LMFDB - Embedded newform 825.4.c.l.199.5

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:	6.0.2230106176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 41x^{4} + 452x^{2} + 676 $ x^6 + 41x^4 + 452x^2 + 676
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.5
Root			$4.59056i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.l.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.59056i q^{2} +3.00000i q^{3} -4.89212 q^{4} -10.7717 q^{6} -16.1465i q^{7} +11.1590i q^{8} -9.00000 q^{9} +11.0000 q^{11} -14.6764i q^{12} +54.1214i q^{13} +57.9749 q^{14} -79.2041 q^{16} -107.010i q^{17} -32.3150i q^{18} -48.7496 q^{19} +48.4394 q^{21} +39.4962i q^{22} -11.9498i q^{23} -33.4771 q^{24} -194.326 q^{26} -27.0000i q^{27} +78.9905i q^{28} -239.733 q^{29} -82.0851 q^{31} -195.115i q^{32} +33.0000i q^{33} +384.224 q^{34} +44.0291 q^{36} -21.7573i q^{37} -175.038i q^{38} -162.364 q^{39} -124.835 q^{41} +173.925i q^{42} -224.459i q^{43} -53.8133 q^{44} +42.9064 q^{46} -186.832i q^{47} -237.612i q^{48} +82.2913 q^{49} +321.029 q^{51} -264.768i q^{52} -233.997i q^{53} +96.9451 q^{54} +180.179 q^{56} -146.249i q^{57} -860.774i q^{58} -232.936 q^{59} +163.849 q^{61} -294.731i q^{62} +145.318i q^{63} +66.9386 q^{64} -118.488 q^{66} -876.918i q^{67} +523.503i q^{68} +35.8493 q^{69} -733.141 q^{71} -100.431i q^{72} -1161.97i q^{73} +78.1208 q^{74} +238.489 q^{76} -177.611i q^{77} -582.978i q^{78} +588.831 q^{79} +81.0000 q^{81} -448.226i q^{82} +1161.06i q^{83} -236.971 q^{84} +805.933 q^{86} -719.198i q^{87} +122.749i q^{88} +1042.16 q^{89} +873.869 q^{91} +58.4597i q^{92} -246.255i q^{93} +670.831 q^{94} +585.345 q^{96} +1546.63i q^{97} +295.472i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) $ 6 * q - 44 * q^4 + 24 * q^6 - 54 * q^9 + 66 * q^11 + 112 * q^14 + 100 * q^16 - 292 * q^19 + 24 * q^21 - 288 * q^24 - 1016 * q^26 - 136 * q^29 - 136 * q^31 + 352 * q^34 + 396 * q^36 - 392 * q^41 - 484 * q^44 + 2320 * q^46 + 314 * q^49 + 1308 * q^51 - 216 * q^54 + 2736 * q^56 + 2088 * q^59 + 1284 * q^61 - 2332 * q^64 + 264 * q^66 - 1200 * q^69 - 1088 * q^71 - 3072 * q^74 + 3992 * q^76 + 3172 * q^79 + 486 * q^81 - 2040 * q^84 + 7136 * q^86 + 4244 * q^89 - 16 * q^91 + 4304 * q^94 + 4128 * 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 $

$2$	3.59056i	1.26945i	0.772736	+	0.634727i	$0.218888\pi$
$2$	3.59056i	1.26945i	−0.772736	+	0.634727i	$0.781112\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−4.89212	−0.611515
$5$	0	0
$5$	0	0
$6$	−10.7717	−0.732920
$7$	− 16.1465i	− 0.871828i	−0.899988	−	0.435914i	$-0.856425\pi$
$7$	− 16.1465i	− 0.871828i	0.899988	−	0.435914i	$-0.143575\pi$
$8$	11.1590i	0.493164i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	− 14.6764i	− 0.353058i
$13$	54.1214i	1.15466i	0.816511	+	0.577329i	$0.195905\pi$
$13$	54.1214i	1.15466i	−0.816511	+	0.577329i	$0.804095\pi$
$14$	57.9749	1.10675
$15$	0	0
$16$	−79.2041	−1.23756
$17$	− 107.010i	− 1.52668i	−0.645995	−	0.763342i	$-0.723557\pi$
$17$	− 107.010i	− 1.52668i	0.645995	−	0.763342i	$-0.276443\pi$
$18$	− 32.3150i	− 0.423152i
$19$	−48.7496	−0.588628	−0.294314	−	0.955709i	$-0.595091\pi$
$19$	−48.7496	−0.588628	−0.294314	+	0.955709i	$0.595091\pi$
$20$	0	0
$21$	48.4394	0.503350
$22$	39.4962i	0.382755i
$23$	− 11.9498i	− 0.108335i	−0.998532	−	0.0541674i	$-0.982750\pi$
$23$	− 11.9498i	− 0.108335i	0.998532	−	0.0541674i	$-0.0172505\pi$
$24$	−33.4771	−0.284729
$25$	0	0
$26$	−194.326	−1.46579
$27$	− 27.0000i	− 0.192450i
$28$	78.9905i	0.533136i
$29$	−239.733	−1.53508	−0.767538	−	0.641003i	$-0.778519\pi$
$29$	−239.733	−1.53508	−0.767538	+	0.641003i	$0.778519\pi$
$30$	0	0
$31$	−82.0851	−0.475578	−0.237789	−	0.971317i	$-0.576423\pi$
$31$	−82.0851	−0.475578	−0.237789	+	0.971317i	$0.576423\pi$
$32$	− 195.115i	− 1.07787i
$33$	33.0000i	0.174078i
$34$	384.224	1.93806
$35$	0	0
$36$	44.0291	0.203838
$37$	− 21.7573i	− 0.0966723i	−0.998831	−	0.0483361i	$-0.984608\pi$
$37$	− 21.7573i	− 0.0966723i	0.998831	−	0.0483361i	$-0.0153919\pi$
$38$	− 175.038i	− 0.747236i
$39$	−162.364	−0.666643
$40$	0	0
$41$	−124.835	−0.475510	−0.237755	−	0.971325i	$-0.576412\pi$
$41$	−124.835	−0.475510	−0.237755	+	0.971325i	$0.576412\pi$
$42$	173.925i	0.638980i
$43$	− 224.459i	− 0.796039i	−0.917377	−	0.398019i	$-0.869698\pi$
$43$	− 224.459i	− 0.796039i	0.917377	−	0.398019i	$-0.130302\pi$
$44$	−53.8133	−0.184379
$45$	0	0
$46$	42.9064	0.137526
$47$	− 186.832i	− 0.579835i	−0.957052	−	0.289917i	$-0.906372\pi$
$47$	− 186.832i	− 0.579835i	0.957052	−	0.289917i	$-0.0936278\pi$
$48$	− 237.612i	− 0.714508i
$49$	82.2913	0.239916
$50$	0	0
$51$	321.029	0.881431
$52$	− 264.768i	− 0.706091i
$53$	− 233.997i	− 0.606453i	−0.952919	−	0.303226i	$-0.901936\pi$
$53$	− 233.997i	− 0.606453i	0.952919	−	0.303226i	$-0.0980638\pi$
$54$	96.9451	0.244307
$55$	0	0
$56$	180.179	0.429954
$57$	− 146.249i	− 0.339844i
$58$	− 860.774i	− 1.94871i
$59$	−232.936	−0.513996	−0.256998	−	0.966412i	$-0.582733\pi$
$59$	−232.936	−0.513996	−0.256998	+	0.966412i	$0.582733\pi$
$60$	0	0
$61$	163.849	0.343913	0.171957	−	0.985105i	$-0.444991\pi$
$61$	163.849	0.343913	0.171957	+	0.985105i	$0.444991\pi$
$62$	− 294.731i	− 0.603725i
$63$	145.318i	0.290609i
$64$	66.9386	0.130739
$65$	0	0
$66$	−118.488	−0.220984
$67$	− 876.918i	− 1.59899i	−0.600670	−	0.799497i	$-0.705100\pi$
$67$	− 876.918i	− 1.59899i	0.600670	−	0.799497i	$-0.294900\pi$
$68$	523.503i	0.933590i
$69$	35.8493	0.0625471
$70$	0	0
$71$	−733.141	−1.22546	−0.612731	−	0.790291i	$-0.709929\pi$
