LMFDB - Embedded newform 176.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,6,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, 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("176.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$176 = 2^{4} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	176.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:	$28.2275522871$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{793})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 198$ x^2 - x - 198
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$14.5801$ of defining polynomial
Character	$\chi$	$=$	176.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-28.5801 q^{3} -76.9006 q^{5} +91.4808 q^{7} +573.824 q^{9} +121.000 q^{11} -463.801 q^{13} +2197.83 q^{15} +1641.94 q^{17} +1693.53 q^{19} -2614.53 q^{21} -75.9391 q^{23} +2788.71 q^{25} -9454.98 q^{27} -2942.33 q^{29} -8443.63 q^{31} -3458.20 q^{33} -7034.93 q^{35} -35.5610 q^{37} +13255.5 q^{39} +9222.25 q^{41} +11516.7 q^{43} -44127.4 q^{45} -6179.00 q^{47} -8438.27 q^{49} -46926.7 q^{51} +25255.1 q^{53} -9304.98 q^{55} -48401.2 q^{57} -40786.8 q^{59} -7368.85 q^{61} +52493.8 q^{63} +35666.6 q^{65} +11024.7 q^{67} +2170.35 q^{69} +46964.9 q^{71} -60727.5 q^{73} -79701.6 q^{75} +11069.2 q^{77} -18386.1 q^{79} +130785. q^{81} +59321.9 q^{83} -126266. q^{85} +84092.1 q^{87} +5070.71 q^{89} -42428.9 q^{91} +241320. q^{93} -130233. q^{95} -130795. q^{97} +69432.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 29 q^{3} - 13 q^{5} + 14 q^{7} + 331 q^{9} + 242 q^{11} - 646 q^{13} + 2171 q^{15} - 208 q^{17} + 2148 q^{19} - 2582 q^{21} - 349 q^{23} + 3747 q^{25} - 9251 q^{27} + 4422 q^{29} - 14381 q^{31} - 3509 q^{33}+ \cdots + 40051 q^{99}+O(q^{100})$ 2 * q - 29 * q^3 - 13 * q^5 + 14 * q^7 + 331 * q^9 + 242 * q^11 - 646 * q^13 + 2171 * q^15 - 208 * q^17 + 2148 * q^19 - 2582 * q^21 - 349 * q^23 + 3747 * q^25 - 9251 * q^27 + 4422 * q^29 - 14381 * q^31 - 3509 * q^33 - 11986 * q^35 - 4267 * q^37 + 13332 * q^39 - 10110 * q^41 + 9798 * q^43 - 59644 * q^45 - 21144 * q^47 - 19242 * q^49 - 46150 * q^51 + 39584 * q^53 - 1573 * q^55 - 48592 * q^57 - 90951 * q^59 - 29550 * q^61 + 71308 * q^63 + 24024 * q^65 + 64149 * q^67 + 2285 * q^69 - 23583 * q^71 - 39058 * q^73 - 80104 * q^75 + 1694 * q^77 + 54974 * q^79 + 189706 * q^81 - 29986 * q^83 - 244478 * q^85 + 81000 * q^87 - 18047 * q^89 - 28312 * q^91 + 243813 * q^93 - 101192 * q^95 - 30309 * q^97 + 40051 * 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
$3$	−28.5801	−1.83342	−0.916708	−	0.399558i	$-0.869164\pi$
$3$	−28.5801	−1.83342	−0.916708	+	0.399558i	$0.869164\pi$
$4$	0	0
$5$	−76.9006	−1.37564	−0.687820	−	0.725881i	$-0.741432\pi$
$5$	−76.9006	−1.37564	−0.687820	+	0.725881i	$0.741432\pi$
$6$	0	0
$7$	91.4808	0.705642	0.352821	−	0.935691i	$-0.385222\pi$
$7$	91.4808	0.705642	0.352821	+	0.935691i	$0.385222\pi$
$8$	0	0
$9$	573.824	2.36141
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−463.801	−0.761156	−0.380578	−	0.924749i	$-0.624275\pi$
$13$	−463.801	−0.761156	−0.380578	+	0.924749i	$0.624275\pi$
$14$	0	0
$15$	2197.83	2.52212
$16$	0	0
$17$	1641.94	1.37795	0.688976	−	0.724784i	$-0.258061\pi$
$17$	1641.94	1.37795	0.688976	+	0.724784i	$0.258061\pi$
$18$	0	0
$19$	1693.53	1.07624	0.538118	−	0.842869i	$-0.319135\pi$
$19$	1693.53	1.07624	0.538118	+	0.842869i	$0.319135\pi$
$20$	0	0
$21$	−2614.53	−1.29374
$22$	0	0
$23$	−75.9391	−0.0299327	−0.0149663	−	0.999888i	$-0.504764\pi$
$23$	−75.9391	−0.0299327	−0.0149663	+	0.999888i	$0.504764\pi$
$24$	0	0
$25$	2788.71	0.892387
$26$	0	0
$27$	−9454.98	−2.49604
$28$	0	0
$29$	−2942.33	−0.649675	−0.324837	−	0.945770i	$-0.605310\pi$
$29$	−2942.33	−0.649675	−0.324837	+	0.945770i	$0.605310\pi$
$30$	0	0
$31$	−8443.63	−1.57807	−0.789033	−	0.614351i	$-0.789418\pi$
$31$	−8443.63	−1.57807	−0.789033	+	0.614351i	$0.789418\pi$
$32$	0	0
$33$	−3458.20	−0.552796
$34$	0	0
$35$	−7034.93	−0.970710
$36$	0	0
$37$	−35.5610	−0.00427041	−0.00213520	−	0.999998i	$-0.500680\pi$
$37$	−35.5610	−0.00427041	−0.00213520	+	0.999998i	$0.500680\pi$
$38$	0	0
$39$	13255.5	1.39552
$40$	0	0
$41$	9222.25	0.856796	0.428398	−	0.903590i	$-0.359078\pi$
$41$	9222.25	0.856796	0.428398	+	0.903590i	$0.359078\pi$
$42$	0	0
$43$	11516.7	0.949851	0.474925	−	0.880026i	$-0.342475\pi$
$43$	11516.7	0.949851	0.474925	+	0.880026i	$0.342475\pi$
$44$	0	0
$45$	−44127.4	−3.24846
$46$	0	0
$47$	−6179.00	−0.408013	−0.204006	−	0.978970i	$-0.565396\pi$
$47$	−6179.00	−0.408013	−0.204006	+	0.978970i	$0.565396\pi$
$48$	0	0
$49$	−8438.27	−0.502069
$50$	0	0
$51$	−46926.7	−2.52636
$52$	0	0
$53$	25255.1	1.23498	0.617489	−	0.786579i	$-0.288150\pi$
