LMFDB - Embedded newform 150.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,6,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	150.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:	$24.0575729719$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +47.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +47.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +222.000 q^{11} +144.000 q^{12} +101.000 q^{13} +188.000 q^{14} +256.000 q^{16} +162.000 q^{17} +324.000 q^{18} +1685.00 q^{19} +423.000 q^{21} +888.000 q^{22} +306.000 q^{23} +576.000 q^{24} +404.000 q^{26} +729.000 q^{27} +752.000 q^{28} +7890.00 q^{29} -8593.00 q^{31} +1024.00 q^{32} +1998.00 q^{33} +648.000 q^{34} +1296.00 q^{36} +8642.00 q^{37} +6740.00 q^{38} +909.000 q^{39} -18168.0 q^{41} +1692.00 q^{42} +14351.0 q^{43} +3552.00 q^{44} +1224.00 q^{46} -1098.00 q^{47} +2304.00 q^{48} -14598.0 q^{49} +1458.00 q^{51} +1616.00 q^{52} +17916.0 q^{53} +2916.00 q^{54} +3008.00 q^{56} +15165.0 q^{57} +31560.0 q^{58} +17610.0 q^{59} -21853.0 q^{61} -34372.0 q^{62} +3807.00 q^{63} +4096.00 q^{64} +7992.00 q^{66} +107.000 q^{67} +2592.00 q^{68} +2754.00 q^{69} -40728.0 q^{71} +5184.00 q^{72} +34706.0 q^{73} +34568.0 q^{74} +26960.0 q^{76} +10434.0 q^{77} +3636.00 q^{78} -69160.0 q^{79} +6561.00 q^{81} -72672.0 q^{82} -108534. q^{83} +6768.00 q^{84} +57404.0 q^{86} +71010.0 q^{87} +14208.0 q^{88} +35040.0 q^{89} +4747.00 q^{91} +4896.00 q^{92} -77337.0 q^{93} -4392.00 q^{94} +9216.00 q^{96} -823.000 q^{97} -58392.0 q^{98} +17982.0 q^{99} +O(q^{100})\)

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	47.0000	0.362537	0.181269	−	0.983434i	$-0.441980\pi$
$7$	47.0000	0.362537	0.181269	+	0.983434i	$0.441980\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	0	0
$11$	222.000	0.553186	0.276593	−	0.960987i	$-0.410795\pi$
$11$	222.000	0.553186	0.276593	+	0.960987i	$0.410795\pi$
$12$	144.000	0.288675
$13$	101.000	0.165754	0.0828768	−	0.996560i	$-0.473589\pi$
$13$	101.000	0.165754	0.0828768	+	0.996560i	$0.473589\pi$
$14$	188.000	0.256353
$15$	0	0
$16$	256.000	0.250000
$17$	162.000	0.135954	0.0679771	−	0.997687i	$-0.478346\pi$
$17$	162.000	0.135954	0.0679771	+	0.997687i	$0.478346\pi$
$18$	324.000	0.235702
$19$	1685.00	1.07082	0.535409	−	0.844593i	$-0.320157\pi$
$19$	1685.00	1.07082	0.535409	+	0.844593i	$0.320157\pi$
$20$	0	0
$21$	423.000	0.209311
$22$	888.000	0.391162
$23$	306.000	0.120615	0.0603076	−	0.998180i	$-0.480792\pi$
$23$	306.000	0.120615	0.0603076	+	0.998180i	$0.480792\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	404.000	0.117206
$27$	729.000	0.192450
$28$	752.000	0.181269
$29$	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	$0.796759\pi$
$32$	1024.00	0.176777
$33$	1998.00	0.319382
$34$	648.000	0.0961342
$35$	0	0
$36$	1296.00	0.166667
$37$	8642.00	1.03779	0.518896	−	0.854838i	$-0.326343\pi$
$37$	8642.00	1.03779	0.518896	+	0.854838i	$0.326343\pi$
$38$	6740.00	0.757183
$39$	909.000	0.0956979
$40$	0	0
$41$	−18168.0	−1.68790	−0.843951	−	0.536420i	$-0.819777\pi$
$41$	−18168.0	−1.68790	−0.843951	+	0.536420i	$0.819777\pi$
$42$	1692.00	0.148005
$43$	14351.0	1.18362	0.591808	−	0.806079i	$-0.298414\pi$
$43$	14351.0	1.18362	0.591808	+	0.806079i	$0.298414\pi$
$44$	3552.00	0.276593
$45$	0	0
$46$	1224.00	0.0852878
$47$	−1098.00	−0.0725033	−0.0362516	−	0.999343i	$-0.511542\pi$
$47$	−1098.00	−0.0725033	−0.0362516	+	0.999343i	$0.511542\pi$
$48$	2304.00	0.144338
$49$	−14598.0	−0.868567
$50$	0	0
$51$	1458.00	0.0784932
$52$	1616.00	0.0828768
$53$	17916.0	0.876095	0.438048	−	0.898952i	$-0.355670\pi$
$53$	17916.0	0.876095	0.438048	+	0.898952i	$0.355670\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	3008.00	0.128176
$57$	15165.0	0.618237
$58$	31560.0	1.23188
$59$	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	$0.622691\pi$
$62$	−34372.0	−1.13560
$63$	3807.00	0.120846
$64$	4096.00	0.125000
$65$	0	0
$66$	7992.00	0.225837
$67$	107.000	0.00291204	0.00145602	−	0.999999i	$-0.499537\pi$
$67$	107.000	0.00291204	0.00145602	+	0.999999i	$0.499537\pi$
$68$	2592.00	0.0679771
$69$	2754.00	0.0696372
$70$	0	0
$71$	−40728.0	−0.958842	−0.479421	−	0.877585i	$-0.659153\pi$
$71$	−40728.0	−0.958842	−0.479421	+	0.877585i	$0.659153\pi$
$72$	5184.00	0.117851
$73$	34706.0	0.762250	0.381125	−	0.924524i	$-0.375537\pi$
$73$	34706.0	0.762250	0.381125	+	0.924524i	$0.375537\pi$
$74$	34568.0	0.733829
$75$	0	0
$76$	26960.0	0.535409
$77$	10434.0	0.200551
$78$	3636.00	0.0676686
$79$	−69160.0	−1.24677	−0.623386	−	0.781914i	$-0.714244\pi$
$79$	−69160.0	−1.24677	−0.623386	+	0.781914i	$0.714244\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−72672.0	−1.19353
