LMFDB - Embedded newform 1350.4.c.z.649.3

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:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.z.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} -4.69694i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} -4.69694i q^{7} -8.00000i q^{8} -14.3939 q^{11} -68.4847i q^{13} +9.39388 q^{14} +16.0000 q^{16} -55.4847i q^{17} -57.4847 q^{19} -28.7878i q^{22} +64.4847i q^{23} +136.969 q^{26} +18.7878i q^{28} -133.485 q^{29} -24.9694 q^{31} +32.0000i q^{32} +110.969 q^{34} +160.909i q^{37} -114.969i q^{38} +121.818 q^{41} -57.1214i q^{43} +57.5755 q^{44} -128.969 q^{46} +433.393i q^{47} +320.939 q^{49} +273.939i q^{52} +310.515i q^{53} -37.5755 q^{56} -266.969i q^{58} -40.7571 q^{59} -8.54592 q^{61} -49.9388i q^{62} -64.0000 q^{64} +474.091i q^{67} +221.939i q^{68} +334.546 q^{71} -518.182i q^{73} -321.818 q^{74} +229.939 q^{76} +67.6072i q^{77} -951.362 q^{79} +243.637i q^{82} +12.1225i q^{83} +114.243 q^{86} +115.151i q^{88} +462.272 q^{89} -321.668 q^{91} -257.939i q^{92} -866.786 q^{94} +449.243i q^{97} +641.878i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} + 60 q^{11} - 80 q^{14} + 64 q^{16} + 64 q^{19} - 40 q^{26} - 240 q^{29} + 488 q^{31} - 144 q^{34} + 840 q^{41} - 240 q^{44} + 72 q^{46} + 108 q^{49} + 320 q^{56} + 660 q^{59} - 916 q^{61} - 256 q^{64} + 2220 q^{71} - 1640 q^{74} - 256 q^{76} - 1160 q^{79} + 1280 q^{86} + 1320 q^{89} - 4520 q^{91} + 648 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 + 60 * q^11 - 80 * q^14 + 64 * q^16 + 64 * q^19 - 40 * q^26 - 240 * q^29 + 488 * q^31 - 144 * q^34 + 840 * q^41 - 240 * q^44 + 72 * q^46 + 108 * q^49 + 320 * q^56 + 660 * q^59 - 916 * q^61 - 256 * q^64 + 2220 * q^71 - 1640 * q^74 - 256 * q^76 - 1160 * q^79 + 1280 * q^86 + 1320 * q^89 - 4520 * q^91 + 648 * 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
$7$	− 4.69694i	− 0.253611i	−0.991928	−	0.126805i	$-0.959528\pi$
$7$	− 4.69694i	− 0.253611i	0.991928	−	0.126805i	$-0.0404724\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−14.3939	−0.394538	−0.197269	−	0.980349i	$-0.563207\pi$
$11$	−14.3939	−0.394538	−0.197269	+	0.980349i	$0.563207\pi$
$12$	0	0
$13$	− 68.4847i	− 1.46110i	−0.682862	−	0.730548i	$-0.739265\pi$
$13$	− 68.4847i	− 1.46110i	0.682862	−	0.730548i	$-0.260735\pi$
$14$	9.39388	0.179330
$15$	0	0
$16$	16.0000	0.250000
$17$	− 55.4847i	− 0.791589i	−0.918339	−	0.395795i	$-0.870469\pi$
$17$	− 55.4847i	− 0.791589i	0.918339	−	0.395795i	$-0.129531\pi$
$18$	0	0
$19$	−57.4847	−0.694100	−0.347050	−	0.937847i	$-0.612817\pi$
$19$	−57.4847	−0.694100	−0.347050	+	0.937847i	$0.612817\pi$
$20$	0	0
$21$	0	0
$22$	− 28.7878i	− 0.278980i
$23$	64.4847i	0.584608i	0.956325	+	0.292304i	$0.0944219\pi$
$23$	64.4847i	0.584608i	−0.956325	+	0.292304i	$0.905578\pi$
$24$	0	0
$25$	0	0
$26$	136.969	1.03315
$27$	0	0
$28$	18.7878i	0.126805i
$29$	−133.485	−0.854741	−0.427370	−	0.904077i	$-0.640560\pi$
$29$	−133.485	−0.854741	−0.427370	+	0.904077i	$0.640560\pi$
$30$	0	0
$31$	−24.9694	−0.144666	−0.0723328	−	0.997381i	$-0.523044\pi$
$31$	−24.9694	−0.144666	−0.0723328	+	0.997381i	$0.523044\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	110.969	0.559738
$35$	0	0
$36$	0	0
$37$	160.909i	0.714954i	0.933922	+	0.357477i	$0.116363\pi$
$37$	160.909i	0.714954i	−0.933922	+	0.357477i	$0.883637\pi$
$38$	− 114.969i	− 0.490803i
$39$	0	0
$40$	0	0
$41$	121.818	0.464020	0.232010	−	0.972713i	$-0.425470\pi$
$41$	121.818	0.464020	0.232010	+	0.972713i	$0.425470\pi$
$42$	0	0
$43$	− 57.1214i	− 0.202580i	−0.994857	−	0.101290i	$-0.967703\pi$
$43$	− 57.1214i	− 0.202580i	0.994857	−	0.101290i	$-0.0322970\pi$
$44$	57.5755	0.197269
$45$	0	0
$46$	−128.969	−0.413380
$47$	433.393i	1.34504i	0.740079	+	0.672520i	$0.234788\pi$
$47$	433.393i	1.34504i	−0.740079	+	0.672520i	$0.765212\pi$
$48$	0	0
$49$	320.939	0.935682
$50$	0	0
$51$	0	0
$52$	273.939i	0.730548i
$53$	310.515i	0.804765i	0.915472	+	0.402383i	$0.131818\pi$
$53$	310.515i	0.804765i	−0.915472	+	0.402383i	$0.868182\pi$
$54$	0	0
$55$	0	0
$56$	−37.5755	−0.0896650
$57$	0	0
$58$	− 266.969i	− 0.604393i
$59$	−40.7571	−0.0899344	−0.0449672	−	0.998988i	$-0.514318\pi$
$59$	−40.7571	−0.0899344	−0.0449672	+	0.998988i	$0.514318\pi$
$60$	0	0
$61$	−8.54592	−0.0179376	−0.00896880	−	0.999960i	$-0.502855\pi$
$61$	−8.54592	−0.0179376	−0.00896880	+	0.999960i	$0.502855\pi$
$62$	− 49.9388i	− 0.102294i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	474.091i	0.864469i	0.901761	+	0.432234i	$0.142275\pi$
$67$	474.091i	0.864469i	−0.901761	+	0.432234i	$0.857725\pi$
$68$	221.939i	0.395795i
$69$	0	0
$70$	0	0
$71$	334.546	0.559201	0.279601	−	0.960116i	$-0.409798\pi$
$71$	334.546	0.559201	0.279601	+	0.960116i	$0.409798\pi$
$72$	0	0
$73$	− 518.182i	− 0.830802i	−0.909638	−	0.415401i	$-0.863641\pi$
$73$	− 518.182i	− 0.830802i	0.909638	−	0.415401i	$-0.136359\pi$
