LMFDB - Embedded newform 112.16.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,16,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.1"); S:= CuspForms(chi, 16); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3864] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$159.816725712$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 911559x^{4} - 33878904x^{3} + 200237882347x^{2} + 5867703981762x - 7046699346580509 $ x^6 - 2x^5 - 911559x^4 - 33878904x^3 + 200237882347x^2 + 5867703981762*x - 7046699346580509
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{6}\cdot 5\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$503.385$ of defining polynomial
Character	$\chi$	$=$	112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1220.73 q^{3} +302431. q^{5} +823543. q^{7} -1.28587e7 q^{9} +4.25717e7 q^{11} -6.17422e7 q^{13} -3.69187e8 q^{15} +2.23749e9 q^{17} -2.64619e9 q^{19} -1.00533e9 q^{21} -2.05625e10 q^{23} +6.09467e10 q^{25} +3.32133e10 q^{27} -1.60930e11 q^{29} -2.41214e11 q^{31} -5.19687e10 q^{33} +2.49065e11 q^{35} -5.81407e11 q^{37} +7.53708e10 q^{39} +1.74800e11 q^{41} -2.72649e12 q^{43} -3.88887e12 q^{45} -5.02446e12 q^{47} +6.78223e11 q^{49} -2.73138e12 q^{51} +1.34108e13 q^{53} +1.28750e13 q^{55} +3.23029e12 q^{57} +1.29682e13 q^{59} -7.88668e12 q^{61} -1.05897e13 q^{63} -1.86727e13 q^{65} +7.41775e13 q^{67} +2.51013e13 q^{69} -2.45011e13 q^{71} +1.14509e14 q^{73} -7.43997e13 q^{75} +3.50596e13 q^{77} -1.61131e14 q^{79} +1.43964e14 q^{81} -8.81127e13 q^{83} +6.76686e14 q^{85} +1.96453e14 q^{87} -2.33877e14 q^{89} -5.08474e13 q^{91} +2.94458e14 q^{93} -8.00289e14 q^{95} -1.22725e15 q^{97} -5.47418e14 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3864 q^{3} + 168532 q^{5} + 4941258 q^{7} + 27735822 q^{9} - 58040072 q^{11} - 15956444 q^{13} + 705059136 q^{15} - 2879979060 q^{17} - 3435500376 q^{19} - 3182170152 q^{21} - 34508782592 q^{23} + 29232294906 q^{25}+ \cdots - 44\!\cdots\!92 q^{99}+O(q^{100}) $ 6 * q - 3864 * q^3 + 168532 * q^5 + 4941258 * q^7 + 27735822 * q^9 - 58040072 * q^11 - 15956444 * q^13 + 705059136 * q^15 - 2879979060 * q^17 - 3435500376 * q^19 - 3182170152 * q^21 - 34508782592 * q^23 + 29232294906 * q^25 - 102516868944 * q^27 - 46458543420 * q^29 - 300765178336 * q^31 - 457356427584 * q^33 + 138793348876 * q^35 - 509722422412 * q^37 + 437968495008 * q^39 + 1619276439676 * q^41 - 2727781008680 * q^43 + 5316894690660 * q^45 + 1567726443808 * q^47 + 4069338437094 * q^49 - 3110310296208 * q^51 + 2612518893588 * q^53 + 4577322926960 * q^55 - 21550151959824 * q^57 - 5112561397384 * q^59 + 4545346033732 * q^61 + 22841642057346 * q^63 + 57444596600712 * q^65 - 74088461612408 * q^67 + 176631933295104 * q^69 - 51092191255952 * q^71 + 294343628983292 * q^73 - 208088876183496 * q^75 - 47798495015096 * q^77 - 56070305620928 * q^79 + 691991522436870 * q^81 - 984559173278520 * q^83 + 396331720481544 * q^85 - 1024554713548464 * q^87 + 1528383063352572 * q^89 - 13140817761092 * q^91 + 1245617896072608 * q^93 - 2532178187158304 * q^95 + 2255703641701580 * q^97 - 4484026592356392 * 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 $

$2$	0	0
$2$	0	0
$3$	−1220.73	−0.322264	−0.161132	−	0.986933i	$-0.551514\pi$
$3$	−1220.73	−0.322264	−0.161132	+	0.986933i	$0.551514\pi$
$4$	0	0
$5$	302431.	1.73121	0.865607	−	0.500724i	$-0.166933\pi$
$5$	302431.	1.73121	0.865607	+	0.500724i	$0.166933\pi$
$6$	0	0
$7$	823543.	0.377964
$7$	823543.	0.377964
$8$	0	0
$9$	−1.28587e7	−0.896146
$10$	0	0
$11$	4.25717e7	0.658682	0.329341	−	0.944211i	$-0.393173\pi$
$11$	4.25717e7	0.658682	0.329341	+	0.944211i	$0.393173\pi$
$12$	0	0
$13$	−6.17422e7	−0.272902	−0.136451	−	0.990647i	$-0.543570\pi$
$13$	−6.17422e7	−0.272902	−0.136451	+	0.990647i	$0.543570\pi$
$14$	0	0
$15$	−3.69187e8	−0.557907
$16$	0	0
$17$	2.23749e9	1.32250	0.661248	−	0.750167i	$-0.270027\pi$
$17$	2.23749e9	1.32250	0.661248	+	0.750167i	$0.270027\pi$
$18$	0	0
$19$	−2.64619e9	−0.679155	−0.339578	−	0.940578i	$-0.610284\pi$
$19$	−2.64619e9	−0.679155	−0.339578	+	0.940578i	$0.610284\pi$
$20$	0	0
$21$	−1.00533e9	−0.121804
$22$	0	0
$23$	−2.05625e10	−1.25927	−0.629633	−	0.776893i	$-0.716795\pi$
$23$	−2.05625e10	−1.25927	−0.629633	+	0.776893i	$0.716795\pi$
$24$	0	0
$25$	6.09467e10	1.99710
$26$	0	0
$27$	3.32133e10	0.611059
$28$	0	0
$29$	−1.60930e11	−1.73242	−0.866209	−	0.499682i	$-0.833450\pi$
$29$	−1.60930e11	−1.73242	−0.866209	+	0.499682i	$0.833450\pi$
$30$	0	0
$31$	−2.41214e11	−1.57467	−0.787335	−	0.616525i	$-0.788540\pi$
$31$	−2.41214e11	−1.57467	−0.787335	+	0.616525i	$0.788540\pi$
$32$	0	0
$33$	−5.19687e10	−0.212269
$34$	0	0
$35$	2.49065e11	0.654337
$36$	0	0
$37$	−5.81407e11	−1.00686	−0.503428	−	0.864037i	$-0.667928\pi$
$37$	−5.81407e11	−1.00686	−0.503428	+	0.864037i	$0.667928\pi$
$38$	0	0
$39$	7.53708e10	0.0879465
$40$	0	0
$41$	1.74800e11	0.140172	0.0700862	−	0.997541i	$-0.477673\pi$
$41$	1.74800e11	0.140172	0.0700862	+	0.997541i	$0.477673\pi$
$42$	0	0
$43$	−2.72649e12	−1.52964	−0.764822	−	0.644242i	$-0.777173\pi$
$43$	−2.72649e12	−1.52964	−0.764822	+	0.644242i	$0.777173\pi$
$44$	0	0
$45$	−3.88887e12	−1.55142