$53$	25255.1	1.23498	0.617489	+	0.786579i	$0.288150\pi$
$54$	0	0
$55$	−9304.98	−0.414771
$56$	0	0
$57$	−48401.2	−1.97319
$58$	0	0
$59$	−40786.8	−1.52542	−0.762711	−	0.646740i	$-0.776132\pi$
$59$	−40786.8	−1.52542	−0.762711	+	0.646740i	$0.776132\pi$
$60$	0	0
$61$	−7368.85	−0.253557	−0.126778	−	0.991931i	$-0.540464\pi$
$61$	−7368.85	−0.253557	−0.126778	+	0.991931i	$0.540464\pi$
$62$	0	0
$63$	52493.8	1.66631
$64$	0	0
$65$	35666.6	1.04708
$66$	0	0
$67$	11024.7	0.300041	0.150020	−	0.988683i	$-0.452066\pi$
$67$	11024.7	0.300041	0.150020	+	0.988683i	$0.452066\pi$
$68$	0	0
$69$	2170.35	0.0548791
$70$	0	0
$71$	46964.9	1.10567	0.552837	−	0.833289i	$-0.313545\pi$
$71$	46964.9	1.10567	0.552837	+	0.833289i	$0.313545\pi$
$72$	0	0
$73$	−60727.5	−1.33376	−0.666880	−	0.745165i	$-0.732371\pi$
$73$	−60727.5	−1.33376	−0.666880	+	0.745165i	$0.732371\pi$
$74$	0	0
$75$	−79701.6	−1.63612
$76$	0	0
$77$	11069.2	0.212759
$78$	0	0
$79$	−18386.1	−0.331452	−0.165726	−	0.986172i	$-0.552997\pi$
$79$	−18386.1	−0.331452	−0.165726	+	0.986172i	$0.552997\pi$
$80$	0	0
$81$	130785.	2.21486
$82$	0	0
$83$	59321.9	0.945192	0.472596	−	0.881279i	$-0.343317\pi$
$83$	59321.9	0.945192	0.472596	+	0.881279i	$0.343317\pi$
$84$	0	0
$85$	−126266.	−1.89557
$86$	0	0
$87$	84092.1	1.19112
$88$	0	0
$89$	5070.71	0.0678568	0.0339284	−	0.999424i	$-0.489198\pi$
$89$	5070.71	0.0678568	0.0339284	+	0.999424i	$0.489198\pi$
$90$	0	0
$91$	−42428.9	−0.537104
$92$	0	0
$93$	241320.	2.89325
$94$	0	0
$95$	−130233.	−1.48051
$96$	0	0
$97$	−130795.	−1.41143	−0.705717	−	0.708494i	$-0.749375\pi$
$97$	−130795.	−1.41143	−0.705717	+	0.708494i	$0.749375\pi$
$98$	0	0
$99$	69432.7	0.711993
$100$	0	0
$101$	−27114.8	−0.264486	−0.132243	−	0.991217i	$-0.542218\pi$
$101$	−27114.8	−0.264486	−0.132243	+	0.991217i	$0.542218\pi$
$102$	0	0
$103$	12802.9	0.118909	0.0594546	−	0.998231i	$-0.481064\pi$
$103$	12802.9	0.118909	0.0594546	+	0.998231i	$0.481064\pi$
$104$	0	0
$105$	201059.	1.77972
$106$	0	0
$107$	64131.5	0.541516	0.270758	−	0.962647i	$-0.412726\pi$
$107$	64131.5	0.541516	0.270758	+	0.962647i	$0.412726\pi$
$108$	0	0
$109$	−126630.	−1.02087	−0.510434	−	0.859917i	$-0.670515\pi$
$109$	−126630.	−1.02087	−0.510434	+	0.859917i	$0.670515\pi$
$110$	0	0
$111$	1016.34	0.00782943
$112$	0	0
$113$	131953.	0.972130	0.486065	−	0.873923i	$-0.338432\pi$
$113$	131953.	0.972130	0.486065	+	0.873923i	$0.338432\pi$
$114$	0	0
$115$	5839.77	0.0411766
$116$	0	0
$117$	−266140.	−1.79740
$118$	0	0
$119$	150206.	0.972341
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−263573.	−1.57086
$124$	0	0
$125$	25861.0	0.148037
$126$	0	0
$127$	−118876.	−0.654013	−0.327006	−	0.945022i	$-0.606040\pi$
$127$	−118876.	−0.654013	−0.327006	+	0.945022i	$0.606040\pi$
$128$	0	0
$129$	−329148.	−1.74147
$130$	0	0
$131$	−205935.	−1.04846	−0.524229	−	0.851577i	$-0.675647\pi$
$131$	−205935.	−1.04846	−0.524229	+	0.851577i	$0.675647\pi$
$132$	0	0
$133$	154925.	0.759438
$134$	0	0
$135$	727094.	3.43365
$136$	0	0
$137$	−196568.	−0.894769	−0.447385	−	0.894342i	$-0.647645\pi$
$137$	−196568.	−0.894769	−0.447385	+	0.894342i	$0.647645\pi$
$138$	0	0
$139$	29936.4	0.131420	0.0657102	−	0.997839i	$-0.479069\pi$
$139$	29936.4	0.131420	0.0657102	+	0.997839i	$0.479069\pi$
$140$	0	0
$141$	176597.	0.748057
$142$	0	0
$143$	−56120.0	−0.229497
$144$	0	0
$145$	226267.	0.893719
$146$	0	0
$147$	241167.	0.920501
$148$	0	0
$149$	−362800.	−1.33876	−0.669379	−	0.742921i	$-0.733440\pi$
$149$	−362800.	−1.33876	−0.669379	+	0.742921i	$0.733440\pi$
$150$	0	0
$151$	−226244.	−0.807485	−0.403742	−	0.914873i	$-0.632291\pi$
$151$	−226244.	−0.807485	−0.403742	+	0.914873i	$0.632291\pi$
$152$	0	0
$153$	942182.	3.25391
$154$	0	0
$155$	649321.	2.17085
$156$	0	0
$157$	392007.	1.26924	0.634621	−	0.772824i	$-0.281156\pi$
$157$	392007.	1.26924	0.634621	+	0.772824i	$0.281156\pi$
$158$	0	0
$159$	−721794.	−2.26423
$160$	0	0
$161$	−6946.97	−0.0211218
$162$	0	0
$163$	84380.4	0.248755	0.124378	−	0.992235i	$-0.460307\pi$
$163$	84380.4	0.248755	0.124378	+	0.992235i	$0.460307\pi$
$164$	0	0
$165$	265937.	0.760448
$166$	0	0
$167$	−543654.	−1.50845	−0.754226	−	0.656615i	$-0.771988\pi$
$167$	−543654.	−1.50845	−0.754226	+	0.656615i	$0.771988\pi$
$168$	0	0
$169$	−156181.	−0.420642
$170$	0	0
$171$	971785.	2.54144
$172$	0	0
$173$	356699.	0.906123	0.453061	−	0.891479i	$-0.350332\pi$
$173$	356699.	0.906123	0.453061	+	0.891479i	$0.350332\pi$
$174$	0	0
$175$	255113.	0.629706
$176$	0	0
$177$	1.16569e6	2.79673
$178$	0	0
$179$	521697.	1.21699	0.608493	−	0.793559i	$-0.291774\pi$
$179$	521697.	1.21699	0.608493	+	0.793559i	$0.291774\pi$
$180$	0	0
$181$	−698048.	−1.58376	−0.791879	−	0.610678i	$-0.790897\pi$
$181$	−698048.	−1.58376	−0.791879	+	0.610678i	$0.790897\pi$
