LMFDB - Embedded newform 882.5.c.c.685.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,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, 5, names="a")

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
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:	$91.1723074400$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.4
Root			$-0.765367i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.5.c.c.685.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +8.00000 q^{4} +27.8477i q^{5} +22.6274 q^{8} +O(q^{10})$	$q+2.82843 q^{2} +8.00000 q^{4} +27.8477i q^{5} +22.6274 q^{8} +78.7652i q^{10} +139.664 q^{11} -101.931i q^{13} +64.0000 q^{16} +542.887i q^{17} +139.953i q^{19} +222.782i q^{20} +395.029 q^{22} -229.647 q^{23} -150.495 q^{25} -288.303i q^{26} +383.113 q^{29} -397.492i q^{31} +181.019 q^{32} +1535.52i q^{34} -898.912 q^{37} +395.848i q^{38} +630.122i q^{40} +657.862i q^{41} +1155.70 q^{43} +1117.31 q^{44} -649.539 q^{46} -1294.51i q^{47} -425.663 q^{50} -815.445i q^{52} +5168.43 q^{53} +3889.32i q^{55} +1083.61 q^{58} +4155.24i q^{59} -1840.09i q^{61} -1124.28i q^{62} +512.000 q^{64} +2838.53 q^{65} -6315.57 q^{67} +4343.09i q^{68} -1263.28 q^{71} +3995.44i q^{73} -2542.51 q^{74} +1119.63i q^{76} -10614.3 q^{79} +1782.25i q^{80} +1860.71i q^{82} +5750.31i q^{83} -15118.2 q^{85} +3268.82 q^{86} +3160.24 q^{88} +4187.05i q^{89} -1837.17 q^{92} -3661.42i q^{94} -3897.38 q^{95} +3814.28i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4}+O(q^{10}) $ 4 * q + 32 * q^4	$ 4 q + 32 q^{4} - 24 q^{11} + 256 q^{16} + 1648 q^{22} - 104 q^{23} + 948 q^{25} + 1408 q^{29} - 3392 q^{37} - 2024 q^{43} - 192 q^{44} - 2304 q^{46} - 4384 q^{50} + 16680 q^{53} + 352 q^{58} + 2048 q^{64} + 6048 q^{65} - 20816 q^{67} + 1984 q^{71} - 576 q^{74} - 29616 q^{79} - 29688 q^{85} + 18800 q^{86} + 13184 q^{88} - 832 q^{92} - 7240 q^{95}+O(q^{100}) $ 4 * q + 32 * q^4 - 24 * q^11 + 256 * q^16 + 1648 * q^22 - 104 * q^23 + 948 * q^25 + 1408 * q^29 - 3392 * q^37 - 2024 * q^43 - 192 * q^44 - 2304 * q^46 - 4384 * q^50 + 16680 * q^53 + 352 * q^58 + 2048 * q^64 + 6048 * q^65 - 20816 * q^67 + 1984 * q^71 - 576 * q^74 - 29616 * q^79 - 29688 * q^85 + 18800 * q^86 + 13184 * q^88 - 832 * q^92 - 7240 * 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 $

$2$	2.82843	0.707107
$2$	2.82843	0.707107
$3$	0	0
$3$	0	0
$4$	8.00000	0.500000
$5$	27.8477i	1.11391i	0.830543	+	0.556954i	$0.188030\pi$
$5$	27.8477i	1.11391i	−0.830543	+	0.556954i	$0.811970\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274	0.353553
$9$	0	0
$10$	78.7652i	0.787652i
$11$	139.664	1.15425	0.577124	−	0.816657i	$-0.304175\pi$
$11$	139.664	1.15425	0.577124	+	0.816657i	$0.304175\pi$
$12$	0	0
$13$	− 101.931i	− 0.603140i	−0.953444	−	0.301570i	$-0.902489\pi$
$13$	− 101.931i	− 0.603140i	0.953444	−	0.301570i	$-0.0975106\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	542.887i	1.87850i	0.343232	+	0.939251i	$0.388478\pi$
$17$	542.887i	1.87850i	−0.343232	+	0.939251i	$0.611522\pi$
$18$	0	0
$19$	139.953i	0.387682i	0.981033	+	0.193841i	$0.0620947\pi$
$19$	139.953i	0.387682i	−0.981033	+	0.193841i	$0.937905\pi$
$20$	222.782i	0.556954i
$21$	0	0
$22$	395.029	0.816177
$23$	−229.647	−0.434115	−0.217057	−	0.976159i	$-0.569646\pi$
$23$	−229.647	−0.434115	−0.217057	+	0.976159i	$0.569646\pi$
$24$	0	0
$25$	−150.495	−0.240791
$26$	− 288.303i	− 0.426484i
$27$	0	0
$28$	0	0
$29$	383.113	0.455544	0.227772	−	0.973714i	$-0.426856\pi$
$29$	383.113	0.455544	0.227772	+	0.973714i	$0.426856\pi$
$30$	0	0
$31$	− 397.492i	− 0.413623i	−0.978381	−	0.206812i	$-0.933691\pi$
$31$	− 397.492i	− 0.413623i	0.978381	−	0.206812i	$-0.0663087\pi$
$32$	181.019	0.176777
$33$	0	0
$34$	1535.52i	1.32830i
$35$	0	0
$36$	0	0
$37$	−898.912	−0.656619	−0.328310	−	0.944570i	$-0.606479\pi$
$37$	−898.912	−0.656619	−0.328310	+	0.944570i	$0.606479\pi$
$38$	395.848i	0.274133i
$39$	0	0
$40$	630.122i	0.393826i
$41$	657.862i	0.391352i	0.980669	+	0.195676i	$0.0626900\pi$
$41$	657.862i	0.391352i	−0.980669	+	0.195676i	$0.937310\pi$
$42$	0	0
$43$	1155.70	0.625041	0.312521	−	0.949911i	$-0.398827\pi$
$43$	1155.70	0.625041	0.312521	+	0.949911i	$0.398827\pi$
$44$	1117.31	0.577124
$45$	0	0
$46$	−649.539	−0.306966
$47$	− 1294.51i	− 0.586015i	−0.956110	−	0.293007i	$-0.905344\pi$
$47$	− 1294.51i	− 0.586015i	0.956110	−	0.293007i	$-0.0946561\pi$
$48$	0	0
$49$	0	0
$50$	−425.663	−0.170265
$51$	0	0
$52$	− 815.445i	− 0.301570i
$53$	5168.43	1.83996	0.919978	−	0.391971i	$-0.128207\pi$
$53$	5168.43	1.83996	0.919978	+	0.391971i	$0.128207\pi$
$54$	0	0
$55$	3889.32i	1.28573i
$56$	0	0
$57$	0	0
$58$	1083.61	0.322118
$59$	4155.24i	1.19369i	0.802356	+	0.596846i	$0.203580\pi$
$59$	4155.24i	1.19369i	−0.802356	+	0.596846i	$0.796420\pi$
$60$	0	0
$61$	− 1840.09i	− 0.494514i	−0.968950	−	0.247257i	$-0.920471\pi$
$61$	− 1840.09i	− 0.494514i	0.968950	−	0.247257i	$-0.0795292\pi$
$62$	− 1124.28i	− 0.292476i
$63$	0	0
$64$	512.000	0.125000
$65$	2838.53	0.671842
$66$	0	0
$67$	−6315.57	−1.40690	−0.703450	−	0.710745i	$-0.748358\pi$
