LMFDB - Embedded newform 450.6.c.e.199.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.e.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} +47.0000i q^{7} +64.0000i q^{8} +O(q^{10})$	$q-4.00000i q^{2} -16.0000 q^{4} +47.0000i q^{7} +64.0000i q^{8} -222.000 q^{11} -101.000i q^{13} +188.000 q^{14} +256.000 q^{16} -162.000i q^{17} -1685.00 q^{19} +888.000i q^{22} +306.000i q^{23} -404.000 q^{26} -752.000i q^{28} +7890.00 q^{29} -8593.00 q^{31} -1024.00i q^{32} -648.000 q^{34} +8642.00i q^{37} +6740.00i q^{38} +18168.0 q^{41} -14351.0i q^{43} +3552.00 q^{44} +1224.00 q^{46} +1098.00i q^{47} +14598.0 q^{49} +1616.00i q^{52} +17916.0i q^{53} -3008.00 q^{56} -31560.0i q^{58} +17610.0 q^{59} -21853.0 q^{61} +34372.0i q^{62} -4096.00 q^{64} +107.000i q^{67} +2592.00i q^{68} +40728.0 q^{71} -34706.0i q^{73} +34568.0 q^{74} +26960.0 q^{76} -10434.0i q^{77} +69160.0 q^{79} -72672.0i q^{82} -108534. i q^{83} -57404.0 q^{86} -14208.0i q^{88} +35040.0 q^{89} +4747.00 q^{91} -4896.00i q^{92} +4392.00 q^{94} -823.000i q^{97} -58392.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 444 q^{11} + 376 q^{14} + 512 q^{16} - 3370 q^{19} - 808 q^{26} + 15780 q^{29} - 17186 q^{31} - 1296 q^{34} + 36336 q^{41} + 7104 q^{44} + 2448 q^{46} + 29196 q^{49} - 6016 q^{56} + 35220 q^{59} - 43706 q^{61} - 8192 q^{64} + 81456 q^{71} + 69136 q^{74} + 53920 q^{76} + 138320 q^{79} - 114808 q^{86} + 70080 q^{89} + 9494 q^{91} + 8784 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 444 * q^11 + 376 * q^14 + 512 * q^16 - 3370 * q^19 - 808 * q^26 + 15780 * q^29 - 17186 * q^31 - 1296 * q^34 + 36336 * q^41 + 7104 * q^44 + 2448 * q^46 + 29196 * q^49 - 6016 * q^56 + 35220 * q^59 - 43706 * q^61 - 8192 * q^64 + 81456 * q^71 + 69136 * q^74 + 53920 * q^76 + 138320 * q^79 - 114808 * q^86 + 70080 * q^89 + 9494 * q^91 + 8784 * 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
$7$	47.0000i	0.362537i	0.983434	+	0.181269i	$0.0580204\pi$
$7$	47.0000i	0.362537i	−0.983434	+	0.181269i	$0.941980\pi$
$8$	64.0000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−222.000	−0.553186	−0.276593	−	0.960987i	$-0.589205\pi$
$11$	−222.000	−0.553186	−0.276593	+	0.960987i	$0.589205\pi$
$12$	0	0
$13$	− 101.000i	− 0.165754i	−0.996560	−	0.0828768i	$-0.973589\pi$
$13$	− 101.000i	− 0.165754i	0.996560	−	0.0828768i	$-0.0264108\pi$
$14$	188.000	0.256353
$15$	0	0
$16$	256.000	0.250000
$17$	− 162.000i	− 0.135954i	−0.997687	−	0.0679771i	$-0.978346\pi$
$17$	− 162.000i	− 0.135954i	0.997687	−	0.0679771i	$-0.0216545\pi$
$18$	0	0
$19$	−1685.00	−1.07082	−0.535409	−	0.844593i	$-0.679843\pi$
$19$	−1685.00	−1.07082	−0.535409	+	0.844593i	$0.679843\pi$
$20$	0	0
$21$	0	0
$22$	888.000i	0.391162i
$23$	306.000i	0.120615i	0.998180	+	0.0603076i	$0.0192082\pi$
$23$	306.000i	0.120615i	−0.998180	+	0.0603076i	$0.980792\pi$
$24$	0	0
$25$	0	0
$26$	−404.000	−0.117206
$27$	0	0
$28$	− 752.000i	− 0.181269i
$29$	7890.00	1.74214	0.871068	−	0.491163i	$-0.163428\pi$
$29$	7890.00	1.74214	0.871068	+	0.491163i	$0.163428\pi$
$30$	0	0
$31$	−8593.00	−1.60598	−0.802991	−	0.595991i	$-0.796759\pi$
$31$	−8593.00	−1.60598	−0.802991	+	0.595991i	$0.796759\pi$
$32$	− 1024.00i	− 0.176777i
$33$	0	0
$34$	−648.000	−0.0961342
$35$	0	0
$36$	0	0
$37$	8642.00i	1.03779i	0.854838	+	0.518896i	$0.173657\pi$
$37$	8642.00i	1.03779i	−0.854838	+	0.518896i	$0.826343\pi$
$38$	6740.00i	0.757183i
$39$	0	0
$40$	0	0
$41$	18168.0	1.68790	0.843951	−	0.536420i	$-0.180223\pi$
$41$	18168.0	1.68790	0.843951	+	0.536420i	$0.180223\pi$
$42$	0	0
$43$	− 14351.0i	− 1.18362i	−0.806079	−	0.591808i	$-0.798414\pi$
$43$	− 14351.0i	− 1.18362i	0.806079	−	0.591808i	$-0.201586\pi$
$44$	3552.00	0.276593
$45$	0	0
$46$	1224.00	0.0852878
$47$	1098.00i	0.0725033i	0.999343	+	0.0362516i	$0.0115418\pi$
$47$	1098.00i	0.0725033i	−0.999343	+	0.0362516i	$0.988458\pi$
$48$	0	0
$49$	14598.0	0.868567
$50$	0	0
$51$	0	0
$52$	1616.00i	0.0828768i
$53$	17916.0i	0.876095i	0.898952	+	0.438048i	$0.144330\pi$
$53$	17916.0i	0.876095i	−0.898952	+	0.438048i	$0.855670\pi$
$54$	0	0
$55$	0	0
$56$	−3008.00	−0.128176
$57$	0	0
$58$	− 31560.0i	− 1.23188i
$59$	17610.0	0.658612	0.329306	−	0.944223i	$-0.393185\pi$
$59$	17610.0	0.658612	0.329306	+	0.944223i	$0.393185\pi$
$60$	0	0
$61$	−21853.0	−0.751946	−0.375973	−	0.926631i	$-0.622691\pi$
$61$	−21853.0	−0.751946	−0.375973	+	0.926631i	$0.622691\pi$
$62$	34372.0i	1.13560i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	107.000i	0.00291204i	0.999999	+	0.00145602i	$0.000463465\pi$
$67$	107.000i	0.00291204i	−0.999999	+	0.00145602i	$0.999537\pi$
$68$	2592.00i	0.0679771i
$69$	0	0
$70$	0	0
$71$	40728.0	0.958842	0.479421	−	0.877585i	$-0.340847\pi$
$71$	40728.0	0.958842	0.479421	+	0.877585i	$0.340847\pi$
$72$	0	0
$73$	− 34706.0i	− 0.762250i	−0.924524	−	0.381125i	$-0.875537\pi$
$73$	− 34706.0i	− 0.762250i	0.924524	−	0.381125i	$-0.124463\pi$
$74$	34568.0	0.733829
$75$	0	0
$76$	26960.0	0.535409
$77$	− 10434.0i	− 0.200551i
$78$	0	0
