LMFDB - Embedded newform 675.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5637.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 23x + 6 $ x^3 - x^2 - 23*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.258712$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.258712 q^{2} -7.93307 q^{4} -14.5174 q^{7} -4.12208 q^{8} +O(q^{10})$	$q+0.258712 q^{2} -7.93307 q^{4} -14.5174 q^{7} -4.12208 q^{8} -49.2845 q^{11} -72.1800 q^{13} -3.75584 q^{14} +62.3981 q^{16} -118.017 q^{17} +123.389 q^{19} -12.7505 q^{22} +91.4883 q^{23} -18.6739 q^{26} +115.168 q^{28} +174.400 q^{29} -46.2956 q^{31} +49.1198 q^{32} -30.5324 q^{34} -154.977 q^{37} +31.9223 q^{38} -364.203 q^{41} -125.714 q^{43} +390.978 q^{44} +23.6692 q^{46} +221.523 q^{47} -132.244 q^{49} +572.609 q^{52} +13.6794 q^{53} +59.8420 q^{56} +45.1195 q^{58} +239.087 q^{59} -54.5457 q^{61} -11.9772 q^{62} -486.477 q^{64} +76.0558 q^{67} +936.235 q^{68} +728.303 q^{71} +501.815 q^{73} -40.0944 q^{74} -978.854 q^{76} +715.485 q^{77} +397.610 q^{79} -94.2237 q^{82} +1369.46 q^{83} -32.5237 q^{86} +203.155 q^{88} +1468.13 q^{89} +1047.87 q^{91} -725.783 q^{92} +57.3108 q^{94} -335.023 q^{97} -34.2133 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8}+O(q^{10}) $ 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8	$ 3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8} + 38 q^{11} - 28 q^{13} - 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 122 q^{22} + 81 q^{23} + 416 q^{26} - 410 q^{28} + 160 q^{29} + 227 q^{31} + 569 q^{32} + 17 q^{34} - 78 q^{37} + 757 q^{38} - 338 q^{41} - 22 q^{43} + 1636 q^{44} - 1425 q^{46} + 472 q^{47} - 197 q^{49} + 1566 q^{52} - 521 q^{53} - 1254 q^{56} + 2096 q^{58} + 140 q^{59} + 595 q^{61} - 1407 q^{62} - 918 q^{64} - 878 q^{67} + 3053 q^{68} - 602 q^{71} - 1294 q^{73} + 2878 q^{74} + 525 q^{76} - 288 q^{77} + 629 q^{79} + 1682 q^{82} + 1287 q^{83} - 3730 q^{86} + 858 q^{88} + 2154 q^{89} - 440 q^{91} - 1959 q^{92} - 1108 q^{94} - 1392 q^{97} + 2693 q^{98}+O(q^{100}) $ 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8 + 38 * q^11 - 28 * q^13 - 108 * q^14 + 191 * q^16 + 19 * q^17 + 187 * q^19 - 122 * q^22 + 81 * q^23 + 416 * q^26 - 410 * q^28 + 160 * q^29 + 227 * q^31 + 569 * q^32 + 17 * q^34 - 78 * q^37 + 757 * q^38 - 338 * q^41 - 22 * q^43 + 1636 * q^44 - 1425 * q^46 + 472 * q^47 - 197 * q^49 + 1566 * q^52 - 521 * q^53 - 1254 * q^56 + 2096 * q^58 + 140 * q^59 + 595 * q^61 - 1407 * q^62 - 918 * q^64 - 878 * q^67 + 3053 * q^68 - 602 * q^71 - 1294 * q^73 + 2878 * q^74 + 525 * q^76 - 288 * q^77 + 629 * q^79 + 1682 * q^82 + 1287 * q^83 - 3730 * q^86 + 858 * q^88 + 2154 * q^89 - 440 * q^91 - 1959 * q^92 - 1108 * q^94 - 1392 * q^97 + 2693 * q^98

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 $

$2$	0.258712	0.0914686	0.0457343	−	0.998954i	$-0.485437\pi$
$2$	0.258712	0.0914686	0.0457343	+	0.998954i	$0.485437\pi$
$3$	0	0
$3$	0	0
$4$	−7.93307	−0.991633
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−14.5174	−0.783867	−0.391934	−	0.919993i	$-0.628194\pi$
$7$	−14.5174	−0.783867	−0.391934	+	0.919993i	$0.628194\pi$
$8$	−4.12208	−0.182172
$9$	0	0
$10$	0	0
$11$	−49.2845	−1.35090	−0.675448	−	0.737408i	$-0.736050\pi$
$11$	−49.2845	−1.35090	−0.675448	+	0.737408i	$0.736050\pi$
$12$	0	0
$13$	−72.1800	−1.53993	−0.769967	−	0.638084i	$-0.779727\pi$
$13$	−72.1800	−1.53993	−0.769967	+	0.638084i	$0.779727\pi$
$14$	−3.75584	−0.0716993
$15$	0	0
$16$	62.3981	0.974970
$17$	−118.017	−1.68372	−0.841861	−	0.539694i	$-0.818540\pi$
$17$	−118.017	−1.68372	−0.841861	+	0.539694i	$0.818540\pi$
$18$	0	0
$19$	123.389	1.48986	0.744932	−	0.667141i	$-0.232482\pi$
$19$	123.389	1.48986	0.744932	+	0.667141i	$0.232482\pi$
$20$	0	0
$21$	0	0
$22$	−12.7505	−0.123565
$23$	91.4883	0.829419	0.414709	−	0.909954i	$-0.363883\pi$
$23$	91.4883	0.829419	0.414709	+	0.909954i	$0.363883\pi$
$24$	0	0
$25$	0	0
$26$	−18.6739	−0.140856
$27$	0	0
$28$	115.168	0.777309
$29$	174.400	1.11673	0.558367	−	0.829594i	$-0.311428\pi$
$29$	174.400	1.11673	0.558367	+	0.829594i	$0.311428\pi$
$30$	0	0
$31$	−46.2956	−0.268224	−0.134112	−	0.990966i	$-0.542818\pi$
$31$	−46.2956	−0.268224	−0.134112	+	0.990966i	$0.542818\pi$
$32$	49.1198	0.271351
$33$	0	0
$34$	−30.5324	−0.154008
$35$	0	0
$36$	0	0
$37$	−154.977	−0.688595	−0.344297	−	0.938861i	$-0.611883\pi$
$37$	−154.977	−0.688595	−0.344297	+	0.938861i	$0.611883\pi$
$38$	31.9223	0.136276
$39$	0	0
$40$	0	0
$41$	−364.203	−1.38729	−0.693645	−	0.720317i	$-0.743996\pi$
$41$	−364.203	−1.38729	−0.693645	+	0.720317i	$0.743996\pi$
$42$	0	0
$43$	−125.714	−0.445841	−0.222921	−	0.974837i	$-0.571559\pi$
$43$	−125.714	−0.445841	−0.222921	+	0.974837i	$0.571559\pi$
$44$	390.978	1.33959
$45$	0	0
$46$	23.6692	0.0758658
$47$	221.523	0.687499	0.343750	−	0.939061i	$-0.388303\pi$
$47$	221.523	0.687499	0.343750	+	0.939061i	$0.388303\pi$
$48$	0	0
$49$	−132.244	−0.385552
$50$	0	0
$51$	0	0
$52$	572.609	1.52705
$53$	13.6794	0.0354530	0.0177265	−	0.999843i	$-0.494357\pi$
