LMFDB - Embedded newform 882.3.c.a.685.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.c (of order $2$, degree $1$, 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:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.2
Root			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.a.685.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +1.21371i q^{5} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +1.21371i q^{5} -2.82843 q^{8} -1.71644i q^{10} -6.34315 q^{11} +8.15640i q^{13} +4.00000 q^{16} +6.88830i q^{17} -30.4608i q^{19} +2.42742i q^{20} +8.97056 q^{22} -36.9706 q^{23} +23.5269 q^{25} -11.5349i q^{26} +34.3848 q^{29} -17.8435i q^{31} -5.65685 q^{32} -9.74153i q^{34} -36.2426 q^{37} +43.0781i q^{38} -3.43289i q^{40} -14.0936i q^{41} -32.6863 q^{43} -12.6863 q^{44} +52.2843 q^{46} -36.4299i q^{47} -33.2721 q^{50} +16.3128i q^{52} +46.6274 q^{53} -7.69873i q^{55} -48.6274 q^{58} -1.94721i q^{59} -78.4800i q^{61} +25.2346i q^{62} +8.00000 q^{64} -9.89949 q^{65} +30.0589 q^{67} +13.7766i q^{68} +48.5685 q^{71} +70.5769i q^{73} +51.2548 q^{74} -60.9217i q^{76} -76.2843 q^{79} +4.85483i q^{80} +19.9314i q^{82} -30.9861i q^{83} -8.36039 q^{85} +46.2254 q^{86} +17.9411 q^{88} -66.0166i q^{89} -73.9411 q^{92} +51.5197i q^{94} +36.9706 q^{95} -155.791i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} - 48 q^{11} + 16 q^{16} - 32 q^{22} - 80 q^{23} - 36 q^{25} + 64 q^{29} - 128 q^{37} - 176 q^{43} - 96 q^{44} + 96 q^{46} - 184 q^{50} + 96 q^{53} - 104 q^{58} + 32 q^{64} + 256 q^{67} - 32 q^{71} + 24 q^{74} - 192 q^{79} - 288 q^{85} - 64 q^{86} - 64 q^{88} - 160 q^{92} + 80 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 - 48 * q^11 + 16 * q^16 - 32 * q^22 - 80 * q^23 - 36 * q^25 + 64 * q^29 - 128 * q^37 - 176 * q^43 - 96 * q^44 + 96 * q^46 - 184 * q^50 + 96 * q^53 - 104 * q^58 + 32 * q^64 + 256 * q^67 - 32 * q^71 + 24 * q^74 - 192 * q^79 - 288 * q^85 - 64 * q^86 - 64 * q^88 - 160 * q^92 + 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	1.21371i	0.242742i	0.992607	+	0.121371i	$0.0387290\pi$
$5$	1.21371i	0.242742i	−0.992607	+	0.121371i	$0.961271\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	0	0
$10$	− 1.71644i	− 0.171644i
$11$	−6.34315	−0.576650	−0.288325	−	0.957533i	$-0.593098\pi$
$11$	−6.34315	−0.576650	−0.288325	+	0.957533i	$0.593098\pi$
$12$	0	0
$13$	8.15640i	0.627416i	0.949520	+	0.313708i	$0.101571\pi$
$13$	8.15640i	0.627416i	−0.949520	+	0.313708i	$0.898429\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	6.88830i	0.405194i	0.979262	+	0.202597i	$0.0649382\pi$
$17$	6.88830i	0.405194i	−0.979262	+	0.202597i	$0.935062\pi$
$18$	0	0
$19$	− 30.4608i	− 1.60320i	−0.597860	−	0.801601i	$-0.703982\pi$
$19$	− 30.4608i	− 1.60320i	0.597860	−	0.801601i	$-0.296018\pi$
$20$	2.42742i	0.121371i
$21$	0	0
$22$	8.97056	0.407753
$23$	−36.9706	−1.60742	−0.803708	−	0.595024i	$-0.797143\pi$
$23$	−36.9706	−1.60742	−0.803708	+	0.595024i	$0.797143\pi$
$24$	0	0
$25$	23.5269	0.941076
$26$	− 11.5349i	− 0.443650i
$27$	0	0
$28$	0	0
$29$	34.3848	1.18568	0.592841	−	0.805320i	$-0.298006\pi$
$29$	34.3848	1.18568	0.592841	+	0.805320i	$0.298006\pi$
$30$	0	0
$31$	− 17.8435i	− 0.575598i	−0.957691	−	0.287799i	$-0.907076\pi$
$31$	− 17.8435i	− 0.575598i	0.957691	−	0.287799i	$-0.0929235\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	− 9.74153i	− 0.286516i
$35$	0	0
$36$	0	0
$37$	−36.2426	−0.979531	−0.489765	−	0.871854i	$-0.662918\pi$
$37$	−36.2426	−0.979531	−0.489765	+	0.871854i	$0.662918\pi$
$38$	43.0781i	1.13363i
$39$	0	0
$40$	− 3.43289i	− 0.0858221i
$41$	− 14.0936i	− 0.343747i	−0.985119	−	0.171874i	$-0.945018\pi$
$41$	− 14.0936i	− 0.343747i	0.985119	−	0.171874i	$-0.0549820\pi$
$42$	0	0
$43$	−32.6863	−0.760146	−0.380073	−	0.924956i	$-0.624101\pi$
$43$	−32.6863	−0.760146	−0.380073	+	0.924956i	$0.624101\pi$
$44$	−12.6863	−0.288325
$45$	0	0
$46$	52.2843	1.13661
$47$	− 36.4299i	− 0.775105i	−0.921848	−	0.387552i	$-0.873321\pi$
$47$	− 36.4299i	− 0.775105i	0.921848	−	0.387552i	$-0.126679\pi$
$48$	0	0
$49$	0	0
$50$	−33.2721	−0.665442
$51$	0	0
$52$	16.3128i	0.313708i
$53$	46.6274	0.879763	0.439881	−	0.898056i	$-0.355021\pi$
$53$	46.6274	0.879763	0.439881	+	0.898056i	$0.355021\pi$
$54$	0	0
$55$	− 7.69873i	− 0.139977i
$56$	0	0
$57$	0	0
$58$	−48.6274	−0.838404
$59$	− 1.94721i	− 0.0330036i	−0.999864	−	0.0165018i	$-0.994747\pi$
$59$	− 1.94721i	− 0.0330036i	0.999864	−	0.0165018i	$-0.00525293\pi$
$60$	0	0
$61$	− 78.4800i	− 1.28656i	−0.765632	−	0.643279i	$-0.777574\pi$
$61$	− 78.4800i	− 1.28656i	0.765632	−	0.643279i	$-0.222426\pi$
$62$	25.2346i	0.407009i
$63$	0	0
$64$	8.00000	0.125000
$65$	−9.89949	−0.152300
$66$	0	0
$67$	30.0589	0.448640	0.224320	−	0.974516i	$-0.427984\pi$
$67$	30.0589	0.448640	0.224320	+	0.974516i	$0.427984\pi$
$68$	13.7766i	0.202597i
$69$	0	0
$70$	0	0
$71$	48.5685	0.684064	0.342032	−	0.939688i	$-0.388885\pi$
$71$	48.5685	0.684064	0.342032	+	0.939688i	$0.388885\pi$
$72$	0	0
$73$	70.5769i	0.966807i	0.875398	+	0.483404i	$0.160600\pi$
$73$	70.5769i	0.966807i	−0.875398	+	0.483404i	$0.839400\pi$
