LMFDB - Embedded newform 2475.4.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.2301792529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 22x^{4} + 101x^{2} - 16 $ x^6 - 22x^4 + 101x^2 - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.94784$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.94784 q^{2} +7.58548 q^{4} -2.30429 q^{7} +1.63647 q^{8} +O(q^{10})$	$q-3.94784 q^{2} +7.58548 q^{4} -2.30429 q^{7} +1.63647 q^{8} +11.0000 q^{11} -20.1150 q^{13} +9.09698 q^{14} -67.1444 q^{16} -39.2897 q^{17} -74.9589 q^{19} -43.4263 q^{22} +160.389 q^{23} +79.4109 q^{26} -17.4791 q^{28} +98.5250 q^{29} +41.7447 q^{31} +251.984 q^{32} +155.110 q^{34} -187.950 q^{37} +295.926 q^{38} +7.39834 q^{41} +534.468 q^{43} +83.4402 q^{44} -633.192 q^{46} -163.480 q^{47} -337.690 q^{49} -152.582 q^{52} -214.277 q^{53} -3.77091 q^{56} -388.961 q^{58} +535.609 q^{59} -698.218 q^{61} -164.801 q^{62} -457.638 q^{64} +461.629 q^{67} -298.031 q^{68} -634.311 q^{71} +733.619 q^{73} +741.999 q^{74} -568.599 q^{76} -25.3472 q^{77} -685.436 q^{79} -29.2075 q^{82} -610.811 q^{83} -2110.00 q^{86} +18.0012 q^{88} -919.083 q^{89} +46.3508 q^{91} +1216.63 q^{92} +645.396 q^{94} +1466.45 q^{97} +1333.15 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4}+O(q^{10}) $ 6 * q - 4 * q^4	$ 6 q - 4 q^{4} + 66 q^{11} + 18 q^{14} - 108 q^{16} - 258 q^{19} + 356 q^{26} + 494 q^{29} - 514 q^{31} - 6 q^{34} + 824 q^{41} - 44 q^{44} - 940 q^{46} - 496 q^{49} + 130 q^{56} + 200 q^{59} - 2210 q^{61} - 676 q^{64} + 270 q^{71} - 266 q^{74} + 422 q^{76} - 824 q^{79} - 2476 q^{86} - 186 q^{89} + 2000 q^{91} + 660 q^{94}+O(q^{100}) $ 6 * q - 4 * q^4 + 66 * q^11 + 18 * q^14 - 108 * q^16 - 258 * q^19 + 356 * q^26 + 494 * q^29 - 514 * q^31 - 6 * q^34 + 824 * q^41 - 44 * q^44 - 940 * q^46 - 496 * q^49 + 130 * q^56 + 200 * q^59 - 2210 * q^61 - 676 * q^64 + 270 * q^71 - 266 * q^74 + 422 * q^76 - 824 * q^79 - 2476 * q^86 - 186 * q^89 + 2000 * q^91 + 660 * q^94

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 $

$2$	−3.94784	−1.39577	−0.697887	−	0.716208i	$-0.745876\pi$
$2$	−3.94784	−1.39577	−0.697887	+	0.716208i	$0.745876\pi$
$3$	0	0
$3$	0	0
$4$	7.58548	0.948185
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.30429	−0.124420	−0.0622100	−	0.998063i	$-0.519815\pi$
$7$	−2.30429	−0.124420	−0.0622100	+	0.998063i	$0.519815\pi$
$8$	1.63647	0.0723226
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−20.1150	−0.429146	−0.214573	−	0.976708i	$-0.568836\pi$
$13$	−20.1150	−0.429146	−0.214573	+	0.976708i	$0.568836\pi$
$14$	9.09698	0.173662
$15$	0	0
$16$	−67.1444	−1.04913
$17$	−39.2897	−0.560538	−0.280269	−	0.959921i	$-0.590424\pi$
$17$	−39.2897	−0.560538	−0.280269	+	0.959921i	$0.590424\pi$
$18$	0	0
$19$	−74.9589	−0.905092	−0.452546	−	0.891741i	$-0.649484\pi$
$19$	−74.9589	−0.905092	−0.452546	+	0.891741i	$0.649484\pi$
$20$	0	0
$21$	0	0
$22$	−43.4263	−0.420842
$23$	160.389	1.45406	0.727032	−	0.686604i	$-0.240899\pi$
$23$	160.389	1.45406	0.727032	+	0.686604i	$0.240899\pi$
$24$	0	0
$25$	0	0
$26$	79.4109	0.598991
$27$	0	0
$28$	−17.4791	−0.117973
$29$	98.5250	0.630884	0.315442	−	0.948945i	$-0.397847\pi$
$29$	98.5250	0.630884	0.315442	+	0.948945i	$0.397847\pi$
$30$	0	0
$31$	41.7447	0.241857	0.120928	−	0.992661i	$-0.461413\pi$
$31$	41.7447	0.241857	0.120928	+	0.992661i	$0.461413\pi$
$32$	251.984	1.39203
$33$	0	0
$34$	155.110	0.782385
$35$	0	0
$36$	0	0
$37$	−187.950	−0.835104	−0.417552	−	0.908653i	$-0.637112\pi$
$37$	−187.950	−0.835104	−0.417552	+	0.908653i	$0.637112\pi$
$38$	295.926	1.26330
$39$	0	0
$40$	0	0
$41$	7.39834	0.0281811	0.0140906	−	0.999901i	$-0.495515\pi$
$41$	7.39834	0.0281811	0.0140906	+	0.999901i	$0.495515\pi$
$42$	0	0
$43$	534.468	1.89548	0.947739	−	0.319047i	$-0.103363\pi$
$43$	534.468	1.89548	0.947739	+	0.319047i	$0.103363\pi$
$44$	83.4402	0.285888
$45$	0	0
$46$	−633.192	−2.02954
$47$	−163.480	−0.507363	−0.253682	−	0.967288i	$-0.581642\pi$
$47$	−163.480	−0.507363	−0.253682	+	0.967288i	$0.581642\pi$
$48$	0	0
$49$	−337.690	−0.984520
$50$	0	0
$51$	0	0
$52$	−152.582	−0.406910
$53$	−214.277	−0.555343	−0.277671	−	0.960676i	$-0.589563\pi$
$53$	−214.277	−0.555343	−0.277671	+	0.960676i	$0.589563\pi$
$54$	0	0
$55$	0	0
$56$	−3.77091	−0.00899838
$57$	0	0
$58$	−388.961	−0.880571
$59$	535.609	1.18187	0.590935	−	0.806719i	$-0.298759\pi$
$59$	535.609	1.18187	0.590935	+	0.806719i	$0.298759\pi$
$60$	0	0
$61$	−698.218	−1.46554	−0.732768	−	0.680479i	$-0.761772\pi$
$61$	−698.218	−1.46554	−0.732768	+	0.680479i	$0.761772\pi$
$62$	−164.801	−0.337577
$63$	0	0
$64$	−457.638	−0.893823
$65$	0	0
$66$	0	0
$67$	461.629	0.841745	0.420873	−	0.907120i	$-0.361724\pi$
$67$	461.629	0.841745	0.420873	+	0.907120i	$0.361724\pi$
$68$	−298.031	−0.531494
$69$	0	0
$70$	0	0
