LMFDB - Embedded newform 5239.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 $ x^8 - x^7 - 11x^6 + 10x^5 + 37x^4 - 33x^3 - 36x^2 + 33x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.656875$ of defining polynomial
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.656875 q^{2} +2.67827 q^{3} -1.56852 q^{4} +1.55873 q^{5} +1.75929 q^{6} -2.17607 q^{7} -2.34407 q^{8} +4.17313 q^{9} +O(q^{10})$	\(q+0.656875 q^{2} +2.67827 q^{3} -1.56852 q^{4} +1.55873 q^{5} +1.75929 q^{6} -2.17607 q^{7} -2.34407 q^{8} +4.17313 q^{9} +1.02389 q^{10} -0.877966 q^{11} -4.20091 q^{12} -1.42941 q^{14} +4.17471 q^{15} +1.59727 q^{16} +4.00763 q^{17} +2.74122 q^{18} +0.601130 q^{19} -2.44490 q^{20} -5.82810 q^{21} -0.576714 q^{22} +7.74079 q^{23} -6.27805 q^{24} -2.57035 q^{25} +3.14195 q^{27} +3.41320 q^{28} +4.35321 q^{29} +2.74226 q^{30} +1.00000 q^{31} +5.73734 q^{32} -2.35143 q^{33} +2.63251 q^{34} -3.39192 q^{35} -6.54561 q^{36} +2.60850 q^{37} +0.394867 q^{38} -3.65378 q^{40} +5.00956 q^{41} -3.82833 q^{42} -2.25813 q^{43} +1.37710 q^{44} +6.50480 q^{45} +5.08473 q^{46} +1.99049 q^{47} +4.27792 q^{48} -2.26472 q^{49} -1.68840 q^{50} +10.7335 q^{51} -3.22380 q^{53} +2.06387 q^{54} -1.36852 q^{55} +5.10086 q^{56} +1.60999 q^{57} +2.85951 q^{58} -6.18276 q^{59} -6.54810 q^{60} +2.46585 q^{61} +0.656875 q^{62} -9.08102 q^{63} +0.574175 q^{64} -1.54459 q^{66} +6.59746 q^{67} -6.28603 q^{68} +20.7319 q^{69} -2.22806 q^{70} +0.570087 q^{71} -9.78210 q^{72} +6.17960 q^{73} +1.71346 q^{74} -6.88408 q^{75} -0.942882 q^{76} +1.91052 q^{77} +9.85609 q^{79} +2.48972 q^{80} -4.10439 q^{81} +3.29065 q^{82} -7.25438 q^{83} +9.14147 q^{84} +6.24683 q^{85} -1.48331 q^{86} +11.6591 q^{87} +2.05801 q^{88} +16.2492 q^{89} +4.27284 q^{90} -12.1415 q^{92} +2.67827 q^{93} +1.30750 q^{94} +0.937002 q^{95} +15.3662 q^{96} +15.9725 q^{97} -1.48764 q^{98} -3.66386 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 7 q^{3} + 7 q^{4} - 11 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q + q^2 + 7 * q^3 + 7 * q^4 - 11 * q^5 + 2 * q^7 + 9 * q^9	\( 8 q + q^{2} + 7 q^{3} + 7 q^{4} - 11 q^{5} + 2 q^{7} + 9 q^{9} - 2 q^{10} + 2 q^{11} + 19 q^{12} + 3 q^{14} - 6 q^{15} - 7 q^{16} + 7 q^{17} + 12 q^{18} + 5 q^{19} - 6 q^{20} - 8 q^{21} - 4 q^{22} + 14 q^{23} + 17 q^{24} + 17 q^{25} + 7 q^{27} + 9 q^{28} + 12 q^{29} - 9 q^{30} + 8 q^{31} + 21 q^{32} - 10 q^{33} + 12 q^{34} - q^{35} + 11 q^{36} - 2 q^{37} + 24 q^{38} - 19 q^{40} - 13 q^{41} - 27 q^{42} - 5 q^{43} - 22 q^{44} - 19 q^{45} - 17 q^{46} - 23 q^{47} + 3 q^{48} + 26 q^{49} + 26 q^{50} + 18 q^{51} + 25 q^{53} + 36 q^{54} - 17 q^{55} + 8 q^{56} + 35 q^{57} + 29 q^{58} + 5 q^{59} - 71 q^{60} - 9 q^{61} + q^{62} + 37 q^{63} - 14 q^{64} - 41 q^{66} - 22 q^{67} - 6 q^{68} - 7 q^{69} + 29 q^{70} + 17 q^{71} + 34 q^{72} + 27 q^{73} + 14 q^{74} - 33 q^{75} + 36 q^{76} + 31 q^{77} - 23 q^{79} - 9 q^{80} - 12 q^{81} + 18 q^{82} + 25 q^{83} + 62 q^{84} - 13 q^{85} - 11 q^{86} + 26 q^{87} + 5 q^{88} - 2 q^{89} - 14 q^{90} - 20 q^{92} + 7 q^{93} - 38 q^{94} + 3 q^{95} + 52 q^{96} + 15 q^{97} - 39 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q + q^2 + 7 * q^3 + 7 * q^4 - 11 * q^5 + 2 * q^7 + 9 * q^9 - 2 * q^10 + 2 * q^11 + 19 * q^12 + 3 * q^14 - 6 * q^15 - 7 * q^16 + 7 * q^17 + 12 * q^18 + 5 * q^19 - 6 * q^20 - 8 * q^21 - 4 * q^22 + 14 * q^23 + 17 * q^24 + 17 * q^25 + 7 * q^27 + 9 * q^28 + 12 * q^29 - 9 * q^30 + 8 * q^31 + 21 * q^32 - 10 * q^33 + 12 * q^34 - q^35 + 11 * q^36 - 2 * q^37 + 24 * q^38 - 19 * q^40 - 13 * q^41 - 27 * q^42 - 5 * q^43 - 22 * q^44 - 19 * q^45 - 17 * q^46 - 23 * q^47 + 3 * q^48 + 26 * q^49 + 26 * q^50 + 18 * q^51 + 25 * q^53 + 36 * q^54 - 17 * q^55 + 8 * q^56 + 35 * q^57 + 29 * q^58 + 5 * q^59 - 71 * q^60 - 9 * q^61 + q^62 + 37 * q^63 - 14 * q^64 - 41 * q^66 - 22 * q^67 - 6 * q^68 - 7 * q^69 + 29 * q^70 + 17 * q^71 + 34 * q^72 + 27 * q^73 + 14 * q^74 - 33 * q^75 + 36 * q^76 + 31 * q^77 - 23 * q^79 - 9 * q^80 - 12 * q^81 + 18 * q^82 + 25 * q^83 + 62 * q^84 - 13 * q^85 - 11 * q^86 + 26 * q^87 + 5 * q^88 - 2 * q^89 - 14 * q^90 - 20 * q^92 + 7 * q^93 - 38 * q^94 + 3 * q^95 + 52 * q^96 + 15 * q^97 - 39 * q^98 + 8 * 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 $

$2$	0.656875	0.464481	0.232240	−	0.972658i	$-0.425394\pi$
$2$	0.656875	0.464481	0.232240	+	0.972658i	$0.425394\pi$
$3$	2.67827	1.54630	0.773150	−	0.634223i	$-0.218680\pi$
$3$	2.67827	1.54630	0.773150	+	0.634223i	$0.218680\pi$
$4$	−1.56852	−0.784258
$5$	1.55873	0.697087	0.348544	−	0.937293i	$-0.386676\pi$
$5$	1.55873	0.697087	0.348544	+	0.937293i	$0.386676\pi$
$6$	1.75929	0.718226
$7$	−2.17607	−0.822477	−0.411239	−	0.911528i	$-0.634904\pi$
$7$	−2.17607	−0.822477	−0.411239	+	0.911528i	$0.634904\pi$
$8$	−2.34407	−0.828753
$9$	4.17313	1.39104
$10$	1.02389	0.323784
$11$	−0.877966	−0.264717	−0.132358	−	0.991202i	$-0.542255\pi$
$11$	−0.877966	−0.264717	−0.132358	+	0.991202i	$0.542255\pi$
$12$	−4.20091	−1.21270
$13$	0	0
$13$	0	0
$14$	−1.42941	−0.382025
$15$	4.17471	1.07791
$16$	1.59727	0.399318
$17$	4.00763	0.971993	0.485996	−	0.873961i	$-0.338457\pi$
$17$	4.00763	0.971993	0.485996	+	0.873961i	$0.338457\pi$
$18$	2.74122	0.646112
$19$	0.601130	0.137909	0.0689543	−	0.997620i	$-0.478034\pi$
$19$	0.601130	0.137909	0.0689543	+	0.997620i	$0.478034\pi$
$20$	−2.44490	−0.546696
$21$	−5.82810	−1.27180
$22$	−0.576714	−0.122956
$23$	7.74079	1.61407	0.807033	−	0.590507i	$-0.201072\pi$
$23$	7.74079	1.61407	0.807033	+	0.590507i	$0.201072\pi$
$24$	−6.27805	−1.28150
$25$	−2.57035	−0.514070
$26$	0	0
$27$	3.14195	0.604669
$28$	3.41320	0.645034
$29$	4.35321	0.808371	0.404185	−	0.914677i	$-0.367555\pi$
$29$	4.35321	0.808371	0.404185	+	0.914677i	$0.367555\pi$
$30$	2.74226	0.500666
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.73734	1.01423
$33$	−2.35143	−0.409331
$34$	2.63251	0.451472
$35$	−3.39192	−0.573338
$36$	−6.54561	−1.09094
$37$	2.60850	0.428834	0.214417	−	0.976742i	$-0.431215\pi$