$71$	−733.141	−1.22546	−0.612731	+	0.790291i	$0.709929\pi$
$72$	− 100.431i	− 0.164388i
$73$	− 1161.97i	− 1.86299i	−0.363750	−	0.931496i	$-0.618504\pi$
$73$	− 1161.97i	− 1.86299i	0.363750	−	0.931496i	$-0.381496\pi$
$74$	78.1208	0.122721
$75$	0	0
$76$	238.489	0.359954
$77$	− 177.611i	− 0.262866i
$78$	− 582.978i	− 0.846272i
$79$	588.831	0.838591	0.419296	−	0.907850i	$-0.362277\pi$
$79$	588.831	0.838591	0.419296	+	0.907850i	$0.362277\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 448.226i	− 0.603638i
$83$	1161.06i	1.53546i	0.640772	+	0.767731i	$0.278614\pi$
$83$	1161.06i	1.53546i	−0.640772	+	0.767731i	$0.721386\pi$
$84$	−236.971	−0.307806
$85$	0	0
$86$	805.933	1.01053
$87$	− 719.198i	− 0.886277i
$88$	122.749i	0.148695i
$89$	1042.16	1.24122	0.620610	−	0.784120i	$-0.286885\pi$
$89$	1042.16	1.24122	0.620610	+	0.784120i	$0.286885\pi$
$90$	0	0
$91$	873.869	1.00666
$92$	58.4597i	0.0662483i
$93$	− 246.255i	− 0.274575i
$94$	670.831	0.736074
$95$	0	0
$96$	585.345	0.622307
$97$	1546.63i	1.61893i	0.587169	+	0.809464i	$0.300242\pi$
$97$	1546.63i	1.61893i	−0.587169	+	0.809464i	$0.699758\pi$
$98$	295.472i	0.304563i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−662.282	−0.652470	−0.326235	−	0.945289i	$-0.605780\pi$
$101$	−662.282	−0.652470	−0.326235	+	0.945289i	$0.605780\pi$
$102$	1152.67i	1.11894i
$103$	− 399.592i	− 0.382262i	−0.981565	−	0.191131i	$-0.938784\pi$
$103$	− 399.592i	− 0.382262i	0.981565	−	0.191131i	$-0.0612156\pi$
$104$	−603.942	−0.569436
$105$	0	0
$106$	840.181	0.769864
$107$	− 1591.22i	− 1.43765i	−0.695189	−	0.718827i	$-0.744679\pi$
$107$	− 1591.22i	− 1.43765i	0.695189	−	0.718827i	$-0.255321\pi$
$108$	132.087i	0.117686i
$109$	−755.128	−0.663561	−0.331780	−	0.943357i	$-0.607649\pi$
$109$	−755.128	−0.663561	−0.331780	+	0.943357i	$0.607649\pi$
$110$	0	0
$111$	65.2718	0.0558138
$112$	1278.87i	1.07894i
$113$	1145.65i	0.953753i	0.878970	+	0.476876i	$0.158231\pi$
$113$	1145.65i	0.953753i	−0.878970	+	0.476876i	$0.841769\pi$
$114$	525.115	0.431417
$115$	0	0
$116$	1172.80	0.938722
$117$	− 487.092i	− 0.384886i
$118$	− 836.372i	− 0.652494i
$119$	−1727.83	−1.33101
$120$	0	0
$121$	121.000	0.0909091
$122$	588.309i	0.436582i
$123$	− 374.504i	− 0.274536i
$124$	401.570	0.290823
$125$	0	0
$126$	−521.774	−0.368915
$127$	1461.29i	1.02101i	0.859875	+	0.510505i	$0.170542\pi$
$127$	1461.29i	1.02101i	−0.859875	+	0.510505i	$0.829458\pi$
$128$	− 1320.57i	− 0.911900i
$129$	673.377	0.459593
$130$	0	0
$131$	−1524.94	−1.01706	−0.508528	−	0.861045i	$-0.669810\pi$
$131$	−1524.94	−1.01706	−0.508528	+	0.861045i	$0.669810\pi$
$132$	− 161.440i	− 0.106451i
$133$	787.134i	0.513182i
$134$	3148.63	2.02985
$135$	0	0
$136$	1194.12	0.752906
$137$	− 2125.68i	− 1.32561i	−0.748792	−	0.662805i	$-0.769366\pi$
$137$	− 2125.68i	− 1.32561i	0.748792	−	0.662805i	$-0.230634\pi$
$138$	128.719i	0.0794007i
$139$	1774.28	1.08268	0.541339	−	0.840805i	$-0.317918\pi$
$139$	1774.28	1.08268	0.541339	+	0.840805i	$0.317918\pi$
$140$	0	0
$141$	560.496	0.334768
$142$	− 2632.39i	− 1.55567i
$143$	595.335i	0.348143i
$144$	712.837	0.412521
$145$	0	0
$146$	4172.13	2.36498
$147$	246.874i	0.138516i
$148$	106.439i	0.0591165i
$149$	−1575.78	−0.866393	−0.433197	−	0.901299i	$-0.642614\pi$
$149$	−1575.78	−0.866393	−0.433197	+	0.901299i	$0.642614\pi$
$150$	0	0
$151$	420.978	0.226879	0.113439	−	0.993545i	$-0.463813\pi$
$151$	420.978	0.226879	0.113439	+	0.993545i	$0.463813\pi$
$152$	− 543.998i	− 0.290290i
$153$	963.086i	0.508895i
$154$	637.724	0.333696
$155$	0	0
$156$	794.304	0.407662
$157$	− 2224.30i	− 1.13069i	−0.824854	−	0.565345i	$-0.808743\pi$
$157$	− 2224.30i	− 1.13069i	0.824854	−	0.565345i	$-0.191257\pi$
$158$	2114.23i	1.06455i
$159$	701.992	0.350136
$160$	0	0
$161$	−192.947	−0.0944492
$162$	290.835i	0.141051i
$163$	3093.37i	1.48645i	0.669040	+	0.743226i	$0.266705\pi$
$163$	3093.37i	1.48645i	−0.669040	+	0.743226i	$0.733295\pi$
$164$	610.706	0.290781
$165$	0	0
$166$	−4168.87	−1.94920
$167$	− 2416.43i	− 1.11970i	−0.828595	−	0.559848i	$-0.810860\pi$
$167$	− 2416.43i	− 1.11970i	0.828595	−	0.559848i	$-0.189140\pi$
$168$	540.537i	0.248234i
$169$	−732.122	−0.333237
$170$	0	0
$171$	438.746	0.196209
$172$	1098.08i	0.486789i
$173$	3758.02i	1.65154i	0.564005	+	0.825771i	$0.309260\pi$
$173$	3758.02i	1.65154i	−0.564005	+	0.825771i	$0.690740\pi$
$174$	2582.32	1.12509
$175$	0	0
$176$	−871.245	−0.373140
$177$	− 698.809i	− 0.296756i
$178$	3741.93i	1.57567i
$179$	−2533.99	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$179$	−2533.99	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$180$	0	0
$181$	−13.8995	−0.00570798	−0.00285399	−	0.999996i	$-0.500908\pi$
$181$	−13.8995	−0.00570798	−0.00285399	+	0.999996i	$0.500908\pi$
$182$	3137.68i	1.27791i
$183$	491.547i	0.198558i
$184$	133.348	0.0534268
$185$	0	0
$186$	884.194	0.348561
$187$	− 1177.10i	− 0.460312i
$188$	914.004i	0.354577i
$189$	−435.955	−0.167783
$190$	0	0
$191$	3495.39	1.32417	0.662087	−	0.749427i	$-0.269671\pi$
$191$	3495.39	1.32417	0.662087	+	0.749427i	$0.269671\pi$
$192$	200.816i	0.0754824i
$193$	− 3469.33i	− 1.29393i	−0.762522	−	0.646963i	$-0.776039\pi$
$193$	− 3469.33i	− 1.29393i	0.762522	−	0.646963i	$-0.223961\pi$
$194$	−5553.25	−2.05516
$195$	0	0
$196$	−402.579	−0.146712
$197$	3638.39i	1.31586i	0.753079	+	0.657930i	$0.228568\pi$
$197$	3638.39i	1.31586i	−0.753079	+	0.657930i	$0.771432\pi$
$198$	− 355.465i	− 0.127585i
$199$	−51.6049	−0.0183828	−0.00919140	−	0.999958i	$-0.502926\pi$
