LMFDB - Embedded newform 495.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.35386$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08127 q^{2} -30.8308 q^{4} -25.0000 q^{5} -139.508 q^{7} +67.9374 q^{8} +O(q^{10})$	\(q-1.08127 q^{2} -30.8308 q^{4} -25.0000 q^{5} -139.508 q^{7} +67.9374 q^{8} +27.0319 q^{10} +121.000 q^{11} -646.331 q^{13} +150.846 q^{14} +913.128 q^{16} +1378.32 q^{17} +1908.90 q^{19} +770.771 q^{20} -130.834 q^{22} -343.726 q^{23} +625.000 q^{25} +698.861 q^{26} +4301.15 q^{28} -53.5092 q^{29} +634.133 q^{31} -3161.34 q^{32} -1490.34 q^{34} +3487.70 q^{35} +11674.2 q^{37} -2064.04 q^{38} -1698.43 q^{40} -18866.5 q^{41} +13379.3 q^{43} -3730.53 q^{44} +371.662 q^{46} +2252.95 q^{47} +2655.48 q^{49} -675.796 q^{50} +19926.9 q^{52} +8900.71 q^{53} -3025.00 q^{55} -9477.80 q^{56} +57.8581 q^{58} +11276.7 q^{59} +11852.4 q^{61} -685.672 q^{62} -25801.8 q^{64} +16158.3 q^{65} -57333.6 q^{67} -42494.7 q^{68} -3771.16 q^{70} -31075.8 q^{71} +56010.8 q^{73} -12623.0 q^{74} -58853.0 q^{76} -16880.5 q^{77} -883.036 q^{79} -22828.2 q^{80} +20399.9 q^{82} -93986.3 q^{83} -34457.9 q^{85} -14466.7 q^{86} +8220.42 q^{88} -21739.9 q^{89} +90168.4 q^{91} +10597.4 q^{92} -2436.06 q^{94} -47722.5 q^{95} -158059. q^{97} -2871.30 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8	$ 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8 + 50 * q^10 + 363 * q^11 - 546 * q^13 + 8 * q^14 - 1360 * q^16 + 314 * q^17 + 1808 * q^19 + 500 * q^20 - 242 * q^22 - 4288 * q^23 + 1875 * q^25 - 812 * q^26 + 5888 * q^28 - 5582 * q^29 + 6328 * q^31 + 736 * q^32 + 11596 * q^34 - 3800 * q^35 + 16866 * q^37 - 9584 * q^38 - 600 * q^40 - 23282 * q^41 + 20572 * q^43 - 2420 * q^44 + 16592 * q^46 - 3432 * q^47 + 11531 * q^49 - 1250 * q^50 + 21816 * q^52 - 16138 * q^53 - 9075 * q^55 - 15648 * q^56 + 17460 * q^58 - 21972 * q^59 + 8322 * q^61 - 5056 * q^62 + 22208 * q^64 + 13650 * q^65 - 84332 * q^67 - 59832 * q^68 - 200 * q^70 - 50528 * q^71 - 53838 * q^73 - 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 6364 * q^79 + 34000 * q^80 - 68020 * q^82 - 96272 * q^83 - 7850 * q^85 - 143152 * q^86 + 2904 * q^88 + 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 49088 * q^94 - 45200 * q^95 - 103242 * q^97 - 9554 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.08127	−0.191144	−0.0955720	−	0.995423i	$-0.530468\pi$
$2$	−1.08127	−0.191144	−0.0955720	+	0.995423i	$0.530468\pi$
$3$	0	0
$3$	0	0
$4$	−30.8308	−0.963464
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−139.508	−1.07610	−0.538052	−	0.842912i	$-0.680839\pi$
$7$	−139.508	−1.07610	−0.538052	+	0.842912i	$0.680839\pi$
$8$	67.9374	0.375304
$9$	0	0
$10$	27.0319	0.0854822
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−646.331	−1.06071	−0.530355	−	0.847776i	$-0.677941\pi$
$13$	−646.331	−1.06071	−0.530355	+	0.847776i	$0.677941\pi$
$14$	150.846	0.205691
$15$	0	0
$16$	913.128	0.891727
$17$	1378.32	1.15672	0.578358	−	0.815783i	$-0.303694\pi$
$17$	1378.32	1.15672	0.578358	+	0.815783i	$0.303694\pi$
$18$	0	0
$19$	1908.90	1.21311	0.606553	−	0.795043i	$-0.292552\pi$
$19$	1908.90	1.21311	0.606553	+	0.795043i	$0.292552\pi$
$20$	770.771	0.430874
$21$	0	0
$22$	−130.834	−0.0576321
$23$	−343.726	−0.135485	−0.0677427	−	0.997703i	$-0.521580\pi$
$23$	−343.726	−0.135485	−0.0677427	+	0.997703i	$0.521580\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	698.861	0.202748
$27$	0	0
$28$	4301.15	1.03679
$29$	−53.5092	−0.0118150	−0.00590750	−	0.999983i	$-0.501880\pi$
$29$	−53.5092	−0.0118150	−0.00590750	+	0.999983i	$0.501880\pi$
$30$	0	0
$31$	634.133	0.118516	0.0592579	−	0.998243i	$-0.481127\pi$
$31$	634.133	0.118516	0.0592579	+	0.998243i	$0.481127\pi$
$32$	−3161.34	−0.545753
$33$	0	0
$34$	−1490.34	−0.221099
$35$	3487.70	0.481248
$36$	0	0
$37$	11674.2	1.40192	0.700960	−	0.713201i	$-0.252755\pi$
$37$	11674.2	1.40192	0.700960	+	0.713201i	$0.252755\pi$
$38$	−2064.04	−0.231878
$39$	0	0
$40$	−1698.43	−0.167841
$41$	−18866.5	−1.75280	−0.876399	−	0.481585i	$-0.840061\pi$
$41$	−18866.5	−1.75280	−0.876399	+	0.481585i	$0.840061\pi$
$42$	0	0
$43$	13379.3	1.10347	0.551736	−	0.834018i	$-0.313965\pi$
$43$	13379.3	1.10347	0.551736	+	0.834018i	$0.313965\pi$
$44$	−3730.53	−0.290495
$45$	0	0
$46$	371.662	0.0258972
$47$	2252.95	0.148767	0.0743835	−	0.997230i	$-0.476301\pi$
$47$	2252.95	0.148767	0.0743835	+	0.997230i	$0.476301\pi$
$48$	0	0
$49$	2655.48	0.157998
$50$	−675.796	−0.0382288
$51$	0	0
$52$	19926.9	1.02196
$53$	8900.71	0.435246	0.217623	−	0.976033i	$-0.430170\pi$
$53$	8900.71	0.435246	0.217623	+	0.976033i	$0.430170\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−9477.80	−0.403866
$57$	0	0
$58$	57.8581	0.00225837
$59$	11276.7	0.421746	0.210873	−	0.977514i	$-0.432369\pi$
$59$	11276.7	0.421746	0.210873	+	0.977514i	$0.432369\pi$
$60$	0	0
$61$	11852.4	0.407834	0.203917	−	0.978988i	$-0.434633\pi$
$61$	11852.4	0.407834	0.203917	+	0.978988i	$0.434633\pi$
$62$	−685.672	−0.0226536
$63$	0	0
$64$	−25801.8	−0.787409
$65$	16158.3	0.474364
$66$	0	0
$67$	−57333.6	−1.56035	−0.780175	−	0.625561i	$-0.784870\pi$
$67$	−57333.6	−1.56035	−0.780175	+	0.625561i	$0.784870\pi$
$68$	−42494.7	−1.11445
$69$	0	0