$79$	69160.0	1.24677	0.623386	−	0.781914i	$-0.285756\pi$
$79$	69160.0	1.24677	0.623386	+	0.781914i	$0.285756\pi$
$80$	0	0
$81$	0	0
$82$	− 72672.0i	− 1.19353i
$83$	− 108534.i	− 1.72930i	−0.502374	−	0.864650i	$-0.667540\pi$
$83$	− 108534.i	− 1.72930i	0.502374	−	0.864650i	$-0.332460\pi$
$84$	0	0
$85$	0	0
$86$	−57404.0	−0.836943
$87$	0	0
$88$	− 14208.0i	− 0.195581i
$89$	35040.0	0.468910	0.234455	−	0.972127i	$-0.424670\pi$
$89$	35040.0	0.468910	0.234455	+	0.972127i	$0.424670\pi$
$90$	0	0
$91$	4747.00	0.0600919
$92$	− 4896.00i	− 0.0603076i
$93$	0	0
$94$	4392.00	0.0512676
$95$	0	0
$96$	0	0
$97$	− 823.000i	− 0.00888118i	−0.999990	−	0.00444059i	$-0.998587\pi$
$97$	− 823.000i	− 0.00888118i	0.999990	−	0.00444059i	$-0.00141349\pi$
$98$	− 58392.0i	− 0.614169i
$99$	0	0
$100$	0	0
$101$	33828.0	0.329969	0.164984	−	0.986296i	$-0.447243\pi$
$101$	33828.0	0.329969	0.164984	+	0.986296i	$0.447243\pi$
$102$	0	0
$103$	133444.i	1.23938i	0.784845	+	0.619692i	$0.212743\pi$
$103$	133444.i	1.23938i	−0.784845	+	0.619692i	$0.787257\pi$
$104$	6464.00	0.0586028
$105$	0	0
$106$	71664.0	0.619493
$107$	− 81252.0i	− 0.686080i	−0.939321	−	0.343040i	$-0.888543\pi$
$107$	− 81252.0i	− 0.686080i	0.939321	−	0.343040i	$-0.111457\pi$
$108$	0	0
$109$	217015.	1.74954	0.874769	−	0.484540i	$-0.161013\pi$
$109$	217015.	1.74954	0.874769	+	0.484540i	$0.161013\pi$
$110$	0	0
$111$	0	0
$112$	12032.0i	0.0906343i
$113$	− 138324.i	− 1.01906i	−0.860452	−	0.509532i	$-0.829819\pi$
$113$	− 138324.i	− 1.01906i	0.860452	−	0.509532i	$-0.170181\pi$
$114$	0	0
$115$	0	0
$116$	−126240.	−0.871068
$117$	0	0
$118$	− 70440.0i	− 0.465709i
$119$	7614.00	0.0492885
$120$	0	0
$121$	−111767.	−0.693985
$122$	87412.0i	0.531706i
$123$	0	0
$124$	137488.	0.802991
$125$	0	0
$126$	0	0
$127$	− 256048.i	− 1.40868i	−0.709863	−	0.704340i	$-0.751243\pi$
$127$	− 256048.i	− 1.40868i	0.709863	−	0.704340i	$-0.248757\pi$
$128$	16384.0i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−118452.	−0.603065	−0.301533	−	0.953456i	$-0.597498\pi$
$131$	−118452.	−0.603065	−0.301533	+	0.953456i	$0.597498\pi$
$132$	0	0
$133$	− 79195.0i	− 0.388212i
$134$	428.000	0.00205912
$135$	0	0
$136$	10368.0	0.0480671
$137$	13218.0i	0.0601678i	0.999547	+	0.0300839i	$0.00957745\pi$
$137$	13218.0i	0.0601678i	−0.999547	+	0.0300839i	$0.990423\pi$
$138$	0	0
$139$	350740.	1.53974	0.769872	−	0.638199i	$-0.220320\pi$
$139$	350740.	1.53974	0.769872	+	0.638199i	$0.220320\pi$
$140$	0	0
$141$	0	0
$142$	− 162912.i	− 0.678004i
$143$	22422.0i	0.0916926i
$144$	0	0
$145$	0	0
$146$	−138824.	−0.538992
$147$	0	0
$148$	− 138272.i	− 0.518896i
$149$	109890.	0.405502	0.202751	−	0.979230i	$-0.435012\pi$
$149$	109890.	0.405502	0.202751	+	0.979230i	$0.435012\pi$
$150$	0	0
$151$	−172603.	−0.616036	−0.308018	−	0.951381i	$-0.599666\pi$
$151$	−172603.	−0.616036	−0.308018	+	0.951381i	$0.599666\pi$
$152$	− 107840.i	− 0.378592i
$153$	0	0
$154$	−41736.0	−0.141811
$155$	0	0
$156$	0	0
$157$	− 349993.i	− 1.13321i	−0.823990	−	0.566605i	$-0.808257\pi$
$157$	− 349993.i	− 1.13321i	0.823990	−	0.566605i	$-0.191743\pi$
$158$	− 276640.i	− 0.881601i
$159$	0	0
$160$	0	0
$161$	−14382.0	−0.0437275
$162$	0	0
$163$	− 192581.i	− 0.567733i	−0.958864	−	0.283867i	$-0.908383\pi$
$163$	− 192581.i	− 0.567733i	0.958864	−	0.283867i	$-0.0916173\pi$
$164$	−290688.	−0.843951
$165$	0	0
$166$	−434136.	−1.22280
$167$	− 580692.i	− 1.61122i	−0.592447	−	0.805610i	$-0.701838\pi$
$167$	− 580692.i	− 1.61122i	0.592447	−	0.805610i	$-0.298162\pi$
$168$	0	0
$169$	361092.	0.972526
$170$	0	0
$171$	0	0
$172$	229616.i	0.591808i
$173$	738126.i	1.87506i	0.347904	+	0.937530i	$0.386894\pi$
$173$	738126.i	1.87506i	−0.347904	+	0.937530i	$0.613106\pi$
$174$	0	0
$175$	0	0
$176$	−56832.0	−0.138297
$177$	0	0
$178$	− 140160.i	− 0.331569i
$179$	497370.	1.16024	0.580119	−	0.814532i	$-0.303006\pi$
$179$	497370.	1.16024	0.580119	+	0.814532i	$0.303006\pi$
$180$	0	0
$181$	−333163.	−0.755893	−0.377947	−	0.925827i	$-0.623370\pi$
$181$	−333163.	−0.755893	−0.377947	+	0.925827i	$0.623370\pi$
$182$	− 18988.0i	− 0.0424914i
$183$	0	0
$184$	−19584.0	−0.0426439
$185$	0	0
$186$	0	0
$187$	35964.0i	0.0752080i
$188$	− 17568.0i	− 0.0362516i
$189$	0	0
$190$	0	0
$191$	40638.0	0.0806026	0.0403013	−	0.999188i	$-0.487168\pi$
$191$	40638.0	0.0806026	0.0403013	+	0.999188i	$0.487168\pi$
$192$	0	0
$193$	− 494651.i	− 0.955885i	−0.878391	−	0.477942i	$-0.841383\pi$
$193$	− 494651.i	− 0.955885i	0.878391	−	0.477942i	$-0.158617\pi$
$194$	−3292.00	−0.00627994
$195$	0	0
$196$	−233568.	−0.434283
$197$	− 552342.i	− 1.01401i	−0.861943	−	0.507005i	$-0.830752\pi$
$197$	− 552342.i	− 1.01401i	0.861943	−	0.507005i	$-0.169248\pi$
$198$	0	0
$199$	−685625.	−1.22731	−0.613655	−	0.789575i	$-0.710301\pi$
$199$	−685625.	−1.22731	−0.613655	+	0.789575i	$0.710301\pi$
$200$	0	0
$201$	0	0
$202$	− 135312.i	− 0.233323i
$203$	370830.i	0.631589i
$204$	0	0
$205$	0	0
$206$	533776.	0.876377
$207$	0	0
$208$	− 25856.0i	− 0.0414384i
$209$	374070.	0.592362
$210$	0	0
