LMFDB - Embedded newform 450.6.a.bc.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 312 $ x^2 - x - 312
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$18.1706$ 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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -233.706 q^{7} +64.0000 q^{8} +O(q^{10})$	$q+4.00000 q^{2} +16.0000 q^{4} -233.706 q^{7} +64.0000 q^{8} +89.7060 q^{11} +209.118 q^{13} -934.824 q^{14} +256.000 q^{16} +226.882 q^{17} -2180.47 q^{19} +358.824 q^{22} +4296.47 q^{23} +836.472 q^{26} -3739.30 q^{28} -3556.06 q^{29} +6902.24 q^{31} +1024.00 q^{32} +907.528 q^{34} +5435.59 q^{37} -8721.89 q^{38} -6484.00 q^{41} +21979.5 q^{43} +1435.30 q^{44} +17185.9 q^{46} -7718.24 q^{47} +37811.5 q^{49} +3345.89 q^{52} +12765.4 q^{53} -14957.2 q^{56} -14224.2 q^{58} +44555.6 q^{59} +11598.2 q^{61} +27608.9 q^{62} +4096.00 q^{64} +3961.29 q^{67} +3630.11 q^{68} -46413.6 q^{71} +61591.2 q^{73} +21742.3 q^{74} -34887.5 q^{76} -20964.8 q^{77} -27877.8 q^{79} -25936.0 q^{82} -50941.0 q^{83} +87918.1 q^{86} +5741.18 q^{88} -5131.19 q^{89} -48872.1 q^{91} +68743.5 q^{92} -30872.9 q^{94} +88435.3 q^{97} +151246. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 114 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 - 114 * q^7 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} - 114 q^{7} + 128 q^{8} - 174 q^{11} - 642 q^{13} - 456 q^{14} + 512 q^{16} + 1514 q^{17} - 120 q^{19} - 696 q^{22} + 4352 q^{23} - 2568 q^{26} - 1824 q^{28} + 2430 q^{29} + 11684 q^{31} + 2048 q^{32} + 6056 q^{34} + 17586 q^{37} - 480 q^{38} - 24984 q^{41} + 24168 q^{43} - 2784 q^{44} + 17408 q^{46} - 13316 q^{47} + 35334 q^{49} - 10272 q^{52} - 13698 q^{53} - 7296 q^{56} + 9720 q^{58} + 23730 q^{59} + 57124 q^{61} + 46736 q^{62} + 8192 q^{64} + 38316 q^{67} + 24224 q^{68} + 11076 q^{71} + 88548 q^{73} + 70344 q^{74} - 1920 q^{76} - 52532 q^{77} + 14220 q^{79} - 99936 q^{82} + 50792 q^{83} + 96672 q^{86} - 11136 q^{88} + 60420 q^{89} - 150756 q^{91} + 69632 q^{92} - 53264 q^{94} + 171216 q^{97} + 141336 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 114 * q^7 + 128 * q^8 - 174 * q^11 - 642 * q^13 - 456 * q^14 + 512 * q^16 + 1514 * q^17 - 120 * q^19 - 696 * q^22 + 4352 * q^23 - 2568 * q^26 - 1824 * q^28 + 2430 * q^29 + 11684 * q^31 + 2048 * q^32 + 6056 * q^34 + 17586 * q^37 - 480 * q^38 - 24984 * q^41 + 24168 * q^43 - 2784 * q^44 + 17408 * q^46 - 13316 * q^47 + 35334 * q^49 - 10272 * q^52 - 13698 * q^53 - 7296 * q^56 + 9720 * q^58 + 23730 * q^59 + 57124 * q^61 + 46736 * q^62 + 8192 * q^64 + 38316 * q^67 + 24224 * q^68 + 11076 * q^71 + 88548 * q^73 + 70344 * q^74 - 1920 * q^76 - 52532 * q^77 + 14220 * q^79 - 99936 * q^82 + 50792 * q^83 + 96672 * q^86 - 11136 * q^88 + 60420 * q^89 - 150756 * q^91 + 69632 * q^92 - 53264 * q^94 + 171216 * q^97 + 141336 * 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
$7$	−233.706	−1.80271	−0.901353	−	0.433086i	$-0.857425\pi$
$7$	−233.706	−1.80271	−0.901353	+	0.433086i	$0.857425\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	89.7060	0.223532	0.111766	−	0.993735i	$-0.464349\pi$
$11$	89.7060	0.223532	0.111766	+	0.993735i	$0.464349\pi$
$12$	0	0
$13$	209.118	0.343189	0.171594	−	0.985168i	$-0.445108\pi$
$13$	209.118	0.343189	0.171594	+	0.985168i	$0.445108\pi$
$14$	−934.824	−1.27471
$15$	0	0
$16$	256.000	0.250000
$17$	226.882	0.190405	0.0952024	−	0.995458i	$-0.469650\pi$
$17$	226.882	0.190405	0.0952024	+	0.995458i	$0.469650\pi$
$18$	0	0
$19$	−2180.47	−1.38569	−0.692846	−	0.721086i	$-0.743643\pi$
$19$	−2180.47	−1.38569	−0.692846	+	0.721086i	$0.743643\pi$
$20$	0	0
$21$	0	0
$22$	358.824	0.158061
$23$	4296.47	1.69353	0.846764	−	0.531969i	$-0.178548\pi$
$23$	4296.47	1.69353	0.846764	+	0.531969i	$0.178548\pi$
$24$	0	0
$25$	0	0
$26$	836.472	0.242671
$27$	0	0
$28$	−3739.30	−0.901353
$29$	−3556.06	−0.785189	−0.392595	−	0.919712i	$-0.628422\pi$
$29$	−3556.06	−0.785189	−0.392595	+	0.919712i	$0.628422\pi$
$30$	0	0
$31$	6902.24	1.28999	0.644994	−	0.764188i	$-0.276860\pi$
$31$	6902.24	1.28999	0.644994	+	0.764188i	$0.276860\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	907.528	0.134637
$35$	0	0
$36$	0	0
$37$	5435.59	0.652743	0.326371	−	0.945242i	$-0.394174\pi$
$37$	5435.59	0.652743	0.326371	+	0.945242i	$0.394174\pi$
$38$	−8721.89	−0.979832
$39$	0	0
$40$	0	0
$41$	−6484.00	−0.602398	−0.301199	−	0.953561i	$-0.597387\pi$
$41$	−6484.00	−0.602398	−0.301199	+	0.953561i	$0.597387\pi$
$42$	0	0
$43$	21979.5	1.81279	0.906395	−	0.422432i	$-0.138823\pi$
$43$	21979.5	1.81279	0.906395	+	0.422432i	$0.138823\pi$
$44$	1435.30	0.111766
$45$	0	0
$46$	17185.9	1.19751
$47$	−7718.24	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$47$	−7718.24	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$48$	0	0
$49$	37811.5	2.24975
$50$	0	0
$51$	0	0
$52$	3345.89	0.171594
$53$	12765.4	0.624228	0.312114	−	0.950045i	$-0.398963\pi$
$53$	12765.4	0.624228	0.312114	+	0.950045i	$0.398963\pi$
$54$	0	0
$55$	0	0
$56$	−14957.2	−0.637353
$57$	0	0
$58$	−14224.2	−0.555212
$59$	44555.6	1.66637	0.833187	−	0.552992i	$-0.186514\pi$
