LMFDB - Embedded newform 1232.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$72.6903531271$
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} - 40x^{2} - 41x + 194 $ x^4 - 40x^2 - 41x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.45730$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.15990 q^{3} -9.75471 q^{5} +7.00000 q^{7} +24.2641 q^{9} +O(q^{10})$	$q-7.15990 q^{3} -9.75471 q^{5} +7.00000 q^{7} +24.2641 q^{9} -11.0000 q^{11} +1.08384 q^{13} +69.8427 q^{15} -95.3803 q^{17} +77.1099 q^{19} -50.1193 q^{21} -114.111 q^{23} -29.8457 q^{25} +19.5886 q^{27} +280.084 q^{29} +233.754 q^{31} +78.7589 q^{33} -68.2830 q^{35} -294.993 q^{37} -7.76020 q^{39} -136.712 q^{41} +312.037 q^{43} -236.689 q^{45} +476.867 q^{47} +49.0000 q^{49} +682.913 q^{51} +283.013 q^{53} +107.302 q^{55} -552.099 q^{57} +706.222 q^{59} -25.3200 q^{61} +169.849 q^{63} -10.5726 q^{65} -1069.14 q^{67} +817.020 q^{69} +869.319 q^{71} +197.692 q^{73} +213.692 q^{75} -77.0000 q^{77} -821.822 q^{79} -795.384 q^{81} +668.476 q^{83} +930.407 q^{85} -2005.37 q^{87} +542.910 q^{89} +7.58690 q^{91} -1673.66 q^{93} -752.185 q^{95} -1702.09 q^{97} -266.905 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9	$ 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9} - 44 q^{11} + 2 q^{13} - 71 q^{15} + 8 q^{17} + 6 q^{19} + 21 q^{21} - 159 q^{23} - 119 q^{25} + 117 q^{27} + 144 q^{29} + 183 q^{31} - 33 q^{33} - 133 q^{35} - 475 q^{37} + 64 q^{39} - 768 q^{41} + 152 q^{43} - 1048 q^{45} + 228 q^{47} + 196 q^{49} + 882 q^{51} + 396 q^{53} + 209 q^{55} - 1286 q^{57} + 733 q^{59} - 1012 q^{61} + 259 q^{63} + 420 q^{65} + 171 q^{67} - 321 q^{69} + 1019 q^{71} - 1836 q^{73} + 1016 q^{75} - 308 q^{77} - 1374 q^{79} - 956 q^{81} + 1248 q^{83} + 46 q^{85} - 2238 q^{87} - 1401 q^{89} + 14 q^{91} - 1859 q^{93} - 238 q^{95} - 2559 q^{97} - 407 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9 - 44 * q^11 + 2 * q^13 - 71 * q^15 + 8 * q^17 + 6 * q^19 + 21 * q^21 - 159 * q^23 - 119 * q^25 + 117 * q^27 + 144 * q^29 + 183 * q^31 - 33 * q^33 - 133 * q^35 - 475 * q^37 + 64 * q^39 - 768 * q^41 + 152 * q^43 - 1048 * q^45 + 228 * q^47 + 196 * q^49 + 882 * q^51 + 396 * q^53 + 209 * q^55 - 1286 * q^57 + 733 * q^59 - 1012 * q^61 + 259 * q^63 + 420 * q^65 + 171 * q^67 - 321 * q^69 + 1019 * q^71 - 1836 * q^73 + 1016 * q^75 - 308 * q^77 - 1374 * q^79 - 956 * q^81 + 1248 * q^83 + 46 * q^85 - 2238 * q^87 - 1401 * q^89 + 14 * q^91 - 1859 * q^93 - 238 * q^95 - 2559 * q^97 - 407 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	−7.15990	−1.37792	−0.688961	−	0.724798i	$-0.741933\pi$
$3$	−7.15990	−1.37792	−0.688961	+	0.724798i	$0.741933\pi$
$4$	0	0
$5$	−9.75471	−0.872488	−0.436244	−	0.899828i	$-0.643691\pi$
$5$	−9.75471	−0.872488	−0.436244	+	0.899828i	$0.643691\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	24.2641	0.898671
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	1.08384	0.0231234	0.0115617	−	0.999933i	$-0.496320\pi$
$13$	1.08384	0.0231234	0.0115617	+	0.999933i	$0.496320\pi$
$14$	0	0
$15$	69.8427	1.20222
$16$	0	0
$17$	−95.3803	−1.36077	−0.680386	−	0.732854i	$-0.738188\pi$
$17$	−95.3803	−1.36077	−0.680386	+	0.732854i	$0.738188\pi$
$18$	0	0
$19$	77.1099	0.931065	0.465532	−	0.885031i	$-0.345863\pi$
$19$	77.1099	0.931065	0.465532	+	0.885031i	$0.345863\pi$
$20$	0	0
$21$	−50.1193	−0.520806
$22$	0	0
$23$	−114.111	−1.03451	−0.517255	−	0.855832i	$-0.673046\pi$
$23$	−114.111	−1.03451	−0.517255	+	0.855832i	$0.673046\pi$
$24$	0	0
$25$	−29.8457	−0.238765
$26$	0	0
$27$	19.5886	0.139623
$28$	0	0
$29$	280.084	1.79346	0.896728	−	0.442582i	$-0.145937\pi$
$29$	280.084	1.79346	0.896728	+	0.442582i	$0.145937\pi$
$30$	0	0
$31$	233.754	1.35431	0.677154	−	0.735842i	$-0.263213\pi$
$31$	233.754	1.35431	0.677154	+	0.735842i	$0.263213\pi$
$32$	0	0
$33$	78.7589	0.415459
$34$	0	0
$35$	−68.2830	−0.329769
$36$	0	0
$37$	−294.993	−1.31072	−0.655360	−	0.755317i	$-0.727483\pi$
$37$	−294.993	−1.31072	−0.655360	+	0.755317i	$0.727483\pi$
$38$	0	0
$39$	−7.76020	−0.0318622
$40$	0	0
$41$	−136.712	−0.520753	−0.260377	−	0.965507i	$-0.583847\pi$
$41$	−136.712	−0.520753	−0.260377	+	0.965507i	$0.583847\pi$
$42$	0	0
$43$	312.037	1.10663	0.553316	−	0.832971i	$-0.313362\pi$
$43$	312.037	1.10663	0.553316	+	0.832971i	$0.313362\pi$
$44$	0	0
$45$	−236.689	−0.784080
$46$	0	0
$47$	476.867	1.47996	0.739981	−	0.672628i	$-0.234835\pi$
$47$	476.867	1.47996	0.739981	+	0.672628i	$0.234835\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	682.913	1.87504
$52$	0	0
$53$	283.013	0.733488	0.366744	−	0.930322i	$-0.380473\pi$
$53$	283.013	0.733488	0.366744	+	0.930322i	$0.380473\pi$
$54$	0	0
$55$	107.302	0.263065
$56$	0	0
$57$	−552.099	−1.28294
$58$	0	0
$59$	706.222	1.55834	0.779172	−	0.626811i	$-0.215640\pi$
