LMFDB - Embedded newform 1350.4.c.u.649.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{401})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 201x^{2} + 10000 $ x^4 + 201*x^2 + 10000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-9.51249i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.u.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +29.5375i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} +29.5375i q^{7} -8.00000i q^{8} -46.5375 q^{11} +92.0750i q^{13} -59.0750 q^{14} +16.0000 q^{16} -4.53748i q^{17} -87.5375 q^{19} -93.0750i q^{22} -160.687i q^{23} -184.150 q^{26} -118.150i q^{28} -241.762 q^{29} -2.68738 q^{31} +32.0000i q^{32} +9.07495 q^{34} +20.6124i q^{37} -175.075i q^{38} +501.225 q^{41} +294.687i q^{43} +186.150 q^{44} +321.375 q^{46} +478.987i q^{47} -529.463 q^{49} -368.300i q^{52} +243.075i q^{53} +236.300 q^{56} -483.525i q^{58} -383.700 q^{59} +132.612 q^{61} -5.37477i q^{62} -64.0000 q^{64} -582.612i q^{67} +18.1499i q^{68} +566.775 q^{71} -839.388i q^{73} -41.2249 q^{74} +350.150 q^{76} -1374.60i q^{77} -451.775 q^{79} +1002.45i q^{82} +301.049i q^{83} -589.375 q^{86} +372.300i q^{88} +739.349 q^{89} -2719.66 q^{91} +642.750i q^{92} -957.974 q^{94} +1146.14i q^{97} -1058.93i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} - 66 q^{11} + 4 q^{14} + 64 q^{16} - 230 q^{19} - 256 q^{26} - 126 q^{29} + 590 q^{31} - 204 q^{34} + 1284 q^{41} + 264 q^{44} + 84 q^{46} - 2238 q^{49} - 16 q^{56} - 2496 q^{59} + 170 q^{61} - 256 q^{64} + 2988 q^{71} + 556 q^{74} + 920 q^{76} - 2528 q^{79} - 1156 q^{86} - 1368 q^{89} - 7154 q^{91} - 708 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 - 66 * q^11 + 4 * q^14 + 64 * q^16 - 230 * q^19 - 256 * q^26 - 126 * q^29 + 590 * q^31 - 204 * q^34 + 1284 * q^41 + 264 * q^44 + 84 * q^46 - 2238 * q^49 - 16 * q^56 - 2496 * q^59 + 170 * q^61 - 256 * q^64 + 2988 * q^71 + 556 * q^74 + 920 * q^76 - 2528 * q^79 - 1156 * q^86 - 1368 * q^89 - 7154 * q^91 - 708 * 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
$7$	29.5375i	1.59487i	0.603402	+	0.797437i	$0.293811\pi$
$7$	29.5375i	1.59487i	−0.603402	+	0.797437i	$0.706189\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−46.5375	−1.27560	−0.637799	−	0.770203i	$-0.720155\pi$
$11$	−46.5375	−1.27560	−0.637799	+	0.770203i	$0.720155\pi$
$12$	0	0
$13$	92.0750i	1.96438i	0.187879	+	0.982192i	$0.439839\pi$
$13$	92.0750i	1.96438i	−0.187879	+	0.982192i	$0.560161\pi$
$14$	−59.0750	−1.12775
$15$	0	0
$16$	16.0000	0.250000
$17$	− 4.53748i	− 0.0647353i	−0.999476	−	0.0323676i	$-0.989695\pi$
$17$	− 4.53748i	− 0.0647353i	0.999476	−	0.0323676i	$-0.0103047\pi$
$18$	0	0
$19$	−87.5375	−1.05697	−0.528486	−	0.848942i	$-0.677240\pi$
$19$	−87.5375	−1.05697	−0.528486	+	0.848942i	$0.677240\pi$
$20$	0	0
$21$	0	0
$22$	− 93.0750i	− 0.901984i
$23$	− 160.687i	− 1.45677i	−0.685170	−	0.728383i	$-0.740272\pi$
$23$	− 160.687i	− 1.45677i	0.685170	−	0.728383i	$-0.259728\pi$
$24$	0	0
$25$	0	0
$26$	−184.150	−1.38903
$27$	0	0
$28$	− 118.150i	− 0.797437i
$29$	−241.762	−1.54807	−0.774037	−	0.633141i	$-0.781766\pi$
$29$	−241.762	−1.54807	−0.774037	+	0.633141i	$0.781766\pi$
$30$	0	0
$31$	−2.68738	−0.0155699	−0.00778497	−	0.999970i	$-0.502478\pi$
$31$	−2.68738	−0.0155699	−0.00778497	+	0.999970i	$0.502478\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	9.07495	0.0457748
$35$	0	0
$36$	0	0
$37$	20.6124i	0.0915855i	0.998951	+	0.0457927i	$0.0145814\pi$
$37$	20.6124i	0.0915855i	−0.998951	+	0.0457927i	$0.985419\pi$
$38$	− 175.075i	− 0.747392i
$39$	0	0
$40$	0	0
$41$	501.225	1.90922	0.954612	−	0.297853i	$-0.0962704\pi$
$41$	501.225	1.90922	0.954612	+	0.297853i	$0.0962704\pi$
$42$	0	0
$43$	294.687i	1.04510i	0.852608	+	0.522551i	$0.175020\pi$
$43$	294.687i	1.04510i	−0.852608	+	0.522551i	$0.824980\pi$
$44$	186.150	0.637799
$45$	0	0
$46$	321.375	1.03009
$47$	478.987i	1.48654i	0.668991	+	0.743271i	$0.266727\pi$
$47$	478.987i	1.48654i	−0.668991	+	0.743271i	$0.733273\pi$
$48$	0	0
$49$	−529.463	−1.54362
$50$	0	0
$51$	0	0
$52$	− 368.300i	− 0.982192i
$53$	243.075i	0.629979i	0.949095	+	0.314990i	$0.102001\pi$
$53$	243.075i	0.629979i	−0.949095	+	0.314990i	$0.897999\pi$
$54$	0	0
$55$	0	0
$56$	236.300	0.563873
$57$	0	0
$58$	− 483.525i	− 1.09465i
$59$	−383.700	−0.846670	−0.423335	−	0.905973i	$-0.639141\pi$
$59$	−383.700	−0.846670	−0.423335	+	0.905973i	$0.639141\pi$
$60$	0	0
$61$	132.612	0.278349	0.139174	−	0.990268i	$-0.455555\pi$
$61$	132.612	0.278349	0.139174	+	0.990268i	$0.455555\pi$
$62$	− 5.37477i	− 0.0110096i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 582.612i	− 1.06235i	−0.847262	−	0.531175i	$-0.821751\pi$
$67$	− 582.612i	− 1.06235i	0.847262	−	0.531175i	$-0.178249\pi$
$68$	18.1499i	0.0323676i
$69$	0	0
$70$	0	0
$71$	566.775	0.947378	0.473689	−	0.880692i	$-0.342922\pi$
$71$	566.775	0.947378	0.473689	+	0.880692i	$0.342922\pi$
$72$	0	0
$73$	− 839.388i	− 1.34579i	−0.739737	−	0.672896i	$-0.765050\pi$