$37$	2.60850	0.428834	0.214417	+	0.976742i	$0.431215\pi$
$38$	0.394867	0.0640559
$39$	0	0
$40$	−3.65378	−0.577713
$41$	5.00956	0.782361	0.391181	−	0.920314i	$-0.372067\pi$
$41$	5.00956	0.782361	0.391181	+	0.920314i	$0.372067\pi$
$42$	−3.82833	−0.590725
$43$	−2.25813	−0.344362	−0.172181	−	0.985065i	$-0.555081\pi$
$43$	−2.25813	−0.344362	−0.172181	+	0.985065i	$0.555081\pi$
$44$	1.37710	0.207606
$45$	6.50480	0.969678
$46$	5.08473	0.749702
$47$	1.99049	0.290342	0.145171	−	0.989407i	$-0.453627\pi$
$47$	1.99049	0.290342	0.145171	+	0.989407i	$0.453627\pi$
$48$	4.27792	0.617465
$49$	−2.26472	−0.323531
$50$	−1.68840	−0.238775
$51$	10.7335	1.50299
$52$	0	0
$53$	−3.22380	−0.442823	−0.221411	−	0.975180i	$-0.571066\pi$
$53$	−3.22380	−0.442823	−0.221411	+	0.975180i	$0.571066\pi$
$54$	2.06387	0.280857
$55$	−1.36852	−0.184531
$56$	5.10086	0.681631
$57$	1.60999	0.213248
$58$	2.85951	0.375473
$59$	−6.18276	−0.804927	−0.402464	−	0.915436i	$-0.631846\pi$
$59$	−6.18276	−0.804927	−0.402464	+	0.915436i	$0.631846\pi$
$60$	−6.54810	−0.845356
$61$	2.46585	0.315720	0.157860	−	0.987462i	$-0.449541\pi$
$61$	2.46585	0.315720	0.157860	+	0.987462i	$0.449541\pi$
$62$	0.656875	0.0834232
$63$	−9.08102	−1.14410
$64$	0.574175	0.0717718
$65$	0	0
$66$	−1.54459	−0.190126
$67$	6.59746	0.806007	0.403004	−	0.915198i	$-0.367966\pi$
$67$	6.59746	0.806007	0.403004	+	0.915198i	$0.367966\pi$
$68$	−6.28603	−0.762293
$69$	20.7319	2.49583
$70$	−2.22806	−0.266305
$71$	0.570087	0.0676569	0.0338284	−	0.999428i	$-0.489230\pi$
$71$	0.570087	0.0676569	0.0338284	+	0.999428i	$0.489230\pi$
$72$	−9.78210	−1.15283
$73$	6.17960	0.723267	0.361633	−	0.932320i	$-0.382219\pi$
$73$	6.17960	0.723267	0.361633	+	0.932320i	$0.382219\pi$
$74$	1.71346	0.199185
$75$	−6.88408	−0.794906
$76$	−0.942882	−0.108156
$77$	1.91052	0.217723
$78$	0	0
$79$	9.85609	1.10890	0.554448	−	0.832218i	$-0.312929\pi$
$79$	9.85609	1.10890	0.554448	+	0.832218i	$0.312929\pi$
$80$	2.48972	0.278359
$81$	−4.10439	−0.456043
$82$	3.29065	0.363392
$83$	−7.25438	−0.796272	−0.398136	−	0.917326i	$-0.630343\pi$
$83$	−7.25438	−0.796272	−0.398136	+	0.917326i	$0.630343\pi$
$84$	9.14147	0.997416
$85$	6.24683	0.677564
$86$	−1.48331	−0.159949
$87$	11.6591	1.24998
$88$	2.05801	0.219385
$89$	16.2492	1.72242	0.861208	−	0.508252i	$-0.169708\pi$
$89$	16.2492	1.72242	0.861208	+	0.508252i	$0.169708\pi$
$90$	4.27284	0.450397
$91$	0	0
$92$	−12.1415	−1.26584
$93$	2.67827	0.277724
$94$	1.30750	0.134858
$95$	0.937002	0.0961344
$96$	15.3662	1.56830
$97$	15.9725	1.62176	0.810882	−	0.585210i	$-0.198988\pi$
$97$	15.9725	1.62176	0.810882	+	0.585210i	$0.198988\pi$
$98$	−1.48764	−0.150274
$99$	−3.66386	−0.368232
$100$	4.03163	0.403163
$101$	0.816483	0.0812431	0.0406215	−	0.999175i	$-0.487066\pi$
$101$	0.816483	0.0812431	0.0406215	+	0.999175i	$0.487066\pi$
$102$	7.05057	0.698111
$103$	−0.363514	−0.0358181	−0.0179090	−	0.999840i	$-0.505701\pi$
$103$	−0.363514	−0.0358181	−0.0179090	+	0.999840i	$0.505701\pi$
$104$	0	0
$105$	−9.08446	−0.886553
$106$	−2.11763	−0.205683
$107$	−17.8288	−1.72357	−0.861787	−	0.507270i	$-0.830655\pi$
$107$	−17.8288	−1.72357	−0.861787	+	0.507270i	$0.830655\pi$
$108$	−4.92820	−0.474216
$109$	10.3157	0.988069	0.494034	−	0.869442i	$-0.335522\pi$
$109$	10.3157	0.988069	0.494034	+	0.869442i	$0.335522\pi$
$110$	−0.898944	−0.0857109
$111$	6.98626	0.663107
$112$	−3.47577	−0.328430
$113$	15.5910	1.46668	0.733339	−	0.679863i	$-0.237961\pi$
$113$	15.5910	1.46668	0.733339	+	0.679863i	$0.237961\pi$
$114$	1.05756	0.0990497
$115$	12.0658	1.12514
$116$	−6.82808	−0.633971
$117$	0	0
$118$	−4.06130	−0.373873
$119$	−8.72088	−0.799442
$120$	−9.78581	−0.893318
$121$	−10.2292	−0.929925
$122$	1.61975	0.146646
$123$	13.4169	1.20976
$124$	−1.56852	−0.140857
$125$	−11.8002	−1.05544
$126$	−5.96509	−0.531413
$127$	−9.71552	−0.862113	−0.431057	−	0.902325i	$-0.641859\pi$
$127$	−9.71552	−0.862113	−0.431057	+	0.902325i	$0.641859\pi$
$128$	−11.0975	−0.980892
$129$	−6.04788	−0.532486
$130$	0	0
$131$	13.5147	1.18078	0.590392	−	0.807117i	$-0.298973\pi$
$131$	13.5147	1.18078	0.590392	+	0.807117i	$0.298973\pi$
$132$	3.68825	0.321021
$133$	−1.30810	−0.113427
$134$	4.33370	0.374375
$135$	4.89747	0.421507
$136$	−9.39415	−0.805542
$137$	5.15662	0.440560	0.220280	−	0.975437i	$-0.429303\pi$
$137$	5.15662	0.440560	0.220280	+	0.975437i	$0.429303\pi$
$138$	13.6183	1.15926
$139$	22.2319	1.88568	0.942842	−	0.333241i	$-0.108142\pi$
$139$	22.2319	1.88568	0.942842	+	0.333241i	$0.108142\pi$
$140$	5.32027	0.449645
$141$	5.33106	0.448956
$142$	0.374476	0.0314253
$143$	0	0
$144$	6.66562	0.555468
$145$	6.78550	0.563505
$146$	4.05922	0.335944
$147$	−6.06552	−0.500276
$148$	−4.09147	−0.336317
$149$	−4.62898	−0.379221	−0.189611	−	0.981859i	$-0.560723\pi$
$149$	−4.62898	−0.379221	−0.189611	+	0.981859i	$0.560723\pi$
$150$	−4.52198	−0.369218
$151$	4.33901	0.353104	0.176552	−	0.984291i	$-0.443506\pi$
$151$	4.33901	0.353104	0.176552	+	0.984291i	$0.443506\pi$
$152$	−1.40909	−0.114292
$153$	16.7243	1.35208
$154$	1.25497	0.101128
$155$	1.55873	0.125201
$156$	0	0
$157$	−12.7698	−1.01914	−0.509572	−	0.860428i	$-0.670196\pi$
$157$	−12.7698	−1.01914	−0.509572	+	0.860428i	$0.670196\pi$
$158$	6.47422	0.515061
$159$	−8.63420	−0.684737
$160$	8.94299	0.707006
$161$	−16.8445	−1.32753
$162$	−2.69607	−0.211823
$163$	19.4359	1.52234	0.761168	−	0.648554i	$-0.224626\pi$
$163$	19.4359	1.52234	0.761168	+	0.648554i	$0.224626\pi$
$164$	−7.85756	−0.613573
$165$	−3.66525	−0.285340
$166$	−4.76522	−0.369853
$167$	11.6238	0.899476	0.449738	−	0.893160i	$-0.351517\pi$
$167$	11.6238	0.899476	0.449738	+	0.893160i	$0.351517\pi$
$168$	13.6615	1.05401
$169$	0	0
$170$	4.10338	0.314715
$171$	2.50859	0.191837
$172$	3.54191	0.270068
$173$	5.18733	0.394385	0.197193	−	0.980365i	$-0.436818\pi$
$173$	5.18733	0.394385	0.197193	+	0.980365i	$0.436818\pi$
$174$	7.65855	0.580593
$175$	5.59326	0.422811
$176$	−1.40235	−0.105706
$177$	−16.5591	−1.24466
$178$	10.6737	0.800029
$179$	20.0692	1.50005	0.750023	−	0.661412i	$-0.230042\pi$