$59$	706.222	1.55834	0.779172	+	0.626811i	$0.215640\pi$
$60$	0	0
$61$	−25.3200	−0.0531458	−0.0265729	−	0.999647i	$-0.508459\pi$
$61$	−25.3200	−0.0531458	−0.0265729	+	0.999647i	$0.508459\pi$
$62$	0	0
$63$	169.849	0.339666
$64$	0	0
$65$	−10.5726	−0.0201749
$66$	0	0
$67$	−1069.14	−1.94949	−0.974745	−	0.223322i	$-0.928310\pi$
$67$	−1069.14	−1.94949	−0.974745	+	0.223322i	$0.928310\pi$
$68$	0	0
$69$	817.020	1.42547
$70$	0	0
$71$	869.319	1.45309	0.726544	−	0.687120i	$-0.241126\pi$
$71$	869.319	1.45309	0.726544	+	0.687120i	$0.241126\pi$
$72$	0	0
$73$	197.692	0.316961	0.158480	−	0.987362i	$-0.449341\pi$
$73$	197.692	0.316961	0.158480	+	0.987362i	$0.449341\pi$
$74$	0	0
$75$	213.692	0.329000
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−821.822	−1.17041	−0.585204	−	0.810886i	$-0.698985\pi$
$79$	−821.822	−1.17041	−0.585204	+	0.810886i	$0.698985\pi$
$80$	0	0
$81$	−795.384	−1.09106
$82$	0	0
$83$	668.476	0.884033	0.442016	−	0.897007i	$-0.354263\pi$
$83$	668.476	0.884033	0.442016	+	0.897007i	$0.354263\pi$
$84$	0	0
$85$	930.407	1.18726
$86$	0	0
$87$	−2005.37	−2.47124
$88$	0	0
$89$	542.910	0.646611	0.323306	−	0.946295i	$-0.395206\pi$
$89$	542.910	0.646611	0.323306	+	0.946295i	$0.395206\pi$
$90$	0	0
$91$	7.58690	0.00873981
$92$	0	0
$93$	−1673.66	−1.86613
$94$	0	0
$95$	−752.185	−0.812343
$96$	0	0
$97$	−1702.09	−1.78166	−0.890832	−	0.454332i	$-0.849878\pi$
$97$	−1702.09	−1.78166	−0.890832	+	0.454332i	$0.849878\pi$
$98$	0	0
$99$	−266.905	−0.270960
$100$	0	0
$101$	332.224	0.327302	0.163651	−	0.986518i	$-0.447673\pi$
$101$	332.224	0.327302	0.163651	+	0.986518i	$0.447673\pi$
$102$	0	0
$103$	−1697.14	−1.62354	−0.811769	−	0.583978i	$-0.801495\pi$
$103$	−1697.14	−1.62354	−0.811769	+	0.583978i	$0.801495\pi$
$104$	0	0
$105$	488.899	0.454397
$106$	0	0
$107$	882.363	0.797208	0.398604	−	0.917123i	$-0.369495\pi$
$107$	882.363	0.797208	0.398604	+	0.917123i	$0.369495\pi$
$108$	0	0
$109$	318.007	0.279445	0.139722	−	0.990191i	$-0.455379\pi$
$109$	318.007	0.279445	0.139722	+	0.990191i	$0.455379\pi$
$110$	0	0
$111$	2112.12	1.80607
$112$	0	0
$113$	−498.088	−0.414656	−0.207328	−	0.978271i	$-0.566477\pi$
$113$	−498.088	−0.414656	−0.207328	+	0.978271i	$0.566477\pi$
$114$	0	0
$115$	1113.12	0.902596
$116$	0	0
$117$	26.2985	0.0207803
$118$	0	0
$119$	−667.662	−0.514323
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	978.846	0.717558
$124$	0	0
$125$	1510.47	1.08081
$126$	0	0
$127$	−2109.76	−1.47410	−0.737050	−	0.675838i	$-0.763782\pi$
$127$	−2109.76	−1.47410	−0.737050	+	0.675838i	$0.763782\pi$
$128$	0	0
$129$	−2234.15	−1.52485
$130$	0	0
$131$	−2173.76	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$131$	−2173.76	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$132$	0	0
$133$	539.770	0.351909
$134$	0	0
$135$	−191.081	−0.121819
$136$	0	0
$137$	−853.116	−0.532019	−0.266010	−	0.963970i	$-0.585705\pi$
$137$	−853.116	−0.532019	−0.266010	+	0.963970i	$0.585705\pi$
$138$	0	0
$139$	2341.95	1.42907	0.714537	−	0.699597i	$-0.246637\pi$
$139$	2341.95	1.42907	0.714537	+	0.699597i	$0.246637\pi$
$140$	0	0
$141$	−3414.32	−2.03927
$142$	0	0
$143$	−11.9223	−0.00697196
$144$	0	0
$145$	−2732.13	−1.56477
$146$	0	0
$147$	−350.835	−0.196846
$148$	0	0
$149$	−1713.82	−0.942292	−0.471146	−	0.882055i	$-0.656160\pi$
$149$	−1713.82	−0.942292	−0.471146	+	0.882055i	$0.656160\pi$
$150$	0	0
$151$	−1790.24	−0.964819	−0.482410	−	0.875946i	$-0.660238\pi$
$151$	−1790.24	−0.964819	−0.482410	+	0.875946i	$0.660238\pi$
$152$	0	0
$153$	−2314.32	−1.22289
$154$	0	0
$155$	−2280.21	−1.18162
$156$	0	0
$157$	1262.21	0.641625	0.320813	−	0.947143i	$-0.396044\pi$
$157$	1262.21	0.641625	0.320813	+	0.947143i	$0.396044\pi$
$158$	0	0
$159$	−2026.34	−1.01069
$160$	0	0
$161$	−798.774	−0.391008
$162$	0	0
$163$	−2492.91	−1.19791	−0.598956	−	0.800782i	$-0.704418\pi$
$163$	−2492.91	−1.19791	−0.598956	+	0.800782i	$0.704418\pi$
$164$	0	0
$165$	−768.270	−0.362483
$166$	0	0
$167$	2792.85	1.29412	0.647059	−	0.762440i	$-0.275999\pi$
$167$	2792.85	1.29412	0.647059	+	0.762440i	$0.275999\pi$
$168$	0	0
$169$	−2195.83	−0.999465
$170$	0	0
$171$	1871.01	0.836721
$172$	0	0
$173$	544.150	0.239139	0.119569	−	0.992826i	$-0.461849\pi$
$173$	544.150	0.239139	0.119569	+	0.992826i	$0.461849\pi$
$174$	0	0
$175$	−208.920	−0.0902448
$176$	0	0
$177$	−5056.48	−2.14728
$178$	0	0
$179$	−1533.67	−0.640402	−0.320201	−	0.947350i	$-0.603750\pi$
$179$	−1533.67	−0.640402	−0.320201	+	0.947350i	$0.603750\pi$
$180$	0	0
$181$	−2738.18	−1.12446	−0.562231	−	0.826980i	$-0.690057\pi$
$181$	−2738.18	−1.12446	−0.562231	+	0.826980i	$0.690057\pi$
$182$	0	0
$183$	181.288	0.0732308
$184$	0	0
$185$	2877.57	1.14359
$186$	0	0
$187$	1049.18	0.410288
$188$	0	0
$189$	137.120	0.0527726
$190$	0	0
$191$	3576.01	1.35472	0.677358	−	0.735653i	$-0.263125\pi$
