LMFDB - Embedded newform 825.6.a.q.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 $ x^8 - x^7 - 176x^6 + 272x^5 + 9055x^4 - 15851x^3 - 118840x^2 + 149572x - 33248
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$5.43413$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.43413 q^{2} -9.00000 q^{3} +9.39797 q^{4} -57.9071 q^{6} -9.96708 q^{7} -145.424 q^{8} +81.0000 q^{9} -121.000 q^{11} -84.5818 q^{12} +813.596 q^{13} -64.1295 q^{14} -1236.41 q^{16} -616.038 q^{17} +521.164 q^{18} +1989.66 q^{19} +89.7037 q^{21} -778.529 q^{22} -545.944 q^{23} +1308.82 q^{24} +5234.78 q^{26} -729.000 q^{27} -93.6704 q^{28} -670.788 q^{29} -1229.55 q^{31} -3301.66 q^{32} +1089.00 q^{33} -3963.67 q^{34} +761.236 q^{36} +9820.34 q^{37} +12801.8 q^{38} -7322.36 q^{39} +5587.84 q^{41} +577.165 q^{42} -6305.91 q^{43} -1137.15 q^{44} -3512.67 q^{46} -10680.8 q^{47} +11127.7 q^{48} -16707.7 q^{49} +5544.35 q^{51} +7646.15 q^{52} +21821.5 q^{53} -4690.48 q^{54} +1449.46 q^{56} -17907.0 q^{57} -4315.93 q^{58} -2157.19 q^{59} -6743.79 q^{61} -7911.10 q^{62} -807.334 q^{63} +18321.9 q^{64} +7006.76 q^{66} -38984.7 q^{67} -5789.51 q^{68} +4913.49 q^{69} +25270.9 q^{71} -11779.4 q^{72} -23958.4 q^{73} +63185.3 q^{74} +18698.8 q^{76} +1206.02 q^{77} -47113.0 q^{78} -101544. q^{79} +6561.00 q^{81} +35952.9 q^{82} -14547.0 q^{83} +843.033 q^{84} -40573.0 q^{86} +6037.09 q^{87} +17596.3 q^{88} -80447.7 q^{89} -8109.18 q^{91} -5130.76 q^{92} +11066.0 q^{93} -68721.6 q^{94} +29715.0 q^{96} -78631.2 q^{97} -107499. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22}+ \cdots - 78408 q^{99}+O(q^{100}) $ 8 * q + 9 * q^2 - 72 * q^3 + 107 * q^4 - 81 * q^6 - 66 * q^7 + 207 * q^8 + 648 * q^9 - 968 * q^11 - 963 * q^12 + 382 * q^13 + 1048 * q^14 - 325 * q^16 + 288 * q^17 + 729 * q^18 - 988 * q^19 + 594 * q^21 - 1089 * q^22 + 5972 * q^23 - 1863 * q^24 - 14579 * q^26 - 5832 * q^27 + 5942 * q^28 + 1032 * q^29 - 4682 * q^31 + 3863 * q^32 + 8712 * q^33 + 2206 * q^34 + 8667 * q^36 - 17200 * q^37 + 11011 * q^38 - 3438 * q^39 - 13220 * q^41 - 9432 * q^42 + 22872 * q^43 - 12947 * q^44 + 9101 * q^46 + 6700 * q^47 + 2925 * q^48 - 43466 * q^49 - 2592 * q^51 + 5009 * q^52 + 6224 * q^53 - 6561 * q^54 + 20992 * q^56 + 8892 * q^57 + 33015 * q^58 - 77556 * q^59 + 11554 * q^61 + 12135 * q^62 - 5346 * q^63 - 149917 * q^64 + 9801 * q^66 + 20894 * q^67 + 91776 * q^68 - 53748 * q^69 - 21648 * q^71 + 16767 * q^72 + 64660 * q^73 - 179522 * q^74 + 24401 * q^76 + 7986 * q^77 + 131211 * q^78 - 22660 * q^79 + 52488 * q^81 - 56080 * q^82 + 100390 * q^83 - 53478 * q^84 + 47271 * q^86 - 9288 * q^87 - 25047 * q^88 - 25578 * q^89 + 73250 * q^91 + 95311 * q^92 + 42138 * q^93 - 170120 * q^94 - 34767 * q^96 + 142828 * q^97 + 303397 * q^98 - 78408 * 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 $

$2$	6.43413	1.13740	0.568702	−	0.822544i	$-0.307446\pi$
$2$	6.43413	1.13740	0.568702	+	0.822544i	$0.307446\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	9.39797	0.293687
$5$	0	0
$5$	0	0
$6$	−57.9071	−0.656680
$7$	−9.96708	−0.0768817	−0.0384408	−	0.999261i	$-0.512239\pi$
$7$	−9.96708	−0.0768817	−0.0384408	+	0.999261i	$0.512239\pi$
$8$	−145.424	−0.803363
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−84.5818	−0.169560
$13$	813.596	1.33521	0.667606	−	0.744515i	$-0.267319\pi$
$13$	813.596	1.33521	0.667606	+	0.744515i	$0.267319\pi$
$14$	−64.1295	−0.0874455
$15$	0	0
$16$	−1236.41	−1.20743
$17$	−616.038	−0.516994	−0.258497	−	0.966012i	$-0.583227\pi$
$17$	−616.038	−0.516994	−0.258497	+	0.966012i	$0.583227\pi$
$18$	521.164	0.379134
$19$	1989.66	1.26443	0.632217	−	0.774792i	$-0.282145\pi$
$19$	1989.66	1.26443	0.632217	+	0.774792i	$0.282145\pi$
$20$	0	0
$21$	89.7037	0.0443877
$22$	−778.529	−0.342940
$23$	−545.944	−0.215193	−0.107597	−	0.994195i	$-0.534315\pi$
$23$	−545.944	−0.215193	−0.107597	+	0.994195i	$0.534315\pi$
$24$	1308.82	0.463822
$25$	0	0
$26$	5234.78	1.51868
$27$	−729.000	−0.192450
$28$	−93.6704	−0.0225791
$29$	−670.788	−0.148112	−0.0740560	−	0.997254i	$-0.523594\pi$
$29$	−670.788	−0.148112	−0.0740560	+	0.997254i	$0.523594\pi$
$30$	0	0
$31$	−1229.55	−0.229796	−0.114898	−	0.993377i	$-0.536654\pi$
$31$	−1229.55	−0.229796	−0.114898	+	0.993377i	$0.536654\pi$
$32$	−3301.66	−0.569977
$33$	1089.00	0.174078
$34$	−3963.67	−0.588031
$35$	0	0
$36$	761.236	0.0978956
$37$	9820.34	1.17929	0.589647	−	0.807661i	$-0.299267\pi$
$37$	9820.34	1.17929	0.589647	+	0.807661i	$0.299267\pi$
$38$	12801.8	1.43817
$39$	−7322.36	−0.770885
$40$	0	0
$41$	5587.84	0.519140	0.259570	−	0.965724i	$-0.416419\pi$
$41$	5587.84	0.519140	0.259570	+	0.965724i	$0.416419\pi$
$42$	577.165	0.0504867
$43$	−6305.91	−0.520088	−0.260044	−	0.965597i	$-0.583737\pi$
$43$	−6305.91	−0.520088	−0.260044	+	0.965597i	$0.583737\pi$
$44$	−1137.15	−0.0885499
$45$	0	0
$46$	−3512.67	−0.244761
$47$	−10680.8	−0.705276	−0.352638	−	0.935760i	$-0.614715\pi$
$47$	−10680.8	−0.705276	−0.352638	+	0.935760i	$0.614715\pi$
$48$	11127.7	0.697113
$49$	−16707.7	−0.994089
$50$	0	0
$51$	5544.35	0.298487
$52$	7646.15	0.392134
$53$	21821.5	1.06708	0.533538	−	0.845776i	$-0.320862\pi$
$53$	21821.5	1.06708	0.533538	+	0.845776i	$0.320862\pi$
$54$	−4690.48	−0.218893
$55$	0	0
$56$	1449.46	0.0617639
$57$	−17907.0	−0.730021
$58$	−4315.93	−0.168463
$59$	−2157.19	−0.0806788	−0.0403394	−	0.999186i	$-0.512844\pi$
$59$	−2157.19	−0.0806788	−0.0403394	+	0.999186i	$0.512844\pi$
$60$	0	0
$61$	−6743.79	−0.232049	−0.116024	−	0.993246i	$-0.537015\pi$
$61$	−6743.79	−0.232049	−0.116024	+	0.993246i	$0.537015\pi$
