LMFDB - Embedded newform 1500.4.d.d.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1500,4,Mod(1249,1500)]

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, 1]))

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

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

chi := DirichletCharacter("1500.1249");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1500 = 2^{2} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1500.d (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:	$88.5028650086$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 399x^{10} + 57331x^{8} + 3540939x^{6} + 85026526x^{4} + 353913300x^{2} + 4100625 $ x^12 + 399x^10 + 57331x^8 + 3540939x^6 + 85026526x^4 + 353913300*x^2 + 4100625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$7.44992i$ of defining polynomial
Character	$\chi$	$=$	1500.1249
Dual form			1500.4.d.d.1249.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +6.61902i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +6.61902i q^{7} -9.00000 q^{9} -17.8465 q^{11} -87.7534i q^{13} +37.3789i q^{17} +73.8745 q^{19} +19.8571 q^{21} -64.2627i q^{23} +27.0000i q^{27} -19.4492 q^{29} +107.814 q^{31} +53.5395i q^{33} -124.462i q^{37} -263.260 q^{39} +69.4637 q^{41} +4.34222i q^{43} +207.730i q^{47} +299.189 q^{49} +112.137 q^{51} +191.341i q^{53} -221.624i q^{57} -783.101 q^{59} -641.716 q^{61} -59.5712i q^{63} +347.308i q^{67} -192.788 q^{69} -282.089 q^{71} -488.902i q^{73} -118.126i q^{77} -77.8718 q^{79} +81.0000 q^{81} -746.043i q^{83} +58.3476i q^{87} -631.566 q^{89} +580.841 q^{91} -323.442i q^{93} -1154.28i q^{97} +160.619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 108 q^{9}+O(q^{10}) $ 12 * q - 108 * q^9	$ 12 q - 108 q^{9} + 122 q^{11} - 290 q^{19} + 138 q^{21} - 272 q^{29} + 248 q^{31} + 90 q^{39} + 390 q^{41} - 642 q^{49} + 408 q^{51} - 342 q^{59} + 344 q^{61} - 462 q^{69} + 1640 q^{71} - 2304 q^{79} + 972 q^{81} - 1910 q^{89} + 1420 q^{91} - 1098 q^{99}+O(q^{100}) $ 12 * q - 108 * q^9 + 122 * q^11 - 290 * q^19 + 138 * q^21 - 272 * q^29 + 248 * q^31 + 90 * q^39 + 390 * q^41 - 642 * q^49 + 408 * q^51 - 342 * q^59 + 344 * q^61 - 462 * q^69 + 1640 * q^71 - 2304 * q^79 + 972 * q^81 - 1910 * q^89 + 1420 * q^91 - 1098 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times$.

$n$	$751$	$877$	$1001$
$\chi(n)$	$1$	$-1$	$1$

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	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.61902i	0.357393i	0.983904	+	0.178697i	$0.0571881\pi$
$7$	6.61902i	0.357393i	−0.983904	+	0.178697i	$0.942812\pi$
$8$	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
$13$	− 87.7534i	− 1.87219i	−0.351753	−	0.936093i	$-0.614414\pi$
$13$	− 87.7534i	− 1.87219i	0.351753	−	0.936093i	$-0.385586\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	37.3789i	0.533278i	0.963797	+	0.266639i	$0.0859131\pi$
$17$	37.3789i	0.533278i	−0.963797	+	0.266639i	$0.914087\pi$
$18$	0	0
$19$	73.8745	0.891999	0.445999	−	0.895033i	$-0.352848\pi$
$19$	73.8745	0.891999	0.445999	+	0.895033i	$0.352848\pi$
$20$	0	0
$21$	19.8571	0.206341
$22$	0	0
$23$	− 64.2627i	− 0.582595i	−0.956632	−	0.291298i	$-0.905913\pi$
$23$	− 64.2627i	− 0.582595i	0.956632	−	0.291298i	$-0.0940870\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−19.4492	−0.124539	−0.0622694	−	0.998059i	$-0.519834\pi$
$29$	−19.4492	−0.124539	−0.0622694	+	0.998059i	$0.519834\pi$
$30$	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
$33$	53.5395i	0.282425i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 124.462i	− 0.553011i	−0.961012	−	0.276505i	$-0.910824\pi$
$37$	− 124.462i	− 0.553011i	0.961012	−	0.276505i	$-0.0891764\pi$
$38$	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
$43$	4.34222i	0.0153996i	0.999970	+	0.00769979i	$0.00245094\pi$
$43$	4.34222i	0.0153996i	−0.999970	+	0.00769979i	$0.997549\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	207.730i	0.644692i	0.946622	+	0.322346i	$0.104471\pi$
$47$	207.730i	0.644692i	−0.946622	+	0.322346i	$0.895529\pi$
$48$	0	0
$49$	299.189	0.872270
$50$	0	0
$51$	112.137	0.307888
$52$	0	0
$53$	191.341i	0.495901i	0.968773	+	0.247951i	$0.0797571\pi$
$53$	191.341i	0.495901i	−0.968773	+	0.247951i	$0.920243\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 221.624i	− 0.514996i
