$239$	9.16880	0.593080	0.296540	+	0.955020i	$0.404167\pi$
$240$	0	0
$241$	20.5876	1.32616	0.663082	−	0.748547i	$-0.269248\pi$
$241$	20.5876	1.32616	0.663082	+	0.748547i	$0.269248\pi$
$242$	0	0
$243$	13.5720	0.870645
$244$	0	0
$245$	7.40607	0.473156
$246$	0	0
$247$	2.49167	0.158541
$248$	0	0
$249$	−16.7314	−1.06031
$250$	0	0
$251$	−21.2624	−1.34207	−0.671035	−	0.741426i	$-0.734150\pi$
$251$	−21.2624	−1.34207	−0.671035	+	0.741426i	$0.734150\pi$
$252$	0	0
$253$	−19.6673	−1.23647
$254$	0	0
$255$	5.79944	0.363175
$256$	0	0
$257$	10.9054	0.680258	0.340129	−	0.940379i	$-0.389529\pi$
$257$	10.9054	0.680258	0.340129	+	0.940379i	$0.389529\pi$
$258$	0	0
$259$	26.1700	1.62612
$260$	0	0
$261$	42.3266	2.61995
$262$	0	0
$263$	9.71952	0.599331	0.299666	−	0.954044i	$-0.403125\pi$
$263$	9.71952	0.599331	0.299666	+	0.954044i	$0.403125\pi$
$264$	0	0
$265$	−14.2815	−0.877309
$266$	0	0
$267$	10.1541	0.621419
$268$	0	0
$269$	30.7820	1.87681	0.938407	−	0.345532i	$-0.112301\pi$
$269$	30.7820	1.87681	0.938407	+	0.345532i	$0.112301\pi$
$270$	0	0
$271$	17.6954	1.07492	0.537460	−	0.843289i	$-0.319384\pi$
$271$	17.6954	1.07492	0.537460	+	0.843289i	$0.319384\pi$
$272$	0	0
$273$	−47.4274	−2.87044
$274$	0	0
$275$	−4.27430	−0.257750
$276$	0	0
$277$	−6.10987	−0.367107	−0.183553	−	0.983010i	$-0.558760\pi$
$277$	−6.10987	−0.367107	−0.183553	+	0.983010i	$0.558760\pi$
$278$	0	0
$279$	35.1256	2.10292
$280$	0	0
$281$	−26.3383	−1.57121	−0.785604	−	0.618729i	$-0.787648\pi$
$281$	−26.3383	−1.57121	−0.785604	+	0.618729i	$0.787648\pi$
$282$	0	0
$283$	14.9487	0.888606	0.444303	−	0.895877i	$-0.353451\pi$
$283$	14.9487	0.888606	0.444303	+	0.895877i	$0.353451\pi$
$284$	0	0
$285$	−2.81831	−0.166942
$286$	0	0
$287$	36.9340	2.18014
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	48.0507	2.81678
$292$	0	0
$293$	−2.32759	−0.135979	−0.0679895	−	0.997686i	$-0.521658\pi$
$293$	−2.32759	−0.135979	−0.0679895	+	0.997686i	$0.521658\pi$
$294$	0	0
$295$	−2.04357	−0.118981
$296$	0	0
$297$	−30.2382	−1.75460
$298$	0	0
$299$	−19.4359	−1.12401
$300$	0	0
$301$	21.4278	1.23508
$302$	0	0
$303$	11.7297	0.673855
$304$	0	0
$305$	25.4779	1.45886
$306$	0	0
$307$	−25.6895	−1.46618	−0.733090	−	0.680132i	$-0.761922\pi$
$307$	−25.6895	−1.46618	−0.733090	+	0.680132i	$0.761922\pi$
$308$	0	0
$309$	23.6225	1.34384
$310$	0	0
$311$	−26.6290	−1.50999	−0.754996	−	0.655729i	$-0.772361\pi$
$311$	−26.6290	−1.50999	−0.754996	+	0.655729i	$0.772361\pi$
$312$	0	0
$313$	7.48658	0.423167	0.211583	−	0.977360i	$-0.432138\pi$
$313$	7.48658	0.423167	0.211583	+	0.977360i	$0.432138\pi$
$314$	0	0
$315$	33.6622	1.89665
$316$	0	0
$317$	1.51242	0.0849457	0.0424729	−	0.999098i	$-0.486476\pi$
$317$	1.51242	0.0849457	0.0424729	+	0.999098i	$0.486476\pi$
$318$	0	0
$319$	43.4543	2.43298
$320$	0	0
$321$	34.0177	1.89868
$322$	0	0
$323$	−0.485963	−0.0270397
$324$	0	0
$325$	−4.22401	−0.234306
$326$	0	0
$327$	−23.3332	−1.29033
$328$	0	0
$329$	−21.8668	−1.20556
$330$	0	0
$331$	13.1902	0.724998	0.362499	−	0.931984i	$-0.381924\pi$
$331$	13.1902	0.724998	0.362499	+	0.931984i	$0.381924\pi$
$332$	0	0
$333$	40.5756	2.22353
$334$	0	0
$335$	−24.8232	−1.35623
$336$	0	0