$182$	0	0
$183$	210603.	0.464875
$184$	0	0
$185$	2734.66	0.00587454
$186$	0	0
$187$	198674.	0.415468
$188$	0	0
$189$	−864949.	−1.76131
$190$	0	0
$191$	−612931.	−1.21571	−0.607853	−	0.794050i	$-0.707969\pi$
$191$	−612931.	−1.21571	−0.607853	+	0.794050i	$0.707969\pi$
$192$	0	0
$193$	−184465.	−0.356467	−0.178234	−	0.983988i	$-0.557038\pi$
$193$	−184465.	−0.356467	−0.178234	+	0.983988i	$0.557038\pi$
$194$	0	0
$195$	−1.01936e6	−1.91973
$196$	0	0
$197$	−1.03211e6	−1.89479	−0.947393	−	0.320074i	$-0.896292\pi$
$197$	−1.03211e6	−1.89479	−0.947393	+	0.320074i	$0.896292\pi$
$198$	0	0
$199$	−244463.	−0.437603	−0.218801	−	0.975769i	$-0.570215\pi$
$199$	−244463.	−0.437603	−0.218801	+	0.975769i	$0.570215\pi$
$200$	0	0
$201$	−315088.	−0.550099
$202$	0	0
$203$	−269166.	−0.458438
$204$	0	0
$205$	−709197.	−1.17864
$206$	0	0
$207$	−43575.7	−0.0706835
$208$	0	0
$209$	204917.	0.324498
$210$	0	0
$211$	−1.12590e6	−1.74098	−0.870488	−	0.492190i	$-0.836197\pi$
$211$	−1.12590e6	−1.74098	−0.870488	+	0.492190i	$0.836197\pi$
$212$	0	0
$213$	−1.34226e6	−2.02716
$214$	0	0
$215$	−885639.	−1.30665
$216$	0	0
$217$	−772430.	−1.11355
$218$	0	0
$219$	1.73560e6	2.44534
$220$	0	0
$221$	−761532.	−1.04884
$222$	0	0
$223$	−72016.8	−0.0969776	−0.0484888	−	0.998824i	$-0.515441\pi$
$223$	−72016.8	−0.0969776	−0.0484888	+	0.998824i	$0.515441\pi$
$224$	0	0
$225$	1.60023e6	2.10729
$226$	0	0
$227$	−1.13240e6	−1.45860	−0.729298	−	0.684196i	$-0.760153\pi$
$227$	−1.13240e6	−1.45860	−0.729298	+	0.684196i	$0.760153\pi$
$228$	0	0
$229$	−34423.7	−0.0433779	−0.0216890	−	0.999765i	$-0.506904\pi$
$229$	−34423.7	−0.0433779	−0.0216890	+	0.999765i	$0.506904\pi$
$230$	0	0
$231$	−316358.	−0.390076
$232$	0	0
$233$	689707.	0.832290	0.416145	−	0.909298i	$-0.363381\pi$
$233$	689707.	0.832290	0.416145	+	0.909298i	$0.363381\pi$
$234$	0	0
$235$	475169.	0.561279
$236$	0	0
$237$	525476.	0.607690
$238$	0	0
$239$	−289978.	−0.328375	−0.164188	−	0.986429i	$-0.552500\pi$
$239$	−289978.	−0.328375	−0.164188	+	0.986429i	$0.552500\pi$
$240$	0	0
$241$	1.40673e6	1.56016	0.780078	−	0.625682i	$-0.215179\pi$
$241$	1.40673e6	1.56016	0.780078	+	0.625682i	$0.215179\pi$
$242$	0	0
$243$	−1.44030e6	−1.56473
$244$	0	0
$245$	648908.	0.690666
$246$	0	0
$247$	−785459.	−0.819184
$248$	0	0
$249$	−1.69543e6	−1.73293
$250$	0	0
$251$	−580131.	−0.581222	−0.290611	−	0.956841i	$-0.593859\pi$
$251$	−580131.	−0.581222	−0.290611	+	0.956841i	$0.593859\pi$
$252$	0	0
$253$	−9188.63	−0.00902505
$254$	0	0
$255$	3.60870e6	3.47536
$256$	0	0
$257$	−407104.	−0.384479	−0.192240	−	0.981348i	$-0.561575\pi$
$257$	−407104.	−0.384479	−0.192240	+	0.981348i	$0.561575\pi$
$258$	0	0
$259$	−3253.14	−0.00301338
$260$	0	0
$261$	−1.68838e6	−1.53415
$262$	0	0
$263$	1.57675e6	1.40564	0.702819	−	0.711369i	$-0.251924\pi$
$263$	1.57675e6	1.40564	0.702819	+	0.711369i	$0.251924\pi$
$264$	0	0
$265$	−1.94213e6	−1.69889
$266$	0	0
$267$	−144921.	−0.124410
$268$	0	0
$269$	−752593.	−0.634132	−0.317066	−	0.948404i	$-0.602698\pi$
$269$	−752593.	−0.634132	−0.317066	+	0.948404i	$0.602698\pi$
$270$	0	0
$271$	208553.	0.172501	0.0862507	−	0.996273i	$-0.472511\pi$
$271$	208553.	0.172501	0.0862507	+	0.996273i	$0.472511\pi$
$272$	0	0
$273$	1.21262e6	0.984735
$274$	0	0
$275$	337434.	0.269065
$276$	0	0
$277$	1.01926e6	0.798151	0.399075	−	0.916918i	$-0.369331\pi$
$277$	1.01926e6	0.798151	0.399075	+	0.916918i	$0.369331\pi$
$278$	0	0
$279$	−4.84516e6	−3.72647
$280$	0	0
$281$	865597.	0.653958	0.326979	−	0.945032i	$-0.393969\pi$
$281$	865597.	0.653958	0.326979	+	0.945032i	$0.393969\pi$
$282$	0	0
$283$	−1.72508e6	−1.28040	−0.640198	−	0.768210i	$-0.721148\pi$
$283$	−1.72508e6	−1.28040	−0.640198	+	0.768210i	$0.721148\pi$
$284$	0	0
$285$	3.72208e6	2.71440
$286$	0	0
$287$	843658.	0.604591
$288$	0	0
$289$	1.27610e6	0.898750
$290$	0	0
$291$	3.73813e6	2.58774
$292$	0	0
$293$	1.25254e6	0.852361	0.426181	−	0.904638i	$-0.359859\pi$
$293$	1.25254e6	0.852361	0.426181	+	0.904638i	$0.359859\pi$
$294$	0	0
$295$	3.13653e6	2.09843
$296$	0	0
$297$	−1.14405e6	−0.752584
$298$	0	0
$299$	35220.7	0.0227834
$300$	0	0
$301$	1.05355e6	0.670255
$302$	0	0
$303$	774945.	0.484914
$304$	0	0
$305$	566669.	0.348803
$306$	0	0
$307$	−2.69819e6	−1.63390	−0.816951	−	0.576707i	$-0.804337\pi$
$307$	−2.69819e6	−1.63390	−0.816951	+	0.576707i	$0.804337\pi$
$308$	0	0
$309$	−365908.	−0.218010
$310$	0	0
$311$	1.74373e6	1.02230	0.511150	−	0.859491i	$-0.329220\pi$
$311$	1.74373e6	1.02230	0.511150	+	0.859491i	$0.329220\pi$
$312$	0	0
$313$	−1.15038e6	−0.663714	−0.331857	−	0.943330i	$-0.607675\pi$
$313$	−1.15038e6	−0.663714	−0.331857	+	0.943330i	$0.607675\pi$
$314$	0	0
$315$	−4.03681e6	−2.29225
$316$	0	0
$317$	−1.16110e6	−0.648967	−0.324484	−	0.945891i	$-0.605190\pi$