$46$	0	0
$47$	−5.02446e12	−1.44662	−0.723311	−	0.690522i	$-0.757381\pi$
$47$	−5.02446e12	−1.44662	−0.723311	+	0.690522i	$0.757381\pi$
$48$	0	0
$49$	6.78223e11	0.142857
$50$	0	0
$51$	−2.73138e12	−0.426193
$52$	0	0
$53$	1.34108e13	1.56814	0.784070	−	0.620672i	$-0.213140\pi$
$53$	1.34108e13	1.56814	0.784070	+	0.620672i	$0.213140\pi$
$54$	0	0
$55$	1.28750e13	1.14032
$56$	0	0
$57$	3.23029e12	0.218867
$58$	0	0
$59$	1.29682e13	0.678406	0.339203	−	0.940713i	$-0.389843\pi$
$59$	1.29682e13	0.678406	0.339203	+	0.940713i	$0.389843\pi$
$60$	0	0
$61$	−7.88668e12	−0.321307	−0.160654	−	0.987011i	$-0.551360\pi$
$61$	−7.88668e12	−0.321307	−0.160654	+	0.987011i	$0.551360\pi$
$62$	0	0
$63$	−1.05897e13	−0.338711
$64$	0	0
$65$	−1.86727e13	−0.472452
$66$	0	0
$67$	7.41775e13	1.49524	0.747621	−	0.664126i	$-0.231196\pi$
$67$	7.41775e13	1.49524	0.747621	+	0.664126i	$0.231196\pi$
$68$	0	0
$69$	2.51013e13	0.405815
$70$	0	0
$71$	−2.45011e13	−0.319704	−0.159852	−	0.987141i	$-0.551102\pi$
$71$	−2.45011e13	−0.319704	−0.159852	+	0.987141i	$0.551102\pi$
$72$	0	0
$73$	1.14509e14	1.21316	0.606581	−	0.795022i	$-0.292541\pi$
$73$	1.14509e14	1.21316	0.606581	+	0.795022i	$0.292541\pi$
$74$	0	0
$75$	−7.43997e13	−0.643593
$76$	0	0
$77$	3.50596e13	0.248958
$78$	0	0
$79$	−1.61131e14	−0.944006	−0.472003	−	0.881597i	$-0.656469\pi$
$79$	−1.61131e14	−0.944006	−0.472003	+	0.881597i	$0.656469\pi$
$80$	0	0
$81$	1.43964e14	0.699224
$82$	0	0
$83$	−8.81127e13	−0.356412	−0.178206	−	0.983993i	$-0.557029\pi$
$83$	−8.81127e13	−0.356412	−0.178206	+	0.983993i	$0.557029\pi$
$84$	0	0
$85$	6.76686e14	2.28953
$86$	0	0
$87$	1.96453e14	0.558295
$88$	0	0
$89$	−2.33877e14	−0.560484	−0.280242	−	0.959929i	$-0.590415\pi$
$89$	−2.33877e14	−0.560484	−0.280242	+	0.959929i	$0.590415\pi$
$90$	0	0
$91$	−5.08474e13	−0.103147
$92$	0	0
$93$	2.94458e14	0.507459
$94$	0	0
$95$	−8.00289e14	−1.17576
$96$	0	0
$97$	−1.22725e15	−1.54222	−0.771109	−	0.636704i	$-0.780297\pi$
$97$	−1.22725e15	−1.54222	−0.771109	+	0.636704i	$0.780297\pi$
$98$	0	0
$99$	−5.47418e14	−0.590276
$100$	0	0
$101$	5.72318e14	0.531162	0.265581	−	0.964089i	$-0.414436\pi$
$101$	5.72318e14	0.531162	0.265581	+	0.964089i	$0.414436\pi$
$102$	0	0
$103$	−9.76230e14	−0.782119	−0.391060	−	0.920365i	$-0.627891\pi$
$103$	−9.76230e14	−0.782119	−0.391060	+	0.920365i	$0.627891\pi$
$104$	0	0
$105$	−3.04042e14	−0.210869
$106$	0	0
$107$	−1.69443e15	−1.02011	−0.510054	−	0.860143i	$-0.670374\pi$
$107$	−1.69443e15	−1.02011	−0.510054	+	0.860143i	$0.670374\pi$
$108$	0	0
$109$	1.31536e14	0.0689204	0.0344602	−	0.999406i	$-0.489029\pi$
$109$	1.31536e14	0.0689204	0.0344602	+	0.999406i	$0.489029\pi$
$110$	0	0
$111$	7.09743e14	0.324473
$112$	0	0
$113$	2.32450e15	0.929483	0.464741	−	0.885447i	$-0.346147\pi$
$113$	2.32450e15	0.929483	0.464741	+	0.885447i	$0.346147\pi$
$114$	0	0
$115$	−6.21873e15	−2.18006
$116$	0	0
$117$	7.93926e14	0.244560
$118$	0	0
$119$	1.84267e15	0.499857
$120$	0	0
$121$	−2.36490e15	−0.566138
$122$	0	0
$123$	−2.13384e14	−0.0451724
$124$	0	0
$125$	9.20270e15	1.72620
$126$	0	0
$127$	−9.45385e15	−1.57428	−0.787138	−	0.616777i	$-0.788438\pi$
$127$	−9.45385e15	−1.57428	−0.787138	+	0.616777i	$0.788438\pi$
$128$	0	0
$129$	3.32831e15	0.492949
$130$	0	0
$131$	2.03796e15	0.268944	0.134472	−	0.990917i	$-0.457066\pi$
$131$	2.03796e15	0.268944	0.134472	+	0.990917i	$0.457066\pi$
$132$	0	0
$133$	−2.17925e15	−0.256696
$134$	0	0
$135$	1.00447e16	1.05787
$136$	0	0
$137$	1.54543e16	1.45763	0.728813	−	0.684713i	$-0.240072\pi$
$137$	1.54543e16	1.45763	0.728813	+	0.684713i	$0.240072\pi$
$138$	0	0
$139$	1.22505e16	1.03644	0.518220	−	0.855247i	$-0.326595\pi$
$139$	1.22505e16	1.03644	0.518220	+	0.855247i	$0.326595\pi$
$140$	0	0
$141$	6.13352e15	0.466194
$142$	0	0
$143$	−2.62847e15	−0.179756
$144$	0	0
$145$	−4.86702e16	−2.99919
$146$	0	0
$147$	−8.27930e14	−0.0460377
$148$	0	0
$149$	−6.25245e15	−0.314162	−0.157081	−	0.987586i	$-0.550208\pi$
$149$	−6.25245e15	−0.314162	−0.157081	+	0.987586i	$0.550208\pi$
$150$	0	0
$151$	−1.00448e16	−0.456683	−0.228342	−	0.973581i	$-0.573330\pi$
$151$	−1.00448e16	−0.456683	−0.228342	+	0.973581i	$0.573330\pi$
$152$	0	0
$153$	−2.87713e16	−1.18515
$154$	0	0
$155$	−7.29505e16	−2.72609
$156$	0	0
$157$	3.06264e16	1.03956	0.519778	−	0.854301i	$-0.326015\pi$
$157$	3.06264e16	1.03956	0.519778	+	0.854301i	$0.326015\pi$
$158$	0	0
$159$	−1.63710e16	−0.505355
$160$	0	0
$161$	−1.69341e16	−0.475957
$162$	0	0
$163$	3.64022e16	0.932654	0.466327	−	0.884612i	$-0.345577\pi$
$163$	3.64022e16	0.932654	0.466327	+	0.884612i	$0.345577\pi$
$164$	0	0
$165$	−1.57169e16	−0.367484
$166$	0	0
$167$	1.74007e16	0.371701	0.185851	−	0.982578i	$-0.440496\pi$
$167$	1.74007e16	0.371701	0.185851	+	0.982578i	$0.440496\pi$
$168$	0	0
$169$	−4.73738e16	−0.925524
$170$	0	0
$171$	3.40266e16	0.608622
$172$	0	0
$173$	−3.73169e16	−0.611730	−0.305865	−	0.952075i	$-0.598946\pi$
$173$	−3.73169e16	−0.611730	−0.305865	+	0.952075i	$0.598946\pi$
$174$	0	0
$175$	5.01922e16	0.754833
$176$	0	0
$177$	−1.58307e16	−0.218626
$178$	0	0
$179$	−5.48513e16	−0.696288	−0.348144	−	0.937441i	$-0.613188\pi$
$179$	−5.48513e16	−0.696288	−0.348144	+	0.937441i	$0.613188\pi$
