LMFDB - Embedded newform 1350.4.c.s.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.s.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} -29.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} -29.0000i q^{7} -8.00000i q^{8} +57.0000 q^{11} +20.0000i q^{13} +58.0000 q^{14} +16.0000 q^{16} -72.0000i q^{17} +106.000 q^{19} +114.000i q^{22} -174.000i q^{23} -40.0000 q^{26} +116.000i q^{28} -210.000 q^{29} +47.0000 q^{31} +32.0000i q^{32} +144.000 q^{34} -2.00000i q^{37} +212.000i q^{38} +6.00000 q^{41} +218.000i q^{43} -228.000 q^{44} +348.000 q^{46} +474.000i q^{47} -498.000 q^{49} -80.0000i q^{52} -81.0000i q^{53} -232.000 q^{56} -420.000i q^{58} +84.0000 q^{59} +56.0000 q^{61} +94.0000i q^{62} -64.0000 q^{64} +142.000i q^{67} +288.000i q^{68} -360.000 q^{71} -1159.00i q^{73} +4.00000 q^{74} -424.000 q^{76} -1653.00i q^{77} +160.000 q^{79} +12.0000i q^{82} -735.000i q^{83} -436.000 q^{86} -456.000i q^{88} -954.000 q^{89} +580.000 q^{91} +696.000i q^{92} -948.000 q^{94} -191.000i q^{97} -996.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} + 114 q^{11} + 116 q^{14} + 32 q^{16} + 212 q^{19} - 80 q^{26} - 420 q^{29} + 94 q^{31} + 288 q^{34} + 12 q^{41} - 456 q^{44} + 696 q^{46} - 996 q^{49} - 464 q^{56} + 168 q^{59} + 112 q^{61} - 128 q^{64} - 720 q^{71} + 8 q^{74} - 848 q^{76} + 320 q^{79} - 872 q^{86} - 1908 q^{89} + 1160 q^{91} - 1896 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 + 114 * q^11 + 116 * q^14 + 32 * q^16 + 212 * q^19 - 80 * q^26 - 420 * q^29 + 94 * q^31 + 288 * q^34 + 12 * q^41 - 456 * q^44 + 696 * q^46 - 996 * q^49 - 464 * q^56 + 168 * q^59 + 112 * q^61 - 128 * q^64 - 720 * q^71 + 8 * q^74 - 848 * q^76 + 320 * q^79 - 872 * q^86 - 1908 * q^89 + 1160 * q^91 - 1896 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 29.0000i	− 1.56585i	−0.622114	−	0.782926i	$-0.713726\pi$
$7$	− 29.0000i	− 1.56585i	0.622114	−	0.782926i	$-0.286274\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	57.0000	1.56238	0.781188	−	0.624295i	$-0.214614\pi$
$11$	57.0000	1.56238	0.781188	+	0.624295i	$0.214614\pi$
$12$	0	0
$13$	20.0000i	0.426692i	0.976977	+	0.213346i	$0.0684362\pi$
$13$	20.0000i	0.426692i	−0.976977	+	0.213346i	$0.931564\pi$
$14$	58.0000	1.10723
$15$	0	0
$16$	16.0000	0.250000
$17$	− 72.0000i	− 1.02721i	−0.858027	−	0.513605i	$-0.828310\pi$
$17$	− 72.0000i	− 1.02721i	0.858027	−	0.513605i	$-0.171690\pi$
$18$	0	0
$19$	106.000	1.27990	0.639949	−	0.768417i	$-0.278955\pi$
$19$	106.000	1.27990	0.639949	+	0.768417i	$0.278955\pi$
$20$	0	0
$21$	0	0
$22$	114.000i	1.10477i
$23$	− 174.000i	− 1.57746i	−0.614742	−	0.788728i	$-0.710740\pi$
$23$	− 174.000i	− 1.57746i	0.614742	−	0.788728i	$-0.289260\pi$
$24$	0	0
$25$	0	0
$26$	−40.0000	−0.301717
$27$	0	0
$28$	116.000i	0.782926i
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	47.0000	0.272305	0.136152	−	0.990688i	$-0.456526\pi$
$31$	47.0000	0.272305	0.136152	+	0.990688i	$0.456526\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	144.000	0.726347
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.00888643i	−0.999990	−	0.00444322i	$-0.998586\pi$
$37$	− 2.00000i	− 0.00888643i	0.999990	−	0.00444322i	$-0.00141432\pi$
$38$	212.000i	0.905025i
$39$	0	0
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	218.000i	0.773132i	0.922262	+	0.386566i	$0.126339\pi$
$43$	218.000i	0.773132i	−0.922262	+	0.386566i	$0.873661\pi$
$44$	−228.000	−0.781188
$45$	0	0
$46$	348.000	1.11543
$47$	474.000i	1.47106i	0.677490	+	0.735532i	$0.263068\pi$
$47$	474.000i	1.47106i	−0.677490	+	0.735532i	$0.736932\pi$
$48$	0	0
$49$	−498.000	−1.45190
$50$	0	0
$51$	0	0
$52$	− 80.0000i	− 0.213346i
$53$	− 81.0000i	− 0.209928i	−0.994476	−	0.104964i	$-0.966527\pi$
$53$	− 81.0000i	− 0.209928i	0.994476	−	0.104964i	$-0.0334728\pi$
$54$	0	0
$55$	0	0
$56$	−232.000	−0.553613
$57$	0	0
$58$	− 420.000i	− 0.950840i
$59$	84.0000	0.185354	0.0926769	−	0.995696i	$-0.470458\pi$
$59$	84.0000	0.185354	0.0926769	+	0.995696i	$0.470458\pi$
$60$	0	0
$61$	56.0000	0.117542	0.0587710	−	0.998271i	$-0.481282\pi$
$61$	56.0000	0.117542	0.0587710	+	0.998271i	$0.481282\pi$
$62$	94.0000i	0.192549i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	142.000i	0.258926i	0.991584	+	0.129463i	$0.0413254\pi$
$67$	142.000i	0.258926i	−0.991584	+	0.129463i	$0.958675\pi$
$68$	288.000i	0.513605i
$69$	0	0
$70$	0	0
$71$	−360.000	−0.601748	−0.300874	−	0.953664i	$-0.597278\pi$
$71$	−360.000	−0.601748	−0.300874	+	0.953664i	$0.597278\pi$
$72$	0	0
$73$	− 1159.00i	− 1.85823i	−0.369793	−	0.929114i	$-0.620571\pi$
$73$	− 1159.00i	− 1.85823i	0.369793	−	0.929114i	$-0.379429\pi$
$74$	4.00000	0.00628366
$75$	0	0
$76$	−424.000	−0.639949
$77$	− 1653.00i	− 2.44645i
$78$	0	0
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	0	0
$82$	12.0000i	0.0161607i
$83$	− 735.000i	− 0.972009i	−0.873956	−	0.486004i	$-0.838454\pi$
$83$	− 735.000i	− 0.972009i	0.873956	−	0.486004i	$-0.161546\pi$
