LMFDB - Embedded newform 225.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.56776$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.56776 q^{2} +41.4066 q^{4} +184.626 q^{7} -80.5934 q^{8} +O(q^{10})$	$q-8.56776 q^{2} +41.4066 q^{4} +184.626 q^{7} -80.5934 q^{8} +495.963 q^{11} +1061.51 q^{13} -1581.84 q^{14} -634.505 q^{16} +635.289 q^{17} +1171.62 q^{19} -4249.30 q^{22} +3829.91 q^{23} -9094.73 q^{26} +7644.75 q^{28} -1722.21 q^{29} -6510.43 q^{31} +8015.28 q^{32} -5443.01 q^{34} -7682.35 q^{37} -10038.1 q^{38} -3967.94 q^{41} -5427.64 q^{43} +20536.1 q^{44} -32813.8 q^{46} -19759.5 q^{47} +17279.9 q^{49} +43953.3 q^{52} +33912.8 q^{53} -14879.7 q^{56} +14755.4 q^{58} +15214.2 q^{59} +9267.31 q^{61} +55779.8 q^{62} -48368.9 q^{64} +2137.34 q^{67} +26305.2 q^{68} -14631.9 q^{71} -32256.9 q^{73} +65820.6 q^{74} +48512.6 q^{76} +91567.9 q^{77} +52690.1 q^{79} +33996.4 q^{82} +28610.0 q^{83} +46502.7 q^{86} -39971.4 q^{88} -33913.3 q^{89} +195982. q^{91} +158584. q^{92} +169295. q^{94} +157623. q^{97} -148050. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} + 16 q^{4} + 102 q^{7} - 228 q^{8}+O(q^{10}) $ 2 * q - 6 * q^2 + 16 * q^4 + 102 * q^7 - 228 * q^8	$ 2 q - 6 q^{2} + 16 q^{4} + 102 q^{7} - 228 q^{8} + 12 q^{11} + 1054 q^{13} - 1794 q^{14} - 200 q^{16} + 1716 q^{17} + 4214 q^{19} - 5492 q^{22} + 444 q^{23} - 9114 q^{26} + 9744 q^{28} - 4068 q^{29} - 2598 q^{31} + 13848 q^{32} - 2668 q^{34} + 4412 q^{37} - 2226 q^{38} - 11232 q^{41} - 8450 q^{43} + 32832 q^{44} - 41508 q^{46} - 2460 q^{47} + 7300 q^{49} + 44144 q^{52} + 65064 q^{53} - 2700 q^{56} + 8732 q^{58} + 63924 q^{59} + 7310 q^{61} + 65826 q^{62} - 47296 q^{64} + 61734 q^{67} - 1152 q^{68} - 98304 q^{71} - 26564 q^{73} + 96876 q^{74} - 28784 q^{76} + 131556 q^{77} + 84000 q^{79} + 15344 q^{82} + 65772 q^{83} + 38742 q^{86} + 31368 q^{88} - 103104 q^{89} + 196602 q^{91} + 244608 q^{92} + 213716 q^{94} + 69374 q^{97} - 173676 q^{98}+O(q^{100}) $ 2 * q - 6 * q^2 + 16 * q^4 + 102 * q^7 - 228 * q^8 + 12 * q^11 + 1054 * q^13 - 1794 * q^14 - 200 * q^16 + 1716 * q^17 + 4214 * q^19 - 5492 * q^22 + 444 * q^23 - 9114 * q^26 + 9744 * q^28 - 4068 * q^29 - 2598 * q^31 + 13848 * q^32 - 2668 * q^34 + 4412 * q^37 - 2226 * q^38 - 11232 * q^41 - 8450 * q^43 + 32832 * q^44 - 41508 * q^46 - 2460 * q^47 + 7300 * q^49 + 44144 * q^52 + 65064 * q^53 - 2700 * q^56 + 8732 * q^58 + 63924 * q^59 + 7310 * q^61 + 65826 * q^62 - 47296 * q^64 + 61734 * q^67 - 1152 * q^68 - 98304 * q^71 - 26564 * q^73 + 96876 * q^74 - 28784 * q^76 + 131556 * q^77 + 84000 * q^79 + 15344 * q^82 + 65772 * q^83 + 38742 * q^86 + 31368 * q^88 - 103104 * q^89 + 196602 * q^91 + 244608 * q^92 + 213716 * q^94 + 69374 * q^97 - 173676 * 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 $

$2$	−8.56776	−1.51458	−0.757291	−	0.653078i	$-0.773477\pi$
$2$	−8.56776	−1.51458	−0.757291	+	0.653078i	$0.773477\pi$
$3$	0	0
$3$	0	0
$4$	41.4066	1.29396
$5$	0	0
$5$	0	0
$6$	0	0
$7$	184.626	1.42413	0.712063	−	0.702115i	$-0.247761\pi$
$7$	184.626	1.42413	0.712063	+	0.702115i	$0.247761\pi$
$8$	−80.5934	−0.445220
$9$	0	0
$10$	0	0
$11$	495.963	1.23586	0.617928	−	0.786235i	$-0.287972\pi$
$11$	495.963	1.23586	0.617928	+	0.786235i	$0.287972\pi$
$12$	0	0
$13$	1061.51	1.74206	0.871031	−	0.491227i	$-0.163452\pi$
$13$	1061.51	1.74206	0.871031	+	0.491227i	$0.163452\pi$
$14$	−1581.84	−2.15696
$15$	0	0
$16$	−634.505	−0.619634
$17$	635.289	0.533150	0.266575	−	0.963814i	$-0.414108\pi$
$17$	635.289	0.533150	0.266575	+	0.963814i	$0.414108\pi$
$18$	0	0
$19$	1171.62	0.744562	0.372281	−	0.928120i	$-0.378576\pi$
$19$	1171.62	0.744562	0.372281	+	0.928120i	$0.378576\pi$
$20$	0	0
$21$	0	0
$22$	−4249.30	−1.87180
$23$	3829.91	1.50963	0.754813	−	0.655941i	$-0.227728\pi$
$23$	3829.91	1.50963	0.754813	+	0.655941i	$0.227728\pi$
$24$	0	0
$25$	0	0
$26$	−9094.73	−2.63850
$27$	0	0
$28$	7644.75	1.84276
$29$	−1722.21	−0.380268	−0.190134	−	0.981758i	$-0.560892\pi$
$29$	−1722.21	−0.380268	−0.190134	+	0.981758i	$0.560892\pi$
$30$	0	0
$31$	−6510.43	−1.21676	−0.608380	−	0.793646i	$-0.708181\pi$
$31$	−6510.43	−1.21676	−0.608380	+	0.793646i	$0.708181\pi$
$32$	8015.28	1.38371
$33$	0	0
$34$	−5443.01	−0.807499
$35$	0	0
$36$	0	0
$37$	−7682.35	−0.922550	−0.461275	−	0.887257i	$-0.652608\pi$
$37$	−7682.35	−0.922550	−0.461275	+	0.887257i	$0.652608\pi$
$38$	−10038.1	−1.12770
$39$	0	0
$40$	0	0
$41$	−3967.94	−0.368643	−0.184321	−	0.982866i	$-0.559009\pi$
$41$	−3967.94	−0.368643	−0.184321	+	0.982866i	$0.559009\pi$
$42$	0	0
$43$	−5427.64	−0.447651	−0.223826	−	0.974629i	$-0.571855\pi$
$43$	−5427.64	−0.447651	−0.223826	+	0.974629i	$0.571855\pi$
$44$	20536.1	1.59914
$45$	0	0
$46$	−32813.8	−2.28645
$47$	−19759.5	−1.30476	−0.652382	−	0.757891i	$-0.726230\pi$
$47$	−19759.5	−1.30476	−0.652382	+	0.757891i	$0.726230\pi$
$48$	0	0
$49$	17279.9	1.02814
$50$	0	0
$51$	0	0
$52$	43953.3	2.25415
$53$	33912.8	1.65834	0.829171	−	0.558995i	$-0.188813\pi$
$53$	33912.8	1.65834	0.829171	+	0.558995i	$0.188813\pi$
$54$	0	0
$55$	0	0
$56$	−14879.7	−0.634049
$57$	0	0
$58$	14755.4	0.575947
