LMFDB - Embedded newform 256.12.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,12,Mod(1,256)] 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("256.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$196.695854223$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 51211 x^{10} + 256110 x^{9} + 921161404 x^{8} - 3686182342 x^{7} + \cdots + 37\!\cdots\!47 $ x^12 - 6x^11 - 51211x^10 + 256110x^9 + 921161404x^8 - 3686182342x^7 - 7189142902783x^6 + 21580331422274x^5 + 27586413649287756x^4 - 55208798818670954x^3 - 51606116089026310579x^2 + 51633724085771990330*x + 37747816615894917324647
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{126}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-139.567$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-784.336 q^{3} +11907.7 q^{5} +21148.3 q^{7} +438037. q^{9} -561003. q^{11} -1.10728e6 q^{13} -9.33962e6 q^{15} +5.37054e6 q^{17} +8.72557e6 q^{19} -1.65874e7 q^{21} -2.29090e7 q^{23} +9.29644e7 q^{25} -2.04625e8 q^{27} +1.74228e8 q^{29} +2.58795e8 q^{31} +4.40015e8 q^{33} +2.51827e8 q^{35} -3.35460e8 q^{37} +8.68483e8 q^{39} +4.92499e8 q^{41} +7.74649e8 q^{43} +5.21599e9 q^{45} -1.19986e9 q^{47} -1.53008e9 q^{49} -4.21231e9 q^{51} -1.79948e9 q^{53} -6.68023e9 q^{55} -6.84378e9 q^{57} -9.05069e8 q^{59} +9.99168e9 q^{61} +9.26374e9 q^{63} -1.31852e10 q^{65} -1.13658e10 q^{67} +1.79684e10 q^{69} +2.40027e10 q^{71} +5.92056e9 q^{73} -7.29153e10 q^{75} -1.18643e10 q^{77} -1.80146e10 q^{79} +8.28982e10 q^{81} -2.45012e10 q^{83} +6.39506e10 q^{85} -1.36654e11 q^{87} +8.73556e10 q^{89} -2.34172e10 q^{91} -2.02982e11 q^{93} +1.03901e11 q^{95} -1.35707e11 q^{97} -2.45740e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 1152028 q^{9} + 26470216 q^{17} + 207502500 q^{25} + 935847712 q^{33} + 2031735176 q^{41} - 2342764308 q^{49} - 23753487776 q^{57} - 39461105920 q^{65} + 43591631448 q^{73} + 232401225292 q^{81}+ \cdots - 285441034488 q^{97}+O(q^{100}) $ 12 * q + 1152028 * q^9 + 26470216 * q^17 + 207502500 * q^25 + 935847712 * q^33 + 2031735176 * q^41 - 2342764308 * q^49 - 23753487776 * q^57 - 39461105920 * q^65 + 43591631448 * q^73 + 232401225292 * q^81 + 209389352088 * q^89 - 285441034488 * q^97

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
$3$	−784.336	−1.86353	−0.931763	−	0.363067i	$-0.881730\pi$
$3$	−784.336	−1.86353	−0.931763	+	0.363067i	$0.881730\pi$
$4$	0	0
$5$	11907.7	1.70409	0.852043	−	0.523472i	$-0.175363\pi$
$5$	11907.7	1.70409	0.852043	+	0.523472i	$0.175363\pi$
$6$	0	0
$7$	21148.3	0.475594	0.237797	−	0.971315i	$-0.423575\pi$
$7$	21148.3	0.475594	0.237797	+	0.971315i	$0.423575\pi$
$8$	0	0
$9$	438037.	2.47273
$10$	0	0
$11$	−561003.	−1.05028	−0.525140	−	0.851016i	$-0.675987\pi$
$11$	−561003.	−1.05028	−0.525140	+	0.851016i	$0.675987\pi$
$12$	0	0
$13$	−1.10728e6	−0.827124	−0.413562	−	0.910476i	$-0.635715\pi$
$13$	−1.10728e6	−0.827124	−0.413562	+	0.910476i	$0.635715\pi$
$14$	0	0
$15$	−9.33962e6	−3.17561
$16$	0	0
$17$	5.37054e6	0.917379	0.458689	−	0.888597i	$-0.348319\pi$
$17$	5.37054e6	0.917379	0.458689	+	0.888597i	$0.348319\pi$
$18$	0	0
$19$	8.72557e6	0.808442	0.404221	−	0.914661i	$-0.367543\pi$
$19$	8.72557e6	0.808442	0.404221	+	0.914661i	$0.367543\pi$
$20$	0	0
$21$	−1.65874e7	−0.886282
$22$	0	0
$23$	−2.29090e7	−0.742171	−0.371085	−	0.928599i	$-0.621014\pi$
$23$	−2.29090e7	−0.742171	−0.371085	+	0.928599i	$0.621014\pi$
$24$	0	0
$25$	9.29644e7	1.90391
$26$	0	0
$27$	−2.04625e8	−2.74447
$28$	0	0
$29$	1.74228e8	1.57736	0.788678	−	0.614806i	$-0.210766\pi$
$29$	1.74228e8	1.57736	0.788678	+	0.614806i	$0.210766\pi$
$30$	0	0
$31$	2.58795e8	1.62355	0.811777	−	0.583968i	$-0.198501\pi$
$31$	2.58795e8	1.62355	0.811777	+	0.583968i	$0.198501\pi$
$32$	0	0
$33$	4.40015e8	1.95723
$34$	0	0
$35$	2.51827e8	0.810453
$36$	0	0
$37$	−3.35460e8	−0.795302	−0.397651	−	0.917537i	$-0.630174\pi$
$37$	−3.35460e8	−0.795302	−0.397651	+	0.917537i	$0.630174\pi$
$38$	0	0
$39$	8.68483e8	1.54137
$40$	0	0
$41$	4.92499e8	0.663888	0.331944	−	0.943299i	$-0.392296\pi$
$41$	4.92499e8	0.663888	0.331944	+	0.943299i	$0.392296\pi$
$42$	0	0
$43$	7.74649e8	0.803579	0.401789	−	0.915732i	$-0.368388\pi$
$43$	7.74649e8	0.803579	0.401789	+	0.915732i	$0.368388\pi$
$44$	0	0
$45$	5.21599e9	4.21375
$46$	0	0
$47$	−1.19986e9	−0.763118	−0.381559	−	0.924344i	$-0.624613\pi$
$47$	−1.19986e9	−0.763118	−0.381559	+	0.924344i	$0.624613\pi$
$48$	0	0
$49$	−1.53008e9	−0.773810
$50$	0	0
$51$	−4.21231e9	−1.70956
$52$	0	0
$53$	−1.79948e9	−0.591058	−0.295529	−	0.955334i	$-0.595496\pi$
$53$	−1.79948e9	−0.591058	−0.295529	+	0.955334i	$0.595496\pi$
$54$	0	0
$55$	−6.68023e9	−1.78977
$56$	0	0
$57$	−6.84378e9	−1.50655
$58$	0	0
$59$	−9.05069e8	−0.164815	−0.0824073	−	0.996599i	$-0.526261\pi$
$59$	−9.05069e8	−0.164815	−0.0824073	+	0.996599i	$0.526261\pi$
$60$	0	0
$61$	9.99168e9	1.51469	0.757346	−	0.653014i	$-0.226496\pi$
$61$	9.99168e9	1.51469	0.757346	+	0.653014i	$0.226496\pi$
$62$	0	0
