LMFDB - Embedded newform 1350.5.d.b.701.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,5,Mod(701,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1350.701"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1350.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-32,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$139.549450163$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.3
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.701
Dual form			1350.5.d.b.701.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} -20.0000 q^{7} -22.6274i q^{8} -109.909i q^{11} +137.868 q^{13} -56.5685i q^{14} +64.0000 q^{16} -310.354i q^{17} +245.737 q^{19} +310.868 q^{22} -301.069i q^{23} +389.951i q^{26} +160.000 q^{28} +1350.84i q^{29} +1509.08 q^{31} +181.019i q^{32} +877.815 q^{34} -1947.37 q^{37} +695.048i q^{38} -813.787i q^{41} -2183.58 q^{43} +879.268i q^{44} +851.552 q^{46} +766.476i q^{47} -2001.00 q^{49} -1102.95 q^{52} -2556.32i q^{53} +452.548i q^{56} -3820.76 q^{58} +278.731i q^{59} +3023.52 q^{61} +4268.32i q^{62} -512.000 q^{64} -302.000 q^{67} +2482.84i q^{68} +2629.17i q^{71} -2993.74 q^{73} -5507.98i q^{74} -1965.89 q^{76} +2198.17i q^{77} +6697.07 q^{79} +2301.74 q^{82} +367.449i q^{83} -6176.09i q^{86} -2486.95 q^{88} +6448.51i q^{89} -2757.37 q^{91} +2408.55i q^{92} -2167.92 q^{94} -9962.26 q^{97} -5659.68i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} - 80 q^{7} + 172 q^{13} + 256 q^{16} + 224 q^{19} + 864 q^{22} + 640 q^{28} + 3380 q^{31} + 96 q^{34} - 200 q^{37} + 1132 q^{43} - 768 q^{46} - 8004 q^{49} - 1376 q^{52} - 8832 q^{58} - 808 q^{61}+ \cdots - 11768 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 - 80 * q^7 + 172 * q^13 + 256 * q^16 + 224 * q^19 + 864 * q^22 + 640 * q^28 + 3380 * q^31 + 96 * q^34 - 200 * q^37 + 1132 * q^43 - 768 * q^46 - 8004 * q^49 - 1376 * q^52 - 8832 * q^58 - 808 * q^61 - 2048 * q^64 - 1208 * q^67 - 11216 * q^73 - 1792 * q^76 - 4708 * q^79 + 8448 * q^82 - 6912 * q^88 - 3440 * q^91 - 11328 * q^94 - 11768 * q^97

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 $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−20.0000	−0.408163	−0.204082	−	0.978954i	$-0.565421\pi$
$7$	−20.0000	−0.408163	−0.204082	+	0.978954i	$0.565421\pi$
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	− 109.909i	− 0.908335i	−0.890916	−	0.454168i	$-0.849937\pi$
$11$	− 109.909i	− 0.908335i	0.890916	−	0.454168i	$-0.150063\pi$
$12$	0	0
$13$	137.868	0.815789	0.407894	−	0.913029i	$-0.366263\pi$
$13$	137.868	0.815789	0.407894	+	0.913029i	$0.366263\pi$
$14$	− 56.5685i	− 0.288615i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 310.354i	− 1.07389i	−0.843617	−	0.536945i	$-0.819578\pi$
$17$	− 310.354i	− 1.07389i	0.843617	−	0.536945i	$-0.180422\pi$
$18$	0	0
$19$	245.737	0.680711	0.340355	−	0.940297i	$-0.389453\pi$
$19$	245.737	0.680711	0.340355	+	0.940297i	$0.389453\pi$
$20$	0	0
$21$	0	0
$22$	310.868	0.642290
$23$	− 301.069i	− 0.569128i	−0.958657	−	0.284564i	$-0.908151\pi$
$23$	− 301.069i	− 0.569128i	0.958657	−	0.284564i	$-0.0918489\pi$
$24$	0	0
$25$	0	0
$26$	389.951i	0.576850i
$27$	0	0
$28$	160.000	0.204082
$29$	1350.84i	1.60623i	0.595821	+	0.803117i	$0.296827\pi$
$29$	1350.84i	1.60623i	−0.595821	+	0.803117i	$0.703173\pi$
$30$	0	0
$31$	1509.08	1.57032	0.785160	−	0.619292i	$-0.212580\pi$
$31$	1509.08	1.57032	0.785160	+	0.619292i	$0.212580\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	877.815	0.759356
$35$	0	0
$36$	0	0
$37$	−1947.37	−1.42247	−0.711237	−	0.702952i	$-0.751865\pi$
$37$	−1947.37	−1.42247	−0.711237	+	0.702952i	$0.751865\pi$
$38$	695.048i	0.481335i
$39$	0	0
$40$	0	0
$41$	− 813.787i	− 0.484109i	−0.970263	−	0.242054i	$-0.922179\pi$
$41$	− 813.787i	− 0.484109i	0.970263	−	0.242054i	$-0.0778212\pi$
$42$	0	0
$43$	−2183.58	−1.18095	−0.590475	−	0.807056i	$-0.701060\pi$
$43$	−2183.58	−1.18095	−0.590475	+	0.807056i	$0.701060\pi$
$44$	879.268i	0.454168i
$45$	0	0
$46$	851.552	0.402435
$47$	766.476i	0.346979i	0.984836	+	0.173489i	$0.0555042\pi$
$47$	766.476i	0.346979i	−0.984836	+	0.173489i	$0.944496\pi$
$48$	0	0
$49$	−2001.00	−0.833403
$50$	0	0
$51$	0	0
$52$	−1102.95	−0.407894
$53$	− 2556.32i	− 0.910046i	−0.890480	−	0.455023i	$-0.849631\pi$
$53$	− 2556.32i	− 0.910046i	0.890480	−	0.455023i	$-0.150369\pi$
$54$	0	0
$55$	0	0
$56$	452.548i	0.144308i
$57$	0	0
$58$	−3820.76	−1.13578
$59$	278.731i	0.0800721i	0.999198	+	0.0400360i	$0.0127473\pi$
$59$	278.731i	0.0800721i	−0.999198	+	0.0400360i	$0.987253\pi$
$60$	0	0
$61$	3023.52	0.812557	0.406278	−	0.913749i	$-0.366826\pi$
$61$	3023.52	0.812557	0.406278	+	0.913749i	$0.366826\pi$
$62$	4268.32i	1.11038i
$63$	0	0
$64$	−512.000	−0.125000
$65$	0	0
$66$	0	0
$67$	−302.000	−0.0672756	−0.0336378	−	0.999434i	$-0.510709\pi$
$67$	−302.000	−0.0672756	−0.0336378	+	0.999434i	$0.510709\pi$
$68$	2482.84i	0.536945i
$69$	0	0
$70$	0	0