$84$	0	0
$85$	0	0
$86$	−436.000	−0.546687
$87$	0	0
$88$	− 456.000i	− 0.552384i
$89$	−954.000	−1.13622	−0.568111	−	0.822952i	$-0.692326\pi$
$89$	−954.000	−1.13622	−0.568111	+	0.822952i	$0.692326\pi$
$90$	0	0
$91$	580.000	0.668138
$92$	696.000i	0.788728i
$93$	0	0
$94$	−948.000	−1.04020
$95$	0	0
$96$	0	0
$97$	− 191.000i	− 0.199929i	−0.994991	−	0.0999645i	$-0.968127\pi$
$97$	− 191.000i	− 0.199929i	0.994991	−	0.0999645i	$-0.0318729\pi$
$98$	− 996.000i	− 1.02664i
$99$	0	0
$100$	0	0
$101$	363.000	0.357622	0.178811	−	0.983883i	$-0.442775\pi$
$101$	363.000	0.357622	0.178811	+	0.983883i	$0.442775\pi$
$102$	0	0
$103$	− 628.000i	− 0.600764i	−0.953819	−	0.300382i	$-0.902886\pi$
$103$	− 628.000i	− 0.600764i	0.953819	−	0.300382i	$-0.0971141\pi$
$104$	160.000	0.150859
$105$	0	0
$106$	162.000	0.148442
$107$	675.000i	0.609857i	0.952375	+	0.304929i	$0.0986326\pi$
$107$	675.000i	0.609857i	−0.952375	+	0.304929i	$0.901367\pi$
$108$	0	0
$109$	−1730.00	−1.52022	−0.760110	−	0.649795i	$-0.774855\pi$
$109$	−1730.00	−1.52022	−0.760110	+	0.649795i	$0.774855\pi$
$110$	0	0
$111$	0	0
$112$	− 464.000i	− 0.391463i
$113$	− 1866.00i	− 1.55344i	−0.629847	−	0.776719i	$-0.716882\pi$
$113$	− 1866.00i	− 1.55344i	0.629847	−	0.776719i	$-0.283118\pi$
$114$	0	0
$115$	0	0
$116$	840.000	0.672345
$117$	0	0
$118$	168.000i	0.131065i
$119$	−2088.00	−1.60846
$120$	0	0
$121$	1918.00	1.44102
$122$	112.000i	0.0831148i
$123$	0	0
$124$	−188.000	−0.136152
$125$	0	0
$126$	0	0
$127$	− 1379.00i	− 0.963515i	−0.876304	−	0.481758i	$-0.839999\pi$
$127$	− 1379.00i	− 0.963515i	0.876304	−	0.481758i	$-0.160001\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−579.000	−0.386164	−0.193082	−	0.981183i	$-0.561848\pi$
$131$	−579.000	−0.386164	−0.193082	+	0.981183i	$0.561848\pi$
$132$	0	0
$133$	− 3074.00i	− 2.00413i
$134$	−284.000	−0.183089
$135$	0	0
$136$	−576.000	−0.363173
$137$	654.000i	0.407847i	0.978987	+	0.203923i	$0.0653693\pi$
$137$	654.000i	0.407847i	−0.978987	+	0.203923i	$0.934631\pi$
$138$	0	0
$139$	3004.00	1.83306	0.916532	−	0.399961i	$-0.130976\pi$
$139$	3004.00	1.83306	0.916532	+	0.399961i	$0.130976\pi$
$140$	0	0
$141$	0	0
$142$	− 720.000i	− 0.425500i
$143$	1140.00i	0.666654i
$144$	0	0
$145$	0	0
$146$	2318.00	1.31397
$147$	0	0
$148$	8.00000i	0.00444322i
$149$	1803.00	0.991326	0.495663	−	0.868515i	$-0.334925\pi$
$149$	1803.00	0.991326	0.495663	+	0.868515i	$0.334925\pi$
$150$	0	0
$151$	2459.00	1.32524	0.662618	−	0.748958i	$-0.269445\pi$
$151$	2459.00	1.32524	0.662618	+	0.748958i	$0.269445\pi$
$152$	− 848.000i	− 0.452512i
$153$	0	0
$154$	3306.00	1.72990
$155$	0	0
$156$	0	0
$157$	196.000i	0.0996338i	0.998758	+	0.0498169i	$0.0158638\pi$
$157$	196.000i	0.0996338i	−0.998758	+	0.0498169i	$0.984136\pi$
$158$	320.000i	0.161126i
$159$	0	0
$160$	0	0
$161$	−5046.00	−2.47007
$162$	0	0
$163$	− 1564.00i	− 0.751546i	−0.926712	−	0.375773i	$-0.877377\pi$
$163$	− 1564.00i	− 0.751546i	0.926712	−	0.375773i	$-0.122623\pi$
$164$	−24.0000	−0.0114273
$165$	0	0
$166$	1470.00	0.687314
$167$	1974.00i	0.914687i	0.889290	+	0.457343i	$0.151199\pi$
$167$	1974.00i	0.914687i	−0.889290	+	0.457343i	$0.848801\pi$
$168$	0	0
$169$	1797.00	0.817934
$170$	0	0
$171$	0	0
$172$	− 872.000i	− 0.386566i
$173$	2217.00i	0.974309i	0.873316	+	0.487154i	$0.161965\pi$
$173$	2217.00i	0.974309i	−0.873316	+	0.487154i	$0.838035\pi$
$174$	0	0
$175$	0	0
$176$	912.000	0.390594
$177$	0	0
$178$	− 1908.00i	− 0.803431i
$179$	−2475.00	−1.03346	−0.516732	−	0.856147i	$-0.672852\pi$
$179$	−2475.00	−1.03346	−0.516732	+	0.856147i	$0.672852\pi$
$180$	0	0
$181$	1568.00	0.643914	0.321957	−	0.946754i	$-0.395659\pi$
$181$	1568.00	0.643914	0.321957	+	0.946754i	$0.395659\pi$
$182$	1160.00i	0.472445i
$183$	0	0
$184$	−1392.00	−0.557715
$185$	0	0
$186$	0	0
$187$	− 4104.00i	− 1.60489i
$188$	− 1896.00i	− 0.735532i
$189$	0	0
$190$	0	0
$191$	−1140.00	−0.431872	−0.215936	−	0.976408i	$-0.569280\pi$
$191$	−1140.00	−0.431872	−0.215936	+	0.976408i	$0.569280\pi$
$192$	0	0
$193$	2045.00i	0.762706i	0.924429	+	0.381353i	$0.124542\pi$
$193$	2045.00i	0.762706i	−0.924429	+	0.381353i	$0.875458\pi$
$194$	382.000	0.141371
$195$	0	0
$196$	1992.00	0.725948
$197$	− 3735.00i	− 1.35080i	−0.737451	−	0.675400i	$-0.763971\pi$
$197$	− 3735.00i	− 1.35080i	0.737451	−	0.675400i	$-0.236029\pi$
$198$	0	0
$199$	−1163.00	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$199$	−1163.00	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$200$	0	0
$201$	0	0
$202$	726.000i	0.252877i
$203$	6090.00i	2.10559i
$204$	0	0
$205$	0	0
$206$	1256.00	0.424804
$207$	0	0
$208$	320.000i	0.106673i
$209$	6042.00	1.99968
$210$	0	0
$211$	2126.00	0.693649	0.346824	−	0.937930i	$-0.387260\pi$
$211$	2126.00	0.693649	0.346824	+	0.937930i	$0.387260\pi$
$212$	324.000i	0.104964i
$213$	0	0
$214$	−1350.00	−0.431234
$215$	0	0
$216$	0	0
$217$	− 1363.00i	− 0.426389i
$218$	− 3460.00i	− 1.07496i
$219$	0	0
$220$	0	0