$83$	−108534.	−1.72930	−0.864650	−	0.502374i	$-0.832460\pi$
$83$	−108534.	−1.72930	−0.864650	+	0.502374i	$0.832460\pi$
$84$	6768.00	0.104656
$85$	0	0
$86$	57404.0	0.836943
$87$	71010.0	1.00582
$88$	14208.0	0.195581
$89$	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
$92$	4896.00	0.0603076
$93$	−77337.0	−0.927214
$94$	−4392.00	−0.0512676
$95$	0	0
$96$	9216.00	0.102062
$97$	−823.000	−0.00888118	−0.00444059	−	0.999990i	$-0.501413\pi$
$97$	−823.000	−0.00888118	−0.00444059	+	0.999990i	$0.501413\pi$
$98$	−58392.0	−0.614169
$99$	17982.0	0.184395
$100$	0	0
$101$	−33828.0	−0.329969	−0.164984	−	0.986296i	$-0.552757\pi$
$101$	−33828.0	−0.329969	−0.164984	+	0.986296i	$0.552757\pi$
$102$	5832.00	0.0555031
$103$	−133444.	−1.23938	−0.619692	−	0.784845i	$-0.712743\pi$
$103$	−133444.	−1.23938	−0.619692	+	0.784845i	$0.712743\pi$
$104$	6464.00	0.0586028
$105$	0	0
$106$	71664.0	0.619493
$107$	81252.0	0.686080	0.343040	−	0.939321i	$-0.388543\pi$
$107$	81252.0	0.686080	0.343040	+	0.939321i	$0.388543\pi$
$108$	11664.0	0.0962250
$109$	−217015.	−1.74954	−0.874769	−	0.484540i	$-0.838987\pi$
$109$	−217015.	−1.74954	−0.874769	+	0.484540i	$0.838987\pi$
$110$	0	0
$111$	77778.0	0.599169
$112$	12032.0	0.0906343
$113$	−138324.	−1.01906	−0.509532	−	0.860452i	$-0.670181\pi$
$113$	−138324.	−1.01906	−0.509532	+	0.860452i	$0.670181\pi$
$114$	60660.0	0.437160
$115$	0	0
$116$	126240.	0.871068
$117$	8181.00	0.0552512
$118$	70440.0	0.465709
$119$	7614.00	0.0492885
$120$	0	0
$121$	−111767.	−0.693985
$122$	−87412.0	−0.531706
$123$	−163512.	−0.974511
$124$	−137488.	−0.802991
$125$	0	0
$126$	15228.0	0.0854509
$127$	−256048.	−1.40868	−0.704340	−	0.709863i	$-0.748757\pi$
$127$	−256048.	−1.40868	−0.704340	+	0.709863i	$0.748757\pi$
$128$	16384.0	0.0883883
$129$	129159.	0.683361
$130$	0	0
$131$	118452.	0.603065	0.301533	−	0.953456i	$-0.402502\pi$
$131$	118452.	0.603065	0.301533	+	0.953456i	$0.402502\pi$
$132$	31968.0	0.159691
$133$	79195.0	0.388212
$134$	428.000	0.00205912
$135$	0	0
$136$	10368.0	0.0480671
$137$	−13218.0	−0.0601678	−0.0300839	−	0.999547i	$-0.509577\pi$
$137$	−13218.0	−0.0601678	−0.0300839	+	0.999547i	$0.509577\pi$
$138$	11016.0	0.0492409
$139$	−350740.	−1.53974	−0.769872	−	0.638199i	$-0.779680\pi$
$139$	−350740.	−1.53974	−0.769872	+	0.638199i	$0.779680\pi$
$140$	0	0
$141$	−9882.00	−0.0418598
$142$	−162912.	−0.678004
$143$	22422.0	0.0916926
$144$	20736.0	0.0833333
$145$	0	0
$146$	138824.	0.538992
$147$	−131382.	−0.501467
$148$	138272.	0.518896
$149$	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.	0.378592
$153$	13122.0	0.0453181
$154$	41736.0	0.141811
$155$	0	0
$156$	14544.0	0.0478489
$157$	−349993.	−1.13321	−0.566605	−	0.823990i	$-0.691743\pi$
$157$	−349993.	−1.13321	−0.566605	+	0.823990i	$0.691743\pi$
$158$	−276640.	−0.881601
$159$	161244.	0.505814
$160$	0	0
$161$	14382.0	0.0437275
$162$	26244.0	0.0785674
$163$	192581.	0.567733	0.283867	−	0.958864i	$-0.408383\pi$
$163$	192581.	0.567733	0.283867	+	0.958864i	$0.408383\pi$
$164$	−290688.	−0.843951
$165$	0	0
$166$	−434136.	−1.22280
$167$	580692.	1.61122	0.805610	−	0.592447i	$-0.201838\pi$
$167$	580692.	1.61122	0.805610	+	0.592447i	$0.201838\pi$
$168$	27072.0	0.0740026
$169$	−361092.	−0.972526
$170$	0	0
$171$	136485.	0.356940
$172$	229616.	0.591808
$173$	738126.	1.87506	0.937530	−	0.347904i	$-0.113106\pi$
$173$	738126.	1.87506	0.937530	+	0.347904i	$0.113106\pi$
$174$	284040.	0.711224
$175$	0	0
$176$	56832.0	0.138297
$177$	158490.	0.380250
$178$	140160.	0.331569
$179$	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	$0.623370\pi$
$182$	18988.0	0.0424914
$183$	−196677.	−0.434136
$184$	19584.0	0.0426439
$185$	0	0
$186$	−309348.	−0.655639
$187$	35964.0	0.0752080
$188$	−17568.0	−0.0362516
$189$	34263.0	0.0697703
$190$	0	0
$191$	−40638.0	−0.0806026	−0.0403013	−	0.999188i	$-0.512832\pi$
$191$	−40638.0	−0.0806026	−0.0403013	+	0.999188i	$0.512832\pi$
$192$	36864.0	0.0721688
$193$	494651.	0.955885	0.477942	−	0.878391i	$-0.341383\pi$
$193$	494651.	0.955885	0.477942	+	0.878391i	$0.341383\pi$
$194$	−3292.00	−0.00627994
$195$	0	0
$196$	−233568.	−0.434283
$197$	552342.	1.01401	0.507005	−	0.861943i	$-0.330752\pi$
$197$	552342.	1.01401	0.507005	+	0.861943i	$0.330752\pi$
$198$	71928.0	0.130387
$199$	685625.	1.22731	0.613655	−	0.789575i	$-0.289699\pi$
$199$	685625.	1.22731	0.613655	+	0.789575i	$0.289699\pi$
$200$	0	0
$201$	963.000	0.00168126
$202$	−135312.	−0.233323
$203$	370830.	0.631589
$204$	23328.0	0.0392466
$205$	0	0
$206$	−533776.	−0.876377
$207$	24786.0	0.0402050