$73$	− 839.388i	− 1.34579i	0.739737	−	0.672896i	$-0.234950\pi$
$74$	−41.2249	−0.0647607
$75$	0	0
$76$	350.150	0.528486
$77$	− 1374.60i	− 2.03442i
$78$	0	0
$79$	−451.775	−0.643401	−0.321700	−	0.946841i	$-0.604254\pi$
$79$	−451.775	−0.643401	−0.321700	+	0.946841i	$0.604254\pi$
$80$	0	0
$81$	0	0
$82$	1002.45i	1.35003i
$83$	301.049i	0.398126i	0.979987	+	0.199063i	$0.0637898\pi$
$83$	301.049i	0.398126i	−0.979987	+	0.199063i	$0.936210\pi$
$84$	0	0
$85$	0	0
$86$	−589.375	−0.738999
$87$	0	0
$88$	372.300i	0.450992i
$89$	739.349	0.880571	0.440286	−	0.897858i	$-0.354877\pi$
$89$	739.349	0.880571	0.440286	+	0.897858i	$0.354877\pi$
$90$	0	0
$91$	−2719.66	−3.13295
$92$	642.750i	0.728383i
$93$	0	0
$94$	−957.974	−1.05114
$95$	0	0
$96$	0	0
$97$	1146.14i	1.19972i	0.800106	+	0.599859i	$0.204777\pi$
$97$	1146.14i	1.19972i	−0.800106	+	0.599859i	$0.795223\pi$
$98$	− 1058.93i	− 1.09151i
$99$	0	0
$100$	0	0
$101$	511.463	0.503885	0.251943	−	0.967742i	$-0.418931\pi$
$101$	511.463	0.503885	0.251943	+	0.967742i	$0.418931\pi$
$102$	0	0
$103$	421.512i	0.403231i	0.979465	+	0.201616i	$0.0646191\pi$
$103$	421.512i	0.403231i	−0.979465	+	0.201616i	$0.935381\pi$
$104$	736.600	0.694515
$105$	0	0
$106$	−486.150	−0.445463
$107$	− 1639.50i	− 1.48127i	−0.671905	−	0.740637i	$-0.734524\pi$
$107$	− 1639.50i	− 1.48127i	0.671905	−	0.740637i	$-0.265476\pi$
$108$	0	0
$109$	690.475	0.606748	0.303374	−	0.952872i	$-0.401887\pi$
$109$	690.475	0.606748	0.303374	+	0.952872i	$0.401887\pi$
$110$	0	0
$111$	0	0
$112$	472.600i	0.398718i
$113$	− 970.238i	− 0.807719i	−0.914821	−	0.403860i	$-0.867668\pi$
$113$	− 970.238i	− 0.807719i	0.914821	−	0.403860i	$-0.132332\pi$
$114$	0	0
$115$	0	0
$116$	967.049	0.774037
$117$	0	0
$118$	− 767.400i	− 0.598686i
$119$	134.026	0.103245
$120$	0	0
$121$	834.737	0.627150
$122$	265.225i	0.196822i
$123$	0	0
$124$	10.7495	0.00778497
$125$	0	0
$126$	0	0
$127$	− 805.024i	− 0.562475i	−0.959638	−	0.281237i	$-0.909255\pi$
$127$	− 805.024i	− 0.562475i	0.959638	−	0.281237i	$-0.0907448\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−214.687	−0.143186	−0.0715928	−	0.997434i	$-0.522808\pi$
$131$	−214.687	−0.143186	−0.0715928	+	0.997434i	$0.522808\pi$
$132$	0	0
$133$	− 2585.64i	− 1.68574i
$134$	1165.22	0.751195
$135$	0	0
$136$	−36.2998	−0.0228874
$137$	367.799i	0.229366i	0.993402	+	0.114683i	$0.0365853\pi$
$137$	367.799i	0.229366i	−0.993402	+	0.114683i	$0.963415\pi$
$138$	0	0
$139$	1427.31	0.870956	0.435478	−	0.900199i	$-0.356579\pi$
$139$	1427.31	0.870956	0.435478	+	0.900199i	$0.356579\pi$
$140$	0	0
$141$	0	0
$142$	1133.55i	0.669897i
$143$	− 4284.94i	− 2.50576i
$144$	0	0
$145$	0	0
$146$	1678.78	0.951619
$147$	0	0
$148$	− 82.4497i	− 0.0457927i
$149$	−1757.74	−0.966439	−0.483219	−	0.875499i	$-0.660533\pi$
$149$	−1757.74	−0.966439	−0.483219	+	0.875499i	$0.660533\pi$
$150$	0	0
$151$	−537.775	−0.289825	−0.144912	−	0.989445i	$-0.546290\pi$
$151$	−537.775	−0.289825	−0.144912	+	0.989445i	$0.546290\pi$
$152$	700.300i	0.373696i
$153$	0	0
$154$	2749.20	1.43855
$155$	0	0
$156$	0	0
$157$	− 671.788i	− 0.341494i	−0.985315	−	0.170747i	$-0.945382\pi$
$157$	− 671.788i	− 0.341494i	0.985315	−	0.170747i	$-0.0546180\pi$
$158$	− 903.550i	− 0.454953i
$159$	0	0
$160$	0	0
$161$	4746.30	2.32336
$162$	0	0
$163$	− 2501.42i	− 1.20200i	−0.799248	−	0.601002i	$-0.794768\pi$
$163$	− 2501.42i	− 1.20200i	0.799248	−	0.601002i	$-0.205232\pi$
$164$	−2004.90	−0.954612
$165$	0	0
$166$	−602.099	−0.281518
$167$	751.649i	0.348289i	0.984720	+	0.174145i	$0.0557161\pi$
$167$	751.649i	0.348289i	−0.984720	+	0.174145i	$0.944284\pi$
$168$	0	0
$169$	−6280.80	−2.85881
$170$	0	0
$171$	0	0
$172$	− 1178.75i	− 0.522551i
$173$	− 2374.27i	− 1.04343i	−0.853121	−	0.521713i	$-0.825293\pi$
$173$	− 2374.27i	− 1.04343i	0.853121	−	0.521713i	$-0.174707\pi$
$174$	0	0
$175$	0	0
$176$	−744.600	−0.318899
$177$	0	0
$178$	1478.70i	0.622658i
$179$	−2424.52	−1.01239	−0.506194	−	0.862420i	$-0.668948\pi$
$179$	−2424.52	−1.01239	−0.506194	+	0.862420i	$0.668948\pi$
$180$	0	0
$181$	1179.46	0.484357	0.242179	−	0.970232i	$-0.422138\pi$
$181$	1179.46	0.484357	0.242179	+	0.970232i	$0.422138\pi$
$182$	− 5439.32i	− 2.21533i
$183$	0	0
$184$	−1285.50	−0.515045
$185$	0	0
$186$	0	0
$187$	211.163i	0.0825762i
$188$	− 1915.95i	− 0.743271i
$189$	0	0
$190$	0	0
$191$	344.475	0.130499	0.0652496	−	0.997869i	$-0.479216\pi$
$191$	344.475	0.130499	0.0652496	+	0.997869i	$0.479216\pi$
$192$	0	0
$193$	1941.84i	0.724231i	0.932133	+	0.362115i	$0.117945\pi$
$193$	1941.84i	0.724231i	−0.932133	+	0.362115i	$0.882055\pi$
$194$	−2292.27	−0.848328
$195$	0	0
$196$	2117.85	0.771811
$197$	− 1780.95i	− 0.644099i	−0.946723	−	0.322049i	$-0.895628\pi$
$197$	− 1780.95i	− 0.644099i	0.946723	−	0.322049i	$-0.104372\pi$
$198$	0	0
$199$	808.375	0.287961	0.143980	−	0.989581i	$-0.454010\pi$
$199$	808.375	0.287961	0.143980	+	0.989581i	$0.454010\pi$
$200$	0	0
$201$	0	0
$202$	1022.93i	0.356301i
$203$	− 7141.05i	− 2.46898i