$58$	0	0
$59$	−783.101	−1.72798	−0.863992	−	0.503505i	$-0.832044\pi$
$59$	−783.101	−1.72798	−0.863992	+	0.503505i	$0.832044\pi$
$60$	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
$63$	− 59.5712i	− 0.119131i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	347.308i	0.633290i	0.948544	+	0.316645i	$0.102556\pi$
$67$	347.308i	0.633290i	−0.948544	+	0.316645i	$0.897444\pi$
$68$	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
$73$	− 488.902i	− 0.783858i	−0.919995	−	0.391929i	$-0.871808\pi$
$73$	− 488.902i	− 0.783858i	0.919995	−	0.391929i	$-0.128192\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 118.126i	− 0.174828i
$78$	0	0
$79$	−77.8718	−0.110902	−0.0554510	−	0.998461i	$-0.517660\pi$
$79$	−77.8718	−0.110902	−0.0554510	+	0.998461i	$0.517660\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 746.043i	− 0.986612i	−0.869856	−	0.493306i	$-0.835788\pi$
$83$	− 746.043i	− 0.986612i	0.869856	−	0.493306i	$-0.164212\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	58.3476i	0.0719025i
$88$	0	0
$89$	−631.566	−0.752201	−0.376101	−	0.926579i	$-0.622735\pi$
$89$	−631.566	−0.752201	−0.376101	+	0.926579i	$0.622735\pi$
$90$	0	0
$91$	580.841	0.669107
$92$	0	0
$93$	− 323.442i	− 0.360638i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1154.28i	− 1.20824i	−0.796892	−	0.604122i	$-0.793524\pi$
$97$	− 1154.28i	− 1.20824i	0.796892	−	0.604122i	$-0.206476\pi$
$98$	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
$103$	− 146.972i	− 0.140598i	−0.997526	−	0.0702990i	$-0.977605\pi$
$103$	− 146.972i	− 0.140598i	0.997526	−	0.0702990i	$-0.0223953\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	274.474i	0.247985i	0.992283	+	0.123992i	$0.0395698\pi$
$107$	274.474i	0.247985i	−0.992283	+	0.123992i	$0.960430\pi$
$108$	0	0
$109$	−1533.63	−1.34766	−0.673832	−	0.738885i	$-0.735353\pi$
$109$	−1533.63	−1.34766	−0.673832	+	0.738885i	$0.735353\pi$
$110$	0	0
$111$	−373.386	−0.319281
$112$	0	0
$113$	− 1759.56i	− 1.46482i	−0.680861	−	0.732412i	$-0.738394\pi$
$113$	− 1759.56i	− 1.46482i	0.680861	−	0.732412i	$-0.261606\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	789.781i	0.624062i
$118$	0	0
$119$	−247.412	−0.190590
$120$	0	0
$121$	−1012.50	−0.760708
$122$	0	0
$123$	− 208.391i	− 0.152764i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 613.918i	− 0.428948i	−0.976730	−	0.214474i	$-0.931196\pi$
$127$	− 613.918i	− 0.428948i	0.976730	−	0.214474i	$-0.0688038\pi$
$128$	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
$133$	488.977i	0.318795i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	104.587i	0.0652225i	0.999468	+	0.0326112i	$0.0103823\pi$
$137$	104.587i	0.0652225i	−0.999468	+	0.0326112i	$0.989618\pi$
$138$	0	0
$139$	−2481.91	−1.51448	−0.757240	−	0.653137i	$-0.773452\pi$
$139$	−2481.91	−1.51448	−0.757240	+	0.653137i	$0.773452\pi$
$140$	0	0
$141$	623.189	0.372213
$142$	0	0
$143$	1566.09i	0.915826i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 897.566i	− 0.503605i
$148$	0	0
$149$	5.99581	0.00329662	0.00164831	−	0.999999i	$-0.499475\pi$
$149$	5.99581	0.00329662	0.00164831	+	0.999999i	$0.499475\pi$
$150$	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
$153$	− 336.410i	− 0.177759i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	748.992i	0.380739i	0.981712	+	0.190370i	$0.0609686\pi$
$157$	748.992i	0.380739i	−0.981712	+	0.190370i	$0.939031\pi$
$158$	0	0
$159$	574.024	0.286309
$160$	0	0
$161$	425.356	0.208216
$162$	0	0
$163$	1939.96i	0.932207i	0.884730	+	0.466104i	$0.154343\pi$
$163$	1939.96i	0.932207i	−0.884730	+	0.466104i	$0.845657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1216.42i	− 0.563650i	−0.959466	−	0.281825i	$-0.909060\pi$
$167$	− 1216.42i	− 0.563650i	0.959466	−	0.281825i	$-0.0909398\pi$
$168$	0	0
$169$	−5503.66	−2.50508
$170$	0	0
$171$	−664.871	−0.297333
$172$	0	0
$173$	4148.53i	1.82316i	0.411123	+	0.911580i	$0.365137\pi$
$173$	4148.53i	1.82316i	−0.411123	+	0.911580i	$0.634863\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2349.30i	0.997653i
$178$	0	0
$179$	−3143.77	−1.31272	−0.656359	−	0.754449i	$-0.727904\pi$
$179$	−3143.77	−1.31272	−0.656359	+	0.754449i	$0.727904\pi$
$180$	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
