LMFDB - Embedded newform 195.6.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,6,Mod(1,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("195.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$195 = 3 \cdot 5 \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	195.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:	$31.2748448635$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 89x^{2} + 82x + 720$ x^4 - 2x^3 - 89x^2 + 82*x + 720
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-8.46798$ of defining polynomial
Character	$\chi$	$=$	195.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.46798 q^{2} -9.00000 q^{3} +57.6426 q^{4} -25.0000 q^{5} -85.2118 q^{6} -87.7314 q^{7} +242.784 q^{8} +81.0000 q^{9} -236.699 q^{10} -70.4184 q^{11} -518.784 q^{12} -169.000 q^{13} -830.639 q^{14} +225.000 q^{15} +454.109 q^{16} -892.889 q^{17} +766.906 q^{18} -1245.32 q^{19} -1441.07 q^{20} +789.583 q^{21} -666.720 q^{22} -1681.71 q^{23} -2185.06 q^{24} +625.000 q^{25} -1600.09 q^{26} -729.000 q^{27} -5057.07 q^{28} -5734.49 q^{29} +2130.30 q^{30} -2679.72 q^{31} -3469.59 q^{32} +633.766 q^{33} -8453.86 q^{34} +2193.29 q^{35} +4669.05 q^{36} +42.9130 q^{37} -11790.7 q^{38} +1521.00 q^{39} -6069.60 q^{40} +15414.3 q^{41} +7475.75 q^{42} +921.195 q^{43} -4059.10 q^{44} -2025.00 q^{45} -15922.4 q^{46} +28502.6 q^{47} -4086.98 q^{48} -9110.20 q^{49} +5917.49 q^{50} +8036.00 q^{51} -9741.60 q^{52} -25572.2 q^{53} -6902.16 q^{54} +1760.46 q^{55} -21299.8 q^{56} +11207.9 q^{57} -54294.0 q^{58} +2102.94 q^{59} +12969.6 q^{60} -19188.5 q^{61} -25371.6 q^{62} -7106.24 q^{63} -47381.5 q^{64} +4225.00 q^{65} +6000.48 q^{66} +13226.2 q^{67} -51468.5 q^{68} +15135.4 q^{69} +20766.0 q^{70} -27465.9 q^{71} +19665.5 q^{72} +85888.9 q^{73} +406.300 q^{74} -5625.00 q^{75} -71783.7 q^{76} +6177.91 q^{77} +14400.8 q^{78} +17713.2 q^{79} -11352.7 q^{80} +6561.00 q^{81} +145942. q^{82} -12373.2 q^{83} +45513.6 q^{84} +22322.2 q^{85} +8721.86 q^{86} +51610.4 q^{87} -17096.5 q^{88} -3584.55 q^{89} -19172.7 q^{90} +14826.6 q^{91} -96938.5 q^{92} +24117.5 q^{93} +269862. q^{94} +31133.1 q^{95} +31226.3 q^{96} +111961. q^{97} -86255.2 q^{98} -5703.89 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{2} - 36 q^{3} + 54 q^{4} - 100 q^{5} - 18 q^{6} + 87 q^{7} + 120 q^{8} + 324 q^{9} - 50 q^{10} - 277 q^{11} - 486 q^{12} - 676 q^{13} + 478 q^{14} + 900 q^{15} + 274 q^{16} + 2017 q^{17} + 162 q^{18}+ \cdots - 22437 q^{99}+O(q^{100})$ 4 * q + 2 * q^2 - 36 * q^3 + 54 * q^4 - 100 * q^5 - 18 * q^6 + 87 * q^7 + 120 * q^8 + 324 * q^9 - 50 * q^10 - 277 * q^11 - 486 * q^12 - 676 * q^13 + 478 * q^14 + 900 * q^15 + 274 * q^16 + 2017 * q^17 + 162 * q^18 - 1132 * q^19 - 1350 * q^20 - 783 * q^21 + 3216 * q^22 + 99 * q^23 - 1080 * q^24 + 2500 * q^25 - 338 * q^26 - 2916 * q^27 - 12462 * q^28 - 242 * q^29 + 450 * q^30 - 10330 * q^31 + 4512 * q^32 + 2493 * q^33 - 29834 * q^34 - 2175 * q^35 + 4374 * q^36 - 19191 * q^37 - 11654 * q^38 + 6084 * q^39 - 3000 * q^40 - 4297 * q^41 - 4302 * q^42 + 10668 * q^43 - 20076 * q^44 - 8100 * q^45 - 32914 * q^46 + 40662 * q^47 - 2466 * q^48 - 5415 * q^49 + 1250 * q^50 - 18153 * q^51 - 9126 * q^52 - 75477 * q^53 - 1458 * q^54 + 6925 * q^55 - 54430 * q^56 + 10188 * q^57 - 96230 * q^58 - 13052 * q^59 + 12150 * q^60 - 58909 * q^61 - 98456 * q^62 + 7047 * q^63 - 75678 * q^64 + 16900 * q^65 - 28944 * q^66 - 89840 * q^67 + 14490 * q^68 - 891 * q^69 - 11950 * q^70 + 25903 * q^71 + 9720 * q^72 + 71242 * q^73 - 8132 * q^74 - 22500 * q^75 - 161750 * q^76 + 74919 * q^77 + 3042 * q^78 - 63609 * q^79 - 6850 * q^80 + 26244 * q^81 + 170460 * q^82 - 67600 * q^83 + 112158 * q^84 - 50425 * q^85 - 256072 * q^86 + 2178 * q^87 - 41116 * q^88 + 45869 * q^89 - 4050 * q^90 - 14703 * q^91 - 13598 * q^92 + 92970 * q^93 + 127944 * q^94 + 28300 * q^95 - 40608 * q^96 + 151169 * q^97 + 180122 * q^98 - 22437 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	9.46798	1.67372	0.836859	−	0.547418i	$-0.184389\pi$
$2$	9.46798	1.67372	0.836859	+	0.547418i	$0.184389\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	57.6426	1.80133
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−85.2118	−0.966322
$7$	−87.7314	−0.676722	−0.338361	−	0.941016i	$-0.609872\pi$
$7$	−87.7314	−0.676722	−0.338361	+	0.941016i	$0.609872\pi$
$8$	242.784	1.34120
$9$	81.0000	0.333333
$10$	−236.699	−0.748509
$11$	−70.4184	−0.175471	−0.0877353	−	0.996144i	$-0.527963\pi$
$11$	−70.4184	−0.175471	−0.0877353	+	0.996144i	$0.527963\pi$
$12$	−518.784	−1.04000
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−830.639	−1.13264
$15$	225.000	0.258199
$16$	454.109	0.443466
$17$	−892.889	−0.749334	−0.374667	−	0.927160i	$-0.622243\pi$
$17$	−892.889	−0.749334	−0.374667	+	0.927160i	$0.622243\pi$
$18$	766.906	0.557906
$19$	−1245.32	−0.791404	−0.395702	−	0.918379i	$-0.629499\pi$
$19$	−1245.32	−0.791404	−0.395702	+	0.918379i	$0.629499\pi$
$20$	−1441.07	−0.805580
$21$	789.583	0.390705
$22$	−666.720	−0.293688
$23$	−1681.71	−0.662877	−0.331438	−	0.943477i	$-0.607534\pi$
$23$	−1681.71	−0.662877	−0.331438	+	0.943477i	$0.607534\pi$
$24$	−2185.06	−0.774345
$25$	625.000	0.200000
$26$	−1600.09	−0.464206
$27$	−729.000	−0.192450
$28$	−5057.07	−1.21900
$29$	−5734.49	−1.26619	−0.633096	−	0.774073i	$-0.718216\pi$
$29$	−5734.49	−1.26619	−0.633096	+	0.774073i	$0.718216\pi$
$30$	2130.30	0.432152
$31$	−2679.72	−0.500824	−0.250412	−	0.968139i	$-0.580566\pi$
$31$	−2679.72	−0.500824	−0.250412	+	0.968139i	$0.580566\pi$
$32$	−3469.59	−0.598968
$33$	633.766	0.101308
$34$	−8453.86	−1.25417
$35$	2193.29	0.302639
$36$	4669.05	0.600444
$37$	42.9130	0.00515329	0.00257665	−	0.999997i	$-0.499180\pi$
$37$	42.9130	0.00515329	0.00257665	+	0.999997i	$0.499180\pi$
$38$	−11790.7	−1.32459
$39$	1521.00	0.160128
