LMFDB - Embedded newform 1575.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.174928.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 13 $ x^4 - 9*x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.34440$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34440 q^{2} -0.192582 q^{4} +1.00000 q^{7} +2.94771 q^{8} +O(q^{10})$	$q-1.34440 q^{2} -0.192582 q^{4} +1.00000 q^{7} +2.94771 q^{8} +5.63652 q^{11} +4.38516 q^{13} -1.34440 q^{14} -3.57775 q^{16} -2.68880 q^{17} +8.38516 q^{19} -7.57775 q^{22} -5.63652 q^{23} -5.89543 q^{26} -0.192582 q^{28} -8.32532 q^{29} +6.00000 q^{31} -1.08549 q^{32} +3.61484 q^{34} -3.00000 q^{37} -11.2730 q^{38} +5.37761 q^{41} -1.38516 q^{43} -1.08549 q^{44} +7.57775 q^{46} -8.58423 q^{47} +1.00000 q^{49} -0.844506 q^{52} +5.37761 q^{53} +2.94771 q^{56} +11.1926 q^{58} +8.58423 q^{59} -8.06641 q^{62} +8.61484 q^{64} -1.38516 q^{67} +0.517816 q^{68} -0.258908 q^{71} +6.38516 q^{73} +4.03321 q^{74} -1.61484 q^{76} +5.63652 q^{77} +5.38516 q^{79} -7.22967 q^{82} +16.6506 q^{83} +1.86222 q^{86} +16.6148 q^{88} -13.9618 q^{89} +4.38516 q^{91} +1.08549 q^{92} +11.5407 q^{94} -10.7703 q^{97} -1.34440 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 4 q^{7}+O(q^{10}) $ 4 * q + 10 * q^4 + 4 * q^7	$ 4 q + 10 q^{4} + 4 q^{7} - 4 q^{13} + 18 q^{16} + 12 q^{19} + 2 q^{22} + 10 q^{28} + 24 q^{31} + 36 q^{34} - 12 q^{37} + 16 q^{43} - 2 q^{46} + 4 q^{49} - 68 q^{52} + 34 q^{58} + 56 q^{64} + 16 q^{67} + 4 q^{73} - 28 q^{76} - 72 q^{82} + 88 q^{88} - 4 q^{91} - 40 q^{94}+O(q^{100}) $ 4 * q + 10 * q^4 + 4 * q^7 - 4 * q^13 + 18 * q^16 + 12 * q^19 + 2 * q^22 + 10 * q^28 + 24 * q^31 + 36 * q^34 - 12 * q^37 + 16 * q^43 - 2 * q^46 + 4 * q^49 - 68 * q^52 + 34 * q^58 + 56 * q^64 + 16 * q^67 + 4 * q^73 - 28 * q^76 - 72 * q^82 + 88 * q^88 - 4 * q^91 - 40 * 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 $

$2$	−1.34440	−0.950636	−0.475318	−	0.879814i	$-0.657667\pi$
$2$	−1.34440	−0.950636	−0.475318	+	0.879814i	$0.657667\pi$
$3$	0	0
$3$	0	0
$4$	−0.192582	−0.0962912
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.94771	1.04217
$9$	0	0
$10$	0	0
$11$	5.63652	1.69947	0.849737	−	0.527207i	$-0.176761\pi$
$11$	5.63652	1.69947	0.849737	+	0.527207i	$0.176761\pi$
$12$	0	0
$13$	4.38516	1.21623	0.608113	−	0.793851i	$-0.291927\pi$
$13$	4.38516	1.21623	0.608113	+	0.793851i	$0.291927\pi$
$14$	−1.34440	−0.359307
$15$	0	0
$16$	−3.57775	−0.894437
$17$	−2.68880	−0.652131	−0.326065	−	0.945347i	$-0.605723\pi$
$17$	−2.68880	−0.652131	−0.326065	+	0.945347i	$0.605723\pi$
$18$	0	0
$19$	8.38516	1.92369	0.961844	−	0.273597i	$-0.0882135\pi$
$19$	8.38516	1.92369	0.961844	+	0.273597i	$0.0882135\pi$
$20$	0	0
$21$	0	0
$22$	−7.57775	−1.61558
$23$	−5.63652	−1.17530	−0.587648	−	0.809117i	$-0.699946\pi$
$23$	−5.63652	−1.17530	−0.587648	+	0.809117i	$0.699946\pi$
$24$	0	0
$25$	0	0
$26$	−5.89543	−1.15619
$27$	0	0
$28$	−0.192582	−0.0363947
$29$	−8.32532	−1.54597	−0.772987	−	0.634422i	$-0.781238\pi$
$29$	−8.32532	−1.54597	−0.772987	+	0.634422i	$0.781238\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.08549	−0.191890
$33$	0	0
$34$	3.61484	0.619939
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−11.2730	−1.82873
$39$	0	0
$40$	0	0
$41$	5.37761	0.839841	0.419921	−	0.907561i	$-0.362058\pi$
$41$	5.37761	0.839841	0.419921	+	0.907561i	$0.362058\pi$
$42$	0	0
$43$	−1.38516	−0.211236	−0.105618	−	0.994407i	$-0.533682\pi$
$43$	−1.38516	−0.211236	−0.105618	+	0.994407i	$0.533682\pi$
$44$	−1.08549	−0.163644
$45$	0	0
$46$	7.57775	1.11728
$47$	−8.58423	−1.25214	−0.626069	−	0.779767i	$-0.715337\pi$
$47$	−8.58423	−1.25214	−0.626069	+	0.779767i	$0.715337\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−0.844506	−0.117112
$53$	5.37761	0.738671	0.369336	−	0.929296i	$-0.379585\pi$
$53$	5.37761	0.738671	0.369336	+	0.929296i	$0.379585\pi$
$54$	0	0
$55$	0	0
$56$	2.94771	0.393905
$57$	0	0
$58$	11.1926	1.46966
$59$	8.58423	1.11757	0.558786	−	0.829312i	$-0.311267\pi$
$59$	8.58423	1.11757	0.558786	+	0.829312i	$0.311267\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−8.06641	−1.02444
$63$	0	0
$64$	8.61484	1.07685
$65$	0	0
$66$	0	0
$67$	−1.38516	−0.169225	−0.0846124	−	0.996414i	$-0.526965\pi$
$67$	−1.38516	−0.169225	−0.0846124	+	0.996414i	$0.526965\pi$
$68$	0.517816	0.0627945
$69$	0	0
$70$	0	0
$71$	−0.258908	−0.0307268	−0.0153634	−	0.999882i	$-0.504891\pi$
$71$	−0.258908	−0.0307268	−0.0153634	+	0.999882i	$0.504891\pi$
$72$	0	0
$73$	6.38516	0.747327	0.373664	−	0.927564i	$-0.378101\pi$
$73$	6.38516	0.747327	0.373664	+	0.927564i	$0.378101\pi$
$74$	4.03321	0.468851
$75$	0	0