$199$	−51.6049	−0.0183828	−0.00919140	+	0.999958i	$0.502926\pi$
$200$	0	0
$201$	2630.75	0.923179
$202$	− 2377.96i	− 0.828281i
$203$	3870.84i	1.33832i
$204$	−1570.51	−0.539008
$205$	0	0
$206$	1434.76	0.485265
$207$	107.548i	0.0361116i
$208$	− 4286.63i	− 1.42896i
$209$	−536.246	−0.177478
$210$	0	0
$211$	2084.57	0.680131	0.340065	−	0.940402i	$-0.389551\pi$
$211$	2084.57	0.680131	0.340065	+	0.940402i	$0.389551\pi$
$212$	1144.74i	0.370855i
$213$	− 2199.42i	− 0.707521i
$214$	5713.37	1.82504
$215$	0	0
$216$	301.294	0.0949095
$217$	1325.38i	0.414622i
$218$	− 2711.33i	− 0.842361i
$219$	3485.91	1.07560
$220$	0	0
$221$	5791.50	1.76280
$222$	234.362i	0.0708531i
$223$	− 1887.36i	− 0.566757i	−0.959008	−	0.283378i	$-0.908545\pi$
$223$	− 1887.36i	− 0.566757i	0.959008	−	0.283378i	$-0.0914552\pi$
$224$	−3150.42	−0.939715
$225$	0	0
$226$	−4113.54	−1.21075
$227$	− 1150.73i	− 0.336462i	−0.985748	−	0.168231i	$-0.946194\pi$
$227$	− 1150.73i	− 0.336462i	0.985748	−	0.168231i	$-0.0538055\pi$
$228$	715.466i	0.207820i
$229$	−4106.79	−1.18508	−0.592542	−	0.805540i	$-0.701875\pi$
$229$	−4106.79	−1.18508	−0.592542	+	0.805540i	$0.701875\pi$
$230$	0	0
$231$	532.834	0.151766
$232$	− 2675.18i	− 0.757045i
$233$	5733.58i	1.61210i	0.591848	+	0.806050i	$0.298399\pi$
$233$	5733.58i	1.61210i	−0.591848	+	0.806050i	$0.701601\pi$
$234$	1748.93	0.488596
$235$	0	0
$236$	1139.55	0.314316
$237$	1766.49i	0.484161i
$238$	− 6203.86i	− 1.68965i
$239$	−6036.18	−1.63367	−0.816837	−	0.576868i	$-0.804275\pi$
$239$	−6036.18	−1.63367	−0.816837	+	0.576868i	$0.804275\pi$
$240$	0	0
$241$	3720.90	0.994540	0.497270	−	0.867596i	$-0.334336\pi$
$241$	3720.90	0.994540	0.497270	+	0.867596i	$0.334336\pi$
$242$	434.458i	0.115405i
$243$	243.000i	0.0641500i
$244$	−801.568	−0.210308
$245$	0	0
$246$	1344.68	0.348511
$247$	− 2638.39i	− 0.679664i
$248$	− 915.990i	− 0.234538i
$249$	−3483.19	−0.886500
$250$	0	0
$251$	−3809.88	−0.958077	−0.479038	−	0.877794i	$-0.659015\pi$
$251$	−3809.88	−0.958077	−0.479038	+	0.877794i	$0.659015\pi$
$252$	− 710.914i	− 0.177712i
$253$	− 131.447i	− 0.0326641i
$254$	−5246.84	−1.29613
$255$	0	0
$256$	5277.10	1.28835
$257$	1225.95i	0.297559i	0.988870	+	0.148780i	$0.0475345\pi$
$257$	1225.95i	0.297559i	−0.988870	+	0.148780i	$0.952465\pi$
$258$	2417.80i	0.583433i
$259$	−351.303	−0.0842816
$260$	0	0
$261$	2157.59	0.511692
$262$	− 5475.38i	− 1.29111i
$263$	− 7397.68i	− 1.73445i	−0.497916	−	0.867225i	$-0.665901\pi$
$263$	− 7397.68i	− 1.73445i	0.497916	−	0.867225i	$-0.334099\pi$
$264$	−368.248	−0.0858489
$265$	0	0
$266$	−2826.25	−0.651461
$267$	3126.47i	0.716618i
$268$	4289.99i	0.977808i
$269$	3214.48	0.728589	0.364295	−	0.931284i	$-0.381310\pi$
$269$	3214.48	0.728589	0.364295	+	0.931284i	$0.381310\pi$
$270$	0	0
$271$	−7377.37	−1.65367	−0.826833	−	0.562448i	$-0.809860\pi$
$271$	−7377.37	−1.65367	−0.826833	+	0.562448i	$0.809860\pi$
$272$	8475.60i	1.88937i
$273$	2621.61i	0.581197i
$274$	7632.36	1.68280
$275$	0	0
$276$	−175.379	−0.0382485
$277$	− 810.606i	− 0.175829i	−0.996128	−	0.0879144i	$-0.971980\pi$
$277$	− 810.606i	− 0.175829i	0.996128	−	0.0879144i	$-0.0280202\pi$
$278$	6370.65i	1.37441i
$279$	738.766	0.158526
$280$	0	0
$281$	1114.72	0.236651	0.118325	−	0.992975i	$-0.462247\pi$
$281$	1114.72	0.236651	0.118325	+	0.992975i	$0.462247\pi$
$282$	2012.49i	0.424972i
$283$	− 2265.40i	− 0.475844i	−0.971284	−	0.237922i	$-0.923534\pi$
$283$	− 2265.40i	− 0.475844i	0.971284	−	0.237922i	$-0.0764663\pi$
$284$	3586.61	0.749388
$285$	0	0
$286$	−2137.59	−0.441951
$287$	2015.64i	0.414563i
$288$	1756.03i	0.359289i
$289$	−6538.04	−1.33076
$290$	0	0
$291$	−4639.88	−0.934689
$292$	5684.50i	1.13925i
$293$	− 3802.06i	− 0.758084i	−0.925380	−	0.379042i	$-0.876254\pi$
$293$	− 3802.06i	− 0.758084i	0.925380	−	0.379042i	$-0.123746\pi$
$294$	−886.416	−0.175840
$295$	0	0
$296$	242.790	0.0476753
$297$	− 297.000i	− 0.0580259i
$298$	− 5657.92i	− 1.09985i
$299$	646.738	0.125090
$300$	0	0
$301$	−3624.22	−0.694009
$302$	1511.55i	0.288013i
$303$	− 1986.85i	− 0.376704i
$304$	3861.17	0.728465
$305$	0	0
$306$	−3458.02	−0.646019
$307$	− 1356.35i	− 0.252153i	−0.992021	−	0.126077i	$-0.959761\pi$
$307$	− 1356.35i	− 0.252153i	0.992021	−	0.126077i	$-0.0402385\pi$
$308$	868.895i	0.160746i
$309$	1198.78	0.220699
$310$	0	0
$311$	−8078.07	−1.47288	−0.736440	−	0.676503i	$-0.763495\pi$
$311$	−8078.07	−1.47288	−0.736440	+	0.676503i	$0.763495\pi$
$312$	− 1811.83i	− 0.328764i
$313$	− 5761.54i	− 1.04045i	−0.854029	−	0.520226i	$-0.825848\pi$
$313$	− 5761.54i	− 1.04045i	0.854029	−	0.520226i	$-0.174152\pi$
$314$	7986.48	1.43536
$315$	0	0
$316$	−2880.63	−0.512811
$317$	5107.00i	0.904851i	0.891802	+	0.452426i	$0.149441\pi$
$317$	5107.00i	0.904851i	−0.891802	+	0.452426i	$0.850559\pi$
$318$	2520.54i	0.444481i
$319$	−2637.06	−0.462843
$320$	0	0
$321$	4773.66	0.830030
$322$	− 692.786i	− 0.119899i
$323$	5216.67i	0.898648i
$324$	−396.262	−0.0679461
$325$	0	0
$326$	−11106.9	−1.88698
$327$	− 2265.38i	− 0.383107i
$328$	− 1393.03i	− 0.234504i
$329$	−3016.68	−0.505516
$330$	0	0
$331$	−2780.94	−0.461796	−0.230898	−	0.972978i	$-0.574166\pi$
$331$	−2780.94	−0.461796	−0.230898	+	0.972978i	$0.574166\pi$
$332$	− 5680.06i	− 0.938958i
$333$	195.816i	0.0322241i
$334$	8676.35	1.42140
$335$	0	0
$336$	−3836.60	−0.622928
$337$	4939.28i	0.798397i	0.916865	+	0.399198i	$0.130712\pi$
$337$	4939.28i	0.798397i	−0.916865	+	0.399198i	$0.869288\pi$
$338$	− 2628.73i	− 0.423029i
$339$	−3436.96	−0.550649
$340$	0	0
$341$	−902.936	−0.143392
$342$	1575.34i	0.249079i