$71$	2629.17i	0.521557i	0.965399	+	0.260779i	$0.0839792\pi$
$71$	2629.17i	0.521557i	−0.965399	+	0.260779i	$0.916021\pi$
$72$	0	0
$73$	−2993.74	−0.561782	−0.280891	−	0.959740i	$-0.590630\pi$
$73$	−2993.74	−0.561782	−0.280891	+	0.959740i	$0.590630\pi$
$74$	− 5507.98i	− 1.00584i
$75$	0	0
$76$	−1965.89	−0.340355
$77$	2198.17i	0.370749i
$78$	0	0
$79$	6697.07	1.07308	0.536538	−	0.843876i	$-0.319732\pi$
$79$	6697.07	1.07308	0.536538	+	0.843876i	$0.319732\pi$
$80$	0	0
$81$	0	0
$82$	2301.74	0.342317
$83$	367.449i	0.0533385i	0.999644	+	0.0266693i	$0.00849010\pi$
$83$	367.449i	0.0533385i	−0.999644	+	0.0266693i	$0.991510\pi$
$84$	0	0
$85$	0	0
$86$	− 6176.09i	− 0.835058i
$87$	0	0
$88$	−2486.95	−0.321145
$89$	6448.51i	0.814103i	0.913405	+	0.407052i	$0.133443\pi$
$89$	6448.51i	0.814103i	−0.913405	+	0.407052i	$0.866557\pi$
$90$	0	0
$91$	−2757.37	−0.332975
$92$	2408.55i	0.284564i
$93$	0	0
$94$	−2167.92	−0.245351
$95$	0	0
$96$	0	0
$97$	−9962.26	−1.05880	−0.529400	−	0.848372i	$-0.677583\pi$
$97$	−9962.26	−1.05880	−0.529400	+	0.848372i	$0.677583\pi$
$98$	− 5659.68i	− 0.589305i
$99$	0	0
$100$	0	0
$101$	− 8045.41i	− 0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$101$	− 8045.41i	− 0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$102$	0	0
$103$	−4628.11	−0.436243	−0.218122	−	0.975922i	$-0.569993\pi$
$103$	−4628.11	−0.436243	−0.218122	+	0.975922i	$0.569993\pi$
$104$	− 3119.60i	− 0.288425i
$105$	0	0
$106$	7230.36	0.643500
$107$	− 3810.10i	− 0.332789i	−0.986059	−	0.166395i	$-0.946787\pi$
$107$	− 3810.10i	− 0.332789i	0.986059	−	0.166395i	$-0.0532125\pi$
$108$	0	0
$109$	−8829.10	−0.743128	−0.371564	−	0.928407i	$-0.621178\pi$
$109$	−8829.10	−0.743128	−0.371564	+	0.928407i	$0.621178\pi$
$110$	0	0
$111$	0	0
$112$	−1280.00	−0.102041
$113$	− 20397.3i	− 1.59741i	−0.601724	−	0.798704i	$-0.705519\pi$
$113$	− 20397.3i	− 1.59741i	0.601724	−	0.798704i	$-0.294481\pi$
$114$	0	0
$115$	0	0
$116$	− 10806.7i	− 0.803117i
$117$	0	0
$118$	−788.370	−0.0566195
$119$	6207.09i	0.438323i
$120$	0	0
$121$	2561.11	0.174927
$122$	8551.82i	0.574564i
$123$	0	0
$124$	−12072.6	−0.785160
$125$	0	0
$126$	0	0
$127$	7374.67	0.457230	0.228615	−	0.973517i	$-0.426580\pi$
$127$	7374.67	0.457230	0.228615	+	0.973517i	$0.426580\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	13516.1i	0.787604i	0.919195	+	0.393802i	$0.128841\pi$
$131$	13516.1i	0.787604i	−0.919195	+	0.393802i	$0.871159\pi$
$132$	0	0
$133$	−4914.73	−0.277841
$134$	− 854.185i	− 0.0475710i
$135$	0	0
$136$	−7022.52	−0.379678
$137$	2252.45i	0.120009i	0.998198	+	0.0600045i	$0.0191115\pi$
$137$	2252.45i	0.120009i	−0.998198	+	0.0600045i	$0.980888\pi$
$138$	0	0
$139$	−34776.7	−1.79994	−0.899972	−	0.435948i	$-0.856413\pi$
$139$	−34776.7	−1.79994	−0.899972	+	0.435948i	$0.856413\pi$
$140$	0	0
$141$	0	0
$142$	−7436.41	−0.368797
$143$	− 15152.9i	− 0.741010i
$144$	0	0
$145$	0	0
$146$	− 8467.57i	− 0.397240i
$147$	0	0
$148$	15578.9	0.711237
$149$	− 18040.9i	− 0.812616i	−0.913736	−	0.406308i	$-0.866816\pi$
$149$	− 18040.9i	− 0.812616i	0.913736	−	0.406308i	$-0.133184\pi$
$150$	0	0
$151$	15746.0	0.690582	0.345291	−	0.938496i	$-0.387780\pi$
$151$	15746.0	0.690582	0.345291	+	0.938496i	$0.387780\pi$
$152$	− 5560.39i	− 0.240668i
$153$	0	0
$154$	−6217.37	−0.262159
$155$	0	0
$156$	0	0
$157$	37616.4	1.52608	0.763040	−	0.646351i	$-0.223706\pi$
$157$	37616.4	1.52608	0.763040	+	0.646351i	$0.223706\pi$
$158$	18942.2i	0.758780i
$159$	0	0
$160$	0	0
$161$	6021.38i	0.232297i
$162$	0	0
$163$	1513.09	0.0569493	0.0284746	−	0.999595i	$-0.490935\pi$
$163$	1513.09	0.0569493	0.0284746	+	0.999595i	$0.490935\pi$
$164$	6510.29i	0.242054i
$165$	0	0
$166$	−1039.30	−0.0377160
$167$	− 20610.9i	− 0.739032i	−0.929224	−	0.369516i	$-0.879524\pi$
$167$	− 20610.9i	− 0.739032i	0.929224	−	0.369516i	$-0.120476\pi$
$168$	0	0
$169$	−9553.32	−0.334488
$170$	0	0
$171$	0	0
$172$	17468.6	0.590475
$173$	− 55591.2i	− 1.85743i	−0.370789	−	0.928717i	$-0.620913\pi$
$173$	− 55591.2i	− 1.85743i	0.370789	−	0.928717i	$-0.379087\pi$
$174$	0	0
$175$	0	0
$176$	− 7034.15i	− 0.227084i
$177$	0	0
$178$	−18239.1	−0.575658
$179$	7524.19i	0.234830i	0.993083	+	0.117415i	$0.0374608\pi$
$179$	7524.19i	0.234830i	−0.993083	+	0.117415i	$0.962539\pi$
$180$	0	0
$181$	−31763.8	−0.969562	−0.484781	−	0.874636i	$-0.661101\pi$
$181$	−31763.8	−0.969562	−0.484781	+	0.874636i	$0.661101\pi$
$182$	− 7799.01i	− 0.235449i
$183$	0	0
$184$	−6812.41	−0.201217
$185$	0	0
$186$	0	0
$187$	−34110.6	−0.975453
$188$	− 6131.81i	− 0.173489i
$189$	0	0
$190$	0	0
$191$	− 62705.0i	− 1.71884i	−0.511271	−	0.859419i	$-0.670825\pi$
$191$	− 62705.0i	− 1.71884i	0.511271	−	0.859419i	$-0.329175\pi$
$192$	0	0
$193$	22650.9	0.608094	0.304047	−	0.952657i	$-0.401662\pi$
$193$	22650.9	0.608094	0.304047	+	0.952657i	$0.401662\pi$
$194$	− 28177.5i	− 0.748685i
$195$	0	0
$196$	16008.0	0.416701
$197$	60169.5i	1.55040i	0.631716	+	0.775200i	$0.282351\pi$
