LMFDB - Embedded newform 1148.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, 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("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 $ x^15 - 3x^14 - 238x^13 + 602x^12 + 21013x^11 - 44923x^10 - 876344x^9 + 1512470x^8 + 18440602x^7 - 23548938x^6 - 192795335x^5 + 157072885x^4 + 930883767x^3 - 339053445x^2 - 1549251522x - 45134496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.89624$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.89624 q^{3} -3.91686 q^{5} -7.00000 q^{7} -3.02684 q^{9} +O(q^{10})$	$q-4.89624 q^{3} -3.91686 q^{5} -7.00000 q^{7} -3.02684 q^{9} -10.6341 q^{11} -31.5104 q^{13} +19.1779 q^{15} +48.2579 q^{17} -38.9335 q^{19} +34.2737 q^{21} -95.1046 q^{23} -109.658 q^{25} +147.019 q^{27} -245.378 q^{29} -128.634 q^{31} +52.0672 q^{33} +27.4180 q^{35} +47.0220 q^{37} +154.283 q^{39} +41.0000 q^{41} -272.222 q^{43} +11.8557 q^{45} -143.632 q^{47} +49.0000 q^{49} -236.282 q^{51} -332.517 q^{53} +41.6523 q^{55} +190.628 q^{57} +170.245 q^{59} +397.351 q^{61} +21.1879 q^{63} +123.422 q^{65} +579.005 q^{67} +465.655 q^{69} +1152.60 q^{71} -999.619 q^{73} +536.913 q^{75} +74.4389 q^{77} -987.932 q^{79} -638.113 q^{81} -1459.16 q^{83} -189.019 q^{85} +1201.43 q^{87} -862.309 q^{89} +220.573 q^{91} +629.821 q^{93} +152.497 q^{95} -120.092 q^{97} +32.1878 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * 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
$3$	−4.89624	−0.942282	−0.471141	−	0.882058i	$-0.656158\pi$
$3$	−4.89624	−0.942282	−0.471141	+	0.882058i	$0.656158\pi$
$4$	0	0
$5$	−3.91686	−0.350334	−0.175167	−	0.984539i	$-0.556047\pi$
$5$	−3.91686	−0.350334	−0.175167	+	0.984539i	$0.556047\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−3.02684	−0.112105
$10$	0	0
$11$	−10.6341	−0.291483	−0.145741	−	0.989323i	$-0.546557\pi$
$11$	−10.6341	−0.291483	−0.145741	+	0.989323i	$0.546557\pi$
$12$	0	0
$13$	−31.5104	−0.672263	−0.336131	−	0.941815i	$-0.609119\pi$
$13$	−31.5104	−0.672263	−0.336131	+	0.941815i	$0.609119\pi$
$14$	0	0
$15$	19.1779	0.330114
$16$	0	0
$17$	48.2579	0.688485	0.344243	−	0.938881i	$-0.388136\pi$
$17$	48.2579	0.688485	0.344243	+	0.938881i	$0.388136\pi$
$18$	0	0
$19$	−38.9335	−0.470103	−0.235051	−	0.971983i	$-0.575526\pi$
$19$	−38.9335	−0.470103	−0.235051	+	0.971983i	$0.575526\pi$
$20$	0	0
$21$	34.2737	0.356149
$22$	0	0
$23$	−95.1046	−0.862204	−0.431102	−	0.902303i	$-0.641875\pi$
$23$	−95.1046	−0.862204	−0.431102	+	0.902303i	$0.641875\pi$
$24$	0	0
$25$	−109.658	−0.877266
$26$	0	0
$27$	147.019	1.04792
$28$	0	0
$29$	−245.378	−1.57123	−0.785614	−	0.618718i	$-0.787653\pi$
$29$	−245.378	−1.57123	−0.785614	+	0.618718i	$0.787653\pi$
$30$	0	0
$31$	−128.634	−0.745267	−0.372633	−	0.927979i	$-0.621545\pi$
$31$	−128.634	−0.745267	−0.372633	+	0.927979i	$0.621545\pi$
$32$	0	0
$33$	52.0672	0.274659
$34$	0	0
$35$	27.4180	0.132414
$36$	0	0
$37$	47.0220	0.208929	0.104464	−	0.994529i	$-0.466687\pi$
$37$	47.0220	0.208929	0.104464	+	0.994529i	$0.466687\pi$
$38$	0	0
$39$	154.283	0.633461
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	−272.222	−0.965429	−0.482715	−	0.875778i	$-0.660349\pi$
$43$	−272.222	−0.965429	−0.482715	+	0.875778i	$0.660349\pi$
$44$	0	0
$45$	11.8557	0.0392743
$46$	0	0
$47$	−143.632	−0.445763	−0.222882	−	0.974846i	$-0.571546\pi$
$47$	−143.632	−0.445763	−0.222882	+	0.974846i	$0.571546\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−236.282	−0.648747
$52$	0	0
$53$	−332.517	−0.861786	−0.430893	−	0.902403i	$-0.641801\pi$
$53$	−332.517	−0.861786	−0.430893	+	0.902403i	$0.641801\pi$
$54$	0	0
$55$	41.6523	0.102116
$56$	0	0
$57$	190.628	0.442969
$58$	0	0
$59$	170.245	0.375662	0.187831	−	0.982201i	$-0.439854\pi$
$59$	170.245	0.375662	0.187831	+	0.982201i	$0.439854\pi$
$60$	0	0
$61$	397.351	0.834026	0.417013	−	0.908901i	$-0.363077\pi$
$61$	397.351	0.834026	0.417013	+	0.908901i	$0.363077\pi$
$62$	0	0
$63$	21.1879	0.0423718
$64$	0	0
$65$	123.422	0.235517
$66$	0	0
$67$	579.005	1.05577	0.527886	−	0.849315i	$-0.322985\pi$
$67$	579.005	1.05577	0.527886	+	0.849315i	$0.322985\pi$
$68$	0	0
$69$	465.655	0.812439
$70$	0	0
$71$	1152.60	1.92660	0.963302	−	0.268420i	$-0.0865015\pi$
$71$	1152.60	1.92660	0.963302	+	0.268420i	$0.0865015\pi$
$72$	0	0
$73$	−999.619	−1.60269	−0.801346	−	0.598201i	$-0.795883\pi$
$73$	−999.619	−1.60269	−0.801346	+	0.598201i	$0.795883\pi$
$74$	0	0
$75$	536.913	0.826632
$76$	0	0
$77$	74.4389	0.110170
$78$	0	0