$59$	15214.2	0.569008	0.284504	−	0.958675i	$-0.408171\pi$
$59$	15214.2	0.569008	0.284504	+	0.958675i	$0.408171\pi$
$60$	0	0
$61$	9267.31	0.318881	0.159441	−	0.987208i	$-0.449031\pi$
$61$	9267.31	0.318881	0.159441	+	0.987208i	$0.449031\pi$
$62$	55779.8	1.84288
$63$	0	0
$64$	−48368.9	−1.47610
$65$	0	0
$66$	0	0
$67$	2137.34	0.0581682	0.0290841	−	0.999577i	$-0.490741\pi$
$67$	2137.34	0.0581682	0.0290841	+	0.999577i	$0.490741\pi$
$68$	26305.2	0.689872
$69$	0	0
$70$	0	0
$71$	−14631.9	−0.344472	−0.172236	−	0.985056i	$-0.555099\pi$
$71$	−14631.9	−0.344472	−0.172236	+	0.985056i	$0.555099\pi$
$72$	0	0
$73$	−32256.9	−0.708461	−0.354231	−	0.935158i	$-0.615257\pi$
$73$	−32256.9	−0.708461	−0.354231	+	0.935158i	$0.615257\pi$
$74$	65820.6	1.39728
$75$	0	0
$76$	48512.6	0.963431
$77$	91567.9	1.76001
$78$	0	0
$79$	52690.1	0.949864	0.474932	−	0.880023i	$-0.342473\pi$
$79$	52690.1	0.949864	0.474932	+	0.880023i	$0.342473\pi$
$80$	0	0
$81$	0	0
$82$	33996.4	0.558339
$83$	28610.0	0.455850	0.227925	−	0.973679i	$-0.426806\pi$
$83$	28610.0	0.455850	0.227925	+	0.973679i	$0.426806\pi$
$84$	0	0
$85$	0	0
$86$	46502.7	0.678004
$87$	0	0
$88$	−39971.4	−0.550228
$89$	−33913.3	−0.453832	−0.226916	−	0.973914i	$-0.572864\pi$
$89$	−33913.3	−0.453832	−0.226916	+	0.973914i	$0.572864\pi$
$90$	0	0
$91$	195982.	2.48092
$92$	158584.	1.95339
$93$	0	0
$94$	169295.	1.97617
$95$	0	0
$96$	0	0
$97$	157623.	1.70095	0.850474	−	0.526017i	$-0.176315\pi$
$97$	157623.	1.70095	0.850474	+	0.526017i	$0.176315\pi$
$98$	−148050.	−1.55720
$99$	0	0
$100$	0	0
$101$	−28550.8	−0.278494	−0.139247	−	0.990258i	$-0.544468\pi$
$101$	−28550.8	−0.278494	−0.139247	+	0.990258i	$0.544468\pi$
$102$	0	0
$103$	118192.	1.09773	0.548866	−	0.835911i	$-0.315060\pi$
$103$	118192.	1.09773	0.548866	+	0.835911i	$0.315060\pi$
$104$	−85550.3	−0.775601
$105$	0	0
$106$	−290557.	−2.51169
$107$	−188732.	−1.59362	−0.796811	−	0.604229i	$-0.793481\pi$
$107$	−188732.	−1.59362	−0.796811	+	0.604229i	$0.793481\pi$
$108$	0	0
$109$	116765.	0.941339	0.470670	−	0.882310i	$-0.344012\pi$
$109$	116765.	0.941339	0.470670	+	0.882310i	$0.344012\pi$
$110$	0	0
$111$	0	0
$112$	−117146.	−0.882437
$113$	−147900.	−1.08961	−0.544805	−	0.838563i	$-0.683396\pi$
$113$	−147900.	−1.08961	−0.544805	+	0.838563i	$0.683396\pi$
$114$	0	0
$115$	0	0
$116$	−71310.6	−0.492050
$117$	0	0
$118$	−130351.	−0.861808
$119$	117291.	0.759273
$120$	0	0
$121$	84928.6	0.527340
$122$	−79400.1	−0.482971
$123$	0	0
$124$	−269575.	−1.57443
$125$	0	0
$126$	0	0
$127$	−305547.	−1.68101	−0.840503	−	0.541807i	$-0.817740\pi$
$127$	−305547.	−1.68101	−0.840503	+	0.541807i	$0.817740\pi$
$128$	157924.	0.851968
$129$	0	0
$130$	0	0
$131$	−227450.	−1.15800	−0.579000	−	0.815328i	$-0.696557\pi$
$131$	−227450.	−1.15800	−0.579000	+	0.815328i	$0.696557\pi$
$132$	0	0
$133$	216311.	1.06035
$134$	−18312.2	−0.0881005
$135$	0	0
$136$	−51200.1	−0.237369
$137$	−118545.	−0.539615	−0.269807	−	0.962914i	$-0.586960\pi$
$137$	−118545.	−0.539615	−0.269807	+	0.962914i	$0.586960\pi$
$138$	0	0
$139$	−203547.	−0.893568	−0.446784	−	0.894642i	$-0.647431\pi$
$139$	−203547.	−0.893568	−0.446784	+	0.894642i	$0.647431\pi$
$140$	0	0
$141$	0	0
$142$	125362.	0.521730
$143$	526468.	2.15294
$144$	0	0
$145$	0	0
$146$	276370.	1.07302
$147$	0	0
$148$	−318100.	−1.19374
$149$	−120690.	−0.445354	−0.222677	−	0.974892i	$-0.571480\pi$
$149$	−120690.	−0.445354	−0.222677	+	0.974892i	$0.571480\pi$
$150$	0	0
$151$	−277850.	−0.991671	−0.495835	−	0.868417i	$-0.665138\pi$
$151$	−277850.	−0.991671	−0.495835	+	0.868417i	$0.665138\pi$
$152$	−94424.5	−0.331494
$153$	0	0
$154$	−784532.	−2.66569
$155$	0	0
$156$	0	0
$157$	−279147.	−0.903825	−0.451913	−	0.892062i	$-0.649258\pi$
$157$	−279147.	−0.903825	−0.451913	+	0.892062i	$0.649258\pi$
$158$	−451436.	−1.43865
$159$	0	0
$160$	0	0
$161$	707103.	2.14990
$162$	0	0
$163$	328623.	0.968789	0.484394	−	0.874850i	$-0.339040\pi$
$163$	328623.	0.968789	0.484394	+	0.874850i	$0.339040\pi$
$164$	−164299.	−0.477007
$165$	0	0
$166$	−245123.	−0.690422
$167$	−189699.	−0.526349	−0.263174	−	0.964748i	$-0.584769\pi$
$167$	−189699.	−0.526349	−0.263174	+	0.964748i	$0.584769\pi$
$168$	0	0
$169$	755501.	2.03478
$170$	0	0
$171$	0	0
$172$	−224740.	−0.579241
$173$	461327.	1.17191	0.585954	−	0.810344i	$-0.300720\pi$
$173$	461327.	1.17191	0.585954	+	0.810344i	$0.300720\pi$
$174$	0	0
$175$	0	0
$176$	−314691.	−0.765778
$177$	0	0
$178$	290561.	0.687366
$179$	−34179.6	−0.0797324	−0.0398662	−	0.999205i	$-0.512693\pi$
$179$	−34179.6	−0.0797324	−0.0398662	+	0.999205i	$0.512693\pi$
$180$	0	0
$181$	292967.	0.664695	0.332348	−	0.943157i	$-0.392159\pi$
$181$	292967.	0.664695	0.332348	+	0.943157i	$0.392159\pi$
$182$	−1.67913e6	−3.75755
$183$	0	0
$184$	−308666.	−0.672115
$185$	0	0
$186$	0	0
$187$	315080.	0.658896
$188$	−818174.	−1.68831
$189$	0	0
$190$	0	0
$191$	351390.	0.696956	0.348478	−	0.937317i	$-0.386699\pi$
$191$	351390.	0.696956	0.348478	+	0.937317i	$0.386699\pi$
$192$	0	0
$193$	−237520.	−0.458995	−0.229497	−	0.973309i	$-0.573708\pi$
$193$	−237520.	−0.458995	−0.229497	+	0.973309i	$0.573708\pi$