$71$	−634.311	−1.06027	−0.530133	−	0.847915i	$-0.677858\pi$
$71$	−634.311	−1.06027	−0.530133	+	0.847915i	$0.677858\pi$
$72$	0	0
$73$	733.619	1.17621	0.588107	−	0.808783i	$-0.299873\pi$
$73$	733.619	1.17621	0.588107	+	0.808783i	$0.299873\pi$
$74$	741.999	1.16562
$75$	0	0
$76$	−568.599	−0.858195
$77$	−25.3472	−0.0375140
$78$	0	0
$79$	−685.436	−0.976172	−0.488086	−	0.872795i	$-0.662305\pi$
$79$	−685.436	−0.976172	−0.488086	+	0.872795i	$0.662305\pi$
$80$	0	0
$81$	0	0
$82$	−29.2075	−0.0393345
$83$	−610.811	−0.807774	−0.403887	−	0.914809i	$-0.632341\pi$
$83$	−610.811	−0.807774	−0.403887	+	0.914809i	$0.632341\pi$
$84$	0	0
$85$	0	0
$86$	−2110.00	−2.64566
$87$	0	0
$88$	18.0012	0.0218061
$89$	−919.083	−1.09464	−0.547318	−	0.836925i	$-0.684351\pi$
$89$	−919.083	−1.09464	−0.547318	+	0.836925i	$0.684351\pi$
$90$	0	0
$91$	46.3508	0.0533944
$92$	1216.63	1.37872
$93$	0	0
$94$	645.396	0.708165
$95$	0	0
$96$	0	0
$97$	1466.45	1.53501	0.767505	−	0.641043i	$-0.221498\pi$
$97$	1466.45	1.53501	0.767505	+	0.641043i	$0.221498\pi$
$98$	1333.15	1.37417
$99$	0	0
$100$	0	0
$101$	836.340	0.823950	0.411975	−	0.911195i	$-0.364839\pi$
$101$	836.340	0.823950	0.411975	+	0.911195i	$0.364839\pi$
$102$	0	0
$103$	−1589.52	−1.52058	−0.760290	−	0.649584i	$-0.774943\pi$
$103$	−1589.52	−1.52058	−0.760290	+	0.649584i	$0.774943\pi$
$104$	−32.9177	−0.0310370
$105$	0	0
$106$	845.931	0.775133
$107$	−63.6397	−0.0574980	−0.0287490	−	0.999587i	$-0.509152\pi$
$107$	−63.6397	−0.0574980	−0.0287490	+	0.999587i	$0.509152\pi$
$108$	0	0
$109$	1865.60	1.63938	0.819688	−	0.572810i	$-0.194147\pi$
$109$	1865.60	1.63938	0.819688	+	0.572810i	$0.194147\pi$
$110$	0	0
$111$	0	0
$112$	154.720	0.130533
$113$	1359.98	1.13218	0.566090	−	0.824344i	$-0.308456\pi$
$113$	1359.98	1.13218	0.566090	+	0.824344i	$0.308456\pi$
$114$	0	0
$115$	0	0
$116$	747.359	0.598194
$117$	0	0
$118$	−2114.50	−1.64962
$119$	90.5349	0.0697422
$120$	0	0
$121$	121.000	0.0909091
$122$	2756.46	2.04556
$123$	0	0
$124$	316.653	0.229325
$125$	0	0
$126$	0	0
$127$	−866.966	−0.605754	−0.302877	−	0.953030i	$-0.597947\pi$
$127$	−866.966	−0.605754	−0.302877	+	0.953030i	$0.597947\pi$
$128$	−209.187	−0.144451
$129$	0	0
$130$	0	0
$131$	−42.8497	−0.0285786	−0.0142893	−	0.999898i	$-0.504549\pi$
$131$	−42.8497	−0.0285786	−0.0142893	+	0.999898i	$0.504549\pi$
$132$	0	0
$133$	172.727	0.112612
$134$	−1822.44	−1.17489
$135$	0	0
$136$	−64.2965	−0.0405396
$137$	3059.42	1.90791	0.953956	−	0.299947i	$-0.0969689\pi$
$137$	3059.42	1.90791	0.953956	+	0.299947i	$0.0969689\pi$
$138$	0	0
$139$	−3238.86	−1.97638	−0.988190	−	0.153234i	$-0.951031\pi$
$139$	−3238.86	−1.97638	−0.988190	+	0.153234i	$0.951031\pi$
$140$	0	0
$141$	0	0
$142$	2504.16	1.47989
$143$	−221.265	−0.129392
$144$	0	0
$145$	0	0
$146$	−2896.21	−1.64173
$147$	0	0
$148$	−1425.69	−0.791833
$149$	1353.74	0.744316	0.372158	−	0.928169i	$-0.378618\pi$
$149$	1353.74	0.744316	0.372158	+	0.928169i	$0.378618\pi$
$150$	0	0
$151$	−434.325	−0.234072	−0.117036	−	0.993128i	$-0.537339\pi$
$151$	−434.325	−0.234072	−0.117036	+	0.993128i	$0.537339\pi$
$152$	−122.668	−0.0654586
$153$	0	0
$154$	100.067	0.0523611
$155$	0	0
$156$	0	0
$157$	−2131.64	−1.08359	−0.541794	−	0.840511i	$-0.682255\pi$
$157$	−2131.64	−1.08359	−0.541794	+	0.840511i	$0.682255\pi$
$158$	2706.00	1.36252
$159$	0	0
$160$	0	0
$161$	−369.583	−0.180915
$162$	0	0
$163$	1442.52	0.693169	0.346585	−	0.938019i	$-0.387341\pi$
$163$	1442.52	0.693169	0.346585	+	0.938019i	$0.387341\pi$
$164$	56.1199	0.0267209
$165$	0	0
$166$	2411.39	1.12747
$167$	−1642.60	−0.761128	−0.380564	−	0.924755i	$-0.624270\pi$
$167$	−1642.60	−0.761128	−0.380564	+	0.924755i	$0.624270\pi$
$168$	0	0
$169$	−1792.39	−0.815834
$170$	0	0
$171$	0	0
$172$	4054.19	1.79726
$173$	2194.38	0.964367	0.482184	−	0.876070i	$-0.339844\pi$
$173$	2194.38	0.964367	0.482184	+	0.876070i	$0.339844\pi$
$174$	0	0
$175$	0	0
$176$	−738.588	−0.316325
$177$	0	0
$178$	3628.39	1.52786
$179$	−1470.87	−0.614179	−0.307090	−	0.951681i	$-0.599355\pi$
$179$	−1470.87	−0.614179	−0.307090	+	0.951681i	$0.599355\pi$
$180$	0	0
$181$	−2215.10	−0.909653	−0.454826	−	0.890580i	$-0.650299\pi$
$181$	−2215.10	−0.909653	−0.454826	+	0.890580i	$0.650299\pi$
$182$	−182.986	−0.0745265
$183$	0	0
$184$	262.473	0.105162
$185$	0	0
$186$	0	0
$187$	−432.187	−0.169009
$188$	−1240.08	−0.481074
$189$	0	0
$190$	0	0
$191$	−873.291	−0.330833	−0.165417	−	0.986224i	$-0.552897\pi$
$191$	−873.291	−0.330833	−0.165417	+	0.986224i	$0.552897\pi$
$192$	0	0
$193$	5197.58	1.93850	0.969249	−	0.246084i	$-0.0791438\pi$
$193$	5197.58	1.93850	0.969249	+	0.246084i	$0.0791438\pi$
$194$	−5789.34	−2.14253
$195$	0	0
$196$	−2561.54	−0.933506
$197$	2858.30	1.03373	0.516866	−	0.856066i	$-0.327099\pi$
$197$	2858.30	1.03373	0.516866	+	0.856066i	$0.327099\pi$
$198$	0	0