$211$	749477.	1.15892	0.579458	−	0.815002i	$-0.303264\pi$
$211$	749477.	1.15892	0.579458	+	0.815002i	$0.303264\pi$
$212$	− 286656.i	− 0.438048i
$213$	0	0
$214$	−325008.	−0.485132
$215$	0	0
$216$	0	0
$217$	− 403871.i	− 0.582228i
$218$	− 868060.i	− 1.23711i
$219$	0	0
$220$	0	0
$221$	−16362.0	−0.0225349
$222$	0	0
$223$	− 169271.i	− 0.227940i	−0.993484	−	0.113970i	$-0.963643\pi$
$223$	− 169271.i	− 0.227940i	0.993484	−	0.113970i	$-0.0363568\pi$
$224$	48128.0	0.0640882
$225$	0	0
$226$	−553296.	−0.720587
$227$	46488.0i	0.0598792i	0.999552	+	0.0299396i	$0.00953150\pi$
$227$	46488.0i	0.0598792i	−0.999552	+	0.0299396i	$0.990468\pi$
$228$	0	0
$229$	90115.0	0.113556	0.0567778	−	0.998387i	$-0.481917\pi$
$229$	90115.0	0.113556	0.0567778	+	0.998387i	$0.481917\pi$
$230$	0	0
$231$	0	0
$232$	504960.i	0.615938i
$233$	1.06414e6i	1.28413i	0.766652	+	0.642063i	$0.221921\pi$
$233$	1.06414e6i	1.28413i	−0.766652	+	0.642063i	$0.778079\pi$
$234$	0	0
$235$	0	0
$236$	−281760.	−0.329306
$237$	0	0
$238$	− 30456.0i	− 0.0348522i
$239$	1.15158e6	1.30407	0.652033	−	0.758191i	$-0.273916\pi$
$239$	1.15158e6	1.30407	0.652033	+	0.758191i	$0.273916\pi$
$240$	0	0
$241$	856217.	0.949601	0.474801	−	0.880093i	$-0.342520\pi$
$241$	856217.	0.949601	0.474801	+	0.880093i	$0.342520\pi$
$242$	447068.i	0.490722i
$243$	0	0
$244$	349648.	0.375973
$245$	0	0
$246$	0	0
$247$	170185.i	0.177492i
$248$	− 549952.i	− 0.567800i
$249$	0	0
$250$	0	0
$251$	207708.	0.208098	0.104049	−	0.994572i	$-0.466820\pi$
$251$	207708.	0.208098	0.104049	+	0.994572i	$0.466820\pi$
$252$	0	0
$253$	− 67932.0i	− 0.0667226i
$254$	−1.02419e6	−0.996087
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.45319e6i	1.37243i	0.727401	+	0.686213i	$0.240728\pi$
$257$	1.45319e6i	1.37243i	−0.727401	+	0.686213i	$0.759272\pi$
$258$	0	0
$259$	−406174.	−0.376238
$260$	0	0
$261$	0	0
$262$	473808.i	0.426431i
$263$	169296.i	0.150924i	0.997149	+	0.0754618i	$0.0240431\pi$
$263$	169296.i	0.150924i	−0.997149	+	0.0754618i	$0.975957\pi$
$264$	0	0
$265$	0	0
$266$	−316780.	−0.274507
$267$	0	0
$268$	− 1712.00i	− 0.00145602i
$269$	−1.58109e6	−1.33222	−0.666110	−	0.745854i	$-0.732042\pi$
$269$	−1.58109e6	−1.33222	−0.666110	+	0.745854i	$0.732042\pi$
$270$	0	0
$271$	822512.	0.680329	0.340165	−	0.940366i	$-0.389517\pi$
$271$	822512.	0.680329	0.340165	+	0.940366i	$0.389517\pi$
$272$	− 41472.0i	− 0.0339886i
$273$	0	0
$274$	52872.0	0.0425451
$275$	0	0
$276$	0	0
$277$	− 546823.i	− 0.428201i	−0.976812	−	0.214100i	$-0.931318\pi$
$277$	− 546823.i	− 0.428201i	0.976812	−	0.214100i	$-0.0686820\pi$
$278$	− 1.40296e6i	− 1.08876i
$279$	0	0
$280$	0	0
$281$	1.09250e6	0.825382	0.412691	−	0.910871i	$-0.364589\pi$
$281$	1.09250e6	0.825382	0.412691	+	0.910871i	$0.364589\pi$
$282$	0	0
$283$	2.48480e6i	1.84427i	0.386865	+	0.922136i	$0.373558\pi$
$283$	2.48480e6i	1.84427i	−0.386865	+	0.922136i	$0.626442\pi$
$284$	−651648.	−0.479421
$285$	0	0
$286$	89688.0	0.0648365
$287$	853896.i	0.611928i
$288$	0	0
$289$	1.39361e6	0.981516
$290$	0	0
$291$	0	0
$292$	555296.i	0.381125i
$293$	− 341394.i	− 0.232320i	−0.993231	−	0.116160i	$-0.962941\pi$
$293$	− 341394.i	− 0.232320i	0.993231	−	0.116160i	$-0.0370586\pi$
$294$	0	0
$295$	0	0
$296$	−553088.	−0.366915
$297$	0	0
$298$	− 439560.i	− 0.286733i
$299$	30906.0	0.0199924
$300$	0	0
$301$	674497.	0.429105
$302$	690412.i	0.435603i
$303$	0	0
$304$	−431360.	−0.267705
$305$	0	0
$306$	0	0
$307$	2.02898e6i	1.22866i	0.789050	+	0.614329i	$0.210573\pi$
$307$	2.02898e6i	1.22866i	−0.789050	+	0.614329i	$0.789427\pi$
$308$	166944.i	0.100275i
$309$	0	0
$310$	0	0
$311$	206598.	0.121123	0.0605613	−	0.998164i	$-0.480711\pi$
$311$	206598.	0.121123	0.0605613	+	0.998164i	$0.480711\pi$
$312$	0	0
$313$	3.34223e6i	1.92830i	0.265352	+	0.964152i	$0.414512\pi$
$313$	3.34223e6i	1.92830i	−0.265352	+	0.964152i	$0.585488\pi$
$314$	−1.39997e6	−0.801300
$315$	0	0
$316$	−1.10656e6	−0.623386
$317$	2.53289e6i	1.41569i	0.706368	+	0.707844i	$0.250332\pi$
$317$	2.53289e6i	1.41569i	−0.706368	+	0.707844i	$0.749668\pi$
$318$	0	0
$319$	−1.75158e6	−0.963725
$320$	0	0
$321$	0	0
$322$	57528.0i	0.0309200i
$323$	272970.i	0.145582i
$324$	0	0
$325$	0	0
$326$	−770324.	−0.401448
$327$	0	0
$328$	1.16275e6i	0.596764i
$329$	−51606.0	−0.0262851
$330$	0	0
$331$	602132.	0.302080	0.151040	−	0.988528i	$-0.451738\pi$
$331$	602132.	0.302080	0.151040	+	0.988528i	$0.451738\pi$
$332$	1.73654e6i	0.864650i
$333$	0	0
$334$	−2.32277e6	−1.13930
$335$	0	0
$336$	0	0
$337$	209777.i	0.100620i	0.998734	+	0.0503099i	$0.0160209\pi$
$337$	209777.i	0.100620i	−0.998734	+	0.0503099i	$0.983979\pi$
$338$	− 1.44437e6i	− 0.687680i
$339$	0	0
$340$	0	0
$341$	1.90765e6	0.888407
$342$	0	0
$343$	1.47603e6i	0.677425i
$344$	918464.	0.418472
$345$	0	0
$346$	2.95250e6	1.32587
$347$	− 4.02166e6i	− 1.79301i	−0.443037	−	0.896503i	$-0.646099\pi$
$347$	− 4.02166e6i	− 1.79301i	0.443037	−	0.896503i	$-0.353901\pi$
$348$	0	0
$349$	−8330.00	−0.00366085	−0.00183042	−	0.999998i	$-0.500583\pi$
$349$	−8330.00	−0.00366085	−0.00183042	+	0.999998i	$0.500583\pi$