$59$	44555.6	1.66637	0.833187	+	0.552992i	$0.186514\pi$
$60$	0	0
$61$	11598.2	0.399086	0.199543	−	0.979889i	$-0.436054\pi$
$61$	11598.2	0.399086	0.199543	+	0.979889i	$0.436054\pi$
$62$	27608.9	0.912159
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	3961.29	0.107808	0.0539038	−	0.998546i	$-0.482834\pi$
$67$	3961.29	0.107808	0.0539038	+	0.998546i	$0.482834\pi$
$68$	3630.11	0.0952024
$69$	0	0
$70$	0	0
$71$	−46413.6	−1.09270	−0.546348	−	0.837559i	$-0.683982\pi$
$71$	−46413.6	−1.09270	−0.546348	+	0.837559i	$0.683982\pi$
$72$	0	0
$73$	61591.2	1.35273	0.676365	−	0.736566i	$-0.263554\pi$
$73$	61591.2	1.35273	0.676365	+	0.736566i	$0.263554\pi$
$74$	21742.3	0.461559
$75$	0	0
$76$	−34887.5	−0.692846
$77$	−20964.8	−0.402962
$78$	0	0
$79$	−27877.8	−0.502563	−0.251281	−	0.967914i	$-0.580852\pi$
$79$	−27877.8	−0.502563	−0.251281	+	0.967914i	$0.580852\pi$
$80$	0	0
$81$	0	0
$82$	−25936.0	−0.425959
$83$	−50941.0	−0.811656	−0.405828	−	0.913949i	$-0.633017\pi$
$83$	−50941.0	−0.811656	−0.405828	+	0.913949i	$0.633017\pi$
$84$	0	0
$85$	0	0
$86$	87918.1	1.28184
$87$	0	0
$88$	5741.18	0.0790305
$89$	−5131.19	−0.0686663	−0.0343331	−	0.999410i	$-0.510931\pi$
$89$	−5131.19	−0.0686663	−0.0343331	+	0.999410i	$0.510931\pi$
$90$	0	0
$91$	−48872.1	−0.618668
$92$	68743.5	0.846764
$93$	0	0
$94$	−30872.9	−0.360378
$95$	0	0
$96$	0	0
$97$	88435.3	0.954325	0.477162	−	0.878815i	$-0.341665\pi$
$97$	88435.3	0.954325	0.477162	+	0.878815i	$0.341665\pi$
$98$	151246.	1.59081
$99$	0	0
$100$	0	0
$101$	116471.	1.13609	0.568047	−	0.822996i	$-0.307699\pi$
$101$	116471.	1.13609	0.568047	+	0.822996i	$0.307699\pi$
$102$	0	0
$103$	−22369.1	−0.207756	−0.103878	−	0.994590i	$-0.533125\pi$
$103$	−22369.1	−0.207756	−0.103878	+	0.994590i	$0.533125\pi$
$104$	13383.5	0.121335
$105$	0	0
$106$	51061.5	0.441396
$107$	−130626.	−1.10299	−0.551495	−	0.834178i	$-0.685942\pi$
$107$	−130626.	−1.10299	−0.551495	+	0.834178i	$0.685942\pi$
$108$	0	0
$109$	−51443.5	−0.414729	−0.207365	−	0.978264i	$-0.566489\pi$
$109$	−51443.5	−0.414729	−0.207365	+	0.978264i	$0.566489\pi$
$110$	0	0
$111$	0	0
$112$	−59828.7	−0.450676
$113$	138259.	1.01858	0.509291	−	0.860594i	$-0.329908\pi$
$113$	138259.	1.01858	0.509291	+	0.860594i	$0.329908\pi$
$114$	0	0
$115$	0	0
$116$	−56897.0	−0.392595
$117$	0	0
$118$	178222.	1.17830
$119$	−53023.7	−0.343244
$120$	0	0
$121$	−153004.	−0.950033
$122$	46392.9	0.282197
$123$	0	0
$124$	110436.	0.644994
$125$	0	0
$126$	0	0
$127$	199662.	1.09847	0.549233	−	0.835669i	$-0.314920\pi$
$127$	199662.	1.09847	0.549233	+	0.835669i	$0.314920\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	314223.	1.59978	0.799889	−	0.600148i	$-0.204891\pi$
$131$	314223.	1.59978	0.799889	+	0.600148i	$0.204891\pi$
$132$	0	0
$133$	509589.	2.49799
$134$	15845.1	0.0762315
$135$	0	0
$136$	14520.5	0.0673183
$137$	256841.	1.16913	0.584564	−	0.811348i	$-0.301266\pi$
$137$	256841.	1.16913	0.584564	+	0.811348i	$0.301266\pi$
$138$	0	0
$139$	−10777.2	−0.0473118	−0.0236559	−	0.999720i	$-0.507531\pi$
$139$	−10777.2	−0.0473118	−0.0236559	+	0.999720i	$0.507531\pi$
$140$	0	0
$141$	0	0
$142$	−185654.	−0.772652
$143$	18759.1	0.0767136
$144$	0	0
$145$	0	0
$146$	246365.	0.956525
$147$	0	0
$148$	86969.4	0.326371
$149$	−4103.27	−0.0151413	−0.00757067	−	0.999971i	$-0.502410\pi$
$149$	−4103.27	−0.0151413	−0.00757067	+	0.999971i	$0.502410\pi$
$150$	0	0
$151$	107449.	0.383495	0.191748	−	0.981444i	$-0.438585\pi$
$151$	107449.	0.383495	0.191748	+	0.981444i	$0.438585\pi$
$152$	−139550.	−0.489916
$153$	0	0
$154$	−83859.3	−0.284937
$155$	0	0
$156$	0	0
$157$	345824.	1.11971	0.559855	−	0.828591i	$-0.310857\pi$
$157$	345824.	1.11971	0.559855	+	0.828591i	$0.310857\pi$
$158$	−111511.	−0.355366
$159$	0	0
$160$	0	0
$161$	−1.00411e6	−3.05293
$162$	0	0
$163$	−58776.3	−0.173274	−0.0866370	−	0.996240i	$-0.527612\pi$
$163$	−58776.3	−0.173274	−0.0866370	+	0.996240i	$0.527612\pi$
$164$	−103744.	−0.301199
$165$	0	0
$166$	−203764.	−0.573928
$167$	154654.	0.429111	0.214556	−	0.976712i	$-0.431170\pi$
$167$	154654.	0.429111	0.214556	+	0.976712i	$0.431170\pi$
$168$	0	0
$169$	−327563.	−0.882222
$170$	0	0
$171$	0	0
$172$	351673.	0.906395
$173$	171600.	0.435916	0.217958	−	0.975958i	$-0.430060\pi$
$173$	171600.	0.435916	0.217958	+	0.975958i	$0.430060\pi$
$174$	0	0
$175$	0	0
$176$	22964.7	0.0558830
$177$	0	0
$178$	−20524.8	−0.0485544
$179$	−595571.	−1.38932	−0.694658	−	0.719340i	$-0.744444\pi$
$179$	−595571.	−1.38932	−0.694658	+	0.719340i	$0.744444\pi$
$180$	0	0
$181$	144518.	0.327889	0.163944	−	0.986470i	$-0.447578\pi$
$181$	144518.	0.327889	0.163944	+	0.986470i	$0.447578\pi$
$182$	−195488.	−0.437464
$183$	0	0
$184$	274974.	0.598753
$185$	0	0
$186$	0	0
$187$	20352.7	0.0425616
$188$	−123492.	−0.254826
$189$	0	0
$190$	0	0
$191$	−815918.	−1.61831	−0.809157	−	0.587592i	$-0.800076\pi$
$191$	−815918.	−1.61831	−0.809157	+	0.587592i	$0.800076\pi$
$192$	0	0
$193$	665030.	1.28513	0.642566	−	0.766230i	$-0.277870\pi$
$193$	665030.	1.28513	0.642566	+	0.766230i	$0.277870\pi$
$194$	353741.	0.674810
$195$	0	0
$196$	604984.	1.12487
$197$	332972.	0.611283	0.305642	−	0.952147i	$-0.401129\pi$
