LMFDB - Embedded newform 450.6.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.1727189158$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.35890$ 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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -17.4356 q^{7} +64.0000 q^{8} +O(q^{10})$	$q+4.00000 q^{2} +16.0000 q^{4} -17.4356 q^{7} +64.0000 q^{8} +645.117 q^{11} -1098.44 q^{13} -69.7424 q^{14} +256.000 q^{16} -1166.00 q^{17} -2244.00 q^{19} +2580.47 q^{22} -500.000 q^{23} -4393.77 q^{26} -278.970 q^{28} -470.761 q^{29} +3856.00 q^{31} +1024.00 q^{32} -4664.00 q^{34} +6991.67 q^{37} -8976.00 q^{38} +9589.58 q^{41} -5300.42 q^{43} +10321.9 q^{44} -2000.00 q^{46} -19900.0 q^{47} -16503.0 q^{49} -17575.1 q^{52} -1146.00 q^{53} -1115.88 q^{56} -1883.04 q^{58} -37364.5 q^{59} -38158.0 q^{61} +15424.0 q^{62} +4096.00 q^{64} -36231.2 q^{67} -18656.0 q^{68} -16633.6 q^{71} +69393.7 q^{73} +27966.7 q^{74} -35904.0 q^{76} -11248.0 q^{77} -20664.0 q^{79} +38358.3 q^{82} -96968.0 q^{83} -21201.7 q^{86} +41287.5 q^{88} +59699.5 q^{89} +19152.0 q^{91} -8000.00 q^{92} -79600.0 q^{94} +5823.49 q^{97} -66012.0 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} + 128 q^{8} + 512 q^{16} - 2332 q^{17} - 4488 q^{19} - 1000 q^{23} + 7712 q^{31} + 2048 q^{32} - 9328 q^{34} - 17952 q^{38} - 4000 q^{46} - 39800 q^{47} - 33006 q^{49} - 2292 q^{53} - 76316 q^{61} + 30848 q^{62} + 8192 q^{64} - 37312 q^{68} - 71808 q^{76} - 22496 q^{77} - 41328 q^{79} - 193936 q^{83} + 38304 q^{91} - 16000 q^{92} - 159200 q^{94} - 132024 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 + 128 * q^8 + 512 * q^16 - 2332 * q^17 - 4488 * q^19 - 1000 * q^23 + 7712 * q^31 + 2048 * q^32 - 9328 * q^34 - 17952 * q^38 - 4000 * q^46 - 39800 * q^47 - 33006 * q^49 - 2292 * q^53 - 76316 * q^61 + 30848 * q^62 + 8192 * q^64 - 37312 * q^68 - 71808 * q^76 - 22496 * q^77 - 41328 * q^79 - 193936 * q^83 + 38304 * q^91 - 16000 * q^92 - 159200 * q^94 - 132024 * 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
$7$	−17.4356	−0.134491	−0.0672453	−	0.997736i	$-0.521421\pi$
$7$	−17.4356	−0.134491	−0.0672453	+	0.997736i	$0.521421\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	645.117	1.60752	0.803761	−	0.594953i	$-0.202829\pi$
$11$	645.117	1.60752	0.803761	+	0.594953i	$0.202829\pi$
$12$	0	0
$13$	−1098.44	−1.80268	−0.901341	−	0.433111i	$-0.857416\pi$
$13$	−1098.44	−1.80268	−0.901341	+	0.433111i	$0.857416\pi$
$14$	−69.7424	−0.0950992
$15$	0	0
$16$	256.000	0.250000
$17$	−1166.00	−0.978535	−0.489267	−	0.872134i	$-0.662736\pi$
$17$	−1166.00	−0.978535	−0.489267	+	0.872134i	$0.662736\pi$
$18$	0	0
$19$	−2244.00	−1.42606	−0.713032	−	0.701132i	$-0.752678\pi$
$19$	−2244.00	−1.42606	−0.713032	+	0.701132i	$0.752678\pi$
$20$	0	0
$21$	0	0
$22$	2580.47	1.13669
$23$	−500.000	−0.197084	−0.0985418	−	0.995133i	$-0.531418\pi$
$23$	−500.000	−0.197084	−0.0985418	+	0.995133i	$0.531418\pi$
$24$	0	0
$25$	0	0
$26$	−4393.77	−1.27469
$27$	0	0
$28$	−278.970	−0.0672453
$29$	−470.761	−0.103945	−0.0519727	−	0.998649i	$-0.516551\pi$
$29$	−470.761	−0.103945	−0.0519727	+	0.998649i	$0.516551\pi$
$30$	0	0
$31$	3856.00	0.720664	0.360332	−	0.932824i	$-0.382663\pi$
$31$	3856.00	0.720664	0.360332	+	0.932824i	$0.382663\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−4664.00	−0.691929
$35$	0	0
$36$	0	0
$37$	6991.67	0.839609	0.419804	−	0.907615i	$-0.362099\pi$
$37$	6991.67	0.839609	0.419804	+	0.907615i	$0.362099\pi$
$38$	−8976.00	−1.00838
$39$	0	0
$40$	0	0
$41$	9589.58	0.890922	0.445461	−	0.895301i	$-0.353040\pi$
$41$	9589.58	0.890922	0.445461	+	0.895301i	$0.353040\pi$
$42$	0	0
$43$	−5300.42	−0.437159	−0.218579	−	0.975819i	$-0.570142\pi$
$43$	−5300.42	−0.437159	−0.218579	+	0.975819i	$0.570142\pi$
$44$	10321.9	0.803761
$45$	0	0
$46$	−2000.00	−0.139359
$47$	−19900.0	−1.31404	−0.657020	−	0.753873i	$-0.728183\pi$
$47$	−19900.0	−1.31404	−0.657020	+	0.753873i	$0.728183\pi$
$48$	0	0
$49$	−16503.0	−0.981912
$50$	0	0
$51$	0	0
$52$	−17575.1	−0.901341
$53$	−1146.00	−0.0560396	−0.0280198	−	0.999607i	$-0.508920\pi$
$53$	−1146.00	−0.0560396	−0.0280198	+	0.999607i	$0.508920\pi$
$54$	0	0
$55$	0	0
$56$	−1115.88	−0.0475496
$57$	0	0
$58$	−1883.04	−0.0735005
$59$	−37364.5	−1.39743	−0.698713	−	0.715402i	$-0.746244\pi$
$59$	−37364.5	−1.39743	−0.698713	+	0.715402i	$0.746244\pi$
$60$	0	0
$61$	−38158.0	−1.31299	−0.656494	−	0.754331i	$-0.727961\pi$
$61$	−38158.0	−1.31299	−0.656494	+	0.754331i	$0.727961\pi$
$62$	15424.0	0.509586
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−36231.2	−0.986042	−0.493021	−	0.870017i	$-0.664107\pi$
$67$	−36231.2	−0.986042	−0.493021	+	0.870017i	$0.664107\pi$
$68$	−18656.0	−0.489267
$69$	0	0
$70$	0	0
$71$	−16633.6	−0.391597	−0.195798	−	0.980644i	$-0.562730\pi$
$71$	−16633.6	−0.391597	−0.195798	+	0.980644i	$0.562730\pi$
$72$	0	0
$73$	69393.7	1.52410	0.762049	−	0.647520i	$-0.224194\pi$