$79$	−987.932	−1.40698	−0.703488	−	0.710707i	$-0.748375\pi$
$79$	−987.932	−1.40698	−0.703488	+	0.710707i	$0.748375\pi$
$80$	0	0
$81$	−638.113	−0.875327
$82$	0	0
$83$	−1459.16	−1.92968	−0.964839	−	0.262842i	$-0.915340\pi$
$83$	−1459.16	−1.92968	−0.964839	+	0.262842i	$0.915340\pi$
$84$	0	0
$85$	−189.019	−0.241200
$86$	0	0
$87$	1201.43	1.48054
$88$	0	0
$89$	−862.309	−1.02702	−0.513509	−	0.858084i	$-0.671655\pi$
$89$	−862.309	−1.02702	−0.513509	+	0.858084i	$0.671655\pi$
$90$	0	0
$91$	220.573	0.254091
$92$	0	0
$93$	629.821	0.702251
$94$	0	0
$95$	152.497	0.164693
$96$	0	0
$97$	−120.092	−0.125706	−0.0628528	−	0.998023i	$-0.520020\pi$
$97$	−120.092	−0.125706	−0.0628528	+	0.998023i	$0.520020\pi$
$98$	0	0
$99$	32.1878	0.0326767
$100$	0	0
$101$	492.916	0.485613	0.242807	−	0.970075i	$-0.421932\pi$
$101$	492.916	0.485613	0.242807	+	0.970075i	$0.421932\pi$
$102$	0	0
$103$	37.1487	0.0355376	0.0177688	−	0.999842i	$-0.494344\pi$
$103$	37.1487	0.0355376	0.0177688	+	0.999842i	$0.494344\pi$
$104$	0	0
$105$	−134.245	−0.124771
$106$	0	0
$107$	1710.45	1.54538	0.772689	−	0.634784i	$-0.218911\pi$
$107$	1710.45	1.54538	0.772689	+	0.634784i	$0.218911\pi$
$108$	0	0
$109$	715.121	0.628405	0.314203	−	0.949356i	$-0.398263\pi$
$109$	715.121	0.628405	0.314203	+	0.949356i	$0.398263\pi$
$110$	0	0
$111$	−230.231	−0.196870
$112$	0	0
$113$	1730.04	1.44025	0.720126	−	0.693844i	$-0.244084\pi$
$113$	1730.04	1.44025	0.720126	+	0.693844i	$0.244084\pi$
$114$	0	0
$115$	372.511	0.302059
$116$	0	0
$117$	95.3771	0.0753642
$118$	0	0
$119$	−337.805	−0.260223
$120$	0	0
$121$	−1217.92	−0.915038
$122$	0	0
$123$	−200.746	−0.147160
$124$	0	0
$125$	919.122	0.657671
$126$	0	0
$127$	1893.36	1.32291	0.661453	−	0.749987i	$-0.269940\pi$
$127$	1893.36	1.32291	0.661453	+	0.749987i	$0.269940\pi$
$128$	0	0
$129$	1332.86	0.909706
$130$	0	0
$131$	−865.294	−0.577108	−0.288554	−	0.957464i	$-0.593174\pi$
$131$	−865.294	−0.577108	−0.288554	+	0.957464i	$0.593174\pi$
$132$	0	0
$133$	272.534	0.177682
$134$	0	0
$135$	−575.851	−0.367121
$136$	0	0
$137$	766.808	0.478196	0.239098	−	0.970995i	$-0.423148\pi$
$137$	766.808	0.478196	0.239098	+	0.970995i	$0.423148\pi$
$138$	0	0
$139$	255.727	0.156047	0.0780234	−	0.996952i	$-0.475139\pi$
$139$	255.727	0.156047	0.0780234	+	0.996952i	$0.475139\pi$
$140$	0	0
$141$	703.256	0.420034
$142$	0	0
$143$	335.086	0.195953
$144$	0	0
$145$	961.111	0.550455
$146$	0	0
$147$	−239.916	−0.134612
$148$	0	0
$149$	1835.53	1.00921	0.504605	−	0.863351i	$-0.331638\pi$
$149$	1835.53	1.00921	0.504605	+	0.863351i	$0.331638\pi$
$150$	0	0
$151$	2651.94	1.42922	0.714608	−	0.699525i	$-0.246605\pi$
$151$	2651.94	1.42922	0.714608	+	0.699525i	$0.246605\pi$
$152$	0	0
$153$	−146.069	−0.0771828
$154$	0	0
$155$	503.839	0.261092
$156$	0	0
$157$	−2764.55	−1.40532	−0.702660	−	0.711526i	$-0.748005\pi$
$157$	−2764.55	−1.40532	−0.702660	+	0.711526i	$0.748005\pi$
$158$	0	0
$159$	1628.08	0.812045
$160$	0	0
$161$	665.732	0.325882
$162$	0	0
$163$	2739.80	1.31655	0.658276	−	0.752777i	$-0.271286\pi$
$163$	2739.80	1.31655	0.658276	+	0.752777i	$0.271286\pi$
$164$	0	0
$165$	−203.940	−0.0962223
$166$	0	0
$167$	−1145.43	−0.530755	−0.265377	−	0.964145i	$-0.585497\pi$
$167$	−1145.43	−0.530755	−0.265377	+	0.964145i	$0.585497\pi$
$168$	0	0
$169$	−1204.09	−0.548063
$170$	0	0
$171$	117.846	0.0527010
$172$	0	0
$173$	1442.46	0.633919	0.316960	−	0.948439i	$-0.397338\pi$
$173$	1442.46	0.633919	0.316960	+	0.948439i	$0.397338\pi$
$174$	0	0
$175$	767.608	0.331575
$176$	0	0
$177$	−833.562	−0.353979
$178$	0	0
$179$	961.334	0.401416	0.200708	−	0.979651i	$-0.435676\pi$
$179$	961.334	0.401416	0.200708	+	0.979651i	$0.435676\pi$
$180$	0	0
$181$	3035.47	1.24655	0.623273	−	0.782005i	$-0.285803\pi$
$181$	3035.47	1.24655	0.623273	+	0.782005i	$0.285803\pi$
$182$	0	0
$183$	−1945.52	−0.785887
$184$	0	0
$185$	−184.178	−0.0731950
$186$	0	0
$187$	−513.180	−0.200681
$188$	0	0
$189$	−1029.13	−0.396075
$190$	0	0
$191$	4071.55	1.54244	0.771222	−	0.636567i	$-0.219646\pi$
$191$	4071.55	1.54244	0.771222	+	0.636567i	$0.219646\pi$
$192$	0	0
$193$	2648.06	0.987624	0.493812	−	0.869569i	$-0.335603\pi$
$193$	2648.06	0.987624	0.493812	+	0.869569i	$0.335603\pi$
$194$	0	0
$195$	−604.302	−0.221923
$196$	0	0
$197$	−754.345	−0.272816	−0.136408	−	0.990653i	$-0.543556\pi$
$197$	−754.345	−0.272816	−0.136408	+	0.990653i	$0.543556\pi$
$198$	0	0
$199$	3789.95	1.35006	0.675031	−	0.737789i	$-0.264130\pi$
$199$	3789.95	1.35006	0.675031	+	0.737789i	$0.264130\pi$
$200$	0	0
