LMFDB - Embedded newform 675.4.b.k.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2033649216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 47x^{4} + 541x^{2} + 36 $ x^6 + 47x^4 + 541x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$0.258712i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.k.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.258712i q^{2} +7.93307 q^{4} +14.5174i q^{7} +4.12208i q^{8} +O(q^{10})$	$q+0.258712i q^{2} +7.93307 q^{4} +14.5174i q^{7} +4.12208i q^{8} +49.2845 q^{11} -72.1800i q^{13} -3.75584 q^{14} +62.3981 q^{16} -118.017i q^{17} -123.389 q^{19} +12.7505i q^{22} -91.4883i q^{23} +18.6739 q^{26} +115.168i q^{28} +174.400 q^{29} -46.2956 q^{31} +49.1198i q^{32} +30.5324 q^{34} +154.977i q^{37} -31.9223i q^{38} +364.203 q^{41} -125.714i q^{43} +390.978 q^{44} +23.6692 q^{46} +221.523i q^{47} +132.244 q^{49} -572.609i q^{52} -13.6794i q^{53} -59.8420 q^{56} +45.1195i q^{58} +239.087 q^{59} -54.5457 q^{61} -11.9772i q^{62} +486.477 q^{64} -76.0558i q^{67} -936.235i q^{68} -728.303 q^{71} +501.815i q^{73} -40.0944 q^{74} -978.854 q^{76} +715.485i q^{77} -397.610 q^{79} +94.2237i q^{82} -1369.46i q^{83} +32.5237 q^{86} +203.155i q^{88} +1468.13 q^{89} +1047.87 q^{91} -725.783i q^{92} -57.3108 q^{94} +335.023i q^{97} +34.2133i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 46 q^{4}+O(q^{10}) $ 6 * q - 46 * q^4	$ 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) $ 6 * q - 46 * q^4 - 76 * q^11 - 216 * q^14 + 382 * q^16 - 374 * q^19 - 832 * q^26 + 320 * q^29 + 454 * q^31 - 34 * q^34 + 676 * q^41 + 3272 * q^44 - 2850 * q^46 + 394 * q^49 + 2508 * q^56 + 280 * q^59 + 1190 * q^61 + 1836 * q^64 + 1204 * q^71 + 5756 * q^74 + 1050 * q^76 - 1258 * q^79 + 7460 * q^86 + 4308 * q^89 - 880 * q^91 + 2216 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.258712i	0.0914686i	0.998954	+	0.0457343i	$0.0145628\pi$
$2$	0.258712i	0.0914686i	−0.998954	+	0.0457343i	$0.985437\pi$
$3$	0	0
$3$	0	0
$4$	7.93307	0.991633
$5$	0	0
$5$	0	0
$6$	0	0
$7$	14.5174i	0.783867i	0.919993	+	0.391934i	$0.128194\pi$
$7$	14.5174i	0.783867i	−0.919993	+	0.391934i	$0.871806\pi$
$8$	4.12208i	0.182172i
$9$	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
$13$	− 72.1800i	− 1.53993i	−0.638084	−	0.769967i	$-0.720273\pi$
$13$	− 72.1800i	− 1.53993i	0.638084	−	0.769967i	$-0.279727\pi$
$14$	−3.75584	−0.0716993
$15$	0	0
$16$	62.3981	0.974970
$17$	− 118.017i	− 1.68372i	−0.539694	−	0.841861i	$-0.681460\pi$
$17$	− 118.017i	− 1.68372i	0.539694	−	0.841861i	$-0.318540\pi$
$18$	0	0
$19$	−123.389	−1.48986	−0.744932	−	0.667141i	$-0.767518\pi$
$19$	−123.389	−1.48986	−0.744932	+	0.667141i	$0.767518\pi$
$20$	0	0
$21$	0	0
$22$	12.7505i	0.123565i
$23$	− 91.4883i	− 0.829419i	−0.909954	−	0.414709i	$-0.863883\pi$
$23$	− 91.4883i	− 0.829419i	0.909954	−	0.414709i	$-0.136117\pi$
$24$	0	0
$25$	0	0
$26$	18.6739	0.140856
$27$	0	0
$28$	115.168i	0.777309i
$29$	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	$0.542818\pi$
$32$	49.1198i	0.271351i
$33$	0	0
$34$	30.5324	0.154008
$35$	0	0
$36$	0	0
$37$	154.977i	0.688595i	0.938861	+	0.344297i	$0.111883\pi$
$37$	154.977i	0.688595i	−0.938861	+	0.344297i	$0.888117\pi$
$38$	− 31.9223i	− 0.136276i
$39$	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
$43$	− 125.714i	− 0.445841i	−0.974837	−	0.222921i	$-0.928441\pi$
$43$	− 125.714i	− 0.445841i	0.974837	−	0.222921i	$-0.0715591\pi$
$44$	390.978	1.33959
$45$	0	0
$46$	23.6692	0.0758658
$47$	221.523i	0.687499i	0.939061	+	0.343750i	$0.111697\pi$
$47$	221.523i	0.687499i	−0.939061	+	0.343750i	$0.888303\pi$
$48$	0	0
$49$	132.244	0.385552
$50$	0	0
$51$	0	0
$52$	− 572.609i	− 1.52705i
$53$	− 13.6794i	− 0.0354530i	−0.999843	−	0.0177265i	$-0.994357\pi$
$53$	− 13.6794i	− 0.0354530i	0.999843	−	0.0177265i	$-0.00564282\pi$
$54$	0	0
$55$	0	0
$56$	−59.8420	−0.142799
$57$	0	0
$58$	45.1195i	0.102146i
$59$	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	$0.518232\pi$
$62$	− 11.9772i	− 0.0245341i
$63$	0	0
$64$	486.477	0.950150
$65$	0	0
$66$	0	0
$67$	− 76.0558i	− 0.138682i	−0.997593	−	0.0693410i	$-0.977910\pi$
$67$	− 76.0558i	− 0.138682i	0.997593	−	0.0693410i	$-0.0220896\pi$
$68$	− 936.235i	− 1.66964i