$73$	69393.7	1.52410	0.762049	+	0.647520i	$0.224194\pi$
$74$	27966.7	0.593693
$75$	0	0
$76$	−35904.0	−0.713032
$77$	−11248.0	−0.216196
$78$	0	0
$79$	−20664.0	−0.372517	−0.186259	−	0.982501i	$-0.559636\pi$
$79$	−20664.0	−0.372517	−0.186259	+	0.982501i	$0.559636\pi$
$80$	0	0
$81$	0	0
$82$	38358.3	0.629977
$83$	−96968.0	−1.54502	−0.772508	−	0.635005i	$-0.780998\pi$
$83$	−96968.0	−1.54502	−0.772508	+	0.635005i	$0.780998\pi$
$84$	0	0
$85$	0	0
$86$	−21201.7	−0.309118
$87$	0	0
$88$	41287.5	0.568345
$89$	59699.5	0.798906	0.399453	−	0.916754i	$-0.369200\pi$
$89$	59699.5	0.798906	0.399453	+	0.916754i	$0.369200\pi$
$90$	0	0
$91$	19152.0	0.242444
$92$	−8000.00	−0.0985418
$93$	0	0
$94$	−79600.0	−0.929166
$95$	0	0
$96$	0	0
$97$	5823.49	0.0628426	0.0314213	−	0.999506i	$-0.489997\pi$
$97$	5823.49	0.0628426	0.0314213	+	0.999506i	$0.489997\pi$
$98$	−66012.0	−0.694317
$99$	0	0
$100$	0	0
$101$	102922.	1.00394	0.501968	−	0.864886i	$-0.332609\pi$
$101$	102922.	1.00394	0.501968	+	0.864886i	$0.332609\pi$
$102$	0	0
$103$	−178558.	−1.65839	−0.829194	−	0.558961i	$-0.811200\pi$
$103$	−178558.	−1.65839	−0.829194	+	0.558961i	$0.811200\pi$
$104$	−70300.3	−0.637344
$105$	0	0
$106$	−4584.00	−0.0396260
$107$	7512.00	0.0634302	0.0317151	−	0.999497i	$-0.489903\pi$
$107$	7512.00	0.0634302	0.0317151	+	0.999497i	$0.489903\pi$
$108$	0	0
$109$	−58238.0	−0.469505	−0.234752	−	0.972055i	$-0.575428\pi$
$109$	−58238.0	−0.469505	−0.234752	+	0.972055i	$0.575428\pi$
$110$	0	0
$111$	0	0
$112$	−4463.51	−0.0336226
$113$	−217314.	−1.60100	−0.800500	−	0.599332i	$-0.795433\pi$
$113$	−217314.	−1.60100	−0.800500	+	0.599332i	$0.795433\pi$
$114$	0	0
$115$	0	0
$116$	−7532.18	−0.0519727
$117$	0	0
$118$	−149458.	−0.988130
$119$	20329.9	0.131604
$120$	0	0
$121$	255125.	1.58413
$122$	−152632.	−0.928423
$123$	0	0
$124$	61696.0	0.360332
$125$	0	0
$126$	0	0
$127$	212348.	1.16826	0.584129	−	0.811660i	$-0.301436\pi$
$127$	212348.	1.16826	0.584129	+	0.811660i	$0.301436\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	−279266.	−1.42180	−0.710902	−	0.703291i	$-0.751713\pi$
$131$	−279266.	−1.42180	−0.710902	+	0.703291i	$0.751713\pi$
$132$	0	0
$133$	39125.5	0.191792
$134$	−144925.	−0.697237
$135$	0	0
$136$	−74624.0	−0.345964
$137$	−337914.	−1.53817	−0.769086	−	0.639146i	$-0.779288\pi$
$137$	−337914.	−1.53817	−0.769086	+	0.639146i	$0.779288\pi$
$138$	0	0
$139$	92620.0	0.406600	0.203300	−	0.979116i	$-0.434833\pi$
$139$	92620.0	0.406600	0.203300	+	0.979116i	$0.434833\pi$
$140$	0	0
$141$	0	0
$142$	−66534.2	−0.276901
$143$	−708624.	−2.89785
$144$	0	0
$145$	0	0
$146$	277575.	1.07770
$147$	0	0
$148$	111867.	0.419804
$149$	29588.2	0.109182	0.0545912	−	0.998509i	$-0.482614\pi$
$149$	29588.2	0.109182	0.0545912	+	0.998509i	$0.482614\pi$
$150$	0	0
$151$	40384.0	0.144134	0.0720671	−	0.997400i	$-0.477040\pi$
$151$	40384.0	0.144134	0.0720671	+	0.997400i	$0.477040\pi$
$152$	−143616.	−0.504190
$153$	0	0
$154$	−44992.0	−0.152874
$155$	0	0
$156$	0	0
$157$	−81476.5	−0.263805	−0.131903	−	0.991263i	$-0.542109\pi$
$157$	−81476.5	−0.263805	−0.131903	+	0.991263i	$0.542109\pi$
$158$	−82656.0	−0.263410
$159$	0	0
$160$	0	0
$161$	8717.80	0.0265059
$162$	0	0
$163$	458382.	1.35132	0.675660	−	0.737213i	$-0.263859\pi$
$163$	458382.	1.35132	0.675660	+	0.737213i	$0.263859\pi$
$164$	153433.	0.445461
$165$	0	0
$166$	−387872.	−1.09249
$167$	194436.	0.539493	0.269746	−	0.962931i	$-0.413060\pi$
$167$	194436.	0.539493	0.269746	+	0.962931i	$0.413060\pi$
$168$	0	0
$169$	835283.	2.24966
$170$	0	0
$171$	0	0
$172$	−84806.7	−0.218579
$173$	−322222.	−0.818540	−0.409270	−	0.912413i	$-0.634217\pi$
$173$	−322222.	−0.818540	−0.409270	+	0.912413i	$0.634217\pi$
$174$	0	0
$175$	0	0
$176$	165150.	0.401880
$177$	0	0
$178$	238798.	0.564912
$179$	430258.	1.00368	0.501842	−	0.864960i	$-0.332656\pi$
$179$	430258.	1.00368	0.501842	+	0.864960i	$0.332656\pi$
$180$	0	0
$181$	242522.	0.550243	0.275122	−	0.961409i	$-0.411282\pi$
$181$	242522.	0.550243	0.275122	+	0.961409i	$0.411282\pi$
$182$	76608.0	0.171433
$183$	0	0
$184$	−32000.0	−0.0696796
$185$	0	0
$186$	0	0
$187$	−752206.	−1.57302
$188$	−318400.	−0.657020
$189$	0	0
$190$	0	0
$191$	−290930.	−0.577040	−0.288520	−	0.957474i	$-0.593163\pi$
$191$	−290930.	−0.577040	−0.288520	+	0.957474i	$0.593163\pi$
$192$	0	0
$193$	−723054.	−1.39726	−0.698631	−	0.715483i	$-0.746207\pi$
$193$	−723054.	−1.39726	−0.698631	+	0.715483i	$0.746207\pi$
$194$	23294.0	0.0444364
$195$	0	0
$196$	−264048.	−0.490956
$197$	−599638.	−1.10084	−0.550419	−	0.834888i	$-0.685532\pi$
$197$	−599638.	−1.10084	−0.550419	+	0.834888i	$0.685532\pi$
$198$	0	0
$199$	650184.	1.16387	0.581934	−	0.813236i	$-0.302296\pi$
$199$	650184.	1.16387	0.581934	+	0.813236i	$0.302296\pi$
$200$	0	0
$201$	0	0
$202$	411689.	0.709890
$203$	8208.00	0.0139797
$204$	0	0
$205$	0	0
$206$	−714232.	−1.17266
$207$	0	0
$208$	−281201.	−0.450670
$209$	−1.44764e6	−2.29243
$210$	0	0
