LMFDB - Embedded newform 825.4.c.n.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.181494784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 12x^{3} + 49x^{2} - 14x + 2 $ x^6 - 2x^5 + 2x^4 + 12x^3 + 49x^2 - 14*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$-1.51966 - 1.51966i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.n.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.03932i q^{2} +3.00000i q^{3} -8.31608 q^{4} -12.1180 q^{6} -6.49923i q^{7} -1.27677i q^{8} -9.00000 q^{9} +11.0000 q^{11} -24.9483i q^{12} -30.5385i q^{13} +26.2524 q^{14} -61.3714 q^{16} -43.1314i q^{17} -36.3539i q^{18} +6.46044 q^{19} +19.4977 q^{21} +44.4325i q^{22} -108.104i q^{23} +3.83030 q^{24} +123.355 q^{26} -27.0000i q^{27} +54.0481i q^{28} +274.857 q^{29} -68.5519 q^{31} -258.113i q^{32} +33.0000i q^{33} +174.221 q^{34} +74.8448 q^{36} -402.200i q^{37} +26.0958i q^{38} +91.6156 q^{39} -268.450 q^{41} +78.7573i q^{42} +30.5334i q^{43} -91.4769 q^{44} +436.666 q^{46} +31.5850i q^{47} -184.114i q^{48} +300.760 q^{49} +129.394 q^{51} +253.961i q^{52} -252.497i q^{53} +109.062 q^{54} -8.29800 q^{56} +19.3813i q^{57} +1110.24i q^{58} +558.331 q^{59} +335.270 q^{61} -276.903i q^{62} +58.4930i q^{63} +551.628 q^{64} -133.297 q^{66} +28.7521i q^{67} +358.684i q^{68} +324.312 q^{69} +89.0081 q^{71} +11.4909i q^{72} +717.622i q^{73} +1624.61 q^{74} -53.7256 q^{76} -71.4915i q^{77} +370.065i q^{78} -200.702 q^{79} +81.0000 q^{81} -1084.36i q^{82} -243.681i q^{83} -162.144 q^{84} -123.334 q^{86} +824.571i q^{87} -14.0444i q^{88} +312.832 q^{89} -198.477 q^{91} +899.002i q^{92} -205.656i q^{93} -127.582 q^{94} +774.338 q^{96} +27.9996i q^{97} +1214.87i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} + 232 q^{14} - 170 q^{16} + 338 q^{19} - 96 q^{21} - 18 q^{24} + 334 q^{26} + 554 q^{29} - 346 q^{31} + 292 q^{34} + 126 q^{36} + 270 q^{39} + 88 q^{41} - 154 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) $ 6 * q - 14 * q^4 - 6 * q^6 - 54 * q^9 + 66 * q^11 + 232 * q^14 - 170 * q^16 + 338 * q^19 - 96 * q^21 - 18 * q^24 + 334 * q^26 + 554 * q^29 - 346 * q^31 + 292 * q^34 + 126 * q^36 + 270 * q^39 + 88 * q^41 - 154 * q^44 + 850 * q^46 + 854 * q^49 + 348 * q^51 + 54 * q^54 + 336 * q^56 + 1368 * q^59 - 2076 * q^61 + 2258 * q^64 - 66 * q^66 + 930 * q^69 - 2918 * q^71 + 3528 * q^74 + 422 * q^76 + 1012 * q^79 + 486 * q^81 + 600 * q^84 - 1054 * q^86 + 1214 * q^89 - 796 * q^91 + 3644 * q^94 - 582 * q^96 - 594 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.03932i	1.42811i	0.700087	+	0.714057i	$0.253144\pi$
$2$	4.03932i	1.42811i	−0.700087	+	0.714057i	$0.746856\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−8.31608	−1.03951
$5$	0	0
$5$	0	0
$6$	−12.1180	−0.824522
$7$	− 6.49923i	− 0.350925i	−0.984486	−	0.175463i	$-0.943858\pi$
$7$	− 6.49923i	− 0.350925i	0.984486	−	0.175463i	$-0.0561421\pi$
$8$	− 1.27677i	− 0.0564257i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	− 24.9483i	− 0.600162i
$13$	− 30.5385i	− 0.651528i	−0.945451	−	0.325764i	$-0.894378\pi$
$13$	− 30.5385i	− 0.651528i	0.945451	−	0.325764i	$-0.105622\pi$
$14$	26.2524	0.501161
$15$	0	0
$16$	−61.3714	−0.958928
$17$	− 43.1314i	− 0.615347i	−0.951492	−	0.307674i	$-0.900450\pi$
$17$	− 43.1314i	− 0.615347i	0.951492	−	0.307674i	$-0.0995505\pi$
$18$	− 36.3539i	− 0.476038i
$19$	6.46044	0.0780066	0.0390033	−	0.999239i	$-0.487582\pi$
$19$	6.46044	0.0780066	0.0390033	+	0.999239i	$0.487582\pi$
$20$	0	0
$21$	19.4977	0.202607
$22$	44.4325i	0.430593i
$23$	− 108.104i	− 0.980054i	−0.871707	−	0.490027i	$-0.836987\pi$
$23$	− 108.104i	− 0.980054i	0.871707	−	0.490027i	$-0.163013\pi$
$24$	3.83030	0.0325774
$25$	0	0
$26$	123.355	0.930457
$27$	− 27.0000i	− 0.192450i
$28$	54.0481i	0.364791i
$29$	274.857	1.75999	0.879995	−	0.474984i	$-0.157546\pi$
$29$	274.857	1.75999	0.879995	+	0.474984i	$0.157546\pi$
$30$	0	0
$31$	−68.5519	−0.397170	−0.198585	−	0.980084i	$-0.563635\pi$
$31$	−68.5519	−0.397170	−0.198585	+	0.980084i	$0.563635\pi$
$32$	− 258.113i	− 1.42588i
$33$	33.0000i	0.174078i
$34$	174.221	0.878786
$35$	0	0
$36$	74.8448	0.346504
$37$	− 402.200i	− 1.78706i	−0.449004	−	0.893530i	$-0.648221\pi$
$37$	− 402.200i	− 1.78706i	0.449004	−	0.893530i	$-0.351779\pi$
$38$	26.0958i	0.111402i
$39$	91.6156	0.376160
$40$	0	0
$41$	−268.450	−1.02256	−0.511280	−	0.859414i	$-0.670828\pi$
$41$	−268.450	−1.02256	−0.511280	+	0.859414i	$0.670828\pi$
$42$	78.7573i	0.289346i
$43$	30.5334i	0.108286i	0.998533	+	0.0541431i	$0.0172427\pi$
$43$	30.5334i	0.108286i	−0.998533	+	0.0541431i	$0.982757\pi$
$44$	−91.4769	−0.313424
$45$	0	0
$46$	436.666	1.39963
$47$	31.5850i	0.0980243i	0.998798	+	0.0490121i	$0.0156073\pi$
$47$	31.5850i	0.0980243i	−0.998798	+	0.0490121i	$0.984393\pi$
$48$	− 184.114i	− 0.553638i
$49$	300.760	0.876851
$50$	0	0
$51$	129.394	0.355271
$52$	253.961i	0.677271i
$53$	− 252.497i	− 0.654398i	−0.944956	−	0.327199i	$-0.893895\pi$
$53$	− 252.497i	− 0.654398i	0.944956	−	0.327199i	$-0.106105\pi$
$54$	109.062	0.274841
$55$	0	0
$56$	−8.29800	−0.0198012
$57$	19.3813i	0.0450372i
$58$	1110.24i	2.51347i
$59$	558.331	1.23201	0.616004	−	0.787743i	$-0.288750\pi$
$59$	558.331	1.23201	0.616004	+	0.787743i	$0.288750\pi$
$60$	0	0
$61$	335.270	0.703719	0.351860	−	0.936053i	$-0.385549\pi$
$61$	335.270	0.703719	0.351860	+	0.936053i	$0.385549\pi$
$62$	− 276.903i	− 0.567205i
$63$	58.4930i	0.116975i
$64$	551.628	1.07740
$65$	0	0
$66$	−133.297	−0.248603
$67$	28.7521i	0.0524273i	0.999656	+	0.0262136i	$0.00834502\pi$
$67$	28.7521i	0.0524273i	−0.999656	+	0.0262136i	$0.991655\pi$
