LMFDB - Embedded newform 1500.4.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1500,4,Mod(1,1500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1500, 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("1500.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1500 = 2^{2} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1500.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:	$88.5028650086$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 199x^{4} + 329x^{3} + 8536x^{2} - 19710x + 2025 $ x^6 - x^5 - 199x^4 + 329x^3 + 8536x^2 - 19710x + 2025
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$7.44992$ of defining polynomial
Character	$\chi$	$=$	1500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -6.61902 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -6.61902 q^{7} +9.00000 q^{9} -17.8465 q^{11} -87.7534 q^{13} -37.3789 q^{17} -73.8745 q^{19} +19.8571 q^{21} -64.2627 q^{23} -27.0000 q^{27} +19.4492 q^{29} +107.814 q^{31} +53.5395 q^{33} +124.462 q^{37} +263.260 q^{39} +69.4637 q^{41} +4.34222 q^{43} -207.730 q^{47} -299.189 q^{49} +112.137 q^{51} +191.341 q^{53} +221.624 q^{57} +783.101 q^{59} -641.716 q^{61} -59.5712 q^{63} -347.308 q^{67} +192.788 q^{69} -282.089 q^{71} -488.902 q^{73} +118.126 q^{77} +77.8718 q^{79} +81.0000 q^{81} -746.043 q^{83} -58.3476 q^{87} +631.566 q^{89} +580.841 q^{91} -323.442 q^{93} +1154.28 q^{97} -160.619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{3} - 23 q^{7} + 54 q^{9}+O(q^{10}) $ 6 * q - 18 * q^3 - 23 * q^7 + 54 * q^9	$ 6 q - 18 q^{3} - 23 q^{7} + 54 q^{9} + 61 q^{11} + 15 q^{13} - 68 q^{17} + 145 q^{19} + 69 q^{21} - 77 q^{23} - 162 q^{27} + 136 q^{29} + 124 q^{31} - 183 q^{33} - 569 q^{37} - 45 q^{39} + 195 q^{41} - 152 q^{43} - 447 q^{47} + 321 q^{49} + 204 q^{51} - 829 q^{53} - 435 q^{57} + 171 q^{59} + 172 q^{61} - 207 q^{63} - 282 q^{67} + 231 q^{69} + 820 q^{71} - 1666 q^{73} - 478 q^{77} + 1152 q^{79} + 486 q^{81} - 152 q^{83} - 408 q^{87} + 955 q^{89} + 710 q^{91} - 372 q^{93} - 756 q^{97} + 549 q^{99}+O(q^{100}) $ 6 * q - 18 * q^3 - 23 * q^7 + 54 * q^9 + 61 * q^11 + 15 * q^13 - 68 * q^17 + 145 * q^19 + 69 * q^21 - 77 * q^23 - 162 * q^27 + 136 * q^29 + 124 * q^31 - 183 * q^33 - 569 * q^37 - 45 * q^39 + 195 * q^41 - 152 * q^43 - 447 * q^47 + 321 * q^49 + 204 * q^51 - 829 * q^53 - 435 * q^57 + 171 * q^59 + 172 * q^61 - 207 * q^63 - 282 * q^67 + 231 * q^69 + 820 * q^71 - 1666 * q^73 - 478 * q^77 + 1152 * q^79 + 486 * q^81 - 152 * q^83 - 408 * q^87 + 955 * q^89 + 710 * q^91 - 372 * q^93 - 756 * q^97 + 549 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.61902	−0.357393	−0.178697	−	0.983904i	$-0.557188\pi$
$7$	−6.61902	−0.357393	−0.178697	+	0.983904i	$0.557188\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−17.8465	−0.489175	−0.244587	−	0.969627i	$-0.578653\pi$
$11$	−17.8465	−0.489175	−0.244587	+	0.969627i	$0.578653\pi$
$12$	0	0
$13$	−87.7534	−1.87219	−0.936093	−	0.351753i	$-0.885586\pi$
$13$	−87.7534	−1.87219	−0.936093	+	0.351753i	$0.885586\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−37.3789	−0.533278	−0.266639	−	0.963797i	$-0.585913\pi$
$17$	−37.3789	−0.533278	−0.266639	+	0.963797i	$0.585913\pi$
$18$	0	0
$19$	−73.8745	−0.891999	−0.445999	−	0.895033i	$-0.647152\pi$
$19$	−73.8745	−0.891999	−0.445999	+	0.895033i	$0.647152\pi$
$20$	0	0
$21$	19.8571	0.206341
$22$	0	0
$23$	−64.2627	−0.582595	−0.291298	−	0.956632i	$-0.594087\pi$
$23$	−64.2627	−0.582595	−0.291298	+	0.956632i	$0.594087\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	19.4492	0.124539	0.0622694	−	0.998059i	$-0.480166\pi$
$29$	19.4492	0.124539	0.0622694	+	0.998059i	$0.480166\pi$
$30$	0	0
$31$	107.814	0.624644	0.312322	−	0.949976i	$-0.398893\pi$
$31$	107.814	0.624644	0.312322	+	0.949976i	$0.398893\pi$
$32$	0	0
$33$	53.5395	0.282425
$34$	0	0
$35$	0	0
$36$	0	0
$37$	124.462	0.553011	0.276505	−	0.961012i	$-0.410824\pi$
$37$	124.462	0.553011	0.276505	+	0.961012i	$0.410824\pi$
$38$	0	0
$39$	263.260	1.08091
$40$	0	0
$41$	69.4637	0.264595	0.132298	−	0.991210i	$-0.457765\pi$
$41$	69.4637	0.264595	0.132298	+	0.991210i	$0.457765\pi$
$42$	0	0
$43$	4.34222	0.0153996	0.00769979	−	0.999970i	$-0.497549\pi$
$43$	4.34222	0.0153996	0.00769979	+	0.999970i	$0.497549\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−207.730	−0.644692	−0.322346	−	0.946622i	$-0.604471\pi$
$47$	−207.730	−0.644692	−0.322346	+	0.946622i	$0.604471\pi$
$48$	0	0
$49$	−299.189	−0.872270
$50$	0	0
$51$	112.137	0.307888
$52$	0	0
$53$	191.341	0.495901	0.247951	−	0.968773i	$-0.420243\pi$
$53$	191.341	0.495901	0.247951	+	0.968773i	$0.420243\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	221.624	0.514996
$58$	0	0
$59$	783.101	1.72798	0.863992	−	0.503505i	$-0.167956\pi$