$67$	−6315.57	−1.40690	−0.703450	+	0.710745i	$0.748358\pi$
$68$	4343.09i	0.939251i
$69$	0	0
$70$	0	0
$71$	−1263.28	−0.250601	−0.125301	−	0.992119i	$-0.539990\pi$
$71$	−1263.28	−0.250601	−0.125301	+	0.992119i	$0.539990\pi$
$72$	0	0
$73$	3995.44i	0.749753i	0.927075	+	0.374877i	$0.122315\pi$
$73$	3995.44i	0.749753i	−0.927075	+	0.374877i	$0.877685\pi$
$74$	−2542.51	−0.464300
$75$	0	0
$76$	1119.63i	0.193841i
$77$	0	0
$78$	0	0
$79$	−10614.3	−1.70073	−0.850366	−	0.526192i	$-0.823619\pi$
$79$	−10614.3	−1.70073	−0.850366	+	0.526192i	$0.823619\pi$
$80$	1782.25i	0.278477i
$81$	0	0
$82$	1860.71i	0.276727i
$83$	5750.31i	0.834709i	0.908744	+	0.417355i	$0.137043\pi$
$83$	5750.31i	0.834709i	−0.908744	+	0.417355i	$0.862957\pi$
$84$	0	0
$85$	−15118.2	−2.09248
$86$	3268.82	0.441971
$87$	0	0
$88$	3160.24	0.408088
$89$	4187.05i	0.528601i	0.964440	+	0.264301i	$0.0851411\pi$
$89$	4187.05i	0.528601i	−0.964440	+	0.264301i	$0.914859\pi$
$90$	0	0
$91$	0	0
$92$	−1837.17	−0.217057
$93$	0	0
$94$	− 3661.42i	− 0.414375i
$95$	−3897.38	−0.431843
$96$	0	0
$97$	3814.28i	0.405386i	0.979242	+	0.202693i	$0.0649694\pi$
$97$	3814.28i	0.405386i	−0.979242	+	0.202693i	$0.935031\pi$
$98$	0	0
$99$	0	0
$100$	−1203.96	−0.120396
$101$	4925.41i	0.482836i	0.970421	+	0.241418i	$0.0776126\pi$
$101$	4925.41i	0.482836i	−0.970421	+	0.241418i	$0.922387\pi$
$102$	0	0
$103$	18892.6i	1.78081i	0.455166	+	0.890407i	$0.349580\pi$
$103$	18892.6i	1.78081i	−0.455166	+	0.890407i	$0.650420\pi$
$104$	− 2306.43i	− 0.213242i
$105$	0	0
$106$	14618.5	1.30104
$107$	8479.39	0.740623	0.370311	−	0.928908i	$-0.379251\pi$
$107$	8479.39	0.740623	0.370311	+	0.928908i	$0.379251\pi$
$108$	0	0
$109$	11678.0	0.982911	0.491455	−	0.870903i	$-0.336465\pi$
$109$	11678.0	0.982911	0.491455	+	0.870903i	$0.336465\pi$
$110$	11000.7i	0.909146i
$111$	0	0
$112$	0	0
$113$	−19683.8	−1.54153	−0.770763	−	0.637121i	$-0.780125\pi$
$113$	−19683.8	−1.54153	−0.770763	+	0.637121i	$0.780125\pi$
$114$	0	0
$115$	− 6395.13i	− 0.483564i
$116$	3064.90	0.227772
$117$	0	0
$118$	11752.8i	0.844067i
$119$	0	0
$120$	0	0
$121$	4865.03	0.332288
$122$	− 5204.55i	− 0.349674i
$123$	0	0
$124$	− 3179.94i	− 0.206812i
$125$	13213.9i	0.845689i
$126$	0	0
$127$	−27965.5	−1.73386	−0.866931	−	0.498428i	$-0.833911\pi$
$127$	−27965.5	−1.73386	−0.866931	+	0.498428i	$0.833911\pi$
$128$	1448.15	0.0883883
$129$	0	0
$130$	8028.58	0.475064
$131$	1169.73i	0.0681622i	0.999419	+	0.0340811i	$0.0108505\pi$
$131$	1169.73i	0.0681622i	−0.999419	+	0.0340811i	$0.989150\pi$
$132$	0	0
$133$	0	0
$134$	−17863.1	−0.994828
$135$	0	0
$136$	12284.1i	0.664150i
$137$	20884.6	1.11272	0.556359	−	0.830942i	$-0.312198\pi$
$137$	20884.6	1.11272	0.556359	+	0.830942i	$0.312198\pi$
$138$	0	0
$139$	3499.87i	0.181143i	0.995890	+	0.0905716i	$0.0288694\pi$
$139$	3499.87i	0.181143i	−0.995890	+	0.0905716i	$0.971131\pi$
$140$	0	0
$141$	0	0
$142$	−3573.10	−0.177202
$143$	− 14236.0i	− 0.696173i
$144$	0	0
$145$	10668.8i	0.507434i
$146$	11300.8i	0.530156i
$147$	0	0
$148$	−7191.29	−0.328310
$149$	−29533.8	−1.33029	−0.665146	−	0.746713i	$-0.731631\pi$
$149$	−29533.8	−1.33029	−0.665146	+	0.746713i	$0.731631\pi$
$150$	0	0
$151$	25780.8	1.13069	0.565344	−	0.824856i	$-0.308744\pi$
$151$	25780.8	1.13069	0.565344	+	0.824856i	$0.308744\pi$
$152$	3166.78i	0.137066i
$153$	0	0
$154$	0	0
$155$	11069.2	0.460738
$156$	0	0
$157$	− 25629.4i	− 1.03977i	−0.854235	−	0.519887i	$-0.825974\pi$
$157$	− 25629.4i	− 1.03977i	0.854235	−	0.519887i	$-0.174026\pi$
$158$	−30021.7	−1.20260
$159$	0	0
$160$	5040.97i	0.196913i
$161$	0	0
$162$	0	0
$163$	−6182.78	−0.232707	−0.116353	−	0.993208i	$-0.537120\pi$
$163$	−6182.78	−0.232707	−0.116353	+	0.993208i	$0.537120\pi$
$164$	5262.90i	0.195676i
$165$	0	0
$166$	16264.3i	0.590229i
$167$	39270.9i	1.40811i	0.710144	+	0.704056i	$0.248630\pi$
$167$	39270.9i	1.40811i	−0.710144	+	0.704056i	$0.751370\pi$
$168$	0	0
$169$	18171.2	0.636223
$170$	−42760.6	−1.47961
$171$	0	0
$172$	9245.61	0.312521
$173$	− 32905.9i	− 1.09946i	−0.835341	−	0.549732i	$-0.814730\pi$
$173$	− 32905.9i	− 1.09946i	0.835341	−	0.549732i	$-0.185270\pi$
$174$	0	0
$175$	0	0
$176$	8938.50	0.288562
$177$	0	0
$178$	11842.8i	0.373777i
$179$	4767.88	0.148806	0.0744029	−	0.997228i	$-0.476295\pi$
$179$	4767.88	0.148806	0.0744029	+	0.997228i	$0.476295\pi$
$180$	0	0
$181$	35195.4i	1.07431i	0.843484	+	0.537154i	$0.180501\pi$
$181$	35195.4i	1.07431i	−0.843484	+	0.537154i	$0.819499\pi$
$182$	0	0
$183$	0	0
$184$	−5196.31	−0.153483
$185$	− 25032.6i	− 0.731413i
$186$	0	0
$187$	75821.7i	2.16826i
$188$	− 10356.1i	− 0.293007i
$189$	0	0
$190$	−11023.5	−0.305359
$191$	−23315.1	−0.639101	−0.319551	−	0.947569i	$-0.603532\pi$
$191$	−23315.1	−0.639101	−0.319551	+	0.947569i	$0.603532\pi$
$192$	0	0
$193$	59183.8	1.58887	0.794434	−	0.607350i	$-0.207768\pi$
$193$	59183.8	1.58887	0.794434	+	0.607350i	$0.207768\pi$
$194$	10788.4i	0.286651i
$195$	0	0
$196$	0	0
$197$	60902.2	1.56928	0.784640	−	0.619952i	$-0.212848\pi$
$197$	60902.2	1.56928	0.784640	+	0.619952i	$0.212848\pi$
$198$	0	0
