LMFDB - Embedded newform 8038.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.26564 q^{3} +1.00000 q^{4} +1.02149 q^{5} +3.26564 q^{6} -2.73396 q^{7} -1.00000 q^{8} +7.66439 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.26564 q^{3} +1.00000 q^{4} +1.02149 q^{5} +3.26564 q^{6} -2.73396 q^{7} -1.00000 q^{8} +7.66439 q^{9} -1.02149 q^{10} +3.38972 q^{11} -3.26564 q^{12} -1.41890 q^{13} +2.73396 q^{14} -3.33582 q^{15} +1.00000 q^{16} -5.21166 q^{17} -7.66439 q^{18} +7.69390 q^{19} +1.02149 q^{20} +8.92812 q^{21} -3.38972 q^{22} +4.49274 q^{23} +3.26564 q^{24} -3.95656 q^{25} +1.41890 q^{26} -15.2322 q^{27} -2.73396 q^{28} +1.91583 q^{29} +3.33582 q^{30} -2.95186 q^{31} -1.00000 q^{32} -11.0696 q^{33} +5.21166 q^{34} -2.79272 q^{35} +7.66439 q^{36} +10.8726 q^{37} -7.69390 q^{38} +4.63361 q^{39} -1.02149 q^{40} -11.0177 q^{41} -8.92812 q^{42} -1.74252 q^{43} +3.38972 q^{44} +7.82911 q^{45} -4.49274 q^{46} -9.14669 q^{47} -3.26564 q^{48} +0.474537 q^{49} +3.95656 q^{50} +17.0194 q^{51} -1.41890 q^{52} +4.22277 q^{53} +15.2322 q^{54} +3.46257 q^{55} +2.73396 q^{56} -25.1255 q^{57} -1.91583 q^{58} -4.83756 q^{59} -3.33582 q^{60} +5.23970 q^{61} +2.95186 q^{62} -20.9541 q^{63} +1.00000 q^{64} -1.44939 q^{65} +11.0696 q^{66} +0.0967700 q^{67} -5.21166 q^{68} -14.6717 q^{69} +2.79272 q^{70} +14.5326 q^{71} -7.66439 q^{72} -6.06251 q^{73} -10.8726 q^{74} +12.9207 q^{75} +7.69390 q^{76} -9.26737 q^{77} -4.63361 q^{78} -8.26016 q^{79} +1.02149 q^{80} +26.7497 q^{81} +11.0177 q^{82} -1.04319 q^{83} +8.92812 q^{84} -5.32366 q^{85} +1.74252 q^{86} -6.25641 q^{87} -3.38972 q^{88} +5.58028 q^{89} -7.82911 q^{90} +3.87921 q^{91} +4.49274 q^{92} +9.63972 q^{93} +9.14669 q^{94} +7.85925 q^{95} +3.26564 q^{96} -17.5707 q^{97} -0.474537 q^{98} +25.9802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.26564	−1.88542	−0.942708	−	0.333618i	$-0.891731\pi$
$3$	−3.26564	−1.88542	−0.942708	+	0.333618i	$0.891731\pi$
$4$	1.00000	0.500000
$5$	1.02149	0.456825	0.228412	−	0.973564i	$-0.426647\pi$
$5$	1.02149	0.456825	0.228412	+	0.973564i	$0.426647\pi$
$6$	3.26564	1.33319
$7$	−2.73396	−1.03334	−0.516670	−	0.856185i	$-0.672829\pi$
$7$	−2.73396	−1.03334	−0.516670	+	0.856185i	$0.672829\pi$
$8$	−1.00000	−0.353553
$9$	7.66439	2.55480
$10$	−1.02149	−0.323024
$11$	3.38972	1.02204	0.511020	−	0.859569i	$-0.329268\pi$
$11$	3.38972	1.02204	0.511020	+	0.859569i	$0.329268\pi$
$12$	−3.26564	−0.942708
$13$	−1.41890	−0.393532	−0.196766	−	0.980451i	$-0.563044\pi$
$13$	−1.41890	−0.393532	−0.196766	+	0.980451i	$0.563044\pi$
$14$	2.73396	0.730682
$15$	−3.33582	−0.861305
$16$	1.00000	0.250000
$17$	−5.21166	−1.26401	−0.632006	−	0.774963i	$-0.717768\pi$
$17$	−5.21166	−1.26401	−0.632006	+	0.774963i	$0.717768\pi$
$18$	−7.66439	−1.80651
$19$	7.69390	1.76510	0.882551	−	0.470218i	$-0.155825\pi$
$19$	7.69390	1.76510	0.882551	+	0.470218i	$0.155825\pi$
$20$	1.02149	0.228412
$21$	8.92812	1.94828
$22$	−3.38972	−0.722692
$23$	4.49274	0.936800	0.468400	−	0.883516i	$-0.344831\pi$
$23$	4.49274	0.936800	0.468400	+	0.883516i	$0.344831\pi$
$24$	3.26564	0.666596
$25$	−3.95656	−0.791311
$26$	1.41890	0.278269
$27$	−15.2322	−2.93144
$28$	−2.73396	−0.516670
$29$	1.91583	0.355761	0.177880	−	0.984052i	$-0.443076\pi$
$29$	1.91583	0.355761	0.177880	+	0.984052i	$0.443076\pi$
$30$	3.33582	0.609035
$31$	−2.95186	−0.530170	−0.265085	−	0.964225i	$-0.585400\pi$
$31$	−2.95186	−0.530170	−0.265085	+	0.964225i	$0.585400\pi$
$32$	−1.00000	−0.176777
$33$	−11.0696	−1.92697
$34$	5.21166	0.893792
$35$	−2.79272	−0.472055
$36$	7.66439	1.27740
$37$	10.8726	1.78744	0.893718	−	0.448629i	$-0.148087\pi$
$37$	10.8726	1.78744	0.893718	+	0.448629i	$0.148087\pi$
$38$	−7.69390	−1.24811
$39$	4.63361	0.741971
$40$	−1.02149	−0.161512
$41$	−11.0177	−1.72067	−0.860334	−	0.509730i	$-0.829745\pi$
$41$	−11.0177	−1.72067	−0.860334	+	0.509730i	$0.829745\pi$
$42$	−8.92812	−1.37764
$43$	−1.74252	−0.265731	−0.132866	−	0.991134i	$-0.542418\pi$
$43$	−1.74252	−0.265731	−0.132866	+	0.991134i	$0.542418\pi$
$44$	3.38972	0.511020
$45$	7.82911	1.16709
$46$	−4.49274	−0.662418
$47$	−9.14669	−1.33418	−0.667091	−	0.744976i	$-0.732461\pi$
$47$	−9.14669	−1.33418	−0.667091	+	0.744976i	$0.732461\pi$
$48$	−3.26564	−0.471354
$49$	0.474537	0.0677910
$50$	3.95656	0.559542
$51$	17.0194	2.38319
$52$	−1.41890	−0.196766
$53$	4.22277	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$53$	4.22277	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$54$	15.2322	2.07284
$55$	3.46257	0.466893
$56$	2.73396	0.365341
$57$	−25.1255	−3.32795
$58$	−1.91583	−0.251561
$59$	−4.83756	−0.629797	−0.314899	−	0.949125i	$-0.601971\pi$
$59$	−4.83756	−0.629797	−0.314899	+	0.949125i	$0.601971\pi$
$60$	−3.33582	−0.430652
$61$	5.23970	0.670874	0.335437	−	0.942063i	$-0.391116\pi$
$61$	5.23970	0.670874	0.335437	+	0.942063i	$0.391116\pi$
$62$	2.95186	0.374887
$63$	−20.9541	−2.63997
$64$	1.00000	0.125000
$65$	−1.44939	−0.179775
$66$	11.0696	1.36258
$67$	0.0967700	0.0118223	0.00591117	−	0.999983i	$-0.498118\pi$
$67$	0.0967700	0.0118223	0.00591117	+	0.999983i	$0.498118\pi$
$68$	−5.21166	−0.632006
$69$	−14.6717	−1.76626
$70$	2.79272	0.333793
$71$	14.5326	1.72471	0.862354	−	0.506306i	$-0.168989\pi$
$71$	14.5326	1.72471	0.862354	+	0.506306i	$0.168989\pi$
$72$	−7.66439	−0.903257
$73$	−6.06251	−0.709564	−0.354782	−	0.934949i	$-0.615445\pi$
$73$	−6.06251	−0.709564	−0.354782	+	0.934949i	$0.615445\pi$
$74$	−10.8726	−1.26391
$75$	12.9207	1.49195
$76$	7.69390	0.882551
$77$	−9.26737	−1.05611