$74$	−321.818	−0.505549
$75$	0	0
$76$	229.939	0.347050
$77$	67.6072i	0.100059i
$78$	0	0
$79$	−951.362	−1.35489	−0.677447	−	0.735572i	$-0.736914\pi$
$79$	−951.362	−1.35489	−0.677447	+	0.735572i	$0.736914\pi$
$80$	0	0
$81$	0	0
$82$	243.637i	0.328112i
$83$	12.1225i	0.0160315i	0.999968	+	0.00801574i	$0.00255152\pi$
$83$	12.1225i	0.0160315i	−0.999968	+	0.00801574i	$0.997448\pi$
$84$	0	0
$85$	0	0
$86$	114.243	0.143246
$87$	0	0
$88$	115.151i	0.139490i
$89$	462.272	0.550571	0.275285	−	0.961363i	$-0.411228\pi$
$89$	462.272	0.550571	0.275285	+	0.961363i	$0.411228\pi$
$90$	0	0
$91$	−321.668	−0.370550
$92$	− 257.939i	− 0.292304i
$93$	0	0
$94$	−866.786	−0.951086
$95$	0	0
$96$	0	0
$97$	449.243i	0.470244i	0.971966	+	0.235122i	$0.0755490\pi$
$97$	449.243i	0.470244i	−0.971966	+	0.235122i	$0.924451\pi$
$98$	641.878i	0.661627i
$99$	0	0
$100$	0	0
$101$	1554.69	1.53166	0.765831	−	0.643042i	$-0.222328\pi$
$101$	1554.69	1.53166	0.765831	+	0.643042i	$0.222328\pi$
$102$	0	0
$103$	− 264.547i	− 0.253074i	−0.991962	−	0.126537i	$-0.959614\pi$
$103$	− 264.547i	− 0.253074i	0.991962	−	0.126537i	$-0.0403862\pi$
$104$	−547.878	−0.516575
$105$	0	0
$106$	−621.031	−0.569055
$107$	2063.79i	1.86461i	0.361668	+	0.932307i	$0.382207\pi$
$107$	2063.79i	1.86461i	−0.361668	+	0.932307i	$0.617793\pi$
$108$	0	0
$109$	−896.061	−0.787405	−0.393702	−	0.919238i	$-0.628806\pi$
$109$	−896.061	−0.787405	−0.393702	+	0.919238i	$0.628806\pi$
$110$	0	0
$111$	0	0
$112$	− 75.1510i	− 0.0634027i
$113$	668.847i	0.556813i	0.960463	+	0.278406i	$0.0898062\pi$
$113$	668.847i	0.556813i	−0.960463	+	0.278406i	$0.910194\pi$
$114$	0	0
$115$	0	0
$116$	533.939	0.427370
$117$	0	0
$118$	− 81.5143i	− 0.0635932i
$119$	−260.608	−0.200756
$120$	0	0
$121$	−1123.82	−0.844340
$122$	− 17.0918i	− 0.0126838i
$123$	0	0
$124$	99.8775	0.0723328
$125$	0	0
$126$	0	0
$127$	2384.54i	1.66609i	0.553202	+	0.833047i	$0.313406\pi$
$127$	2384.54i	1.66609i	−0.553202	+	0.833047i	$0.686594\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−340.151	−0.226864	−0.113432	−	0.993546i	$-0.536184\pi$
$131$	−340.151	−0.226864	−0.113432	+	0.993546i	$0.536184\pi$
$132$	0	0
$133$	270.002i	0.176031i
$134$	−948.182	−0.611272
$135$	0	0
$136$	−443.878	−0.279869
$137$	− 1600.09i	− 0.997844i	−0.866647	−	0.498922i	$-0.833729\pi$
$137$	− 1600.09i	− 0.997844i	0.866647	−	0.498922i	$-0.166271\pi$
$138$	0	0
$139$	−123.730	−0.0755008	−0.0377504	−	0.999287i	$-0.512019\pi$
$139$	−123.730	−0.0755008	−0.0377504	+	0.999287i	$0.512019\pi$
$140$	0	0
$141$	0	0
$142$	669.092i	0.395415i
$143$	985.760i	0.576457i
$144$	0	0
$145$	0	0
$146$	1036.36	0.587466
$147$	0	0
$148$	− 643.637i	− 0.357477i
$149$	986.363	0.542322	0.271161	−	0.962534i	$-0.412592\pi$
$149$	986.363	0.542322	0.271161	+	0.962534i	$0.412592\pi$
$150$	0	0
$151$	−2978.27	−1.60509	−0.802544	−	0.596593i	$-0.796521\pi$
$151$	−2978.27	−1.60509	−0.802544	+	0.596593i	$0.796521\pi$
$152$	459.878i	0.245401i
$153$	0	0
$154$	−135.214	−0.0707525
$155$	0	0
$156$	0	0
$157$	3515.76i	1.78718i	0.448880	+	0.893592i	$0.351823\pi$
$157$	3515.76i	1.78718i	−0.448880	+	0.893592i	$0.648177\pi$
$158$	− 1902.72i	− 0.958055i
$159$	0	0
$160$	0	0
$161$	302.881	0.148263
$162$	0	0
$163$	897.573i	0.431309i	0.976470	+	0.215655i	$0.0691885\pi$
$163$	897.573i	0.431309i	−0.976470	+	0.215655i	$0.930811\pi$
$164$	−487.273	−0.232010
$165$	0	0
$166$	−24.2449	−0.0113360
$167$	526.423i	0.243927i	0.992535	+	0.121964i	$0.0389191\pi$
$167$	526.423i	0.243927i	−0.992535	+	0.121964i	$0.961081\pi$
$168$	0	0
$169$	−2493.15	−1.13480
$170$	0	0
$171$	0	0
$172$	228.486i	0.101290i
$173$	853.056i	0.374894i	0.982275	+	0.187447i	$0.0600213\pi$
$173$	853.056i	0.374894i	−0.982275	+	0.187447i	$0.939979\pi$
$174$	0	0
$175$	0	0
$176$	−230.302	−0.0986345
$177$	0	0
$178$	924.545i	0.389312i
$179$	−692.571	−0.289191	−0.144596	−	0.989491i	$-0.546188\pi$
$179$	−692.571	−0.289191	−0.144596	+	0.989491i	$0.546188\pi$
$180$	0	0
$181$	−3176.24	−1.30435	−0.652177	−	0.758067i	$-0.726144\pi$
$181$	−3176.24	−1.30435	−0.652177	+	0.758067i	$0.726144\pi$
$182$	− 643.337i	− 0.262018i
$183$	0	0
$184$	515.878	0.206690
$185$	0	0
$186$	0	0
$187$	798.640i	0.312312i
$188$	− 1733.57i	− 0.672520i
$189$	0	0
$190$	0	0
$191$	−1868.79	−0.707961	−0.353981	−	0.935253i	$-0.615172\pi$
$191$	−1868.79	−0.707961	−0.353981	+	0.935253i	$0.615172\pi$
$192$	0	0
$193$	− 4413.63i	− 1.64612i	−0.567958	−	0.823058i	$-0.692266\pi$
$193$	− 4413.63i	− 1.64612i	0.567958	−	0.823058i	$-0.307734\pi$
$194$	−898.486	−0.332513
$195$	0	0
$196$	−1283.76	−0.467841
$197$	2172.12i	0.785570i	0.919630	+	0.392785i	$0.128488\pi$
$197$	2172.12i	0.785570i	−0.919630	+	0.392785i	$0.871512\pi$
$198$	0	0
$199$	−3769.51	−1.34278	−0.671391	−	0.741104i	$-0.734303\pi$
$199$	−3769.51	−1.34278	−0.671391	+	0.741104i	$0.734303\pi$
$200$	0	0
$201$	0	0
$202$	3109.39i	1.08305i
$203$	626.969i	0.216772i
$204$	0	0
$205$	0	0
$206$	529.094	0.178950