$197$	332972.	0.611283	0.305642	+	0.952147i	$0.401129\pi$
$198$	0	0
$199$	−19565.3	−0.0350231	−0.0175115	−	0.999847i	$-0.505574\pi$
$199$	−19565.3	−0.0350231	−0.0175115	+	0.999847i	$0.505574\pi$
$200$	0	0
$201$	0	0
$202$	465884.	0.803339
$203$	831073.	1.41546
$204$	0	0
$205$	0	0
$206$	−89476.2	−0.146906
$207$	0	0
$208$	53534.2	0.0857971
$209$	−195601.	−0.309746
$210$	0	0
$211$	−669474.	−1.03521	−0.517604	−	0.855620i	$-0.673176\pi$
$211$	−669474.	−1.03521	−0.517604	+	0.855620i	$0.673176\pi$
$212$	204246.	0.312114
$213$	0	0
$214$	−522506.	−0.779932
$215$	0	0
$216$	0	0
$217$	−1.61309e6	−2.32547
$218$	−205774.	−0.293258
$219$	0	0
$220$	0	0
$221$	47445.1	0.0653448
$222$	0	0
$223$	−841915.	−1.13372	−0.566860	−	0.823814i	$-0.691842\pi$
$223$	−841915.	−1.13372	−0.566860	+	0.823814i	$0.691842\pi$
$224$	−239315.	−0.318676
$225$	0	0
$226$	553034.	0.720246
$227$	−106296.	−0.136916	−0.0684580	−	0.997654i	$-0.521808\pi$
$227$	−106296.	−0.136916	−0.0684580	+	0.997654i	$0.521808\pi$
$228$	0	0
$229$	1.52931e6	1.92711	0.963553	−	0.267518i	$-0.0862033\pi$
$229$	1.52931e6	1.92711	0.963553	+	0.267518i	$0.0862033\pi$
$230$	0	0
$231$	0	0
$232$	−227588.	−0.277606
$233$	−362144.	−0.437011	−0.218505	−	0.975836i	$-0.570118\pi$
$233$	−362144.	−0.437011	−0.218505	+	0.975836i	$0.570118\pi$
$234$	0	0
$235$	0	0
$236$	712890.	0.833187
$237$	0	0
$238$	−212095.	−0.242710
$239$	−38495.6	−0.0435929	−0.0217965	−	0.999762i	$-0.506939\pi$
$239$	−38495.6	−0.0435929	−0.0217965	+	0.999762i	$0.506939\pi$
$240$	0	0
$241$	567433.	0.629320	0.314660	−	0.949204i	$-0.398109\pi$
$241$	567433.	0.629320	0.314660	+	0.949204i	$0.398109\pi$
$242$	−612015.	−0.671775
$243$	0	0
$244$	185572.	0.199543
$245$	0	0
$246$	0	0
$247$	−455976.	−0.475553
$248$	441743.	0.456080
$249$	0	0
$250$	0	0
$251$	−19067.0	−0.0191028	−0.00955141	−	0.999954i	$-0.503040\pi$
$251$	−19067.0	−0.0191028	−0.00955141	+	0.999954i	$0.503040\pi$
$252$	0	0
$253$	385419.	0.378558
$254$	798649.	0.776733
$255$	0	0
$256$	65536.0	0.0625000
$257$	−854054.	−0.806589	−0.403295	−	0.915070i	$-0.632135\pi$
$257$	−854054.	−0.806589	−0.403295	+	0.915070i	$0.632135\pi$
$258$	0	0
$259$	−1.27033e6	−1.17670
$260$	0	0
$261$	0	0
$262$	1.25689e6	1.13121
$263$	−755220.	−0.673262	−0.336631	−	0.941637i	$-0.609288\pi$
$263$	−755220.	−0.673262	−0.336631	+	0.941637i	$0.609288\pi$
$264$	0	0
$265$	0	0
$266$	2.03836e6	1.76635
$267$	0	0
$268$	63380.6	0.0539038
$269$	256615.	0.216222	0.108111	−	0.994139i	$-0.465520\pi$
$269$	256615.	0.216222	0.108111	+	0.994139i	$0.465520\pi$
$270$	0	0
$271$	598518.	0.495056	0.247528	−	0.968881i	$-0.420382\pi$
$271$	598518.	0.495056	0.247528	+	0.968881i	$0.420382\pi$
$272$	58081.8	0.0476012
$273$	0	0
$274$	1.02736e6	0.826698
$275$	0	0
$276$	0	0
$277$	425045.	0.332840	0.166420	−	0.986055i	$-0.446779\pi$
$277$	425045.	0.332840	0.166420	+	0.986055i	$0.446779\pi$
$278$	−43108.9	−0.0334545
$279$	0	0
$280$	0	0
$281$	949963.	0.717696	0.358848	−	0.933396i	$-0.383170\pi$
$281$	949963.	0.717696	0.358848	+	0.933396i	$0.383170\pi$
$282$	0	0
$283$	−2.12422e6	−1.57664	−0.788321	−	0.615264i	$-0.789050\pi$
$283$	−2.12422e6	−1.57664	−0.788321	+	0.615264i	$0.789050\pi$
$284$	−742617.	−0.546348
$285$	0	0
$286$	75036.5	0.0542447
$287$	1.51535e6	1.08595
$288$	0	0
$289$	−1.36838e6	−0.963746
$290$	0	0
$291$	0	0
$292$	985459.	0.676365
$293$	−1.17556e6	−0.799977	−0.399988	−	0.916520i	$-0.630986\pi$
$293$	−1.17556e6	−0.799977	−0.399988	+	0.916520i	$0.630986\pi$
$294$	0	0
$295$	0	0
$296$	347878.	0.230779
$297$	0	0
$298$	−16413.1	−0.0107065
$299$	898469.	0.581199
$300$	0	0
$301$	−5.13675e6	−3.26792
$302$	429796.	0.271172
$303$	0	0
$304$	−558201.	−0.346423
$305$	0	0
$306$	0	0
$307$	−588520.	−0.356382	−0.178191	−	0.983996i	$-0.557024\pi$
$307$	−588520.	−0.356382	−0.178191	+	0.983996i	$0.557024\pi$
$308$	−335437.	−0.201481
$309$	0	0
$310$	0	0
$311$	−1.61863e6	−0.948958	−0.474479	−	0.880267i	$-0.657363\pi$
$311$	−1.61863e6	−0.948958	−0.474479	+	0.880267i	$0.657363\pi$
$312$	0	0
$313$	1.58901e6	0.916783	0.458392	−	0.888750i	$-0.348426\pi$
$313$	1.58901e6	0.916783	0.458392	+	0.888750i	$0.348426\pi$
$314$	1.38329e6	0.791754
$315$	0	0
$316$	−446045.	−0.251281
$317$	−2.35137e6	−1.31423	−0.657117	−	0.753789i	$-0.728224\pi$
$317$	−2.35137e6	−1.31423	−0.657117	+	0.753789i	$0.728224\pi$
$318$	0	0
$319$	−319000.	−0.175515
$320$	0	0
$321$	0	0
$322$	−4.01644e6	−2.15875
$323$	−494710.	−0.263842
$324$	0	0
$325$	0	0
$326$	−235105.	−0.122523
$327$	0	0
$328$	−414976.	−0.212980
$329$	1.80380e6	0.918752
$330$	0	0
$331$	480922.	0.241271	0.120635	−	0.992697i	$-0.461507\pi$
$331$	480922.	0.241271	0.120635	+	0.992697i	$0.461507\pi$
$332$	−815056.	−0.405828
$333$	0	0
$334$	618616.	0.303427
$335$	0	0
$336$	0	0
$337$	981449.	0.470753	0.235376	−	0.971904i	$-0.424368\pi$
$337$	981449.	0.470753	0.235376	+	0.971904i	$0.424368\pi$
$338$	−1.31025e6	−0.623825
$339$	0	0
$340$	0	0
$341$	619172.	0.288353
$342$	0	0
$343$	−4.90887e6	−2.25292
$344$	1.40669e6	0.640918
$345$	0	0
$346$	686400.	0.308239
$347$	1.91647e6	0.854432	0.427216	−	0.904150i	$-0.359494\pi$
$347$	1.91647e6	0.854432	0.427216	+	0.904150i	$0.359494\pi$
