LMFDB - Embedded newform 1216.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

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:	yes
Analytic conductor:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27492 q^{3} +6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} +O(q^{10})$	$q+3.27492 q^{3} +6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} -33.9244 q^{11} +3.07558 q^{13} +20.5498 q^{15} -14.2508 q^{17} -19.0000 q^{19} +59.2749 q^{21} -114.072 q^{23} -85.6254 q^{25} -141.722 q^{27} +34.5257 q^{29} -107.698 q^{31} -111.100 q^{33} +113.574 q^{35} +181.698 q^{37} +10.0723 q^{39} -444.743 q^{41} +120.323 q^{43} -102.124 q^{45} -306.371 q^{47} -15.4020 q^{49} -46.6703 q^{51} -115.825 q^{53} -212.873 q^{55} -62.2234 q^{57} +161.680 q^{59} -274.571 q^{61} -294.571 q^{63} +19.2990 q^{65} +81.6254 q^{67} -373.577 q^{69} -773.492 q^{71} +148.794 q^{73} -280.416 q^{75} -614.021 q^{77} +557.341 q^{79} -24.7043 q^{81} +768.337 q^{83} -89.4228 q^{85} +113.069 q^{87} +457.884 q^{89} +55.6670 q^{91} -352.701 q^{93} -119.223 q^{95} -1500.82 q^{97} +552.117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 5 q^{5} + 6 q^{7} - 25 q^{9}+O(q^{10}) $ 2 * q - q^3 + 5 * q^5 + 6 * q^7 - 25 * q^9	$ 2 q - q^{3} + 5 q^{5} + 6 q^{7} - 25 q^{9} - 15 q^{11} + 59 q^{13} + 26 q^{15} - 104 q^{17} - 38 q^{19} + 111 q^{21} + 21 q^{23} - 209 q^{25} + 11 q^{27} + 137 q^{29} - 4 q^{31} - 192 q^{33} + 129 q^{35} + 152 q^{37} - 229 q^{39} - 210 q^{41} + 67 q^{43} - 91 q^{45} - 273 q^{47} - 212 q^{49} + 337 q^{51} - 209 q^{53} - 237 q^{55} + 19 q^{57} + 799 q^{59} - 149 q^{61} - 189 q^{63} - 52 q^{65} + 201 q^{67} - 951 q^{69} - 792 q^{71} - 246 q^{73} + 247 q^{75} - 843 q^{77} + 254 q^{79} - 442 q^{81} + 374 q^{83} + 25 q^{85} - 325 q^{87} - 564 q^{89} - 621 q^{91} - 796 q^{93} - 95 q^{95} - 178 q^{97} + 387 q^{99}+O(q^{100}) $ 2 * q - q^3 + 5 * q^5 + 6 * q^7 - 25 * q^9 - 15 * q^11 + 59 * q^13 + 26 * q^15 - 104 * q^17 - 38 * q^19 + 111 * q^21 + 21 * q^23 - 209 * q^25 + 11 * q^27 + 137 * q^29 - 4 * q^31 - 192 * q^33 + 129 * q^35 + 152 * q^37 - 229 * q^39 - 210 * q^41 + 67 * q^43 - 91 * q^45 - 273 * q^47 - 212 * q^49 + 337 * q^51 - 209 * q^53 - 237 * q^55 + 19 * q^57 + 799 * q^59 - 149 * q^61 - 189 * q^63 - 52 * q^65 + 201 * q^67 - 951 * q^69 - 792 * q^71 - 246 * q^73 + 247 * q^75 - 843 * q^77 + 254 * q^79 - 442 * q^81 + 374 * q^83 + 25 * q^85 - 325 * q^87 - 564 * q^89 - 621 * q^91 - 796 * q^93 - 95 * q^95 - 178 * q^97 + 387 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	3.27492	0.630258	0.315129	−	0.949049i	$-0.397952\pi$
$3$	3.27492	0.630258	0.315129	+	0.949049i	$0.397952\pi$
$4$	0	0
$5$	6.27492	0.561246	0.280623	−	0.959818i	$-0.409459\pi$
$5$	6.27492	0.561246	0.280623	+	0.959818i	$0.409459\pi$
$6$	0	0
$7$	18.0997	0.977290	0.488645	−	0.872483i	$-0.337491\pi$
$7$	18.0997	0.977290	0.488645	+	0.872483i	$0.337491\pi$
$8$	0	0
$9$	−16.2749	−0.602775
$10$	0	0
$11$	−33.9244	−0.929873	−0.464936	−	0.885344i	$-0.653923\pi$
$11$	−33.9244	−0.929873	−0.464936	+	0.885344i	$0.653923\pi$
$12$	0	0
$13$	3.07558	0.0656163	0.0328082	−	0.999462i	$-0.489555\pi$
$13$	3.07558	0.0656163	0.0328082	+	0.999462i	$0.489555\pi$
$14$	0	0
$15$	20.5498	0.353730
$16$	0	0
$17$	−14.2508	−0.203314	−0.101657	−	0.994820i	$-0.532414\pi$
$17$	−14.2508	−0.203314	−0.101657	+	0.994820i	$0.532414\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	59.2749	0.615945
$22$	0	0
$23$	−114.072	−1.03416	−0.517081	−	0.855937i	$-0.672981\pi$
$23$	−114.072	−1.03416	−0.517081	+	0.855937i	$0.672981\pi$
$24$	0	0
$25$	−85.6254	−0.685003
$26$	0	0
$27$	−141.722	−1.01016
$28$	0	0
$29$	34.5257	0.221078	0.110539	−	0.993872i	$-0.464742\pi$
$29$	34.5257	0.221078	0.110539	+	0.993872i	$0.464742\pi$
$30$	0	0
$31$	−107.698	−0.623970	−0.311985	−	0.950087i	$-0.600994\pi$
$31$	−107.698	−0.623970	−0.311985	+	0.950087i	$0.600994\pi$
$32$	0	0
$33$	−111.100	−0.586060
$34$	0	0
$35$	113.574	0.548500
$36$	0	0
$37$	181.698	0.807322	0.403661	−	0.914909i	$-0.367738\pi$
$37$	181.698	0.807322	0.403661	+	0.914909i	$0.367738\pi$
$38$	0	0
$39$	10.0723	0.0413552
$40$	0	0
$41$	−444.743	−1.69408	−0.847038	−	0.531532i	$-0.821616\pi$
$41$	−444.743	−1.69408	−0.847038	+	0.531532i	$0.821616\pi$
$42$	0	0
$43$	120.323	0.426723	0.213362	−	0.976973i	$-0.431559\pi$
$43$	120.323	0.426723	0.213362	+	0.976973i	$0.431559\pi$
$44$	0	0
$45$	−102.124	−0.338305
$46$	0	0
$47$	−306.371	−0.950826	−0.475413	−	0.879763i	$-0.657701\pi$
$47$	−306.371	−0.950826	−0.475413	+	0.879763i	$0.657701\pi$
$48$	0	0
$49$	−15.4020	−0.0449038
$50$	0	0
$51$	−46.6703	−0.128140
$52$	0	0
$53$	−115.825	−0.300184	−0.150092	−	0.988672i	$-0.547957\pi$
$53$	−115.825	−0.300184	−0.150092	+	0.988672i	$0.547957\pi$
$54$	0	0
$55$	−212.873	−0.521887
$56$	0	0
$57$	−62.2234	−0.144591
$58$	0	0
