LMFDB - Embedded newform 825.6.a.q.1.8

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			1.8
Root			$8.70245$ 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)

$f(q)$	$=$	\(q+9.70245 q^{2} -9.00000 q^{3} +62.1375 q^{4} -87.3220 q^{6} -79.4231 q^{7} +292.408 q^{8} +81.0000 q^{9} -121.000 q^{11} -559.238 q^{12} -780.381 q^{13} -770.598 q^{14} +848.670 q^{16} +1974.55 q^{17} +785.898 q^{18} +2318.66 q^{19} +714.808 q^{21} -1174.00 q^{22} +1282.47 q^{23} -2631.67 q^{24} -7571.61 q^{26} -729.000 q^{27} -4935.15 q^{28} -1278.69 q^{29} -6820.62 q^{31} -1122.87 q^{32} +1089.00 q^{33} +19158.0 q^{34} +5033.14 q^{36} -14719.4 q^{37} +22496.6 q^{38} +7023.43 q^{39} -8353.18 q^{41} +6935.38 q^{42} +13380.3 q^{43} -7518.64 q^{44} +12443.1 q^{46} +1363.31 q^{47} -7638.03 q^{48} -10499.0 q^{49} -17770.9 q^{51} -48490.9 q^{52} -35100.9 q^{53} -7073.08 q^{54} -23223.9 q^{56} -20867.9 q^{57} -12406.4 q^{58} -34493.5 q^{59} -45097.0 q^{61} -66176.7 q^{62} -6433.27 q^{63} -38052.0 q^{64} +10566.0 q^{66} +24151.0 q^{67} +122694. q^{68} -11542.3 q^{69} -18253.3 q^{71} +23685.0 q^{72} +32455.4 q^{73} -142815. q^{74} +144075. q^{76} +9610.19 q^{77} +68144.5 q^{78} -28488.4 q^{79} +6561.00 q^{81} -81046.3 q^{82} -44345.4 q^{83} +44416.4 q^{84} +129821. q^{86} +11508.2 q^{87} -35381.3 q^{88} -82213.9 q^{89} +61980.3 q^{91} +79689.6 q^{92} +61385.6 q^{93} +13227.5 q^{94} +10105.8 q^{96} -20136.3 q^{97} -101866. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	9.70245	1.71517	0.857583	−	0.514345i	$-0.171965\pi$
$2$	9.70245	1.71517	0.857583	+	0.514345i	$0.171965\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	62.1375	1.94180
$5$	0	0
$5$	0	0
$6$	−87.3220	−0.990252
$7$	−79.4231	−0.612635	−0.306317	−	0.951929i	$-0.599097\pi$
$7$	−79.4231	−0.612635	−0.306317	+	0.951929i	$0.599097\pi$
$8$	292.408	1.61534
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−559.238	−1.12110
$13$	−780.381	−1.28070	−0.640352	−	0.768082i	$-0.721211\pi$
$13$	−780.381	−1.28070	−0.640352	+	0.768082i	$0.721211\pi$
$14$	−770.598	−1.05077
$15$	0	0
$16$	848.670	0.828779
$17$	1974.55	1.65709	0.828544	−	0.559924i	$-0.189170\pi$
$17$	1974.55	1.65709	0.828544	+	0.559924i	$0.189170\pi$
$18$	785.898	0.571722
$19$	2318.66	1.47351	0.736754	−	0.676161i	$-0.236358\pi$
$19$	2318.66	1.47351	0.736754	+	0.676161i	$0.236358\pi$
$20$	0	0
$21$	714.808	0.353705
$22$	−1174.00	−0.517142
$23$	1282.47	0.505509	0.252754	−	0.967531i	$-0.418664\pi$
$23$	1282.47	0.505509	0.252754	+	0.967531i	$0.418664\pi$
$24$	−2631.67	−0.932616
$25$	0	0
$26$	−7571.61	−2.19662
$27$	−729.000	−0.192450
$28$	−4935.15	−1.18961
$29$	−1278.69	−0.282338	−0.141169	−	0.989986i	$-0.545086\pi$
$29$	−1278.69	−0.282338	−0.141169	+	0.989986i	$0.545086\pi$
$30$	0	0
$31$	−6820.62	−1.27473	−0.637367	−	0.770560i	$-0.719976\pi$
$31$	−6820.62	−1.27473	−0.637367	+	0.770560i	$0.719976\pi$
$32$	−1122.87	−0.193845
$33$	1089.00	0.174078
$34$	19158.0	2.84218
$35$	0	0
$36$	5033.14	0.647266
$37$	−14719.4	−1.76761	−0.883807	−	0.467853i	$-0.845028\pi$
$37$	−14719.4	−1.76761	−0.883807	+	0.467853i	$0.845028\pi$
$38$	22496.6	2.52731
$39$	7023.43	0.739414
$40$	0	0
$41$	−8353.18	−0.776055	−0.388027	−	0.921648i	$-0.626843\pi$
$41$	−8353.18	−0.776055	−0.388027	+	0.921648i	$0.626843\pi$
$42$	6935.38	0.606663
$43$	13380.3	1.10355	0.551777	−	0.833991i	$-0.313950\pi$
$43$	13380.3	1.10355	0.551777	+	0.833991i	$0.313950\pi$
$44$	−7518.64	−0.585474
$45$	0	0
$46$	12443.1	0.867031
$47$	1363.31	0.0900225	0.0450112	−	0.998986i	$-0.485668\pi$
$47$	1363.31	0.0900225	0.0450112	+	0.998986i	$0.485668\pi$
$48$	−7638.03	−0.478496
$49$	−10499.0	−0.624679
$50$	0	0
$51$	−17770.9	−0.956720
$52$	−48490.9	−2.48687
$53$	−35100.9	−1.71644	−0.858220	−	0.513282i	$-0.828430\pi$
$53$	−35100.9	−1.71644	−0.858220	+	0.513282i	$0.828430\pi$
$54$	−7073.08	−0.330084
$55$	0	0
$56$	−23223.9	−0.989613
$57$	−20867.9	−0.850730
$58$	−12406.4	−0.484256
$59$	−34493.5	−1.29005	−0.645025	−	0.764161i	$-0.723153\pi$
$59$	−34493.5	−1.29005	−0.645025	+	0.764161i	$0.723153\pi$
$60$	0	0
$61$	−45097.0	−1.55176	−0.775878	−	0.630883i	$-0.782693\pi$
$61$	−45097.0	−1.55176	−0.775878	+	0.630883i	$0.782693\pi$
$62$	−66176.7	−2.18638
$63$	−6433.27	−0.204212
$64$	−38052.0	−1.16126
$65$	0	0
$66$	10566.0	0.298572
$67$	24151.0	0.657275	0.328638	−	0.944456i	$-0.393411\pi$
$67$	24151.0	0.657275	0.328638	+	0.944456i	$0.393411\pi$
$68$	122694.	3.21773
$69$	−11542.3	−0.291855
$70$	0	0
$71$	−18253.3	−0.429731	−0.214865	−	0.976644i	$-0.568931\pi$
$71$	−18253.3	−0.429731	−0.214865	+	0.976644i	$0.568931\pi$
$72$	23685.0	0.538446
$73$	32455.4	0.712819	0.356410	−	0.934330i	$-0.384001\pi$
$73$	32455.4	0.712819	0.356410	+	0.934330i	$0.384001\pi$
$74$	−142815.	−3.03175
$75$	0	0
$76$	144075.	2.86125
$77$	9610.19	0.184716
$78$	68144.5	1.26822
$79$	−28488.4	−0.513570	−0.256785	−	0.966469i	$-0.582663\pi$
$79$	−28488.4	−0.513570	−0.256785	+	0.966469i	$0.582663\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−81046.3	−1.33106
$83$	−44345.4	−0.706566	−0.353283	−	0.935516i	$-0.614935\pi$
$83$	−44345.4	−0.706566	−0.353283	+	0.935516i	$0.614935\pi$
$84$	44416.4	0.686823
$85$	0	0
$86$	129821.	1.89278
$87$	11508.2	0.163008
$88$	−35381.3	−0.487043
