LMFDB - Embedded newform 1216.4.a.n.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{93}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 23 $ x^2 - x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.32183$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +21.2873 q^{5} +31.6437 q^{7} -26.0000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +21.2873 q^{5} +31.6437 q^{7} -26.0000 q^{9} -0.712698 q^{11} +38.9310 q^{13} +21.2873 q^{15} -54.1492 q^{17} -19.0000 q^{19} +31.6437 q^{21} -61.7817 q^{23} +328.149 q^{25} -53.0000 q^{27} +225.229 q^{29} +131.586 q^{31} -0.712698 q^{33} +673.608 q^{35} -94.1381 q^{37} +38.9310 q^{39} -108.873 q^{41} +205.287 q^{43} -553.470 q^{45} +523.448 q^{47} +658.321 q^{49} -54.1492 q^{51} -560.240 q^{53} -15.1714 q^{55} -19.0000 q^{57} +498.425 q^{59} -179.011 q^{61} -822.735 q^{63} +828.735 q^{65} -985.022 q^{67} -61.7817 q^{69} +904.713 q^{71} -724.149 q^{73} +328.149 q^{75} -22.5524 q^{77} +1168.20 q^{79} +649.000 q^{81} -1131.93 q^{83} -1152.69 q^{85} +225.229 q^{87} +208.370 q^{89} +1231.92 q^{91} +131.586 q^{93} -404.459 q^{95} -683.470 q^{97} +18.5302 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9} - 40 q^{11} + 20 q^{13} + 4 q^{15} + 46 q^{17} - 38 q^{19} + 44 q^{21} - 220 q^{23} + 502 q^{25} - 106 q^{27} + 84 q^{29} - 84 q^{31} - 40 q^{33} + 460 q^{35} - 304 q^{37} + 20 q^{39} + 168 q^{41} + 372 q^{43} - 104 q^{45} + 584 q^{47} + 468 q^{49} + 46 q^{51} - 484 q^{53} + 664 q^{55} - 38 q^{57} + 1074 q^{59} - 88 q^{61} - 1144 q^{63} + 1156 q^{65} - 1430 q^{67} - 220 q^{69} + 1848 q^{71} - 1294 q^{73} + 502 q^{75} - 508 q^{77} + 832 q^{79} + 1298 q^{81} - 528 q^{83} - 2884 q^{85} + 84 q^{87} + 1844 q^{89} + 998 q^{91} - 84 q^{93} - 76 q^{95} - 364 q^{97} + 1040 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9 - 40 * q^11 + 20 * q^13 + 4 * q^15 + 46 * q^17 - 38 * q^19 + 44 * q^21 - 220 * q^23 + 502 * q^25 - 106 * q^27 + 84 * q^29 - 84 * q^31 - 40 * q^33 + 460 * q^35 - 304 * q^37 + 20 * q^39 + 168 * q^41 + 372 * q^43 - 104 * q^45 + 584 * q^47 + 468 * q^49 + 46 * q^51 - 484 * q^53 + 664 * q^55 - 38 * q^57 + 1074 * q^59 - 88 * q^61 - 1144 * q^63 + 1156 * q^65 - 1430 * q^67 - 220 * q^69 + 1848 * q^71 - 1294 * q^73 + 502 * q^75 - 508 * q^77 + 832 * q^79 + 1298 * q^81 - 528 * q^83 - 2884 * q^85 + 84 * q^87 + 1844 * q^89 + 998 * q^91 - 84 * q^93 - 76 * q^95 - 364 * q^97 + 1040 * 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
$3$	1.00000	0.192450	0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000	0.192450	0.0962250	+	0.995360i	$0.469323\pi$
$4$	0	0
$5$	21.2873	1.90399	0.951997	−	0.306107i	$-0.0990266\pi$
$5$	21.2873	1.90399	0.951997	+	0.306107i	$0.0990266\pi$
$6$	0	0
$7$	31.6437	1.70860	0.854298	−	0.519783i	$-0.173987\pi$
$7$	31.6437	1.70860	0.854298	+	0.519783i	$0.173987\pi$
$8$	0	0
$9$	−26.0000	−0.962963
$10$	0	0
$11$	−0.712698	−0.0195352	−0.00976758	−	0.999952i	$-0.503109\pi$
$11$	−0.712698	−0.0195352	−0.00976758	+	0.999952i	$0.503109\pi$
$12$	0	0
$13$	38.9310	0.830577	0.415289	−	0.909690i	$-0.363681\pi$
$13$	38.9310	0.830577	0.415289	+	0.909690i	$0.363681\pi$
$14$	0	0
$15$	21.2873	0.366424
$16$	0	0
$17$	−54.1492	−0.772536	−0.386268	−	0.922387i	$-0.626236\pi$
$17$	−54.1492	−0.772536	−0.386268	+	0.922387i	$0.626236\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	31.6437	0.328820
$22$	0	0
$23$	−61.7817	−0.560104	−0.280052	−	0.959985i	$-0.590352\pi$
$23$	−61.7817	−0.560104	−0.280052	+	0.959985i	$0.590352\pi$
$24$	0	0
$25$	328.149	2.62519
$26$	0	0
$27$	−53.0000	−0.377772
$28$	0	0
$29$	225.229	1.44221	0.721104	−	0.692827i	$-0.243635\pi$
$29$	225.229	1.44221	0.721104	+	0.692827i	$0.243635\pi$
$30$	0	0
$31$	131.586	0.762371	0.381185	−	0.924499i	$-0.375516\pi$
$31$	131.586	0.762371	0.381185	+	0.924499i	$0.375516\pi$
$32$	0	0
$33$	−0.712698	−0.00375954
$34$	0	0
$35$	673.608	3.25316
$36$	0	0
$37$	−94.1381	−0.418276	−0.209138	−	0.977886i	$-0.567066\pi$
$37$	−94.1381	−0.418276	−0.209138	+	0.977886i	$0.567066\pi$
$38$	0	0
$39$	38.9310	0.159845
$40$	0	0
$41$	−108.873	−0.414710	−0.207355	−	0.978266i	$-0.566486\pi$
$41$	−108.873	−0.414710	−0.207355	+	0.978266i	$0.566486\pi$
$42$	0	0
$43$	205.287	0.728047	0.364023	−	0.931390i	$-0.381403\pi$
$43$	205.287	0.728047	0.364023	+	0.931390i	$0.381403\pi$
$44$	0	0
$45$	−553.470	−1.83348
$46$	0	0
$47$	523.448	1.62453	0.812263	−	0.583292i	$-0.198236\pi$
$47$	523.448	1.62453	0.812263	+	0.583292i	$0.198236\pi$
$48$	0	0
$49$	658.321	1.91930
$50$	0	0
$51$	−54.1492	−0.148675
$52$	0	0
$53$	−560.240	−1.45198	−0.725990	−	0.687705i	$-0.758618\pi$
$53$	−560.240	−1.45198	−0.725990	+	0.687705i	$0.758618\pi$
$54$	0	0
$55$	−15.1714	−0.0371948
$56$	0	0
$57$	−19.0000	−0.0441511
$58$	0	0
$59$	498.425	1.09982	0.549911	−	0.835223i	$-0.314662\pi$