$76$	−1.61484	−0.185234
$77$	5.63652	0.642341
$78$	0	0
$79$	5.38516	0.605878	0.302939	−	0.953010i	$-0.402032\pi$
$79$	5.38516	0.605878	0.302939	+	0.953010i	$0.402032\pi$
$80$	0	0
$81$	0	0
$82$	−7.22967	−0.798384
$83$	16.6506	1.82765	0.913823	−	0.406113i	$-0.133116\pi$
$83$	16.6506	1.82765	0.913823	+	0.406113i	$0.133116\pi$
$84$	0	0
$85$	0	0
$86$	1.86222	0.200808
$87$	0	0
$88$	16.6148	1.77115
$89$	−13.9618	−1.47995	−0.739976	−	0.672633i	$-0.765163\pi$
$89$	−13.9618	−1.47995	−0.739976	+	0.672633i	$0.765163\pi$
$90$	0	0
$91$	4.38516	0.459690
$92$	1.08549	0.113171
$93$	0	0
$94$	11.5407	1.19033
$95$	0	0
$96$	0	0
$97$	−10.7703	−1.09356	−0.546781	−	0.837276i	$-0.684147\pi$
$97$	−10.7703	−1.09356	−0.546781	+	0.837276i	$0.684147\pi$
$98$	−1.34440	−0.135805
$99$	0	0
$100$	0	0
$101$	−2.68880	−0.267546	−0.133773	−	0.991012i	$-0.542709\pi$
$101$	−2.68880	−0.267546	−0.133773	+	0.991012i	$0.542709\pi$
$102$	0	0
$103$	−16.3852	−1.61448	−0.807239	−	0.590225i	$-0.799039\pi$
$103$	−16.3852	−1.61448	−0.807239	+	0.590225i	$0.799039\pi$
$104$	12.9262	1.26752
$105$	0	0
$106$	−7.22967	−0.702208
$107$	16.6506	1.60968	0.804839	−	0.593493i	$-0.202251\pi$
$107$	16.6506	1.60968	0.804839	+	0.593493i	$0.202251\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	−3.57775	−0.338065
$113$	−8.32532	−0.783180	−0.391590	−	0.920140i	$-0.628075\pi$
$113$	−8.32532	−0.783180	−0.391590	+	0.920140i	$0.628075\pi$
$114$	0	0
$115$	0	0
$116$	1.60331	0.148864
$117$	0	0
$118$	−11.5407	−1.06240
$119$	−2.68880	−0.246482
$120$	0	0
$121$	20.7703	1.88821
$122$	0	0
$123$	0	0
$124$	−1.15549	−0.103766
$125$	0	0
$126$	0	0
$127$	15.3852	1.36521	0.682606	−	0.730786i	$-0.260846\pi$
$127$	15.3852	1.36521	0.682606	+	0.730786i	$0.260846\pi$
$128$	−9.41082	−0.831806
$129$	0	0
$130$	0	0
$131$	−2.68880	−0.234922	−0.117461	−	0.993077i	$-0.537476\pi$
$131$	−2.68880	−0.234922	−0.117461	+	0.993077i	$0.537476\pi$
$132$	0	0
$133$	8.38516	0.727086
$134$	1.86222	0.160871
$135$	0	0
$136$	−7.92582	−0.679634
$137$	5.37761	0.459440	0.229720	−	0.973257i	$-0.426219\pi$
$137$	5.37761	0.459440	0.229720	+	0.973257i	$0.426219\pi$
$138$	0	0
$139$	6.38516	0.541583	0.270791	−	0.962638i	$-0.412715\pi$
$139$	6.38516	0.541583	0.270791	+	0.962638i	$0.412715\pi$
$140$	0	0
$141$	0	0
$142$	0.348077	0.0292100
$143$	24.7171	2.06694
$144$	0	0
$145$	0	0
$146$	−8.58423	−0.710436
$147$	0	0
$148$	0.577747	0.0474905
$149$	−13.7029	−1.12259	−0.561294	−	0.827617i	$-0.689696\pi$
$149$	−13.7029	−1.12259	−0.561294	+	0.827617i	$0.689696\pi$
$150$	0	0
$151$	0.614835	0.0500346	0.0250173	−	0.999687i	$-0.492036\pi$
$151$	0.614835	0.0500346	0.0250173	+	0.999687i	$0.492036\pi$
$152$	24.7171	2.00482
$153$	0	0
$154$	−7.57775	−0.610632
$155$	0	0
$156$	0	0
$157$	4.77033	0.380714	0.190357	−	0.981715i	$-0.439035\pi$
$157$	4.77033	0.380714	0.190357	+	0.981715i	$0.439035\pi$
$158$	−7.23983	−0.575970
$159$	0	0
$160$	0	0
$161$	−5.63652	−0.444220
$162$	0	0
$163$	4.77033	0.373641	0.186821	−	0.982394i	$-0.440182\pi$
$163$	4.77033	0.373641	0.186821	+	0.982394i	$0.440182\pi$
$164$	−1.03563	−0.0808693
$165$	0	0
$166$	−22.3852	−1.73743
$167$	−13.9618	−1.08040	−0.540200	−	0.841537i	$-0.681651\pi$
$167$	−13.9618	−1.08040	−0.540200	+	0.841537i	$0.681651\pi$
$168$	0	0
$169$	6.22967	0.479205
$170$	0	0
$171$	0	0
$172$	0.266758	0.0203401
$173$	−3.20662	−0.243795	−0.121897	−	0.992543i	$-0.538898\pi$
$173$	−3.20662	−0.243795	−0.121897	+	0.992543i	$0.538898\pi$
$174$	0	0
$175$	0	0
$176$	−20.1660	−1.52007
$177$	0	0
$178$	18.7703	1.40690
$179$	16.6506	1.24453	0.622264	−	0.782808i	$-0.286213\pi$
$179$	16.6506	1.24453	0.622264	+	0.782808i	$0.286213\pi$
$180$	0	0
$181$	9.61484	0.714665	0.357333	−	0.933977i	$-0.383686\pi$
$181$	9.61484	0.714665	0.357333	+	0.933977i	$0.383686\pi$
$182$	−5.89543	−0.436998
$183$	0	0
$184$	−16.6148	−1.22486
$185$	0	0
$186$	0	0
$187$	−15.1555	−1.10828
$188$	1.65317	0.120570
$189$	0	0
$190$	0	0
$191$	−11.2730	−0.815688	−0.407844	−	0.913052i	$-0.633719\pi$
$191$	−11.2730	−0.815688	−0.407844	+	0.913052i	$0.633719\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	14.4797	1.03958
$195$	0	0
$196$	−0.192582	−0.0137559
$197$	19.0805	1.35943	0.679716	−	0.733475i	$-0.262103\pi$
$197$	19.0805	1.35943	0.679716	+	0.733475i	$0.262103\pi$
$198$	0	0
$199$	−20.7703	−1.47237	−0.736185	−	0.676781i	$-0.763375\pi$
$199$	−20.7703	−1.47237	−0.736185	+	0.676781i	$0.763375\pi$
$200$	0	0
$201$	0	0
$202$	3.61484	0.254339
$203$	−8.32532	−0.584323
$204$	0	0
$205$	0	0
$206$	22.0283	1.53478
$207$	0	0
$208$	−15.6890	−1.08784
$209$	47.2631	3.26926
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−1.03563	−0.0711276
