LMFDB - Embedded newform 8024.2.a.ba.1.2

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:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.78964 q^{3} +3.12966 q^{5} +4.93262 q^{7} +4.78208 q^{9} +O(q^{10})$	$q-2.78964 q^{3} +3.12966 q^{5} +4.93262 q^{7} +4.78208 q^{9} +2.03439 q^{11} +4.58503 q^{13} -8.73062 q^{15} -1.00000 q^{17} +4.73453 q^{19} -13.7602 q^{21} -1.83954 q^{23} +4.79478 q^{25} -4.97134 q^{27} -4.03737 q^{29} +4.92448 q^{31} -5.67521 q^{33} +15.4374 q^{35} -10.0209 q^{37} -12.7906 q^{39} -2.17449 q^{41} +6.41294 q^{43} +14.9663 q^{45} +12.5461 q^{47} +17.3307 q^{49} +2.78964 q^{51} +10.9390 q^{53} +6.36695 q^{55} -13.2076 q^{57} +1.00000 q^{59} -8.91219 q^{61} +23.5881 q^{63} +14.3496 q^{65} +10.4601 q^{67} +5.13166 q^{69} +5.53947 q^{71} -9.35947 q^{73} -13.3757 q^{75} +10.0349 q^{77} -12.5391 q^{79} -0.477980 q^{81} -14.7473 q^{83} -3.12966 q^{85} +11.2628 q^{87} +9.14799 q^{89} +22.6162 q^{91} -13.7375 q^{93} +14.8175 q^{95} -2.28860 q^{97} +9.72861 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	−2.78964	−1.61060	−0.805299	−	0.592869i	$-0.797995\pi$
$3$	−2.78964	−1.61060	−0.805299	+	0.592869i	$0.797995\pi$
$4$	0	0
$5$	3.12966	1.39963	0.699814	−	0.714326i	$-0.253266\pi$
$5$	3.12966	1.39963	0.699814	+	0.714326i	$0.253266\pi$
$6$	0	0
$7$	4.93262	1.86435	0.932177	−	0.362003i	$-0.117907\pi$
$7$	4.93262	1.86435	0.932177	+	0.362003i	$0.117907\pi$
$8$	0	0
$9$	4.78208	1.59403
$10$	0	0
$11$	2.03439	0.613392	0.306696	−	0.951808i	$-0.400777\pi$
$11$	2.03439	0.613392	0.306696	+	0.951808i	$0.400777\pi$
$12$	0	0
$13$	4.58503	1.27166	0.635829	−	0.771830i	$-0.280658\pi$
$13$	4.58503	1.27166	0.635829	+	0.771830i	$0.280658\pi$
$14$	0	0
$15$	−8.73062	−2.25424
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.73453	1.08618	0.543088	−	0.839676i	$-0.317255\pi$
$19$	4.73453	1.08618	0.543088	+	0.839676i	$0.317255\pi$
$20$	0	0
$21$	−13.7602	−3.00272
$22$	0	0
$23$	−1.83954	−0.383571	−0.191786	−	0.981437i	$-0.561428\pi$
$23$	−1.83954	−0.383571	−0.191786	+	0.981437i	$0.561428\pi$
$24$	0	0
$25$	4.79478	0.958956
$26$	0	0
$27$	−4.97134	−0.956736
$28$	0	0
$29$	−4.03737	−0.749720	−0.374860	−	0.927081i	$-0.622309\pi$
$29$	−4.03737	−0.749720	−0.374860	+	0.927081i	$0.622309\pi$
$30$	0	0
$31$	4.92448	0.884463	0.442232	−	0.896901i	$-0.354187\pi$
$31$	4.92448	0.884463	0.442232	+	0.896901i	$0.354187\pi$
$32$	0	0
$33$	−5.67521	−0.987928
$34$	0	0
$35$	15.4374	2.60940
$36$	0	0
$37$	−10.0209	−1.64743	−0.823713	−	0.567007i	$-0.808101\pi$
$37$	−10.0209	−1.64743	−0.823713	+	0.567007i	$0.808101\pi$
$38$	0	0
$39$	−12.7906	−2.04813
$40$	0	0
$41$	−2.17449	−0.339599	−0.169799	−	0.985479i	$-0.554312\pi$
$41$	−2.17449	−0.339599	−0.169799	+	0.985479i	$0.554312\pi$
$42$	0	0
$43$	6.41294	0.977964	0.488982	−	0.872294i	$-0.337368\pi$
$43$	6.41294	0.977964	0.488982	+	0.872294i	$0.337368\pi$
$44$	0	0
$45$	14.9663	2.23104
$46$	0	0
$47$	12.5461	1.83004	0.915020	−	0.403408i	$-0.132175\pi$
$47$	12.5461	1.83004	0.915020	+	0.403408i	$0.132175\pi$
$48$	0	0
$49$	17.3307	2.47582
$50$	0	0
$51$	2.78964	0.390627
$52$	0	0
$53$	10.9390	1.50258	0.751291	−	0.659971i	$-0.229432\pi$
$53$	10.9390	1.50258	0.751291	+	0.659971i	$0.229432\pi$
$54$	0	0
$55$	6.36695	0.858520
$56$	0	0
$57$	−13.2076	−1.74939
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−8.91219	−1.14109	−0.570544	−	0.821267i	$-0.693268\pi$
$61$	−8.91219	−1.14109	−0.570544	+	0.821267i	$0.693268\pi$
$62$	0	0
$63$	23.5881	2.97183
$64$	0	0
$65$	14.3496	1.77985
$66$	0	0
$67$	10.4601	1.27790	0.638951	−	0.769247i	$-0.279369\pi$
$67$	10.4601	1.27790	0.638951	+	0.769247i	$0.279369\pi$
$68$	0	0
$69$	5.13166	0.617779
$70$	0	0
$71$	5.53947	0.657414	0.328707	−	0.944432i	$-0.393387\pi$
$71$	5.53947	0.657414	0.328707	+	0.944432i	$0.393387\pi$
$72$	0	0
$73$	−9.35947	−1.09544	−0.547722	−	0.836661i	$-0.684505\pi$
$73$	−9.35947	−1.09544	−0.547722	+	0.836661i	$0.684505\pi$
$74$	0	0
$75$	−13.3757	−1.54449
$76$	0	0
$77$	10.0349	1.14358
$78$	0	0
$79$	−12.5391	−1.41076	−0.705379	−	0.708830i	$-0.749223\pi$
$79$	−12.5391	−1.41076	−0.705379	+	0.708830i	$0.749223\pi$
$80$	0	0
$81$	−0.477980	−0.0531089
$82$	0	0
$83$	−14.7473	−1.61873	−0.809365	−	0.587305i	$-0.800189\pi$
$83$	−14.7473	−1.61873	−0.809365	+	0.587305i	$0.800189\pi$
$84$	0	0
$85$	−3.12966	−0.339459
$86$	0	0
$87$	11.2628	1.20750
$88$	0	0
$89$	9.14799	0.969685	0.484842	−	0.874602i	$-0.338877\pi$
