LMFDB - Embedded newform 325.6.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.838151$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.83815 q^{2} +11.9262 q^{3} -28.6212 q^{4} +21.9221 q^{6} +112.158 q^{7} -111.431 q^{8} -100.766 q^{9} +162.916 q^{11} -341.342 q^{12} -169.000 q^{13} +206.164 q^{14} +711.052 q^{16} +379.413 q^{17} -185.224 q^{18} -284.116 q^{19} +1337.62 q^{21} +299.464 q^{22} -4188.02 q^{23} -1328.94 q^{24} -310.647 q^{26} -4099.82 q^{27} -3210.10 q^{28} +6279.50 q^{29} -7559.74 q^{31} +4872.81 q^{32} +1942.96 q^{33} +697.419 q^{34} +2884.05 q^{36} +7060.39 q^{37} -522.249 q^{38} -2015.52 q^{39} +4166.01 q^{41} +2458.75 q^{42} -21505.5 q^{43} -4662.85 q^{44} -7698.22 q^{46} -22686.6 q^{47} +8480.13 q^{48} -4227.52 q^{49} +4524.95 q^{51} +4836.98 q^{52} -4774.01 q^{53} -7536.08 q^{54} -12497.9 q^{56} -3388.42 q^{57} +11542.7 q^{58} -48311.4 q^{59} +3650.37 q^{61} -13895.9 q^{62} -11301.8 q^{63} -13796.7 q^{64} +3571.46 q^{66} +15803.8 q^{67} -10859.3 q^{68} -49947.1 q^{69} -37247.8 q^{71} +11228.5 q^{72} +29323.4 q^{73} +12978.1 q^{74} +8131.75 q^{76} +18272.4 q^{77} -3704.84 q^{78} -44954.4 q^{79} -24409.0 q^{81} +7657.75 q^{82} -23704.5 q^{83} -38284.3 q^{84} -39530.3 q^{86} +74890.4 q^{87} -18153.9 q^{88} +561.767 q^{89} -18954.7 q^{91} +119866. q^{92} -90158.8 q^{93} -41701.3 q^{94} +58114.0 q^{96} +31260.4 q^{97} -7770.82 q^{98} -16416.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * q^99

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 $

$2$	1.83815	0.324942	0.162471	−	0.986713i	$-0.448054\pi$
$2$	1.83815	0.324942	0.162471	+	0.986713i	$0.448054\pi$
$3$	11.9262	0.765065	0.382532	−	0.923942i	$-0.375052\pi$
$3$	11.9262	0.765065	0.382532	+	0.923942i	$0.375052\pi$
$4$	−28.6212	−0.894413
$5$	0	0
$5$	0	0
$6$	21.9221	0.248602
$7$	112.158	0.865140	0.432570	−	0.901600i	$-0.357607\pi$
$7$	112.158	0.865140	0.432570	+	0.901600i	$0.357607\pi$
$8$	−111.431	−0.615575
$9$	−100.766	−0.414676
$10$	0	0
$11$	162.916	0.405958	0.202979	−	0.979183i	$-0.434938\pi$
$11$	162.916	0.405958	0.202979	+	0.979183i	$0.434938\pi$
$12$	−341.342	−0.684284
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	206.164	0.281120
$15$	0	0
$16$	711.052	0.694386
$17$	379.413	0.318413	0.159206	−	0.987245i	$-0.449107\pi$
$17$	379.413	0.318413	0.159206	+	0.987245i	$0.449107\pi$
$18$	−185.224	−0.134746
$19$	−284.116	−0.180556	−0.0902781	−	0.995917i	$-0.528776\pi$
$19$	−284.116	−0.180556	−0.0902781	+	0.995917i	$0.528776\pi$
$20$	0	0
$21$	1337.62	0.661888
$22$	299.464	0.131913
$23$	−4188.02	−1.65078	−0.825391	−	0.564562i	$-0.809045\pi$
$23$	−4188.02	−1.65078	−0.825391	+	0.564562i	$0.809045\pi$
$24$	−1328.94	−0.470954
$25$	0	0
$26$	−310.647	−0.0901227
$27$	−4099.82	−1.08232
$28$	−3210.10	−0.773792
$29$	6279.50	1.38653	0.693266	−	0.720682i	$-0.256171\pi$
$29$	6279.50	1.38653	0.693266	+	0.720682i	$0.256171\pi$
$30$	0	0
$31$	−7559.74	−1.41287	−0.706436	−	0.707777i	$-0.749698\pi$
$31$	−7559.74	−1.41287	−0.706436	+	0.707777i	$0.749698\pi$
$32$	4872.81	0.841210
$33$	1942.96	0.310584
$34$	697.419	0.103466
$35$	0	0
$36$	2884.05	0.370891
$37$	7060.39	0.847860	0.423930	−	0.905695i	$-0.360650\pi$
$37$	7060.39	0.847860	0.423930	+	0.905695i	$0.360650\pi$
$38$	−522.249	−0.0586703
$39$	−2015.52	−0.212191
$40$	0	0
$41$	4166.01	0.387044	0.193522	−	0.981096i	$-0.438009\pi$
$41$	4166.01	0.387044	0.193522	+	0.981096i	$0.438009\pi$
$42$	2458.75	0.215075
$43$	−21505.5	−1.77369	−0.886845	−	0.462067i	$-0.847108\pi$
$43$	−21505.5	−1.77369	−0.886845	+	0.462067i	$0.847108\pi$
$44$	−4662.85	−0.363094
$45$	0	0
$46$	−7698.22	−0.536409
$47$	−22686.6	−1.49804	−0.749021	−	0.662546i	$-0.769476\pi$
$47$	−22686.6	−1.49804	−0.749021	+	0.662546i	$0.769476\pi$
$48$	8480.13	0.531251
$49$	−4227.52	−0.251533
$50$	0	0
$51$	4524.95	0.243606
$52$	4836.98	0.248065
$53$	−4774.01	−0.233450	−0.116725	−	0.993164i	$-0.537240\pi$
$53$	−4774.01	−0.233450	−0.116725	+	0.993164i	$0.537240\pi$
$54$	−7536.08	−0.351691
$55$	0	0
$56$	−12497.9	−0.532558
$57$	−3388.42	−0.138137
$58$	11542.7	0.450543
$59$	−48311.4	−1.80684	−0.903420	−	0.428756i	$-0.858952\pi$
$59$	−48311.4	−1.80684	−0.903420	+	0.428756i	$0.858952\pi$
$60$	0	0
$61$	3650.37	0.125606	0.0628032	−	0.998026i	$-0.479996\pi$
$61$	3650.37	0.125606	0.0628032	+	0.998026i	$0.479996\pi$
$62$	−13895.9	−0.459102
$63$	−11301.8	−0.358753
$64$	−13796.7	−0.421042
$65$	0	0
$66$	3571.46	0.100922
$67$	15803.8	0.430106	0.215053	−	0.976602i	$-0.431008\pi$
$67$	15803.8	0.430106	0.215053	+	0.976602i	$0.431008\pi$
$68$	−10859.3	−0.284792
$69$	−49947.1	−1.26295
$70$	0	0
$71$	−37247.8	−0.876909	−0.438455	−	0.898753i	$-0.644474\pi$
$71$	−37247.8	−0.876909	−0.438455	+	0.898753i	$0.644474\pi$
$72$	11228.5	0.255264
$73$	29323.4	0.644033	0.322016	−	0.946734i	$-0.395639\pi$
$73$	29323.4	0.644033	0.322016	+	0.946734i	$0.395639\pi$
$74$	12978.1	0.275506
$75$	0	0
$76$	8131.75	0.161492
$77$	18272.4	0.351211
$78$	−3704.84	−0.0689497
$79$	−44954.4	−0.810409	−0.405204	−	0.914226i	$-0.632800\pi$
$79$	−44954.4	−0.810409	−0.405204	+	0.914226i	$0.632800\pi$
$80$	0	0
$81$	−24409.0	−0.413368
$82$	7657.75	0.125767
$83$	−23704.5	−0.377690	−0.188845	−	0.982007i	$-0.560474\pi$
$83$	−23704.5	−0.377690	−0.188845	+	0.982007i	$0.560474\pi$