$53$	13.6794	0.0354530	0.0177265	+	0.999843i	$0.494357\pi$
$54$	0	0
$55$	0	0
$56$	59.8420	0.142799
$57$	0	0
$58$	45.1195	0.102146
$59$	239.087	0.527567	0.263784	−	0.964582i	$-0.415030\pi$
$59$	239.087	0.527567	0.263784	+	0.964582i	$0.415030\pi$
$60$	0	0
$61$	−54.5457	−0.114490	−0.0572448	−	0.998360i	$-0.518232\pi$
$61$	−54.5457	−0.114490	−0.0572448	+	0.998360i	$0.518232\pi$
$62$	−11.9772	−0.0245341
$63$	0	0
$64$	−486.477	−0.950150
$65$	0	0
$66$	0	0
$67$	76.0558	0.138682	0.0693410	−	0.997593i	$-0.477910\pi$
$67$	76.0558	0.138682	0.0693410	+	0.997593i	$0.477910\pi$
$68$	936.235	1.66964
$69$	0	0
$70$	0	0
$71$	728.303	1.21738	0.608688	−	0.793410i	$-0.291696\pi$
$71$	728.303	1.21738	0.608688	+	0.793410i	$0.291696\pi$
$72$	0	0
$73$	501.815	0.804562	0.402281	−	0.915516i	$-0.368218\pi$
$73$	501.815	0.804562	0.402281	+	0.915516i	$0.368218\pi$
$74$	−40.0944	−0.0629848
$75$	0	0
$76$	−978.854	−1.47740
$77$	715.485	1.05892
$78$	0	0
$79$	397.610	0.566261	0.283130	−	0.959081i	$-0.408627\pi$
$79$	397.610	0.566261	0.283130	+	0.959081i	$0.408627\pi$
$80$	0	0
$81$	0	0
$82$	−94.2237	−0.126894
$83$	1369.46	1.81106	0.905530	−	0.424283i	$-0.139474\pi$
$83$	1369.46	1.81106	0.905530	+	0.424283i	$0.139474\pi$
$84$	0	0
$85$	0	0
$86$	−32.5237	−0.0407805
$87$	0	0
$88$	203.155	0.246095
$89$	1468.13	1.74856	0.874278	−	0.485425i	$-0.161335\pi$
$89$	1468.13	1.74856	0.874278	+	0.485425i	$0.161335\pi$
$90$	0	0
$91$	1047.87	1.20710
$92$	−725.783	−0.822480
$93$	0	0
$94$	57.3108	0.0628846
$95$	0	0
$96$	0	0
$97$	−335.023	−0.350685	−0.175343	−	0.984507i	$-0.556103\pi$
$97$	−335.023	−0.350685	−0.175343	+	0.984507i	$0.556103\pi$
$98$	−34.2133	−0.0352659
$99$	0	0
$100$	0	0
$101$	−1206.09	−1.18822	−0.594109	−	0.804384i	$-0.702495\pi$
$101$	−1206.09	−1.18822	−0.594109	+	0.804384i	$0.702495\pi$
$102$	0	0
$103$	−1061.11	−1.01509	−0.507545	−	0.861625i	$-0.669447\pi$
$103$	−1061.11	−1.01509	−0.507545	+	0.861625i	$0.669447\pi$
$104$	297.532	0.280533
$105$	0	0
$106$	3.53903	0.00324284
$107$	475.578	0.429681	0.214841	−	0.976649i	$-0.431077\pi$
$107$	475.578	0.429681	0.214841	+	0.976649i	$0.431077\pi$
$108$	0	0
$109$	1320.42	1.16030	0.580152	−	0.814508i	$-0.302993\pi$
$109$	1320.42	1.16030	0.580152	+	0.814508i	$0.302993\pi$
$110$	0	0
$111$	0	0
$112$	−905.860	−0.764247
$113$	68.1750	0.0567555	0.0283777	−	0.999597i	$-0.490966\pi$
$113$	68.1750	0.0567555	0.0283777	+	0.999597i	$0.490966\pi$
$114$	0	0
$115$	0	0
$116$	−1383.53	−1.10739
$117$	0	0
$118$	61.8547	0.0482559
$119$	1713.30	1.31981
$120$	0	0
$121$	1097.97	0.824918
$122$	−14.1117	−0.0104722
$123$	0	0
$124$	367.266	0.265980
$125$	0	0
$126$	0	0
$127$	593.009	0.414339	0.207170	−	0.978305i	$-0.433575\pi$
$127$	593.009	0.414339	0.207170	+	0.978305i	$0.433575\pi$
$128$	−518.816	−0.358260
$129$	0	0
$130$	0	0
$131$	338.937	0.226054	0.113027	−	0.993592i	$-0.463945\pi$
$131$	338.937	0.226054	0.113027	+	0.993592i	$0.463945\pi$
$132$	0	0
$133$	−1791.29	−1.16785
$134$	19.6766	0.0126850
$135$	0	0
$136$	486.475	0.306727
$137$	811.442	0.506030	0.253015	−	0.967462i	$-0.418578\pi$
$137$	811.442	0.506030	0.253015	+	0.967462i	$0.418578\pi$
$138$	0	0
$139$	−3106.13	−1.89538	−0.947691	−	0.319189i	$-0.896590\pi$
$139$	−3106.13	−1.89538	−0.947691	+	0.319189i	$0.896590\pi$
$140$	0	0
$141$	0	0
$142$	188.421	0.111352
$143$	3557.36	2.08029
$144$	0	0
$145$	0	0
$146$	129.826	0.0735922
$147$	0	0
$148$	1229.44	0.682834
$149$	2541.01	1.39710	0.698550	−	0.715561i	$-0.253829\pi$
$149$	2541.01	1.39710	0.698550	+	0.715561i	$0.253829\pi$
$150$	0	0
$151$	−1125.37	−0.606499	−0.303249	−	0.952911i	$-0.598072\pi$
$151$	−1125.37	−0.606499	−0.303249	+	0.952911i	$0.598072\pi$
$152$	−508.620	−0.271411
$153$	0	0
$154$	185.105	0.0968582
$155$	0	0
$156$	0	0
$157$	−3230.05	−1.64195	−0.820975	−	0.570963i	$-0.806570\pi$
$157$	−3230.05	−1.64195	−0.820975	+	0.570963i	$0.806570\pi$
$158$	102.867	0.0517951
$159$	0	0
$160$	0	0
$161$	−1328.17	−0.650154
$162$	0	0
$163$	694.054	0.333512	0.166756	−	0.985998i	$-0.446671\pi$
$163$	694.054	0.333512	0.166756	+	0.985998i	$0.446671\pi$
$164$	2889.24	1.37568
$165$	0	0
$166$	354.297	0.165655
$167$	−3216.04	−1.49021	−0.745103	−	0.666950i	$-0.767600\pi$
$167$	−3216.04	−1.49021	−0.745103	+	0.666950i	$0.767600\pi$
$168$	0	0
$169$	3012.95	1.37139
$170$	0	0
$171$	0	0
$172$	997.296	0.442111
$173$	−297.546	−0.130763	−0.0653816	−	0.997860i	$-0.520826\pi$
$173$	−297.546	−0.130763	−0.0653816	+	0.997860i	$0.520826\pi$
$174$	0	0
$175$	0	0
$176$	−3075.26	−1.31708
$177$	0	0
$178$	379.823	0.159938
$179$	−3450.12	−1.44064	−0.720320	−	0.693642i	$-0.756005\pi$
$179$	−3450.12	−1.44064	−0.720320	+	0.693642i	$0.756005\pi$
$180$	0	0
$181$	−3089.75	−1.26883	−0.634417	−	0.772991i	$-0.718760\pi$
$181$	−3089.75	−1.26883	−0.634417	+	0.772991i	$0.718760\pi$
$182$	271.096	0.110412
$183$	0	0
$184$	−377.122	−0.151097
$185$	0	0
$186$	0	0
$187$	5816.41	2.27453
$188$	−1757.36	−0.681748
$189$	0	0
$190$	0	0
$191$	1532.11	0.580419	0.290209	−	0.956963i	$-0.406275\pi$
$191$	1532.11	0.580419	0.290209	+	0.956963i	$0.406275\pi$