$348$	0	0
$349$	1.22795e6	0.539658	0.269829	−	0.962908i	$-0.413033\pi$
$349$	1.22795e6	0.539658	0.269829	+	0.962908i	$0.413033\pi$
$350$	0	0
$351$	0	0
$352$	91858.9	0.0395152
$353$	−547823.	−0.233993	−0.116997	−	0.993132i	$-0.537327\pi$
$353$	−547823.	−0.233993	−0.116997	+	0.993132i	$0.537327\pi$
$354$	0	0
$355$	0	0
$356$	−82099.1	−0.0343331
$357$	0	0
$358$	−2.38228e6	−0.982395
$359$	−1.02314e6	−0.418984	−0.209492	−	0.977810i	$-0.567181\pi$
$359$	−1.02314e6	−0.418984	−0.209492	+	0.977810i	$0.567181\pi$
$360$	0	0
$361$	2.27836e6	0.920140
$362$	578073.	0.231852
$363$	0	0
$364$	−781954.	−0.309334
$365$	0	0
$366$	0	0
$367$	3.05305e6	1.18323	0.591614	−	0.806221i	$-0.298491\pi$
$367$	3.05305e6	1.18323	0.591614	+	0.806221i	$0.298491\pi$
$368$	1.09990e6	0.423382
$369$	0	0
$370$	0	0
$371$	−2.98334e6	−1.12530
$372$	0	0
$373$	−3.69411e6	−1.37479	−0.687397	−	0.726282i	$-0.741247\pi$
$373$	−3.69411e6	−1.37479	−0.687397	+	0.726282i	$0.741247\pi$
$374$	81410.7	0.0300956
$375$	0	0
$376$	−493967.	−0.180189
$377$	−743636.	−0.269468
$378$	0	0
$379$	4.48524e6	1.60394	0.801969	−	0.597365i	$-0.203786\pi$
$379$	4.48524e6	1.60394	0.801969	+	0.597365i	$0.203786\pi$
$380$	0	0
$381$	0	0
$382$	−3.26367e6	−1.14432
$383$	4.53062e6	1.57819	0.789097	−	0.614269i	$-0.210549\pi$
$383$	4.53062e6	1.57819	0.789097	+	0.614269i	$0.210549\pi$
$384$	0	0
$385$	0	0
$386$	2.66012e6	0.908726
$387$	0	0
$388$	1.41496e6	0.477162
$389$	−2.73557e6	−0.916588	−0.458294	−	0.888801i	$-0.651539\pi$
$389$	−2.73557e6	−0.916588	−0.458294	+	0.888801i	$0.651539\pi$
$390$	0	0
$391$	974792.	0.322456
$392$	2.41993e6	0.795405
$393$	0	0
$394$	1.33189e6	0.432242
$395$	0	0
$396$	0	0
$397$	−1.54948e6	−0.493413	−0.246707	−	0.969090i	$-0.579348\pi$
$397$	−1.54948e6	−0.493413	−0.246707	+	0.969090i	$0.579348\pi$
$398$	−78261.3	−0.0247651
$399$	0	0
$400$	0	0
$401$	−4.46136e6	−1.38550	−0.692749	−	0.721179i	$-0.743601\pi$
$401$	−4.46136e6	−1.38550	−0.692749	+	0.721179i	$0.743601\pi$
$402$	0	0
$403$	1.44338e6	0.442709
$404$	1.86353e6	0.568047
$405$	0	0
$406$	3.32429e6	1.00088
$407$	487605.	0.145909
$408$	0	0
$409$	−2.36815e6	−0.700003	−0.350002	−	0.936749i	$-0.613819\pi$
$409$	−2.36815e6	−0.700003	−0.350002	+	0.936749i	$0.613819\pi$
$410$	0	0
$411$	0	0
$412$	−357905.	−0.103878
$413$	−1.04129e7	−3.00398
$414$	0	0
$415$	0	0
$416$	214137.	0.0606677
$417$	0	0
$418$	−782405.	−0.219024
$419$	−5.76363e6	−1.60384	−0.801920	−	0.597432i	$-0.796188\pi$
$419$	−5.76363e6	−1.60384	−0.801920	+	0.597432i	$0.796188\pi$
$420$	0	0
$421$	3.26572e6	0.897995	0.448998	−	0.893533i	$-0.351781\pi$
$421$	3.26572e6	0.897995	0.448998	+	0.893533i	$0.351781\pi$
$422$	−2.67789e6	−0.732002
$423$	0	0
$424$	816983.	0.220698
$425$	0	0
$426$	0	0
$427$	−2.71057e6	−0.719435
$428$	−2.09002e6	−0.551495
$429$	0	0
$430$	0	0
$431$	5.62110e6	1.45756	0.728782	−	0.684745i	$-0.240087\pi$
$431$	5.62110e6	1.45756	0.728782	+	0.684745i	$0.240087\pi$
$432$	0	0
$433$	3.18656e6	0.816774	0.408387	−	0.912809i	$-0.366091\pi$
$433$	3.18656e6	0.816774	0.408387	+	0.912809i	$0.366091\pi$
$434$	−6.45237e6	−1.64435
$435$	0	0
$436$	−823097.	−0.207365
$437$	−9.36833e6	−2.34671
$438$	0	0
$439$	−7.41102e6	−1.83534	−0.917671	−	0.397342i	$-0.869933\pi$
$439$	−7.41102e6	−1.83534	−0.917671	+	0.397342i	$0.869933\pi$
$440$	0	0
$441$	0	0
$442$	189780.	0.0462057
$443$	2.62862e6	0.636382	0.318191	−	0.948027i	$-0.396925\pi$
$443$	2.62862e6	0.636382	0.318191	+	0.948027i	$0.396925\pi$
$444$	0	0
$445$	0	0
$446$	−3.36766e6	−0.801662
$447$	0	0
$448$	−957260.	−0.225338
$449$	−71764.4	−0.0167994	−0.00839969	−	0.999965i	$-0.502674\pi$
$449$	−71764.4	−0.0167994	−0.00839969	+	0.999965i	$0.502674\pi$
$450$	0	0
$451$	−581653.	−0.134655
$452$	2.21214e6	0.509291
$453$	0	0
$454$	−425186.	−0.0968142
$455$	0	0
$456$	0	0
$457$	3.51471e6	0.787225	0.393613	−	0.919276i	$-0.371225\pi$
$457$	3.51471e6	0.787225	0.393613	+	0.919276i	$0.371225\pi$
$458$	6.11722e6	1.36267
$459$	0	0
$460$	0	0
$461$	−3.35569e6	−0.735410	−0.367705	−	0.929943i	$-0.619856\pi$
$461$	−3.35569e6	−0.735410	−0.367705	+	0.929943i	$0.619856\pi$
$462$	0	0
$463$	3.47269e6	0.752860	0.376430	−	0.926445i	$-0.377152\pi$
$463$	3.47269e6	0.752860	0.376430	+	0.926445i	$0.377152\pi$
$464$	−910352.	−0.196297
$465$	0	0
$466$	−1.44858e6	−0.309013
$467$	−2.35495e6	−0.499678	−0.249839	−	0.968287i	$-0.580378\pi$
$467$	−2.35495e6	−0.499678	−0.249839	+	0.968287i	$0.580378\pi$
$468$	0	0
$469$	−925776.	−0.194345
$470$	0	0
$471$	0	0
$472$	2.85156e6	0.589152
$473$	1.97170e6	0.405216
$474$	0	0
$475$	0	0
$476$	−848379.	−0.171622
$477$	0	0
$478$	−153982.	−0.0308249
$479$	187771.	0.0373929	0.0186964	−	0.999825i	$-0.494048\pi$
$479$	187771.	0.0373929	0.0186964	+	0.999825i	$0.494048\pi$
$480$	0	0
$481$	1.13668e6	0.224014
$482$	2.26973e6	0.444997
$483$	0	0
$484$	−2.44806e6	−0.475017
$485$	0	0
$486$	0	0
$487$	−2.01505e6	−0.385002	−0.192501	−	0.981297i	$-0.561660\pi$
$487$	−2.01505e6	−0.385002	−0.192501	+	0.981297i	$0.561660\pi$
$488$	742287.	0.141098
$489$	0	0
$490$	0	0
$491$	−5.80814e6	−1.08726	−0.543630	−	0.839325i	$-0.682951\pi$