$63$	9.26374e9	1.17602
$64$	0	0
$65$	−1.31852e10	−1.40949
$66$	0	0
$67$	−1.13658e10	−1.02846	−0.514230	−	0.857652i	$-0.671922\pi$
$67$	−1.13658e10	−1.02846	−0.514230	+	0.857652i	$0.671922\pi$
$68$	0	0
$69$	1.79684e10	1.38305
$70$	0	0
$71$	2.40027e10	1.57884	0.789421	−	0.613852i	$-0.210381\pi$
$71$	2.40027e10	1.57884	0.789421	+	0.613852i	$0.210381\pi$
$72$	0	0
$73$	5.92056e9	0.334262	0.167131	−	0.985935i	$-0.446550\pi$
$73$	5.92056e9	0.334262	0.167131	+	0.985935i	$0.446550\pi$
$74$	0	0
$75$	−7.29153e10	−3.54799
$76$	0	0
$77$	−1.18643e10	−0.499507
$78$	0	0
$79$	−1.80146e10	−0.658682	−0.329341	−	0.944211i	$-0.606827\pi$
$79$	−1.80146e10	−0.658682	−0.329341	+	0.944211i	$0.606827\pi$
$80$	0	0
$81$	8.28982e10	2.64166
$82$	0	0
$83$	−2.45012e10	−0.682743	−0.341372	−	0.939928i	$-0.610891\pi$
$83$	−2.45012e10	−0.682743	−0.341372	+	0.939928i	$0.610891\pi$
$84$	0	0
$85$	6.39506e10	1.56329
$86$	0	0
$87$	−1.36654e11	−2.93944
$88$	0	0
$89$	8.73556e10	1.65823	0.829117	−	0.559075i	$-0.188844\pi$
$89$	8.73556e10	1.65823	0.829117	+	0.559075i	$0.188844\pi$
$90$	0	0
$91$	−2.34172e10	−0.393375
$92$	0	0
$93$	−2.02982e11	−3.02553
$94$	0	0
$95$	1.03901e11	1.37765
$96$	0	0
$97$	−1.35707e11	−1.60456	−0.802281	−	0.596947i	$-0.796380\pi$
$97$	−1.35707e11	−1.60456	−0.802281	+	0.596947i	$0.796380\pi$
$98$	0	0
$99$	−2.45740e11	−2.59706
$100$	0	0
$101$	−1.17395e11	−1.11143	−0.555716	−	0.831372i	$-0.687556\pi$
$101$	−1.17395e11	−1.11143	−0.555716	+	0.831372i	$0.687556\pi$
$102$	0	0
$103$	1.84106e10	0.156481	0.0782407	−	0.996934i	$-0.475070\pi$
$103$	1.84106e10	0.156481	0.0782407	+	0.996934i	$0.475070\pi$
$104$	0	0
$105$	−1.97517e11	−1.51030
$106$	0	0
$107$	−1.08175e11	−0.745616	−0.372808	−	0.927909i	$-0.621605\pi$
$107$	−1.08175e11	−0.745616	−0.372808	+	0.927909i	$0.621605\pi$
$108$	0	0
$109$	−1.80324e9	−0.0112256	−0.00561279	−	0.999984i	$-0.501787\pi$
$109$	−1.80324e9	−0.0112256	−0.00561279	+	0.999984i	$0.501787\pi$
$110$	0	0
$111$	2.63114e11	1.48207
$112$	0	0
$113$	1.50400e11	0.767921	0.383960	−	0.923350i	$-0.374560\pi$
$113$	1.50400e11	0.767921	0.383960	+	0.923350i	$0.374560\pi$
$114$	0	0
$115$	−2.72793e11	−1.26472
$116$	0	0
$117$	−4.85031e11	−2.04525
$118$	0	0
$119$	1.13578e11	0.436300
$120$	0	0
$121$	2.94126e10	0.103089
$122$	0	0
$123$	−3.86285e11	−1.23717
$124$	0	0
$125$	5.25560e11	1.54034
$126$	0	0
$127$	4.09141e11	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$127$	4.09141e11	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$128$	0	0
$129$	−6.07585e11	−1.49749
$130$	0	0
$131$	2.82610e11	0.640023	0.320012	−	0.947414i	$-0.396313\pi$
$131$	2.82610e11	0.640023	0.320012	+	0.947414i	$0.396313\pi$
$132$	0	0
$133$	1.84531e11	0.384490
$134$	0	0
$135$	−2.43661e12	−4.67682
$136$	0	0
$137$	5.04027e9	0.00892258	0.00446129	−	0.999990i	$-0.498580\pi$
$137$	5.04027e9	0.00892258	0.00446129	+	0.999990i	$0.498580\pi$
$138$	0	0
$139$	−5.20421e11	−0.850695	−0.425347	−	0.905030i	$-0.639848\pi$
$139$	−5.20421e11	−0.850695	−0.425347	+	0.905030i	$0.639848\pi$
$140$	0	0
$141$	9.41093e11	1.42209
$142$	0	0
$143$	6.21189e11	0.868712
$144$	0	0
$145$	2.07465e12	2.68795
$146$	0	0
$147$	1.20009e12	1.44202
$148$	0	0
$149$	−3.21703e11	−0.358865	−0.179432	−	0.983770i	$-0.557426\pi$
$149$	−3.21703e11	−0.358865	−0.179432	+	0.983770i	$0.557426\pi$
$150$	0	0
$151$	6.64644e11	0.688995	0.344497	−	0.938787i	$-0.388049\pi$
$151$	6.64644e11	0.688995	0.344497	+	0.938787i	$0.388049\pi$
$152$	0	0
$153$	2.35249e12	2.26843
$154$	0	0
$155$	3.08164e12	2.76668
$156$	0	0
$157$	−1.91656e11	−0.160352	−0.0801760	−	0.996781i	$-0.525548\pi$
$157$	−1.91656e11	−0.160352	−0.0801760	+	0.996781i	$0.525548\pi$
$158$	0	0
$159$	1.41140e12	1.10145
$160$	0	0
$161$	−4.84488e11	−0.352972
$162$	0	0
$163$	1.19385e12	0.812679	0.406339	−	0.913722i	$-0.366805\pi$
$163$	1.19385e12	0.812679	0.406339	+	0.913722i	$0.366805\pi$
$164$	0	0
$165$	5.23955e12	3.33528
$166$	0	0
$167$	3.53708e11	0.210719	0.105360	−	0.994434i	$-0.466401\pi$
$167$	3.53708e11	0.210719	0.105360	+	0.994434i	$0.466401\pi$
$168$	0	0
$169$	−5.66083e11	−0.315866
$170$	0	0
$171$	3.82212e12	1.99906
$172$	0	0
$173$	−2.41578e12	−1.18523	−0.592617	−	0.805484i	$-0.701905\pi$
$173$	−2.41578e12	−1.18523	−0.592617	+	0.805484i	$0.701905\pi$
$174$	0	0
$175$	1.96604e12	0.905488
$176$	0	0
$177$	7.09879e11	0.307136
$178$	0	0
$179$	−2.07221e12	−0.842834	−0.421417	−	0.906867i	$-0.638467\pi$
$179$	−2.07221e12	−0.842834	−0.421417	+	0.906867i	$0.638467\pi$
$180$	0	0
$181$	−8.77418e11	−0.335718	−0.167859	−	0.985811i	$-0.553685\pi$
$181$	−8.77418e11	−0.335718	−0.167859	+	0.985811i	$0.553685\pi$
$182$	0	0
$183$	−7.83684e12	−2.82267
$184$	0	0
$185$	−3.99455e12	−1.35526
$186$	0	0
$187$	−3.01289e12	−0.963505
$188$	0	0
$189$	−4.32748e12	−1.30525
$190$	0	0
$191$	2.74196e12	0.780510	0.390255	−	0.920707i	$-0.372387\pi$
$191$	2.74196e12	0.780510	0.390255	+	0.920707i	$0.372387\pi$
$192$	0	0
$193$	5.39488e11	0.145016	0.0725081	−	0.997368i	$-0.476900\pi$