$62$	−7911.10	−0.261371
$63$	−807.334	−0.0256272
$64$	18321.9	0.559141
$65$	0	0
$66$	7006.76	0.197997
$67$	−38984.7	−1.06098	−0.530490	−	0.847691i	$-0.677992\pi$
$67$	−38984.7	−1.06098	−0.530490	+	0.847691i	$0.677992\pi$
$68$	−5789.51	−0.151834
$69$	4913.49	0.124242
$70$	0	0
$71$	25270.9	0.594941	0.297471	−	0.954731i	$-0.403857\pi$
$71$	25270.9	0.594941	0.297471	+	0.954731i	$0.403857\pi$
$72$	−11779.4	−0.267788
$73$	−23958.4	−0.526201	−0.263100	−	0.964768i	$-0.584745\pi$
$73$	−23958.4	−0.526201	−0.263100	+	0.964768i	$0.584745\pi$
$74$	63185.3	1.34133
$75$	0	0
$76$	18698.8	0.371347
$77$	1206.02	0.0231807
$78$	−47113.0	−0.876808
$79$	−101544.	−1.83057	−0.915284	−	0.402810i	$-0.868034\pi$
$79$	−101544.	−1.83057	−0.915284	+	0.402810i	$0.868034\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	35952.9	0.590472
$83$	−14547.0	−0.231781	−0.115891	−	0.993262i	$-0.536972\pi$
$83$	−14547.0	−0.231781	−0.115891	+	0.993262i	$0.536972\pi$
$84$	843.033	0.0130361
$85$	0	0
$86$	−40573.0	−0.591550
$87$	6037.09	0.0855125
$88$	17596.3	0.242223
$89$	−80447.7	−1.07656	−0.538281	−	0.842766i	$-0.680926\pi$
$89$	−80447.7	−1.07656	−0.538281	+	0.842766i	$0.680926\pi$
$90$	0	0
$91$	−8109.18	−0.102653
$92$	−5130.76	−0.0631993
$93$	11066.0	0.132673
$94$	−68721.6	−0.802184
$95$	0	0
$96$	29715.0	0.329077
$97$	−78631.2	−0.848527	−0.424264	−	0.905539i	$-0.639467\pi$
$97$	−78631.2	−0.848527	−0.424264	+	0.905539i	$0.639467\pi$
$98$	−107499.	−1.13068
$99$	−9801.00	−0.100504
$100$	0	0
$101$	22024.1	0.214830	0.107415	−	0.994214i	$-0.465743\pi$
$101$	22024.1	0.214830	0.107415	+	0.994214i	$0.465743\pi$
$102$	35673.0	0.339500
$103$	−85005.4	−0.789503	−0.394751	−	0.918788i	$-0.629169\pi$
$103$	−85005.4	−0.789503	−0.394751	+	0.918788i	$0.629169\pi$
$104$	−118317.	−1.07266
$105$	0	0
$106$	140403.	1.21370
$107$	−129131.	−1.09036	−0.545181	−	0.838319i	$-0.683539\pi$
$107$	−129131.	−1.09036	−0.545181	+	0.838319i	$0.683539\pi$
$108$	−6851.12	−0.0565200
$109$	−153961.	−1.24121	−0.620605	−	0.784124i	$-0.713113\pi$
$109$	−153961.	−1.24121	−0.620605	+	0.784124i	$0.713113\pi$
$110$	0	0
$111$	−88383.0	−0.680866
$112$	12323.4	0.0928296
$113$	139946.	1.03102	0.515508	−	0.856885i	$-0.327603\pi$
$113$	139946.	1.03102	0.515508	+	0.856885i	$0.327603\pi$
$114$	−115216.	−0.830328
$115$	0	0
$116$	−6304.04	−0.0434985
$117$	65901.3	0.445071
$118$	−13879.7	−0.0917643
$119$	6140.10	0.0397474
$120$	0	0
$121$	14641.0	0.0909091
$122$	−43390.4	−0.263933
$123$	−50290.6	−0.299726
$124$	−11555.3	−0.0674881
$125$	0	0
$126$	−5194.49	−0.0291485
$127$	−119781.	−0.658992	−0.329496	−	0.944157i	$-0.606879\pi$
$127$	−119781.	−0.658992	−0.329496	+	0.944157i	$0.606879\pi$
$128$	223539.	1.20595
$129$	56753.2	0.300273
$130$	0	0
$131$	167797.	0.854293	0.427147	−	0.904182i	$-0.359519\pi$
$131$	167797.	0.854293	0.427147	+	0.904182i	$0.359519\pi$
$132$	10234.4	0.0511243
$133$	−19831.2	−0.0972118
$134$	−250832.	−1.20676
$135$	0	0
$136$	89586.9	0.415334
$137$	23349.3	0.106285	0.0531425	−	0.998587i	$-0.483076\pi$
$137$	23349.3	0.106285	0.0531425	+	0.998587i	$0.483076\pi$
$138$	31614.0	0.141313
$139$	−108551.	−0.476537	−0.238269	−	0.971199i	$-0.576580\pi$
$139$	−108551.	−0.476537	−0.238269	+	0.971199i	$0.576580\pi$
$140$	0	0
$141$	96127.2	0.407191
$142$	162596.	0.676688
$143$	−98445.1	−0.402582
$144$	−100149.	−0.402478
$145$	0	0
$146$	−154152.	−0.598502
$147$	150369.	0.573938
$148$	92291.3	0.346343
$149$	−276956.	−1.02199	−0.510993	−	0.859585i	$-0.670722\pi$
$149$	−276956.	−1.02199	−0.510993	+	0.859585i	$0.670722\pi$
$150$	0	0
$151$	326462.	1.16517	0.582587	−	0.812769i	$-0.302041\pi$
$151$	326462.	1.16517	0.582587	+	0.812769i	$0.302041\pi$
$152$	−289346.	−1.01580
$153$	−49899.1	−0.172331
$154$	7759.66	0.0263658
$155$	0	0
$156$	−68815.4	−0.226399
$157$	−366289.	−1.18597	−0.592986	−	0.805213i	$-0.702051\pi$
$157$	−366289.	−1.18597	−0.592986	+	0.805213i	$0.702051\pi$
$158$	−653346.	−2.08209
$159$	−196394.	−0.616077
$160$	0	0
$161$	5441.46	0.0165444
$162$	42214.3	0.126378
$163$	−642723.	−1.89476	−0.947381	−	0.320107i	$-0.896281\pi$
$163$	−642723.	−1.89476	−0.947381	+	0.320107i	$0.896281\pi$
$164$	52514.4	0.152464
$165$	0	0
$166$	−93597.3	−0.263629
$167$	259189.	0.719159	0.359579	−	0.933115i	$-0.382920\pi$
$167$	259189.	0.719159	0.359579	+	0.933115i	$0.382920\pi$
$168$	−13045.1	−0.0356594
$169$	290645.	0.782792
$170$	0	0
$171$	161163.	0.421478
$172$	−59262.8	−0.152743
$173$	242721.	0.616585	0.308292	−	0.951292i	$-0.400242\pi$
$173$	242721.	0.616585	0.308292	+	0.951292i	$0.400242\pi$
$174$	38843.4	0.0972622
$175$	0	0
$176$	149606.	0.364055
$177$	19414.8	0.0465799
$178$	−517611.	−1.22448
$179$	−214184.	−0.499638	−0.249819	−	0.968293i	$-0.580371\pi$
$179$	−214184.	−0.499638	−0.249819	+	0.968293i	$0.580371\pi$
$180$	0	0
$181$	−247705.	−0.562003	−0.281002	−	0.959707i	$-0.590667\pi$
$181$	−247705.	−0.562003	−0.281002	+	0.959707i	$0.590667\pi$
$182$	−52175.5	−0.116758
$183$	60694.1	0.133973
$184$	79393.4	0.172878
$185$	0	0
$186$	71199.9	0.150903
$187$	74540.6	0.155880
$188$	−100378.	−0.207130
$189$	7266.00	0.0147959
$190$	0	0
$191$	−410762.	−0.814718	−0.407359	−	0.913268i	$-0.633550\pi$
$191$	−410762.	−0.814718	−0.407359	+	0.913268i	$0.633550\pi$
$192$	−164897.	−0.322820
$193$	150990.	0.291780	0.145890	−	0.989301i	$-0.453395\pi$
$193$	150990.	0.291780	0.145890	+	0.989301i	$0.453395\pi$
$194$	−505923.	−0.965118
$195$	0	0
$196$	−157018.	−0.291951
$197$	151624.	0.278357	0.139178	−	0.990267i	$-0.455554\pi$
$197$	151624.	0.278357	0.139178	+	0.990267i	$0.455554\pi$