$221$	1440.00	0.438303
$222$	0	0
$223$	− 2752.00i	− 0.826402i	−0.910640	−	0.413201i	$-0.864411\pi$
$223$	− 2752.00i	− 0.826402i	0.910640	−	0.413201i	$-0.135589\pi$
$224$	928.000	0.276806
$225$	0	0
$226$	3732.00	1.09845
$227$	− 3972.00i	− 1.16137i	−0.814128	−	0.580685i	$-0.802785\pi$
$227$	− 3972.00i	− 1.16137i	0.814128	−	0.580685i	$-0.197215\pi$
$228$	0	0
$229$	−4502.00	−1.29913	−0.649564	−	0.760307i	$-0.725049\pi$
$229$	−4502.00	−1.29913	−0.649564	+	0.760307i	$0.725049\pi$
$230$	0	0
$231$	0	0
$232$	1680.00i	0.475420i
$233$	− 4842.00i	− 1.36142i	−0.732555	−	0.680708i	$-0.761672\pi$
$233$	− 4842.00i	− 1.36142i	0.732555	−	0.680708i	$-0.238328\pi$
$234$	0	0
$235$	0	0
$236$	−336.000	−0.0926769
$237$	0	0
$238$	− 4176.00i	− 1.13735i
$239$	−5334.00	−1.44363	−0.721815	−	0.692086i	$-0.756692\pi$
$239$	−5334.00	−1.44363	−0.721815	+	0.692086i	$0.756692\pi$
$240$	0	0
$241$	−3994.00	−1.06754	−0.533768	−	0.845631i	$-0.679224\pi$
$241$	−3994.00	−1.06754	−0.533768	+	0.845631i	$0.679224\pi$
$242$	3836.00i	1.01896i
$243$	0	0
$244$	−224.000	−0.0587710
$245$	0	0
$246$	0	0
$247$	2120.00i	0.546123i
$248$	− 376.000i	− 0.0962743i
$249$	0	0
$250$	0	0
$251$	1008.00	0.253484	0.126742	−	0.991936i	$-0.459548\pi$
$251$	1008.00	0.253484	0.126742	+	0.991936i	$0.459548\pi$
$252$	0	0
$253$	− 9918.00i	− 2.46458i
$254$	2758.00	0.681308
$255$	0	0
$256$	256.000	0.0625000
$257$	− 924.000i	− 0.224271i	−0.993693	−	0.112135i	$-0.964231\pi$
$257$	− 924.000i	− 0.224271i	0.993693	−	0.112135i	$-0.0357690\pi$
$258$	0	0
$259$	−58.0000	−0.0139148
$260$	0	0
$261$	0	0
$262$	− 1158.00i	− 0.273059i
$263$	− 1014.00i	− 0.237741i	−0.992910	−	0.118871i	$-0.962073\pi$
$263$	− 1014.00i	− 0.237741i	0.992910	−	0.118871i	$-0.0379274\pi$
$264$	0	0
$265$	0	0
$266$	6148.00	1.41714
$267$	0	0
$268$	− 568.000i	− 0.129463i
$269$	−2970.00	−0.673175	−0.336588	−	0.941652i	$-0.609273\pi$
$269$	−2970.00	−0.673175	−0.336588	+	0.941652i	$0.609273\pi$
$270$	0	0
$271$	245.000	0.0549177	0.0274588	−	0.999623i	$-0.491258\pi$
$271$	245.000	0.0549177	0.0274588	+	0.999623i	$0.491258\pi$
$272$	− 1152.00i	− 0.256802i
$273$	0	0
$274$	−1308.00	−0.288391
$275$	0	0
$276$	0	0
$277$	− 4376.00i	− 0.949200i	−0.880202	−	0.474600i	$-0.842593\pi$
$277$	− 4376.00i	− 0.949200i	0.880202	−	0.474600i	$-0.157407\pi$
$278$	6008.00i	1.29617i
$279$	0	0
$280$	0	0
$281$	−240.000	−0.0509509	−0.0254754	−	0.999675i	$-0.508110\pi$
$281$	−240.000	−0.0509509	−0.0254754	+	0.999675i	$0.508110\pi$
$282$	0	0
$283$	− 6838.00i	− 1.43631i	−0.695881	−	0.718157i	$-0.744986\pi$
$283$	− 6838.00i	− 1.43631i	0.695881	−	0.718157i	$-0.255014\pi$
$284$	1440.00	0.300874
$285$	0	0
$286$	−2280.00	−0.471396
$287$	− 174.000i	− 0.0357871i
$288$	0	0
$289$	−271.000	−0.0551598
$290$	0	0
$291$	0	0
$292$	4636.00i	0.929114i
$293$	5118.00i	1.02047i	0.860036	+	0.510233i	$0.170441\pi$
$293$	5118.00i	1.02047i	−0.860036	+	0.510233i	$0.829559\pi$
$294$	0	0
$295$	0	0
$296$	−16.0000	−0.00314183
$297$	0	0
$298$	3606.00i	0.700973i
$299$	3480.00	0.673089
$300$	0	0
$301$	6322.00	1.21061
$302$	4918.00i	0.937083i
$303$	0	0
$304$	1696.00	0.319975
$305$	0	0
$306$	0	0
$307$	5560.00i	1.03364i	0.856096	+	0.516818i	$0.172883\pi$
$307$	5560.00i	1.03364i	−0.856096	+	0.516818i	$0.827117\pi$
$308$	6612.00i	1.22323i
$309$	0	0
$310$	0	0
$311$	−7662.00	−1.39702	−0.698508	−	0.715602i	$-0.746152\pi$
$311$	−7662.00	−1.39702	−0.698508	+	0.715602i	$0.746152\pi$
$312$	0	0
$313$	3485.00i	0.629341i	0.949201	+	0.314671i	$0.101894\pi$
$313$	3485.00i	0.629341i	−0.949201	+	0.314671i	$0.898106\pi$
$314$	−392.000	−0.0704517
$315$	0	0
$316$	−640.000	−0.113933
$317$	− 7059.00i	− 1.25070i	−0.780343	−	0.625352i	$-0.784956\pi$
$317$	− 7059.00i	− 1.25070i	0.780343	−	0.625352i	$-0.215044\pi$
$318$	0	0
$319$	−11970.0	−2.10091
$320$	0	0
$321$	0	0
$322$	− 10092.0i	− 1.74660i
$323$	− 7632.00i	− 1.31472i
$324$	0	0
$325$	0	0
$326$	3128.00	0.531423
$327$	0	0
$328$	− 48.0000i	− 0.00808036i
$329$	13746.0	2.30347
$330$	0	0
$331$	9290.00	1.54267	0.771336	−	0.636428i	$-0.219589\pi$
$331$	9290.00	1.54267	0.771336	+	0.636428i	$0.219589\pi$
$332$	2940.00i	0.486004i
$333$	0	0
$334$	−3948.00	−0.646781
$335$	0	0
$336$	0	0
$337$	3814.00i	0.616504i	0.951305	+	0.308252i	$0.0997440\pi$
$337$	3814.00i	0.616504i	−0.951305	+	0.308252i	$0.900256\pi$
$338$	3594.00i	0.578366i
$339$	0	0
$340$	0	0
$341$	2679.00	0.425443
$342$	0	0
$343$	4495.00i	0.707601i
$344$	1744.00	0.273344
$345$	0	0
$346$	−4434.00	−0.688940
$347$	1929.00i	0.298427i	0.988805	+	0.149213i	$0.0476742\pi$
$347$	1929.00i	0.298427i	−0.988805	+	0.149213i	$0.952326\pi$
$348$	0	0
$349$	6586.00	1.01014	0.505072	−	0.863077i	$-0.331466\pi$
$349$	6586.00	1.01014	0.505072	+	0.863077i	$0.331466\pi$
$350$	0	0
$351$	0	0
$352$	1824.00i	0.276192i
$353$	6042.00i	0.911001i	0.890236	+	0.455500i	$0.150540\pi$
$353$	6042.00i	0.911001i	−0.890236	+	0.455500i	$0.849460\pi$
$354$	0	0
$355$	0	0
$356$	3816.00	0.568111
