LMFDB - Embedded newform 1216.4.a.r.1.3

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:	$3$
Coefficient field:	3.3.7057.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 22x + 32 $ x^3 - x^2 - 22*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.50686$ 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)

$f(q)$	$=$	$q+5.61812 q^{3} +13.8269 q^{5} -9.22252 q^{7} +4.56325 q^{9} +O(q^{10})$	$q+5.61812 q^{3} +13.8269 q^{5} -9.22252 q^{7} +4.56325 q^{9} -59.8818 q^{11} -38.2034 q^{13} +77.6813 q^{15} -47.0714 q^{17} +19.0000 q^{19} -51.8132 q^{21} +179.632 q^{23} +66.1837 q^{25} -126.052 q^{27} +96.6484 q^{29} +125.918 q^{31} -336.423 q^{33} -127.519 q^{35} -407.764 q^{37} -214.631 q^{39} -220.654 q^{41} -100.777 q^{43} +63.0957 q^{45} -213.749 q^{47} -257.945 q^{49} -264.453 q^{51} -19.8284 q^{53} -827.981 q^{55} +106.744 q^{57} +97.7827 q^{59} -266.365 q^{61} -42.0847 q^{63} -528.235 q^{65} -627.332 q^{67} +1009.19 q^{69} -644.105 q^{71} +807.966 q^{73} +371.828 q^{75} +552.261 q^{77} +785.715 q^{79} -831.385 q^{81} -1174.90 q^{83} -650.853 q^{85} +542.982 q^{87} -234.248 q^{89} +352.331 q^{91} +707.421 q^{93} +262.711 q^{95} -1348.51 q^{97} -273.256 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9	$ 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9} - 103 q^{11} - 32 q^{13} + 122 q^{15} + 11 q^{17} + 57 q^{19} - 114 q^{21} + 316 q^{23} + 162 q^{25} - 178 q^{27} + 138 q^{29} + 420 q^{31} - 330 q^{33} - 333 q^{35} - 102 q^{37} + 164 q^{39} - 370 q^{41} - 431 q^{43} + 429 q^{45} - 199 q^{47} - 802 q^{49} - 272 q^{51} + 308 q^{53} + 85 q^{55} - 76 q^{57} + 188 q^{59} + 609 q^{61} - 61 q^{63} - 1536 q^{65} + 246 q^{67} + 892 q^{69} - 954 q^{71} - 629 q^{73} + 1074 q^{75} + 71 q^{77} + 452 q^{79} - 361 q^{81} + 780 q^{83} - 1883 q^{85} - 96 q^{87} - 1356 q^{89} + 670 q^{91} - 1092 q^{93} - 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9 - 103 * q^11 - 32 * q^13 + 122 * q^15 + 11 * q^17 + 57 * q^19 - 114 * q^21 + 316 * q^23 + 162 * q^25 - 178 * q^27 + 138 * q^29 + 420 * q^31 - 330 * q^33 - 333 * q^35 - 102 * q^37 + 164 * q^39 - 370 * q^41 - 431 * q^43 + 429 * q^45 - 199 * q^47 - 802 * q^49 - 272 * q^51 + 308 * q^53 + 85 * q^55 - 76 * q^57 + 188 * q^59 + 609 * q^61 - 61 * q^63 - 1536 * q^65 + 246 * q^67 + 892 * q^69 - 954 * q^71 - 629 * q^73 + 1074 * q^75 + 71 * q^77 + 452 * q^79 - 361 * q^81 + 780 * q^83 - 1883 * q^85 - 96 * q^87 - 1356 * q^89 + 670 * q^91 - 1092 * q^93 - 133 * q^95 - 548 * q^97 + 1305 * 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
$3$	5.61812	1.08121	0.540604	−	0.841277i	$-0.318196\pi$
$3$	5.61812	1.08121	0.540604	+	0.841277i	$0.318196\pi$
$4$	0	0
$5$	13.8269	1.23672	0.618359	−	0.785896i	$-0.287798\pi$
$5$	13.8269	1.23672	0.618359	+	0.785896i	$0.287798\pi$
$6$	0	0
$7$	−9.22252	−0.497969	−0.248985	−	0.968507i	$-0.580097\pi$
$7$	−9.22252	−0.497969	−0.248985	+	0.968507i	$0.580097\pi$
$8$	0	0
$9$	4.56325	0.169009
$10$	0	0
$11$	−59.8818	−1.64137	−0.820684	−	0.571383i	$-0.806407\pi$
$11$	−59.8818	−1.64137	−0.820684	+	0.571383i	$0.806407\pi$
$12$	0	0
$13$	−38.2034	−0.815055	−0.407527	−	0.913193i	$-0.633609\pi$
$13$	−38.2034	−0.815055	−0.407527	+	0.913193i	$0.633609\pi$
$14$	0	0
$15$	77.6813	1.33715
$16$	0	0
$17$	−47.0714	−0.671559	−0.335779	−	0.941941i	$-0.609000\pi$
$17$	−47.0714	−0.671559	−0.335779	+	0.941941i	$0.609000\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−51.8132	−0.538408
$22$	0	0
$23$	179.632	1.62851	0.814256	−	0.580506i	$-0.197145\pi$
$23$	179.632	1.62851	0.814256	+	0.580506i	$0.197145\pi$
$24$	0	0
$25$	66.1837	0.529470
$26$	0	0
$27$	−126.052	−0.898473
$28$	0	0
$29$	96.6484	0.618868	0.309434	−	0.950921i	$-0.399860\pi$
$29$	96.6484	0.618868	0.309434	+	0.950921i	$0.399860\pi$
$30$	0	0
$31$	125.918	0.729532	0.364766	−	0.931099i	$-0.381149\pi$
$31$	125.918	0.729532	0.364766	+	0.931099i	$0.381149\pi$
$32$	0	0
$33$	−336.423	−1.77466
$34$	0	0
$35$	−127.519	−0.615847
$36$	0	0
$37$	−407.764	−1.81178	−0.905891	−	0.423510i	$-0.860798\pi$
$37$	−407.764	−1.81178	−0.905891	+	0.423510i	$0.860798\pi$
$38$	0	0
$39$	−214.631	−0.881243
$40$	0	0
$41$	−220.654	−0.840495	−0.420248	−	0.907409i	$-0.638057\pi$
$41$	−220.654	−0.840495	−0.420248	+	0.907409i	$0.638057\pi$
$42$	0	0
$43$	−100.777	−0.357403	−0.178702	−	0.983903i	$-0.557190\pi$
$43$	−100.777	−0.357403	−0.178702	+	0.983903i	$0.557190\pi$
$44$	0	0
$45$	63.0957	0.209017
$46$	0	0
$47$	−213.749	−0.663372	−0.331686	−	0.943390i	$-0.607617\pi$
$47$	−213.749	−0.663372	−0.331686	+	0.943390i	$0.607617\pi$
$48$	0	0
$49$	−257.945	−0.752027
$50$	0	0
$51$	−264.453	−0.726095
$52$	0	0
$53$	−19.8284	−0.0513895	−0.0256948	−	0.999670i	$-0.508180\pi$
$53$	−19.8284	−0.0513895	−0.0256948	+	0.999670i	$0.508180\pi$
$54$	0	0
$55$	−827.981	−2.02991
$56$	0	0