$192$	0	0
$193$	5194.42	1.93732	0.968660	−	0.248389i	$-0.0799012\pi$
$193$	5194.42	1.93732	0.968660	+	0.248389i	$0.0799012\pi$
$194$	−86.6747	−0.0320767
$195$	0	0
$196$	1049.10	0.382326
$197$	2005.61	0.725349	0.362674	−	0.931916i	$-0.381864\pi$
$197$	2005.61	0.725349	0.362674	+	0.931916i	$0.381864\pi$
$198$	0	0
$199$	−2874.68	−1.02402	−0.512011	−	0.858979i	$-0.671099\pi$
$199$	−2874.68	−1.02402	−0.512011	+	0.858979i	$0.671099\pi$
$200$	0	0
$201$	0	0
$202$	−312.029	−0.108685
$203$	−2531.84	−0.875372
$204$	0	0
$205$	0	0
$206$	−274.522	−0.0928488
$207$	0	0
$208$	−4503.90	−1.50139
$209$	−6081.18	−2.01265
$210$	0	0
$211$	−2749.94	−0.897220	−0.448610	−	0.893728i	$-0.648081\pi$
$211$	−2749.94	−0.897220	−0.448610	+	0.893728i	$0.648081\pi$
$212$	−108.520	−0.0351564
$213$	0	0
$214$	123.038	0.0393023
$215$	0	0
$216$	0	0
$217$	672.093	0.210252
$218$	341.609	0.106131
$219$	0	0
$220$	0	0
$221$	8518.45	2.59282
$222$	0	0
$223$	783.727	0.235346	0.117673	−	0.993052i	$-0.462456\pi$
$223$	783.727	0.235346	0.117673	+	0.993052i	$0.462456\pi$
$224$	−713.093	−0.212703
$225$	0	0
$226$	17.6377	0.00519134
$227$	145.665	0.0425909	0.0212955	−	0.999773i	$-0.493221\pi$
$227$	145.665	0.0425909	0.0212955	+	0.999773i	$0.493221\pi$
$228$	0	0
$229$	−3411.82	−0.984539	−0.492270	−	0.870443i	$-0.663833\pi$
$229$	−3411.82	−0.984539	−0.492270	+	0.870443i	$0.663833\pi$
$230$	0	0
$231$	0	0
$232$	−718.892	−0.203438
$233$	−134.977	−0.0379511	−0.0189756	−	0.999820i	$-0.506040\pi$
$233$	−134.977	−0.0379511	−0.0189756	+	0.999820i	$0.506040\pi$
$234$	0	0
$235$	0	0
$236$	−1896.69	−0.523153
$237$	0	0
$238$	443.252	0.120722
$239$	−2245.32	−0.607690	−0.303845	−	0.952722i	$-0.598270\pi$
$239$	−2245.32	−0.607690	−0.303845	+	0.952722i	$0.598270\pi$
$240$	0	0
$241$	4158.54	1.11151	0.555757	−	0.831345i	$-0.312428\pi$
$241$	4158.54	1.11151	0.555757	+	0.831345i	$0.312428\pi$
$242$	284.057	0.0754542
$243$	0	0
$244$	432.715	0.113532
$245$	0	0
$246$	0	0
$247$	−8906.22	−2.29429
$248$	190.834	0.0488629
$249$	0	0
$250$	0	0
$251$	3946.14	0.992343	0.496171	−	0.868225i	$-0.334739\pi$
$251$	3946.14	0.992343	0.496171	+	0.868225i	$0.334739\pi$
$252$	0	0
$253$	−4508.96	−1.12046
$254$	153.419	0.0378990
$255$	0	0
$256$	3757.59	0.917381
$257$	5695.84	1.38248	0.691239	−	0.722626i	$-0.257065\pi$
$257$	5695.84	1.38248	0.691239	+	0.722626i	$0.257065\pi$
$258$	0	0
$259$	2249.86	0.539767
$260$	0	0
$261$	0	0
$262$	87.6873	0.0206769
$263$	−2814.06	−0.659781	−0.329891	−	0.944019i	$-0.607012\pi$
$263$	−2814.06	−0.659781	−0.329891	+	0.944019i	$0.607012\pi$
$264$	0	0
$265$	0	0
$266$	−463.429	−0.106822
$267$	0	0
$268$	−603.356	−0.137522
$269$	200.985	0.0455548	0.0227774	−	0.999741i	$-0.492749\pi$
$269$	200.985	0.0455548	0.0227774	+	0.999741i	$0.492749\pi$
$270$	0	0
$271$	−2406.05	−0.539326	−0.269663	−	0.962955i	$-0.586912\pi$
$271$	−2406.05	−0.539326	−0.269663	+	0.962955i	$0.586912\pi$
$272$	−7364.03	−1.64158
$273$	0	0
$274$	209.930	0.0462859
$275$	0	0
$276$	0	0
$277$	8429.33	1.82841	0.914205	−	0.405253i	$-0.132816\pi$
$277$	8429.33	1.82841	0.914205	+	0.405253i	$0.132816\pi$
$278$	−803.593	−0.173368
$279$	0	0
$280$	0	0
$281$	3974.26	0.843717	0.421859	−	0.906662i	$-0.361378\pi$
$281$	3974.26	0.843717	0.421859	+	0.906662i	$0.361378\pi$
$282$	0	0
$283$	3072.41	0.645356	0.322678	−	0.946509i	$-0.395417\pi$
$283$	3072.41	0.645356	0.322678	+	0.946509i	$0.395417\pi$
$284$	−5777.68	−1.20719
$285$	0	0
$286$	920.333	0.190281
$287$	5287.28	1.08745
$288$	0	0
$289$	9014.97	1.83492
$290$	0	0
$291$	0	0
$292$	−3980.93	−0.797831
$293$	−3982.21	−0.794004	−0.397002	−	0.917818i	$-0.629949\pi$
$293$	−3982.21	−0.794004	−0.397002	+	0.917818i	$0.629949\pi$
$294$	0	0
$295$	0	0
$296$	638.826	0.125443
$297$	0	0
$298$	657.391	0.127791
$299$	−6603.63	−1.27725
$300$	0	0
$301$	1825.04	0.349480
$302$	−291.147	−0.0554756
$303$	0	0
$304$	7699.25	1.45257
$305$	0	0
$306$	0	0
$307$	2996.06	0.556984	0.278492	−	0.960439i	$-0.410165\pi$
$307$	2996.06	0.556984	0.278492	+	0.960439i	$0.410165\pi$
$308$	−5675.99	−1.05006
$309$	0	0
$310$	0	0
$311$	−3079.94	−0.561567	−0.280783	−	0.959771i	$-0.590594\pi$
$311$	−3079.94	−0.561567	−0.280783	+	0.959771i	$0.590594\pi$
$312$	0	0
$313$	−7953.65	−1.43632	−0.718158	−	0.695880i	$-0.755014\pi$
$313$	−7953.65	−1.43632	−0.718158	+	0.695880i	$0.755014\pi$
$314$	−835.655	−0.150187
$315$	0	0
$316$	−3154.26	−0.561523
$317$	6832.98	1.21066	0.605328	−	0.795976i	$-0.293042\pi$
$317$	6832.98	1.21066	0.605328	+	0.795976i	$0.293042\pi$
$318$	0	0
$319$	−8595.23	−1.50859
$320$	0	0
$321$	0	0
$322$	−343.615	−0.0594687
$323$	−14562.0	−2.50852
$324$	0	0
$325$	0	0
$326$	179.560	0.0305059
$327$	0	0
$328$	1501.27	0.252725
$329$	−3215.95	−0.538908
$330$	0	0
$331$	−2296.57	−0.381363	−0.190682	−	0.981652i	$-0.561070\pi$
$331$	−2296.57	−0.381363	−0.190682	+	0.981652i	$0.561070\pi$
$332$	−10864.0	−1.79591
$333$	0	0
$334$	−832.028	−0.136307
$335$	0	0
$336$	0	0
$337$	−7261.48	−1.17376	−0.586881	−	0.809673i	$-0.699645\pi$