$201$	−2834.95	−0.994835
$202$	0	0
$203$	1717.65	0.593868
$204$	0	0
$205$	−160.591	−0.0547130
$206$	0	0
$207$	287.867	0.0966576
$208$	0	0
$209$	414.023	0.137027
$210$	0	0
$211$	1094.92	0.357238	0.178619	−	0.983918i	$-0.442837\pi$
$211$	1094.92	0.357238	0.178619	+	0.983918i	$0.442837\pi$
$212$	0	0
$213$	−5643.42	−1.81540
$214$	0	0
$215$	1066.25	0.338223
$216$	0	0
$217$	900.435	0.281684
$218$	0	0
$219$	4894.37	1.51019
$220$	0	0
$221$	−1520.63	−0.462843
$222$	0	0
$223$	−2564.47	−0.770089	−0.385044	−	0.922898i	$-0.625814\pi$
$223$	−2564.47	−0.770089	−0.385044	+	0.922898i	$0.625814\pi$
$224$	0	0
$225$	331.918	0.0983461
$226$	0	0
$227$	5108.07	1.49354	0.746772	−	0.665080i	$-0.231603\pi$
$227$	5108.07	1.49354	0.746772	+	0.665080i	$0.231603\pi$
$228$	0	0
$229$	−6198.49	−1.78868	−0.894340	−	0.447387i	$-0.852355\pi$
$229$	−6198.49	−1.78868	−0.894340	+	0.447387i	$0.852355\pi$
$230$	0	0
$231$	−364.470	−0.103811
$232$	0	0
$233$	−5504.07	−1.54757	−0.773785	−	0.633449i	$-0.781639\pi$
$233$	−5504.07	−1.54757	−0.773785	+	0.633449i	$0.781639\pi$
$234$	0	0
$235$	562.585	0.156166
$236$	0	0
$237$	4837.15	1.32577
$238$	0	0
$239$	−4509.19	−1.22040	−0.610199	−	0.792248i	$-0.708911\pi$
$239$	−4509.19	−1.22040	−0.610199	+	0.792248i	$0.708911\pi$
$240$	0	0
$241$	7088.03	1.89452	0.947262	−	0.320461i	$-0.103838\pi$
$241$	7088.03	1.89452	0.947262	+	0.320461i	$0.103838\pi$
$242$	0	0
$243$	−845.146	−0.223112
$244$	0	0
$245$	−191.926	−0.0500477
$246$	0	0
$247$	1226.81	0.316033
$248$	0	0
$249$	7144.38	1.81830
$250$	0	0
$251$	2336.05	0.587452	0.293726	−	0.955890i	$-0.405105\pi$
$251$	2336.05	0.587452	0.293726	+	0.955890i	$0.405105\pi$
$252$	0	0
$253$	1011.35	0.251317
$254$	0	0
$255$	925.483	0.227278
$256$	0	0
$257$	−6143.60	−1.49116	−0.745578	−	0.666418i	$-0.767827\pi$
$257$	−6143.60	−1.49116	−0.745578	+	0.666418i	$0.767827\pi$
$258$	0	0
$259$	−329.154	−0.0789677
$260$	0	0
$261$	742.721	0.176143
$262$	0	0
$263$	2092.35	0.490570	0.245285	−	0.969451i	$-0.421118\pi$
$263$	2092.35	0.490570	0.245285	+	0.969451i	$0.421118\pi$
$264$	0	0
$265$	1302.42	0.301913
$266$	0	0
$267$	4222.07	0.967740
$268$	0	0
$269$	5509.36	1.24874	0.624371	−	0.781128i	$-0.285355\pi$
$269$	5509.36	1.24874	0.624371	+	0.781128i	$0.285355\pi$
$270$	0	0
$271$	493.746	0.110675	0.0553375	−	0.998468i	$-0.482377\pi$
$271$	493.746	0.110675	0.0553375	+	0.998468i	$0.482377\pi$
$272$	0	0
$273$	−1079.98	−0.239426
$274$	0	0
$275$	1166.12	0.255708
$276$	0	0
$277$	827.013	0.179388	0.0896938	−	0.995969i	$-0.471411\pi$
$277$	827.013	0.179388	0.0896938	+	0.995969i	$0.471411\pi$
$278$	0	0
$279$	389.353	0.0835483
$280$	0	0
$281$	−6775.86	−1.43848	−0.719242	−	0.694760i	$-0.755511\pi$
$281$	−6775.86	−1.43848	−0.719242	+	0.694760i	$0.755511\pi$
$282$	0	0
$283$	−5716.73	−1.20079	−0.600397	−	0.799702i	$-0.704991\pi$
$283$	−5716.73	−1.20079	−0.600397	+	0.799702i	$0.704991\pi$
$284$	0	0
$285$	−746.661	−0.155187
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	−2584.18	−0.525988
$290$	0	0
$291$	587.997	0.118450
$292$	0	0
$293$	−699.262	−0.139424	−0.0697121	−	0.997567i	$-0.522208\pi$
$293$	−699.262	−0.139424	−0.0697121	+	0.997567i	$0.522208\pi$
$294$	0	0
$295$	−666.827	−0.131607
$296$	0	0
$297$	−1563.41	−0.305449
$298$	0	0
$299$	2996.79	0.579627
$300$	0	0
$301$	1905.55	0.364898
$302$	0	0
$303$	−2413.43	−0.457585
$304$	0	0
$305$	−1556.37	−0.292188
$306$	0	0
$307$	4666.59	0.867545	0.433772	−	0.901022i	$-0.357182\pi$
$307$	4666.59	0.867545	0.433772	+	0.901022i	$0.357182\pi$
$308$	0	0
$309$	−181.889	−0.0334864
$310$	0	0
$311$	433.202	0.0789860	0.0394930	−	0.999220i	$-0.487426\pi$
$311$	433.202	0.0789860	0.0394930	+	0.999220i	$0.487426\pi$
$312$	0	0
$313$	−2973.73	−0.537014	−0.268507	−	0.963278i	$-0.586530\pi$
$313$	−2973.73	−0.537014	−0.268507	+	0.963278i	$0.586530\pi$
$314$	0	0
$315$	−82.9899	−0.0148443
$316$	0	0
$317$	122.947	0.0217837	0.0108918	−	0.999941i	$-0.496533\pi$
$317$	122.947	0.0217837	0.0108918	+	0.999941i	$0.496533\pi$
$318$	0	0
$319$	2609.38	0.457985
$320$	0	0
$321$	−8374.78	−1.45618
$322$	0	0
$323$	−1878.85	−0.323659
$324$	0	0
$325$	3455.38	0.589753
$326$	0	0
$327$	−3501.40	−0.592135
$328$	0	0
$329$	1005.42	0.168483
$330$	0	0
$331$	−5607.19	−0.931115	−0.465558	−	0.885018i	$-0.654146\pi$
$331$	−5607.19	−0.931115	−0.465558	+	0.885018i	$0.654146\pi$
$332$	0	0
$333$	−142.328	−0.0234220
$334$	0	0
$335$	−2267.88	−0.369873
$336$	0	0
$337$	5919.34	0.956816	0.478408	−	0.878138i	$-0.341214\pi$
$337$	5919.34	0.956816	0.478408	+	0.878138i	$0.341214\pi$
