LMFDB - Embedded newform 288.8.d.d.145.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(145,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("288.145");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$89.9668873394$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 $ x^14 - 6x^13 - 52x^12 + 300x^11 - 1005x^10 - 23250x^9 + 349930x^8 + 2867784x^7 - 20463993x^6 - 78987210x^5 + 94608296x^4 - 6477861300x^3 - 21982316987x^2 + 613270661346*x + 3813237677250
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			145.3
Root			$8.85262 - 1.52851i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.8.d.d.145.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-425.308i q^{5} -1664.03 q^{7} +O(q^{10})$	$q-425.308i q^{5} -1664.03 q^{7} +2467.01i q^{11} -3767.59i q^{13} -16241.0 q^{17} -4869.30i q^{19} -108549. q^{23} -102762. q^{25} +89066.4i q^{29} +69483.3 q^{31} +707727. i q^{35} -418478. i q^{37} -274307. q^{41} +462897. i q^{43} +153501. q^{47} +1.94546e6 q^{49} -181473. i q^{53} +1.04924e6 q^{55} +647716. i q^{59} -2.26966e6i q^{61} -1.60239e6 q^{65} -2.31785e6i q^{67} +2.77599e6 q^{71} -4.45276e6 q^{73} -4.10518e6i q^{77} +3.14911e6 q^{79} +3.33510e6i q^{83} +6.90743e6i q^{85} +5.00420e6 q^{89} +6.26939e6i q^{91} -2.07096e6 q^{95} +8.06376e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 1372 q^{7}+O(q^{10}) $ 14 * q - 1372 * q^7	$ 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) $ 14 * q - 1372 * q^7 + 2908 * q^17 - 143416 * q^23 - 202626 * q^25 + 89468 * q^31 + 441284 * q^41 - 1056408 * q^47 + 2158134 * q^49 - 4757504 * q^55 + 2520464 * q^65 + 5172696 * q^71 - 5446196 * q^73 + 14373548 * q^79 + 11952620 * q^89 - 69327376 * q^95 + 133732 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 425.308i	− 1.52163i	−0.648969	−	0.760815i	$-0.724800\pi$
$5$	− 425.308i	− 1.52163i	0.648969	−	0.760815i	$-0.275200\pi$
$6$	0	0
$7$	−1664.03	−1.83366	−0.916829	−	0.399279i	$-0.869260\pi$
$7$	−1664.03	−1.83366	−0.916829	+	0.399279i	$0.869260\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2467.01i	0.558852i	0.960167	+	0.279426i	$0.0901441\pi$
$11$	2467.01i	0.558852i	−0.960167	+	0.279426i	$0.909856\pi$
$12$	0	0
$13$	− 3767.59i	− 0.475622i	−0.971311	−	0.237811i	$-0.923570\pi$
$13$	− 3767.59i	− 0.475622i	0.971311	−	0.237811i	$-0.0764298\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−16241.0	−0.801754	−0.400877	−	0.916132i	$-0.631295\pi$
$17$	−16241.0	−0.801754	−0.400877	+	0.916132i	$0.631295\pi$
$18$	0	0
$19$	− 4869.30i	− 0.162866i	−0.996679	−	0.0814328i	$-0.974050\pi$
$19$	− 4869.30i	− 0.162866i	0.996679	−	0.0814328i	$-0.0259496\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−108549.	−1.86027	−0.930137	−	0.367212i	$-0.880313\pi$
$23$	−108549.	−1.86027	−0.930137	+	0.367212i	$0.880313\pi$
$24$	0	0
$25$	−102762.	−1.31536
$26$	0	0
$27$	0	0
$28$	0	0
$29$	89066.4i	0.678142i	0.940761	+	0.339071i	$0.110113\pi$
$29$	89066.4i	0.678142i	−0.940761	+	0.339071i	$0.889887\pi$
$30$	0	0
$31$	69483.3	0.418904	0.209452	−	0.977819i	$-0.432832\pi$
$31$	69483.3	0.418904	0.209452	+	0.977819i	$0.432832\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	707727.i	2.79015i
$36$	0	0
$37$	− 418478.i	− 1.35821i	−0.734042	−	0.679104i	$-0.762369\pi$
$37$	− 418478.i	− 1.35821i	0.734042	−	0.679104i	$-0.237631\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−274307.	−0.621574	−0.310787	−	0.950480i	$-0.600593\pi$
$41$	−274307.	−0.621574	−0.310787	+	0.950480i	$0.600593\pi$
$42$	0	0
$43$	462897.i	0.887861i	0.896061	+	0.443930i	$0.146416\pi$
$43$	462897.i	0.887861i	−0.896061	+	0.443930i	$0.853584\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	153501.	0.215660	0.107830	−	0.994169i	$-0.465610\pi$
$47$	153501.	0.215660	0.107830	+	0.994169i	$0.465610\pi$
$48$	0	0
$49$	1.94546e6	2.36230
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 181473.i	− 0.167435i	−0.996490	−	0.0837176i	$-0.973321\pi$
$53$	− 181473.i	− 0.167435i	0.996490	−	0.0837176i	$-0.0266794\pi$
$54$	0	0
$55$	1.04924e6	0.850365
$56$	0	0
$57$	0	0
$58$	0	0
$59$	647716.i	0.410584i	0.978701	+	0.205292i	$0.0658145\pi$
$59$	647716.i	0.410584i	−0.978701	+	0.205292i	$0.934186\pi$
$60$	0	0
$61$	− 2.26966e6i	− 1.28028i	−0.768257	−	0.640141i	$-0.778876\pi$
$61$	− 2.26966e6i	− 1.28028i	0.768257	−	0.640141i	$-0.221124\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.60239e6	−0.723720
$66$	0	0
$67$	− 2.31785e6i	− 0.941506i	−0.882265	−	0.470753i	$-0.843982\pi$
$67$	− 2.31785e6i	− 0.941506i	0.882265	−	0.470753i	$-0.156018\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.77599e6	0.920480	0.460240	−	0.887795i	$-0.347763\pi$
$71$	2.77599e6	0.920480	0.460240	+	0.887795i	$0.347763\pi$
$72$	0	0
$73$	−4.45276e6	−1.33967	−0.669837	−	0.742508i	$-0.733636\pi$