$337$	−13.2325	−0.720820	−0.360410	−	0.932794i	$-0.617363\pi$
$337$	−13.2325	−0.720820	−0.360410	+	0.932794i	$0.617363\pi$
$338$	0	0
$339$	−1.74946	−0.0950179
$340$	0	0
$341$	36.0615	1.95284
$342$	0	0
$343$	−11.0037	−0.594142
$344$	0	0
$345$	21.9838	1.18357
$346$	0	0
$347$	−5.26271	−0.282517	−0.141258	−	0.989973i	$-0.545115\pi$
$347$	−5.26271	−0.282517	−0.141258	+	0.989973i	$0.545115\pi$
$348$	0	0
$349$	21.8436	1.16926	0.584630	−	0.811300i	$-0.301240\pi$
$349$	21.8436	1.16926	0.584630	+	0.811300i	$0.301240\pi$
$350$	0	0
$351$	−29.8824	−1.59501
$352$	0	0
$353$	−28.8270	−1.53431	−0.767154	−	0.641463i	$-0.778328\pi$
$353$	−28.8270	−1.53431	−0.767154	+	0.641463i	$0.778328\pi$
$354$	0	0
$355$	3.30347	0.175330
$356$	0	0
$357$	9.25002	0.489563
$358$	0	0
$359$	10.7383	0.566743	0.283372	−	0.959010i	$-0.408547\pi$
$359$	10.7383	0.566743	0.283372	+	0.959010i	$0.408547\pi$
$360$	0	0
$361$	−18.7638	−0.987571
$362$	0	0
$363$	−45.1756	−2.37111
$364$	0	0
$365$	24.4764	1.28115
$366$	0	0
$367$	5.65966	0.295432	0.147716	−	0.989030i	$-0.452808\pi$
$367$	5.65966	0.295432	0.147716	+	0.989030i	$0.452808\pi$
$368$	0	0
$369$	57.2648	2.98108
$370$	0	0
$371$	−22.7789	−1.18262
$372$	0	0
$373$	8.42449	0.436204	0.218102	−	0.975926i	$-0.430014\pi$
$373$	8.42449	0.436204	0.218102	+	0.975926i	$0.430014\pi$
$374$	0	0
$375$	33.7750	1.74413
$376$	0	0
$377$	42.9431	2.21168
$378$	0	0
$379$	−37.3815	−1.92016	−0.960079	−	0.279728i	$-0.909756\pi$
$379$	−37.3815	−1.92016	−0.960079	+	0.279728i	$0.909756\pi$
$380$	0	0
$381$	33.7229	1.72768
$382$	0	0
$383$	−6.21648	−0.317647	−0.158824	−	0.987307i	$-0.550770\pi$
$383$	−6.21648	−0.317647	−0.158824	+	0.987307i	$0.550770\pi$
$384$	0	0
$385$	34.5590	1.76129
$386$	0	0
$387$	33.2231	1.68882
$388$	0	0
$389$	−15.8169	−0.801948	−0.400974	−	0.916089i	$-0.631328\pi$
$389$	−15.8169	−0.801948	−0.400974	+	0.916089i	$0.631328\pi$
$390$	0	0
$391$	3.79068	0.191703
$392$	0	0
$393$	39.3565	1.98527
$394$	0	0
$395$	33.7927	1.70029
$396$	0	0
$397$	−37.4181	−1.87796	−0.938980	−	0.343970i	$-0.888228\pi$
$397$	−37.4181	−1.87796	−0.938980	+	0.343970i	$0.888228\pi$
$398$	0	0
$399$	−4.49517	−0.225040
$400$	0	0
$401$	38.2311	1.90917	0.954586	−	0.297936i	$-0.0962981\pi$
$401$	38.2311	1.90917	0.954586	+	0.297936i	$0.0962981\pi$
$402$	0	0
$403$	35.6372	1.77521
$404$	0	0
$405$	2.81730	0.139993
$406$	0	0
$407$	41.6567	2.06485
$408$	0	0
$409$	−11.4496	−0.566146	−0.283073	−	0.959098i	$-0.591354\pi$
$409$	−11.4496	−0.566146	−0.283073	+	0.959098i	$0.591354\pi$
$410$	0	0
$411$	−26.7978	−1.32184
$412$	0	0
$413$	−3.25946	−0.160388
$414$	0	0
$415$	12.0483	0.591426
$416$	0	0
$417$	17.7785	0.870616
$418$	0	0
$419$	−14.9152	−0.728656	−0.364328	−	0.931271i	$-0.618701\pi$
$419$	−14.9152	−0.728656	−0.364328	+	0.931271i	$0.618701\pi$
$420$	0	0
$421$	16.6122	0.809628	0.404814	−	0.914399i	$-0.367336\pi$
$421$	16.6122	0.809628	0.404814	+	0.914399i	$0.367336\pi$
$422$	0	0
$423$	−33.9037	−1.64846
$424$	0	0
$425$	0.823831	0.0399617
$426$	0	0
$427$	40.6369	1.96656
$428$	0	0
$429$	−75.4937	−3.64487
$430$	0	0
$431$	28.5203	1.37378	0.686888	−	0.726764i	$-0.258976\pi$
$431$	28.5203	1.37378	0.686888	+	0.726764i	$0.258976\pi$
$432$	0	0
