LMFDB - Embedded newform 784.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 13 $ x^3 - x^2 - 10*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.90222$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80445 q^{3} +6.69159 q^{5} -19.1351 q^{9} +O(q^{10})$	$q+2.80445 q^{3} +6.69159 q^{5} -19.1351 q^{9} -31.4881 q^{11} -18.6837 q^{13} +18.7662 q^{15} +87.9538 q^{17} +13.0970 q^{19} +9.37681 q^{23} -80.2226 q^{25} -129.383 q^{27} -5.11923 q^{29} -257.412 q^{31} -88.3067 q^{33} -380.215 q^{37} -52.3973 q^{39} +217.959 q^{41} -377.049 q^{43} -128.044 q^{45} -357.710 q^{47} +246.662 q^{51} +764.390 q^{53} -210.706 q^{55} +36.7298 q^{57} -450.671 q^{59} +174.078 q^{61} -125.023 q^{65} +497.234 q^{67} +26.2967 q^{69} -350.238 q^{71} +1062.69 q^{73} -224.980 q^{75} -560.449 q^{79} +153.799 q^{81} -1105.27 q^{83} +588.551 q^{85} -14.3566 q^{87} -1206.71 q^{89} -721.897 q^{93} +87.6398 q^{95} -1442.99 q^{97} +602.528 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) $ 3 * q - 7 * q^3 + 3 * q^5 + 18 * q^9	$ 3 q - 7 q^{3} + 3 q^{5} + 18 q^{9} + 3 q^{11} + 26 q^{13} - 127 q^{15} + 31 q^{17} - 89 q^{19} + 201 q^{23} + 300 q^{25} - 469 q^{27} + 190 q^{29} - 339 q^{31} + 105 q^{33} - 535 q^{37} + 134 q^{39} - 58 q^{41} - 268 q^{43} + 1410 q^{45} - 205 q^{47} + 965 q^{51} + 757 q^{53} - 1653 q^{55} + 261 q^{57} - 1799 q^{59} - 625 q^{61} - 1750 q^{65} + 495 q^{67} - 973 q^{69} - 640 q^{71} + 443 q^{73} - 1484 q^{75} - 79 q^{79} + 2523 q^{81} - 2372 q^{83} - 977 q^{85} - 910 q^{87} - 821 q^{89} - 1321 q^{93} + 1327 q^{95} + 342 q^{97} - 2310 q^{99}+O(q^{100}) $ 3 * q - 7 * q^3 + 3 * q^5 + 18 * q^9 + 3 * q^11 + 26 * q^13 - 127 * q^15 + 31 * q^17 - 89 * q^19 + 201 * q^23 + 300 * q^25 - 469 * q^27 + 190 * q^29 - 339 * q^31 + 105 * q^33 - 535 * q^37 + 134 * q^39 - 58 * q^41 - 268 * q^43 + 1410 * q^45 - 205 * q^47 + 965 * q^51 + 757 * q^53 - 1653 * q^55 + 261 * q^57 - 1799 * q^59 - 625 * q^61 - 1750 * q^65 + 495 * q^67 - 973 * q^69 - 640 * q^71 + 443 * q^73 - 1484 * q^75 - 79 * q^79 + 2523 * q^81 - 2372 * q^83 - 977 * q^85 - 910 * q^87 - 821 * q^89 - 1321 * q^93 + 1327 * q^95 + 342 * q^97 - 2310 * 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
$3$	2.80445	0.539716	0.269858	−	0.962900i	$-0.413023\pi$
$3$	2.80445	0.539716	0.269858	+	0.962900i	$0.413023\pi$
$4$	0	0
$5$	6.69159	0.598514	0.299257	−	0.954173i	$-0.403261\pi$
$5$	6.69159	0.598514	0.299257	+	0.954173i	$0.403261\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−19.1351	−0.708707
$10$	0	0
$11$	−31.4881	−0.863093	−0.431546	−	0.902091i	$-0.642032\pi$
$11$	−31.4881	−0.863093	−0.431546	+	0.902091i	$0.642032\pi$
$12$	0	0
$13$	−18.6837	−0.398609	−0.199304	−	0.979938i	$-0.563868\pi$
$13$	−18.6837	−0.398609	−0.199304	+	0.979938i	$0.563868\pi$
$14$	0	0
$15$	18.7662	0.323028
$16$	0	0
$17$	87.9538	1.25482	0.627410	−	0.778689i	$-0.284115\pi$
$17$	87.9538	1.25482	0.627410	+	0.778689i	$0.284115\pi$
$18$	0	0
$19$	13.0970	0.158140	0.0790700	−	0.996869i	$-0.474805\pi$
$19$	13.0970	0.158140	0.0790700	+	0.996869i	$0.474805\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.37681	0.0850087	0.0425043	−	0.999096i	$-0.486466\pi$
$23$	9.37681	0.0850087	0.0425043	+	0.999096i	$0.486466\pi$
$24$	0	0
$25$	−80.2226	−0.641781
$26$	0	0
$27$	−129.383	−0.922216
$28$	0	0
$29$	−5.11923	−0.0327799	−0.0163900	−	0.999866i	$-0.505217\pi$
$29$	−5.11923	−0.0327799	−0.0163900	+	0.999866i	$0.505217\pi$
$30$	0	0
$31$	−257.412	−1.49137	−0.745685	−	0.666298i	$-0.767878\pi$
$31$	−257.412	−1.49137	−0.745685	+	0.666298i	$0.767878\pi$
$32$	0	0
$33$	−88.3067	−0.465825
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−380.215	−1.68938	−0.844688	−	0.535259i	$-0.820214\pi$
$37$	−380.215	−1.68938	−0.844688	+	0.535259i	$0.820214\pi$
$38$	0	0
$39$	−52.3973	−0.215136
$40$	0	0
$41$	217.959	0.830230	0.415115	−	0.909769i	$-0.363741\pi$
$41$	217.959	0.830230	0.415115	+	0.909769i	$0.363741\pi$
$42$	0	0
$43$	−377.049	−1.33720	−0.668598	−	0.743624i	$-0.733105\pi$
$43$	−377.049	−1.33720	−0.668598	+	0.743624i	$0.733105\pi$
$44$	0	0
$45$	−128.044	−0.424171
$46$	0	0
$47$	−357.710	−1.11016	−0.555079	−	0.831798i	$-0.687312\pi$
$47$	−357.710	−1.11016	−0.555079	+	0.831798i	$0.687312\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	246.662	0.677246
$52$	0	0
$53$	764.390	1.98108	0.990538	−	0.137240i	$-0.0438233\pi$
$53$	764.390	1.98108	0.990538	+	0.137240i	$0.0438233\pi$
$54$	0	0
$55$	−210.706	−0.516573
$56$	0	0
$57$	36.7298	0.0853506
$58$	0	0
$59$	−450.671	−0.994447	−0.497224	−	0.867622i	$-0.665647\pi$
$59$	−450.671	−0.994447	−0.497224	+	0.867622i	$0.665647\pi$
$60$	0	0
$61$	174.078	0.365383	0.182691	−	0.983170i	$-0.441519\pi$
$61$	174.078	0.365383	0.182691	+	0.983170i	$0.441519\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−125.023	−0.238573
$66$	0	0