$337$	−7261.48	−1.17376	−0.586881	+	0.809673i	$0.699645\pi$
$338$	779.488	0.125440
$339$	0	0
$340$	0	0
$341$	2281.66	0.362342
$342$	0	0
$343$	6899.32	1.08609
$344$	518.203	0.0812198
$345$	0	0
$346$	−76.9789	−0.0119607
$347$	−7425.22	−1.14872	−0.574361	−	0.818602i	$-0.694749\pi$
$347$	−7425.22	−1.14872	−0.574361	+	0.818602i	$0.694749\pi$
$348$	0	0
$349$	−478.160	−0.0733390	−0.0366695	−	0.999327i	$-0.511675\pi$
$349$	−478.160	−0.0733390	−0.0366695	+	0.999327i	$0.511675\pi$
$350$	0	0
$351$	0	0
$352$	−2420.85	−0.366567
$353$	4993.09	0.752847	0.376424	−	0.926448i	$-0.377154\pi$
$353$	4993.09	0.752847	0.376424	+	0.926448i	$0.377154\pi$
$354$	0	0
$355$	0	0
$356$	−11646.8	−1.73393
$357$	0	0
$358$	−892.590	−0.131773
$359$	−6873.09	−1.01044	−0.505219	−	0.862991i	$-0.668588\pi$
$359$	−6873.09	−1.01044	−0.505219	+	0.862991i	$0.668588\pi$
$360$	0	0
$361$	8365.87	1.21969
$362$	−799.356	−0.116059
$363$	0	0
$364$	−8312.81	−1.19700
$365$	0	0
$366$	0	0
$367$	8688.72	1.23582	0.617912	−	0.786247i	$-0.287979\pi$
$367$	8688.72	1.23582	0.617912	+	0.786247i	$0.287979\pi$
$368$	5708.70	0.808659
$369$	0	0
$370$	0	0
$371$	−198.590	−0.0277904
$372$	0	0
$373$	−3494.54	−0.485095	−0.242548	−	0.970140i	$-0.577983\pi$
$373$	−3494.54	−0.485095	−0.242548	+	0.970140i	$0.577983\pi$
$374$	1504.78	0.208048
$375$	0	0
$376$	−913.137	−0.125243
$377$	−12588.2	−1.71970
$378$	0	0
$379$	−5802.83	−0.786468	−0.393234	−	0.919438i	$-0.628644\pi$
$379$	−5802.83	−0.786468	−0.393234	+	0.919438i	$0.628644\pi$
$380$	0	0
$381$	0	0
$382$	396.377	0.0530901
$383$	3358.56	0.448080	0.224040	−	0.974580i	$-0.428075\pi$
$383$	3358.56	0.448080	0.224040	+	0.974580i	$0.428075\pi$
$384$	0	0
$385$	0	0
$386$	1343.86	0.177204
$387$	0	0
$388$	2657.76	0.347751
$389$	19.1370	0.00249430	0.00124715	−	0.999999i	$-0.499603\pi$
$389$	19.1370	0.00249430	0.00124715	+	0.999999i	$0.499603\pi$
$390$	0	0
$391$	−10797.2	−1.39651
$392$	545.122	0.0702368
$393$	0	0
$394$	518.876	0.0663467
$395$	0	0
$396$	0	0
$397$	4348.59	0.549747	0.274873	−	0.961480i	$-0.411364\pi$
$397$	4348.59	0.549747	0.274873	+	0.961480i	$0.411364\pi$
$398$	−743.715	−0.0936659
$399$	0	0
$400$	0	0
$401$	8501.61	1.05873	0.529364	−	0.848395i	$-0.322430\pi$
$401$	8501.61	1.05873	0.529364	+	0.848395i	$0.322430\pi$
$402$	0	0
$403$	3341.62	0.413047
$404$	9567.96	1.17828
$405$	0	0
$406$	−655.019	−0.0800690
$407$	7637.95	0.930219
$408$	0	0
$409$	−2810.67	−0.339801	−0.169900	−	0.985461i	$-0.554345\pi$
$409$	−2810.67	−0.339801	−0.169900	+	0.985461i	$0.554345\pi$
$410$	0	0
$411$	0	0
$412$	8417.85	1.00660
$413$	−3470.93	−0.413543
$414$	0	0
$415$	0	0
$416$	−3545.47	−0.417863
$417$	0	0
$418$	−1573.28	−0.184094
$419$	−16355.4	−1.90696	−0.953478	−	0.301461i	$-0.902526\pi$
$419$	−16355.4	−1.90696	−0.953478	+	0.301461i	$0.902526\pi$
$420$	0	0
$421$	−4510.90	−0.522204	−0.261102	−	0.965311i	$-0.584086\pi$
$421$	−4510.90	−0.522204	−0.261102	+	0.965311i	$0.584086\pi$
$422$	−711.443	−0.0820675
$423$	0	0
$424$	−56.3876	−0.00645854
$425$	0	0
$426$	0	0
$427$	791.864	0.0897447
$428$	−3772.79	−0.426086
$429$	0	0
$430$	0	0
$431$	5850.47	0.653845	0.326923	−	0.945051i	$-0.393988\pi$
$431$	5850.47	0.653845	0.326923	+	0.945051i	$0.393988\pi$
$432$	0	0
$433$	3836.82	0.425833	0.212916	−	0.977070i	$-0.431704\pi$
$433$	3836.82	0.425833	0.212916	+	0.977070i	$0.431704\pi$
$434$	173.879	0.0192314
$435$	0	0
$436$	−10475.0	−1.15060
$437$	11288.7	1.23572
$438$	0	0
$439$	16227.3	1.76421	0.882106	−	0.471052i	$-0.156125\pi$
$439$	16227.3	1.76421	0.882106	+	0.471052i	$0.156125\pi$
$440$	0	0
$441$	0	0
$442$	2203.83	0.237162
$443$	−6705.13	−0.719120	−0.359560	−	0.933122i	$-0.617073\pi$
$443$	−6705.13	−0.719120	−0.359560	+	0.933122i	$0.617073\pi$
$444$	0	0
$445$	0	0
$446$	202.760	0.0215268
$447$	0	0
$448$	7062.39	0.744792
$449$	213.100	0.0223982	0.0111991	−	0.999937i	$-0.496435\pi$
$449$	213.100	0.0223982	0.0111991	+	0.999937i	$0.496435\pi$
$450$	0	0
$451$	17949.6	1.87408
$452$	−540.837	−0.0562806
$453$	0	0
$454$	37.6854	0.00389573
$455$	0	0
$456$	0	0
$457$	−16462.1	−1.68504	−0.842520	−	0.538665i	$-0.818929\pi$
$457$	−16462.1	−1.68504	−0.842520	+	0.538665i	$0.818929\pi$
$458$	−882.680	−0.0900545
$459$	0	0
$460$	0	0
$461$	1562.06	0.157814	0.0789071	−	0.996882i	$-0.474857\pi$
$461$	1562.06	0.157814	0.0789071	+	0.996882i	$0.474857\pi$
$462$	0	0
$463$	5924.27	0.594653	0.297326	−	0.954776i	$-0.403905\pi$
$463$	5924.27	0.594653	0.297326	+	0.954776i	$0.403905\pi$
$464$	10882.2	1.08878
$465$	0	0
$466$	−34.9201	−0.00347134
$467$	17905.1	1.77420	0.887098	−	0.461582i	$-0.152718\pi$
$467$	17905.1	1.77420	0.887098	+	0.461582i	$0.152718\pi$
$468$	0	0
$469$	−1104.13	−0.108708
$470$	0	0
$471$	0	0
$472$	−985.536	−0.0961080
$473$	6195.75	0.602285
$474$	0	0
$475$	0	0
$476$	−13591.7	−1.30877
$477$	0	0
$478$	−580.893	−0.0555846
$479$	9915.44	0.945820	0.472910	−	0.881111i	$-0.343204\pi$
$479$	9915.44	0.945820	0.472910	+	0.881111i	$0.343204\pi$
$480$	0	0
$481$	11186.2	1.06039
$482$	1075.87	0.101669
$483$	0	0
$484$	−8710.24	−0.818017
$485$	0	0