$40$	−6069.60	−0.599805
$41$	15414.3	1.43207	0.716034	−	0.698066i	$-0.245956\pi$
$41$	15414.3	1.43207	0.716034	+	0.698066i	$0.245956\pi$
$42$	7475.75	0.653931
$43$	921.195	0.0759767	0.0379884	−	0.999278i	$-0.487905\pi$
$43$	921.195	0.0759767	0.0379884	+	0.999278i	$0.487905\pi$
$44$	−4059.10	−0.316081
$45$	−2025.00	−0.149071
$46$	−15922.4	−1.10947
$47$	28502.6	1.88209	0.941045	−	0.338282i	$-0.109846\pi$
$47$	28502.6	1.88209	0.941045	+	0.338282i	$0.109846\pi$
$48$	−4086.98	−0.256035
$49$	−9110.20	−0.542048
$50$	5917.49	0.334744
$51$	8036.00	0.432628
$52$	−9741.60	−0.499600
$53$	−25572.2	−1.25048	−0.625241	−	0.780432i	$-0.714999\pi$
$53$	−25572.2	−1.25048	−0.625241	+	0.780432i	$0.714999\pi$
$54$	−6902.16	−0.322107
$55$	1760.46	0.0784728
$56$	−21299.8	−0.907622
$57$	11207.9	0.456917
$58$	−54294.0	−2.11925
$59$	2102.94	0.0786495	0.0393247	−	0.999226i	$-0.487479\pi$
$59$	2102.94	0.0786495	0.0393247	+	0.999226i	$0.487479\pi$
$60$	12969.6	0.465102
$61$	−19188.5	−0.660262	−0.330131	−	0.943935i	$-0.607093\pi$
$61$	−19188.5	−0.660262	−0.330131	+	0.943935i	$0.607093\pi$
$62$	−25371.6	−0.838239
$63$	−7106.24	−0.225574
$64$	−47381.5	−1.44597
$65$	4225.00	0.124035
$66$	6000.48	0.169561
$67$	13226.2	0.359955	0.179978	−	0.983671i	$-0.442397\pi$
$67$	13226.2	0.359955	0.179978	+	0.983671i	$0.442397\pi$
$68$	−51468.5	−1.34980
$69$	15135.4	0.382712
$70$	20766.0	0.506533
$71$	−27465.9	−0.646618	−0.323309	−	0.946293i	$-0.604795\pi$
$71$	−27465.9	−0.646618	−0.323309	+	0.946293i	$0.604795\pi$
$72$	19665.5	0.447068
$73$	85888.9	1.88638	0.943192	−	0.332249i	$-0.107807\pi$
$73$	85888.9	1.88638	0.943192	+	0.332249i	$0.107807\pi$
$74$	406.300	0.00862516
$75$	−5625.00	−0.115470
$76$	−71783.7	−1.42558
$77$	6177.91	0.118745
$78$	14400.8	0.268009
$79$	17713.2	0.319323	0.159661	−	0.987172i	$-0.448960\pi$
$79$	17713.2	0.319323	0.159661	+	0.987172i	$0.448960\pi$
$80$	−11352.7	−0.198324
$81$	6561.00	0.111111
$82$	145942.	2.39688
$83$	−12373.2	−0.197145	−0.0985724	−	0.995130i	$-0.531428\pi$
$83$	−12373.2	−0.197145	−0.0985724	+	0.995130i	$0.531428\pi$
$84$	45513.6	0.703790
$85$	22322.2	0.335112
$86$	8721.86	0.127164
$87$	51610.4	0.731037
$88$	−17096.5	−0.235342
$89$	−3584.55	−0.0479689	−0.0239844	−	0.999712i	$-0.507635\pi$
$89$	−3584.55	−0.0479689	−0.0239844	+	0.999712i	$0.507635\pi$
$90$	−19172.7	−0.249503
$91$	14826.6	0.187689
$92$	−96938.5	−1.19406
$93$	24117.5	0.289151
$94$	269862.	3.15009
$95$	31133.1	0.353927
$96$	31226.3	0.345814
$97$	111961.	1.20820	0.604099	−	0.796910i	$-0.293533\pi$
$97$	111961.	1.20820	0.604099	+	0.796910i	$0.293533\pi$
$98$	−86255.2	−0.907235
$99$	−5703.89	−0.0584902
$100$	36026.6	0.360266
$101$	−71533.6	−0.697761	−0.348880	−	0.937167i	$-0.613438\pi$
$101$	−71533.6	−0.697761	−0.348880	+	0.937167i	$0.613438\pi$
$102$	76084.7	0.724097
$103$	−179.050	−0.00166296	−0.000831479	−	1.00000i	$-0.500265\pi$
$103$	−179.050	−0.00166296	−0.000831479		1.00000i	$0.500265\pi$
$104$	−41030.5	−0.371983
$105$	−19739.6	−0.174729
$106$	−242117.	−2.09296
$107$	93692.3	0.791124	0.395562	−	0.918439i	$-0.370550\pi$
$107$	93692.3	0.791124	0.395562	+	0.918439i	$0.370550\pi$
$108$	−42021.5	−0.346667
$109$	−21332.0	−0.171975	−0.0859875	−	0.996296i	$-0.527405\pi$
$109$	−21332.0	−0.171975	−0.0859875	+	0.996296i	$0.527405\pi$
$110$	16668.0	0.131341
$111$	−386.217	−0.00297525
$112$	−39839.6	−0.300103
$113$	21860.1	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$113$	21860.1	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$114$	106116.	0.764751
$115$	42042.9	0.296447
$116$	−330551.	−2.28083
$117$	−13689.0	−0.0924500
$118$	19910.6	0.131637
$119$	78334.4	0.507090
$120$	54626.4	0.346297
$121$	−156092.	−0.969210
$122$	−181676.	−1.10509
$123$	−138728.	−0.826804
$124$	−154466.	−0.902151
$125$	−15625.0	−0.0894427
$126$	−67281.8	−0.377547
$127$	33539.1	0.184520	0.0922598	−	0.995735i	$-0.470591\pi$
$127$	33539.1	0.184520	0.0922598	+	0.995735i	$0.470591\pi$
$128$	−337580.	−1.82118
$129$	−8290.76	−0.0438652
$130$	40002.2	0.207599
$131$	87609.1	0.446037	0.223019	−	0.974814i	$-0.428409\pi$
$131$	87609.1	0.446037	0.223019	+	0.974814i	$0.428409\pi$
$132$	36531.9	0.182489
$133$	109254.	0.535560
$134$	125225.	0.602463
$135$	18225.0	0.0860663
$136$	−216779.	−1.00501
$137$	−225234.	−1.02526	−0.512629	−	0.858610i	$-0.671328\pi$
$137$	−225234.	−1.02526	−0.512629	+	0.858610i	$0.671328\pi$
$138$	143302.	0.640552
$139$	72865.4	0.319878	0.159939	−	0.987127i	$-0.448870\pi$
$139$	72865.4	0.319878	0.159939	+	0.987127i	$0.448870\pi$
$140$	126427.	0.545154
$141$	−256524.	−1.08662
$142$	−260046.	−1.08226
$143$	11900.7	0.0486668
$144$	36782.8	0.147822
$145$	143362.	0.566259
$146$	813195.	3.15727
$147$	81991.8	0.312952
$148$	2473.62	0.00928279
$149$	200346.	0.739289	0.369645	−	0.929173i	$-0.379479\pi$
$149$	200346.	0.739289	0.369645	+	0.929173i	$0.379479\pi$
$150$	−53257.4	−0.193264
$151$	−268794.	−0.959350	−0.479675	−	0.877446i	$-0.659245\pi$
$151$	−268794.	−0.959350	−0.479675	+	0.877446i	$0.659245\pi$
$152$	−302345.	−1.06143
$153$	−72324.0	−0.249778
$154$	58492.3	0.198745
$155$	66993.1	0.223976
$156$	87674.4	0.288444
$157$	−187764.	−0.607945	−0.303972	−	0.952681i	$-0.598313\pi$
$157$	−187764.	−0.607945	−0.303972	+	0.952681i	$0.598313\pi$
$158$	167708.	0.534456
$159$	230149.	0.721966
$160$	86739.8	0.267867
$161$	147539.	0.448583
$162$	62119.4	0.185969
$163$	−312637.	−0.921660	−0.460830	−	0.887488i	$-0.652448\pi$
$163$	−312637.	−0.921660	−0.460830	+	0.887488i	$0.652448\pi$
$164$	888519.	2.57963
$165$	−15844.1	−0.0453063
$166$	−117149.	−0.329965
$167$	−303566.	−0.842292	−0.421146	−	0.906993i	$-0.638372\pi$
$167$	−303566.	−0.842292	−0.421146	+	0.906993i	$0.638372\pi$
$168$	191698.	0.524016
$169$	28561.0	0.0769231
$170$	211346.	0.560883
$171$	−100871.	−0.263801
$172$	53100.1	0.136859