$211$	986316.	1.52514	0.762570	−	0.646905i	$-0.223937\pi$
$211$	986316.	1.52514	0.762570	+	0.646905i	$0.223937\pi$
$212$	−18336.0	−0.0280198
$213$	0	0
$214$	30048.0	0.0448519
$215$	0	0
$216$	0	0
$217$	−67231.7	−0.0969225
$218$	−232952.	−0.331990
$219$	0	0
$220$	0	0
$221$	1.28078e6	1.76399
$222$	0	0
$223$	694407.	0.935087	0.467544	−	0.883970i	$-0.345139\pi$
$223$	694407.	0.935087	0.467544	+	0.883970i	$0.345139\pi$
$224$	−17854.1	−0.0237748
$225$	0	0
$226$	−869256.	−1.13208
$227$	435400.	0.560820	0.280410	−	0.959880i	$-0.409530\pi$
$227$	435400.	0.560820	0.280410	+	0.959880i	$0.409530\pi$
$228$	0	0
$229$	−1.16123e6	−1.46328	−0.731641	−	0.681690i	$-0.761245\pi$
$229$	−1.16123e6	−1.46328	−0.731641	+	0.681690i	$0.761245\pi$
$230$	0	0
$231$	0	0
$232$	−30128.7	−0.0367503
$233$	1.21303e6	1.46380	0.731902	−	0.681409i	$-0.238633\pi$
$233$	1.21303e6	1.46380	0.731902	+	0.681409i	$0.238633\pi$
$234$	0	0
$235$	0	0
$236$	−597832.	−0.698713
$237$	0	0
$238$	81319.6	0.0930578
$239$	1.40053e6	1.58598	0.792991	−	0.609234i	$-0.208523\pi$
$239$	1.40053e6	1.58598	0.792991	+	0.609234i	$0.208523\pi$
$240$	0	0
$241$	471490.	0.522914	0.261457	−	0.965215i	$-0.415797\pi$
$241$	471490.	0.522914	0.261457	+	0.965215i	$0.415797\pi$
$242$	1.02050e6	1.12015
$243$	0	0
$244$	−610528.	−0.656494
$245$	0	0
$246$	0	0
$247$	2.46491e6	2.57074
$248$	246784.	0.254793
$249$	0	0
$250$	0	0
$251$	601371.	0.602502	0.301251	−	0.953545i	$-0.402596\pi$
$251$	601371.	0.602502	0.301251	+	0.953545i	$0.402596\pi$
$252$	0	0
$253$	−322559.	−0.316816
$254$	849392.	0.826084
$255$	0	0
$256$	65536.0	0.0625000
$257$	124206.	0.117303	0.0586516	−	0.998279i	$-0.481320\pi$
$257$	124206.	0.117303	0.0586516	+	0.998279i	$0.481320\pi$
$258$	0	0
$259$	−121904.	−0.112919
$260$	0	0
$261$	0	0
$262$	−1.11706e6	−1.00537
$263$	−429956.	−0.383296	−0.191648	−	0.981464i	$-0.561383\pi$
$263$	−429956.	−0.383296	−0.191648	+	0.981464i	$0.561383\pi$
$264$	0	0
$265$	0	0
$266$	156502.	0.135617
$267$	0	0
$268$	−579699.	−0.493021
$269$	−1.43755e6	−1.21127	−0.605636	−	0.795742i	$-0.707081\pi$
$269$	−1.43755e6	−1.21127	−0.605636	+	0.795742i	$0.707081\pi$
$270$	0	0
$271$	1.54473e6	1.27770	0.638850	−	0.769331i	$-0.279411\pi$
$271$	1.54473e6	1.27770	0.638850	+	0.769331i	$0.279411\pi$
$272$	−298496.	−0.244634
$273$	0	0
$274$	−1.35166e6	−1.08765
$275$	0	0
$276$	0	0
$277$	−211128.	−0.165328	−0.0826639	−	0.996577i	$-0.526343\pi$
$277$	−211128.	−0.165328	−0.0826639	+	0.996577i	$0.526343\pi$
$278$	370480.	0.287510
$279$	0	0
$280$	0	0
$281$	−1.29051e6	−0.974982	−0.487491	−	0.873128i	$-0.662088\pi$
$281$	−1.29051e6	−0.974982	−0.487491	+	0.873128i	$0.662088\pi$
$282$	0	0
$283$	−1.69959e6	−1.26147	−0.630736	−	0.775998i	$-0.717247\pi$
$283$	−1.69959e6	−1.26147	−0.630736	+	0.775998i	$0.717247\pi$
$284$	−266137.	−0.195798
$285$	0	0
$286$	−2.83450e6	−2.04909
$287$	−167200.	−0.119821
$288$	0	0
$289$	−60301.0	−0.0424698
$290$	0	0
$291$	0	0
$292$	1.11030e6	0.762049
$293$	2.49148e6	1.69546	0.847732	−	0.530424i	$-0.177967\pi$
$293$	2.49148e6	1.69546	0.847732	+	0.530424i	$0.177967\pi$
$294$	0	0
$295$	0	0
$296$	447467.	0.296846
$297$	0	0
$298$	118353.	0.0772037
$299$	549221.	0.355279
$300$	0	0
$301$	92416.0	0.0587937
$302$	161536.	0.101918
$303$	0	0
$304$	−574464.	−0.356516
$305$	0	0
$306$	0	0
$307$	−3.16791e6	−1.91834	−0.959172	−	0.282822i	$-0.908729\pi$
$307$	−3.16791e6	−1.91834	−0.959172	+	0.282822i	$0.908729\pi$
$308$	−179968.	−0.108098
$309$	0	0
$310$	0	0
$311$	−2.23653e6	−1.31122	−0.655608	−	0.755101i	$-0.727588\pi$
$311$	−2.23653e6	−1.31122	−0.655608	+	0.755101i	$0.727588\pi$
$312$	0	0
$313$	2.37351e6	1.36940	0.684699	−	0.728826i	$-0.259934\pi$
$313$	2.37351e6	1.36940	0.684699	+	0.728826i	$0.259934\pi$
$314$	−325906.	−0.186538
$315$	0	0
$316$	−330624.	−0.186259
$317$	987502.	0.551937	0.275969	−	0.961167i	$-0.411001\pi$
$317$	987502.	0.551937	0.275969	+	0.961167i	$0.411001\pi$
$318$	0	0
$319$	−303696.	−0.167095
$320$	0	0
$321$	0	0
$322$	34871.2	0.0187425
$323$	2.61650e6	1.39545
$324$	0	0
$325$	0	0
$326$	1.83353e6	0.955528
$327$	0	0
$328$	613733.	0.314989
$329$	346968.	0.176726
$330$	0	0
$331$	−918644.	−0.460869	−0.230434	−	0.973088i	$-0.574015\pi$
$331$	−918644.	−0.460869	−0.230434	+	0.973088i	$0.574015\pi$
$332$	−1.55149e6	−0.772508
$333$	0	0
$334$	777744.	0.381479
$335$	0	0
$336$	0	0
$337$	−355547.	−0.170538	−0.0852691	−	0.996358i	$-0.527175\pi$
$337$	−355547.	−0.170538	−0.0852691	+	0.996358i	$0.527175\pi$
$338$	3.34113e6	1.59075
$339$	0	0
$340$	0	0
$341$	2.48757e6	1.15848
$342$	0	0
$343$	580780.	0.266548
$344$	−339227.	−0.154559
$345$	0	0
$346$	−1.28889e6	−0.578795
$347$	1.85760e6	0.828187	0.414094	−	0.910234i	$-0.364099\pi$
$347$	1.85760e6	0.828187	0.414094	+	0.910234i	$0.364099\pi$
$348$	0	0
$349$	950578.	0.417757	0.208879	−	0.977942i	$-0.433019\pi$
$349$	950578.	0.417757	0.208879	+	0.977942i	$0.433019\pi$
$350$	0	0
$351$	0	0
$352$	660600.	0.284172