$208$	25856.0	0.0414384
$209$	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.	0.438048
$213$	−366552.	−0.553588
$214$	325008.	0.485132
$215$	0	0
$216$	46656.0	0.0680414
$217$	−403871.	−0.582228
$218$	−868060.	−1.23711
$219$	312354.	0.440085
$220$	0	0
$221$	16362.0	0.0225349
$222$	311112.	0.423676
$223$	169271.	0.227940	0.113970	−	0.993484i	$-0.463643\pi$
$223$	169271.	0.227940	0.113970	+	0.993484i	$0.463643\pi$
$224$	48128.0	0.0640882
$225$	0	0
$226$	−553296.	−0.720587
$227$	−46488.0	−0.0598792	−0.0299396	−	0.999552i	$-0.509532\pi$
$227$	−46488.0	−0.0598792	−0.0299396	+	0.999552i	$0.509532\pi$
$228$	242640.	0.309119
$229$	−90115.0	−0.113556	−0.0567778	−	0.998387i	$-0.518083\pi$
$229$	−90115.0	−0.113556	−0.0567778	+	0.998387i	$0.518083\pi$
$230$	0	0
$231$	93906.0	0.115788
$232$	504960.	0.615938
$233$	1.06414e6	1.28413	0.642063	−	0.766652i	$-0.278079\pi$
$233$	1.06414e6	1.28413	0.642063	+	0.766652i	$0.278079\pi$
$234$	32724.0	0.0390685
$235$	0	0
$236$	281760.	0.329306
$237$	−622440.	−0.719825
$238$	30456.0	0.0348522
$239$	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.	−0.490722
$243$	59049.0	0.0641500
$244$	−349648.	−0.375973
$245$	0	0
$246$	−654048.	−0.689084
$247$	170185.	0.177492
$248$	−549952.	−0.567800
$249$	−976806.	−0.998412
$250$	0	0
$251$	−207708.	−0.208098	−0.104049	−	0.994572i	$-0.533180\pi$
$251$	−207708.	−0.208098	−0.104049	+	0.994572i	$0.533180\pi$
$252$	60912.0	0.0604229
$253$	67932.0	0.0667226
$254$	−1.02419e6	−0.996087
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.45319e6	−1.37243	−0.686213	−	0.727401i	$-0.740728\pi$
$257$	−1.45319e6	−1.37243	−0.686213	+	0.727401i	$0.740728\pi$
$258$	516636.	0.483209
$259$	406174.	0.376238
$260$	0	0
$261$	639090.	0.580712
$262$	473808.	0.426431
$263$	169296.	0.150924	0.0754618	−	0.997149i	$-0.475957\pi$
$263$	169296.	0.150924	0.0754618	+	0.997149i	$0.475957\pi$
$264$	127872.	0.112919
$265$	0	0
$266$	316780.	0.274507
$267$	315360.	0.270725
$268$	1712.00	0.00145602
$269$	−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	$0.389517\pi$
$272$	41472.0	0.0339886
$273$	42723.0	0.0346941
$274$	−52872.0	−0.0425451
$275$	0	0
$276$	44064.0	0.0348186
$277$	−546823.	−0.428201	−0.214100	−	0.976812i	$-0.568682\pi$
$277$	−546823.	−0.428201	−0.214100	+	0.976812i	$0.568682\pi$
$278$	−1.40296e6	−1.08876
$279$	−696033.	−0.535327
$280$	0	0
$281$	−1.09250e6	−0.825382	−0.412691	−	0.910871i	$-0.635411\pi$
$281$	−1.09250e6	−0.825382	−0.412691	+	0.910871i	$0.635411\pi$
$282$	−39528.0	−0.0295993
$283$	−2.48480e6	−1.84427	−0.922136	−	0.386865i	$-0.873558\pi$
$283$	−2.48480e6	−1.84427	−0.922136	+	0.386865i	$0.873558\pi$
$284$	−651648.	−0.479421
$285$	0	0
$286$	89688.0	0.0648365
$287$	−853896.	−0.611928
$288$	82944.0	0.0589256
$289$	−1.39361e6	−0.981516
$290$	0	0
$291$	−7407.00	−0.00512755
$292$	555296.	0.381125
$293$	−341394.	−0.232320	−0.116160	−	0.993231i	$-0.537059\pi$
$293$	−341394.	−0.232320	−0.116160	+	0.993231i	$0.537059\pi$
$294$	−525528.	−0.354591
$295$	0	0
$296$	553088.	0.366915
$297$	161838.	0.106461
$298$	439560.	0.286733
$299$	30906.0	0.0199924
$300$	0	0
$301$	674497.	0.429105
$302$	−690412.	−0.435603
$303$	−304452.	−0.190508
$304$	431360.	0.267705
$305$	0	0
$306$	52488.0	0.0320447
$307$	2.02898e6	1.22866	0.614329	−	0.789050i	$-0.289427\pi$
$307$	2.02898e6	1.22866	0.614329	+	0.789050i	$0.289427\pi$
$308$	166944.	0.100275
$309$	−1.20100e6	−0.715559
$310$	0	0
$311$	−206598.	−0.121123	−0.0605613	−	0.998164i	$-0.519289\pi$
$311$	−206598.	−0.121123	−0.0605613	+	0.998164i	$0.519289\pi$
$312$	58176.0	0.0338343
$313$	−3.34223e6	−1.92830	−0.964152	−	0.265352i	$-0.914512\pi$
$313$	−3.34223e6	−1.92830	−0.964152	+	0.265352i	$0.914512\pi$
$314$	−1.39997e6	−0.801300
$315$	0	0
$316$	−1.10656e6	−0.623386
$317$	−2.53289e6	−1.41569	−0.707844	−	0.706368i	$-0.750332\pi$
$317$	−2.53289e6	−1.41569	−0.707844	+	0.706368i	$0.750332\pi$
$318$	644976.	0.357664
$319$	1.75158e6	0.963725
$320$	0	0
$321$	731268.	0.396108
$322$	57528.0	0.0309200
$323$	272970.	0.145582
$324$	104976.	0.0555556
$325$	0	0
$326$	770324.	0.401448
$327$	−1.95314e6	−1.01010
$328$	−1.16275e6	−0.596764
$329$	−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	$0.451738\pi$
$332$	−1.73654e6	−0.864650
$333$	700002.	0.345930
$334$	2.32277e6	1.13930
$335$	0	0
$336$	108288.	0.0523278
$337$	209777.	0.100620	0.0503099	−	0.998734i	$-0.483979\pi$
$337$	209777.	0.100620	0.0503099	+	0.998734i	$0.483979\pi$
$338$	−1.44437e6	−0.687680
$339$	−1.24492e6	−0.588357
$340$	0	0
$341$	−1.90765e6	−0.888407
$342$	545940.	0.252394
$343$	−1.47603e6	−0.677425
$344$	918464.	0.418472
