LMFDB - Embedded newform 450.6.c.p.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(199,450)]

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, 1]))

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

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{4081})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2041x^{2} + 1040400 $ x^4 + 2041*x^2 + 1040400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$-31.4414i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.p.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} +241.648i q^{7} -64.0000i q^{8} +O(q^{10})$	$q+4.00000i q^{2} -16.0000 q^{4} +241.648i q^{7} -64.0000i q^{8} -653.296 q^{11} +828.296i q^{13} -966.592 q^{14} +256.000 q^{16} -2162.48i q^{17} -1254.24 q^{19} -2613.18i q^{22} +3746.48i q^{23} -3313.18 q^{26} -3866.37i q^{28} +2466.70 q^{29} -1895.76 q^{31} +1024.00i q^{32} +8649.92 q^{34} -1050.45i q^{37} -5016.96i q^{38} -1960.56 q^{41} +11017.6i q^{43} +10452.7 q^{44} -14985.9 q^{46} +23064.8i q^{47} -41586.8 q^{49} -13252.7i q^{52} -27329.4i q^{53} +15465.5 q^{56} +9866.82i q^{58} +35225.7 q^{59} +2685.33 q^{61} -7583.04i q^{62} -4096.00 q^{64} -48125.1i q^{67} +34599.7i q^{68} -27237.7 q^{71} -6547.77i q^{73} +4201.78 q^{74} +20067.8 q^{76} -157868. i q^{77} +65037.3 q^{79} -7842.23i q^{82} -62567.2i q^{83} -44070.3 q^{86} +41811.0i q^{88} +17926.2 q^{89} -200156. q^{91} -59943.7i q^{92} -92259.2 q^{94} -95974.9i q^{97} -166347. i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4}+O(q^{10}) $ 4 * q - 64 * q^4	$ 4 q - 64 q^{4} - 1080 q^{11} - 800 q^{14} + 1024 q^{16} - 1184 q^{19} - 7120 q^{26} + 11400 q^{29} - 11416 q^{31} + 3936 q^{34} - 30840 q^{41} + 17280 q^{44} - 29280 q^{46} - 89688 q^{49} + 12800 q^{56} + 101040 q^{59} - 58252 q^{61} - 16384 q^{64} - 12360 q^{71} + 90400 q^{74} + 18944 q^{76} - 15824 q^{79} - 50560 q^{86} + 329280 q^{89} - 382832 q^{91} - 62400 q^{94}+O(q^{100}) $ 4 * q - 64 * q^4 - 1080 * q^11 - 800 * q^14 + 1024 * q^16 - 1184 * q^19 - 7120 * q^26 + 11400 * q^29 - 11416 * q^31 + 3936 * q^34 - 30840 * q^41 + 17280 * q^44 - 29280 * q^46 - 89688 * q^49 + 12800 * q^56 + 101040 * q^59 - 58252 * q^61 - 16384 * q^64 - 12360 * q^71 + 90400 * q^74 + 18944 * q^76 - 15824 * q^79 - 50560 * q^86 + 329280 * q^89 - 382832 * q^91 - 62400 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	241.648i	1.86397i	0.362500	+	0.931984i	$0.381923\pi$
$7$	241.648i	1.86397i	−0.362500	+	0.931984i	$0.618077\pi$
$8$	− 64.0000i	− 0.353553i
$9$	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
$13$	828.296i	1.35934i	0.733519	+	0.679669i	$0.237876\pi$
$13$	828.296i	1.35934i	−0.733519	+	0.679669i	$0.762124\pi$
$14$	−966.592	−1.31802
$15$	0	0
$16$	256.000	0.250000
$17$	− 2162.48i	− 1.81481i	−0.420262	−	0.907403i	$-0.638062\pi$
$17$	− 2162.48i	− 1.81481i	0.420262	−	0.907403i	$-0.361938\pi$
$18$	0	0
$19$	−1254.24	−0.797071	−0.398535	−	0.917153i	$-0.630481\pi$
$19$	−1254.24	−0.797071	−0.398535	+	0.917153i	$0.630481\pi$
$20$	0	0
$21$	0	0
$22$	− 2613.18i	− 1.15110i
$23$	3746.48i	1.47674i	0.674396	+	0.738370i	$0.264404\pi$
$23$	3746.48i	1.47674i	−0.674396	+	0.738370i	$0.735596\pi$
$24$	0	0
$25$	0	0
$26$	−3313.18	−0.961197
$27$	0	0
$28$	− 3866.37i	− 0.931984i
$29$	2466.70	0.544656	0.272328	−	0.962205i	$-0.412206\pi$
$29$	2466.70	0.544656	0.272328	+	0.962205i	$0.412206\pi$
$30$	0	0
$31$	−1895.76	−0.354306	−0.177153	−	0.984183i	$-0.556689\pi$
$31$	−1895.76	−0.354306	−0.177153	+	0.984183i	$0.556689\pi$
$32$	1024.00i	0.176777i
$33$	0	0
$34$	8649.92	1.28326
$35$	0	0
$36$	0	0
$37$	− 1050.45i	− 0.126145i	−0.998009	−	0.0630724i	$-0.979910\pi$
$37$	− 1050.45i	− 0.126145i	0.998009	−	0.0630724i	$-0.0200899\pi$
$38$	− 5016.96i	− 0.563614i
$39$	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
$43$	11017.6i	0.908688i	0.890826	+	0.454344i	$0.150126\pi$
$43$	11017.6i	0.908688i	−0.890826	+	0.454344i	$0.849874\pi$
$44$	10452.7	0.813951
$45$	0	0
$46$	−14985.9	−1.04421
$47$	23064.8i	1.52302i	0.648154	+	0.761509i	$0.275541\pi$
$47$	23064.8i	1.52302i	−0.648154	+	0.761509i	$0.724459\pi$
$48$	0	0
$49$	−41586.8	−2.47437
$50$	0	0
$51$	0	0
$52$	− 13252.7i	− 0.679669i
$53$	− 27329.4i	− 1.33641i	−0.743977	−	0.668205i	$-0.767063\pi$
$53$	− 27329.4i	− 1.33641i	0.743977	−	0.668205i	$-0.232937\pi$
$54$	0	0
$55$	0	0
$56$	15465.5	0.659012
$57$	0	0
$58$	9866.82i	0.385130i
$59$	35225.7	1.31744	0.658718	−	0.752390i	$-0.271099\pi$
$59$	35225.7	1.31744	0.658718	+	0.752390i	$0.271099\pi$
$60$	0	0
$61$	2685.33	0.0924002	0.0462001	−	0.998932i	$-0.485289\pi$
$61$	2685.33	0.0924002	0.0462001	+	0.998932i	$0.485289\pi$
