LMFDB - Embedded newform 882.5.b.g.197.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(197,882)]

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([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.b (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:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 7932x^{3} + 185683x^{2} + 6753864x + 64548144 $ x^6 - 32x^4 - 7932x^3 + 185683x^2 + 6753864x + 64548144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$21.9137 + 13.5986i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.g.197.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -25.7831i q^{5} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} -25.7831i q^{5} +22.6274i q^{8} -72.9255 q^{10} +2.36859i q^{11} +268.521 q^{13} +64.0000 q^{16} +258.563i q^{17} -527.634 q^{19} +206.265i q^{20} +6.69938 q^{22} -128.190i q^{23} -39.7667 q^{25} -759.491i q^{26} -1457.13i q^{29} +792.400 q^{31} -181.019i q^{32} +731.327 q^{34} +2004.12 q^{37} +1492.37i q^{38} +583.404 q^{40} -1014.13i q^{41} +2491.96 q^{43} -18.9487i q^{44} -362.577 q^{46} -655.817i q^{47} +112.477i q^{50} -2148.17 q^{52} +2820.60i q^{53} +61.0695 q^{55} -4121.38 q^{58} -1915.65i q^{59} +3243.35 q^{61} -2241.25i q^{62} -512.000 q^{64} -6923.29i q^{65} -2699.22 q^{67} -2068.51i q^{68} +4680.05i q^{71} +939.107 q^{73} -5668.52i q^{74} +4221.07 q^{76} -5391.03 q^{79} -1650.12i q^{80} -2868.39 q^{82} -7991.44i q^{83} +6666.55 q^{85} -7048.34i q^{86} -53.5951 q^{88} -4128.97i q^{89} +1025.52i q^{92} -1854.93 q^{94} +13604.0i q^{95} -3972.54 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{4}+O(q^{10}) $ 6 * q - 48 * q^4	$ 6 q - 48 q^{4} + 40 q^{10} - 278 q^{13} + 384 q^{16} - 654 q^{19} + 808 q^{22} - 1786 q^{25} + 3790 q^{31} + 2768 q^{34} - 4542 q^{37} - 320 q^{40} + 8142 q^{43} - 1136 q^{46} + 2224 q^{52} + 11432 q^{55} - 832 q^{58} + 4024 q^{61} - 3072 q^{64} - 7466 q^{67} - 5506 q^{73} + 5232 q^{76} - 3778 q^{79} - 5496 q^{82} + 37312 q^{85} - 6464 q^{88} - 31224 q^{94} + 30692 q^{97}+O(q^{100}) $ 6 * q - 48 * q^4 + 40 * q^10 - 278 * q^13 + 384 * q^16 - 654 * q^19 + 808 * q^22 - 1786 * q^25 + 3790 * q^31 + 2768 * q^34 - 4542 * q^37 - 320 * q^40 + 8142 * q^43 - 1136 * q^46 + 2224 * q^52 + 11432 * q^55 - 832 * q^58 + 4024 * q^61 - 3072 * q^64 - 7466 * q^67 - 5506 * q^73 + 5232 * q^76 - 3778 * q^79 - 5496 * q^82 + 37312 * q^85 - 6464 * q^88 - 31224 * q^94 + 30692 * q^97

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 25.7831i	− 1.03132i	−0.856792	−	0.515661i	$-0.827546\pi$
$5$	− 25.7831i	− 1.03132i	0.856792	−	0.515661i	$-0.172454\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−72.9255	−0.729255
$11$	2.36859i	0.0195751i	0.999952	+	0.00978756i	$0.00311553\pi$
$11$	2.36859i	0.0195751i	−0.999952	+	0.00978756i	$0.996884\pi$
$12$	0	0
$13$	268.521	1.58888	0.794440	−	0.607343i	$-0.207765\pi$
$13$	268.521	1.58888	0.794440	+	0.607343i	$0.207765\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	258.563i	0.894682i	0.894363	+	0.447341i	$0.147629\pi$
$17$	258.563i	0.894682i	−0.894363	+	0.447341i	$0.852371\pi$
$18$	0	0
$19$	−527.634	−1.46159	−0.730795	−	0.682597i	$-0.760850\pi$
$19$	−527.634	−1.46159	−0.730795	+	0.682597i	$0.760850\pi$
$20$	206.265i	0.515661i
$21$	0	0
$22$	6.69938	0.0138417
$23$	− 128.190i	− 0.242326i	−0.992633	−	0.121163i	$-0.961338\pi$
$23$	− 128.190i	− 0.242326i	0.992633	−	0.121163i	$-0.0386623\pi$
$24$	0	0
$25$	−39.7667	−0.0636267
$26$	− 759.491i	− 1.12351i
$27$	0	0
$28$	0	0
$29$	− 1457.13i	− 1.73261i	−0.499514	−	0.866306i	$-0.666488\pi$
$29$	− 1457.13i	− 1.73261i	0.499514	−	0.866306i	$-0.333512\pi$
$30$	0	0
$31$	792.400	0.824558	0.412279	−	0.911058i	$-0.364733\pi$
$31$	792.400	0.824558	0.412279	+	0.911058i	$0.364733\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	731.327	0.632636
$35$	0	0
$36$	0	0
$37$	2004.12	1.46393	0.731966	−	0.681341i	$-0.238603\pi$
$37$	2004.12	1.46393	0.731966	+	0.681341i	$0.238603\pi$
$38$	1492.37i	1.03350i
$39$	0	0
$40$	583.404	0.364628
$41$	− 1014.13i	− 0.603289i	−0.953420	−	0.301645i	$-0.902464\pi$
$41$	− 1014.13i	− 0.603289i	0.953420	−	0.301645i	$-0.0975356\pi$
$42$	0	0
$43$	2491.96	1.34774	0.673868	−	0.738852i	$-0.264632\pi$
$43$	2491.96	1.34774	0.673868	+	0.738852i	$0.264632\pi$
$44$	− 18.9487i	− 0.00978756i
$45$	0	0
$46$	−362.577	−0.171350
$47$	− 655.817i	− 0.296884i	−0.988921	−	0.148442i	$-0.952574\pi$
$47$	− 655.817i	− 0.296884i	0.988921	−	0.148442i	$-0.0474258\pi$
$48$	0	0
$49$	0	0
$50$	112.477i	0.0449909i
$51$	0	0
$52$	−2148.17	−0.794440
$53$	2820.60i	1.00413i	0.864830	+	0.502065i	$0.167426\pi$
$53$	2820.60i	1.00413i	−0.864830	+	0.502065i	$0.832574\pi$
$54$	0	0
$55$	61.0695	0.0201883
$56$	0	0
$57$	0	0
$58$	−4121.38	−1.22514
$59$	− 1915.65i	− 0.550315i	−0.961399	−	0.275158i	$-0.911270\pi$
$59$	− 1915.65i	− 0.550315i	0.961399	−	0.275158i	$-0.0887300\pi$
$60$	0	0
$61$	3243.35	0.871635	0.435818	−	0.900035i	$-0.356459\pi$
$61$	3243.35	0.871635	0.435818	+	0.900035i	$0.356459\pi$
$62$	− 2241.25i	− 0.583051i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 6923.29i	− 1.63865i