$197$	60169.5i	1.55040i	−0.631716	+	0.775200i	$0.717649\pi$
$198$	0	0
$199$	−65875.7	−1.66349	−0.831743	−	0.555160i	$-0.812657\pi$
$199$	−65875.7	−1.66349	−0.831743	+	0.555160i	$0.812657\pi$
$200$	0	0
$201$	0	0
$202$	22755.9	0.557687
$203$	− 27016.9i	− 0.655606i
$204$	0	0
$205$	0	0
$206$	− 13090.3i	− 0.308471i
$207$	0	0
$208$	8823.57	0.203947
$209$	− 27008.6i	− 0.618314i
$210$	0	0
$211$	−66484.5	−1.49333	−0.746664	−	0.665201i	$-0.768346\pi$
$211$	−66484.5	−1.49333	−0.746664	+	0.665201i	$0.768346\pi$
$212$	20450.6i	0.455023i
$213$	0	0
$214$	10776.6	0.235317
$215$	0	0
$216$	0	0
$217$	−30181.6	−0.640947
$218$	− 24972.5i	− 0.525471i
$219$	0	0
$220$	0	0
$221$	− 42788.1i	− 0.876068i
$222$	0	0
$223$	−84740.9	−1.70405	−0.852027	−	0.523497i	$-0.824627\pi$
$223$	−84740.9	−1.70405	−0.852027	+	0.523497i	$0.824627\pi$
$224$	− 3620.39i	− 0.0721538i
$225$	0	0
$226$	57692.3	1.12954
$227$	− 15941.8i	− 0.309375i	−0.987963	−	0.154687i	$-0.950563\pi$
$227$	− 15941.8i	− 0.309375i	0.987963	−	0.154687i	$-0.0494370\pi$
$228$	0	0
$229$	73930.4	1.40978	0.704891	−	0.709315i	$-0.250996\pi$
$229$	73930.4	1.40978	0.704891	+	0.709315i	$0.250996\pi$
$230$	0	0
$231$	0	0
$232$	30566.1	0.567890
$233$	65527.1i	1.20700i	0.797361	+	0.603502i	$0.206229\pi$
$233$	65527.1i	1.20700i	−0.797361	+	0.603502i	$0.793771\pi$
$234$	0	0
$235$	0	0
$236$	− 2229.85i	− 0.0400360i
$237$	0	0
$238$	−17556.3	−0.309941
$239$	51958.0i	0.909613i	0.890590	+	0.454806i	$0.150291\pi$
$239$	51958.0i	0.909613i	−0.890590	+	0.454806i	$0.849709\pi$
$240$	0	0
$241$	32900.8	0.566464	0.283232	−	0.959051i	$-0.408593\pi$
$241$	32900.8	0.566464	0.283232	+	0.959051i	$0.408593\pi$
$242$	7243.91i	0.123692i
$243$	0	0
$244$	−24188.2	−0.406278
$245$	0	0
$246$	0	0
$247$	33879.3	0.555316
$248$	− 34146.5i	− 0.555192i
$249$	0	0
$250$	0	0
$251$	− 123616.i	− 1.96212i	−0.193693	−	0.981062i	$-0.562046\pi$
$251$	− 123616.i	− 1.96212i	0.193693	−	0.981062i	$-0.437954\pi$
$252$	0	0
$253$	−33090.1	−0.516959
$254$	20858.7i	0.323311i
$255$	0	0
$256$	4096.00	0.0625000
$257$	21822.4i	0.330397i	0.986260	+	0.165199i	$0.0528265\pi$
$257$	21822.4i	0.330397i	−0.986260	+	0.165199i	$0.947174\pi$
$258$	0	0
$259$	38947.3	0.580602
$260$	0	0
$261$	0	0
$262$	−38229.2	−0.556920
$263$	− 93885.8i	− 1.35734i	−0.734444	−	0.678670i	$-0.762557\pi$
$263$	− 93885.8i	− 1.35734i	0.734444	−	0.678670i	$-0.237443\pi$
$264$	0	0
$265$	0	0
$266$	− 13901.0i	− 0.196463i
$267$	0	0
$268$	2416.00	0.0336378
$269$	− 104848.i	− 1.44895i	−0.689298	−	0.724477i	$-0.742081\pi$
$269$	− 104848.i	− 1.44895i	0.689298	−	0.724477i	$-0.257919\pi$
$270$	0	0
$271$	33957.6	0.462380	0.231190	−	0.972909i	$-0.425738\pi$
$271$	33957.6	0.462380	0.231190	+	0.972909i	$0.425738\pi$
$272$	− 19862.7i	− 0.268473i
$273$	0	0
$274$	−6370.89	−0.0848592
$275$	0	0
$276$	0	0
$277$	−97542.3	−1.27126	−0.635628	−	0.771995i	$-0.719259\pi$
$277$	−97542.3	−1.27126	−0.635628	+	0.771995i	$0.719259\pi$
$278$	− 98363.4i	− 1.27275i
$279$	0	0
$280$	0	0
$281$	− 88207.1i	− 1.11710i	−0.829472	−	0.558549i	$-0.811358\pi$
$281$	− 88207.1i	− 1.11710i	0.829472	−	0.558549i	$-0.188642\pi$
$282$	0	0
$283$	−3373.87	−0.0421265	−0.0210633	−	0.999778i	$-0.506705\pi$
$283$	−3373.87	−0.0421265	−0.0210633	+	0.999778i	$0.506705\pi$
$284$	− 21033.4i	− 0.260779i
$285$	0	0
$286$	42858.9	0.523973
$287$	16275.7i	0.197595i
$288$	0	0
$289$	−12798.9	−0.153242
$290$	0	0
$291$	0	0
$292$	23949.9	0.280891
$293$	− 147945.i	− 1.72331i	−0.507490	−	0.861657i	$-0.669427\pi$
$293$	− 147945.i	− 1.72331i	0.507490	−	0.861657i	$-0.330573\pi$
$294$	0	0
$295$	0	0
$296$	44063.9i	0.502920i
$297$	0	0
$298$	51027.3	0.574606
$299$	− 41507.9i	− 0.464289i
$300$	0	0
$301$	43671.5	0.482020
$302$	44536.3i	0.488315i
$303$	0	0
$304$	15727.1	0.170178
$305$	0	0
$306$	0	0
$307$	−147577.	−1.56583	−0.782913	−	0.622131i	$-0.786267\pi$
$307$	−147577.	−1.56583	−0.782913	+	0.622131i	$0.786267\pi$
$308$	− 17585.4i	− 0.185375i
$309$	0	0
$310$	0	0
$311$	54500.0i	0.563477i	0.959491	+	0.281738i	$0.0909110\pi$
$311$	54500.0i	0.563477i	−0.959491	+	0.281738i	$0.909089\pi$
$312$	0	0
$313$	98960.3	1.01012	0.505059	−	0.863085i	$-0.331471\pi$
$313$	98960.3	1.01012	0.505059	+	0.863085i	$0.331471\pi$
$314$	106395.i	1.07910i
$315$	0	0
$316$	−53576.6	−0.536538
$317$	20926.8i	0.208250i	0.994564	+	0.104125i	$0.0332042\pi$
$317$	20926.8i	0.208250i	−0.994564	+	0.104125i	$0.966796\pi$
$318$	0	0
$319$	148469.	1.45900
$320$	0	0
$321$	0	0
$322$	−17031.0	−0.164259
$323$	− 76265.5i	− 0.731009i
$324$	0	0
$325$	0	0
$326$	4279.65i	0.0402692i
$327$	0	0
$328$	−18413.9	−0.171158
$329$	− 15329.5i	− 0.141624i
$330$	0	0
$331$	−139684.	−1.27494	−0.637470	−	0.770475i	$-0.720019\pi$
$331$	−139684.	−1.27494	−0.637470	+	0.770475i	$0.720019\pi$
$332$	− 2939.59i	− 0.0266693i
$333$	0	0
$334$	58296.3	0.522574
$335$	0	0
$336$	0	0
$337$	16542.7	0.145662	0.0728311	−	0.997344i	$-0.476797\pi$
$337$	16542.7	0.145662	0.0728311	+	0.997344i	$0.476797\pi$