$193$	5.39488e11	0.145016	0.0725081	+	0.997368i	$0.476900\pi$
$194$	0	0
$195$	1.03416e13	2.62662
$196$	0	0
$197$	−6.83190e11	−0.164050	−0.0820252	−	0.996630i	$-0.526139\pi$
$197$	−6.83190e11	−0.164050	−0.0820252	+	0.996630i	$0.526139\pi$
$198$	0	0
$199$	4.65463e12	1.05729	0.528644	−	0.848843i	$-0.322701\pi$
$199$	4.65463e12	1.05729	0.528644	+	0.848843i	$0.322701\pi$
$200$	0	0
$201$	8.91459e12	1.91656
$202$	0	0
$203$	3.68463e12	0.750181
$204$	0	0
$205$	5.86452e12	1.13132
$206$	0	0
$207$	−1.00350e13	−1.83519
$208$	0	0
$209$	−4.89507e12	−0.849091
$210$	0	0
$211$	2.06896e12	0.340565	0.170282	−	0.985395i	$-0.445532\pi$
$211$	2.06896e12	0.340565	0.170282	+	0.985395i	$0.445532\pi$
$212$	0	0
$213$	−1.88262e13	−2.94221
$214$	0	0
$215$	9.22426e12	1.36937
$216$	0	0
$217$	5.47308e12	0.772153
$218$	0	0
$219$	−4.64371e12	−0.622906
$220$	0	0
$221$	−5.94671e12	−0.758786
$222$	0	0
$223$	4.03814e12	0.490348	0.245174	−	0.969479i	$-0.421155\pi$
$223$	4.03814e12	0.490348	0.245174	+	0.969479i	$0.421155\pi$
$224$	0	0
$225$	4.07218e13	4.70786
$226$	0	0
$227$	−1.10735e13	−1.21939	−0.609694	−	0.792637i	$-0.708708\pi$
$227$	−1.10735e13	−1.21939	−0.609694	+	0.792637i	$0.708708\pi$
$228$	0	0
$229$	−1.38640e12	−0.145477	−0.0727383	−	0.997351i	$-0.523174\pi$
$229$	−1.38640e12	−0.145477	−0.0727383	+	0.997351i	$0.523174\pi$
$230$	0	0
$231$	9.30557e12	0.930845
$232$	0	0
$233$	1.76073e13	1.67972	0.839859	−	0.542805i	$-0.182638\pi$
$233$	1.76073e13	1.67972	0.839859	+	0.542805i	$0.182638\pi$
$234$	0	0
$235$	−1.42875e13	−1.30042
$236$	0	0
$237$	1.41295e13	1.22747
$238$	0	0
$239$	−4.02394e11	−0.0333782	−0.0166891	−	0.999861i	$-0.505313\pi$
$239$	−4.02394e11	−0.0333782	−0.0166891	+	0.999861i	$0.505313\pi$
$240$	0	0
$241$	−6.87906e12	−0.545049	−0.272524	−	0.962149i	$-0.587859\pi$
$241$	−6.87906e12	−0.545049	−0.272524	+	0.962149i	$0.587859\pi$
$242$	0	0
$243$	−2.87713e13	−2.17834
$244$	0	0
$245$	−1.82196e13	−1.31864
$246$	0	0
$247$	−9.66168e12	−0.668681
$248$	0	0
$249$	1.92172e13	1.27231
$250$	0	0
$251$	2.50600e13	1.58772	0.793862	−	0.608097i	$-0.208067\pi$
$251$	2.50600e13	1.58772	0.793862	+	0.608097i	$0.208067\pi$
$252$	0	0
$253$	1.28520e13	0.779488
$254$	0	0
$255$	−5.01588e13	−2.91324
$256$	0	0
$257$	2.61595e12	0.145545	0.0727724	−	0.997349i	$-0.476815\pi$
$257$	2.61595e12	0.145545	0.0727724	+	0.997349i	$0.476815\pi$
$258$	0	0
$259$	−7.09442e12	−0.378241
$260$	0	0
$261$	7.63184e13	3.90038
$262$	0	0
$263$	−1.35684e12	−0.0664924	−0.0332462	−	0.999447i	$-0.510585\pi$
$263$	−1.35684e12	−0.0664924	−0.0332462	+	0.999447i	$0.510585\pi$
$264$	0	0
$265$	−2.14276e13	−1.00721
$266$	0	0
$267$	−6.85162e13	−3.09016
$268$	0	0
$269$	6.55358e12	0.283688	0.141844	−	0.989889i	$-0.454697\pi$
$269$	6.55358e12	0.283688	0.141844	+	0.989889i	$0.454697\pi$
$270$	0	0
$271$	−2.39710e13	−0.996220	−0.498110	−	0.867114i	$-0.665972\pi$
$271$	−2.39710e13	−0.996220	−0.498110	+	0.867114i	$0.665972\pi$
$272$	0	0
$273$	1.83669e13	0.733065
$274$	0	0
$275$	−5.21533e13	−1.99964
$276$	0	0
$277$	2.54063e13	0.936058	0.468029	−	0.883713i	$-0.344964\pi$
$277$	2.54063e13	0.936058	0.468029	+	0.883713i	$0.344964\pi$
$278$	0	0
$279$	1.13362e14	4.01461
$280$	0	0
$281$	9.10879e12	0.310153	0.155076	−	0.987902i	$-0.450438\pi$
$281$	9.10879e12	0.310153	0.155076	+	0.987902i	$0.450438\pi$
$282$	0	0
$283$	−3.57642e13	−1.17118	−0.585589	−	0.810608i	$-0.699137\pi$
$283$	−3.57642e13	−1.17118	−0.585589	+	0.810608i	$0.699137\pi$
$284$	0	0
$285$	−8.14934e13	−2.56730
$286$	0	0
$287$	1.04155e13	0.315741
$288$	0	0
$289$	−5.42922e12	−0.158416
$290$	0	0
$291$	1.06440e14	2.99014
$292$	0	0
$293$	8.24513e11	0.0223062	0.0111531	−	0.999938i	$-0.496450\pi$
$293$	8.24513e11	0.0223062	0.0111531	+	0.999938i	$0.496450\pi$
$294$	0	0
$295$	−1.07773e13	−0.280858
$296$	0	0
$297$	1.14795e14	2.88246
$298$	0	0
$299$	2.53668e13	0.613867
$300$	0	0
$301$	1.63825e13	0.382177
$302$	0	0
$303$	9.20774e13	2.07118
$304$	0	0
$305$	1.18978e14	2.58117
$306$	0	0
$307$	−2.09611e13	−0.438685	−0.219342	−	0.975648i	$-0.570391\pi$
$307$	−2.09611e13	−0.438685	−0.219342	+	0.975648i	$0.570391\pi$
$308$	0	0
$309$	−1.44401e13	−0.291607
$310$	0	0
$311$	−7.82526e13	−1.52516	−0.762582	−	0.646892i	$-0.776069\pi$
$311$	−7.82526e13	−1.52516	−0.762582	+	0.646892i	$0.776069\pi$
$312$	0	0
$313$	3.47387e13	0.653611	0.326806	−	0.945092i	$-0.394028\pi$
$313$	3.47387e13	0.653611	0.326806	+	0.945092i	$0.394028\pi$
$314$	0	0
$315$	1.10309e14	2.00403
$316$	0	0
$317$	9.15418e13	1.60618	0.803088	−	0.595860i	$-0.203189\pi$
$317$	9.15418e13	1.60618	0.803088	+	0.595860i	$0.203189\pi$
$318$	0	0
$319$	−9.77426e13	−1.65667
$320$	0	0
$321$	8.48454e13	1.38947
$322$	0	0
$323$	4.68610e13	0.741647
$324$	0	0
$325$	−1.02938e14	−1.57477
$326$	0	0
$327$	1.41435e12	0.0209192
$328$	0	0
$329$	−2.53750e13	−0.362935
$330$	0	0
$331$	3.51219e13	0.485875	0.242937	−	0.970042i	$-0.421889\pi$
$331$	3.51219e13	0.485875	0.242937	+	0.970042i	$0.421889\pi$
$332$	0	0