$173$	199740.	0.507398	0.253699	−	0.967283i	$-0.418353\pi$
$173$	199740.	0.507398	0.253699	+	0.967283i	$0.418353\pi$
$174$	488646.	1.22355
$175$	−54832.1	−0.135344
$176$	−31977.6	−0.0778152
$177$	−18926.4	−0.0454083
$178$	−33938.4	−0.0802864
$179$	159284.	0.371570	0.185785	−	0.982590i	$-0.440517\pi$
$179$	159284.	0.371570	0.185785	+	0.982590i	$0.440517\pi$
$180$	−116726.	−0.268527
$181$	109312.	0.248011	0.124005	−	0.992282i	$-0.460426\pi$
$181$	109312.	0.248011	0.124005	+	0.992282i	$0.460426\pi$
$182$	140378.	0.314138
$183$	172696.	0.381202
$184$	−408293.	−0.889053
$185$	−1072.83	−0.00230462
$186$	228344.	0.483957
$187$	62875.8	0.131486
$188$	1.64297e6	3.39027
$189$	63956.2	0.130235
$190$	294767.	0.592373
$191$	614947.	1.21970	0.609851	−	0.792516i	$-0.291229\pi$
$191$	614947.	1.21970	0.609851	+	0.792516i	$0.291229\pi$
$192$	426434.	0.834831
$193$	−636703.	−1.23039	−0.615196	−	0.788374i	$-0.710923\pi$
$193$	−636703.	−1.23039	−0.615196	+	0.788374i	$0.710923\pi$
$194$	1.06005e6	2.02218
$195$	−38025.0	−0.0716115
$196$	−525136.	−0.976408
$197$	−227727.	−0.418069	−0.209035	−	0.977908i	$-0.567032\pi$
$197$	−227727.	−0.418069	−0.209035	+	0.977908i	$0.567032\pi$
$198$	−54004.3	−0.0978961
$199$	693103.	1.24069	0.620347	−	0.784327i	$-0.286992\pi$
$199$	693103.	1.24069	0.620347	+	0.784327i	$0.286992\pi$
$200$	151740.	0.268241
$201$	−119036.	−0.207820
$202$	−677278.	−1.16785
$203$	503095.	0.856860
$204$	463216.	0.779307
$205$	−385357.	−0.640440
$206$	−1695.24	−0.00278332
$207$	−136219.	−0.220959
$208$	−76744.4	−0.122995
$209$	87693.7	0.138868
$210$	−186894.	−0.292447
$211$	−131810.	−0.203818	−0.101909	−	0.994794i	$-0.532495\pi$
$211$	−131810.	−0.203818	−0.101909	+	0.994794i	$0.532495\pi$
$212$	−1.47405e6	−2.25253
$213$	247193.	0.373325
$214$	887077.	1.32412
$215$	−23029.9	−0.0339778
$216$	−176989.	−0.258115
$217$	235096.	0.338919
$218$	−201971.	−0.287838
$219$	−773000.	−1.08910
$220$	101478.	0.141356
$221$	150898.	0.207828
$222$	−3656.70	−0.00497974
$223$	−1.20295e6	−1.61989	−0.809946	−	0.586505i	$-0.800503\pi$
$223$	−1.20295e6	−1.61989	−0.809946	+	0.586505i	$0.800503\pi$
$224$	304392.	0.405334
$225$	50625.0	0.0666667
$226$	206971.	0.269550
$227$	1.32636e6	1.70843	0.854215	−	0.519920i	$-0.174038\pi$
$227$	1.32636e6	1.70843	0.854215	+	0.519920i	$0.174038\pi$
$228$	646054.	0.823060
$229$	−1.31617e6	−1.65853	−0.829267	−	0.558853i	$-0.811241\pi$
$229$	−1.31617e6	−1.65853	−0.829267	+	0.558853i	$0.811241\pi$
$230$	398061.	0.496169
$231$	−55601.1	−0.0685573
$232$	−1.39224e6	−1.69822
$233$	−1.25449e6	−1.51383	−0.756913	−	0.653515i	$-0.773293\pi$
$233$	−1.25449e6	−1.51383	−0.756913	+	0.653515i	$0.773293\pi$
$234$	−129607.	−0.154735
$235$	−712566.	−0.841696
$236$	121219.	0.141674
$237$	−159419.	−0.184361
$238$	741669.	0.848726
$239$	−1.36373e6	−1.54430	−0.772151	−	0.635439i	$-0.780819\pi$
$239$	−1.36373e6	−1.54430	−0.772151	+	0.635439i	$0.780819\pi$
$240$	102174.	0.114502
$241$	−1.52021e6	−1.68602	−0.843009	−	0.537899i	$-0.819218\pi$
$241$	−1.52021e6	−1.68602	−0.843009	+	0.537899i	$0.819218\pi$
$242$	−1.47788e6	−1.62218
$243$	−59049.0	−0.0641500
$244$	−1.10608e6	−1.18935
$245$	227755.	0.242411
$246$	−1.31348e6	−1.38384
$247$	210460.	0.219496
$248$	−650593.	−0.671708
$249$	111358.	0.113822
$250$	−147937.	−0.149702
$251$	−1.15822e6	−1.16040	−0.580198	−	0.814476i	$-0.697025\pi$
$251$	−1.15822e6	−1.16040	−0.580198	+	0.814476i	$0.697025\pi$
$252$	−409623.	−0.406333
$253$	118424.	0.116315
$254$	317548.	0.308834
$255$	−200900.	−0.193477
$256$	−1.67999e6	−1.60217
$257$	203996.	0.192659	0.0963296	−	0.995349i	$-0.469290\pi$
$257$	203996.	0.192659	0.0963296	+	0.995349i	$0.469290\pi$
$258$	−78496.7	−0.0734180
$259$	−3764.82	−0.00348734
$260$	243540.	0.223428
$261$	−464494.	−0.422064
$262$	829481.	0.746540
$263$	726824.	0.647948	0.323974	−	0.946066i	$-0.394981\pi$
$263$	726824.	0.647948	0.323974	+	0.946066i	$0.394981\pi$
$264$	153868.	0.135875
$265$	639304.	0.559233
$266$	1.03441e6	0.896377
$267$	32260.9	0.0276948
$268$	762393.	0.648399
$269$	−488140.	−0.411304	−0.205652	−	0.978625i	$-0.565932\pi$
$269$	−488140.	−0.411304	−0.205652	+	0.978625i	$0.565932\pi$
$270$	172554.	0.144051
$271$	−1.25773e6	−1.04031	−0.520157	−	0.854071i	$-0.674127\pi$
$271$	−1.25773e6	−1.04031	−0.520157	+	0.854071i	$0.674127\pi$
$272$	−405469.	−0.332304
$273$	−133439.	−0.108362
$274$	−2.13251e6	−1.71599
$275$	−44011.5	−0.0350941
$276$	872446.	0.689391
$277$	1.35396e6	1.06025	0.530123	−	0.847921i	$-0.322146\pi$
$277$	1.35396e6	1.06025	0.530123	+	0.847921i	$0.322146\pi$
$278$	689889.	0.535386
$279$	−217057.	−0.166941
$280$	532494.	0.405901
$281$	1.17153e6	0.885092	0.442546	−	0.896746i	$-0.354075\pi$
$281$	1.17153e6	0.885092	0.442546	+	0.896746i	$0.354075\pi$
$282$	−2.42876e6	−1.81870
$283$	−1.86173e6	−1.38182	−0.690909	−	0.722942i	$-0.742789\pi$
$283$	−1.86173e6	−1.38182	−0.690909	+	0.722942i	$0.742789\pi$
$284$	−1.58321e6	−1.16477
$285$	−280198.	−0.204340
$286$	112676.	0.0814545
$287$	−1.35232e6	−0.969111
$288$	−281037.	−0.199656
$289$	−622606.	−0.438499
$290$	1.35735e6	0.947757
$291$	−1.00765e6	−0.697553
$292$	4.95086e6	3.39800
$293$	1.17629e6	0.800470	0.400235	−	0.916413i	$-0.368929\pi$
$293$	1.17629e6	0.800470	0.400235	+	0.916413i	$0.368929\pi$
$294$	776297.	0.523793
$295$	−52573.4	−0.0351731
$296$	10418.6	0.00691162
$297$	51335.0	0.0337693
$298$	1.89687e6	1.23736
$299$	284210.	0.183849
$300$	−324240.	−0.208000
$301$	−80817.8	−0.0514151
$302$	−2.54494e6	−1.60568
$303$	643802.	0.402852
$304$	−565512.	−0.350960
$305$	479712.	0.295278
$306$	−684762.	−0.418058
$307$	17636.1	0.0106797	0.00533983	−	0.999986i	$-0.498300\pi$
$307$	17636.1	0.0106797	0.00533983	+	0.999986i	$0.498300\pi$
$308$	356111.	0.213899
$309$	1611.45	0.000960109 0
$310$	634289.	0.374872
