LMFDB - Embedded newform 678.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [678,2,Mod(1,678)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("678.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(678, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 678 = 2 \cdot 3 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	678.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,-3,3,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.41385725704$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	678.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.84822 q^{5} -1.00000 q^{6} +1.68397 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.84822 q^{10} +1.31603 q^{11} -1.00000 q^{12} -3.84822 q^{13} +1.68397 q^{14} -3.84822 q^{15} +1.00000 q^{16} -6.01247 q^{17} +1.00000 q^{18} +5.64453 q^{19} +3.84822 q^{20} -1.68397 q^{21} +1.31603 q^{22} +2.83575 q^{23} -1.00000 q^{24} +9.80877 q^{25} -3.84822 q^{26} -1.00000 q^{27} +1.68397 q^{28} -10.6570 q^{29} -3.84822 q^{30} -8.12481 q^{31} +1.00000 q^{32} -1.31603 q^{33} -6.01247 q^{34} +6.48028 q^{35} +1.00000 q^{36} -0.683969 q^{37} +5.64453 q^{38} +3.84822 q^{39} +3.84822 q^{40} +9.69643 q^{41} -1.68397 q^{42} +5.53219 q^{43} +1.31603 q^{44} +3.84822 q^{45} +2.83575 q^{46} +6.16425 q^{47} -1.00000 q^{48} -4.16425 q^{49} +9.80877 q^{50} +6.01247 q^{51} -3.84822 q^{52} +7.12481 q^{53} -1.00000 q^{54} +5.06437 q^{55} +1.68397 q^{56} -5.64453 q^{57} -10.6570 q^{58} -3.84822 q^{59} -3.84822 q^{60} -10.4803 q^{61} -8.12481 q^{62} +1.68397 q^{63} +1.00000 q^{64} -14.8088 q^{65} -1.31603 q^{66} +15.4533 q^{67} -6.01247 q^{68} -2.83575 q^{69} +6.48028 q^{70} -0.0124650 q^{71} +1.00000 q^{72} -4.00000 q^{73} -0.683969 q^{74} -9.80877 q^{75} +5.64453 q^{76} +2.21616 q^{77} +3.84822 q^{78} -9.69643 q^{79} +3.84822 q^{80} +1.00000 q^{81} +9.69643 q^{82} +1.12481 q^{83} -1.68397 q^{84} -23.1373 q^{85} +5.53219 q^{86} +10.6570 q^{87} +1.31603 q^{88} -1.09135 q^{89} +3.84822 q^{90} -6.48028 q^{91} +2.83575 q^{92} +8.12481 q^{93} +6.16425 q^{94} +21.7214 q^{95} -1.00000 q^{96} -8.26806 q^{97} -4.16425 q^{98} +1.31603 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 7 q^{11} - 3 q^{12} - q^{13} + 2 q^{14} - q^{15} + 3 q^{16} + 3 q^{18} + 5 q^{19} + q^{20} - 2 q^{21} + 7 q^{22}+ \cdots + 7 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + q^5 - 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 7 * q^11 - 3 * q^12 - q^13 + 2 * q^14 - q^15 + 3 * q^16 + 3 * q^18 + 5 * q^19 + q^20 - 2 * q^21 + 7 * q^22 + 16 * q^23 - 3 * q^24 + 10 * q^25 - q^26 - 3 * q^27 + 2 * q^28 - 2 * q^29 - q^30 - 8 * q^31 + 3 * q^32 - 7 * q^33 + 15 * q^35 + 3 * q^36 + q^37 + 5 * q^38 + q^39 + q^40 + 8 * q^41 - 2 * q^42 + 3 * q^43 + 7 * q^44 + q^45 + 16 * q^46 + 11 * q^47 - 3 * q^48 - 5 * q^49 + 10 * q^50 - q^52 + 5 * q^53 - 3 * q^54 - 12 * q^55 + 2 * q^56 - 5 * q^57 - 2 * q^58 - q^59 - q^60 - 27 * q^61 - 8 * q^62 + 2 * q^63 + 3 * q^64 - 25 * q^65 - 7 * q^66 + 15 * q^67 - 16 * q^69 + 15 * q^70 + 18 * q^71 + 3 * q^72 - 12 * q^73 + q^74 - 10 * q^75 + 5 * q^76 - 10 * q^77 + q^78 - 8 * q^79 + q^80 + 3 * q^81 + 8 * q^82 - 13 * q^83 - 2 * q^84 - 35 * q^85 + 3 * q^86 + 2 * q^87 + 7 * q^88 - 3 * q^89 + q^90 - 15 * q^91 + 16 * q^92 + 8 * q^93 + 11 * q^94 + 8 * q^95 - 3 * q^96 + q^97 - 5 * q^98 + 7 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.84822	1.72097	0.860487	−	0.509472i	$-0.170159\pi$
$5$	3.84822	1.72097	0.860487	+	0.509472i	$0.170159\pi$
$6$	−1.00000	−0.408248
$7$	1.68397	0.636481	0.318240	−	0.948010i	$-0.396908\pi$
$7$	1.68397	0.636481	0.318240	+	0.948010i	$0.396908\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.84822	1.21691
$11$	1.31603	0.396798	0.198399	−	0.980121i	$-0.436426\pi$
$11$	1.31603	0.396798	0.198399	+	0.980121i	$0.436426\pi$
$12$	−1.00000	−0.288675
$13$	−3.84822	−1.06730	−0.533652	−	0.845704i	$-0.679181\pi$
$13$	−3.84822	−1.06730	−0.533652	+	0.845704i	$0.679181\pi$
$14$	1.68397	0.450060
$15$	−3.84822	−0.993605
$16$	1.00000	0.250000
$17$	−6.01247	−1.45824	−0.729118	−	0.684387i	$-0.760070\pi$
$17$	−6.01247	−1.45824	−0.729118	+	0.684387i	$0.760070\pi$
$18$	1.00000	0.235702
$19$	5.64453	1.29494	0.647472	−	0.762090i	$-0.275826\pi$
$19$	5.64453	1.29494	0.647472	+	0.762090i	$0.275826\pi$
$20$	3.84822	0.860487
$21$	−1.68397	−0.367472
$22$	1.31603	0.280579
$23$	2.83575	0.591295	0.295648	−	0.955297i	$-0.404465\pi$
$23$	2.83575	0.591295	0.295648	+	0.955297i	$0.404465\pi$
$24$	−1.00000	−0.204124
$25$	9.80877	1.96175
$26$	−3.84822	−0.754697
$27$	−1.00000	−0.192450
$28$	1.68397	0.318240
$29$	−10.6570	−1.97895	−0.989477	−	0.144691i	$-0.953781\pi$
$29$	−10.6570	−1.97895	−0.989477	+	0.144691i	$0.953781\pi$
$30$	−3.84822	−0.702585
$31$	−8.12481	−1.45926	−0.729629	−	0.683843i	$-0.760307\pi$
$31$	−8.12481	−1.45926	−0.729629	+	0.683843i	$0.760307\pi$
$32$	1.00000	0.176777
$33$	−1.31603	−0.229092
$34$	−6.01247	−1.03113
$35$	6.48028	1.09537
$36$	1.00000	0.166667
$37$	−0.683969	−0.112444	−0.0562219	−	0.998418i	$-0.517905\pi$
$37$	−0.683969	−0.112444	−0.0562219	+	0.998418i	$0.517905\pi$
$38$	5.64453	0.915663
$39$	3.84822	0.616208
$40$	3.84822	0.608457
$41$	9.69643	1.51433	0.757164	−	0.653224i	$-0.226584\pi$
$41$	9.69643	1.51433	0.757164	+	0.653224i	$0.226584\pi$
$42$	−1.68397	−0.259842
$43$	5.53219	0.843650	0.421825	−	0.906677i	$-0.361390\pi$
$43$	5.53219	0.843650	0.421825	+	0.906677i	$0.361390\pi$
$44$	1.31603	0.198399
$45$	3.84822	0.573658
$46$	2.83575	0.418109
$47$	6.16425	0.899148	0.449574	−	0.893243i	$-0.351576\pi$
$47$	6.16425	0.899148	0.449574	+	0.893243i	$0.351576\pi$
$48$	−1.00000	−0.144338
$49$	−4.16425	−0.594893
$50$	9.80877	1.38717
$51$	6.01247	0.841914
$52$	−3.84822	−0.533652
$53$	7.12481	0.978667	0.489334	−	0.872097i	$-0.337240\pi$
$53$	7.12481	0.978667	0.489334	+	0.872097i	$0.337240\pi$