$486$	0	0
$487$	−11910.8	−1.10828	−0.554138	−	0.832425i	$-0.686952\pi$
$487$	−11910.8	−1.10828	−0.554138	+	0.832425i	$0.686952\pi$
$488$	224.842	0.0208568
$489$	0	0
$490$	0	0
$491$	11063.8	1.01691	0.508453	−	0.861090i	$-0.330218\pi$
$491$	11063.8	1.01691	0.508453	+	0.861090i	$0.330218\pi$
$492$	0	0
$493$	−20582.2	−1.88027
$494$	−2304.15	−0.209856
$495$	0	0
$496$	−2888.76	−0.261510
$497$	−10573.1	−0.954261
$498$	0	0
$499$	9347.25	0.838557	0.419279	−	0.907858i	$-0.362283\pi$
$499$	9347.25	0.838557	0.419279	+	0.907858i	$0.362283\pi$
$500$	0	0
$501$	0	0
$502$	1020.91	0.0907682
$503$	19474.2	1.72627	0.863135	−	0.504973i	$-0.168498\pi$
$503$	19474.2	1.72627	0.863135	+	0.504973i	$0.168498\pi$
$504$	0	0
$505$	0	0
$506$	−1166.52	−0.102487
$507$	0	0
$508$	−4704.38	−0.410873
$509$	22164.1	1.93007	0.965035	−	0.262121i	$-0.0844221\pi$
$509$	22164.1	1.93007	0.965035	+	0.262121i	$0.0844221\pi$
$510$	0	0
$511$	−7285.06	−0.630670
$512$	5122.66	0.442172
$513$	0	0
$514$	1473.58	0.126453
$515$	0	0
$516$	0	0
$517$	−10917.7	−0.928740
$518$	582.067	0.0493717
$519$	0	0
$520$	0	0
$521$	−254.564	−0.0214062	−0.0107031	−	0.999943i	$-0.503407\pi$
$521$	−254.564	−0.0214062	−0.0107031	+	0.999943i	$0.503407\pi$
$522$	0	0
$523$	−4049.92	−0.338606	−0.169303	−	0.985564i	$-0.554152\pi$
$523$	−4049.92	−0.338606	−0.169303	+	0.985564i	$0.554152\pi$
$524$	−2688.81	−0.224163
$525$	0	0
$526$	−728.033	−0.0603493
$527$	5463.66	0.451614
$528$	0	0
$529$	−3796.89	−0.312064
$530$	0	0
$531$	0	0
$532$	14210.4	1.15808
$533$	26288.1	2.13633
$534$	0	0
$535$	0	0
$536$	−313.508	−0.0252640
$537$	0	0
$538$	51.9972	0.00416684
$539$	6517.60	0.520841
$540$	0	0
$541$	−4085.88	−0.324705	−0.162353	−	0.986733i	$-0.551908\pi$
$541$	−4085.88	−0.324705	−0.162353	+	0.986733i	$0.551908\pi$
$542$	−622.475	−0.0493314
$543$	0	0
$544$	−5796.96	−0.456880
$545$	0	0
$546$	0	0
$547$	15392.2	1.20315	0.601575	−	0.798816i	$-0.294540\pi$
$547$	15392.2	1.20315	0.601575	+	0.798816i	$0.294540\pi$
$548$	−6437.22	−0.501796
$549$	0	0
$550$	0	0
$551$	21519.1	1.66378
$552$	0	0
$553$	−5772.27	−0.443873
$554$	2180.77	0.167242
$555$	0	0
$556$	24641.1	1.87952
$557$	10897.6	0.828987	0.414493	−	0.910052i	$-0.363959\pi$
$557$	10897.6	0.828987	0.414493	+	0.910052i	$0.363959\pi$
$558$	0	0
$559$	9074.02	0.686566
$560$	0	0
$561$	0	0
$562$	1028.19	0.0771737
$563$	1551.69	0.116156	0.0580781	−	0.998312i	$-0.481503\pi$
$563$	1551.69	0.116156	0.0580781	+	0.998312i	$0.481503\pi$
$564$	0	0
$565$	0	0
$566$	794.870	0.0590298
$567$	0	0
$568$	−3002.12	−0.221772
$569$	−1246.95	−0.0918715	−0.0459357	−	0.998944i	$-0.514627\pi$
$569$	−1246.95	−0.0918715	−0.0459357	+	0.998944i	$0.514627\pi$
$570$	0	0
$571$	4196.58	0.307568	0.153784	−	0.988104i	$-0.450854\pi$
$571$	4196.58	0.307568	0.153784	+	0.988104i	$0.450854\pi$
$572$	−28220.8	−2.06288
$573$	0	0
$574$	1367.89	0.0994677
$575$	0	0
$576$	0	0
$577$	−20585.1	−1.48521	−0.742607	−	0.669728i	$-0.766411\pi$
$577$	−20585.1	−1.48521	−0.742607	+	0.669728i	$0.766411\pi$
$578$	2332.28	0.167838
$579$	0	0
$580$	0	0
$581$	−19881.1	−1.41963
$582$	0	0
$583$	−674.183	−0.0478933
$584$	−2068.52	−0.146569
$585$	0	0
$586$	−1030.25	−0.0726265
$587$	4855.78	0.341430	0.170715	−	0.985320i	$-0.445392\pi$
$587$	4855.78	0.341430	0.170715	+	0.985320i	$0.445392\pi$
$588$	0	0
$589$	−5712.37	−0.399617
$590$	0	0
$591$	0	0
$592$	−9670.25	−0.671360
$593$	23965.6	1.65961	0.829804	−	0.558055i	$-0.188452\pi$
$593$	23965.6	1.65961	0.829804	+	0.558055i	$0.188452\pi$
$594$	0	0
$595$	0	0
$596$	−20158.0	−1.38541
$597$	0	0
$598$	−1708.44	−0.116828
$599$	14229.1	0.970595	0.485297	−	0.874349i	$-0.338711\pi$
$599$	14229.1	0.970595	0.485297	+	0.874349i	$0.338711\pi$
$600$	0	0
$601$	−8877.97	−0.602562	−0.301281	−	0.953535i	$-0.597414\pi$
$601$	−8877.97	−0.602562	−0.301281	+	0.953535i	$0.597414\pi$
$602$	472.161	0.0319665
$603$	0	0
$604$	8927.64	0.601424
$605$	0	0
$606$	0	0
$607$	10876.7	0.727302	0.363651	−	0.931535i	$-0.381530\pi$
$607$	10876.7	0.727302	0.363651	+	0.931535i	$0.381530\pi$
$608$	6060.85	0.404276
$609$	0	0
$610$	0	0
$611$	−15989.5	−1.05870
$612$	0	0
$613$	19544.8	1.28778	0.643890	−	0.765118i	$-0.277320\pi$
$613$	19544.8	1.28778	0.643890	+	0.765118i	$0.277320\pi$
$614$	775.118	0.0509466
$615$	0	0
$616$	−2949.29	−0.192906
$617$	−5041.75	−0.328968	−0.164484	−	0.986380i	$-0.552596\pi$
$617$	−5041.75	−0.328968	−0.164484	+	0.986380i	$0.552596\pi$
$618$	0	0
$619$	−5208.05	−0.338173	−0.169087	−	0.985601i	$-0.554082\pi$
$619$	−5208.05	−0.338173	−0.169087	+	0.985601i	$0.554082\pi$
$620$	0	0
$621$	0	0
$622$	−796.818	−0.0513657
$623$	−21313.5	−1.37064
$624$	0	0
$625$	0	0
$626$	−2057.71	−0.131378
$627$	0	0
$628$	25624.2	1.62821
$629$	18289.9	1.15940
$630$	0	0
$631$	−20284.6	−1.27974	−0.639872	−	0.768482i	$-0.721013\pi$
$631$	−20284.6	−1.27974	−0.639872	+	0.768482i	$0.721013\pi$
$632$	−1638.98	−0.103157
$633$	0	0
$634$	1767.78	0.110737
$635$	0	0
$636$	0	0
$637$	9545.40	0.593724
$638$	−2223.69	−0.137989
$639$	0	0
$640$	0	0
$641$	−20852.4	−1.28490	−0.642449	−	0.766329i	$-0.722081\pi$