$204$	0	0
$205$	0	0
$206$	−843.024	−0.285127
$207$	0	0
$208$	1473.20i	0.491096i
$209$	4073.77	1.34827
$210$	0	0
$211$	1223.04	0.399039	0.199520	−	0.979894i	$-0.436062\pi$
$211$	1223.04	0.399039	0.199520	+	0.979894i	$0.436062\pi$
$212$	− 972.300i	− 0.314990i
$213$	0	0
$214$	3279.00	1.04742
$215$	0	0
$216$	0	0
$217$	− 79.3785i	− 0.0248321i
$218$	1380.95i	0.429036i
$219$	0	0
$220$	0	0
$221$	417.788	0.127165
$222$	0	0
$223$	1061.00i	0.318610i	0.987229	+	0.159305i	$0.0509253\pi$
$223$	1061.00i	0.318610i	−0.987229	+	0.159305i	$0.949075\pi$
$224$	−945.199	−0.281937
$225$	0	0
$226$	1940.48	0.571144
$227$	− 3573.07i	− 1.04473i	−0.852723	−	0.522364i	$-0.825050\pi$
$227$	− 3573.07i	− 1.04473i	0.852723	−	0.522364i	$-0.174950\pi$
$228$	0	0
$229$	−1001.82	−0.289094	−0.144547	−	0.989498i	$-0.546172\pi$
$229$	−1001.82	−0.289094	−0.144547	+	0.989498i	$0.546172\pi$
$230$	0	0
$231$	0	0
$232$	1934.10i	0.547327i
$233$	6082.42i	1.71018i	0.518477	+	0.855091i	$0.326499\pi$
$233$	6082.42i	1.71018i	−0.518477	+	0.855091i	$0.673501\pi$
$234$	0	0
$235$	0	0
$236$	1534.80	0.423335
$237$	0	0
$238$	268.051i	0.0730050i
$239$	212.399	0.0574850	0.0287425	−	0.999587i	$-0.490850\pi$
$239$	212.399	0.0574850	0.0287425	+	0.999587i	$0.490850\pi$
$240$	0	0
$241$	−2944.45	−0.787006	−0.393503	−	0.919323i	$-0.628737\pi$
$241$	−2944.45	−0.787006	−0.393503	+	0.919323i	$0.628737\pi$
$242$	1669.47i	0.443462i
$243$	0	0
$244$	−530.450	−0.139174
$245$	0	0
$246$	0	0
$247$	− 8060.01i	− 2.07630i
$248$	21.4991i	0.00550481i
$249$	0	0
$250$	0	0
$251$	1130.36	0.284254	0.142127	−	0.989848i	$-0.454606\pi$
$251$	1130.36	0.284254	0.142127	+	0.989848i	$0.454606\pi$
$252$	0	0
$253$	7477.99i	1.85825i
$254$	1610.05	0.397730
$255$	0	0
$256$	256.000	0.0625000
$257$	− 1218.34i	− 0.295711i	−0.989009	−	0.147856i	$-0.952763\pi$
$257$	− 1218.34i	− 0.295711i	0.989009	−	0.147856i	$-0.0472371\pi$
$258$	0	0
$259$	−608.839	−0.146067
$260$	0	0
$261$	0	0
$262$	− 429.375i	− 0.101248i
$263$	− 6937.50i	− 1.62656i	−0.581875	−	0.813279i	$-0.697681\pi$
$263$	− 6937.50i	− 1.62656i	0.581875	−	0.813279i	$-0.302319\pi$
$264$	0	0
$265$	0	0
$266$	5171.27	1.19200
$267$	0	0
$268$	2330.45i	0.531175i
$269$	−515.214	−0.116778	−0.0583888	−	0.998294i	$-0.518596\pi$
$269$	−515.214	−0.116778	−0.0583888	+	0.998294i	$0.518596\pi$
$270$	0	0
$271$	−652.411	−0.146240	−0.0731202	−	0.997323i	$-0.523296\pi$
$271$	−652.411	−0.146240	−0.0731202	+	0.997323i	$0.523296\pi$
$272$	− 72.5996i	− 0.0161838i
$273$	0	0
$274$	−735.598	−0.162186
$275$	0	0
$276$	0	0
$277$	− 2938.85i	− 0.637467i	−0.947844	−	0.318733i	$-0.896743\pi$
$277$	− 2938.85i	− 0.637467i	0.947844	−	0.318733i	$-0.103257\pi$
$278$	2854.62i	0.615859i
$279$	0	0
$280$	0	0
$281$	5351.77	1.13616	0.568078	−	0.822974i	$-0.307687\pi$
$281$	5351.77	1.13616	0.568078	+	0.822974i	$0.307687\pi$
$282$	0	0
$283$	1479.95i	0.310862i	0.987847	+	0.155431i	$0.0496766\pi$
$283$	1479.95i	0.310862i	−0.987847	+	0.155431i	$0.950323\pi$
$284$	−2267.10	−0.473689
$285$	0	0
$286$	8569.87	1.77184
$287$	14804.9i	3.04497i
$288$	0	0
$289$	4892.41	0.995809
$290$	0	0
$291$	0	0
$292$	3357.55i	0.672896i
$293$	− 5178.90i	− 1.03261i	−0.856405	−	0.516305i	$-0.827307\pi$
$293$	− 5178.90i	− 1.03261i	0.856405	−	0.516305i	$-0.172693\pi$
$294$	0	0
$295$	0	0
$296$	164.899	0.0323804
$297$	0	0
$298$	− 3515.47i	− 0.683376i
$299$	14795.3	2.86165
$300$	0	0
$301$	−8704.32	−1.66681
$302$	− 1075.55i	− 0.204937i
$303$	0	0
$304$	−1400.60	−0.264243
$305$	0	0
$306$	0	0
$307$	− 10306.0i	− 1.91594i	−0.286868	−	0.957970i	$-0.592614\pi$
$307$	− 10306.0i	− 1.91594i	0.286868	−	0.957970i	$-0.407386\pi$
$308$	5498.40i	1.01721i
$309$	0	0
$310$	0	0
$311$	2062.65	0.376084	0.188042	−	0.982161i	$-0.439786\pi$
$311$	2062.65	0.376084	0.188042	+	0.982161i	$0.439786\pi$
$312$	0	0
$313$	9247.96i	1.67005i	0.550212	+	0.835025i	$0.314547\pi$
$313$	9247.96i	1.67005i	−0.550212	+	0.835025i	$0.685453\pi$
$314$	1343.58	0.241473
$315$	0	0
$316$	1807.10	0.321700
$317$	− 610.347i	− 0.108140i	−0.998537	−	0.0540702i	$-0.982781\pi$
$317$	− 610.347i	− 0.108140i	0.998537	−	0.0540702i	$-0.0172195\pi$
$318$	0	0
$319$	11251.0	1.97472
$320$	0	0
$321$	0	0
$322$	9492.60i	1.64286i
$323$	397.199i	0.0684234i
$324$	0	0
$325$	0	0
$326$	5002.85	0.849945
$327$	0	0
$328$	− 4009.80i	− 0.675013i
$329$	−14148.1	−2.37085
$330$	0	0
$331$	−2089.26	−0.346937	−0.173469	−	0.984839i	$-0.555498\pi$
$331$	−2089.26	−0.346937	−0.173469	+	0.984839i	$0.555498\pi$
$332$	− 1204.20i	− 0.199063i
$333$	0	0
$334$	−1503.30	−0.246278
$335$	0	0
$336$	0	0
$337$	9426.36i	1.52370i	0.647754	+	0.761849i	$0.275708\pi$
$337$	9426.36i	1.52370i	−0.647754	+	0.761849i	$0.724292\pi$
$338$	− 12561.6i	− 2.02148i
$339$	0	0
$340$	0	0
$341$	125.064	0.0198610
$342$	0	0
$343$	− 5507.63i	− 0.867009i
$344$	2357.50	0.369500
$345$	0	0
$346$	4748.55	0.737814
$347$	− 1221.22i	− 0.188930i	−0.995528	−	0.0944651i	$-0.969886\pi$
$347$	− 1221.22i	− 0.188930i	0.995528	−	0.0944651i	$-0.0301141\pi$