$353$	767570.	0.327855	0.163927	−	0.986472i	$-0.447584\pi$
$353$	767570.	0.327855	0.163927	+	0.986472i	$0.447584\pi$
$354$	0	0
$355$	0	0
$356$	955192.	0.399453
$357$	0	0
$358$	1.72103e6	0.709711
$359$	160896.	0.0658883	0.0329441	−	0.999457i	$-0.489512\pi$
$359$	160896.	0.0658883	0.0329441	+	0.999457i	$0.489512\pi$
$360$	0	0
$361$	2.55944e6	1.03366
$362$	970088.	0.389081
$363$	0	0
$364$	306432.	0.121222
$365$	0	0
$366$	0	0
$367$	−202131.	−0.0783371	−0.0391685	−	0.999233i	$-0.512471\pi$
$367$	−202131.	−0.0783371	−0.0391685	+	0.999233i	$0.512471\pi$
$368$	−128000.	−0.0492709
$369$	0	0
$370$	0	0
$371$	19981.2	0.00753679
$372$	0	0
$373$	1.48933e6	0.554267	0.277134	−	0.960831i	$-0.410616\pi$
$373$	1.48933e6	0.554267	0.277134	+	0.960831i	$0.410616\pi$
$374$	−3.00883e6	−1.11229
$375$	0	0
$376$	−1.27360e6	−0.464583
$377$	517104.	0.187381
$378$	0	0
$379$	2.69222e6	0.962748	0.481374	−	0.876515i	$-0.340138\pi$
$379$	2.69222e6	0.962748	0.481374	+	0.876515i	$0.340138\pi$
$380$	0	0
$381$	0	0
$382$	−1.16372e6	−0.408029
$383$	565100.	0.196847	0.0984234	−	0.995145i	$-0.468620\pi$
$383$	565100.	0.196847	0.0984234	+	0.995145i	$0.468620\pi$
$384$	0	0
$385$	0	0
$386$	−2.89222e6	−0.988013
$387$	0	0
$388$	93175.8	0.0314213
$389$	5.46587e6	1.83141	0.915704	−	0.401853i	$-0.131634\pi$
$389$	5.46587e6	1.83141	0.915704	+	0.401853i	$0.131634\pi$
$390$	0	0
$391$	583000.	0.192853
$392$	−1.05619e6	−0.347158
$393$	0	0
$394$	−2.39855e6	−0.778410
$395$	0	0
$396$	0	0
$397$	4.00762e6	1.27618	0.638088	−	0.769963i	$-0.279726\pi$
$397$	4.00762e6	1.27618	0.638088	+	0.769963i	$0.279726\pi$
$398$	2.60074e6	0.822979
$399$	0	0
$400$	0	0
$401$	−3.92967e6	−1.22038	−0.610190	−	0.792255i	$-0.708907\pi$
$401$	−3.92967e6	−1.22038	−0.610190	+	0.792255i	$0.708907\pi$
$402$	0	0
$403$	−4.23559e6	−1.29913
$404$	1.64676e6	0.501968
$405$	0	0
$406$	32832.0	0.00988513
$407$	4.51045e6	1.34969
$408$	0	0
$409$	−1.15028e6	−0.340013	−0.170007	−	0.985443i	$-0.554379\pi$
$409$	−1.15028e6	−0.340013	−0.170007	+	0.985443i	$0.554379\pi$
$410$	0	0
$411$	0	0
$412$	−2.85693e6	−0.829194
$413$	651472.	0.187941
$414$	0	0
$415$	0	0
$416$	−1.12481e6	−0.318672
$417$	0	0
$418$	−5.79057e6	−1.62099
$419$	−4.95898e6	−1.37993	−0.689965	−	0.723842i	$-0.742374\pi$
$419$	−4.95898e6	−1.37993	−0.689965	+	0.723842i	$0.742374\pi$
$420$	0	0
$421$	2.89077e6	0.794893	0.397447	−	0.917625i	$-0.369896\pi$
$421$	2.89077e6	0.794893	0.397447	+	0.917625i	$0.369896\pi$
$422$	3.94526e6	1.07844
$423$	0	0
$424$	−73344.0	−0.0198130
$425$	0	0
$426$	0	0
$427$	665307.	0.176585
$428$	120192.	0.0317151
$429$	0	0
$430$	0	0
$431$	1.98417e6	0.514501	0.257250	−	0.966345i	$-0.417184\pi$
$431$	1.98417e6	0.514501	0.257250	+	0.966345i	$0.417184\pi$
$432$	0	0
$433$	−3.75622e6	−0.962789	−0.481395	−	0.876504i	$-0.659870\pi$
$433$	−3.75622e6	−0.962789	−0.481395	+	0.876504i	$0.659870\pi$
$434$	−268927.	−0.0685345
$435$	0	0
$436$	−931808.	−0.234752
$437$	1.12200e6	0.281054
$438$	0	0
$439$	1.94897e6	0.482662	0.241331	−	0.970443i	$-0.422416\pi$
$439$	1.94897e6	0.482662	0.241331	+	0.970443i	$0.422416\pi$
$440$	0	0
$441$	0	0
$442$	5.12314e6	1.24733
$443$	5.74198e6	1.39012	0.695060	−	0.718952i	$-0.255378\pi$
$443$	5.74198e6	1.39012	0.695060	+	0.718952i	$0.255378\pi$
$444$	0	0
$445$	0	0
$446$	2.77763e6	0.661207
$447$	0	0
$448$	−71416.2	−0.0168113
$449$	−538655.	−0.126094	−0.0630471	−	0.998011i	$-0.520082\pi$
$449$	−538655.	−0.126094	−0.0630471	+	0.998011i	$0.520082\pi$
$450$	0	0
$451$	6.18640e6	1.43218
$452$	−3.47702e6	−0.800500
$453$	0	0
$454$	1.74160e6	0.396560
$455$	0	0
$456$	0	0
$457$	1.55459e6	0.348198	0.174099	−	0.984728i	$-0.444299\pi$
$457$	1.55459e6	0.348198	0.174099	+	0.984728i	$0.444299\pi$
$458$	−4.64490e6	−1.03470
$459$	0	0
$460$	0	0
$461$	4.22445e6	0.925802	0.462901	−	0.886410i	$-0.346809\pi$
$461$	4.22445e6	0.925802	0.462901	+	0.886410i	$0.346809\pi$
$462$	0	0
$463$	4.77347e6	1.03486	0.517430	−	0.855726i	$-0.326889\pi$
$463$	4.77347e6	1.03486	0.517430	+	0.855726i	$0.326889\pi$
$464$	−120515.	−0.0259864
$465$	0	0
$466$	4.85214e6	1.03507
$467$	−9.11683e6	−1.93442	−0.967212	−	0.253970i	$-0.918264\pi$
$467$	−9.11683e6	−1.93442	−0.967212	+	0.253970i	$0.918264\pi$
$468$	0	0
$469$	631712.	0.132613
$470$	0	0
$471$	0	0
$472$	−2.39133e6	−0.494065
$473$	−3.41939e6	−0.702742
$474$	0	0
$475$	0	0
$476$	325278.	0.0658018
$477$	0	0
$478$	5.60213e6	1.12146
$479$	−4.22182e6	−0.840739	−0.420369	−	0.907353i	$-0.638099\pi$
$479$	−4.22182e6	−0.840739	−0.420369	+	0.907353i	$0.638099\pi$
$480$	0	0
$481$	−7.67995e6	−1.51355
$482$	1.88596e6	0.369756
$483$	0	0
$484$	4.08200e6	0.792063
$485$	0	0
$486$	0	0
$487$	501465.	0.0958117	0.0479058	−	0.998852i	$-0.484745\pi$
$487$	501465.	0.0958117	0.0479058	+	0.998852i	$0.484745\pi$
$488$	−2.44211e6	−0.464212
$489$	0	0
$490$	0	0
$491$	2.61435e6	0.489395	0.244697	−	0.969600i	$-0.421311\pi$
$491$	2.61435e6	0.489395	0.244697	+	0.969600i	$0.421311\pi$