$183$	1925.15i	0.777656i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 667.083i	− 0.260866i
$188$	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
$193$	− 1210.99i	− 0.451654i	−0.974167	−	0.225827i	$-0.927492\pi$
$193$	− 1210.99i	− 0.451654i	0.974167	−	0.225827i	$-0.0725084\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4170.97i	1.50848i	0.656602	+	0.754238i	$0.271993\pi$
$197$	4170.97i	1.50848i	−0.656602	+	0.754238i	$0.728007\pi$
$198$	0	0
$199$	−3097.25	−1.10331	−0.551653	−	0.834074i	$-0.686003\pi$
$199$	−3097.25	−1.10331	−0.551653	+	0.834074i	$0.686003\pi$
$200$	0	0
$201$	1041.92	0.365630
$202$	0	0
$203$	− 128.735i	− 0.0445094i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	578.364i	0.194198i
$208$	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
$213$	846.267i	0.272231i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	713.623i	0.223244i
$218$	0	0
$219$	−1466.71	−0.452561
$220$	0	0
$221$	3280.13	0.998394
$222$	0	0
$223$	− 1597.31i	− 0.479659i	−0.970815	−	0.239830i	$-0.922908\pi$
$223$	− 1597.31i	− 0.479659i	0.970815	−	0.239830i	$-0.0770916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4478.36i	− 1.30942i	−0.755879	−	0.654712i	$-0.772790\pi$
$227$	− 4478.36i	− 1.30942i	0.755879	−	0.654712i	$-0.227210\pi$
$228$	0	0
$229$	−963.658	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$229$	−963.658	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$230$	0	0
$231$	−354.379	−0.100937
$232$	0	0
$233$	− 3920.17i	− 1.10223i	−0.834430	−	0.551114i	$-0.814203\pi$
$233$	− 3920.17i	− 1.10223i	0.834430	−	0.551114i	$-0.185797\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	233.615i	0.0640293i
$238$	0	0
$239$	−3397.65	−0.919563	−0.459782	−	0.888032i	$-0.652072\pi$
$239$	−3397.65	−0.919563	−0.459782	+	0.888032i	$0.652072\pi$
$240$	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
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6482.74i	− 1.66999i
$248$	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
$253$	1146.86i	0.284991i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3842.32i	0.932598i	0.884627	+	0.466299i	$0.154413\pi$
$257$	3842.32i	0.932598i	−0.884627	+	0.466299i	$0.845587\pi$
$258$	0	0
$259$	823.815	0.197642
$260$	0	0
$261$	175.043	0.0415130
$262$	0	0
$263$	− 1505.17i	− 0.352900i	−0.984310	−	0.176450i	$-0.943539\pi$
$263$	− 1505.17i	− 0.352900i	0.984310	−	0.176450i	$-0.0564615\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1894.70i	0.434284i
$268$	0	0
$269$	−6931.71	−1.57113	−0.785565	−	0.618779i	$-0.787627\pi$
$269$	−6931.71	−1.57113	−0.785565	+	0.618779i	$0.787627\pi$
$270$	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
$273$	− 1742.52i	− 0.386309i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 485.763i	− 0.105367i	−0.998611	−	0.0526835i	$-0.983223\pi$
$277$	− 485.763i	− 0.105367i	0.998611	−	0.0526835i	$-0.0167775\pi$
$278$	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
$283$	15.6822i	0.00329404i	0.999999	+	0.00164702i	$0.000524262\pi$
$283$	15.6822i	0.00329404i	−0.999999	+	0.00164702i	$0.999476\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	459.782i	0.0945647i
$288$	0	0
$289$	3515.82	0.715615
$290$	0	0
$291$	−3462.85	−0.697580
$292$	0	0
$293$	7856.99i	1.56659i	0.621651	+	0.783294i	$0.286462\pi$
$293$	7856.99i	1.56659i	−0.621651	+	0.783294i	$0.713538\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 481.856i	− 0.0941417i
$298$	0	0
$299$	−5639.27	−1.09073
$300$	0	0
$301$	−28.7412	−0.00550371
$302$	0	0
$303$	− 1243.44i	− 0.235755i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5306.16i	− 0.986446i	−0.869903	−	0.493223i	$-0.835819\pi$
$307$	− 5306.16i	− 0.986446i	0.869903	−	0.493223i	$-0.164181\pi$
$308$	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
$313$	4602.96i	0.831230i	0.909541	+	0.415615i	$0.136434\pi$
$313$	4602.96i	0.831230i	−0.909541	+	0.415615i	$0.863566\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1173.27i	0.207878i	0.994584	+	0.103939i	$0.0331447\pi$
$317$	1173.27i	0.207878i	−0.994584	+	0.103939i	$0.966855\pi$
$318$	0	0
$319$	347.100	0.0609213
$320$	0	0
$321$	823.421	0.143174
$322$	0	0
$323$	2761.35i	0.475683i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4600.90i	0.778074i
$328$	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