$73$	−4.45276e6	−1.33967	−0.669837	+	0.742508i	$0.733636\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.10518e6i	− 1.02474i
$78$	0	0
$79$	3.14911e6	0.718610	0.359305	−	0.933220i	$-0.383014\pi$
$79$	3.14911e6	0.718610	0.359305	+	0.933220i	$0.383014\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.33510e6i	0.640230i	0.947379	+	0.320115i	$0.103721\pi$
$83$	3.33510e6i	0.640230i	−0.947379	+	0.320115i	$0.896279\pi$
$84$	0	0
$85$	6.90743e6i	1.21997i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.00420e6	0.752435	0.376218	−	0.926531i	$-0.377224\pi$
$89$	5.00420e6	0.752435	0.376218	+	0.926531i	$0.377224\pi$
$90$	0	0
$91$	6.26939e6i	0.872128i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.07096e6	−0.247821
$96$	0	0
$97$	8.06376e6	0.897091	0.448545	−	0.893760i	$-0.351942\pi$
$97$	8.06376e6	0.897091	0.448545	+	0.893760i	$0.351942\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.31452e6i	− 0.126953i	−0.997983	−	0.0634764i	$-0.979781\pi$
$101$	− 1.31452e6i	− 0.126953i	0.997983	−	0.0634764i	$-0.0202187\pi$
$102$	0	0
$103$	3.14528e6	0.283615	0.141807	−	0.989894i	$-0.454709\pi$
$103$	3.14528e6	0.283615	0.141807	+	0.989894i	$0.454709\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.01610e7i	0.801850i	0.916111	+	0.400925i	$0.131311\pi$
$107$	1.01610e7i	0.801850i	−0.916111	+	0.400925i	$0.868689\pi$
$108$	0	0
$109$	1.48744e7i	1.10014i	0.835119	+	0.550070i	$0.185399\pi$
$109$	1.48744e7i	1.10014i	−0.835119	+	0.550070i	$0.814601\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.17317e7	0.764866	0.382433	−	0.923983i	$-0.375086\pi$
$113$	1.17317e7	0.764866	0.382433	+	0.923983i	$0.375086\pi$
$114$	0	0
$115$	4.61667e7i	2.83065i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.70255e7	1.47014
$120$	0	0
$121$	1.34010e7	0.687685
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.04784e7i	0.479856i
$126$	0	0
$127$	1.58694e7	0.687460	0.343730	−	0.939068i	$-0.388309\pi$
$127$	1.58694e7	0.687460	0.343730	+	0.939068i	$0.388309\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.35876e7i	− 0.916717i	−0.888768	−	0.458358i	$-0.848438\pi$
$131$	− 2.35876e7i	− 0.916717i	0.888768	−	0.458358i	$-0.151562\pi$
$132$	0	0
$133$	8.10268e6i	0.298640i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.16838e7	−1.05272	−0.526362	−	0.850260i	$-0.676444\pi$
$137$	−3.16838e7	−1.05272	−0.526362	+	0.850260i	$0.676444\pi$
$138$	0	0
$139$	− 3.89914e7i	− 1.23145i	−0.787961	−	0.615726i	$-0.788863\pi$
$139$	− 3.89914e7i	− 1.23145i	0.787961	−	0.615726i	$-0.211137\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.29468e6	0.265802
$144$	0	0
$145$	3.78807e7	1.03188
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.87705e6i	0.170314i	0.996368	+	0.0851570i	$0.0271392\pi$
$149$	6.87705e6i	0.170314i	−0.996368	+	0.0851570i	$0.972861\pi$
$150$	0	0
$151$	−5.09509e6	−0.120429	−0.0602147	−	0.998185i	$-0.519179\pi$
$151$	−5.09509e6	−0.120429	−0.0602147	+	0.998185i	$0.519179\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.95518e7i	− 0.637417i
$156$	0	0
$157$	− 3.94930e7i	− 0.814463i	−0.913325	−	0.407232i	$-0.866494\pi$
$157$	− 3.94930e7i	− 0.814463i	0.913325	−	0.407232i	$-0.133506\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.80628e8	3.41111
$162$	0	0
$163$	− 4.90817e6i	− 0.0887693i	−0.999015	−	0.0443847i	$-0.985867\pi$
$163$	− 4.90817e6i	− 0.0887693i	0.999015	−	0.0443847i	$-0.0141327\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.43526e7	−1.06920	−0.534600	−	0.845105i	$-0.679538\pi$
$167$	−6.43526e7	−1.06920	−0.534600	+	0.845105i	$0.679538\pi$
$168$	0	0
$169$	4.85538e7	0.773784
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.25082e7i	1.35837i	0.733966	+	0.679186i	$0.237667\pi$
$173$	9.25082e7i	1.35837i	−0.733966	+	0.679186i	$0.762333\pi$
$174$	0	0
$175$	1.71000e8	2.41192
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.35773e8i	1.76941i	0.466150	+	0.884706i	$0.345641\pi$
$179$	1.35773e8i	1.76941i	−0.466150	+	0.884706i	$0.654359\pi$
$180$	0	0
$181$	1.52234e8i	1.90826i	0.299400	+	0.954128i	$0.403213\pi$
$181$	1.52234e8i	1.90826i	−0.299400	+	0.954128i	$0.596787\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.77982e8	−2.06669
$186$	0	0
$187$	− 4.00667e7i	− 0.448061i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.14883e7	−0.119299	−0.0596497	−	0.998219i	$-0.518998\pi$
$191$	−1.14883e7	−0.119299	−0.0596497	+	0.998219i	$0.518998\pi$
$192$	0	0
$193$	1.30726e8	1.30892	0.654458	−	0.756098i	$-0.272897\pi$
$193$	1.30726e8	1.30892	0.654458	+	0.756098i	$0.272897\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.47421e8i	1.37381i	0.726745	+	0.686907i	$0.241032\pi$
$197$	1.47421e8i	1.37381i	−0.726745	+	0.686907i	$0.758968\pi$
$198$	0	0
$199$	7.23843e7	0.651116	0.325558	−	0.945522i	$-0.394448\pi$