$207$	0	0
$208$	− 1095.76i	− 0.365274i
$209$	827.428	0.273849
$210$	0	0
$211$	1186.60	0.387152	0.193576	−	0.981085i	$-0.437991\pi$
$211$	1186.60	0.387152	0.193576	+	0.981085i	$0.437991\pi$
$212$	− 1242.06i	− 0.402383i
$213$	0	0
$214$	−4127.57	−1.31848
$215$	0	0
$216$	0	0
$217$	117.280i	0.0366888i
$218$	− 1792.12i	− 0.556779i
$219$	0	0
$220$	0	0
$221$	−3799.85	−1.15659
$222$	0	0
$223$	− 32.8765i	− 0.00987253i	−0.999988	−	0.00493626i	$-0.998429\pi$
$223$	− 32.8765i	− 0.00987253i	0.999988	−	0.00493626i	$-0.00157127\pi$
$224$	150.302	0.0448325
$225$	0	0
$226$	−1337.69	−0.393726
$227$	713.908i	0.208739i	0.994539	+	0.104369i	$0.0332825\pi$
$227$	713.908i	0.208739i	−0.994539	+	0.104369i	$0.966718\pi$
$228$	0	0
$229$	−2777.75	−0.801567	−0.400784	−	0.916173i	$-0.631262\pi$
$229$	−2777.75	−0.801567	−0.400784	+	0.916173i	$0.631262\pi$
$230$	0	0
$231$	0	0
$232$	1067.88i	0.302197i
$233$	5667.54i	1.59353i	0.604289	+	0.796765i	$0.293457\pi$
$233$	5667.54i	1.59353i	−0.604289	+	0.796765i	$0.706543\pi$
$234$	0	0
$235$	0	0
$236$	163.029	0.0449672
$237$	0	0
$238$	− 521.216i	− 0.141956i
$239$	5786.66	1.56614	0.783071	−	0.621932i	$-0.213652\pi$
$239$	5786.66	1.56614	0.783071	+	0.621932i	$0.213652\pi$
$240$	0	0
$241$	−1561.43	−0.417346	−0.208673	−	0.977985i	$-0.566914\pi$
$241$	−1561.43	−0.417346	−0.208673	+	0.977985i	$0.566914\pi$
$242$	− 2247.63i	− 0.597038i
$243$	0	0
$244$	34.1837	0.00896880
$245$	0	0
$246$	0	0
$247$	3936.82i	1.01415i
$248$	199.755i	0.0511470i
$249$	0	0
$250$	0	0
$251$	4708.32	1.18401	0.592005	−	0.805934i	$-0.298336\pi$
$251$	4708.32	1.18401	0.592005	+	0.805934i	$0.298336\pi$
$252$	0	0
$253$	− 928.185i	− 0.230650i
$254$	−4769.09	−1.17811
$255$	0	0
$256$	256.000	0.0625000
$257$	− 2086.58i	− 0.506448i	−0.967408	−	0.253224i	$-0.918509\pi$
$257$	− 2086.58i	− 0.506448i	0.967408	−	0.253224i	$-0.0814909\pi$
$258$	0	0
$259$	755.781	0.181320
$260$	0	0
$261$	0	0
$262$	− 680.302i	− 0.160417i
$263$	− 3810.66i	− 0.893442i	−0.894673	−	0.446721i	$-0.852592\pi$
$263$	− 3810.66i	− 0.893442i	0.894673	−	0.446721i	$-0.147408\pi$
$264$	0	0
$265$	0	0
$266$	−540.004	−0.124473
$267$	0	0
$268$	− 1896.36i	− 0.432234i
$269$	5942.11	1.34683	0.673414	−	0.739265i	$-0.264827\pi$
$269$	5942.11	1.34683	0.673414	+	0.739265i	$0.264827\pi$
$270$	0	0
$271$	829.082	0.185842	0.0929209	−	0.995673i	$-0.470380\pi$
$271$	829.082	0.185842	0.0929209	+	0.995673i	$0.470380\pi$
$272$	− 887.755i	− 0.197897i
$273$	0	0
$274$	3200.17	0.705582
$275$	0	0
$276$	0	0
$277$	6519.68i	1.41419i	0.707120	+	0.707093i	$0.249994\pi$
$277$	6519.68i	1.41419i	−0.707120	+	0.707093i	$0.750006\pi$
$278$	− 247.459i	− 0.0533871i
$279$	0	0
$280$	0	0
$281$	−8745.29	−1.85658	−0.928292	−	0.371851i	$-0.878723\pi$
$281$	−8745.29	−1.85658	−0.928292	+	0.371851i	$0.878723\pi$
$282$	0	0
$283$	− 8485.60i	− 1.78239i	−0.453619	−	0.891196i	$-0.649867\pi$
$283$	− 8485.60i	− 1.78239i	0.453619	−	0.891196i	$-0.350133\pi$
$284$	−1338.18	−0.279601
$285$	0	0
$286$	−1971.52	−0.407617
$287$	− 572.173i	− 0.117681i
$288$	0	0
$289$	1834.45	0.373387
$290$	0	0
$291$	0	0
$292$	2072.73i	0.415401i
$293$	− 9719.71i	− 1.93799i	−0.247078	−	0.968996i	$-0.579470\pi$
$293$	− 9719.71i	− 1.93799i	0.247078	−	0.968996i	$-0.420530\pi$
$294$	0	0
$295$	0	0
$296$	1287.27	0.252774
$297$	0	0
$298$	1972.73i	0.383480i
$299$	4416.21	0.854168
$300$	0	0
$301$	−268.296	−0.0513765
$302$	− 5956.54i	− 1.13497i
$303$	0	0
$304$	−919.755	−0.173525
$305$	0	0
$306$	0	0
$307$	8585.44i	1.59608i	0.602604	+	0.798040i	$0.294130\pi$
$307$	8585.44i	1.59608i	−0.602604	+	0.798040i	$0.705870\pi$
$308$	− 270.429i	− 0.0500295i
$309$	0	0
$310$	0	0
$311$	−7574.24	−1.38102	−0.690508	−	0.723325i	$-0.742613\pi$
$311$	−7574.24	−1.38102	−0.690508	+	0.723325i	$0.742613\pi$
$312$	0	0
$313$	9543.01i	1.72333i	0.507476	+	0.861666i	$0.330579\pi$
$313$	9543.01i	1.72333i	−0.507476	+	0.861666i	$0.669421\pi$
$314$	−7031.51	−1.26373
$315$	0	0
$316$	3805.45	0.677447
$317$	4743.11i	0.840377i	0.907437	+	0.420188i	$0.138036\pi$
$317$	4743.11i	0.840377i	−0.907437	+	0.420188i	$0.861964\pi$
$318$	0	0
$319$	1921.36	0.337228
$320$	0	0
$321$	0	0
$322$	605.761i	0.104838i
$323$	3189.52i	0.549442i
$324$	0	0
$325$	0	0
$326$	−1795.15	−0.304982
$327$	0	0
$328$	− 974.547i	− 0.164056i
$329$	2035.62	0.341117
$330$	0	0
$331$	−4659.08	−0.773675	−0.386837	−	0.922148i	$-0.626432\pi$
$331$	−4659.08	−0.773675	−0.386837	+	0.922148i	$0.626432\pi$
$332$	− 48.4898i	− 0.00801574i
$333$	0	0
$334$	−1052.85	−0.172483
$335$	0	0
$336$	0	0
$337$	− 464.839i	− 0.0751376i	−0.999294	−	0.0375688i	$-0.988039\pi$
$337$	− 464.839i	− 0.0751376i	0.999294	−	0.0375688i	$-0.0119613\pi$
$338$	− 4986.31i	− 0.802424i
$339$	0	0
$340$	0	0
$341$	359.406	0.0570761
$342$	0	0
$343$	− 3118.48i	− 0.490910i
$344$	−456.971	−0.0716228
$345$	0	0
$346$	−1706.11	−0.265090
$347$	− 2829.64i	− 0.437761i	−0.975752	−	0.218881i	$-0.929759\pi$