$54$	−1.00000	−0.136083
$55$	5.06437	0.682880
$56$	1.68397	0.225030
$57$	−5.64453	−0.747636
$58$	−10.6570	−1.39933
$59$	−3.84822	−0.500995	−0.250498	−	0.968117i	$-0.580594\pi$
$59$	−3.84822	−0.500995	−0.250498	+	0.968117i	$0.580594\pi$
$60$	−3.84822	−0.496803
$61$	−10.4803	−1.34186	−0.670931	−	0.741520i	$-0.734105\pi$
$61$	−10.4803	−1.34186	−0.670931	+	0.741520i	$0.734105\pi$
$62$	−8.12481	−1.03185
$63$	1.68397	0.212160
$64$	1.00000	0.125000
$65$	−14.8088	−1.83680
$66$	−1.31603	−0.161992
$67$	15.4533	1.88792	0.943961	−	0.330058i	$-0.107068\pi$
$67$	15.4533	1.88792	0.943961	+	0.330058i	$0.107068\pi$
$68$	−6.01247	−0.729118
$69$	−2.83575	−0.341384
$70$	6.48028	0.774541
$71$	−0.0124650	−0.00147933	−0.000739663	−	1.00000i	$-0.500235\pi$
$71$	−0.0124650	−0.00147933	−0.000739663		1.00000i	$0.500235\pi$
$72$	1.00000	0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−0.683969	−0.0795098
$75$	−9.80877	−1.13262
$76$	5.64453	0.647472
$77$	2.21616	0.252554
$78$	3.84822	0.435725
$79$	−9.69643	−1.09093	−0.545467	−	0.838132i	$-0.683648\pi$
$79$	−9.69643	−1.09093	−0.545467	+	0.838132i	$0.683648\pi$
$80$	3.84822	0.430244
$81$	1.00000	0.111111
$82$	9.69643	1.07079
$83$	1.12481	0.123463	0.0617317	−	0.998093i	$-0.480338\pi$
$83$	1.12481	0.123463	0.0617317	+	0.998093i	$0.480338\pi$
$84$	−1.68397	−0.183736
$85$	−23.1373	−2.50959
$86$	5.53219	0.596551
$87$	10.6570	1.14255
$88$	1.31603	0.140289
$89$	−1.09135	−0.115683	−0.0578414	−	0.998326i	$-0.518422\pi$
$89$	−1.09135	−0.115683	−0.0578414	+	0.998326i	$0.518422\pi$
$90$	3.84822	0.405638
$91$	−6.48028	−0.679318
$92$	2.83575	0.295648
$93$	8.12481	0.842503
$94$	6.16425	0.635794
$95$	21.7214	2.22856
$96$	−1.00000	−0.102062
$97$	−8.26806	−0.839495	−0.419747	−	0.907641i	$-0.637881\pi$
$97$	−8.26806	−0.839495	−0.419747	+	0.907641i	$0.637881\pi$
$98$	−4.16425	−0.420653
$99$	1.31603	0.132266
$100$	9.80877	0.980877
$101$	−5.11234	−0.508697	−0.254348	−	0.967113i	$-0.581861\pi$
$101$	−5.11234	−0.508697	−0.254348	+	0.967113i	$0.581861\pi$
$102$	6.01247	0.595323
$103$	−16.7608	−1.65149	−0.825746	−	0.564043i	$-0.809245\pi$
$103$	−16.7608	−1.65149	−0.825746	+	0.564043i	$0.809245\pi$
$104$	−3.84822	−0.377349
$105$	−6.48028	−0.632410
$106$	7.12481	0.692022
$107$	−12.5052	−1.20892	−0.604462	−	0.796634i	$-0.706612\pi$
$107$	−12.5052	−1.20892	−0.604462	+	0.796634i	$0.706612\pi$
$108$	−1.00000	−0.0962250
$109$	−19.7693	−1.89356	−0.946779	−	0.321883i	$-0.895684\pi$
$109$	−19.7693	−1.89356	−0.946779	+	0.321883i	$0.895684\pi$
$110$	5.06437	0.482869
$111$	0.683969	0.0649195
$112$	1.68397	0.159120
$113$	−1.00000	−0.0940721
$113$	−1.00000	−0.0940721
$114$	−5.64453	−0.528658
$115$	10.9126	1.01760
$116$	−10.6570	−0.989477
$117$	−3.84822	−0.355768
$118$	−3.84822	−0.354257
$119$	−10.1248	−0.928139
$120$	−3.84822	−0.351293
$121$	−9.26806	−0.842551
$122$	−10.4803	−0.948840
$123$	−9.69643	−0.874298
$124$	−8.12481	−0.729629
$125$	18.5052	1.65516
$126$	1.68397	0.150020
$127$	6.27659	0.556957	0.278479	−	0.960442i	$-0.410170\pi$
$127$	6.27659	0.556957	0.278479	+	0.960442i	$0.410170\pi$
$128$	1.00000	0.0883883
$129$	−5.53219	−0.487082
$130$	−14.8088	−1.29882
$131$	8.16425	0.713314	0.356657	−	0.934235i	$-0.383917\pi$
$131$	8.16425	0.713314	0.356657	+	0.934235i	$0.383917\pi$
$132$	−1.31603	−0.114546
$133$	9.50521	0.824206
$134$	15.4533	1.33496
$135$	−3.84822	−0.331202
$136$	−6.01247	−0.515565
$137$	19.5447	1.66981	0.834906	−	0.550392i	$-0.185522\pi$
$137$	19.5447	1.66981	0.834906	+	0.550392i	$0.185522\pi$
$138$	−2.83575	−0.241395
$139$	−5.51972	−0.468177	−0.234088	−	0.972215i	$-0.575210\pi$
$139$	−5.51972	−0.468177	−0.234088	+	0.972215i	$0.575210\pi$
$140$	6.48028	0.547684
$141$	−6.16425	−0.519123
$142$	−0.0124650	−0.00104604
$143$	−5.06437	−0.423504
$144$	1.00000	0.0833333
$145$	−41.0104	−3.40573
$146$	−4.00000	−0.331042
$147$	4.16425	0.343461
$148$	−0.683969	−0.0562219
$149$	−2.02698	−0.166056	−0.0830282	−	0.996547i	$-0.526459\pi$
$149$	−2.02698	−0.166056	−0.0830282	+	0.996547i	$0.526459\pi$
$150$	−9.80877	−0.800883
$151$	2.73588	0.222642	0.111321	−	0.993784i	$-0.464492\pi$
$151$	2.73588	0.222642	0.111321	+	0.993784i	$0.464492\pi$
$152$	5.64453	0.457832
$153$	−6.01247	−0.486079
$154$	2.21616	0.178583
$155$	−31.2660	−2.51135
$156$	3.84822	0.308104
$157$	18.7608	1.49728	0.748638	−	0.662979i	$-0.230708\pi$
$157$	18.7608	1.49728	0.748638	+	0.662979i	$0.230708\pi$
$158$	−9.69643	−0.771407
$159$	−7.12481	−0.565034
$160$	3.84822	0.304228
$161$	4.77532	0.376348
$162$	1.00000	0.0785674
$163$	−10.0729	−0.788970	−0.394485	−	0.918902i	$-0.629077\pi$
$163$	−10.0729	−0.788970	−0.394485	+	0.918902i	$0.629077\pi$
$164$	9.69643	0.757164
$165$	−5.06437	−0.394261
$166$	1.12481	0.0873018
$167$	1.19123	0.0921798	0.0460899	−	0.998937i	$-0.485324\pi$
$167$	1.19123	0.0921798	0.0460899	+	0.998937i	$0.485324\pi$
$168$	−1.68397	−0.129921
$169$	1.80877	0.139137
$170$	−23.1373	−1.77455
$171$	5.64453	0.431648
$172$	5.53219	0.421825
$173$	13.3659	1.01619	0.508095	−	0.861301i	$-0.330350\pi$
$173$	13.3659	1.01619	0.508095	+	0.861301i	$0.330350\pi$
$174$	10.6570	0.807904
$175$	16.5177	1.24862
$176$	1.31603	0.0991996
$177$	3.84822	0.289250
$178$	−1.09135	−0.0818001
$179$	−4.17671	−0.312182	−0.156091	−	0.987743i	$-0.549889\pi$
$179$	−4.17671	−0.312182	−0.156091	+	0.987743i	$0.549889\pi$
$180$	3.84822	0.286829
$181$	−14.5716	−1.08310	−0.541550	−	0.840668i	$-0.682162\pi$
$181$	−14.5716	−1.08310	−0.541550	+	0.840668i	$0.682162\pi$
$182$	−6.48028	−0.480350
$183$	10.4803	0.774725
$184$	2.83575	0.209054
$185$	−2.63206	−0.193513
$186$	8.12481	0.595740
$187$	−7.91259	−0.578626
$188$	6.16425	0.449574
$189$	−1.68397	−0.122491
$190$	21.7214	1.57583
$191$	14.2286	1.02955	0.514773	−	0.857326i	$-0.327876\pi$
$191$	14.2286	1.02955	0.514773	+	0.857326i	$0.327876\pi$
$192$	−1.00000	−0.0721688