$348$	0	0
$349$	4188.29	0.642389	0.321195	−	0.947013i	$-0.395916\pi$
$349$	4188.29	0.642389	0.321195	+	0.947013i	$0.395916\pi$
$350$	0	0
$351$	0	0
$352$	− 1489.20i	− 0.225496i
$353$	4469.36i	0.673882i	0.941526	+	0.336941i	$0.109392\pi$
$353$	4469.36i	0.673882i	−0.941526	+	0.336941i	$0.890608\pi$
$354$	0	0
$355$	0	0
$356$	−2957.40	−0.440286
$357$	0	0
$358$	− 4849.05i	− 0.715866i
$359$	−643.572	−0.0946140	−0.0473070	−	0.998880i	$-0.515064\pi$
$359$	−643.572	−0.0946140	−0.0473070	+	0.998880i	$0.515064\pi$
$360$	0	0
$361$	803.810	0.117191
$362$	2358.92i	0.342492i
$363$	0	0
$364$	10878.6	1.56647
$365$	0	0
$366$	0	0
$367$	5934.74i	0.844116i	0.906569	+	0.422058i	$0.138692\pi$
$367$	5934.74i	0.844116i	−0.906569	+	0.422058i	$0.861308\pi$
$368$	− 2571.00i	− 0.364192i
$369$	0	0
$370$	0	0
$371$	−7179.82	−1.00474
$372$	0	0
$373$	− 6187.83i	− 0.858964i	−0.903075	−	0.429482i	$-0.858696\pi$
$373$	− 6187.83i	− 0.858964i	0.903075	−	0.429482i	$-0.141304\pi$
$374$	−422.325	−0.0583902
$375$	0	0
$376$	3831.90	0.525572
$377$	− 22260.3i	− 3.04101i
$378$	0	0
$379$	−10428.5	−1.41340	−0.706699	−	0.707515i	$-0.749816\pi$
$379$	−10428.5	−1.41340	−0.706699	+	0.707515i	$0.749816\pi$
$380$	0	0
$381$	0	0
$382$	688.951i	0.0922769i
$383$	10246.7i	1.36705i	0.729926	+	0.683526i	$0.239554\pi$
$383$	10246.7i	1.36705i	−0.729926	+	0.683526i	$0.760446\pi$
$384$	0	0
$385$	0	0
$386$	−3883.67	−0.512108
$387$	0	0
$388$	− 4584.55i	− 0.599859i
$389$	−8508.48	−1.10899	−0.554495	−	0.832187i	$-0.687088\pi$
$389$	−8508.48	−1.10899	−0.554495	+	0.832187i	$0.687088\pi$
$390$	0	0
$391$	−729.115	−0.0943042
$392$	4235.70i	0.545753i
$393$	0	0
$394$	3561.90	0.455447
$395$	0	0
$396$	0	0
$397$	− 3775.96i	− 0.477356i	−0.971099	−	0.238678i	$-0.923286\pi$
$397$	− 3775.96i	− 0.477356i	0.971099	−	0.238678i	$-0.0767140\pi$
$398$	1616.75i	0.203619i
$399$	0	0
$400$	0	0
$401$	−7556.02	−0.940971	−0.470486	−	0.882408i	$-0.655921\pi$
$401$	−7556.02	−0.940971	−0.470486	+	0.882408i	$0.655921\pi$
$402$	0	0
$403$	− 247.441i	− 0.0305854i
$404$	−2045.85	−0.251943
$405$	0	0
$406$	14282.1	1.74583
$407$	− 959.250i	− 0.116826i
$408$	0	0
$409$	−8963.22	−1.08362	−0.541812	−	0.840499i	$-0.682262\pi$
$409$	−8963.22	−1.08362	−0.541812	+	0.840499i	$0.682262\pi$
$410$	0	0
$411$	0	0
$412$	− 1686.05i	− 0.201616i
$413$	− 11333.5i	− 1.35033i
$414$	0	0
$415$	0	0
$416$	−2946.40	−0.347257
$417$	0	0
$418$	8147.55i	0.953372i
$419$	−14153.8	−1.65026	−0.825130	−	0.564943i	$-0.808898\pi$
$419$	−14153.8	−1.65026	−0.825130	+	0.564943i	$0.808898\pi$
$420$	0	0
$421$	−14727.3	−1.70490	−0.852451	−	0.522807i	$-0.824885\pi$
$421$	−14727.3	−1.70490	−0.852451	+	0.522807i	$0.824885\pi$
$422$	2446.07i	0.282163i
$423$	0	0
$424$	1944.60	0.222731
$425$	0	0
$426$	0	0
$427$	3917.04i	0.443931i
$428$	6558.00i	0.740637i
$429$	0	0
$430$	0	0
$431$	−3048.98	−0.340752	−0.170376	−	0.985379i	$-0.554498\pi$
$431$	−3048.98	−0.340752	−0.170376	+	0.985379i	$0.554498\pi$
$432$	0	0
$433$	17218.4i	1.91100i	0.294990	+	0.955500i	$0.404684\pi$
$433$	17218.4i	1.91100i	−0.294990	+	0.955500i	$0.595316\pi$
$434$	158.757	0.0175589
$435$	0	0
$436$	−2761.90	−0.303374
$437$	14066.2i	1.53976i
$438$	0	0
$439$	−1667.91	−0.181332	−0.0906660	−	0.995881i	$-0.528900\pi$
$439$	−1667.91	−0.181332	−0.0906660	+	0.995881i	$0.528900\pi$
$440$	0	0
$441$	0	0
$442$	835.576i	0.0899192i
$443$	9223.20i	0.989182i	0.869126	+	0.494591i	$0.164682\pi$
$443$	9223.20i	0.989182i	−0.869126	+	0.494591i	$0.835318\pi$
$444$	0	0
$445$	0	0
$446$	−2122.00	−0.225291
$447$	0	0
$448$	− 1890.40i	− 0.199359i
$449$	−2636.18	−0.277080	−0.138540	−	0.990357i	$-0.544241\pi$
$449$	−2636.18	−0.277080	−0.138540	+	0.990357i	$0.544241\pi$
$450$	0	0
$451$	−23325.7	−2.43540
$452$	3880.95i	0.403860i
$453$	0	0
$454$	7146.14	0.738734
$455$	0	0
$456$	0	0
$457$	3859.67i	0.395072i	0.980296	+	0.197536i	$0.0632940\pi$
$457$	3859.67i	0.395072i	−0.980296	+	0.197536i	$0.936706\pi$
$458$	− 2003.65i	− 0.204420i
$459$	0	0
$460$	0	0
$461$	6615.07	0.668318	0.334159	−	0.942517i	$-0.391548\pi$
$461$	6615.07	0.668318	0.334159	+	0.942517i	$0.391548\pi$
$462$	0	0
$463$	− 17571.7i	− 1.76377i	−0.471460	−	0.881887i	$-0.656273\pi$
$463$	− 17571.7i	− 1.76377i	0.471460	−	0.881887i	$-0.343727\pi$
$464$	−3868.20	−0.387018
$465$	0	0
$466$	−12164.8	−1.20928
$467$	− 10076.8i	− 0.998502i	−0.866457	−	0.499251i	$-0.833609\pi$
$467$	− 10076.8i	− 0.998502i	0.866457	−	0.499251i	$-0.166391\pi$
$468$	0	0
$469$	17208.9	1.69431
$470$	0	0
$471$	0	0
$472$	3069.60i	0.299343i
$473$	− 13714.0i	− 1.33313i
$474$	0	0
$475$	0	0
$476$	−536.102	−0.0516223
$477$	0	0
$478$	424.797i	0.0406480i
$479$	17945.8	1.71182	0.855912	−	0.517122i	$-0.172997\pi$
$479$	17945.8	1.71182	0.855912	+	0.517122i	$0.172997\pi$
$480$	0	0
$481$	−1897.89	−0.179909
$482$	− 5888.90i	− 0.556498i
$483$	0	0
$484$	−3338.95	−0.313575
$485$	0	0
$486$	0	0
$487$	− 17356.9i	− 1.61502i	−0.589853	−	0.807511i	$-0.700814\pi$