$199$	7.23843e7	0.651116	0.325558	+	0.945522i	$0.394448\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.48209e8i	− 1.24348i
$204$	0	0
$205$	1.16665e8i	0.945805i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.20126e7	0.0910177
$210$	0	0
$211$	1.13987e8i	0.835349i	0.908597	+	0.417675i	$0.137155\pi$
$211$	1.13987e8i	0.835349i	−0.908597	+	0.417675i	$0.862845\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.96874e8	1.35099
$216$	0	0
$217$	−1.15622e8	−0.768127
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.11893e7i	0.381332i
$222$	0	0
$223$	−3.25338e8	−1.96457	−0.982285	−	0.187395i	$-0.939995\pi$
$223$	−3.25338e8	−1.96457	−0.982285	+	0.187395i	$0.939995\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.77241e8i	1.00571i	0.864370	+	0.502857i	$0.167718\pi$
$227$	1.77241e8i	1.00571i	−0.864370	+	0.502857i	$0.832282\pi$
$228$	0	0
$229$	− 2.59923e8i	− 1.43028i	−0.698983	−	0.715139i	$-0.746364\pi$
$229$	− 2.59923e8i	− 1.43028i	0.698983	−	0.715139i	$-0.253636\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.27541e7	0.221428	0.110714	−	0.993852i	$-0.464686\pi$
$233$	4.27541e7	0.221428	0.110714	+	0.993852i	$0.464686\pi$
$234$	0	0
$235$	− 6.52854e7i	− 0.328154i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.31118e6	−0.0204269	−0.0102135	−	0.999948i	$-0.503251\pi$
$239$	−4.31118e6	−0.0204269	−0.0102135	+	0.999948i	$0.503251\pi$
$240$	0	0
$241$	−1.51908e8	−0.699069	−0.349534	−	0.936924i	$-0.613660\pi$
$241$	−1.51908e8	−0.699069	−0.349534	+	0.936924i	$0.613660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 8.27420e8i	− 3.59455i
$246$	0	0
$247$	−1.83455e7	−0.0774624
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.71376e7i	− 0.188152i	−0.995565	−	0.0940761i	$-0.970010\pi$
$251$	− 4.71376e7i	− 0.188152i	0.995565	−	0.0940761i	$-0.0299897\pi$
$252$	0	0
$253$	− 2.67791e8i	− 1.03962i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.54383e7	−0.130229	−0.0651143	−	0.997878i	$-0.520741\pi$
$257$	−3.54383e7	−0.130229	−0.0651143	+	0.997878i	$0.520741\pi$
$258$	0	0
$259$	6.96360e8i	2.49049i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.32493e8	−0.788069	−0.394035	−	0.919096i	$-0.628921\pi$
$263$	−2.32493e8	−0.788069	−0.394035	+	0.919096i	$0.628921\pi$
$264$	0	0
$265$	−7.71820e7	−0.254774
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.41676e8i	− 0.757008i	−0.925600	−	0.378504i	$-0.876439\pi$
$269$	− 2.41676e8i	− 0.757008i	0.925600	−	0.378504i	$-0.123561\pi$
$270$	0	0
$271$	−3.99206e7	−0.121844	−0.0609220	−	0.998143i	$-0.519404\pi$
$271$	−3.99206e7	−0.121844	−0.0609220	+	0.998143i	$0.519404\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.53515e8i	− 0.735089i
$276$	0	0
$277$	4.59198e8i	1.29814i	0.760730	+	0.649069i	$0.224841\pi$
$277$	4.59198e8i	1.29814i	−0.760730	+	0.649069i	$0.775159\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.66896e8	1.25530	0.627651	−	0.778495i	$-0.284017\pi$
$281$	4.66896e8	1.25530	0.627651	+	0.778495i	$0.284017\pi$
$282$	0	0
$283$	− 7.16866e8i	− 1.88012i	−0.341011	−	0.940059i	$-0.610769\pi$
$283$	− 7.16866e8i	− 1.88012i	0.341011	−	0.940059i	$-0.389231\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.56455e8	1.13975
$288$	0	0
$289$	−1.46569e8	−0.357190
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.97911e7i	0.138867i	0.997587	+	0.0694336i	$0.0221192\pi$
$293$	5.97911e7i	0.138867i	−0.997587	+	0.0694336i	$0.977881\pi$
$294$	0	0
$295$	2.75479e8	0.624757
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.08967e8i	0.884787i
$300$	0	0
$301$	− 7.70275e8i	− 1.62803i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.65304e8	−1.94812
$306$	0	0
$307$	− 1.89764e8i	− 0.374308i	−0.982331	−	0.187154i	$-0.940074\pi$
$307$	− 1.89764e8i	− 0.374308i	0.982331	−	0.187154i	$-0.0599263\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.09202e8	−0.394371	−0.197185	−	0.980366i	$-0.563180\pi$
$311$	−2.09202e8	−0.394371	−0.197185	+	0.980366i	$0.563180\pi$
$312$	0	0
$313$	−6.33548e8	−1.16782	−0.583908	−	0.811820i	$-0.698477\pi$
$313$	−6.33548e8	−1.16782	−0.583908	+	0.811820i	$0.698477\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.69116e8i	1.35608i	0.735027	+	0.678038i	$0.237170\pi$
$317$	7.69116e8i	1.35608i	−0.735027	+	0.678038i	$0.762830\pi$
$318$	0	0
$319$	−2.19728e8	−0.378981
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.90823e7i	0.130578i
$324$	0	0
$325$	3.87166e8i	0.625612i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.55431e8	−0.395447
$330$	0	0
$331$	− 4.85538e8i	− 0.735911i	−0.929843	−	0.367955i	$-0.880058\pi$
$331$	− 4.85538e8i	− 0.735911i	0.929843	−	0.367955i	$-0.119942\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.85800e8	−1.43262
$336$	0	0
$337$	−2.04774e7	−0.0291454	−0.0145727	−	0.999894i	$-0.504639\pi$
$337$	−2.04774e7	−0.0291454	−0.0145727	+	0.999894i	$0.504639\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.71416e8i	0.234105i