$193$	8.02493	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$193$	8.02493	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$194$	−8.26806	−0.593612
$195$	14.8088	1.06048
$196$	−4.16425	−0.297446
$197$	1.34301	0.0956854	0.0478427	−	0.998855i	$-0.484765\pi$
$197$	1.34301	0.0956854	0.0478427	+	0.998855i	$0.484765\pi$
$198$	1.31603	0.0935262
$199$	21.0893	1.49498	0.747491	−	0.664272i	$-0.231259\pi$
$199$	21.0893	1.49498	0.747491	+	0.664272i	$0.231259\pi$
$200$	9.80877	0.693585
$201$	−15.4533	−1.08999
$202$	−5.11234	−0.359703
$203$	−17.9460	−1.25957
$204$	6.01247	0.420957
$205$	37.3140	2.60612
$206$	−16.7608	−1.16778
$207$	2.83575	0.197098
$208$	−3.84822	−0.266826
$209$	7.42837	0.513831
$210$	−6.48028	−0.447182
$211$	−15.1373	−1.04209	−0.521046	−	0.853528i	$-0.674458\pi$
$211$	−15.1373	−1.04209	−0.521046	+	0.853528i	$0.674458\pi$
$212$	7.12481	0.489334
$213$	0.0124650	0.000854089 0
$214$	−12.5052	−0.854838
$215$	21.2891	1.45190
$216$	−1.00000	−0.0680414
$217$	−13.6819	−0.928789
$218$	−19.7693	−1.33895
$219$	4.00000	0.270295
$220$	5.06437	0.341440
$221$	23.1373	1.55638
$222$	0.683969	0.0459050
$223$	−16.5781	−1.11015	−0.555076	−	0.831800i	$-0.687311\pi$
$223$	−16.5781	−1.11015	−0.555076	+	0.831800i	$0.687311\pi$
$224$	1.68397	0.112515
$225$	9.80877	0.653918
$226$	−1.00000	−0.0665190
$227$	29.4782	1.95654	0.978269	−	0.207338i	$-0.0664798\pi$
$227$	29.4782	1.95654	0.978269	+	0.207338i	$0.0664798\pi$
$228$	−5.64453	−0.373818
$229$	19.0104	1.25624	0.628122	−	0.778115i	$-0.283824\pi$
$229$	19.0104	1.25624	0.628122	+	0.778115i	$0.283824\pi$
$230$	10.9126	0.719555
$231$	−2.21616	−0.145812
$232$	−10.6570	−0.699666
$233$	8.40738	0.550786	0.275393	−	0.961332i	$-0.411192\pi$
$233$	8.40738	0.550786	0.275393	+	0.961332i	$0.411192\pi$
$234$	−3.84822	−0.251566
$235$	23.7214	1.54741
$236$	−3.84822	−0.250498
$237$	9.69643	0.629851
$238$	−10.1248	−0.656294
$239$	−3.26412	−0.211139	−0.105569	−	0.994412i	$-0.533666\pi$
$239$	−3.26412	−0.211139	−0.105569	+	0.994412i	$0.533666\pi$
$240$	−3.84822	−0.248401
$241$	1.31603	0.0847730	0.0423865	−	0.999101i	$-0.486504\pi$
$241$	1.31603	0.0847730	0.0423865	+	0.999101i	$0.486504\pi$
$242$	−9.26806	−0.595774
$243$	−1.00000	−0.0641500
$244$	−10.4803	−0.670931
$245$	−16.0249	−1.02380
$246$	−9.69643	−0.618222
$247$	−21.7214	−1.38210
$248$	−8.12481	−0.515926
$249$	−1.12481	−0.0712817
$250$	18.5052	1.17037
$251$	4.35547	0.274915	0.137458	−	0.990508i	$-0.456107\pi$
$251$	4.35547	0.274915	0.137458	+	0.990508i	$0.456107\pi$
$252$	1.68397	0.106080
$253$	3.73194	0.234625
$254$	6.27659	0.393828
$255$	23.1373	1.44891
$256$	1.00000	0.0625000
$257$	−27.4178	−1.71028	−0.855138	−	0.518401i	$-0.826527\pi$
$257$	−27.4178	−1.71028	−0.855138	+	0.518401i	$0.826527\pi$
$258$	−5.53219	−0.344419
$259$	−1.15178	−0.0715683
$260$	−14.8088	−0.918401
$261$	−10.6570	−0.659651
$262$	8.16425	0.504389
$263$	−3.62354	−0.223437	−0.111718	−	0.993740i	$-0.535635\pi$
$263$	−3.62354	−0.223437	−0.111718	+	0.993740i	$0.535635\pi$
$264$	−1.31603	−0.0809961
$265$	27.4178	1.68426
$266$	9.50521	0.582802
$267$	1.09135	0.0667895
$268$	15.4533	0.943961
$269$	−8.78384	−0.535560	−0.267780	−	0.963480i	$-0.586290\pi$
$269$	−8.78384	−0.535560	−0.267780	+	0.963480i	$0.586290\pi$
$270$	−3.84822	−0.234195
$271$	−7.80025	−0.473831	−0.236916	−	0.971530i	$-0.576137\pi$
$271$	−7.80025	−0.473831	−0.236916	+	0.971530i	$0.576137\pi$
$272$	−6.01247	−0.364559
$273$	6.48028	0.392204
$274$	19.5447	1.18074
$275$	12.9087	0.778421
$276$	−2.83575	−0.170692
$277$	15.6235	0.938727	0.469364	−	0.883005i	$-0.344483\pi$
$277$	15.6235	0.938727	0.469364	+	0.883005i	$0.344483\pi$
$278$	−5.51972	−0.331051
$279$	−8.12481	−0.486419
$280$	6.48028	0.387271
$281$	22.4782	1.34094	0.670469	−	0.741937i	$-0.266093\pi$
$281$	22.4782	1.34094	0.670469	+	0.741937i	$0.266093\pi$
$282$	−6.16425	−0.367076
$283$	−20.5052	−1.21891	−0.609454	−	0.792821i	$-0.708611\pi$
$283$	−20.5052	−1.21891	−0.609454	+	0.792821i	$0.708611\pi$
$284$	−0.0124650	−0.000739663 0
$285$	−21.7214	−1.28666
$286$	−5.06437	−0.299463
$287$	16.3285	0.963841
$288$	1.00000	0.0589256
$289$	19.1497	1.12646
$290$	−41.0104	−2.40821
$291$	8.26806	0.484682
$292$	−4.00000	−0.234082
$293$	−2.04797	−0.119644	−0.0598218	−	0.998209i	$-0.519053\pi$
$293$	−2.04797	−0.119644	−0.0598218	+	0.998209i	$0.519053\pi$
$294$	4.16425	0.242864
$295$	−14.8088	−0.862200
$296$	−0.683969	−0.0397549
$297$	−1.31603	−0.0763639
$298$	−2.02698	−0.117420
$299$	−10.9126	−0.631091
$300$	−9.80877	−0.566310
$301$	9.31603	0.536967
$302$	2.73588	0.157432
$303$	5.11234	0.293696
$304$	5.64453	0.323736
$305$	−40.3304	−2.30931
$306$	−6.01247	−0.343710
$307$	−10.2805	−0.586741	−0.293370	−	0.955999i	$-0.594777\pi$
$307$	−10.2805	−0.586741	−0.293370	+	0.955999i	$0.594777\pi$
$308$	2.21616	0.126277
$309$	16.7608	0.953489
$310$	−31.2660	−1.77579
$311$	22.0249	1.24892	0.624460	−	0.781057i	$-0.285319\pi$
$311$	22.0249	1.24892	0.624460	+	0.781057i	$0.285319\pi$
$312$	3.84822	0.217862
$313$	−11.7174	−0.662308	−0.331154	−	0.943577i	$-0.607438\pi$
$313$	−11.7174	−0.662308	−0.331154	+	0.943577i	$0.607438\pi$
$314$	18.7608	1.05873
$315$	6.48028	0.365122
$316$	−9.69643	−0.545467
$317$	−8.60508	−0.483310	−0.241655	−	0.970362i	$-0.577690\pi$
$317$	−8.60508	−0.483310	−0.241655	+	0.970362i	$0.577690\pi$
$318$	−7.12481	−0.399539
$319$	−14.0249	−0.785245
$320$	3.84822	0.215122
$321$	12.5052	0.697973
$322$	4.77532	0.266118
$323$	−33.9375	−1.88833
$324$	1.00000	0.0555556
$325$	−37.7463	−2.09379
$326$	−10.0729	−0.557886
$327$	19.7693	1.09325
$328$	9.69643	0.535396
$329$	10.3804	0.572290
$330$	−5.06437	−0.278785
$331$	20.4014	1.12136	0.560681	−	0.828032i	$-0.310539\pi$
$331$	20.4014	1.12136	0.560681	+	0.828032i	$0.310539\pi$
$332$	1.12481	0.0617317
$333$	−0.683969	−0.0374813
$334$	1.19123	0.0651809
$335$	59.4677	3.24907