$78$	−4.63361	−0.524653
$79$	−8.26016	−0.929340	−0.464670	−	0.885484i	$-0.653827\pi$
$79$	−8.26016	−0.929340	−0.464670	+	0.885484i	$0.653827\pi$
$80$	1.02149	0.114206
$81$	26.7497	2.97219
$82$	11.0177	1.21670
$83$	−1.04319	−0.114505	−0.0572527	−	0.998360i	$-0.518234\pi$
$83$	−1.04319	−0.114505	−0.0572527	+	0.998360i	$0.518234\pi$
$84$	8.92812	0.974138
$85$	−5.32366	−0.577432
$86$	1.74252	0.187900
$87$	−6.25641	−0.670758
$88$	−3.38972	−0.361346
$89$	5.58028	0.591509	0.295754	−	0.955264i	$-0.404429\pi$
$89$	5.58028	0.591509	0.295754	+	0.955264i	$0.404429\pi$
$90$	−7.82911	−0.825260
$91$	3.87921	0.406652
$92$	4.49274	0.468400
$93$	9.63972	0.999592
$94$	9.14669	0.943409
$95$	7.85925	0.806342
$96$	3.26564	0.333298
$97$	−17.5707	−1.78403	−0.892016	−	0.452003i	$-0.850710\pi$
$97$	−17.5707	−1.78403	−0.892016	+	0.452003i	$0.850710\pi$
$98$	−0.474537	−0.0479355
$99$	25.9802	2.61111
$100$	−3.95656	−0.395656
$101$	8.82045	0.877668	0.438834	−	0.898568i	$-0.355392\pi$
$101$	8.82045	0.877668	0.438834	+	0.898568i	$0.355392\pi$
$102$	−17.0194	−1.68517
$103$	−0.495690	−0.0488418	−0.0244209	−	0.999702i	$-0.507774\pi$
$103$	−0.495690	−0.0488418	−0.0244209	+	0.999702i	$0.507774\pi$
$104$	1.41890	0.139134
$105$	9.12000	0.890021
$106$	−4.22277	−0.410152
$107$	6.20044	0.599419	0.299710	−	0.954030i	$-0.403110\pi$
$107$	6.20044	0.599419	0.299710	+	0.954030i	$0.403110\pi$
$108$	−15.2322	−1.46572
$109$	−17.0538	−1.63346	−0.816728	−	0.577023i	$-0.804214\pi$
$109$	−17.0538	−1.63346	−0.816728	+	0.577023i	$0.804214\pi$
$110$	−3.46257	−0.330143
$111$	−35.5058	−3.37006
$112$	−2.73396	−0.258335
$113$	−16.3496	−1.53804	−0.769020	−	0.639224i	$-0.779256\pi$
$113$	−16.3496	−1.53804	−0.769020	+	0.639224i	$0.779256\pi$
$114$	25.1255	2.35322
$115$	4.58929	0.427954
$116$	1.91583	0.177880
$117$	−10.8750	−1.00539
$118$	4.83756	0.445334
$119$	14.2485	1.30615
$120$	3.33582	0.304517
$121$	0.490231	0.0445665
$122$	−5.23970	−0.474380
$123$	35.9797	3.24418
$124$	−2.95186	−0.265085
$125$	−9.14904	−0.818315
$126$	20.9541	1.86674
$127$	6.52360	0.578876	0.289438	−	0.957197i	$-0.406532\pi$
$127$	6.52360	0.578876	0.289438	+	0.957197i	$0.406532\pi$
$128$	−1.00000	−0.0883883
$129$	5.69043	0.501014
$130$	1.44939	0.127120
$131$	−17.4232	−1.52228	−0.761138	−	0.648590i	$-0.775359\pi$
$131$	−17.4232	−1.52228	−0.761138	+	0.648590i	$0.775359\pi$
$132$	−11.0696	−0.963486
$133$	−21.0348	−1.82395
$134$	−0.0967700	−0.00835965
$135$	−15.5596	−1.33915
$136$	5.21166	0.446896
$137$	10.8650	0.928261	0.464130	−	0.885767i	$-0.346367\pi$
$137$	10.8650	0.928261	0.464130	+	0.885767i	$0.346367\pi$
$138$	14.6717	1.24893
$139$	21.6211	1.83388	0.916939	−	0.399028i	$-0.130653\pi$
$139$	21.6211	1.83388	0.916939	+	0.399028i	$0.130653\pi$
$140$	−2.79272	−0.236028
$141$	29.8698	2.51549
$142$	−14.5326	−1.21955
$143$	−4.80968	−0.402205
$144$	7.66439	0.638699
$145$	1.95700	0.162520
$146$	6.06251	0.501737
$147$	−1.54967	−0.127814
$148$	10.8726	0.893718
$149$	6.16228	0.504834	0.252417	−	0.967619i	$-0.418775\pi$
$149$	6.16228	0.504834	0.252417	+	0.967619i	$0.418775\pi$
$150$	−12.9207	−1.05497
$151$	13.6654	1.11207	0.556036	−	0.831158i	$-0.312321\pi$
$151$	13.6654	1.11207	0.556036	+	0.831158i	$0.312321\pi$
$152$	−7.69390	−0.624057
$153$	−39.9442	−3.22930
$154$	9.26737	0.746786
$155$	−3.01530	−0.242195
$156$	4.63361	0.370986
$157$	20.3583	1.62477	0.812384	−	0.583123i	$-0.198170\pi$
$157$	20.3583	1.62477	0.812384	+	0.583123i	$0.198170\pi$
$158$	8.26016	0.657143
$159$	−13.7900	−1.09362
$160$	−1.02149	−0.0807560
$161$	−12.2830	−0.968033
$162$	−26.7497	−2.10166
$163$	14.1368	1.10728	0.553638	−	0.832757i	$-0.313239\pi$
$163$	14.1368	1.10728	0.553638	+	0.832757i	$0.313239\pi$
$164$	−11.0177	−0.860334
$165$	−11.3075	−0.880288
$166$	1.04319	0.0809676
$167$	20.2124	1.56408	0.782040	−	0.623228i	$-0.214179\pi$
$167$	20.2124	1.56408	0.782040	+	0.623228i	$0.214179\pi$
$168$	−8.92812	−0.688820
$169$	−10.9867	−0.845133
$170$	5.32366	0.408306
$171$	58.9690	4.50947
$172$	−1.74252	−0.132866
$173$	−20.8134	−1.58242	−0.791208	−	0.611547i	$-0.790548\pi$
$173$	−20.8134	−1.58242	−0.791208	+	0.611547i	$0.790548\pi$
$174$	6.25641	0.474297
$175$	10.8171	0.817693
$176$	3.38972	0.255510
$177$	15.7977	1.18743
$178$	−5.58028	−0.418260
$179$	2.02069	0.151034	0.0755168	−	0.997145i	$-0.475939\pi$
$179$	2.02069	0.151034	0.0755168	+	0.997145i	$0.475939\pi$
$180$	7.82911	0.583547
$181$	5.83387	0.433628	0.216814	−	0.976213i	$-0.430434\pi$
$181$	5.83387	0.433628	0.216814	+	0.976213i	$0.430434\pi$
$182$	−3.87921	−0.287546
$183$	−17.1109	−1.26488
$184$	−4.49274	−0.331209
$185$	11.1062	0.816545
$186$	−9.63972	−0.706818
$187$	−17.6661	−1.29187
$188$	−9.14669	−0.667091
$189$	41.6442	3.02917
$190$	−7.85925	−0.570170
$191$	−9.31463	−0.673983	−0.336992	−	0.941508i	$-0.609409\pi$
$191$	−9.31463	−0.673983	−0.336992	+	0.941508i	$0.609409\pi$
$192$	−3.26564	−0.235677
$193$	11.3478	0.816831	0.408416	−	0.912796i	$-0.366081\pi$
$193$	11.3478	0.816831	0.408416	+	0.912796i	$0.366081\pi$
$194$	17.5707	1.26150
$195$	4.73319	0.338951
$196$	0.474537	0.0338955
$197$	0.684224	0.0487489	0.0243745	−	0.999703i	$-0.492241\pi$
$197$	0.684224	0.0487489	0.0243745	+	0.999703i	$0.492241\pi$
$198$	−25.9802	−1.84633
$199$	−14.1312	−1.00174	−0.500869	−	0.865523i	$-0.666986\pi$
$199$	−14.1312	−1.00174	−0.500869	+	0.865523i	$0.666986\pi$
$200$	3.95656	0.279771
$201$	−0.316016	−0.0222900
$202$	−8.82045	−0.620605
$203$	−5.23781	−0.367622
$204$	17.0194	1.19160
$205$	−11.2544	−0.786044