$74$	51.2548	0.692633
$75$	0	0
$76$	− 60.9217i	− 0.801601i
$77$	0	0
$78$	0	0
$79$	−76.2843	−0.965624	−0.482812	−	0.875724i	$-0.660384\pi$
$79$	−76.2843	−0.965624	−0.482812	+	0.875724i	$0.660384\pi$
$80$	4.85483i	0.0606854i
$81$	0	0
$82$	19.9314i	0.243066i
$83$	− 30.9861i	− 0.373326i	−0.982424	−	0.186663i	$-0.940233\pi$
$83$	− 30.9861i	− 0.373326i	0.982424	−	0.186663i	$-0.0597673\pi$
$84$	0	0
$85$	−8.36039	−0.0983575
$86$	46.2254	0.537505
$87$	0	0
$88$	17.9411	0.203876
$89$	− 66.0166i	− 0.741759i	−0.928681	−	0.370880i	$-0.879056\pi$
$89$	− 66.0166i	− 0.741759i	0.928681	−	0.370880i	$-0.120944\pi$
$90$	0	0
$91$	0	0
$92$	−73.9411	−0.803708
$93$	0	0
$94$	51.5197i	0.548082i
$95$	36.9706	0.389164
$96$	0	0
$97$	− 155.791i	− 1.60610i	−0.595914	−	0.803049i	$-0.703210\pi$
$97$	− 155.791i	− 1.60610i	0.595914	−	0.803049i	$-0.296790\pi$
$98$	0	0
$99$	0	0
$100$	47.0538	0.470538
$101$	− 187.027i	− 1.85175i	−0.377828	−	0.925876i	$-0.623329\pi$
$101$	− 187.027i	− 1.85175i	0.377828	−	0.925876i	$-0.376671\pi$
$102$	0	0
$103$	− 141.173i	− 1.37061i	−0.728258	−	0.685304i	$-0.759670\pi$
$103$	− 141.173i	− 1.37061i	0.728258	−	0.685304i	$-0.240330\pi$
$104$	− 23.0698i	− 0.221825i
$105$	0	0
$106$	−65.9411	−0.622086
$107$	−67.4802	−0.630656	−0.315328	−	0.948983i	$-0.602115\pi$
$107$	−67.4802	−0.630656	−0.315328	+	0.948983i	$0.602115\pi$
$108$	0	0
$109$	−93.6152	−0.858855	−0.429428	−	0.903101i	$-0.641285\pi$
$109$	−93.6152	−0.858855	−0.429428	+	0.903101i	$0.641285\pi$
$110$	10.8876i	0.0989786i
$111$	0	0
$112$	0	0
$113$	−65.9411	−0.583550	−0.291775	−	0.956487i	$-0.594246\pi$
$113$	−65.9411	−0.583550	−0.291775	+	0.956487i	$0.594246\pi$
$114$	0	0
$115$	− 44.8715i	− 0.390187i
$116$	68.7696	0.592841
$117$	0	0
$118$	2.75377i	0.0233371i
$119$	0	0
$120$	0	0
$121$	−80.7645	−0.667475
$122$	110.988i	0.909734i
$123$	0	0
$124$	− 35.6871i	− 0.287799i
$125$	58.8975i	0.471180i
$126$	0	0
$127$	33.6569	0.265015	0.132507	−	0.991182i	$-0.457697\pi$
$127$	33.6569	0.265015	0.132507	+	0.991182i	$0.457697\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	14.0000	0.107692
$131$	− 119.153i	− 0.909567i	−0.890602	−	0.454783i	$-0.849717\pi$
$131$	− 119.153i	− 0.909567i	0.890602	−	0.454783i	$-0.150283\pi$
$132$	0	0
$133$	0	0
$134$	−42.5097	−0.317236
$135$	0	0
$136$	− 19.4831i	− 0.143258i
$137$	−107.924	−0.787766	−0.393883	−	0.919161i	$-0.628868\pi$
$137$	−107.924	−0.787766	−0.393883	+	0.919161i	$0.628868\pi$
$138$	0	0
$139$	− 45.4605i	− 0.327054i	−0.986539	−	0.163527i	$-0.947713\pi$
$139$	− 45.4605i	− 0.327054i	0.986539	−	0.163527i	$-0.0522870\pi$
$140$	0	0
$141$	0	0
$142$	−68.6863	−0.483706
$143$	− 51.7373i	− 0.361799i
$144$	0	0
$145$	41.7331i	0.287814i
$146$	− 99.8109i	− 0.683636i
$147$	0	0
$148$	−72.4853	−0.489765
$149$	184.000	1.23490	0.617450	−	0.786610i	$-0.288166\pi$
$149$	184.000	1.23490	0.617450	+	0.786610i	$0.288166\pi$
$150$	0	0
$151$	159.716	1.05772	0.528860	−	0.848709i	$-0.322620\pi$
$151$	159.716	1.05772	0.528860	+	0.848709i	$0.322620\pi$
$152$	86.1562i	0.566817i
$153$	0	0
$154$	0	0
$155$	21.6569	0.139722
$156$	0	0
$157$	− 168.998i	− 1.07642i	−0.842811	−	0.538209i	$-0.819101\pi$
$157$	− 168.998i	− 1.07642i	0.842811	−	0.538209i	$-0.180899\pi$
$158$	107.882	0.682799
$159$	0	0
$160$	− 6.86577i	− 0.0429111i
$161$	0	0
$162$	0	0
$163$	135.078	0.828701	0.414350	−	0.910117i	$-0.364009\pi$
$163$	135.078	0.828701	0.414350	+	0.910117i	$0.364009\pi$
$164$	− 28.1873i	− 0.171874i
$165$	0	0
$166$	43.8210i	0.263982i
$167$	− 238.614i	− 1.42883i	−0.699723	−	0.714414i	$-0.746693\pi$
$167$	− 238.614i	− 1.42883i	0.699723	−	0.714414i	$-0.253307\pi$
$168$	0	0
$169$	102.473	0.606350
$170$	11.8234	0.0695493
$171$	0	0
$172$	−65.3726	−0.380073
$173$	− 58.2409i	− 0.336653i	−0.985731	−	0.168326i	$-0.946164\pi$
$173$	− 58.2409i	− 0.336653i	0.985731	−	0.168326i	$-0.0538363\pi$
$174$	0	0
$175$	0	0
$176$	−25.3726	−0.144162
$177$	0	0
$178$	93.3616i	0.524503i
$179$	−321.872	−1.79817	−0.899084	−	0.437776i	$-0.855766\pi$
$179$	−321.872	−1.79817	−0.899084	+	0.437776i	$0.855766\pi$
$180$	0	0
$181$	113.891i	0.629234i	0.949219	+	0.314617i	$0.101876\pi$
$181$	113.891i	0.629234i	−0.949219	+	0.314617i	$0.898124\pi$
$182$	0	0
$183$	0	0
$184$	104.569	0.568307
$185$	− 43.9880i	− 0.237773i
$186$	0	0
$187$	− 43.6935i	− 0.233655i
$188$	− 72.8598i	− 0.387552i
$189$	0	0
$190$	−52.2843	−0.275180
$191$	−89.2061	−0.467047	−0.233524	−	0.972351i	$-0.575026\pi$
$191$	−89.2061	−0.467047	−0.233524	+	0.972351i	$0.575026\pi$
$192$	0	0
$193$	−140.451	−0.727724	−0.363862	−	0.931453i	$-0.618542\pi$
$193$	−140.451	−0.727724	−0.363862	+	0.931453i	$0.618542\pi$
$194$	220.322i	1.13568i
$195$	0	0
$196$	0	0
$197$	140.510	0.713247	0.356624	−	0.934248i	$-0.383928\pi$
$197$	140.510	0.713247	0.356624	+	0.934248i	$0.383928\pi$
$198$	0	0
$199$	− 222.872i	− 1.11996i	−0.828507	−	0.559979i	$-0.810809\pi$
$199$	− 222.872i	− 1.11996i	0.828507	−	0.559979i	$-0.189191\pi$
$200$	−66.5442	−0.332721
$201$	0	0
$202$	264.496i	1.30939i
$203$	0	0
$204$	0	0
$205$	17.1056	0.0834417