$213$	0	0
$214$	−22.3852	−1.53022
$215$	0	0
$216$	0	0
$217$	6.00000	0.407307
$218$	−9.41082	−0.637381
$219$	0	0
$220$	0	0
$221$	−11.7909	−0.793139
$222$	0	0
$223$	−14.3852	−0.963302	−0.481651	−	0.876363i	$-0.659963\pi$
$223$	−14.3852	−0.963302	−0.481651	+	0.876363i	$0.659963\pi$
$224$	−1.08549	−0.0725276
$225$	0	0
$226$	11.1926	0.744520
$227$	8.58423	0.569755	0.284878	−	0.958564i	$-0.408047\pi$
$227$	8.58423	0.569755	0.284878	+	0.958564i	$0.408047\pi$
$228$	0	0
$229$	27.5407	1.81994	0.909969	−	0.414676i	$-0.136105\pi$
$229$	27.5407	1.81994	0.909969	+	0.414676i	$0.136105\pi$
$230$	0	0
$231$	0	0
$232$	−24.5407	−1.61117
$233$	19.5984	1.28393	0.641966	−	0.766733i	$-0.278119\pi$
$233$	19.5984	1.28393	0.641966	+	0.766733i	$0.278119\pi$
$234$	0	0
$235$	0	0
$236$	−1.65317	−0.107612
$237$	0	0
$238$	3.61484	0.234315
$239$	−11.2730	−0.729192	−0.364596	−	0.931166i	$-0.618793\pi$
$239$	−11.2730	−0.729192	−0.364596	+	0.931166i	$0.618793\pi$
$240$	0	0
$241$	7.61484	0.490515	0.245257	−	0.969458i	$-0.421128\pi$
$241$	7.61484	0.490515	0.245257	+	0.969458i	$0.421128\pi$
$242$	−27.9237	−1.79500
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.7703	2.33964
$248$	17.6863	1.12308
$249$	0	0
$250$	0	0
$251$	30.6125	1.93224	0.966121	−	0.258088i	$-0.0830924\pi$
$251$	30.6125	1.93224	0.966121	+	0.258088i	$0.0830924\pi$
$252$	0	0
$253$	−31.7703	−1.99738
$254$	−20.6839	−1.29782
$255$	0	0
$256$	−4.57775	−0.286109
$257$	5.89543	0.367747	0.183873	−	0.982950i	$-0.441136\pi$
$257$	5.89543	0.367747	0.183873	+	0.982950i	$0.441136\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	3.61484	0.223325
$263$	11.5319	0.711090	0.355545	−	0.934659i	$-0.384295\pi$
$263$	11.5319	0.711090	0.355545	+	0.934659i	$0.384295\pi$
$264$	0	0
$265$	0	0
$266$	−11.2730	−0.691194
$267$	0	0
$268$	0.266758	0.0162949
$269$	−16.6506	−1.01521	−0.507604	−	0.861591i	$-0.669469\pi$
$269$	−16.6506	−1.01521	−0.507604	+	0.861591i	$0.669469\pi$
$270$	0	0
$271$	−17.1555	−1.04212	−0.521061	−	0.853519i	$-0.674464\pi$
$271$	−17.1555	−1.04212	−0.521061	+	0.853519i	$0.674464\pi$
$272$	9.61986	0.583290
$273$	0	0
$274$	−7.22967	−0.436760
$275$	0	0
$276$	0	0
$277$	19.5407	1.17408	0.587042	−	0.809556i	$-0.300292\pi$
$277$	19.5407	1.17408	0.587042	+	0.809556i	$0.300292\pi$
$278$	−8.58423	−0.514848
$279$	0	0
$280$	0	0
$281$	14.2207	0.848339	0.424169	−	0.905583i	$-0.360566\pi$
$281$	14.2207	0.848339	0.424169	+	0.905583i	$0.360566\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0.0498612	0.00295872
$285$	0	0
$286$	−33.2297	−1.96491
$287$	5.37761	0.317430
$288$	0	0
$289$	−9.77033	−0.574725
$290$	0	0
$291$	0	0
$292$	−1.22967	−0.0719610
$293$	−8.58423	−0.501496	−0.250748	−	0.968052i	$-0.580677\pi$
$293$	−8.58423	−0.501496	−0.250748	+	0.968052i	$0.580677\pi$
$294$	0	0
$295$	0	0
$296$	−8.84314	−0.513997
$297$	0	0
$298$	18.4223	1.06717
$299$	−24.7171	−1.42942
$300$	0	0
$301$	−1.38516	−0.0798396
$302$	−0.826586	−0.0475647
$303$	0	0
$304$	−30.0000	−1.72062
$305$	0	0
$306$	0	0
$307$	19.1555	1.09326	0.546631	−	0.837374i	$-0.315910\pi$
$307$	19.1555	1.09326	0.546631	+	0.837374i	$0.315910\pi$
$308$	−1.08549	−0.0618518
$309$	0	0
$310$	0	0
$311$	−13.4440	−0.762341	−0.381170	−	0.924505i	$-0.624479\pi$
$311$	−13.4440	−0.762341	−0.381170	+	0.924505i	$0.624479\pi$
$312$	0	0
$313$	−29.9258	−1.69151	−0.845754	−	0.533573i	$-0.820849\pi$
$313$	−29.9258	−1.69151	−0.845754	+	0.533573i	$0.820849\pi$
$314$	−6.41324	−0.361920
$315$	0	0
$316$	−1.03709	−0.0583408
$317$	2.94771	0.165560	0.0827800	−	0.996568i	$-0.473620\pi$
$317$	2.94771	0.165560	0.0827800	+	0.996568i	$0.473620\pi$
$318$	0	0
$319$	−46.9258	−2.62734
$320$	0	0
$321$	0	0
$322$	7.57775	0.422291
$323$	−22.5461	−1.25450
$324$	0	0
$325$	0	0
$326$	−6.41324	−0.355197
$327$	0	0
$328$	15.8516	0.875261
$329$	−8.58423	−0.473264
$330$	0	0
$331$	−22.1555	−1.21778	−0.608888	−	0.793256i	$-0.708384\pi$
$331$	−22.1555	−1.21778	−0.608888	+	0.793256i	$0.708384\pi$
$332$	−3.20662	−0.175986
$333$	0	0
$334$	18.7703	1.02707
$335$	0	0
$336$	0	0
$337$	−10.7703	−0.586697	−0.293349	−	0.956006i	$-0.594770\pi$
$337$	−10.7703	−0.586697	−0.293349	+	0.956006i	$0.594770\pi$
$338$	−8.37518	−0.455550
$339$	0	0
$340$	0	0
$341$	33.8191	1.83141
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.08307	−0.220144
$345$	0	0
$346$	4.31099	0.231760
$347$	−5.11870	−0.274786	−0.137393	−	0.990517i	$-0.543872\pi$
$347$	−5.11870	−0.274786	−0.137393	+	0.990517i	$0.543872\pi$
$348$	0	0
$349$	−8.38516	−0.448848	−0.224424	−	0.974492i	$-0.572050\pi$
$349$	−8.38516	−0.448848	−0.224424	+	0.974492i	$0.572050\pi$
$350$	0	0
