LMFDB - Embedded newform 304.8.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	304.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:	$94.9650477472$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6373x^{4} - 12403x^{3} + 11165936x^{2} + 51537728x - 4683020288 $ x^6 - x^5 - 6373x^4 - 12403x^3 + 11165936x^2 + 51537728x - 4683020288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-60.0909$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-65.0909 q^{3} +165.215 q^{5} +774.961 q^{7} +2049.82 q^{9} +O(q^{10})$	$q-65.0909 q^{3} +165.215 q^{5} +774.961 q^{7} +2049.82 q^{9} +3411.22 q^{11} -10405.3 q^{13} -10754.0 q^{15} -21097.0 q^{17} +6859.00 q^{19} -50442.9 q^{21} +2513.10 q^{23} -50828.9 q^{25} +8929.03 q^{27} -58605.6 q^{29} +198151. q^{31} -222039. q^{33} +128035. q^{35} +556808. q^{37} +677289. q^{39} +323753. q^{41} -599201. q^{43} +338662. q^{45} -289931. q^{47} -222978. q^{49} +1.37322e6 q^{51} +119297. q^{53} +563585. q^{55} -446458. q^{57} +595531. q^{59} -364583. q^{61} +1.58853e6 q^{63} -1.71911e6 q^{65} +146835. q^{67} -163580. q^{69} -2.82634e6 q^{71} +3.60391e6 q^{73} +3.30850e6 q^{75} +2.64356e6 q^{77} +151550. q^{79} -5.06416e6 q^{81} +7.67819e6 q^{83} -3.48554e6 q^{85} +3.81469e6 q^{87} -9.88564e6 q^{89} -8.06369e6 q^{91} -1.28978e7 q^{93} +1.13321e6 q^{95} -1.03471e7 q^{97} +6.99239e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 29 q^{3} - 43 q^{5} + 908 q^{7} - 235 q^{9}+O(q^{10}) $ 6 * q - 29 * q^3 - 43 * q^5 + 908 * q^7 - 235 * q^9	$ 6 q - 29 q^{3} - 43 q^{5} + 908 q^{7} - 235 q^{9} - 3935 q^{11} + 6449 q^{13} - 8422 q^{15} - 6920 q^{17} + 41154 q^{19} - 54471 q^{21} + 43633 q^{23} - 48003 q^{25} - 8705 q^{27} - 134097 q^{29} + 95448 q^{31} - 212700 q^{33} + 202017 q^{35} - 57516 q^{37} + 683975 q^{39} - 399254 q^{41} + 359107 q^{43} - 668039 q^{45} + 590285 q^{47} - 1140196 q^{49} + 2380541 q^{51} - 2926531 q^{53} + 1799465 q^{55} - 198911 q^{57} + 2933563 q^{59} - 4738005 q^{61} + 2126109 q^{63} - 7312896 q^{65} + 4389837 q^{67} - 5697861 q^{69} + 751308 q^{71} - 7310018 q^{73} + 8992379 q^{75} - 12493471 q^{77} + 2495758 q^{79} - 18423454 q^{81} + 4583082 q^{83} - 13253407 q^{85} + 4493999 q^{87} - 21914280 q^{89} + 8775919 q^{91} - 18623792 q^{93} - 294937 q^{95} - 21931958 q^{97} + 2897895 q^{99}+O(q^{100}) $ 6 * q - 29 * q^3 - 43 * q^5 + 908 * q^7 - 235 * q^9 - 3935 * q^11 + 6449 * q^13 - 8422 * q^15 - 6920 * q^17 + 41154 * q^19 - 54471 * q^21 + 43633 * q^23 - 48003 * q^25 - 8705 * q^27 - 134097 * q^29 + 95448 * q^31 - 212700 * q^33 + 202017 * q^35 - 57516 * q^37 + 683975 * q^39 - 399254 * q^41 + 359107 * q^43 - 668039 * q^45 + 590285 * q^47 - 1140196 * q^49 + 2380541 * q^51 - 2926531 * q^53 + 1799465 * q^55 - 198911 * q^57 + 2933563 * q^59 - 4738005 * q^61 + 2126109 * q^63 - 7312896 * q^65 + 4389837 * q^67 - 5697861 * q^69 + 751308 * q^71 - 7310018 * q^73 + 8992379 * q^75 - 12493471 * q^77 + 2495758 * q^79 - 18423454 * q^81 + 4583082 * q^83 - 13253407 * q^85 + 4493999 * q^87 - 21914280 * q^89 + 8775919 * q^91 - 18623792 * q^93 - 294937 * q^95 - 21931958 * q^97 + 2897895 * 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
$3$	−65.0909	−1.39186	−0.695930	−	0.718109i	$-0.745008\pi$
$3$	−65.0909	−1.39186	−0.695930	+	0.718109i	$0.745008\pi$
$4$	0	0
$5$	165.215	0.591092	0.295546	−	0.955328i	$-0.404498\pi$
$5$	165.215	0.591092	0.295546	+	0.955328i	$0.404498\pi$
$6$	0	0
$7$	774.961	0.853958	0.426979	−	0.904261i	$-0.359578\pi$
$7$	774.961	0.853958	0.426979	+	0.904261i	$0.359578\pi$
$8$	0	0
$9$	2049.82	0.937276
$10$	0	0
$11$	3411.22	0.772743	0.386372	−	0.922343i	$-0.373728\pi$
$11$	3411.22	0.772743	0.386372	+	0.922343i	$0.373728\pi$
$12$	0	0
$13$	−10405.3	−1.31357	−0.656784	−	0.754079i	$-0.728084\pi$
$13$	−10405.3	−1.31357	−0.656784	+	0.754079i	$0.728084\pi$
$14$	0	0
$15$	−10754.0	−0.822718
$16$	0	0
$17$	−21097.0	−1.04147	−0.520737	−	0.853717i	$-0.674343\pi$
$17$	−21097.0	−1.04147	−0.520737	+	0.853717i	$0.674343\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	−50442.9	−1.18859
$22$	0	0
$23$	2513.10	0.0430687	0.0215344	−	0.999768i	$-0.493145\pi$