$59$	498.425	1.09982	0.549911	+	0.835223i	$0.314662\pi$
$60$	0	0
$61$	−179.011	−0.375738	−0.187869	−	0.982194i	$-0.560158\pi$
$61$	−179.011	−0.375738	−0.187869	+	0.982194i	$0.560158\pi$
$62$	0	0
$63$	−822.735	−1.64532
$64$	0	0
$65$	828.735	1.58141
$66$	0	0
$67$	−985.022	−1.79611	−0.898057	−	0.439879i	$-0.855021\pi$
$67$	−985.022	−1.79611	−0.898057	+	0.439879i	$0.855021\pi$
$68$	0	0
$69$	−61.7817	−0.107792
$70$	0	0
$71$	904.713	1.51225	0.756124	−	0.654428i	$-0.227091\pi$
$71$	904.713	1.51225	0.756124	+	0.654428i	$0.227091\pi$
$72$	0	0
$73$	−724.149	−1.16103	−0.580515	−	0.814249i	$-0.697149\pi$
$73$	−724.149	−1.16103	−0.580515	+	0.814249i	$0.697149\pi$
$74$	0	0
$75$	328.149	0.505219
$76$	0	0
$77$	−22.5524	−0.0333777
$78$	0	0
$79$	1168.20	1.66371	0.831856	−	0.554991i	$-0.187278\pi$
$79$	1168.20	1.66371	0.831856	+	0.554991i	$0.187278\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	−1131.93	−1.49693	−0.748466	−	0.663174i	$-0.769209\pi$
$83$	−1131.93	−1.49693	−0.748466	+	0.663174i	$0.769209\pi$
$84$	0	0
$85$	−1152.69	−1.47090
$86$	0	0
$87$	225.229	0.277553
$88$	0	0
$89$	208.370	0.248170	0.124085	−	0.992272i	$-0.460400\pi$
$89$	208.370	0.248170	0.124085	+	0.992272i	$0.460400\pi$
$90$	0	0
$91$	1231.92	1.41912
$92$	0	0
$93$	131.586	0.146718
$94$	0	0
$95$	−404.459	−0.436806
$96$	0	0
$97$	−683.470	−0.715421	−0.357711	−	0.933833i	$-0.616443\pi$
$97$	−683.470	−0.715421	−0.357711	+	0.933833i	$0.616443\pi$
$98$	0	0
$99$	18.5302	0.0188116
$100$	0	0
$101$	−360.806	−0.355461	−0.177731	−	0.984079i	$-0.556876\pi$
$101$	−360.806	−0.355461	−0.177731	+	0.984079i	$0.556876\pi$
$102$	0	0
$103$	−155.514	−0.148770	−0.0743848	−	0.997230i	$-0.523699\pi$
$103$	−155.514	−0.148770	−0.0743848	+	0.997230i	$0.523699\pi$
$104$	0	0
$105$	673.608	0.626071
$106$	0	0
$107$	1443.09	1.30382	0.651910	−	0.758297i	$-0.273968\pi$
$107$	1443.09	1.30382	0.651910	+	0.758297i	$0.273968\pi$
$108$	0	0
$109$	−980.953	−0.862003	−0.431001	−	0.902351i	$-0.641840\pi$
$109$	−980.953	−0.862003	−0.431001	+	0.902351i	$0.641840\pi$
$110$	0	0
$111$	−94.1381	−0.0804972
$112$	0	0
$113$	−842.917	−0.701726	−0.350863	−	0.936427i	$-0.614112\pi$
$113$	−842.917	−0.701726	−0.350863	+	0.936427i	$0.614112\pi$
$114$	0	0
$115$	−1315.17	−1.06643
$116$	0	0
$117$	−1012.20	−0.799815
$118$	0	0
$119$	−1713.48	−1.31995
$120$	0	0
$121$	−1330.49	−0.999618
$122$	0	0
$123$	−108.873	−0.0798110
$124$	0	0
$125$	4324.50	3.09436
$126$	0	0
$127$	−489.310	−0.341883	−0.170942	−	0.985281i	$-0.554681\pi$
$127$	−489.310	−0.341883	−0.170942	+	0.985281i	$0.554681\pi$
$128$	0	0
$129$	205.287	0.140113
$130$	0	0
$131$	−927.310	−0.618469	−0.309234	−	0.950986i	$-0.600073\pi$
$131$	−927.310	−0.618469	−0.309234	+	0.950986i	$0.600073\pi$
$132$	0	0
$133$	−601.229	−0.391979
$134$	0	0
$135$	−1128.23	−0.719276
$136$	0	0
$137$	1897.02	1.18302	0.591509	−	0.806298i	$-0.298532\pi$
$137$	1897.02	1.18302	0.591509	+	0.806298i	$0.298532\pi$
$138$	0	0
$139$	−590.713	−0.360458	−0.180229	−	0.983625i	$-0.557684\pi$
$139$	−590.713	−0.360458	−0.180229	+	0.983625i	$0.557684\pi$
$140$	0	0
$141$	523.448	0.312640
$142$	0	0
$143$	−27.7460	−0.0162255
$144$	0	0
$145$	4794.53	2.74596
$146$	0	0
$147$	658.321	0.369370
$148$	0	0
$149$	−824.641	−0.453404	−0.226702	−	0.973964i	$-0.572794\pi$
$149$	−824.641	−0.453404	−0.226702	+	0.973964i	$0.572794\pi$
$150$	0	0
$151$	1424.71	0.767822	0.383911	−	0.923370i	$-0.374577\pi$
$151$	1424.71	0.767822	0.383911	+	0.923370i	$0.374577\pi$
$152$	0	0
$153$	1407.88	0.743924
$154$	0	0
$155$	2801.10	1.45155
$156$	0	0
$157$	−2499.51	−1.27059	−0.635295	−	0.772270i	$-0.719121\pi$
$157$	−2499.51	−1.27059	−0.635295	+	0.772270i	$0.719121\pi$
$158$	0	0
$159$	−560.240	−0.279434
$160$	0	0
$161$	−1955.00	−0.956991
$162$	0	0
$163$	1102.62	0.529841	0.264921	−	0.964270i	$-0.414654\pi$
$163$	1102.62	0.529841	0.264921	+	0.964270i	$0.414654\pi$
$164$	0	0
$165$	−15.1714	−0.00715815
$166$	0	0
$167$	161.613	0.0748860	0.0374430	−	0.999299i	$-0.488079\pi$
$167$	161.613	0.0748860	0.0374430	+	0.999299i	$0.488079\pi$
$168$	0	0
$169$	−681.381	−0.310142
$170$	0	0
$171$	494.000	0.220919
$172$	0	0
$173$	1829.59	0.804051	0.402026	−	0.915628i	$-0.368306\pi$
$173$	1829.59	0.804051	0.402026	+	0.915628i	$0.368306\pi$
$174$	0	0
$175$	10383.8	4.48540
$176$	0	0
$177$	498.425	0.211661
$178$	0	0
$179$	1048.00	0.437604	0.218802	−	0.975769i	$-0.429785\pi$
$179$	1048.00	0.437604	0.218802	+	0.975769i	$0.429785\pi$
$180$	0	0
$181$	−897.354	−0.368507	−0.184254	−	0.982879i	$-0.558987\pi$
$181$	−897.354	−0.368507	−0.184254	+	0.982879i	$0.558987\pi$
$182$	0	0
$183$	−179.011	−0.0723108
$184$	0	0
$185$	−2003.95	−0.796395
$186$	0	0
$187$	38.5921	0.0150916
$188$	0	0
$189$	−1677.11	−0.645461
$190$	0	0
$191$	4113.98	1.55852	0.779259	−	0.626702i	$-0.215596\pi$