$59$	783.101	1.72798	0.863992	+	0.503505i	$0.167956\pi$
$60$	0	0
$61$	−641.716	−1.34694	−0.673470	−	0.739215i	$-0.735197\pi$
$61$	−641.716	−1.34694	−0.673470	+	0.739215i	$0.735197\pi$
$62$	0	0
$63$	−59.5712	−0.119131
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−347.308	−0.633290	−0.316645	−	0.948544i	$-0.602556\pi$
$67$	−347.308	−0.633290	−0.316645	+	0.948544i	$0.602556\pi$
$68$	0	0
$69$	192.788	0.336362
$70$	0	0
$71$	−282.089	−0.471518	−0.235759	−	0.971812i	$-0.575758\pi$
$71$	−282.089	−0.471518	−0.235759	+	0.971812i	$0.575758\pi$
$72$	0	0
$73$	−488.902	−0.783858	−0.391929	−	0.919995i	$-0.628192\pi$
$73$	−488.902	−0.783858	−0.391929	+	0.919995i	$0.628192\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	118.126	0.174828
$78$	0	0
$79$	77.8718	0.110902	0.0554510	−	0.998461i	$-0.482340\pi$
$79$	77.8718	0.110902	0.0554510	+	0.998461i	$0.482340\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−746.043	−0.986612	−0.493306	−	0.869856i	$-0.664212\pi$
$83$	−746.043	−0.986612	−0.493306	+	0.869856i	$0.664212\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−58.3476	−0.0719025
$88$	0	0
$89$	631.566	0.752201	0.376101	−	0.926579i	$-0.377265\pi$
$89$	631.566	0.752201	0.376101	+	0.926579i	$0.377265\pi$
$90$	0	0
$91$	580.841	0.669107
$92$	0	0
$93$	−323.442	−0.360638
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1154.28	1.20824	0.604122	−	0.796892i	$-0.293524\pi$
$97$	1154.28	1.20824	0.604122	+	0.796892i	$0.293524\pi$
$98$	0	0
$99$	−160.619	−0.163058
$100$	0	0
$101$	414.479	0.408339	0.204169	−	0.978936i	$-0.434551\pi$
$101$	414.479	0.408339	0.204169	+	0.978936i	$0.434551\pi$
$102$	0	0
$103$	−146.972	−0.140598	−0.0702990	−	0.997526i	$-0.522395\pi$
$103$	−146.972	−0.140598	−0.0702990	+	0.997526i	$0.522395\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−274.474	−0.247985	−0.123992	−	0.992283i	$-0.539570\pi$
$107$	−274.474	−0.247985	−0.123992	+	0.992283i	$0.539570\pi$
$108$	0	0
$109$	1533.63	1.34766	0.673832	−	0.738885i	$-0.264647\pi$
$109$	1533.63	1.34766	0.673832	+	0.738885i	$0.264647\pi$
$110$	0	0
$111$	−373.386	−0.319281
$112$	0	0
$113$	−1759.56	−1.46482	−0.732412	−	0.680861i	$-0.761606\pi$
$113$	−1759.56	−1.46482	−0.732412	+	0.680861i	$0.761606\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−789.781	−0.624062
$118$	0	0
$119$	247.412	0.190590
$120$	0	0
$121$	−1012.50	−0.760708
$122$	0	0
$123$	−208.391	−0.152764
$124$	0	0
$125$	0	0
$126$	0	0
$127$	613.918	0.428948	0.214474	−	0.976730i	$-0.431196\pi$
$127$	613.918	0.428948	0.214474	+	0.976730i	$0.431196\pi$
$128$	0	0
$129$	−13.0267	−0.00889095
$130$	0	0
$131$	1287.66	0.858804	0.429402	−	0.903114i	$-0.358724\pi$
$131$	1287.66	0.858804	0.429402	+	0.903114i	$0.358724\pi$
$132$	0	0
$133$	488.977	0.318795
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−104.587	−0.0652225	−0.0326112	−	0.999468i	$-0.510382\pi$
$137$	−104.587	−0.0652225	−0.0326112	+	0.999468i	$0.510382\pi$
$138$	0	0
$139$	2481.91	1.51448	0.757240	−	0.653137i	$-0.226548\pi$
$139$	2481.91	1.51448	0.757240	+	0.653137i	$0.226548\pi$
$140$	0	0
$141$	623.189	0.372213
$142$	0	0
$143$	1566.09	0.915826
$144$	0	0
$145$	0	0
$146$	0	0
$147$	897.566	0.503605
$148$	0	0
$149$	−5.99581	−0.00329662	−0.00164831	−	0.999999i	$-0.500525\pi$
$149$	−5.99581	−0.00329662	−0.00164831	+	0.999999i	$0.500525\pi$
$150$	0	0
$151$	520.615	0.280576	0.140288	−	0.990111i	$-0.455197\pi$
$151$	520.615	0.280576	0.140288	+	0.990111i	$0.455197\pi$
$152$	0	0
$153$	−336.410	−0.177759
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−748.992	−0.380739	−0.190370	−	0.981712i	$-0.560969\pi$
$157$	−748.992	−0.380739	−0.190370	+	0.981712i	$0.560969\pi$
$158$	0	0
$159$	−574.024	−0.286309
$160$	0	0
$161$	425.356	0.208216
$162$	0	0
$163$	1939.96	0.932207	0.466104	−	0.884730i	$-0.345657\pi$
$163$	1939.96	0.932207	0.466104	+	0.884730i	$0.345657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1216.42	0.563650	0.281825	−	0.959466i	$-0.409060\pi$
$167$	1216.42	0.563650	0.281825	+	0.959466i	$0.409060\pi$
$168$	0	0
$169$	5503.66	2.50508
$170$	0	0
$171$	−664.871	−0.297333
$172$	0	0
$173$	4148.53	1.82316	0.911580	−	0.411123i	$-0.134863\pi$
$173$	4148.53	1.82316	0.911580	+	0.411123i	$0.134863\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2349.30	−0.997653
$178$	0	0
$179$	3143.77	1.31272	0.656359	−	0.754449i	$-0.272096\pi$
$179$	3143.77	1.31272	0.656359	+	0.754449i	$0.272096\pi$
$180$	0	0
$181$	−2271.01	−0.932611	−0.466305	−	0.884624i	$-0.654415\pi$
$181$	−2271.01	−0.932611	−0.466305	+	0.884624i	$0.654415\pi$
$182$	0	0
$183$	1925.15	0.777656
$184$	0	0
$185$	0	0
$186$	0	0
$187$	667.083	0.260866
$188$	0	0
$189$	178.714	0.0687804
$190$	0	0
$191$	−4157.29	−1.57493	−0.787463	−	0.616362i	$-0.788606\pi$
$191$	−4157.29	−1.57493	−0.787463	+	0.616362i	$0.788606\pi$
