LMFDB - Embedded newform 675.4.a.q.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:	$1$
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	$-0.514563\pi$
$2$	−0.258712	−0.0914686	−0.0457343	+	0.998954i	$0.514563\pi$
$3$	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	$-0.263950\pi$
$11$	49.2845	1.35090	0.675448	+	0.737408i	$0.263950\pi$
$12$	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	$-0.181460\pi$
$17$	118.017	1.68372	0.841861	+	0.539694i	$0.181460\pi$
$18$	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	$-0.636117\pi$
$23$	−91.4883	−0.829419	−0.414709	+	0.909954i	$0.636117\pi$
$24$	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	$-0.688572\pi$
$29$	−174.400	−1.11673	−0.558367	+	0.829594i	$0.688572\pi$
$30$	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	$-0.256004\pi$
$41$	364.203	1.38729	0.693645	+	0.720317i	$0.256004\pi$
$42$	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	$-0.611697\pi$
$47$	−221.523	−0.687499	−0.343750	+	0.939061i	$0.611697\pi$
$48$	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	$-0.505643\pi$
$53$	−13.6794	−0.0354530	−0.0177265	+	0.999843i	$0.505643\pi$
$54$	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	$-0.584970\pi$
$59$	−239.087	−0.527567	−0.263784	+	0.964582i	$0.584970\pi$
$60$	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	$-0.708304\pi$
$71$	−728.303	−1.21738	−0.608688	+	0.793410i	$0.708304\pi$
$72$	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	$-0.860526\pi$
$83$	−1369.46	−1.81106	−0.905530	+	0.424283i	$0.860526\pi$
$84$	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	$-0.838665\pi$
$89$	−1468.13	−1.74856	−0.874278	+	0.485425i	$0.838665\pi$
$90$	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	$-0.297505\pi$
$101$	1206.09	1.18822	0.594109	+	0.804384i	$0.297505\pi$
$102$	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	$-0.568923\pi$
$107$	−475.578	−0.429681	−0.214841	+	0.976649i	$0.568923\pi$
$108$	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	$-0.509034\pi$
$113$	−68.1750	−0.0567555	−0.0283777	+	0.999597i	$0.509034\pi$
$114$	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	$-0.536055\pi$
$131$	−338.937	−0.226054	−0.113027	+	0.993592i	$0.536055\pi$
$132$	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	$-0.581422\pi$
$137$	−811.442	−0.506030	−0.253015	+	0.967462i	$0.581422\pi$
$138$	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	$-0.746171\pi$
$149$	−2541.01	−1.39710	−0.698550	+	0.715561i	$0.746171\pi$
$150$	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	$-0.232400\pi$
$167$	3216.04	1.49021	0.745103	+	0.666950i	$0.232400\pi$
$168$	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	$-0.479174\pi$
$173$	297.546	0.130763	0.0653816	+	0.997860i	$0.479174\pi$
$174$	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	$-0.243995\pi$
$179$	3450.12	1.44064	0.720320	+	0.693642i	$0.243995\pi$
$180$	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	$-0.593725\pi$
$191$	−1532.11	−0.580419	−0.290209	+	0.956963i	$0.593725\pi$
$192$	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	$-0.618136\pi$
$197$	−2005.61	−0.725349	−0.362674	+	0.931916i	$0.618136\pi$
$198$	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	$-0.506779\pi$
$227$	−145.665	−0.0425909	−0.0212955	+	0.999773i	$0.506779\pi$
$228$	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	$-0.493960\pi$
$233$	134.977	0.0379511	0.0189756	+	0.999820i	$0.493960\pi$
$234$	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	$-0.401730\pi$
$239$	2245.32	0.607690	0.303845	+	0.952722i	$0.401730\pi$
$240$	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	$-0.665261\pi$
$251$	−3946.14	−0.992343	−0.496171	+	0.868225i	$0.665261\pi$
$252$	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	$-0.742935\pi$
$257$	−5695.84	−1.38248	−0.691239	+	0.722626i	$0.742935\pi$
$258$	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	$-0.392988\pi$
$263$	2814.06	0.659781	0.329891	+	0.944019i	$0.392988\pi$
$264$	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	$-0.507251\pi$