$191$	4113.98	1.55852	0.779259	+	0.626702i	$0.215596\pi$
$192$	0	0
$193$	3511.87	1.30979	0.654897	−	0.755718i	$-0.272712\pi$
$193$	3511.87	1.30979	0.654897	+	0.755718i	$0.272712\pi$
$194$	0	0
$195$	828.735	0.304343
$196$	0	0
$197$	2488.28	0.899910	0.449955	−	0.893051i	$-0.351440\pi$
$197$	2488.28	0.899910	0.449955	+	0.893051i	$0.351440\pi$
$198$	0	0
$199$	−386.556	−0.137700	−0.0688499	−	0.997627i	$-0.521933\pi$
$199$	−386.556	−0.137700	−0.0688499	+	0.997627i	$0.521933\pi$
$200$	0	0
$201$	−985.022	−0.345662
$202$	0	0
$203$	7127.08	2.46415
$204$	0	0
$205$	−2317.61	−0.789605
$206$	0	0
$207$	1606.33	0.539359
$208$	0	0
$209$	13.5413	0.00448167
$210$	0	0
$211$	4851.06	1.58275	0.791376	−	0.611330i	$-0.209365\pi$
$211$	4851.06	1.58275	0.791376	+	0.611330i	$0.209365\pi$
$212$	0	0
$213$	904.713	0.291032
$214$	0	0
$215$	4370.01	1.38620
$216$	0	0
$217$	4163.85	1.30258
$218$	0	0
$219$	−724.149	−0.223440
$220$	0	0
$221$	−2108.08	−0.641651
$222$	0	0
$223$	−802.187	−0.240890	−0.120445	−	0.992720i	$-0.538432\pi$
$223$	−802.187	−0.240890	−0.120445	+	0.992720i	$0.538432\pi$
$224$	0	0
$225$	−8531.88	−2.52796
$226$	0	0
$227$	−4889.70	−1.42969	−0.714847	−	0.699281i	$-0.753504\pi$
$227$	−4889.70	−1.42969	−0.714847	+	0.699281i	$0.753504\pi$
$228$	0	0
$229$	−569.132	−0.164233	−0.0821163	−	0.996623i	$-0.526168\pi$
$229$	−569.132	−0.164233	−0.0821163	+	0.996623i	$0.526168\pi$
$230$	0	0
$231$	−22.5524	−0.00642354
$232$	0	0
$233$	−3835.88	−1.07853	−0.539264	−	0.842137i	$-0.681297\pi$
$233$	−3835.88	−1.07853	−0.539264	+	0.842137i	$0.681297\pi$
$234$	0	0
$235$	11142.8	3.09309
$236$	0	0
$237$	1168.20	0.320182
$238$	0	0
$239$	527.821	0.142853	0.0714266	−	0.997446i	$-0.477245\pi$
$239$	527.821	0.142853	0.0714266	+	0.997446i	$0.477245\pi$
$240$	0	0
$241$	−5523.03	−1.47622	−0.738111	−	0.674679i	$-0.764282\pi$
$241$	−5523.03	−1.47622	−0.738111	+	0.674679i	$0.764282\pi$
$242$	0	0
$243$	2080.00	0.549103
$244$	0	0
$245$	14013.9	3.65434
$246$	0	0
$247$	−739.688	−0.190547
$248$	0	0
$249$	−1131.93	−0.288085
$250$	0	0
$251$	−7573.11	−1.90442	−0.952212	−	0.305438i	$-0.901197\pi$
$251$	−7573.11	−1.90442	−0.952212	+	0.305438i	$0.901197\pi$
$252$	0	0
$253$	44.0318	0.0109417
$254$	0	0
$255$	−1152.69	−0.283076
$256$	0	0
$257$	−6349.60	−1.54116	−0.770578	−	0.637346i	$-0.780032\pi$
$257$	−6349.60	−1.54116	−0.770578	+	0.637346i	$0.780032\pi$
$258$	0	0
$259$	−2978.87	−0.714665
$260$	0	0
$261$	−5855.96	−1.38879
$262$	0	0
$263$	−3915.06	−0.917921	−0.458960	−	0.888457i	$-0.651778\pi$
$263$	−3915.06	−0.917921	−0.458960	+	0.888457i	$0.651778\pi$
$264$	0	0
$265$	−11926.0	−2.76456
$266$	0	0
$267$	208.370	0.0477604
$268$	0	0
$269$	7171.13	1.62540	0.812699	−	0.582684i	$-0.197998\pi$
$269$	7171.13	1.62540	0.812699	+	0.582684i	$0.197998\pi$
$270$	0	0
$271$	−1100.04	−0.246577	−0.123289	−	0.992371i	$-0.539344\pi$
$271$	−1100.04	−0.246577	−0.123289	+	0.992371i	$0.539344\pi$
$272$	0	0
$273$	1231.92	0.273110
$274$	0	0
$275$	−233.871	−0.0512836
$276$	0	0
$277$	−1361.51	−0.295326	−0.147663	−	0.989038i	$-0.547175\pi$
$277$	−1361.51	−0.295326	−0.147663	+	0.989038i	$0.547175\pi$
$278$	0	0
$279$	−3421.23	−0.734135
$280$	0	0
$281$	4438.51	0.942275	0.471138	−	0.882060i	$-0.343843\pi$
$281$	4438.51	0.942275	0.471138	+	0.882060i	$0.343843\pi$
$282$	0	0
$283$	9190.64	1.93048	0.965241	−	0.261360i	$-0.0841710\pi$
$283$	9190.64	1.93048	0.965241	+	0.261360i	$0.0841710\pi$
$284$	0	0
$285$	−404.459	−0.0840634
$286$	0	0
$287$	−3445.14	−0.708572
$288$	0	0
$289$	−1980.86	−0.403188
$290$	0	0
$291$	−683.470	−0.137683
$292$	0	0
$293$	−3316.50	−0.661269	−0.330635	−	0.943759i	$-0.607263\pi$
$293$	−3316.50	−0.661269	−0.330635	+	0.943759i	$0.607263\pi$
$294$	0	0
$295$	10610.1	2.09405
$296$	0	0
$297$	37.7730	0.00737984
$298$	0	0
$299$	−2405.22	−0.465209
$300$	0	0
$301$	6496.04	1.24394
$302$	0	0
$303$	−360.806	−0.0684085
$304$	0	0
$305$	−3810.66	−0.715403
$306$	0	0
$307$	−1574.75	−0.292755	−0.146378	−	0.989229i	$-0.546761\pi$
$307$	−1574.75	−0.292755	−0.146378	+	0.989229i	$0.546761\pi$
$308$	0	0
$309$	−155.514	−0.0286307
$310$	0	0
$311$	4614.64	0.841389	0.420695	−	0.907202i	$-0.361786\pi$
$311$	4614.64	0.841389	0.420695	+	0.907202i	$0.361786\pi$
$312$	0	0
$313$	4738.42	0.855690	0.427845	−	0.903852i	$-0.359273\pi$
$313$	4738.42	0.855690	0.427845	+	0.903852i	$0.359273\pi$
$314$	0	0
$315$	−17513.8	−3.13267
$316$	0	0
$317$	−1039.66	−0.184206	−0.0921028	−	0.995750i	$-0.529359\pi$
$317$	−1039.66	−0.184206	−0.0921028	+	0.995750i	$0.529359\pi$
$318$	0	0
$319$	−160.521	−0.0281738
$320$	0	0
$321$	1443.09	0.250920
$322$	0	0
$323$	1028.83	0.177232
$324$	0	0
$325$	12775.2	2.18043
$326$	0	0
$327$	−980.953	−0.165892
$328$	0	0
$329$	16563.8	2.77566
$330$	0	0
$331$	−4634.52	−0.769597	−0.384798	−	0.923001i	$-0.625729\pi$
$331$	−4634.52	−0.769597	−0.384798	+	0.923001i	$0.625729\pi$