$57$	106.744	0.248046
$58$	0	0
$59$	97.7827	0.215767	0.107883	−	0.994164i	$-0.465593\pi$
$59$	97.7827	0.215767	0.107883	+	0.994164i	$0.465593\pi$
$60$	0	0
$61$	−266.365	−0.559091	−0.279545	−	0.960132i	$-0.590184\pi$
$61$	−266.365	−0.559091	−0.279545	+	0.960132i	$0.590184\pi$
$62$	0	0
$63$	−42.0847	−0.0841614
$64$	0	0
$65$	−528.235	−1.00799
$66$	0	0
$67$	−627.332	−1.14389	−0.571946	−	0.820291i	$-0.693811\pi$
$67$	−627.332	−1.14389	−0.571946	+	0.820291i	$0.693811\pi$
$68$	0	0
$69$	1009.19	1.76076
$70$	0	0
$71$	−644.105	−1.07664	−0.538318	−	0.842741i	$-0.680940\pi$
$71$	−644.105	−1.07664	−0.538318	+	0.842741i	$0.680940\pi$
$72$	0	0
$73$	807.966	1.29541	0.647707	−	0.761889i	$-0.275728\pi$
$73$	807.966	1.29541	0.647707	+	0.761889i	$0.275728\pi$
$74$	0	0
$75$	371.828	0.572467
$76$	0	0
$77$	552.261	0.817350
$78$	0	0
$79$	785.715	1.11899	0.559493	−	0.828835i	$-0.310996\pi$
$79$	785.715	1.11899	0.559493	+	0.828835i	$0.310996\pi$
$80$	0	0
$81$	−831.385	−1.14045
$82$	0	0
$83$	−1174.90	−1.55376	−0.776879	−	0.629650i	$-0.783198\pi$
$83$	−1174.90	−1.55376	−0.776879	+	0.629650i	$0.783198\pi$
$84$	0	0
$85$	−650.853	−0.830529
$86$	0	0
$87$	542.982	0.669124
$88$	0	0
$89$	−234.248	−0.278991	−0.139496	−	0.990223i	$-0.544548\pi$
$89$	−234.248	−0.278991	−0.139496	+	0.990223i	$0.544548\pi$
$90$	0	0
$91$	352.331	0.405872
$92$	0	0
$93$	707.421	0.788775
$94$	0	0
$95$	262.711	0.283722
$96$	0	0
$97$	−1348.51	−1.41155	−0.705774	−	0.708437i	$-0.749401\pi$
$97$	−1348.51	−1.41155	−0.705774	+	0.708437i	$0.749401\pi$
$98$	0	0
$99$	−273.256	−0.277406
$100$	0	0
$101$	−867.044	−0.854199	−0.427100	−	0.904205i	$-0.640465\pi$
$101$	−867.044	−0.854199	−0.427100	+	0.904205i	$0.640465\pi$
$102$	0	0
$103$	492.373	0.471019	0.235509	−	0.971872i	$-0.424324\pi$
$103$	492.373	0.471019	0.235509	+	0.971872i	$0.424324\pi$
$104$	0	0
$105$	−716.417	−0.665859
$106$	0	0
$107$	1307.06	1.18092	0.590460	−	0.807067i	$-0.298946\pi$
$107$	1307.06	1.18092	0.590460	+	0.807067i	$0.298946\pi$
$108$	0	0
$109$	−1435.58	−1.26150	−0.630749	−	0.775987i	$-0.717252\pi$
$109$	−1435.58	−1.26150	−0.630749	+	0.775987i	$0.717252\pi$
$110$	0	0
$111$	−2290.87	−1.95891
$112$	0	0
$113$	1779.17	1.48116	0.740578	−	0.671970i	$-0.234552\pi$
$113$	1779.17	1.48116	0.740578	+	0.671970i	$0.234552\pi$
$114$	0	0
$115$	2483.75	2.01401
$116$	0	0
$117$	−174.332	−0.137752
$118$	0	0
$119$	434.117	0.334416
$120$	0	0
$121$	2254.83	1.69409
$122$	0	0
$123$	−1239.66	−0.908750
$124$	0	0
$125$	−813.248	−0.581913
$126$	0	0
$127$	1334.73	0.932582	0.466291	−	0.884631i	$-0.345590\pi$
$127$	1334.73	0.932582	0.466291	+	0.884631i	$0.345590\pi$
$128$	0	0
$129$	−566.177	−0.386427
$130$	0	0
$131$	−52.5550	−0.0350516	−0.0175258	−	0.999846i	$-0.505579\pi$
$131$	−52.5550	−0.0350516	−0.0175258	+	0.999846i	$0.505579\pi$
$132$	0	0
$133$	−175.228	−0.114242
$134$	0	0
$135$	−1742.92	−1.11116
$136$	0	0
$137$	852.662	0.531736	0.265868	−	0.964009i	$-0.414341\pi$
$137$	852.662	0.531736	0.265868	+	0.964009i	$0.414341\pi$
$138$	0	0
$139$	812.834	0.495998	0.247999	−	0.968760i	$-0.420227\pi$
$139$	812.834	0.495998	0.247999	+	0.968760i	$0.420227\pi$
$140$	0	0
$141$	−1200.87	−0.717243
$142$	0	0
$143$	2287.69	1.33780
$144$	0	0
$145$	1336.35	0.765364
$146$	0	0
$147$	−1449.17	−0.813097
$148$	0	0
$149$	−3103.16	−1.70618	−0.853089	−	0.521765i	$-0.825274\pi$
$149$	−3103.16	−1.70618	−0.853089	+	0.521765i	$0.825274\pi$
$150$	0	0
$151$	109.068	0.0587803	0.0293902	−	0.999568i	$-0.490643\pi$
$151$	109.068	0.0587803	0.0293902	+	0.999568i	$0.490643\pi$
$152$	0	0
$153$	−214.799	−0.113500
$154$	0	0
$155$	1741.05	0.902225
$156$	0	0
$157$	−1949.74	−0.991123	−0.495561	−	0.868573i	$-0.665038\pi$
$157$	−1949.74	−0.991123	−0.495561	+	0.868573i	$0.665038\pi$
$158$	0	0
$159$	−111.398	−0.0555627
$160$	0	0
$161$	−1656.66	−0.810949
$162$	0	0
$163$	−2089.53	−1.00408	−0.502040	−	0.864845i	$-0.667417\pi$
$163$	−2089.53	−1.00408	−0.502040	+	0.864845i	$0.667417\pi$
$164$	0	0
$165$	−4651.69	−2.19475
$166$	0	0
$167$	−506.705	−0.234790	−0.117395	−	0.993085i	$-0.537454\pi$
$167$	−506.705	−0.234790	−0.117395	+	0.993085i	$0.537454\pi$
$168$	0	0
$169$	−737.502	−0.335686
$170$	0	0
$171$	86.7018	0.0387734
$172$	0	0
$173$	3668.57	1.61223	0.806116	−	0.591758i	$-0.201566\pi$
$173$	3668.57	1.61223	0.806116	+	0.591758i	$0.201566\pi$
$174$	0	0
$175$	−610.381	−0.263660
$176$	0	0
$177$	549.355	0.233288
$178$	0	0
$179$	3873.56	1.61745	0.808725	−	0.588187i	$-0.200158\pi$
$179$	3873.56	1.61745	0.808725	+	0.588187i	$0.200158\pi$
$180$	0	0
$181$	−559.567	−0.229791	−0.114896	−	0.993378i	$-0.536653\pi$
$181$	−559.567	−0.229791	−0.114896	+	0.993378i	$0.536653\pi$
$182$	0	0
$183$	−1496.47	−0.604493
$184$	0	0
$185$	−5638.12	−2.24066
$186$	0	0
$187$	2818.72	1.10227
$188$	0	0
$189$	1162.52	0.447412
$190$	0	0