$317$	−1.16110e6	−0.648967	−0.324484	+	0.945891i	$0.605190\pi$
$318$	0	0
$319$	−356022.	−0.195884
$320$	0	0
$321$	−1.83289e6	−0.992825
$322$	0	0
$323$	2.78066e6	1.48300
$324$	0	0
$325$	−1.29341e6	−0.679245
$326$	0	0
$327$	3.61910e6	1.87168
$328$	0	0
$329$	−565260.	−0.287911
$330$	0	0
$331$	2.31915e6	1.16348	0.581740	−	0.813375i	$-0.302372\pi$
$331$	2.31915e6	1.16348	0.581740	+	0.813375i	$0.302372\pi$
$332$	0	0
$333$	−20405.7	−0.0100842
$334$	0	0
$335$	−847807.	−0.412748
$336$	0	0
$337$	1.43786e6	0.689671	0.344835	−	0.938663i	$-0.387935\pi$
$337$	1.43786e6	0.689671	0.344835	+	0.938663i	$0.387935\pi$
$338$	0	0
$339$	−3.77124e6	−1.78232
$340$	0	0
$341$	−1.02168e6	−0.475805
$342$	0	0
$343$	−2.30946e6	−1.05992
$344$	0	0
$345$	−166901.	−0.0754939
$346$	0	0
$347$	−575629.	−0.256637	−0.128318	−	0.991733i	$-0.540958\pi$
$347$	−575629.	−0.256637	−0.128318	+	0.991733i	$0.540958\pi$
$348$	0	0
$349$	−2.77702e6	−1.22044	−0.610219	−	0.792233i	$-0.708919\pi$
$349$	−2.77702e6	−1.22044	−0.610219	+	0.792233i	$0.708919\pi$
$350$	0	0
$351$	4.38523e6	1.89987
$352$	0	0
$353$	2.18172e6	0.931885	0.465942	−	0.884815i	$-0.345715\pi$
$353$	2.18172e6	0.931885	0.465942	+	0.884815i	$0.345715\pi$
$354$	0	0
$355$	−3.61163e6	−1.52101
$356$	0	0
$357$	−4.29289e6	−1.78271
$358$	0	0
$359$	2.14848e6	0.879822	0.439911	−	0.898041i	$-0.355010\pi$
$359$	2.14848e6	0.879822	0.439911	+	0.898041i	$0.355010\pi$
$360$	0	0
$361$	391930.	0.158285
$362$	0	0
$363$	−418442.	−0.166674
$364$	0	0
$365$	4.66998e6	1.83478
$366$	0	0
$367$	−15838.0	−0.00613813	−0.00306907	−	0.999995i	$-0.500977\pi$
$367$	−15838.0	−0.00613813	−0.00306907	+	0.999995i	$0.500977\pi$
$368$	0	0
$369$	5.29195e6	2.02325
$370$	0	0
$371$	2.31035e6	0.871453
$372$	0	0
$373$	−394343.	−0.146758	−0.0733791	−	0.997304i	$-0.523378\pi$
$373$	−394343.	−0.146758	−0.0733791	+	0.997304i	$0.523378\pi$
$374$	0	0
$375$	−739112.	−0.271414
$376$	0	0
$377$	1.36465e6	0.494504
$378$	0	0
$379$	−1.68501e6	−0.602567	−0.301283	−	0.953535i	$-0.597415\pi$
$379$	−1.68501e6	−0.602567	−0.301283	+	0.953535i	$0.597415\pi$
$380$	0	0
$381$	3.39750e6	1.19908
$382$	0	0
$383$	−3.94284e6	−1.37345	−0.686724	−	0.726918i	$-0.740952\pi$
$383$	−3.94284e6	−1.37345	−0.686724	+	0.726918i	$0.740952\pi$
$384$	0	0
$385$	−851226.	−0.292680
$386$	0	0
$387$	6.60853e6	2.24299
$388$	0	0
$389$	−571118.	−0.191360	−0.0956801	−	0.995412i	$-0.530503\pi$
$389$	−571118.	−0.191360	−0.0956801	+	0.995412i	$0.530503\pi$
$390$	0	0
$391$	−124687.	−0.0412458
$392$	0	0
$393$	5.88564e6	1.92226
$394$	0	0
$395$	1.41390e6	0.455959
$396$	0	0
$397$	6.17025e6	1.96484	0.982419	−	0.186689i	$-0.0597756\pi$
$397$	6.17025e6	1.96484	0.982419	+	0.186689i	$0.0597756\pi$
$398$	0	0
$399$	−4.42778e6	−1.39237
$400$	0	0
$401$	1.82838e6	0.567814	0.283907	−	0.958852i	$-0.408369\pi$
$401$	1.82838e6	0.567814	0.283907	+	0.958852i	$0.408369\pi$
$402$	0	0
$403$	3.91617e6	1.20115
$404$	0	0
$405$	−1.00575e7	−3.04686
$406$	0	0
$407$	−4302.88	−0.00128758
$408$	0	0
$409$	−5.94235e6	−1.75651	−0.878254	−	0.478195i	$-0.841291\pi$
$409$	−5.94235e6	−1.75651	−0.878254	+	0.478195i	$0.841291\pi$
$410$	0	0
$411$	5.61794e6	1.64048
$412$	0	0
$413$	−3.73121e6	−1.07640
$414$	0	0
$415$	−4.56189e6	−1.30024
$416$	0	0
$417$	−855586.	−0.240948
$418$	0	0
$419$	2.80473e6	0.780468	0.390234	−	0.920716i	$-0.372394\pi$
$419$	2.80473e6	0.780468	0.390234	+	0.920716i	$0.372394\pi$
$420$	0	0
$421$	3.57571e6	0.983234	0.491617	−	0.870812i	$-0.336406\pi$
$421$	3.57571e6	0.983234	0.491617	+	0.870812i	$0.336406\pi$
$422$	0	0
$423$	−3.54566e6	−0.963487
$424$	0	0
$425$	4.57888e6	1.22967
$426$	0	0
$427$	−674108.	−0.178920
$428$	0	0
$429$	1.60392e6	0.420764
$430$	0	0
$431$	2.82711e6	0.733078	0.366539	−	0.930403i	$-0.380543\pi$
$431$	2.82711e6	0.733078	0.366539	+	0.930403i	$0.380543\pi$
$432$	0	0
$433$	−765652.	−0.196251	−0.0981254	−	0.995174i	$-0.531285\pi$
$433$	−765652.	−0.196251	−0.0981254	+	0.995174i	$0.531285\pi$
$434$	0	0
$435$	−6.46673e6	−1.63856
$436$	0	0
$437$	−128605.	−0.0322147
$438$	0	0
$439$	−4.65788e6	−1.15352	−0.576762	−	0.816912i	$-0.695684\pi$
$439$	−4.65788e6	−1.15352	−0.576762	+	0.816912i	$0.695684\pi$
$440$	0	0
$441$	−4.84208e6	−1.18559
$442$	0	0
$443$	4.99700e6	1.20976	0.604881	−	0.796316i	$-0.293221\pi$
$443$	4.99700e6	1.20976	0.604881	+	0.796316i	$0.293221\pi$
$444$	0	0
$445$	−389941.	−0.0933466
$446$	0	0
$447$	1.03689e7	2.45450
$448$	0	0
$449$	165769.	0.0388050	0.0194025	−	0.999812i	$-0.493824\pi$
$449$	165769.	0.0388050	0.0194025	+	0.999812i	$0.493824\pi$
$450$	0	0
$451$	1.11589e6	0.258334
$452$	0	0
$453$	6.46608e6	1.48046
$454$	0	0
$455$	3.26281e6	0.738862
$456$	0	0
$457$	−8.07259e6	−1.80810	−0.904050	−	0.427428i	$-0.859420\pi$