$333$	−1.46944e14	−1.96657
$334$	0	0
$335$	−1.35340e14	−1.75259
$336$	0	0
$337$	5.42414e13	0.679777	0.339889	−	0.940466i	$-0.389611\pi$
$337$	5.42414e13	0.679777	0.339889	+	0.940466i	$0.389611\pi$
$338$	0	0
$339$	−1.17964e14	−1.43104
$340$	0	0
$341$	−1.45185e14	−1.70519
$342$	0	0
$343$	−7.41756e13	−0.843614
$344$	0	0
$345$	2.13962e14	2.35684
$346$	0	0
$347$	1.38780e14	1.48086	0.740430	−	0.672134i	$-0.234622\pi$
$347$	1.38780e14	1.48086	0.740430	+	0.672134i	$0.234622\pi$
$348$	0	0
$349$	1.70059e14	1.75816	0.879080	−	0.476674i	$-0.158158\pi$
$349$	1.70059e14	1.75816	0.879080	+	0.476674i	$0.158158\pi$
$350$	0	0
$351$	2.26578e14	2.27002
$352$	0	0
$353$	−1.71901e14	−1.66924	−0.834620	−	0.550826i	$-0.814313\pi$
$353$	−1.71901e14	−1.66924	−0.834620	+	0.550826i	$0.814313\pi$
$354$	0	0
$355$	2.85816e14	2.69048
$356$	0	0
$357$	−8.90832e13	−0.813056
$358$	0	0
$359$	1.27135e14	1.12524	0.562619	−	0.826716i	$-0.309794\pi$
$359$	1.27135e14	1.12524	0.562619	+	0.826716i	$0.309794\pi$
$360$	0	0
$361$	−4.03548e13	−0.346422
$362$	0	0
$363$	−2.30694e13	−0.192110
$364$	0	0
$365$	7.05000e13	0.569611
$366$	0	0
$367$	2.04239e14	1.60131	0.800656	−	0.599124i	$-0.204485\pi$
$367$	2.04239e14	1.60131	0.800656	+	0.599124i	$0.204485\pi$
$368$	0	0
$369$	2.15733e14	1.64161
$370$	0	0
$371$	−3.80560e13	−0.281104
$372$	0	0
$373$	1.16113e14	0.832688	0.416344	−	0.909207i	$-0.363311\pi$
$373$	1.16113e14	0.832688	0.416344	+	0.909207i	$0.363311\pi$
$374$	0	0
$375$	−4.12216e14	−2.87047
$376$	0	0
$377$	−1.92920e14	−1.30467
$378$	0	0
$379$	3.86376e13	0.253802	0.126901	−	0.991915i	$-0.459497\pi$
$379$	3.86376e13	0.253802	0.126901	+	0.991915i	$0.459497\pi$
$380$	0	0
$381$	−3.20904e14	−2.04780
$382$	0	0
$383$	2.83128e14	1.75545	0.877727	−	0.479161i	$-0.159059\pi$
$383$	2.83128e14	1.75545	0.877727	+	0.479161i	$0.159059\pi$
$384$	0	0
$385$	−1.41276e14	−0.851203
$386$	0	0
$387$	3.39325e14	1.98703
$388$	0	0
$389$	−5.49647e13	−0.312868	−0.156434	−	0.987688i	$-0.550000\pi$
$389$	−5.49647e13	−0.312868	−0.156434	+	0.987688i	$0.550000\pi$
$390$	0	0
$391$	−1.23034e14	−0.680852
$392$	0	0
$393$	−2.21661e14	−1.19270
$394$	0	0
$395$	−2.14512e14	−1.12245
$396$	0	0
$397$	2.00590e14	1.02085	0.510423	−	0.859923i	$-0.329489\pi$
$397$	2.00590e14	1.02085	0.510423	+	0.859923i	$0.329489\pi$
$398$	0	0
$399$	−1.44734e14	−0.716508
$400$	0	0
$401$	−2.05103e14	−0.987819	−0.493910	−	0.869513i	$-0.664433\pi$
$401$	−2.05103e14	−0.987819	−0.493910	+	0.869513i	$0.664433\pi$
$402$	0	0
$403$	−2.86560e14	−1.34288
$404$	0	0
$405$	9.87124e14	4.50162
$406$	0	0
$407$	1.88194e14	0.835290
$408$	0	0
$409$	4.18830e14	1.80950	0.904751	−	0.425940i	$-0.140057\pi$
$409$	4.18830e14	1.80950	0.904751	+	0.425940i	$0.140057\pi$
$410$	0	0
$411$	−3.95326e12	−0.0166275
$412$	0	0
$413$	−1.91407e13	−0.0783849
$414$	0	0
$415$	−2.91752e14	−1.16345
$416$	0	0
$417$	4.08185e14	1.58529
$418$	0	0
$419$	2.95630e14	1.11833	0.559166	−	0.829056i	$-0.311121\pi$
$419$	2.95630e14	1.11833	0.559166	+	0.829056i	$0.311121\pi$
$420$	0	0
$421$	7.24565e13	0.267009	0.133504	−	0.991048i	$-0.457377\pi$
$421$	7.24565e13	0.267009	0.133504	+	0.991048i	$0.457377\pi$
$422$	0	0
$423$	−5.25582e14	−1.88699
$424$	0	0
$425$	4.99269e14	1.74661
$426$	0	0
$427$	2.11307e14	0.720379
$428$	0	0
$429$	−4.87221e14	−1.61887
$430$	0	0
$431$	4.24283e14	1.37414	0.687069	−	0.726592i	$-0.258897\pi$
$431$	4.24283e14	1.37414	0.687069	+	0.726592i	$0.258897\pi$
$432$	0	0
$433$	1.62175e14	0.512038	0.256019	−	0.966672i	$-0.417589\pi$
$433$	1.62175e14	0.512038	0.256019	+	0.966672i	$0.417589\pi$
$434$	0	0
$435$	−1.62723e15	−5.00907
$436$	0	0
$437$	−1.99894e14	−0.600002
$438$	0	0
$439$	6.41896e14	1.87893	0.939464	−	0.342648i	$-0.111324\pi$
$439$	6.41896e14	1.87893	0.939464	+	0.342648i	$0.111324\pi$
$440$	0	0
$441$	−6.70229e14	−1.91342
$442$	0	0
$443$	−4.43845e14	−1.23598	−0.617989	−	0.786187i	$-0.712052\pi$
$443$	−4.43845e14	−1.23598	−0.617989	+	0.786187i	$0.712052\pi$
$444$	0	0
$445$	1.04020e15	2.82577
$446$	0	0
$447$	2.52323e14	0.668754
$448$	0	0
$449$	3.08017e14	0.796562	0.398281	−	0.917263i	$-0.369607\pi$
$449$	3.08017e14	0.796562	0.398281	+	0.917263i	$0.369607\pi$
$450$	0	0
$451$	−2.76294e14	−0.697268
$452$	0	0
$453$	−5.21304e14	−1.28396
$454$	0	0
$455$	−2.78844e14	−0.670345
$456$	0	0
$457$	−2.49484e14	−0.585469	−0.292734	−	0.956194i	$-0.594565\pi$
$457$	−2.49484e14	−0.585469	−0.292734	+	0.956194i	$0.594565\pi$
$458$	0	0
$459$	−1.09895e15	−2.51772
$460$	0	0
$461$	−2.25843e14	−0.505186	−0.252593	−	0.967573i	$-0.581283\pi$
$461$	−2.25843e14	−0.505186	−0.252593	+	0.967573i	$0.581283\pi$
$462$	0	0
$463$	4.88011e14	1.06594	0.532971	−	0.846133i	$-0.321075\pi$
$463$	4.88011e14	1.06594	0.532971	+	0.846133i	$0.321075\pi$
$464$	0	0
$465$	−2.41705e15	−5.15577
$466$	0	0
$467$	4.26504e14	0.888547	0.444274	−	0.895891i	$-0.353462\pi$
$467$	4.26504e14	0.888547	0.444274	+	0.895891i	$0.353462\pi$
$468$	0	0
$469$	−2.40367e14	−0.489130
$470$	0	0