$191$	−1333.34	−0.505115	−0.252558	−	0.967582i	$-0.581272\pi$
$191$	−1333.34	−0.505115	−0.252558	+	0.967582i	$0.581272\pi$
$192$	0	0
$193$	−3723.10	−1.38857	−0.694287	−	0.719698i	$-0.744280\pi$
$193$	−3723.10	−1.38857	−0.694287	+	0.719698i	$0.744280\pi$
$194$	0	0
$195$	−2967.69	−1.08985
$196$	0	0
$197$	4989.58	1.80453	0.902267	−	0.431178i	$-0.141902\pi$
$197$	4989.58	1.80453	0.902267	+	0.431178i	$0.141902\pi$
$198$	0	0
$199$	5308.31	1.89094	0.945468	−	0.325714i	$-0.105605\pi$
$199$	5308.31	1.89094	0.945468	+	0.325714i	$0.105605\pi$
$200$	0	0
$201$	−3524.43	−1.23679
$202$	0	0
$203$	−891.342	−0.308177
$204$	0	0
$205$	−3050.96	−1.03946
$206$	0	0
$207$	819.704	0.275234
$208$	0	0
$209$	−1137.75	−0.376555
$210$	0	0
$211$	1738.60	0.567251	0.283626	−	0.958935i	$-0.408463\pi$
$211$	1738.60	0.567251	0.283626	+	0.958935i	$0.408463\pi$
$212$	0	0
$213$	−3618.66	−1.16407
$214$	0	0
$215$	−1393.44	−0.442007
$216$	0	0
$217$	−1161.28	−0.363284
$218$	0	0
$219$	4539.25	1.40061
$220$	0	0
$221$	1798.29	0.547357
$222$	0	0
$223$	1278.02	0.383777	0.191889	−	0.981417i	$-0.438539\pi$
$223$	1278.02	0.383777	0.191889	+	0.981417i	$0.438539\pi$
$224$	0	0
$225$	302.013	0.0894853
$226$	0	0
$227$	−1022.70	−0.299027	−0.149514	−	0.988760i	$-0.547771\pi$
$227$	−1022.70	−0.299027	−0.149514	+	0.988760i	$0.547771\pi$
$228$	0	0
$229$	−600.407	−0.173258	−0.0866289	−	0.996241i	$-0.527609\pi$
$229$	−600.407	−0.173258	−0.0866289	+	0.996241i	$0.527609\pi$
$230$	0	0
$231$	3102.67	0.883725
$232$	0	0
$233$	6067.68	1.70604	0.853019	−	0.521879i	$-0.174769\pi$
$233$	6067.68	1.70604	0.853019	+	0.521879i	$0.174769\pi$
$234$	0	0
$235$	−2955.49	−0.820404
$236$	0	0
$237$	4414.24	1.20986
$238$	0	0
$239$	11.8896	0.00321788	0.00160894	−	0.999999i	$-0.499488\pi$
$239$	11.8896	0.00321788	0.00160894	+	0.999999i	$0.499488\pi$
$240$	0	0
$241$	−2816.80	−0.752888	−0.376444	−	0.926439i	$-0.622853\pi$
$241$	−2816.80	−0.752888	−0.376444	+	0.926439i	$0.622853\pi$
$242$	0	0
$243$	−1267.40	−0.334584
$244$	0	0
$245$	−3566.59	−0.930044
$246$	0	0
$247$	−725.864	−0.186986
$248$	0	0
$249$	−6600.72	−1.67993
$250$	0	0
$251$	3973.52	0.999228	0.499614	−	0.866248i	$-0.333475\pi$
$251$	3973.52	0.999228	0.499614	+	0.866248i	$0.333475\pi$
$252$	0	0
$253$	−10756.7	−2.67299
$254$	0	0
$255$	−3656.57	−0.897974
$256$	0	0
$257$	−7356.09	−1.78545	−0.892724	−	0.450603i	$-0.851209\pi$
$257$	−7356.09	−1.78545	−0.892724	+	0.450603i	$0.851209\pi$
$258$	0	0
$259$	3760.61	0.902212
$260$	0	0
$261$	441.031	0.104594
$262$	0	0
$263$	4092.74	0.959578	0.479789	−	0.877384i	$-0.340713\pi$
$263$	4092.74	0.959578	0.479789	+	0.877384i	$0.340713\pi$
$264$	0	0
$265$	−274.166	−0.0635543
$266$	0	0
$267$	−1316.03	−0.301647
$268$	0	0
$269$	3662.93	0.830234	0.415117	−	0.909768i	$-0.363741\pi$
$269$	3662.93	0.830234	0.415117	+	0.909768i	$0.363741\pi$
$270$	0	0
$271$	5576.56	1.25001	0.625003	−	0.780622i	$-0.285098\pi$
$271$	5576.56	1.25001	0.625003	+	0.780622i	$0.285098\pi$
$272$	0	0
$273$	1979.44	0.438832
$274$	0	0
$275$	−3963.20	−0.869055
$276$	0	0
$277$	−764.590	−0.165848	−0.0829238	−	0.996556i	$-0.526426\pi$
$277$	−764.590	−0.165848	−0.0829238	+	0.996556i	$0.526426\pi$
$278$	0	0
$279$	574.594	0.123298
$280$	0	0
$281$	−2805.51	−0.595597	−0.297798	−	0.954629i	$-0.596252\pi$
$281$	−2805.51	−0.595597	−0.297798	+	0.954629i	$0.596252\pi$
$282$	0	0
$283$	2564.76	0.538725	0.269362	−	0.963039i	$-0.413187\pi$
$283$	2564.76	0.538725	0.269362	+	0.963039i	$0.413187\pi$
$284$	0	0
$285$	1475.94	0.306763
$286$	0	0
$287$	2034.98	0.418541
$288$	0	0
$289$	−2697.28	−0.549009
$290$	0	0
$291$	−7576.07	−1.52618
$292$	0	0
$293$	6350.30	1.26617	0.633086	−	0.774081i	$-0.281788\pi$
$293$	6350.30	1.26617	0.633086	+	0.774081i	$0.281788\pi$
$294$	0	0
$295$	1352.03	0.266842
$296$	0	0
$297$	7548.24	1.47472
$298$	0	0
$299$	−6862.54	−1.32733
$300$	0	0
$301$	929.418	0.177976
$302$	0	0
$303$	−4871.16	−0.923567
$304$	0	0
$305$	−3683.01	−0.691437
$306$	0	0
$307$	−7556.96	−1.40488	−0.702441	−	0.711742i	$-0.747906\pi$
$307$	−7556.96	−1.40488	−0.702441	+	0.711742i	$0.747906\pi$
$308$	0	0
$309$	2766.21	0.509269
$310$	0	0
$311$	−7200.87	−1.31294	−0.656470	−	0.754352i	$-0.727951\pi$
$311$	−7200.87	−1.31294	−0.656470	+	0.754352i	$0.727951\pi$
$312$	0	0
$313$	7971.84	1.43960	0.719800	−	0.694181i	$-0.244233\pi$
$313$	7971.84	1.43960	0.719800	+	0.694181i	$0.244233\pi$
$314$	0	0
$315$	−581.901	−0.104084
$316$	0	0
$317$	−6328.95	−1.12135	−0.560677	−	0.828035i	$-0.689459\pi$
$317$	−6328.95	−1.12135	−0.560677	+	0.828035i	$0.689459\pi$
$318$	0	0
$319$	−5787.48	−1.01579
$320$	0	0
$321$	7343.23	1.27682
$322$	0	0
$323$	−894.358	−0.154066
$324$	0	0
$325$	−2528.44	−0.431547
$326$	0	0
$327$	−8065.24	−1.36394
$328$	0	0
$329$	1971.30	0.330339
$330$	0	0
$331$	−9467.70	−1.57218	−0.786090	−	0.618112i	$-0.787898\pi$