$332$	0	0
$333$	2447.59	0.402784
$334$	0	0
$335$	−20968.5	−3.41979
$336$	0	0
$337$	5219.30	0.843660	0.421830	−	0.906675i	$-0.361388\pi$
$337$	5219.30	0.843660	0.421830	+	0.906675i	$0.361388\pi$
$338$	0	0
$339$	−842.917	−0.135047
$340$	0	0
$341$	−93.7809	−0.0148930
$342$	0	0
$343$	9977.90	1.57072
$344$	0	0
$345$	−1315.17	−0.205235
$346$	0	0
$347$	−7019.77	−1.08600	−0.542999	−	0.839734i	$-0.682711\pi$
$347$	−7019.77	−1.08600	−0.542999	+	0.839734i	$0.682711\pi$
$348$	0	0
$349$	−6552.97	−1.00508	−0.502539	−	0.864554i	$-0.667601\pi$
$349$	−6552.97	−1.00508	−0.502539	+	0.864554i	$0.667601\pi$
$350$	0	0
$351$	−2063.34	−0.313769
$352$	0	0
$353$	−8837.11	−1.33244	−0.666221	−	0.745754i	$-0.732089\pi$
$353$	−8837.11	−1.33244	−0.666221	+	0.745754i	$0.732089\pi$
$354$	0	0
$355$	19258.9	2.87931
$356$	0	0
$357$	−1713.48	−0.254025
$358$	0	0
$359$	2521.14	0.370643	0.185321	−	0.982678i	$-0.440667\pi$
$359$	2521.14	0.370643	0.185321	+	0.982678i	$0.440667\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−1330.49	−0.192377
$364$	0	0
$365$	−15415.2	−2.21060
$366$	0	0
$367$	−5652.41	−0.803960	−0.401980	−	0.915648i	$-0.631678\pi$
$367$	−5652.41	−0.803960	−0.401980	+	0.915648i	$0.631678\pi$
$368$	0	0
$369$	2830.70	0.399350
$370$	0	0
$371$	−17728.1	−2.48085
$372$	0	0
$373$	−6864.49	−0.952895	−0.476447	−	0.879203i	$-0.658076\pi$
$373$	−6864.49	−0.952895	−0.476447	+	0.879203i	$0.658076\pi$
$374$	0	0
$375$	4324.50	0.595510
$376$	0	0
$377$	8768.39	1.19787
$378$	0	0
$379$	−6393.09	−0.866467	−0.433234	−	0.901282i	$-0.642627\pi$
$379$	−6393.09	−0.866467	−0.433234	+	0.901282i	$0.642627\pi$
$380$	0	0
$381$	−489.310	−0.0657955
$382$	0	0
$383$	7948.16	1.06040	0.530198	−	0.847874i	$-0.322118\pi$
$383$	7948.16	1.06040	0.530198	+	0.847874i	$0.322118\pi$
$384$	0	0
$385$	−480.079	−0.0635509
$386$	0	0
$387$	−5337.47	−0.701082
$388$	0	0
$389$	8896.48	1.15956	0.579781	−	0.814773i	$-0.303138\pi$
$389$	8896.48	1.15956	0.579781	+	0.814773i	$0.303138\pi$
$390$	0	0
$391$	3345.43	0.432700
$392$	0	0
$393$	−927.310	−0.119024
$394$	0	0
$395$	24867.9	3.16770
$396$	0	0
$397$	−2865.66	−0.362276	−0.181138	−	0.983458i	$-0.557978\pi$
$397$	−2865.66	−0.362276	−0.181138	+	0.983458i	$0.557978\pi$
$398$	0	0
$399$	−601.229	−0.0754364
$400$	0	0
$401$	10635.1	1.32442	0.662208	−	0.749320i	$-0.269619\pi$
$401$	10635.1	1.32442	0.662208	+	0.749320i	$0.269619\pi$
$402$	0	0
$403$	5122.76	0.633208
$404$	0	0
$405$	13815.5	1.69505
$406$	0	0
$407$	67.0921	0.00817108
$408$	0	0
$409$	9261.74	1.11972	0.559858	−	0.828589i	$-0.310856\pi$
$409$	9261.74	1.11972	0.559858	+	0.828589i	$0.310856\pi$
$410$	0	0
$411$	1897.02	0.227672
$412$	0	0
$413$	15772.0	1.87915
$414$	0	0
$415$	−24095.7	−2.85015
$416$	0	0
$417$	−590.713	−0.0693701
$418$	0	0
$419$	−3241.64	−0.377959	−0.188979	−	0.981981i	$-0.560518\pi$
$419$	−3241.64	−0.377959	−0.188979	+	0.981981i	$0.560518\pi$
$420$	0	0
$421$	3075.85	0.356075	0.178038	−	0.984024i	$-0.443025\pi$
$421$	3075.85	0.356075	0.178038	+	0.984024i	$0.443025\pi$
$422$	0	0
$423$	−13609.6	−1.56436
$424$	0	0
$425$	−17769.0	−2.02806
$426$	0	0
$427$	−5664.57	−0.641985
$428$	0	0
$429$	−27.7460	−0.00312259
$430$	0	0
$431$	−4730.26	−0.528651	−0.264326	−	0.964433i	$-0.585149\pi$
$431$	−4730.26	−0.528651	−0.264326	+	0.964433i	$0.585149\pi$
$432$	0	0
$433$	−8846.65	−0.981854	−0.490927	−	0.871201i	$-0.663342\pi$
$433$	−8846.65	−0.981854	−0.490927	+	0.871201i	$0.663342\pi$
$434$	0	0
$435$	4794.53	0.528460
$436$	0	0
$437$	1173.85	0.128497
$438$	0	0
$439$	−14485.9	−1.57489	−0.787443	−	0.616387i	$-0.788596\pi$
$439$	−14485.9	−1.57489	−0.787443	+	0.616387i	$0.788596\pi$
$440$	0	0
$441$	−17116.3	−1.84822
$442$	0	0
$443$	−1271.64	−0.136383	−0.0681914	−	0.997672i	$-0.521723\pi$
$443$	−1271.64	−0.136383	−0.0681914	+	0.997672i	$0.521723\pi$
$444$	0	0
$445$	4435.63	0.472515
$446$	0	0
$447$	−824.641	−0.0872577
$448$	0	0
$449$	4706.71	0.494706	0.247353	−	0.968925i	$-0.420439\pi$
$449$	4706.71	0.494706	0.247353	+	0.968925i	$0.420439\pi$
$450$	0	0
$451$	77.5936	0.00810142
$452$	0	0
$453$	1424.71	0.147767
$454$	0	0
$455$	26224.2	2.70200
$456$	0	0
$457$	4850.91	0.496534	0.248267	−	0.968692i	$-0.420139\pi$
$457$	4850.91	0.496534	0.248267	+	0.968692i	$0.420139\pi$
$458$	0	0
$459$	2869.91	0.291843
$460$	0	0
$461$	−17447.0	−1.76267	−0.881333	−	0.472495i	$-0.843353\pi$
$461$	−17447.0	−1.76267	−0.881333	+	0.472495i	$0.843353\pi$
$462$	0	0
$463$	−5892.21	−0.591435	−0.295717	−	0.955275i	$-0.595559\pi$
$463$	−5892.21	−0.591435	−0.295717	+	0.955275i	$0.595559\pi$
$464$	0	0
$465$	2801.10	0.279351
$466$	0	0
$467$	−11880.9	−1.17727	−0.588634	−	0.808400i	$-0.700334\pi$
$467$	−11880.9	−1.17727	−0.588634	+	0.808400i	$0.700334\pi$
$468$	0	0
$469$	−31169.7	−3.06883
$470$	0	0
$471$	−2499.51	−0.244525
$472$	0	0