$68$	358.684i	0.639660i
$69$	324.312	0.565834
$70$	0	0
$71$	89.0081	0.148779	0.0743895	−	0.997229i	$-0.476299\pi$
$71$	89.0081	0.148779	0.0743895	+	0.997229i	$0.476299\pi$
$72$	11.4909i	0.0188086i
$73$	717.622i	1.15057i	0.817954	+	0.575283i	$0.195108\pi$
$73$	717.622i	1.15057i	−0.817954	+	0.575283i	$0.804892\pi$
$74$	1624.61	2.55213
$75$	0	0
$76$	−53.7256	−0.0810887
$77$	− 71.4915i	− 0.105808i
$78$	370.065i	0.537200i
$79$	−200.702	−0.285832	−0.142916	−	0.989735i	$-0.545648\pi$
$79$	−200.702	−0.285832	−0.142916	+	0.989735i	$0.545648\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 1084.36i	− 1.46033i
$83$	− 243.681i	− 0.322258i	−0.986933	−	0.161129i	$-0.948486\pi$
$83$	− 243.681i	− 0.322258i	0.986933	−	0.161129i	$-0.0515136\pi$
$84$	−162.144	−0.210612
$85$	0	0
$86$	−123.334	−0.154645
$87$	824.571i	1.01613i
$88$	− 14.0444i	− 0.0170130i
$89$	312.832	0.372585	0.186293	−	0.982494i	$-0.440353\pi$
$89$	312.832	0.372585	0.186293	+	0.982494i	$0.440353\pi$
$90$	0	0
$91$	−198.477	−0.228638
$92$	899.002i	1.01878i
$93$	− 205.656i	− 0.229306i
$94$	−127.582	−0.139990
$95$	0	0
$96$	774.338	0.823235
$97$	27.9996i	0.0293085i	0.999893	+	0.0146543i	$0.00466476\pi$
$97$	27.9996i	0.0293085i	−0.999893	+	0.0146543i	$0.995335\pi$
$98$	1214.87i	1.25224i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−228.278	−0.224896	−0.112448	−	0.993658i	$-0.535869\pi$
$101$	−228.278	−0.224896	−0.112448	+	0.993658i	$0.535869\pi$
$102$	522.664i	0.507367i
$103$	800.328i	0.765618i	0.923827	+	0.382809i	$0.125043\pi$
$103$	800.328i	0.765618i	−0.923827	+	0.382809i	$0.874957\pi$
$104$	−38.9906	−0.0367629
$105$	0	0
$106$	1019.91	0.934555
$107$	− 1046.89i	− 0.945857i	−0.881101	−	0.472929i	$-0.843197\pi$
$107$	− 1046.89i	− 0.945857i	0.881101	−	0.472929i	$-0.156803\pi$
$108$	224.534i	0.200054i
$109$	−317.023	−0.278581	−0.139291	−	0.990252i	$-0.544482\pi$
$109$	−317.023	−0.278581	−0.139291	+	0.990252i	$0.544482\pi$
$110$	0	0
$111$	1206.60	1.03176
$112$	398.867i	0.336512i
$113$	− 1333.82i	− 1.11040i	−0.831718	−	0.555198i	$-0.812642\pi$
$113$	− 1333.82i	− 1.11040i	0.831718	−	0.555198i	$-0.187358\pi$
$114$	−78.2873	−0.0643182
$115$	0	0
$116$	−2285.74	−1.82953
$117$	274.847i	0.217176i
$118$	2255.28i	1.75945i
$119$	−280.321	−0.215941
$120$	0	0
$121$	121.000	0.0909091
$122$	1354.26i	1.00499i
$123$	− 805.351i	− 0.590375i
$124$	570.083	0.412863
$125$	0	0
$126$	−236.272	−0.167054
$127$	564.888i	0.394691i	0.980334	+	0.197345i	$0.0632320\pi$
$127$	564.888i	0.394691i	−0.980334	+	0.197345i	$0.936768\pi$
$128$	163.299i	0.112763i
$129$	−91.6003	−0.0625190
$130$	0	0
$131$	−214.082	−0.142782	−0.0713908	−	0.997448i	$-0.522744\pi$
$131$	−214.082	−0.142782	−0.0713908	+	0.997448i	$0.522744\pi$
$132$	− 274.431i	− 0.180956i
$133$	− 41.9879i	− 0.0273745i
$134$	−116.139	−0.0748721
$135$	0	0
$136$	−55.0688	−0.0347214
$137$	− 65.2288i	− 0.0406779i	−0.999793	−	0.0203389i	$-0.993525\pi$
$137$	− 65.2288i	− 0.0406779i	0.999793	−	0.0203389i	$-0.00647453\pi$
$138$	1310.00i	0.808076i
$139$	407.084	0.248406	0.124203	−	0.992257i	$-0.460363\pi$
$139$	407.084	0.248406	0.124203	+	0.992257i	$0.460363\pi$
$140$	0	0
$141$	−94.7549	−0.0565943
$142$	359.532i	0.212474i
$143$	− 335.924i	− 0.196443i
$144$	552.343	0.319643
$145$	0	0
$146$	−2898.70	−1.64314
$147$	902.280i	0.506250i
$148$	3344.73i	1.85767i
$149$	−792.092	−0.435508	−0.217754	−	0.976004i	$-0.569873\pi$
$149$	−792.092	−0.435508	−0.217754	+	0.976004i	$0.569873\pi$
$150$	0	0
$151$	−2861.80	−1.54232	−0.771159	−	0.636642i	$-0.780323\pi$
$151$	−2861.80	−1.54232	−0.771159	+	0.636642i	$0.780323\pi$
$152$	− 8.24848i	− 0.00440158i
$153$	388.183i	0.205116i
$154$	288.777	0.151106
$155$	0	0
$156$	−761.883	−0.391022
$157$	2991.55i	1.52071i	0.649507	+	0.760355i	$0.274975\pi$
$157$	2991.55i	1.52071i	−0.649507	+	0.760355i	$0.725025\pi$
$158$	− 810.698i	− 0.408201i
$159$	757.490	0.377817
$160$	0	0
$161$	−702.592	−0.343926
$162$	327.185i	0.158679i
$163$	− 2907.60i	− 1.39718i	−0.715520	−	0.698592i	$-0.753810\pi$
$163$	− 2907.60i	− 1.39718i	0.715520	−	0.698592i	$-0.246190\pi$
$164$	2232.46	1.06296
$165$	0	0
$166$	984.304	0.460222
$167$	1100.39i	0.509885i	0.966956	+	0.254942i	$0.0820565\pi$
$167$	1100.39i	0.509885i	−0.966956	+	0.254942i	$0.917944\pi$
$168$	− 24.8940i	− 0.0114322i
$169$	1264.40	0.575511
$170$	0	0
$171$	−58.1439	−0.0260022
$172$	− 253.919i	− 0.112565i
$173$	− 3754.75i	− 1.65011i	−0.565054	−	0.825054i	$-0.691145\pi$
$173$	− 3754.75i	− 1.65011i	0.565054	−	0.825054i	$-0.308855\pi$
$174$	−3330.71	−1.45115
$175$	0	0
$176$	−675.086	−0.289128
$177$	1674.99i	0.711301i
$178$	1263.63i	0.532094i
$179$	3218.79	1.34404	0.672021	−	0.740532i	$-0.265426\pi$
$179$	3218.79	1.34404	0.672021	+	0.740532i	$0.265426\pi$
$180$	0	0
$181$	−456.885	−0.187624	−0.0938121	−	0.995590i	$-0.529905\pi$
$181$	−456.885	−0.187624	−0.0938121	+	0.995590i	$0.529905\pi$
$182$	− 801.711i	− 0.326521i
$183$	1005.81i	0.406292i
$184$	−138.024	−0.0553002
$185$	0	0
$186$	830.709	0.327476
$187$	− 474.445i	− 0.185534i
$188$	− 262.663i	− 0.101897i
$189$	−175.479	−0.0675356
$190$	0	0
$191$	−4907.11	−1.85899	−0.929493	−	0.368841i	$-0.879755\pi$
$191$	−4907.11	−1.85899	−0.929493	+	0.368841i	$0.879755\pi$
$192$	1654.88i	0.622036i
$193$	− 1462.17i	− 0.545333i	−0.962109	−	0.272667i	$-0.912094\pi$
$193$	− 1462.17i	− 0.545333i	0.962109	−	0.272667i	$-0.0879056\pi$
$194$	−113.099	−0.0418559
$195$	0	0
$196$	−2501.15	−0.911496
$197$	2210.10i	0.799307i	0.916666	+	0.399653i	$0.130869\pi$
$197$	2210.10i	0.799307i	−0.916666	+	0.399653i	$0.869131\pi$
