LMFDB - Embedded newform 450.6.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4081}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1020 $ x^2 - x - 1020
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$32.4414$ of defining polynomial
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -241.648 q^{7} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} -241.648 q^{7} -64.0000 q^{8} -653.296 q^{11} +828.296 q^{13} +966.592 q^{14} +256.000 q^{16} +2162.48 q^{17} +1254.24 q^{19} +2613.18 q^{22} +3746.48 q^{23} -3313.18 q^{26} -3866.37 q^{28} -2466.70 q^{29} -1895.76 q^{31} -1024.00 q^{32} -8649.92 q^{34} +1050.45 q^{37} -5016.96 q^{38} -1960.56 q^{41} +11017.6 q^{43} -10452.7 q^{44} -14985.9 q^{46} -23064.8 q^{47} +41586.8 q^{49} +13252.7 q^{52} -27329.4 q^{53} +15465.5 q^{56} +9866.82 q^{58} -35225.7 q^{59} +2685.33 q^{61} +7583.04 q^{62} +4096.00 q^{64} +48125.1 q^{67} +34599.7 q^{68} -27237.7 q^{71} -6547.77 q^{73} -4201.78 q^{74} +20067.8 q^{76} +157868. q^{77} -65037.3 q^{79} +7842.23 q^{82} -62567.2 q^{83} -44070.3 q^{86} +41811.0 q^{88} -17926.2 q^{89} -200156. q^{91} +59943.7 q^{92} +92259.2 q^{94} +95974.9 q^{97} -166347. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 100 q^{7} - 128 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 32 * q^4 - 100 * q^7 - 128 * q^8	$ 2 q - 8 q^{2} + 32 q^{4} - 100 q^{7} - 128 q^{8} - 540 q^{11} + 890 q^{13} + 400 q^{14} + 512 q^{16} + 492 q^{17} + 592 q^{19} + 2160 q^{22} + 3660 q^{23} - 3560 q^{26} - 1600 q^{28} - 5700 q^{29} - 5708 q^{31} - 2048 q^{32} - 1968 q^{34} + 11300 q^{37} - 2368 q^{38} - 15420 q^{41} + 6320 q^{43} - 8640 q^{44} - 14640 q^{46} - 7800 q^{47} + 44844 q^{49} + 14240 q^{52} - 27828 q^{53} + 6400 q^{56} + 22800 q^{58} - 50520 q^{59} - 29126 q^{61} + 22832 q^{62} + 8192 q^{64} + 97400 q^{67} + 7872 q^{68} - 6180 q^{71} + 32900 q^{73} - 45200 q^{74} + 9472 q^{76} + 173916 q^{77} + 7912 q^{79} + 61680 q^{82} - 163464 q^{83} - 25280 q^{86} + 34560 q^{88} - 164640 q^{89} - 191416 q^{91} + 58560 q^{92} + 31200 q^{94} + 52430 q^{97} - 179376 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 100 * q^7 - 128 * q^8 - 540 * q^11 + 890 * q^13 + 400 * q^14 + 512 * q^16 + 492 * q^17 + 592 * q^19 + 2160 * q^22 + 3660 * q^23 - 3560 * q^26 - 1600 * q^28 - 5700 * q^29 - 5708 * q^31 - 2048 * q^32 - 1968 * q^34 + 11300 * q^37 - 2368 * q^38 - 15420 * q^41 + 6320 * q^43 - 8640 * q^44 - 14640 * q^46 - 7800 * q^47 + 44844 * q^49 + 14240 * q^52 - 27828 * q^53 + 6400 * q^56 + 22800 * q^58 - 50520 * q^59 - 29126 * q^61 + 22832 * q^62 + 8192 * q^64 + 97400 * q^67 + 7872 * q^68 - 6180 * q^71 + 32900 * q^73 - 45200 * q^74 + 9472 * q^76 + 173916 * q^77 + 7912 * q^79 + 61680 * q^82 - 163464 * q^83 - 25280 * q^86 + 34560 * q^88 - 164640 * q^89 - 191416 * q^91 + 58560 * q^92 + 31200 * q^94 + 52430 * q^97 - 179376 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−241.648	−1.86397	−0.931984	−	0.362500i	$-0.881923\pi$
$7$	−241.648	−1.86397	−0.931984	+	0.362500i	$0.881923\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	−653.296	−1.62790	−0.813951	−	0.580933i	$-0.802688\pi$
$11$	−653.296	−1.62790	−0.813951	+	0.580933i	$0.802688\pi$
$12$	0	0
$13$	828.296	1.35934	0.679669	−	0.733519i	$-0.262124\pi$
$13$	828.296	1.35934	0.679669	+	0.733519i	$0.262124\pi$
$14$	966.592	1.31802
$15$	0	0
$16$	256.000	0.250000
$17$	2162.48	1.81481	0.907403	−	0.420262i	$-0.138062\pi$
$17$	2162.48	1.81481	0.907403	+	0.420262i	$0.138062\pi$
$18$	0	0
$19$	1254.24	0.797071	0.398535	−	0.917153i	$-0.369519\pi$
$19$	1254.24	0.797071	0.398535	+	0.917153i	$0.369519\pi$
$20$	0	0
$21$	0	0
$22$	2613.18	1.15110
$23$	3746.48	1.47674	0.738370	−	0.674396i	$-0.235596\pi$
$23$	3746.48	1.47674	0.738370	+	0.674396i	$0.235596\pi$
$24$	0	0
$25$	0	0
$26$	−3313.18	−0.961197
$27$	0	0
$28$	−3866.37	−0.931984
$29$	−2466.70	−0.544656	−0.272328	−	0.962205i	$-0.587794\pi$
$29$	−2466.70	−0.544656	−0.272328	+	0.962205i	$0.587794\pi$
$30$	0	0
$31$	−1895.76	−0.354306	−0.177153	−	0.984183i	$-0.556689\pi$
$31$	−1895.76	−0.354306	−0.177153	+	0.984183i	$0.556689\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−8649.92	−1.28326
$35$	0	0
$36$	0	0
$37$	1050.45	0.126145	0.0630724	−	0.998009i	$-0.479910\pi$
$37$	1050.45	0.126145	0.0630724	+	0.998009i	$0.479910\pi$
$38$	−5016.96	−0.563614
$39$	0	0
$40$	0	0
$41$	−1960.56	−0.182146	−0.0910730	−	0.995844i	$-0.529030\pi$
$41$	−1960.56	−0.182146	−0.0910730	+	0.995844i	$0.529030\pi$
$42$	0	0
$43$	11017.6	0.908688	0.454344	−	0.890826i	$-0.349874\pi$
$43$	11017.6	0.908688	0.454344	+	0.890826i	$0.349874\pi$
$44$	−10452.7	−0.813951
$45$	0	0
$46$	−14985.9	−1.04421
$47$	−23064.8	−1.52302	−0.761509	−	0.648154i	$-0.775541\pi$
$47$	−23064.8	−1.52302	−0.761509	+	0.648154i	$0.775541\pi$
$48$	0	0
$49$	41586.8	2.47437
$50$	0	0
$51$	0	0
$52$	13252.7	0.679669
$53$	−27329.4	−1.33641	−0.668205	−	0.743977i	$-0.732937\pi$
$53$	−27329.4	−1.33641	−0.668205	+	0.743977i	$0.732937\pi$
$54$	0	0
$55$	0	0
$56$	15465.5	0.659012
$57$	0	0
$58$	9866.82	0.385130