$206$	0.495690	0.0345364
$207$	34.4341	2.39333
$208$	−1.41890	−0.0983829
$209$	26.0802	1.80400
$210$	−9.12000	−0.629340
$211$	4.20867	0.289737	0.144868	−	0.989451i	$-0.453724\pi$
$211$	4.20867	0.289737	0.144868	+	0.989451i	$0.453724\pi$
$212$	4.22277	0.290021
$213$	−47.4584	−3.25179
$214$	−6.20044	−0.423853
$215$	−1.77997	−0.121393
$216$	15.2322	1.03642
$217$	8.07028	0.547846
$218$	17.0538	1.15503
$219$	19.7980	1.33782
$220$	3.46257	0.233447
$221$	7.39482	0.497429
$222$	35.5058	2.38299
$223$	3.71545	0.248805	0.124402	−	0.992232i	$-0.460299\pi$
$223$	3.71545	0.248805	0.124402	+	0.992232i	$0.460299\pi$
$224$	2.73396	0.182670
$225$	−30.3246	−2.02164
$226$	16.3496	1.08756
$227$	19.7688	1.31210	0.656050	−	0.754717i	$-0.272226\pi$
$227$	19.7688	1.31210	0.656050	+	0.754717i	$0.272226\pi$
$228$	−25.1255	−1.66398
$229$	−10.2732	−0.678870	−0.339435	−	0.940630i	$-0.610236\pi$
$229$	−10.2732	−0.678870	−0.339435	+	0.940630i	$0.610236\pi$
$230$	−4.58929	−0.302609
$231$	30.2639	1.99122
$232$	−1.91583	−0.125780
$233$	−10.2062	−0.668631	−0.334315	−	0.942461i	$-0.608505\pi$
$233$	−10.2062	−0.668631	−0.334315	+	0.942461i	$0.608505\pi$
$234$	10.8750	0.710921
$235$	−9.34326	−0.609487
$236$	−4.83756	−0.314899
$237$	26.9747	1.75219
$238$	−14.2485	−0.923591
$239$	−16.1069	−1.04187	−0.520934	−	0.853597i	$-0.674416\pi$
$239$	−16.1069	−1.04187	−0.520934	+	0.853597i	$0.674416\pi$
$240$	−3.33582	−0.215326
$241$	−17.8111	−1.14731	−0.573655	−	0.819097i	$-0.694475\pi$
$241$	−17.8111	−1.14731	−0.573655	+	0.819097i	$0.694475\pi$
$242$	−0.490231	−0.0315133
$243$	−41.6582	−2.67238
$244$	5.23970	0.335437
$245$	0.484735	0.0309686
$246$	−35.9797	−2.29398
$247$	−10.9169	−0.694623
$248$	2.95186	0.187444
$249$	3.40669	0.215890
$250$	9.14904	0.578636
$251$	23.9525	1.51187	0.755935	−	0.654646i	$-0.227182\pi$
$251$	23.9525	1.51187	0.755935	+	0.654646i	$0.227182\pi$
$252$	−20.9541	−1.31999
$253$	15.2291	0.957448
$254$	−6.52360	−0.409327
$255$	17.3852	1.08870
$256$	1.00000	0.0625000
$257$	−13.3320	−0.831630	−0.415815	−	0.909449i	$-0.636504\pi$
$257$	−13.3320	−0.831630	−0.415815	+	0.909449i	$0.636504\pi$
$258$	−5.69043	−0.354271
$259$	−29.7251	−1.84703
$260$	−1.44939	−0.0898875
$261$	14.6837	0.908897
$262$	17.4232	1.07641
$263$	19.4617	1.20006	0.600029	−	0.799979i	$-0.295156\pi$
$263$	19.4617	1.20006	0.600029	+	0.799979i	$0.295156\pi$
$264$	11.0696	0.681288
$265$	4.31352	0.264978
$266$	21.0348	1.28973
$267$	−18.2232	−1.11524
$268$	0.0967700	0.00591117
$269$	23.3698	1.42488	0.712440	−	0.701733i	$-0.247590\pi$
$269$	23.3698	1.42488	0.712440	+	0.701733i	$0.247590\pi$
$270$	15.5596	0.946925
$271$	5.42058	0.329277	0.164638	−	0.986354i	$-0.447354\pi$
$271$	5.42058	0.329277	0.164638	+	0.986354i	$0.447354\pi$
$272$	−5.21166	−0.316003
$273$	−12.6681	−0.766708
$274$	−10.8650	−0.656380
$275$	−13.4116	−0.808752
$276$	−14.6717	−0.883130
$277$	19.6237	1.17908	0.589538	−	0.807740i	$-0.299310\pi$
$277$	19.6237	1.17908	0.589538	+	0.807740i	$0.299310\pi$
$278$	−21.6211	−1.29675
$279$	−22.6242	−1.35448
$280$	2.79272	0.166897
$281$	−26.1004	−1.55702	−0.778510	−	0.627632i	$-0.784024\pi$
$281$	−26.1004	−1.55702	−0.778510	+	0.627632i	$0.784024\pi$
$282$	−29.8698	−1.77872
$283$	−14.4153	−0.856901	−0.428451	−	0.903565i	$-0.640940\pi$
$283$	−14.4153	−0.856901	−0.428451	+	0.903565i	$0.640940\pi$
$284$	14.5326	0.862354
$285$	−25.6655	−1.52029
$286$	4.80968	0.284402
$287$	30.1218	1.77804
$288$	−7.66439	−0.451629
$289$	10.1614	0.597728
$290$	−1.95700	−0.114919
$291$	57.3795	3.36365
$292$	−6.06251	−0.354782
$293$	33.1436	1.93627	0.968135	−	0.250431i	$-0.0805722\pi$
$293$	33.1436	1.93627	0.968135	+	0.250431i	$0.0805722\pi$
$294$	1.54967	0.0903784
$295$	−4.94153	−0.287707
$296$	−10.8726	−0.631954
$297$	−51.6330	−2.99605
$298$	−6.16228	−0.356972
$299$	−6.37474	−0.368661
$300$	12.9207	0.745976
$301$	4.76397	0.274591
$302$	−13.6654	−0.786354
$303$	−28.8044	−1.65477
$304$	7.69390	0.441275
$305$	5.35230	0.306472
$306$	39.9442	2.28346
$307$	−20.4310	−1.16606	−0.583029	−	0.812451i	$-0.698133\pi$
$307$	−20.4310	−1.16606	−0.583029	+	0.812451i	$0.698133\pi$
$308$	−9.26737	−0.528057
$309$	1.61874	0.0920872
$310$	3.01530	0.171258
$311$	−20.8797	−1.18398	−0.591990	−	0.805946i	$-0.701657\pi$
$311$	−20.8797	−1.18398	−0.591990	+	0.805946i	$0.701657\pi$
$312$	−4.63361	−0.262326
$313$	11.2211	0.634256	0.317128	−	0.948383i	$-0.397282\pi$
$313$	11.2211	0.634256	0.317128	+	0.948383i	$0.397282\pi$
$314$	−20.3583	−1.14888
$315$	−21.4045	−1.20600
$316$	−8.26016	−0.464670
$317$	−19.1133	−1.07351	−0.536755	−	0.843738i	$-0.680350\pi$
$317$	−19.1133	−1.07351	−0.536755	+	0.843738i	$0.680350\pi$
$318$	13.7900	0.773307
$319$	6.49414	0.363602
$320$	1.02149	0.0571031
$321$	−20.2484	−1.13016
$322$	12.2830	0.684503
$323$	−40.0980	−2.23111
$324$	26.7497	1.48609
$325$	5.61395	0.311406
$326$	−14.1368	−0.782962
$327$	55.6915	3.07974
$328$	11.0177	0.608348
$329$	25.0067	1.37866
$330$	11.3075	0.622458
$331$	22.1234	1.21601	0.608006	−	0.793933i	$-0.291970\pi$
$331$	22.1234	1.21601	0.608006	+	0.793933i	$0.291970\pi$
$332$	−1.04319	−0.0572527
$333$	83.3315	4.56654
$334$	−20.2124	−1.10597
$335$	0.0988497	0.00540074
$336$	8.92812	0.487069
$337$	−13.5693	−0.739166	−0.369583	−	0.929198i	$-0.620500\pi$
$337$	−13.5693	−0.739166	−0.369583	+	0.929198i	$0.620500\pi$
$338$	10.9867	0.597599
$339$	53.3919	2.89985
$340$	−5.32366	−0.288716
$341$	−10.0060	−0.541856
$342$	−58.9690	−3.18868
$343$	17.8404	0.963289
$344$	1.74252	0.0939502
$345$	−14.9870	−0.806871
$346$	20.8134	1.11894