$191$	3576.01	1.35472	0.677358	+	0.735653i	$0.263125\pi$
$192$	0	0
$193$	986.164	0.367801	0.183901	−	0.982945i	$-0.441128\pi$
$193$	986.164	0.367801	0.183901	+	0.982945i	$0.441128\pi$
$194$	0	0
$195$	75.6985	0.0277994
$196$	0	0
$197$	−423.615	−0.153205	−0.0766024	−	0.997062i	$-0.524407\pi$
$197$	−423.615	−0.153205	−0.0766024	+	0.997062i	$0.524407\pi$
$198$	0	0
$199$	−2740.86	−0.976355	−0.488178	−	0.872744i	$-0.662338\pi$
$199$	−2740.86	−0.976355	−0.488178	+	0.872744i	$0.662338\pi$
$200$	0	0
$201$	7654.91	2.68625
$202$	0	0
$203$	1960.59	0.677863
$204$	0	0
$205$	1333.59	0.454351
$206$	0	0
$207$	−2768.79	−0.929684
$208$	0	0
$209$	−848.209	−0.280727
$210$	0	0
$211$	2367.28	0.772370	0.386185	−	0.922421i	$-0.373793\pi$
$211$	2367.28	0.772370	0.386185	+	0.922421i	$0.373793\pi$
$212$	0	0
$213$	−6224.23	−2.00224
$214$	0	0
$215$	−3043.83	−0.965523
$216$	0	0
$217$	1636.28	0.511880
$218$	0	0
$219$	−1415.46	−0.436747
$220$	0	0
$221$	−103.377	−0.0314656
$222$	0	0
$223$	−1052.10	−0.315935	−0.157968	−	0.987444i	$-0.550494\pi$
$223$	−1052.10	−0.315935	−0.157968	+	0.987444i	$0.550494\pi$
$224$	0	0
$225$	−724.179	−0.214572
$226$	0	0
$227$	−4055.84	−1.18588	−0.592941	−	0.805246i	$-0.702033\pi$
$227$	−4055.84	−1.18588	−0.592941	+	0.805246i	$0.702033\pi$
$228$	0	0
$229$	−833.574	−0.240542	−0.120271	−	0.992741i	$-0.538376\pi$
$229$	−833.574	−0.240542	−0.120271	+	0.992741i	$0.538376\pi$
$230$	0	0
$231$	551.312	0.157029
$232$	0	0
$233$	−1994.64	−0.560829	−0.280414	−	0.959879i	$-0.590472\pi$
$233$	−1994.64	−0.560829	−0.280414	+	0.959879i	$0.590472\pi$
$234$	0	0
$235$	−4651.70	−1.29125
$236$	0	0
$237$	5884.16	1.61273
$238$	0	0
$239$	899.226	0.243373	0.121686	−	0.992569i	$-0.461170\pi$
$239$	899.226	0.243373	0.121686	+	0.992569i	$0.461170\pi$
$240$	0	0
$241$	3812.38	1.01899	0.509496	−	0.860473i	$-0.329832\pi$
$241$	3812.38	1.01899	0.509496	+	0.860473i	$0.329832\pi$
$242$	0	0
$243$	5165.97	1.36378
$244$	0	0
$245$	−477.981	−0.124641
$246$	0	0
$247$	83.5750	0.0215294
$248$	0	0
$249$	−4786.22	−1.21813
$250$	0	0
$251$	−1704.40	−0.428610	−0.214305	−	0.976767i	$-0.568749\pi$
$251$	−1704.40	−0.428610	−0.214305	+	0.976767i	$0.568749\pi$
$252$	0	0
$253$	1255.22	0.311916
$254$	0	0
$255$	−6661.62	−1.63595
$256$	0	0
$257$	−2353.79	−0.571304	−0.285652	−	0.958333i	$-0.592210\pi$
$257$	−2353.79	−0.571304	−0.285652	+	0.958333i	$0.592210\pi$
$258$	0	0
$259$	−2064.95	−0.495405
$260$	0	0
$261$	6795.99	1.61173
$262$	0	0
$263$	−866.133	−0.203072	−0.101536	−	0.994832i	$-0.532376\pi$
$263$	−866.133	−0.203072	−0.101536	+	0.994832i	$0.532376\pi$
$264$	0	0
$265$	−2760.71	−0.639959
$266$	0	0
$267$	−3887.18	−0.890980
$268$	0	0
$269$	4617.66	1.04663	0.523315	−	0.852139i	$-0.324695\pi$
$269$	4617.66	1.04663	0.523315	+	0.852139i	$0.324695\pi$
$270$	0	0
$271$	−6168.39	−1.38267	−0.691333	−	0.722536i	$-0.742976\pi$
$271$	−6168.39	−1.38267	−0.691333	+	0.722536i	$0.742976\pi$
$272$	0	0
$273$	−54.3214	−0.0120428
$274$	0	0
$275$	328.302	0.0719904
$276$	0	0
$277$	2189.29	0.474879	0.237439	−	0.971402i	$-0.423692\pi$
$277$	2189.29	0.474879	0.237439	+	0.971402i	$0.423692\pi$
$278$	0	0
$279$	5671.85	1.21708
$280$	0	0
$281$	−6761.80	−1.43550	−0.717749	−	0.696302i	$-0.754828\pi$
$281$	−6761.80	−1.43550	−0.717749	+	0.696302i	$0.754828\pi$
$282$	0	0
$283$	−7928.96	−1.66547	−0.832734	−	0.553673i	$-0.813226\pi$
$283$	−7928.96	−1.66547	−0.832734	+	0.553673i	$0.813226\pi$
$284$	0	0
$285$	5385.57	1.11935
$286$	0	0
$287$	−956.986	−0.196826
$288$	0	0
$289$	4184.40	0.851700
$290$	0	0
$291$	12186.8	2.45500
$292$	0	0
$293$	1919.31	0.382687	0.191343	−	0.981523i	$-0.438716\pi$
$293$	1919.31	0.382687	0.191343	+	0.981523i	$0.438716\pi$
$294$	0	0
$295$	−6888.99	−1.35964
$296$	0	0
$297$	−215.474	−0.0420979
$298$	0	0
$299$	−123.678	−0.0239213
$300$	0	0
$301$	2184.26	0.418268
$302$	0	0
$303$	−2378.69	−0.450997
$304$	0	0
$305$	246.989	0.0463690
$306$	0	0
$307$	−4337.93	−0.806445	−0.403222	−	0.915102i	$-0.632110\pi$
$307$	−4337.93	−0.806445	−0.403222	+	0.915102i	$0.632110\pi$
$308$	0	0
$309$	12151.4	2.23711
$310$	0	0
$311$	3519.75	0.641757	0.320879	−	0.947120i	$-0.396022\pi$
$311$	3519.75	0.641757	0.320879	+	0.947120i	$0.396022\pi$
$312$	0	0
$313$	5826.02	1.05210	0.526048	−	0.850455i	$-0.323673\pi$
$313$	5826.02	1.05210	0.526048	+	0.850455i	$0.323673\pi$
$314$	0	0
$315$	−1656.83	−0.296354
$316$	0	0
$317$	1563.64	0.277044	0.138522	−	0.990359i	$-0.455765\pi$
$317$	1563.64	0.277044	0.138522	+	0.990359i	$0.455765\pi$
$318$	0	0
$319$	−3080.92	−0.540747
$320$	0	0
$321$	−6317.63	−1.09849
$322$	0	0
$323$	−7354.77	−1.26697
$324$	0	0
$325$	−32.3480	−0.00552106
$326$	0	0
$327$	−2276.89	−0.385054
$328$	0	0
$329$	3338.07	0.559373
$330$	0	0
$331$	5218.91	0.866639	0.433319	−	0.901240i	$-0.357342\pi$
$331$	5218.91	0.866639	0.433319	+	0.901240i	$0.357342\pi$