$59$	−35225.7	−1.31744	−0.658718	−	0.752390i	$-0.728901\pi$
$59$	−35225.7	−1.31744	−0.658718	+	0.752390i	$0.728901\pi$
$60$	0	0
$61$	2685.33	0.0924002	0.0462001	−	0.998932i	$-0.485289\pi$
$61$	2685.33	0.0924002	0.0462001	+	0.998932i	$0.485289\pi$
$62$	7583.04	0.250532
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	48125.1	1.30974	0.654869	−	0.755743i	$-0.272724\pi$
$67$	48125.1	1.30974	0.654869	+	0.755743i	$0.272724\pi$
$68$	34599.7	0.907403
$69$	0	0
$70$	0	0
$71$	−27237.7	−0.641245	−0.320622	−	0.947207i	$-0.603892\pi$
$71$	−27237.7	−0.641245	−0.320622	+	0.947207i	$0.603892\pi$
$72$	0	0
$73$	−6547.77	−0.143809	−0.0719046	−	0.997412i	$-0.522908\pi$
$73$	−6547.77	−0.143809	−0.0719046	+	0.997412i	$0.522908\pi$
$74$	−4201.78	−0.0891978
$75$	0	0
$76$	20067.8	0.398535
$77$	157868.	3.03436
$78$	0	0
$79$	−65037.3	−1.17245	−0.586226	−	0.810148i	$-0.699387\pi$
$79$	−65037.3	−1.17245	−0.586226	+	0.810148i	$0.699387\pi$
$80$	0	0
$81$	0	0
$82$	7842.23	0.128797
$83$	−62567.2	−0.996900	−0.498450	−	0.866919i	$-0.666097\pi$
$83$	−62567.2	−0.996900	−0.498450	+	0.866919i	$0.666097\pi$
$84$	0	0
$85$	0	0
$86$	−44070.3	−0.642539
$87$	0	0
$88$	41811.0	0.575551
$89$	−17926.2	−0.239891	−0.119946	−	0.992780i	$-0.538272\pi$
$89$	−17926.2	−0.239891	−0.119946	+	0.992780i	$0.538272\pi$
$90$	0	0
$91$	−200156.	−2.53376
$92$	59943.7	0.738370
$93$	0	0
$94$	92259.2	1.07694
$95$	0	0
$96$	0	0
$97$	95974.9	1.03569	0.517843	−	0.855475i	$-0.326735\pi$
$97$	95974.9	1.03569	0.517843	+	0.855475i	$0.326735\pi$
$98$	−166347.	−1.74965
$99$	0	0
$100$	0	0
$101$	−41533.7	−0.405133	−0.202567	−	0.979269i	$-0.564928\pi$
$101$	−41533.7	−0.405133	−0.202567	+	0.979269i	$0.564928\pi$
$102$	0	0
$103$	−41380.4	−0.384328	−0.192164	−	0.981363i	$-0.561551\pi$
$103$	−41380.4	−0.384328	−0.192164	+	0.981363i	$0.561551\pi$
$104$	−53011.0	−0.480598
$105$	0	0
$106$	109317.	0.944985
$107$	11403.4	0.0962885	0.0481442	−	0.998840i	$-0.484669\pi$
$107$	11403.4	0.0962885	0.0481442	+	0.998840i	$0.484669\pi$
$108$	0	0
$109$	98362.4	0.792981	0.396490	−	0.918039i	$-0.370228\pi$
$109$	98362.4	0.792981	0.396490	+	0.918039i	$0.370228\pi$
$110$	0	0
$111$	0	0
$112$	−61861.9	−0.465992
$113$	−10076.8	−0.0742377	−0.0371189	−	0.999311i	$-0.511818\pi$
$113$	−10076.8	−0.0742377	−0.0371189	+	0.999311i	$0.511818\pi$
$114$	0	0
$115$	0	0
$116$	−39467.3	−0.272328
$117$	0	0
$118$	140903.	0.931568
$119$	−522559.	−3.38274
$120$	0	0
$121$	265745.	1.65007
$122$	−10741.3	−0.0653368
$123$	0	0
$124$	−30332.2	−0.177153
$125$	0	0
$126$	0	0
$127$	−255129.	−1.40362	−0.701812	−	0.712362i	$-0.747626\pi$
$127$	−255129.	−1.40362	−0.701812	+	0.712362i	$0.747626\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	65433.3	0.333135	0.166568	−	0.986030i	$-0.446732\pi$
$131$	65433.3	0.333135	0.166568	+	0.986030i	$0.446732\pi$
$132$	0	0
$133$	−303085.	−1.48571
$134$	−192500.	−0.926124
$135$	0	0
$136$	−138399.	−0.641631
$137$	67354.0	0.306593	0.153296	−	0.988180i	$-0.451011\pi$
$137$	67354.0	0.306593	0.153296	+	0.988180i	$0.451011\pi$
$138$	0	0
$139$	−93837.8	−0.411946	−0.205973	−	0.978558i	$-0.566036\pi$
$139$	−93837.8	−0.411946	−0.205973	+	0.978558i	$0.566036\pi$
$140$	0	0
$141$	0	0
$142$	108951.	0.453429
$143$	−541123.	−2.21287
$144$	0	0
$145$	0	0
$146$	26191.1	0.101688
$147$	0	0
$148$	16807.1	0.0630724
$149$	−432235.	−1.59498	−0.797489	−	0.603334i	$-0.793839\pi$
$149$	−432235.	−1.59498	−0.797489	+	0.603334i	$0.793839\pi$
$150$	0	0
$151$	−255822.	−0.913053	−0.456527	−	0.889710i	$-0.650907\pi$
$151$	−255822.	−0.913053	−0.456527	+	0.889710i	$0.650907\pi$
$152$	−80271.4	−0.281807
$153$	0	0
$154$	−631471.	−2.14561
$155$	0	0
$156$	0	0
$157$	−58837.3	−0.190504	−0.0952519	−	0.995453i	$-0.530366\pi$
$157$	−58837.3	−0.190504	−0.0952519	+	0.995453i	$0.530366\pi$
$158$	260149.	0.829048
$159$	0	0
$160$	0	0
$161$	−905330.	−2.75259
$162$	0	0
$163$	155006.	0.456962	0.228481	−	0.973548i	$-0.426624\pi$
$163$	155006.	0.456962	0.228481	+	0.973548i	$0.426624\pi$
$164$	−31368.9	−0.0910730
$165$	0	0
$166$	250269.	0.704914
$167$	588656.	1.63332	0.816658	−	0.577121i	$-0.195824\pi$
$167$	588656.	1.63332	0.816658	+	0.577121i	$0.195824\pi$
$168$	0	0
$169$	314782.	0.847798
$170$	0	0
$171$	0	0
$172$	176281.	0.454344
$173$	−192049.	−0.487862	−0.243931	−	0.969793i	$-0.578437\pi$
$173$	−192049.	−0.487862	−0.243931	+	0.969793i	$0.578437\pi$
$174$	0	0
$175$	0	0
$176$	−167244.	−0.406976
$177$	0	0
$178$	71704.9	0.169629
$179$	350715.	0.818130	0.409065	−	0.912505i	$-0.365855\pi$
$179$	350715.	0.818130	0.409065	+	0.912505i	$0.365855\pi$
$180$	0	0
$181$	524637.	1.19032	0.595158	−	0.803608i	$-0.297089\pi$
$181$	524637.	1.19032	0.595158	+	0.803608i	$0.297089\pi$
$182$	800625.	1.79164
$183$	0	0
$184$	−239775.	−0.522106
$185$	0	0
$186$	0	0
$187$	−1.41274e6	−2.95433
$188$	−369037.	−0.761509
$189$	0	0
$190$	0	0
$191$	−63662.7	−0.126270	−0.0631352	−	0.998005i	$-0.520110\pi$
$191$	−63662.7	−0.126270	−0.0631352	+	0.998005i	$0.520110\pi$
$192$	0	0
$193$	−717477.	−1.38648	−0.693242	−	0.720705i	$-0.743818\pi$
$193$	−717477.	−1.38648	−0.693242	+	0.720705i	$0.743818\pi$