$194$	−1.35048e6	−2.57622
$195$	0	0
$196$	715501.	1.33036
$197$	577353.	1.05993	0.529964	−	0.848020i	$-0.322206\pi$
$197$	577353.	1.05993	0.529964	+	0.848020i	$0.322206\pi$
$198$	0	0
$199$	13847.4	0.0247877	0.0123939	−	0.999923i	$-0.496055\pi$
$199$	13847.4	0.0247877	0.0123939	+	0.999923i	$0.496055\pi$
$200$	0	0
$201$	0	0
$202$	244617.	0.421801
$203$	−317964.	−0.541550
$204$	0	0
$205$	0	0
$206$	−1.01264e6	−1.66260
$207$	0	0
$208$	−673531.	−1.07944
$209$	581078.	0.920172
$210$	0	0
$211$	−207222.	−0.320427	−0.160214	−	0.987082i	$-0.551218\pi$
$211$	−207222.	−0.320427	−0.160214	+	0.987082i	$0.551218\pi$
$212$	1.40421e6	2.14582
$213$	0	0
$214$	1.61701e6	2.41367
$215$	0	0
$216$	0	0
$217$	−1.20200e6	−1.73282
$218$	−1.00041e6	−1.42573
$219$	0	0
$220$	0	0
$221$	674363.	0.928781
$222$	0	0
$223$	−413619.	−0.556978	−0.278489	−	0.960439i	$-0.589834\pi$
$223$	−413619.	−0.556978	−0.278489	+	0.960439i	$0.589834\pi$
$224$	1.47983e6	1.97057
$225$	0	0
$226$	1.26717e6	1.65030
$227$	830006.	1.06910	0.534548	−	0.845138i	$-0.320482\pi$
$227$	830006.	1.06910	0.534548	+	0.845138i	$0.320482\pi$
$228$	0	0
$229$	−484052.	−0.609963	−0.304981	−	0.952358i	$-0.598650\pi$
$229$	−484052.	−0.609963	−0.304981	+	0.952358i	$0.598650\pi$
$230$	0	0
$231$	0	0
$232$	138798.	0.169303
$233$	17893.7	0.0215928	0.0107964	−	0.999942i	$-0.496563\pi$
$233$	17893.7	0.0215928	0.0107964	+	0.999942i	$0.496563\pi$
$234$	0	0
$235$	0	0
$236$	629967.	0.736271
$237$	0	0
$238$	−1.00492e6	−1.14998
$239$	−1.43474e6	−1.62472	−0.812361	−	0.583155i	$-0.801818\pi$
$239$	−1.43474e6	−1.62472	−0.812361	+	0.583155i	$0.801818\pi$
$240$	0	0
$241$	1.10585e6	1.22647	0.613233	−	0.789902i	$-0.289869\pi$
$241$	1.10585e6	1.22647	0.613233	+	0.789902i	$0.289869\pi$
$242$	−727648.	−0.798698
$243$	0	0
$244$	383728.	0.412618
$245$	0	0
$246$	0	0
$247$	1.24368e6	1.29707
$248$	524698.	0.541726
$249$	0	0
$250$	0	0
$251$	933824.	0.935580	0.467790	−	0.883840i	$-0.345050\pi$
$251$	933824.	0.935580	0.467790	+	0.883840i	$0.345050\pi$
$252$	0	0
$253$	1.89950e6	1.86568
$254$	2.61786e6	2.54602
$255$	0	0
$256$	194748.	0.185726
$257$	703126.	0.664050	0.332025	−	0.943271i	$-0.392268\pi$
$257$	703126.	0.664050	0.332025	+	0.943271i	$0.392268\pi$
$258$	0	0
$259$	−1.41836e6	−1.31383
$260$	0	0
$261$	0	0
$262$	1.94874e6	1.75388
$263$	1.82712e6	1.62884	0.814420	−	0.580276i	$-0.197055\pi$
$263$	1.82712e6	1.62884	0.814420	+	0.580276i	$0.197055\pi$
$264$	0	0
$265$	0	0
$266$	−1.85330e6	−1.60599
$267$	0	0
$268$	88499.8	0.0752671
$269$	−2.17857e6	−1.83566	−0.917828	−	0.396978i	$-0.870059\pi$
$269$	−2.17857e6	−1.83566	−0.917828	+	0.396978i	$0.870059\pi$
$270$	0	0
$271$	−1.86789e6	−1.54500	−0.772498	−	0.635017i	$-0.780993\pi$
$271$	−1.86789e6	−1.54500	−0.772498	+	0.635017i	$0.780993\pi$
$272$	−403095.	−0.330358
$273$	0	0
$274$	1.01567e6	0.817290
$275$	0	0
$276$	0	0
$277$	937721.	0.734301	0.367151	−	0.930162i	$-0.380333\pi$
$277$	937721.	0.734301	0.367151	+	0.930162i	$0.380333\pi$
$278$	1.74394e6	1.35338
$279$	0	0
$280$	0	0
$281$	−601808.	−0.454666	−0.227333	−	0.973817i	$-0.573001\pi$
$281$	−601808.	−0.454666	−0.227333	+	0.973817i	$0.573001\pi$
$282$	0	0
$283$	−332051.	−0.246456	−0.123228	−	0.992378i	$-0.539325\pi$
$283$	−332051.	−0.246456	−0.123228	+	0.992378i	$0.539325\pi$
$284$	−605855.	−0.445731
$285$	0	0
$286$	−4.51065e6	−3.26080
$287$	−732587.	−0.524994
$288$	0	0
$289$	−1.01626e6	−0.715751
$290$	0	0
$291$	0	0
$292$	−1.33565e6	−0.916717
$293$	−1.87340e6	−1.27486	−0.637430	−	0.770508i	$-0.720002\pi$
$293$	−1.87340e6	−1.27486	−0.637430	+	0.770508i	$0.720002\pi$
$294$	0	0
$295$	0	0
$296$	619147.	0.410737
$297$	0	0
$298$	1.03404e6	0.674525
$299$	4.06547e6	2.62986
$300$	0	0
$301$	−1.00208e6	−0.637512
$302$	2.38055e6	1.50197
$303$	0	0
$304$	−743396.	−0.461356
$305$	0	0
$306$	0	0
$307$	1.54740e6	0.937039	0.468519	−	0.883453i	$-0.344788\pi$
$307$	1.54740e6	0.937039	0.468519	+	0.883453i	$0.344788\pi$
$308$	3.79151e6	2.27738
$309$	0	0
$310$	0	0
$311$	−1.32330e6	−0.775813	−0.387907	−	0.921699i	$-0.626802\pi$
$311$	−1.32330e6	−0.775813	−0.387907	+	0.921699i	$0.626802\pi$
$312$	0	0
$313$	513658.	0.296356	0.148178	−	0.988961i	$-0.452659\pi$
$313$	513658.	0.296356	0.148178	+	0.988961i	$0.452659\pi$
$314$	2.39167e6	1.36892
$315$	0	0
$316$	2.18172e6	1.22908
$317$	177343.	0.0991208	0.0495604	−	0.998771i	$-0.484218\pi$
$317$	177343.	0.0991208	0.0495604	+	0.998771i	$0.484218\pi$
$318$	0	0
$319$	−854151.	−0.469957
$320$	0	0
$321$	0	0
$322$	−6.05829e6	−3.25619
$323$	744315.	0.396963
$324$	0	0
$325$	0	0
$326$	−2.81557e6	−1.46731
$327$	0	0
$328$	319790.	0.164127
$329$	−3.64813e6	−1.85815
$330$	0	0
$331$	1.94448e6	0.975514	0.487757	−	0.872979i	$-0.337815\pi$
$331$	1.94448e6	0.975514	0.487757	+	0.872979i	$0.337815\pi$
$332$	1.18464e6	0.589850
$333$	0	0
$334$	1.62530e6	0.797198
$335$	0	0
$336$	0	0
$337$	−788301.	−0.378109	−0.189055	−	0.981967i	$-0.560542\pi$
$337$	−788301.	−0.378109	−0.189055	+	0.981967i	$0.560542\pi$
$338$	−6.47295e6	−3.08184
$339$	0	0
$340$	0	0
$341$	−3.22893e6	−1.50374
$342$	0	0
$343$	87307.4	0.0400697
$344$	437432.	0.199303