$351$	0	0
$352$	−6.11841	−0.326112
$353$	−5.89543	−0.313782	−0.156891	−	0.987616i	$-0.550147\pi$
$353$	−5.89543	−0.313782	−0.156891	+	0.987616i	$0.550147\pi$
$354$	0	0
$355$	0	0
$356$	2.68880	0.142506
$357$	0	0
$358$	−22.3852	−1.18309
$359$	−16.3917	−0.865123	−0.432561	−	0.901604i	$-0.642390\pi$
$359$	−16.3917	−0.865123	−0.432561	+	0.901604i	$0.642390\pi$
$360$	0	0
$361$	51.3110	2.70058
$362$	−12.9262	−0.679386
$363$	0	0
$364$	−0.844506	−0.0442641
$365$	0	0
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	20.1660	1.05123
$369$	0	0
$370$	0	0
$371$	5.37761	0.279192
$372$	0	0
$373$	25.0000	1.29445	0.647225	−	0.762299i	$-0.275929\pi$
$373$	25.0000	1.29445	0.647225	+	0.762299i	$0.275929\pi$
$374$	20.3751	1.05357
$375$	0	0
$376$	−25.3038	−1.30495
$377$	−36.5079	−1.88025
$378$	0	0
$379$	−0.614835	−0.0315820	−0.0157910	−	0.999875i	$-0.505027\pi$
$379$	−0.614835	−0.0315820	−0.0157910	+	0.999875i	$0.505027\pi$
$380$	0	0
$381$	0	0
$382$	15.1555	0.775423
$383$	−19.3394	−0.988200	−0.494100	−	0.869405i	$-0.664502\pi$
$383$	−19.3394	−0.988200	−0.494100	+	0.869405i	$0.664502\pi$
$384$	0	0
$385$	0	0
$386$	1.34440	0.0684283
$387$	0	0
$388$	2.07418	0.105300
$389$	30.8714	1.56524	0.782621	−	0.622499i	$-0.213882\pi$
$389$	30.8714	1.56524	0.782621	+	0.622499i	$0.213882\pi$
$390$	0	0
$391$	15.1555	0.766446
$392$	2.94771	0.148882
$393$	0	0
$394$	−25.6519	−1.29233
$395$	0	0
$396$	0	0
$397$	22.7703	1.14281	0.571405	−	0.820668i	$-0.306399\pi$
$397$	22.7703	1.14281	0.571405	+	0.820668i	$0.306399\pi$
$398$	27.9237	1.39969
$399$	0	0
$400$	0	0
$401$	−13.7029	−0.684292	−0.342146	−	0.939647i	$-0.611154\pi$
$401$	−13.7029	−0.684292	−0.342146	+	0.939647i	$0.611154\pi$
$402$	0	0
$403$	26.3110	1.31064
$404$	0.517816	0.0257623
$405$	0	0
$406$	11.1926	0.555479
$407$	−16.9096	−0.838175
$408$	0	0
$409$	11.6148	0.574317	0.287158	−	0.957883i	$-0.407289\pi$
$409$	11.6148	0.574317	0.287158	+	0.957883i	$0.407289\pi$
$410$	0	0
$411$	0	0
$412$	3.15549	0.155460
$413$	8.58423	0.422402
$414$	0	0
$415$	0	0
$416$	−4.76007	−0.233382
$417$	0	0
$418$	−63.5407	−3.10788
$419$	−5.89543	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$419$	−5.89543	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$420$	0	0
$421$	−39.3110	−1.91590	−0.957950	−	0.286935i	$-0.907364\pi$
$421$	−39.3110	−1.91590	−0.957950	+	0.286935i	$0.907364\pi$
$422$	0	0
$423$	0	0
$424$	15.8516	0.769824
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−3.20662	−0.154998
$429$	0	0
$430$	0	0
$431$	27.4059	1.32009	0.660047	−	0.751224i	$-0.270536\pi$
$431$	27.4059	1.32009	0.660047	+	0.751224i	$0.270536\pi$
$432$	0	0
$433$	−33.5407	−1.61186	−0.805931	−	0.592010i	$-0.798335\pi$
$433$	−33.5407	−1.61186	−0.805931	+	0.592010i	$0.798335\pi$
$434$	−8.06641	−0.387200
$435$	0	0
$436$	−1.34808	−0.0645612
$437$	−47.2631	−2.26090
$438$	0	0
$439$	−13.1555	−0.627877	−0.313939	−	0.949443i	$-0.601649\pi$
$439$	−13.1555	−0.627877	−0.313939	+	0.949443i	$0.601649\pi$
$440$	0	0
$441$	0	0
$442$	15.8516	0.753986
$443$	−33.8191	−1.60679	−0.803397	−	0.595444i	$-0.796976\pi$
$443$	−33.8191	−1.60679	−0.803397	+	0.595444i	$0.796976\pi$
$444$	0	0
$445$	0	0
$446$	19.3394	0.915749
$447$	0	0
$448$	8.61484	0.407013
$449$	−13.7029	−0.646681	−0.323341	−	0.946283i	$-0.604806\pi$
$449$	−13.7029	−0.646681	−0.323341	+	0.946283i	$0.604806\pi$
$450$	0	0
$451$	30.3110	1.42729
$452$	1.60331	0.0754134
$453$	0	0
$454$	−11.5407	−0.541630
$455$	0	0
$456$	0	0
$457$	−14.5407	−0.680183	−0.340092	−	0.940392i	$-0.610458\pi$
$457$	−14.5407	−0.680183	−0.340092	+	0.940392i	$0.610458\pi$
$458$	−37.0257	−1.73010
$459$	0	0
$460$	0	0
$461$	−13.9618	−0.650268	−0.325134	−	0.945668i	$-0.605409\pi$
$461$	−13.9618	−0.650268	−0.325134	+	0.945668i	$0.605409\pi$
$462$	0	0
$463$	−12.7703	−0.593488	−0.296744	−	0.954957i	$-0.595901\pi$
$463$	−12.7703	−0.593488	−0.296744	+	0.954957i	$0.595901\pi$
$464$	29.7859	1.38278
$465$	0	0
$466$	−26.3481	−1.22055
$467$	11.2730	0.521654	0.260827	−	0.965386i	$-0.416005\pi$
$467$	11.2730	0.521654	0.260827	+	0.965386i	$0.416005\pi$
$468$	0	0
$469$	−1.38516	−0.0639610
$470$	0	0
$471$	0	0
$472$	25.3038	1.16470
$473$	−7.80751	−0.358989
$474$	0	0
$475$	0	0
$476$	0.517816	0.0237341
$477$	0	0
$478$	15.1555	0.693196
$479$	2.17099	0.0991950	0.0495975	−	0.998769i	$-0.484206\pi$
$479$	2.17099	0.0991950	0.0495975	+	0.998769i	$0.484206\pi$
$480$	0	0
$481$	−13.1555	−0.599839
$482$	−10.2374	−0.466301
$483$	0	0
$484$	−4.00000	−0.181818
$485$	0	0
$486$	0	0
$487$	2.61484	0.118489	0.0592447	−	0.998243i	$-0.481131\pi$
