LMFDB - Embedded newform 882.5.b.c.197.4

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-53})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 52x^{2} + 729 $ x^4 - 52*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$-5.14782 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.c.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +29.1204i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} +29.1204i q^{5} -22.6274i q^{8} -82.3650 q^{10} -1.41421i q^{11} +82.3650 q^{13} +64.0000 q^{16} +262.084i q^{17} +453.008 q^{19} -232.964i q^{20} +4.00000 q^{22} +63.6396i q^{23} -223.000 q^{25} +232.964i q^{26} -502.046i q^{29} +1276.66 q^{31} +181.019i q^{32} -741.285 q^{34} +1560.00 q^{37} +1281.30i q^{38} +658.920 q^{40} -1310.42i q^{41} +328.000 q^{43} +11.3137i q^{44} -180.000 q^{46} +4135.10i q^{47} -630.739i q^{50} -658.920 q^{52} +4012.12i q^{53} +41.1825 q^{55} +1420.00 q^{58} -757.131i q^{59} -1194.29 q^{61} +3610.93i q^{62} -512.000 q^{64} +2398.51i q^{65} -1130.00 q^{67} -2096.67i q^{68} +422.850i q^{71} +3006.32 q^{73} +4412.35i q^{74} -3624.06 q^{76} +8650.00 q^{79} +1863.71i q^{80} +3706.43 q^{82} -2970.28i q^{83} -7632.00 q^{85} +927.724i q^{86} -32.0000 q^{88} -13308.0i q^{89} -509.117i q^{92} -11695.8 q^{94} +13191.8i q^{95} +13137.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} + 256 q^{16} + 16 q^{22} - 892 q^{25} + 6240 q^{37} + 1312 q^{43} - 720 q^{46} + 5680 q^{58} - 2048 q^{64} - 4520 q^{67} + 34600 q^{79} - 30528 q^{85} - 128 q^{88}+O(q^{100}) $ 4 * q - 32 * q^4 + 256 * q^16 + 16 * q^22 - 892 * q^25 + 6240 * q^37 + 1312 * q^43 - 720 * q^46 + 5680 * q^58 - 2048 * q^64 - 4520 * q^67 + 34600 * q^79 - 30528 * q^85 - 128 * q^88

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
$5$	29.1204i	1.16482i	0.812896	+	0.582409i	$0.197890\pi$
$5$	29.1204i	1.16482i	−0.812896	+	0.582409i	$0.802110\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−82.3650	−0.823650
$11$	− 1.41421i	− 0.0116877i	−0.999983	−	0.00584386i	$-0.998140\pi$
$11$	− 1.41421i	− 0.0116877i	0.999983	−	0.00584386i	$-0.00186017\pi$
$12$	0	0
$13$	82.3650	0.487367	0.243684	−	0.969855i	$-0.421644\pi$
$13$	82.3650	0.487367	0.243684	+	0.969855i	$0.421644\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	262.084i	0.906865i	0.891291	+	0.453432i	$0.149801\pi$
$17$	262.084i	0.906865i	−0.891291	+	0.453432i	$0.850199\pi$
$18$	0	0
$19$	453.008	1.25487	0.627435	−	0.778669i	$-0.284105\pi$
$19$	453.008	1.25487	0.627435	+	0.778669i	$0.284105\pi$
$20$	− 232.964i	− 0.582409i
$21$	0	0
$22$	4.00000	0.00826446
$23$	63.6396i	0.120302i	0.998189	+	0.0601509i	$0.0191582\pi$
$23$	63.6396i	0.120302i	−0.998189	+	0.0601509i	$0.980842\pi$
$24$	0	0
$25$	−223.000	−0.356800
$26$	232.964i	0.344621i
$27$	0	0
$28$	0	0
$29$	− 502.046i	− 0.596963i	−0.954415	−	0.298481i	$-0.903520\pi$
$29$	− 502.046i	− 0.596963i	0.954415	−	0.298481i	$-0.0964801\pi$
$30$	0	0
$31$	1276.66	1.32847	0.664234	−	0.747525i	$-0.268758\pi$
$31$	1276.66	1.32847	0.664234	+	0.747525i	$0.268758\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	−741.285	−0.641250
$35$	0	0
$36$	0	0
$37$	1560.00	1.13952	0.569759	−	0.821812i	$-0.307037\pi$
$37$	1560.00	1.13952	0.569759	+	0.821812i	$0.307037\pi$
$38$	1281.30i	0.887326i
$39$	0	0
$40$	658.920	0.411825
$41$	− 1310.42i	− 0.779548i	−0.920911	−	0.389774i	$-0.872553\pi$
$41$	− 1310.42i	− 0.779548i	0.920911	−	0.389774i	$-0.127447\pi$
$42$	0	0
$43$	328.000	0.177393	0.0886966	−	0.996059i	$-0.471730\pi$
$43$	328.000	0.177393	0.0886966	+	0.996059i	$0.471730\pi$
$44$	11.3137i	0.00584386i
$45$	0	0
$46$	−180.000	−0.0850662
$47$	4135.10i	1.87193i	0.352088	+	0.935967i	$0.385472\pi$
$47$	4135.10i	1.87193i	−0.352088	+	0.935967i	$0.614528\pi$
$48$	0	0
$49$	0	0
$50$	− 630.739i	− 0.252296i
$51$	0	0
$52$	−658.920	−0.243684
$53$	4012.12i	1.42831i	0.699987	+	0.714155i	$0.253189\pi$
$53$	4012.12i	1.42831i	−0.699987	+	0.714155i	$0.746811\pi$
$54$	0	0
$55$	41.1825	0.0136141
$56$	0	0
$57$	0	0
$58$	1420.00	0.422117
$59$	− 757.131i	− 0.217504i	−0.994069	−	0.108752i	$-0.965315\pi$
$59$	− 757.131i	− 0.217504i	0.994069	−	0.108752i	$-0.0346854\pi$
$60$	0	0
$61$	−1194.29	−0.320960	−0.160480	−	0.987039i	$-0.551304\pi$
$61$	−1194.29	−0.320960	−0.160480	+	0.987039i	$0.551304\pi$
$62$	3610.93i	0.939369i
$63$	0	0
$64$	−512.000	−0.125000
$65$	2398.51i	0.567694i
$66$	0	0
$67$	−1130.00	−0.251726	−0.125863	−	0.992048i	$-0.540170\pi$
$67$	−1130.00	−0.251726	−0.125863	+	0.992048i	$0.540170\pi$
$68$	− 2096.67i	− 0.453432i
$69$	0	0
$70$	0	0
$71$	422.850i	0.0838821i	0.999120	+	0.0419411i	$0.0133542\pi$
$71$	422.850i	0.0838821i	−0.999120	+	0.0419411i	$0.986646\pi$
$72$	0	0
$73$	3006.32	0.564144	0.282072	−	0.959393i	$-0.408978\pi$
$73$	3006.32	0.564144	0.282072	+	0.959393i	$0.408978\pi$
$74$	4412.35i	0.805761i
$75$	0	0
$76$	−3624.06	−0.627435
$77$	0	0
$78$	0	0