$343$	− 6866.96i	− 1.08099i
$344$	2504.74	0.392578
$345$	0	0
$346$	−13493.4	−2.09656
$347$	− 2711.58i	− 0.419496i	−0.977755	−	0.209748i	$-0.932736\pi$
$347$	− 2711.58i	− 0.419496i	0.977755	−	0.209748i	$-0.0672644\pi$
$348$	3518.40i	0.541971i
$349$	−5496.03	−0.842967	−0.421484	−	0.906836i	$-0.638490\pi$
$349$	−5496.03	−0.842967	−0.421484	+	0.906836i	$0.638490\pi$
$350$	0	0
$351$	1461.28	0.222214
$352$	− 2146.26i	− 0.324989i
$353$	− 6372.23i	− 0.960792i	−0.877052	−	0.480396i	$-0.840493\pi$
$353$	− 6372.23i	− 0.960792i	0.877052	−	0.480396i	$-0.159507\pi$
$354$	2509.12	0.376718
$355$	0	0
$356$	−5098.36	−0.759024
$357$	− 5183.48i	− 0.768456i
$358$	− 9098.46i	− 1.34321i
$359$	−630.622	−0.0927101	−0.0463551	−	0.998925i	$-0.514761\pi$
$359$	−630.622	−0.0927101	−0.0463551	+	0.998925i	$0.514761\pi$
$360$	0	0
$361$	−4482.48	−0.653518
$362$	− 49.9071i	− 0.00724602i
$363$	363.000i	0.0524864i
$364$	−4275.07	−0.615590
$365$	0	0
$366$	−1764.93	−0.252061
$367$	− 5374.49i	− 0.764431i	−0.924073	−	0.382216i	$-0.875161\pi$
$367$	− 5374.49i	− 0.764431i	0.924073	−	0.382216i	$-0.124839\pi$
$368$	946.471i	0.134071i
$369$	1123.51	0.158503
$370$	0	0
$371$	−3778.23	−0.528722
$372$	1204.71i	0.167907i
$373$	− 7520.12i	− 1.04391i	−0.852974	−	0.521953i	$-0.825204\pi$
$373$	− 7520.12i	− 1.04391i	0.852974	−	0.521953i	$-0.174796\pi$
$374$	4226.46	0.584346
$375$	0	0
$376$	2084.86	0.285954
$377$	− 12974.7i	− 1.77249i
$378$	− 1565.32i	− 0.212993i
$379$	12509.1	1.69538	0.847690	−	0.530492i	$-0.177993\pi$
$379$	12509.1	1.69538	0.847690	+	0.530492i	$0.177993\pi$
$380$	0	0
$381$	−4383.86	−0.589481
$382$	12550.4i	1.68098i
$383$	11149.9i	1.48755i	0.668430	+	0.743775i	$0.266967\pi$
$383$	11149.9i	1.48755i	−0.668430	+	0.743775i	$0.733033\pi$
$384$	3961.72	0.526486
$385$	0	0
$386$	12456.8	1.64258
$387$	2020.13i	0.265346i
$388$	− 7566.28i	− 0.989999i
$389$	−3194.22	−0.416333	−0.208166	−	0.978093i	$-0.566750\pi$
$389$	−3194.22	−0.416333	−0.208166	+	0.978093i	$0.566750\pi$
$390$	0	0
$391$	−1278.74	−0.165393
$392$	918.292i	0.118318i
$393$	− 4574.81i	− 0.587198i
$394$	−13063.8	−1.67042
$395$	0	0
$396$	484.320	0.0614596
$397$	− 584.410i	− 0.0738809i	−0.999317	−	0.0369404i	$-0.988239\pi$
$397$	− 584.410i	− 0.0738809i	0.999317	−	0.0369404i	$-0.0117612\pi$
$398$	− 185.291i	− 0.0233361i
$399$	−2361.40	−0.296286
$400$	0	0
$401$	−6951.24	−0.865657	−0.432829	−	0.901476i	$-0.642484\pi$
$401$	−6951.24	−0.865657	−0.432829	+	0.901476i	$0.642484\pi$
$402$	9445.88i	1.17193i
$403$	− 4442.56i	− 0.549130i
$404$	3239.96	0.398995
$405$	0	0
$406$	−13898.5	−1.69894
$407$	− 239.330i	− 0.0291478i
$408$	3582.37i	0.434690i
$409$	−11754.1	−1.42104	−0.710519	−	0.703678i	$-0.751540\pi$
$409$	−11754.1	−1.42104	−0.710519	+	0.703678i	$0.751540\pi$
$410$	0	0
$411$	6377.03	0.765342
$412$	1954.85i	0.233759i
$413$	3761.10i	0.448116i
$414$	−386.157	−0.0458420
$415$	0	0
$416$	10559.9	1.24457
$417$	5322.83i	0.625084i
$418$	− 1925.42i	− 0.225300i
$419$	−11829.8	−1.37929	−0.689646	−	0.724146i	$-0.742234\pi$
$419$	−11829.8	−1.37929	−0.689646	+	0.724146i	$0.742234\pi$
$420$	0	0
$421$	−2000.07	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$421$	−2000.07	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$422$	7484.76i	0.863395i
$423$	1681.49i	0.193278i
$424$	2611.18	0.299081
$425$	0	0
$426$	7897.16	0.898166
$427$	− 2645.58i	− 0.299833i
$428$	7784.43i	0.879147i
$429$	−1786.00	−0.201000
$430$	0	0
$431$	−8064.11	−0.901240	−0.450620	−	0.892716i	$-0.648797\pi$
$431$	−8064.11	−0.901240	−0.450620	+	0.892716i	$0.648797\pi$
$432$	2138.51i	0.238169i
$433$	10710.3i	1.18869i	0.804210	+	0.594345i	$0.202589\pi$
$433$	10710.3i	1.18869i	−0.804210	+	0.594345i	$0.797411\pi$
$434$	−4758.87	−0.526344
$435$	0	0
$436$	3694.18	0.405777
$437$	582.546i	0.0637688i
$438$	12516.4i	1.36542i
$439$	−4658.08	−0.506419	−0.253210	−	0.967411i	$-0.581486\pi$
$439$	−4658.08	−0.506419	−0.253210	+	0.967411i	$0.581486\pi$
$440$	0	0
$441$	−740.622	−0.0799721
$442$	20794.7i	2.23779i
$443$	− 2094.73i	− 0.224658i	−0.993671	−	0.112329i	$-0.964169\pi$
$443$	− 2094.73i	− 0.224658i	0.993671	−	0.112329i	$-0.0358310\pi$
$444$	−319.318	−0.0341310
$445$	0	0
$446$	6776.67	0.719472
$447$	− 4727.33i	− 0.500212i
$448$	− 1080.82i	− 0.113982i
$449$	1402.21	0.147382	0.0736909	−	0.997281i	$-0.476522\pi$
$449$	1402.21	0.147382	0.0736909	+	0.997281i	$0.476522\pi$
$450$	0	0
$451$	−1373.18	−0.143372
$452$	− 5604.67i	− 0.583234i
$453$	1262.93i	0.130989i
$454$	4131.78	0.427124
$455$	0	0
$456$	1632.00	0.167599
$457$	− 4914.19i	− 0.503011i	−0.967856	−	0.251506i	$-0.919074\pi$
$457$	− 4914.19i	− 0.503011i	0.967856	−	0.251506i	$-0.0809257\pi$
$458$	− 14745.7i	− 1.50441i
$459$	−2889.26	−0.293810
$460$	0	0
$461$	−2214.08	−0.223688	−0.111844	−	0.993726i	$-0.535676\pi$
$461$	−2214.08	−0.223688	−0.111844	+	0.993726i	$0.535676\pi$
$462$	1913.17i	0.192660i
$463$	5567.02i	0.558793i	0.960176	+	0.279396i	$0.0901344\pi$
$463$	5567.02i	0.558793i	−0.960176	+	0.279396i	$0.909866\pi$
$464$	18987.8	1.89976
$465$	0	0
$466$	−20586.8	−2.04649
$467$	497.054i	0.0492525i	0.999697	+	0.0246263i	$0.00783957\pi$
$467$	497.054i	0.0492525i	−0.999697	+	0.0246263i	$0.992160\pi$
$468$	2382.91i	0.235364i
$469$	−14159.1	−1.39405
$470$	0	0
$471$	6672.90	0.652804
$472$	− 2599.35i	− 0.253484i
$473$	− 2469.05i	− 0.240015i
$474$	−6342.70	−0.614620
$475$	0	0
$476$	8452.73	0.813929
$477$	2105.97i	0.202151i
$478$	− 21673.3i	− 2.07388i
$479$	9349.28	0.891815	0.445908	−	0.895079i	$-0.352881\pi$