$192$	0	0
$193$	−1210.99	−0.451654	−0.225827	−	0.974167i	$-0.572508\pi$
$193$	−1210.99	−0.451654	−0.225827	+	0.974167i	$0.572508\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4170.97	−1.50848	−0.754238	−	0.656602i	$-0.771993\pi$
$197$	−4170.97	−1.50848	−0.754238	+	0.656602i	$0.771993\pi$
$198$	0	0
$199$	3097.25	1.10331	0.551653	−	0.834074i	$-0.313997\pi$
$199$	3097.25	1.10331	0.551653	+	0.834074i	$0.313997\pi$
$200$	0	0
$201$	1041.92	0.365630
$202$	0	0
$203$	−128.735	−0.0445094
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−578.364	−0.194198
$208$	0	0
$209$	1318.40	0.436343
$210$	0	0
$211$	5358.63	1.74836	0.874178	−	0.485606i	$-0.161401\pi$
$211$	5358.63	1.74836	0.874178	+	0.485606i	$0.161401\pi$
$212$	0	0
$213$	846.267	0.272231
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−713.623	−0.223244
$218$	0	0
$219$	1466.71	0.452561
$220$	0	0
$221$	3280.13	0.998394
$222$	0	0
$223$	−1597.31	−0.479659	−0.239830	−	0.970815i	$-0.577092\pi$
$223$	−1597.31	−0.479659	−0.239830	+	0.970815i	$0.577092\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4478.36	1.30942	0.654712	−	0.755879i	$-0.272790\pi$
$227$	4478.36	1.30942	0.654712	+	0.755879i	$0.272790\pi$
$228$	0	0
$229$	963.658	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$229$	963.658	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$230$	0	0
$231$	−354.379	−0.100937
$232$	0	0
$233$	−3920.17	−1.10223	−0.551114	−	0.834430i	$-0.685797\pi$
$233$	−3920.17	−1.10223	−0.551114	+	0.834430i	$0.685797\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−233.615	−0.0640293
$238$	0	0
$239$	3397.65	0.919563	0.459782	−	0.888032i	$-0.347928\pi$
$239$	3397.65	0.919563	0.459782	+	0.888032i	$0.347928\pi$
$240$	0	0
$241$	−2703.30	−0.722550	−0.361275	−	0.932459i	$-0.617659\pi$
$241$	−2703.30	−0.722550	−0.361275	+	0.932459i	$0.617659\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6482.74	1.66999
$248$	0	0
$249$	2238.13	0.569621
$250$	0	0
$251$	−1511.36	−0.380065	−0.190032	−	0.981778i	$-0.560859\pi$
$251$	−1511.36	−0.380065	−0.190032	+	0.981778i	$0.560859\pi$
$252$	0	0
$253$	1146.86	0.284991
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3842.32	−0.932598	−0.466299	−	0.884627i	$-0.654413\pi$
$257$	−3842.32	−0.932598	−0.466299	+	0.884627i	$0.654413\pi$
$258$	0	0
$259$	−823.815	−0.197642
$260$	0	0
$261$	175.043	0.0415130
$262$	0	0
$263$	−1505.17	−0.352900	−0.176450	−	0.984310i	$-0.556461\pi$
$263$	−1505.17	−0.352900	−0.176450	+	0.984310i	$0.556461\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1894.70	−0.434284
$268$	0	0
$269$	6931.71	1.57113	0.785565	−	0.618779i	$-0.212373\pi$
$269$	6931.71	1.57113	0.785565	+	0.618779i	$0.212373\pi$
$270$	0	0
$271$	2099.72	0.470660	0.235330	−	0.971916i	$-0.424383\pi$
$271$	2099.72	0.470660	0.235330	+	0.971916i	$0.424383\pi$
$272$	0	0
$273$	−1742.52	−0.386309
$274$	0	0
$275$	0	0
$276$	0	0
$277$	485.763	0.105367	0.0526835	−	0.998611i	$-0.483223\pi$
$277$	485.763	0.105367	0.0526835	+	0.998611i	$0.483223\pi$
$278$	0	0
$279$	970.326	0.208215
$280$	0	0
$281$	−7507.38	−1.59378	−0.796891	−	0.604123i	$-0.793523\pi$
$281$	−7507.38	−1.59378	−0.796891	+	0.604123i	$0.793523\pi$
$282$	0	0
$283$	15.6822	0.00329404	0.00164702	−	0.999999i	$-0.499476\pi$
$283$	15.6822	0.00329404	0.00164702	+	0.999999i	$0.499476\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−459.782	−0.0945647
$288$	0	0
$289$	−3515.82	−0.715615
$290$	0	0
$291$	−3462.85	−0.697580
$292$	0	0
$293$	7856.99	1.56659	0.783294	−	0.621651i	$-0.213538\pi$
$293$	7856.99	1.56659	0.783294	+	0.621651i	$0.213538\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	481.856	0.0941417
$298$	0	0
$299$	5639.27	1.09073
$300$	0	0
$301$	−28.7412	−0.00550371
$302$	0	0
$303$	−1243.44	−0.235755
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5306.16	0.986446	0.493223	−	0.869903i	$-0.335819\pi$
$307$	5306.16	0.986446	0.493223	+	0.869903i	$0.335819\pi$
$308$	0	0
$309$	440.916	0.0811743
$310$	0	0
$311$	9348.25	1.70447	0.852236	−	0.523157i	$-0.175246\pi$
$311$	9348.25	1.70447	0.852236	+	0.523157i	$0.175246\pi$
$312$	0	0
$313$	4602.96	0.831230	0.415615	−	0.909541i	$-0.363566\pi$
$313$	4602.96	0.831230	0.415615	+	0.909541i	$0.363566\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1173.27	−0.207878	−0.103939	−	0.994584i	$-0.533145\pi$
$317$	−1173.27	−0.207878	−0.103939	+	0.994584i	$0.533145\pi$
$318$	0	0
$319$	−347.100	−0.0609213
$320$	0	0
$321$	823.421	0.143174
$322$	0	0
$323$	2761.35	0.475683
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4600.90	−0.778074
$328$	0	0
$329$	1374.97	0.230409
$330$	0	0
$331$	1147.77	0.190596	0.0952980	−	0.995449i	$-0.469620\pi$
$331$	1147.77	0.190596	0.0952980	+	0.995449i	$0.469620\pi$
$332$	0	0
$333$	1120.16	0.184337