$84$	−38284.3	−0.592001
$85$	0	0
$86$	−39530.3	−0.576347
$87$	74890.4	1.06079
$88$	−18153.9	−0.249898
$89$	561.767	0.00751763	0.00375882	−	0.999993i	$-0.498804\pi$
$89$	561.767	0.00751763	0.00375882	+	0.999993i	$0.498804\pi$
$90$	0	0
$91$	−18954.7	−0.239947
$92$	119866.	1.47648
$93$	−90158.8	−1.08094
$94$	−41701.3	−0.486777
$95$	0	0
$96$	58114.0	0.643580
$97$	31260.4	0.337338	0.168669	−	0.985673i	$-0.446053\pi$
$97$	31260.4	0.337338	0.168669	+	0.985673i	$0.446053\pi$
$98$	−7770.82	−0.0817338
$99$	−16416.4	−0.168341
$100$	0	0
$101$	104822.	1.02246	0.511231	−	0.859443i	$-0.329190\pi$
$101$	104822.	1.02246	0.511231	+	0.859443i	$0.329190\pi$
$102$	8317.54	0.0791579
$103$	−17867.6	−0.165948	−0.0829742	−	0.996552i	$-0.526442\pi$
$103$	−17867.6	−0.165948	−0.0829742	+	0.996552i	$0.526442\pi$
$104$	18831.8	0.170730
$105$	0	0
$106$	−8775.35	−0.0758577
$107$	13003.9	0.109803	0.0549015	−	0.998492i	$-0.482516\pi$
$107$	13003.9	0.109803	0.0549015	+	0.998492i	$0.482516\pi$
$108$	117342.	0.968039
$109$	−205680.	−1.65816	−0.829080	−	0.559130i	$-0.811135\pi$
$109$	−205680.	−1.65816	−0.829080	+	0.559130i	$0.811135\pi$
$110$	0	0
$111$	84203.5	0.648668
$112$	79750.3	0.600741
$113$	−119392.	−0.879585	−0.439792	−	0.898100i	$-0.644948\pi$
$113$	−119392.	−0.879585	−0.439792	+	0.898100i	$0.644948\pi$
$114$	−6228.43	−0.0448866
$115$	0	0
$116$	−179727.	−1.24013
$117$	17029.5	0.115010
$118$	−88803.7	−0.587119
$119$	42554.3	0.275471
$120$	0	0
$121$	−134509.	−0.835198
$122$	6709.92	0.0408148
$123$	49684.5	0.296114
$124$	216369.	1.26369
$125$	0	0
$126$	−20774.4	−0.116574
$127$	213727.	1.17584	0.587922	−	0.808918i	$-0.299946\pi$
$127$	213727.	1.17584	0.587922	+	0.808918i	$0.299946\pi$
$128$	−181290.	−0.978024
$129$	−256478.	−1.35699
$130$	0	0
$131$	176742.	0.899833	0.449916	−	0.893071i	$-0.351454\pi$
$131$	176742.	0.899833	0.449916	+	0.893071i	$0.351454\pi$
$132$	−55609.9	−0.277791
$133$	−31866.0	−0.156206
$134$	29049.8	0.139759
$135$	0	0
$136$	−42278.4	−0.196007
$137$	−133457.	−0.607493	−0.303746	−	0.952753i	$-0.598238\pi$
$137$	−133457.	−0.607493	−0.303746	+	0.952753i	$0.598238\pi$
$138$	−91810.3	−0.410387
$139$	−179302.	−0.787132	−0.393566	−	0.919296i	$-0.628759\pi$
$139$	−179302.	−0.787132	−0.393566	+	0.919296i	$0.628759\pi$
$140$	0	0
$141$	−270564.	−1.14610
$142$	−68467.0	−0.284945
$143$	−27532.8	−0.112593
$144$	−71650.0	−0.287945
$145$	0	0
$146$	53900.9	0.209273
$147$	−50418.2	−0.192439
$148$	−202077.	−0.758337
$149$	69993.7	0.258281	0.129141	−	0.991626i	$-0.458778\pi$
$149$	69993.7	0.258281	0.129141	+	0.991626i	$0.458778\pi$
$150$	0	0
$151$	−360861.	−1.28795	−0.643973	−	0.765048i	$-0.722715\pi$
$151$	−360861.	−1.28795	−0.643973	+	0.765048i	$0.722715\pi$
$152$	31659.3	0.111146
$153$	−38232.0	−0.132038
$154$	33587.3	0.114123
$155$	0	0
$156$	57686.7	0.189786
$157$	74024.2	0.239676	0.119838	−	0.992793i	$-0.461762\pi$
$157$	74024.2	0.239676	0.119838	+	0.992793i	$0.461762\pi$
$158$	−82632.9	−0.263336
$159$	−56935.7	−0.178604
$160$	0	0
$161$	−469722.	−1.42816
$162$	−44867.3	−0.134321
$163$	395684.	1.16649	0.583243	−	0.812298i	$-0.301783\pi$
$163$	395684.	1.16649	0.583243	+	0.812298i	$0.301783\pi$
$164$	−119236.	−0.346177
$165$	0	0
$166$	−43572.4	−0.122727
$167$	−243331.	−0.675160	−0.337580	−	0.941297i	$-0.609608\pi$
$167$	−243331.	−0.675160	−0.337580	+	0.941297i	$0.609608\pi$
$168$	−149052.	−0.407441
$169$	28561.0	0.0769231
$170$	0	0
$171$	28629.3	0.0748723
$172$	615512.	1.58641
$173$	12828.9	0.0325892	0.0162946	−	0.999867i	$-0.494813\pi$
$173$	12828.9	0.0325892	0.0162946	+	0.999867i	$0.494813\pi$
$174$	137660.	0.344694
$175$	0	0
$176$	115842.	0.281892
$177$	−576171.	−1.38235
$178$	1032.61	0.00244280
$179$	643719.	1.50163	0.750817	−	0.660510i	$-0.229660\pi$
$179$	643719.	1.50163	0.750817	+	0.660510i	$0.229660\pi$
$180$	0	0
$181$	598308.	1.35746	0.678732	−	0.734386i	$-0.262530\pi$
$181$	598308.	1.35746	0.678732	+	0.734386i	$0.262530\pi$
$182$	−34841.7	−0.0779688
$183$	43534.9	0.0960970
$184$	466675.	1.01618
$185$	0	0
$186$	−165726.	−0.351242
$187$	61812.4	0.129262
$188$	649317.	1.33987
$189$	−459829.	−0.936357
$190$	0	0
$191$	364186.	0.722337	0.361169	−	0.932501i	$-0.382378\pi$
$191$	364186.	0.722337	0.361169	+	0.932501i	$0.382378\pi$
$192$	−164542.	−0.322124
$193$	533579.	1.03111	0.515555	−	0.856856i	$-0.327586\pi$
$193$	533579.	1.03111	0.515555	+	0.856856i	$0.327586\pi$
$194$	57461.4	0.109615
$195$	0	0
$196$	120997.	0.224975
$197$	−295693.	−0.542845	−0.271422	−	0.962460i	$-0.587494\pi$
$197$	−295693.	−0.542845	−0.271422	+	0.962460i	$0.587494\pi$
$198$	−30175.8	−0.0547012
$199$	−581653.	−1.04119	−0.520597	−	0.853803i	$-0.674290\pi$
$199$	−581653.	−1.04119	−0.520597	+	0.853803i	$0.674290\pi$
$200$	0	0
$201$	188479.	0.329059
$202$	192678.	0.332241
$203$	704297.	1.19954
$204$	−129509.	−0.217884
$205$	0	0
$206$	−32843.3	−0.0539236
$207$	422012.	0.684539
$208$	−120168.	−0.192588
$209$	−46287.1	−0.0732983
$210$	0	0
$211$	452629.	0.699900	0.349950	−	0.936768i	$-0.386198\pi$
$211$	452629.	0.699900	0.349950	+	0.936768i	$0.386198\pi$
$212$	136638.	0.208801
$213$	−444224.	−0.670892
$214$	23903.1	0.0356796
$215$	0	0
$216$	456846.	0.666248
$217$	−847888.	−1.22233
$218$	−378071.	−0.538806
