LMFDB - Embedded newform 152.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [152,6,Mod(1,152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(152, 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("152.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	152.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:	$24.3783406116$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.976277.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 267x + 1118 $ x^3 - x^2 - 267*x + 1118
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.44154$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.4044 q^{3} -30.4472 q^{5} +33.8943 q^{7} +352.577 q^{9} +571.898 q^{11} +129.707 q^{13} +743.046 q^{15} -541.255 q^{17} +361.000 q^{19} -827.172 q^{21} -1003.53 q^{23} -2197.97 q^{25} -2674.17 q^{27} +703.497 q^{29} +4148.79 q^{31} -13956.8 q^{33} -1031.99 q^{35} -6961.06 q^{37} -3165.43 q^{39} -14908.2 q^{41} -1794.32 q^{43} -10735.0 q^{45} -2809.37 q^{47} -15658.2 q^{49} +13209.0 q^{51} -17905.9 q^{53} -17412.7 q^{55} -8810.01 q^{57} +23048.5 q^{59} +42078.8 q^{61} +11950.4 q^{63} -3949.21 q^{65} -46086.7 q^{67} +24490.6 q^{69} -29169.5 q^{71} +10368.1 q^{73} +53640.2 q^{75} +19384.1 q^{77} -39465.2 q^{79} -20414.6 q^{81} +74100.3 q^{83} +16479.7 q^{85} -17168.5 q^{87} +49090.3 q^{89} +4396.33 q^{91} -101249. q^{93} -10991.4 q^{95} -57350.4 q^{97} +201638. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{3} + 58 q^{5} - 197 q^{7} + 80 q^{9} + 476 q^{11} - 1417 q^{13} + 988 q^{15} - 2427 q^{17} + 1083 q^{19} - 1787 q^{21} - 2407 q^{23} - 93 q^{25} - 8197 q^{27} + 4227 q^{29} + 10692 q^{31} - 19274 q^{33}+ \cdots + 136690 q^{99}+O(q^{100}) $ 3 * q - 7 * q^3 + 58 * q^5 - 197 * q^7 + 80 * q^9 + 476 * q^11 - 1417 * q^13 + 988 * q^15 - 2427 * q^17 + 1083 * q^19 - 1787 * q^21 - 2407 * q^23 - 93 * q^25 - 8197 * q^27 + 4227 * q^29 + 10692 * q^31 - 19274 * q^33 - 20130 * q^35 - 13130 * q^37 - 14993 * q^39 - 22440 * q^41 - 8230 * q^43 - 31922 * q^45 - 26630 * q^47 - 4842 * q^49 + 11995 * q^51 - 75557 * q^53 + 16306 * q^55 - 2527 * q^57 + 40877 * q^59 + 24024 * q^61 + 61684 * q^63 - 86172 * q^65 - 93135 * q^67 - 31679 * q^69 + 5738 * q^71 - 19661 * q^73 + 25547 * q^75 - 45464 * q^77 + 82854 * q^79 - 16357 * q^81 + 66886 * q^83 - 195598 * q^85 + 80595 * q^87 + 92268 * q^89 + 210603 * q^91 - 16048 * q^93 + 20938 * q^95 + 47310 * q^97 + 136690 * 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 $

$2$	0	0
$2$	0	0
$3$	−24.4044	−1.56555	−0.782773	−	0.622307i	$-0.786195\pi$
$3$	−24.4044	−1.56555	−0.782773	+	0.622307i	$0.786195\pi$
$4$	0	0
$5$	−30.4472	−0.544655	−0.272328	−	0.962205i	$-0.587793\pi$
$5$	−30.4472	−0.544655	−0.272328	+	0.962205i	$0.587793\pi$
$6$	0	0
$7$	33.8943	0.261446	0.130723	−	0.991419i	$-0.458270\pi$
$7$	33.8943	0.261446	0.130723	+	0.991419i	$0.458270\pi$
$8$	0	0
$9$	352.577	1.45093
$10$	0	0
$11$	571.898	1.42507	0.712536	−	0.701636i	$-0.247547\pi$
$11$	571.898	1.42507	0.712536	+	0.701636i	$0.247547\pi$
$12$	0	0
$13$	129.707	0.212865	0.106433	−	0.994320i	$-0.466057\pi$
$13$	129.707	0.212865	0.106433	+	0.994320i	$0.466057\pi$
$14$	0	0
$15$	743.046	0.852683
$16$	0	0
$17$	−541.255	−0.454234	−0.227117	−	0.973867i	$-0.572930\pi$
$17$	−541.255	−0.454234	−0.227117	+	0.973867i	$0.572930\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−827.172	−0.409306
$22$	0	0
$23$	−1003.53	−0.395558	−0.197779	−	0.980247i	$-0.563373\pi$
$23$	−1003.53	−0.395558	−0.197779	+	0.980247i	$0.563373\pi$
$24$	0	0
$25$	−2197.97	−0.703350
$26$	0	0
$27$	−2674.17	−0.705959
$28$	0	0
$29$	703.497	0.155334	0.0776671	−	0.996979i	$-0.475253\pi$
$29$	703.497	0.155334	0.0776671	+	0.996979i	$0.475253\pi$
$30$	0	0
$31$	4148.79	0.775384	0.387692	−	0.921789i	$-0.373272\pi$
$31$	4148.79	0.775384	0.387692	+	0.921789i	$0.373272\pi$
$32$	0	0
$33$	−13956.8	−2.23101
$34$	0	0
$35$	−1031.99	−0.142398
$36$	0	0
$37$	−6961.06	−0.835932	−0.417966	−	0.908463i	$-0.637257\pi$
$37$	−6961.06	−0.835932	−0.417966	+	0.908463i	$0.637257\pi$
$38$	0	0
$39$	−3165.43	−0.333251
$40$	0	0
$41$	−14908.2	−1.38505	−0.692524	−	0.721395i	$-0.743501\pi$
$41$	−14908.2	−1.38505	−0.692524	+	0.721395i	$0.743501\pi$
$42$	0	0
$43$	−1794.32	−0.147989	−0.0739945	−	0.997259i	$-0.523575\pi$
$43$	−1794.32	−0.147989	−0.0739945	+	0.997259i	$0.523575\pi$
$44$	0	0
$45$	−10735.0	−0.790259
$46$	0	0
$47$	−2809.37	−0.185509	−0.0927545	−	0.995689i	$-0.529567\pi$
$47$	−2809.37	−0.185509	−0.0927545	+	0.995689i	$0.529567\pi$
$48$	0	0
$49$	−15658.2	−0.931646
$50$	0	0
$51$	13209.0	0.711124
$52$	0	0
$53$	−17905.9	−0.875600	−0.437800	−	0.899072i	$-0.644242\pi$
$53$	−17905.9	−0.875600	−0.437800	+	0.899072i	$0.644242\pi$
$54$	0	0
$55$	−17412.7	−0.776173
$56$	0	0
$57$	−8810.01	−0.359161
$58$	0	0
$59$	23048.5	0.862012	0.431006	−	0.902349i	$-0.358159\pi$