$433$	15.2705	0.733852	0.366926	−	0.930250i	$-0.380410\pi$
$433$	15.2705	0.733852	0.366926	+	0.930250i	$0.380410\pi$
$434$	0	0
$435$	−48.5727	−2.32888
$436$	0	0
$437$	−1.84213	−0.0881210
$438$	0	0
$439$	26.5876	1.26896	0.634479	−	0.772940i	$-0.281215\pi$
$439$	26.5876	1.26896	0.634479	+	0.772940i	$0.281215\pi$
$440$	0	0
$441$	18.3150	0.872141
$442$	0	0
$443$	−5.61039	−0.266558	−0.133279	−	0.991079i	$-0.542551\pi$
$443$	−5.61039	−0.266558	−0.133279	+	0.991079i	$0.542551\pi$
$444$	0	0
$445$	−7.31193	−0.346619
$446$	0	0
$447$	13.4143	0.634476
$448$	0	0
$449$	−5.94529	−0.280576	−0.140288	−	0.990111i	$-0.544803\pi$
$449$	−5.94529	−0.280576	−0.140288	+	0.990111i	$0.544803\pi$
$450$	0	0
$451$	58.7905	2.76833
$452$	0	0
$453$	46.0493	2.16358
$454$	0	0
$455$	34.1524	1.60109
$456$	0	0
$457$	8.56130	0.400481	0.200240	−	0.979747i	$-0.435828\pi$
$457$	8.56130	0.400481	0.200240	+	0.979747i	$0.435828\pi$
$458$	0	0
$459$	5.82813	0.272034
$460$	0	0
$461$	−13.2251	−0.615953	−0.307977	−	0.951394i	$-0.599652\pi$
$461$	−13.2251	−0.615953	−0.307977	+	0.951394i	$0.599652\pi$
$462$	0	0
$463$	−38.1152	−1.77136	−0.885681	−	0.464294i	$-0.846308\pi$
$463$	−38.1152	−1.77136	−0.885681	+	0.464294i	$0.846308\pi$
$464$	0	0
$465$	−40.3091	−1.86929
$466$	0	0
$467$	−27.1071	−1.25437	−0.627183	−	0.778872i	$-0.715792\pi$
$467$	−27.1071	−1.25437	−0.627183	+	0.778872i	$0.715792\pi$
$468$	0	0
$469$	−39.5926	−1.82822
$470$	0	0
$471$	−30.7822	−1.41837
$472$	0	0
$473$	34.1082	1.56830
$474$	0	0
$475$	−0.400351	−0.0183694
$476$	0	0
$477$	−35.3178	−1.61709
$478$	0	0
$479$	−16.0870	−0.735036	−0.367518	−	0.930016i	$-0.619792\pi$
$479$	−16.0870	−0.735036	−0.367518	+	0.930016i	$0.619792\pi$
$480$	0	0
$481$	41.1666	1.87703
$482$	0	0
$483$	35.0639	1.59546
$484$	0	0
$485$	−34.6013	−1.57116
$486$	0	0
$487$	−15.2061	−0.689055	−0.344527	−	0.938776i	$-0.611961\pi$
$487$	−15.2061	−0.689055	−0.344527	+	0.938776i	$0.611961\pi$
$488$	0	0
$489$	15.3343	0.693443
$490$	0	0
$491$	−26.2588	−1.18504	−0.592520	−	0.805555i	$-0.701867\pi$
$491$	−26.2588	−1.18504	−0.592520	+	0.805555i	$0.701867\pi$
$492$	0	0
$493$	−8.37542	−0.377210
$494$	0	0
$495$	53.5825	2.40836
$496$	0	0
$497$	5.26899	0.236346
$498$	0	0
$499$	5.60180	0.250771	0.125386	−	0.992108i	$-0.459983\pi$
$499$	5.60180	0.250771	0.125386	+	0.992108i	$0.459983\pi$
$500$	0	0
$501$	−4.29027	−0.191675
$502$	0	0
$503$	27.8976	1.24389	0.621946	−	0.783060i	$-0.286342\pi$
$503$	27.8976	1.24389	0.621946	+	0.783060i	$0.286342\pi$
$504$	0	0
$505$	−8.44655	−0.375867
$506$	0	0
$507$	−37.7128	−1.67488
$508$	0	0
$509$	9.58363	0.424787	0.212393	−	0.977184i	$-0.431874\pi$
$509$	9.58363	0.424787	0.212393	+	0.977184i	$0.431874\pi$
$510$	0	0
$511$	39.0395	1.72701
$512$	0	0
$513$	−2.83225	−0.125047
$514$	0	0
$515$	−17.0105	−0.749574
$516$	0	0
$517$	−34.8070	−1.53081
$518$	0	0
$519$	57.0439	2.50395
$520$	0	0
$521$	−3.71312	−0.162675	−0.0813373	−	0.996687i	$-0.525919\pi$
$521$	−3.71312	−0.162675	−0.0813373	+	0.996687i	$0.525919\pi$
$522$	0	0
$523$	25.2254	1.10303	0.551515	−	0.834165i	$-0.314050\pi$
$523$	25.2254	1.10303	0.551515	+	0.834165i	$0.314050\pi$
$524$	0	0
$525$	7.62046	0.332584
$526$	0	0
$527$	−6.95051	−0.302769
$528$	0	0