$333$	1120.16i	0.184337i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9017.73i	− 1.45765i	−0.684701	−	0.728824i	$-0.740067\pi$
$337$	− 9017.73i	− 1.45765i	0.684701	−	0.728824i	$-0.259933\pi$
$338$	0	0
$339$	−5278.67	−0.845717
$340$	0	0
$341$	−1924.10	−0.305560
$342$	0	0
$343$	4250.66i	0.669137i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10857.8i	1.67976i	0.542769	+	0.839882i	$0.317376\pi$
$347$	10857.8i	1.67976i	−0.542769	+	0.839882i	$0.682624\pi$
$348$	0	0
$349$	−2757.87	−0.422996	−0.211498	−	0.977378i	$-0.567834\pi$
$349$	−2757.87	−0.422996	−0.211498	+	0.977378i	$0.567834\pi$
$350$	0	0
$351$	2369.34	0.360302
$352$	0	0
$353$	− 3107.17i	− 0.468493i	−0.972177	−	0.234247i	$-0.924738\pi$
$353$	− 3107.17i	− 0.468493i	0.972177	−	0.234247i	$-0.0752623\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	742.235i	0.110037i
$358$	0	0
$359$	1905.22	0.280093	0.140047	−	0.990145i	$-0.455275\pi$
$359$	1905.22	0.280093	0.140047	+	0.990145i	$0.455275\pi$
$360$	0	0
$361$	−1401.55	−0.204338
$362$	0	0
$363$	3037.51i	0.439195i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4388.28i	− 0.624159i	−0.950056	−	0.312080i	$-0.898974\pi$
$367$	− 4388.28i	− 0.624159i	0.950056	−	0.312080i	$-0.101026\pi$
$368$	0	0
$369$	−625.174	−0.0881985
$370$	0	0
$371$	−1266.49	−0.177232
$372$	0	0
$373$	1738.19i	0.241287i	0.992696	+	0.120644i	$0.0384959\pi$
$373$	1738.19i	0.241287i	−0.992696	+	0.120644i	$0.961504\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1706.73i	0.233160i
$378$	0	0
$379$	6776.90	0.918486	0.459243	−	0.888311i	$-0.348121\pi$
$379$	6776.90	0.918486	0.459243	+	0.888311i	$0.348121\pi$
$380$	0	0
$381$	−1841.75	−0.247653
$382$	0	0
$383$	− 9792.38i	− 1.30644i	−0.757168	−	0.653221i	$-0.773417\pi$
$383$	− 9792.38i	− 1.30644i	0.757168	−	0.653221i	$-0.226583\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 39.0800i	− 0.00513319i
$388$	0	0
$389$	−4944.39	−0.644448	−0.322224	−	0.946663i	$-0.604430\pi$
$389$	−4944.39	−0.644448	−0.322224	+	0.946663i	$0.604430\pi$
$390$	0	0
$391$	2402.07	0.310685
$392$	0	0
$393$	− 3862.98i	− 0.495830i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4874.09i	− 0.616180i	−0.951357	−	0.308090i	$-0.900310\pi$
$397$	− 4874.09i	− 0.616180i	0.951357	−	0.308090i	$-0.0996898\pi$
$398$	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
$403$	− 9461.04i	− 1.16945i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2221.21i	0.270519i
$408$	0	0
$409$	−8894.04	−1.07526	−0.537631	−	0.843181i	$-0.680680\pi$
$409$	−8894.04	−1.07526	−0.537631	+	0.843181i	$0.680680\pi$
$410$	0	0
$411$	313.761	0.0376562
$412$	0	0
$413$	− 5183.36i	− 0.617570i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7445.72i	0.874385i
$418$	0	0
$419$	2347.48	0.273704	0.136852	−	0.990591i	$-0.456301\pi$
$419$	2347.48	0.273704	0.136852	+	0.990591i	$0.456301\pi$
$420$	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
$423$	− 1869.57i	− 0.214897i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4247.53i	− 0.481387i
$428$	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
$433$	− 8295.66i	− 0.920702i	−0.887737	−	0.460351i	$-0.847724\pi$
$433$	− 8295.66i	− 0.920702i	0.887737	−	0.460351i	$-0.152276\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4747.37i	− 0.519674i
$438$	0	0
$439$	−13403.7	−1.45723	−0.728613	−	0.684925i	$-0.759835\pi$
$439$	−13403.7	−1.45723	−0.728613	+	0.684925i	$0.759835\pi$
$440$	0	0
$441$	−2692.70	−0.290757
$442$	0	0
$443$	− 320.338i	− 0.0343560i	−0.999852	−	0.0171780i	$-0.994532\pi$
$443$	− 320.338i	− 0.0343560i	0.999852	−	0.0171780i	$-0.00546819\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 17.9874i	− 0.00190330i
$448$	0	0
$449$	11716.3	1.23146	0.615732	−	0.787956i	$-0.288860\pi$
$449$	11716.3	1.23146	0.615732	+	0.787956i	$0.288860\pi$
$450$	0	0
$451$	−1239.68	−0.129433
$452$	0	0
$453$	− 1561.84i	− 0.161991i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1055.01i	− 0.107990i	−0.998541	−	0.0539948i	$-0.982805\pi$
$457$	− 1055.01i	− 0.107990i	0.998541	−	0.0539948i	$-0.0171954\pi$
$458$	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
$463$	− 12977.6i	− 1.30264i	−0.758805	−	0.651318i	$-0.774216\pi$
