LMFDB - Embedded newform 1575.2.a.v.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} +1.00000 q^{7} +7.47214 q^{8} +O(q^{10})$	$q+2.61803 q^{2} +4.85410 q^{4} +1.00000 q^{7} +7.47214 q^{8} +5.47214 q^{11} -0.763932 q^{13} +2.61803 q^{14} +9.85410 q^{16} -7.70820 q^{17} -3.23607 q^{19} +14.3262 q^{22} +5.00000 q^{23} -2.00000 q^{26} +4.85410 q^{28} -4.70820 q^{29} +4.47214 q^{31} +10.8541 q^{32} -20.1803 q^{34} -5.47214 q^{37} -8.47214 q^{38} +8.00000 q^{41} -8.23607 q^{43} +26.5623 q^{44} +13.0902 q^{46} +7.23607 q^{47} +1.00000 q^{49} -3.70820 q^{52} +0.472136 q^{53} +7.47214 q^{56} -12.3262 q^{58} +0.763932 q^{59} -15.4164 q^{61} +11.7082 q^{62} +8.70820 q^{64} +2.70820 q^{67} -37.4164 q^{68} +0.527864 q^{71} -1.23607 q^{73} -14.3262 q^{74} -15.7082 q^{76} +5.47214 q^{77} +1.76393 q^{79} +20.9443 q^{82} +12.4721 q^{83} -21.5623 q^{86} +40.8885 q^{88} +5.70820 q^{89} -0.763932 q^{91} +24.2705 q^{92} +18.9443 q^{94} -12.4721 q^{97} +2.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8} + 2 q^{11} - 6 q^{13} + 3 q^{14} + 13 q^{16} - 2 q^{17} - 2 q^{19} + 13 q^{22} + 10 q^{23} - 4 q^{26} + 3 q^{28} + 4 q^{29} + 15 q^{32} - 18 q^{34} - 2 q^{37} - 8 q^{38} + 16 q^{41} - 12 q^{43} + 33 q^{44} + 15 q^{46} + 10 q^{47} + 2 q^{49} + 6 q^{52} - 8 q^{53} + 6 q^{56} - 9 q^{58} + 6 q^{59} - 4 q^{61} + 10 q^{62} + 4 q^{64} - 8 q^{67} - 48 q^{68} + 10 q^{71} + 2 q^{73} - 13 q^{74} - 18 q^{76} + 2 q^{77} + 8 q^{79} + 24 q^{82} + 16 q^{83} - 23 q^{86} + 46 q^{88} - 2 q^{89} - 6 q^{91} + 15 q^{92} + 20 q^{94} - 16 q^{97} + 3 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8 + 2 * q^11 - 6 * q^13 + 3 * q^14 + 13 * q^16 - 2 * q^17 - 2 * q^19 + 13 * q^22 + 10 * q^23 - 4 * q^26 + 3 * q^28 + 4 * q^29 + 15 * q^32 - 18 * q^34 - 2 * q^37 - 8 * q^38 + 16 * q^41 - 12 * q^43 + 33 * q^44 + 15 * q^46 + 10 * q^47 + 2 * q^49 + 6 * q^52 - 8 * q^53 + 6 * q^56 - 9 * q^58 + 6 * q^59 - 4 * q^61 + 10 * q^62 + 4 * q^64 - 8 * q^67 - 48 * q^68 + 10 * q^71 + 2 * q^73 - 13 * q^74 - 18 * q^76 + 2 * q^77 + 8 * q^79 + 24 * q^82 + 16 * q^83 - 23 * q^86 + 46 * q^88 - 2 * q^89 - 6 * q^91 + 15 * q^92 + 20 * q^94 - 16 * q^97 + 3 * q^98

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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	7.47214	2.64180
$9$	0	0
$10$	0	0
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	0	0
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	2.61803	0.699699
$15$	0	0
$16$	9.85410	2.46353
$17$	−7.70820	−1.86951	−0.934757	−	0.355288i	$-0.884383\pi$
$17$	−7.70820	−1.86951	−0.934757	+	0.355288i	$0.884383\pi$
$18$	0	0
$19$	−3.23607	−0.742405	−0.371202	−	0.928552i	$-0.621054\pi$
$19$	−3.23607	−0.742405	−0.371202	+	0.928552i	$0.621054\pi$
$20$	0	0
$21$	0	0
$22$	14.3262	3.05436
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	4.85410	0.917339
$29$	−4.70820	−0.874292	−0.437146	−	0.899391i	$-0.644011\pi$
$29$	−4.70820	−0.874292	−0.437146	+	0.899391i	$0.644011\pi$
$30$	0	0
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	−20.1803	−3.46090
$35$	0	0
$36$	0	0
$37$	−5.47214	−0.899614	−0.449807	−	0.893126i	$-0.648507\pi$
$37$	−5.47214	−0.899614	−0.449807	+	0.893126i	$0.648507\pi$
$38$	−8.47214	−1.37436
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−8.23607	−1.25599	−0.627994	−	0.778218i	$-0.716124\pi$
$43$	−8.23607	−1.25599	−0.627994	+	0.778218i	$0.716124\pi$
$44$	26.5623	4.00442
$45$	0	0
$46$	13.0902	1.93004
$47$	7.23607	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$47$	7.23607	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−3.70820	−0.514235
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	0	0
$56$	7.47214	0.998506
$57$	0	0
$58$	−12.3262	−1.61851
$59$	0.763932	0.0994555	0.0497277	−	0.998763i	$-0.484165\pi$
$59$	0.763932	0.0994555	0.0497277	+	0.998763i	$0.484165\pi$
$60$	0	0
$61$	−15.4164	−1.97387	−0.986934	−	0.161123i	$-0.948489\pi$
$61$	−15.4164	−1.97387	−0.986934	+	0.161123i	$0.948489\pi$
$62$	11.7082	1.48694
$63$	0	0
$64$	8.70820	1.08853
$65$	0	0
$66$	0	0
$67$	2.70820	0.330860	0.165430	−	0.986222i	$-0.447099\pi$
$67$	2.70820	0.330860	0.165430	+	0.986222i	$0.447099\pi$
$68$	−37.4164	−4.53741
$69$	0	0
$70$	0	0
$71$	0.527864	0.0626459	0.0313230	−	0.999509i	$-0.490028\pi$
$71$	0.527864	0.0626459	0.0313230	+	0.999509i	$0.490028\pi$
$72$	0	0
$73$	−1.23607	−0.144671	−0.0723354	−	0.997380i	$-0.523045\pi$
$73$	−1.23607	−0.144671	−0.0723354	+	0.997380i	$0.523045\pi$
$74$	−14.3262	−1.66539
$75$	0	0
$76$	−15.7082	−1.80185
$77$	5.47214	0.623608
$78$	0	0
$79$	1.76393	0.198458	0.0992289	−	0.995065i	$-0.468362\pi$
$79$	1.76393	0.198458	0.0992289	+	0.995065i	$0.468362\pi$
$80$	0	0
$81$	0	0
$82$	20.9443	2.31291