$70$	−3771.16	−0.0919877
$71$	−31075.8	−0.731605	−0.365803	−	0.930692i	$-0.619206\pi$
$71$	−31075.8	−0.731605	−0.365803	+	0.930692i	$0.619206\pi$
$72$	0	0
$73$	56010.8	1.23017	0.615084	−	0.788462i	$-0.289122\pi$
$73$	56010.8	1.23017	0.615084	+	0.788462i	$0.289122\pi$
$74$	−12623.0	−0.267969
$75$	0	0
$76$	−58853.0	−1.16878
$77$	−16880.5	−0.324457
$78$	0	0
$79$	−883.036	−0.0159188	−0.00795941	−	0.999968i	$-0.502534\pi$
$79$	−883.036	−0.0159188	−0.00795941	+	0.999968i	$0.502534\pi$
$80$	−22828.2	−0.398792
$81$	0	0
$82$	20399.9	0.335037
$83$	−93986.3	−1.49751	−0.748754	−	0.662848i	$-0.769348\pi$
$83$	−93986.3	−1.49751	−0.748754	+	0.662848i	$0.769348\pi$
$84$	0	0
$85$	−34457.9	−0.517299
$86$	−14466.7	−0.210922
$87$	0	0
$88$	8220.42	0.113159
$89$	−21739.9	−0.290926	−0.145463	−	0.989364i	$-0.546467\pi$
$89$	−21739.9	−0.290926	−0.145463	+	0.989364i	$0.546467\pi$
$90$	0	0
$91$	90168.4	1.14143
$92$	10597.4	0.130535
$93$	0	0
$94$	−2436.06	−0.0284359
$95$	−47722.5	−0.542518
$96$	0	0
$97$	−158059.	−1.70565	−0.852825	−	0.522197i	$-0.825112\pi$
$97$	−158059.	−1.70565	−0.852825	+	0.522197i	$0.825112\pi$
$98$	−2871.30	−0.0302004
$99$	0	0
$100$	−19269.3	−0.192693
$101$	156968.	1.53111	0.765556	−	0.643369i	$-0.222464\pi$
$101$	156968.	1.53111	0.765556	+	0.643369i	$0.222464\pi$
$102$	0	0
$103$	169126.	1.57078	0.785391	−	0.618999i	$-0.212462\pi$
$103$	169126.	1.57078	0.785391	+	0.618999i	$0.212462\pi$
$104$	−43910.0	−0.398089
$105$	0	0
$106$	−9624.11	−0.0831948
$107$	13538.5	0.114317	0.0571587	−	0.998365i	$-0.481796\pi$
$107$	13538.5	0.114317	0.0571587	+	0.998365i	$0.481796\pi$
$108$	0	0
$109$	−74736.4	−0.602513	−0.301256	−	0.953543i	$-0.597406\pi$
$109$	−74736.4	−0.602513	−0.301256	+	0.953543i	$0.597406\pi$
$110$	3270.85	0.0257739
$111$	0	0
$112$	−127389.	−0.959590
$113$	205914.	1.51701	0.758507	−	0.651665i	$-0.225929\pi$
$113$	205914.	1.51701	0.758507	+	0.651665i	$0.225929\pi$
$114$	0	0
$115$	8593.15	0.0605909
$116$	1649.73	0.0113833
$117$	0	0
$118$	−12193.2	−0.0806142
$119$	−192286.	−1.24475
$120$	0	0
$121$	14641.0	0.0909091
$122$	−12815.7	−0.0779550
$123$	0	0
$124$	−19550.9	−0.114186
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−214682.	−1.18110	−0.590550	−	0.807001i	$-0.701089\pi$
$127$	−214682.	−1.18110	−0.590550	+	0.807001i	$0.701089\pi$
$128$	129062.	0.696261
$129$	0	0
$130$	−17471.5	−0.0906719
$131$	−253067.	−1.28842	−0.644209	−	0.764849i	$-0.722813\pi$
$131$	−253067.	−1.28842	−0.644209	+	0.764849i	$0.722813\pi$
$132$	0	0
$133$	−266307.	−1.30543
$134$	61993.3	0.298252
$135$	0	0
$136$	93639.2	0.434121
$137$	−239398.	−1.08973	−0.544865	−	0.838524i	$-0.683419\pi$
$137$	−239398.	−1.08973	−0.544865	+	0.838524i	$0.683419\pi$
$138$	0	0
$139$	−166398.	−0.730484	−0.365242	−	0.930913i	$-0.619014\pi$
$139$	−166398.	−0.730484	−0.365242	+	0.930913i	$0.619014\pi$
$140$	−107529.	−0.463665
$141$	0	0
$142$	33601.5	0.139842
$143$	−78206.1	−0.319816
$144$	0	0
$145$	1337.73	0.00528383
$146$	−60563.0	−0.235139
$147$	0	0
$148$	−359926.	−1.35070
$149$	−295512.	−1.09046	−0.545229	−	0.838287i	$-0.683557\pi$
$149$	−295512.	−1.09046	−0.545229	+	0.838287i	$0.683557\pi$
$150$	0	0
$151$	428373.	1.52890	0.764452	−	0.644681i	$-0.223010\pi$
$151$	428373.	1.52890	0.764452	+	0.644681i	$0.223010\pi$
$152$	129686.	0.455284
$153$	0	0
$154$	18252.4	0.0620181
$155$	−15853.3	−0.0530019
$156$	0	0
$157$	−289003.	−0.935735	−0.467867	−	0.883799i	$-0.654978\pi$
$157$	−289003.	−0.935735	−0.467867	+	0.883799i	$0.654978\pi$
$158$	954.804	0.00304279
$159$	0	0
$160$	79033.4	0.244068
$161$	47952.5	0.145796
$162$	0	0
$163$	288223.	0.849688	0.424844	−	0.905267i	$-0.360329\pi$
$163$	288223.	0.849688	0.424844	+	0.905267i	$0.360329\pi$
$164$	581671.	1.68876
$165$	0	0
$166$	101625.	0.286240
$167$	180912.	0.501969	0.250985	−	0.967991i	$-0.419246\pi$
$167$	180912.	0.501969	0.250985	+	0.967991i	$0.419246\pi$
$168$	0	0
$169$	46451.1	0.125106
$170$	37258.5	0.0988787
$171$	0	0
$172$	−412495.	−1.06316
$173$	−218579.	−0.555255	−0.277628	−	0.960689i	$-0.589548\pi$
$173$	−218579.	−0.555255	−0.277628	+	0.960689i	$0.589548\pi$
$174$	0	0
$175$	−87192.5	−0.215221
$176$	110489.	0.268866
$177$	0	0
$178$	23506.8	0.0556089
$179$	−310916.	−0.725287	−0.362643	−	0.931928i	$-0.618126\pi$
$179$	−310916.	−0.725287	−0.362643	+	0.931928i	$0.618126\pi$
$180$	0	0
$181$	−423637.	−0.961163	−0.480582	−	0.876950i	$-0.659574\pi$
$181$	−423637.	−0.961163	−0.480582	+	0.876950i	$0.659574\pi$
$182$	−97496.7	−0.218178
$183$	0	0
$184$	−23351.8	−0.0508483
$185$	−291855.	−0.626957
$186$	0	0
$187$	166776.	0.348763
$188$	−69460.3	−0.143332
$189$	0	0
$190$	51601.1	0.103699
$191$	−861089.	−1.70791	−0.853954	−	0.520348i	$-0.825802\pi$
$191$	−861089.	−1.70791	−0.853954	+	0.520348i	$0.825802\pi$
$192$	0	0
$193$	430583.	0.832077	0.416038	−	0.909347i	$-0.363418\pi$
$193$	430583.	0.832077	0.416038	+	0.909347i	$0.363418\pi$
$194$	170905.	0.326025
$195$	0	0
$196$	−81870.6	−0.152226
$197$	184755.	0.339181	0.169590	−	0.985515i	$-0.445756\pi$
$197$	184755.	0.339181	0.169590	+	0.985515i	$0.445756\pi$
$198$	0	0
$199$	322993.	0.578177	0.289089	−	0.957302i	$-0.406648\pi$
$199$	322993.	0.578177	0.289089	+	0.957302i	$0.406648\pi$