$62$	− 7583.04i	− 0.250532i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	− 48125.1i	− 1.30974i	−0.755743	−	0.654869i	$-0.772724\pi$
$67$	− 48125.1i	− 1.30974i	0.755743	−	0.654869i	$-0.227276\pi$
$68$	34599.7i	0.907403i
$69$	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
$73$	− 6547.77i	− 0.143809i	−0.997412	−	0.0719046i	$-0.977092\pi$
$73$	− 6547.77i	− 0.143809i	0.997412	−	0.0719046i	$-0.0229077\pi$
$74$	4201.78	0.0891978
$75$	0	0
$76$	20067.8	0.398535
$77$	− 157868.i	− 3.03436i
$78$	0	0
$79$	65037.3	1.17245	0.586226	−	0.810148i	$-0.300613\pi$
$79$	65037.3	1.17245	0.586226	+	0.810148i	$0.300613\pi$
$80$	0	0
$81$	0	0
$82$	− 7842.23i	− 0.128797i
$83$	− 62567.2i	− 0.996900i	−0.866919	−	0.498450i	$-0.833903\pi$
$83$	− 62567.2i	− 0.996900i	0.866919	−	0.498450i	$-0.166097\pi$
$84$	0	0
$85$	0	0
$86$	−44070.3	−0.642539
$87$	0	0
$88$	41811.0i	0.575551i
$89$	17926.2	0.239891	0.119946	−	0.992780i	$-0.461728\pi$
$89$	17926.2	0.239891	0.119946	+	0.992780i	$0.461728\pi$
$90$	0	0
$91$	−200156.	−2.53376
$92$	− 59943.7i	− 0.738370i
$93$	0	0
$94$	−92259.2	−1.07694
$95$	0	0
$96$	0	0
$97$	− 95974.9i	− 1.03569i	−0.855475	−	0.517843i	$-0.826735\pi$
$97$	− 95974.9i	− 1.03569i	0.855475	−	0.517843i	$-0.173265\pi$
$98$	− 166347.i	− 1.74965i
$99$	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
$103$	− 41380.4i	− 0.384328i	−0.981363	−	0.192164i	$-0.938449\pi$
$103$	− 41380.4i	− 0.384328i	0.981363	−	0.192164i	$-0.0615506\pi$
$104$	53011.0	0.480598
$105$	0	0
$106$	109317.	0.944985
$107$	− 11403.4i	− 0.0962885i	−0.998840	−	0.0481442i	$-0.984669\pi$
$107$	− 11403.4i	− 0.0962885i	0.998840	−	0.0481442i	$-0.0153307\pi$
$108$	0	0
$109$	−98362.4	−0.792981	−0.396490	−	0.918039i	$-0.629772\pi$
$109$	−98362.4	−0.792981	−0.396490	+	0.918039i	$0.629772\pi$
$110$	0	0
$111$	0	0
$112$	61861.9i	0.465992i
$113$	− 10076.8i	− 0.0742377i	−0.999311	−	0.0371189i	$-0.988182\pi$
$113$	− 10076.8i	− 0.0742377i	0.999311	−	0.0371189i	$-0.0118180\pi$
$114$	0	0
$115$	0	0
$116$	−39467.3	−0.272328
$117$	0	0
$118$	140903.i	0.931568i
$119$	522559.	3.38274
$120$	0	0
$121$	265745.	1.65007
$122$	10741.3i	0.0653368i
$123$	0	0
$124$	30332.2	0.177153
$125$	0	0
$126$	0	0
$127$	255129.i	1.40362i	0.712362	+	0.701812i	$0.247626\pi$
$127$	255129.i	1.40362i	−0.712362	+	0.701812i	$0.752374\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	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.i	− 1.48571i
$134$	192500.	0.926124
$135$	0	0
$136$	−138399.	−0.641631
$137$	− 67354.0i	− 0.306593i	−0.988180	−	0.153296i	$-0.951011\pi$
$137$	− 67354.0i	− 0.306593i	0.988180	−	0.153296i	$-0.0489889\pi$
$138$	0	0
$139$	93837.8	0.411946	0.205973	−	0.978558i	$-0.433964\pi$
$139$	93837.8	0.411946	0.205973	+	0.978558i	$0.433964\pi$
$140$	0	0
$141$	0	0
$142$	− 108951.i	− 0.453429i
$143$	− 541123.i	− 2.21287i
$144$	0	0
$145$	0	0
$146$	26191.1	0.101688
$147$	0	0
$148$	16807.1i	0.0630724i
$149$	432235.	1.59498	0.797489	−	0.603334i	$-0.206161\pi$
$149$	432235.	1.59498	0.797489	+	0.603334i	$0.206161\pi$
$150$	0	0
$151$	−255822.	−0.913053	−0.456527	−	0.889710i	$-0.650907\pi$
$151$	−255822.	−0.913053	−0.456527	+	0.889710i	$0.650907\pi$
$152$	80271.4i	0.281807i
$153$	0	0
$154$	631471.	2.14561
$155$	0	0
$156$	0	0
$157$	58837.3i	0.190504i	0.995453	+	0.0952519i	$0.0303656\pi$
$157$	58837.3i	0.190504i	−0.995453	+	0.0952519i	$0.969634\pi$
$158$	260149.i	0.829048i
$159$	0	0
$160$	0	0
$161$	−905330.	−2.75259
$162$	0	0
$163$	155006.i	0.456962i	0.973548	+	0.228481i	$0.0733758\pi$
$163$	155006.i	0.456962i	−0.973548	+	0.228481i	$0.926624\pi$
$164$	31368.9	0.0910730
$165$	0	0
$166$	250269.	0.704914
$167$	− 588656.i	− 1.63332i	−0.577121	−	0.816658i	$-0.695824\pi$
$167$	− 588656.i	− 1.63332i	0.577121	−	0.816658i	$-0.304176\pi$
$168$	0	0
$169$	−314782.	−0.847798
$170$	0	0
$171$	0	0
$172$	− 176281.i	− 0.454344i
$173$	− 192049.i	− 0.487862i	−0.969793	−	0.243931i	$-0.921563\pi$
$173$	− 192049.i	− 0.487862i	0.969793	−	0.243931i	$-0.0784370\pi$
$174$	0	0
$175$	0	0
$176$	−167244.	−0.406976
$177$	0	0
$178$	71704.9i	0.169629i
$179$	−350715.	−0.818130	−0.409065	−	0.912505i	$-0.634145\pi$
$179$	−350715.	−0.818130	−0.409065	+	0.912505i	$0.634145\pi$
$180$	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.i	− 1.79164i
$183$	0	0
$184$	239775.	0.522106
$185$	0	0
$186$	0	0
$187$	1.41274e6i	2.95433i
$188$	− 369037.i	− 0.761509i
$189$	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.i	− 1.38648i	−0.720705	−	0.693242i	$-0.756182\pi$
$193$	− 717477.i	− 1.38648i	0.720705	−	0.693242i	$-0.243818\pi$
$194$	383900.	0.732341
$195$	0	0
$196$	665389.	1.23719
$197$	− 287465.i	− 0.527740i	−0.964558	−	0.263870i	$-0.915001\pi$