$206$	199.648i	0.969166i
$207$	0	0
$208$	32.6256i	0.156854i
$209$	193.217i	0.924486i
$210$	0	0
$211$	20.2843	0.0961340	0.0480670	−	0.998844i	$-0.484694\pi$
$211$	20.2843	0.0961340	0.0480670	+	0.998844i	$0.484694\pi$
$212$	93.2548	0.439881
$213$	0	0
$214$	95.4315	0.445941
$215$	− 39.6716i	− 0.184519i
$216$	0	0
$217$	0	0
$218$	132.392	0.607302
$219$	0	0
$220$	− 15.3975i	− 0.0699884i
$221$	−56.1838	−0.254225
$222$	0	0
$223$	140.737i	0.631109i	0.948907	+	0.315555i	$0.102191\pi$
$223$	140.737i	0.631109i	−0.948907	+	0.315555i	$0.897809\pi$
$224$	0	0
$225$	0	0
$226$	93.2548	0.412632
$227$	10.2085i	0.0449715i	0.999747	+	0.0224858i	$0.00715804\pi$
$227$	10.2085i	0.0449715i	−0.999747	+	0.0224858i	$0.992842\pi$
$228$	0	0
$229$	431.995i	1.88644i	0.332168	+	0.943220i	$0.392220\pi$
$229$	431.995i	1.88644i	−0.332168	+	0.943220i	$0.607780\pi$
$230$	63.4579i	0.275904i
$231$	0	0
$232$	−97.2548	−0.419202
$233$	−144.610	−0.620645	−0.310322	−	0.950631i	$-0.600437\pi$
$233$	−144.610	−0.620645	−0.310322	+	0.950631i	$0.600437\pi$
$234$	0	0
$235$	44.2153	0.188150
$236$	− 3.89443i	− 0.0165018i
$237$	0	0
$238$	0	0
$239$	−213.470	−0.893180	−0.446590	−	0.894739i	$-0.647362\pi$
$239$	−213.470	−0.893180	−0.446590	+	0.894739i	$0.647362\pi$
$240$	0	0
$241$	65.4012i	0.271374i	0.990752	+	0.135687i	$0.0433242\pi$
$241$	65.4012i	0.271374i	−0.990752	+	0.135687i	$0.956676\pi$
$242$	114.218	0.471976
$243$	0	0
$244$	− 156.960i	− 0.643279i
$245$	0	0
$246$	0	0
$247$	248.451	1.00587
$248$	50.4692i	0.203505i
$249$	0	0
$250$	− 83.2937i	− 0.333175i
$251$	74.8070i	0.298036i	0.988834	+	0.149018i	$0.0476112\pi$
$251$	74.8070i	0.298036i	−0.988834	+	0.149018i	$0.952389\pi$
$252$	0	0
$253$	234.510	0.926916
$254$	−47.5980	−0.187394
$255$	0	0
$256$	16.0000	0.0625000
$257$	429.222i	1.67013i	0.550154	+	0.835063i	$0.314569\pi$
$257$	429.222i	1.67013i	−0.550154	+	0.835063i	$0.685431\pi$
$258$	0	0
$259$	0	0
$260$	−19.7990	−0.0761500
$261$	0	0
$262$	168.508i	0.643161i
$263$	−113.539	−0.431708	−0.215854	−	0.976426i	$-0.569253\pi$
$263$	−113.539	−0.431708	−0.215854	+	0.976426i	$0.569253\pi$
$264$	0	0
$265$	56.5921i	0.213555i
$266$	0	0
$267$	0	0
$268$	60.1177	0.224320
$269$	− 11.2629i	− 0.0418696i	−0.999781	−	0.0209348i	$-0.993336\pi$
$269$	− 11.2629i	− 0.0418696i	0.999781	−	0.0209348i	$-0.00666425\pi$
$270$	0	0
$271$	283.358i	1.04560i	0.852455	+	0.522801i	$0.175113\pi$
$271$	283.358i	1.04560i	−0.852455	+	0.522801i	$0.824887\pi$
$272$	27.5532i	0.101299i
$273$	0	0
$274$	152.627	0.557034
$275$	−149.235	−0.542671
$276$	0	0
$277$	−217.765	−0.786153	−0.393077	−	0.919506i	$-0.628589\pi$
$277$	−217.765	−0.786153	−0.393077	+	0.919506i	$0.628589\pi$
$278$	64.2908i	0.231262i
$279$	0	0
$280$	0	0
$281$	317.782	1.13090	0.565448	−	0.824784i	$-0.308703\pi$
$281$	317.782	1.13090	0.565448	+	0.824784i	$0.308703\pi$
$282$	0	0
$283$	240.164i	0.848635i	0.905514	+	0.424317i	$0.139486\pi$
$283$	240.164i	0.848635i	−0.905514	+	0.424317i	$0.860514\pi$
$284$	97.1371	0.342032
$285$	0	0
$286$	73.1675i	0.255831i
$287$	0	0
$288$	0	0
$289$	241.551	0.835818
$290$	− 59.0195i	− 0.203516i
$291$	0	0
$292$	141.154i	0.483404i
$293$	437.887i	1.49450i	0.664546	+	0.747248i	$0.268625\pi$
$293$	437.887i	1.49450i	−0.664546	+	0.747248i	$0.731375\pi$
$294$	0	0
$295$	2.36335	0.00801135
$296$	102.510	0.346316
$297$	0	0
$298$	−260.215	−0.873206
$299$	− 301.547i	− 1.00852i
$300$	0	0
$301$	0	0
$302$	−225.872	−0.747921
$303$	0	0
$304$	− 121.843i	− 0.400800i
$305$	95.2519	0.312301
$306$	0	0
$307$	476.947i	1.55357i	0.629764	+	0.776787i	$0.283152\pi$
$307$	476.947i	1.55357i	−0.629764	+	0.776787i	$0.716848\pi$
$308$	0	0
$309$	0	0
$310$	−30.6274	−0.0987981
$311$	330.586i	1.06298i	0.847066	+	0.531488i	$0.178367\pi$
$311$	330.586i	1.06298i	−0.847066	+	0.531488i	$0.821633\pi$
$312$	0	0
$313$	− 162.421i	− 0.518917i	−0.965754	−	0.259458i	$-0.916456\pi$
$313$	− 162.421i	− 0.518917i	0.965754	−	0.259458i	$-0.0835440\pi$
$314$	238.999i	0.761143i
$315$	0	0
$316$	−152.569	−0.482812
$317$	434.548	1.37081	0.685407	−	0.728160i	$-0.259624\pi$
$317$	434.548	1.37081	0.685407	+	0.728160i	$0.259624\pi$
$318$	0	0
$319$	−218.108	−0.683723
$320$	9.70967i	0.0303427i
$321$	0	0
$322$	0	0
$323$	209.823	0.649608
$324$	0	0
$325$	191.895i	0.590446i
$326$	−191.029	−0.585980
$327$	0	0
$328$	39.8628i	0.121533i
$329$	0	0
$330$	0	0
$331$	−460.666	−1.39174	−0.695870	−	0.718168i	$-0.744981\pi$
$331$	−460.666	−1.39174	−0.695870	+	0.718168i	$0.744981\pi$
$332$	− 61.9722i	− 0.186663i
$333$	0	0
$334$	337.451i	1.01033i
$335$	36.4827i	0.108904i
$336$	0	0
$337$	−61.8650	−0.183576	−0.0917878	−	0.995779i	$-0.529258\pi$
$337$	−61.8650	−0.183576	−0.0917878	+	0.995779i	$0.529258\pi$
$338$	−144.919	−0.428754
$339$	0	0
$340$	−16.7208	−0.0491788
$341$	113.184i	0.331918i
$342$	0	0
$343$	0	0
$344$	92.4508	0.268752
$345$	0	0
$346$	82.3651i	0.238049i
$347$	536.950	1.54741	0.773704	−	0.633548i	$-0.218402\pi$
$347$	536.950	1.54741	0.773704	+	0.633548i	$0.218402\pi$
$348$	0	0
$349$	− 86.8633i	− 0.248892i	−0.992226	−	0.124446i	$-0.960285\pi$