$59$	161.680	0.356762	0.178381	−	0.983961i	$-0.442914\pi$
$59$	161.680	0.356762	0.178381	+	0.983961i	$0.442914\pi$
$60$	0	0
$61$	−274.571	−0.576314	−0.288157	−	0.957583i	$-0.593043\pi$
$61$	−274.571	−0.576314	−0.288157	+	0.957583i	$0.593043\pi$
$62$	0	0
$63$	−294.571	−0.589086
$64$	0	0
$65$	19.2990	0.0368269
$66$	0	0
$67$	81.6254	0.148838	0.0744189	−	0.997227i	$-0.476290\pi$
$67$	81.6254	0.148838	0.0744189	+	0.997227i	$0.476290\pi$
$68$	0	0
$69$	−373.577	−0.651789
$70$	0	0
$71$	−773.492	−1.29291	−0.646455	−	0.762952i	$-0.723749\pi$
$71$	−773.492	−1.29291	−0.646455	+	0.762952i	$0.723749\pi$
$72$	0	0
$73$	148.794	0.238562	0.119281	−	0.992861i	$-0.461941\pi$
$73$	148.794	0.238562	0.119281	+	0.992861i	$0.461941\pi$
$74$	0	0
$75$	−280.416	−0.431729
$76$	0	0
$77$	−614.021	−0.908755
$78$	0	0
$79$	557.341	0.793743	0.396872	−	0.917874i	$-0.370096\pi$
$79$	557.341	0.793743	0.396872	+	0.917874i	$0.370096\pi$
$80$	0	0
$81$	−24.7043	−0.0338879
$82$	0	0
$83$	768.337	1.01610	0.508048	−	0.861329i	$-0.330367\pi$
$83$	768.337	1.01610	0.508048	+	0.861329i	$0.330367\pi$
$84$	0	0
$85$	−89.4228	−0.114109
$86$	0	0
$87$	113.069	0.139336
$88$	0	0
$89$	457.884	0.545344	0.272672	−	0.962107i	$-0.412093\pi$
$89$	457.884	0.545344	0.272672	+	0.962107i	$0.412093\pi$
$90$	0	0
$91$	55.6670	0.0641262
$92$	0	0
$93$	−352.701	−0.393262
$94$	0	0
$95$	−119.223	−0.128759
$96$	0	0
$97$	−1500.82	−1.57098	−0.785490	−	0.618874i	$-0.787589\pi$
$97$	−1500.82	−1.57098	−0.785490	+	0.618874i	$0.787589\pi$
$98$	0	0
$99$	552.117	0.560504
$100$	0	0
$101$	1731.06	1.70541	0.852707	−	0.522389i	$-0.174959\pi$
$101$	1731.06	1.70541	0.852707	+	0.522389i	$0.174959\pi$
$102$	0	0
$103$	1887.77	1.80590	0.902951	−	0.429743i	$-0.141396\pi$
$103$	1887.77	1.80590	0.902951	+	0.429743i	$0.141396\pi$
$104$	0	0
$105$	371.945	0.345697
$106$	0	0
$107$	−411.468	−0.371758	−0.185879	−	0.982573i	$-0.559513\pi$
$107$	−411.468	−0.371758	−0.185879	+	0.982573i	$0.559513\pi$
$108$	0	0
$109$	−1186.86	−1.04294	−0.521469	−	0.853270i	$-0.674616\pi$
$109$	−1186.86	−1.04294	−0.521469	+	0.853270i	$0.674616\pi$
$110$	0	0
$111$	595.045	0.508821
$112$	0	0
$113$	509.478	0.424139	0.212069	−	0.977255i	$-0.431980\pi$
$113$	509.478	0.424139	0.212069	+	0.977255i	$0.431980\pi$
$114$	0	0
$115$	−715.794	−0.580419
$116$	0	0
$117$	−50.0548	−0.0395519
$118$	0	0
$119$	−257.935	−0.198697
$120$	0	0
$121$	−180.134	−0.135337
$122$	0	0
$123$	−1456.50	−1.06771
$124$	0	0
$125$	−1321.66	−0.945701
$126$	0	0
$127$	1169.71	0.817284	0.408642	−	0.912695i	$-0.366002\pi$
$127$	1169.71	0.817284	0.408642	+	0.912695i	$0.366002\pi$
$128$	0	0
$129$	394.048	0.268946
$130$	0	0
$131$	−1052.31	−0.701838	−0.350919	−	0.936406i	$-0.614131\pi$
$131$	−1052.31	−0.701838	−0.350919	+	0.936406i	$0.614131\pi$
$132$	0	0
$133$	−343.894	−0.224206
$134$	0	0
$135$	−889.292	−0.566949
$136$	0	0
$137$	−2116.92	−1.32015	−0.660075	−	0.751200i	$-0.729476\pi$
$137$	−2116.92	−1.32015	−0.660075	+	0.751200i	$0.729476\pi$
$138$	0	0
$139$	−1115.90	−0.680929	−0.340464	−	0.940257i	$-0.610584\pi$
$139$	−1115.90	−0.680929	−0.340464	+	0.940257i	$0.610584\pi$
$140$	0	0
$141$	−1003.34	−0.599266
$142$	0	0
$143$	−104.337	−0.0610148
$144$	0	0
$145$	216.646	0.124079
$146$	0	0
$147$	−50.4402	−0.0283010
$148$	0	0
$149$	−472.693	−0.259896	−0.129948	−	0.991521i	$-0.541481\pi$
$149$	−472.693	−0.259896	−0.129948	+	0.991521i	$0.541481\pi$
$150$	0	0
$151$	1635.40	0.881372	0.440686	−	0.897661i	$-0.354735\pi$
$151$	1635.40	0.881372	0.440686	+	0.897661i	$0.354735\pi$
$152$	0	0
$153$	231.931	0.122552
$154$	0	0
$155$	−675.794	−0.350201
$156$	0	0
$157$	785.904	0.399503	0.199751	−	0.979847i	$-0.435987\pi$
$157$	785.904	0.399503	0.199751	+	0.979847i	$0.435987\pi$
$158$	0	0
$159$	−379.316	−0.189193
$160$	0	0
$161$	−2064.67	−1.01068
$162$	0	0
$163$	1224.95	0.588623	0.294311	−	0.955710i	$-0.404910\pi$
$163$	1224.95	0.588623	0.294311	+	0.955710i	$0.404910\pi$
$164$	0	0
$165$	−697.141	−0.328923
$166$	0	0
$167$	−1071.26	−0.496386	−0.248193	−	0.968711i	$-0.579837\pi$
$167$	−1071.26	−0.496386	−0.248193	+	0.968711i	$0.579837\pi$
$168$	0	0
$169$	−2187.54	−0.995694
$170$	0	0
$171$	309.223	0.138286
$172$	0	0
$173$	−825.821	−0.362925	−0.181462	−	0.983398i	$-0.558083\pi$
$173$	−825.821	−0.362925	−0.181462	+	0.983398i	$0.558083\pi$
$174$	0	0
$175$	−1549.79	−0.669447
$176$	0	0
$177$	529.489	0.224852
$178$	0	0
$179$	−465.802	−0.194501	−0.0972506	−	0.995260i	$-0.531005\pi$
$179$	−465.802	−0.194501	−0.0972506	+	0.995260i	$0.531005\pi$
$180$	0	0
$181$	1214.74	0.498847	0.249423	−	0.968395i	$-0.419759\pi$
$181$	1214.74	0.498847	0.249423	+	0.968395i	$0.419759\pi$
$182$	0	0
$183$	−899.196	−0.363227
$184$	0	0
$185$	1140.14	0.453106
$186$	0	0
$187$	483.451	0.189056
$188$	0	0
$189$	−2565.12	−0.987221
$190$	0	0
$191$	−154.943	−0.0586980	−0.0293490	−	0.999569i	$-0.509343\pi$