$67$	497.234	0.906669	0.453334	−	0.891340i	$-0.350234\pi$
$67$	497.234	0.906669	0.453334	+	0.891340i	$0.350234\pi$
$68$	0	0
$69$	26.2967	0.0458805
$70$	0	0
$71$	−350.238	−0.585432	−0.292716	−	0.956199i	$-0.594559\pi$
$71$	−350.238	−0.585432	−0.292716	+	0.956199i	$0.594559\pi$
$72$	0	0
$73$	1062.69	1.70381	0.851903	−	0.523699i	$-0.175448\pi$
$73$	1062.69	1.70381	0.851903	+	0.523699i	$0.175448\pi$
$74$	0	0
$75$	−224.980	−0.346379
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−560.449	−0.798170	−0.399085	−	0.916914i	$-0.630672\pi$
$79$	−560.449	−0.798170	−0.399085	+	0.916914i	$0.630672\pi$
$80$	0	0
$81$	153.799	0.210972
$82$	0	0
$83$	−1105.27	−1.46168	−0.730840	−	0.682549i	$-0.760871\pi$
$83$	−1105.27	−1.46168	−0.730840	+	0.682549i	$0.760871\pi$
$84$	0	0
$85$	588.551	0.751027
$86$	0	0
$87$	−14.3566	−0.0176918
$88$	0	0
$89$	−1206.71	−1.43721	−0.718604	−	0.695420i	$-0.755218\pi$
$89$	−1206.71	−1.43721	−0.718604	+	0.695420i	$0.755218\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−721.897	−0.804916
$94$	0	0
$95$	87.6398	0.0946490
$96$	0	0
$97$	−1442.99	−1.51045	−0.755226	−	0.655465i	$-0.772473\pi$
$97$	−1442.99	−1.51045	−0.755226	+	0.655465i	$0.772473\pi$
$98$	0	0
$99$	602.528	0.611680
$100$	0	0
$101$	−311.922	−0.307301	−0.153651	−	0.988125i	$-0.549103\pi$
$101$	−311.922	−0.307301	−0.153651	+	0.988125i	$0.549103\pi$
$102$	0	0
$103$	−581.632	−0.556407	−0.278203	−	0.960522i	$-0.589739\pi$
$103$	−581.632	−0.556407	−0.278203	+	0.960522i	$0.589739\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1175.48	1.06203	0.531016	−	0.847361i	$-0.321810\pi$
$107$	1175.48	1.06203	0.531016	+	0.847361i	$0.321810\pi$
$108$	0	0
$109$	337.052	0.296181	0.148091	−	0.988974i	$-0.452687\pi$
$109$	337.052	0.296181	0.148091	+	0.988974i	$0.452687\pi$
$110$	0	0
$111$	−1066.29	−0.911783
$112$	0	0
$113$	−1168.16	−0.972487	−0.486243	−	0.873823i	$-0.661633\pi$
$113$	−1168.16	−0.972487	−0.486243	+	0.873823i	$0.661633\pi$
$114$	0	0
$115$	62.7457	0.0508789
$116$	0	0
$117$	357.513	0.282497
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−339.499	−0.255071
$122$	0	0
$123$	611.253	0.448088
$124$	0	0
$125$	−1373.27	−0.982629
$126$	0	0
$127$	−23.4734	−0.0164010	−0.00820049	−	0.999966i	$-0.502610\pi$
$127$	−23.4734	−0.0164010	−0.00820049	+	0.999966i	$0.502610\pi$
$128$	0	0
$129$	−1057.41	−0.721706
$130$	0	0
$131$	−373.982	−0.249427	−0.124713	−	0.992193i	$-0.539801\pi$
$131$	−373.982	−0.249427	−0.124713	+	0.992193i	$0.539801\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−865.780	−0.551959
$136$	0	0
$137$	1138.14	0.709764	0.354882	−	0.934911i	$-0.384521\pi$
$137$	1138.14	0.709764	0.354882	+	0.934911i	$0.384521\pi$
$138$	0	0
$139$	−1229.46	−0.750224	−0.375112	−	0.926979i	$-0.622396\pi$
$139$	−1229.46	−0.750224	−0.375112	+	0.926979i	$0.622396\pi$
$140$	0	0
$141$	−1003.18	−0.599170
$142$	0	0
$143$	588.313	0.344037
$144$	0	0
$145$	−34.2558	−0.0196192
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2962.76	1.62898	0.814492	−	0.580175i	$-0.197016\pi$
$149$	2962.76	1.62898	0.814492	+	0.580175i	$0.197016\pi$
$150$	0	0
$151$	2597.70	1.39999	0.699993	−	0.714149i	$-0.253186\pi$
$151$	2597.70	1.39999	0.699993	+	0.714149i	$0.253186\pi$
$152$	0	0
$153$	−1683.00	−0.889299
$154$	0	0
$155$	−1722.49	−0.892606
$156$	0	0
$157$	−1364.98	−0.693870	−0.346935	−	0.937889i	$-0.612778\pi$
$157$	−1364.98	−0.693870	−0.346935	+	0.937889i	$0.612778\pi$
$158$	0	0
$159$	2143.69	1.06922
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3317.48	−1.59414	−0.797071	−	0.603886i	$-0.793618\pi$
$163$	−3317.48	−1.59414	−0.797071	+	0.603886i	$0.793618\pi$
$164$	0	0
$165$	−590.912	−0.278803
$166$	0	0
$167$	3803.96	1.76263	0.881315	−	0.472530i	$-0.156659\pi$
$167$	3803.96	1.76263	0.881315	+	0.472530i	$0.156659\pi$
$168$	0	0
$169$	−1847.92	−0.841111
$170$	0	0
$171$	−250.612	−0.112075
$172$	0	0
$173$	−2522.30	−1.10848	−0.554239	−	0.832358i	$-0.686991\pi$
$173$	−2522.30	−1.10848	−0.554239	+	0.832358i	$0.686991\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1263.88	−0.536719
$178$	0	0
$179$	187.036	0.0780992	0.0390496	−	0.999237i	$-0.487567\pi$
$179$	187.036	0.0780992	0.0390496	+	0.999237i	$0.487567\pi$
$180$	0	0
$181$	457.654	0.187940	0.0939701	−	0.995575i	$-0.470044\pi$
$181$	457.654	0.187940	0.0939701	+	0.995575i	$0.470044\pi$
$182$	0	0
$183$	488.191	0.197203
$184$	0	0
$185$	−2544.24	−1.01112
$186$	0	0
$187$	−2769.50	−1.08303
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2891.60	−1.09544	−0.547719	−	0.836663i	$-0.684504\pi$
$191$	−2891.60	−1.09544	−0.547719	+	0.836663i	$0.684504\pi$
$192$	0	0
$193$	1147.89	0.428120	0.214060	−	0.976821i	$-0.431331\pi$
$193$	1147.89	0.428120	0.214060	+	0.976821i	$0.431331\pi$
$194$	0	0
$195$	−350.621	−0.128762
$196$	0	0
$197$	1638.22	0.592481	0.296240	−	0.955113i	$-0.404267\pi$