$194$	−383900.	−0.732341
$195$	0	0
$196$	665389.	1.23719
$197$	287465.	0.527740	0.263870	−	0.964558i	$-0.415001\pi$
$197$	287465.	0.527740	0.263870	+	0.964558i	$0.415001\pi$
$198$	0	0
$199$	643619.	1.15212	0.576058	−	0.817409i	$-0.304590\pi$
$199$	643619.	1.15212	0.576058	+	0.817409i	$0.304590\pi$
$200$	0	0
$201$	0	0
$202$	166135.	0.286472
$203$	596074.	1.01522
$204$	0	0
$205$	0	0
$206$	165522.	0.271761
$207$	0	0
$208$	212044.	0.339834
$209$	−819391.	−1.29755
$210$	0	0
$211$	−873827.	−1.35120	−0.675600	−	0.737269i	$-0.736115\pi$
$211$	−873827.	−1.35120	−0.675600	+	0.737269i	$0.736115\pi$
$212$	−437270.	−0.668205
$213$	0	0
$214$	−45613.5	−0.0680862
$215$	0	0
$216$	0	0
$217$	458107.	0.660416
$218$	−393449.	−0.560722
$219$	0	0
$220$	0	0
$221$	1.79117e6	2.46693
$222$	0	0
$223$	−288496.	−0.388488	−0.194244	−	0.980953i	$-0.562225\pi$
$223$	−288496.	−0.388488	−0.194244	+	0.980953i	$0.562225\pi$
$224$	247448.	0.329506
$225$	0	0
$226$	40307.0	0.0524940
$227$	−125358.	−0.161469	−0.0807343	−	0.996736i	$-0.525727\pi$
$227$	−125358.	−0.161469	−0.0807343	+	0.996736i	$0.525727\pi$
$228$	0	0
$229$	−698183.	−0.879793	−0.439897	−	0.898048i	$-0.644985\pi$
$229$	−698183.	−0.879793	−0.439897	+	0.898048i	$0.644985\pi$
$230$	0	0
$231$	0	0
$232$	157869.	0.192565
$233$	−984588.	−1.18813	−0.594066	−	0.804416i	$-0.702478\pi$
$233$	−984588.	−1.18813	−0.594066	+	0.804416i	$0.702478\pi$
$234$	0	0
$235$	0	0
$236$	−563611.	−0.658718
$237$	0	0
$238$	2.09024e6	2.39196
$239$	−131049.	−0.148402	−0.0742011	−	0.997243i	$-0.523641\pi$
$239$	−131049.	−0.148402	−0.0742011	+	0.997243i	$0.523641\pi$
$240$	0	0
$241$	−1.23632e6	−1.37116	−0.685579	−	0.727998i	$-0.740451\pi$
$241$	−1.23632e6	−1.37116	−0.685579	+	0.727998i	$0.740451\pi$
$242$	−1.06298e6	−1.16677
$243$	0	0
$244$	42965.3	0.0462001
$245$	0	0
$246$	0	0
$247$	1.03888e6	1.08349
$248$	121329.	0.125266
$249$	0	0
$250$	0	0
$251$	−611918.	−0.613068	−0.306534	−	0.951860i	$-0.599169\pi$
$251$	−611918.	−0.613068	−0.306534	+	0.951860i	$0.599169\pi$
$252$	0	0
$253$	−2.44756e6	−2.40399
$254$	1.02052e6	0.992513
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.64515e6	−1.55372	−0.776858	−	0.629675i	$-0.783188\pi$
$257$	−1.64515e6	−1.55372	−0.776858	+	0.629675i	$0.783188\pi$
$258$	0	0
$259$	−253838.	−0.235130
$260$	0	0
$261$	0	0
$262$	−261733.	−0.235562
$263$	268273.	0.239159	0.119580	−	0.992825i	$-0.461845\pi$
$263$	268273.	0.239159	0.119580	+	0.992825i	$0.461845\pi$
$264$	0	0
$265$	0	0
$266$	1.21234e6	1.05056
$267$	0	0
$268$	770001.	0.654869
$269$	947177.	0.798087	0.399044	−	0.916932i	$-0.369342\pi$
$269$	947177.	0.798087	0.399044	+	0.916932i	$0.369342\pi$
$270$	0	0
$271$	609159.	0.503857	0.251929	−	0.967746i	$-0.418935\pi$
$271$	609159.	0.503857	0.251929	+	0.967746i	$0.418935\pi$
$272$	553595.	0.453701
$273$	0	0
$274$	−269416.	−0.216794
$275$	0	0
$276$	0	0
$277$	741976.	0.581019	0.290510	−	0.956872i	$-0.406175\pi$
$277$	741976.	0.581019	0.290510	+	0.956872i	$0.406175\pi$
$278$	375351.	0.291290
$279$	0	0
$280$	0	0
$281$	423084.	0.319639	0.159820	−	0.987146i	$-0.448909\pi$
$281$	423084.	0.319639	0.159820	+	0.987146i	$0.448909\pi$
$282$	0	0
$283$	−445130.	−0.330385	−0.165192	−	0.986261i	$-0.552825\pi$
$283$	−445130.	−0.330385	−0.165192	+	0.986261i	$0.552825\pi$
$284$	−435803.	−0.320622
$285$	0	0
$286$	2.16449e6	1.56473
$287$	473765.	0.339514
$288$	0	0
$289$	3.25647e6	2.29352
$290$	0	0
$291$	0	0
$292$	−104764.	−0.0719046
$293$	2.45992e6	1.67399	0.836994	−	0.547211i	$-0.184311\pi$
$293$	2.45992e6	1.67399	0.836994	+	0.547211i	$0.184311\pi$
$294$	0	0
$295$	0	0
$296$	−67228.5	−0.0445989
$297$	0	0
$298$	1.72894e6	1.12782
$299$	3.10320e6	2.00739
$300$	0	0
$301$	−2.66238e6	−1.69376
$302$	1.02329e6	0.645626
$303$	0	0
$304$	321086.	0.199268
$305$	0	0
$306$	0	0
$307$	374253.	0.226631	0.113315	−	0.993559i	$-0.463853\pi$
$307$	374253.	0.226631	0.113315	+	0.993559i	$0.463853\pi$
$308$	2.52588e6	1.51718
$309$	0	0
$310$	0	0
$311$	−2.22568e6	−1.30485	−0.652427	−	0.757851i	$-0.726249\pi$
$311$	−2.22568e6	−1.30485	−0.652427	+	0.757851i	$0.726249\pi$
$312$	0	0
$313$	3.05869e6	1.76471	0.882357	−	0.470580i	$-0.155955\pi$
$313$	3.05869e6	1.76471	0.882357	+	0.470580i	$0.155955\pi$
$314$	235349.	0.134706
$315$	0	0
$316$	−1.04060e6	−0.586226
$317$	−520610.	−0.290981	−0.145490	−	0.989360i	$-0.546476\pi$
$317$	−520610.	−0.290981	−0.145490	+	0.989360i	$0.546476\pi$
$318$	0	0
$319$	1.61149e6	0.886646
$320$	0	0
$321$	0	0
$322$	3.62132e6	1.94638
$323$	2.71227e6	1.44653
$324$	0	0
$325$	0	0
$326$	−620025.	−0.323121
$327$	0	0
$328$	125476.	0.0643983
$329$	5.57357e6	2.83886
$330$	0	0
$331$	−2.12755e6	−1.06736	−0.533679	−	0.845687i	$-0.679191\pi$
$331$	−2.12755e6	−1.06736	−0.533679	+	0.845687i	$0.679191\pi$
$332$	−1.00108e6	−0.498450
$333$	0	0
$334$	−2.35462e6	−1.15493
$335$	0	0
$336$	0	0
$337$	−3.08383e6	−1.47916	−0.739581	−	0.673067i	$-0.764976\pi$
$337$	−3.08383e6	−1.47916	−0.739581	+	0.673067i	$0.764976\pi$
$338$	−1.25913e6	−0.599484
$339$	0	0
$340$	0	0
$341$	1.23849e6	0.576776
$342$	0	0
$343$	−5.98799e6	−2.74819
$344$	−705125.	−0.321270