$191$	−154.943	−0.0586980	−0.0293490	+	0.999569i	$0.509343\pi$
$192$	0	0
$193$	−2776.64	−1.03558	−0.517790	−	0.855508i	$-0.673245\pi$
$193$	−2776.64	−1.03558	−0.517790	+	0.855508i	$0.673245\pi$
$194$	0	0
$195$	63.2026	0.0232104
$196$	0	0
$197$	−4865.09	−1.75951	−0.879754	−	0.475429i	$-0.842293\pi$
$197$	−4865.09	−1.75951	−0.879754	+	0.475429i	$0.842293\pi$
$198$	0	0
$199$	3014.43	1.07381	0.536903	−	0.843644i	$-0.319594\pi$
$199$	3014.43	1.07381	0.536903	+	0.843644i	$0.319594\pi$
$200$	0	0
$201$	267.316	0.0938062
$202$	0	0
$203$	624.905	0.216058
$204$	0	0
$205$	−2790.72	−0.950793
$206$	0	0
$207$	1856.52	0.623366
$208$	0	0
$209$	644.564	0.213327
$210$	0	0
$211$	3434.46	1.12056	0.560280	−	0.828303i	$-0.310693\pi$
$211$	3434.46	1.12056	0.560280	+	0.828303i	$0.310693\pi$
$212$	0	0
$213$	−2533.12	−0.814867
$214$	0	0
$215$	755.017	0.239497
$216$	0	0
$217$	−1949.29	−0.609800
$218$	0	0
$219$	487.288	0.150356
$220$	0	0
$221$	−43.8296	−0.0133407
$222$	0	0
$223$	2247.73	0.674975	0.337488	−	0.941330i	$-0.390423\pi$
$223$	2247.73	0.674975	0.337488	+	0.941330i	$0.390423\pi$
$224$	0	0
$225$	1393.55	0.412903
$226$	0	0
$227$	1883.81	0.550806	0.275403	−	0.961329i	$-0.411189\pi$
$227$	1883.81	0.550806	0.275403	+	0.961329i	$0.411189\pi$
$228$	0	0
$229$	−3594.49	−1.03725	−0.518626	−	0.855001i	$-0.673556\pi$
$229$	−3594.49	−1.03725	−0.518626	+	0.855001i	$0.673556\pi$
$230$	0	0
$231$	−2010.87	−0.572750
$232$	0	0
$233$	−5530.01	−1.55486	−0.777431	−	0.628968i	$-0.783478\pi$
$233$	−5530.01	−1.55486	−0.777431	+	0.628968i	$0.783478\pi$
$234$	0	0
$235$	−1922.45	−0.533647
$236$	0	0
$237$	1825.24	0.500263
$238$	0	0
$239$	−2587.82	−0.700387	−0.350193	−	0.936677i	$-0.613884\pi$
$239$	−2587.82	−0.700387	−0.350193	+	0.936677i	$0.613884\pi$
$240$	0	0
$241$	−3576.45	−0.955932	−0.477966	−	0.878378i	$-0.658626\pi$
$241$	−3576.45	−0.955932	−0.477966	+	0.878378i	$0.658626\pi$
$242$	0	0
$243$	3745.58	0.988804
$244$	0	0
$245$	−96.6462	−0.0252020
$246$	0	0
$247$	−58.4360	−0.0150534
$248$	0	0
$249$	2516.24	0.640403
$250$	0	0
$251$	5966.87	1.50050	0.750251	−	0.661154i	$-0.229933\pi$
$251$	5966.87	1.50050	0.750251	+	0.661154i	$0.229933\pi$
$252$	0	0
$253$	3869.84	0.961638
$254$	0	0
$255$	−292.852	−0.0719181
$256$	0	0
$257$	3841.01	0.932278	0.466139	−	0.884712i	$-0.345645\pi$
$257$	3841.01	0.932278	0.466139	+	0.884712i	$0.345645\pi$
$258$	0	0
$259$	3288.67	0.788988
$260$	0	0
$261$	−561.904	−0.133260
$262$	0	0
$263$	−3956.41	−0.927614	−0.463807	−	0.885936i	$-0.653517\pi$
$263$	−3956.41	−0.927614	−0.463807	+	0.885936i	$0.653517\pi$
$264$	0	0
$265$	−726.791	−0.168477
$266$	0	0
$267$	1499.53	0.343707
$268$	0	0
$269$	−2158.66	−0.489279	−0.244639	−	0.969614i	$-0.578670\pi$
$269$	−2158.66	−0.489279	−0.244639	+	0.969614i	$0.578670\pi$
$270$	0	0
$271$	2321.77	0.520435	0.260217	−	0.965550i	$-0.416206\pi$
$271$	2321.77	0.520435	0.260217	+	0.965550i	$0.416206\pi$
$272$	0	0
$273$	182.305	0.0404161
$274$	0	0
$275$	2904.79	0.636966
$276$	0	0
$277$	6184.24	1.34143	0.670713	−	0.741717i	$-0.265988\pi$
$277$	6184.24	1.34143	0.670713	+	0.741717i	$0.265988\pi$
$278$	0	0
$279$	1752.77	0.376113
$280$	0	0
$281$	−7981.21	−1.69437	−0.847187	−	0.531294i	$-0.821706\pi$
$281$	−7981.21	−1.69437	−0.847187	+	0.531294i	$0.821706\pi$
$282$	0	0
$283$	−4851.71	−1.01910	−0.509548	−	0.860442i	$-0.670187\pi$
$283$	−4851.71	−1.01910	−0.509548	+	0.860442i	$0.670187\pi$
$284$	0	0
$285$	−390.447	−0.0811511
$286$	0	0
$287$	−8049.69	−1.65560
$288$	0	0
$289$	−4709.91	−0.958664
$290$	0	0
$291$	−4915.06	−0.990123
$292$	0	0
$293$	1269.75	0.253172	0.126586	−	0.991956i	$-0.459598\pi$
$293$	1269.75	0.253172	0.126586	+	0.991956i	$0.459598\pi$
$294$	0	0
$295$	1014.53	0.200231
$296$	0	0
$297$	4807.83	0.939322
$298$	0	0
$299$	−350.838	−0.0678579
$300$	0	0
$301$	2177.81	0.417032
$302$	0	0
$303$	5669.08	1.07485
$304$	0	0
$305$	−1722.91	−0.323454
$306$	0	0
$307$	−9699.35	−1.80316	−0.901581	−	0.432610i	$-0.857593\pi$
$307$	−9699.35	−1.80316	−0.901581	+	0.432610i	$0.857593\pi$
$308$	0	0
$309$	6182.30	1.13818
$310$	0	0
$311$	323.284	0.0589446	0.0294723	−	0.999566i	$-0.490617\pi$
$311$	323.284	0.0589446	0.0294723	+	0.999566i	$0.490617\pi$
$312$	0	0
$313$	−3251.29	−0.587137	−0.293569	−	0.955938i	$-0.594843\pi$
$313$	−3251.29	−0.587137	−0.293569	+	0.955938i	$0.594843\pi$
$314$	0	0
$315$	−1848.41	−0.330622
$316$	0	0
$317$	7512.25	1.33101	0.665505	−	0.746393i	$-0.268216\pi$
$317$	7512.25	1.33101	0.665505	+	0.746393i	$0.268216\pi$
$318$	0	0
$319$	−1171.27	−0.205575
$320$	0	0
$321$	−1347.52	−0.234303
$322$	0	0
$323$	270.766	0.0466434
$324$	0	0
$325$	−263.348	−0.0449474
$326$	0	0
$327$	−3886.86	−0.657320
$328$	0	0
$329$	−5545.22	−0.929233
$330$	0	0
$331$	2857.14	0.474449	0.237225	−	0.971455i	$-0.423762\pi$
$331$	2857.14	0.474449	0.237225	+	0.971455i	$0.423762\pi$
$332$	0	0