$197$	1638.22	0.592481	0.296240	+	0.955113i	$0.404267\pi$
$198$	0	0
$199$	5311.28	1.89199	0.945996	−	0.324179i	$-0.105088\pi$
$199$	5311.28	1.89199	0.945996	+	0.324179i	$0.105088\pi$
$200$	0	0
$201$	1394.47	0.489344
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1458.49	0.496904
$206$	0	0
$207$	−179.426	−0.0602462
$208$	0	0
$209$	−412.400	−0.136489
$210$	0	0
$211$	1505.74	0.491277	0.245639	−	0.969361i	$-0.421002\pi$
$211$	1505.74	0.491277	0.245639	+	0.969361i	$0.421002\pi$
$212$	0	0
$213$	−982.225	−0.315967
$214$	0	0
$215$	−2523.06	−0.800331
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2980.24	0.919571
$220$	0	0
$221$	−1643.30	−0.500182
$222$	0	0
$223$	−2520.23	−0.756802	−0.378401	−	0.925642i	$-0.623526\pi$
$223$	−2520.23	−0.756802	−0.378401	+	0.925642i	$0.623526\pi$
$224$	0	0
$225$	1535.07	0.454834
$226$	0	0
$227$	−1538.04	−0.449705	−0.224853	−	0.974393i	$-0.572190\pi$
$227$	−1538.04	−0.449705	−0.224853	+	0.974393i	$0.572190\pi$
$228$	0	0
$229$	−571.683	−0.164969	−0.0824845	−	0.996592i	$-0.526285\pi$
$229$	−571.683	−0.164969	−0.0824845	+	0.996592i	$0.526285\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1604.20	−0.451051	−0.225525	−	0.974237i	$-0.572410\pi$
$233$	−1604.20	−0.451051	−0.225525	+	0.974237i	$0.572410\pi$
$234$	0	0
$235$	−2393.65	−0.664445
$236$	0	0
$237$	−1571.75	−0.430785
$238$	0	0
$239$	4699.38	1.27187	0.635936	−	0.771742i	$-0.280614\pi$
$239$	4699.38	1.27187	0.635936	+	0.771742i	$0.280614\pi$
$240$	0	0
$241$	872.607	0.233235	0.116617	−	0.993177i	$-0.462795\pi$
$241$	872.607	0.233235	0.116617	+	0.993177i	$0.462795\pi$
$242$	0	0
$243$	3924.67	1.03608
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−244.700	−0.0630360
$248$	0	0
$249$	−3099.68	−0.788892
$250$	0	0
$251$	3126.30	0.786176	0.393088	−	0.919501i	$-0.371407\pi$
$251$	3126.30	0.786176	0.393088	+	0.919501i	$0.371407\pi$
$252$	0	0
$253$	−295.258	−0.0733704
$254$	0	0
$255$	1650.56	0.405341
$256$	0	0
$257$	−904.015	−0.219420	−0.109710	−	0.993964i	$-0.534992\pi$
$257$	−904.015	−0.219420	−0.109710	+	0.993964i	$0.534992\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	97.9569	0.0232313
$262$	0	0
$263$	92.6368	0.0217195	0.0108598	−	0.999941i	$-0.496543\pi$
$263$	92.6368	0.0217195	0.0108598	+	0.999941i	$0.496543\pi$
$264$	0	0
$265$	5114.98	1.18570
$266$	0	0
$267$	−3384.16	−0.775683
$268$	0	0
$269$	4051.78	0.918368	0.459184	−	0.888341i	$-0.348142\pi$
$269$	4051.78	0.918368	0.459184	+	0.888341i	$0.348142\pi$
$270$	0	0
$271$	2791.30	0.625680	0.312840	−	0.949806i	$-0.398720\pi$
$271$	2791.30	0.625680	0.312840	+	0.949806i	$0.398720\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2526.06	0.553917
$276$	0	0
$277$	−2580.66	−0.559773	−0.279886	−	0.960033i	$-0.590297\pi$
$277$	−2580.66	−0.559773	−0.279886	+	0.960033i	$0.590297\pi$
$278$	0	0
$279$	4925.59	1.05694
$280$	0	0
$281$	−919.487	−0.195203	−0.0976014	−	0.995226i	$-0.531117\pi$
$281$	−919.487	−0.195203	−0.0976014	+	0.995226i	$0.531117\pi$
$282$	0	0
$283$	−8272.00	−1.73752	−0.868762	−	0.495229i	$-0.835084\pi$
$283$	−8272.00	−1.73752	−0.868762	+	0.495229i	$0.835084\pi$
$284$	0	0
$285$	245.781	0.0510835
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2822.88	0.574573
$290$	0	0
$291$	−4046.80	−0.815215
$292$	0	0
$293$	2859.38	0.570126	0.285063	−	0.958509i	$-0.407986\pi$
$293$	2859.38	0.570126	0.285063	+	0.958509i	$0.407986\pi$
$294$	0	0
$295$	−3015.71	−0.595191
$296$	0	0
$297$	4074.04	0.795958
$298$	0	0
$299$	−175.193	−0.0338852
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−874.770	−0.165855
$304$	0	0
$305$	1164.86	0.218687
$306$	0	0
$307$	−3542.86	−0.658638	−0.329319	−	0.944219i	$-0.606819\pi$
$307$	−3542.86	−0.658638	−0.329319	+	0.944219i	$0.606819\pi$
$308$	0	0
$309$	−1631.15	−0.300301
$310$	0	0
$311$	−622.549	−0.113510	−0.0567548	−	0.998388i	$-0.518075\pi$
$311$	−622.549	−0.113510	−0.0567548	+	0.998388i	$0.518075\pi$
$312$	0	0
$313$	7064.85	1.27581	0.637905	−	0.770115i	$-0.279801\pi$
$313$	7064.85	1.27581	0.637905	+	0.770115i	$0.279801\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6459.21	−1.14443	−0.572217	−	0.820103i	$-0.693916\pi$
$317$	−6459.21	−1.14443	−0.572217	+	0.820103i	$0.693916\pi$
$318$	0	0
$319$	161.195	0.0282921
$320$	0	0
$321$	3296.56	0.573196
$322$	0	0
$323$	1151.93	0.198437
$324$	0	0
$325$	1498.85	0.255820
$326$	0	0
$327$	945.245	0.159854
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8335.22	1.38412	0.692062	−	0.721838i	$-0.256702\pi$
$331$	8335.22	1.38412	0.692062	+	0.721838i	$0.256702\pi$
$332$	0	0
$333$	7275.44	1.19727
$334$	0	0
$335$	3327.29	0.542654
$336$	0	0
$337$	5853.18	0.946122	0.473061	−	0.881030i	$-0.343149\pi$
$337$	5853.18	0.946122	0.473061	+	0.881030i	$0.343149\pi$
$338$	0	0
$339$	−3276.03	−0.524866
$340$	0	0
$341$	8105.41	1.28719
$342$	0	0
$343$	0	0
$344$	0	0
$345$	175.967	0.0274601