$311$	1.30118e6	0.762844	0.381422	−	0.924401i	$-0.375435\pi$
$311$	1.30118e6	0.762844	0.381422	+	0.924401i	$0.375435\pi$
$312$	369274.	0.214765
$313$	−841322.	−0.485402	−0.242701	−	0.970101i	$-0.578033\pi$
$313$	−841322.	−0.485402	−0.242701	+	0.970101i	$0.578033\pi$
$314$	−1.77775e6	−1.01753
$315$	177656.	0.100880
$316$	1.02104e6	0.575206
$317$	169458.	0.0947137	0.0473568	−	0.998878i	$-0.484920\pi$
$317$	169458.	0.0947137	0.0473568	+	0.998878i	$0.484920\pi$
$318$	2.17905e6	1.20837
$319$	403814.	0.222180
$320$	1.18454e6	0.646657
$321$	−843231.	−0.456755
$322$	1.39690e6	0.750801
$323$	1.11194e6	0.593026
$324$	378193.	0.200148
$325$	−105625.	−0.0554700
$326$	−2.96004e6	−1.54260
$327$	191988.	0.0992899
$328$	3.74234e6	1.92069
$329$	−2.50058e6	−1.27365
$330$	−150012.	−0.0758300
$331$	−2.64106e6	−1.32498	−0.662489	−	0.749071i	$-0.730500\pi$
$331$	−2.64106e6	−1.32498	−0.662489	+	0.749071i	$0.730500\pi$
$332$	−713221.	−0.355123
$333$	3475.95	0.00171776
$334$	−2.87416e6	−1.40976
$335$	−330655.	−0.160977
$336$	358556.	0.173264
$337$	1.80911e6	0.867740	0.433870	−	0.900975i	$-0.357148\pi$
$337$	1.80911e6	0.867740	0.433870	+	0.900975i	$0.357148\pi$
$338$	270415.	0.128748
$339$	−196741.	−0.0929813
$340$	1.28671e6	0.603648
$341$	188702.	0.0878800
$342$	−955047.	−0.441529
$343$	2.27375e6	1.04354
$344$	223651.	0.101900
$345$	−378386.	−0.171154
$346$	1.89113e6	0.849242
$347$	−1.16255e6	−0.518309	−0.259155	−	0.965836i	$-0.583444\pi$
$347$	−1.16255e6	−0.518309	−0.259155	+	0.965836i	$0.583444\pi$
$348$	2.97496e6	1.31684
$349$	−3.29498e6	−1.44807	−0.724034	−	0.689764i	$-0.757714\pi$
$349$	−3.29498e6	−1.44807	−0.724034	+	0.689764i	$0.757714\pi$
$350$	−519149.	−0.226528
$351$	123201.	0.0533761
$352$	244323.	0.105101
$353$	1.42638e6	0.609253	0.304627	−	0.952472i	$-0.401468\pi$
$353$	1.42638e6	0.609253	0.304627	+	0.952472i	$0.401468\pi$
$354$	−179195.	−0.0760007
$355$	686647.	0.289176
$356$	−206623.	−0.0864079
$357$	−705010.	−0.292769
$358$	1.50810e6	0.621903
$359$	−895639.	−0.366773	−0.183386	−	0.983041i	$-0.558706\pi$
$359$	−895639.	−0.366773	−0.183386	+	0.983041i	$0.558706\pi$
$360$	−491637.	−0.199935
$361$	−925268.	−0.373680
$362$	1.03496e6	0.415100
$363$	1.40483e6	0.559574
$364$	854645.	0.338090
$365$	−2.14722e6	−0.843616
$366$	1.63509e6	0.638025
$367$	715834.	0.277426	0.138713	−	0.990333i	$-0.455703\pi$
$367$	715834.	0.277426	0.138713	+	0.990333i	$0.455703\pi$
$368$	−763681.	−0.293963
$369$	1.24856e6	0.477356
$370$	−10157.5	−0.00385729
$371$	2.24348e6	0.846229
$372$	1.39020e6	0.520857
$373$	−2.73534e6	−1.01798	−0.508990	−	0.860773i	$-0.669981\pi$
$373$	−2.73534e6	−1.01798	−0.508990	+	0.860773i	$0.669981\pi$
$374$	595307.	0.220071
$375$	140625.	0.0516398
$376$	6.91998e6	2.52427
$377$	969129.	0.351179
$378$	605536.	0.217977
$379$	−1.85409e6	−0.663029	−0.331515	−	0.943450i	$-0.607560\pi$
$379$	−1.85409e6	−0.663029	−0.331515	+	0.943450i	$0.607560\pi$
$380$	1.79459e6	0.637539
$381$	−301852.	−0.106532
$382$	5.82230e6	2.04144
$383$	1.41765e6	0.493825	0.246912	−	0.969038i	$-0.420584\pi$
$383$	1.41765e6	0.493825	0.246912	+	0.969038i	$0.420584\pi$
$384$	3.03822e6	1.05146
$385$	−154448.	−0.0531043
$386$	−6.02829e6	−2.05933
$387$	74616.8	0.0253256
$388$	6.45373e6	2.17636
$389$	−620930.	−0.208050	−0.104025	−	0.994575i	$-0.533172\pi$
$389$	−620930.	−0.208050	−0.104025	+	0.994575i	$0.533172\pi$
$390$	−360020.	−0.119857
$391$	1.50158e6	0.496716
$392$	−2.21181e6	−0.726997
$393$	−788482.	−0.257520
$394$	−2.15611e6	−0.699730
$395$	−442831.	−0.142805
$396$	−328787.	−0.105360
$397$	−17627.9	−0.00561339	−0.00280670	−	0.999996i	$-0.500893\pi$
$397$	−17627.9	−0.00561339	−0.00280670	+	0.999996i	$0.500893\pi$
$398$	6.56228e6	2.07657
$399$	−983286.	−0.309206
$400$	283818.	0.0886931
$401$	−3.13600e6	−0.973903	−0.486951	−	0.873429i	$-0.661891\pi$
$401$	−3.13600e6	−0.973903	−0.486951	+	0.873429i	$0.661891\pi$
$402$	−1.12703e6	−0.347832
$403$	452873.	0.138904
$404$	−4.12338e6	−1.25690
$405$	−164025.	−0.0496904
$406$	4.76329e6	1.43414
$407$	−3021.87	−0.000904251 0
$408$	1.95101e6	0.580242
$409$	−4.54221e6	−1.34264	−0.671319	−	0.741169i	$-0.734272\pi$
$409$	−4.54221e6	−1.34264	−0.671319	+	0.741169i	$0.734272\pi$
$410$	−3.64855e6	−1.07192
$411$	2.02711e6	0.591933
$412$	−10320.9	−0.00299554
$413$	−184494.	−0.0532238
$414$	−1.28972e6	−0.369823
$415$	309329.	0.0881658
$416$	586361.	0.166124
$417$	−655789.	−0.184682
$418$	830282.	0.232426
$419$	6.13384e6	1.70686	0.853430	−	0.521208i	$-0.174519\pi$
$419$	6.13384e6	1.70686	0.853430	+	0.521208i	$0.174519\pi$
$420$	−1.13784e6	−0.314745
$421$	441743.	0.121469	0.0607343	−	0.998154i	$-0.480656\pi$
$421$	441743.	0.121469	0.0607343	+	0.998154i	$0.480656\pi$
$422$	−1.24798e6	−0.341135
$423$	2.30871e6	0.627363
$424$	−6.20851e6	−1.67715
$425$	−558056.	−0.149867
$426$	2.34042e6	0.624841
$427$	1.68343e6	0.446814
$428$	5.40067e6	1.42508
$429$	−107106.	−0.0280978
$430$	−218046.	−0.0568693
$431$	4.05311e6	1.05098	0.525491	−	0.850799i	$-0.323882\pi$
$431$	4.05311e6	1.05098	0.525491	+	0.850799i	$0.323882\pi$
$432$	−331045.	−0.0853450
$433$	2.15043e6	0.551195	0.275597	−	0.961273i	$-0.411124\pi$
$433$	2.15043e6	0.551195	0.275597	+	0.961273i	$0.411124\pi$
$434$	2.22588e6	0.567254
$435$	−1.29026e6	−0.326930
$436$	−1.22963e6	−0.309784
$437$	2.09428e6	0.524603
$438$	−7.31875e6	−1.82285
$439$	−2.96948e6	−0.735391	−0.367696	−	0.929946i	$-0.619853\pi$
$439$	−2.96948e6	−0.735391	−0.367696	+	0.929946i	$0.619853\pi$
$440$	427411.	0.105248
$441$	−737926.	−0.180683
$442$	1.42870e6	0.347845
$443$	222455.	0.0538558	0.0269279	−	0.999637i	$-0.491428\pi$
$443$	222455.	0.0538558	0.0269279	+	0.999637i	$0.491428\pi$
$444$	−22262.6	−0.00535942
$445$	89613.7	0.0214523
$446$	−1.13895e7	−2.71124
$447$	−1.80311e6	−0.426829
$448$	4.15685e6	0.978518