$197$	− 287465.i	− 0.527740i	0.964558	−	0.263870i	$-0.0849990\pi$
$198$	0	0
$199$	−643619.	−1.15212	−0.576058	−	0.817409i	$-0.695410\pi$
$199$	−643619.	−1.15212	−0.576058	+	0.817409i	$0.695410\pi$
$200$	0	0
$201$	0	0
$202$	− 166135.i	− 0.286472i
$203$	596074.i	1.01522i
$204$	0	0
$205$	0	0
$206$	165522.	0.271761
$207$	0	0
$208$	212044.i	0.339834i
$209$	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.i	0.668205i
$213$	0	0
$214$	45613.5	0.0680862
$215$	0	0
$216$	0	0
$217$	− 458107.i	− 0.660416i
$218$	− 393449.i	− 0.560722i
$219$	0	0
$220$	0	0
$221$	1.79117e6	2.46693
$222$	0	0
$223$	− 288496.i	− 0.388488i	−0.980953	−	0.194244i	$-0.937775\pi$
$223$	− 288496.i	− 0.388488i	0.980953	−	0.194244i	$-0.0622254\pi$
$224$	−247448.	−0.329506
$225$	0	0
$226$	40307.0	0.0524940
$227$	125358.i	0.161469i	0.996736	+	0.0807343i	$0.0257265\pi$
$227$	125358.i	0.161469i	−0.996736	+	0.0807343i	$0.974273\pi$
$228$	0	0
$229$	698183.	0.879793	0.439897	−	0.898048i	$-0.355015\pi$
$229$	698183.	0.879793	0.439897	+	0.898048i	$0.355015\pi$
$230$	0	0
$231$	0	0
$232$	− 157869.i	− 0.192565i
$233$	− 984588.i	− 1.18813i	−0.804416	−	0.594066i	$-0.797522\pi$
$233$	− 984588.i	− 1.18813i	0.804416	−	0.594066i	$-0.202478\pi$
$234$	0	0
$235$	0	0
$236$	−563611.	−0.658718
$237$	0	0
$238$	2.09024e6i	2.39196i
$239$	131049.	0.148402	0.0742011	−	0.997243i	$-0.476359\pi$
$239$	131049.	0.148402	0.0742011	+	0.997243i	$0.476359\pi$
$240$	0	0
$241$	−1.23632e6	−1.37116	−0.685579	−	0.727998i	$-0.740451\pi$
$241$	−1.23632e6	−1.37116	−0.685579	+	0.727998i	$0.740451\pi$
$242$	1.06298e6i	1.16677i
$243$	0	0
$244$	−42965.3	−0.0462001
$245$	0	0
$246$	0	0
$247$	− 1.03888e6i	− 1.08349i
$248$	121329.i	0.125266i
$249$	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
$253$	− 2.44756e6i	− 2.40399i
$254$	−1.02052e6	−0.992513
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.64515e6i	1.55372i	0.629675	+	0.776858i	$0.283188\pi$
$257$	1.64515e6i	1.55372i	−0.629675	+	0.776858i	$0.716812\pi$
$258$	0	0
$259$	253838.	0.235130
$260$	0	0
$261$	0	0
$262$	261733.i	0.235562i
$263$	268273.i	0.239159i	0.992825	+	0.119580i	$0.0381547\pi$
$263$	268273.i	0.239159i	−0.992825	+	0.119580i	$0.961845\pi$
$264$	0	0
$265$	0	0
$266$	1.21234e6	1.05056
$267$	0	0
$268$	770001.i	0.654869i
$269$	−947177.	−0.798087	−0.399044	−	0.916932i	$-0.630658\pi$
$269$	−947177.	−0.798087	−0.399044	+	0.916932i	$0.630658\pi$
$270$	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.i	− 0.453701i
$273$	0	0
$274$	269416.	0.216794
$275$	0	0
$276$	0	0
$277$	− 741976.i	− 0.581019i	−0.956872	−	0.290510i	$-0.906175\pi$
$277$	− 741976.i	− 0.581019i	0.956872	−	0.290510i	$-0.0938249\pi$
$278$	375351.i	0.291290i
$279$	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.i	− 0.330385i	−0.986261	−	0.165192i	$-0.947175\pi$
$283$	− 445130.i	− 0.330385i	0.986261	−	0.165192i	$-0.0528245\pi$
$284$	435803.	0.320622
$285$	0	0
$286$	2.16449e6	1.56473
$287$	− 473765.i	− 0.339514i
$288$	0	0
$289$	−3.25647e6	−2.29352
$290$	0	0
$291$	0	0
$292$	104764.i	0.0719046i
$293$	2.45992e6i	1.67399i	0.547211	+	0.836994i	$0.315689\pi$
$293$	2.45992e6i	1.67399i	−0.547211	+	0.836994i	$0.684311\pi$
$294$	0	0
$295$	0	0
$296$	−67228.5	−0.0445989
$297$	0	0
$298$	1.72894e6i	1.12782i
$299$	−3.10320e6	−2.00739
$300$	0	0
$301$	−2.66238e6	−1.69376
$302$	− 1.02329e6i	− 0.645626i
$303$	0	0
$304$	−321086.	−0.199268
$305$	0	0
$306$	0	0
$307$	− 374253.i	− 0.226631i	−0.993559	−	0.113315i	$-0.963853\pi$
$307$	− 374253.i	− 0.226631i	0.993559	−	0.113315i	$-0.0361470\pi$
$308$	2.52588e6i	1.51718i
$309$	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
$313$	3.05869e6i	1.76471i	0.470580	+	0.882357i	$0.344045\pi$
$313$	3.05869e6i	1.76471i	−0.470580	+	0.882357i	$0.655955\pi$
$314$	−235349.	−0.134706
$315$	0	0
$316$	−1.04060e6	−0.586226
$317$	520610.i	0.290981i	0.989360	+	0.145490i	$0.0464760\pi$
$317$	520610.i	0.290981i	−0.989360	+	0.145490i	$0.953524\pi$
$318$	0	0
$319$	−1.61149e6	−0.886646
$320$	0	0
$321$	0	0
$322$	− 3.62132e6i	− 1.94638i
$323$	2.71227e6i	1.44653i
$324$	0	0
$325$	0	0
$326$	−620025.	−0.323121
$327$	0	0
$328$	125476.i	0.0643983i
$329$	−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	$0.679191\pi$
$332$	1.00108e6i	0.498450i
$333$	0	0
$334$	2.35462e6	1.15493
$335$	0	0
$336$	0	0
$337$	3.08383e6i	1.47916i	0.673067	+	0.739581i	$0.264976\pi$
$337$	3.08383e6i	1.47916i	−0.673067	+	0.739581i	$0.735024\pi$
$338$	− 1.25913e6i	− 0.599484i
$339$	0	0
$340$	0	0
$341$	1.23849e6	0.576776
$342$	0	0
$343$	− 5.98799e6i	− 2.74819i
$344$	705125.	0.321270
$345$	0	0
$346$	768196.	0.344970
$347$	4.04955e6i	1.80544i	0.430231	+	0.902719i	$0.358432\pi$