$89$	−82213.9	−1.10020	−0.550098	−	0.835100i	$-0.685410\pi$
$89$	−82213.9	−1.10020	−0.550098	+	0.835100i	$0.685410\pi$
$90$	0	0
$91$	61980.3	0.784603
$92$	79689.6	0.981595
$93$	61385.6	0.735968
$94$	13227.5	0.154404
$95$	0	0
$96$	10105.8	0.111917
$97$	−20136.3	−0.217295	−0.108648	−	0.994080i	$-0.534652\pi$
$97$	−20136.3	−0.217295	−0.108648	+	0.994080i	$0.534652\pi$
$98$	−101866.	−1.07143
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−39455.8	−0.384865	−0.192432	−	0.981310i	$-0.561638\pi$
$101$	−39455.8	−0.384865	−0.192432	+	0.981310i	$0.561638\pi$
$102$	−172422.	−1.64093
$103$	100304.	0.931587	0.465793	−	0.884893i	$-0.345769\pi$
$103$	100304.	0.931587	0.465793	+	0.884893i	$0.345769\pi$
$104$	−228189.	−2.06877
$105$	0	0
$106$	−340565.	−2.94398
$107$	172037.	1.45265	0.726325	−	0.687351i	$-0.241227\pi$
$107$	172037.	1.45265	0.726325	+	0.687351i	$0.241227\pi$
$108$	−45298.2	−0.373699
$109$	217939.	1.75699	0.878495	−	0.477752i	$-0.158548\pi$
$109$	217939.	1.75699	0.878495	+	0.477752i	$0.158548\pi$
$110$	0	0
$111$	132475.	1.02053
$112$	−67403.9	−0.507739
$113$	−49169.5	−0.362243	−0.181121	−	0.983461i	$-0.557973\pi$
$113$	−49169.5	−0.362243	−0.181121	+	0.983461i	$0.557973\pi$
$114$	−202470.	−1.45914
$115$	0	0
$116$	−79454.3	−0.548242
$117$	−63210.9	−0.426901
$118$	−334671.	−2.21265
$119$	−156825.	−1.01519
$120$	0	0
$121$	14641.0	0.0909091
$122$	−437552.	−2.66152
$123$	75178.6	0.448055
$124$	−423816.	−2.47528
$125$	0	0
$126$	−62418.5	−0.350257
$127$	−55222.2	−0.303812	−0.151906	−	0.988395i	$-0.548541\pi$
$127$	−55222.2	−0.303812	−0.151906	+	0.988395i	$0.548541\pi$
$128$	−333266.	−1.79790
$129$	−120422.	−0.637138
$130$	0	0
$131$	56033.9	0.285281	0.142640	−	0.989775i	$-0.454441\pi$
$131$	56033.9	0.285281	0.142640	+	0.989775i	$0.454441\pi$
$132$	67667.7	0.338023
$133$	−184155.	−0.902722
$134$	234323.	1.12734
$135$	0	0
$136$	577373.	2.67676
$137$	−370432.	−1.68619	−0.843096	−	0.537763i	$-0.819269\pi$
$137$	−370432.	−1.68619	−0.843096	+	0.537763i	$0.819269\pi$
$138$	−111988.	−0.500581
$139$	−144540.	−0.634527	−0.317264	−	0.948337i	$-0.602764\pi$
$139$	−144540.	−0.634527	−0.317264	+	0.948337i	$0.602764\pi$
$140$	0	0
$141$	−12269.8	−0.0519745
$142$	−177102.	−0.737060
$143$	94426.1	0.386147
$144$	68742.2	0.276260
$145$	0	0
$146$	314897.	1.22260
$147$	94490.8	0.360658
$148$	−914630.	−3.43235
$149$	63090.5	0.232808	0.116404	−	0.993202i	$-0.462863\pi$
$149$	63090.5	0.232808	0.116404	+	0.993202i	$0.462863\pi$
$150$	0	0
$151$	400569.	1.42967	0.714834	−	0.699294i	$-0.246502\pi$
$151$	400569.	1.42967	0.714834	+	0.699294i	$0.246502\pi$
$152$	677993.	2.38021
$153$	159938.	0.552363
$154$	93242.4	0.316819
$155$	0	0
$156$	436419.	1.43579
$157$	−119299.	−0.386267	−0.193133	−	0.981173i	$-0.561865\pi$
$157$	−119299.	−0.386267	−0.193133	+	0.981173i	$0.561865\pi$
$158$	−276407.	−0.880858
$159$	315908.	0.990987
$160$	0	0
$161$	−101858.	−0.309692
$162$	63657.8	0.190574
$163$	−496732.	−1.46438	−0.732189	−	0.681101i	$-0.761501\pi$
$163$	−496732.	−1.46438	−0.732189	+	0.681101i	$0.761501\pi$
$164$	−519046.	−1.50694
$165$	0	0
$166$	−430259.	−1.21188
$167$	−402617.	−1.11712	−0.558562	−	0.829463i	$-0.688647\pi$
$167$	−402617.	−1.11712	−0.558562	+	0.829463i	$0.688647\pi$
$168$	209015.	0.571353
$169$	237702.	0.640200
$170$	0	0
$171$	187811.	0.491169
$172$	831417.	2.14288
$173$	642176.	1.63132	0.815660	−	0.578532i	$-0.196374\pi$
$173$	642176.	1.63132	0.815660	+	0.578532i	$0.196374\pi$
$174$	111657.	0.279585
$175$	0	0
$176$	−102689.	−0.249886
$177$	310441.	0.744811
$178$	−797676.	−1.88702
$179$	−628315.	−1.46570	−0.732850	−	0.680390i	$-0.761810\pi$
$179$	−628315.	−1.46570	−0.732850	+	0.680390i	$0.761810\pi$
$180$	0	0
$181$	227359.	0.515842	0.257921	−	0.966166i	$-0.416963\pi$
$181$	227359.	0.515842	0.257921	+	0.966166i	$0.416963\pi$
$182$	601360.	1.34573
$183$	405873.	0.895906
$184$	375005.	0.816568
$185$	0	0
$186$	595590.	1.26231
$187$	−238920.	−0.499631
$188$	84712.9	0.174805
$189$	57899.4	0.117902
$190$	0	0
$191$	232254.	0.460659	0.230329	−	0.973113i	$-0.426020\pi$
$191$	232254.	0.460659	0.230329	+	0.973113i	$0.426020\pi$
$192$	342468.	0.670451
$193$	199404.	0.385336	0.192668	−	0.981264i	$-0.438286\pi$
$193$	199404.	0.385336	0.192668	+	0.981264i	$0.438286\pi$
$194$	−195371.	−0.372697
$195$	0	0
$196$	−652380.	−1.21300
$197$	−66319.8	−0.121752	−0.0608762	−	0.998145i	$-0.519389\pi$
$197$	−66319.8	−0.121752	−0.0608762	+	0.998145i	$0.519389\pi$
$198$	−95093.7	−0.172381
$199$	−472509.	−0.845819	−0.422909	−	0.906172i	$-0.638991\pi$
$199$	−472509.	−0.845819	−0.422909	+	0.906172i	$0.638991\pi$
$200$	0	0
$201$	−217359.	−0.379478
$202$	−382818.	−0.660107
$203$	101557.	0.172970
$204$	−1.10424e6	−1.85776
$205$	0	0
$206$	973190.	1.59783
$207$	103880.	0.168503
$208$	−662286.	−1.06142
$209$	−280557.	−0.444279
$210$	0	0
$211$	−287964.	−0.445280	−0.222640	−	0.974901i	$-0.571467\pi$
$211$	−287964.	−0.445280	−0.222640	+	0.974901i	$0.571467\pi$
$212$	−2.18108e6	−3.33298
$213$	164280.	0.248105
$214$	1.66918e6	2.49154
$215$	0	0
$216$	−213165.	−0.310872
$217$	541715.	0.780946
$218$	2.11454e6	3.01353
$219$	−292098.	−0.411546
$220$	0	0
$221$	−1.54090e6	−2.12224
$222$	1.28533e6	1.75038