$331$	−9467.70	−1.57218	−0.786090	+	0.618112i	$0.787898\pi$
$332$	0	0
$333$	−1860.73	−0.306208
$334$	0	0
$335$	−8674.07	−1.41467
$336$	0	0
$337$	−10979.0	−1.77466	−0.887332	−	0.461131i	$-0.847444\pi$
$337$	−10979.0	−1.77466	−0.887332	+	0.461131i	$0.847444\pi$
$338$	0	0
$339$	9995.61	1.60144
$340$	0	0
$341$	−7540.18	−1.19743
$342$	0	0
$343$	5542.23	0.872455
$344$	0	0
$345$	13954.0	2.17756
$346$	0	0
$347$	−8731.23	−1.35077	−0.675385	−	0.737465i	$-0.736023\pi$
$347$	−8731.23	−1.35077	−0.675385	+	0.737465i	$0.736023\pi$
$348$	0	0
$349$	7156.72	1.09768	0.548840	−	0.835927i	$-0.315070\pi$
$349$	7156.72	1.09768	0.548840	+	0.835927i	$0.315070\pi$
$350$	0	0
$351$	4815.62	0.732305
$352$	0	0
$353$	5017.02	0.756456	0.378228	−	0.925713i	$-0.376534\pi$
$353$	5017.02	0.756456	0.378228	+	0.925713i	$0.376534\pi$
$354$	0	0
$355$	−8905.99	−1.33150
$356$	0	0
$357$	2438.92	0.361573
$358$	0	0
$359$	13325.3	1.95901	0.979503	−	0.201428i	$-0.0645584\pi$
$359$	13325.3	1.95901	0.979503	+	0.201428i	$0.0645584\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	12667.9	1.83166
$364$	0	0
$365$	11171.7	1.60206
$366$	0	0
$367$	−4706.78	−0.669460	−0.334730	−	0.942314i	$-0.608645\pi$
$367$	−4706.78	−0.669460	−0.334730	+	0.942314i	$0.608645\pi$
$368$	0	0
$369$	−1006.90	−0.142052
$370$	0	0
$371$	182.868	0.0255904
$372$	0	0
$373$	−2524.08	−0.350381	−0.175190	−	0.984535i	$-0.556054\pi$
$373$	−2524.08	−0.350381	−0.175190	+	0.984535i	$0.556054\pi$
$374$	0	0
$375$	−4568.92	−0.629168
$376$	0	0
$377$	−3692.30	−0.504411
$378$	0	0
$379$	2987.03	0.404838	0.202419	−	0.979299i	$-0.435120\pi$
$379$	2987.03	0.404838	0.202419	+	0.979299i	$0.435120\pi$
$380$	0	0
$381$	7498.66	1.00831
$382$	0	0
$383$	8661.02	1.15550	0.577751	−	0.816213i	$-0.303930\pi$
$383$	8661.02	1.15550	0.577751	+	0.816213i	$0.303930\pi$
$384$	0	0
$385$	7636.07	1.01083
$386$	0	0
$387$	−459.871	−0.0604045
$388$	0	0
$389$	1880.29	0.245076	0.122538	−	0.992464i	$-0.460897\pi$
$389$	1880.29	0.245076	0.122538	+	0.992464i	$0.460897\pi$
$390$	0	0
$391$	−8455.52	−1.09364
$392$	0	0
$393$	−295.260	−0.0378980
$394$	0	0
$395$	10864.0	1.38387
$396$	0	0
$397$	4453.25	0.562978	0.281489	−	0.959564i	$-0.409172\pi$
$397$	4453.25	0.562978	0.281489	+	0.959564i	$0.409172\pi$
$398$	0	0
$399$	−984.451	−0.123519
$400$	0	0
$401$	3404.95	0.424027	0.212014	−	0.977267i	$-0.431998\pi$
$401$	3404.95	0.424027	0.212014	+	0.977267i	$0.431998\pi$
$402$	0	0
$403$	−4810.48	−0.594608
$404$	0	0
$405$	−11495.5	−1.41041
$406$	0	0
$407$	24417.6	2.97380
$408$	0	0
$409$	3784.84	0.457576	0.228788	−	0.973476i	$-0.426524\pi$
$409$	3784.84	0.457576	0.228788	+	0.973476i	$0.426524\pi$
$410$	0	0
$411$	4790.36	0.574917
$412$	0	0
$413$	−901.803	−0.107445
$414$	0	0
$415$	−16245.2	−1.92156
$416$	0	0
$417$	4566.60	0.536276
$418$	0	0
$419$	5688.12	0.663205	0.331602	−	0.943419i	$-0.392411\pi$
$419$	5688.12	0.663205	0.331602	+	0.943419i	$0.392411\pi$
$420$	0	0
$421$	−5133.25	−0.594250	−0.297125	−	0.954839i	$-0.596028\pi$
$421$	−5133.25	−0.594250	−0.297125	+	0.954839i	$0.596028\pi$
$422$	0	0
$423$	−975.390	−0.112116
$424$	0	0
$425$	−3115.36	−0.355570
$426$	0	0
$427$	2456.56	0.278410
$428$	0	0
$429$	12852.5	1.44644
$430$	0	0
$431$	4240.52	0.473918	0.236959	−	0.971520i	$-0.423849\pi$
$431$	4240.52	0.473918	0.236959	+	0.971520i	$0.423849\pi$
$432$	0	0
$433$	536.405	0.0595334	0.0297667	−	0.999557i	$-0.490524\pi$
$433$	536.405	0.0595334	0.0297667	+	0.999557i	$0.490524\pi$
$434$	0	0
$435$	7507.77	0.827518
$436$	0	0
$437$	3413.00	0.373606
$438$	0	0
$439$	−2099.31	−0.228234	−0.114117	−	0.993467i	$-0.536404\pi$
$439$	−2099.31	−0.228234	−0.114117	+	0.993467i	$0.536404\pi$
$440$	0	0
$441$	−1177.07	−0.127099
$442$	0	0
$443$	2468.83	0.264780	0.132390	−	0.991198i	$-0.457735\pi$
$443$	2468.83	0.264780	0.132390	+	0.991198i	$0.457735\pi$
$444$	0	0
$445$	−3238.92	−0.345033
$446$	0	0
$447$	−17433.9	−1.84473
$448$	0	0
$449$	−18561.5	−1.95094	−0.975468	−	0.220141i	$-0.929348\pi$
$449$	−18561.5	−1.95094	−0.975468	+	0.220141i	$0.929348\pi$
$450$	0	0
$451$	13213.1	1.37956
$452$	0	0
$453$	612.757	0.0635537
$454$	0	0
$455$	4871.66	0.501949
$456$	0	0
$457$	−4128.97	−0.422637	−0.211319	−	0.977417i	$-0.567776\pi$
$457$	−4128.97	−0.422637	−0.211319	+	0.977417i	$0.567776\pi$
$458$	0	0
$459$	5933.46	0.603378
$460$	0	0
$461$	15795.0	1.59576	0.797879	−	0.602818i	$-0.205955\pi$
$461$	15795.0	1.59576	0.797879	+	0.602818i	$0.205955\pi$
$462$	0	0
$463$	−16293.3	−1.63545	−0.817725	−	0.575610i	$-0.804765\pi$
$463$	−16293.3	−1.63545	−0.817725	+	0.575610i	$0.804765\pi$
$464$	0	0
$465$	9781.45	0.975492
$466$	0	0
$467$	−13787.7	−1.36621	−0.683106	−	0.730320i	$-0.739371\pi$
$467$	−13787.7	−1.36621	−0.683106	+	0.730320i	$0.739371\pi$
$468$	0	0
$469$	5785.58	0.569623
$470$	0	0
$471$	−10953.9	−1.07161
$472$	0	0
$473$	6034.71	0.586630
$474$	0	0