$69$	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
$73$	501.815i	0.804562i	0.915516	+	0.402281i	$0.131782\pi$
$73$	501.815i	0.804562i	−0.915516	+	0.402281i	$0.868218\pi$
$74$	−40.0944	−0.0629848
$75$	0	0
$76$	−978.854	−1.47740
$77$	715.485i	1.05892i
$78$	0	0
$79$	−397.610	−0.566261	−0.283130	−	0.959081i	$-0.591373\pi$
$79$	−397.610	−0.566261	−0.283130	+	0.959081i	$0.591373\pi$
$80$	0	0
$81$	0	0
$82$	94.2237i	0.126894i
$83$	− 1369.46i	− 1.81106i	−0.424283	−	0.905530i	$-0.639474\pi$
$83$	− 1369.46i	− 1.81106i	0.424283	−	0.905530i	$-0.360526\pi$
$84$	0	0
$85$	0	0
$86$	32.5237	0.0407805
$87$	0	0
$88$	203.155i	0.246095i
$89$	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
$92$	− 725.783i	− 0.822480i
$93$	0	0
$94$	−57.3108	−0.0628846
$95$	0	0
$96$	0	0
$97$	335.023i	0.350685i	0.984507	+	0.175343i	$0.0561033\pi$
$97$	335.023i	0.350685i	−0.984507	+	0.175343i	$0.943897\pi$
$98$	34.2133i	0.0352659i
$99$	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
$103$	− 1061.11i	− 1.01509i	−0.861625	−	0.507545i	$-0.830553\pi$
$103$	− 1061.11i	− 1.01509i	0.861625	−	0.507545i	$-0.169447\pi$
$104$	297.532	0.280533
$105$	0	0
$106$	3.53903	0.00324284
$107$	475.578i	0.429681i	0.976649	+	0.214841i	$0.0689232\pi$
$107$	475.578i	0.429681i	−0.976649	+	0.214841i	$0.931077\pi$
$108$	0	0
$109$	−1320.42	−1.16030	−0.580152	−	0.814508i	$-0.697007\pi$
$109$	−1320.42	−1.16030	−0.580152	+	0.814508i	$0.697007\pi$
$110$	0	0
$111$	0	0
$112$	905.860i	0.764247i
$113$	− 68.1750i	− 0.0567555i	−0.999597	−	0.0283777i	$-0.990966\pi$
$113$	− 68.1750i	− 0.0567555i	0.999597	−	0.0283777i	$-0.00903412\pi$
$114$	0	0
$115$	0	0
$116$	1383.53	1.10739
$117$	0	0
$118$	61.8547i	0.0482559i
$119$	1713.30	1.31981
$120$	0	0
$121$	1097.97	0.824918
$122$	− 14.1117i	− 0.0104722i
$123$	0	0
$124$	−367.266	−0.265980
$125$	0	0
$126$	0	0
$127$	− 593.009i	− 0.414339i	−0.978305	−	0.207170i	$-0.933575\pi$
$127$	− 593.009i	− 0.414339i	0.978305	−	0.207170i	$-0.0664252\pi$
$128$	518.816i	0.358260i
$129$	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
$133$	− 1791.29i	− 1.16785i
$134$	19.6766	0.0126850
$135$	0	0
$136$	486.475	0.306727
$137$	811.442i	0.506030i	0.967462	+	0.253015i	$0.0814222\pi$
$137$	811.442i	0.506030i	−0.967462	+	0.253015i	$0.918578\pi$
$138$	0	0
$139$	3106.13	1.89538	0.947691	−	0.319189i	$-0.103410\pi$
$139$	3106.13	1.89538	0.947691	+	0.319189i	$0.103410\pi$
$140$	0	0
$141$	0	0
$142$	− 188.421i	− 0.111352i
$143$	− 3557.36i	− 2.08029i
$144$	0	0
$145$	0	0
$146$	−129.826	−0.0735922
$147$	0	0
$148$	1229.44i	0.682834i
$149$	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	$0.598072\pi$
$152$	− 508.620i	− 0.271411i
$153$	0	0
$154$	−185.105	−0.0968582
$155$	0	0
$156$	0	0
$157$	3230.05i	1.64195i	0.570963	+	0.820975i	$0.306570\pi$
$157$	3230.05i	1.64195i	−0.570963	+	0.820975i	$0.693430\pi$
$158$	− 102.867i	− 0.0517951i
$159$	0	0
$160$	0	0
$161$	1328.17	0.650154
$162$	0	0
$163$	694.054i	0.333512i	0.985998	+	0.166756i	$0.0533293\pi$
$163$	694.054i	0.333512i	−0.985998	+	0.166756i	$0.946671\pi$
$164$	2889.24	1.37568
$165$	0	0
$166$	354.297	0.165655
$167$	− 3216.04i	− 1.49021i	−0.666950	−	0.745103i	$-0.732400\pi$
$167$	− 3216.04i	− 1.49021i	0.666950	−	0.745103i	$-0.267600\pi$
$168$	0	0
$169$	−3012.95	−1.37139
$170$	0	0
$171$	0	0
$172$	− 997.296i	− 0.442111i
$173$	297.546i	0.130763i	0.997860	+	0.0653816i	$0.0208265\pi$
$173$	297.546i	0.130763i	−0.997860	+	0.0653816i	$0.979174\pi$
$174$	0	0
$175$	0	0
$176$	3075.26	1.31708
$177$	0	0
$178$	379.823i	0.159938i
$179$	−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	$0.718760\pi$
$182$	271.096i	0.110412i
$183$	0	0
$184$	377.122	0.151097
$185$	0	0
$186$	0	0
$187$	− 5816.41i	− 2.27453i
$188$	1757.36i	0.681748i
$189$	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
$193$	5194.42i	1.93732i	0.248389	+	0.968660i	$0.420099\pi$
$193$	5194.42i	1.93732i	−0.248389	+	0.968660i	$0.579901\pi$
$194$	−86.6747	−0.0320767
$195$	0	0
$196$	1049.10	0.382326
$197$	2005.61i	0.725349i	0.931916	+	0.362674i	$0.118136\pi$
$197$	2005.61i	0.725349i	−0.931916	+	0.362674i	$0.881864\pi$
$198$	0	0
$199$	2874.68	1.02402	0.512011	−	0.858979i	$-0.328901\pi$
$199$	2874.68	1.02402	0.512011	+	0.858979i	$0.328901\pi$
$200$	0	0
$201$	0	0