$198$	− 399.892i	− 0.143531i
$199$	395.272	0.140804	0.0704022	−	0.997519i	$-0.477572\pi$
$199$	395.272	0.140804	0.0704022	+	0.997519i	$0.477572\pi$
$200$	0	0
$201$	−86.2563	−0.0302689
$202$	− 922.086i	− 0.321177i
$203$	− 1786.36i	− 0.617625i
$204$	−1076.05	−0.369308
$205$	0	0
$206$	−3232.78	−1.09339
$207$	972.936i	0.326685i
$208$	1874.19i	0.624769i
$209$	71.0648	0.0235199
$210$	0	0
$211$	1559.98	0.508974	0.254487	−	0.967076i	$-0.418093\pi$
$211$	1559.98	0.508974	0.254487	+	0.967076i	$0.418093\pi$
$212$	2099.78i	0.680253i
$213$	267.024i	0.0858976i
$214$	4228.72	1.35079
$215$	0	0
$216$	−34.4727	−0.0108591
$217$	445.534i	0.139377i
$218$	− 1280.56i	− 0.397846i
$219$	−2152.87	−0.664280
$220$	0	0
$221$	−1317.17	−0.400916
$222$	4873.84i	1.47347i
$223$	− 756.215i	− 0.227085i	−0.993533	−	0.113542i	$-0.963780\pi$
$223$	− 756.215i	− 0.227085i	0.993533	−	0.113542i	$-0.0362198\pi$
$224$	−1677.53	−0.500379
$225$	0	0
$226$	5387.70	1.58577
$227$	− 5758.69i	− 1.68378i	−0.539649	−	0.841890i	$-0.681443\pi$
$227$	− 5758.69i	− 1.68378i	0.539649	−	0.841890i	$-0.318557\pi$
$228$	− 161.177i	− 0.0468166i
$229$	1912.63	0.551921	0.275961	−	0.961169i	$-0.411004\pi$
$229$	1912.63	0.551921	0.275961	+	0.961169i	$0.411004\pi$
$230$	0	0
$231$	214.474	0.0610882
$232$	− 350.929i	− 0.0993086i
$233$	− 1034.86i	− 0.290970i	−0.989360	−	0.145485i	$-0.953526\pi$
$233$	− 1034.86i	− 0.290970i	0.989360	−	0.145485i	$-0.0464742\pi$
$234$	−1110.19	−0.310152
$235$	0	0
$236$	−4643.13	−1.28069
$237$	− 602.105i	− 0.165025i
$238$	− 1132.30i	− 0.308388i
$239$	−2647.23	−0.716465	−0.358232	−	0.933632i	$-0.616620\pi$
$239$	−2647.23	−0.716465	−0.358232	+	0.933632i	$0.616620\pi$
$240$	0	0
$241$	6734.00	1.79990	0.899948	−	0.435996i	$-0.143604\pi$
$241$	6734.00	1.79990	0.899948	+	0.435996i	$0.143604\pi$
$242$	488.757i	0.129829i
$243$	243.000i	0.0641500i
$244$	−2788.13	−0.731523
$245$	0	0
$246$	3253.07	0.843123
$247$	− 197.292i	− 0.0508235i
$248$	87.5248i	0.0224106i
$249$	731.043	0.186056
$250$	0	0
$251$	−5377.57	−1.35231	−0.676154	−	0.736761i	$-0.736354\pi$
$251$	−5377.57	−1.35231	−0.676154	+	0.736761i	$0.736354\pi$
$252$	− 486.433i	− 0.121597i
$253$	− 1189.14i	− 0.295497i
$254$	−2281.76	−0.563663
$255$	0	0
$256$	3753.41	0.916360
$257$	5882.33i	1.42774i	0.700278	+	0.713871i	$0.253060\pi$
$257$	5882.33i	1.42774i	−0.700278	+	0.713871i	$0.746940\pi$
$258$	− 370.003i	− 0.0892843i
$259$	−2613.99	−0.627124
$260$	0	0
$261$	−2473.71	−0.586663
$262$	− 864.744i	− 0.203909i
$263$	4001.67i	0.938227i	0.883138	+	0.469113i	$0.155426\pi$
$263$	4001.67i	0.938227i	−0.883138	+	0.469113i	$0.844574\pi$
$264$	42.1333	0.00982245
$265$	0	0
$266$	169.602	0.0390939
$267$	938.495i	0.215112i
$268$	− 239.105i	− 0.0544987i
$269$	1758.29	0.398530	0.199265	−	0.979946i	$-0.436144\pi$
$269$	1758.29	0.398530	0.199265	+	0.979946i	$0.436144\pi$
$270$	0	0
$271$	6018.44	1.34906	0.674528	−	0.738249i	$-0.264347\pi$
$271$	6018.44	1.34906	0.674528	+	0.738249i	$0.264347\pi$
$272$	2647.03i	0.590074i
$273$	− 595.431i	− 0.132004i
$274$	263.480	0.0580927
$275$	0	0
$276$	−2697.01	−0.588191
$277$	2910.30i	0.631275i	0.948880	+	0.315637i	$0.102218\pi$
$277$	2910.30i	0.631275i	−0.948880	+	0.315637i	$0.897782\pi$
$278$	1644.34i	0.354752i
$279$	616.967	0.132390
$280$	0	0
$281$	6893.19	1.46339	0.731697	−	0.681631i	$-0.238729\pi$
$281$	6893.19	1.46339	0.731697	+	0.681631i	$0.238729\pi$
$282$	− 382.745i	− 0.0808232i
$283$	− 3120.16i	− 0.655385i	−0.944784	−	0.327693i	$-0.893729\pi$
$283$	− 3120.16i	− 0.655385i	0.944784	−	0.327693i	$-0.106271\pi$
$284$	−740.199	−0.154657
$285$	0	0
$286$	1356.90	0.280543
$287$	1744.72i	0.358842i
$288$	2323.01i	0.475295i
$289$	3052.68	0.621348
$290$	0	0
$291$	−83.9988	−0.0169213
$292$	− 5967.81i	− 1.19603i
$293$	− 8869.62i	− 1.76849i	−0.467020	−	0.884247i	$-0.654672\pi$
$293$	− 8869.62i	− 1.76849i	0.467020	−	0.884247i	$-0.345328\pi$
$294$	−3644.60	−0.722983
$295$	0	0
$296$	−513.515	−0.100836
$297$	− 297.000i	− 0.0580259i
$298$	− 3199.51i	− 0.621955i
$299$	−3301.34	−0.638533
$300$	0	0
$301$	198.444	0.0380003
$302$	− 11559.7i	− 2.20261i
$303$	− 684.833i	− 0.129844i
$304$	−396.486	−0.0748028
$305$	0	0
$306$	−1567.99	−0.292929
$307$	− 5713.42i	− 1.06216i	−0.847323	−	0.531078i	$-0.821787\pi$
$307$	− 5713.42i	− 1.06216i	0.847323	−	0.531078i	$-0.178213\pi$
$308$	594.529i	0.109988i
$309$	−2400.98	−0.442030
$310$	0	0
$311$	1984.01	0.361746	0.180873	−	0.983506i	$-0.442108\pi$
$311$	1984.01	0.361746	0.180873	+	0.983506i	$0.442108\pi$
$312$	− 116.972i	− 0.0212251i
$313$	2675.50i	0.483156i	0.970381	+	0.241578i	$0.0776650\pi$
$313$	2675.50i	0.483156i	−0.970381	+	0.241578i	$0.922335\pi$
$314$	−12083.8	−2.17175
$315$	0	0
$316$	1669.05	0.297125
$317$	− 7083.76i	− 1.25509i	−0.778580	−	0.627545i	$-0.784060\pi$
$317$	− 7083.76i	− 1.25509i	0.778580	−	0.627545i	$-0.215940\pi$
$318$	3059.74i	0.539565i
$319$	3023.43	0.530657
$320$	0	0
$321$	3140.67	0.546091
$322$	− 2837.99i	− 0.491165i
$323$	− 278.648i	− 0.0480012i
$324$	−673.603	−0.115501
$325$	0	0
$326$	11744.7	1.99534
$327$	− 951.070i	− 0.160839i
$328$	342.749i	0.0576986i
$329$	205.278	0.0343992
$330$	0	0
$331$	7812.39	1.29730	0.648652	−	0.761085i	$-0.275333\pi$
$331$	7812.39	1.29730	0.648652	+	0.761085i	$0.275333\pi$
$332$	2026.47i	0.334991i
$333$	3619.80i	0.595686i
$334$	−4444.82	−0.728173
$335$	0	0
$336$	−1196.60	−0.194285
$337$	− 11866.4i	− 1.91811i	−0.283218	−	0.959056i	$-0.591402\pi$
$337$	− 11866.4i	− 1.91811i	0.283218	−	0.959056i	$-0.408598\pi$
$338$	5107.30i	0.821895i
$339$	4001.45	0.641088