$345$	0	0
$346$	−3.95254e6	−1.77495
$347$	3.04940e6	1.35953	0.679767	−	0.733428i	$-0.262081\pi$
$347$	3.04940e6	1.35953	0.679767	+	0.733428i	$0.262081\pi$
$348$	0	0
$349$	−668763.	−0.293906	−0.146953	−	0.989143i	$-0.546947\pi$
$349$	−668763.	−0.293906	−0.146953	+	0.989143i	$0.546947\pi$
$350$	0	0
$351$	0	0
$352$	3.97529e6	1.71006
$353$	369464.	0.157810	0.0789052	−	0.996882i	$-0.474858\pi$
$353$	369464.	0.157810	0.0789052	+	0.996882i	$0.474858\pi$
$354$	0	0
$355$	0	0
$356$	−1.40423e6	−0.587239
$357$	0	0
$358$	292843.	0.120761
$359$	1.55381e6	0.636300	0.318150	−	0.948040i	$-0.396938\pi$
$359$	1.55381e6	0.636300	0.318150	+	0.948040i	$0.396938\pi$
$360$	0	0
$361$	−1.10342e6	−0.445627
$362$	−2.51007e6	−1.00673
$363$	0	0
$364$	8.11494e6	3.21020
$365$	0	0
$366$	0	0
$367$	−62220.2	−0.0241138	−0.0120569	−	0.999927i	$-0.503838\pi$
$367$	−62220.2	−0.0241138	−0.0120569	+	0.999927i	$0.503838\pi$
$368$	−2.43010e6	−0.935415
$369$	0	0
$370$	0	0
$371$	6.26120e6	2.36169
$372$	0	0
$373$	−379126.	−0.141095	−0.0705475	−	0.997508i	$-0.522475\pi$
$373$	−379126.	−0.141095	−0.0705475	+	0.997508i	$0.522475\pi$
$374$	−2.69953e6	−0.997952
$375$	0	0
$376$	1.59249e6	0.580907
$377$	−1.82813e6	−0.662451
$378$	0	0
$379$	−2.83443e6	−1.01360	−0.506801	−	0.862063i	$-0.669172\pi$
$379$	−2.83443e6	−1.01360	−0.506801	+	0.862063i	$0.669172\pi$
$380$	0	0
$381$	0	0
$382$	−3.01062e6	−1.05560
$383$	4.74712e6	1.65361	0.826806	−	0.562487i	$-0.190155\pi$
$383$	4.74712e6	1.65361	0.826806	+	0.562487i	$0.190155\pi$
$384$	0	0
$385$	0	0
$386$	2.03502e6	0.695185
$387$	0	0
$388$	6.52664e6	2.20095
$389$	−5.71361e6	−1.91442	−0.957209	−	0.289399i	$-0.906545\pi$
$389$	−5.71361e6	−1.91442	−0.957209	+	0.289399i	$0.906545\pi$
$390$	0	0
$391$	2.43310e6	0.804856
$392$	−1.39265e6	−0.457747
$393$	0	0
$394$	−4.94663e6	−1.60535
$395$	0	0
$396$	0	0
$397$	2.88320e6	0.918119	0.459060	−	0.888405i	$-0.348186\pi$
$397$	2.88320e6	0.918119	0.459060	+	0.888405i	$0.348186\pi$
$398$	−118642.	−0.0375430
$399$	0	0
$400$	0	0
$401$	−1.12893e6	−0.350595	−0.175298	−	0.984515i	$-0.556089\pi$
$401$	−1.12893e6	−0.350595	−0.175298	+	0.984515i	$0.556089\pi$
$402$	0	0
$403$	−6.91085e6	−2.11967
$404$	−1.18219e6	−0.360358
$405$	0	0
$406$	2.72424e6	0.820221
$407$	−3.81016e6	−1.14014
$408$	0	0
$409$	4.97494e6	1.47055	0.735274	−	0.677770i	$-0.237053\pi$
$409$	4.97494e6	1.47055	0.735274	+	0.677770i	$0.237053\pi$
$410$	0	0
$411$	0	0
$412$	4.89394e6	1.42042
$413$	2.80894e6	0.810339
$414$	0	0
$415$	0	0
$416$	8.50826e6	2.41050
$417$	0	0
$418$	−4.97854e6	−1.39367
$419$	−2.40911e6	−0.670381	−0.335191	−	0.942150i	$-0.608801\pi$
$419$	−2.40911e6	−0.670381	−0.335191	+	0.942150i	$0.608801\pi$
$420$	0	0
$421$	6.97118e6	1.91691	0.958454	−	0.285247i	$-0.0920757\pi$
$421$	6.97118e6	1.91691	0.958454	+	0.285247i	$0.0920757\pi$
$422$	1.77543e6	0.485313
$423$	0	0
$424$	−2.73315e6	−0.738327
$425$	0	0
$426$	0	0
$427$	1.71099e6	0.454127
$428$	−7.81473e6	−2.06208
$429$	0	0
$430$	0	0
$431$	−387837.	−0.100567	−0.0502836	−	0.998735i	$-0.516013\pi$
$431$	−387837.	−0.100567	−0.0502836	+	0.998735i	$0.516013\pi$
$432$	0	0
$433$	223145.	0.0571963	0.0285981	−	0.999591i	$-0.490896\pi$
$433$	223145.	0.0571963	0.0285981	+	0.999591i	$0.490896\pi$
$434$	1.02984e7	2.62450
$435$	0	0
$436$	4.83484e6	1.21805
$437$	4.48718e6	1.12401
$438$	0	0
$439$	−2.60244e6	−0.644496	−0.322248	−	0.946655i	$-0.604438\pi$
$439$	−2.60244e6	−0.644496	−0.322248	+	0.946655i	$0.604438\pi$
$440$	0	0
$441$	0	0
$442$	−5.77778e6	−1.40671
$443$	61785.1	0.0149580	0.00747901	−	0.999972i	$-0.497619\pi$
$443$	61785.1	0.0149580	0.00747901	+	0.999972i	$0.497619\pi$
$444$	0	0
$445$	0	0
$446$	3.54379e6	0.843589
$447$	0	0
$448$	−8.93017e6	−2.10215
$449$	2.53882e6	0.594314	0.297157	−	0.954829i	$-0.403962\pi$
$449$	2.53882e6	0.594314	0.297157	+	0.954829i	$0.403962\pi$
$450$	0	0
$451$	−1.96795e6	−0.455589
$452$	−6.12402e6	−1.40991
$453$	0	0
$454$	−7.11130e6	−1.61923
$455$	0	0
$456$	0	0
$457$	−2.84094e6	−0.636315	−0.318158	−	0.948038i	$-0.603064\pi$
$457$	−2.84094e6	−0.636315	−0.318158	+	0.948038i	$0.603064\pi$
$458$	4.14724e6	0.923838
$459$	0	0
$460$	0	0
$461$	−736656.	−0.161441	−0.0807203	−	0.996737i	$-0.525722\pi$
$461$	−736656.	−0.161441	−0.0807203	+	0.996737i	$0.525722\pi$
$462$	0	0
$463$	−606838.	−0.131559	−0.0657794	−	0.997834i	$-0.520953\pi$
$463$	−606838.	−0.131559	−0.0657794	+	0.997834i	$0.520953\pi$
$464$	1.09275e6	0.235627
$465$	0	0
$466$	−153309.	−0.0327041
$467$	275727.	0.0585043	0.0292522	−	0.999572i	$-0.490687\pi$
$467$	275727.	0.0585043	0.0292522	+	0.999572i	$0.490687\pi$
$468$	0	0
$469$	394609.	0.0828389
$470$	0	0
$471$	0	0
$472$	−1.22616e6	−0.253334
$473$	−2.69191e6	−0.553232
$474$	0	0
$475$	0	0
$476$	4.85663e6	0.982466
$477$	0	0
$478$	1.22925e7	2.46077
$479$	1.70157e6	0.338853	0.169426	−	0.985543i	$-0.445808\pi$
$479$	1.70157e6	0.338853	0.169426	+	0.985543i	$0.445808\pi$
$480$	0	0
$481$	−8.15486e6	−1.60714
$482$	−9.47470e6	−1.85758
$483$	0	0
$484$	3.51660e6	0.682354
$485$	0	0
$486$	0	0
$487$	7.50609e6	1.43414	0.717070	−	0.697002i	$-0.245483\pi$