$332$	0	0
$333$	−7157.76	−1.17791
$334$	0	0
$335$	10429.1	1.70091
$336$	0	0
$337$	−777.400	−0.125661	−0.0628304	−	0.998024i	$-0.520013\pi$
$337$	−777.400	−0.125661	−0.0628304	+	0.998024i	$0.520013\pi$
$338$	0	0
$339$	3566.26	0.571364
$340$	0	0
$341$	−2571.30	−0.408339
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−7969.80	−1.24371
$346$	0	0
$347$	−10578.7	−1.63658	−0.818291	−	0.574804i	$-0.805078\pi$
$347$	−10578.7	−1.63658	−0.818291	+	0.574804i	$0.805078\pi$
$348$	0	0
$349$	1560.01	0.239270	0.119635	−	0.992818i	$-0.461828\pi$
$349$	1560.01	0.239270	0.119635	+	0.992818i	$0.461828\pi$
$350$	0	0
$351$	21.2309	0.00322856
$352$	0	0
$353$	5901.06	0.889750	0.444875	−	0.895593i	$-0.353248\pi$
$353$	5901.06	0.889750	0.444875	+	0.895593i	$0.353248\pi$
$354$	0	0
$355$	−8479.95	−1.26780
$356$	0	0
$357$	4780.39	0.708698
$358$	0	0
$359$	−8015.27	−1.17836	−0.589178	−	0.808003i	$-0.700548\pi$
$359$	−8015.27	−1.17836	−0.589178	+	0.808003i	$0.700548\pi$
$360$	0	0
$361$	−913.057	−0.133118
$362$	0	0
$363$	−866.348	−0.125266
$364$	0	0
$365$	−1928.43	−0.276544
$366$	0	0
$367$	11606.1	1.65077	0.825383	−	0.564573i	$-0.190959\pi$
$367$	11606.1	1.65077	0.825383	+	0.564573i	$0.190959\pi$
$368$	0	0
$369$	−3317.21	−0.467986
$370$	0	0
$371$	1981.09	0.277232
$372$	0	0
$373$	−5056.08	−0.701860	−0.350930	−	0.936402i	$-0.614134\pi$
$373$	−5056.08	−0.701860	−0.350930	+	0.936402i	$0.614134\pi$
$374$	0	0
$375$	−10814.8	−1.48927
$376$	0	0
$377$	303.567	0.0414708
$378$	0	0
$379$	10463.1	1.41809	0.709044	−	0.705164i	$-0.249127\pi$
$379$	10463.1	1.41809	0.709044	+	0.705164i	$0.249127\pi$
$380$	0	0
$381$	15105.7	2.03120
$382$	0	0
$383$	3909.72	0.521612	0.260806	−	0.965391i	$-0.416012\pi$
$383$	3909.72	0.521612	0.260806	+	0.965391i	$0.416012\pi$
$384$	0	0
$385$	751.113	0.0994292
$386$	0	0
$387$	7571.30	0.994498
$388$	0	0
$389$	−14945.5	−1.94799	−0.973993	−	0.226577i	$-0.927247\pi$
$389$	−14945.5	−1.94799	−0.973993	+	0.226577i	$0.927247\pi$
$390$	0	0
$391$	10883.9	1.40773
$392$	0	0
$393$	15563.9	1.99769
$394$	0	0
$395$	8016.63	1.02117
$396$	0	0
$397$	−7882.94	−0.996558	−0.498279	−	0.867017i	$-0.666034\pi$
$397$	−7882.94	−0.996558	−0.498279	+	0.867017i	$0.666034\pi$
$398$	0	0
$399$	−3864.69	−0.484904
$400$	0	0
$401$	−848.714	−0.105693	−0.0528463	−	0.998603i	$-0.516829\pi$
$401$	−848.714	−0.105693	−0.0528463	+	0.998603i	$0.516829\pi$
$402$	0	0
$403$	253.353	0.0313161
$404$	0	0
$405$	7758.74	0.951937
$406$	0	0
$407$	3244.93	0.395197
$408$	0	0
$409$	−9172.72	−1.10895	−0.554477	−	0.832199i	$-0.687081\pi$
$409$	−9172.72	−1.10895	−0.554477	+	0.832199i	$0.687081\pi$
$410$	0	0
$411$	6108.22	0.733081
$412$	0	0
$413$	4943.55	0.588998
$414$	0	0
$415$	−6520.78	−0.771308
$416$	0	0
$417$	−16768.1	−1.96915
$418$	0	0
$419$	7743.61	0.902865	0.451432	−	0.892305i	$-0.350913\pi$
$419$	7743.61	0.902865	0.451432	+	0.892305i	$0.350913\pi$
$420$	0	0
$421$	13818.3	1.59967	0.799835	−	0.600219i	$-0.204920\pi$
$421$	13818.3	1.59967	0.799835	+	0.600219i	$0.204920\pi$
$422$	0	0
$423$	11570.8	1.33000
$424$	0	0
$425$	2846.69	0.324905
$426$	0	0
$427$	−177.240	−0.0200872
$428$	0	0
$429$	85.3622	0.00960682
$430$	0	0
$431$	733.023	0.0819223	0.0409611	−	0.999161i	$-0.486958\pi$
$431$	733.023	0.0819223	0.0409611	+	0.999161i	$0.486958\pi$
$432$	0	0
$433$	−4372.26	−0.485259	−0.242630	−	0.970119i	$-0.578010\pi$
$433$	−4372.26	−0.485259	−0.242630	+	0.970119i	$0.578010\pi$
$434$	0	0
$435$	19561.8	2.15613
$436$	0	0
$437$	−8799.06	−0.963195
$438$	0	0
$439$	−36.1532	−0.00393052	−0.00196526	−	0.999998i	$-0.500626\pi$
$439$	−36.1532	−0.00393052	−0.00196526	+	0.999998i	$0.500626\pi$
$440$	0	0
$441$	1188.94	0.128382
$442$	0	0
$443$	15714.0	1.68532	0.842658	−	0.538449i	$-0.180990\pi$
$443$	15714.0	1.68532	0.842658	+	0.538449i	$0.180990\pi$
$444$	0	0
$445$	−5295.93	−0.564160
$446$	0	0
$447$	12270.8	1.29841
$448$	0	0
$449$	10988.2	1.15493	0.577466	−	0.816415i	$-0.304042\pi$
$449$	10988.2	1.15493	0.577466	+	0.816415i	$0.304042\pi$
$450$	0	0
$451$	1503.84	0.157013
$452$	0	0
$453$	12817.9	1.32945
$454$	0	0
$455$	−74.0080	−0.00762538
$456$	0	0
$457$	1944.56	0.199043	0.0995214	−	0.995035i	$-0.468269\pi$
$457$	1944.56	0.199043	0.0995214	+	0.995035i	$0.468269\pi$
$458$	0	0
$459$	−1868.36	−0.189995
$460$	0	0
$461$	−13379.0	−1.35168	−0.675839	−	0.737049i	$-0.736219\pi$
$461$	−13379.0	−1.35168	−0.675839	+	0.737049i	$0.736219\pi$
$462$	0	0
$463$	−4430.09	−0.444674	−0.222337	−	0.974970i	$-0.571368\pi$
$463$	−4430.09	−0.444674	−0.222337	+	0.974970i	$0.571368\pi$
$464$	0	0
$465$	16326.0	1.62818
$466$	0	0
$467$	14258.8	1.41289	0.706445	−	0.707768i	$-0.250298\pi$
$467$	14258.8	1.41289	0.706445	+	0.707768i	$0.250298\pi$
$468$	0	0
$469$	−7483.96	−0.736838
$470$	0	0
$471$	−9037.28	−0.884110
$472$	0	0