$340$	0	0
$341$	−754.071	−0.119751
$342$	− 234.862i	− 0.0371341i
$343$	− 4183.94i	− 0.658635i
$344$	38.9841	0.00611012
$345$	0	0
$346$	15166.6	2.35654
$347$	3485.50i	0.539225i	0.962969	+	0.269613i	$0.0868957\pi$
$347$	3485.50i	0.539225i	−0.962969	+	0.269613i	$0.913104\pi$
$348$	− 6857.21i	− 1.05628i
$349$	6366.76	0.976518	0.488259	−	0.872699i	$-0.337632\pi$
$349$	6366.76	0.976518	0.488259	+	0.872699i	$0.337632\pi$
$350$	0	0
$351$	−824.541	−0.125387
$352$	− 2839.24i	− 0.429920i
$353$	− 8223.15i	− 1.23987i	−0.784653	−	0.619935i	$-0.787159\pi$
$353$	− 8223.15i	− 1.23987i	0.784653	−	0.619935i	$-0.212841\pi$
$354$	−6765.83	−1.01582
$355$	0	0
$356$	−2601.54	−0.387306
$357$	− 840.962i	− 0.124674i
$358$	13001.7i	1.91945i
$359$	9377.54	1.37863	0.689314	−	0.724462i	$-0.257912\pi$
$359$	9377.54	1.37863	0.689314	+	0.724462i	$0.257912\pi$
$360$	0	0
$361$	−6817.26	−0.993915
$362$	− 1845.50i	− 0.267949i
$363$	363.000i	0.0524864i
$364$	1650.55	0.237671
$365$	0	0
$366$	−4062.78	−0.580232
$367$	− 310.895i	− 0.0442196i	−0.999756	−	0.0221098i	$-0.992962\pi$
$367$	− 310.895i	− 0.0442196i	0.999756	−	0.0221098i	$-0.00703834\pi$
$368$	6634.49i	0.939801i
$369$	2416.05	0.340853
$370$	0	0
$371$	−1641.03	−0.229645
$372$	1710.25i	0.238366i
$373$	4738.51i	0.657777i	0.944369	+	0.328889i	$0.106674\pi$
$373$	4738.51i	0.657777i	−0.944369	+	0.328889i	$0.893326\pi$
$374$	1916.44	0.264964
$375$	0	0
$376$	40.3267	0.00553109
$377$	− 8393.74i	− 1.14668i
$378$	− 708.816i	− 0.0964486i
$379$	4735.47	0.641806	0.320903	−	0.947112i	$-0.396014\pi$
$379$	4735.47	0.641806	0.320903	+	0.947112i	$0.396014\pi$
$380$	0	0
$381$	−1694.66	−0.227875
$382$	− 19821.4i	− 2.65484i
$383$	− 3632.14i	− 0.484579i	−0.970204	−	0.242289i	$-0.922102\pi$
$383$	− 3632.14i	− 0.484579i	0.970204	−	0.242289i	$-0.0778984\pi$
$384$	−489.896	−0.0651039
$385$	0	0
$386$	5906.17	0.778798
$387$	− 274.801i	− 0.0360954i
$388$	− 232.847i	− 0.0304665i
$389$	−6557.50	−0.854701	−0.427350	−	0.904086i	$-0.640553\pi$
$389$	−6557.50	−0.854701	−0.427350	+	0.904086i	$0.640553\pi$
$390$	0	0
$391$	−4662.68	−0.603073
$392$	− 384.001i	− 0.0494769i
$393$	− 642.245i	− 0.0824350i
$394$	−8927.31	−1.14150
$395$	0	0
$396$	823.292	0.104475
$397$	4084.97i	0.516420i	0.966089	+	0.258210i	$0.0831327\pi$
$397$	4084.97i	0.516420i	−0.966089	+	0.258210i	$0.916867\pi$
$398$	1596.63i	0.201085i
$399$	125.964	0.0158047
$400$	0	0
$401$	−6676.04	−0.831386	−0.415693	−	0.909505i	$-0.636461\pi$
$401$	−6676.04	−0.831386	−0.415693	+	0.909505i	$0.636461\pi$
$402$	− 348.417i	− 0.0432275i
$403$	2093.47i	0.258768i
$404$	1898.38	0.233782
$405$	0	0
$406$	7215.67	0.882039
$407$	− 4424.20i	− 0.538819i
$408$	− 165.206i	− 0.0200464i
$409$	−1718.56	−0.207769	−0.103884	−	0.994589i	$-0.533127\pi$
$409$	−1718.56	−0.207769	−0.103884	+	0.994589i	$0.533127\pi$
$410$	0	0
$411$	195.686	0.0234854
$412$	− 6655.60i	− 0.795868i
$413$	− 3628.72i	− 0.432343i
$414$	−3930.00	−0.466543
$415$	0	0
$416$	−7882.39	−0.929004
$417$	1221.25i	0.143417i
$418$	287.053i	0.0335891i
$419$	−5732.23	−0.668348	−0.334174	−	0.942511i	$-0.608457\pi$
$419$	−5732.23	−0.668348	−0.334174	+	0.942511i	$0.608457\pi$
$420$	0	0
$421$	−9647.38	−1.11683	−0.558414	−	0.829563i	$-0.688590\pi$
$421$	−9647.38	−1.11683	−0.558414	+	0.829563i	$0.688590\pi$
$422$	6301.26i	0.726873i
$423$	− 284.265i	− 0.0326748i
$424$	−322.379	−0.0369248
$425$	0	0
$426$	−1078.60	−0.122672
$427$	− 2178.99i	− 0.246953i
$428$	8706.03i	0.983229i
$429$	1007.77	0.113417
$430$	0	0
$431$	−3054.57	−0.341376	−0.170688	−	0.985325i	$-0.554599\pi$
$431$	−3054.57	−0.341376	−0.170688	+	0.985325i	$0.554599\pi$
$432$	1657.03i	0.184546i
$433$	− 11022.8i	− 1.22337i	−0.791100	−	0.611686i	$-0.790491\pi$
$433$	− 11022.8i	− 1.22337i	0.791100	−	0.611686i	$-0.209509\pi$
$434$	−1799.65	−0.199046
$435$	0	0
$436$	2636.39	0.289588
$437$	− 698.399i	− 0.0764507i
$438$	− 8696.11i	− 0.948667i
$439$	7589.30	0.825097	0.412548	−	0.910936i	$-0.364639\pi$
$439$	7589.30	0.825097	0.412548	+	0.910936i	$0.364639\pi$
$440$	0	0
$441$	−2706.84	−0.292284
$442$	− 5320.47i	− 0.572554i
$443$	− 4043.30i	− 0.433641i	−0.976211	−	0.216821i	$-0.930431\pi$
$443$	− 4043.30i	− 0.433641i	0.976211	−	0.216821i	$-0.0695687\pi$
$444$	−10034.2	−1.07252
$445$	0	0
$446$	3054.59	0.324303
$447$	− 2376.28i	− 0.251441i
$448$	− 3585.16i	− 0.378086i
$449$	−14619.3	−1.53658	−0.768291	−	0.640101i	$-0.778893\pi$
$449$	−14619.3	−1.53658	−0.768291	+	0.640101i	$0.778893\pi$
$450$	0	0
$451$	−2952.96	−0.308313
$452$	11092.1i	1.15427i
$453$	− 8585.40i	− 0.890458i
$454$	23261.2	2.40463
$455$	0	0
$456$	24.7454	0.00254125
$457$	− 10910.8i	− 1.11682i	−0.829565	−	0.558411i	$-0.811411\pi$
$457$	− 10910.8i	− 1.11682i	0.829565	−	0.558411i	$-0.188589\pi$
$458$	7725.71i	0.788207i
$459$	−1164.55	−0.118424
$460$	0	0
$461$	16371.0	1.65395	0.826977	−	0.562236i	$-0.190059\pi$
$461$	16371.0	1.65395	0.826977	+	0.562236i	$0.190059\pi$
$462$	866.331i	0.0872410i
$463$	16023.5i	1.60837i	0.594377	+	0.804186i	$0.297399\pi$
$463$	16023.5i	1.60837i	−0.594377	+	0.804186i	$0.702601\pi$
$464$	−16868.4	−1.68770
$465$	0	0
$466$	4180.13	0.415538
$467$	− 15097.3i	− 1.49597i	−0.663716	−	0.747985i	$-0.731022\pi$
$467$	− 15097.3i	− 1.49597i	0.663716	−	0.747985i	$-0.268978\pi$
$468$	− 2285.65i	− 0.225757i
$469$	186.866	0.0183981
$470$	0	0
$471$	−8974.65	−0.877983
$472$	− 712.859i	− 0.0695170i
$473$	335.868i	0.0326495i
$474$	2432.09	0.235675
$475$	0	0
$476$	2331.17	0.224473
$477$	2272.47i	0.218133i
$478$	− 10693.0i	− 1.02319i
$479$	10459.0	0.997667	0.498834	−	0.866698i	$-0.333762\pi$