$23$	2513.10	0.0430687	0.0215344	+	0.999768i	$0.493145\pi$
$24$	0	0
$25$	−50828.9	−0.650610
$26$	0	0
$27$	8929.03	0.0873034
$28$	0	0
$29$	−58605.6	−0.446217	−0.223109	−	0.974794i	$-0.571620\pi$
$29$	−58605.6	−0.446217	−0.223109	+	0.974794i	$0.571620\pi$
$30$	0	0
$31$	198151.	1.19462	0.597312	−	0.802009i	$-0.296236\pi$
$31$	198151.	1.19462	0.597312	+	0.802009i	$0.296236\pi$
$32$	0	0
$33$	−222039.	−1.07555
$34$	0	0
$35$	128035.	0.504768
$36$	0	0
$37$	556808.	1.80717	0.903586	−	0.428406i	$-0.140925\pi$
$37$	556808.	1.80717	0.903586	+	0.428406i	$0.140925\pi$
$38$	0	0
$39$	677289.	1.82830
$40$	0	0
$41$	323753.	0.733619	0.366809	−	0.930296i	$-0.380450\pi$
$41$	323753.	0.733619	0.366809	+	0.930296i	$0.380450\pi$
$42$	0	0
$43$	−599201.	−1.14930	−0.574649	−	0.818400i	$-0.694862\pi$
$43$	−599201.	−1.14930	−0.574649	+	0.818400i	$0.694862\pi$
$44$	0	0
$45$	338662.	0.554016
$46$	0	0
$47$	−289931.	−0.407336	−0.203668	−	0.979040i	$-0.565286\pi$
$47$	−289931.	−0.407336	−0.203668	+	0.979040i	$0.565286\pi$
$48$	0	0
$49$	−222978.	−0.270755
$50$	0	0
$51$	1.37322e6	1.44959
$52$	0	0
$53$	119297.	0.110069	0.0550345	−	0.998484i	$-0.482473\pi$
$53$	119297.	0.110069	0.0550345	+	0.998484i	$0.482473\pi$
$54$	0	0
$55$	563585.	0.456762
$56$	0	0
$57$	−446458.	−0.319315
$58$	0	0
$59$	595531.	0.377505	0.188752	−	0.982025i	$-0.439556\pi$
$59$	595531.	0.377505	0.188752	+	0.982025i	$0.439556\pi$
$60$	0	0
$61$	−364583.	−0.205656	−0.102828	−	0.994699i	$-0.532789\pi$
$61$	−364583.	−0.205656	−0.102828	+	0.994699i	$0.532789\pi$
$62$	0	0
$63$	1.58853e6	0.800395
$64$	0	0
$65$	−1.71911e6	−0.776439
$66$	0	0
$67$	146835.	0.0596443	0.0298221	−	0.999555i	$-0.490506\pi$
$67$	146835.	0.0596443	0.0298221	+	0.999555i	$0.490506\pi$
$68$	0	0
$69$	−163580.	−0.0599456
$70$	0	0
$71$	−2.82634e6	−0.937175	−0.468588	−	0.883417i	$-0.655237\pi$
$71$	−2.82634e6	−0.937175	−0.468588	+	0.883417i	$0.655237\pi$
$72$	0	0
$73$	3.60391e6	1.08429	0.542144	−	0.840286i	$-0.317613\pi$
$73$	3.60391e6	1.08429	0.542144	+	0.840286i	$0.317613\pi$
$74$	0	0
$75$	3.30850e6	0.905559
$76$	0	0
$77$	2.64356e6	0.659891
$78$	0	0
$79$	151550.	0.0345829	0.0172915	−	0.999850i	$-0.494496\pi$
$79$	151550.	0.0345829	0.0172915	+	0.999850i	$0.494496\pi$
$80$	0	0
$81$	−5.06416e6	−1.05879
$82$	0	0
$83$	7.67819e6	1.47396	0.736979	−	0.675915i	$-0.236251\pi$
$83$	7.67819e6	1.47396	0.736979	+	0.675915i	$0.236251\pi$
$84$	0	0
$85$	−3.48554e6	−0.615607
$86$	0	0
$87$	3.81469e6	0.621072
$88$	0	0
$89$	−9.88564e6	−1.48641	−0.743207	−	0.669062i	$-0.766696\pi$
$89$	−9.88564e6	−1.48641	−0.743207	+	0.669062i	$0.766696\pi$
$90$	0	0
$91$	−8.06369e6	−1.12173
$92$	0	0
$93$	−1.28978e7	−1.66275
$94$	0	0
$95$	1.13321e6	0.135606
$96$	0	0
$97$	−1.03471e7	−1.15111	−0.575556	−	0.817763i	$-0.695214\pi$
$97$	−1.03471e7	−1.15111	−0.575556	+	0.817763i	$0.695214\pi$
$98$	0	0
$99$	6.99239e6	0.724273
$100$	0	0
$101$	−9.56905e6	−0.924153	−0.462076	−	0.886840i	$-0.652895\pi$
$101$	−9.56905e6	−0.924153	−0.462076	+	0.886840i	$0.652895\pi$
$102$	0	0
$103$	−8.56696e6	−0.772497	−0.386248	−	0.922395i	$-0.626229\pi$
$103$	−8.56696e6	−0.772497	−0.386248	+	0.922395i	$0.626229\pi$
$104$	0	0
$105$	−8.33394e6	−0.702567
$106$	0	0
$107$	2.09406e7	1.65252	0.826259	−	0.563290i	$-0.190465\pi$
$107$	2.09406e7	1.65252	0.826259	+	0.563290i	$0.190465\pi$
$108$	0	0
$109$	−6.42242e6	−0.475014	−0.237507	−	0.971386i	$-0.576330\pi$
$109$	−6.42242e6	−0.475014	−0.237507	+	0.971386i	$0.576330\pi$
$110$	0	0
$111$	−3.62431e7	−2.51533
$112$	0	0
$113$	1.27156e7	0.829014	0.414507	−	0.910046i	$-0.363954\pi$
$113$	1.27156e7	0.829014	0.414507	+	0.910046i	$0.363954\pi$
$114$	0	0
$115$	415202.	0.0254576
$116$	0	0
$117$	−2.13290e7	−1.23118
$118$	0	0
$119$	−1.63493e7	−0.889376
$120$	0	0
$121$	−7.85076e6	−0.402868
$122$	0	0
$123$	−2.10734e7	−1.02109
$124$	0	0
$125$	−2.13052e7	−0.975663
$126$	0	0
$127$	−1.02338e7	−0.443327	−0.221664	−	0.975123i	$-0.571149\pi$
$127$	−1.02338e7	−0.443327	−0.221664	+	0.975123i	$0.571149\pi$
$128$	0	0
$129$	3.90025e7	1.59966
$130$	0	0
$131$	−2.92044e7	−1.13501	−0.567503	−	0.823371i	$-0.692091\pi$
$131$	−2.92044e7	−1.13501	−0.567503	+	0.823371i	$0.692091\pi$