$338$	− 27020.9i	− 0.236519i
$339$	0	0
$340$	0	0
$341$	− 165861.i	− 1.42638i
$342$	0	0
$343$	88040.0	0.748328
$344$	49408.7i	0.417529i
$345$	0	0
$346$	157236.	1.31340
$347$	153854.i	1.27776i	0.769305	+	0.638882i	$0.220603\pi$
$347$	153854.i	1.27776i	−0.769305	+	0.638882i	$0.779397\pi$
$348$	0	0
$349$	96083.9	0.788860	0.394430	−	0.918926i	$-0.370942\pi$
$349$	96083.9	0.788860	0.394430	+	0.918926i	$0.370942\pi$
$350$	0	0
$351$	0	0
$352$	19895.6	0.160572
$353$	− 4899.48i	− 0.0393189i	−0.999807	−	0.0196594i	$-0.993742\pi$
$353$	− 4899.48i	− 0.0393189i	0.999807	−	0.0196594i	$-0.00625820\pi$
$354$	0	0
$355$	0	0
$356$	− 51588.1i	− 0.407052i
$357$	0	0
$358$	−21281.6	−0.166050
$359$	141577.i	1.09851i	0.835654	+	0.549256i	$0.185089\pi$
$359$	141577.i	1.09851i	−0.835654	+	0.549256i	$0.814911\pi$
$360$	0	0
$361$	−69934.5	−0.536633
$362$	− 89841.6i	− 0.685584i
$363$	0	0
$364$	22058.9	0.166488
$365$	0	0
$366$	0	0
$367$	27267.4	0.202447	0.101224	−	0.994864i	$-0.467724\pi$
$367$	27267.4	0.202447	0.101224	+	0.994864i	$0.467724\pi$
$368$	− 19268.4i	− 0.142282i
$369$	0	0
$370$	0	0
$371$	51126.4i	0.371447i
$372$	0	0
$373$	−231918.	−1.66692	−0.833462	−	0.552577i	$-0.813645\pi$
$373$	−231918.	−1.66692	−0.833462	+	0.552577i	$0.813645\pi$
$374$	− 96479.4i	− 0.689749i
$375$	0	0
$376$	17343.4	0.122676
$377$	186238.i	1.31035i
$378$	0	0
$379$	163207.	1.13622	0.568108	−	0.822954i	$-0.307676\pi$
$379$	163207.	1.13622	0.568108	+	0.822954i	$0.307676\pi$
$380$	0	0
$381$	0	0
$382$	177356.	1.21540
$383$	− 255232.i	− 1.73996i	−0.493091	−	0.869978i	$-0.664133\pi$
$383$	− 255232.i	− 1.73996i	0.493091	−	0.869978i	$-0.335867\pi$
$384$	0	0
$385$	0	0
$386$	64066.4i	0.429988i
$387$	0	0
$388$	79698.1	0.529400
$389$	− 209273.i	− 1.38297i	−0.722389	−	0.691487i	$-0.756956\pi$
$389$	− 209273.i	− 1.38297i	0.722389	−	0.691487i	$-0.243044\pi$
$390$	0	0
$391$	−93438.1	−0.611182
$392$	45277.5i	0.294652i
$393$	0	0
$394$	−170185.	−1.09630
$395$	0	0
$396$	0	0
$397$	93872.0	0.595600	0.297800	−	0.954628i	$-0.403747\pi$
$397$	93872.0	0.595600	0.297800	+	0.954628i	$0.403747\pi$
$398$	− 186325.i	− 1.17626i
$399$	0	0
$400$	0	0
$401$	− 180410.i	− 1.12195i	−0.827833	−	0.560974i	$-0.810427\pi$
$401$	− 180410.i	− 1.12195i	0.827833	−	0.560974i	$-0.189573\pi$
$402$	0	0
$403$	208054.	1.28105
$404$	64363.3i	0.394344i
$405$	0	0
$406$	76415.2	0.463583
$407$	214032.i	1.29208i
$408$	0	0
$409$	39669.6	0.237143	0.118572	−	0.992945i	$-0.462168\pi$
$409$	39669.6	0.237143	0.118572	+	0.992945i	$0.462168\pi$
$410$	0	0
$411$	0	0
$412$	37024.9	0.218122
$413$	− 5574.62i	− 0.0326825i
$414$	0	0
$415$	0	0
$416$	24956.8i	0.144212i
$417$	0	0
$418$	76391.7	0.437214
$419$	56877.2i	0.323974i	0.986793	+	0.161987i	$0.0517903\pi$
$419$	56877.2i	0.323974i	−0.986793	+	0.161987i	$0.948210\pi$
$420$	0	0
$421$	−223978.	−1.26369	−0.631845	−	0.775095i	$-0.717702\pi$
$421$	−223978.	−1.26369	−0.631845	+	0.775095i	$0.717702\pi$
$422$	− 188047.i	− 1.05594i
$423$	0	0
$424$	−57842.9	−0.321750
$425$	0	0
$426$	0	0
$427$	−60470.5	−0.331656
$428$	30480.8i	0.166395i
$429$	0	0
$430$	0	0
$431$	− 158431.i	− 0.852875i	−0.904517	−	0.426438i	$-0.859768\pi$
$431$	− 158431.i	− 0.852875i	0.904517	−	0.426438i	$-0.140232\pi$
$432$	0	0
$433$	94181.0	0.502328	0.251164	−	0.967945i	$-0.419187\pi$
$433$	94181.0	0.502328	0.251164	+	0.967945i	$0.419187\pi$
$434$	− 85366.4i	− 0.453218i
$435$	0	0
$436$	70632.8	0.371564
$437$	− 73983.7i	− 0.387412i
$438$	0	0
$439$	−293847.	−1.52473	−0.762365	−	0.647148i	$-0.775962\pi$
$439$	−293847.	−1.52473	−0.762365	+	0.647148i	$0.775962\pi$
$440$	0	0
$441$	0	0
$442$	121023.	0.619474
$443$	− 57235.5i	− 0.291647i	−0.989311	−	0.145824i	$-0.953417\pi$
$443$	− 57235.5i	− 0.291647i	0.989311	−	0.145824i	$-0.0465832\pi$
$444$	0	0
$445$	0	0
$446$	− 239684.i	− 1.20495i
$447$	0	0
$448$	10240.0	0.0510204
$449$	201872.i	1.00134i	0.865637	+	0.500672i	$0.166914\pi$
$449$	201872.i	1.00134i	−0.865637	+	0.500672i	$0.833086\pi$
$450$	0	0
$451$	−89442.1	−0.439733
$452$	163178.i	0.798704i
$453$	0	0
$454$	45090.1	0.218761
$455$	0	0
$456$	0	0
$457$	124221.	0.594789	0.297395	−	0.954755i	$-0.403882\pi$
$457$	124221.	0.594789	0.297395	+	0.954755i	$0.403882\pi$
$458$	209107.i	0.996867i
$459$	0	0
$460$	0	0
$461$	− 190748.i	− 0.897548i	−0.893645	−	0.448774i	$-0.851861\pi$
$461$	− 190748.i	− 0.897548i	0.893645	−	0.448774i	$-0.148139\pi$
$462$	0	0
$463$	109479.	0.510704	0.255352	−	0.966848i	$-0.417809\pi$
$463$	109479.	0.510704	0.255352	+	0.966848i	$0.417809\pi$
$464$	86454.0i	0.401559i
$465$	0	0
$466$	−185339.	−0.853481
$467$	− 173847.i	− 0.797140i	−0.917138	−	0.398570i	$-0.869507\pi$
$467$	− 173847.i	− 0.797140i	0.917138	−	0.398570i	$-0.130493\pi$
$468$	0	0
$469$	6040.00	0.0274594
$470$	0	0
$471$	0	0
$472$	6306.96	0.0283098
$473$	239994.i	1.07270i
$474$	0	0
$475$	0	0
$476$	− 49656.7i	− 0.219161i
$477$	0	0
$478$	−146959.	−0.643193