$473$	−146.308	−0.0142225
$474$	0	0
$475$	−6234.83	−0.602261
$476$	0	0
$477$	14566.3	1.39820
$478$	0	0
$479$	9454.03	0.901807	0.450904	−	0.892573i	$-0.351102\pi$
$479$	9454.03	0.901807	0.450904	+	0.892573i	$0.351102\pi$
$480$	0	0
$481$	−3664.89	−0.347410
$482$	0	0
$483$	−1955.00	−0.184173
$484$	0	0
$485$	−14549.2	−1.36216
$486$	0	0
$487$	17015.1	1.58322	0.791608	−	0.611029i	$-0.209244\pi$
$487$	17015.1	1.58322	0.791608	+	0.611029i	$0.209244\pi$
$488$	0	0
$489$	1102.62	0.101968
$490$	0	0
$491$	4075.78	0.374618	0.187309	−	0.982301i	$-0.440023\pi$
$491$	4075.78	0.374618	0.187309	+	0.982301i	$0.440023\pi$
$492$	0	0
$493$	−12196.0	−1.11416
$494$	0	0
$495$	394.457	0.0358172
$496$	0	0
$497$	28628.4	2.58382
$498$	0	0
$499$	−6632.38	−0.595002	−0.297501	−	0.954721i	$-0.596153\pi$
$499$	−6632.38	−0.595002	−0.297501	+	0.954721i	$0.596153\pi$
$500$	0	0
$501$	161.613	0.0144118
$502$	0	0
$503$	−9052.55	−0.802452	−0.401226	−	0.915979i	$-0.631416\pi$
$503$	−9052.55	−0.802452	−0.401226	+	0.915979i	$0.631416\pi$
$504$	0	0
$505$	−7680.59	−0.676796
$506$	0	0
$507$	−681.381	−0.0596868
$508$	0	0
$509$	−8676.14	−0.755527	−0.377763	−	0.925902i	$-0.623307\pi$
$509$	−8676.14	−0.755527	−0.377763	+	0.925902i	$0.623307\pi$
$510$	0	0
$511$	−22914.7	−1.98373
$512$	0	0
$513$	1007.00	0.0866669
$514$	0	0
$515$	−3310.48	−0.283257
$516$	0	0
$517$	−373.060	−0.0317353
$518$	0	0
$519$	1829.59	0.154740
$520$	0	0
$521$	−2967.88	−0.249568	−0.124784	−	0.992184i	$-0.539824\pi$
$521$	−2967.88	−0.249568	−0.124784	+	0.992184i	$0.539824\pi$
$522$	0	0
$523$	9815.59	0.820661	0.410330	−	0.911937i	$-0.365413\pi$
$523$	9815.59	0.820661	0.410330	+	0.911937i	$0.365413\pi$
$524$	0	0
$525$	10383.8	0.863215
$526$	0	0
$527$	−7125.26	−0.588959
$528$	0	0
$529$	−8350.02	−0.686284
$530$	0	0
$531$	−12959.1	−1.05909
$532$	0	0
$533$	−4238.53	−0.344449
$534$	0	0
$535$	30719.5	2.48246
$536$	0	0
$537$	1048.00	0.0842170
$538$	0	0
$539$	−469.184	−0.0374939
$540$	0	0
$541$	−4090.55	−0.325077	−0.162538	−	0.986702i	$-0.551968\pi$
$541$	−4090.55	−0.325077	−0.162538	+	0.986702i	$0.551968\pi$
$542$	0	0
$543$	−897.354	−0.0709192
$544$	0	0
$545$	−20881.8	−1.64125
$546$	0	0
$547$	2001.12	0.156420	0.0782100	−	0.996937i	$-0.475080\pi$
$547$	2001.12	0.156420	0.0782100	+	0.996937i	$0.475080\pi$
$548$	0	0
$549$	4654.29	0.361822
$550$	0	0
$551$	−4279.36	−0.330865
$552$	0	0
$553$	36966.3	2.84261
$554$	0	0
$555$	−2003.95	−0.153266
$556$	0	0
$557$	6727.24	0.511746	0.255873	−	0.966710i	$-0.417637\pi$
$557$	6727.24	0.511746	0.255873	+	0.966710i	$0.417637\pi$
$558$	0	0
$559$	7992.03	0.604699
$560$	0	0
$561$	38.5921	0.00290438
$562$	0	0
$563$	−2255.21	−0.168820	−0.0844100	−	0.996431i	$-0.526901\pi$
$563$	−2255.21	−0.168820	−0.0844100	+	0.996431i	$0.526901\pi$
$564$	0	0
$565$	−17943.4	−1.33608
$566$	0	0
$567$	20536.7	1.52110
$568$	0	0
$569$	12206.7	0.899349	0.449675	−	0.893192i	$-0.351540\pi$
$569$	12206.7	0.899349	0.449675	+	0.893192i	$0.351540\pi$
$570$	0	0
$571$	−19976.5	−1.46408	−0.732042	−	0.681259i	$-0.761433\pi$
$571$	−19976.5	−1.46408	−0.732042	+	0.681259i	$0.761433\pi$
$572$	0	0
$573$	4113.98	0.299937
$574$	0	0
$575$	−20273.6	−1.47038
$576$	0	0
$577$	7025.87	0.506916	0.253458	−	0.967346i	$-0.418432\pi$
$577$	7025.87	0.506916	0.253458	+	0.967346i	$0.418432\pi$
$578$	0	0
$579$	3511.87	0.252070
$580$	0	0
$581$	−35818.4	−2.55765
$582$	0	0
$583$	399.283	0.0283647
$584$	0	0
$585$	−21547.1	−1.52284
$586$	0	0
$587$	15464.1	1.08735	0.543674	−	0.839297i	$-0.317033\pi$
$587$	15464.1	1.08735	0.543674	+	0.839297i	$0.317033\pi$
$588$	0	0
$589$	−2500.13	−0.174900
$590$	0	0
$591$	2488.28	0.173188
$592$	0	0
$593$	15847.9	1.09746	0.548731	−	0.835999i	$-0.315111\pi$
$593$	15847.9	1.09746	0.548731	+	0.835999i	$0.315111\pi$
$594$	0	0
$595$	−36475.3	−2.51318
$596$	0	0
$597$	−386.556	−0.0265003
$598$	0	0
$599$	−19135.1	−1.30524	−0.652621	−	0.757684i	$-0.726331\pi$
$599$	−19135.1	−1.30524	−0.652621	+	0.757684i	$0.726331\pi$
$600$	0	0
$601$	−2997.59	−0.203451	−0.101726	−	0.994812i	$-0.532436\pi$
$601$	−2997.59	−0.203451	−0.101726	+	0.994812i	$0.532436\pi$
$602$	0	0
$603$	25610.6	1.72959
$604$	0	0
$605$	−28322.6	−1.90327
$606$	0	0
$607$	−18978.6	−1.26906	−0.634528	−	0.772900i	$-0.718805\pi$
$607$	−18978.6	−1.26906	−0.634528	+	0.772900i	$0.718805\pi$
$608$	0	0
$609$	7127.08	0.474226
$610$	0	0
$611$	20378.3	1.34929
$612$	0	0
$613$	11248.7	0.741156	0.370578	−	0.928801i	$-0.379160\pi$
$613$	11248.7	0.741156	0.370578	+	0.928801i	$0.379160\pi$
$614$	0	0
$615$	−2317.61	−0.151960
$616$	0	0
$617$	−4803.53	−0.313425	−0.156712	−	0.987644i	$-0.550090\pi$
$617$	−4803.53	−0.313425	−0.156712	+	0.987644i	$0.550090\pi$
$618$	0	0
$619$	11306.6	0.734168	0.367084	−	0.930188i	$-0.380356\pi$
$619$	11306.6	0.734168	0.367084	+	0.930188i	$0.380356\pi$
$620$	0	0