$350$	0	0
$351$	0	0
$352$	227328.i	0.0977904i
$353$	1.95001e6i	0.832912i	0.909156	+	0.416456i	$0.136728\pi$
$353$	1.95001e6i	0.832912i	−0.909156	+	0.416456i	$0.863272\pi$
$354$	0	0
$355$	0	0
$356$	−560640.	−0.234455
$357$	0	0
$358$	− 1.98948e6i	− 0.820412i
$359$	2.27088e6	0.929947	0.464973	−	0.885325i	$-0.346064\pi$
$359$	2.27088e6	0.929947	0.464973	+	0.885325i	$0.346064\pi$
$360$	0	0
$361$	363126.	0.146652
$362$	1.33265e6i	0.534497i
$363$	0	0
$364$	−75952.0	−0.0300459
$365$	0	0
$366$	0	0
$367$	2.86154e6i	1.10901i	0.832181	+	0.554503i	$0.187092\pi$
$367$	2.86154e6i	1.10901i	−0.832181	+	0.554503i	$0.812908\pi$
$368$	78336.0i	0.0301538i
$369$	0	0
$370$	0	0
$371$	−842052.	−0.317617
$372$	0	0
$373$	− 615311.i	− 0.228993i	−0.993424	−	0.114497i	$-0.963474\pi$
$373$	− 615311.i	− 0.228993i	0.993424	−	0.114497i	$-0.0365255\pi$
$374$	143856.	0.0531801
$375$	0	0
$376$	−70272.0	−0.0256338
$377$	− 796890.i	− 0.288765i
$378$	0	0
$379$	−5.39878e6	−1.93062	−0.965311	−	0.261103i	$-0.915914\pi$
$379$	−5.39878e6	−1.93062	−0.965311	+	0.261103i	$0.915914\pi$
$380$	0	0
$381$	0	0
$382$	− 162552.i	− 0.0569946i
$383$	1.08688e6i	0.378602i	0.981919	+	0.189301i	$0.0606222\pi$
$383$	1.08688e6i	0.378602i	−0.981919	+	0.189301i	$0.939378\pi$
$384$	0	0
$385$	0	0
$386$	−1.97860e6	−0.675913
$387$	0	0
$388$	13168.0i	0.00444059i
$389$	−3.48432e6	−1.16747	−0.583733	−	0.811946i	$-0.698408\pi$
$389$	−3.48432e6	−1.16747	−0.583733	+	0.811946i	$0.698408\pi$
$390$	0	0
$391$	49572.0	0.0163981
$392$	934272.i	0.307085i
$393$	0	0
$394$	−2.20937e6	−0.717014
$395$	0	0
$396$	0	0
$397$	3.26591e6i	1.03999i	0.854170	+	0.519993i	$0.174065\pi$
$397$	3.26591e6i	1.03999i	−0.854170	+	0.519993i	$0.825935\pi$
$398$	2.74250e6i	0.867839i
$399$	0	0
$400$	0	0
$401$	4.27319e6	1.32706	0.663531	−	0.748149i	$-0.269057\pi$
$401$	4.27319e6	1.32706	0.663531	+	0.748149i	$0.269057\pi$
$402$	0	0
$403$	867893.i	0.266197i
$404$	−541248.	−0.164984
$405$	0	0
$406$	1.48332e6	0.446601
$407$	− 1.91852e6i	− 0.574092i
$408$	0	0
$409$	1.45188e6	0.429162	0.214581	−	0.976706i	$-0.431161\pi$
$409$	1.45188e6	0.429162	0.214581	+	0.976706i	$0.431161\pi$
$410$	0	0
$411$	0	0
$412$	− 2.13510e6i	− 0.619692i
$413$	827670.i	0.238771i
$414$	0	0
$415$	0	0
$416$	−103424.	−0.0293014
$417$	0	0
$418$	− 1.49628e6i	− 0.418863i
$419$	559380.	0.155658	0.0778291	−	0.996967i	$-0.475201\pi$
$419$	559380.	0.155658	0.0778291	+	0.996967i	$0.475201\pi$
$420$	0	0
$421$	−3.91470e6	−1.07645	−0.538224	−	0.842802i	$-0.680905\pi$
$421$	−3.91470e6	−1.07645	−0.538224	+	0.842802i	$0.680905\pi$
$422$	− 2.99791e6i	− 0.819478i
$423$	0	0
$424$	−1.14662e6	−0.309746
$425$	0	0
$426$	0	0
$427$	− 1.02709e6i	− 0.272608i
$428$	1.30003e6i	0.343040i
$429$	0	0
$430$	0	0
$431$	3.57500e6	0.927006	0.463503	−	0.886095i	$-0.346592\pi$
$431$	3.57500e6	0.927006	0.463503	+	0.886095i	$0.346592\pi$
$432$	0	0
$433$	− 7.15969e6i	− 1.83516i	−0.397548	−	0.917581i	$-0.630139\pi$
$433$	− 7.15969e6i	− 1.83516i	0.397548	−	0.917581i	$-0.369861\pi$
$434$	−1.61548e6	−0.411698
$435$	0	0
$436$	−3.47224e6	−0.874769
$437$	− 515610.i	− 0.129157i
$438$	0	0
$439$	−1.71790e6	−0.425437	−0.212719	−	0.977114i	$-0.568232\pi$
$439$	−1.71790e6	−0.425437	−0.212719	+	0.977114i	$0.568232\pi$
$440$	0	0
$441$	0	0
$442$	65448.0i	0.0159346i
$443$	3.39670e6i	0.822332i	0.911560	+	0.411166i	$0.134878\pi$
$443$	3.39670e6i	0.822332i	−0.911560	+	0.411166i	$0.865122\pi$
$444$	0	0
$445$	0	0
$446$	−677084.	−0.161178
$447$	0	0
$448$	− 192512.i	− 0.0453172i
$449$	−3.39606e6	−0.794986	−0.397493	−	0.917605i	$-0.630120\pi$
$449$	−3.39606e6	−0.794986	−0.397493	+	0.917605i	$0.630120\pi$
$450$	0	0
$451$	−4.03330e6	−0.933724
$452$	2.21318e6i	0.509532i
$453$	0	0
$454$	185952.	0.0423410
$455$	0	0
$456$	0	0
$457$	− 4.52814e6i	− 1.01421i	−0.861883	−	0.507106i	$-0.830715\pi$
$457$	− 4.52814e6i	− 1.01421i	0.861883	−	0.507106i	$-0.169285\pi$
$458$	− 360460.i	− 0.0802959i
$459$	0	0
$460$	0	0
$461$	1.27895e6	0.280285	0.140143	−	0.990131i	$-0.455244\pi$
$461$	1.27895e6	0.280285	0.140143	+	0.990131i	$0.455244\pi$
$462$	0	0
$463$	7.19862e6i	1.56062i	0.625393	+	0.780310i	$0.284939\pi$
$463$	7.19862e6i	1.56062i	−0.625393	+	0.780310i	$0.715061\pi$
$464$	2.01984e6	0.435534
$465$	0	0
$466$	4.25654e6	0.908014
$467$	− 4.83034e6i	− 1.02491i	−0.858714	−	0.512455i	$-0.828736\pi$
$467$	− 4.83034e6i	− 1.02491i	0.858714	−	0.512455i	$-0.171264\pi$
$468$	0	0
$469$	−5029.00	−0.00105572
$470$	0	0
$471$	0	0
$472$	1.12704e6i	0.232854i
$473$	3.18592e6i	0.654760i
$474$	0	0
$475$	0	0
$476$	−121824.	−0.0246442
$477$	0	0
$478$	− 4.60632e6i	− 0.922113i
$479$	748650.	0.149087	0.0745435	−	0.997218i	$-0.476250\pi$
$479$	748650.	0.149087	0.0745435	+	0.997218i	$0.476250\pi$
$480$	0	0
$481$	872842.	0.172018
$482$	− 3.42487e6i	− 0.671469i
$483$	0	0
$484$	1.78827e6	0.346993
$485$	0	0
$486$	0	0
$487$	− 5.16394e6i	− 0.986641i	−0.869848	−	0.493320i	$-0.835783\pi$
$487$	− 5.16394e6i	− 0.986641i	0.869848	−	0.493320i	$-0.164217\pi$