$269$	−200.985	−0.0455548	−0.0227774	+	0.999741i	$0.507251\pi$
$270$	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	$-0.638622\pi$
$281$	−3974.26	−0.843717	−0.421859	+	0.906662i	$0.638622\pi$
$282$	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	$-0.370051\pi$
$293$	3982.21	0.794004	0.397002	+	0.917818i	$0.370051\pi$
$294$	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	$-0.409406\pi$
$311$	3079.94	0.561567	0.280783	+	0.959771i	$0.409406\pi$
$312$	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	$-0.706958\pi$
$317$	−6832.98	−1.21066	−0.605328	+	0.795976i	$0.706958\pi$
$318$	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	$-0.305251\pi$
$347$	7425.22	1.14872	0.574361	+	0.818602i	$0.305251\pi$
$348$	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	$-0.622846\pi$
$353$	−4993.09	−0.752847	−0.376424	+	0.926448i	$0.622846\pi$
$354$	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	$-0.331412\pi$
$359$	6873.09	1.01044	0.505219	+	0.862991i	$0.331412\pi$
$360$	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	$-0.571925\pi$
$383$	−3358.56	−0.448080	−0.224040	+	0.974580i	$0.571925\pi$
$384$	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	$-0.500397\pi$
$389$	−19.1370	−0.00249430	−0.00124715	+	0.999999i	$0.500397\pi$
$390$	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	$-0.677570\pi$
$401$	−8501.61	−1.05873	−0.529364	+	0.848395i	$0.677570\pi$
$402$	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	$-0.0974745\pi$
$419$	16355.4	1.90696	0.953478	+	0.301461i	$0.0974745\pi$
$420$	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	$-0.606012\pi$
$431$	−5850.47	−0.653845	−0.326923	+	0.945051i	$0.606012\pi$
$432$	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	$-0.382927\pi$
$443$	6705.13	0.719120	0.359560	+	0.933122i	$0.382927\pi$
$444$	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	$-0.503565\pi$
$449$	−213.100	−0.0223982	−0.0111991	+	0.999937i	$0.503565\pi$
$450$	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	$-0.525143\pi$
$461$	−1562.06	−0.157814	−0.0789071	+	0.996882i	$0.525143\pi$
$462$	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	$-0.847282\pi$
$467$	−17905.1	−1.77420	−0.887098	+	0.461582i	$0.847282\pi$
$468$	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	$-0.656796\pi$
$479$	−9915.44	−0.945820	−0.472910	+	0.881111i	$0.656796\pi$
$480$	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	$-0.669782\pi$
$491$	−11063.8	−1.01691	−0.508453	+	0.861090i	$0.669782\pi$
$492$	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	$-0.831502\pi$
$503$	−19474.2	−1.72627	−0.863135	+	0.504973i	$0.831502\pi$
$504$	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	$-0.915578\pi$
$509$	−22164.1	−1.93007	−0.965035	+	0.262121i	$0.915578\pi$
$510$	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	$-0.496593\pi$
$521$	254.564	0.0214062	0.0107031	+	0.999943i	$0.496593\pi$
$522$	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	$-0.636041\pi$
$557$	−10897.6	−0.828987	−0.414493	+	0.910052i	$0.636041\pi$
$558$	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	$-0.518497\pi$
$563$	−1551.69	−0.116156	−0.0580781	+	0.998312i	$0.518497\pi$
$564$	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	$-0.485373\pi$
$569$	1246.95	0.0918715	0.0459357	+	0.998944i	$0.485373\pi$
$570$	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	$-0.554608\pi$
$587$	−4855.78	−0.341430	−0.170715	+	0.985320i	$0.554608\pi$
$588$	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	$-0.811548\pi$
$593$	−23965.6	−1.65961	−0.829804	+	0.558055i	$0.811548\pi$
$594$	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	$-0.661289\pi$
$599$	−14229.1	−0.970595	−0.485297	+	0.874349i	$0.661289\pi$
$600$	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	$-0.447404\pi$
$617$	5041.75	0.328968	0.164484	+	0.986380i	$0.447404\pi$
$618$	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	$-0.277919\pi$
$641$	20852.4	1.28490	0.642449	+	0.766329i	$0.277919\pi$
$642$	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	$-0.326740\pi$