$449$	6.44827e6	1.50948	0.754740	−	0.656024i	$-0.227763\pi$
$449$	6.44827e6	1.50948	0.754740	+	0.656024i	$0.227763\pi$
$450$	479316.	0.111581
$451$	−1.08545e6	−0.251286
$452$	1.26007e6	0.290102
$453$	2.41915e6	0.553881
$454$	1.25580e7	2.85943
$455$	−370665.	−0.0839370
$456$	2.72110e6	0.612819
$457$	4.52847e6	1.01429	0.507144	−	0.861862i	$-0.330701\pi$
$457$	4.52847e6	1.01429	0.507144	+	0.861862i	$0.330701\pi$
$458$	−1.24615e7	−2.77592
$459$	650916.	0.144209
$460$	2.42346e6	0.534000
$461$	−8.97112e6	−1.96605	−0.983024	−	0.183475i	$-0.941265\pi$
$461$	−8.97112e6	−1.96605	−0.983024	+	0.183475i	$0.941265\pi$
$462$	−526431.	−0.114746
$463$	5.51660e6	1.19597	0.597984	−	0.801508i	$-0.295969\pi$
$463$	5.51660e6	1.19597	0.597984	+	0.801508i	$0.295969\pi$
$464$	−2.60408e6	−0.561513
$465$	−602937.	−0.129312
$466$	−1.18774e7	−2.53372
$467$	6.47012e6	1.37284	0.686421	−	0.727205i	$-0.259181\pi$
$467$	6.47012e6	1.37284	0.686421	+	0.727205i	$0.259181\pi$
$468$	−789070.	−0.166533
$469$	−1.16035e6	−0.243589
$470$	−6.74656e6	−1.40876
$471$	1.68988e6	0.350997
$472$	510559.	0.105485
$473$	−64869.1	−0.0133317
$474$	−1.50938e6	−0.308568
$475$	−778327.	−0.158281
$476$	4.51540e6	0.913438
$477$	−2.07134e6	−0.416828
$478$	−1.29117e7	−2.58473
$479$	7.88628e6	1.57048	0.785242	−	0.619189i	$-0.212538\pi$
$479$	7.88628e6	1.57048	0.785242	+	0.619189i	$0.212538\pi$
$480$	−780658.	−0.154653
$481$	−7252.30	−0.00142927
$482$	−1.43934e7	−2.82192
$483$	−1.32785e6	−0.258989
$484$	−8.99757e6	−1.74587
$485$	−2.79903e6	−0.540322
$486$	−559075.	−0.107369
$487$	−4.37472e6	−0.835848	−0.417924	−	0.908482i	$-0.637242\pi$
$487$	−4.37472e6	−0.835848	−0.417924	+	0.908482i	$0.637242\pi$
$488$	−4.65866e6	−0.885546
$489$	2.81373e6	0.532121
$490$	2.15638e6	0.405728
$491$	−285109.	−0.0533711	−0.0266856	−	0.999644i	$-0.508495\pi$
$491$	−285109.	−0.0533711	−0.0266856	+	0.999644i	$0.508495\pi$
$492$	−7.99667e6	−1.48935
$493$	5.12026e6	0.948801
$494$	1.99263e6	0.367374
$495$	142597.	0.0261576
$496$	−1.21689e6	−0.222098
$497$	2.40962e6	0.437580
$498$	1.05434e6	0.190505
$499$	6.31747e6	1.13577	0.567887	−	0.823106i	$-0.307761\pi$
$499$	6.31747e6	1.13577	0.567887	+	0.823106i	$0.307761\pi$
$500$	−900666.	−0.161116
$501$	2.73210e6	0.486297
$502$	−1.09660e7	−1.94217
$503$	−5.68821e6	−1.00243	−0.501217	−	0.865322i	$-0.667114\pi$
$503$	−5.68821e6	−1.00243	−0.501217	+	0.865322i	$0.667114\pi$
$504$	−1.72528e6	−0.302541
$505$	1.78834e6	0.312048
$506$	1.12123e6	0.194679
$507$	−257049.	−0.0444116
$508$	1.93328e6	0.332381
$509$	3.61839e6	0.619043	0.309522	−	0.950892i	$-0.399831\pi$
$509$	3.61839e6	0.619043	0.309522	+	0.950892i	$0.399831\pi$
$510$	−1.90212e6	−0.323826
$511$	−7.53516e6	−1.27656
$512$	−5.10358e6	−0.860400
$513$	907841.	0.152306
$514$	1.93143e6	0.322457
$515$	4476.25	0.000743697 0
$516$	−477901.	−0.0790158
$517$	−2.00711e6	−0.330251
$518$	−35645.2	−0.00583683
$519$	−1.79766e6	−0.292947
$520$	1.02576e6	0.166356
$521$	−1.67303e6	−0.270028	−0.135014	−	0.990844i	$-0.543108\pi$
$521$	−1.67303e6	−0.270028	−0.135014	+	0.990844i	$0.543108\pi$
$522$	−4.39782e6	−0.706417
$523$	1.31354e6	0.209986	0.104993	−	0.994473i	$-0.466518\pi$
$523$	1.31354e6	0.209986	0.104993	+	0.994473i	$0.466518\pi$
$524$	5.05002e6	0.803461
$525$	493489.	0.0781411
$526$	6.88156e6	1.08448
$527$	2.39269e6	0.375285
$528$	287798.	0.0449266
$529$	−3.60818e6	−0.560595
$530$	6.05292e6	0.935998
$531$	170338.	0.0262165
$532$	6.29769e6	0.964722
$533$	−2.60501e6	−0.397184
$534$	305446.	0.0463533
$535$	−2.34231e6	−0.353801
$536$	3.21111e6	0.482773
$537$	−1.43356e6	−0.214526
$538$	−4.62169e6	−0.688407
$539$	641526.	0.0951135
$540$	1.05054e6	0.155034
$541$	229926.	0.0337750	0.0168875	−	0.999857i	$-0.494624\pi$
$541$	229926.	0.0337750	0.0168875	+	0.999857i	$0.494624\pi$
$542$	−1.19082e7	−1.74119
$543$	−983807.	−0.143189
$544$	3.09796e6	0.448827
$545$	533300.	0.0769096
$546$	−1.26340e6	−0.181368
$547$	7.70690e6	1.10132	0.550658	−	0.834731i	$-0.314377\pi$
$547$	7.70690e6	1.10132	0.550658	+	0.834731i	$0.314377\pi$
$548$	−1.29831e7	−1.84683
$549$	−1.55427e6	−0.220087
$550$	−416700.	−0.0587377
$551$	7.14130e6	1.00207
$552$	3.67464e6	0.513295
$553$	−1.55401e6	−0.216093
$554$	1.28193e7	1.77455
$555$	9655.43	0.00133057
$556$	4.20016e6	0.576207
$557$	−8.73237e6	−1.19260	−0.596299	−	0.802763i	$-0.703363\pi$
$557$	−8.73237e6	−1.19260	−0.596299	+	0.802763i	$0.703363\pi$
$558$	−2.05510e6	−0.279413
$559$	−155682.	−0.0210722
$560$	995990.	0.134210
$561$	−565882.	−0.0759135
$562$	1.10920e7	1.48139
$563$	1.04109e7	1.38426	0.692128	−	0.721775i	$-0.256673\pi$
$563$	1.04109e7	1.38426	0.692128	+	0.721775i	$0.256673\pi$
$564$	−1.47867e7	−1.95737
$565$	−546503.	−0.0720230
$566$	−1.76268e7	−2.31277
$567$	−575606.	−0.0751913
$568$	−6.66828e6	−0.867247
$569$	1.00943e7	1.30706	0.653531	−	0.756900i	$-0.273287\pi$
$569$	1.00943e7	1.30706	0.653531	+	0.756900i	$0.273287\pi$
$570$	−2.65291e6	−0.342007
$571$	1.22705e7	1.57497	0.787483	−	0.616337i	$-0.211384\pi$
$571$	1.22705e7	1.57497	0.787483	+	0.616337i	$0.211384\pi$
$572$	685988.	0.0876651
$573$	−5.53452e6	−0.704196
$574$	−1.28037e7	−1.62202
$575$	−1.05107e6	−0.132575
$576$	−3.83790e6	−0.481990
$577$	2.77864e6	0.347451	0.173725	−	0.984794i	$-0.444419\pi$
$577$	2.77864e6	0.347451	0.173725	+	0.984794i	$0.444419\pi$
$578$	−5.89482e6	−0.733924
$579$	5.73033e6	0.710368
$580$	8.26378e6	1.02002
$581$	1.08551e6	0.133412
$582$	−9.54041e6	−1.16751
$583$	1.80075e6	0.219423
$584$	2.08524e7	2.53003
$585$	342225.	0.0413449
$586$	1.11371e7	1.33976
$587$	1.45815e7	1.74666	0.873328	−	0.487133i	$-0.161957\pi$
$587$	1.45815e7	1.74666	0.873328	+	0.487133i	$0.161957\pi$
$588$	4.72622e6	0.563730
$589$	3.33712e6	0.396355
$590$	−497764.	−0.0588699
$591$	2.04954e6	0.241372
$592$	19487.2	0.00228531
$593$	1.24857e7	1.45806	0.729031	−	0.684480i	$-0.239971\pi$