$487$	7.50609e6	1.43414	0.717070	+	0.697002i	$0.245483\pi$
$488$	−746884.	−0.141972
$489$	0	0
$490$	0	0
$491$	−2.15246e6	−0.402932	−0.201466	−	0.979496i	$-0.564571\pi$
$491$	−2.15246e6	−0.402932	−0.201466	+	0.979496i	$0.564571\pi$
$492$	0	0
$493$	−1.09410e6	−0.202740
$494$	−1.06555e7	−1.96452
$495$	0	0
$496$	4.13090e6	0.753947
$497$	−2.70143e6	−0.490571
$498$	0	0
$499$	2.44200e6	0.439029	0.219515	−	0.975609i	$-0.429553\pi$
$499$	2.44200e6	0.439029	0.219515	+	0.975609i	$0.429553\pi$
$500$	0	0
$501$	0	0
$502$	−8.00079e6	−1.41701
$503$	−1.08947e7	−1.91998	−0.959988	−	0.280041i	$-0.909652\pi$
$503$	−1.08947e7	−1.91998	−0.959988	+	0.280041i	$0.909652\pi$
$504$	0	0
$505$	0	0
$506$	−1.62744e7	−2.82572
$507$	0	0
$508$	−1.26517e7	−2.17515
$509$	−5.50060e6	−0.941057	−0.470529	−	0.882385i	$-0.655937\pi$
$509$	−5.50060e6	−0.941057	−0.470529	+	0.882385i	$0.655937\pi$
$510$	0	0
$511$	−5.95548e6	−1.00894
$512$	−6.72212e6	−1.13327
$513$	0	0
$514$	−6.02422e6	−1.00576
$515$	0	0
$516$	0	0
$517$	−9.80000e6	−1.61250
$518$	1.21522e7	1.98990
$519$	0	0
$520$	0	0
$521$	2.82214e6	0.455496	0.227748	−	0.973720i	$-0.426864\pi$
$521$	2.82214e6	0.455496	0.227748	+	0.973720i	$0.426864\pi$
$522$	0	0
$523$	4.45597e6	0.712341	0.356171	−	0.934421i	$-0.384082\pi$
$523$	4.45597e6	0.712341	0.356171	+	0.934421i	$0.384082\pi$
$524$	−9.41794e6	−1.49840
$525$	0	0
$526$	−1.56544e7	−2.46701
$527$	−4.13601e6	−0.648716
$528$	0	0
$529$	8.23188e6	1.27897
$530$	0	0
$531$	0	0
$532$	8.95670e6	1.37205
$533$	−4.21199e6	−0.642199
$534$	0	0
$535$	0	0
$536$	−172255.	−0.0258976
$537$	0	0
$538$	1.86655e7	2.78025
$539$	8.57019e6	1.27063
$540$	0	0
$541$	7.33120e6	1.07692	0.538458	−	0.842652i	$-0.319007\pi$
$541$	7.33120e6	1.07692	0.538458	+	0.842652i	$0.319007\pi$
$542$	1.60036e7	2.34002
$543$	0	0
$544$	5.09202e6	0.737723
$545$	0	0
$546$	0	0
$547$	−6.84249e6	−0.977791	−0.488896	−	0.872342i	$-0.662600\pi$
$547$	−6.84249e6	−0.977791	−0.488896	+	0.872342i	$0.662600\pi$
$548$	−4.90856e6	−0.698237
$549$	0	0
$550$	0	0
$551$	−2.01776e6	−0.283133
$552$	0	0
$553$	9.72798e6	1.35273
$554$	−8.03417e6	−1.11216
$555$	0	0
$556$	−8.42818e6	−1.15624
$557$	−4.86219e6	−0.664040	−0.332020	−	0.943272i	$-0.607730\pi$
$557$	−4.86219e6	−0.664040	−0.332020	+	0.943272i	$0.607730\pi$
$558$	0	0
$559$	−5.76147e6	−0.779836
$560$	0	0
$561$	0	0
$562$	5.15615e6	0.688629
$563$	6.23070e6	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$563$	6.23070e6	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$564$	0	0
$565$	0	0
$566$	2.84494e6	0.373277
$567$	0	0
$568$	1.17923e6	0.153366
$569$	8.88975e6	1.15109	0.575544	−	0.817771i	$-0.304790\pi$
$569$	8.88975e6	1.15109	0.575544	+	0.817771i	$0.304790\pi$
$570$	0	0
$571$	4.97693e6	0.638809	0.319405	−	0.947618i	$-0.396517\pi$
$571$	4.97693e6	0.638809	0.319405	+	0.947618i	$0.396517\pi$
$572$	2.17992e7	2.78581
$573$	0	0
$574$	6.27663e6	0.795146
$575$	0	0
$576$	0	0
$577$	−1.49645e7	−1.87121	−0.935606	−	0.353046i	$-0.885146\pi$
$577$	−1.49645e7	−1.87121	−0.935606	+	0.353046i	$0.885146\pi$
$578$	8.70711e6	1.08406
$579$	0	0
$580$	0	0
$581$	5.28215e6	0.649188
$582$	0	0
$583$	1.68195e7	2.04947
$584$	2.59970e6	0.315421
$585$	0	0
$586$	1.60509e7	1.93088
$587$	1.11647e7	1.33737	0.668686	−	0.743545i	$-0.266857\pi$
$587$	1.11647e7	1.33737	0.668686	+	0.743545i	$0.266857\pi$
$588$	0	0
$589$	−7.62772e6	−0.905955
$590$	0	0
$591$	0	0
$592$	4.87449e6	0.571643
$593$	6.85159e6	0.800119	0.400059	−	0.916489i	$-0.368990\pi$
$593$	6.85159e6	0.800119	0.400059	+	0.916489i	$0.368990\pi$
$594$	0	0
$595$	0	0
$596$	−4.99736e6	−0.576269
$597$	0	0
$598$	−3.48320e7	−3.98314
$599$	1.41540e7	1.61180	0.805901	−	0.592050i	$-0.201681\pi$
$599$	1.41540e7	1.61180	0.805901	+	0.592050i	$0.201681\pi$
$600$	0	0
$601$	−1.38634e6	−0.156561	−0.0782807	−	0.996931i	$-0.524943\pi$
$601$	−1.38634e6	−0.156561	−0.0782807	+	0.996931i	$0.524943\pi$
$602$	8.58563e6	0.965563
$603$	0	0
$604$	−1.15048e7	−1.28318
$605$	0	0
$606$	0	0
$607$	1.10063e6	0.121247	0.0606236	−	0.998161i	$-0.480691\pi$
$607$	1.10063e6	0.121247	0.0606236	+	0.998161i	$0.480691\pi$
$608$	9.39083e6	1.03026
$609$	0	0
$610$	0	0
$611$	−2.09748e7	−2.27298
$612$	0	0
$613$	1.30801e6	0.140592	0.0702961	−	0.997526i	$-0.477606\pi$
$613$	1.30801e6	0.140592	0.0702961	+	0.997526i	$0.477606\pi$
$614$	−1.32578e7	−1.41922
$615$	0	0
$616$	−7.37977e6	−0.783594
$617$	−1.32716e7	−1.40350	−0.701749	−	0.712424i	$-0.747597\pi$
$617$	−1.32716e7	−1.40350	−0.701749	+	0.712424i	$0.747597\pi$
$618$	0	0
$619$	1.08901e7	1.14236	0.571181	−	0.820824i	$-0.306485\pi$
$619$	1.08901e7	1.14236	0.571181	+	0.820824i	$0.306485\pi$
$620$	0	0
$621$	0	0
$622$	1.13377e7	1.17503
$623$	−6.26129e6	−0.646314
$624$	0	0
$625$	0	0
$626$	−4.40090e6	−0.448855
$627$	0	0
$628$	−1.15585e7	−1.16951
$629$	−4.88052e6	−0.491857
$630$	0	0
$631$	−5.08129e6	−0.508043	−0.254022	−	0.967199i	$-0.581753\pi$
$631$	−5.08129e6	−0.508043	−0.254022	+	0.967199i	$0.581753\pi$
$632$	−4.24648e6	−0.422898
$633$	0	0
$634$	−1.51943e6	−0.150126
$635$	0	0
$636$	0	0
$637$	1.83427e7	1.79108
$638$	7.31816e6	0.711787
$639$	0	0
$640$	0	0