$491$	−5.80814e6	−1.08726	−0.543630	+	0.839325i	$0.682951\pi$
$492$	0	0
$493$	−806807.	−0.149504
$494$	−1.82390e6	−0.336267
$495$	0	0
$496$	1.76697e6	0.322497
$497$	1.08471e7	1.96981
$498$	0	0
$499$	580611.	0.104384	0.0521920	−	0.998637i	$-0.483379\pi$
$499$	580611.	0.104384	0.0521920	+	0.998637i	$0.483379\pi$
$500$	0	0
$501$	0	0
$502$	−76267.9	−0.0135077
$503$	557751.	0.0982926	0.0491463	−	0.998792i	$-0.484350\pi$
$503$	557751.	0.0982926	0.0491463	+	0.998792i	$0.484350\pi$
$504$	0	0
$505$	0	0
$506$	1.54168e6	0.267681
$507$	0	0
$508$	3.19460e6	0.549233
$509$	2.63318e6	0.450491	0.225246	−	0.974302i	$-0.427682\pi$
$509$	2.63318e6	0.450491	0.225246	+	0.974302i	$0.427682\pi$
$510$	0	0
$511$	−1.43942e7	−2.43858
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.41621e6	−0.570345
$515$	0	0
$516$	0	0
$517$	−692372.	−0.113923
$518$	−5.08132e6	−0.832055
$519$	0	0
$520$	0	0
$521$	6.78129e6	1.09451	0.547253	−	0.836967i	$-0.315674\pi$
$521$	6.78129e6	1.09451	0.547253	+	0.836967i	$0.315674\pi$
$522$	0	0
$523$	1.55863e6	0.249166	0.124583	−	0.992209i	$-0.460241\pi$
$523$	1.55863e6	0.249166	0.124583	+	0.992209i	$0.460241\pi$
$524$	5.02757e6	0.799889
$525$	0	0
$526$	−3.02088e6	−0.476068
$527$	1.56599e6	0.245620
$528$	0	0
$529$	1.20233e7	1.86804
$530$	0	0
$531$	0	0
$532$	8.15343e6	1.24900
$533$	−1.35592e6	−0.206736
$534$	0	0
$535$	0	0
$536$	253522.	0.0381157
$537$	0	0
$538$	1.02646e6	0.152892
$539$	3.39192e6	0.502890
$540$	0	0
$541$	7.29465e6	1.07155	0.535774	−	0.844362i	$-0.320020\pi$
$541$	7.29465e6	1.07155	0.535774	+	0.844362i	$0.320020\pi$
$542$	2.39407e6	0.350057
$543$	0	0
$544$	232327.	0.0336591
$545$	0	0
$546$	0	0
$547$	−5.15944e6	−0.737283	−0.368642	−	0.929572i	$-0.620177\pi$
$547$	−5.15944e6	−0.737283	−0.368642	+	0.929572i	$0.620177\pi$
$548$	4.10945e6	0.584564
$549$	0	0
$550$	0	0
$551$	7.75389e6	1.08803
$552$	0	0
$553$	6.51520e6	0.905973
$554$	1.70018e6	0.235353
$555$	0	0
$556$	−172436.	−0.0236559
$557$	2.18378e6	0.298244	0.149122	−	0.988819i	$-0.452355\pi$
$557$	2.18378e6	0.298244	0.149122	+	0.988819i	$0.452355\pi$
$558$	0	0
$559$	4.59631e6	0.622129
$560$	0	0
$561$	0	0
$562$	3.79985e6	0.507488
$563$	1.13003e7	1.50252	0.751259	−	0.660007i	$-0.229447\pi$
$563$	1.13003e7	1.50252	0.751259	+	0.660007i	$0.229447\pi$
$564$	0	0
$565$	0	0
$566$	−8.49688e6	−1.11485
$567$	0	0
$568$	−2.97047e6	−0.386326
$569$	1.01686e7	1.31669	0.658343	−	0.752718i	$-0.271258\pi$
$569$	1.01686e7	1.31669	0.658343	+	0.752718i	$0.271258\pi$
$570$	0	0
$571$	−284871.	−0.0365644	−0.0182822	−	0.999833i	$-0.505820\pi$
$571$	−284871.	−0.0365644	−0.0182822	+	0.999833i	$0.505820\pi$
$572$	300146.	0.0383568
$573$	0	0
$574$	6.06140e6	0.767879
$575$	0	0
$576$	0	0
$577$	9.14119e6	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$577$	9.14119e6	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$578$	−5.47353e6	−0.681471
$579$	0	0
$580$	0	0
$581$	1.19052e7	1.46318
$582$	0	0
$583$	1.14513e6	0.139535
$584$	3.94184e6	0.478263
$585$	0	0
$586$	−4.70226e6	−0.565669
$587$	1.85347e6	0.222019	0.111010	−	0.993819i	$-0.464592\pi$
$587$	1.85347e6	0.222019	0.111010	+	0.993819i	$0.464592\pi$
$588$	0	0
$589$	−1.50501e7	−1.78752
$590$	0	0
$591$	0	0
$592$	1.39151e6	0.163186
$593$	1.30967e7	1.52941	0.764705	−	0.644380i	$-0.222884\pi$
$593$	1.30967e7	1.52941	0.764705	+	0.644380i	$0.222884\pi$
$594$	0	0
$595$	0	0
$596$	−65652.3	−0.00757067
$597$	0	0
$598$	3.59388e6	0.410970
$599$	7.13874e6	0.812933	0.406466	−	0.913666i	$-0.366761\pi$
$599$	7.13874e6	0.812933	0.406466	+	0.913666i	$0.366761\pi$
$600$	0	0
$601$	−7.27856e6	−0.821976	−0.410988	−	0.911641i	$-0.634816\pi$
$601$	−7.27856e6	−0.821976	−0.410988	+	0.911641i	$0.634816\pi$
$602$	−2.05470e7	−2.31077
$603$	0	0
$604$	1.71918e6	0.191748
$605$	0	0
$606$	0	0
$607$	−466696.	−0.0514117	−0.0257058	−	0.999670i	$-0.508183\pi$
$607$	−466696.	−0.0514117	−0.0257058	+	0.999670i	$0.508183\pi$
$608$	−2.23280e6	−0.244958
$609$	0	0
$610$	0	0
$611$	−1.61402e6	−0.174907
$612$	0	0
$613$	9.21342e6	0.990307	0.495153	−	0.868806i	$-0.335112\pi$
$613$	9.21342e6	0.990307	0.495153	+	0.868806i	$0.335112\pi$
$614$	−2.35408e6	−0.252000
$615$	0	0
$616$	−1.34175e6	−0.142469
$617$	8.53638e6	0.902736	0.451368	−	0.892338i	$-0.350936\pi$
$617$	8.53638e6	0.902736	0.451368	+	0.892338i	$0.350936\pi$
$618$	0	0
$619$	7.72476e6	0.810323	0.405162	−	0.914245i	$-0.367215\pi$
$619$	7.72476e6	0.810323	0.405162	+	0.914245i	$0.367215\pi$
$620$	0	0
$621$	0	0
$622$	−6.47452e6	−0.671014
$623$	1.19919e6	0.123785
$624$	0	0
$625$	0	0
$626$	6.35605e6	0.648264
$627$	0	0
$628$	5.53318e6	0.559855
$629$	1.23324e6	0.124285
$630$	0	0
$631$	1.50119e7	1.50094	0.750470	−	0.660904i	$-0.229827\pi$
$631$	1.50119e7	1.50094	0.750470	+	0.660904i	$0.229827\pi$
$632$	−1.78418e6	−0.177683
$633$	0	0
$634$	−9.40547e6	−0.929303
$635$	0	0
$636$	0	0
$637$	7.90706e6	0.772087
$638$	−1.27600e6	−0.124108
$639$	0	0
$640$	0	0
$641$	8.60718e6	0.827401	0.413700	−	0.910413i	$-0.364236\pi$
$641$	8.60718e6	0.827401	0.413700	+	0.910413i	$0.364236\pi$
$642$	0	0
$643$	1.97196e7	1.88092	0.940460	−	0.339904i	$-0.110395\pi$