$66$	0	0
$67$	−2699.22	−0.601295	−0.300648	−	0.953735i	$-0.597203\pi$
$67$	−2699.22	−0.601295	−0.300648	+	0.953735i	$0.597203\pi$
$68$	− 2068.51i	− 0.447341i
$69$	0	0
$70$	0	0
$71$	4680.05i	0.928397i	0.885731	+	0.464199i	$0.153658\pi$
$71$	4680.05i	0.928397i	−0.885731	+	0.464199i	$0.846342\pi$
$72$	0	0
$73$	939.107	0.176226	0.0881129	−	0.996110i	$-0.471916\pi$
$73$	939.107	0.176226	0.0881129	+	0.996110i	$0.471916\pi$
$74$	− 5668.52i	− 1.03516i
$75$	0	0
$76$	4221.07	0.730795
$77$	0	0
$78$	0	0
$79$	−5391.03	−0.863808	−0.431904	−	0.901920i	$-0.642158\pi$
$79$	−5391.03	−0.863808	−0.431904	+	0.901920i	$0.642158\pi$
$80$	− 1650.12i	− 0.257831i
$81$	0	0
$82$	−2868.39	−0.426590
$83$	− 7991.44i	− 1.16003i	−0.814606	−	0.580014i	$-0.803047\pi$
$83$	− 7991.44i	− 1.16003i	0.814606	−	0.580014i	$-0.196953\pi$
$84$	0	0
$85$	6666.55	0.922706
$86$	− 7048.34i	− 0.952993i
$87$	0	0
$88$	−53.5951	−0.00692085
$89$	− 4128.97i	− 0.521269i	−0.965438	−	0.260634i	$-0.916068\pi$
$89$	− 4128.97i	− 0.521269i	0.965438	−	0.260634i	$-0.0839317\pi$
$90$	0	0
$91$	0	0
$92$	1025.52i	0.121163i
$93$	0	0
$94$	−1854.93	−0.209929
$95$	13604.0i	1.50737i
$96$	0	0
$97$	−3972.54	−0.422206	−0.211103	−	0.977464i	$-0.567706\pi$
$97$	−3972.54	−0.422206	−0.211103	+	0.977464i	$0.567706\pi$
$98$	0	0
$99$	0	0
$100$	318.134	0.0318134
$101$	− 13514.6i	− 1.32483i	−0.749136	−	0.662416i	$-0.769531\pi$
$101$	− 13514.6i	− 1.32483i	0.749136	−	0.662416i	$-0.230469\pi$
$102$	0	0
$103$	−10215.6	−0.962918	−0.481459	−	0.876469i	$-0.659893\pi$
$103$	−10215.6	−0.962918	−0.481459	+	0.876469i	$0.659893\pi$
$104$	6075.93i	0.561754i
$105$	0	0
$106$	7977.87	0.710027
$107$	21890.2i	1.91197i	0.293410	+	0.955987i	$0.405210\pi$
$107$	21890.2i	1.91197i	−0.293410	+	0.955987i	$0.594790\pi$
$108$	0	0
$109$	−8385.96	−0.705830	−0.352915	−	0.935655i	$-0.614809\pi$
$109$	−8385.96	−0.705830	−0.352915	+	0.935655i	$0.614809\pi$
$110$	− 172.731i	− 0.0142753i
$111$	0	0
$112$	0	0
$113$	8691.80i	0.680695i	0.940300	+	0.340348i	$0.110545\pi$
$113$	8691.80i	0.680695i	−0.940300	+	0.340348i	$0.889455\pi$
$114$	0	0
$115$	−3305.14	−0.249916
$116$	11657.0i	0.866306i
$117$	0	0
$118$	−5418.27	−0.389132
$119$	0	0
$120$	0	0
$121$	14635.4	0.999617
$122$	− 9173.59i	− 0.616339i
$123$	0	0
$124$	−6339.20	−0.412279
$125$	− 15089.1i	− 0.965703i
$126$	0	0
$127$	−238.060	−0.0147598	−0.00737989	−	0.999973i	$-0.502349\pi$
$127$	−238.060	−0.0147598	−0.00737989	+	0.999973i	$0.502349\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−19582.0	−1.15870
$131$	− 32201.2i	− 1.87642i	−0.346073	−	0.938208i	$-0.612485\pi$
$131$	− 32201.2i	− 1.87642i	0.346073	−	0.938208i	$-0.387515\pi$
$132$	0	0
$133$	0	0
$134$	7634.53i	0.425180i
$135$	0	0
$136$	−5850.62	−0.316318
$137$	− 35602.4i	− 1.89687i	−0.316970	−	0.948436i	$-0.602665\pi$
$137$	− 35602.4i	− 1.89687i	0.316970	−	0.948436i	$-0.397335\pi$
$138$	0	0
$139$	11433.9	0.591786	0.295893	−	0.955221i	$-0.404383\pi$
$139$	11433.9	0.591786	0.295893	+	0.955221i	$0.404383\pi$
$140$	0	0
$141$	0	0
$142$	13237.2	0.656476
$143$	636.015i	0.0311025i
$144$	0	0
$145$	−37569.2	−1.78688
$146$	− 2656.20i	− 0.124610i
$147$	0	0
$148$	−16033.0	−0.731966
$149$	− 21519.8i	− 0.969318i	−0.874703	−	0.484659i	$-0.838944\pi$
$149$	− 21519.8i	− 0.969318i	0.874703	−	0.484659i	$-0.161056\pi$
$150$	0	0
$151$	−9444.26	−0.414204	−0.207102	−	0.978319i	$-0.566403\pi$
$151$	−9444.26	−0.414204	−0.207102	+	0.978319i	$0.566403\pi$
$152$	− 11939.0i	− 0.516750i
$153$	0	0
$154$	0	0
$155$	− 20430.5i	− 0.850386i
$156$	0	0
$157$	−33876.2	−1.37435	−0.687173	−	0.726494i	$-0.741148\pi$
$157$	−33876.2	−1.37435	−0.687173	+	0.726494i	$0.741148\pi$
$158$	15248.1i	0.610804i
$159$	0	0
$160$	−4667.23	−0.182314
$161$	0	0
$162$	0	0
$163$	−40735.0	−1.53318	−0.766589	−	0.642138i	$-0.778048\pi$
$163$	−40735.0	−1.53318	−0.766589	+	0.642138i	$0.778048\pi$
$164$	8113.03i	0.301645i
$165$	0	0
$166$	−22603.2	−0.820264
$167$	− 8580.60i	− 0.307670i	−0.988097	−	0.153835i	$-0.950838\pi$
$167$	− 8580.60i	− 0.307670i	0.988097	−	0.153835i	$-0.0491624\pi$
$168$	0	0
$169$	43542.4	1.52454
$170$	− 18855.9i	− 0.652452i
$171$	0	0
$172$	−19935.7	−0.673868
$173$	− 18416.1i	− 0.615326i	−0.951495	−	0.307663i	$-0.900453\pi$
$173$	− 18416.1i	− 0.615326i	0.951495	−	0.307663i	$-0.0995469\pi$
$174$	0	0
$175$	0	0
$176$	151.590i	0.00489378i
$177$	0	0
$178$	−11678.5	−0.368593
$179$	− 53015.7i	− 1.65462i	−0.561744	−	0.827311i	$-0.689870\pi$
$179$	− 53015.7i	− 1.65462i	0.561744	−	0.827311i	$-0.310130\pi$
$180$	0	0
$181$	23866.0	0.728487	0.364243	−	0.931304i	$-0.381328\pi$
$181$	23866.0	0.728487	0.364243	+	0.931304i	$0.381328\pi$
$182$	0	0
$183$	0	0
$184$	2900.62	0.0856751
$185$	− 51672.5i	− 1.50979i
$186$	0	0
$187$	−612.430	−0.0175135
$188$	5246.53i	0.148442i
$189$	0	0
$190$	38478.0	1.06587
$191$	− 5915.71i	− 0.162159i	−0.996708	−	0.0810793i	$-0.974163\pi$
$191$	− 5915.71i	− 0.162159i	0.996708	−	0.0810793i	$-0.0258367\pi$
$192$	0	0
$193$	2029.50	0.0544847	0.0272424	−	0.999629i	$-0.491327\pi$