$59$	23048.5	0.862012	0.431006	+	0.902349i	$0.358159\pi$
$60$	0	0
$61$	42078.8	1.44790	0.723951	−	0.689852i	$-0.242324\pi$
$61$	42078.8	1.44790	0.723951	+	0.689852i	$0.242324\pi$
$62$	0	0
$63$	11950.4	0.379341
$64$	0	0
$65$	−3949.21	−0.115938
$66$	0	0
$67$	−46086.7	−1.25426	−0.627132	−	0.778913i	$-0.715771\pi$
$67$	−46086.7	−1.25426	−0.627132	+	0.778913i	$0.715771\pi$
$68$	0	0
$69$	24490.6	0.619264
$70$	0	0
$71$	−29169.5	−0.686726	−0.343363	−	0.939203i	$-0.611566\pi$
$71$	−29169.5	−0.686726	−0.343363	+	0.939203i	$0.611566\pi$
$72$	0	0
$73$	10368.1	0.227716	0.113858	−	0.993497i	$-0.463679\pi$
$73$	10368.1	0.227716	0.113858	+	0.993497i	$0.463679\pi$
$74$	0	0
$75$	53640.2	1.10113
$76$	0	0
$77$	19384.1	0.372579
$78$	0	0
$79$	−39465.2	−0.711454	−0.355727	−	0.934590i	$-0.615767\pi$
$79$	−39465.2	−0.711454	−0.355727	+	0.934590i	$0.615767\pi$
$80$	0	0
$81$	−20414.6	−0.345724
$82$	0	0
$83$	74100.3	1.18066	0.590330	−	0.807162i	$-0.298998\pi$
$83$	74100.3	1.18066	0.590330	+	0.807162i	$0.298998\pi$
$84$	0	0
$85$	16479.7	0.247401
$86$	0	0
$87$	−17168.5	−0.243183
$88$	0	0
$89$	49090.3	0.656932	0.328466	−	0.944516i	$-0.393468\pi$
$89$	49090.3	0.656932	0.328466	+	0.944516i	$0.393468\pi$
$90$	0	0
$91$	4396.33	0.0556528
$92$	0	0
$93$	−101249.	−1.21390
$94$	0	0
$95$	−10991.4	−0.124953
$96$	0	0
$97$	−57350.4	−0.618880	−0.309440	−	0.950919i	$-0.600142\pi$
$97$	−57350.4	−0.618880	−0.309440	+	0.950919i	$0.600142\pi$
$98$	0	0
$99$	201638.	2.06768
$100$	0	0
$101$	−19098.4	−0.186292	−0.0931458	−	0.995652i	$-0.529692\pi$
$101$	−19098.4	−0.186292	−0.0931458	+	0.995652i	$0.529692\pi$
$102$	0	0
$103$	−123041.	−1.14276	−0.571382	−	0.820684i	$-0.693593\pi$
$103$	−123041.	−1.14276	−0.571382	+	0.820684i	$0.693593\pi$
$104$	0	0
$105$	25185.1	0.222931
$106$	0	0
$107$	82833.7	0.699436	0.349718	−	0.936855i	$-0.386277\pi$
$107$	82833.7	0.699436	0.349718	+	0.936855i	$0.386277\pi$
$108$	0	0
$109$	−228830.	−1.84479	−0.922395	−	0.386247i	$-0.873771\pi$
$109$	−228830.	−1.84479	−0.922395	+	0.386247i	$0.873771\pi$
$110$	0	0
$111$	169881.	1.30869
$112$	0	0
$113$	−160540.	−1.18273	−0.591367	−	0.806402i	$-0.701412\pi$
$113$	−160540.	−1.18273	−0.591367	+	0.806402i	$0.701412\pi$
$114$	0	0
$115$	30554.6	0.215443
$116$	0	0
$117$	45731.7	0.308854
$118$	0	0
$119$	−18345.5	−0.118758
$120$	0	0
$121$	166016.	1.03083
$122$	0	0
$123$	363826.	2.16836
$124$	0	0
$125$	162069.	0.927739
$126$	0	0
$127$	−53082.2	−0.292038	−0.146019	−	0.989282i	$-0.546646\pi$
$127$	−53082.2	−0.292038	−0.146019	+	0.989282i	$0.546646\pi$
$128$	0	0
$129$	43789.4	0.231683
$130$	0	0
$131$	−69344.8	−0.353050	−0.176525	−	0.984296i	$-0.556486\pi$
$131$	−69344.8	−0.353050	−0.176525	+	0.984296i	$0.556486\pi$
$132$	0	0
$133$	12235.9	0.0599798
$134$	0	0
$135$	81420.8	0.384504
$136$	0	0
$137$	−44631.1	−0.203159	−0.101580	−	0.994827i	$-0.532390\pi$
$137$	−44631.1	−0.203159	−0.101580	+	0.994827i	$0.532390\pi$
$138$	0	0
$139$	151826.	0.666512	0.333256	−	0.942836i	$-0.391853\pi$
$139$	151826.	0.666512	0.333256	+	0.942836i	$0.391853\pi$
$140$	0	0
$141$	68561.2	0.290423
$142$	0	0
$143$	74179.1	0.303348
$144$	0	0
$145$	−21419.5	−0.0846036
$146$	0	0
$147$	382129.	1.45853
$148$	0	0
$149$	93033.5	0.343300	0.171650	−	0.985158i	$-0.445090\pi$
$149$	93033.5	0.343300	0.171650	+	0.985158i	$0.445090\pi$
$150$	0	0
$151$	346369.	1.23622	0.618111	−	0.786091i	$-0.287898\pi$
$151$	346369.	1.23622	0.618111	+	0.786091i	$0.287898\pi$
$152$	0	0
$153$	−190834.	−0.659064
$154$	0	0
$155$	−126319.	−0.422317
$156$	0	0
$157$	6828.38	0.0221090	0.0110545	−	0.999939i	$-0.496481\pi$
$157$	6828.38	0.0221090	0.0110545	+	0.999939i	$0.496481\pi$
$158$	0	0
$159$	436983.	1.37079
$160$	0	0
$161$	−34013.9	−0.103417
$162$	0	0
$163$	−603266.	−1.77844	−0.889222	−	0.457476i	$-0.848754\pi$
$163$	−603266.	−1.77844	−0.889222	+	0.457476i	$0.848754\pi$
$164$	0	0
$165$	424946.	1.21513
$166$	0	0
$167$	−517596.	−1.43615	−0.718075	−	0.695966i	$-0.754976\pi$
$167$	−517596.	−1.43615	−0.718075	+	0.695966i	$0.754976\pi$
$168$	0	0
$169$	−354469.	−0.954688
$170$	0	0
$171$	127280.	0.332867
$172$	0	0
$173$	−666652.	−1.69349	−0.846747	−	0.531995i	$-0.821442\pi$
$173$	−666652.	−1.69349	−0.846747	+	0.531995i	$0.821442\pi$
$174$	0	0
$175$	−74498.7	−0.183888
$176$	0	0
$177$	−562487.	−1.34952
$178$	0	0
$179$	−329846.	−0.769446	−0.384723	−	0.923032i	$-0.625703\pi$
$179$	−329846.	−0.769446	−0.384723	+	0.923032i	$0.625703\pi$
$180$	0	0
$181$	355404.	0.806355	0.403177	−	0.915122i	$-0.367906\pi$
$181$	355404.	0.806355	0.403177	+	0.915122i	$0.367906\pi$
$182$	0	0
$183$	−1.02691e6	−2.26676
$184$	0	0
$185$	211945.	0.455295
$186$	0	0
$187$	−309542.	−0.647316
$188$	0	0
$189$	−90639.1	−0.184570
$190$	0	0
$191$	−870721.	−1.72701	−0.863506	−	0.504338i	$-0.831737\pi$
$191$	−870721.	−1.72701	−0.863506	+	0.504338i	$0.831737\pi$
$192$	0	0
$193$	−378718.	−0.731850	−0.365925	−	0.930644i	$-0.619247\pi$