$199$	−1026.01	−0.365488	−0.182744	−	0.983161i	$-0.558498\pi$
$199$	−1026.01	−0.365488	−0.182744	+	0.983161i	$0.558498\pi$
$200$	0	0
$201$	0	0
$202$	−3301.74	−1.15005
$203$	−227.030	−0.0784946
$204$	0	0
$205$	0	0
$206$	6275.17	2.12239
$207$	0	0
$208$	1350.61	0.450230
$209$	−824.548	−0.272896
$210$	0	0
$211$	4370.58	1.42599	0.712993	−	0.701171i	$-0.247339\pi$
$211$	4370.58	1.42599	0.712993	+	0.701171i	$0.247339\pi$
$212$	−1625.39	−0.526567
$213$	0	0
$214$	251.240	0.0802542
$215$	0	0
$216$	0	0
$217$	−96.1918	−0.0300918
$218$	−7365.10	−2.28820
$219$	0	0
$220$	0	0
$221$	790.312	0.240553
$222$	0	0
$223$	−4818.50	−1.44695	−0.723477	−	0.690349i	$-0.757457\pi$
$223$	−4818.50	−1.44695	−0.723477	+	0.690349i	$0.757457\pi$
$224$	−580.644	−0.173196
$225$	0	0
$226$	−5369.00	−1.58027
$227$	2700.77	0.789677	0.394839	−	0.918751i	$-0.370801\pi$
$227$	2700.77	0.789677	0.394839	+	0.918751i	$0.370801\pi$
$228$	0	0
$229$	−2117.92	−0.611162	−0.305581	−	0.952166i	$-0.598851\pi$
$229$	−2117.92	−0.611162	−0.305581	+	0.952166i	$0.598851\pi$
$230$	0	0
$231$	0	0
$232$	161.234	0.0456271
$233$	5126.91	1.44152	0.720761	−	0.693183i	$-0.243792\pi$
$233$	5126.91	1.44152	0.720761	+	0.693183i	$0.243792\pi$
$234$	0	0
$235$	0	0
$236$	4062.85	1.12063
$237$	0	0
$238$	−357.418	−0.0973443
$239$	−3209.84	−0.868733	−0.434367	−	0.900736i	$-0.643028\pi$
$239$	−3209.84	−0.868733	−0.434367	+	0.900736i	$0.643028\pi$
$240$	0	0
$241$	−3718.28	−0.993840	−0.496920	−	0.867796i	$-0.665536\pi$
$241$	−3718.28	−0.993840	−0.496920	+	0.867796i	$0.665536\pi$
$242$	−477.689	−0.126889
$243$	0	0
$244$	−5296.32	−1.38960
$245$	0	0
$246$	0	0
$247$	1507.80	0.388417
$248$	68.3140	0.0174917
$249$	0	0
$250$	0	0
$251$	6076.19	1.52799	0.763995	−	0.645222i	$-0.223235\pi$
$251$	6076.19	1.52799	0.763995	+	0.645222i	$0.223235\pi$
$252$	0	0
$253$	1764.28	0.438417
$254$	3422.65	0.845496
$255$	0	0
$256$	4486.94	1.09544
$257$	5699.82	1.38344	0.691722	−	0.722164i	$-0.256852\pi$
$257$	5699.82	1.38344	0.691722	+	0.722164i	$0.256852\pi$
$258$	0	0
$259$	433.092	0.103904
$260$	0	0
$261$	0	0
$262$	169.164	0.0398892
$263$	−3176.74	−0.744815	−0.372407	−	0.928069i	$-0.621468\pi$
$263$	−3176.74	−0.744815	−0.372407	+	0.928069i	$0.621468\pi$
$264$	0	0
$265$	0	0
$266$	−681.900	−0.157180
$267$	0	0
$268$	3501.67	0.798130
$269$	−59.2650	−0.0134329	−0.00671645	−	0.999977i	$-0.502138\pi$
$269$	−59.2650	−0.0134329	−0.00671645	+	0.999977i	$0.502138\pi$
$270$	0	0
$271$	7410.76	1.66115	0.830575	−	0.556907i	$-0.188012\pi$
$271$	7410.76	1.66115	0.830575	+	0.556907i	$0.188012\pi$
$272$	2638.08	0.588078
$273$	0	0
$274$	−12078.1	−2.66301
$275$	0	0
$276$	0	0
$277$	−4481.41	−0.972064	−0.486032	−	0.873941i	$-0.661556\pi$
$277$	−4481.41	−0.972064	−0.486032	+	0.873941i	$0.661556\pi$
$278$	12786.5	2.75858
$279$	0	0
$280$	0	0
$281$	911.898	0.193592	0.0967958	−	0.995304i	$-0.469141\pi$
$281$	911.898	0.193592	0.0967958	+	0.995304i	$0.469141\pi$
$282$	0	0
$283$	−1560.73	−0.327829	−0.163914	−	0.986475i	$-0.552412\pi$
$283$	−1560.73	−0.327829	−0.163914	+	0.986475i	$0.552412\pi$
$284$	−4811.55	−1.00533
$285$	0	0
$286$	873.520	0.180603
$287$	−17.0479	−0.00350630
$288$	0	0
$289$	−3369.32	−0.685797
$290$	0	0
$291$	0	0
$292$	5564.85	1.11527
$293$	−4143.09	−0.826082	−0.413041	−	0.910712i	$-0.635533\pi$
$293$	−4143.09	−0.826082	−0.413041	+	0.910712i	$0.635533\pi$
$294$	0	0
$295$	0	0
$296$	−307.576	−0.0603969
$297$	0	0
$298$	−5344.37	−1.03890
$299$	−3226.23	−0.624006
$300$	0	0
$301$	−1231.57	−0.235835
$302$	1714.65	0.326711
$303$	0	0
$304$	5033.07	0.949560
$305$	0	0
$306$	0	0
$307$	257.195	0.0478139	0.0239070	−	0.999714i	$-0.492389\pi$
$307$	257.195	0.0478139	0.0239070	+	0.999714i	$0.492389\pi$
$308$	−192.271	−0.0355702
$309$	0	0
$310$	0	0
$311$	3847.11	0.701445	0.350723	−	0.936479i	$-0.385936\pi$
$311$	3847.11	0.701445	0.350723	+	0.936479i	$0.385936\pi$
$312$	0	0
$313$	−6353.12	−1.14728	−0.573641	−	0.819107i	$-0.694470\pi$
$313$	−6353.12	−1.14728	−0.573641	+	0.819107i	$0.694470\pi$
$314$	8415.38	1.51244
$315$	0	0
$316$	−5199.36	−0.925592
$317$	−1079.18	−0.191207	−0.0956034	−	0.995420i	$-0.530478\pi$
$317$	−1079.18	−0.191207	−0.0956034	+	0.995420i	$0.530478\pi$
$318$	0	0
$319$	1083.77	0.190219
$320$	0	0
$321$	0	0
$322$	1459.06	0.252516
$323$	2945.11	0.507339
$324$	0	0
$325$	0	0
$326$	−5694.83	−0.967507
$327$	0	0
$328$	12.1072	0.00203813
$329$	376.707	0.0631262
$330$	0	0
$331$	8878.10	1.47427	0.737136	−	0.675744i	$-0.236177\pi$
$331$	8878.10	1.47427	0.737136	+	0.675744i	$0.236177\pi$
$332$	−4633.29	−0.765919
$333$	0	0
$334$	6484.73	1.06236
$335$	0	0
$336$	0	0
$337$	3776.97	0.610518	0.305259	−	0.952269i	$-0.401257\pi$
$337$	3776.97	0.610518	0.305259	+	0.952269i	$0.401257\pi$
$338$	7076.06	1.13872
$339$	0	0
$340$	0	0
$341$	459.191	0.0729226
$342$	0	0
$343$	1568.51	0.246914