$83$	12.4721	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$83$	12.4721	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$84$	0	0
$85$	0	0
$86$	−21.5623	−2.32512
$87$	0	0
$88$	40.8885	4.35873
$89$	5.70820	0.605068	0.302534	−	0.953139i	$-0.402167\pi$
$89$	5.70820	0.605068	0.302534	+	0.953139i	$0.402167\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	24.2705	2.53038
$93$	0	0
$94$	18.9443	1.95395
$95$	0	0
$96$	0	0
$97$	−12.4721	−1.26635	−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721	−1.26635	−0.633177	+	0.774007i	$0.718249\pi$
$98$	2.61803	0.264461
$99$	0	0
$100$	0	0
$101$	4.29180	0.427050	0.213525	−	0.976938i	$-0.431506\pi$
$101$	4.29180	0.427050	0.213525	+	0.976938i	$0.431506\pi$
$102$	0	0
$103$	−9.70820	−0.956578	−0.478289	−	0.878203i	$-0.658743\pi$
$103$	−9.70820	−0.956578	−0.478289	+	0.878203i	$0.658743\pi$
$104$	−5.70820	−0.559735
$105$	0	0
$106$	1.23607	0.120058
$107$	−4.94427	−0.477981	−0.238990	−	0.971022i	$-0.576816\pi$
$107$	−4.94427	−0.477981	−0.238990	+	0.971022i	$0.576816\pi$
$108$	0	0
$109$	−6.41641	−0.614580	−0.307290	−	0.951616i	$-0.599422\pi$
$109$	−6.41641	−0.614580	−0.307290	+	0.951616i	$0.599422\pi$
$110$	0	0
$111$	0	0
$112$	9.85410	0.931125
$113$	−16.2361	−1.52736	−0.763680	−	0.645594i	$-0.776610\pi$
$113$	−16.2361	−1.52736	−0.763680	+	0.645594i	$0.776610\pi$
$114$	0	0
$115$	0	0
$116$	−22.8541	−2.12195
$117$	0	0
$118$	2.00000	0.184115
$119$	−7.70820	−0.706610
$120$	0	0
$121$	18.9443	1.72221
$122$	−40.3607	−3.65408
$123$	0	0
$124$	21.7082	1.94945
$125$	0	0
$126$	0	0
$127$	12.2361	1.08578	0.542888	−	0.839805i	$-0.317331\pi$
$127$	12.2361	1.08578	0.542888	+	0.839805i	$0.317331\pi$
$128$	1.09017	0.0963583
$129$	0	0
$130$	0	0
$131$	4.29180	0.374976	0.187488	−	0.982267i	$-0.439965\pi$
$131$	4.29180	0.374976	0.187488	+	0.982267i	$0.439965\pi$
$132$	0	0
$133$	−3.23607	−0.280603
$134$	7.09017	0.612497
$135$	0	0
$136$	−57.5967	−4.93888
$137$	−0.472136	−0.0403373	−0.0201686	−	0.999797i	$-0.506420\pi$
$137$	−0.472136	−0.0403373	−0.0201686	+	0.999797i	$0.506420\pi$
$138$	0	0
$139$	3.70820	0.314526	0.157263	−	0.987557i	$-0.449733\pi$
$139$	3.70820	0.314526	0.157263	+	0.987557i	$0.449733\pi$
$140$	0	0
$141$	0	0
$142$	1.38197	0.115972
$143$	−4.18034	−0.349578
$144$	0	0
$145$	0	0
$146$	−3.23607	−0.267819
$147$	0	0
$148$	−26.5623	−2.18341
$149$	−8.23607	−0.674725	−0.337362	−	0.941375i	$-0.609535\pi$
$149$	−8.23607	−0.674725	−0.337362	+	0.941375i	$0.609535\pi$
$150$	0	0
$151$	1.29180	0.105125	0.0525624	−	0.998618i	$-0.483261\pi$
$151$	1.29180	0.105125	0.0525624	+	0.998618i	$0.483261\pi$
$152$	−24.1803	−1.96128
$153$	0	0
$154$	14.3262	1.15444
$155$	0	0
$156$	0	0
$157$	−16.9443	−1.35230	−0.676150	−	0.736764i	$-0.736353\pi$
$157$	−16.9443	−1.35230	−0.676150	+	0.736764i	$0.736353\pi$
$158$	4.61803	0.367391
$159$	0	0
$160$	0	0
$161$	5.00000	0.394055
$162$	0	0
$163$	−18.4721	−1.44685	−0.723425	−	0.690403i	$-0.757433\pi$
$163$	−18.4721	−1.44685	−0.723425	+	0.690403i	$0.757433\pi$
$164$	38.8328	3.03233
$165$	0	0
$166$	32.6525	2.53432
$167$	−6.29180	−0.486874	−0.243437	−	0.969917i	$-0.578275\pi$
$167$	−6.29180	−0.486874	−0.243437	+	0.969917i	$0.578275\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	0	0
$172$	−39.9787	−3.04835
$173$	−3.81966	−0.290403	−0.145202	−	0.989402i	$-0.546383\pi$
$173$	−3.81966	−0.290403	−0.145202	+	0.989402i	$0.546383\pi$
$174$	0	0
$175$	0	0
$176$	53.9230	4.06460
$177$	0	0
$178$	14.9443	1.12012
$179$	−4.94427	−0.369552	−0.184776	−	0.982781i	$-0.559156\pi$
$179$	−4.94427	−0.369552	−0.184776	+	0.982781i	$0.559156\pi$
$180$	0	0
$181$	4.65248	0.345816	0.172908	−	0.984938i	$-0.444684\pi$
$181$	4.65248	0.345816	0.172908	+	0.984938i	$0.444684\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	37.3607	2.75427
$185$	0	0
$186$	0	0
$187$	−42.1803	−3.08453
$188$	35.1246	2.56173
$189$	0	0
$190$	0	0
$191$	11.4164	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$191$	11.4164	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$192$	0	0
$193$	−0.416408	−0.0299737	−0.0149868	−	0.999888i	$-0.504771\pi$
$193$	−0.416408	−0.0299737	−0.0149868	+	0.999888i	$0.504771\pi$
$194$	−32.6525	−2.34431
$195$	0	0
$196$	4.85410	0.346722
$197$	−5.76393	−0.410663	−0.205332	−	0.978692i	$-0.565827\pi$
$197$	−5.76393	−0.410663	−0.205332	+	0.978692i	$0.565827\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	11.2361	0.790567
$203$	−4.70820	−0.330451
$204$	0	0
$205$	0	0
$206$	−25.4164	−1.77085
$207$	0	0
$208$	−7.52786	−0.521963
$209$	−17.7082	−1.22490
$210$	0	0
$211$	12.9443	0.891120	0.445560	−	0.895252i	$-0.353005\pi$
$211$	12.9443	0.891120	0.445560	+	0.895252i	$0.353005\pi$
$212$	2.29180	0.157401