$492$	0	0
$493$	548907.	0.101714
$494$	9.85962e6	1.81779
$495$	0	0
$496$	987136.	0.180166
$497$	290016.	0.0526661
$498$	0	0
$499$	5.68240e6	1.02160	0.510799	−	0.859700i	$-0.329350\pi$
$499$	5.68240e6	1.02160	0.510799	+	0.859700i	$0.329350\pi$
$500$	0	0
$501$	0	0
$502$	2.40548e6	0.426033
$503$	−3.36030e6	−0.592186	−0.296093	−	0.955159i	$-0.595684\pi$
$503$	−3.36030e6	−0.592186	−0.296093	+	0.955159i	$0.595684\pi$
$504$	0	0
$505$	0	0
$506$	−1.29023e6	−0.224023
$507$	0	0
$508$	3.39757e6	0.584129
$509$	5.60242e6	0.958476	0.479238	−	0.877685i	$-0.340913\pi$
$509$	5.60242e6	0.958476	0.479238	+	0.877685i	$0.340913\pi$
$510$	0	0
$511$	−1.20992e6	−0.204977
$512$	262144.	0.0441942
$513$	0	0
$514$	496824.	0.0829459
$515$	0	0
$516$	0	0
$517$	−1.28378e7	−2.11235
$518$	−487616.	−0.0798461
$519$	0	0
$520$	0	0
$521$	−8.49019e6	−1.37032	−0.685162	−	0.728391i	$-0.740269\pi$
$521$	−8.49019e6	−1.37032	−0.685162	+	0.728391i	$0.740269\pi$
$522$	0	0
$523$	3.31350e6	0.529703	0.264851	−	0.964289i	$-0.414677\pi$
$523$	3.31350e6	0.529703	0.264851	+	0.964289i	$0.414677\pi$
$524$	−4.46826e6	−0.710902
$525$	0	0
$526$	−1.71982e6	−0.271031
$527$	−4.49610e6	−0.705195
$528$	0	0
$529$	−6.18634e6	−0.961158
$530$	0	0
$531$	0	0
$532$	626008.	0.0958960
$533$	−1.05336e7	−1.60605
$534$	0	0
$535$	0	0
$536$	−2.31879e6	−0.348618
$537$	0	0
$538$	−5.75019e6	−0.856498
$539$	−1.06464e7	−1.57845
$540$	0	0
$541$	−4.76958e6	−0.700627	−0.350314	−	0.936632i	$-0.613925\pi$
$541$	−4.76958e6	−0.700627	−0.350314	+	0.936632i	$0.613925\pi$
$542$	6.17891e6	0.903470
$543$	0	0
$544$	−1.19398e6	−0.172982
$545$	0	0
$546$	0	0
$547$	1.10402e6	0.157765	0.0788823	−	0.996884i	$-0.474865\pi$
$547$	1.10402e6	0.157765	0.0788823	+	0.996884i	$0.474865\pi$
$548$	−5.40662e6	−0.769086
$549$	0	0
$550$	0	0
$551$	1.05639e6	0.148233
$552$	0	0
$553$	360289.	0.0501001
$554$	−844511.	−0.116904
$555$	0	0
$556$	1.48192e6	0.203300
$557$	−5.14466e6	−0.702617	−0.351308	−	0.936260i	$-0.614263\pi$
$557$	−5.14466e6	−0.702617	−0.351308	+	0.936260i	$0.614263\pi$
$558$	0	0
$559$	5.82221e6	0.788058
$560$	0	0
$561$	0	0
$562$	−5.16205e6	−0.689416
$563$	−9.51894e6	−1.26566	−0.632831	−	0.774290i	$-0.718107\pi$
$563$	−9.51894e6	−1.26566	−0.632831	+	0.774290i	$0.718107\pi$
$564$	0	0
$565$	0	0
$566$	−6.79835e6	−0.891995
$567$	0	0
$568$	−1.06455e6	−0.138450
$569$	1.29027e6	0.167070	0.0835352	−	0.996505i	$-0.473379\pi$
$569$	1.29027e6	0.167070	0.0835352	+	0.996505i	$0.473379\pi$
$570$	0	0
$571$	−1.11726e7	−1.43405	−0.717024	−	0.697048i	$-0.754496\pi$
$571$	−1.11726e7	−1.43405	−0.717024	+	0.697048i	$0.754496\pi$
$572$	−1.13380e7	−1.44892
$573$	0	0
$574$	−668800.	−0.0847260
$575$	0	0
$576$	0	0
$577$	−129791.	−0.0162294	−0.00811472	−	0.999967i	$-0.502583\pi$
$577$	−129791.	−0.0162294	−0.00811472	+	0.999967i	$0.502583\pi$
$578$	−241204.	−0.0300307
$579$	0	0
$580$	0	0
$581$	1.69069e6	0.207790
$582$	0	0
$583$	−739304.	−0.0900848
$584$	4.44119e6	0.538850
$585$	0	0
$586$	9.96593e6	1.19887
$587$	−9.30142e6	−1.11418	−0.557088	−	0.830453i	$-0.688081\pi$
$587$	−9.30142e6	−1.11418	−0.557088	+	0.830453i	$0.688081\pi$
$588$	0	0
$589$	−8.65286e6	−1.02771
$590$	0	0
$591$	0	0
$592$	1.78987e6	0.209902
$593$	−6.91802e6	−0.807876	−0.403938	−	0.914786i	$-0.632359\pi$
$593$	−6.91802e6	−0.807876	−0.403938	+	0.914786i	$0.632359\pi$
$594$	0	0
$595$	0	0
$596$	473411.	0.0545912
$597$	0	0
$598$	2.19689e6	0.251220
$599$	1.57759e7	1.79650	0.898252	−	0.439481i	$-0.144838\pi$
$599$	1.57759e7	1.79650	0.898252	+	0.439481i	$0.144838\pi$
$600$	0	0
$601$	−6.95761e6	−0.785731	−0.392866	−	0.919596i	$-0.628516\pi$
$601$	−6.95761e6	−0.785731	−0.392866	+	0.919596i	$0.628516\pi$
$602$	369664.	0.0415734
$603$	0	0
$604$	646144.	0.0720671
$605$	0	0
$606$	0	0
$607$	6.13874e6	0.676251	0.338125	−	0.941101i	$-0.390207\pi$
$607$	6.13874e6	0.676251	0.338125	+	0.941101i	$0.390207\pi$
$608$	−2.29786e6	−0.252095
$609$	0	0
$610$	0	0
$611$	2.18590e7	2.36879
$612$	0	0
$613$	−5.89321e6	−0.633433	−0.316717	−	0.948520i	$-0.602580\pi$
$613$	−5.89321e6	−0.633433	−0.316717	+	0.948520i	$0.602580\pi$
$614$	−1.26716e7	−1.35647
$615$	0	0
$616$	−719872.	−0.0764370
$617$	9.14737e6	0.967349	0.483675	−	0.875248i	$-0.339302\pi$
$617$	9.14737e6	0.967349	0.483675	+	0.875248i	$0.339302\pi$
$618$	0	0
$619$	−1.23318e7	−1.29360	−0.646799	−	0.762661i	$-0.723893\pi$
$619$	−1.23318e7	−1.29360	−0.646799	+	0.762661i	$0.723893\pi$
$620$	0	0
$621$	0	0
$622$	−8.94613e6	−0.927170
$623$	−1.04090e6	−0.107445
$624$	0	0
$625$	0	0
$626$	9.49403e6	0.968311
$627$	0	0
$628$	−1.30362e6	−0.131903
$629$	−8.15229e6	−0.821586
$630$	0	0
$631$	6.87452e6	0.687336	0.343668	−	0.939091i	$-0.388331\pi$
$631$	6.87452e6	0.687336	0.343668	+	0.939091i	$0.388331\pi$
$632$	−1.32250e6	−0.131705
$633$	0	0
$634$	3.95001e6	0.390279
$635$	0	0
$636$	0	0
$637$	1.81276e7	1.77007
$638$	−1.21478e6	−0.118154
$639$	0	0
$640$	0	0