$198$	−63060.9	−0.114313
$199$	464676.	0.831797	0.415898	−	0.909411i	$-0.363467\pi$
$199$	464676.	0.831797	0.415898	+	0.909411i	$0.363467\pi$
$200$	0	0
$201$	350862.	0.612557
$202$	141706.	0.244349
$203$	6685.80	0.0113871
$204$	52105.6	0.0876615
$205$	0	0
$206$	−546936.	−0.897983
$207$	−44221.4	−0.0717310
$208$	−1.00594e6	−1.61218
$209$	−240749.	−0.381241
$210$	0	0
$211$	−1.17253e6	−1.81308	−0.906541	−	0.422117i	$-0.861287\pi$
$211$	−1.17253e6	−1.81308	−0.906541	+	0.422117i	$0.861287\pi$
$212$	205078.	0.313386
$213$	−227438.	−0.343490
$214$	−830844.	−1.24018
$215$	0	0
$216$	106014.	0.154607
$217$	12255.1	0.0176671
$218$	−990606.	−1.41176
$219$	215626.	0.303802
$220$	0	0
$221$	−501206.	−0.690297
$222$	−568668.	−0.774419
$223$	494387.	0.665740	0.332870	−	0.942973i	$-0.391983\pi$
$223$	494387.	0.665740	0.332870	+	0.942973i	$0.391983\pi$
$224$	32907.9	0.0438208
$225$	0	0
$226$	900432.	1.17268
$227$	219419.	0.282624	0.141312	−	0.989965i	$-0.454868\pi$
$227$	219419.	0.282624	0.141312	+	0.989965i	$0.454868\pi$
$228$	−168289.	−0.214397
$229$	−1.23652e6	−1.55816	−0.779080	−	0.626925i	$-0.784313\pi$
$229$	−1.23652e6	−1.55816	−0.779080	+	0.626925i	$0.784313\pi$
$230$	0	0
$231$	−10854.2	−0.0133834
$232$	97548.8	0.118988
$233$	−84155.4	−0.101553	−0.0507764	−	0.998710i	$-0.516170\pi$
$233$	−84155.4	−0.101553	−0.0507764	+	0.998710i	$0.516170\pi$
$234$	424017.	0.506225
$235$	0	0
$236$	−20273.3	−0.0236943
$237$	913894.	1.05688
$238$	39506.2	0.0452088
$239$	−113266.	−0.128264	−0.0641320	−	0.997941i	$-0.520428\pi$
$239$	−113266.	−0.128264	−0.0641320	+	0.997941i	$0.520428\pi$
$240$	0	0
$241$	−616377.	−0.683603	−0.341801	−	0.939772i	$-0.611037\pi$
$241$	−616377.	−0.683603	−0.341801	+	0.939772i	$0.611037\pi$
$242$	94202.0	0.103400
$243$	−59049.0	−0.0641500
$244$	−63378.0	−0.0681497
$245$	0	0
$246$	−323576.	−0.340909
$247$	1.61878e6	1.68829
$248$	178807.	0.184610
$249$	130923.	0.133819
$250$	0	0
$251$	−1.34829e6	−1.35082	−0.675412	−	0.737441i	$-0.736034\pi$
$251$	−1.34829e6	−1.35082	−0.675412	+	0.737441i	$0.736034\pi$
$252$	−7587.30	−0.00752638
$253$	66059.2	0.0648831
$254$	−770689.	−0.749540
$255$	0	0
$256$	851975.	0.812506
$257$	−715746.	−0.675968	−0.337984	−	0.941152i	$-0.609745\pi$
$257$	−715746.	−0.675968	−0.337984	+	0.941152i	$0.609745\pi$
$258$	365157.	0.341531
$259$	−97880.1	−0.0906661
$260$	0	0
$261$	−54333.8	−0.0493707
$262$	1.07963e6	0.971676
$263$	2.10033e6	1.87239	0.936197	−	0.351475i	$-0.114320\pi$
$263$	2.10033e6	1.87239	0.936197	+	0.351475i	$0.114320\pi$
$264$	−158367.	−0.139848
$265$	0	0
$266$	−127596.	−0.110569
$267$	724029.	0.621553
$268$	−366377.	−0.311596
$269$	−1.41437e6	−1.19174	−0.595871	−	0.803080i	$-0.703193\pi$
$269$	−1.41437e6	−1.19174	−0.595871	+	0.803080i	$0.703193\pi$
$270$	0	0
$271$	190480.	0.157553	0.0787763	−	0.996892i	$-0.474899\pi$
$271$	190480.	0.157553	0.0787763	+	0.996892i	$0.474899\pi$
$272$	761678.	0.624236
$273$	72982.6	0.0592670
$274$	150232.	0.120889
$275$	0	0
$276$	46176.9	0.0364881
$277$	998049.	0.781542	0.390771	−	0.920488i	$-0.372208\pi$
$277$	998049.	0.781542	0.390771	+	0.920488i	$0.372208\pi$
$278$	−698430.	−0.542015
$279$	−99593.8	−0.0765988
$280$	0	0
$281$	−1.43088e6	−1.08103	−0.540515	−	0.841334i	$-0.681771\pi$
$281$	−1.43088e6	−1.08103	−0.540515	+	0.841334i	$0.681771\pi$
$282$	618495.	0.463141
$283$	−2.40496e6	−1.78501	−0.892506	−	0.451035i	$-0.851055\pi$
$283$	−2.40496e6	−1.78501	−0.892506	+	0.451035i	$0.851055\pi$
$284$	237495.	0.174726
$285$	0	0
$286$	−633408.	−0.457898
$287$	−55694.5	−0.0399124
$288$	−267435.	−0.189992
$289$	−1.04035e6	−0.732717
$290$	0	0
$291$	707681.	0.489897
$292$	−225161.	−0.154538
$293$	689179.	0.468990	0.234495	−	0.972117i	$-0.424656\pi$
$293$	689179.	0.468990	0.234495	+	0.972117i	$0.424656\pi$
$294$	967492.	0.652799
$295$	0	0
$296$	−1.42812e6	−0.947401
$297$	88209.0	0.0580259
$298$	−1.78197e6	−1.16241
$299$	−444177.	−0.287328
$300$	0	0
$301$	62851.5	0.0399852
$302$	2.10050e6	1.32527
$303$	−198217.	−0.124032
$304$	−2.46005e6	−1.52672
$305$	0	0
$306$	−321057.	−0.196010
$307$	−986212.	−0.597206	−0.298603	−	0.954377i	$-0.596521\pi$
$307$	−986212.	−0.597206	−0.298603	+	0.954377i	$0.596521\pi$
$308$	11334.1	0.00680786
$309$	765049.	0.455819
$310$	0	0
$311$	−1.70281e6	−0.998308	−0.499154	−	0.866513i	$-0.666356\pi$
$311$	−1.70281e6	−0.998308	−0.499154	+	0.866513i	$0.666356\pi$
$312$	1.06485e6	0.619301
$313$	2.92333e6	1.68662	0.843309	−	0.537428i	$-0.180604\pi$
$313$	2.92333e6	1.68662	0.843309	+	0.537428i	$0.180604\pi$
$314$	−2.35675e6	−1.34893
$315$	0	0
$316$	−954306.	−0.537613
$317$	1.69795e6	0.949022	0.474511	−	0.880250i	$-0.342625\pi$
$317$	1.69795e6	0.949022	0.474511	+	0.880250i	$0.342625\pi$
$318$	−1.26362e6	−0.700728
$319$	81165.3	0.0446574
$320$	0	0
$321$	1.16218e6	0.629520
$322$	35011.1	0.0188177
$323$	−1.22571e6	−0.653704
$324$	61660.1	0.0326319
$325$	0	0
$326$	−4.13536e6	−2.15511
$327$	1.38565e6	0.716612
$328$	−812608.	−0.417058
$329$	106456.	0.0542228
$330$	0	0
$331$	1.44074e6	0.722795	0.361397	−	0.932412i	$-0.382300\pi$
$331$	1.44074e6	0.722795	0.361397	+	0.932412i	$0.382300\pi$
$332$	−136712.	−0.0680711
$333$	795447.	0.393098
$334$	1.66765e6	0.817973
$335$	0	0
$336$	−110911.	−0.0535952
$337$	967187.	0.463912	0.231956	−	0.972726i	$-0.425487\pi$
$337$	967187.	0.463912	0.231956	+	0.972726i	$0.425487\pi$
$338$	1.87005e6	0.890351
$339$	−1.25952e6	−0.595257
$340$	0	0
$341$	148776.	0.0692862
$342$	1.03694e6	0.479390
$343$	334043.	0.153309
$344$	917032.	0.417819