$179$	20.0692	1.50005	0.750023	+	0.661412i	$0.230042\pi$
$180$	−10.2029	−0.760477
$181$	17.1393	1.27395	0.636976	−	0.770883i	$-0.280185\pi$
$181$	17.1393	1.27395	0.636976	+	0.770883i	$0.280185\pi$
$182$	0	0
$183$	6.60421	0.488197
$184$	−18.1449	−1.33766
$185$	4.06596	0.298935
$186$	1.75929	0.128997
$187$	−3.51856	−0.257303
$188$	−3.12211	−0.227703
$189$	−6.83711	−0.497326
$190$	0.615493	0.0446526
$191$	0.321161	0.0232384	0.0116192	−	0.999932i	$-0.496301\pi$
$191$	0.321161	0.0232384	0.0116192	+	0.999932i	$0.496301\pi$
$192$	1.53779	0.110981
$193$	4.54684	0.327289	0.163644	−	0.986519i	$-0.447675\pi$
$193$	4.54684	0.327289	0.163644	+	0.986519i	$0.447675\pi$
$194$	10.4920	0.753278
$195$	0	0
$196$	3.55224	0.253732
$197$	−9.75213	−0.694811	−0.347405	−	0.937715i	$-0.612937\pi$
$197$	−9.75213	−0.694811	−0.347405	+	0.937715i	$0.612937\pi$
$198$	−2.40670	−0.171037
$199$	−7.30281	−0.517683	−0.258841	−	0.965920i	$-0.583341\pi$
$199$	−7.30281	−0.517683	−0.258841	+	0.965920i	$0.583341\pi$
$200$	6.02507	0.426037
$201$	17.6698	1.24633
$202$	0.536327	0.0377358
$203$	−9.47289	−0.664867
$204$	−16.8357	−1.17873
$205$	7.80856	0.545374
$206$	−0.238783	−0.0166368
$207$	32.3033	2.24523
$208$	0	0
$209$	−0.527772	−0.0365067
$210$	−5.96736	−0.411787
$211$	−14.7678	−1.01666	−0.508330	−	0.861162i	$-0.669737\pi$
$211$	−14.7678	−1.01666	−0.508330	+	0.861162i	$0.669737\pi$
$212$	5.05658	0.347287
$213$	1.52685	0.104618
$214$	−11.7113	−0.800567
$215$	−3.51983	−0.240050
$216$	−7.36495	−0.501121
$217$	−2.17607	−0.147721
$218$	6.77615	0.458939
$219$	16.5506	1.11839
$220$	2.14654	0.144720
$221$	0	0
$222$	4.58910	0.308000
$223$	−24.6628	−1.65155	−0.825773	−	0.564003i	$-0.809261\pi$
$223$	−24.6628	−1.65155	−0.825773	+	0.564003i	$0.809261\pi$
$224$	−12.4849	−0.834180
$225$	−10.7264	−0.715093
$226$	10.2413	0.681244
$227$	−1.39159	−0.0923628	−0.0461814	−	0.998933i	$-0.514705\pi$
$227$	−1.39159	−0.0923628	−0.0461814	+	0.998933i	$0.514705\pi$
$228$	−2.52529	−0.167242
$229$	−24.9011	−1.64551	−0.822756	−	0.568394i	$-0.807565\pi$
$229$	−24.9011	−1.64551	−0.822756	+	0.568394i	$0.807565\pi$
$230$	7.92574	0.522608
$231$	5.11688	0.336666
$232$	−10.2042	−0.669940
$233$	−22.4331	−1.46964	−0.734820	−	0.678263i	$-0.762733\pi$
$233$	−22.4331	−1.46964	−0.734820	+	0.678263i	$0.762733\pi$
$234$	0	0
$235$	3.10264	0.202394
$236$	9.69776	0.631270
$237$	26.3973	1.71469
$238$	−5.72853	−0.371325
$239$	−9.04754	−0.585237	−0.292618	−	0.956229i	$-0.594527\pi$
$239$	−9.04754	−0.585237	−0.292618	+	0.956229i	$0.594527\pi$
$240$	6.66814	0.430427
$241$	−4.46843	−0.287837	−0.143918	−	0.989590i	$-0.545970\pi$
$241$	−4.46843	−0.287837	−0.143918	+	0.989590i	$0.545970\pi$
$242$	−6.71929	−0.431932
$243$	−20.4185	−1.30985
$244$	−3.86772	−0.247606
$245$	−3.53009	−0.225529
$246$	8.81325	0.561912
$247$	0	0
$248$	−2.34407	−0.148848
$249$	−19.4292	−1.23127
$250$	−7.75123	−0.490231
$251$	−16.7302	−1.05600	−0.528001	−	0.849244i	$-0.677058\pi$
$251$	−16.7302	−1.05600	−0.528001	+	0.849244i	$0.677058\pi$
$252$	14.2437	0.897270
$253$	−6.79615	−0.427270
$254$	−6.38188	−0.400435
$255$	16.7307	1.04772
$256$	−8.43804	−0.527377
$257$	13.0642	0.814921	0.407461	−	0.913223i	$-0.366414\pi$
$257$	13.0642	0.814921	0.407461	+	0.913223i	$0.366414\pi$
$258$	−3.97270	−0.247330
$259$	−5.67628	−0.352707
$260$	0	0
$261$	18.1665	1.12448
$262$	8.87746	0.548451
$263$	18.6866	1.15227	0.576134	−	0.817355i	$-0.304561\pi$
$263$	18.6866	1.15227	0.576134	+	0.817355i	$0.304561\pi$
$264$	5.51191	0.339235
$265$	−5.02505	−0.308686
$266$	−0.859259	−0.0526845
$267$	43.5199	2.66337
$268$	−10.3482	−0.632118
$269$	−3.71975	−0.226797	−0.113398	−	0.993550i	$-0.536174\pi$
$269$	−3.71975	−0.226797	−0.113398	+	0.993550i	$0.536174\pi$
$270$	3.21702	0.195782
$271$	25.4488	1.54591	0.772953	−	0.634464i	$-0.218779\pi$
$271$	25.4488	1.54591	0.772953	+	0.634464i	$0.218779\pi$
$272$	6.40127	0.388134
$273$	0	0
$274$	3.38726	0.204632
$275$	2.25668	0.136083
$276$	−32.5183	−1.95737
$277$	16.2270	0.974986	0.487493	−	0.873127i	$-0.337911\pi$
$277$	16.2270	0.974986	0.487493	+	0.873127i	$0.337911\pi$
$278$	14.6036	0.875864
$279$	4.17313	0.249839
$280$	7.95088	0.475156
$281$	10.9457	0.652966	0.326483	−	0.945203i	$-0.394136\pi$
$281$	10.9457	0.652966	0.326483	+	0.945203i	$0.394136\pi$
$282$	3.50184	0.208531
$283$	−27.2032	−1.61706	−0.808532	−	0.588452i	$-0.799738\pi$
$283$	−27.2032	−1.61706	−0.808532	+	0.588452i	$0.799738\pi$
$284$	−0.894190	−0.0530604
$285$	2.50954	0.148653
$286$	0	0
$287$	−10.9011	−0.643474
$288$	23.9427	1.41084
$289$	−0.938913	−0.0552302
$290$	4.45722	0.261737
$291$	42.7787	2.50773
$292$	−9.69279	−0.567228
$293$	−32.5419	−1.90112	−0.950559	−	0.310544i	$-0.899489\pi$
$293$	−32.5419	−1.90112	−0.950559	+	0.310544i	$0.899489\pi$
$294$	−3.98429	−0.232369
$295$	−9.63728	−0.561104
$296$	−6.11450	−0.355398
$297$	−2.75853	−0.160066
$298$	−3.04066	−0.176141
$299$	0	0
$300$	10.7978	0.623411
$301$	4.91385	0.283230
$302$	2.85019	0.164010
$303$	2.18676	0.125626
$304$	0.960168	0.0550694
$305$	3.84360	0.220084
$306$	10.9858	0.628017
$307$	−27.5404	−1.57181	−0.785906	−	0.618346i	$-0.787803\pi$
$307$	−27.5404	−1.57181	−0.785906	+	0.618346i	$0.787803\pi$
$308$	−2.99667	−0.170751
$309$	−0.973588	−0.0553855
$310$	1.02389	0.0581532
$311$	−27.8161	−1.57731	−0.788653	−	0.614838i	$-0.789221\pi$
$311$	−27.8161	−1.57731	−0.788653	+	0.614838i	$0.789221\pi$
$312$	0	0
$313$	13.7900	0.779458	0.389729	−	0.920930i	$-0.372569\pi$
$313$	13.7900	0.779458	0.389729	+	0.920930i	$0.372569\pi$
$314$	−8.38818	−0.473372
$315$	−14.1549	−0.797538
$316$	−15.4594	−0.869661
$317$	−5.05640	−0.283995	−0.141998	−	0.989867i	$-0.545353\pi$
$317$	−5.05640	−0.283995	−0.141998	+	0.989867i	$0.545353\pi$
$318$	−5.67159	−0.318047
$319$	−3.82197	−0.213989
$320$	0.894986	0.0500312
$321$	−47.7503	−2.66516
$322$	−11.0647	−0.616613
$323$	2.40911	0.134046
$324$	6.43780	0.357655
$325$	0	0
$326$	12.7670	0.707096
$327$	27.6283	1.52785
$328$	−11.7427	−0.648384