$338$	0	0
$339$	−8470.69	−1.35712
$340$	0	0
$341$	1367.90	0.217232
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1823.90	−0.284625
$346$	0	0
$347$	736.243	0.113901	0.0569504	−	0.998377i	$-0.481862\pi$
$347$	736.243	0.113901	0.0569504	+	0.998377i	$0.481862\pi$
$348$	0	0
$349$	−8169.26	−1.25298	−0.626491	−	0.779429i	$-0.715509\pi$
$349$	−8169.26	−1.25298	−0.626491	+	0.779429i	$0.715509\pi$
$350$	0	0
$351$	−4632.62	−0.704475
$352$	0	0
$353$	−7523.27	−1.13434	−0.567171	−	0.823600i	$-0.691962\pi$
$353$	−7523.27	−1.13434	−0.567171	+	0.823600i	$0.691962\pi$
$354$	0	0
$355$	−4514.58	−0.674955
$356$	0	0
$357$	1653.97	0.245203
$358$	0	0
$359$	−12964.1	−1.90590	−0.952950	−	0.303129i	$-0.901969\pi$
$359$	−12964.1	−1.90590	−0.952950	+	0.303129i	$0.901969\pi$
$360$	0	0
$361$	−5343.18	−0.779003
$362$	0	0
$363$	5963.21	0.862223
$364$	0	0
$365$	3915.36	0.561478
$366$	0	0
$367$	7259.57	1.03255	0.516276	−	0.856422i	$-0.327318\pi$
$367$	7259.57	1.03255	0.516276	+	0.856422i	$0.327318\pi$
$368$	0	0
$369$	−124.101	−0.0175079
$370$	0	0
$371$	2327.62	0.325725
$372$	0	0
$373$	10980.4	1.52424	0.762122	−	0.647433i	$-0.224158\pi$
$373$	10980.4	1.52424	0.762122	+	0.647433i	$0.224158\pi$
$374$	0	0
$375$	−4500.24	−0.619711
$376$	0	0
$377$	7731.97	1.05628
$378$	0	0
$379$	14476.5	1.96202	0.981010	−	0.193958i	$-0.0621325\pi$
$379$	14476.5	1.96202	0.981010	+	0.193958i	$0.0621325\pi$
$380$	0	0
$381$	−9270.37	−1.24655
$382$	0	0
$383$	1881.97	0.251081	0.125541	−	0.992088i	$-0.459933\pi$
$383$	1881.97	0.251081	0.125541	+	0.992088i	$0.459933\pi$
$384$	0	0
$385$	−291.566	−0.0385963
$386$	0	0
$387$	823.973	0.108230
$388$	0	0
$389$	−6584.15	−0.858174	−0.429087	−	0.903263i	$-0.641165\pi$
$389$	−6584.15	−0.858174	−0.429087	+	0.903263i	$0.641165\pi$
$390$	0	0
$391$	−4589.55	−0.593615
$392$	0	0
$393$	4236.69	0.543798
$394$	0	0
$395$	3869.59	0.492912
$396$	0	0
$397$	−14960.2	−1.89126	−0.945631	−	0.325241i	$-0.894554\pi$
$397$	−14960.2	−1.89126	−0.945631	+	0.325241i	$0.894554\pi$
$398$	0	0
$399$	−1334.39	−0.167427
$400$	0	0
$401$	−11147.5	−1.38823	−0.694115	−	0.719864i	$-0.744204\pi$
$401$	−11147.5	−1.38823	−0.694115	+	0.719864i	$0.744204\pi$
$402$	0	0
$403$	4053.30	0.501015
$404$	0	0
$405$	2499.40	0.306657
$406$	0	0
$407$	−500.038	−0.0608992
$408$	0	0
$409$	6308.65	0.762696	0.381348	−	0.924432i	$-0.375460\pi$
$409$	6308.65	0.762696	0.381348	+	0.924432i	$0.375460\pi$
$410$	0	0
$411$	−3754.48	−0.450595
$412$	0	0
$413$	−1191.72	−0.141987
$414$	0	0
$415$	5715.31	0.676032
$416$	0	0
$417$	−1252.10	−0.147040
$418$	0	0
$419$	−14643.8	−1.70739	−0.853696	−	0.520771i	$-0.825645\pi$
$419$	−14643.8	−1.70739	−0.853696	+	0.520771i	$0.825645\pi$
$420$	0	0
$421$	2926.53	0.338789	0.169394	−	0.985548i	$-0.445819\pi$
$421$	2926.53	0.338789	0.169394	+	0.985548i	$0.445819\pi$
$422$	0	0
$423$	434.751	0.0499724
$424$	0	0
$425$	−5291.87	−0.603985
$426$	0	0
$427$	−2781.46	−0.315232
$428$	0	0
$429$	−1640.66	−0.184643
$430$	0	0
$431$	10130.7	1.13220	0.566102	−	0.824335i	$-0.308451\pi$
$431$	10130.7	1.13220	0.566102	+	0.824335i	$0.308451\pi$
$432$	0	0
$433$	−6668.69	−0.740132	−0.370066	−	0.929006i	$-0.620665\pi$
$433$	−6668.69	−0.740132	−0.370066	+	0.929006i	$0.620665\pi$
$434$	0	0
$435$	−4705.83	−0.518683
$436$	0	0
$437$	3702.75	0.405324
$438$	0	0
$439$	−8131.89	−0.884087	−0.442043	−	0.896994i	$-0.645746\pi$
$439$	−8131.89	−0.884087	−0.442043	+	0.896994i	$0.645746\pi$
$440$	0	0
$441$	−148.315	−0.0160150
$442$	0	0
$443$	13962.7	1.49748	0.748742	−	0.662861i	$-0.230658\pi$
$443$	13962.7	1.49748	0.748742	+	0.662861i	$0.230658\pi$
$444$	0	0
$445$	3377.54	0.359799
$446$	0	0
$447$	−8987.18	−0.950959
$448$	0	0
$449$	−15160.6	−1.59348	−0.796739	−	0.604323i	$-0.793443\pi$
$449$	−15160.6	−1.59348	−0.796739	+	0.604323i	$0.793443\pi$
$450$	0	0
$451$	−435.999	−0.0455219
$452$	0	0
$453$	−12984.5	−1.34672
$454$	0	0
$455$	−863.952	−0.0890169
$456$	0	0
$457$	13508.6	1.38273	0.691365	−	0.722506i	$-0.257010\pi$
$457$	13508.6	1.38273	0.691365	+	0.722506i	$0.257010\pi$
$458$	0	0
$459$	7094.80	0.721475
$460$	0	0
$461$	−11510.0	−1.16285	−0.581426	−	0.813599i	$-0.697505\pi$
$461$	−11510.0	−1.16285	−0.581426	+	0.813599i	$0.697505\pi$
$462$	0	0
$463$	9699.21	0.973565	0.486782	−	0.873523i	$-0.338170\pi$
$463$	9699.21	0.973565	0.486782	+	0.873523i	$0.338170\pi$
$464$	0	0
$465$	−2466.92	−0.246023
$466$	0	0
$467$	−3859.51	−0.382434	−0.191217	−	0.981548i	$-0.561243\pi$
$467$	−3859.51	−0.382434	−0.191217	+	0.981548i	$0.561243\pi$