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9017.73	1.45765	0.728824	−	0.684701i	$-0.240067\pi$
$337$	9017.73	1.45765	0.728824	+	0.684701i	$0.240067\pi$
$338$	0	0
$339$	5278.67	0.845717
$340$	0	0
$341$	−1924.10	−0.305560
$342$	0	0
$343$	4250.66	0.669137
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10857.8	−1.67976	−0.839882	−	0.542769i	$-0.817376\pi$
$347$	−10857.8	−1.67976	−0.839882	+	0.542769i	$0.817376\pi$
$348$	0	0
$349$	2757.87	0.422996	0.211498	−	0.977378i	$-0.432166\pi$
$349$	2757.87	0.422996	0.211498	+	0.977378i	$0.432166\pi$
$350$	0	0
$351$	2369.34	0.360302
$352$	0	0
$353$	−3107.17	−0.468493	−0.234247	−	0.972177i	$-0.575262\pi$
$353$	−3107.17	−0.468493	−0.234247	+	0.972177i	$0.575262\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−742.235	−0.110037
$358$	0	0
$359$	−1905.22	−0.280093	−0.140047	−	0.990145i	$-0.544725\pi$
$359$	−1905.22	−0.280093	−0.140047	+	0.990145i	$0.544725\pi$
$360$	0	0
$361$	−1401.55	−0.204338
$362$	0	0
$363$	3037.51	0.439195
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4388.28	0.624159	0.312080	−	0.950056i	$-0.398974\pi$
$367$	4388.28	0.624159	0.312080	+	0.950056i	$0.398974\pi$
$368$	0	0
$369$	625.174	0.0881985
$370$	0	0
$371$	−1266.49	−0.177232
$372$	0	0
$373$	1738.19	0.241287	0.120644	−	0.992696i	$-0.461504\pi$
$373$	1738.19	0.241287	0.120644	+	0.992696i	$0.461504\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1706.73	−0.233160
$378$	0	0
$379$	−6776.90	−0.918486	−0.459243	−	0.888311i	$-0.651879\pi$
$379$	−6776.90	−0.918486	−0.459243	+	0.888311i	$0.651879\pi$
$380$	0	0
$381$	−1841.75	−0.247653
$382$	0	0
$383$	−9792.38	−1.30644	−0.653221	−	0.757168i	$-0.726583\pi$
$383$	−9792.38	−1.30644	−0.653221	+	0.757168i	$0.726583\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	39.0800	0.00513319
$388$	0	0
$389$	4944.39	0.644448	0.322224	−	0.946663i	$-0.395570\pi$
$389$	4944.39	0.644448	0.322224	+	0.946663i	$0.395570\pi$
$390$	0	0
$391$	2402.07	0.310685
$392$	0	0
$393$	−3862.98	−0.495830
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4874.09	0.616180	0.308090	−	0.951357i	$-0.400310\pi$
$397$	4874.09	0.616180	0.308090	+	0.951357i	$0.400310\pi$
$398$	0	0
$399$	−1466.93	−0.184056
$400$	0	0
$401$	7855.50	0.978266	0.489133	−	0.872209i	$-0.337313\pi$
$401$	7855.50	0.978266	0.489133	+	0.872209i	$0.337313\pi$
$402$	0	0
$403$	−9461.04	−1.16945
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2221.21	−0.270519
$408$	0	0
$409$	8894.04	1.07526	0.537631	−	0.843181i	$-0.319320\pi$
$409$	8894.04	1.07526	0.537631	+	0.843181i	$0.319320\pi$
$410$	0	0
$411$	313.761	0.0376562
$412$	0	0
$413$	−5183.36	−0.617570
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7445.72	−0.874385
$418$	0	0
$419$	−2347.48	−0.273704	−0.136852	−	0.990591i	$-0.543699\pi$
$419$	−2347.48	−0.273704	−0.136852	+	0.990591i	$0.543699\pi$
$420$	0	0
$421$	−106.781	−0.0123615	−0.00618075	−	0.999981i	$-0.501967\pi$
$421$	−106.781	−0.0123615	−0.00618075	+	0.999981i	$0.501967\pi$
$422$	0	0
$423$	−1869.57	−0.214897
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4247.53	0.481387
$428$	0	0
$429$	−4698.27	−0.528752
$430$	0	0
$431$	1648.76	0.184265	0.0921324	−	0.995747i	$-0.470632\pi$
$431$	1648.76	0.184265	0.0921324	+	0.995747i	$0.470632\pi$
$432$	0	0
$433$	−8295.66	−0.920702	−0.460351	−	0.887737i	$-0.652276\pi$
$433$	−8295.66	−0.920702	−0.460351	+	0.887737i	$0.652276\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4747.37	0.519674
$438$	0	0
$439$	13403.7	1.45723	0.728613	−	0.684925i	$-0.240165\pi$
$439$	13403.7	1.45723	0.728613	+	0.684925i	$0.240165\pi$
$440$	0	0
$441$	−2692.70	−0.290757
$442$	0	0
$443$	−320.338	−0.0343560	−0.0171780	−	0.999852i	$-0.505468\pi$
$443$	−320.338	−0.0343560	−0.0171780	+	0.999852i	$0.505468\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.9874	0.00190330
$448$	0	0
$449$	−11716.3	−1.23146	−0.615732	−	0.787956i	$-0.711140\pi$
$449$	−11716.3	−1.23146	−0.615732	+	0.787956i	$0.711140\pi$
$450$	0	0
$451$	−1239.68	−0.129433
$452$	0	0
$453$	−1561.84	−0.161991
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1055.01	0.107990	0.0539948	−	0.998541i	$-0.482805\pi$
$457$	1055.01	0.107990	0.0539948	+	0.998541i	$0.482805\pi$
$458$	0	0
$459$	1009.23	0.102629
$460$	0	0
$461$	−17020.5	−1.71957	−0.859785	−	0.510656i	$-0.829403\pi$
$461$	−17020.5	−1.71957	−0.859785	+	0.510656i	$0.829403\pi$
$462$	0	0
$463$	−12977.6	−1.30264	−0.651318	−	0.758805i	$-0.725784\pi$
$463$	−12977.6	−1.30264	−0.651318	+	0.758805i	$0.725784\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15404.4	−1.52640	−0.763201	−	0.646162i	$-0.776373\pi$
$467$	−15404.4	−1.52640	−0.763201	+	0.646162i	$0.776373\pi$
$468$	0	0
$469$	2298.84	0.226334
$470$	0	0
$471$	2246.98	0.219820
$472$	0	0
$473$	−77.4934	−0.00753309