$647$	17044.1	1.03566	0.517831	+	0.855483i	$0.326740\pi$
$648$	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	$-0.418279\pi$
$653$	8474.26	0.507846	0.253923	+	0.967224i	$0.418279\pi$
$654$	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	$-0.772581\pi$
$659$	−25560.2	−1.51090	−0.755450	+	0.655207i	$0.772581\pi$
$660$	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	$-0.576862\pi$
$677$	−8424.49	−0.478256	−0.239128	+	0.970988i	$0.576862\pi$
$678$	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	$-0.666449\pi$
$683$	−17828.6	−0.998817	−0.499408	+	0.866367i	$0.666449\pi$
$684$	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	$-0.669202\pi$
$701$	−18815.5	−1.01377	−0.506883	+	0.862015i	$0.669202\pi$
$702$	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	$-0.429494\pi$
$719$	8471.10	0.439386	0.219693	+	0.975569i	$0.429494\pi$
$720$	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	$-0.703095\pi$
$743$	−24125.9	−1.19125	−0.595623	+	0.803264i	$0.703095\pi$
$744$	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	$-0.339400\pi$
$761$	20296.3	0.966809	0.483404	+	0.875397i	$0.339400\pi$
$762$	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	$-0.264974\pi$
$773$	28930.9	1.34615	0.673073	+	0.739576i	$0.264974\pi$
$774$	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	$-0.519577\pi$
$797$	−2765.87	−0.122926	−0.0614632	+	0.998109i	$0.519577\pi$
$798$	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	$-0.618627\pi$
$809$	−16756.6	−0.728220	−0.364110	+	0.931356i	$0.618627\pi$
$810$	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	$-0.454123\pi$
$821$	6757.48	0.287256	0.143628	+	0.989632i	$0.454123\pi$
$822$	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	$-0.507842\pi$
$827$	−1171.74	−0.0492688	−0.0246344	+	0.999697i	$0.507842\pi$
$828$	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	$-0.243581\pi$
$839$	35054.3	1.44244	0.721222	+	0.692704i	$0.243581\pi$
$840$	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	$-0.522246\pi$
$857$	−3503.93	−0.139664	−0.0698319	+	0.997559i	$0.522246\pi$
$858$	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	$-0.326173\pi$
$863$	26333.6	1.03871	0.519354	+	0.854559i	$0.326173\pi$
$864$	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	$-0.750356\pi$
$881$	−37022.4	−1.41579	−0.707897	+	0.706315i	$0.750356\pi$
$882$	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	$-0.491172\pi$
$887$	1465.05	0.0554584	0.0277292	+	0.999615i	$0.491172\pi$
$888$	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	$-0.656704\pi$
$911$	−25992.7	−0.945311	−0.472655	+	0.881247i	$0.656704\pi$
$912$	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	$-0.510812\pi$
$929$	−1923.20	−0.0679204	−0.0339602	+	0.999423i	$0.510812\pi$
$930$	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	$-0.462153\pi$
$941$	6848.16	0.237241	0.118620	+	0.992940i	$0.462153\pi$
$942$	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	$-0.188527\pi$
$947$	48357.3	1.65935	0.829673	+	0.558250i	$0.188527\pi$
$948$	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	$-0.271512\pi$
$953$	38701.1	1.31548	0.657740	+	0.753245i	$0.271512\pi$
$954$	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	$-0.287499\pi$
$971$	37464.3	1.23820	0.619098	+	0.785314i	$0.287499\pi$
$972$	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	$-0.483388\pi$
$977$	3186.09	0.104332	0.0521659	+	0.998638i	$0.483388\pi$
$978$	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	$-0.336154\pi$
$983$	30345.6	0.984614	0.492307	+	0.870422i	$0.336154\pi$
$984$	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.q.1.2		3
3.2	odd	2		675.4.a.r.1.2		3
5.2	odd	4		675.4.b.k.649.3		6
5.3	odd	4		675.4.b.k.649.4		6
5.4	even	2		135.4.a.g.1.2	yes	3
15.2	even	4		675.4.b.l.649.4		6
15.8	even	4		675.4.b.l.649.3		6
15.14	odd	2		135.4.a.f.1.2	✓	3
20.19	odd	2		2160.4.a.be.1.2		3
45.4	even	6		405.4.e.r.136.2		6
45.14	odd	6		405.4.e.t.136.2		6
45.29	odd	6		405.4.e.t.271.2		6
45.34	even	6		405.4.e.r.271.2		6
60.59	even	2		2160.4.a.bm.1.2		3

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