$345$	0	0
$346$	768196.	0.344970
$347$	−4.04955e6	−1.80544	−0.902719	−	0.430231i	$-0.858432\pi$
$347$	−4.04955e6	−1.80544	−0.902719	+	0.430231i	$0.858432\pi$
$348$	0	0
$349$	−493135.	−0.216722	−0.108361	−	0.994112i	$-0.534560\pi$
$349$	−493135.	−0.216722	−0.108361	+	0.994112i	$0.534560\pi$
$350$	0	0
$351$	0	0
$352$	668975.	0.287775
$353$	−1.34815e6	−0.575841	−0.287920	−	0.957654i	$-0.592964\pi$
$353$	−1.34815e6	−0.575841	−0.287920	+	0.957654i	$0.592964\pi$
$354$	0	0
$355$	0	0
$356$	−286820.	−0.119946
$357$	0	0
$358$	−1.40286e6	−0.578505
$359$	−1.64749e6	−0.674664	−0.337332	−	0.941386i	$-0.609524\pi$
$359$	−1.64749e6	−0.674664	−0.337332	+	0.941386i	$0.609524\pi$
$360$	0	0
$361$	−902980.	−0.364678
$362$	−2.09855e6	−0.841681
$363$	0	0
$364$	−3.20250e6	−1.26688
$365$	0	0
$366$	0	0
$367$	−27918.4	−0.0108199	−0.00540997	−	0.999985i	$-0.501722\pi$
$367$	−27918.4	−0.0108199	−0.00540997	+	0.999985i	$0.501722\pi$
$368$	959099.	0.369185
$369$	0	0
$370$	0	0
$371$	6.60409e6	2.49103
$372$	0	0
$373$	2.80536e6	1.04404	0.522019	−	0.852934i	$-0.325179\pi$
$373$	2.80536e6	1.04404	0.522019	+	0.852934i	$0.325179\pi$
$374$	5.65096e6	2.08902
$375$	0	0
$376$	1.47615e6	0.538468
$377$	−2.04316e6	−0.740371
$378$	0	0
$379$	1.70014e6	0.607976	0.303988	−	0.952676i	$-0.401682\pi$
$379$	1.70014e6	0.607976	0.303988	+	0.952676i	$0.401682\pi$
$380$	0	0
$381$	0	0
$382$	254651.	0.0892867
$383$	−627913.	−0.218727	−0.109363	−	0.994002i	$-0.534881\pi$
$383$	−627913.	−0.218727	−0.109363	+	0.994002i	$0.534881\pi$
$384$	0	0
$385$	0	0
$386$	2.86991e6	0.980392
$387$	0	0
$388$	1.53560e6	0.517843
$389$	−3.14339e6	−1.05323	−0.526616	−	0.850103i	$-0.676540\pi$
$389$	−3.14339e6	−1.05323	−0.526616	+	0.850103i	$0.676540\pi$
$390$	0	0
$391$	8.10169e6	2.68000
$392$	−2.66156e6	−0.874823
$393$	0	0
$394$	−1.14986e6	−0.373169
$395$	0	0
$396$	0	0
$397$	−1.10773e6	−0.352743	−0.176371	−	0.984324i	$-0.556436\pi$
$397$	−1.10773e6	−0.352743	−0.176371	+	0.984324i	$0.556436\pi$
$398$	−2.57448e6	−0.814669
$399$	0	0
$400$	0	0
$401$	−2.41563e6	−0.750187	−0.375093	−	0.926987i	$-0.622389\pi$
$401$	−2.41563e6	−0.750187	−0.375093	+	0.926987i	$0.622389\pi$
$402$	0	0
$403$	−1.57025e6	−0.481622
$404$	−664540.	−0.202567
$405$	0	0
$406$	−2.38430e6	−0.717869
$407$	−686252.	−0.205351
$408$	0	0
$409$	3.69830e6	1.09319	0.546593	−	0.837398i	$-0.315925\pi$
$409$	3.69830e6	1.09319	0.546593	+	0.837398i	$0.315925\pi$
$410$	0	0
$411$	0	0
$412$	−662087.	−0.192164
$413$	8.51222e6	2.45566
$414$	0	0
$415$	0	0
$416$	−848175.	−0.240299
$417$	0	0
$418$	3.27756e6	0.917509
$419$	3.45478e6	0.961357	0.480679	−	0.876897i	$-0.340390\pi$
$419$	3.45478e6	0.961357	0.480679	+	0.876897i	$0.340390\pi$
$420$	0	0
$421$	−4.83980e6	−1.33083	−0.665415	−	0.746474i	$-0.731745\pi$
$421$	−4.83980e6	−1.33083	−0.665415	+	0.746474i	$0.731745\pi$
$422$	3.49531e6	0.955442
$423$	0	0
$424$	1.74908e6	0.472493
$425$	0	0
$426$	0	0
$427$	−648905.	−0.172231
$428$	182454.	0.0481442
$429$	0	0
$430$	0	0
$431$	5.54923e6	1.43893	0.719464	−	0.694529i	$-0.244387\pi$
$431$	5.54923e6	1.43893	0.719464	+	0.694529i	$0.244387\pi$
$432$	0	0
$433$	−814302.	−0.208721	−0.104360	−	0.994540i	$-0.533280\pi$
$433$	−814302.	−0.208721	−0.104360	+	0.994540i	$0.533280\pi$
$434$	−1.83243e6	−0.466984
$435$	0	0
$436$	1.57380e6	0.396490
$437$	4.69899e6	1.17707
$438$	0	0
$439$	−4.02462e6	−0.996697	−0.498349	−	0.866977i	$-0.666060\pi$
$439$	−4.02462e6	−0.996697	−0.498349	+	0.866977i	$0.666060\pi$
$440$	0	0
$441$	0	0
$442$	−7.16470e6	−1.74438
$443$	−4.46382e6	−1.08068	−0.540341	−	0.841446i	$-0.681705\pi$
$443$	−4.46382e6	−1.08068	−0.540341	+	0.841446i	$0.681705\pi$
$444$	0	0
$445$	0	0
$446$	1.15399e6	0.274703
$447$	0	0
$448$	−989791.	−0.232996
$449$	4.66889e6	1.09294	0.546472	−	0.837477i	$-0.315970\pi$
$449$	4.66889e6	1.09294	0.546472	+	0.837477i	$0.315970\pi$
$450$	0	0
$451$	1.28082e6	0.296516
$452$	−161228.	−0.0371189
$453$	0	0
$454$	501433.	0.114176
$455$	0	0
$456$	0	0
$457$	−4.71340e6	−1.05571	−0.527854	−	0.849335i	$-0.677003\pi$
$457$	−4.71340e6	−1.05571	−0.527854	+	0.849335i	$0.677003\pi$
$458$	2.79273e6	0.622108
$459$	0	0
$460$	0	0
$461$	−6.23975e6	−1.36746	−0.683731	−	0.729734i	$-0.739644\pi$
$461$	−6.23975e6	−1.36746	−0.683731	+	0.729734i	$0.739644\pi$
$462$	0	0
$463$	−3.33992e6	−0.724076	−0.362038	−	0.932163i	$-0.617919\pi$
$463$	−3.33992e6	−0.724076	−0.362038	+	0.932163i	$0.617919\pi$
$464$	−631476.	−0.136164
$465$	0	0
$466$	3.93835e6	0.840136
$467$	722497.	0.153301	0.0766503	−	0.997058i	$-0.475578\pi$
$467$	722497.	0.153301	0.0766503	+	0.997058i	$0.475578\pi$
$468$	0	0
$469$	−1.16293e7	−2.44131
$470$	0	0
$471$	0	0
$472$	2.25444e6	0.465784
$473$	−7.19774e6	−1.47926
$474$	0	0
$475$	0	0
$476$	−8.36095e6	−1.69137
$477$	0	0
$478$	524197.	0.104936
$479$	5.37714e6	1.07081	0.535405	−	0.844595i	$-0.320159\pi$
$479$	5.37714e6	1.07081	0.535405	+	0.844595i	$0.320159\pi$
$480$	0	0
$481$	870080.	0.171473
$482$	4.94527e6	0.969556
$483$	0	0
$484$	4.25192e6	0.825034
$485$	0	0
$486$	0	0
$487$	−1.72084e6	−0.328790	−0.164395	−	0.986395i	$-0.552567\pi$