$347$	− 2829.64i	− 0.437761i	0.975752	−	0.218881i	$-0.0702405\pi$
$348$	0	0
$349$	−3160.36	−0.484728	−0.242364	−	0.970185i	$-0.577923\pi$
$349$	−3160.36	−0.484728	−0.242364	+	0.970185i	$0.577923\pi$
$350$	0	0
$351$	0	0
$352$	− 460.604i	− 0.0697451i
$353$	4164.26i	0.627879i	0.949443	+	0.313939i	$0.101649\pi$
$353$	4164.26i	0.627879i	−0.949443	+	0.313939i	$0.898351\pi$
$354$	0	0
$355$	0	0
$356$	−1849.09	−0.275285
$357$	0	0
$358$	− 1385.14i	− 0.204489i
$359$	−812.428	−0.119438	−0.0597191	−	0.998215i	$-0.519020\pi$
$359$	−812.428	−0.119438	−0.0597191	+	0.998215i	$0.519020\pi$
$360$	0	0
$361$	−3554.51	−0.518226
$362$	− 6352.48i	− 0.922317i
$363$	0	0
$364$	1286.67	0.185275
$365$	0	0
$366$	0	0
$367$	− 11482.7i	− 1.63322i	−0.577188	−	0.816611i	$-0.695850\pi$
$367$	− 11482.7i	− 1.63322i	0.577188	−	0.816611i	$-0.304150\pi$
$368$	1031.76i	0.146152i
$369$	0	0
$370$	0	0
$371$	1458.47	0.204097
$372$	0	0
$373$	− 4599.69i	− 0.638507i	−0.947669	−	0.319254i	$-0.896568\pi$
$373$	− 4599.69i	− 0.638507i	0.947669	−	0.319254i	$-0.103432\pi$
$374$	−1597.28	−0.220838
$375$	0	0
$376$	3467.14	0.475543
$377$	9141.66i	1.24886i
$378$	0	0
$379$	−3866.12	−0.523983	−0.261991	−	0.965070i	$-0.584379\pi$
$379$	−3866.12	−0.523983	−0.261991	+	0.965070i	$0.584379\pi$
$380$	0	0
$381$	0	0
$382$	− 3737.57i	− 0.500604i
$383$	3545.92i	0.473076i	0.971622	+	0.236538i	$0.0760128\pi$
$383$	3545.92i	0.473076i	−0.971622	+	0.236538i	$0.923987\pi$
$384$	0	0
$385$	0	0
$386$	8827.27	1.16398
$387$	0	0
$388$	− 1796.97i	− 0.235122i
$389$	6344.70	0.826965	0.413482	−	0.910512i	$-0.364312\pi$
$389$	6344.70	0.826965	0.413482	+	0.910512i	$0.364312\pi$
$390$	0	0
$391$	3577.91	0.462769
$392$	− 2567.51i	− 0.330813i
$393$	0	0
$394$	−4344.24	−0.555482
$395$	0	0
$396$	0	0
$397$	− 1712.11i	− 0.216445i	−0.994127	−	0.108222i	$-0.965484\pi$
$397$	− 1712.11i	− 0.216445i	0.994127	−	0.108222i	$-0.0345158\pi$
$398$	− 7539.02i	− 0.949490i
$399$	0	0
$400$	0	0
$401$	−14444.1	−1.79876	−0.899381	−	0.437167i	$-0.855982\pi$
$401$	−14444.1	−1.79876	−0.899381	+	0.437167i	$0.855982\pi$
$402$	0	0
$403$	1710.02i	0.211370i
$404$	−6218.78	−0.765831
$405$	0	0
$406$	−1253.94	−0.153281
$407$	− 2316.11i	− 0.282077i
$408$	0	0
$409$	4386.88	0.530360	0.265180	−	0.964199i	$-0.414569\pi$
$409$	4386.88	0.530360	0.265180	+	0.964199i	$0.414569\pi$
$410$	0	0
$411$	0	0
$412$	1058.19i	0.126537i
$413$	191.434i	0.0228083i
$414$	0	0
$415$	0	0
$416$	2191.51	0.258288
$417$	0	0
$418$	1654.86i	0.193640i
$419$	6940.60	0.809238	0.404619	−	0.914485i	$-0.367404\pi$
$419$	6940.60	0.809238	0.404619	+	0.914485i	$0.367404\pi$
$420$	0	0
$421$	8573.26	0.992482	0.496241	−	0.868185i	$-0.334713\pi$
$421$	8573.26	0.992482	0.496241	+	0.868185i	$0.334713\pi$
$422$	2373.20i	0.273758i
$423$	0	0
$424$	2484.12	0.284527
$425$	0	0
$426$	0	0
$427$	40.1397i	0.00454917i
$428$	− 8255.14i	− 0.932307i
$429$	0	0
$430$	0	0
$431$	13533.6	1.51251	0.756256	−	0.654276i	$-0.227027\pi$
$431$	13533.6	1.51251	0.756256	+	0.654276i	$0.227027\pi$
$432$	0	0
$433$	− 1073.47i	− 0.119140i	−0.998224	−	0.0595700i	$-0.981027\pi$
$433$	− 1073.47i	− 0.119140i	0.998224	−	0.0595700i	$-0.0189730\pi$
$434$	−234.559	−0.0259429
$435$	0	0
$436$	3584.24	0.393702
$437$	− 3706.88i	− 0.405776i
$438$	0	0
$439$	1626.28	0.176807	0.0884034	−	0.996085i	$-0.471824\pi$
$439$	1626.28	0.176807	0.0884034	+	0.996085i	$0.471824\pi$
$440$	0	0
$441$	0	0
$442$	− 7599.70i	− 0.817830i
$443$	13538.4i	1.45198i	0.687704	+	0.725991i	$0.258619\pi$
$443$	13538.4i	1.45198i	−0.687704	+	0.725991i	$0.741381\pi$
$444$	0	0
$445$	0	0
$446$	65.7530	0.00698093
$447$	0	0
$448$	300.604i	0.0317014i
$449$	−12721.5	−1.33711	−0.668557	−	0.743661i	$-0.733088\pi$
$449$	−12721.5	−1.33711	−0.668557	+	0.743661i	$0.733088\pi$
$450$	0	0
$451$	−1753.44	−0.183074
$452$	− 2675.39i	− 0.278406i
$453$	0	0
$454$	−1427.82	−0.147601
$455$	0	0
$456$	0	0
$457$	− 11790.5i	− 1.20686i	−0.797417	−	0.603429i	$-0.793801\pi$
$457$	− 11790.5i	− 1.20686i	0.797417	−	0.603429i	$-0.206199\pi$
$458$	− 5555.50i	− 0.566794i
$459$	0	0
$460$	0	0
$461$	−3420.92	−0.345615	−0.172807	−	0.984956i	$-0.555284\pi$
$461$	−3420.92	−0.345615	−0.172807	+	0.984956i	$0.555284\pi$
$462$	0	0
$463$	11016.8i	1.10582i	0.833241	+	0.552909i	$0.186482\pi$
$463$	11016.8i	1.10582i	−0.833241	+	0.552909i	$0.813518\pi$
$464$	−2135.76	−0.213685
$465$	0	0
$466$	−11335.1	−1.12680
$467$	13350.3i	1.32286i	0.750006	+	0.661432i	$0.230051\pi$
$467$	13350.3i	1.32286i	−0.750006	+	0.661432i	$0.769949\pi$
$468$	0	0
$469$	2226.78	0.219239
$470$	0	0
$471$	0	0
$472$	326.057i	0.0317966i
$473$	822.199i	0.0799255i
$474$	0	0
$475$	0	0
$476$	1042.43	0.100378
$477$	0	0
$478$	11573.3i	1.10743i
$479$	17045.1	1.62591	0.812956	−	0.582325i	$-0.197857\pi$
$479$	17045.1	1.62591	0.812956	+	0.582325i	$0.197857\pi$
$480$	0	0
$481$	11019.8	1.04462
$482$	− 3122.86i	− 0.295108i
$483$	0	0
$484$	4495.27	0.422170
$485$	0	0