$529$	−8.63075	−0.375250
$530$	0	0
$531$	−5.05368	−0.219311
$532$	0	0
$533$	58.0987	2.51654
$534$	0	0
$535$	−24.4961	−1.05906
$536$	0	0
$537$	40.3413	1.74086
$538$	0	0
$539$	18.8029	0.809899
$540$	0	0
$541$	−20.9665	−0.901419	−0.450710	−	0.892671i	$-0.648829\pi$
$541$	−20.9665	−0.901419	−0.450710	+	0.892671i	$0.648829\pi$
$542$	0	0
$543$	73.9163	3.17205
$544$	0	0
$545$	16.8022	0.719726
$546$	0	0
$547$	7.59168	0.324597	0.162298	−	0.986742i	$-0.448109\pi$
$547$	7.59168	0.324597	0.162298	+	0.986742i	$0.448109\pi$
$548$	0	0
$549$	63.0060	2.68903
$550$	0	0
$551$	4.07014	0.173394
$552$	0	0
$553$	53.8988	2.29201
$554$	0	0
$555$	−46.5633	−1.97650
$556$	0	0
$557$	5.45869	0.231292	0.115646	−	0.993290i	$-0.463106\pi$
$557$	5.45869	0.231292	0.115646	+	0.993290i	$0.463106\pi$
$558$	0	0
$559$	33.7069	1.42565
$560$	0	0
$561$	14.7239	0.621645
$562$	0	0
$563$	−16.9200	−0.713095	−0.356547	−	0.934277i	$-0.616046\pi$
$563$	−16.9200	−0.713095	−0.356547	+	0.934277i	$0.616046\pi$
$564$	0	0
$565$	1.25979	0.0529996
$566$	0	0
$567$	4.49355	0.188711
$568$	0	0
$569$	−25.9215	−1.08668	−0.543342	−	0.839511i	$-0.682841\pi$
$569$	−25.9215	−1.08668	−0.543342	+	0.839511i	$0.682841\pi$
$570$	0	0
$571$	12.7321	0.532823	0.266411	−	0.963859i	$-0.414162\pi$
$571$	12.7321	0.532823	0.266411	+	0.963859i	$0.414162\pi$
$572$	0	0
$573$	−7.59522	−0.317295
$574$	0	0
$575$	3.12288	0.130233
$576$	0	0
$577$	12.9107	0.537481	0.268740	−	0.963213i	$-0.413393\pi$
$577$	12.9107	0.537481	0.268740	+	0.963213i	$0.413393\pi$
$578$	0	0
$579$	−17.8534	−0.741960
$580$	0	0
$581$	19.2168	0.797247
$582$	0	0
$583$	−36.2588	−1.50168
$584$	0	0
$585$	52.9521	2.18930
$586$	0	0
$587$	43.3136	1.78774	0.893872	−	0.448321i	$-0.147978\pi$
$587$	43.3136	1.78774	0.893872	+	0.448321i	$0.147978\pi$
$588$	0	0
$589$	3.37769	0.139175
$590$	0	0
$591$	−17.2034	−0.707655
$592$	0	0
$593$	23.6825	0.972522	0.486261	−	0.873814i	$-0.338360\pi$
$593$	23.6825	0.972522	0.486261	+	0.873814i	$0.338360\pi$
$594$	0	0
$595$	−6.66093	−0.273071
$596$	0	0
$597$	40.9107	1.67436
$598$	0	0
$599$	−31.0987	−1.27066	−0.635329	−	0.772242i	$-0.719135\pi$
$599$	−31.0987	−1.27066	−0.635329	+	0.772242i	$0.719135\pi$
$600$	0	0
$601$	−31.2847	−1.27613	−0.638064	−	0.769983i	$-0.720264\pi$
$601$	−31.2847	−1.27613	−0.638064	+	0.769983i	$0.720264\pi$
$602$	0	0
$603$	−61.3869	−2.49987
$604$	0	0
$605$	32.5309	1.32257
$606$	0	0
$607$	15.4140	0.625633	0.312817	−	0.949814i	$-0.398727\pi$
$607$	15.4140	0.625633	0.312817	+	0.949814i	$0.398727\pi$
$608$	0	0
$609$	−77.4728	−3.13936
$610$	0	0
$611$	−34.3975	−1.39157
$612$	0	0
$613$	−29.8030	−1.20373	−0.601866	−	0.798597i	$-0.705576\pi$
$613$	−29.8030	−1.20373	−0.601866	+	0.798597i	$0.705576\pi$
$614$	0	0
$615$	−65.7152	−2.64989
$616$	0	0
$617$	1.63297	0.0657409	0.0328705	−	0.999460i	$-0.489535\pi$
$617$	1.63297	0.0657409	0.0328705	+	0.999460i	$0.489535\pi$
$618$	0	0
$619$	−27.8806	−1.12062	−0.560309	−	0.828284i	$-0.689317\pi$
$619$	−27.8806	−1.12062	−0.560309	+	0.828284i	$0.689317\pi$
$620$	0	0
$621$	22.0926	0.886544
$622$	0	0
$623$	−11.6624	−0.467245
$624$	0	0
$625$	−20.2021	−0.808086
$626$	0	0
$627$	−7.15529	−0.285755
$628$	0	0
$629$	−8.02893	−0.320135
$630$	0	0