$219$	349717.	0.492727
$220$	0	0
$221$	−64120.8	−0.0883117
$222$	154779.	0.210780
$223$	−430237.	−0.579356	−0.289678	−	0.957124i	$-0.593548\pi$
$223$	−430237.	−0.579356	−0.289678	+	0.957124i	$0.593548\pi$
$224$	546526.	0.727764
$225$	0	0
$226$	−219460.	−0.285814
$227$	−694207.	−0.894178	−0.447089	−	0.894489i	$-0.647539\pi$
$227$	−694207.	−0.894178	−0.447089	+	0.894489i	$0.647539\pi$
$228$	96980.7	0.123552
$229$	1.29181e6	1.62784	0.813920	−	0.580977i	$-0.197330\pi$
$229$	1.29181e6	1.62784	0.813920	+	0.580977i	$0.197330\pi$
$230$	0	0
$231$	217919.	0.268699
$232$	−699730.	−0.853514
$233$	604845.	0.729885	0.364942	−	0.931030i	$-0.381089\pi$
$233$	604845.	0.729885	0.364942	+	0.931030i	$0.381089\pi$
$234$	31302.8	0.0373717
$235$	0	0
$236$	1.38273e6	1.61606
$237$	−536134.	−0.620015
$238$	78221.3	0.0895122
$239$	−182757.	−0.206957	−0.103478	−	0.994632i	$-0.532997\pi$
$239$	−182757.	−0.206957	−0.103478	+	0.994632i	$0.532997\pi$
$240$	0	0
$241$	1.34821e6	1.49525	0.747627	−	0.664118i	$-0.231193\pi$
$241$	1.34821e6	1.49525	0.747627	+	0.664118i	$0.231193\pi$
$242$	−247249.	−0.271391
$243$	705150.	0.766065
$244$	−104478.	−0.112344
$245$	0	0
$246$	91327.6	0.0962198
$247$	48015.7	0.0500773
$248$	842389.	0.869728
$249$	−282704.	−0.288957
$250$	0	0
$251$	−597169.	−0.598291	−0.299146	−	0.954207i	$-0.596702\pi$
$251$	−597169.	−0.598291	−0.299146	+	0.954207i	$0.596702\pi$
$252$	323470.	0.320873
$253$	−682295.	−0.670149
$254$	392862.	0.382081
$255$	0	0
$256$	108255.	0.103240
$257$	230980.	0.218144	0.109072	−	0.994034i	$-0.465212\pi$
$257$	230980.	0.218144	0.109072	+	0.994034i	$0.465212\pi$
$258$	−471445.	−0.440943
$259$	791881.	0.733518
$260$	0	0
$261$	−632761.	−0.574961
$262$	324879.	0.292394
$263$	1.89928e6	1.69317	0.846585	−	0.532254i	$-0.178655\pi$
$263$	1.89928e6	1.69317	0.846585	+	0.532254i	$0.178655\pi$
$264$	−216506.	−0.191188
$265$	0	0
$266$	−58574.5	−0.0507580
$267$	6699.73	0.00575148
$268$	−452324.	−0.384692
$269$	2.30772e6	1.94448	0.972239	−	0.233991i	$-0.0751785\pi$
$269$	2.30772e6	1.94448	0.972239	+	0.233991i	$0.0751785\pi$
$270$	0	0
$271$	−2.18852e6	−1.81020	−0.905101	−	0.425197i	$-0.860205\pi$
$271$	−2.18852e6	−1.81020	−0.905101	+	0.425197i	$0.860205\pi$
$272$	269782.	0.221101
$273$	−226058.	−0.183575
$274$	−245315.	−0.197400
$275$	0	0
$276$	1.42955e6	1.12960
$277$	24533.4	0.0192114	0.00960569	−	0.999954i	$-0.496942\pi$
$277$	24533.4	0.0192114	0.00960569	+	0.999954i	$0.496942\pi$
$278$	−329583.	−0.255772
$279$	761767.	0.585884
$280$	0	0
$281$	−237327.	−0.179300	−0.0896501	−	0.995973i	$-0.528575\pi$
$281$	−237327.	−0.179300	−0.0896501	+	0.995973i	$0.528575\pi$
$282$	−497338.	−0.372416
$283$	−1.21567e6	−0.902297	−0.451148	−	0.892449i	$-0.648985\pi$
$283$	−1.21567e6	−0.902297	−0.451148	+	0.892449i	$0.648985\pi$
$284$	1.06608e6	0.784318
$285$	0	0
$286$	−50609.4	−0.0365861
$287$	467252.	0.334847
$288$	−491015.	−0.348830
$289$	−1.27590e6	−0.898613
$290$	0	0
$291$	372818.	0.258086
$292$	−839272.	−0.576031
$293$	2.18776e6	1.48878	0.744391	−	0.667744i	$-0.232740\pi$
$293$	2.18776e6	1.48878	0.744391	+	0.667744i	$0.232740\pi$
$294$	−92676.2	−0.0625316
$295$	0	0
$296$	−786746.	−0.521921
$297$	−667925.	−0.439376
$298$	128659.	0.0839265
$299$	707776.	0.457844
$300$	0	0
$301$	−2.41202e6	−1.53449
$302$	−663317.	−0.418508
$303$	1.25012e6	0.782249
$304$	−202021.	−0.125376
$305$	0	0
$306$	−70276.3	−0.0429047
$307$	460260.	0.278713	0.139357	−	0.990242i	$-0.455497\pi$
$307$	460260.	0.278713	0.139357	+	0.990242i	$0.455497\pi$
$308$	−522977.	−0.314127
$309$	−213092.	−0.126961
$310$	0	0
$311$	−1.54256e6	−0.904362	−0.452181	−	0.891926i	$-0.649354\pi$
$311$	−1.54256e6	−0.904362	−0.452181	+	0.891926i	$0.649354\pi$
$312$	224592.	0.130619
$313$	1.20176e6	0.693357	0.346679	−	0.937984i	$-0.387309\pi$
$313$	1.20176e6	0.693357	0.346679	+	0.937984i	$0.387309\pi$
$314$	136068.	0.0778808
$315$	0	0
$316$	1.28665e6	0.724840
$317$	512204.	0.286282	0.143141	−	0.989702i	$-0.454280\pi$
$317$	512204.	0.286282	0.143141	+	0.989702i	$0.454280\pi$
$318$	−104656.	−0.0580361
$319$	1.02303e6	0.562874
$320$	0	0
$321$	155087.	0.0840064
$322$	−863419.	−0.464068
$323$	−107797.	−0.0574913
$324$	698614.	0.369721
$325$	0	0
$326$	727327.	0.379041
$327$	−2.45298e6	−1.26860
$328$	−464222.	−0.238254
$329$	−2.54449e6	−1.29602
$330$	0	0
$331$	−135150.	−0.0678026	−0.0339013	−	0.999425i	$-0.510793\pi$
$331$	−135150.	−0.0678026	−0.0339013	+	0.999425i	$0.510793\pi$
$332$	678451.	0.337811
$333$	−711449.	−0.351587
$334$	−447279.	−0.219388
$335$	0	0
$336$	951117.	0.459606
$337$	−1.40146e6	−0.672214	−0.336107	−	0.941824i	$-0.609110\pi$
$337$	−1.40146e6	−0.672214	−0.336107	+	0.941824i	$0.609110\pi$
$338$	52499.4	0.0249956
$339$	−1.42389e6	−0.672939
$340$	0	0
$341$	−1.23160e6	−0.573567
$342$	52625.0	0.0243292
$343$	−2.35920e6	−1.08275
$344$	2.39637e6	1.09184
$345$	0	0
$346$	23581.4	0.0105896
$347$	−59106.2	−0.0263517	−0.0131759	−	0.999913i	$-0.504194\pi$
$347$	−59106.2	−0.0263517	−0.0131759	+	0.999913i	$0.504194\pi$
$348$	−2.14345e6	−0.948781
$349$	2.58914e6	1.13787	0.568934	−	0.822383i	$-0.307356\pi$
$349$	2.58914e6	1.13787	0.568934	+	0.822383i	$0.307356\pi$
$350$	0	0
$351$	692869.	0.300181
$352$	793858.	0.341496
$353$	1.08860e6	0.464978	0.232489	−	0.972599i	$-0.425313\pi$