$641$	−20852.4	−1.28490	−0.642449	+	0.766329i	$0.722081\pi$
$642$	0	0
$643$	2187.22	0.134146	0.0670729	−	0.997748i	$-0.478634\pi$
$643$	2187.22	0.134146	0.0670729	+	0.997748i	$0.478634\pi$
$644$	10536.5	0.644715
$645$	0	0
$646$	−3767.37	−0.229451
$647$	−17044.1	−1.03566	−0.517831	−	0.855483i	$-0.673260\pi$
$647$	−17044.1	−1.03566	−0.517831	+	0.855483i	$0.673260\pi$
$648$	0	0
$649$	−11783.3	−0.712688
$650$	0	0
$651$	0	0
$652$	−5505.98	−0.330722
$653$	−8474.26	−0.507846	−0.253923	−	0.967224i	$-0.581721\pi$
$653$	−8474.26	−0.507846	−0.253923	+	0.967224i	$0.581721\pi$
$654$	0	0
$655$	0	0
$656$	−22725.6	−1.35257
$657$	0	0
$658$	−832.005	−0.0492932
$659$	25560.2	1.51090	0.755450	−	0.655207i	$-0.227419\pi$
$659$	25560.2	1.51090	0.755450	+	0.655207i	$0.227419\pi$
$660$	0	0
$661$	1209.59	0.0711766	0.0355883	−	0.999367i	$-0.488670\pi$
$661$	1209.59	0.0711766	0.0355883	+	0.999367i	$0.488670\pi$
$662$	−594.152	−0.0348828
$663$	0	0
$664$	−5645.03	−0.329924
$665$	0	0
$666$	0	0
$667$	15955.6	0.926241
$668$	25513.0	1.47774
$669$	0	0
$670$	0	0
$671$	2688.26	0.154664
$672$	0	0
$673$	8698.21	0.498204	0.249102	−	0.968477i	$-0.419865\pi$
$673$	8698.21	0.498204	0.249102	+	0.968477i	$0.419865\pi$
$674$	−1878.64	−0.107362
$675$	0	0
$676$	−23902.0	−1.35992
$677$	8424.49	0.478256	0.239128	−	0.970988i	$-0.423138\pi$
$677$	8424.49	0.478256	0.239128	+	0.970988i	$0.423138\pi$
$678$	0	0
$679$	4863.68	0.274891
$680$	0	0
$681$	0	0
$682$	590.293	0.0331430
$683$	17828.6	0.998817	0.499408	−	0.866367i	$-0.333551\pi$
$683$	17828.6	0.998817	0.499408	+	0.866367i	$0.333551\pi$
$684$	0	0
$685$	0	0
$686$	1784.94	0.0993431
$687$	0	0
$688$	−7844.30	−0.434682
$689$	−987.378	−0.0545952
$690$	0	0
$691$	14525.1	0.799652	0.399826	−	0.916591i	$-0.369070\pi$
$691$	14525.1	0.799652	0.399826	+	0.916591i	$0.369070\pi$
$692$	2360.45	0.129669
$693$	0	0
$694$	−1921.00	−0.105072
$695$	0	0
$696$	0	0
$697$	42982.0	2.33581
$698$	−123.706	−0.00670822
$699$	0	0
$700$	0	0
$701$	18815.5	1.01377	0.506883	−	0.862015i	$-0.330798\pi$
$701$	18815.5	1.01377	0.506883	+	0.862015i	$0.330798\pi$
$702$	0	0
$703$	−19122.4	−1.02591
$704$	23975.8	1.28355
$705$	0	0
$706$	1291.77	0.0688619
$707$	17509.3	0.931405
$708$	0	0
$709$	−12934.4	−0.685137	−0.342569	−	0.939493i	$-0.611297\pi$
$709$	−12934.4	−0.685137	−0.342569	+	0.939493i	$0.611297\pi$
$710$	0	0
$711$	0	0
$712$	−6051.75	−0.318538
$713$	−4235.51	−0.222470
$714$	0	0
$715$	0	0
$716$	27370.1	1.42859
$717$	0	0
$718$	−1778.15	−0.0924235
$719$	−8471.10	−0.439386	−0.219693	−	0.975569i	$-0.570506\pi$
$719$	−8471.10	−0.439386	−0.219693	+	0.975569i	$0.570506\pi$
$720$	0	0
$721$	15404.6	0.795695
$722$	2164.35	0.111564
$723$	0	0
$724$	24511.2	1.25822
$725$	0	0
$726$	0	0
$727$	−24369.5	−1.24321	−0.621605	−	0.783331i	$-0.713519\pi$
$727$	−24369.5	−1.24321	−0.621605	+	0.783331i	$0.713519\pi$
$728$	−4319.40	−0.219900
$729$	0	0
$730$	0	0
$731$	14836.3	0.750673
$732$	0	0
$733$	35411.8	1.78440	0.892199	−	0.451642i	$-0.149162\pi$
$733$	35411.8	1.78440	0.892199	+	0.451642i	$0.149162\pi$
$734$	2247.88	0.113039
$735$	0	0
$736$	4493.89	0.225064
$737$	−3748.37	−0.187345
$738$	0	0
$739$	−24447.0	−1.21691	−0.608456	−	0.793588i	$-0.708211\pi$
$739$	−24447.0	−1.21691	−0.608456	+	0.793588i	$0.708211\pi$
$740$	0	0
$741$	0	0
$742$	−51.3776	−0.00254195
$743$	24125.9	1.19125	0.595623	−	0.803264i	$-0.296905\pi$
$743$	24125.9	1.19125	0.595623	+	0.803264i	$0.296905\pi$
$744$	0	0
$745$	0	0
$746$	−904.081	−0.0443710
$747$	0	0
$748$	−46141.9	−2.25550
$749$	−6904.17	−0.336813
$750$	0	0
$751$	11882.4	0.577356	0.288678	−	0.957426i	$-0.406784\pi$
$751$	11882.4	0.577356	0.288678	+	0.957426i	$0.406784\pi$
$752$	13822.6	0.670292
$753$	0	0
$754$	−3256.72	−0.157298
$755$	0	0
$756$	0	0
$757$	14601.3	0.701049	0.350525	−	0.936554i	$-0.386003\pi$
$757$	14601.3	0.701049	0.350525	+	0.936554i	$0.386003\pi$
$758$	−1501.26	−0.0719372
$759$	0	0
$760$	0	0
$761$	−20296.3	−0.966809	−0.483404	−	0.875397i	$-0.660600\pi$
$761$	−20296.3	−0.966809	−0.483404	+	0.875397i	$0.660600\pi$
$762$	0	0
$763$	−19169.1	−0.909524
$764$	−12154.4	−0.575562
$765$	0	0
$766$	868.902	0.0409853
$767$	−17257.3	−0.812418
$768$	0	0
$769$	36322.0	1.70326	0.851629	−	0.524146i	$-0.175615\pi$
$769$	36322.0	1.70326	0.851629	+	0.524146i	$0.175615\pi$
$770$	0	0
$771$	0	0
$772$	−41207.7	−1.92111
$773$	−28930.9	−1.34615	−0.673073	−	0.739576i	$-0.735026\pi$
$773$	−28930.9	−1.34615	−0.673073	+	0.739576i	$0.735026\pi$
$774$	0	0
$775$	0	0
$776$	1380.99	0.0638850
$777$	0	0
$778$	4.95098	0.000228150 0
$779$	−44938.6	−2.06687
$780$	0	0
$781$	−35894.1	−1.64455
$782$	−2793.36	−0.127737
$783$	0	0
$784$	−8251.80	−0.375902
$785$	0	0
$786$	0	0
$787$	−21128.3	−0.956978	−0.478489	−	0.878094i	$-0.658815\pi$
$787$	−21128.3	−0.956978	−0.478489	+	0.878094i	$0.658815\pi$
$788$	−15910.6	−0.719280
$789$	0	0
$790$	0	0
$791$	−989.726	−0.0444887
$792$	0	0
$793$	3937.11	0.176306
$794$	1125.03	0.0502846
$795$	0	0
$796$	22805.0	1.01546