$333$	−2957.11	−0.486633
$334$	0	0
$335$	512.193	0.0835346
$336$	0	0
$337$	10635.9	1.71921	0.859606	−	0.510958i	$-0.170709\pi$
$337$	10635.9	1.71921	0.859606	+	0.510958i	$0.170709\pi$
$338$	0	0
$339$	1668.50	0.267317
$340$	0	0
$341$	3653.58	0.580213
$342$	0	0
$343$	−6486.96	−1.02117
$344$	0	0
$345$	−2344.17	−0.365814
$346$	0	0
$347$	−8357.08	−1.29289	−0.646443	−	0.762962i	$-0.723744\pi$
$347$	−8357.08	−1.29289	−0.646443	+	0.762962i	$0.723744\pi$
$348$	0	0
$349$	−7741.28	−1.18734	−0.593669	−	0.804709i	$-0.702321\pi$
$349$	−7741.28	−1.18734	−0.593669	+	0.804709i	$0.702321\pi$
$350$	0	0
$351$	−435.877	−0.0662831
$352$	0	0
$353$	2480.46	0.373999	0.186999	−	0.982360i	$-0.440124\pi$
$353$	2480.46	0.373999	0.186999	+	0.982360i	$0.440124\pi$
$354$	0	0
$355$	−4853.60	−0.725640
$356$	0	0
$357$	−844.717	−0.125230
$358$	0	0
$359$	−4310.17	−0.633655	−0.316827	−	0.948483i	$-0.602618\pi$
$359$	−4310.17	−0.633655	−0.316827	+	0.948483i	$0.602618\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−589.923	−0.0852973
$364$	0	0
$365$	933.670	0.133892
$366$	0	0
$367$	8484.71	1.20681	0.603404	−	0.797436i	$-0.293811\pi$
$367$	8484.71	1.20681	0.603404	+	0.797436i	$0.293811\pi$
$368$	0	0
$369$	7238.15	1.02115
$370$	0	0
$371$	−2096.39	−0.293367
$372$	0	0
$373$	−8039.16	−1.11596	−0.557978	−	0.829855i	$-0.688423\pi$
$373$	−8039.16	−1.11596	−0.557978	+	0.829855i	$0.688423\pi$
$374$	0	0
$375$	−4328.32	−0.596036
$376$	0	0
$377$	106.187	0.0145063
$378$	0	0
$379$	−2976.22	−0.403372	−0.201686	−	0.979450i	$-0.564642\pi$
$379$	−2976.22	−0.403372	−0.201686	+	0.979450i	$0.564642\pi$
$380$	0	0
$381$	3830.71	0.515100
$382$	0	0
$383$	5118.15	0.682834	0.341417	−	0.939912i	$-0.389093\pi$
$383$	5118.15	0.682834	0.341417	+	0.939912i	$0.389093\pi$
$384$	0	0
$385$	−3852.93	−0.510035
$386$	0	0
$387$	−1958.25	−0.257218
$388$	0	0
$389$	6758.74	0.880929	0.440465	−	0.897770i	$-0.354814\pi$
$389$	6758.74	0.880929	0.440465	+	0.897770i	$0.354814\pi$
$390$	0	0
$391$	1625.62	0.210259
$392$	0	0
$393$	−3446.23	−0.442339
$394$	0	0
$395$	3497.27	0.445485
$396$	0	0
$397$	−14529.0	−1.83675	−0.918373	−	0.395716i	$-0.870496\pi$
$397$	−14529.0	−1.83675	−0.918373	+	0.395716i	$0.870496\pi$
$398$	0	0
$399$	−1126.22	−0.141308
$400$	0	0
$401$	12416.4	1.54625	0.773124	−	0.634255i	$-0.218693\pi$
$401$	12416.4	1.54625	0.773124	+	0.634255i	$0.218693\pi$
$402$	0	0
$403$	−331.233	−0.0409426
$404$	0	0
$405$	−155.017	−0.0190195
$406$	0	0
$407$	−6163.99	−0.750707
$408$	0	0
$409$	11615.8	1.40432	0.702159	−	0.712020i	$-0.252220\pi$
$409$	11615.8	1.40432	0.702159	+	0.712020i	$0.252220\pi$
$410$	0	0
$411$	−6932.73	−0.832035
$412$	0	0
$413$	2926.36	0.348660
$414$	0	0
$415$	4821.25	0.570279
$416$	0	0
$417$	−3654.47	−0.429161
$418$	0	0
$419$	16739.9	1.95179	0.975893	−	0.218248i	$-0.0700341\pi$
$419$	16739.9	1.95179	0.975893	+	0.218248i	$0.0700341\pi$
$420$	0	0
$421$	−6261.41	−0.724852	−0.362426	−	0.932013i	$-0.618051\pi$
$421$	−6261.41	−0.724852	−0.362426	+	0.932013i	$0.618051\pi$
$422$	0	0
$423$	4986.17	0.573134
$424$	0	0
$425$	1220.23	0.139271
$426$	0	0
$427$	−4969.64	−0.563226
$428$	0	0
$429$	−341.696	−0.0384551
$430$	0	0
$431$	8893.01	0.993878	0.496939	−	0.867786i	$-0.334457\pi$
$431$	8893.01	0.993878	0.496939	+	0.867786i	$0.334457\pi$
$432$	0	0
$433$	6326.61	0.702166	0.351083	−	0.936344i	$-0.385814\pi$
$433$	6326.61	0.702166	0.351083	+	0.936344i	$0.385814\pi$
$434$	0	0
$435$	709.498	0.0782019
$436$	0	0
$437$	2167.37	0.237253
$438$	0	0
$439$	−15823.4	−1.72030	−0.860149	−	0.510043i	$-0.829630\pi$
$439$	−15823.4	−1.72030	−0.860149	+	0.510043i	$0.829630\pi$
$440$	0	0
$441$	250.666	0.0270668
$442$	0	0
$443$	−14056.3	−1.50753	−0.753763	−	0.657147i	$-0.771763\pi$
$443$	−14056.3	−1.50753	−0.753763	+	0.657147i	$0.771763\pi$
$444$	0	0
$445$	2873.18	0.306072
$446$	0	0
$447$	−1548.03	−0.163802
$448$	0	0
$449$	15412.9	1.62000	0.809999	−	0.586431i	$-0.199468\pi$
$449$	15412.9	1.62000	0.809999	+	0.586431i	$0.199468\pi$
$450$	0	0
$451$	15087.6	1.57527
$452$	0	0
$453$	5355.81	0.555492
$454$	0	0
$455$	349.306	0.0359906
$456$	0	0
$457$	13336.9	1.36515	0.682573	−	0.730817i	$-0.260861\pi$
$457$	13336.9	1.36515	0.682573	+	0.730817i	$0.260861\pi$
$458$	0	0
$459$	2019.65	0.205380
$460$	0	0
$461$	10226.9	1.03322	0.516612	−	0.856220i	$-0.327193\pi$
$461$	10226.9	1.03322	0.516612	+	0.856220i	$0.327193\pi$
$462$	0	0
$463$	17797.4	1.78642	0.893212	−	0.449636i	$-0.148446\pi$
$463$	17797.4	1.78642	0.893212	+	0.449636i	$0.148446\pi$
$464$	0	0
$465$	−2213.17	−0.220717
$466$	0	0
$467$	−3100.03	−0.307179	−0.153589	−	0.988135i	$-0.549083\pi$
$467$	−3100.03	−0.307179	−0.153589	+	0.988135i	$0.549083\pi$
$468$	0	0
$469$	1477.39	0.145458
$470$	0	0
$471$	2573.77	0.251790
$472$	0	0
$473$	−4081.89	−0.396798
$474$	0	0
$475$	1626.88	0.157151
$476$	0	0
$477$	1885.04	0.180943
$478$	0	0