$473$	−3432.41	−0.333662
$474$	0	0
$475$	−2301.40	−0.222306
$476$	0	0
$477$	6867.07	0.659164
$478$	0	0
$479$	−8237.70	−0.785783	−0.392891	−	0.919585i	$-0.628525\pi$
$479$	−8237.70	−0.785783	−0.392891	+	0.919585i	$0.628525\pi$
$480$	0	0
$481$	−319.726	−0.0303083
$482$	0	0
$483$	5719.14	0.538778
$484$	0	0
$485$	16603.4	1.55448
$486$	0	0
$487$	−8259.29	−0.768509	−0.384255	−	0.923227i	$-0.625542\pi$
$487$	−8259.29	−0.768509	−0.384255	+	0.923227i	$0.625542\pi$
$488$	0	0
$489$	17849.0	1.65063
$490$	0	0
$491$	−12076.1	−1.10996	−0.554978	−	0.831865i	$-0.687273\pi$
$491$	−12076.1	−1.10996	−0.554978	+	0.831865i	$0.687273\pi$
$492$	0	0
$493$	−26714.5	−2.44048
$494$	0	0
$495$	2603.58	0.236409
$496$	0	0
$497$	6085.23	0.549215
$498$	0	0
$499$	−16936.7	−1.51942	−0.759708	−	0.650264i	$-0.774658\pi$
$499$	−16936.7	−1.51942	−0.759708	+	0.650264i	$0.774658\pi$
$500$	0	0
$501$	−19996.5	−1.78319
$502$	0	0
$503$	−5105.04	−0.452529	−0.226265	−	0.974066i	$-0.572651\pi$
$503$	−5105.04	−0.452529	−0.226265	+	0.974066i	$0.572651\pi$
$504$	0	0
$505$	−3240.75	−0.285567
$506$	0	0
$507$	15721.9	1.37719
$508$	0	0
$509$	−7260.32	−0.632236	−0.316118	−	0.948720i	$-0.602379\pi$
$509$	−7260.32	−0.632236	−0.316118	+	0.948720i	$0.602379\pi$
$510$	0	0
$511$	1383.85	0.119800
$512$	0	0
$513$	1510.47	0.129998
$514$	0	0
$515$	16555.1	1.41652
$516$	0	0
$517$	−5245.54	−0.446225
$518$	0	0
$519$	−3896.06	−0.329514
$520$	0	0
$521$	−5577.37	−0.469000	−0.234500	−	0.972116i	$-0.575345\pi$
$521$	−5577.37	−0.469000	−0.234500	+	0.972116i	$0.575345\pi$
$522$	0	0
$523$	−4182.37	−0.349679	−0.174840	−	0.984597i	$-0.555941\pi$
$523$	−4182.37	−0.349679	−0.174840	+	0.984597i	$0.555941\pi$
$524$	0	0
$525$	1495.84	0.124350
$526$	0	0
$527$	−22295.6	−1.84290
$528$	0	0
$529$	854.234	0.0702091
$530$	0	0
$531$	17135.9	1.40044
$532$	0	0
$533$	−148.175	−0.0120416
$534$	0	0
$535$	−8607.20	−0.695554
$536$	0	0
$537$	10980.9	0.882425
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	2659.40	0.211343	0.105671	−	0.994401i	$-0.466301\pi$
$541$	2659.40	0.211343	0.105671	+	0.994401i	$0.466301\pi$
$542$	0	0
$543$	19605.1	1.54942
$544$	0	0
$545$	−3102.06	−0.243812
$546$	0	0
$547$	19521.7	1.52594	0.762969	−	0.646435i	$-0.223741\pi$
$547$	19521.7	1.52594	0.762969	+	0.646435i	$0.223741\pi$
$548$	0	0
$549$	−614.367	−0.0477606
$550$	0	0
$551$	21597.2	1.66982
$552$	0	0
$553$	−5752.75	−0.442372
$554$	0	0
$555$	−20603.1	−1.57577
$556$	0	0
$557$	−4690.05	−0.356775	−0.178388	−	0.983960i	$-0.557088\pi$
$557$	−4690.05	−0.356775	−0.178388	+	0.983960i	$0.557088\pi$
$558$	0	0
$559$	338.199	0.0255891
$560$	0	0
$561$	−7512.04	−0.565345
$562$	0	0
$563$	15363.0	1.15004	0.575020	−	0.818139i	$-0.304994\pi$
$563$	15363.0	1.15004	0.575020	+	0.818139i	$0.304994\pi$
$564$	0	0
$565$	4858.70	0.361782
$566$	0	0
$567$	−5567.69	−0.412382
$568$	0	0
$569$	17909.7	1.31954	0.659768	−	0.751470i	$-0.270655\pi$
$569$	17909.7	1.31954	0.659768	+	0.751470i	$0.270655\pi$
$570$	0	0
$571$	−18507.7	−1.35643	−0.678216	−	0.734863i	$-0.737247\pi$
$571$	−18507.7	−1.35643	−0.678216	+	0.734863i	$0.737247\pi$
$572$	0	0
$573$	−25603.9	−1.86670
$574$	0	0
$575$	3405.71	0.247005
$576$	0	0
$577$	−3224.97	−0.232682	−0.116341	−	0.993209i	$-0.537117\pi$
$577$	−3224.97	−0.232682	−0.116341	+	0.993209i	$0.537117\pi$
$578$	0	0
$579$	−7060.83	−0.506802
$580$	0	0
$581$	4679.33	0.334133
$582$	0	0
$583$	−3113.14	−0.221155
$584$	0	0
$585$	−256.534	−0.0181306
$586$	0	0
$587$	7462.89	0.524747	0.262373	−	0.964966i	$-0.415495\pi$
$587$	7462.89	0.524747	0.262373	+	0.964966i	$0.415495\pi$
$588$	0	0
$589$	18024.8	1.26095
$590$	0	0
$591$	3033.04	0.211104
$592$	0	0
$593$	−3750.47	−0.259719	−0.129859	−	0.991532i	$-0.541453\pi$
$593$	−3750.47	−0.259719	−0.129859	+	0.991532i	$0.541453\pi$
$594$	0	0
$595$	6512.85	0.448741
$596$	0	0
$597$	19624.3	1.34534
$598$	0	0
$599$	−10841.7	−0.739532	−0.369766	−	0.929125i	$-0.620562\pi$
$599$	−10841.7	−0.739532	−0.369766	+	0.929125i	$0.620562\pi$
$600$	0	0
$601$	6270.54	0.425592	0.212796	−	0.977097i	$-0.431743\pi$
$601$	6270.54	0.425592	0.212796	+	0.977097i	$0.431743\pi$
$602$	0	0
$603$	−25941.7	−1.75195
$604$	0	0
$605$	−1180.32	−0.0793171
$606$	0	0
$607$	−6955.15	−0.465075	−0.232538	−	0.972587i	$-0.574703\pi$
$607$	−6955.15	−0.465075	−0.232538	+	0.972587i	$0.574703\pi$
$608$	0	0
$609$	−14037.6	−0.934043
$610$	0	0
$611$	516.849	0.0342217
$612$	0	0
$613$	−16284.7	−1.07297	−0.536487	−	0.843908i	$-0.680249\pi$
$613$	−16284.7	−1.07297	−0.536487	+	0.843908i	$0.680249\pi$
$614$	0	0
$615$	−9548.36	−0.626060
$616$	0	0
$617$	−437.340	−0.0285359	−0.0142680	−	0.999898i	$-0.504542\pi$
$617$	−437.340	−0.0285359	−0.0142680	+	0.999898i	$0.504542\pi$
$618$	0	0
$619$	11211.3	0.727984	0.363992	−	0.931402i	$-0.381414\pi$
$619$	11211.3	0.727984	0.363992	+	0.931402i	$0.381414\pi$