$797$	2765.87	0.122926	0.0614632	−	0.998109i	$-0.480423\pi$
$797$	2765.87	0.122926	0.0614632	+	0.998109i	$0.480423\pi$
$798$	0	0
$799$	−26143.5	−1.15756
$800$	0	0
$801$	0	0
$802$	2199.47	0.0968405
$803$	−24731.7	−1.08688
$804$	0	0
$805$	0	0
$806$	864.518	0.0377808
$807$	0	0
$808$	4971.59	0.216460
$809$	16756.6	0.728220	0.364110	−	0.931356i	$-0.381373\pi$
$809$	16756.6	0.728220	0.364110	+	0.931356i	$0.381373\pi$
$810$	0	0
$811$	17829.6	0.771987	0.385993	−	0.922502i	$-0.373859\pi$
$811$	17829.6	0.771987	0.385993	+	0.922502i	$0.373859\pi$
$812$	20085.3	0.868048
$813$	0	0
$814$	1976.03	0.0850859
$815$	0	0
$816$	0	0
$817$	−15511.7	−0.664243
$818$	−727.154	−0.0310811
$819$	0	0
$820$	0	0
$821$	−6757.48	−0.287256	−0.143628	−	0.989632i	$-0.545877\pi$
$821$	−6757.48	−0.287256	−0.143628	+	0.989632i	$0.545877\pi$
$822$	0	0
$823$	−7121.28	−0.301619	−0.150809	−	0.988563i	$-0.548188\pi$
$823$	−7121.28	−0.301619	−0.150809	+	0.988563i	$0.548188\pi$
$824$	4373.98	0.184921
$825$	0	0
$826$	−897.972	−0.0378262
$827$	1171.74	0.0492688	0.0246344	−	0.999697i	$-0.492158\pi$
$827$	1171.74	0.0492688	0.0246344	+	0.999697i	$0.492158\pi$
$828$	0	0
$829$	23617.8	0.989483	0.494742	−	0.869040i	$-0.335263\pi$
$829$	23617.8	0.989483	0.494742	+	0.869040i	$0.335263\pi$
$830$	0	0
$831$	0	0
$832$	35113.9	1.46317
$833$	15607.1	0.649163
$834$	0	0
$835$	0	0
$836$	48242.4	1.99581
$837$	0	0
$838$	−4231.35	−0.174427
$839$	−35054.3	−1.44244	−0.721222	−	0.692704i	$-0.756419\pi$
$839$	−35054.3	−1.44244	−0.721222	+	0.692704i	$0.756419\pi$
$840$	0	0
$841$	6026.42	0.247096
$842$	−1167.02	−0.0477652
$843$	0	0
$844$	21815.4	0.889714
$845$	0	0
$846$	0	0
$847$	−15939.6	−0.646627
$848$	853.568	0.0345656
$849$	0	0
$850$	0	0
$851$	−14178.6	−0.571133
$852$	0	0
$853$	32772.3	1.31548	0.657740	−	0.753245i	$-0.271513\pi$
$853$	32772.3	1.31548	0.657740	+	0.753245i	$0.271513\pi$
$854$	204.865	0.00820882
$855$	0	0
$856$	−1960.37	−0.0782758
$857$	3503.93	0.139664	0.0698319	−	0.997559i	$-0.477754\pi$
$857$	3503.93	0.139664	0.0698319	+	0.997559i	$0.477754\pi$
$858$	0	0
$859$	31044.1	1.23307	0.616537	−	0.787326i	$-0.288535\pi$
$859$	31044.1	1.23307	0.616537	+	0.787326i	$0.288535\pi$
$860$	0	0
$861$	0	0
$862$	1513.59	0.0598063
$863$	−26333.6	−1.03871	−0.519354	−	0.854559i	$-0.673827\pi$
$863$	−26333.6	−1.03871	−0.519354	+	0.854559i	$0.673827\pi$
$864$	0	0
$865$	0	0
$866$	992.632	0.0389504
$867$	0	0
$868$	−5331.76	−0.208493
$869$	−19596.0	−0.764959
$870$	0	0
$871$	−5489.71	−0.213561
$872$	−5442.87	−0.211375
$873$	0	0
$874$	2920.52	0.113030
$875$	0	0
$876$	0	0
$877$	40977.3	1.57777	0.788886	−	0.614540i	$-0.210658\pi$
$877$	40977.3	1.57777	0.788886	+	0.614540i	$0.210658\pi$
$878$	4198.21	0.161370
$879$	0	0
$880$	0	0
$881$	37022.4	1.41579	0.707897	−	0.706315i	$-0.249644\pi$
$881$	37022.4	1.41579	0.707897	+	0.706315i	$0.249644\pi$
$882$	0	0
$883$	36037.9	1.37347	0.686734	−	0.726909i	$-0.259044\pi$
$883$	36037.9	1.37347	0.686734	+	0.726909i	$0.259044\pi$
$884$	−67577.5	−2.57113
$885$	0	0
$886$	−1734.70	−0.0657769
$887$	−1465.05	−0.0554584	−0.0277292	−	0.999615i	$-0.508828\pi$
$887$	−1465.05	−0.0554584	−0.0277292	+	0.999615i	$0.508828\pi$
$888$	0	0
$889$	−8608.97	−0.324787
$890$	0	0
$891$	0	0
$892$	−6217.36	−0.233377
$893$	27333.5	1.02428
$894$	0	0
$895$	0	0
$896$	7531.87	0.280828
$897$	0	0
$898$	55.1315	0.00204873
$899$	−8073.96	−0.299535
$900$	0	0
$901$	−1614.40	−0.0596930
$902$	4643.77	0.171420
$903$	0	0
$904$	−281.023	−0.0103393
$905$	0	0
$906$	0	0
$907$	−33660.8	−1.23229	−0.616146	−	0.787632i	$-0.711307\pi$
$907$	−33660.8	−1.23229	−0.616146	+	0.787632i	$0.711307\pi$
$908$	−1155.57	−0.0422346
$909$	0	0
$910$	0	0
$911$	25992.7	0.945311	0.472655	−	0.881247i	$-0.343296\pi$
$911$	25992.7	0.945311	0.472655	+	0.881247i	$0.343296\pi$
$912$	0	0
$913$	−67493.3	−2.44655
$914$	−4258.94	−0.154128
$915$	0	0
$916$	27066.2	0.976302
$917$	−4920.50	−0.177196
$918$	0	0
$919$	1149.54	0.0412620	0.0206310	−	0.999787i	$-0.493432\pi$
$919$	1149.54	0.0412620	0.0206310	+	0.999787i	$0.493432\pi$
$920$	0	0
$921$	0	0
$922$	404.124	0.0144350
$923$	−52568.9	−1.87468
$924$	0	0
$925$	0	0
$926$	1532.68	0.0543921
$927$	0	0
$928$	8566.50	0.303027
$929$	1923.20	0.0679204	0.0339602	−	0.999423i	$-0.489188\pi$
$929$	1923.20	0.0679204	0.0339602	+	0.999423i	$0.489188\pi$
$930$	0	0
$931$	−16317.5	−0.574420
$932$	1070.78	0.0376336
$933$	0	0
$934$	4632.27	0.162283
$935$	0	0
$936$	0	0
$937$	−3511.90	−0.122443	−0.0612213	−	0.998124i	$-0.519500\pi$
$937$	−3511.90	−0.122443	−0.0612213	+	0.998124i	$0.519500\pi$
$938$	−285.653	−0.00994339
$939$	0	0
$940$	0	0
$941$	−6848.16	−0.237241	−0.118620	−	0.992940i	$-0.537847\pi$
$941$	−6848.16	−0.237241	−0.118620	+	0.992940i	$0.537847\pi$
$942$	0	0
$943$	−33320.3	−1.15064
$944$	14918.6	0.514363
$945$	0	0
$946$	1602.92	0.0550902
$947$	−48357.3	−1.65935	−0.829673	−	0.558250i	$-0.811473\pi$
$947$	−48357.3	−1.65935	−0.829673	+	0.558250i	$0.811473\pi$
$948$	0	0
$949$	−36221.0	−1.23897
$950$	0	0
$951$	0	0
$952$	−7062.36	−0.240433