$193$	−378718.	−0.731850	−0.365925	+	0.930644i	$0.619247\pi$
$194$	0	0
$195$	96378.3	0.181507
$196$	0	0
$197$	411889.	0.756162	0.378081	−	0.925773i	$-0.376584\pi$
$197$	411889.	0.756162	0.378081	+	0.925773i	$0.376584\pi$
$198$	0	0
$199$	−933002.	−1.67013	−0.835064	−	0.550153i	$-0.814569\pi$
$199$	−933002.	−1.67013	−0.835064	+	0.550153i	$0.814569\pi$
$200$	0	0
$201$	1.12472e6	1.96361
$202$	0	0
$203$	23844.6	0.0406115
$204$	0	0
$205$	453912.	0.754374
$206$	0	0
$207$	−353821.	−0.573929
$208$	0	0
$209$	206455.	0.326934
$210$	0	0
$211$	−1.02774e6	−1.58920	−0.794598	−	0.607136i	$-0.792318\pi$
$211$	−1.02774e6	−1.58920	−0.794598	+	0.607136i	$0.792318\pi$
$212$	0	0
$213$	711866.	1.07510
$214$	0	0
$215$	54632.0	0.0806030
$216$	0	0
$217$	140620.	0.202721
$218$	0	0
$219$	−253028.	−0.356499
$220$	0	0
$221$	−70204.5	−0.0966907
$222$	0	0
$223$	328104.	0.441824	0.220912	−	0.975294i	$-0.429097\pi$
$223$	328104.	0.441824	0.220912	+	0.975294i	$0.429097\pi$
$224$	0	0
$225$	−774954.	−1.02052
$226$	0	0
$227$	506685.	0.652640	0.326320	−	0.945259i	$-0.394191\pi$
$227$	506685.	0.652640	0.326320	+	0.945259i	$0.394191\pi$
$228$	0	0
$229$	−697491.	−0.878921	−0.439460	−	0.898262i	$-0.644830\pi$
$229$	−697491.	−0.878921	−0.439460	+	0.898262i	$0.644830\pi$
$230$	0	0
$231$	−473058.	−0.583290
$232$	0	0
$233$	−404940.	−0.488654	−0.244327	−	0.969693i	$-0.578567\pi$
$233$	−404940.	−0.488654	−0.244327	+	0.969693i	$0.578567\pi$
$234$	0	0
$235$	85537.5	0.101038
$236$	0	0
$237$	963127.	1.11381
$238$	0	0
$239$	1.00621e6	1.13945	0.569726	−	0.821835i	$-0.307049\pi$
$239$	1.00621e6	1.13945	0.569726	+	0.821835i	$0.307049\pi$
$240$	0	0
$241$	487434.	0.540597	0.270298	−	0.962777i	$-0.412878\pi$
$241$	487434.	0.540597	0.270298	+	0.962777i	$0.412878\pi$
$242$	0	0
$243$	1.14803e6	1.24720
$244$	0	0
$245$	476747.	0.507426
$246$	0	0
$247$	46824.2	0.0488347
$248$	0	0
$249$	−1.80838e6	−1.84838
$250$	0	0
$251$	442749.	0.443582	0.221791	−	0.975094i	$-0.428810\pi$
$251$	442749.	0.443582	0.221791	+	0.975094i	$0.428810\pi$
$252$	0	0
$253$	−573916.	−0.563698
$254$	0	0
$255$	−402177.	−0.387318
$256$	0	0
$257$	1.95848e6	1.84963	0.924817	−	0.380413i	$-0.124218\pi$
$257$	1.95848e6	1.84963	0.924817	+	0.380413i	$0.124218\pi$
$258$	0	0
$259$	−235940.	−0.218551
$260$	0	0
$261$	248037.	0.225380
$262$	0	0
$263$	1.42289e6	1.26848	0.634240	−	0.773137i	$-0.281313\pi$
$263$	1.42289e6	1.26848	0.634240	+	0.773137i	$0.281313\pi$
$264$	0	0
$265$	545183.	0.476900
$266$	0	0
$267$	−1.19802e6	−1.02846
$268$	0	0
$269$	6218.27	0.00523949	0.00261975	−	0.999997i	$-0.499166\pi$
$269$	6218.27	0.00523949	0.00261975	+	0.999997i	$0.499166\pi$
$270$	0	0
$271$	1.38862e6	1.14858	0.574290	−	0.818652i	$-0.305278\pi$
$271$	1.38862e6	1.14858	0.574290	+	0.818652i	$0.305278\pi$
$272$	0	0
$273$	−107290.	−0.0871270
$274$	0	0
$275$	−1.25701e6	−1.00232
$276$	0	0
$277$	1.61815e6	1.26713	0.633564	−	0.773690i	$-0.281591\pi$
$277$	1.61815e6	1.26713	0.633564	+	0.773690i	$0.281591\pi$
$278$	0	0
$279$	1.46277e6	1.12503
$280$	0	0
$281$	909534.	0.687152	0.343576	−	0.939125i	$-0.388362\pi$
$281$	909534.	0.687152	0.343576	+	0.939125i	$0.388362\pi$
$282$	0	0
$283$	−2.02610e6	−1.50382	−0.751909	−	0.659266i	$-0.770867\pi$
$283$	−2.02610e6	−1.50382	−0.751909	+	0.659266i	$0.770867\pi$
$284$	0	0
$285$	268240.	0.195619
$286$	0	0
$287$	−505303.	−0.362115
$288$	0	0
$289$	−1.12690e6	−0.793672
$290$	0	0
$291$	1.39960e6	0.968886
$292$	0	0
$293$	−941373.	−0.640609	−0.320304	−	0.947315i	$-0.603785\pi$
$293$	−941373.	−0.640609	−0.320304	+	0.947315i	$0.603785\pi$
$294$	0	0
$295$	−701763.	−0.469499
$296$	0	0
$297$	−1.52935e6	−1.00604
$298$	0	0
$299$	−130165.	−0.0842006
$300$	0	0
$301$	−60817.3	−0.0386911
$302$	0	0
$303$	466086.	0.291648
$304$	0	0
$305$	−1.28118e6	−0.788607
$306$	0	0
$307$	−2.52468e6	−1.52884	−0.764418	−	0.644721i	$-0.776974\pi$
$307$	−2.52468e6	−1.52884	−0.764418	+	0.644721i	$0.776974\pi$
$308$	0	0
$309$	3.00275e6	1.78905
$310$	0	0
$311$	1.94680e6	1.14135	0.570676	−	0.821175i	$-0.306681\pi$
$311$	1.94680e6	1.14135	0.570676	+	0.821175i	$0.306681\pi$
$312$	0	0
$313$	−834960.	−0.481731	−0.240866	−	0.970558i	$-0.577431\pi$
$313$	−834960.	−0.481731	−0.240866	+	0.970558i	$0.577431\pi$
$314$	0	0
$315$	−363855.	−0.206610
$316$	0	0
$317$	1.73920e6	0.972081	0.486041	−	0.873936i	$-0.338441\pi$
$317$	1.73920e6	0.972081	0.486041	+	0.873936i	$0.338441\pi$
$318$	0	0
$319$	402328.	0.221362
$320$	0	0
$321$	−2.02151e6	−1.09500
$322$	0	0
$323$	−195393.	−0.104208
$324$	0	0
$325$	−285092.	−0.149719
$326$	0	0
$327$	5.58447e6	2.88810
$328$	0	0
$329$	−95221.9	−0.0485006
$330$	0	0
$331$	−2.10505e6	−1.05607	−0.528034	−	0.849224i	$-0.677070\pi$
$331$	−2.10505e6	−1.05607	−0.528034	+	0.849224i	$0.677070\pi$
$332$	0	0
$333$	−2.45431e6	−1.21288
$334$	0	0
$335$	1.40321e6	0.683141
$336$	0	0