$620$	0	0
$621$	−2235.26	−0.144441
$622$	0	0
$623$	3800.37	0.244396
$624$	0	0
$625$	−11003.5	−0.704226
$626$	0	0
$627$	6073.09	0.386820
$628$	0	0
$629$	28136.6	1.78359
$630$	0	0
$631$	3651.85	0.230393	0.115196	−	0.993343i	$-0.463250\pi$
$631$	3651.85	0.230393	0.115196	+	0.993343i	$0.463250\pi$
$632$	0	0
$633$	−16949.5	−1.06427
$634$	0	0
$635$	20580.1	1.28613
$636$	0	0
$637$	53.1083	0.00330334
$638$	0	0
$639$	21093.3	1.30585
$640$	0	0
$641$	−16162.1	−0.995889	−0.497944	−	0.867209i	$-0.665912\pi$
$641$	−16162.1	−0.995889	−0.497944	+	0.867209i	$0.665912\pi$
$642$	0	0
$643$	30147.3	1.84898	0.924489	−	0.381210i	$-0.124492\pi$
$643$	30147.3	1.84898	0.924489	+	0.381210i	$0.124492\pi$
$644$	0	0
$645$	21793.5	1.33042
$646$	0	0
$647$	11426.9	0.694338	0.347169	−	0.937803i	$-0.387143\pi$
$647$	11426.9	0.694338	0.347169	+	0.937803i	$0.387143\pi$
$648$	0	0
$649$	−7768.44	−0.469858
$650$	0	0
$651$	−11715.6	−0.705331
$652$	0	0
$653$	−31258.9	−1.87329	−0.936644	−	0.350283i	$-0.886085\pi$
$653$	−31258.9	−1.87329	−0.936644	+	0.350283i	$0.886085\pi$
$654$	0	0
$655$	21204.3	1.26492
$656$	0	0
$657$	4796.83	0.284843
$658$	0	0
$659$	2428.13	0.143530	0.0717650	−	0.997422i	$-0.477137\pi$
$659$	2428.13	0.143530	0.0717650	+	0.997422i	$0.477137\pi$
$660$	0	0
$661$	15005.5	0.882976	0.441488	−	0.897267i	$-0.354451\pi$
$661$	15005.5	0.882976	0.441488	+	0.897267i	$0.354451\pi$
$662$	0	0
$663$	740.170	0.0433572
$664$	0	0
$665$	−5265.30	−0.307037
$666$	0	0
$667$	−31960.5	−1.85535
$668$	0	0
$669$	7532.90	0.435334
$670$	0	0
$671$	278.520	0.0160240
$672$	0	0
$673$	28297.9	1.62081	0.810405	−	0.585871i	$-0.199247\pi$
$673$	28297.9	1.62081	0.810405	+	0.585871i	$0.199247\pi$
$674$	0	0
$675$	−584.634	−0.0333371
$676$	0	0
$677$	18256.0	1.03639	0.518193	−	0.855264i	$-0.326605\pi$
$677$	18256.0	1.03639	0.518193	+	0.855264i	$0.326605\pi$
$678$	0	0
$679$	−11914.7	−0.673406
$680$	0	0
$681$	29039.4	1.63405
$682$	0	0
$683$	15756.8	0.882751	0.441376	−	0.897322i	$-0.354491\pi$
$683$	15756.8	0.882751	0.441376	+	0.897322i	$0.354491\pi$
$684$	0	0
$685$	8321.90	0.464180
$686$	0	0
$687$	5968.30	0.331448
$688$	0	0
$689$	306.742	0.0169607
$690$	0	0
$691$	795.332	0.0437856	0.0218928	−	0.999760i	$-0.493031\pi$
$691$	795.332	0.0437856	0.0218928	+	0.999760i	$0.493031\pi$
$692$	0	0
$693$	−1868.34	−0.102413
$694$	0	0
$695$	−22845.0	−1.24685
$696$	0	0
$697$	13039.7	0.708626
$698$	0	0
$699$	14281.4	0.772779
$700$	0	0
$701$	−32670.8	−1.76029	−0.880143	−	0.474708i	$-0.842554\pi$
$701$	−32670.8	−1.76029	−0.880143	+	0.474708i	$0.842554\pi$
$702$	0	0
$703$	−22746.9	−1.22036
$704$	0	0
$705$	33305.7	1.77924
$706$	0	0
$707$	2325.57	0.123709
$708$	0	0
$709$	−26577.0	−1.40779	−0.703893	−	0.710306i	$-0.748556\pi$
$709$	−26577.0	−1.40779	−0.703893	+	0.710306i	$0.748556\pi$
$710$	0	0
$711$	−19940.8	−1.05181
$712$	0	0
$713$	−26673.9	−1.40104
$714$	0	0
$715$	116.298	0.00608295
$716$	0	0
$717$	−6438.37	−0.335349
$718$	0	0
$719$	−21991.9	−1.14070	−0.570348	−	0.821403i	$-0.693192\pi$
$719$	−21991.9	−1.14070	−0.570348	+	0.821403i	$0.693192\pi$
$720$	0	0
$721$	−11880.0	−0.613640
$722$	0	0
$723$	−27296.2	−1.40409
$724$	0	0
$725$	−8359.28	−0.428215
$726$	0	0
$727$	−6400.89	−0.326542	−0.163271	−	0.986581i	$-0.552204\pi$
$727$	−6400.89	−0.326542	−0.163271	+	0.986581i	$0.552204\pi$
$728$	0	0
$729$	−15512.5	−0.788116
$730$	0	0
$731$	−29762.2	−1.50587
$732$	0	0
$733$	−28093.4	−1.41563	−0.707813	−	0.706400i	$-0.750318\pi$
$733$	−28093.4	−1.41563	−0.707813	+	0.706400i	$0.750318\pi$
$734$	0	0
$735$	3422.29	0.171746
$736$	0	0
$737$	11760.5	0.587793
$738$	0	0
$739$	−20645.8	−1.02770	−0.513848	−	0.857881i	$-0.671781\pi$
$739$	−20645.8	−1.02770	−0.513848	+	0.857881i	$0.671781\pi$
$740$	0	0
$741$	−598.389	−0.0296658
$742$	0	0
$743$	32127.5	1.58633	0.793165	−	0.609007i	$-0.208432\pi$
$743$	32127.5	1.58633	0.793165	+	0.609007i	$0.208432\pi$
$744$	0	0
$745$	16717.8	0.822138
$746$	0	0
$747$	16220.0	0.794455
$748$	0	0
$749$	6176.54	0.301316
$750$	0	0
$751$	18959.0	0.921205	0.460602	−	0.887607i	$-0.347633\pi$
$751$	18959.0	0.921205	0.460602	+	0.887607i	$0.347633\pi$
$752$	0	0
$753$	12203.4	0.590591
$754$	0	0
$755$	17463.3	0.841793
$756$	0	0
$757$	14050.2	0.674588	0.337294	−	0.941399i	$-0.390488\pi$
$757$	14050.2	0.674588	0.337294	+	0.941399i	$0.390488\pi$
$758$	0	0
$759$	−8987.22	−0.429796
$760$	0	0
$761$	7949.74	0.378683	0.189342	−	0.981911i	$-0.439365\pi$
$761$	7949.74	0.378683	0.189342	+	0.981911i	$0.439365\pi$
$762$	0	0
$763$	2226.05	0.105620
$764$	0	0
$765$	22575.5	1.06695
$766$	0	0
$767$	765.433	0.0360341
$768$	0	0
$769$	−7113.87	−0.333593	−0.166796	−	0.985991i	$-0.553342\pi$
$769$	−7113.87	−0.333593	−0.166796	+	0.985991i	$0.553342\pi$
$770$	0	0
$771$	16852.9	0.787213
$772$	0	0