$641$	−9.05120e6	−0.870084	−0.435042	−	0.900410i	$-0.643266\pi$
$641$	−9.05120e6	−0.870084	−0.435042	+	0.900410i	$0.643266\pi$
$642$	0	0
$643$	8.76840e6	0.836359	0.418179	−	0.908364i	$-0.362668\pi$
$643$	8.76840e6	0.836359	0.418179	+	0.908364i	$0.362668\pi$
$644$	139485.	0.0132529
$645$	0	0
$646$	1.04660e7	0.986734
$647$	−1.49567e6	−0.140467	−0.0702335	−	0.997531i	$-0.522374\pi$
$647$	−1.49567e6	−0.140467	−0.0702335	+	0.997531i	$0.522374\pi$
$648$	0	0
$649$	−2.41045e7	−2.24639
$650$	0	0
$651$	0	0
$652$	7.33411e6	0.675660
$653$	−1.17639e7	−1.07962	−0.539809	−	0.841788i	$-0.681503\pi$
$653$	−1.17639e7	−1.07962	−0.539809	+	0.841788i	$0.681503\pi$
$654$	0	0
$655$	0	0
$656$	2.45493e6	0.222731
$657$	0	0
$658$	1.38787e6	0.124964
$659$	3.36184e6	0.301553	0.150777	−	0.988568i	$-0.451823\pi$
$659$	3.36184e6	0.301553	0.150777	+	0.988568i	$0.451823\pi$
$660$	0	0
$661$	−1.28140e7	−1.14072	−0.570361	−	0.821394i	$-0.693197\pi$
$661$	−1.28140e7	−1.14072	−0.570361	+	0.821394i	$0.693197\pi$
$662$	−3.67458e6	−0.325883
$663$	0	0
$664$	−6.20595e6	−0.546246
$665$	0	0
$666$	0	0
$667$	235381.	0.0204859
$668$	3.11098e6	0.269746
$669$	0	0
$670$	0	0
$671$	−2.46164e7	−2.11066
$672$	0	0
$673$	−1.07843e7	−0.917813	−0.458907	−	0.888485i	$-0.651759\pi$
$673$	−1.07843e7	−0.917813	−0.458907	+	0.888485i	$0.651759\pi$
$674$	−1.42219e6	−0.120589
$675$	0	0
$676$	1.33645e7	1.12483
$677$	1.72031e7	1.44257	0.721283	−	0.692640i	$-0.243553\pi$
$677$	1.72031e7	1.44257	0.721283	+	0.692640i	$0.243553\pi$
$678$	0	0
$679$	−101536.	−0.00845173
$680$	0	0
$681$	0	0
$682$	9.95029e6	0.819171
$683$	−5.61037e6	−0.460193	−0.230096	−	0.973168i	$-0.573904\pi$
$683$	−5.61037e6	−0.460193	−0.230096	+	0.973168i	$0.573904\pi$
$684$	0	0
$685$	0	0
$686$	2.32312e6	0.188478
$687$	0	0
$688$	−1.35691e6	−0.109290
$689$	1.25882e6	0.101022
$690$	0	0
$691$	−3.89064e6	−0.309974	−0.154987	−	0.987916i	$-0.549534\pi$
$691$	−3.89064e6	−0.309974	−0.154987	+	0.987916i	$0.549534\pi$
$692$	−5.15555e6	−0.409270
$693$	0	0
$694$	7.43040e6	0.585617
$695$	0	0
$696$	0	0
$697$	−1.11814e7	−0.871798
$698$	3.80231e6	0.295399
$699$	0	0
$700$	0	0
$701$	−2.02753e7	−1.55838	−0.779190	−	0.626788i	$-0.784369\pi$
$701$	−2.02753e7	−1.55838	−0.779190	+	0.626788i	$0.784369\pi$
$702$	0	0
$703$	−1.56893e7	−1.19734
$704$	2.64240e6	0.200940
$705$	0	0
$706$	3.07028e6	0.231828
$707$	−1.79451e6	−0.135020
$708$	0	0
$709$	−1.60535e6	−0.119938	−0.0599688	−	0.998200i	$-0.519100\pi$
$709$	−1.60535e6	−0.119938	−0.0599688	+	0.998200i	$0.519100\pi$
$710$	0	0
$711$	0	0
$712$	3.82077e6	0.282456
$713$	−1.92800e6	−0.142031
$714$	0	0
$715$	0	0
$716$	6.88413e6	0.501842
$717$	0	0
$718$	643583.	0.0465901
$719$	1.04392e7	0.753090	0.376545	−	0.926398i	$-0.377112\pi$
$719$	1.04392e7	0.753090	0.376545	+	0.926398i	$0.377112\pi$
$720$	0	0
$721$	3.11326e6	0.223037
$722$	1.02377e7	0.730906
$723$	0	0
$724$	3.88035e6	0.275122
$725$	0	0
$726$	0	0
$727$	−2.36879e7	−1.66223	−0.831114	−	0.556102i	$-0.812297\pi$
$727$	−2.36879e7	−1.66223	−0.831114	+	0.556102i	$0.812297\pi$
$728$	1.22573e6	0.0857167
$729$	0	0
$730$	0	0
$731$	6.18029e6	0.427775
$732$	0	0
$733$	1.14116e7	0.784487	0.392244	−	0.919861i	$-0.371699\pi$
$733$	1.14116e7	0.784487	0.392244	+	0.919861i	$0.371699\pi$
$734$	−808523.	−0.0553927
$735$	0	0
$736$	−512000.	−0.0348398
$737$	−2.33733e7	−1.58508
$738$	0	0
$739$	2.57126e7	1.73195	0.865975	−	0.500088i	$-0.166699\pi$
$739$	2.57126e7	1.73195	0.865975	+	0.500088i	$0.166699\pi$
$740$	0	0
$741$	0	0
$742$	79924.8	0.00532932
$743$	−7.87830e6	−0.523553	−0.261776	−	0.965129i	$-0.584308\pi$
$743$	−7.87830e6	−0.523553	−0.261776	+	0.965129i	$0.584308\pi$
$744$	0	0
$745$	0	0
$746$	5.95732e6	0.391926
$747$	0	0
$748$	−1.20353e7	−0.786508
$749$	−130976.	−0.00853076
$750$	0	0
$751$	−2.69649e7	−1.74461	−0.872306	−	0.488961i	$-0.837376\pi$
$751$	−2.69649e7	−1.74461	−0.872306	+	0.488961i	$0.837376\pi$
$752$	−5.09440e6	−0.328510
$753$	0	0
$754$	2.06842e6	0.132498
$755$	0	0
$756$	0	0
$757$	−3.92135e6	−0.248712	−0.124356	−	0.992238i	$-0.539686\pi$
$757$	−3.92135e6	−0.248712	−0.124356	+	0.992238i	$0.539686\pi$
$758$	1.07689e7	0.680765
$759$	0	0
$760$	0	0
$761$	2.84849e7	1.78301	0.891504	−	0.453013i	$-0.149651\pi$
$761$	2.84849e7	1.78301	0.891504	+	0.453013i	$0.149651\pi$
$762$	0	0
$763$	1.01541e6	0.0631440
$764$	−4.65489e6	−0.288520
$765$	0	0
$766$	2.26040e6	0.139192
$767$	4.10427e7	2.51911
$768$	0	0
$769$	−5.18935e6	−0.316444	−0.158222	−	0.987404i	$-0.550576\pi$
$769$	−5.18935e6	−0.316444	−0.158222	+	0.987404i	$0.550576\pi$
$770$	0	0
$771$	0	0
$772$	−1.15689e7	−0.698631
$773$	1.88767e7	1.13626	0.568130	−	0.822939i	$-0.307667\pi$
$773$	1.88767e7	1.13626	0.568130	+	0.822939i	$0.307667\pi$
$774$	0	0
$775$	0	0
$776$	372703.	0.0222182
$777$	0	0
$778$	2.18635e7	1.29500
$779$	−2.15190e7	−1.27051
$780$	0	0
$781$	−1.07306e7	−0.629501
$782$	2.33200e6	0.136368
$783$	0	0
$784$	−4.22477e6	−0.245478
$785$	0	0