$471$	1.50323e14	0.298820
$472$	0	0
$473$	−4.34580e14	−0.843983
$474$	0	0
$475$	8.11167e14	1.53920
$476$	0	0
$477$	−7.88238e14	−1.46153
$478$	0	0
$479$	−8.00213e14	−1.44997	−0.724987	−	0.688762i	$-0.758154\pi$
$479$	−8.00213e14	−1.44997	−0.724987	+	0.688762i	$0.758154\pi$
$480$	0	0
$481$	3.71450e14	0.657813
$482$	0	0
$483$	3.80001e14	0.657773
$484$	0	0
$485$	−1.61595e15	−2.73431
$486$	0	0
$487$	2.70351e14	0.447217	0.223608	−	0.974679i	$-0.428216\pi$
$487$	2.70351e14	0.447217	0.223608	+	0.974679i	$0.428216\pi$
$488$	0	0
$489$	−9.36382e14	−1.51445
$490$	0	0
$491$	−1.09128e15	−1.72578	−0.862892	−	0.505388i	$-0.831349\pi$
$491$	−1.09128e15	−1.72578	−0.862892	+	0.505388i	$0.831349\pi$
$492$	0	0
$493$	9.35700e14	1.44703
$494$	0	0
$495$	−2.92619e15	−4.42561
$496$	0	0
$497$	5.07616e14	0.750888
$498$	0	0
$499$	5.65887e14	0.818798	0.409399	−	0.912355i	$-0.365738\pi$
$499$	5.65887e14	0.818798	0.409399	+	0.912355i	$0.365738\pi$
$500$	0	0
$501$	−2.77426e14	−0.392681
$502$	0	0
$503$	3.29370e14	0.456100	0.228050	−	0.973649i	$-0.426765\pi$
$503$	3.29370e14	0.456100	0.228050	+	0.973649i	$0.426765\pi$
$504$	0	0
$505$	−1.39790e15	−1.89398
$506$	0	0
$507$	4.44000e14	0.588625
$508$	0	0
$509$	1.64942e14	0.213985	0.106992	−	0.994260i	$-0.465878\pi$
$509$	1.64942e14	0.213985	0.106992	+	0.994260i	$0.465878\pi$
$510$	0	0
$511$	1.25210e14	0.158973
$512$	0	0
$513$	−1.78547e15	−2.21874
$514$	0	0
$515$	2.19227e14	0.266658
$516$	0	0
$517$	6.73124e14	0.801488
$518$	0	0
$519$	1.89479e15	2.20871
$520$	0	0
$521$	2.94308e14	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$521$	2.94308e14	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$522$	0	0
$523$	6.57551e14	0.734802	0.367401	−	0.930063i	$-0.380248\pi$
$523$	6.57551e14	0.734802	0.367401	+	0.930063i	$0.380248\pi$
$524$	0	0
$525$	−1.54204e15	−1.68740
$526$	0	0
$527$	1.38987e15	1.48941
$528$	0	0
$529$	−4.27985e14	−0.449182
$530$	0	0
$531$	−3.96454e14	−0.407542
$532$	0	0
$533$	−5.45337e14	−0.549117
$534$	0	0
$535$	−1.28811e15	−1.27059
$536$	0	0
$537$	1.62531e15	1.57064
$538$	0	0
$539$	8.58377e14	0.812718
$540$	0	0
$541$	4.39506e14	0.407736	0.203868	−	0.978998i	$-0.434649\pi$
$541$	4.39506e14	0.407736	0.203868	+	0.978998i	$0.434649\pi$
$542$	0	0
$543$	6.88191e14	0.625619
$544$	0	0
$545$	−2.14724e13	−0.0191294
$546$	0	0
$547$	3.67457e14	0.320831	0.160416	−	0.987050i	$-0.448717\pi$
$547$	3.67457e14	0.320831	0.160416	+	0.987050i	$0.448717\pi$
$548$	0	0
$549$	4.37672e15	3.74543
$550$	0	0
$551$	1.52024e15	1.27520
$552$	0	0
$553$	−3.80979e14	−0.313265
$554$	0	0
$555$	3.13307e15	2.52557
$556$	0	0
$557$	−3.41815e13	−0.0270139	−0.0135070	−	0.999909i	$-0.504300\pi$
$557$	−3.41815e13	−0.0270139	−0.0135070	+	0.999909i	$0.504300\pi$
$558$	0	0
$559$	−8.57756e14	−0.664659
$560$	0	0
$561$	2.36312e15	1.79552
$562$	0	0
$563$	1.57295e15	1.17198	0.585988	−	0.810320i	$-0.300707\pi$
$563$	1.57295e15	1.17198	0.585988	+	0.810320i	$0.300707\pi$
$564$	0	0
$565$	1.79091e15	1.30860
$566$	0	0
$567$	1.75316e15	1.25636
$568$	0	0
$569$	−4.74133e14	−0.333260	−0.166630	−	0.986020i	$-0.553289\pi$
$569$	−4.74133e14	−0.333260	−0.166630	+	0.986020i	$0.553289\pi$
$570$	0	0
$571$	−2.54823e14	−0.175687	−0.0878436	−	0.996134i	$-0.527998\pi$
$571$	−2.54823e14	−0.175687	−0.0878436	+	0.996134i	$0.527998\pi$
$572$	0	0
$573$	−2.15062e15	−1.45450
$574$	0	0
$575$	−2.12972e15	−1.41303
$576$	0	0
$577$	−1.70605e15	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$577$	−1.70605e15	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$578$	0	0
$579$	−4.23140e14	−0.270241
$580$	0	0
$581$	−5.18158e14	−0.324709
$582$	0	0
$583$	1.00951e15	0.620776
$584$	0	0
$585$	−5.77559e15	−3.48529
$586$	0	0
$587$	2.17525e15	1.28825	0.644123	−	0.764922i	$-0.277222\pi$
$587$	2.17525e15	1.28825	0.644123	+	0.764922i	$0.277222\pi$
$588$	0	0
$589$	2.25813e15	1.31255
$590$	0	0
$591$	5.35851e14	0.305712
$592$	0	0
$593$	−3.32178e15	−1.86025	−0.930123	−	0.367249i	$-0.880300\pi$
$593$	−3.32178e15	−1.86025	−0.930123	+	0.367249i	$0.880300\pi$
$594$	0	0
$595$	1.35245e15	0.743493
$596$	0	0
$597$	−3.65080e15	−1.97028
$598$	0	0
$599$	−3.39864e15	−1.80077	−0.900384	−	0.435096i	$-0.856715\pi$
$599$	−3.39864e15	−1.80077	−0.900384	+	0.435096i	$0.856715\pi$
$600$	0	0
$601$	2.62640e15	1.36632	0.683159	−	0.730269i	$-0.260605\pi$
$601$	2.62640e15	1.36632	0.683159	+	0.730269i	$0.260605\pi$
$602$	0	0
$603$	−4.97863e15	−2.54311
$604$	0	0
$605$	3.50235e14	0.175673
$606$	0	0
$607$	−3.25639e15	−1.60398	−0.801989	−	0.597339i	$-0.796225\pi$
$607$	−3.25639e15	−1.60398	−0.801989	+	0.597339i	$0.796225\pi$
$608$	0	0
$609$	−2.88999e15	−1.39798
$610$	0	0
$611$	1.32858e15	0.631193
$612$	0	0
$613$	3.86533e15	1.80366	0.901829	−	0.432092i	$-0.142224\pi$
$613$	3.86533e15	1.80366	0.901829	+	0.432092i	$0.142224\pi$
$614$	0	0
$615$	−4.59976e15	−2.10825
$616$	0	0
$617$	−2.76303e14	−0.124399	−0.0621994	−	0.998064i	$-0.519811\pi$
$617$	−2.76303e14	−0.124399	−0.0621994	+	0.998064i	$0.519811\pi$