$643$	1.97196e7	1.88092	0.940460	+	0.339904i	$0.110395\pi$
$644$	−1.60658e7	−1.52647
$645$	0	0
$646$	−1.97884e6	−0.186565
$647$	−541311.	−0.0508378	−0.0254189	−	0.999677i	$-0.508092\pi$
$647$	−541311.	−0.0508378	−0.0254189	+	0.999677i	$0.508092\pi$
$648$	0	0
$649$	3.99690e6	0.372488
$650$	0	0
$651$	0	0
$652$	−940421.	−0.0866370
$653$	−9.28231e6	−0.851870	−0.425935	−	0.904754i	$-0.640055\pi$
$653$	−9.28231e6	−0.851870	−0.425935	+	0.904754i	$0.640055\pi$
$654$	0	0
$655$	0	0
$656$	−1.65990e6	−0.150599
$657$	0	0
$658$	7.21519e6	0.649655
$659$	−1.79964e7	−1.61425	−0.807127	−	0.590378i	$-0.798979\pi$
$659$	−1.79964e7	−1.61425	−0.807127	+	0.590378i	$0.798979\pi$
$660$	0	0
$661$	7.78607e6	0.693130	0.346565	−	0.938026i	$-0.387348\pi$
$661$	7.78607e6	0.693130	0.346565	+	0.938026i	$0.387348\pi$
$662$	1.92369e6	0.170604
$663$	0	0
$664$	−3.26022e6	−0.286964
$665$	0	0
$666$	0	0
$667$	−1.52785e7	−1.32974
$668$	2.47446e6	0.214556
$669$	0	0
$670$	0	0
$671$	1.04043e6	0.0892086
$672$	0	0
$673$	−1.35540e7	−1.15353	−0.576766	−	0.816910i	$-0.695685\pi$
$673$	−1.35540e7	−1.15353	−0.576766	+	0.816910i	$0.695685\pi$
$674$	3.92579e6	0.332872
$675$	0	0
$676$	−5.24100e6	−0.441111
$677$	1.46008e7	1.22435	0.612173	−	0.790724i	$-0.290296\pi$
$677$	1.46008e7	1.22435	0.612173	+	0.790724i	$0.290296\pi$
$678$	0	0
$679$	−2.06679e7	−1.72037
$680$	0	0
$681$	0	0
$682$	2.47669e6	0.203897
$683$	−9.71085e6	−0.796536	−0.398268	−	0.917269i	$-0.630389\pi$
$683$	−9.71085e6	−0.796536	−0.398268	+	0.917269i	$0.630389\pi$
$684$	0	0
$685$	0	0
$686$	−1.96355e7	−1.59306
$687$	0	0
$688$	5.62676e6	0.453197
$689$	2.66947e6	0.214228
$690$	0	0
$691$	1.30382e7	1.03878	0.519389	−	0.854538i	$-0.326159\pi$
$691$	1.30382e7	1.03878	0.519389	+	0.854538i	$0.326159\pi$
$692$	2.74560e6	0.217958
$693$	0	0
$694$	7.66587e6	0.604175
$695$	0	0
$696$	0	0
$697$	−1.47110e6	−0.114699
$698$	4.91182e6	0.381596
$699$	0	0
$700$	0	0
$701$	−4.65952e6	−0.358134	−0.179067	−	0.983837i	$-0.557308\pi$
$701$	−4.65952e6	−0.358134	−0.179067	+	0.983837i	$0.557308\pi$
$702$	0	0
$703$	−1.18521e7	−0.904500
$704$	367436.	0.0279415
$705$	0	0
$706$	−2.19129e6	−0.165458
$707$	−2.72199e7	−2.04804
$708$	0	0
$709$	1.94059e7	1.44983	0.724916	−	0.688837i	$-0.241878\pi$
$709$	1.94059e7	1.44983	0.724916	+	0.688837i	$0.241878\pi$
$710$	0	0
$711$	0	0
$712$	−328396.	−0.0242772
$713$	2.96553e7	2.18463
$714$	0	0
$715$	0	0
$716$	−9.52914e6	−0.694658
$717$	0	0
$718$	−4.09254e6	−0.296266
$719$	−1.20100e7	−0.866405	−0.433202	−	0.901297i	$-0.642616\pi$
$719$	−1.20100e7	−0.866405	−0.433202	+	0.901297i	$0.642616\pi$
$720$	0	0
$721$	5.22778e6	0.374524
$722$	9.11343e6	0.650637
$723$	0	0
$724$	2.31229e6	0.163944
$725$	0	0
$726$	0	0
$727$	1.14993e7	0.806927	0.403463	−	0.914996i	$-0.367806\pi$
$727$	1.14993e7	0.806927	0.403463	+	0.914996i	$0.367806\pi$
$728$	−3.12781e6	−0.218732
$729$	0	0
$730$	0	0
$731$	4.98676e6	0.345164
$732$	0	0
$733$	−2.04473e7	−1.40565	−0.702823	−	0.711364i	$-0.748078\pi$
$733$	−2.04473e7	−1.40565	−0.702823	+	0.711364i	$0.748078\pi$
$734$	1.22122e7	0.836668
$735$	0	0
$736$	4.39959e6	0.299376
$737$	355351.	0.0240984
$738$	0	0
$739$	220621.	0.0148606	0.00743030	−	0.999972i	$-0.497635\pi$
$739$	220621.	0.0148606	0.00743030	+	0.999972i	$0.497635\pi$
$740$	0	0
$741$	0	0
$742$	−1.19334e7	−0.795707
$743$	−2.51435e7	−1.67091	−0.835456	−	0.549558i	$-0.814796\pi$
$743$	−2.51435e7	−1.67091	−0.835456	+	0.549558i	$0.814796\pi$
$744$	0	0
$745$	0	0
$746$	−1.47764e7	−0.972126
$747$	0	0
$748$	325643.	0.0212808
$749$	3.05282e7	1.98837
$750$	0	0
$751$	−667547.	−0.0431899	−0.0215950	−	0.999767i	$-0.506874\pi$
$751$	−667547.	−0.0431899	−0.0215950	+	0.999767i	$0.506874\pi$
$752$	−1.97587e6	−0.127413
$753$	0	0
$754$	−2.97454e6	−0.190543
$755$	0	0
$756$	0	0
$757$	1.62490e7	1.03059	0.515297	−	0.857012i	$-0.327682\pi$
$757$	1.62490e7	1.03059	0.515297	+	0.857012i	$0.327682\pi$
$758$	1.79410e7	1.13416
$759$	0	0
$760$	0	0
$761$	1.43662e7	0.899253	0.449626	−	0.893217i	$-0.351557\pi$
$761$	1.43662e7	0.899253	0.449626	+	0.893217i	$0.351557\pi$
$762$	0	0
$763$	1.20227e7	0.747634
$764$	−1.30547e7	−0.809157
$765$	0	0
$766$	1.81225e7	1.11595
$767$	9.31737e6	0.571880
$768$	0	0
$769$	1.92705e7	1.17511	0.587553	−	0.809185i	$-0.300091\pi$
$769$	1.92705e7	1.17511	0.587553	+	0.809185i	$0.300091\pi$
$770$	0	0
$771$	0	0
$772$	1.06405e7	0.642566
$773$	−2.40693e7	−1.44882	−0.724410	−	0.689370i	$-0.757888\pi$
$773$	−2.40693e7	−1.44882	−0.724410	+	0.689370i	$0.757888\pi$
$774$	0	0
$775$	0	0
$776$	5.65986e6	0.337405
$777$	0	0
$778$	−1.09423e7	−0.648125
$779$	1.41382e7	0.834737
$780$	0	0
$781$	−4.16357e6	−0.244252
$782$	3.89917e6	0.228011
$783$	0	0
$784$	9.67974e6	0.562436
$785$	0	0
$786$	0	0
$787$	3.76223e6	0.216525	0.108263	−	0.994122i	$-0.465471\pi$
$787$	3.76223e6	0.216525	0.108263	+	0.994122i	$0.465471\pi$
$788$	5.32756e6	0.305642
$789$	0	0
$790$	0	0
$791$	−3.23119e7	−1.83620
$792$	0	0
$793$	2.42540e6	0.136962
$794$	−6.19794e6	−0.348896
$795$	0	0
$796$	−313045.	−0.0175115
$797$	−9.10423e6	−0.507689	−0.253844	−	0.967245i	$-0.581695\pi$
$797$	−9.10423e6	−0.507689	−0.253844	+	0.967245i	$0.581695\pi$