$468$	0	0
$469$	−4053.04	−0.399045
$470$	0	0
$471$	13535.9	1.32421
$472$	0	0
$473$	2894.84	0.281406
$474$	0	0
$475$	4269.38	0.412405
$476$	0	0
$477$	1006.48	0.0966108
$478$	0	0
$479$	−2452.52	−0.233942	−0.116971	−	0.993135i	$-0.537319\pi$
$479$	−2452.52	−0.233942	−0.116971	+	0.993135i	$0.537319\pi$
$480$	0	0
$481$	−1481.68	−0.140455
$482$	0	0
$483$	−3259.58	−0.307073
$484$	0	0
$485$	470.381	0.0440390
$486$	0	0
$487$	1138.30	0.105917	0.0529583	−	0.998597i	$-0.483135\pi$
$487$	1138.30	0.105917	0.0529583	+	0.998597i	$0.483135\pi$
$488$	0	0
$489$	−13414.7	−1.24056
$490$	0	0
$491$	16028.2	1.47320	0.736601	−	0.676328i	$-0.236430\pi$
$491$	16028.2	1.47320	0.736601	+	0.676328i	$0.236430\pi$
$492$	0	0
$493$	−11841.4	−1.08177
$494$	0	0
$495$	−126.075	−0.0114478
$496$	0	0
$497$	−8068.23	−0.728188
$498$	0	0
$499$	−728.423	−0.0653481	−0.0326740	−	0.999466i	$-0.510402\pi$
$499$	−728.423	−0.0653481	−0.0326740	+	0.999466i	$0.510402\pi$
$500$	0	0
$501$	5608.30	0.500121
$502$	0	0
$503$	10295.1	0.912594	0.456297	−	0.889828i	$-0.349175\pi$
$503$	10295.1	0.912594	0.456297	+	0.889828i	$0.349175\pi$
$504$	0	0
$505$	−1930.68	−0.170127
$506$	0	0
$507$	5895.53	0.516429
$508$	0	0
$509$	8212.04	0.715113	0.357556	−	0.933892i	$-0.383610\pi$
$509$	8212.04	0.715113	0.357556	+	0.933892i	$0.383610\pi$
$510$	0	0
$511$	6997.33	0.605761
$512$	0	0
$513$	−5723.95	−0.492628
$514$	0	0
$515$	−145.506	−0.0124500
$516$	0	0
$517$	1527.40	0.129932
$518$	0	0
$519$	−7062.62	−0.597330
$520$	0	0
$521$	−1395.22	−0.117324	−0.0586618	−	0.998278i	$-0.518683\pi$
$521$	−1395.22	−0.117324	−0.0586618	+	0.998278i	$0.518683\pi$
$522$	0	0
$523$	6284.42	0.525428	0.262714	−	0.964874i	$-0.415382\pi$
$523$	6284.42	0.525428	0.262714	+	0.964874i	$0.415382\pi$
$524$	0	0
$525$	−3758.39	−0.312437
$526$	0	0
$527$	−6207.58	−0.513105
$528$	0	0
$529$	−3122.11	−0.256605
$530$	0	0
$531$	−515.306	−0.0421137
$532$	0	0
$533$	−1291.93	−0.104990
$534$	0	0
$535$	−6699.59	−0.541399
$536$	0	0
$537$	−4706.92	−0.378247
$538$	0	0
$539$	−521.072	−0.0416404
$540$	0	0
$541$	−17196.3	−1.36659	−0.683296	−	0.730142i	$-0.739454\pi$
$541$	−17196.3	−1.36659	−0.683296	+	0.730142i	$0.739454\pi$
$542$	0	0
$543$	−14862.4	−1.17460
$544$	0	0
$545$	−2801.03	−0.220152
$546$	0	0
$547$	−17692.6	−1.38297	−0.691483	−	0.722393i	$-0.743042\pi$
$547$	−17692.6	−1.38297	−0.691483	+	0.722393i	$0.743042\pi$
$548$	0	0
$549$	−1202.72	−0.0934987
$550$	0	0
$551$	9553.43	0.738638
$552$	0	0
$553$	6915.53	0.531787
$554$	0	0
$555$	901.782	0.0689703
$556$	0	0
$557$	13751.9	1.04612	0.523059	−	0.852297i	$-0.324791\pi$
$557$	13751.9	1.04612	0.523059	+	0.852297i	$0.324791\pi$
$558$	0	0
$559$	8577.83	0.649022
$560$	0	0
$561$	2512.65	0.189098
$562$	0	0
$563$	23166.1	1.73417	0.867083	−	0.498164i	$-0.165992\pi$
$563$	23166.1	1.73417	0.867083	+	0.498164i	$0.165992\pi$
$564$	0	0
$565$	−6776.32	−0.504569
$566$	0	0
$567$	4466.79	0.330843
$568$	0	0
$569$	−3977.25	−0.293031	−0.146516	−	0.989208i	$-0.546806\pi$
$569$	−3977.25	−0.293031	−0.146516	+	0.989208i	$0.546806\pi$
$570$	0	0
$571$	−25015.7	−1.83340	−0.916702	−	0.399572i	$-0.869159\pi$
$571$	−25015.7	−1.83340	−0.916702	+	0.399572i	$0.869159\pi$
$572$	0	0
$573$	−19935.3	−1.45342
$574$	0	0
$575$	10429.0	0.756382
$576$	0	0
$577$	17573.2	1.26791	0.633953	−	0.773372i	$-0.281431\pi$
$577$	17573.2	1.26791	0.633953	+	0.773372i	$0.281431\pi$
$578$	0	0
$579$	−12965.5	−0.930620
$580$	0	0
$581$	10214.1	0.729350
$582$	0	0
$583$	3536.02	0.251196
$584$	0	0
$585$	−373.578	−0.0264027
$586$	0	0
$587$	−26758.1	−1.88147	−0.940737	−	0.339136i	$-0.889865\pi$
$587$	−26758.1	−1.88147	−0.940737	+	0.339136i	$0.889865\pi$
$588$	0	0
$589$	5008.15	0.350352
$590$	0	0
$591$	3693.45	0.257070
$592$	0	0
$593$	21764.6	1.50719	0.753595	−	0.657339i	$-0.228318\pi$
$593$	21764.6	1.50719	0.753595	+	0.657339i	$0.228318\pi$
$594$	0	0
$595$	1323.13	0.0911650
$596$	0	0
$597$	−18556.5	−1.27214
$598$	0	0
$599$	1756.03	0.119782	0.0598909	−	0.998205i	$-0.480925\pi$
$599$	1756.03	0.119782	0.0598909	+	0.998205i	$0.480925\pi$
$600$	0	0
$601$	−13261.8	−0.900102	−0.450051	−	0.893003i	$-0.648594\pi$
$601$	−13261.8	−0.900102	−0.450051	+	0.893003i	$0.648594\pi$
$602$	0	0
$603$	−1752.56	−0.118358
$604$	0	0
$605$	4770.40	0.320569
$606$	0	0
$607$	−24570.4	−1.64297	−0.821485	−	0.570230i	$-0.806854\pi$
$607$	−24570.4	−1.64297	−0.821485	+	0.570230i	$0.806854\pi$
$608$	0	0
$609$	−8410.01	−0.559591
$610$	0	0
$611$	4525.90	0.299670
$612$	0	0