$345$	0	0
$346$	2.95250e6	1.32587
$347$	4.02166e6	1.79301	0.896503	−	0.443037i	$-0.146099\pi$
$347$	4.02166e6	1.79301	0.896503	+	0.443037i	$0.146099\pi$
$348$	1.13616e6	0.502911
$349$	8330.00	0.00366085	0.00183042	−	0.999998i	$-0.499417\pi$
$349$	8330.00	0.00366085	0.00183042	+	0.999998i	$0.499417\pi$
$350$	0	0
$351$	73629.0	0.0318993
$352$	227328.	0.0977904
$353$	1.95001e6	0.832912	0.416456	−	0.909156i	$-0.363272\pi$
$353$	1.95001e6	0.832912	0.416456	+	0.909156i	$0.363272\pi$
$354$	633960.	0.268877
$355$	0	0
$356$	560640.	0.234455
$357$	68526.0	0.0284567
$358$	1.98948e6	0.820412
$359$	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
$362$	−1.33265e6	−0.534497
$363$	−1.00590e6	−0.400673
$364$	75952.0	0.0300459
$365$	0	0
$366$	−786708.	−0.306981
$367$	2.86154e6	1.10901	0.554503	−	0.832181i	$-0.312908\pi$
$367$	2.86154e6	1.10901	0.554503	+	0.832181i	$0.312908\pi$
$368$	78336.0	0.0301538
$369$	−1.47161e6	−0.562634
$370$	0	0
$371$	842052.	0.317617
$372$	−1.23739e6	−0.463607
$373$	615311.	0.228993	0.114497	−	0.993424i	$-0.463474\pi$
$373$	615311.	0.228993	0.114497	+	0.993424i	$0.463474\pi$
$374$	143856.	0.0531801
$375$	0	0
$376$	−70272.0	−0.0256338
$377$	796890.	0.288765
$378$	137052.	0.0493351
$379$	5.39878e6	1.93062	0.965311	−	0.261103i	$-0.0840863\pi$
$379$	5.39878e6	1.93062	0.965311	+	0.261103i	$0.0840863\pi$
$380$	0	0
$381$	−2.30443e6	−0.813301
$382$	−162552.	−0.0569946
$383$	1.08688e6	0.378602	0.189301	−	0.981919i	$-0.439378\pi$
$383$	1.08688e6	0.378602	0.189301	+	0.981919i	$0.439378\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	1.97860e6	0.675913
$387$	1.16243e6	0.394539
$388$	−13168.0	−0.00444059
$389$	−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.	−0.307085
$393$	1.06607e6	0.348180
$394$	2.20937e6	0.717014
$395$	0	0
$396$	287712.	0.0921977
$397$	3.26591e6	1.03999	0.519993	−	0.854170i	$-0.325935\pi$
$397$	3.26591e6	1.03999	0.519993	+	0.854170i	$0.325935\pi$
$398$	2.74250e6	0.867839
$399$	712755.	0.224134
$400$	0	0
$401$	−4.27319e6	−1.32706	−0.663531	−	0.748149i	$-0.730943\pi$
$401$	−4.27319e6	−1.32706	−0.663531	+	0.748149i	$0.730943\pi$
$402$	3852.00	0.00118883
$403$	−867893.	−0.266197
$404$	−541248.	−0.164984
$405$	0	0
$406$	1.48332e6	0.446601
$407$	1.91852e6	0.574092
$408$	93312.0	0.0277515
$409$	−1.45188e6	−0.429162	−0.214581	−	0.976706i	$-0.568839\pi$
$409$	−1.45188e6	−0.429162	−0.214581	+	0.976706i	$0.568839\pi$
$410$	0	0
$411$	−118962.	−0.0347379
$412$	−2.13510e6	−0.619692
$413$	827670.	0.238771
$414$	99144.0	0.0284293
$415$	0	0
$416$	103424.	0.0293014
$417$	−3.15666e6	−0.888971
$418$	1.49628e6	0.418863
$419$	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	$0.680905\pi$
$422$	2.99791e6	0.819478
$423$	−88938.0	−0.0241678
$424$	1.14662e6	0.309746
$425$	0	0
$426$	−1.46621e6	−0.391446
$427$	−1.02709e6	−0.272608
$428$	1.30003e6	0.343040
$429$	201798.	0.0529387
$430$	0	0
$431$	−3.57500e6	−0.927006	−0.463503	−	0.886095i	$-0.653408\pi$
$431$	−3.57500e6	−0.927006	−0.463503	+	0.886095i	$0.653408\pi$
$432$	186624.	0.0481125
$433$	7.15969e6	1.83516	0.917581	−	0.397548i	$-0.130139\pi$
$433$	7.15969e6	1.83516	0.917581	+	0.397548i	$0.130139\pi$
$434$	−1.61548e6	−0.411698
$435$	0	0
$436$	−3.47224e6	−0.874769
$437$	515610.	0.129157
$438$	1.24942e6	0.311187
$439$	1.71790e6	0.425437	0.212719	−	0.977114i	$-0.431768\pi$
$439$	1.71790e6	0.425437	0.212719	+	0.977114i	$0.431768\pi$
$440$	0	0
$441$	−1.18244e6	−0.289522
$442$	65448.0	0.0159346
$443$	3.39670e6	0.822332	0.411166	−	0.911560i	$-0.365122\pi$
$443$	3.39670e6	0.822332	0.411166	+	0.911560i	$0.365122\pi$
$444$	1.24445e6	0.299584
$445$	0	0
$446$	677084.	0.161178
$447$	989010.	0.234116
$448$	192512.	0.0453172
$449$	−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
$452$	−2.21318e6	−0.509532
$453$	−1.55343e6	−0.355668
$454$	−185952.	−0.0423410
$455$	0	0
$456$	970560.	0.218580
$457$	−4.52814e6	−1.01421	−0.507106	−	0.861883i	$-0.669285\pi$
$457$	−4.52814e6	−1.01421	−0.507106	+	0.861883i	$0.669285\pi$
$458$	−360460.	−0.0802959
$459$	118098.	0.0261644
$460$	0	0
$461$	−1.27895e6	−0.280285	−0.140143	−	0.990131i	$-0.544756\pi$
$461$	−1.27895e6	−0.280285	−0.140143	+	0.990131i	$0.544756\pi$
$462$	375624.	0.0818744
$463$	−7.19862e6	−1.56062	−0.780310	−	0.625393i	$-0.784939\pi$
$463$	−7.19862e6	−1.56062	−0.780310	+	0.625393i	$0.784939\pi$
$464$	2.01984e6	0.435534
$465$	0	0
$466$	4.25654e6	0.908014
$467$	4.83034e6	1.02491	0.512455	−	0.858714i	$-0.328736\pi$
$467$	4.83034e6	1.02491	0.512455	+	0.858714i	$0.328736\pi$
$468$	130896.	0.0276256
$469$	5029.00	0.00105572
$470$	0	0
$471$	−3.14994e6	−0.654259