$344$	874.642	0.137086
$345$	0	0
$346$	−8663.07	−1.34604
$347$	−4835.41	−0.748065	−0.374032	−	0.927416i	$-0.622025\pi$
$347$	−4835.41	−0.748065	−0.374032	+	0.927416i	$0.622025\pi$
$348$	0	0
$349$	553.458	0.0848880	0.0424440	−	0.999099i	$-0.486486\pi$
$349$	553.458	0.0848880	0.0424440	+	0.999099i	$0.486486\pi$
$350$	0	0
$351$	0	0
$352$	2771.82	0.419712
$353$	−12221.2	−1.84269	−0.921345	−	0.388746i	$-0.872908\pi$
$353$	−12221.2	−1.84269	−0.921345	+	0.388746i	$0.872908\pi$
$354$	0	0
$355$	0	0
$356$	−6971.68	−1.03792
$357$	0	0
$358$	5806.77	0.857255
$359$	−6178.88	−0.908381	−0.454190	−	0.890905i	$-0.650071\pi$
$359$	−6178.88	−0.908381	−0.454190	+	0.890905i	$0.650071\pi$
$360$	0	0
$361$	−1240.16	−0.180808
$362$	8744.87	1.26967
$363$	0	0
$364$	351.593	0.0506277
$365$	0	0
$366$	0	0
$367$	8064.21	1.14700	0.573499	−	0.819206i	$-0.305586\pi$
$367$	8064.21	1.14700	0.573499	+	0.819206i	$0.305586\pi$
$368$	−10769.2	−1.52550
$369$	0	0
$370$	0	0
$371$	493.756	0.0690957
$372$	0	0
$373$	−7822.80	−1.08592	−0.542962	−	0.839758i	$-0.682697\pi$
$373$	−7822.80	−1.08592	−0.542962	+	0.839758i	$0.682697\pi$
$374$	1706.21	0.235898
$375$	0	0
$376$	−267.531	−0.0366938
$377$	−1981.83	−0.270741
$378$	0	0
$379$	−43.5702	−0.00590515	−0.00295257	−	0.999996i	$-0.500940\pi$
$379$	−43.5702	−0.00590515	−0.00295257	+	0.999996i	$0.500940\pi$
$380$	0	0
$381$	0	0
$382$	3447.62	0.461768
$383$	−9301.21	−1.24091	−0.620457	−	0.784241i	$-0.713053\pi$
$383$	−9301.21	−1.24091	−0.620457	+	0.784241i	$0.713053\pi$
$384$	0	0
$385$	0	0
$386$	−20519.2	−2.70570
$387$	0	0
$388$	11123.8	1.45547
$389$	2533.64	0.330233	0.165117	−	0.986274i	$-0.447200\pi$
$389$	2533.64	0.330233	0.165117	+	0.986274i	$0.447200\pi$
$390$	0	0
$391$	−6301.64	−0.815058
$392$	−552.621	−0.0712030
$393$	0	0
$394$	−11284.1	−1.44286
$395$	0	0
$396$	0	0
$397$	190.291	0.0240566	0.0120283	−	0.999928i	$-0.496171\pi$
$397$	190.291	0.0240566	0.0120283	+	0.999928i	$0.496171\pi$
$398$	4050.53	0.510138
$399$	0	0
$400$	0	0
$401$	−11651.8	−1.45103	−0.725517	−	0.688204i	$-0.758399\pi$
$401$	−11651.8	−1.45103	−0.725517	+	0.688204i	$0.758399\pi$
$402$	0	0
$403$	−839.694	−0.103792
$404$	6344.04	0.781256
$405$	0	0
$406$	896.280	0.109561
$407$	−2067.45	−0.251793
$408$	0	0
$409$	−10441.3	−1.26232	−0.631160	−	0.775653i	$-0.717421\pi$
$409$	−10441.3	−1.26232	−0.631160	+	0.775653i	$0.717421\pi$
$410$	0	0
$411$	0	0
$412$	−12057.2	−1.44179
$413$	−1234.20	−0.147048
$414$	0	0
$415$	0	0
$416$	−5068.65	−0.597383
$417$	0	0
$418$	3255.19	0.380901
$419$	−4345.03	−0.506607	−0.253304	−	0.967387i	$-0.581517\pi$
$419$	−4345.03	−0.506607	−0.253304	+	0.967387i	$0.581517\pi$
$420$	0	0
$421$	−7168.31	−0.829838	−0.414919	−	0.909858i	$-0.636190\pi$
$421$	−7168.31	−0.829838	−0.414919	+	0.909858i	$0.636190\pi$
$422$	−17254.4	−1.99035
$423$	0	0
$424$	−350.658	−0.0401638
$425$	0	0
$426$	0	0
$427$	1608.90	0.182342
$428$	−482.738	−0.0545187
$429$	0	0
$430$	0	0
$431$	−5606.60	−0.626590	−0.313295	−	0.949656i	$-0.601433\pi$
$431$	−5606.60	−0.626590	−0.313295	+	0.949656i	$0.601433\pi$
$432$	0	0
$433$	−1399.82	−0.155360	−0.0776799	−	0.996978i	$-0.524751\pi$
$433$	−1399.82	−0.155360	−0.0776799	+	0.996978i	$0.524751\pi$
$434$	379.750	0.0420014
$435$	0	0
$436$	14151.5	1.55443
$437$	−12022.6	−1.31606
$438$	0	0
$439$	2535.47	0.275652	0.137826	−	0.990456i	$-0.455988\pi$
$439$	2535.47	0.275652	0.137826	+	0.990456i	$0.455988\pi$
$440$	0	0
$441$	0	0
$442$	−3120.03	−0.335757
$443$	−7272.01	−0.779917	−0.389959	−	0.920832i	$-0.627511\pi$
$443$	−7272.01	−0.779917	−0.389959	+	0.920832i	$0.627511\pi$
$444$	0	0
$445$	0	0
$446$	19022.7	2.01962
$447$	0	0
$448$	1054.53	0.111210
$449$	4638.58	0.487545	0.243773	−	0.969832i	$-0.421615\pi$
$449$	4638.58	0.487545	0.243773	+	0.969832i	$0.421615\pi$
$450$	0	0
$451$	81.3817	0.00849693
$452$	10316.1	1.07352
$453$	0	0
$454$	−10662.2	−1.10221
$455$	0	0
$456$	0	0
$457$	−9452.39	−0.967536	−0.483768	−	0.875196i	$-0.660732\pi$
$457$	−9452.39	−0.967536	−0.483768	+	0.875196i	$0.660732\pi$
$458$	8361.22	0.853044
$459$	0	0
$460$	0	0
$461$	9401.91	0.949872	0.474936	−	0.880020i	$-0.342471\pi$
$461$	9401.91	0.949872	0.474936	+	0.880020i	$0.342471\pi$
$462$	0	0
$463$	−979.537	−0.0983217	−0.0491608	−	0.998791i	$-0.515655\pi$
$463$	−979.537	−0.0983217	−0.0491608	+	0.998791i	$0.515655\pi$
$464$	−6615.40	−0.661879
$465$	0	0
$466$	−20240.2	−2.01204
$467$	−7645.72	−0.757605	−0.378803	−	0.925477i	$-0.623664\pi$
$467$	−7645.72	−0.757605	−0.378803	+	0.925477i	$0.623664\pi$
$468$	0	0
$469$	−1063.73	−0.104730
$470$	0	0
$471$	0	0
$472$	876.509	0.0854759
$473$	5879.14	0.571508
$474$	0	0
$475$	0	0
$476$	686.750	0.0661284
$477$	0	0
$478$	12671.9	1.21255
$479$	−4925.34	−0.469822	−0.234911	−	0.972017i	$-0.575480\pi$
$479$	−4925.34	−0.469822	−0.234911	+	0.972017i	$0.575480\pi$