$786$	0	0
$787$	1.73159e7	0.996570	0.498285	−	0.867013i	$-0.333963\pi$
$787$	1.73159e7	0.996570	0.498285	+	0.867013i	$0.333963\pi$
$788$	−9.59421e6	−0.550419
$789$	0	0
$790$	0	0
$791$	3.78900e6	0.215319
$792$	0	0
$793$	4.19144e7	2.36690
$794$	1.60305e7	0.902393
$795$	0	0
$796$	1.04029e7	0.581934
$797$	−2.62525e7	−1.46394	−0.731972	−	0.681335i	$-0.761400\pi$
$797$	−2.62525e7	−1.46394	−0.731972	+	0.681335i	$0.761400\pi$
$798$	0	0
$799$	2.32034e7	1.28583
$800$	0	0
$801$	0	0
$802$	−1.57187e7	−0.862939
$803$	4.47670e7	2.45002
$804$	0	0
$805$	0	0
$806$	−1.69424e7	−0.918622
$807$	0	0
$808$	6.58703e6	0.354945
$809$	−3.02992e6	−0.162765	−0.0813824	−	0.996683i	$-0.525933\pi$
$809$	−3.02992e6	−0.162765	−0.0813824	+	0.996683i	$0.525933\pi$
$810$	0	0
$811$	9.81951e6	0.524249	0.262124	−	0.965034i	$-0.415577\pi$
$811$	9.81951e6	0.524249	0.262124	+	0.965034i	$0.415577\pi$
$812$	131328.	0.00698984
$813$	0	0
$814$	1.80418e7	0.954374
$815$	0	0
$816$	0	0
$817$	1.18941e7	0.623416
$818$	−4.60113e6	−0.240426
$819$	0	0
$820$	0	0
$821$	5.80196e6	0.300411	0.150206	−	0.988655i	$-0.452006\pi$
$821$	5.80196e6	0.300411	0.150206	+	0.988655i	$0.452006\pi$
$822$	0	0
$823$	−1.31473e7	−0.676606	−0.338303	−	0.941037i	$-0.609853\pi$
$823$	−1.31473e7	−0.676606	−0.338303	+	0.941037i	$0.609853\pi$
$824$	−1.14277e7	−0.586329
$825$	0	0
$826$	2.60589e6	0.132894
$827$	−2.77989e7	−1.41340	−0.706698	−	0.707515i	$-0.749816\pi$
$827$	−2.77989e7	−1.41340	−0.706698	+	0.707515i	$0.749816\pi$
$828$	0	0
$829$	−1.08971e7	−0.550714	−0.275357	−	0.961342i	$-0.588796\pi$
$829$	−1.08971e7	−0.550714	−0.275357	+	0.961342i	$0.588796\pi$
$830$	0	0
$831$	0	0
$832$	−4.49922e6	−0.225335
$833$	1.92425e7	0.960835
$834$	0	0
$835$	0	0
$836$	−2.31623e7	−1.14621
$837$	0	0
$838$	−1.98359e7	−0.975758
$839$	1.13976e7	0.558995	0.279497	−	0.960146i	$-0.409832\pi$
$839$	1.13976e7	0.558995	0.279497	+	0.960146i	$0.409832\pi$
$840$	0	0
$841$	−2.02895e7	−0.989195
$842$	1.15631e7	0.562075
$843$	0	0
$844$	1.57811e7	0.762570
$845$	0	0
$846$	0	0
$847$	−4.44826e6	−0.213050
$848$	−293376.	−0.0140099
$849$	0	0
$850$	0	0
$851$	−3.49584e6	−0.165473
$852$	0	0
$853$	4.97544e6	0.234131	0.117065	−	0.993124i	$-0.462651\pi$
$853$	4.97544e6	0.234131	0.117065	+	0.993124i	$0.462651\pi$
$854$	2.66123e6	0.124864
$855$	0	0
$856$	480768.	0.0224260
$857$	2.38632e7	1.10988	0.554942	−	0.831889i	$-0.312741\pi$
$857$	2.38632e7	1.10988	0.554942	+	0.831889i	$0.312741\pi$
$858$	0	0
$859$	1.49286e7	0.690298	0.345149	−	0.938548i	$-0.387828\pi$
$859$	1.49286e7	0.690298	0.345149	+	0.938548i	$0.387828\pi$
$860$	0	0
$861$	0	0
$862$	7.93668e6	0.363807
$863$	−9.90376e6	−0.452661	−0.226330	−	0.974051i	$-0.572673\pi$
$863$	−9.90376e6	−0.452661	−0.226330	+	0.974051i	$0.572673\pi$
$864$	0	0
$865$	0	0
$866$	−1.50249e7	−0.680795
$867$	0	0
$868$	−1.07571e6	−0.0484612
$869$	−1.33307e7	−0.598830
$870$	0	0
$871$	3.97979e7	1.77752
$872$	−3.72723e6	−0.165995
$873$	0	0
$874$	4.48800e6	0.198735
$875$	0	0
$876$	0	0
$877$	−3.84422e6	−0.168775	−0.0843877	−	0.996433i	$-0.526893\pi$
$877$	−3.84422e6	−0.168775	−0.0843877	+	0.996433i	$0.526893\pi$
$878$	7.79587e6	0.341294
$879$	0	0
$880$	0	0
$881$	2.94025e7	1.27627	0.638137	−	0.769922i	$-0.279705\pi$
$881$	2.94025e7	1.27627	0.638137	+	0.769922i	$0.279705\pi$
$882$	0	0
$883$	−6.24156e6	−0.269396	−0.134698	−	0.990887i	$-0.543006\pi$
$883$	−6.24156e6	−0.269396	−0.134698	+	0.990887i	$0.543006\pi$
$884$	2.04925e7	0.881993
$885$	0	0
$886$	2.29679e7	0.982963
$887$	1.67076e7	0.713025	0.356512	−	0.934291i	$-0.383966\pi$
$887$	1.67076e7	0.713025	0.356512	+	0.934291i	$0.383966\pi$
$888$	0	0
$889$	−3.70242e6	−0.157120
$890$	0	0
$891$	0	0
$892$	1.11105e7	0.467544
$893$	4.46556e7	1.87390
$894$	0	0
$895$	0	0
$896$	−285665.	−0.0118874
$897$	0	0
$898$	−2.15462e6	−0.0891621
$899$	−1.81525e6	−0.0749098
$900$	0	0
$901$	1.33624e6	0.0548367
$902$	2.47456e7	1.01270
$903$	0	0
$904$	−1.39081e7	−0.566039
$905$	0	0
$906$	0	0
$907$	7.14197e6	0.288270	0.144135	−	0.989558i	$-0.453960\pi$
$907$	7.14197e6	0.288270	0.144135	+	0.989558i	$0.453960\pi$
$908$	6.96640e6	0.280410
$909$	0	0
$910$	0	0
$911$	−3.51083e7	−1.40157	−0.700784	−	0.713374i	$-0.747166\pi$
$911$	−3.51083e7	−1.40157	−0.700784	+	0.713374i	$0.747166\pi$
$912$	0	0
$913$	−6.25557e7	−2.48365
$914$	6.21837e6	0.246213
$915$	0	0
$916$	−1.85796e7	−0.731641
$917$	4.86917e6	0.191219
$918$	0	0
$919$	2.66789e7	1.04203	0.521014	−	0.853548i	$-0.325554\pi$
$919$	2.66789e7	1.04203	0.521014	+	0.853548i	$0.325554\pi$
$920$	0	0
$921$	0	0
$922$	1.68978e7	0.654641
$923$	1.82710e7	0.705925
$924$	0	0
$925$	0	0
$926$	1.90939e7	0.731756
$927$	0	0
$928$	−482059.	−0.0183751
$929$	6.04426e6	0.229776	0.114888	−	0.993378i	$-0.463349\pi$
$929$	6.04426e6	0.229776	0.114888	+	0.993378i	$0.463349\pi$
$930$	0	0
$931$	3.70327e7	1.40027
$932$	1.94085e7	0.731902
$933$	0	0
$934$	−3.64673e7	−1.36784
$935$	0	0
$936$	0	0