$479$	9349.28	0.891815	0.445908	+	0.895079i	$0.352881\pi$
$480$	0	0
$481$	1177.53	0.111624
$482$	13360.1i	1.26252i
$483$	− 578.840i	− 0.0545303i
$484$	−591.946	−0.0555923
$485$	0	0
$486$	−872.506	−0.0814355
$487$	197.750i	0.0184003i	0.999958	+	0.00920013i	$0.00292853\pi$
$487$	197.750i	0.0184003i	−0.999958	+	0.00920013i	$0.997071\pi$
$488$	1828.40i	0.169606i
$489$	−9280.12	−0.858204
$490$	0	0
$491$	−9997.05	−0.918861	−0.459430	−	0.888214i	$-0.651946\pi$
$491$	−9997.05	−0.918861	−0.459430	+	0.888214i	$0.651946\pi$
$492$	1832.12i	0.167883i
$493$	25653.7i	2.34358i
$494$	9473.31	0.862803
$495$	0	0
$496$	6501.48	0.588558
$497$	11837.6i	1.06839i
$498$	− 12506.6i	− 1.12537i
$499$	8714.73	0.781813	0.390907	−	0.920430i	$-0.372162\pi$
$499$	8714.73	0.781813	0.390907	+	0.920430i	$0.372162\pi$
$500$	0	0
$501$	7249.30	0.646457
$502$	− 13679.6i	− 1.21623i
$503$	− 5978.53i	− 0.529959i	−0.964254	−	0.264979i	$-0.914635\pi$
$503$	− 5978.53i	− 0.529959i	0.964254	−	0.264979i	$-0.0853652\pi$
$504$	−1621.61	−0.143318
$505$	0	0
$506$	471.970	0.0414657
$507$	− 2196.36i	− 0.192394i
$508$	− 7148.79i	− 0.624363i
$509$	8205.79	0.714569	0.357284	−	0.933996i	$-0.383703\pi$
$509$	8205.79	0.714569	0.357284	+	0.933996i	$0.383703\pi$
$510$	0	0
$511$	−18761.7	−1.62421
$512$	8383.17i	0.723608i
$513$	1316.24i	0.113281i
$514$	−4401.85	−0.377738
$515$	0	0
$516$	−3294.24	−0.281048
$517$	− 2055.15i	− 0.174827i
$518$	− 1261.38i	− 0.106992i
$519$	−11274.1	−0.953519
$520$	0	0
$521$	−5266.06	−0.442822	−0.221411	−	0.975181i	$-0.571066\pi$
$521$	−5266.06	−0.442822	−0.221411	+	0.975181i	$0.571066\pi$
$522$	7746.97i	0.649570i
$523$	22398.6i	1.87270i	0.351068	+	0.936350i	$0.385819\pi$
$523$	22398.6i	1.87270i	−0.351068	+	0.936350i	$0.614181\pi$
$524$	7460.18	0.621945
$525$	0	0
$526$	26561.8	2.20181
$527$	8783.89i	0.726057i
$528$	− 2613.74i	− 0.215432i
$529$	12024.2	0.988264
$530$	0	0
$531$	2096.43	0.171332
$532$	− 3850.75i	− 0.313818i
$533$	− 6756.22i	− 0.549052i
$534$	−11225.8	−0.909714
$535$	0	0
$536$	9785.56	0.788566
$537$	− 7601.98i	− 0.610893i
$538$	11541.8i	0.924911i
$539$	905.205	0.0723375
$540$	0	0
$541$	−13030.2	−1.03551	−0.517757	−	0.855528i	$-0.673233\pi$
$541$	−13030.2	−1.03551	−0.517757	+	0.855528i	$0.673233\pi$
$542$	− 26488.9i	− 2.09925i
$543$	− 41.6986i	− 0.00329550i
$544$	−20879.1	−1.64556
$545$	0	0
$546$	−9413.04	−0.737804
$547$	10448.9i	0.816747i	0.912815	+	0.408374i	$0.133904\pi$
$547$	10448.9i	0.816747i	−0.912815	+	0.408374i	$0.866096\pi$
$548$	10399.1i	0.810631i
$549$	−1474.64	−0.114638
$550$	0	0
$551$	11686.9	0.903588
$552$	400.044i	0.0308460i
$553$	− 9507.55i	− 0.731107i
$554$	2910.53	0.223207
$555$	0	0
$556$	−8679.97	−0.662073
$557$	− 11448.7i	− 0.870911i	−0.900210	−	0.435456i	$-0.856587\pi$
$557$	− 11448.7i	− 0.870911i	0.900210	−	0.435456i	$-0.143413\pi$
$558$	2652.58i	0.201242i
$559$	12148.0	0.919153
$560$	0	0
$561$	3531.31	0.265762
$562$	4002.49i	0.300418i
$563$	26035.8i	1.94898i	0.224422	+	0.974492i	$0.427951\pi$
$563$	26035.8i	1.94898i	−0.224422	+	0.974492i	$0.572049\pi$
$564$	−2742.01	−0.204715
$565$	0	0
$566$	8134.05	0.604063
$567$	− 1307.86i	− 0.0968697i
$568$	− 8181.15i	− 0.604354i
$569$	−25075.0	−1.84745	−0.923725	−	0.383057i	$-0.874871\pi$
$569$	−25075.0	−1.84745	−0.923725	+	0.383057i	$0.874871\pi$
$570$	0	0
$571$	19056.6	1.39666	0.698330	−	0.715776i	$-0.253927\pi$
$571$	19056.6	1.39666	0.698330	+	0.715776i	$0.253927\pi$
$572$	− 2912.45i	− 0.212894i
$573$	10486.2i	0.764512i
$574$	−7237.28	−0.526268
$575$	0	0
$576$	−602.447	−0.0435798
$577$	− 6482.55i	− 0.467716i	−0.972271	−	0.233858i	$-0.924865\pi$
$577$	− 6482.55i	− 0.467716i	0.972271	−	0.233858i	$-0.0751351\pi$
$578$	− 23475.2i	− 1.68934i
$579$	10408.0	0.747048
$580$	0	0
$581$	18747.1	1.33866
$582$	− 16659.8i	− 1.18654i
$583$	− 2573.97i	− 0.182852i
$584$	12966.5	0.918762
$585$	0	0
$586$	13651.5	0.962353
$587$	24650.3i	1.73327i	0.498946	+	0.866633i	$0.333720\pi$
$587$	24650.3i	1.73327i	−0.498946	+	0.866633i	$0.666280\pi$
$588$	− 1207.74i	− 0.0847045i
$589$	4001.61	0.279938
$590$	0	0
$591$	−10915.2	−0.759712
$592$	1723.27i	0.119638i
$593$	− 23163.2i	− 1.60405i	−0.597292	−	0.802024i	$-0.703757\pi$
$593$	− 23163.2i	− 1.60405i	0.597292	−	0.802024i	$-0.296243\pi$
$594$	1066.40	0.0736612
$595$	0	0
$596$	7708.88	0.529812
$597$	− 154.815i	− 0.0106133i
$598$	2322.15i	0.158796i
$599$	−5355.44	−0.365304	−0.182652	−	0.983178i	$-0.558468\pi$
$599$	−5355.44	−0.365304	−0.182652	+	0.983178i	$0.558468\pi$
$600$	0	0
$601$	−20417.6	−1.38578	−0.692889	−	0.721045i	$-0.743662\pi$
$601$	−20417.6	−1.38578	−0.692889	+	0.721045i	$0.743662\pi$
$602$	− 13013.0i	− 0.881012i
$603$	7892.26i	0.532998i
$604$	−2059.48	−0.138740
$605$	0	0
$606$	7133.89	0.478208
$607$	− 6749.81i	− 0.451345i	−0.974203	−	0.225673i	$-0.927542\pi$
$607$	− 6749.81i	− 0.451345i	0.974203	−	0.225673i	$-0.0724580\pi$
$608$	9511.77i	0.634463i
$609$	−11612.5	−0.772681
$610$	0	0
$611$	10111.6	0.669511
$612$	− 4711.53i	− 0.311197i
$613$	30321.0i	1.99780i	0.0468613	+	0.998901i	$0.485078\pi$
$613$	30321.0i	1.99780i	−0.0468613	+	0.998901i	$0.514922\pi$
$614$	4870.06	0.320097
$615$	0	0
$616$	1981.97	0.129636
$617$	15236.8i	0.994182i	0.867699	+	0.497091i	$0.165598\pi$
$617$	15236.8i	0.994182i	−0.867699	+	0.497091i	$0.834402\pi$
$618$	4304.28i	0.280168i
$619$	20875.4	1.35550	0.677749	−	0.735293i	$-0.262956\pi$
$619$	20875.4	1.35550	0.677749	+	0.735293i	$0.262956\pi$
$620$	0	0
$621$	−322.644	−0.0208490
$622$	− 29004.8i	− 1.86975i