$213$	0	0
$214$	−12.9443	−0.884852
$215$	0	0
$216$	0	0
$217$	4.47214	0.303588
$218$	−16.7984	−1.13773
$219$	0	0
$220$	0	0
$221$	5.88854	0.396106
$222$	0	0
$223$	1.23607	0.0827732	0.0413866	−	0.999143i	$-0.486822\pi$
$223$	1.23607	0.0827732	0.0413866	+	0.999143i	$0.486822\pi$
$224$	10.8541	0.725220
$225$	0	0
$226$	−42.5066	−2.82750
$227$	−4.76393	−0.316193	−0.158097	−	0.987424i	$-0.550536\pi$
$227$	−4.76393	−0.316193	−0.158097	+	0.987424i	$0.550536\pi$
$228$	0	0
$229$	22.3607	1.47764	0.738818	−	0.673905i	$-0.235384\pi$
$229$	22.3607	1.47764	0.738818	+	0.673905i	$0.235384\pi$
$230$	0	0
$231$	0	0
$232$	−35.1803	−2.30970
$233$	7.29180	0.477701	0.238851	−	0.971056i	$-0.423229\pi$
$233$	7.29180	0.477701	0.238851	+	0.971056i	$0.423229\pi$
$234$	0	0
$235$	0	0
$236$	3.70820	0.241384
$237$	0	0
$238$	−20.1803	−1.30810
$239$	28.3607	1.83450	0.917250	−	0.398312i	$-0.130404\pi$
$239$	28.3607	1.83450	0.917250	+	0.398312i	$0.130404\pi$
$240$	0	0
$241$	19.2361	1.23910	0.619552	−	0.784956i	$-0.287314\pi$
$241$	19.2361	1.23910	0.619552	+	0.784956i	$0.287314\pi$
$242$	49.5967	3.18820
$243$	0	0
$244$	−74.8328	−4.79068
$245$	0	0
$246$	0	0
$247$	2.47214	0.157298
$248$	33.4164	2.12194
$249$	0	0
$250$	0	0
$251$	−2.76393	−0.174458	−0.0872289	−	0.996188i	$-0.527801\pi$
$251$	−2.76393	−0.174458	−0.0872289	+	0.996188i	$0.527801\pi$
$252$	0	0
$253$	27.3607	1.72015
$254$	32.0344	2.01002
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−8.47214	−0.528477	−0.264239	−	0.964457i	$-0.585121\pi$
$257$	−8.47214	−0.528477	−0.264239	+	0.964457i	$0.585121\pi$
$258$	0	0
$259$	−5.47214	−0.340022
$260$	0	0
$261$	0	0
$262$	11.2361	0.694167
$263$	2.05573	0.126762	0.0633808	−	0.997989i	$-0.479812\pi$
$263$	2.05573	0.126762	0.0633808	+	0.997989i	$0.479812\pi$
$264$	0	0
$265$	0	0
$266$	−8.47214	−0.519460
$267$	0	0
$268$	13.1459	0.803014
$269$	28.4721	1.73598	0.867988	−	0.496584i	$-0.165413\pi$
$269$	28.4721	1.73598	0.867988	+	0.496584i	$0.165413\pi$
$270$	0	0
$271$	23.2361	1.41149	0.705745	−	0.708466i	$-0.250612\pi$
$271$	23.2361	1.41149	0.705745	+	0.708466i	$0.250612\pi$
$272$	−75.9574	−4.60560
$273$	0	0
$274$	−1.23607	−0.0746736
$275$	0	0
$276$	0	0
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	9.70820	0.582259
$279$	0	0
$280$	0	0
$281$	−13.6525	−0.814438	−0.407219	−	0.913330i	$-0.633501\pi$
$281$	−13.6525	−0.814438	−0.407219	+	0.913330i	$0.633501\pi$
$282$	0	0
$283$	13.4164	0.797523	0.398761	−	0.917055i	$-0.369440\pi$
$283$	13.4164	0.797523	0.398761	+	0.917055i	$0.369440\pi$
$284$	2.56231	0.152045
$285$	0	0
$286$	−10.9443	−0.647148
$287$	8.00000	0.472225
$288$	0	0
$289$	42.4164	2.49508
$290$	0	0
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−26.6525	−1.55705	−0.778527	−	0.627611i	$-0.784033\pi$
$293$	−26.6525	−1.55705	−0.778527	+	0.627611i	$0.784033\pi$
$294$	0	0
$295$	0	0
$296$	−40.8885	−2.37660
$297$	0	0
$298$	−21.5623	−1.24907
$299$	−3.81966	−0.220897
$300$	0	0
$301$	−8.23607	−0.474719
$302$	3.38197	0.194610
$303$	0	0
$304$	−31.8885	−1.82893
$305$	0	0
$306$	0	0
$307$	−24.6525	−1.40699	−0.703496	−	0.710700i	$-0.748379\pi$
$307$	−24.6525	−1.40699	−0.703496	+	0.710700i	$0.748379\pi$
$308$	26.5623	1.51353
$309$	0	0
$310$	0	0
$311$	−13.2361	−0.750549	−0.375274	−	0.926914i	$-0.622451\pi$
$311$	−13.2361	−0.750549	−0.375274	+	0.926914i	$0.622451\pi$
$312$	0	0
$313$	5.70820	0.322647	0.161323	−	0.986902i	$-0.448424\pi$
$313$	5.70820	0.322647	0.161323	+	0.986902i	$0.448424\pi$
$314$	−44.3607	−2.50342
$315$	0	0
$316$	8.56231	0.481667
$317$	17.6525	0.991462	0.495731	−	0.868476i	$-0.334900\pi$
$317$	17.6525	0.991462	0.495731	+	0.868476i	$0.334900\pi$
$318$	0	0
$319$	−25.7639	−1.44250
$320$	0	0
$321$	0	0
$322$	13.0902	0.729487
$323$	24.9443	1.38794
$324$	0	0
$325$	0	0
$326$	−48.3607	−2.67845
$327$	0	0
$328$	59.7771	3.30064
$329$	7.23607	0.398937
$330$	0	0
$331$	13.7639	0.756534	0.378267	−	0.925697i	$-0.376520\pi$
$331$	13.7639	0.756534	0.378267	+	0.925697i	$0.376520\pi$
$332$	60.5410	3.32262
$333$	0	0
$334$	−16.4721	−0.901315
$335$	0	0
$336$	0	0
$337$	28.4721	1.55098	0.775488	−	0.631362i	$-0.217504\pi$
$337$	28.4721	1.55098	0.775488	+	0.631362i	$0.217504\pi$
$338$	−32.5066	−1.76812
$339$	0	0
$340$	0	0
$341$	24.4721	1.32524
$342$	0	0
$343$	1.00000	0.0539949
$344$	−61.5410	−3.31807
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−13.0000	−0.697877	−0.348938	−	0.937146i	$-0.613458\pi$
$347$	−13.0000	−0.697877	−0.348938	+	0.937146i	$0.613458\pi$
$348$	0	0
$349$	0.763932	0.0408923	0.0204462	−	0.999791i	$-0.493491\pi$
$349$	0.763932	0.0408923	0.0204462	+	0.999791i	$0.493491\pi$
$350$	0	0
$351$	0	0
$352$	59.3951	3.16577
$353$	1.05573	0.0561907	0.0280954	−	0.999605i	$-0.491056\pi$