$353$	1.08860e6	0.464978	0.232489	+	0.972599i	$0.425313\pi$
$354$	−1.05909e6	−0.449184
$355$	0	0
$356$	−16078.4	−0.00672387
$357$	507511.	0.210753
$358$	1.18325e6	0.487944
$359$	−3.81776e6	−1.56341	−0.781705	−	0.623648i	$-0.785650\pi$
$359$	−3.81776e6	−1.56341	−0.781705	+	0.623648i	$0.785650\pi$
$360$	0	0
$361$	−2.39538e6	−0.967399
$362$	1.09978e6	0.441097
$363$	−1.60418e6	−0.638980
$364$	542508.	0.214611
$365$	0	0
$366$	80023.7	0.0312260
$367$	4.74165e6	1.83766	0.918828	−	0.394658i	$-0.129137\pi$
$367$	4.74165e6	1.83766	0.918828	+	0.394658i	$0.129137\pi$
$368$	−2.97790e6	−1.14628
$369$	−419793.	−0.160498
$370$	0	0
$371$	−535445.	−0.201967
$372$	2.58045e6	0.966805
$373$	705178.	0.262438	0.131219	−	0.991353i	$-0.458111\pi$
$373$	705178.	0.262438	0.131219	+	0.991353i	$0.458111\pi$
$374$	113621.	0.0420028
$375$	0	0
$376$	2.52798e6	0.922157
$377$	−1.06123e6	−0.384555
$378$	−845234.	−0.304262
$379$	615406.	0.220071	0.110036	−	0.993928i	$-0.464903\pi$
$379$	615406.	0.220071	0.110036	+	0.993928i	$0.464903\pi$
$380$	0	0
$381$	2.54894e6	0.899597
$382$	669429.	0.234718
$383$	3.77847e6	1.31619	0.658096	−	0.752934i	$-0.271362\pi$
$383$	3.77847e6	1.31619	0.658096	+	0.752934i	$0.271362\pi$
$384$	−2.16210e6	−0.748252
$385$	0	0
$386$	980798.	0.335051
$387$	2.16703e6	0.735507
$388$	−894712.	−0.301720
$389$	−1.92639e6	−0.645461	−0.322730	−	0.946491i	$-0.604601\pi$
$389$	−1.92639e6	−0.645461	−0.322730	+	0.946491i	$0.604601\pi$
$390$	0	0
$391$	−1.58899e6	−0.525630
$392$	471076.	0.154838
$393$	2.10786e6	0.688430
$394$	−543528.	−0.176393
$395$	0	0
$396$	469858.	0.150566
$397$	3.30989e6	1.05399	0.526996	−	0.849868i	$-0.323318\pi$
$397$	3.30989e6	1.05399	0.526996	+	0.849868i	$0.323318\pi$
$398$	−1.06917e6	−0.338328
$399$	−380040.	−0.119508
$400$	0	0
$401$	2.63190e6	0.817350	0.408675	−	0.912680i	$-0.365991\pi$
$401$	2.63190e6	0.817350	0.408675	+	0.912680i	$0.365991\pi$
$402$	346453.	0.106925
$403$	1.27760e6	0.391860
$404$	−3.00012e6	−0.914502
$405$	0	0
$406$	1.29460e6	0.389782
$407$	1.15025e6	0.344196
$408$	−504219.	−0.149958
$409$	575874.	0.170223	0.0851116	−	0.996371i	$-0.472875\pi$
$409$	575874.	0.170223	0.0851116	+	0.996371i	$0.472875\pi$
$410$	0	0
$411$	−1.59164e6	−0.464771
$412$	511392.	0.148426
$413$	−5.41853e6	−1.56317
$414$	775721.	0.222436
$415$	0	0
$416$	−823505.	−0.233310
$417$	−2.13838e6	−0.602207
$418$	−85082.6	−0.0238177
$419$	−5.72386e6	−1.59277	−0.796386	−	0.604788i	$-0.793258\pi$
$419$	−5.72386e6	−1.59277	−0.796386	+	0.604788i	$0.793258\pi$
$420$	0	0
$421$	−6.93158e6	−1.90602	−0.953009	−	0.302942i	$-0.902031\pi$
$421$	−6.93158e6	−1.90602	−0.953009	+	0.302942i	$0.902031\pi$
$422$	832000.	0.227427
$423$	2.28604e6	0.621202
$424$	531972.	0.143706
$425$	0	0
$426$	−816550.	−0.218001
$427$	409419.	0.108667
$428$	−372187.	−0.0982092
$429$	−328361.	−0.0861406
$430$	0	0
$431$	−215036.	−0.0557594	−0.0278797	−	0.999611i	$-0.508876\pi$
$431$	−215036.	−0.0557594	−0.0278797	+	0.999611i	$0.508876\pi$
$432$	−2.91518e6	−0.751547
$433$	−5.17858e6	−1.32737	−0.663684	−	0.748013i	$-0.731008\pi$
$433$	−5.17858e6	−1.32737	−0.663684	+	0.748013i	$0.731008\pi$
$434$	−1.55855e6	−0.397187
$435$	0	0
$436$	5.88682e6	1.48308
$437$	1.18989e6	0.298059
$438$	642832.	0.160108
$439$	−7.60133e6	−1.88247	−0.941235	−	0.337753i	$-0.890333\pi$
$439$	−7.60133e6	−1.88247	−0.941235	+	0.337753i	$0.890333\pi$
$440$	0	0
$441$	425991.	0.104305
$442$	−117864.	−0.0286962
$443$	7.13928e6	1.72840	0.864201	−	0.503146i	$-0.167824\pi$
$443$	7.13928e6	1.72840	0.864201	+	0.503146i	$0.167824\pi$
$444$	−2.41000e6	−0.580177
$445$	0	0
$446$	−790840.	−0.188257
$447$	834757.	0.197602
$448$	−1.54741e6	−0.364260
$449$	−848355.	−0.198592	−0.0992960	−	0.995058i	$-0.531659\pi$
$449$	−848355.	−0.198592	−0.0992960	+	0.995058i	$0.531659\pi$
$450$	0	0
$451$	678708.	0.157124
$452$	3.41713e6	0.786712
$453$	−4.30369e6	−0.985362
$454$	−1.27606e6	−0.290556
$455$	0	0
$456$	377575.	0.0850337
$457$	−3.35404e6	−0.751238	−0.375619	−	0.926774i	$-0.622570\pi$
$457$	−3.35404e6	−0.751238	−0.375619	+	0.926774i	$0.622570\pi$
$458$	2.37455e6	0.528954
$459$	−1.55552e6	−0.344624
$460$	0	0
$461$	−3.57354e6	−0.783152	−0.391576	−	0.920146i	$-0.628070\pi$
$461$	−3.57354e6	−0.783152	−0.391576	+	0.920146i	$0.628070\pi$
$462$	400569.	0.0873116
$463$	1.44187e6	0.312590	0.156295	−	0.987710i	$-0.450045\pi$
$463$	1.44187e6	0.312590	0.156295	+	0.987710i	$0.450045\pi$
$464$	4.46505e6	0.962789
$465$	0	0
$466$	1.11180e6	0.237170
$467$	−7.03571e6	−1.49285	−0.746424	−	0.665471i	$-0.768231\pi$
$467$	−7.03571e6	−1.49285	−0.746424	+	0.665471i	$0.768231\pi$
$468$	−487405.	−0.102867
$469$	1.77253e6	0.372101
$470$	0	0
$471$	882825.	0.183368
$472$	5.38339e6	1.11225
$473$	−3.50358e6	−0.720044
$474$	−985494.	−0.201469
$475$	0	0
$476$	−1.21796e6	−0.246385
$477$	481059.	0.0968061
$478$	−335935.	−0.0672489
$479$	−1.40107e6	−0.279010	−0.139505	−	0.990221i	$-0.544551\pi$
$479$	−1.40107e6	−0.279010	−0.139505	+	0.990221i	$0.544551\pi$
$480$	0	0
$481$	−1.19321e6	−0.235154
$482$	2.47821e6	0.485871
$483$	−5.60198e6	−1.09263
$484$	3.84982e6	0.747011
$485$	0	0
$486$	1.29617e6	0.248927
$487$	−8.45449e6	−1.61534	−0.807672	−	0.589632i	$-0.799273\pi$
$487$	−8.45449e6	−1.61534	−0.807672	+	0.589632i	$0.799273\pi$