$463$	− 12977.6i	− 1.30264i	0.758805	−	0.651318i	$-0.225784\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15404.4i	1.52640i	0.646162	+	0.763201i	$0.276373\pi$
$467$	15404.4i	1.52640i	−0.646162	+	0.763201i	$0.723627\pi$
$468$	0	0
$469$	−2298.84	−0.226334
$470$	0	0
$471$	2246.98	0.219820
$472$	0	0
$473$	− 77.4934i	− 0.00753309i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1722.07i	− 0.165300i
$478$	0	0
$479$	15235.6	1.45330	0.726650	−	0.687008i	$-0.241076\pi$
$479$	15235.6	1.45330	0.726650	+	0.687008i	$0.241076\pi$
$480$	0	0
$481$	−10921.9	−1.03534
$482$	0	0
$483$	− 1276.07i	− 0.120213i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12958.2i	− 1.20573i	−0.797843	−	0.602866i	$-0.794025\pi$
$487$	− 12958.2i	− 1.20573i	0.797843	−	0.602866i	$-0.205975\pi$
$488$	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
$493$	− 726.990i	− 0.0664138i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1867.15i	− 0.168517i
$498$	0	0
$499$	10604.2	0.951317	0.475658	−	0.879630i	$-0.342210\pi$
$499$	10604.2	0.951317	0.475658	+	0.879630i	$0.342210\pi$
$500$	0	0
$501$	−3649.27	−0.325424
$502$	0	0
$503$	3814.90i	0.338167i	0.985602	+	0.169084i	$0.0540808\pi$
$503$	3814.90i	0.338167i	−0.985602	+	0.169084i	$0.945919\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16511.0i	1.44631i
$508$	0	0
$509$	8697.29	0.757368	0.378684	−	0.925526i	$-0.376377\pi$
$509$	8697.29	0.757368	0.378684	+	0.925526i	$0.376377\pi$
$510$	0	0
$511$	3236.05	0.280146
$512$	0	0
$513$	1994.61i	0.171665i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3707.25i	− 0.315367i
$518$	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
$523$	13787.3i	1.15273i	0.817194	+	0.576363i	$0.195529\pi$
$523$	13787.3i	1.15273i	−0.817194	+	0.576363i	$0.804471\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4029.97i	0.333108i
$528$	0	0
$529$	8037.31	0.660583
$530$	0	0
$531$	7047.91	0.575995
$532$	0	0
$533$	− 6095.68i	− 0.495372i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9431.31i	0.757898i
$538$	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
$543$	6813.02i	0.538443i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 22548.0i	− 1.76249i	−0.472655	−	0.881247i	$-0.656704\pi$
$547$	− 22548.0i	− 1.76249i	0.472655	−	0.881247i	$-0.343296\pi$
$548$	0	0
$549$	5775.45	0.448980
$550$	0	0
$551$	−1436.80	−0.111089
$552$	0	0
$553$	− 515.435i	− 0.0396357i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18784.7i	1.42896i	0.699655	+	0.714481i	$0.253337\pi$
$557$	18784.7i	1.42896i	−0.699655	+	0.714481i	$0.746663\pi$
$558$	0	0
$559$	381.044	0.0288309
$560$	0	0
$561$	−2001.25	−0.150611
$562$	0	0
$563$	12105.9i	0.906221i	0.891454	+	0.453111i	$0.149686\pi$
$563$	12105.9i	0.906221i	−0.891454	+	0.453111i	$0.850314\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	536.141i	0.0397104i
$568$	0	0
$569$	6348.65	0.467749	0.233874	−	0.972267i	$-0.424860\pi$
$569$	6348.65	0.467749	0.233874	+	0.972267i	$0.424860\pi$
$570$	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
$573$	12471.9i	0.909284i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 16003.1i	− 1.15462i	−0.816524	−	0.577312i	$-0.804102\pi$
$577$	− 16003.1i	− 1.15462i	0.816524	−	0.577312i	$-0.195898\pi$
$578$	0	0
$579$	−3632.98	−0.260763
$580$	0	0
$581$	4938.07	0.352609
$582$	0	0
$583$	− 3414.78i	− 0.242582i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11097.1i	− 0.780282i	−0.920755	−	0.390141i	$-0.872426\pi$
$587$	− 11097.1i	− 0.780282i	0.920755	−	0.390141i	$-0.127574\pi$
$588$	0	0
$589$	7964.70	0.557182
$590$	0	0
$591$	12512.9	0.870918
$592$	0	0
$593$	14101.0i	0.976491i	0.872706	+	0.488245i	$0.162363\pi$
$593$	14101.0i	0.976491i	−0.872706	+	0.488245i	$0.837637\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9291.74i	0.636994i
$598$	0	0
$599$	15179.9	1.03545	0.517726	−	0.855547i	$-0.326779\pi$
$599$	15179.9	1.03545	0.517726	+	0.855547i	$0.326779\pi$
$600$	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
$603$	− 3125.77i	− 0.211097i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	296.900i	0.0198530i	0.999951	+	0.00992651i	$0.00315976\pi$
$607$	296.900i	0.0198530i	−0.999951	+	0.00992651i	$0.996840\pi$
$608$	0	0
$609$	−386.204	−0.0256975
$610$	0	0
$611$	18229.0	1.20698
$612$	0	0
$613$	3440.96i	0.226720i	0.993554	+	0.113360i	$0.0361613\pi$