$89$	9.14799	0.969685	0.484842	+	0.874602i	$0.338877\pi$
$90$	0	0
$91$	22.6162	2.37082
$92$	0	0
$93$	−13.7375	−1.42451
$94$	0	0
$95$	14.8175	1.52024
$96$	0	0
$97$	−2.28860	−0.232372	−0.116186	−	0.993227i	$-0.537067\pi$
$97$	−2.28860	−0.232372	−0.116186	+	0.993227i	$0.537067\pi$
$98$	0	0
$99$	9.72861	0.977762
$100$	0	0
$101$	12.9401	1.28759	0.643794	−	0.765199i	$-0.277359\pi$
$101$	12.9401	1.28759	0.643794	+	0.765199i	$0.277359\pi$
$102$	0	0
$103$	13.1607	1.29676	0.648382	−	0.761315i	$-0.275446\pi$
$103$	13.1607	1.29676	0.648382	+	0.761315i	$0.275446\pi$
$104$	0	0
$105$	−43.0648	−4.20269
$106$	0	0
$107$	2.93045	0.283297	0.141649	−	0.989917i	$-0.454760\pi$
$107$	2.93045	0.283297	0.141649	+	0.989917i	$0.454760\pi$
$108$	0	0
$109$	16.8264	1.61168	0.805838	−	0.592136i	$-0.201715\pi$
$109$	16.8264	1.61168	0.805838	+	0.592136i	$0.201715\pi$
$110$	0	0
$111$	27.9547	2.65334
$112$	0	0
$113$	−15.3673	−1.44563	−0.722816	−	0.691040i	$-0.757153\pi$
$113$	−15.3673	−1.44563	−0.722816	+	0.691040i	$0.757153\pi$
$114$	0	0
$115$	−5.75714	−0.536856
$116$	0	0
$117$	21.9260	2.02706
$118$	0	0
$119$	−4.93262	−0.452172
$120$	0	0
$121$	−6.86125	−0.623750
$122$	0	0
$123$	6.06605	0.546957
$124$	0	0
$125$	−0.642265	−0.0574459
$126$	0	0
$127$	12.5597	1.11450	0.557248	−	0.830346i	$-0.311857\pi$
$127$	12.5597	1.11450	0.557248	+	0.830346i	$0.311857\pi$
$128$	0	0
$129$	−17.8898	−1.57511
$130$	0	0
$131$	−19.3890	−1.69403	−0.847014	−	0.531571i	$-0.821602\pi$
$131$	−19.3890	−1.69403	−0.847014	+	0.531571i	$0.821602\pi$
$132$	0	0
$133$	23.3536	2.02502
$134$	0	0
$135$	−15.5586	−1.33907
$136$	0	0
$137$	−23.0227	−1.96697	−0.983483	−	0.181000i	$-0.942067\pi$
$137$	−23.0227	−1.96697	−0.983483	+	0.181000i	$0.942067\pi$
$138$	0	0
$139$	−6.09361	−0.516853	−0.258427	−	0.966031i	$-0.583204\pi$
$139$	−6.09361	−0.516853	−0.258427	+	0.966031i	$0.583204\pi$
$140$	0	0
$141$	−34.9991	−2.94746
$142$	0	0
$143$	9.32775	0.780025
$144$	0	0
$145$	−12.6356	−1.04933
$146$	0	0
$147$	−48.3464	−3.98754
$148$	0	0
$149$	−0.465610	−0.0381442	−0.0190721	−	0.999818i	$-0.506071\pi$
$149$	−0.465610	−0.0381442	−0.0190721	+	0.999818i	$0.506071\pi$
$150$	0	0
$151$	−14.5739	−1.18601	−0.593003	−	0.805201i	$-0.702058\pi$
$151$	−14.5739	−1.18601	−0.593003	+	0.805201i	$0.702058\pi$
$152$	0	0
$153$	−4.78208	−0.386608
$154$	0	0
$155$	15.4120	1.23792
$156$	0	0
$157$	−4.68104	−0.373588	−0.186794	−	0.982399i	$-0.559810\pi$
$157$	−4.68104	−0.373588	−0.186794	+	0.982399i	$0.559810\pi$
$158$	0	0
$159$	−30.5157	−2.42005
$160$	0	0
$161$	−9.07376	−0.715112
$162$	0	0
$163$	8.70795	0.682059	0.341030	−	0.940053i	$-0.389224\pi$
$163$	8.70795	0.682059	0.341030	+	0.940053i	$0.389224\pi$
$164$	0	0
$165$	−17.7615	−1.38273
$166$	0	0
$167$	−3.77557	−0.292162	−0.146081	−	0.989273i	$-0.546666\pi$
$167$	−3.77557	−0.292162	−0.146081	+	0.989273i	$0.546666\pi$
$168$	0	0
$169$	8.02251	0.617116
$170$	0	0
$171$	22.6409	1.73139
$172$	0	0
$173$	−11.5253	−0.876249	−0.438124	−	0.898914i	$-0.644357\pi$
$173$	−11.5253	−0.876249	−0.438124	+	0.898914i	$0.644357\pi$
$174$	0	0
$175$	23.6508	1.78783
$176$	0	0
$177$	−2.78964	−0.209682
$178$	0	0
$179$	15.2243	1.13792	0.568960	−	0.822365i	$-0.307346\pi$
$179$	15.2243	1.13792	0.568960	+	0.822365i	$0.307346\pi$
$180$	0	0
$181$	−4.04469	−0.300639	−0.150320	−	0.988637i	$-0.548030\pi$
$181$	−4.04469	−0.300639	−0.150320	+	0.988637i	$0.548030\pi$
$182$	0	0
$183$	24.8618	1.83784
$184$	0	0
$185$	−31.3620	−2.30578
$186$	0	0
$187$	−2.03439	−0.148769
$188$	0	0
$189$	−24.5217	−1.78369
$190$	0	0
$191$	−9.80262	−0.709292	−0.354646	−	0.935001i	$-0.615399\pi$
$191$	−9.80262	−0.709292	−0.354646	+	0.935001i	$0.615399\pi$
$192$	0	0
$193$	13.5761	0.977227	0.488613	−	0.872500i	$-0.337503\pi$
$193$	13.5761	0.977227	0.488613	+	0.872500i	$0.337503\pi$
$194$	0	0
$195$	−40.0302	−2.86662
$196$	0	0
$197$	15.9120	1.13369	0.566843	−	0.823826i	$-0.308165\pi$
$197$	15.9120	1.13369	0.566843	+	0.823826i	$0.308165\pi$
$198$	0	0
$199$	−20.8813	−1.48023	−0.740117	−	0.672478i	$-0.765230\pi$
$199$	−20.8813	−1.48023	−0.740117	+	0.672478i	$0.765230\pi$
$200$	0	0
$201$	−29.1798	−2.05819
$202$	0	0
$203$	−19.9148	−1.39774
$204$	0	0
$205$	−6.80543	−0.475312
$206$	0	0
$207$	−8.79683	−0.611422
$208$	0	0
$209$	9.63188	0.666251
$210$	0	0
$211$	22.9109	1.57725	0.788625	−	0.614874i	$-0.210793\pi$
$211$	22.9109	1.57725	0.788625	+	0.614874i	$0.210793\pi$
$212$	0	0
$213$	−15.4531	−1.05883