$357$	0	0
$358$	− 4950.00i	− 0.730770i
$359$	−3762.00	−0.553066	−0.276533	−	0.961004i	$-0.589186\pi$
$359$	−3762.00	−0.553066	−0.276533	+	0.961004i	$0.589186\pi$
$360$	0	0
$361$	4377.00	0.638140
$362$	3136.00i	0.455316i
$363$	0	0
$364$	−2320.00	−0.334069
$365$	0	0
$366$	0	0
$367$	7261.00i	1.03276i	0.856361	+	0.516378i	$0.172720\pi$
$367$	7261.00i	1.03276i	−0.856361	+	0.516378i	$0.827280\pi$
$368$	− 2784.00i	− 0.394364i
$369$	0	0
$370$	0	0
$371$	−2349.00	−0.328717
$372$	0	0
$373$	1640.00i	0.227657i	0.993500	+	0.113828i	$0.0363114\pi$
$373$	1640.00i	0.227657i	−0.993500	+	0.113828i	$0.963689\pi$
$374$	8208.00	1.13483
$375$	0	0
$376$	3792.00	0.520100
$377$	− 4200.00i	− 0.573769i
$378$	0	0
$379$	7396.00	1.00239	0.501197	−	0.865333i	$-0.332893\pi$
$379$	7396.00	1.00239	0.501197	+	0.865333i	$0.332893\pi$
$380$	0	0
$381$	0	0
$382$	− 2280.00i	− 0.305379i
$383$	4992.00i	0.666003i	0.942926	+	0.333002i	$0.108061\pi$
$383$	4992.00i	0.666003i	−0.942926	+	0.333002i	$0.891939\pi$
$384$	0	0
$385$	0	0
$386$	−4090.00	−0.539315
$387$	0	0
$388$	764.000i	0.0999645i
$389$	−9453.00	−1.23210	−0.616049	−	0.787708i	$-0.711268\pi$
$389$	−9453.00	−1.23210	−0.616049	+	0.787708i	$0.711268\pi$
$390$	0	0
$391$	−12528.0	−1.62038
$392$	3984.00i	0.513322i
$393$	0	0
$394$	7470.00	0.955160
$395$	0	0
$396$	0	0
$397$	− 8588.00i	− 1.08569i	−0.839833	−	0.542846i	$-0.817347\pi$
$397$	− 8588.00i	− 1.08569i	0.839833	−	0.542846i	$-0.182653\pi$
$398$	− 2326.00i	− 0.292944i
$399$	0	0
$400$	0	0
$401$	1716.00	0.213698	0.106849	−	0.994275i	$-0.465924\pi$
$401$	1716.00	0.213698	0.106849	+	0.994275i	$0.465924\pi$
$402$	0	0
$403$	940.000i	0.116190i
$404$	−1452.00	−0.178811
$405$	0	0
$406$	−12180.0	−1.48888
$407$	− 114.000i	− 0.0138840i
$408$	0	0
$409$	9889.00	1.19555	0.597775	−	0.801664i	$-0.296052\pi$
$409$	9889.00	1.19555	0.597775	+	0.801664i	$0.296052\pi$
$410$	0	0
$411$	0	0
$412$	2512.00i	0.300382i
$413$	− 2436.00i	− 0.290237i
$414$	0	0
$415$	0	0
$416$	−640.000	−0.0754293
$417$	0	0
$418$	12084.0i	1.41399i
$419$	5556.00	0.647800	0.323900	−	0.946091i	$-0.395006\pi$
$419$	5556.00	0.647800	0.323900	+	0.946091i	$0.395006\pi$
$420$	0	0
$421$	−2104.00	−0.243569	−0.121785	−	0.992557i	$-0.538862\pi$
$421$	−2104.00	−0.243569	−0.121785	+	0.992557i	$0.538862\pi$
$422$	4252.00i	0.490484i
$423$	0	0
$424$	−648.000	−0.0742209
$425$	0	0
$426$	0	0
$427$	− 1624.00i	− 0.184054i
$428$	− 2700.00i	− 0.304929i
$429$	0	0
$430$	0	0
$431$	−7614.00	−0.850936	−0.425468	−	0.904973i	$-0.639891\pi$
$431$	−7614.00	−0.850936	−0.425468	+	0.904973i	$0.639891\pi$
$432$	0	0
$433$	7805.00i	0.866246i	0.901335	+	0.433123i	$0.142588\pi$
$433$	7805.00i	0.866246i	−0.901335	+	0.433123i	$0.857412\pi$
$434$	2726.00	0.301503
$435$	0	0
$436$	6920.00	0.760110
$437$	− 18444.0i	− 2.01898i
$438$	0	0
$439$	5209.00	0.566314	0.283157	−	0.959074i	$-0.408618\pi$
$439$	5209.00	0.566314	0.283157	+	0.959074i	$0.408618\pi$
$440$	0	0
$441$	0	0
$442$	2880.00i	0.309927i
$443$	− 4236.00i	− 0.454308i	−0.973859	−	0.227154i	$-0.927058\pi$
$443$	− 4236.00i	− 0.454308i	0.973859	−	0.227154i	$-0.0729421\pi$
$444$	0	0
$445$	0	0
$446$	5504.00	0.584354
$447$	0	0
$448$	1856.00i	0.195732i
$449$	−16002.0	−1.68192	−0.840959	−	0.541099i	$-0.818008\pi$
$449$	−16002.0	−1.68192	−0.840959	+	0.541099i	$0.818008\pi$
$450$	0	0
$451$	342.000	0.0357077
$452$	7464.00i	0.776719i
$453$	0	0
$454$	7944.00	0.821212
$455$	0	0
$456$	0	0
$457$	− 7319.00i	− 0.749165i	−0.927194	−	0.374582i	$-0.877786\pi$
$457$	− 7319.00i	− 0.749165i	0.927194	−	0.374582i	$-0.122214\pi$
$458$	− 9004.00i	− 0.918623i
$459$	0	0
$460$	0	0
$461$	−9483.00	−0.958064	−0.479032	−	0.877798i	$-0.659012\pi$
$461$	−9483.00	−0.958064	−0.479032	+	0.877798i	$0.659012\pi$
$462$	0	0
$463$	10793.0i	1.08335i	0.840586	+	0.541677i	$0.182211\pi$
$463$	10793.0i	1.08335i	−0.840586	+	0.541677i	$0.817789\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	9684.00	0.962667
$467$	2583.00i	0.255946i	0.991778	+	0.127973i	$0.0408472\pi$
$467$	2583.00i	0.255946i	−0.991778	+	0.127973i	$0.959153\pi$
$468$	0	0
$469$	4118.00	0.405440
$470$	0	0
$471$	0	0
$472$	− 672.000i	− 0.0655324i
$473$	12426.0i	1.20792i
$474$	0	0
$475$	0	0
$476$	8352.00	0.804230
$477$	0	0
$478$	− 10668.0i	− 1.02080i
$479$	−1254.00	−0.119617	−0.0598087	−	0.998210i	$-0.519049\pi$
$479$	−1254.00	−0.119617	−0.0598087	+	0.998210i	$0.519049\pi$
$480$	0	0
$481$	40.0000	0.00379177
$482$	− 7988.00i	− 0.754862i
$483$	0	0
$484$	−7672.00	−0.720511
$485$	0	0
$486$	0	0
$487$	− 17336.0i	− 1.61308i	−0.591181	−	0.806539i	$-0.701338\pi$
$487$	− 17336.0i	− 1.61308i	0.591181	−	0.806539i	$-0.298662\pi$
$488$	− 448.000i	− 0.0415574i
$489$	0	0
$490$	0	0
$491$	15171.0	1.39441	0.697207	−	0.716869i	$-0.254426\pi$
$491$	15171.0	1.39441	0.697207	+	0.716869i	$0.254426\pi$
$492$	0	0
$493$	15120.0i	1.38128i
$494$	−4240.00	−0.386167
$495$	0	0
$496$	752.000	0.0680762
$497$	10440.0i	0.942249i