$132$	0	0
$133$	5.31546e6	0.195911
$134$	0	0
$135$	1.47521e6	0.0516043
$136$	0	0
$137$	2.96546e7	0.985304	0.492652	−	0.870226i	$-0.336028\pi$
$137$	2.96546e7	0.985304	0.492652	+	0.870226i	$0.336028\pi$
$138$	0	0
$139$	−3.24201e7	−1.02391	−0.511957	−	0.859011i	$-0.671079\pi$
$139$	−3.24201e7	−1.02391	−0.511957	+	0.859011i	$0.671079\pi$
$140$	0	0
$141$	1.88719e7	0.566955
$142$	0	0
$143$	−3.54947e7	−1.01505
$144$	0	0
$145$	−9.68254e6	−0.263755
$146$	0	0
$147$	1.45139e7	0.376853
$148$	0	0
$149$	−4.77590e7	−1.18278	−0.591389	−	0.806387i	$-0.701420\pi$
$149$	−4.77590e7	−1.18278	−0.591389	+	0.806387i	$0.701420\pi$
$150$	0	0
$151$	−3.87182e7	−0.915157	−0.457578	−	0.889169i	$-0.651283\pi$
$151$	−3.87182e7	−0.915157	−0.457578	+	0.889169i	$0.651283\pi$
$152$	0	0
$153$	−4.32450e7	−0.976148
$154$	0	0
$155$	3.27376e7	0.706132
$156$	0	0
$157$	3.28145e7	0.676731	0.338366	−	0.941015i	$-0.390126\pi$
$157$	3.28145e7	0.676731	0.338366	+	0.941015i	$0.390126\pi$
$158$	0	0
$159$	−7.76517e6	−0.153201
$160$	0	0
$161$	1.94755e6	0.0367789
$162$	0	0
$163$	−6.19904e6	−0.112116	−0.0560580	−	0.998428i	$-0.517853\pi$
$163$	−6.19904e6	−0.112116	−0.0560580	+	0.998428i	$0.517853\pi$
$164$	0	0
$165$	−3.66843e7	−0.635750
$166$	0	0
$167$	−3.65382e7	−0.607071	−0.303536	−	0.952820i	$-0.598167\pi$
$167$	−3.65382e7	−0.607071	−0.303536	+	0.952820i	$0.598167\pi$
$168$	0	0
$169$	4.55216e7	0.725460
$170$	0	0
$171$	1.40597e7	0.215026
$172$	0	0
$173$	3.89600e7	0.572081	0.286040	−	0.958218i	$-0.407661\pi$
$173$	3.89600e7	0.572081	0.286040	+	0.958218i	$0.407661\pi$
$174$	0	0
$175$	−3.93904e7	−0.555594
$176$	0	0
$177$	−3.87637e7	−0.525434
$178$	0	0
$179$	1.99920e7	0.260538	0.130269	−	0.991479i	$-0.458416\pi$
$179$	1.99920e7	0.260538	0.130269	+	0.991479i	$0.458416\pi$
$180$	0	0
$181$	8.72640e7	1.09386	0.546928	−	0.837180i	$-0.315797\pi$
$181$	8.72640e7	1.09386	0.546928	+	0.837180i	$0.315797\pi$
$182$	0	0
$183$	2.37310e7	0.286245
$184$	0	0
$185$	9.19932e7	1.06821
$186$	0	0
$187$	−7.19663e7	−0.804792
$188$	0	0
$189$	6.91965e6	0.0745535
$190$	0	0
$191$	−7.61798e6	−0.0791084	−0.0395542	−	0.999217i	$-0.512594\pi$
$191$	−7.61798e6	−0.0791084	−0.0395542	+	0.999217i	$0.512594\pi$
$192$	0	0
$193$	−1.91424e8	−1.91666	−0.958330	−	0.285664i	$-0.907786\pi$
$193$	−1.91424e8	−1.91666	−0.958330	+	0.285664i	$0.907786\pi$
$194$	0	0
$195$	1.11899e8	1.08070
$196$	0	0
$197$	−1.38828e8	−1.29373	−0.646866	−	0.762603i	$-0.723921\pi$
$197$	−1.38828e8	−1.29373	−0.646866	+	0.762603i	$0.723921\pi$
$198$	0	0
$199$	2.41077e7	0.216856	0.108428	−	0.994104i	$-0.465418\pi$
$199$	2.41077e7	0.216856	0.108428	+	0.994104i	$0.465418\pi$
$200$	0	0
$201$	−9.55764e6	−0.0830165
$202$	0	0
$203$	−4.54171e7	−0.381051
$204$	0	0
$205$	5.34889e7	0.433636
$206$	0	0
$207$	5.15140e6	0.0403673
$208$	0	0
$209$	2.33975e7	0.177279
$210$	0	0
$211$	5.12714e7	0.375740	0.187870	−	0.982194i	$-0.439842\pi$
$211$	5.12714e7	0.375740	0.187870	+	0.982194i	$0.439842\pi$
$212$	0	0
$213$	1.83969e8	1.30442
$214$	0	0
$215$	−9.89972e7	−0.679341
$216$	0	0
$217$	1.53560e8	1.02016
$218$	0	0
$219$	−2.34582e8	−1.50918
$220$	0	0
$221$	2.19520e8	1.36805
$222$	0	0
$223$	−6.90522e7	−0.416976	−0.208488	−	0.978025i	$-0.566854\pi$
$223$	−6.90522e7	−0.416976	−0.208488	+	0.978025i	$0.566854\pi$
$224$	0	0
$225$	−1.04190e8	−0.609801
$226$	0	0
$227$	1.54103e8	0.874420	0.437210	−	0.899359i	$-0.355967\pi$
$227$	1.54103e8	0.874420	0.437210	+	0.899359i	$0.355967\pi$
$228$	0	0
$229$	−3.07698e8	−1.69317	−0.846586	−	0.532252i	$-0.821346\pi$
$229$	−3.07698e8	−1.69317	−0.846586	+	0.532252i	$0.821346\pi$
$230$	0	0
$231$	−1.72072e8	−0.918476
$232$	0	0
$233$	−1.29029e8	−0.668252	−0.334126	−	0.942528i	$-0.608441\pi$
$233$	−1.29029e8	−0.668252	−0.334126	+	0.942528i	$0.608441\pi$
$234$	0	0
$235$	−4.79011e7	−0.240773
$236$	0	0
$237$	−9.86454e6	−0.0481346
$238$	0	0
$239$	−3.62543e8	−1.71778	−0.858888	−	0.512163i	$-0.828844\pi$
$239$	−3.62543e8	−1.71778	−0.858888	+	0.512163i	$0.828844\pi$
$240$	0	0
$241$	9.46614e7	0.435626	0.217813	−	0.975991i	$-0.430108\pi$
$241$	9.46614e7	0.435626	0.217813	+	0.975991i	$0.430108\pi$
$242$	0	0
$243$	3.10103e8	1.38638