$641$	1.72774e7	1.66086	0.830430	−	0.557123i	$-0.188095\pi$
$641$	1.72774e7	1.66086	0.830430	+	0.557123i	$0.188095\pi$
$642$	0	0
$643$	−9.83084e6	−0.937698	−0.468849	−	0.883278i	$-0.655331\pi$
$643$	−9.83084e6	−0.937698	−0.468849	+	0.883278i	$0.655331\pi$
$644$	2.92787e7	2.78187
$645$	0	0
$646$	−6.37712e6	−0.601233
$647$	1.08236e7	1.01651	0.508256	−	0.861206i	$-0.330290\pi$
$647$	1.08236e7	1.01651	0.508256	+	0.861206i	$0.330290\pi$
$648$	0	0
$649$	7.54567e6	0.703211
$650$	0	0
$651$	0	0
$652$	1.36072e7	1.25357
$653$	−1.91712e7	−1.75940	−0.879702	−	0.475525i	$-0.842258\pi$
$653$	−1.91712e7	−1.75940	−0.879702	+	0.475525i	$0.842258\pi$
$654$	0	0
$655$	0	0
$656$	2.51768e6	0.228424
$657$	0	0
$658$	3.12563e7	2.81432
$659$	3.87175e6	0.347292	0.173646	−	0.984808i	$-0.444445\pi$
$659$	3.87175e6	0.347292	0.173646	+	0.984808i	$0.444445\pi$
$660$	0	0
$661$	−3.99889e6	−0.355989	−0.177994	−	0.984032i	$-0.556961\pi$
$661$	−3.99889e6	−0.355989	−0.177994	+	0.984032i	$0.556961\pi$
$662$	−1.66598e7	−1.47750
$663$	0	0
$664$	−2.30577e6	−0.202954
$665$	0	0
$666$	0	0
$667$	−6.59589e6	−0.574062
$668$	−7.85478e6	−0.681072
$669$	0	0
$670$	0	0
$671$	4.59624e6	0.394091
$672$	0	0
$673$	−1.85986e7	−1.58286	−0.791431	−	0.611259i	$-0.790663\pi$
$673$	−1.85986e7	−1.58286	−0.791431	+	0.611259i	$0.790663\pi$
$674$	6.75398e6	0.572677
$675$	0	0
$676$	3.12827e7	2.63292
$677$	491412.	0.0412073	0.0206036	−	0.999788i	$-0.493441\pi$
$677$	491412.	0.0412073	0.0206036	+	0.999788i	$0.493441\pi$
$678$	0	0
$679$	2.91014e7	2.42236
$680$	0	0
$681$	0	0
$682$	2.76647e7	2.27754
$683$	1.18161e7	0.969216	0.484608	−	0.874731i	$-0.338962\pi$
$683$	1.18161e7	0.969216	0.484608	+	0.874731i	$0.338962\pi$
$684$	0	0
$685$	0	0
$686$	−748029.	−0.0606888
$687$	0	0
$688$	3.44386e6	0.277380
$689$	3.59986e7	2.88894
$690$	0	0
$691$	−1.58549e7	−1.26319	−0.631594	−	0.775299i	$-0.717599\pi$
$691$	−1.58549e7	−1.26319	−0.631594	+	0.775299i	$0.717599\pi$
$692$	1.91020e7	1.51640
$693$	0	0
$694$	−2.61265e7	−2.05912
$695$	0	0
$696$	0	0
$697$	−2.52079e6	−0.196542
$698$	5.72981e6	0.445145
$699$	0	0
$700$	0	0
$701$	2.49504e6	0.191770	0.0958852	−	0.995392i	$-0.469432\pi$
$701$	2.49504e6	0.191770	0.0958852	+	0.995392i	$0.469432\pi$
$702$	0	0
$703$	−9.00076e6	−0.686896
$704$	−2.39892e7	−1.82425
$705$	0	0
$706$	−3.16548e6	−0.239017
$707$	−5.27123e6	−0.396610
$708$	0	0
$709$	4.48130e6	0.334802	0.167401	−	0.985889i	$-0.446463\pi$
$709$	4.48130e6	0.334802	0.167401	+	0.985889i	$0.446463\pi$
$710$	0	0
$711$	0	0
$712$	2.73319e6	0.202055
$713$	−2.49344e7	−1.83685
$714$	0	0
$715$	0	0
$716$	−1.41526e6	−0.103170
$717$	0	0
$718$	−1.33127e7	−0.963729
$719$	−1.65946e7	−1.19714	−0.598571	−	0.801070i	$-0.704264\pi$
$719$	−1.65946e7	−1.19714	−0.598571	+	0.801070i	$0.704264\pi$
$720$	0	0
$721$	2.18214e7	1.56331
$722$	9.45381e6	0.674938
$723$	0	0
$724$	1.21308e7	0.860086
$725$	0	0
$726$	0	0
$727$	−1.36570e6	−0.0958341	−0.0479171	−	0.998851i	$-0.515258\pi$
$727$	−1.36570e6	−0.0958341	−0.0479171	+	0.998851i	$0.515258\pi$
$728$	−1.57948e7	−1.10455
$729$	0	0
$730$	0	0
$731$	−3.44812e6	−0.238665
$732$	0	0
$733$	−2.16084e7	−1.48547	−0.742733	−	0.669588i	$-0.766471\pi$
$733$	−2.16084e7	−1.48547	−0.742733	+	0.669588i	$0.766471\pi$
$734$	533088.	0.0365224
$735$	0	0
$736$	3.06978e7	2.08888
$737$	1.06004e6	0.0718875
$738$	0	0
$739$	−7.39638e6	−0.498205	−0.249102	−	0.968477i	$-0.580136\pi$
$739$	−7.39638e6	−0.498205	−0.249102	+	0.968477i	$0.580136\pi$
$740$	0	0
$741$	0	0
$742$	−5.36445e7	−3.57697
$743$	1.51861e7	1.00919	0.504596	−	0.863356i	$-0.331641\pi$
$743$	1.51861e7	1.00919	0.504596	+	0.863356i	$0.331641\pi$
$744$	0	0
$745$	0	0
$746$	3.24827e6	0.213700
$747$	0	0
$748$	1.30464e7	0.852583
$749$	−3.48448e7	−2.26952
$750$	0	0
$751$	1.03400e6	0.0668990	0.0334495	−	0.999440i	$-0.489351\pi$
$751$	1.03400e6	0.0668990	0.0334495	+	0.999440i	$0.489351\pi$
$752$	1.25375e7	0.808476
$753$	0	0
$754$	1.56630e7	1.00334
$755$	0	0
$756$	0	0
$757$	−1.46243e7	−0.927549	−0.463774	−	0.885953i	$-0.653505\pi$
$757$	−1.46243e7	−0.927549	−0.463774	+	0.885953i	$0.653505\pi$
$758$	2.42847e7	1.53518
$759$	0	0
$760$	0	0
$761$	3.36209e6	0.210450	0.105225	−	0.994448i	$-0.466444\pi$
$761$	3.36209e6	0.210450	0.105225	+	0.994448i	$0.466444\pi$
$762$	0	0
$763$	2.15579e7	1.34059
$764$	1.45498e7	0.901831
$765$	0	0
$766$	−4.06722e7	−2.50453
$767$	1.61499e7	0.991247
$768$	0	0
$769$	−2.73574e7	−1.66824	−0.834121	−	0.551582i	$-0.814024\pi$
$769$	−2.73574e7	−1.66824	−0.834121	+	0.551582i	$0.814024\pi$
$770$	0	0
$771$	0	0
$772$	−9.83491e6	−0.593919
$773$	8.13760e6	0.489833	0.244916	−	0.969544i	$-0.421239\pi$
$773$	8.13760e6	0.489833	0.244916	+	0.969544i	$0.421239\pi$
$774$	0	0
$775$	0	0
$776$	−1.27034e7	−0.757296
$777$	0	0
$778$	4.89529e7	2.89954
$779$	−4.64890e6	−0.274478
$780$	0	0
$781$	−7.25687e6	−0.425717
$782$	−2.08462e7	−1.21902
$783$	0	0
$784$	−1.09642e7	−0.637068
$785$	0	0
$786$	0	0
$787$	1.63248e7	0.939532	0.469766	−	0.882791i	$-0.344338\pi$
$787$	1.63248e7	0.939532	0.469766	+	0.882791i	$0.344338\pi$
$788$	2.39062e7	1.37150
$789$	0	0
$790$	0	0