$347$	4.04955e6i	1.80544i	−0.430231	+	0.902719i	$0.641568\pi$
$348$	0	0
$349$	493135.	0.216722	0.108361	−	0.994112i	$-0.465440\pi$
$349$	493135.	0.216722	0.108361	+	0.994112i	$0.465440\pi$
$350$	0	0
$351$	0	0
$352$	− 668975.i	− 0.287775i
$353$	− 1.34815e6i	− 0.575841i	−0.957654	−	0.287920i	$-0.907036\pi$
$353$	− 1.34815e6i	− 0.575841i	0.957654	−	0.287920i	$-0.0929638\pi$
$354$	0	0
$355$	0	0
$356$	−286820.	−0.119946
$357$	0	0
$358$	− 1.40286e6i	− 0.578505i
$359$	1.64749e6	0.674664	0.337332	−	0.941386i	$-0.390476\pi$
$359$	1.64749e6	0.674664	0.337332	+	0.941386i	$0.390476\pi$
$360$	0	0
$361$	−902980.	−0.364678
$362$	2.09855e6i	0.841681i
$363$	0	0
$364$	3.20250e6	1.26688
$365$	0	0
$366$	0	0
$367$	27918.4i	0.0108199i	0.999985	+	0.00540997i	$0.00172206\pi$
$367$	27918.4i	0.0108199i	−0.999985	+	0.00540997i	$0.998278\pi$
$368$	959099.i	0.369185i
$369$	0	0
$370$	0	0
$371$	6.60409e6	2.49103
$372$	0	0
$373$	2.80536e6i	1.04404i	0.852934	+	0.522019i	$0.174821\pi$
$373$	2.80536e6i	1.04404i	−0.852934	+	0.522019i	$0.825179\pi$
$374$	−5.65096e6	−2.08902
$375$	0	0
$376$	1.47615e6	0.538468
$377$	2.04316e6i	0.740371i
$378$	0	0
$379$	−1.70014e6	−0.607976	−0.303988	−	0.952676i	$-0.598318\pi$
$379$	−1.70014e6	−0.607976	−0.303988	+	0.952676i	$0.598318\pi$
$380$	0	0
$381$	0	0
$382$	− 254651.i	− 0.0892867i
$383$	− 627913.i	− 0.218727i	−0.994002	−	0.109363i	$-0.965119\pi$
$383$	− 627913.i	− 0.218727i	0.994002	−	0.109363i	$-0.0348812\pi$
$384$	0	0
$385$	0	0
$386$	2.86991e6	0.980392
$387$	0	0
$388$	1.53560e6i	0.517843i
$389$	3.14339e6	1.05323	0.526616	−	0.850103i	$-0.323460\pi$
$389$	3.14339e6	1.05323	0.526616	+	0.850103i	$0.323460\pi$
$390$	0	0
$391$	8.10169e6	2.68000
$392$	2.66156e6i	0.874823i
$393$	0	0
$394$	1.14986e6	0.373169
$395$	0	0
$396$	0	0
$397$	1.10773e6i	0.352743i	0.984324	+	0.176371i	$0.0564359\pi$
$397$	1.10773e6i	0.352743i	−0.984324	+	0.176371i	$0.943564\pi$
$398$	− 2.57448e6i	− 0.814669i
$399$	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
$403$	− 1.57025e6i	− 0.481622i
$404$	664540.	0.202567
$405$	0	0
$406$	−2.38430e6	−0.717869
$407$	686252.i	0.205351i
$408$	0	0
$409$	−3.69830e6	−1.09319	−0.546593	−	0.837398i	$-0.684075\pi$
$409$	−3.69830e6	−1.09319	−0.546593	+	0.837398i	$0.684075\pi$
$410$	0	0
$411$	0	0
$412$	662087.i	0.192164i
$413$	8.51222e6i	2.45566i
$414$	0	0
$415$	0	0
$416$	−848175.	−0.240299
$417$	0	0
$418$	3.27756e6i	0.917509i
$419$	−3.45478e6	−0.961357	−0.480679	−	0.876897i	$-0.659610\pi$
$419$	−3.45478e6	−0.961357	−0.480679	+	0.876897i	$0.659610\pi$
$420$	0	0
$421$	−4.83980e6	−1.33083	−0.665415	−	0.746474i	$-0.731745\pi$
$421$	−4.83980e6	−1.33083	−0.665415	+	0.746474i	$0.731745\pi$
$422$	− 3.49531e6i	− 0.955442i
$423$	0	0
$424$	−1.74908e6	−0.472493
$425$	0	0
$426$	0	0
$427$	648905.i	0.172231i
$428$	182454.i	0.0481442i
$429$	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.i	− 0.208721i	−0.994540	−	0.104360i	$-0.966720\pi$
$433$	− 814302.i	− 0.208721i	0.994540	−	0.104360i	$-0.0332795\pi$
$434$	1.83243e6	0.466984
$435$	0	0
$436$	1.57380e6	0.396490
$437$	− 4.69899e6i	− 1.17707i
$438$	0	0
$439$	4.02462e6	0.996697	0.498349	−	0.866977i	$-0.333940\pi$
$439$	4.02462e6	0.996697	0.498349	+	0.866977i	$0.333940\pi$
$440$	0	0
$441$	0	0
$442$	7.16470e6i	1.74438i
$443$	− 4.46382e6i	− 1.08068i	−0.841446	−	0.540341i	$-0.818295\pi$
$443$	− 4.46382e6i	− 1.08068i	0.841446	−	0.540341i	$-0.181705\pi$
$444$	0	0
$445$	0	0
$446$	1.15399e6	0.274703
$447$	0	0
$448$	− 989791.i	− 0.232996i
$449$	−4.66889e6	−1.09294	−0.546472	−	0.837477i	$-0.684030\pi$
$449$	−4.66889e6	−1.09294	−0.546472	+	0.837477i	$0.684030\pi$
$450$	0	0
$451$	1.28082e6	0.296516
$452$	161228.i	0.0371189i
$453$	0	0
$454$	−501433.	−0.114176
$455$	0	0
$456$	0	0
$457$	4.71340e6i	1.05571i	0.849335	+	0.527854i	$0.177003\pi$
$457$	4.71340e6i	1.05571i	−0.849335	+	0.527854i	$0.822997\pi$
$458$	2.79273e6i	0.622108i
$459$	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
$463$	− 3.33992e6i	− 0.724076i	−0.932163	−	0.362038i	$-0.882081\pi$
$463$	− 3.33992e6i	− 0.724076i	0.932163	−	0.362038i	$-0.117919\pi$
$464$	631476.	0.136164
$465$	0	0
$466$	3.93835e6	0.840136
$467$	− 722497.i	− 0.153301i	−0.997058	−	0.0766503i	$-0.975578\pi$
$467$	− 722497.i	− 0.153301i	0.997058	−	0.0766503i	$-0.0244225\pi$
$468$	0	0
$469$	1.16293e7	2.44131
$470$	0	0
$471$	0	0
$472$	− 2.25444e6i	− 0.465784i
$473$	− 7.19774e6i	− 1.47926i
$474$	0	0
$475$	0	0
$476$	−8.36095e6	−1.69137
$477$	0	0
$478$	524197.i	0.104936i
$479$	−5.37714e6	−1.07081	−0.535405	−	0.844595i	$-0.679841\pi$
$479$	−5.37714e6	−1.07081	−0.535405	+	0.844595i	$0.679841\pi$
$480$	0	0
$481$	870080.	0.171473