$223$	−1.05057e6	−1.41469	−0.707347	−	0.706867i	$-0.750108\pi$
$223$	−1.05057e6	−1.41469	−0.707347	+	0.706867i	$0.750108\pi$
$224$	89181.8	0.118756
$225$	0	0
$226$	−477065.	−0.621307
$227$	304726.	0.392505	0.196253	−	0.980553i	$-0.437123\pi$
$227$	304726.	0.392505	0.196253	+	0.980553i	$0.437123\pi$
$228$	−1.29668e6	−1.65194
$229$	−612466.	−0.771780	−0.385890	−	0.922545i	$-0.626106\pi$
$229$	−612466.	−0.771780	−0.385890	+	0.922545i	$0.626106\pi$
$230$	0	0
$231$	−86491.7	−0.106646
$232$	−373897.	−0.456071
$233$	1.51321e6	1.82604	0.913021	−	0.407914i	$-0.133744\pi$
$233$	1.51321e6	1.82604	0.913021	+	0.407914i	$0.133744\pi$
$234$	−613300.	−0.732206
$235$	0	0
$236$	−2.14334e6	−2.50502
$237$	256395.	0.296510
$238$	−1.52158e6	−1.74122
$239$	−749979.	−0.849287	−0.424644	−	0.905361i	$-0.639601\pi$
$239$	−749979.	−0.849287	−0.424644	+	0.905361i	$0.639601\pi$
$240$	0	0
$241$	251133.	0.278523	0.139261	−	0.990256i	$-0.455527\pi$
$241$	251133.	0.278523	0.139261	+	0.990256i	$0.455527\pi$
$242$	142054.	0.155924
$243$	−59049.0	−0.0641500
$244$	−2.80222e6	−3.01319
$245$	0	0
$246$	729417.	0.768490
$247$	−1.80944e6	−1.88713
$248$	−1.99440e6	−2.05913
$249$	399108.	0.407936
$250$	0	0
$251$	901997.	0.903692	0.451846	−	0.892096i	$-0.350766\pi$
$251$	901997.	0.903692	0.451846	+	0.892096i	$0.350766\pi$
$252$	−399747.	−0.396537
$253$	−155179.	−0.152417
$254$	−535791.	−0.521088
$255$	0	0
$256$	−2.01583e6	−1.92245
$257$	1.08538e6	1.02506	0.512528	−	0.858671i	$-0.328709\pi$
$257$	1.08538e6	1.02506	0.512528	+	0.858671i	$0.328709\pi$
$258$	−1.16839e6	−1.09280
$259$	1.16906e6	1.08290
$260$	0	0
$261$	−103574.	−0.0941126
$262$	543666.	0.489304
$263$	522831.	0.466092	0.233046	−	0.972466i	$-0.425131\pi$
$263$	522831.	0.466092	0.233046	+	0.972466i	$0.425131\pi$
$264$	318432.	0.281194
$265$	0	0
$266$	−1.78675e6	−1.54832
$267$	739925.	0.635199
$268$	1.50068e6	1.27630
$269$	1.48280e6	1.24940	0.624699	−	0.780866i	$-0.285222\pi$
$269$	1.48280e6	1.24940	0.624699	+	0.780866i	$0.285222\pi$
$270$	0	0
$271$	−1.16789e6	−0.966002	−0.483001	−	0.875620i	$-0.660453\pi$
$271$	−1.16789e6	−0.966002	−0.483001	+	0.875620i	$0.660453\pi$
$272$	1.67574e6	1.37336
$273$	−557822.	−0.452991
$274$	−3.59410e6	−2.89210
$275$	0	0
$276$	−717207.	−0.566724
$277$	793314.	0.621220	0.310610	−	0.950537i	$-0.399467\pi$
$277$	793314.	0.621220	0.310610	+	0.950537i	$0.399467\pi$
$278$	−1.40239e6	−1.08832
$279$	−552470.	−0.424911
$280$	0	0
$281$	−1.79487e6	−1.35602	−0.678010	−	0.735053i	$-0.737157\pi$
$281$	−1.79487e6	−1.35602	−0.678010	+	0.735053i	$0.737157\pi$
$282$	−119047.	−0.0891449
$283$	1.36160e6	1.01061	0.505306	−	0.862940i	$-0.331379\pi$
$283$	1.36160e6	1.01061	0.505306	+	0.862940i	$0.331379\pi$
$284$	−1.13422e6	−0.834450
$285$	0	0
$286$	916165.	0.662306
$287$	663435.	0.475438
$288$	−90952.5	−0.0646150
$289$	2.47899e6	1.74594
$290$	0	0
$291$	181227.	0.125455
$292$	2.01670e6	1.38415
$293$	353801.	0.240763	0.120382	−	0.992728i	$-0.461588\pi$
$293$	353801.	0.240763	0.120382	+	0.992728i	$0.461588\pi$
$294$	916792.	0.618589
$295$	0	0
$296$	−4.30408e6	−2.85529
$297$	88209.0	0.0580259
$298$	612132.	0.399305
$299$	−1.00082e6	−0.647406
$300$	0	0
$301$	−1.06270e6	−0.676076
$302$	3.88650e6	2.45212
$303$	355103.	0.222202
$304$	1.96777e6	1.22121
$305$	0	0
$306$	1.55179e6	0.947394
$307$	2.53723e6	1.53644	0.768218	−	0.640189i	$-0.221144\pi$
$307$	2.53723e6	1.53644	0.768218	+	0.640189i	$0.221144\pi$
$308$	597153.	0.358682
$309$	−902732.	−0.537852
$310$	0	0
$311$	1.13460e6	0.665183	0.332592	−	0.943071i	$-0.392077\pi$
$311$	1.13460e6	0.665183	0.332592	+	0.943071i	$0.392077\pi$
$312$	2.05370e6	1.19440
$313$	−1.91840e6	−1.10682	−0.553412	−	0.832908i	$-0.686675\pi$
$313$	−1.91840e6	−1.10682	−0.553412	+	0.832908i	$0.686675\pi$
$314$	−1.15749e6	−0.662511
$315$	0	0
$316$	−1.77020e6	−0.997249
$317$	−1.47779e6	−0.825970	−0.412985	−	0.910738i	$-0.635514\pi$
$317$	−1.47779e6	−0.825970	−0.412985	+	0.910738i	$0.635514\pi$
$318$	3.06508e6	1.69971
$319$	154721.	0.0851280
$320$	0	0
$321$	−1.54833e6	−0.838688
$322$	−988271.	−0.531173
$323$	4.57830e6	2.44173
$324$	407684.	0.215755
$325$	0	0
$326$	−4.81952e6	−2.51165
$327$	−1.96145e6	−1.01440
$328$	−2.44253e6	−1.25359
$329$	−108278.	−0.0551509
$330$	0	0
$331$	−1.74625e6	−0.876066	−0.438033	−	0.898959i	$-0.644325\pi$
$331$	−1.74625e6	−0.876066	−0.438033	+	0.898959i	$0.644325\pi$
$332$	−2.75551e6	−1.37201
$333$	−1.19228e6	−0.589204
$334$	−3.90637e6	−1.91605
$335$	0	0
$336$	606635.	0.293143
$337$	3.76831e6	1.80747	0.903736	−	0.428090i	$-0.140813\pi$
$337$	3.76831e6	1.80747	0.903736	+	0.428090i	$0.140813\pi$
$338$	2.30629e6	1.09805
$339$	442526.	0.209141
$340$	0	0
$341$	825295.	0.384347
$342$	1.82223e6	0.842437
$343$	2.16872e6	0.995335
$344$	3.91249e6	1.78261
$345$	0	0
$346$	6.23068e6	2.79799
$347$	−1.08132e6	−0.482094	−0.241047	−	0.970513i	$-0.577491\pi$
$347$	−1.08132e6	−0.482094	−0.241047	+	0.970513i	$0.577491\pi$
$348$	715089.	0.316528
$349$	2.13960e6	0.940304	0.470152	−	0.882586i	$-0.344199\pi$
$349$	2.13960e6	0.940304	0.470152	+	0.882586i	$0.344199\pi$
$350$	0	0
$351$	568898.	0.246471
$352$	135867.	0.0584465
$353$	−1.07749e6	−0.460234	−0.230117	−	0.973163i	$-0.573911\pi$
$353$	−1.07749e6	−0.460234	−0.230117	+	0.973163i	$0.573911\pi$