$336$	−1.68397	−0.0918680
$337$	13.9336	0.759010	0.379505	−	0.925190i	$-0.376094\pi$
$337$	13.9336	0.759010	0.379505	+	0.925190i	$0.376094\pi$
$338$	1.80877	0.0983844
$339$	1.00000	0.0543125
$340$	−23.1373	−1.25479
$341$	−10.6925	−0.579031
$342$	5.64453	0.305221
$343$	−18.8002	−1.01512
$344$	5.53219	0.298275
$345$	−10.9126	−0.587514
$346$	13.3659	0.718555
$347$	−16.6570	−0.894194	−0.447097	−	0.894485i	$-0.647542\pi$
$347$	−16.6570	−0.894194	−0.447097	+	0.894485i	$0.647542\pi$
$348$	10.6570	0.571275
$349$	−33.2870	−1.78181	−0.890906	−	0.454187i	$-0.849930\pi$
$349$	−33.2870	−1.78181	−0.890906	+	0.454187i	$0.849930\pi$
$350$	16.5177	0.882907
$351$	3.84822	0.205403
$352$	1.31603	0.0701447
$353$	5.69643	0.303191	0.151595	−	0.988443i	$-0.451559\pi$
$353$	5.69643	0.303191	0.151595	+	0.988443i	$0.451559\pi$
$354$	3.84822	0.204530
$355$	−0.0479681	−0.00254588
$356$	−1.09135	−0.0578414
$357$	10.1248	0.535862
$358$	−4.17671	−0.220746
$359$	36.9500	1.95015	0.975073	−	0.221885i	$-0.0712210\pi$
$359$	36.9500	1.95015	0.975073	+	0.221885i	$0.0712210\pi$
$360$	3.84822	0.202819
$361$	12.8607	0.676878
$362$	−14.5716	−0.765868
$363$	9.26806	0.486447
$364$	−6.48028	−0.339659
$365$	−15.3929	−0.805700
$366$	10.4803	0.547813
$367$	2.75687	0.143907	0.0719536	−	0.997408i	$-0.477077\pi$
$367$	2.75687	0.143907	0.0719536	+	0.997408i	$0.477077\pi$
$368$	2.83575	0.147824
$369$	9.69643	0.504776
$370$	−2.63206	−0.136834
$371$	11.9980	0.622903
$372$	8.12481	0.421252
$373$	−2.30357	−0.119274	−0.0596371	−	0.998220i	$-0.518994\pi$
$373$	−2.30357	−0.119274	−0.0596371	+	0.998220i	$0.518994\pi$
$374$	−7.91259	−0.409150
$375$	−18.5052	−0.955605
$376$	6.16425	0.317897
$377$	41.0104	2.11214
$378$	−1.68397	−0.0866140
$379$	−6.05191	−0.310866	−0.155433	−	0.987846i	$-0.549677\pi$
$379$	−6.05191	−0.310866	−0.155433	+	0.987846i	$0.549677\pi$
$380$	21.7214	1.11428
$381$	−6.27659	−0.321559
$382$	14.2286	0.728000
$383$	−14.1458	−0.722816	−0.361408	−	0.932408i	$-0.617704\pi$
$383$	−14.1458	−0.722816	−0.361408	+	0.932408i	$0.617704\pi$
$384$	−1.00000	−0.0510310
$385$	8.52825	0.434640
$386$	8.02493	0.408458
$387$	5.53219	0.281217
$388$	−8.26806	−0.419747
$389$	−28.7857	−1.45949	−0.729747	−	0.683717i	$-0.760362\pi$
$389$	−28.7857	−1.45949	−0.729747	+	0.683717i	$0.760362\pi$
$390$	14.8088	0.749871
$391$	−17.0499	−0.862248
$392$	−4.16425	−0.210326
$393$	−8.16425	−0.411832
$394$	1.34301	0.0676598
$395$	−37.3140	−1.87747
$396$	1.31603	0.0661330
$397$	9.28258	0.465879	0.232940	−	0.972491i	$-0.425166\pi$
$397$	9.28258	0.465879	0.232940	+	0.972491i	$0.425166\pi$
$398$	21.0893	1.05711
$399$	−9.50521	−0.475856
$400$	9.80877	0.490439
$401$	17.9460	0.896183	0.448091	−	0.893988i	$-0.352104\pi$
$401$	17.9460	0.896183	0.448091	+	0.893988i	$0.352104\pi$
$402$	−15.4533	−0.770741
$403$	31.2660	1.55747
$404$	−5.11234	−0.254348
$405$	3.84822	0.191219
$406$	−17.9460	−0.890647
$407$	−0.900124	−0.0446175
$408$	6.01247	0.297661
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	37.3140	1.84281
$411$	−19.5447	−0.964067
$412$	−16.7608	−0.825746
$413$	−6.48028	−0.318874
$414$	2.83575	0.139370
$415$	4.32850	0.212478
$416$	−3.84822	−0.188674
$417$	5.51972	0.270302
$418$	7.42837	0.363334
$419$	−26.7299	−1.30584	−0.652920	−	0.757427i	$-0.726456\pi$
$419$	−26.7299	−1.30584	−0.652920	+	0.757427i	$0.726456\pi$
$420$	−6.48028	−0.316205
$421$	19.5197	0.951333	0.475667	−	0.879626i	$-0.342207\pi$
$421$	19.5197	0.951333	0.475667	+	0.879626i	$0.342207\pi$
$422$	−15.1373	−0.736871
$423$	6.16425	0.299716
$424$	7.12481	0.346011
$425$	−58.9749	−2.86070
$426$	0.0124650	0.000603932 0
$427$	−17.6485	−0.854069
$428$	−12.5052	−0.604462
$429$	5.06437	0.244510
$430$	21.2891	1.02665
$431$	−1.63001	−0.0785150	−0.0392575	−	0.999229i	$-0.512499\pi$
$431$	−1.63001	−0.0785150	−0.0392575	+	0.999229i	$0.512499\pi$
$432$	−1.00000	−0.0481125
$433$	32.4323	1.55860	0.779299	−	0.626653i	$-0.215576\pi$
$433$	32.4323	1.55860	0.779299	+	0.626653i	$0.215576\pi$
$434$	−13.6819	−0.656753
$435$	41.0104	1.96630
$436$	−19.7693	−0.946779
$437$	16.0065	0.765694
$438$	4.00000	0.191127
$439$	−5.93957	−0.283480	−0.141740	−	0.989904i	$-0.545270\pi$
$439$	−5.93957	−0.283480	−0.141740	+	0.989904i	$0.545270\pi$
$440$	5.06437	0.241434
$441$	−4.16425	−0.198298
$442$	23.1373	1.10053
$443$	7.98155	0.379215	0.189607	−	0.981860i	$-0.439278\pi$
$443$	7.98155	0.379215	0.189607	+	0.981860i	$0.439278\pi$
$444$	0.683969	0.0324597
$445$	−4.19975	−0.199087
$446$	−16.5781	−0.784996
$447$	2.02698	0.0958727
$448$	1.68397	0.0795601
$449$	19.3889	0.915020	0.457510	−	0.889204i	$-0.348741\pi$
$449$	19.3889	0.915020	0.457510	+	0.889204i	$0.348741\pi$
$450$	9.80877	0.462390
$451$	12.7608	0.600883
$452$	−1.00000	−0.0470360
$453$	−2.73588	−0.128543
$454$	29.4782	1.38348
$455$	−24.9375	−1.16909
$456$	−5.64453	−0.264329
$457$	−5.84223	−0.273288	−0.136644	−	0.990620i	$-0.543632\pi$
$457$	−5.84223	−0.273288	−0.136644	+	0.990620i	$0.543632\pi$
$458$	19.0104	0.888298
$459$	6.01247	0.280638
$460$	10.9126	0.508802
$461$	−36.2930	−1.69033	−0.845167	−	0.534503i	$-0.820499\pi$
$461$	−36.2930	−1.69033	−0.845167	+	0.534503i	$0.820499\pi$
$462$	−2.21616	−0.103105
$463$	12.7234	0.591307	0.295654	−	0.955295i	$-0.404463\pi$
$463$	12.7234	0.591307	0.295654	+	0.955295i	$0.404463\pi$
$464$	−10.6570	−0.494738
$465$	31.2660	1.44993
$466$	8.40738	0.389464
$467$	17.8252	0.824851	0.412425	−	0.910991i	$-0.364682\pi$
$467$	17.8252	0.824851	0.412425	+	0.910991i	$0.364682\pi$
$468$	−3.84822	−0.177884
$469$	26.0229	1.20163
$470$	23.7214	1.09418
$471$	−18.7608	−0.864452
$472$	−3.84822	−0.177129
$473$	7.28053	0.334759
$474$	9.69643	0.445372
$475$	55.3659	2.54036
$476$	−10.1248	−0.464070
$477$	7.12481	0.326222
$478$	−3.26412	−0.149298
$479$	−23.3659	−1.06761	−0.533807	−	0.845606i	$-0.679239\pi$
$479$	−23.3659	−1.06761	−0.533807	+	0.845606i	$0.679239\pi$