$621$	3274.43	0.211592
$622$	0	0
$623$	6593.58	0.424023
$624$	0	0
$625$	51038.3	3.26645
$626$	0	0
$627$	13.5413	0.000862498 0
$628$	0	0
$629$	5097.50	0.323133
$630$	0	0
$631$	28148.8	1.77589	0.887945	−	0.459950i	$-0.152133\pi$
$631$	28148.8	1.77589	0.887945	+	0.459950i	$0.152133\pi$
$632$	0	0
$633$	4851.06	0.304601
$634$	0	0
$635$	−10416.1	−0.650944
$636$	0	0
$637$	25629.0	1.59413
$638$	0	0
$639$	−23522.5	−1.45624
$640$	0	0
$641$	−28222.5	−1.73904	−0.869519	−	0.493900i	$-0.835571\pi$
$641$	−28222.5	−1.73904	−0.869519	+	0.493900i	$0.835571\pi$
$642$	0	0
$643$	−11441.5	−0.701723	−0.350861	−	0.936427i	$-0.614111\pi$
$643$	−11441.5	−0.701723	−0.350861	+	0.936427i	$0.614111\pi$
$644$	0	0
$645$	4370.01	0.266774
$646$	0	0
$647$	−26346.6	−1.60091	−0.800457	−	0.599390i	$-0.795410\pi$
$647$	−26346.6	−1.60091	−0.800457	+	0.599390i	$0.795410\pi$
$648$	0	0
$649$	−355.227	−0.0214852
$650$	0	0
$651$	4163.85	0.250682
$652$	0	0
$653$	−29759.4	−1.78342	−0.891710	−	0.452607i	$-0.850494\pi$
$653$	−29759.4	−1.78342	−0.891710	+	0.452607i	$0.850494\pi$
$654$	0	0
$655$	−19739.9	−1.17756
$656$	0	0
$657$	18827.9	1.11803
$658$	0	0
$659$	−15959.4	−0.943386	−0.471693	−	0.881763i	$-0.656357\pi$
$659$	−15959.4	−0.943386	−0.471693	+	0.881763i	$0.656357\pi$
$660$	0	0
$661$	−3165.29	−0.186256	−0.0931282	−	0.995654i	$-0.529687\pi$
$661$	−3165.29	−0.186256	−0.0931282	+	0.995654i	$0.529687\pi$
$662$	0	0
$663$	−2108.08	−0.123486
$664$	0	0
$665$	−12798.6	−0.746326
$666$	0	0
$667$	−13915.1	−0.807786
$668$	0	0
$669$	−802.187	−0.0463593
$670$	0	0
$671$	127.581	0.00734010
$672$	0	0
$673$	29152.9	1.66978	0.834890	−	0.550416i	$-0.185531\pi$
$673$	29152.9	1.66978	0.834890	+	0.550416i	$0.185531\pi$
$674$	0	0
$675$	−17391.9	−0.991726
$676$	0	0
$677$	15830.4	0.898687	0.449343	−	0.893359i	$-0.351658\pi$
$677$	15830.4	0.898687	0.449343	+	0.893359i	$0.351658\pi$
$678$	0	0
$679$	−21627.5	−1.22237
$680$	0	0
$681$	−4889.70	−0.275145
$682$	0	0
$683$	−9998.31	−0.560139	−0.280069	−	0.959980i	$-0.590357\pi$
$683$	−9998.31	−0.560139	−0.280069	+	0.959980i	$0.590357\pi$
$684$	0	0
$685$	40382.5	2.25246
$686$	0	0
$687$	−569.132	−0.0316066
$688$	0	0
$689$	−21810.7	−1.20598
$690$	0	0
$691$	−7331.53	−0.403625	−0.201812	−	0.979424i	$-0.564683\pi$
$691$	−7331.53	−0.403625	−0.201812	+	0.979424i	$0.564683\pi$
$692$	0	0
$693$	586.362	0.0321415
$694$	0	0
$695$	−12574.7	−0.686309
$696$	0	0
$697$	5895.39	0.320378
$698$	0	0
$699$	−3835.88	−0.207563
$700$	0	0
$701$	−9928.34	−0.534933	−0.267467	−	0.963567i	$-0.586187\pi$
$701$	−9928.34	−0.534933	−0.267467	+	0.963567i	$0.586187\pi$
$702$	0	0
$703$	1788.62	0.0959591
$704$	0	0
$705$	11142.8	0.595265
$706$	0	0
$707$	−11417.2	−0.607340
$708$	0	0
$709$	−1613.88	−0.0854874	−0.0427437	−	0.999086i	$-0.513610\pi$
$709$	−1613.88	−0.0854874	−0.0427437	+	0.999086i	$0.513610\pi$
$710$	0	0
$711$	−30373.3	−1.60209
$712$	0	0
$713$	−8129.60	−0.427007
$714$	0	0
$715$	−590.638	−0.0308932
$716$	0	0
$717$	527.821	0.0274921
$718$	0	0
$719$	26007.1	1.34896	0.674480	−	0.738293i	$-0.264368\pi$
$719$	26007.1	1.34896	0.674480	+	0.738293i	$0.264368\pi$
$720$	0	0
$721$	−4921.04	−0.254187
$722$	0	0
$723$	−5523.03	−0.284099
$724$	0	0
$725$	73908.8	3.78608
$726$	0	0
$727$	−18096.2	−0.923180	−0.461590	−	0.887093i	$-0.652721\pi$
$727$	−18096.2	−0.923180	−0.461590	+	0.887093i	$0.652721\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	−11116.1	−0.562442
$732$	0	0
$733$	−25273.4	−1.27353	−0.636764	−	0.771059i	$-0.719727\pi$
$733$	−25273.4	−1.27353	−0.636764	+	0.771059i	$0.719727\pi$
$734$	0	0
$735$	14013.9	0.703278
$736$	0	0
$737$	702.024	0.0350874
$738$	0	0
$739$	15373.0	0.765229	0.382615	−	0.923908i	$-0.375024\pi$
$739$	15373.0	0.765229	0.382615	+	0.923908i	$0.375024\pi$
$740$	0	0
$741$	−739.688	−0.0366709
$742$	0	0
$743$	25357.0	1.25203	0.626016	−	0.779810i	$-0.284685\pi$
$743$	25357.0	1.25203	0.626016	+	0.779810i	$0.284685\pi$
$744$	0	0
$745$	−17554.4	−0.863279
$746$	0	0
$747$	29430.1	1.44149
$748$	0	0
$749$	45664.6	2.22770
$750$	0	0
$751$	2019.97	0.0981486	0.0490743	−	0.998795i	$-0.484373\pi$
$751$	2019.97	0.0981486	0.0490743	+	0.998795i	$0.484373\pi$
$752$	0	0
$753$	−7573.11	−0.366506
$754$	0	0
$755$	30328.2	1.46193
$756$	0	0
$757$	−38997.9	−1.87239	−0.936197	−	0.351477i	$-0.885680\pi$
$757$	−38997.9	−1.87239	−0.936197	+	0.351477i	$0.885680\pi$
$758$	0	0
$759$	44.0318	0.00210573
$760$	0	0
$761$	−4691.83	−0.223493	−0.111747	−	0.993737i	$-0.535645\pi$
$761$	−4691.83	−0.223493	−0.111747	+	0.993737i	$0.535645\pi$
$762$	0	0
$763$	−31040.9	−1.47281
$764$	0	0
$765$	29970.0	1.41643
$766$	0	0
$767$	19404.2	0.913487
$768$	0	0
$769$	−18338.5	−0.859952	−0.429976	−	0.902840i	$-0.641478\pi$
$769$	−18338.5	−0.859952	−0.429976	+	0.902840i	$0.641478\pi$
$770$	0	0
$771$	−6349.60	−0.296596