$487$	2.61484	0.118489	0.0592447	+	0.998243i	$0.481131\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.4963	0.473692	0.236846	−	0.971547i	$-0.423886\pi$
$491$	10.4963	0.473692	0.236846	+	0.971547i	$0.423886\pi$
$492$	0	0
$493$	22.3852	1.00818
$494$	−49.4341	−2.22415
$495$	0	0
$496$	−21.4665	−0.963874
$497$	−0.258908	−0.0116136
$498$	0	0
$499$	−13.5407	−0.606163	−0.303082	−	0.952965i	$-0.598015\pi$
$499$	−13.5407	−0.606163	−0.303082	+	0.952965i	$0.598015\pi$
$500$	0	0
$501$	0	0
$502$	−41.1555	−1.83686
$503$	23.0639	1.02837	0.514184	−	0.857680i	$-0.328095\pi$
$503$	23.0639	1.02837	0.514184	+	0.857680i	$0.328095\pi$
$504$	0	0
$505$	0	0
$506$	42.7121	1.89878
$507$	0	0
$508$	−2.96291	−0.131458
$509$	19.3394	0.857206	0.428603	−	0.903493i	$-0.359006\pi$
$509$	19.3394	0.857206	0.428603	+	0.903493i	$0.359006\pi$
$510$	0	0
$511$	6.38516	0.282463
$512$	24.9760	1.10379
$513$	0	0
$514$	−7.92582	−0.349593
$515$	0	0
$516$	0	0
$517$	−48.3852	−2.12798
$518$	4.03321	0.177209
$519$	0	0
$520$	0	0
$521$	−32.7835	−1.43627	−0.718135	−	0.695904i	$-0.755004\pi$
$521$	−32.7835	−1.43627	−0.718135	+	0.695904i	$0.755004\pi$
$522$	0	0
$523$	−40.7703	−1.78276	−0.891381	−	0.453255i	$-0.850263\pi$
$523$	−40.7703	−1.78276	−0.891381	+	0.453255i	$0.850263\pi$
$524$	0.517816	0.0226209
$525$	0	0
$526$	−15.5036	−0.675988
$527$	−16.1328	−0.702757
$528$	0	0
$529$	8.77033	0.381319
$530$	0	0
$531$	0	0
$532$	−1.61484	−0.0700120
$533$	23.5817	1.02144
$534$	0	0
$535$	0	0
$536$	−4.08307	−0.176362
$537$	0	0
$538$	22.3852	0.965093
$539$	5.63652	0.242782
$540$	0	0
$541$	28.5407	1.22706	0.613529	−	0.789672i	$-0.289749\pi$
$541$	28.5407	1.22706	0.613529	+	0.789672i	$0.289749\pi$
$542$	23.0639	0.990679
$543$	0	0
$544$	2.91868	0.125137
$545$	0	0
$546$	0	0
$547$	23.3852	0.999877	0.499939	−	0.866061i	$-0.333356\pi$
$547$	23.3852	0.999877	0.499939	+	0.866061i	$0.333356\pi$
$548$	−1.03563	−0.0442400
$549$	0	0
$550$	0	0
$551$	−69.8092	−2.97397
$552$	0	0
$553$	5.38516	0.229001
$554$	−26.2705	−1.11613
$555$	0	0
$556$	−1.22967	−0.0521496
$557$	−2.94771	−0.124899	−0.0624493	−	0.998048i	$-0.519891\pi$
$557$	−2.94771	−0.124899	−0.0624493	+	0.998048i	$0.519891\pi$
$558$	0	0
$559$	−6.07418	−0.256910
$560$	0	0
$561$	0	0
$562$	−19.1184	−0.806461
$563$	−16.6506	−0.701741	−0.350870	−	0.936424i	$-0.614114\pi$
$563$	−16.6506	−0.701741	−0.350870	+	0.936424i	$0.614114\pi$
$564$	0	0
$565$	0	0
$566$	18.8216	0.791132
$567$	0	0
$568$	−0.763187	−0.0320226
$569$	−20.1162	−0.843314	−0.421657	−	0.906755i	$-0.638551\pi$
$569$	−20.1162	−0.843314	−0.421657	+	0.906755i	$0.638551\pi$
$570$	0	0
$571$	24.6148	1.03010	0.515049	−	0.857160i	$-0.327774\pi$
$571$	24.6148	1.03010	0.515049	+	0.857160i	$0.327774\pi$
$572$	−4.76007	−0.199029
$573$	0	0
$574$	−7.22967	−0.301761
$575$	0	0
$576$	0	0
$577$	14.8445	0.617985	0.308992	−	0.951064i	$-0.400008\pi$
$577$	14.8445	0.617985	0.308992	+	0.951064i	$0.400008\pi$
$578$	13.1353	0.546355
$579$	0	0
$580$	0	0
$581$	16.6506	0.690785
$582$	0	0
$583$	30.3110	1.25535
$584$	18.8216	0.778845
$585$	0	0
$586$	11.5407	0.476740
$587$	35.9901	1.48547	0.742735	−	0.669585i	$-0.233528\pi$
$587$	35.9901	1.48547	0.742735	+	0.669585i	$0.233528\pi$
$588$	0	0
$589$	50.3110	2.07303
$590$	0	0
$591$	0	0
$592$	10.7332	0.441134
$593$	11.7909	0.484192	0.242096	−	0.970252i	$-0.422165\pi$
$593$	11.7909	0.484192	0.242096	+	0.970252i	$0.422165\pi$
$594$	0	0
$595$	0	0
$596$	2.63894	0.108095
$597$	0	0
$598$	33.2297	1.35886
$599$	−22.8050	−0.931786	−0.465893	−	0.884841i	$-0.654267\pi$
$599$	−22.8050	−0.931786	−0.465893	+	0.884841i	$0.654267\pi$
$600$	0	0
$601$	−1.54066	−0.0628448	−0.0314224	−	0.999506i	$-0.510004\pi$
$601$	−1.54066	−0.0628448	−0.0314224	+	0.999506i	$0.510004\pi$
$602$	1.86222	0.0758984
$603$	0	0
$604$	−0.118406	−0.00481789
$605$	0	0
$606$	0	0
$607$	−9.15549	−0.371610	−0.185805	−	0.982587i	$-0.559489\pi$
$607$	−9.15549	−0.371610	−0.185805	+	0.982587i	$0.559489\pi$
$608$	−9.10205	−0.369137
$609$	0	0
$610$	0	0
$611$	−37.6433	−1.52288
$612$	0	0
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	−25.7527	−1.03929
$615$	0	0
$616$	16.6148	0.669431
$617$	30.8714	1.24284	0.621418	−	0.783479i	$-0.286557\pi$
$617$	30.8714	1.24284	0.621418	+	0.783479i	$0.286557\pi$
$618$	0	0
$619$	5.15549	0.207217	0.103608	−	0.994618i	$-0.466961\pi$
$619$	5.15549	0.207217	0.103608	+	0.994618i	$0.466961\pi$
$620$	0	0
$621$	0	0
$622$	18.0742	0.724708
$623$	−13.9618	−0.559369
$624$	0	0
$625$	0	0
$626$	40.2323	1.60801
$627$	0	0
$628$	−0.918682	−0.0366594
$629$	8.06641	0.321629
$630$	0	0
$631$	5.38516	0.214380	0.107190	−	0.994239i	$-0.465815\pi$
$631$	5.38516	0.214380	0.107190	+	0.994239i	$0.465815\pi$