$79$	8650.00	1.38600	0.692998	−	0.720940i	$-0.256289\pi$
$79$	8650.00	1.38600	0.692998	+	0.720940i	$0.256289\pi$
$80$	1863.71i	0.291204i
$81$	0	0
$82$	3706.43	0.551224
$83$	− 2970.28i	− 0.431163i	−0.976486	−	0.215582i	$-0.930835\pi$
$83$	− 2970.28i	− 0.431163i	0.976486	−	0.215582i	$-0.0691648\pi$
$84$	0	0
$85$	−7632.00	−1.05633
$86$	927.724i	0.125436i
$87$	0	0
$88$	−32.0000	−0.00413223
$89$	− 13308.0i	− 1.68010i	−0.542512	−	0.840048i	$-0.682527\pi$
$89$	− 13308.0i	− 1.68010i	0.542512	−	0.840048i	$-0.317473\pi$
$90$	0	0
$91$	0	0
$92$	− 509.117i	− 0.0601509i
$93$	0	0
$94$	−11695.8	−1.32366
$95$	13191.8i	1.46169i
$96$	0	0
$97$	13137.2	1.39624	0.698120	−	0.715981i	$-0.254020\pi$
$97$	13137.2	1.39624	0.698120	+	0.715981i	$0.254020\pi$
$98$	0	0
$99$	0	0
$100$	1784.00	0.178400
$101$	− 12026.7i	− 1.17898i	−0.807777	−	0.589488i	$-0.799329\pi$
$101$	− 12026.7i	− 1.17898i	0.807777	−	0.589488i	$-0.200671\pi$
$102$	0	0
$103$	−10913.4	−1.02869	−0.514345	−	0.857583i	$-0.671965\pi$
$103$	−10913.4	−1.02869	−0.514345	+	0.857583i	$0.671965\pi$
$104$	− 1863.71i	− 0.172310i
$105$	0	0
$106$	−11348.0	−1.00997
$107$	12703.9i	1.10961i	0.831982	+	0.554803i	$0.187206\pi$
$107$	12703.9i	1.10961i	−0.831982	+	0.554803i	$0.812794\pi$
$108$	0	0
$109$	−9752.00	−0.820806	−0.410403	−	0.911904i	$-0.634612\pi$
$109$	−9752.00	−0.820806	−0.410403	+	0.911904i	$0.634612\pi$
$110$	116.482i	0.00962659i
$111$	0	0
$112$	0	0
$113$	11303.8i	0.885254i	0.896706	+	0.442627i	$0.145953\pi$
$113$	11303.8i	0.885254i	−0.896706	+	0.442627i	$0.854047\pi$
$114$	0	0
$115$	−1853.21	−0.140130
$116$	4016.37i	0.298481i
$117$	0	0
$118$	2141.49	0.153799
$119$	0	0
$120$	0	0
$121$	14639.0	0.999863
$122$	− 3377.97i	− 0.226953i
$123$	0	0
$124$	−10213.3	−0.664234
$125$	11706.4i	0.749211i
$126$	0	0
$127$	−30922.0	−1.91717	−0.958584	−	0.284810i	$-0.908069\pi$
$127$	−30922.0	−1.91717	−0.958584	+	0.284810i	$0.908069\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−6784.00	−0.401420
$131$	5940.57i	0.346167i	0.984907	+	0.173083i	$0.0553730\pi$
$131$	5940.57i	0.346167i	−0.984907	+	0.173083i	$0.944627\pi$
$132$	0	0
$133$	0	0
$134$	− 3196.12i	− 0.177997i
$135$	0	0
$136$	5930.28	0.320625
$137$	18929.2i	1.00854i	0.863547	+	0.504269i	$0.168238\pi$
$137$	18929.2i	1.00854i	−0.863547	+	0.504269i	$0.831762\pi$
$138$	0	0
$139$	−10872.2	−0.562713	−0.281357	−	0.959603i	$-0.590784\pi$
$139$	−10872.2	−0.562713	−0.281357	+	0.959603i	$0.590784\pi$
$140$	0	0
$141$	0	0
$142$	−1196.00	−0.0593136
$143$	− 116.482i	− 0.00569621i
$144$	0	0
$145$	14619.8	0.695353
$146$	8503.17i	0.398910i
$147$	0	0
$148$	−12480.0	−0.569759
$149$	7448.66i	0.335510i	0.985829	+	0.167755i	$0.0536518\pi$
$149$	7448.66i	0.335510i	−0.985829	+	0.167755i	$0.946348\pi$
$150$	0	0
$151$	−35792.0	−1.56976	−0.784878	−	0.619651i	$-0.787274\pi$
$151$	−35792.0	−1.56976	−0.784878	+	0.619651i	$0.787274\pi$
$152$	− 10250.4i	− 0.443663i
$153$	0	0
$154$	0	0
$155$	37176.8i	1.54742i
$156$	0	0
$157$	2676.86	0.108599	0.0542996	−	0.998525i	$-0.482707\pi$
$157$	2676.86	0.108599	0.0542996	+	0.998525i	$0.482707\pi$
$158$	24465.9i	0.980047i
$159$	0	0
$160$	−5271.36	−0.205913
$161$	0	0
$162$	0	0
$163$	19254.0	0.724679	0.362340	−	0.932046i	$-0.381978\pi$
$163$	19254.0	0.724679	0.362340	+	0.932046i	$0.381978\pi$
$164$	10483.4i	0.389774i
$165$	0	0
$166$	8401.23	0.304879
$167$	− 12521.8i	− 0.448987i	−0.974476	−	0.224493i	$-0.927927\pi$
$167$	− 12521.8i	− 0.448987i	0.974476	−	0.224493i	$-0.0720727\pi$
$168$	0	0
$169$	−21777.0	−0.762473
$170$	− 21586.6i	− 0.746940i
$171$	0	0
$172$	−2624.00	−0.0886966
$173$	23558.4i	0.787144i	0.919294	+	0.393572i	$0.128761\pi$
$173$	23558.4i	0.787144i	−0.919294	+	0.393572i	$0.871239\pi$
$174$	0	0
$175$	0	0
$176$	− 90.5097i	− 0.00292193i
$177$	0	0
$178$	37640.8	1.18801
$179$	25511.0i	0.796199i	0.917342	+	0.398099i	$0.130330\pi$
$179$	25511.0i	0.796199i	−0.917342	+	0.398099i	$0.869670\pi$
$180$	0	0
$181$	−38876.3	−1.18666	−0.593332	−	0.804958i	$-0.702188\pi$
$181$	−38876.3	−1.18666	−0.593332	+	0.804958i	$0.702188\pi$
$182$	0	0
$183$	0	0
$184$	1440.00	0.0425331
$185$	45427.9i	1.32733i
$186$	0	0
$187$	370.643	0.0105992
$188$	− 33080.8i	− 0.935967i
$189$	0	0
$190$	−37312.0	−1.03357
$191$	− 34352.7i	− 0.941659i	−0.882224	−	0.470829i	$-0.843955\pi$
$191$	− 34352.7i	− 0.941659i	0.882224	−	0.470829i	$-0.156045\pi$
$192$	0	0
$193$	−65824.0	−1.76713	−0.883567	−	0.468304i	$-0.844865\pi$
$193$	−65824.0	−1.76713	−0.883567	+	0.468304i	$0.844865\pi$
$194$	37157.7i	0.987291i
$195$	0	0
$196$	0	0
$197$	64831.8i	1.67053i	0.549844	+	0.835267i	$0.314687\pi$
$197$	64831.8i	1.67053i	−0.549844	+	0.835267i	$0.685313\pi$
$198$	0	0
$199$	−46536.2	−1.17513	−0.587564	−	0.809178i	$-0.699913\pi$
$199$	−46536.2	−1.17513	−0.587564	+	0.809178i	$0.699913\pi$
$200$	5045.91i	0.126148i
$201$	0	0
$202$	34016.8	0.833662
$203$	0	0
$204$	0	0
$205$	38160.0	0.908031
$206$	− 30867.7i	− 0.727393i