$214$	0	0
$215$	20.0703	1.36879
$216$	0	0
$217$	24.2906	1.64895
$218$	0	0
$219$	26.1095	1.76432
$220$	0	0
$221$	−4.58503	−0.308423
$222$	0	0
$223$	2.28282	0.152869	0.0764344	−	0.997075i	$-0.475646\pi$
$223$	2.28282	0.152869	0.0764344	+	0.997075i	$0.475646\pi$
$224$	0	0
$225$	22.9290	1.52860
$226$	0	0
$227$	−19.4951	−1.29394	−0.646969	−	0.762516i	$-0.723964\pi$
$227$	−19.4951	−1.29394	−0.646969	+	0.762516i	$0.723964\pi$
$228$	0	0
$229$	−23.0429	−1.52272	−0.761360	−	0.648329i	$-0.775468\pi$
$229$	−23.0429	−1.52272	−0.761360	+	0.648329i	$0.775468\pi$
$230$	0	0
$231$	−27.9936	−1.84185
$232$	0	0
$233$	−12.4996	−0.818874	−0.409437	−	0.912338i	$-0.634275\pi$
$233$	−12.4996	−0.818874	−0.409437	+	0.912338i	$0.634275\pi$
$234$	0	0
$235$	39.2651	2.56137
$236$	0	0
$237$	34.9795	2.27217
$238$	0	0
$239$	−5.39702	−0.349104	−0.174552	−	0.984648i	$-0.555848\pi$
$239$	−5.39702	−0.349104	−0.174552	+	0.984648i	$0.555848\pi$
$240$	0	0
$241$	18.0726	1.16416	0.582081	−	0.813131i	$-0.302239\pi$
$241$	18.0726	1.16416	0.582081	+	0.813131i	$0.302239\pi$
$242$	0	0
$243$	16.2474	1.04227
$244$	0	0
$245$	54.2393	3.46522
$246$	0	0
$247$	21.7080	1.38124
$248$	0	0
$249$	41.1397	2.60712
$250$	0	0
$251$	−28.5200	−1.80017	−0.900084	−	0.435717i	$-0.856495\pi$
$251$	−28.5200	−1.80017	−0.900084	+	0.435717i	$0.856495\pi$
$252$	0	0
$253$	−3.74235	−0.235279
$254$	0	0
$255$	8.73062	0.546733
$256$	0	0
$257$	−16.2078	−1.01102	−0.505509	−	0.862822i	$-0.668695\pi$
$257$	−16.2078	−1.01102	−0.505509	+	0.862822i	$0.668695\pi$
$258$	0	0
$259$	−49.4293	−3.07139
$260$	0	0
$261$	−19.3070	−1.19507
$262$	0	0
$263$	−23.8234	−1.46901	−0.734505	−	0.678603i	$-0.762586\pi$
$263$	−23.8234	−1.46901	−0.734505	+	0.678603i	$0.762586\pi$
$264$	0	0
$265$	34.2352	2.10305
$266$	0	0
$267$	−25.5196	−1.56177
$268$	0	0
$269$	10.3583	0.631557	0.315779	−	0.948833i	$-0.397734\pi$
$269$	10.3583	0.631557	0.315779	+	0.948833i	$0.397734\pi$
$270$	0	0
$271$	−10.1337	−0.615581	−0.307791	−	0.951454i	$-0.599590\pi$
$271$	−10.1337	−0.615581	−0.307791	+	0.951454i	$0.599590\pi$
$272$	0	0
$273$	−63.0910	−3.81844
$274$	0	0
$275$	9.75446	0.588216
$276$	0	0
$277$	−25.8687	−1.55430	−0.777150	−	0.629315i	$-0.783336\pi$
$277$	−25.8687	−1.55430	−0.777150	+	0.629315i	$0.783336\pi$
$278$	0	0
$279$	23.5492	1.40986
$280$	0	0
$281$	6.58234	0.392670	0.196335	−	0.980537i	$-0.437096\pi$
$281$	6.58234	0.392670	0.196335	+	0.980537i	$0.437096\pi$
$282$	0	0
$283$	5.12678	0.304755	0.152378	−	0.988322i	$-0.451307\pi$
$283$	5.12678	0.304755	0.152378	+	0.988322i	$0.451307\pi$
$284$	0	0
$285$	−41.3354	−2.44850
$286$	0	0
$287$	−10.7259	−0.633132
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.38437	0.374259
$292$	0	0
$293$	−29.1252	−1.70151	−0.850756	−	0.525560i	$-0.823856\pi$
$293$	−29.1252	−1.70151	−0.850756	+	0.525560i	$0.823856\pi$
$294$	0	0
$295$	3.12966	0.182216
$296$	0	0
$297$	−10.1137	−0.586854
$298$	0	0
$299$	−8.43436	−0.487772
$300$	0	0
$301$	31.6326	1.82327
$302$	0	0
$303$	−36.0982	−2.07379
$304$	0	0
$305$	−27.8921	−1.59710
$306$	0	0
$307$	−1.25812	−0.0718048	−0.0359024	−	0.999355i	$-0.511431\pi$
$307$	−1.25812	−0.0718048	−0.0359024	+	0.999355i	$0.511431\pi$
$308$	0	0
$309$	−36.7136	−2.08856
$310$	0	0
$311$	−1.10715	−0.0627810	−0.0313905	−	0.999507i	$-0.509994\pi$
$311$	−1.10715	−0.0627810	−0.0313905	+	0.999507i	$0.509994\pi$
$312$	0	0
$313$	28.9772	1.63789	0.818943	−	0.573874i	$-0.194560\pi$
$313$	28.9772	1.63789	0.818943	+	0.573874i	$0.194560\pi$
$314$	0	0
$315$	73.8229	4.15945
$316$	0	0
$317$	−1.70294	−0.0956465	−0.0478232	−	0.998856i	$-0.515228\pi$
$317$	−1.70294	−0.0956465	−0.0478232	+	0.998856i	$0.515228\pi$
$318$	0	0
$319$	−8.21358	−0.459872
$320$	0	0
$321$	−8.17489	−0.456278
$322$	0	0
$323$	−4.73453	−0.263436
$324$	0	0
$325$	21.9842	1.21947
$326$	0	0
$327$	−46.9395	−2.59576
$328$	0	0
$329$	61.8852	3.41184
$330$	0	0
$331$	−2.09441	−0.115119	−0.0575596	−	0.998342i	$-0.518332\pi$
$331$	−2.09441	−0.115119	−0.0575596	+	0.998342i	$0.518332\pi$
$332$	0	0
$333$	−47.9207	−2.62604
$334$	0	0
$335$	32.7365	1.78859
$336$	0	0
$337$	−24.3314	−1.32542	−0.662709	−	0.748877i	$-0.730593\pi$
$337$	−24.3314	−1.32542	−0.662709	+	0.748877i	$0.730593\pi$
$338$	0	0
$339$	42.8691	2.32833
$340$	0	0
$341$	10.0183	0.542522
$342$	0	0
$343$	50.9575	2.75144
$344$	0	0
$345$	16.0603	0.864660
$346$	0	0
$347$	−1.96828	−0.105663	−0.0528315	−	0.998603i	$-0.516825\pi$