$953$	−38701.1	−1.31548	−0.657740	−	0.753245i	$-0.728488\pi$
$953$	−38701.1	−1.31548	−0.657740	+	0.753245i	$0.728488\pi$
$954$	0	0
$955$	0	0
$956$	17812.3	0.602606
$957$	0	0
$958$	2565.25	0.0865129
$959$	−11780.0	−0.396660
$960$	0	0
$961$	−27647.7	−0.928056
$962$	2894.01	0.0969924
$963$	0	0
$964$	−32990.0	−1.10222
$965$	0	0
$966$	0	0
$967$	24312.7	0.808526	0.404263	−	0.914643i	$-0.367528\pi$
$967$	24312.7	0.808526	0.404263	+	0.914643i	$0.367528\pi$
$968$	−4525.91	−0.150277
$969$	0	0
$970$	0	0
$971$	−37464.3	−1.23820	−0.619098	−	0.785314i	$-0.712501\pi$
$971$	−37464.3	−1.23820	−0.619098	+	0.785314i	$0.712501\pi$
$972$	0	0
$973$	45092.9	1.48573
$974$	−3081.47	−0.101373
$975$	0	0
$976$	−3403.55	−0.111624
$977$	−3186.09	−0.104332	−0.0521659	−	0.998638i	$-0.516612\pi$
$977$	−3186.09	−0.104332	−0.0521659	+	0.998638i	$0.516612\pi$
$978$	0	0
$979$	−72356.1	−2.36212
$980$	0	0
$981$	0	0
$982$	2862.33	0.0930150
$983$	−30345.6	−0.984614	−0.492307	−	0.870422i	$-0.663846\pi$
$983$	−30345.6	−0.984614	−0.492307	+	0.870422i	$0.663846\pi$
$984$	0	0
$985$	0	0
$986$	−5324.86	−0.171986
$987$	0	0
$988$	70653.7	2.27509
$989$	−11501.3	−0.369789
$990$	0	0
$991$	3443.75	0.110388	0.0551940	−	0.998476i	$-0.482422\pi$
$991$	3443.75	0.110388	0.0551940	+	0.998476i	$0.482422\pi$
$992$	−2274.03	−0.0727828
$993$	0	0
$994$	−2735.39	−0.0872849
$995$	0	0
$996$	0	0
$997$	4567.89	0.145102	0.0725510	−	0.997365i	$-0.476886\pi$
$997$	4567.89	0.145102	0.0725510	+	0.997365i	$0.476886\pi$
$998$	2418.25	0.0767017
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.r.1.2		3
3.2	odd	2		675.4.a.q.1.2		3
5.2	odd	4		675.4.b.l.649.4		6
5.3	odd	4		675.4.b.l.649.3		6
5.4	even	2		135.4.a.f.1.2	✓	3
15.2	even	4		675.4.b.k.649.3		6
15.8	even	4		675.4.b.k.649.4		6
15.14	odd	2		135.4.a.g.1.2	yes	3
20.19	odd	2		2160.4.a.bm.1.2		3
45.4	even	6		405.4.e.t.136.2		6
45.14	odd	6		405.4.e.r.136.2		6
45.29	odd	6		405.4.e.r.271.2		6
45.34	even	6		405.4.e.t.271.2		6
60.59	even	2		2160.4.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.2	✓	3	5.4	even	2
135.4.a.g.1.2	yes	3	15.14	odd	2
405.4.e.r.136.2		6	45.14	odd	6
405.4.e.r.271.2		6	45.29	odd	6
405.4.e.t.136.2		6	45.4	even	6
405.4.e.t.271.2		6	45.34	even	6
675.4.a.q.1.2		3	3.2	odd	2
675.4.a.r.1.2		3	1.1	even	1	trivial
675.4.b.k.649.3		6	15.2	even	4
675.4.b.k.649.4		6	15.8	even	4
675.4.b.l.649.3		6	5.3	odd	4
675.4.b.l.649.4		6	5.2	odd	4
2160.4.a.be.1.2		3	60.59	even	2
2160.4.a.bm.1.2		3	20.19	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+0.258712 q^{2} -7.93307 q^{4} -14.5174 q^{7} -4.12208 q^{8} +O(q^{10})\)	\(q+0.258712 q^{2} -7.93307 q^{4} -14.5174 q^{7} -4.12208 q^{8} -49.2845 q^{11} -72.1800 q^{13} -3.75584 q^{14} +62.3981 q^{16} -118.017 q^{17} +123.389 q^{19} -12.7505 q^{22} +91.4883 q^{23} -18.6739 q^{26} +115.168 q^{28} +174.400 q^{29} -46.2956 q^{31} +49.1198 q^{32} -30.5324 q^{34} -154.977 q^{37} +31.9223 q^{38} -364.203 q^{41} -125.714 q^{43} +390.978 q^{44} +23.6692 q^{46} +221.523 q^{47} -132.244 q^{49} +572.609 q^{52} +13.6794 q^{53} +59.8420 q^{56} +45.1195 q^{58} +239.087 q^{59} -54.5457 q^{61} -11.9772 q^{62} -486.477 q^{64} +76.0558 q^{67} +936.235 q^{68} +728.303 q^{71} +501.815 q^{73} -40.0944 q^{74} -978.854 q^{76} +715.485 q^{77} +397.610 q^{79} -94.2237 q^{82} +1369.46 q^{83} -32.5237 q^{86} +203.155 q^{88} +1468.13 q^{89} +1047.87 q^{91} -725.783 q^{92} +57.3108 q^{94} -335.023 q^{97} -34.2133 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8}+O(q^{10}) \) 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8	\( 3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8} + 38 q^{11} - 28 q^{13} - 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 122 q^{22} + 81 q^{23} + 416 q^{26} - 410 q^{28} + 160 q^{29} + 227 q^{31} + 569 q^{32} + 17 q^{34} - 78 q^{37} + 757 q^{38} - 338 q^{41} - 22 q^{43} + 1636 q^{44} - 1425 q^{46} + 472 q^{47} - 197 q^{49} + 1566 q^{52} - 521 q^{53} - 1254 q^{56} + 2096 q^{58} + 140 q^{59} + 595 q^{61} - 1407 q^{62} - 918 q^{64} - 878 q^{67} + 3053 q^{68} - 602 q^{71} - 1294 q^{73} + 2878 q^{74} + 525 q^{76} - 288 q^{77} + 629 q^{79} + 1682 q^{82} + 1287 q^{83} - 3730 q^{86} + 858 q^{88} + 2154 q^{89} - 440 q^{91} - 1959 q^{92} - 1108 q^{94} - 1392 q^{97} + 2693 q^{98}+O(q^{100}) \) 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8 + 38 * q^11 - 28 * q^13 - 108 * q^14 + 191 * q^16 + 19 * q^17 + 187 * q^19 - 122 * q^22 + 81 * q^23 + 416 * q^26 - 410 * q^28 + 160 * q^29 + 227 * q^31 + 569 * q^32 + 17 * q^34 - 78 * q^37 + 757 * q^38 - 338 * q^41 - 22 * q^43 + 1636 * q^44 - 1425 * q^46 + 472 * q^47 - 197 * q^49 + 1566 * q^52 - 521 * q^53 - 1254 * q^56 + 2096 * q^58 + 140 * q^59 + 595 * q^61 - 1407 * q^62 - 918 * q^64 - 878 * q^67 + 3053 * q^68 - 602 * q^71 - 1294 * q^73 + 2878 * q^74 + 525 * q^76 - 288 * q^77 + 629 * q^79 + 1682 * q^82 + 1287 * q^83 - 3730 * q^86 + 858 * q^88 + 2154 * q^89 - 440 * q^91 - 1959 * q^92 - 1108 * q^94 - 1392 * q^97 + 2693 * q^98

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