$487$	− 17356.9i	− 1.61502i	0.589853	−	0.807511i	$-0.299186\pi$
$488$	− 1060.90i	− 0.0984112i
$489$	0	0
$490$	0	0
$491$	−8393.17	−0.771443	−0.385722	−	0.922615i	$-0.626047\pi$
$491$	−8393.17	−0.771443	−0.385722	+	0.922615i	$0.626047\pi$
$492$	0	0
$493$	1096.99i	0.100215i
$494$	16120.0	1.46817
$495$	0	0
$496$	−42.9981	−0.00389249
$497$	16741.1i	1.51095i
$498$	0	0
$499$	−16421.1	−1.47317	−0.736585	−	0.676345i	$-0.763563\pi$
$499$	−16421.1	−1.47317	−0.736585	+	0.676345i	$0.763563\pi$
$500$	0	0
$501$	0	0
$502$	2260.72i	0.200998i
$503$	16014.9i	1.41962i	0.704395	+	0.709808i	$0.251218\pi$
$503$	16014.9i	1.41962i	−0.704395	+	0.709808i	$0.748782\pi$
$504$	0	0
$505$	0	0
$506$	−14956.0	−1.31398
$507$	0	0
$508$	3220.09i	0.281237i
$509$	13695.9	1.19265	0.596327	−	0.802741i	$-0.296626\pi$
$509$	13695.9	1.19265	0.596327	+	0.802741i	$0.296626\pi$
$510$	0	0
$511$	24793.4	2.14637
$512$	512.000i	0.0441942i
$513$	0	0
$514$	2436.67	0.209099
$515$	0	0
$516$	0	0
$517$	− 22290.9i	− 1.89623i
$518$	− 1217.68i	− 0.103285i
$519$	0	0
$520$	0	0
$521$	5887.12	0.495047	0.247523	−	0.968882i	$-0.420383\pi$
$521$	5887.12	0.495047	0.247523	+	0.968882i	$0.420383\pi$
$522$	0	0
$523$	8544.57i	0.714394i	0.934029	+	0.357197i	$0.116267\pi$
$523$	8544.57i	0.714394i	−0.934029	+	0.357197i	$0.883733\pi$
$524$	858.750	0.0715928
$525$	0	0
$526$	13875.0	1.15015
$527$	12.1939i	0.00100792i
$528$	0	0
$529$	−13653.4	−1.12217
$530$	0	0
$531$	0	0
$532$	10342.5i	0.842869i
$533$	46150.3i	3.75045i
$534$	0	0
$535$	0	0
$536$	−4660.90	−0.375597
$537$	0	0
$538$	− 1030.43i	− 0.0825742i
$539$	24639.8	1.96904
$540$	0	0
$541$	10236.5	0.813499	0.406749	−	0.913540i	$-0.366662\pi$
$541$	10236.5	0.813499	0.406749	+	0.913540i	$0.366662\pi$
$542$	− 1304.82i	− 0.103408i
$543$	0	0
$544$	145.199	0.0114437
$545$	0	0
$546$	0	0
$547$	4785.31i	0.374050i	0.982355	+	0.187025i	$0.0598845\pi$
$547$	4785.31i	0.374050i	−0.982355	+	0.187025i	$0.940115\pi$
$548$	− 1471.20i	− 0.114683i
$549$	0	0
$550$	0	0
$551$	21163.3	1.63627
$552$	0	0
$553$	− 13344.3i	− 1.02614i
$554$	5877.70	0.450757
$555$	0	0
$556$	−5709.24	−0.435478
$557$	2031.75i	0.154557i	0.997010	+	0.0772783i	$0.0246230\pi$
$557$	2031.75i	0.154557i	−0.997010	+	0.0772783i	$0.975377\pi$
$558$	0	0
$559$	−27133.3	−2.05298
$560$	0	0
$561$	0	0
$562$	10703.5i	0.803384i
$563$	− 12162.4i	− 0.910449i	−0.890377	−	0.455224i	$-0.849559\pi$
$563$	− 12162.4i	− 0.910449i	0.890377	−	0.455224i	$-0.150441\pi$
$564$	0	0
$565$	0	0
$566$	−2959.90	−0.219812
$567$	0	0
$568$	− 4534.20i	− 0.334949i
$569$	144.753	0.0106650	0.00533249	−	0.999986i	$-0.498303\pi$
$569$	144.753	0.0106650	0.00533249	+	0.999986i	$0.498303\pi$
$570$	0	0
$571$	−7479.28	−0.548158	−0.274079	−	0.961707i	$-0.588373\pi$
$571$	−7479.28	−0.548158	−0.274079	+	0.961707i	$0.588373\pi$
$572$	17139.7i	1.25288i
$573$	0	0
$574$	−29609.8	−2.15312
$575$	0	0
$576$	0	0
$577$	6562.94i	0.473516i	0.971569	+	0.236758i	$0.0760848\pi$
$577$	6562.94i	0.473516i	−0.971569	+	0.236758i	$0.923915\pi$
$578$	9784.82i	0.704144i
$579$	0	0
$580$	0	0
$581$	−8892.24	−0.634961
$582$	0	0
$583$	− 11312.1i	− 0.803601i
$584$	−6715.10	−0.475810
$585$	0	0
$586$	10357.8	0.730165
$587$	4227.89i	0.297281i	0.988891	+	0.148640i	$0.0474897\pi$
$587$	4227.89i	0.297281i	−0.988891	+	0.148640i	$0.952510\pi$
$588$	0	0
$589$	235.247	0.0164570
$590$	0	0
$591$	0	0
$592$	329.799i	0.0228964i
$593$	28746.9i	1.99071i	0.0962641	+	0.995356i	$0.469311\pi$
$593$	28746.9i	1.99071i	−0.0962641	+	0.995356i	$0.530689\pi$
$594$	0	0
$595$	0	0
$596$	7030.95	0.483219
$597$	0	0
$598$	29590.6i	2.02349i
$599$	−21714.9	−1.48121	−0.740607	−	0.671939i	$-0.765462\pi$
$599$	−21714.9	−1.48121	−0.740607	+	0.671939i	$0.765462\pi$
$600$	0	0
$601$	6055.82	0.411018	0.205509	−	0.978655i	$-0.434115\pi$
$601$	6055.82	0.411018	0.205509	+	0.978655i	$0.434115\pi$
$602$	− 17408.6i	− 1.17861i
$603$	0	0
$604$	2151.10	0.144912
$605$	0	0
$606$	0	0
$607$	− 18260.0i	− 1.22100i	−0.792015	−	0.610502i	$-0.790968\pi$
$607$	− 18260.0i	− 1.22100i	0.792015	−	0.610502i	$-0.209032\pi$
$608$	− 2801.20i	− 0.186848i
$609$	0	0
$610$	0	0
$611$	−44102.7	−2.92014
$612$	0	0
$613$	− 16207.4i	− 1.06788i	−0.845521	−	0.533941i	$-0.820710\pi$
$613$	− 16207.4i	− 1.06788i	0.845521	−	0.533941i	$-0.179290\pi$
$614$	20612.0	1.35477
$615$	0	0
$616$	−10996.8	−0.719275
$617$	− 14003.1i	− 0.913686i	−0.889547	−	0.456843i	$-0.848980\pi$
$617$	− 14003.1i	− 0.913686i	0.889547	−	0.456843i	$-0.151020\pi$
$618$	0	0
$619$	269.684	0.0175113	0.00875565	−	0.999962i	$-0.497213\pi$
$619$	269.684	0.0175113	0.00875565	+	0.999962i	$0.497213\pi$
$620$	0	0
$621$	0	0
$622$	4125.29i	0.265931i
$623$	21838.5i	1.40440i
$624$	0	0
$625$	0	0
$626$	−18495.9	−1.18090
$627$	0	0
$628$	2687.15i	0.170747i
$629$	93.5284	0.00592881
$630$	0	0
$631$	−21240.2	−1.34003	−0.670016	−	0.742347i	$-0.733713\pi$
$631$	−21240.2	−1.34003	−0.670016	+	0.742347i	$0.733713\pi$
$632$	3614.20i	0.227477i
$633$	0	0
$634$	1220.69	0.0764668