$488$	− 1.39859e6i	− 0.265853i
$489$	0	0
$490$	0	0
$491$	−8.54287e6	−1.59919	−0.799595	−	0.600539i	$-0.794953\pi$
$491$	−8.54287e6	−1.59919	−0.799595	+	0.600539i	$0.794953\pi$
$492$	0	0
$493$	− 1.27818e6i	− 0.236851i
$494$	680740.	0.125506
$495$	0	0
$496$	−2.19981e6	−0.401495
$497$	1.91422e6i	0.347616i
$498$	0	0
$499$	4.20588e6	0.756145	0.378072	−	0.925776i	$-0.376587\pi$
$499$	4.20588e6	0.756145	0.378072	+	0.925776i	$0.376587\pi$
$500$	0	0
$501$	0	0
$502$	− 830832.i	− 0.147148i
$503$	− 8.18342e6i	− 1.44217i	−0.692849	−	0.721083i	$-0.743645\pi$
$503$	− 8.18342e6i	− 1.44217i	0.692849	−	0.721083i	$-0.256355\pi$
$504$	0	0
$505$	0	0
$506$	−271728.	−0.0471800
$507$	0	0
$508$	4.09677e6i	0.704340i
$509$	3.85923e6	0.660247	0.330123	−	0.943938i	$-0.392910\pi$
$509$	3.85923e6	0.660247	0.330123	+	0.943938i	$0.392910\pi$
$510$	0	0
$511$	1.63118e6	0.276344
$512$	− 262144.i	− 0.0441942i
$513$	0	0
$514$	5.81275e6	0.970452
$515$	0	0
$516$	0	0
$517$	− 243756.i	− 0.0401078i
$518$	1.62470e6i	0.266040i
$519$	0	0
$520$	0	0
$521$	−4.55410e6	−0.735036	−0.367518	−	0.930016i	$-0.619792\pi$
$521$	−4.55410e6	−0.735036	−0.367518	+	0.930016i	$0.619792\pi$
$522$	0	0
$523$	− 4.82224e6i	− 0.770894i	−0.922730	−	0.385447i	$-0.874047\pi$
$523$	− 4.82224e6i	− 0.770894i	0.922730	−	0.385447i	$-0.125953\pi$
$524$	1.89523e6	0.301533
$525$	0	0
$526$	677184.	0.106719
$527$	1.39207e6i	0.218340i
$528$	0	0
$529$	6.34271e6	0.985452
$530$	0	0
$531$	0	0
$532$	1.26712e6i	0.194106i
$533$	− 1.83497e6i	− 0.279776i
$534$	0	0
$535$	0	0
$536$	−6848.00	−0.00102956
$537$	0	0
$538$	6.32436e6i	0.942022i
$539$	−3.24076e6	−0.480479
$540$	0	0
$541$	362537.	0.0532549	0.0266274	−	0.999645i	$-0.491523\pi$
$541$	362537.	0.0532549	0.0266274	+	0.999645i	$0.491523\pi$
$542$	− 3.29005e6i	− 0.481065i
$543$	0	0
$544$	−165888.	−0.0240335
$545$	0	0
$546$	0	0
$547$	3.11439e6i	0.445046i	0.974927	+	0.222523i	$0.0714293\pi$
$547$	3.11439e6i	0.445046i	−0.974927	+	0.222523i	$0.928571\pi$
$548$	− 211488.i	− 0.0300839i
$549$	0	0
$550$	0	0
$551$	−1.32947e7	−1.86551
$552$	0	0
$553$	3.25052e6i	0.452002i
$554$	−2.18729e6	−0.302784
$555$	0	0
$556$	−5.61184e6	−0.769872
$557$	7.99304e6i	1.09163i	0.837907	+	0.545813i	$0.183779\pi$
$557$	7.99304e6i	1.09163i	−0.837907	+	0.545813i	$0.816221\pi$
$558$	0	0
$559$	−1.44945e6	−0.196189
$560$	0	0
$561$	0	0
$562$	− 4.36999e6i	− 0.583633i
$563$	− 1.23236e7i	− 1.63857i	−0.573385	−	0.819286i	$-0.694370\pi$
$563$	− 1.23236e7i	− 1.63857i	0.573385	−	0.819286i	$-0.305630\pi$
$564$	0	0
$565$	0	0
$566$	9.93920e6	1.30410
$567$	0	0
$568$	2.60659e6i	0.339002i
$569$	1.01364e7	1.31252	0.656258	−	0.754537i	$-0.272138\pi$
$569$	1.01364e7	1.31252	0.656258	+	0.754537i	$0.272138\pi$
$570$	0	0
$571$	6.53084e6	0.838260	0.419130	−	0.907926i	$-0.362335\pi$
$571$	6.53084e6	0.838260	0.419130	+	0.907926i	$0.362335\pi$
$572$	− 358752.i	− 0.0458463i
$573$	0	0
$574$	3.41558e6	0.432698
$575$	0	0
$576$	0	0
$577$	− 1.24453e6i	− 0.155621i	−0.996968	−	0.0778103i	$-0.975207\pi$
$577$	− 1.24453e6i	− 0.155621i	0.996968	−	0.0778103i	$-0.0247928\pi$
$578$	− 5.57445e6i	− 0.694037i
$579$	0	0
$580$	0	0
$581$	5.10110e6	0.626936
$582$	0	0
$583$	− 3.97735e6i	− 0.484644i
$584$	2.22118e6	0.269496
$585$	0	0
$586$	−1.36558e6	−0.164275
$587$	1.33403e6i	0.159797i	0.996803	+	0.0798987i	$0.0254597\pi$
$587$	1.33403e6i	0.159797i	−0.996803	+	0.0798987i	$0.974540\pi$
$588$	0	0
$589$	1.44792e7	1.71972
$590$	0	0
$591$	0	0
$592$	2.21235e6i	0.259448i
$593$	− 1.19401e7i	− 1.39435i	−0.716899	−	0.697177i	$-0.754439\pi$
$593$	− 1.19401e7i	− 1.39435i	0.716899	−	0.697177i	$-0.245561\pi$
$594$	0	0
$595$	0	0
$596$	−1.75824e6	−0.202751
$597$	0	0
$598$	− 123624.i	− 0.0141368i
$599$	−7.16430e6	−0.815843	−0.407922	−	0.913017i	$-0.633746\pi$
$599$	−7.16430e6	−0.815843	−0.407922	+	0.913017i	$0.633746\pi$
$600$	0	0
$601$	1.15163e6	0.130055	0.0650273	−	0.997883i	$-0.479287\pi$
$601$	1.15163e6	0.130055	0.0650273	+	0.997883i	$0.479287\pi$
$602$	− 2.69799e6i	− 0.303423i
$603$	0	0
$604$	2.76165e6	0.308018
$605$	0	0
$606$	0	0
$607$	− 1.34268e7i	− 1.47911i	−0.673097	−	0.739554i	$-0.735037\pi$
$607$	− 1.34268e7i	− 1.47911i	0.673097	−	0.739554i	$-0.264963\pi$
$608$	1.72544e6i	0.189296i
$609$	0	0
$610$	0	0
$611$	110898.	0.0120177
$612$	0	0
$613$	− 1.20184e7i	− 1.29180i	−0.763422	−	0.645900i	$-0.776482\pi$
$613$	− 1.20184e7i	− 1.29180i	0.763422	−	0.645900i	$-0.223518\pi$
$614$	8.11591e6	0.868793
$615$	0	0
$616$	667776.	0.0709054
$617$	6.98519e6i	0.738695i	0.929291	+	0.369348i	$0.120419\pi$
$617$	6.98519e6i	0.738695i	−0.929291	+	0.369348i	$0.879581\pi$
$618$	0	0
$619$	8.20625e6	0.860832	0.430416	−	0.902631i	$-0.358367\pi$
$619$	8.20625e6	0.860832	0.430416	+	0.902631i	$0.358367\pi$
$620$	0	0
$621$	0	0
$622$	− 826392.i	− 0.0856466i
$623$	1.64688e6i	0.169997i
$624$	0	0
$625$	0	0
$626$	1.33689e7	1.36352
$627$	0	0
$628$	5.59989e6i	0.566605i
$629$	1.40000e6	0.141092
$630$	0	0
$631$	−1.07686e7	−1.07668	−0.538338	−	0.842729i	$-0.680947\pi$