$475$	1257.49	0.121469
$476$	0	0
$477$	−90.4821	−0.00868531
$478$	0	0
$479$	−8123.95	−0.774932	−0.387466	−	0.921884i	$-0.626650\pi$
$479$	−8123.95	−0.774932	−0.387466	+	0.921884i	$0.626650\pi$
$480$	0	0
$481$	15578.0	1.47670
$482$	0	0
$483$	−9307.29	−0.876804
$484$	0	0
$485$	−18645.7	−1.74569
$486$	0	0
$487$	−1100.55	−0.102404	−0.0512018	−	0.998688i	$-0.516305\pi$
$487$	−1100.55	−0.102404	−0.0512018	+	0.998688i	$0.516305\pi$
$488$	0	0
$489$	−11739.3	−1.08562
$490$	0	0
$491$	−18720.2	−1.72063	−0.860315	−	0.509762i	$-0.829733\pi$
$491$	−18720.2	−1.72063	−0.860315	+	0.509762i	$0.829733\pi$
$492$	0	0
$493$	−4549.38	−0.415606
$494$	0	0
$495$	−3778.28	−0.343073
$496$	0	0
$497$	5940.27	0.536132
$498$	0	0
$499$	6309.11	0.566001	0.283001	−	0.959120i	$-0.408670\pi$
$499$	6309.11	0.566001	0.283001	+	0.959120i	$0.408670\pi$
$500$	0	0
$501$	−2846.73	−0.253857
$502$	0	0
$503$	8091.31	0.717244	0.358622	−	0.933483i	$-0.383247\pi$
$503$	8091.31	0.717244	0.358622	+	0.933483i	$0.383247\pi$
$504$	0	0
$505$	−11988.6	−1.05640
$506$	0	0
$507$	−4143.37	−0.362946
$508$	0	0
$509$	−11338.8	−0.987398	−0.493699	−	0.869633i	$-0.664356\pi$
$509$	−11338.8	−0.987398	−0.493699	+	0.869633i	$0.664356\pi$
$510$	0	0
$511$	−7451.48	−0.645077
$512$	0	0
$513$	−2394.99	−0.206124
$514$	0	0
$515$	6808.00	0.582517
$516$	0	0
$517$	12799.7	1.08884
$518$	0	0
$519$	20610.4	1.74316
$520$	0	0
$521$	12848.2	1.08041	0.540203	−	0.841535i	$-0.318348\pi$
$521$	12848.2	1.08041	0.540203	+	0.841535i	$0.318348\pi$
$522$	0	0
$523$	14224.9	1.18932	0.594658	−	0.803978i	$-0.297287\pi$
$523$	14224.9	1.18932	0.594658	+	0.803978i	$0.297287\pi$
$524$	0	0
$525$	−3429.19	−0.285071
$526$	0	0
$527$	−5927.13	−0.489924
$528$	0	0
$529$	20100.5	1.65205
$530$	0	0
$531$	446.207	0.0364666
$532$	0	0
$533$	8429.71	0.685050
$534$	0	0
$535$	18072.6	1.46046
$536$	0	0
$537$	21762.1	1.74880
$538$	0	0
$539$	15446.2	1.23435
$540$	0	0
$541$	−24551.3	−1.95109	−0.975547	−	0.219789i	$-0.929463\pi$
$541$	−24551.3	−1.95109	−0.975547	+	0.219789i	$0.929463\pi$
$542$	0	0
$543$	−3143.71	−0.248452
$544$	0	0
$545$	−19849.6	−1.56012
$546$	0	0
$547$	−2290.42	−0.179034	−0.0895169	−	0.995985i	$-0.528532\pi$
$547$	−2290.42	−0.179034	−0.0895169	+	0.995985i	$0.528532\pi$
$548$	0	0
$549$	−1215.49	−0.0944915
$550$	0	0
$551$	1836.32	0.141978
$552$	0	0
$553$	−7246.27	−0.557220
$554$	0	0
$555$	−31675.6	−2.42262
$556$	0	0
$557$	−11590.3	−0.881683	−0.440841	−	0.897585i	$-0.645320\pi$
$557$	−11590.3	−0.881683	−0.440841	+	0.897585i	$0.645320\pi$
$558$	0	0
$559$	3850.02	0.291303
$560$	0	0
$561$	15835.9	1.19179
$562$	0	0
$563$	15650.3	1.17155	0.585773	−	0.810475i	$-0.300791\pi$
$563$	15650.3	1.17155	0.585773	+	0.810475i	$0.300791\pi$
$564$	0	0
$565$	24600.5	1.83177
$566$	0	0
$567$	7667.46	0.567907
$568$	0	0
$569$	745.409	0.0549194	0.0274597	−	0.999623i	$-0.491258\pi$
$569$	745.409	0.0549194	0.0274597	+	0.999623i	$0.491258\pi$
$570$	0	0
$571$	13663.3	1.00138	0.500691	−	0.865626i	$-0.333079\pi$
$571$	13663.3	1.00138	0.500691	+	0.865626i	$0.333079\pi$
$572$	0	0
$573$	−7490.86	−0.546134
$574$	0	0
$575$	11888.7	0.862248
$576$	0	0
$577$	12816.7	0.924721	0.462361	−	0.886692i	$-0.347003\pi$
$577$	12816.7	0.924721	0.462361	+	0.886692i	$0.347003\pi$
$578$	0	0
$579$	−20916.8	−1.50134
$580$	0	0
$581$	10835.5	0.773724
$582$	0	0
$583$	1187.36	0.0843491
$584$	0	0
$585$	−2410.47	−0.170360
$586$	0	0
$587$	−23688.4	−1.66563	−0.832817	−	0.553549i	$-0.813273\pi$
$587$	−23688.4	−1.66563	−0.832817	+	0.553549i	$0.813273\pi$
$588$	0	0
$589$	2392.44	0.167366
$590$	0	0
$591$	28032.1	1.95108
$592$	0	0
$593$	−4251.84	−0.294439	−0.147219	−	0.989104i	$-0.547032\pi$
$593$	−4251.84	−0.294439	−0.147219	+	0.989104i	$0.547032\pi$
$594$	0	0
$595$	6002.51	0.413578
$596$	0	0
$597$	29822.7	2.04449
$598$	0	0
$599$	−3852.14	−0.262761	−0.131381	−	0.991332i	$-0.541941\pi$
$599$	−3852.14	−0.262761	−0.131381	+	0.991332i	$0.541941\pi$
$600$	0	0
$601$	4596.36	0.311963	0.155981	−	0.987760i	$-0.450146\pi$
$601$	4596.36	0.311963	0.155981	+	0.987760i	$0.450146\pi$
$602$	0	0
$603$	−2862.67	−0.193328
$604$	0	0
$605$	31177.3	2.09511
$606$	0	0
$607$	7814.06	0.522509	0.261255	−	0.965270i	$-0.415864\pi$
$607$	7814.06	0.522509	0.261255	+	0.965270i	$0.415864\pi$
$608$	0	0
$609$	−5007.66	−0.333203
$610$	0	0
$611$	8165.93	0.540685
$612$	0	0
$613$	10013.8	0.659795	0.329897	−	0.944017i	$-0.392986\pi$
$613$	10013.8	0.659795	0.329897	+	0.944017i	$0.392986\pi$
$614$	0	0
$615$	−17140.7	−1.12387
$616$	0	0
$617$	−18316.8	−1.19515	−0.597576	−	0.801813i	$-0.703869\pi$
$617$	−18316.8	−1.19515	−0.597576	+	0.801813i	$0.703869\pi$
$618$	0	0
$619$	16755.3	1.08797	0.543985	−	0.839095i	$-0.316915\pi$
$619$	16755.3	1.08797	0.543985	+	0.839095i	$0.316915\pi$
$620$	0	0
$621$	−22643.0	−1.46317
$622$	0	0
$623$	2160.35	0.138929
$624$	0	0