$472$	1.12704e6	0.232854
$473$	3.18592e6	0.654760
$474$	−2.48976e6	−0.508993
$475$	0	0
$476$	121824.	0.0246442
$477$	1.45120e6	0.292032
$478$	4.60632e6	0.922113
$479$	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
$482$	3.42487e6	0.671469
$483$	129438.	0.0252461
$484$	−1.78827e6	−0.346993
$485$	0	0
$486$	236196.	0.0453609
$487$	−5.16394e6	−0.986641	−0.493320	−	0.869848i	$-0.664217\pi$
$487$	−5.16394e6	−0.986641	−0.493320	+	0.869848i	$0.664217\pi$
$488$	−1.39859e6	−0.265853
$489$	1.73323e6	0.327781
$490$	0	0
$491$	8.54287e6	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$491$	8.54287e6	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$492$	−2.61619e6	−0.487256
$493$	1.27818e6	0.236851
$494$	680740.	0.125506
$495$	0	0
$496$	−2.19981e6	−0.401495
$497$	−1.91422e6	−0.347616
$498$	−3.90722e6	−0.705984
$499$	−4.20588e6	−0.756145	−0.378072	−	0.925776i	$-0.623413\pi$
$499$	−4.20588e6	−0.756145	−0.378072	+	0.925776i	$0.623413\pi$
$500$	0	0
$501$	5.22623e6	0.930238
$502$	−830832.	−0.147148
$503$	−8.18342e6	−1.44217	−0.721083	−	0.692849i	$-0.756355\pi$
$503$	−8.18342e6	−1.44217	−0.721083	+	0.692849i	$0.756355\pi$
$504$	243648.	0.0427254
$505$	0	0
$506$	271728.	0.0471800
$507$	−3.24983e6	−0.561488
$508$	−4.09677e6	−0.704340
$509$	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.	0.0441942
$513$	1.22837e6	0.206079
$514$	−5.81275e6	−0.970452
$515$	0	0
$516$	2.06654e6	0.341681
$517$	−243756.	−0.0401078
$518$	1.62470e6	0.266040
$519$	6.64313e6	1.08257
$520$	0	0
$521$	4.55410e6	0.735036	0.367518	−	0.930016i	$-0.380208\pi$
$521$	4.55410e6	0.735036	0.367518	+	0.930016i	$0.380208\pi$
$522$	2.55636e6	0.410625
$523$	4.82224e6	0.770894	0.385447	−	0.922730i	$-0.374047\pi$
$523$	4.82224e6	0.770894	0.385447	+	0.922730i	$0.374047\pi$
$524$	1.89523e6	0.301533
$525$	0	0
$526$	677184.	0.106719
$527$	−1.39207e6	−0.218340
$528$	511488.	0.0798455
$529$	−6.34271e6	−0.985452
$530$	0	0
$531$	1.42641e6	0.219537
$532$	1.26712e6	0.194106
$533$	−1.83497e6	−0.279776
$534$	1.26144e6	0.191432
$535$	0	0
$536$	6848.00	0.00102956
$537$	4.47633e6	0.669864
$538$	−6.32436e6	−0.942022
$539$	−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	$0.491523\pi$
$542$	3.29005e6	0.481065
$543$	−2.99847e6	−0.436415
$544$	165888.	0.0240335
$545$	0	0
$546$	170892.	0.0245324
$547$	3.11439e6	0.445046	0.222523	−	0.974927i	$-0.428571\pi$
$547$	3.11439e6	0.445046	0.222523	+	0.974927i	$0.428571\pi$
$548$	−211488.	−0.0300839
$549$	−1.77009e6	−0.250649
$550$	0	0
$551$	1.32947e7	1.86551
$552$	176256.	0.0246205
$553$	−3.25052e6	−0.452002
$554$	−2.18729e6	−0.302784
$555$	0	0
$556$	−5.61184e6	−0.769872
$557$	−7.99304e6	−1.09163	−0.545813	−	0.837907i	$-0.683779\pi$
$557$	−7.99304e6	−1.09163	−0.545813	+	0.837907i	$0.683779\pi$
$558$	−2.78413e6	−0.378534
$559$	1.44945e6	0.196189
$560$	0	0
$561$	323676.	0.0434214
$562$	−4.36999e6	−0.583633
$563$	−1.23236e7	−1.63857	−0.819286	−	0.573385i	$-0.805630\pi$
$563$	−1.23236e7	−1.63857	−0.819286	+	0.573385i	$0.805630\pi$
$564$	−158112.	−0.0209299
$565$	0	0
$566$	−9.93920e6	−1.30410
$567$	308367.	0.0402819
$568$	−2.60659e6	−0.339002
$569$	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.	0.0458463
$573$	−365742.	−0.0465359
$574$	−3.41558e6	−0.432698
$575$	0	0
$576$	331776.	0.0416667
$577$	−1.24453e6	−0.155621	−0.0778103	−	0.996968i	$-0.524793\pi$
$577$	−1.24453e6	−0.155621	−0.0778103	+	0.996968i	$0.524793\pi$
$578$	−5.57445e6	−0.694037
$579$	4.45186e6	0.551880
$580$	0	0
$581$	−5.10110e6	−0.626936
$582$	−29628.0	−0.00362573
$583$	3.97735e6	0.484644
$584$	2.22118e6	0.269496
$585$	0	0
$586$	−1.36558e6	−0.164275
$587$	−1.33403e6	−0.159797	−0.0798987	−	0.996803i	$-0.525460\pi$
$587$	−1.33403e6	−0.159797	−0.0798987	+	0.996803i	$0.525460\pi$
$588$	−2.10211e6	−0.250734
$589$	−1.44792e7	−1.71972
$590$	0	0
$591$	4.97108e6	0.585439
$592$	2.21235e6	0.259448
$593$	−1.19401e7	−1.39435	−0.697177	−	0.716899i	$-0.745561\pi$
$593$	−1.19401e7	−1.39435	−0.697177	+	0.716899i	$0.745561\pi$
$594$	647352.	0.0752791
$595$	0	0
$596$	1.75824e6	0.202751
$597$	6.17062e6	0.708587
$598$	123624.	0.0141368
$599$	−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	$0.479287\pi$
$602$	2.69799e6	0.303423
$603$	8667.00	0.000970679 0
$604$	−2.76165e6	−0.308018
$605$	0	0
$606$	−1.21781e6	−0.134709
$607$	−1.34268e7	−1.47911	−0.739554	−	0.673097i	$-0.764963\pi$
$607$	−1.34268e7	−1.47911	−0.739554	+	0.673097i	$0.764963\pi$
$608$	1.72544e6	0.189296
$609$	3.33747e6	0.364648
$610$	0	0
$611$	−110898.	−0.0120177
$612$	209952.	0.0226590
$613$	1.20184e7	1.29180	0.645900	−	0.763422i	$-0.276482\pi$