$244$	0	0
$245$	−3.68394e7	−0.160041
$246$	0	0
$247$	−7.13699e7	−0.301353
$248$	0	0
$249$	−4.99780e8	−2.05154
$250$	0	0
$251$	−3.24944e8	−1.29703	−0.648516	−	0.761201i	$-0.724610\pi$
$251$	−3.24944e8	−1.29703	−0.648516	+	0.761201i	$0.724610\pi$
$252$	0	0
$253$	8.57272e6	0.0332810
$254$	0	0
$255$	2.26877e8	0.856839
$256$	0	0
$257$	−2.16885e8	−0.797008	−0.398504	−	0.917166i	$-0.630471\pi$
$257$	−2.16885e8	−0.797008	−0.398504	+	0.917166i	$0.630471\pi$
$258$	0	0
$259$	4.31505e8	1.54325
$260$	0	0
$261$	−1.20131e8	−0.418228
$262$	0	0
$263$	−3.00135e8	−1.01735	−0.508676	−	0.860958i	$-0.669865\pi$
$263$	−3.00135e8	−1.01735	−0.508676	+	0.860958i	$0.669865\pi$
$264$	0	0
$265$	1.97097e7	0.0650609
$266$	0	0
$267$	6.43465e8	2.06888
$268$	0	0
$269$	1.54514e8	0.483988	0.241994	−	0.970278i	$-0.422199\pi$
$269$	1.54514e8	0.483988	0.241994	+	0.970278i	$0.422199\pi$
$270$	0	0
$271$	8.08530e7	0.246776	0.123388	−	0.992358i	$-0.460624\pi$
$271$	8.08530e7	0.246776	0.123388	+	0.992358i	$0.460624\pi$
$272$	0	0
$273$	5.24873e8	1.56129
$274$	0	0
$275$	−1.73389e8	−0.502755
$276$	0	0
$277$	−4.37320e8	−1.23629	−0.618144	−	0.786065i	$-0.712115\pi$
$277$	−4.37320e8	−1.23629	−0.618144	+	0.786065i	$0.712115\pi$
$278$	0	0
$279$	4.06175e8	1.11969
$280$	0	0
$281$	−4.96748e8	−1.33556	−0.667781	−	0.744357i	$-0.732756\pi$
$281$	−4.96748e8	−1.33556	−0.667781	+	0.744357i	$0.732756\pi$
$282$	0	0
$283$	−3.11985e8	−0.818240	−0.409120	−	0.912481i	$-0.634164\pi$
$283$	−3.11985e8	−0.818240	−0.409120	+	0.912481i	$0.634164\pi$
$284$	0	0
$285$	−7.37617e7	−0.188744
$286$	0	0
$287$	2.50896e8	0.626480
$288$	0	0
$289$	3.47427e7	0.0846682
$290$	0	0
$291$	6.73501e8	1.60219
$292$	0	0
$293$	7.62569e7	0.177110	0.0885549	−	0.996071i	$-0.471775\pi$
$293$	7.62569e7	0.177110	0.0885549	+	0.996071i	$0.471775\pi$
$294$	0	0
$295$	9.83909e7	0.223140
$296$	0	0
$297$	3.04589e7	0.0674631
$298$	0	0
$299$	−2.61495e7	−0.0565737
$300$	0	0
$301$	−4.64357e8	−0.981453
$302$	0	0
$303$	6.22858e8	1.28629
$304$	0	0
$305$	−6.02346e7	−0.121562
$306$	0	0
$307$	6.84432e8	1.35004	0.675019	−	0.737800i	$-0.264135\pi$
$307$	6.84432e8	1.35004	0.675019	+	0.737800i	$0.264135\pi$
$308$	0	0
$309$	5.57631e8	1.07521
$310$	0	0
$311$	−7.85903e8	−1.48152	−0.740760	−	0.671770i	$-0.765534\pi$
$311$	−7.85903e8	−1.48152	−0.740760	+	0.671770i	$0.765534\pi$
$312$	0	0
$313$	1.40680e8	0.259315	0.129657	−	0.991559i	$-0.458612\pi$
$313$	1.40680e8	0.259315	0.129657	+	0.991559i	$0.458612\pi$
$314$	0	0
$315$	2.62450e8	0.473107
$316$	0	0
$317$	4.01045e8	0.707108	0.353554	−	0.935414i	$-0.384973\pi$
$317$	4.01045e8	0.707108	0.353554	+	0.935414i	$0.384973\pi$
$318$	0	0
$319$	−1.99917e8	−0.344811
$320$	0	0
$321$	−1.36304e9	−2.30007
$322$	0	0
$323$	−1.44704e8	−0.238931
$324$	0	0
$325$	5.28890e8	0.854621
$326$	0	0
$327$	4.18041e8	0.661153
$328$	0	0
$329$	−2.24685e8	−0.347848
$330$	0	0
$331$	6.72089e8	1.01866	0.509329	−	0.860572i	$-0.329893\pi$
$331$	6.72089e8	1.01866	0.509329	+	0.860572i	$0.329893\pi$
$332$	0	0
$333$	1.14136e9	1.69382
$334$	0	0
$335$	2.42594e7	0.0352553
$336$	0	0
$337$	7.40374e8	1.05377	0.526886	−	0.849936i	$-0.323360\pi$
$337$	7.40374e8	1.05377	0.526886	+	0.849936i	$0.323360\pi$
$338$	0	0
$339$	−8.27669e8	−1.15387
$340$	0	0
$341$	6.75937e8	0.923137
$342$	0	0
$343$	−8.11013e8	−1.08517
$344$	0	0
$345$	−2.70259e7	−0.0354334
$346$	0	0
$347$	1.18124e9	1.51770	0.758851	−	0.651264i	$-0.225761\pi$
$347$	1.18124e9	1.51770	0.758851	+	0.651264i	$0.225761\pi$
$348$	0	0
$349$	1.12031e9	1.41074	0.705371	−	0.708838i	$-0.250780\pi$
$349$	1.12031e9	1.41074	0.705371	+	0.708838i	$0.250780\pi$
$350$	0	0
$351$	−9.29091e7	−0.114679
$352$	0	0
$353$	5.86760e8	0.709985	0.354993	−	0.934869i	$-0.384483\pi$
$353$	5.86760e8	0.709985	0.354993	+	0.934869i	$0.384483\pi$
$354$	0	0
$355$	−4.66955e8	−0.553957
$356$	0	0
$357$	1.06419e9	1.23789
$358$	0	0
$359$	1.17182e9	1.33669	0.668347	−	0.743850i	$-0.267002\pi$
$359$	1.17182e9	1.33669	0.668347	+	0.743850i	$0.267002\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	5.11013e8	0.560736
$364$	0	0
$365$	5.95422e8	0.640914
$366$	0	0
$367$	−6.79549e8	−0.717611	−0.358806	−	0.933412i	$-0.616816\pi$