$635$	0	0
$636$	0	0
$637$	− 48750.2i	− 3.03227i
$638$	22502.0i	1.39634i
$639$	0	0
$640$	0	0
$641$	−19210.9	−1.18375	−0.591875	−	0.806030i	$-0.701612\pi$
$641$	−19210.9	−1.18375	−0.591875	+	0.806030i	$0.701612\pi$
$642$	0	0
$643$	7731.07i	0.474158i	0.971490	+	0.237079i	$0.0761901\pi$
$643$	7731.07i	0.474158i	−0.971490	+	0.237079i	$0.923810\pi$
$644$	−18985.2	−1.16168
$645$	0	0
$646$	−794.399	−0.0483826
$647$	23347.4i	1.41867i	0.704870	+	0.709337i	$0.251005\pi$
$647$	23347.4i	1.41867i	−0.704870	+	0.709337i	$0.748995\pi$
$648$	0	0
$649$	17856.4	1.08001
$650$	0	0
$651$	0	0
$652$	10005.7i	0.601002i
$653$	− 7667.71i	− 0.459511i	−0.973248	−	0.229755i	$-0.926207\pi$
$653$	− 7667.71i	− 0.459511i	0.973248	−	0.229755i	$-0.0737926\pi$
$654$	0	0
$655$	0	0
$656$	8019.60	0.477306
$657$	0	0
$658$	− 28296.1i	− 1.67644i
$659$	−14673.6	−0.867378	−0.433689	−	0.901063i	$-0.642788\pi$
$659$	−14673.6	−0.867378	−0.433689	+	0.901063i	$0.642788\pi$
$660$	0	0
$661$	25446.9	1.49738	0.748691	−	0.662919i	$-0.230683\pi$
$661$	25446.9	1.49738	0.748691	+	0.662919i	$0.230683\pi$
$662$	− 4178.53i	− 0.245322i
$663$	0	0
$664$	2408.39	0.140759
$665$	0	0
$666$	0	0
$667$	38848.2i	2.25518i
$668$	− 3006.60i	− 0.174145i
$669$	0	0
$670$	0	0
$671$	−6171.45	−0.355061
$672$	0	0
$673$	− 12239.5i	− 0.701035i	−0.936556	−	0.350518i	$-0.886006\pi$
$673$	− 12239.5i	− 0.701035i	0.936556	−	0.350518i	$-0.113994\pi$
$674$	−18852.7	−1.07742
$675$	0	0
$676$	25123.2	1.42940
$677$	− 10457.5i	− 0.593672i	−0.954929	−	0.296836i	$-0.904069\pi$
$677$	− 10457.5i	− 0.593672i	0.954929	−	0.296836i	$-0.0959314\pi$
$678$	0	0
$679$	−33854.0	−1.91340
$680$	0	0
$681$	0	0
$682$	250.128i	0.0140438i
$683$	− 304.720i	− 0.0170714i	−0.999964	−	0.00853572i	$-0.997283\pi$
$683$	− 304.720i	− 0.0170714i	0.999964	−	0.00853572i	$-0.00271704\pi$
$684$	0	0
$685$	0	0
$686$	11015.3	0.613068
$687$	0	0
$688$	4715.00i	0.261276i
$689$	−22381.1	−1.23752
$690$	0	0
$691$	−14937.8	−0.822373	−0.411186	−	0.911551i	$-0.634885\pi$
$691$	−14937.8	−0.822373	−0.411186	+	0.911551i	$0.634885\pi$
$692$	9497.10i	0.521713i
$693$	0	0
$694$	2442.45	0.133594
$695$	0	0
$696$	0	0
$697$	− 2274.30i	− 0.123594i
$698$	8376.57i	0.454238i
$699$	0	0
$700$	0	0
$701$	−13752.9	−0.741000	−0.370500	−	0.928832i	$-0.620814\pi$
$701$	−13752.9	−0.741000	−0.370500	+	0.928832i	$0.620814\pi$
$702$	0	0
$703$	− 1804.36i	− 0.0968033i
$704$	2978.40	0.159450
$705$	0	0
$706$	−8938.73	−0.476506
$707$	15107.3i	0.803634i
$708$	0	0
$709$	31225.5	1.65402	0.827008	−	0.562190i	$-0.190041\pi$
$709$	31225.5	1.65402	0.827008	+	0.562190i	$0.190041\pi$
$710$	0	0
$711$	0	0
$712$	− 5914.79i	− 0.311329i
$713$	431.829i	0.0226818i
$714$	0	0
$715$	0	0
$716$	9698.09	0.506194
$717$	0	0
$718$	− 1287.14i	− 0.0669022i
$719$	13945.5	0.723337	0.361669	−	0.932307i	$-0.382207\pi$
$719$	13945.5	0.723337	0.361669	+	0.932307i	$0.382207\pi$
$720$	0	0
$721$	−12450.4	−0.643103
$722$	1607.62i	0.0828662i
$723$	0	0
$724$	−4717.84	−0.242179
$725$	0	0
$726$	0	0
$727$	37585.1i	1.91741i	0.284407	+	0.958704i	$0.408203\pi$
$727$	37585.1i	1.91741i	−0.284407	+	0.958704i	$0.591797\pi$
$728$	21757.3i	1.10766i
$729$	0	0
$730$	0	0
$731$	1337.14	0.0676550
$732$	0	0
$733$	2848.70i	0.143546i	0.997421	+	0.0717729i	$0.0228657\pi$
$733$	2848.70i	0.143546i	−0.997421	+	0.0717729i	$0.977134\pi$
$734$	−11869.5	−0.596880
$735$	0	0
$736$	5142.00	0.257522
$737$	27113.3i	1.35513i
$738$	0	0
$739$	−1828.99	−0.0910428	−0.0455214	−	0.998963i	$-0.514495\pi$
$739$	−1828.99	−0.0910428	−0.0455214	+	0.998963i	$0.514495\pi$
$740$	0	0
$741$	0	0
$742$	− 14359.6i	− 0.710457i
$743$	− 32798.9i	− 1.61948i	−0.586789	−	0.809740i	$-0.699608\pi$
$743$	− 32798.9i	− 1.61948i	0.586789	−	0.809740i	$-0.300392\pi$
$744$	0	0
$745$	0	0
$746$	12375.7	0.607379
$747$	0	0
$748$	− 844.651i	− 0.0412881i
$749$	48426.7	2.36245
$750$	0	0
$751$	−16225.2	−0.788371	−0.394186	−	0.919031i	$-0.628973\pi$
$751$	−16225.2	−0.788371	−0.394186	+	0.919031i	$0.628973\pi$
$752$	7663.80i	0.371635i
$753$	0	0
$754$	44520.5	2.15032
$755$	0	0
$756$	0	0
$757$	14461.0i	0.694312i	0.937807	+	0.347156i	$0.112853\pi$
$757$	14461.0i	0.694312i	−0.937807	+	0.347156i	$0.887147\pi$
$758$	− 20857.1i	− 0.999423i
$759$	0	0
$760$	0	0
$761$	−22060.5	−1.05084	−0.525422	−	0.850842i	$-0.676093\pi$
$761$	−22060.5	−1.05084	−0.525422	+	0.850842i	$0.676093\pi$
$762$	0	0
$763$	20394.9i	0.967687i
$764$	−1377.90	−0.0652496
$765$	0	0
$766$	−20493.4	−0.966652
$767$	− 35329.2i	− 1.66318i
$768$	0	0
$769$	1675.74	0.0785810	0.0392905	−	0.999228i	$-0.487490\pi$
$769$	1675.74	0.0785810	0.0392905	+	0.999228i	$0.487490\pi$
$770$	0	0
$771$	0	0
$772$	− 7767.35i	− 0.362115i
$773$	− 10878.6i	− 0.506179i	−0.967443	−	0.253089i	$-0.918553\pi$
$773$	− 10878.6i	− 0.506179i	0.967443	−	0.253089i	$-0.0814467\pi$
$774$	0	0
$775$	0	0
$776$	9169.10	0.424164
$777$	0	0
$778$	− 17017.0i	− 0.784174i
$779$	−43876.0	−2.01800
$780$	0	0
$781$	−26376.3	−1.20847