$353$	1.05573	0.0561907	0.0280954	+	0.999605i	$0.491056\pi$
$354$	0	0
$355$	0	0
$356$	27.7082	1.46853
$357$	0	0
$358$	−12.9443	−0.684126
$359$	−25.9443	−1.36929	−0.684643	−	0.728878i	$-0.740042\pi$
$359$	−25.9443	−1.36929	−0.684643	+	0.728878i	$0.740042\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	12.1803	0.640184
$363$	0	0
$364$	−3.70820	−0.194363
$365$	0	0
$366$	0	0
$367$	8.94427	0.466887	0.233444	−	0.972370i	$-0.425001\pi$
$367$	8.94427	0.466887	0.233444	+	0.972370i	$0.425001\pi$
$368$	49.2705	2.56840
$369$	0	0
$370$	0	0
$371$	0.472136	0.0245121
$372$	0	0
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	−110.430	−5.71018
$375$	0	0
$376$	54.0689	2.78839
$377$	3.59675	0.185242
$378$	0	0
$379$	16.5967	0.852518	0.426259	−	0.904601i	$-0.359831\pi$
$379$	16.5967	0.852518	0.426259	+	0.904601i	$0.359831\pi$
$380$	0	0
$381$	0	0
$382$	29.8885	1.52923
$383$	25.1246	1.28381	0.641904	−	0.766785i	$-0.278145\pi$
$383$	25.1246	1.28381	0.641904	+	0.766785i	$0.278145\pi$
$384$	0	0
$385$	0	0
$386$	−1.09017	−0.0554882
$387$	0	0
$388$	−60.5410	−3.07350
$389$	1.76393	0.0894349	0.0447175	−	0.999000i	$-0.485761\pi$
$389$	1.76393	0.0894349	0.0447175	+	0.999000i	$0.485761\pi$
$390$	0	0
$391$	−38.5410	−1.94910
$392$	7.47214	0.377400
$393$	0	0
$394$	−15.0902	−0.760232
$395$	0	0
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−41.8885	−2.09968
$399$	0	0
$400$	0	0
$401$	26.7082	1.33374	0.666872	−	0.745172i	$-0.267633\pi$
$401$	26.7082	1.33374	0.666872	+	0.745172i	$0.267633\pi$
$402$	0	0
$403$	−3.41641	−0.170183
$404$	20.8328	1.03647
$405$	0	0
$406$	−12.3262	−0.611741
$407$	−29.9443	−1.48428
$408$	0	0
$409$	−8.18034	−0.404492	−0.202246	−	0.979335i	$-0.564824\pi$
$409$	−8.18034	−0.404492	−0.202246	+	0.979335i	$0.564824\pi$
$410$	0	0
$411$	0	0
$412$	−47.1246	−2.32166
$413$	0.763932	0.0375906
$414$	0	0
$415$	0	0
$416$	−8.29180	−0.406539
$417$	0	0
$418$	−46.3607	−2.26757
$419$	−9.05573	−0.442401	−0.221201	−	0.975228i	$-0.570998\pi$
$419$	−9.05573	−0.442401	−0.221201	+	0.975228i	$0.570998\pi$
$420$	0	0
$421$	−0.416408	−0.0202945	−0.0101472	−	0.999949i	$-0.503230\pi$
$421$	−0.416408	−0.0202945	−0.0101472	+	0.999949i	$0.503230\pi$
$422$	33.8885	1.64967
$423$	0	0
$424$	3.52786	0.171328
$425$	0	0
$426$	0	0
$427$	−15.4164	−0.746052
$428$	−24.0000	−1.16008
$429$	0	0
$430$	0	0
$431$	−33.8885	−1.63235	−0.816177	−	0.577802i	$-0.803911\pi$
$431$	−33.8885	−1.63235	−0.816177	+	0.577802i	$0.803911\pi$
$432$	0	0
$433$	−37.3050	−1.79276	−0.896381	−	0.443285i	$-0.853813\pi$
$433$	−37.3050	−1.79276	−0.896381	+	0.443285i	$0.853813\pi$
$434$	11.7082	0.562012
$435$	0	0
$436$	−31.1459	−1.49162
$437$	−16.1803	−0.774011
$438$	0	0
$439$	−11.2361	−0.536268	−0.268134	−	0.963382i	$-0.586407\pi$
$439$	−11.2361	−0.536268	−0.268134	+	0.963382i	$0.586407\pi$
$440$	0	0
$441$	0	0
$442$	15.4164	0.733284
$443$	12.5836	0.597865	0.298932	−	0.954274i	$-0.403370\pi$
$443$	12.5836	0.597865	0.298932	+	0.954274i	$0.403370\pi$
$444$	0	0
$445$	0	0
$446$	3.23607	0.153232
$447$	0	0
$448$	8.70820	0.411424
$449$	−3.29180	−0.155349	−0.0776747	−	0.996979i	$-0.524750\pi$
$449$	−3.29180	−0.155349	−0.0776747	+	0.996979i	$0.524750\pi$
$450$	0	0
$451$	43.7771	2.06138
$452$	−78.8115	−3.70698
$453$	0	0
$454$	−12.4721	−0.585346
$455$	0	0
$456$	0	0
$457$	29.4721	1.37865	0.689324	−	0.724453i	$-0.257908\pi$
$457$	29.4721	1.37865	0.689324	+	0.724453i	$0.257908\pi$
$458$	58.5410	2.73544
$459$	0	0
$460$	0	0
$461$	−10.6525	−0.496135	−0.248068	−	0.968743i	$-0.579796\pi$
$461$	−10.6525	−0.496135	−0.248068	+	0.968743i	$0.579796\pi$
$462$	0	0
$463$	−28.3607	−1.31803	−0.659016	−	0.752129i	$-0.729027\pi$
$463$	−28.3607	−1.31803	−0.659016	+	0.752129i	$0.729027\pi$
$464$	−46.3951	−2.15384
$465$	0	0
$466$	19.0902	0.884335
$467$	23.8885	1.10543	0.552715	−	0.833370i	$-0.313592\pi$
$467$	23.8885	1.10543	0.552715	+	0.833370i	$0.313592\pi$
$468$	0	0
$469$	2.70820	0.125053
$470$	0	0
$471$	0	0
$472$	5.70820	0.262741
$473$	−45.0689	−2.07227
$474$	0	0
$475$	0	0
$476$	−37.4164	−1.71498
$477$	0	0
$478$	74.2492	3.39608
$479$	23.2361	1.06168	0.530842	−	0.847471i	$-0.321876\pi$
$479$	23.2361	1.06168	0.530842	+	0.847471i	$0.321876\pi$
$480$	0	0
$481$	4.18034	0.190607
$482$	50.3607	2.29387
$483$	0	0
$484$	91.9574	4.17988
$485$	0	0
$486$	0	0
$487$	6.81966	0.309028	0.154514	−	0.987991i	$-0.450619\pi$
$487$	6.81966	0.309028	0.154514	+	0.987991i	$0.450619\pi$
$488$	−115.193	−5.21456
$489$	0	0
$490$	0	0
$491$	0.527864	0.0238222	0.0119111	−	0.999929i	$-0.496208\pi$
$491$	0.527864	0.0238222	0.0119111	+	0.999929i	$0.496208\pi$
$492$	0	0
$493$	36.2918	1.63450
$494$	6.47214	0.291195
$495$	0	0
$496$	44.0689	1.97875