$193$	2029.50	0.0544847	0.0272424	+	0.999629i	$0.491327\pi$
$194$	11236.0i	0.298545i
$195$	0	0
$196$	0	0
$197$	65625.9i	1.69100i	0.533978	+	0.845499i	$0.320697\pi$
$197$	65625.9i	1.69100i	−0.533978	+	0.845499i	$0.679303\pi$
$198$	0	0
$199$	−40607.3	−1.02541	−0.512705	−	0.858565i	$-0.671357\pi$
$199$	−40607.3	−1.02541	−0.512705	+	0.858565i	$0.671357\pi$
$200$	− 899.818i	− 0.0224954i
$201$	0	0
$202$	−38225.1	−0.936798
$203$	0	0
$204$	0	0
$205$	−26147.4	−0.622186
$206$	28894.1i	0.680886i
$207$	0	0
$208$	17185.3	0.397220
$209$	− 1249.75i	− 0.0286108i
$210$	0	0
$211$	79629.9	1.78859	0.894295	−	0.447477i	$-0.147677\pi$
$211$	79629.9	1.78859	0.894295	+	0.447477i	$0.147677\pi$
$212$	− 22564.8i	− 0.502065i
$213$	0	0
$214$	61914.8	1.35197
$215$	− 64250.5i	− 1.38995i
$216$	0	0
$217$	0	0
$218$	23719.1i	0.499097i
$219$	0	0
$220$	−488.556	−0.0100941
$221$	69429.6i	1.42154i
$222$	0	0
$223$	−76804.2	−1.54446	−0.772228	−	0.635346i	$-0.780858\pi$
$223$	−76804.2	−1.54446	−0.772228	+	0.635346i	$0.780858\pi$
$224$	0	0
$225$	0	0
$226$	24584.1	0.481324
$227$	21098.1i	0.409441i	0.978820	+	0.204720i	$0.0656285\pi$
$227$	21098.1i	0.409441i	−0.978820	+	0.204720i	$0.934371\pi$
$228$	0	0
$229$	−45785.6	−0.873088	−0.436544	−	0.899683i	$-0.643798\pi$
$229$	−45785.6	−0.873088	−0.436544	+	0.899683i	$0.643798\pi$
$230$	9348.35i	0.176717i
$231$	0	0
$232$	32971.0	0.612571
$233$	− 4191.34i	− 0.0772042i	−0.999255	−	0.0386021i	$-0.987710\pi$
$233$	− 4191.34i	− 0.0772042i	0.999255	−	0.0386021i	$-0.0122905\pi$
$234$	0	0
$235$	−16909.0	−0.306183
$236$	15325.2i	0.275158i
$237$	0	0
$238$	0	0
$239$	− 30870.6i	− 0.540442i	−0.962798	−	0.270221i	$-0.912903\pi$
$239$	− 30870.6i	− 0.540442i	0.962798	−	0.270221i	$-0.0870967\pi$
$240$	0	0
$241$	33963.1	0.584754	0.292377	−	0.956303i	$-0.405554\pi$
$241$	33963.1	0.584754	0.292377	+	0.956303i	$0.405554\pi$
$242$	− 41395.1i	− 0.706836i
$243$	0	0
$244$	−25946.8	−0.435818
$245$	0	0
$246$	0	0
$247$	−141681.	−2.32229
$248$	17930.0i	0.291525i
$249$	0	0
$250$	−42678.5	−0.682855
$251$	32972.4i	0.523364i	0.965154	+	0.261682i	$0.0842771\pi$
$251$	32972.4i	0.523364i	−0.965154	+	0.261682i	$0.915723\pi$
$252$	0	0
$253$	303.630	0.00474356
$254$	673.336i	0.0104367i
$255$	0	0
$256$	4096.00	0.0625000
$257$	84545.3i	1.28004i	0.768359	+	0.640019i	$0.221074\pi$
$257$	84545.3i	1.28004i	−0.768359	+	0.640019i	$0.778926\pi$
$258$	0	0
$259$	0	0
$260$	55386.3i	0.819324i
$261$	0	0
$262$	−91078.6	−1.32683
$263$	− 103271.i	− 1.49303i	−0.665371	−	0.746513i	$-0.731727\pi$
$263$	− 103271.i	− 1.49303i	0.665371	−	0.746513i	$-0.268273\pi$
$264$	0	0
$265$	72723.8	1.03558
$266$	0	0
$267$	0	0
$268$	21593.7	0.300648
$269$	− 115242.i	− 1.59259i	−0.604907	−	0.796296i	$-0.706790\pi$
$269$	− 115242.i	− 1.59259i	0.604907	−	0.796296i	$-0.293210\pi$
$270$	0	0
$271$	−119513.	−1.62734	−0.813668	−	0.581330i	$-0.802533\pi$
$271$	−119513.	−1.62734	−0.813668	+	0.581330i	$0.802533\pi$
$272$	16548.0i	0.223671i
$273$	0	0
$274$	−100699.	−1.34129
$275$	− 94.1910i	− 0.00124550i
$276$	0	0
$277$	−35918.2	−0.468117	−0.234059	−	0.972222i	$-0.575201\pi$
$277$	−35918.2	−0.468117	−0.234059	+	0.972222i	$0.575201\pi$
$278$	− 32339.9i	− 0.418456i
$279$	0	0
$280$	0	0
$281$	107567.i	1.36227i	0.732156	+	0.681137i	$0.238514\pi$
$281$	107567.i	1.36227i	−0.732156	+	0.681137i	$0.761486\pi$
$282$	0	0
$283$	74573.0	0.931127	0.465563	−	0.885015i	$-0.345852\pi$
$283$	74573.0	0.931127	0.465563	+	0.885015i	$0.345852\pi$
$284$	− 37440.4i	− 0.464199i
$285$	0	0
$286$	1798.92	0.0219928
$287$	0	0
$288$	0	0
$289$	16666.1	0.199544
$290$	106262.i	1.26352i
$291$	0	0
$292$	−7512.86	−0.0881129
$293$	31081.0i	0.362042i	0.983479	+	0.181021i	$0.0579403\pi$
$293$	31081.0i	0.362042i	−0.983479	+	0.181021i	$0.942060\pi$
$294$	0	0
$295$	−49391.3	−0.567552
$296$	45348.1i	0.517578i
$297$	0	0
$298$	−60867.3	−0.685411
$299$	− 34421.7i	− 0.385026i
$300$	0	0
$301$	0	0
$302$	26712.4i	0.292886i
$303$	0	0
$304$	−33768.6	−0.365397
$305$	− 83623.6i	− 0.898937i
$306$	0	0
$307$	−19922.2	−0.211378	−0.105689	−	0.994399i	$-0.533705\pi$
$307$	−19922.2	−0.211378	−0.105689	+	0.994399i	$0.533705\pi$
$308$	0	0
$309$	0	0
$310$	−57786.2	−0.601314
$311$	− 174219.i	− 1.80125i	−0.434596	−	0.900626i	$-0.643109\pi$
$311$	− 174219.i	− 1.80125i	0.434596	−	0.900626i	$-0.356891\pi$
$312$	0	0
$313$	−85378.5	−0.871485	−0.435743	−	0.900071i	$-0.643514\pi$
$313$	−85378.5	−0.871485	−0.435743	+	0.900071i	$0.643514\pi$
$314$	95816.5i	0.971809i
$315$	0	0
$316$	43128.2	0.431904
$317$	177007.i	1.76146i	0.473623	+	0.880728i	$0.342946\pi$
$317$	177007.i	1.76146i	−0.473623	+	0.880728i	$0.657054\pi$
$318$	0	0
$319$	3451.33	0.0339161
$320$	13200.9i	0.128915i
$321$	0	0
$322$	0	0
$323$	− 136427.i	− 1.30766i
$324$	0	0
$325$	−10678.2	−0.101095
$326$	115216.i	1.08412i
$327$	0	0
$328$	22947.1	0.213295
$329$	0	0
$330$	0	0
$331$	137003.	1.25047	0.625234	−	0.780437i	$-0.285003\pi$
$331$	137003.	1.25047	0.625234	+	0.780437i	$0.285003\pi$
$332$	63931.5i	0.580014i
$333$	0	0
$334$	−24269.6	−0.217555
$335$	69594.1i	0.620130i
$336$	0	0