$347$	19.0553	1.02294	0.511470	−	0.859301i	$-0.329101\pi$
$347$	19.0553	1.02294	0.511470	+	0.859301i	$0.329101\pi$
$348$	−6.25641	−0.335379
$349$	−16.6426	−0.890860	−0.445430	−	0.895317i	$-0.646949\pi$
$349$	−16.6426	−0.890860	−0.445430	+	0.895317i	$0.646949\pi$
$350$	−10.8171	−0.578196
$351$	21.6130	1.15361
$352$	−3.38972	−0.180673
$353$	−2.78151	−0.148045	−0.0740224	−	0.997257i	$-0.523584\pi$
$353$	−2.78151	−0.148045	−0.0740224	+	0.997257i	$0.523584\pi$
$354$	−15.7977	−0.839640
$355$	14.8450	0.787889
$356$	5.58028	0.295754
$357$	−46.5303	−2.46265
$358$	−2.02069	−0.106797
$359$	30.9988	1.63605	0.818026	−	0.575181i	$-0.195068\pi$
$359$	30.9988	1.63605	0.818026	+	0.575181i	$0.195068\pi$
$360$	−7.82911	−0.412630
$361$	40.1961	2.11558
$362$	−5.83387	−0.306621
$363$	−1.60092	−0.0840264
$364$	3.87921	0.203326
$365$	−6.19280	−0.324146
$366$	17.1109	0.894404
$367$	14.6171	0.763006	0.381503	−	0.924368i	$-0.375407\pi$
$367$	14.6171	0.763006	0.381503	+	0.924368i	$0.375407\pi$
$368$	4.49274	0.234200
$369$	−84.4436	−4.39596
$370$	−11.1062	−0.577385
$371$	−11.5449	−0.599380
$372$	9.63972	0.499796
$373$	−8.29292	−0.429391	−0.214696	−	0.976681i	$-0.568876\pi$
$373$	−8.29292	−0.429391	−0.214696	+	0.976681i	$0.568876\pi$
$374$	17.6661	0.913492
$375$	29.8775	1.54287
$376$	9.14669	0.471705
$377$	−2.71837	−0.140003
$378$	−41.6442	−2.14195
$379$	24.7077	1.26915	0.634574	−	0.772862i	$-0.281176\pi$
$379$	24.7077	1.26915	0.634574	+	0.772862i	$0.281176\pi$
$380$	7.85925	0.403171
$381$	−21.3037	−1.09142
$382$	9.31463	0.476578
$383$	13.4284	0.686157	0.343078	−	0.939307i	$-0.388530\pi$
$383$	13.4284	0.686157	0.343078	+	0.939307i	$0.388530\pi$
$384$	3.26564	0.166649
$385$	−9.46654	−0.482459
$386$	−11.3478	−0.577587
$387$	−13.3553	−0.678890
$388$	−17.5707	−0.892016
$389$	29.9533	1.51869	0.759345	−	0.650688i	$-0.225519\pi$
$389$	29.9533	1.51869	0.759345	+	0.650688i	$0.225519\pi$
$390$	−4.73319	−0.239674
$391$	−23.4146	−1.18413
$392$	−0.474537	−0.0239677
$393$	56.8980	2.87012
$394$	−0.684224	−0.0344707
$395$	−8.43768	−0.424546
$396$	25.9802	1.30555
$397$	−25.9227	−1.30102	−0.650511	−	0.759497i	$-0.725445\pi$
$397$	−25.9227	−1.30102	−0.650511	+	0.759497i	$0.725445\pi$
$398$	14.1312	0.708335
$399$	68.6921	3.43890
$400$	−3.95656	−0.197828
$401$	−18.9425	−0.945941	−0.472971	−	0.881078i	$-0.656818\pi$
$401$	−18.9425	−0.945941	−0.472971	+	0.881078i	$0.656818\pi$
$402$	0.316016	0.0157614
$403$	4.18840	0.208639
$404$	8.82045	0.438834
$405$	27.3246	1.35777
$406$	5.23781	0.259948
$407$	36.8550	1.82683
$408$	−17.0194	−0.842585
$409$	6.03525	0.298424	0.149212	−	0.988805i	$-0.452326\pi$
$409$	6.03525	0.298424	0.149212	+	0.988805i	$0.452326\pi$
$410$	11.2544	0.555817
$411$	−35.4812	−1.75016
$412$	−0.495690	−0.0244209
$413$	13.2257	0.650795
$414$	−34.4341	−1.69234
$415$	−1.06561	−0.0523089
$416$	1.41890	0.0695672
$417$	−70.6067	−3.45762
$418$	−26.0802	−1.27562
$419$	−4.28616	−0.209393	−0.104696	−	0.994504i	$-0.533387\pi$
$419$	−4.28616	−0.209393	−0.104696	+	0.994504i	$0.533387\pi$
$420$	9.12000	0.445010
$421$	−33.4069	−1.62815	−0.814076	−	0.580758i	$-0.802756\pi$
$421$	−33.4069	−1.62815	−0.814076	+	0.580758i	$0.802756\pi$
$422$	−4.20867	−0.204875
$423$	−70.1038	−3.40856
$424$	−4.22277	−0.205076
$425$	20.6202	1.00023
$426$	47.4584	2.29937
$427$	−14.3251	−0.693241
$428$	6.20044	0.299710
$429$	15.7067	0.758325
$430$	1.77997	0.0858376
$431$	−26.4848	−1.27573	−0.637864	−	0.770149i	$-0.720182\pi$
$431$	−26.4848	−1.27573	−0.637864	+	0.770149i	$0.720182\pi$
$432$	−15.2322	−0.732860
$433$	−24.5641	−1.18047	−0.590237	−	0.807230i	$-0.700966\pi$
$433$	−24.5641	−1.18047	−0.590237	+	0.807230i	$0.700966\pi$
$434$	−8.07028	−0.387386
$435$	−6.39087	−0.306419
$436$	−17.0538	−0.816728
$437$	34.5667	1.65355
$438$	−19.7980	−0.945984
$439$	−33.5262	−1.60012	−0.800059	−	0.599921i	$-0.795199\pi$
$439$	−33.5262	−1.60012	−0.800059	+	0.599921i	$0.795199\pi$
$440$	−3.46257	−0.165072
$441$	3.63704	0.173192
$442$	−7.39482	−0.351736
$443$	−18.0573	−0.857929	−0.428965	−	0.903321i	$-0.641122\pi$
$443$	−18.0573	−0.857929	−0.428965	+	0.903321i	$0.641122\pi$
$444$	−35.5058	−1.68503
$445$	5.70021	0.270216
$446$	−3.71545	−0.175932
$447$	−20.1238	−0.951823
$448$	−2.73396	−0.129167
$449$	2.49238	0.117623	0.0588114	−	0.998269i	$-0.481269\pi$
$449$	2.49238	0.117623	0.0588114	+	0.998269i	$0.481269\pi$
$450$	30.3246	1.42951
$451$	−37.3468	−1.75859
$452$	−16.3496	−0.769020
$453$	−44.6262	−2.09672
$454$	−19.7688	−0.927795
$455$	3.96258	0.185769
$456$	25.1255	1.17661
$457$	0.898240	0.0420179	0.0210090	−	0.999779i	$-0.493312\pi$
$457$	0.898240	0.0420179	0.0210090	+	0.999779i	$0.493312\pi$
$458$	10.2732	0.480033
$459$	79.3851	3.70538
$460$	4.58929	0.213977
$461$	−15.4189	−0.718128	−0.359064	−	0.933313i	$-0.616904\pi$
$461$	−15.4189	−0.718128	−0.359064	+	0.933313i	$0.616904\pi$
$462$	−30.2639	−1.40800
$463$	−13.8731	−0.644737	−0.322369	−	0.946614i	$-0.604479\pi$
$463$	−13.8731	−0.644737	−0.322369	+	0.946614i	$0.604479\pi$
$464$	1.91583	0.0889402
$465$	9.84689	0.456638
$466$	10.2062	0.472793
$467$	−38.6731	−1.78958	−0.894789	−	0.446490i	$-0.852674\pi$
$467$	−38.6731	−1.78958	−0.894789	+	0.446490i	$0.852674\pi$
$468$	−10.8750	−0.502697
$469$	−0.264565	−0.0122165
$470$	9.34326	0.430973
$471$	−66.4828	−3.06336
$472$	4.83756	0.222667
$473$	−5.90665	−0.271588
$474$	−26.9747	−1.23899
$475$	−30.4413	−1.39674
$476$	14.2485	0.653077
$477$	32.3650	1.48189
$478$	16.1069	0.736712
$479$	6.07751	0.277689	0.138844	−	0.990314i	$-0.455661\pi$