$631$	23.7644	0.946044	0.473022	−	0.881051i	$-0.343163\pi$
$631$	23.7644	0.946044	0.473022	+	0.881051i	$0.343163\pi$
$632$	0	0
$633$	29.7266	1.18152
$634$	0	0
$635$	−24.2838	−0.963674
$636$	0	0
$637$	18.5817	0.736234
$638$	0	0
$639$	8.16937	0.323175
$640$	0	0
$641$	−13.5179	−0.533925	−0.266962	−	0.963707i	$-0.586020\pi$
$641$	−13.5179	−0.533925	−0.266962	+	0.963707i	$0.586020\pi$
$642$	0	0
$643$	−13.6544	−0.538476	−0.269238	−	0.963074i	$-0.586772\pi$
$643$	−13.6544	−0.538476	−0.269238	+	0.963074i	$0.586772\pi$
$644$	0	0
$645$	−38.1258	−1.50120
$646$	0	0
$647$	−5.63429	−0.221507	−0.110753	−	0.993848i	$-0.535326\pi$
$647$	−5.63429	−0.221507	−0.110753	+	0.993848i	$0.535326\pi$
$648$	0	0
$649$	−5.18832	−0.203659
$650$	0	0
$651$	−64.2924	−2.51982
$652$	0	0
$653$	−10.8537	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$653$	−10.8537	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$654$	0	0
$655$	−28.3405	−1.10736
$656$	0	0
$657$	60.5294	2.36148
$658$	0	0
$659$	−13.9491	−0.543380	−0.271690	−	0.962385i	$-0.587582\pi$
$659$	−13.9491	−0.543380	−0.271690	+	0.962385i	$0.587582\pi$
$660$	0	0
$661$	26.7855	1.04183	0.520917	−	0.853607i	$-0.325590\pi$
$661$	26.7855	1.04183	0.520917	+	0.853607i	$0.325590\pi$
$662$	0	0
$663$	14.5507	0.565102
$664$	0	0
$665$	3.23696	0.125524
$666$	0	0
$667$	−31.7485	−1.22931
$668$	0	0
$669$	48.3215	1.86822
$670$	0	0
$671$	64.6847	2.49712
$672$	0	0
$673$	20.3253	0.783482	0.391741	−	0.920076i	$-0.371873\pi$
$673$	20.3253	0.783482	0.391741	+	0.920076i	$0.371873\pi$
$674$	0	0
$675$	4.80139	0.184806
$676$	0	0
$677$	−39.1260	−1.50373	−0.751867	−	0.659315i	$-0.770846\pi$
$677$	−39.1260	−1.50373	−0.751867	+	0.659315i	$0.770846\pi$
$678$	0	0
$679$	−55.1885	−2.11794
$680$	0	0
$681$	28.8075	1.10391
$682$	0	0
$683$	−28.8753	−1.10488	−0.552441	−	0.833552i	$-0.686304\pi$
$683$	−28.8753	−1.10488	−0.552441	+	0.833552i	$0.686304\pi$
$684$	0	0
$685$	19.2971	0.737304
$686$	0	0
$687$	47.1029	1.79709
$688$	0	0
$689$	−35.8322	−1.36510
$690$	0	0
$691$	13.4916	0.513244	0.256622	−	0.966512i	$-0.417390\pi$
$691$	13.4916	0.513244	0.256622	+	0.966512i	$0.417390\pi$
$692$	0	0
$693$	85.4634	3.24649
$694$	0	0
$695$	−12.8023	−0.485617
$696$	0	0
$697$	−11.3313	−0.429204
$698$	0	0
$699$	55.3189	2.09235
$700$	0	0
$701$	−25.0728	−0.946986	−0.473493	−	0.880798i	$-0.657007\pi$
$701$	−25.0728	−0.946986	−0.473493	+	0.880798i	$0.657007\pi$
$702$	0	0
$703$	3.90176	0.147158
$704$	0	0
$705$	38.9069	1.46532
$706$	0	0
$707$	−13.4721	−0.506672
$708$	0	0
$709$	39.7376	1.49238	0.746189	−	0.665734i	$-0.231881\pi$
$709$	39.7376	1.49238	0.746189	+	0.665734i	$0.231881\pi$
$710$	0	0
$711$	83.5682	3.13405
$712$	0	0
$713$	−26.3472	−0.986709
$714$	0	0
$715$	54.3629	2.03306
$716$	0	0
$717$	−26.0201	−0.971740
$718$	0	0
$719$	24.6814	0.920460	0.460230	−	0.887800i	$-0.347767\pi$
$719$	24.6814	0.920460	0.460230	+	0.887800i	$0.347767\pi$
$720$	0	0
$721$	−27.1316	−1.01043
$722$	0	0
$723$	−58.4256	−2.17287
$724$	0	0
$725$	−6.89993	−0.256257
$726$	0	0
$727$	7.47197	0.277120	0.138560	−	0.990354i	$-0.455753\pi$
$727$	7.47197	0.277120	0.138560	+	0.990354i	$0.455753\pi$
$728$	0	0
$729$	−42.6519	−1.57970
$730$	0	0