$488$	−406764.	−0.0773201
$489$	4.71900e6	0.892438
$490$	0	0
$491$	−7.81764e6	−1.46343	−0.731715	−	0.681611i	$-0.761280\pi$
$491$	−7.81764e6	−1.46343	−0.731715	+	0.681611i	$0.761280\pi$
$492$	−1.42203e6	−0.264848
$493$	2.38252e6	0.441489
$494$	88260.0	0.0162722
$495$	0	0
$496$	−5.37537e6	−0.981079
$497$	−4.17765e6	−0.758649
$498$	−519652.	−0.0938944
$499$	−6.37828e6	−1.14671	−0.573353	−	0.819308i	$-0.694358\pi$
$499$	−6.37828e6	−1.14671	−0.573353	+	0.819308i	$0.694358\pi$
$500$	0	0
$501$	−2.90201e6	−0.516541
$502$	−1.09769e6	−0.194410
$503$	−2.99032e6	−0.526985	−0.263492	−	0.964661i	$-0.584874\pi$
$503$	−2.99032e6	−0.526985	−0.263492	+	0.964661i	$0.584874\pi$
$504$	1.25937e6	0.220839
$505$	0	0
$506$	−1.25416e6	−0.217760
$507$	340624.	0.0588511
$508$	−6.11712e6	−1.05169
$509$	−5.75011e6	−0.983743	−0.491871	−	0.870668i	$-0.663687\pi$
$509$	−5.75011e6	−0.983743	−0.491871	+	0.870668i	$0.663687\pi$
$510$	0	0
$511$	3.28887e6	0.557178
$512$	6.00028e6	1.01157
$513$	1.16483e6	0.195419
$514$	424577.	0.0708840
$515$	0	0
$516$	7.34071e6	1.21371
$517$	−3.69600e6	−0.608143
$518$	1.45560e6	0.238351
$519$	153000.	0.0249329
$520$	0	0
$521$	−1.06342e7	−1.71638	−0.858188	−	0.513336i	$-0.828409\pi$
$521$	−1.06342e7	−1.71638	−0.858188	+	0.513336i	$0.828409\pi$
$522$	−1.16311e6	−0.186829
$523$	−5.82469e6	−0.931148	−0.465574	−	0.885009i	$-0.654152\pi$
$523$	−5.82469e6	−0.931148	−0.465574	+	0.885009i	$0.654152\pi$
$524$	−5.05857e6	−0.804822
$525$	0	0
$526$	3.49117e6	0.550182
$527$	−2.86827e6	−0.449876
$528$	1.38155e6	0.215666
$529$	1.11032e7	1.72508
$530$	0	0
$531$	4.86816e6	0.749253
$532$	912043.	0.139713
$533$	−704055.	−0.107347
$534$	12315.1	0.00186890
$535$	0	0
$536$	−1.76103e6	−0.264762
$537$	7.67711e6	1.14885
$538$	4.24194e6	0.631843
$539$	−688730.	−0.102112
$540$	0	0
$541$	249946.	0.0367158	0.0183579	−	0.999831i	$-0.494156\pi$
$541$	249946.	0.0367158	0.0183579	+	0.999831i	$0.494156\pi$
$542$	−4.02283e6	−0.588211
$543$	7.13552e6	1.03855
$544$	1.84881e6	0.267852
$545$	0	0
$546$	−415528.	−0.0596512
$547$	9.11755e6	1.30290	0.651448	−	0.758693i	$-0.274162\pi$
$547$	9.11755e6	1.30290	0.651448	+	0.758693i	$0.274162\pi$
$548$	3.81971e6	0.543349
$549$	−367834.	−0.0520860
$550$	0	0
$551$	−1.78411e6	−0.250347
$552$	5.56565e6	0.777443
$553$	−5.04200e6	−0.701117
$554$	45096.1	0.00624259
$555$	0	0
$556$	5.13183e6	0.704020
$557$	−6.02542e6	−0.822905	−0.411452	−	0.911431i	$-0.634978\pi$
$557$	−6.02542e6	−0.822905	−0.411452	+	0.911431i	$0.634978\pi$
$558$	1.40024e6	0.190378
$559$	3.63442e6	0.491933
$560$	0	0
$561$	737186.	0.0988940
$562$	−436242.	−0.0582622
$563$	−1.08154e6	−0.143804	−0.0719021	−	0.997412i	$-0.522907\pi$
$563$	−1.08154e6	−0.143804	−0.0719021	+	0.997412i	$0.522907\pi$
$564$	7.74387e6	1.02509
$565$	0	0
$566$	−2.23458e6	−0.293194
$567$	−2.73767e6	−0.357621
$568$	4.15055e6	0.539803
$569$	745292.	0.0965041	0.0482520	−	0.998835i	$-0.484635\pi$
$569$	745292.	0.0965041	0.0482520	+	0.998835i	$0.484635\pi$
$570$	0	0
$571$	4.58349e6	0.588309	0.294155	−	0.955758i	$-0.404962\pi$
$571$	4.58349e6	0.588309	0.294155	+	0.955758i	$0.404962\pi$
$572$	788021.	0.100704
$573$	4.34335e6	0.552635
$574$	858880.	0.108806
$575$	0	0
$576$	1.39024e6	0.174596
$577$	1.17307e7	1.46685	0.733424	−	0.679772i	$-0.237921\pi$
$577$	1.17307e7	1.46685	0.733424	+	0.679772i	$0.237921\pi$
$578$	−2.34530e6	−0.291997
$579$	6.36355e6	0.788866
$580$	0	0
$581$	−2.65865e6	−0.326754
$582$	685295.	0.0838629
$583$	−777762.	−0.0947710
$584$	−3.26754e6	−0.396450
$585$	0	0
$586$	4.02144e6	0.483768
$587$	−8.72678e6	−1.04534	−0.522672	−	0.852534i	$-0.675065\pi$
$587$	−8.72678e6	−1.04534	−0.522672	+	0.852534i	$0.675065\pi$
$588$	1.44303e6	0.172120
$589$	2.14785e6	0.255103
$590$	0	0
$591$	−3.52649e6	−0.415311
$592$	5.02030e6	0.588743
$593$	500982.	0.0585040	0.0292520	−	0.999572i	$-0.490687\pi$
$593$	500982.	0.0585040	0.0292520	+	0.999572i	$0.490687\pi$
$594$	−1.22775e6	−0.142772
$595$	0	0
$596$	−2.00330e6	−0.231010
$597$	−6.93690e6	−0.796580
$598$	1.30100e6	0.148773
$599$	−1.04449e6	−0.118943	−0.0594714	−	0.998230i	$-0.518942\pi$
$599$	−1.04449e6	−0.118943	−0.0594714	+	0.998230i	$0.518942\pi$
$600$	0	0
$601$	3.03678e6	0.342948	0.171474	−	0.985189i	$-0.445147\pi$
$601$	3.03678e6	0.342948	0.171474	+	0.985189i	$0.445147\pi$
$602$	−4.43365e6	−0.498620
$603$	−1.59249e6	−0.178354
$604$	1.03283e7	1.15195
$605$	0	0
$606$	2.29791e6	0.254186
$607$	809264.	0.0891494	0.0445747	−	0.999006i	$-0.485807\pi$
$607$	809264.	0.0891494	0.0445747	+	0.999006i	$0.485807\pi$
$608$	−1.38444e6	−0.151886
$609$	8.39958e6	0.917728
$610$	0	0
$611$	3.83403e6	0.415482
$612$	1.09425e6	0.118096
$613$	1.04363e7	1.12175	0.560875	−	0.827900i	$-0.310465\pi$
$613$	1.04363e7	1.12175	0.560875	+	0.827900i	$0.310465\pi$
$614$	846028.	0.0905657
$615$	0	0
$616$	−2.03611e6	−0.216196
$617$	−2.56343e6	−0.271086	−0.135543	−	0.990771i	$-0.543278\pi$
$617$	−2.56343e6	−0.271086	−0.135543	+	0.990771i	$0.543278\pi$
$618$	−391695.	−0.0412551
$619$	6.77494e6	0.710687	0.355344	−	0.934736i	$-0.384364\pi$
$619$	6.77494e6	0.710687	0.355344	+	0.934736i	$0.384364\pi$
$620$	0	0
$621$	1.71701e7	1.78667
$622$	−2.83547e6	−0.293865
$623$	63006.8	0.00650380
$624$	−1.43314e6	−0.147342
$625$	0	0