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1722.07	0.165300
$478$	0	0
$479$	−15235.6	−1.45330	−0.726650	−	0.687008i	$-0.758924\pi$
$479$	−15235.6	−1.45330	−0.726650	+	0.687008i	$0.758924\pi$
$480$	0	0
$481$	−10921.9	−1.03534
$482$	0	0
$483$	−1276.07	−0.120213
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12958.2	1.20573	0.602866	−	0.797843i	$-0.294025\pi$
$487$	12958.2	1.20573	0.602866	+	0.797843i	$0.294025\pi$
$488$	0	0
$489$	−5819.89	−0.538210
$490$	0	0
$491$	18790.2	1.72707	0.863534	−	0.504290i	$-0.168246\pi$
$491$	18790.2	1.72707	0.863534	+	0.504290i	$0.168246\pi$
$492$	0	0
$493$	−726.990	−0.0664138
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1867.15	0.168517
$498$	0	0
$499$	−10604.2	−0.951317	−0.475658	−	0.879630i	$-0.657790\pi$
$499$	−10604.2	−0.951317	−0.475658	+	0.879630i	$0.657790\pi$
$500$	0	0
$501$	−3649.27	−0.325424
$502$	0	0
$503$	3814.90	0.338167	0.169084	−	0.985602i	$-0.445919\pi$
$503$	3814.90	0.338167	0.169084	+	0.985602i	$0.445919\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−16511.0	−1.44631
$508$	0	0
$509$	−8697.29	−0.757368	−0.378684	−	0.925526i	$-0.623623\pi$
$509$	−8697.29	−0.757368	−0.378684	+	0.925526i	$0.623623\pi$
$510$	0	0
$511$	3236.05	0.280146
$512$	0	0
$513$	1994.61	0.171665
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3707.25	0.315367
$518$	0	0
$519$	−12445.6	−1.05260
$520$	0	0
$521$	−17493.3	−1.47101	−0.735504	−	0.677520i	$-0.763055\pi$
$521$	−17493.3	−1.47101	−0.735504	+	0.677520i	$0.763055\pi$
$522$	0	0
$523$	13787.3	1.15273	0.576363	−	0.817194i	$-0.304471\pi$
$523$	13787.3	1.15273	0.576363	+	0.817194i	$0.304471\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4029.97	−0.333108
$528$	0	0
$529$	−8037.31	−0.660583
$530$	0	0
$531$	7047.91	0.575995
$532$	0	0
$533$	−6095.68	−0.495372
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9431.31	−0.757898
$538$	0	0
$539$	5339.47	0.426692
$540$	0	0
$541$	3355.09	0.266629	0.133315	−	0.991074i	$-0.457438\pi$
$541$	3355.09	0.266629	0.133315	+	0.991074i	$0.457438\pi$
$542$	0	0
$543$	6813.02	0.538443
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22548.0	1.76249	0.881247	−	0.472655i	$-0.156704\pi$
$547$	22548.0	1.76249	0.881247	+	0.472655i	$0.156704\pi$
$548$	0	0
$549$	−5775.45	−0.448980
$550$	0	0
$551$	−1436.80	−0.111089
$552$	0	0
$553$	−515.435	−0.0396357
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18784.7	−1.42896	−0.714481	−	0.699655i	$-0.753337\pi$
$557$	−18784.7	−1.42896	−0.714481	+	0.699655i	$0.753337\pi$
$558$	0	0
$559$	−381.044	−0.0288309
$560$	0	0
$561$	−2001.25	−0.150611
$562$	0	0
$563$	12105.9	0.906221	0.453111	−	0.891454i	$-0.350314\pi$
$563$	12105.9	0.906221	0.453111	+	0.891454i	$0.350314\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−536.141	−0.0397104
$568$	0	0
$569$	−6348.65	−0.467749	−0.233874	−	0.972267i	$-0.575140\pi$
$569$	−6348.65	−0.467749	−0.233874	+	0.972267i	$0.575140\pi$
$570$	0	0
$571$	−33.9275	−0.00248655	−0.00124328	−	0.999999i	$-0.500396\pi$
$571$	−33.9275	−0.00248655	−0.00124328	+	0.999999i	$0.500396\pi$
$572$	0	0
$573$	12471.9	0.909284
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16003.1	1.15462	0.577312	−	0.816524i	$-0.304102\pi$
$577$	16003.1	1.15462	0.577312	+	0.816524i	$0.304102\pi$
$578$	0	0
$579$	3632.98	0.260763
$580$	0	0
$581$	4938.07	0.352609
$582$	0	0
$583$	−3414.78	−0.242582
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11097.1	0.780282	0.390141	−	0.920755i	$-0.372426\pi$
$587$	11097.1	0.780282	0.390141	+	0.920755i	$0.372426\pi$
$588$	0	0
$589$	−7964.70	−0.557182
$590$	0	0
$591$	12512.9	0.870918
$592$	0	0
$593$	14101.0	0.976491	0.488245	−	0.872706i	$-0.337637\pi$
$593$	14101.0	0.976491	0.488245	+	0.872706i	$0.337637\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9291.74	−0.636994
$598$	0	0
$599$	−15179.9	−1.03545	−0.517726	−	0.855547i	$-0.673221\pi$
$599$	−15179.9	−1.03545	−0.517726	+	0.855547i	$0.673221\pi$
$600$	0	0
$601$	−1690.64	−0.114747	−0.0573733	−	0.998353i	$-0.518273\pi$
$601$	−1690.64	−0.114747	−0.0573733	+	0.998353i	$0.518273\pi$
$602$	0	0
$603$	−3125.77	−0.211097
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−296.900	−0.0198530	−0.00992651	−	0.999951i	$-0.503160\pi$
$607$	−296.900	−0.0198530	−0.00992651	+	0.999951i	$0.503160\pi$
$608$	0	0
$609$	386.204	0.0256975
$610$	0	0
$611$	18229.0	1.20698
$612$	0	0
$613$	3440.96	0.226720	0.113360	−	0.993554i	$-0.463839\pi$
$613$	3440.96	0.226720	0.113360	+	0.993554i	$0.463839\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12232.6	−0.798161	−0.399080	−	0.916916i	$-0.630671\pi$
$617$	−12232.6	−0.798161	−0.399080	+	0.916916i	$0.630671\pi$
$618$	0	0
$619$	−16154.1	−1.04893	−0.524467	−	0.851431i	$-0.675735\pi$
$619$	−16154.1	−1.04893	−0.524467	+	0.851431i	$0.675735\pi$