$623$	− 16827.2i	− 1.08213i
$624$	12859.9	0.825013
$625$	0	0
$626$	20687.2	1.32081
$627$	− 1608.74i	− 0.102467i
$628$	10881.5i	0.691434i
$629$	−2328.24	−0.147588
$630$	0	0
$631$	27966.1	1.76437	0.882183	−	0.470907i	$-0.156073\pi$
$631$	27966.1	1.76437	0.882183	+	0.470907i	$0.156073\pi$
$632$	6570.79i	0.413563i
$633$	6253.70i	0.392674i
$634$	−18337.0	−1.14867
$635$	0	0
$636$	−3434.23	−0.214113
$637$	4453.72i	0.277022i
$638$	− 9468.52i	− 0.587558i
$639$	6598.27	0.408487
$640$	0	0
$641$	−17992.0	−1.10865	−0.554323	−	0.832301i	$-0.687023\pi$
$641$	−17992.0	−1.10865	−0.554323	+	0.832301i	$0.687023\pi$
$642$	17140.1i	1.05369i
$643$	9448.64i	0.579499i	0.957102	+	0.289750i	$0.0935721\pi$
$643$	9448.64i	0.579499i	−0.957102	+	0.289750i	$0.906428\pi$
$644$	943.918	0.0577571
$645$	0	0
$646$	−18730.8	−1.14079
$647$	− 7429.22i	− 0.451426i	−0.974194	−	0.225713i	$-0.927529\pi$
$647$	− 7429.22i	− 0.451426i	0.974194	−	0.225713i	$-0.0724712\pi$
$648$	903.882i	0.0547960i
$649$	−2562.30	−0.154976
$650$	0	0
$651$	−3976.15	−0.239382
$652$	− 15133.2i	− 0.908988i
$653$	4488.63i	0.268995i	0.990914	+	0.134497i	$0.0429420\pi$
$653$	4488.63i	0.268995i	−0.990914	+	0.134497i	$0.957058\pi$
$654$	8134.00	0.486337
$655$	0	0
$656$	9887.42	0.588474
$657$	10457.7i	0.620998i
$658$	− 10831.6i	− 0.641730i
$659$	25326.7	1.49710	0.748550	−	0.663078i	$-0.230750\pi$
$659$	25326.7	1.49710	0.748550	+	0.663078i	$0.230750\pi$
$660$	0	0
$661$	15192.3	0.893969	0.446984	−	0.894542i	$-0.352498\pi$
$661$	15192.3	0.893969	0.446984	+	0.894542i	$0.352498\pi$
$662$	− 9985.15i	− 0.586229i
$663$	17374.5i	1.01775i
$664$	−12956.4	−0.757235
$665$	0	0
$666$	−703.087	−0.0409070
$667$	2864.75i	0.166302i
$668$	11821.5i	0.684711i
$669$	5662.07	0.327217
$670$	0	0
$671$	1802.34	0.103694
$672$	− 9451.25i	− 0.542545i
$673$	− 11718.2i	− 0.671180i	−0.942008	−	0.335590i	$-0.891064\pi$
$673$	− 11718.2i	− 0.671180i	0.942008	−	0.335590i	$-0.108936\pi$
$674$	−17734.8	−1.01353
$675$	0	0
$676$	3581.63	0.203779
$677$	− 15649.1i	− 0.888397i	−0.895928	−	0.444198i	$-0.853489\pi$
$677$	− 15649.1i	− 0.888397i	0.895928	−	0.444198i	$-0.146511\pi$
$678$	− 12340.6i	− 0.699024i
$679$	24972.6	1.41143
$680$	0	0
$681$	3452.20	0.194257
$682$	− 3242.05i	− 0.182030i
$683$	18162.8i	1.01754i	0.860903	+	0.508770i	$0.169899\pi$
$683$	18162.8i	1.01754i	−0.860903	+	0.508770i	$0.830101\pi$
$684$	−2146.40	−0.119985
$685$	0	0
$686$	24656.2	1.37227
$687$	− 12320.4i	− 0.684208i
$688$	17778.1i	0.985149i
$689$	12664.2	0.700246
$690$	0	0
$691$	29606.7	1.62995	0.814973	−	0.579500i	$-0.196752\pi$
$691$	29606.7	1.62995	0.814973	+	0.579500i	$0.196752\pi$
$692$	− 18384.7i	− 1.00994i
$693$	1598.50i	0.0876220i
$694$	9736.08	0.532531
$695$	0	0
$696$	8025.55	0.437080
$697$	13358.5i	0.725953i
$698$	− 19733.8i	− 1.07011i
$699$	−17200.7	−0.930746
$700$	0	0
$701$	−26164.7	−1.40974	−0.704870	−	0.709336i	$-0.748995\pi$
$701$	−26164.7	−1.40974	−0.704870	+	0.709336i	$0.748995\pi$
$702$	5246.80i	0.282091i
$703$	1060.66i	0.0569040i
$704$	736.324	0.0394194
$705$	0	0
$706$	22879.9	1.21968
$707$	10693.5i	0.568842i
$708$	3418.66i	0.181470i
$709$	14508.9	0.768535	0.384268	−	0.923222i	$-0.374454\pi$
$709$	14508.9	0.768535	0.384268	+	0.923222i	$0.374454\pi$
$710$	0	0
$711$	−5299.48	−0.279530
$712$	11629.5i	0.612125i
$713$	980.898i	0.0515216i
$714$	18611.6	0.975520
$715$	0	0
$716$	12396.6	0.647043
$717$	− 18108.5i	− 0.943202i
$718$	− 2264.28i	− 0.117691i
$719$	2545.80	0.132048	0.0660239	−	0.997818i	$-0.478969\pi$
$719$	2545.80	0.132048	0.0660239	+	0.997818i	$0.478969\pi$
$720$	0	0
$721$	−6452.01	−0.333267
$722$	− 16094.6i	− 0.829611i
$723$	11162.7i	0.574198i
$724$	67.9982	0.00349051
$725$	0	0
$726$	−1303.37	−0.0666291
$727$	− 18984.8i	− 0.968511i	−0.874927	−	0.484255i	$-0.839091\pi$
$727$	− 18984.8i	− 0.968511i	0.874927	−	0.484255i	$-0.160909\pi$
$728$	9751.54i	0.496451i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−24019.2	−1.21530
$732$	− 2404.70i	− 0.121421i
$733$	− 36740.4i	− 1.85134i	−0.378326	−	0.925672i	$-0.623500\pi$
$733$	− 36740.4i	− 1.85134i	0.378326	−	0.925672i	$-0.376500\pi$
$734$	19297.4	0.970411
$735$	0	0
$736$	−2331.58	−0.116770
$737$	− 9646.10i	− 0.482115i
$738$	4034.04i	0.201213i
$739$	−1755.17	−0.0873682	−0.0436841	−	0.999045i	$-0.513910\pi$
$739$	−1755.17	−0.0873682	−0.0436841	+	0.999045i	$0.513910\pi$
$740$	0	0
$741$	7915.18	0.392404
$742$	− 13566.0i	− 0.671189i
$743$	− 17972.6i	− 0.887419i	−0.896171	−	0.443709i	$-0.853662\pi$
$743$	− 17972.6i	− 0.887419i	0.896171	−	0.443709i	$-0.146338\pi$
$744$	2747.97	0.135411
$745$	0	0
$746$	27001.4	1.32519
$747$	− 10449.6i	− 0.511821i
$748$	5758.54i	0.281488i
$749$	−25692.6	−1.25339
$750$	0	0
$751$	22704.9	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$751$	22704.9	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$752$	14797.9i	0.717583i
$753$	− 11429.6i	− 0.553146i
$754$	46586.3	2.25010
$755$	0	0
$756$	2132.74	0.102602
$757$	39860.3i	1.91380i	0.290414	+	0.956901i	$0.406207\pi$
$757$	39860.3i	1.91380i	−0.290414	+	0.956901i	$0.593793\pi$
$758$	44914.6i	2.15221i
$759$	394.342	0.0188587
$760$	0	0
$761$	19631.4	0.935135	0.467568	−	0.883957i	$-0.345130\pi$
$761$	19631.4	0.935135	0.467568	+	0.883957i	$0.345130\pi$
$762$	− 15740.5i	− 0.748319i
$763$	12192.7i	0.578511i
$764$	−17099.8	−0.809752
$765$	0	0
$766$	−40034.3	−1.88838
$767$	− 12606.8i	− 0.593490i
$768$	15831.3i	0.743832i
$769$	−35029.9	−1.64267	−0.821333	−	0.570449i	$-0.806769\pi$