$480$	0	0
$481$	3780.62	0.358382
$482$	14679.2	1.38718
$483$	0	0
$484$	917.843	0.0861986
$485$	0	0
$486$	0	0
$487$	−11574.7	−1.07700	−0.538500	−	0.842625i	$-0.681009\pi$
$487$	−11574.7	−1.07700	−0.538500	+	0.842625i	$0.681009\pi$
$488$	−1142.62	−0.105991
$489$	0	0
$490$	0	0
$491$	−10050.7	−0.923795	−0.461898	−	0.886933i	$-0.652831\pi$
$491$	−10050.7	−0.923795	−0.461898	+	0.886933i	$0.652831\pi$
$492$	0	0
$493$	−3871.02	−0.353634
$494$	−5952.56	−0.542142
$495$	0	0
$496$	−2802.92	−0.253739
$497$	1461.64	0.131918
$498$	0	0
$499$	8155.68	0.731660	0.365830	−	0.930682i	$-0.380785\pi$
$499$	8155.68	0.731660	0.365830	+	0.930682i	$0.380785\pi$
$500$	0	0
$501$	0	0
$502$	−23987.8	−2.13273
$503$	−6912.90	−0.612785	−0.306393	−	0.951905i	$-0.599122\pi$
$503$	−6912.90	−0.612785	−0.306393	+	0.951905i	$0.599122\pi$
$504$	0	0
$505$	0	0
$506$	−6965.11	−0.611931
$507$	0	0
$508$	−6576.35	−0.574367
$509$	−13413.1	−1.16802	−0.584012	−	0.811745i	$-0.698518\pi$
$509$	−13413.1	−1.16802	−0.584012	+	0.811745i	$0.698518\pi$
$510$	0	0
$511$	−1690.47	−0.146345
$512$	−16040.2	−1.38454
$513$	0	0
$514$	−22502.0	−1.93098
$515$	0	0
$516$	0	0
$517$	−1798.29	−0.152976
$518$	−1709.78	−0.145026
$519$	0	0
$520$	0	0
$521$	−3875.80	−0.325915	−0.162958	−	0.986633i	$-0.552103\pi$
$521$	−3875.80	−0.325915	−0.162958	+	0.986633i	$0.552103\pi$
$522$	0	0
$523$	−13489.2	−1.12780	−0.563902	−	0.825842i	$-0.690700\pi$
$523$	−13489.2	−1.12780	−0.563902	+	0.825842i	$0.690700\pi$
$524$	−325.035	−0.0270978
$525$	0	0
$526$	12541.3	1.03959
$527$	−1640.13	−0.135570
$528$	0	0
$529$	13557.7	1.11430
$530$	0	0
$531$	0	0
$532$	1310.22	0.106777
$533$	−148.818	−0.0120938
$534$	0	0
$535$	0	0
$536$	755.443	0.0608772
$537$	0	0
$538$	233.969	0.0187493
$539$	−3714.59	−0.296844
$540$	0	0
$541$	−13351.4	−1.06104	−0.530519	−	0.847673i	$-0.678003\pi$
$541$	−13351.4	−1.06104	−0.530519	+	0.847673i	$0.678003\pi$
$542$	−29256.5	−2.31859
$543$	0	0
$544$	−9900.36	−0.780284
$545$	0	0
$546$	0	0
$547$	3372.01	0.263577	0.131789	−	0.991278i	$-0.457928\pi$
$547$	3372.01	0.263577	0.131789	+	0.991278i	$0.457928\pi$
$548$	23207.2	1.80905
$549$	0	0
$550$	0	0
$551$	−7385.33	−0.571008
$552$	0	0
$553$	1579.44	0.121455
$554$	17691.9	1.35678
$555$	0	0
$556$	−24568.3	−1.87397
$557$	−2097.07	−0.159525	−0.0797627	−	0.996814i	$-0.525416\pi$
$557$	−2097.07	−0.159525	−0.0797627	+	0.996814i	$0.525416\pi$
$558$	0	0
$559$	−10750.8	−0.813437
$560$	0	0
$561$	0	0
$562$	−3600.03	−0.270210
$563$	9822.60	0.735298	0.367649	−	0.929965i	$-0.380163\pi$
$563$	9822.60	0.735298	0.367649	+	0.929965i	$0.380163\pi$
$564$	0	0
$565$	0	0
$566$	6161.51	0.457575
$567$	0	0
$568$	−1038.03	−0.0766812
$569$	15129.7	1.11471	0.557357	−	0.830273i	$-0.311816\pi$
$569$	15129.7	1.11471	0.557357	+	0.830273i	$0.311816\pi$
$570$	0	0
$571$	−10003.0	−0.733119	−0.366559	−	0.930395i	$-0.619464\pi$
$571$	−10003.0	−0.733119	−0.366559	+	0.930395i	$0.619464\pi$
$572$	−1678.40	−0.122688
$573$	0	0
$574$	67.3026	0.00489400
$575$	0	0
$576$	0	0
$577$	12271.5	0.885391	0.442695	−	0.896672i	$-0.354022\pi$
$577$	12271.5	0.885391	0.442695	+	0.896672i	$0.354022\pi$
$578$	13301.6	0.957217
$579$	0	0
$580$	0	0
$581$	1407.49	0.100503
$582$	0	0
$583$	−2357.04	−0.167442
$584$	1200.55	0.0850668
$585$	0	0
$586$	16356.3	1.15302
$587$	5222.36	0.367206	0.183603	−	0.983000i	$-0.441224\pi$
$587$	5222.36	0.367206	0.183603	+	0.983000i	$0.441224\pi$
$588$	0	0
$589$	−3129.13	−0.218903
$590$	0	0
$591$	0	0
$592$	12619.8	0.876133
$593$	−19444.7	−1.34654	−0.673270	−	0.739397i	$-0.735111\pi$
$593$	−19444.7	−1.34654	−0.673270	+	0.739397i	$0.735111\pi$
$594$	0	0
$595$	0	0
$596$	10268.8	0.705749
$597$	0	0
$598$	12736.7	0.870971
$599$	−16371.1	−1.11671	−0.558353	−	0.829603i	$-0.688567\pi$
$599$	−16371.1	−1.11671	−0.558353	+	0.829603i	$0.688567\pi$
$600$	0	0
$601$	−8639.68	−0.586389	−0.293195	−	0.956053i	$-0.594718\pi$
$601$	−8639.68	−0.586389	−0.293195	+	0.956053i	$0.594718\pi$
$602$	4862.04	0.329173
$603$	0	0
$604$	−3294.56	−0.221943
$605$	0	0
$606$	0	0
$607$	−26727.6	−1.78722	−0.893608	−	0.448847i	$-0.851835\pi$
$607$	−26727.6	−1.78722	−0.893608	+	0.448847i	$0.851835\pi$
$608$	−18888.4	−1.25991
$609$	0	0
$610$	0	0
$611$	3288.41	0.217733
$612$	0	0
$613$	24639.7	1.62347	0.811734	−	0.584027i	$-0.198524\pi$
$613$	24639.7	1.62347	0.811734	+	0.584027i	$0.198524\pi$
$614$	−1015.36	−0.0667374
$615$	0	0
$616$	−41.4800	−0.00271311
$617$	−18931.0	−1.23522	−0.617612	−	0.786483i	$-0.711900\pi$
$617$	−18931.0	−1.23522	−0.617612	+	0.786483i	$0.711900\pi$
$618$	0	0
$619$	15181.4	0.985768	0.492884	−	0.870095i	$-0.335943\pi$
$619$	15181.4	0.985768	0.492884	+	0.870095i	$0.335943\pi$
$620$	0	0
$621$	0	0
$622$	−15187.8	−0.979059
$623$	2117.83	0.136195
$624$	0	0
$625$	0	0
$626$	25081.1	1.60135
$627$	0	0
$628$	−16169.5	−1.02744