$200$	42460.9	0.0750609
$201$	0	0
$202$	−169725.	−0.292663
$203$	7464.96	0.0127142
$204$	0	0
$205$	471663.	0.783875
$206$	−182871.	−0.300246
$207$	0	0
$208$	−590183.	−0.945864
$209$	230977.	0.365765
$210$	0	0
$211$	−394106.	−0.609406	−0.304703	−	0.952447i	$-0.598557\pi$
$211$	−394106.	−0.609406	−0.304703	+	0.952447i	$0.598557\pi$
$212$	−274417.	−0.419344
$213$	0	0
$214$	−14638.9	−0.0218511
$215$	−334482.	−0.493488
$216$	0	0
$217$	−88466.6	−0.127535
$218$	80810.6	0.115167
$219$	0	0
$220$	93263.3	0.129913
$221$	−890849.	−1.22694
$222$	0	0
$223$	445038.	0.599287	0.299643	−	0.954051i	$-0.403132\pi$
$223$	445038.	0.599287	0.299643	+	0.954051i	$0.403132\pi$
$224$	441032.	0.587286
$225$	0	0
$226$	−222649.	−0.289968
$227$	−1.15887e6	−1.49270	−0.746349	−	0.665555i	$-0.768195\pi$
$227$	−1.15887e6	−1.49270	−0.746349	+	0.665555i	$0.768195\pi$
$228$	0	0
$229$	885853.	1.11628	0.558140	−	0.829747i	$-0.311515\pi$
$229$	885853.	1.11628	0.558140	+	0.829747i	$0.311515\pi$
$230$	−9291.55	−0.0115816
$231$	0	0
$232$	−3635.27	−0.00443422
$233$	−750392.	−0.905520	−0.452760	−	0.891632i	$-0.649561\pi$
$233$	−750392.	−0.905520	−0.452760	+	0.891632i	$0.649561\pi$
$234$	0	0
$235$	−56323.7	−0.0665306
$236$	−347669.	−0.406337
$237$	0	0
$238$	207914.	0.237926
$239$	−617415.	−0.699169	−0.349585	−	0.936905i	$-0.613677\pi$
$239$	−617415.	−0.699169	−0.349585	+	0.936905i	$0.613677\pi$
$240$	0	0
$241$	1.20275e6	1.33393	0.666966	−	0.745088i	$-0.267593\pi$
$241$	1.20275e6	1.33393	0.666966	+	0.745088i	$0.267593\pi$
$242$	−15830.9	−0.0173767
$243$	0	0
$244$	−365421.	−0.392933
$245$	−66386.9	−0.0706589
$246$	0	0
$247$	−1.23378e6	−1.28675
$248$	43081.3	0.0444795
$249$	0	0
$250$	16894.9	0.0170964
$251$	−1.55775e6	−1.56068	−0.780338	−	0.625358i	$-0.784953\pi$
$251$	−1.55775e6	−1.56068	−0.780338	+	0.625358i	$0.784953\pi$
$252$	0	0
$253$	−41590.8	−0.0408504
$254$	232130.	0.225760
$255$	0	0
$256$	686107.	0.654323
$257$	−92156.0	−0.0870344	−0.0435172	−	0.999053i	$-0.513856\pi$
$257$	−92156.0	−0.0870344	−0.0435172	+	0.999053i	$0.513856\pi$
$258$	0	0
$259$	−1.62865e6	−1.50861
$260$	−498173.	−0.457033
$261$	0	0
$262$	273635.	0.246274
$263$	862122.	0.768563	0.384281	−	0.923216i	$-0.374449\pi$
$263$	862122.	0.768563	0.384281	+	0.923216i	$0.374449\pi$
$264$	0	0
$265$	−222518.	−0.194648
$266$	287950.	0.249525
$267$	0	0
$268$	1.76764e6	1.50334
$269$	1.25052e6	1.05368	0.526841	−	0.849964i	$-0.323376\pi$
$269$	1.25052e6	1.05368	0.526841	+	0.849964i	$0.323376\pi$
$270$	0	0
$271$	844654.	0.698644	0.349322	−	0.937003i	$-0.386412\pi$
$271$	844654.	0.698644	0.349322	+	0.937003i	$0.386412\pi$
$272$	1.25858e6	1.03147
$273$	0	0
$274$	258855.	0.208295
$275$	75625.0	0.0603023
$276$	0	0
$277$	−1.52990e6	−1.19802	−0.599008	−	0.800743i	$-0.704438\pi$
$277$	−1.52990e6	−1.19802	−0.599008	+	0.800743i	$0.704438\pi$
$278$	179922.	0.139628
$279$	0	0
$280$	236945.	0.180615
$281$	−1.13284e6	−0.855858	−0.427929	−	0.903812i	$-0.640757\pi$
$281$	−1.13284e6	−0.855858	−0.427929	+	0.903812i	$0.640757\pi$
$282$	0	0
$283$	−411138.	−0.305156	−0.152578	−	0.988291i	$-0.548758\pi$
$283$	−411138.	−0.305156	−0.152578	+	0.988291i	$0.548758\pi$
$284$	958094.	0.704875
$285$	0	0
$286$	84562.2	0.0611310
$287$	2.63203e6	1.88619
$288$	0	0
$289$	479901.	0.337993
$290$	−1446.45	−0.00100997
$291$	0	0
$292$	−1.72686e6	−1.18522
$293$	−314013.	−0.213687	−0.106843	−	0.994276i	$-0.534074\pi$
$293$	−314013.	−0.213687	−0.106843	+	0.994276i	$0.534074\pi$
$294$	0	0
$295$	−281917.	−0.188610
$296$	793115.	0.526147
$297$	0	0
$298$	319529.	0.208435
$299$	222161.	0.143711
$300$	0	0
$301$	−1.86652e6	−1.18745
$302$	−463189.	−0.292241
$303$	0	0
$304$	1.74307e6	1.08176
$305$	−296311.	−0.182389
$306$	0	0
$307$	797600.	0.482991	0.241496	−	0.970402i	$-0.422362\pi$
$307$	797600.	0.482991	0.241496	+	0.970402i	$0.422362\pi$
$308$	520439.	0.312603
$309$	0	0
$310$	17141.8	0.0101310
$311$	618368.	0.362532	0.181266	−	0.983434i	$-0.441980\pi$
$311$	618368.	0.362532	0.181266	+	0.983434i	$0.441980\pi$
$312$	0	0
$313$	−2.60175e6	−1.50108	−0.750541	−	0.660824i	$-0.770207\pi$
$313$	−2.60175e6	−1.50108	−0.750541	+	0.660824i	$0.770207\pi$
$314$	312491.	0.178860
$315$	0	0
$316$	27224.7	0.0153372
$317$	729701.	0.407846	0.203923	−	0.978987i	$-0.434631\pi$
$317$	729701.	0.407846	0.203923	+	0.978987i	$0.434631\pi$
$318$	0	0
$319$	−6474.61	−0.00356235
$320$	645046.	0.352140
$321$	0	0
$322$	−51849.8	−0.0278681
$323$	2.63107e6	1.40322
$324$	0	0
$325$	−403957.	−0.212142
$326$	−311648.	−0.162413
$327$	0	0
$328$	−1.28174e6	−0.657833
$329$	−314304.	−0.160089
$330$	0	0
$331$	2.87129e6	1.44048	0.720239	−	0.693726i	$-0.244032\pi$
$331$	2.87129e6	1.44048	0.720239	+	0.693726i	$0.244032\pi$
$332$	2.89768e6	1.44280
$333$	0	0
$334$	−195616.	−0.0959484
$335$	1.43334e6	0.697810
$336$	0	0
$337$	2.16558e6	1.03872	0.519361	−	0.854555i	$-0.326170\pi$
$337$	2.16558e6	1.03872	0.519361	+	0.854555i	$0.326170\pi$
$338$	−50226.3	−0.0239133
$339$	0	0
$340$	1.06237e6	0.498399
$341$	76730.1	0.0357338
$342$	0	0
$343$	1.97425e6	0.906081
$344$	908953.	0.414138
$345$	0	0
$346$	236343.	0.106134
$347$	−392930.	−0.175183	−0.0875913	−	0.996156i	$-0.527917\pi$