$202$	312.029i	0.108685i
$203$	2531.84i	0.875372i
$204$	0	0
$205$	0	0
$206$	274.522	0.0928488
$207$	0	0
$208$	− 4503.90i	− 1.50139i
$209$	−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	$0.648081\pi$
$212$	− 108.520i	− 0.0351564i
$213$	0	0
$214$	−123.038	−0.0393023
$215$	0	0
$216$	0	0
$217$	− 672.093i	− 0.210252i
$218$	− 341.609i	− 0.106131i
$219$	0	0
$220$	0	0
$221$	−8518.45	−2.59282
$222$	0	0
$223$	783.727i	0.235346i	0.993052	+	0.117673i	$0.0375435\pi$
$223$	783.727i	0.235346i	−0.993052	+	0.117673i	$0.962456\pi$
$224$	−713.093	−0.212703
$225$	0	0
$226$	17.6377	0.00519134
$227$	145.665i	0.0425909i	0.999773	+	0.0212955i	$0.00677907\pi$
$227$	145.665i	0.0425909i	−0.999773	+	0.0212955i	$0.993221\pi$
$228$	0	0
$229$	3411.82	0.984539	0.492270	−	0.870443i	$-0.336167\pi$
$229$	3411.82	0.984539	0.492270	+	0.870443i	$0.336167\pi$
$230$	0	0
$231$	0	0
$232$	718.892i	0.203438i
$233$	134.977i	0.0379511i	0.999820	+	0.0189756i	$0.00604047\pi$
$233$	134.977i	0.0379511i	−0.999820	+	0.0189756i	$0.993960\pi$
$234$	0	0
$235$	0	0
$236$	1896.69	0.523153
$237$	0	0
$238$	443.252i	0.120722i
$239$	−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	$0.312428\pi$
$242$	284.057i	0.0754542i
$243$	0	0
$244$	−432.715	−0.113532
$245$	0	0
$246$	0	0
$247$	8906.22i	2.29429i
$248$	− 190.834i	− 0.0488629i
$249$	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
$253$	− 4508.96i	− 1.12046i
$254$	153.419	0.0378990
$255$	0	0
$256$	3757.59	0.917381
$257$	5695.84i	1.38248i	0.722626	+	0.691239i	$0.242935\pi$
$257$	5695.84i	1.38248i	−0.722626	+	0.691239i	$0.757065\pi$
$258$	0	0
$259$	−2249.86	−0.539767
$260$	0	0
$261$	0	0
$262$	− 87.6873i	− 0.0206769i
$263$	2814.06i	0.659781i	0.944019	+	0.329891i	$0.107012\pi$
$263$	2814.06i	0.659781i	−0.944019	+	0.329891i	$0.892988\pi$
$264$	0	0
$265$	0	0
$266$	463.429	0.106822
$267$	0	0
$268$	− 603.356i	− 0.137522i
$269$	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	$0.586912\pi$
$272$	− 7364.03i	− 1.64158i
$273$	0	0
$274$	−209.930	−0.0462859
$275$	0	0
$276$	0	0
$277$	− 8429.33i	− 1.82841i	−0.405253	−	0.914205i	$-0.632816\pi$
$277$	− 8429.33i	− 1.82841i	0.405253	−	0.914205i	$-0.367184\pi$
$278$	803.593i	0.173368i
$279$	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
$283$	3072.41i	0.645356i	0.946509	+	0.322678i	$0.104583\pi$
$283$	3072.41i	0.645356i	−0.946509	+	0.322678i	$0.895417\pi$
$284$	−5777.68	−1.20719
$285$	0	0
$286$	920.333	0.190281
$287$	5287.28i	1.08745i
$288$	0	0
$289$	−9014.97	−1.83492
$290$	0	0
$291$	0	0
$292$	3980.93i	0.797831i
$293$	3982.21i	0.794004i	0.917818	+	0.397002i	$0.129949\pi$
$293$	3982.21i	0.794004i	−0.917818	+	0.397002i	$0.870051\pi$
$294$	0	0
$295$	0	0
$296$	−638.826	−0.125443
$297$	0	0
$298$	657.391i	0.127791i
$299$	−6603.63	−1.27725
$300$	0	0
$301$	1825.04	0.349480
$302$	− 291.147i	− 0.0554756i
$303$	0	0
$304$	−7699.25	−1.45257
$305$	0	0
$306$	0	0
$307$	− 2996.06i	− 0.556984i	−0.960439	−	0.278492i	$-0.910165\pi$
$307$	− 2996.06i	− 0.556984i	0.960439	−	0.278492i	$-0.0898346\pi$
$308$	5675.99i	1.05006i
$309$	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
$313$	− 7953.65i	− 1.43632i	−0.695880	−	0.718158i	$-0.744986\pi$
$313$	− 7953.65i	− 1.43632i	0.695880	−	0.718158i	$-0.255014\pi$
$314$	−835.655	−0.150187
$315$	0	0
$316$	−3154.26	−0.561523
$317$	6832.98i	1.21066i	0.795976	+	0.605328i	$0.206958\pi$
$317$	6832.98i	1.21066i	−0.795976	+	0.605328i	$0.793042\pi$
$318$	0	0
$319$	8595.23	1.50859
$320$	0	0
$321$	0	0
$322$	343.615i	0.0594687i
$323$	14562.0i	2.50852i
$324$	0	0
$325$	0	0
$326$	−179.560	−0.0305059
$327$	0	0
$328$	1501.27i	0.252725i
$329$	−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	$0.561070\pi$
$332$	− 10864.0i	− 1.79591i
$333$	0	0
$334$	832.028	0.136307
$335$	0	0
$336$	0	0
$337$	7261.48i	1.17376i	0.809673	+	0.586881i	$0.199645\pi$
$337$	7261.48i	1.17376i	−0.809673	+	0.586881i	$0.800355\pi$
$338$	− 779.488i	− 0.125440i
$339$	0	0
$340$	0	0
$341$	−2281.66	−0.362342
$342$	0	0
$343$	6899.32i	1.08609i
$344$	518.203	0.0812198
$345$	0	0
$346$	−76.9789	−0.0119607
$347$	− 7425.22i	− 1.14872i	−0.818602	−	0.574361i	$-0.805251\pi$
$347$	− 7425.22i	− 1.14872i	0.818602	−	0.574361i	$-0.194749\pi$
$348$	0	0
$349$	478.160	0.0733390	0.0366695	−	0.999327i	$-0.488325\pi$