$731$	−6.57404	−0.243150
$732$	0	0
$733$	5.79188	0.213928	0.106964	−	0.994263i	$-0.465887\pi$
$733$	5.79188	0.213928	0.106964	+	0.994263i	$0.465887\pi$
$734$	0	0
$735$	−21.0177	−0.775249
$736$	0	0
$737$	−63.0224	−2.32146
$738$	0	0
$739$	−51.7460	−1.90351	−0.951753	−	0.306866i	$-0.900720\pi$
$739$	−51.7460	−1.90351	−0.951753	+	0.306866i	$0.900720\pi$
$740$	0	0
$741$	−7.07110	−0.259763
$742$	0	0
$743$	11.5244	0.422790	0.211395	−	0.977401i	$-0.432199\pi$
$743$	11.5244	0.422790	0.211395	+	0.977401i	$0.432199\pi$
$744$	0	0
$745$	−9.65964	−0.353902
$746$	0	0
$747$	29.7950	1.09014
$748$	0	0
$749$	−39.0709	−1.42762
$750$	0	0
$751$	38.3579	1.39970	0.699850	−	0.714289i	$-0.253250\pi$
$751$	38.3579	1.39970	0.699850	+	0.714289i	$0.253250\pi$
$752$	0	0
$753$	60.3405	2.19893
$754$	0	0
$755$	−33.1600	−1.20682
$756$	0	0
$757$	31.5669	1.14732	0.573659	−	0.819094i	$-0.305523\pi$
$757$	31.5669	1.14732	0.573659	+	0.819094i	$0.305523\pi$
$758$	0	0
$759$	55.8137	2.02591
$760$	0	0
$761$	12.0239	0.435866	0.217933	−	0.975964i	$-0.430069\pi$
$761$	12.0239	0.435866	0.217933	+	0.975964i	$0.430069\pi$
$762$	0	0
$763$	26.7992	0.970197
$764$	0	0
$765$	−10.3275	−0.373393
$766$	0	0
$767$	−5.12728	−0.185135
$768$	0	0
$769$	−45.1914	−1.62964	−0.814821	−	0.579712i	$-0.803165\pi$
$769$	−45.1914	−1.62964	−0.814821	+	0.579712i	$0.803165\pi$
$770$	0	0
$771$	−30.9483	−1.11458
$772$	0	0
$773$	5.58287	0.200802	0.100401	−	0.994947i	$-0.467987\pi$
$773$	5.58287	0.200802	0.100401	+	0.994947i	$0.467987\pi$
$774$	0	0
$775$	−5.72604	−0.205686
$776$	0	0
$777$	−74.2678	−2.66434
$778$	0	0
$779$	5.50659	0.197294
$780$	0	0
$781$	8.38703	0.300112
$782$	0	0
$783$	−48.8130	−1.74443
$784$	0	0
$785$	22.1662	0.791147
$786$	0	0
$787$	−7.23365	−0.257852	−0.128926	−	0.991654i	$-0.541153\pi$
$787$	−7.23365	−0.257852	−0.128926	+	0.991654i	$0.541153\pi$
$788$	0	0
$789$	−27.5830	−0.981982
$790$	0	0
$791$	2.00934	0.0714440
$792$	0	0
$793$	63.9236	2.26999
$794$	0	0
$795$	40.5296	1.43744
$796$	0	0
$797$	41.0190	1.45297	0.726484	−	0.687184i	$-0.241153\pi$
$797$	41.0190	1.45297	0.726484	+	0.687184i	$0.241153\pi$
$798$	0	0
$799$	6.70873	0.237338
$800$	0	0
$801$	−18.0822	−0.638902
$802$	0	0
$803$	62.1420	2.19295
$804$	0	0
$805$	−25.2495	−0.889926
$806$	0	0
$807$	−87.3564	−3.07509
$808$	0	0
$809$	−0.932729	−0.0327930	−0.0163965	−	0.999866i	$-0.505219\pi$
$809$	−0.932729	−0.0327930	−0.0163965	+	0.999866i	$0.505219\pi$
$810$	0	0
$811$	19.3267	0.678652	0.339326	−	0.940669i	$-0.389801\pi$
$811$	19.3267	0.678652	0.339326	+	0.940669i	$0.389801\pi$
$812$	0	0
$813$	−50.2179	−1.76122
$814$	0	0
$815$	−11.0422	−0.386793
$816$	0	0
$817$	3.19474	0.111770
$818$	0	0
$819$	84.4579	2.95120
$820$	0	0
$821$	−9.21563	−0.321628	−0.160814	−	0.986985i	$-0.551412\pi$
$821$	−9.21563	−0.321628	−0.160814	+	0.986985i	$0.551412\pi$
$822$	0	0
$823$	38.1410	1.32951	0.664756	−	0.747060i	$-0.268535\pi$
$823$	38.1410	1.32951	0.664756	+	0.747060i	$0.268535\pi$
$824$	0	0
$825$	12.1300	0.422314
$826$	0	0
$827$	−6.45460	−0.224448	−0.112224	−	0.993683i	$-0.535797\pi$
$827$	−6.45460	−0.224448	−0.112224	+	0.993683i	$0.535797\pi$
$828$	0	0
$829$	−37.8072	−1.31310	−0.656549	−	0.754283i	$-0.727984\pi$