$337$	56842.6	0.500511	0.250256	−	0.968180i	$-0.419485\pi$
$337$	56842.6	0.500511	0.250256	+	0.968180i	$0.419485\pi$
$338$	− 123156.i	− 1.07801i
$339$	0	0
$340$	−53332.4	−0.461353
$341$	1876.87i	0.0161408i
$342$	0	0
$343$	0	0
$344$	56386.7i	0.476497i
$345$	0	0
$346$	−52088.6	−0.435101
$347$	51452.6i	0.427315i	0.976909	+	0.213658i	$0.0685377\pi$
$347$	51452.6i	0.427315i	−0.976909	+	0.213658i	$0.931462\pi$
$348$	0	0
$349$	−69068.5	−0.567060	−0.283530	−	0.958963i	$-0.591506\pi$
$349$	−69068.5	−0.567060	−0.283530	+	0.958963i	$0.591506\pi$
$350$	0	0
$351$	0	0
$352$	428.761	0.00346043
$353$	− 20365.1i	− 0.163432i	−0.996656	−	0.0817161i	$-0.973960\pi$
$353$	− 20365.1i	− 0.163432i	0.996656	−	0.0817161i	$-0.0260401\pi$
$354$	0	0
$355$	120666.	0.957477
$356$	33031.8i	0.260634i
$357$	0	0
$358$	−149951.	−1.16999
$359$	134562.i	1.04408i	0.852922	+	0.522039i	$0.174828\pi$
$359$	134562.i	1.04408i	−0.852922	+	0.522039i	$0.825172\pi$
$360$	0	0
$361$	148076.	1.13624
$362$	− 67503.1i	− 0.515118i
$363$	0	0
$364$	0	0
$365$	− 24213.1i	− 0.181746i
$366$	0	0
$367$	199428.	1.48066	0.740329	−	0.672244i	$-0.234670\pi$
$367$	199428.	1.48066	0.740329	+	0.672244i	$0.234670\pi$
$368$	− 8204.18i	− 0.0605814i
$369$	0	0
$370$	−146152.	−1.06758
$371$	0	0
$372$	0	0
$373$	12186.3	0.0875898	0.0437949	−	0.999041i	$-0.486055\pi$
$373$	12186.3	0.0875898	0.0437949	+	0.999041i	$0.486055\pi$
$374$	1732.21i	0.0123839i
$375$	0	0
$376$	14839.4	0.104964
$377$	− 391269.i	− 2.75291i
$378$	0	0
$379$	−47966.9	−0.333936	−0.166968	−	0.985962i	$-0.553398\pi$
$379$	−47966.9	−0.333936	−0.166968	+	0.985962i	$0.553398\pi$
$380$	− 108832.i	− 0.753685i
$381$	0	0
$382$	−16732.1	−0.114663
$383$	7540.04i	0.0514015i	0.999670	+	0.0257008i	$0.00818170\pi$
$383$	7540.04i	0.0514015i	−0.999670	+	0.0257008i	$0.991818\pi$
$384$	0	0
$385$	0	0
$386$	− 5740.30i	− 0.0385265i
$387$	0	0
$388$	31780.3	0.211103
$389$	30745.8i	0.203183i	0.994826	+	0.101591i	$0.0323935\pi$
$389$	30745.8i	0.203183i	−0.994826	+	0.101591i	$0.967607\pi$
$390$	0	0
$391$	33145.3	0.216804
$392$	0	0
$393$	0	0
$394$	185618.	1.19572
$395$	138997.i	0.890865i
$396$	0	0
$397$	261023.	1.65614	0.828070	−	0.560625i	$-0.189439\pi$
$397$	261023.	1.65614	0.828070	+	0.560625i	$0.189439\pi$
$398$	114855.i	0.725074i
$399$	0	0
$400$	−2545.07	−0.0159067
$401$	10220.1i	0.0635574i	0.999495	+	0.0317787i	$0.0101172\pi$
$401$	10220.1i	0.0635574i	−0.999495	+	0.0317787i	$0.989883\pi$
$402$	0	0
$403$	212776.	1.31012
$404$	108117.i	0.662416i
$405$	0	0
$406$	0	0
$407$	4746.95i	0.0286567i
$408$	0	0
$409$	−135962.	−0.812776	−0.406388	−	0.913701i	$-0.633212\pi$
$409$	−135962.	−0.812776	−0.406388	+	0.913701i	$0.633212\pi$
$410$	73955.9i	0.439952i
$411$	0	0
$412$	81724.8	0.481459
$413$	0	0
$414$	0	0
$415$	−206044.	−1.19636
$416$	− 48607.4i	− 0.280877i
$417$	0	0
$418$	−3534.82	−0.0202309
$419$	− 239307.i	− 1.36310i	−0.731773	−	0.681549i	$-0.761307\pi$
$419$	− 239307.i	− 1.36310i	0.731773	−	0.681549i	$-0.238693\pi$
$420$	0	0
$421$	221785.	1.25132	0.625659	−	0.780097i	$-0.284830\pi$
$421$	221785.	1.25132	0.625659	+	0.780097i	$0.284830\pi$
$422$	− 225227.i	− 1.26472i
$423$	0	0
$424$	−63822.9	−0.355014
$425$	− 10282.2i	− 0.0569257i
$426$	0	0
$427$	0	0
$428$	− 175121.i	− 0.955987i
$429$	0	0
$430$	−181728.	−0.982844
$431$	25402.7i	0.136750i	0.997660	+	0.0683748i	$0.0217814\pi$
$431$	25402.7i	0.136750i	−0.997660	+	0.0683748i	$0.978219\pi$
$432$	0	0
$433$	277393.	1.47952	0.739759	−	0.672872i	$-0.234939\pi$
$433$	277393.	1.47952	0.739759	+	0.672872i	$0.234939\pi$
$434$	0	0
$435$	0	0
$436$	67087.7	0.352915
$437$	67637.5i	0.354181i
$438$	0	0
$439$	245813.	1.27549	0.637743	−	0.770249i	$-0.279868\pi$
$439$	245813.	1.27549	0.637743	+	0.770249i	$0.279868\pi$
$440$	1381.85i	0.00713763i
$441$	0	0
$442$	196376.	1.00518
$443$	− 49514.3i	− 0.252304i	−0.992011	−	0.126152i	$-0.959737\pi$
$443$	− 49514.3i	− 0.252304i	0.992011	−	0.126152i	$-0.0402626\pi$
$444$	0	0
$445$	−106458.	−0.537596
$446$	217235.i	1.09210i
$447$	0	0
$448$	0	0
$449$	− 315550.i	− 1.56522i	−0.622512	−	0.782610i	$-0.713888\pi$
$449$	− 315550.i	− 1.56522i	0.622512	−	0.782610i	$-0.286112\pi$
$450$	0	0
$451$	2402.06	0.0118095
$452$	− 69534.4i	− 0.340348i
$453$	0	0
$454$	59674.4	0.289518
$455$	0	0
$456$	0	0
$457$	47052.2	0.225293	0.112646	−	0.993635i	$-0.464067\pi$
$457$	47052.2	0.225293	0.112646	+	0.993635i	$0.464067\pi$
$458$	129501.i	0.617367i
$459$	0	0
$460$	26441.1	0.124958
$461$	144717.i	0.680956i	0.940252	+	0.340478i	$0.110589\pi$
$461$	144717.i	0.680956i	−0.940252	+	0.340478i	$0.889411\pi$
$462$	0	0
$463$	−314156.	−1.46549	−0.732746	−	0.680502i	$-0.761762\pi$
$463$	−314156.	−1.46549	−0.732746	+	0.680502i	$0.761762\pi$
$464$	− 93256.1i	− 0.433153i
$465$	0	0
$466$	−11854.9	−0.0545916
$467$	71579.3i	0.328211i	0.986443	+	0.164106i	$0.0524738\pi$
$467$	71579.3i	0.328211i	−0.986443	+	0.164106i	$0.947526\pi$
$468$	0	0
$469$	0	0
$470$	47825.8i	0.216504i
$471$	0	0
$472$	43346.1	0.194566
$473$	5902.44i	0.0263821i
$474$	0	0
$475$	20982.3	0.0929962
$476$	0	0
$477$	0	0
$478$	−87315.2	−0.382150