$626$	2.20902e6	0.225301
$627$	−552028.	−0.0560779
$628$	−2.11866e6	−0.214369
$629$	2.67881e6	0.269969
$630$	0	0
$631$	1.07855e7	1.07837	0.539185	−	0.842187i	$-0.318732\pi$
$631$	1.07855e7	1.07837	0.539185	+	0.842187i	$0.318732\pi$
$632$	5.00930e6	0.498867
$633$	5.39814e6	0.535469
$634$	941508.	0.0930252
$635$	0	0
$636$	1.62957e6	0.159746
$637$	714451.	0.0697628
$638$	1.88048e6	0.182902
$639$	3.75332e6	0.363633
$640$	0	0
$641$	329201.	0.0316458	0.0158229	−	0.999875i	$-0.494963\pi$
$641$	329201.	0.0316458	0.0158229	+	0.999875i	$0.494963\pi$
$642$	285073.	0.0272972
$643$	1.85214e7	1.76663	0.883317	−	0.468777i	$-0.155305\pi$
$643$	1.85214e7	1.76663	0.883317	+	0.468777i	$0.155305\pi$
$644$	1.34440e7	1.27736
$645$	0	0
$646$	−198148.	−0.0186814
$647$	4.85885e6	0.456324	0.228162	−	0.973623i	$-0.426728\pi$
$647$	4.85885e6	0.456324	0.228162	+	0.973623i	$0.426728\pi$
$648$	2.71991e6	0.254459
$649$	−7.87069e6	−0.733502
$650$	0	0
$651$	−1.01121e7	−0.935163
$652$	−1.13250e7	−1.04332
$653$	−4.56792e6	−0.419214	−0.209607	−	0.977786i	$-0.567218\pi$
$653$	−4.56792e6	−0.419214	−0.209607	+	0.977786i	$0.567218\pi$
$654$	−4.50895e6	−0.412222
$655$	0	0
$656$	2.96225e6	0.268758
$657$	−2.95481e6	−0.267065
$658$	−4.67715e6	−0.421130
$659$	−9.51262e6	−0.853270	−0.426635	−	0.904424i	$-0.640301\pi$
$659$	−9.51262e6	−0.853270	−0.426635	+	0.904424i	$0.640301\pi$
$660$	0	0
$661$	−1.91227e6	−0.170233	−0.0851167	−	0.996371i	$-0.527126\pi$
$661$	−1.91227e6	−0.170233	−0.0851167	+	0.996371i	$0.527126\pi$
$662$	−248426.	−0.0220319
$663$	−764716.	−0.0675642
$664$	2.64141e6	0.232496
$665$	0	0
$666$	−1.30775e6	−0.114246
$667$	−2.62987e7	−2.28886
$668$	6.96443e6	0.603872
$669$	−5.13108e6	−0.443245
$670$	0	0
$671$	594702.	0.0509910
$672$	6.51797e6	0.556787
$673$	5.38551e6	0.458341	0.229171	−	0.973386i	$-0.426399\pi$
$673$	5.38551e6	0.458341	0.229171	+	0.973386i	$0.426399\pi$
$674$	−2.57610e6	−0.218431
$675$	0	0
$676$	−817450.	−0.0688010
$677$	−1.65232e7	−1.38555	−0.692775	−	0.721154i	$-0.743612\pi$
$677$	−1.65232e7	−1.38555	−0.692775	+	0.721154i	$0.743612\pi$
$678$	−2.61732e6	−0.218666
$679$	3.50612e6	0.291845
$680$	0	0
$681$	−8.27923e6	−0.684104
$682$	−2.26387e6	−0.186376
$683$	8.83329e6	0.724554	0.362277	−	0.932070i	$-0.381999\pi$
$683$	8.83329e6	0.724554	0.362277	+	0.932070i	$0.381999\pi$
$684$	−819406.	−0.0669667
$685$	0	0
$686$	−4.33656e6	−0.351832
$687$	1.54064e7	1.24540
$688$	−1.52915e7	−1.23163
$689$	806808.	0.0647474
$690$	0	0
$691$	1.77549e7	1.41457	0.707284	−	0.706929i	$-0.249920\pi$
$691$	1.77549e7	1.41457	0.707284	+	0.706929i	$0.249920\pi$
$692$	−367179.	−0.0291482
$693$	−1.84124e6	−0.145639
$694$	−108646.	−0.00856279
$695$	0	0
$696$	−8.34510e6	−0.652993
$697$	1.58064e6	0.123240
$698$	4.75923e6	0.369741
$699$	7.21349e6	0.558409
$700$	0	0
$701$	1.32222e7	1.01627	0.508133	−	0.861279i	$-0.330336\pi$
$701$	1.32222e7	1.01627	0.508133	+	0.861279i	$0.330336\pi$
$702$	1.27360e6	0.0975415
$703$	−2.00597e6	−0.153086
$704$	−2.24770e6	−0.170925
$705$	0	0
$706$	2.00101e6	0.151091
$707$	1.17566e7	0.884572
$708$	1.64907e7	1.23639
$709$	2.51312e7	1.87758	0.938790	−	0.344490i	$-0.111948\pi$
$709$	2.51312e7	1.87758	0.938790	+	0.344490i	$0.111948\pi$
$710$	0	0
$711$	4.52988e6	0.336057
$712$	−62598.2	−0.00462766
$713$	3.16604e7	2.33234
$714$	932881.	0.0684827
$715$	0	0
$716$	−1.84240e7	−1.34308
$717$	−2.17959e6	−0.158335
$718$	−7.01763e6	−0.508018
$719$	2.01931e6	0.145674	0.0728369	−	0.997344i	$-0.476795\pi$
$719$	2.01931e6	0.145674	0.0728369	+	0.997344i	$0.476795\pi$
$720$	0	0
$721$	−2.00400e6	−0.143568
$722$	−4.40306e6	−0.314349
$723$	1.60790e7	1.14397
$724$	−1.71243e7	−1.21413
$725$	0	0
$726$	−2.94873e6	−0.207632
$727$	−2.25046e6	−0.157919	−0.0789597	−	0.996878i	$-0.525160\pi$
$727$	−2.25046e6	−0.157919	−0.0789597	+	0.996878i	$0.525160\pi$
$728$	2.11214e6	0.147705
$729$	1.43411e7	0.999458
$730$	0	0
$731$	−8.15946e6	−0.564765
$732$	−1.24602e6	−0.0859504
$733$	−5.24333e6	−0.360452	−0.180226	−	0.983625i	$-0.557683\pi$
$733$	−5.24333e6	−0.360452	−0.180226	+	0.983625i	$0.557683\pi$
$734$	8.71586e6	0.597132
$735$	0	0
$736$	−2.04074e7	−1.38865
$737$	2.57469e6	0.174605
$738$	−771642.	−0.0521525
$739$	−1.56695e7	−1.05547	−0.527733	−	0.849411i	$-0.676958\pi$
$739$	−1.56695e7	−1.05547	−0.527733	+	0.849411i	$0.676958\pi$
$740$	0	0
$741$	572643.	0.0383123
$742$	−984228.	−0.0656275
$743$	−6.91028e6	−0.459223	−0.229611	−	0.973282i	$-0.573746\pi$
$743$	−6.91028e6	−0.459223	−0.229611	+	0.973282i	$0.573746\pi$
$744$	1.00465e7	0.665398
$745$	0	0
$746$	1.29622e6	0.0852772
$747$	2.38861e6	0.156619
$748$	−1.76915e6	−0.115614
$749$	1.45850e6	0.0949949
$750$	0	0
$751$	9.35145e6	0.605033	0.302517	−	0.953144i	$-0.402173\pi$
$751$	9.35145e6	0.605033	0.302517	+	0.953144i	$0.402173\pi$
$752$	−1.61313e7	−1.04022
$753$	−7.12194e6	−0.457731
$754$	−1.95071e6	−0.124958
$755$	0	0
$756$	1.31608e7	0.837489
$757$	−1.84763e7	−1.17186	−0.585929	−	0.810362i	$-0.699270\pi$
$757$	−1.84763e7	−1.17186	−0.585929	+	0.810362i	$0.699270\pi$
$758$	1.13121e6	0.0715105
$759$	−8.13718e6	−0.512707
$760$	0	0
$761$	−1.84001e7	−1.15175	−0.575876	−	0.817537i	$-0.695339\pi$
$761$	−1.84001e7	−1.15175	−0.575876	+	0.817537i	$0.695339\pi$
$762$	4.68534e6	0.292317