$613$	3440.96i	0.226720i	−0.993554	+	0.113360i	$0.963839\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12232.6i	0.798161i	0.916916	+	0.399080i	$0.130671\pi$
$617$	12232.6i	0.798161i	−0.916916	+	0.399080i	$0.869329\pi$
$618$	0	0
$619$	16154.1	1.04893	0.524467	−	0.851431i	$-0.324265\pi$
$619$	16154.1	1.04893	0.524467	+	0.851431i	$0.324265\pi$
$620$	0	0
$621$	1735.09	0.112121
$622$	0	0
$623$	− 4180.35i	− 0.268832i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3955.21i	0.251923i
$628$	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
$633$	− 16075.9i	− 1.00941i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 26254.8i	− 1.63305i
$638$	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
$643$	10978.4i	0.673320i	0.941626	+	0.336660i	$0.109297\pi$
$643$	10978.4i	0.673320i	−0.941626	+	0.336660i	$0.890703\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26974.4i	− 1.63906i	−0.573033	−	0.819532i	$-0.694233\pi$
$647$	− 26974.4i	− 1.63906i	0.573033	−	0.819532i	$-0.305767\pi$
$648$	0	0
$649$	13975.6	0.845287
$650$	0	0
$651$	2140.87	0.128890
$652$	0	0
$653$	− 11307.1i	− 0.677611i	−0.940856	−	0.338806i	$-0.889977\pi$
$653$	− 11307.1i	− 0.677611i	0.940856	−	0.338806i	$-0.110023\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4400.12i	0.261286i
$658$	0	0
$659$	4996.63	0.295358	0.147679	−	0.989035i	$-0.452820\pi$
$659$	4996.63	0.295358	0.147679	+	0.989035i	$0.452820\pi$
$660$	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
$663$	− 9840.38i	− 0.576423i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1249.86i	0.0725558i
$668$	0	0
$669$	−4791.94	−0.276931
$670$	0	0
$671$	11452.4	0.658889
$672$	0	0
$673$	25166.8i	1.44147i	0.693211	+	0.720735i	$0.256195\pi$
$673$	25166.8i	1.44147i	−0.693211	+	0.720735i	$0.743805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19293.3i	1.09527i	0.836716	+	0.547637i	$0.184472\pi$
$677$	19293.3i	1.09527i	−0.836716	+	0.547637i	$0.815528\pi$
$678$	0	0
$679$	7640.22	0.431819
$680$	0	0
$681$	−13435.1	−0.755996
$682$	0	0
$683$	3548.69i	0.198809i	0.995047	+	0.0994047i	$0.0316939\pi$
$683$	3548.69i	0.198809i	−0.995047	+	0.0994047i	$0.968306\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2890.97i	0.160550i
$688$	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
$693$	1063.14i	0.0582759i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2596.48i	0.141103i
$698$	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
$703$	− 9194.56i	− 0.493285i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2743.45i	0.145938i
$708$	0	0
$709$	32203.6	1.70583	0.852914	−	0.522051i	$-0.174833\pi$
$709$	32203.6	1.70583	0.852914	+	0.522051i	$0.174833\pi$
$710$	0	0
$711$	700.846	0.0369674
$712$	0	0
$713$	− 6928.41i	− 0.363915i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10192.9i	0.530910i
$718$	0	0
$719$	−10510.9	−0.545190	−0.272595	−	0.962129i	$-0.587882\pi$
$719$	−10510.9	−0.545190	−0.272595	+	0.962129i	$0.587882\pi$
$720$	0	0
$721$	972.811	0.0502488
$722$	0	0
$723$	8109.89i	0.417165i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23405.1i	− 1.19401i	−0.802237	−	0.597006i	$-0.796357\pi$
$727$	− 23405.1i	− 1.19401i	0.802237	−	0.597006i	$-0.203643\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−162.307	−0.00821225
$732$	0	0
$733$	− 21709.7i	− 1.09395i	−0.837149	−	0.546974i	$-0.815779\pi$
$733$	− 21709.7i	− 1.09395i	0.837149	−	0.546974i	$-0.184221\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6198.24i	− 0.309790i
$738$	0	0
$739$	29065.2	1.44679	0.723397	−	0.690432i	$-0.242580\pi$
$739$	29065.2	1.44679	0.723397	+	0.690432i	$0.242580\pi$
$740$	0	0
$741$	−19448.2	−0.964168
$742$	0	0
$743$	9515.59i	0.469843i	0.972014	+	0.234921i	$0.0754832\pi$
$743$	9515.59i	0.469843i	−0.972014	+	0.234921i	$0.924517\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6714.38i	0.328871i
$748$	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
$753$	4534.08i	0.219431i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 16014.6i	− 0.768903i	−0.923145	−	0.384451i	$-0.874391\pi$
$757$	− 16014.6i	− 0.768903i	0.923145	−	0.384451i	$-0.125609\pi$
$758$	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
$763$	− 10151.1i	− 0.481646i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	68719.8i	3.23511i