$367$	−6.79549e8	−0.717611	−0.358806	+	0.933412i	$0.616816\pi$
$368$	0	0
$369$	6.63636e8	0.687603
$370$	0	0
$371$	9.24508e7	0.0939944
$372$	0	0
$373$	−1.65814e9	−1.65440	−0.827199	−	0.561909i	$-0.810067\pi$
$373$	−1.65814e9	−1.65440	−0.827199	+	0.561909i	$0.810067\pi$
$374$	0	0
$375$	1.38677e9	1.35799
$376$	0	0
$377$	6.09808e8	0.586136
$378$	0	0
$379$	−1.08999e9	−1.02846	−0.514230	−	0.857652i	$-0.671922\pi$
$379$	−1.08999e9	−1.02846	−0.514230	+	0.857652i	$0.671922\pi$
$380$	0	0
$381$	6.66128e8	0.617050
$382$	0	0
$383$	1.10982e7	0.0100939	0.00504693	−	0.999987i	$-0.498394\pi$
$383$	1.10982e7	0.0100939	0.00504693	+	0.999987i	$0.498394\pi$
$384$	0	0
$385$	4.36757e8	0.390056
$386$	0	0
$387$	−1.22826e9	−1.07721
$388$	0	0
$389$	9.82638e8	0.846389	0.423195	−	0.906039i	$-0.360909\pi$
$389$	9.82638e8	0.846389	0.423195	+	0.906039i	$0.360909\pi$
$390$	0	0
$391$	−5.30187e7	−0.0448549
$392$	0	0
$393$	1.90094e9	1.57977
$394$	0	0
$395$	2.50384e7	0.0204417
$396$	0	0
$397$	−1.51430e9	−1.21463	−0.607316	−	0.794460i	$-0.707754\pi$
$397$	−1.51430e9	−1.21463	−0.607316	+	0.794460i	$0.707754\pi$
$398$	0	0
$399$	−3.45988e8	−0.272681
$400$	0	0
$401$	−1.31500e9	−1.01841	−0.509203	−	0.860647i	$-0.670060\pi$
$401$	−1.31500e9	−1.01841	−0.509203	+	0.860647i	$0.670060\pi$
$402$	0	0
$403$	−2.06182e9	−1.56922
$404$	0	0
$405$	−8.36676e8	−0.625842
$406$	0	0
$407$	1.89939e9	1.39648
$408$	0	0
$409$	−2.37384e8	−0.171562	−0.0857809	−	0.996314i	$-0.527338\pi$
$409$	−2.37384e8	−0.171562	−0.0857809	+	0.996314i	$0.527338\pi$
$410$	0	0
$411$	−1.93024e9	−1.37141
$412$	0	0
$413$	4.61514e8	0.322373
$414$	0	0
$415$	1.26855e9	0.871245
$416$	0	0
$417$	2.11026e9	1.42514
$418$	0	0
$419$	4.34834e8	0.288785	0.144393	−	0.989520i	$-0.453877\pi$
$419$	4.34834e8	0.288785	0.144393	+	0.989520i	$0.453877\pi$
$420$	0	0
$421$	−2.36164e9	−1.54251	−0.771253	−	0.636528i	$-0.780370\pi$
$421$	−2.36164e9	−1.54251	−0.771253	+	0.636528i	$0.780370\pi$
$422$	0	0
$423$	−5.94307e8	−0.381786
$424$	0	0
$425$	1.07234e9	0.677594
$426$	0	0
$427$	−2.82537e8	−0.175622
$428$	0	0
$429$	2.31038e9	1.41281
$430$	0	0
$431$	−1.13403e9	−0.682265	−0.341132	−	0.940015i	$-0.610810\pi$
$431$	−1.13403e9	−0.682265	−0.341132	+	0.940015i	$0.610810\pi$
$432$	0	0
$433$	−6.53938e7	−0.0387105	−0.0193553	−	0.999813i	$-0.506161\pi$
$433$	−6.53938e7	−0.0387105	−0.0193553	+	0.999813i	$0.506161\pi$
$434$	0	0
$435$	6.30245e8	0.367111
$436$	0	0
$437$	1.72373e7	0.00988064
$438$	0	0
$439$	2.47000e9	1.39338	0.696692	−	0.717370i	$-0.254654\pi$
$439$	2.47000e9	1.39338	0.696692	+	0.717370i	$0.254654\pi$
$440$	0	0
$441$	−4.57066e8	−0.253772
$442$	0	0
$443$	−1.25272e9	−0.684604	−0.342302	−	0.939590i	$-0.611207\pi$
$443$	−1.25272e9	−0.684604	−0.342302	+	0.939590i	$0.611207\pi$
$444$	0	0
$445$	−1.63326e9	−0.878608
$446$	0	0
$447$	3.10867e9	1.64626
$448$	0	0
$449$	−1.02696e9	−0.535416	−0.267708	−	0.963500i	$-0.586266\pi$
$449$	−1.02696e9	−0.535416	−0.267708	+	0.963500i	$0.586266\pi$
$450$	0	0
$451$	1.10439e9	0.566899
$452$	0	0
$453$	2.52020e9	1.27377
$454$	0	0
$455$	−1.33225e9	−0.663047
$456$	0	0
$457$	−1.62331e9	−0.795601	−0.397800	−	0.917472i	$-0.630226\pi$
$457$	−1.62331e9	−0.795601	−0.397800	+	0.917472i	$0.630226\pi$
$458$	0	0
$459$	−1.88375e8	−0.0909242
$460$	0	0
$461$	6.81445e8	0.323950	0.161975	−	0.986795i	$-0.448214\pi$
$461$	6.81445e8	0.323950	0.161975	+	0.986795i	$0.448214\pi$
$462$	0	0
$463$	1.21102e9	0.567046	0.283523	−	0.958965i	$-0.408497\pi$
$463$	1.21102e9	0.567046	0.283523	+	0.958965i	$0.408497\pi$
$464$	0	0
$465$	−2.13092e9	−0.982838
$466$	0	0
$467$	−2.44385e9	−1.11036	−0.555182	−	0.831729i	$-0.687351\pi$
$467$	−2.44385e9	−1.11036	−0.555182	+	0.831729i	$0.687351\pi$
$468$	0	0
$469$	1.13792e8	0.0509337
$470$	0	0
$471$	−2.13592e9	−0.941916
$472$	0	0
$473$	−2.04401e9	−0.888113
$474$	0	0
$475$	−3.48636e8	−0.149260
$476$	0	0
$477$	2.44538e8	0.103165
$478$	0	0
$479$	−2.57537e9	−1.07069	−0.535347	−	0.844632i	$-0.679819\pi$
$479$	−2.57537e9	−1.07069	−0.535347	+	0.844632i	$0.679819\pi$
$480$	0	0
$481$	−5.79375e9	−2.37384
$482$	0	0
$483$	−1.26768e8	−0.0511911
$484$	0	0
$485$	−1.70950e9	−0.680413