$763$	−2.30688e7	−1.43454
$764$	−1.04234e7	−0.646067
$765$	0	0
$766$	6.94540e6	0.427686
$767$	8.16463e6	0.501127
$768$	1.29107e6	0.0789856
$769$	−1.14513e7	−0.698295	−0.349148	−	0.937068i	$-0.613529\pi$
$769$	−1.14513e7	−0.698295	−0.349148	+	0.937068i	$0.613529\pi$
$770$	0	0
$771$	2.75471e6	0.166894
$772$	−1.52717e7	−0.922238
$773$	−1.86929e7	−1.12519	−0.562597	−	0.826731i	$-0.690198\pi$
$773$	−1.86929e7	−1.12519	−0.562597	+	0.826731i	$0.690198\pi$
$774$	3.98332e6	0.238997
$775$	0	0
$776$	−3.48338e6	−0.207657
$777$	9.44412e6	0.561189
$778$	−3.54099e6	−0.209737
$779$	−1.18363e6	−0.0698831
$780$	0	0
$781$	−6.06825e6	−0.355989
$782$	−2.92081e6	−0.170799
$783$	−2.57448e7	−1.50067
$784$	−3.00599e6	−0.174661
$785$	0	0
$786$	3.87456e6	0.223700
$787$	2.53432e7	1.45856	0.729281	−	0.684214i	$-0.239855\pi$
$787$	2.53432e7	1.45856	0.729281	+	0.684214i	$0.239855\pi$
$788$	8.46309e6	0.485527
$789$	2.26512e7	1.29538
$790$	0	0
$791$	−1.33908e7	−0.760964
$792$	1.82930e6	0.103627
$793$	−616912.	−0.0348370
$794$	6.08408e6	0.342487
$795$	0	0
$796$	1.66476e7	0.931256
$797$	−2.38962e7	−1.33255	−0.666276	−	0.745706i	$-0.732113\pi$
$797$	−2.38962e7	−1.33255	−0.666276	+	0.745706i	$0.732113\pi$
$798$	−698570.	−0.0388332
$799$	−8.60758e6	−0.476995
$800$	0	0
$801$	−56607.1	−0.00311738
$802$	4.83783e6	0.265592
$803$	4.77725e6	0.261450
$804$	−5.39450e6	−0.294314
$805$	0	0
$806$	2.34841e6	0.127332
$807$	2.75223e7	1.48765
$808$	−1.16804e7	−0.629401
$809$	−1.73141e7	−0.930096	−0.465048	−	0.885285i	$-0.653963\pi$
$809$	−1.73141e7	−0.930096	−0.465048	+	0.885285i	$0.653963\pi$
$810$	0	0
$811$	2.18577e7	1.16695	0.583474	−	0.812131i	$-0.301693\pi$
$811$	2.18577e7	1.16695	0.583474	+	0.812131i	$0.301693\pi$
$812$	−2.01578e7	−1.07289
$813$	−2.61007e7	−1.38492
$814$	2.11433e6	0.111844
$815$	0	0
$816$	3.21747e6	0.169157
$817$	6.11006e6	0.320251
$818$	1.05854e6	0.0553127
$819$	1.91000e6	0.0995001
$820$	0	0
$821$	7.12188e6	0.368754	0.184377	−	0.982856i	$-0.440973\pi$
$821$	7.12188e6	0.368754	0.184377	+	0.982856i	$0.440973\pi$
$822$	−2.92567e6	−0.151024
$823$	−2.08032e7	−1.07061	−0.535303	−	0.844660i	$-0.679803\pi$
$823$	−2.08032e7	−1.07061	−0.535303	+	0.844660i	$0.679803\pi$
$824$	1.99100e6	0.102154
$825$	0	0
$826$	−9.96007e6	−0.507940
$827$	−1.96032e6	−0.0996698	−0.0498349	−	0.998757i	$-0.515870\pi$
$827$	−1.96032e6	−0.0996698	−0.0498349	+	0.998757i	$0.515870\pi$
$828$	−1.20785e7	−0.612261
$829$	1.66913e7	0.843535	0.421768	−	0.906704i	$-0.361410\pi$
$829$	1.66913e7	0.843535	0.421768	+	0.906704i	$0.361410\pi$
$830$	0	0
$831$	292590.	0.0146979
$832$	2.33164e6	0.116776
$833$	−1.60398e6	−0.0800914
$834$	−3.93067e6	−0.195682
$835$	0	0
$836$	1.32479e6	0.0655589
$837$	3.09936e7	1.52918
$838$	−1.05213e7	−0.517559
$839$	−2.47847e7	−1.21557	−0.607783	−	0.794103i	$-0.707941\pi$
$839$	−2.47847e7	−1.21557	−0.607783	+	0.794103i	$0.707941\pi$
$840$	0	0
$841$	1.89209e7	0.922470
$842$	−1.27413e7	−0.619346
$843$	−2.83040e6	−0.137176
$844$	−1.29548e7	−0.626000
$845$	0	0
$846$	4.20209e6	0.201855
$847$	−1.50863e7	−0.722563
$848$	−3.39457e6	−0.162104
$849$	−1.44983e7	−0.690315
$850$	0	0
$851$	−2.95691e7	−1.39963
$852$	1.27142e7	0.600054
$853$	1.16070e7	0.546193	0.273096	−	0.961987i	$-0.411952\pi$
$853$	1.16070e7	0.546193	0.273096	+	0.961987i	$0.411952\pi$
$854$	752573.	0.0353105
$855$	0	0
$856$	−1.44904e6	−0.0675919
$857$	−2.85621e7	−1.32843	−0.664214	−	0.747543i	$-0.731233\pi$
$857$	−2.85621e7	−1.32843	−0.664214	+	0.747543i	$0.731233\pi$
$858$	−603577.	−0.0279907
$859$	−2.72393e6	−0.125955	−0.0629773	−	0.998015i	$-0.520060\pi$
$859$	−2.72393e6	−0.125955	−0.0629773	+	0.998015i	$0.520060\pi$
$860$	0	0
$861$	5.57253e6	0.256180
$862$	−395269.	−0.0181186
$863$	4.18556e7	1.91305	0.956525	−	0.291651i	$-0.0942045\pi$
$863$	4.18556e7	1.91305	0.956525	+	0.291651i	$0.0942045\pi$
$864$	−1.99776e7	−0.910457
$865$	0	0
$866$	−9.51902e6	−0.431318
$867$	−1.52166e7	−0.687497
$868$	2.42676e7	1.09327
$869$	−7.32378e6	−0.328992
$870$	0	0
$871$	−2.67085e6	−0.119290
$872$	2.29191e7	1.02072
$873$	−3.15000e6	−0.139886
$874$	2.18719e6	0.0968519
$875$	0	0
$876$	−1.00093e7	−0.440701
$877$	3.60953e7	1.58472	0.792359	−	0.610055i	$-0.208853\pi$
$877$	3.60953e7	1.58472	0.792359	+	0.610055i	$0.208853\pi$
$878$	−1.39724e7	−0.611694
$879$	2.60917e7	1.13901
$880$	0	0
$881$	3.31393e7	1.43848	0.719240	−	0.694761i	$-0.244490\pi$
$881$	3.31393e7	1.43848	0.719240	+	0.694761i	$0.244490\pi$
$882$	783036.	0.0338930
$883$	−9.64225e6	−0.416176	−0.208088	−	0.978110i	$-0.566724\pi$
$883$	−9.64225e6	−0.416176	−0.208088	+	0.978110i	$0.566724\pi$
$884$	1.83522e6	0.0789871
$885$	0	0
$886$	1.31231e7	0.561631
$887$	3.59387e6	0.153375	0.0766873	−	0.997055i	$-0.475566\pi$
$887$	3.59387e6	0.153375	0.0766873	+	0.997055i	$0.475566\pi$
$888$	−9.38287e6	−0.399304
$889$	2.39712e7	1.01727
$890$	0	0
$891$	−3.97661e6	−0.167810
$892$	1.23139e7	0.518183
$893$	6.44563e6	0.270481
$894$	1.53441e6	0.0642092
$895$	0	0
$896$	−2.03332e7	−0.846128
$897$	8.44106e6	0.350281
$898$	−1.55940e6	−0.0645309
$899$	−4.74714e7	−1.95899
$900$	0	0
$901$	−1.81132e6	−0.0743334
$902$	1.24757e6	0.0510561
$903$	−2.87661e7	−1.17398
$904$	1.33039e7	0.541450
$905$	0	0
$906$	−7.91083e6	−0.320186
$907$	4.23884e7	1.71092	0.855458	−	0.517872i	$-0.173276\pi$