$773$	−32090.0	−1.49314	−0.746571	−	0.665306i	$-0.768301\pi$
$773$	−32090.0	−1.49314	−0.746571	+	0.665306i	$0.768301\pi$
$774$	0	0
$775$	−6976.55	−0.323362
$776$	0	0
$777$	14784.9	0.682630
$778$	0	0
$779$	−10541.9	−0.484855
$780$	0	0
$781$	−9562.51	−0.438122
$782$	0	0
$783$	5486.44	0.250408
$784$	0	0
$785$	−12312.5	−0.559810
$786$	0	0
$787$	16618.2	0.752701	0.376351	−	0.926477i	$-0.377179\pi$
$787$	16618.2	0.752701	0.376351	+	0.926477i	$0.377179\pi$
$788$	0	0
$789$	6201.42	0.279818
$790$	0	0
$791$	−3486.61	−0.156725
$792$	0	0
$793$	−27.4429	−0.00122891
$794$	0	0
$795$	19766.4	0.881814
$796$	0	0
$797$	−40615.3	−1.80510	−0.902552	−	0.430581i	$-0.858309\pi$
$797$	−40615.3	−1.80510	−0.902552	+	0.430581i	$0.858309\pi$
$798$	0	0
$799$	−45483.7	−2.01389
$800$	0	0
$801$	13173.2	0.581091
$802$	0	0
$803$	−2174.61	−0.0955672
$804$	0	0
$805$	7791.81	0.341149
$806$	0	0
$807$	−33062.0	−1.44218
$808$	0	0
$809$	−9592.65	−0.416885	−0.208442	−	0.978035i	$-0.566839\pi$
$809$	−9592.65	−0.416885	−0.208442	+	0.978035i	$0.566839\pi$
$810$	0	0
$811$	26807.1	1.16070	0.580349	−	0.814368i	$-0.302916\pi$
$811$	26807.1	1.16070	0.580349	+	0.814368i	$0.302916\pi$
$812$	0	0
$813$	44165.0	1.90521
$814$	0	0
$815$	24317.6	1.04516
$816$	0	0
$817$	24061.1	1.03035
$818$	0	0
$819$	184.089	0.00785422
$820$	0	0
$821$	−19266.7	−0.819016	−0.409508	−	0.912307i	$-0.634300\pi$
$821$	−19266.7	−0.819016	−0.409508	+	0.912307i	$0.634300\pi$
$822$	0	0
$823$	−23426.1	−0.992201	−0.496101	−	0.868265i	$-0.665235\pi$
$823$	−23426.1	−0.992201	−0.496101	+	0.868265i	$0.665235\pi$
$824$	0	0
$825$	−2350.61	−0.0991973
$826$	0	0
$827$	18937.9	0.796294	0.398147	−	0.917322i	$-0.369653\pi$
$827$	18937.9	0.796294	0.398147	+	0.917322i	$0.369653\pi$
$828$	0	0
$829$	−36892.2	−1.54562	−0.772810	−	0.634637i	$-0.781150\pi$
$829$	−36892.2	−1.54562	−0.772810	+	0.634637i	$0.781150\pi$
$830$	0	0
$831$	−15675.1	−0.654346
$832$	0	0
$833$	−4673.63	−0.194396
$834$	0	0
$835$	−27243.5	−1.12910
$836$	0	0
$837$	4578.92	0.189093
$838$	0	0
$839$	−89.1613	−0.00366888	−0.00183444	−	0.999998i	$-0.500584\pi$
$839$	−89.1613	−0.00366888	−0.00183444	+	0.999998i	$0.500584\pi$
$840$	0	0
$841$	54057.9	2.21649
$842$	0	0
$843$	48413.8	1.97801
$844$	0	0
$845$	21419.6	0.872021
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	56770.5	2.29489
$850$	0	0
$851$	33661.9	1.35595
$852$	0	0
$853$	−19646.3	−0.788603	−0.394301	−	0.918981i	$-0.629013\pi$
$853$	−19646.3	−0.788603	−0.394301	+	0.918981i	$0.629013\pi$
$854$	0	0
$855$	−18251.1	−0.730029
$856$	0	0
$857$	−35839.9	−1.42855	−0.714275	−	0.699865i	$-0.753243\pi$
$857$	−35839.9	−1.42855	−0.714275	+	0.699865i	$0.753243\pi$
$858$	0	0
$859$	18987.2	0.754174	0.377087	−	0.926178i	$-0.376926\pi$
$859$	18987.2	0.754174	0.377087	+	0.926178i	$0.376926\pi$
$860$	0	0
$861$	6851.92	0.271211
$862$	0	0
$863$	−14677.9	−0.578958	−0.289479	−	0.957184i	$-0.593482\pi$
$863$	−14677.9	−0.578958	−0.289479	+	0.957184i	$0.593482\pi$
$864$	0	0
$865$	−5308.03	−0.208645
$866$	0	0
$867$	−29959.9	−1.17358
$868$	0	0
$869$	9040.04	0.352891
$870$	0	0
$871$	−1158.78	−0.0450788
$872$	0	0
$873$	−41299.8	−1.60113
$874$	0	0
$875$	10573.3	0.408507
$876$	0	0
$877$	25680.2	0.988780	0.494390	−	0.869240i	$-0.335391\pi$
$877$	25680.2	0.988780	0.494390	+	0.869240i	$0.335391\pi$
$878$	0	0
$879$	−13742.0	−0.527313
$880$	0	0
$881$	−17037.7	−0.651551	−0.325775	−	0.945447i	$-0.605625\pi$
$881$	−17037.7	−0.651551	−0.325775	+	0.945447i	$0.605625\pi$
$882$	0	0
$883$	26504.8	1.01015	0.505073	−	0.863077i	$-0.331466\pi$
$883$	26504.8	1.01015	0.505073	+	0.863077i	$0.331466\pi$
$884$	0	0
$885$	49324.4	1.87347
$886$	0	0
$887$	−13417.5	−0.507909	−0.253955	−	0.967216i	$-0.581731\pi$
$887$	−13417.5	−0.507909	−0.253955	+	0.967216i	$0.581731\pi$
$888$	0	0
$889$	−14768.3	−0.557158
$890$	0	0
$891$	8749.22	0.328967
$892$	0	0
$893$	36771.2	1.37794
$894$	0	0
$895$	14960.5	0.558743
$896$	0	0
$897$	885.521	0.0329618
$898$	0	0
$899$	65470.8	2.42889
$900$	0	0
$901$	−26993.9	−0.998109
$902$	0	0
$903$	−15639.1	−0.576340
$904$	0	0
$905$	26710.2	0.981079
$906$	0	0
$907$	18383.6	0.673006	0.336503	−	0.941682i	$-0.390756\pi$
$907$	18383.6	0.673006	0.336503	+	0.941682i	$0.390756\pi$
$908$	0	0
$909$	8061.13	0.294137
$910$	0	0
$911$	−4746.26	−0.172613	−0.0863066	−	0.996269i	$-0.527506\pi$
$911$	−4746.26	−0.172613	−0.0863066	+	0.996269i	$0.527506\pi$
$912$	0	0
$913$	−7353.23	−0.266546
$914$	0	0
$915$	−1768.42	−0.0638929
$916$	0	0
$917$	−15216.3	−0.547967
$918$	0	0
$919$	52237.6	1.87504	0.937519	−	0.347933i	$-0.113116\pi$
$919$	52237.6	1.87504	0.937519	+	0.347933i	$0.113116\pi$
$920$	0	0
$921$	31059.1	1.11122
$922$	0	0
$923$	942.205	0.0336003
$924$	0	0
$925$	8804.27	0.312954
$926$	0	0
$927$	−41179.7	−1.45903
$928$	0	0
$929$	42088.2	1.48640	0.743202	−	0.669068i	$-0.233306\pi$