$207$	0	0
$208$	5271.36	0.121842
$209$	− 640.650i	− 0.0146666i
$210$	0	0
$211$	8504.00	0.191011	0.0955055	−	0.995429i	$-0.469553\pi$
$211$	8504.00	0.191011	0.0955055	+	0.995429i	$0.469553\pi$
$212$	− 32097.0i	− 0.714155i
$213$	0	0
$214$	−35932.0	−0.784610
$215$	9551.50i	0.206631i
$216$	0	0
$217$	0	0
$218$	− 27582.8i	− 0.580398i
$219$	0	0
$220$	−329.460	−0.00680703
$221$	21586.6i	0.441976i
$222$	0	0
$223$	56749.5	1.14118	0.570588	−	0.821237i	$-0.306715\pi$
$223$	56749.5	1.14118	0.570588	+	0.821237i	$0.306715\pi$
$224$	0	0
$225$	0	0
$226$	−31972.0	−0.625969
$227$	55969.5i	1.08617i	0.839676	+	0.543087i	$0.182745\pi$
$227$	55969.5i	1.08617i	−0.839676	+	0.543087i	$0.817255\pi$
$228$	0	0
$229$	84588.9	1.61303	0.806515	−	0.591214i	$-0.201351\pi$
$229$	84588.9	1.61303	0.806515	+	0.591214i	$0.201351\pi$
$230$	− 5241.68i	− 0.0990866i
$231$	0	0
$232$	−11360.0	−0.211058
$233$	− 35469.9i	− 0.653353i	−0.945136	−	0.326677i	$-0.894071\pi$
$233$	− 35469.9i	− 0.653353i	0.945136	−	0.326677i	$-0.105929\pi$
$234$	0	0
$235$	−120416.	−2.18046
$236$	6057.05i	0.108752i
$237$	0	0
$238$	0	0
$239$	57330.8i	1.00367i	0.864963	+	0.501836i	$0.167342\pi$
$239$	57330.8i	1.00367i	−0.864963	+	0.501836i	$0.832658\pi$
$240$	0	0
$241$	85783.2	1.47696	0.738479	−	0.674277i	$-0.235544\pi$
$241$	85783.2	1.47696	0.738479	+	0.674277i	$0.235544\pi$
$242$	41405.3i	0.707010i
$243$	0	0
$244$	9554.34	0.160480
$245$	0	0
$246$	0	0
$247$	37312.0	0.611582
$248$	− 28887.5i	− 0.469685i
$249$	0	0
$250$	−33110.7	−0.529772
$251$	25276.5i	0.401209i	0.979672	+	0.200604i	$0.0642905\pi$
$251$	25276.5i	0.401209i	−0.979672	+	0.200604i	$0.935709\pi$
$252$	0	0
$253$	90.0000	0.00140605
$254$	− 87460.6i	− 1.35564i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 87215.7i	− 1.32047i	−0.751059	−	0.660235i	$-0.770457\pi$
$257$	− 87215.7i	− 1.32047i	0.751059	−	0.660235i	$-0.229543\pi$
$258$	0	0
$259$	0	0
$260$	− 19188.0i	− 0.283847i
$261$	0	0
$262$	−16802.5	−0.244777
$263$	− 60747.5i	− 0.878248i	−0.898426	−	0.439124i	$-0.855289\pi$
$263$	− 60747.5i	− 0.878248i	0.898426	−	0.439124i	$-0.144711\pi$
$264$	0	0
$265$	−116835.	−1.66372
$266$	0	0
$267$	0	0
$268$	9040.00	0.125863
$269$	− 109871.i	− 1.51838i	−0.650870	−	0.759189i	$-0.725596\pi$
$269$	− 109871.i	− 1.51838i	0.650870	−	0.759189i	$-0.274404\pi$
$270$	0	0
$271$	−115188.	−1.56844	−0.784218	−	0.620485i	$-0.786936\pi$
$271$	−115188.	−1.56844	−0.784218	+	0.620485i	$0.786936\pi$
$272$	16773.4i	0.226716i
$273$	0	0
$274$	−53540.0	−0.713144
$275$	315.370i	0.00417018i
$276$	0	0
$277$	−109310.	−1.42462	−0.712312	−	0.701863i	$-0.752352\pi$
$277$	−109310.	−1.42462	−0.712312	+	0.701863i	$0.752352\pi$
$278$	− 30751.2i	− 0.397898i
$279$	0	0
$280$	0	0
$281$	103576.i	1.31173i	0.754878	+	0.655865i	$0.227696\pi$
$281$	103576.i	1.31173i	−0.754878	+	0.655865i	$0.772304\pi$
$282$	0	0
$283$	−144015.	−1.79819	−0.899095	−	0.437753i	$-0.855774\pi$
$283$	−144015.	−1.79819	−0.899095	+	0.437753i	$0.855774\pi$
$284$	− 3382.80i	− 0.0419411i
$285$	0	0
$286$	329.460	0.00402783
$287$	0	0
$288$	0	0
$289$	14833.0	0.177596
$290$	41351.0i	0.491689i
$291$	0	0
$292$	−24050.6	−0.282072
$293$	42370.2i	0.493544i	0.969074	+	0.246772i	$0.0793698\pi$
$293$	42370.2i	0.493544i	−0.969074	+	0.246772i	$0.920630\pi$
$294$	0	0
$295$	22048.0	0.253352
$296$	− 35298.8i	− 0.402880i
$297$	0	0
$298$	−21068.0	−0.237242
$299$	5241.68i	0.0586311i
$300$	0	0
$301$	0	0
$302$	− 101235.i	− 1.10998i
$303$	0	0
$304$	28992.5	0.313717
$305$	− 34778.3i	− 0.373860i
$306$	0	0
$307$	64533.0	0.684708	0.342354	−	0.939571i	$-0.388776\pi$
$307$	64533.0	0.684708	0.342354	+	0.939571i	$0.388776\pi$
$308$	0	0
$309$	0	0
$310$	−105152.	−1.09419
$311$	− 33838.0i	− 0.349851i	−0.984582	−	0.174926i	$-0.944032\pi$
$311$	− 33838.0i	− 0.349851i	0.984582	−	0.174926i	$-0.0559685\pi$
$312$	0	0
$313$	−53619.6	−0.547312	−0.273656	−	0.961828i	$-0.588233\pi$
$313$	−53619.6	−0.547312	−0.273656	+	0.961828i	$0.588233\pi$
$314$	7571.31i	0.0767913i
$315$	0	0
$316$	−69200.0	−0.692998
$317$	− 25426.1i	− 0.253024i	−0.991965	−	0.126512i	$-0.959622\pi$
$317$	− 25426.1i	− 0.253024i	0.991965	−	0.126512i	$-0.0403783\pi$
$318$	0	0
$319$	−710.000	−0.00697713
$320$	− 14909.7i	− 0.145602i
$321$	0	0
$322$	0	0
$323$	118726.i	1.13800i
$324$	0	0
$325$	−18367.4	−0.173893
$326$	54458.5i	0.512426i
$327$	0	0
$328$	−29651.4	−0.275612
$329$	0	0
$330$	0	0
$331$	24392.0	0.222634	0.111317	−	0.993785i	$-0.464493\pi$
$331$	24392.0	0.222634	0.111317	+	0.993785i	$0.464493\pi$
$332$	23762.3i	0.215582i
$333$	0	0
$334$	35417.0	0.317482
$335$	− 32906.1i	− 0.293215i
$336$	0	0
$337$	105296.	0.927154	0.463577	−	0.886057i	$-0.346566\pi$
$337$	105296.	0.927154	0.463577	+	0.886057i	$0.346566\pi$
$338$	− 61594.7i	− 0.539150i
$339$	0	0
$340$	61056.0	0.528166
$341$	− 1805.47i	− 0.0155268i
$342$	0	0
$343$	0	0
$344$	− 7421.79i	− 0.0627180i
$345$	0	0
$346$	−66633.3	−0.556595