$479$	31394.1i	0.136829i	0.997657	+	0.0684144i	$0.0217940\pi$
$479$	31394.1i	0.136829i	−0.997657	+	0.0684144i	$0.978206\pi$
$480$	0	0
$481$	538149.	2.32601
$482$	− 96062.2i	− 0.413484i
$483$	0	0
$484$	−117083.	−0.499808
$485$	102424.i	0.435431i
$486$	0	0
$487$	−162986.	−0.687214	−0.343607	−	0.939114i	$-0.611649\pi$
$487$	−162986.	−0.687214	−0.343607	+	0.939114i	$0.611649\pi$
$488$	73388.7i	0.308170i
$489$	0	0
$490$	0	0
$491$	− 213254.i	− 0.884573i	−0.896874	−	0.442286i	$-0.854167\pi$
$491$	− 213254.i	− 0.884573i	0.896874	−	0.442286i	$-0.145833\pi$
$492$	0	0
$493$	376759.	1.55014
$494$	400733.i	1.64211i
$495$	0	0
$496$	50713.6	0.206140
$497$	0	0
$498$	0	0
$499$	170422.	0.684421	0.342210	−	0.939623i	$-0.388824\pi$
$499$	170422.	0.684421	0.342210	+	0.939623i	$0.388824\pi$
$500$	120713.i	0.482852i
$501$	0	0
$502$	93260.1	0.370074
$503$	− 144506.i	− 0.571151i	−0.958356	−	0.285575i	$-0.907815\pi$
$503$	− 144506.i	− 0.571151i	0.958356	−	0.285575i	$-0.0921847\pi$
$504$	0	0
$505$	−348448.	−1.36633
$506$	− 858.796i	− 0.00335420i
$507$	0	0
$508$	1904.48	0.00737989
$509$	− 423335.i	− 1.63399i	−0.576647	−	0.816993i	$-0.695639\pi$
$509$	− 423335.i	− 1.63399i	0.576647	−	0.816993i	$-0.304361\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	239130.	0.905124
$515$	263390.i	0.993080i
$516$	0	0
$517$	1553.36	0.00581154
$518$	0	0
$519$	0	0
$520$	156656.	0.579350
$521$	− 221150.i	− 0.814726i	−0.913266	−	0.407363i	$-0.866448\pi$
$521$	− 221150.i	− 0.814726i	0.913266	−	0.407363i	$-0.133552\pi$
$522$	0	0
$523$	−377707.	−1.38087	−0.690434	−	0.723396i	$-0.742580\pi$
$523$	−377707.	−1.38087	−0.690434	+	0.723396i	$0.742580\pi$
$524$	257609.i	0.938208i
$525$	0	0
$526$	−292095.	−1.05573
$527$	204886.i	0.737718i
$528$	0	0
$529$	263408.	0.941278
$530$	− 205694.i	− 0.732267i
$531$	0	0
$532$	0	0
$533$	− 272315.i	− 0.958554i
$534$	0	0
$535$	564396.	1.97186
$536$	− 61076.3i	− 0.212590i
$537$	0	0
$538$	−325952.	−1.12613
$539$	0	0
$540$	0	0
$541$	135542.	0.463105	0.231553	−	0.972822i	$-0.425619\pi$
$541$	135542.	0.463105	0.231553	+	0.972822i	$0.425619\pi$
$542$	338034.i	1.15070i
$543$	0	0
$544$	46804.9	0.158159
$545$	216216.i	0.727938i
$546$	0	0
$547$	314274.	1.05035	0.525175	−	0.850994i	$-0.324000\pi$
$547$	314274.	1.05035	0.525175	+	0.850994i	$0.324000\pi$
$548$	284819.i	0.948436i
$549$	0	0
$550$	−266.412	−0.000880702 0
$551$	768829.i	2.53237i
$552$	0	0
$553$	0	0
$554$	101592.i	0.331009i
$555$	0	0
$556$	−91471.2	−0.295893
$557$	− 59258.7i	− 0.191004i	−0.995429	−	0.0955019i	$-0.969554\pi$
$557$	− 59258.7i	− 0.191004i	0.995429	−	0.0955019i	$-0.0304456\pi$
$558$	0	0
$559$	669144.	2.14139
$560$	0	0
$561$	0	0
$562$	304244.	0.963273
$563$	214575.i	0.676958i	0.940974	+	0.338479i	$0.109912\pi$
$563$	214575.i	0.676958i	−0.940974	+	0.338479i	$0.890088\pi$
$564$	0	0
$565$	224101.	0.702016
$566$	− 210924.i	− 0.658406i
$567$	0	0
$568$	−105897.	−0.328238
$569$	123955.i	0.382861i	0.981506	+	0.191430i	$0.0613126\pi$
$569$	123955.i	0.382861i	−0.981506	+	0.191430i	$0.938687\pi$
$570$	0	0
$571$	169974.	0.521327	0.260664	−	0.965430i	$-0.416059\pi$
$571$	169974.	0.521327	0.260664	+	0.965430i	$0.416059\pi$
$572$	− 5088.12i	− 0.0155513i
$573$	0	0
$574$	0	0
$575$	5097.71i	0.0154184i
$576$	0	0
$577$	336708.	1.01135	0.505675	−	0.862724i	$-0.331243\pi$
$577$	336708.	1.01135	0.505675	+	0.862724i	$0.331243\pi$
$578$	− 47138.8i	− 0.141099i
$579$	0	0
$580$	300554.	0.893441
$581$	0	0
$582$	0	0
$583$	−6680.85	−0.0196560
$584$	21249.6i	0.0623052i
$585$	0	0
$586$	87910.3	0.256003
$587$	− 216616.i	− 0.628658i	−0.949314	−	0.314329i	$-0.898220\pi$
$587$	− 216616.i	− 0.628658i	0.949314	−	0.314329i	$-0.101780\pi$
$588$	0	0
$589$	−418097.	−1.20517
$590$	139700.i	0.401320i
$591$	0	0
$592$	128264.	0.365983
$593$	186592.i	0.530619i	0.964163	+	0.265310i	$0.0854742\pi$
$593$	186592.i	0.530619i	−0.964163	+	0.265310i	$0.914526\pi$
$594$	0	0
$595$	0	0
$596$	172159.i	0.484659i
$597$	0	0
$598$	−97359.4	−0.272255
$599$	239406.i	0.667238i	0.942708	+	0.333619i	$0.108270\pi$
$599$	239406.i	0.667238i	−0.942708	+	0.333619i	$0.891730\pi$
$600$	0	0
$601$	51508.8	0.142604	0.0713021	−	0.997455i	$-0.477285\pi$
$601$	51508.8	0.142604	0.0713021	+	0.997455i	$0.477285\pi$
$602$	0	0
$603$	0	0
$604$	75554.1	0.207102
$605$	− 377345.i	− 1.03093i
$606$	0	0
$607$	−24554.5	−0.0666429	−0.0333214	−	0.999445i	$-0.510609\pi$
$607$	−24554.5	−0.0666429	−0.0333214	+	0.999445i	$0.510609\pi$
$608$	95511.9i	0.258375i
$609$	0	0
$610$	−236523.	−0.635645
$611$	− 176100.i	− 0.471713i
$612$	0	0
$613$	169561.	0.451237	0.225618	−	0.974216i	$-0.427560\pi$
$613$	169561.	0.451237	0.225618	+	0.974216i	$0.427560\pi$
$614$	56348.5i	0.149467i
$615$	0	0
$616$	0	0
$617$	624837.i	1.64133i	0.571409	+	0.820665i	$0.306397\pi$
$617$	624837.i	1.64133i	−0.571409	+	0.820665i	$0.693603\pi$
$618$	0	0
$619$	−334855.	−0.873928	−0.436964	−	0.899479i	$-0.643946\pi$
$619$	−334855.	−0.873928	−0.436964	+	0.899479i	$0.643946\pi$
$620$	163444.i	0.425193i
$621$	0	0
$622$	−492765.	−1.27368
$623$	0	0
$624$	0	0