$349$	478.160	0.0733390	0.0366695	+	0.999327i	$0.488325\pi$
$350$	0	0
$351$	0	0
$352$	2420.85i	0.366567i
$353$	− 4993.09i	− 0.752847i	−0.926448	−	0.376424i	$-0.877154\pi$
$353$	− 4993.09i	− 0.752847i	0.926448	−	0.376424i	$-0.122846\pi$
$354$	0	0
$355$	0	0
$356$	11646.8	1.73393
$357$	0	0
$358$	− 892.590i	− 0.131773i
$359$	−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
$362$	− 799.356i	− 0.116059i
$363$	0	0
$364$	8312.81	1.19700
$365$	0	0
$366$	0	0
$367$	− 8688.72i	− 1.23582i	−0.786247	−	0.617912i	$-0.787979\pi$
$367$	− 8688.72i	− 1.23582i	0.786247	−	0.617912i	$-0.212021\pi$
$368$	− 5708.70i	− 0.808659i
$369$	0	0
$370$	0	0
$371$	198.590	0.0277904
$372$	0	0
$373$	− 3494.54i	− 0.485095i	−0.970140	−	0.242548i	$-0.922017\pi$
$373$	− 3494.54i	− 0.485095i	0.970140	−	0.242548i	$-0.0779830\pi$
$374$	1504.78	0.208048
$375$	0	0
$376$	−913.137	−0.125243
$377$	− 12588.2i	− 1.71970i
$378$	0	0
$379$	5802.83	0.786468	0.393234	−	0.919438i	$-0.371356\pi$
$379$	5802.83	0.786468	0.393234	+	0.919438i	$0.371356\pi$
$380$	0	0
$381$	0	0
$382$	− 396.377i	− 0.0530901i
$383$	− 3358.56i	− 0.448080i	−0.974580	−	0.224040i	$-0.928075\pi$
$383$	− 3358.56i	− 0.448080i	0.974580	−	0.224040i	$-0.0719246\pi$
$384$	0	0
$385$	0	0
$386$	−1343.86	−0.177204
$387$	0	0
$388$	2657.76i	0.347751i
$389$	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
$392$	545.122i	0.0702368i
$393$	0	0
$394$	−518.876	−0.0663467
$395$	0	0
$396$	0	0
$397$	− 4348.59i	− 0.549747i	−0.961480	−	0.274873i	$-0.911364\pi$
$397$	− 4348.59i	− 0.549747i	0.961480	−	0.274873i	$-0.0886359\pi$
$398$	743.715i	0.0936659i
$399$	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
$403$	3341.62i	0.413047i
$404$	9567.96	1.17828
$405$	0	0
$406$	−655.019	−0.0800690
$407$	7637.95i	0.930219i
$408$	0	0
$409$	2810.67	0.339801	0.169900	−	0.985461i	$-0.445655\pi$
$409$	2810.67	0.339801	0.169900	+	0.985461i	$0.445655\pi$
$410$	0	0
$411$	0	0
$412$	− 8417.85i	− 1.00660i
$413$	3470.93i	0.413543i
$414$	0	0
$415$	0	0
$416$	3545.47	0.417863
$417$	0	0
$418$	− 1573.28i	− 0.184094i
$419$	−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	$0.584086\pi$
$422$	− 711.443i	− 0.0820675i
$423$	0	0
$424$	56.3876	0.00645854
$425$	0	0
$426$	0	0
$427$	− 791.864i	− 0.0897447i
$428$	3772.79i	0.426086i
$429$	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
$433$	3836.82i	0.425833i	0.977070	+	0.212916i	$0.0682962\pi$
$433$	3836.82i	0.425833i	−0.977070	+	0.212916i	$0.931704\pi$
$434$	173.879	0.0192314
$435$	0	0
$436$	−10475.0	−1.15060
$437$	11288.7i	1.23572i
$438$	0	0
$439$	−16227.3	−1.76421	−0.882106	−	0.471052i	$-0.843875\pi$
$439$	−16227.3	−1.76421	−0.882106	+	0.471052i	$0.843875\pi$
$440$	0	0
$441$	0	0
$442$	− 2203.83i	− 0.237162i
$443$	6705.13i	0.719120i	0.933122	+	0.359560i	$0.117073\pi$
$443$	6705.13i	0.719120i	−0.933122	+	0.359560i	$0.882927\pi$
$444$	0	0
$445$	0	0
$446$	−202.760	−0.0215268
$447$	0	0
$448$	7062.39i	0.744792i
$449$	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
$452$	− 540.837i	− 0.0562806i
$453$	0	0
$454$	−37.6854	−0.00389573
$455$	0	0
$456$	0	0
$457$	16462.1i	1.68504i	0.538665	+	0.842520i	$0.318929\pi$
$457$	16462.1i	1.68504i	−0.538665	+	0.842520i	$0.681071\pi$
$458$	882.680i	0.0900545i
$459$	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
$463$	5924.27i	0.594653i	0.954776	+	0.297326i	$0.0960950\pi$
$463$	5924.27i	0.594653i	−0.954776	+	0.297326i	$0.903905\pi$
$464$	10882.2	1.08878
$465$	0	0
$466$	−34.9201	−0.00347134
$467$	17905.1i	1.77420i	0.461582	+	0.887098i	$0.347282\pi$
$467$	17905.1i	1.77420i	−0.461582	+	0.887098i	$0.652718\pi$
$468$	0	0
$469$	1104.13	0.108708
$470$	0	0
$471$	0	0
$472$	985.536i	0.0961080i
$473$	− 6195.75i	− 0.602285i
$474$	0	0
$475$	0	0
$476$	13591.7	1.30877
$477$	0	0
$478$	− 580.893i	− 0.0555846i
$479$	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
$482$	1075.87i	0.101669i
$483$	0	0
$484$	8710.24	0.818017
$485$	0	0
$486$	0	0
$487$	11910.8i	1.10828i	0.832425	+	0.554138i	$0.186952\pi$
$487$	11910.8i	1.10828i	−0.832425	+	0.554138i	$0.813048\pi$
$488$	− 224.842i	− 0.0208568i
$489$	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
$493$	− 20582.2i	− 1.88027i
$494$	−2304.15	−0.209856
$495$	0	0