$480$	−3.84822	−0.175646
$481$	2.63206	0.120012
$482$	1.31603	0.0599436
$483$	−4.77532	−0.217285
$484$	−9.26806	−0.421276
$485$	−31.8173	−1.44475
$486$	−1.00000	−0.0453609
$487$	39.0104	1.76773	0.883865	−	0.467741i	$-0.154932\pi$
$487$	39.0104	1.76773	0.883865	+	0.467741i	$0.154932\pi$
$488$	−10.4803	−0.474420
$489$	10.0729	0.455512
$490$	−16.0249	−0.723933
$491$	2.04797	0.0924235	0.0462118	−	0.998932i	$-0.485285\pi$
$491$	2.04797	0.0924235	0.0462118	+	0.998932i	$0.485285\pi$
$492$	−9.69643	−0.437149
$493$	64.0748	2.88578
$494$	−21.7214	−0.977290
$495$	5.06437	0.227627
$496$	−8.12481	−0.364815
$497$	−0.0209907	−0.000941562 0
$498$	−1.12481	−0.0504037
$499$	4.54670	0.203538	0.101769	−	0.994808i	$-0.467550\pi$
$499$	4.54670	0.203538	0.101769	+	0.994808i	$0.467550\pi$
$500$	18.5052	0.827578
$501$	−1.19123	−0.0532200
$502$	4.35547	0.194394
$503$	35.8252	1.59737	0.798683	−	0.601752i	$-0.205530\pi$
$503$	35.8252	1.59737	0.798683	+	0.601752i	$0.205530\pi$
$504$	1.68397	0.0750099
$505$	−19.6734	−0.875455
$506$	3.73194	0.165905
$507$	−1.80877	−0.0803305
$508$	6.27659	0.278479
$509$	21.6780	0.960860	0.480430	−	0.877033i	$-0.340481\pi$
$509$	21.6780	0.960860	0.480430	+	0.877033i	$0.340481\pi$
$510$	23.1373	1.02454
$511$	−6.73588	−0.297978
$512$	1.00000	0.0441942
$513$	−5.64453	−0.249212
$514$	−27.4178	−1.20935
$515$	−64.4992	−2.84218
$516$	−5.53219	−0.243541
$517$	8.11234	0.356780
$518$	−1.15178	−0.0506064
$519$	−13.3659	−0.586697
$520$	−14.8088	−0.649408
$521$	−12.4074	−0.543577	−0.271789	−	0.962357i	$-0.587615\pi$
$521$	−12.4074	−0.543577	−0.271789	+	0.962357i	$0.587615\pi$
$522$	−10.6570	−0.466444
$523$	16.2226	0.709366	0.354683	−	0.934987i	$-0.384589\pi$
$523$	16.2226	0.709366	0.354683	+	0.934987i	$0.384589\pi$
$524$	8.16425	0.356657
$525$	−16.5177	−0.720890
$526$	−3.62354	−0.157994
$527$	48.8501	2.12794
$528$	−1.31603	−0.0572729
$529$	−14.9585	−0.650370
$530$	27.4178	1.19095
$531$	−3.84822	−0.166998
$532$	9.50521	0.412103
$533$	−37.3140	−1.61625
$534$	1.09135	0.0472273
$535$	−48.1228	−2.08053
$536$	15.4533	0.667481
$537$	4.17671	0.180238
$538$	−8.78384	−0.378698
$539$	−5.48028	−0.236052
$540$	−3.84822	−0.165601
$541$	−28.0314	−1.20516	−0.602582	−	0.798057i	$-0.705861\pi$
$541$	−28.0314	−1.20516	−0.602582	+	0.798057i	$0.705861\pi$
$542$	−7.80025	−0.335049
$543$	14.5716	0.625328
$544$	−6.01247	−0.257782
$545$	−76.0767	−3.25877
$546$	6.48028	0.277330
$547$	10.5112	0.449426	0.224713	−	0.974425i	$-0.427856\pi$
$547$	10.5112	0.449426	0.224713	+	0.974425i	$0.427856\pi$
$548$	19.5447	0.834906
$549$	−10.4803	−0.447287
$550$	12.9087	0.550427
$551$	−60.1537	−2.56263
$552$	−2.83575	−0.120698
$553$	−16.3285	−0.694358
$554$	15.6235	0.663780
$555$	2.63206	0.111725
$556$	−5.51972	−0.234088
$557$	15.5407	0.658481	0.329241	−	0.944246i	$-0.393207\pi$
$557$	15.5407	0.658481	0.329241	+	0.944246i	$0.393207\pi$
$558$	−8.12481	−0.343950
$559$	−21.2891	−0.900431
$560$	6.48028	0.273842
$561$	7.91259	0.334070
$562$	22.4782	0.948187
$563$	−9.92316	−0.418211	−0.209106	−	0.977893i	$-0.567055\pi$
$563$	−9.92316	−0.418211	−0.209106	+	0.977893i	$0.567055\pi$
$564$	−6.16425	−0.259562
$565$	−3.84822	−0.161896
$566$	−20.5052	−0.861898
$567$	1.68397	0.0707201
$568$	−0.0124650	−0.000523020 0
$569$	9.41780	0.394815	0.197407	−	0.980322i	$-0.436748\pi$
$569$	9.41780	0.394815	0.197407	+	0.980322i	$0.436748\pi$
$570$	−21.7214	−0.909808
$571$	15.7193	0.657833	0.328916	−	0.944359i	$-0.393317\pi$
$571$	15.7193	0.657833	0.328916	+	0.944359i	$0.393317\pi$
$572$	−5.06437	−0.211752
$573$	−14.2286	−0.594409
$574$	16.3285	0.681538
$575$	27.8153	1.15998
$576$	1.00000	0.0416667
$577$	4.60713	0.191797	0.0958987	−	0.995391i	$-0.469428\pi$
$577$	4.60713	0.191797	0.0958987	+	0.995391i	$0.469428\pi$
$578$	19.1497	0.796524
$579$	−8.02493	−0.333505
$580$	−41.0104	−1.70286
$581$	1.89414	0.0785821
$582$	8.26806	0.342722
$583$	9.37646	0.388333
$584$	−4.00000	−0.165521
$585$	−14.8088	−0.612267
$586$	−2.04797	−0.0846008
$587$	8.46781	0.349504	0.174752	−	0.984612i	$-0.444088\pi$
$587$	8.46781	0.349504	0.174752	+	0.984612i	$0.444088\pi$
$588$	4.16425	0.171731
$589$	−45.8607	−1.88966
$590$	−14.8088	−0.609668
$591$	−1.34301	−0.0552440
$592$	−0.683969	−0.0281110
$593$	−33.0644	−1.35779	−0.678896	−	0.734235i	$-0.737541\pi$
$593$	−33.0644	−1.35779	−0.678896	+	0.734235i	$0.737541\pi$
$594$	−1.31603	−0.0539974
$595$	−38.9624	−1.59730
$596$	−2.02698	−0.0830282
$597$	−21.0893	−0.863128
$598$	−10.9126	−0.446249
$599$	24.7069	1.00949	0.504747	−	0.863267i	$-0.331586\pi$
$599$	24.7069	1.00949	0.504747	+	0.863267i	$0.331586\pi$
$600$	−9.80877	−0.400442
$601$	21.7424	0.886889	0.443445	−	0.896302i	$-0.353756\pi$
$601$	21.7424	0.886889	0.443445	+	0.896302i	$0.353756\pi$
$602$	9.31603	0.379693
$603$	15.4533	0.629307
$604$	2.73588	0.111321
$605$	−35.6655	−1.45001
$606$	5.11234	0.207675
$607$	22.4743	0.912203	0.456102	−	0.889928i	$-0.349245\pi$
$607$	22.4743	0.912203	0.456102	+	0.889928i	$0.349245\pi$
$608$	5.64453	0.228916
$609$	17.9460	0.727210
$610$	−40.3304	−1.63293
$611$	−23.7214	−0.959664
$612$	−6.01247	−0.243039
$613$	25.7818	1.04132	0.520658	−	0.853765i	$-0.325686\pi$
$613$	25.7818	1.04132	0.520658	+	0.853765i	$0.325686\pi$
$614$	−10.2805	−0.414888
$615$	−37.3140	−1.50465
$616$	2.21616	0.0892914
$617$	28.5361	1.14882	0.574411	−	0.818567i	$-0.305231\pi$
$617$	28.5361	1.14882	0.574411	+	0.818567i	$0.305231\pi$
$618$	16.7608	0.674219
$619$	29.5820	1.18900	0.594501	−	0.804095i	$-0.297349\pi$
$619$	29.5820	1.18900	0.594501	+	0.804095i	$0.297349\pi$
$620$	−31.2660	−1.25567
$621$	−2.83575	−0.113795
$622$	22.0249	0.883119
$623$	−1.83780	−0.0736299
$624$	3.84822	0.154052
$625$	22.1682	0.886727
$626$	−11.7174	−0.468322
$627$	−7.42837	−0.296661
$628$	18.7608	0.748638
$629$	4.11234	0.163970
$630$	6.48028	0.258180