$347$	−1.96828	−0.105663	−0.0528315	+	0.998603i	$0.516825\pi$
$348$	0	0
$349$	28.7496	1.53893	0.769466	−	0.638687i	$-0.220522\pi$
$349$	28.7496	1.53893	0.769466	+	0.638687i	$0.220522\pi$
$350$	0	0
$351$	−22.7938	−1.21664
$352$	0	0
$353$	−11.3751	−0.605434	−0.302717	−	0.953080i	$-0.597894\pi$
$353$	−11.3751	−0.605434	−0.302717	+	0.953080i	$0.597894\pi$
$354$	0	0
$355$	17.3367	0.920135
$356$	0	0
$357$	13.7602	0.728268
$358$	0	0
$359$	33.2805	1.75648	0.878239	−	0.478222i	$-0.158719\pi$
$359$	33.2805	1.75648	0.878239	+	0.478222i	$0.158719\pi$
$360$	0	0
$361$	3.41576	0.179777
$362$	0	0
$363$	19.1404	1.00461
$364$	0	0
$365$	−29.2920	−1.53321
$366$	0	0
$367$	−25.0766	−1.30899	−0.654495	−	0.756067i	$-0.727119\pi$
$367$	−25.0766	−1.30899	−0.654495	+	0.756067i	$0.727119\pi$
$368$	0	0
$369$	−10.3986	−0.541329
$370$	0	0
$371$	53.9577	2.80134
$372$	0	0
$373$	18.7248	0.969535	0.484768	−	0.874643i	$-0.338904\pi$
$373$	18.7248	0.969535	0.484768	+	0.874643i	$0.338904\pi$
$374$	0	0
$375$	1.79169	0.0925223
$376$	0	0
$377$	−18.5115	−0.953388
$378$	0	0
$379$	11.1264	0.571526	0.285763	−	0.958300i	$-0.407753\pi$
$379$	11.1264	0.571526	0.285763	+	0.958300i	$0.407753\pi$
$380$	0	0
$381$	−35.0371	−1.79500
$382$	0	0
$383$	−4.32756	−0.221128	−0.110564	−	0.993869i	$-0.535266\pi$
$383$	−4.32756	−0.221128	−0.110564	+	0.993869i	$0.535266\pi$
$384$	0	0
$385$	31.4058	1.60059
$386$	0	0
$387$	30.6672	1.55890
$388$	0	0
$389$	37.7833	1.91569	0.957844	−	0.287289i	$-0.0927541\pi$
$389$	37.7833	1.91569	0.957844	+	0.287289i	$0.0927541\pi$
$390$	0	0
$391$	1.83954	0.0930296
$392$	0	0
$393$	54.0883	2.72840
$394$	0	0
$395$	−39.2431	−1.97454
$396$	0	0
$397$	−7.60377	−0.381622	−0.190811	−	0.981627i	$-0.561112\pi$
$397$	−7.60377	−0.381622	−0.190811	+	0.981627i	$0.561112\pi$
$398$	0	0
$399$	−65.1481	−3.26149
$400$	0	0
$401$	−2.67470	−0.133568	−0.0667841	−	0.997767i	$-0.521274\pi$
$401$	−2.67470	−0.133568	−0.0667841	+	0.997767i	$0.521274\pi$
$402$	0	0
$403$	22.5789	1.12474
$404$	0	0
$405$	−1.49592	−0.0743327
$406$	0	0
$407$	−20.3864	−1.01052
$408$	0	0
$409$	−13.6240	−0.673665	−0.336833	−	0.941565i	$-0.609356\pi$
$409$	−13.6240	−0.673665	−0.336833	+	0.941565i	$0.609356\pi$
$410$	0	0
$411$	64.2251	3.16799
$412$	0	0
$413$	4.93262	0.242718
$414$	0	0
$415$	−46.1542	−2.26562
$416$	0	0
$417$	16.9990	0.832443
$418$	0	0
$419$	−30.9632	−1.51265	−0.756325	−	0.654196i	$-0.773007\pi$
$419$	−30.9632	−1.51265	−0.756325	+	0.654196i	$0.773007\pi$
$420$	0	0
$421$	−23.2951	−1.13534	−0.567668	−	0.823258i	$-0.692154\pi$
$421$	−23.2951	−1.13534	−0.567668	+	0.823258i	$0.692154\pi$
$422$	0	0
$423$	59.9965	2.91713
$424$	0	0
$425$	−4.79478	−0.232581
$426$	0	0
$427$	−43.9604	−2.12739
$428$	0	0
$429$	−26.0210	−1.25631
$430$	0	0
$431$	−29.0062	−1.39718	−0.698591	−	0.715522i	$-0.746189\pi$
$431$	−29.0062	−1.39718	−0.698591	+	0.715522i	$0.746189\pi$
$432$	0	0
$433$	11.1266	0.534711	0.267355	−	0.963598i	$-0.413850\pi$
$433$	11.1266	0.534711	0.267355	+	0.963598i	$0.413850\pi$
$434$	0	0
$435$	35.2487	1.69005
$436$	0	0
$437$	−8.70936	−0.416625
$438$	0	0
$439$	−7.51477	−0.358661	−0.179330	−	0.983789i	$-0.557393\pi$
$439$	−7.51477	−0.358661	−0.179330	+	0.983789i	$0.557393\pi$
$440$	0	0
$441$	82.8768	3.94651
$442$	0	0
$443$	10.3839	0.493352	0.246676	−	0.969098i	$-0.420662\pi$
$443$	10.3839	0.493352	0.246676	+	0.969098i	$0.420662\pi$
$444$	0	0
$445$	28.6301	1.35720
$446$	0	0
$447$	1.29888	0.0614350
$448$	0	0
$449$	−8.48884	−0.400613	−0.200306	−	0.979733i	$-0.564194\pi$
$449$	−8.48884	−0.400613	−0.200306	+	0.979733i	$0.564194\pi$
$450$	0	0
$451$	−4.42377	−0.208307
$452$	0	0
$453$	40.6558	1.91018
$454$	0	0
$455$	70.7811	3.31827
$456$	0	0
$457$	−22.8954	−1.07100	−0.535501	−	0.844535i	$-0.679877\pi$
$457$	−22.8954	−1.07100	−0.535501	+	0.844535i	$0.679877\pi$
$458$	0	0
$459$	4.97134	0.232042
$460$	0	0
$461$	23.5799	1.09823	0.549114	−	0.835748i	$-0.314965\pi$
$461$	23.5799	1.09823	0.549114	+	0.835748i	$0.314965\pi$
$462$	0	0
$463$	23.9814	1.11451	0.557255	−	0.830341i	$-0.311854\pi$
$463$	23.9814	1.11451	0.557255	+	0.830341i	$0.311854\pi$
$464$	0	0
$465$	−42.9938	−1.99379
$466$	0	0
$467$	29.4829	1.36431	0.682154	−	0.731208i	$-0.261043\pi$
$467$	29.4829	1.36431	0.682154	+	0.731208i	$0.261043\pi$
$468$	0	0
$469$	51.5956	2.38246
$470$	0	0
$471$	13.0584	0.601700
$472$	0	0
$473$	13.0464	0.599875
$474$	0	0
$475$	22.7010	1.04159
$476$	0	0
$477$	52.3109	2.39515