$907$	4.23884e7	1.71092	0.855458	+	0.517872i	$0.173276\pi$
$908$	1.98690e7	0.799764
$909$	−1.05625e7	−0.423990
$910$	0	0
$911$	−2.39933e7	−0.957841	−0.478921	−	0.877858i	$-0.658972\pi$
$911$	−2.39933e7	−0.957841	−0.478921	+	0.877858i	$0.658972\pi$
$912$	−2.40934e6	−0.0959205
$913$	−3.86184e6	−0.153326
$914$	−6.16522e6	−0.244109
$915$	0	0
$916$	−3.69733e7	−1.45596
$917$	1.98231e7	0.778481
$918$	−2.85929e6	−0.111983
$919$	−2.53438e7	−0.989883	−0.494941	−	0.868926i	$-0.664810\pi$
$919$	−2.53438e7	−0.989883	−0.494941	+	0.868926i	$0.664810\pi$
$920$	0	0
$921$	5.48915e6	0.213234
$922$	−6.56870e6	−0.254479
$923$	6.29488e6	0.243211
$924$	−6.23712e6	−0.240328
$925$	0	0
$926$	2.65038e6	0.101574
$927$	1.80045e6	0.0688148
$928$	3.05988e7	1.16636
$929$	−2.04548e7	−0.777601	−0.388800	−	0.921322i	$-0.627110\pi$
$929$	−2.04548e7	−0.777601	−0.388800	+	0.921322i	$0.627110\pi$
$930$	0	0
$931$	1.20111e6	0.0454159
$932$	−1.73114e7	−0.652818
$933$	−1.83969e7	−0.691896
$934$	−1.29327e7	−0.485089
$935$	0	0
$936$	−1.89761e6	−0.0707975
$937$	1.98261e7	0.737716	0.368858	−	0.929486i	$-0.379749\pi$
$937$	1.98261e7	0.737716	0.368858	+	0.929486i	$0.379749\pi$
$938$	3.25818e6	0.120911
$939$	1.43324e7	0.530463
$940$	0	0
$941$	−3.32679e7	−1.22476	−0.612380	−	0.790564i	$-0.709788\pi$
$941$	−3.32679e7	−1.22476	−0.612380	+	0.790564i	$0.709788\pi$
$942$	1.62277e6	0.0595839
$943$	−1.74473e7	−0.638925
$944$	−3.43519e7	−1.25465
$945$	0	0
$946$	−6.44011e6	−0.233973
$947$	4.75820e7	1.72412	0.862060	−	0.506806i	$-0.169174\pi$
$947$	4.75820e7	1.72412	0.862060	+	0.506806i	$0.169174\pi$
$948$	1.53448e7	0.554549
$949$	−4.95566e6	−0.178623
$950$	0	0
$951$	6.10863e6	0.219025
$952$	−4.74187e6	−0.169573
$953$	4.16316e7	1.48488	0.742440	−	0.669913i	$-0.233669\pi$
$953$	4.16316e7	1.48488	0.742440	+	0.669913i	$0.233669\pi$
$954$	884259.	0.0314564
$955$	0	0
$956$	5.23073e6	0.185105
$957$	1.22008e7	0.430635
$958$	−2.57537e6	−0.0906623
$959$	−1.49683e7	−0.525566
$960$	0	0
$961$	2.85206e7	0.996207
$962$	−2.19329e6	−0.0764115
$963$	−1.31035e6	−0.0455327
$964$	−3.85874e7	−1.33737
$965$	0	0
$966$	−1.02973e7	−0.355042
$967$	3.70062e7	1.27265	0.636324	−	0.771422i	$-0.280454\pi$
$967$	3.70062e7	1.27265	0.636324	+	0.771422i	$0.280454\pi$
$968$	1.49885e7	0.514127
$969$	−1.28561e6	−0.0439846
$970$	0	0
$971$	846545.	0.0288139	0.0144070	−	0.999896i	$-0.495414\pi$
$971$	846545.	0.0288139	0.0144070	+	0.999896i	$0.495414\pi$
$972$	−2.01822e7	−0.685179
$973$	−2.01102e7	−0.680979
$974$	−1.55406e7	−0.524894
$975$	0	0
$976$	2.59560e6	0.0872194
$977$	−5.86464e7	−1.96564	−0.982822	−	0.184554i	$-0.940916\pi$
$977$	−5.86464e7	−1.96564	−0.982822	+	0.184554i	$0.940916\pi$
$978$	8.67423e6	0.289991
$979$	91520.7	0.00305185
$980$	0	0
$981$	2.07256e7	0.687599
$982$	−1.43700e7	−0.475530
$983$	1.17394e7	0.387492	0.193746	−	0.981052i	$-0.437936\pi$
$983$	1.17394e7	0.387492	0.193746	+	0.981052i	$0.437936\pi$
$984$	−5.53639e6	−0.182280
$985$	0	0
$986$	4.37944e6	0.143458
$987$	−3.03460e7	−0.991536
$988$	−1.37427e6	−0.0447897
$989$	9.00654e7	2.92798
$990$	0	0
$991$	−2.34339e7	−0.757984	−0.378992	−	0.925400i	$-0.623729\pi$
$991$	−2.34339e7	−0.757984	−0.378992	+	0.925400i	$0.623729\pi$
$992$	−3.68372e7	−1.18852
$993$	−1.61182e6	−0.0518734
$994$	−7.67914e6	−0.246517
$995$	0	0
$996$	8.09133e6	0.258447
$997$	3.77262e7	1.20200	0.601001	−	0.799248i	$-0.294769\pi$
$997$	3.77262e7	1.20200	0.601001	+	0.799248i	$0.294769\pi$
$998$	−1.17242e7	−0.372613
$999$	−2.89463e7	−0.917655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.i.1.6	yes	9
5.2	odd	4		325.6.b.h.274.11		18
5.3	odd	4		325.6.b.h.274.8		18
5.4	even	2		325.6.a.h.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.4	✓	9	5.4	even	2
325.6.a.i.1.6	yes	9	1.1	even	1	trivial
325.6.b.h.274.8		18	5.3	odd	4
325.6.b.h.274.11		18	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83815 q^{2} +11.9262 q^{3} -28.6212 q^{4} +21.9221 q^{6} +112.158 q^{7} -111.431 q^{8} -100.766 q^{9} +162.916 q^{11} -341.342 q^{12} -169.000 q^{13} +206.164 q^{14} +711.052 q^{16} +379.413 q^{17} -185.224 q^{18} -284.116 q^{19} +1337.62 q^{21} +299.464 q^{22} -4188.02 q^{23} -1328.94 q^{24} -310.647 q^{26} -4099.82 q^{27} -3210.10 q^{28} +6279.50 q^{29} -7559.74 q^{31} +4872.81 q^{32} +1942.96 q^{33} +697.419 q^{34} +2884.05 q^{36} +7060.39 q^{37} -522.249 q^{38} -2015.52 q^{39} +4166.01 q^{41} +2458.75 q^{42} -21505.5 q^{43} -4662.85 q^{44} -7698.22 q^{46} -22686.6 q^{47} +8480.13 q^{48} -4227.52 q^{49} +4524.95 q^{51} +4836.98 q^{52} -4774.01 q^{53} -7536.08 q^{54} -12497.9 q^{56} -3388.42 q^{57} +11542.7 q^{58} -48311.4 q^{59} +3650.37 q^{61} -13895.9 q^{62} -11301.8 q^{63} -13796.7 q^{64} +3571.46 q^{66} +15803.8 q^{67} -10859.3 q^{68} -49947.1 q^{69} -37247.8 q^{71} +11228.5 q^{72} +29323.4 q^{73} +12978.1 q^{74} +8131.75 q^{76} +18272.4 q^{77} -3704.84 q^{78} -44954.4 q^{79} -24409.0 q^{81} +7657.75 q^{82} -23704.5 q^{83} -38284.3 q^{84} -39530.3 q^{86} +74890.4 q^{87} -18153.9 q^{88} +561.767 q^{89} -18954.7 q^{91} +119866. q^{92} -90158.8 q^{93} -41701.3 q^{94} +58114.0 q^{96} +31260.4 q^{97} -7770.82 q^{98} -16416.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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