$345$	0	0
$346$	1.56170e6	0.701306
$347$	2.81126e6	1.25336	0.626682	−	0.779275i	$-0.284413\pi$
$347$	2.81126e6	1.25336	0.626682	+	0.779275i	$0.284413\pi$
$348$	56736.4	0.0251139
$349$	−1.64221e6	−0.721713	−0.360856	−	0.932621i	$-0.617516\pi$
$349$	−1.64221e6	−0.721713	−0.360856	+	0.932621i	$0.617516\pi$
$350$	0	0
$351$	−593111.	−0.256962
$352$	399501.	0.171855
$353$	2.89812e6	1.23788	0.618941	−	0.785438i	$-0.287562\pi$
$353$	2.89812e6	1.23788	0.618941	+	0.785438i	$0.287562\pi$
$354$	124917.	0.0529802
$355$	0	0
$356$	−756045.	−0.316172
$357$	−55260.9	−0.0229482
$358$	−1.37809e6	−0.568290
$359$	3.03267e6	1.24191	0.620954	−	0.783847i	$-0.286745\pi$
$359$	3.03267e6	1.24191	0.620954	+	0.783847i	$0.286745\pi$
$360$	0	0
$361$	1.48267e6	0.598792
$362$	−1.59377e6	−0.639224
$363$	−131769.	−0.0524864
$364$	−76209.8	−0.0301479
$365$	0	0
$366$	390514.	0.152382
$367$	1.16998e6	0.453431	0.226716	−	0.973961i	$-0.427201\pi$
$367$	1.16998e6	0.453431	0.226716	+	0.973961i	$0.427201\pi$
$368$	675012.	0.259832
$369$	452615.	0.173047
$370$	0	0
$371$	−217497.	−0.0820387
$372$	103998.	0.0389643
$373$	1.47636e6	0.549438	0.274719	−	0.961525i	$-0.411415\pi$
$373$	1.47636e6	0.549438	0.274719	+	0.961525i	$0.411415\pi$
$374$	479604.	0.177298
$375$	0	0
$376$	1.55325e6	0.566593
$377$	−545750.	−0.197761
$378$	46750.4	0.0168289
$379$	4.95103e6	1.77051	0.885253	−	0.465110i	$-0.153985\pi$
$379$	4.95103e6	1.77051	0.885253	+	0.465110i	$0.153985\pi$
$380$	0	0
$381$	1.07803e6	0.380469
$382$	−2.64290e6	−0.926663
$383$	1.41125e6	0.491594	0.245797	−	0.969321i	$-0.420950\pi$
$383$	1.41125e6	0.491594	0.245797	+	0.969321i	$0.420950\pi$
$384$	−2.01185e6	−0.696253
$385$	0	0
$386$	971492.	0.331872
$387$	−510779.	−0.173363
$388$	−738974.	−0.249201
$389$	1.35721e6	0.454750	0.227375	−	0.973807i	$-0.426986\pi$
$389$	1.35721e6	0.454750	0.227375	+	0.973807i	$0.426986\pi$
$390$	0	0
$391$	336322.	0.111253
$392$	2.42970e6	0.798615
$393$	−1.51018e6	−0.493226
$394$	975566.	0.316604
$395$	0	0
$396$	−92109.5	−0.0295166
$397$	1.77467e6	0.565122	0.282561	−	0.959249i	$-0.408816\pi$
$397$	1.77467e6	0.565122	0.282561	+	0.959249i	$0.408816\pi$
$398$	2.98978e6	0.946089
$399$	178480.	0.0561252
$400$	0	0
$401$	3.69680e6	1.14806	0.574030	−	0.818834i	$-0.305379\pi$
$401$	3.69680e6	1.14806	0.574030	+	0.818834i	$0.305379\pi$
$402$	2.25749e6	0.696725
$403$	−1.00036e6	−0.306827
$404$	206982.	0.0630928
$405$	0	0
$406$	43017.3	0.0129517
$407$	−1.18826e6	−0.355571
$408$	−806282.	−0.239793
$409$	−824495.	−0.243714	−0.121857	−	0.992548i	$-0.538885\pi$
$409$	−824495.	−0.243714	−0.121857	+	0.992548i	$0.538885\pi$
$410$	0	0
$411$	−210144.	−0.0613637
$412$	−798879.	−0.231866
$413$	21500.9	0.00620272
$414$	−284526.	−0.0815871
$415$	0	0
$416$	−2.68622e6	−0.761041
$417$	976959.	0.275129
$418$	−1.54901e6	−0.433625
$419$	−5.87905e6	−1.63596	−0.817979	−	0.575249i	$-0.804905\pi$
$419$	−5.87905e6	−1.63596	−0.817979	+	0.575249i	$0.804905\pi$
$420$	0	0
$421$	3.48720e6	0.958895	0.479448	−	0.877570i	$-0.340837\pi$
$421$	3.48720e6	0.958895	0.479448	+	0.877570i	$0.340837\pi$
$422$	−7.54420e6	−2.06221
$423$	−865145.	−0.235092
$424$	−3.17338e6	−0.857250
$425$	0	0
$426$	−1.46336e6	−0.390686
$427$	67215.9	0.0178403
$428$	−1.21357e6	−0.320225
$429$	886006.	0.232431
$430$	0	0
$431$	−4.41226e6	−1.14411	−0.572055	−	0.820215i	$-0.693854\pi$
$431$	−4.41226e6	−1.14411	−0.572055	+	0.820215i	$0.693854\pi$
$432$	901345.	0.232371
$433$	989722.	0.253684	0.126842	−	0.991923i	$-0.459516\pi$
$433$	989722.	0.253684	0.126842	+	0.991923i	$0.459516\pi$
$434$	78850.6	0.0200947
$435$	0	0
$436$	−1.44692e6	−0.364527
$437$	−1.08624e6	−0.272097
$438$	1.38736e6	0.345545
$439$	−2.69582e6	−0.667621	−0.333811	−	0.942640i	$-0.608335\pi$
$439$	−2.69582e6	−0.667621	−0.333811	+	0.942640i	$0.608335\pi$
$440$	0	0
$441$	−1.35332e6	−0.331363
$442$	−3.22482e6	−0.785146
$443$	1.84727e6	0.447221	0.223610	−	0.974679i	$-0.428216\pi$
$443$	1.84727e6	0.447221	0.223610	+	0.974679i	$0.428216\pi$
$444$	−830621.	−0.199961
$445$	0	0
$446$	3.18095e6	0.757215
$447$	2.49260e6	0.590044
$448$	−182616.	−0.0429877
$449$	−806296.	−0.188746	−0.0943732	−	0.995537i	$-0.530085\pi$
$449$	−806296.	−0.188746	−0.0943732	+	0.995537i	$0.530085\pi$
$450$	0	0
$451$	−676129.	−0.156527
$452$	1.31521e6	0.302796
$453$	−2.93816e6	−0.672713
$454$	1.41177e6	0.321457
$455$	0	0
$456$	2.60411e6	0.586472
$457$	4.78463e6	1.07166	0.535830	−	0.844326i	$-0.319999\pi$
$457$	4.78463e6	1.07166	0.535830	+	0.844326i	$0.319999\pi$
$458$	−7.95592e6	−1.77226
$459$	449092.	0.0994955
$460$	0	0
$461$	−8.05029e6	−1.76425	−0.882123	−	0.471019i	$-0.843886\pi$
$461$	−8.05029e6	−1.76425	−0.882123	+	0.471019i	$0.843886\pi$
$462$	−69837.0	−0.0152223
$463$	−646347.	−0.140124	−0.0700622	−	0.997543i	$-0.522320\pi$
$463$	−646347.	−0.140124	−0.0700622	+	0.997543i	$0.522320\pi$
$464$	829371.	0.178836
$465$	0	0
$466$	−541466.	−0.115507
$467$	−5.97613e6	−1.26803	−0.634013	−	0.773323i	$-0.718593\pi$
$467$	−5.97613e6	−1.26803	−0.634013	+	0.773323i	$0.718593\pi$
$468$	619338.	0.130711
$469$	388564.	0.0815699
$470$	0	0
$471$	3.29660e6	0.684721
$472$	313709.	0.0648144
$473$	763015.	0.156812
$474$	5.88011e6	1.20210
$475$	0	0
$476$	57704.5	0.0116733
$477$	1.76754e6	0.355692
$478$	−728768.	−0.145888
$479$	−7.53887e6	−1.50130	−0.750650	−	0.660700i	$-0.770260\pi$
$479$	−7.53887e6	−1.50130	−0.750650	+	0.660700i	$0.770260\pi$
$480$	0	0
$481$	7.98979e6	1.57461
$482$	−3.96585e6	−0.777532
$483$	−48973.2	−0.00955191
$484$	137596.	0.0266988
$485$	0	0
$486$	−379929.	−0.0729645