$498$	0	0
$499$	−8930.00	−0.801126	−0.400563	−	0.916269i	$-0.631185\pi$
$499$	−8930.00	−0.801126	−0.400563	+	0.916269i	$0.631185\pi$
$500$	0	0
$501$	0	0
$502$	2016.00i	0.179240i
$503$	− 15210.0i	− 1.34827i	−0.738608	−	0.674136i	$-0.764516\pi$
$503$	− 15210.0i	− 1.34827i	0.738608	−	0.674136i	$-0.235484\pi$
$504$	0	0
$505$	0	0
$506$	19836.0	1.74272
$507$	0	0
$508$	5516.00i	0.481758i
$509$	19641.0	1.71036	0.855179	−	0.518333i	$-0.173447\pi$
$509$	19641.0	1.71036	0.855179	+	0.518333i	$0.173447\pi$
$510$	0	0
$511$	−33611.0	−2.90971
$512$	512.000i	0.0441942i
$513$	0	0
$514$	1848.00	0.158583
$515$	0	0
$516$	0	0
$517$	27018.0i	2.29836i
$518$	− 116.000i	− 0.00983928i
$519$	0	0
$520$	0	0
$521$	−22428.0	−1.88597	−0.942983	−	0.332840i	$-0.891993\pi$
$521$	−22428.0	−1.88597	−0.942983	+	0.332840i	$0.891993\pi$
$522$	0	0
$523$	− 8152.00i	− 0.681572i	−0.940141	−	0.340786i	$-0.889307\pi$
$523$	− 8152.00i	− 0.681572i	0.940141	−	0.340786i	$-0.110693\pi$
$524$	2316.00	0.193082
$525$	0	0
$526$	2028.00	0.168108
$527$	− 3384.00i	− 0.279714i
$528$	0	0
$529$	−18109.0	−1.48837
$530$	0	0
$531$	0	0
$532$	12296.0i	1.00207i
$533$	120.000i	0.00975193i
$534$	0	0
$535$	0	0
$536$	1136.00	0.0915443
$537$	0	0
$538$	− 5940.00i	− 0.476007i
$539$	−28386.0	−2.26841
$540$	0	0
$541$	−2860.00	−0.227285	−0.113642	−	0.993522i	$-0.536252\pi$
$541$	−2860.00	−0.227285	−0.113642	+	0.993522i	$0.536252\pi$
$542$	490.000i	0.0388327i
$543$	0	0
$544$	2304.00	0.181587
$545$	0	0
$546$	0	0
$547$	9664.00i	0.755398i	0.925928	+	0.377699i	$0.123285\pi$
$547$	9664.00i	0.755398i	−0.925928	+	0.377699i	$0.876715\pi$
$548$	− 2616.00i	− 0.203923i
$549$	0	0
$550$	0	0
$551$	−22260.0	−1.72107
$552$	0	0
$553$	− 4640.00i	− 0.356804i
$554$	8752.00	0.671186
$555$	0	0
$556$	−12016.0	−0.916532
$557$	14859.0i	1.13033i	0.824977	+	0.565167i	$0.191188\pi$
$557$	14859.0i	1.13033i	−0.824977	+	0.565167i	$0.808812\pi$
$558$	0	0
$559$	−4360.00	−0.329890
$560$	0	0
$561$	0	0
$562$	− 480.000i	− 0.0360277i
$563$	− 8193.00i	− 0.613310i	−0.951821	−	0.306655i	$-0.900790\pi$
$563$	− 8193.00i	− 0.613310i	0.951821	−	0.306655i	$-0.0992099\pi$
$564$	0	0
$565$	0	0
$566$	13676.0	1.01563
$567$	0	0
$568$	2880.00i	0.212750i
$569$	−16572.0	−1.22097	−0.610487	−	0.792026i	$-0.709026\pi$
$569$	−16572.0	−1.22097	−0.610487	+	0.792026i	$0.709026\pi$
$570$	0	0
$571$	−6244.00	−0.457624	−0.228812	−	0.973471i	$-0.573484\pi$
$571$	−6244.00	−0.457624	−0.228812	+	0.973471i	$0.573484\pi$
$572$	− 4560.00i	− 0.333327i
$573$	0	0
$574$	348.000	0.0253053
$575$	0	0
$576$	0	0
$577$	14794.0i	1.06739i	0.845678	+	0.533693i	$0.179196\pi$
$577$	14794.0i	1.06739i	−0.845678	+	0.533693i	$0.820804\pi$
$578$	− 542.000i	− 0.0390039i
$579$	0	0
$580$	0	0
$581$	−21315.0	−1.52202
$582$	0	0
$583$	− 4617.00i	− 0.327987i
$584$	−9272.00	−0.656983
$585$	0	0
$586$	−10236.0	−0.721579
$587$	− 26769.0i	− 1.88224i	−0.338073	−	0.941120i	$-0.609775\pi$
$587$	− 26769.0i	− 1.88224i	0.338073	−	0.941120i	$-0.390225\pi$
$588$	0	0
$589$	4982.00	0.348522
$590$	0	0
$591$	0	0
$592$	− 32.0000i	− 0.00222161i
$593$	3078.00i	0.213151i	0.994305	+	0.106575i	$0.0339885\pi$
$593$	3078.00i	0.213151i	−0.994305	+	0.106575i	$0.966011\pi$
$594$	0	0
$595$	0	0
$596$	−7212.00	−0.495663
$597$	0	0
$598$	6960.00i	0.475946i
$599$	1002.00	0.0683483	0.0341741	−	0.999416i	$-0.489120\pi$
$599$	1002.00	0.0683483	0.0341741	+	0.999416i	$0.489120\pi$
$600$	0	0
$601$	−20653.0	−1.40175	−0.700876	−	0.713283i	$-0.747208\pi$
$601$	−20653.0	−1.40175	−0.700876	+	0.713283i	$0.747208\pi$
$602$	12644.0i	0.856032i
$603$	0	0
$604$	−9836.00	−0.662618
$605$	0	0
$606$	0	0
$607$	− 27128.0i	− 1.81399i	−0.421142	−	0.906995i	$-0.638371\pi$
$607$	− 27128.0i	− 1.81399i	0.421142	−	0.906995i	$-0.361629\pi$
$608$	3392.00i	0.226256i
$609$	0	0
$610$	0	0
$611$	−9480.00	−0.627692
$612$	0	0
$613$	24518.0i	1.61545i	0.589557	+	0.807727i	$0.299302\pi$
$613$	24518.0i	1.61545i	−0.589557	+	0.807727i	$0.700698\pi$
$614$	−11120.0	−0.730890
$615$	0	0
$616$	−13224.0	−0.864952
$617$	− 474.000i	− 0.0309279i	−0.999880	−	0.0154640i	$-0.995077\pi$
$617$	− 474.000i	− 0.0309279i	0.999880	−	0.0154640i	$-0.00492253\pi$
$618$	0	0
$619$	1132.00	0.0735039	0.0367520	−	0.999324i	$-0.488299\pi$
$619$	1132.00	0.0735039	0.0367520	+	0.999324i	$0.488299\pi$
$620$	0	0
$621$	0	0
$622$	− 15324.0i	− 0.987840i
$623$	27666.0i	1.77916i
$624$	0	0
$625$	0	0
$626$	−6970.00	−0.445012
$627$	0	0
$628$	− 784.000i	− 0.0498169i
$629$	−144.000	−0.00912823
$630$	0	0
$631$	6725.00	0.424276	0.212138	−	0.977240i	$-0.431957\pi$
$631$	6725.00	0.424276	0.212138	+	0.977240i	$0.431957\pi$
$632$	− 1280.00i	− 0.0805628i
$633$	0	0
$634$	14118.0	0.884381
$635$	0	0
$636$	0	0
$637$	− 9960.00i	− 0.619513i
$638$	− 23940.0i	− 1.48557i
$639$	0	0
$640$	0	0
$641$	21126.0	1.30176	0.650879	−	0.759182i	$-0.274401\pi$
$641$	21126.0	1.30176	0.650879	+	0.759182i	$0.274401\pi$
$642$	0	0