$631$	23.3389	0.929107	0.464554	−	0.885545i	$-0.346215\pi$
$631$	23.3389	0.929107	0.464554	+	0.885545i	$0.346215\pi$
$632$	−9.69643	−0.385703
$633$	15.1373	0.601652
$634$	−8.60508	−0.341752
$635$	24.1537	0.958509
$636$	−7.12481	−0.282517
$637$	16.0249	0.634931
$638$	−14.0249	−0.555252
$639$	−0.0124650	−0.000493108 0
$640$	3.84822	0.152114
$641$	15.4842	0.611590	0.305795	−	0.952097i	$-0.401078\pi$
$641$	15.4842	0.611590	0.305795	+	0.952097i	$0.401078\pi$
$642$	12.5052	0.493541
$643$	−30.3015	−1.19498	−0.597488	−	0.801878i	$-0.703834\pi$
$643$	−30.3015	−1.19498	−0.597488	+	0.801878i	$0.703834\pi$
$644$	4.77532	0.188174
$645$	−21.2891	−0.838256
$646$	−33.9375	−1.33525
$647$	−8.76081	−0.344423	−0.172211	−	0.985060i	$-0.555091\pi$
$647$	−8.76081	−0.344423	−0.172211	+	0.985060i	$0.555091\pi$
$648$	1.00000	0.0392837
$649$	−5.06437	−0.198794
$650$	−37.7463	−1.48053
$651$	13.6819	0.536237
$652$	−10.0729	−0.394485
$653$	−39.0289	−1.52732	−0.763659	−	0.645620i	$-0.776599\pi$
$653$	−39.0289	−1.52732	−0.763659	+	0.645620i	$0.776599\pi$
$654$	19.7693	0.773042
$655$	31.4178	1.22759
$656$	9.69643	0.378582
$657$	−4.00000	−0.156055
$658$	10.3804	0.404670
$659$	38.4572	1.49808	0.749041	−	0.662524i	$-0.230515\pi$
$659$	38.4572	1.49808	0.749041	+	0.662524i	$0.230515\pi$
$660$	−5.06437	−0.197130
$661$	−43.6695	−1.69855	−0.849273	−	0.527955i	$-0.822959\pi$
$661$	−43.6695	−1.69855	−0.849273	+	0.527955i	$0.822959\pi$
$662$	20.4014	0.792923
$663$	−23.1373	−0.898577
$664$	1.12481	0.0436509
$665$	36.5781	1.41844
$666$	−0.683969	−0.0265033
$667$	−30.2206	−1.17015
$668$	1.19123	0.0460899
$669$	16.5781	0.640947
$670$	59.4677	2.29744
$671$	−13.7924	−0.532449
$672$	−1.68397	−0.0649605
$673$	14.3825	0.554403	0.277201	−	0.960812i	$-0.410593\pi$
$673$	14.3825	0.554403	0.277201	+	0.960812i	$0.410593\pi$
$674$	13.9336	0.536701
$675$	−9.80877	−0.377540
$676$	1.80877	0.0695683
$677$	21.4427	0.824111	0.412055	−	0.911159i	$-0.364811\pi$
$677$	21.4427	0.824111	0.412055	+	0.911159i	$0.364811\pi$
$678$	1.00000	0.0384048
$679$	−13.9232	−0.534322
$680$	−23.1373	−0.887274
$681$	−29.4782	−1.12961
$682$	−10.6925	−0.409437
$683$	40.6589	1.55577	0.777884	−	0.628407i	$-0.216293\pi$
$683$	40.6589	1.55577	0.777884	+	0.628407i	$0.216293\pi$
$684$	5.64453	0.215824
$685$	75.2121	2.87371
$686$	−18.8002	−0.717797
$687$	−19.0104	−0.725293
$688$	5.53219	0.210913
$689$	−27.4178	−1.04453
$690$	−10.9126	−0.415435
$691$	−9.46986	−0.360250	−0.180125	−	0.983644i	$-0.557650\pi$
$691$	−9.46986	−0.360250	−0.180125	+	0.983644i	$0.557650\pi$
$692$	13.3659	0.508095
$693$	2.21616	0.0841848
$694$	−16.6570	−0.632291
$695$	−21.2411	−0.805720
$696$	10.6570	0.403952
$697$	−58.2995	−2.20825
$698$	−33.2870	−1.25993
$699$	−8.40738	−0.317996
$700$	16.5177	0.624309
$701$	11.3679	0.429361	0.214681	−	0.976684i	$-0.431129\pi$
$701$	11.3679	0.429361	0.214681	+	0.976684i	$0.431129\pi$
$702$	3.84822	0.145242
$703$	−3.86068	−0.145608
$704$	1.31603	0.0495998
$705$	−23.7214	−0.893398
$706$	5.69643	0.214388
$707$	−8.60902	−0.323776
$708$	3.84822	0.144625
$709$	−10.3594	−0.389056	−0.194528	−	0.980897i	$-0.562317\pi$
$709$	−10.3594	−0.389056	−0.194528	+	0.980897i	$0.562317\pi$
$710$	−0.0479681	−0.00180021
$711$	−9.69643	−0.363645
$712$	−1.09135	−0.0409001
$713$	−23.0399	−0.862852
$714$	10.1248	0.378911
$715$	−19.4888	−0.728840
$716$	−4.17671	−0.156091
$717$	3.26412	0.121901
$718$	36.9500	1.37896
$719$	42.9066	1.60015	0.800073	−	0.599902i	$-0.204794\pi$
$719$	42.9066	1.60015	0.800073	+	0.599902i	$0.204794\pi$
$720$	3.84822	0.143415
$721$	−28.2247	−1.05114
$722$	12.8607	0.478625
$723$	−1.31603	−0.0489437
$724$	−14.5716	−0.541550
$725$	−104.532	−3.88222
$726$	9.26806	0.343970
$727$	41.4487	1.53725	0.768624	−	0.639701i	$-0.220942\pi$
$727$	41.4487	1.53725	0.768624	+	0.639701i	$0.220942\pi$
$728$	−6.48028	−0.240175
$729$	1.00000	0.0370370
$730$	−15.3929	−0.569716
$731$	−33.2621	−1.23024
$732$	10.4803	0.387362
$733$	3.16819	0.117020	0.0585098	−	0.998287i	$-0.481365\pi$
$733$	3.16819	0.117020	0.0585098	+	0.998287i	$0.481365\pi$
$734$	2.75687	0.101758
$735$	16.0249	0.591088
$736$	2.83575	0.104527
$737$	20.3370	0.749124
$738$	9.69643	0.356931
$739$	12.2076	0.449065	0.224532	−	0.974467i	$-0.427915\pi$
$739$	12.2076	0.449065	0.224532	+	0.974467i	$0.427915\pi$
$740$	−2.63206	−0.0967565
$741$	21.7214	0.797954
$742$	11.9980	0.440459
$743$	−18.1557	−0.666069	−0.333034	−	0.942915i	$-0.608073\pi$
$743$	−18.1557	−0.666069	−0.333034	+	0.942915i	$0.608073\pi$
$744$	8.12481	0.297870
$745$	−7.80025	−0.285779
$746$	−2.30357	−0.0843395
$747$	1.12481	0.0411545
$748$	−7.91259	−0.289313
$749$	−21.0584	−0.769457
$750$	−18.5052	−0.675715
$751$	−7.64657	−0.279027	−0.139514	−	0.990220i	$-0.544554\pi$
$751$	−7.64657	−0.279027	−0.139514	+	0.990220i	$0.544554\pi$
$752$	6.16425	0.224787
$753$	−4.35547	−0.158722
$754$	41.0104	1.49351
$755$	10.5282	0.383162
$756$	−1.68397	−0.0612454
$757$	17.6445	0.641301	0.320651	−	0.947198i	$-0.396098\pi$
$757$	17.6445	0.641301	0.320651	+	0.947198i	$0.396098\pi$
$758$	−6.05191	−0.219815
$759$	−3.73194	−0.135461
$760$	21.7214	0.787917
$761$	−19.9211	−0.722140	−0.361070	−	0.932539i	$-0.617588\pi$
$761$	−19.9211	−0.722140	−0.361070	+	0.932539i	$0.617588\pi$
$762$	−6.27659	−0.227377
$763$	−33.2909	−1.20521
$764$	14.2286	0.514773
$765$	−23.1373	−0.836530
$766$	−14.1458	−0.511108
$767$	14.8088	0.534714
$768$	−1.00000	−0.0360844
$769$	−32.2056	−1.16136	−0.580682	−	0.814131i	$-0.697214\pi$
$769$	−32.2056	−1.16136	−0.580682	+	0.814131i	$0.697214\pi$
$770$	8.52825	0.307337
$771$	27.4178	0.987428
$772$	8.02493	0.288824
$773$	−34.6300	−1.24556	−0.622778	−	0.782399i	$-0.713996\pi$
$773$	−34.6300	−1.24556	−0.622778	+	0.782399i	$0.713996\pi$
$774$	5.53219	0.198850
$775$	−79.6944	−2.86271
$776$	−8.26806	−0.296806
$777$	1.15178	0.0413200
$778$	−28.7857	−1.03202