$354$	3.01204e6	1.27748
$355$	0	0
$356$	−5.10857e6	−2.13636
$357$	1.41142e6	0.586120
$358$	−6.09620e6	−2.51392
$359$	−3.88382e6	−1.59046	−0.795230	−	0.606308i	$-0.792650\pi$
$359$	−3.88382e6	−1.59046	−0.795230	+	0.606308i	$0.792650\pi$
$360$	0	0
$361$	2.90007e6	1.17122
$362$	2.20594e6	0.884755
$363$	−131769.	−0.0524864
$364$	3.85130e6	1.52354
$365$	0	0
$366$	3.93796e6	1.53663
$367$	1.46017e6	0.565899	0.282949	−	0.959135i	$-0.408687\pi$
$367$	1.46017e6	0.565899	0.282949	+	0.959135i	$0.408687\pi$
$368$	1.08840e6	0.418955
$369$	−676608.	−0.258685
$370$	0	0
$371$	2.78782e6	1.05155
$372$	3.81435e6	1.42910
$373$	4.78602e6	1.78116	0.890579	−	0.454829i	$-0.150300\pi$
$373$	4.78602e6	1.78116	0.890579	+	0.454829i	$0.150300\pi$
$374$	−2.31811e6	−0.856950
$375$	0	0
$376$	398643.	0.145417
$377$	997862.	0.361591
$378$	561766.	0.202221
$379$	−1.63705e6	−0.585414	−0.292707	−	0.956202i	$-0.594556\pi$
$379$	−1.63705e6	−0.585414	−0.292707	+	0.956202i	$0.594556\pi$
$380$	0	0
$381$	497000.	0.175406
$382$	2.25343e6	0.790107
$383$	−4.09383e6	−1.42604	−0.713022	−	0.701142i	$-0.752674\pi$
$383$	−4.09383e6	−1.42604	−0.713022	+	0.701142i	$0.752674\pi$
$384$	2.99939e6	1.03802
$385$	0	0
$386$	1.93470e6	0.660916
$387$	1.08380e6	0.367852
$388$	−1.25122e6	−0.421943
$389$	−181312.	−0.0607509	−0.0303755	−	0.999539i	$-0.509670\pi$
$389$	−181312.	−0.0607509	−0.0303755	+	0.999539i	$0.509670\pi$
$390$	0	0
$391$	2.53230e6	0.837672
$392$	−3.06998e6	−1.00907
$393$	−504305.	−0.164707
$394$	−643464.	−0.208826
$395$	0	0
$396$	−609010.	−0.195158
$397$	907.679	0.000289039 0	0.000144519	−	1.00000i	$-0.499954\pi$
$397$	907.679	0.000289039 0	0.000144519		1.00000i	$0.499954\pi$
$398$	−4.58449e6	−1.45072
$399$	1.65739e6	0.521187
$400$	0	0
$401$	−4.23119e6	−1.31402	−0.657009	−	0.753883i	$-0.728179\pi$
$401$	−4.23119e6	−1.31402	−0.657009	+	0.753883i	$0.728179\pi$
$402$	−2.10891e6	−0.650868
$403$	5.32268e6	1.63256
$404$	−2.45169e6	−0.747329
$405$	0	0
$406$	985353.	0.296672
$407$	1.78105e6	0.532955
$408$	−5.19636e6	−1.54543
$409$	−2.62102e6	−0.774750	−0.387375	−	0.921922i	$-0.626618\pi$
$409$	−2.62102e6	−0.774750	−0.387375	+	0.921922i	$0.626618\pi$
$410$	0	0
$411$	3.33389e6	0.973523
$412$	6.23261e6	1.80895
$413$	2.73958e6	0.790330
$414$	1.00789e6	0.289010
$415$	0	0
$416$	876267.	0.248258
$417$	1.30086e6	0.366345
$418$	−2.72209e6	−0.762013
$419$	2.02046e6	0.562233	0.281116	−	0.959674i	$-0.409295\pi$
$419$	2.02046e6	0.562233	0.281116	+	0.959674i	$0.409295\pi$
$420$	0	0
$421$	5.46228e6	1.50200	0.750998	−	0.660304i	$-0.229573\pi$
$421$	5.46228e6	1.50200	0.750998	+	0.660304i	$0.229573\pi$
$422$	−2.79396e6	−0.763729
$423$	110428.	0.0300075
$424$	−1.02638e7	−2.77263
$425$	0	0
$426$	1.59392e6	0.425542
$427$	3.58174e6	0.950659
$428$	1.06899e7	2.82075
$429$	−849835.	−0.222942
$430$	0	0
$431$	−640161.	−0.165995	−0.0829977	−	0.996550i	$-0.526449\pi$
$431$	−640161.	−0.165995	−0.0829977	+	0.996550i	$0.526449\pi$
$432$	−618680.	−0.159499
$433$	5.52690e6	1.41665	0.708324	−	0.705888i	$-0.249452\pi$
$433$	5.52690e6	1.41665	0.708324	+	0.705888i	$0.249452\pi$
$434$	5.25596e6	1.33945
$435$	0	0
$436$	1.35422e7	3.41172
$437$	2.97361e6	0.744870
$438$	−2.83407e6	−0.705871
$439$	3.50611e6	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$439$	3.50611e6	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$440$	0	0
$441$	−850417.	−0.208226
$442$	−1.49505e7	−3.63999
$443$	3.37380e6	0.816790	0.408395	−	0.912805i	$-0.366089\pi$
$443$	3.37380e6	0.816790	0.408395	+	0.912805i	$0.366089\pi$
$444$	8.23167e6	1.98167
$445$	0	0
$446$	−1.01931e7	−2.42644
$447$	−567814.	−0.134412
$448$	3.02221e6	0.711425
$449$	−1.55483e6	−0.363970	−0.181985	−	0.983301i	$-0.558252\pi$
$449$	−1.55483e6	−0.363970	−0.181985	+	0.983301i	$0.558252\pi$
$450$	0	0
$451$	1.01073e6	0.233989
$452$	−3.05527e6	−0.703402
$453$	−3.60512e6	−0.825419
$454$	2.95659e6	0.673212
$455$	0	0
$456$	−6.10193e6	−1.37422
$457$	−6.22946e6	−1.39527	−0.697637	−	0.716451i	$-0.745765\pi$
$457$	−6.22946e6	−1.39527	−0.697637	+	0.716451i	$0.745765\pi$
$458$	−5.94242e6	−1.32373
$459$	−1.43945e6	−0.318907
$460$	0	0
$461$	458075.	0.100389	0.0501943	−	0.998739i	$-0.484016\pi$
$461$	458075.	0.100389	0.0501943	+	0.998739i	$0.484016\pi$
$462$	−839181.	−0.182916
$463$	−5.20799e6	−1.12906	−0.564531	−	0.825412i	$-0.690943\pi$
$463$	−5.20799e6	−1.12906	−0.564531	+	0.825412i	$0.690943\pi$
$464$	−1.08518e6	−0.233996
$465$	0	0
$466$	1.46819e7	3.13197
$467$	−1.67532e6	−0.355473	−0.177736	−	0.984078i	$-0.556877\pi$
$467$	−1.67532e6	−0.355473	−0.177736	+	0.984078i	$0.556877\pi$
$468$	−3.92777e6	−0.828955
$469$	−1.91814e6	−0.402670
$470$	0	0
$471$	1.07369e6	0.223011
$472$	−1.00862e7	−2.08387
$473$	−1.61901e6	−0.332734
$474$	2.48766e6	0.508564
$475$	0	0
$476$	−9.74470e6	−1.97129
$477$	−2.84317e6	−0.572147
$478$	−7.27664e6	−1.45667
$479$	2.61462e6	0.520679	0.260340	−	0.965517i	$-0.416166\pi$
$479$	2.61462e6	0.520679	0.260340	+	0.965517i	$0.416166\pi$
$480$	0	0
$481$	1.14868e7	2.26379
$482$	2.43660e6	0.477713
$483$	916721.	0.178801
$484$	909755.	0.176527
$485$	0	0
$486$	−572920.	−0.110028
$487$	6.20282e6	1.18513	0.592566	−	0.805522i	$-0.298115\pi$
$487$	6.20282e6	1.18513	0.592566	+	0.805522i	$0.298115\pi$