$782$	− 1458.23i	− 0.0666831i
$783$	0	0
$784$	−8471.40	−0.385906
$785$	0	0
$786$	0	0
$787$	27293.2i	1.23621i	0.786096	+	0.618105i	$0.212099\pi$
$787$	27293.2i	1.23621i	−0.786096	+	0.618105i	$0.787901\pi$
$788$	7123.80i	0.322049i
$789$	0	0
$790$	0	0
$791$	28658.4	1.28821
$792$	0	0
$793$	12210.3i	0.546784i
$794$	7551.93	0.337541
$795$	0	0
$796$	−3233.50	−0.143980
$797$	38925.4i	1.73000i	0.501772	+	0.865000i	$0.332682\pi$
$797$	38925.4i	1.73000i	−0.501772	+	0.865000i	$0.667318\pi$
$798$	0	0
$799$	2173.39	0.0962317
$800$	0	0
$801$	0	0
$802$	− 15112.0i	− 0.665367i
$803$	39063.0i	1.71669i
$804$	0	0
$805$	0	0
$806$	494.881	0.0216271
$807$	0	0
$808$	− 4091.70i	− 0.178150i
$809$	−22806.9	−0.991159	−0.495579	−	0.868563i	$-0.665044\pi$
$809$	−22806.9	−0.991159	−0.495579	+	0.868563i	$0.665044\pi$
$810$	0	0
$811$	−42523.0	−1.84117	−0.920583	−	0.390547i	$-0.872286\pi$
$811$	−42523.0	−1.84117	−0.920583	+	0.390547i	$0.872286\pi$
$812$	28564.2i	1.23449i
$813$	0	0
$814$	1918.50	0.0826086
$815$	0	0
$816$	0	0
$817$	− 25796.2i	− 1.10464i
$818$	− 17926.4i	− 0.766238i
$819$	0	0
$820$	0	0
$821$	−32644.4	−1.38770	−0.693848	−	0.720122i	$-0.744086\pi$
$821$	−32644.4	−1.38770	−0.693848	+	0.720122i	$0.744086\pi$
$822$	0	0
$823$	4471.81i	0.189402i	0.995506	+	0.0947008i	$0.0301894\pi$
$823$	4471.81i	0.189402i	−0.995506	+	0.0947008i	$0.969811\pi$
$824$	3372.09	0.142564
$825$	0	0
$826$	22667.1	0.954828
$827$	29667.5i	1.24745i	0.781644	+	0.623725i	$0.214381\pi$
$827$	29667.5i	1.24745i	−0.781644	+	0.623725i	$0.785619\pi$
$828$	0	0
$829$	−5130.74	−0.214955	−0.107478	−	0.994207i	$-0.534277\pi$
$829$	−5130.74	−0.214955	−0.107478	+	0.994207i	$0.534277\pi$
$830$	0	0
$831$	0	0
$832$	− 5892.80i	− 0.245548i
$833$	2402.42i	0.0999268i
$834$	0	0
$835$	0	0
$836$	−16295.1	−0.674136
$837$	0	0
$838$	− 28307.6i	− 1.16691i
$839$	24205.9	0.996042	0.498021	−	0.867165i	$-0.334060\pi$
$839$	24205.9	0.996042	0.498021	+	0.867165i	$0.334060\pi$
$840$	0	0
$841$	34060.0	1.39653
$842$	− 29454.6i	− 1.20555i
$843$	0	0
$844$	−4892.15	−0.199520
$845$	0	0
$846$	0	0
$847$	24656.0i	1.00023i
$848$	3889.20i	0.157495i
$849$	0	0
$850$	0	0
$851$	3312.16	0.133419
$852$	0	0
$853$	21843.2i	0.876784i	0.898784	+	0.438392i	$0.144452\pi$
$853$	21843.2i	0.876784i	−0.898784	+	0.438392i	$0.855548\pi$
$854$	−7834.07	−0.313907
$855$	0	0
$856$	−13116.0	−0.523710
$857$	2770.50i	0.110430i	0.998474	+	0.0552151i	$0.0175844\pi$
$857$	2770.50i	0.110430i	−0.998474	+	0.0552151i	$0.982416\pi$
$858$	0	0
$859$	18439.6	0.732421	0.366211	−	0.930532i	$-0.380655\pi$
$859$	18439.6	0.732421	0.366211	+	0.930532i	$0.380655\pi$
$860$	0	0
$861$	0	0
$862$	− 6097.95i	− 0.240948i
$863$	− 22073.1i	− 0.870659i	−0.900271	−	0.435329i	$-0.856632\pi$
$863$	− 22073.1i	− 0.870659i	0.900271	−	0.435329i	$-0.143368\pi$
$864$	0	0
$865$	0	0
$866$	−34436.8	−1.35128
$867$	0	0
$868$	317.514i	0.0124160i
$869$	21024.5	0.820721
$870$	0	0
$871$	53644.0	2.08686
$872$	− 5523.80i	− 0.214518i
$873$	0	0
$874$	−28132.3	−1.08878
$875$	0	0
$876$	0	0
$877$	8446.67i	0.325227i	0.986690	+	0.162613i	$0.0519923\pi$
$877$	8446.67i	0.325227i	−0.986690	+	0.162613i	$0.948008\pi$
$878$	− 3335.81i	− 0.128221i
$879$	0	0
$880$	0	0
$881$	16665.8	0.637325	0.318663	−	0.947868i	$-0.396766\pi$
$881$	16665.8	0.637325	0.318663	+	0.947868i	$0.396766\pi$
$882$	0	0
$883$	37742.6i	1.43844i	0.694784	+	0.719219i	$0.255500\pi$
$883$	37742.6i	1.43844i	−0.694784	+	0.719219i	$0.744500\pi$
$884$	−1671.15	−0.0635825
$885$	0	0
$886$	−18446.4	−0.699457
$887$	− 19447.2i	− 0.736158i	−0.929795	−	0.368079i	$-0.880016\pi$
$887$	− 19447.2i	− 0.736158i	0.929795	−	0.368079i	$-0.119984\pi$
$888$	0	0
$889$	23778.4	0.897076
$890$	0	0
$891$	0	0
$892$	− 4244.01i	− 0.159305i
$893$	− 41929.3i	− 1.57123i
$894$	0	0
$895$	0	0
$896$	3780.80	0.140968
$897$	0	0
$898$	− 5272.35i	− 0.195925i
$899$	649.708	0.0241034
$900$	0	0
$901$	1102.95	0.0407819
$902$	− 46651.5i	− 1.72209i
$903$	0	0
$904$	−7761.90	−0.285572
$905$	0	0
$906$	0	0
$907$	− 15428.9i	− 0.564838i	−0.959291	−	0.282419i	$-0.908863\pi$
$907$	− 15428.9i	− 0.564838i	0.959291	−	0.282419i	$-0.0911369\pi$
$908$	14292.3i	0.522364i
$909$	0	0
$910$	0	0
$911$	−688.190	−0.0250283	−0.0125141	−	0.999922i	$-0.503983\pi$
$911$	−688.190	−0.0250283	−0.0125141	+	0.999922i	$0.503983\pi$
$912$	0	0
$913$	− 14010.1i	− 0.507849i
$914$	−7719.35	−0.279358
$915$	0	0
$916$	4007.30	0.144547
$917$	− 6341.32i	− 0.228363i
$918$	0	0
$919$	−6809.05	−0.244407	−0.122203	−	0.992505i	$-0.538996\pi$
$919$	−6809.05	−0.244407	−0.122203	+	0.992505i	$0.538996\pi$
$920$	0	0
$921$	0	0
$922$	13230.1i	0.472572i
$923$	52185.8i	1.86101i
$924$	0	0
$925$	0	0
$926$	35143.5	1.24718
$927$	0	0
$928$	− 7736.39i	− 0.273663i
$929$	47682.2	1.68396	0.841982	−	0.539506i	$-0.181389\pi$
$929$	47682.2	1.68396	0.841982	+	0.539506i	$0.181389\pi$
$930$	0	0
$931$	46347.8	1.63157
$932$	− 24329.7i	− 0.855091i
$933$	0	0
$934$	20153.7	0.706048
$935$	0	0
$936$	0	0