$625$	−413898.	−1.05958
$626$	241487.i	0.616233i
$627$	0	0
$628$	271010.	0.687173
$629$	518193.i	1.30975i
$630$	0	0
$631$	−154411.	−0.387809	−0.193905	−	0.981020i	$-0.562115\pi$
$631$	−154411.	−0.387809	−0.193905	+	0.981020i	$0.562115\pi$
$632$	− 121985.i	− 0.305402i
$633$	0	0
$634$	500651.	1.24554
$635$	6137.93i	0.0152221i
$636$	0	0
$637$	0	0
$638$	− 9761.85i	− 0.0239823i
$639$	0	0
$640$	37337.9	0.0911569
$641$	− 350351.i	− 0.852682i	−0.904562	−	0.426341i	$-0.859802\pi$
$641$	− 350351.i	− 0.852682i	0.904562	−	0.426341i	$-0.140198\pi$
$642$	0	0
$643$	79292.9	0.191784	0.0958920	−	0.995392i	$-0.469430\pi$
$643$	79292.9	0.191784	0.0958920	+	0.995392i	$0.469430\pi$
$644$	0	0
$645$	0	0
$646$	−385873.	−0.924654
$647$	− 711211.i	− 1.69899i	−0.527600	−	0.849493i	$-0.676908\pi$
$647$	− 711211.i	− 1.69899i	0.527600	−	0.849493i	$-0.323092\pi$
$648$	0	0
$649$	4537.38	0.0107725
$650$	30202.5i	0.0714851i
$651$	0	0
$652$	325880.	0.766589
$653$	− 33637.9i	− 0.0788864i	−0.999222	−	0.0394432i	$-0.987442\pi$
$653$	− 33637.9i	− 0.0788864i	0.999222	−	0.0394432i	$-0.0125584\pi$
$654$	0	0
$655$	−830245.	−1.93519
$656$	− 64904.3i	− 0.150822i
$657$	0	0
$658$	0	0
$659$	− 358536.i	− 0.825585i	−0.910825	−	0.412793i	$-0.864553\pi$
$659$	− 358536.i	− 0.825585i	0.910825	−	0.412793i	$-0.135447\pi$
$660$	0	0
$661$	584224.	1.33714	0.668569	−	0.743650i	$-0.266907\pi$
$661$	584224.	1.33714	0.668569	+	0.743650i	$0.266907\pi$
$662$	− 387502.i	− 0.884215i
$663$	0	0
$664$	180826.	0.410132
$665$	0	0
$666$	0	0
$667$	−186789.	−0.419856
$668$	68644.8i	0.153835i
$669$	0	0
$670$	196842.	0.438498
$671$	7682.18i	0.0170624i
$672$	0	0
$673$	−175181.	−0.386773	−0.193387	−	0.981123i	$-0.561947\pi$
$673$	−175181.	−0.386773	−0.193387	+	0.981123i	$0.561947\pi$
$674$	− 160775.i	− 0.353915i
$675$	0	0
$676$	−348339.	−0.762269
$677$	61931.0i	0.135124i	0.997715	+	0.0675618i	$0.0215220\pi$
$677$	61931.0i	0.135124i	−0.997715	+	0.0675618i	$0.978478\pi$
$678$	0	0
$679$	0	0
$680$	150847.i	0.326226i
$681$	0	0
$682$	5308.60	0.0114133
$683$	206583.i	0.442846i	0.975178	+	0.221423i	$0.0710702\pi$
$683$	206583.i	0.442846i	−0.975178	+	0.221423i	$0.928930\pi$
$684$	0	0
$685$	−917939.	−1.95629
$686$	0	0
$687$	0	0
$688$	159486.	0.336934
$689$	757390.i	1.59544i
$690$	0	0
$691$	140378.	0.293997	0.146998	−	0.989137i	$-0.453039\pi$
$691$	140378.	0.293997	0.146998	+	0.989137i	$0.453039\pi$
$692$	147329.i	0.307663i
$693$	0	0
$694$	145530.	0.302157
$695$	− 294801.i	− 0.610322i
$696$	0	0
$697$	262216.	0.539752
$698$	195355.i	0.400972i
$699$	0	0
$700$	0	0
$701$	− 798257.i	− 1.62445i	−0.583343	−	0.812226i	$-0.698256\pi$
$701$	− 798257.i	− 1.62445i	0.583343	−	0.812226i	$-0.301744\pi$
$702$	0	0
$703$	−1.05744e6	−2.13967
$704$	− 1212.72i	− 0.00244689i
$705$	0	0
$706$	−57601.3	−0.115564
$707$	0	0
$708$	0	0
$709$	−409921.	−0.815470	−0.407735	−	0.913100i	$-0.633681\pi$
$709$	−409921.	−0.815470	−0.407735	+	0.913100i	$0.633681\pi$
$710$	− 341295.i	− 0.677039i
$711$	0	0
$712$	93427.9	0.184296
$713$	− 101578.i	− 0.199812i
$714$	0	0
$715$	16398.4	0.0320767
$716$	424126.i	0.827311i
$717$	0	0
$718$	380598.	0.738274
$719$	721642.i	1.39593i	0.716131	+	0.697966i	$0.245911\pi$
$719$	721642.i	1.39593i	−0.716131	+	0.697966i	$0.754089\pi$
$720$	0	0
$721$	0	0
$722$	− 418823.i	− 0.803445i
$723$	0	0
$724$	−190928.	−0.364243
$725$	57945.1i	0.110240i
$726$	0	0
$727$	−695893.	−1.31666	−0.658330	−	0.752729i	$-0.728737\pi$
$727$	−695893.	−1.31666	−0.658330	+	0.752729i	$0.728737\pi$
$728$	0	0
$729$	0	0
$730$	−68484.9	−0.128514
$731$	644330.i	1.20580i
$732$	0	0
$733$	344991.	0.642096	0.321048	−	0.947063i	$-0.395965\pi$
$733$	344991.	0.642096	0.321048	+	0.947063i	$0.395965\pi$
$734$	− 564069.i	− 1.04698i
$735$	0	0
$736$	−23204.9	−0.0428375
$737$	− 6393.33i	− 0.0117704i
$738$	0	0
$739$	553173.	1.01291	0.506457	−	0.862266i	$-0.330955\pi$
$739$	553173.	1.01291	0.506457	+	0.862266i	$0.330955\pi$
$740$	413380.i	0.754893i
$741$	0	0
$742$	0	0
$743$	331872.i	0.601163i	0.953756	+	0.300582i	$0.0971808\pi$
$743$	331872.i	0.601163i	−0.953756	+	0.300582i	$0.902819\pi$
$744$	0	0
$745$	−554847.	−0.999680
$746$	− 34468.0i	− 0.0619353i
$747$	0	0
$748$	4899.44	0.00875676
$749$	0	0
$750$	0	0
$751$	908007.	1.60994	0.804969	−	0.593316i	$-0.202182\pi$
$751$	908007.	1.60994	0.804969	+	0.593316i	$0.202182\pi$
$752$	− 41972.3i	− 0.0742210i
$753$	0	0
$754$	−1.10667e6	−1.94660
$755$	243502.i	0.427178i
$756$	0	0
$757$	906297.	1.58153	0.790767	−	0.612117i	$-0.209682\pi$
$757$	906297.	1.58153	0.790767	+	0.612117i	$0.209682\pi$
$758$	135671.i	0.236129i
$759$	0	0
$760$	−307824.	−0.532936
$761$	− 972723.i	− 1.67966i	−0.542853	−	0.839828i	$-0.682656\pi$
$761$	− 972723.i	− 1.67966i	0.542853	−	0.839828i	$-0.317344\pi$
$762$	0	0
$763$	0	0
$764$	47325.7i	0.0810793i
$765$	0	0
$766$	21326.4	0.0363464
$767$	− 514391.i	− 0.874384i
$768$	0	0
$769$	−91810.6	−0.155253	−0.0776265	−	0.996983i	$-0.524734\pi$
$769$	−91810.6	−0.155253	−0.0776265	+	0.996983i	$0.524734\pi$
$770$	0	0
$771$	0	0
$772$	−16236.0	−0.0272424