$342$	0	0
$343$	−1.86690e9	−2.49800
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.82874e8i	− 0.877379i	−0.898639	−	0.438690i	$-0.855443\pi$
$347$	− 6.82874e8i	− 0.877379i	0.898639	−	0.438690i	$-0.144557\pi$
$348$	0	0
$349$	− 9.14155e8i	− 1.15115i	−0.817750	−	0.575573i	$-0.804779\pi$
$349$	− 9.14155e8i	− 1.15115i	0.817750	−	0.575573i	$-0.195221\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.60893e8	−1.04169	−0.520844	−	0.853652i	$-0.674383\pi$
$353$	−8.60893e8	−1.04169	−0.520844	+	0.853652i	$0.674383\pi$
$354$	0	0
$355$	− 1.18065e9i	− 1.40063i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.38426e9	−1.57902	−0.789512	−	0.613736i	$-0.789666\pi$
$359$	−1.38426e9	−1.57902	−0.789512	+	0.613736i	$0.789666\pi$
$360$	0	0
$361$	8.70162e8	0.973475
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.89380e9i	2.03849i
$366$	0	0
$367$	−3.12309e8	−0.329802	−0.164901	−	0.986310i	$-0.552731\pi$
$367$	−3.12309e8	−0.329802	−0.164901	+	0.986310i	$0.552731\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.01977e8i	0.307019i
$372$	0	0
$373$	1.03487e8i	0.103253i	0.998666	+	0.0516266i	$0.0164406\pi$
$373$	1.03487e8i	0.103253i	−0.998666	+	0.0516266i	$0.983559\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.35565e8	0.322539
$378$	0	0
$379$	− 1.96626e8i	− 0.185525i	−0.995688	−	0.0927626i	$-0.970430\pi$
$379$	− 1.96626e8i	− 0.185525i	0.995688	−	0.0927626i	$-0.0295697\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.32285e9	1.20313	0.601567	−	0.798822i	$-0.294543\pi$
$383$	1.32285e9	1.20313	0.601567	+	0.798822i	$0.294543\pi$
$384$	0	0
$385$	−1.74597e9	−1.55928
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.23494e9i	− 1.92505i	−0.271189	−	0.962526i	$-0.587417\pi$
$389$	− 2.23494e9i	− 1.92505i	0.271189	−	0.962526i	$-0.412583\pi$
$390$	0	0
$391$	1.76294e9	1.49148
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.33934e9i	− 1.09346i
$396$	0	0
$397$	− 1.06152e8i	− 0.0851456i	−0.999093	−	0.0425728i	$-0.986445\pi$
$397$	− 1.06152e8i	− 0.0851456i	0.999093	−	0.0425728i	$-0.0135554\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.69252e9	1.31078	0.655388	−	0.755292i	$-0.272505\pi$
$401$	1.69252e9	1.31078	0.655388	+	0.755292i	$0.272505\pi$
$402$	0	0
$403$	− 2.61784e8i	− 0.199240i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.03239e9	0.759037
$408$	0	0
$409$	8.80345e8	0.636240	0.318120	−	0.948050i	$-0.396948\pi$
$409$	8.80345e8	0.636240	0.318120	+	0.948050i	$0.396948\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.07782e9i	− 0.752872i
$414$	0	0
$415$	1.41845e9	0.974192
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.24808e9i	− 0.828886i	−0.910075	−	0.414443i	$-0.863976\pi$
$419$	− 1.24808e9i	− 0.828886i	0.910075	−	0.414443i	$-0.136024\pi$
$420$	0	0
$421$	1.98536e9i	1.29674i	0.761326	+	0.648370i	$0.224549\pi$
$421$	1.98536e9i	1.29674i	−0.761326	+	0.648370i	$0.775451\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.66896e9	1.05459
$426$	0	0
$427$	3.77678e9i	2.34760i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.92811e9	−1.16001	−0.580005	−	0.814613i	$-0.696949\pi$
$431$	−1.92811e9	−1.16001	−0.580005	+	0.814613i	$0.696949\pi$
$432$	0	0
$433$	7.31630e6	0.00433096	0.00216548	−	0.999998i	$-0.499311\pi$
$433$	7.31630e6	0.00433096	0.00216548	+	0.999998i	$0.499311\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.28556e8i	0.302975i
$438$	0	0
$439$	−1.45798e9	−0.822481	−0.411241	−	0.911527i	$-0.634904\pi$
$439$	−1.45798e9	−0.822481	−0.411241	+	0.911527i	$0.634904\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.48722e8i	0.409173i	0.978848	+	0.204587i	$0.0655850\pi$
$443$	7.48722e8i	0.409173i	−0.978848	+	0.204587i	$0.934415\pi$
$444$	0	0
$445$	− 2.12833e9i	− 1.14493i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.02570e8	0.0534760	0.0267380	−	0.999642i	$-0.491488\pi$
$449$	1.02570e8	0.0534760	0.0267380	+	0.999642i	$0.491488\pi$
$450$	0	0
$451$	− 6.76717e8i	− 0.347368i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.66642e9	1.32706
$456$	0	0
$457$	1.79099e9	0.877784	0.438892	−	0.898540i	$-0.355371\pi$
$457$	1.79099e9	0.877784	0.438892	+	0.898540i	$0.355371\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.38726e9i	0.659482i	0.944071	+	0.329741i	$0.106961\pi$
$461$	1.38726e9i	0.659482i	−0.944071	+	0.329741i	$0.893039\pi$
$462$	0	0
$463$	−2.51864e9	−1.17932	−0.589662	−	0.807650i	$-0.700739\pi$
$463$	−2.51864e9	−1.17932	−0.589662	+	0.807650i	$0.700739\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.98627e8i	0.226551i	0.993564	+	0.113276i	$0.0361343\pi$
$467$	4.98627e8i	0.226551i	−0.993564	+	0.113276i	$0.963866\pi$
$468$	0	0
$469$	3.85697e9i	1.72640i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.14197e9	−0.496182
$474$	0	0