$479$	6.07751	0.277689	0.138844	+	0.990314i	$0.455661\pi$
$480$	3.33582	0.152259
$481$	−15.4270	−0.703413
$482$	17.8111	0.811271
$483$	40.1117	1.82515
$484$	0.490231	0.0222832
$485$	−17.9483	−0.814990
$486$	41.6582	1.88966
$487$	−25.7440	−1.16657	−0.583285	−	0.812267i	$-0.698233\pi$
$487$	−25.7440	−1.16657	−0.583285	+	0.812267i	$0.698233\pi$
$488$	−5.23970	−0.237190
$489$	−46.1655	−2.08768
$490$	−0.484735	−0.0218981
$491$	−10.7406	−0.484718	−0.242359	−	0.970187i	$-0.577921\pi$
$491$	−10.7406	−0.484718	−0.242359	+	0.970187i	$0.577921\pi$
$492$	35.9797	1.62209
$493$	−9.98466	−0.449686
$494$	10.9169	0.491173
$495$	26.5385	1.19282
$496$	−2.95186	−0.132543
$497$	−39.7317	−1.78221
$498$	−3.40669	−0.152658
$499$	2.81013	0.125799	0.0628994	−	0.998020i	$-0.479965\pi$
$499$	2.81013	0.125799	0.0628994	+	0.998020i	$0.479965\pi$
$500$	−9.14904	−0.409158
$501$	−66.0063	−2.94894
$502$	−23.9525	−1.06905
$503$	0.236592	0.0105491	0.00527457	−	0.999986i	$-0.498321\pi$
$503$	0.236592	0.0105491	0.00527457	+	0.999986i	$0.498321\pi$
$504$	20.9541	0.933371
$505$	9.01001	0.400940
$506$	−15.2291	−0.677018
$507$	35.8787	1.59343
$508$	6.52360	0.289438
$509$	1.47259	0.0652715	0.0326357	−	0.999467i	$-0.489610\pi$
$509$	1.47259	0.0652715	0.0326357	+	0.999467i	$0.489610\pi$
$510$	−17.3852	−0.769828
$511$	16.5747	0.733220
$512$	−1.00000	−0.0441942
$513$	−117.195	−5.17429
$514$	13.3320	0.588051
$515$	−0.506343	−0.0223121
$516$	5.69043	0.250507
$517$	−31.0048	−1.36359
$518$	29.7251	1.30605
$519$	67.9691	2.98351
$520$	1.44939	0.0635601
$521$	−12.7828	−0.560025	−0.280013	−	0.959996i	$-0.590339\pi$
$521$	−12.7828	−0.560025	−0.280013	+	0.959996i	$0.590339\pi$
$522$	−14.6837	−0.642687
$523$	10.3989	0.454714	0.227357	−	0.973811i	$-0.426992\pi$
$523$	10.3989	0.454714	0.227357	+	0.973811i	$0.426992\pi$
$524$	−17.4232	−0.761138
$525$	−35.3246	−1.54169
$526$	−19.4617	−0.848569
$527$	15.3841	0.670142
$528$	−11.0696	−0.481743
$529$	−2.81532	−0.122405
$530$	−4.31352	−0.187367
$531$	−37.0770	−1.60900
$532$	−21.0348	−0.911975
$533$	15.6329	0.677138
$534$	18.2232	0.788594
$535$	6.33370	0.273830
$536$	−0.0967700	−0.00417983
$537$	−6.59885	−0.284761
$538$	−23.3698	−1.00754
$539$	1.60855	0.0692851
$540$	−15.5596	−0.669577
$541$	21.7575	0.935429	0.467715	−	0.883880i	$-0.345077\pi$
$541$	21.7575	0.935429	0.467715	+	0.883880i	$0.345077\pi$
$542$	−5.42058	−0.232834
$543$	−19.0513	−0.817569
$544$	5.21166	0.223448
$545$	−17.4203	−0.746203
$546$	12.6681	0.542145
$547$	−45.5147	−1.94607	−0.973033	−	0.230665i	$-0.925910\pi$
$547$	−45.5147	−1.94607	−0.973033	+	0.230665i	$0.925910\pi$
$548$	10.8650	0.464130
$549$	40.1591	1.71395
$550$	13.4116	0.571874
$551$	14.7402	0.627954
$552$	14.6717	0.624467
$553$	22.5829	0.960324
$554$	−19.6237	−0.833733
$555$	−36.2689	−1.53953
$556$	21.6211	0.916939
$557$	34.8948	1.47854	0.739271	−	0.673408i	$-0.235170\pi$
$557$	34.8948	1.47854	0.739271	+	0.673408i	$0.235170\pi$
$558$	22.6242	0.957760
$559$	2.47246	0.104574
$560$	−2.79272	−0.118014
$561$	57.6910	2.43572
$562$	26.1004	1.10098
$563$	−35.6612	−1.50294	−0.751470	−	0.659768i	$-0.770655\pi$
$563$	−35.6612	−1.50294	−0.751470	+	0.659768i	$0.770655\pi$
$564$	29.8698	1.25774
$565$	−16.7010	−0.702615
$566$	14.4153	0.605921
$567$	−73.1326	−3.07128
$568$	−14.5326	−0.609776
$569$	−20.6650	−0.866324	−0.433162	−	0.901316i	$-0.642602\pi$
$569$	−20.6650	−0.866324	−0.433162	+	0.901316i	$0.642602\pi$
$570$	25.6655	1.07501
$571$	37.5094	1.56972	0.784859	−	0.619674i	$-0.212735\pi$
$571$	37.5094	1.56972	0.784859	+	0.619674i	$0.212735\pi$
$572$	−4.80968	−0.201103
$573$	30.4182	1.27074
$574$	−30.1218	−1.25726
$575$	−17.7758	−0.741301
$576$	7.66439	0.319350
$577$	−39.3760	−1.63924	−0.819622	−	0.572904i	$-0.805817\pi$
$577$	−39.3760	−1.63924	−0.819622	+	0.572904i	$0.805817\pi$
$578$	−10.1614	−0.422658
$579$	−37.0577	−1.54007
$580$	1.95700	0.0812602
$581$	2.85205	0.118323
$582$	−57.3795	−2.37846
$583$	14.3140	0.592826
$584$	6.06251	0.250869
$585$	−11.1087	−0.459289
$586$	−33.1436	−1.36915
$587$	−34.4224	−1.42076	−0.710382	−	0.703816i	$-0.751478\pi$
$587$	−34.4224	−1.42076	−0.710382	+	0.703816i	$0.751478\pi$
$588$	−1.54967	−0.0639072
$589$	−22.7113	−0.935804
$590$	4.94153	0.203440
$591$	−2.23443	−0.0919120
$592$	10.8726	0.446859
$593$	7.33602	0.301254	0.150627	−	0.988591i	$-0.451871\pi$
$593$	7.33602	0.301254	0.150627	+	0.988591i	$0.451871\pi$
$594$	51.6330	2.11853
$595$	14.5547	0.596684
$596$	6.16228	0.252417
$597$	46.1475	1.88869
$598$	6.37474	0.260682
$599$	8.57835	0.350502	0.175251	−	0.984524i	$-0.443926\pi$
$599$	8.57835	0.350502	0.175251	+	0.984524i	$0.443926\pi$
$600$	−12.9207	−0.527485
$601$	28.1402	1.14786	0.573931	−	0.818903i	$-0.305418\pi$
$601$	28.1402	1.14786	0.573931	+	0.818903i	$0.305418\pi$
$602$	−4.76397	−0.194165
$603$	0.741683	0.0302037
$604$	13.6654	0.556036
$605$	0.500767	0.0203591
$606$	28.8044	1.17010
$607$	42.8429	1.73894	0.869471	−	0.493984i	$-0.164460\pi$
$607$	42.8429	1.73894	0.869471	+	0.493984i	$0.164460\pi$
$608$	−7.69390	−0.312029
$609$	17.1048	0.693120
$610$	−5.35230	−0.216708
$611$	12.9782	0.525043
$612$	−39.9442	−1.61465
$613$	10.2098	0.412369	0.206185	−	0.978513i	$-0.433895\pi$
$613$	10.2098	0.412369	0.206185	+	0.978513i	$0.433895\pi$
$614$	20.4310	0.824528
$615$	36.7529	1.48202
$616$	9.26737	0.373393
$617$	6.93721	0.279281	0.139641	−	0.990202i	$-0.455405\pi$
$617$	6.93721	0.279281	0.139641	+	0.990202i	$0.455405\pi$
$618$	−1.61874	−0.0651155
$619$	−36.7230	−1.47602	−0.738012	−	0.674788i	$-0.764235\pi$