$199$	40116.7i	1.01302i	0.862233	+	0.506512i	$0.169065\pi$
$199$	40116.7i	1.01302i	−0.862233	+	0.506512i	$0.830935\pi$
$200$	−3405.30	−0.0851326
$201$	0	0
$202$	13931.2i	0.341417i
$203$	0	0
$204$	0	0
$205$	−18319.9	−0.435930
$206$	53436.5i	1.25923i
$207$	0	0
$208$	− 6523.56i	− 0.150785i
$209$	19546.4i	0.447482i
$210$	0	0
$211$	21929.0	0.492554	0.246277	−	0.969200i	$-0.420793\pi$
$211$	21929.0	0.492554	0.246277	+	0.969200i	$0.420793\pi$
$212$	41347.5	0.919978
$213$	0	0
$214$	23983.3	0.523699
$215$	32183.6i	0.696238i
$216$	0	0
$217$	0	0
$218$	33030.3	0.695023
$219$	0	0
$220$	31114.6i	0.642863i
$221$	55336.8	1.13300
$222$	0	0
$223$	− 64358.0i	− 1.29418i	−0.762416	−	0.647088i	$-0.775987\pi$
$223$	− 64358.0i	− 1.29418i	0.762416	−	0.647088i	$-0.224013\pi$
$224$	0	0
$225$	0	0
$226$	−55674.1	−1.09002
$227$	− 43774.8i	− 0.849517i	−0.905307	−	0.424759i	$-0.860359\pi$
$227$	− 43774.8i	− 0.849517i	0.905307	−	0.424759i	$-0.139641\pi$
$228$	0	0
$229$	28506.4i	0.543590i	0.962355	+	0.271795i	$0.0876173\pi$
$229$	28506.4i	0.543590i	−0.962355	+	0.271795i	$0.912383\pi$
$230$	− 18088.2i	− 0.341931i
$231$	0	0
$232$	8668.85	0.161059
$233$	2194.89	0.0404298	0.0202149	−	0.999796i	$-0.493565\pi$
$233$	2194.89	0.0404298	0.0202149	+	0.999796i	$0.493565\pi$
$234$	0	0
$235$	36049.0	0.652767
$236$	33241.9i	0.596846i
$237$	0	0
$238$	0	0
$239$	−80942.6	−1.41704	−0.708519	−	0.705692i	$-0.750636\pi$
$239$	−80942.6	−1.41704	−0.708519	+	0.705692i	$0.750636\pi$
$240$	0	0
$241$	− 43016.6i	− 0.740631i	−0.928906	−	0.370316i	$-0.879249\pi$
$241$	− 43016.6i	− 0.740631i	0.928906	−	0.370316i	$-0.120751\pi$
$242$	13760.4	0.234963
$243$	0	0
$244$	− 14720.7i	− 0.247257i
$245$	0	0
$246$	0	0
$247$	14265.5	0.233827
$248$	− 8994.21i	− 0.146238i
$249$	0	0
$250$	37374.5i	0.597992i
$251$	− 31568.5i	− 0.501080i	−0.968106	−	0.250540i	$-0.919392\pi$
$251$	− 31568.5i	− 0.501080i	0.968106	−	0.250540i	$-0.0806082\pi$
$252$	0	0
$253$	−32073.4	−0.501076
$254$	−79098.3	−1.22603
$255$	0	0
$256$	4096.00	0.0625000
$257$	68105.9i	1.03114i	0.856847	+	0.515571i	$0.172420\pi$
$257$	68105.9i	1.03114i	−0.856847	+	0.515571i	$0.827580\pi$
$258$	0	0
$259$	0	0
$260$	22708.3	0.335921
$261$	0	0
$262$	3308.50i	0.0481980i
$263$	8347.56	0.120684	0.0603418	−	0.998178i	$-0.480781\pi$
$263$	8347.56	0.120684	0.0603418	+	0.998178i	$0.480781\pi$
$264$	0	0
$265$	143929.i	2.04954i
$266$	0	0
$267$	0	0
$268$	−50524.6	−0.703450
$269$	− 91921.0i	− 1.27031i	−0.772384	−	0.635156i	$-0.780936\pi$
$269$	− 91921.0i	− 1.27031i	0.772384	−	0.635156i	$-0.219064\pi$
$270$	0	0
$271$	− 5134.36i	− 0.0699114i	−0.999389	−	0.0349557i	$-0.988871\pi$
$271$	− 5134.36i	− 0.0699114i	0.999389	−	0.0349557i	$-0.0111290\pi$
$272$	34744.8i	0.469625i
$273$	0	0
$274$	59070.5	0.786810
$275$	−21018.7	−0.277933
$276$	0	0
$277$	−66254.2	−0.863483	−0.431741	−	0.901997i	$-0.642101\pi$
$277$	−66254.2	−0.863483	−0.431741	+	0.901997i	$0.642101\pi$
$278$	9899.12i	0.128088i
$279$	0	0
$280$	0	0
$281$	−38156.4	−0.483230	−0.241615	−	0.970372i	$-0.577677\pi$
$281$	−38156.4	−0.483230	−0.241615	+	0.970372i	$0.577677\pi$
$282$	0	0
$283$	− 110233.i	− 1.37638i	−0.725528	−	0.688192i	$-0.758405\pi$
$283$	− 110233.i	− 1.37638i	0.725528	−	0.688192i	$-0.241595\pi$
$284$	−10106.3	−0.125301
$285$	0	0
$286$	− 40265.6i	− 0.492268i
$287$	0	0
$288$	0	0
$289$	−211205.	−2.52877
$290$	30175.9i	0.358810i
$291$	0	0
$292$	31963.5i	0.374877i
$293$	− 79011.7i	− 0.920357i	−0.887826	−	0.460179i	$-0.847785\pi$
$293$	− 79011.7i	− 0.920357i	0.887826	−	0.460179i	$-0.152215\pi$
$294$	0	0
$295$	−115714.	−1.32966
$296$	−20340.0	−0.232150
$297$	0	0
$298$	−83534.2	−0.940659
$299$	23408.0i	0.261832i
$300$	0	0
$301$	0	0
$302$	72919.1	0.799517
$303$	0	0
$304$	8957.01i	0.0969206i
$305$	51242.2	0.550843
$306$	0	0
$307$	− 2801.57i	− 0.0297252i	−0.999890	−	0.0148626i	$-0.995269\pi$
$307$	− 2801.57i	− 0.0297252i	0.999890	−	0.0148626i	$-0.00473109\pi$
$308$	0	0
$309$	0	0
$310$	31308.5	0.325791
$311$	− 20659.3i	− 0.213597i	−0.994281	−	0.106799i	$-0.965940\pi$
$311$	− 20659.3i	− 0.213597i	0.994281	−	0.106799i	$-0.0340600\pi$
$312$	0	0
$313$	− 37376.1i	− 0.381509i	−0.981638	−	0.190755i	$-0.938907\pi$
$313$	− 37376.1i	− 0.381509i	0.981638	−	0.190755i	$-0.0610935\pi$
$314$	− 72490.8i	− 0.735231i
$315$	0	0
$316$	−84914.1	−0.850366
$317$	106666.	1.06147	0.530735	−	0.847538i	$-0.321916\pi$
$317$	106666.	1.06147	0.530735	+	0.847538i	$0.321916\pi$
$318$	0	0
$319$	53507.1	0.525811
$320$	14258.0i	0.139239i
$321$	0	0
$322$	0	0
$323$	−75978.8	−0.728262
$324$	0	0
$325$	15340.0i	0.145231i
$326$	−17487.5	−0.164548
$327$	0	0
$328$	14885.7i	0.138364i
$329$	0	0
$330$	0	0
$331$	108363.	0.989063	0.494532	−	0.869160i	$-0.335340\pi$
$331$	108363.	0.989063	0.494532	+	0.869160i	$0.335340\pi$
$332$	46002.5i	0.417355i
$333$	0	0
$334$	111075.i	0.995686i
$335$	− 175874.i	− 1.56716i
$336$	0	0
$337$	186946.	1.64610	0.823050	−	0.567968i	$-0.192270\pi$
$337$	186946.	1.64610	0.823050	+	0.567968i	$0.192270\pi$
$338$	51395.8	0.449877
$339$	0	0
$340$	−120945.	−1.04624