$613$	29703.0	1.95708	0.978542	−	0.206047i	$-0.0660599\pi$
$613$	29703.0	1.95708	0.978542	+	0.206047i	$0.0660599\pi$
$614$	0	0
$615$	786.292	0.0515551
$616$	0	0
$617$	−6686.96	−0.436316	−0.218158	−	0.975913i	$-0.570005\pi$
$617$	−6686.96	−0.436316	−0.218158	+	0.975913i	$0.570005\pi$
$618$	0	0
$619$	6430.48	0.417549	0.208775	−	0.977964i	$-0.433053\pi$
$619$	6430.48	0.417549	0.208775	+	0.977964i	$0.433053\pi$
$620$	0	0
$621$	−13982.1	−0.903517
$622$	0	0
$623$	6036.16	0.388176
$624$	0	0
$625$	10107.2	0.646861
$626$	0	0
$627$	−2027.16	−0.129118
$628$	0	0
$629$	2269.18	0.143845
$630$	0	0
$631$	11522.2	0.726929	0.363464	−	0.931608i	$-0.381594\pi$
$631$	11522.2	0.726929	0.363464	+	0.931608i	$0.381594\pi$
$632$	0	0
$633$	−5360.97	−0.336618
$634$	0	0
$635$	−7416.04	−0.463459
$636$	0	0
$637$	−1544.01	−0.0960375
$638$	0	0
$639$	−3488.75	−0.215982
$640$	0	0
$641$	−4997.08	−0.307914	−0.153957	−	0.988078i	$-0.549202\pi$
$641$	−4997.08	−0.307914	−0.153957	+	0.988078i	$0.549202\pi$
$642$	0	0
$643$	14213.8	0.871753	0.435877	−	0.900006i	$-0.356438\pi$
$643$	14213.8	0.871753	0.435877	+	0.900006i	$0.356438\pi$
$644$	0	0
$645$	−5220.63	−0.318701
$646$	0	0
$647$	9859.64	0.599107	0.299554	−	0.954079i	$-0.403162\pi$
$647$	9859.64	0.599107	0.299554	+	0.954079i	$0.403162\pi$
$648$	0	0
$649$	−1810.41	−0.109499
$650$	0	0
$651$	−4408.74	−0.265426
$652$	0	0
$653$	−6314.74	−0.378430	−0.189215	−	0.981936i	$-0.560594\pi$
$653$	−6314.74	−0.378430	−0.189215	+	0.981936i	$0.560594\pi$
$654$	0	0
$655$	3389.23	0.202181
$656$	0	0
$657$	3025.69	0.179670
$658$	0	0
$659$	−20840.2	−1.23189	−0.615946	−	0.787788i	$-0.711226\pi$
$659$	−20840.2	−1.23189	−0.615946	+	0.787788i	$0.711226\pi$
$660$	0	0
$661$	27271.7	1.60476	0.802379	−	0.596814i	$-0.203567\pi$
$661$	27271.7	1.60476	0.802379	+	0.596814i	$0.203567\pi$
$662$	0	0
$663$	7445.34	0.436129
$664$	0	0
$665$	−1067.48	−0.0622481
$666$	0	0
$667$	23336.6	1.35472
$668$	0	0
$669$	12556.3	0.725641
$670$	0	0
$671$	−4225.48	−0.243104
$672$	0	0
$673$	3226.98	0.184831	0.0924153	−	0.995721i	$-0.470541\pi$
$673$	3226.98	0.184831	0.0924153	+	0.995721i	$0.470541\pi$
$674$	0	0
$675$	−16121.8	−0.919301
$676$	0	0
$677$	−22631.8	−1.28480	−0.642400	−	0.766369i	$-0.722061\pi$
$677$	−22631.8	−1.28480	−0.642400	+	0.766369i	$0.722061\pi$
$678$	0	0
$679$	840.641	0.0475123
$680$	0	0
$681$	−25010.3	−1.40734
$682$	0	0
$683$	−18040.0	−1.01066	−0.505330	−	0.862926i	$-0.668629\pi$
$683$	−18040.0	−1.01066	−0.505330	+	0.862926i	$0.668629\pi$
$684$	0	0
$685$	−3003.48	−0.167528
$686$	0	0
$687$	30349.3	1.68544
$688$	0	0
$689$	10477.7	0.579347
$690$	0	0
$691$	1348.88	0.0742605	0.0371302	−	0.999310i	$-0.488178\pi$
$691$	1348.88	0.0742605	0.0371302	+	0.999310i	$0.488178\pi$
$692$	0	0
$693$	−225.315	−0.0123506
$694$	0	0
$695$	−1001.65	−0.0546686
$696$	0	0
$697$	1978.57	0.107523
$698$	0	0
$699$	26949.2	1.45825
$700$	0	0
$701$	−23278.9	−1.25425	−0.627127	−	0.778917i	$-0.715769\pi$
$701$	−23278.9	−1.25425	−0.627127	+	0.778917i	$0.715769\pi$
$702$	0	0
$703$	−1830.73	−0.0982181
$704$	0	0
$705$	−2754.55	−0.147152
$706$	0	0
$707$	−3450.41	−0.183545
$708$	0	0
$709$	30407.6	1.61069	0.805346	−	0.592805i	$-0.201979\pi$
$709$	30407.6	1.61069	0.805346	+	0.592805i	$0.201979\pi$
$710$	0	0
$711$	2990.31	0.157729
$712$	0	0
$713$	12233.6	0.642572
$714$	0	0
$715$	−1312.48	−0.0686490
$716$	0	0
$717$	22078.1	1.14996
$718$	0	0
$719$	−21022.5	−1.09041	−0.545206	−	0.838302i	$-0.683549\pi$
$719$	−21022.5	−1.09041	−0.545206	+	0.838302i	$0.683549\pi$
$720$	0	0
$721$	−260.041	−0.0134319
$722$	0	0
$723$	−34704.7	−1.78517
$724$	0	0
$725$	26907.7	1.37838
$726$	0	0
$727$	−17489.8	−0.892245	−0.446123	−	0.894972i	$-0.647196\pi$
$727$	−17489.8	−0.892245	−0.446123	+	0.894972i	$0.647196\pi$
$728$	0	0
$729$	21367.1	1.08556
$730$	0	0
$731$	−13136.8	−0.664684
$732$	0	0
$733$	3294.16	0.165992	0.0829962	−	0.996550i	$-0.473551\pi$
$733$	3294.16	0.165992	0.0829962	+	0.996550i	$0.473551\pi$
$734$	0	0
$735$	939.715	0.0471591
$736$	0	0
$737$	−6157.21	−0.307739
$738$	0	0
$739$	−4752.95	−0.236590	−0.118295	−	0.992978i	$-0.537743\pi$
$739$	−4752.95	−0.236590	−0.118295	+	0.992978i	$0.537743\pi$
$740$	0	0
$741$	−6006.76	−0.297792
$742$	0	0
$743$	−6046.77	−0.298566	−0.149283	−	0.988795i	$-0.547697\pi$
$743$	−6046.77	−0.298566	−0.149283	+	0.988795i	$0.547697\pi$
$744$	0	0
$745$	−7189.49	−0.353561
$746$	0	0
$747$	4416.64	0.216327
$748$	0	0
$749$	−11973.2	−0.584098
$750$	0	0
$751$	−13128.8	−0.637920	−0.318960	−	0.947768i	$-0.603334\pi$