$475$	5.00380e8i	0.214226i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.38769e9	0.992667	0.496334	−	0.868132i	$-0.334679\pi$
$479$	2.38769e9	0.992667	0.496334	+	0.868132i	$0.334679\pi$
$480$	0	0
$481$	−1.57665e9	−0.645993
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.42958e9i	− 1.36504i
$486$	0	0
$487$	−3.95272e8	−0.155076	−0.0775380	−	0.996989i	$-0.524706\pi$
$487$	−3.95272e8	−0.155076	−0.0775380	+	0.996989i	$0.524706\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.21639e9i	1.60752i	0.594956	+	0.803759i	$0.297170\pi$
$491$	4.21639e9i	1.60752i	−0.594956	+	0.803759i	$0.702830\pi$
$492$	0	0
$493$	− 1.44653e9i	− 0.543703i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.61934e9	−1.68785
$498$	0	0
$499$	3.48633e9i	1.25608i	0.778181	+	0.628039i	$0.216142\pi$
$499$	3.48633e9i	1.25608i	−0.778181	+	0.628039i	$0.783858\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.97546e9	−0.692117	−0.346058	−	0.938213i	$-0.612480\pi$
$503$	−1.97546e9	−0.692117	−0.346058	+	0.938213i	$0.612480\pi$
$504$	0	0
$505$	−5.59076e8	−0.193175
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.00763e9i	0.674796i	0.941362	+	0.337398i	$0.109547\pi$
$509$	2.00763e9i	0.674796i	−0.941362	+	0.337398i	$0.890453\pi$
$510$	0	0
$511$	7.40953e9	2.45651
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.33771e9i	− 0.431557i
$516$	0	0
$517$	3.78689e8i	0.120522i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.23829e8	0.131298	0.0656491	−	0.997843i	$-0.479088\pi$
$521$	4.23829e8	0.131298	0.0656491	+	0.997843i	$0.479088\pi$
$522$	0	0
$523$	− 2.04172e9i	− 0.624081i	−0.950069	−	0.312040i	$-0.898988\pi$
$523$	− 2.04172e9i	− 0.624081i	0.950069	−	0.312040i	$-0.101012\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.12848e9	−0.335858
$528$	0	0
$529$	8.37799e9	2.46062
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.03347e9i	0.295634i
$534$	0	0
$535$	4.32156e9	1.22012
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.79947e9i	1.32018i
$540$	0	0
$541$	4.11067e9i	1.11615i	0.829791	+	0.558075i	$0.188460\pi$
$541$	4.11067e9i	1.11615i	−0.829791	+	0.558075i	$0.811540\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.32622e9	1.67400
$546$	0	0
$547$	− 3.47958e9i	− 0.909016i	−0.890743	−	0.454508i	$-0.849815\pi$
$547$	− 3.47958e9i	− 0.909016i	0.890743	−	0.454508i	$-0.150185\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.33691e8	0.110446
$552$	0	0
$553$	−5.24022e9	−1.31769
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.06054e9i	1.48600i	0.669293	+	0.742998i	$0.266597\pi$
$557$	6.06054e9i	1.48600i	−0.669293	+	0.742998i	$0.733403\pi$
$558$	0	0
$559$	1.74401e9	0.422286
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.86742e9i	0.441024i	0.975384	+	0.220512i	$0.0707728\pi$
$563$	1.86742e9i	0.441024i	−0.975384	+	0.220512i	$0.929227\pi$
$564$	0	0
$565$	− 4.98958e9i	− 1.16384i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.20147e8	0.0956111	0.0478056	−	0.998857i	$-0.484777\pi$
$569$	4.20147e8	0.0956111	0.0478056	+	0.998857i	$0.484777\pi$
$570$	0	0
$571$	7.99582e9i	1.79737i	0.438599	+	0.898683i	$0.355475\pi$
$571$	7.99582e9i	1.79737i	−0.438599	+	0.898683i	$0.644525\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.11547e10	2.44693
$576$	0	0
$577$	9.06331e9	1.96414	0.982068	−	0.188527i	$-0.0603712\pi$
$577$	9.06331e9	1.96414	0.982068	+	0.188527i	$0.0603712\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.54971e9i	− 1.17396i
$582$	0	0
$583$	4.47696e8	0.0935714
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.24696e9i	− 0.254460i	−0.991873	−	0.127230i	$-0.959391\pi$
$587$	− 1.24696e9i	− 0.254460i	0.991873	−	0.127230i	$-0.0406086\pi$
$588$	0	0
$589$	− 3.38335e8i	− 0.0682250i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.61574e9	0.908970	0.454485	−	0.890754i	$-0.349823\pi$
$593$	4.61574e9	0.908970	0.454485	+	0.890754i	$0.349823\pi$
$594$	0	0
$595$	− 1.14942e10i	− 2.23701i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.32636e9	−1.77304	−0.886520	−	0.462690i	$-0.846884\pi$
$599$	−9.32636e9	−1.77304	−0.886520	+	0.462690i	$0.846884\pi$
$600$	0	0
$601$	2.37970e8	0.0447159	0.0223580	−	0.999750i	$-0.492883\pi$
$601$	2.37970e8	0.0447159	0.0223580	+	0.999750i	$0.492883\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 5.69957e9i	− 1.04640i
$606$	0	0
$607$	7.73975e8	0.140464	0.0702322	−	0.997531i	$-0.477626\pi$
$607$	7.73975e8	0.140464	0.0702322	+	0.997531i	$0.477626\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5.78329e8i	− 0.102573i
$612$	0	0
$613$	− 1.51299e9i	− 0.265291i	−0.991164	−	0.132646i	$-0.957653\pi$
$613$	− 1.51299e9i	− 0.265291i	0.991164	−	0.132646i	$-0.0423472\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.14910e10	−1.96952	−0.984760	−	0.173917i	$-0.944357\pi$
$617$	−1.14910e10	−1.96952	−0.984760	+	0.173917i	$0.944357\pi$