$479$	10459.0	0.997667	0.498834	+	0.866698i	$0.333762\pi$
$480$	0	0
$481$	−12282.6	−1.16432
$482$	27200.8i	2.57046i
$483$	− 2107.78i	− 0.198566i
$484$	−1006.25	−0.0945010
$485$	0	0
$486$	−981.554	−0.0916136
$487$	− 7939.86i	− 0.738788i	−0.929273	−	0.369394i	$-0.879565\pi$
$487$	− 7939.86i	− 0.738788i	0.929273	−	0.369394i	$-0.120435\pi$
$488$	− 428.061i	− 0.0397078i
$489$	8722.81	0.806665
$490$	0	0
$491$	11492.7	1.05633	0.528167	−	0.849141i	$-0.322880\pi$
$491$	11492.7	1.05633	0.528167	+	0.849141i	$0.322880\pi$
$492$	6697.37i	0.613701i
$493$	− 11855.0i	− 1.08300i
$494$	796.927	0.0725818
$495$	0	0
$496$	4207.13	0.380858
$497$	− 578.484i	− 0.0522103i
$498$	2952.91i	0.265709i
$499$	−4813.54	−0.431831	−0.215915	−	0.976412i	$-0.569274\pi$
$499$	−4813.54	−0.431831	−0.215915	+	0.976412i	$0.569274\pi$
$500$	0	0
$501$	−3301.17	−0.294382
$502$	− 21721.7i	− 1.93125i
$503$	− 17756.3i	− 1.57399i	−0.616962	−	0.786993i	$-0.711637\pi$
$503$	− 17756.3i	− 1.57399i	0.616962	−	0.786993i	$-0.288363\pi$
$504$	74.6820	0.00660040
$505$	0	0
$506$	4803.33	0.422004
$507$	3793.19i	0.332271i
$508$	− 4697.66i	− 0.410285i
$509$	−3183.05	−0.277183	−0.138592	−	0.990350i	$-0.544258\pi$
$509$	−3183.05	−0.277183	−0.138592	+	0.990350i	$0.544258\pi$
$510$	0	0
$511$	4663.99	0.403763
$512$	16467.6i	1.42143i
$513$	− 174.432i	− 0.0150124i
$514$	−23760.6	−2.03898
$515$	0	0
$516$	761.756	0.0649892
$517$	347.435i	0.0295554i
$518$	− 10558.7i	− 0.895605i
$519$	11264.3	0.952690
$520$	0	0
$521$	7246.60	0.609365	0.304683	−	0.952454i	$-0.401450\pi$
$521$	7246.60	0.609365	0.304683	+	0.952454i	$0.401450\pi$
$522$	− 9992.12i	− 0.837822i
$523$	17985.0i	1.50369i	0.659340	+	0.751845i	$0.270836\pi$
$523$	17985.0i	1.50369i	−0.659340	+	0.751845i	$0.729164\pi$
$524$	1780.32	0.148423
$525$	0	0
$526$	−16164.0	−1.33990
$527$	2956.74i	0.244398i
$528$	− 2025.26i	− 0.166928i
$529$	480.530	0.0394945
$530$	0	0
$531$	−5024.98	−0.410670
$532$	349.175i	0.0284561i
$533$	8198.09i	0.666226i
$534$	−3790.88	−0.307205
$535$	0	0
$536$	36.7097	0.00295825
$537$	9656.36i	0.775983i
$538$	7102.28i	0.569147i
$539$	3308.36	0.264381
$540$	0	0
$541$	−14549.3	−1.15624	−0.578119	−	0.815952i	$-0.696213\pi$
$541$	−14549.3	−1.15624	−0.578119	+	0.815952i	$0.696213\pi$
$542$	24310.4i	1.92661i
$543$	− 1370.65i	− 0.108325i
$544$	−11132.8	−0.877414
$545$	0	0
$546$	2405.13	0.188517
$547$	5487.29i	0.428920i	0.976733	+	0.214460i	$0.0687992\pi$
$547$	5487.29i	0.428920i	−0.976733	+	0.214460i	$0.931201\pi$
$548$	542.448i	0.0422851i
$549$	−3017.43	−0.234573
$550$	0	0
$551$	1775.70	0.137291
$552$	− 414.071i	− 0.0319276i
$553$	1304.41i	0.100306i
$554$	−11755.6	−0.901532
$555$	0	0
$556$	−3385.35	−0.258221
$557$	− 7366.60i	− 0.560382i	−0.959944	−	0.280191i	$-0.909602\pi$
$557$	− 7366.60i	− 0.560382i	0.959944	−	0.280191i	$-0.0903978\pi$
$558$	2492.13i	0.189068i
$559$	932.446	0.0705515
$560$	0	0
$561$	1423.34	0.107118
$562$	27843.8i	2.08989i
$563$	4516.14i	0.338068i	0.985610	+	0.169034i	$0.0540648\pi$
$563$	4516.14i	0.338068i	−0.985610	+	0.169034i	$0.945935\pi$
$564$	787.990	0.0588304
$565$	0	0
$566$	12603.3	0.935965
$567$	− 526.437i	− 0.0389917i
$568$	− 113.643i	− 0.00839496i
$569$	−2174.35	−0.160200	−0.0800999	−	0.996787i	$-0.525524\pi$
$569$	−2174.35	−0.160200	−0.0800999	+	0.996787i	$0.525524\pi$
$570$	0	0
$571$	−25339.7	−1.85715	−0.928575	−	0.371145i	$-0.878965\pi$
$571$	−25339.7	−1.85715	−0.928575	+	0.371145i	$0.878965\pi$
$572$	2793.57i	0.204205i
$573$	− 14721.3i	− 1.07329i
$574$	−7047.48	−0.512467
$575$	0	0
$576$	−4964.65	−0.359133
$577$	− 21909.1i	− 1.58074i	−0.612630	−	0.790370i	$-0.709888\pi$
$577$	− 21909.1i	− 1.58074i	0.612630	−	0.790370i	$-0.290112\pi$
$578$	12330.8i	0.887356i
$579$	4386.51	0.314848
$580$	0	0
$581$	−1583.74	−0.113089
$582$	− 339.298i	− 0.0241655i
$583$	− 2777.46i	− 0.197308i
$584$	916.237	0.0649215
$585$	0	0
$586$	35827.2	2.52561
$587$	− 922.235i	− 0.0648462i	−0.999474	−	0.0324231i	$-0.989678\pi$
$587$	− 922.235i	− 0.0648462i	0.999474	−	0.0324231i	$-0.0103224\pi$
$588$	− 7503.44i	− 0.526253i
$589$	−442.875	−0.0309819
$590$	0	0
$591$	−6630.31	−0.461480
$592$	24683.6i	1.71366i
$593$	9075.86i	0.628501i	0.949340	+	0.314250i	$0.101753\pi$
$593$	9075.86i	0.628501i	−0.949340	+	0.314250i	$0.898247\pi$
$594$	1199.68	0.0828676
$595$	0	0
$596$	6587.10	0.452715
$597$	1185.82i	0.0812935i
$598$	− 13335.2i	− 0.911898i
$599$	−19987.0	−1.36335	−0.681675	−	0.731655i	$-0.738748\pi$
$599$	−19987.0	−1.36335	−0.681675	+	0.731655i	$0.738748\pi$
$600$	0	0
$601$	11116.7	0.754509	0.377255	−	0.926110i	$-0.376868\pi$
$601$	11116.7	0.754509	0.377255	+	0.926110i	$0.376868\pi$
$602$	801.577i	0.0542688i
$603$	− 258.769i	− 0.0174758i
$604$	23799.0	1.60326
$605$	0	0
$606$	2766.26	0.185432
$607$	19355.0i	1.29423i	0.762393	+	0.647114i	$0.224024\pi$
$607$	19355.0i	1.29423i	−0.762393	+	0.647114i	$0.775976\pi$
$608$	− 1667.52i	− 0.111228i
$609$	5359.08	0.356586
$610$	0	0
$611$	964.559	0.0638656
$612$	− 3228.16i	− 0.213220i
$613$	12325.5i	0.812111i	0.913848	+	0.406056i	$0.133096\pi$
$613$	12325.5i	0.812111i	−0.913848	+	0.406056i	$0.866904\pi$
$614$	23078.3	1.51688
$615$	0	0
$616$	−91.2780	−0.00597029
$617$	28207.0i	1.84047i	0.391363	+	0.920236i	$0.372004\pi$
$617$	28207.0i	1.84047i	−0.391363	+	0.920236i	$0.627996\pi$
$618$	− 9698.34i	− 0.631269i
$619$	−17236.9	−1.11924	−0.559621	−	0.828748i	$-0.689053\pi$
$619$	−17236.9	−1.11924	−0.559621	+	0.828748i	$0.689053\pi$
$620$	0	0
$621$	−2918.81	−0.188611
$622$	8014.06i	0.516615i
$623$	− 2033.16i	− 0.130750i