$486$	0	0
$487$	− 8391.04i	− 0.780768i	−0.920652	−	0.390384i	$-0.872342\pi$
$487$	− 8391.04i	− 0.780768i	0.920652	−	0.390384i	$-0.127658\pi$
$488$	68.3674i	0.00634190i
$489$	0	0
$490$	0	0
$491$	−13006.6	−1.19548	−0.597741	−	0.801689i	$-0.703935\pi$
$491$	−13006.6	−1.19548	−0.597741	+	0.801689i	$0.703935\pi$
$492$	0	0
$493$	7406.36i	0.676604i
$494$	−7873.64	−0.717109
$495$	0	0
$496$	−399.510	−0.0361664
$497$	− 1571.34i	− 0.141820i
$498$	0	0
$499$	12113.6	1.08673	0.543367	−	0.839495i	$-0.317149\pi$
$499$	12113.6	1.08673	0.543367	+	0.839495i	$0.317149\pi$
$500$	0	0
$501$	0	0
$502$	9416.64i	0.837222i
$503$	18776.7i	1.66443i	0.554450	+	0.832217i	$0.312928\pi$
$503$	18776.7i	1.66443i	−0.554450	+	0.832217i	$0.687072\pi$
$504$	0	0
$505$	0	0
$506$	1856.37	0.163094
$507$	0	0
$508$	− 9538.17i	− 0.833047i
$509$	14595.4	1.27099	0.635493	−	0.772107i	$-0.280797\pi$
$509$	14595.4	1.27099	0.635493	+	0.772107i	$0.280797\pi$
$510$	0	0
$511$	−2433.87	−0.210700
$512$	512.000i	0.0441942i
$513$	0	0
$514$	4173.15	0.358113
$515$	0	0
$516$	0	0
$517$	− 6238.20i	− 0.530669i
$518$	1511.56i	0.128213i
$519$	0	0
$520$	0	0
$521$	14706.8	1.23669	0.618346	−	0.785906i	$-0.287803\pi$
$521$	14706.8	1.23669	0.618346	+	0.785906i	$0.287803\pi$
$522$	0	0
$523$	16225.4i	1.35657i	0.734797	+	0.678287i	$0.237277\pi$
$523$	16225.4i	1.35657i	−0.734797	+	0.678287i	$0.762723\pi$
$524$	1360.60	0.113432
$525$	0	0
$526$	7621.32	0.631759
$527$	1385.42i	0.114516i
$528$	0	0
$529$	8008.72	0.658233
$530$	0	0
$531$	0	0
$532$	− 1080.01i	− 0.0880156i
$533$	− 8342.69i	− 0.677978i
$534$	0	0
$535$	0	0
$536$	3792.73	0.305636
$537$	0	0
$538$	11884.2i	0.952352i
$539$	−4619.55	−0.369162
$540$	0	0
$541$	−3313.93	−0.263359	−0.131679	−	0.991292i	$-0.542037\pi$
$541$	−3313.93	−0.263359	−0.131679	+	0.991292i	$0.542037\pi$
$542$	1658.16i	0.131410i
$543$	0	0
$544$	1775.51	0.139935
$545$	0	0
$546$	0	0
$547$	− 16599.1i	− 1.29749i	−0.761007	−	0.648744i	$-0.775295\pi$
$547$	− 16599.1i	− 1.29749i	0.761007	−	0.648744i	$-0.224705\pi$
$548$	6400.35i	0.498922i
$549$	0	0
$550$	0	0
$551$	7673.33	0.593275
$552$	0	0
$553$	4468.49i	0.343616i
$554$	−13039.4	−0.999981
$555$	0	0
$556$	494.918	0.0377504
$557$	10258.1i	0.780339i	0.920743	+	0.390170i	$0.127584\pi$
$557$	10258.1i	0.780339i	−0.920743	+	0.390170i	$0.872416\pi$
$558$	0	0
$559$	−3911.94	−0.295989
$560$	0	0
$561$	0	0
$562$	− 17490.6i	− 1.31280i
$563$	− 7368.77i	− 0.551610i	−0.961214	−	0.275805i	$-0.911056\pi$
$563$	− 7368.77i	− 0.551610i	0.961214	−	0.275805i	$-0.0889444\pi$
$564$	0	0
$565$	0	0
$566$	16971.2	1.26034
$567$	0	0
$568$	− 2676.37i	− 0.197708i
$569$	695.761	0.0512616	0.0256308	−	0.999671i	$-0.491841\pi$
$569$	695.761	0.0512616	0.0256308	+	0.999671i	$0.491841\pi$
$570$	0	0
$571$	2369.34	0.173649	0.0868246	−	0.996224i	$-0.472328\pi$
$571$	2369.34	0.173649	0.0868246	+	0.996224i	$0.472328\pi$
$572$	− 3943.04i	− 0.288229i
$573$	0	0
$574$	1144.35	0.0832127
$575$	0	0
$576$	0	0
$577$	− 2915.01i	− 0.210318i	−0.994455	−	0.105159i	$-0.966465\pi$
$577$	− 2915.01i	− 0.210318i	0.994455	−	0.105159i	$-0.0335352\pi$
$578$	3668.90i	0.264024i
$579$	0	0
$580$	0	0
$581$	56.9385	0.00406576
$582$	0	0
$583$	− 4469.52i	− 0.317510i
$584$	−4145.45	−0.293733
$585$	0	0
$586$	19439.4	1.37037
$587$	20041.4i	1.40919i	0.709607	+	0.704597i	$0.248872\pi$
$587$	20041.4i	1.40919i	−0.709607	+	0.704597i	$0.751128\pi$
$588$	0	0
$589$	1435.36	0.100412
$590$	0	0
$591$	0	0
$592$	2574.55i	0.178739i
$593$	18221.3i	1.26182i	0.775855	+	0.630911i	$0.217319\pi$
$593$	18221.3i	1.26182i	−0.775855	+	0.630911i	$0.782681\pi$
$594$	0	0
$595$	0	0
$596$	−3945.45	−0.271161
$597$	0	0
$598$	8832.43i	0.603988i
$599$	−8431.78	−0.575147	−0.287574	−	0.957759i	$-0.592849\pi$
$599$	−8431.78	−0.575147	−0.287574	+	0.957759i	$0.592849\pi$
$600$	0	0
$601$	26508.5	1.79918	0.899588	−	0.436740i	$-0.143867\pi$
$601$	26508.5	1.79918	0.899588	+	0.436740i	$0.143867\pi$
$602$	− 536.592i	− 0.0363286i
$603$	0	0
$604$	11913.1	0.802544
$605$	0	0
$606$	0	0
$607$	− 5229.11i	− 0.349659i	−0.984599	−	0.174829i	$-0.944063\pi$
$607$	− 5229.11i	− 0.349659i	0.984599	−	0.174829i	$-0.0559374\pi$
$608$	− 1839.51i	− 0.122701i
$609$	0	0
$610$	0	0
$611$	29680.8	1.96523
$612$	0	0
$613$	− 12910.0i	− 0.850618i	−0.905048	−	0.425309i	$-0.860165\pi$
$613$	− 12910.0i	− 0.850618i	0.905048	−	0.425309i	$-0.139835\pi$
$614$	−17170.9	−1.12860
$615$	0	0
$616$	540.857	0.0353762
$617$	− 12437.6i	− 0.811540i	−0.913975	−	0.405770i	$-0.867003\pi$
$617$	− 12437.6i	− 0.811540i	0.913975	−	0.405770i	$-0.132997\pi$
$618$	0	0
$619$	8969.29	0.582401	0.291200	−	0.956662i	$-0.405945\pi$
$619$	8969.29	0.582401	0.291200	+	0.956662i	$0.405945\pi$
$620$	0	0
$621$	0	0
$622$	− 15148.5i	− 0.976526i
$623$	− 2171.27i	− 0.139631i
$624$	0	0
$625$	0	0
$626$	−19086.0	−1.21858
$627$	0	0
$628$	− 14063.0i	− 0.893592i
$629$	8928.00	0.565950
$630$	0	0