$347$	− 39045.0i	− 0.324270i	−0.986769	−	0.162135i	$-0.948162\pi$
$347$	− 39045.0i	− 0.324270i	0.986769	−	0.162135i	$-0.0518380\pi$
$348$	0	0
$349$	188410.	1.54687	0.773434	−	0.633877i	$-0.218537\pi$
$349$	188410.	1.54687	0.773434	+	0.633877i	$0.218537\pi$
$350$	0	0
$351$	0	0
$352$	256.000	0.00206612
$353$	− 124140.i	− 0.996240i	−0.867108	−	0.498120i	$-0.834024\pi$
$353$	− 124140.i	− 0.996240i	0.867108	−	0.498120i	$-0.165976\pi$
$354$	0	0
$355$	−12313.6	−0.0977074
$356$	106464.i	0.840048i
$357$	0	0
$358$	−72156.0	−0.562997
$359$	202443.i	1.57078i	0.619003	+	0.785388i	$0.287537\pi$
$359$	202443.i	1.57078i	−0.619003	+	0.785388i	$0.712463\pi$
$360$	0	0
$361$	74895.0	0.574696
$362$	− 109959.i	− 0.839098i
$363$	0	0
$364$	0	0
$365$	87545.5i	0.657125i
$366$	0	0
$367$	228728.	1.69819	0.849096	−	0.528239i	$-0.177147\pi$
$367$	228728.	1.69819	0.849096	+	0.528239i	$0.177147\pi$
$368$	4072.94i	0.0300754i
$369$	0	0
$370$	−128489.	−0.938564
$371$	0	0
$372$	0	0
$373$	−49874.0	−0.358473	−0.179237	−	0.983806i	$-0.557363\pi$
$373$	−49874.0	−0.358473	−0.179237	+	0.983806i	$0.557363\pi$
$374$	1048.34i	0.00749475i
$375$	0	0
$376$	93566.7	0.661829
$377$	− 41351.0i	− 0.290940i
$378$	0	0
$379$	16632.0	0.115789	0.0578943	−	0.998323i	$-0.481561\pi$
$379$	16632.0	0.115789	0.0578943	+	0.998323i	$0.481561\pi$
$380$	− 105534.i	− 0.730847i
$381$	0	0
$382$	97164.0	0.665853
$383$	128596.i	0.876656i	0.898815	+	0.438328i	$0.144429\pi$
$383$	128596.i	0.876656i	−0.898815	+	0.438328i	$0.855571\pi$
$384$	0	0
$385$	0	0
$386$	− 186178.i	− 1.24955i
$387$	0	0
$388$	−105098.	−0.698120
$389$	− 82370.9i	− 0.544345i	−0.962248	−	0.272173i	$-0.912258\pi$
$389$	− 82370.9i	− 0.544345i	0.962248	−	0.272173i	$-0.0877422\pi$
$390$	0	0
$391$	−16678.9	−0.109097
$392$	0	0
$393$	0	0
$394$	−183372.	−1.18125
$395$	251892.i	1.61443i
$396$	0	0
$397$	208013.	1.31980	0.659902	−	0.751352i	$-0.270598\pi$
$397$	208013.	1.31980	0.659902	+	0.751352i	$0.270598\pi$
$398$	− 131624.i	− 0.830941i
$399$	0	0
$400$	−14272.0	−0.0892000
$401$	274325.i	1.70599i	0.521919	+	0.852995i	$0.325216\pi$
$401$	274325.i	1.70599i	−0.521919	+	0.852995i	$0.674784\pi$
$402$	0	0
$403$	105152.	0.647452
$404$	96213.9i	0.589488i
$405$	0	0
$406$	0	0
$407$	− 2206.17i	− 0.0133184i
$408$	0	0
$409$	88830.7	0.531027	0.265513	−	0.964107i	$-0.414459\pi$
$409$	88830.7	0.531027	0.265513	+	0.964107i	$0.414459\pi$
$410$	107933.i	0.642075i
$411$	0	0
$412$	87306.9	0.514345
$413$	0	0
$414$	0	0
$415$	86496.0	0.502227
$416$	14909.7i	0.0861551i
$417$	0	0
$418$	1812.03	0.0103708
$419$	− 228071.i	− 1.29910i	−0.760319	−	0.649550i	$-0.774957\pi$
$419$	− 228071.i	− 1.29910i	0.760319	−	0.649550i	$-0.225043\pi$
$420$	0	0
$421$	149694.	0.844579	0.422289	−	0.906461i	$-0.361227\pi$
$421$	149694.	0.844579	0.422289	+	0.906461i	$0.361227\pi$
$422$	24052.9i	0.135065i
$423$	0	0
$424$	90784.0	0.504984
$425$	− 58444.7i	− 0.323569i
$426$	0	0
$427$	0	0
$428$	− 101631.i	− 0.554803i
$429$	0	0
$430$	−27015.7	−0.146110
$431$	− 161643.i	− 0.870168i	−0.900390	−	0.435084i	$-0.856719\pi$
$431$	− 161643.i	− 0.870168i	0.900390	−	0.435084i	$-0.143281\pi$
$432$	0	0
$433$	−112099.	−0.597895	−0.298948	−	0.954269i	$-0.596636\pi$
$433$	−112099.	−0.597895	−0.298948	+	0.954269i	$0.596636\pi$
$434$	0	0
$435$	0	0
$436$	78016.0	0.410403
$437$	28829.2i	0.150963i
$438$	0	0
$439$	−185321.	−0.961604	−0.480802	−	0.876829i	$-0.659654\pi$
$439$	−185321.	−0.961604	−0.480802	+	0.876829i	$0.659654\pi$
$440$	− 931.854i	− 0.00481330i
$441$	0	0
$442$	−61056.0	−0.312524
$443$	129337.i	0.659045i	0.944148	+	0.329522i	$0.106888\pi$
$443$	129337.i	0.659045i	−0.944148	+	0.329522i	$0.893112\pi$
$444$	0	0
$445$	387536.	1.95701
$446$	160512.i	0.806933i
$447$	0	0
$448$	0	0
$449$	4323.25i	0.0214446i	0.999943	+	0.0107223i	$0.00341308\pi$
$449$	4323.25i	0.0214446i	−0.999943	+	0.0107223i	$0.996587\pi$
$450$	0	0
$451$	−1853.21	−0.00911113
$452$	− 90430.5i	− 0.442627i
$453$	0	0
$454$	−158306.	−0.768041
$455$	0	0
$456$	0	0
$457$	133376.	0.638624	0.319312	−	0.947650i	$-0.396548\pi$
$457$	133376.	0.638624	0.319312	+	0.947650i	$0.396548\pi$
$458$	239254.i	1.14058i
$459$	0	0
$460$	14825.7	0.0700648
$461$	111735.i	0.525760i	0.964828	+	0.262880i	$0.0846724\pi$
$461$	111735.i	0.525760i	−0.964828	+	0.262880i	$0.915328\pi$
$462$	0	0
$463$	324458.	1.51355	0.756775	−	0.653676i	$-0.226774\pi$
$463$	324458.	1.51355	0.756775	+	0.653676i	$0.226774\pi$
$464$	− 32130.9i	− 0.149241i
$465$	0	0
$466$	100324.	0.461990
$467$	330109.i	1.51364i	0.653621	+	0.756822i	$0.273249\pi$
$467$	330109.i	1.51364i	−0.653621	+	0.756822i	$0.726751\pi$
$468$	0	0
$469$	0	0
$470$	− 340588.i	− 1.54182i
$471$	0	0
$472$	−17131.9	−0.0768993
$473$	− 463.862i	− 0.00207332i
$474$	0	0
$475$	−101021.	−0.447737
$476$	0	0
$477$	0	0
$478$	−162156.	−0.709704
$479$	− 142224.i	− 0.619873i	−0.950757	−	0.309936i	$-0.899692\pi$