$337$	1.09640e6	0.525888	0.262944	−	0.964811i	$-0.415307\pi$
$337$	1.09640e6	0.525888	0.262944	+	0.964811i	$0.415307\pi$
$338$	0	0
$339$	3.91789e6	1.85162
$340$	0	0
$341$	2.37268e6	1.10498
$342$	0	0
$343$	−1.10039e6	−0.505021
$344$	0	0
$345$	−745668.	−0.337286
$346$	0	0
$347$	2.41074e6	1.07480	0.537398	−	0.843328i	$-0.319407\pi$
$347$	2.41074e6	1.07480	0.537398	+	0.843328i	$0.319407\pi$
$348$	0	0
$349$	4.17252e6	1.83373	0.916864	−	0.399199i	$-0.130712\pi$
$349$	4.17252e6	1.83373	0.916864	+	0.399199i	$0.130712\pi$
$350$	0	0
$351$	−346858.	−0.150274
$352$	0	0
$353$	−956427.	−0.408522	−0.204261	−	0.978916i	$-0.565479\pi$
$353$	−956427.	−0.408522	−0.204261	+	0.978916i	$0.565479\pi$
$354$	0	0
$355$	888129.	0.374029
$356$	0	0
$357$	447711.	0.185921
$358$	0	0
$359$	781647.	0.320092	0.160046	−	0.987110i	$-0.448836\pi$
$359$	781647.	0.320092	0.160046	+	0.987110i	$0.448836\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−4.05153e6	−1.61381
$364$	0	0
$365$	−315680.	−0.124027
$366$	0	0
$367$	2.26060e6	0.876110	0.438055	−	0.898948i	$-0.355668\pi$
$367$	2.26060e6	0.876110	0.438055	+	0.898948i	$0.355668\pi$
$368$	0	0
$369$	−5.25628e6	−2.00961
$370$	0	0
$371$	−606907.	−0.228922
$372$	0	0
$373$	−1.64441e6	−0.611980	−0.305990	−	0.952035i	$-0.598987\pi$
$373$	−1.64441e6	−0.611980	−0.305990	+	0.952035i	$0.598987\pi$
$374$	0	0
$375$	−3.95521e6	−1.45242
$376$	0	0
$377$	91248.5	0.0330653
$378$	0	0
$379$	−1.02041e6	−0.364904	−0.182452	−	0.983215i	$-0.558403\pi$
$379$	−1.02041e6	−0.364904	−0.182452	+	0.983215i	$0.558403\pi$
$380$	0	0
$381$	1.29544e6	0.457199
$382$	0	0
$383$	−1.50919e6	−0.525709	−0.262855	−	0.964835i	$-0.584664\pi$
$383$	−1.50919e6	−0.525709	−0.262855	+	0.964835i	$0.584664\pi$
$384$	0	0
$385$	−590190.	−0.202927
$386$	0	0
$387$	−632637.	−0.214722
$388$	0	0
$389$	5.57933e6	1.86942	0.934712	−	0.355405i	$-0.115657\pi$
$389$	5.57933e6	1.86942	0.934712	+	0.355405i	$0.115657\pi$
$390$	0	0
$391$	543165.	0.179676
$392$	0	0
$393$	1.69232e6	0.552716
$394$	0	0
$395$	1.20160e6	0.387497
$396$	0	0
$397$	−2.89931e6	−0.923249	−0.461625	−	0.887075i	$-0.652733\pi$
$397$	−2.89931e6	−0.923249	−0.461625	+	0.887075i	$0.652733\pi$
$398$	0	0
$399$	−298609.	−0.0939012
$400$	0	0
$401$	−2.19254e6	−0.680904	−0.340452	−	0.940262i	$-0.610580\pi$
$401$	−2.19254e6	−0.680904	−0.340452	+	0.940262i	$0.610580\pi$
$402$	0	0
$403$	538127.	0.165052
$404$	0	0
$405$	621568.	0.188300
$406$	0	0
$407$	−3.98101e6	−1.19126
$408$	0	0
$409$	−1.43210e6	−0.423317	−0.211658	−	0.977344i	$-0.567886\pi$
$409$	−1.43210e6	−0.423317	−0.211658	+	0.977344i	$0.567886\pi$
$410$	0	0
$411$	1.08920e6	0.318055
$412$	0	0
$413$	781215.	0.225370
$414$	0	0
$415$	−2.25614e6	−0.643053
$416$	0	0
$417$	−3.70522e6	−1.04346
$418$	0	0
$419$	4.93286e6	1.37266	0.686331	−	0.727290i	$-0.259220\pi$
$419$	4.93286e6	1.37266	0.686331	+	0.727290i	$0.259220\pi$
$420$	0	0
$421$	1.71380e6	0.471255	0.235628	−	0.971843i	$-0.424285\pi$
$421$	1.71380e6	0.471255	0.235628	+	0.971843i	$0.424285\pi$
$422$	0	0
$423$	−990521.	−0.269161
$424$	0	0
$425$	1.18966e6	0.319486
$426$	0	0
$427$	1.42623e6	0.378548
$428$	0	0
$429$	−1.81030e6	−0.474906
$430$	0	0
$431$	3.25703e6	0.844557	0.422278	−	0.906466i	$-0.361230\pi$
$431$	3.25703e6	0.844557	0.422278	+	0.906466i	$0.361230\pi$
$432$	0	0
$433$	−6.32543e6	−1.62133	−0.810663	−	0.585513i	$-0.800893\pi$
$433$	−6.32543e6	−1.62133	−0.810663	+	0.585513i	$0.800893\pi$
$434$	0	0
$435$	522731.	0.132451
$436$	0	0
$437$	−362274.	−0.0907472
$438$	0	0
$439$	−2.91464e6	−0.721812	−0.360906	−	0.932602i	$-0.617532\pi$
$439$	−2.91464e6	−0.721812	−0.360906	+	0.932602i	$0.617532\pi$
$440$	0	0
$441$	−5.52071e6	−1.35176
$442$	0	0
$443$	−1.83551e6	−0.444373	−0.222187	−	0.975004i	$-0.571319\pi$
$443$	−1.83551e6	−0.444373	−0.222187	+	0.975004i	$0.571319\pi$
$444$	0	0
$445$	−1.49466e6	−0.357802
$446$	0	0
$447$	−2.27043e6	−0.537452
$448$	0	0
$449$	−6.08239e6	−1.42383	−0.711916	−	0.702265i	$-0.752172\pi$
$449$	−6.08239e6	−1.42383	−0.711916	+	0.702265i	$0.752172\pi$
$450$	0	0
$451$	−8.52595e6	−1.97379
$452$	0	0
$453$	−8.45294e6	−1.93536
$454$	0	0
$455$	−133856.	−0.0303116
$456$	0	0
$457$	−3.80437e6	−0.852102	−0.426051	−	0.904699i	$-0.640096\pi$
$457$	−3.80437e6	−0.852102	−0.426051	+	0.904699i	$0.640096\pi$
$458$	0	0
$459$	1.44741e6	0.320670
$460$	0	0
$461$	3.33825e6	0.731589	0.365794	−	0.930696i	$-0.380797\pi$
$461$	3.33825e6	0.731589	0.365794	+	0.930696i	$0.380797\pi$
$462$	0	0
$463$	−3.14227e6	−0.681226	−0.340613	−	0.940204i	$-0.610635\pi$
$463$	−3.14227e6	−0.681226	−0.340613	+	0.940204i	$0.610635\pi$
$464$	0	0
$465$	3.08274e6	0.661157
$466$	0	0
$467$	6.63621e6	1.40808	0.704041	−	0.710159i	$-0.251377\pi$
$467$	6.63621e6	1.40808	0.704041	+	0.710159i	$0.251377\pi$
$468$	0	0
$469$	−1.56208e6	−0.327922
$470$	0	0
$471$	−166643.	−0.0346126
$472$	0	0
$473$	−1.02617e6	−0.210895