$486$	0	0
$487$	−7.15399e8	−0.280671	−0.140335	−	0.990104i	$-0.544818\pi$
$487$	−7.15399e8	−0.280671	−0.140335	+	0.990104i	$0.544818\pi$
$488$	0	0
$489$	4.03501e8	0.156050
$490$	0	0
$491$	−5.53548e8	−0.211043	−0.105521	−	0.994417i	$-0.533651\pi$
$491$	−5.53548e8	−0.211043	−0.105521	+	0.994417i	$0.533651\pi$
$492$	0	0
$493$	1.23640e9	0.464724
$494$	0	0
$495$	1.15525e9	0.428112
$496$	0	0
$497$	−2.19031e9	−0.800308
$498$	0	0
$499$	−2.94359e9	−1.06054	−0.530268	−	0.847830i	$-0.677909\pi$
$499$	−2.94359e9	−1.06054	−0.530268	+	0.847830i	$0.677909\pi$
$500$	0	0
$501$	2.37830e9	0.844958
$502$	0	0
$503$	3.25069e9	1.13890	0.569452	−	0.822025i	$-0.307155\pi$
$503$	3.25069e9	1.13890	0.569452	+	0.822025i	$0.307155\pi$
$504$	0	0
$505$	−1.58095e9	−0.546260
$506$	0	0
$507$	−2.96304e9	−1.00974
$508$	0	0
$509$	4.28218e9	1.43930	0.719651	−	0.694335i	$-0.244302\pi$
$509$	4.28218e9	1.43930	0.719651	+	0.694335i	$0.244302\pi$
$510$	0	0
$511$	2.79289e9	0.925936
$512$	0	0
$513$	6.12442e7	0.0200288
$514$	0	0
$515$	−1.41539e9	−0.456617
$516$	0	0
$517$	−9.89019e8	−0.314766
$518$	0	0
$519$	−2.53594e9	−0.796257
$520$	0	0
$521$	−6.10596e9	−1.89157	−0.945784	−	0.324795i	$-0.894705\pi$
$521$	−6.10596e9	−1.89157	−0.945784	+	0.324795i	$0.894705\pi$
$522$	0	0
$523$	−2.70645e9	−0.827263	−0.413631	−	0.910444i	$-0.635740\pi$
$523$	−2.70645e9	−0.827263	−0.413631	+	0.910444i	$0.635740\pi$
$524$	0	0
$525$	2.56396e9	0.773309
$526$	0	0
$527$	−4.18039e9	−1.24417
$528$	0	0
$529$	−3.39851e9	−0.998145
$530$	0	0
$531$	1.22073e9	0.353826
$532$	0	0
$533$	−3.36874e9	−0.963658
$534$	0	0
$535$	3.45971e9	0.976790
$536$	0	0
$537$	−1.30130e9	−0.362632
$538$	0	0
$539$	−7.60628e8	−0.209224
$540$	0	0
$541$	−3.93980e9	−1.06975	−0.534877	−	0.844930i	$-0.679642\pi$
$541$	−3.93980e9	−1.06975	−0.534877	+	0.844930i	$0.679642\pi$
$542$	0	0
$543$	−5.68009e9	−1.52249
$544$	0	0
$545$	−1.06108e9	−0.280777
$546$	0	0
$547$	1.23208e9	0.321871	0.160936	−	0.986965i	$-0.448549\pi$
$547$	1.23208e9	0.321871	0.160936	+	0.986965i	$0.448549\pi$
$548$	0	0
$549$	−7.47329e8	−0.192756
$550$	0	0
$551$	−4.01976e8	−0.102369
$552$	0	0
$553$	1.17446e8	0.0295324
$554$	0	0
$555$	−5.98792e9	−1.48679
$556$	0	0
$557$	−1.28810e8	−0.0315831	−0.0157916	−	0.999875i	$-0.505027\pi$
$557$	−1.28810e8	−0.0315831	−0.0157916	+	0.999875i	$0.505027\pi$
$558$	0	0
$559$	6.23486e9	1.50968
$560$	0	0
$561$	4.68435e9	1.12016
$562$	0	0
$563$	2.97889e9	0.703518	0.351759	−	0.936091i	$-0.385584\pi$
$563$	2.97889e9	0.703518	0.351759	+	0.936091i	$0.385584\pi$
$564$	0	0
$565$	2.10081e9	0.490023
$566$	0	0
$567$	−3.92453e9	−0.904163
$568$	0	0
$569$	−6.25314e9	−1.42300	−0.711500	−	0.702686i	$-0.751984\pi$
$569$	−6.25314e9	−1.42300	−0.711500	+	0.702686i	$0.751984\pi$
$570$	0	0
$571$	8.21409e9	1.84643	0.923216	−	0.384281i	$-0.125551\pi$
$571$	8.21409e9	1.84643	0.923216	+	0.384281i	$0.125551\pi$
$572$	0	0
$573$	4.95861e8	0.110108
$574$	0	0
$575$	−1.27738e8	−0.0280209
$576$	0	0
$577$	4.79043e9	1.03815	0.519074	−	0.854729i	$-0.326277\pi$
$577$	4.79043e9	1.03815	0.519074	+	0.854729i	$0.326277\pi$
$578$	0	0
$579$	1.24599e10	2.66772
$580$	0	0
$581$	5.95029e9	1.25870
$582$	0	0
$583$	4.06949e8	0.0850551
$584$	0	0
$585$	−3.52388e9	−0.727738
$586$	0	0
$587$	3.18914e9	0.650789	0.325394	−	0.945578i	$-0.394503\pi$
$587$	3.18914e9	0.650789	0.325394	+	0.945578i	$0.394503\pi$
$588$	0	0
$589$	1.35912e9	0.274065
$590$	0	0
$591$	9.03642e9	1.80070
$592$	0	0
$593$	−6.00794e9	−1.18313	−0.591567	−	0.806256i	$-0.701491\pi$
$593$	−6.00794e9	−1.18313	−0.591567	+	0.806256i	$0.701491\pi$
$594$	0	0
$595$	−2.70116e9	−0.525703
$596$	0	0
$597$	−1.56919e9	−0.301833
$598$	0	0
$599$	1.10866e9	0.210768	0.105384	−	0.994432i	$-0.466393\pi$
$599$	1.10866e9	0.210768	0.105384	+	0.994432i	$0.466393\pi$
$600$	0	0
$601$	−7.07923e9	−1.33023	−0.665113	−	0.746743i	$-0.731616\pi$
$601$	−7.07923e9	−1.33023	−0.665113	+	0.746743i	$0.731616\pi$
$602$	0	0
$603$	3.00986e8	0.0559032
$604$	0	0
$605$	−1.29706e9	−0.238132
$606$	0	0
$607$	9.69356e9	1.75923	0.879615	−	0.475686i	$-0.157800\pi$
$607$	9.69356e9	1.75923	0.879615	+	0.475686i	$0.157800\pi$
$608$	0	0