$346$	0	0
$347$	−7175.81	−1.11014	−0.555069	−	0.831805i	$-0.687308\pi$
$347$	−7175.81	−1.11014	−0.555069	+	0.831805i	$0.687308\pi$
$348$	0	0
$349$	10302.4	1.58016	0.790081	−	0.613003i	$-0.210039\pi$
$349$	10302.4	1.58016	0.790081	+	0.613003i	$0.210039\pi$
$350$	0	0
$351$	2417.35	0.367604
$352$	0	0
$353$	−1101.12	−0.166025	−0.0830124	−	0.996549i	$-0.526454\pi$
$353$	−1101.12	−0.166025	−0.0830124	+	0.996549i	$0.526454\pi$
$354$	0	0
$355$	−2343.65	−0.350389
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5632.93	0.828119	0.414060	−	0.910250i	$-0.364111\pi$
$359$	5632.93	0.828119	0.414060	+	0.910250i	$0.364111\pi$
$360$	0	0
$361$	−6687.47	−0.974992
$362$	0	0
$363$	−952.106	−0.137666
$364$	0	0
$365$	7111.05	1.01975
$366$	0	0
$367$	3196.46	0.454643	0.227322	−	0.973820i	$-0.427003\pi$
$367$	3196.46	0.454643	0.227322	+	0.973820i	$0.427003\pi$
$368$	0	0
$369$	−4170.66	−0.588390
$370$	0	0
$371$	0	0
$372$	0	0
$373$	915.394	0.127071	0.0635353	−	0.997980i	$-0.479762\pi$
$373$	915.394	0.127071	0.0635353	+	0.997980i	$0.479762\pi$
$374$	0	0
$375$	−3851.25	−0.530340
$376$	0	0
$377$	95.6460	0.0130664
$378$	0	0
$379$	−11767.5	−1.59487	−0.797437	−	0.603402i	$-0.793812\pi$
$379$	−11767.5	−1.59487	−0.797437	+	0.603402i	$0.793812\pi$
$380$	0	0
$381$	−65.8298	−0.00885187
$382$	0	0
$383$	8415.12	1.12270	0.561348	−	0.827580i	$-0.310283\pi$
$383$	8415.12	1.12270	0.561348	+	0.827580i	$0.310283\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7214.86	0.947680
$388$	0	0
$389$	−3362.38	−0.438251	−0.219125	−	0.975697i	$-0.570320\pi$
$389$	−3362.38	−0.438251	−0.219125	+	0.975697i	$0.570320\pi$
$390$	0	0
$391$	824.726	0.106671
$392$	0	0
$393$	−1048.81	−0.134620
$394$	0	0
$395$	−3750.29	−0.477716
$396$	0	0
$397$	4863.44	0.614834	0.307417	−	0.951575i	$-0.400535\pi$
$397$	4863.44	0.614834	0.307417	+	0.951575i	$0.400535\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3738.85	0.465609	0.232805	−	0.972524i	$-0.425210\pi$
$401$	3738.85	0.465609	0.232805	+	0.972524i	$0.425210\pi$
$402$	0	0
$403$	4809.39	0.594474
$404$	0	0
$405$	1029.16	0.126270
$406$	0	0
$407$	11972.2	1.45809
$408$	0	0
$409$	14013.3	1.69416	0.847081	−	0.531464i	$-0.178358\pi$
$409$	14013.3	1.69416	0.847081	+	0.531464i	$0.178358\pi$
$410$	0	0
$411$	3191.85	0.383071
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7396.03	−0.874836
$416$	0	0
$417$	−3447.95	−0.404908
$418$	0	0
$419$	−8043.46	−0.937826	−0.468913	−	0.883244i	$-0.655354\pi$
$419$	−8043.46	−0.937826	−0.468913	+	0.883244i	$0.655354\pi$
$420$	0	0
$421$	1832.27	0.212112	0.106056	−	0.994360i	$-0.466178\pi$
$421$	1832.27	0.212112	0.106056	+	0.994360i	$0.466178\pi$
$422$	0	0
$423$	6844.82	0.786777
$424$	0	0
$425$	−7055.89	−0.805319
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1649.89	0.185682
$430$	0	0
$431$	−2863.25	−0.319996	−0.159998	−	0.987117i	$-0.551149\pi$
$431$	−2863.25	−0.319996	−0.159998	+	0.987117i	$0.551149\pi$
$432$	0	0
$433$	6856.23	0.760946	0.380473	−	0.924792i	$-0.375761\pi$
$433$	6856.23	0.760946	0.380473	+	0.924792i	$0.375761\pi$
$434$	0	0
$435$	−96.0685	−0.0105888
$436$	0	0
$437$	122.808	0.0134433
$438$	0	0
$439$	−4074.42	−0.442964	−0.221482	−	0.975164i	$-0.571089\pi$
$439$	−4074.42	−0.442964	−0.221482	+	0.975164i	$0.571089\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2497.92	0.267901	0.133950	−	0.990988i	$-0.457234\pi$
$443$	2497.92	0.267901	0.133950	+	0.990988i	$0.457234\pi$
$444$	0	0
$445$	−8074.84	−0.860189
$446$	0	0
$447$	8308.89	0.879188
$448$	0	0
$449$	9529.68	1.00163	0.500817	−	0.865553i	$-0.333033\pi$
$449$	9529.68	1.00163	0.500817	+	0.865553i	$0.333033\pi$
$450$	0	0
$451$	−6863.11	−0.716566
$452$	0	0
$453$	7285.11	0.755595
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10551.9	1.08008	0.540041	−	0.841639i	$-0.318409\pi$
$457$	10551.9	1.08008	0.540041	+	0.841639i	$0.318409\pi$
$458$	0	0
$459$	−11379.8	−1.15722
$460$	0	0
$461$	−13103.5	−1.32384	−0.661922	−	0.749573i	$-0.730259\pi$
$461$	−13103.5	−1.32384	−0.661922	+	0.749573i	$0.730259\pi$
$462$	0	0
$463$	816.035	0.0819101	0.0409550	−	0.999161i	$-0.486960\pi$
$463$	816.035	0.0819101	0.0409550	+	0.999161i	$0.486960\pi$
$464$	0	0
$465$	−4830.64	−0.481754
$466$	0	0
$467$	−4588.33	−0.454652	−0.227326	−	0.973819i	$-0.572998\pi$
$467$	−4588.33	−0.454652	−0.227326	+	0.973819i	$0.572998\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3828.03	−0.374493
$472$	0	0
$473$	11872.6	1.15412
$474$	0	0
$475$	−1050.68	−0.101491
$476$	0	0
$477$	−14626.7	−1.40400
$478$	0	0
$479$	17381.5	1.65800	0.829001	−	0.559247i	$-0.188910\pi$
$479$	17381.5	1.65800	0.829001	+	0.559247i	$0.188910\pi$
$480$	0	0
$481$	7103.80	0.673400
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9655.93	−0.904027
$486$	0	0
$487$	16352.5	1.52156	0.760781	−	0.649008i	$-0.224816\pi$
$487$	16352.5	1.52156	0.760781	+	0.649008i	$0.224816\pi$