$487$	−6.72146e6	−1.28423	−0.642113	−	0.766610i	$-0.721942\pi$
$487$	−6.72146e6	−1.28423	−0.642113	+	0.766610i	$0.721942\pi$
$488$	980711.	0.186420
$489$	5.78451e6	1.09394
$490$	0	0
$491$	−2.38900e6	−0.447212	−0.223606	−	0.974680i	$-0.571783\pi$
$491$	−2.38900e6	−0.447212	−0.223606	+	0.974680i	$0.571783\pi$
$492$	−472629.	−0.0880254
$493$	413231.	0.0765730
$494$	1.04155e7	1.92026
$495$	0	0
$496$	1.52024e6	0.277464
$497$	−251877.	−0.0457401
$498$	842376.	0.152206
$499$	−4.78068e6	−0.859485	−0.429742	−	0.902952i	$-0.641396\pi$
$499$	−4.78068e6	−0.859485	−0.429742	+	0.902952i	$0.641396\pi$
$500$	0	0
$501$	−2.33270e6	−0.415206
$502$	−8.67506e6	−1.53643
$503$	7.98014e6	1.40634	0.703170	−	0.711021i	$-0.251767\pi$
$503$	7.98014e6	1.40634	0.703170	+	0.711021i	$0.251767\pi$
$504$	117406.	0.0205880
$505$	0	0
$506$	425033.	0.0737983
$507$	−2.61581e6	−0.451945
$508$	−1.12570e6	−0.193537
$509$	888027.	0.151926	0.0759630	−	0.997111i	$-0.475797\pi$
$509$	888027.	0.151926	0.0759630	+	0.997111i	$0.475797\pi$
$510$	0	0
$511$	238796.	0.0404552
$512$	−1.67153e6	−0.281798
$513$	−1.45047e6	−0.243340
$514$	−4.60520e6	−0.768848
$515$	0	0
$516$	533365.	0.0881861
$517$	1.29238e6	0.212649
$518$	−629773.	−0.103124
$519$	−2.18449e6	−0.355985
$520$	0	0
$521$	4.40559e6	0.711066	0.355533	−	0.934664i	$-0.384299\pi$
$521$	4.40559e6	0.711066	0.355533	+	0.934664i	$0.384299\pi$
$522$	−349591.	−0.0561543
$523$	3.95891e6	0.632879	0.316440	−	0.948613i	$-0.397513\pi$
$523$	3.95891e6	0.632879	0.316440	+	0.948613i	$0.397513\pi$
$524$	1.57696e6	0.250895
$525$	0	0
$526$	1.35138e7	2.12967
$527$	757452.	0.118803
$528$	−1.34645e6	−0.210187
$529$	−6.13829e6	−0.953692
$530$	0	0
$531$	−174733.	−0.0268929
$532$	−186373.	−0.0285498
$533$	4.54625e6	0.693162
$534$	4.65850e6	0.706956
$535$	0	0
$536$	5.66932e6	0.852352
$537$	1.92766e6	0.288466
$538$	−9.10023e6	−1.35549
$539$	2.02163e6	0.299729
$540$	0	0
$541$	8.12527e6	1.19356	0.596780	−	0.802405i	$-0.296446\pi$
$541$	8.12527e6	1.19356	0.596780	+	0.802405i	$0.296446\pi$
$542$	1.22557e6	0.179201
$543$	2.22935e6	0.324473
$544$	2.03395e6	0.294675
$545$	0	0
$546$	469579.	0.0674104
$547$	−1.49316e6	−0.213373	−0.106686	−	0.994293i	$-0.534024\pi$
$547$	−1.49316e6	−0.213373	−0.106686	+	0.994293i	$0.534024\pi$
$548$	219436.	0.0312145
$549$	−546247.	−0.0773496
$550$	0	0
$551$	−1.33464e6	−0.187278
$552$	−714541.	−0.0998112
$553$	1.01210e6	0.140737
$554$	6.42157e6	0.888929
$555$	0	0
$556$	−1.02016e6	−0.139953
$557$	3.54818e6	0.484582	0.242291	−	0.970204i	$-0.422101\pi$
$557$	3.54818e6	0.484582	0.242291	+	0.970204i	$0.422101\pi$
$558$	−640799.	−0.0871237
$559$	−5.13046e6	−0.694428
$560$	0	0
$561$	−670866.	−0.0899971
$562$	−9.20647e6	−1.22957
$563$	6.70997e6	0.892174	0.446087	−	0.894990i	$-0.352817\pi$
$563$	6.70997e6	0.892174	0.446087	+	0.894990i	$0.352817\pi$
$564$	903401.	0.119587
$565$	0	0
$566$	−1.54738e7	−2.03028
$567$	−65394.0	−0.00854241
$568$	−3.67500e6	−0.477954
$569$	6.17393e6	0.799431	0.399716	−	0.916639i	$-0.369109\pi$
$569$	6.17393e6	0.799431	0.399716	+	0.916639i	$0.369109\pi$
$570$	0	0
$571$	1.82693e6	0.234494	0.117247	−	0.993103i	$-0.462593\pi$
$571$	1.82693e6	0.234494	0.117247	+	0.993103i	$0.462593\pi$
$572$	−925184.	−0.118233
$573$	3.69686e6	0.470377
$574$	−358345.	−0.0453965
$575$	0	0
$576$	1.48408e6	0.186380
$577$	−7.62349e6	−0.953267	−0.476633	−	0.879102i	$-0.658143\pi$
$577$	−7.62349e6	−0.953267	−0.476633	+	0.879102i	$0.658143\pi$
$578$	−6.69377e6	−0.833395
$579$	−1.35891e6	−0.168460
$580$	0	0
$581$	144991.	0.0178197
$582$	4.55331e6	0.557211
$583$	−2.64041e6	−0.321736
$584$	3.48414e6	0.422730
$585$	0	0
$586$	4.43427e6	0.533430
$587$	−1.53545e7	−1.83925	−0.919623	−	0.392803i	$-0.871505\pi$
$587$	−1.53545e7	−1.83925	−0.919623	+	0.392803i	$0.871505\pi$
$588$	1.41316e6	0.168558
$589$	−2.44640e6	−0.290562
$590$	0	0
$591$	−1.36461e6	−0.160709
$592$	−1.21420e7	−1.42392
$593$	8.62212e6	1.00688	0.503439	−	0.864031i	$-0.332068\pi$
$593$	8.62212e6	1.00688	0.503439	+	0.864031i	$0.332068\pi$
$594$	567548.	0.0659988
$595$	0	0
$596$	−2.60282e6	−0.300144
$597$	−4.18208e6	−0.480238
$598$	−2.85789e6	−0.326808
$599$	−2.18726e6	−0.249077	−0.124539	−	0.992215i	$-0.539745\pi$
$599$	−2.18726e6	−0.249077	−0.124539	+	0.992215i	$0.539745\pi$
$600$	0	0
$601$	−3.77734e6	−0.426579	−0.213290	−	0.976989i	$-0.568418\pi$
$601$	−3.77734e6	−0.426579	−0.213290	+	0.976989i	$0.568418\pi$
$602$	404395.	0.0454793
$603$	−3.15776e6	−0.353660
$604$	3.06808e6	0.342196
$605$	0	0
$606$	−1.27535e6	−0.141075
$607$	−1.39714e7	−1.53910	−0.769550	−	0.638587i	$-0.779519\pi$
$607$	−1.39714e7	−1.53910	−0.769550	+	0.638587i	$0.779519\pi$
$608$	−6.56920e6	−0.720698
$609$	−60172.2	−0.00657434
$610$	0	0
$611$	−8.68986e6	−0.941694
$612$	−468950.	−0.0506114
$613$	−8.72631e6	−0.937949	−0.468974	−	0.883212i	$-0.655376\pi$
$613$	−8.72631e6	−0.937949	−0.468974	+	0.883212i	$0.655376\pi$
$614$	−6.34541e6	−0.679264
$615$	0	0
$616$	−175384.	−0.0186225
$617$	−8.27444e6	−0.875035	−0.437518	−	0.899210i	$-0.644142\pi$
$617$	−8.27444e6	−0.875035	−0.437518	+	0.899210i	$0.644142\pi$
$618$	4.92242e6	0.518451
$619$	1.25043e7	1.31169	0.655846	−	0.754894i	$-0.272312\pi$
$619$	1.25043e7	1.31169	0.655846	+	0.754894i	$0.272312\pi$
$620$	0	0
$621$	397993.	0.0414139
$622$	−1.09561e7	−1.13548
$623$	801829.	0.0827678
$624$	9.05347e6	0.930794
$625$	0	0
$626$	1.88091e7	1.91837
$627$	2.16675e6	0.220110
$628$	−3.44237e6	−0.348304
$629$	−6.04970e6	−0.609688
$630$	0	0
$631$	1.59310e7	1.59283	0.796416	−	0.604750i	$-0.206727\pi$