$496$	−2888.76	−0.261510
$497$	− 10573.1i	− 0.954261i
$498$	0	0
$499$	−9347.25	−0.838557	−0.419279	−	0.907858i	$-0.637717\pi$
$499$	−9347.25	−0.838557	−0.419279	+	0.907858i	$0.637717\pi$
$500$	0	0
$501$	0	0
$502$	− 1020.91i	− 0.0907682i
$503$	− 19474.2i	− 1.72627i	−0.504973	−	0.863135i	$-0.668498\pi$
$503$	− 19474.2i	− 1.72627i	0.504973	−	0.863135i	$-0.331502\pi$
$504$	0	0
$505$	0	0
$506$	1166.52	0.102487
$507$	0	0
$508$	− 4704.38i	− 0.410873i
$509$	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
$512$	5122.66i	0.442172i
$513$	0	0
$514$	−1473.58	−0.126453
$515$	0	0
$516$	0	0
$517$	10917.7i	0.928740i
$518$	− 582.067i	− 0.0493717i
$519$	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
$523$	− 4049.92i	− 0.338606i	−0.985564	−	0.169303i	$-0.945848\pi$
$523$	− 4049.92i	− 0.338606i	0.985564	−	0.169303i	$-0.0541516\pi$
$524$	−2688.81	−0.224163
$525$	0	0
$526$	−728.033	−0.0603493
$527$	5463.66i	0.451614i
$528$	0	0
$529$	3796.89	0.312064
$530$	0	0
$531$	0	0
$532$	− 14210.4i	− 1.15808i
$533$	− 26288.1i	− 2.13633i
$534$	0	0
$535$	0	0
$536$	313.508	0.0252640
$537$	0	0
$538$	51.9972i	0.00416684i
$539$	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	$0.551908\pi$
$542$	− 622.475i	− 0.0493314i
$543$	0	0
$544$	5796.96	0.456880
$545$	0	0
$546$	0	0
$547$	− 15392.2i	− 1.20315i	−0.798816	−	0.601575i	$-0.794540\pi$
$547$	− 15392.2i	− 1.20315i	0.798816	−	0.601575i	$-0.205460\pi$
$548$	6437.22i	0.501796i
$549$	0	0
$550$	0	0
$551$	−21519.1	−1.66378
$552$	0	0
$553$	− 5772.27i	− 0.443873i
$554$	2180.77	0.167242
$555$	0	0
$556$	24641.1	1.87952
$557$	10897.6i	0.828987i	0.910052	+	0.414493i	$0.136041\pi$
$557$	10897.6i	0.828987i	−0.910052	+	0.414493i	$0.863959\pi$
$558$	0	0
$559$	−9074.02	−0.686566
$560$	0	0
$561$	0	0
$562$	− 1028.19i	− 0.0771737i
$563$	− 1551.69i	− 0.116156i	−0.998312	−	0.0580781i	$-0.981503\pi$
$563$	− 1551.69i	− 0.116156i	0.998312	−	0.0580781i	$-0.0184972\pi$
$564$	0	0
$565$	0	0
$566$	−794.870	−0.0590298
$567$	0	0
$568$	− 3002.12i	− 0.221772i
$569$	−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	$0.450854\pi$
$572$	− 28220.8i	− 2.06288i
$573$	0	0
$574$	−1367.89	−0.0994677
$575$	0	0
$576$	0	0
$577$	20585.1i	1.48521i	0.669728	+	0.742607i	$0.266411\pi$
$577$	20585.1i	1.48521i	−0.669728	+	0.742607i	$0.733589\pi$
$578$	− 2332.28i	− 0.167838i
$579$	0	0
$580$	0	0
$581$	19881.1	1.41963
$582$	0	0
$583$	− 674.183i	− 0.0478933i
$584$	−2068.52	−0.146569
$585$	0	0
$586$	−1030.25	−0.0726265
$587$	4855.78i	0.341430i	0.985320	+	0.170715i	$0.0546077\pi$
$587$	4855.78i	0.341430i	−0.985320	+	0.170715i	$0.945392\pi$
$588$	0	0
$589$	5712.37	0.399617
$590$	0	0
$591$	0	0
$592$	9670.25i	0.671360i
$593$	− 23965.6i	− 1.65961i	−0.558055	−	0.829804i	$-0.688452\pi$
$593$	− 23965.6i	− 1.65961i	0.558055	−	0.829804i	$-0.311548\pi$
$594$	0	0
$595$	0	0
$596$	20158.0	1.38541
$597$	0	0
$598$	− 1708.44i	− 0.116828i
$599$	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	$0.597414\pi$
$602$	472.161i	0.0319665i
$603$	0	0
$604$	−8927.64	−0.601424
$605$	0	0
$606$	0	0
$607$	− 10876.7i	− 0.727302i	−0.931535	−	0.363651i	$-0.881530\pi$
$607$	− 10876.7i	− 0.727302i	0.931535	−	0.363651i	$-0.118470\pi$
$608$	− 6060.85i	− 0.404276i
$609$	0	0
$610$	0	0
$611$	15989.5	1.05870
$612$	0	0
$613$	19544.8i	1.28778i	0.765118	+	0.643890i	$0.222680\pi$
$613$	19544.8i	1.28778i	−0.765118	+	0.643890i	$0.777320\pi$
$614$	775.118	0.0509466
$615$	0	0
$616$	−2949.29	−0.192906
$617$	− 5041.75i	− 0.328968i	−0.986380	−	0.164484i	$-0.947404\pi$
$617$	− 5041.75i	− 0.328968i	0.986380	−	0.164484i	$-0.0525959\pi$
$618$	0	0
$619$	5208.05	0.338173	0.169087	−	0.985601i	$-0.445918\pi$
$619$	5208.05	0.338173	0.169087	+	0.985601i	$0.445918\pi$
$620$	0	0
$621$	0	0
$622$	796.818i	0.0513657i
$623$	21313.5i	1.37064i
$624$	0	0
$625$	0	0
$626$	2057.71	0.131378
$627$	0	0
$628$	25624.2i	1.62821i
$629$	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	$0.721013\pi$
$632$	− 1638.98i	− 0.103157i
$633$	0	0
$634$	−1767.78	−0.110737
$635$	0	0
$636$	0	0
$637$	− 9545.40i	− 0.593724i
$638$	2223.69i	0.137989i
$639$	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
$643$	2187.22i	0.134146i	0.997748	+	0.0670729i	$0.0213660\pi$