$488$	0	0
$489$	−9303.70	−0.860384
$490$	0	0
$491$	13094.7	1.20358	0.601789	−	0.798655i	$-0.294455\pi$
$491$	13094.7	1.20358	0.601789	+	0.798655i	$0.294455\pi$
$492$	0	0
$493$	−450.256	−0.0411329
$494$	0	0
$495$	4031.87	0.366099
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9761.05	0.875681	0.437840	−	0.899053i	$-0.355744\pi$
$499$	9761.05	0.875681	0.437840	+	0.899053i	$0.355744\pi$
$500$	0	0
$501$	10668.0	0.951319
$502$	0	0
$503$	14229.0	1.26131	0.630657	−	0.776061i	$-0.282785\pi$
$503$	14229.0	1.26131	0.630657	+	0.776061i	$0.282785\pi$
$504$	0	0
$505$	−2087.26	−0.183924
$506$	0	0
$507$	−5182.39	−0.453961
$508$	0	0
$509$	−20320.3	−1.76951	−0.884757	−	0.466053i	$-0.845676\pi$
$509$	−20320.3	−1.76951	−0.884757	+	0.466053i	$0.845676\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1694.53	−0.145839
$514$	0	0
$515$	−3892.04	−0.333017
$516$	0	0
$517$	11263.6	0.958170
$518$	0	0
$519$	−7073.65	−0.598263
$520$	0	0
$521$	−20185.4	−1.69738	−0.848692	−	0.528888i	$-0.822609\pi$
$521$	−20185.4	−1.69738	−0.848692	+	0.528888i	$0.822609\pi$
$522$	0	0
$523$	−2710.67	−0.226634	−0.113317	−	0.993559i	$-0.536148\pi$
$523$	−2710.67	−0.226634	−0.113317	+	0.993559i	$0.536148\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22640.3	−1.87140
$528$	0	0
$529$	−12079.1	−0.992774
$530$	0	0
$531$	8623.63	0.704771
$532$	0	0
$533$	−4072.27	−0.330937
$534$	0	0
$535$	7865.80	0.635642
$536$	0	0
$537$	524.533	0.0421514
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3656.67	0.290596	0.145298	−	0.989388i	$-0.453586\pi$
$541$	3656.67	0.290596	0.145298	+	0.989388i	$0.453586\pi$
$542$	0	0
$543$	1283.47	0.101434
$544$	0	0
$545$	2255.42	0.177269
$546$	0	0
$547$	−787.130	−0.0615269	−0.0307635	−	0.999527i	$-0.509794\pi$
$547$	−787.130	−0.0615269	−0.0307635	+	0.999527i	$0.509794\pi$
$548$	0	0
$549$	−3330.99	−0.258949
$550$	0	0
$551$	−67.0466	−0.00518381
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7135.19	−0.545715
$556$	0	0
$557$	−12457.4	−0.947640	−0.473820	−	0.880622i	$-0.657125\pi$
$557$	−12457.4	−0.947640	−0.473820	+	0.880622i	$0.657125\pi$
$558$	0	0
$559$	7044.66	0.533018
$560$	0	0
$561$	−7766.91	−0.584526
$562$	0	0
$563$	−15918.7	−1.19164	−0.595820	−	0.803118i	$-0.703173\pi$
$563$	−15918.7	−1.19164	−0.595820	+	0.803118i	$0.703173\pi$
$564$	0	0
$565$	−7816.83	−0.582047
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7816.50	−0.575896	−0.287948	−	0.957646i	$-0.592973\pi$
$569$	−7816.50	−0.575896	−0.287948	+	0.957646i	$0.592973\pi$
$570$	0	0
$571$	21837.8	1.60050	0.800248	−	0.599670i	$-0.204701\pi$
$571$	21837.8	1.60050	0.800248	+	0.599670i	$0.204701\pi$
$572$	0	0
$573$	−8109.32	−0.591225
$574$	0	0
$575$	−752.232	−0.0545569
$576$	0	0
$577$	−15184.6	−1.09557	−0.547785	−	0.836619i	$-0.684529\pi$
$577$	−15184.6	−1.09557	−0.547785	+	0.836619i	$0.684529\pi$
$578$	0	0
$579$	3219.20	0.231063
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24069.2	−1.70985
$584$	0	0
$585$	2392.33	0.169078
$586$	0	0
$587$	−21861.6	−1.53718	−0.768591	−	0.639740i	$-0.779042\pi$
$587$	−21861.6	−1.53718	−0.768591	+	0.639740i	$0.779042\pi$
$588$	0	0
$589$	−3371.32	−0.235845
$590$	0	0
$591$	4594.31	0.319771
$592$	0	0
$593$	273.502	0.0189399	0.00946997	−	0.999955i	$-0.496986\pi$
$593$	273.502	0.0189399	0.00946997	+	0.999955i	$0.496986\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14895.2	1.02114
$598$	0	0
$599$	2443.63	0.166684	0.0833421	−	0.996521i	$-0.473441\pi$
$599$	2443.63	0.166684	0.0833421	+	0.996521i	$0.473441\pi$
$600$	0	0
$601$	1107.14	0.0751431	0.0375716	−	0.999294i	$-0.488038\pi$
$601$	1107.14	0.0751431	0.0375716	+	0.999294i	$0.488038\pi$
$602$	0	0
$603$	−9514.62	−0.642562
$604$	0	0
$605$	−2271.79	−0.152663
$606$	0	0
$607$	−1418.86	−0.0948759	−0.0474379	−	0.998874i	$-0.515106\pi$
$607$	−1418.86	−0.0948759	−0.0474379	+	0.998874i	$0.515106\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6683.34	0.442519
$612$	0	0
$613$	9402.98	0.619548	0.309774	−	0.950810i	$-0.399747\pi$
$613$	9402.98	0.619548	0.309774	+	0.950810i	$0.399747\pi$
$614$	0	0
$615$	4090.26	0.268187
$616$	0	0
$617$	−11223.0	−0.732284	−0.366142	−	0.930559i	$-0.619322\pi$
$617$	−11223.0	−0.732284	−0.366142	+	0.930559i	$0.619322\pi$
$618$	0	0
$619$	15879.1	1.03108	0.515538	−	0.856867i	$-0.327592\pi$
$619$	15879.1	1.03108	0.515538	+	0.856867i	$0.327592\pi$
$620$	0	0
$621$	−1213.20	−0.0783964
$622$	0	0
$623$	0	0
$624$	0	0
$625$	838.493	0.0536636
$626$	0	0
$627$	−1156.55	−0.0736655
$628$	0	0
$629$	−33441.3	−2.11986
$630$	0	0
$631$	9079.04	0.572791	0.286395	−	0.958112i	$-0.407543\pi$
$631$	9079.04	0.572791	0.286395	+	0.958112i	$0.407543\pi$
$632$	0	0
$633$	4222.77	0.265150
$634$	0	0
$635$	−157.074	−0.00981622
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6701.84	0.414900
$640$	0	0