$791$	−2.73062e7	−1.55174
$792$	0	0
$793$	9.83730e6	0.555511
$794$	−2.47026e7	−1.39057
$795$	0	0
$796$	573375.	0.0320742
$797$	−2.88660e7	−1.60969	−0.804843	−	0.593488i	$-0.797751\pi$
$797$	−2.88660e7	−1.60969	−0.804843	+	0.593488i	$0.797751\pi$
$798$	0	0
$799$	−1.25530e7	−0.695634
$800$	0	0
$801$	0	0
$802$	9.67241e6	0.531005
$803$	−1.59983e7	−0.875556
$804$	0	0
$805$	0	0
$806$	5.92106e7	3.21042
$807$	0	0
$808$	2.30101e6	0.123991
$809$	−2.63497e7	−1.41548	−0.707741	−	0.706472i	$-0.750286\pi$
$809$	−2.63497e7	−1.41548	−0.707741	+	0.706472i	$0.750286\pi$
$810$	0	0
$811$	−4.76400e6	−0.254343	−0.127171	−	0.991881i	$-0.540590\pi$
$811$	−4.76400e6	−0.254343	−0.127171	+	0.991881i	$0.540590\pi$
$812$	−1.31658e7	−0.700742
$813$	0	0
$814$	3.26446e7	1.72683
$815$	0	0
$816$	0	0
$817$	−6.35910e6	−0.333304
$818$	−4.26241e7	−2.22727
$819$	0	0
$820$	0	0
$821$	−2.76433e7	−1.43130	−0.715652	−	0.698458i	$-0.753870\pi$
$821$	−2.76433e7	−1.43130	−0.715652	+	0.698458i	$0.753870\pi$
$822$	0	0
$823$	−1.95121e7	−1.00416	−0.502082	−	0.864820i	$-0.667433\pi$
$823$	−1.95121e7	−1.00416	−0.502082	+	0.864820i	$0.667433\pi$
$824$	−9.52552e6	−0.488732
$825$	0	0
$826$	−2.40663e7	−1.22732
$827$	−2.45155e7	−1.24645	−0.623227	−	0.782041i	$-0.714179\pi$
$827$	−2.45155e7	−1.24645	−0.623227	+	0.782041i	$0.714179\pi$
$828$	0	0
$829$	−9.47691e6	−0.478939	−0.239470	−	0.970904i	$-0.576974\pi$
$829$	−9.47691e6	−0.478939	−0.239470	+	0.970904i	$0.576974\pi$
$830$	0	0
$831$	0	0
$832$	−5.13438e7	−2.57146
$833$	1.09777e7	0.548151
$834$	0	0
$835$	0	0
$836$	2.40605e7	1.19066
$837$	0	0
$838$	2.06407e7	1.01535
$839$	5.17992e6	0.254050	0.127025	−	0.991900i	$-0.459457\pi$
$839$	5.17992e6	0.254050	0.127025	+	0.991900i	$0.459457\pi$
$840$	0	0
$841$	−1.75452e7	−0.855396
$842$	−5.97275e7	−2.90331
$843$	0	0
$844$	−8.58036e6	−0.414619
$845$	0	0
$846$	0	0
$847$	1.56800e7	0.750998
$848$	−2.15179e7	−1.02757
$849$	0	0
$850$	0	0
$851$	−2.94227e7	−1.39270
$852$	0	0
$853$	−2.24589e7	−1.05685	−0.528427	−	0.848979i	$-0.677218\pi$
$853$	−2.24589e7	−1.05685	−0.528427	+	0.848979i	$0.677218\pi$
$854$	−1.46593e7	−0.687812
$855$	0	0
$856$	1.52105e7	0.709512
$857$	2.22945e6	0.103692	0.0518461	−	0.998655i	$-0.483489\pi$
$857$	2.22945e6	0.103692	0.0518461	+	0.998655i	$0.483489\pi$
$858$	0	0
$859$	2.55828e7	1.18295	0.591473	−	0.806325i	$-0.298547\pi$
$859$	2.55828e7	1.18295	0.591473	+	0.806325i	$0.298547\pi$
$860$	0	0
$861$	0	0
$862$	3.32290e6	0.152317
$863$	−6.99750e6	−0.319827	−0.159914	−	0.987131i	$-0.551122\pi$
$863$	−6.99750e6	−0.319827	−0.159914	+	0.987131i	$0.551122\pi$
$864$	0	0
$865$	0	0
$866$	−1.91186e6	−0.0866284
$867$	0	0
$868$	−4.97706e7	−2.24219
$869$	2.61324e7	1.17389
$870$	0	0
$871$	2.26879e6	0.101333
$872$	−9.41048e6	−0.419103
$873$	0	0
$874$	−3.84451e7	−1.70240
$875$	0	0
$876$	0	0
$877$	3.48299e7	1.52916	0.764580	−	0.644529i	$-0.222946\pi$
$877$	3.48299e7	1.52916	0.764580	+	0.644529i	$0.222946\pi$
$878$	2.22971e7	0.976141
$879$	0	0
$880$	0	0
$881$	2.67201e7	1.15984	0.579921	−	0.814673i	$-0.303083\pi$
$881$	2.67201e7	1.15984	0.579921	+	0.814673i	$0.303083\pi$
$882$	0	0
$883$	1.89467e7	0.817771	0.408885	−	0.912586i	$-0.365918\pi$
$883$	1.89467e7	0.817771	0.408885	+	0.912586i	$0.365918\pi$
$884$	2.79231e7	1.20180
$885$	0	0
$886$	−529360.	−0.0226551
$887$	3.20464e7	1.36763	0.683817	−	0.729654i	$-0.260319\pi$
$887$	3.20464e7	1.36763	0.683817	+	0.729654i	$0.260319\pi$
$888$	0	0
$889$	−5.64121e7	−2.39396
$890$	0	0
$891$	0	0
$892$	−1.71266e7	−0.720706
$893$	−2.31506e7	−0.971478
$894$	0	0
$895$	0	0
$896$	2.91569e7	1.21331
$897$	0	0
$898$	−2.17520e7	−0.900137
$899$	1.12123e7	0.462695
$900$	0	0
$901$	2.15444e7	0.884145
$902$	1.68610e7	0.690027
$903$	0	0
$904$	1.19197e7	0.485116
$905$	0	0
$906$	0	0
$907$	−3.80296e7	−1.53498	−0.767492	−	0.641059i	$-0.778496\pi$
$907$	−3.80296e7	−1.53498	−0.767492	+	0.641059i	$0.778496\pi$
$908$	3.43677e7	1.38336
$909$	0	0
$910$	0	0
$911$	2.62362e7	1.04738	0.523690	−	0.851909i	$-0.324555\pi$
$911$	2.62362e7	1.04738	0.523690	+	0.851909i	$0.324555\pi$
$912$	0	0
$913$	1.41895e7	0.563365
$914$	2.43405e7	0.963751
$915$	0	0
$916$	−2.00429e7	−0.789265
$917$	−4.19933e7	−1.64914
$918$	0	0
$919$	3.04906e7	1.19091	0.595453	−	0.803390i	$-0.296972\pi$
$919$	3.04906e7	1.19091	0.595453	+	0.803390i	$0.296972\pi$
$920$	0	0
$921$	0	0
$922$	6.31150e6	0.244515
$923$	−1.55318e7	−0.600092
$924$	0	0
$925$	0	0
$926$	5.19924e6	0.199257
$927$	0	0
$928$	−1.38040e7	−0.526179
$929$	3.40521e7	1.29451	0.647253	−	0.762275i	$-0.275918\pi$
$929$	3.40521e7	1.29451	0.647253	+	0.762275i	$0.275918\pi$
$930$	0	0
$931$	2.02454e7	0.765512
$932$	740915.	0.0279402
$933$	0	0
$934$	−2.36237e6	−0.0886095
$935$	0	0
$936$	0	0
$937$	4.93780e7	1.83732	0.918659	−	0.395051i	$-0.129273\pi$
$937$	4.93780e7	1.83732	0.918659	+	0.395051i	$0.129273\pi$
$938$	−3.38091e6	−0.125466
$939$	0	0
$940$	0	0
$941$	−3.88046e7	−1.42859	−0.714297	−	0.699842i	$-0.753254\pi$
$941$	−3.88046e7	−1.42859	−0.714297	+	0.699842i	$0.753254\pi$
$942$	0	0
$943$	−1.51969e7	−0.556512
$944$	−9.65347e6	−0.352577
$945$	0	0