$347$	−392930.	−0.175183	−0.0875913	+	0.996156i	$0.527917\pi$
$348$	0	0
$349$	247840.	0.108920	0.0544601	−	0.998516i	$-0.482656\pi$
$349$	247840.	0.108920	0.0544601	+	0.998516i	$0.482656\pi$
$350$	94279.0	0.0411381
$351$	0	0
$352$	−382522.	−0.164551
$353$	2.90847e6	1.24230	0.621151	−	0.783691i	$-0.286665\pi$
$353$	2.90847e6	1.24230	0.621151	+	0.783691i	$0.286665\pi$
$354$	0	0
$355$	776896.	0.327184
$356$	670261.	0.280297
$357$	0	0
$358$	336185.	0.138634
$359$	−758403.	−0.310573	−0.155287	−	0.987869i	$-0.549630\pi$
$359$	−758403.	−0.310573	−0.155287	+	0.987869i	$0.549630\pi$
$360$	0	0
$361$	1.16779e6	0.471627
$362$	458067.	0.183721
$363$	0	0
$364$	−2.77997e6	−1.09973
$365$	−1.40027e6	−0.550148
$366$	0	0
$367$	−2.29924e6	−0.891085	−0.445542	−	0.895261i	$-0.646989\pi$
$367$	−2.29924e6	−0.891085	−0.445542	+	0.895261i	$0.646989\pi$
$368$	−313866.	−0.120816
$369$	0	0
$370$	315575.	0.119839
$371$	−1.24172e6	−0.468370
$372$	0	0
$373$	−19163.9	−0.00713203	−0.00356601	−	0.999994i	$-0.501135\pi$
$373$	−19163.9	−0.00713203	−0.00356601	+	0.999994i	$0.501135\pi$
$374$	−180331.	−0.0666640
$375$	0	0
$376$	153059.	0.0558329
$377$	34584.7	0.0125323
$378$	0	0
$379$	346535.	0.123922	0.0619610	−	0.998079i	$-0.480265\pi$
$379$	346535.	0.123922	0.0619610	+	0.998079i	$0.480265\pi$
$380$	1.47132e6	0.522696
$381$	0	0
$382$	931073.	0.326457
$383$	5.25066e6	1.82901	0.914507	−	0.404571i	$-0.132579\pi$
$383$	5.25066e6	1.82901	0.914507	+	0.404571i	$0.132579\pi$
$384$	0	0
$385$	422012.	0.145102
$386$	−465578.	−0.159047
$387$	0	0
$388$	4.87309e6	1.64333
$389$	4.34474e6	1.45576	0.727880	−	0.685704i	$-0.240506\pi$
$389$	4.34474e6	1.45576	0.727880	+	0.685704i	$0.240506\pi$
$390$	0	0
$391$	−473763.	−0.156718
$392$	180406.	0.0592974
$393$	0	0
$394$	−199771.	−0.0648324
$395$	22075.9	0.00711911
$396$	0	0
$397$	3.90748e6	1.24429	0.622144	−	0.782903i	$-0.286262\pi$
$397$	3.90748e6	1.24429	0.622144	+	0.782903i	$0.286262\pi$
$398$	−349244.	−0.110515
$399$	0	0
$400$	570705.	0.178345
$401$	3.05315e6	0.948171	0.474085	−	0.880479i	$-0.342779\pi$
$401$	3.05315e6	0.948171	0.474085	+	0.880479i	$0.342779\pi$
$402$	0	0
$403$	−409860.	−0.125711
$404$	−4.83945e6	−1.47517
$405$	0	0
$406$	−8071.67	−0.00243023
$407$	1.41258e6	0.422695
$408$	0	0
$409$	−5.46050e6	−1.61408	−0.807038	−	0.590499i	$-0.798931\pi$
$409$	−5.46050e6	−1.61408	−0.807038	+	0.590499i	$0.798931\pi$
$410$	−509997.	−0.149833
$411$	0	0
$412$	−5.21428e6	−1.51339
$413$	−1.57318e6	−0.453842
$414$	0	0
$415$	2.34966e6	0.669706
$416$	2.04327e6	0.578886
$417$	0	0
$418$	−249749.	−0.0699139
$419$	−3.02804e6	−0.842609	−0.421305	−	0.906919i	$-0.638428\pi$
$419$	−3.02804e6	−0.842609	−0.421305	+	0.906919i	$0.638428\pi$
$420$	0	0
$421$	−3.31510e6	−0.911572	−0.455786	−	0.890089i	$-0.650642\pi$
$421$	−3.31510e6	−0.911572	−0.455786	+	0.890089i	$0.650642\pi$
$422$	426136.	0.116484
$423$	0	0
$424$	604691.	0.163350
$425$	861448.	0.231343
$426$	0	0
$427$	−1.65351e6	−0.438871
$428$	−417404.	−0.110141
$429$	0	0
$430$	361667.	0.0943273
$431$	3.02139e6	0.783455	0.391728	−	0.920081i	$-0.371878\pi$
$431$	3.02139e6	0.783455	0.391728	+	0.920081i	$0.371878\pi$
$432$	0	0
$433$	−739663.	−0.189589	−0.0947947	−	0.995497i	$-0.530219\pi$
$433$	−739663.	−0.189589	−0.0947947	+	0.995497i	$0.530219\pi$
$434$	95656.7	0.0243776
$435$	0	0
$436$	2.30419e6	0.580499
$437$	−656138.	−0.164358
$438$	0	0
$439$	89371.4	0.0221329	0.0110664	−	0.999939i	$-0.496477\pi$
$439$	89371.4	0.0221329	0.0110664	+	0.999939i	$0.496477\pi$
$440$	−205511.	−0.0506060
$441$	0	0
$442$	963252.	0.234522
$443$	−5.06164e6	−1.22541	−0.612706	−	0.790311i	$-0.709919\pi$
$443$	−5.06164e6	−1.22541	−0.612706	+	0.790311i	$0.709919\pi$
$444$	0	0
$445$	543498.	0.130106
$446$	−481208.	−0.114550
$447$	0	0
$448$	3.59956e6	0.847334
$449$	−6.64598e6	−1.55576	−0.777881	−	0.628411i	$-0.783706\pi$
$449$	−6.64598e6	−1.55576	−0.777881	+	0.628411i	$0.783706\pi$
$450$	0	0
$451$	−2.28285e6	−0.528489
$452$	−6.34850e6	−1.46159
$453$	0	0
$454$	1.25306e6	0.285320
$455$	−2.25421e6	−0.510465
$456$	0	0
$457$	−176313.	−0.0394905	−0.0197453	−	0.999805i	$-0.506286\pi$
$457$	−176313.	−0.0394905	−0.0197453	+	0.999805i	$0.506286\pi$
$458$	−957850.	−0.213370
$459$	0	0
$460$	−264934.	−0.0583772
$461$	−8.43740e6	−1.84908	−0.924542	−	0.381081i	$-0.875552\pi$
$461$	−8.43740e6	−1.84908	−0.924542	+	0.381081i	$0.875552\pi$
$462$	0	0
$463$	5.65739e6	1.22649	0.613244	−	0.789893i	$-0.289864\pi$
$463$	5.65739e6	1.22649	0.613244	+	0.789893i	$0.289864\pi$
$464$	−48860.8	−0.0105357
$465$	0	0
$466$	811379.	0.173085
$467$	2.54278e6	0.539532	0.269766	−	0.962926i	$-0.413054\pi$
$467$	2.54278e6	0.539532	0.269766	+	0.962926i	$0.413054\pi$
$468$	0	0
$469$	7.99850e6	1.67910
$470$	60901.4	0.0127169
$471$	0	0
$472$	766107.	0.158283
$473$	1.61889e6	0.332710
$474$	0	0
$475$	1.19306e6	0.242621
$476$	5.92835e6	1.19927
$477$	0	0
$478$	667594.	0.133642
$479$	−6.89136e6	−1.37235	−0.686177	−	0.727434i	$-0.740713\pi$
$479$	−6.89136e6	−1.37235	−0.686177	+	0.727434i	$0.740713\pi$
$480$	0	0
$481$	−7.54540e6	−1.48703
$482$	−1.30051e6	−0.254973
$483$	0	0
$484$	−451394.	−0.0875876
$485$	3.95147e6	0.762790
$486$	0	0
$487$	7.89949e6	1.50930	0.754652	−	0.656125i	$-0.227806\pi$