$641$	9935.15	0.612192	0.306096	−	0.952001i	$-0.400977\pi$
$641$	9935.15	0.612192	0.306096	+	0.952001i	$0.400977\pi$
$642$	0	0
$643$	−21282.8	−1.30531	−0.652653	−	0.757657i	$-0.726344\pi$
$643$	−21282.8	−1.30531	−0.652653	+	0.757657i	$0.726344\pi$
$644$	0	0
$645$	−7075.78	−0.431951
$646$	0	0
$647$	6876.14	0.417819	0.208910	−	0.977935i	$-0.433009\pi$
$647$	6876.14	0.417819	0.208910	+	0.977935i	$0.433009\pi$
$648$	0	0
$649$	14190.8	0.858300
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1402.24	−0.0840335	−0.0420168	−	0.999117i	$-0.513378\pi$
$653$	−1402.24	−0.0840335	−0.0420168	+	0.999117i	$0.513378\pi$
$654$	0	0
$655$	−2502.53	−0.149286
$656$	0	0
$657$	−20334.6	−1.20750
$658$	0	0
$659$	−19975.7	−1.18079	−0.590395	−	0.807114i	$-0.701028\pi$
$659$	−19975.7	−1.18079	−0.590395	+	0.807114i	$0.701028\pi$
$660$	0	0
$661$	−656.831	−0.0386502	−0.0193251	−	0.999813i	$-0.506152\pi$
$661$	−656.831	−0.0386502	−0.0193251	+	0.999813i	$0.506152\pi$
$662$	0	0
$663$	−4608.55	−0.269956
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−48.0020	−0.00278658
$668$	0	0
$669$	−7067.84	−0.408458
$670$	0	0
$671$	−5481.37	−0.315359
$672$	0	0
$673$	14668.1	0.840140	0.420070	−	0.907492i	$-0.362006\pi$
$673$	14668.1	0.840140	0.420070	+	0.907492i	$0.362006\pi$
$674$	0	0
$675$	10379.5	0.591861
$676$	0	0
$677$	−19210.1	−1.09055	−0.545276	−	0.838257i	$-0.683575\pi$
$677$	−19210.1	−1.09055	−0.545276	+	0.838257i	$0.683575\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4313.34	−0.242713
$682$	0	0
$683$	−18446.0	−1.03341	−0.516704	−	0.856164i	$-0.672841\pi$
$683$	−18446.0	−1.03341	−0.516704	+	0.856164i	$0.672841\pi$
$684$	0	0
$685$	7615.95	0.424804
$686$	0	0
$687$	−1603.25	−0.0890364
$688$	0	0
$689$	−14281.6	−0.789674
$690$	0	0
$691$	−24902.0	−1.37093	−0.685467	−	0.728103i	$-0.740402\pi$
$691$	−24902.0	−1.37093	−0.685467	+	0.728103i	$0.740402\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8227.02	−0.449020
$696$	0	0
$697$	19170.3	1.04179
$698$	0	0
$699$	−4498.90	−0.243439
$700$	0	0
$701$	22637.7	1.21970	0.609852	−	0.792515i	$-0.291229\pi$
$701$	22637.7	1.21970	0.609852	+	0.792515i	$0.291229\pi$
$702$	0	0
$703$	−4979.67	−0.267158
$704$	0	0
$705$	−6712.87	−0.358612
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.98987	0.000370254 0	0.000185127	−	1.00000i	$-0.499941\pi$
$709$	6.98987	0.000370254 0	0.000185127		1.00000i	$0.499941\pi$
$710$	0	0
$711$	10724.2	0.565668
$712$	0	0
$713$	−2413.70	−0.126779
$714$	0	0
$715$	3936.75	0.205911
$716$	0	0
$717$	13179.1	0.686449
$718$	0	0
$719$	−19019.6	−0.986525	−0.493262	−	0.869881i	$-0.664196\pi$
$719$	−19019.6	−0.986525	−0.493262	+	0.869881i	$0.664196\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2447.18	0.125880
$724$	0	0
$725$	410.678	0.0210375
$726$	0	0
$727$	−33880.6	−1.72842	−0.864210	−	0.503130i	$-0.832182\pi$
$727$	−33880.6	−1.72842	−0.864210	+	0.503130i	$0.832182\pi$
$728$	0	0
$729$	6853.96	0.348217
$730$	0	0
$731$	−33162.9	−1.67794
$732$	0	0
$733$	10445.4	0.526344	0.263172	−	0.964749i	$-0.415231\pi$
$733$	10445.4	0.526344	0.263172	+	0.964749i	$0.415231\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15657.0	−0.782540
$738$	0	0
$739$	25674.5	1.27801	0.639007	−	0.769201i	$-0.279345\pi$
$739$	25674.5	1.27801	0.639007	+	0.769201i	$0.279345\pi$
$740$	0	0
$741$	−686.248	−0.0340215
$742$	0	0
$743$	30646.2	1.51319	0.756596	−	0.653883i	$-0.226861\pi$
$743$	30646.2	1.51319	0.756596	+	0.653883i	$0.226861\pi$
$744$	0	0
$745$	19825.6	0.974970
$746$	0	0
$747$	21149.5	1.03590
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8711.90	0.423305	0.211652	−	0.977345i	$-0.432116\pi$
$751$	8711.90	0.423305	0.211652	+	0.977345i	$0.432116\pi$
$752$	0	0
$753$	8767.54	0.424312
$754$	0	0
$755$	17382.8	0.837912
$756$	0	0
$757$	−18830.7	−0.904114	−0.452057	−	0.891989i	$-0.649310\pi$
$757$	−18830.7	−0.904114	−0.452057	+	0.891989i	$0.649310\pi$
$758$	0	0
$759$	−828.035	−0.0395992
$760$	0	0
$761$	−21800.6	−1.03846	−0.519232	−	0.854633i	$-0.673782\pi$
$761$	−21800.6	−1.03846	−0.519232	+	0.854633i	$0.673782\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−11262.0	−0.532258
$766$	0	0
$767$	8420.19	0.396395
$768$	0	0
$769$	−4412.21	−0.206903	−0.103452	−	0.994634i	$-0.532989\pi$
$769$	−4412.21	−0.206903	−0.103452	+	0.994634i	$0.532989\pi$
$770$	0	0
$771$	−2535.26	−0.118424
$772$	0	0
$773$	30446.8	1.41668	0.708342	−	0.705870i	$-0.249444\pi$
$773$	30446.8	1.41668	0.708342	+	0.705870i	$0.249444\pi$
$774$	0	0
$775$	20650.2	0.957133
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2854.61	0.131292
$780$	0	0
$781$	11028.3	0.505282
$782$	0	0
$783$	662.343	0.0302302
$784$	0	0
$785$	−9133.92	−0.415291
$786$	0	0
$787$	−11212.0	−0.507834	−0.253917	−	0.967226i	$-0.581719\pi$
$787$	−11212.0	−0.507834	−0.253917	+	0.967226i	$0.581719\pi$
$788$	0	0
$789$	259.795	0.0117224