$479$	− 92549.8i	− 0.403371i	−0.979450	−	0.201685i	$-0.935358\pi$
$479$	− 92549.8i	− 0.403371i	0.979450	−	0.201685i	$-0.0646418\pi$
$480$	0	0
$481$	−268480.	−1.16044
$482$	93057.5i	0.400550i
$483$	0	0
$484$	−20488.9	−0.0874636
$485$	0	0
$486$	0	0
$487$	−356922.	−1.50493	−0.752463	−	0.658635i	$-0.771134\pi$
$487$	−356922.	−1.50493	−0.752463	+	0.658635i	$0.771134\pi$
$488$	− 68414.5i	− 0.287282i
$489$	0	0
$490$	0	0
$491$	415365.i	1.72293i	0.507821	+	0.861463i	$0.330451\pi$
$491$	415365.i	1.72293i	−0.507821	+	0.861463i	$0.669549\pi$
$492$	0	0
$493$	419240.	1.72492
$494$	95825.1i	0.392668i
$495$	0	0
$496$	96581.0	0.392580
$497$	− 52583.4i	− 0.212880i
$498$	0	0
$499$	353939.	1.42144	0.710719	−	0.703476i	$-0.248370\pi$
$499$	353939.	1.42144	0.710719	+	0.703476i	$0.248370\pi$
$500$	0	0
$501$	0	0
$502$	349638.	1.38743
$503$	116774.i	0.461539i	0.973008	+	0.230770i	$0.0741244\pi$
$503$	116774.i	0.461539i	−0.973008	+	0.230770i	$0.925876\pi$
$504$	0	0
$505$	0	0
$506$	− 93592.8i	− 0.365545i
$507$	0	0
$508$	−58997.4	−0.228615
$509$	− 296669.i	− 1.14508i	−0.819876	−	0.572541i	$-0.805958\pi$
$509$	− 296669.i	− 1.14508i	0.819876	−	0.572541i	$-0.194042\pi$
$510$	0	0
$511$	59874.7	0.229299
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−61723.1	−0.233626
$515$	0	0
$516$	0	0
$517$	84242.3	0.315173
$518$	110160.i	0.410547i
$519$	0	0
$520$	0	0
$521$	19300.7i	0.0711045i	0.999368	+	0.0355523i	$0.0113190\pi$
$521$	19300.7i	0.0711045i	−0.999368	+	0.0355523i	$0.988681\pi$
$522$	0	0
$523$	59751.6	0.218447	0.109223	−	0.994017i	$-0.465164\pi$
$523$	59751.6	0.218447	0.109223	+	0.994017i	$0.465164\pi$
$524$	− 108129.i	− 0.393802i
$525$	0	0
$526$	265549.	0.959784
$527$	− 468349.i	− 1.68635i
$528$	0	0
$529$	189198.	0.676093
$530$	0	0
$531$	0	0
$532$	39317.9	0.138921
$533$	− 112195.i	− 0.394931i
$534$	0	0
$535$	0	0
$536$	6833.48i	0.0237855i
$537$	0	0
$538$	296554.	1.02457
$539$	219927.i	0.757009i
$540$	0	0
$541$	−286545.	−0.979034	−0.489517	−	0.871994i	$-0.662827\pi$
$541$	−286545.	−0.979034	−0.489517	+	0.871994i	$0.662827\pi$
$542$	96046.7i	0.326952i
$543$	0	0
$544$	56180.2	0.189839
$545$	0	0
$546$	0	0
$547$	−238579.	−0.797366	−0.398683	−	0.917089i	$-0.630533\pi$
$547$	−238579.	−0.797366	−0.398683	+	0.917089i	$0.630533\pi$
$548$	− 18019.6i	− 0.0600045i
$549$	0	0
$550$	0	0
$551$	331952.i	1.09338i
$552$	0	0
$553$	−133941.	−0.437990
$554$	− 275891.i	− 0.898914i
$555$	0	0
$556$	278214.	0.899972
$557$	425747.i	1.37227i	0.727472	+	0.686137i	$0.240695\pi$
$557$	425747.i	1.37227i	−0.727472	+	0.686137i	$0.759305\pi$
$558$	0	0
$559$	−301046.	−0.963406
$560$	0	0
$561$	0	0
$562$	249487.	0.789907
$563$	− 530975.i	− 1.67516i	−0.546312	−	0.837582i	$-0.683969\pi$
$563$	− 530975.i	− 1.67516i	0.546312	−	0.837582i	$-0.316031\pi$
$564$	0	0
$565$	0	0
$566$	− 9542.75i	− 0.0297880i
$567$	0	0
$568$	59491.3	0.184398
$569$	54492.4i	0.168311i	0.996453	+	0.0841553i	$0.0268192\pi$
$569$	54492.4i	0.168311i	−0.996453	+	0.0841553i	$0.973181\pi$
$570$	0	0
$571$	95526.6	0.292989	0.146495	−	0.989211i	$-0.453201\pi$
$571$	95526.6	0.292989	0.146495	+	0.989211i	$0.453201\pi$
$572$	121223.i	0.370505i
$573$	0	0
$574$	−46034.7	−0.139721
$575$	0	0
$576$	0	0
$577$	−643813.	−1.93378	−0.966892	−	0.255187i	$-0.917863\pi$
$577$	−643813.	−1.93378	−0.966892	+	0.255187i	$0.917863\pi$
$578$	− 36200.7i	− 0.108358i
$579$	0	0
$580$	0	0
$581$	− 7348.98i	− 0.0217708i
$582$	0	0
$583$	−280961.	−0.826627
$584$	67740.5i	0.198620i
$585$	0	0
$586$	418451.	1.21857
$587$	− 414403.i	− 1.20267i	−0.798996	−	0.601336i	$-0.794635\pi$
$587$	− 414403.i	− 1.20267i	0.798996	−	0.601336i	$-0.205365\pi$
$588$	0	0
$589$	370836.	1.06893
$590$	0	0
$591$	0	0
$592$	−124631.	−0.355618
$593$	507625.i	1.44355i	0.692125	+	0.721777i	$0.256675\pi$
$593$	507625.i	1.44355i	−0.692125	+	0.721777i	$0.743325\pi$
$594$	0	0
$595$	0	0
$596$	144327.i	0.406308i
$597$	0	0
$598$	117402.	0.328302
$599$	256614.i	0.715199i	0.933875	+	0.357600i	$0.116405\pi$
$599$	256614.i	0.715199i	−0.933875	+	0.357600i	$0.883595\pi$
$600$	0	0
$601$	−429439.	−1.18892	−0.594460	−	0.804126i	$-0.702634\pi$
$601$	−429439.	−1.18892	−0.594460	+	0.804126i	$0.702634\pi$
$602$	123522.i	0.340840i
$603$	0	0
$604$	−125968.	−0.345291
$605$	0	0
$606$	0	0
$607$	121093.	0.328657	0.164329	−	0.986406i	$-0.447454\pi$
$607$	121093.	0.328657	0.164329	+	0.986406i	$0.447454\pi$
$608$	44483.1i	0.120334i
$609$	0	0
$610$	0	0
$611$	105673.i	0.283061i
$612$	0	0
$613$	−478350.	−1.27299	−0.636495	−	0.771281i	$-0.719616\pi$
$613$	−478350.	−1.27299	−0.636495	+	0.771281i	$0.719616\pi$
$614$	− 417412.i	− 1.10721i
$615$	0	0
$616$	49738.9	0.131080
$617$	− 373275.i	− 0.980524i	−0.871575	−	0.490262i	$-0.836901\pi$
$617$	− 373275.i	− 0.980524i	0.871575	−	0.490262i	$-0.163099\pi$
$618$	0	0
$619$	−459667.	−1.19967	−0.599835	−	0.800123i	$-0.704767\pi$
$619$	−459667.	−1.19967	−0.599835	+	0.800123i	$0.704767\pi$