$349$	− 86.8633i	− 0.248892i	0.992226	−	0.124446i	$-0.0397154\pi$
$350$	0	0
$351$	0	0
$352$	35.8823	0.101938
$353$	− 575.197i	− 1.62945i	−0.579845	−	0.814727i	$-0.696887\pi$
$353$	− 575.197i	− 1.62945i	0.579845	−	0.814727i	$-0.303113\pi$
$354$	0	0
$355$	58.9480i	0.166051i
$356$	− 132.033i	− 0.370880i
$357$	0	0
$358$	455.196	1.27150
$359$	50.0589	0.139440	0.0697199	−	0.997567i	$-0.477789\pi$
$359$	50.0589	0.139440	0.0697199	+	0.997567i	$0.477789\pi$
$360$	0	0
$361$	−566.862	−1.57025
$362$	− 161.067i	− 0.444935i
$363$	0	0
$364$	0	0
$365$	−85.6598	−0.234684
$366$	0	0
$367$	− 141.788i	− 0.386343i	−0.981165	−	0.193172i	$-0.938123\pi$
$367$	− 141.788i	− 0.386343i	0.981165	−	0.193172i	$-0.0618774\pi$
$368$	−147.882	−0.401854
$369$	0	0
$370$	62.2084i	0.168131i
$371$	0	0
$372$	0	0
$373$	62.1564	0.166639	0.0833196	−	0.996523i	$-0.473448\pi$
$373$	62.1564	0.166639	0.0833196	+	0.996523i	$0.473448\pi$
$374$	61.7919i	0.165219i
$375$	0	0
$376$	103.039i	0.274041i
$377$	280.456i	0.743915i
$378$	0	0
$379$	103.245	0.272414	0.136207	−	0.990680i	$-0.456509\pi$
$379$	103.245	0.272414	0.136207	+	0.990680i	$0.456509\pi$
$380$	73.9411	0.194582
$381$	0	0
$382$	126.156	0.330252
$383$	355.077i	0.927095i	0.886072	+	0.463548i	$0.153424\pi$
$383$	355.077i	0.927095i	−0.886072	+	0.463548i	$0.846576\pi$
$384$	0	0
$385$	0	0
$386$	198.627	0.514579
$387$	0	0
$388$	− 311.583i	− 0.803049i
$389$	−404.909	−1.04090	−0.520448	−	0.853893i	$-0.674235\pi$
$389$	−404.909	−1.04090	−0.520448	+	0.853893i	$0.674235\pi$
$390$	0	0
$391$	− 254.664i	− 0.651316i
$392$	0	0
$393$	0	0
$394$	−198.711	−0.504342
$395$	− 92.5869i	− 0.234397i
$396$	0	0
$397$	540.926i	1.36254i	0.732035	+	0.681268i	$0.238571\pi$
$397$	540.926i	1.36254i	−0.732035	+	0.681268i	$0.761429\pi$
$398$	315.188i	0.791930i
$399$	0	0
$400$	94.1076	0.235269
$401$	467.688	1.16631	0.583153	−	0.812363i	$-0.301819\pi$
$401$	467.688	1.16631	0.583153	+	0.812363i	$0.301819\pi$
$402$	0	0
$403$	145.539	0.361139
$404$	− 374.054i	− 0.925876i
$405$	0	0
$406$	0	0
$407$	229.892	0.564846
$408$	0	0
$409$	− 96.2280i	− 0.235276i	−0.993057	−	0.117638i	$-0.962468\pi$
$409$	− 96.2280i	− 0.235276i	0.993057	−	0.117638i	$-0.0375323\pi$
$410$	−24.1909	−0.0590022
$411$	0	0
$412$	− 282.345i	− 0.685304i
$413$	0	0
$414$	0	0
$415$	37.6081	0.0906219
$416$	− 46.1396i	− 0.110912i
$417$	0	0
$418$	− 273.251i	− 0.653710i
$419$	− 433.126i	− 1.03371i	−0.856072	−	0.516857i	$-0.827102\pi$
$419$	− 433.126i	− 1.03371i	0.856072	−	0.516857i	$-0.172898\pi$
$420$	0	0
$421$	−516.627	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$421$	−516.627	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$422$	−28.6863	−0.0679770
$423$	0	0
$424$	−131.882	−0.311043
$425$	162.060i	0.381319i
$426$	0	0
$427$	0	0
$428$	−134.960	−0.315328
$429$	0	0
$430$	56.1042i	0.130475i
$431$	366.745	0.850917	0.425458	−	0.904978i	$-0.360113\pi$
$431$	366.745	0.850917	0.425458	+	0.904978i	$0.360113\pi$
$432$	0	0
$433$	449.480i	1.03806i	0.854756	+	0.519030i	$0.173707\pi$
$433$	449.480i	1.03806i	−0.854756	+	0.519030i	$0.826293\pi$
$434$	0	0
$435$	0	0
$436$	−187.230	−0.429428
$437$	1126.15i	2.57701i
$438$	0	0
$439$	− 661.824i	− 1.50757i	−0.657120	−	0.753786i	$-0.728226\pi$
$439$	− 661.824i	− 1.50757i	0.657120	−	0.753786i	$-0.271774\pi$
$440$	21.7753i	0.0494893i
$441$	0	0
$442$	79.4558	0.179764
$443$	−787.647	−1.77798	−0.888992	−	0.457923i	$-0.848594\pi$
$443$	−787.647	−1.77798	−0.888992	+	0.457923i	$0.848594\pi$
$444$	0	0
$445$	80.1249	0.180056
$446$	− 199.033i	− 0.446262i
$447$	0	0
$448$	0	0
$449$	−281.373	−0.626665	−0.313332	−	0.949643i	$-0.601445\pi$
$449$	−281.373	−0.626665	−0.313332	+	0.949643i	$0.601445\pi$
$450$	0	0
$451$	89.3979i	0.198222i
$452$	−131.882	−0.291775
$453$	0	0
$454$	− 14.4371i	− 0.0317997i
$455$	0	0
$456$	0	0
$457$	397.117	0.868965	0.434482	−	0.900680i	$-0.356931\pi$
$457$	397.117	0.868965	0.434482	+	0.900680i	$0.356931\pi$
$458$	− 610.933i	− 1.33391i
$459$	0	0
$460$	− 89.7430i	− 0.195093i
$461$	− 433.589i	− 0.940541i	−0.882522	−	0.470270i	$-0.844156\pi$
$461$	− 433.589i	− 0.940541i	0.882522	−	0.470270i	$-0.155844\pi$
$462$	0	0
$463$	−107.813	−0.232858	−0.116429	−	0.993199i	$-0.537145\pi$
$463$	−107.813	−0.232858	−0.116429	+	0.993199i	$0.537145\pi$
$464$	137.539	0.296420
$465$	0	0
$466$	204.510	0.438862
$467$	− 163.372i	− 0.349833i	−0.984583	−	0.174917i	$-0.944034\pi$
$467$	− 163.372i	− 0.349833i	0.984583	−	0.174917i	$-0.0559656\pi$
$468$	0	0
$469$	0	0
$470$	−62.5299	−0.133042
$471$	0	0
$472$	5.50755i	0.0116685i
$473$	207.334	0.438338
$474$	0	0
$475$	− 716.649i	− 1.50874i
$476$	0	0
$477$	0	0
$478$	301.892	0.631574
$479$	− 599.724i	− 1.25203i	−0.779809	−	0.626017i	$-0.784684\pi$
$479$	− 599.724i	− 1.25203i	0.779809	−	0.626017i	$-0.215316\pi$
$480$	0	0
$481$	− 295.610i	− 0.614573i
$482$	− 92.4913i	− 0.191891i
$483$	0	0
$484$	−161.529	−0.333738
$485$	189.085	0.389867
$486$	0	0
$487$	453.608	0.931433	0.465717	−	0.884934i	$-0.345797\pi$
$487$	453.608	0.931433	0.465717	+	0.884934i	$0.345797\pi$
$488$	221.975i	0.454867i