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3252.41	−0.145645
$794$	0	0
$795$	14344.7	0.639942
$796$	0	0
$797$	11809.3	0.524852	0.262426	−	0.964952i	$-0.415477\pi$
$797$	11809.3	0.524852	0.262426	+	0.964952i	$0.415477\pi$
$798$	0	0
$799$	−31462.0	−1.39305
$800$	0	0
$801$	23090.6	1.01856
$802$	0	0
$803$	−33461.9	−1.47054
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11363.0	0.495658
$808$	0	0
$809$	14327.7	0.622662	0.311331	−	0.950302i	$-0.399225\pi$
$809$	14327.7	0.622662	0.311331	+	0.950302i	$0.399225\pi$
$810$	0	0
$811$	−32413.9	−1.40346	−0.701730	−	0.712443i	$-0.747589\pi$
$811$	−32413.9	−1.40346	−0.701730	+	0.712443i	$0.747589\pi$
$812$	0	0
$813$	7828.05	0.337690
$814$	0	0
$815$	−22199.2	−0.954117
$816$	0	0
$817$	−4938.21	−0.211464
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2406.16	0.102285	0.0511423	−	0.998691i	$-0.483714\pi$
$821$	2406.16	0.102285	0.0511423	+	0.998691i	$0.483714\pi$
$822$	0	0
$823$	1960.36	0.0830303	0.0415152	−	0.999138i	$-0.486782\pi$
$823$	1960.36	0.0830303	0.0415152	+	0.999138i	$0.486782\pi$
$824$	0	0
$825$	7084.19	0.298958
$826$	0	0
$827$	−17517.2	−0.736557	−0.368278	−	0.929716i	$-0.620053\pi$
$827$	−17517.2	−0.736557	−0.368278	+	0.929716i	$0.620053\pi$
$828$	0	0
$829$	−33498.7	−1.40345	−0.701724	−	0.712449i	$-0.747586\pi$
$829$	−33498.7	−1.40345	−0.701724	+	0.712449i	$0.747586\pi$
$830$	0	0
$831$	−7237.33	−0.302118
$832$	0	0
$833$	0	0
$834$	0	0
$835$	25454.5	1.05496
$836$	0	0
$837$	33304.8	1.37537
$838$	0	0
$839$	−1609.45	−0.0662268	−0.0331134	−	0.999452i	$-0.510542\pi$
$839$	−1609.45	−0.0662268	−0.0331134	+	0.999452i	$0.510542\pi$
$840$	0	0
$841$	−24362.8	−0.998925
$842$	0	0
$843$	−2578.65	−0.105354
$844$	0	0
$845$	−12365.5	−0.503417
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−23198.4	−0.937770
$850$	0	0
$851$	−3565.20	−0.143612
$852$	0	0
$853$	20446.9	0.820739	0.410369	−	0.911919i	$-0.365400\pi$
$853$	20446.9	0.820739	0.410369	+	0.911919i	$0.365400\pi$
$854$	0	0
$855$	−1676.99	−0.0670784
$856$	0	0
$857$	5784.22	0.230554	0.115277	−	0.993333i	$-0.463224\pi$
$857$	5784.22	0.230554	0.115277	+	0.993333i	$0.463224\pi$
$858$	0	0
$859$	32361.3	1.28539	0.642697	−	0.766120i	$-0.277815\pi$
$859$	32361.3	1.28539	0.642697	+	0.766120i	$0.277815\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28831.3	1.13723	0.568614	−	0.822604i	$-0.307480\pi$
$863$	28831.3	1.13723	0.568614	+	0.822604i	$0.307480\pi$
$864$	0	0
$865$	−16878.2	−0.663440
$866$	0	0
$867$	7916.60	0.310106
$868$	0	0
$869$	17647.5	0.688895
$870$	0	0
$871$	−9290.16	−0.361406
$872$	0	0
$873$	27611.8	1.07047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13105.4	−0.504604	−0.252302	−	0.967649i	$-0.581188\pi$
$877$	−13105.4	−0.504604	−0.252302	+	0.967649i	$0.581188\pi$
$878$	0	0
$879$	8018.98	0.307706
$880$	0	0
$881$	3089.10	0.118132	0.0590661	−	0.998254i	$-0.481188\pi$
$881$	3089.10	0.118132	0.0590661	+	0.998254i	$0.481188\pi$
$882$	0	0
$883$	−1601.96	−0.0610536	−0.0305268	−	0.999534i	$-0.509718\pi$
$883$	−1601.96	−0.0610536	−0.0305268	+	0.999534i	$0.509718\pi$
$884$	0	0
$885$	−8457.39	−0.321234
$886$	0	0
$887$	−22570.4	−0.854387	−0.427194	−	0.904160i	$-0.640498\pi$
$887$	−22570.4	−0.854387	−0.427194	+	0.904160i	$0.640498\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4842.83	−0.182089
$892$	0	0
$893$	−4684.93	−0.175560
$894$	0	0
$895$	1251.57	0.0467435
$896$	0	0
$897$	−491.320	−0.0182884
$898$	0	0
$899$	1317.75	0.0488870
$900$	0	0
$901$	67231.0	2.48589
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3062.44	0.112485
$906$	0	0
$907$	−48344.3	−1.76984	−0.884920	−	0.465743i	$-0.845787\pi$
$907$	−48344.3	−1.76984	−0.884920	+	0.465743i	$0.845787\pi$
$908$	0	0
$909$	5968.66	0.217787
$910$	0	0
$911$	−14697.9	−0.534539	−0.267269	−	0.963622i	$-0.586121\pi$
$911$	−14697.9	−0.534539	−0.267269	+	0.963622i	$0.586121\pi$
$912$	0	0
$913$	34802.9	1.26157
$914$	0	0
$915$	3266.78	0.118029
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−15330.6	−0.550282	−0.275141	−	0.961404i	$-0.588724\pi$
$919$	−15330.6	−0.550282	−0.275141	+	0.961404i	$0.588724\pi$
$920$	0	0
$921$	−9935.77	−0.355477
$922$	0	0
$923$	6543.74	0.233358
$924$	0	0
$925$	30501.8	1.08421
$926$	0	0
$927$	11129.6	0.394329
$928$	0	0
$929$	−55910.3	−1.97455	−0.987275	−	0.159020i	$-0.949166\pi$
$929$	−55910.3	−1.97455	−0.987275	+	0.159020i	$0.949166\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1745.90	−0.0612629
$934$	0	0
$935$	−18532.4	−0.648206
$936$	0	0
$937$	−18627.7	−0.649457	−0.324728	−	0.945807i	$-0.605273\pi$
$937$	−18627.7	−0.649457	−0.324728	+	0.945807i	$0.605273\pi$
$938$	0	0
$939$	19813.0	0.688575
$940$	0	0
$941$	−40564.7	−1.40528	−0.702641	−	0.711545i	$-0.747996\pi$
$941$	−40564.7	−1.40528	−0.702641	+	0.711545i	$0.747996\pi$
$942$	0	0
$943$	2043.76	0.0705767