$620$	0	0
$621$	0	0
$622$	−154149.	−0.398438
$623$	− 128970.i	− 0.332287i
$624$	0	0
$625$	0	0
$626$	279902.i	0.714262i
$627$	0	0
$628$	−300931.	−0.763040
$629$	604374.i	1.52758i
$630$	0	0
$631$	−299678.	−0.752655	−0.376328	−	0.926487i	$-0.622813\pi$
$631$	−299678.	−0.752655	−0.376328	+	0.926487i	$0.622813\pi$
$632$	− 151537.i	− 0.379390i
$633$	0	0
$634$	−59189.9	−0.147255
$635$	0	0
$636$	0	0
$637$	−275875.	−0.679881
$638$	419934.i	1.03167i
$639$	0	0
$640$	0	0
$641$	688194.i	1.67492i	0.546496	+	0.837462i	$0.315961\pi$
$641$	688194.i	1.67492i	−0.546496	+	0.837462i	$0.684039\pi$
$642$	0	0
$643$	−664047.	−1.60612	−0.803058	−	0.595901i	$-0.796795\pi$
$643$	−664047.	−1.60612	−0.803058	+	0.595901i	$0.796795\pi$
$644$	− 48171.0i	− 0.116149i
$645$	0	0
$646$	215711.	0.516902
$647$	− 771328.i	− 1.84260i	−0.388856	−	0.921299i	$-0.627130\pi$
$647$	− 771328.i	− 1.84260i	0.388856	−	0.921299i	$-0.372870\pi$
$648$	0	0
$649$	30634.9	0.0727323
$650$	0	0
$651$	0	0
$652$	−12104.7	−0.0284746
$653$	404203.i	0.947923i	0.880545	+	0.473962i	$0.157177\pi$
$653$	404203.i	0.947923i	−0.880545	+	0.473962i	$0.842823\pi$
$654$	0	0
$655$	0	0
$656$	− 52082.4i	− 0.121027i
$657$	0	0
$658$	43358.4	0.100143
$659$	− 708853.i	− 1.63225i	−0.577878	−	0.816123i	$-0.696119\pi$
$659$	− 708853.i	− 1.63225i	0.577878	−	0.816123i	$-0.303881\pi$
$660$	0	0
$661$	427048.	0.977404	0.488702	−	0.872451i	$-0.337471\pi$
$661$	427048.	0.977404	0.488702	+	0.872451i	$0.337471\pi$
$662$	− 395085.i	− 0.901519i
$663$	0	0
$664$	8314.42	0.0188580
$665$	0	0
$666$	0	0
$667$	406697.	0.914154
$668$	164887.i	0.369516i
$669$	0	0
$670$	0	0
$671$	− 332311.i	− 0.738074i
$672$	0	0
$673$	−380564.	−0.840228	−0.420114	−	0.907471i	$-0.638010\pi$
$673$	−380564.	−0.840228	−0.420114	+	0.907471i	$0.638010\pi$
$674$	46789.8i	0.102999i
$675$	0	0
$676$	76426.6	0.167244
$677$	390917.i	0.852919i	0.904507	+	0.426459i	$0.140239\pi$
$677$	390917.i	0.852919i	−0.904507	+	0.426459i	$0.859761\pi$
$678$	0	0
$679$	199245.	0.432164
$680$	0	0
$681$	0	0
$682$	469125.	1.00860
$683$	767834.i	1.64598i	0.568052	+	0.822992i	$0.307697\pi$
$683$	767834.i	1.64598i	−0.568052	+	0.822992i	$0.692303\pi$
$684$	0	0
$685$	0	0
$686$	249015.i	0.529148i
$687$	0	0
$688$	−139749.	−0.295238
$689$	− 352435.i	− 0.742406i
$690$	0	0
$691$	376742.	0.789019	0.394510	−	0.918892i	$-0.370915\pi$
$691$	376742.	0.789019	0.394510	+	0.918892i	$0.370915\pi$
$692$	444729.i	0.928717i
$693$	0	0
$694$	−435166.	−0.903516
$695$	0	0
$696$	0	0
$697$	−252562.	−0.519880
$698$	271766.i	0.557808i
$699$	0	0
$700$	0	0
$701$	− 499780.i	− 1.01705i	−0.861047	−	0.508526i	$-0.830191\pi$
$701$	− 499780.i	− 1.01705i	0.861047	−	0.508526i	$-0.169809\pi$
$702$	0	0
$703$	−478539.	−0.968294
$704$	56273.2i	0.113542i
$705$	0	0
$706$	13857.8	0.0278026
$707$	160908.i	0.321914i
$708$	0	0
$709$	307648.	0.612013	0.306007	−	0.952029i	$-0.401007\pi$
$709$	307648.	0.612013	0.306007	+	0.952029i	$0.401007\pi$
$710$	0	0
$711$	0	0
$712$	145913.	0.287829
$713$	− 454337.i	− 0.893714i
$714$	0	0
$715$	0	0
$716$	− 60193.6i	− 0.117415i
$717$	0	0
$718$	−400441.	−0.776765
$719$	376870.i	0.729010i	0.931201	+	0.364505i	$0.118762\pi$
$719$	376870.i	0.729010i	−0.931201	+	0.364505i	$0.881238\pi$
$720$	0	0
$721$	92562.1	0.178059
$722$	− 197805.i	− 0.379457i
$723$	0	0
$724$	254111.	0.484781
$725$	0	0
$726$	0	0
$727$	−647353.	−1.22482	−0.612410	−	0.790541i	$-0.709800\pi$
$727$	−647353.	−1.22482	−0.612410	+	0.790541i	$0.709800\pi$
$728$	62392.1i	0.117724i
$729$	0	0
$730$	0	0
$731$	677683.i	1.26821i
$732$	0	0
$733$	329866.	0.613944	0.306972	−	0.951718i	$-0.400684\pi$
$733$	329866.	0.613944	0.306972	+	0.951718i	$0.400684\pi$
$734$	77124.0i	0.143152i
$735$	0	0
$736$	54499.3	0.100609
$737$	33192.4i	0.0611088i
$738$	0	0
$739$	257939.	0.472311	0.236155	−	0.971715i	$-0.424113\pi$
$739$	257939.	0.472311	0.236155	+	0.971715i	$0.424113\pi$
$740$	0	0
$741$	0	0
$742$	−144607.	−0.262653
$743$	77366.1i	0.140144i	0.997542	+	0.0700718i	$0.0223228\pi$
$743$	77366.1i	0.140144i	−0.997542	+	0.0700718i	$0.977677\pi$
$744$	0	0
$745$	0	0
$746$	− 655962.i	− 1.17869i
$747$	0	0
$748$	272885.	0.487726
$749$	76202.1i	0.135832i
$750$	0	0
$751$	−648770.	−1.15030	−0.575150	−	0.818048i	$-0.695056\pi$
$751$	−648770.	−1.15030	−0.575150	+	0.818048i	$0.695056\pi$
$752$	49054.5i	0.0867447i
$753$	0	0
$754$	−526762.	−0.926556
$755$	0	0
$756$	0	0
$757$	−130710.	−0.228095	−0.114048	−	0.993475i	$-0.536382\pi$
$757$	−130710.	−0.228095	−0.114048	+	0.993475i	$0.536382\pi$
$758$	461620.i	0.803426i
$759$	0	0
$760$	0	0
$761$	667900.i	1.15330i	0.816991	+	0.576650i	$0.195640\pi$
$761$	667900.i	1.15330i	−0.816991	+	0.576650i	$0.804360\pi$
$762$	0	0
$763$	176582.	0.303318
$764$	501640.i	0.859419i
$765$	0	0
$766$	721906.	1.23033
$767$	38428.2i	0.0653219i
$768$	0	0
$769$	−920955.	−1.55735	−0.778674	−	0.627428i	$-0.784108\pi$
$769$	−920955.	−1.55735	−0.778674	+	0.627428i	$0.784108\pi$