$643$	2187.22i	0.134146i	−0.997748	+	0.0670729i	$0.978634\pi$
$644$	10536.5	0.644715
$645$	0	0
$646$	−3767.37	−0.229451
$647$	− 17044.1i	− 1.03566i	−0.855483	−	0.517831i	$-0.826740\pi$
$647$	− 17044.1i	− 1.03566i	0.855483	−	0.517831i	$-0.173260\pi$
$648$	0	0
$649$	11783.3	0.712688
$650$	0	0
$651$	0	0
$652$	5505.98i	0.330722i
$653$	8474.26i	0.507846i	0.967224	+	0.253923i	$0.0817210\pi$
$653$	8474.26i	0.507846i	−0.967224	+	0.253923i	$0.918279\pi$
$654$	0	0
$655$	0	0
$656$	22725.6	1.35257
$657$	0	0
$658$	− 832.005i	− 0.0492932i
$659$	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	$0.488670\pi$
$662$	− 594.152i	− 0.0348828i
$663$	0	0
$664$	5645.03	0.329924
$665$	0	0
$666$	0	0
$667$	− 15955.6i	− 0.926241i
$668$	− 25513.0i	− 1.47774i
$669$	0	0
$670$	0	0
$671$	−2688.26	−0.154664
$672$	0	0
$673$	8698.21i	0.498204i	0.968477	+	0.249102i	$0.0801355\pi$
$673$	8698.21i	0.498204i	−0.968477	+	0.249102i	$0.919865\pi$
$674$	−1878.64	−0.107362
$675$	0	0
$676$	−23902.0	−1.35992
$677$	8424.49i	0.478256i	0.970988	+	0.239128i	$0.0768616\pi$
$677$	8424.49i	0.478256i	−0.970988	+	0.239128i	$0.923138\pi$
$678$	0	0
$679$	−4863.68	−0.274891
$680$	0	0
$681$	0	0
$682$	− 590.293i	− 0.0331430i
$683$	− 17828.6i	− 0.998817i	−0.866367	−	0.499408i	$-0.833551\pi$
$683$	− 17828.6i	− 0.998817i	0.866367	−	0.499408i	$-0.166449\pi$
$684$	0	0
$685$	0	0
$686$	−1784.94	−0.0993431
$687$	0	0
$688$	− 7844.30i	− 0.434682i
$689$	−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	$0.369070\pi$
$692$	2360.45i	0.129669i
$693$	0	0
$694$	1921.00	0.105072
$695$	0	0
$696$	0	0
$697$	− 42982.0i	− 2.33581i
$698$	123.706i	0.00670822i
$699$	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
$703$	− 19122.4i	− 1.02591i
$704$	23975.8	1.28355
$705$	0	0
$706$	1291.77	0.0688619
$707$	17509.3i	0.931405i
$708$	0	0
$709$	12934.4	0.685137	0.342569	−	0.939493i	$-0.388703\pi$
$709$	12934.4	0.685137	0.342569	+	0.939493i	$0.388703\pi$
$710$	0	0
$711$	0	0
$712$	6051.75i	0.318538i
$713$	4235.51i	0.222470i
$714$	0	0
$715$	0	0
$716$	−27370.1	−1.42859
$717$	0	0
$718$	− 1778.15i	− 0.0924235i
$719$	−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
$722$	2164.35i	0.111564i
$723$	0	0
$724$	−24511.2	−1.25822
$725$	0	0
$726$	0	0
$727$	24369.5i	1.24321i	0.783331	+	0.621605i	$0.213519\pi$
$727$	24369.5i	1.24321i	−0.783331	+	0.621605i	$0.786481\pi$
$728$	4319.40i	0.219900i
$729$	0	0
$730$	0	0
$731$	−14836.3	−0.750673
$732$	0	0
$733$	35411.8i	1.78440i	0.451642	+	0.892199i	$0.350838\pi$
$733$	35411.8i	1.78440i	−0.451642	+	0.892199i	$0.649162\pi$
$734$	2247.88	0.113039
$735$	0	0
$736$	4493.89	0.225064
$737$	− 3748.37i	− 0.187345i
$738$	0	0
$739$	24447.0	1.21691	0.608456	−	0.793588i	$-0.291789\pi$
$739$	24447.0	1.21691	0.608456	+	0.793588i	$0.291789\pi$
$740$	0	0
$741$	0	0
$742$	51.3776i	0.00254195i
$743$	− 24125.9i	− 1.19125i	−0.803264	−	0.595623i	$-0.796905\pi$
$743$	− 24125.9i	− 1.19125i	0.803264	−	0.595623i	$-0.203095\pi$
$744$	0	0
$745$	0	0
$746$	904.081	0.0443710
$747$	0	0
$748$	− 46141.9i	− 2.25550i
$749$	−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	$0.406784\pi$
$752$	13822.6i	0.670292i
$753$	0	0
$754$	3256.72	0.157298
$755$	0	0
$756$	0	0
$757$	− 14601.3i	− 0.701049i	−0.936554	−	0.350525i	$-0.886003\pi$
$757$	− 14601.3i	− 0.701049i	0.936554	−	0.350525i	$-0.113997\pi$
$758$	1501.26i	0.0719372i
$759$	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
$763$	− 19169.1i	− 0.909524i
$764$	−12154.4	−0.575562
$765$	0	0
$766$	868.902	0.0409853
$767$	− 17257.3i	− 0.812418i
$768$	0	0
$769$	−36322.0	−1.70326	−0.851629	−	0.524146i	$-0.824385\pi$
$769$	−36322.0	−1.70326	−0.851629	+	0.524146i	$0.824385\pi$
$770$	0	0
$771$	0	0
$772$	41207.7i	1.92111i
$773$	28930.9i	1.34615i	0.739576	+	0.673073i	$0.235026\pi$
$773$	28930.9i	1.34615i	−0.739576	+	0.673073i	$0.764974\pi$
$774$	0	0
$775$	0	0
$776$	−1380.99	−0.0638850
$777$	0	0
$778$	4.95098i	0 0.000228150i
$779$	−44938.6	−2.06687
$780$	0	0
$781$	−35894.1	−1.64455
$782$	− 2793.36i	− 0.127737i
$783$	0	0
$784$	8251.80	0.375902
$785$	0	0
$786$	0	0
$787$	21128.3i	0.956978i	0.878094	+	0.478489i	$0.158815\pi$
$787$	21128.3i	0.956978i	−0.878094	+	0.478489i	$0.841185\pi$
$788$	15910.6i	0.719280i
$789$	0	0
$790$	0	0
$791$	989.726	0.0444887
$792$	0	0