$593$	1.24857e7	1.45806	0.729031	+	0.684480i	$0.239971\pi$
$594$	486039.	0.0565204
$595$	−1.95836e6	−0.226778
$596$	1.15485e7	1.33171
$597$	−6.23792e6	−0.716315
$598$	2.69089e6	0.307711
$599$	1.85120e6	0.210807	0.105404	−	0.994430i	$-0.466387\pi$
$599$	1.85120e6	0.210807	0.105404	+	0.994430i	$0.466387\pi$
$600$	−1.36566e6	−0.154869
$601$	5.47054e6	0.617795	0.308897	−	0.951095i	$-0.400040\pi$
$601$	5.47054e6	0.617795	0.308897	+	0.951095i	$0.400040\pi$
$602$	−765181.	−0.0860544
$603$	1.07132e6	0.119985
$604$	−1.54940e7	−1.72811
$605$	3.90231e6	0.433444
$606$	6.09550e6	0.674261
$607$	−127671.	−0.0140644	−0.00703220	−	0.999975i	$-0.502238\pi$
$607$	−127671.	−0.0140644	−0.00703220	+	0.999975i	$0.502238\pi$
$608$	4.32077e6	0.474026
$609$	−4.52785e6	−0.494708
$610$	4.54191e6	0.494212
$611$	−4.81694e6	−0.521998
$612$	−4.16895e6	−0.449933
$613$	−1.25347e7	−1.34730	−0.673648	−	0.739052i	$-0.735274\pi$
$613$	−1.25347e7	−1.34730	−0.673648	+	0.739052i	$0.735274\pi$
$614$	166979.	0.0178748
$615$	3.46821e6	0.369758
$616$	1.49990e6	0.159261
$617$	1.44144e7	1.52434	0.762172	−	0.647374i	$-0.224133\pi$
$617$	1.44144e7	1.52434	0.762172	+	0.647374i	$0.224133\pi$
$618$	15257.2	0.00160695
$619$	−9.76407e6	−1.02425	−0.512123	−	0.858912i	$-0.671141\pi$
$619$	−9.76407e6	−1.02425	−0.512123	+	0.858912i	$0.671141\pi$
$620$	3.86166e6	0.403454
$621$	1.22597e6	0.127571
$622$	1.23195e7	1.27679
$623$	314478.	0.0324616
$624$	690699.	0.0710113
$625$	390625.	0.0400000
$626$	−7.96562e6	−0.812426
$627$	−789243.	−0.0801756
$628$	−1.08232e7	−1.09511
$629$	−38316.6	−0.00386153
$630$	1.68204e6	0.168844
$631$	−1.89186e7	−1.89154	−0.945768	−	0.324843i	$-0.894688\pi$
$631$	−1.89186e7	−1.89154	−0.945768	+	0.324843i	$0.894688\pi$
$632$	4.30049e6	0.428277
$633$	1.18629e6	0.117675
$634$	1.60442e6	0.158524
$635$	−838478.	−0.0825197
$636$	1.32664e7	1.30050
$637$	1.53962e6	0.150337
$638$	3.82330e6	0.371866
$639$	−2.22474e6	−0.215539
$640$	8.43951e6	0.814455
$641$	−1.68562e7	−1.62038	−0.810188	−	0.586171i	$-0.800635\pi$
$641$	−1.68562e7	−1.62038	−0.810188	+	0.586171i	$0.800635\pi$
$642$	−7.98369e6	−0.764480
$643$	1.31206e6	0.125149	0.0625744	−	0.998040i	$-0.480069\pi$
$643$	1.31206e6	0.125149	0.0625744	+	0.998040i	$0.480069\pi$
$644$	8.50455e6	0.808047
$645$	207269.	0.0196171
$646$	1.05278e7	0.992558
$647$	1.84547e7	1.73319	0.866596	−	0.499010i	$-0.166303\pi$
$647$	1.84547e7	1.73319	0.866596	+	0.499010i	$0.166303\pi$
$648$	1.59291e6	0.149023
$649$	−148085.	−0.0138007
$650$	−1.00006e6	−0.0928412
$651$	−2.11586e6	−0.195675
$652$	−1.80212e7	−1.66022
$653$	−7.98759e6	−0.733048	−0.366524	−	0.930409i	$-0.619452\pi$
$653$	−7.98759e6	−0.733048	−0.366524	+	0.930409i	$0.619452\pi$
$654$	1.81774e6	0.166183
$655$	−2.19023e6	−0.199474
$656$	6.99976e6	0.635073
$657$	6.95700e6	0.628795
$658$	−2.36754e7	−2.13173
$659$	−1.16048e7	−1.04094	−0.520468	−	0.853881i	$-0.674243\pi$
$659$	−1.16048e7	−1.04094	−0.520468	+	0.853881i	$0.674243\pi$
$660$	−913298.	−0.0816117
$661$	−9.83454e6	−0.875489	−0.437744	−	0.899099i	$-0.644223\pi$
$661$	−9.83454e6	−0.875489	−0.437744	+	0.899099i	$0.644223\pi$
$662$	−2.50055e7	−2.21764
$663$	−1.35808e6	−0.119989
$664$	−3.00400e6	−0.264411
$665$	−2.73135e6	−0.239510
$666$	32910.3	0.00287505
$667$	9.64378e6	0.839330
$668$	−1.74984e7	−1.51725
$669$	1.08266e7	0.935245
$670$	−3.13064e6	−0.269430
$671$	1.35122e6	0.115857
$672$	−2.73953e6	−0.234020
$673$	−2.05002e7	−1.74470	−0.872351	−	0.488881i	$-0.837405\pi$
$673$	−2.05002e7	−1.74470	−0.872351	+	0.488881i	$0.837405\pi$
$674$	1.71286e7	1.45235
$675$	−455625.	−0.0384900
$676$	1.64633e6	0.138564
$677$	5.19490e6	0.435618	0.217809	−	0.975991i	$-0.430109\pi$
$677$	5.19490e6	0.435618	0.217809	+	0.975991i	$0.430109\pi$
$678$	−1.86274e6	−0.155625
$679$	−9.82251e6	−0.817613
$680$	5.41948e6	0.449454
$681$	−1.19372e7	−0.986362
$682$	1.78662e6	0.147086
$683$	−1.91454e7	−1.57041	−0.785206	−	0.619234i	$-0.787443\pi$
$683$	−1.91454e7	−1.57041	−0.785206	+	0.619234i	$0.787443\pi$
$684$	−5.81448e6	−0.475194
$685$	5.63086e6	0.458509
$686$	2.15278e7	1.74659
$687$	1.18456e7	0.957554
$688$	418323.	0.0336931
$689$	4.32169e6	0.346821
$690$	−3.58255e6	−0.286464
$691$	−1.95284e7	−1.55586	−0.777932	−	0.628349i	$-0.783731\pi$
$691$	−1.95284e7	−1.55586	−0.777932	+	0.628349i	$0.783731\pi$
$692$	1.15135e7	0.913993
$693$	500410.	0.0395816
$694$	−1.10070e7	−0.867503
$695$	−1.82164e6	−0.143054
$696$	1.25302e7	0.980470
$697$	−1.37632e7	−1.07310
$698$	−3.11968e7	−2.42366
$699$	1.12904e7	0.874008
$700$	−3.16067e6	−0.243800
$701$	4.01022e6	0.308229	0.154114	−	0.988053i	$-0.450748\pi$
$701$	4.01022e6	0.308229	0.154114	+	0.988053i	$0.450748\pi$
$702$	1.16646e6	0.0893365
$703$	−53440.6	−0.00407834
$704$	3.33653e6	0.253725
$705$	6.41309e6	0.485953
$706$	1.35049e7	1.01972
$707$	6.27574e6	0.472190
$708$	−1.09097e6	−0.0817955
$709$	1.14260e7	0.853647	0.426823	−	0.904335i	$-0.359633\pi$
$709$	1.14260e7	0.853647	0.426823	+	0.904335i	$0.359633\pi$
$710$	6.50116e6	0.484000
$711$	1.43477e6	0.106441
$712$	−870271.	−0.0643360
$713$	4.50653e6	0.331985
$714$	−6.67502e6	−0.490012
$715$	−297518.	−0.0217645
$716$	9.18156e6	0.669320
$717$	1.22735e7	0.891604
$718$	−8.47989e6	−0.613874
$719$	1.79890e7	1.29773	0.648865	−	0.760903i	$-0.275244\pi$
$719$	1.79890e7	1.29773	0.648865	+	0.760903i	$0.275244\pi$
$720$	−919570.	−0.0661079
$721$	15708.3	0.00112536
$722$	−8.76042e6	−0.625434
$723$	1.36819e7	0.973423
$724$	6.30102e6	0.446750
$725$	−3.58406e6	−0.253239
$726$	1.33009e7	0.936569
$727$	8.92635e6	0.626380	0.313190	−	0.949690i	$-0.398602\pi$
$727$	8.92635e6	0.626380	0.313190	+	0.949690i	$0.398602\pi$
$728$	3.59966e6	0.251729
$729$	531441.	0.0370370
$730$	−2.03299e7	−1.41198
$731$	−822525.	−0.0569319
$732$	9.95468e6	0.686672
$733$	−2.67429e7	−1.83844	−0.919219	−	0.393746i	$-0.871179\pi$