$629$	7384.51	0.468108
$630$	0	0
$631$	−6904.58	−0.435605	−0.217803	−	0.975993i	$-0.569889\pi$
$631$	−6904.58	−0.435605	−0.217803	+	0.975993i	$0.569889\pi$
$632$	−1121.70	−0.0705993
$633$	0	0
$634$	4260.42	0.266881
$635$	0	0
$636$	0	0
$637$	6792.64	0.422503
$638$	−4278.57	−0.265502
$639$	0	0
$640$	0	0
$641$	822.316	0.0506701	0.0253350	−	0.999679i	$-0.491935\pi$
$641$	822.316	0.0506701	0.0253350	+	0.999679i	$0.491935\pi$
$642$	0	0
$643$	−13868.0	−0.850544	−0.425272	−	0.905066i	$-0.639822\pi$
$643$	−13868.0	−0.850544	−0.425272	+	0.905066i	$0.639822\pi$
$644$	−2803.47	−0.171540
$645$	0	0
$646$	−11626.8	−0.708130
$647$	5629.79	0.342086	0.171043	−	0.985264i	$-0.445286\pi$
$647$	5629.79	0.342086	0.171043	+	0.985264i	$0.445286\pi$
$648$	0	0
$649$	5891.69	0.356347
$650$	0	0
$651$	0	0
$652$	10942.2	0.657252
$653$	−24233.9	−1.45229	−0.726145	−	0.687541i	$-0.758690\pi$
$653$	−24233.9	−1.45229	−0.726145	+	0.687541i	$0.758690\pi$
$654$	0	0
$655$	0	0
$656$	−496.757	−0.0295657
$657$	0	0
$658$	−1487.18	−0.0881098
$659$	−8443.55	−0.499111	−0.249555	−	0.968361i	$-0.580284\pi$
$659$	−8443.55	−0.499111	−0.249555	+	0.968361i	$0.580284\pi$
$660$	0	0
$661$	−21347.6	−1.25617	−0.628084	−	0.778146i	$-0.716161\pi$
$661$	−21347.6	−1.25617	−0.628084	+	0.778146i	$0.716161\pi$
$662$	−35049.3	−2.05775
$663$	0	0
$664$	−999.576	−0.0584203
$665$	0	0
$666$	0	0
$667$	15802.3	0.917345
$668$	−12459.9	−0.721689
$669$	0	0
$670$	0	0
$671$	−7680.40	−0.441876
$672$	0	0
$673$	−12563.9	−0.719618	−0.359809	−	0.933026i	$-0.617158\pi$
$673$	−12563.9	−0.719618	−0.359809	+	0.933026i	$0.617158\pi$
$674$	−14910.9	−0.852145
$675$	0	0
$676$	−13596.1	−0.773561
$677$	6343.46	0.360116	0.180058	−	0.983656i	$-0.442371\pi$
$677$	6343.46	0.360116	0.180058	+	0.983656i	$0.442371\pi$
$678$	0	0
$679$	−3379.14	−0.190986
$680$	0	0
$681$	0	0
$682$	−1812.82	−0.101783
$683$	−9149.59	−0.512590	−0.256295	−	0.966599i	$-0.582502\pi$
$683$	−9149.59	−0.512590	−0.256295	+	0.966599i	$0.582502\pi$
$684$	0	0
$685$	0	0
$686$	−6192.23	−0.344636
$687$	0	0
$688$	−35886.5	−1.98860
$689$	4310.18	0.238323
$690$	0	0
$691$	1877.16	0.103344	0.0516720	−	0.998664i	$-0.483545\pi$
$691$	1877.16	0.103344	0.0516720	+	0.998664i	$0.483545\pi$
$692$	16645.4	0.914398
$693$	0	0
$694$	19089.4	1.04413
$695$	0	0
$696$	0	0
$697$	−290.678	−0.0157966
$698$	−2184.97	−0.118484
$699$	0	0
$700$	0	0
$701$	17782.6	0.958115	0.479057	−	0.877784i	$-0.340979\pi$
$701$	17782.6	0.958115	0.479057	+	0.877784i	$0.340979\pi$
$702$	0	0
$703$	14088.6	0.755846
$704$	−5034.01	−0.269498
$705$	0	0
$706$	48247.4	2.57198
$707$	−1927.17	−0.102516
$708$	0	0
$709$	−10174.5	−0.538946	−0.269473	−	0.963008i	$-0.586849\pi$
$709$	−10174.5	−0.538946	−0.269473	+	0.963008i	$0.586849\pi$
$710$	0	0
$711$	0	0
$712$	−1504.05	−0.0791669
$713$	6695.39	0.351675
$714$	0	0
$715$	0	0
$716$	−11157.3	−0.582355
$717$	0	0
$718$	24393.2	1.26789
$719$	−6729.89	−0.349072	−0.174536	−	0.984651i	$-0.555843\pi$
$719$	−6729.89	−0.349072	−0.174536	+	0.984651i	$0.555843\pi$
$720$	0	0
$721$	3662.71	0.189191
$722$	4895.96	0.252367
$723$	0	0
$724$	−16802.6	−0.862519
$725$	0	0
$726$	0	0
$727$	2080.04	0.106113	0.0530566	−	0.998592i	$-0.483104\pi$
$727$	2080.04	0.106113	0.0530566	+	0.998592i	$0.483104\pi$
$728$	75.8519	0.00386162
$729$	0	0
$730$	0	0
$731$	−20999.1	−1.06249
$732$	0	0
$733$	−22835.7	−1.15069	−0.575345	−	0.817911i	$-0.695132\pi$
$733$	−22835.7	−1.15069	−0.575345	+	0.817911i	$0.695132\pi$
$734$	−31836.2	−1.60095
$735$	0	0
$736$	40415.5	2.02409
$737$	5077.92	0.253796
$738$	0	0
$739$	−31701.0	−1.57800	−0.788998	−	0.614396i	$-0.789400\pi$
$739$	−31701.0	−1.57800	−0.788998	+	0.614396i	$0.789400\pi$
$740$	0	0
$741$	0	0
$742$	−1949.27	−0.0964420
$743$	9827.85	0.485261	0.242631	−	0.970119i	$-0.421990\pi$
$743$	9827.85	0.485261	0.242631	+	0.970119i	$0.421990\pi$
$744$	0	0
$745$	0	0
$746$	30883.2	1.51570
$747$	0	0
$748$	−3278.34	−0.160251
$749$	146.644	0.00715390
$750$	0	0
$751$	34412.8	1.67209	0.836046	−	0.548660i	$-0.184862\pi$
$751$	34412.8	1.67209	0.836046	+	0.548660i	$0.184862\pi$
$752$	10976.8	0.532290
$753$	0	0
$754$	7823.96	0.377894
$755$	0	0
$756$	0	0
$757$	−531.633	−0.0255251	−0.0127626	−	0.999919i	$-0.504063\pi$
$757$	−531.633	−0.0255251	−0.0127626	+	0.999919i	$0.504063\pi$
$758$	172.008	0.00824225
$759$	0	0
$760$	0	0
$761$	−7686.79	−0.366157	−0.183079	−	0.983098i	$-0.558606\pi$
$761$	−7686.79	−0.366157	−0.183079	+	0.983098i	$0.558606\pi$
$762$	0	0
$763$	−4298.88	−0.203971
$764$	−6624.33	−0.313691
$765$	0	0
$766$	36719.7	1.73203
$767$	−10773.8	−0.507195
$768$	0	0
$769$	−20016.3	−0.938631	−0.469316	−	0.883031i	$-0.655499\pi$
$769$	−20016.3	−0.938631	−0.469316	+	0.883031i	$0.655499\pi$
$770$	0	0
$771$	0	0
$772$	39426.1	1.83805
$773$	−18442.8	−0.858137	−0.429069	−	0.903272i	$-0.641158\pi$