$329$	−4.33144	−0.238800
$330$	−2.40761	−0.132535
$331$	−31.5996	−1.73687	−0.868436	−	0.495801i	$-0.834875\pi$
$331$	−31.5996	−1.73687	−0.868436	+	0.495801i	$0.834875\pi$
$332$	11.3786	0.624482
$333$	10.8856	0.596527
$334$	7.63538	0.417789
$335$	10.2837	0.561857
$336$	−9.30906	−0.507851
$337$	17.6494	0.961425	0.480713	−	0.876878i	$-0.340378\pi$
$337$	17.6494	0.961425	0.480713	+	0.876878i	$0.340378\pi$
$338$	0	0
$339$	41.7569	2.26792
$340$	−9.79825	−0.531384
$341$	−0.877966	−0.0475445
$342$	1.64783	0.0891045
$343$	20.1607	1.08857
$344$	5.29321	0.285391
$345$	32.3155	1.73981
$346$	3.40743	0.183184
$347$	−6.21085	−0.333416	−0.166708	−	0.986006i	$-0.553314\pi$
$347$	−6.21085	−0.333416	−0.166708	+	0.986006i	$0.553314\pi$
$348$	−18.2874	−0.980309
$349$	28.3495	1.51752	0.758758	−	0.651373i	$-0.225807\pi$
$349$	28.3495	1.51752	0.758758	+	0.651373i	$0.225807\pi$
$350$	3.67407	0.196387
$351$	0	0
$352$	−5.03719	−0.268483
$353$	−29.8862	−1.59068	−0.795341	−	0.606162i	$-0.792708\pi$
$353$	−29.8862	−1.59068	−0.795341	+	0.606162i	$0.792708\pi$
$354$	−10.8773	−0.578120
$355$	0.888614	0.0471627
$356$	−25.4872	−1.35082
$357$	−23.3569	−1.23618
$358$	13.1830	0.696742
$359$	−12.6922	−0.669869	−0.334934	−	0.942241i	$-0.608714\pi$
$359$	−12.6922	−0.669869	−0.334934	+	0.942241i	$0.608714\pi$
$360$	−15.2477	−0.803624
$361$	−18.6386	−0.980981
$362$	11.2584	0.591727
$363$	−27.3965	−1.43794
$364$	0	0
$365$	9.63235	0.504180
$366$	4.33814	0.226758
$367$	6.55762	0.342305	0.171152	−	0.985245i	$-0.445251\pi$
$367$	6.55762	0.342305	0.171152	+	0.985245i	$0.445251\pi$
$368$	12.3641	0.644525
$369$	20.9055	1.08830
$370$	2.67082	0.138850
$371$	7.01521	0.364212
$372$	−4.20091	−0.217807
$373$	2.47432	0.128115	0.0640577	−	0.997946i	$-0.479596\pi$
$373$	2.47432	0.128115	0.0640577	+	0.997946i	$0.479596\pi$
$374$	−2.31125	−0.119512
$375$	−31.6040	−1.63202
$376$	−4.66584	−0.240622
$377$	0	0
$378$	−4.49112	−0.230999
$379$	−19.7360	−1.01377	−0.506885	−	0.862014i	$-0.669203\pi$
$379$	−19.7360	−1.01377	−0.506885	+	0.862014i	$0.669203\pi$
$380$	−1.46970	−0.0753941
$381$	−26.0208	−1.33309
$382$	0.210962	0.0107938
$383$	19.3671	0.989614	0.494807	−	0.869003i	$-0.335239\pi$
$383$	19.3671	0.989614	0.494807	+	0.869003i	$0.335239\pi$
$384$	−29.7222	−1.51675
$385$	2.97799	0.151772
$386$	2.98671	0.152019
$387$	−9.42347	−0.479022
$388$	−25.0532	−1.27188
$389$	−31.2291	−1.58338	−0.791688	−	0.610925i	$-0.790798\pi$
$389$	−31.2291	−1.58338	−0.791688	+	0.610925i	$0.790798\pi$
$390$	0	0
$391$	31.0222	1.56886
$392$	5.30865	0.268127
$393$	36.1960	1.82585
$394$	−6.40593	−0.322726
$395$	15.3630	0.772998
$396$	5.74683	0.288789
$397$	−8.14879	−0.408976	−0.204488	−	0.978869i	$-0.565553\pi$
$397$	−8.14879	−0.408976	−0.204488	+	0.978869i	$0.565553\pi$
$398$	−4.79704	−0.240454
$399$	−3.50345	−0.175392
$400$	−4.10554	−0.205277
$401$	−6.37211	−0.318208	−0.159104	−	0.987262i	$-0.550861\pi$
$401$	−6.37211	−0.318208	−0.159104	+	0.987262i	$0.550861\pi$
$402$	11.6068	0.578896
$403$	0	0
$404$	−1.28067	−0.0637155
$405$	−6.39765	−0.317902
$406$	−6.22250	−0.308818
$407$	−2.29017	−0.113520
$408$	−25.1601	−1.24561
$409$	9.57256	0.473332	0.236666	−	0.971591i	$-0.423945\pi$
$409$	9.57256	0.473332	0.236666	+	0.971591i	$0.423945\pi$
$410$	5.12925	0.253316
$411$	13.8108	0.681238
$412$	0.570177	0.0280906
$413$	13.4541	0.662034
$414$	21.2192	1.04287
$415$	−11.3077	−0.555071
$416$	0	0
$417$	59.5430	2.91583
$418$	−0.346680	−0.0169567
$419$	−22.4308	−1.09582	−0.547908	−	0.836538i	$-0.684576\pi$
$419$	−22.4308	−1.09582	−0.547908	+	0.836538i	$0.684576\pi$
$420$	14.2491	0.695286
$421$	−27.5293	−1.34169	−0.670847	−	0.741596i	$-0.734069\pi$
$421$	−27.5293	−1.34169	−0.670847	+	0.741596i	$0.734069\pi$
$422$	−9.70062	−0.472219
$423$	8.30655	0.403878
$424$	7.55680	0.366991
$425$	−10.3010	−0.499672
$426$	1.00295	0.0485929
$427$	−5.36586	−0.259672
$428$	27.9647	1.35173
$429$	0	0
$430$	−2.31209	−0.111499
$431$	−14.2990	−0.688759	−0.344379	−	0.938831i	$-0.611911\pi$
$431$	−14.2990	−0.688759	−0.344379	+	0.938831i	$0.611911\pi$
$432$	5.01855	0.241455
$433$	−6.73207	−0.323523	−0.161761	−	0.986830i	$-0.551717\pi$
$433$	−6.73207	−0.323523	−0.161761	+	0.986830i	$0.551717\pi$
$434$	−1.42941	−0.0686137
$435$	18.1734	0.871347
$436$	−16.1804	−0.774901
$437$	4.65322	0.222594
$438$	10.8717	0.519469
$439$	−20.3619	−0.971819	−0.485910	−	0.874009i	$-0.661512\pi$
$439$	−20.3619	−0.971819	−0.485910	+	0.874009i	$0.661512\pi$
$440$	3.20789	0.152930
$441$	−9.45096	−0.450045
$442$	0	0
$443$	17.6424	0.838216	0.419108	−	0.907936i	$-0.362343\pi$
$443$	17.6424	0.838216	0.419108	+	0.907936i	$0.362343\pi$
$444$	−10.9581	−0.520046
$445$	25.3283	1.20067
$446$	−16.2004	−0.767111
$447$	−12.3977	−0.586389
$448$	−1.24944	−0.0590307
$449$	−19.1462	−0.903565	−0.451782	−	0.892128i	$-0.649212\pi$
$449$	−19.1462	−0.903565	−0.451782	+	0.892128i	$0.649212\pi$
$450$	−7.04590	−0.332147
$451$	−4.39822	−0.207104
$452$	−24.4547	−1.15025
$453$	11.6210	0.546004
$454$	−0.914097	−0.0429007
$455$	0	0
$456$	−3.77392	−0.176730
$457$	26.0477	1.21846	0.609230	−	0.792994i	$-0.291479\pi$
$457$	26.0477	1.21846	0.609230	+	0.792994i	$0.291479\pi$
$458$	−16.3569	−0.764309
$459$	12.5918	0.587734
$460$	−18.9254	−0.882403
$461$	−4.57637	−0.213143	−0.106571	−	0.994305i	$-0.533987\pi$
$461$	−4.57637	−0.213143	−0.106571	+	0.994305i	$0.533987\pi$
$462$	3.36115	0.156375
$463$	−10.6466	−0.494790	−0.247395	−	0.968915i	$-0.579574\pi$
$463$	−10.6466	−0.494790	−0.247395	+	0.968915i	$0.579574\pi$
$464$	6.95326	0.322797
$465$	4.17471	0.193598
$466$	−14.7357	−0.682619
$467$	−15.0034	−0.694275	−0.347138	−	0.937814i	$-0.612846\pi$
$467$	−15.0034	−0.694275	−0.347138	+	0.937814i	$0.612846\pi$
$468$	0	0
$469$	−14.3565	−0.662923
$470$	2.03805	0.0940080
$471$	−34.2010	−1.57590
$472$	14.4928	0.667086
$473$	1.98256	0.0911583
$474$	17.3397	0.796439
$475$	−1.54511	−0.0708947
$476$	13.6788	0.626968
$477$	−13.4533	−0.615985
$478$	−5.94310	−0.271831
$479$	37.1477	1.69732	0.848661	−	0.528937i	$-0.177409\pi$