$609$	2.95624e9	0.530370
$610$	0	0
$611$	3.01682e9	0.535063
$612$	0	0
$613$	3.80418e9	0.667036	0.333518	−	0.942744i	$-0.391764\pi$
$613$	3.80418e9	0.667036	0.333518	+	0.942744i	$0.391764\pi$
$614$	0	0
$615$	−3.48164e9	−0.603561
$616$	0	0
$617$	7.03189e9	1.20524	0.602621	−	0.798028i	$-0.294123\pi$
$617$	7.03189e9	1.20524	0.602621	+	0.798028i	$0.294123\pi$
$618$	0	0
$619$	1.00612e10	1.70503	0.852515	−	0.522703i	$-0.175076\pi$
$619$	1.00612e10	1.70503	0.852515	+	0.522703i	$0.175076\pi$
$620$	0	0
$621$	2.24395e7	0.00376004
$622$	0	0
$623$	−7.66099e9	−1.26934
$624$	0	0
$625$	4.51073e8	0.0739038
$626$	0	0
$627$	−1.52297e9	−0.246748
$628$	0	0
$629$	−1.17470e10	−1.88212
$630$	0	0
$631$	−2.85699e9	−0.452696	−0.226348	−	0.974046i	$-0.572679\pi$
$631$	−2.85699e9	−0.452696	−0.226348	+	0.974046i	$0.572679\pi$
$632$	0	0
$633$	−3.33730e9	−0.522977
$634$	0	0
$635$	−1.69078e9	−0.262047
$636$	0	0
$637$	2.32016e9	0.355655
$638$	0	0
$639$	−5.79350e9	−0.878391
$640$	0	0
$641$	1.04843e9	0.157230	0.0786151	−	0.996905i	$-0.474950\pi$
$641$	1.04843e9	0.157230	0.0786151	+	0.996905i	$0.474950\pi$
$642$	0	0
$643$	−5.72938e9	−0.849903	−0.424951	−	0.905216i	$-0.639709\pi$
$643$	−5.72938e9	−0.849903	−0.424951	+	0.905216i	$0.639709\pi$
$644$	0	0
$645$	6.44381e9	0.945549
$646$	0	0
$647$	−1.03797e10	−1.50667	−0.753336	−	0.657636i	$-0.771557\pi$
$647$	−1.03797e10	−1.50667	−0.753336	+	0.657636i	$0.771557\pi$
$648$	0	0
$649$	2.03149e9	0.291714
$650$	0	0
$651$	−9.99532e9	−1.41992
$652$	0	0
$653$	8.00890e9	1.12558	0.562790	−	0.826600i	$-0.309728\pi$
$653$	8.00890e9	1.12558	0.562790	+	0.826600i	$0.309728\pi$
$654$	0	0
$655$	−4.82501e9	−0.670893
$656$	0	0
$657$	7.38738e9	1.01628
$658$	0	0
$659$	−1.45006e9	−0.197373	−0.0986863	−	0.995119i	$-0.531464\pi$
$659$	−1.45006e9	−0.197373	−0.0986863	+	0.995119i	$0.531464\pi$
$660$	0	0
$661$	6.16554e9	0.830359	0.415180	−	0.909740i	$-0.363719\pi$
$661$	6.16554e9	0.830359	0.415180	+	0.909740i	$0.363719\pi$
$662$	0	0
$663$	−1.42887e10	−1.90413
$664$	0	0
$665$	8.78195e8	0.115802
$666$	0	0
$667$	−1.47282e8	−0.0192180
$668$	0	0
$669$	4.49467e9	0.580372
$670$	0	0
$671$	−1.24367e9	−0.158919
$672$	0	0
$673$	−9.21834e9	−1.16574	−0.582868	−	0.812567i	$-0.698069\pi$
$673$	−9.21834e9	−1.16574	−0.582868	+	0.812567i	$0.698069\pi$
$674$	0	0
$675$	−4.53853e8	−0.0568005
$676$	0	0
$677$	−8.79564e9	−1.08945	−0.544725	−	0.838615i	$-0.683366\pi$
$677$	−8.79564e9	−1.08945	−0.544725	+	0.838615i	$0.683366\pi$
$678$	0	0
$679$	−8.01859e9	−0.983001
$680$	0	0
$681$	−1.00307e10	−1.21707
$682$	0	0
$683$	−1.11750e10	−1.34207	−0.671036	−	0.741425i	$-0.734151\pi$
$683$	−1.11750e10	−1.34207	−0.671036	+	0.741425i	$0.734151\pi$
$684$	0	0
$685$	4.89939e9	0.582405
$686$	0	0
$687$	2.00284e10	2.35666
$688$	0	0
$689$	−1.24132e9	−0.144583
$690$	0	0
$691$	−5.87182e9	−0.677017	−0.338509	−	0.940963i	$-0.609922\pi$
$691$	−5.87182e9	−0.677017	−0.338509	+	0.940963i	$0.609922\pi$
$692$	0	0
$693$	5.41883e9	0.618499
$694$	0	0
$695$	−5.35630e9	−0.605227
$696$	0	0
$697$	−6.83020e9	−0.764045
$698$	0	0
$699$	8.39858e9	0.930113
$700$	0	0
$701$	1.66708e10	1.82787	0.913933	−	0.405865i	$-0.133030\pi$
$701$	1.66708e10	1.82787	0.913933	+	0.405865i	$0.133030\pi$
$702$	0	0
$703$	3.81915e9	0.414594
$704$	0	0
$705$	3.11792e9	0.335122
$706$	0	0
$707$	−7.41564e9	−0.789188
$708$	0	0
$709$	−1.24511e10	−1.31203	−0.656017	−	0.754746i	$-0.727760\pi$
$709$	−1.24511e10	−1.31203	−0.656017	+	0.754746i	$0.727760\pi$
$710$	0	0
$711$	3.10651e8	0.0324137
$712$	0	0
$713$	4.97973e8	0.0514509
$714$	0	0
$715$	−5.86427e9	−0.599988
$716$	0	0
$717$	2.35982e10	2.39090
$718$	0	0
$719$	−1.45709e10	−1.46196	−0.730980	−	0.682399i	$-0.760937\pi$
$719$	−1.45709e10	−1.46196	−0.730980	+	0.682399i	$0.760937\pi$
$720$	0	0
$721$	−6.63906e9	−0.659680
$722$	0	0
$723$	−6.16160e9	−0.606330
$724$	0	0
$725$	2.97886e9	0.290313
$726$	0	0
$727$	1.12315e10	1.08410	0.542050	−	0.840346i	$-0.317648\pi$
$727$	1.12315e10	1.08410	0.542050	+	0.840346i	$0.317648\pi$
$728$	0	0
$729$	−9.10954e9	−0.870864
$730$	0	0
$731$	1.26413e10	1.19697
$732$	0	0
$733$	8.47485e8	0.0794819	0.0397410	−	0.999210i	$-0.487347\pi$