$618$	0	0
$619$	−1.13832e15	−0.503463	−0.251731	−	0.967797i	$-0.581000\pi$
$619$	−1.13832e15	−0.503463	−0.251731	+	0.967797i	$0.581000\pi$
$620$	0	0
$621$	4.68777e15	2.03687
$622$	0	0
$623$	1.84742e15	0.788646
$624$	0	0
$625$	1.71891e15	0.720964
$626$	0	0
$627$	3.83938e15	1.58230
$628$	0	0
$629$	−1.80160e15	−0.729593
$630$	0	0
$631$	1.38413e15	0.550828	0.275414	−	0.961326i	$-0.411185\pi$
$631$	1.38413e15	0.550828	0.275414	+	0.961326i	$0.411185\pi$
$632$	0	0
$633$	−1.62276e15	−0.634651
$634$	0	0
$635$	4.87191e15	1.87259
$636$	0	0
$637$	1.69423e15	0.640037
$638$	0	0
$639$	1.05141e16	3.90405
$640$	0	0
$641$	1.75717e15	0.641351	0.320675	−	0.947189i	$-0.396090\pi$
$641$	1.75717e15	0.641351	0.320675	+	0.947189i	$0.396090\pi$
$642$	0	0
$643$	−2.92244e15	−1.04854	−0.524271	−	0.851552i	$-0.675662\pi$
$643$	−2.92244e15	−1.04854	−0.524271	+	0.851552i	$0.675662\pi$
$644$	0	0
$645$	−7.23492e15	−2.55185
$646$	0	0
$647$	−2.78639e15	−0.966203	−0.483102	−	0.875564i	$-0.660490\pi$
$647$	−2.78639e15	−0.966203	−0.483102	+	0.875564i	$0.660490\pi$
$648$	0	0
$649$	5.07746e14	0.173102
$650$	0	0
$651$	−4.29273e15	−1.43893
$652$	0	0
$653$	−4.07215e15	−1.34215	−0.671075	−	0.741390i	$-0.734167\pi$
$653$	−4.07215e15	−1.34215	−0.671075	+	0.741390i	$0.734167\pi$
$654$	0	0
$655$	3.36523e15	1.09065
$656$	0	0
$657$	2.59342e15	0.826539
$658$	0	0
$659$	−3.54311e15	−1.11049	−0.555246	−	0.831686i	$-0.687376\pi$
$659$	−3.54311e15	−1.11049	−0.555246	+	0.831686i	$0.687376\pi$
$660$	0	0
$661$	−3.95683e15	−1.21966	−0.609831	−	0.792531i	$-0.708763\pi$
$661$	−3.95683e15	−1.21966	−0.609831	+	0.792531i	$0.708763\pi$
$662$	0	0
$663$	4.66422e15	1.41402
$664$	0	0
$665$	2.19733e15	0.655204
$666$	0	0
$667$	−3.99140e15	−1.17067
$668$	0	0
$669$	−3.16726e15	−0.913777
$670$	0	0
$671$	−5.60536e15	−1.59085
$672$	0	0
$673$	−6.68109e14	−0.186537	−0.0932684	−	0.995641i	$-0.529731\pi$
$673$	−6.68109e14	−0.186537	−0.0932684	+	0.995641i	$0.529731\pi$
$674$	0	0
$675$	−1.90229e16	−5.22523
$676$	0	0
$677$	1.84247e15	0.497924	0.248962	−	0.968513i	$-0.419911\pi$
$677$	1.84247e15	0.497924	0.248962	+	0.968513i	$0.419911\pi$
$678$	0	0
$679$	−2.86996e15	−0.763120
$680$	0	0
$681$	8.68534e15	2.27236
$682$	0	0
$683$	3.44924e15	0.887993	0.443997	−	0.896028i	$-0.353560\pi$
$683$	3.44924e15	0.887993	0.443997	+	0.896028i	$0.353560\pi$
$684$	0	0
$685$	6.00178e13	0.0152048
$686$	0	0
$687$	1.08740e15	0.271099
$688$	0	0
$689$	1.99254e15	0.488878
$690$	0	0
$691$	6.31976e15	1.52606	0.763030	−	0.646363i	$-0.223711\pi$
$691$	6.31976e15	1.52606	0.763030	+	0.646363i	$0.223711\pi$
$692$	0	0
$693$	−5.19698e15	−1.23515
$694$	0	0
$695$	−6.19700e15	−1.44966
$696$	0	0
$697$	2.64499e15	0.609036
$698$	0	0
$699$	−1.38101e16	−3.13020
$700$	0	0
$701$	−4.54606e15	−1.01435	−0.507173	−	0.861844i	$-0.669310\pi$
$701$	−4.54606e15	−1.01435	−0.507173	+	0.861844i	$0.669310\pi$
$702$	0	0
$703$	−2.92708e15	−0.642955
$704$	0	0
$705$	1.12062e16	2.42337
$706$	0	0
$707$	−2.48271e15	−0.528591
$708$	0	0
$709$	5.71404e15	1.19781	0.598906	−	0.800819i	$-0.295602\pi$
$709$	5.71404e15	1.19781	0.598906	+	0.800819i	$0.295602\pi$
$710$	0	0
$711$	−7.89106e15	−1.62874
$712$	0	0
$713$	−5.92875e15	−1.20495
$714$	0	0
$715$	7.39692e15	1.48036
$716$	0	0
$717$	3.15612e14	0.0622012
$718$	0	0
$719$	3.36530e15	0.653153	0.326577	−	0.945171i	$-0.394105\pi$
$719$	3.36530e15	0.653153	0.326577	+	0.945171i	$0.394105\pi$
$720$	0	0
$721$	3.89353e14	0.0744217
$722$	0	0
$723$	5.39550e15	1.01571
$724$	0	0
$725$	1.61970e16	3.00314
$726$	0	0
$727$	5.08263e14	0.0928216	0.0464108	−	0.998922i	$-0.485222\pi$
$727$	5.08263e14	0.0928216	0.0464108	+	0.998922i	$0.485222\pi$
$728$	0	0
$729$	7.88123e15	1.41773
$730$	0	0
$731$	4.16028e15	0.737186
$732$	0	0
$733$	−8.76006e15	−1.52910	−0.764548	−	0.644566i	$-0.777038\pi$
$733$	−8.76006e15	−1.52910	−0.764548	+	0.644566i	$0.777038\pi$
$734$	0	0
$735$	1.42903e16	2.45732
$736$	0	0
$737$	6.37623e15	1.08017
$738$	0	0
$739$	2.63072e15	0.439066	0.219533	−	0.975605i	$-0.429547\pi$
$739$	2.63072e15	0.439066	0.219533	+	0.975605i	$0.429547\pi$
$740$	0	0
$741$	7.57801e15	1.24611
$742$	0	0
$743$	9.60892e15	1.55681	0.778406	−	0.627761i	$-0.216029\pi$
$743$	9.60892e15	1.55681	0.778406	+	0.627761i	$0.216029\pi$
$744$	0	0
$745$	−3.83073e15	−0.611536
$746$	0	0
$747$	−1.07324e16	−1.68824
$748$	0	0
$749$	−2.28771e15	−0.354611
$750$	0	0
$751$	1.83390e15	0.280128	0.140064	−	0.990142i	$-0.455269\pi$
$751$	1.83390e15	0.280128	0.140064	+	0.990142i	$0.455269\pi$
$752$	0	0
$753$	−1.96555e16	−2.95877
$754$	0	0
$755$	7.91436e15	1.17411
$756$	0	0
$757$	−3.59036e14	−0.0524942	−0.0262471	−	0.999655i	$-0.508356\pi$
$757$	−3.59036e14	−0.0524942	−0.0262471	+	0.999655i	$0.508356\pi$
$758$	0	0
$759$	−1.00803e16	−1.45260
$760$	0	0
$761$	6.83481e15	0.970757	0.485379	−	0.874304i	$-0.338682\pi$
$761$	6.83481e15	0.970757	0.485379	+	0.874304i	$0.338682\pi$
$762$	0	0
$763$	−3.81356e13	−0.00533882
$764$	0	0
$765$	2.80127e16	3.86560