$341$	− 55515.3i	− 0.477424i
$342$	0	0
$343$	0	0
$344$	26150.5	0.220985
$345$	0	0
$346$	− 93071.9i	− 0.777439i
$347$	63227.3	0.525105	0.262552	−	0.964918i	$-0.415436\pi$
$347$	63227.3	0.525105	0.262552	+	0.964918i	$0.415436\pi$
$348$	0	0
$349$	93605.0i	0.768508i	0.923227	+	0.384254i	$0.125541\pi$
$349$	93605.0i	0.768508i	−0.923227	+	0.384254i	$0.874459\pi$
$350$	0	0
$351$	0	0
$352$	25281.9	0.204044
$353$	143251.i	1.14960i	0.818294	+	0.574800i	$0.194920\pi$
$353$	143251.i	1.14960i	−0.818294	+	0.574800i	$0.805080\pi$
$354$	0	0
$355$	− 35179.5i	− 0.279147i
$356$	33496.4i	0.264301i
$357$	0	0
$358$	13485.6	0.105222
$359$	−102340.	−0.794065	−0.397032	−	0.917805i	$-0.629960\pi$
$359$	−102340.	−0.794065	−0.397032	+	0.917805i	$0.629960\pi$
$360$	0	0
$361$	110734.	0.849702
$362$	99547.7i	0.759651i
$363$	0	0
$364$	0	0
$365$	−111264.	−0.835156
$366$	0	0
$367$	− 15366.4i	− 0.114088i	−0.998372	−	0.0570442i	$-0.981832\pi$
$367$	− 15366.4i	− 0.114088i	0.998372	−	0.0570442i	$-0.0181676\pi$
$368$	−14697.4	−0.108529
$369$	0	0
$370$	− 70803.0i	− 0.517187i
$371$	0	0
$372$	0	0
$373$	97490.8	0.700722	0.350361	−	0.936615i	$-0.386059\pi$
$373$	97490.8	0.700722	0.350361	+	0.936615i	$0.386059\pi$
$374$	214456.i	1.53319i
$375$	0	0
$376$	− 29291.3i	− 0.207188i
$377$	− 39050.9i	− 0.274757i
$378$	0	0
$379$	−66090.7	−0.460110	−0.230055	−	0.973178i	$-0.573891\pi$
$379$	−66090.7	−0.460110	−0.230055	+	0.973178i	$0.573891\pi$
$380$	−31179.0	−0.215921
$381$	0	0
$382$	−65944.9	−0.451913
$383$	− 208901.i	− 1.42411i	−0.702126	−	0.712053i	$-0.747766\pi$
$383$	− 208901.i	− 1.42411i	0.702126	−	0.712053i	$-0.252234\pi$
$384$	0	0
$385$	0	0
$386$	167397.	1.12350
$387$	0	0
$388$	30514.2i	0.202693i
$389$	143999.	0.951614	0.475807	−	0.879550i	$-0.342156\pi$
$389$	143999.	0.951614	0.475807	+	0.879550i	$0.342156\pi$
$390$	0	0
$391$	− 124672.i	− 0.815485i
$392$	0	0
$393$	0	0
$394$	172257.	1.10965
$395$	− 295583.i	− 1.89446i
$396$	0	0
$397$	− 284577.i	− 1.80559i	−0.430070	−	0.902796i	$-0.641511\pi$
$397$	− 284577.i	− 1.80559i	0.430070	−	0.902796i	$-0.358489\pi$
$398$	113467.i	0.716316i
$399$	0	0
$400$	−9631.65	−0.0601978
$401$	107846.	0.670680	0.335340	−	0.942097i	$-0.391149\pi$
$401$	107846.	0.670680	0.335340	+	0.942097i	$0.391149\pi$
$402$	0	0
$403$	−40516.6	−0.249472
$404$	39403.3i	0.241418i
$405$	0	0
$406$	0	0
$407$	−125546.	−0.757901
$408$	0	0
$409$	− 84932.2i	− 0.507722i	−0.967241	−	0.253861i	$-0.918299\pi$
$409$	− 84932.2i	− 0.507722i	0.967241	−	0.253861i	$-0.0817005\pi$
$410$	−51816.6	−0.308249
$411$	0	0
$412$	151141.i	0.890407i
$413$	0	0
$414$	0	0
$415$	−160133.	−0.929789
$416$	− 18451.4i	− 0.106621i
$417$	0	0
$418$	55285.7i	0.316417i
$419$	331905.i	1.89054i	0.326292	+	0.945269i	$0.394201\pi$
$419$	331905.i	1.89054i	−0.326292	+	0.945269i	$0.605799\pi$
$420$	0	0
$421$	101490.	0.572609	0.286304	−	0.958139i	$-0.407573\pi$
$421$	101490.	0.572609	0.286304	+	0.958139i	$0.407573\pi$
$422$	62024.5	0.348288
$423$	0	0
$424$	116948.	0.650522
$425$	− 81701.5i	− 0.452327i
$426$	0	0
$427$	0	0
$428$	67835.1	0.370311
$429$	0	0
$430$	91029.0i	0.492315i
$431$	−229798.	−1.23706	−0.618530	−	0.785761i	$-0.712272\pi$
$431$	−229798.	−1.23706	−0.618530	+	0.785761i	$0.712272\pi$
$432$	0	0
$433$	− 114264.i	− 0.609442i	−0.952442	−	0.304721i	$-0.901437\pi$
$433$	− 114264.i	− 0.609442i	0.952442	−	0.304721i	$-0.0985632\pi$
$434$	0	0
$435$	0	0
$436$	93423.7	0.491455
$437$	− 32139.8i	− 0.168299i
$438$	0	0
$439$	− 352339.i	− 1.82823i	−0.405453	−	0.914116i	$-0.632886\pi$
$439$	− 352339.i	− 1.82823i	0.405453	−	0.914116i	$-0.367114\pi$
$440$	88005.3i	0.454573i
$441$	0	0
$442$	156516.	0.801151
$443$	196316.	1.00034	0.500172	−	0.865926i	$-0.333270\pi$
$443$	196316.	1.00034	0.500172	+	0.865926i	$0.333270\pi$
$444$	0	0
$445$	−116600.	−0.588813
$446$	− 182032.i	− 0.915120i
$447$	0	0
$448$	0	0
$449$	−77520.3	−0.384523	−0.192262	−	0.981344i	$-0.561582\pi$
$449$	−77520.3	−0.384523	−0.192262	+	0.981344i	$0.561582\pi$
$450$	0	0
$451$	91879.6i	0.451717i
$452$	−157470.	−0.770763
$453$	0	0
$454$	− 123814.i	− 0.600699i
$455$	0	0
$456$	0	0
$457$	189872.	0.909137	0.454568	−	0.890712i	$-0.349794\pi$
$457$	189872.	0.909137	0.454568	+	0.890712i	$0.349794\pi$
$458$	80628.3i	0.384376i
$459$	0	0
$460$	− 51161.1i	− 0.241782i
$461$	52599.9i	0.247505i	0.992313	+	0.123752i	$0.0394928\pi$
$461$	52599.9i	0.247505i	−0.992313	+	0.123752i	$0.960507\pi$
$462$	0	0
$463$	−244268.	−1.13947	−0.569736	−	0.821828i	$-0.692955\pi$
$463$	−244268.	−1.13947	−0.569736	+	0.821828i	$0.692955\pi$
$464$	24519.2	0.113886
$465$	0	0
$466$	6208.09	0.0285882
$467$	− 297615.i	− 1.36465i	−0.731050	−	0.682324i	$-0.760969\pi$
$467$	− 297615.i	− 1.36465i	0.731050	−	0.682324i	$-0.239031\pi$
$468$	0	0
$469$	0	0
$470$	101962.	0.461576
$471$	0	0
$472$	94022.3i	0.422034i
$473$	161410.	0.721452
$474$	0	0
$475$	− 21062.2i	− 0.0933505i
$476$	0	0
$477$	0	0
$478$	−228940.	−1.00200
$479$	− 46870.8i	− 0.204283i	−0.994770	−	0.102141i	$-0.967431\pi$
$479$	− 46870.8i	− 0.204283i	0.994770	−	0.102141i	$-0.0325694\pi$