$479$	−6280.67	−0.599105	−0.299552	−	0.954080i	$-0.596837\pi$
$479$	−6280.67	−0.599105	−0.299552	+	0.954080i	$0.596837\pi$
$480$	0	0
$481$	558.826	0.0529735
$482$	0	0
$483$	−6761.62	−0.636987
$484$	0	0
$485$	−9417.52	−0.881706
$486$	0	0
$487$	−7940.07	−0.738807	−0.369403	−	0.929269i	$-0.620438\pi$
$487$	−7940.07	−0.738807	−0.369403	+	0.929269i	$0.620438\pi$
$488$	0	0
$489$	4011.61	0.370984
$490$	0	0
$491$	17276.3	1.58792	0.793959	−	0.607971i	$-0.208016\pi$
$491$	17276.3	1.58792	0.793959	+	0.607971i	$0.208016\pi$
$492$	0	0
$493$	−492.020	−0.0449482
$494$	0	0
$495$	3464.49	0.314580
$496$	0	0
$497$	−13999.9	−1.26355
$498$	0	0
$499$	14939.6	1.34026	0.670128	−	0.742246i	$-0.266239\pi$
$499$	14939.6	1.34026	0.670128	+	0.742246i	$0.266239\pi$
$500$	0	0
$501$	−3508.28	−0.312851
$502$	0	0
$503$	5559.39	0.492805	0.246402	−	0.969168i	$-0.420752\pi$
$503$	5559.39	0.492805	0.246402	+	0.969168i	$0.420752\pi$
$504$	0	0
$505$	10862.3	0.957157
$506$	0	0
$507$	−7164.02	−0.627545
$508$	0	0
$509$	−11029.4	−0.960452	−0.480226	−	0.877145i	$-0.659445\pi$
$509$	−11029.4	−0.960452	−0.480226	+	0.877145i	$0.659445\pi$
$510$	0	0
$511$	2693.12	0.233144
$512$	0	0
$513$	2692.71	0.231747
$514$	0	0
$515$	11845.6	1.01355
$516$	0	0
$517$	10393.5	0.884147
$518$	0	0
$519$	−2704.49	−0.228736
$520$	0	0
$521$	−7802.02	−0.656070	−0.328035	−	0.944665i	$-0.606386\pi$
$521$	−7802.02	−0.656070	−0.328035	+	0.944665i	$0.606386\pi$
$522$	0	0
$523$	−23625.2	−1.97525	−0.987627	−	0.156820i	$-0.949876\pi$
$523$	−23625.2	−1.97525	−0.987627	+	0.156820i	$0.949876\pi$
$524$	0	0
$525$	−5075.44	−0.421924
$526$	0	0
$527$	1534.78	0.126862
$528$	0	0
$529$	845.482	0.0694898
$530$	0	0
$531$	−2631.33	−0.215047
$532$	0	0
$533$	−1367.84	−0.111159
$534$	0	0
$535$	−2581.93	−0.208647
$536$	0	0
$537$	−1525.46	−0.122586
$538$	0	0
$539$	522.503	0.0417548
$540$	0	0
$541$	24753.3	1.96715	0.983573	−	0.180509i	$-0.0577744\pi$
$541$	24753.3	1.96715	0.983573	+	0.180509i	$0.0577744\pi$
$542$	0	0
$543$	3978.19	0.314402
$544$	0	0
$545$	−7447.43	−0.585344
$546$	0	0
$547$	5655.38	0.442060	0.221030	−	0.975267i	$-0.429058\pi$
$547$	5655.38	0.442060	0.221030	+	0.975267i	$0.429058\pi$
$548$	0	0
$549$	4468.61	0.347388
$550$	0	0
$551$	−655.989	−0.0507188
$552$	0	0
$553$	10087.7	0.775717
$554$	0	0
$555$	3733.86	0.285574
$556$	0	0
$557$	14716.3	1.11948	0.559741	−	0.828668i	$-0.310901\pi$
$557$	14716.3	1.11948	0.559741	+	0.828668i	$0.310901\pi$
$558$	0	0
$559$	370.063	0.0280000
$560$	0	0
$561$	1583.26	0.119154
$562$	0	0
$563$	−4400.91	−0.329443	−0.164721	−	0.986340i	$-0.552672\pi$
$563$	−4400.91	−0.329443	−0.164721	+	0.986340i	$0.552672\pi$
$564$	0	0
$565$	3196.94	0.238046
$566$	0	0
$567$	−447.140	−0.0331183
$568$	0	0
$569$	10176.3	0.749758	0.374879	−	0.927074i	$-0.377684\pi$
$569$	10176.3	0.749758	0.374879	+	0.927074i	$0.377684\pi$
$570$	0	0
$571$	6775.74	0.496595	0.248298	−	0.968684i	$-0.420129\pi$
$571$	6775.74	0.496595	0.248298	+	0.968684i	$0.420129\pi$
$572$	0	0
$573$	−507.427	−0.0369949
$574$	0	0
$575$	9767.49	0.708404
$576$	0	0
$577$	−3714.70	−0.268016	−0.134008	−	0.990980i	$-0.542785\pi$
$577$	−3714.70	−0.268016	−0.134008	+	0.990980i	$0.542785\pi$
$578$	0	0
$579$	−9093.26	−0.652683
$580$	0	0
$581$	13906.6	0.993021
$582$	0	0
$583$	3929.29	0.279133
$584$	0	0
$585$	−314.090	−0.0221983
$586$	0	0
$587$	−9362.24	−0.658298	−0.329149	−	0.944278i	$-0.606762\pi$
$587$	−9362.24	−0.658298	−0.329149	+	0.944278i	$0.606762\pi$
$588$	0	0
$589$	2046.26	0.143149
$590$	0	0
$591$	−15932.8	−1.10894
$592$	0	0
$593$	−14437.5	−0.999793	−0.499897	−	0.866085i	$-0.666629\pi$
$593$	−14437.5	−0.999793	−0.499897	+	0.866085i	$0.666629\pi$
$594$	0	0
$595$	−1618.52	−0.111518
$596$	0	0
$597$	9872.02	0.676775
$598$	0	0
$599$	19992.7	1.36374	0.681871	−	0.731473i	$-0.261167\pi$
$599$	19992.7	1.36374	0.681871	+	0.731473i	$0.261167\pi$
$600$	0	0
$601$	17371.4	1.17902	0.589511	−	0.807760i	$-0.299320\pi$
$601$	17371.4	1.17902	0.589511	+	0.807760i	$0.299320\pi$
$602$	0	0
$603$	−1328.45	−0.0897157
$604$	0	0
$605$	−1130.32	−0.0759574
$606$	0	0
$607$	14626.3	0.978025	0.489013	−	0.872277i	$-0.337357\pi$
$607$	14626.3	0.978025	0.489013	+	0.872277i	$0.337357\pi$
$608$	0	0
$609$	2046.51	0.136172
$610$	0	0
$611$	−942.269	−0.0623897
$612$	0	0
$613$	21052.2	1.38710	0.693549	−	0.720409i	$-0.256046\pi$
$613$	21052.2	1.38710	0.693549	+	0.720409i	$0.256046\pi$
$614$	0	0
$615$	−9139.39	−0.599245
$616$	0	0
$617$	1605.72	0.104771	0.0523855	−	0.998627i	$-0.483318\pi$
$617$	1605.72	0.104771	0.0523855	+	0.998627i	$0.483318\pi$
$618$	0	0
$619$	21457.6	1.39330	0.696652	−	0.717410i	$-0.254672\pi$
$619$	21457.6	1.39330	0.696652	+	0.717410i	$0.254672\pi$
$620$	0	0
$621$	16166.5	1.04467
$622$	0	0
$623$	8287.54	0.532959
$624$	0	0
$625$	2409.89	0.154233
$626$	0	0
$627$	2110.89	0.134451
$628$	0	0
$629$	−2589.34	−0.164140