$613$	1.20184e7	1.29180	0.645900	+	0.763422i	$0.276482\pi$
$614$	8.11591e6	0.868793
$615$	0	0
$616$	667776.	0.0709054
$617$	−6.98519e6	−0.738695	−0.369348	−	0.929291i	$-0.620419\pi$
$617$	−6.98519e6	−0.738695	−0.369348	+	0.929291i	$0.620419\pi$
$618$	−4.80398e6	−0.505977
$619$	−8.20625e6	−0.860832	−0.430416	−	0.902631i	$-0.641633\pi$
$619$	−8.20625e6	−0.860832	−0.430416	+	0.902631i	$0.641633\pi$
$620$	0	0
$621$	223074.	0.0232124
$622$	−826392.	−0.0856466
$623$	1.64688e6	0.169997
$624$	232704.	0.0239245
$625$	0	0
$626$	−1.33689e7	−1.36352
$627$	3.36663e6	0.342000
$628$	−5.59989e6	−0.566605
$629$	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	$0.680947\pi$
$632$	−4.42624e6	−0.440801
$633$	6.74529e6	0.669101
$634$	−1.01316e7	−1.00104
$635$	0	0
$636$	2.57990e6	0.252907
$637$	−1.47440e6	−0.143968
$638$	7.00632e6	0.681457
$639$	−3.29897e6	−0.319614
$640$	0	0
$641$	1.92571e7	1.85117	0.925585	−	0.378539i	$-0.123573\pi$
$641$	1.92571e7	1.85117	0.925585	+	0.378539i	$0.123573\pi$
$642$	2.92507e6	0.280091
$643$	1.00999e7	0.963364	0.481682	−	0.876346i	$-0.340026\pi$
$643$	1.00999e7	0.963364	0.481682	+	0.876346i	$0.340026\pi$
$644$	230112.	0.0218637
$645$	0	0
$646$	1.09188e6	0.102942
$647$	7.52113e6	0.706354	0.353177	−	0.935556i	$-0.385101\pi$
$647$	7.52113e6	0.706354	0.353177	+	0.935556i	$0.385101\pi$
$648$	419904.	0.0392837
$649$	3.90942e6	0.364335
$650$	0	0
$651$	−3.63484e6	−0.336150
$652$	3.08130e6	0.283867
$653$	−2.67197e6	−0.245216	−0.122608	−	0.992455i	$-0.539126\pi$
$653$	−2.67197e6	−0.245216	−0.122608	+	0.992455i	$0.539126\pi$
$654$	−7.81254e6	−0.714246
$655$	0	0
$656$	−4.65101e6	−0.421976
$657$	2.81119e6	0.254083
$658$	−206424.	−0.0185864
$659$	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	$0.494217\pi$
$662$	2.40853e6	0.213603
$663$	147258.	0.0130105
$664$	−6.94618e6	−0.611400
$665$	0	0
$666$	2.80001e6	0.244610
$667$	2.41434e6	0.210128
$668$	9.29107e6	0.805610
$669$	1.52344e6	0.131601
$670$	0	0
$671$	−4.85137e6	−0.415966
$672$	433152.	0.0370013
$673$	1.74939e7	1.48885	0.744423	−	0.667709i	$-0.232725\pi$
$673$	1.74939e7	1.48885	0.744423	+	0.667709i	$0.232725\pi$
$674$	839108.	0.0711489
$675$	0	0
$676$	−5.77747e6	−0.486263
$677$	−8.67440e6	−0.727391	−0.363695	−	0.931518i	$-0.618485\pi$
$677$	−8.67440e6	−0.727391	−0.363695	+	0.931518i	$0.618485\pi$
$678$	−4.97966e6	−0.416031
$679$	−38681.0	−0.00321976
$680$	0	0
$681$	−418392.	−0.0345713
$682$	−7.63058e6	−0.628198
$683$	1.18478e7	0.971822	0.485911	−	0.874008i	$-0.338488\pi$
$683$	1.18478e7	0.971822	0.485911	+	0.874008i	$0.338488\pi$
$684$	2.18376e6	0.178470
$685$	0	0
$686$	−5.90414e6	−0.479012
$687$	−811035.	−0.0655613
$688$	3.67386e6	0.295904
$689$	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	$0.376765\pi$
$692$	1.18100e7	0.937530
$693$	845154.	0.0668502
$694$	1.60866e7	1.26785
$695$	0	0
$696$	4.54464e6	0.355612
$697$	−2.94322e6	−0.229478
$698$	33320.0	0.00258861
$699$	9.57722e6	0.741390
$700$	0	0
$701$	−2.28147e7	−1.75355	−0.876777	−	0.480898i	$-0.840311\pi$
$701$	−2.28147e7	−1.75355	−0.876777	+	0.480898i	$0.840311\pi$
$702$	294516.	0.0225562
$703$	1.45618e7	1.11129
$704$	909312.	0.0691483
$705$	0	0
$706$	7.80002e6	0.588958
$707$	−1.58992e6	−0.119626
$708$	2.53584e6	0.190125
$709$	1.27436e7	0.952090	0.476045	−	0.879421i	$-0.342070\pi$
$709$	1.27436e7	0.952090	0.476045	+	0.879421i	$0.342070\pi$
$710$	0	0
$711$	−5.60196e6	−0.415591
$712$	2.24256e6	0.165785
$713$	−2.62946e6	−0.193706
$714$	274104.	0.0201219
$715$	0	0
$716$	7.95792e6	0.580119
$717$	1.03642e7	0.752902
$718$	9.08352e6	0.657572
$719$	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
$722$	1.45250e6	0.103699
$723$	7.70595e6	0.548252
$724$	−5.33061e6	−0.377947
$725$	0	0
$726$	−4.02361e6	−0.283318
$727$	−415033.	−0.0291237	−0.0145619	−	0.999894i	$-0.504635\pi$
$727$	−415033.	−0.0291237	−0.0145619	+	0.999894i	$0.504635\pi$
$728$	303808.	0.0212457
$729$	531441.	0.0370370
$730$	0	0
$731$	2.32486e6	0.160918
$732$	−3.14683e6	−0.217068
$733$	−1.72877e7	−1.18844	−0.594221	−	0.804302i	$-0.702539\pi$
$733$	−1.72877e7	−1.18844	−0.594221	+	0.804302i	$0.702539\pi$
$734$	1.14461e7	0.784186
$735$	0	0
$736$	313344.	0.0213219
$737$	23754.0	0.00161090
$738$	−5.88643e6	−0.397843
$739$	5.18834e6	0.349476	0.174738	−	0.984615i	$-0.444092\pi$
$739$	5.18834e6	0.349476	0.174738	+	0.984615i	$0.444092\pi$
$740$	0	0
$741$	1.53166e6	0.102475
$742$	3.36821e6	0.224589
$743$	4.79572e6	0.318700	0.159350	−	0.987222i	$-0.449060\pi$
$743$	4.79572e6	0.318700	0.159350	+	0.987222i	$0.449060\pi$
$744$	−4.94957e6	−0.327820
$745$	0	0
$746$	2.46124e6	0.161923