$479$	37.1477	1.69732	0.848661	+	0.528937i	$0.177409\pi$
$480$	23.9517	1.09324
$481$	0	0
$482$	−2.93520	−0.133695
$483$	−45.1141	−2.05276
$484$	16.0446	0.729301
$485$	24.8969	1.13051
$486$	−13.4124	−0.608399
$487$	23.1824	1.05049	0.525247	−	0.850950i	$-0.323973\pi$
$487$	23.1824	1.05049	0.525247	+	0.850950i	$0.323973\pi$
$488$	−5.78012	−0.261654
$489$	52.0546	2.35399
$490$	−2.31883	−0.104754
$491$	−26.1739	−1.18121	−0.590606	−	0.806960i	$-0.701111\pi$
$491$	−26.1739	−1.18121	−0.590606	+	0.806960i	$0.701111\pi$
$492$	−21.0447	−0.948767
$493$	17.4460	0.785730
$494$	0	0
$495$	−5.71099	−0.256690
$496$	1.59727	0.0717196
$497$	−1.24055	−0.0556462
$498$	−12.7625	−0.571903
$499$	38.5295	1.72482	0.862408	−	0.506215i	$-0.168956\pi$
$499$	38.5295	1.72482	0.862408	+	0.506215i	$0.168956\pi$
$500$	18.5087	0.827736
$501$	31.1317	1.39086
$502$	−10.9897	−0.490492
$503$	−23.8172	−1.06195	−0.530977	−	0.847386i	$-0.678175\pi$
$503$	−23.8172	−1.06195	−0.530977	+	0.847386i	$0.678175\pi$
$504$	21.2865	0.948177
$505$	1.27268	0.0566335
$506$	−4.46422	−0.198459
$507$	0	0
$508$	15.2389	0.676119
$509$	10.5621	0.468156	0.234078	−	0.972218i	$-0.424793\pi$
$509$	10.5621	0.468156	0.234078	+	0.972218i	$0.424793\pi$
$510$	10.9900	0.486644
$511$	−13.4472	−0.594871
$512$	16.6523	0.735935
$513$	1.88872	0.0833891
$514$	8.58153	0.378515
$515$	−0.566621	−0.0249683
$516$	9.48620	0.417607
$517$	−1.74758	−0.0768584
$518$	−3.72860	−0.163825
$519$	13.8931	0.609838
$520$	0	0
$521$	−28.8357	−1.26331	−0.631657	−	0.775248i	$-0.717625\pi$
$521$	−28.8357	−1.26331	−0.631657	+	0.775248i	$0.717625\pi$
$522$	11.9331	0.522298
$523$	−13.2684	−0.580188	−0.290094	−	0.956998i	$-0.593687\pi$
$523$	−13.2684	−0.580188	−0.290094	+	0.956998i	$0.593687\pi$
$524$	−21.1980	−0.926039
$525$	14.9803	0.653792
$526$	12.2748	0.535206
$527$	4.00763	0.174575
$528$	−3.75587	−0.163453
$529$	36.9198	1.60521
$530$	−3.30083	−0.143379
$531$	−25.8015	−1.11969
$532$	2.05178	0.0889558
$533$	0	0
$534$	28.5871	1.23708
$535$	−27.7903	−1.20148
$536$	−15.4649	−0.667981
$537$	53.7508	2.31952
$538$	−2.44341	−0.105343
$539$	1.98834	0.0856441
$540$	−7.68175	−0.330570
$541$	7.22402	0.310585	0.155292	−	0.987869i	$-0.450368\pi$
$541$	7.22402	0.310585	0.155292	+	0.987869i	$0.450368\pi$
$542$	16.7167	0.718043
$543$	45.9036	1.96991
$544$	22.9931	0.985823
$545$	16.0795	0.688770
$546$	0	0
$547$	−10.7901	−0.461352	−0.230676	−	0.973031i	$-0.574094\pi$
$547$	−10.7901	−0.461352	−0.230676	+	0.973031i	$0.574094\pi$
$548$	−8.08824	−0.345513
$549$	10.2903	0.439179
$550$	1.48235	0.0632078
$551$	2.61685	0.111481
$552$	−48.5970	−2.06843
$553$	−21.4475	−0.912042
$554$	10.6591	0.452862
$555$	10.8897	0.462243
$556$	−34.8710	−1.47886
$557$	12.8194	0.543175	0.271588	−	0.962414i	$-0.412451\pi$
$557$	12.8194	0.543175	0.271588	+	0.962414i	$0.412451\pi$
$558$	2.74122	0.116045
$559$	0	0
$560$	−5.41781	−0.228944
$561$	−9.42366	−0.397867
$562$	7.18996	0.303290
$563$	−33.1561	−1.39736	−0.698680	−	0.715434i	$-0.746229\pi$
$563$	−33.1561	−1.39736	−0.698680	+	0.715434i	$0.746229\pi$
$564$	−8.36185	−0.352097
$565$	24.3022	1.02240
$566$	−17.8691	−0.751095
$567$	8.93144	0.375085
$568$	−1.33632	−0.0560709
$569$	−14.1575	−0.593512	−0.296756	−	0.954953i	$-0.595905\pi$
$569$	−14.1575	−0.593512	−0.296756	+	0.954953i	$0.595905\pi$
$570$	1.64846	0.0690462
$571$	−5.28503	−0.221172	−0.110586	−	0.993867i	$-0.535273\pi$
$571$	−5.28503	−0.221172	−0.110586	+	0.993867i	$0.535273\pi$
$572$	0	0
$573$	0.860155	0.0359335
$574$	−7.16069	−0.298881
$575$	−19.8965	−0.829742
$576$	2.39610	0.0998377
$577$	4.65661	0.193857	0.0969286	−	0.995291i	$-0.469098\pi$
$577$	4.65661	0.193857	0.0969286	+	0.995291i	$0.469098\pi$
$578$	−0.616748	−0.0256534
$579$	12.1777	0.506086
$580$	−10.6432	−0.441933
$581$	15.7860	0.654916
$582$	28.1003	1.16479
$583$	2.83039	0.117223
$584$	−14.4854	−0.599410
$585$	0	0
$586$	−21.3760	−0.883033
$587$	7.30288	0.301422	0.150711	−	0.988578i	$-0.451844\pi$
$587$	7.30288	0.301422	0.150711	+	0.988578i	$0.451844\pi$
$588$	9.51387	0.392345
$589$	0.601130	0.0247691
$590$	−6.33049	−0.260622
$591$	−26.1188	−1.07439
$592$	4.16648	0.171241
$593$	−11.8112	−0.485030	−0.242515	−	0.970148i	$-0.577972\pi$
$593$	−11.8112	−0.485030	−0.242515	+	0.970148i	$0.577972\pi$
$594$	−1.81201	−0.0743475
$595$	−13.5935	−0.557281
$596$	7.26063	0.297407
$597$	−19.5589	−0.800493
$598$	0	0
$599$	−20.1734	−0.824262	−0.412131	−	0.911125i	$-0.635215\pi$
$599$	−20.1734	−0.824262	−0.412131	+	0.911125i	$0.635215\pi$
$600$	16.1368	0.658781
$601$	35.7716	1.45916	0.729578	−	0.683898i	$-0.239717\pi$
$601$	35.7716	1.45916	0.729578	+	0.683898i	$0.239717\pi$
$602$	3.22779	0.131555
$603$	27.5320	1.12119
$604$	−6.80580	−0.276924
$605$	−15.9446	−0.648239
$606$	1.43643	0.0583509
$607$	29.9711	1.21649	0.608245	−	0.793750i	$-0.291874\pi$
$607$	29.9711	1.21649	0.608245	+	0.793750i	$0.291874\pi$
$608$	3.44889	0.139871
$609$	−25.3710	−1.02808
$610$	2.52477	0.102225
$611$	0	0
$612$	−26.2324	−1.06038
$613$	15.7324	0.635425	0.317713	−	0.948187i	$-0.397085\pi$
$613$	15.7324	0.635425	0.317713	+	0.948187i	$0.397085\pi$
$614$	−18.0906	−0.730076
$615$	20.9134	0.843311
$616$	−4.47838	−0.180439
$617$	−46.9097	−1.88851	−0.944257	−	0.329209i	$-0.893218\pi$
$617$	−46.9097	−1.88851	−0.944257	+	0.329209i	$0.893218\pi$
$618$	−0.639525	−0.0257255
$619$	−23.5719	−0.947436	−0.473718	−	0.880677i	$-0.657088\pi$
$619$	−23.5719	−0.947436	−0.473718	+	0.880677i	$0.657088\pi$
$620$	−2.44490	−0.0981895
$621$	24.3212	0.975975
$622$	−18.2717	−0.732628
$623$	−35.3595	−1.41665
$624$	0	0
$625$	−5.54157	−0.221663
$626$	9.05832	0.362043
$627$	−1.41351	−0.0564503
$628$	20.0297	0.799271
$629$	10.4539	0.416824
$630$	−9.29800	−0.370441
$631$	−35.1311	−1.39855	−0.699274	−	0.714854i	$-0.746493\pi$
$631$	−35.1311	−1.39855	−0.699274	+	0.714854i	$0.746493\pi$
$632$	−23.1033	−0.919002
$633$	−39.5522	−1.57206
$634$	−3.32142	−0.131910
$635$	−15.1439	−0.600968
$636$	13.5429	0.537010
$637$	0	0
$638$	−2.51056	−0.0993939