$631$	1.59310e7	1.59283	0.796416	+	0.604750i	$0.206727\pi$
$632$	1.47669e7	1.47061
$633$	1.05528e7	1.04678
$634$	1.09248e7	1.07942
$635$	0	0
$636$	−1.84570e6	−0.180934
$637$	−1.35933e7	−1.32732
$638$	522228.	0.0507935
$639$	2.04694e6	0.198314
$640$	0	0
$641$	−5.06724e6	−0.487109	−0.243555	−	0.969887i	$-0.578313\pi$
$641$	−5.06724e6	−0.487109	−0.243555	+	0.969887i	$0.578313\pi$
$642$	7.47759e6	0.716019
$643$	5.24812e6	0.500583	0.250292	−	0.968170i	$-0.419474\pi$
$643$	5.24812e6	0.500583	0.250292	+	0.968170i	$0.419474\pi$
$644$	51138.7	0.00485887
$645$	0	0
$646$	−7.88637e6	−0.743526
$647$	1.34119e7	1.25959	0.629797	−	0.776760i	$-0.283138\pi$
$647$	1.34119e7	1.25959	0.629797	+	0.776760i	$0.283138\pi$
$648$	−954129.	−0.0892626
$649$	261021.	0.0243256
$650$	0	0
$651$	−110296.	−0.0102001
$652$	−6.04029e6	−0.556467
$653$	4.94854e6	0.454144	0.227072	−	0.973878i	$-0.427085\pi$
$653$	4.94854e6	0.454144	0.227072	+	0.973878i	$0.427085\pi$
$654$	8.91545e6	0.815077
$655$	0	0
$656$	−6.90888e6	−0.626828
$657$	−1.94063e6	−0.175400
$658$	684954.	0.0616732
$659$	1.09443e7	0.981687	0.490844	−	0.871248i	$-0.336689\pi$
$659$	1.09443e7	0.981687	0.490844	+	0.871248i	$0.336689\pi$
$660$	0	0
$661$	−9.35940e6	−0.833191	−0.416595	−	0.909092i	$-0.636777\pi$
$661$	−9.35940e6	−0.833191	−0.416595	+	0.909092i	$0.636777\pi$
$662$	9.26989e6	0.822109
$663$	4.51086e6	0.398543
$664$	2.11549e6	0.186205
$665$	0	0
$666$	5.11801e6	0.447111
$667$	366212.	0.0318727
$668$	2.43585e6	0.211207
$669$	−4.44948e6	−0.384365
$670$	0	0
$671$	815999.	0.0699654
$672$	−296171.	−0.0253000
$673$	−2.66957e6	−0.227197	−0.113599	−	0.993527i	$-0.536238\pi$
$673$	−2.66957e6	−0.227197	−0.113599	+	0.993527i	$0.536238\pi$
$674$	6.22300e6	0.527655
$675$	0	0
$676$	2.73148e6	0.229896
$677$	−1.36620e7	−1.14563	−0.572814	−	0.819685i	$-0.694148\pi$
$677$	−1.36620e7	−1.14563	−0.572814	+	0.819685i	$0.694148\pi$
$678$	−8.10389e6	−0.677048
$679$	783724.	0.0652362
$680$	0	0
$681$	−1.97477e6	−0.163173
$682$	957243.	0.0788064
$683$	−5.01005e6	−0.410951	−0.205475	−	0.978662i	$-0.565874\pi$
$683$	−5.01005e6	−0.410951	−0.205475	+	0.978662i	$0.565874\pi$
$684$	1.51460e6	0.123782
$685$	0	0
$686$	2.14928e6	0.174374
$687$	1.11287e7	0.899604
$688$	7.79671e6	0.627972
$689$	1.77539e7	1.42477
$690$	0	0
$691$	1.37818e6	0.109802	0.0549009	−	0.998492i	$-0.482516\pi$
$691$	1.37818e6	0.109802	0.0549009	+	0.998492i	$0.482516\pi$
$692$	2.28109e6	0.181083
$693$	97687.4	0.00772690
$694$	1.80880e7	1.42558
$695$	0	0
$696$	−877939.	−0.0686976
$697$	−3.44232e6	−0.268392
$698$	−1.05662e7	−0.820879
$699$	757399.	0.0586316
$700$	0	0
$701$	1.98776e7	1.52781	0.763906	−	0.645328i	$-0.223279\pi$
$701$	1.98776e7	1.52781	0.763906	+	0.645328i	$0.223279\pi$
$702$	−3.81615e6	−0.292269
$703$	1.95392e7	1.49114
$704$	−2.21695e6	−0.168587
$705$	0	0
$706$	1.86468e7	1.40797
$707$	−219516.	−0.0165165
$708$	182459.	0.0136799
$709$	−5.68213e6	−0.424518	−0.212259	−	0.977213i	$-0.568082\pi$
$709$	−5.68213e6	−0.424518	−0.212259	+	0.977213i	$0.568082\pi$
$710$	0	0
$711$	−8.22505e6	−0.610189
$712$	1.16991e7	0.864870
$713$	671267.	0.0494506
$714$	−355556.	−0.0261013
$715$	0	0
$716$	−2.01290e6	−0.146737
$717$	1.01939e6	0.0740533
$718$	1.95126e7	1.41255
$719$	−4.90403e6	−0.353778	−0.176889	−	0.984231i	$-0.556603\pi$
$719$	−4.90403e6	−0.353778	−0.176889	+	0.984231i	$0.556603\pi$
$720$	0	0
$721$	847256.	0.0606983
$722$	9.53967e6	0.681068
$723$	5.54739e6	0.394678
$724$	−2.32793e6	−0.165053
$725$	0	0
$726$	−847818.	−0.0596982
$727$	−2.17098e7	−1.52342	−0.761711	−	0.647917i	$-0.775641\pi$
$727$	−2.17098e7	−1.52342	−0.761711	+	0.647917i	$0.775641\pi$
$728$	1.17927e6	0.0824680
$729$	531441.	0.0370370
$730$	0	0
$731$	3.88468e6	0.268882
$732$	570402.	0.0393462
$733$	1.22771e7	0.843985	0.421992	−	0.906599i	$-0.361331\pi$
$733$	1.22771e7	0.843985	0.421992	+	0.906599i	$0.361331\pi$
$734$	7.52777e6	0.515734
$735$	0	0
$736$	1.80252e6	0.122655
$737$	4.71715e6	0.319898
$738$	2.91218e6	0.196824
$739$	−1.86424e7	−1.25571	−0.627857	−	0.778329i	$-0.716068\pi$
$739$	−1.86424e7	−1.25571	−0.627857	+	0.778329i	$0.716068\pi$
$740$	0	0
$741$	−1.45690e7	−0.974733
$742$	−1.39940e6	−0.0933111
$743$	9.40769e6	0.625188	0.312594	−	0.949887i	$-0.398802\pi$
$743$	9.40769e6	0.625188	0.312594	+	0.949887i	$0.398802\pi$
$744$	−1.60926e6	−0.106585
$745$	0	0
$746$	9.49906e6	0.624933
$747$	−1.17831e6	−0.0772605
$748$	700531.	0.0457797
$749$	1.28706e6	0.0838288
$750$	0	0
$751$	−3.32683e6	−0.215244	−0.107622	−	0.994192i	$-0.534324\pi$
$751$	−3.32683e6	−0.215244	−0.107622	+	0.994192i	$0.534324\pi$
$752$	1.32059e7	0.851575
$753$	1.21346e7	0.779899
$754$	−3.51142e6	−0.224934
$755$	0	0
$756$	68285.7	0.00434535
$757$	−3.52278e6	−0.223432	−0.111716	−	0.993740i	$-0.535635\pi$
$757$	−3.52278e6	−0.223432	−0.111716	+	0.993740i	$0.535635\pi$
$758$	3.18555e7	2.01378
$759$	−594533.	−0.0374603
$760$	0	0
$761$	3.43213e6	0.214833	0.107417	−	0.994214i	$-0.465742\pi$
$761$	3.43213e6	0.214833	0.107417	+	0.994214i	$0.465742\pi$
$762$	6.93620e6	0.432747
$763$	1.53454e6	0.0954263
$764$	−3.86033e6	−0.239272
$765$	0	0
$766$	9.08014e6	0.559140
$767$	−1.75508e6	−0.107723
$768$	−7.66777e6	−0.469101
$769$	−1.40196e7	−0.854908	−0.427454	−	0.904037i	$-0.640589\pi$
$769$	−1.40196e7	−0.854908	−0.427454	+	0.904037i	$0.640589\pi$
$770$	0	0
$771$	6.44171e6	0.390270
$772$	1.41900e6	0.0856920
$773$	−3.40453e6	−0.204931	−0.102466	−	0.994737i	$-0.532673\pi$
$773$	−3.40453e6	−0.204931	−0.102466	+	0.994737i	$0.532673\pi$
$774$	−3.28641e6	−0.197183
$775$	0	0