$773$	−18442.8	−0.858137	−0.429069	+	0.903272i	$0.641158\pi$
$774$	0	0
$775$	0	0
$776$	2399.81	0.111016
$777$	0	0
$778$	−10002.4	−0.460931
$779$	−554.572	−0.0255065
$780$	0	0
$781$	−6977.42	−0.319682
$782$	24877.9	1.13764
$783$	0	0
$784$	22674.0	1.03289
$785$	0	0
$786$	0	0
$787$	−35237.5	−1.59604	−0.798018	−	0.602634i	$-0.794118\pi$
$787$	−35237.5	−1.59604	−0.798018	+	0.602634i	$0.794118\pi$
$788$	21681.5	0.980168
$789$	0	0
$790$	0	0
$791$	−3133.79	−0.140866
$792$	0	0
$793$	14044.7	0.628929
$794$	−751.241	−0.0335775
$795$	0	0
$796$	−7782.78	−0.346550
$797$	−8033.29	−0.357031	−0.178515	−	0.983937i	$-0.557129\pi$
$797$	−8033.29	−0.357031	−0.178515	+	0.983937i	$0.557129\pi$
$798$	0	0
$799$	6423.10	0.284397
$800$	0	0
$801$	0	0
$802$	45999.6	2.02532
$803$	8069.81	0.354642
$804$	0	0
$805$	0	0
$806$	3314.98	0.144870
$807$	0	0
$808$	1368.65	0.0595902
$809$	23620.9	1.02653	0.513267	−	0.858229i	$-0.328435\pi$
$809$	23620.9	1.02653	0.513267	+	0.858229i	$0.328435\pi$
$810$	0	0
$811$	−3766.34	−0.163075	−0.0815377	−	0.996670i	$-0.525983\pi$
$811$	−3766.34	−0.163075	−0.0815377	+	0.996670i	$0.525983\pi$
$812$	−1722.13	−0.0744273
$813$	0	0
$814$	8161.99	0.351447
$815$	0	0
$816$	0	0
$817$	−40063.1	−1.71558
$818$	41220.6	1.76191
$819$	0	0
$820$	0	0
$821$	39820.4	1.69274	0.846372	−	0.532592i	$-0.178782\pi$
$821$	39820.4	1.69274	0.846372	+	0.532592i	$0.178782\pi$
$822$	0	0
$823$	−16817.3	−0.712290	−0.356145	−	0.934431i	$-0.615909\pi$
$823$	−16817.3	−0.712290	−0.356145	+	0.934431i	$0.615909\pi$
$824$	−2601.20	−0.109972
$825$	0	0
$826$	4872.42	0.205246
$827$	34390.4	1.44604	0.723018	−	0.690829i	$-0.242754\pi$
$827$	34390.4	1.44604	0.723018	+	0.690829i	$0.242754\pi$
$828$	0	0
$829$	16197.9	0.678622	0.339311	−	0.940674i	$-0.389806\pi$
$829$	16197.9	0.678622	0.339311	+	0.940674i	$0.389806\pi$
$830$	0	0
$831$	0	0
$832$	9205.39	0.383581
$833$	13267.7	0.551861
$834$	0	0
$835$	0	0
$836$	−6254.59	−0.258755
$837$	0	0
$838$	17153.5	0.707109
$839$	43513.2	1.79052	0.895258	−	0.445549i	$-0.146992\pi$
$839$	43513.2	1.79052	0.895258	+	0.445549i	$0.146992\pi$
$840$	0	0
$841$	−14681.8	−0.601986
$842$	28299.4	1.15827
$843$	0	0
$844$	33152.9	1.35210
$845$	0	0
$846$	0	0
$847$	−278.819	−0.0113109
$848$	14387.5	0.582627
$849$	0	0
$850$	0	0
$851$	−30145.2	−1.21429
$852$	0	0
$853$	2454.18	0.0985107	0.0492554	−	0.998786i	$-0.484315\pi$
$853$	2454.18	0.0985107	0.0492554	+	0.998786i	$0.484315\pi$
$854$	−6351.68	−0.254508
$855$	0	0
$856$	−104.145	−0.00415840
$857$	−34642.8	−1.38083	−0.690417	−	0.723412i	$-0.742573\pi$
$857$	−34642.8	−1.38083	−0.690417	+	0.723412i	$0.742573\pi$
$858$	0	0
$859$	−20430.9	−0.811515	−0.405758	−	0.913981i	$-0.632992\pi$
$859$	−20430.9	−0.811515	−0.405758	+	0.913981i	$0.632992\pi$
$860$	0	0
$861$	0	0
$862$	22134.0	0.874578
$863$	−24873.4	−0.981113	−0.490556	−	0.871409i	$-0.663206\pi$
$863$	−24873.4	−0.981113	−0.490556	+	0.871409i	$0.663206\pi$
$864$	0	0
$865$	0	0
$866$	5526.25	0.216847
$867$	0	0
$868$	−729.661	−0.0285326
$869$	−7539.80	−0.294327
$870$	0	0
$871$	−9285.67	−0.361232
$872$	3053.00	0.118564
$873$	0	0
$874$	47463.4	1.83692
$875$	0	0
$876$	0	0
$877$	−4452.21	−0.171426	−0.0857130	−	0.996320i	$-0.527317\pi$
$877$	−4452.21	−0.171426	−0.0857130	+	0.996320i	$0.527317\pi$
$878$	−10009.6	−0.384748
$879$	0	0
$880$	0	0
$881$	8112.64	0.310241	0.155120	−	0.987896i	$-0.450423\pi$
$881$	8112.64	0.310241	0.155120	+	0.987896i	$0.450423\pi$
$882$	0	0
$883$	105.027	0.00400277	0.00200139	−	0.999998i	$-0.499363\pi$
$883$	105.027	0.00400277	0.00200139	+	0.999998i	$0.499363\pi$
$884$	5994.90	0.228088
$885$	0	0
$886$	28708.7	1.08859
$887$	19771.3	0.748429	0.374215	−	0.927342i	$-0.377912\pi$
$887$	19771.3	0.748429	0.374215	+	0.927342i	$0.377912\pi$
$888$	0	0
$889$	1997.74	0.0753679
$890$	0	0
$891$	0	0
$892$	−36550.6	−1.37198
$893$	12254.3	0.459211
$894$	0	0
$895$	0	0
$896$	482.029	0.0179726
$897$	0	0
$898$	−18312.4	−0.680503
$899$	4112.89	0.152584
$900$	0	0
$901$	8418.86	0.311291
$902$	−321.282	−0.0118598
$903$	0	0
$904$	2225.57	0.0818822
$905$	0	0
$906$	0	0
$907$	−42333.8	−1.54980	−0.774900	−	0.632083i	$-0.782200\pi$
$907$	−42333.8	−1.54980	−0.774900	+	0.632083i	$0.782200\pi$
$908$	20486.7	0.748760
$909$	0	0
$910$	0	0
$911$	−19460.4	−0.707741	−0.353870	−	0.935294i	$-0.615135\pi$
$911$	−19460.4	−0.707741	−0.353870	+	0.935294i	$0.615135\pi$
$912$	0	0
$913$	−6718.92	−0.243553
$914$	37316.6	1.35046
$915$	0	0
$916$	−16065.4	−0.579495
$917$	98.7381	0.00355575
$918$	0	0
$919$	41427.8	1.48703	0.743514	−	0.668720i	$-0.233158\pi$
$919$	41427.8	1.48703	0.743514	+	0.668720i	$0.233158\pi$
$920$	0	0
$921$	0	0
$922$	−37117.3	−1.32581
$923$	12759.2	0.455009
$924$	0	0
$925$	0	0
$926$	3867.06	0.137235
$927$	0	0
$928$	24826.7	0.878207