$618$	0	0
$619$	8.37828e9i	1.41983i	0.704285	+	0.709917i	$0.251268\pi$
$619$	8.37828e9i	1.41983i	−0.704285	+	0.709917i	$0.748732\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.32714e9	−1.37971
$624$	0	0
$625$	−3.57174e9	−0.585193
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.79649e9i	1.08895i
$630$	0	0
$631$	−3.05024e9	−0.483316	−0.241658	−	0.970362i	$-0.577691\pi$
$631$	−3.05024e9	−0.483316	−0.241658	+	0.970362i	$0.577691\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 6.74939e9i	− 1.04606i
$636$	0	0
$637$	− 7.32969e9i	− 1.12356i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.51079e9	0.676471	0.338236	−	0.941061i	$-0.390170\pi$
$641$	4.51079e9	0.676471	0.338236	+	0.941061i	$0.390170\pi$
$642$	0	0
$643$	7.40844e9i	1.09898i	0.835502	+	0.549488i	$0.185177\pi$
$643$	7.40844e9i	1.09898i	−0.835502	+	0.549488i	$0.814823\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.43537e9	1.07929	0.539644	−	0.841893i	$-0.318559\pi$
$647$	7.43537e9	1.07929	0.539644	+	0.841893i	$0.318559\pi$
$648$	0	0
$649$	−1.59792e9	−0.229456
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 4.56618e9i	− 0.641737i	−0.947124	−	0.320868i	$-0.896025\pi$
$653$	− 4.56618e9i	− 0.641737i	0.947124	−	0.320868i	$-0.103975\pi$
$654$	0	0
$655$	−1.00320e10	−1.39490
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 6.99163e9i	− 0.951655i	−0.879539	−	0.475828i	$-0.842149\pi$
$659$	− 6.99163e9i	− 0.951655i	0.879539	−	0.475828i	$-0.157851\pi$
$660$	0	0
$661$	1.28732e10i	1.73374i	0.498537	+	0.866868i	$0.333871\pi$
$661$	1.28732e10i	1.73374i	−0.498537	+	0.866868i	$0.666129\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.44614e9	0.454419
$666$	0	0
$667$	− 9.66804e9i	− 1.26153i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.59926e9	0.715488
$672$	0	0
$673$	−1.25348e10	−1.58513	−0.792563	−	0.609790i	$-0.791254\pi$
$673$	−1.25348e10	−1.58513	−0.792563	+	0.609790i	$0.791254\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.18534e10i	1.46819i	0.679049	+	0.734093i	$0.262392\pi$
$677$	1.18534e10i	1.46819i	−0.679049	+	0.734093i	$0.737608\pi$
$678$	0	0
$679$	−1.34183e10	−1.64496
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.84949e9i	− 0.342211i	−0.985253	−	0.171106i	$-0.945266\pi$
$683$	− 2.84949e9i	− 0.342211i	0.985253	−	0.171106i	$-0.0547339\pi$
$684$	0	0
$685$	1.34754e10i	1.60186i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.83716e8	−0.0796358
$690$	0	0
$691$	− 1.25711e9i	− 0.144944i	−0.997370	−	0.0724720i	$-0.976911\pi$
$691$	− 1.25711e9i	− 0.144944i	0.997370	−	0.0724720i	$-0.0230888\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.65834e10	−1.87381
$696$	0	0
$697$	4.45501e9	0.498349
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.39464e9i	0.920426i	0.887808	+	0.460213i	$0.152227\pi$
$701$	8.39464e9i	0.920426i	−0.887808	+	0.460213i	$0.847773\pi$
$702$	0	0
$703$	−2.03769e9	−0.221205
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.18740e9i	0.232788i
$708$	0	0
$709$	− 9.69833e9i	− 1.02196i	−0.859592	−	0.510982i	$-0.829282\pi$
$709$	− 9.69833e9i	− 1.02196i	0.859592	−	0.510982i	$-0.170718\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.54232e9	−0.779277
$714$	0	0
$715$	− 3.95310e9i	− 0.404452i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.10435e10	1.10804	0.554021	−	0.832503i	$-0.313093\pi$
$719$	1.10435e10	1.10804	0.554021	+	0.832503i	$0.313093\pi$
$720$	0	0
$721$	−5.23385e9	−0.520053
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.15266e9i	− 0.891999i
$726$	0	0
$727$	8.43847e9	0.814504	0.407252	−	0.913316i	$-0.366487\pi$
$727$	8.43847e9	0.814504	0.407252	+	0.913316i	$0.366487\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 7.51790e9i	− 0.711846i
$732$	0	0
$733$	− 2.39430e9i	− 0.224551i	−0.993677	−	0.112275i	$-0.964186\pi$
$733$	− 2.39430e9i	− 0.224551i	0.993677	−	0.112275i	$-0.0358139\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.71815e9	0.526162
$738$	0	0
$739$	6.87252e9i	0.626412i	0.949685	+	0.313206i	$0.101403\pi$
$739$	6.87252e9i	0.626412i	−0.949685	+	0.313206i	$0.898597\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.79987e9	−0.429307	−0.214654	−	0.976690i	$-0.568862\pi$
$743$	−4.79987e9	−0.429307	−0.214654	+	0.976690i	$0.568862\pi$
$744$	0	0
$745$	2.92487e9	0.259155
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.69082e10i	− 1.47032i
$750$	0	0
$751$	1.02345e10	0.881711	0.440855	−	0.897578i	$-0.354675\pi$
$751$	1.02345e10	0.881711	0.440855	+	0.897578i	$0.354675\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.16698e9i	0.183249i
$756$	0	0
$757$	8.82502e9i	0.739401i	0.929151	+	0.369701i	$0.120540\pi$
$757$	8.82502e9i	0.739401i	−0.929151	+	0.369701i	$0.879460\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.02572e9	0.660143	0.330071	−	0.943956i	$-0.392927\pi$