$776$	1.14349e7	0.681675
$777$	880921.	0.0523461
$778$	8.73246e6	0.517235
$779$	1.11179e7	0.656418
$780$	0	0
$781$	−3.05777e6	−0.179382
$782$	2.16394e6	0.126540
$783$	489004.	0.0285042
$784$	2.06576e7	1.20030
$785$	0	0
$786$	−9.71667e6	−0.560998
$787$	5.77229e6	0.332209	0.166105	−	0.986108i	$-0.446881\pi$
$787$	5.77229e6	0.332209	0.166105	+	0.986108i	$0.446881\pi$
$788$	1.42496e6	0.0817497
$789$	−1.89029e7	−1.08103
$790$	0	0
$791$	−1.39486e6	−0.0792662
$792$	1.42530e6	0.0807410
$793$	−5.48672e6	−0.309835
$794$	1.14185e7	0.642771
$795$	0	0
$796$	4.36701e6	0.244288
$797$	3.55488e7	1.98234	0.991171	−	0.132589i	$-0.0423291\pi$
$797$	3.55488e7	1.98234	0.991171	+	0.132589i	$0.0423291\pi$
$798$	1.14837e6	0.0638370
$799$	6.57978e6	0.364624
$800$	0	0
$801$	−6.51626e6	−0.358854
$802$	2.37857e7	1.30581
$803$	2.89897e6	0.158655
$804$	3.29739e6	0.179900
$805$	0	0
$806$	−6.43644e6	−0.348986
$807$	1.27293e7	0.688052
$808$	−3.20284e6	−0.172587
$809$	8.35527e6	0.448838	0.224419	−	0.974493i	$-0.427952\pi$
$809$	8.35527e6	0.448838	0.224419	+	0.974493i	$0.427952\pi$
$810$	0	0
$811$	3.83660e6	0.204831	0.102415	−	0.994742i	$-0.467343\pi$
$811$	3.83660e6	0.204831	0.102415	+	0.994742i	$0.467343\pi$
$812$	62832.9	0.00334424
$813$	−1.71432e6	−0.0909630
$814$	−7.64542e6	−0.404427
$815$	0	0
$816$	−6.85510e6	−0.360403
$817$	−1.25466e7	−0.657616
$818$	−5.30491e6	−0.277201
$819$	−656843.	−0.0342178
$820$	0	0
$821$	−9.99705e6	−0.517624	−0.258812	−	0.965928i	$-0.583331\pi$
$821$	−9.99705e6	−0.517624	−0.258812	+	0.965928i	$0.583331\pi$
$822$	−1.35209e6	−0.0697953
$823$	−2.67331e7	−1.37578	−0.687890	−	0.725815i	$-0.741463\pi$
$823$	−2.67331e7	−1.37578	−0.687890	+	0.725815i	$0.741463\pi$
$824$	1.23619e7	0.634257
$825$	0	0
$826$	138340.	0.00705500
$827$	951161.	0.0483605	0.0241802	−	0.999708i	$-0.492302\pi$
$827$	951161.	0.0483605	0.0241802	+	0.999708i	$0.492302\pi$
$828$	−415592.	−0.0210664
$829$	−1.44000e7	−0.727741	−0.363870	−	0.931450i	$-0.618545\pi$
$829$	−1.44000e7	−0.727741	−0.363870	+	0.931450i	$0.618545\pi$
$830$	0	0
$831$	−8.98244e6	−0.451223
$832$	1.49066e7	0.746572
$833$	1.02926e7	0.513938
$834$	6.28587e6	0.312932
$835$	0	0
$836$	−2.26256e6	−0.111965
$837$	896344.	0.0442243
$838$	−3.78265e7	−1.86074
$839$	3.42903e6	0.168177	0.0840884	−	0.996458i	$-0.473202\pi$
$839$	3.42903e6	0.168177	0.0840884	+	0.996458i	$0.473202\pi$
$840$	0	0
$841$	−2.00612e7	−0.978063
$842$	2.24371e7	1.09065
$843$	1.28779e7	0.624133
$844$	−1.10194e7	−0.532478
$845$	0	0
$846$	−5.56645e6	−0.267395
$847$	−145928.	−0.00698924
$848$	−2.69804e7	−1.28843
$849$	2.16446e7	1.03058
$850$	0	0
$851$	−5.36135e6	−0.253776
$852$	−2.13745e6	−0.100878
$853$	1.57990e7	0.743460	0.371730	−	0.928341i	$-0.378765\pi$
$853$	1.57990e7	0.743460	0.371730	+	0.928341i	$0.378765\pi$
$854$	432476.	0.0202916
$855$	0	0
$856$	1.87788e7	0.875956
$857$	5.06147e6	0.235410	0.117705	−	0.993049i	$-0.462446\pi$
$857$	5.06147e6	0.235410	0.117705	+	0.993049i	$0.462446\pi$
$858$	5.70067e6	0.264367
$859$	−2.48302e7	−1.14815	−0.574074	−	0.818803i	$-0.694638\pi$
$859$	−2.48302e7	−1.14815	−0.574074	+	0.818803i	$0.694638\pi$
$860$	0	0
$861$	501250.	0.0230434
$862$	−2.83890e7	−1.30131
$863$	1.03123e7	0.471336	0.235668	−	0.971834i	$-0.424272\pi$
$863$	1.03123e7	0.471336	0.235668	+	0.971834i	$0.424272\pi$
$864$	2.40691e6	0.109692
$865$	0	0
$866$	6.36800e6	0.288541
$867$	9.36318e6	0.423035
$868$	115173.	0.00518860
$869$	1.22868e7	0.551937
$870$	0	0
$871$	−3.17178e7	−1.41663
$872$	2.23897e7	0.997142
$873$	−6.36913e6	−0.282842
$874$	−6.98904e6	−0.309484
$875$	0	0
$876$	2.02645e6	0.0892226
$877$	1.04098e7	0.457027	0.228514	−	0.973541i	$-0.426613\pi$
$877$	1.04098e7	0.457027	0.228514	+	0.973541i	$0.426613\pi$
$878$	−1.73453e7	−0.759355
$879$	−6.20261e6	−0.270771
$880$	0	0
$881$	−1.99961e7	−0.867970	−0.433985	−	0.900920i	$-0.642893\pi$
$881$	−1.99961e7	−0.867970	−0.433985	+	0.900920i	$0.642893\pi$
$882$	−8.70743e6	−0.376894
$883$	−2.42565e7	−1.04695	−0.523475	−	0.852041i	$-0.675364\pi$
$883$	−2.42565e7	−1.04695	−0.523475	+	0.852041i	$0.675364\pi$
$884$	−4.71032e6	−0.202731
$885$	0	0
$886$	1.18856e7	0.508671
$887$	3.28464e7	1.40178	0.700888	−	0.713271i	$-0.252787\pi$
$887$	3.28464e7	1.40178	0.700888	+	0.713271i	$0.252787\pi$
$888$	1.28530e7	0.546983
$889$	1.19387e6	0.0506644
$890$	0	0
$891$	−793881.	−0.0335013
$892$	4.64624e6	0.195519
$893$	−2.12512e7	−0.891775
$894$	1.60377e7	0.671118
$895$	0	0
$896$	−2.22803e6	−0.0927151
$897$	3.99760e6	0.165889
$898$	−5.18781e6	−0.214681
$899$	824769.	0.0340356
$900$	0	0
$901$	−1.34429e7	−0.551672
$902$	−4.35030e6	−0.178034
$903$	−565664.	−0.0230855
$904$	−2.03516e7	−0.828280
$905$	0	0
$906$	−1.89045e7	−0.765146
$907$	−1.54289e7	−0.622753	−0.311377	−	0.950287i	$-0.600790\pi$
$907$	−1.54289e7	−0.622753	−0.311377	+	0.950287i	$0.600790\pi$
$908$	2.06209e6	0.0830029
$909$	1.78395e6	0.0716101
$910$	0	0
$911$	3.82548e7	1.52718	0.763590	−	0.645701i	$-0.223435\pi$
$911$	3.82548e7	1.52718	0.763590	+	0.645701i	$0.223435\pi$
$912$	2.21404e7	0.881453
$913$	1.76019e6	0.0698847
$914$	3.07849e7	1.21891
$915$	0	0
$916$	−1.16208e7	−0.457611
$917$	−1.67245e6	−0.0656795
$918$	2.88951e6	0.113167
$919$	3.21575e7	1.25601	0.628005	−	0.778209i	$-0.283872\pi$
$919$	3.21575e7	1.25601	0.628005	+	0.778209i	$0.283872\pi$
$920$	0	0
$921$	8.87591e6	0.344797
$922$	−5.17966e7	−2.00666
$923$	2.05603e7	0.794373
$924$	−102007.	−0.00393052
$925$	0	0
$926$	−4.15868e6	−0.159378
$927$	−6.88544e6	−0.263168
$928$	2.21471e6	0.0844204
$929$	2.16438e7	0.822802	0.411401	−	0.911455i	$-0.365040\pi$