$937$	30516.3i	1.06395i	0.846759	+	0.531977i	$0.178551\pi$
$937$	30516.3i	1.06395i	−0.846759	+	0.531977i	$0.821449\pi$
$938$	34417.8i	1.19806i
$939$	0	0
$940$	0	0
$941$	−31348.8	−1.08602	−0.543009	−	0.839727i	$-0.682715\pi$
$941$	−31348.8	−1.08602	−0.543009	+	0.839727i	$0.682715\pi$
$942$	0	0
$943$	− 80540.5i	− 2.78129i
$944$	−6139.20	−0.211667
$945$	0	0
$946$	27428.0	0.942666
$947$	− 34598.6i	− 1.18723i	−0.804750	−	0.593614i	$-0.797701\pi$
$947$	− 34598.6i	− 1.18723i	0.804750	−	0.593614i	$-0.202299\pi$
$948$	0	0
$949$	77286.6	2.64365
$950$	0	0
$951$	0	0
$952$	− 1072.20i	− 0.0365025i
$953$	26766.7i	0.909820i	0.890537	+	0.454910i	$0.150329\pi$
$953$	26766.7i	0.909820i	−0.890537	+	0.454910i	$0.849671\pi$
$954$	0	0
$955$	0	0
$956$	−849.594	−0.0287425
$957$	0	0
$958$	35891.5i	1.21044i
$959$	−10863.9	−0.365810
$960$	0	0
$961$	−29783.8	−0.999758
$962$	− 3795.78i	− 0.127215i
$963$	0	0
$964$	11777.8	0.393503
$965$	0	0
$966$	0	0
$967$	− 8764.26i	− 0.291458i	−0.989325	−	0.145729i	$-0.953447\pi$
$967$	− 8764.26i	− 0.291458i	0.989325	−	0.145729i	$-0.0465527\pi$
$968$	− 6677.89i	− 0.221731i
$969$	0	0
$970$	0	0
$971$	35417.3	1.17054	0.585270	−	0.810838i	$-0.300988\pi$
$971$	35417.3	1.17054	0.585270	+	0.810838i	$0.300988\pi$
$972$	0	0
$973$	42159.2i	1.38907i
$974$	34713.8	1.14199
$975$	0	0
$976$	2121.80	0.0695872
$977$	− 26629.0i	− 0.871993i	−0.899948	−	0.435997i	$-0.856396\pi$
$977$	− 26629.0i	− 0.871993i	0.899948	−	0.435997i	$-0.143604\pi$
$978$	0	0
$979$	−34407.4	−1.12326
$980$	0	0
$981$	0	0
$982$	− 16786.3i	− 0.545493i
$983$	− 16809.2i	− 0.545402i	−0.962099	−	0.272701i	$-0.912083\pi$
$983$	− 16809.2i	− 0.545402i	0.962099	−	0.272701i	$-0.0879169\pi$
$984$	0	0
$985$	0	0
$986$	−2193.98	−0.0708627
$987$	0	0
$988$	32240.0i	1.03815i
$989$	47352.5	1.52247
$990$	0	0
$991$	53758.7	1.72321	0.861606	−	0.507578i	$-0.169459\pi$
$991$	53758.7	1.72321	0.861606	+	0.507578i	$0.169459\pi$
$992$	− 85.9963i	− 0.00275240i
$993$	0	0
$994$	−33482.2	−1.06840
$995$	0	0
$996$	0	0
$997$	− 41575.1i	− 1.32066i	−0.750976	−	0.660330i	$-0.770417\pi$
$997$	− 41575.1i	− 1.32066i	0.750976	−	0.660330i	$-0.229583\pi$
$998$	− 32842.3i	− 1.04169i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.u.649.4		4
3.2	odd	2		1350.4.c.bb.649.2		4
5.2	odd	4		270.4.a.m.1.1	✓	2
5.3	odd	4		1350.4.a.bm.1.2		2
5.4	even	2	inner	1350.4.c.u.649.1		4
15.2	even	4		270.4.a.n.1.1	yes	2
15.8	even	4		1350.4.a.bf.1.2		2
15.14	odd	2		1350.4.c.bb.649.3		4
20.7	even	4		2160.4.a.w.1.2		2
45.2	even	12		810.4.e.z.271.2		4
45.7	odd	12		810.4.e.bd.271.2		4
45.22	odd	12		810.4.e.bd.541.2		4
45.32	even	12		810.4.e.z.541.2		4
60.47	odd	4		2160.4.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.m.1.1	✓	2	5.2	odd	4
270.4.a.n.1.1	yes	2	15.2	even	4
810.4.e.z.271.2		4	45.2	even	12
810.4.e.z.541.2		4	45.32	even	12
810.4.e.bd.271.2		4	45.7	odd	12
810.4.e.bd.541.2		4	45.22	odd	12
1350.4.a.bf.1.2		2	15.8	even	4
1350.4.a.bm.1.2		2	5.3	odd	4
1350.4.c.u.649.1		4	5.4	even	2	inner
1350.4.c.u.649.4		4	1.1	even	1	trivial
1350.4.c.bb.649.2		4	3.2	odd	2
1350.4.c.bb.649.3		4	15.14	odd	2
2160.4.a.w.1.2		2	20.7	even	4
2160.4.a.bb.1.2		2	60.47	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +29.5375i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +29.5375i q^{7} -8.00000i q^{8} -46.5375 q^{11} +92.0750i q^{13} -59.0750 q^{14} +16.0000 q^{16} -4.53748i q^{17} -87.5375 q^{19} -93.0750i q^{22} -160.687i q^{23} -184.150 q^{26} -118.150i q^{28} -241.762 q^{29} -2.68738 q^{31} +32.0000i q^{32} +9.07495 q^{34} +20.6124i q^{37} -175.075i q^{38} +501.225 q^{41} +294.687i q^{43} +186.150 q^{44} +321.375 q^{46} +478.987i q^{47} -529.463 q^{49} -368.300i q^{52} +243.075i q^{53} +236.300 q^{56} -483.525i q^{58} -383.700 q^{59} +132.612 q^{61} -5.37477i q^{62} -64.0000 q^{64} -582.612i q^{67} +18.1499i q^{68} +566.775 q^{71} -839.388i q^{73} -41.2249 q^{74} +350.150 q^{76} -1374.60i q^{77} -451.775 q^{79} +1002.45i q^{82} +301.049i q^{83} -589.375 q^{86} +372.300i q^{88} +739.349 q^{89} -2719.66 q^{91} +642.750i q^{92} -957.974 q^{94} +1146.14i q^{97} -1058.93i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} - 66 q^{11} + 4 q^{14} + 64 q^{16} - 230 q^{19} - 256 q^{26} - 126 q^{29} + 590 q^{31} - 204 q^{34} + 1284 q^{41} + 264 q^{44} + 84 q^{46} - 2238 q^{49} - 16 q^{56} - 2496 q^{59} + 170 q^{61} - 256 q^{64} + 2988 q^{71} + 556 q^{74} + 920 q^{76} - 2528 q^{79} - 1156 q^{86} - 1368 q^{89} - 7154 q^{91} - 708 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 - 66 * q^11 + 4 * q^14 + 64 * q^16 - 230 * q^19 - 256 * q^26 - 126 * q^29 + 590 * q^31 - 204 * q^34 + 1284 * q^41 + 264 * q^44 + 84 * q^46 - 2238 * q^49 - 16 * q^56 - 2496 * q^59 + 170 * q^61 - 256 * q^64 + 2988 * q^71 + 556 * q^74 + 920 * q^76 - 2528 * q^79 - 1156 * q^86 - 1368 * q^89 - 7154 * q^91 - 708 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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