$937$	3.60422e6	0.134110	0.0670551	−	0.997749i	$-0.478640\pi$
$937$	3.60422e6	0.134110	0.0670551	+	0.997749i	$0.478640\pi$
$938$	2.52685e6	0.0937718
$939$	0	0
$940$	0	0
$941$	1.30244e7	0.479495	0.239747	−	0.970835i	$-0.422935\pi$
$941$	1.30244e7	0.479495	0.239747	+	0.970835i	$0.422935\pi$
$942$	0	0
$943$	−4.79479e6	−0.175586
$944$	−9.56531e6	−0.349357
$945$	0	0
$946$	−1.36776e7	−0.496914
$947$	3.17698e7	1.15117	0.575585	−	0.817742i	$-0.304774\pi$
$947$	3.17698e7	1.15117	0.575585	+	0.817742i	$0.304774\pi$
$948$	0	0
$949$	−7.62250e7	−2.74746
$950$	0	0
$951$	0	0
$952$	1.30111e6	0.0465289
$953$	2.25254e7	0.803417	0.401708	−	0.915768i	$-0.368417\pi$
$953$	2.25254e7	0.803417	0.401708	+	0.915768i	$0.368417\pi$
$954$	0	0
$955$	0	0
$956$	2.24085e7	0.792991
$957$	0	0
$958$	−1.68873e7	−0.594492
$959$	5.89173e6	0.206869
$960$	0	0
$961$	−1.37604e7	−0.480643
$962$	−3.07198e7	−1.07024
$963$	0	0
$964$	7.54384e6	0.261457
$965$	0	0
$966$	0	0
$967$	169805.	0.00583963	0.00291981	−	0.999996i	$-0.499071\pi$
$967$	169805.	0.00583963	0.00291981	+	0.999996i	$0.499071\pi$
$968$	1.63280e7	0.560073
$969$	0	0
$970$	0	0
$971$	164296.	0.00559214	0.00279607	−	0.999996i	$-0.499110\pi$
$971$	164296.	0.00559214	0.00279607	+	0.999996i	$0.499110\pi$
$972$	0	0
$973$	−1.61488e6	−0.0546839
$974$	2.00586e6	0.0677491
$975$	0	0
$976$	−9.76845e6	−0.328247
$977$	−2.00989e7	−0.673653	−0.336827	−	0.941567i	$-0.609354\pi$
$977$	−2.00989e7	−0.673653	−0.336827	+	0.941567i	$0.609354\pi$
$978$	0	0
$979$	3.85132e7	1.28426
$980$	0	0
$981$	0	0
$982$	1.04574e7	0.346054
$983$	−1.39716e7	−0.461172	−0.230586	−	0.973052i	$-0.574064\pi$
$983$	−1.39716e7	−0.461172	−0.230586	+	0.973052i	$0.574064\pi$
$984$	0	0
$985$	0	0
$986$	2.19563e6	0.0719228
$987$	0	0
$988$	3.94385e7	1.28537
$989$	2.65021e6	0.0861568
$990$	0	0
$991$	140392.	0.00454107	0.00227054	−	0.999997i	$-0.499277\pi$
$991$	140392.	0.00454107	0.00227054	+	0.999997i	$0.499277\pi$
$992$	3.94854e6	0.127397
$993$	0	0
$994$	1.16006e6	0.0372405
$995$	0	0
$996$	0	0
$997$	−1.79809e7	−0.572894	−0.286447	−	0.958096i	$-0.592474\pi$
$997$	−1.79809e7	−0.572894	−0.286447	+	0.958096i	$0.592474\pi$
$998$	2.27296e7	0.722379
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.be.1.1		2
3.2	odd	2		450.6.a.z.1.1		2
5.2	odd	4		90.6.c.d.19.3	yes	4
5.3	odd	4		90.6.c.d.19.1	✓	4
5.4	even	2		450.6.a.z.1.2		2
15.2	even	4		90.6.c.d.19.2	yes	4
15.8	even	4		90.6.c.d.19.4	yes	4
15.14	odd	2	inner	450.6.a.be.1.2		2
20.3	even	4		720.6.f.j.289.2		4
20.7	even	4		720.6.f.j.289.1		4
60.23	odd	4		720.6.f.j.289.3		4
60.47	odd	4		720.6.f.j.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.c.d.19.1	✓	4	5.3	odd	4
90.6.c.d.19.2	yes	4	15.2	even	4
90.6.c.d.19.3	yes	4	5.2	odd	4
90.6.c.d.19.4	yes	4	15.8	even	4
450.6.a.z.1.1		2	3.2	odd	2
450.6.a.z.1.2		2	5.4	even	2
450.6.a.be.1.1		2	1.1	even	1	trivial
450.6.a.be.1.2		2	15.14	odd	2	inner
720.6.f.j.289.1		4	20.7	even	4
720.6.f.j.289.2		4	20.3	even	4
720.6.f.j.289.3		4	60.23	odd	4
720.6.f.j.289.4		4	60.47	odd	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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -17.4356 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -17.4356 q^{7} +64.0000 q^{8} +645.117 q^{11} -1098.44 q^{13} -69.7424 q^{14} +256.000 q^{16} -1166.00 q^{17} -2244.00 q^{19} +2580.47 q^{22} -500.000 q^{23} -4393.77 q^{26} -278.970 q^{28} -470.761 q^{29} +3856.00 q^{31} +1024.00 q^{32} -4664.00 q^{34} +6991.67 q^{37} -8976.00 q^{38} +9589.58 q^{41} -5300.42 q^{43} +10321.9 q^{44} -2000.00 q^{46} -19900.0 q^{47} -16503.0 q^{49} -17575.1 q^{52} -1146.00 q^{53} -1115.88 q^{56} -1883.04 q^{58} -37364.5 q^{59} -38158.0 q^{61} +15424.0 q^{62} +4096.00 q^{64} -36231.2 q^{67} -18656.0 q^{68} -16633.6 q^{71} +69393.7 q^{73} +27966.7 q^{74} -35904.0 q^{76} -11248.0 q^{77} -20664.0 q^{79} +38358.3 q^{82} -96968.0 q^{83} -21201.7 q^{86} +41287.5 q^{88} +59699.5 q^{89} +19152.0 q^{91} -8000.00 q^{92} -79600.0 q^{94} +5823.49 q^{97} -66012.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} + 128 q^{8} + 512 q^{16} - 2332 q^{17} - 4488 q^{19} - 1000 q^{23} + 7712 q^{31} + 2048 q^{32} - 9328 q^{34} - 17952 q^{38} - 4000 q^{46} - 39800 q^{47} - 33006 q^{49} - 2292 q^{53} - 76316 q^{61} + 30848 q^{62} + 8192 q^{64} - 37312 q^{68} - 71808 q^{76} - 22496 q^{77} - 41328 q^{79} - 193936 q^{83} + 38304 q^{91} - 16000 q^{92} - 159200 q^{94} - 132024 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 + 128 * q^8 + 512 * q^16 - 2332 * q^17 - 4488 * q^19 - 1000 * q^23 + 7712 * q^31 + 2048 * q^32 - 9328 * q^34 - 17952 * q^38 - 4000 * q^46 - 39800 * q^47 - 33006 * q^49 - 2292 * q^53 - 76316 * q^61 + 30848 * q^62 + 8192 * q^64 - 37312 * q^68 - 71808 * q^76 - 22496 * q^77 - 41328 * q^79 - 193936 * q^83 + 38304 * q^91 - 16000 * q^92 - 159200 * q^94 - 132024 * q^98

Introduction

L-functions

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