$487$	7.89949e6	1.50930	0.754652	+	0.656125i	$0.227806\pi$
$488$	805223.	0.153062
$489$	0	0
$490$	71782.4	0.0135060
$491$	2.29000e6	0.428679	0.214340	−	0.976759i	$-0.431240\pi$
$491$	2.29000e6	0.428679	0.214340	+	0.976759i	$0.431240\pi$
$492$	0	0
$493$	−73752.6	−0.0136666
$494$	1.33406e6	0.245955
$495$	0	0
$496$	579045.	0.105684
$497$	4.33533e6	0.787283
$498$	0	0
$499$	−5.96310e6	−1.07206	−0.536032	−	0.844198i	$-0.680077\pi$
$499$	−5.96310e6	−1.07206	−0.536032	+	0.844198i	$0.680077\pi$
$500$	481732.	0.0861748
$501$	0	0
$502$	1.68435e6	0.298314
$503$	−2.17453e6	−0.383218	−0.191609	−	0.981471i	$-0.561371\pi$
$503$	−2.17453e6	−0.383218	−0.191609	+	0.981471i	$0.561371\pi$
$504$	0	0
$505$	−3.92420e6	−0.684734
$506$	44971.1	0.00780831
$507$	0	0
$508$	6.61883e6	1.13795
$509$	3.33531e6	0.570613	0.285307	−	0.958436i	$-0.407905\pi$
$509$	3.33531e6	0.570613	0.285307	+	0.958436i	$0.407905\pi$
$510$	0	0
$511$	−7.81395e6	−1.32379
$512$	−4.87184e6	−0.821331
$513$	0	0
$514$	99645.9	0.0166361
$515$	−4.22814e6	−0.702475
$516$	0	0
$517$	272607.	0.0448549
$518$	1.76101e6	0.288362
$519$	0	0
$520$	1.09775e6	0.178031
$521$	−9.81327e6	−1.58387	−0.791935	−	0.610605i	$-0.790926\pi$
$521$	−9.81327e6	−1.58387	−0.791935	+	0.610605i	$0.790926\pi$
$522$	0	0
$523$	7.96466e6	1.27325	0.636624	−	0.771175i	$-0.280330\pi$
$523$	7.96466e6	1.27325	0.636624	+	0.771175i	$0.280330\pi$
$524$	7.80227e6	1.24134
$525$	0	0
$526$	−932190.	−0.146906
$527$	874036.	0.137089
$528$	0	0
$529$	−6.31820e6	−0.981644
$530$	240603.	0.0372058
$531$	0	0
$532$	8.21046e6	1.25773
$533$	1.21940e7	1.85921
$534$	0	0
$535$	−338463.	−0.0511243
$536$	−3.89509e6	−0.585607
$537$	0	0
$538$	−1.35215e6	−0.201405
$539$	321313.	0.0476383
$540$	0	0
$541$	−5.50616e6	−0.808827	−0.404413	−	0.914576i	$-0.632524\pi$
$541$	−5.50616e6	−0.808827	−0.404413	+	0.914576i	$0.632524\pi$
$542$	−913303.	−0.133542
$543$	0	0
$544$	−4.35733e6	−0.631281
$545$	1.86841e6	0.269452
$546$	0	0
$547$	2.92918e6	0.418580	0.209290	−	0.977854i	$-0.432885\pi$
$547$	2.92918e6	0.418580	0.209290	+	0.977854i	$0.432885\pi$
$548$	7.38084e6	1.04992
$549$	0	0
$550$	−81771.4	−0.0115264
$551$	−102144.	−0.0143328
$552$	0	0
$553$	123191.	0.0171303
$554$	1.65424e6	0.228994
$555$	0	0
$556$	5.13018e6	0.703795
$557$	−4.05453e6	−0.553736	−0.276868	−	0.960908i	$-0.589296\pi$
$557$	−4.05453e6	−0.553736	−0.276868	+	0.960908i	$0.589296\pi$
$558$	0	0
$559$	−8.64745e6	−1.17047
$560$	3.18472e6	0.429142
$561$	0	0
$562$	1.22491e6	0.163592
$563$	1.65845e6	0.220512	0.110256	−	0.993903i	$-0.464833\pi$
$563$	1.65845e6	0.220512	0.110256	+	0.993903i	$0.464833\pi$
$564$	0	0
$565$	−5.14785e6	−0.678430
$566$	444553.	0.0583288
$567$	0	0
$568$	−2.11121e6	−0.274575
$569$	−1.05475e7	−1.36575	−0.682873	−	0.730537i	$-0.739270\pi$
$569$	−1.05475e7	−1.36575	−0.682873	+	0.730537i	$0.739270\pi$
$570$	0	0
$571$	−1.46769e7	−1.88384	−0.941918	−	0.335843i	$-0.890979\pi$
$571$	−1.46769e7	−1.88384	−0.941918	+	0.335843i	$0.890979\pi$
$572$	2.41116e6	0.308131
$573$	0	0
$574$	−2.84595e6	−0.360534
$575$	−214829.	−0.0270971
$576$	0	0
$577$	−8.25271e6	−1.03195	−0.515973	−	0.856605i	$-0.672570\pi$
$577$	−8.25271e6	−1.03195	−0.515973	+	0.856605i	$0.672570\pi$
$578$	−518905.	−0.0646053
$579$	0	0
$580$	−41243.3	−0.00509078
$581$	1.31118e7	1.61147
$582$	0	0
$583$	1.07699e6	0.131232
$584$	3.80522e6	0.461687
$585$	0	0
$586$	339534.	0.0408450
$587$	−1.01063e7	−1.21059	−0.605296	−	0.796000i	$-0.706945\pi$
$587$	−1.01063e7	−1.21059	−0.605296	+	0.796000i	$0.706945\pi$
$588$	0	0
$589$	1.21050e6	0.143772
$590$	304829.	0.0360517
$591$	0	0
$592$	1.06600e7	1.25013
$593$	−1.48922e7	−1.73909	−0.869543	−	0.493857i	$-0.835587\pi$
$593$	−1.48922e7	−1.73909	−0.869543	+	0.493857i	$0.835587\pi$
$594$	0	0
$595$	4.80716e6	0.556667
$596$	9.11088e6	1.05062
$597$	0	0
$598$	−240217.	−0.0274695
$599$	−1.59013e7	−1.81078	−0.905390	−	0.424580i	$-0.860422\pi$
$599$	−1.59013e7	−1.81078	−0.905390	+	0.424580i	$0.860422\pi$
$600$	0	0
$601$	−2.84828e6	−0.321660	−0.160830	−	0.986982i	$-0.551417\pi$
$601$	−2.84828e6	−0.321660	−0.160830	+	0.986982i	$0.551417\pi$
$602$	2.01822e6	0.226974
$603$	0	0
$604$	−1.32071e7	−1.47304
$605$	−366025.	−0.0406558
$606$	0	0
$607$	1.32016e6	0.145430	0.0727149	−	0.997353i	$-0.476834\pi$
$607$	1.32016e6	0.145430	0.0727149	+	0.997353i	$0.476834\pi$
$608$	−6.03467e6	−0.662056
$609$	0	0
$610$	320393.	0.0348625
$611$	−1.45615e6	−0.157799
$612$	0	0
$613$	−978527.	−0.105177	−0.0525886	−	0.998616i	$-0.516747\pi$
$613$	−978527.	−0.105177	−0.0525886	+	0.998616i	$0.516747\pi$
$614$	−862424.	−0.0923209
$615$	0	0
$616$	−1.14681e6	−0.121770
$617$	−1.29643e7	−1.37100	−0.685500	−	0.728073i	$-0.740416\pi$
$617$	−1.29643e7	−1.37100	−0.685500	+	0.728073i	$0.740416\pi$
$618$	0	0
$619$	1.27406e7	1.33648	0.668240	−	0.743946i	$-0.267048\pi$
$619$	1.27406e7	1.33648	0.668240	+	0.743946i	$0.267048\pi$
$620$	488771.	0.0510654
$621$	0	0
$622$	−668626.	−0.0692958
$623$	3.03289e6	0.313067
$624$	0	0
$625$	390625.	0.0400000
$626$	2.81320e6	0.286923
$627$	0	0
$628$	8.91020e6	0.901547
$629$	1.60908e7	1.62162
$630$	0	0
$631$	−1.93988e7	−1.93955	−0.969775	−	0.243999i	$-0.921541\pi$