$488$	−1.31867e7	−2.50661
$489$	4.47059e6	0.845459
$490$	0	0
$491$	−6.40953e6	−1.19984	−0.599919	−	0.800061i	$-0.704801\pi$
$491$	−6.40953e6	−1.19984	−0.599919	+	0.800061i	$0.704801\pi$
$492$	4.67141e6	0.870032
$493$	−2.52483e6	−0.467858
$494$	−1.75560e7	−3.23673
$495$	0	0
$496$	−5.78845e6	−1.05647
$497$	1.44974e6	0.263268
$498$	3.87233e6	0.699679
$499$	−8.63006e6	−1.55154	−0.775769	−	0.631017i	$-0.782638\pi$
$499$	−8.63006e6	−1.55154	−0.775769	+	0.631017i	$0.782638\pi$
$500$	0	0
$501$	3.62355e6	0.644971
$502$	8.75158e6	1.54998
$503$	−3.86443e6	−0.681028	−0.340514	−	0.940239i	$-0.610601\pi$
$503$	−3.86443e6	−0.681028	−0.340514	+	0.940239i	$0.610601\pi$
$504$	−1.88114e6	−0.329871
$505$	0	0
$506$	−1.50562e6	−0.261420
$507$	−2.13932e6	−0.369620
$508$	−3.43137e6	−0.589941
$509$	3.14562e6	0.538160	0.269080	−	0.963118i	$-0.413280\pi$
$509$	3.14562e6	0.538160	0.269080	+	0.963118i	$0.413280\pi$
$510$	0	0
$511$	−2.57771e6	−0.436698
$512$	−8.89398e6	−1.49941
$513$	−1.69030e6	−0.283577
$514$	1.05308e7	1.75814
$515$	0	0
$516$	−7.48275e6	−1.23719
$517$	−164961.	−0.0271428
$518$	1.13428e7	1.85736
$519$	−5.77959e6	−0.941843
$520$	0	0
$521$	−870807.	−0.140549	−0.0702745	−	0.997528i	$-0.522388\pi$
$521$	−870807.	−0.140549	−0.0702745	+	0.997528i	$0.522388\pi$
$522$	−1.00492e6	−0.161419
$523$	3.73764e6	0.597508	0.298754	−	0.954330i	$-0.403429\pi$
$523$	3.73764e6	0.597508	0.298754	+	0.954330i	$0.403429\pi$
$524$	3.48181e6	0.553958
$525$	0	0
$526$	5.07274e6	0.799426
$527$	−1.34676e7	−2.11235
$528$	924201.	0.144272
$529$	−4.79161e6	−0.744461
$530$	0	0
$531$	−2.79397e6	−0.430017
$532$	−1.14429e7	−1.75290
$533$	6.51867e6	0.993895
$534$	7.17909e6	1.08947
$535$	0	0
$536$	7.06192e6	1.06172
$537$	5.65484e6	0.846222
$538$	1.43868e7	2.14293
$539$	1.27038e6	0.188348
$540$	0	0
$541$	−3.18691e6	−0.468142	−0.234071	−	0.972220i	$-0.575205\pi$
$541$	−3.18691e6	−0.468142	−0.234071	+	0.972220i	$0.575205\pi$
$542$	−1.13314e7	−1.65685
$543$	−2.04623e6	−0.297821
$544$	−2.21716e6	−0.321218
$545$	0	0
$546$	−5.41224e6	−0.776955
$547$	−3.82413e6	−0.546467	−0.273234	−	0.961948i	$-0.588093\pi$
$547$	−3.82413e6	−0.546467	−0.273234	+	0.961948i	$0.588093\pi$
$548$	−2.30177e7	−3.27424
$549$	−3.65286e6	−0.517252
$550$	0	0
$551$	−2.96483e6	−0.416027
$552$	−3.37504e6	−0.471446
$553$	2.26263e6	0.314631
$554$	7.69709e6	1.06550
$555$	0	0
$556$	−8.98134e6	−1.23212
$557$	−7.46292e6	−1.01923	−0.509613	−	0.860404i	$-0.670211\pi$
$557$	−7.46292e6	−1.01923	−0.509613	+	0.860404i	$0.670211\pi$
$558$	−5.36031e6	−0.728794
$559$	−1.04417e7	−1.41333
$560$	0	0
$561$	2.15028e6	0.288462
$562$	−1.74146e7	−2.32580
$563$	5.50547e6	0.732021	0.366011	−	0.930611i	$-0.380723\pi$
$563$	5.50547e6	0.732021	0.366011	+	0.930611i	$0.380723\pi$
$564$	−762416.	−0.100924
$565$	0	0
$566$	1.32109e7	1.73337
$567$	−521095.	−0.0680705
$568$	−5.33742e6	−0.694161
$569$	−1.28692e6	−0.166637	−0.0833183	−	0.996523i	$-0.526552\pi$
$569$	−1.28692e6	−0.166637	−0.0833183	+	0.996523i	$0.526552\pi$
$570$	0	0
$571$	−1.08430e7	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$571$	−1.08430e7	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$572$	5.86740e6	0.749818
$573$	−2.09028e6	−0.265961
$574$	6.43695e6	0.815455
$575$	0	0
$576$	−3.08221e6	−0.387085
$577$	−2.33126e6	−0.291508	−0.145754	−	0.989321i	$-0.546561\pi$
$577$	−2.33126e6	−0.291508	−0.145754	+	0.989321i	$0.546561\pi$
$578$	2.40522e7	2.99458
$579$	−1.79463e6	−0.222474
$580$	0	0
$581$	3.52205e6	0.432867
$582$	1.75834e6	0.215177
$583$	4.24721e6	0.517526
$584$	9.49020e6	1.15144
$585$	0	0
$586$	3.43274e6	0.412949
$587$	2.84062e6	0.340266	0.170133	−	0.985421i	$-0.445580\pi$
$587$	2.84062e6	0.340266	0.170133	+	0.985421i	$0.445580\pi$
$588$	5.87142e6	0.700326
$589$	−1.58147e7	−1.87833
$590$	0	0
$591$	596878.	0.0702938
$592$	−1.24919e7	−1.46496
$593$	1.53492e7	1.79246	0.896231	−	0.443587i	$-0.146294\pi$
$593$	1.53492e7	1.79246	0.896231	+	0.443587i	$0.146294\pi$
$594$	855843.	0.0995241
$595$	0	0
$596$	3.92029e6	0.452066
$597$	4.25258e6	0.488334
$598$	−9.71038e6	−1.11041
$599$	1.21107e7	1.37912	0.689560	−	0.724228i	$-0.257804\pi$
$599$	1.21107e7	1.37912	0.689560	+	0.724228i	$0.257804\pi$
$600$	0	0
$601$	1.27801e7	1.44327	0.721635	−	0.692273i	$-0.243391\pi$
$601$	1.27801e7	1.44327	0.721635	+	0.692273i	$0.243391\pi$
$602$	−1.03108e7	−1.15958
$603$	1.95623e6	0.219092
$604$	2.48904e7	2.77612
$605$	0	0
$606$	3.44536e6	0.381113
$607$	−1.21041e7	−1.33341	−0.666703	−	0.745323i	$-0.732295\pi$
$607$	−1.21041e7	−1.33341	−0.666703	+	0.745323i	$0.732295\pi$
$608$	−2.60355e6	−0.285632
$609$	−914014.	−0.0998642
$610$	0	0
$611$	−1.06390e6	−0.115292
$612$	9.93818e6	1.07258
$613$	3.32675e6	0.357577	0.178788	−	0.983888i	$-0.442782\pi$
$613$	3.32675e6	0.357577	0.178788	+	0.983888i	$0.442782\pi$
$614$	2.46174e7	2.63524
$615$	0	0
$616$	2.81009e6	0.298379
$617$	−6.60697e6	−0.698698	−0.349349	−	0.936993i	$-0.613597\pi$
$617$	−6.60697e6	−0.698698	−0.349349	+	0.936993i	$0.613597\pi$
$618$	−8.75871e6	−0.922506
$619$	−6.09619e6	−0.639487	−0.319743	−	0.947504i	$-0.603597\pi$
$619$	−6.09619e6	−0.639487	−0.319743	+	0.947504i	$0.603597\pi$
$620$	0	0
$621$	−934922.	−0.0972852
$622$	1.10084e7	1.14090
$623$	6.52968e6	0.674018