$946$	2.30636e7	0.837915
$947$	4.98230e7	1.80532	0.902662	−	0.430350i	$-0.141610\pi$
$947$	4.98230e7	1.80532	0.902662	+	0.430350i	$0.141610\pi$
$948$	0	0
$949$	−3.42409e7	−1.23418
$950$	0	0
$951$	0	0
$952$	−9.45290e6	−0.338043
$953$	1.81555e7	0.647553	0.323776	−	0.946134i	$-0.395047\pi$
$953$	1.81555e7	0.647553	0.323776	+	0.946134i	$0.395047\pi$
$954$	0	0
$955$	0	0
$956$	−5.94078e7	−2.10232
$957$	0	0
$958$	−1.45787e7	−0.513220
$959$	−2.18866e7	−0.768479
$960$	0	0
$961$	1.37565e7	0.480507
$962$	6.98689e7	2.43414
$963$	0	0
$964$	4.57897e7	1.58699
$965$	0	0
$966$	0	0
$967$	−1.84269e6	−0.0633704	−0.0316852	−	0.999498i	$-0.510087\pi$
$967$	−1.84269e6	−0.0633704	−0.0316852	+	0.999498i	$0.510087\pi$
$968$	−6.84468e6	−0.234782
$969$	0	0
$970$	0	0
$971$	−2.07031e7	−0.704673	−0.352337	−	0.935873i	$-0.614613\pi$
$971$	−2.07031e7	−0.704673	−0.352337	+	0.935873i	$0.614613\pi$
$972$	0	0
$973$	−3.75801e7	−1.27255
$974$	−6.43104e7	−2.17212
$975$	0	0
$976$	−5.88016e6	−0.197590
$977$	−4.80424e7	−1.61023	−0.805116	−	0.593118i	$-0.797897\pi$
$977$	−4.80424e7	−1.61023	−0.805116	+	0.593118i	$0.797897\pi$
$978$	0	0
$979$	−1.68198e7	−0.560871
$980$	0	0
$981$	0	0
$982$	1.84418e7	0.610273
$983$	−3.53714e6	−0.116753	−0.0583765	−	0.998295i	$-0.518592\pi$
$983$	−3.53714e6	−0.116753	−0.0583765	+	0.998295i	$0.518592\pi$
$984$	0	0
$985$	0	0
$986$	9.37398e6	0.307066
$987$	0	0
$988$	5.14964e7	1.67836
$989$	−2.07874e7	−0.675785
$990$	0	0
$991$	4.12366e7	1.33382	0.666912	−	0.745137i	$-0.267616\pi$
$991$	4.12366e7	1.33382	0.666912	+	0.745137i	$0.267616\pi$
$992$	−5.21829e7	−1.68364
$993$	0	0
$994$	2.31452e7	0.743010
$995$	0	0
$996$	0	0
$997$	−1.48434e7	−0.472928	−0.236464	−	0.971640i	$-0.575989\pi$
$997$	−1.48434e7	−0.472928	−0.236464	+	0.971640i	$0.575989\pi$
$998$	−2.09224e7	−0.664946
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.j.1.1		2
3.2	odd	2		75.6.a.i.1.2	yes	2
5.2	odd	4		225.6.b.l.199.1		4
5.3	odd	4		225.6.b.l.199.4		4
5.4	even	2		225.6.a.t.1.2		2
15.2	even	4		75.6.b.f.49.4		4
15.8	even	4		75.6.b.f.49.1		4
15.14	odd	2		75.6.a.g.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.a.g.1.1	✓	2	15.14	odd	2
75.6.a.i.1.2	yes	2	3.2	odd	2
75.6.b.f.49.1		4	15.8	even	4
75.6.b.f.49.4		4	15.2	even	4
225.6.a.j.1.1		2	1.1	even	1	trivial
225.6.a.t.1.2		2	5.4	even	2
225.6.b.l.199.1		4	5.2	odd	4
225.6.b.l.199.4		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-8.56776 q^{2} +41.4066 q^{4} +184.626 q^{7} -80.5934 q^{8} +O(q^{10})\)	\(q-8.56776 q^{2} +41.4066 q^{4} +184.626 q^{7} -80.5934 q^{8} +495.963 q^{11} +1061.51 q^{13} -1581.84 q^{14} -634.505 q^{16} +635.289 q^{17} +1171.62 q^{19} -4249.30 q^{22} +3829.91 q^{23} -9094.73 q^{26} +7644.75 q^{28} -1722.21 q^{29} -6510.43 q^{31} +8015.28 q^{32} -5443.01 q^{34} -7682.35 q^{37} -10038.1 q^{38} -3967.94 q^{41} -5427.64 q^{43} +20536.1 q^{44} -32813.8 q^{46} -19759.5 q^{47} +17279.9 q^{49} +43953.3 q^{52} +33912.8 q^{53} -14879.7 q^{56} +14755.4 q^{58} +15214.2 q^{59} +9267.31 q^{61} +55779.8 q^{62} -48368.9 q^{64} +2137.34 q^{67} +26305.2 q^{68} -14631.9 q^{71} -32256.9 q^{73} +65820.6 q^{74} +48512.6 q^{76} +91567.9 q^{77} +52690.1 q^{79} +33996.4 q^{82} +28610.0 q^{83} +46502.7 q^{86} -39971.4 q^{88} -33913.3 q^{89} +195982. q^{91} +158584. q^{92} +169295. q^{94} +157623. q^{97} -148050. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 16 q^{4} + 102 q^{7} - 228 q^{8}+O(q^{10}) \) 2 * q - 6 * q^2 + 16 * q^4 + 102 * q^7 - 228 * q^8	\( 2 q - 6 q^{2} + 16 q^{4} + 102 q^{7} - 228 q^{8} + 12 q^{11} + 1054 q^{13} - 1794 q^{14} - 200 q^{16} + 1716 q^{17} + 4214 q^{19} - 5492 q^{22} + 444 q^{23} - 9114 q^{26} + 9744 q^{28} - 4068 q^{29} - 2598 q^{31} + 13848 q^{32} - 2668 q^{34} + 4412 q^{37} - 2226 q^{38} - 11232 q^{41} - 8450 q^{43} + 32832 q^{44} - 41508 q^{46} - 2460 q^{47} + 7300 q^{49} + 44144 q^{52} + 65064 q^{53} - 2700 q^{56} + 8732 q^{58} + 63924 q^{59} + 7310 q^{61} + 65826 q^{62} - 47296 q^{64} + 61734 q^{67} - 1152 q^{68} - 98304 q^{71} - 26564 q^{73} + 96876 q^{74} - 28784 q^{76} + 131556 q^{77} + 84000 q^{79} + 15344 q^{82} + 65772 q^{83} + 38742 q^{86} + 31368 q^{88} - 103104 q^{89} + 196602 q^{91} + 244608 q^{92} + 213716 q^{94} + 69374 q^{97} - 173676 q^{98}+O(q^{100}) \) 2 * q - 6 * q^2 + 16 * q^4 + 102 * q^7 - 228 * q^8 + 12 * q^11 + 1054 * q^13 - 1794 * q^14 - 200 * q^16 + 1716 * q^17 + 4214 * q^19 - 5492 * q^22 + 444 * q^23 - 9114 * q^26 + 9744 * q^28 - 4068 * q^29 - 2598 * q^31 + 13848 * q^32 - 2668 * q^34 + 4412 * q^37 - 2226 * q^38 - 11232 * q^41 - 8450 * q^43 + 32832 * q^44 - 41508 * q^46 - 2460 * q^47 + 7300 * q^49 + 44144 * q^52 + 65064 * q^53 - 2700 * q^56 + 8732 * q^58 + 63924 * q^59 + 7310 * q^61 + 65826 * q^62 - 47296 * q^64 + 61734 * q^67 - 1152 * q^68 - 98304 * q^71 - 26564 * q^73 + 96876 * q^74 - 28784 * q^76 + 131556 * q^77 + 84000 * q^79 + 15344 * q^82 + 65772 * q^83 + 38742 * q^86 + 31368 * q^88 - 103104 * q^89 + 196602 * q^91 + 244608 * q^92 + 213716 * q^94 + 69374 * q^97 - 173676 * q^98

Introduction

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