$180$	0	0
$181$	−1.34105e17	−1.56623	−0.783115	−	0.621876i	$-0.786371\pi$
$181$	−1.34105e17	−1.56623	−0.783115	+	0.621876i	$0.786371\pi$
$182$	0	0
$183$	9.62754e15	0.103546
$184$	0	0
$185$	−1.75835e17	−1.74308
$186$	0	0
$187$	9.52539e16	0.871105
$188$	0	0
$189$	2.73525e16	0.230959
$190$	0	0
$191$	2.31192e16	0.180394	0.0901972	−	0.995924i	$-0.471250\pi$
$191$	2.31192e16	0.180394	0.0901972	+	0.995924i	$0.471250\pi$
$192$	0	0
$193$	−5.81946e16	−0.419955	−0.209978	−	0.977706i	$-0.567339\pi$
$193$	−5.81946e16	−0.419955	−0.209978	+	0.977706i	$0.567339\pi$
$194$	0	0
$195$	2.27944e16	0.152254
$196$	0	0
$197$	2.37279e17	1.46812	0.734062	−	0.679083i	$-0.237622\pi$
$197$	2.37279e17	1.46812	0.734062	+	0.679083i	$0.237622\pi$
$198$	0	0
$199$	−2.47253e17	−1.41822	−0.709109	−	0.705099i	$-0.750903\pi$
$199$	−2.47253e17	−1.41822	−0.709109	+	0.705099i	$0.750903\pi$
$200$	0	0
$201$	−9.05510e16	−0.481862
$202$	0	0
$203$	−1.32533e17	−0.654792
$204$	0	0
$205$	5.28648e16	0.242668
$206$	0	0
$207$	2.64407e17	1.12849
$208$	0	0
$209$	−1.12653e17	−0.447347
$210$	0	0
$211$	1.16852e17	0.432033	0.216016	−	0.976390i	$-0.430694\pi$
$211$	1.16852e17	0.432033	0.216016	+	0.976390i	$0.430694\pi$
$212$	0	0
$213$	2.99094e16	0.103029
$214$	0	0
$215$	−8.24574e17	−2.64814
$216$	0	0
$217$	−1.98650e17	−0.595170
$218$	0	0
$219$	−1.39785e17	−0.390958
$220$	0	0
$221$	−1.38148e17	−0.360913
$222$	0	0
$223$	−1.41126e17	−0.344605	−0.172302	−	0.985044i	$-0.555121\pi$
$223$	−1.41126e17	−0.344605	−0.172302	+	0.985044i	$0.555121\pi$
$224$	0	0
$225$	−7.83696e17	−1.78969
$226$	0	0
$227$	3.52597e16	0.0753502	0.0376751	−	0.999290i	$-0.488005\pi$
$227$	3.52597e16	0.0753502	0.0376751	+	0.999290i	$0.488005\pi$
$228$	0	0
$229$	−4.56467e17	−0.913363	−0.456681	−	0.889630i	$-0.650962\pi$
$229$	−4.56467e17	−0.913363	−0.456681	+	0.889630i	$0.650962\pi$
$230$	0	0
$231$	−4.27985e16	−0.0802303
$232$	0	0
$233$	4.84605e17	0.851566	0.425783	−	0.904825i	$-0.359999\pi$
$233$	4.84605e17	0.851566	0.425783	+	0.904825i	$0.359999\pi$
$234$	0	0
$235$	−1.51955e18	−2.50441
$236$	0	0
$237$	1.96697e17	0.304219
$238$	0	0
$239$	5.93213e17	0.861442	0.430721	−	0.902485i	$-0.358259\pi$
$239$	5.93213e17	0.861442	0.430721	+	0.902485i	$0.358259\pi$
$240$	0	0
$241$	−1.15303e18	−1.57294	−0.786469	−	0.617630i	$-0.788093\pi$
$241$	−1.15303e18	−1.57294	−0.786469	+	0.617630i	$0.788093\pi$
$242$	0	0
$243$	−6.52316e17	−0.836394
$244$	0	0
$245$	2.05115e17	0.247316
$246$	0	0
$247$	1.63382e17	0.185343
$248$	0	0
$249$	1.07562e17	0.114859
$250$	0	0
$251$	−1.34578e18	−1.35339	−0.676694	−	0.736264i	$-0.736588\pi$
$251$	−1.34578e18	−1.35339	−0.676694	+	0.736264i	$0.736588\pi$
$252$	0	0
$253$	−8.75381e17	−0.829456
$254$	0	0
$255$	−8.26053e17	−0.737831
$256$	0	0
$257$	1.61953e18	1.36424	0.682120	−	0.731240i	$-0.261058\pi$
$257$	1.61953e18	1.36424	0.682120	+	0.731240i	$0.261058\pi$
$258$	0	0
$259$	−4.78814e17	−0.380556
$260$	0	0
$261$	2.06936e18	1.55250
$262$	0	0
$263$	−1.34009e18	−0.949435	−0.474718	−	0.880138i	$-0.657450\pi$
$263$	−1.34009e18	−0.949435	−0.474718	+	0.880138i	$0.657450\pi$
$264$	0	0
$265$	4.05583e18	2.71479
$266$	0	0
$267$	2.85502e17	0.180624
$268$	0	0
$269$	−3.03300e18	−1.81439	−0.907194	−	0.420713i	$-0.861780\pi$
$269$	−3.03300e18	−1.81439	−0.907194	+	0.420713i	$0.861780\pi$
$270$	0	0
$271$	−1.00397e18	−0.568134	−0.284067	−	0.958804i	$-0.591684\pi$
$271$	−1.00397e18	−0.568134	−0.284067	+	0.958804i	$0.591684\pi$
$272$	0	0
$273$	6.20711e16	0.0332407
$274$	0	0
$275$	2.59461e18	1.31546
$276$	0	0
$277$	−2.40652e18	−1.15556	−0.577778	−	0.816194i	$-0.696080\pi$
$277$	−2.40652e18	−1.15556	−0.577778	+	0.816194i	$0.696080\pi$
$278$	0	0
$279$	3.10170e18	1.41114
$280$	0	0
$281$	1.20055e18	0.517704	0.258852	−	0.965917i	$-0.416656\pi$
$281$	1.20055e18	0.517704	0.258852	+	0.965917i	$0.416656\pi$
$282$	0	0
$283$	−1.89415e18	−0.774492	−0.387246	−	0.921976i	$-0.626574\pi$
$283$	−1.89415e18	−0.774492	−0.387246	+	0.921976i	$0.626574\pi$
$284$	0	0
$285$	9.76939e17	0.378906
$286$	0	0
$287$	1.43955e17	0.0529802
$288$	0	0
$289$	2.14395e18	0.748998
$290$	0	0
$291$	1.49815e18	0.497001
$292$	0	0
$293$	−4.14463e18	−1.30611	−0.653053	−	0.757312i	$-0.726512\pi$
$293$	−4.14463e18	−1.30611	−0.653053	+	0.757312i	$0.726512\pi$
$294$	0	0
$295$	3.92198e18	1.17447
$296$	0	0
$297$	1.41395e18	0.402494
$298$	0	0
$299$	1.26958e18	0.343656
$300$	0	0
$301$	−2.24538e18	−0.578151
$302$	0	0
$303$	−6.98647e17	−0.171174
$304$	0	0
$305$	−2.38517e18	−0.556252
$306$	0	0
$307$	1.65491e18	0.367483	0.183742	−	0.982975i	$-0.441179\pi$
$307$	1.65491e18	0.367483	0.183742	+	0.982975i	$0.441179\pi$
$308$	0	0
$309$	1.19172e18	0.252049
$310$	0	0
$311$	1.24756e18	0.251395	0.125698	−	0.992069i	$-0.459883\pi$
$311$	1.24756e18	0.251395	0.125698	+	0.992069i	$0.459883\pi$
$312$	0	0
$313$	−5.26739e18	−1.01161	−0.505804	−	0.862648i	$-0.668804\pi$
$313$	−5.26739e18	−1.01161	−0.505804	+	0.862648i	$0.668804\pi$
$314$	0	0
$315$	−3.20265e18	−0.586382
$316$	0	0
$317$	−7.85722e17	−0.137191	−0.0685953	−	0.997645i	$-0.521852\pi$
$317$	−7.85722e17	−0.137191	−0.0685953	+	0.997645i	$0.521852\pi$
$318$	0	0