$489$	0	0
$490$	0	0
$491$	−507.578	−1.03376	−0.516882	−	0.856057i	$-0.672907\pi$
$491$	−507.578	−1.03376	−0.516882	+	0.856057i	$0.672907\pi$
$492$	0	0
$493$	236.853i	0.480431i
$494$	−351.362	−0.711260
$495$	0	0
$496$	− 71.3742i	− 0.143900i
$497$	0	0
$498$	0	0
$499$	303.598	0.608413	0.304206	−	0.952606i	$-0.401609\pi$
$499$	303.598	0.608413	0.304206	+	0.952606i	$0.401609\pi$
$500$	117.795i	0.235590i
$501$	0	0
$502$	− 105.793i	− 0.210743i
$503$	− 884.733i	− 1.75891i	−0.475979	−	0.879456i	$-0.657906\pi$
$503$	− 884.733i	− 1.75891i	0.475979	−	0.879456i	$-0.342094\pi$
$504$	0	0
$505$	226.996	0.449497
$506$	−331.647	−0.655428
$507$	0	0
$508$	67.3137	0.132507
$509$	646.067i	1.26929i	0.772805	+	0.634643i	$0.218853\pi$
$509$	646.067i	1.26929i	−0.772805	+	0.634643i	$0.781147\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	− 607.012i	− 1.18096i
$515$	171.342	0.332703
$516$	0	0
$517$	231.080i	0.446964i
$518$	0	0
$519$	0	0
$520$	28.0000	0.0538462
$521$	− 53.1872i	− 0.102087i	−0.998696	−	0.0510434i	$-0.983745\pi$
$521$	− 53.1872i	− 0.102087i	0.998696	−	0.0510434i	$-0.0162547\pi$
$522$	0	0
$523$	− 283.947i	− 0.542920i	−0.962450	−	0.271460i	$-0.912494\pi$
$523$	− 283.947i	− 0.542920i	0.962450	−	0.271460i	$-0.0875065\pi$
$524$	− 238.307i	− 0.454783i
$525$	0	0
$526$	160.569	0.305263
$527$	122.912	0.233229
$528$	0	0
$529$	837.823	1.58379
$530$	− 80.0333i	− 0.151006i
$531$	0	0
$532$	0	0
$533$	114.953	0.215672
$534$	0	0
$535$	− 81.9013i	− 0.153087i
$536$	−85.0193	−0.158618
$537$	0	0
$538$	15.9282i	0.0296063i
$539$	0	0
$540$	0	0
$541$	−678.392	−1.25396	−0.626980	−	0.779036i	$-0.715709\pi$
$541$	−678.392	−1.25396	−0.626980	+	0.779036i	$0.715709\pi$
$542$	− 400.729i	− 0.739353i
$543$	0	0
$544$	− 38.9661i	− 0.0716289i
$545$	− 113.622i	− 0.208480i
$546$	0	0
$547$	−517.470	−0.946015	−0.473007	−	0.881058i	$-0.656832\pi$
$547$	−517.470	−0.946015	−0.473007	+	0.881058i	$0.656832\pi$
$548$	−215.848	−0.393883
$549$	0	0
$550$	211.050	0.383727
$551$	− 1047.39i	− 1.90089i
$552$	0	0
$553$	0	0
$554$	307.966	0.555894
$555$	0	0
$556$	− 90.9209i	− 0.163527i
$557$	−583.373	−1.04735	−0.523674	−	0.851919i	$-0.675439\pi$
$557$	−583.373	−1.04735	−0.523674	+	0.851919i	$0.675439\pi$
$558$	0	0
$559$	− 266.603i	− 0.476928i
$560$	0	0
$561$	0	0
$562$	−449.411	−0.799664
$563$	1047.67i	1.86087i	0.366456	+	0.930435i	$0.380571\pi$
$563$	1047.67i	1.86087i	−0.366456	+	0.930435i	$0.619429\pi$
$564$	0	0
$565$	− 80.0333i	− 0.141652i
$566$	− 339.643i	− 0.600075i
$567$	0	0
$568$	−137.373	−0.241853
$569$	−334.336	−0.587585	−0.293793	−	0.955869i	$-0.594918\pi$
$569$	−334.336	−0.587585	−0.293793	+	0.955869i	$0.594918\pi$
$570$	0	0
$571$	413.401	0.723995	0.361998	−	0.932179i	$-0.382095\pi$
$571$	413.401	0.723995	0.361998	+	0.932179i	$0.382095\pi$
$572$	− 103.475i	− 0.180899i
$573$	0	0
$574$	0	0
$575$	−869.803	−1.51270
$576$	0	0
$577$	398.531i	0.690695i	0.938475	+	0.345347i	$0.112239\pi$
$577$	398.531i	0.690695i	−0.938475	+	0.345347i	$0.887761\pi$
$578$	−341.605	−0.591012
$579$	0	0
$580$	83.4662i	0.143907i
$581$	0	0
$582$	0	0
$583$	−295.765	−0.507315
$584$	− 199.622i	− 0.341818i
$585$	0	0
$586$	− 619.266i	− 1.05677i
$587$	236.810i	0.403424i	0.979445	+	0.201712i	$0.0646505\pi$
$587$	236.810i	0.403424i	−0.979445	+	0.201712i	$0.935349\pi$
$588$	0	0
$589$	−543.529	−0.922800
$590$	−3.34228	−0.00566488
$591$	0	0
$592$	−144.971	−0.244883
$593$	597.691i	1.00791i	0.863730	+	0.503955i	$0.168122\pi$
$593$	597.691i	1.00791i	−0.863730	+	0.503955i	$0.831878\pi$
$594$	0	0
$595$	0	0
$596$	368.000	0.617450
$597$	0	0
$598$	426.452i	0.713130i
$599$	821.754	1.37188	0.685939	−	0.727659i	$-0.259392\pi$
$599$	821.754	1.37188	0.685939	+	0.727659i	$0.259392\pi$
$600$	0	0
$601$	− 130.321i	− 0.216840i	−0.994105	−	0.108420i	$-0.965421\pi$
$601$	− 130.321i	− 0.216840i	0.994105	−	0.108420i	$-0.0345791\pi$
$602$	0	0
$603$	0	0
$604$	319.431	0.528860
$605$	− 98.0246i	− 0.162024i
$606$	0	0
$607$	− 46.0121i	− 0.0758025i	−0.999281	−	0.0379013i	$-0.987933\pi$
$607$	− 46.0121i	− 0.0758025i	0.999281	−	0.0379013i	$-0.0120672\pi$
$608$	172.312i	0.283409i
$609$	0	0
$610$	−134.706	−0.220830
$611$	297.137	0.486313
$612$	0	0
$613$	513.360	0.837454	0.418727	−	0.908112i	$-0.362476\pi$
$613$	513.360	0.837454	0.418727	+	0.908112i	$0.362476\pi$
$614$	− 674.505i	− 1.09854i
$615$	0	0
$616$	0	0
$617$	−1018.23	−1.65030	−0.825148	−	0.564917i	$-0.808908\pi$
$617$	−1018.23	−1.65030	−0.825148	+	0.564917i	$0.808908\pi$
$618$	0	0
$619$	− 62.7414i	− 0.101359i	−0.998715	−	0.0506797i	$-0.983861\pi$
$619$	− 62.7414i	− 0.101359i	0.998715	−	0.0506797i	$-0.0161388\pi$
$620$	43.3137	0.0698608
$621$	0	0
$622$	− 467.519i	− 0.751638i
$623$	0	0
$624$	0	0
$625$	516.688	0.826701
$626$	229.698i	0.366930i
$627$	0	0
$628$	− 337.995i	− 0.538209i
$629$	− 249.650i	− 0.396900i
$630$	0	0
$631$	−264.735	−0.419548	−0.209774	−	0.977750i	$-0.567273\pi$
$631$	−264.735	−0.419548	−0.209774	+	0.977750i	$0.567273\pi$
$632$	215.765	0.341400
$633$	0	0
$634$	−614.544	−0.969313
$635$	40.8496i	0.0643301i
$636$	0	0
$637$	0	0