$632$	15.8739	0.631431
$633$	0	0
$634$	−3.96291	−0.157387
$635$	0	0
$636$	0	0
$637$	4.38516	0.173747
$638$	63.0872	2.49765
$639$	0	0
$640$	0	0
$641$	25.4938	1.00694	0.503472	−	0.864012i	$-0.332056\pi$
$641$	25.4938	1.00694	0.503472	+	0.864012i	$0.332056\pi$
$642$	0	0
$643$	−49.5407	−1.95369	−0.976846	−	0.213942i	$-0.931370\pi$
$643$	−49.5407	−1.95369	−0.976846	+	0.213942i	$0.931370\pi$
$644$	1.08549	0.0427745
$645$	0	0
$646$	30.3110	1.19257
$647$	−4.85979	−0.191058	−0.0955291	−	0.995427i	$-0.530454\pi$
$647$	−4.85979	−0.191058	−0.0955291	+	0.995427i	$0.530454\pi$
$648$	0	0
$649$	48.3852	1.89928
$650$	0	0
$651$	0	0
$652$	−0.918682	−0.0359783
$653$	5.37761	0.210442	0.105221	−	0.994449i	$-0.466445\pi$
$653$	5.37761	0.210442	0.105221	+	0.994449i	$0.466445\pi$
$654$	0	0
$655$	0	0
$656$	−19.2397	−0.751185
$657$	0	0
$658$	11.5407	0.449902
$659$	38.1611	1.48654	0.743272	−	0.668989i	$-0.233273\pi$
$659$	38.1611	1.48654	0.743272	+	0.668989i	$0.233273\pi$
$660$	0	0
$661$	−21.9258	−0.852816	−0.426408	−	0.904531i	$-0.640221\pi$
$661$	−21.9258	−0.852816	−0.426408	+	0.904531i	$0.640221\pi$
$662$	29.7859	1.15766
$663$	0	0
$664$	49.0813	1.90472
$665$	0	0
$666$	0	0
$667$	46.9258	1.81698
$668$	2.68880	0.104033
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	39.5407	1.52418	0.762090	−	0.647471i	$-0.224173\pi$
$673$	39.5407	1.52418	0.762090	+	0.647471i	$0.224173\pi$
$674$	14.4797	0.557736
$675$	0	0
$676$	−1.19972	−0.0461433
$677$	−11.2730	−0.433258	−0.216629	−	0.976254i	$-0.569506\pi$
$677$	−11.2730	−0.433258	−0.216629	+	0.976254i	$0.569506\pi$
$678$	0	0
$679$	−10.7703	−0.413327
$680$	0	0
$681$	0	0
$682$	−45.4665	−1.74100
$683$	−16.9096	−0.647026	−0.323513	−	0.946224i	$-0.604864\pi$
$683$	−16.9096	−0.647026	−0.323513	+	0.946224i	$0.604864\pi$
$684$	0	0
$685$	0	0
$686$	−1.34440	−0.0513295
$687$	0	0
$688$	4.95577	0.188937
$689$	23.5817	0.898391
$690$	0	0
$691$	−36.7703	−1.39881	−0.699405	−	0.714726i	$-0.746551\pi$
$691$	−36.7703	−1.39881	−0.699405	+	0.714726i	$0.746551\pi$
$692$	0.617539	0.0234753
$693$	0	0
$694$	6.88159	0.261222
$695$	0	0
$696$	0	0
$697$	−14.4593	−0.547687
$698$	11.2730	0.426691
$699$	0	0
$700$	0	0
$701$	−6.41324	−0.242225	−0.121112	−	0.992639i	$-0.538646\pi$
$701$	−6.41324	−0.242225	−0.121112	+	0.992639i	$0.538646\pi$
$702$	0	0
$703$	−25.1555	−0.948757
$704$	48.5577	1.83009
$705$	0	0
$706$	7.92582	0.298292
$707$	−2.68880	−0.101123
$708$	0	0
$709$	−3.54066	−0.132972	−0.0664861	−	0.997787i	$-0.521179\pi$
$709$	−3.54066	−0.132972	−0.0664861	+	0.997787i	$0.521179\pi$
$710$	0	0
$711$	0	0
$712$	−41.1555	−1.54237
$713$	−33.8191	−1.26654
$714$	0	0
$715$	0	0
$716$	−3.20662	−0.119837
$717$	0	0
$718$	22.0371	0.822417
$719$	38.6789	1.44248	0.721240	−	0.692686i	$-0.243573\pi$
$719$	38.6789	1.44248	0.721240	+	0.692686i	$0.243573\pi$
$720$	0	0
$721$	−16.3852	−0.610215
$722$	−68.9826	−2.56727
$723$	0	0
$724$	−1.85165	−0.0688160
$725$	0	0
$726$	0	0
$727$	−21.6148	−0.801650	−0.400825	−	0.916155i	$-0.631276\pi$
$727$	−21.6148	−0.801650	−0.400825	+	0.916155i	$0.631276\pi$
$728$	12.9262	0.479077
$729$	0	0
$730$	0	0
$731$	3.72444	0.137753
$732$	0	0
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	−5.37761	−0.198491
$735$	0	0
$736$	6.11841	0.225527
$737$	−7.80751	−0.287593
$738$	0	0
$739$	−32.6148	−1.19976	−0.599878	−	0.800091i	$-0.704784\pi$
$739$	−32.6148	−1.19976	−0.599878	+	0.800091i	$0.704784\pi$
$740$	0	0
$741$	0	0
$742$	−7.22967	−0.265410
$743$	−22.0283	−0.808138	−0.404069	−	0.914728i	$-0.632404\pi$
$743$	−22.0283	−0.808138	−0.404069	+	0.914728i	$0.632404\pi$
$744$	0	0
$745$	0	0
$746$	−33.6101	−1.23055
$747$	0	0
$748$	2.91868	0.106718
$749$	16.6506	0.608401
$750$	0	0
$751$	−7.22967	−0.263814	−0.131907	−	0.991262i	$-0.542110\pi$
$751$	−7.22967	−0.263814	−0.131907	+	0.991262i	$0.542110\pi$
$752$	30.7122	1.11996
$753$	0	0
$754$	49.0813	1.78744
$755$	0	0
$756$	0	0
$757$	−53.3110	−1.93762	−0.968810	−	0.247803i	$-0.920291\pi$
$757$	−53.3110	−1.93762	−0.968810	+	0.247803i	$0.920291\pi$
$758$	0.826586	0.0300230
$759$	0	0
$760$	0	0
$761$	−22.5461	−0.817294	−0.408647	−	0.912692i	$-0.633999\pi$
$761$	−22.5461	−0.817294	−0.408647	+	0.912692i	$0.633999\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	2.17099	0.0785436
$765$	0	0
$766$	26.0000	0.939418
$767$	37.6433	1.35922
$768$	0	0
$769$	23.2297	0.837683	0.418842	−	0.908059i	$-0.362436\pi$
$769$	23.2297	0.837683	0.418842	+	0.908059i	$0.362436\pi$
$770$	0	0
$771$	0	0
$772$	0.192582	0.00693119
$773$	22.0283	0.792301	0.396151	−	0.918186i	$-0.370346\pi$
$773$	22.0283	0.792301	0.396151	+	0.918186i	$0.370346\pi$
$774$	0	0
$775$	0	0