$631$	15378.9	0.970245	0.485123	−	0.874446i	$-0.338775\pi$
$631$	15378.9	0.970245	0.485123	+	0.874446i	$0.338775\pi$
$632$	7610.90i	0.479027i
$633$	0	0
$634$	−9486.21	−0.594236
$635$	0	0
$636$	0	0
$637$	− 21979.4i	− 1.36712i
$638$	3842.72i	0.238456i
$639$	0	0
$640$	0	0
$641$	−15305.9	−0.943129	−0.471565	−	0.881831i	$-0.656311\pi$
$641$	−15305.9	−0.943129	−0.471565	+	0.881831i	$0.656311\pi$
$642$	0	0
$643$	4869.07i	0.298627i	0.988790	+	0.149314i	$0.0477064\pi$
$643$	4869.07i	0.298627i	−0.988790	+	0.149314i	$0.952294\pi$
$644$	−1211.52	−0.0741315
$645$	0	0
$646$	−6379.04	−0.388514
$647$	21541.1i	1.30892i	0.756098	+	0.654459i	$0.227104\pi$
$647$	21541.1i	1.30892i	−0.756098	+	0.654459i	$0.772896\pi$
$648$	0	0
$649$	586.653	0.0354825
$650$	0	0
$651$	0	0
$652$	− 3590.29i	− 0.215655i
$653$	− 9977.40i	− 0.597926i	−0.954265	−	0.298963i	$-0.903359\pi$
$653$	− 9977.40i	− 0.597926i	0.954265	−	0.298963i	$-0.0966408\pi$
$654$	0	0
$655$	0	0
$656$	1949.09	0.116005
$657$	0	0
$658$	4071.24i	0.241206i
$659$	−16162.1	−0.955365	−0.477683	−	0.878532i	$-0.658523\pi$
$659$	−16162.1	−0.955365	−0.477683	+	0.878532i	$0.658523\pi$
$660$	0	0
$661$	−8334.71	−0.490443	−0.245221	−	0.969467i	$-0.578861\pi$
$661$	−8334.71	−0.490443	−0.245221	+	0.969467i	$0.578861\pi$
$662$	− 9318.16i	− 0.547071i
$663$	0	0
$664$	96.9797	0.00566798
$665$	0	0
$666$	0	0
$667$	− 8607.72i	− 0.499688i
$668$	− 2105.69i	− 0.121964i
$669$	0	0
$670$	0	0
$671$	123.009	0.00707706
$672$	0	0
$673$	1129.53i	0.0646959i	0.999477	+	0.0323480i	$0.0102985\pi$
$673$	1129.53i	0.0646959i	−0.999477	+	0.0323480i	$0.989702\pi$
$674$	929.677	0.0531303
$675$	0	0
$676$	9972.61	0.567399
$677$	− 5378.66i	− 0.305345i	−0.988277	−	0.152673i	$-0.951212\pi$
$677$	− 5378.66i	− 0.305345i	0.988277	−	0.152673i	$-0.0487880\pi$
$678$	0	0
$679$	2110.07	0.119259
$680$	0	0
$681$	0	0
$682$	718.812i	0.0403589i
$683$	14261.7i	0.798986i	0.916736	+	0.399493i	$0.130814\pi$
$683$	14261.7i	0.798986i	−0.916736	+	0.399493i	$0.869186\pi$
$684$	0	0
$685$	0	0
$686$	6236.96	0.347126
$687$	0	0
$688$	− 913.943i	− 0.0506450i
$689$	21265.5	1.17584
$690$	0	0
$691$	26200.0	1.44239	0.721197	−	0.692730i	$-0.243592\pi$
$691$	26200.0	1.44239	0.721197	+	0.692730i	$0.243592\pi$
$692$	− 3412.22i	− 0.187447i
$693$	0	0
$694$	5659.29	0.309544
$695$	0	0
$696$	0	0
$697$	− 6759.05i	− 0.367313i
$698$	− 6320.71i	− 0.342754i
$699$	0	0
$700$	0	0
$701$	27273.7	1.46949	0.734747	−	0.678341i	$-0.237301\pi$
$701$	27273.7	1.46949	0.734747	+	0.678341i	$0.237301\pi$
$702$	0	0
$703$	− 9249.81i	− 0.496249i
$704$	921.208	0.0493172
$705$	0	0
$706$	−8328.52	−0.443977
$707$	− 7302.31i	− 0.388446i
$708$	0	0
$709$	26749.8	1.41694	0.708471	−	0.705740i	$-0.249385\pi$
$709$	26749.8	1.41694	0.708471	+	0.705740i	$0.249385\pi$
$710$	0	0
$711$	0	0
$712$	− 3698.18i	− 0.194656i
$713$	− 1610.14i	− 0.0845727i
$714$	0	0
$715$	0	0
$716$	2770.29	0.144596
$717$	0	0
$718$	− 1624.86i	− 0.0844555i
$719$	−13255.4	−0.687544	−0.343772	−	0.939053i	$-0.611705\pi$
$719$	−13255.4	−0.687544	−0.343772	+	0.939053i	$0.611705\pi$
$720$	0	0
$721$	−1242.56	−0.0641822
$722$	− 7109.02i	− 0.366441i
$723$	0	0
$724$	12705.0	0.652177
$725$	0	0
$726$	0	0
$727$	− 8880.92i	− 0.453060i	−0.974004	−	0.226530i	$-0.927262\pi$
$727$	− 8880.92i	− 0.453060i	0.974004	−	0.226530i	$-0.0727382\pi$
$728$	2573.35i	0.131009i
$729$	0	0
$730$	0	0
$731$	−3169.37	−0.160360
$732$	0	0
$733$	21591.8i	1.08801i	0.839082	+	0.544005i	$0.183093\pi$
$733$	21591.8i	1.08801i	−0.839082	+	0.544005i	$0.816907\pi$
$734$	22965.4	1.15486
$735$	0	0
$736$	−2063.51	−0.103345
$737$	− 6824.00i	− 0.341066i
$738$	0	0
$739$	26087.3	1.29856	0.649281	−	0.760549i	$-0.275070\pi$
$739$	26087.3	1.29856	0.649281	+	0.760549i	$0.275070\pi$
$740$	0	0
$741$	0	0
$742$	2916.94i	0.144319i
$743$	655.087i	0.0323456i	0.999869	+	0.0161728i	$0.00514819\pi$
$743$	655.087i	0.0323456i	−0.999869	+	0.0161728i	$0.994852\pi$
$744$	0	0
$745$	0	0
$746$	9199.39	0.451493
$747$	0	0
$748$	− 3194.56i	− 0.156156i
$749$	9693.47	0.472886
$750$	0	0
$751$	−3100.89	−0.150670	−0.0753349	−	0.997158i	$-0.524003\pi$
$751$	−3100.89	−0.150670	−0.0753349	+	0.997158i	$0.524003\pi$
$752$	6934.29i	0.336260i
$753$	0	0
$754$	−18283.3	−0.883076
$755$	0	0
$756$	0	0
$757$	16626.3i	0.798276i	0.916891	+	0.399138i	$0.130691\pi$
$757$	16626.3i	0.798276i	−0.916891	+	0.399138i	$0.869309\pi$
$758$	− 7732.24i	− 0.370512i
$759$	0	0
$760$	0	0
$761$	−34426.1	−1.63987	−0.819937	−	0.572454i	$-0.805991\pi$
$761$	−34426.1	−1.63987	−0.819937	+	0.572454i	$0.805991\pi$
$762$	0	0
$763$	4208.74i	0.199694i
$764$	7475.14	0.353981
$765$	0	0
$766$	−7091.85	−0.334515
$767$	2791.24i	0.131403i
$768$	0	0
$769$	−657.531	−0.0308338	−0.0154169	−	0.999881i	$-0.504908\pi$
$769$	−657.531	−0.0308338	−0.0154169	+	0.999881i	$0.504908\pi$
$770$	0	0
$771$	0	0
$772$	17654.5i	0.823058i
$773$	6917.68i	0.321878i	0.986964	+	0.160939i	$0.0514522\pi$