$779$	54.7318	1.96097
$780$	14.8088	0.530239
$781$	−0.0164043	−0.000586994 0
$782$	−17.0499	−0.609702
$783$	10.6570	0.380850
$784$	−4.16425	−0.148723
$785$	72.1957	2.57677
$786$	−8.16425	−0.291209
$787$	−18.8397	−0.671562	−0.335781	−	0.941940i	$-0.609000\pi$
$787$	−18.8397	−0.671562	−0.335781	+	0.941940i	$0.609000\pi$
$788$	1.34301	0.0478427
$789$	3.62354	0.129001
$790$	−37.3140	−1.32757
$791$	−1.68397	−0.0598750
$792$	1.31603	0.0467631
$793$	40.3304	1.43217
$794$	9.28258	0.329426
$795$	−27.4178	−0.972409
$796$	21.0893	0.747491
$797$	31.0353	1.09933	0.549664	−	0.835386i	$-0.314756\pi$
$797$	31.0353	1.09933	0.549664	+	0.835386i	$0.314756\pi$
$798$	−9.50521	−0.336481
$799$	−37.0623	−1.31117
$800$	9.80877	0.346793
$801$	−1.09135	−0.0385609
$802$	17.9460	0.633697
$803$	−5.26412	−0.185767
$804$	−15.4533	−0.544996
$805$	18.3765	0.647685
$806$	31.2660	1.10130
$807$	8.78384	0.309206
$808$	−5.11234	−0.179852
$809$	2.48627	0.0874124	0.0437062	−	0.999044i	$-0.486083\pi$
$809$	2.48627	0.0874124	0.0437062	+	0.999044i	$0.486083\pi$
$810$	3.84822	0.135213
$811$	−37.9355	−1.33209	−0.666047	−	0.745910i	$-0.732015\pi$
$811$	−37.9355	−1.33209	−0.666047	+	0.745910i	$0.732015\pi$
$812$	−17.9460	−0.629783
$813$	7.80025	0.273567
$814$	−0.900124	−0.0315493
$815$	−38.7627	−1.35780
$816$	6.01247	0.210478
$817$	31.2266	1.09248
$818$	16.0000	0.559427
$819$	−6.48028	−0.226439
$820$	37.3140	1.30306
$821$	−7.33244	−0.255904	−0.127952	−	0.991780i	$-0.540840\pi$
$821$	−7.33244	−0.255904	−0.127952	+	0.991780i	$0.540840\pi$
$822$	−19.5447	−0.681698
$823$	−53.9809	−1.88166	−0.940828	−	0.338885i	$-0.889950\pi$
$823$	−53.9809	−1.88166	−0.940828	+	0.338885i	$0.889950\pi$
$824$	−16.7608	−0.583890
$825$	−12.9087	−0.449422
$826$	−6.48028	−0.225478
$827$	20.3719	0.708400	0.354200	−	0.935170i	$-0.384753\pi$
$827$	20.3719	0.708400	0.354200	+	0.935170i	$0.384753\pi$
$828$	2.83575	0.0985492
$829$	−49.6859	−1.72566	−0.862831	−	0.505493i	$-0.831311\pi$
$829$	−49.6859	−1.72566	−0.862831	+	0.505493i	$0.831311\pi$
$830$	4.32850	0.150244
$831$	−15.6235	−0.541974
$832$	−3.84822	−0.133413
$833$	25.0374	0.867494
$834$	5.51972	0.191132
$835$	4.58409	0.158639
$836$	7.42837	0.256916
$837$	8.12481	0.280834
$838$	−26.7299	−0.923369
$839$	51.1622	1.76632	0.883158	−	0.469076i	$-0.155413\pi$
$839$	51.1622	1.76632	0.883158	+	0.469076i	$0.155413\pi$
$840$	−6.48028	−0.223591
$841$	84.5715	2.91626
$842$	19.5197	0.672694
$843$	−22.4782	−0.774191
$844$	−15.1373	−0.521046
$845$	6.96056	0.239450
$846$	6.16425	0.211931
$847$	−15.6071	−0.536267
$848$	7.12481	0.244667
$849$	20.5052	0.703737
$850$	−58.9749	−2.02282
$851$	−1.93957	−0.0664875
$852$	0.0124650	0.000427044 0
$853$	1.46387	0.0501221	0.0250611	−	0.999686i	$-0.492022\pi$
$853$	1.46387	0.0501221	0.0250611	+	0.999686i	$0.492022\pi$
$854$	−17.6485	−0.603918
$855$	21.7214	0.742855
$856$	−12.5052	−0.427419
$857$	−0.515782	−0.0176188	−0.00880939	−	0.999961i	$-0.502804\pi$
$857$	−0.515782	−0.0176188	−0.00880939	+	0.999961i	$0.502804\pi$
$858$	5.06437	0.172895
$859$	−56.3429	−1.92239	−0.961197	−	0.275864i	$-0.911036\pi$
$859$	−56.3429	−1.92239	−0.961197	+	0.275864i	$0.911036\pi$
$860$	21.2891	0.725951
$861$	−16.3285	−0.556474
$862$	−1.63001	−0.0555185
$863$	−38.9315	−1.32524	−0.662622	−	0.748954i	$-0.730557\pi$
$863$	−38.9315	−1.32524	−0.662622	+	0.748954i	$0.730557\pi$
$864$	−1.00000	−0.0340207
$865$	51.4348	1.74884
$866$	32.4323	1.10209
$867$	−19.1497	−0.650359
$868$	−13.6819	−0.464395
$869$	−12.7608	−0.432881
$870$	41.0104	1.39038
$871$	−59.4677	−2.01498
$872$	−19.7693	−0.669474
$873$	−8.26806	−0.279832
$874$	16.0065	0.541427
$875$	31.1622	1.05347
$876$	4.00000	0.135147
$877$	−17.6360	−0.595525	−0.297763	−	0.954640i	$-0.596240\pi$
$877$	−17.6360	−0.595525	−0.297763	+	0.954640i	$0.596240\pi$
$878$	−5.93957	−0.200451
$879$	2.04797	0.0690762
$880$	5.06437	0.170720
$881$	47.5656	1.60253	0.801264	−	0.598311i	$-0.204161\pi$
$881$	47.5656	1.60253	0.801264	+	0.598311i	$0.204161\pi$
$882$	−4.16425	−0.140218
$883$	11.2662	0.379137	0.189568	−	0.981868i	$-0.439291\pi$
$883$	11.2662	0.379137	0.189568	+	0.981868i	$0.439291\pi$
$884$	23.1373	0.778191
$885$	14.8088	0.497792
$886$	7.98155	0.268145
$887$	−51.0813	−1.71514	−0.857571	−	0.514366i	$-0.828027\pi$
$887$	−51.0813	−1.71514	−0.857571	+	0.514366i	$0.828027\pi$
$888$	0.683969	0.0229525
$889$	10.5696	0.354492
$890$	−4.19975	−0.140776
$891$	1.31603	0.0440887
$892$	−16.5781	−0.555076
$893$	34.7943	1.16435
$894$	2.02698	0.0677923
$895$	−16.0729	−0.537258
$896$	1.68397	0.0562575
$897$	10.9126	0.364361
$898$	19.3889	0.647017
$899$	86.5860	2.88780
$900$	9.80877	0.326959
$901$	−42.8376	−1.42713
$902$	12.7608	0.424888
$903$	−9.31603	−0.310018
$904$	−1.00000	−0.0332595
$905$	−56.0748	−1.86399
$906$	−2.73588	−0.0908934
$907$	19.5506	0.649168	0.324584	−	0.945857i	$-0.394776\pi$
$907$	19.5506	0.649168	0.324584	+	0.945857i	$0.394776\pi$
$908$	29.4782	0.978269
$909$	−5.11234	−0.169566
$910$	−24.9375	−0.826671
$911$	13.8791	0.459836	0.229918	−	0.973210i	$-0.426154\pi$
$911$	13.8791	0.459836	0.229918	+	0.973210i	$0.426154\pi$
$912$	−5.64453	−0.186909
$913$	1.48028	0.0489901
$914$	−5.84223	−0.193244
$915$	40.3304	1.33328
$916$	19.0104	0.628122
$917$	13.7483	0.454010
$918$	6.01247	0.198441
$919$	50.3179	1.65984	0.829918	−	0.557886i	$-0.188387\pi$
$919$	50.3179	1.65984	0.829918	+	0.557886i	$0.188387\pi$
$920$	10.9126	0.359777
$921$	10.2805	0.338755
$922$	−36.2930	−1.19525
$923$	0.0479681	0.00157889
$924$	−2.21616	−0.0729062
$925$	−6.70890	−0.220587
$926$	12.7234	0.418117
$927$	−16.7608	−0.550497
$928$	−10.6570	−0.349833
$929$	3.67150	0.120458	0.0602291	−	0.998185i	$-0.480817\pi$
$929$	3.67150	0.120458	0.0602291	+	0.998185i	$0.480817\pi$
$930$	31.2660	1.02525
$931$	−23.5052	−0.770352
$932$	8.40738	0.275393
$933$	−22.0249	−0.721064
$934$	17.8252	0.583258