$751$	−13128.8	−0.637920	−0.318960	+	0.947768i	$0.603334\pi$
$752$	0	0
$753$	−11437.9	−0.553545
$754$	0	0
$755$	−10387.3	−0.500703
$756$	0	0
$757$	−1103.75	−0.0529942	−0.0264971	−	0.999649i	$-0.508435\pi$
$757$	−1103.75	−0.0529942	−0.0264971	+	0.999649i	$0.508435\pi$
$758$	0	0
$759$	−4951.83	−0.236812
$760$	0	0
$761$	−25268.0	−1.20363	−0.601816	−	0.798635i	$-0.705556\pi$
$761$	−25268.0	−1.20363	−0.601816	+	0.798635i	$0.705556\pi$
$762$	0	0
$763$	−5005.85	−0.237515
$764$	0	0
$765$	572.131	0.0270398
$766$	0	0
$767$	−5364.50	−0.252544
$768$	0	0
$769$	−24891.9	−1.16726	−0.583632	−	0.812018i	$-0.698369\pi$
$769$	−24891.9	−1.16726	−0.583632	+	0.812018i	$0.698369\pi$
$770$	0	0
$771$	30080.5	1.40509
$772$	0	0
$773$	−24275.1	−1.12951	−0.564756	−	0.825258i	$-0.691030\pi$
$773$	−24275.1	−1.12951	−0.564756	+	0.825258i	$0.691030\pi$
$774$	0	0
$775$	14105.7	0.653797
$776$	0	0
$777$	1611.62	0.0744098
$778$	0	0
$779$	−1596.27	−0.0734177
$780$	0	0
$781$	−12256.9	−0.561572
$782$	0	0
$783$	−36075.2	−1.64651
$784$	0	0
$785$	10828.4	0.492332
$786$	0	0
$787$	23472.9	1.06318	0.531588	−	0.847003i	$-0.321595\pi$
$787$	23472.9	1.06318	0.531588	+	0.847003i	$0.321595\pi$
$788$	0	0
$789$	−10244.7	−0.462255
$790$	0	0
$791$	−12110.3	−0.544364
$792$	0	0
$793$	−12520.7	−0.560684
$794$	0	0
$795$	−6376.96	−0.284487
$796$	0	0
$797$	36868.7	1.63859	0.819294	−	0.573373i	$-0.194365\pi$
$797$	36868.7	1.63859	0.819294	+	0.573373i	$0.194365\pi$
$798$	0	0
$799$	−6931.37	−0.306901
$800$	0	0
$801$	2610.07	0.115134
$802$	0	0
$803$	10630.1	0.467157
$804$	0	0
$805$	−2607.58	−0.114168
$806$	0	0
$807$	−26975.2	−1.17667
$808$	0	0
$809$	−18964.3	−0.824167	−0.412083	−	0.911146i	$-0.635199\pi$
$809$	−18964.3	−0.824167	−0.412083	+	0.911146i	$0.635199\pi$
$810$	0	0
$811$	39735.7	1.72048	0.860239	−	0.509891i	$-0.170314\pi$
$811$	39735.7	1.72048	0.860239	+	0.509891i	$0.170314\pi$
$812$	0	0
$813$	−2417.50	−0.104287
$814$	0	0
$815$	−10731.4	−0.461233
$816$	0	0
$817$	10598.5	0.453851
$818$	0	0
$819$	−667.639	−0.0284850
$820$	0	0
$821$	31839.1	1.35346	0.676730	−	0.736231i	$-0.263396\pi$
$821$	31839.1	1.35346	0.676730	+	0.736231i	$0.263396\pi$
$822$	0	0
$823$	25798.7	1.09269	0.546346	−	0.837560i	$-0.316018\pi$
$823$	25798.7	1.09269	0.546346	+	0.837560i	$0.316018\pi$
$824$	0	0
$825$	−5709.60	−0.240949
$826$	0	0
$827$	46229.8	1.94386	0.971928	−	0.235279i	$-0.0756005\pi$
$827$	46229.8	1.94386	0.971928	+	0.235279i	$0.0756005\pi$
$828$	0	0
$829$	−11354.5	−0.475702	−0.237851	−	0.971302i	$-0.576443\pi$
$829$	−11354.5	−0.475702	−0.237851	+	0.971302i	$0.576443\pi$
$830$	0	0
$831$	−4049.25	−0.169034
$832$	0	0
$833$	2364.64	0.0983550
$834$	0	0
$835$	4486.48	0.185942
$836$	0	0
$837$	−18911.5	−0.780977
$838$	0	0
$839$	−24086.7	−0.991138	−0.495569	−	0.868569i	$-0.665040\pi$
$839$	−24086.7	−0.991138	−0.495569	+	0.868569i	$0.665040\pi$
$840$	0	0
$841$	35821.5	1.46875
$842$	0	0
$843$	33176.2	1.35546
$844$	0	0
$845$	4716.26	0.192005
$846$	0	0
$847$	8525.41	0.345852
$848$	0	0
$849$	27990.5	1.13149
$850$	0	0
$851$	−4472.01	−0.180139
$852$	0	0
$853$	−24127.5	−0.968476	−0.484238	−	0.874936i	$-0.660903\pi$
$853$	−24127.5	−0.968476	−0.484238	+	0.874936i	$0.660903\pi$
$854$	0	0
$855$	−461.584	−0.0184630
$856$	0	0
$857$	18800.9	0.749390	0.374695	−	0.927148i	$-0.377747\pi$
$857$	18800.9	0.749390	0.374695	+	0.927148i	$0.377747\pi$
$858$	0	0
$859$	35326.7	1.40318	0.701590	−	0.712581i	$-0.252474\pi$
$859$	35326.7	1.40318	0.701590	+	0.712581i	$0.252474\pi$
$860$	0	0
$861$	1405.22	0.0556211
$862$	0	0
$863$	7391.65	0.291558	0.145779	−	0.989317i	$-0.453431\pi$
$863$	7391.65	0.291558	0.145779	+	0.989317i	$0.453431\pi$
$864$	0	0
$865$	−5649.90	−0.222084
$866$	0	0
$867$	12652.8	0.495629
$868$	0	0
$869$	10505.8	0.410109
$870$	0	0
$871$	−18244.7	−0.709757
$872$	0	0
$873$	363.498	0.0140923
$874$	0	0
$875$	−6433.86	−0.248576
$876$	0	0
$877$	−45662.5	−1.75817	−0.879084	−	0.476667i	$-0.841845\pi$
$877$	−45662.5	−1.75817	−0.879084	+	0.476667i	$0.841845\pi$
$878$	0	0
$879$	3423.75	0.131377
$880$	0	0
$881$	−13733.6	−0.525195	−0.262598	−	0.964905i	$-0.584579\pi$
$881$	−13733.6	−0.525195	−0.262598	+	0.964905i	$0.584579\pi$
$882$	0	0
$883$	28521.5	1.08701	0.543503	−	0.839407i	$-0.317098\pi$
$883$	28521.5	1.08701	0.543503	+	0.839407i	$0.317098\pi$
$884$	0	0
$885$	3264.94	0.124011
$886$	0	0
$887$	4142.71	0.156819	0.0784095	−	0.996921i	$-0.475016\pi$
$887$	4142.71	0.156819	0.0784095	+	0.996921i	$0.475016\pi$
$888$	0	0