$829$	−37.8072	−1.31310	−0.656549	+	0.754283i	$0.727984\pi$
$830$	0	0
$831$	17.3392	0.601491
$832$	0	0
$833$	−3.62409	−0.125567
$834$	0	0
$835$	3.08941	0.106914
$836$	0	0
$837$	−40.5085	−1.40018
$838$	0	0
$839$	13.4860	0.465588	0.232794	−	0.972526i	$-0.425213\pi$
$839$	13.4860	0.465588	0.232794	+	0.972526i	$0.425213\pi$
$840$	0	0
$841$	41.1476	1.41888
$842$	0	0
$843$	74.7454	2.57437
$844$	0	0
$845$	27.1569	0.934226
$846$	0	0
$847$	51.8863	1.78283
$848$	0	0
$849$	−42.4228	−1.45595
$850$	0	0
$851$	−30.4351	−1.04330
$852$	0	0
$853$	39.2045	1.34233	0.671167	−	0.741306i	$-0.265793\pi$
$853$	39.2045	1.34233	0.671167	+	0.741306i	$0.265793\pi$
$854$	0	0
$855$	5.01880	0.171639
$856$	0	0
$857$	−41.7022	−1.42452	−0.712261	−	0.701915i	$-0.752329\pi$
$857$	−41.7022	−1.42452	−0.712261	+	0.701915i	$0.752329\pi$
$858$	0	0
$859$	13.6205	0.464724	0.232362	−	0.972629i	$-0.425355\pi$
$859$	13.6205	0.464724	0.232362	+	0.972629i	$0.425355\pi$
$860$	0	0
$861$	−104.815	−3.57208
$862$	0	0
$863$	49.4124	1.68202	0.841009	−	0.541022i	$-0.181962\pi$
$863$	49.4124	1.68202	0.841009	+	0.541022i	$0.181962\pi$
$864$	0	0
$865$	−41.0772	−1.39667
$866$	0	0
$867$	−2.83790	−0.0963801
$868$	0	0
$869$	85.7947	2.91039
$870$	0	0
$871$	−62.2809	−2.11031
$872$	0	0
$873$	−85.5678	−2.89603
$874$	0	0
$875$	−38.7921	−1.31141
$876$	0	0
$877$	15.7312	0.531203	0.265602	−	0.964083i	$-0.414429\pi$
$877$	15.7312	0.531203	0.265602	+	0.964083i	$0.414429\pi$
$878$	0	0
$879$	6.60546	0.222797
$880$	0	0
$881$	49.8777	1.68042	0.840212	−	0.542259i	$-0.182431\pi$
$881$	49.8777	1.68042	0.840212	+	0.542259i	$0.182431\pi$
$882$	0	0
$883$	9.33539	0.314161	0.157080	−	0.987586i	$-0.449792\pi$
$883$	9.33539	0.314161	0.157080	+	0.987586i	$0.449792\pi$
$884$	0	0
$885$	5.79944	0.194946
$886$	0	0
$887$	19.6204	0.658788	0.329394	−	0.944193i	$-0.393156\pi$
$887$	19.6204	0.658788	0.329394	+	0.944193i	$0.393156\pi$
$888$	0	0
$889$	−38.7324	−1.29904
$890$	0	0
$891$	7.15271	0.239625
$892$	0	0
$893$	−3.26019	−0.109098
$894$	0	0
$895$	−29.0497	−0.971026
$896$	0	0
$897$	55.1570	1.84164
$898$	0	0
$899$	58.2134	1.94153
$900$	0	0
$901$	6.98854	0.232822
$902$	0	0
$903$	−60.8100	−2.02363
$904$	0	0
$905$	−53.2270	−1.76933
$906$	0	0
$907$	8.75037	0.290551	0.145276	−	0.989391i	$-0.453593\pi$
$907$	8.75037	0.290551	0.145276	+	0.989391i	$0.453593\pi$
$908$	0	0
$909$	−20.8881	−0.692813
$910$	0	0
$911$	59.2304	1.96239	0.981195	−	0.193020i	$-0.0618281\pi$
$911$	59.2304	1.96239	0.981195	+	0.193020i	$0.0618281\pi$
$912$	0	0
$913$	30.5888	1.01234
$914$	0	0
$915$	−72.3037	−2.39029
$916$	0	0
$917$	−45.2028	−1.49273
$918$	0	0
$919$	44.0121	1.45183	0.725913	−	0.687787i	$-0.241418\pi$
$919$	44.0121	1.45183	0.725913	+	0.687787i	$0.241418\pi$
$920$	0	0
$921$	72.9043	2.40228
$922$	0	0
$923$	8.28835	0.272814
$924$	0	0
$925$	−6.61448	−0.217483
$926$	0	0
$927$	−42.0665	−1.38165
$928$	0	0
$929$	−17.1597	−0.562990	−0.281495	−	0.959563i	$-0.590830\pi$
$929$	−17.1597	−0.562990	−0.281495	+	0.959563i	$0.590830\pi$
$930$	0	0
$931$	1.76117	0.0577200
$932$	0	0
$933$	75.5705	2.47407
$934$	0	0
$935$	−10.6027	−0.346745
$936$	0	0
$937$	16.3016	0.532549	0.266274	−	0.963897i	$-0.414207\pi$