$733$	−2.67429e7	−1.83844	−0.919219	+	0.393746i	$0.871179\pi$
$734$	6.77750e6	0.464333
$735$	−2.04979e6	−0.139956
$736$	5.83486e6	0.397042
$737$	−931368.	−0.0631615
$738$	1.18213e7	0.798959
$739$	−1.52087e6	−0.102443	−0.0512213	−	0.998687i	$-0.516311\pi$
$739$	−1.52087e6	−0.102443	−0.0512213	+	0.998687i	$0.516311\pi$
$740$	−61840.5	−0.00415139
$741$	−1.89414e6	−0.126726
$742$	2.12412e7	1.41635
$743$	1.77630e7	1.18044	0.590222	−	0.807241i	$-0.299040\pi$
$743$	1.77630e7	1.18044	0.590222	+	0.807241i	$0.299040\pi$
$744$	5.85534e6	0.387811
$745$	−5.00864e6	−0.330620
$746$	−2.58981e7	−1.70381
$747$	−1.00223e6	−0.0657149
$748$	3.62433e6	0.236850
$749$	−8.21976e6	−0.535370
$750$	1.33143e6	0.0864304
$751$	1.81110e6	0.117177	0.0585884	−	0.998282i	$-0.481340\pi$
$751$	1.81110e6	0.117177	0.0585884	+	0.998282i	$0.481340\pi$
$752$	1.29433e7	0.834642
$753$	1.04240e7	0.669955
$754$	9.17569e6	0.587774
$755$	6.71985e6	0.429034
$756$	3.68660e6	0.234597
$757$	1.79449e7	1.13815	0.569077	−	0.822285i	$-0.307301\pi$
$757$	1.79449e7	1.13815	0.569077	+	0.822285i	$0.307301\pi$
$758$	−1.75545e7	−1.10972
$759$	−1.06581e6	−0.0671547
$760$	7.55861e6	0.474688
$761$	−2.89207e7	−1.81029	−0.905143	−	0.425108i	$-0.860236\pi$
$761$	−2.89207e7	−1.81029	−0.905143	+	0.425108i	$0.860236\pi$
$762$	−2.85793e6	−0.178305
$763$	1.87149e6	0.116379
$764$	3.54471e7	2.19709
$765$	1.80810e6	0.111704
$766$	1.34223e7	0.826524
$767$	−355396.	−0.0218134
$768$	1.51199e7	0.925012
$769$	1.41345e7	0.861914	0.430957	−	0.902373i	$-0.358176\pi$
$769$	1.41345e7	0.861914	0.430957	+	0.902373i	$0.358176\pi$
$770$	−1.46231e6	−0.0888816
$771$	−1.83597e6	−0.111232
$772$	−3.67012e7	−2.21635
$773$	−2.44736e7	−1.47316	−0.736578	−	0.676353i	$-0.763560\pi$
$773$	−2.44736e7	−1.47316	−0.736578	+	0.676353i	$0.763560\pi$
$774$	706471.	0.0423879
$775$	−1.67483e6	−0.100165
$776$	2.71824e7	1.62044
$777$	33883.4	0.00201342
$778$	−5.87895e6	−0.348218
$779$	−1.91958e7	−1.13334
$780$	−2.19186e6	−0.128996
$781$	1.93410e6	0.113462
$782$	1.42170e7	0.831362
$783$	4.18044e6	0.243679
$784$	−4.13702e6	−0.240380
$785$	4.69411e6	0.271881
$786$	−7.46533e6	−0.431015
$787$	2.64173e7	1.52038	0.760189	−	0.649702i	$-0.225106\pi$
$787$	2.64173e7	1.52038	0.760189	+	0.649702i	$0.225106\pi$
$788$	−1.31268e7	−0.753082
$789$	−6.54142e6	−0.374093
$790$	−4.19271e6	−0.239016
$791$	−1.91782e6	−0.108985
$792$	−1.38481e6	−0.0784473
$793$	3.24286e6	0.183124
$794$	−166901.	−0.00939523
$795$	−5.75373e6	−0.322873
$796$	3.99523e7	2.23490
$797$	1.16049e7	0.647134	0.323567	−	0.946205i	$-0.395118\pi$
$797$	1.16049e7	0.647134	0.323567	+	0.946205i	$0.395118\pi$
$798$	−9.30973e6	−0.517523
$799$	−2.54497e7	−1.41031
$800$	−2.16850e6	−0.119794
$801$	−290348.	−0.0159896
$802$	−2.96916e7	−1.63004
$803$	−6.04816e6	−0.331005
$804$	−6.86154e6	−0.374353
$805$	−3.68848e6	−0.200612
$806$	4.28779e6	0.232486
$807$	4.39326e6	0.237467
$808$	−1.73672e7	−0.935839
$809$	−1.74431e7	−0.937029	−0.468515	−	0.883456i	$-0.655211\pi$
$809$	−1.74431e7	−0.937029	−0.468515	+	0.883456i	$0.655211\pi$
$810$	−1.55299e6	−0.0831677
$811$	6.63388e6	0.354173	0.177087	−	0.984195i	$-0.443333\pi$
$811$	6.63388e6	0.354173	0.177087	+	0.984195i	$0.443333\pi$
$812$	2.89997e7	1.54349
$813$	1.13196e7	0.600626
$814$	−28611.0	−0.00151346
$815$	7.81591e6	0.412179
$816$	3.64922e6	0.191856
$817$	−1.14719e6	−0.0601283
$818$	−4.30055e7	−2.24720
$819$	1.20096e6	0.0625629
$820$	−2.22130e7	−1.15365
$821$	−1.50323e7	−0.778339	−0.389169	−	0.921166i	$-0.627238\pi$
$821$	−1.50323e7	−0.778339	−0.389169	+	0.921166i	$0.627238\pi$
$822$	1.91926e7	0.990729
$823$	1.50676e7	0.775433	0.387716	−	0.921779i	$-0.373264\pi$
$823$	1.50676e7	0.775433	0.387716	+	0.921779i	$0.373264\pi$
$824$	−43470.4	−0.00223037
$825$	396103.	0.0202616
$826$	−1.74678e6	−0.0890817
$827$	−3.56260e7	−1.81135	−0.905677	−	0.423969i	$-0.860636\pi$
$827$	−3.56260e7	−1.81135	−0.905677	+	0.423969i	$0.860636\pi$
$828$	−7.85202e6	−0.398020
$829$	−5.83029e6	−0.294648	−0.147324	−	0.989088i	$-0.547066\pi$
$829$	−5.83029e6	−0.294648	−0.147324	+	0.989088i	$0.547066\pi$
$830$	2.92872e6	0.147565
$831$	−1.21857e7	−0.612134
$832$	8.00748e6	0.401040
$833$	8.13440e6	0.406175
$834$	−6.20900e6	−0.309105
$835$	7.58916e6	0.376684
$836$	5.05490e6	0.250148
$837$	1.95352e6	0.0963837
$838$	5.80751e7	2.85680
$839$	−3.42304e7	−1.67883	−0.839415	−	0.543492i	$-0.817102\pi$
$839$	−3.42304e7	−1.67883	−0.839415	+	0.543492i	$0.817102\pi$
$840$	−4.79245e6	−0.234347
$841$	1.23732e7	0.603244
$842$	4.18241e6	0.203304
$843$	−1.05438e7	−0.511008
$844$	−7.59790e6	−0.367145
$845$	−714025.	−0.0344010
$846$	2.18588e7	1.05003
$847$	1.36942e7	0.655885
$848$	−1.16125e7	−0.554546
$849$	1.67556e7	0.797793
$850$	−5.28366e6	−0.250835
$851$	−72167.4	−0.00341600
$852$	1.42489e7	0.672482
$853$	1.15393e7	0.543011	0.271505	−	0.962437i	$-0.412479\pi$
$853$	1.15393e7	0.543011	0.271505	+	0.962437i	$0.412479\pi$
$854$	1.59387e7	0.747840
$855$	2.52178e6	0.117976
$856$	2.27470e7	1.06106
$857$	−2.89877e7	−1.34822	−0.674111	−	0.738630i	$-0.735473\pi$
$857$	−2.89877e7	−1.34822	−0.674111	+	0.738630i	$0.735473\pi$
$858$	−1.01408e6	−0.0470278
$859$	1.73688e7	0.803132	0.401566	−	0.915830i	$-0.368466\pi$
$859$	1.73688e7	0.803132	0.401566	+	0.915830i	$0.368466\pi$
$860$	−1.32750e6	−0.0612054
$861$	1.21708e7	0.559516
$862$	3.83748e7	1.75905
$863$	3.46809e7	1.58512	0.792562	−	0.609791i	$-0.208747\pi$
$863$	3.46809e7	1.58512	0.792562	+	0.609791i	$0.208747\pi$
$864$	2.52933e6	0.115271
$865$	−4.99349e6	−0.226915
$866$	2.03602e7	0.922545
$867$	5.60345e6	0.253168
$868$	1.35515e7	0.610505
$869$	−1.24734e6	−0.0560318
$870$	−1.22162e7	−0.547188
$871$	−2.23523e6	−0.0998336
$872$	−5.17907e6	−0.230654
$873$	9.06885e6	0.402732
$874$	1.98286e7	0.878038
$875$	1.37080e6	0.0605278