$638$	308.451	0.483465
$639$	0	0
$640$	− 13.7315i	− 0.0214555i
$641$	−422.350	−0.658893	−0.329446	−	0.944174i	$-0.606862\pi$
$641$	−422.350	−0.658893	−0.329446	+	0.944174i	$0.606862\pi$
$642$	0	0
$643$	− 827.350i	− 1.28670i	−0.765571	−	0.643351i	$-0.777543\pi$
$643$	− 827.350i	− 1.28670i	0.765571	−	0.643351i	$-0.222457\pi$
$644$	0	0
$645$	0	0
$646$	−296.735	−0.459342
$647$	336.979i	0.520833i	0.965496	+	0.260417i	$0.0838599\pi$
$647$	336.979i	0.520833i	−0.965496	+	0.260417i	$0.916140\pi$
$648$	0	0
$649$	12.3515i	0.0190315i
$650$	− 271.380i	− 0.417508i
$651$	0	0
$652$	270.156	0.414350
$653$	47.7776	0.0731663	0.0365831	−	0.999331i	$-0.488353\pi$
$653$	47.7776	0.0731663	0.0365831	+	0.999331i	$0.488353\pi$
$654$	0	0
$655$	144.617	0.220790
$656$	− 56.3745i	− 0.0859368i
$657$	0	0
$658$	0	0
$659$	557.803	0.846439	0.423219	−	0.906027i	$-0.360900\pi$
$659$	557.803	0.846439	0.423219	+	0.906027i	$0.360900\pi$
$660$	0	0
$661$	− 950.853i	− 1.43851i	−0.694748	−	0.719253i	$-0.744484\pi$
$661$	− 950.853i	− 1.43851i	0.694748	−	0.719253i	$-0.255516\pi$
$662$	651.480	0.984109
$663$	0	0
$664$	87.6419i	0.131991i
$665$	0	0
$666$	0	0
$667$	−1271.22	−1.90588
$668$	− 477.228i	− 0.714414i
$669$	0	0
$670$	− 51.5943i	− 0.0770065i
$671$	497.810i	0.741893i
$672$	0	0
$673$	753.747	1.11998	0.559991	−	0.828499i	$-0.310805\pi$
$673$	753.747	1.11998	0.559991	+	0.828499i	$0.310805\pi$
$674$	87.4903	0.129808
$675$	0	0
$676$	204.946	0.303175
$677$	− 254.029i	− 0.375227i	−0.982243	−	0.187614i	$-0.939925\pi$
$677$	− 254.029i	− 0.375227i	0.982243	−	0.187614i	$-0.0600753\pi$
$678$	0	0
$679$	0	0
$680$	23.6468	0.0347746
$681$	0	0
$682$	− 160.067i	− 0.234702i
$683$	792.500	1.16032	0.580161	−	0.814502i	$-0.302990\pi$
$683$	792.500	1.16032	0.580161	+	0.814502i	$0.302990\pi$
$684$	0	0
$685$	− 130.988i	− 0.191224i
$686$	0	0
$687$	0	0
$688$	−130.745	−0.190037
$689$	380.312i	0.551977i
$690$	0	0
$691$	− 55.0054i	− 0.0796025i	−0.999208	−	0.0398013i	$-0.987328\pi$
$691$	− 55.0054i	− 0.0796025i	0.999208	−	0.0398013i	$-0.0126725\pi$
$692$	− 116.482i	− 0.168326i
$693$	0	0
$694$	−759.362	−1.09418
$695$	55.1758	0.0793896
$696$	0	0
$697$	97.0812	0.139284
$698$	122.843i	0.175993i
$699$	0	0
$700$	0	0
$701$	829.894	1.18387	0.591936	−	0.805985i	$-0.298364\pi$
$701$	829.894	1.18387	0.591936	+	0.805985i	$0.298364\pi$
$702$	0	0
$703$	1103.98i	1.57039i
$704$	−50.7452	−0.0720812
$705$	0	0
$706$	813.451i	1.15220i
$707$	0	0
$708$	0	0
$709$	−302.503	−0.426661	−0.213330	−	0.976980i	$-0.568431\pi$
$709$	−302.503	−0.426661	−0.213330	+	0.976980i	$0.568431\pi$
$710$	− 83.3651i	− 0.117416i
$711$	0	0
$712$	186.723i	0.262252i
$713$	659.686i	0.925225i
$714$	0	0
$715$	62.7939	0.0878237
$716$	−643.744	−0.899084
$717$	0	0
$718$	−70.7939	−0.0985988
$719$	− 1241.26i	− 1.72637i	−0.504889	−	0.863184i	$-0.668467\pi$
$719$	− 1241.26i	− 1.72637i	0.504889	−	0.863184i	$-0.331533\pi$
$720$	0	0
$721$	0	0
$722$	801.664	1.11034
$723$	0	0
$724$	227.783i	0.314617i
$725$	808.968	1.11582
$726$	0	0
$727$	− 546.321i	− 0.751474i	−0.926726	−	0.375737i	$-0.877390\pi$
$727$	− 546.321i	− 0.751474i	0.926726	−	0.375737i	$-0.122610\pi$
$728$	0	0
$729$	0	0
$730$	121.141	0.165947
$731$	− 225.153i	− 0.308007i
$732$	0	0
$733$	− 406.633i	− 0.554751i	−0.960762	−	0.277376i	$-0.910535\pi$
$733$	− 406.633i	− 0.554751i	0.960762	−	0.277376i	$-0.0894647\pi$
$734$	200.518i	0.273186i
$735$	0	0
$736$	209.137	0.284154
$737$	−190.668	−0.258708
$738$	0	0
$739$	−382.412	−0.517472	−0.258736	−	0.965948i	$-0.583306\pi$
$739$	−382.412	−0.517472	−0.258736	+	0.965948i	$0.583306\pi$
$740$	− 87.9760i	− 0.118886i
$741$	0	0
$742$	0	0
$743$	−1015.04	−1.36613	−0.683067	−	0.730356i	$-0.739354\pi$
$743$	−1015.04	−1.36613	−0.683067	+	0.730356i	$0.739354\pi$
$744$	0	0
$745$	223.322i	0.299762i
$746$	−87.9025	−0.117832
$747$	0	0
$748$	− 87.3870i	− 0.116828i
$749$	0	0
$750$	0	0
$751$	−761.961	−1.01460	−0.507298	−	0.861771i	$-0.669356\pi$
$751$	−761.961	−1.01460	−0.507298	+	0.861771i	$0.669356\pi$
$752$	− 145.720i	− 0.193776i
$753$	0	0
$754$	− 396.625i	− 0.526028i
$755$	193.848i	0.256753i
$756$	0	0
$757$	223.307	0.294989	0.147494	−	0.989063i	$-0.452879\pi$
$757$	223.307	0.294989	0.147494	+	0.989063i	$0.452879\pi$
$758$	−146.010	−0.192625
$759$	0	0
$760$	−104.569	−0.137590
$761$	817.907i	1.07478i	0.843334	+	0.537389i	$0.180589\pi$
$761$	817.907i	1.07478i	−0.843334	+	0.537389i	$0.819411\pi$
$762$	0	0
$763$	0	0
$764$	−178.412	−0.233524
$765$	0	0
$766$	− 502.155i	− 0.655555i
$767$	15.8823	0.0207070
$768$	0	0
$769$	− 726.318i	− 0.944496i	−0.881466	−	0.472248i	$-0.843443\pi$
$769$	− 726.318i	− 0.944496i	0.881466	−	0.472248i	$-0.156557\pi$
$770$	0	0
$771$	0	0
$772$	−280.902	−0.363862
$773$	− 466.557i	− 0.603566i	−0.953377	−	0.301783i	$-0.902418\pi$
$773$	− 466.557i	− 0.603566i	0.953377	−	0.301783i	$-0.0975819\pi$
$774$	0	0
$775$	− 419.803i	− 0.541682i
$776$	440.645i	0.567841i
$777$	0	0
$778$	572.627	0.736025
$779$	−429.304	−0.551096
$780$	0	0
$781$	−308.077	−0.394465
$782$	360.150i	0.460550i
$783$	0	0
$784$	0	0
$785$	205.114	0.261292