$889$	−13253.6	−0.500011
$890$	0	0
$891$	6785.78	0.255143
$892$	0	0
$893$	5592.09	0.209554
$894$	0	0
$895$	−3765.41	−0.140630
$896$	0	0
$897$	−14673.0	−0.546172
$898$	0	0
$899$	31563.9	1.17098
$900$	0	0
$901$	−16046.5	−0.593327
$902$	0	0
$903$	−9330.05	−0.343837
$904$	0	0
$905$	−11889.5	−0.436707
$906$	0	0
$907$	28934.4	1.05926	0.529631	−	0.848228i	$-0.322330\pi$
$907$	28934.4	1.05926	0.529631	+	0.848228i	$0.322330\pi$
$908$	0	0
$909$	−1491.98	−0.0544398
$910$	0	0
$911$	35767.0	1.30078	0.650391	−	0.759600i	$-0.274605\pi$
$911$	35767.0	1.30078	0.650391	+	0.759600i	$0.274605\pi$
$912$	0	0
$913$	15516.9	0.562467
$914$	0	0
$915$	7620.34	0.275323
$916$	0	0
$917$	6057.06	0.218126
$918$	0	0
$919$	−6293.70	−0.225908	−0.112954	−	0.993600i	$-0.536031\pi$
$919$	−6293.70	−0.225908	−0.112954	+	0.993600i	$0.536031\pi$
$920$	0	0
$921$	−22848.7	−0.817472
$922$	0	0
$923$	−36319.0	−1.29518
$924$	0	0
$925$	−5156.35	−0.183286
$926$	0	0
$927$	−112.443	−0.00398395
$928$	0	0
$929$	−38346.0	−1.35424	−0.677121	−	0.735872i	$-0.736773\pi$
$929$	−38346.0	−1.35424	−0.677121	+	0.735872i	$0.736773\pi$
$930$	0	0
$931$	−1907.74	−0.0671575
$932$	0	0
$933$	−2121.06	−0.0744271
$934$	0	0
$935$	2010.05	0.0703056
$936$	0	0
$937$	12639.1	0.440663	0.220332	−	0.975425i	$-0.429286\pi$
$937$	12639.1	0.440663	0.220332	+	0.975425i	$0.429286\pi$
$938$	0	0
$939$	14560.1	0.506019
$940$	0	0
$941$	22136.7	0.766882	0.383441	−	0.923565i	$-0.374739\pi$
$941$	22136.7	0.766882	0.383441	+	0.923565i	$0.374739\pi$
$942$	0	0
$943$	−3899.29	−0.134654
$944$	0	0
$945$	4030.95	0.138759
$946$	0	0
$947$	3674.10	0.126074	0.0630370	−	0.998011i	$-0.479921\pi$
$947$	3674.10	0.126074	0.0630370	+	0.998011i	$0.479921\pi$
$948$	0	0
$949$	31498.4	1.07743
$950$	0	0
$951$	−601.980	−0.0205263
$952$	0	0
$953$	2175.54	0.0739482	0.0369741	−	0.999316i	$-0.488228\pi$
$953$	2175.54	0.0739482	0.0369741	+	0.999316i	$0.488228\pi$
$954$	0	0
$955$	−15947.7	−0.540371
$956$	0	0
$957$	−12776.2	−0.431551
$958$	0	0
$959$	−5367.66	−0.180741
$960$	0	0
$961$	−13244.4	−0.444578
$962$	0	0
$963$	−5177.27	−0.173245
$964$	0	0
$965$	−10372.1	−0.345999
$966$	0	0
$967$	3394.80	0.112895	0.0564474	−	0.998406i	$-0.482023\pi$
$967$	3394.80	0.112895	0.0564474	+	0.998406i	$0.482023\pi$
$968$	0	0
$969$	9199.28	0.304978
$970$	0	0
$971$	−37028.8	−1.22380	−0.611900	−	0.790935i	$-0.709594\pi$
$971$	−37028.8	−1.22380	−0.611900	+	0.790935i	$0.709594\pi$
$972$	0	0
$973$	−1790.09	−0.0589802
$974$	0	0
$975$	−16918.4	−0.555714
$976$	0	0
$977$	−29572.1	−0.968369	−0.484185	−	0.874966i	$-0.660884\pi$
$977$	−29572.1	−0.968369	−0.484185	+	0.874966i	$0.660884\pi$
$978$	0	0
$979$	9169.90	0.299358
$980$	0	0
$981$	−2164.56	−0.0704475
$982$	0	0
$983$	10108.3	0.327981	0.163991	−	0.986462i	$-0.447563\pi$
$983$	10108.3	0.327981	0.163991	+	0.986462i	$0.447563\pi$
$984$	0	0
$985$	2954.66	0.0955769
$986$	0	0
$987$	−4922.79	−0.158758
$988$	0	0
$989$	25889.6	0.832397
$990$	0	0
$991$	30369.5	0.973481	0.486740	−	0.873547i	$-0.338186\pi$
$991$	30369.5	0.973481	0.486740	+	0.873547i	$0.338186\pi$
$992$	0	0
$993$	27454.1	0.877373
$994$	0	0
$995$	−14844.7	−0.472973
$996$	0	0
$997$	3901.69	0.123940	0.0619698	−	0.998078i	$-0.480262\pi$
$997$	3901.69	0.123940	0.0619698	+	0.998078i	$0.480262\pi$
$998$	0	0
$999$	6913.11	0.218940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.d.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.d.1.3	✓	15	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-4.89624 q^{3} -3.91686 q^{5} -7.00000 q^{7} -3.02684 q^{9} +O(q^{10})\)	\(q-4.89624 q^{3} -3.91686 q^{5} -7.00000 q^{7} -3.02684 q^{9} -10.6341 q^{11} -31.5104 q^{13} +19.1779 q^{15} +48.2579 q^{17} -38.9335 q^{19} +34.2737 q^{21} -95.1046 q^{23} -109.658 q^{25} +147.019 q^{27} -245.378 q^{29} -128.634 q^{31} +52.0672 q^{33} +27.4180 q^{35} +47.0220 q^{37} +154.283 q^{39} +41.0000 q^{41} -272.222 q^{43} +11.8557 q^{45} -143.632 q^{47} +49.0000 q^{49} -236.282 q^{51} -332.517 q^{53} +41.6523 q^{55} +190.628 q^{57} +170.245 q^{59} +397.351 q^{61} +21.1879 q^{63} +123.422 q^{65} +579.005 q^{67} +465.655 q^{69} +1152.60 q^{71} -999.619 q^{73} +536.913 q^{75} +74.4389 q^{77} -987.932 q^{79} -638.113 q^{81} -1459.16 q^{83} -189.019 q^{85} +1201.43 q^{87} -862.309 q^{89} +220.573 q^{91} +629.821 q^{93} +152.497 q^{95} -120.092 q^{97} +32.1878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Introduction

L-functions

Modular forms

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