$937$	16.3016	0.532549	0.266274	+	0.963897i	$0.414207\pi$
$938$	0	0
$939$	−21.2462	−0.693342
$940$	0	0
$941$	41.5617	1.35487	0.677436	−	0.735581i	$-0.263091\pi$
$941$	41.5617	1.35487	0.677436	+	0.735581i	$0.263091\pi$
$942$	0	0
$943$	−42.9534	−1.39875
$944$	0	0
$945$	−38.8208	−1.26284
$946$	0	0
$947$	29.7965	0.968256	0.484128	−	0.874997i	$-0.339137\pi$
$947$	29.7965	0.968256	0.484128	+	0.874997i	$0.339137\pi$
$948$	0	0
$949$	61.4109	1.99348
$950$	0	0
$951$	−4.29208	−0.139180
$952$	0	0
$953$	−25.6281	−0.830175	−0.415088	−	0.909781i	$-0.636249\pi$
$953$	−25.6281	−0.830175	−0.415088	+	0.909781i	$0.636249\pi$
$954$	0	0
$955$	5.46931	0.176983
$956$	0	0
$957$	−123.319	−3.98634
$958$	0	0
$959$	30.7786	0.993892
$960$	0	0
$961$	17.3096	0.558373
$962$	0	0
$963$	−60.5780	−1.95210
$964$	0	0
$965$	12.8562	0.413855
$966$	0	0
$967$	20.1560	0.648175	0.324087	−	0.946027i	$-0.394943\pi$
$967$	20.1560	0.648175	0.324087	+	0.946027i	$0.394943\pi$
$968$	0	0
$969$	1.37911	0.0443035
$970$	0	0
$971$	−30.2113	−0.969527	−0.484763	−	0.874645i	$-0.661094\pi$
$971$	−30.2113	−0.969527	−0.484763	+	0.874645i	$0.661094\pi$
$972$	0	0
$973$	−20.4194	−0.654616
$974$	0	0
$975$	11.9873	0.383901
$976$	0	0
$977$	−54.4938	−1.74341	−0.871705	−	0.490031i	$-0.836985\pi$
$977$	−54.4938	−1.74341	−0.871705	+	0.490031i	$0.836985\pi$
$978$	0	0
$979$	−18.5639	−0.593305
$980$	0	0
$981$	41.5512	1.32663
$982$	0	0
$983$	43.2120	1.37825	0.689124	−	0.724643i	$-0.257995\pi$
$983$	43.2120	1.37825	0.689124	+	0.724643i	$0.257995\pi$
$984$	0	0
$985$	12.3882	0.394720
$986$	0	0
$987$	62.0559	1.97526
$988$	0	0
$989$	−24.9201	−0.792413
$990$	0	0
$991$	−18.0401	−0.573063	−0.286531	−	0.958071i	$-0.592502\pi$
$991$	−18.0401	−0.573063	−0.286531	+	0.958071i	$0.592502\pi$
$992$	0	0
$993$	−37.4324	−1.18788
$994$	0	0
$995$	−29.4597	−0.933936
$996$	0	0
$997$	−44.5178	−1.40989	−0.704947	−	0.709260i	$-0.749029\pi$
$997$	−44.5178	−1.40989	−0.704947	+	0.709260i	$0.749029\pi$
$998$	0	0
$999$	−46.7937	−1.48049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.4	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.4	✓	33	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.83790 q^{3} +2.04357 q^{5} +3.25946 q^{7} +5.05368 q^{9} +O(q^{10})\)	\(q-2.83790 q^{3} +2.04357 q^{5} +3.25946 q^{7} +5.05368 q^{9} +5.18832 q^{11} +5.12728 q^{13} -5.79944 q^{15} -1.00000 q^{17} +0.485963 q^{19} -9.25002 q^{21} -3.79068 q^{23} -0.823831 q^{25} -5.82813 q^{27} +8.37542 q^{29} +6.95051 q^{31} -14.7239 q^{33} +6.66093 q^{35} +8.02893 q^{37} -14.5507 q^{39} +11.3313 q^{41} +6.57404 q^{43} +10.3275 q^{45} -6.70873 q^{47} +3.62409 q^{49} +2.83790 q^{51} -6.98854 q^{53} +10.6027 q^{55} -1.37911 q^{57} -1.00000 q^{59} +12.4674 q^{61} +16.4723 q^{63} +10.4779 q^{65} -12.1470 q^{67} +10.7576 q^{69} +1.61652 q^{71} +11.9773 q^{73} +2.33795 q^{75} +16.9111 q^{77} +16.5361 q^{79} +1.37862 q^{81} +5.89570 q^{83} -2.04357 q^{85} -23.7686 q^{87} -3.57802 q^{89} +16.7122 q^{91} -19.7248 q^{93} +0.993098 q^{95} -16.9318 q^{97} +26.2201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 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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