$482$	− 4.94527e6i	− 0.969556i
$483$	0	0
$484$	−4.25192e6	−0.825034
$485$	0	0
$486$	0	0
$487$	1.72084e6i	0.328790i	0.986395	+	0.164395i	$0.0525672\pi$
$487$	1.72084e6i	0.328790i	−0.986395	+	0.164395i	$0.947433\pi$
$488$	− 171861.i	− 0.0326684i
$489$	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
$493$	− 5.33420e6i	− 0.988444i
$494$	4.15553e6	0.766142
$495$	0	0
$496$	−485314.	−0.0885766
$497$	− 6.58193e6i	− 1.19526i
$498$	0	0
$499$	7.29878e6	1.31220	0.656098	−	0.754676i	$-0.272206\pi$
$499$	7.29878e6	1.31220	0.656098	+	0.754676i	$0.272206\pi$
$500$	0	0
$501$	0	0
$502$	− 2.44767e6i	− 0.433505i
$503$	923591.i	0.162764i	0.996683	+	0.0813822i	$0.0259334\pi$
$503$	923591.i	0.162764i	−0.996683	+	0.0813822i	$0.974067\pi$
$504$	0	0
$505$	0	0
$506$	9.79025e6	1.69988
$507$	0	0
$508$	− 4.08207e6i	− 0.701812i
$509$	8.81318e6	1.50778	0.753891	−	0.657000i	$-0.228175\pi$
$509$	8.81318e6	1.50778	0.753891	+	0.657000i	$0.228175\pi$
$510$	0	0
$511$	1.58226e6	0.268056
$512$	262144.i	0.0441942i
$513$	0	0
$514$	−6.58059e6	−1.09864
$515$	0	0
$516$	0	0
$517$	− 1.50682e7i	− 2.47933i
$518$	1.01535e6i	0.166262i
$519$	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
$523$	2.85438e6i	0.456308i	0.973625	+	0.228154i	$0.0732689\pi$
$523$	2.85438e6i	0.456308i	−0.973625	+	0.228154i	$0.926731\pi$
$524$	−1.04693e6	−0.166568
$525$	0	0
$526$	−1.07309e6	−0.169111
$527$	4.09954e6i	0.642997i
$528$	0	0
$529$	−7.59978e6	−1.18076
$530$	0	0
$531$	0	0
$532$	4.84936e6i	0.742857i
$533$	− 1.62392e6i	− 0.247598i
$534$	0	0
$535$	0	0
$536$	−3.08000e6	−0.463062
$537$	0	0
$538$	− 3.78871e6i	− 0.564333i
$539$	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	$0.695794\pi$
$542$	2.43664e6i	0.356281i
$543$	0	0
$544$	2.21438e6	0.320815
$545$	0	0
$546$	0	0
$547$	1.07748e7i	1.53972i	0.638213	+	0.769860i	$0.279674\pi$
$547$	1.07748e7i	1.53972i	−0.638213	+	0.769860i	$0.720326\pi$
$548$	1.07766e6i	0.153296i
$549$	0	0
$550$	0	0
$551$	−3.09384e6	−0.434129
$552$	0	0
$553$	1.57161e7i	2.18541i
$554$	2.96791e6	0.410843
$555$	0	0
$556$	−1.50140e6	−0.205973
$557$	7.62158e6i	1.04090i	0.853893	+	0.520448i	$0.174235\pi$
$557$	7.62158e6i	1.04090i	−0.853893	+	0.520448i	$0.825765\pi$
$558$	0	0
$559$	−9.12581e6	−1.23521
$560$	0	0
$561$	0	0
$562$	1.69233e6i	0.226019i
$563$	− 4.25551e6i	− 0.565823i	−0.959146	−	0.282911i	$-0.908700\pi$
$563$	− 4.25551e6i	− 0.565823i	0.959146	−	0.282911i	$-0.0913003\pi$
$564$	0	0
$565$	0	0
$566$	1.78052e6	0.233617
$567$	0	0
$568$	1.74321e6i	0.226714i
$569$	5.99899e6	0.776779	0.388389	−	0.921495i	$-0.373032\pi$
$569$	5.99899e6	0.776779	0.388389	+	0.921495i	$0.373032\pi$
$570$	0	0
$571$	9.20283e6	1.18122	0.590611	−	0.806957i	$-0.298887\pi$
$571$	9.20283e6	1.18122	0.590611	+	0.806957i	$0.298887\pi$
$572$	8.65796e6i	1.10643i
$573$	0	0
$574$	1.89506e6	0.240073
$575$	0	0
$576$	0	0
$577$	− 4.03879e6i	− 0.505024i	−0.967594	−	0.252512i	$-0.918743\pi$
$577$	− 4.03879e6i	− 0.505024i	0.967594	−	0.252512i	$-0.0812568\pi$
$578$	− 1.30259e7i	− 1.62176i
$579$	0	0
$580$	0	0
$581$	1.51192e7	1.85819
$582$	0	0
$583$	1.78542e7i	2.17555i
$584$	−419058.	−0.0508442
$585$	0	0
$586$	−9.83969e6	−1.18369
$587$	− 1.65997e7i	− 1.98840i	−0.107544	−	0.994200i	$-0.534299\pi$
$587$	− 1.65997e7i	− 1.98840i	0.107544	−	0.994200i	$-0.465701\pi$
$588$	0	0
$589$	2.37774e6	0.282407
$590$	0	0
$591$	0	0
$592$	− 268914.i	− 0.0315362i
$593$	− 1.09900e7i	− 1.28340i	−0.766957	−	0.641698i	$-0.778230\pi$
$593$	− 1.09900e7i	− 1.28340i	0.766957	−	0.641698i	$-0.221770\pi$
$594$	0	0
$595$	0	0
$596$	−6.91577e6	−0.797489
$597$	0	0
$598$	− 1.24128e7i	− 1.41944i
$599$	4.47167e6	0.509217	0.254608	−	0.967044i	$-0.418053\pi$
$599$	4.47167e6	0.509217	0.254608	+	0.967044i	$0.418053\pi$
$600$	0	0
$601$	1.60311e7	1.81041	0.905206	−	0.424974i	$-0.139717\pi$
$601$	1.60311e7	1.81041	0.905206	+	0.424974i	$0.139717\pi$
$602$	− 1.06495e7i	− 1.19767i
$603$	0	0
$604$	4.09316e6	0.456527
$605$	0	0
$606$	0	0
$607$	720250.i	0.0793435i	0.999213	+	0.0396718i	$0.0126312\pi$
$607$	720250.i	0.0793435i	−0.999213	+	0.0396718i	$0.987369\pi$
$608$	− 1.28434e6i	− 0.140904i
$609$	0	0
$610$	0	0
$611$	−1.91045e7	−2.07030
$612$	0	0
$613$	− 1.46218e7i	− 1.57163i	−0.618464	−	0.785813i	$-0.712245\pi$
$613$	− 1.46218e7i	− 1.57163i	0.618464	−	0.785813i	$-0.287755\pi$
$614$	1.49701e6	0.160252
$615$	0	0
$616$	−1.01035e7	−1.07281
$617$	− 4.57269e6i	− 0.483569i	−0.970330	−	0.241784i	$-0.922267\pi$
$617$	− 4.57269e6i	− 0.483569i	0.970330	−	0.241784i	$-0.0777327\pi$
$618$	0	0
$619$	4.62486e6	0.485145	0.242573	−	0.970133i	$-0.422009\pi$
$619$	4.62486e6	0.485145	0.242573	+	0.970133i	$0.422009\pi$
$620$	0	0
$621$	0	0