$773$	6917.68i	0.321878i	−0.986964	+	0.160939i	$0.948548\pi$
$774$	0	0
$775$	0	0
$776$	3593.94	0.166256
$777$	0	0
$778$	12689.4i	0.584752i
$779$	−7002.69	−0.322076
$780$	0	0
$781$	−4815.41	−0.220626
$782$	7155.83i	0.327227i
$783$	0	0
$784$	5135.02	0.233920
$785$	0	0
$786$	0	0
$787$	− 10032.7i	− 0.454419i	−0.973846	−	0.227210i	$-0.927040\pi$
$787$	− 10032.7i	− 0.454419i	0.973846	−	0.227210i	$-0.0729603\pi$
$788$	− 8688.49i	− 0.392785i
$789$	0	0
$790$	0	0
$791$	3141.53	0.141214
$792$	0	0
$793$	585.265i	0.0262085i
$794$	3424.23	0.153049
$795$	0	0
$796$	15078.0	0.671391
$797$	− 17537.1i	− 0.779416i	−0.920938	−	0.389708i	$-0.872576\pi$
$797$	− 17537.1i	− 0.779416i	0.920938	−	0.389708i	$-0.127424\pi$
$798$	0	0
$799$	24046.7	1.06472
$800$	0	0
$801$	0	0
$802$	− 28888.2i	− 1.27192i
$803$	7458.64i	0.327783i
$804$	0	0
$805$	0	0
$806$	−3420.04	−0.149461
$807$	0	0
$808$	− 12437.6i	− 0.541525i
$809$	−16817.7	−0.730875	−0.365437	−	0.930836i	$-0.619081\pi$
$809$	−16817.7	−0.730875	−0.365437	+	0.930836i	$0.619081\pi$
$810$	0	0
$811$	−14187.8	−0.614305	−0.307152	−	0.951660i	$-0.599376\pi$
$811$	−14187.8	−0.614305	−0.307152	+	0.951660i	$0.599376\pi$
$812$	− 2507.88i	− 0.108386i
$813$	0	0
$814$	4632.21	0.199458
$815$	0	0
$816$	0	0
$817$	3283.61i	0.140611i
$818$	8773.76i	0.375021i
$819$	0	0
$820$	0	0
$821$	−8934.84	−0.379815	−0.189908	−	0.981802i	$-0.560819\pi$
$821$	−8934.84	−0.379815	−0.189908	+	0.981802i	$0.560819\pi$
$822$	0	0
$823$	− 7056.18i	− 0.298861i	−0.988772	−	0.149431i	$-0.952256\pi$
$823$	− 7056.18i	− 0.298861i	0.988772	−	0.149431i	$-0.0477441\pi$
$824$	−2116.38	−0.0894750
$825$	0	0
$826$	−382.868	−0.0161279
$827$	36134.5i	1.51937i	0.650291	+	0.759686i	$0.274647\pi$
$827$	36134.5i	1.51937i	−0.650291	+	0.759686i	$0.725353\pi$
$828$	0	0
$829$	−3670.19	−0.153765	−0.0768823	−	0.997040i	$-0.524497\pi$
$829$	−3670.19	−0.153765	−0.0768823	+	0.997040i	$0.524497\pi$
$830$	0	0
$831$	0	0
$832$	4383.02i	0.182637i
$833$	− 17807.2i	− 0.740675i
$834$	0	0
$835$	0	0
$836$	−3309.71	−0.136924
$837$	0	0
$838$	13881.2i	0.572218i
$839$	−23058.2	−0.948815	−0.474408	−	0.880305i	$-0.657338\pi$
$839$	−23058.2	−0.948815	−0.474408	+	0.880305i	$0.657338\pi$
$840$	0	0
$841$	−6570.84	−0.269418
$842$	17146.5i	0.701791i
$843$	0	0
$844$	−4746.41	−0.193576
$845$	0	0
$846$	0	0
$847$	5278.50i	0.214134i
$848$	4968.24i	0.201191i
$849$	0	0
$850$	0	0
$851$	−10376.2	−0.417968
$852$	0	0
$853$	− 12100.0i	− 0.485691i	−0.970065	−	0.242846i	$-0.921919\pi$
$853$	− 12100.0i	− 0.485691i	0.970065	−	0.242846i	$-0.0780809\pi$
$854$	−80.2794	−0.00321675
$855$	0	0
$856$	16510.3	0.659241
$857$	− 44738.2i	− 1.78323i	−0.452793	−	0.891616i	$-0.649572\pi$
$857$	− 44738.2i	− 1.78323i	0.452793	−	0.891616i	$-0.350428\pi$
$858$	0	0
$859$	−43598.6	−1.73174	−0.865870	−	0.500269i	$-0.833235\pi$
$859$	−43598.6	−1.73174	−0.865870	+	0.500269i	$0.833235\pi$
$860$	0	0
$861$	0	0
$862$	27067.3i	1.06951i
$863$	16790.6i	0.662294i	0.943579	+	0.331147i	$0.107436\pi$
$863$	16790.6i	0.662294i	−0.943579	+	0.331147i	$0.892564\pi$
$864$	0	0
$865$	0	0
$866$	2146.94	0.0842447
$867$	0	0
$868$	− 469.119i	− 0.0183444i
$869$	13693.8	0.534557
$870$	0	0
$871$	32468.0	1.26307
$872$	7168.49i	0.278390i
$873$	0	0
$874$	7413.77	0.286927
$875$	0	0
$876$	0	0
$877$	− 29794.2i	− 1.14718i	−0.819142	−	0.573590i	$-0.805550\pi$
$877$	− 29794.2i	− 1.14718i	0.819142	−	0.573590i	$-0.194450\pi$
$878$	3252.56i	0.125021i
$879$	0	0
$880$	0	0
$881$	−10626.3	−0.406368	−0.203184	−	0.979141i	$-0.565129\pi$
$881$	−10626.3	−0.406368	−0.203184	+	0.979141i	$0.565129\pi$
$882$	0	0
$883$	2286.20i	0.0871310i	0.999051	+	0.0435655i	$0.0138717\pi$
$883$	2286.20i	0.0871310i	−0.999051	+	0.0435655i	$0.986128\pi$
$884$	15199.4	0.578293
$885$	0	0
$886$	−27076.8	−1.02671
$887$	− 4335.96i	− 0.164135i	−0.996627	−	0.0820673i	$-0.973848\pi$
$887$	− 4335.96i	− 0.164135i	0.996627	−	0.0820673i	$-0.0261523\pi$
$888$	0	0
$889$	11200.1	0.422540
$890$	0	0
$891$	0	0
$892$	131.506i	0.00493626i
$893$	− 24913.5i	− 0.933591i
$894$	0	0
$895$	0	0
$896$	−601.208	−0.0224162
$897$	0	0
$898$	− 25443.0i	− 0.945483i
$899$	3333.03	0.123652
$900$	0	0
$901$	17228.8	0.637043
$902$	− 3506.88i	− 0.129453i
$903$	0	0
$904$	5350.78	0.196863
$905$	0	0
$906$	0	0
$907$	− 6341.37i	− 0.232152i	−0.993240	−	0.116076i	$-0.962968\pi$
$907$	− 6341.37i	− 0.232152i	0.993240	−	0.116076i	$-0.0370316\pi$
$908$	− 2855.63i	− 0.104369i
$909$	0	0
$910$	0	0
$911$	11659.1	0.424020	0.212010	−	0.977267i	$-0.431999\pi$
$911$	11659.1	0.424020	0.212010	+	0.977267i	$0.431999\pi$
$912$	0	0
$913$	− 174.489i	− 0.00632503i
$914$	23580.9	0.853378
$915$	0	0
$916$	11111.0	0.400784
$917$	1597.67i	0.0575351i
$918$	0	0
$919$	−45780.0	−1.64325	−0.821623	−	0.570031i	$-0.806931\pi$
$919$	−45780.0	−1.64325	−0.821623	+	0.570031i	$0.806931\pi$
$920$	0	0
$921$	0	0
$922$	− 6841.85i	− 0.244386i
$923$	− 22911.3i	− 0.817046i
$924$	0	0
$925$	0	0