$776$	−31.7478	−1.13968
$777$	0	0
$778$	−41.5036	−1.48798
$779$	45.0921	1.61559
$780$	0	0
$781$	−1.45934	−0.0522193
$782$	−20.3751	−0.728611
$783$	0	0
$784$	−3.57775	−0.127777
$785$	0	0
$786$	0	0
$787$	−10.3852	−0.370191	−0.185096	−	0.982721i	$-0.559259\pi$
$787$	−10.3852	−0.370191	−0.185096	+	0.982721i	$0.559259\pi$
$788$	−3.67458	−0.130901
$789$	0	0
$790$	0	0
$791$	−8.32532	−0.296014
$792$	0	0
$793$	0	0
$794$	−30.6125	−1.08640
$795$	0	0
$796$	4.00000	0.141776
$797$	42.4033	1.50200	0.751002	−	0.660300i	$-0.229571\pi$
$797$	42.4033	1.50200	0.751002	+	0.660300i	$0.229571\pi$
$798$	0	0
$799$	23.0813	0.816558
$800$	0	0
$801$	0	0
$802$	18.4223	0.650512
$803$	35.9901	1.27006
$804$	0	0
$805$	0	0
$806$	−35.3726	−1.24595
$807$	0	0
$808$	−7.92582	−0.278830
$809$	−3.46553	−0.121842	−0.0609208	−	0.998143i	$-0.519404\pi$
$809$	−3.46553	−0.121842	−0.0609208	+	0.998143i	$0.519404\pi$
$810$	0	0
$811$	9.22967	0.324098	0.162049	−	0.986783i	$-0.448190\pi$
$811$	9.22967	0.324098	0.162049	+	0.986783i	$0.448190\pi$
$812$	1.60331	0.0562652
$813$	0	0
$814$	22.7332	0.796800
$815$	0	0
$816$	0	0
$817$	−11.6148	−0.406352
$818$	−15.6150	−0.545966
$819$	0	0
$820$	0	0
$821$	34.3369	1.19837	0.599183	−	0.800612i	$-0.295492\pi$
$821$	34.3369	1.19837	0.599183	+	0.800612i	$0.295492\pi$
$822$	0	0
$823$	28.9258	1.00829	0.504145	−	0.863619i	$-0.331808\pi$
$823$	28.9258	1.00829	0.504145	+	0.863619i	$0.331808\pi$
$824$	−48.2988	−1.68257
$825$	0	0
$826$	−11.5407	−0.401551
$827$	−0.258908	−0.00900312	−0.00450156	−	0.999990i	$-0.501433\pi$
$827$	−0.258908	−0.00900312	−0.00450156	+	0.999990i	$0.501433\pi$
$828$	0	0
$829$	2.38516	0.0828402	0.0414201	−	0.999142i	$-0.486812\pi$
$829$	2.38516	0.0828402	0.0414201	+	0.999142i	$0.486812\pi$
$830$	0	0
$831$	0	0
$832$	37.7775	1.30970
$833$	−2.68880	−0.0931616
$834$	0	0
$835$	0	0
$836$	−9.10205	−0.314801
$837$	0	0
$838$	7.92582	0.273793
$839$	−35.9901	−1.24252	−0.621258	−	0.783606i	$-0.713378\pi$
$839$	−35.9901	−1.24252	−0.621258	+	0.783606i	$0.713378\pi$
$840$	0	0
$841$	40.3110	1.39003
$842$	52.8498	1.82132
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.7703	0.713677
$848$	−19.2397	−0.660695
$849$	0	0
$850$	0	0
$851$	16.9096	0.579652
$852$	0	0
$853$	34.3110	1.17479	0.587393	−	0.809302i	$-0.300154\pi$
$853$	34.3110	1.17479	0.587393	+	0.809302i	$0.300154\pi$
$854$	0	0
$855$	0	0
$856$	49.0813	1.67756
$857$	39.7145	1.35662	0.678311	−	0.734775i	$-0.262712\pi$
$857$	39.7145	1.35662	0.678311	+	0.734775i	$0.262712\pi$
$858$	0	0
$859$	15.5407	0.530240	0.265120	−	0.964215i	$-0.414588\pi$
$859$	15.5407	0.530240	0.265120	+	0.964215i	$0.414588\pi$
$860$	0	0
$861$	0	0
$862$	−36.8445	−1.25493
$863$	22.2872	0.758664	0.379332	−	0.925261i	$-0.376154\pi$
$863$	22.2872	0.758664	0.379332	+	0.925261i	$0.376154\pi$
$864$	0	0
$865$	0	0
$866$	45.0921	1.53229
$867$	0	0
$868$	−1.15549	−0.0392200
$869$	30.3536	1.02967
$870$	0	0
$871$	−6.07418	−0.205816
$872$	20.6340	0.698755
$873$	0	0
$874$	63.5407	2.14929
$875$	0	0
$876$	0	0
$877$	23.5407	0.794912	0.397456	−	0.917621i	$-0.369893\pi$
$877$	23.5407	0.794912	0.397456	+	0.917621i	$0.369893\pi$
$878$	17.6863	0.596883
$879$	0	0
$880$	0	0
$881$	−23.0639	−0.777042	−0.388521	−	0.921440i	$-0.627014\pi$
$881$	−23.0639	−0.777042	−0.388521	+	0.921440i	$0.627014\pi$
$882$	0	0
$883$	−19.3852	−0.652363	−0.326181	−	0.945307i	$-0.605762\pi$
$883$	−19.3852	−0.652363	−0.326181	+	0.945307i	$0.605762\pi$
$884$	2.27071	0.0763723
$885$	0	0
$886$	45.4665	1.52748
$887$	−5.37761	−0.180562	−0.0902812	−	0.995916i	$-0.528777\pi$
$887$	−5.37761	−0.180562	−0.0902812	+	0.995916i	$0.528777\pi$
$888$	0	0
$889$	15.3852	0.516002
$890$	0	0
$891$	0	0
$892$	2.77033	0.0927575
$893$	−71.9802	−2.40873
$894$	0	0
$895$	0	0
$896$	−9.41082	−0.314393
$897$	0	0
$898$	18.4223	0.614759
$899$	−49.9519	−1.66599
$900$	0	0
$901$	−14.4593	−0.481710
$902$	−40.7502	−1.35683
$903$	0	0
$904$	−24.5407	−0.816210
$905$	0	0
$906$	0	0
$907$	−42.3110	−1.40491	−0.702457	−	0.711727i	$-0.747914\pi$
$907$	−42.3110	−1.40491	−0.702457	+	0.711727i	$0.747914\pi$
$908$	−1.65317	−0.0548624
$909$	0	0
$910$	0	0
$911$	−38.9378	−1.29007	−0.645034	−	0.764154i	$-0.723157\pi$
$911$	−38.9378	−1.29007	−0.645034	+	0.764154i	$0.723157\pi$
$912$	0	0
$913$	93.8516	3.10604
$914$	19.5485	0.646607
$915$	0	0
$916$	−5.30385	−0.175244
$917$	−2.68880	−0.0887922
$918$	0	0
$919$	25.6962	0.847638	0.423819	−	0.905747i	$-0.360689\pi$
$919$	25.6962	0.847638	0.423819	+	0.905747i	$0.360689\pi$
$920$	0	0
$921$	0	0
$922$	18.7703	0.618168
$923$	−1.13536	−0.0373707
$924$	0	0
$925$	0	0
$926$	17.1685	0.564191
$927$	0	0