$768$	0	0
$769$	31525.4	1.47833	0.739164	−	0.673526i	$-0.235221\pi$
$769$	31525.4	1.47833	0.739164	+	0.673526i	$0.235221\pi$
$770$	0	0
$771$	11527.0	0.538435
$772$	0	0
$773$	− 15720.8i	− 0.731483i	−0.930716	−	0.365742i	$-0.880815\pi$
$773$	− 15720.8i	− 0.731483i	0.930716	−	0.365742i	$-0.119185\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2471.45i	− 0.114109i
$778$	0	0
$779$	5131.60	0.236019
$780$	0	0
$781$	5034.30	0.230655
$782$	0	0
$783$	− 525.129i	− 0.0239675i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29815.1i	− 1.35044i	−0.737618	−	0.675219i	$-0.764049\pi$
$787$	− 29815.1i	− 1.35044i	0.737618	−	0.675219i	$-0.235951\pi$
$788$	0	0
$789$	−4515.51	−0.203747
$790$	0	0
$791$	11646.5	0.523519
$792$	0	0
$793$	56312.8i	2.52172i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11434.5i	− 0.508193i	−0.967179	−	0.254097i	$-0.918222\pi$
$797$	− 11434.5i	− 0.508193i	0.967179	−	0.254097i	$-0.0817781\pi$
$798$	0	0
$799$	−7764.71	−0.343800
$800$	0	0
$801$	5684.10	0.250734
$802$	0	0
$803$	8725.19i	0.383444i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20795.1i	0.907092i
$808$	0	0
$809$	15795.6	0.686457	0.343229	−	0.939252i	$-0.388479\pi$
$809$	15795.6	0.686457	0.343229	+	0.939252i	$0.388479\pi$
$810$	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
$813$	− 6299.16i	− 0.271736i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	320.779i	0.0137364i
$818$	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
$823$	5000.26i	0.211784i	0.994378	+	0.105892i	$0.0337698\pi$
$823$	5000.26i	0.211784i	−0.994378	+	0.105892i	$0.966230\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17629.0i	0.741256i	0.928781	+	0.370628i	$0.120858\pi$
$827$	17629.0i	0.741256i	−0.928781	+	0.370628i	$0.879142\pi$
$828$	0	0
$829$	−14297.7	−0.599010	−0.299505	−	0.954095i	$-0.596822\pi$
$829$	−14297.7	−0.599010	−0.299505	+	0.954095i	$0.596822\pi$
$830$	0	0
$831$	−1457.29	−0.0608337
$832$	0	0
$833$	11183.3i	0.465162i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2910.98i	0.120213i
$838$	0	0
$839$	39759.2	1.63604	0.818021	−	0.575188i	$-0.195071\pi$
$839$	39759.2	1.63604	0.818021	+	0.575188i	$0.195071\pi$
$840$	0	0
$841$	−24010.7	−0.984490
$842$	0	0
$843$	22522.1i	0.920170i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6701.77i	− 0.271872i
$848$	0	0
$849$	47.0467	0.00190181
$850$	0	0
$851$	−7998.25	−0.322182
$852$	0	0
$853$	− 12468.0i	− 0.500464i	−0.968186	−	0.250232i	$-0.919493\pi$
$853$	− 12468.0i	− 0.500464i	0.968186	−	0.250232i	$-0.0805068\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4494.37i	0.179142i	0.995980	+	0.0895710i	$0.0285496\pi$
$857$	4494.37i	0.179142i	−0.995980	+	0.0895710i	$0.971450\pi$
$858$	0	0
$859$	19659.9	0.780891	0.390446	−	0.920626i	$-0.372321\pi$
$859$	19659.9	0.780891	0.390446	+	0.920626i	$0.372321\pi$
$860$	0	0
$861$	1379.35	0.0545969
$862$	0	0
$863$	29270.1i	1.15454i	0.816555	+	0.577268i	$0.195881\pi$
$863$	29270.1i	1.15454i	−0.816555	+	0.577268i	$0.804119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 10547.5i	− 0.413161i
$868$	0	0
$869$	1389.74	0.0542505
$870$	0	0
$871$	30477.5	1.18564
$872$	0	0
$873$	10388.6i	0.402748i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 43860.4i	− 1.68878i	−0.535727	−	0.844391i	$-0.679962\pi$
$877$	− 43860.4i	− 1.68878i	0.535727	−	0.844391i	$-0.320038\pi$
$878$	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
$883$	− 33837.6i	− 1.28961i	−0.764347	−	0.644805i	$-0.776939\pi$
$883$	− 33837.6i	− 1.28961i	0.764347	−	0.644805i	$-0.223061\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 5957.96i	− 0.225534i	−0.993621	−	0.112767i	$-0.964029\pi$
$887$	− 5957.96i	− 0.225534i	0.993621	−	0.112767i	$-0.0359714\pi$
$888$	0	0
$889$	4063.54	0.153303
$890$	0	0
$891$	−1445.57	−0.0543528
$892$	0	0
$893$	15345.9i	0.575064i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16917.8i	0.629731i
$898$	0	0
$899$	−2096.90	−0.0777924
$900$	0	0
$901$	−7152.14	−0.264453
$902$	0	0
$903$	86.2237i	0.00317757i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52622.5i	1.92646i	0.268672	+	0.963232i	$0.413415\pi$
$907$	52622.5i	1.92646i	−0.268672	+	0.963232i	$0.586585\pi$
$908$	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