$631$	−1.93988e7	−1.93955	−0.969775	+	0.243999i	$0.921541\pi$
$632$	−59991.1	−0.00597440
$633$	0	0
$634$	−789006.	−0.0779574
$635$	5.36706e6	0.528204
$636$	0	0
$637$	−1.71632e6	−0.167590
$638$	7000.83	0.000680923 0
$639$	0	0
$640$	−3.22654e6	−0.311378
$641$	−6.77844e6	−0.651605	−0.325803	−	0.945438i	$-0.605634\pi$
$641$	−6.77844e6	−0.651605	−0.325803	+	0.945438i	$0.605634\pi$
$642$	0	0
$643$	−1.74257e7	−1.66212	−0.831060	−	0.556182i	$-0.812266\pi$
$643$	−1.74257e7	−1.66212	−0.831060	+	0.556182i	$0.812266\pi$
$644$	−1.47842e6	−0.140470
$645$	0	0
$646$	−2.84491e6	−0.268217
$647$	−792517.	−0.0744300	−0.0372150	−	0.999307i	$-0.511849\pi$
$647$	−792517.	−0.0744300	−0.0372150	+	0.999307i	$0.511849\pi$
$648$	0	0
$649$	1.36448e6	0.127161
$650$	436788.	0.0405497
$651$	0	0
$652$	−8.88616e6	−0.818644
$653$	9.09663e6	0.834829	0.417414	−	0.908716i	$-0.362936\pi$
$653$	9.09663e6	0.834829	0.417414	+	0.908716i	$0.362936\pi$
$654$	0	0
$655$	6.32667e6	0.576198
$656$	−1.72275e7	−1.56302
$657$	0	0
$658$	339849.	0.0306000
$659$	8.15992e6	0.731935	0.365968	−	0.930628i	$-0.380738\pi$
$659$	8.15992e6	0.731935	0.365968	+	0.930628i	$0.380738\pi$
$660$	0	0
$661$	−1.85172e6	−0.164843	−0.0824216	−	0.996598i	$-0.526265\pi$
$661$	−1.85172e6	−0.164843	−0.0824216	+	0.996598i	$0.526265\pi$
$662$	−3.10465e6	−0.275339
$663$	0	0
$664$	−6.38518e6	−0.562022
$665$	6.65766e6	0.583805
$666$	0	0
$667$	18392.5	0.00160076
$668$	−5.57768e6	−0.483629
$669$	0	0
$670$	−1.54983e6	−0.133382
$671$	1.43414e6	0.122966
$672$	0	0
$673$	1.42740e6	0.121481	0.0607405	−	0.998154i	$-0.480654\pi$
$673$	1.42740e6	0.121481	0.0607405	+	0.998154i	$0.480654\pi$
$674$	−2.34158e6	−0.198545
$675$	0	0
$676$	−1.43213e6	−0.120535
$677$	−941868.	−0.0789802	−0.0394901	−	0.999220i	$-0.512573\pi$
$677$	−941868.	−0.0789802	−0.0394901	+	0.999220i	$0.512573\pi$
$678$	0	0
$679$	2.20505e7	1.83545
$680$	−2.34098e6	−0.194145
$681$	0	0
$682$	−82966.3	−0.00683031
$683$	−9.56089e6	−0.784236	−0.392118	−	0.919915i	$-0.628257\pi$
$683$	−9.56089e6	−0.784236	−0.392118	+	0.919915i	$0.628257\pi$
$684$	0	0
$685$	5.98494e6	0.487342
$686$	−2.13471e6	−0.173192
$687$	0	0
$688$	1.22170e7	0.983996
$689$	−5.75281e6	−0.461670
$690$	0	0
$691$	−1.53765e7	−1.22507	−0.612536	−	0.790443i	$-0.709851\pi$
$691$	−1.53765e7	−1.22507	−0.612536	+	0.790443i	$0.709851\pi$
$692$	6.73897e6	0.534968
$693$	0	0
$694$	424865.	0.0334851
$695$	4.15994e6	0.326682
$696$	0	0
$697$	−2.60040e7	−2.02749
$698$	−267983.	−0.0208194
$699$	0	0
$700$	2.68822e6	0.207357
$701$	−1.72261e7	−1.32401	−0.662006	−	0.749499i	$-0.730295\pi$
$701$	−1.72261e7	−1.32401	−0.662006	+	0.749499i	$0.730295\pi$
$702$	0	0
$703$	2.22849e7	1.70068
$704$	−3.12202e6	−0.237413
$705$	0	0
$706$	−3.14485e6	−0.237459
$707$	−2.18983e7	−1.64764
$708$	0	0
$709$	2.47878e7	1.85192	0.925959	−	0.377625i	$-0.123259\pi$
$709$	2.47878e7	1.85192	0.925959	+	0.377625i	$0.123259\pi$
$710$	−840037.	−0.0625393
$711$	0	0
$712$	−1.47695e6	−0.109186
$713$	−217968.	−0.0160572
$714$	0	0
$715$	1.95515e6	0.143026
$716$	9.58579e6	0.698788
$717$	0	0
$718$	820042.	0.0593642
$719$	−7.44502e6	−0.537086	−0.268543	−	0.963268i	$-0.586542\pi$
$719$	−7.44502e6	−0.537086	−0.268543	+	0.963268i	$0.586542\pi$
$720$	0	0
$721$	−2.35944e7	−1.69032
$722$	−1.26271e6	−0.0901486
$723$	0	0
$724$	1.30611e7	0.926046
$725$	−33443.2	−0.00236300
$726$	0	0
$727$	3.56658e6	0.250274	0.125137	−	0.992139i	$-0.460063\pi$
$727$	3.56658e6	0.250274	0.125137	+	0.992139i	$0.460063\pi$
$728$	6.12580e6	0.428385
$729$	0	0
$730$	1.51407e6	0.105157
$731$	1.84409e7	1.27641
$732$	0	0
$733$	1.38091e7	0.949305	0.474652	−	0.880173i	$-0.342574\pi$
$733$	1.38091e7	0.949305	0.474652	+	0.880173i	$0.342574\pi$
$734$	2.48611e6	0.170326
$735$	0	0
$736$	1.08663e6	0.0739416
$737$	−6.93737e6	−0.470463
$738$	0	0
$739$	−7.18607e6	−0.484039	−0.242020	−	0.970271i	$-0.577810\pi$
$739$	−7.18607e6	−0.484039	−0.242020	+	0.970271i	$0.577810\pi$
$740$	8.99814e6	0.604051
$741$	0	0
$742$	1.34264e6	0.0895261
$743$	−2.18843e6	−0.145432	−0.0727161	−	0.997353i	$-0.523167\pi$
$743$	−2.18843e6	−0.145432	−0.0727161	+	0.997353i	$0.523167\pi$
$744$	0	0
$745$	7.38780e6	0.487668
$746$	20721.5	0.00136324
$747$	0	0
$748$	−5.14186e6	−0.336021
$749$	−1.88873e6	−0.123017
$750$	0	0
$751$	−1.41771e6	−0.0917250	−0.0458625	−	0.998948i	$-0.514604\pi$
$751$	−1.41771e6	−0.0917250	−0.0458625	+	0.998948i	$0.514604\pi$
$752$	2.05723e6	0.132660
$753$	0	0
$754$	−37395.5	−0.00239547
$755$	−1.07093e7	−0.683746
$756$	0	0
$757$	−2.05513e7	−1.30347	−0.651734	−	0.758448i	$-0.725958\pi$
$757$	−2.05513e7	−1.30347	−0.651734	+	0.758448i	$0.725958\pi$
$758$	−374699.	−0.0236870
$759$	0	0
$760$	−3.24214e6	−0.203609
$761$	1.30587e7	0.817409	0.408705	−	0.912667i	$-0.365981\pi$
$761$	1.30587e7	0.817409	0.408705	+	0.912667i	$0.365981\pi$
$762$	0	0
$763$	1.04263e7	0.648366
$764$	2.65481e7	1.64551
$765$	0	0
$766$	−5.67740e6	−0.349605
$767$	−7.28846e6	−0.447350
$768$	0	0
$769$	1.82667e7	1.11389	0.556946	−	0.830549i	$-0.311973\pi$
$769$	1.82667e7	1.11389	0.556946	+	0.830549i	$0.311973\pi$
$770$	−456310.	−0.0277353
$771$	0	0
$772$	−1.32752e7	−0.801676
$773$	1.49182e7	0.897981	0.448990	−	0.893537i	$-0.351784\pi$