$928$	9.03709	0.296657
$929$	8.06641	0.264650	0.132325	−	0.991206i	$-0.457756\pi$
$929$	8.06641	0.264650	0.132325	+	0.991206i	$0.457756\pi$
$930$	0	0
$931$	8.38516	0.274813
$932$	−3.77430	−0.123631
$933$	0	0
$934$	−15.1555	−0.495903
$935$	0	0
$936$	0	0
$937$	−40.7703	−1.33191	−0.665954	−	0.745993i	$-0.731975\pi$
$937$	−40.7703	−1.33191	−0.665954	+	0.745993i	$0.731975\pi$
$938$	1.86222	0.0608036
$939$	0	0
$940$	0	0
$941$	−10.2374	−0.333730	−0.166865	−	0.985980i	$-0.553364\pi$
$941$	−10.2374	−0.333730	−0.166865	+	0.985980i	$0.553364\pi$
$942$	0	0
$943$	−30.3110	−0.987062
$944$	−30.7122	−0.999597
$945$	0	0
$946$	10.4964	0.341268
$947$	−50.9876	−1.65687	−0.828437	−	0.560083i	$-0.810769\pi$
$947$	−50.9876	−1.65687	−0.828437	+	0.560083i	$0.810769\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	0	0
$952$	−7.92582	−0.256877
$953$	−36.2490	−1.17422	−0.587110	−	0.809507i	$-0.699734\pi$
$953$	−36.2490	−1.17422	−0.587110	+	0.809507i	$0.699734\pi$
$954$	0	0
$955$	0	0
$956$	2.17099	0.0702148
$957$	0	0
$958$	−2.91868	−0.0942983
$959$	5.37761	0.173652
$960$	0	0
$961$	5.00000	0.161290
$962$	17.6863	0.570228
$963$	0	0
$964$	−1.46648	−0.0472322
$965$	0	0
$966$	0	0
$967$	−5.54066	−0.178176	−0.0890878	−	0.996024i	$-0.528395\pi$
$967$	−5.54066	−0.178176	−0.0890878	+	0.996024i	$0.528395\pi$
$968$	61.2250	1.96784
$969$	0	0
$970$	0	0
$971$	41.8855	1.34417	0.672085	−	0.740474i	$-0.265399\pi$
$971$	41.8855	1.34417	0.672085	+	0.740474i	$0.265399\pi$
$972$	0	0
$973$	6.38516	0.204699
$974$	−3.51539	−0.112640
$975$	0	0
$976$	0	0
$977$	−48.0399	−1.53693	−0.768466	−	0.639891i	$-0.778979\pi$
$977$	−48.0399	−1.53693	−0.768466	+	0.639891i	$0.778979\pi$
$978$	0	0
$979$	−78.6962	−2.51514
$980$	0	0
$981$	0	0
$982$	−14.1113	−0.450309
$983$	32.7835	1.04563	0.522815	−	0.852446i	$-0.324882\pi$
$983$	32.7835	1.04563	0.522815	+	0.852446i	$0.324882\pi$
$984$	0	0
$985$	0	0
$986$	−30.0947	−0.958409
$987$	0	0
$988$	−7.08132	−0.225287
$989$	7.80751	0.248264
$990$	0	0
$991$	−35.6962	−1.13393	−0.566963	−	0.823743i	$-0.691882\pi$
$991$	−35.6962	−1.13393	−0.566963	+	0.823743i	$0.691882\pi$
$992$	−6.51296	−0.206787
$993$	0	0
$994$	0.348077	0.0110403
$995$	0	0
$996$	0	0
$997$	−57.1555	−1.81013	−0.905066	−	0.425270i	$-0.860179\pi$
$997$	−57.1555	−1.81013	−0.905066	+	0.425270i	$0.860179\pi$
$998$	18.2041	0.576241
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.z.1.2	yes	4
3.2	odd	2	inner	1575.2.a.z.1.3	yes	4
5.2	odd	4		1575.2.d.l.1324.4		8
5.3	odd	4		1575.2.d.l.1324.5		8
5.4	even	2		1575.2.a.y.1.3	yes	4
15.2	even	4		1575.2.d.l.1324.6		8
15.8	even	4		1575.2.d.l.1324.3		8
15.14	odd	2		1575.2.a.y.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.a.y.1.2	✓	4	15.14	odd	2
1575.2.a.y.1.3	yes	4	5.4	even	2
1575.2.a.z.1.2	yes	4	1.1	even	1	trivial
1575.2.a.z.1.3	yes	4	3.2	odd	2	inner
1575.2.d.l.1324.3		8	15.8	even	4
1575.2.d.l.1324.4		8	5.2	odd	4
1575.2.d.l.1324.5		8	5.3	odd	4
1575.2.d.l.1324.6		8	15.2	even	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34440 q^{2} -0.192582 q^{4} +1.00000 q^{7} +2.94771 q^{8} +O(q^{10})\)	\(q-1.34440 q^{2} -0.192582 q^{4} +1.00000 q^{7} +2.94771 q^{8} +5.63652 q^{11} +4.38516 q^{13} -1.34440 q^{14} -3.57775 q^{16} -2.68880 q^{17} +8.38516 q^{19} -7.57775 q^{22} -5.63652 q^{23} -5.89543 q^{26} -0.192582 q^{28} -8.32532 q^{29} +6.00000 q^{31} -1.08549 q^{32} +3.61484 q^{34} -3.00000 q^{37} -11.2730 q^{38} +5.37761 q^{41} -1.38516 q^{43} -1.08549 q^{44} +7.57775 q^{46} -8.58423 q^{47} +1.00000 q^{49} -0.844506 q^{52} +5.37761 q^{53} +2.94771 q^{56} +11.1926 q^{58} +8.58423 q^{59} -8.06641 q^{62} +8.61484 q^{64} -1.38516 q^{67} +0.517816 q^{68} -0.258908 q^{71} +6.38516 q^{73} +4.03321 q^{74} -1.61484 q^{76} +5.63652 q^{77} +5.38516 q^{79} -7.22967 q^{82} +16.6506 q^{83} +1.86222 q^{86} +16.6148 q^{88} -13.9618 q^{89} +4.38516 q^{91} +1.08549 q^{92} +11.5407 q^{94} -10.7703 q^{97} -1.34440 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 4 q^{7}+O(q^{10}) \) 4 * q + 10 * q^4 + 4 * q^7	\( 4 q + 10 q^{4} + 4 q^{7} - 4 q^{13} + 18 q^{16} + 12 q^{19} + 2 q^{22} + 10 q^{28} + 24 q^{31} + 36 q^{34} - 12 q^{37} + 16 q^{43} - 2 q^{46} + 4 q^{49} - 68 q^{52} + 34 q^{58} + 56 q^{64} + 16 q^{67} + 4 q^{73} - 28 q^{76} - 72 q^{82} + 88 q^{88} - 4 q^{91} - 40 q^{94}+O(q^{100}) \) 4 * q + 10 * q^4 + 4 * q^7 - 4 * q^13 + 18 * q^16 + 12 * q^19 + 2 * q^22 + 10 * q^28 + 24 * q^31 + 36 * q^34 - 12 * q^37 + 16 * q^43 - 2 * q^46 + 4 * q^49 - 68 * q^52 + 34 * q^58 + 56 * q^64 + 16 * q^67 + 4 * q^73 - 28 * q^76 - 72 * q^82 + 88 * q^88 - 4 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more