$319$	−6.85108e18	−1.14111
$320$	0	0
$321$	2.06845e18	0.328744
$322$	0	0
$323$	−5.92083e18	−0.898180
$324$	0	0
$325$	−3.76299e18	−0.545014
$326$	0	0
$327$	−1.60571e17	−0.0222105
$328$	0	0
$329$	−4.13786e18	−0.546772
$330$	0	0
$331$	1.18635e19	1.49796	0.748982	−	0.662591i	$-0.230543\pi$
$331$	1.18635e19	1.49796	0.748982	+	0.662591i	$0.230543\pi$
$332$	0	0
$333$	7.47615e18	0.902290
$334$	0	0
$335$	2.24336e19	2.58858
$336$	0	0
$337$	2.39450e18	0.264235	0.132117	−	0.991234i	$-0.457822\pi$
$337$	2.39450e18	0.264235	0.132117	+	0.991234i	$0.457822\pi$
$338$	0	0
$339$	−2.83759e18	−0.299538
$340$	0	0
$341$	−1.02689e19	−1.03721
$342$	0	0
$343$	5.58546e17	0.0539949
$344$	0	0
$345$	7.59141e18	0.702553
$346$	0	0
$347$	1.72965e19	1.53280	0.766402	−	0.642361i	$-0.222045\pi$
$347$	1.72965e19	1.53280	0.766402	+	0.642361i	$0.222045\pi$
$348$	0	0
$349$	3.45029e18	0.292863	0.146432	−	0.989221i	$-0.453221\pi$
$349$	3.45029e18	0.292863	0.146432	+	0.989221i	$0.453221\pi$
$350$	0	0
$351$	−2.05066e18	−0.166759
$352$	0	0
$353$	4.60031e18	0.358490	0.179245	−	0.983805i	$-0.442635\pi$
$353$	4.60031e18	0.358490	0.179245	+	0.983805i	$0.442635\pi$
$354$	0	0
$355$	−7.40990e18	−0.553477
$356$	0	0
$357$	−2.24941e18	−0.161086
$358$	0	0
$359$	2.31291e19	1.58836	0.794182	−	0.607680i	$-0.207900\pi$
$359$	2.31291e19	1.58836	0.794182	+	0.607680i	$0.207900\pi$
$360$	0	0
$361$	−8.17881e18	−0.538748
$362$	0	0
$363$	2.88691e18	0.182446
$364$	0	0
$365$	3.46311e19	2.10024
$366$	0	0
$367$	−1.24556e17	−0.00725055	−0.00362527	−	0.999993i	$-0.501154\pi$
$367$	−1.24556e17	−0.00725055	−0.00362527	+	0.999993i	$0.501154\pi$
$368$	0	0
$369$	−2.24770e18	−0.125615
$370$	0	0
$371$	1.10444e19	0.592701
$372$	0	0
$373$	2.66080e19	1.37150	0.685750	−	0.727837i	$-0.259474\pi$
$373$	2.66080e19	1.37150	0.685750	+	0.727837i	$0.259474\pi$
$374$	0	0
$375$	−1.12340e19	−0.556290
$376$	0	0
$377$	9.93619e18	0.472781
$378$	0	0
$379$	−8.87782e18	−0.405987	−0.202994	−	0.979180i	$-0.565067\pi$
$379$	−8.87782e18	−0.405987	−0.202994	+	0.979180i	$0.565067\pi$
$380$	0	0
$381$	1.15406e19	0.507332
$382$	0	0
$383$	2.48372e19	1.04981	0.524905	−	0.851161i	$-0.324101\pi$
$383$	2.48372e19	1.04981	0.524905	+	0.851161i	$0.324101\pi$
$384$	0	0
$385$	1.06031e19	0.431000
$386$	0	0
$387$	3.50591e19	1.37078
$388$	0	0
$389$	2.53209e19	0.952484	0.476242	−	0.879314i	$-0.341999\pi$
$389$	2.53209e19	0.952484	0.476242	+	0.879314i	$0.341999\pi$
$390$	0	0
$391$	−4.60084e19	−1.66537
$392$	0	0
$393$	−2.48781e18	−0.0866708
$394$	0	0
$395$	−4.87308e19	−1.63428
$396$	0	0
$397$	−1.49391e19	−0.482386	−0.241193	−	0.970477i	$-0.577539\pi$
$397$	−1.49391e19	−0.482386	−0.241193	+	0.970477i	$0.577539\pi$
$398$	0	0
$399$	2.66028e18	0.0827240
$400$	0	0
$401$	−1.41395e19	−0.423498	−0.211749	−	0.977324i	$-0.567916\pi$
$401$	−1.41395e19	−0.423498	−0.211749	+	0.977324i	$0.567916\pi$
$402$	0	0
$403$	1.48931e19	0.429731
$404$	0	0
$405$	4.35391e19	1.21051
$406$	0	0
$407$	−2.47515e19	−0.663198
$408$	0	0
$409$	−4.08338e19	−1.05462	−0.527309	−	0.849674i	$-0.676799\pi$
$409$	−4.08338e19	−1.05462	−0.527309	+	0.849674i	$0.676799\pi$
$410$	0	0
$411$	−1.88656e19	−0.469740
$412$	0	0
$413$	1.06799e19	0.256413
$414$	0	0
$415$	−2.66480e19	−0.617026
$416$	0	0
$417$	−1.49546e19	−0.334007
$418$	0	0
$419$	5.34957e18	0.115269	0.0576347	−	0.998338i	$-0.481644\pi$
$419$	5.34957e18	0.115269	0.0576347	+	0.998338i	$0.481644\pi$
$420$	0	0
$421$	−5.14341e19	−1.06939	−0.534695	−	0.845045i	$-0.679573\pi$
$421$	−5.14341e19	−1.06939	−0.534695	+	0.845045i	$0.679573\pi$
$422$	0	0
$423$	6.46081e19	1.29639
$424$	0	0
$425$	1.36368e20	2.64116
$426$	0	0
$427$	−6.49502e18	−0.121443
$428$	0	0
$429$	3.20866e18	0.0579288
$430$	0	0
$431$	−1.84398e19	−0.321497	−0.160748	−	0.986995i	$-0.551391\pi$
$431$	−1.84398e19	−0.321497	−0.160748	+	0.986995i	$0.551391\pi$
$432$	0	0
$433$	−6.42340e19	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$433$	−6.42340e19	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$434$	0	0
$435$	5.94134e19	0.966529
$436$	0	0
$437$	5.44123e19	0.855236
$438$	0	0
$439$	−8.62412e19	−1.30988	−0.654939	−	0.755681i	$-0.727306\pi$
$439$	−8.62412e19	−1.30988	−0.654939	+	0.755681i	$0.727306\pi$
$440$	0	0
$441$	−8.72108e18	−0.128021
$442$	0	0
$443$	−1.33710e20	−1.89730	−0.948652	−	0.316321i	$-0.897552\pi$
$443$	−1.33710e20	−1.89730	−0.948652	+	0.316321i	$0.897552\pi$
$444$	0	0
$445$	−7.07316e19	−0.970317
$446$	0	0
$447$	7.63257e18	0.101243
$448$	0	0
$449$	−6.40137e19	−0.821155	−0.410578	−	0.911826i	$-0.634673\pi$
$449$	−6.40137e19	−0.821155	−0.410578	+	0.911826i	$0.634673\pi$
$450$	0	0
$451$	7.44153e18	0.0923290
$452$	0	0
$453$	1.22621e19	0.147172
$454$	0	0
$455$	−1.53778e19	−0.178570
$456$	0	0
$457$	8.37052e18	0.0940548	0.0470274	−	0.998894i	$-0.485025\pi$
$457$	8.37052e18	0.0940548	0.0470274	+	0.998894i	$0.485025\pi$
$458$	0	0
$459$	7.43144e19	0.808124
$460$	0	0
$461$	6.38758e19	0.672325	0.336163	−	0.941804i	$-0.390871\pi$
$461$	6.38758e19	0.672325	0.336163	+	0.941804i	$0.390871\pi$
$462$	0	0
$463$	−8.02771e19	−0.817964	−0.408982	−	0.912542i	$-0.634116\pi$
$463$	−8.02771e19	−0.817964	−0.408982	+	0.912542i	$0.634116\pi$