$766$	0	0
$767$	1.00217e15	0.136322
$768$	0	0
$769$	4.00775e15	0.537410	0.268705	−	0.963223i	$-0.413404\pi$
$769$	4.00775e15	0.537410	0.268705	+	0.963223i	$0.413404\pi$
$770$	0	0
$771$	−2.05178e15	−0.271227
$772$	0	0
$773$	−6.76108e15	−0.881108	−0.440554	−	0.897726i	$-0.645218\pi$
$773$	−6.76108e15	−0.881108	−0.440554	+	0.897726i	$0.645218\pi$
$774$	0	0
$775$	2.40587e16	3.09110
$776$	0	0
$777$	5.56441e15	0.704862
$778$	0	0
$779$	4.29734e15	0.536715
$780$	0	0
$781$	−1.34656e16	−1.65823
$782$	0	0
$783$	−3.56515e16	−4.32901
$784$	0	0
$785$	−2.28218e15	−0.273254
$786$	0	0
$787$	1.39564e16	1.64783	0.823915	−	0.566714i	$-0.191785\pi$
$787$	1.39564e16	1.64783	0.823915	+	0.566714i	$0.191785\pi$
$788$	0	0
$789$	1.06422e15	0.123910
$790$	0	0
$791$	3.18070e15	0.365218
$792$	0	0
$793$	−1.10636e16	−1.25284
$794$	0	0
$795$	1.68065e16	1.87697
$796$	0	0
$797$	1.39940e16	1.54143	0.770713	−	0.637183i	$-0.219900\pi$
$797$	1.39940e16	1.54143	0.770713	+	0.637183i	$0.219900\pi$
$798$	0	0
$799$	−6.44389e15	−0.700069
$800$	0	0
$801$	3.82650e16	4.10036
$802$	0	0
$803$	−3.32145e15	−0.351069
$804$	0	0
$805$	−5.76912e15	−0.601495
$806$	0	0
$807$	−5.14021e15	−0.528660
$808$	0	0
$809$	7.03097e13	0.00713343	0.00356671	−	0.999994i	$-0.498865\pi$
$809$	7.03097e13	0.00713343	0.00356671	+	0.999994i	$0.498865\pi$
$810$	0	0
$811$	1.76449e16	1.76605	0.883026	−	0.469324i	$-0.155502\pi$
$811$	1.76449e16	1.76605	0.883026	+	0.469324i	$0.155502\pi$
$812$	0	0
$813$	1.88013e16	1.85648
$814$	0	0
$815$	1.42160e16	1.38487
$816$	0	0
$817$	6.75925e15	0.649647
$818$	0	0
$819$	−1.02576e16	−0.972711
$820$	0	0
$821$	5.86886e15	0.549118	0.274559	−	0.961570i	$-0.411468\pi$
$821$	5.86886e15	0.549118	0.274559	+	0.961570i	$0.411468\pi$
$822$	0	0
$823$	−9.75691e15	−0.900769	−0.450384	−	0.892835i	$-0.648713\pi$
$823$	−9.75691e15	−0.900769	−0.450384	+	0.892835i	$0.648713\pi$
$824$	0	0
$825$	4.09057e16	3.72638
$826$	0	0
$827$	4.32365e15	0.388661	0.194330	−	0.980936i	$-0.437747\pi$
$827$	4.32365e15	0.388661	0.194330	+	0.980936i	$0.437747\pi$
$828$	0	0
$829$	−1.50130e16	−1.33174	−0.665868	−	0.746069i	$-0.731939\pi$
$829$	−1.50130e16	−1.33174	−0.665868	+	0.746069i	$0.731939\pi$
$830$	0	0
$831$	−1.99271e16	−1.74437
$832$	0	0
$833$	−8.21733e15	−0.709877
$834$	0	0
$835$	4.21184e15	0.359084
$836$	0	0
$837$	−5.29560e16	−4.45580
$838$	0	0
$839$	−1.30045e16	−1.07995	−0.539974	−	0.841681i	$-0.681566\pi$
$839$	−1.30045e16	−1.07995	−0.539974	+	0.841681i	$0.681566\pi$
$840$	0	0
$841$	1.81550e16	1.48805
$842$	0	0
$843$	−7.14435e15	−0.577978
$844$	0	0
$845$	−6.74073e15	−0.538263
$846$	0	0
$847$	6.22026e14	0.0490287
$848$	0	0
$849$	2.80512e16	2.18252
$850$	0	0
$851$	7.68508e15	0.590250
$852$	0	0
$853$	1.16924e16	0.886510	0.443255	−	0.896395i	$-0.353824\pi$
$853$	1.16924e16	0.886510	0.443255	+	0.896395i	$0.353824\pi$
$854$	0	0
$855$	4.55125e16	3.40657
$856$	0	0
$857$	6.40223e15	0.473083	0.236541	−	0.971621i	$-0.423986\pi$
$857$	6.40223e15	0.473083	0.236541	+	0.971621i	$0.423986\pi$
$858$	0	0
$859$	1.33379e16	0.973030	0.486515	−	0.873672i	$-0.338268\pi$
$859$	1.33379e16	0.973030	0.486515	+	0.873672i	$0.338268\pi$
$860$	0	0
$861$	−8.16928e15	−0.588392
$862$	0	0
$863$	1.01919e16	0.724759	0.362380	−	0.932031i	$-0.381964\pi$
$863$	1.01919e16	0.724759	0.362380	+	0.932031i	$0.381964\pi$
$864$	0	0
$865$	−2.87663e16	−2.01974
$866$	0	0
$867$	4.25833e15	0.295212
$868$	0	0
$869$	1.01062e16	0.691801
$870$	0	0
$871$	1.25851e16	0.850664
$872$	0	0
$873$	−5.94445e16	−3.96765
$874$	0	0
$875$	1.11147e16	0.732577
$876$	0	0
$877$	1.14838e16	0.747460	0.373730	−	0.927537i	$-0.378079\pi$
$877$	1.14838e16	0.747460	0.373730	+	0.927537i	$0.378079\pi$
$878$	0	0
$879$	−6.46695e14	−0.0415682
$880$	0	0
$881$	−1.96603e15	−0.124802	−0.0624011	−	0.998051i	$-0.519876\pi$
$881$	−1.96603e15	−0.124802	−0.0624011	+	0.998051i	$0.519876\pi$
$882$	0	0
$883$	2.77835e16	1.74182	0.870910	−	0.491442i	$-0.163530\pi$
$883$	2.77835e16	1.74182	0.870910	+	0.491442i	$0.163530\pi$
$884$	0	0
$885$	8.45300e15	0.523387
$886$	0	0
$887$	2.98288e16	1.82413	0.912064	−	0.410048i	$-0.134488\pi$
$887$	2.98288e16	1.82413	0.912064	+	0.410048i	$0.134488\pi$
$888$	0	0
$889$	8.65263e15	0.522623
$890$	0	0
$891$	−4.65061e16	−2.77449
$892$	0	0
$893$	−1.04694e16	−0.616937
$894$	0	0
$895$	−2.46752e16	−1.43626
$896$	0	0
$897$	−1.98961e16	−1.14396
$898$	0	0
$899$	4.50894e16	2.56092
$900$	0	0
$901$	−9.66418e15	−0.542224
$902$	0	0
$903$	−1.28494e16	−0.712197
$904$	0	0
$905$	−1.04480e16	−0.572092
$906$	0	0
$907$	6.55702e15	0.354704	0.177352	−	0.984147i	$-0.443247\pi$
$907$	6.55702e15	0.354704	0.177352	+	0.984147i	$0.443247\pi$
$908$	0	0
$909$	−5.14234e16	−2.74827
$910$	0	0
$911$	−6.38239e14	−0.0337002	−0.0168501	−	0.999858i	$-0.505364\pi$
$911$	−6.38239e14	−0.0337002	−0.0168501	+	0.999858i	$0.505364\pi$
$912$	0	0
$913$	1.37452e16	0.717072
$914$	0	0
$915$	−9.33185e16	−4.81007
$916$	0	0
$917$	5.97673e15	0.304391
$918$	0	0