$630$	0	0
$631$	−9732.44	−0.614013	−0.307007	−	0.951707i	$-0.599327\pi$
$631$	−9732.44	−0.614013	−0.307007	+	0.951707i	$0.599327\pi$
$632$	0	0
$633$	11247.6	0.706242
$634$	0	0
$635$	7339.84	0.458697
$636$	0	0
$637$	−47.3700	−0.00294642
$638$	0	0
$639$	12588.5	0.779333
$640$	0	0
$641$	−25594.0	−1.57707	−0.788536	−	0.614989i	$-0.789161\pi$
$641$	−25594.0	−1.57707	−0.788536	+	0.614989i	$0.789161\pi$
$642$	0	0
$643$	−11938.5	−0.732207	−0.366104	−	0.930574i	$-0.619308\pi$
$643$	−11938.5	−0.732207	−0.366104	+	0.930574i	$0.619308\pi$
$644$	0	0
$645$	2472.62	0.150945
$646$	0	0
$647$	−15685.4	−0.953105	−0.476552	−	0.879146i	$-0.658114\pi$
$647$	−15685.4	−0.953105	−0.476552	+	0.879146i	$0.658114\pi$
$648$	0	0
$649$	−5484.91	−0.331743
$650$	0	0
$651$	−6383.77	−0.384331
$652$	0	0
$653$	−11414.3	−0.684037	−0.342019	−	0.939693i	$-0.611111\pi$
$653$	−11414.3	−0.684037	−0.342019	+	0.939693i	$0.611111\pi$
$654$	0	0
$655$	−6603.16	−0.393903
$656$	0	0
$657$	−2421.61	−0.143799
$658$	0	0
$659$	24500.2	1.44824	0.724122	−	0.689672i	$-0.242245\pi$
$659$	24500.2	1.44824	0.724122	+	0.689672i	$0.242245\pi$
$660$	0	0
$661$	−24821.0	−1.46055	−0.730276	−	0.683152i	$-0.760609\pi$
$661$	−24821.0	−1.46055	−0.730276	+	0.683152i	$0.760609\pi$
$662$	0	0
$663$	−143.538	−0.00840808
$664$	0	0
$665$	−2157.90	−0.125835
$666$	0	0
$667$	−3938.43	−0.228631
$668$	0	0
$669$	7361.14	0.425408
$670$	0	0
$671$	9314.65	0.535899
$672$	0	0
$673$	−14862.9	−0.851297	−0.425649	−	0.904888i	$-0.639954\pi$
$673$	−14862.9	−0.851297	−0.425649	+	0.904888i	$0.639954\pi$
$674$	0	0
$675$	12135.0	0.691964
$676$	0	0
$677$	1218.79	0.0691902	0.0345951	−	0.999401i	$-0.488986\pi$
$677$	1218.79	0.0691902	0.0345951	+	0.999401i	$0.488986\pi$
$678$	0	0
$679$	−27164.3	−1.53530
$680$	0	0
$681$	6169.33	0.347150
$682$	0	0
$683$	−10180.0	−0.570319	−0.285159	−	0.958480i	$-0.592047\pi$
$683$	−10180.0	−0.570319	−0.285159	+	0.958480i	$0.592047\pi$
$684$	0	0
$685$	−13283.5	−0.740929
$686$	0	0
$687$	−11771.7	−0.653736
$688$	0	0
$689$	−356.228	−0.0196970
$690$	0	0
$691$	13712.9	0.754941	0.377470	−	0.926022i	$-0.376794\pi$
$691$	13712.9	0.754941	0.377470	+	0.926022i	$0.376794\pi$
$692$	0	0
$693$	9993.14	0.547775
$694$	0	0
$695$	−7002.16	−0.382168
$696$	0	0
$697$	6337.95	0.344429
$698$	0	0
$699$	−18110.3	−0.979964
$700$	0	0
$701$	29853.2	1.60848	0.804238	−	0.594307i	$-0.202574\pi$
$701$	29853.2	1.60848	0.804238	+	0.594307i	$0.202574\pi$
$702$	0	0
$703$	−3452.26	−0.185212
$704$	0	0
$705$	−6295.88	−0.336335
$706$	0	0
$707$	31331.6	1.66669
$708$	0	0
$709$	11488.4	0.608541	0.304271	−	0.952586i	$-0.401587\pi$
$709$	11488.4	0.608541	0.304271	+	0.952586i	$0.401587\pi$
$710$	0	0
$711$	−9070.67	−0.478448
$712$	0	0
$713$	12285.3	0.645286
$714$	0	0
$715$	−654.708	−0.0342443
$716$	0	0
$717$	−8474.91	−0.441424
$718$	0	0
$719$	17536.5	0.909596	0.454798	−	0.890595i	$-0.349711\pi$
$719$	17536.5	0.909596	0.454798	+	0.890595i	$0.349711\pi$
$720$	0	0
$721$	34168.1	1.76489
$722$	0	0
$723$	−11712.6	−0.602484
$724$	0	0
$725$	−2956.28	−0.151439
$726$	0	0
$727$	−27607.0	−1.40837	−0.704185	−	0.710016i	$-0.748688\pi$
$727$	−27607.0	−1.40837	−0.704185	+	0.710016i	$0.748688\pi$
$728$	0	0
$729$	12933.5	0.657089
$730$	0	0
$731$	−1714.70	−0.0867587
$732$	0	0
$733$	3519.63	0.177354	0.0886770	−	0.996060i	$-0.471736\pi$
$733$	3519.63	0.177354	0.0886770	+	0.996060i	$0.471736\pi$
$734$	0	0
$735$	−316.508	−0.0158838
$736$	0	0
$737$	−2769.09	−0.138400
$738$	0	0
$739$	4557.41	0.226857	0.113428	−	0.993546i	$-0.463817\pi$
$739$	4557.41	0.226857	0.113428	+	0.993546i	$0.463817\pi$
$740$	0	0
$741$	−191.373	−0.00948754
$742$	0	0
$743$	−5330.53	−0.263201	−0.131600	−	0.991303i	$-0.542012\pi$
$743$	−5330.53	−0.263201	−0.131600	+	0.991303i	$0.542012\pi$
$744$	0	0
$745$	−2966.11	−0.145866
$746$	0	0
$747$	−12504.6	−0.612477
$748$	0	0
$749$	−7447.43	−0.363315
$750$	0	0
$751$	−22386.8	−1.08776	−0.543879	−	0.839163i	$-0.683045\pi$
$751$	−22386.8	−1.08776	−0.543879	+	0.839163i	$0.683045\pi$
$752$	0	0
$753$	19541.0	0.945703
$754$	0	0
$755$	10262.0	0.494666
$756$	0	0
$757$	−25315.6	−1.21547	−0.607736	−	0.794139i	$-0.707922\pi$
$757$	−25315.6	−1.21547	−0.607736	+	0.794139i	$0.707922\pi$
$758$	0	0
$759$	12673.4	0.606080
$760$	0	0
$761$	−18715.0	−0.891485	−0.445742	−	0.895161i	$-0.647060\pi$
$761$	−18715.0	−0.891485	−0.445742	+	0.895161i	$0.647060\pi$
$762$	0	0
$763$	−21481.7	−1.01925
$764$	0	0
$765$	1455.35	0.0687820
$766$	0	0
$767$	497.260	0.0234094
$768$	0	0
$769$	20985.3	0.984070	0.492035	−	0.870575i	$-0.336253\pi$
$769$	20985.3	0.984070	0.492035	+	0.870575i	$0.336253\pi$
$770$	0	0
$771$	12579.0	0.587576
$772$	0	0
$773$	−6302.27	−0.293243	−0.146622	−	0.989193i	$-0.546840\pi$
$773$	−6302.27	−0.293243	−0.146622	+	0.989193i	$0.546840\pi$
$774$	0	0
$775$	9221.66	0.427422
$776$	0	0
$777$	10770.1	0.497266