$487$	−1.72084e6	−0.328790	−0.164395	+	0.986395i	$0.552567\pi$
$488$	−171861.	−0.0326684
$489$	0	0
$490$	0	0
$491$	−7.31449e6	−1.36924	−0.684621	−	0.728899i	$-0.740032\pi$
$491$	−7.31449e6	−1.36924	−0.684621	+	0.728899i	$0.740032\pi$
$492$	0	0
$493$	−5.33420e6	−0.988444
$494$	−4.15553e6	−0.766142
$495$	0	0
$496$	−485314.	−0.0885766
$497$	6.58193e6	1.19526
$498$	0	0
$499$	−7.29878e6	−1.31220	−0.656098	−	0.754676i	$-0.727794\pi$
$499$	−7.29878e6	−1.31220	−0.656098	+	0.754676i	$0.727794\pi$
$500$	0	0
$501$	0	0
$502$	2.44767e6	0.433505
$503$	923591.	0.162764	0.0813822	−	0.996683i	$-0.474067\pi$
$503$	923591.	0.162764	0.0813822	+	0.996683i	$0.474067\pi$
$504$	0	0
$505$	0	0
$506$	9.79025e6	1.69988
$507$	0	0
$508$	−4.08207e6	−0.701812
$509$	−8.81318e6	−1.50778	−0.753891	−	0.657000i	$-0.771825\pi$
$509$	−8.81318e6	−1.50778	−0.753891	+	0.657000i	$0.771825\pi$
$510$	0	0
$511$	1.58226e6	0.268056
$512$	−262144.	−0.0441942
$513$	0	0
$514$	6.58059e6	1.09864
$515$	0	0
$516$	0	0
$517$	1.50682e7	2.47933
$518$	1.01535e6	0.166262
$519$	0	0
$520$	0	0
$521$	−1.66995e6	−0.269531	−0.134765	−	0.990878i	$-0.543028\pi$
$521$	−1.66995e6	−0.269531	−0.134765	+	0.990878i	$0.543028\pi$
$522$	0	0
$523$	2.85438e6	0.456308	0.228154	−	0.973625i	$-0.426731\pi$
$523$	2.85438e6	0.456308	0.228154	+	0.973625i	$0.426731\pi$
$524$	1.04693e6	0.166568
$525$	0	0
$526$	−1.07309e6	−0.169111
$527$	−4.09954e6	−0.642997
$528$	0	0
$529$	7.59978e6	1.18076
$530$	0	0
$531$	0	0
$532$	−4.84936e6	−0.742857
$533$	−1.62392e6	−0.247598
$534$	0	0
$535$	0	0
$536$	−3.08000e6	−0.463062
$537$	0	0
$538$	−3.78871e6	−0.564333
$539$	−2.71685e7	−4.02804
$540$	0	0
$541$	−7.85655e6	−1.15409	−0.577044	−	0.816713i	$-0.695794\pi$
$541$	−7.85655e6	−1.15409	−0.577044	+	0.816713i	$0.695794\pi$
$542$	−2.43664e6	−0.356281
$543$	0	0
$544$	−2.21438e6	−0.320815
$545$	0	0
$546$	0	0
$547$	−1.07748e7	−1.53972	−0.769860	−	0.638213i	$-0.779674\pi$
$547$	−1.07748e7	−1.53972	−0.769860	+	0.638213i	$0.779674\pi$
$548$	1.07766e6	0.153296
$549$	0	0
$550$	0	0
$551$	−3.09384e6	−0.434129
$552$	0	0
$553$	1.57161e7	2.18541
$554$	−2.96791e6	−0.410843
$555$	0	0
$556$	−1.50140e6	−0.205973
$557$	−7.62158e6	−1.04090	−0.520448	−	0.853893i	$-0.674235\pi$
$557$	−7.62158e6	−1.04090	−0.520448	+	0.853893i	$0.674235\pi$
$558$	0	0
$559$	9.12581e6	1.23521
$560$	0	0
$561$	0	0
$562$	−1.69233e6	−0.226019
$563$	−4.25551e6	−0.565823	−0.282911	−	0.959146i	$-0.591300\pi$
$563$	−4.25551e6	−0.565823	−0.282911	+	0.959146i	$0.591300\pi$
$564$	0	0
$565$	0	0
$566$	1.78052e6	0.233617
$567$	0	0
$568$	1.74321e6	0.226714
$569$	−5.99899e6	−0.776779	−0.388389	−	0.921495i	$-0.626968\pi$
$569$	−5.99899e6	−0.776779	−0.388389	+	0.921495i	$0.626968\pi$
$570$	0	0
$571$	9.20283e6	1.18122	0.590611	−	0.806957i	$-0.298887\pi$
$571$	9.20283e6	1.18122	0.590611	+	0.806957i	$0.298887\pi$
$572$	−8.65796e6	−1.10643
$573$	0	0
$574$	−1.89506e6	−0.240073
$575$	0	0
$576$	0	0
$577$	4.03879e6	0.505024	0.252512	−	0.967594i	$-0.418743\pi$
$577$	4.03879e6	0.505024	0.252512	+	0.967594i	$0.418743\pi$
$578$	−1.30259e7	−1.62176
$579$	0	0
$580$	0	0
$581$	1.51192e7	1.85819
$582$	0	0
$583$	1.78542e7	2.17555
$584$	419058.	0.0508442
$585$	0	0
$586$	−9.83969e6	−1.18369
$587$	1.65997e7	1.98840	0.994200	−	0.107544i	$-0.0342987\pi$
$587$	1.65997e7	1.98840	0.994200	+	0.107544i	$0.0342987\pi$
$588$	0	0
$589$	−2.37774e6	−0.282407
$590$	0	0
$591$	0	0
$592$	268914.	0.0315362
$593$	−1.09900e7	−1.28340	−0.641698	−	0.766957i	$-0.721770\pi$
$593$	−1.09900e7	−1.28340	−0.641698	+	0.766957i	$0.721770\pi$
$594$	0	0
$595$	0	0
$596$	−6.91577e6	−0.797489
$597$	0	0
$598$	−1.24128e7	−1.41944
$599$	−4.47167e6	−0.509217	−0.254608	−	0.967044i	$-0.581947\pi$
$599$	−4.47167e6	−0.509217	−0.254608	+	0.967044i	$0.581947\pi$
$600$	0	0
$601$	1.60311e7	1.81041	0.905206	−	0.424974i	$-0.139717\pi$
$601$	1.60311e7	1.81041	0.905206	+	0.424974i	$0.139717\pi$
$602$	1.06495e7	1.19767
$603$	0	0
$604$	−4.09316e6	−0.456527
$605$	0	0
$606$	0	0
$607$	−720250.	−0.0793435	−0.0396718	−	0.999213i	$-0.512631\pi$
$607$	−720250.	−0.0793435	−0.0396718	+	0.999213i	$0.512631\pi$
$608$	−1.28434e6	−0.140904
$609$	0	0
$610$	0	0
$611$	−1.91045e7	−2.07030
$612$	0	0
$613$	−1.46218e7	−1.57163	−0.785813	−	0.618464i	$-0.787755\pi$
$613$	−1.46218e7	−1.57163	−0.785813	+	0.618464i	$0.787755\pi$
$614$	−1.49701e6	−0.160252
$615$	0	0
$616$	−1.01035e7	−1.07281
$617$	4.57269e6	0.483569	0.241784	−	0.970330i	$-0.422267\pi$
$617$	4.57269e6	0.483569	0.241784	+	0.970330i	$0.422267\pi$
$618$	0	0
$619$	−4.62486e6	−0.485145	−0.242573	−	0.970133i	$-0.577991\pi$
$619$	−4.62486e6	−0.485145	−0.242573	+	0.970133i	$0.577991\pi$
$620$	0	0
$621$	0	0
$622$	8.90273e6	0.922671
$623$	4.33184e6	0.447149
$624$	0	0
$625$	0	0
$626$	−1.22348e7	−1.24784
$627$	0	0
$628$	−941397.	−0.0952519
$629$	2.27157e6	0.228928
$630$	0	0
$631$	−1.03670e7	−1.03653	−0.518263	−	0.855221i	$-0.673421\pi$
$631$	−1.03670e7	−1.03653	−0.518263	+	0.855221i	$0.673421\pi$
$632$	4.16239e6	0.414524
$633$	0	0
$634$	2.08244e6	0.205755
$635$	0	0
$636$	0	0
$637$	3.44462e7	3.36351
$638$	−6.44595e6	−0.626954