$480$	0	0
$481$	91626.6i	0.396033i
$482$	− 121669.i	− 0.523706i
$483$	0	0
$484$	38920.3	0.166144
$485$	−106219.	−0.451563
$486$	0	0
$487$	107184.	0.451930	0.225965	−	0.974135i	$-0.427446\pi$
$487$	107184.	0.451930	0.225965	+	0.974135i	$0.427446\pi$
$488$	− 41636.4i	− 0.174837i
$489$	0	0
$490$	0	0
$491$	116367.	0.482687	0.241344	−	0.970440i	$-0.422412\pi$
$491$	116367.	0.482687	0.241344	+	0.970440i	$0.422412\pi$
$492$	0	0
$493$	207987.i	0.855740i
$494$	40349.0	0.165340
$495$	0	0
$496$	− 25439.5i	− 0.103406i
$497$	0	0
$498$	0	0
$499$	442908.	1.77874	0.889369	−	0.457190i	$-0.151144\pi$
$499$	442908.	1.77874	0.889369	+	0.457190i	$0.151144\pi$
$500$	105711.i	0.422844i
$501$	0	0
$502$	− 89289.3i	− 0.354317i
$503$	68955.1i	0.272540i	0.990672	+	0.136270i	$0.0435115\pi$
$503$	68955.1i	0.272540i	−0.990672	+	0.136270i	$0.956489\pi$
$504$	0	0
$505$	−137161.	−0.537835
$506$	−90717.2	−0.354314
$507$	0	0
$508$	−223724.	−0.866931
$509$	92773.5i	0.358087i	0.983841	+	0.179044i	$0.0573003\pi$
$509$	92773.5i	0.358087i	−0.983841	+	0.179044i	$0.942700\pi$
$510$	0	0
$511$	0	0
$512$	11585.2	0.0441942
$513$	0	0
$514$	192632.i	0.729127i
$515$	−526117.	−1.98366
$516$	0	0
$517$	− 180796.i	− 0.676406i
$518$	0	0
$519$	0	0
$520$	64228.7	0.237532
$521$	142912.i	0.526495i	0.964728	+	0.263248i	$0.0847936\pi$
$521$	142912.i	0.526495i	−0.964728	+	0.263248i	$0.915206\pi$
$522$	0	0
$523$	22203.5i	0.0811742i	0.999176	+	0.0405871i	$0.0129228\pi$
$523$	22203.5i	0.0811742i	−0.999176	+	0.0405871i	$0.987077\pi$
$524$	9357.85i	0.0340811i
$525$	0	0
$526$	23610.5	0.0853361
$527$	215793.	0.776992
$528$	0	0
$529$	−227103.	−0.811544
$530$	407093.i	1.44924i
$531$	0	0
$532$	0	0
$533$	67056.3	0.236040
$534$	0	0
$535$	236132.i	0.824986i
$536$	−142905.	−0.497414
$537$	0	0
$538$	− 259992.i	− 0.898246i
$539$	0	0
$540$	0	0
$541$	211209.	0.721636	0.360818	−	0.932636i	$-0.382498\pi$
$541$	211209.	0.721636	0.360818	+	0.932636i	$0.382498\pi$
$542$	− 14522.2i	− 0.0494348i
$543$	0	0
$544$	98273.0i	0.332075i
$545$	325204.i	1.09487i
$546$	0	0
$547$	254227.	0.849664	0.424832	−	0.905272i	$-0.360333\pi$
$547$	254227.	0.849664	0.424832	+	0.905272i	$0.360333\pi$
$548$	167077.	0.556359
$549$	0	0
$550$	−59449.8	−0.196528
$551$	53617.9i	0.176606i
$552$	0	0
$553$	0	0
$554$	−187395.	−0.610575
$555$	0	0
$556$	27998.9i	0.0905716i
$557$	−46139.2	−0.148717	−0.0743584	−	0.997232i	$-0.523691\pi$
$557$	−46139.2	−0.148717	−0.0743584	+	0.997232i	$0.523691\pi$
$558$	0	0
$559$	− 117801.i	− 0.376987i
$560$	0	0
$561$	0	0
$562$	−107922.	−0.341695
$563$	− 349629.i	− 1.10304i	−0.834162	−	0.551519i	$-0.814048\pi$
$563$	− 349629.i	− 1.10304i	0.834162	−	0.551519i	$-0.185952\pi$
$564$	0	0
$565$	− 548147.i	− 1.71712i
$566$	− 311787.i	− 0.973251i
$567$	0	0
$568$	−28584.8	−0.0886010
$569$	176196.	0.544217	0.272109	−	0.962267i	$-0.412279\pi$
$569$	176196.	0.544217	0.272109	+	0.962267i	$0.412279\pi$
$570$	0	0
$571$	−176527.	−0.541426	−0.270713	−	0.962660i	$-0.587259\pi$
$571$	−176527.	−0.541426	−0.270713	+	0.962660i	$0.587259\pi$
$572$	− 113888.i	− 0.348086i
$573$	0	0
$574$	0	0
$575$	34560.6	0.104531
$576$	0	0
$577$	− 226863.i	− 0.681416i	−0.940169	−	0.340708i	$-0.889333\pi$
$577$	− 226863.i	− 0.681416i	0.940169	−	0.340708i	$-0.110667\pi$
$578$	−597378.	−1.78811
$579$	0	0
$580$	85350.5i	0.253717i
$581$	0	0
$582$	0	0
$583$	721844.	2.12376
$584$	90406.4i	0.265078i
$585$	0	0
$586$	− 223479.i	− 0.650791i
$587$	− 349659.i	− 1.01477i	−0.861719	−	0.507386i	$-0.830612\pi$
$587$	− 349659.i	− 1.01477i	0.861719	−	0.507386i	$-0.169388\pi$
$588$	0	0
$589$	55630.3	0.160354
$590$	−327288.	−0.940213
$591$	0	0
$592$	−57530.3	−0.164155
$593$	− 137755.i	− 0.391739i	−0.980630	−	0.195870i	$-0.937247\pi$
$593$	− 137755.i	− 0.391739i	0.980630	−	0.195870i	$-0.0627529\pi$
$594$	0	0
$595$	0	0
$596$	−236271.	−0.665146
$597$	0	0
$598$	66207.9i	0.185143i
$599$	−80877.2	−0.225410	−0.112705	−	0.993629i	$-0.535951\pi$
$599$	−80877.2	−0.225410	−0.112705	+	0.993629i	$0.535951\pi$
$600$	0	0
$601$	− 228798.i	− 0.633438i	−0.948519	−	0.316719i	$-0.897419\pi$
$601$	− 228798.i	− 0.633438i	0.948519	−	0.316719i	$-0.102581\pi$
$602$	0	0
$603$	0	0
$604$	206246.	0.565344
$605$	135480.i	0.370139i
$606$	0	0
$607$	265951.i	0.721812i	0.932602	+	0.360906i	$0.117532\pi$
$607$	265951.i	0.721812i	−0.932602	+	0.360906i	$0.882468\pi$
$608$	25334.3i	0.0685332i
$609$	0	0
$610$	144935.	0.389505
$611$	−131950.	−0.353449
$612$	0	0
$613$	−306522.	−0.815718	−0.407859	−	0.913045i	$-0.633725\pi$
$613$	−306522.	−0.815718	−0.407859	+	0.913045i	$0.633725\pi$
$614$	− 7924.04i	− 0.0210189i
$615$	0	0
$616$	0	0
$617$	−491906.	−1.29215	−0.646074	−	0.763275i	$-0.723590\pi$
$617$	−491906.	−1.29215	−0.646074	+	0.763275i	$0.723590\pi$
$618$	0	0
$619$	− 115230.i	− 0.300735i	−0.988630	−	0.150368i	$-0.951954\pi$
$619$	− 115230.i	− 0.300735i	0.988630	−	0.150368i	$-0.0480457\pi$
$620$	88553.9	0.230369
$621$	0	0
$622$	− 58433.5i	− 0.151036i
$623$	0	0
$624$	0	0
$625$	−462035.	−1.18281
$626$	− 105716.i	− 0.269768i
$627$	0	0
$628$	− 205035.i	− 0.519887i