$625$	−19517.7	−1.24913
$626$	0	0
$627$	−6392.04	−0.407135
$628$	0	0
$629$	19194.0	1.21672
$630$	0	0
$631$	27586.7	1.74042	0.870212	−	0.492677i	$-0.163982\pi$
$631$	27586.7	1.74042	0.870212	+	0.492677i	$0.163982\pi$
$632$	0	0
$633$	9767.64	0.613316
$634$	0	0
$635$	18455.2	1.15334
$636$	0	0
$637$	9854.38	0.612943
$638$	0	0
$639$	−2939.21	−0.181962
$640$	0	0
$641$	17669.0	1.08874	0.544371	−	0.838845i	$-0.316769\pi$
$641$	17669.0	1.08874	0.544371	+	0.838845i	$0.316769\pi$
$642$	0	0
$643$	−15999.3	−0.981259	−0.490630	−	0.871368i	$-0.663233\pi$
$643$	−15999.3	−0.981259	−0.490630	+	0.871368i	$0.663233\pi$
$644$	0	0
$645$	−7828.49	−0.477901
$646$	0	0
$647$	9922.65	0.602936	0.301468	−	0.953476i	$-0.402523\pi$
$647$	9922.65	0.602936	0.301468	+	0.953476i	$0.402523\pi$
$648$	0	0
$649$	−5855.40	−0.354152
$650$	0	0
$651$	−6524.20	−0.392786
$652$	0	0
$653$	−3000.87	−0.179836	−0.0899181	−	0.995949i	$-0.528661\pi$
$653$	−3000.87	−0.179836	−0.0899181	+	0.995949i	$0.528661\pi$
$654$	0	0
$655$	−726.674	−0.0433489
$656$	0	0
$657$	3686.95	0.218937
$658$	0	0
$659$	−23808.3	−1.40734	−0.703672	−	0.710525i	$-0.748457\pi$
$659$	−23808.3	−1.40734	−0.703672	+	0.710525i	$0.748457\pi$
$660$	0	0
$661$	19729.7	1.16096	0.580482	−	0.814273i	$-0.302864\pi$
$661$	19729.7	1.16096	0.580482	+	0.814273i	$0.302864\pi$
$662$	0	0
$663$	10103.0	0.591807
$664$	0	0
$665$	−2422.86	−0.141285
$666$	0	0
$667$	17361.1	1.00783
$668$	0	0
$669$	7180.05	0.414943
$670$	0	0
$671$	15950.4	0.917673
$672$	0	0
$673$	20300.0	1.16272	0.581358	−	0.813648i	$-0.302522\pi$
$673$	20300.0	1.16272	0.581358	+	0.813648i	$0.302522\pi$
$674$	0	0
$675$	−8342.61	−0.475715
$676$	0	0
$677$	−16523.6	−0.938040	−0.469020	−	0.883188i	$-0.655393\pi$
$677$	−16523.6	−0.938040	−0.469020	+	0.883188i	$0.655393\pi$
$678$	0	0
$679$	12436.6	0.702908
$680$	0	0
$681$	−5745.66	−0.323310
$682$	0	0
$683$	5359.32	0.300247	0.150123	−	0.988667i	$-0.452033\pi$
$683$	5359.32	0.300247	0.150123	+	0.988667i	$0.452033\pi$
$684$	0	0
$685$	11789.7	0.657607
$686$	0	0
$687$	−3373.16	−0.187328
$688$	0	0
$689$	757.513	0.0418853
$690$	0	0
$691$	−19231.8	−1.05877	−0.529385	−	0.848382i	$-0.677577\pi$
$691$	−19231.8	−1.05877	−0.529385	+	0.848382i	$0.677577\pi$
$692$	0	0
$693$	2520.11	0.138140
$694$	0	0
$695$	11239.0	0.613409
$696$	0	0
$697$	10386.5	0.564442
$698$	0	0
$699$	34089.0	1.84458
$700$	0	0
$701$	−6552.10	−0.353023	−0.176512	−	0.984299i	$-0.556481\pi$
$701$	−6552.10	−0.353023	−0.176512	+	0.984299i	$0.556481\pi$
$702$	0	0
$703$	−7747.51	−0.415651
$704$	0	0
$705$	−16604.3	−0.887027
$706$	0	0
$707$	7996.33	0.425365
$708$	0	0
$709$	27985.5	1.48240	0.741198	−	0.671287i	$-0.234258\pi$
$709$	27985.5	1.48240	0.741198	+	0.671287i	$0.234258\pi$
$710$	0	0
$711$	3585.42	0.189119
$712$	0	0
$713$	22618.8	1.18805
$714$	0	0
$715$	31631.7	1.65449
$716$	0	0
$717$	66.7971	0.00347920
$718$	0	0
$719$	−20268.3	−1.05129	−0.525646	−	0.850703i	$-0.676177\pi$
$719$	−20268.3	−1.05129	−0.525646	+	0.850703i	$0.676177\pi$
$720$	0	0
$721$	−4540.92	−0.234553
$722$	0	0
$723$	−15825.1	−0.814029
$724$	0	0
$725$	6396.55	0.327672
$726$	0	0
$727$	2068.98	0.105549	0.0527745	−	0.998606i	$-0.483194\pi$
$727$	2068.98	0.105549	0.0527745	+	0.998606i	$0.483194\pi$
$728$	0	0
$729$	15327.0	0.778690
$730$	0	0
$731$	4743.72	0.240017
$732$	0	0
$733$	−28074.9	−1.41469	−0.707347	−	0.706866i	$-0.750108\pi$
$733$	−28074.9	−1.41469	−0.707347	+	0.706866i	$0.750108\pi$
$734$	0	0
$735$	−20037.5	−1.00557
$736$	0	0
$737$	37565.8	1.87755
$738$	0	0
$739$	17218.0	0.857068	0.428534	−	0.903526i	$-0.359030\pi$
$739$	17218.0	0.857068	0.428534	+	0.903526i	$0.359030\pi$
$740$	0	0
$741$	−4077.99	−0.202171
$742$	0	0
$743$	−6618.16	−0.326779	−0.163390	−	0.986562i	$-0.552243\pi$
$743$	−6618.16	−0.326779	−0.163390	+	0.986562i	$0.552243\pi$
$744$	0	0
$745$	−42907.1	−2.11006
$746$	0	0
$747$	−5361.36	−0.262600
$748$	0	0
$749$	−12054.4	−0.588062
$750$	0	0
$751$	−22945.4	−1.11490	−0.557450	−	0.830210i	$-0.688220\pi$
$751$	−22945.4	−1.11490	−0.557450	+	0.830210i	$0.688220\pi$
$752$	0	0
$753$	22323.7	1.08037
$754$	0	0
$755$	1508.07	0.0726947
$756$	0	0
$757$	30316.2	1.45556	0.727782	−	0.685808i	$-0.240551\pi$
$757$	30316.2	1.45556	0.727782	+	0.685808i	$0.240551\pi$
$758$	0	0
$759$	−60432.2	−2.89005
$760$	0	0
$761$	−22856.6	−1.08877	−0.544384	−	0.838836i	$-0.683236\pi$
$761$	−22856.6	−1.08877	−0.544384	+	0.838836i	$0.683236\pi$
$762$	0	0
$763$	13239.6	0.628187
$764$	0	0
$765$	−2970.01	−0.140367
$766$	0	0
$767$	−3735.63	−0.175862
$768$	0	0
$769$	−28279.6	−1.32612	−0.663061	−	0.748565i	$-0.730743\pi$
$769$	−28279.6	−1.32612	−0.663061	+	0.748565i	$0.730743\pi$
$770$	0	0
$771$	−41327.4	−1.93044
$772$	0	0
$773$	−7705.64	−0.358542	−0.179271	−	0.983800i	$-0.557374\pi$
$773$	−7705.64	−0.358542	−0.179271	+	0.983800i	$0.557374\pi$