$786$	0	0
$787$	723.552i	0.919380i	0.888079	+	0.459690i	$0.152040\pi$
$787$	723.552i	0.919380i	−0.888079	+	0.459690i	$0.847960\pi$
$788$	281.019	0.356624
$789$	0	0
$790$	130.938i	0.165744i
$791$	0	0
$792$	0	0
$793$	640.115	0.807207
$794$	− 764.985i	− 0.963458i
$795$	0	0
$796$	− 445.743i	− 0.559979i
$797$	296.985i	0.372628i	0.982490	+	0.186314i	$0.0596542\pi$
$797$	296.985i	0.372628i	−0.982490	+	0.186314i	$0.940346\pi$
$798$	0	0
$799$	250.940	0.314068
$800$	−133.088	−0.166360
$801$	0	0
$802$	−661.411	−0.824702
$803$	− 447.680i	− 0.557509i
$804$	0	0
$805$	0	0
$806$	−205.823	−0.255364
$807$	0	0
$808$	528.992i	0.654693i
$809$	−1176.18	−1.45386	−0.726932	−	0.686709i	$-0.759055\pi$
$809$	−1176.18	−1.45386	−0.726932	+	0.686709i	$0.759055\pi$
$810$	0	0
$811$	907.076i	1.11847i	0.829011	+	0.559233i	$0.188904\pi$
$811$	907.076i	1.11847i	−0.829011	+	0.559233i	$0.811096\pi$
$812$	0	0
$813$	0	0
$814$	−325.117	−0.399406
$815$	163.946i	0.201160i
$816$	0	0
$817$	995.652i	1.21867i
$818$	136.087i	0.166365i
$819$	0	0
$820$	34.2111	0.0417209
$821$	584.156	0.711518	0.355759	−	0.934578i	$-0.384222\pi$
$821$	584.156	0.711518	0.355759	+	0.934578i	$0.384222\pi$
$822$	0	0
$823$	−220.077	−0.267409	−0.133704	−	0.991021i	$-0.542687\pi$
$823$	−220.077	−0.267409	−0.133704	+	0.991021i	$0.542687\pi$
$824$	399.296i	0.484583i
$825$	0	0
$826$	0	0
$827$	69.2548	0.0837422	0.0418711	−	0.999123i	$-0.486668\pi$
$827$	69.2548	0.0837422	0.0418711	+	0.999123i	$0.486668\pi$
$828$	0	0
$829$	− 844.672i	− 1.01890i	−0.860499	−	0.509452i	$-0.829848\pi$
$829$	− 844.672i	− 1.01890i	0.860499	−	0.509452i	$-0.170152\pi$
$830$	−53.1859	−0.0640793
$831$	0	0
$832$	65.2512i	0.0784270i
$833$	0	0
$834$	0	0
$835$	289.608	0.346836
$836$	386.435i	0.462243i
$837$	0	0
$838$	612.533i	0.730946i
$839$	− 428.770i	− 0.511049i	−0.966803	−	0.255525i	$-0.917752\pi$
$839$	− 428.770i	− 0.511049i	0.966803	−	0.255525i	$-0.0822482\pi$
$840$	0	0
$841$	341.313	0.405842
$842$	730.621	0.867721
$843$	0	0
$844$	40.5685	0.0480670
$845$	124.372i	0.147186i
$846$	0	0
$847$	0	0
$848$	186.510	0.219941
$849$	0	0
$850$	− 229.188i	− 0.269633i
$851$	1339.91	1.57451
$852$	0	0
$853$	1386.09i	1.62496i	0.582989	+	0.812480i	$0.301883\pi$
$853$	1386.09i	1.62496i	−0.582989	+	0.812480i	$0.698117\pi$
$854$	0	0
$855$	0	0
$856$	190.863	0.222971
$857$	326.586i	0.381081i	0.981679	+	0.190540i	$0.0610240\pi$
$857$	326.586i	0.381081i	−0.981679	+	0.190540i	$0.938976\pi$
$858$	0	0
$859$	360.882i	0.420118i	0.977689	+	0.210059i	$0.0673657\pi$
$859$	360.882i	0.420118i	−0.977689	+	0.210059i	$0.932634\pi$
$860$	− 79.3433i	− 0.0922596i
$861$	0	0
$862$	−518.656	−0.601689
$863$	−690.705	−0.800353	−0.400177	−	0.916438i	$-0.631051\pi$
$863$	−690.705	−0.800353	−0.400177	+	0.916438i	$0.631051\pi$
$864$	0	0
$865$	70.6875	0.0817197
$866$	− 635.661i	− 0.734020i
$867$	0	0
$868$	0	0
$869$	483.882	0.556827
$870$	0	0
$871$	245.172i	0.281484i
$872$	264.784	0.303651
$873$	0	0
$874$	− 1592.62i	− 1.82222i
$875$	0	0
$876$	0	0
$877$	617.150	0.703706	0.351853	−	0.936055i	$-0.385552\pi$
$877$	617.150	0.703706	0.351853	+	0.936055i	$0.385552\pi$
$878$	935.961i	1.06601i
$879$	0	0
$880$	− 30.7949i	− 0.0349942i
$881$	1015.48i	1.15264i	0.817224	+	0.576320i	$0.195512\pi$
$881$	1015.48i	1.15264i	−0.817224	+	0.576320i	$0.804488\pi$
$882$	0	0
$883$	1266.30	1.43409	0.717046	−	0.697026i	$-0.245494\pi$
$883$	1266.30	1.43409	0.717046	+	0.697026i	$0.245494\pi$
$884$	−112.368	−0.127113
$885$	0	0
$886$	1113.90	1.25722
$887$	893.482i	1.00731i	0.863905	+	0.503654i	$0.168011\pi$
$887$	893.482i	1.00731i	−0.863905	+	0.503654i	$0.831989\pi$
$888$	0	0
$889$	0	0
$890$	−113.314	−0.127319
$891$	0	0
$892$	281.475i	0.315555i
$893$	−1109.69	−1.24265
$894$	0	0
$895$	− 390.659i	− 0.436490i
$896$	0	0
$897$	0	0
$898$	397.921	0.443119
$899$	− 613.546i	− 0.682476i
$900$	0	0
$901$	321.184i	0.356475i
$902$	− 126.428i	− 0.140164i
$903$	0	0
$904$	186.510	0.206316
$905$	−138.231	−0.152741
$906$	0	0
$907$	−337.706	−0.372333	−0.186166	−	0.982518i	$-0.559606\pi$
$907$	−337.706	−0.372333	−0.186166	+	0.982518i	$0.559606\pi$
$908$	20.4171i	0.0224858i
$909$	0	0
$910$	0	0
$911$	482.461	0.529595	0.264797	−	0.964304i	$-0.414695\pi$
$911$	482.461	0.529595	0.264797	+	0.964304i	$0.414695\pi$
$912$	0	0
$913$	196.549i	0.215279i
$914$	−561.608	−0.614451
$915$	0	0
$916$	863.990i	0.943220i
$917$	0	0
$918$	0	0
$919$	658.940	0.717019	0.358509	−	0.933526i	$-0.383285\pi$
$919$	658.940	0.717019	0.358509	+	0.933526i	$0.383285\pi$
$920$	126.916i	0.137952i
$921$	0	0
$922$	613.188i	0.665063i
$923$	396.145i	0.429192i
$924$	0	0
$925$	−852.677	−0.921813
$926$	152.471	0.164656
$927$	0	0
$928$	−194.510	−0.209601
$929$	595.770i	0.641303i	0.947197	+	0.320651i	$0.103902\pi$
$929$	595.770i	0.641303i	−0.947197	+	0.320651i	$0.896098\pi$
$930$	0	0
$931$	0	0
$932$	−289.220	−0.310322
$933$	0	0
$934$	231.043i	0.247369i
$935$	53.0312	0.0567178
$936$	0	0
$937$	1397.12i	1.49106i	0.666473	+	0.745529i	$0.267803\pi$
$937$	1397.12i	1.49106i	−0.666473	+	0.745529i	$0.732197\pi$
$938$	0	0
$939$	0	0