$479$	− 142224.i	− 0.619873i	0.950757	−	0.309936i	$-0.100308\pi$
$480$	0	0
$481$	128489.	0.555364
$482$	242632.i	1.04437i
$483$	0	0
$484$	−117112.	−0.499932
$485$	382562.i	1.62637i
$486$	0	0
$487$	35920.0	0.151453	0.0757266	−	0.997129i	$-0.475872\pi$
$487$	35920.0	0.151453	0.0757266	+	0.997129i	$0.475872\pi$
$488$	27023.8i	0.113477i
$489$	0	0
$490$	0	0
$491$	− 347986.i	− 1.44344i	−0.692186	−	0.721719i	$-0.743352\pi$
$491$	− 347986.i	− 1.44344i	0.692186	−	0.721719i	$-0.256648\pi$
$492$	0	0
$493$	131578.	0.541365
$494$	105534.i	0.432454i
$495$	0	0
$496$	81706.1	0.332117
$497$	0	0
$498$	0	0
$499$	12120.0	0.0486745	0.0243373	−	0.999704i	$-0.492252\pi$
$499$	12120.0	0.0486745	0.0243373	+	0.999704i	$0.492252\pi$
$500$	− 93651.3i	− 0.374605i
$501$	0	0
$502$	−71492.9	−0.283697
$503$	222655.i	0.880028i	0.897991	+	0.440014i	$0.145026\pi$
$503$	222655.i	0.880028i	−0.897991	+	0.440014i	$0.854974\pi$
$504$	0	0
$505$	350224.	1.37329
$506$	254.558i	0 0.000994229i
$507$	0	0
$508$	247376.	0.958584
$509$	− 232760.i	− 0.898405i	−0.893430	−	0.449203i	$-0.851708\pi$
$509$	− 232760.i	− 0.898405i	0.893430	−	0.449203i	$-0.148292\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	246683.	0.933713
$515$	− 317802.i	− 1.19824i
$516$	0	0
$517$	5847.92	0.0218786
$518$	0	0
$519$	0	0
$520$	54272.0	0.200710
$521$	− 37245.0i	− 0.137212i	−0.997644	−	0.0686061i	$-0.978145\pi$
$521$	− 37245.0i	− 0.137212i	0.997644	−	0.0686061i	$-0.0218552\pi$
$522$	0	0
$523$	112016.	0.409523	0.204762	−	0.978812i	$-0.434358\pi$
$523$	112016.	0.409523	0.204762	+	0.978812i	$0.434358\pi$
$524$	− 47524.6i	− 0.173083i
$525$	0	0
$526$	171820.	0.621015
$527$	334592.i	1.20474i
$528$	0	0
$529$	275791.	0.985527
$530$	− 330459.i	− 1.17643i
$531$	0	0
$532$	0	0
$533$	− 107933.i	− 0.379926i
$534$	0	0
$535$	−369943.	−1.29249
$536$	25569.0i	0.0889987i
$537$	0	0
$538$	310763.	1.07366
$539$	0	0
$540$	0	0
$541$	128194.	0.437999	0.219000	−	0.975725i	$-0.429721\pi$
$541$	128194.	0.437999	0.219000	+	0.975725i	$0.429721\pi$
$542$	− 325799.i	− 1.10905i
$543$	0	0
$544$	−47442.3	−0.160313
$545$	− 283983.i	− 0.956090i
$546$	0	0
$547$	186442.	0.623116	0.311558	−	0.950227i	$-0.399149\pi$
$547$	186442.	0.623116	0.311558	+	0.950227i	$0.399149\pi$
$548$	− 151434.i	− 0.504269i
$549$	0	0
$550$	−892.000	−0.00294876
$551$	− 227431.i	− 0.749110i
$552$	0	0
$553$	0	0
$554$	− 309175.i	− 1.00736i
$555$	0	0
$556$	86977.5	0.281357
$557$	− 14318.9i	− 0.0461530i	−0.999734	−	0.0230765i	$-0.992654\pi$
$557$	− 14318.9i	− 0.0461530i	0.999734	−	0.0230765i	$-0.00734613\pi$
$558$	0	0
$559$	27015.7	0.0864556
$560$	0	0
$561$	0	0
$562$	−292956.	−0.927534
$563$	− 66336.4i	− 0.209283i	−0.994510	−	0.104642i	$-0.966630\pi$
$563$	− 66336.4i	− 0.209283i	0.994510	−	0.104642i	$-0.0333696\pi$
$564$	0	0
$565$	−329172.	−1.03116
$566$	− 407337.i	− 1.27151i
$567$	0	0
$568$	9568.00	0.0296568
$569$	− 208920.i	− 0.645292i	−0.946520	−	0.322646i	$-0.895428\pi$
$569$	− 208920.i	− 0.645292i	0.946520	−	0.322646i	$-0.104572\pi$
$570$	0	0
$571$	515862.	1.58220	0.791100	−	0.611687i	$-0.209509\pi$
$571$	515862.	1.58220	0.791100	+	0.611687i	$0.209509\pi$
$572$	931.854i	0.00284810i
$573$	0	0
$574$	0	0
$575$	− 14191.6i	− 0.0429237i
$576$	0	0
$577$	−352440.	−1.05860	−0.529302	−	0.848433i	$-0.677546\pi$
$577$	−352440.	−1.05860	−0.529302	+	0.848433i	$0.677546\pi$
$578$	41954.1i	0.125579i
$579$	0	0
$580$	−116958.	−0.347676
$581$	0	0
$582$	0	0
$583$	5674.00	0.0166937
$584$	− 68025.3i	− 0.199455i
$585$	0	0
$586$	−119841.	−0.348988
$587$	296621.i	0.860846i	0.902627	+	0.430423i	$0.141636\pi$
$587$	296621.i	0.860846i	−0.902627	+	0.430423i	$0.858364\pi$
$588$	0	0
$589$	578336.	1.66705
$590$	62361.2i	0.179147i
$591$	0	0
$592$	99840.0	0.284879
$593$	22626.6i	0.0643442i	0.999482	+	0.0321721i	$0.0102425\pi$
$593$	22626.6i	0.0643442i	−0.999482	+	0.0321721i	$0.989758\pi$
$594$	0	0
$595$	0	0
$596$	− 59589.3i	− 0.167755i
$597$	0	0
$598$	−14825.7	−0.0414584
$599$	− 146528.i	− 0.408383i	−0.978931	−	0.204191i	$-0.934544\pi$
$599$	− 146528.i	− 0.408383i	0.978931	−	0.204191i	$-0.0654565\pi$
$600$	0	0
$601$	136479.	0.377847	0.188924	−	0.981992i	$-0.439500\pi$
$601$	136479.	0.377847	0.188924	+	0.981992i	$0.439500\pi$
$602$	0	0
$603$	0	0
$604$	286336.	0.784878
$605$	426294.i	1.16466i
$606$	0	0
$607$	−79317.5	−0.215274	−0.107637	−	0.994190i	$-0.534328\pi$
$607$	−79317.5	−0.215274	−0.107637	+	0.994190i	$0.534328\pi$
$608$	82003.2i	0.221832i
$609$	0	0
$610$	98368.0	0.264359
$611$	340588.i	0.912319i
$612$	0	0
$613$	13742.0	0.0365703	0.0182852	−	0.999833i	$-0.494179\pi$
$613$	13742.0	0.0365703	0.0182852	+	0.999833i	$0.494179\pi$
$614$	182527.i	0.484161i
$615$	0	0
$616$	0	0
$617$	125015.i	0.328392i	0.986428	+	0.164196i	$0.0525029\pi$
$617$	125015.i	0.328392i	−0.986428	+	0.164196i	$0.947497\pi$
$618$	0	0
$619$	−318341.	−0.830828	−0.415414	−	0.909632i	$-0.636363\pi$
$619$	−318341.	−0.830828	−0.415414	+	0.909632i	$0.636363\pi$