$935$	−30.4494	−0.995801
$936$	−3.84822	−0.125783
$937$	34.0918	1.11373	0.556866	−	0.830602i	$-0.312004\pi$
$937$	34.0918	1.11373	0.556866	+	0.830602i	$0.312004\pi$
$938$	26.0229	0.849677
$939$	11.7174	0.382384
$940$	23.7214	0.773706
$941$	−23.3620	−0.761578	−0.380789	−	0.924662i	$-0.624348\pi$
$941$	−23.3620	−0.761578	−0.380789	+	0.924662i	$0.624348\pi$
$942$	−18.7608	−0.611260
$943$	27.4967	0.895415
$944$	−3.84822	−0.125249
$945$	−6.48028	−0.210803
$946$	7.28053	0.236710
$947$	−6.07888	−0.197537	−0.0987686	−	0.995110i	$-0.531490\pi$
$947$	−6.07888	−0.197537	−0.0987686	+	0.995110i	$0.531490\pi$
$948$	9.69643	0.314926
$949$	15.3929	0.499674
$950$	55.3659	1.79631
$951$	8.60508	0.279039
$952$	−10.1248	−0.328147
$953$	32.1038	1.03994	0.519972	−	0.854183i	$-0.325942\pi$
$953$	32.1038	1.03994	0.519972	+	0.854183i	$0.325942\pi$
$954$	7.12481	0.230674
$955$	54.7548	1.77182
$956$	−3.26412	−0.105569
$957$	14.0249	0.453362
$958$	−23.3659	−0.754918
$959$	32.9126	1.06280
$960$	−3.84822	−0.124201
$961$	35.0125	1.12943
$962$	2.63206	0.0848611
$963$	−12.5052	−0.402975
$964$	1.31603	0.0423865
$965$	30.8817	0.994116
$966$	−4.77532	−0.153643
$967$	−49.2909	−1.58509	−0.792545	−	0.609813i	$-0.791244\pi$
$967$	−49.2909	−1.58509	−0.792545	+	0.609813i	$0.791244\pi$
$968$	−9.26806	−0.297887
$969$	33.9375	1.09023
$970$	−31.8173	−1.02159
$971$	−32.6759	−1.04862	−0.524310	−	0.851527i	$-0.675677\pi$
$971$	−32.6759	−1.04862	−0.524310	+	0.851527i	$0.675677\pi$
$972$	−1.00000	−0.0320750
$973$	−9.29504	−0.297985
$974$	39.0104	1.24997
$975$	37.7463	1.20885
$976$	−10.4803	−0.335466
$977$	4.73588	0.151514	0.0757571	−	0.997126i	$-0.475863\pi$
$977$	4.73588	0.151514	0.0757571	+	0.997126i	$0.475863\pi$
$978$	10.0729	0.322096
$979$	−1.43625	−0.0459028
$980$	−16.0249	−0.511898
$981$	−19.7693	−0.631186
$982$	2.04797	0.0653533
$983$	21.3015	0.679413	0.339706	−	0.940532i	$-0.389672\pi$
$983$	21.3015	0.679413	0.339706	+	0.940532i	$0.389672\pi$
$984$	−9.69643	−0.309111
$985$	5.16819	0.164672
$986$	64.0748	2.04056
$987$	−10.3804	−0.330412
$988$	−21.7214	−0.691049
$989$	15.6879	0.498846
$990$	5.06437	0.160956
$991$	43.0727	1.36825	0.684125	−	0.729364i	$-0.260184\pi$
$991$	43.0727	1.36825	0.684125	+	0.729364i	$0.260184\pi$
$992$	−8.12481	−0.257963
$993$	−20.4014	−0.647419
$994$	−0.0209907	−0.000665785 0
$995$	81.1562	2.57282
$996$	−1.12481	−0.0356408
$997$	8.65051	0.273965	0.136982	−	0.990574i	$-0.456260\pi$
$997$	8.65051	0.273965	0.136982	+	0.990574i	$0.456260\pi$
$998$	4.54670	0.143923
$999$	0.683969	0.0216398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	678.2.a.j.1.3	✓	3
3.2	odd	2		2034.2.a.q.1.1		3
4.3	odd	2		5424.2.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
678.2.a.j.1.3	✓	3	1.1	even	1	trivial
2034.2.a.q.1.1		3	3.2	odd	2
5424.2.a.bk.1.3		3	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 678 = 2 \cdot 3 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	678.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.84822 q^{5} -1.00000 q^{6} +1.68397 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.84822 q^{10} +1.31603 q^{11} -1.00000 q^{12} -3.84822 q^{13} +1.68397 q^{14} -3.84822 q^{15} +1.00000 q^{16} -6.01247 q^{17} +1.00000 q^{18} +5.64453 q^{19} +3.84822 q^{20} -1.68397 q^{21} +1.31603 q^{22} +2.83575 q^{23} -1.00000 q^{24} +9.80877 q^{25} -3.84822 q^{26} -1.00000 q^{27} +1.68397 q^{28} -10.6570 q^{29} -3.84822 q^{30} -8.12481 q^{31} +1.00000 q^{32} -1.31603 q^{33} -6.01247 q^{34} +6.48028 q^{35} +1.00000 q^{36} -0.683969 q^{37} +5.64453 q^{38} +3.84822 q^{39} +3.84822 q^{40} +9.69643 q^{41} -1.68397 q^{42} +5.53219 q^{43} +1.31603 q^{44} +3.84822 q^{45} +2.83575 q^{46} +6.16425 q^{47} -1.00000 q^{48} -4.16425 q^{49} +9.80877 q^{50} +6.01247 q^{51} -3.84822 q^{52} +7.12481 q^{53} -1.00000 q^{54} +5.06437 q^{55} +1.68397 q^{56} -5.64453 q^{57} -10.6570 q^{58} -3.84822 q^{59} -3.84822 q^{60} -10.4803 q^{61} -8.12481 q^{62} +1.68397 q^{63} +1.00000 q^{64} -14.8088 q^{65} -1.31603 q^{66} +15.4533 q^{67} -6.01247 q^{68} -2.83575 q^{69} +6.48028 q^{70} -0.0124650 q^{71} +1.00000 q^{72} -4.00000 q^{73} -0.683969 q^{74} -9.80877 q^{75} +5.64453 q^{76} +2.21616 q^{77} +3.84822 q^{78} -9.69643 q^{79} +3.84822 q^{80} +1.00000 q^{81} +9.69643 q^{82} +1.12481 q^{83} -1.68397 q^{84} -23.1373 q^{85} +5.53219 q^{86} +10.6570 q^{87} +1.31603 q^{88} -1.09135 q^{89} +3.84822 q^{90} -6.48028 q^{91} +2.83575 q^{92} +8.12481 q^{93} +6.16425 q^{94} +21.7214 q^{95} -1.00000 q^{96} -8.26806 q^{97} -4.16425 q^{98} +1.31603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 7 q^{11} - 3 q^{12} - q^{13} + 2 q^{14} - q^{15} + 3 q^{16} + 3 q^{18} + 5 q^{19} + q^{20} - 2 q^{21} + 7 q^{22}+ \cdots + 7 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + q^5 - 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 7 * q^11 - 3 * q^12 - q^13 + 2 * q^14 - q^15 + 3 * q^16 + 3 * q^18 + 5 * q^19 + q^20 - 2 * q^21 + 7 * q^22 + 16 * q^23 - 3 * q^24 + 10 * q^25 - q^26 - 3 * q^27 + 2 * q^28 - 2 * q^29 - q^30 - 8 * q^31 + 3 * q^32 - 7 * q^33 + 15 * q^35 + 3 * q^36 + q^37 + 5 * q^38 + q^39 + q^40 + 8 * q^41 - 2 * q^42 + 3 * q^43 + 7 * q^44 + q^45 + 16 * q^46 + 11 * q^47 - 3 * q^48 - 5 * q^49 + 10 * q^50 - q^52 + 5 * q^53 - 3 * q^54 - 12 * q^55 + 2 * q^56 - 5 * q^57 - 2 * q^58 - q^59 - q^60 - 27 * q^61 - 8 * q^62 + 2 * q^63 + 3 * q^64 - 25 * q^65 - 7 * q^66 + 15 * q^67 - 16 * q^69 + 15 * q^70 + 18 * q^71 + 3 * q^72 - 12 * q^73 + q^74 - 10 * q^75 + 5 * q^76 - 10 * q^77 + q^78 - 8 * q^79 + q^80 + 3 * q^81 + 8 * q^82 - 13 * q^83 - 2 * q^84 - 35 * q^85 + 3 * q^86 + 2 * q^87 + 7 * q^88 - 3 * q^89 + q^90 - 15 * q^91 + 16 * q^92 + 8 * q^93 + 11 * q^94 + 8 * q^95 - 3 * q^96 + q^97 - 5 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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