$798$	0	0
$799$	−1.75113e6	−0.0970401
$800$	0	0
$801$	0	0
$802$	−1.78454e7	−0.979695
$803$	5.52510e6	0.302379
$804$	0	0
$805$	0	0
$806$	5.77352e6	0.313043
$807$	0	0
$808$	7.45414e6	0.401670
$809$	−1.20232e7	−0.645877	−0.322938	−	0.946420i	$-0.604671\pi$
$809$	−1.20232e7	−0.645877	−0.322938	+	0.946420i	$0.604671\pi$
$810$	0	0
$811$	−5.50728e6	−0.294026	−0.147013	−	0.989135i	$-0.546966\pi$
$811$	−5.50728e6	−0.294026	−0.147013	+	0.989135i	$0.546966\pi$
$812$	1.32972e7	0.707732
$813$	0	0
$814$	1.95042e6	0.103173
$815$	0	0
$816$	0	0
$817$	−4.79258e7	−2.51197
$818$	−9.47258e6	−0.494977
$819$	0	0
$820$	0	0
$821$	−1.33971e7	−0.693672	−0.346836	−	0.937926i	$-0.612744\pi$
$821$	−1.33971e7	−0.693672	−0.346836	+	0.937926i	$0.612744\pi$
$822$	0	0
$823$	−1.41005e7	−0.725664	−0.362832	−	0.931855i	$-0.618190\pi$
$823$	−1.41005e7	−0.725664	−0.362832	+	0.931855i	$0.618190\pi$
$824$	−1.43162e6	−0.0734530
$825$	0	0
$826$	−4.16516e7	−2.12413
$827$	−1.28958e7	−0.655669	−0.327834	−	0.944735i	$-0.606319\pi$
$827$	−1.28958e7	−0.655669	−0.327834	+	0.944735i	$0.606319\pi$
$828$	0	0
$829$	−372726.	−0.0188366	−0.00941831	−	0.999956i	$-0.502998\pi$
$829$	−372726.	−0.0188366	−0.00941831	+	0.999956i	$0.502998\pi$
$830$	0	0
$831$	0	0
$832$	856547.	0.0428986
$833$	8.57875e6	0.428362
$834$	0	0
$835$	0	0
$836$	−3.12962e6	−0.154873
$837$	0	0
$838$	−2.30545e7	−1.13409
$839$	−1.57724e7	−0.773560	−0.386780	−	0.922172i	$-0.626413\pi$
$839$	−1.57724e7	−0.773560	−0.386780	+	0.922172i	$0.626413\pi$
$840$	0	0
$841$	−7.86558e6	−0.383478
$842$	1.30629e7	0.634978
$843$	0	0
$844$	−1.07116e7	−0.517604
$845$	0	0
$846$	0	0
$847$	3.57579e7	1.71263
$848$	3.26793e6	0.156057
$849$	0	0
$850$	0	0
$851$	2.33538e7	1.10544
$852$	0	0
$853$	3.39555e6	0.159786	0.0798928	−	0.996803i	$-0.474542\pi$
$853$	3.39555e6	0.159786	0.0798928	+	0.996803i	$0.474542\pi$
$854$	−1.08423e7	−0.508718
$855$	0	0
$856$	−8.36009e6	−0.389966
$857$	4.62981e6	0.215333	0.107667	−	0.994187i	$-0.465662\pi$
$857$	4.62981e6	0.215333	0.107667	+	0.994187i	$0.465662\pi$
$858$	0	0
$859$	−1.57703e7	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$859$	−1.57703e7	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$860$	0	0
$861$	0	0
$862$	2.24844e7	1.03065
$863$	−3.26217e7	−1.49101	−0.745503	−	0.666502i	$-0.767791\pi$
$863$	−3.26217e7	−1.49101	−0.745503	+	0.666502i	$0.767791\pi$
$864$	0	0
$865$	0	0
$866$	1.27462e7	0.577546
$867$	0	0
$868$	−2.58095e7	−1.16273
$869$	−2.50080e6	−0.112339
$870$	0	0
$871$	828376.	0.0369983
$872$	−3.29239e6	−0.146629
$873$	0	0
$874$	−3.74733e7	−1.65937
$875$	0	0
$876$	0	0
$877$	−3.88139e6	−0.170407	−0.0852037	−	0.996364i	$-0.527154\pi$
$877$	−3.88139e6	−0.170407	−0.0852037	+	0.996364i	$0.527154\pi$
$878$	−2.96441e7	−1.29778
$879$	0	0
$880$	0	0
$881$	2.11909e7	0.919834	0.459917	−	0.887962i	$-0.347879\pi$
$881$	2.11909e7	0.919834	0.459917	+	0.887962i	$0.347879\pi$
$882$	0	0
$883$	−257162.	−0.0110995	−0.00554976	−	0.999985i	$-0.501767\pi$
$883$	−257162.	−0.0110995	−0.00554976	+	0.999985i	$0.501767\pi$
$884$	759122.	0.0326724
$885$	0	0
$886$	1.05145e7	0.449990
$887$	−1.02364e7	−0.436856	−0.218428	−	0.975853i	$-0.570093\pi$
$887$	−1.02364e7	−0.436856	−0.218428	+	0.975853i	$0.570093\pi$
$888$	0	0
$889$	−4.66623e7	−1.98021
$890$	0	0
$891$	0	0
$892$	−1.34706e7	−0.566860
$893$	1.68294e7	0.706220
$894$	0	0
$895$	0	0
$896$	−3.82904e6	−0.159338
$897$	0	0
$898$	−287058.	−0.0118790
$899$	−2.45448e7	−1.01288
$900$	0	0
$901$	2.89623e6	0.118856
$902$	−2.32661e6	−0.0952155
$903$	0	0
$904$	8.84855e6	0.360123
$905$	0	0
$906$	0	0
$907$	−1.30652e7	−0.527350	−0.263675	−	0.964612i	$-0.584935\pi$
$907$	−1.30652e7	−0.527350	−0.263675	+	0.964612i	$0.584935\pi$
$908$	−1.70074e6	−0.0684580
$909$	0	0
$910$	0	0
$911$	9.81608e6	0.391870	0.195935	−	0.980617i	$-0.437226\pi$
$911$	9.81608e6	0.391870	0.195935	+	0.980617i	$0.437226\pi$
$912$	0	0
$913$	−4.56971e6	−0.181431
$914$	1.40588e7	0.556652
$915$	0	0
$916$	2.44689e7	0.963553
$917$	−7.34358e7	−2.88393
$918$	0	0
$919$	2.71414e6	0.106009	0.0530045	−	0.998594i	$-0.483120\pi$
$919$	2.71414e6	0.106009	0.0530045	+	0.998594i	$0.483120\pi$
$920$	0	0
$921$	0	0
$922$	−1.34228e7	−0.520013
$923$	−9.70591e6	−0.375000
$924$	0	0
$925$	0	0
$926$	1.38908e7	0.532352
$927$	0	0
$928$	−3.64141e6	−0.138803
$929$	4.31963e6	0.164213	0.0821064	−	0.996624i	$-0.473835\pi$
$929$	4.31963e6	0.164213	0.0821064	+	0.996624i	$0.473835\pi$
$930$	0	0
$931$	−8.24469e7	−3.11745
$932$	−5.79431e6	−0.218505
$933$	0	0
$934$	−9.41981e6	−0.353326
$935$	0	0
$936$	0	0
$937$	−2.17380e7	−0.808854	−0.404427	−	0.914570i	$-0.632529\pi$
$937$	−2.17380e7	−0.808854	−0.404427	+	0.914570i	$0.632529\pi$
$938$	−3.70311e6	−0.137423
$939$	0	0
$940$	0	0
$941$	2.17794e7	0.801811	0.400905	−	0.916119i	$-0.368696\pi$
$941$	2.17794e7	0.801811	0.400905	+	0.916119i	$0.368696\pi$
$942$	0	0
$943$	−2.78583e7	−1.02018
$944$	1.14062e7	0.416593
$945$	0	0
$946$	7.88678e6	0.286531
$947$	4.93262e7	1.78732	0.893660	−	0.448745i	$-0.148129\pi$
$947$	4.93262e7	1.78732	0.893660	+	0.448745i	$0.148129\pi$
$948$	0	0
$949$	1.28798e7	0.464242
$950$	0	0
$951$	0	0