$620$	− 297415.i	− 0.773712i
$621$	0	0
$622$	95708.2	0.247382
$623$	0	0
$624$	0	0
$625$	−480271.	−1.22949
$626$	− 151659.i	− 0.387008i
$627$	0	0
$628$	−21414.9	−0.0542996
$629$	408851.i	1.03339i
$630$	0	0
$631$	−12618.0	−0.0316907	−0.0158453	−	0.999874i	$-0.505044\pi$
$631$	−12618.0	−0.0316907	−0.0158453	+	0.999874i	$0.505044\pi$
$632$	− 195727.i	− 0.490024i
$633$	0	0
$634$	71916.0	0.178915
$635$	− 900462.i	− 2.23315i
$636$	0	0
$637$	0	0
$638$	− 2008.18i	− 0.00493358i
$639$	0	0
$640$	42170.9	0.102956
$641$	− 88644.3i	− 0.215742i	−0.994165	−	0.107871i	$-0.965597\pi$
$641$	− 88644.3i	− 0.215742i	0.994165	−	0.107871i	$-0.0344034\pi$
$642$	0	0
$643$	195329.	0.472437	0.236219	−	0.971700i	$-0.424092\pi$
$643$	195329.	0.472437	0.236219	+	0.971700i	$0.424092\pi$
$644$	0	0
$645$	0	0
$646$	−335808.	−0.804685
$647$	− 441524.i	− 1.05474i	−0.849635	−	0.527371i	$-0.823178\pi$
$647$	− 441524.i	− 1.05474i	0.849635	−	0.527371i	$-0.176822\pi$
$648$	0	0
$649$	−1070.75	−0.00254212
$650$	− 51950.9i	− 0.122961i
$651$	0	0
$652$	−154032.	−0.362340
$653$	− 346158.i	− 0.811799i	−0.913918	−	0.405900i	$-0.866958\pi$
$653$	− 346158.i	− 0.811799i	0.913918	−	0.405900i	$-0.133042\pi$
$654$	0	0
$655$	−172992.	−0.403221
$656$	− 83866.9i	− 0.194887i
$657$	0	0
$658$	0	0
$659$	− 255677.i	− 0.588737i	−0.955692	−	0.294368i	$-0.904891\pi$
$659$	− 255677.i	− 0.588737i	0.955692	−	0.294368i	$-0.0951092\pi$
$660$	0	0
$661$	124742.	0.285502	0.142751	−	0.989759i	$-0.454405\pi$
$661$	124742.	0.285502	0.142751	+	0.989759i	$0.454405\pi$
$662$	68991.0i	0.157426i
$663$	0	0
$664$	−67209.9	−0.152439
$665$	0	0
$666$	0	0
$667$	31950.0	0.0718157
$668$	100174.i	0.224493i
$669$	0	0
$670$	93072.5	0.207335
$671$	1688.99i	0.00375129i
$672$	0	0
$673$	−168784.	−0.372650	−0.186325	−	0.982488i	$-0.559658\pi$
$673$	−168784.	−0.372650	−0.186325	+	0.982488i	$0.559658\pi$
$674$	297822.i	0.655597i
$675$	0	0
$676$	174216.	0.381237
$677$	709287.i	1.54755i	0.633461	+	0.773774i	$0.281634\pi$
$677$	709287.i	1.54755i	−0.633461	+	0.773774i	$0.718366\pi$
$678$	0	0
$679$	0	0
$680$	172692.i	0.373470i
$681$	0	0
$682$	5106.63	0.0109791
$683$	− 746025.i	− 1.59923i	−0.600511	−	0.799616i	$-0.705036\pi$
$683$	− 746025.i	− 1.59923i	0.600511	−	0.799616i	$-0.294964\pi$
$684$	0	0
$685$	−551228.	−1.17476
$686$	0	0
$687$	0	0
$688$	20992.0	0.0443483
$689$	330459.i	0.696112i
$690$	0	0
$691$	901815.	1.88869	0.944346	−	0.328953i	$-0.106696\pi$
$691$	901815.	1.88869	0.944346	+	0.328953i	$0.106696\pi$
$692$	− 188467.i	− 0.393572i
$693$	0	0
$694$	110436.	0.229293
$695$	− 316603.i	− 0.655458i
$696$	0	0
$697$	343440.	0.706944
$698$	532904.i	1.09380i
$699$	0	0
$700$	0	0
$701$	571146.i	1.16228i	0.813803	+	0.581140i	$0.197393\pi$
$701$	571146.i	1.16228i	−0.813803	+	0.581140i	$0.802607\pi$
$702$	0	0
$703$	706692.	1.42995
$704$	724.077i	0.00146096i
$705$	0	0
$706$	351122.	0.704448
$707$	0	0
$708$	0	0
$709$	−403688.	−0.803070	−0.401535	−	0.915844i	$-0.631523\pi$
$709$	−403688.	−0.803070	−0.401535	+	0.915844i	$0.631523\pi$
$710$	− 34828.0i	− 0.0690896i
$711$	0	0
$712$	−301127.	−0.594004
$713$	81246.0i	0.159817i
$714$	0	0
$715$	3392.00	0.00663504
$716$	− 204088.i	− 0.398099i
$717$	0	0
$718$	−572596.	−1.11071
$719$	365170.i	0.706379i	0.935552	+	0.353189i	$0.114903\pi$
$719$	365170.i	0.706379i	−0.935552	+	0.353189i	$0.885097\pi$
$720$	0	0
$721$	0	0
$722$	211835.i	0.406372i
$723$	0	0
$724$	311010.	0.593332
$725$	111956.i	0.212996i
$726$	0	0
$727$	−941968.	−1.78224	−0.891122	−	0.453764i	$-0.850081\pi$
$727$	−941968.	−1.78224	−0.891122	+	0.453764i	$0.850081\pi$
$728$	0	0
$729$	0	0
$730$	−247616.	−0.464658
$731$	85963.5i	0.160872i
$732$	0	0
$733$	−418909.	−0.779671	−0.389835	−	0.920885i	$-0.627468\pi$
$733$	−418909.	−0.779671	−0.389835	+	0.920885i	$0.627468\pi$
$734$	646940.i	1.20080i
$735$	0	0
$736$	−11520.0	−0.0212665
$737$	1598.06i	0.00294211i
$738$	0	0
$739$	−1.04476e6	−1.91305	−0.956526	−	0.291647i	$-0.905797\pi$
$739$	−1.04476e6	−1.91305	−0.956526	+	0.291647i	$0.905797\pi$
$740$	− 363423.i	− 0.663665i
$741$	0	0
$742$	0	0
$743$	− 232297.i	− 0.420791i	−0.977616	−	0.210396i	$-0.932525\pi$
$743$	− 232297.i	− 0.420791i	0.977616	−	0.210396i	$-0.0674752\pi$
$744$	0	0
$745$	−216908.	−0.390808
$746$	− 141065.i	− 0.253479i
$747$	0	0
$748$	−2965.14	−0.00529959
$749$	0	0
$750$	0	0
$751$	−465408.	−0.825190	−0.412595	−	0.910915i	$-0.635378\pi$
$751$	−465408.	−0.825190	−0.412595	+	0.910915i	$0.635378\pi$
$752$	264647.i	0.467984i
$753$	0	0
$754$	116958.	0.205726
$755$	− 1.04228e6i	− 1.82848i
$756$	0	0
$757$	−496088.	−0.865699	−0.432850	−	0.901466i	$-0.642492\pi$
$757$	−496088.	−0.865699	−0.432850	+	0.901466i	$0.642492\pi$
$758$	47042.4i	0.0818750i
$759$	0	0
$760$	298496.	0.516787
$761$	738407.i	1.27505i	0.770431	+	0.637524i	$0.220041\pi$
$761$	738407.i	1.27505i	−0.770431	+	0.637524i	$0.779959\pi$
$762$	0	0