$497$	0.527864	0.0236779
$498$	0	0
$499$	−31.7771	−1.42254	−0.711269	−	0.702920i	$-0.751879\pi$
$499$	−31.7771	−1.42254	−0.711269	+	0.702920i	$0.751879\pi$
$500$	0	0
$501$	0	0
$502$	−7.23607	−0.322962
$503$	17.4164	0.776559	0.388280	−	0.921542i	$-0.373069\pi$
$503$	17.4164	0.776559	0.388280	+	0.921542i	$0.373069\pi$
$504$	0	0
$505$	0	0
$506$	71.6312	3.18439
$507$	0	0
$508$	59.3951	2.63523
$509$	30.6525	1.35865	0.679324	−	0.733839i	$-0.262273\pi$
$509$	30.6525	1.35865	0.679324	+	0.733839i	$0.262273\pi$
$510$	0	0
$511$	−1.23607	−0.0546804
$512$	−40.3050	−1.78124
$513$	0	0
$514$	−22.1803	−0.978333
$515$	0	0
$516$	0	0
$517$	39.5967	1.74146
$518$	−14.3262	−0.629459
$519$	0	0
$520$	0	0
$521$	37.7771	1.65504	0.827522	−	0.561433i	$-0.189750\pi$
$521$	37.7771	1.65504	0.827522	+	0.561433i	$0.189750\pi$
$522$	0	0
$523$	35.7771	1.56442	0.782211	−	0.623013i	$-0.214092\pi$
$523$	35.7771	1.56442	0.782211	+	0.623013i	$0.214092\pi$
$524$	20.8328	0.910086
$525$	0	0
$526$	5.38197	0.234665
$527$	−34.4721	−1.50163
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	−15.7082	−0.681037
$533$	−6.11146	−0.264717
$534$	0	0
$535$	0	0
$536$	20.2361	0.874065
$537$	0	0
$538$	74.5410	3.21369
$539$	5.47214	0.235702
$540$	0	0
$541$	−17.9443	−0.771485	−0.385742	−	0.922607i	$-0.626055\pi$
$541$	−17.9443	−0.771485	−0.385742	+	0.922607i	$0.626055\pi$
$542$	60.8328	2.61299
$543$	0	0
$544$	−83.6656	−3.58713
$545$	0	0
$546$	0	0
$547$	35.0689	1.49944	0.749719	−	0.661757i	$-0.230189\pi$
$547$	35.0689	1.49944	0.749719	+	0.661757i	$0.230189\pi$
$548$	−2.29180	−0.0979007
$549$	0	0
$550$	0	0
$551$	15.2361	0.649078
$552$	0	0
$553$	1.76393	0.0750100
$554$	41.5967	1.76728
$555$	0	0
$556$	18.0000	0.763370
$557$	−7.65248	−0.324246	−0.162123	−	0.986771i	$-0.551834\pi$
$557$	−7.65248	−0.324246	−0.162123	+	0.986771i	$0.551834\pi$
$558$	0	0
$559$	6.29180	0.266115
$560$	0	0
$561$	0	0
$562$	−35.7426	−1.50771
$563$	35.3050	1.48793	0.743963	−	0.668221i	$-0.232944\pi$
$563$	35.3050	1.48793	0.743963	+	0.668221i	$0.232944\pi$
$564$	0	0
$565$	0	0
$566$	35.1246	1.47640
$567$	0	0
$568$	3.94427	0.165498
$569$	12.2361	0.512963	0.256481	−	0.966549i	$-0.417437\pi$
$569$	12.2361	0.512963	0.256481	+	0.966549i	$0.417437\pi$
$570$	0	0
$571$	−40.7082	−1.70359	−0.851793	−	0.523879i	$-0.824484\pi$
$571$	−40.7082	−1.70359	−0.851793	+	0.523879i	$0.824484\pi$
$572$	−20.2918	−0.848443
$573$	0	0
$574$	20.9443	0.874197
$575$	0	0
$576$	0	0
$577$	−11.2361	−0.467764	−0.233882	−	0.972265i	$-0.575143\pi$
$577$	−11.2361	−0.467764	−0.233882	+	0.972265i	$0.575143\pi$
$578$	111.048	4.61897
$579$	0	0
$580$	0	0
$581$	12.4721	0.517431
$582$	0	0
$583$	2.58359	0.107001
$584$	−9.23607	−0.382191
$585$	0	0
$586$	−69.7771	−2.88246
$587$	−7.12461	−0.294064	−0.147032	−	0.989132i	$-0.546972\pi$
$587$	−7.12461	−0.294064	−0.147032	+	0.989132i	$0.546972\pi$
$588$	0	0
$589$	−14.4721	−0.596314
$590$	0	0
$591$	0	0
$592$	−53.9230	−2.21622
$593$	25.3050	1.03915	0.519575	−	0.854425i	$-0.326090\pi$
$593$	25.3050	1.03915	0.519575	+	0.854425i	$0.326090\pi$
$594$	0	0
$595$	0	0
$596$	−39.9787	−1.63759
$597$	0	0
$598$	−10.0000	−0.408930
$599$	45.9443	1.87723	0.938616	−	0.344964i	$-0.112109\pi$
$599$	45.9443	1.87723	0.938616	+	0.344964i	$0.112109\pi$
$600$	0	0
$601$	−28.3607	−1.15686	−0.578428	−	0.815733i	$-0.696334\pi$
$601$	−28.3607	−1.15686	−0.578428	+	0.815733i	$0.696334\pi$
$602$	−21.5623	−0.878814
$603$	0	0
$604$	6.27051	0.255143
$605$	0	0
$606$	0	0
$607$	−2.29180	−0.0930211	−0.0465106	−	0.998918i	$-0.514810\pi$
$607$	−2.29180	−0.0930211	−0.0465106	+	0.998918i	$0.514810\pi$
$608$	−35.1246	−1.42449
$609$	0	0
$610$	0	0
$611$	−5.52786	−0.223633
$612$	0	0
$613$	−10.0557	−0.406147	−0.203074	−	0.979163i	$-0.565093\pi$
$613$	−10.0557	−0.406147	−0.203074	+	0.979163i	$0.565093\pi$
$614$	−64.5410	−2.60466
$615$	0	0
$616$	40.8885	1.64745
$617$	−21.6525	−0.871696	−0.435848	−	0.900020i	$-0.643551\pi$
$617$	−21.6525	−0.871696	−0.435848	+	0.900020i	$0.643551\pi$
$618$	0	0
$619$	25.1246	1.00984	0.504922	−	0.863165i	$-0.331521\pi$
$619$	25.1246	1.00984	0.504922	+	0.863165i	$0.331521\pi$
$620$	0	0
$621$	0	0
$622$	−34.6525	−1.38944
$623$	5.70820	0.228694
$624$	0	0
$625$	0	0
$626$	14.9443	0.597293
$627$	0	0
$628$	−82.2492	−3.28210
$629$	42.1803	1.68184
$630$	0	0
$631$	−40.1246	−1.59734	−0.798668	−	0.601772i	$-0.794462\pi$
$631$	−40.1246	−1.59734	−0.798668	+	0.601772i	$0.794462\pi$
$632$	13.1803	0.524286
$633$	0	0
$634$	46.2148	1.83542
$635$	0	0
$636$	0	0
$637$	−0.763932	−0.0302681
$638$	−67.4508	−2.67040
$639$	0	0
$640$	0	0
$641$	15.2918	0.603990	0.301995	−	0.953310i	$-0.402347\pi$
$641$	15.2918	0.603990	0.301995	+	0.953310i	$0.402347\pi$