$774$	0	0
$775$	8333.70	0.386265
$776$	0	0
$777$	21127.5	0.975478
$778$	0	0
$779$	−4192.42	−0.192823
$780$	0	0
$781$	38570.2	1.76716
$782$	0	0
$783$	−12182.8	−0.556036
$784$	0	0
$785$	−26958.9	−1.22574
$786$	0	0
$787$	42608.4	1.92989	0.964946	−	0.262448i	$-0.0845300\pi$
$787$	42608.4	1.92989	0.964946	+	0.262448i	$0.0845300\pi$
$788$	0	0
$789$	22993.5	1.03750
$790$	0	0
$791$	−16408.5	−0.737570
$792$	0	0
$793$	10176.0	0.455690
$794$	0	0
$795$	−1540.30	−0.0687154
$796$	0	0
$797$	−42744.6	−1.89974	−0.949869	−	0.312647i	$-0.898784\pi$
$797$	−42744.6	−1.89974	−0.949869	+	0.312647i	$0.898784\pi$
$798$	0	0
$799$	10061.5	0.445494
$800$	0	0
$801$	−1068.93	−0.0471521
$802$	0	0
$803$	−48382.4	−2.12625
$804$	0	0
$805$	−22906.5	−1.00291
$806$	0	0
$807$	20578.8	0.897656
$808$	0	0
$809$	17770.3	0.772277	0.386139	−	0.922441i	$-0.373809\pi$
$809$	17770.3	0.772277	0.386139	+	0.922441i	$0.373809\pi$
$810$	0	0
$811$	−17635.6	−0.763588	−0.381794	−	0.924247i	$-0.624694\pi$
$811$	−17635.6	−0.763588	−0.381794	+	0.924247i	$0.624694\pi$
$812$	0	0
$813$	31329.8	1.35152
$814$	0	0
$815$	−28891.8	−1.24176
$816$	0	0
$817$	−1914.76	−0.0819940
$818$	0	0
$819$	1607.78	0.0685962
$820$	0	0
$821$	−6447.65	−0.274086	−0.137043	−	0.990565i	$-0.543760\pi$
$821$	−6447.65	−0.274086	−0.137043	+	0.990565i	$0.543760\pi$
$822$	0	0
$823$	−21513.1	−0.911179	−0.455590	−	0.890190i	$-0.650572\pi$
$823$	−21513.1	−0.911179	−0.455590	+	0.890190i	$0.650572\pi$
$824$	0	0
$825$	−22265.7	−0.939628
$826$	0	0
$827$	42712.8	1.79597	0.897987	−	0.440022i	$-0.145030\pi$
$827$	42712.8	1.79597	0.897987	+	0.440022i	$0.145030\pi$
$828$	0	0
$829$	−28421.8	−1.19075	−0.595374	−	0.803449i	$-0.702996\pi$
$829$	−28421.8	−1.19075	−0.595374	+	0.803449i	$0.702996\pi$
$830$	0	0
$831$	−4295.56	−0.179316
$832$	0	0
$833$	12141.9	0.505030
$834$	0	0
$835$	−7006.17	−0.290369
$836$	0	0
$837$	−15872.2	−0.655465
$838$	0	0
$839$	−12288.2	−0.505645	−0.252823	−	0.967513i	$-0.581359\pi$
$839$	−12288.2	−0.505645	−0.252823	+	0.967513i	$0.581359\pi$
$840$	0	0
$841$	−15048.1	−0.617003
$842$	0	0
$843$	−15761.7	−0.643963
$844$	0	0
$845$	−10197.4	−0.415149
$846$	0	0
$847$	−20795.2	−0.843603
$848$	0	0
$849$	14409.1	0.582473
$850$	0	0
$851$	−73247.3	−2.95051
$852$	0	0
$853$	−23447.1	−0.941166	−0.470583	−	0.882356i	$-0.655956\pi$
$853$	−23447.1	−0.941166	−0.470583	+	0.882356i	$0.655956\pi$
$854$	0	0
$855$	1198.82	0.0479517
$856$	0	0
$857$	−34878.3	−1.39022	−0.695111	−	0.718902i	$-0.744645\pi$
$857$	−34878.3	−1.39022	−0.695111	+	0.718902i	$0.744645\pi$
$858$	0	0
$859$	−31553.6	−1.25331	−0.626656	−	0.779296i	$-0.715577\pi$
$859$	−31553.6	−1.25331	−0.626656	+	0.779296i	$0.715577\pi$
$860$	0	0
$861$	11432.8	0.452529
$862$	0	0
$863$	−3137.99	−0.123776	−0.0618879	−	0.998083i	$-0.519712\pi$
$863$	−3137.99	−0.123776	−0.0618879	+	0.998083i	$0.519712\pi$
$864$	0	0
$865$	50725.0	1.99387
$866$	0	0
$867$	−15153.6	−0.593592
$868$	0	0
$869$	−47050.0	−1.83667
$870$	0	0
$871$	23966.2	0.932335
$872$	0	0
$873$	−6153.58	−0.238565
$874$	0	0
$875$	7500.19	0.289775
$876$	0	0
$877$	26872.9	1.03470	0.517351	−	0.855773i	$-0.326918\pi$
$877$	26872.9	1.03470	0.517351	+	0.855773i	$0.326918\pi$
$878$	0	0
$879$	35676.8	1.36900
$880$	0	0
$881$	10220.7	0.390856	0.195428	−	0.980718i	$-0.437390\pi$
$881$	10220.7	0.390856	0.195428	+	0.980718i	$0.437390\pi$
$882$	0	0
$883$	18978.1	0.723290	0.361645	−	0.932316i	$-0.382215\pi$
$883$	18978.1	0.723290	0.361645	+	0.932316i	$0.382215\pi$
$884$	0	0
$885$	7595.89	0.288512
$886$	0	0
$887$	−25106.7	−0.950395	−0.475197	−	0.879879i	$-0.657623\pi$
$887$	−25106.7	−0.950395	−0.475197	+	0.879879i	$0.657623\pi$
$888$	0	0
$889$	−12309.5	−0.464397
$890$	0	0
$891$	49784.8	1.87189
$892$	0	0
$893$	−4061.23	−0.152188
$894$	0	0
$895$	53559.4	2.00033
$896$	0	0
$897$	−38554.5	−1.43512
$898$	0	0
$899$	12169.7	0.451484
$900$	0	0
$901$	933.353	0.0345111
$902$	0	0
$903$	5221.58	0.192429
$904$	0	0
$905$	−7737.08	−0.284187
$906$	0	0
$907$	−40789.2	−1.49325	−0.746627	−	0.665243i	$-0.768328\pi$
$907$	−40789.2	−1.49325	−0.746627	+	0.665243i	$0.768328\pi$
$908$	0	0
$909$	−3956.54	−0.144368
$910$	0	0
$911$	8653.99	0.314731	0.157365	−	0.987540i	$-0.449700\pi$
$911$	8653.99	0.314731	0.157365	+	0.987540i	$0.449700\pi$
$912$	0	0
$913$	70355.1	2.55029
$914$	0	0
$915$	−20691.6	−0.747587
$916$	0	0
$917$	484.690	0.0174546
$918$	0	0
$919$	−9025.35	−0.323960	−0.161980	−	0.986794i	$-0.551788\pi$
$919$	−9025.35	−0.323960	−0.161980	+	0.986794i	$0.551788\pi$
$920$	0	0
$921$	−42455.9	−1.51897
$922$	0	0
$923$	24607.0	0.877518
$924$	0	0
$925$	−26987.3	−0.959284
$926$	0	0
$927$	2246.82	0.0796066
$928$	0	0
$929$	−30560.6	−1.07929	−0.539646	−	0.841892i	$-0.681442\pi$
$929$	−30560.6	−1.07929	−0.539646	+	0.841892i	$0.681442\pi$
$930$	0	0
$931$	−4900.96	−0.172527
$932$	0	0