$622$	− 8.90273e6i	− 0.922671i
$623$	4.33184e6i	0.447149i
$624$	0	0
$625$	0	0
$626$	−1.22348e7	−1.24784
$627$	0	0
$628$	− 941397.i	− 0.0952519i
$629$	−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	$0.673421\pi$
$632$	− 4.16239e6i	− 0.414524i
$633$	0	0
$634$	−2.08244e6	−0.205755
$635$	0	0
$636$	0	0
$637$	− 3.44462e7i	− 3.36351i
$638$	− 6.44595e6i	− 0.626954i
$639$	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
$643$	1.13460e7i	1.08222i	0.840951	+	0.541111i	$0.181996\pi$
$643$	1.13460e7i	1.08222i	−0.840951	+	0.541111i	$0.818004\pi$
$644$	1.44853e7	1.37630
$645$	0	0
$646$	−1.08491e7	−1.02285
$647$	− 1.10482e7i	− 1.03760i	−0.854895	−	0.518800i	$-0.826379\pi$
$647$	− 1.10482e7i	− 1.03760i	0.854895	−	0.518800i	$-0.173621\pi$
$648$	0	0
$649$	−2.30128e7	−2.14466
$650$	0	0
$651$	0	0
$652$	− 2.48010e6i	− 0.228481i
$653$	− 1.37432e6i	− 0.126126i	−0.998010	−	0.0630631i	$-0.979913\pi$
$653$	− 1.37432e6i	− 0.126126i	0.998010	−	0.0630631i	$-0.0200869\pi$
$654$	0	0
$655$	0	0
$656$	−501902.	−0.0455365
$657$	0	0
$658$	− 2.22943e7i	− 2.00738i
$659$	−1.28061e7	−1.14869	−0.574344	−	0.818614i	$-0.694743\pi$
$659$	−1.28061e7	−1.14869	−0.574344	+	0.818614i	$0.694743\pi$
$660$	0	0
$661$	−104794.	−0.00932894	−0.00466447	−	0.999989i	$-0.501485\pi$
$661$	−104794.	−0.00932894	−0.00466447	+	0.999989i	$0.501485\pi$
$662$	− 8.51021e6i	− 0.754737i
$663$	0	0
$664$	−4.00430e6	−0.352457
$665$	0	0
$666$	0	0
$667$	9.24146e6i	0.804315i
$668$	9.41850e6i	0.816658i
$669$	0	0
$670$	0	0
$671$	−1.75432e6	−0.150419
$672$	0	0
$673$	1.80490e7i	1.53608i	0.640400	+	0.768042i	$0.278769\pi$
$673$	1.80490e7i	1.53608i	−0.640400	+	0.768042i	$0.721231\pi$
$674$	−1.23353e7	−1.04593
$675$	0	0
$676$	5.03651e6	0.423899
$677$	− 1.29914e7i	− 1.08939i	−0.838633	−	0.544697i	$-0.816645\pi$
$677$	− 1.29914e7i	− 1.08939i	0.838633	−	0.544697i	$-0.183355\pi$
$678$	0	0
$679$	2.31922e7	1.93049
$680$	0	0
$681$	0	0
$682$	4.95397e6i	0.407842i
$683$	− 3.91843e6i	− 0.321411i	−0.987002	−	0.160705i	$-0.948623\pi$
$683$	− 3.91843e6i	− 0.321411i	0.987002	−	0.160705i	$-0.0513769\pi$
$684$	0	0
$685$	0	0
$686$	2.39520e7	1.94326
$687$	0	0
$688$	2.82050e6i	0.227172i
$689$	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	$0.539547\pi$
$692$	3.07278e6i	0.243931i
$693$	0	0
$694$	−1.61982e7	−1.27664
$695$	0	0
$696$	0	0
$697$	4.23967e6i	0.330560i
$698$	1.97254e6i	0.153245i
$699$	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
$703$	1.31751e6i	0.100546i
$704$	2.67590e6	0.203488
$705$	0	0
$706$	5.39261e6	0.407181
$707$	− 1.00366e7i	− 0.755155i
$708$	0	0
$709$	8.13693e6	0.607918	0.303959	−	0.952685i	$-0.401691\pi$
$709$	8.13693e6	0.607918	0.303959	+	0.952685i	$0.401691\pi$
$710$	0	0
$711$	0	0
$712$	− 1.14728e6i	− 0.0848143i
$713$	− 7.10243e6i	− 0.523218i
$714$	0	0
$715$	0	0
$716$	5.61144e6	0.409065
$717$	0	0
$718$	6.58997e6i	0.477059i
$719$	−1.14993e7	−0.829566	−0.414783	−	0.909920i	$-0.636143\pi$
$719$	−1.14993e7	−0.829566	−0.414783	+	0.909920i	$0.636143\pi$
$720$	0	0
$721$	9.99951e6	0.716375
$722$	− 3.61192e6i	− 0.257867i
$723$	0	0
$724$	−8.39419e6	−0.595158
$725$	0	0
$726$	0	0
$727$	2.10897e7i	1.47991i	0.672658	+	0.739953i	$0.265152\pi$
$727$	2.10897e7i	1.47991i	−0.672658	+	0.739953i	$0.734848\pi$
$728$	1.28100e7i	0.895820i
$729$	0	0
$730$	0	0
$731$	2.38253e7	1.64909
$732$	0	0
$733$	2.20274e7i	1.51427i	0.653259	+	0.757134i	$0.273401\pi$
$733$	2.20274e7i	1.51427i	−0.653259	+	0.757134i	$0.726599\pi$
$734$	−111673.	−0.00765085
$735$	0	0
$736$	−3.83640e6	−0.261053
$737$	3.14399e7i	2.13213i
$738$	0	0
$739$	1.26088e7	0.849305	0.424653	−	0.905356i	$-0.360396\pi$
$739$	1.26088e7	0.849305	0.424653	+	0.905356i	$0.360396\pi$
$740$	0	0
$741$	0	0
$742$	2.64164e7i	1.76142i
$743$	− 5.90968e6i	− 0.392728i	−0.980531	−	0.196364i	$-0.937087\pi$
$743$	− 5.90968e6i	− 0.392728i	0.980531	−	0.196364i	$-0.0629134\pi$
$744$	0	0
$745$	0	0
$746$	−1.12214e7	−0.738247
$747$	0	0
$748$	− 2.26039e7i	− 1.47716i
$749$	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	$0.621985\pi$
$752$	5.90459e6i	0.380755i
$753$	0	0
$754$	−8.17265e6	−0.523521
$755$	0	0
$756$	0	0
$757$	− 2.64988e7i	− 1.68068i	−0.542057	−	0.840342i	$-0.682354\pi$
$757$	− 2.64988e7i	− 1.68068i	0.542057	−	0.840342i	$-0.317646\pi$
$758$	− 6.80056e6i	− 0.429904i
$759$	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
$763$	− 2.37691e7i	− 1.47809i
$764$	1.01860e6	0.0631352
$765$	0	0
$766$	2.51165e6	0.154663
$767$	2.91773e7i	1.79084i
$768$	0	0
$769$	−1.16191e7	−0.708529	−0.354265	−	0.935145i	$-0.615269\pi$
$769$	−1.16191e7	−0.708529	−0.354265	+	0.935145i	$0.615269\pi$
$770$	0	0
$771$	0	0
$772$	1.14796e7i	0.693242i