$772$	0	0
$773$	8104.70	0.377110	0.188555	−	0.982063i	$-0.439620\pi$
$773$	8104.70	0.377110	0.188555	+	0.982063i	$0.439620\pi$
$774$	0	0
$775$	43179.7	2.00137
$776$	0	0
$777$	−2978.87	−0.137537
$778$	0	0
$779$	2068.59	0.0951410
$780$	0	0
$781$	−644.787	−0.0295420
$782$	0	0
$783$	−11937.2	−0.544827
$784$	0	0
$785$	−53207.8	−2.41920
$786$	0	0
$787$	−31649.8	−1.43354	−0.716769	−	0.697311i	$-0.754380\pi$
$787$	−31649.8	−1.43354	−0.716769	+	0.697311i	$0.754380\pi$
$788$	0	0
$789$	−3915.06	−0.176654
$790$	0	0
$791$	−26673.0	−1.19897
$792$	0	0
$793$	−6969.07	−0.312079
$794$	0	0
$795$	−11926.0	−0.532040
$796$	0	0
$797$	−1846.41	−0.0820615	−0.0410308	−	0.999158i	$-0.513064\pi$
$797$	−1846.41	−0.0820615	−0.0410308	+	0.999158i	$0.513064\pi$
$798$	0	0
$799$	−28344.3	−1.25500
$800$	0	0
$801$	−5417.62	−0.238979
$802$	0	0
$803$	516.100	0.0226809
$804$	0	0
$805$	−41616.7	−1.82211
$806$	0	0
$807$	7171.13	0.312808
$808$	0	0
$809$	−1277.18	−0.0555047	−0.0277523	−	0.999615i	$-0.508835\pi$
$809$	−1277.18	−0.0555047	−0.0277523	+	0.999615i	$0.508835\pi$
$810$	0	0
$811$	7935.98	0.343613	0.171806	−	0.985131i	$-0.445040\pi$
$811$	7935.98	0.343613	0.171806	+	0.985131i	$0.445040\pi$
$812$	0	0
$813$	−1100.04	−0.0474538
$814$	0	0
$815$	23471.9	1.00882
$816$	0	0
$817$	−3900.46	−0.167025
$818$	0	0
$819$	−32029.9	−1.36656
$820$	0	0
$821$	19972.8	0.849034	0.424517	−	0.905420i	$-0.360444\pi$
$821$	19972.8	0.849034	0.424517	+	0.905420i	$0.360444\pi$
$822$	0	0
$823$	−10986.0	−0.465307	−0.232654	−	0.972560i	$-0.574741\pi$
$823$	−10986.0	−0.465307	−0.232654	+	0.972560i	$0.574741\pi$
$824$	0	0
$825$	−233.871	−0.00986953
$826$	0	0
$827$	9581.30	0.402871	0.201436	−	0.979502i	$-0.435439\pi$
$827$	9581.30	0.402871	0.201436	+	0.979502i	$0.435439\pi$
$828$	0	0
$829$	23501.8	0.984621	0.492311	−	0.870420i	$-0.336152\pi$
$829$	23501.8	0.984621	0.492311	+	0.870420i	$0.336152\pi$
$830$	0	0
$831$	−1361.51	−0.0568354
$832$	0	0
$833$	−35647.5	−1.48273
$834$	0	0
$835$	3440.30	0.142583
$836$	0	0
$837$	−6974.04	−0.288003
$838$	0	0
$839$	14939.3	0.614733	0.307367	−	0.951591i	$-0.400552\pi$
$839$	14939.3	0.614733	0.307367	+	0.951591i	$0.400552\pi$
$840$	0	0
$841$	26339.3	1.07997
$842$	0	0
$843$	4438.51	0.181341
$844$	0	0
$845$	−14504.8	−0.590508
$846$	0	0
$847$	−42101.6	−1.70794
$848$	0	0
$849$	9190.64	0.371522
$850$	0	0
$851$	5816.02	0.234278
$852$	0	0
$853$	14952.6	0.600196	0.300098	−	0.953908i	$-0.402981\pi$
$853$	14952.6	0.600196	0.300098	+	0.953908i	$0.402981\pi$
$854$	0	0
$855$	10515.9	0.420628
$856$	0	0
$857$	9152.49	0.364811	0.182406	−	0.983223i	$-0.441612\pi$
$857$	9152.49	0.364811	0.182406	+	0.983223i	$0.441612\pi$
$858$	0	0
$859$	37436.7	1.48699	0.743494	−	0.668743i	$-0.233167\pi$
$859$	37436.7	1.48699	0.743494	+	0.668743i	$0.233167\pi$
$860$	0	0
$861$	−3445.14	−0.136365
$862$	0	0
$863$	29723.0	1.17240	0.586200	−	0.810166i	$-0.300623\pi$
$863$	29723.0	1.17240	0.586200	+	0.810166i	$0.300623\pi$
$864$	0	0
$865$	38946.9	1.53091
$866$	0	0
$867$	−1980.86	−0.0775936
$868$	0	0
$869$	−832.578	−0.0325009
$870$	0	0
$871$	−38347.9	−1.49181
$872$	0	0
$873$	17770.2	0.688924
$874$	0	0
$875$	136843.	5.28701
$876$	0	0
$877$	−42664.1	−1.64272	−0.821360	−	0.570411i	$-0.806784\pi$
$877$	−42664.1	−1.64272	−0.821360	+	0.570411i	$0.806784\pi$
$878$	0	0
$879$	−3316.50	−0.127261
$880$	0	0
$881$	−42198.0	−1.61372	−0.806859	−	0.590743i	$-0.798835\pi$
$881$	−42198.0	−1.61372	−0.806859	+	0.590743i	$0.798835\pi$
$882$	0	0
$883$	−41031.1	−1.56377	−0.781884	−	0.623424i	$-0.785741\pi$
$883$	−41031.1	−1.56377	−0.781884	+	0.623424i	$0.785741\pi$
$884$	0	0
$885$	10610.1	0.403001
$886$	0	0
$887$	42729.3	1.61749	0.808743	−	0.588162i	$-0.200148\pi$
$887$	42729.3	1.61749	0.808743	+	0.588162i	$0.200148\pi$
$888$	0	0
$889$	−15483.5	−0.584141
$890$	0	0
$891$	−462.541	−0.0173914
$892$	0	0
$893$	−9945.50	−0.372692
$894$	0	0
$895$	22309.1	0.833196
$896$	0	0
$897$	−2405.22	−0.0895296
$898$	0	0
$899$	29637.0	1.09950
$900$	0	0
$901$	30336.6	1.12171
$902$	0	0
$903$	6496.04	0.239396
$904$	0	0
$905$	−19102.2	−0.701635
$906$	0	0
$907$	−39315.7	−1.43931	−0.719656	−	0.694330i	$-0.755701\pi$
$907$	−39315.7	−1.43931	−0.719656	+	0.694330i	$0.755701\pi$
$908$	0	0
$909$	9380.97	0.342296
$910$	0	0
$911$	−5025.05	−0.182752	−0.0913761	−	0.995816i	$-0.529127\pi$
$911$	−5025.05	−0.182752	−0.0913761	+	0.995816i	$0.529127\pi$
$912$	0	0
$913$	806.724	0.0292428
$914$	0	0
$915$	−3810.66	−0.137679
$916$	0	0
$917$	−29343.5	−1.05671
$918$	0	0
$919$	14259.7	0.511843	0.255922	−	0.966698i	$-0.417621\pi$
$919$	14259.7	0.511843	0.255922	+	0.966698i	$0.417621\pi$
$920$	0	0
$921$	−1574.75	−0.0563408
$922$	0	0
$923$	35221.3	1.25604
$924$	0	0
$925$	−30891.3	−1.09806
$926$	0	0
$927$	4043.37	0.143260
$928$	0	0
$929$	27668.1	0.977137	0.488568	−	0.872526i	$-0.337519\pi$