$624$	−5622.58	−0.360711
$625$	0	0
$626$	−10807.2	−0.690003
$627$	213.194i	0.0135792i
$628$	− 24878.0i	− 1.58080i
$629$	−17347.4	−1.09966
$630$	0	0
$631$	−23582.7	−1.48782	−0.743908	−	0.668282i	$-0.767030\pi$
$631$	−23582.7	−1.48782	−0.743908	+	0.668282i	$0.767030\pi$
$632$	256.250i	0.0161283i
$633$	4679.94i	0.293856i
$634$	28613.6	1.79241
$635$	0	0
$636$	−6299.35	−0.392744
$637$	− 9184.77i	− 0.571294i
$638$	12212.6i	0.757839i
$639$	−801.073	−0.0495930
$640$	0	0
$641$	−13003.1	−0.801233	−0.400617	−	0.916246i	$-0.631204\pi$
$641$	−13003.1	−0.801233	−0.400617	+	0.916246i	$0.631204\pi$
$642$	12686.2i	0.779880i
$643$	21269.1i	1.30446i	0.758019	+	0.652232i	$0.226167\pi$
$643$	21269.1i	1.30446i	−0.758019	+	0.652232i	$0.773833\pi$
$644$	5842.82	0.357514
$645$	0	0
$646$	1125.55	0.0685511
$647$	29247.2i	1.77716i	0.458717	+	0.888582i	$0.348309\pi$
$647$	29247.2i	1.77716i	−0.458717	+	0.888582i	$0.651691\pi$
$648$	− 103.418i	− 0.00626952i
$649$	6141.64	0.371465
$650$	0	0
$651$	−1336.60	−0.0804694
$652$	24179.9i	1.45239i
$653$	1678.30i	0.100577i	0.998735	+	0.0502885i	$0.0160141\pi$
$653$	1678.30i	0.100577i	−0.998735	+	0.0502885i	$0.983986\pi$
$654$	3841.67	0.229696
$655$	0	0
$656$	16475.2	0.980561
$657$	− 6458.60i	− 0.383522i
$658$	829.182i	0.0491260i
$659$	7620.41	0.450454	0.225227	−	0.974306i	$-0.427688\pi$
$659$	7620.41	0.450454	0.225227	+	0.974306i	$0.427688\pi$
$660$	0	0
$661$	−22762.2	−1.33941	−0.669703	−	0.742629i	$-0.733579\pi$
$661$	−22762.2	−1.33941	−0.669703	+	0.742629i	$0.733579\pi$
$662$	31556.7i	1.85270i
$663$	− 3951.51i	− 0.231469i
$664$	−311.124	−0.0181837
$665$	0	0
$666$	−14621.5	−0.850708
$667$	− 29713.2i	− 1.72488i
$668$	− 9150.93i	− 0.530030i
$669$	2268.65	0.131107
$670$	0	0
$671$	3687.96	0.212179
$672$	− 5032.60i	− 0.288894i
$673$	− 698.528i	− 0.0400093i	−0.999800	−	0.0200047i	$-0.993632\pi$
$673$	− 698.528i	− 0.0400093i	0.999800	−	0.0200047i	$-0.00636811\pi$
$674$	47932.1	2.73928
$675$	0	0
$676$	−10514.8	−0.598250
$677$	− 10860.1i	− 0.616524i	−0.951302	−	0.308262i	$-0.900253\pi$
$677$	− 10860.1i	− 0.616524i	0.951302	−	0.308262i	$-0.0997473\pi$
$678$	16163.1i	0.915547i
$679$	181.976	0.0102851
$680$	0	0
$681$	17276.1	0.972131
$682$	− 3045.93i	− 0.171019i
$683$	6534.95i	0.366110i	0.983103	+	0.183055i	$0.0585986\pi$
$683$	6534.95i	0.366110i	−0.983103	+	0.183055i	$0.941401\pi$
$684$	483.530	0.0270296
$685$	0	0
$686$	16900.3	0.940606
$687$	5737.88i	0.318652i
$688$	− 1873.88i	− 0.103839i
$689$	−7710.88	−0.426359
$690$	0	0
$691$	−10915.3	−0.600924	−0.300462	−	0.953794i	$-0.597141\pi$
$691$	−10915.3	−0.600924	−0.300462	+	0.953794i	$0.597141\pi$
$692$	31224.9i	1.71530i
$693$	643.423i	0.0352693i
$694$	−14079.0	−0.770076
$695$	0	0
$696$	1052.79	0.0573359
$697$	11578.6i	0.629229i
$698$	25717.3i	1.39458i
$699$	3104.58	0.167991
$700$	0	0
$701$	18405.4	0.991674	0.495837	−	0.868416i	$-0.334861\pi$
$701$	18405.4	0.991674	0.495837	+	0.868416i	$0.334861\pi$
$702$	− 3330.58i	− 0.179067i
$703$	− 2598.39i	− 0.139403i
$704$	6067.91	0.324848
$705$	0	0
$706$	33215.9	1.77068
$707$	1483.63i	0.0789217i
$708$	− 13929.4i	− 0.739405i
$709$	21436.2	1.13547	0.567737	−	0.823210i	$-0.307819\pi$
$709$	21436.2	1.13547	0.567737	+	0.823210i	$0.307819\pi$
$710$	0	0
$711$	1806.32	0.0952773
$712$	− 399.413i	− 0.0210234i
$713$	7410.73i	0.389248i
$714$	3396.91	0.178048
$715$	0	0
$716$	−26767.7	−1.39715
$717$	− 7941.69i	− 0.413651i
$718$	37878.9i	1.96884i
$719$	−28641.7	−1.48561	−0.742807	−	0.669506i	$-0.766506\pi$
$719$	−28641.7	−1.48561	−0.742807	+	0.669506i	$0.766506\pi$
$720$	0	0
$721$	5201.51	0.268675
$722$	− 27537.1i	− 1.41942i
$723$	20202.0i	1.03917i
$724$	3799.49	0.195037
$725$	0	0
$726$	−1466.27	−0.0749566
$727$	− 1723.13i	− 0.0879055i	−0.999034	−	0.0439527i	$-0.986005\pi$
$727$	− 1723.13i	− 0.0879055i	0.999034	−	0.0439527i	$-0.0139951\pi$
$728$	253.409i	0.0129010i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1316.95	0.0666335
$732$	− 8364.39i	− 0.422345i
$733$	18813.4i	0.948005i	0.880523	+	0.474003i	$0.157191\pi$
$733$	18813.4i	0.948005i	−0.880523	+	0.474003i	$0.842809\pi$
$734$	1255.80	0.0631506
$735$	0	0
$736$	−27903.0	−1.39744
$737$	316.273i	0.0158074i
$738$	9759.21i	0.486777i
$739$	10893.3	0.542242	0.271121	−	0.962545i	$-0.412606\pi$
$739$	10893.3	0.542242	0.271121	+	0.962545i	$0.412606\pi$
$740$	0	0
$741$	591.877	0.0293430
$742$	− 6628.65i	− 0.327959i
$743$	− 11888.6i	− 0.587013i	−0.955957	−	0.293507i	$-0.905178\pi$
$743$	− 11888.6i	− 0.587013i	0.955957	−	0.293507i	$-0.0948223\pi$
$744$	−262.574	−0.0129388
$745$	0	0
$746$	−19140.4	−0.939381
$747$	2193.13i	0.107419i
$748$	3945.53i	0.192865i
$749$	−6803.98	−0.331925
$750$	0	0
$751$	−32134.8	−1.56140	−0.780702	−	0.624903i	$-0.785138\pi$
$751$	−32134.8	−1.56140	−0.780702	+	0.624903i	$0.785138\pi$
$752$	− 1938.41i	− 0.0939982i
$753$	− 16132.7i	− 0.780755i
$754$	33905.0	1.63759
$755$	0	0
$756$	1459.30	0.0702040
$757$	17366.1i	0.833794i	0.908954	+	0.416897i	$0.136882\pi$
$757$	17366.1i	0.833794i	−0.908954	+	0.416897i	$0.863118\pi$
$758$	19128.1i	0.916573i
$759$	3567.43	0.170605
$760$	0	0
$761$	−27393.9	−1.30490	−0.652450	−	0.757832i	$-0.726259\pi$
$761$	−27393.9	−1.30490	−0.652450	+	0.757832i	$0.726259\pi$
$762$	− 6845.28i	− 0.325431i
$763$	2060.41i	0.0977611i
$764$	40808.0	1.93243
$765$	0	0
$766$	14671.4	0.692034
$767$	− 17050.6i	− 0.802689i
$768$	11260.2i	0.529060i
$769$	12091.3	0.567000	0.283500	−	0.958972i	$-0.408504\pi$
$769$	12091.3	0.567000	0.283500	+	0.958972i	$0.408504\pi$
$770$	0	0
$771$	−17647.0	−0.824307
$772$	12159.5i	0.566879i