$639$	2.37905	0.0941136
$640$	−17.2981	−0.683767
$641$	17.3741	0.686235	0.343118	−	0.939292i	$-0.388517\pi$
$641$	17.3741	0.686235	0.343118	+	0.939292i	$0.388517\pi$
$642$	−31.3660	−1.23792
$643$	−43.7159	−1.72399	−0.861993	−	0.506920i	$-0.830784\pi$
$643$	−43.7159	−1.72399	−0.861993	+	0.506920i	$0.830784\pi$
$644$	26.4208	1.04113
$645$	−9.42704	−0.371189
$646$	1.58248	0.0622619
$647$	31.6896	1.24585	0.622924	−	0.782283i	$-0.285945\pi$
$647$	31.6896	1.24585	0.622924	+	0.782283i	$0.285945\pi$
$648$	9.62097	0.377947
$649$	5.42826	0.213078
$650$	0	0
$651$	−5.82810	−0.228421
$652$	−30.4855	−1.19390
$653$	1.10032	0.0430587	0.0215293	−	0.999768i	$-0.493146\pi$
$653$	1.10032	0.0430587	0.0215293	+	0.999768i	$0.493146\pi$
$654$	18.1484	0.709657
$655$	21.0658	0.823109
$656$	8.00162	0.312411
$657$	25.7882	1.00610
$658$	−2.84521	−0.110918
$659$	26.0015	1.01288	0.506438	−	0.862277i	$-0.330962\pi$
$659$	26.0015	1.01288	0.506438	+	0.862277i	$0.330962\pi$
$660$	5.74901	0.223780
$661$	−9.05504	−0.352200	−0.176100	−	0.984372i	$-0.556348\pi$
$661$	−9.05504	−0.352200	−0.176100	+	0.984372i	$0.556348\pi$
$662$	−20.7570	−0.806744
$663$	0	0
$664$	17.0048	0.659913
$665$	−2.03898	−0.0790683
$666$	7.15047	0.277075
$667$	33.6973	1.30476
$668$	−18.2321	−0.705421
$669$	−66.0537	−2.55378
$670$	6.75509	0.260972
$671$	−2.16493	−0.0835763
$672$	−33.4378	−1.28989
$673$	32.2884	1.24463	0.622313	−	0.782769i	$-0.286193\pi$
$673$	32.2884	1.24463	0.622313	+	0.782769i	$0.286193\pi$
$674$	11.5935	0.446564
$675$	−8.07591	−0.310842
$676$	0	0
$677$	−45.1998	−1.73717	−0.868585	−	0.495540i	$-0.834970\pi$
$677$	−45.1998	−1.73717	−0.868585	+	0.495540i	$0.834970\pi$
$678$	27.4291	1.05341
$679$	−34.7573	−1.33386
$680$	−14.6430	−0.561533
$681$	−3.72704	−0.142820
$682$	−0.576714	−0.0220835
$683$	−13.3857	−0.512189	−0.256094	−	0.966652i	$-0.582436\pi$
$683$	−13.3857	−0.512189	−0.256094	+	0.966652i	$0.582436\pi$
$684$	−3.93477	−0.150450
$685$	8.03780	0.307109
$686$	13.2430	0.505622
$687$	−66.6919	−2.54446
$688$	−3.60685	−0.137510
$689$	0	0
$690$	21.2273	0.808108
$691$	0.244786	0.00931211	0.00465605	−	0.999989i	$-0.498518\pi$
$691$	0.244786	0.00931211	0.00465605	+	0.999989i	$0.498518\pi$
$692$	−8.13641	−0.309300
$693$	7.97283	0.302863
$694$	−4.07975	−0.154865
$695$	34.6536	1.31449
$696$	−27.3297	−1.03593
$697$	20.0764	0.760449
$698$	18.6221	0.704857
$699$	−60.0818	−2.27250
$700$	−8.77311	−0.331592
$701$	21.6400	0.817330	0.408665	−	0.912684i	$-0.365994\pi$
$701$	21.6400	0.817330	0.408665	+	0.912684i	$0.365994\pi$
$702$	0	0
$703$	1.56805	0.0591400
$704$	−0.504106	−0.0189992
$705$	8.30970	0.312962
$706$	−19.6315	−0.738841
$707$	−1.77672	−0.0668206
$708$	25.9732	0.976133
$709$	−3.51683	−0.132077	−0.0660387	−	0.997817i	$-0.521036\pi$
$709$	−3.51683	−0.132077	−0.0660387	+	0.997817i	$0.521036\pi$
$710$	0.583708	0.0219062
$711$	41.1307	1.54252
$712$	−38.0893	−1.42746
$713$	7.74079	0.289895
$714$	−15.3425	−0.574180
$715$	0	0
$716$	−31.4789	−1.17642
$717$	−24.2318	−0.904951
$718$	−8.33719	−0.311141
$719$	3.98915	0.148770	0.0743852	−	0.997230i	$-0.476301\pi$
$719$	3.98915	0.148770	0.0743852	+	0.997230i	$0.476301\pi$
$720$	10.3899	0.387210
$721$	0.791031	0.0294596
$722$	−12.2433	−0.455647
$723$	−11.9677	−0.445082
$724$	−26.8832	−0.999107
$725$	−11.1893	−0.415559
$726$	−17.9961	−0.667897
$727$	−2.47671	−0.0918562	−0.0459281	−	0.998945i	$-0.514625\pi$
$727$	−2.47671	−0.0918562	−0.0459281	+	0.998945i	$0.514625\pi$
$728$	0	0
$729$	−42.3731	−1.56937
$730$	6.32725	0.234182
$731$	−9.04975	−0.334717
$732$	−10.3588	−0.382872
$733$	−25.9259	−0.957593	−0.478797	−	0.877926i	$-0.658927\pi$
$733$	−25.9259	−0.957593	−0.478797	+	0.877926i	$0.658927\pi$
$734$	4.30753	0.158994
$735$	−9.45454	−0.348736
$736$	44.4115	1.63703
$737$	−5.79234	−0.213364
$738$	13.7323	0.505493
$739$	43.5409	1.60168	0.800840	−	0.598879i	$-0.204387\pi$
$739$	43.5409	1.60168	0.800840	+	0.598879i	$0.204387\pi$
$740$	−6.37751	−0.234442
$741$	0	0
$742$	4.60812	0.169169
$743$	−16.0866	−0.590161	−0.295081	−	0.955472i	$-0.595347\pi$
$743$	−16.0866	−0.590161	−0.295081	+	0.955472i	$0.595347\pi$
$744$	−6.27805	−0.230164
$745$	−7.21535	−0.264350
$746$	1.62532	0.0595072
$747$	−30.2735	−1.10765
$748$	5.51892	0.201792
$749$	38.7967	1.41760
$750$	−20.7599	−0.758044
$751$	37.4826	1.36776	0.683879	−	0.729595i	$-0.260292\pi$
$751$	37.4826	1.36776	0.683879	+	0.729595i	$0.260292\pi$
$752$	3.17935	0.115939
$753$	−44.8080	−1.63290
$754$	0	0
$755$	6.76336	0.246144
$756$	10.7241	0.390032
$757$	10.6822	0.388253	0.194126	−	0.980977i	$-0.437813\pi$
$757$	10.6822	0.388253	0.194126	+	0.980977i	$0.437813\pi$
$758$	−12.9641	−0.470877
$759$	−18.2019	−0.660687
$760$	−2.19640	−0.0796717
$761$	−16.4984	−0.598068	−0.299034	−	0.954242i	$-0.596664\pi$
$761$	−16.4984	−0.598068	−0.299034	+	0.954242i	$0.596664\pi$
$762$	−17.0924	−0.619193
$763$	−22.4478	−0.812664
$764$	−0.503745	−0.0182249
$765$	26.0688	0.942520
$766$	12.7218	0.459657
$767$	0	0
$768$	−22.5993	−0.815483
$769$	38.3588	1.38325	0.691627	−	0.722255i	$-0.256894\pi$
$769$	38.3588	1.38325	0.691627	+	0.722255i	$0.256894\pi$
$770$	1.95616	0.0704953
$771$	34.9894	1.26011
$772$	−7.13179	−0.256679
$773$	10.2453	0.368498	0.184249	−	0.982880i	$-0.441015\pi$
$773$	10.2453	0.368498	0.184249	+	0.982880i	$0.441015\pi$
$774$	−6.19004	−0.222496
$775$	−2.57035	−0.0923296
$776$	−37.4407	−1.34404
$777$	−15.2026	−0.545390
$778$	−20.5136	−0.735448
$779$	3.01139	0.107894
$780$	0	0
$781$	−0.500517	−0.0179099
$782$	20.3777	0.728705
$783$	13.6776	0.488797
$784$	−3.61737	−0.129192
$785$	−19.9048	−0.710432
$786$	23.7762	0.848070
$787$	−21.3288	−0.760291	−0.380145	−	0.924927i	$-0.624126\pi$
$787$	−21.3288	−0.760291	−0.380145	+	0.924927i	$0.624126\pi$
$788$	15.2964	0.544911
$789$	50.0479	1.78175
$790$	10.0916	0.359042
$791$	−33.9271	−1.20631
$792$	8.58835	0.305174
$793$	0	0
$794$	−5.35274	−0.189962
$795$	−13.4584	−0.477321
$796$	11.4546	0.405997
$797$	52.2408	1.85046	0.925231	−	0.379403i	$-0.123871\pi$