$478$	0	0
$479$	−18.1482	−0.829210	−0.414605	−	0.910001i	$-0.636080\pi$
$479$	−18.1482	−0.829210	−0.414605	+	0.910001i	$0.636080\pi$
$480$	0	0
$481$	−45.9462	−2.09496
$482$	0	0
$483$	25.3125	1.15176
$484$	0	0
$485$	−7.16255	−0.325235
$486$	0	0
$487$	27.2318	1.23399	0.616995	−	0.786967i	$-0.288350\pi$
$487$	27.2318	1.23399	0.616995	+	0.786967i	$0.288350\pi$
$488$	0	0
$489$	−24.2920	−1.09852
$490$	0	0
$491$	19.4054	0.875751	0.437876	−	0.899036i	$-0.355731\pi$
$491$	19.4054	0.875751	0.437876	+	0.899036i	$0.355731\pi$
$492$	0	0
$493$	4.03737	0.181834
$494$	0	0
$495$	30.4473	1.36850
$496$	0	0
$497$	27.3241	1.22565
$498$	0	0
$499$	−11.4554	−0.512812	−0.256406	−	0.966569i	$-0.582538\pi$
$499$	−11.4554	−0.512812	−0.256406	+	0.966569i	$0.582538\pi$
$500$	0	0
$501$	10.5325	0.470556
$502$	0	0
$503$	0.101611	0.00453062	0.00226531	−	0.999997i	$-0.499279\pi$
$503$	0.101611	0.00453062	0.00226531	+	0.999997i	$0.499279\pi$
$504$	0	0
$505$	40.4981	1.80214
$506$	0	0
$507$	−22.3799	−0.993926
$508$	0	0
$509$	29.3581	1.30127	0.650637	−	0.759389i	$-0.274502\pi$
$509$	29.3581	1.30127	0.650637	+	0.759389i	$0.274502\pi$
$510$	0	0
$511$	−46.1667	−2.04229
$512$	0	0
$513$	−23.5370	−1.03918
$514$	0	0
$515$	41.1886	1.81499
$516$	0	0
$517$	25.5237	1.12253
$518$	0	0
$519$	32.1513	1.41128
$520$	0	0
$521$	0.529446	0.0231955	0.0115977	−	0.999933i	$-0.496308\pi$
$521$	0.529446	0.0231955	0.0115977	+	0.999933i	$0.496308\pi$
$522$	0	0
$523$	7.83028	0.342394	0.171197	−	0.985237i	$-0.445236\pi$
$523$	7.83028	0.342394	0.171197	+	0.985237i	$0.445236\pi$
$524$	0	0
$525$	−65.9772	−2.87948
$526$	0	0
$527$	−4.92448	−0.214514
$528$	0	0
$529$	−19.6161	−0.852873
$530$	0	0
$531$	4.78208	0.207524
$532$	0	0
$533$	−9.97012	−0.431854
$534$	0	0
$535$	9.17131	0.396510
$536$	0	0
$537$	−42.4704	−1.83273
$538$	0	0
$539$	35.2574	1.51865
$540$	0	0
$541$	−4.43588	−0.190713	−0.0953566	−	0.995443i	$-0.530399\pi$
$541$	−4.43588	−0.190713	−0.0953566	+	0.995443i	$0.530399\pi$
$542$	0	0
$543$	11.2832	0.484209
$544$	0	0
$545$	52.6609	2.25575
$546$	0	0
$547$	−8.73022	−0.373277	−0.186639	−	0.982429i	$-0.559759\pi$
$547$	−8.73022	−0.373277	−0.186639	+	0.982429i	$0.559759\pi$
$548$	0	0
$549$	−42.6188	−1.81892
$550$	0	0
$551$	−19.1150	−0.814328
$552$	0	0
$553$	−61.8506	−2.63015
$554$	0	0
$555$	87.4887	3.71369
$556$	0	0
$557$	5.05912	0.214362	0.107181	−	0.994240i	$-0.465818\pi$
$557$	5.05912	0.214362	0.107181	+	0.994240i	$0.465818\pi$
$558$	0	0
$559$	29.4035	1.24364
$560$	0	0
$561$	5.67521	0.239608
$562$	0	0
$563$	−0.0556546	−0.00234556	−0.00117278	−	0.999999i	$-0.500373\pi$
$563$	−0.0556546	−0.00234556	−0.00117278	+	0.999999i	$0.500373\pi$
$564$	0	0
$565$	−48.0944	−2.02335
$566$	0	0
$567$	−2.35769	−0.0990138
$568$	0	0
$569$	−32.7325	−1.37222	−0.686110	−	0.727498i	$-0.740683\pi$
$569$	−32.7325	−1.37222	−0.686110	+	0.727498i	$0.740683\pi$
$570$	0	0
$571$	19.3180	0.808433	0.404216	−	0.914663i	$-0.367544\pi$
$571$	19.3180	0.808433	0.404216	+	0.914663i	$0.367544\pi$
$572$	0	0
$573$	27.3457	1.14238
$574$	0	0
$575$	−8.82020	−0.367828
$576$	0	0
$577$	23.6716	0.985460	0.492730	−	0.870182i	$-0.335999\pi$
$577$	23.6716	0.985460	0.492730	+	0.870182i	$0.335999\pi$
$578$	0	0
$579$	−37.8723	−1.57392
$580$	0	0
$581$	−72.7430	−3.01789
$582$	0	0
$583$	22.2541	0.921672
$584$	0	0
$585$	68.6209	2.83712
$586$	0	0
$587$	4.57889	0.188991	0.0944954	−	0.995525i	$-0.469876\pi$
$587$	4.57889	0.188991	0.0944954	+	0.995525i	$0.469876\pi$
$588$	0	0
$589$	23.3151	0.960682
$590$	0	0
$591$	−44.3888	−1.82591
$592$	0	0
$593$	−19.6414	−0.806577	−0.403288	−	0.915073i	$-0.632133\pi$
$593$	−19.6414	−0.806577	−0.403288	+	0.915073i	$0.632133\pi$
$594$	0	0
$595$	−15.4374	−0.632873
$596$	0	0
$597$	58.2511	2.38406
$598$	0	0
$599$	−22.8452	−0.933430	−0.466715	−	0.884408i	$-0.654563\pi$
$599$	−22.8452	−0.933430	−0.466715	+	0.884408i	$0.654563\pi$
$600$	0	0
$601$	−12.2336	−0.499018	−0.249509	−	0.968372i	$-0.580269\pi$
$601$	−12.2336	−0.499018	−0.249509	+	0.968372i	$0.580269\pi$
$602$	0	0
$603$	50.0209	2.03701
$604$	0	0
$605$	−21.4734	−0.873018
$606$	0	0
$607$	4.07922	0.165570	0.0827852	−	0.996567i	$-0.473618\pi$
$607$	4.07922	0.165570	0.0827852	+	0.996567i	$0.473618\pi$
$608$	0	0
$609$	55.5550	2.25120
$610$	0	0
$611$	57.5244	2.32719
$612$	0	0
$613$	−21.0032	−0.848313	−0.424156	−	0.905589i	$-0.639429\pi$
$613$	−21.0032	−0.848313	−0.424156	+	0.905589i	$0.639429\pi$
$614$	0	0