$620$	0	0
$621$	1735.09	0.112121
$622$	0	0
$623$	−4180.35	−0.268832
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3955.21	−0.251923
$628$	0	0
$629$	−4652.25	−0.294908
$630$	0	0
$631$	937.305	0.0591340	0.0295670	−	0.999563i	$-0.490587\pi$
$631$	937.305	0.0591340	0.0295670	+	0.999563i	$0.490587\pi$
$632$	0	0
$633$	−16075.9	−1.00941
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26254.8	1.63305
$638$	0	0
$639$	−2538.80	−0.157173
$640$	0	0
$641$	−13871.3	−0.854730	−0.427365	−	0.904079i	$-0.640558\pi$
$641$	−13871.3	−0.854730	−0.427365	+	0.904079i	$0.640558\pi$
$642$	0	0
$643$	10978.4	0.673320	0.336660	−	0.941626i	$-0.390703\pi$
$643$	10978.4	0.673320	0.336660	+	0.941626i	$0.390703\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26974.4	1.63906	0.819532	−	0.573033i	$-0.194233\pi$
$647$	26974.4	1.63906	0.819532	+	0.573033i	$0.194233\pi$
$648$	0	0
$649$	−13975.6	−0.845287
$650$	0	0
$651$	2140.87	0.128890
$652$	0	0
$653$	−11307.1	−0.677611	−0.338806	−	0.940856i	$-0.610023\pi$
$653$	−11307.1	−0.677611	−0.338806	+	0.940856i	$0.610023\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4400.12	−0.261286
$658$	0	0
$659$	−4996.63	−0.295358	−0.147679	−	0.989035i	$-0.547180\pi$
$659$	−4996.63	−0.295358	−0.147679	+	0.989035i	$0.547180\pi$
$660$	0	0
$661$	−2785.67	−0.163918	−0.0819590	−	0.996636i	$-0.526118\pi$
$661$	−2785.67	−0.163918	−0.0819590	+	0.996636i	$0.526118\pi$
$662$	0	0
$663$	−9840.38	−0.576423
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1249.86	−0.0725558
$668$	0	0
$669$	4791.94	0.276931
$670$	0	0
$671$	11452.4	0.658889
$672$	0	0
$673$	25166.8	1.44147	0.720735	−	0.693211i	$-0.243805\pi$
$673$	25166.8	1.44147	0.720735	+	0.693211i	$0.243805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19293.3	−1.09527	−0.547637	−	0.836716i	$-0.684472\pi$
$677$	−19293.3	−1.09527	−0.547637	+	0.836716i	$0.684472\pi$
$678$	0	0
$679$	−7640.22	−0.431819
$680$	0	0
$681$	−13435.1	−0.755996
$682$	0	0
$683$	3548.69	0.198809	0.0994047	−	0.995047i	$-0.468306\pi$
$683$	3548.69	0.198809	0.0994047	+	0.995047i	$0.468306\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2890.97	−0.160550
$688$	0	0
$689$	−16790.9	−0.928419
$690$	0	0
$691$	−10769.1	−0.592872	−0.296436	−	0.955053i	$-0.595798\pi$
$691$	−10769.1	−0.592872	−0.296436	+	0.955053i	$0.595798\pi$
$692$	0	0
$693$	1063.14	0.0582759
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2596.48	−0.141103
$698$	0	0
$699$	11760.5	0.636372
$700$	0	0
$701$	−15997.9	−0.861960	−0.430980	−	0.902361i	$-0.641832\pi$
$701$	−15997.9	−0.861960	−0.430980	+	0.902361i	$0.641832\pi$
$702$	0	0
$703$	−9194.56	−0.493285
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2743.45	−0.145938
$708$	0	0
$709$	−32203.6	−1.70583	−0.852914	−	0.522051i	$-0.825167\pi$
$709$	−32203.6	−1.70583	−0.852914	+	0.522051i	$0.825167\pi$
$710$	0	0
$711$	700.846	0.0369674
$712$	0	0
$713$	−6928.41	−0.363915
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10192.9	−0.530910
$718$	0	0
$719$	10510.9	0.545190	0.272595	−	0.962129i	$-0.412118\pi$
$719$	10510.9	0.545190	0.272595	+	0.962129i	$0.412118\pi$
$720$	0	0
$721$	972.811	0.0502488
$722$	0	0
$723$	8109.89	0.417165
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23405.1	1.19401	0.597006	−	0.802237i	$-0.296357\pi$
$727$	23405.1	1.19401	0.597006	+	0.802237i	$0.296357\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−162.307	−0.00821225
$732$	0	0
$733$	−21709.7	−1.09395	−0.546974	−	0.837149i	$-0.684221\pi$
$733$	−21709.7	−1.09395	−0.546974	+	0.837149i	$0.684221\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6198.24	0.309790
$738$	0	0
$739$	−29065.2	−1.44679	−0.723397	−	0.690432i	$-0.757420\pi$
$739$	−29065.2	−1.44679	−0.723397	+	0.690432i	$0.757420\pi$
$740$	0	0
$741$	−19448.2	−0.964168
$742$	0	0
$743$	9515.59	0.469843	0.234921	−	0.972014i	$-0.424517\pi$
$743$	9515.59	0.469843	0.234921	+	0.972014i	$0.424517\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6714.38	−0.328871
$748$	0	0
$749$	1816.75	0.0886281
$750$	0	0
$751$	16374.7	0.795633	0.397817	−	0.917465i	$-0.369768\pi$
$751$	16374.7	0.795633	0.397817	+	0.917465i	$0.369768\pi$
$752$	0	0
$753$	4534.08	0.219431
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16014.6	0.768903	0.384451	−	0.923145i	$-0.374391\pi$
$757$	16014.6	0.768903	0.384451	+	0.923145i	$0.374391\pi$
$758$	0	0
$759$	−3440.59	−0.164540
$760$	0	0
$761$	16205.3	0.771935	0.385967	−	0.922512i	$-0.373868\pi$
$761$	16205.3	0.771935	0.385967	+	0.922512i	$0.373868\pi$
$762$	0	0
$763$	−10151.1	−0.481646
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−68719.8	−3.23511
$768$	0	0
$769$	−31525.4	−1.47833	−0.739164	−	0.673526i	$-0.764779\pi$
$769$	−31525.4	−1.47833	−0.739164	+	0.673526i	$0.764779\pi$