$642$	0	0
$643$	44.7214	1.76364	0.881819	−	0.471588i	$-0.156319\pi$
$643$	44.7214	1.76364	0.881819	+	0.471588i	$0.156319\pi$
$644$	24.2705	0.956392
$645$	0	0
$646$	65.3050	2.56939
$647$	−1.05573	−0.0415050	−0.0207525	−	0.999785i	$-0.506606\pi$
$647$	−1.05573	−0.0415050	−0.0207525	+	0.999785i	$0.506606\pi$
$648$	0	0
$649$	4.18034	0.164093
$650$	0	0
$651$	0	0
$652$	−89.6656	−3.51158
$653$	25.4164	0.994621	0.497310	−	0.867573i	$-0.334321\pi$
$653$	25.4164	0.994621	0.497310	+	0.867573i	$0.334321\pi$
$654$	0	0
$655$	0	0
$656$	78.8328	3.07790
$657$	0	0
$658$	18.9443	0.738525
$659$	−40.9443	−1.59496	−0.797481	−	0.603344i	$-0.793835\pi$
$659$	−40.9443	−1.59496	−0.797481	+	0.603344i	$0.793835\pi$
$660$	0	0
$661$	8.18034	0.318178	0.159089	−	0.987264i	$-0.449144\pi$
$661$	8.18034	0.318178	0.159089	+	0.987264i	$0.449144\pi$
$662$	36.0344	1.40052
$663$	0	0
$664$	93.1935	3.61661
$665$	0	0
$666$	0	0
$667$	−23.5410	−0.911512
$668$	−30.5410	−1.18167
$669$	0	0
$670$	0	0
$671$	−84.3607	−3.25671
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	74.5410	2.87121
$675$	0	0
$676$	−60.2705	−2.31810
$677$	43.3050	1.66434	0.832172	−	0.554517i	$-0.187097\pi$
$677$	43.3050	1.66434	0.832172	+	0.554517i	$0.187097\pi$
$678$	0	0
$679$	−12.4721	−0.478637
$680$	0	0
$681$	0	0
$682$	64.0689	2.45332
$683$	5.11146	0.195584	0.0977922	−	0.995207i	$-0.468822\pi$
$683$	5.11146	0.195584	0.0977922	+	0.995207i	$0.468822\pi$
$684$	0	0
$685$	0	0
$686$	2.61803	0.0999570
$687$	0	0
$688$	−81.1591	−3.09416
$689$	−0.360680	−0.0137408
$690$	0	0
$691$	−12.3607	−0.470222	−0.235111	−	0.971968i	$-0.575545\pi$
$691$	−12.3607	−0.470222	−0.235111	+	0.971968i	$0.575545\pi$
$692$	−18.5410	−0.704824
$693$	0	0
$694$	−34.0344	−1.29193
$695$	0	0
$696$	0	0
$697$	−61.6656	−2.33575
$698$	2.00000	0.0757011
$699$	0	0
$700$	0	0
$701$	−33.4164	−1.26212	−0.631060	−	0.775734i	$-0.717380\pi$
$701$	−33.4164	−1.26212	−0.631060	+	0.775734i	$0.717380\pi$
$702$	0	0
$703$	17.7082	0.667878
$704$	47.6525	1.79597
$705$	0	0
$706$	2.76393	0.104022
$707$	4.29180	0.161410
$708$	0	0
$709$	43.8885	1.64827	0.824134	−	0.566394i	$-0.191662\pi$
$709$	43.8885	1.64827	0.824134	+	0.566394i	$0.191662\pi$
$710$	0	0
$711$	0	0
$712$	42.6525	1.59847
$713$	22.3607	0.837414
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−67.9230	−2.53486
$719$	−42.2492	−1.57563	−0.787815	−	0.615912i	$-0.788788\pi$
$719$	−42.2492	−1.57563	−0.787815	+	0.615912i	$0.788788\pi$
$720$	0	0
$721$	−9.70820	−0.361552
$722$	−22.3262	−0.830897
$723$	0	0
$724$	22.5836	0.839313
$725$	0	0
$726$	0	0
$727$	18.5410	0.687648	0.343824	−	0.939034i	$-0.388278\pi$
$727$	18.5410	0.687648	0.343824	+	0.939034i	$0.388278\pi$
$728$	−5.70820	−0.211560
$729$	0	0
$730$	0	0
$731$	63.4853	2.34809
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	23.4164	0.864315
$735$	0	0
$736$	54.2705	2.00044
$737$	14.8197	0.545889
$738$	0	0
$739$	3.76393	0.138458	0.0692292	−	0.997601i	$-0.477946\pi$
$739$	3.76393	0.138458	0.0692292	+	0.997601i	$0.477946\pi$
$740$	0	0
$741$	0	0
$742$	1.23607	0.0453775
$743$	−30.2492	−1.10974	−0.554868	−	0.831938i	$-0.687231\pi$
$743$	−30.2492	−1.10974	−0.554868	+	0.831938i	$0.687231\pi$
$744$	0	0
$745$	0	0
$746$	39.2705	1.43780
$747$	0	0
$748$	−204.748	−7.48632
$749$	−4.94427	−0.180660
$750$	0	0
$751$	40.3607	1.47278	0.736391	−	0.676556i	$-0.236528\pi$
$751$	40.3607	1.47278	0.736391	+	0.676556i	$0.236528\pi$
$752$	71.3050	2.60022
$753$	0	0
$754$	9.41641	0.342925
$755$	0	0
$756$	0	0
$757$	13.9443	0.506813	0.253407	−	0.967360i	$-0.418449\pi$
$757$	13.9443	0.506813	0.253407	+	0.967360i	$0.418449\pi$
$758$	43.4508	1.57821
$759$	0	0
$760$	0	0
$761$	−9.30495	−0.337304	−0.168652	−	0.985676i	$-0.553941\pi$
$761$	−9.30495	−0.337304	−0.168652	+	0.985676i	$0.553941\pi$
$762$	0	0
$763$	−6.41641	−0.232290
$764$	55.4164	2.00490
$765$	0	0
$766$	65.7771	2.37662
$767$	−0.583592	−0.0210723
$768$	0	0
$769$	35.4164	1.27715	0.638574	−	0.769560i	$-0.279525\pi$
$769$	35.4164	1.27715	0.638574	+	0.769560i	$0.279525\pi$
$770$	0	0
$771$	0	0
$772$	−2.02129	−0.0727477
$773$	−34.9443	−1.25686	−0.628429	−	0.777867i	$-0.716302\pi$
$773$	−34.9443	−1.25686	−0.628429	+	0.777867i	$0.716302\pi$
$774$	0	0
$775$	0	0
$776$	−93.1935	−3.34545
$777$	0	0
$778$	4.61803	0.165565
$779$	−25.8885	−0.927553
$780$	0	0
$781$	2.88854	0.103360
$782$	−100.902	−3.60824
$783$	0	0
$784$	9.85410	0.351932
$785$	0	0
$786$	0	0
$787$	−38.7639	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$787$	−38.7639	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$788$	−27.9787	−0.996700
$789$	0	0
$790$	0	0
$791$	−16.2361	−0.577288
$792$	0	0
$793$	11.7771	0.418217
$794$	47.1246	1.67239
$795$	0	0