$457$	−8.07259e6	−1.80810	−0.904050	+	0.427428i	$0.859420\pi$
$458$	0	0
$459$	−1.55245e7	−3.43942
$460$	0	0
$461$	−5.87512e6	−1.28755	−0.643776	−	0.765214i	$-0.722633\pi$
$461$	−5.87512e6	−1.28755	−0.643776	+	0.765214i	$0.722633\pi$
$462$	0	0
$463$	1.02289e6	0.221757	0.110878	−	0.993834i	$-0.464634\pi$
$463$	1.02289e6	0.221757	0.110878	+	0.993834i	$0.464634\pi$
$464$	0	0
$465$	−1.85577e7	−3.98007
$466$	0	0
$467$	6.17201e6	1.30959	0.654794	−	0.755807i	$-0.272755\pi$
$467$	6.17201e6	1.30959	0.654794	+	0.755807i	$0.272755\pi$
$468$	0	0
$469$	1.00855e6	0.211721
$470$	0	0
$471$	−1.12036e7	−2.32705
$472$	0	0
$473$	1.39352e6	0.286391
$474$	0	0
$475$	4.72275e6	0.960419
$476$	0	0
$477$	1.44920e7	2.91629
$478$	0	0
$479$	7.00244e6	1.39447	0.697237	−	0.716841i	$-0.254412\pi$
$479$	7.00244e6	1.39447	0.697237	+	0.716841i	$0.254412\pi$
$480$	0	0
$481$	16493.2	0.00325044
$482$	0	0
$483$	198545.	0.0387250
$484$	0	0
$485$	1.00582e7	1.94163
$486$	0	0
$487$	−365769.	−0.0698852	−0.0349426	−	0.999389i	$-0.511125\pi$
$487$	−365769.	−0.0698852	−0.0349426	+	0.999389i	$0.511125\pi$
$488$	0	0
$489$	−2.41160e6	−0.456072
$490$	0	0
$491$	−5.00170e6	−0.936297	−0.468149	−	0.883650i	$-0.655079\pi$
$491$	−5.00170e6	−0.936297	−0.468149	+	0.883650i	$0.655079\pi$
$492$	0	0
$493$	−4.83111e6	−0.895220
$494$	0	0
$495$	−5.33942e6	−0.979447
$496$	0	0
$497$	4.29638e6	0.780211
$498$	0	0
$499$	1.63517e6	0.293976	0.146988	−	0.989138i	$-0.453042\pi$
$499$	1.63517e6	0.293976	0.146988	+	0.989138i	$0.453042\pi$
$500$	0	0
$501$	1.55377e7	2.76562
$502$	0	0
$503$	−5.20393e6	−0.917088	−0.458544	−	0.888672i	$-0.651629\pi$
$503$	−5.20393e6	−0.917088	−0.458544	+	0.888672i	$0.651629\pi$
$504$	0	0
$505$	2.08515e6	0.363838
$506$	0	0
$507$	4.46368e6	0.771212
$508$	0	0
$509$	−8.62390e6	−1.47540	−0.737699	−	0.675130i	$-0.764088\pi$
$509$	−8.62390e6	−1.47540	−0.737699	+	0.675130i	$0.764088\pi$
$510$	0	0
$511$	−5.55539e6	−0.941158
$512$	0	0
$513$	−1.60123e7	−2.68633
$514$	0	0
$515$	−984551.	−0.163576
$516$	0	0
$517$	−747659.	−0.123020
$518$	0	0
$519$	−1.01945e7	−1.66130
$520$	0	0
$521$	140764.	0.0227194	0.0113597	−	0.999935i	$-0.496384\pi$
$521$	140764.	0.0227194	0.0113597	+	0.999935i	$0.496384\pi$
$522$	0	0
$523$	9.29900e6	1.48656	0.743279	−	0.668981i	$-0.233269\pi$
$523$	9.29900e6	1.48656	0.743279	+	0.668981i	$0.233269\pi$
$524$	0	0
$525$	−7.29117e6	−1.15451
$526$	0	0
$527$	−1.38639e7	−2.17450
$528$	0	0
$529$	−6.43058e6	−0.999104
$530$	0	0
$531$	−2.34044e7	−3.60215
$532$	0	0
$533$	−4.27729e6	−0.652155
$534$	0	0
$535$	−4.93175e6	−0.744932
$536$	0	0
$537$	−1.49102e7	−2.23124
$538$	0	0
$539$	−1.02103e6	−0.151379
$540$	0	0
$541$	−4.98590e6	−0.732403	−0.366202	−	0.930536i	$-0.619342\pi$
$541$	−4.98590e6	−0.732403	−0.366202	+	0.930536i	$0.619342\pi$
$542$	0	0
$543$	1.99503e7	2.90369
$544$	0	0
$545$	9.73792e6	1.40435
$546$	0	0
$547$	3.57048e6	0.510221	0.255111	−	0.966912i	$-0.417888\pi$
$547$	3.57048e6	0.510221	0.255111	+	0.966912i	$0.417888\pi$
$548$	0	0
$549$	−4.22842e6	−0.598753
$550$	0	0
$551$	−4.98291e6	−0.699204
$552$	0	0
$553$	−1.68197e6	−0.233887
$554$	0	0
$555$	−78156.9	−0.0107705
$556$	0	0
$557$	1.24799e7	1.70441	0.852206	−	0.523207i	$-0.175264\pi$
$557$	1.24799e7	1.70441	0.852206	+	0.523207i	$0.175264\pi$
$558$	0	0
$559$	−5.34144e6	−0.722984
$560$	0	0
$561$	−5.67814e6	−0.761726
$562$	0	0
$563$	5.12581e6	0.681540	0.340770	−	0.940147i	$-0.389312\pi$
$563$	5.12581e6	0.681540	0.340770	+	0.940147i	$0.389312\pi$
$564$	0	0
$565$	−1.01473e7	−1.33730
$566$	0	0
$567$	1.19644e7	1.56290
$568$	0	0
$569$	−1.09610e6	−0.141929	−0.0709645	−	0.997479i	$-0.522608\pi$
$569$	−1.09610e6	−0.141929	−0.0709645	+	0.997479i	$0.522608\pi$
$570$	0	0
$571$	1.08787e7	1.39632	0.698162	−	0.715940i	$-0.254002\pi$
$571$	1.08787e7	1.39632	0.698162	+	0.715940i	$0.254002\pi$
$572$	0	0
$573$	1.75176e7	2.22889
$574$	0	0
$575$	−211772.	−0.0267115
$576$	0	0
$577$	1.37408e7	1.71820	0.859101	−	0.511807i	$-0.171024\pi$
$577$	1.37408e7	1.71820	0.859101	+	0.511807i	$0.171024\pi$
$578$	0	0
$579$	5.27202e6	0.653553
$580$	0	0
$581$	5.42681e6	0.666967
$582$	0	0
$583$	3.05587e6	0.372360
$584$	0	0
$585$	2.04663e7	2.47258
$586$	0	0
$587$	−7.50167e6	−0.898593	−0.449296	−	0.893383i	$-0.648325\pi$
$587$	−7.50167e6	−0.898593	−0.449296	+	0.893383i	$0.648325\pi$
$588$	0	0
$589$	−1.42995e7	−1.69837
$590$	0	0
$591$	2.94978e7	3.47393
$592$	0	0
$593$	1.23281e7	1.43966	0.719829	−	0.694151i	$-0.244220\pi$
$593$	1.23281e7	1.43966	0.719829	+	0.694151i	$0.244220\pi$
$594$	0	0
$595$	−1.15509e7	−1.33759
$596$	0	0
$597$	6.98677e6	0.802308
$598$	0	0
$599$	−3.71959e6	−0.423573	−0.211786	−	0.977316i	$-0.567928\pi$
$599$	−3.71959e6	−0.423573	−0.211786	+	0.977316i	$0.567928\pi$