$474$	0	0
$475$	−793467.	−0.161360
$476$	0	0
$477$	−6.31320e6	−1.27044
$478$	0	0
$479$	1.08458e6	0.215984	0.107992	−	0.994152i	$-0.465558\pi$
$479$	1.08458e6	0.215984	0.107992	+	0.994152i	$0.465558\pi$
$480$	0	0
$481$	−902898.	−0.177941
$482$	0	0
$483$	830091.	0.161904
$484$	0	0
$485$	1.74616e6	0.337077
$486$	0	0
$487$	7.50934e6	1.43476	0.717380	−	0.696682i	$-0.245341\pi$
$487$	7.50934e6	1.43476	0.717380	+	0.696682i	$0.245341\pi$
$488$	0	0
$489$	1.47224e7	2.78424
$490$	0	0
$491$	4.81580e6	0.901498	0.450749	−	0.892651i	$-0.351157\pi$
$491$	4.81580e6	0.901498	0.450749	+	0.892651i	$0.351157\pi$
$492$	0	0
$493$	−380771.	−0.0705581
$494$	0	0
$495$	−6.13931e6	−1.12618
$496$	0	0
$497$	−988681.	−0.179542
$498$	0	0
$499$	2.98843e6	0.537269	0.268634	−	0.963242i	$-0.413428\pi$
$499$	2.98843e6	0.537269	0.268634	+	0.963242i	$0.413428\pi$
$500$	0	0
$501$	1.26316e7	2.24836
$502$	0	0
$503$	−547996.	−0.0965733	−0.0482867	−	0.998834i	$-0.515376\pi$
$503$	−547996.	−0.0965733	−0.0482867	+	0.998834i	$0.515376\pi$
$504$	0	0
$505$	581492.	0.101465
$506$	0	0
$507$	8.65062e6	1.49461
$508$	0	0
$509$	−1.14090e7	−1.95187	−0.975937	−	0.218052i	$-0.930030\pi$
$509$	−1.14090e7	−1.95187	−0.975937	+	0.218052i	$0.930030\pi$
$510$	0	0
$511$	351421.	0.0595354
$512$	0	0
$513$	−965375.	−0.161958
$514$	0	0
$515$	3.74625e6	0.622413
$516$	0	0
$517$	−1.60667e6	−0.264364
$518$	0	0
$519$	1.62693e7	2.65124
$520$	0	0
$521$	1.10664e7	1.78612	0.893059	−	0.449940i	$-0.148554\pi$
$521$	1.10664e7	1.78612	0.893059	+	0.449940i	$0.148554\pi$
$522$	0	0
$523$	−5.16013e6	−0.824909	−0.412455	−	0.910978i	$-0.635328\pi$
$523$	−5.16013e6	−0.824909	−0.412455	+	0.910978i	$0.635328\pi$
$524$	0	0
$525$	1.81810e6	0.287885
$526$	0	0
$527$	−2.24555e6	−0.352206
$528$	0	0
$529$	−5.42927e6	−0.843534
$530$	0	0
$531$	8.12638e6	1.25072
$532$	0	0
$533$	−1.93370e6	−0.294829
$534$	0	0
$535$	−2.52205e6	−0.380951
$536$	0	0
$537$	8.04970e6	1.20460
$538$	0	0
$539$	−8.95487e6	−1.32766
$540$	0	0
$541$	3.02590e6	0.444489	0.222244	−	0.974991i	$-0.428662\pi$
$541$	3.02590e6	0.444489	0.222244	+	0.974991i	$0.428662\pi$
$542$	0	0
$543$	−8.67344e6	−1.26239
$544$	0	0
$545$	6.96723e6	1.00478
$546$	0	0
$547$	5.28748e6	0.755580	0.377790	−	0.925891i	$-0.376684\pi$
$547$	5.28748e6	0.755580	0.377790	+	0.925891i	$0.376684\pi$
$548$	0	0
$549$	1.48360e7	2.10081
$550$	0	0
$551$	253962.	0.0356361
$552$	0	0
$553$	−1.33765e6	−0.186007
$554$	0	0
$555$	−5.17239e6	−0.712785
$556$	0	0
$557$	−1.42555e7	−1.94691	−0.973454	−	0.228883i	$-0.926493\pi$
$557$	−1.42555e7	−1.94691	−0.973454	+	0.228883i	$0.926493\pi$
$558$	0	0
$559$	−232736.	−0.0315017
$560$	0	0
$561$	7.55421e6	1.01340
$562$	0	0
$563$	−8.00522e6	−1.06439	−0.532197	−	0.846621i	$-0.678633\pi$
$563$	−8.00522e6	−1.06439	−0.532197	+	0.846621i	$0.678633\pi$
$564$	0	0
$565$	4.88799e6	0.644183
$566$	0	0
$567$	−691940.	−0.0903880
$568$	0	0
$569$	2.92084e6	0.378205	0.189103	−	0.981957i	$-0.439442\pi$
$569$	2.92084e6	0.378205	0.189103	+	0.981957i	$0.439442\pi$
$570$	0	0
$571$	2.94994e6	0.378637	0.189319	−	0.981916i	$-0.439372\pi$
$571$	2.94994e6	0.378637	0.189319	+	0.981916i	$0.439372\pi$
$572$	0	0
$573$	2.12495e7	2.70372
$574$	0	0
$575$	2.20573e6	0.278216
$576$	0	0
$577$	−1.42530e7	−1.78224	−0.891121	−	0.453765i	$-0.850081\pi$
$577$	−1.42530e7	−1.78224	−0.891121	+	0.453765i	$0.850081\pi$
$578$	0	0
$579$	9.24239e6	1.14574
$580$	0	0
$581$	2.51158e6	0.308679
$582$	0	0
$583$	−1.02403e7	−1.24779
$584$	0	0
$585$	−1.39240e6	−0.168219
$586$	0	0
$587$	1.19711e7	1.43397	0.716983	−	0.697090i	$-0.245522\pi$
$587$	1.19711e7	1.43397	0.716983	+	0.697090i	$0.245522\pi$
$588$	0	0
$589$	1.49771e6	0.177885
$590$	0	0
$591$	−1.00519e7	−1.18381
$592$	0	0
$593$	−1.09738e7	−1.28150	−0.640750	−	0.767749i	$-0.721377\pi$
$593$	−1.09738e7	−1.28150	−0.640750	+	0.767749i	$0.721377\pi$
$594$	0	0
$595$	558568.	0.0646820
$596$	0	0
$597$	2.27694e7	2.61466
$598$	0	0
$599$	1.95514e6	0.222644	0.111322	−	0.993784i	$-0.464492\pi$
$599$	1.95514e6	0.222644	0.111322	+	0.993784i	$0.464492\pi$
$600$	0	0
$601$	−4.98555e6	−0.563024	−0.281512	−	0.959558i	$-0.590836\pi$
$601$	−4.98555e6	−0.563024	−0.281512	+	0.959558i	$0.590836\pi$
$602$	0	0
$603$	−1.62491e7	−1.81985
$604$	0	0
$605$	−5.05471e6	−0.561446
$606$	0	0
$607$	−1.50635e7	−1.65941	−0.829706	−	0.558200i	$-0.811492\pi$
$607$	−1.50635e7	−1.65941	−0.829706	+	0.558200i	$0.811492\pi$
$608$	0	0
$609$	−581913.	−0.0635792
$610$	0	0
$611$	−364396.	−0.0394884
$612$	0	0
$613$	3.60822e6	0.387830	0.193915	−	0.981018i	$-0.437881\pi$
$613$	3.60822e6	0.387830	0.193915	+	0.981018i	$0.437881\pi$
$614$	0	0
$615$	−1.10775e7	−1.18101
$616$	0	0
$617$	−4.12560e6	−0.436289	−0.218144	−	0.975916i	$-0.570000\pi$
$617$	−4.12560e6	−0.436289	−0.218144	+	0.975916i	$0.570000\pi$
$618$	0	0
$619$	−7.18926e6	−0.754150	−0.377075	−	0.926183i	$-0.623070\pi$