$631$	−1.07686e7	−1.07668	−0.538338	+	0.842729i	$0.680947\pi$
$632$	4.42624e6i	0.440801i
$633$	0	0
$634$	1.01316e7	1.00104
$635$	0	0
$636$	0	0
$637$	− 1.47440e6i	− 0.143968i
$638$	7.00632e6i	0.681457i
$639$	0	0
$640$	0	0
$641$	−1.92571e7	−1.85117	−0.925585	−	0.378539i	$-0.876427\pi$
$641$	−1.92571e7	−1.85117	−0.925585	+	0.378539i	$0.876427\pi$
$642$	0	0
$643$	− 1.00999e7i	− 0.963364i	−0.876346	−	0.481682i	$-0.840026\pi$
$643$	− 1.00999e7i	− 0.963364i	0.876346	−	0.481682i	$-0.159974\pi$
$644$	230112.	0.0218637
$645$	0	0
$646$	1.09188e6	0.102942
$647$	− 7.52113e6i	− 0.706354i	−0.935556	−	0.353177i	$-0.885101\pi$
$647$	− 7.52113e6i	− 0.706354i	0.935556	−	0.353177i	$-0.114899\pi$
$648$	0	0
$649$	−3.90942e6	−0.364335
$650$	0	0
$651$	0	0
$652$	3.08130e6i	0.283867i
$653$	− 2.67197e6i	− 0.245216i	−0.992455	−	0.122608i	$-0.960874\pi$
$653$	− 2.67197e6i	− 0.245216i	0.992455	−	0.122608i	$-0.0391258\pi$
$654$	0	0
$655$	0	0
$656$	4.65101e6	0.421976
$657$	0	0
$658$	206424.i	0.0185864i
$659$	6.99948e6	0.627845	0.313922	−	0.949449i	$-0.398357\pi$
$659$	6.99948e6	0.627845	0.313922	+	0.949449i	$0.398357\pi$
$660$	0	0
$661$	408122.	0.0363318	0.0181659	−	0.999835i	$-0.494217\pi$
$661$	408122.	0.0363318	0.0181659	+	0.999835i	$0.494217\pi$
$662$	− 2.40853e6i	− 0.213603i
$663$	0	0
$664$	6.94618e6	0.611400
$665$	0	0
$666$	0	0
$667$	2.41434e6i	0.210128i
$668$	9.29107e6i	0.805610i
$669$	0	0
$670$	0	0
$671$	4.85137e6	0.415966
$672$	0	0
$673$	− 1.74939e7i	− 1.48885i	−0.667709	−	0.744423i	$-0.732725\pi$
$673$	− 1.74939e7i	− 1.48885i	0.667709	−	0.744423i	$-0.267275\pi$
$674$	839108.	0.0711489
$675$	0	0
$676$	−5.77747e6	−0.486263
$677$	8.67440e6i	0.727391i	0.931518	+	0.363695i	$0.118485\pi$
$677$	8.67440e6i	0.727391i	−0.931518	+	0.363695i	$0.881515\pi$
$678$	0	0
$679$	38681.0	0.00321976
$680$	0	0
$681$	0	0
$682$	− 7.63058e6i	− 0.628198i
$683$	1.18478e7i	0.971822i	0.874008	+	0.485911i	$0.161512\pi$
$683$	1.18478e7i	0.971822i	−0.874008	+	0.485911i	$0.838488\pi$
$684$	0	0
$685$	0	0
$686$	5.90414e6	0.479012
$687$	0	0
$688$	− 3.67386e6i	− 0.295904i
$689$	1.80952e6	0.145216
$690$	0	0
$691$	9.47775e6	0.755110	0.377555	−	0.925987i	$-0.376765\pi$
$691$	9.47775e6	0.755110	0.377555	+	0.925987i	$0.376765\pi$
$692$	− 1.18100e7i	− 0.937530i
$693$	0	0
$694$	−1.60866e7	−1.26785
$695$	0	0
$696$	0	0
$697$	− 2.94322e6i	− 0.229478i
$698$	33320.0i	0.00258861i
$699$	0	0
$700$	0	0
$701$	2.28147e7	1.75355	0.876777	−	0.480898i	$-0.159689\pi$
$701$	2.28147e7	1.75355	0.876777	+	0.480898i	$0.159689\pi$
$702$	0	0
$703$	− 1.45618e7i	− 1.11129i
$704$	909312.	0.0691483
$705$	0	0
$706$	7.80002e6	0.588958
$707$	1.58992e6i	0.119626i
$708$	0	0
$709$	−1.27436e7	−0.952090	−0.476045	−	0.879421i	$-0.657930\pi$
$709$	−1.27436e7	−0.952090	−0.476045	+	0.879421i	$0.657930\pi$
$710$	0	0
$711$	0	0
$712$	2.24256e6i	0.165785i
$713$	− 2.62946e6i	− 0.193706i
$714$	0	0
$715$	0	0
$716$	−7.95792e6	−0.580119
$717$	0	0
$718$	− 9.08352e6i	− 0.657572i
$719$	2.44929e6	0.176692	0.0883462	−	0.996090i	$-0.471842\pi$
$719$	2.44929e6	0.176692	0.0883462	+	0.996090i	$0.471842\pi$
$720$	0	0
$721$	−6.27187e6	−0.449323
$722$	− 1.45250e6i	− 0.103699i
$723$	0	0
$724$	5.33061e6	0.377947
$725$	0	0
$726$	0	0
$727$	− 415033.i	− 0.0291237i	−0.999894	−	0.0145619i	$-0.995365\pi$
$727$	− 415033.i	− 0.0291237i	0.999894	−	0.0145619i	$-0.00463535\pi$
$728$	303808.i	0.0212457i
$729$	0	0
$730$	0	0
$731$	−2.32486e6	−0.160918
$732$	0	0
$733$	1.72877e7i	1.18844i	0.804302	+	0.594221i	$0.202539\pi$
$733$	1.72877e7i	1.18844i	−0.804302	+	0.594221i	$0.797461\pi$
$734$	1.14461e7	0.784186
$735$	0	0
$736$	313344.	0.0213219
$737$	− 23754.0i	− 0.00161090i
$738$	0	0
$739$	−5.18834e6	−0.349476	−0.174738	−	0.984615i	$-0.555908\pi$
$739$	−5.18834e6	−0.349476	−0.174738	+	0.984615i	$0.555908\pi$
$740$	0	0
$741$	0	0
$742$	3.36821e6i	0.224589i
$743$	4.79572e6i	0.318700i	0.987222	+	0.159350i	$0.0509398\pi$
$743$	4.79572e6i	0.318700i	−0.987222	+	0.159350i	$0.949060\pi$
$744$	0	0
$745$	0	0
$746$	−2.46124e6	−0.161923
$747$	0	0
$748$	− 575424.i	− 0.0376040i
$749$	3.81884e6	0.248730
$750$	0	0
$751$	−1.85654e7	−1.20117	−0.600585	−	0.799561i	$-0.705066\pi$
$751$	−1.85654e7	−1.20117	−0.600585	+	0.799561i	$0.705066\pi$
$752$	281088.i	0.0181258i
$753$	0	0
$754$	−3.18756e6	−0.204188
$755$	0	0
$756$	0	0
$757$	2.82068e7i	1.78902i	0.447053	+	0.894508i	$0.352474\pi$
$757$	2.82068e7i	1.78902i	−0.447053	+	0.894508i	$0.647526\pi$
$758$	2.15951e7i	1.36516i
$759$	0	0
$760$	0	0
$761$	−6.56161e6	−0.410723	−0.205361	−	0.978686i	$-0.565837\pi$
$761$	−6.56161e6	−0.410723	−0.205361	+	0.978686i	$0.565837\pi$
$762$	0	0
$763$	1.01997e7i	0.634273i
$764$	−650208.	−0.0403013
$765$	0	0
$766$	4.34750e6	0.267712
$767$	− 1.77861e6i	− 0.109167i
$768$	0	0
$769$	−2.20930e7	−1.34722	−0.673610	−	0.739087i	$-0.735257\pi$
$769$	−2.20930e7	−1.34722	−0.673610	+	0.739087i	$0.735257\pi$
$770$	0	0
$771$	0	0
$772$	7.91442e6i	0.477942i
$773$	− 3.00787e7i	− 1.81055i	−0.424824	−	0.905276i	$-0.639664\pi$