$600$	0	0
$601$	−8.89207e6	−1.00419	−0.502096	−	0.864812i	$-0.667437\pi$
$601$	−8.89207e6	−1.00419	−0.502096	+	0.864812i	$0.667437\pi$
$602$	0	0
$603$	6.32624e6	0.708520
$604$	0	0
$605$	−1.12590e6	−0.125058
$606$	0	0
$607$	−1.18740e7	−1.30805	−0.654027	−	0.756471i	$-0.726922\pi$
$607$	−1.18740e7	−1.30805	−0.654027	+	0.756471i	$0.726922\pi$
$608$	0	0
$609$	7.69281e6	0.840508
$610$	0	0
$611$	2.86583e6	0.310561
$612$	0	0
$613$	9.08062e6	0.976033	0.488016	−	0.872834i	$-0.337721\pi$
$613$	9.08062e6	0.976033	0.488016	+	0.872834i	$0.337721\pi$
$614$	0	0
$615$	2.02689e7	2.16094
$616$	0	0
$617$	−9.98190e6	−1.05560	−0.527801	−	0.849368i	$-0.676983\pi$
$617$	−9.98190e6	−1.05560	−0.527801	+	0.849368i	$0.676983\pi$
$618$	0	0
$619$	3.86586e6	0.405527	0.202763	−	0.979228i	$-0.435008\pi$
$619$	3.86586e6	0.405527	0.202763	+	0.979228i	$0.435008\pi$
$620$	0	0
$621$	718003.	0.0747132
$622$	0	0
$623$	463872.	0.0478827
$624$	0	0
$625$	−1.07034e7	−1.09603
$626$	0	0
$627$	−5.85654e6	−0.594939
$628$	0	0
$629$	−58388.8	−0.00588441
$630$	0	0
$631$	−1.45626e7	−1.45601	−0.728007	−	0.685570i	$-0.759553\pi$
$631$	−1.45626e7	−1.45601	−0.728007	+	0.685570i	$0.759553\pi$
$632$	0	0
$633$	3.21783e7	3.19193
$634$	0	0
$635$	9.14167e6	0.899686
$636$	0	0
$637$	3.91368e6	0.382153
$638$	0	0
$639$	2.69496e7	2.61096
$640$	0	0
$641$	−1.34201e7	−1.29006	−0.645032	−	0.764155i	$-0.723156\pi$
$641$	−1.34201e7	−1.29006	−0.645032	+	0.764155i	$0.723156\pi$
$642$	0	0
$643$	−1.50895e7	−1.43929	−0.719644	−	0.694344i	$-0.755695\pi$
$643$	−1.50895e7	−1.43929	−0.719644	+	0.694344i	$0.755695\pi$
$644$	0	0
$645$	2.53117e7	2.39564
$646$	0	0
$647$	−1.42974e7	−1.34275	−0.671377	−	0.741116i	$-0.734297\pi$
$647$	−1.42974e7	−1.34275	−0.671377	+	0.741116i	$0.734297\pi$
$648$	0	0
$649$	−4.93520e6	−0.459932
$650$	0	0
$651$	2.20761e7	2.04160
$652$	0	0
$653$	−1.31607e6	−0.120780	−0.0603900	−	0.998175i	$-0.519234\pi$
$653$	−1.31607e6	−0.120780	−0.0603900	+	0.998175i	$0.519234\pi$
$654$	0	0
$655$	1.58365e7	1.44230
$656$	0	0
$657$	−3.48469e7	−3.14956
$658$	0	0
$659$	−3.75917e6	−0.337193	−0.168596	−	0.985685i	$-0.553923\pi$
$659$	−3.75917e6	−0.337193	−0.168596	+	0.985685i	$0.553923\pi$
$660$	0	0
$661$	−8.95170e6	−0.796896	−0.398448	−	0.917191i	$-0.630451\pi$
$661$	−8.95170e6	−0.796896	−0.398448	+	0.917191i	$0.630451\pi$
$662$	0	0
$663$	2.17647e7	1.92295
$664$	0	0
$665$	−1.19138e7	−1.04471
$666$	0	0
$667$	223438.	0.0194465
$668$	0	0
$669$	2.05825e6	0.177800
$670$	0	0
$671$	−891631.	−0.0764503
$672$	0	0
$673$	−519050.	−0.0441745	−0.0220873	−	0.999756i	$-0.507031\pi$
$673$	−519050.	−0.0441745	−0.0220873	+	0.999756i	$0.507031\pi$
$674$	0	0
$675$	−2.63672e7	−2.22743
$676$	0	0
$677$	−4.12272e6	−0.345711	−0.172855	−	0.984947i	$-0.555299\pi$
$677$	−4.12272e6	−0.345711	−0.172855	+	0.984947i	$0.555299\pi$
$678$	0	0
$679$	−1.19652e7	−0.995967
$680$	0	0
$681$	3.23641e7	2.67421
$682$	0	0
$683$	−8.35625e6	−0.685425	−0.342712	−	0.939440i	$-0.611346\pi$
$683$	−8.35625e6	−0.685425	−0.342712	+	0.939440i	$0.611346\pi$
$684$	0	0
$685$	1.51162e7	1.23088
$686$	0	0
$687$	983834.	0.0795298
$688$	0	0
$689$	−1.17133e7	−0.940011
$690$	0	0
$691$	−9.40353e6	−0.749197	−0.374598	−	0.927187i	$-0.622219\pi$
$691$	−9.40353e6	−0.749197	−0.374598	+	0.927187i	$0.622219\pi$
$692$	0	0
$693$	6.35175e6	0.502413
$694$	0	0
$695$	−2.30213e6	−0.180787
$696$	0	0
$697$	1.51423e7	1.18062
$698$	0	0
$699$	−1.97119e7	−1.52593
$700$	0	0
$701$	−1.76888e7	−1.35957	−0.679787	−	0.733410i	$-0.737928\pi$
$701$	−1.76888e7	−1.35957	−0.679787	+	0.733410i	$0.737928\pi$
$702$	0	0
$703$	−60223.4	−0.00459597
$704$	0	0
$705$	−1.35804e7	−1.02906
$706$	0	0
$707$	−2.48049e6	−0.186633
$708$	0	0
$709$	−1.11343e7	−0.831853	−0.415926	−	0.909398i	$-0.636543\pi$
$709$	−1.11343e7	−0.831853	−0.415926	+	0.909398i	$0.636543\pi$
$710$	0	0
$711$	−1.05504e7	−0.782696
$712$	0	0
$713$	641202.	0.0472358
$714$	0	0
$715$	4.31566e6	0.315706
$716$	0	0
$717$	8.28761e6	0.602048
$718$	0	0
$719$	9.55010e6	0.688947	0.344474	−	0.938796i	$-0.388057\pi$
$719$	9.55010e6	0.688947	0.344474	+	0.938796i	$0.388057\pi$
$720$	0	0
$721$	1.17122e6	0.0839073
$722$	0	0
$723$	−4.02045e7	−2.86042
$724$	0	0
$725$	−8.20529e6	−0.579761
$726$	0	0
$727$	1.89994e7	1.33322	0.666612	−	0.745405i	$-0.267744\pi$
$727$	1.89994e7	1.33322	0.666612	+	0.745405i	$0.267744\pi$
$728$	0	0
$729$	9.38322e6	0.653933
$730$	0	0
$731$	1.89096e7	1.30885
$732$	0	0
$733$	−2.68111e7	−1.84312	−0.921561	−	0.388233i	$-0.873086\pi$
$733$	−2.68111e7	−1.84312	−0.921561	+	0.388233i	$0.873086\pi$
$734$	0	0
$735$	−1.85459e7	−1.26628
$736$	0	0
$737$	1.33399e6	0.0904657
$738$	0	0
$739$	6.41899e6	0.432370	0.216185	−	0.976352i	$-0.430639\pi$