$619$	−7.18926e6	−0.754150	−0.377075	+	0.926183i	$0.623070\pi$
$620$	0	0
$621$	2.68360e6	0.279248
$622$	0	0
$623$	1.66388e6	0.171752
$624$	0	0
$625$	1.93410e6	0.198052
$626$	0	0
$627$	−5.03842e6	−0.511830
$628$	0	0
$629$	3.76771e6	0.379709
$630$	0	0
$631$	−781858.	−0.0781726	−0.0390863	−	0.999236i	$-0.512445\pi$
$631$	−781858.	−0.0781726	−0.0390863	+	0.999236i	$0.512445\pi$
$632$	0	0
$633$	2.50814e7	2.48796
$634$	0	0
$635$	1.61620e6	0.159060
$636$	0	0
$637$	−2.03097e6	−0.198315
$638$	0	0
$639$	−1.02845e7	−0.996394
$640$	0	0
$641$	−7.16700e6	−0.688958	−0.344479	−	0.938794i	$-0.611944\pi$
$641$	−7.16700e6	−0.688958	−0.344479	+	0.938794i	$0.611944\pi$
$642$	0	0
$643$	7.30767e6	0.697030	0.348515	−	0.937303i	$-0.386686\pi$
$643$	7.30767e6	0.697030	0.348515	+	0.937303i	$0.386686\pi$
$644$	0	0
$645$	−1.33326e6	−0.126188
$646$	0	0
$647$	−9.04810e6	−0.849761	−0.424880	−	0.905250i	$-0.639684\pi$
$647$	−9.04810e6	−0.849761	−0.424880	+	0.905250i	$0.639684\pi$
$648$	0	0
$649$	1.31814e7	1.22843
$650$	0	0
$651$	−3.43176e6	−0.317369
$652$	0	0
$653$	−1.14059e7	−1.04676	−0.523381	−	0.852099i	$-0.675329\pi$
$653$	−1.14059e7	−1.04676	−0.523381	+	0.852099i	$0.675329\pi$
$654$	0	0
$655$	2.11135e6	0.192290
$656$	0	0
$657$	3.65556e6	0.330401
$658$	0	0
$659$	−1.24205e7	−1.11410	−0.557050	−	0.830479i	$-0.688067\pi$
$659$	−1.24205e7	−1.11410	−0.557050	+	0.830479i	$0.688067\pi$
$660$	0	0
$661$	5.96950e6	0.531416	0.265708	−	0.964054i	$-0.414394\pi$
$661$	5.96950e6	0.531416	0.265708	+	0.964054i	$0.414394\pi$
$662$	0	0
$663$	1.71330e6	0.151374
$664$	0	0
$665$	−372547.	−0.0326683
$666$	0	0
$667$	−705979.	−0.0614437
$668$	0	0
$669$	−8.00719e6	−0.691695
$670$	0	0
$671$	2.40648e7	2.06336
$672$	0	0
$673$	9.35412e6	0.796096	0.398048	−	0.917365i	$-0.369688\pi$
$673$	9.35412e6	0.796096	0.398048	+	0.917365i	$0.369688\pi$
$674$	0	0
$675$	5.87774e6	0.496536
$676$	0	0
$677$	−1.16862e7	−0.979946	−0.489973	−	0.871738i	$-0.662993\pi$
$677$	−1.16862e7	−0.979946	−0.489973	+	0.871738i	$0.662993\pi$
$678$	0	0
$679$	−1.94385e6	−0.161804
$680$	0	0
$681$	−1.23654e7	−1.02174
$682$	0	0
$683$	2.10063e7	1.72305	0.861525	−	0.507716i	$-0.169510\pi$
$683$	2.10063e7	1.72305	0.861525	+	0.507716i	$0.169510\pi$
$684$	0	0
$685$	1.35889e6	0.110652
$686$	0	0
$687$	1.70219e7	1.37599
$688$	0	0
$689$	−2.32252e6	−0.186385
$690$	0	0
$691$	−2.13781e7	−1.70323	−0.851616	−	0.524167i	$-0.824377\pi$
$691$	−2.13781e7	−1.70323	−0.851616	+	0.524167i	$0.824377\pi$
$692$	0	0
$693$	6.83438e6	0.540588
$694$	0	0
$695$	−4.62266e6	−0.363020
$696$	0	0
$697$	8.06913e6	0.629136
$698$	0	0
$699$	9.88235e6	0.765010
$700$	0	0
$701$	1.39960e7	1.07574	0.537872	−	0.843026i	$-0.319228\pi$
$701$	1.39960e7	1.07574	0.537872	+	0.843026i	$0.319228\pi$
$702$	0	0
$703$	−2.51294e6	−0.191776
$704$	0	0
$705$	−2.08750e6	−0.158180
$706$	0	0
$707$	−647327.	−0.0487052
$708$	0	0
$709$	2.54356e6	0.190032	0.0950158	−	0.995476i	$-0.469710\pi$
$709$	2.54356e6	0.190032	0.0950158	+	0.995476i	$0.469710\pi$
$710$	0	0
$711$	−1.39145e7	−1.03227
$712$	0	0
$713$	−4.16343e6	−0.306709
$714$	0	0
$715$	−2.25854e6	−0.165220
$716$	0	0
$717$	−2.45561e7	−1.78386
$718$	0	0
$719$	2.26551e7	1.63434	0.817172	−	0.576394i	$-0.195541\pi$
$719$	2.26551e7	1.63434	0.817172	+	0.576394i	$0.195541\pi$
$720$	0	0
$721$	−4.17039e6	−0.298771
$722$	0	0
$723$	−1.18956e7	−0.846329
$724$	0	0
$725$	−1.54627e6	−0.109254
$726$	0	0
$727$	1.33692e7	0.938144	0.469072	−	0.883160i	$-0.344588\pi$
$727$	1.33692e7	0.938144	0.469072	+	0.883160i	$0.344588\pi$
$728$	0	0
$729$	−2.30563e7	−1.60683
$730$	0	0
$731$	971185.	0.0672216
$732$	0	0
$733$	−1.62537e7	−1.11736	−0.558679	−	0.829384i	$-0.688692\pi$
$733$	−1.62537e7	−1.11736	−0.558679	+	0.829384i	$0.688692\pi$
$734$	0	0
$735$	−1.16347e7	−0.794399
$736$	0	0
$737$	−2.63569e7	−1.78741
$738$	0	0
$739$	−1.55893e7	−1.05006	−0.525030	−	0.851084i	$-0.675946\pi$
$739$	−1.55893e7	−1.05006	−0.525030	+	0.851084i	$0.675946\pi$
$740$	0	0
$741$	−1.14272e6	−0.0764529
$742$	0	0
$743$	2.88846e7	1.91953	0.959765	−	0.280806i	$-0.0906017\pi$
$743$	2.88846e7	1.91953	0.959765	+	0.280806i	$0.0906017\pi$
$744$	0	0
$745$	−2.83260e6	−0.186980
$746$	0	0
$747$	2.61261e7	1.71306
$748$	0	0
$749$	2.80759e6	0.182865
$750$	0	0
$751$	−4.86310e6	−0.314639	−0.157320	−	0.987548i	$-0.550285\pi$
$751$	−4.86310e6	−0.314639	−0.157320	+	0.987548i	$0.550285\pi$
$752$	0	0
$753$	−1.08051e7	−0.694448
$754$	0	0
$755$	−1.05460e7	−0.673315
$756$	0	0
$757$	−1.49268e7	−0.946730	−0.473365	−	0.880866i	$-0.656961\pi$
$757$	−1.49268e7	−0.946730	−0.473365	+	0.880866i	$0.656961\pi$
$758$	0	0
$759$	1.40061e7	0.882496
$760$	0	0
$761$	−9.73797e6	−0.609547	−0.304773	−	0.952425i	$-0.598581\pi$
$761$	−9.73797e6	−0.609547	−0.304773	+	0.952425i	$0.598581\pi$
$762$	0	0
$763$	−7.75604e6	−0.482313
$764$	0	0
$765$	5.81036e6	0.358963
$766$	0	0