$778$	0	0
$779$	8450.11	0.388648
$780$	0	0
$781$	26240.3	1.20224
$782$	0	0
$783$	−4893.05	−0.223325
$784$	0	0
$785$	4931.48	0.224219
$786$	0	0
$787$	4124.73	0.186825	0.0934123	−	0.995628i	$-0.470223\pi$
$787$	4124.73	0.186825	0.0934123	+	0.995628i	$0.470223\pi$
$788$	0	0
$789$	−12956.9	−0.584636
$790$	0	0
$791$	9221.39	0.414507
$792$	0	0
$793$	−844.464	−0.0378156
$794$	0	0
$795$	−2380.18	−0.106184
$796$	0	0
$797$	−8090.48	−0.359573	−0.179786	−	0.983706i	$-0.557541\pi$
$797$	−8090.48	−0.359573	−0.179786	+	0.983706i	$0.557541\pi$
$798$	0	0
$799$	4366.04	0.193316
$800$	0	0
$801$	−7452.02	−0.328719
$802$	0	0
$803$	−5047.75	−0.221832
$804$	0	0
$805$	−12955.6	−0.567237
$806$	0	0
$807$	−7069.44	−0.308372
$808$	0	0
$809$	−16735.9	−0.727323	−0.363662	−	0.931531i	$-0.618474\pi$
$809$	−16735.9	−0.727323	−0.363662	+	0.931531i	$0.618474\pi$
$810$	0	0
$811$	8476.59	0.367020	0.183510	−	0.983018i	$-0.441254\pi$
$811$	8476.59	0.367020	0.183510	+	0.983018i	$0.441254\pi$
$812$	0	0
$813$	7603.62	0.328008
$814$	0	0
$815$	7686.46	0.330362
$816$	0	0
$817$	−2286.14	−0.0978970
$818$	0	0
$819$	−905.975	−0.0386537
$820$	0	0
$821$	−902.399	−0.0383605	−0.0191802	−	0.999816i	$-0.506106\pi$
$821$	−902.399	−0.0383605	−0.0191802	+	0.999816i	$0.506106\pi$
$822$	0	0
$823$	23266.2	0.985431	0.492716	−	0.870190i	$-0.336004\pi$
$823$	23266.2	0.985431	0.492716	+	0.870190i	$0.336004\pi$
$824$	0	0
$825$	9512.96	0.401453
$826$	0	0
$827$	−17103.5	−0.719160	−0.359580	−	0.933114i	$-0.617080\pi$
$827$	−17103.5	−0.719160	−0.359580	+	0.933114i	$0.617080\pi$
$828$	0	0
$829$	6125.30	0.256623	0.128312	−	0.991734i	$-0.459044\pi$
$829$	6125.30	0.256623	0.128312	+	0.991734i	$0.459044\pi$
$830$	0	0
$831$	20252.9	0.845445
$832$	0	0
$833$	219.491	0.00912955
$834$	0	0
$835$	−6722.05	−0.278594
$836$	0	0
$837$	15263.1	0.630311
$838$	0	0
$839$	−19918.5	−0.819620	−0.409810	−	0.912171i	$-0.634405\pi$
$839$	−19918.5	−0.819620	−0.409810	+	0.912171i	$0.634405\pi$
$840$	0	0
$841$	−23197.0	−0.951124
$842$	0	0
$843$	−26137.8	−1.06789
$844$	0	0
$845$	−13726.6	−0.558829
$846$	0	0
$847$	−3260.36	−0.132264
$848$	0	0
$849$	−15889.0	−0.642294
$850$	0	0
$851$	−20726.7	−0.834901
$852$	0	0
$853$	−18785.3	−0.754042	−0.377021	−	0.926205i	$-0.623052\pi$
$853$	−18785.3	−0.754042	−0.377021	+	0.926205i	$0.623052\pi$
$854$	0	0
$855$	1940.35	0.0776124
$856$	0	0
$857$	−33483.4	−1.33462	−0.667311	−	0.744779i	$-0.732555\pi$
$857$	−33483.4	−1.33462	−0.667311	+	0.744779i	$0.732555\pi$
$858$	0	0
$859$	7322.74	0.290860	0.145430	−	0.989369i	$-0.453543\pi$
$859$	7322.74	0.290860	0.145430	+	0.989369i	$0.453543\pi$
$860$	0	0
$861$	−26362.1	−1.04346
$862$	0	0
$863$	−905.563	−0.0357193	−0.0178596	−	0.999841i	$-0.505685\pi$
$863$	−905.563	−0.0357193	−0.0178596	+	0.999841i	$0.505685\pi$
$864$	0	0
$865$	−5181.96	−0.203690
$866$	0	0
$867$	−15424.6	−0.604205
$868$	0	0
$869$	−18907.5	−0.738080
$870$	0	0
$871$	251.045	0.00976619
$872$	0	0
$873$	24425.7	0.946947
$874$	0	0
$875$	−23921.6	−0.924224
$876$	0	0
$877$	27155.7	1.04559	0.522795	−	0.852459i	$-0.324889\pi$
$877$	27155.7	1.04559	0.522795	+	0.852459i	$0.324889\pi$
$878$	0	0
$879$	4158.32	0.159564
$880$	0	0
$881$	26094.1	0.997881	0.498941	−	0.866636i	$-0.333723\pi$
$881$	26094.1	0.997881	0.498941	+	0.866636i	$0.333723\pi$
$882$	0	0
$883$	29674.6	1.13095	0.565476	−	0.824765i	$-0.308693\pi$
$883$	29674.6	1.13095	0.565476	+	0.824765i	$0.308693\pi$
$884$	0	0
$885$	3322.50	0.126197
$886$	0	0
$887$	1734.72	0.0656665	0.0328333	−	0.999461i	$-0.489547\pi$
$887$	1734.72	0.0656665	0.0328333	+	0.999461i	$0.489547\pi$
$888$	0	0
$889$	21171.4	0.798724
$890$	0	0
$891$	838.079	0.0315115
$892$	0	0
$893$	5821.05	0.218135
$894$	0	0
$895$	−2922.87	−0.109163
$896$	0	0
$897$	−1148.97	−0.0427680
$898$	0	0
$899$	−3718.34	−0.137946
$900$	0	0
$901$	1650.60	0.0610315
$902$	0	0
$903$	7132.14	0.262838
$904$	0	0
$905$	7622.42	0.279975
$906$	0	0
$907$	6755.01	0.247295	0.123647	−	0.992326i	$-0.460541\pi$
$907$	6755.01	0.247295	0.123647	+	0.992326i	$0.460541\pi$
$908$	0	0
$909$	−28172.9	−1.02798
$910$	0	0
$911$	35150.7	1.27837	0.639184	−	0.769054i	$-0.279272\pi$
$911$	35150.7	1.27837	0.639184	+	0.769054i	$0.279272\pi$
$912$	0	0
$913$	−26065.4	−0.944840
$914$	0	0
$915$	−5642.38	−0.203859
$916$	0	0
$917$	−19046.5	−0.685899
$918$	0	0
$919$	28684.2	1.02960	0.514802	−	0.857309i	$-0.327866\pi$
$919$	28684.2	1.02960	0.514802	+	0.857309i	$0.327866\pi$
$920$	0	0
$921$	−31764.6	−1.13646
$922$	0	0
$923$	−2378.94	−0.0848360
$924$	0	0
$925$	−15557.9	−0.553018
$926$	0	0
$927$	−30723.4	−1.08855
$928$	0	0
$929$	−3762.58	−0.132881	−0.0664405	−	0.997790i	$-0.521164\pi$
$929$	−3762.58	−0.132881	−0.0664405	+	0.997790i	$0.521164\pi$
$930$	0	0
$931$	292.638	0.0103016
$932$	0	0
$933$	1058.73	0.0371503
$934$	0	0
$935$	3033.62	0.106107
$936$	0	0