$739$	6.41899e6	0.432370	0.216185	+	0.976352i	$0.430639\pi$
$740$	0	0
$741$	2.24485e7	1.50190
$742$	0	0
$743$	−4.66177e6	−0.309798	−0.154899	−	0.987930i	$-0.549505\pi$
$743$	−4.66177e6	−0.309798	−0.154899	+	0.987930i	$0.549505\pi$
$744$	0	0
$745$	2.78996e7	1.84165
$746$	0	0
$747$	3.40403e7	2.23199
$748$	0	0
$749$	5.86679e6	0.382117
$750$	0	0
$751$	−2.39792e7	−1.55144	−0.775721	−	0.631076i	$-0.782614\pi$
$751$	−2.39792e7	−1.55144	−0.775721	+	0.631076i	$0.782614\pi$
$752$	0	0
$753$	1.65802e7	1.06562
$754$	0	0
$755$	1.73983e7	1.11081
$756$	0	0
$757$	4.65197e6	0.295051	0.147526	−	0.989058i	$-0.452869\pi$
$757$	4.65197e6	0.295051	0.147526	+	0.989058i	$0.452869\pi$
$758$	0	0
$759$	262612.	0.0165467
$760$	0	0
$761$	−1.91952e7	−1.20152	−0.600760	−	0.799429i	$-0.705135\pi$
$761$	−1.91952e7	−1.20152	−0.600760	+	0.799429i	$0.705135\pi$
$762$	0	0
$763$	−1.15842e7	−0.720368
$764$	0	0
$765$	−7.24544e7	−4.47622
$766$	0	0
$767$	1.89170e7	1.16108
$768$	0	0
$769$	1.47548e7	0.899743	0.449871	−	0.893093i	$-0.351470\pi$
$769$	1.47548e7	0.899743	0.449871	+	0.893093i	$0.351470\pi$
$770$	0	0
$771$	1.16351e7	0.704910
$772$	0	0
$773$	1.48823e7	0.895820	0.447910	−	0.894079i	$-0.352169\pi$
$773$	1.48823e7	0.895820	0.447910	+	0.894079i	$0.352169\pi$
$774$	0	0
$775$	−2.35468e7	−1.40824
$776$	0	0
$777$	92975.2	0.00552478
$778$	0	0
$779$	1.56181e7	0.922115
$780$	0	0
$781$	5.68275e6	0.333373
$782$	0	0
$783$	2.78197e7	1.62161
$784$	0	0
$785$	−3.01456e7	−1.74602
$786$	0	0
$787$	−1.56041e7	−0.898054	−0.449027	−	0.893518i	$-0.648229\pi$
$787$	−1.56041e7	−0.898054	−0.449027	+	0.893518i	$0.648229\pi$
$788$	0	0
$789$	−4.50637e7	−2.57712
$790$	0	0
$791$	1.20712e7	0.685976
$792$	0	0
$793$	3.41768e6	0.192996
$794$	0	0
$795$	5.55064e7	3.11476
$796$	0	0
$797$	2.19852e7	1.22599	0.612993	−	0.790088i	$-0.289965\pi$
$797$	2.19852e7	1.22599	0.612993	+	0.790088i	$0.289965\pi$
$798$	0	0
$799$	−1.01455e7	−0.562221
$800$	0	0
$801$	2.90969e6	0.160238
$802$	0	0
$803$	−7.34802e6	−0.402144
$804$	0	0
$805$	534226.	0.0290560
$806$	0	0
$807$	2.15092e7	1.16263
$808$	0	0
$809$	2.91738e7	1.56719	0.783596	−	0.621271i	$-0.213383\pi$
$809$	2.91738e7	1.56719	0.783596	+	0.621271i	$0.213383\pi$
$810$	0	0
$811$	−2.59750e7	−1.38677	−0.693383	−	0.720569i	$-0.743881\pi$
$811$	−2.59750e7	−1.38677	−0.693383	+	0.720569i	$0.743881\pi$
$812$	0	0
$813$	−5.96046e6	−0.316267
$814$	0	0
$815$	−6.48891e6	−0.342198
$816$	0	0
$817$	1.95038e7	1.02226
$818$	0	0
$819$	−2.43467e7	−1.26832
$820$	0	0
$821$	3.42686e7	1.77435	0.887174	−	0.461436i	$-0.152666\pi$
$821$	3.42686e7	1.77435	0.887174	+	0.461436i	$0.152666\pi$
$822$	0	0
$823$	2.37132e7	1.22037	0.610184	−	0.792259i	$-0.291095\pi$
$823$	2.37132e7	1.22037	0.610184	+	0.792259i	$0.291095\pi$
$824$	0	0
$825$	−9.64390e6	−0.493308
$826$	0	0
$827$	−1.83274e7	−0.931833	−0.465916	−	0.884829i	$-0.654275\pi$
$827$	−1.83274e7	−0.931833	−0.465916	+	0.884829i	$0.654275\pi$
$828$	0	0
$829$	−2.66383e7	−1.34623	−0.673115	−	0.739538i	$-0.735044\pi$
$829$	−2.66383e7	−1.34623	−0.673115	+	0.739538i	$0.735044\pi$
$830$	0	0
$831$	−2.91305e7	−1.46334
$832$	0	0
$833$	−1.38551e7	−0.691826
$834$	0	0
$835$	4.18074e7	2.07509
$836$	0	0
$837$	7.98344e7	3.93891
$838$	0	0
$839$	−1.46709e7	−0.719537	−0.359768	−	0.933042i	$-0.617144\pi$
$839$	−1.46709e7	−0.719537	−0.359768	+	0.933042i	$0.617144\pi$
$840$	0	0
$841$	−1.18539e7	−0.577923
$842$	0	0
$843$	−2.47389e7	−1.19898
$844$	0	0
$845$	1.20104e7	0.578652
$846$	0	0
$847$	1.33937e6	0.0641493
$848$	0	0
$849$	4.93031e7	2.34750
$850$	0	0
$851$	2700.47	0.000127825 0
$852$	0	0
$853$	−1.74709e7	−0.822133	−0.411066	−	0.911605i	$-0.634844\pi$
$853$	−1.74709e7	−0.822133	−0.411066	+	0.911605i	$0.634844\pi$
$854$	0	0
$855$	−7.47309e7	−3.49611
$856$	0	0
$857$	−2.49580e7	−1.16080	−0.580402	−	0.814330i	$-0.697104\pi$
$857$	−2.49580e7	−1.16080	−0.580402	+	0.814330i	$0.697104\pi$
$858$	0	0
$859$	1.42612e7	0.659439	0.329719	−	0.944079i	$-0.393046\pi$
$859$	1.42612e7	0.659439	0.329719	+	0.944079i	$0.393046\pi$
$860$	0	0
$861$	−2.41119e7	−1.10847
$862$	0	0
$863$	−494514.	−0.0226022	−0.0113011	−	0.999936i	$-0.503597\pi$
$863$	−494514.	−0.0226022	−0.0113011	+	0.999936i	$0.503597\pi$
$864$	0	0
$865$	−2.74304e7	−1.24650
$866$	0	0
$867$	−3.64710e7	−1.64778
$868$	0	0
$869$	−2.22471e6	−0.0999366
$870$	0	0
$871$	−5.11327e6	−0.228378
$872$	0	0
$873$	−7.50530e7	−3.33298
$874$	0	0
$875$	2.36579e6	0.104461
$876$	0	0
$877$	−8.07539e6	−0.354539	−0.177270	−	0.984162i	$-0.556726\pi$
$877$	−8.07539e6	−0.354539	−0.177270	+	0.984162i	$0.556726\pi$
$878$	0	0
$879$	−3.57979e7	−1.56273
$880$	0	0
$881$	−5.24166e6	−0.227525	−0.113763	−	0.993508i	$-0.536290\pi$
$881$	−5.24166e6	−0.227525	−0.113763	+	0.993508i	$0.536290\pi$