$793$	3937.11i	0.176306i
$794$	1125.03	0.0502846
$795$	0	0
$796$	22805.0	1.01546
$797$	2765.87i	0.122926i	0.998109	+	0.0614632i	$0.0195767\pi$
$797$	2765.87i	0.122926i	−0.998109	+	0.0614632i	$0.980423\pi$
$798$	0	0
$799$	26143.5	1.15756
$800$	0	0
$801$	0	0
$802$	− 2199.47i	− 0.0968405i
$803$	24731.7i	1.08688i
$804$	0	0
$805$	0	0
$806$	−864.518	−0.0377808
$807$	0	0
$808$	4971.59i	0.216460i
$809$	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	$0.373859\pi$
$812$	20085.3i	0.868048i
$813$	0	0
$814$	−1976.03	−0.0850859
$815$	0	0
$816$	0	0
$817$	15511.7i	0.664243i
$818$	727.154i	0.0310811i
$819$	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
$823$	− 7121.28i	− 0.301619i	−0.988563	−	0.150809i	$-0.951812\pi$
$823$	− 7121.28i	− 0.301619i	0.988563	−	0.150809i	$-0.0481880\pi$
$824$	4373.98	0.184921
$825$	0	0
$826$	−897.972	−0.0378262
$827$	1171.74i	0.0492688i	0.999697	+	0.0246344i	$0.00784217\pi$
$827$	1171.74i	0.0492688i	−0.999697	+	0.0246344i	$0.992158\pi$
$828$	0	0
$829$	−23617.8	−0.989483	−0.494742	−	0.869040i	$-0.664737\pi$
$829$	−23617.8	−0.989483	−0.494742	+	0.869040i	$0.664737\pi$
$830$	0	0
$831$	0	0
$832$	− 35113.9i	− 1.46317i
$833$	− 15607.1i	− 0.649163i
$834$	0	0
$835$	0	0
$836$	−48242.4	−1.99581
$837$	0	0
$838$	− 4231.35i	− 0.174427i
$839$	−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
$842$	− 1167.02i	− 0.0477652i
$843$	0	0
$844$	−21815.4	−0.889714
$845$	0	0
$846$	0	0
$847$	15939.6i	0.646627i
$848$	− 853.568i	− 0.0345656i
$849$	0	0
$850$	0	0
$851$	14178.6	0.571133
$852$	0	0
$853$	32772.3i	1.31548i	0.753245	+	0.657740i	$0.228487\pi$
$853$	32772.3i	1.31548i	−0.753245	+	0.657740i	$0.771513\pi$
$854$	204.865	0.00820882
$855$	0	0
$856$	−1960.37	−0.0782758
$857$	3503.93i	0.139664i	0.997559	+	0.0698319i	$0.0222463\pi$
$857$	3503.93i	0.139664i	−0.997559	+	0.0698319i	$0.977754\pi$
$858$	0	0
$859$	−31044.1	−1.23307	−0.616537	−	0.787326i	$-0.711465\pi$
$859$	−31044.1	−1.23307	−0.616537	+	0.787326i	$0.711465\pi$
$860$	0	0
$861$	0	0
$862$	− 1513.59i	− 0.0598063i
$863$	26333.6i	1.03871i	0.854559	+	0.519354i	$0.173827\pi$
$863$	26333.6i	1.03871i	−0.854559	+	0.519354i	$0.826173\pi$
$864$	0	0
$865$	0	0
$866$	−992.632	−0.0389504
$867$	0	0
$868$	− 5331.76i	− 0.208493i
$869$	−19596.0	−0.764959
$870$	0	0
$871$	−5489.71	−0.213561
$872$	− 5442.87i	− 0.211375i
$873$	0	0
$874$	−2920.52	−0.113030
$875$	0	0
$876$	0	0
$877$	− 40977.3i	− 1.57777i	−0.614540	−	0.788886i	$-0.710658\pi$
$877$	− 40977.3i	− 1.57777i	0.614540	−	0.788886i	$-0.289342\pi$
$878$	− 4198.21i	− 0.161370i
$879$	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
$883$	36037.9i	1.37347i	0.726909	+	0.686734i	$0.240956\pi$
$883$	36037.9i	1.37347i	−0.726909	+	0.686734i	$0.759044\pi$
$884$	−67577.5	−2.57113
$885$	0	0
$886$	−1734.70	−0.0657769
$887$	− 1465.05i	− 0.0554584i	−0.999615	−	0.0277292i	$-0.991172\pi$
$887$	− 1465.05i	− 0.0554584i	0.999615	−	0.0277292i	$-0.00882761\pi$
$888$	0	0
$889$	8608.97	0.324787
$890$	0	0
$891$	0	0
$892$	6217.36i	0.233377i
$893$	− 27333.5i	− 1.02428i
$894$	0	0
$895$	0	0
$896$	−7531.87	−0.280828
$897$	0	0
$898$	55.1315i	0.00204873i
$899$	−8073.96	−0.299535
$900$	0	0
$901$	−1614.40	−0.0596930
$902$	4643.77i	0.171420i
$903$	0	0
$904$	281.023	0.0103393
$905$	0	0
$906$	0	0
$907$	33660.8i	1.23229i	0.787632	+	0.616146i	$0.211307\pi$
$907$	33660.8i	1.23229i	−0.787632	+	0.616146i	$0.788693\pi$
$908$	1155.57i	0.0422346i
$909$	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
$913$	− 67493.3i	− 2.44655i
$914$	−4258.94	−0.154128
$915$	0	0
$916$	27066.2	0.976302
$917$	− 4920.50i	− 0.177196i
$918$	0	0
$919$	−1149.54	−0.0412620	−0.0206310	−	0.999787i	$-0.506568\pi$
$919$	−1149.54	−0.0412620	−0.0206310	+	0.999787i	$0.506568\pi$
$920$	0	0
$921$	0	0
$922$	− 404.124i	− 0.0144350i
$923$	52568.9i	1.87468i
$924$	0	0
$925$	0	0
$926$	−1532.68	−0.0543921
$927$	0	0
$928$	8566.50i	0.303027i
$929$	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
$932$	1070.78i	0.0376336i
$933$	0	0
$934$	−4632.27	−0.162283
$935$	0	0
$936$	0	0
$937$	3511.90i	0.122443i	0.998124	+	0.0612213i	$0.0194995\pi$
$937$	3511.90i	0.122443i	−0.998124	+	0.0612213i	$0.980500\pi$
$938$	285.653i	0.00994339i
$939$	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