$797$	52.2408	1.85046	0.925231	+	0.379403i	$0.123871\pi$
$798$	−2.30133	−0.0814661
$799$	7.97713	0.282211
$800$	−14.7470	−0.521384
$801$	67.8102	2.39595
$802$	−4.18568	−0.147802
$803$	−5.42547	−0.191461
$804$	−27.7153	−0.977443
$805$	−26.2561	−0.925406
$806$	0	0
$807$	−9.96248	−0.350696
$808$	−1.91389	−0.0673305
$809$	41.0611	1.44363	0.721815	−	0.692086i	$-0.243308\pi$
$809$	41.0611	1.44363	0.721815	+	0.692086i	$0.243308\pi$
$810$	−4.20246	−0.147659
$811$	12.9575	0.455001	0.227500	−	0.973778i	$-0.426945\pi$
$811$	12.9575	0.455001	0.227500	+	0.973778i	$0.426945\pi$
$812$	14.8584	0.521427
$813$	68.1588	2.39043
$814$	−1.50436	−0.0527277
$815$	30.2954	1.06120
$816$	17.1443	0.600171
$817$	−1.35743	−0.0474905
$818$	6.28797	0.219854
$819$	0	0
$820$	−12.2479	−0.427714
$821$	21.9928	0.767554	0.383777	−	0.923426i	$-0.374623\pi$
$821$	21.9928	0.767554	0.383777	+	0.923426i	$0.374623\pi$
$822$	9.07198	0.316422
$823$	2.80755	0.0978650	0.0489325	−	0.998802i	$-0.484418\pi$
$823$	2.80755	0.0978650	0.0489325	+	0.998802i	$0.484418\pi$
$824$	0.852101	0.0296843
$825$	6.04399	0.210425
$826$	8.83768	0.307502
$827$	33.6672	1.17072	0.585362	−	0.810772i	$-0.300952\pi$
$827$	33.6672	1.17072	0.585362	+	0.810772i	$0.300952\pi$
$828$	−50.6682	−1.76084
$829$	31.8293	1.10548	0.552738	−	0.833355i	$-0.313583\pi$
$829$	31.8293	1.10548	0.552738	+	0.833355i	$0.313583\pi$
$830$	−7.42771	−0.257820
$831$	43.4603	1.50762
$832$	0	0
$833$	−9.07615	−0.314470
$834$	39.1123	1.35435
$835$	18.1184	0.627013
$836$	0.827818	0.0286307
$837$	3.14195	0.108602
$838$	−14.7342	−0.508986
$839$	−5.70452	−0.196942	−0.0984709	−	0.995140i	$-0.531395\pi$
$839$	−5.70452	−0.196942	−0.0984709	+	0.995140i	$0.531395\pi$
$840$	21.2946	0.734734
$841$	−10.0496	−0.346537
$842$	−18.0833	−0.623191
$843$	29.3155	1.00968
$844$	23.1636	0.797323
$845$	0	0
$846$	5.45637	0.187594
$847$	22.2594	0.764842
$848$	−5.14928	−0.176827
$849$	−72.8576	−2.50047
$850$	−6.76647	−0.232088
$851$	20.1918	0.692167
$852$	−2.39488	−0.0820473
$853$	5.83724	0.199863	0.0999316	−	0.994994i	$-0.468138\pi$
$853$	5.83724	0.199863	0.0999316	+	0.994994i	$0.468138\pi$
$854$	−3.52470	−0.120613
$855$	3.91023	0.133727
$856$	41.7919	1.42842
$857$	−1.10228	−0.0376530	−0.0188265	−	0.999823i	$-0.505993\pi$
$857$	−1.10228	−0.0376530	−0.0188265	+	0.999823i	$0.505993\pi$
$858$	0	0
$859$	13.0961	0.446834	0.223417	−	0.974723i	$-0.428279\pi$
$859$	13.0961	0.446834	0.223417	+	0.974723i	$0.428279\pi$
$860$	5.52090	0.188261
$861$	−29.1962	−0.995004
$862$	−9.39266	−0.319915
$863$	0.0939544	0.00319824	0.00159912	−	0.999999i	$-0.499491\pi$
$863$	0.0939544	0.00319824	0.00159912	+	0.999999i	$0.499491\pi$
$864$	18.0265	0.613273
$865$	8.08567	0.274921
$866$	−4.42213	−0.150270
$867$	−2.51466	−0.0854024
$868$	3.41320	0.115852
$869$	−8.65331	−0.293543
$870$	11.9376	0.404724
$871$	0	0
$872$	−24.1808	−0.818865
$873$	66.6554	2.25594
$874$	3.05658	0.103390
$875$	25.6780	0.868074
$876$	−25.9599	−0.877104
$877$	−52.5489	−1.77445	−0.887226	−	0.461335i	$-0.847371\pi$
$877$	−52.5489	−1.77445	−0.887226	+	0.461335i	$0.847371\pi$
$878$	−13.3752	−0.451391
$879$	−87.1560	−2.93970
$880$	−2.18589	−0.0736864
$881$	58.3414	1.96557	0.982786	−	0.184746i	$-0.0591463\pi$
$881$	58.3414	1.96557	0.982786	+	0.184746i	$0.0591463\pi$
$882$	−6.20810	−0.209037
$883$	−38.7524	−1.30412	−0.652061	−	0.758166i	$-0.726095\pi$
$883$	−38.7524	−1.30412	−0.652061	+	0.758166i	$0.726095\pi$
$884$	0	0
$885$	−25.8112	−0.867635
$886$	11.5888	0.389335
$887$	−16.4399	−0.551998	−0.275999	−	0.961158i	$-0.589009\pi$
$887$	−16.4399	−0.551998	−0.275999	+	0.961158i	$0.589009\pi$
$888$	−16.3763	−0.549552
$889$	21.1417	0.709069
$890$	16.6375	0.557690
$891$	3.60351	0.120722
$892$	38.6840	1.29524
$893$	1.19654	0.0400407
$894$	−8.14371	−0.272367
$895$	31.2826	1.04566
$896$	24.1490	0.806761
$897$	0	0
$898$	−12.5767	−0.419688
$899$	4.35321	0.145188
$900$	16.8245	0.560817
$901$	−12.9198	−0.430421
$902$	−2.88908	−0.0961958
$903$	13.1606	0.437958
$904$	−36.5464	−1.21551
$905$	26.7156	0.888056
$906$	7.63357	0.253608
$907$	40.2105	1.33517	0.667584	−	0.744534i	$-0.267328\pi$
$907$	40.2105	1.33517	0.667584	+	0.744534i	$0.267328\pi$
$908$	2.18272	0.0724362
$909$	3.40729	0.113013
$910$	0	0
$911$	−24.4143	−0.808881	−0.404441	−	0.914564i	$-0.632534\pi$
$911$	−24.4143	−0.808881	−0.404441	+	0.914564i	$0.632534\pi$
$912$	2.57159	0.0851538
$913$	6.36910	0.210786
$914$	17.1101	0.565951
$915$	10.2942	0.340316
$916$	39.0578	1.29051
$917$	−29.4089	−0.971168
$918$	8.27122	0.272991
$919$	−2.91104	−0.0960264	−0.0480132	−	0.998847i	$-0.515289\pi$
$919$	−2.91104	−0.0960264	−0.0480132	+	0.998847i	$0.515289\pi$
$920$	−28.2831	−0.932467
$921$	−73.7606	−2.43049
$922$	−3.00610	−0.0990007
$923$	0	0
$924$	−8.02590	−0.264033
$925$	−6.70475	−0.220451
$926$	−6.99349	−0.229820
$927$	−1.51699	−0.0498245
$928$	24.9759	0.819873
$929$	23.9908	0.787112	0.393556	−	0.919301i	$-0.371245\pi$
$929$	23.9908	0.787112	0.393556	+	0.919301i	$0.371245\pi$
$930$	2.74226	0.0899223
$931$	−1.36139	−0.0446177
$932$	35.1866	1.15258
$933$	−74.4990	−2.43899
$934$	−9.85537	−0.322478
$935$	−5.48450	−0.179362
$936$	0	0
$937$	31.4017	1.02585	0.512924	−	0.858434i	$-0.328562\pi$
$937$	31.4017	1.02585	0.512924	+	0.858434i	$0.328562\pi$
$938$	−9.43044	−0.307915
$939$	36.9334	1.20528
$940$	−4.86654	−0.158729
$941$	−33.3661	−1.08771	−0.543853	−	0.839181i	$-0.683035\pi$
$941$	−33.3661	−1.08771	−0.543853	+	0.839181i	$0.683035\pi$
$942$	−22.4658	−0.731976
$943$	38.7779	1.26278
$944$	−9.87555	−0.321422
$945$	−10.6572	−0.346680
$946$	1.30230	0.0423413
$947$	47.4687	1.54253	0.771263	−	0.636517i	$-0.219625\pi$
$947$	47.4687	1.54253	0.771263	+	0.636517i	$0.219625\pi$
$948$	−41.4045	−1.34476
$949$	0	0
$950$	−1.01495	−0.0329292
$951$	−13.5424	−0.439142
$952$	20.4423	0.662540
$953$	−22.1379	−0.717118	−0.358559	−	0.933507i	$-0.616732\pi$
$953$	−22.1379	−0.717118	−0.358559	+	0.933507i	$0.616732\pi$
$954$	−8.83715	−0.286113
$955$	0.500604	0.0161992