$796$	−77.6656	−2.75279
$797$	−40.0689	−1.41931	−0.709656	−	0.704548i	$-0.751150\pi$
$797$	−40.0689	−1.41931	−0.709656	+	0.704548i	$0.751150\pi$
$798$	0	0
$799$	−55.7771	−1.97325
$800$	0	0
$801$	0	0
$802$	69.9230	2.46907
$803$	−6.76393	−0.238694
$804$	0	0
$805$	0	0
$806$	−8.94427	−0.315049
$807$	0	0
$808$	32.0689	1.12818
$809$	6.81966	0.239766	0.119883	−	0.992788i	$-0.461748\pi$
$809$	6.81966	0.239766	0.119883	+	0.992788i	$0.461748\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	−22.8541	−0.802022
$813$	0	0
$814$	−78.3951	−2.74775
$815$	0	0
$816$	0	0
$817$	26.6525	0.932452
$818$	−21.4164	−0.748807
$819$	0	0
$820$	0	0
$821$	10.9443	0.381958	0.190979	−	0.981594i	$-0.438834\pi$
$821$	10.9443	0.381958	0.190979	+	0.981594i	$0.438834\pi$
$822$	0	0
$823$	44.0132	1.53420	0.767101	−	0.641526i	$-0.221698\pi$
$823$	44.0132	1.53420	0.767101	+	0.641526i	$0.221698\pi$
$824$	−72.5410	−2.52709
$825$	0	0
$826$	2.00000	0.0695889
$827$	31.9443	1.11081	0.555406	−	0.831580i	$-0.312563\pi$
$827$	31.9443	1.11081	0.555406	+	0.831580i	$0.312563\pi$
$828$	0	0
$829$	26.7639	0.929550	0.464775	−	0.885429i	$-0.346135\pi$
$829$	26.7639	0.929550	0.464775	+	0.885429i	$0.346135\pi$
$830$	0	0
$831$	0	0
$832$	−6.65248	−0.230633
$833$	−7.70820	−0.267073
$834$	0	0
$835$	0	0
$836$	−85.9574	−2.97290
$837$	0	0
$838$	−23.7082	−0.818986
$839$	−27.1246	−0.936446	−0.468223	−	0.883610i	$-0.655106\pi$
$839$	−27.1246	−0.936446	−0.468223	+	0.883610i	$0.655106\pi$
$840$	0	0
$841$	−6.83282	−0.235614
$842$	−1.09017	−0.0375697
$843$	0	0
$844$	62.8328	2.16279
$845$	0	0
$846$	0	0
$847$	18.9443	0.650933
$848$	4.65248	0.159767
$849$	0	0
$850$	0	0
$851$	−27.3607	−0.937912
$852$	0	0
$853$	−52.3607	−1.79280	−0.896398	−	0.443251i	$-0.853825\pi$
$853$	−52.3607	−1.79280	−0.896398	+	0.443251i	$0.853825\pi$
$854$	−40.3607	−1.38111
$855$	0	0
$856$	−36.9443	−1.26273
$857$	39.4164	1.34644	0.673219	−	0.739443i	$-0.264911\pi$
$857$	39.4164	1.34644	0.673219	+	0.739443i	$0.264911\pi$
$858$	0	0
$859$	−0.472136	−0.0161091	−0.00805454	−	0.999968i	$-0.502564\pi$
$859$	−0.472136	−0.0161091	−0.00805454	+	0.999968i	$0.502564\pi$
$860$	0	0
$861$	0	0
$862$	−88.7214	−3.02186
$863$	2.05573	0.0699778	0.0349889	−	0.999388i	$-0.488860\pi$
$863$	2.05573	0.0699778	0.0349889	+	0.999388i	$0.488860\pi$
$864$	0	0
$865$	0	0
$866$	−97.6656	−3.31881
$867$	0	0
$868$	21.7082	0.736824
$869$	9.65248	0.327438
$870$	0	0
$871$	−2.06888	−0.0701015
$872$	−47.9443	−1.62360
$873$	0	0
$874$	−42.3607	−1.43287
$875$	0	0
$876$	0	0
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−29.4164	−0.992756
$879$	0	0
$880$	0	0
$881$	38.3607	1.29240	0.646202	−	0.763166i	$-0.276356\pi$
$881$	38.3607	1.29240	0.646202	+	0.763166i	$0.276356\pi$
$882$	0	0
$883$	−34.2361	−1.15214	−0.576068	−	0.817402i	$-0.695414\pi$
$883$	−34.2361	−1.15214	−0.576068	+	0.817402i	$0.695414\pi$
$884$	28.5836	0.961370
$885$	0	0
$886$	32.9443	1.10678
$887$	−15.4164	−0.517632	−0.258816	−	0.965927i	$-0.583332\pi$
$887$	−15.4164	−0.517632	−0.258816	+	0.965927i	$0.583332\pi$
$888$	0	0
$889$	12.2361	0.410385
$890$	0	0
$891$	0	0
$892$	6.00000	0.200895
$893$	−23.4164	−0.783600
$894$	0	0
$895$	0	0
$896$	1.09017	0.0364200
$897$	0	0
$898$	−8.61803	−0.287588
$899$	−21.0557	−0.702248
$900$	0	0
$901$	−3.63932	−0.121243
$902$	114.610	3.81609
$903$	0	0
$904$	−121.318	−4.03498
$905$	0	0
$906$	0	0
$907$	16.3607	0.543247	0.271624	−	0.962404i	$-0.412439\pi$
$907$	16.3607	0.543247	0.271624	+	0.962404i	$0.412439\pi$
$908$	−23.1246	−0.767417
$909$	0	0
$910$	0	0
$911$	−7.58359	−0.251256	−0.125628	−	0.992077i	$-0.540095\pi$
$911$	−7.58359	−0.251256	−0.125628	+	0.992077i	$0.540095\pi$
$912$	0	0
$913$	68.2492	2.25872
$914$	77.1591	2.55219
$915$	0	0
$916$	108.541	3.58630
$917$	4.29180	0.141728
$918$	0	0
$919$	36.0132	1.18796	0.593982	−	0.804478i	$-0.297555\pi$
$919$	36.0132	1.18796	0.593982	+	0.804478i	$0.297555\pi$
$920$	0	0
$921$	0	0
$922$	−27.8885	−0.918460
$923$	−0.403252	−0.0132732
$924$	0	0
$925$	0	0
$926$	−74.2492	−2.43998
$927$	0	0
$928$	−51.1033	−1.67755
$929$	30.1803	0.990185	0.495092	−	0.868840i	$-0.335134\pi$
$929$	30.1803	0.990185	0.495092	+	0.868840i	$0.335134\pi$
$930$	0	0
$931$	−3.23607	−0.106058
$932$	35.3951	1.15941
$933$	0	0
$934$	62.5410	2.04640
$935$	0	0
$936$	0	0
$937$	−4.36068	−0.142457	−0.0712286	−	0.997460i	$-0.522692\pi$
$937$	−4.36068	−0.142457	−0.0712286	+	0.997460i	$0.522692\pi$
$938$	7.09017	0.231502
$939$	0	0
$940$	0	0
$941$	−44.8328	−1.46151	−0.730754	−	0.682641i	$-0.760831\pi$
$941$	−44.8328	−1.46151	−0.730754	+	0.682641i	$0.760831\pi$
$942$	0	0
$943$	40.0000	1.30258
$944$	7.52786	0.245011
$945$	0	0
$946$	−117.992	−3.83625
$947$	32.9443	1.07054	0.535272	−	0.844679i	$-0.320209\pi$