$933$	−40455.4	−1.41956
$934$	0	0
$935$	38974.3	1.36320
$936$	0	0
$937$	15242.6	0.531436	0.265718	−	0.964051i	$-0.414391\pi$
$937$	15242.6	0.531436	0.265718	+	0.964051i	$0.414391\pi$
$938$	0	0
$939$	44786.7	1.55651
$940$	0	0
$941$	6241.91	0.216238	0.108119	−	0.994138i	$-0.465517\pi$
$941$	6241.91	0.216238	0.108119	+	0.994138i	$0.465517\pi$
$942$	0	0
$943$	−39636.4	−1.36876
$944$	0	0
$945$	16074.1	0.553322
$946$	0	0
$947$	−19299.8	−0.662259	−0.331130	−	0.943585i	$-0.607430\pi$
$947$	−19299.8	−0.662259	−0.331130	+	0.943585i	$0.607430\pi$
$948$	0	0
$949$	−30867.0	−1.05583
$950$	0	0
$951$	−35556.8	−1.21242
$952$	0	0
$953$	1598.11	0.0543209	0.0271605	−	0.999631i	$-0.491353\pi$
$953$	1598.11	0.0543209	0.0271605	+	0.999631i	$0.491353\pi$
$954$	0	0
$955$	−18436.0	−0.624685
$956$	0	0
$957$	−32514.7	−1.09828
$958$	0	0
$959$	−7863.70	−0.264788
$960$	0	0
$961$	−13935.7	−0.467783
$962$	0	0
$963$	5964.45	0.199586
$964$	0	0
$965$	−51479.1	−1.71727
$966$	0	0
$967$	31860.4	1.05952	0.529762	−	0.848146i	$-0.322281\pi$
$967$	31860.4	1.05952	0.529762	+	0.848146i	$0.322281\pi$
$968$	0	0
$969$	−5024.61	−0.166578
$970$	0	0
$971$	3196.13	0.105632	0.0528160	−	0.998604i	$-0.483180\pi$
$971$	3196.13	0.105632	0.0528160	+	0.998604i	$0.483180\pi$
$972$	0	0
$973$	−7496.38	−0.246992
$974$	0	0
$975$	−14205.1	−0.466592
$976$	0	0
$977$	−36786.9	−1.20462	−0.602312	−	0.798261i	$-0.705754\pi$
$977$	−36786.9	−1.20462	−0.602312	+	0.798261i	$0.705754\pi$
$978$	0	0
$979$	14027.2	0.457927
$980$	0	0
$981$	−6550.90	−0.213205
$982$	0	0
$983$	−25120.5	−0.815076	−0.407538	−	0.913188i	$-0.633613\pi$
$983$	−25120.5	−0.815076	−0.407538	+	0.913188i	$0.633613\pi$
$984$	0	0
$985$	68990.6	2.23170
$986$	0	0
$987$	11075.0	0.357165
$988$	0	0
$989$	−18102.7	−0.582036
$990$	0	0
$991$	−9526.01	−0.305352	−0.152676	−	0.988276i	$-0.548789\pi$
$991$	−9526.01	−0.305352	−0.152676	+	0.988276i	$0.548789\pi$
$992$	0	0
$993$	−53190.6	−1.69985
$994$	0	0
$995$	73397.6	2.33855
$996$	0	0
$997$	−13162.4	−0.418111	−0.209056	−	0.977904i	$-0.567039\pi$
$997$	−13162.4	−0.418111	−0.209056	+	0.977904i	$0.567039\pi$
$998$	0	0
$999$	51399.6	1.62784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.r.1.3		3
4.3	odd	2		1216.4.a.w.1.1		3
8.3	odd	2		304.4.a.g.1.3		3
8.5	even	2		152.4.a.c.1.1	✓	3
24.5	odd	2		1368.4.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.c.1.1	✓	3	8.5	even	2
304.4.a.g.1.3		3	8.3	odd	2
1216.4.a.r.1.3		3	1.1	even	1	trivial
1216.4.a.w.1.1		3	4.3	odd	2
1368.4.a.d.1.3		3	24.5	odd	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)

\(f(q)\)	\(=\)	\(q+5.61812 q^{3} +13.8269 q^{5} -9.22252 q^{7} +4.56325 q^{9} +O(q^{10})\)	\(q+5.61812 q^{3} +13.8269 q^{5} -9.22252 q^{7} +4.56325 q^{9} -59.8818 q^{11} -38.2034 q^{13} +77.6813 q^{15} -47.0714 q^{17} +19.0000 q^{19} -51.8132 q^{21} +179.632 q^{23} +66.1837 q^{25} -126.052 q^{27} +96.6484 q^{29} +125.918 q^{31} -336.423 q^{33} -127.519 q^{35} -407.764 q^{37} -214.631 q^{39} -220.654 q^{41} -100.777 q^{43} +63.0957 q^{45} -213.749 q^{47} -257.945 q^{49} -264.453 q^{51} -19.8284 q^{53} -827.981 q^{55} +106.744 q^{57} +97.7827 q^{59} -266.365 q^{61} -42.0847 q^{63} -528.235 q^{65} -627.332 q^{67} +1009.19 q^{69} -644.105 q^{71} +807.966 q^{73} +371.828 q^{75} +552.261 q^{77} +785.715 q^{79} -831.385 q^{81} -1174.90 q^{83} -650.853 q^{85} +542.982 q^{87} -234.248 q^{89} +352.331 q^{91} +707.421 q^{93} +262.711 q^{95} -1348.51 q^{97} -273.256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9	\( 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9} - 103 q^{11} - 32 q^{13} + 122 q^{15} + 11 q^{17} + 57 q^{19} - 114 q^{21} + 316 q^{23} + 162 q^{25} - 178 q^{27} + 138 q^{29} + 420 q^{31} - 330 q^{33} - 333 q^{35} - 102 q^{37} + 164 q^{39} - 370 q^{41} - 431 q^{43} + 429 q^{45} - 199 q^{47} - 802 q^{49} - 272 q^{51} + 308 q^{53} + 85 q^{55} - 76 q^{57} + 188 q^{59} + 609 q^{61} - 61 q^{63} - 1536 q^{65} + 246 q^{67} + 892 q^{69} - 954 q^{71} - 629 q^{73} + 1074 q^{75} + 71 q^{77} + 452 q^{79} - 361 q^{81} + 780 q^{83} - 1883 q^{85} - 96 q^{87} - 1356 q^{89} + 670 q^{91} - 1092 q^{93} - 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9 - 103 * q^11 - 32 * q^13 + 122 * q^15 + 11 * q^17 + 57 * q^19 - 114 * q^21 + 316 * q^23 + 162 * q^25 - 178 * q^27 + 138 * q^29 + 420 * q^31 - 330 * q^33 - 333 * q^35 - 102 * q^37 + 164 * q^39 - 370 * q^41 - 431 * q^43 + 429 * q^45 - 199 * q^47 - 802 * q^49 - 272 * q^51 + 308 * q^53 + 85 * q^55 - 76 * q^57 + 188 * q^59 + 609 * q^61 - 61 * q^63 - 1536 * q^65 + 246 * q^67 + 892 * q^69 - 954 * q^71 - 629 * q^73 + 1074 * q^75 + 71 * q^77 + 452 * q^79 - 361 * q^81 + 780 * q^83 - 1883 * q^85 - 96 * q^87 - 1356 * q^89 + 670 * q^91 - 1092 * q^93 - 133 * q^95 - 548 * q^97 + 1305 * q^99

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