$767$	2.98956e6	0.183492
$768$	0	0
$769$	2.81150e7	1.71444	0.857221	−	0.514948i	$-0.172189\pi$
$769$	2.81150e7	1.71444	0.857221	+	0.514948i	$0.172189\pi$
$770$	0	0
$771$	−4.77956e7	−2.89569
$772$	0	0
$773$	−1.68487e7	−1.01419	−0.507094	−	0.861891i	$-0.669280\pi$
$773$	−1.68487e7	−1.01419	−0.507094	+	0.861891i	$0.669280\pi$
$774$	0	0
$775$	−9.11891e6	−0.545367
$776$	0	0
$777$	5.75800e6	0.342152
$778$	0	0
$779$	−5.38185e6	−0.317752
$780$	0	0
$781$	−1.66820e7	−0.978633
$782$	0	0
$783$	−1.88127e6	−0.109660
$784$	0	0
$785$	−207905.	−0.0120418
$786$	0	0
$787$	2.85828e7	1.64501	0.822505	−	0.568758i	$-0.192576\pi$
$787$	2.85828e7	1.64501	0.822505	+	0.568758i	$0.192576\pi$
$788$	0	0
$789$	−3.47250e7	−1.98586
$790$	0	0
$791$	−5.44140e6	−0.309221
$792$	0	0
$793$	5.45792e6	0.308208
$794$	0	0
$795$	−1.33049e7	−0.746609
$796$	0	0
$797$	−6.38542e6	−0.356077	−0.178039	−	0.984024i	$-0.556975\pi$
$797$	−6.38542e6	−0.356077	−0.178039	+	0.984024i	$0.556975\pi$
$798$	0	0
$799$	1.52059e6	0.0842645
$800$	0	0
$801$	1.73081e7	0.953165
$802$	0	0
$803$	5.92951e6	0.324511
$804$	0	0
$805$	1.03563e6	0.0563266
$806$	0	0
$807$	−151754.	−0.00820267
$808$	0	0
$809$	2.09935e7	1.12775	0.563875	−	0.825860i	$-0.309310\pi$
$809$	2.09935e7	1.12775	0.563875	+	0.825860i	$0.309310\pi$
$810$	0	0
$811$	1.53184e6	0.0817824	0.0408912	−	0.999164i	$-0.486980\pi$
$811$	1.53184e6	0.0817824	0.0408912	+	0.999164i	$0.486980\pi$
$812$	0	0
$813$	−3.38886e7	−1.79815
$814$	0	0
$815$	1.83678e7	0.968639
$816$	0	0
$817$	−647750.	−0.0339510
$818$	0	0
$819$	1.55005e6	0.0807486
$820$	0	0
$821$	−1.40514e7	−0.727549	−0.363775	−	0.931487i	$-0.618512\pi$
$821$	−1.40514e7	−0.727549	−0.363775	+	0.931487i	$0.618512\pi$
$822$	0	0
$823$	3.17951e7	1.63629	0.818145	−	0.575012i	$-0.195002\pi$
$823$	3.17951e7	1.63629	0.818145	+	0.575012i	$0.195002\pi$
$824$	0	0
$825$	3.06767e7	1.56919
$826$	0	0
$827$	−1.36913e7	−0.696117	−0.348058	−	0.937473i	$-0.613159\pi$
$827$	−1.36913e7	−0.696117	−0.348058	+	0.937473i	$0.613159\pi$
$828$	0	0
$829$	1.01811e7	0.514526	0.257263	−	0.966341i	$-0.417179\pi$
$829$	1.01811e7	0.514526	0.257263	+	0.966341i	$0.417179\pi$
$830$	0	0
$831$	−3.94902e7	−1.98375
$832$	0	0
$833$	8.47506e6	0.423185
$834$	0	0
$835$	1.57593e7	0.782207
$836$	0	0
$837$	−1.10946e7	−0.547389
$838$	0	0
$839$	3.91248e7	1.91888	0.959438	−	0.281921i	$-0.0909717\pi$
$839$	3.91248e7	1.91888	0.959438	+	0.281921i	$0.0909717\pi$
$840$	0	0
$841$	−2.00162e7	−0.975871
$842$	0	0
$843$	−2.21967e7	−1.07577
$844$	0	0
$845$	1.07926e7	0.519976
$846$	0	0
$847$	5.62700e6	0.269506
$848$	0	0
$849$	4.94459e7	2.35430
$850$	0	0
$851$	6.98562e6	0.330660
$852$	0	0
$853$	1.33752e7	0.629399	0.314700	−	0.949191i	$-0.398096\pi$
$853$	1.33752e7	0.629399	0.314700	+	0.949191i	$0.398096\pi$
$854$	0	0
$855$	−3.87532e6	−0.181298
$856$	0	0
$857$	−2.76888e6	−0.128781	−0.0643905	−	0.997925i	$-0.520510\pi$
$857$	−2.76888e6	−0.128781	−0.0643905	+	0.997925i	$0.520510\pi$
$858$	0	0
$859$	−2.16792e7	−1.00244	−0.501222	−	0.865319i	$-0.667116\pi$
$859$	−2.16792e7	−1.00244	−0.501222	+	0.865319i	$0.667116\pi$
$860$	0	0
$861$	1.23316e7	0.566908
$862$	0	0
$863$	−1.80747e7	−0.826122	−0.413061	−	0.910703i	$-0.635540\pi$
$863$	−1.80747e7	−0.826122	−0.413061	+	0.910703i	$0.635540\pi$
$864$	0	0
$865$	2.02977e7	0.922371
$866$	0	0
$867$	2.75014e7	1.24253
$868$	0	0
$869$	−2.25701e7	−1.01387
$870$	0	0
$871$	−5.97777e6	−0.266989
$872$	0	0
$873$	−2.02204e7	−0.897955
$874$	0	0
$875$	5.49323e6	0.242554
$876$	0	0
$877$	3.29573e7	1.44695	0.723474	−	0.690352i	$-0.242544\pi$
$877$	3.29573e7	1.44695	0.723474	+	0.690352i	$0.242544\pi$
$878$	0	0
$879$	2.29737e7	1.00290
$880$	0	0
$881$	−1.79044e7	−0.777175	−0.388588	−	0.921412i	$-0.627037\pi$
$881$	−1.79044e7	−0.777175	−0.388588	+	0.921412i	$0.627037\pi$
$882$	0	0
$883$	2.41771e7	1.04352	0.521761	−	0.853092i	$-0.325275\pi$
$883$	2.41771e7	1.04352	0.521761	+	0.853092i	$0.325275\pi$
$884$	0	0
$885$	1.71261e7	0.735023
$886$	0	0
$887$	3.43324e7	1.46519	0.732597	−	0.680662i	$-0.238308\pi$
$887$	3.43324e7	1.46519	0.732597	+	0.680662i	$0.238308\pi$
$888$	0	0
$889$	−1.79918e6	−0.0763521
$890$	0	0
$891$	−1.16751e7	−0.492681
$892$	0	0
$893$	−1.01418e6	−0.0425587
$894$	0	0
$895$	1.00429e7	0.419083
$896$	0	0
$897$	3.17660e6	0.131820
$898$	0	0
$899$	2.91866e6	0.120444
$900$	0	0
$901$	9.69164e6	0.397727
$902$	0	0
$903$	1.48421e6	0.0605727
$904$	0	0
$905$	−1.08211e7	−0.439186
$906$	0	0
$907$	−1.21213e7	−0.489250	−0.244625	−	0.969618i	$-0.578665\pi$
$907$	−1.21213e7	−0.489250	−0.244625	+	0.969618i	$0.578665\pi$
$908$	0	0
$909$	−6.73366e6	−0.270297
$910$	0	0
$911$	−5.70879e6	−0.227902	−0.113951	−	0.993486i	$-0.536351\pi$
$911$	−5.70879e6	−0.227902	−0.113951	+	0.993486i	$0.536351\pi$
$912$	0	0
$913$	4.23778e7	1.68252
$914$	0	0
$915$	3.12665e7	1.23460
$916$	0	0
$917$	−2.35040e6	−0.0923034
$918$	0	0