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26690.1	−0.915852	−0.457926	−	0.888990i	$-0.651408\pi$
$947$	−26690.1	−0.915852	−0.457926	+	0.888990i	$0.651408\pi$
$948$	0	0
$949$	−19854.9	−0.679153
$950$	0	0
$951$	−18114.5	−0.617669
$952$	0	0
$953$	−7182.06	−0.244123	−0.122062	−	0.992523i	$-0.538951\pi$
$953$	−7182.06	−0.244123	−0.122062	+	0.992523i	$0.538951\pi$
$954$	0	0
$955$	−19349.4	−0.655635
$956$	0	0
$957$	452.062	0.0152697
$958$	0	0
$959$	0	0
$960$	0	0
$961$	36469.7	1.22419
$962$	0	0
$963$	−22492.8	−0.752670
$964$	0	0
$965$	7681.22	0.256236
$966$	0	0
$967$	−18129.4	−0.602898	−0.301449	−	0.953482i	$-0.597470\pi$
$967$	−18129.4	−0.602898	−0.301449	+	0.953482i	$0.597470\pi$
$968$	0	0
$969$	3230.53	0.107100
$970$	0	0
$971$	−20976.0	−0.693257	−0.346629	−	0.938002i	$-0.612674\pi$
$971$	−20976.0	−0.693257	−0.346629	+	0.938002i	$0.612674\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4203.45	0.138070
$976$	0	0
$977$	−44694.7	−1.46357	−0.731786	−	0.681534i	$-0.761313\pi$
$977$	−44694.7	−1.46357	−0.731786	+	0.681534i	$0.761313\pi$
$978$	0	0
$979$	37997.1	1.24044
$980$	0	0
$981$	−6449.52	−0.209906
$982$	0	0
$983$	15426.2	0.500529	0.250265	−	0.968177i	$-0.419482\pi$
$983$	15426.2	0.500529	0.250265	+	0.968177i	$0.419482\pi$
$984$	0	0
$985$	10962.3	0.354608
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3535.51	−0.113673
$990$	0	0
$991$	−28248.1	−0.905479	−0.452739	−	0.891643i	$-0.649553\pi$
$991$	−28248.1	−0.905479	−0.452739	+	0.891643i	$0.649553\pi$
$992$	0	0
$993$	23375.7	0.747034
$994$	0	0
$995$	35540.9	1.13238
$996$	0	0
$997$	37121.5	1.17919	0.589594	−	0.807700i	$-0.299288\pi$
$997$	37121.5	1.17919	0.589594	+	0.807700i	$0.299288\pi$
$998$	0	0
$999$	49193.4	1.55797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bb.1.3		3
4.3	odd	2		392.4.a.l.1.1		3
7.3	odd	6		112.4.i.e.65.3		6
7.5	odd	6		112.4.i.e.81.3		6
7.6	odd	2		784.4.a.be.1.1		3
28.3	even	6		56.4.i.b.9.1	✓	6
28.11	odd	6		392.4.i.m.177.3		6
28.19	even	6		56.4.i.b.25.1	yes	6
28.23	odd	6		392.4.i.m.361.3		6
28.27	even	2		392.4.a.i.1.3		3
56.3	even	6		448.4.i.j.65.3		6
56.5	odd	6		448.4.i.m.193.1		6
56.19	even	6		448.4.i.j.193.3		6
56.45	odd	6		448.4.i.m.65.1		6
84.47	odd	6		504.4.s.h.361.2		6
84.59	odd	6		504.4.s.h.289.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.i.b.9.1	✓	6	28.3	even	6
56.4.i.b.25.1	yes	6	28.19	even	6
112.4.i.e.65.3		6	7.3	odd	6
112.4.i.e.81.3		6	7.5	odd	6
392.4.a.i.1.3		3	28.27	even	2
392.4.a.l.1.1		3	4.3	odd	2
392.4.i.m.177.3		6	28.11	odd	6
392.4.i.m.361.3		6	28.23	odd	6
448.4.i.j.65.3		6	56.3	even	6
448.4.i.j.193.3		6	56.19	even	6
448.4.i.m.65.1		6	56.45	odd	6
448.4.i.m.193.1		6	56.5	odd	6
504.4.s.h.289.2		6	84.59	odd	6
504.4.s.h.361.2		6	84.47	odd	6
784.4.a.bb.1.3		3	1.1	even	1	trivial
784.4.a.be.1.1		3	7.6	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80445 q^{3} +6.69159 q^{5} -19.1351 q^{9} +O(q^{10})\)	\(q+2.80445 q^{3} +6.69159 q^{5} -19.1351 q^{9} -31.4881 q^{11} -18.6837 q^{13} +18.7662 q^{15} +87.9538 q^{17} +13.0970 q^{19} +9.37681 q^{23} -80.2226 q^{25} -129.383 q^{27} -5.11923 q^{29} -257.412 q^{31} -88.3067 q^{33} -380.215 q^{37} -52.3973 q^{39} +217.959 q^{41} -377.049 q^{43} -128.044 q^{45} -357.710 q^{47} +246.662 q^{51} +764.390 q^{53} -210.706 q^{55} +36.7298 q^{57} -450.671 q^{59} +174.078 q^{61} -125.023 q^{65} +497.234 q^{67} +26.2967 q^{69} -350.238 q^{71} +1062.69 q^{73} -224.980 q^{75} -560.449 q^{79} +153.799 q^{81} -1105.27 q^{83} +588.551 q^{85} -14.3566 q^{87} -1206.71 q^{89} -721.897 q^{93} +87.6398 q^{95} -1442.99 q^{97} +602.528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) \) 3 * q - 7 * q^3 + 3 * q^5 + 18 * q^9	\( 3 q - 7 q^{3} + 3 q^{5} + 18 q^{9} + 3 q^{11} + 26 q^{13} - 127 q^{15} + 31 q^{17} - 89 q^{19} + 201 q^{23} + 300 q^{25} - 469 q^{27} + 190 q^{29} - 339 q^{31} + 105 q^{33} - 535 q^{37} + 134 q^{39} - 58 q^{41} - 268 q^{43} + 1410 q^{45} - 205 q^{47} + 965 q^{51} + 757 q^{53} - 1653 q^{55} + 261 q^{57} - 1799 q^{59} - 625 q^{61} - 1750 q^{65} + 495 q^{67} - 973 q^{69} - 640 q^{71} + 443 q^{73} - 1484 q^{75} - 79 q^{79} + 2523 q^{81} - 2372 q^{83} - 977 q^{85} - 910 q^{87} - 821 q^{89} - 1321 q^{93} + 1327 q^{95} + 342 q^{97} - 2310 q^{99}+O(q^{100}) \) 3 * q - 7 * q^3 + 3 * q^5 + 18 * q^9 + 3 * q^11 + 26 * q^13 - 127 * q^15 + 31 * q^17 - 89 * q^19 + 201 * q^23 + 300 * q^25 - 469 * q^27 + 190 * q^29 - 339 * q^31 + 105 * q^33 - 535 * q^37 + 134 * q^39 - 58 * q^41 - 268 * q^43 + 1410 * q^45 - 205 * q^47 + 965 * q^51 + 757 * q^53 - 1653 * q^55 + 261 * q^57 - 1799 * q^59 - 625 * q^61 - 1750 * q^65 + 495 * q^67 - 973 * q^69 - 640 * q^71 + 443 * q^73 - 1484 * q^75 - 79 * q^79 + 2523 * q^81 - 2372 * q^83 - 977 * q^85 - 910 * q^87 - 821 * q^89 - 1321 * q^93 + 1327 * q^95 + 342 * q^97 - 2310 * q^99

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