$773$	− 18895.8i	− 0.879216i	−0.898190	−	0.439608i	$-0.855117\pi$
$773$	− 18895.8i	− 0.879216i	0.898190	−	0.439608i	$-0.144883\pi$
$774$	1110.01	0.0515483
$775$	0	0
$776$	35.7490	0.00165375
$777$	− 7841.96i	− 0.362070i
$778$	− 26487.8i	− 1.22061i
$779$	−1734.31	−0.0797664
$780$	0	0
$781$	979.089	0.0448586
$782$	− 18834.0i	− 0.861258i
$783$	− 7421.14i	− 0.338710i
$784$	−18458.1	−0.840838
$785$	0	0
$786$	2594.23	0.117727
$787$	23705.9i	1.07373i	0.843669	+	0.536863i	$0.180391\pi$
$787$	23705.9i	1.07373i	−0.843669	+	0.536863i	$0.819609\pi$
$788$	− 18379.4i	− 0.830888i
$789$	−12005.0	−0.541685
$790$	0	0
$791$	−8668.77	−0.389666
$792$	126.400i	0.00567100i
$793$	− 10238.6i	− 0.458493i
$794$	−16500.5	−0.737507
$795$	0	0
$796$	−3287.11	−0.146368
$797$	− 37248.0i	− 1.65545i	−0.561135	−	0.827725i	$-0.689635\pi$
$797$	− 37248.0i	− 1.65545i	0.561135	−	0.827725i	$-0.310365\pi$
$798$	508.807i	0.0225709i
$799$	1362.30	0.0603189
$800$	0	0
$801$	−2815.49	−0.124195
$802$	− 26966.7i	− 1.18731i
$803$	7893.84i	0.346909i
$804$	717.315	0.0314648
$805$	0	0
$806$	−8456.21	−0.369550
$807$	5274.86i	0.230092i
$808$	291.458i	0.0126899i
$809$	24861.0	1.08043	0.540214	−	0.841528i	$-0.318343\pi$
$809$	24861.0	1.08043	0.540214	+	0.841528i	$0.318343\pi$
$810$	0	0
$811$	4131.24	0.178875	0.0894375	−	0.995992i	$-0.471493\pi$
$811$	4131.24	0.178875	0.0894375	+	0.995992i	$0.471493\pi$
$812$	14855.5i	0.642027i
$813$	18055.3i	0.778878i
$814$	17870.7	0.769495
$815$	0	0
$816$	−7941.10	−0.340679
$817$	197.259i	0.00844704i
$818$	− 6941.81i	− 0.296717i
$819$	1786.29	0.0762126
$820$	0	0
$821$	25297.3	1.07537	0.537687	−	0.843145i	$-0.319298\pi$
$821$	25297.3	1.07537	0.537687	+	0.843145i	$0.319298\pi$
$822$	790.439i	0.0335398i
$823$	18155.5i	0.768969i	0.923131	+	0.384484i	$0.125621\pi$
$823$	18155.5i	0.768969i	−0.923131	+	0.384484i	$0.874379\pi$
$824$	1021.83	0.0432005
$825$	0	0
$826$	14657.6	0.617435
$827$	− 29566.8i	− 1.24321i	−0.783329	−	0.621607i	$-0.786480\pi$
$827$	− 29566.8i	− 1.24321i	0.783329	−	0.621607i	$-0.213520\pi$
$828$	− 8091.02i	− 0.339592i
$829$	−27249.4	−1.14163	−0.570814	−	0.821079i	$-0.693372\pi$
$829$	−27249.4	−1.14163	−0.570814	+	0.821079i	$0.693372\pi$
$830$	0	0
$831$	−8730.90	−0.364467
$832$	− 16845.9i	− 0.701956i
$833$	− 12972.2i	− 0.539568i
$834$	−4933.02	−0.204816
$835$	0	0
$836$	−590.981	−0.0244492
$837$	1850.90i	0.0764355i
$838$	− 23154.3i	− 0.954478i
$839$	15506.1	0.638058	0.319029	−	0.947745i	$-0.396643\pi$
$839$	15506.1	0.638058	0.319029	+	0.947745i	$0.396643\pi$
$840$	0	0
$841$	51157.5	2.09756
$842$	− 38968.8i	− 1.59496i
$843$	20679.6i	0.844890i
$844$	−12972.9	−0.529084
$845$	0	0
$846$	1148.24	0.0466633
$847$	− 786.406i	− 0.0319023i
$848$	15496.1i	0.627520i
$849$	9360.47	0.378387
$850$	0	0
$851$	−43479.4	−1.75141
$852$	− 2220.60i	− 0.0892915i
$853$	− 8698.16i	− 0.349143i	−0.984644	−	0.174572i	$-0.944146\pi$
$853$	− 8698.16i	− 0.349143i	0.984644	−	0.174572i	$-0.0558541\pi$
$854$	8801.64	0.352677
$855$	0	0
$856$	−1336.64	−0.0533707
$857$	− 28468.8i	− 1.13474i	−0.823462	−	0.567372i	$-0.807960\pi$
$857$	− 28468.8i	− 1.13474i	0.823462	−	0.567372i	$-0.192040\pi$
$858$	4070.71i	0.161972i
$859$	21154.3	0.840251	0.420125	−	0.907466i	$-0.361986\pi$
$859$	21154.3	0.840251	0.420125	+	0.907466i	$0.361986\pi$
$860$	0	0
$861$	−5234.16	−0.207177
$862$	− 12338.4i	− 0.487525i
$863$	45959.5i	1.81284i	0.422378	+	0.906420i	$0.361195\pi$
$863$	45959.5i	1.81284i	−0.422378	+	0.906420i	$0.638805\pi$
$864$	−6969.04	−0.274412
$865$	0	0
$866$	44524.5	1.74712
$867$	9158.05i	0.358735i
$868$	− 3705.10i	− 0.144884i
$869$	−2207.72	−0.0861815
$870$	0	0
$871$	878.047	0.0341579
$872$	404.765i	0.0157191i
$873$	− 251.996i	− 0.00976951i
$874$	2821.06	0.109180
$875$	0	0
$876$	17903.4	0.690526
$877$	− 7315.10i	− 0.281657i	−0.990034	−	0.140829i	$-0.955023\pi$
$877$	− 7315.10i	− 0.281657i	0.990034	−	0.140829i	$-0.0449767\pi$
$878$	30655.6i	1.17833i
$879$	26608.9	1.02104
$880$	0	0
$881$	4652.19	0.177907	0.0889536	−	0.996036i	$-0.471648\pi$
$881$	4652.19	0.177907	0.0889536	+	0.996036i	$0.471648\pi$
$882$	− 10933.8i	− 0.417415i
$883$	38954.3i	1.48462i	0.670058	+	0.742309i	$0.266269\pi$
$883$	38954.3i	1.48462i	−0.670058	+	0.742309i	$0.733731\pi$
$884$	10953.7	0.416756
$885$	0	0
$886$	16332.2	0.619289
$887$	− 16917.5i	− 0.640398i	−0.947350	−	0.320199i	$-0.896250\pi$
$887$	− 16917.5i	− 0.640398i	0.947350	−	0.320199i	$-0.103750\pi$
$888$	− 1540.55i	− 0.0582177i
$889$	3671.33	0.138507
$890$	0	0
$891$	891.000	0.0335013
$892$	6288.75i	0.236057i
$893$	204.053i	0.00764654i
$894$	9598.53	0.359086
$895$	0	0
$896$	1061.32	0.0395715
$897$	− 9904.01i	− 0.368657i
$898$	− 59051.8i	− 2.19441i
$899$	−18842.0	−0.699016
$900$	0	0
$901$	−10890.5	−0.402682
$902$	− 11927.9i	− 0.440306i
$903$	595.331i	0.0219395i
$904$	−1702.97	−0.0626549
$905$	0	0
$906$	34679.2	1.27168
$907$	38985.4i	1.42722i	0.700543	+	0.713610i	$0.252941\pi$
$907$	38985.4i	1.42722i	−0.700543	+	0.713610i	$0.747059\pi$
$908$	47889.8i	1.75031i
$909$	2054.50	0.0749653
$910$	0	0
$911$	27759.8	1.00957	0.504787	−	0.863244i	$-0.331571\pi$
$911$	27759.8	1.00957	0.504787	+	0.863244i	$0.331571\pi$
$912$	− 1189.46i	− 0.0431874i
$913$	− 2680.49i	− 0.0971646i
$914$	44072.3	1.59495
$915$	0	0
$916$	−15905.6	−0.573728
$917$	1391.37i	0.0501057i
$918$	− 4703.98i	− 0.169122i
$919$	−29705.5	−1.06626	−0.533130	−	0.846033i	$-0.678984\pi$
$919$	−29705.5	−1.06626	−0.533130	+	0.846033i	$0.678984\pi$
$920$	0	0
$921$	17140.3	0.613236
$922$	66127.6i	2.36204i
$923$	− 2718.18i	− 0.0969338i