$937$	25465.7	0.887863	0.443932	−	0.896061i	$-0.353583\pi$
$937$	25465.7	0.887863	0.443932	+	0.896061i	$0.353583\pi$
$938$	0	0
$939$	−10647.7	−0.370048
$940$	0	0
$941$	−40865.6	−1.41571	−0.707854	−	0.706359i	$-0.750337\pi$
$941$	−40865.6	−1.41571	−0.707854	+	0.706359i	$0.750337\pi$
$942$	0	0
$943$	50732.8	1.75195
$944$	0	0
$945$	−16095.9	−0.554074
$946$	0	0
$947$	−8047.62	−0.276148	−0.138074	−	0.990422i	$-0.544091\pi$
$947$	−8047.62	−0.276148	−0.138074	+	0.990422i	$0.544091\pi$
$948$	0	0
$949$	457.628	0.0156536
$950$	0	0
$951$	24602.0	0.838880
$952$	0	0
$953$	−21562.2	−0.732915	−0.366458	−	0.930435i	$-0.619430\pi$
$953$	−21562.2	−0.732915	−0.366458	+	0.930435i	$0.619430\pi$
$954$	0	0
$955$	−972.257	−0.0329440
$956$	0	0
$957$	−3835.80	−0.129565
$958$	0	0
$959$	−38315.5	−1.29017
$960$	0	0
$961$	−18192.2	−0.610661
$962$	0	0
$963$	6696.60	0.224086
$964$	0	0
$965$	−17423.2	−0.581215
$966$	0	0
$967$	14667.4	0.487767	0.243883	−	0.969805i	$-0.421579\pi$
$967$	14667.4	0.487767	0.243883	+	0.969805i	$0.421579\pi$
$968$	0	0
$969$	886.735	0.0293974
$970$	0	0
$971$	−60154.7	−1.98811	−0.994056	−	0.108867i	$-0.965278\pi$
$971$	−60154.7	−1.98811	−0.994056	+	0.108867i	$0.965278\pi$
$972$	0	0
$973$	−20197.4	−0.665465
$974$	0	0
$975$	−862.442	−0.0283285
$976$	0	0
$977$	−29474.6	−0.965176	−0.482588	−	0.875848i	$-0.660303\pi$
$977$	−29474.6	−0.965176	−0.482588	+	0.875848i	$0.660303\pi$
$978$	0	0
$979$	−15533.4	−0.507100
$980$	0	0
$981$	19316.0	0.628657
$982$	0	0
$983$	−6813.42	−0.221073	−0.110536	−	0.993872i	$-0.535257\pi$
$983$	−6813.42	−0.221073	−0.110536	+	0.993872i	$0.535257\pi$
$984$	0	0
$985$	−30528.0	−0.987517
$986$	0	0
$987$	−18160.1	−0.585657
$988$	0	0
$989$	−13725.5	−0.441301
$990$	0	0
$991$	11660.9	0.373786	0.186893	−	0.982380i	$-0.440158\pi$
$991$	11660.9	0.373786	0.186893	+	0.982380i	$0.440158\pi$
$992$	0	0
$993$	9356.91	0.299026
$994$	0	0
$995$	18915.3	0.602669
$996$	0	0
$997$	37537.3	1.19240	0.596198	−	0.802838i	$-0.296677\pi$
$997$	37537.3	1.19240	0.596198	+	0.802838i	$0.296677\pi$
$998$	0	0
$999$	−25750.5	−0.815526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.k.1.2		2
4.3	odd	2		1216.4.a.m.1.1		2
8.3	odd	2		152.4.a.a.1.2	✓	2
8.5	even	2		304.4.a.e.1.1		2
24.11	even	2		1368.4.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.a.1.2	✓	2	8.3	odd	2
304.4.a.e.1.1		2	8.5	even	2
1216.4.a.k.1.2		2	1.1	even	1	trivial
1216.4.a.m.1.1		2	4.3	odd	2
1368.4.a.a.1.2		2	24.11	even	2

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+3.27492 q^{3} +6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} +O(q^{10})\)	\(q+3.27492 q^{3} +6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} -33.9244 q^{11} +3.07558 q^{13} +20.5498 q^{15} -14.2508 q^{17} -19.0000 q^{19} +59.2749 q^{21} -114.072 q^{23} -85.6254 q^{25} -141.722 q^{27} +34.5257 q^{29} -107.698 q^{31} -111.100 q^{33} +113.574 q^{35} +181.698 q^{37} +10.0723 q^{39} -444.743 q^{41} +120.323 q^{43} -102.124 q^{45} -306.371 q^{47} -15.4020 q^{49} -46.6703 q^{51} -115.825 q^{53} -212.873 q^{55} -62.2234 q^{57} +161.680 q^{59} -274.571 q^{61} -294.571 q^{63} +19.2990 q^{65} +81.6254 q^{67} -373.577 q^{69} -773.492 q^{71} +148.794 q^{73} -280.416 q^{75} -614.021 q^{77} +557.341 q^{79} -24.7043 q^{81} +768.337 q^{83} -89.4228 q^{85} +113.069 q^{87} +457.884 q^{89} +55.6670 q^{91} -352.701 q^{93} -119.223 q^{95} -1500.82 q^{97} +552.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 5 q^{5} + 6 q^{7} - 25 q^{9}+O(q^{10}) \) 2 * q - q^3 + 5 * q^5 + 6 * q^7 - 25 * q^9	\( 2 q - q^{3} + 5 q^{5} + 6 q^{7} - 25 q^{9} - 15 q^{11} + 59 q^{13} + 26 q^{15} - 104 q^{17} - 38 q^{19} + 111 q^{21} + 21 q^{23} - 209 q^{25} + 11 q^{27} + 137 q^{29} - 4 q^{31} - 192 q^{33} + 129 q^{35} + 152 q^{37} - 229 q^{39} - 210 q^{41} + 67 q^{43} - 91 q^{45} - 273 q^{47} - 212 q^{49} + 337 q^{51} - 209 q^{53} - 237 q^{55} + 19 q^{57} + 799 q^{59} - 149 q^{61} - 189 q^{63} - 52 q^{65} + 201 q^{67} - 951 q^{69} - 792 q^{71} - 246 q^{73} + 247 q^{75} - 843 q^{77} + 254 q^{79} - 442 q^{81} + 374 q^{83} + 25 q^{85} - 325 q^{87} - 564 q^{89} - 621 q^{91} - 796 q^{93} - 95 q^{95} - 178 q^{97} + 387 q^{99}+O(q^{100}) \) 2 * q - q^3 + 5 * q^5 + 6 * q^7 - 25 * q^9 - 15 * q^11 + 59 * q^13 + 26 * q^15 - 104 * q^17 - 38 * q^19 + 111 * q^21 + 21 * q^23 - 209 * q^25 + 11 * q^27 + 137 * q^29 - 4 * q^31 - 192 * q^33 + 129 * q^35 + 152 * q^37 - 229 * q^39 - 210 * q^41 + 67 * q^43 - 91 * q^45 - 273 * q^47 - 212 * q^49 + 337 * q^51 - 209 * q^53 - 237 * q^55 + 19 * q^57 + 799 * q^59 - 149 * q^61 - 189 * q^63 - 52 * q^65 + 201 * q^67 - 951 * q^69 - 792 * q^71 - 246 * q^73 + 247 * q^75 - 843 * q^77 + 254 * q^79 - 442 * q^81 + 374 * q^83 + 25 * q^85 - 325 * q^87 - 564 * q^89 - 621 * q^91 - 796 * q^93 - 95 * q^95 - 178 * q^97 + 387 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more