$773$	− 3.00787e7i	− 1.81055i	0.424824	−	0.905276i	$-0.360336\pi$
$774$	0	0
$775$	0	0
$776$	52672.0	0.00313997
$777$	0	0
$778$	1.39373e7i	0.825523i
$779$	−3.06131e7	−1.80744
$780$	0	0
$781$	−9.04162e6	−0.530418
$782$	− 198288.i	− 0.0115952i
$783$	0	0
$784$	3.73709e6	0.217142
$785$	0	0
$786$	0	0
$787$	− 3.28954e6i	− 0.189321i	−0.995510	−	0.0946605i	$-0.969823\pi$
$787$	− 3.28954e6i	− 0.189321i	0.995510	−	0.0946605i	$-0.0301766\pi$
$788$	8.83747e6i	0.507005i
$789$	0	0
$790$	0	0
$791$	6.50123e6	0.369449
$792$	0	0
$793$	2.20715e6i	0.124638i
$794$	1.30636e7	0.735381
$795$	0	0
$796$	1.09700e7	0.613655
$797$	− 6.71053e6i	− 0.374206i	−0.982340	−	0.187103i	$-0.940090\pi$
$797$	− 6.71053e6i	− 0.374206i	0.982340	−	0.187103i	$-0.0599099\pi$
$798$	0	0
$799$	177876.	0.00985713
$800$	0	0
$801$	0	0
$802$	− 1.70928e7i	− 0.938374i
$803$	7.70473e6i	0.421666i
$804$	0	0
$805$	0	0
$806$	3.47157e6	0.188230
$807$	0	0
$808$	2.16499e6i	0.116662i
$809$	8.74254e6	0.469641	0.234821	−	0.972039i	$-0.424550\pi$
$809$	8.74254e6	0.469641	0.234821	+	0.972039i	$0.424550\pi$
$810$	0	0
$811$	−2.48410e7	−1.32622	−0.663112	−	0.748520i	$-0.730765\pi$
$811$	−2.48410e7	−1.32622	−0.663112	+	0.748520i	$0.730765\pi$
$812$	− 5.93328e6i	− 0.315795i
$813$	0	0
$814$	−7.67410e6	−0.405944
$815$	0	0
$816$	0	0
$817$	2.41814e7i	1.26744i
$818$	− 5.80750e6i	− 0.303463i
$819$	0	0
$820$	0	0
$821$	−2.12219e7	−1.09882	−0.549409	−	0.835554i	$-0.685147\pi$
$821$	−2.12219e7	−1.09882	−0.549409	+	0.835554i	$0.685147\pi$
$822$	0	0
$823$	8.70659e6i	0.448073i	0.974581	+	0.224036i	$0.0719234\pi$
$823$	8.70659e6i	0.448073i	−0.974581	+	0.224036i	$0.928077\pi$
$824$	−8.54042e6	−0.438189
$825$	0	0
$826$	3.31068e6	0.168837
$827$	− 3.71184e7i	− 1.88723i	−0.331040	−	0.943617i	$-0.607400\pi$
$827$	− 3.71184e7i	− 1.88723i	0.331040	−	0.943617i	$-0.392600\pi$
$828$	0	0
$829$	−1.01765e6	−0.0514295	−0.0257147	−	0.999669i	$-0.508186\pi$
$829$	−1.01765e6	−0.0514295	−0.0257147	+	0.999669i	$0.508186\pi$
$830$	0	0
$831$	0	0
$832$	413696.i	0.0207192i
$833$	− 2.36488e6i	− 0.118085i
$834$	0	0
$835$	0	0
$836$	−5.98512e6	−0.296181
$837$	0	0
$838$	− 2.23752e6i	− 0.110067i
$839$	−3.36194e7	−1.64887	−0.824433	−	0.565960i	$-0.808506\pi$
$839$	−3.36194e7	−1.64887	−0.824433	+	0.565960i	$0.808506\pi$
$840$	0	0
$841$	4.17410e7	2.03504
$842$	1.56588e7i	0.761164i
$843$	0	0
$844$	−1.19916e7	−0.579458
$845$	0	0
$846$	0	0
$847$	− 5.25305e6i	− 0.251596i
$848$	4.58650e6i	0.219024i
$849$	0	0
$850$	0	0
$851$	−2.64445e6	−0.125173
$852$	0	0
$853$	3.52574e7i	1.65912i	0.558419	+	0.829559i	$0.311408\pi$
$853$	3.52574e7i	1.65912i	−0.558419	+	0.829559i	$0.688592\pi$
$854$	−4.10836e6	−0.192763
$855$	0	0
$856$	5.20013e6	0.242566
$857$	3.14941e7i	1.46480i	0.680877	+	0.732398i	$0.261599\pi$
$857$	3.14941e7i	1.46480i	−0.680877	+	0.732398i	$0.738401\pi$
$858$	0	0
$859$	−1.19344e7	−0.551848	−0.275924	−	0.961180i	$-0.588984\pi$
$859$	−1.19344e7	−0.551848	−0.275924	+	0.961180i	$0.588984\pi$
$860$	0	0
$861$	0	0
$862$	− 1.43000e7i	− 0.655492i
$863$	8.70442e6i	0.397844i	0.980015	+	0.198922i	$0.0637440\pi$
$863$	8.70442e6i	0.397844i	−0.980015	+	0.198922i	$0.936256\pi$
$864$	0	0
$865$	0	0
$866$	−2.86388e7	−1.29766
$867$	0	0
$868$	6.46194e6i	0.291114i
$869$	−1.53535e7	−0.689697
$870$	0	0
$871$	10807.0	0.000482681 0
$872$	1.38890e7i	0.618555i
$873$	0	0
$874$	−2.06244e6	−0.0913277
$875$	0	0
$876$	0	0
$877$	1.17999e7i	0.518059i	0.965869	+	0.259029i	$0.0834026\pi$
$877$	1.17999e7i	0.518059i	−0.965869	+	0.259029i	$0.916597\pi$
$878$	6.87158e6i	0.300829i
$879$	0	0
$880$	0	0
$881$	2.73840e7	1.18866	0.594330	−	0.804221i	$-0.297417\pi$
$881$	2.73840e7	1.18866	0.594330	+	0.804221i	$0.297417\pi$
$882$	0	0
$883$	8.80577e6i	0.380072i	0.981777	+	0.190036i	$0.0608604\pi$
$883$	8.80577e6i	0.380072i	−0.981777	+	0.190036i	$0.939140\pi$
$884$	261792.	0.0112675
$885$	0	0
$886$	1.35868e7	0.581477
$887$	− 250122.i	− 0.0106744i	−0.999986	−	0.00533719i	$-0.998301\pi$
$887$	− 250122.i	− 0.0106744i	0.999986	−	0.00533719i	$-0.00169889\pi$
$888$	0	0
$889$	1.20343e7	0.510699
$890$	0	0
$891$	0	0
$892$	2.70834e6i	0.113970i
$893$	− 1.85013e6i	− 0.0776379i
$894$	0	0
$895$	0	0
$896$	−770048.	−0.0320441
$897$	0	0
$898$	1.35842e7i	0.562140i
$899$	−6.77988e7	−2.79784
$900$	0	0
$901$	2.90239e6	0.119109
$902$	1.61332e7i	0.660243i
$903$	0	0
$904$	8.85274e6	0.360294
$905$	0	0
$906$	0	0
$907$	− 3.24955e7i	− 1.31161i	−0.754929	−	0.655806i	$-0.772329\pi$
$907$	− 3.24955e7i	− 1.31161i	0.754929	−	0.655806i	$-0.227671\pi$
$908$	− 743808.i	− 0.0299396i
$909$	0	0
$910$	0	0
$911$	−4.24595e7	−1.69504	−0.847518	−	0.530766i	$-0.821904\pi$
$911$	−4.24595e7	−1.69504	−0.847518	+	0.530766i	$0.821904\pi$
$912$	0	0
$913$	2.40945e7i	0.956625i
$914$	−1.81126e7	−0.717157
$915$	0	0
$916$	−1.44184e6	−0.0567778
$917$	− 5.56724e6i	− 0.218634i
$918$	0	0
$919$	−1.41629e7	−0.553176	−0.276588	−	0.960989i	$-0.589204\pi$
$919$	−1.41629e7	−0.553176	−0.276588	+	0.960989i	$0.589204\pi$
$920$	0	0
$921$	0	0