$926$	−22033.6	−0.781932
$927$	0	0
$928$	− 4271.51i	− 0.151098i
$929$	19361.4	0.683777	0.341888	−	0.939741i	$-0.388934\pi$
$929$	19361.4	0.683777	0.341888	+	0.939741i	$0.388934\pi$
$930$	0	0
$931$	−18449.1	−0.649456
$932$	− 22670.1i	− 0.796765i
$933$	0	0
$934$	−26700.6	−0.935405
$935$	0	0
$936$	0	0
$937$	29517.1i	1.02912i	0.857456	+	0.514558i	$0.172044\pi$
$937$	29517.1i	1.02912i	−0.857456	+	0.514558i	$0.827956\pi$
$938$	4453.55i	0.155025i
$939$	0	0
$940$	0	0
$941$	−47940.1	−1.66079	−0.830395	−	0.557175i	$-0.811885\pi$
$941$	−47940.1	−1.66079	−0.830395	+	0.557175i	$0.811885\pi$
$942$	0	0
$943$	7855.42i	0.271270i
$944$	−652.114	−0.0224836
$945$	0	0
$946$	−1644.40	−0.0565158
$947$	− 33531.1i	− 1.15060i	−0.817944	−	0.575298i	$-0.804886\pi$
$947$	− 33531.1i	− 1.15060i	0.817944	−	0.575298i	$-0.195114\pi$
$948$	0	0
$949$	−35487.5	−1.21388
$950$	0	0
$951$	0	0
$952$	2084.87i	0.0709778i
$953$	− 39678.4i	− 1.34870i	−0.738413	−	0.674349i	$-0.764425\pi$
$953$	− 39678.4i	− 1.34870i	0.738413	−	0.674349i	$-0.235575\pi$
$954$	0	0
$955$	0	0
$956$	−23146.6	−0.783071
$957$	0	0
$958$	34090.2i	1.14969i
$959$	−7515.51	−0.253064
$960$	0	0
$961$	−29167.5	−0.979072
$962$	22039.6i	0.738655i
$963$	0	0
$964$	6245.71	0.208673
$965$	0	0
$966$	0	0
$967$	42548.1i	1.41495i	0.706739	+	0.707474i	$0.250165\pi$
$967$	42548.1i	1.41495i	−0.706739	+	0.707474i	$0.749835\pi$
$968$	8990.53i	0.298519i
$969$	0	0
$970$	0	0
$971$	2929.89	0.0968329	0.0484164	−	0.998827i	$-0.484583\pi$
$971$	2929.89	0.0968329	0.0484164	+	0.998827i	$0.484583\pi$
$972$	0	0
$973$	581.150i	0.0191478i
$974$	16782.1	0.552087
$975$	0	0
$976$	−136.735	−0.00448440
$977$	− 2755.87i	− 0.0902436i	−0.998981	−	0.0451218i	$-0.985632\pi$
$977$	− 2755.87i	− 0.0902436i	0.998981	−	0.0451218i	$-0.0143676\pi$
$978$	0	0
$979$	−6653.89	−0.217221
$980$	0	0
$981$	0	0
$982$	− 26013.3i	− 0.845333i
$983$	− 16404.0i	− 0.532256i	−0.963938	−	0.266128i	$-0.914256\pi$
$983$	− 16404.0i	− 0.532256i	0.963938	−	0.266128i	$-0.0857444\pi$
$984$	0	0
$985$	0	0
$986$	−14812.7	−0.478431
$987$	0	0
$988$	− 15747.3i	− 0.507073i
$989$	3683.46	0.118430
$990$	0	0
$991$	−43815.3	−1.40448	−0.702239	−	0.711941i	$-0.747816\pi$
$991$	−43815.3	−1.40448	−0.702239	+	0.711941i	$0.747816\pi$
$992$	− 799.020i	− 0.0255735i
$993$	0	0
$994$	3142.68	0.100282
$995$	0	0
$996$	0	0
$997$	− 17304.5i	− 0.549688i	−0.961489	−	0.274844i	$-0.911374\pi$
$997$	− 17304.5i	− 0.549688i	0.961489	−	0.274844i	$-0.0886262\pi$
$998$	24227.2i	0.768437i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.z.649.3		4
3.2	odd	2		1350.4.c.w.649.1		4
5.2	odd	4		1350.4.a.bc.1.2	✓	2
5.3	odd	4		1350.4.a.bp.1.1	yes	2
5.4	even	2	inner	1350.4.c.z.649.2		4
15.2	even	4		1350.4.a.bj.1.2	yes	2
15.8	even	4		1350.4.a.bi.1.1	yes	2
15.14	odd	2		1350.4.c.w.649.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.bc.1.2	✓	2	5.2	odd	4
1350.4.a.bi.1.1	yes	2	15.8	even	4
1350.4.a.bj.1.2	yes	2	15.2	even	4
1350.4.a.bp.1.1	yes	2	5.3	odd	4
1350.4.c.w.649.1		4	3.2	odd	2
1350.4.c.w.649.4		4	15.14	odd	2
1350.4.c.z.649.2		4	5.4	even	2	inner
1350.4.c.z.649.3		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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} -4.69694i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} -4.69694i q^{7} -8.00000i q^{8} -14.3939 q^{11} -68.4847i q^{13} +9.39388 q^{14} +16.0000 q^{16} -55.4847i q^{17} -57.4847 q^{19} -28.7878i q^{22} +64.4847i q^{23} +136.969 q^{26} +18.7878i q^{28} -133.485 q^{29} -24.9694 q^{31} +32.0000i q^{32} +110.969 q^{34} +160.909i q^{37} -114.969i q^{38} +121.818 q^{41} -57.1214i q^{43} +57.5755 q^{44} -128.969 q^{46} +433.393i q^{47} +320.939 q^{49} +273.939i q^{52} +310.515i q^{53} -37.5755 q^{56} -266.969i q^{58} -40.7571 q^{59} -8.54592 q^{61} -49.9388i q^{62} -64.0000 q^{64} +474.091i q^{67} +221.939i q^{68} +334.546 q^{71} -518.182i q^{73} -321.818 q^{74} +229.939 q^{76} +67.6072i q^{77} -951.362 q^{79} +243.637i q^{82} +12.1225i q^{83} +114.243 q^{86} +115.151i q^{88} +462.272 q^{89} -321.668 q^{91} -257.939i q^{92} -866.786 q^{94} +449.243i q^{97} +641.878i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} + 60 q^{11} - 80 q^{14} + 64 q^{16} + 64 q^{19} - 40 q^{26} - 240 q^{29} + 488 q^{31} - 144 q^{34} + 840 q^{41} - 240 q^{44} + 72 q^{46} + 108 q^{49} + 320 q^{56} + 660 q^{59} - 916 q^{61} - 256 q^{64} + 2220 q^{71} - 1640 q^{74} - 256 q^{76} - 1160 q^{79} + 1280 q^{86} + 1320 q^{89} - 4520 q^{91} + 648 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 + 60 * q^11 - 80 * q^14 + 64 * q^16 + 64 * q^19 - 40 * q^26 - 240 * q^29 + 488 * q^31 - 144 * q^34 + 840 * q^41 - 240 * q^44 + 72 * q^46 + 108 * q^49 + 320 * q^56 + 660 * q^59 - 916 * q^61 - 256 * q^64 + 2220 * q^71 - 1640 * q^74 - 256 * q^76 - 1160 * q^79 + 1280 * q^86 + 1320 * q^89 - 4520 * q^91 + 648 * q^94

Introduction

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