$643$	19460.0i	1.19351i	0.802423	+	0.596755i	$0.203544\pi$
$643$	19460.0i	1.19351i	−0.802423	+	0.596755i	$0.796456\pi$
$644$	20184.0	1.23503
$645$	0	0
$646$	15264.0	0.929650
$647$	− 11664.0i	− 0.708747i	−0.935104	−	0.354373i	$-0.884694\pi$
$647$	− 11664.0i	− 0.708747i	0.935104	−	0.354373i	$-0.115306\pi$
$648$	0	0
$649$	4788.00	0.289592
$650$	0	0
$651$	0	0
$652$	6256.00i	0.375773i
$653$	3345.00i	0.200459i	0.994964	+	0.100230i	$0.0319578\pi$
$653$	3345.00i	0.200459i	−0.994964	+	0.100230i	$0.968042\pi$
$654$	0	0
$655$	0	0
$656$	96.0000	0.00571367
$657$	0	0
$658$	27492.0i	1.62880i
$659$	9393.00	0.555234	0.277617	−	0.960692i	$-0.410455\pi$
$659$	9393.00	0.555234	0.277617	+	0.960692i	$0.410455\pi$
$660$	0	0
$661$	−1762.00	−0.103682	−0.0518410	−	0.998655i	$-0.516509\pi$
$661$	−1762.00	−0.103682	−0.0518410	+	0.998655i	$0.516509\pi$
$662$	18580.0i	1.09083i
$663$	0	0
$664$	−5880.00	−0.343657
$665$	0	0
$666$	0	0
$667$	36540.0i	2.12119i
$668$	− 7896.00i	− 0.457343i
$669$	0	0
$670$	0	0
$671$	3192.00	0.183645
$672$	0	0
$673$	25517.0i	1.46153i	0.682630	+	0.730764i	$0.260836\pi$
$673$	25517.0i	1.46153i	−0.682630	+	0.730764i	$0.739164\pi$
$674$	−7628.00	−0.435934
$675$	0	0
$676$	−7188.00	−0.408967
$677$	26898.0i	1.52699i	0.645812	+	0.763496i	$0.276519\pi$
$677$	26898.0i	1.52699i	−0.645812	+	0.763496i	$0.723481\pi$
$678$	0	0
$679$	−5539.00	−0.313059
$680$	0	0
$681$	0	0
$682$	5358.00i	0.300833i
$683$	− 23940.0i	− 1.34120i	−0.741820	−	0.670599i	$-0.766037\pi$
$683$	− 23940.0i	− 1.34120i	0.741820	−	0.670599i	$-0.233963\pi$
$684$	0	0
$685$	0	0
$686$	−8990.00	−0.500350
$687$	0	0
$688$	3488.00i	0.193283i
$689$	1620.00	0.0895749
$690$	0	0
$691$	23060.0	1.26953	0.634764	−	0.772706i	$-0.281097\pi$
$691$	23060.0	1.26953	0.634764	+	0.772706i	$0.281097\pi$
$692$	− 8868.00i	− 0.487154i
$693$	0	0
$694$	−3858.00	−0.211020
$695$	0	0
$696$	0	0
$697$	− 432.000i	− 0.0234766i
$698$	13172.0i	0.714280i
$699$	0	0
$700$	0	0
$701$	14175.0	0.763741	0.381870	−	0.924216i	$-0.375280\pi$
$701$	14175.0	0.763741	0.381870	+	0.924216i	$0.375280\pi$
$702$	0	0
$703$	− 212.000i	− 0.0113737i
$704$	−3648.00	−0.195297
$705$	0	0
$706$	−12084.0	−0.644175
$707$	− 10527.0i	− 0.559984i
$708$	0	0
$709$	8692.00	0.460416	0.230208	−	0.973141i	$-0.426059\pi$
$709$	8692.00	0.460416	0.230208	+	0.973141i	$0.426059\pi$
$710$	0	0
$711$	0	0
$712$	7632.00i	0.401715i
$713$	− 8178.00i	− 0.429549i
$714$	0	0
$715$	0	0
$716$	9900.00	0.516732
$717$	0	0
$718$	− 7524.00i	− 0.391077i
$719$	−29556.0	−1.53304	−0.766518	−	0.642223i	$-0.778012\pi$
$719$	−29556.0	−1.53304	−0.766518	+	0.642223i	$0.778012\pi$
$720$	0	0
$721$	−18212.0	−0.940708
$722$	8754.00i	0.451233i
$723$	0	0
$724$	−6272.00	−0.321957
$725$	0	0
$726$	0	0
$727$	36691.0i	1.87179i	0.352274	+	0.935897i	$0.385408\pi$
$727$	36691.0i	1.87179i	−0.352274	+	0.935897i	$0.614592\pi$
$728$	− 4640.00i	− 0.236222i
$729$	0	0
$730$	0	0
$731$	15696.0	0.794169
$732$	0	0
$733$	− 19798.0i	− 0.997620i	−0.866711	−	0.498810i	$-0.833770\pi$
$733$	− 19798.0i	− 0.997620i	0.866711	−	0.498810i	$-0.166230\pi$
$734$	−14522.0	−0.730268
$735$	0	0
$736$	5568.00	0.278858
$737$	8094.00i	0.404540i
$738$	0	0
$739$	21976.0	1.09391	0.546955	−	0.837162i	$-0.315787\pi$
$739$	21976.0	1.09391	0.546955	+	0.837162i	$0.315787\pi$
$740$	0	0
$741$	0	0
$742$	− 4698.00i	− 0.232438i
$743$	13236.0i	0.653542i	0.945104	+	0.326771i	$0.105961\pi$
$743$	13236.0i	0.653542i	−0.945104	+	0.326771i	$0.894039\pi$
$744$	0	0
$745$	0	0
$746$	−3280.00	−0.160978
$747$	0	0
$748$	16416.0i	0.802444i
$749$	19575.0	0.954947
$750$	0	0
$751$	−6325.00	−0.307327	−0.153663	−	0.988123i	$-0.549107\pi$
$751$	−6325.00	−0.307327	−0.153663	+	0.988123i	$0.549107\pi$
$752$	7584.00i	0.367766i
$753$	0	0
$754$	8400.00	0.405716
$755$	0	0
$756$	0	0
$757$	3238.00i	0.155465i	0.996974	+	0.0777326i	$0.0247680\pi$
$757$	3238.00i	0.155465i	−0.996974	+	0.0777326i	$0.975232\pi$
$758$	14792.0i	0.708799i
$759$	0	0
$760$	0	0
$761$	40416.0	1.92520	0.962601	−	0.270923i	$-0.0873288\pi$
$761$	40416.0	1.92520	0.962601	+	0.270923i	$0.0873288\pi$
$762$	0	0
$763$	50170.0i	2.38044i
$764$	4560.00	0.215936
$765$	0	0
$766$	−9984.00	−0.470935
$767$	1680.00i	0.0790890i
$768$	0	0
$769$	4759.00	0.223165	0.111583	−	0.993755i	$-0.464408\pi$
$769$	4759.00	0.223165	0.111583	+	0.993755i	$0.464408\pi$
$770$	0	0
$771$	0	0
$772$	− 8180.00i	− 0.381353i
$773$	− 27414.0i	− 1.27557i	−0.770216	−	0.637783i	$-0.779852\pi$
$773$	− 27414.0i	− 1.27557i	0.770216	−	0.637783i	$-0.220148\pi$
$774$	0	0
$775$	0	0
$776$	−1528.00	−0.0706856
$777$	0	0
$778$	− 18906.0i	− 0.871225i
$779$	636.000	0.0292517
$780$	0	0
$781$	−20520.0	−0.940158
$782$	− 25056.0i	− 1.14578i
$783$	0	0
$784$	−7968.00	−0.362974
$785$	0	0
$786$	0	0
$787$	− 6176.00i	− 0.279734i	−0.990170	−	0.139867i	$-0.955333\pi$
$787$	− 6176.00i	− 0.279734i	0.990170	−	0.139867i	$-0.0446675\pi$
$788$	14940.0i	0.675400i
$789$	0	0
$790$	0	0
$791$	−54114.0	−2.43246