$770$	0	0
$771$	11527.0	0.538435
$772$	0	0
$773$	−15720.8	−0.731483	−0.365742	−	0.930716i	$-0.619185\pi$
$773$	−15720.8	−0.731483	−0.365742	+	0.930716i	$0.619185\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2471.45	0.114109
$778$	0	0
$779$	−5131.60	−0.236019
$780$	0	0
$781$	5034.30	0.230655
$782$	0	0
$783$	−525.129	−0.0239675
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29815.1	1.35044	0.675219	−	0.737618i	$-0.264049\pi$
$787$	29815.1	1.35044	0.675219	+	0.737618i	$0.264049\pi$
$788$	0	0
$789$	4515.51	0.203747
$790$	0	0
$791$	11646.5	0.523519
$792$	0	0
$793$	56312.8	2.52172
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11434.5	0.508193	0.254097	−	0.967179i	$-0.418222\pi$
$797$	11434.5	0.508193	0.254097	+	0.967179i	$0.418222\pi$
$798$	0	0
$799$	7764.71	0.343800
$800$	0	0
$801$	5684.10	0.250734
$802$	0	0
$803$	8725.19	0.383444
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20795.1	−0.907092
$808$	0	0
$809$	−15795.6	−0.686457	−0.343229	−	0.939252i	$-0.611521\pi$
$809$	−15795.6	−0.686457	−0.343229	+	0.939252i	$0.611521\pi$
$810$	0	0
$811$	−8624.65	−0.373431	−0.186715	−	0.982414i	$-0.559784\pi$
$811$	−8624.65	−0.373431	−0.186715	+	0.982414i	$0.559784\pi$
$812$	0	0
$813$	−6299.16	−0.271736
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−320.779	−0.0137364
$818$	0	0
$819$	5227.57	0.223036
$820$	0	0
$821$	−13499.1	−0.573838	−0.286919	−	0.957955i	$-0.592631\pi$
$821$	−13499.1	−0.573838	−0.286919	+	0.957955i	$0.592631\pi$
$822$	0	0
$823$	5000.26	0.211784	0.105892	−	0.994378i	$-0.466230\pi$
$823$	5000.26	0.211784	0.105892	+	0.994378i	$0.466230\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17629.0	−0.741256	−0.370628	−	0.928781i	$-0.620858\pi$
$827$	−17629.0	−0.741256	−0.370628	+	0.928781i	$0.620858\pi$
$828$	0	0
$829$	14297.7	0.599010	0.299505	−	0.954095i	$-0.403178\pi$
$829$	14297.7	0.599010	0.299505	+	0.954095i	$0.403178\pi$
$830$	0	0
$831$	−1457.29	−0.0608337
$832$	0	0
$833$	11183.3	0.465162
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2910.98	−0.120213
$838$	0	0
$839$	−39759.2	−1.63604	−0.818021	−	0.575188i	$-0.804929\pi$
$839$	−39759.2	−1.63604	−0.818021	+	0.575188i	$0.804929\pi$
$840$	0	0
$841$	−24010.7	−0.984490
$842$	0	0
$843$	22522.1	0.920170
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6701.77	0.271872
$848$	0	0
$849$	−47.0467	−0.00190181
$850$	0	0
$851$	−7998.25	−0.322182
$852$	0	0
$853$	−12468.0	−0.500464	−0.250232	−	0.968186i	$-0.580507\pi$
$853$	−12468.0	−0.500464	−0.250232	+	0.968186i	$0.580507\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4494.37	−0.179142	−0.0895710	−	0.995980i	$-0.528550\pi$
$857$	−4494.37	−0.179142	−0.0895710	+	0.995980i	$0.528550\pi$
$858$	0	0
$859$	−19659.9	−0.780891	−0.390446	−	0.920626i	$-0.627679\pi$
$859$	−19659.9	−0.780891	−0.390446	+	0.920626i	$0.627679\pi$
$860$	0	0
$861$	1379.35	0.0545969
$862$	0	0
$863$	29270.1	1.15454	0.577268	−	0.816555i	$-0.304119\pi$
$863$	29270.1	1.15454	0.577268	+	0.816555i	$0.304119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10547.5	0.413161
$868$	0	0
$869$	−1389.74	−0.0542505
$870$	0	0
$871$	30477.5	1.18564
$872$	0	0
$873$	10388.6	0.402748
$874$	0	0
$875$	0	0
$876$	0	0
$877$	43860.4	1.68878	0.844391	−	0.535727i	$-0.179962\pi$
$877$	43860.4	1.68878	0.844391	+	0.535727i	$0.179962\pi$
$878$	0	0
$879$	−23571.0	−0.904470
$880$	0	0
$881$	−6060.07	−0.231747	−0.115873	−	0.993264i	$-0.536967\pi$
$881$	−6060.07	−0.231747	−0.115873	+	0.993264i	$0.536967\pi$
$882$	0	0
$883$	−33837.6	−1.28961	−0.644805	−	0.764347i	$-0.723061\pi$
$883$	−33837.6	−1.28961	−0.644805	+	0.764347i	$0.723061\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5957.96	0.225534	0.112767	−	0.993621i	$-0.464029\pi$
$887$	5957.96	0.225534	0.112767	+	0.993621i	$0.464029\pi$
$888$	0	0
$889$	−4063.54	−0.153303
$890$	0	0
$891$	−1445.57	−0.0543528
$892$	0	0
$893$	15345.9	0.575064
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16917.8	−0.629731
$898$	0	0
$899$	2096.90	0.0777924
$900$	0	0
$901$	−7152.14	−0.264453
$902$	0	0
$903$	86.2237	0.00317757
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−52622.5	−1.92646	−0.963232	−	0.268672i	$-0.913415\pi$
$907$	−52622.5	−1.92646	−0.963232	+	0.268672i	$0.913415\pi$
$908$	0	0
$909$	3730.31	0.136113
$910$	0	0
$911$	10593.3	0.385262	0.192631	−	0.981271i	$-0.438298\pi$
$911$	10593.3	0.385262	0.192631	+	0.981271i	$0.438298\pi$
$912$	0	0
$913$	13314.3	0.482626
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8523.04	−0.306931
$918$	0	0
$919$	23964.9	0.860206	0.430103	−	0.902780i	$-0.358477\pi$
$919$	23964.9	0.860206	0.430103	+	0.902780i	$0.358477\pi$
$920$	0	0
$921$	−15918.5	−0.569525
$922$	0	0
$923$	24754.3	0.882769
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1322.75	−0.0468660