$761$	8.02572e9	0.660143	0.330071	+	0.943956i	$0.392927\pi$
$762$	0	0
$763$	− 2.47515e10i	− 2.01728i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.44033e9	0.195283
$768$	0	0
$769$	−1.84781e10	−1.46526	−0.732631	−	0.680626i	$-0.761708\pi$
$769$	−1.84781e10	−1.46526	−0.732631	+	0.680626i	$0.761708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.79885e9i	0.140077i	0.997544	+	0.0700387i	$0.0223123\pi$
$773$	1.79885e9i	0.140077i	−0.997544	+	0.0700387i	$0.977688\pi$
$774$	0	0
$775$	−7.14026e9	−0.551008
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.33568e9i	0.101233i
$780$	0	0
$781$	6.84841e9i	0.514412i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.67967e10	−1.23931
$786$	0	0
$787$	− 1.00314e9i	− 0.0733584i	−0.999327	−	0.0366792i	$-0.988322\pi$
$787$	− 1.00314e9i	− 0.0733584i	0.999327	−	0.0366792i	$-0.0116780\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.95219e10	−1.40250
$792$	0	0
$793$	−8.55113e9	−0.608930
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.35708e10i	− 0.949511i	−0.880118	−	0.474756i	$-0.842536\pi$
$797$	− 1.35708e10i	− 0.949511i	0.880118	−	0.474756i	$-0.157464\pi$
$798$	0	0
$799$	−2.49301e9	−0.172906
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.09850e10i	− 0.748679i
$804$	0	0
$805$	− 7.68228e10i	− 5.19044i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.33727e9	0.553610	0.276805	−	0.960926i	$-0.410724\pi$
$809$	8.33727e9	0.553610	0.276805	+	0.960926i	$0.410724\pi$
$810$	0	0
$811$	− 2.27829e10i	− 1.49981i	−0.661545	−	0.749906i	$-0.730099\pi$
$811$	− 2.27829e10i	− 1.49981i	0.661545	−	0.749906i	$-0.269901\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.08749e9	−0.135074
$816$	0	0
$817$	2.25399e9	0.144602
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.35551e10i	0.854871i	0.904046	+	0.427436i	$0.140583\pi$
$821$	1.35551e10i	0.854871i	−0.904046	+	0.427436i	$0.859417\pi$
$822$	0	0
$823$	−1.12181e10	−0.701487	−0.350743	−	0.936472i	$-0.614071\pi$
$823$	−1.12181e10	−0.701487	−0.350743	+	0.936472i	$0.614071\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.37157e10i	1.45803i	0.684496	+	0.729016i	$0.260022\pi$
$827$	2.37157e10i	1.45803i	−0.684496	+	0.729016i	$0.739978\pi$
$828$	0	0
$829$	− 1.17633e10i	− 0.717113i	−0.933508	−	0.358557i	$-0.883269\pi$
$829$	− 1.17633e10i	− 0.717113i	0.933508	−	0.358557i	$-0.116731\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.15962e10	−1.89399
$834$	0	0
$835$	2.73697e10i	1.62693i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.79741e9	0.397353	0.198677	−	0.980065i	$-0.436336\pi$
$839$	6.79741e9	0.397353	0.198677	+	0.980065i	$0.436336\pi$
$840$	0	0
$841$	9.31706e9	0.540123
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.06503e10i	− 1.17741i
$846$	0	0
$847$	−2.22997e10	−1.26098
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.54252e10i	2.52664i
$852$	0	0
$853$	2.68751e10i	1.48262i	0.671165	+	0.741308i	$0.265794\pi$
$853$	2.68751e10i	1.48262i	−0.671165	+	0.741308i	$0.734206\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.07409e10	−1.12563	−0.562814	−	0.826584i	$-0.690281\pi$
$857$	−2.07409e10	−1.12563	−0.562814	+	0.826584i	$0.690281\pi$
$858$	0	0
$859$	1.85518e10i	0.998642i	0.866417	+	0.499321i	$0.166417\pi$
$859$	1.85518e10i	0.998642i	−0.866417	+	0.499321i	$0.833583\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.60997e10	1.38229	0.691144	−	0.722717i	$-0.257107\pi$
$863$	2.60997e10	1.38229	0.691144	+	0.722717i	$0.257107\pi$
$864$	0	0
$865$	3.93445e10	2.06694
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.76889e9i	0.401596i
$870$	0	0
$871$	−8.73269e9	−0.447801
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.74364e10i	− 0.879892i
$876$	0	0
$877$	− 2.82084e9i	− 0.141215i	−0.997504	−	0.0706074i	$-0.977506\pi$
$877$	− 2.82084e9i	− 0.141215i	0.997504	−	0.0706074i	$-0.0224937\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.92677e9	−0.489094	−0.244547	−	0.969637i	$-0.578639\pi$
$881$	−9.92677e9	−0.489094	−0.244547	+	0.969637i	$0.578639\pi$
$882$	0	0
$883$	8.85510e9i	0.432844i	0.976300	+	0.216422i	$0.0694387\pi$
$883$	8.85510e9i	0.432844i	−0.976300	+	0.216422i	$0.930561\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.69747e9	−0.322239	−0.161120	−	0.986935i	$-0.551510\pi$
$887$	−6.69747e9	−0.322239	−0.161120	+	0.986935i	$0.551510\pi$
$888$	0	0
$889$	−2.64072e10	−1.26057
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7.47444e8i	− 0.0351236i
$894$	0	0
$895$	5.77455e10	2.69239
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.18863e9i	0.284076i
$900$	0	0
$901$	2.94730e9i	0.134242i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.47464e10	2.90366
$906$	0	0
$907$	4.12904e9i	0.183749i	0.995771	+	0.0918743i	$0.0292858\pi$