$624$	5.96057e6	0.612811
$625$	0	0
$626$	−1.86132e7	−1.89839
$627$	2.52502e6	0.256505
$628$	−7.41293e6	−0.750051
$629$	−2.90643e7	−2.92909
$630$	0	0
$631$	−1.61232e7	−1.61205	−0.806026	−	0.591881i	$-0.798386\pi$
$631$	−1.61232e7	−1.61205	−0.806026	+	0.591881i	$0.798386\pi$
$632$	−8.33021e6	−0.829590
$633$	2.59168e6	0.257082
$634$	−1.43382e7	−1.41668
$635$	0	0
$636$	1.96297e7	1.92430
$637$	8.19320e6	0.800028
$638$	1.50117e6	0.146009
$639$	−1.47852e6	−0.143244
$640$	0	0
$641$	8.95786e6	0.861111	0.430555	−	0.902564i	$-0.358318\pi$
$641$	8.95786e6	0.861111	0.430555	+	0.902564i	$0.358318\pi$
$642$	−1.50226e7	−1.43849
$643$	1.99681e7	1.90462	0.952310	−	0.305131i	$-0.0987003\pi$
$643$	1.99681e7	1.90462	0.952310	+	0.305131i	$0.0987003\pi$
$644$	−6.32920e6	−0.601359
$645$	0	0
$646$	4.44207e7	4.18798
$647$	7.36055e6	0.691273	0.345636	−	0.938369i	$-0.387663\pi$
$647$	7.36055e6	0.691273	0.345636	+	0.938369i	$0.387663\pi$
$648$	1.91849e6	0.179482
$649$	4.17371e6	0.388965
$650$	0	0
$651$	−4.87543e6	−0.450880
$652$	−3.08657e7	−2.84353
$653$	1.03621e7	0.950961	0.475481	−	0.879726i	$-0.342274\pi$
$653$	1.03621e7	0.950961	0.475481	+	0.879726i	$0.342274\pi$
$654$	−1.90309e7	−1.73986
$655$	0	0
$656$	−7.08909e6	−0.643178
$657$	2.62889e6	0.237606
$658$	−1.05057e6	−0.0945930
$659$	−1.14611e7	−1.02805	−0.514023	−	0.857777i	$-0.671845\pi$
$659$	−1.14611e7	−1.02805	−0.514023	+	0.857777i	$0.671845\pi$
$660$	0	0
$661$	−1.23869e7	−1.10271	−0.551353	−	0.834272i	$-0.685888\pi$
$661$	−1.23869e7	−1.10271	−0.551353	+	0.834272i	$0.685888\pi$
$662$	−1.69429e7	−1.50260
$663$	1.38681e7	1.22527
$664$	−1.29669e7	−1.14134
$665$	0	0
$666$	−1.15680e7	−1.01058
$667$	−1.63988e6	−0.142724
$668$	−2.50176e7	−2.16923
$669$	9.45512e6	0.816774
$670$	0	0
$671$	5.45674e6	0.467872
$672$	−802637.	−0.0685639
$673$	1.92657e7	1.63964	0.819818	−	0.572624i	$-0.194075\pi$
$673$	1.92657e7	1.63964	0.819818	+	0.572624i	$0.194075\pi$
$674$	3.65618e7	3.10012
$675$	0	0
$676$	1.47702e7	1.24314
$677$	−1.13886e6	−0.0954988	−0.0477494	−	0.998859i	$-0.515205\pi$
$677$	−1.13886e6	−0.0954988	−0.0477494	+	0.998859i	$0.515205\pi$
$678$	4.29358e6	0.358712
$679$	1.59929e6	0.133123
$680$	0	0
$681$	−2.74254e6	−0.226613
$682$	8.00738e6	0.659219
$683$	1.17487e7	0.963691	0.481845	−	0.876256i	$-0.339967\pi$
$683$	1.17487e7	0.963691	0.481845	+	0.876256i	$0.339967\pi$
$684$	1.16701e7	0.953751
$685$	0	0
$686$	2.10419e7	1.70716
$687$	5.51220e6	0.445587
$688$	1.13554e7	0.914603
$689$	2.73921e7	2.19825
$690$	0	0
$691$	1.25506e7	0.999932	0.499966	−	0.866045i	$-0.333346\pi$
$691$	1.25506e7	0.999932	0.499966	+	0.866045i	$0.333346\pi$
$692$	3.99032e7	3.16769
$693$	778425.	0.0615721
$694$	−1.04915e7	−0.826872
$695$	0	0
$696$	3.36508e6	0.263313
$697$	−1.64938e7	−1.28599
$698$	2.07593e7	1.61278
$699$	−1.36189e7	−1.05427
$700$	0	0
$701$	−2.34000e7	−1.79854	−0.899272	−	0.437391i	$-0.855903\pi$
$701$	−2.34000e7	−1.79854	−0.899272	+	0.437391i	$0.855903\pi$
$702$	5.51970e6	0.422740
$703$	−3.41293e7	−2.60459
$704$	4.60429e6	0.350132
$705$	0	0
$706$	−1.04543e7	−0.789377
$707$	3.13370e6	0.235781
$708$	1.92900e7	1.44627
$709$	1.66095e7	1.24091	0.620455	−	0.784242i	$-0.286948\pi$
$709$	1.66095e7	1.24091	0.620455	+	0.784242i	$0.286948\pi$
$710$	0	0
$711$	−2.30756e6	−0.171190
$712$	−2.40400e7	−1.77719
$713$	−8.74726e6	−0.644389
$714$	1.36943e7	1.00529
$715$	0	0
$716$	−3.90420e7	−2.84609
$717$	6.74981e6	0.490336
$718$	−3.76825e7	−2.72790
$719$	1.98028e7	1.42858	0.714291	−	0.699848i	$-0.246749\pi$
$719$	1.98028e7	1.42858	0.714291	+	0.699848i	$0.246749\pi$
$720$	0	0
$721$	−7.96642e6	−0.570722
$722$	2.81377e7	2.00884
$723$	−2.26020e6	−0.160805
$724$	1.41275e7	1.00166
$725$	0	0
$726$	−1.27848e6	−0.0900229
$727$	−2.73066e7	−1.91616	−0.958079	−	0.286504i	$-0.907507\pi$
$727$	−2.73066e7	−1.91616	−0.958079	+	0.286504i	$0.907507\pi$
$728$	1.81235e7	1.26740
$729$	531441.	0.0370370
$730$	0	0
$731$	2.64200e7	1.82869
$732$	2.52199e7	1.73967
$733$	−1.08525e7	−0.746052	−0.373026	−	0.927821i	$-0.621680\pi$
$733$	−1.08525e7	−0.746052	−0.373026	+	0.927821i	$0.621680\pi$
$734$	1.41672e7	0.970610
$735$	0	0
$736$	−1.44005e6	−0.0979903
$737$	−2.92227e6	−0.198176
$738$	−6.56475e6	−0.443688
$739$	687400.	0.0463019	0.0231509	−	0.999732i	$-0.492630\pi$
$739$	687400.	0.0463019	0.0231509	+	0.999732i	$0.492630\pi$
$740$	0	0
$741$	1.62849e7	1.08953
$742$	2.70487e7	1.80359
$743$	−4.30995e6	−0.286418	−0.143209	−	0.989692i	$-0.545742\pi$
$743$	−4.30995e6	−0.286418	−0.143209	+	0.989692i	$0.545742\pi$
$744$	1.79496e7	1.18884
$745$	0	0
$746$	4.64361e7	3.05498
$747$	−3.59197e6	−0.235522
$748$	−1.48459e7	−0.970182
$749$	−1.36637e7	−0.889944
$750$	0	0
$751$	−9.07828e6	−0.587359	−0.293680	−	0.955904i	$-0.594880\pi$
$751$	−9.07828e6	−0.587359	−0.293680	+	0.955904i	$0.594880\pi$
$752$	1.15700e6	0.0746087
$753$	−8.11797e6	−0.521747
$754$	9.68171e6	0.620188
$755$	0	0
$756$	3.59773e6	0.228941
$757$	1.01884e7	0.646197	0.323098	−	0.946365i	$-0.395276\pi$
$757$	1.01884e7	0.646197	0.323098	+	0.946365i	$0.395276\pi$
$758$	−1.58834e7	−1.00408
$759$	1.39661e6	0.0879977
$760$	0	0
$761$	1.93077e7	1.20856	0.604282	−	0.796771i	$-0.293460\pi$
$761$	1.93077e7	1.20856	0.604282	+	0.796771i	$0.293460\pi$
$762$	4.82212e6	0.300850