$763$	0	0
$764$	274821.i	0.470829i
$765$	0	0
$766$	−363724.	−0.619890
$767$	− 62361.2i	− 0.106004i
$768$	0	0
$769$	527136.	0.891395	0.445698	−	0.895184i	$-0.352956\pi$
$769$	527136.	0.891395	0.445698	+	0.895184i	$0.352956\pi$
$770$	0	0
$771$	0	0
$772$	526592.	0.883567
$773$	− 100378.i	− 0.167989i	−0.996466	−	0.0839944i	$-0.973232\pi$
$773$	− 100378.i	− 0.167989i	0.996466	−	0.0839944i	$-0.0267678\pi$
$774$	0	0
$775$	−284695.	−0.473998
$776$	− 297261.i	− 0.493645i
$777$	0	0
$778$	232980.	0.384910
$779$	− 593630.i	− 0.978230i
$780$	0	0
$781$	598.000	0.000980391 0
$782$	− 47175.1i	− 0.0771435i
$783$	0	0
$784$	0	0
$785$	77951.5i	0.126498i
$786$	0	0
$787$	715752.	1.15562	0.577808	−	0.816173i	$-0.303908\pi$
$787$	715752.	1.15562	0.577808	+	0.816173i	$0.303908\pi$
$788$	− 518654.i	− 0.835267i
$789$	0	0
$790$	−712458.	−1.14158
$791$	0	0
$792$	0	0
$793$	−98368.0	−0.156425
$794$	588349.i	0.933242i
$795$	0	0
$796$	372290.	0.587564
$797$	− 921924.i	− 1.45137i	−0.688027	−	0.725686i	$-0.741523\pi$
$797$	− 921924.i	− 1.45137i	0.688027	−	0.725686i	$-0.258477\pi$
$798$	0	0
$799$	−1.08374e6	−1.69759
$800$	− 40367.3i	− 0.0630739i
$801$	0	0
$802$	−775908.	−1.20632
$803$	− 4251.58i	− 0.00659356i
$804$	0	0
$805$	0	0
$806$	297415.i	0.457818i
$807$	0	0
$808$	−272134.	−0.416831
$809$	601594.i	0.919192i	0.888128	+	0.459596i	$0.152006\pi$
$809$	601594.i	0.919192i	−0.888128	+	0.459596i	$0.847994\pi$
$810$	0	0
$811$	1.22221e6	1.85826	0.929129	−	0.369757i	$-0.120559\pi$
$811$	1.22221e6	1.85826	0.929129	+	0.369757i	$0.120559\pi$
$812$	0	0
$813$	0	0
$814$	6240.00	0.00941750
$815$	560685.i	0.844119i
$816$	0	0
$817$	148587.	0.222605
$818$	251251.i	0.375493i
$819$	0	0
$820$	−305280.	−0.454015
$821$	− 911304.i	− 1.35200i	−0.736901	−	0.676000i	$-0.763712\pi$
$821$	− 911304.i	− 1.35200i	0.736901	−	0.676000i	$-0.236288\pi$
$822$	0	0
$823$	−207542.	−0.306412	−0.153206	−	0.988194i	$-0.548960\pi$
$823$	−207542.	−0.306412	−0.153206	+	0.988194i	$0.548960\pi$
$824$	246941.i	0.363697i
$825$	0	0
$826$	0	0
$827$	746785.i	1.09190i	0.837816	+	0.545952i	$0.183832\pi$
$827$	746785.i	1.09190i	−0.837816	+	0.545952i	$0.816168\pi$
$828$	0	0
$829$	−233093.	−0.339172	−0.169586	−	0.985515i	$-0.554243\pi$
$829$	−233093.	−0.339172	−0.169586	+	0.985515i	$0.554243\pi$
$830$	244648.i	0.355128i
$831$	0	0
$832$	−42170.9	−0.0609209
$833$	0	0
$834$	0	0
$835$	364640.	0.522988
$836$	5125.20i	0.00733328i
$837$	0	0
$838$	645083.	0.918602
$839$	798482.i	1.13434i	0.823602	+	0.567168i	$0.191961\pi$
$839$	798482.i	1.13434i	−0.823602	+	0.567168i	$0.808039\pi$
$840$	0	0
$841$	455231.	0.643635
$842$	423399.i	0.597207i
$843$	0	0
$844$	−68032.0	−0.0955055
$845$	− 634156.i	− 0.888142i
$846$	0	0
$847$	0	0
$848$	256776.i	0.357078i
$849$	0	0
$850$	165307.	0.228798
$851$	99277.8i	0.137086i
$852$	0	0
$853$	432128.	0.593902	0.296951	−	0.954893i	$-0.404030\pi$
$853$	432128.	0.593902	0.296951	+	0.954893i	$0.404030\pi$
$854$	0	0
$855$	0	0
$856$	287456.	0.392305
$857$	− 466247.i	− 0.634826i	−0.948287	−	0.317413i	$-0.897186\pi$
$857$	− 466247.i	− 0.634826i	0.948287	−	0.317413i	$-0.102814\pi$
$858$	0	0
$859$	610449.	0.827299	0.413650	−	0.910436i	$-0.364254\pi$
$859$	610449.	0.827299	0.413650	+	0.910436i	$0.364254\pi$
$860$	− 76412.0i	− 0.103315i
$861$	0	0
$862$	457196.	0.615301
$863$	− 198424.i	− 0.266424i	−0.991088	−	0.133212i	$-0.957471\pi$
$863$	− 198424.i	− 0.266424i	0.991088	−	0.133212i	$-0.0425291\pi$
$864$	0	0
$865$	−686032.	−0.916879
$866$	− 317063.i	− 0.422776i
$867$	0	0
$868$	0	0
$869$	− 12232.9i	− 0.0161991i
$870$	0	0
$871$	−93072.5	−0.122683
$872$	220663.i	0.290199i
$873$	0	0
$874$	−81541.4	−0.106747
$875$	0	0
$876$	0	0
$877$	−33678.0	−0.0437872	−0.0218936	−	0.999760i	$-0.506970\pi$
$877$	−33678.0	−0.0437872	−0.0218936	+	0.999760i	$0.506970\pi$
$878$	− 524168.i	− 0.679957i
$879$	0	0
$880$	2635.68	0.00340351
$881$	1.11930e6i	1.44210i	0.692883	+	0.721050i	$0.256340\pi$
$881$	1.11930e6i	1.44210i	−0.692883	+	0.721050i	$0.743660\pi$
$882$	0	0
$883$	−1.28444e6	−1.64737	−0.823687	−	0.567044i	$-0.808087\pi$
$883$	−1.28444e6	−1.64737	−0.823687	+	0.567044i	$0.808087\pi$
$884$	− 172692.i	− 0.220988i
$885$	0	0
$886$	−365820.	−0.466015
$887$	− 111997.i	− 0.142351i	−0.997464	−	0.0711754i	$-0.977325\pi$
$887$	− 111997.i	− 0.142351i	0.997464	−	0.0711754i	$-0.0226750\pi$
$888$	0	0
$889$	0	0
$890$	1.09612e6i	1.38381i
$891$	0	0
$892$	−453996.	−0.570588
$893$	1.87323e6i	2.34903i
$894$	0	0
$895$	−742891.	−0.927426
$896$	0	0
$897$	0	0
$898$	−12228.0	−0.0151636
$899$	− 640941.i	− 0.793046i
$900$	0	0
$901$	−1.05151e6	−1.29528
$902$	− 5241.68i	− 0.00644254i
$903$	0	0
$904$	255776.	0.312985
$905$	− 1.13209e6i	− 1.38225i
$906$	0	0
$907$	−1.49953e6	−1.82280	−0.911402	−	0.411517i	$-0.864999\pi$
$907$	−1.49953e6	−1.82280	−0.911402	+	0.411517i	$0.864999\pi$
$908$	− 447756.i	− 0.543087i
$909$	0	0
$910$	0	0