$907$	4.12904e9i	0.183749i	−0.995771	+	0.0918743i	$0.970714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.66278e10	−0.728653	−0.364326	−	0.931271i	$-0.618701\pi$
$911$	−1.66278e10	−0.728653	−0.364326	+	0.931271i	$0.618701\pi$
$912$	0	0
$913$	−8.22773e9	−0.357793
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.92506e10i	1.68095i
$918$	0	0
$919$	−1.15132e10	−0.489317	−0.244658	−	0.969609i	$-0.578676\pi$
$919$	−1.15132e10	−0.489317	−0.244658	+	0.969609i	$0.578676\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.04588e10i	− 0.437800i
$924$	0	0
$925$	4.30037e10i	1.78653i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.46627e10	−0.600010	−0.300005	−	0.953938i	$-0.596988\pi$
$929$	−1.46627e10	−0.600010	−0.300005	+	0.953938i	$0.596988\pi$
$930$	0	0
$931$	− 9.47303e9i	− 0.384738i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.70407e10	−0.681784
$936$	0	0
$937$	−9.78575e9	−0.388602	−0.194301	−	0.980942i	$-0.562244\pi$
$937$	−9.78575e9	−0.388602	−0.194301	+	0.980942i	$0.562244\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.76197e10i	0.689341i	0.938724	+	0.344670i	$0.112009\pi$
$941$	1.76197e10i	0.689341i	−0.938724	+	0.344670i	$0.887991\pi$
$942$	0	0
$943$	2.97756e10	1.15630
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.90489e9i	0.0728862i	0.999336	+	0.0364431i	$0.0116028\pi$
$947$	1.90489e9i	0.0728862i	−0.999336	+	0.0364431i	$0.988397\pi$
$948$	0	0
$949$	1.67762e10i	0.637178i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.36695e10	0.885857	0.442928	−	0.896557i	$-0.353940\pi$
$953$	2.36695e10	0.885857	0.442928	+	0.896557i	$0.353940\pi$
$954$	0	0
$955$	4.88606e9i	0.181529i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.27228e10	1.93034
$960$	0	0
$961$	−2.26847e10	−0.824519
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 5.55990e10i	− 1.99169i
$966$	0	0
$967$	1.02607e10	0.364907	0.182454	−	0.983214i	$-0.441596\pi$
$967$	1.02607e10	0.364907	0.182454	+	0.983214i	$0.441596\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 4.79387e10i	− 1.68042i	−0.542258	−	0.840212i	$-0.682430\pi$
$971$	− 4.79387e10i	− 1.68042i	0.542258	−	0.840212i	$-0.317570\pi$
$972$	0	0
$973$	6.48830e10i	2.25806i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.78771e10	0.956349	0.478174	−	0.878265i	$-0.341299\pi$
$977$	2.78771e10	0.956349	0.478174	+	0.878265i	$0.341299\pi$
$978$	0	0
$979$	1.23454e10i	0.420500i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.77775e9	−0.294745	−0.147372	−	0.989081i	$-0.547082\pi$
$983$	−8.77775e9	−0.294745	−0.147372	+	0.989081i	$0.547082\pi$
$984$	0	0
$985$	6.26995e10	2.09044
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 5.02469e10i	− 1.65166i
$990$	0	0
$991$	−4.78690e8	−0.0156242	−0.00781208	−	0.999969i	$-0.502487\pi$
$991$	−4.78690e8	−0.0156242	−0.00781208	+	0.999969i	$0.502487\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.07856e10i	− 0.990757i
$996$	0	0
$997$	− 4.09532e10i	− 1.30874i	−0.756173	−	0.654371i	$-0.772933\pi$
$997$	− 4.09532e10i	− 1.30874i	0.756173	−	0.654371i	$-0.227067\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.8.d.d.145.3		14
3.2	odd	2		96.8.d.a.49.6		14
4.3	odd	2		72.8.d.d.37.2		14
8.3	odd	2		72.8.d.d.37.1		14
8.5	even	2	inner	288.8.d.d.145.12		14
12.11	even	2		24.8.d.a.13.13	✓	14
24.5	odd	2		96.8.d.a.49.9		14
24.11	even	2		24.8.d.a.13.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.d.a.13.13	✓	14	12.11	even	2
24.8.d.a.13.14	yes	14	24.11	even	2
72.8.d.d.37.1		14	8.3	odd	2
72.8.d.d.37.2		14	4.3	odd	2
96.8.d.a.49.6		14	3.2	odd	2
96.8.d.a.49.9		14	24.5	odd	2
288.8.d.d.145.3		14	1.1	even	1	trivial
288.8.d.d.145.12		14	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-425.308i q^{5} -1664.03 q^{7} +O(q^{10})\)	\(q-425.308i q^{5} -1664.03 q^{7} +2467.01i q^{11} -3767.59i q^{13} -16241.0 q^{17} -4869.30i q^{19} -108549. q^{23} -102762. q^{25} +89066.4i q^{29} +69483.3 q^{31} +707727. i q^{35} -418478. i q^{37} -274307. q^{41} +462897. i q^{43} +153501. q^{47} +1.94546e6 q^{49} -181473. i q^{53} +1.04924e6 q^{55} +647716. i q^{59} -2.26966e6i q^{61} -1.60239e6 q^{65} -2.31785e6i q^{67} +2.77599e6 q^{71} -4.45276e6 q^{73} -4.10518e6i q^{77} +3.14911e6 q^{79} +3.33510e6i q^{83} +6.90743e6i q^{85} +5.00420e6 q^{89} +6.26939e6i q^{91} -2.07096e6 q^{95} +8.06376e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 1372 q^{7}+O(q^{10}) \) 14 * q - 1372 * q^7	\( 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) \) 14 * q - 1372 * q^7 + 2908 * q^17 - 143416 * q^23 - 202626 * q^25 + 89468 * q^31 + 441284 * q^41 - 1056408 * q^47 + 2158134 * q^49 - 4757504 * q^55 + 2520464 * q^65 + 5172696 * q^71 - 5446196 * q^73 + 14373548 * q^79 + 11952620 * q^89 - 69327376 * q^95 + 133732 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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