$929$	2.16438e7	0.822802	0.411401	+	0.911455i	$0.365040\pi$
$930$	0	0
$931$	−3.32426e7	−1.25696
$932$	−790890.	−0.0298247
$933$	1.53253e7	0.576374
$934$	−3.84512e7	−1.44226
$935$	0	0
$936$	−9.58364e6	−0.357554
$937$	−2.80993e7	−1.04555	−0.522776	−	0.852470i	$-0.675104\pi$
$937$	−2.80993e7	−1.04555	−0.522776	+	0.852470i	$0.675104\pi$
$938$	2.50007e6	0.0927779
$939$	−2.63100e7	−0.973770
$940$	0	0
$941$	−1.13300e7	−0.417116	−0.208558	−	0.978010i	$-0.566877\pi$
$941$	−1.13300e7	−0.417116	−0.208558	+	0.978010i	$0.566877\pi$
$942$	2.12107e7	0.778804
$943$	−3.05065e6	−0.111715
$944$	2.66718e6	0.0974144
$945$	0	0
$946$	4.90933e6	0.178359
$947$	−3.04381e7	−1.10292	−0.551458	−	0.834203i	$-0.685928\pi$
$947$	−3.04381e7	−1.10292	−0.551458	+	0.834203i	$0.685928\pi$
$948$	8.58875e6	0.310391
$949$	−1.94925e7	−0.702589
$950$	0	0
$951$	−1.52815e7	−0.547918
$952$	−892920.	−0.0319316
$953$	2.97688e7	1.06177	0.530883	−	0.847445i	$-0.321860\pi$
$953$	2.97688e7	1.06177	0.530883	+	0.847445i	$0.321860\pi$
$954$	1.13726e7	0.404566
$955$	0	0
$956$	−1.06447e6	−0.0376694
$957$	−730488.	−0.0257830
$958$	−4.85060e7	−1.70758
$959$	−232724.	−0.00817137
$960$	0	0
$961$	−2.71174e7	−0.947194
$962$	5.14073e7	1.79096
$963$	−1.04596e7	−0.363454
$964$	−5.79270e6	−0.200765
$965$	0	0
$966$	−315100.	−0.0108644
$967$	1.65127e7	0.567875	0.283937	−	0.958843i	$-0.408359\pi$
$967$	1.65127e7	0.567875	0.283937	+	0.958843i	$0.408359\pi$
$968$	−2.12916e6	−0.0730330
$969$	1.10314e7	0.377416
$970$	0	0
$971$	3.43022e7	1.16755	0.583773	−	0.811917i	$-0.301576\pi$
$971$	3.43022e7	1.16755	0.583773	+	0.811917i	$0.301576\pi$
$972$	−554941.	−0.0188400
$973$	1.08194e6	0.0366370
$974$	−4.32467e7	−1.46068
$975$	0	0
$976$	8.33811e6	0.280184
$977$	−4.76980e7	−1.59869	−0.799344	−	0.600873i	$-0.794820\pi$
$977$	−4.76980e7	−1.59869	−0.799344	+	0.600873i	$0.794820\pi$
$978$	3.72182e7	1.24425
$979$	9.73417e6	0.324595
$980$	0	0
$981$	−1.24709e7	−0.413736
$982$	−1.53711e7	−0.508660
$983$	−4.43331e6	−0.146334	−0.0731669	−	0.997320i	$-0.523311\pi$
$983$	−4.43331e6	−0.146334	−0.0731669	+	0.997320i	$0.523311\pi$
$984$	7.31347e6	0.240789
$985$	0	0
$986$	2.65878e6	0.0870944
$987$	−958108.	−0.0313056
$988$	1.52133e7	0.495827
$989$	3.44267e6	0.111919
$990$	0	0
$991$	−9.98431e6	−0.322949	−0.161474	−	0.986877i	$-0.551625\pi$
$991$	−9.98431e6	−0.322949	−0.161474	+	0.986877i	$0.551625\pi$
$992$	4.05957e6	0.130979
$993$	−1.29666e7	−0.417306
$994$	−1.62061e6	−0.0520249
$995$	0	0
$996$	1.23041e6	0.0393009
$997$	3.98385e7	1.26930	0.634652	−	0.772798i	$-0.281144\pi$
$997$	3.98385e7	1.26930	0.634652	+	0.772798i	$0.281144\pi$
$998$	−3.07595e7	−0.977581
$999$	−7.15903e6	−0.226955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.q.1.6	yes	8
5.4	even	2		825.6.a.p.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.3	✓	8	5.4	even	2
825.6.a.q.1.6	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+6.43413 q^{2} -9.00000 q^{3} +9.39797 q^{4} -57.9071 q^{6} -9.96708 q^{7} -145.424 q^{8} +81.0000 q^{9} -121.000 q^{11} -84.5818 q^{12} +813.596 q^{13} -64.1295 q^{14} -1236.41 q^{16} -616.038 q^{17} +521.164 q^{18} +1989.66 q^{19} +89.7037 q^{21} -778.529 q^{22} -545.944 q^{23} +1308.82 q^{24} +5234.78 q^{26} -729.000 q^{27} -93.6704 q^{28} -670.788 q^{29} -1229.55 q^{31} -3301.66 q^{32} +1089.00 q^{33} -3963.67 q^{34} +761.236 q^{36} +9820.34 q^{37} +12801.8 q^{38} -7322.36 q^{39} +5587.84 q^{41} +577.165 q^{42} -6305.91 q^{43} -1137.15 q^{44} -3512.67 q^{46} -10680.8 q^{47} +11127.7 q^{48} -16707.7 q^{49} +5544.35 q^{51} +7646.15 q^{52} +21821.5 q^{53} -4690.48 q^{54} +1449.46 q^{56} -17907.0 q^{57} -4315.93 q^{58} -2157.19 q^{59} -6743.79 q^{61} -7911.10 q^{62} -807.334 q^{63} +18321.9 q^{64} +7006.76 q^{66} -38984.7 q^{67} -5789.51 q^{68} +4913.49 q^{69} +25270.9 q^{71} -11779.4 q^{72} -23958.4 q^{73} +63185.3 q^{74} +18698.8 q^{76} +1206.02 q^{77} -47113.0 q^{78} -101544. q^{79} +6561.00 q^{81} +35952.9 q^{82} -14547.0 q^{83} +843.033 q^{84} -40573.0 q^{86} +6037.09 q^{87} +17596.3 q^{88} -80447.7 q^{89} -8109.18 q^{91} -5130.76 q^{92} +11066.0 q^{93} -68721.6 q^{94} +29715.0 q^{96} -78631.2 q^{97} -107499. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22}+ \cdots - 78408 q^{99}+O(q^{100}) \) 8 * q + 9 * q^2 - 72 * q^3 + 107 * q^4 - 81 * q^6 - 66 * q^7 + 207 * q^8 + 648 * q^9 - 968 * q^11 - 963 * q^12 + 382 * q^13 + 1048 * q^14 - 325 * q^16 + 288 * q^17 + 729 * q^18 - 988 * q^19 + 594 * q^21 - 1089 * q^22 + 5972 * q^23 - 1863 * q^24 - 14579 * q^26 - 5832 * q^27 + 5942 * q^28 + 1032 * q^29 - 4682 * q^31 + 3863 * q^32 + 8712 * q^33 + 2206 * q^34 + 8667 * q^36 - 17200 * q^37 + 11011 * q^38 - 3438 * q^39 - 13220 * q^41 - 9432 * q^42 + 22872 * q^43 - 12947 * q^44 + 9101 * q^46 + 6700 * q^47 + 2925 * q^48 - 43466 * q^49 - 2592 * q^51 + 5009 * q^52 + 6224 * q^53 - 6561 * q^54 + 20992 * q^56 + 8892 * q^57 + 33015 * q^58 - 77556 * q^59 + 11554 * q^61 + 12135 * q^62 - 5346 * q^63 - 149917 * q^64 + 9801 * q^66 + 20894 * q^67 + 91776 * q^68 - 53748 * q^69 - 21648 * q^71 + 16767 * q^72 + 64660 * q^73 - 179522 * q^74 + 24401 * q^76 + 7986 * q^77 + 131211 * q^78 - 22660 * q^79 + 52488 * q^81 - 56080 * q^82 + 100390 * q^83 - 53478 * q^84 + 47271 * q^86 - 9288 * q^87 - 25047 * q^88 - 25578 * q^89 + 73250 * q^91 + 95311 * q^92 + 42138 * q^93 - 170120 * q^94 - 34767 * q^96 + 142828 * q^97 + 303397 * q^98 - 78408 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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