$747$	−8.79125e6	−0.576434
$748$	575424.	0.0376040
$749$	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.	−0.0181258
$753$	−1.86937e6	−0.120146
$754$	3.18756e6	0.204188
$755$	0	0
$756$	548208.	0.0348852
$757$	2.82068e7	1.78902	0.894508	−	0.447053i	$-0.147526\pi$
$757$	2.82068e7	1.78902	0.894508	+	0.447053i	$0.147526\pi$
$758$	2.15951e7	1.36516
$759$	611388.	0.0385223
$760$	0	0
$761$	6.56161e6	0.410723	0.205361	−	0.978686i	$-0.434163\pi$
$761$	6.56161e6	0.410723	0.205361	+	0.978686i	$0.434163\pi$
$762$	−9.21773e6	−0.575091
$763$	−1.01997e7	−0.634273
$764$	−650208.	−0.0403013
$765$	0	0
$766$	4.34750e6	0.267712
$767$	1.77861e6	0.109167
$768$	589824.	0.0360844
$769$	2.20930e7	1.34722	0.673610	−	0.739087i	$-0.264743\pi$
$769$	2.20930e7	1.34722	0.673610	+	0.739087i	$0.264743\pi$
$770$	0	0
$771$	−1.30787e7	−0.792371
$772$	7.91442e6	0.477942
$773$	−3.00787e7	−1.81055	−0.905276	−	0.424824i	$-0.860336\pi$
$773$	−3.00787e7	−1.81055	−0.905276	+	0.424824i	$0.860336\pi$
$774$	4.64972e6	0.278981
$775$	0	0
$776$	−52672.0	−0.00313997
$777$	3.65557e6	0.217221
$778$	−1.39373e7	−0.825523
$779$	−3.06131e7	−1.80744
$780$	0	0
$781$	−9.04162e6	−0.530418
$782$	198288.	0.0115952
$783$	5.75181e6	0.335274
$784$	−3.73709e6	−0.217142
$785$	0	0
$786$	4.26427e6	0.246200
$787$	−3.28954e6	−0.189321	−0.0946605	−	0.995510i	$-0.530177\pi$
$787$	−3.28954e6	−0.189321	−0.0946605	+	0.995510i	$0.530177\pi$
$788$	8.83747e6	0.507005
$789$	1.52366e6	0.0871358
$790$	0	0
$791$	−6.50123e6	−0.369449
$792$	1.15085e6	0.0651936
$793$	−2.20715e6	−0.124638
$794$	1.30636e7	0.735381
$795$	0	0
$796$	1.09700e7	0.613655
$797$	6.71053e6	0.374206	0.187103	−	0.982340i	$-0.440090\pi$
$797$	6.71053e6	0.374206	0.187103	+	0.982340i	$0.440090\pi$
$798$	2.85102e6	0.158487
$799$	−177876.	−0.00985713
$800$	0	0
$801$	2.83824e6	0.156303
$802$	−1.70928e7	−0.938374
$803$	7.70473e6	0.421666
$804$	15408.0	0.000840632 0
$805$	0	0
$806$	−3.47157e6	−0.188230
$807$	−1.42298e7	−0.769157
$808$	−2.16499e6	−0.116662
$809$	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	$0.730765\pi$
$812$	5.93328e6	0.315795
$813$	7.40261e6	0.392788
$814$	7.67410e6	0.405944
$815$	0	0
$816$	373248.	0.0196233
$817$	2.41814e7	1.26744
$818$	−5.80750e6	−0.303463
$819$	384507.	0.0200306
$820$	0	0
$821$	2.12219e7	1.09882	0.549409	−	0.835554i	$-0.314853\pi$
$821$	2.12219e7	1.09882	0.549409	+	0.835554i	$0.314853\pi$
$822$	−475848.	−0.0245634
$823$	−8.70659e6	−0.448073	−0.224036	−	0.974581i	$-0.571923\pi$
$823$	−8.70659e6	−0.448073	−0.224036	+	0.974581i	$0.571923\pi$
$824$	−8.54042e6	−0.438189
$825$	0	0
$826$	3.31068e6	0.168837
$827$	3.71184e7	1.88723	0.943617	−	0.331040i	$-0.107400\pi$
$827$	3.71184e7	1.88723	0.943617	+	0.331040i	$0.107400\pi$
$828$	396576.	0.0201025
$829$	1.01765e6	0.0514295	0.0257147	−	0.999669i	$-0.491814\pi$
$829$	1.01765e6	0.0514295	0.0257147	+	0.999669i	$0.491814\pi$
$830$	0	0
$831$	−4.92141e6	−0.247222
$832$	413696.	0.0207192
$833$	−2.36488e6	−0.118085
$834$	−1.26266e7	−0.628598
$835$	0	0
$836$	5.98512e6	0.296181
$837$	−6.26430e6	−0.309071
$838$	2.23752e6	0.110067
$839$	−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
$842$	−1.56588e7	−0.761164
$843$	−9.83248e6	−0.476534
$844$	1.19916e7	0.579458
$845$	0	0
$846$	−355752.	−0.0170892
$847$	−5.25305e6	−0.251596
$848$	4.58650e6	0.219024
$849$	−2.23632e7	−1.06479
$850$	0	0
$851$	2.64445e6	0.125173
$852$	−5.86483e6	−0.276794
$853$	−3.52574e7	−1.65912	−0.829559	−	0.558419i	$-0.811408\pi$
$853$	−3.52574e7	−1.65912	−0.829559	+	0.558419i	$0.811408\pi$
$854$	−4.10836e6	−0.192763
$855$	0	0
$856$	5.20013e6	0.242566
$857$	−3.14941e7	−1.46480	−0.732398	−	0.680877i	$-0.761599\pi$
$857$	−3.14941e7	−1.46480	−0.732398	+	0.680877i	$0.761599\pi$
$858$	807192.	0.0374333
$859$	1.19344e7	0.551848	0.275924	−	0.961180i	$-0.411016\pi$
$859$	1.19344e7	0.551848	0.275924	+	0.961180i	$0.411016\pi$
$860$	0	0
$861$	−7.68506e6	−0.353297
$862$	−1.43000e7	−0.655492
$863$	8.70442e6	0.397844	0.198922	−	0.980015i	$-0.436256\pi$
$863$	8.70442e6	0.397844	0.198922	+	0.980015i	$0.436256\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	2.86388e7	1.29766
$867$	−1.25425e7	−0.566679
$868$	−6.46194e6	−0.291114
$869$	−1.53535e7	−0.689697
$870$	0	0
$871$	10807.0	0.000482681 0
$872$	−1.38890e7	−0.618555
$873$	−66663.0	−0.00296039
$874$	2.06244e6	0.0913277
$875$	0	0
$876$	4.99766e6	0.220043
$877$	1.17999e7	0.518059	0.259029	−	0.965869i	$-0.416597\pi$
$877$	1.17999e7	0.518059	0.259029	+	0.965869i	$0.416597\pi$
$878$	6.87158e6	0.300829
$879$	−3.07255e6	−0.134130
$880$	0	0
$881$	−2.73840e7	−1.18866	−0.594330	−	0.804221i	$-0.702583\pi$