$619$	−36.7230	−1.47602	−0.738012	+	0.674788i	$0.764235\pi$
$620$	−3.01530	−0.121097
$621$	−68.4343	−2.74617
$622$	20.8797	0.837200
$623$	−15.2563	−0.611230
$624$	4.63361	0.185493
$625$	10.4371	0.417485
$626$	−11.2211	−0.448486
$627$	−85.1685	−3.40130
$628$	20.3583	0.812384
$629$	−56.6640	−2.25934
$630$	21.4045	0.852774
$631$	−23.8905	−0.951067	−0.475534	−	0.879698i	$-0.657745\pi$
$631$	−23.8905	−0.951067	−0.475534	+	0.879698i	$0.657745\pi$
$632$	8.26016	0.328571
$633$	−13.7440	−0.546274
$634$	19.1133	0.759086
$635$	6.66380	0.264445
$636$	−13.7900	−0.546811
$637$	−0.673320	−0.0266779
$638$	−6.49414	−0.257105
$639$	111.384	4.40628
$640$	−1.02149	−0.0403780
$641$	−35.5666	−1.40480	−0.702399	−	0.711784i	$-0.747888\pi$
$641$	−35.5666	−1.40480	−0.702399	+	0.711784i	$0.747888\pi$
$642$	20.2484	0.799140
$643$	−15.9325	−0.628317	−0.314158	−	0.949371i	$-0.601722\pi$
$643$	−15.9325	−0.628317	−0.314158	+	0.949371i	$0.601722\pi$
$644$	−12.2830	−0.484017
$645$	5.81272	0.228876
$646$	40.0980	1.57763
$647$	−26.3334	−1.03527	−0.517636	−	0.855601i	$-0.673188\pi$
$647$	−26.3334	−1.03527	−0.517636	+	0.855601i	$0.673188\pi$
$648$	−26.7497	−1.05083
$649$	−16.3980	−0.643678
$650$	−5.61395	−0.220197
$651$	−26.3546	−1.03292
$652$	14.1368	0.553638
$653$	−33.0437	−1.29310	−0.646550	−	0.762872i	$-0.723789\pi$
$653$	−33.0437	−1.29310	−0.646550	+	0.762872i	$0.723789\pi$
$654$	−55.6915	−2.17771
$655$	−17.7977	−0.695413
$656$	−11.0177	−0.430167
$657$	−46.4655	−1.81279
$658$	−25.0067	−0.974862
$659$	−17.6412	−0.687205	−0.343602	−	0.939115i	$-0.611647\pi$
$659$	−17.6412	−0.687205	−0.343602	+	0.939115i	$0.611647\pi$
$660$	−11.3075	−0.440144
$661$	−41.7557	−1.62411	−0.812055	−	0.583581i	$-0.801651\pi$
$661$	−41.7557	−1.62411	−0.812055	+	0.583581i	$0.801651\pi$
$662$	−22.1234	−0.859850
$663$	−24.1488	−0.937861
$664$	1.04319	0.0404838
$665$	−21.4869	−0.833225
$666$	−83.3315	−3.22903
$667$	8.60732	0.333277
$668$	20.2124	0.782040
$669$	−12.1333	−0.469101
$670$	−0.0988497	−0.00381890
$671$	17.7611	0.685661
$672$	−8.92812	−0.344410
$673$	−36.0764	−1.39064	−0.695321	−	0.718700i	$-0.744738\pi$
$673$	−36.0764	−1.39064	−0.695321	+	0.718700i	$0.744738\pi$
$674$	13.5693	0.522670
$675$	60.2671	2.31968
$676$	−10.9867	−0.422566
$677$	−9.12204	−0.350588	−0.175294	−	0.984516i	$-0.556088\pi$
$677$	−9.12204	−0.350588	−0.175294	+	0.984516i	$0.556088\pi$
$678$	−53.3919	−2.05050
$679$	48.0375	1.84351
$680$	5.32366	0.204153
$681$	−64.5577	−2.47386
$682$	10.0060	0.383150
$683$	−49.6981	−1.90164	−0.950822	−	0.309737i	$-0.899759\pi$
$683$	−49.6981	−1.90164	−0.950822	+	0.309737i	$0.899759\pi$
$684$	58.9690	2.25474
$685$	11.0985	0.424053
$686$	−17.8404	−0.681148
$687$	33.5484	1.27995
$688$	−1.74252	−0.0664328
$689$	−5.99168	−0.228265
$690$	14.9870	0.570544
$691$	26.1227	0.993754	0.496877	−	0.867821i	$-0.334480\pi$
$691$	26.1227	0.993754	0.496877	+	0.867821i	$0.334480\pi$
$692$	−20.8134	−0.791208
$693$	−71.0287	−2.69816
$694$	−19.0553	−0.723328
$695$	22.0858	0.837760
$696$	6.25641	0.237149
$697$	57.4203	2.17495
$698$	16.6426	0.629933
$699$	33.3298	1.26065
$700$	10.8171	0.408847
$701$	20.0813	0.758461	0.379230	−	0.925302i	$-0.376189\pi$
$701$	20.0813	0.758461	0.379230	+	0.925302i	$0.376189\pi$
$702$	−21.6130	−0.815729
$703$	83.6523	3.15501
$704$	3.38972	0.127755
$705$	30.5117	1.14914
$706$	2.78151	0.104683
$707$	−24.1148	−0.906929
$708$	15.7977	0.593715
$709$	17.2861	0.649194	0.324597	−	0.945852i	$-0.394771\pi$
$709$	17.2861	0.649194	0.324597	+	0.945852i	$0.394771\pi$
$710$	−14.8450	−0.557122
$711$	−63.3091	−2.37428
$712$	−5.58028	−0.209130
$713$	−13.2619	−0.496664
$714$	46.5303	1.74135
$715$	−4.91304	−0.183737
$716$	2.02069	0.0755168
$717$	52.5992	1.96435
$718$	−30.9988	−1.15686
$719$	−24.3275	−0.907261	−0.453631	−	0.891190i	$-0.649871\pi$
$719$	−24.3275	−0.907261	−0.453631	+	0.891190i	$0.649871\pi$
$720$	7.82911	0.291774
$721$	1.35520	0.0504702
$722$	−40.1961	−1.49594
$723$	58.1644	2.16316
$724$	5.83387	0.216814
$725$	−7.58009	−0.281518
$726$	1.60092	0.0594156
$727$	12.4082	0.460194	0.230097	−	0.973168i	$-0.426096\pi$
$727$	12.4082	0.460194	0.230097	+	0.973168i	$0.426096\pi$
$728$	−3.87921	−0.143773
$729$	55.7916	2.06635
$730$	6.19280	0.229206
$731$	9.08140	0.335888
$732$	−17.1109	−0.632439
$733$	31.6423	1.16873	0.584367	−	0.811490i	$-0.301343\pi$
$733$	31.6423	1.16873	0.584367	+	0.811490i	$0.301343\pi$
$734$	−14.6171	−0.539527
$735$	−1.58297	−0.0583887
$736$	−4.49274	−0.165604
$737$	0.328024	0.0120829
$738$	84.4436	3.10841
$739$	−8.25093	−0.303515	−0.151758	−	0.988418i	$-0.548493\pi$
$739$	−8.25093	−0.303515	−0.151758	+	0.988418i	$0.548493\pi$
$740$	11.1062	0.408273
$741$	35.6505	1.30965
$742$	11.5449	0.423826
$743$	23.7186	0.870151	0.435075	−	0.900394i	$-0.356722\pi$
$743$	23.7186	0.870151	0.435075	+	0.900394i	$0.356722\pi$
$744$	−9.63972	−0.353409
$745$	6.29472	0.230621
$746$	8.29292	0.303625
$747$	−7.99545	−0.292538
$748$	−17.6661	−0.645936
$749$	−16.9518	−0.619404
$750$	−29.8775	−1.09097
$751$	−3.13200	−0.114288	−0.0571441	−	0.998366i	$-0.518199\pi$
$751$	−3.13200	−0.114288	−0.0571441	+	0.998366i	$0.518199\pi$
$752$	−9.14669	−0.333546
$753$	−78.2203	−2.85051
$754$	2.71837	0.0989972
$755$	13.9591	0.508022
$756$	41.6442	1.51459
$757$	−19.7507	−0.717853	−0.358926	−	0.933366i	$-0.616857\pi$
$757$	−19.7507	−0.717853	−0.358926	+	0.933366i	$0.616857\pi$
$758$	−24.7077	−0.897423
$759$	−49.7329	−1.80519
$760$	−7.85925	−0.285085
$761$	−23.5934	−0.855260	−0.427630	−	0.903954i	$-0.640651\pi$