$464$	0	0
$465$	8.90531e19	0.878521
$466$	0	0
$467$	1.29790e20	1.23984	0.619918	−	0.784667i	$-0.287166\pi$
$467$	1.29790e20	1.23984	0.619918	+	0.784667i	$0.287166\pi$
$468$	0	0
$469$	6.10884e19	0.565148
$470$	0	0
$471$	−3.73866e19	−0.335011
$472$	0	0
$473$	−1.16071e20	−1.00755
$474$	0	0
$475$	−1.61277e20	−1.35634
$476$	0	0
$477$	−1.72445e20	−1.40528
$478$	0	0
$479$	1.91078e20	1.50902	0.754510	−	0.656288i	$-0.227874\pi$
$479$	1.91078e20	1.50902	0.754510	+	0.656288i	$0.227874\pi$
$480$	0	0
$481$	3.58974e19	0.274773
$482$	0	0
$483$	2.06720e19	0.153384
$484$	0	0
$485$	−3.71158e20	−2.66991
$486$	0	0
$487$	2.18542e20	1.52429	0.762146	−	0.647405i	$-0.224146\pi$
$487$	2.18542e20	1.52429	0.762146	+	0.647405i	$0.224146\pi$
$488$	0	0
$489$	−4.44374e19	−0.300561
$490$	0	0
$491$	−3.62782e19	−0.237977	−0.118988	−	0.992896i	$-0.537965\pi$
$491$	−3.62782e19	−0.237977	−0.118988	+	0.992896i	$0.537965\pi$
$492$	0	0
$493$	−3.60080e20	−2.29112
$494$	0	0
$495$	−1.65556e20	−1.02189
$496$	0	0
$497$	−2.01777e19	−0.120837
$498$	0	0
$499$	1.48147e20	0.860870	0.430435	−	0.902622i	$-0.358360\pi$
$499$	1.48147e20	0.860870	0.430435	+	0.902622i	$0.358360\pi$
$500$	0	0
$501$	−2.12417e19	−0.119786
$502$	0	0
$503$	2.03777e19	0.111531	0.0557655	−	0.998444i	$-0.482240\pi$
$503$	2.03777e19	0.111531	0.0557655	+	0.998444i	$0.482240\pi$
$504$	0	0
$505$	1.73086e20	0.919555
$506$	0	0
$507$	5.78308e19	0.298263
$508$	0	0
$509$	1.16832e20	0.585030	0.292515	−	0.956261i	$-0.405508\pi$
$509$	1.16832e20	0.585030	0.292515	+	0.956261i	$0.405508\pi$
$510$	0	0
$511$	9.43032e19	0.458532
$512$	0	0
$513$	−8.78886e19	−0.415004
$514$	0	0
$515$	−2.95242e20	−1.35402
$516$	0	0
$517$	−2.13900e20	−0.952865
$518$	0	0
$519$	4.55540e19	0.197138
$520$	0	0
$521$	2.18946e19	0.0920564	0.0460282	−	0.998940i	$-0.485344\pi$
$521$	2.18946e19	0.0920564	0.0460282	+	0.998940i	$0.485344\pi$
$522$	0	0
$523$	1.87375e18	0.00765508	0.00382754	−	0.999993i	$-0.498782\pi$
$523$	1.87375e18	0.00765508	0.00382754	+	0.999993i	$0.498782\pi$
$524$	0	0
$525$	−6.12713e19	−0.243255
$526$	0	0
$527$	−5.39714e20	−2.08250
$528$	0	0
$529$	1.56181e20	0.585749
$530$	0	0
$531$	−1.66754e20	−0.607951
$532$	0	0
$533$	−1.07925e19	−0.0382534
$534$	0	0
$535$	−5.12448e20	−1.76602
$536$	0	0
$537$	6.69588e19	0.224389
$538$	0	0
$539$	2.88731e19	0.0940975
$540$	0	0
$541$	−3.61601e20	−1.14617	−0.573087	−	0.819495i	$-0.694254\pi$
$541$	−3.61601e20	−1.14617	−0.573087	+	0.819495i	$0.694254\pi$
$542$	0	0
$543$	1.63707e20	0.504739
$544$	0	0
$545$	3.97807e19	0.119316
$546$	0	0
$547$	−3.18732e20	−0.930082	−0.465041	−	0.885289i	$-0.653960\pi$
$547$	−3.18732e20	−0.930082	−0.465041	+	0.885289i	$0.653960\pi$
$548$	0	0
$549$	1.01413e20	0.287938
$550$	0	0
$551$	4.25852e20	1.17658
$552$	0	0
$553$	−1.32698e20	−0.356801
$554$	0	0
$555$	2.14648e20	0.561733
$556$	0	0
$557$	−1.20750e20	−0.307590	−0.153795	−	0.988103i	$-0.549149\pi$
$557$	−1.20750e20	−0.307590	−0.153795	+	0.988103i	$0.549149\pi$
$558$	0	0
$559$	1.68339e20	0.417443
$560$	0	0
$561$	−1.16280e20	−0.280726
$562$	0	0
$563$	6.57121e20	1.54466	0.772329	−	0.635223i	$-0.219092\pi$
$563$	6.57121e20	1.54466	0.772329	+	0.635223i	$0.219092\pi$
$564$	0	0
$565$	7.03000e20	1.60913
$566$	0	0
$567$	1.18561e20	0.264282
$568$	0	0
$569$	−4.00408e20	−0.869283	−0.434641	−	0.900604i	$-0.643125\pi$
$569$	−4.00408e20	−0.869283	−0.434641	+	0.900604i	$0.643125\pi$
$570$	0	0
$571$	−3.47283e20	−0.734367	−0.367183	−	0.930149i	$-0.619678\pi$
$571$	−3.47283e20	−0.734367	−0.367183	+	0.930149i	$0.619678\pi$
$572$	0	0
$573$	−2.82224e19	−0.0581346
$574$	0	0
$575$	−1.25322e21	−2.51488
$576$	0	0
$577$	7.89925e20	1.54443	0.772213	−	0.635363i	$-0.219150\pi$
$577$	7.89925e20	1.54443	0.772213	+	0.635363i	$0.219150\pi$
$578$	0	0
$579$	7.10401e19	0.135336
$580$	0	0
$581$	−7.25646e19	−0.134711
$582$	0	0
$583$	5.70920e20	1.03291
$584$	0	0
$585$	2.40108e20	0.423386
$586$	0	0
$587$	5.92951e20	1.01914	0.509569	−	0.860430i	$-0.329805\pi$
$587$	5.92951e20	1.01914	0.509569	+	0.860430i	$0.329805\pi$
$588$	0	0
$589$	6.38298e20	1.06945
$590$	0	0
$591$	−2.89654e20	−0.473123
$592$	0	0
$593$	6.80453e20	1.08365	0.541823	−	0.840492i	$-0.317734\pi$
$593$	6.80453e20	1.08365	0.541823	+	0.840492i	$0.317734\pi$
$594$	0	0
$595$	5.57280e20	0.865359
$596$	0	0
$597$	3.01830e20	0.457040
$598$	0	0
$599$	8.61174e20	1.27171	0.635857	−	0.771807i	$-0.280647\pi$
$599$	8.61174e20	1.27171	0.635857	+	0.771807i	$0.280647\pi$
$600$	0	0
$601$	1.20125e21	1.73011	0.865055	−	0.501677i	$-0.167283\pi$
$601$	1.20125e21	1.73011	0.865055	+	0.501677i	$0.167283\pi$
$602$	0	0
$603$	−9.53828e20	−1.33995
$604$	0	0
$605$	−7.15217e20	−0.980105
$606$	0	0
$607$	−1.15372e21	−1.54235	−0.771177	−	0.636621i	$-0.780331\pi$
$607$	−1.15372e21	−1.54235	−0.771177	+	0.636621i	$0.780331\pi$
$608$	0	0
$609$	1.61787e20	0.211016
$610$	0	0
$611$	3.10221e20	0.394787
$612$	0	0
$613$	4.46037e19	0.0553883	0.0276941	−	0.999616i	$-0.491184\pi$
$613$	4.46037e19	0.0553883	0.0276941	+	0.999616i	$0.491184\pi$
$614$	0	0
$615$	−6.45339e19	−0.0782032
$616$	0	0
$617$	5.14656e20	0.608665	0.304332	−	0.952566i	$-0.401567\pi$