$919$	7.76358e15	0.390685	0.195343	−	0.980735i	$-0.437418\pi$
$919$	7.76358e15	0.390685	0.195343	+	0.980735i	$0.437418\pi$
$920$	0	0
$921$	1.64405e16	0.817500
$922$	0	0
$923$	−2.65778e16	−1.30590
$924$	0	0
$925$	−3.11859e16	−1.51418
$926$	0	0
$927$	8.06451e15	0.386936
$928$	0	0
$929$	−1.74081e16	−0.825399	−0.412699	−	0.910867i	$-0.635414\pi$
$929$	−1.74081e16	−0.825399	−0.412699	+	0.910867i	$0.635414\pi$
$930$	0	0
$931$	−1.33508e16	−0.625581
$932$	0	0
$933$	6.13763e16	2.84218
$934$	0	0
$935$	−3.58765e16	−1.64190
$936$	0	0
$937$	−2.93626e16	−1.32809	−0.664043	−	0.747694i	$-0.731161\pi$
$937$	−2.93626e16	−1.32809	−0.664043	+	0.747694i	$0.731161\pi$
$938$	0	0
$939$	−2.72468e16	−1.21802
$940$	0	0
$941$	8.19312e15	0.361998	0.180999	−	0.983483i	$-0.442067\pi$
$941$	8.19312e15	0.361998	0.180999	+	0.983483i	$0.442067\pi$
$942$	0	0
$943$	−1.12827e16	−0.492718
$944$	0	0
$945$	−5.15302e16	−2.22427
$946$	0	0
$947$	−2.85510e13	−0.00121814	−0.000609070	−	1.00000i	$-0.500194\pi$
$947$	−2.85510e13	−0.00121814	−0.000609070		1.00000i	$0.500194\pi$
$948$	0	0
$949$	−6.55574e15	−0.276476
$950$	0	0
$951$	−7.17996e16	−2.99315
$952$	0	0
$953$	1.18686e16	0.489090	0.244545	−	0.969638i	$-0.421361\pi$
$953$	1.18686e16	0.489090	0.244545	+	0.969638i	$0.421361\pi$
$954$	0	0
$955$	3.26504e16	1.33006
$956$	0	0
$957$	7.66631e16	3.08724
$958$	0	0
$959$	1.06593e14	0.00424353
$960$	0	0
$961$	4.15664e16	1.63593
$962$	0	0
$963$	−4.73845e16	−1.84371
$964$	0	0
$965$	6.42404e15	0.247120
$966$	0	0
$967$	1.82443e16	0.693877	0.346939	−	0.937888i	$-0.387221\pi$
$967$	1.82443e16	0.693877	0.346939	+	0.937888i	$0.387221\pi$
$968$	0	0
$969$	−3.67548e16	−1.38208
$970$	0	0
$971$	−4.17627e16	−1.55268	−0.776342	−	0.630312i	$-0.782927\pi$
$971$	−4.17627e16	−1.55268	−0.776342	+	0.630312i	$0.782927\pi$
$972$	0	0
$973$	−1.10060e16	−0.404585
$974$	0	0
$975$	8.07380e16	2.93462
$976$	0	0
$977$	3.92505e16	1.41067	0.705335	−	0.708874i	$-0.250797\pi$
$977$	3.92505e16	1.41067	0.705335	+	0.708874i	$0.250797\pi$
$978$	0	0
$979$	−4.90068e16	−1.74161
$980$	0	0
$981$	−7.89887e14	−0.0277578
$982$	0	0
$983$	−4.75481e16	−1.65230	−0.826150	−	0.563451i	$-0.809473\pi$
$983$	−4.75481e16	−1.65230	−0.826150	+	0.563451i	$0.809473\pi$
$984$	0	0
$985$	−8.13519e15	−0.279556
$986$	0	0
$987$	1.99025e16	0.676338
$988$	0	0
$989$	−1.77465e16	−0.596393
$990$	0	0
$991$	−2.53768e16	−0.843396	−0.421698	−	0.906736i	$-0.638566\pi$
$991$	−2.53768e16	−0.843396	−0.421698	+	0.906736i	$0.638566\pi$
$992$	0	0
$993$	−2.75474e16	−0.905440
$994$	0	0
$995$	5.54258e16	1.80171
$996$	0	0
$997$	−4.92972e16	−1.58489	−0.792444	−	0.609944i	$-0.791192\pi$
$997$	−4.92972e16	−1.58489	−0.792444	+	0.609944i	$0.791192\pi$
$998$	0	0
$999$	6.86437e16	2.18268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.12.a.q.1.2		12
4.3	odd	2	inner	256.12.a.q.1.12		12
8.3	odd	2	inner	256.12.a.q.1.1		12
8.5	even	2	inner	256.12.a.q.1.11		12
16.3	odd	4		128.12.b.e.65.11	yes	12
16.5	even	4		128.12.b.e.65.12	yes	12
16.11	odd	4		128.12.b.e.65.2	yes	12
16.13	even	4		128.12.b.e.65.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.b.e.65.1	✓	12	16.13	even	4
128.12.b.e.65.2	yes	12	16.11	odd	4
128.12.b.e.65.11	yes	12	16.3	odd	4
128.12.b.e.65.12	yes	12	16.5	even	4
256.12.a.q.1.1		12	8.3	odd	2	inner
256.12.a.q.1.2		12	1.1	even	1	trivial
256.12.a.q.1.11		12	8.5	even	2	inner
256.12.a.q.1.12		12	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q-784.336 q^{3} +11907.7 q^{5} +21148.3 q^{7} +438037. q^{9} -561003. q^{11} -1.10728e6 q^{13} -9.33962e6 q^{15} +5.37054e6 q^{17} +8.72557e6 q^{19} -1.65874e7 q^{21} -2.29090e7 q^{23} +9.29644e7 q^{25} -2.04625e8 q^{27} +1.74228e8 q^{29} +2.58795e8 q^{31} +4.40015e8 q^{33} +2.51827e8 q^{35} -3.35460e8 q^{37} +8.68483e8 q^{39} +4.92499e8 q^{41} +7.74649e8 q^{43} +5.21599e9 q^{45} -1.19986e9 q^{47} -1.53008e9 q^{49} -4.21231e9 q^{51} -1.79948e9 q^{53} -6.68023e9 q^{55} -6.84378e9 q^{57} -9.05069e8 q^{59} +9.99168e9 q^{61} +9.26374e9 q^{63} -1.31852e10 q^{65} -1.13658e10 q^{67} +1.79684e10 q^{69} +2.40027e10 q^{71} +5.92056e9 q^{73} -7.29153e10 q^{75} -1.18643e10 q^{77} -1.80146e10 q^{79} +8.28982e10 q^{81} -2.45012e10 q^{83} +6.39506e10 q^{85} -1.36654e11 q^{87} +8.73556e10 q^{89} -2.34172e10 q^{91} -2.02982e11 q^{93} +1.03901e11 q^{95} -1.35707e11 q^{97} -2.45740e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 1152028 q^{9} + 26470216 q^{17} + 207502500 q^{25} + 935847712 q^{33} + 2031735176 q^{41} - 2342764308 q^{49} - 23753487776 q^{57} - 39461105920 q^{65} + 43591631448 q^{73} + 232401225292 q^{81}+ \cdots - 285441034488 q^{97}+O(q^{100}) \) 12 * q + 1152028 * q^9 + 26470216 * q^17 + 207502500 * q^25 + 935847712 * q^33 + 2031735176 * q^41 - 2342764308 * q^49 - 23753487776 * q^57 - 39461105920 * q^65 + 43591631448 * q^73 + 232401225292 * q^81 + 209389352088 * q^89 - 285441034488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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