$913$	13314.3i	0.482626i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8523.04i	0.306931i
$918$	0	0
$919$	−23964.9	−0.860206	−0.430103	−	0.902780i	$-0.641523\pi$
$919$	−23964.9	−0.860206	−0.430103	+	0.902780i	$0.641523\pi$
$920$	0	0
$921$	−15918.5	−0.569525
$922$	0	0
$923$	24754.3i	0.882769i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1322.75i	0.0468660i
$928$	0	0
$929$	−33770.8	−1.19266	−0.596332	−	0.802738i	$-0.703376\pi$
$929$	−33770.8	−1.19266	−0.596332	+	0.802738i	$0.703376\pi$
$930$	0	0
$931$	22102.4	0.778064
$932$	0	0
$933$	− 28044.8i	− 0.984078i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 54057.5i	− 1.88472i	−0.334602	−	0.942359i	$-0.608602\pi$
$937$	− 54057.5i	− 1.88472i	0.334602	−	0.942359i	$-0.391398\pi$
$938$	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
$943$	− 4463.93i	− 0.154152i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11559.4i	0.396653i	0.980136	+	0.198327i	$0.0635507\pi$
$947$	11559.4i	0.396653i	−0.980136	+	0.198327i	$0.936449\pi$
$948$	0	0
$949$	−42902.8	−1.46753
$950$	0	0
$951$	3519.81	0.120019
$952$	0	0
$953$	− 36942.1i	− 1.25569i	−0.778338	−	0.627845i	$-0.783937\pi$
$953$	− 36942.1i	− 1.25569i	0.778338	−	0.627845i	$-0.216063\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1041.30i	− 0.0351729i
$958$	0	0
$959$	−692.264	−0.0233101
$960$	0	0
$961$	−18167.2	−0.609820
$962$	0	0
$963$	− 2470.26i	− 0.0826616i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 14131.8i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$967$	− 14131.8i	− 0.469956i	0.972001	−	0.234978i	$-0.0755018\pi$
$968$	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
$973$	− 16427.8i	− 0.541265i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3174.68i	0.103958i	0.998648	+	0.0519791i	$0.0165529\pi$
$977$	3174.68i	0.103958i	−0.998648	+	0.0519791i	$0.983447\pi$
$978$	0	0
$979$	11271.3	0.367958
$980$	0	0
$981$	13802.7	0.449221
$982$	0	0
$983$	− 53701.2i	− 1.74242i	−0.490908	−	0.871211i	$-0.663335\pi$
$983$	− 53701.2i	− 1.74242i	0.490908	−	0.871211i	$-0.336665\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4124.90i	0.133026i
$988$	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
$993$	− 3443.32i	− 0.110041i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 42205.7i	− 1.34069i	−0.742049	−	0.670346i	$-0.766146\pi$
$997$	− 42205.7i	− 1.34069i	0.742049	−	0.670346i	$-0.233854\pi$
$998$	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.d.d.1249.3		12
5.2	odd	4		1500.4.a.b.1.4	✓	6
5.3	odd	4		1500.4.a.g.1.3	yes	6
5.4	even	2	inner	1500.4.d.d.1249.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.4.a.b.1.4	✓	6	5.2	odd	4
1500.4.a.g.1.3	yes	6	5.3	odd	4
1500.4.d.d.1249.3		12	1.1	even	1	trivial
1500.4.d.d.1249.10		12	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:	\( N \)	\(=\)	\( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +6.61902i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +6.61902i q^{7} -9.00000 q^{9} -17.8465 q^{11} -87.7534i q^{13} +37.3789i q^{17} +73.8745 q^{19} +19.8571 q^{21} -64.2627i q^{23} +27.0000i q^{27} -19.4492 q^{29} +107.814 q^{31} +53.5395i q^{33} -124.462i q^{37} -263.260 q^{39} +69.4637 q^{41} +4.34222i q^{43} +207.730i q^{47} +299.189 q^{49} +112.137 q^{51} +191.341i q^{53} -221.624i q^{57} -783.101 q^{59} -641.716 q^{61} -59.5712i q^{63} +347.308i q^{67} -192.788 q^{69} -282.089 q^{71} -488.902i q^{73} -118.126i q^{77} -77.8718 q^{79} +81.0000 q^{81} -746.043i q^{83} +58.3476i q^{87} -631.566 q^{89} +580.841 q^{91} -323.442i q^{93} -1154.28i q^{97} +160.619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 108 q^{9}+O(q^{10}) \) 12 * q - 108 * q^9	\( 12 q - 108 q^{9} + 122 q^{11} - 290 q^{19} + 138 q^{21} - 272 q^{29} + 248 q^{31} + 90 q^{39} + 390 q^{41} - 642 q^{49} + 408 q^{51} - 342 q^{59} + 344 q^{61} - 462 q^{69} + 1640 q^{71} - 2304 q^{79} + 972 q^{81} - 1910 q^{89} + 1420 q^{91} - 1098 q^{99}+O(q^{100}) \) 12 * q - 108 * q^9 + 122 * q^11 - 290 * q^19 + 138 * q^21 - 272 * q^29 + 248 * q^31 + 90 * q^39 + 390 * q^41 - 642 * q^49 + 408 * q^51 - 342 * q^59 + 344 * q^61 - 462 * q^69 + 1640 * q^71 - 2304 * q^79 + 972 * q^81 - 1910 * q^89 + 1420 * q^91 - 1098 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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