$940$	88.4306	0.0940751
$941$	1863.58i	1.98043i	0.139550	+	0.990215i	$0.455434\pi$
$941$	1863.58i	1.98043i	−0.139550	+	0.990215i	$0.544566\pi$
$942$	0	0
$943$	521.049i	0.552544i
$944$	− 7.78885i	− 0.00825090i
$945$	0	0
$946$	−293.214	−0.309952
$947$	−390.412	−0.412262	−0.206131	−	0.978524i	$-0.566087\pi$
$947$	−390.412	−0.412262	−0.206131	+	0.978524i	$0.566087\pi$
$948$	0	0
$949$	−575.654	−0.606590
$950$	1013.50i	1.06684i
$951$	0	0
$952$	0	0
$953$	232.432	0.243895	0.121948	−	0.992537i	$-0.461086\pi$
$953$	232.432	0.243895	0.121948	+	0.992537i	$0.461086\pi$
$954$	0	0
$955$	− 108.270i	− 0.113372i
$956$	−426.940	−0.446590
$957$	0	0
$958$	848.138i	0.885322i
$959$	0	0
$960$	0	0
$961$	642.608	0.668687
$962$	418.055i	0.434569i
$963$	0	0
$964$	130.802i	0.135687i
$965$	− 170.466i	− 0.176649i
$966$	0	0
$967$	−1734.97	−1.79418	−0.897088	−	0.441851i	$-0.854322\pi$
$967$	−1734.97	−1.79418	−0.897088	+	0.441851i	$0.854322\pi$
$968$	228.437	0.235988
$969$	0	0
$970$	−267.407	−0.275677
$971$	− 113.158i	− 0.116537i	−0.998301	−	0.0582687i	$-0.981442\pi$
$971$	− 113.158i	− 0.116537i	0.998301	−	0.0582687i	$-0.0185580\pi$
$972$	0	0
$973$	0	0
$974$	−641.499	−0.658623
$975$	0	0
$976$	− 313.920i	− 0.321639i
$977$	−484.409	−0.495813	−0.247906	−	0.968784i	$-0.579743\pi$
$977$	−484.409	−0.495813	−0.247906	+	0.968784i	$0.579743\pi$
$978$	0	0
$979$	418.753i	0.427735i
$980$	0	0
$981$	0	0
$982$	717.823	0.730981
$983$	− 1786.00i	− 1.81689i	−0.418003	−	0.908446i	$-0.637270\pi$
$983$	− 1786.00i	− 1.81689i	0.418003	−	0.908446i	$-0.362730\pi$
$984$	0	0
$985$	170.538i	0.173135i
$986$	− 334.960i	− 0.339716i
$987$	0	0
$988$	496.902	0.502937
$989$	1208.43	1.22187
$990$	0	0
$991$	321.568	0.324488	0.162244	−	0.986751i	$-0.448127\pi$
$991$	321.568	0.324488	0.162244	+	0.986751i	$0.448127\pi$
$992$	100.938i	0.101752i
$993$	0	0
$994$	0	0
$995$	270.501	0.271861
$996$	0	0
$997$	− 785.969i	− 0.788334i	−0.919039	−	0.394167i	$-0.871033\pi$
$997$	− 785.969i	− 0.788334i	0.919039	−	0.394167i	$-0.128967\pi$
$998$	−429.352	−0.430213
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.a.685.2		4
3.2	odd	2		98.3.b.a.97.4	yes	4
7.2	even	3		882.3.n.j.325.4		8
7.3	odd	6		882.3.n.j.19.4		8
7.4	even	3		882.3.n.j.19.3		8
7.5	odd	6		882.3.n.j.325.3		8
7.6	odd	2	inner	882.3.c.a.685.1		4
12.11	even	2		784.3.c.b.97.1		4
21.2	odd	6		98.3.d.b.31.1		8
21.5	even	6		98.3.d.b.31.2		8
21.11	odd	6		98.3.d.b.19.2		8
21.17	even	6		98.3.d.b.19.1		8
21.20	even	2		98.3.b.a.97.3	✓	4
84.11	even	6		784.3.s.j.705.1		8
84.23	even	6		784.3.s.j.129.4		8
84.47	odd	6		784.3.s.j.129.1		8
84.59	odd	6		784.3.s.j.705.4		8
84.83	odd	2		784.3.c.b.97.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.3.b.a.97.3	✓	4	21.20	even	2
98.3.b.a.97.4	yes	4	3.2	odd	2
98.3.d.b.19.1		8	21.17	even	6
98.3.d.b.19.2		8	21.11	odd	6
98.3.d.b.31.1		8	21.2	odd	6
98.3.d.b.31.2		8	21.5	even	6
784.3.c.b.97.1		4	12.11	even	2
784.3.c.b.97.4		4	84.83	odd	2
784.3.s.j.129.1		8	84.47	odd	6
784.3.s.j.129.4		8	84.23	even	6
784.3.s.j.705.1		8	84.11	even	6
784.3.s.j.705.4		8	84.59	odd	6
882.3.c.a.685.1		4	7.6	odd	2	inner
882.3.c.a.685.2		4	1.1	even	1	trivial
882.3.n.j.19.3		8	7.4	even	3
882.3.n.j.19.4		8	7.3	odd	6
882.3.n.j.325.3		8	7.5	odd	6
882.3.n.j.325.4		8	7.2	even	3

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +1.21371i q^{5} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +1.21371i q^{5} -2.82843 q^{8} -1.71644i q^{10} -6.34315 q^{11} +8.15640i q^{13} +4.00000 q^{16} +6.88830i q^{17} -30.4608i q^{19} +2.42742i q^{20} +8.97056 q^{22} -36.9706 q^{23} +23.5269 q^{25} -11.5349i q^{26} +34.3848 q^{29} -17.8435i q^{31} -5.65685 q^{32} -9.74153i q^{34} -36.2426 q^{37} +43.0781i q^{38} -3.43289i q^{40} -14.0936i q^{41} -32.6863 q^{43} -12.6863 q^{44} +52.2843 q^{46} -36.4299i q^{47} -33.2721 q^{50} +16.3128i q^{52} +46.6274 q^{53} -7.69873i q^{55} -48.6274 q^{58} -1.94721i q^{59} -78.4800i q^{61} +25.2346i q^{62} +8.00000 q^{64} -9.89949 q^{65} +30.0589 q^{67} +13.7766i q^{68} +48.5685 q^{71} +70.5769i q^{73} +51.2548 q^{74} -60.9217i q^{76} -76.2843 q^{79} +4.85483i q^{80} +19.9314i q^{82} -30.9861i q^{83} -8.36039 q^{85} +46.2254 q^{86} +17.9411 q^{88} -66.0166i q^{89} -73.9411 q^{92} +51.5197i q^{94} +36.9706 q^{95} -155.791i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} - 48 q^{11} + 16 q^{16} - 32 q^{22} - 80 q^{23} - 36 q^{25} + 64 q^{29} - 128 q^{37} - 176 q^{43} - 96 q^{44} + 96 q^{46} - 184 q^{50} + 96 q^{53} - 104 q^{58} + 32 q^{64} + 256 q^{67} - 32 q^{71} + 24 q^{74} - 192 q^{79} - 288 q^{85} - 64 q^{86} - 64 q^{88} - 160 q^{92} + 80 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 - 48 * q^11 + 16 * q^16 - 32 * q^22 - 80 * q^23 - 36 * q^25 + 64 * q^29 - 128 * q^37 - 176 * q^43 - 96 * q^44 + 96 * q^46 - 184 * q^50 + 96 * q^53 - 104 * q^58 + 32 * q^64 + 256 * q^67 - 32 * q^71 + 24 * q^74 - 192 * q^79 - 288 * q^85 - 64 * q^86 - 64 * q^88 - 160 * q^92 + 80 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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