$639$	0	0
$640$	0	0
$641$	4.62432e6	0.444532	0.222266	−	0.974986i	$-0.428655\pi$
$641$	4.62432e6	0.444532	0.222266	+	0.974986i	$0.428655\pi$
$642$	0	0
$643$	1.13460e7	1.08222	0.541111	−	0.840951i	$-0.318004\pi$
$643$	1.13460e7	1.08222	0.541111	+	0.840951i	$0.318004\pi$
$644$	−1.44853e7	−1.37630
$645$	0	0
$646$	−1.08491e7	−1.02285
$647$	1.10482e7	1.03760	0.518800	−	0.854895i	$-0.326379\pi$
$647$	1.10482e7	1.03760	0.518800	+	0.854895i	$0.326379\pi$
$648$	0	0
$649$	2.30128e7	2.14466
$650$	0	0
$651$	0	0
$652$	2.48010e6	0.228481
$653$	−1.37432e6	−0.126126	−0.0630631	−	0.998010i	$-0.520087\pi$
$653$	−1.37432e6	−0.126126	−0.0630631	+	0.998010i	$0.520087\pi$
$654$	0	0
$655$	0	0
$656$	−501902.	−0.0455365
$657$	0	0
$658$	−2.22943e7	−2.00738
$659$	1.28061e7	1.14869	0.574344	−	0.818614i	$-0.305257\pi$
$659$	1.28061e7	1.14869	0.574344	+	0.818614i	$0.305257\pi$
$660$	0	0
$661$	−104794.	−0.00932894	−0.00466447	−	0.999989i	$-0.501485\pi$
$661$	−104794.	−0.00932894	−0.00466447	+	0.999989i	$0.501485\pi$
$662$	8.51021e6	0.754737
$663$	0	0
$664$	4.00430e6	0.352457
$665$	0	0
$666$	0	0
$667$	−9.24146e6	−0.804315
$668$	9.41850e6	0.816658
$669$	0	0
$670$	0	0
$671$	−1.75432e6	−0.150419
$672$	0	0
$673$	1.80490e7	1.53608	0.768042	−	0.640400i	$-0.221231\pi$
$673$	1.80490e7	1.53608	0.768042	+	0.640400i	$0.221231\pi$
$674$	1.23353e7	1.04593
$675$	0	0
$676$	5.03651e6	0.423899
$677$	1.29914e7	1.08939	0.544697	−	0.838633i	$-0.316645\pi$
$677$	1.29914e7	1.08939	0.544697	+	0.838633i	$0.316645\pi$
$678$	0	0
$679$	−2.31922e7	−1.93049
$680$	0	0
$681$	0	0
$682$	−4.95397e6	−0.407842
$683$	−3.91843e6	−0.321411	−0.160705	−	0.987002i	$-0.551377\pi$
$683$	−3.91843e6	−0.321411	−0.160705	+	0.987002i	$0.551377\pi$
$684$	0	0
$685$	0	0
$686$	2.39520e7	1.94326
$687$	0	0
$688$	2.82050e6	0.227172
$689$	−2.26368e7	−1.81663
$690$	0	0
$691$	−3.11077e6	−0.247840	−0.123920	−	0.992292i	$-0.539547\pi$
$691$	−3.11077e6	−0.247840	−0.123920	+	0.992292i	$0.539547\pi$
$692$	−3.07278e6	−0.243931
$693$	0	0
$694$	1.61982e7	1.27664
$695$	0	0
$696$	0	0
$697$	−4.23967e6	−0.330560
$698$	1.97254e6	0.153245
$699$	0	0
$700$	0	0
$701$	−1.94916e7	−1.49814	−0.749070	−	0.662490i	$-0.769500\pi$
$701$	−1.94916e7	−1.49814	−0.749070	+	0.662490i	$0.769500\pi$
$702$	0	0
$703$	1.31751e6	0.100546
$704$	−2.67590e6	−0.203488
$705$	0	0
$706$	5.39261e6	0.407181
$707$	1.00366e7	0.755155
$708$	0	0
$709$	−8.13693e6	−0.607918	−0.303959	−	0.952685i	$-0.598309\pi$
$709$	−8.13693e6	−0.607918	−0.303959	+	0.952685i	$0.598309\pi$
$710$	0	0
$711$	0	0
$712$	1.14728e6	0.0848143
$713$	−7.10243e6	−0.523218
$714$	0	0
$715$	0	0
$716$	5.61144e6	0.409065
$717$	0	0
$718$	6.58997e6	0.477059
$719$	1.14993e7	0.829566	0.414783	−	0.909920i	$-0.363857\pi$
$719$	1.14993e7	0.829566	0.414783	+	0.909920i	$0.363857\pi$
$720$	0	0
$721$	9.99951e6	0.716375
$722$	3.61192e6	0.257867
$723$	0	0
$724$	8.39419e6	0.595158
$725$	0	0
$726$	0	0
$727$	−2.10897e7	−1.47991	−0.739953	−	0.672658i	$-0.765152\pi$
$727$	−2.10897e7	−1.47991	−0.739953	+	0.672658i	$0.765152\pi$
$728$	1.28100e7	0.895820
$729$	0	0
$730$	0	0
$731$	2.38253e7	1.64909
$732$	0	0
$733$	2.20274e7	1.51427	0.757134	−	0.653259i	$-0.226599\pi$
$733$	2.20274e7	1.51427	0.757134	+	0.653259i	$0.226599\pi$
$734$	111673.	0.00765085
$735$	0	0
$736$	−3.83640e6	−0.261053
$737$	−3.14399e7	−2.13213
$738$	0	0
$739$	−1.26088e7	−0.849305	−0.424653	−	0.905356i	$-0.639604\pi$
$739$	−1.26088e7	−0.849305	−0.424653	+	0.905356i	$0.639604\pi$
$740$	0	0
$741$	0	0
$742$	−2.64164e7	−1.76142
$743$	−5.90968e6	−0.392728	−0.196364	−	0.980531i	$-0.562913\pi$
$743$	−5.90968e6	−0.392728	−0.196364	+	0.980531i	$0.562913\pi$
$744$	0	0
$745$	0	0
$746$	−1.12214e7	−0.738247
$747$	0	0
$748$	−2.26039e7	−1.47716
$749$	−2.75561e6	−0.179479
$750$	0	0
$751$	−1.15585e7	−0.747830	−0.373915	−	0.927463i	$-0.621985\pi$
$751$	−1.15585e7	−0.747830	−0.373915	+	0.927463i	$0.621985\pi$
$752$	−5.90459e6	−0.380755
$753$	0	0
$754$	8.17265e6	0.523521
$755$	0	0
$756$	0	0
$757$	2.64988e7	1.68068	0.840342	−	0.542057i	$-0.182354\pi$
$757$	2.64988e7	1.68068	0.840342	+	0.542057i	$0.182354\pi$
$758$	−6.80056e6	−0.429904
$759$	0	0
$760$	0	0
$761$	−1.54716e6	−0.0968442	−0.0484221	−	0.998827i	$-0.515419\pi$
$761$	−1.54716e6	−0.0968442	−0.0484221	+	0.998827i	$0.515419\pi$
$762$	0	0
$763$	−2.37691e7	−1.47809
$764$	−1.01860e6	−0.0631352
$765$	0	0
$766$	2.51165e6	0.154663
$767$	−2.91773e7	−1.79084
$768$	0	0
$769$	1.16191e7	0.708529	0.354265	−	0.935145i	$-0.384731\pi$
$769$	1.16191e7	0.708529	0.354265	+	0.935145i	$0.384731\pi$
$770$	0	0
$771$	0	0
$772$	−1.14796e7	−0.693242
$773$	−1.16233e7	−0.699649	−0.349824	−	0.936815i	$-0.613759\pi$
$773$	−1.16233e7	−0.699649	−0.349824	+	0.936815i	$0.613759\pi$
$774$	0	0
$775$	0	0
$776$	−6.14239e6	−0.366171
$777$	0	0
$778$	1.25736e7	0.744748
$779$	−2.45901e6	−0.145183
$780$	0	0
$781$	1.77943e7	1.04388
$782$	−3.24068e7	−1.89504
$783$	0	0
$784$	1.06462e7	0.618594
$785$	0	0
$786$	0	0
$787$	2.03314e7	1.17012	0.585059	−	0.810991i	$-0.301071\pi$
$787$	2.03314e7	1.17012	0.585059	+	0.810991i	$0.301071\pi$