$929$	27668.1	0.977137	0.488568	+	0.872526i	$0.337519\pi$
$930$	0	0
$931$	−12508.1	−0.440318
$932$	0	0
$933$	4614.64	0.161925
$934$	0	0
$935$	821.521	0.0287343
$936$	0	0
$937$	−25122.4	−0.875893	−0.437947	−	0.899001i	$-0.644294\pi$
$937$	−25122.4	−0.875893	−0.437947	+	0.899001i	$0.644294\pi$
$938$	0	0
$939$	4738.42	0.164678
$940$	0	0
$941$	19335.4	0.669836	0.334918	−	0.942247i	$-0.391291\pi$
$941$	19335.4	0.669836	0.334918	+	0.942247i	$0.391291\pi$
$942$	0	0
$943$	6726.36	0.232281
$944$	0	0
$945$	−35701.2	−1.22895
$946$	0	0
$947$	6752.27	0.231700	0.115850	−	0.993267i	$-0.463041\pi$
$947$	6752.27	0.231700	0.115850	+	0.993267i	$0.463041\pi$
$948$	0	0
$949$	−28191.8	−0.964326
$950$	0	0
$951$	−1039.66	−0.0354504
$952$	0	0
$953$	42620.6	1.44870	0.724352	−	0.689430i	$-0.242139\pi$
$953$	42620.6	1.44870	0.724352	+	0.689430i	$0.242139\pi$
$954$	0	0
$955$	87575.5	2.96741
$956$	0	0
$957$	−160.521	−0.00542204
$958$	0	0
$959$	60028.7	2.02130
$960$	0	0
$961$	−12476.2	−0.418791
$962$	0	0
$963$	−37520.3	−1.25553
$964$	0	0
$965$	74758.3	2.49384
$966$	0	0
$967$	−17814.9	−0.592441	−0.296220	−	0.955120i	$-0.595726\pi$
$967$	−17814.9	−0.592441	−0.296220	+	0.955120i	$0.595726\pi$
$968$	0	0
$969$	1028.83	0.0341083
$970$	0	0
$971$	49040.2	1.62078	0.810388	−	0.585893i	$-0.199256\pi$
$971$	49040.2	1.62078	0.810388	+	0.585893i	$0.199256\pi$
$972$	0	0
$973$	−18692.3	−0.615876
$974$	0	0
$975$	12775.2	0.419623
$976$	0	0
$977$	−4853.73	−0.158940	−0.0794702	−	0.996837i	$-0.525323\pi$
$977$	−4853.73	−0.158940	−0.0794702	+	0.996837i	$0.525323\pi$
$978$	0	0
$979$	−148.505	−0.00484805
$980$	0	0
$981$	25504.8	0.830076
$982$	0	0
$983$	37770.7	1.22553	0.612767	−	0.790264i	$-0.290056\pi$
$983$	37770.7	1.22553	0.612767	+	0.790264i	$0.290056\pi$
$984$	0	0
$985$	52968.7	1.71342
$986$	0	0
$987$	16563.8	0.534176
$988$	0	0
$989$	−12683.0	−0.407782
$990$	0	0
$991$	−46048.7	−1.47607	−0.738036	−	0.674762i	$-0.764246\pi$
$991$	−46048.7	−1.47607	−0.738036	+	0.674762i	$0.764246\pi$
$992$	0	0
$993$	−4634.52	−0.148109
$994$	0	0
$995$	−8228.74	−0.262180
$996$	0	0
$997$	43033.7	1.36699	0.683495	−	0.729955i	$-0.260459\pi$
$997$	43033.7	1.36699	0.683495	+	0.729955i	$0.260459\pi$
$998$	0	0
$999$	4989.32	0.158013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.n.1.2		2
4.3	odd	2		1216.4.a.i.1.2		2
8.3	odd	2		608.4.a.d.1.1	yes	2
8.5	even	2		608.4.a.c.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.c.1.1	✓	2	8.5	even	2
608.4.a.d.1.1	yes	2	8.3	odd	2
1216.4.a.i.1.2		2	4.3	odd	2
1216.4.a.n.1.2		2	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +21.2873 q^{5} +31.6437 q^{7} -26.0000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +21.2873 q^{5} +31.6437 q^{7} -26.0000 q^{9} -0.712698 q^{11} +38.9310 q^{13} +21.2873 q^{15} -54.1492 q^{17} -19.0000 q^{19} +31.6437 q^{21} -61.7817 q^{23} +328.149 q^{25} -53.0000 q^{27} +225.229 q^{29} +131.586 q^{31} -0.712698 q^{33} +673.608 q^{35} -94.1381 q^{37} +38.9310 q^{39} -108.873 q^{41} +205.287 q^{43} -553.470 q^{45} +523.448 q^{47} +658.321 q^{49} -54.1492 q^{51} -560.240 q^{53} -15.1714 q^{55} -19.0000 q^{57} +498.425 q^{59} -179.011 q^{61} -822.735 q^{63} +828.735 q^{65} -985.022 q^{67} -61.7817 q^{69} +904.713 q^{71} -724.149 q^{73} +328.149 q^{75} -22.5524 q^{77} +1168.20 q^{79} +649.000 q^{81} -1131.93 q^{83} -1152.69 q^{85} +225.229 q^{87} +208.370 q^{89} +1231.92 q^{91} +131.586 q^{93} -404.459 q^{95} -683.470 q^{97} +18.5302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9} - 40 q^{11} + 20 q^{13} + 4 q^{15} + 46 q^{17} - 38 q^{19} + 44 q^{21} - 220 q^{23} + 502 q^{25} - 106 q^{27} + 84 q^{29} - 84 q^{31} - 40 q^{33} + 460 q^{35} - 304 q^{37} + 20 q^{39} + 168 q^{41} + 372 q^{43} - 104 q^{45} + 584 q^{47} + 468 q^{49} + 46 q^{51} - 484 q^{53} + 664 q^{55} - 38 q^{57} + 1074 q^{59} - 88 q^{61} - 1144 q^{63} + 1156 q^{65} - 1430 q^{67} - 220 q^{69} + 1848 q^{71} - 1294 q^{73} + 502 q^{75} - 508 q^{77} + 832 q^{79} + 1298 q^{81} - 528 q^{83} - 2884 q^{85} + 84 q^{87} + 1844 q^{89} + 998 q^{91} - 84 q^{93} - 76 q^{95} - 364 q^{97} + 1040 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9 - 40 * q^11 + 20 * q^13 + 4 * q^15 + 46 * q^17 - 38 * q^19 + 44 * q^21 - 220 * q^23 + 502 * q^25 - 106 * q^27 + 84 * q^29 - 84 * q^31 - 40 * q^33 + 460 * q^35 - 304 * q^37 + 20 * q^39 + 168 * q^41 + 372 * q^43 - 104 * q^45 + 584 * q^47 + 468 * q^49 + 46 * q^51 - 484 * q^53 + 664 * q^55 - 38 * q^57 + 1074 * q^59 - 88 * q^61 - 1144 * q^63 + 1156 * q^65 - 1430 * q^67 - 220 * q^69 + 1848 * q^71 - 1294 * q^73 + 502 * q^75 - 508 * q^77 + 832 * q^79 + 1298 * q^81 - 528 * q^83 - 2884 * q^85 + 84 * q^87 + 1844 * q^89 + 998 * q^91 - 84 * q^93 - 76 * q^95 - 364 * q^97 + 1040 * q^99

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