$770$	0	0
$771$	0	0
$772$	−181207.	−0.304047
$773$	1.15971e6i	1.94084i	0.241426	+	0.970419i	$0.422385\pi$
$773$	1.15971e6i	1.94084i	−0.241426	+	0.970419i	$0.577615\pi$
$774$	0	0
$775$	0	0
$776$	225420.i	0.374343i
$777$	0	0
$778$	591913.	0.977910
$779$	− 199977.i	− 0.329538i
$780$	0	0
$781$	288968.	0.473749
$782$	− 264283.i	− 0.432171i
$783$	0	0
$784$	−128064.	−0.208351
$785$	0	0
$786$	0	0
$787$	268849.	0.434070	0.217035	−	0.976164i	$-0.430361\pi$
$787$	268849.	0.434070	0.217035	+	0.976164i	$0.430361\pi$
$788$	− 481356.i	− 0.775200i
$789$	0	0
$790$	0	0
$791$	407946.i	0.652003i
$792$	0	0
$793$	416848.	0.662875
$794$	265510.i	0.421153i
$795$	0	0
$796$	527006.	0.831743
$797$	333311.i	0.524726i	0.964969	+	0.262363i	$0.0845018\pi$
$797$	333311.i	0.524726i	−0.964969	+	0.262363i	$0.915498\pi$
$798$	0	0
$799$	237879.	0.372617
$800$	0	0
$801$	0	0
$802$	510278.	0.793338
$803$	329037.i	0.510286i
$804$	0	0
$805$	0	0
$806$	588466.i	0.905839i
$807$	0	0
$808$	−182047.	−0.278844
$809$	874807.i	1.33664i	0.743873	+	0.668321i	$0.232987\pi$
$809$	874807.i	1.33664i	−0.743873	+	0.668321i	$0.767013\pi$
$810$	0	0
$811$	−73147.8	−0.111214	−0.0556070	−	0.998453i	$-0.517709\pi$
$811$	−73147.8	−0.111214	−0.0556070	+	0.998453i	$0.517709\pi$
$812$	216135.i	0.327803i
$813$	0	0
$814$	−605375.	−0.913641
$815$	0	0
$816$	0	0
$817$	−536585.	−0.803886
$818$	112203.i	0.167686i
$819$	0	0
$820$	0	0
$821$	670202.i	0.994305i	0.867663	+	0.497152i	$0.165621\pi$
$821$	670202.i	0.994305i	−0.867663	+	0.497152i	$0.834379\pi$
$822$	0	0
$823$	−380481.	−0.561737	−0.280868	−	0.959746i	$-0.590622\pi$
$823$	−380481.	−0.561737	−0.280868	+	0.959746i	$0.590622\pi$
$824$	104722.i	0.154235i
$825$	0	0
$826$	15767.4	0.0231100
$827$	605946.i	0.885979i	0.896527	+	0.442989i	$0.146082\pi$
$827$	605946.i	0.885979i	−0.896527	+	0.442989i	$0.853918\pi$
$828$	0	0
$829$	−27092.3	−0.0394218	−0.0197109	−	0.999806i	$-0.506275\pi$
$829$	−27092.3	−0.0394218	−0.0197109	+	0.999806i	$0.506275\pi$
$830$	0	0
$831$	0	0
$832$	−70588.6	−0.101974
$833$	621019.i	0.894984i
$834$	0	0
$835$	0	0
$836$	216068.i	0.309157i
$837$	0	0
$838$	−160873.	−0.229084
$839$	− 634418.i	− 0.901262i	−0.892710	−	0.450631i	$-0.851199\pi$
$839$	− 634418.i	− 0.901262i	0.892710	−	0.450631i	$-0.148801\pi$
$840$	0	0
$841$	−1.11750e6	−1.57999
$842$	− 633505.i	− 0.893564i
$843$	0	0
$844$	531876.	0.746664
$845$	0	0
$846$	0	0
$847$	−51222.2	−0.0713989
$848$	− 163604.i	− 0.227512i
$849$	0	0
$850$	0	0
$851$	586292.i	0.809570i
$852$	0	0
$853$	122650.	0.168566	0.0842830	−	0.996442i	$-0.473140\pi$
$853$	122650.	0.168566	0.0842830	+	0.996442i	$0.473140\pi$
$854$	− 171036.i	− 0.234516i
$855$	0	0
$856$	−86212.8	−0.117659
$857$	− 863050.i	− 1.17510i	−0.809188	−	0.587550i	$-0.800093\pi$
$857$	− 863050.i	− 1.17510i	0.809188	−	0.587550i	$-0.199907\pi$
$858$	0	0
$859$	−1.12393e6	−1.52318	−0.761590	−	0.648059i	$-0.775581\pi$
$859$	−1.12393e6	−1.52318	−0.761590	+	0.648059i	$0.775581\pi$
$860$	0	0
$861$	0	0
$862$	448110.	0.603074
$863$	1.34495e6i	1.80586i	0.429787	+	0.902930i	$0.358589\pi$
$863$	1.34495e6i	1.80586i	−0.429787	+	0.902930i	$0.641411\pi$
$864$	0	0
$865$	0	0
$866$	266384.i	0.355200i
$867$	0	0
$868$	241453.	0.320474
$869$	− 736065.i	− 0.974713i
$870$	0	0
$871$	−41636.2	−0.0548827
$872$	199780.i	0.262735i
$873$	0	0
$874$	209257.	0.273942
$875$	0	0
$876$	0	0
$877$	604935.	0.786520	0.393260	−	0.919427i	$-0.371347\pi$
$877$	604935.	0.786520	0.393260	+	0.919427i	$0.371347\pi$
$878$	− 831126.i	− 1.07815i
$879$	0	0
$880$	0	0
$881$	462965.i	0.596480i	0.954491	+	0.298240i	$0.0963997\pi$
$881$	462965.i	0.596480i	−0.954491	+	0.298240i	$0.903600\pi$
$882$	0	0
$883$	724515.	0.929236	0.464618	−	0.885511i	$-0.346192\pi$
$883$	724515.	0.929236	0.464618	+	0.885511i	$0.346192\pi$
$884$	342304.i	0.438034i
$885$	0	0
$886$	161886.	0.206226
$887$	523890.i	0.665876i	0.942949	+	0.332938i	$0.108040\pi$
$887$	523890.i	0.665876i	−0.942949	+	0.332938i	$0.891960\pi$
$888$	0	0
$889$	−147493.	−0.186625
$890$	0	0
$891$	0	0
$892$	677928.	0.852027
$893$	188351.i	0.236192i
$894$	0	0
$895$	0	0
$896$	28963.1i	0.0360769i
$897$	0	0
$898$	−570980.	−0.708057
$899$	2.03853e6i	2.52230i
$900$	0	0
$901$	−793365.	−0.977290
$902$	− 252981.i	− 0.310938i
$903$	0	0
$904$	−461538.	−0.564769
$905$	0	0
$906$	0	0
$907$	−93625.2	−0.113809	−0.0569047	−	0.998380i	$-0.518123\pi$
$907$	−93625.2	−0.113809	−0.0569047	+	0.998380i	$0.518123\pi$
$908$	127534.i	0.154687i
$909$	0	0
$910$	0	0
$911$	− 714859.i	− 0.861358i	−0.902505	−	0.430679i	$-0.858274\pi$
$911$	− 714859.i	− 0.861358i	0.902505	−	0.430679i	$-0.141726\pi$
$912$	0	0
$913$	40385.8	0.0484492
$914$	351350.i	0.420580i
$915$	0	0
$916$	−591443.	−0.704891
$917$	− 270322.i	− 0.321471i
$918$	0	0
$919$	−392960.	−0.465283	−0.232642	−	0.972563i	$-0.574737\pi$
$919$	−392960.	−0.465283	−0.232642	+	0.972563i	$0.574737\pi$
$920$	0	0