$761$	−23.5934	−0.855260	−0.427630	+	0.903954i	$0.640651\pi$
$762$	21.3037	0.771752
$763$	46.6243	1.68791
$764$	−9.31463	−0.336992
$765$	−40.8026	−1.47522
$766$	−13.4284	−0.485186
$767$	6.86401	0.247845
$768$	−3.26564	−0.117839
$769$	−5.26834	−0.189981	−0.0949905	−	0.995478i	$-0.530282\pi$
$769$	−5.26834	−0.189981	−0.0949905	+	0.995478i	$0.530282\pi$
$770$	9.46654	0.341150
$771$	43.5376	1.56797
$772$	11.3478	0.408416
$773$	−17.6266	−0.633986	−0.316993	−	0.948428i	$-0.602673\pi$
$773$	−17.6266	−0.633986	−0.316993	+	0.948428i	$0.602673\pi$
$774$	13.3553	0.480047
$775$	11.6792	0.419530
$776$	17.5707	0.630751
$777$	97.0715	3.48242
$778$	−29.9533	−1.07388
$779$	−84.7687	−3.03715
$780$	4.73319	0.169475
$781$	49.2617	1.76272
$782$	23.4146	0.837305
$783$	−29.1823	−1.04289
$784$	0.474537	0.0169478
$785$	20.7958	0.742234
$786$	−56.8980	−2.02948
$787$	18.0375	0.642968	0.321484	−	0.946915i	$-0.395818\pi$
$787$	18.0375	0.642968	0.321484	+	0.946915i	$0.395818\pi$
$788$	0.684224	0.0243745
$789$	−63.5547	−2.26261
$790$	8.43768	0.300199
$791$	44.6991	1.58932
$792$	−25.9802	−0.923165
$793$	−7.43460	−0.264010
$794$	25.9227	0.919962
$795$	−14.0864	−0.499593
$796$	−14.1312	−0.500869
$797$	−42.2391	−1.49618	−0.748092	−	0.663595i	$-0.769030\pi$
$797$	−42.2391	−1.49618	−0.748092	+	0.663595i	$0.769030\pi$
$798$	−68.6921	−2.43167
$799$	47.6694	1.68642
$800$	3.95656	0.139885
$801$	42.7695	1.51118
$802$	18.9425	0.668882
$803$	−20.5503	−0.725203
$804$	−0.316016	−0.0111450
$805$	−12.5469	−0.442221
$806$	−4.18840	−0.147530
$807$	−76.3172	−2.68649
$808$	−8.82045	−0.310302
$809$	−42.4391	−1.49208	−0.746039	−	0.665902i	$-0.768047\pi$
$809$	−42.4391	−1.49208	−0.746039	+	0.665902i	$0.768047\pi$
$810$	−27.3246	−0.960088
$811$	1.93384	0.0679063	0.0339532	−	0.999423i	$-0.489190\pi$
$811$	1.93384	0.0679063	0.0339532	+	0.999423i	$0.489190\pi$
$812$	−5.23781	−0.183811
$813$	−17.7016	−0.620824
$814$	−36.8550	−1.29177
$815$	14.4406	0.505831
$816$	17.0194	0.595798
$817$	−13.4067	−0.469043
$818$	−6.03525	−0.211017
$819$	29.7318	1.03891
$820$	−11.2544	−0.393022
$821$	−22.6695	−0.791169	−0.395585	−	0.918429i	$-0.629458\pi$
$821$	−22.6695	−0.791169	−0.395585	+	0.918429i	$0.629458\pi$
$822$	35.4812	1.23755
$823$	−32.0936	−1.11871	−0.559356	−	0.828927i	$-0.688952\pi$
$823$	−32.0936	−1.11871	−0.559356	+	0.828927i	$0.688952\pi$
$824$	0.495690	0.0172682
$825$	43.7975	1.52483
$826$	−13.2257	−0.460181
$827$	55.8791	1.94311	0.971554	−	0.236818i	$-0.0761045\pi$
$827$	55.8791	1.94311	0.971554	+	0.236818i	$0.0761045\pi$
$828$	34.4341	1.19667
$829$	34.2902	1.19095	0.595474	−	0.803374i	$-0.296964\pi$
$829$	34.2902	1.19095	0.595474	+	0.803374i	$0.296964\pi$
$830$	1.06561	0.0369880
$831$	−64.0840	−2.22305
$832$	−1.41890	−0.0491915
$833$	−2.47312	−0.0856887
$834$	70.6067	2.44491
$835$	20.6468	0.714511
$836$	26.0802	0.902002
$837$	44.9634	1.55416
$838$	4.28616	0.148063
$839$	−28.3669	−0.979333	−0.489666	−	0.871910i	$-0.662881\pi$
$839$	−28.3669	−0.979333	−0.489666	+	0.871910i	$0.662881\pi$
$840$	−9.12000	−0.314670
$841$	−25.3296	−0.873434
$842$	33.4069	1.15128
$843$	85.2345	2.93563
$844$	4.20867	0.144868
$845$	−11.2228	−0.386078
$846$	70.1038	2.41022
$847$	−1.34027	−0.0460523
$848$	4.22277	0.145011
$849$	47.0752	1.61562
$850$	−20.6202	−0.707268
$851$	48.8475	1.67447
$852$	−47.4584	−1.62590
$853$	−12.8956	−0.441537	−0.220768	−	0.975326i	$-0.570857\pi$
$853$	−12.8956	−0.441537	−0.220768	+	0.975326i	$0.570857\pi$
$854$	14.3251	0.490195
$855$	60.2363	2.06004
$856$	−6.20044	−0.211927
$857$	20.2391	0.691356	0.345678	−	0.938353i	$-0.387649\pi$
$857$	20.2391	0.691356	0.345678	+	0.938353i	$0.387649\pi$
$858$	−15.7067	−0.536216
$859$	54.8146	1.87025	0.935125	−	0.354318i	$-0.115287\pi$
$859$	54.8146	1.87025	0.935125	+	0.354318i	$0.115287\pi$
$860$	−1.77997	−0.0606963
$861$	−98.3670	−3.35234
$862$	26.4848	0.902076
$863$	−26.6385	−0.906787	−0.453393	−	0.891311i	$-0.649787\pi$
$863$	−26.6385	−0.906787	−0.453393	+	0.891311i	$0.649787\pi$
$864$	15.2322	0.518210
$865$	−21.2607	−0.722887
$866$	24.5641	0.834722
$867$	−33.1834	−1.12697
$868$	8.07028	0.273923
$869$	−27.9997	−0.949823
$870$	6.39087	0.216671
$871$	−0.137307	−0.00465246
$872$	17.0538	0.577514
$873$	−134.669	−4.55784
$874$	−34.5667	−1.16923
$875$	25.0131	0.845598
$876$	19.7980	0.668912
$877$	20.2820	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$877$	20.2820	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$878$	33.5262	1.13145
$879$	−108.235	−3.65067
$880$	3.46257	0.116723
$881$	34.7049	1.16924	0.584619	−	0.811308i	$-0.301244\pi$
$881$	34.7049	1.16924	0.584619	+	0.811308i	$0.301244\pi$
$882$	−3.63704	−0.122465
$883$	4.24774	0.142948	0.0714740	−	0.997442i	$-0.477230\pi$
$883$	4.24774	0.142948	0.0714740	+	0.997442i	$0.477230\pi$
$884$	7.39482	0.248715
$885$	16.1372	0.542448
$886$	18.0573	0.606648
$887$	−20.3858	−0.684488	−0.342244	−	0.939611i	$-0.611187\pi$
$887$	−20.3858	−0.684488	−0.342244	+	0.939611i	$0.611187\pi$
$888$	35.5058	1.19150
$889$	−17.8353	−0.598175
$890$	−5.70021	−0.191071
$891$	90.6741	3.03770
$892$	3.71545	0.124402
$893$	−70.3737	−2.35497
$894$	20.1238	0.673040
$895$	2.06412	0.0689959
$896$	2.73396	0.0913352
$897$	20.8176	0.695079
$898$	−2.49238	−0.0831719
$899$	−5.65527	−0.188614
$900$	−30.3246	−1.01082
$901$	−22.0076	−0.733181
$902$	37.3468	1.24351
$903$	−15.5574	−0.517718
$904$	16.3496	0.543780
$905$	5.95924	0.198092
$906$	44.6262	1.48261
$907$	41.2680	1.37028	0.685140	−	0.728411i	$-0.259741\pi$