$876$	−4.45578e7	−1.96184
$877$	5.12506e6	0.225009	0.112504	−	0.993651i	$-0.464113\pi$
$877$	5.12506e6	0.225009	0.112504	+	0.993651i	$0.464113\pi$
$878$	−2.81149e7	−1.23084
$879$	−1.05866e7	−0.462151
$880$	799440.	0.0348000
$881$	1.54340e7	0.669946	0.334973	−	0.942228i	$-0.391273\pi$
$881$	1.54340e7	0.669946	0.334973	+	0.942228i	$0.391273\pi$
$882$	−6.98667e6	−0.302412
$883$	−4.05392e7	−1.74974	−0.874870	−	0.484358i	$-0.839053\pi$
$883$	−4.05392e7	−1.74974	−0.874870	+	0.484358i	$0.839053\pi$
$884$	8.69817e6	0.374367
$885$	473161.	0.0203072
$886$	2.10620e6	0.0901394
$887$	−2.14344e7	−0.914748	−0.457374	−	0.889275i	$-0.651210\pi$
$887$	−2.14344e7	−0.914748	−0.457374	+	0.889275i	$0.651210\pi$
$888$	−93767.3	−0.00399042
$889$	−2.94244e6	−0.124868
$890$	848461.	0.0359052
$891$	−462015.	−0.0194967
$892$	−6.93413e7	−2.91796
$893$	−3.54950e7	−1.48949
$894$	−1.70718e7	−0.714391
$895$	−3.98210e6	−0.166171
$896$	2.96164e7	1.23243
$897$	−2.55789e6	−0.106145
$898$	6.10521e7	2.52644
$899$	1.53668e7	0.634140
$900$	2.91816e6	0.120089
$901$	2.28331e7	0.937029
$902$	−1.02770e7	−0.420582
$903$	727360.	0.0296845
$904$	5.30728e6	0.215999
$905$	−2.73280e6	−0.110914
$906$	2.29044e7	0.927041
$907$	−4.74313e7	−1.91446	−0.957231	−	0.289325i	$-0.906569\pi$
$907$	−4.74313e7	−1.91446	−0.957231	+	0.289325i	$0.906569\pi$
$908$	7.64549e7	3.07745
$909$	−5.79422e6	−0.232587
$910$	−3.50945e6	−0.140487
$911$	−2.73179e7	−1.09056	−0.545282	−	0.838253i	$-0.683577\pi$
$911$	−2.73179e7	−1.09056	−0.545282	+	0.838253i	$0.683577\pi$
$912$	5.08961e6	0.202627
$913$	871298.	0.0345931
$914$	4.28755e7	1.69763
$915$	−4.31741e6	−0.170479
$916$	−7.58677e7	−2.98757
$917$	−7.68607e6	−0.301843
$918$	6.16286e6	0.241366
$919$	1.66893e7	0.651853	0.325926	−	0.945395i	$-0.394324\pi$
$919$	1.66893e7	0.651853	0.325926	+	0.945395i	$0.394324\pi$
$920$	1.02073e7	0.397597
$921$	−158725.	−0.00616591
$922$	−8.49384e7	−3.29061
$923$	4.64173e6	0.179340
$924$	−3.20500e6	−0.123495
$925$	26820.6	0.00103066
$926$	5.22311e7	2.00171
$927$	−14503.0	−0.000554319 0
$928$	1.98963e7	0.758409
$929$	−4.29509e7	−1.63280	−0.816401	−	0.577486i	$-0.804034\pi$
$929$	−4.29509e7	−1.63280	−0.816401	+	0.577486i	$0.804034\pi$
$930$	−5.70860e6	−0.216432
$931$	1.13451e7	0.428979
$932$	−7.23119e7	−2.72690
$933$	−1.17106e7	−0.440428
$934$	6.12590e7	2.29775
$935$	−1.57190e6	−0.0588023
$936$	−3.32347e6	−0.123994
$937$	1.68472e7	0.626872	0.313436	−	0.949609i	$-0.398520\pi$
$937$	1.68472e7	0.626872	0.313436	+	0.949609i	$0.398520\pi$
$938$	−1.09862e7	−0.407700
$939$	7.57190e6	0.280247
$940$	−4.10742e7	−1.51617
$941$	3.62444e6	0.133434	0.0667171	−	0.997772i	$-0.478747\pi$
$941$	3.62444e6	0.133434	0.0667171	+	0.997772i	$0.478747\pi$
$942$	1.59997e7	0.587470
$943$	−2.59224e7	−0.949284
$944$	954962.	0.0348783
$945$	−1.59890e6	−0.0582429
$946$	−614179.	−0.0223135
$947$	4.52224e7	1.63862	0.819311	−	0.573349i	$-0.194356\pi$
$947$	4.52224e7	1.63862	0.819311	+	0.573349i	$0.194356\pi$
$948$	−9.18933e6	−0.332096
$949$	−1.45152e7	−0.523189
$950$	−7.36919e6	−0.264917
$951$	−1.52512e6	−0.0546830
$952$	1.90183e7	0.680112
$953$	−2.74542e7	−0.979210	−0.489605	−	0.871944i	$-0.662859\pi$
$953$	−2.74542e7	−0.979210	−0.489605	+	0.871944i	$0.662859\pi$
$954$	−1.96114e7	−0.697652
$955$	−1.53737e7	−0.545468
$956$	−7.86088e7	−2.78180
$957$	−3.63432e6	−0.128275
$958$	7.46672e7	2.62855
$959$	1.97601e7	0.693814
$960$	−1.06608e7	−0.373348
$961$	−2.14482e7	−0.749175
$962$	−68664.6	−0.00239219
$963$	7.58908e6	0.263708
$964$	−8.76292e7	−3.03708
$965$	1.59176e7	0.550248
$966$	−1.25721e7	−0.433475
$967$	4.89480e7	1.68333	0.841663	−	0.540003i	$-0.181577\pi$
$967$	4.89480e7	1.68333	0.841663	+	0.540003i	$0.181577\pi$
$968$	−3.78967e7	−1.29991
$969$	−1.00074e7	−0.342384
$970$	−2.65011e7	−0.904347
$971$	1.24516e7	0.423817	0.211908	−	0.977290i	$-0.432032\pi$
$971$	1.24516e7	0.423817	0.211908	+	0.977290i	$0.432032\pi$
$972$	−3.40374e6	−0.115556
$973$	−6.39259e6	−0.216468
$974$	−4.14197e7	−1.39897
$975$	950625.	0.0320256
$976$	−8.71366e6	−0.292804
$977$	−3.68879e7	−1.23637	−0.618183	−	0.786034i	$-0.712131\pi$
$977$	−3.68879e7	−1.23637	−0.618183	+	0.786034i	$0.712131\pi$
$978$	2.66403e7	0.890620
$979$	252418.	0.00841713
$980$	1.31284e7	0.436663
$981$	−1.72789e6	−0.0573250
$982$	−2.69940e6	−0.0893282
$983$	−4.90111e7	−1.61775	−0.808874	−	0.587982i	$-0.799923\pi$
$983$	−4.90111e7	−1.61775	−0.808874	+	0.587982i	$0.799923\pi$
$984$	−3.36810e7	−1.10891
$985$	5.69317e6	0.186966
$986$	4.84785e7	1.58802
$987$	2.25052e7	0.735342
$988$	1.21315e7	0.395385
$989$	−1.54919e6	−0.0503632
$990$	1.35011e6	0.0437805
$991$	−5.67096e6	−0.183431	−0.0917154	−	0.995785i	$-0.529235\pi$
$991$	−5.67096e6	−0.183431	−0.0917154	+	0.995785i	$0.529235\pi$
$992$	9.29754e6	0.299978
$993$	2.37696e7	0.764977
$994$	2.28142e7	0.732386
$995$	−1.73276e7	−0.554855
$996$	6.41899e6	0.205031
$997$	−4.31243e7	−1.37399	−0.686995	−	0.726662i	$-0.741071\pi$
$997$	−4.31243e7	−1.37399	−0.686995	+	0.726662i	$0.741071\pi$
$998$	5.98137e7	1.90097
$999$	−31283.6	−0.000991751 0

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	195.6.a.e.1.4	✓	4
3.2	odd	2		585.6.a.d.1.1		4
5.4	even	2		975.6.a.f.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.6.a.e.1.4	✓	4	1.1	even	1	trivial
585.6.a.d.1.1		4	3.2	odd	2
975.6.a.f.1.1		4	5.4	even	2

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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 $\F_{q}$
Belyi maps

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Fields

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Groups

Database

Properties

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	195.6.a.e.1.4

Level	$195$
Weight	$6$
Character	195.1
Self dual	yes
Analytic conductor	$31.275$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$