$788$	4.59945e6	0.263870
$789$	0	0
$790$	0	0
$791$	2.43503e6	0.138377
$792$	0	0
$793$	2.22425e6	0.125603
$794$	4.43092e6	0.249427
$795$	0	0
$796$	1.02979e7	0.576058
$797$	−6.32170e6	−0.352523	−0.176262	−	0.984343i	$-0.556401\pi$
$797$	−6.32170e6	−0.352523	−0.176262	+	0.984343i	$0.556401\pi$
$798$	0	0
$799$	−4.98772e7	−2.76398
$800$	0	0
$801$	0	0
$802$	9.66252e6	0.530462
$803$	4.27764e6	0.234107
$804$	0	0
$805$	0	0
$806$	6.28100e6	0.340558
$807$	0	0
$808$	2.65816e6	0.143236
$809$	1.84549e7	0.991380	0.495690	−	0.868499i	$-0.334915\pi$
$809$	1.84549e7	0.991380	0.495690	+	0.868499i	$0.334915\pi$
$810$	0	0
$811$	3.00263e7	1.60306	0.801530	−	0.597954i	$-0.204020\pi$
$811$	3.00263e7	1.60306	0.801530	+	0.597954i	$0.204020\pi$
$812$	9.53719e6	0.507610
$813$	0	0
$814$	2.74501e6	0.145205
$815$	0	0
$816$	0	0
$817$	1.38187e7	0.724289
$818$	−1.47932e7	−0.772999
$819$	0	0
$820$	0	0
$821$	2.51609e7	1.30277	0.651387	−	0.758746i	$-0.274188\pi$
$821$	2.51609e7	1.30277	0.651387	+	0.758746i	$0.274188\pi$
$822$	0	0
$823$	−7.97388e6	−0.410365	−0.205182	−	0.978724i	$-0.565779\pi$
$823$	−7.97388e6	−0.410365	−0.205182	+	0.978724i	$0.565779\pi$
$824$	2.64835e6	0.135880
$825$	0	0
$826$	−3.40489e7	−1.73641
$827$	−1.37135e7	−0.697243	−0.348622	−	0.937264i	$-0.613350\pi$
$827$	−1.37135e7	−0.697243	−0.348622	+	0.937264i	$0.613350\pi$
$828$	0	0
$829$	−1.62566e7	−0.821569	−0.410785	−	0.911732i	$-0.634745\pi$
$829$	−1.62566e7	−0.821569	−0.410785	+	0.911732i	$0.634745\pi$
$830$	0	0
$831$	0	0
$832$	3.39270e6	0.169917
$833$	8.99307e7	4.49051
$834$	0	0
$835$	0	0
$836$	−1.31103e7	−0.648777
$837$	0	0
$838$	−1.38191e7	−0.679782
$839$	2.36734e7	1.16106	0.580532	−	0.814237i	$-0.302844\pi$
$839$	2.36734e7	1.16106	0.580532	+	0.814237i	$0.302844\pi$
$840$	0	0
$841$	−1.44265e7	−0.703350
$842$	1.93592e7	0.941039
$843$	0	0
$844$	−1.39812e7	−0.675600
$845$	0	0
$846$	0	0
$847$	−6.42168e7	−3.07567
$848$	−6.99632e6	−0.334103
$849$	0	0
$850$	0	0
$851$	3.93547e6	0.186283
$852$	0	0
$853$	−2.53714e7	−1.19391	−0.596956	−	0.802274i	$-0.703623\pi$
$853$	−2.53714e7	−1.19391	−0.596956	+	0.802274i	$0.703623\pi$
$854$	2.59562e6	0.121786
$855$	0	0
$856$	−729817.	−0.0340431
$857$	2.37850e7	1.10624	0.553122	−	0.833100i	$-0.313436\pi$
$857$	2.37850e7	1.10624	0.553122	+	0.833100i	$0.313436\pi$
$858$	0	0
$859$	−8.55700e6	−0.395675	−0.197838	−	0.980235i	$-0.563392\pi$
$859$	−8.55700e6	−0.395675	−0.197838	+	0.980235i	$0.563392\pi$
$860$	0	0
$861$	0	0
$862$	−2.21969e7	−1.01748
$863$	3.91623e7	1.78995	0.894977	−	0.446113i	$-0.147192\pi$
$863$	3.91623e7	1.78995	0.894977	+	0.446113i	$0.147192\pi$
$864$	0	0
$865$	0	0
$866$	3.25721e6	0.147588
$867$	0	0
$868$	7.32971e6	0.330208
$869$	4.24886e7	1.90864
$870$	0	0
$871$	3.98618e7	1.78038
$872$	−6.29519e6	−0.280361
$873$	0	0
$874$	−1.87960e7	−0.832311
$875$	0	0
$876$	0	0
$877$	−2.35047e7	−1.03194	−0.515972	−	0.856605i	$-0.672569\pi$
$877$	−2.35047e7	−1.03194	−0.515972	+	0.856605i	$0.672569\pi$
$878$	1.60985e7	0.704771
$879$	0	0
$880$	0	0
$881$	−4.50902e7	−1.95723	−0.978617	−	0.205691i	$-0.934056\pi$
$881$	−4.50902e7	−1.95723	−0.978617	+	0.205691i	$0.934056\pi$
$882$	0	0
$883$	1.95876e7	0.845434	0.422717	−	0.906262i	$-0.361076\pi$
$883$	1.95876e7	0.845434	0.422717	+	0.906262i	$0.361076\pi$
$884$	2.86588e7	1.23347
$885$	0	0
$886$	1.78553e7	0.764157
$887$	9.54399e6	0.407306	0.203653	−	0.979043i	$-0.434719\pi$
$887$	9.54399e6	0.407306	0.203653	+	0.979043i	$0.434719\pi$
$888$	0	0
$889$	6.16515e7	2.61631
$890$	0	0
$891$	0	0
$892$	−4.61594e6	−0.194244
$893$	−2.89288e7	−1.21395
$894$	0	0
$895$	0	0
$896$	3.95916e6	0.164753
$897$	0	0
$898$	−1.86756e7	−0.772828
$899$	4.67628e6	0.192975
$900$	0	0
$901$	−5.90992e7	−2.42532
$902$	−5.12330e6	−0.209668
$903$	0	0
$904$	644912.	0.0262470
$905$	0	0
$906$	0	0
$907$	−1.08034e7	−0.436054	−0.218027	−	0.975943i	$-0.569962\pi$
$907$	−1.08034e7	−0.436054	−0.218027	+	0.975943i	$0.569962\pi$
$908$	−2.00573e6	−0.0807343
$909$	0	0
$910$	0	0
$911$	3.88914e7	1.55259	0.776296	−	0.630368i	$-0.217096\pi$
$911$	3.88914e7	1.55259	0.776296	+	0.630368i	$0.217096\pi$
$912$	0	0
$913$	4.08749e7	1.62286
$914$	1.88536e7	0.746498
$915$	0	0
$916$	−1.11709e7	−0.439897
$917$	−1.58118e7	−0.620953
$918$	0	0
$919$	2.89714e7	1.13157	0.565785	−	0.824553i	$-0.308573\pi$
$919$	2.89714e7	1.13157	0.565785	+	0.824553i	$0.308573\pi$
$920$	0	0
$921$	0	0
$922$	2.49590e7	0.966941
$923$	−2.25609e7	−0.871668
$924$	0	0
$925$	0	0
$926$	1.33597e7	0.511999
$927$	0	0
$928$	2.52590e6	0.0962824
$929$	3.61023e7	1.37245	0.686224	−	0.727391i	$-0.259267\pi$
$929$	3.61023e7	1.37245	0.686224	+	0.727391i	$0.259267\pi$
$930$	0	0
$931$	5.21599e7	1.97225
$932$	−1.57534e7	−0.594066
$933$	0	0
$934$	−2.88999e6	−0.108400
$935$	0	0
$936$	0	0
$937$	1.02966e7	0.383129	0.191565	−	0.981480i	$-0.438644\pi$
$937$	1.02966e7	0.383129	0.191565	+	0.981480i	$0.438644\pi$
$938$	4.65173e7	1.72627
$939$	0	0
$940$	0	0
$941$	−1.55696e7	−0.573195	−0.286597	−	0.958051i	$-0.592524\pi$
$941$	−1.55696e7	−0.573195	−0.286597	+	0.958051i	$0.592524\pi$
$942$	0	0
$943$	−7.34519e6	−0.268982