$617$	5.14656e20	0.608665	0.304332	+	0.952566i	$0.401567\pi$
$618$	0	0
$619$	−4.83739e20	−0.558382	−0.279191	−	0.960236i	$-0.590066\pi$
$619$	−4.83739e20	−0.558382	−0.279191	+	0.960236i	$0.590066\pi$
$620$	0	0
$621$	−6.82948e20	−0.769485
$622$	0	0
$623$	−1.92608e20	−0.211843
$624$	0	0
$625$	9.23233e20	0.991314
$626$	0	0
$627$	1.37519e20	0.144164
$628$	0	0
$629$	−1.30089e21	−1.33156
$630$	0	0
$631$	1.32672e21	1.32605	0.663023	−	0.748599i	$-0.269273\pi$
$631$	1.32672e21	1.32605	0.663023	+	0.748599i	$0.269273\pi$
$632$	0	0
$633$	−1.42645e20	−0.139228
$634$	0	0
$635$	−2.85913e21	−2.72541
$636$	0	0
$637$	−4.18750e19	−0.0389861
$638$	0	0
$639$	3.15053e20	0.286502
$640$	0	0
$641$	−1.82165e21	−1.61819	−0.809095	−	0.587678i	$-0.800042\pi$
$641$	−1.82165e21	−1.61819	−0.809095	+	0.587678i	$0.800042\pi$
$642$	0	0
$643$	−1.04749e21	−0.909007	−0.454504	−	0.890745i	$-0.650183\pi$
$643$	−1.04749e21	−0.909007	−0.454504	+	0.890745i	$0.650183\pi$
$644$	0	0
$645$	1.00658e21	0.853399
$646$	0	0
$647$	1.86271e21	1.54299	0.771493	−	0.636238i	$-0.219510\pi$
$647$	1.86271e21	1.54299	0.771493	+	0.636238i	$0.219510\pi$
$648$	0	0
$649$	5.52078e20	0.446854
$650$	0	0
$651$	2.42499e20	0.191802
$652$	0	0
$653$	1.23330e21	0.953276	0.476638	−	0.879100i	$-0.341855\pi$
$653$	1.23330e21	0.953276	0.476638	+	0.879100i	$0.341855\pi$
$654$	0	0
$655$	6.16342e20	0.465599
$656$	0	0
$657$	−1.47244e21	−1.08717
$658$	0	0
$659$	2.12813e21	1.53588	0.767942	−	0.640519i	$-0.221281\pi$
$659$	2.12813e21	1.53588	0.767942	+	0.640519i	$0.221281\pi$
$660$	0	0
$661$	−1.65672e21	−1.16879	−0.584396	−	0.811469i	$-0.698669\pi$
$661$	−1.65672e21	−1.16879	−0.584396	+	0.811469i	$0.698669\pi$
$662$	0	0
$663$	1.68642e20	0.116309
$664$	0	0
$665$	−6.59072e20	−0.444397
$666$	0	0
$667$	3.30913e21	2.18157
$668$	0	0
$669$	1.72278e20	0.111054
$670$	0	0
$671$	−3.35750e20	−0.211639
$672$	0	0
$673$	7.21487e19	0.0444750	0.0222375	−	0.999753i	$-0.492921\pi$
$673$	7.21487e19	0.0444750	0.0222375	+	0.999753i	$0.492921\pi$
$674$	0	0
$675$	2.02424e21	1.22035
$676$	0	0
$677$	2.69291e21	1.58784	0.793919	−	0.608023i	$-0.208037\pi$
$677$	2.69291e21	1.58784	0.793919	+	0.608023i	$0.208037\pi$
$678$	0	0
$679$	−1.01069e21	−0.582903
$680$	0	0
$681$	−4.30427e19	−0.0242826
$682$	0	0
$683$	4.18367e20	0.230889	0.115444	−	0.993314i	$-0.463171\pi$
$683$	4.18367e20	0.230889	0.115444	+	0.993314i	$0.463171\pi$
$684$	0	0
$685$	4.67386e21	2.52346
$686$	0	0
$687$	5.57225e20	0.294344
$688$	0	0
$689$	−8.28011e20	−0.427949
$690$	0	0
$691$	−5.76803e20	−0.291704	−0.145852	−	0.989306i	$-0.546592\pi$
$691$	−5.76803e20	−0.291704	−0.145852	+	0.989306i	$0.546592\pi$
$692$	0	0
$693$	−4.50822e20	−0.223103
$694$	0	0
$695$	3.70494e21	1.79430
$696$	0	0
$697$	3.91113e20	0.185377
$698$	0	0
$699$	−5.91573e20	−0.274429
$700$	0	0
$701$	−1.87893e21	−0.853148	−0.426574	−	0.904453i	$-0.640280\pi$
$701$	−1.87893e21	−0.853148	−0.426574	+	0.904453i	$0.640280\pi$
$702$	0	0
$703$	1.53851e21	0.683812
$704$	0	0
$705$	1.85497e21	0.807081
$706$	0	0
$707$	4.71328e20	0.200760
$708$	0	0
$709$	−1.70478e21	−0.710920	−0.355460	−	0.934691i	$-0.615676\pi$
$709$	−1.70478e21	−0.710920	−0.355460	+	0.934691i	$0.615676\pi$
$710$	0	0
$711$	2.07193e21	0.845967
$712$	0	0
$713$	4.95996e21	1.98293
$714$	0	0
$715$	−7.94931e20	−0.311196
$716$	0	0
$717$	−7.24154e20	−0.277612
$718$	0	0
$719$	−4.57543e21	−1.71777	−0.858885	−	0.512168i	$-0.828842\pi$
$719$	−4.57543e21	−1.71777	−0.858885	+	0.512168i	$0.828842\pi$
$720$	0	0
$721$	−8.03967e20	−0.295613
$722$	0	0
$723$	1.40754e21	0.506901
$724$	0	0
$725$	−9.80817e21	−3.45981
$726$	0	0
$727$	4.62999e21	1.59982	0.799912	−	0.600117i	$-0.204879\pi$
$727$	4.62999e21	1.59982	0.799912	+	0.600117i	$0.204879\pi$
$728$	0	0
$729$	−1.26942e21	−0.429685
$730$	0	0
$731$	−6.10050e21	−2.02295
$732$	0	0
$733$	−9.19943e20	−0.298869	−0.149435	−	0.988772i	$-0.547745\pi$
$733$	−9.19943e20	−0.298869	−0.149435	+	0.988772i	$0.547745\pi$
$734$	0	0
$735$	−2.50391e20	−0.0797011
$736$	0	0
$737$	3.15786e21	0.984889
$738$	0	0
$739$	4.11365e21	1.25717	0.628585	−	0.777740i	$-0.283634\pi$
$739$	4.11365e21	1.25717	0.628585	+	0.777740i	$0.283634\pi$
$740$	0	0
$741$	−1.99445e20	−0.0597293
$742$	0	0
$743$	−1.25170e21	−0.367352	−0.183676	−	0.982987i	$-0.558800\pi$
$743$	−1.25170e21	−0.367352	−0.183676	+	0.982987i	$0.558800\pi$
$744$	0	0
$745$	−1.89093e21	−0.543881
$746$	0	0
$747$	1.13302e21	0.319397
$748$	0	0
$749$	−1.39544e21	−0.385564
$750$	0	0
$751$	−3.42484e21	−0.927558	−0.463779	−	0.885951i	$-0.653507\pi$
$751$	−3.42484e21	−0.927558	−0.463779	+	0.885951i	$0.653507\pi$
$752$	0	0
$753$	1.64284e21	0.436148
$754$	0	0
$755$	−3.03786e21	−0.790617
$756$	0	0
$757$	3.16879e21	0.808489	0.404244	−	0.914651i	$-0.367535\pi$
$757$	3.16879e21	0.808489	0.404244	+	0.914651i	$0.367535\pi$
$758$	0	0
$759$	1.06861e21	0.267303
$760$	0	0
$761$	5.62493e21	1.37953	0.689767	−	0.724032i	$-0.257713\pi$
$761$	5.62493e21	1.37953	0.689767	+	0.724032i	$0.257713\pi$
$762$	0	0
$763$	1.08326e20	0.0260494
$764$	0	0
$765$	−8.70132e21	−2.05175
$766$	0	0
$767$	−8.00685e20	−0.185138
$768$	0	0
$769$	4.66019e20	0.105671	0.0528355	−	0.998603i	$-0.483174\pi$