$929$	42088.2	1.48640	0.743202	+	0.669068i	$0.233306\pi$
$930$	0	0
$931$	3778.39	0.133009
$932$	0	0
$933$	−25201.0	−0.884292
$934$	0	0
$935$	−10234.5	−0.357971
$936$	0	0
$937$	29249.6	1.01979	0.509895	−	0.860236i	$-0.329684\pi$
$937$	29249.6	1.01979	0.509895	+	0.860236i	$0.329684\pi$
$938$	0	0
$939$	−41713.7	−1.44971
$940$	0	0
$941$	−21311.4	−0.738293	−0.369146	−	0.929371i	$-0.620350\pi$
$941$	−21311.4	−0.738293	−0.369146	+	0.929371i	$0.620350\pi$
$942$	0	0
$943$	15600.3	0.538724
$944$	0	0
$945$	−1337.57	−0.0460434
$946$	0	0
$947$	−32058.0	−1.10005	−0.550023	−	0.835149i	$-0.685381\pi$
$947$	−32058.0	−1.10005	−0.550023	+	0.835149i	$0.685381\pi$
$948$	0	0
$949$	214.267	0.00732920
$950$	0	0
$951$	−11195.5	−0.381745
$952$	0	0
$953$	−36892.5	−1.25400	−0.627002	−	0.779017i	$-0.715718\pi$
$953$	−36892.5	−1.25400	−0.627002	+	0.779017i	$0.715718\pi$
$954$	0	0
$955$	−34882.9	−1.18197
$956$	0	0
$957$	22059.1	0.745108
$958$	0	0
$959$	−5971.81	−0.201084
$960$	0	0
$961$	24850.1	0.834148
$962$	0	0
$963$	21409.8	0.716428
$964$	0	0
$965$	−9619.74	−0.320902
$966$	0	0
$967$	−38041.2	−1.26507	−0.632534	−	0.774532i	$-0.717985\pi$
$967$	−38041.2	−1.26507	−0.632534	+	0.774532i	$0.717985\pi$
$968$	0	0
$969$	52659.4	1.74578
$970$	0	0
$971$	−15970.6	−0.527828	−0.263914	−	0.964546i	$-0.585013\pi$
$971$	−15970.6	−0.527828	−0.263914	+	0.964546i	$0.585013\pi$
$972$	0	0
$973$	16393.6	0.540139
$974$	0	0
$975$	231.608	0.00760759
$976$	0	0
$977$	24177.5	0.791715	0.395857	−	0.918312i	$-0.370447\pi$
$977$	24177.5	0.791715	0.395857	+	0.918312i	$0.370447\pi$
$978$	0	0
$979$	−5972.01	−0.194961
$980$	0	0
$981$	7716.15	0.251129
$982$	0	0
$983$	−51157.2	−1.65988	−0.829939	−	0.557854i	$-0.811625\pi$
$983$	−51157.2	−1.65988	−0.829939	+	0.557854i	$0.811625\pi$
$984$	0	0
$985$	4132.24	0.133669
$986$	0	0
$987$	−23900.2	−0.770773
$988$	0	0
$989$	−35606.7	−1.14482
$990$	0	0
$991$	−9810.81	−0.314481	−0.157240	−	0.987560i	$-0.550260\pi$
$991$	−9810.81	−0.314481	−0.157240	+	0.987560i	$0.550260\pi$
$992$	0	0
$993$	−37366.9	−1.19416
$994$	0	0
$995$	26736.3	0.851858
$996$	0	0
$997$	23382.0	0.742745	0.371372	−	0.928484i	$-0.378887\pi$
$997$	23382.0	0.742745	0.371372	+	0.928484i	$0.378887\pi$
$998$	0	0
$999$	−5778.50	−0.183007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.t.1.1		4
4.3	odd	2		616.4.a.e.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.e.1.4	✓	4	4.3	odd	2
1232.4.a.t.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-7.15990 q^{3} -9.75471 q^{5} +7.00000 q^{7} +24.2641 q^{9} +O(q^{10})\)	\(q-7.15990 q^{3} -9.75471 q^{5} +7.00000 q^{7} +24.2641 q^{9} -11.0000 q^{11} +1.08384 q^{13} +69.8427 q^{15} -95.3803 q^{17} +77.1099 q^{19} -50.1193 q^{21} -114.111 q^{23} -29.8457 q^{25} +19.5886 q^{27} +280.084 q^{29} +233.754 q^{31} +78.7589 q^{33} -68.2830 q^{35} -294.993 q^{37} -7.76020 q^{39} -136.712 q^{41} +312.037 q^{43} -236.689 q^{45} +476.867 q^{47} +49.0000 q^{49} +682.913 q^{51} +283.013 q^{53} +107.302 q^{55} -552.099 q^{57} +706.222 q^{59} -25.3200 q^{61} +169.849 q^{63} -10.5726 q^{65} -1069.14 q^{67} +817.020 q^{69} +869.319 q^{71} +197.692 q^{73} +213.692 q^{75} -77.0000 q^{77} -821.822 q^{79} -795.384 q^{81} +668.476 q^{83} +930.407 q^{85} -2005.37 q^{87} +542.910 q^{89} +7.58690 q^{91} -1673.66 q^{93} -752.185 q^{95} -1702.09 q^{97} -266.905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9	\( 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9} - 44 q^{11} + 2 q^{13} - 71 q^{15} + 8 q^{17} + 6 q^{19} + 21 q^{21} - 159 q^{23} - 119 q^{25} + 117 q^{27} + 144 q^{29} + 183 q^{31} - 33 q^{33} - 133 q^{35} - 475 q^{37} + 64 q^{39} - 768 q^{41} + 152 q^{43} - 1048 q^{45} + 228 q^{47} + 196 q^{49} + 882 q^{51} + 396 q^{53} + 209 q^{55} - 1286 q^{57} + 733 q^{59} - 1012 q^{61} + 259 q^{63} + 420 q^{65} + 171 q^{67} - 321 q^{69} + 1019 q^{71} - 1836 q^{73} + 1016 q^{75} - 308 q^{77} - 1374 q^{79} - 956 q^{81} + 1248 q^{83} + 46 q^{85} - 2238 q^{87} - 1401 q^{89} + 14 q^{91} - 1859 q^{93} - 238 q^{95} - 2559 q^{97} - 407 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9 - 44 * q^11 + 2 * q^13 - 71 * q^15 + 8 * q^17 + 6 * q^19 + 21 * q^21 - 159 * q^23 - 119 * q^25 + 117 * q^27 + 144 * q^29 + 183 * q^31 - 33 * q^33 - 133 * q^35 - 475 * q^37 + 64 * q^39 - 768 * q^41 + 152 * q^43 - 1048 * q^45 + 228 * q^47 + 196 * q^49 + 882 * q^51 + 396 * q^53 + 209 * q^55 - 1286 * q^57 + 733 * q^59 - 1012 * q^61 + 259 * q^63 + 420 * q^65 + 171 * q^67 - 321 * q^69 + 1019 * q^71 - 1836 * q^73 + 1016 * q^75 - 308 * q^77 - 1374 * q^79 - 956 * q^81 + 1248 * q^83 + 46 * q^85 - 2238 * q^87 - 1401 * q^89 + 14 * q^91 - 1859 * q^93 - 238 * q^95 - 2559 * q^97 - 407 * q^99

Introduction

L-functions

Modular forms

Varieties

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