$773$	− 1.16233e7i	− 0.699649i	−0.936815	−	0.349824i	$-0.886241\pi$
$773$	− 1.16233e7i	− 0.699649i	0.936815	−	0.349824i	$-0.113759\pi$
$774$	0	0
$775$	0	0
$776$	−6.14239e6	−0.366171
$777$	0	0
$778$	1.25736e7i	0.744748i
$779$	2.45901e6	0.145183
$780$	0	0
$781$	1.77943e7	1.04388
$782$	3.24068e7i	1.89504i
$783$	0	0
$784$	−1.06462e7	−0.618594
$785$	0	0
$786$	0	0
$787$	− 2.03314e7i	− 1.17012i	−0.810991	−	0.585059i	$-0.801071\pi$
$787$	− 2.03314e7i	− 1.17012i	0.810991	−	0.585059i	$-0.198929\pi$
$788$	4.59945e6i	0.263870i
$789$	0	0
$790$	0	0
$791$	2.43503e6	0.138377
$792$	0	0
$793$	2.22425e6i	0.125603i
$794$	−4.43092e6	−0.249427
$795$	0	0
$796$	1.02979e7	0.576058
$797$	6.32170e6i	0.352523i	0.984343	+	0.176262i	$0.0564005\pi$
$797$	6.32170e6i	0.352523i	−0.984343	+	0.176262i	$0.943599\pi$
$798$	0	0
$799$	4.98772e7	2.76398
$800$	0	0
$801$	0	0
$802$	− 9.66252e6i	− 0.530462i
$803$	4.27764e6i	0.234107i
$804$	0	0
$805$	0	0
$806$	6.28100e6	0.340558
$807$	0	0
$808$	2.65816e6i	0.143236i
$809$	−1.84549e7	−0.991380	−0.495690	−	0.868499i	$-0.665085\pi$
$809$	−1.84549e7	−0.991380	−0.495690	+	0.868499i	$0.665085\pi$
$810$	0	0
$811$	3.00263e7	1.60306	0.801530	−	0.597954i	$-0.204020\pi$
$811$	3.00263e7	1.60306	0.801530	+	0.597954i	$0.204020\pi$
$812$	− 9.53719e6i	− 0.507610i
$813$	0	0
$814$	−2.74501e6	−0.145205
$815$	0	0
$816$	0	0
$817$	− 1.38187e7i	− 0.724289i
$818$	− 1.47932e7i	− 0.772999i
$819$	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
$823$	− 7.97388e6i	− 0.410365i	−0.978724	−	0.205182i	$-0.934221\pi$
$823$	− 7.97388e6i	− 0.410365i	0.978724	−	0.205182i	$-0.0657788\pi$
$824$	−2.64835e6	−0.135880
$825$	0	0
$826$	−3.40489e7	−1.73641
$827$	1.37135e7i	0.697243i	0.937264	+	0.348622i	$0.113350\pi$
$827$	1.37135e7i	0.697243i	−0.937264	+	0.348622i	$0.886650\pi$
$828$	0	0
$829$	1.62566e7	0.821569	0.410785	−	0.911732i	$-0.365255\pi$
$829$	1.62566e7	0.821569	0.410785	+	0.911732i	$0.365255\pi$
$830$	0	0
$831$	0	0
$832$	− 3.39270e6i	− 0.169917i
$833$	8.99307e7i	4.49051i
$834$	0	0
$835$	0	0
$836$	−1.31103e7	−0.648777
$837$	0	0
$838$	− 1.38191e7i	− 0.679782i
$839$	−2.36734e7	−1.16106	−0.580532	−	0.814237i	$-0.697156\pi$
$839$	−2.36734e7	−1.16106	−0.580532	+	0.814237i	$0.697156\pi$
$840$	0	0
$841$	−1.44265e7	−0.703350
$842$	− 1.93592e7i	− 0.941039i
$843$	0	0
$844$	1.39812e7	0.675600
$845$	0	0
$846$	0	0
$847$	6.42168e7i	3.07567i
$848$	− 6.99632e6i	− 0.334103i
$849$	0	0
$850$	0	0
$851$	3.93547e6	0.186283
$852$	0	0
$853$	− 2.53714e7i	− 1.19391i	−0.802274	−	0.596956i	$-0.796377\pi$
$853$	− 2.53714e7i	− 1.19391i	0.802274	−	0.596956i	$-0.203623\pi$
$854$	−2.59562e6	−0.121786
$855$	0	0
$856$	−729817.	−0.0340431
$857$	− 2.37850e7i	− 1.10624i	−0.833100	−	0.553122i	$-0.813436\pi$
$857$	− 2.37850e7i	− 1.10624i	0.833100	−	0.553122i	$-0.186564\pi$
$858$	0	0
$859$	8.55700e6	0.395675	0.197838	−	0.980235i	$-0.436608\pi$
$859$	8.55700e6	0.395675	0.197838	+	0.980235i	$0.436608\pi$
$860$	0	0
$861$	0	0
$862$	2.21969e7i	1.01748i
$863$	3.91623e7i	1.78995i	0.446113	+	0.894977i	$0.352808\pi$
$863$	3.91623e7i	1.78995i	−0.446113	+	0.894977i	$0.647192\pi$
$864$	0	0
$865$	0	0
$866$	3.25721e6	0.147588
$867$	0	0
$868$	7.32971e6i	0.330208i
$869$	−4.24886e7	−1.90864
$870$	0	0
$871$	3.98618e7	1.78038
$872$	6.29519e6i	0.280361i
$873$	0	0
$874$	1.87960e7	0.832311
$875$	0	0
$876$	0	0
$877$	2.35047e7i	1.03194i	0.856605	+	0.515972i	$0.172569\pi$
$877$	2.35047e7i	1.03194i	−0.856605	+	0.515972i	$0.827431\pi$
$878$	1.60985e7i	0.704771i
$879$	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
$883$	1.95876e7i	0.845434i	0.906262	+	0.422717i	$0.138924\pi$
$883$	1.95876e7i	0.845434i	−0.906262	+	0.422717i	$0.861076\pi$
$884$	−2.86588e7	−1.23347
$885$	0	0
$886$	1.78553e7	0.764157
$887$	− 9.54399e6i	− 0.407306i	−0.979043	−	0.203653i	$-0.934719\pi$
$887$	− 9.54399e6i	− 0.407306i	0.979043	−	0.203653i	$-0.0652815\pi$
$888$	0	0
$889$	−6.16515e7	−2.61631
$890$	0	0
$891$	0	0
$892$	4.61594e6i	0.194244i
$893$	− 2.89288e7i	− 1.21395i
$894$	0	0
$895$	0	0
$896$	3.95916e6	0.164753
$897$	0	0
$898$	− 1.86756e7i	− 0.772828i
$899$	−4.67628e6	−0.192975
$900$	0	0
$901$	−5.90992e7	−2.42532
$902$	5.12330e6i	0.209668i
$903$	0	0
$904$	−644912.	−0.0262470
$905$	0	0
$906$	0	0
$907$	1.08034e7i	0.436054i	0.975943	+	0.218027i	$0.0699621\pi$
$907$	1.08034e7i	0.436054i	−0.975943	+	0.218027i	$0.930038\pi$
$908$	− 2.00573e6i	− 0.0807343i
$909$	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
$913$	4.08749e7i	1.62286i
$914$	−1.88536e7	−0.746498
$915$	0	0
$916$	−1.11709e7	−0.439897
$917$	1.58118e7i	0.620953i
$918$	0	0
$919$	−2.89714e7	−1.13157	−0.565785	−	0.824553i	$-0.691427\pi$