$792$	0	0
$793$	1120.00i	0.0501543i
$794$	17176.0	0.767700
$795$	0	0
$796$	4652.00	0.207143
$797$	6879.00i	0.305730i	0.988247	+	0.152865i	$0.0488499\pi$
$797$	6879.00i	0.305730i	−0.988247	+	0.152865i	$0.951150\pi$
$798$	0	0
$799$	34128.0	1.51109
$800$	0	0
$801$	0	0
$802$	3432.00i	0.151107i
$803$	− 66063.0i	− 2.90325i
$804$	0	0
$805$	0	0
$806$	−1880.00	−0.0821590
$807$	0	0
$808$	− 2904.00i	− 0.126439i
$809$	−16902.0	−0.734540	−0.367270	−	0.930114i	$-0.619707\pi$
$809$	−16902.0	−0.734540	−0.367270	+	0.930114i	$0.619707\pi$
$810$	0	0
$811$	24086.0	1.04288	0.521439	−	0.853289i	$-0.325395\pi$
$811$	24086.0	1.04288	0.521439	+	0.853289i	$0.325395\pi$
$812$	− 24360.0i	− 1.05279i
$813$	0	0
$814$	228.000	0.00981744
$815$	0	0
$816$	0	0
$817$	23108.0i	0.989531i
$818$	19778.0i	0.845381i
$819$	0	0
$820$	0	0
$821$	−7854.00	−0.333869	−0.166935	−	0.985968i	$-0.553387\pi$
$821$	−7854.00	−0.333869	−0.166935	+	0.985968i	$0.553387\pi$
$822$	0	0
$823$	5771.00i	0.244428i	0.992504	+	0.122214i	$0.0389995\pi$
$823$	5771.00i	0.244428i	−0.992504	+	0.122214i	$0.961001\pi$
$824$	−5024.00	−0.212402
$825$	0	0
$826$	4872.00	0.205228
$827$	− 17568.0i	− 0.738693i	−0.929292	−	0.369347i	$-0.879582\pi$
$827$	− 17568.0i	− 0.738693i	0.929292	−	0.369347i	$-0.120418\pi$
$828$	0	0
$829$	−31322.0	−1.31225	−0.656127	−	0.754651i	$-0.727806\pi$
$829$	−31322.0	−1.31225	−0.656127	+	0.754651i	$0.727806\pi$
$830$	0	0
$831$	0	0
$832$	− 1280.00i	− 0.0533366i
$833$	35856.0i	1.49140i
$834$	0	0
$835$	0	0
$836$	−24168.0	−0.999842
$837$	0	0
$838$	11112.0i	0.458064i
$839$	41856.0	1.72232	0.861162	−	0.508331i	$-0.169737\pi$
$839$	41856.0	1.72232	0.861162	+	0.508331i	$0.169737\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	− 4208.00i	− 0.172230i
$843$	0	0
$844$	−8504.00	−0.346824
$845$	0	0
$846$	0	0
$847$	− 55622.0i	− 2.25643i
$848$	− 1296.00i	− 0.0524821i
$849$	0	0
$850$	0	0
$851$	−348.000	−0.0140180
$852$	0	0
$853$	15662.0i	0.628671i	0.949312	+	0.314336i	$0.101782\pi$
$853$	15662.0i	0.628671i	−0.949312	+	0.314336i	$0.898218\pi$
$854$	3248.00	0.130146
$855$	0	0
$856$	5400.00	0.215617
$857$	− 39864.0i	− 1.58895i	−0.607298	−	0.794474i	$-0.707747\pi$
$857$	− 39864.0i	− 1.58895i	0.607298	−	0.794474i	$-0.292253\pi$
$858$	0	0
$859$	9160.00	0.363836	0.181918	−	0.983314i	$-0.441769\pi$
$859$	9160.00	0.363836	0.181918	+	0.983314i	$0.441769\pi$
$860$	0	0
$861$	0	0
$862$	− 15228.0i	− 0.601703i
$863$	− 5076.00i	− 0.200219i	−0.994976	−	0.100110i	$-0.968081\pi$
$863$	− 5076.00i	− 0.200219i	0.994976	−	0.100110i	$-0.0319193\pi$
$864$	0	0
$865$	0	0
$866$	−15610.0	−0.612528
$867$	0	0
$868$	5452.00i	0.213195i
$869$	9120.00	0.356012
$870$	0	0
$871$	−2840.00	−0.110482
$872$	13840.0i	0.537479i
$873$	0	0
$874$	36888.0	1.42764
$875$	0	0
$876$	0	0
$877$	− 14978.0i	− 0.576706i	−0.957524	−	0.288353i	$-0.906892\pi$
$877$	− 14978.0i	− 0.576706i	0.957524	−	0.288353i	$-0.0931076\pi$
$878$	10418.0i	0.400445i
$879$	0	0
$880$	0	0
$881$	22860.0	0.874203	0.437102	−	0.899412i	$-0.356005\pi$
$881$	22860.0	0.874203	0.437102	+	0.899412i	$0.356005\pi$
$882$	0	0
$883$	− 32506.0i	− 1.23886i	−0.785052	−	0.619430i	$-0.787364\pi$
$883$	− 32506.0i	− 1.23886i	0.785052	−	0.619430i	$-0.212636\pi$
$884$	−5760.00	−0.219151
$885$	0	0
$886$	8472.00	0.321244
$887$	35868.0i	1.35776i	0.734251	+	0.678878i	$0.237533\pi$
$887$	35868.0i	1.35776i	−0.734251	+	0.678878i	$0.762467\pi$
$888$	0	0
$889$	−39991.0	−1.50872
$890$	0	0
$891$	0	0
$892$	11008.0i	0.413201i
$893$	50244.0i	1.88281i
$894$	0	0
$895$	0	0
$896$	−3712.00	−0.138403
$897$	0	0
$898$	− 32004.0i	− 1.18930i
$899$	−9870.00	−0.366166
$900$	0	0
$901$	−5832.00	−0.215640
$902$	684.000i	0.0252491i
$903$	0	0
$904$	−14928.0	−0.549223
$905$	0	0
$906$	0	0
$907$	33586.0i	1.22955i	0.788701	+	0.614777i	$0.210754\pi$
$907$	33586.0i	1.22955i	−0.788701	+	0.614777i	$0.789246\pi$
$908$	15888.0i	0.580685i
$909$	0	0
$910$	0	0
$911$	28902.0	1.05112	0.525558	−	0.850758i	$-0.323857\pi$
$911$	28902.0	1.05112	0.525558	+	0.850758i	$0.323857\pi$
$912$	0	0
$913$	− 41895.0i	− 1.51864i
$914$	14638.0	0.529740
$915$	0	0
$916$	18008.0	0.649564
$917$	16791.0i	0.604676i
$918$	0	0
$919$	−28271.0	−1.01477	−0.507385	−	0.861719i	$-0.669388\pi$
$919$	−28271.0	−1.01477	−0.507385	+	0.861719i	$0.669388\pi$
$920$	0	0
$921$	0	0
$922$	− 18966.0i	− 0.677453i
$923$	− 7200.00i	− 0.256762i
$924$	0	0
$925$	0	0
$926$	−21586.0	−0.766047
$927$	0	0
$928$	− 6720.00i	− 0.237710i
$929$	19140.0	0.675956	0.337978	−	0.941154i	$-0.390257\pi$
$929$	19140.0	0.675956	0.337978	+	0.941154i	$0.390257\pi$
$930$	0	0
$931$	−52788.0	−1.85828
$932$	19368.0i	0.680708i
$933$	0	0
$934$	−5166.00	−0.180981
$935$	0	0
$936$	0	0
$937$	− 31619.0i	− 1.10240i	−0.834374	−	0.551199i	$-0.814170\pi$
$937$	− 31619.0i	− 1.10240i	0.834374	−	0.551199i	$-0.185830\pi$
$938$	8236.00i	0.286690i
$939$	0	0
$940$	0	0
$941$	20913.0	0.724489	0.362245	−	0.932083i	$-0.382011\pi$