$763$	−1.73094e7	−1.07639
$764$	1.44317e7	0.894506
$765$	0	0
$766$	−3.97202e7	−2.44590
$767$	2.69181e7	1.65217
$768$	1.81425e7	1.10992
$769$	−4.54597e6	−0.277211	−0.138605	−	0.990348i	$-0.544262\pi$
$769$	−4.54597e6	−0.277211	−0.138605	+	0.990348i	$0.544262\pi$
$770$	0	0
$771$	−9.76838e6	−0.591816
$772$	1.23904e7	0.748245
$773$	2.23550e7	1.34563	0.672814	−	0.739811i	$-0.265085\pi$
$773$	2.23550e7	1.34563	0.672814	+	0.739811i	$0.265085\pi$
$774$	1.05155e7	0.630927
$775$	0	0
$776$	−5.88800e6	−0.351005
$777$	−1.05216e7	−0.625213
$778$	−1.75917e6	−0.104198
$779$	−1.93682e7	−1.14352
$780$	0	0
$781$	2.20866e6	0.129569
$782$	2.45696e7	1.43675
$783$	932162.	0.0543359
$784$	−8.91016e6	−0.517721
$785$	0	0
$786$	−4.89299e6	−0.282500
$787$	−7.13003e6	−0.410350	−0.205175	−	0.978725i	$-0.565776\pi$
$787$	−7.13003e6	−0.410350	−0.205175	+	0.978725i	$0.565776\pi$
$788$	−4.12094e6	−0.236418
$789$	−4.70548e6	−0.269099
$790$	0	0
$791$	3.90519e6	0.221922
$792$	−2.86589e6	−0.162348
$793$	3.51929e7	1.98734
$794$	8806.71	0.000495750 0
$795$	0	0
$796$	−2.93605e7	−1.64241
$797$	2.47446e7	1.37986	0.689931	−	0.723875i	$-0.257641\pi$
$797$	2.47446e7	1.37986	0.689931	+	0.723875i	$0.257641\pi$
$798$	1.60808e7	0.893922
$799$	2.69193e6	0.149175
$800$	0	0
$801$	−6.65933e6	−0.366732
$802$	−4.10529e7	−2.25376
$803$	−3.92710e6	−0.214923
$804$	−1.35061e7	−0.736869
$805$	0	0
$806$	5.16431e7	2.80011
$807$	−1.33452e7	−0.721340
$808$	−1.15372e7	−0.621687
$809$	−7.53550e6	−0.404800	−0.202400	−	0.979303i	$-0.564874\pi$
$809$	−7.53550e6	−0.404800	−0.202400	+	0.979303i	$0.564874\pi$
$810$	0	0
$811$	1.62999e7	0.870229	0.435114	−	0.900375i	$-0.356708\pi$
$811$	1.62999e7	0.870229	0.435114	+	0.900375i	$0.356708\pi$
$812$	6.31051e6	0.335872
$813$	1.05110e7	0.557721
$814$	1.72806e7	0.914107
$815$	0	0
$816$	−1.50817e7	−0.792909
$817$	3.10242e7	1.62610
$818$	−2.54303e7	−1.32882
$819$	5.02040e6	0.261534
$820$	0	0
$821$	−2.27539e7	−1.17814	−0.589071	−	0.808081i	$-0.700506\pi$
$821$	−2.27539e7	−1.17814	−0.589071	+	0.808081i	$0.700506\pi$
$822$	3.23469e7	1.66975
$823$	−3.55290e7	−1.82845	−0.914226	−	0.405204i	$-0.867200\pi$
$823$	−3.55290e7	−1.82845	−0.914226	+	0.405204i	$0.867200\pi$
$824$	2.93295e7	1.50483
$825$	0	0
$826$	2.65806e7	1.35555
$827$	−4.75143e6	−0.241580	−0.120790	−	0.992678i	$-0.538543\pi$
$827$	−4.75143e6	−0.241580	−0.120790	+	0.992678i	$0.538543\pi$
$828$	6.45486e6	0.327198
$829$	−3.32259e6	−0.167915	−0.0839577	−	0.996469i	$-0.526756\pi$
$829$	−3.32259e6	−0.167915	−0.0839577	+	0.996469i	$0.526756\pi$
$830$	0	0
$831$	−7.13983e6	−0.358662
$832$	2.96951e7	1.48722
$833$	−2.07307e7	−1.03515
$834$	1.26215e7	0.628342
$835$	0	0
$836$	−1.74331e7	−0.862700
$837$	4.97223e6	0.245323
$838$	1.96034e7	0.964323
$839$	−2.00974e7	−0.985679	−0.492840	−	0.870120i	$-0.664041\pi$
$839$	−2.00974e7	−0.985679	−0.492840	+	0.870120i	$0.664041\pi$
$840$	0	0
$841$	−1.88761e7	−0.920285
$842$	5.29975e7	2.57617
$843$	1.61538e7	0.782899
$844$	−1.78934e7	−0.864643
$845$	0	0
$846$	1.07143e6	0.0514678
$847$	−1.16283e6	−0.0556941
$848$	−2.97891e7	−1.42255
$849$	−1.22544e7	−0.583478
$850$	0	0
$851$	−1.88773e7	−0.893543
$852$	1.02080e7	0.481770
$853$	1.62177e7	0.763161	0.381581	−	0.924336i	$-0.375380\pi$
$853$	1.62177e7	0.763161	0.381581	+	0.924336i	$0.375380\pi$
$854$	3.47517e7	1.63054
$855$	0	0
$856$	5.03048e7	2.34652
$857$	−2.68814e7	−1.25026	−0.625129	−	0.780521i	$-0.714954\pi$
$857$	−2.68814e7	−1.25026	−0.625129	+	0.780521i	$0.714954\pi$
$858$	−8.24548e6	−0.382382
$859$	3.23807e7	1.49728	0.748640	−	0.662977i	$-0.230707\pi$
$859$	3.23807e7	1.49728	0.748640	+	0.662977i	$0.230707\pi$
$860$	0	0
$861$	−5.97092e6	−0.274494
$862$	−6.21113e6	−0.284710
$863$	−6.65793e6	−0.304307	−0.152154	−	0.988357i	$-0.548621\pi$
$863$	−6.65793e6	−0.304307	−0.152154	+	0.988357i	$0.548621\pi$
$864$	818573.	0.0373055
$865$	0	0
$866$	5.36245e7	2.42979
$867$	−2.23109e7	−1.00802
$868$	3.36608e7	1.51644
$869$	3.44709e6	0.154847
$870$	0	0
$871$	−1.88469e7	−0.841774
$872$	6.37271e7	2.83813
$873$	−1.63104e6	−0.0724317
$874$	2.88513e7	1.27758
$875$	0	0
$876$	−1.81503e7	−0.799140
$877$	−1.18215e6	−0.0519006	−0.0259503	−	0.999663i	$-0.508261\pi$
$877$	−1.18215e6	−0.0519006	−0.0259503	+	0.999663i	$0.508261\pi$
$878$	3.40179e7	1.48926
$879$	−3.18421e6	−0.139005
$880$	0	0
$881$	3.03739e7	1.31844	0.659222	−	0.751949i	$-0.270886\pi$
$881$	3.03739e7	1.31844	0.659222	+	0.751949i	$0.270886\pi$
$882$	−8.25113e6	−0.357143
$883$	2.06314e7	0.890485	0.445243	−	0.895410i	$-0.353117\pi$
$883$	2.06314e7	0.890485	0.445243	+	0.895410i	$0.353117\pi$
$884$	−9.57477e7	−4.12095
$885$	0	0
$886$	3.27342e7	1.40093
$887$	3.19552e7	1.36374	0.681871	−	0.731473i	$-0.261167\pi$
$887$	3.19552e7	1.36374	0.681871	+	0.731473i	$0.261167\pi$
$888$	3.87367e7	1.64851
$889$	4.38592e6	0.186126
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−6.52797e7	−2.74705
$893$	3.16105e6	0.132649
$894$	−5.50919e6	−0.230539
$895$	0	0
$896$	2.64690e7	1.10146
$897$	9.00736e6	0.373780
$898$	−1.50856e7	−0.624270
$899$	8.72143e6	0.359905
$900$	0	0
$901$	−6.93085e7	−2.84429
$902$	9.80660e6	0.401331
$903$	9.56432e6	0.390333
$904$	−1.43775e7	−0.585145
$905$	0	0
$906$	−3.49785e7	−1.41573