$769$	−35029.9	−1.64267	−0.821333	+	0.570449i	$0.806769\pi$
$770$	0	0
$771$	−3677.86	−0.171796
$772$	16972.4i	0.791255i
$773$	26736.9i	1.24406i	0.782992	+	0.622032i	$0.213693\pi$
$773$	26736.9i	1.24406i	−0.782992	+	0.622032i	$0.786307\pi$
$774$	−7253.40	−0.336845
$775$	0	0
$776$	−17258.8	−0.798398
$777$	− 1053.91i	− 0.0486600i
$778$	− 11469.0i	− 0.528515i
$779$	6085.64	0.279898
$780$	0	0
$781$	−8064.55	−0.369491
$782$	− 4591.39i	− 0.209959i
$783$	6472.78i	0.295426i
$784$	−6517.81	−0.296912
$785$	0	0
$786$	16426.1	0.745421
$787$	4799.45i	0.217385i	0.994075	+	0.108693i	$0.0346664\pi$
$787$	4799.45i	0.217385i	−0.994075	+	0.108693i	$0.965334\pi$
$788$	− 17799.4i	− 0.804668i
$789$	22193.0	1.00139
$790$	0	0
$791$	18498.3	0.831508
$792$	− 1104.74i	− 0.0495649i
$793$	8867.72i	0.397102i
$794$	2098.36	0.0937884
$795$	0	0
$796$	252.457	0.0112413
$797$	38438.9i	1.70838i	0.519963	+	0.854189i	$0.325946\pi$
$797$	38438.9i	1.70838i	−0.519963	+	0.854189i	$0.674054\pi$
$798$	− 8478.76i	− 0.376121i
$799$	−19992.8	−0.885224
$800$	0	0
$801$	−9379.42	−0.413740
$802$	− 24958.9i	− 1.09891i
$803$	− 12781.7i	− 0.561713i
$804$	−12870.0	−0.564538
$805$	0	0
$806$	15951.3	0.697096
$807$	9643.45i	0.420651i
$808$	− 7390.42i	− 0.321775i
$809$	−9960.91	−0.432889	−0.216444	−	0.976295i	$-0.569446\pi$
$809$	−9960.91	−0.432889	−0.216444	+	0.976295i	$0.569446\pi$
$810$	0	0
$811$	−34199.9	−1.48079	−0.740396	−	0.672171i	$-0.765362\pi$
$811$	−34199.9	−1.48079	−0.740396	+	0.672171i	$0.765362\pi$
$812$	− 18936.6i	− 0.818404i
$813$	− 22132.1i	− 0.954744i
$814$	859.329	0.0370018
$815$	0	0
$816$	−25426.8	−1.09083
$817$	10942.3i	0.468570i
$818$	− 42203.9i	− 1.80394i
$819$	−7864.82	−0.335555
$820$	0	0
$821$	−31796.4	−1.35165	−0.675823	−	0.737064i	$-0.736212\pi$
$821$	−31796.4	−1.35165	−0.675823	+	0.737064i	$0.736212\pi$
$822$	22897.1i	0.971567i
$823$	10783.6i	0.456733i	0.973575	+	0.228366i	$0.0733384\pi$
$823$	10783.6i	0.456733i	−0.973575	+	0.228366i	$0.926662\pi$
$824$	4459.07	0.188518
$825$	0	0
$826$	−13504.5	−0.568862
$827$	44197.5i	1.85840i	0.369576	+	0.929201i	$0.379503\pi$
$827$	44197.5i	1.85840i	−0.369576	+	0.929201i	$0.620497\pi$
$828$	− 526.137i	− 0.0220828i
$829$	−22487.7	−0.942137	−0.471069	−	0.882097i	$-0.656132\pi$
$829$	−22487.7	−0.942137	−0.471069	+	0.882097i	$0.656132\pi$
$830$	0	0
$831$	2431.82	0.101515
$832$	3622.81i	0.150959i
$833$	− 8805.96i	− 0.366276i
$834$	−19111.9	−0.793516
$835$	0	0
$836$	2623.38	0.108530
$837$	2216.30i	0.0915250i
$838$	− 42475.6i	− 1.75095i
$839$	−22491.0	−0.925477	−0.462738	−	0.886495i	$-0.653133\pi$
$839$	−22491.0	−0.925477	−0.462738	+	0.886495i	$0.653133\pi$
$840$	0	0
$841$	33082.7	1.35646
$842$	− 7181.37i	− 0.293927i
$843$	3344.17i	0.136630i
$844$	−10198.0	−0.415910
$845$	0	0
$846$	−6037.48	−0.245358
$847$	− 1953.72i	− 0.0792571i
$848$	18533.5i	0.750524i
$849$	6796.19	0.274729
$850$	0	0
$851$	−259.994	−0.0104730
$852$	10759.8i	0.432660i
$853$	− 22327.9i	− 0.896241i	−0.893973	−	0.448120i	$-0.852094\pi$
$853$	− 22327.9i	− 0.896241i	0.893973	−	0.448120i	$-0.147906\pi$
$854$	9499.12	0.380624
$855$	0	0
$856$	17756.5	0.708999
$857$	− 14505.9i	− 0.578193i	−0.957300	−	0.289096i	$-0.906645\pi$
$857$	− 14505.9i	− 0.578193i	0.957300	−	0.289096i	$-0.0933548\pi$
$858$	− 6412.76i	− 0.255161i
$859$	8411.45	0.334104	0.167052	−	0.985948i	$-0.446575\pi$
$859$	8411.45	0.334104	0.167052	+	0.985948i	$0.446575\pi$
$860$	0	0
$861$	−6046.92	−0.239348
$862$	− 28954.7i	− 1.14408i
$863$	− 32499.6i	− 1.28192i	−0.767573	−	0.640961i	$-0.778536\pi$
$863$	− 32499.6i	− 1.28192i	0.767573	−	0.640961i	$-0.221464\pi$
$864$	−5268.10	−0.207436
$865$	0	0
$866$	−38455.9	−1.50899
$867$	− 19614.1i	− 0.768316i
$868$	− 6483.94i	− 0.253548i
$869$	6477.15	0.252845
$870$	0	0
$871$	47460.0	1.84629
$872$	− 8426.50i	− 0.327245i
$873$	− 13919.6i	− 0.539643i
$874$	−2091.67	−0.0809516
$875$	0	0
$876$	−17053.5	−0.657745
$877$	− 32183.1i	− 1.23916i	−0.784933	−	0.619581i	$-0.787303\pi$
$877$	− 32183.1i	− 1.23916i	0.784933	−	0.619581i	$-0.212697\pi$
$878$	− 16725.1i	− 0.642876i
$879$	11406.2	0.437680
$880$	0	0
$881$	−6246.34	−0.238870	−0.119435	−	0.992842i	$-0.538108\pi$
$881$	−6246.34	−0.238870	−0.119435	+	0.992842i	$0.538108\pi$
$882$	− 2659.25i	− 0.101521i
$883$	11801.1i	0.449762i	0.974386	+	0.224881i	$0.0721993\pi$
$883$	11801.1i	0.449762i	−0.974386	+	0.224881i	$0.927801\pi$
$884$	−28332.7	−1.07798
$885$	0	0
$886$	7521.24	0.285193
$887$	32375.1i	1.22553i	0.790264	+	0.612767i	$0.209944\pi$
$887$	32375.1i	1.22553i	−0.790264	+	0.612767i	$0.790056\pi$
$888$	728.371i	0.0275254i
$889$	23594.6	0.890145
$890$	0	0
$891$	891.000	0.0335013
$892$	9233.17i	0.346580i
$893$	9107.98i	0.341307i
$894$	16973.8	0.634997
$895$	0	0
$896$	−21322.6	−0.795020
$897$	1940.21i	0.0722205i
$898$	5034.72i	0.187095i
$899$	19678.5	0.730049
$900$	0	0
$901$	−25039.9	−0.925861
$902$	− 4930.49i	− 0.182004i
$903$	− 10872.7i	− 0.400686i
$904$	−12784.4	−0.470357
$905$	0	0
$906$	−4534.64	−0.166284
$907$	− 19592.8i	− 0.717277i	−0.933477	−	0.358638i	$-0.883241\pi$
$907$	− 19592.8i	− 0.717277i	0.933477	−	0.358638i	$-0.116759\pi$
$908$	5629.53i	0.205752i
$909$	5960.54	0.217490
$910$	0	0
$911$	18673.8	0.679133	0.339567	−	0.940582i	$-0.389720\pi$
$911$	18673.8	0.679133	0.339567	+	0.940582i	$0.389720\pi$
$912$	11583.5i	0.420579i
$913$	12771.7i	0.462959i
$914$	17644.7	0.638550
$915$	0	0
$916$	20090.9	0.724696
$917$	24622.4i	0.886698i
$918$	− 10374.1i	− 0.372979i
$919$	4572.90	0.164142	0.0820708	−	0.996627i	$-0.473847\pi$