$941$	20913.0	0.724489	0.362245	+	0.932083i	$0.382011\pi$
$942$	0	0
$943$	− 1044.00i	− 0.0360523i
$944$	1344.00	0.0463384
$945$	0	0
$946$	−24852.0	−0.854131
$947$	17529.0i	0.601495i	0.953704	+	0.300748i	$0.0972362\pi$
$947$	17529.0i	0.601495i	−0.953704	+	0.300748i	$0.902764\pi$
$948$	0	0
$949$	23180.0	0.792892
$950$	0	0
$951$	0	0
$952$	16704.0i	0.568676i
$953$	53604.0i	1.82204i	0.412362	+	0.911020i	$0.364704\pi$
$953$	53604.0i	1.82204i	−0.412362	+	0.911020i	$0.635296\pi$
$954$	0	0
$955$	0	0
$956$	21336.0	0.721815
$957$	0	0
$958$	− 2508.00i	− 0.0845823i
$959$	18966.0	0.638628
$960$	0	0
$961$	−27582.0	−0.925850
$962$	80.0000i	0.00268119i
$963$	0	0
$964$	15976.0	0.533768
$965$	0	0
$966$	0	0
$967$	− 11117.0i	− 0.369699i	−0.982767	−	0.184849i	$-0.940820\pi$
$967$	− 11117.0i	− 0.369699i	0.982767	−	0.184849i	$-0.0591797\pi$
$968$	− 15344.0i	− 0.509478i
$969$	0	0
$970$	0	0
$971$	27297.0	0.902165	0.451083	−	0.892482i	$-0.351038\pi$
$971$	27297.0	0.902165	0.451083	+	0.892482i	$0.351038\pi$
$972$	0	0
$973$	− 87116.0i	− 2.87031i
$974$	34672.0	1.14062
$975$	0	0
$976$	896.000	0.0293855
$977$	25086.0i	0.821466i	0.911756	+	0.410733i	$0.134727\pi$
$977$	25086.0i	0.821466i	−0.911756	+	0.410733i	$0.865273\pi$
$978$	0	0
$979$	−54378.0	−1.77521
$980$	0	0
$981$	0	0
$982$	30342.0i	0.986000i
$983$	20982.0i	0.680795i	0.940282	+	0.340398i	$0.110562\pi$
$983$	20982.0i	0.680795i	−0.940282	+	0.340398i	$0.889438\pi$
$984$	0	0
$985$	0	0
$986$	−30240.0	−0.976712
$987$	0	0
$988$	− 8480.00i	− 0.273061i
$989$	37932.0	1.21958
$990$	0	0
$991$	11477.0	0.367890	0.183945	−	0.982937i	$-0.441113\pi$
$991$	11477.0	0.367890	0.183945	+	0.982937i	$0.441113\pi$
$992$	1504.00i	0.0481371i
$993$	0	0
$994$	−20880.0	−0.666271
$995$	0	0
$996$	0	0
$997$	− 8588.00i	− 0.272803i	−0.990654	−	0.136402i	$-0.956446\pi$
$997$	− 8588.00i	− 0.272803i	0.990654	−	0.136402i	$-0.0435537\pi$
$998$	− 17860.0i	− 0.566481i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.s.649.2		2
3.2	odd	2		1350.4.c.b.649.1		2
5.2	odd	4		54.4.a.b.1.1	✓	1
5.3	odd	4		1350.4.a.o.1.1		1
5.4	even	2	inner	1350.4.c.s.649.1		2
15.2	even	4		54.4.a.c.1.1	yes	1
15.8	even	4		1350.4.a.a.1.1		1
15.14	odd	2		1350.4.c.b.649.2		2
20.7	even	4		432.4.a.e.1.1		1
40.27	even	4		1728.4.a.u.1.1		1
40.37	odd	4		1728.4.a.v.1.1		1
45.2	even	12		162.4.c.b.109.1		2
45.7	odd	12		162.4.c.g.109.1		2
45.22	odd	12		162.4.c.g.55.1		2
45.32	even	12		162.4.c.b.55.1		2
60.47	odd	4		432.4.a.j.1.1		1
120.77	even	4		1728.4.a.l.1.1		1
120.107	odd	4		1728.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.a.b.1.1	✓	1	5.2	odd	4
54.4.a.c.1.1	yes	1	15.2	even	4
162.4.c.b.55.1		2	45.32	even	12
162.4.c.b.109.1		2	45.2	even	12
162.4.c.g.55.1		2	45.22	odd	12
162.4.c.g.109.1		2	45.7	odd	12
432.4.a.e.1.1		1	20.7	even	4
432.4.a.j.1.1		1	60.47	odd	4
1350.4.a.a.1.1		1	15.8	even	4
1350.4.a.o.1.1		1	5.3	odd	4
1350.4.c.b.649.1		2	3.2	odd	2
1350.4.c.b.649.2		2	15.14	odd	2
1350.4.c.s.649.1		2	5.4	even	2	inner
1350.4.c.s.649.2		2	1.1	even	1	trivial
1728.4.a.k.1.1		1	120.107	odd	4
1728.4.a.l.1.1		1	120.77	even	4
1728.4.a.u.1.1		1	40.27	even	4
1728.4.a.v.1.1		1	40.37	odd	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} -29.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} -29.0000i q^{7} -8.00000i q^{8} +57.0000 q^{11} +20.0000i q^{13} +58.0000 q^{14} +16.0000 q^{16} -72.0000i q^{17} +106.000 q^{19} +114.000i q^{22} -174.000i q^{23} -40.0000 q^{26} +116.000i q^{28} -210.000 q^{29} +47.0000 q^{31} +32.0000i q^{32} +144.000 q^{34} -2.00000i q^{37} +212.000i q^{38} +6.00000 q^{41} +218.000i q^{43} -228.000 q^{44} +348.000 q^{46} +474.000i q^{47} -498.000 q^{49} -80.0000i q^{52} -81.0000i q^{53} -232.000 q^{56} -420.000i q^{58} +84.0000 q^{59} +56.0000 q^{61} +94.0000i q^{62} -64.0000 q^{64} +142.000i q^{67} +288.000i q^{68} -360.000 q^{71} -1159.00i q^{73} +4.00000 q^{74} -424.000 q^{76} -1653.00i q^{77} +160.000 q^{79} +12.0000i q^{82} -735.000i q^{83} -436.000 q^{86} -456.000i q^{88} -954.000 q^{89} +580.000 q^{91} +696.000i q^{92} -948.000 q^{94} -191.000i q^{97} -996.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} + 114 q^{11} + 116 q^{14} + 32 q^{16} + 212 q^{19} - 80 q^{26} - 420 q^{29} + 94 q^{31} + 288 q^{34} + 12 q^{41} - 456 q^{44} + 696 q^{46} - 996 q^{49} - 464 q^{56} + 168 q^{59} + 112 q^{61} - 128 q^{64} - 720 q^{71} + 8 q^{74} - 848 q^{76} + 320 q^{79} - 872 q^{86} - 1908 q^{89} + 1160 q^{91} - 1896 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 + 114 * q^11 + 116 * q^14 + 32 * q^16 + 212 * q^19 - 80 * q^26 - 420 * q^29 + 94 * q^31 + 288 * q^34 + 12 * q^41 - 456 * q^44 + 696 * q^46 - 996 * q^49 - 464 * q^56 + 168 * q^59 + 112 * q^61 - 128 * q^64 - 720 * q^71 + 8 * q^74 - 848 * q^76 + 320 * q^79 - 872 * q^86 - 1908 * q^89 + 1160 * q^91 - 1896 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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