$944$	−9.01778e6	−0.329359
$945$	0	0
$946$	2.87910e7	1.04599
$947$	−3.54260e7	−1.28365	−0.641826	−	0.766850i	$-0.721823\pi$
$947$	−3.54260e7	−1.28365	−0.641826	+	0.766850i	$0.721823\pi$
$948$	0	0
$949$	−5.42350e6	−0.195485
$950$	0	0
$951$	0	0
$952$	3.34438e7	1.19598
$953$	5.47490e6	0.195274	0.0976369	−	0.995222i	$-0.468872\pi$
$953$	5.47490e6	0.195274	0.0976369	+	0.995222i	$0.468872\pi$
$954$	0	0
$955$	0	0
$956$	−2.09679e6	−0.0742011
$957$	0	0
$958$	−2.15086e7	−0.757177
$959$	−1.62760e7	−0.571479
$960$	0	0
$961$	−2.50352e7	−0.874467
$962$	−3.48032e6	−0.121250
$963$	0	0
$964$	−1.97811e7	−0.685579
$965$	0	0
$966$	0	0
$967$	1.65533e7	0.569269	0.284635	−	0.958636i	$-0.408128\pi$
$967$	1.65533e7	0.569269	0.284635	+	0.958636i	$0.408128\pi$
$968$	−1.70077e7	−0.583387
$969$	0	0
$970$	0	0
$971$	−1.83751e6	−0.0625435	−0.0312718	−	0.999511i	$-0.509956\pi$
$971$	−1.83751e6	−0.0625435	−0.0312718	+	0.999511i	$0.509956\pi$
$972$	0	0
$973$	2.26757e7	0.767855
$974$	6.88338e6	0.232490
$975$	0	0
$976$	687445.	0.0231001
$977$	516874.	0.0173240	0.00866200	−	0.999962i	$-0.497243\pi$
$977$	516874.	0.0173240	0.00866200	+	0.999962i	$0.497243\pi$
$978$	0	0
$979$	1.17111e7	0.390519
$980$	0	0
$981$	0	0
$982$	2.92580e7	0.968201
$983$	−4.15798e7	−1.37246	−0.686229	−	0.727386i	$-0.740735\pi$
$983$	−4.15798e7	−1.37246	−0.686229	+	0.727386i	$0.740735\pi$
$984$	0	0
$985$	0	0
$986$	2.13368e7	0.698935
$987$	0	0
$988$	1.66221e7	0.541744
$989$	4.12771e7	1.34190
$990$	0	0
$991$	−1.61859e7	−0.523544	−0.261772	−	0.965130i	$-0.584307\pi$
$991$	−1.61859e7	−0.523544	−0.261772	+	0.965130i	$0.584307\pi$
$992$	1.94126e6	0.0626331
$993$	0	0
$994$	−2.63277e7	−0.845176
$995$	0	0
$996$	0	0
$997$	2.74426e7	0.874355	0.437178	−	0.899375i	$-0.355978\pi$
$997$	2.74426e7	0.874355	0.437178	+	0.899375i	$0.355978\pi$
$998$	2.91951e7	0.927863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.y.1.1	✓	2
3.2	odd	2		450.6.a.bd.1.1	yes	2
5.2	odd	4		450.6.c.p.199.1		4
5.3	odd	4		450.6.c.p.199.4		4
5.4	even	2		450.6.a.bf.1.2	yes	2
15.2	even	4		450.6.c.q.199.3		4
15.8	even	4		450.6.c.q.199.2		4
15.14	odd	2		450.6.a.ba.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.a.y.1.1	✓	2	1.1	even	1	trivial
450.6.a.ba.1.2	yes	2	15.14	odd	2
450.6.a.bd.1.1	yes	2	3.2	odd	2
450.6.a.bf.1.2	yes	2	5.4	even	2
450.6.c.p.199.1		4	5.2	odd	4
450.6.c.p.199.4		4	5.3	odd	4
450.6.c.q.199.2		4	15.8	even	4
450.6.c.q.199.3		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -241.648 q^{7} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} -241.648 q^{7} -64.0000 q^{8} -653.296 q^{11} +828.296 q^{13} +966.592 q^{14} +256.000 q^{16} +2162.48 q^{17} +1254.24 q^{19} +2613.18 q^{22} +3746.48 q^{23} -3313.18 q^{26} -3866.37 q^{28} -2466.70 q^{29} -1895.76 q^{31} -1024.00 q^{32} -8649.92 q^{34} +1050.45 q^{37} -5016.96 q^{38} -1960.56 q^{41} +11017.6 q^{43} -10452.7 q^{44} -14985.9 q^{46} -23064.8 q^{47} +41586.8 q^{49} +13252.7 q^{52} -27329.4 q^{53} +15465.5 q^{56} +9866.82 q^{58} -35225.7 q^{59} +2685.33 q^{61} +7583.04 q^{62} +4096.00 q^{64} +48125.1 q^{67} +34599.7 q^{68} -27237.7 q^{71} -6547.77 q^{73} -4201.78 q^{74} +20067.8 q^{76} +157868. q^{77} -65037.3 q^{79} +7842.23 q^{82} -62567.2 q^{83} -44070.3 q^{86} +41811.0 q^{88} -17926.2 q^{89} -200156. q^{91} +59943.7 q^{92} +92259.2 q^{94} +95974.9 q^{97} -166347. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 100 q^{7} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 - 100 * q^7 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} - 100 q^{7} - 128 q^{8} - 540 q^{11} + 890 q^{13} + 400 q^{14} + 512 q^{16} + 492 q^{17} + 592 q^{19} + 2160 q^{22} + 3660 q^{23} - 3560 q^{26} - 1600 q^{28} - 5700 q^{29} - 5708 q^{31} - 2048 q^{32} - 1968 q^{34} + 11300 q^{37} - 2368 q^{38} - 15420 q^{41} + 6320 q^{43} - 8640 q^{44} - 14640 q^{46} - 7800 q^{47} + 44844 q^{49} + 14240 q^{52} - 27828 q^{53} + 6400 q^{56} + 22800 q^{58} - 50520 q^{59} - 29126 q^{61} + 22832 q^{62} + 8192 q^{64} + 97400 q^{67} + 7872 q^{68} - 6180 q^{71} + 32900 q^{73} - 45200 q^{74} + 9472 q^{76} + 173916 q^{77} + 7912 q^{79} + 61680 q^{82} - 163464 q^{83} - 25280 q^{86} + 34560 q^{88} - 164640 q^{89} - 191416 q^{91} + 58560 q^{92} + 31200 q^{94} + 52430 q^{97} - 179376 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 100 * q^7 - 128 * q^8 - 540 * q^11 + 890 * q^13 + 400 * q^14 + 512 * q^16 + 492 * q^17 + 592 * q^19 + 2160 * q^22 + 3660 * q^23 - 3560 * q^26 - 1600 * q^28 - 5700 * q^29 - 5708 * q^31 - 2048 * q^32 - 1968 * q^34 + 11300 * q^37 - 2368 * q^38 - 15420 * q^41 + 6320 * q^43 - 8640 * q^44 - 14640 * q^46 - 7800 * q^47 + 44844 * q^49 + 14240 * q^52 - 27828 * q^53 + 6400 * q^56 + 22800 * q^58 - 50520 * q^59 - 29126 * q^61 + 22832 * q^62 + 8192 * q^64 + 97400 * q^67 + 7872 * q^68 - 6180 * q^71 + 32900 * q^73 - 45200 * q^74 + 9472 * q^76 + 173916 * q^77 + 7912 * q^79 + 61680 * q^82 - 163464 * q^83 - 25280 * q^86 + 34560 * q^88 - 164640 * q^89 - 191416 * q^91 + 58560 * q^92 + 31200 * q^94 + 52430 * q^97 - 179376 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more