$919$	5.01516e7	1.95883	0.979414	−	0.201862i	$-0.0646994\pi$
$919$	5.01516e7	1.95883	0.979414	+	0.201862i	$0.0646994\pi$
$920$	0	0
$921$	6.16135e7	2.39346
$922$	0	0
$923$	−3.78349e6	−0.146180
$924$	0	0
$925$	1.53002e7	0.587953
$926$	0	0
$927$	−4.33814e7	−1.65808
$928$	0	0
$929$	−1.93147e6	−0.0734257	−0.0367128	−	0.999326i	$-0.511689\pi$
$929$	−1.93147e6	−0.0734257	−0.0367128	+	0.999326i	$0.511689\pi$
$930$	0	0
$931$	−5.65260e6	−0.213734
$932$	0	0
$933$	−4.75105e7	−1.78684
$934$	0	0
$935$	9.42469e6	0.352564
$936$	0	0
$937$	−85219.9	−0.00317097	−0.00158548	−	0.999999i	$-0.500505\pi$
$937$	−85219.9	−0.00317097	−0.00158548	+	0.999999i	$0.500505\pi$
$938$	0	0
$939$	2.03767e7	0.754172
$940$	0	0
$941$	3.94849e6	0.145364	0.0726821	−	0.997355i	$-0.476844\pi$
$941$	3.94849e6	0.145364	0.0726821	+	0.997355i	$0.476844\pi$
$942$	0	0
$943$	1.49608e7	0.547867
$944$	0	0
$945$	2.75970e6	0.100527
$946$	0	0
$947$	−2.41655e7	−0.875630	−0.437815	−	0.899065i	$-0.644248\pi$
$947$	−2.41655e7	−0.875630	−0.437815	+	0.899065i	$0.644248\pi$
$948$	0	0
$949$	1.34482e6	0.0484728
$950$	0	0
$951$	−4.24443e7	−1.52184
$952$	0	0
$953$	−7.02709e6	−0.250636	−0.125318	−	0.992117i	$-0.539995\pi$
$953$	−7.02709e6	−0.250636	−0.125318	+	0.992117i	$0.539995\pi$
$954$	0	0
$955$	2.65110e7	0.940627
$956$	0	0
$957$	−9.81860e6	−0.346553
$958$	0	0
$959$	−1.51274e6	−0.0531152
$960$	0	0
$961$	−1.14167e7	−0.398779
$962$	0	0
$963$	2.92053e7	1.01484
$964$	0	0
$965$	1.15309e7	0.398606
$966$	0	0
$967$	2.42724e7	0.834731	0.417365	−	0.908739i	$-0.362954\pi$
$967$	2.42724e7	0.834731	0.417365	+	0.908739i	$0.362954\pi$
$968$	0	0
$969$	4.76846e6	0.163143
$970$	0	0
$971$	−3.06986e7	−1.04489	−0.522444	−	0.852673i	$-0.674980\pi$
$971$	−3.06986e7	−1.04489	−0.522444	+	0.852673i	$0.674980\pi$
$972$	0	0
$973$	5.14603e6	0.174257
$974$	0	0
$975$	6.95752e6	0.234392
$976$	0	0
$977$	−2.73274e7	−0.915931	−0.457965	−	0.888970i	$-0.651422\pi$
$977$	−2.73274e7	−0.915931	−0.457965	+	0.888970i	$0.651422\pi$
$978$	0	0
$979$	2.80746e7	0.936175
$980$	0	0
$981$	−8.06803e7	−2.67667
$982$	0	0
$983$	8.28346e6	0.273419	0.136709	−	0.990611i	$-0.456347\pi$
$983$	8.28346e6	0.273419	0.136709	+	0.990611i	$0.456347\pi$
$984$	0	0
$985$	−1.25409e7	−0.411847
$986$	0	0
$987$	2.32384e6	0.0759299
$988$	0	0
$989$	1.80065e6	0.0585382
$990$	0	0
$991$	−3.63848e7	−1.17689	−0.588445	−	0.808537i	$-0.700260\pi$
$991$	−3.63848e7	−1.17689	−0.588445	+	0.808537i	$0.700260\pi$
$992$	0	0
$993$	5.13725e7	1.65332
$994$	0	0
$995$	2.84073e7	0.909644
$996$	0	0
$997$	−1.41746e7	−0.451619	−0.225809	−	0.974172i	$-0.572503\pi$
$997$	−1.41746e7	−0.451619	−0.225809	+	0.974172i	$0.572503\pi$
$998$	0	0
$999$	1.86150e7	0.590134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.6.a.b.1.1	✓	3
4.3	odd	2		304.6.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.a.b.1.1	✓	3	1.1	even	1	trivial
304.6.a.i.1.3		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q-24.4044 q^{3} -30.4472 q^{5} +33.8943 q^{7} +352.577 q^{9} +571.898 q^{11} +129.707 q^{13} +743.046 q^{15} -541.255 q^{17} +361.000 q^{19} -827.172 q^{21} -1003.53 q^{23} -2197.97 q^{25} -2674.17 q^{27} +703.497 q^{29} +4148.79 q^{31} -13956.8 q^{33} -1031.99 q^{35} -6961.06 q^{37} -3165.43 q^{39} -14908.2 q^{41} -1794.32 q^{43} -10735.0 q^{45} -2809.37 q^{47} -15658.2 q^{49} +13209.0 q^{51} -17905.9 q^{53} -17412.7 q^{55} -8810.01 q^{57} +23048.5 q^{59} +42078.8 q^{61} +11950.4 q^{63} -3949.21 q^{65} -46086.7 q^{67} +24490.6 q^{69} -29169.5 q^{71} +10368.1 q^{73} +53640.2 q^{75} +19384.1 q^{77} -39465.2 q^{79} -20414.6 q^{81} +74100.3 q^{83} +16479.7 q^{85} -17168.5 q^{87} +49090.3 q^{89} +4396.33 q^{91} -101249. q^{93} -10991.4 q^{95} -57350.4 q^{97} +201638. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{3} + 58 q^{5} - 197 q^{7} + 80 q^{9} + 476 q^{11} - 1417 q^{13} + 988 q^{15} - 2427 q^{17} + 1083 q^{19} - 1787 q^{21} - 2407 q^{23} - 93 q^{25} - 8197 q^{27} + 4227 q^{29} + 10692 q^{31} - 19274 q^{33}+ \cdots + 136690 q^{99}+O(q^{100}) \) 3 * q - 7 * q^3 + 58 * q^5 - 197 * q^7 + 80 * q^9 + 476 * q^11 - 1417 * q^13 + 988 * q^15 - 2427 * q^17 + 1083 * q^19 - 1787 * q^21 - 2407 * q^23 - 93 * q^25 - 8197 * q^27 + 4227 * q^29 + 10692 * q^31 - 19274 * q^33 - 20130 * q^35 - 13130 * q^37 - 14993 * q^39 - 22440 * q^41 - 8230 * q^43 - 31922 * q^45 - 26630 * q^47 - 4842 * q^49 + 11995 * q^51 - 75557 * q^53 + 16306 * q^55 - 2527 * q^57 + 40877 * q^59 + 24024 * q^61 + 61684 * q^63 - 86172 * q^65 - 93135 * q^67 - 31679 * q^69 + 5738 * q^71 - 19661 * q^73 + 25547 * q^75 - 45464 * q^77 + 82854 * q^79 - 16357 * q^81 + 66886 * q^83 - 195598 * q^85 + 80595 * q^87 + 92268 * q^89 + 210603 * q^91 - 16048 * q^93 + 20938 * q^95 + 47310 * q^97 + 136690 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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