$929$	−440.804	−0.0155676	−0.00778381	−	0.999970i	$-0.502478\pi$
$929$	−440.804	−0.0155676	−0.00778381	+	0.999970i	$0.502478\pi$
$930$	0	0
$931$	25312.9	0.891081
$932$	38890.0	1.36683
$933$	0	0
$934$	30184.1	1.05745
$935$	0	0
$936$	0	0
$937$	12417.4	0.432932	0.216466	−	0.976290i	$-0.430547\pi$
$937$	12417.4	0.432932	0.216466	+	0.976290i	$0.430547\pi$
$938$	4199.43	0.146179
$939$	0	0
$940$	0	0
$941$	−27882.2	−0.965923	−0.482961	−	0.875642i	$-0.660439\pi$
$941$	−27882.2	−0.965923	−0.482961	+	0.875642i	$0.660439\pi$
$942$	0	0
$943$	1186.61	0.0409772
$944$	−35963.1	−1.23994
$945$	0	0
$946$	−23209.9	−0.797696
$947$	−38254.3	−1.31267	−0.656335	−	0.754469i	$-0.727894\pi$
$947$	−38254.3	−1.31267	−0.656335	+	0.754469i	$0.727894\pi$
$948$	0	0
$949$	−14756.8	−0.504768
$950$	0	0
$951$	0	0
$952$	148.158	0.00504393
$953$	−42519.1	−1.44526	−0.722628	−	0.691237i	$-0.757066\pi$
$953$	−42519.1	−1.44526	−0.722628	+	0.691237i	$0.757066\pi$
$954$	0	0
$955$	0	0
$956$	−24348.2	−0.823719
$957$	0	0
$958$	19444.5	0.655765
$959$	−7049.79	−0.237382
$960$	0	0
$961$	−28048.4	−0.941505
$962$	−14925.3	−0.500220
$963$	0	0
$964$	−28204.9	−0.942344
$965$	0	0
$966$	0	0
$967$	31049.7	1.03257	0.516283	−	0.856418i	$-0.327315\pi$
$967$	31049.7	1.03257	0.516283	+	0.856418i	$0.327315\pi$
$968$	198.013	0.00657478
$969$	0	0
$970$	0	0
$971$	23656.4	0.781843	0.390921	−	0.920424i	$-0.372156\pi$
$971$	23656.4	0.781843	0.390921	+	0.920424i	$0.372156\pi$
$972$	0	0
$973$	7463.28	0.245901
$974$	45695.1	1.50325
$975$	0	0
$976$	46881.4	1.53754
$977$	−37501.7	−1.22803	−0.614016	−	0.789294i	$-0.710447\pi$
$977$	−37501.7	−1.22803	−0.614016	+	0.789294i	$0.710447\pi$
$978$	0	0
$979$	−10109.9	−0.330045
$980$	0	0
$981$	0	0
$982$	39678.8	1.28941
$983$	6473.06	0.210029	0.105015	−	0.994471i	$-0.466511\pi$
$983$	6473.06	0.210029	0.105015	+	0.994471i	$0.466511\pi$
$984$	0	0
$985$	0	0
$986$	15282.2	0.493594
$987$	0	0
$988$	11437.4	0.368291
$989$	85722.9	2.75615
$990$	0	0
$991$	13607.0	0.436167	0.218083	−	0.975930i	$-0.430020\pi$
$991$	13607.0	0.436167	0.218083	+	0.975930i	$0.430020\pi$
$992$	10519.0	0.336671
$993$	0	0
$994$	−5770.32	−0.184128
$995$	0	0
$996$	0	0
$997$	−8421.39	−0.267511	−0.133755	−	0.991014i	$-0.542704\pi$
$997$	−8421.39	−0.267511	−0.133755	+	0.991014i	$0.542704\pi$
$998$	−32197.4	−1.02123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bn.1.1		6
3.2	odd	2		275.4.a.j.1.6		6
5.2	odd	4		495.4.c.a.199.1		6
5.3	odd	4		495.4.c.a.199.6		6
5.4	even	2	inner	2475.4.a.bn.1.6		6
15.2	even	4		55.4.b.a.34.6	yes	6
15.8	even	4		55.4.b.a.34.1	✓	6
15.14	odd	2		275.4.a.j.1.1		6
60.23	odd	4		880.4.b.f.529.5		6
60.47	odd	4		880.4.b.f.529.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.b.a.34.1	✓	6	15.8	even	4
55.4.b.a.34.6	yes	6	15.2	even	4
275.4.a.j.1.1		6	15.14	odd	2
275.4.a.j.1.6		6	3.2	odd	2
495.4.c.a.199.1		6	5.2	odd	4
495.4.c.a.199.6		6	5.3	odd	4
880.4.b.f.529.2		6	60.47	odd	4
880.4.b.f.529.5		6	60.23	odd	4
2475.4.a.bn.1.1		6	1.1	even	1	trivial
2475.4.a.bn.1.6		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-3.94784 q^{2} +7.58548 q^{4} -2.30429 q^{7} +1.63647 q^{8} +O(q^{10})\)	\(q-3.94784 q^{2} +7.58548 q^{4} -2.30429 q^{7} +1.63647 q^{8} +11.0000 q^{11} -20.1150 q^{13} +9.09698 q^{14} -67.1444 q^{16} -39.2897 q^{17} -74.9589 q^{19} -43.4263 q^{22} +160.389 q^{23} +79.4109 q^{26} -17.4791 q^{28} +98.5250 q^{29} +41.7447 q^{31} +251.984 q^{32} +155.110 q^{34} -187.950 q^{37} +295.926 q^{38} +7.39834 q^{41} +534.468 q^{43} +83.4402 q^{44} -633.192 q^{46} -163.480 q^{47} -337.690 q^{49} -152.582 q^{52} -214.277 q^{53} -3.77091 q^{56} -388.961 q^{58} +535.609 q^{59} -698.218 q^{61} -164.801 q^{62} -457.638 q^{64} +461.629 q^{67} -298.031 q^{68} -634.311 q^{71} +733.619 q^{73} +741.999 q^{74} -568.599 q^{76} -25.3472 q^{77} -685.436 q^{79} -29.2075 q^{82} -610.811 q^{83} -2110.00 q^{86} +18.0012 q^{88} -919.083 q^{89} +46.3508 q^{91} +1216.63 q^{92} +645.396 q^{94} +1466.45 q^{97} +1333.15 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4}+O(q^{10}) \) 6 * q - 4 * q^4	\( 6 q - 4 q^{4} + 66 q^{11} + 18 q^{14} - 108 q^{16} - 258 q^{19} + 356 q^{26} + 494 q^{29} - 514 q^{31} - 6 q^{34} + 824 q^{41} - 44 q^{44} - 940 q^{46} - 496 q^{49} + 130 q^{56} + 200 q^{59} - 2210 q^{61} - 676 q^{64} + 270 q^{71} - 266 q^{74} + 422 q^{76} - 824 q^{79} - 2476 q^{86} - 186 q^{89} + 2000 q^{91} + 660 q^{94}+O(q^{100}) \) 6 * q - 4 * q^4 + 66 * q^11 + 18 * q^14 - 108 * q^16 - 258 * q^19 + 356 * q^26 + 494 * q^29 - 514 * q^31 - 6 * q^34 + 824 * q^41 - 44 * q^44 - 940 * q^46 - 496 * q^49 + 130 * q^56 + 200 * q^59 - 2210 * q^61 - 676 * q^64 + 270 * q^71 - 266 * q^74 + 422 * q^76 - 824 * q^79 - 2476 * q^86 - 186 * q^89 + 2000 * q^91 + 660 * q^94

Introduction

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