$882$	0	0
$883$	4.02805e7	1.73857	0.869287	−	0.494308i	$-0.164578\pi$
$883$	4.02805e7	1.73857	0.869287	+	0.494308i	$0.164578\pi$
$884$	0	0
$885$	−8.96425e7	−3.84730
$886$	0	0
$887$	−1.38423e7	−0.590745	−0.295373	−	0.955382i	$-0.595444\pi$
$887$	−1.38423e7	−0.590745	−0.295373	+	0.955382i	$0.595444\pi$
$888$	0	0
$889$	−1.08749e7	−0.461499
$890$	0	0
$891$	1.58250e7	0.667807
$892$	0	0
$893$	−1.04643e7	−0.439118
$894$	0	0
$895$	−4.01188e7	−1.67413
$896$	0	0
$897$	−1.00661e6	−0.0417715
$898$	0	0
$899$	2.48439e7	1.02523
$900$	0	0
$901$	4.14672e7	1.70174
$902$	0	0
$903$	−3.01107e7	−1.22886
$904$	0	0
$905$	5.36803e7	2.17868
$906$	0	0
$907$	5.53908e6	0.223573	0.111786	−	0.993732i	$-0.464343\pi$
$907$	5.53908e6	0.223573	0.111786	+	0.993732i	$0.464343\pi$
$908$	0	0
$909$	−1.55591e7	−0.624562
$910$	0	0
$911$	1.76513e7	0.704660	0.352330	−	0.935876i	$-0.385389\pi$
$911$	1.76513e7	0.704660	0.352330	+	0.935876i	$0.385389\pi$
$912$	0	0
$913$	7.17795e6	0.284986
$914$	0	0
$915$	−1.61955e7	−0.639501
$916$	0	0
$917$	−1.88391e7	−0.739837
$918$	0	0
$919$	2.80718e7	1.09643	0.548215	−	0.836338i	$-0.315308\pi$
$919$	2.80718e7	1.09643	0.548215	+	0.836338i	$0.315308\pi$
$920$	0	0
$921$	7.71145e7	2.99562
$922$	0	0
$923$	−2.17824e7	−0.841591
$924$	0	0
$925$	−99169.1	−0.00381085
$926$	0	0
$927$	7.34660e6	0.280794
$928$	0	0
$929$	−3.51046e6	−0.133452	−0.0667259	−	0.997771i	$-0.521255\pi$
$929$	−3.51046e6	−0.133452	−0.0667259	+	0.997771i	$0.521255\pi$
$930$	0	0
$931$	−1.42904e7	−0.540345
$932$	0	0
$933$	−4.98361e7	−1.87430
$934$	0	0
$935$	−1.52782e7	−0.571535
$936$	0	0
$937$	−3.60265e7	−1.34052	−0.670260	−	0.742126i	$-0.733818\pi$
$937$	−3.60265e7	−1.34052	−0.670260	+	0.742126i	$0.733818\pi$
$938$	0	0
$939$	3.28780e7	1.21686
$940$	0	0
$941$	2.51917e7	0.927435	0.463717	−	0.885983i	$-0.346515\pi$
$941$	2.51917e7	0.927435	0.463717	+	0.885983i	$0.346515\pi$
$942$	0	0
$943$	−700329.	−0.0256462
$944$	0	0
$945$	6.65151e7	2.42293
$946$	0	0
$947$	−2.44815e6	−0.0887082	−0.0443541	−	0.999016i	$-0.514123\pi$
$947$	−2.44815e6	−0.0887082	−0.0443541	+	0.999016i	$0.514123\pi$
$948$	0	0
$949$	2.81655e7	1.01520
$950$	0	0
$951$	3.31845e7	1.18983
$952$	0	0
$953$	−4.51778e7	−1.61136	−0.805681	−	0.592349i	$-0.798200\pi$
$953$	−4.51778e7	−1.61136	−0.805681	+	0.592349i	$0.798200\pi$
$954$	0	0
$955$	4.71348e7	1.67237
$956$	0	0
$957$	1.01751e7	0.359137
$958$	0	0
$959$	−1.79822e7	−0.631387
$960$	0	0
$961$	4.26658e7	1.49029
$962$	0	0
$963$	3.68001e7	1.27874
$964$	0	0
$965$	1.41855e7	0.490371
$966$	0	0
$967$	1.09821e7	0.377676	0.188838	−	0.982008i	$-0.439528\pi$
$967$	1.09821e7	0.377676	0.188838	+	0.982008i	$0.439528\pi$
$968$	0	0
$969$	−7.94716e7	−2.71896
$970$	0	0
$971$	−8.62580e6	−0.293597	−0.146798	−	0.989166i	$-0.546897\pi$
$971$	−8.62580e6	−0.293597	−0.146798	+	0.989166i	$0.546897\pi$
$972$	0	0
$973$	2.73861e6	0.0927358
$974$	0	0
$975$	3.69657e7	1.24534
$976$	0	0
$977$	1.06350e7	0.356451	0.178226	−	0.983990i	$-0.442964\pi$
$977$	1.06350e7	0.356451	0.178226	+	0.983990i	$0.442964\pi$
$978$	0	0
$979$	613556.	0.0204596
$980$	0	0
$981$	−7.26632e7	−2.41069
$982$	0	0
$983$	−2.66620e7	−0.880053	−0.440026	−	0.897985i	$-0.645031\pi$
$983$	−2.66620e7	−0.880053	−0.440026	+	0.897985i	$0.645031\pi$
$984$	0	0
$985$	7.93698e7	2.60654
$986$	0	0
$987$	1.61552e7	0.527861
$988$	0	0
$989$	−874565.	−0.0284316
$990$	0	0
$991$	−2.56353e7	−0.829190	−0.414595	−	0.910006i	$-0.636077\pi$
$991$	−2.56353e7	−0.829190	−0.414595	+	0.910006i	$0.636077\pi$
$992$	0	0
$993$	−6.62816e7	−2.13314
$994$	0	0
$995$	1.87993e7	0.601984
$996$	0	0
$997$	−588022.	−0.0187351	−0.00936754	−	0.999956i	$-0.502982\pi$
$997$	−588022.	−0.0187351	−0.00936754	+	0.999956i	$0.502982\pi$
$998$	0	0
$999$	336228.	0.0106591

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.6.a.f.1.1		2
4.3	odd	2		22.6.a.d.1.2	✓	2
8.3	odd	2		704.6.a.k.1.1		2
8.5	even	2		704.6.a.p.1.2		2
12.11	even	2		198.6.a.k.1.2		2
20.3	even	4		550.6.b.j.199.2		4
20.7	even	4		550.6.b.j.199.3		4
20.19	odd	2		550.6.a.h.1.1		2
28.27	even	2		1078.6.a.h.1.1		2
44.43	even	2		242.6.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.d.1.2	✓	2	4.3	odd	2
176.6.a.f.1.1		2	1.1	even	1	trivial
198.6.a.k.1.2		2	12.11	even	2
242.6.a.g.1.2		2	44.43	even	2
550.6.a.h.1.1		2	20.19	odd	2
550.6.b.j.199.2		4	20.3	even	4
550.6.b.j.199.3		4	20.7	even	4
704.6.a.k.1.1		2	8.3	odd	2
704.6.a.p.1.2		2	8.5	even	2
1078.6.a.h.1.1		2	28.27	even	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

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	176.6.a.f.1.1

Level	$176$
Weight	$6$
Character	176.1
Self dual	yes
Analytic conductor	$28.228$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$