$956$	14.1912	0.458976
$957$	−10.2363	−0.330891
$958$	24.4014	0.788373
$959$	−11.2212	−0.362351
$960$	2.39701	0.0773633
$961$	1.00000	0.0322581
$962$	0	0
$963$	−74.4018	−2.39757
$964$	7.00880	0.225738
$965$	7.08732	0.228149
$966$	−29.6343	−0.953469
$967$	−25.1227	−0.807891	−0.403946	−	0.914783i	$-0.632362\pi$
$967$	−25.1227	−0.807891	−0.403946	+	0.914783i	$0.632362\pi$
$968$	23.9779	0.770678
$969$	6.45224	0.207276
$970$	16.3542	0.525101
$971$	51.4528	1.65120	0.825600	−	0.564256i	$-0.190837\pi$
$971$	51.4528	1.65120	0.825600	+	0.564256i	$0.190837\pi$
$972$	32.0268	1.02726
$973$	−48.3781	−1.55093
$974$	15.2279	0.487934
$975$	0	0
$976$	3.93863	0.126072
$977$	14.7721	0.472601	0.236300	−	0.971680i	$-0.424065\pi$
$977$	14.7721	0.472601	0.236300	+	0.971680i	$0.424065\pi$
$978$	34.1933	1.09338
$979$	−14.2663	−0.455952
$980$	5.53700	0.176873
$981$	43.0489	1.37445
$982$	−17.1930	−0.548650
$983$	−47.9404	−1.52906	−0.764530	−	0.644588i	$-0.777029\pi$
$983$	−47.9404	−1.52906	−0.764530	+	0.644588i	$0.777029\pi$
$984$	−31.4502	−1.00260
$985$	−15.2010	−0.484344
$986$	11.4599	0.364957
$987$	−11.6008	−0.369256
$988$	0	0
$989$	−17.4797	−0.555822
$990$	−3.75141	−0.119228
$991$	−9.92275	−0.315206	−0.157603	−	0.987503i	$-0.550377\pi$
$991$	−9.92275	−0.315206	−0.157603	+	0.987503i	$0.550377\pi$
$992$	5.73734	0.182161
$993$	−84.6323	−2.68573
$994$	−0.814886	−0.0258466
$995$	−11.3831	−0.360870
$996$	30.4750	0.965637
$997$	8.72610	0.276358	0.138179	−	0.990407i	$-0.455875\pi$
$997$	8.72610	0.276358	0.138179	+	0.990407i	$0.455875\pi$
$998$	25.3090	0.801143
$999$	8.19578	0.259303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.i.1.5		8
13.12	even	2		403.2.a.e.1.4	✓	8
39.38	odd	2		3627.2.a.p.1.5		8
52.51	odd	2		6448.2.a.bd.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.e.1.4	✓	8	13.12	even	2
3627.2.a.p.1.5		8	39.38	odd	2
5239.2.a.i.1.5		8	1.1	even	1	trivial
6448.2.a.bd.1.2		8	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q+0.656875 q^{2} +2.67827 q^{3} -1.56852 q^{4} +1.55873 q^{5} +1.75929 q^{6} -2.17607 q^{7} -2.34407 q^{8} +4.17313 q^{9} +O(q^{10})\)	\(q+0.656875 q^{2} +2.67827 q^{3} -1.56852 q^{4} +1.55873 q^{5} +1.75929 q^{6} -2.17607 q^{7} -2.34407 q^{8} +4.17313 q^{9} +1.02389 q^{10} -0.877966 q^{11} -4.20091 q^{12} -1.42941 q^{14} +4.17471 q^{15} +1.59727 q^{16} +4.00763 q^{17} +2.74122 q^{18} +0.601130 q^{19} -2.44490 q^{20} -5.82810 q^{21} -0.576714 q^{22} +7.74079 q^{23} -6.27805 q^{24} -2.57035 q^{25} +3.14195 q^{27} +3.41320 q^{28} +4.35321 q^{29} +2.74226 q^{30} +1.00000 q^{31} +5.73734 q^{32} -2.35143 q^{33} +2.63251 q^{34} -3.39192 q^{35} -6.54561 q^{36} +2.60850 q^{37} +0.394867 q^{38} -3.65378 q^{40} +5.00956 q^{41} -3.82833 q^{42} -2.25813 q^{43} +1.37710 q^{44} +6.50480 q^{45} +5.08473 q^{46} +1.99049 q^{47} +4.27792 q^{48} -2.26472 q^{49} -1.68840 q^{50} +10.7335 q^{51} -3.22380 q^{53} +2.06387 q^{54} -1.36852 q^{55} +5.10086 q^{56} +1.60999 q^{57} +2.85951 q^{58} -6.18276 q^{59} -6.54810 q^{60} +2.46585 q^{61} +0.656875 q^{62} -9.08102 q^{63} +0.574175 q^{64} -1.54459 q^{66} +6.59746 q^{67} -6.28603 q^{68} +20.7319 q^{69} -2.22806 q^{70} +0.570087 q^{71} -9.78210 q^{72} +6.17960 q^{73} +1.71346 q^{74} -6.88408 q^{75} -0.942882 q^{76} +1.91052 q^{77} +9.85609 q^{79} +2.48972 q^{80} -4.10439 q^{81} +3.29065 q^{82} -7.25438 q^{83} +9.14147 q^{84} +6.24683 q^{85} -1.48331 q^{86} +11.6591 q^{87} +2.05801 q^{88} +16.2492 q^{89} +4.27284 q^{90} -12.1415 q^{92} +2.67827 q^{93} +1.30750 q^{94} +0.937002 q^{95} +15.3662 q^{96} +15.9725 q^{97} -1.48764 q^{98} -3.66386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 7 q^{3} + 7 q^{4} - 11 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q + q^2 + 7 * q^3 + 7 * q^4 - 11 * q^5 + 2 * q^7 + 9 * q^9	\( 8 q + q^{2} + 7 q^{3} + 7 q^{4} - 11 q^{5} + 2 q^{7} + 9 q^{9} - 2 q^{10} + 2 q^{11} + 19 q^{12} + 3 q^{14} - 6 q^{15} - 7 q^{16} + 7 q^{17} + 12 q^{18} + 5 q^{19} - 6 q^{20} - 8 q^{21} - 4 q^{22} + 14 q^{23} + 17 q^{24} + 17 q^{25} + 7 q^{27} + 9 q^{28} + 12 q^{29} - 9 q^{30} + 8 q^{31} + 21 q^{32} - 10 q^{33} + 12 q^{34} - q^{35} + 11 q^{36} - 2 q^{37} + 24 q^{38} - 19 q^{40} - 13 q^{41} - 27 q^{42} - 5 q^{43} - 22 q^{44} - 19 q^{45} - 17 q^{46} - 23 q^{47} + 3 q^{48} + 26 q^{49} + 26 q^{50} + 18 q^{51} + 25 q^{53} + 36 q^{54} - 17 q^{55} + 8 q^{56} + 35 q^{57} + 29 q^{58} + 5 q^{59} - 71 q^{60} - 9 q^{61} + q^{62} + 37 q^{63} - 14 q^{64} - 41 q^{66} - 22 q^{67} - 6 q^{68} - 7 q^{69} + 29 q^{70} + 17 q^{71} + 34 q^{72} + 27 q^{73} + 14 q^{74} - 33 q^{75} + 36 q^{76} + 31 q^{77} - 23 q^{79} - 9 q^{80} - 12 q^{81} + 18 q^{82} + 25 q^{83} + 62 q^{84} - 13 q^{85} - 11 q^{86} + 26 q^{87} + 5 q^{88} - 2 q^{89} - 14 q^{90} - 20 q^{92} + 7 q^{93} - 38 q^{94} + 3 q^{95} + 52 q^{96} + 15 q^{97} - 39 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q + q^2 + 7 * q^3 + 7 * q^4 - 11 * q^5 + 2 * q^7 + 9 * q^9 - 2 * q^10 + 2 * q^11 + 19 * q^12 + 3 * q^14 - 6 * q^15 - 7 * q^16 + 7 * q^17 + 12 * q^18 + 5 * q^19 - 6 * q^20 - 8 * q^21 - 4 * q^22 + 14 * q^23 + 17 * q^24 + 17 * q^25 + 7 * q^27 + 9 * q^28 + 12 * q^29 - 9 * q^30 + 8 * q^31 + 21 * q^32 - 10 * q^33 + 12 * q^34 - q^35 + 11 * q^36 - 2 * q^37 + 24 * q^38 - 19 * q^40 - 13 * q^41 - 27 * q^42 - 5 * q^43 - 22 * q^44 - 19 * q^45 - 17 * q^46 - 23 * q^47 + 3 * q^48 + 26 * q^49 + 26 * q^50 + 18 * q^51 + 25 * q^53 + 36 * q^54 - 17 * q^55 + 8 * q^56 + 35 * q^57 + 29 * q^58 + 5 * q^59 - 71 * q^60 - 9 * q^61 + q^62 + 37 * q^63 - 14 * q^64 - 41 * q^66 - 22 * q^67 - 6 * q^68 - 7 * q^69 + 29 * q^70 + 17 * q^71 + 34 * q^72 + 27 * q^73 + 14 * q^74 - 33 * q^75 + 36 * q^76 + 31 * q^77 - 23 * q^79 - 9 * q^80 - 12 * q^81 + 18 * q^82 + 25 * q^83 + 62 * q^84 - 13 * q^85 - 11 * q^86 + 26 * q^87 + 5 * q^88 - 2 * q^89 - 14 * q^90 - 20 * q^92 + 7 * q^93 - 38 * q^94 + 3 * q^95 + 52 * q^96 + 15 * q^97 - 39 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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