$773$	1.49182e7	0.897981	0.448990	+	0.893537i	$0.351784\pi$
$774$	0	0
$775$	396333.	0.0237032
$776$	−1.07381e7	−0.640138
$777$	0	0
$778$	−4.69786e6	−0.278260
$779$	−3.60143e7	−2.12633
$780$	0	0
$781$	−3.76018e6	−0.220587
$782$	512268.	0.0299558
$783$	0	0
$784$	2.42479e6	0.140891
$785$	7.22507e6	0.418473
$786$	0	0
$787$	−1.60565e7	−0.924087	−0.462043	−	0.886857i	$-0.652884\pi$
$787$	−1.60565e7	−0.924087	−0.462043	+	0.886857i	$0.652884\pi$
$788$	−5.69616e6	−0.326788
$789$	0	0
$790$	−23870.1	−0.00136078
$791$	−2.87266e7	−1.63246
$792$	0	0
$793$	−7.66060e6	−0.432593
$794$	−4.22506e6	−0.237838
$795$	0	0
$796$	−9.95816e6	−0.557053
$797$	−2.10741e7	−1.17518	−0.587588	−	0.809160i	$-0.699922\pi$
$797$	−2.10741e7	−1.17518	−0.587588	+	0.809160i	$0.699922\pi$
$798$	0	0
$799$	3.10528e6	0.172081
$800$	−1.97584e6	−0.109151
$801$	0	0
$802$	−3.30129e6	−0.181237
$803$	6.77730e6	0.370909
$804$	0	0
$805$	−1.19881e6	−0.0652021
$806$	443171.	0.0240289
$807$	0	0
$808$	1.06640e7	0.574633
$809$	−2.38181e7	−1.27949	−0.639744	−	0.768588i	$-0.720960\pi$
$809$	−2.38181e7	−1.27949	−0.639744	+	0.768588i	$0.720960\pi$
$810$	0	0
$811$	2.93046e7	1.56453	0.782264	−	0.622947i	$-0.214065\pi$
$811$	2.93046e7	1.56453	0.782264	+	0.622947i	$0.214065\pi$
$812$	−230151.	−0.0122496
$813$	0	0
$814$	−1.52739e6	−0.0807956
$815$	−7.20557e6	−0.379992
$816$	0	0
$817$	2.55397e7	1.33863
$818$	5.90430e6	0.308521
$819$	0	0
$820$	−1.45418e7	−0.755236
$821$	3.14594e6	0.162889	0.0814446	−	0.996678i	$-0.474047\pi$
$821$	3.14594e6	0.162889	0.0814446	+	0.996678i	$0.474047\pi$
$822$	0	0
$823$	2.44990e7	1.26081	0.630404	−	0.776267i	$-0.282889\pi$
$823$	2.44990e7	1.26081	0.630404	+	0.776267i	$0.282889\pi$
$824$	1.14899e7	0.589522
$825$	0	0
$826$	1.70104e6	0.0867492
$827$	−1.34163e7	−0.682135	−0.341067	−	0.940039i	$-0.610788\pi$
$827$	−1.34163e7	−0.682135	−0.341067	+	0.940039i	$0.610788\pi$
$828$	0	0
$829$	1.42349e7	0.719394	0.359697	−	0.933069i	$-0.382880\pi$
$829$	1.42349e7	0.719394	0.359697	+	0.933069i	$0.382880\pi$
$830$	−2.54062e6	−0.128010
$831$	0	0
$832$	1.66765e7	0.835213
$833$	3.66009e6	0.182759
$834$	0	0
$835$	−4.52281e6	−0.224487
$836$	−7.12121e6	−0.352402
$837$	0	0
$838$	3.27414e6	0.161060
$839$	−1.15794e7	−0.567913	−0.283957	−	0.958837i	$-0.591647\pi$
$839$	−1.15794e7	−0.567913	−0.283957	+	0.958837i	$0.591647\pi$
$840$	0	0
$841$	−2.05083e7	−0.999860
$842$	3.58453e6	0.174242
$843$	0	0
$844$	1.21506e7	0.587140
$845$	−1.16128e6	−0.0559492
$846$	0	0
$847$	−2.04254e6	−0.0978276
$848$	8.12749e6	0.388121
$849$	0	0
$850$	−931462.	−0.0442199
$851$	−4.01273e6	−0.189940
$852$	0	0
$853$	6.61595e6	0.311329	0.155664	−	0.987810i	$-0.450248\pi$
$853$	6.61595e6	0.311329	0.155664	+	0.987810i	$0.450248\pi$
$854$	1.78790e6	0.0838876
$855$	0	0
$856$	919772.	0.0429038
$857$	9.97147e6	0.463775	0.231887	−	0.972743i	$-0.425510\pi$
$857$	9.97147e6	0.463775	0.231887	+	0.972743i	$0.425510\pi$
$858$	0	0
$859$	−3.79757e7	−1.75599	−0.877996	−	0.478667i	$-0.841120\pi$
$859$	−3.79757e7	−1.75599	−0.877996	+	0.478667i	$0.841120\pi$
$860$	1.03124e7	0.475458
$861$	0	0
$862$	−3.26695e6	−0.149753
$863$	−8.18531e6	−0.374118	−0.187059	−	0.982349i	$-0.559896\pi$
$863$	−8.18531e6	−0.374118	−0.187059	+	0.982349i	$0.559896\pi$
$864$	0	0
$865$	5.46447e6	0.248318
$866$	799779.	0.0362389
$867$	0	0
$868$	2.72750e6	0.122876
$869$	−106847.	−0.00479970
$870$	0	0
$871$	3.70565e7	1.65508
$872$	−5.07740e6	−0.226126
$873$	0	0
$874$	709465.	0.0314161
$875$	2.17981e6	0.0962496
$876$	0	0
$877$	−3.02517e7	−1.32816	−0.664081	−	0.747660i	$-0.731177\pi$
$877$	−3.02517e7	−1.32816	−0.664081	+	0.747660i	$0.731177\pi$
$878$	−96635.0	−0.00423056
$879$	0	0
$880$	−2.76221e6	−0.120240
$881$	−6.11414e6	−0.265397	−0.132698	−	0.991156i	$-0.542364\pi$
$881$	−6.11414e6	−0.265397	−0.132698	+	0.991156i	$0.542364\pi$
$882$	0	0
$883$	−7.72672e6	−0.333498	−0.166749	−	0.985999i	$-0.553327\pi$
$883$	−7.72672e6	−0.333498	−0.166749	+	0.985999i	$0.553327\pi$
$884$	2.74656e7	1.18211
$885$	0	0
$886$	5.47302e6	0.234230
$887$	2.48274e7	1.05955	0.529775	−	0.848138i	$-0.322276\pi$
$887$	2.48274e7	1.05955	0.529775	+	0.848138i	$0.322276\pi$
$888$	0	0
$889$	2.99499e7	1.27099
$890$	−587671.	−0.0248690
$891$	0	0
$892$	−1.37209e7	−0.577391
$893$	4.30065e6	0.180470
$894$	0	0
$895$	7.77289e6	0.324358
$896$	−1.80051e7	−0.749249
$897$	0	0
$898$	7.18613e6	0.297375
$899$	−33931.9	−0.00140026
$900$	0	0
$901$	1.22680e7	0.503457
$902$	2.46838e6	0.101017
$903$	0	0
$904$	1.39893e7	0.569342
$905$	1.05909e7	0.429845
$906$	0	0
$907$	−3.64608e6	−0.147166	−0.0735831	−	0.997289i	$-0.523443\pi$
$907$	−3.64608e6	−0.147166	−0.0735831	+	0.997289i	$0.523443\pi$
$908$	3.57291e7	1.43816
$909$	0	0
$910$	2.43742e6	0.0975723
$911$	1.15898e7	0.462678	0.231339	−	0.972873i	$-0.425689\pi$
$911$	1.15898e7	0.462678	0.231339	+	0.972873i	$0.425689\pi$
$912$	0	0
$913$	−1.13723e7	−0.451516
$914$	190642.	0.00754838
$915$	0	0
$916$	−2.73116e7	−1.07550
$917$	3.53048e7	1.38647
$918$	0	0
$919$	−3.48544e7	−1.36135	−0.680674	−	0.732586i	$-0.738313\pi$
$919$	−3.48544e7	−1.36135	−0.680674	+	0.732586i	$0.738313\pi$
$920$	583796.	0.0227401
$921$	0	0
$922$	9.12314e6	0.353441
$923$	2.00853e7	0.776021