$629$	− 488007.i	− 1.23346i
$630$	0	0
$631$	72815.9	0.182880	0.0914402	−	0.995811i	$-0.470853\pi$
$631$	72815.9	0.182880	0.0914402	+	0.995811i	$0.470853\pi$
$632$	−240173.	−0.601299
$633$	0	0
$634$	301697.	0.750572
$635$	− 778774.i	− 1.93136i
$636$	0	0
$637$	0	0
$638$	151341.	0.371805
$639$	0	0
$640$	40327.8i	0.0984565i
$641$	757794.	1.84431	0.922157	−	0.386815i	$-0.126425\pi$
$641$	757794.	1.84431	0.922157	+	0.386815i	$0.126425\pi$
$642$	0	0
$643$	− 353082.i	− 0.853992i	−0.904254	−	0.426996i	$-0.859572\pi$
$643$	− 353082.i	− 0.853992i	0.904254	−	0.426996i	$-0.140428\pi$
$644$	0	0
$645$	0	0
$646$	−214901.	−0.514959
$647$	214561.i	0.512557i	0.966603	+	0.256279i	$0.0824965\pi$
$647$	214561.i	0.512557i	−0.966603	+	0.256279i	$0.917504\pi$
$648$	0	0
$649$	580337.i	1.37782i
$650$	43388.1i	0.102694i
$651$	0	0
$652$	−49462.3	−0.116353
$653$	497591.	1.16693	0.583467	−	0.812137i	$-0.301696\pi$
$653$	497591.	1.16693	0.583467	+	0.812137i	$0.301696\pi$
$654$	0	0
$655$	−32574.3	−0.0759264
$656$	42103.2i	0.0978379i
$657$	0	0
$658$	0	0
$659$	197464.	0.454692	0.227346	−	0.973814i	$-0.426995\pi$
$659$	197464.	0.454692	0.227346	+	0.973814i	$0.426995\pi$
$660$	0	0
$661$	− 668423.i	− 1.52985i	−0.644121	−	0.764924i	$-0.722777\pi$
$661$	− 668423.i	− 1.52985i	0.644121	−	0.764924i	$-0.277223\pi$
$662$	306496.	0.699373
$663$	0	0
$664$	130115.i	0.295114i
$665$	0	0
$666$	0	0
$667$	−87980.6	−0.197759
$668$	314167.i	0.704056i
$669$	0	0
$670$	− 497447.i	− 1.10815i
$671$	− 256994.i	− 0.570792i
$672$	0	0
$673$	−404534.	−0.893151	−0.446575	−	0.894746i	$-0.647357\pi$
$673$	−404534.	−0.893151	−0.446575	+	0.894746i	$0.647357\pi$
$674$	528763.	1.16397
$675$	0	0
$676$	145369.	0.318111
$677$	211169.i	0.460737i	0.973103	+	0.230369i	$0.0739932\pi$
$677$	211169.i	0.460737i	−0.973103	+	0.230369i	$0.926007\pi$
$678$	0	0
$679$	0	0
$680$	−342085.	−0.739803
$681$	0	0
$682$	− 157021.i	− 0.337590i
$683$	170109.	0.364657	0.182329	−	0.983238i	$-0.441637\pi$
$683$	170109.	0.364657	0.182329	+	0.983238i	$0.441637\pi$
$684$	0	0
$685$	581588.i	1.23946i
$686$	0	0
$687$	0	0
$688$	73964.9	0.156260
$689$	− 526822.i	− 1.10975i
$690$	0	0
$691$	− 168630.i	− 0.353167i	−0.984286	−	0.176583i	$-0.943496\pi$
$691$	− 168630.i	− 0.353167i	0.984286	−	0.176583i	$-0.0565045\pi$
$692$	− 263247.i	− 0.549732i
$693$	0	0
$694$	178834.	0.371305
$695$	−97463.3	−0.201777
$696$	0	0
$697$	−357145.	−0.735154
$698$	264755.i	0.543417i
$699$	0	0
$700$	0	0
$701$	487646.	0.992359	0.496179	−	0.868220i	$-0.334736\pi$
$701$	487646.	0.992359	0.496179	+	0.868220i	$0.334736\pi$
$702$	0	0
$703$	− 125806.i	− 0.254560i
$704$	71508.0	0.144281
$705$	0	0
$706$	405174.i	0.812890i
$707$	0	0
$708$	0	0
$709$	−386929.	−0.769731	−0.384865	−	0.922973i	$-0.625752\pi$
$709$	−386929.	−0.769731	−0.384865	+	0.922973i	$0.625752\pi$
$710$	− 99502.6i	− 0.197387i
$711$	0	0
$712$	94742.1i	0.186889i
$713$	91282.7i	0.179560i
$714$	0	0
$715$	396441.	0.775472
$716$	38143.1	0.0744029
$717$	0	0
$718$	−289461.	−0.561489
$719$	188374.i	0.364388i	0.983263	+	0.182194i	$0.0583199\pi$
$719$	188374.i	0.364388i	−0.983263	+	0.182194i	$0.941680\pi$
$720$	0	0
$721$	0	0
$722$	313203.	0.600830
$723$	0	0
$724$	281563.i	0.537154i
$725$	−57656.4	−0.109691
$726$	0	0
$727$	− 901538.i	− 1.70575i	−0.522115	−	0.852875i	$-0.674857\pi$
$727$	− 901538.i	− 1.70575i	0.522115	−	0.852875i	$-0.325143\pi$
$728$	0	0
$729$	0	0
$730$	−314701.	−0.590545
$731$	627415.i	1.17414i
$732$	0	0
$733$	− 402482.i	− 0.749098i	−0.927207	−	0.374549i	$-0.877798\pi$
$733$	− 402482.i	− 0.749098i	0.927207	−	0.374549i	$-0.122202\pi$
$734$	− 43462.9i	− 0.0806726i
$735$	0	0
$736$	−41570.5	−0.0767414
$737$	−882058.	−1.62391
$738$	0	0
$739$	−37301.1	−0.0683020	−0.0341510	−	0.999417i	$-0.510873\pi$
$739$	−37301.1	−0.0683020	−0.0341510	+	0.999417i	$0.510873\pi$
$740$	− 200261.i	− 0.365707i
$741$	0	0
$742$	0	0
$743$	831221.	1.50570	0.752851	−	0.658191i	$-0.228678\pi$
$743$	831221.	1.50570	0.752851	+	0.658191i	$0.228678\pi$
$744$	0	0
$745$	− 822449.i	− 1.48182i
$746$	275746.	0.495486
$747$	0	0
$748$	606574.i	1.08413i
$749$	0	0
$750$	0	0
$751$	−251098.	−0.445209	−0.222605	−	0.974909i	$-0.571456\pi$
$751$	−251098.	−0.445209	−0.222605	+	0.974909i	$0.571456\pi$
$752$	− 82848.4i	− 0.146504i
$753$	0	0
$754$	− 110453.i	− 0.194282i
$755$	717936.i	1.25948i
$756$	0	0
$757$	−422399.	−0.737108	−0.368554	−	0.929606i	$-0.620147\pi$
$757$	−422399.	−0.737108	−0.368554	+	0.929606i	$0.620147\pi$
$758$	−186933.	−0.325347
$759$	0	0
$760$	−88187.6	−0.152679
$761$	247203.i	0.426859i	0.976958	+	0.213430i	$0.0684634\pi$
$761$	247203.i	0.426859i	−0.976958	+	0.213430i	$0.931537\pi$
$762$	0	0
$763$	0	0
$764$	−186520.	−0.319551
$765$	0	0
$766$	− 590860.i	− 1.00699i
$767$	423546.	0.719962
$768$	0	0
$769$	903767.i	1.52828i	0.645049	+	0.764141i	$0.276837\pi$
$769$	903767.i	1.52828i	−0.645049	+	0.764141i	$0.723163\pi$
$770$	0	0
$771$	0	0
$772$	473470.	0.794434
$773$	− 244667.i	− 0.409464i	−0.978818	−	0.204732i	$-0.934368\pi$
$773$	− 244667.i	− 0.409464i	0.978818	−	0.204732i	$-0.0656323\pi$