$881$	−2.73840e7	−1.18866	−0.594330	+	0.804221i	$0.702583\pi$
$882$	−4.72975e6	−0.204723
$883$	−8.80577e6	−0.380072	−0.190036	−	0.981777i	$-0.560860\pi$
$883$	−8.80577e6	−0.380072	−0.190036	+	0.981777i	$0.560860\pi$
$884$	261792.	0.0112675
$885$	0	0
$886$	1.35868e7	0.581477
$887$	250122.	0.0106744	0.00533719	−	0.999986i	$-0.498301\pi$
$887$	250122.	0.0106744	0.00533719	+	0.999986i	$0.498301\pi$
$888$	4.97779e6	0.211838
$889$	−1.20343e7	−0.510699
$890$	0	0
$891$	1.45654e6	0.0614651
$892$	2.70834e6	0.113970
$893$	−1.85013e6	−0.0776379
$894$	3.95604e6	0.165545
$895$	0	0
$896$	770048.	0.0320441
$897$	278154.	0.0115426
$898$	−1.35842e7	−0.562140
$899$	−6.77988e7	−2.79784
$900$	0	0
$901$	2.90239e6	0.119109
$902$	−1.61332e7	−0.660243
$903$	6.07047e6	0.247744
$904$	−8.85274e6	−0.360294
$905$	0	0
$906$	−6.21371e6	−0.251496
$907$	−3.24955e7	−1.31161	−0.655806	−	0.754929i	$-0.727671\pi$
$907$	−3.24955e7	−1.31161	−0.655806	+	0.754929i	$0.727671\pi$
$908$	−743808.	−0.0299396
$909$	−2.74007e6	−0.109990
$910$	0	0
$911$	4.24595e7	1.69504	0.847518	−	0.530766i	$-0.178096\pi$
$911$	4.24595e7	1.69504	0.847518	+	0.530766i	$0.178096\pi$
$912$	3.88224e6	0.154559
$913$	−2.40945e7	−0.956625
$914$	−1.81126e7	−0.717157
$915$	0	0
$916$	−1.44184e6	−0.0567778
$917$	5.56724e6	0.218634
$918$	472392.	0.0185010
$919$	1.41629e7	0.553176	0.276588	−	0.960989i	$-0.410796\pi$
$919$	1.41629e7	0.553176	0.276588	+	0.960989i	$0.410796\pi$
$920$	0	0
$921$	1.82608e7	0.709366
$922$	−5.11579e6	−0.198192
$923$	−4.11353e6	−0.158932
$924$	1.50250e6	0.0578940
$925$	0	0
$926$	−2.87945e7	−1.10352
$927$	−1.08090e7	−0.413128
$928$	8.07936e6	0.307969
$929$	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
$932$	1.70262e7	0.642063
$933$	−1.85938e6	−0.0699302
$934$	1.93214e7	0.724721
$935$	0	0
$936$	523584.	0.0195343
$937$	5.73509e6	0.213398	0.106699	−	0.994291i	$-0.465972\pi$
$937$	5.73509e6	0.213398	0.106699	+	0.994291i	$0.465972\pi$
$938$	20116.0	0.000746508 0
$939$	−3.00801e7	−1.11331
$940$	0	0
$941$	−3.37395e7	−1.24212	−0.621061	−	0.783762i	$-0.713298\pi$
$941$	−3.37395e7	−1.24212	−0.621061	+	0.783762i	$0.713298\pi$
$942$	−1.25997e7	−0.462631
$943$	−5.55941e6	−0.203587
$944$	4.50816e6	0.164653
$945$	0	0
$946$	1.27437e7	0.462985
$947$	−3.07342e7	−1.11365	−0.556823	−	0.830631i	$-0.687980\pi$
$947$	−3.07342e7	−1.11365	−0.556823	+	0.830631i	$0.687980\pi$
$948$	−9.95904e6	−0.359912
$949$	3.50531e6	0.126346
$950$	0	0
$951$	−2.27960e7	−0.817348
$952$	487296.	0.0174261
$953$	−2.51847e7	−0.898264	−0.449132	−	0.893465i	$-0.648267\pi$
$953$	−2.51847e7	−0.898264	−0.449132	+	0.893465i	$0.648267\pi$
$954$	5.80478e6	0.206498
$955$	0	0
$956$	1.84253e7	0.652033
$957$	1.57642e7	0.556407
$958$	2.99460e6	0.105420
$959$	−621246.	−0.0218131
$960$	0	0
$961$	4.52105e7	1.57918
$962$	3.49137e6	0.121635
$963$	6.58141e6	0.228693
$964$	1.36995e7	0.474801
$965$	0	0
$966$	517752.	0.0178517
$967$	1.44556e7	0.497130	0.248565	−	0.968615i	$-0.420041\pi$
$967$	1.44556e7	0.497130	0.248565	+	0.968615i	$0.420041\pi$
$968$	−7.15309e6	−0.245361
$969$	2.45673e6	0.0840520
$970$	0	0
$971$	−1.06974e7	−0.364109	−0.182054	−	0.983288i	$-0.558275\pi$
$971$	−1.06974e7	−0.364109	−0.182054	+	0.983288i	$0.558275\pi$
$972$	944784.	0.0320750
$973$	−1.64848e7	−0.558214
$974$	−2.06558e7	−0.697660
$975$	0	0
$976$	−5.59437e6	−0.187986
$977$	−8.41568e6	−0.282067	−0.141034	−	0.990005i	$-0.545043\pi$
$977$	−8.41568e6	−0.282067	−0.141034	+	0.990005i	$0.545043\pi$
$978$	6.93292e6	0.231776
$979$	7.77888e6	0.259394
$980$	0	0
$981$	−1.75782e7	−0.583180
$982$	3.41715e7	1.13080
$983$	3.89409e7	1.28535	0.642676	−	0.766138i	$-0.277824\pi$
$983$	3.89409e7	1.28535	0.642676	+	0.766138i	$0.277824\pi$
$984$	−1.04648e7	−0.344542
$985$	0	0
$986$	5.11272e6	0.167479
$987$	−464454.	−0.0151757
$988$	2.72296e6	0.0887460
$989$	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	$0.213318\pi$
$992$	−8.79923e6	−0.283900
$993$	5.41919e6	0.174406
$994$	−7.65686e6	−0.245802
$995$	0	0
$996$	−1.56289e7	−0.499206
$997$	3.84733e7	1.22581	0.612903	−	0.790158i	$-0.290002\pi$
$997$	3.84733e7	1.22581	0.612903	+	0.790158i	$0.290002\pi$
$998$	−1.68235e7	−0.534675
$999$	6.30002e6	0.199723

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.a.1.1	✓	1	5.4	even	2
150.6.a.m.1.1	yes	1	1.1	even	1	trivial
150.6.c.g.49.1		2	5.3	odd	4
150.6.c.g.49.2		2	5.2	odd	4
450.6.a.i.1.1		1	3.2	odd	2
450.6.a.p.1.1		1	15.14	odd	2
450.6.c.e.199.1		2	15.2	even	4
450.6.c.e.199.2		2	15.8	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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