$911$	− 1.62993e6i	− 1.96396i	−0.188997	−	0.981978i	$-0.560524\pi$
$911$	− 1.62993e6i	− 1.96396i	0.188997	−	0.981978i	$-0.439476\pi$
$912$	0	0
$913$	−4200.62	−0.00503932
$914$	377244.i	0.451575i
$915$	0	0
$916$	−676711.	−0.806515
$917$	0	0
$918$	0	0
$919$	503520.	0.596191	0.298096	−	0.954536i	$-0.403649\pi$
$919$	503520.	0.596191	0.298096	+	0.954536i	$0.403649\pi$
$920$	41933.4i	0.0495433i
$921$	0	0
$922$	−316035.	−0.371769
$923$	34828.0i	0.0408814i
$924$	0	0
$925$	−347880.	−0.406580
$926$	917706.i	1.07024i
$927$	0	0
$928$	90880.0	0.105529
$929$	71898.4i	0.0833082i	0.999132	+	0.0416541i	$0.0132627\pi$
$929$	71898.4i	0.0833082i	−0.999132	+	0.0416541i	$0.986737\pi$
$930$	0	0
$931$	0	0
$932$	283759.i	0.326677i
$933$	0	0
$934$	−933690.	−1.07031
$935$	10793.3i	0.0123461i
$936$	0	0
$937$	679017.	0.773396	0.386698	−	0.922206i	$-0.373616\pi$
$937$	679017.	0.773396	0.386698	+	0.922206i	$0.373616\pi$
$938$	0	0
$939$	0	0
$940$	963328.	1.09023
$941$	− 364908.i	− 0.412102i	−0.978541	−	0.206051i	$-0.933939\pi$
$941$	− 364908.i	− 0.412102i	0.978541	−	0.206051i	$-0.0660612\pi$
$942$	0	0
$943$	83394.6	0.0937809
$944$	− 48456.4i	− 0.0543760i
$945$	0	0
$946$	1312.00	0.00146606
$947$	− 186576.i	− 0.208044i	−0.994575	−	0.104022i	$-0.966829\pi$
$947$	− 186576.i	− 0.208044i	0.994575	−	0.104022i	$-0.0331712\pi$
$948$	0	0
$949$	247616.	0.274945
$950$	− 285730.i	− 0.316598i
$951$	0	0
$952$	0	0
$953$	1.32111e6i	1.45463i	0.686302	+	0.727317i	$0.259233\pi$
$953$	1.32111e6i	1.45463i	−0.686302	+	0.727317i	$0.740767\pi$
$954$	0	0
$955$	1.00036e6	1.09686
$956$	− 458646.i	− 0.501836i
$957$	0	0
$958$	402271.	0.438316
$959$	0	0
$960$	0	0
$961$	706335.	0.764828
$962$	363423.i	0.392701i
$963$	0	0
$964$	−686266.	−0.738479
$965$	− 1.91682e6i	− 2.05839i
$966$	0	0
$967$	−688022.	−0.735782	−0.367891	−	0.929869i	$-0.619920\pi$
$967$	−688022.	−0.735782	−0.367891	+	0.929869i	$0.619920\pi$
$968$	− 331243.i	− 0.353505i
$969$	0	0
$970$	−1.08205e6	−1.15001
$971$	− 485088.i	− 0.514496i	−0.966345	−	0.257248i	$-0.917184\pi$
$971$	− 485088.i	− 0.514496i	0.966345	−	0.257248i	$-0.0828158\pi$
$972$	0	0
$973$	0	0
$974$	101597.i	0.107094i
$975$	0	0
$976$	−76434.8	−0.0802401
$977$	1.70595e6i	1.78721i	0.448850	+	0.893607i	$0.351834\pi$
$977$	1.70595e6i	1.78721i	−0.448850	+	0.893607i	$0.648166\pi$
$978$	0	0
$979$	−18820.4	−0.0196365
$980$	0	0
$981$	0	0
$982$	984252.	1.02067
$983$	− 964527.i	− 0.998177i	−0.866551	−	0.499088i	$-0.833668\pi$
$983$	− 964527.i	− 0.998177i	0.866551	−	0.499088i	$-0.166332\pi$
$984$	0	0
$985$	−1.88793e6	−1.94587
$986$	372159.i	0.382803i
$987$	0	0
$988$	−298496.	−0.305791
$989$	20873.8i	0.0213407i
$990$	0	0
$991$	−639856.	−0.651531	−0.325765	−	0.945451i	$-0.605622\pi$
$991$	−639856.	−0.651531	−0.325765	+	0.945451i	$0.605622\pi$
$992$	231100.i	0.234842i
$993$	0	0
$994$	0	0
$995$	− 1.35516e6i	− 1.36881i
$996$	0	0
$997$	−1.60838e6	−1.61808	−0.809039	−	0.587755i	$-0.800012\pi$
$997$	−1.60838e6	−1.61808	−0.809039	+	0.587755i	$0.800012\pi$
$998$	34280.5i	0.0344181i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.c.197.4	yes	4
3.2	odd	2	inner	882.5.b.c.197.1	✓	4
7.6	odd	2	inner	882.5.b.c.197.3	yes	4
21.20	even	2	inner	882.5.b.c.197.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.5.b.c.197.1	✓	4	3.2	odd	2	inner
882.5.b.c.197.2	yes	4	21.20	even	2	inner
882.5.b.c.197.3	yes	4	7.6	odd	2	inner
882.5.b.c.197.4	yes	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +29.1204i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +29.1204i q^{5} -22.6274i q^{8} -82.3650 q^{10} -1.41421i q^{11} +82.3650 q^{13} +64.0000 q^{16} +262.084i q^{17} +453.008 q^{19} -232.964i q^{20} +4.00000 q^{22} +63.6396i q^{23} -223.000 q^{25} +232.964i q^{26} -502.046i q^{29} +1276.66 q^{31} +181.019i q^{32} -741.285 q^{34} +1560.00 q^{37} +1281.30i q^{38} +658.920 q^{40} -1310.42i q^{41} +328.000 q^{43} +11.3137i q^{44} -180.000 q^{46} +4135.10i q^{47} -630.739i q^{50} -658.920 q^{52} +4012.12i q^{53} +41.1825 q^{55} +1420.00 q^{58} -757.131i q^{59} -1194.29 q^{61} +3610.93i q^{62} -512.000 q^{64} +2398.51i q^{65} -1130.00 q^{67} -2096.67i q^{68} +422.850i q^{71} +3006.32 q^{73} +4412.35i q^{74} -3624.06 q^{76} +8650.00 q^{79} +1863.71i q^{80} +3706.43 q^{82} -2970.28i q^{83} -7632.00 q^{85} +927.724i q^{86} -32.0000 q^{88} -13308.0i q^{89} -509.117i q^{92} -11695.8 q^{94} +13191.8i q^{95} +13137.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 256 q^{16} + 16 q^{22} - 892 q^{25} + 6240 q^{37} + 1312 q^{43} - 720 q^{46} + 5680 q^{58} - 2048 q^{64} - 4520 q^{67} + 34600 q^{79} - 30528 q^{85} - 128 q^{88}+O(q^{100}) \) 4 * q - 32 * q^4 + 256 * q^16 + 16 * q^22 - 892 * q^25 + 6240 * q^37 + 1312 * q^43 - 720 * q^46 + 5680 * q^58 - 2048 * q^64 - 4520 * q^67 + 34600 * q^79 - 30528 * q^85 - 128 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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