$943$	− 33320.3i	− 1.15064i
$944$	14918.6	0.514363
$945$	0	0
$946$	1602.92	0.0550902
$947$	− 48357.3i	− 1.65935i	−0.558250	−	0.829673i	$-0.688527\pi$
$947$	− 48357.3i	− 1.65935i	0.558250	−	0.829673i	$-0.311473\pi$
$948$	0	0
$949$	36221.0	1.23897
$950$	0	0
$951$	0	0
$952$	7062.36i	0.240433i
$953$	38701.1i	1.31548i	0.753245	+	0.657740i	$0.228488\pi$
$953$	38701.1i	1.31548i	−0.753245	+	0.657740i	$0.771512\pi$
$954$	0	0
$955$	0	0
$956$	−17812.3	−0.602606
$957$	0	0
$958$	2565.25i	0.0865129i
$959$	−11780.0	−0.396660
$960$	0	0
$961$	−27647.7	−0.928056
$962$	2894.01i	0.0969924i
$963$	0	0
$964$	32990.0	1.10222
$965$	0	0
$966$	0	0
$967$	− 24312.7i	− 0.808526i	−0.914643	−	0.404263i	$-0.867528\pi$
$967$	− 24312.7i	− 0.808526i	0.914643	−	0.404263i	$-0.132472\pi$
$968$	4525.91i	0.150277i
$969$	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
$973$	45092.9i	1.48573i
$974$	−3081.47	−0.101373
$975$	0	0
$976$	−3403.55	−0.111624
$977$	− 3186.09i	− 0.104332i	−0.998638	−	0.0521659i	$-0.983388\pi$
$977$	− 3186.09i	− 0.104332i	0.998638	−	0.0521659i	$-0.0166125\pi$
$978$	0	0
$979$	72356.1	2.36212
$980$	0	0
$981$	0	0
$982$	− 2862.33i	− 0.0930150i
$983$	30345.6i	0.984614i	0.870422	+	0.492307i	$0.163846\pi$
$983$	30345.6i	0.984614i	−0.870422	+	0.492307i	$0.836154\pi$
$984$	0	0
$985$	0	0
$986$	5324.86	0.171986
$987$	0	0
$988$	70653.7i	2.27509i
$989$	−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	$0.482422\pi$
$992$	− 2274.03i	− 0.0727828i
$993$	0	0
$994$	2735.39	0.0872849
$995$	0	0
$996$	0	0
$997$	− 4567.89i	− 0.145102i	−0.997365	−	0.0725510i	$-0.976886\pi$
$997$	− 4567.89i	− 0.145102i	0.997365	−	0.0725510i	$-0.0231140\pi$
$998$	− 2418.25i	− 0.0767017i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.k.649.4		6
3.2	odd	2		675.4.b.l.649.3		6
5.2	odd	4		675.4.a.q.1.2		3
5.3	odd	4		135.4.a.g.1.2	yes	3
5.4	even	2	inner	675.4.b.k.649.3		6
15.2	even	4		675.4.a.r.1.2		3
15.8	even	4		135.4.a.f.1.2	✓	3
15.14	odd	2		675.4.b.l.649.4		6
20.3	even	4		2160.4.a.be.1.2		3
45.13	odd	12		405.4.e.r.136.2		6
45.23	even	12		405.4.e.t.136.2		6
45.38	even	12		405.4.e.t.271.2		6
45.43	odd	12		405.4.e.r.271.2		6
60.23	odd	4		2160.4.a.bm.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.2	✓	3	15.8	even	4
135.4.a.g.1.2	yes	3	5.3	odd	4
405.4.e.r.136.2		6	45.13	odd	12
405.4.e.r.271.2		6	45.43	odd	12
405.4.e.t.136.2		6	45.23	even	12
405.4.e.t.271.2		6	45.38	even	12
675.4.a.q.1.2		3	5.2	odd	4
675.4.a.r.1.2		3	15.2	even	4
675.4.b.k.649.3		6	5.4	even	2	inner
675.4.b.k.649.4		6	1.1	even	1	trivial
675.4.b.l.649.3		6	3.2	odd	2
675.4.b.l.649.4		6	15.14	odd	2
2160.4.a.be.1.2		3	20.3	even	4
2160.4.a.bm.1.2		3	60.23	odd	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.258712i q^{2} +7.93307 q^{4} +14.5174i q^{7} +4.12208i q^{8} +O(q^{10})\)	\(q+0.258712i q^{2} +7.93307 q^{4} +14.5174i q^{7} +4.12208i q^{8} +49.2845 q^{11} -72.1800i q^{13} -3.75584 q^{14} +62.3981 q^{16} -118.017i q^{17} -123.389 q^{19} +12.7505i q^{22} -91.4883i q^{23} +18.6739 q^{26} +115.168i q^{28} +174.400 q^{29} -46.2956 q^{31} +49.1198i q^{32} +30.5324 q^{34} +154.977i q^{37} -31.9223i q^{38} +364.203 q^{41} -125.714i q^{43} +390.978 q^{44} +23.6692 q^{46} +221.523i q^{47} +132.244 q^{49} -572.609i q^{52} -13.6794i q^{53} -59.8420 q^{56} +45.1195i q^{58} +239.087 q^{59} -54.5457 q^{61} -11.9772i q^{62} +486.477 q^{64} -76.0558i q^{67} -936.235i q^{68} -728.303 q^{71} +501.815i q^{73} -40.0944 q^{74} -978.854 q^{76} +715.485i q^{77} -397.610 q^{79} +94.2237i q^{82} -1369.46i q^{83} +32.5237 q^{86} +203.155i q^{88} +1468.13 q^{89} +1047.87 q^{91} -725.783i q^{92} -57.3108 q^{94} +335.023i q^{97} +34.2133i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 46 q^{4}+O(q^{10}) \) 6 * q - 46 * q^4	\( 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) \) 6 * q - 46 * q^4 - 76 * q^11 - 216 * q^14 + 382 * q^16 - 374 * q^19 - 832 * q^26 + 320 * q^29 + 454 * q^31 - 34 * q^34 + 676 * q^41 + 3272 * q^44 - 2850 * q^46 + 394 * q^49 + 2508 * q^56 + 280 * q^59 + 1190 * q^61 + 1836 * q^64 + 1204 * q^71 + 5756 * q^74 + 1050 * q^76 - 1258 * q^79 + 7460 * q^86 + 4308 * q^89 - 880 * q^91 + 2216 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more