$773$	− 493582.i	− 0.826039i	−0.910722	−	0.413020i	$-0.864474\pi$
$773$	− 493582.i	− 0.826039i	0.910722	−	0.413020i	$-0.135526\pi$
$774$	0	0
$775$	−31511.2	−0.0524639
$776$	− 89888.3i	− 0.149272i
$777$	0	0
$778$	86962.4	0.143672
$779$	535089.i	0.881761i
$780$	0	0
$781$	−11085.1	−0.0181735
$782$	− 93749.0i	− 0.153304i
$783$	0	0
$784$	0	0
$785$	873433.i	1.41739i
$786$	0	0
$787$	172472.	0.278465	0.139232	−	0.990260i	$-0.455537\pi$
$787$	172472.	0.278465	0.139232	+	0.990260i	$0.455537\pi$
$788$	− 525007.i	− 0.845499i
$789$	0	0
$790$	393143.	0.629937
$791$	0	0
$792$	0	0
$793$	870908.	1.38492
$794$	− 738283.i	− 1.17107i
$795$	0	0
$796$	324858.	0.512705
$797$	48077.3i	0.0756873i	0.999284	+	0.0378437i	$0.0120489\pi$
$797$	48077.3i	0.0756873i	−0.999284	+	0.0378437i	$0.987951\pi$
$798$	0	0
$799$	169570.	0.265617
$800$	7198.54i	0.0112477i
$801$	0	0
$802$	28906.8	0.0449419
$803$	2224.36i	0.00344964i
$804$	0	0
$805$	0	0
$806$	− 601821.i	− 0.926398i
$807$	0	0
$808$	305801.	0.468399
$809$	− 692003.i	− 1.05733i	−0.848830	−	0.528666i	$-0.822693\pi$
$809$	− 692003.i	− 1.05733i	0.848830	−	0.528666i	$-0.177307\pi$
$810$	0	0
$811$	327815.	0.498410	0.249205	−	0.968451i	$-0.419831\pi$
$811$	327815.	0.498410	0.249205	+	0.968451i	$0.419831\pi$
$812$	0	0
$813$	0	0
$814$	13426.4	0.0202633
$815$	1.05027e6i	1.58120i
$816$	0	0
$817$	−1.31484e6	−1.96984
$818$	384559.i	0.574720i
$819$	0	0
$820$	209179.	0.311093
$821$	− 327066.i	− 0.485232i	−0.970122	−	0.242616i	$-0.921995\pi$
$821$	− 327066.i	− 0.485232i	0.970122	−	0.242616i	$-0.0780055\pi$
$822$	0	0
$823$	378351.	0.558593	0.279296	−	0.960205i	$-0.409899\pi$
$823$	378351.	0.558593	0.279296	+	0.960205i	$0.409899\pi$
$824$	− 231153.i	− 0.340443i
$825$	0	0
$826$	0	0
$827$	555342.i	0.811988i	0.913876	+	0.405994i	$0.133075\pi$
$827$	555342.i	0.811988i	−0.913876	+	0.405994i	$0.866925\pi$
$828$	0	0
$829$	−519799.	−0.756356	−0.378178	−	0.925733i	$-0.623449\pi$
$829$	−519799.	−0.756356	−0.378178	+	0.925733i	$0.623449\pi$
$830$	582780.i	0.845957i
$831$	0	0
$832$	−137483.	−0.198610
$833$	0	0
$834$	0	0
$835$	−221234.	−0.317307
$836$	9997.98i	0.0143054i
$837$	0	0
$838$	−676862.	−0.963855
$839$	836509.i	1.18836i	0.804333	+	0.594178i	$0.202523\pi$
$839$	836509.i	1.18836i	−0.804333	+	0.594178i	$0.797477\pi$
$840$	0	0
$841$	−1.41594e6	−2.00194
$842$	− 627302.i	− 0.884816i
$843$	0	0
$844$	−637039.	−0.894295
$845$	− 1.12266e6i	− 1.57229i
$846$	0	0
$847$	0	0
$848$	180518.i	0.251033i
$849$	0	0
$850$	−29082.5	−0.0402526
$851$	− 256909.i	− 0.354748i
$852$	0	0
$853$	1.03133e6	1.41742	0.708711	−	0.705499i	$-0.249277\pi$
$853$	1.03133e6	1.41742	0.708711	+	0.705499i	$0.249277\pi$
$854$	0	0
$855$	0	0
$856$	−495318.	−0.675985
$857$	781985.i	1.06472i	0.846517	+	0.532362i	$0.178695\pi$
$857$	781985.i	1.06472i	−0.846517	+	0.532362i	$0.821305\pi$
$858$	0	0
$859$	−1.06243e6	−1.43984	−0.719921	−	0.694056i	$-0.755822\pi$
$859$	−1.06243e6	−1.43984	−0.719921	+	0.694056i	$0.755822\pi$
$860$	514004.i	0.694975i
$861$	0	0
$862$	71849.8	0.0966966
$863$	839858.i	1.12768i	0.825886	+	0.563838i	$0.190676\pi$
$863$	839858.i	1.12768i	−0.825886	+	0.563838i	$0.809324\pi$
$864$	0	0
$865$	−474823.	−0.634600
$866$	− 784587.i	− 1.04618i
$867$	0	0
$868$	0	0
$869$	− 12769.1i	− 0.0169091i
$870$	0	0
$871$	−724795.	−0.955386
$872$	− 189753.i	− 0.249548i
$873$	0	0
$874$	191308.	0.250444
$875$	0	0
$876$	0	0
$877$	−532982.	−0.692968	−0.346484	−	0.938056i	$-0.612625\pi$
$877$	−532982.	−0.692968	−0.346484	+	0.938056i	$0.612625\pi$
$878$	− 695264.i	− 0.901905i
$879$	0	0
$880$	3908.45	0.00504707
$881$	− 1.36059e6i	− 1.75297i	−0.481430	−	0.876484i	$-0.659883\pi$
$881$	− 1.36059e6i	− 1.75297i	0.481430	−	0.876484i	$-0.340117\pi$
$882$	0	0
$883$	1.42318e6	1.82531	0.912657	−	0.408727i	$-0.134027\pi$
$883$	1.42318e6	1.82531	0.912657	+	0.408727i	$0.134027\pi$
$884$	− 555436.i	− 0.710771i
$885$	0	0
$886$	−140048.	−0.178406
$887$	158940.i	0.202016i	0.994886	+	0.101008i	$0.0322068\pi$
$887$	158940.i	0.202016i	−0.994886	+	0.101008i	$0.967793\pi$
$888$	0	0
$889$	0	0
$890$	301107.i	0.380138i
$891$	0	0
$892$	614434.	0.772228
$893$	346031.i	0.433923i
$894$	0	0
$895$	−1.36691e6	−1.70645
$896$	0	0
$897$	0	0
$898$	−892510.	−1.10678
$899$	− 1.15463e6i	− 1.42864i
$900$	0	0
$901$	−729304.	−0.898377
$902$	− 6794.04i	− 0.00835055i
$903$	0	0
$904$	−196673.	−0.240662
$905$	− 615338.i	− 0.751305i
$906$	0	0
$907$	−683535.	−0.830895	−0.415447	−	0.909617i	$-0.636375\pi$
$907$	−683535.	−0.830895	−0.415447	+	0.909617i	$0.636375\pi$
$908$	− 168785.i	− 0.204720i
$909$	0	0
$910$	0	0
$911$	− 16125.4i	− 0.0194300i	−0.999953	−	0.00971500i	$-0.996908\pi$
$911$	− 16125.4i	− 0.0194300i	0.999953	−	0.00971500i	$-0.00309243\pi$
$912$	0	0
$913$	18928.4	0.0227077
$914$	− 133084.i	− 0.159306i
$915$	0	0
$916$	366285.	0.436544
$917$	0	0
$918$	0	0
$919$	−418627.	−0.495674	−0.247837	−	0.968802i	$-0.579720\pi$
$919$	−418627.	−0.495674	−0.247837	+	0.968802i	$0.579720\pi$
$920$	− 74786.8i	− 0.0883587i
$921$	0	0
$922$	409323.	0.481509
$923$	1.25669e6i	1.47511i
$924$	0	0
$925$	−79697.4	−0.0931452