$615$	18.9847	0.765536
$616$	0	0
$617$	−22.3253	−0.898783	−0.449391	−	0.893335i	$-0.648359\pi$
$617$	−22.3253	−0.898783	−0.449391	+	0.893335i	$0.648359\pi$
$618$	0	0
$619$	6.94477	0.279134	0.139567	−	0.990213i	$-0.455429\pi$
$619$	6.94477	0.279134	0.139567	+	0.990213i	$0.455429\pi$
$620$	0	0
$621$	9.14500	0.366976
$622$	0	0
$623$	45.1235	1.80784
$624$	0	0
$625$	−25.9840	−1.03936
$626$	0	0
$627$	−26.8695	−1.07306
$628$	0	0
$629$	10.0209	0.399560
$630$	0	0
$631$	44.0031	1.75174	0.875868	−	0.482550i	$-0.160289\pi$
$631$	44.0031	1.75174	0.875868	+	0.482550i	$0.160289\pi$
$632$	0	0
$633$	−63.9130	−2.54032
$634$	0	0
$635$	39.3077	1.55988
$636$	0	0
$637$	79.4619	3.14839
$638$	0	0
$639$	26.4902	1.04794
$640$	0	0
$641$	11.9194	0.470787	0.235394	−	0.971900i	$-0.424362\pi$
$641$	11.9194	0.470787	0.235394	+	0.971900i	$0.424362\pi$
$642$	0	0
$643$	−5.50875	−0.217244	−0.108622	−	0.994083i	$-0.534644\pi$
$643$	−5.50875	−0.217244	−0.108622	+	0.994083i	$0.534644\pi$
$644$	0	0
$645$	−55.9889	−2.20456
$646$	0	0
$647$	−5.41042	−0.212706	−0.106353	−	0.994328i	$-0.533917\pi$
$647$	−5.41042	−0.212706	−0.106353	+	0.994328i	$0.533917\pi$
$648$	0	0
$649$	2.03439	0.0798568
$650$	0	0
$651$	−67.7619	−2.65580
$652$	0	0
$653$	−8.48444	−0.332022	−0.166011	−	0.986124i	$-0.553089\pi$
$653$	−8.48444	−0.332022	−0.166011	+	0.986124i	$0.553089\pi$
$654$	0	0
$655$	−60.6811	−2.37101
$656$	0	0
$657$	−44.7577	−1.74616
$658$	0	0
$659$	10.1258	0.394445	0.197222	−	0.980359i	$-0.436808\pi$
$659$	10.1258	0.394445	0.197222	+	0.980359i	$0.436808\pi$
$660$	0	0
$661$	−13.3636	−0.519782	−0.259891	−	0.965638i	$-0.583687\pi$
$661$	−13.3636	−0.519782	−0.259891	+	0.965638i	$0.583687\pi$
$662$	0	0
$663$	12.7906	0.496745
$664$	0	0
$665$	73.0889	2.83427
$666$	0	0
$667$	7.42691	0.287571
$668$	0	0
$669$	−6.36823	−0.246210
$670$	0	0
$671$	−18.1309	−0.699935
$672$	0	0
$673$	43.9042	1.69238	0.846192	−	0.532878i	$-0.178890\pi$
$673$	43.9042	1.69238	0.846192	+	0.532878i	$0.178890\pi$
$674$	0	0
$675$	−23.8365	−0.917468
$676$	0	0
$677$	−13.4145	−0.515562	−0.257781	−	0.966203i	$-0.582991\pi$
$677$	−13.4145	−0.515562	−0.257781	+	0.966203i	$0.582991\pi$
$678$	0	0
$679$	−11.2888	−0.433225
$680$	0	0
$681$	54.3844	2.08401
$682$	0	0
$683$	44.4393	1.70042	0.850212	−	0.526441i	$-0.176474\pi$
$683$	44.4393	1.70042	0.850212	+	0.526441i	$0.176474\pi$
$684$	0	0
$685$	−72.0534	−2.75302
$686$	0	0
$687$	64.2814	2.45249
$688$	0	0
$689$	50.1555	1.91077
$690$	0	0
$691$	25.7250	0.978625	0.489312	−	0.872109i	$-0.337248\pi$
$691$	25.7250	0.978625	0.489312	+	0.872109i	$0.337248\pi$
$692$	0	0
$693$	47.9875	1.82289
$694$	0	0
$695$	−19.0709	−0.723402
$696$	0	0
$697$	2.17449	0.0823648
$698$	0	0
$699$	34.8692	1.31888
$700$	0	0
$701$	21.4428	0.809883	0.404942	−	0.914343i	$-0.367292\pi$
$701$	21.4428	0.809883	0.404942	+	0.914343i	$0.367292\pi$
$702$	0	0
$703$	−47.4443	−1.78939
$704$	0	0
$705$	−109.535	−4.12534
$706$	0	0
$707$	63.8286	2.40052
$708$	0	0
$709$	−30.2167	−1.13481	−0.567406	−	0.823438i	$-0.692053\pi$
$709$	−30.2167	−1.13481	−0.567406	+	0.823438i	$0.692053\pi$
$710$	0	0
$711$	−59.9629	−2.24879
$712$	0	0
$713$	−9.05879	−0.339254
$714$	0	0
$715$	29.1927	1.09174
$716$	0	0
$717$	15.0557	0.562266
$718$	0	0
$719$	−19.9399	−0.743632	−0.371816	−	0.928306i	$-0.621265\pi$
$719$	−19.9399	−0.743632	−0.371816	+	0.928306i	$0.621265\pi$
$720$	0	0
$721$	64.9168	2.41763
$722$	0	0
$723$	−50.4161	−1.87500
$724$	0	0
$725$	−19.3583	−0.718949
$726$	0	0
$727$	7.59634	0.281733	0.140866	−	0.990029i	$-0.455011\pi$
$727$	7.59634	0.281733	0.140866	+	0.990029i	$0.455011\pi$
$728$	0	0
$729$	−43.8905	−1.62557
$730$	0	0
$731$	−6.41294	−0.237191
$732$	0	0
$733$	12.2742	0.453356	0.226678	−	0.973970i	$-0.427213\pi$
$733$	12.2742	0.453356	0.226678	+	0.973970i	$0.427213\pi$
$734$	0	0
$735$	−151.308	−5.58108
$736$	0	0
$737$	21.2799	0.783855
$738$	0	0
$739$	15.8824	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$739$	15.8824	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$740$	0	0
$741$	−60.5573	−2.22463
$742$	0	0
$743$	−21.5724	−0.791414	−0.395707	−	0.918377i	$-0.629500\pi$
$743$	−21.5724	−0.791414	−0.395707	+	0.918377i	$0.629500\pi$
$744$	0	0
$745$	−1.45720	−0.0533877
$746$	0	0
$747$	−70.5229	−2.58030
$748$	0	0
$749$	14.4548	0.528166
$750$	0	0
$751$	−1.33196	−0.0486038	−0.0243019	−	0.999705i	$-0.507736\pi$
$751$	−1.33196	−0.0486038	−0.0243019	+	0.999705i	$0.507736\pi$