$952$	−3.39352e6	−0.121355
$953$	3.37247e7	1.20286	0.601432	−	0.798924i	$-0.294597\pi$
$953$	3.37247e7	1.20286	0.601432	+	0.798924i	$0.294597\pi$
$954$	0	0
$955$	0	0
$956$	−615929.	−0.0217965
$957$	0	0
$958$	751082.	0.0264407
$959$	−6.00252e7	−2.10759
$960$	0	0
$961$	1.90117e7	0.664068
$962$	4.54671e6	0.158402
$963$	0	0
$964$	9.07892e6	0.314660
$965$	0	0
$966$	0	0
$967$	2.77310e7	0.953673	0.476836	−	0.878992i	$-0.341783\pi$
$967$	2.77310e7	0.953673	0.476836	+	0.878992i	$0.341783\pi$
$968$	−9.79225e6	−0.335888
$969$	0	0
$970$	0	0
$971$	6.67841e6	0.227314	0.113657	−	0.993520i	$-0.463744\pi$
$971$	6.67841e6	0.227314	0.113657	+	0.993520i	$0.463744\pi$
$972$	0	0
$973$	2.51870e6	0.0852893
$974$	−8.06018e6	−0.272237
$975$	0	0
$976$	2.96915e6	0.0997716
$977$	4.34946e7	1.45780	0.728902	−	0.684618i	$-0.240031\pi$
$977$	4.34946e7	1.45780	0.728902	+	0.684618i	$0.240031\pi$
$978$	0	0
$979$	−460299.	−0.0153491
$980$	0	0
$981$	0	0
$982$	−2.32326e7	−0.768809
$983$	−1.24213e7	−0.410000	−0.205000	−	0.978762i	$-0.565719\pi$
$983$	−1.24213e7	−0.410000	−0.205000	+	0.978762i	$0.565719\pi$
$984$	0	0
$985$	0	0
$986$	−3.22723e6	−0.105715
$987$	0	0
$988$	−7.29561e6	−0.237777
$989$	9.44344e7	3.07001
$990$	0	0
$991$	5.93479e7	1.91965	0.959824	−	0.280604i	$-0.0905348\pi$
$991$	5.93479e7	1.91965	0.959824	+	0.280604i	$0.0905348\pi$
$992$	7.06789e6	0.228040
$993$	0	0
$994$	4.33885e7	1.39286
$995$	0	0
$996$	0	0
$997$	−2.44456e7	−0.778867	−0.389434	−	0.921055i	$-0.627329\pi$
$997$	−2.44456e7	−0.778867	−0.389434	+	0.921055i	$0.627329\pi$
$998$	2.32244e6	0.0738107
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.bc.1.1		2
3.2	odd	2		150.6.a.n.1.1		2
5.2	odd	4		90.6.c.c.19.3		4
5.3	odd	4		90.6.c.c.19.1		4
5.4	even	2		450.6.a.bb.1.2		2
15.2	even	4		30.6.c.b.19.2	✓	4
15.8	even	4		30.6.c.b.19.4	yes	4
15.14	odd	2		150.6.a.o.1.2		2
20.3	even	4		720.6.f.i.289.1		4
20.7	even	4		720.6.f.i.289.2		4
60.23	odd	4		240.6.f.b.49.4		4
60.47	odd	4		240.6.f.b.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.c.b.19.2	✓	4	15.2	even	4
30.6.c.b.19.4	yes	4	15.8	even	4
90.6.c.c.19.1		4	5.3	odd	4
90.6.c.c.19.3		4	5.2	odd	4
150.6.a.n.1.1		2	3.2	odd	2
150.6.a.o.1.2		2	15.14	odd	2
240.6.f.b.49.2		4	60.47	odd	4
240.6.f.b.49.4		4	60.23	odd	4
450.6.a.bb.1.2		2	5.4	even	2
450.6.a.bc.1.1		2	1.1	even	1	trivial
720.6.f.i.289.1		4	20.3	even	4
720.6.f.i.289.2		4	20.7	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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -233.706 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -233.706 q^{7} +64.0000 q^{8} +89.7060 q^{11} +209.118 q^{13} -934.824 q^{14} +256.000 q^{16} +226.882 q^{17} -2180.47 q^{19} +358.824 q^{22} +4296.47 q^{23} +836.472 q^{26} -3739.30 q^{28} -3556.06 q^{29} +6902.24 q^{31} +1024.00 q^{32} +907.528 q^{34} +5435.59 q^{37} -8721.89 q^{38} -6484.00 q^{41} +21979.5 q^{43} +1435.30 q^{44} +17185.9 q^{46} -7718.24 q^{47} +37811.5 q^{49} +3345.89 q^{52} +12765.4 q^{53} -14957.2 q^{56} -14224.2 q^{58} +44555.6 q^{59} +11598.2 q^{61} +27608.9 q^{62} +4096.00 q^{64} +3961.29 q^{67} +3630.11 q^{68} -46413.6 q^{71} +61591.2 q^{73} +21742.3 q^{74} -34887.5 q^{76} -20964.8 q^{77} -27877.8 q^{79} -25936.0 q^{82} -50941.0 q^{83} +87918.1 q^{86} +5741.18 q^{88} -5131.19 q^{89} -48872.1 q^{91} +68743.5 q^{92} -30872.9 q^{94} +88435.3 q^{97} +151246. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 114 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 - 114 * q^7 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} - 114 q^{7} + 128 q^{8} - 174 q^{11} - 642 q^{13} - 456 q^{14} + 512 q^{16} + 1514 q^{17} - 120 q^{19} - 696 q^{22} + 4352 q^{23} - 2568 q^{26} - 1824 q^{28} + 2430 q^{29} + 11684 q^{31} + 2048 q^{32} + 6056 q^{34} + 17586 q^{37} - 480 q^{38} - 24984 q^{41} + 24168 q^{43} - 2784 q^{44} + 17408 q^{46} - 13316 q^{47} + 35334 q^{49} - 10272 q^{52} - 13698 q^{53} - 7296 q^{56} + 9720 q^{58} + 23730 q^{59} + 57124 q^{61} + 46736 q^{62} + 8192 q^{64} + 38316 q^{67} + 24224 q^{68} + 11076 q^{71} + 88548 q^{73} + 70344 q^{74} - 1920 q^{76} - 52532 q^{77} + 14220 q^{79} - 99936 q^{82} + 50792 q^{83} + 96672 q^{86} - 11136 q^{88} + 60420 q^{89} - 150756 q^{91} + 69632 q^{92} - 53264 q^{94} + 171216 q^{97} + 141336 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 114 * q^7 + 128 * q^8 - 174 * q^11 - 642 * q^13 - 456 * q^14 + 512 * q^16 + 1514 * q^17 - 120 * q^19 - 696 * q^22 + 4352 * q^23 - 2568 * q^26 - 1824 * q^28 + 2430 * q^29 + 11684 * q^31 + 2048 * q^32 + 6056 * q^34 + 17586 * q^37 - 480 * q^38 - 24984 * q^41 + 24168 * q^43 - 2784 * q^44 + 17408 * q^46 - 13316 * q^47 + 35334 * q^49 - 10272 * q^52 - 13698 * q^53 - 7296 * q^56 + 9720 * q^58 + 23730 * q^59 + 57124 * q^61 + 46736 * q^62 + 8192 * q^64 + 38316 * q^67 + 24224 * q^68 + 11076 * q^71 + 88548 * q^73 + 70344 * q^74 - 1920 * q^76 - 52532 * q^77 + 14220 * q^79 - 99936 * q^82 + 50792 * q^83 + 96672 * q^86 - 11136 * q^88 + 60420 * q^89 - 150756 * q^91 + 69632 * q^92 - 53264 * q^94 + 171216 * q^97 + 141336 * q^98

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