$926$	888567.i	1.03626i
$927$	0	0
$928$	−263768.	−0.306285
$929$	− 268591.i	− 0.311214i	−0.987819	−	0.155607i	$-0.950267\pi$
$929$	− 268591.i	− 0.311214i	0.987819	−	0.155607i	$-0.0497334\pi$
$930$	0	0
$931$	0	0
$932$	33530.7i	0.0386021i
$933$	0	0
$934$	202457.	0.232080
$935$	15790.3i	0.0180621i
$936$	0	0
$937$	−543470.	−0.619008	−0.309504	−	0.950898i	$-0.600163\pi$
$937$	−543470.	−0.619008	−0.309504	+	0.950898i	$0.600163\pi$
$938$	0	0
$939$	0	0
$940$	135272.	0.153092
$941$	− 434266.i	− 0.490430i	−0.969469	−	0.245215i	$-0.921141\pi$
$941$	− 434266.i	− 0.490430i	0.969469	−	0.245215i	$-0.0788585\pi$
$942$	0	0
$943$	−130002.	−0.146192
$944$	− 122601.i	− 0.137579i
$945$	0	0
$946$	16694.6	0.0186550
$947$	− 873995.i	− 0.974560i	−0.873246	−	0.487280i	$-0.837989\pi$
$947$	− 873995.i	− 0.974560i	0.873246	−	0.487280i	$-0.162011\pi$
$948$	0	0
$949$	252170.	0.280001
$950$	− 59346.8i	− 0.0657582i
$951$	0	0
$952$	0	0
$953$	1.51221e6i	1.66504i	0.553994	+	0.832521i	$0.313103\pi$
$953$	1.51221e6i	1.66504i	−0.553994	+	0.832521i	$0.686897\pi$
$954$	0	0
$955$	−152525.	−0.167238
$956$	246965.i	0.270221i
$957$	0	0
$958$	88796.1	0.0967526
$959$	0	0
$960$	0	0
$961$	−295622.	−0.320104
$962$	− 1.52211e6i	− 1.64474i
$963$	0	0
$964$	−271705.	−0.292377
$965$	− 52326.8i	− 0.0561913i
$966$	0	0
$967$	1.49916e6	1.60322	0.801612	−	0.597844i	$-0.203976\pi$
$967$	1.49916e6	1.60322	0.801612	+	0.597844i	$0.203976\pi$
$968$	331161.i	0.353418i
$969$	0	0
$970$	289699.	0.307896
$971$	470558.i	0.499085i	0.968364	+	0.249543i	$0.0802803\pi$
$971$	470558.i	0.499085i	−0.968364	+	0.249543i	$0.919720\pi$
$972$	0	0
$973$	0	0
$974$	460994.i	0.485934i
$975$	0	0
$976$	207575.	0.217909
$977$	126944.i	0.132992i	0.997787	+	0.0664959i	$0.0211819\pi$
$977$	126944.i	0.132992i	−0.997787	+	0.0664959i	$0.978818\pi$
$978$	0	0
$979$	9779.84	0.0102039
$980$	0	0
$981$	0	0
$982$	−603173.	−0.625487
$983$	218289.i	0.225904i	0.993600	+	0.112952i	$0.0360306\pi$
$983$	218289.i	0.225904i	−0.993600	+	0.112952i	$0.963969\pi$
$984$	0	0
$985$	1.69204e6	1.74396
$986$	− 1.06564e6i	− 1.09611i
$987$	0	0
$988$	1.13344e6	1.16114
$989$	− 319446.i	− 0.326591i
$990$	0	0
$991$	−203403.	−0.207115	−0.103557	−	0.994623i	$-0.533023\pi$
$991$	−203403.	−0.207115	−0.103557	+	0.994623i	$0.533023\pi$
$992$	− 143440.i	− 0.145763i
$993$	0	0
$994$	0	0
$995$	1.04698e6i	1.05753i
$996$	0	0
$997$	461568.	0.464350	0.232175	−	0.972674i	$-0.425416\pi$
$997$	461568.	0.464350	0.232175	+	0.972674i	$0.425416\pi$
$998$	− 482025.i	− 0.483959i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.g.197.1		6
3.2	odd	2	inner	882.5.b.g.197.6		6
7.3	odd	6		126.5.s.a.107.4	yes	12
7.5	odd	6		126.5.s.a.53.3	✓	12
7.6	odd	2		882.5.b.f.197.3		6
21.5	even	6		126.5.s.a.53.4	yes	12
21.17	even	6		126.5.s.a.107.3	yes	12
21.20	even	2		882.5.b.f.197.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.s.a.53.3	✓	12	7.5	odd	6
126.5.s.a.53.4	yes	12	21.5	even	6
126.5.s.a.107.3	yes	12	21.17	even	6
126.5.s.a.107.4	yes	12	7.3	odd	6
882.5.b.f.197.3		6	7.6	odd	2
882.5.b.f.197.4		6	21.20	even	2
882.5.b.g.197.1		6	1.1	even	1	trivial
882.5.b.g.197.6		6	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -25.7831i q^{5} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -25.7831i q^{5} +22.6274i q^{8} -72.9255 q^{10} +2.36859i q^{11} +268.521 q^{13} +64.0000 q^{16} +258.563i q^{17} -527.634 q^{19} +206.265i q^{20} +6.69938 q^{22} -128.190i q^{23} -39.7667 q^{25} -759.491i q^{26} -1457.13i q^{29} +792.400 q^{31} -181.019i q^{32} +731.327 q^{34} +2004.12 q^{37} +1492.37i q^{38} +583.404 q^{40} -1014.13i q^{41} +2491.96 q^{43} -18.9487i q^{44} -362.577 q^{46} -655.817i q^{47} +112.477i q^{50} -2148.17 q^{52} +2820.60i q^{53} +61.0695 q^{55} -4121.38 q^{58} -1915.65i q^{59} +3243.35 q^{61} -2241.25i q^{62} -512.000 q^{64} -6923.29i q^{65} -2699.22 q^{67} -2068.51i q^{68} +4680.05i q^{71} +939.107 q^{73} -5668.52i q^{74} +4221.07 q^{76} -5391.03 q^{79} -1650.12i q^{80} -2868.39 q^{82} -7991.44i q^{83} +6666.55 q^{85} -7048.34i q^{86} -53.5951 q^{88} -4128.97i q^{89} +1025.52i q^{92} -1854.93 q^{94} +13604.0i q^{95} -3972.54 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{4}+O(q^{10}) \) 6 * q - 48 * q^4	\( 6 q - 48 q^{4} + 40 q^{10} - 278 q^{13} + 384 q^{16} - 654 q^{19} + 808 q^{22} - 1786 q^{25} + 3790 q^{31} + 2768 q^{34} - 4542 q^{37} - 320 q^{40} + 8142 q^{43} - 1136 q^{46} + 2224 q^{52} + 11432 q^{55} - 832 q^{58} + 4024 q^{61} - 3072 q^{64} - 7466 q^{67} - 5506 q^{73} + 5232 q^{76} - 3778 q^{79} - 5496 q^{82} + 37312 q^{85} - 6464 q^{88} - 31224 q^{94} + 30692 q^{97}+O(q^{100}) \) 6 * q - 48 * q^4 + 40 * q^10 - 278 * q^13 + 384 * q^16 - 654 * q^19 + 808 * q^22 - 1786 * q^25 + 3790 * q^31 + 2768 * q^34 - 4542 * q^37 - 320 * q^40 + 8142 * q^43 - 1136 * q^46 + 2224 * q^52 + 11432 * q^55 - 832 * q^58 + 4024 * q^61 - 3072 * q^64 - 7466 * q^67 - 5506 * q^73 + 5232 * q^76 - 3778 * q^79 - 5496 * q^82 + 37312 * q^85 - 6464 * q^88 - 31224 * q^94 + 30692 * q^97

Introduction

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