$921$	0	0
$922$	539516.	0.634663
$923$	362479.i	0.425480i
$924$	0	0
$925$	0	0
$926$	309654.i	0.361122i
$927$	0	0
$928$	−244529.	−0.283945
$929$	− 452464.i	− 0.524267i	−0.965032	−	0.262134i	$-0.915574\pi$
$929$	− 452464.i	− 0.524267i	0.965032	−	0.262134i	$-0.0844261\pi$
$930$	0	0
$931$	−491719.	−0.567306
$932$	− 524217.i	− 0.603502i
$933$	0	0
$934$	491715.	0.563663
$935$	0	0
$936$	0	0
$937$	256644.	0.292316	0.146158	−	0.989261i	$-0.453309\pi$
$937$	256644.	0.292316	0.146158	+	0.989261i	$0.453309\pi$
$938$	17083.7i	0.0194167i
$939$	0	0
$940$	0	0
$941$	− 636601.i	− 0.718932i	−0.933158	−	0.359466i	$-0.882959\pi$
$941$	− 636601.i	− 0.718932i	0.933158	−	0.359466i	$-0.117041\pi$
$942$	0	0
$943$	−245006.	−0.275520
$944$	17838.8i	0.0200180i
$945$	0	0
$946$	−678805.	−0.758512
$947$	40671.6i	0.0453515i	0.999743	+	0.0226757i	$0.00721853\pi$
$947$	40671.6i	0.0453515i	−0.999743	+	0.0226757i	$0.992781\pi$
$948$	0	0
$949$	−412741.	−0.458296
$950$	0	0
$951$	0	0
$952$	140450.	0.154971
$953$	311912.i	0.343436i	0.985146	+	0.171718i	$0.0549318\pi$
$953$	311912.i	0.343436i	−0.985146	+	0.171718i	$0.945068\pi$
$954$	0	0
$955$	0	0
$956$	− 415664.i	− 0.454806i
$957$	0	0
$958$	261770.	0.285226
$959$	− 45049.0i	− 0.0489833i
$960$	0	0
$961$	1.35380e6	1.46591
$962$	− 759377.i	− 0.820554i
$963$	0	0
$964$	−263206.	−0.283232
$965$	0	0
$966$	0	0
$967$	−767779.	−0.821076	−0.410538	−	0.911843i	$-0.634659\pi$
$967$	−767779.	−0.821076	−0.410538	+	0.911843i	$0.634659\pi$
$968$	− 57951.3i	− 0.0618461i
$969$	0	0
$970$	0	0
$971$	888897.i	0.942786i	0.881923	+	0.471393i	$0.156249\pi$
$971$	888897.i	0.942786i	−0.881923	+	0.471393i	$0.843751\pi$
$972$	0	0
$973$	695534.	0.734671
$974$	− 1.00953e6i	− 1.06414i
$975$	0	0
$976$	193505.	0.203139
$977$	− 94327.8i	− 0.0988214i	−0.998779	−	0.0494107i	$-0.984266\pi$
$977$	− 94327.8i	− 0.0988214i	0.998779	−	0.0494107i	$-0.0157343\pi$
$978$	0	0
$979$	708747.	0.739479
$980$	0	0
$981$	0	0
$982$	−1.17483e6	−1.21829
$983$	− 271176.i	− 0.280637i	−0.990106	−	0.140318i	$-0.955187\pi$
$983$	− 271176.i	− 0.280637i	0.990106	−	0.140318i	$-0.0448126\pi$
$984$	0	0
$985$	0	0
$986$	1.18579e6i	1.21970i
$987$	0	0
$988$	−271034.	−0.277658
$989$	657407.i	0.672112i
$990$	0	0
$991$	−417334.	−0.424949	−0.212474	−	0.977167i	$-0.568152\pi$
$991$	−417334.	−0.424949	−0.212474	+	0.977167i	$0.568152\pi$
$992$	273172.i	0.277596i
$993$	0	0
$994$	148728.	0.150529
$995$	0	0
$996$	0	0
$997$	−152052.	−0.152969	−0.0764843	−	0.997071i	$-0.524369\pi$
$997$	−152052.	−0.152969	−0.0764843	+	0.997071i	$0.524369\pi$
$998$	1.00109e6i	1.00511i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.5.d.b.701.3		4
3.2	odd	2	inner	1350.5.d.b.701.2		4
5.2	odd	4		1350.5.b.e.1349.1		8
5.3	odd	4		1350.5.b.e.1349.7		8
5.4	even	2		270.5.d.a.161.2	✓	4
15.2	even	4		1350.5.b.e.1349.6		8
15.8	even	4		1350.5.b.e.1349.4		8
15.14	odd	2		270.5.d.a.161.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.5.d.a.161.2	✓	4	5.4	even	2
270.5.d.a.161.3	yes	4	15.14	odd	2
1350.5.b.e.1349.1		8	5.2	odd	4
1350.5.b.e.1349.4		8	15.8	even	4
1350.5.b.e.1349.6		8	15.2	even	4
1350.5.b.e.1349.7		8	5.3	odd	4
1350.5.d.b.701.2		4	3.2	odd	2	inner
1350.5.d.b.701.3		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1350.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} -20.0000 q^{7} -22.6274i q^{8} -109.909i q^{11} +137.868 q^{13} -56.5685i q^{14} +64.0000 q^{16} -310.354i q^{17} +245.737 q^{19} +310.868 q^{22} -301.069i q^{23} +389.951i q^{26} +160.000 q^{28} +1350.84i q^{29} +1509.08 q^{31} +181.019i q^{32} +877.815 q^{34} -1947.37 q^{37} +695.048i q^{38} -813.787i q^{41} -2183.58 q^{43} +879.268i q^{44} +851.552 q^{46} +766.476i q^{47} -2001.00 q^{49} -1102.95 q^{52} -2556.32i q^{53} +452.548i q^{56} -3820.76 q^{58} +278.731i q^{59} +3023.52 q^{61} +4268.32i q^{62} -512.000 q^{64} -302.000 q^{67} +2482.84i q^{68} +2629.17i q^{71} -2993.74 q^{73} -5507.98i q^{74} -1965.89 q^{76} +2198.17i q^{77} +6697.07 q^{79} +2301.74 q^{82} +367.449i q^{83} -6176.09i q^{86} -2486.95 q^{88} +6448.51i q^{89} -2757.37 q^{91} +2408.55i q^{92} -2167.92 q^{94} -9962.26 q^{97} -5659.68i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} - 80 q^{7} + 172 q^{13} + 256 q^{16} + 224 q^{19} + 864 q^{22} + 640 q^{28} + 3380 q^{31} + 96 q^{34} - 200 q^{37} + 1132 q^{43} - 768 q^{46} - 8004 q^{49} - 1376 q^{52} - 8832 q^{58} - 808 q^{61}+ \cdots - 11768 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 - 80 * q^7 + 172 * q^13 + 256 * q^16 + 224 * q^19 + 864 * q^22 + 640 * q^28 + 3380 * q^31 + 96 * q^34 - 200 * q^37 + 1132 * q^43 - 768 * q^46 - 8004 * q^49 - 1376 * q^52 - 8832 * q^58 - 808 * q^61 - 2048 * q^64 - 1208 * q^67 - 11216 * q^73 - 1792 * q^76 - 4708 * q^79 + 8448 * q^82 - 6912 * q^88 - 3440 * q^91 - 11328 * q^94 - 11768 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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