$752$	0	0
$753$	79.5605	2.89935
$754$	0	0
$755$	−45.6113	−1.65997
$756$	0	0
$757$	−47.3507	−1.72099	−0.860495	−	0.509459i	$-0.829846\pi$
$757$	−47.3507	−1.72099	−0.860495	+	0.509459i	$0.829846\pi$
$758$	0	0
$759$	10.4398	0.378940
$760$	0	0
$761$	8.62156	0.312531	0.156266	−	0.987715i	$-0.450054\pi$
$761$	8.62156	0.312531	0.156266	+	0.987715i	$0.450054\pi$
$762$	0	0
$763$	82.9982	3.00474
$764$	0	0
$765$	−14.9663	−0.541107
$766$	0	0
$767$	4.58503	0.165556
$768$	0	0
$769$	−4.27744	−0.154248	−0.0771241	−	0.997021i	$-0.524574\pi$
$769$	−4.27744	−0.154248	−0.0771241	+	0.997021i	$0.524574\pi$
$770$	0	0
$771$	45.2140	1.62834
$772$	0	0
$773$	15.5558	0.559503	0.279751	−	0.960072i	$-0.409748\pi$
$773$	15.5558	0.559503	0.279751	+	0.960072i	$0.409748\pi$
$774$	0	0
$775$	23.6118	0.848161
$776$	0	0
$777$	137.890	4.94677
$778$	0	0
$779$	−10.2952	−0.368864
$780$	0	0
$781$	11.2695	0.403253
$782$	0	0
$783$	20.0711	0.717284
$784$	0	0
$785$	−14.6501	−0.522884
$786$	0	0
$787$	−4.71990	−0.168246	−0.0841232	−	0.996455i	$-0.526809\pi$
$787$	−4.71990	−0.168246	−0.0841232	+	0.996455i	$0.526809\pi$
$788$	0	0
$789$	66.4585	2.36599
$790$	0	0
$791$	−75.8009	−2.69517
$792$	0	0
$793$	−40.8627	−1.45108
$794$	0	0
$795$	−95.5039	−3.38717
$796$	0	0
$797$	24.7311	0.876022	0.438011	−	0.898970i	$-0.355683\pi$
$797$	24.7311	0.876022	0.438011	+	0.898970i	$0.355683\pi$
$798$	0	0
$799$	−12.5461	−0.443850
$800$	0	0
$801$	43.7464	1.54570
$802$	0	0
$803$	−19.0408	−0.671936
$804$	0	0
$805$	−28.3978	−1.00089
$806$	0	0
$807$	−28.8959	−1.01718
$808$	0	0
$809$	19.2417	0.676501	0.338251	−	0.941056i	$-0.390165\pi$
$809$	19.2417	0.676501	0.338251	+	0.941056i	$0.390165\pi$
$810$	0	0
$811$	5.97082	0.209664	0.104832	−	0.994490i	$-0.466570\pi$
$811$	5.97082	0.209664	0.104832	+	0.994490i	$0.466570\pi$
$812$	0	0
$813$	28.2695	0.991453
$814$	0	0
$815$	27.2529	0.954629
$816$	0	0
$817$	30.3622	1.06224
$818$	0	0
$819$	108.152	3.77915
$820$	0	0
$821$	44.0987	1.53906	0.769528	−	0.638613i	$-0.220491\pi$
$821$	44.0987	1.53906	0.769528	+	0.638613i	$0.220491\pi$
$822$	0	0
$823$	−22.1200	−0.771055	−0.385528	−	0.922696i	$-0.625981\pi$
$823$	−22.1200	−0.771055	−0.385528	+	0.922696i	$0.625981\pi$
$824$	0	0
$825$	−27.2114	−0.947379
$826$	0	0
$827$	−33.4730	−1.16397	−0.581984	−	0.813200i	$-0.697724\pi$
$827$	−33.4730	−1.16397	−0.581984	+	0.813200i	$0.697724\pi$
$828$	0	0
$829$	−34.8471	−1.21029	−0.605144	−	0.796116i	$-0.706885\pi$
$829$	−34.8471	−1.21029	−0.605144	+	0.796116i	$0.706885\pi$
$830$	0	0
$831$	72.1644	2.50335
$832$	0	0
$833$	−17.3307	−0.600474
$834$	0	0
$835$	−11.8162	−0.408918
$836$	0	0
$837$	−24.4813	−0.846197
$838$	0	0
$839$	9.55129	0.329747	0.164874	−	0.986315i	$-0.447278\pi$
$839$	9.55129	0.329747	0.164874	+	0.986315i	$0.447278\pi$
$840$	0	0
$841$	−12.6997	−0.437920
$842$	0	0
$843$	−18.3623	−0.632433
$844$	0	0
$845$	25.1077	0.863733
$846$	0	0
$847$	−33.8439	−1.16289
$848$	0	0
$849$	−14.3019	−0.490838
$850$	0	0
$851$	18.4339	0.631905
$852$	0	0
$853$	1.07945	0.0369595	0.0184798	−	0.999829i	$-0.494117\pi$
$853$	1.07945	0.0369595	0.0184798	+	0.999829i	$0.494117\pi$
$854$	0	0
$855$	70.8583	2.42330
$856$	0	0
$857$	30.6344	1.04645	0.523225	−	0.852194i	$-0.324729\pi$
$857$	30.6344	1.04645	0.523225	+	0.852194i	$0.324729\pi$
$858$	0	0
$859$	28.6190	0.976466	0.488233	−	0.872713i	$-0.337642\pi$
$859$	28.6190	0.976466	0.488233	+	0.872713i	$0.337642\pi$
$860$	0	0
$861$	29.9215	1.01972
$862$	0	0
$863$	−9.21810	−0.313788	−0.156894	−	0.987615i	$-0.550148\pi$
$863$	−9.21810	−0.313788	−0.156894	+	0.987615i	$0.550148\pi$
$864$	0	0
$865$	−36.0701	−1.22642
$866$	0	0
$867$	−2.78964	−0.0947410
$868$	0	0
$869$	−25.5094	−0.865348
$870$	0	0
$871$	47.9598	1.62506
$872$	0	0
$873$	−10.9443	−0.370408
$874$	0	0
$875$	−3.16805	−0.107100
$876$	0	0
$877$	−16.9566	−0.572585	−0.286293	−	0.958142i	$-0.592423\pi$
$877$	−16.9566	−0.572585	−0.286293	+	0.958142i	$0.592423\pi$
$878$	0	0
$879$	81.2488	2.74045
$880$	0	0
$881$	−0.254390	−0.00857062	−0.00428531	−	0.999991i	$-0.501364\pi$
$881$	−0.254390	−0.00857062	−0.00428531	+	0.999991i	$0.501364\pi$
$882$	0	0
$883$	18.6390	0.627251	0.313626	−	0.949547i	$-0.398456\pi$
$883$	18.6390	0.627251	0.313626	+	0.949547i	$0.398456\pi$
$884$	0	0
$885$	−8.73062	−0.293477
$886$	0	0
$887$	24.1736	0.811669	0.405835	−	0.913947i	$-0.366981\pi$
$887$	24.1736	0.811669	0.405835	+	0.913947i	$0.366981\pi$
$888$	0	0