$769$	4.66019e20	0.105671	0.0528355	+	0.998603i	$0.483174\pi$
$770$	0	0
$771$	−1.97702e21	−0.439645
$772$	0	0
$773$	−5.17003e21	−1.12758	−0.563789	−	0.825919i	$-0.690657\pi$
$773$	−5.17003e21	−1.12758	−0.563789	+	0.825919i	$0.690657\pi$
$774$	0	0
$775$	−1.47012e22	−3.14478
$776$	0	0
$777$	5.84504e20	0.122639
$778$	0	0
$779$	−4.62554e20	−0.0951987
$780$	0	0
$781$	−1.04306e21	−0.210584
$782$	0	0
$783$	−5.34502e21	−1.05861
$784$	0	0
$785$	9.26235e21	1.79969
$786$	0	0
$787$	−3.11711e21	−0.594213	−0.297107	−	0.954844i	$-0.596022\pi$
$787$	−3.11711e21	−0.594213	−0.297107	+	0.954844i	$0.596022\pi$
$788$	0	0
$789$	1.63589e21	0.305969
$790$	0	0
$791$	1.91433e21	0.351311
$792$	0	0
$793$	4.86942e20	0.0876855
$794$	0	0
$795$	−4.95109e21	−0.874877
$796$	0	0
$797$	−6.71528e21	−1.16447	−0.582233	−	0.813022i	$-0.697821\pi$
$797$	−6.71528e21	−1.16447	−0.582233	+	0.813022i	$0.697821\pi$
$798$	0	0
$799$	−1.12422e22	−1.91315
$800$	0	0
$801$	3.00736e21	0.502275
$802$	0	0
$803$	4.87485e21	0.799088
$804$	0	0
$805$	−5.12139e21	−0.823984
$806$	0	0
$807$	3.70248e21	0.584711
$808$	0	0
$809$	1.74357e20	0.0270287	0.0135143	−	0.999909i	$-0.495698\pi$
$809$	1.74357e20	0.0270287	0.0135143	+	0.999909i	$0.495698\pi$
$810$	0	0
$811$	−1.93632e21	−0.294660	−0.147330	−	0.989087i	$-0.547068\pi$
$811$	−1.93632e21	−0.294660	−0.147330	+	0.989087i	$0.547068\pi$
$812$	0	0
$813$	1.22558e21	0.183089
$814$	0	0
$815$	1.10091e22	1.61462
$816$	0	0
$817$	7.21480e21	1.03887
$818$	0	0
$819$	6.53832e20	0.0924351
$820$	0	0
$821$	−7.16469e21	−0.994543	−0.497272	−	0.867595i	$-0.665665\pi$
$821$	−7.16469e21	−0.994543	−0.497272	+	0.867595i	$0.665665\pi$
$822$	0	0
$823$	−1.29282e21	−0.176214	−0.0881071	−	0.996111i	$-0.528082\pi$
$823$	−1.29282e21	−0.176214	−0.0881071	+	0.996111i	$0.528082\pi$
$824$	0	0
$825$	−3.16732e21	−0.423924
$826$	0	0
$827$	−1.17850e22	−1.54896	−0.774478	−	0.632600i	$-0.781988\pi$
$827$	−1.17850e22	−1.54896	−0.774478	+	0.632600i	$0.781988\pi$
$828$	0	0
$829$	7.96306e21	1.02783	0.513915	−	0.857841i	$-0.328195\pi$
$829$	7.96306e21	1.02783	0.513915	+	0.857841i	$0.328195\pi$
$830$	0	0
$831$	2.93772e21	0.372393
$832$	0	0
$833$	1.51752e21	0.188928
$834$	0	0
$835$	5.26251e21	0.643494
$836$	0	0
$837$	−8.01150e21	−0.962217
$838$	0	0
$839$	−1.20963e21	−0.142705	−0.0713523	−	0.997451i	$-0.522731\pi$
$839$	−1.20963e21	−0.142705	−0.0713523	+	0.997451i	$0.522731\pi$
$840$	0	0
$841$	1.72694e22	2.00127
$842$	0	0
$843$	−1.46555e21	−0.166837
$844$	0	0
$845$	−1.43273e22	−1.60228
$846$	0	0
$847$	−1.94759e21	−0.213980
$848$	0	0
$849$	2.31226e21	0.249591
$850$	0	0
$851$	1.19552e22	1.26790
$852$	0	0
$853$	−1.02169e22	−1.06463	−0.532317	−	0.846545i	$-0.678678\pi$
$853$	−1.02169e22	−1.06463	−0.532317	+	0.846545i	$0.678678\pi$
$854$	0	0
$855$	1.02907e22	1.05366
$856$	0	0
$857$	9.00299e21	0.905797	0.452898	−	0.891562i	$-0.350390\pi$
$857$	9.00299e21	0.905797	0.452898	+	0.891562i	$0.350390\pi$
$858$	0	0
$859$	−7.59468e21	−0.750863	−0.375431	−	0.926850i	$-0.622505\pi$
$859$	−7.59468e21	−0.750863	−0.375431	+	0.926850i	$0.622505\pi$
$860$	0	0
$861$	−1.75731e20	−0.0170736
$862$	0	0
$863$	−2.46635e21	−0.235491	−0.117745	−	0.993044i	$-0.537567\pi$
$863$	−2.46635e21	−0.235491	−0.117745	+	0.993044i	$0.537567\pi$
$864$	0	0
$865$	−1.12858e22	−1.05903
$866$	0	0
$867$	−2.61719e21	−0.241375
$868$	0	0
$869$	−6.85961e21	−0.621800
$870$	0	0
$871$	−4.57989e21	−0.408055
$872$	0	0
$873$	1.57809e22	1.38205
$874$	0	0
$875$	7.57882e21	0.652441
$876$	0	0
$877$	−8.87790e21	−0.751300	−0.375650	−	0.926762i	$-0.622580\pi$
$877$	−8.87790e21	−0.751300	−0.375650	+	0.926762i	$0.622580\pi$
$878$	0	0
$879$	5.05949e21	0.420911
$880$	0	0
$881$	5.81040e21	0.475211	0.237605	−	0.971362i	$-0.423637\pi$
$881$	5.81040e21	0.475211	0.237605	+	0.971362i	$0.423637\pi$
$882$	0	0
$883$	−2.06901e22	−1.66363	−0.831817	−	0.555049i	$-0.812699\pi$
$883$	−2.06901e22	−1.66363	−0.831817	+	0.555049i	$0.812699\pi$
$884$	0	0
$885$	−4.78769e21	−0.378488
$886$	0	0
$887$	−1.30662e22	−1.01560	−0.507798	−	0.861476i	$-0.669540\pi$
$887$	−1.30662e22	−1.01560	−0.507798	+	0.861476i	$0.669540\pi$
$888$	0	0
$889$	−7.78565e21	−0.595020
$890$	0	0
$891$	6.12879e21	0.460566
$892$	0	0
$893$	1.32957e22	0.982481
$894$	0	0
$895$	−1.65887e22	−1.20542
$896$	0	0
$897$	−1.54981e21	−0.110748
$898$	0	0
$899$	3.88186e22	2.72799
$900$	0	0
$901$	3.00065e22	2.07386
$902$	0	0
$903$	2.74101e21	0.186317
$904$	0	0
$905$	−4.05575e22	−2.71148
$906$	0	0
$907$	−1.17651e22	−0.773643	−0.386821	−	0.922155i	$-0.626427\pi$
$907$	−1.17651e22	−0.773643	−0.386821	+	0.922155i	$0.626427\pi$
$908$	0	0
$909$	−7.35927e21	−0.475999
$910$	0	0
$911$	2.01178e22	1.27995	0.639976	−	0.768395i	$-0.278944\pi$
$911$	2.01178e22	1.27995	0.639976	+	0.768395i	$0.278944\pi$
$912$	0	0
$913$	−3.75111e21	−0.234762
$914$	0	0
$915$	2.91166e21	0.179260
$916$	0	0
$917$	1.67835e21	0.101651
$918$	0	0
$919$	−2.15948e22	−1.28672	−0.643358	−	0.765566i	$-0.722459\pi$
$919$	−2.15948e22	−1.28672	−0.643358	+	0.765566i	$0.722459\pi$
$920$	0	0
$921$	−2.02021e21	−0.118426
$922$	0	0
$923$	1.51276e21	0.0872481
$924$	0	0
$925$	−3.54349e22	−2.01079
$926$	0	0
$927$	1.25531e22	0.700893