$922$	− 5.11579e6i	− 0.198192i
$923$	− 4.11353e6i	− 0.158932i
$924$	0	0
$925$	0	0
$926$	2.87945e7	1.10352
$927$	0	0
$928$	− 8.07936e6i	− 0.307969i
$929$	4.37292e7	1.66239	0.831194	−	0.555982i	$-0.187658\pi$
$929$	4.37292e7	1.66239	0.831194	+	0.555982i	$0.187658\pi$
$930$	0	0
$931$	−2.45976e7	−0.930077
$932$	− 1.70262e7i	− 0.642063i
$933$	0	0
$934$	−1.93214e7	−0.724721
$935$	0	0
$936$	0	0
$937$	5.73509e6i	0.213398i	0.994291	+	0.106699i	$0.0340282\pi$
$937$	5.73509e6i	0.213398i	−0.994291	+	0.106699i	$0.965972\pi$
$938$	20116.0i	0 0.000746508i
$939$	0	0
$940$	0	0
$941$	3.37395e7	1.24212	0.621061	−	0.783762i	$-0.286702\pi$
$941$	3.37395e7	1.24212	0.621061	+	0.783762i	$0.286702\pi$
$942$	0	0
$943$	5.55941e6i	0.203587i
$944$	4.50816e6	0.164653
$945$	0	0
$946$	1.27437e7	0.462985
$947$	3.07342e7i	1.11365i	0.830631	+	0.556823i	$0.187980\pi$
$947$	3.07342e7i	1.11365i	−0.830631	+	0.556823i	$0.812020\pi$
$948$	0	0
$949$	−3.50531e6	−0.126346
$950$	0	0
$951$	0	0
$952$	487296.i	0.0174261i
$953$	− 2.51847e7i	− 0.898264i	−0.893465	−	0.449132i	$-0.851733\pi$
$953$	− 2.51847e7i	− 0.898264i	0.893465	−	0.449132i	$-0.148267\pi$
$954$	0	0
$955$	0	0
$956$	−1.84253e7	−0.652033
$957$	0	0
$958$	− 2.99460e6i	− 0.105420i
$959$	−621246.	−0.0218131
$960$	0	0
$961$	4.52105e7	1.57918
$962$	− 3.49137e6i	− 0.121635i
$963$	0	0
$964$	−1.36995e7	−0.474801
$965$	0	0
$966$	0	0
$967$	1.44556e7i	0.497130i	0.968615	+	0.248565i	$0.0799589\pi$
$967$	1.44556e7i	0.497130i	−0.968615	+	0.248565i	$0.920041\pi$
$968$	− 7.15309e6i	− 0.245361i
$969$	0	0
$970$	0	0
$971$	1.06974e7	0.364109	0.182054	−	0.983288i	$-0.441725\pi$
$971$	1.06974e7	0.364109	0.182054	+	0.983288i	$0.441725\pi$
$972$	0	0
$973$	1.64848e7i	0.558214i
$974$	−2.06558e7	−0.697660
$975$	0	0
$976$	−5.59437e6	−0.187986
$977$	8.41568e6i	0.282067i	0.990005	+	0.141034i	$0.0450426\pi$
$977$	8.41568e6i	0.282067i	−0.990005	+	0.141034i	$0.954957\pi$
$978$	0	0
$979$	−7.77888e6	−0.259394
$980$	0	0
$981$	0	0
$982$	3.41715e7i	1.13080i
$983$	3.89409e7i	1.28535i	0.766138	+	0.642676i	$0.222176\pi$
$983$	3.89409e7i	1.28535i	−0.766138	+	0.642676i	$0.777824\pi$
$984$	0	0
$985$	0	0
$986$	−5.11272e6	−0.167479
$987$	0	0
$988$	− 2.72296e6i	− 0.0887460i
$989$	4.39141e6	0.142762
$990$	0	0
$991$	4.84592e7	1.56745	0.783723	−	0.621111i	$-0.213318\pi$
$991$	4.84592e7	1.56745	0.783723	+	0.621111i	$0.213318\pi$
$992$	8.79923e6i	0.283900i
$993$	0	0
$994$	7.65686e6	0.245802
$995$	0	0
$996$	0	0
$997$	3.84733e7i	1.22581i	0.790158	+	0.612903i	$0.209998\pi$
$997$	3.84733e7i	1.22581i	−0.790158	+	0.612903i	$0.790002\pi$
$998$	− 1.68235e7i	− 0.534675i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.e.199.1		2
3.2	odd	2		150.6.c.g.49.2		2
5.2	odd	4		450.6.a.p.1.1		1
5.3	odd	4		450.6.a.i.1.1		1
5.4	even	2	inner	450.6.c.e.199.2		2
15.2	even	4		150.6.a.a.1.1	✓	1
15.8	even	4		150.6.a.m.1.1	yes	1
15.14	odd	2		150.6.c.g.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.a.1.1	✓	1	15.2	even	4
150.6.a.m.1.1	yes	1	15.8	even	4
150.6.c.g.49.1		2	15.14	odd	2
150.6.c.g.49.2		2	3.2	odd	2
450.6.a.i.1.1		1	5.3	odd	4
450.6.a.p.1.1		1	5.2	odd	4
450.6.c.e.199.1		2	1.1	even	1	trivial
450.6.c.e.199.2		2	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} +47.0000i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} +47.0000i q^{7} +64.0000i q^{8} -222.000 q^{11} -101.000i q^{13} +188.000 q^{14} +256.000 q^{16} -162.000i q^{17} -1685.00 q^{19} +888.000i q^{22} +306.000i q^{23} -404.000 q^{26} -752.000i q^{28} +7890.00 q^{29} -8593.00 q^{31} -1024.00i q^{32} -648.000 q^{34} +8642.00i q^{37} +6740.00i q^{38} +18168.0 q^{41} -14351.0i q^{43} +3552.00 q^{44} +1224.00 q^{46} +1098.00i q^{47} +14598.0 q^{49} +1616.00i q^{52} +17916.0i q^{53} -3008.00 q^{56} -31560.0i q^{58} +17610.0 q^{59} -21853.0 q^{61} +34372.0i q^{62} -4096.00 q^{64} +107.000i q^{67} +2592.00i q^{68} +40728.0 q^{71} -34706.0i q^{73} +34568.0 q^{74} +26960.0 q^{76} -10434.0i q^{77} +69160.0 q^{79} -72672.0i q^{82} -108534. i q^{83} -57404.0 q^{86} -14208.0i q^{88} +35040.0 q^{89} +4747.00 q^{91} -4896.00i q^{92} +4392.00 q^{94} -823.000i q^{97} -58392.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 444 q^{11} + 376 q^{14} + 512 q^{16} - 3370 q^{19} - 808 q^{26} + 15780 q^{29} - 17186 q^{31} - 1296 q^{34} + 36336 q^{41} + 7104 q^{44} + 2448 q^{46} + 29196 q^{49} - 6016 q^{56} + 35220 q^{59} - 43706 q^{61} - 8192 q^{64} + 81456 q^{71} + 69136 q^{74} + 53920 q^{76} + 138320 q^{79} - 114808 q^{86} + 70080 q^{89} + 9494 q^{91} + 8784 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 444 * q^11 + 376 * q^14 + 512 * q^16 - 3370 * q^19 - 808 * q^26 + 15780 * q^29 - 17186 * q^31 - 1296 * q^34 + 36336 * q^41 + 7104 * q^44 + 2448 * q^46 + 29196 * q^49 - 6016 * q^56 + 35220 * q^59 - 43706 * q^61 - 8192 * q^64 + 81456 * q^71 + 69136 * q^74 + 53920 * q^76 + 138320 * q^79 - 114808 * q^86 + 70080 * q^89 + 9494 * q^91 + 8784 * q^94

Introduction

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