$919$	−2.89714e7	−1.13157	−0.565785	+	0.824553i	$0.691427\pi$
$920$	0	0
$921$	0	0
$922$	− 2.49590e7i	− 0.966941i
$923$	− 2.25609e7i	− 0.871668i
$924$	0	0
$925$	0	0
$926$	1.33597e7	0.511999
$927$	0	0
$928$	2.52590e6i	0.0962824i
$929$	−3.61023e7	−1.37245	−0.686224	−	0.727391i	$-0.740733\pi$
$929$	−3.61023e7	−1.37245	−0.686224	+	0.727391i	$0.740733\pi$
$930$	0	0
$931$	5.21599e7	1.97225
$932$	1.57534e7i	0.594066i
$933$	0	0
$934$	2.88999e6	0.108400
$935$	0	0
$936$	0	0
$937$	− 1.02966e7i	− 0.383129i	−0.981480	−	0.191565i	$-0.938644\pi$
$937$	− 1.02966e7i	− 0.383129i	0.981480	−	0.191565i	$-0.0613562\pi$
$938$	4.65173e7i	1.72627i
$939$	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
$943$	− 7.34519e6i	− 0.268982i
$944$	9.01778e6	0.329359
$945$	0	0
$946$	2.87910e7	1.04599
$947$	3.54260e7i	1.28365i	0.766850	+	0.641826i	$0.221823\pi$
$947$	3.54260e7i	1.28365i	−0.766850	+	0.641826i	$0.778177\pi$
$948$	0	0
$949$	5.42350e6	0.195485
$950$	0	0
$951$	0	0
$952$	− 3.34438e7i	− 1.19598i
$953$	5.47490e6i	0.195274i	0.995222	+	0.0976369i	$0.0311284\pi$
$953$	5.47490e6i	0.195274i	−0.995222	+	0.0976369i	$0.968872\pi$
$954$	0	0
$955$	0	0
$956$	−2.09679e6	−0.0742011
$957$	0	0
$958$	− 2.15086e7i	− 0.757177i
$959$	1.62760e7	0.571479
$960$	0	0
$961$	−2.50352e7	−0.874467
$962$	3.48032e6i	0.121250i
$963$	0	0
$964$	1.97811e7	0.685579
$965$	0	0
$966$	0	0
$967$	− 1.65533e7i	− 0.569269i	−0.958636	−	0.284635i	$-0.908128\pi$
$967$	− 1.65533e7i	− 0.569269i	0.958636	−	0.284635i	$-0.0918723\pi$
$968$	− 1.70077e7i	− 0.583387i
$969$	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
$973$	2.26757e7i	0.767855i
$974$	−6.88338e6	−0.232490
$975$	0	0
$976$	687445.	0.0231001
$977$	− 516874.i	− 0.0173240i	−0.999962	−	0.00866200i	$-0.997243\pi$
$977$	− 516874.i	− 0.0173240i	0.999962	−	0.00866200i	$-0.00275723\pi$
$978$	0	0
$979$	−1.17111e7	−0.390519
$980$	0	0
$981$	0	0
$982$	− 2.92580e7i	− 0.968201i
$983$	− 4.15798e7i	− 1.37246i	−0.727386	−	0.686229i	$-0.759265\pi$
$983$	− 4.15798e7i	− 1.37246i	0.727386	−	0.686229i	$-0.240735\pi$
$984$	0	0
$985$	0	0
$986$	2.13368e7	0.698935
$987$	0	0
$988$	1.66221e7i	0.541744i
$989$	−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	$0.584307\pi$
$992$	− 1.94126e6i	− 0.0626331i
$993$	0	0
$994$	2.63277e7	0.845176
$995$	0	0
$996$	0	0
$997$	− 2.74426e7i	− 0.874355i	−0.899375	−	0.437178i	$-0.855978\pi$
$997$	− 2.74426e7i	− 0.874355i	0.899375	−	0.437178i	$-0.144022\pi$
$998$	2.91951e7i	0.927863i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.a.y.1.1	✓	2	5.2	odd	4
450.6.a.ba.1.2	yes	2	15.8	even	4
450.6.a.bd.1.1	yes	2	15.2	even	4
450.6.a.bf.1.2	yes	2	5.3	odd	4
450.6.c.p.199.1		4	5.4	even	2	inner
450.6.c.p.199.4		4	1.1	even	1	trivial
450.6.c.q.199.2		4	3.2	odd	2
450.6.c.q.199.3		4	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} +241.648i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} +241.648i q^{7} -64.0000i q^{8} -653.296 q^{11} +828.296i q^{13} -966.592 q^{14} +256.000 q^{16} -2162.48i q^{17} -1254.24 q^{19} -2613.18i q^{22} +3746.48i q^{23} -3313.18 q^{26} -3866.37i q^{28} +2466.70 q^{29} -1895.76 q^{31} +1024.00i q^{32} +8649.92 q^{34} -1050.45i q^{37} -5016.96i q^{38} -1960.56 q^{41} +11017.6i q^{43} +10452.7 q^{44} -14985.9 q^{46} +23064.8i q^{47} -41586.8 q^{49} -13252.7i q^{52} -27329.4i q^{53} +15465.5 q^{56} +9866.82i q^{58} +35225.7 q^{59} +2685.33 q^{61} -7583.04i q^{62} -4096.00 q^{64} -48125.1i q^{67} +34599.7i q^{68} -27237.7 q^{71} -6547.77i q^{73} +4201.78 q^{74} +20067.8 q^{76} -157868. i q^{77} +65037.3 q^{79} -7842.23i q^{82} -62567.2i q^{83} -44070.3 q^{86} +41811.0i q^{88} +17926.2 q^{89} -200156. q^{91} -59943.7i q^{92} -92259.2 q^{94} -95974.9i q^{97} -166347. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4}+O(q^{10}) \) 4 * q - 64 * q^4	\( 4 q - 64 q^{4} - 1080 q^{11} - 800 q^{14} + 1024 q^{16} - 1184 q^{19} - 7120 q^{26} + 11400 q^{29} - 11416 q^{31} + 3936 q^{34} - 30840 q^{41} + 17280 q^{44} - 29280 q^{46} - 89688 q^{49} + 12800 q^{56} + 101040 q^{59} - 58252 q^{61} - 16384 q^{64} - 12360 q^{71} + 90400 q^{74} + 18944 q^{76} - 15824 q^{79} - 50560 q^{86} + 329280 q^{89} - 382832 q^{91} - 62400 q^{94}+O(q^{100}) \) 4 * q - 64 * q^4 - 1080 * q^11 - 800 * q^14 + 1024 * q^16 - 1184 * q^19 - 7120 * q^26 + 11400 * q^29 - 11416 * q^31 + 3936 * q^34 - 30840 * q^41 + 17280 * q^44 - 29280 * q^46 - 89688 * q^49 + 12800 * q^56 + 101040 * q^59 - 58252 * q^61 - 16384 * q^64 - 12360 * q^71 + 90400 * q^74 + 18944 * q^76 - 15824 * q^79 - 50560 * q^86 + 329280 * q^89 - 382832 * q^91 - 62400 * q^94

Introduction

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