$928$	0	0
$929$	−2.32545e22	−1.27758	−0.638792	−	0.769379i	$-0.720566\pi$
$929$	−2.32545e22	−1.27758	−0.638792	+	0.769379i	$0.720566\pi$
$930$	0	0
$931$	−1.79471e21	−0.0970222
$932$	0	0
$933$	−1.52293e21	−0.0810156
$934$	0	0
$935$	2.88077e22	1.50807
$936$	0	0
$937$	1.62658e22	0.837972	0.418986	−	0.907993i	$-0.362386\pi$
$937$	1.62658e22	0.837972	0.418986	+	0.907993i	$0.362386\pi$
$938$	0	0
$939$	6.43007e21	0.326005
$940$	0	0
$941$	6.87043e21	0.342816	0.171408	−	0.985200i	$-0.445168\pi$
$941$	6.87043e21	0.342816	0.171408	+	0.985200i	$0.445168\pi$
$942$	0	0
$943$	−3.59432e21	−0.176514
$944$	0	0
$945$	8.27225e21	0.399839
$946$	0	0
$947$	2.40772e21	0.114547	0.0572733	−	0.998359i	$-0.481759\pi$
$947$	2.40772e21	0.114547	0.0572733	+	0.998359i	$0.481759\pi$
$948$	0	0
$949$	−7.07005e21	−0.331075
$950$	0	0
$951$	9.59157e20	0.0442116
$952$	0	0
$953$	−9.71778e21	−0.440931	−0.220465	−	0.975395i	$-0.570758\pi$
$953$	−9.71778e21	−0.440931	−0.220465	+	0.975395i	$0.570758\pi$
$954$	0	0
$955$	6.99196e21	0.312301
$956$	0	0
$957$	8.36334e21	0.367739
$958$	0	0
$959$	1.27273e22	0.550931
$960$	0	0
$961$	3.47189e22	1.47959
$962$	0	0
$963$	2.17882e22	0.914165
$964$	0	0
$965$	−1.75998e22	−0.727032
$966$	0	0
$967$	4.35136e22	1.76981	0.884904	−	0.465774i	$-0.154224\pi$
$967$	4.35136e22	1.76981	0.884904	+	0.465774i	$0.154224\pi$
$968$	0	0
$969$	7.22775e21	0.289451
$970$	0	0
$971$	8.57014e21	0.337943	0.168972	−	0.985621i	$-0.445955\pi$
$971$	8.57014e21	0.337943	0.168972	+	0.985621i	$0.445955\pi$
$972$	0	0
$973$	1.00888e22	0.391737
$974$	0	0
$975$	4.59360e21	0.175638
$976$	0	0
$977$	3.43574e22	1.29363	0.646817	−	0.762645i	$-0.276100\pi$
$977$	3.43574e22	1.29363	0.646817	+	0.762645i	$0.276100\pi$
$978$	0	0
$979$	−9.95656e21	−0.369181
$980$	0	0
$981$	−1.69139e21	−0.0617627
$982$	0	0
$983$	2.47556e22	0.890272	0.445136	−	0.895463i	$-0.353155\pi$
$983$	2.47556e22	0.890272	0.445136	+	0.895463i	$0.353155\pi$
$984$	0	0
$985$	7.17604e22	2.54164
$986$	0	0
$987$	5.05122e21	0.176205
$988$	0	0
$989$	5.60634e22	1.92623
$990$	0	0
$991$	−2.14427e22	−0.725650	−0.362825	−	0.931857i	$-0.618188\pi$
$991$	−2.14427e22	−0.725650	−0.362825	+	0.931857i	$0.618188\pi$
$992$	0	0
$993$	−1.44821e22	−0.482739
$994$	0	0
$995$	−7.47768e22	−2.45524
$996$	0	0
$997$	1.22404e22	0.395897	0.197948	−	0.980212i	$-0.436572\pi$
$997$	1.22404e22	0.395897	0.197948	+	0.980212i	$0.436572\pi$
$998$	0	0
$999$	−1.93104e22	−0.615249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.16.a.k.1.3		6
4.3	odd	2		56.16.a.d.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.16.a.d.1.4	✓	6	4.3	odd	2
112.16.a.k.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-1220.73 q^{3} +302431. q^{5} +823543. q^{7} -1.28587e7 q^{9} +4.25717e7 q^{11} -6.17422e7 q^{13} -3.69187e8 q^{15} +2.23749e9 q^{17} -2.64619e9 q^{19} -1.00533e9 q^{21} -2.05625e10 q^{23} +6.09467e10 q^{25} +3.32133e10 q^{27} -1.60930e11 q^{29} -2.41214e11 q^{31} -5.19687e10 q^{33} +2.49065e11 q^{35} -5.81407e11 q^{37} +7.53708e10 q^{39} +1.74800e11 q^{41} -2.72649e12 q^{43} -3.88887e12 q^{45} -5.02446e12 q^{47} +6.78223e11 q^{49} -2.73138e12 q^{51} +1.34108e13 q^{53} +1.28750e13 q^{55} +3.23029e12 q^{57} +1.29682e13 q^{59} -7.88668e12 q^{61} -1.05897e13 q^{63} -1.86727e13 q^{65} +7.41775e13 q^{67} +2.51013e13 q^{69} -2.45011e13 q^{71} +1.14509e14 q^{73} -7.43997e13 q^{75} +3.50596e13 q^{77} -1.61131e14 q^{79} +1.43964e14 q^{81} -8.81127e13 q^{83} +6.76686e14 q^{85} +1.96453e14 q^{87} -2.33877e14 q^{89} -5.08474e13 q^{91} +2.94458e14 q^{93} -8.00289e14 q^{95} -1.22725e15 q^{97} -5.47418e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3864 q^{3} + 168532 q^{5} + 4941258 q^{7} + 27735822 q^{9} - 58040072 q^{11} - 15956444 q^{13} + 705059136 q^{15} - 2879979060 q^{17} - 3435500376 q^{19} - 3182170152 q^{21} - 34508782592 q^{23} + 29232294906 q^{25}+ \cdots - 44\!\cdots\!92 q^{99}+O(q^{100}) \) 6 * q - 3864 * q^3 + 168532 * q^5 + 4941258 * q^7 + 27735822 * q^9 - 58040072 * q^11 - 15956444 * q^13 + 705059136 * q^15 - 2879979060 * q^17 - 3435500376 * q^19 - 3182170152 * q^21 - 34508782592 * q^23 + 29232294906 * q^25 - 102516868944 * q^27 - 46458543420 * q^29 - 300765178336 * q^31 - 457356427584 * q^33 + 138793348876 * q^35 - 509722422412 * q^37 + 437968495008 * q^39 + 1619276439676 * q^41 - 2727781008680 * q^43 + 5316894690660 * q^45 + 1567726443808 * q^47 + 4069338437094 * q^49 - 3110310296208 * q^51 + 2612518893588 * q^53 + 4577322926960 * q^55 - 21550151959824 * q^57 - 5112561397384 * q^59 + 4545346033732 * q^61 + 22841642057346 * q^63 + 57444596600712 * q^65 - 74088461612408 * q^67 + 176631933295104 * q^69 - 51092191255952 * q^71 + 294343628983292 * q^73 - 208088876183496 * q^75 - 47798495015096 * q^77 - 56070305620928 * q^79 + 691991522436870 * q^81 - 984559173278520 * q^83 + 396331720481544 * q^85 - 1024554713548464 * q^87 + 1528383063352572 * q^89 - 13140817761092 * q^91 + 1245617896072608 * q^93 - 2532178187158304 * q^95 + 2255703641701580 * q^97 - 4484026592356392 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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