$907$	41.2680	1.37028	0.685140	+	0.728411i	$0.259741\pi$
$908$	19.7688	0.656050
$909$	67.6034	2.24226
$910$	−3.96258	−0.131358
$911$	−29.7728	−0.986417	−0.493208	−	0.869911i	$-0.664176\pi$
$911$	−29.7728	−0.986417	−0.493208	+	0.869911i	$0.664176\pi$
$912$	−25.1255	−0.831988
$913$	−3.53614	−0.117029
$914$	−0.898240	−0.0297112
$915$	−17.4787	−0.577827
$916$	−10.2732	−0.339435
$917$	47.6344	1.57303
$918$	−79.3851	−2.62010
$919$	−6.72639	−0.221883	−0.110942	−	0.993827i	$-0.535387\pi$
$919$	−6.72639	−0.221883	−0.110942	+	0.993827i	$0.535387\pi$
$920$	−4.58929	−0.151304
$921$	66.7202	2.19851
$922$	15.4189	0.507793
$923$	−20.6204	−0.678727
$924$	30.2639	0.995608
$925$	−43.0179	−1.41442
$926$	13.8731	0.455898
$927$	−3.79916	−0.124781
$928$	−1.91583	−0.0628902
$929$	30.7957	1.01038	0.505188	−	0.863010i	$-0.331423\pi$
$929$	30.7957	1.01038	0.505188	+	0.863010i	$0.331423\pi$
$930$	−9.84689	−0.322892
$931$	3.65104	0.119658
$932$	−10.2062	−0.334315
$933$	68.1855	2.23229
$934$	38.6731	1.26542
$935$	−18.0457	−0.590159
$936$	10.8750	0.355460
$937$	−6.71103	−0.219240	−0.109620	−	0.993974i	$-0.534963\pi$
$937$	−6.71103	−0.219240	−0.109620	+	0.993974i	$0.534963\pi$
$938$	0.264565	0.00863836
$939$	−36.6441	−1.19584
$940$	−9.34326	−0.304744
$941$	−18.2768	−0.595806	−0.297903	−	0.954596i	$-0.596287\pi$
$941$	−18.2768	−0.595806	−0.297903	+	0.954596i	$0.596287\pi$
$942$	66.4828	2.16613
$943$	−49.4994	−1.61192
$944$	−4.83756	−0.157449
$945$	42.5392	1.38380
$946$	5.90665	0.192042
$947$	−27.4235	−0.891143	−0.445571	−	0.895246i	$-0.646999\pi$
$947$	−27.4235	−0.891143	−0.445571	+	0.895246i	$0.646999\pi$
$948$	26.9747	0.876097
$949$	8.60209	0.279236
$950$	30.4413	0.987647
$951$	62.4171	2.02401
$952$	−14.2485	−0.461795
$953$	22.2363	0.720304	0.360152	−	0.932894i	$-0.382725\pi$
$953$	22.2363	0.720304	0.360152	+	0.932894i	$0.382725\pi$
$954$	−32.3650	−1.04785
$955$	−9.51481	−0.307892
$956$	−16.1069	−0.520934
$957$	−21.2075	−0.685541
$958$	−6.07751	−0.196356
$959$	−29.7045	−0.959209
$960$	−3.33582	−0.107663
$961$	−22.2865	−0.718919
$962$	15.4270	0.497388
$963$	47.5226	1.53139
$964$	−17.8111	−0.573655
$965$	11.5917	0.373149
$966$	−40.1117	−1.29057
$967$	49.0588	1.57763	0.788813	−	0.614634i	$-0.210696\pi$
$967$	49.0588	1.57763	0.788813	+	0.614634i	$0.210696\pi$
$968$	−0.490231	−0.0157566
$969$	130.945	4.20657
$970$	17.9483	0.576285
$971$	6.16501	0.197845	0.0989224	−	0.995095i	$-0.468460\pi$
$971$	6.16501	0.197845	0.0989224	+	0.995095i	$0.468460\pi$
$972$	−41.6582	−1.33619
$973$	−59.1112	−1.89502
$974$	25.7440	0.824890
$975$	−18.3331	−0.587130
$976$	5.23970	0.167719
$977$	−4.89846	−0.156716	−0.0783579	−	0.996925i	$-0.524968\pi$
$977$	−4.89846	−0.156716	−0.0783579	+	0.996925i	$0.524968\pi$
$978$	46.1655	1.47621
$979$	18.9156	0.604546
$980$	0.484735	0.0154843
$981$	−130.707	−4.17315
$982$	10.7406	0.342748
$983$	−2.32744	−0.0742336	−0.0371168	−	0.999311i	$-0.511817\pi$
$983$	−2.32744	−0.0742336	−0.0371168	+	0.999311i	$0.511817\pi$
$984$	−35.9797	−1.14699
$985$	0.698928	0.0222697
$986$	9.98466	0.317976
$987$	−81.6628	−2.59936
$988$	−10.9169	−0.347312
$989$	−7.82867	−0.248937
$990$	−26.5385	−0.843449
$991$	11.0866	0.352178	0.176089	−	0.984374i	$-0.443655\pi$
$991$	11.0866	0.352178	0.176089	+	0.984374i	$0.443655\pi$
$992$	2.95186	0.0937218
$993$	−72.2470	−2.29269
$994$	39.7317	1.26021
$995$	−14.4349	−0.457618
$996$	3.40669	0.107945
$997$	44.3125	1.40339	0.701696	−	0.712477i	$-0.252427\pi$
$997$	44.3125	1.40339	0.701696	+	0.712477i	$0.252427\pi$
$998$	−2.81013	−0.0889532
$999$	−165.613	−5.23976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.4	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.4	✓	84	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.26564 q^{3} +1.00000 q^{4} +1.02149 q^{5} +3.26564 q^{6} -2.73396 q^{7} -1.00000 q^{8} +7.66439 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.26564 q^{3} +1.00000 q^{4} +1.02149 q^{5} +3.26564 q^{6} -2.73396 q^{7} -1.00000 q^{8} +7.66439 q^{9} -1.02149 q^{10} +3.38972 q^{11} -3.26564 q^{12} -1.41890 q^{13} +2.73396 q^{14} -3.33582 q^{15} +1.00000 q^{16} -5.21166 q^{17} -7.66439 q^{18} +7.69390 q^{19} +1.02149 q^{20} +8.92812 q^{21} -3.38972 q^{22} +4.49274 q^{23} +3.26564 q^{24} -3.95656 q^{25} +1.41890 q^{26} -15.2322 q^{27} -2.73396 q^{28} +1.91583 q^{29} +3.33582 q^{30} -2.95186 q^{31} -1.00000 q^{32} -11.0696 q^{33} +5.21166 q^{34} -2.79272 q^{35} +7.66439 q^{36} +10.8726 q^{37} -7.69390 q^{38} +4.63361 q^{39} -1.02149 q^{40} -11.0177 q^{41} -8.92812 q^{42} -1.74252 q^{43} +3.38972 q^{44} +7.82911 q^{45} -4.49274 q^{46} -9.14669 q^{47} -3.26564 q^{48} +0.474537 q^{49} +3.95656 q^{50} +17.0194 q^{51} -1.41890 q^{52} +4.22277 q^{53} +15.2322 q^{54} +3.46257 q^{55} +2.73396 q^{56} -25.1255 q^{57} -1.91583 q^{58} -4.83756 q^{59} -3.33582 q^{60} +5.23970 q^{61} +2.95186 q^{62} -20.9541 q^{63} +1.00000 q^{64} -1.44939 q^{65} +11.0696 q^{66} +0.0967700 q^{67} -5.21166 q^{68} -14.6717 q^{69} +2.79272 q^{70} +14.5326 q^{71} -7.66439 q^{72} -6.06251 q^{73} -10.8726 q^{74} +12.9207 q^{75} +7.69390 q^{76} -9.26737 q^{77} -4.63361 q^{78} -8.26016 q^{79} +1.02149 q^{80} +26.7497 q^{81} +11.0177 q^{82} -1.04319 q^{83} +8.92812 q^{84} -5.32366 q^{85} +1.74252 q^{86} -6.25641 q^{87} -3.38972 q^{88} +5.58028 q^{89} -7.82911 q^{90} +3.87921 q^{91} +4.49274 q^{92} +9.63972 q^{93} +9.14669 q^{94} +7.85925 q^{95} +3.26564 q^{96} -17.5707 q^{97} -0.474537 q^{98} +25.9802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more