$907$	3.27668e7	1.32256	0.661280	−	0.750139i	$-0.270014\pi$
$907$	3.27668e7	1.32256	0.661280	+	0.750139i	$0.270014\pi$
$908$	1.89349e7	0.762165
$909$	−3.19592e6	−0.128288
$910$	0	0
$911$	1.89497e7	0.756495	0.378248	−	0.925704i	$-0.376527\pi$
$911$	1.89497e7	0.756495	0.378248	+	0.925704i	$0.376527\pi$
$912$	−1.77100e7	−0.705067
$913$	5.36579e6	0.213038
$914$	−6.04410e7	−2.39313
$915$	0	0
$916$	−3.80571e7	−1.49864
$917$	−4.45038e6	−0.174773
$918$	−1.39662e7	−0.546978
$919$	−5.37819e6	−0.210062	−0.105031	−	0.994469i	$-0.533494\pi$
$919$	−5.37819e6	−0.210062	−0.105031	+	0.994469i	$0.533494\pi$
$920$	0	0
$921$	−2.28351e7	−0.887061
$922$	4.44445e6	0.172183
$923$	1.42446e7	0.550358
$924$	−5.37438e6	−0.207085
$925$	0	0
$926$	−5.05303e7	−1.93653
$927$	8.12459e6	0.310529
$928$	1.43580e6	0.0547298
$929$	−1.66380e7	−0.632503	−0.316251	−	0.948675i	$-0.602424\pi$
$929$	−1.66380e7	−0.632503	−0.316251	+	0.948675i	$0.602424\pi$
$930$	0	0
$931$	−2.43435e7	−0.920469
$932$	9.40273e7	3.54580
$933$	−1.02114e7	−0.384044
$934$	−1.62547e7	−0.609695
$935$	0	0
$936$	−1.84833e7	−0.689590
$937$	−3.18707e7	−1.18588	−0.592942	−	0.805245i	$-0.702034\pi$
$937$	−3.18707e7	−1.18588	−0.592942	+	0.805245i	$0.702034\pi$
$938$	−1.86107e7	−0.690645
$939$	1.72656e7	0.639025
$940$	0	0
$941$	−2.37797e7	−0.875452	−0.437726	−	0.899108i	$-0.644216\pi$
$941$	−2.37797e7	−0.875452	−0.437726	+	0.899108i	$0.644216\pi$
$942$	1.04174e7	0.382501
$943$	−1.07127e7	−0.392302
$944$	−2.92736e7	−1.06917
$945$	0	0
$946$	−1.57084e7	−0.570695
$947$	−5.44978e7	−1.97471	−0.987357	−	0.158510i	$-0.949331\pi$
$947$	−5.44978e7	−1.97471	−0.987357	+	0.158510i	$0.949331\pi$
$948$	1.59318e7	0.575762
$949$	−2.53276e7	−0.912910
$950$	0	0
$951$	1.33001e7	0.476874
$952$	−4.58567e7	−1.63988
$953$	2.08795e7	0.744710	0.372355	−	0.928090i	$-0.378550\pi$
$953$	2.08795e7	0.744710	0.372355	+	0.928090i	$0.378550\pi$
$954$	−2.75857e7	−0.981327
$955$	0	0
$956$	−4.66019e7	−1.64914
$957$	−1.39249e6	−0.0491487
$958$	2.53682e7	0.893051
$959$	2.94208e7	1.03302
$960$	0	0
$961$	1.78917e7	0.624947
$962$	1.11450e8	3.88277
$963$	1.39350e7	0.484217
$964$	1.56048e7	0.540835
$965$	0	0
$966$	8.89444e6	0.306673
$967$	−4.72474e7	−1.62484	−0.812422	−	0.583070i	$-0.801851\pi$
$967$	−4.72474e7	−1.62484	−0.812422	+	0.583070i	$0.801851\pi$
$968$	4.28114e6	0.146849
$969$	−4.12047e7	−1.40973
$970$	0	0
$971$	−1.81878e7	−0.619059	−0.309530	−	0.950890i	$-0.600172\pi$
$971$	−1.81878e7	−0.619059	−0.309530	+	0.950890i	$0.600172\pi$
$972$	−3.66916e6	−0.124566
$973$	1.14798e7	0.388734
$974$	6.01825e7	2.03270
$975$	0	0
$976$	−3.82725e7	−1.28606
$977$	−4.40237e7	−1.47554	−0.737769	−	0.675054i	$-0.764121\pi$
$977$	−4.40237e7	−1.47554	−0.737769	+	0.675054i	$0.764121\pi$
$978$	4.33757e7	1.45010
$979$	9.94788e6	0.331722
$980$	0	0
$981$	1.76531e7	0.585663
$982$	−6.21882e7	−2.05792
$983$	6.68153e6	0.220542	0.110271	−	0.993902i	$-0.464828\pi$
$983$	6.68153e6	0.220542	0.110271	+	0.993902i	$0.464828\pi$
$984$	2.19828e7	0.723761
$985$	0	0
$986$	−2.44970e7	−0.802455
$987$	974506.	0.0318414
$988$	−1.12434e8	−3.66441
$989$	1.71598e7	0.557856
$990$	0	0
$991$	−3.06716e7	−0.992092	−0.496046	−	0.868296i	$-0.665215\pi$
$991$	−3.06716e7	−0.992092	−0.496046	+	0.868296i	$0.665215\pi$
$992$	7.65867e6	0.247101
$993$	1.57163e7	0.505797
$994$	1.40660e7	0.451549
$995$	0	0
$996$	2.47996e7	0.792130
$997$	2.18670e7	0.696707	0.348354	−	0.937363i	$-0.386741\pi$
$997$	2.18670e7	0.696707	0.348354	+	0.937363i	$0.386741\pi$
$998$	−8.37327e7	−2.66115
$999$	1.07305e7	0.340177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.q.1.8	yes	8
5.4	even	2		825.6.a.p.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.1	✓	8	5.4	even	2
825.6.a.q.1.8	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)

\(f(q)\)	\(=\)	\(q+9.70245 q^{2} -9.00000 q^{3} +62.1375 q^{4} -87.3220 q^{6} -79.4231 q^{7} +292.408 q^{8} +81.0000 q^{9} -121.000 q^{11} -559.238 q^{12} -780.381 q^{13} -770.598 q^{14} +848.670 q^{16} +1974.55 q^{17} +785.898 q^{18} +2318.66 q^{19} +714.808 q^{21} -1174.00 q^{22} +1282.47 q^{23} -2631.67 q^{24} -7571.61 q^{26} -729.000 q^{27} -4935.15 q^{28} -1278.69 q^{29} -6820.62 q^{31} -1122.87 q^{32} +1089.00 q^{33} +19158.0 q^{34} +5033.14 q^{36} -14719.4 q^{37} +22496.6 q^{38} +7023.43 q^{39} -8353.18 q^{41} +6935.38 q^{42} +13380.3 q^{43} -7518.64 q^{44} +12443.1 q^{46} +1363.31 q^{47} -7638.03 q^{48} -10499.0 q^{49} -17770.9 q^{51} -48490.9 q^{52} -35100.9 q^{53} -7073.08 q^{54} -23223.9 q^{56} -20867.9 q^{57} -12406.4 q^{58} -34493.5 q^{59} -45097.0 q^{61} -66176.7 q^{62} -6433.27 q^{63} -38052.0 q^{64} +10566.0 q^{66} +24151.0 q^{67} +122694. q^{68} -11542.3 q^{69} -18253.3 q^{71} +23685.0 q^{72} +32455.4 q^{73} -142815. q^{74} +144075. q^{76} +9610.19 q^{77} +68144.5 q^{78} -28488.4 q^{79} +6561.00 q^{81} -81046.3 q^{82} -44345.4 q^{83} +44416.4 q^{84} +129821. q^{86} +11508.2 q^{87} -35381.3 q^{88} -82213.9 q^{89} +61980.3 q^{91} +79689.6 q^{92} +61385.6 q^{93} +13227.5 q^{94} +10105.8 q^{96} -20136.3 q^{97} -101866. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22}+ \cdots - 78408 q^{99}+O(q^{100}) \) 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

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