$774$	0	0
$775$	59820.4i	0.0995968i
$776$	86307.3i	0.143326i
$777$	0	0
$778$	407291.	0.672893
$779$	−92070.0	−0.151720
$780$	0	0
$781$	−176435.	−0.289256
$782$	− 352626.i	− 0.576635i
$783$	0	0
$784$	0	0
$785$	713719.	1.15821
$786$	0	0
$787$	950431.i	1.53452i	0.641339	+	0.767258i	$0.278379\pi$
$787$	950431.i	1.53452i	−0.641339	+	0.767258i	$0.721621\pi$
$788$	487218.	0.784640
$789$	0	0
$790$	− 836035.i	− 1.33958i
$791$	0	0
$792$	0	0
$793$	−187561.	−0.298261
$794$	− 804907.i	− 1.27675i
$795$	0	0
$796$	320934.i	0.506512i
$797$	− 560412.i	− 0.882248i	−0.897446	−	0.441124i	$-0.854580\pi$
$797$	− 560412.i	− 0.882248i	0.897446	−	0.441124i	$-0.145420\pi$
$798$	0	0
$799$	702771.	1.10083
$800$	−27242.4	−0.0425663
$801$	0	0
$802$	305035.	0.474242
$803$	558019.i	0.865401i
$804$	0	0
$805$	0	0
$806$	−114598.	−0.176404
$807$	0	0
$808$	111449.i	0.170708i
$809$	51864.2	0.0792447	0.0396224	−	0.999215i	$-0.487385\pi$
$809$	51864.2	0.0792447	0.0396224	+	0.999215i	$0.487385\pi$
$810$	0	0
$811$	978879.i	1.48829i	0.668019	+	0.744145i	$0.267143\pi$
$811$	978879.i	1.48829i	−0.668019	+	0.744145i	$0.732857\pi$
$812$	0	0
$813$	0	0
$814$	−355097.	−0.535917
$815$	− 172176.i	− 0.259214i
$816$	0	0
$817$	161744.i	0.242317i
$818$	− 240224.i	− 0.359013i
$819$	0	0
$820$	−146560.	−0.217965
$821$	−148798.	−0.220755	−0.110377	−	0.993890i	$-0.535206\pi$
$821$	−148798.	−0.220755	−0.110377	+	0.993890i	$0.535206\pi$
$822$	0	0
$823$	1.33907e6	1.97698	0.988490	−	0.151287i	$-0.0483416\pi$
$823$	1.33907e6	1.97698	0.988490	+	0.151287i	$0.0483416\pi$
$824$	427492.i	0.629613i
$825$	0	0
$826$	0	0
$827$	127284.	0.186108	0.0930538	−	0.995661i	$-0.470337\pi$
$827$	127284.	0.186108	0.0930538	+	0.995661i	$0.470337\pi$
$828$	0	0
$829$	445598.i	0.648386i	0.945991	+	0.324193i	$0.105093\pi$
$829$	445598.i	0.648386i	−0.945991	+	0.324193i	$0.894907\pi$
$830$	−452924.	−0.657460
$831$	0	0
$832$	− 52188.5i	− 0.0753924i
$833$	0	0
$834$	0	0
$835$	−1.09360e6	−1.56851
$836$	156372.i	0.223741i
$837$	0	0
$838$	938768.i	1.33681i
$839$	− 552232.i	− 0.784509i	−0.919857	−	0.392254i	$-0.871695\pi$
$839$	− 552232.i	− 0.784509i	0.919857	−	0.392254i	$-0.128305\pi$
$840$	0	0
$841$	−560506.	−0.792479
$842$	287056.	0.404896
$843$	0	0
$844$	175432.	0.246277
$845$	506025.i	0.708694i
$846$	0	0
$847$	0	0
$848$	330780.	0.459989
$849$	0	0
$850$	− 231087.i	− 0.319843i
$851$	206432.	0.285048
$852$	0	0
$853$	− 753308.i	− 1.03532i	−0.855586	−	0.517660i	$-0.826803\pi$
$853$	− 753308.i	− 1.03532i	0.855586	−	0.517660i	$-0.173197\pi$
$854$	0	0
$855$	0	0
$856$	191867.	0.261850
$857$	1.30117e6i	1.77163i	0.464037	+	0.885816i	$0.346400\pi$
$857$	1.30117e6i	1.77163i	−0.464037	+	0.885816i	$0.653600\pi$
$858$	0	0
$859$	− 982372.i	− 1.33134i	−0.746245	−	0.665671i	$-0.768145\pi$
$859$	− 982372.i	− 1.33134i	0.746245	−	0.665671i	$-0.231855\pi$
$860$	257469.i	0.348119i
$861$	0	0
$862$	−649966.	−0.874734
$863$	721435.	0.968670	0.484335	−	0.874883i	$-0.339062\pi$
$863$	721435.	0.968670	0.484335	+	0.874883i	$0.339062\pi$
$864$	0	0
$865$	916353.	1.22470
$866$	− 323186.i	− 0.430940i
$867$	0	0
$868$	0	0
$869$	−1.48243e6	−1.96307
$870$	0	0
$871$	643750.i	0.848557i
$872$	264242.	0.347511
$873$	0	0
$874$	− 90905.2i	− 0.119005i
$875$	0	0
$876$	0	0
$877$	−717008.	−0.932234	−0.466117	−	0.884723i	$-0.654347\pi$
$877$	−717008.	−0.932234	−0.466117	+	0.884723i	$0.654347\pi$
$878$	− 996564.i	− 1.29276i
$879$	0	0
$880$	248917.i	0.321432i
$881$	79128.9i	0.101949i	0.998700	+	0.0509745i	$0.0162327\pi$
$881$	79128.9i	0.101949i	−0.998700	+	0.0509745i	$0.983767\pi$
$882$	0	0
$883$	−286591.	−0.367571	−0.183786	−	0.982966i	$-0.558835\pi$
$883$	−286591.	−0.367571	−0.183786	+	0.982966i	$0.558835\pi$
$884$	442694.	0.566499
$885$	0	0
$886$	555267.	0.707350
$887$	− 837465.i	− 1.06444i	−0.846607	−	0.532218i	$-0.821359\pi$
$887$	− 837465.i	− 1.06444i	0.846607	−	0.532218i	$-0.178641\pi$
$888$	0	0
$889$	0	0
$890$	−329794.	−0.416354
$891$	0	0
$892$	− 514864.i	− 0.647088i
$893$	181171.	0.227188
$894$	0	0
$895$	132775.i	0.165756i
$896$	0	0
$897$	0	0
$898$	−219261.	−0.271899
$899$	− 152284.i	− 0.188424i
$900$	0	0
$901$	2.80588e6i	3.45636i
$902$	259875.i	0.319412i
$903$	0	0
$904$	−445393.	−0.545012
$905$	−980112.	−1.19668
$906$	0	0
$907$	−846569.	−1.02908	−0.514538	−	0.857467i	$-0.672037\pi$
$907$	−846569.	−1.02908	−0.514538	+	0.857467i	$0.672037\pi$
$908$	− 350198.i	− 0.424759i
$909$	0	0
$910$	0	0
$911$	−167017.	−0.201244	−0.100622	−	0.994925i	$-0.532083\pi$
$911$	−167017.	−0.201244	−0.100622	+	0.994925i	$0.532083\pi$
$912$	0	0
$913$	803112.i	0.963461i
$914$	537040.	0.642857
$915$	0	0
$916$	228051.i	0.271795i
$917$	0	0
$918$	0	0
$919$	1.12203e6	1.32854	0.664268	−	0.747494i	$-0.268743\pi$
$919$	1.12203e6	1.32854	0.664268	+	0.747494i	$0.268743\pi$
$920$	− 144705.i	− 0.170966i
$921$	0	0
$922$	148775.i	0.175012i
$923$	128767.i	0.151148i
$924$	0	0
$925$	135281.	0.158108
$926$	−690893.	−0.805729
$927$	0	0
$928$	69350.8	0.0805296
$929$	1.42663e6i	1.65302i	0.562921	+	0.826511i	$0.309677\pi$