$924$	−1783.59	−0.0635019
$925$	0	0
$926$	−64724.1	−2.29694
$927$	− 7202.95i	− 0.255206i
$928$	− 70944.1i	− 2.50954i
$929$	−2940.01	−0.103831	−0.0519153	−	0.998651i	$-0.516533\pi$
$929$	−2940.01	−0.103831	−0.0519153	+	0.998651i	$0.516533\pi$
$930$	0	0
$931$	1943.04	0.0684002
$932$	8605.99i	0.302466i
$933$	5952.04i	0.208854i
$934$	60982.6	2.13642
$935$	0	0
$936$	350.916	0.0122543
$937$	− 20041.9i	− 0.698762i	−0.936981	−	0.349381i	$-0.886392\pi$
$937$	− 20041.9i	− 0.698762i	0.936981	−	0.349381i	$-0.113608\pi$
$938$	754.813i	0.0262745i
$939$	−8026.49	−0.278951
$940$	0	0
$941$	−14446.1	−0.500457	−0.250229	−	0.968187i	$-0.580506\pi$
$941$	−14446.1	−0.500457	−0.250229	+	0.968187i	$0.580506\pi$
$942$	− 36251.4i	− 1.25386i
$943$	29020.6i	1.00216i
$944$	−34265.6	−1.18141
$945$	0	0
$946$	−1356.68	−0.0466272
$947$	3901.33i	0.133871i	0.997757	+	0.0669357i	$0.0213223\pi$
$947$	3901.33i	0.133871i	−0.997757	+	0.0669357i	$0.978678\pi$
$948$	5007.16i	0.171545i
$949$	21915.1	0.749626
$950$	0	0
$951$	21251.3	0.724627
$952$	357.904i	0.0121846i
$953$	− 613.963i	− 0.0208691i	−0.999946	−	0.0104345i	$-0.996679\pi$
$953$	− 613.963i	− 0.0208691i	0.999946	−	0.0104345i	$-0.00332148\pi$
$954$	−9179.22	−0.311518
$955$	0	0
$956$	22014.6	0.744773
$957$	9070.29i	0.306375i
$958$	42247.1i	1.42478i
$959$	−423.937	−0.0142749
$960$	0	0
$961$	−25091.6	−0.842256
$962$	− 49613.3i	− 1.66278i
$963$	9422.02i	0.315286i
$964$	−56000.5	−1.87101
$965$	0	0
$966$	8513.98	0.283574
$967$	− 20617.4i	− 0.685637i	−0.939402	−	0.342819i	$-0.888618\pi$
$967$	− 20617.4i	− 0.685637i	0.939402	−	0.342819i	$-0.111382\pi$
$968$	− 154.489i	− 0.00512961i
$969$	835.943	0.0277135
$970$	0	0
$971$	22816.0	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$971$	22816.0	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$972$	− 2020.81i	− 0.0666846i
$973$	− 2645.73i	− 0.0871719i
$974$	32071.6	1.05507
$975$	0	0
$976$	−20576.0	−0.674816
$977$	22650.2i	0.741702i	0.928692	+	0.370851i	$0.120934\pi$
$977$	22650.2i	0.741702i	−0.928692	+	0.370851i	$0.879066\pi$
$978$	35234.2i	1.15201i
$979$	3441.15	0.112339
$980$	0	0
$981$	2853.21	0.0928604
$982$	46422.8i	1.50857i
$983$	8963.01i	0.290820i	0.989372	+	0.145410i	$0.0464501\pi$
$983$	8963.01i	0.290820i	−0.989372	+	0.145410i	$0.953550\pi$
$984$	−1028.25	−0.0333123
$985$	0	0
$986$	47886.0	1.54665
$987$	615.834i	0.0198604i
$988$	1640.70i	0.0528316i
$989$	3300.78	0.106126
$990$	0	0
$991$	422.204	0.0135336	0.00676678	−	0.999977i	$-0.497846\pi$
$991$	422.204	0.0135336	0.00676678	+	0.999977i	$0.497846\pi$
$992$	17694.1i	0.566319i
$993$	23437.2i	0.748999i
$994$	2336.68	0.0745623
$995$	0	0
$996$	−6079.41	−0.193407
$997$	− 12577.3i	− 0.399524i	−0.979844	−	0.199762i	$-0.935983\pi$
$997$	− 12577.3i	− 0.399524i	0.979844	−	0.199762i	$-0.0640169\pi$
$998$	− 19443.4i	− 0.616704i
$999$	−10859.4	−0.343920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.n.199.6		6
5.2	odd	4		825.4.a.o.1.1	✓	3
5.3	odd	4		825.4.a.q.1.3	yes	3
5.4	even	2	inner	825.4.c.n.199.1		6
15.2	even	4		2475.4.a.y.1.3		3
15.8	even	4		2475.4.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.1	✓	3	5.2	odd	4
825.4.a.q.1.3	yes	3	5.3	odd	4
825.4.c.n.199.1		6	5.4	even	2	inner
825.4.c.n.199.6		6	1.1	even	1	trivial
2475.4.a.v.1.1		3	15.8	even	4
2475.4.a.y.1.3		3	15.2	even	4

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.03932i q^{2} +3.00000i q^{3} -8.31608 q^{4} -12.1180 q^{6} -6.49923i q^{7} -1.27677i q^{8} -9.00000 q^{9} +11.0000 q^{11} -24.9483i q^{12} -30.5385i q^{13} +26.2524 q^{14} -61.3714 q^{16} -43.1314i q^{17} -36.3539i q^{18} +6.46044 q^{19} +19.4977 q^{21} +44.4325i q^{22} -108.104i q^{23} +3.83030 q^{24} +123.355 q^{26} -27.0000i q^{27} +54.0481i q^{28} +274.857 q^{29} -68.5519 q^{31} -258.113i q^{32} +33.0000i q^{33} +174.221 q^{34} +74.8448 q^{36} -402.200i q^{37} +26.0958i q^{38} +91.6156 q^{39} -268.450 q^{41} +78.7573i q^{42} +30.5334i q^{43} -91.4769 q^{44} +436.666 q^{46} +31.5850i q^{47} -184.114i q^{48} +300.760 q^{49} +129.394 q^{51} +253.961i q^{52} -252.497i q^{53} +109.062 q^{54} -8.29800 q^{56} +19.3813i q^{57} +1110.24i q^{58} +558.331 q^{59} +335.270 q^{61} -276.903i q^{62} +58.4930i q^{63} +551.628 q^{64} -133.297 q^{66} +28.7521i q^{67} +358.684i q^{68} +324.312 q^{69} +89.0081 q^{71} +11.4909i q^{72} +717.622i q^{73} +1624.61 q^{74} -53.7256 q^{76} -71.4915i q^{77} +370.065i q^{78} -200.702 q^{79} +81.0000 q^{81} -1084.36i q^{82} -243.681i q^{83} -162.144 q^{84} -123.334 q^{86} +824.571i q^{87} -14.0444i q^{88} +312.832 q^{89} -198.477 q^{91} +899.002i q^{92} -205.656i q^{93} -127.582 q^{94} +774.338 q^{96} +27.9996i q^{97} +1214.87i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} + 232 q^{14} - 170 q^{16} + 338 q^{19} - 96 q^{21} - 18 q^{24} + 334 q^{26} + 554 q^{29} - 346 q^{31} + 292 q^{34} + 126 q^{36} + 270 q^{39} + 88 q^{41} - 154 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) \) 6 * q - 14 * q^4 - 6 * q^6 - 54 * q^9 + 66 * q^11 + 232 * q^14 - 170 * q^16 + 338 * q^19 - 96 * q^21 - 18 * q^24 + 334 * q^26 + 554 * q^29 - 346 * q^31 + 292 * q^34 + 126 * q^36 + 270 * q^39 + 88 * q^41 - 154 * q^44 + 850 * q^46 + 854 * q^49 + 348 * q^51 + 54 * q^54 + 336 * q^56 + 1368 * q^59 - 2076 * q^61 + 2258 * q^64 - 66 * q^66 + 930 * q^69 - 2918 * q^71 + 3528 * q^74 + 422 * q^76 + 1012 * q^79 + 486 * q^81 + 600 * q^84 - 1054 * q^86 + 1214 * q^89 - 796 * q^91 + 3644 * q^94 - 582 * q^96 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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