$947$	32.9443	1.07054	0.535272	+	0.844679i	$0.320209\pi$
$948$	0	0
$949$	0.944272	0.0306524
$950$	0	0
$951$	0	0
$952$	−57.5967	−1.86672
$953$	16.1246	0.522327	0.261164	−	0.965295i	$-0.415894\pi$
$953$	16.1246	0.522327	0.261164	+	0.965295i	$0.415894\pi$
$954$	0	0
$955$	0	0
$956$	137.666	4.45242
$957$	0	0
$958$	60.8328	1.96542
$959$	−0.472136	−0.0152461
$960$	0	0
$961$	−11.0000	−0.354839
$962$	10.9443	0.352857
$963$	0	0
$964$	93.3738	3.00737
$965$	0	0
$966$	0	0
$967$	2.11146	0.0678999	0.0339499	−	0.999424i	$-0.489191\pi$
$967$	2.11146	0.0678999	0.0339499	+	0.999424i	$0.489191\pi$
$968$	141.554	4.54972
$969$	0	0
$970$	0	0
$971$	−11.5967	−0.372157	−0.186079	−	0.982535i	$-0.559578\pi$
$971$	−11.5967	−0.372157	−0.186079	+	0.982535i	$0.559578\pi$
$972$	0	0
$973$	3.70820	0.118880
$974$	17.8541	0.572082
$975$	0	0
$976$	−151.915	−4.86268
$977$	3.29180	0.105314	0.0526569	−	0.998613i	$-0.483231\pi$
$977$	3.29180	0.105314	0.0526569	+	0.998613i	$0.483231\pi$
$978$	0	0
$979$	31.2361	0.998309
$980$	0	0
$981$	0	0
$982$	1.38197	0.0441003
$983$	−2.58359	−0.0824038	−0.0412019	−	0.999151i	$-0.513119\pi$
$983$	−2.58359	−0.0824038	−0.0412019	+	0.999151i	$0.513119\pi$
$984$	0	0
$985$	0	0
$986$	95.0132	3.02584
$987$	0	0
$988$	12.0000	0.381771
$989$	−41.1803	−1.30946
$990$	0	0
$991$	8.59675	0.273085	0.136542	−	0.990634i	$-0.456401\pi$
$991$	8.59675	0.273085	0.136542	+	0.990634i	$0.456401\pi$
$992$	48.5410	1.54118
$993$	0	0
$994$	1.38197	0.0438333
$995$	0	0
$996$	0	0
$997$	−52.5410	−1.66399	−0.831995	−	0.554783i	$-0.812801\pi$
$997$	−52.5410	−1.66399	−0.831995	+	0.554783i	$0.812801\pi$
$998$	−83.1935	−2.63344
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.v.1.2		2
3.2	odd	2		525.2.a.e.1.1	✓	2
5.2	odd	4		1575.2.d.f.1324.4		4
5.3	odd	4		1575.2.d.f.1324.1		4
5.4	even	2		1575.2.a.l.1.1		2
12.11	even	2		8400.2.a.da.1.2		2
15.2	even	4		525.2.d.e.274.1		4
15.8	even	4		525.2.d.e.274.4		4
15.14	odd	2		525.2.a.i.1.2	yes	2
21.20	even	2		3675.2.a.r.1.1		2
60.59	even	2		8400.2.a.cy.1.2		2
105.104	even	2		3675.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.e.1.1	✓	2	3.2	odd	2
525.2.a.i.1.2	yes	2	15.14	odd	2
525.2.d.e.274.1		4	15.2	even	4
525.2.d.e.274.4		4	15.8	even	4
1575.2.a.l.1.1		2	5.4	even	2
1575.2.a.v.1.2		2	1.1	even	1	trivial
1575.2.d.f.1324.1		4	5.3	odd	4
1575.2.d.f.1324.4		4	5.2	odd	4
3675.2.a.r.1.1		2	21.20	even	2
3675.2.a.bh.1.2		2	105.104	even	2
8400.2.a.cy.1.2		2	60.59	even	2
8400.2.a.da.1.2		2	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} +1.00000 q^{7} +7.47214 q^{8} +O(q^{10})\)	\(q+2.61803 q^{2} +4.85410 q^{4} +1.00000 q^{7} +7.47214 q^{8} +5.47214 q^{11} -0.763932 q^{13} +2.61803 q^{14} +9.85410 q^{16} -7.70820 q^{17} -3.23607 q^{19} +14.3262 q^{22} +5.00000 q^{23} -2.00000 q^{26} +4.85410 q^{28} -4.70820 q^{29} +4.47214 q^{31} +10.8541 q^{32} -20.1803 q^{34} -5.47214 q^{37} -8.47214 q^{38} +8.00000 q^{41} -8.23607 q^{43} +26.5623 q^{44} +13.0902 q^{46} +7.23607 q^{47} +1.00000 q^{49} -3.70820 q^{52} +0.472136 q^{53} +7.47214 q^{56} -12.3262 q^{58} +0.763932 q^{59} -15.4164 q^{61} +11.7082 q^{62} +8.70820 q^{64} +2.70820 q^{67} -37.4164 q^{68} +0.527864 q^{71} -1.23607 q^{73} -14.3262 q^{74} -15.7082 q^{76} +5.47214 q^{77} +1.76393 q^{79} +20.9443 q^{82} +12.4721 q^{83} -21.5623 q^{86} +40.8885 q^{88} +5.70820 q^{89} -0.763932 q^{91} +24.2705 q^{92} +18.9443 q^{94} -12.4721 q^{97} +2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8} + 2 q^{11} - 6 q^{13} + 3 q^{14} + 13 q^{16} - 2 q^{17} - 2 q^{19} + 13 q^{22} + 10 q^{23} - 4 q^{26} + 3 q^{28} + 4 q^{29} + 15 q^{32} - 18 q^{34} - 2 q^{37} - 8 q^{38} + 16 q^{41} - 12 q^{43} + 33 q^{44} + 15 q^{46} + 10 q^{47} + 2 q^{49} + 6 q^{52} - 8 q^{53} + 6 q^{56} - 9 q^{58} + 6 q^{59} - 4 q^{61} + 10 q^{62} + 4 q^{64} - 8 q^{67} - 48 q^{68} + 10 q^{71} + 2 q^{73} - 13 q^{74} - 18 q^{76} + 2 q^{77} + 8 q^{79} + 24 q^{82} + 16 q^{83} - 23 q^{86} + 46 q^{88} - 2 q^{89} - 6 q^{91} + 15 q^{92} + 20 q^{94} - 16 q^{97} + 3 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8 + 2 * q^11 - 6 * q^13 + 3 * q^14 + 13 * q^16 - 2 * q^17 - 2 * q^19 + 13 * q^22 + 10 * q^23 - 4 * q^26 + 3 * q^28 + 4 * q^29 + 15 * q^32 - 18 * q^34 - 2 * q^37 - 8 * q^38 + 16 * q^41 - 12 * q^43 + 33 * q^44 + 15 * q^46 + 10 * q^47 + 2 * q^49 + 6 * q^52 - 8 * q^53 + 6 * q^56 - 9 * q^58 + 6 * q^59 - 4 * q^61 + 10 * q^62 + 4 * q^64 - 8 * q^67 - 48 * q^68 + 10 * q^71 + 2 * q^73 - 13 * q^74 - 18 * q^76 + 2 * q^77 + 8 * q^79 + 24 * q^82 + 16 * q^83 - 23 * q^86 + 46 * q^88 - 2 * q^89 - 6 * q^91 + 15 * q^92 + 20 * q^94 - 16 * q^97 + 3 * q^98

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