$928$	0	0
$929$	33770.8	1.19266	0.596332	−	0.802738i	$-0.296624\pi$
$929$	33770.8	1.19266	0.596332	+	0.802738i	$0.296624\pi$
$930$	0	0
$931$	22102.4	0.778064
$932$	0	0
$933$	−28044.8	−0.984078
$934$	0	0
$935$	0	0
$936$	0	0
$937$	54057.5	1.88472	0.942359	−	0.334602i	$-0.108602\pi$
$937$	54057.5	1.88472	0.942359	+	0.334602i	$0.108602\pi$
$938$	0	0
$939$	−13808.9	−0.479911
$940$	0	0
$941$	−35376.6	−1.22555	−0.612775	−	0.790257i	$-0.709947\pi$
$941$	−35376.6	−1.22555	−0.612775	+	0.790257i	$0.709947\pi$
$942$	0	0
$943$	−4463.93	−0.154152
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11559.4	−0.396653	−0.198327	−	0.980136i	$-0.563551\pi$
$947$	−11559.4	−0.396653	−0.198327	+	0.980136i	$0.563551\pi$
$948$	0	0
$949$	42902.8	1.46753
$950$	0	0
$951$	3519.81	0.120019
$952$	0	0
$953$	−36942.1	−1.25569	−0.627845	−	0.778338i	$-0.716063\pi$
$953$	−36942.1	−1.25569	−0.627845	+	0.778338i	$0.716063\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1041.30	0.0351729
$958$	0	0
$959$	692.264	0.0233101
$960$	0	0
$961$	−18167.2	−0.609820
$962$	0	0
$963$	−2470.26	−0.0826616
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14131.8	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$967$	14131.8	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$968$	0	0
$969$	−8284.05	−0.274636
$970$	0	0
$971$	27189.2	0.898604	0.449302	−	0.893380i	$-0.351673\pi$
$971$	27189.2	0.898604	0.449302	+	0.893380i	$0.351673\pi$
$972$	0	0
$973$	−16427.8	−0.541265
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3174.68	−0.103958	−0.0519791	−	0.998648i	$-0.516553\pi$
$977$	−3174.68	−0.103958	−0.0519791	+	0.998648i	$0.516553\pi$
$978$	0	0
$979$	−11271.3	−0.367958
$980$	0	0
$981$	13802.7	0.449221
$982$	0	0
$983$	−53701.2	−1.74242	−0.871211	−	0.490908i	$-0.836665\pi$
$983$	−53701.2	−1.74242	−0.871211	+	0.490908i	$0.836665\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4124.90	−0.133026
$988$	0	0
$989$	−279.043	−0.00897173
$990$	0	0
$991$	27418.6	0.878889	0.439445	−	0.898270i	$-0.355175\pi$
$991$	27418.6	0.878889	0.439445	+	0.898270i	$0.355175\pi$
$992$	0	0
$993$	−3443.32	−0.110041
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42205.7	1.34069	0.670346	−	0.742049i	$-0.266146\pi$
$997$	42205.7	1.34069	0.670346	+	0.742049i	$0.266146\pi$
$998$	0	0
$999$	−3360.47	−0.106427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1500.4.a.b.1.4	✓	6
5.2	odd	4		1500.4.d.d.1249.10		12
5.3	odd	4		1500.4.d.d.1249.3		12
5.4	even	2		1500.4.a.g.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.4.a.b.1.4	✓	6	1.1	even	1	trivial
1500.4.a.g.1.3	yes	6	5.4	even	2
1500.4.d.d.1249.3		12	5.3	odd	4
1500.4.d.d.1249.10		12	5.2	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:	\( N \)	\(=\)	\( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1500.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -6.61902 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -6.61902 q^{7} +9.00000 q^{9} -17.8465 q^{11} -87.7534 q^{13} -37.3789 q^{17} -73.8745 q^{19} +19.8571 q^{21} -64.2627 q^{23} -27.0000 q^{27} +19.4492 q^{29} +107.814 q^{31} +53.5395 q^{33} +124.462 q^{37} +263.260 q^{39} +69.4637 q^{41} +4.34222 q^{43} -207.730 q^{47} -299.189 q^{49} +112.137 q^{51} +191.341 q^{53} +221.624 q^{57} +783.101 q^{59} -641.716 q^{61} -59.5712 q^{63} -347.308 q^{67} +192.788 q^{69} -282.089 q^{71} -488.902 q^{73} +118.126 q^{77} +77.8718 q^{79} +81.0000 q^{81} -746.043 q^{83} -58.3476 q^{87} +631.566 q^{89} +580.841 q^{91} -323.442 q^{93} +1154.28 q^{97} -160.619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{3} - 23 q^{7} + 54 q^{9}+O(q^{10}) \) 6 * q - 18 * q^3 - 23 * q^7 + 54 * q^9	\( 6 q - 18 q^{3} - 23 q^{7} + 54 q^{9} + 61 q^{11} + 15 q^{13} - 68 q^{17} + 145 q^{19} + 69 q^{21} - 77 q^{23} - 162 q^{27} + 136 q^{29} + 124 q^{31} - 183 q^{33} - 569 q^{37} - 45 q^{39} + 195 q^{41} - 152 q^{43} - 447 q^{47} + 321 q^{49} + 204 q^{51} - 829 q^{53} - 435 q^{57} + 171 q^{59} + 172 q^{61} - 207 q^{63} - 282 q^{67} + 231 q^{69} + 820 q^{71} - 1666 q^{73} - 478 q^{77} + 1152 q^{79} + 486 q^{81} - 152 q^{83} - 408 q^{87} + 955 q^{89} + 710 q^{91} - 372 q^{93} - 756 q^{97} + 549 q^{99}+O(q^{100}) \) 6 * q - 18 * q^3 - 23 * q^7 + 54 * q^9 + 61 * q^11 + 15 * q^13 - 68 * q^17 + 145 * q^19 + 69 * q^21 - 77 * q^23 - 162 * q^27 + 136 * q^29 + 124 * q^31 - 183 * q^33 - 569 * q^37 - 45 * q^39 + 195 * q^41 - 152 * q^43 - 447 * q^47 + 321 * q^49 + 204 * q^51 - 829 * q^53 - 435 * q^57 + 171 * q^59 + 172 * q^61 - 207 * q^63 - 282 * q^67 + 231 * q^69 + 820 * q^71 - 1666 * q^73 - 478 * q^77 + 1152 * q^79 + 486 * q^81 - 152 * q^83 - 408 * q^87 + 955 * q^89 + 710 * q^91 - 372 * q^93 - 756 * q^97 + 549 * q^99

Introduction

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