$889$	61.9523	2.07781
$890$	0	0
$891$	−0.972398	−0.0325766
$892$	0	0
$893$	59.4000	1.98774
$894$	0	0
$895$	47.6470	1.59266
$896$	0	0
$897$	23.5288	0.785604
$898$	0	0
$899$	−19.8819	−0.663100
$900$	0	0
$901$	−10.9390	−0.364430
$902$	0	0
$903$	−88.2434	−2.93656
$904$	0	0
$905$	−12.6585	−0.420783
$906$	0	0
$907$	0.563050	0.0186958	0.00934788	−	0.999956i	$-0.497024\pi$
$907$	0.563050	0.0186958	0.00934788	+	0.999956i	$0.497024\pi$
$908$	0	0
$909$	61.8805	2.05245
$910$	0	0
$911$	−53.8502	−1.78414	−0.892068	−	0.451901i	$-0.850746\pi$
$911$	−53.8502	−1.78414	−0.892068	+	0.451901i	$0.850746\pi$
$912$	0	0
$913$	−30.0019	−0.992916
$914$	0	0
$915$	77.8089	2.57228
$916$	0	0
$917$	−95.6386	−3.15827
$918$	0	0
$919$	6.55833	0.216339	0.108170	−	0.994132i	$-0.465501\pi$
$919$	6.55833	0.216339	0.108170	+	0.994132i	$0.465501\pi$
$920$	0	0
$921$	3.50971	0.115649
$922$	0	0
$923$	25.3987	0.836007
$924$	0	0
$925$	−48.0480	−1.57981
$926$	0	0
$927$	62.9355	2.06707
$928$	0	0
$929$	−23.0846	−0.757382	−0.378691	−	0.925523i	$-0.623626\pi$
$929$	−23.0846	−0.757382	−0.378691	+	0.925523i	$0.623626\pi$
$930$	0	0
$931$	82.0528	2.68917
$932$	0	0
$933$	3.08856	0.101115
$934$	0	0
$935$	−6.36695	−0.208222
$936$	0	0
$937$	−2.21658	−0.0724126	−0.0362063	−	0.999344i	$-0.511527\pi$
$937$	−2.21658	−0.0724126	−0.0362063	+	0.999344i	$0.511527\pi$
$938$	0	0
$939$	−80.8358	−2.63798
$940$	0	0
$941$	9.28577	0.302708	0.151354	−	0.988480i	$-0.451637\pi$
$941$	9.28577	0.302708	0.151354	+	0.988480i	$0.451637\pi$
$942$	0	0
$943$	4.00007	0.130260
$944$	0	0
$945$	−76.7447	−2.49651
$946$	0	0
$947$	−35.0400	−1.13865	−0.569324	−	0.822113i	$-0.692795\pi$
$947$	−35.0400	−1.13865	−0.569324	+	0.822113i	$0.692795\pi$
$948$	0	0
$949$	−42.9135	−1.39303
$950$	0	0
$951$	4.75058	0.154048
$952$	0	0
$953$	18.0169	0.583626	0.291813	−	0.956475i	$-0.405742\pi$
$953$	18.0169	0.583626	0.291813	+	0.956475i	$0.405742\pi$
$954$	0	0
$955$	−30.6789	−0.992745
$956$	0	0
$957$	22.9129	0.740669
$958$	0	0
$959$	−113.562	−3.66712
$960$	0	0
$961$	−6.74948	−0.217725
$962$	0	0
$963$	14.0136	0.451583
$964$	0	0
$965$	42.4885	1.36775
$966$	0	0
$967$	27.8884	0.896831	0.448416	−	0.893825i	$-0.351988\pi$
$967$	27.8884	0.896831	0.448416	+	0.893825i	$0.351988\pi$
$968$	0	0
$969$	13.2076	0.424290
$970$	0	0
$971$	36.1632	1.16053	0.580266	−	0.814427i	$-0.302949\pi$
$971$	36.1632	1.16053	0.580266	+	0.814427i	$0.302949\pi$
$972$	0	0
$973$	−30.0574	−0.963597
$974$	0	0
$975$	−61.3280	−1.96407
$976$	0	0
$977$	−49.3233	−1.57799	−0.788997	−	0.614397i	$-0.789399\pi$
$977$	−49.3233	−1.57799	−0.788997	+	0.614397i	$0.789399\pi$
$978$	0	0
$979$	18.6106	0.594797
$980$	0	0
$981$	80.4651	2.56905
$982$	0	0
$983$	−33.6049	−1.07183	−0.535914	−	0.844272i	$-0.680033\pi$
$983$	−33.6049	−1.07183	−0.535914	+	0.844272i	$0.680033\pi$
$984$	0	0
$985$	49.7993	1.58674
$986$	0	0
$987$	−172.637	−5.49511
$988$	0	0
$989$	−11.7969	−0.375119
$990$	0	0
$991$	4.13651	0.131400	0.0657002	−	0.997839i	$-0.479072\pi$
$991$	4.13651	0.131400	0.0657002	+	0.997839i	$0.479072\pi$
$992$	0	0
$993$	5.84265	0.185411
$994$	0	0
$995$	−65.3513	−2.07177
$996$	0	0
$997$	−38.9439	−1.23336	−0.616682	−	0.787212i	$-0.711524\pi$
$997$	−38.9439	−1.23336	−0.616682	+	0.787212i	$0.711524\pi$
$998$	0	0
$999$	49.8174	1.57615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.2	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.2	✓	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.78964 q^{3} +3.12966 q^{5} +4.93262 q^{7} +4.78208 q^{9} +O(q^{10})\)	\(q-2.78964 q^{3} +3.12966 q^{5} +4.93262 q^{7} +4.78208 q^{9} +2.03439 q^{11} +4.58503 q^{13} -8.73062 q^{15} -1.00000 q^{17} +4.73453 q^{19} -13.7602 q^{21} -1.83954 q^{23} +4.79478 q^{25} -4.97134 q^{27} -4.03737 q^{29} +4.92448 q^{31} -5.67521 q^{33} +15.4374 q^{35} -10.0209 q^{37} -12.7906 q^{39} -2.17449 q^{41} +6.41294 q^{43} +14.9663 q^{45} +12.5461 q^{47} +17.3307 q^{49} +2.78964 q^{51} +10.9390 q^{53} +6.36695 q^{55} -13.2076 q^{57} +1.00000 q^{59} -8.91219 q^{61} +23.5881 q^{63} +14.3496 q^{65} +10.4601 q^{67} +5.13166 q^{69} +5.53947 q^{71} -9.35947 q^{73} -13.3757 q^{75} +10.0349 q^{77} -12.5391 q^{79} -0.477980 q^{81} -14.7473 q^{83} -3.12966 q^{85} +11.2628 q^{87} +9.14799 q^{89} +22.6162 q^{91} -13.7375 q^{93} +14.8175 q^{95} -2.28860 q^{97} +9.72861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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