$924$	0	0
$925$	7.29638e6	0.280384
$926$	−6.11719e6	−0.234436
$927$	0	0
$928$	169161.	0.00644806
$929$	4.11213e7	1.56325	0.781624	−	0.623750i	$-0.214392\pi$
$929$	4.11213e7	1.56325	0.781624	+	0.623750i	$0.214392\pi$
$930$	0	0
$931$	5.06903e6	0.191669
$932$	2.31352e7	0.872436
$933$	0	0
$934$	−2.74945e6	−0.103128
$935$	−4.16941e6	−0.155972
$936$	0	0
$937$	−1.53222e6	−0.0570126	−0.0285063	−	0.999594i	$-0.509075\pi$
$937$	−1.53222e6	−0.0570126	−0.0285063	+	0.999594i	$0.509075\pi$
$938$	−8.64857e6	−0.320950
$939$	0	0
$940$	1.73651e6	0.0640999
$941$	1.91551e7	0.705196	0.352598	−	0.935775i	$-0.385298\pi$
$941$	1.91551e7	0.705196	0.352598	+	0.935775i	$0.385298\pi$
$942$	0	0
$943$	6.48491e6	0.237479
$944$	1.02970e7	0.376082
$945$	0	0
$946$	−1.75047e6	−0.0635955
$947$	−2.83184e7	−1.02611	−0.513054	−	0.858356i	$-0.671486\pi$
$947$	−2.83184e7	−1.02611	−0.513054	+	0.858356i	$0.671486\pi$
$948$	0	0
$949$	−3.62015e7	−1.30485
$950$	−1.29003e6	−0.0463756
$951$	0	0
$952$	−1.30634e7	−0.467159
$953$	5.47682e7	1.95342	0.976711	−	0.214559i	$-0.0688313\pi$
$953$	5.47682e7	1.95342	0.976711	+	0.214559i	$0.0688313\pi$
$954$	0	0
$955$	2.15272e7	0.763800
$956$	1.90354e7	0.673624
$957$	0	0
$958$	7.45145e6	0.262318
$959$	3.33979e7	1.17266
$960$	0	0
$961$	−2.82270e7	−0.985954
$962$	8.15865e6	0.284237
$963$	0	0
$964$	−3.70819e7	−1.28520
$965$	−1.07646e7	−0.372116
$966$	0	0
$967$	2.29671e7	0.789843	0.394922	−	0.918715i	$-0.370772\pi$
$967$	2.29671e7	0.789843	0.394922	+	0.918715i	$0.370772\pi$
$968$	994671.	0.0341186
$969$	0	0
$970$	−4.27263e6	−0.145803
$971$	−1.46281e7	−0.497898	−0.248949	−	0.968517i	$-0.580085\pi$
$971$	−1.46281e7	−0.497898	−0.248949	+	0.968517i	$0.580085\pi$
$972$	0	0
$973$	2.32138e7	0.786076
$974$	−8.54152e6	−0.288495
$975$	0	0
$976$	1.08228e7	0.363676
$977$	1.54714e7	0.518552	0.259276	−	0.965803i	$-0.416516\pi$
$977$	1.54714e7	0.518552	0.259276	+	0.965803i	$0.416516\pi$
$978$	0	0
$979$	−2.63053e6	−0.0877176
$980$	2.04676e6	0.0680773
$981$	0	0
$982$	−2.47612e6	−0.0819395
$983$	−2.02694e7	−0.669047	−0.334524	−	0.942387i	$-0.608575\pi$
$983$	−2.02694e7	−0.669047	−0.334524	+	0.942387i	$0.608575\pi$
$984$	0	0
$985$	−4.61888e6	−0.151686
$986$	79746.8	0.00261229
$987$	0	0
$988$	3.80385e7	1.23974
$989$	−4.59881e6	−0.149505
$990$	0	0
$991$	−4.43452e7	−1.43437	−0.717187	−	0.696881i	$-0.754570\pi$
$991$	−4.43452e7	−1.43437	−0.717187	+	0.696881i	$0.754570\pi$
$992$	−2.00471e6	−0.0646803
$993$	0	0
$994$	−4.68768e6	−0.150484
$995$	−8.07483e6	−0.258569
$996$	0	0
$997$	2.32293e7	0.740112	0.370056	−	0.929009i	$-0.379338\pi$
$997$	2.32293e7	0.740112	0.370056	+	0.929009i	$0.379338\pi$
$998$	6.44775e6	0.204919
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.c.1.2		3
3.2	odd	2		165.6.a.d.1.2	✓	3
15.14	odd	2		825.6.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.d.1.2	✓	3	3.2	odd	2
495.6.a.c.1.2		3	1.1	even	1	trivial
825.6.a.h.1.2		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-1.08127 q^{2} -30.8308 q^{4} -25.0000 q^{5} -139.508 q^{7} +67.9374 q^{8} +O(q^{10})\)	\(q-1.08127 q^{2} -30.8308 q^{4} -25.0000 q^{5} -139.508 q^{7} +67.9374 q^{8} +27.0319 q^{10} +121.000 q^{11} -646.331 q^{13} +150.846 q^{14} +913.128 q^{16} +1378.32 q^{17} +1908.90 q^{19} +770.771 q^{20} -130.834 q^{22} -343.726 q^{23} +625.000 q^{25} +698.861 q^{26} +4301.15 q^{28} -53.5092 q^{29} +634.133 q^{31} -3161.34 q^{32} -1490.34 q^{34} +3487.70 q^{35} +11674.2 q^{37} -2064.04 q^{38} -1698.43 q^{40} -18866.5 q^{41} +13379.3 q^{43} -3730.53 q^{44} +371.662 q^{46} +2252.95 q^{47} +2655.48 q^{49} -675.796 q^{50} +19926.9 q^{52} +8900.71 q^{53} -3025.00 q^{55} -9477.80 q^{56} +57.8581 q^{58} +11276.7 q^{59} +11852.4 q^{61} -685.672 q^{62} -25801.8 q^{64} +16158.3 q^{65} -57333.6 q^{67} -42494.7 q^{68} -3771.16 q^{70} -31075.8 q^{71} +56010.8 q^{73} -12623.0 q^{74} -58853.0 q^{76} -16880.5 q^{77} -883.036 q^{79} -22828.2 q^{80} +20399.9 q^{82} -93986.3 q^{83} -34457.9 q^{85} -14466.7 q^{86} +8220.42 q^{88} -21739.9 q^{89} +90168.4 q^{91} +10597.4 q^{92} -2436.06 q^{94} -47722.5 q^{95} -158059. q^{97} -2871.30 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8	\( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8 + 50 * q^10 + 363 * q^11 - 546 * q^13 + 8 * q^14 - 1360 * q^16 + 314 * q^17 + 1808 * q^19 + 500 * q^20 - 242 * q^22 - 4288 * q^23 + 1875 * q^25 - 812 * q^26 + 5888 * q^28 - 5582 * q^29 + 6328 * q^31 + 736 * q^32 + 11596 * q^34 - 3800 * q^35 + 16866 * q^37 - 9584 * q^38 - 600 * q^40 - 23282 * q^41 + 20572 * q^43 - 2420 * q^44 + 16592 * q^46 - 3432 * q^47 + 11531 * q^49 - 1250 * q^50 + 21816 * q^52 - 16138 * q^53 - 9075 * q^55 - 15648 * q^56 + 17460 * q^58 - 21972 * q^59 + 8322 * q^61 - 5056 * q^62 + 22208 * q^64 + 13650 * q^65 - 84332 * q^67 - 59832 * q^68 - 200 * q^70 - 50528 * q^71 - 53838 * q^73 - 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 6364 * q^79 + 34000 * q^80 - 68020 * q^82 - 96272 * q^83 - 7850 * q^85 - 143152 * q^86 + 2904 * q^88 + 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 49088 * q^94 - 45200 * q^95 - 103242 * q^97 - 9554 * q^98

Introduction

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