$919$	4572.90	0.164142	0.0820708	+	0.996627i	$0.473847\pi$
$920$	0	0
$921$	4069.05	0.145581
$922$	− 7949.78i	− 0.283961i
$923$	− 39678.6i	− 1.41499i
$924$	−2606.69	−0.0928070
$925$	0	0
$926$	−19988.7	−0.709362
$927$	3596.33i	0.127421i
$928$	46775.4i	1.65461i
$929$	44222.1	1.56176	0.780882	−	0.624679i	$-0.214770\pi$
$929$	44222.1	1.56176	0.780882	+	0.624679i	$0.214770\pi$
$930$	0	0
$931$	−4011.67	−0.141221
$932$	− 28049.3i	− 0.985823i
$933$	− 24234.2i	− 0.850367i
$934$	−1784.70	−0.0625238
$935$	0	0
$936$	5435.48	0.189812
$937$	9218.62i	0.321408i	0.987003	+	0.160704i	$0.0513764\pi$
$937$	9218.62i	0.321408i	−0.987003	+	0.160704i	$0.948624\pi$
$938$	− 50839.2i	− 1.76968i
$939$	17284.6	0.600705
$940$	0	0
$941$	−26484.9	−0.917516	−0.458758	−	0.888561i	$-0.651706\pi$
$941$	−26484.9	−0.917516	−0.458758	+	0.888561i	$0.651706\pi$
$942$	23959.4i	0.828705i
$943$	1491.75i	0.0515142i
$944$	18449.5	0.636103
$945$	0	0
$946$	8865.26	0.304688
$947$	− 44972.4i	− 1.54320i	−0.636110	−	0.771599i	$-0.719457\pi$
$947$	− 44972.4i	− 1.54320i	0.636110	−	0.771599i	$-0.280543\pi$
$948$	− 8641.90i	− 0.296072i
$949$	62887.5	2.15112
$950$	0	0
$951$	−15321.0	−0.522416
$952$	− 19280.9i	− 0.656404i
$953$	2052.50i	0.0697659i	0.999391	+	0.0348829i	$0.0111058\pi$
$953$	2052.50i	0.0697659i	−0.999391	+	0.0348829i	$0.988894\pi$
$954$	−7561.63	−0.256621
$955$	0	0
$956$	29529.7	0.999016
$957$	− 7911.18i	− 0.267223i
$958$	33569.2i	1.13212i
$959$	−34322.2	−1.15570
$960$	0	0
$961$	−23053.0	−0.773826
$962$	4228.00i	0.141701i
$963$	14321.0i	0.479218i
$964$	−18203.1	−0.608176
$965$	0	0
$966$	2078.36	0.0692237
$967$	14950.9i	0.497195i	0.968607	+	0.248598i	$0.0799697\pi$
$967$	14950.9i	0.497195i	−0.968607	+	0.248598i	$0.920030\pi$
$968$	1350.24i	0.0448331i
$969$	−15650.0	−0.518835
$970$	0	0
$971$	26751.9	0.884151	0.442076	−	0.896978i	$-0.354242\pi$
$971$	26751.9	0.884151	0.442076	+	0.896978i	$0.354242\pi$
$972$	− 1188.78i	− 0.0392287i
$973$	− 28648.3i	− 0.943908i
$974$	−710.034	−0.0233583
$975$	0	0
$976$	−12977.5	−0.425615
$977$	− 56893.7i	− 1.86304i	−0.363690	−	0.931520i	$-0.618483\pi$
$977$	− 56893.7i	− 1.86304i	0.363690	−	0.931520i	$-0.381517\pi$
$978$	− 33320.8i	− 1.08945i
$979$	11463.7	0.374242
$980$	0	0
$981$	6796.15	0.221187
$982$	− 35895.0i	− 1.16645i
$983$	34633.1i	1.12373i	0.827230	+	0.561863i	$0.189915\pi$
$983$	34633.1i	1.12373i	−0.827230	+	0.561863i	$0.810085\pi$
$984$	4179.10	0.135391
$985$	0	0
$986$	−92111.0	−2.97506
$987$	− 9050.03i	− 0.291860i
$988$	12907.3i	0.415625i
$989$	−2682.23	−0.0862386
$990$	0	0
$991$	1961.79	0.0628843	0.0314422	−	0.999506i	$-0.489990\pi$
$991$	1961.79	0.0628843	0.0314422	+	0.999506i	$0.489990\pi$
$992$	16016.0i	0.512610i
$993$	− 8342.83i	− 0.266618i
$994$	−42503.8	−1.35628
$995$	0	0
$996$	17040.2	0.542108
$997$	13096.5i	0.416017i	0.978127	+	0.208009i	$0.0666982\pi$
$997$	13096.5i	0.416017i	−0.978127	+	0.208009i	$0.933302\pi$
$998$	31290.8i	0.992476i
$999$	−587.447	−0.0186046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.l.199.5		6
5.2	odd	4		825.4.a.s.1.1		3
5.3	odd	4		165.4.a.d.1.3	✓	3
5.4	even	2	inner	825.4.c.l.199.2		6
15.2	even	4		2475.4.a.s.1.3		3
15.8	even	4		495.4.a.l.1.1		3
55.43	even	4		1815.4.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.3	✓	3	5.3	odd	4
495.4.a.l.1.1		3	15.8	even	4
825.4.a.s.1.1		3	5.2	odd	4
825.4.c.l.199.2		6	5.4	even	2	inner
825.4.c.l.199.5		6	1.1	even	1	trivial
1815.4.a.s.1.1		3	55.43	even	4
2475.4.a.s.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)

\(f(q)\)	\(=\)	\(q+3.59056i q^{2} +3.00000i q^{3} -4.89212 q^{4} -10.7717 q^{6} -16.1465i q^{7} +11.1590i q^{8} -9.00000 q^{9} +11.0000 q^{11} -14.6764i q^{12} +54.1214i q^{13} +57.9749 q^{14} -79.2041 q^{16} -107.010i q^{17} -32.3150i q^{18} -48.7496 q^{19} +48.4394 q^{21} +39.4962i q^{22} -11.9498i q^{23} -33.4771 q^{24} -194.326 q^{26} -27.0000i q^{27} +78.9905i q^{28} -239.733 q^{29} -82.0851 q^{31} -195.115i q^{32} +33.0000i q^{33} +384.224 q^{34} +44.0291 q^{36} -21.7573i q^{37} -175.038i q^{38} -162.364 q^{39} -124.835 q^{41} +173.925i q^{42} -224.459i q^{43} -53.8133 q^{44} +42.9064 q^{46} -186.832i q^{47} -237.612i q^{48} +82.2913 q^{49} +321.029 q^{51} -264.768i q^{52} -233.997i q^{53} +96.9451 q^{54} +180.179 q^{56} -146.249i q^{57} -860.774i q^{58} -232.936 q^{59} +163.849 q^{61} -294.731i q^{62} +145.318i q^{63} +66.9386 q^{64} -118.488 q^{66} -876.918i q^{67} +523.503i q^{68} +35.8493 q^{69} -733.141 q^{71} -100.431i q^{72} -1161.97i q^{73} +78.1208 q^{74} +238.489 q^{76} -177.611i q^{77} -582.978i q^{78} +588.831 q^{79} +81.0000 q^{81} -448.226i q^{82} +1161.06i q^{83} -236.971 q^{84} +805.933 q^{86} -719.198i q^{87} +122.749i q^{88} +1042.16 q^{89} +873.869 q^{91} +58.4597i q^{92} -246.255i q^{93} +670.831 q^{94} +585.345 q^{96} +1546.63i q^{97} +295.472i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) \) 6 * q - 44 * q^4 + 24 * q^6 - 54 * q^9 + 66 * q^11 + 112 * q^14 + 100 * q^16 - 292 * q^19 + 24 * q^21 - 288 * q^24 - 1016 * q^26 - 136 * q^29 - 136 * q^31 + 352 * q^34 + 396 * q^36 - 392 * q^41 - 484 * q^44 + 2320 * q^46 + 314 * q^49 + 1308 * q^51 - 216 * q^54 + 2736 * q^56 + 2088 * q^59 + 1284 * q^61 - 2332 * q^64 + 264 * q^66 - 1200 * q^69 - 1088 * q^71 - 3072 * q^74 + 3992 * q^76 + 3172 * q^79 + 486 * q^81 - 2040 * q^84 + 7136 * q^86 + 4244 * q^89 - 16 * q^91 + 4304 * q^94 + 4128 * q^96 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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