$929$	1.42663e6i	1.65302i	−0.562921	+	0.826511i	$0.690323\pi$
$930$	0	0
$931$	0	0
$932$	17559.1	0.0202149
$933$	0	0
$934$	− 841781.i	− 0.964952i
$935$	−2.11146e6	−2.41524
$936$	0	0
$937$	265587.i	0.302502i	0.988495	+	0.151251i	$0.0483301\pi$
$937$	265587.i	0.302502i	−0.988495	+	0.151251i	$0.951670\pi$
$938$	0	0
$939$	0	0
$940$	288392.	0.326383
$941$	− 598591.i	− 0.676007i	−0.941145	−	0.338003i	$-0.890249\pi$
$941$	− 598591.i	− 0.676007i	0.941145	−	0.338003i	$-0.109751\pi$
$942$	0	0
$943$	− 151076.i	− 0.169892i
$944$	265935.i	0.298423i
$945$	0	0
$946$	456536.	0.510144
$947$	−1.15085e6	−1.28328	−0.641639	−	0.767007i	$-0.721745\pi$
$947$	−1.15085e6	−1.28328	−0.641639	+	0.767007i	$0.721745\pi$
$948$	0	0
$949$	407257.	0.452206
$950$	− 59572.9i	− 0.0660088i
$951$	0	0
$952$	0	0
$953$	−464604.	−0.511561	−0.255781	−	0.966735i	$-0.582332\pi$
$953$	−464604.	−0.511561	−0.255781	+	0.966735i	$0.582332\pi$
$954$	0	0
$955$	− 649271.i	− 0.711900i
$956$	−647541.	−0.708519
$957$	0	0
$958$	− 132571.i	− 0.144450i
$959$	0	0
$960$	0	0
$961$	765521.	0.828916
$962$	259159.i	0.280038i
$963$	0	0
$964$	− 344133.i	− 0.370316i
$965$	1.64813e6i	1.76985i
$966$	0	0
$967$	101298.	0.108330	0.0541648	−	0.998532i	$-0.482750\pi$
$967$	101298.	0.108330	0.0541648	+	0.998532i	$0.482750\pi$
$968$	110083.	0.117482
$969$	0	0
$970$	−300432.	−0.319303
$971$	326280.i	0.346060i	0.984917	+	0.173030i	$0.0553558\pi$
$971$	326280.i	0.346060i	−0.984917	+	0.173030i	$0.944644\pi$
$972$	0	0
$973$	0	0
$974$	303162.	0.319563
$975$	0	0
$976$	− 117765.i	− 0.123628i
$977$	−741928.	−0.777272	−0.388636	−	0.921391i	$-0.627054\pi$
$977$	−741928.	−0.777272	−0.388636	+	0.921391i	$0.627054\pi$
$978$	0	0
$979$	584780.i	0.610137i
$980$	0	0
$981$	0	0
$982$	329135.	0.341311
$983$	584497.i	0.604888i	0.953167	+	0.302444i	$0.0978026\pi$
$983$	584497.i	0.604888i	−0.953167	+	0.302444i	$0.902197\pi$
$984$	0	0
$985$	1.69599e6i	1.74803i
$986$	588276.i	0.605100i
$987$	0	0
$988$	114124.	0.116913
$989$	−265403.	−0.271340
$990$	0	0
$991$	582343.	0.592969	0.296484	−	0.955038i	$-0.404186\pi$
$991$	582343.	0.592969	0.296484	+	0.955038i	$0.404186\pi$
$992$	− 71953.7i	− 0.0731189i
$993$	0	0
$994$	0	0
$995$	−1.11716e6	−1.12842
$996$	0	0
$997$	1.36934e6i	1.37759i	0.724954	+	0.688797i	$0.241861\pi$
$997$	1.36934e6i	1.37759i	−0.724954	+	0.688797i	$0.758139\pi$
$998$	1.25273e6	1.25776
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.c.c.685.4		4
3.2	odd	2		98.5.b.a.97.2	yes	4
7.6	odd	2	inner	882.5.c.c.685.3		4
12.11	even	2		784.5.c.a.97.1		4
21.2	odd	6		98.5.d.c.31.3		8
21.5	even	6		98.5.d.c.31.4		8
21.11	odd	6		98.5.d.c.19.4		8
21.17	even	6		98.5.d.c.19.3		8
21.20	even	2		98.5.b.a.97.1	✓	4
84.83	odd	2		784.5.c.a.97.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.5.b.a.97.1	✓	4	21.20	even	2
98.5.b.a.97.2	yes	4	3.2	odd	2
98.5.d.c.19.3		8	21.17	even	6
98.5.d.c.19.4		8	21.11	odd	6
98.5.d.c.31.3		8	21.2	odd	6
98.5.d.c.31.4		8	21.5	even	6
784.5.c.a.97.1		4	12.11	even	2
784.5.c.a.97.4		4	84.83	odd	2
882.5.c.c.685.3		4	7.6	odd	2	inner
882.5.c.c.685.4		4	1.1	even	1	trivial

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +8.00000 q^{4} +27.8477i q^{5} +22.6274 q^{8} +O(q^{10})\)	\(q+2.82843 q^{2} +8.00000 q^{4} +27.8477i q^{5} +22.6274 q^{8} +78.7652i q^{10} +139.664 q^{11} -101.931i q^{13} +64.0000 q^{16} +542.887i q^{17} +139.953i q^{19} +222.782i q^{20} +395.029 q^{22} -229.647 q^{23} -150.495 q^{25} -288.303i q^{26} +383.113 q^{29} -397.492i q^{31} +181.019 q^{32} +1535.52i q^{34} -898.912 q^{37} +395.848i q^{38} +630.122i q^{40} +657.862i q^{41} +1155.70 q^{43} +1117.31 q^{44} -649.539 q^{46} -1294.51i q^{47} -425.663 q^{50} -815.445i q^{52} +5168.43 q^{53} +3889.32i q^{55} +1083.61 q^{58} +4155.24i q^{59} -1840.09i q^{61} -1124.28i q^{62} +512.000 q^{64} +2838.53 q^{65} -6315.57 q^{67} +4343.09i q^{68} -1263.28 q^{71} +3995.44i q^{73} -2542.51 q^{74} +1119.63i q^{76} -10614.3 q^{79} +1782.25i q^{80} +1860.71i q^{82} +5750.31i q^{83} -15118.2 q^{85} +3268.82 q^{86} +3160.24 q^{88} +4187.05i q^{89} -1837.17 q^{92} -3661.42i q^{94} -3897.38 q^{95} +3814.28i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4}+O(q^{10}) \) 4 * q + 32 * q^4	\( 4 q + 32 q^{4} - 24 q^{11} + 256 q^{16} + 1648 q^{22} - 104 q^{23} + 948 q^{25} + 1408 q^{29} - 3392 q^{37} - 2024 q^{43} - 192 q^{44} - 2304 q^{46} - 4384 q^{50} + 16680 q^{53} + 352 q^{58} + 2048 q^{64} + 6048 q^{65} - 20816 q^{67} + 1984 q^{71} - 576 q^{74} - 29616 q^{79} - 29688 q^{85} + 18800 q^{86} + 13184 q^{88} - 832 q^{92} - 7240 q^{95}+O(q^{100}) \) 4 * q + 32 * q^4 - 24 * q^11 + 256 * q^16 + 1648 * q^22 - 104 * q^23 + 948 * q^25 + 1408 * q^29 - 3392 * q^37 - 2024 * q^43 - 192 * q^44 - 2304 * q^46 - 4384 * q^50 + 16680 * q^53 + 352 * q^58 + 2048 * q^64 + 6048 * q^65 - 20816 * q^67 + 1984 * q^71 - 576 * q^74 - 29616 * q^79 - 29688 * q^85 + 18800 * q^86 + 13184 * q^88 - 832 * q^92 - 7240 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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