$733$	8.47485e8	0.0794819	0.0397410	+	0.999210i	$0.487347\pi$
$734$	0	0
$735$	2.39791e9	0.222755
$736$	0	0
$737$	5.00888e8	0.0460897
$738$	0	0
$739$	1.96528e10	1.79130	0.895652	−	0.444757i	$-0.146710\pi$
$739$	1.96528e10	1.79130	0.895652	+	0.444757i	$0.146710\pi$
$740$	0	0
$741$	4.64553e9	0.419442
$742$	0	0
$743$	−2.10434e10	−1.88216	−0.941078	−	0.338189i	$-0.890186\pi$
$743$	−2.10434e10	−1.88216	−0.941078	+	0.338189i	$0.890186\pi$
$744$	0	0
$745$	−7.89051e9	−0.699130
$746$	0	0
$747$	1.57389e10	1.38151
$748$	0	0
$749$	1.62282e10	1.41118
$750$	0	0
$751$	5.53354e9	0.476720	0.238360	−	0.971177i	$-0.423390\pi$
$751$	5.53354e9	0.476720	0.238360	+	0.971177i	$0.423390\pi$
$752$	0	0
$753$	2.11509e10	1.80529
$754$	0	0
$755$	−6.39683e9	−0.540942
$756$	0	0
$757$	−9.53196e9	−0.798632	−0.399316	−	0.916813i	$-0.630752\pi$
$757$	−9.53196e9	−0.798632	−0.399316	+	0.916813i	$0.630752\pi$
$758$	0	0
$759$	−5.58006e8	−0.0463226
$760$	0	0
$761$	−2.57729e9	−0.211991	−0.105995	−	0.994367i	$-0.533803\pi$
$761$	−2.57729e9	−0.211991	−0.105995	+	0.994367i	$0.533803\pi$
$762$	0	0
$763$	−4.97713e9	−0.405642
$764$	0	0
$765$	−7.14473e9	−0.576994
$766$	0	0
$767$	−6.19668e9	−0.495878
$768$	0	0
$769$	1.81272e10	1.43743	0.718717	−	0.695303i	$-0.244730\pi$
$769$	1.81272e10	1.43743	0.718717	+	0.695303i	$0.244730\pi$
$770$	0	0
$771$	1.41172e10	1.10932
$772$	0	0
$773$	7.82920e9	0.609662	0.304831	−	0.952406i	$-0.401400\pi$
$773$	7.82920e9	0.609662	0.304831	+	0.952406i	$0.401400\pi$
$774$	0	0
$775$	−1.00718e10	−0.777234
$776$	0	0
$777$	−2.80870e10	−2.14799
$778$	0	0
$779$	2.22062e9	0.168304
$780$	0	0
$781$	−9.64128e9	−0.724196
$782$	0	0
$783$	−5.23291e8	−0.0389563
$784$	0	0
$785$	5.42145e9	0.400011
$786$	0	0
$787$	2.57850e10	1.88563	0.942813	−	0.333323i	$-0.108170\pi$
$787$	2.57850e10	1.88563	0.942813	+	0.333323i	$0.108170\pi$
$788$	0	0
$789$	1.95360e10	1.41601
$790$	0	0
$791$	9.85408e9	0.707943
$792$	0	0
$793$	3.79359e9	0.270143
$794$	0	0
$795$	−1.28292e9	−0.0905557
$796$	0	0
$797$	−4.79964e9	−0.335819	−0.167909	−	0.985802i	$-0.553702\pi$
$797$	−4.79964e9	−0.335819	−0.167909	+	0.985802i	$0.553702\pi$
$798$	0	0
$799$	6.11667e9	0.424230
$800$	0	0
$801$	−2.02638e10	−1.39318
$802$	0	0
$803$	1.22937e10	0.837876
$804$	0	0
$805$	3.21765e8	0.0217397
$806$	0	0
$807$	−1.00574e10	−0.673643
$808$	0	0
$809$	−5.26908e9	−0.349876	−0.174938	−	0.984579i	$-0.555973\pi$
$809$	−5.26908e9	−0.349876	−0.174938	+	0.984579i	$0.555973\pi$
$810$	0	0
$811$	7.97328e9	0.524885	0.262442	−	0.964948i	$-0.415472\pi$
$811$	7.97328e9	0.524885	0.262442	+	0.964948i	$0.415472\pi$
$812$	0	0
$813$	−5.26279e9	−0.343478
$814$	0	0
$815$	−1.02418e9	−0.0662709
$816$	0	0
$817$	−4.10992e9	−0.263667
$818$	0	0
$819$	−1.65291e10	−1.05137
$820$	0	0
$821$	−7.53735e9	−0.475355	−0.237677	−	0.971344i	$-0.576386\pi$
$821$	−7.53735e9	−0.475355	−0.237677	+	0.971344i	$0.576386\pi$
$822$	0	0
$823$	2.38435e10	1.49098	0.745489	−	0.666518i	$-0.232216\pi$
$823$	2.38435e10	1.49098	0.745489	+	0.666518i	$0.232216\pi$
$824$	0	0
$825$	1.12860e10	0.699764
$826$	0	0
$827$	1.63705e10	1.00645	0.503226	−	0.864155i	$-0.332146\pi$
$827$	1.63705e10	1.00645	0.503226	+	0.864155i	$0.332146\pi$
$828$	0	0
$829$	2.89876e10	1.76714	0.883570	−	0.468300i	$-0.155133\pi$
$829$	2.89876e10	1.76714	0.883570	+	0.468300i	$0.155133\pi$
$830$	0	0
$831$	2.84655e10	1.72074
$832$	0	0
$833$	4.70416e9	0.281984
$834$	0	0
$835$	−6.03667e9	−0.358835
$836$	0	0
$837$	1.76930e9	0.104295
$838$	0	0
$839$	2.33866e10	1.36710	0.683551	−	0.729902i	$-0.260435\pi$
$839$	2.33866e10	1.36710	0.683551	+	0.729902i	$0.260435\pi$
$840$	0	0
$841$	−1.38153e10	−0.800890
$842$	0	0
$843$	3.23338e10	1.85892
$844$	0	0
$845$	7.52086e9	0.428814
$846$	0	0
$847$	−6.08403e9	−0.344032
$848$	0	0
$849$	2.03074e10	1.13888
$850$	0	0
$851$	1.39931e9	0.0778326
$852$	0	0
$853$	1.42926e10	0.788478	0.394239	−	0.919008i	$-0.371008\pi$
$853$	1.42926e10	0.788478	0.394239	+	0.919008i	$0.371008\pi$
$854$	0	0
$855$	2.32288e9	0.127100
$856$	0	0
$857$	−8.17620e9	−0.443730	−0.221865	−	0.975077i	$-0.571214\pi$
$857$	−8.17620e9	−0.443730	−0.221865	+	0.975077i	$0.571214\pi$
$858$	0	0
$859$	−1.32272e10	−0.712020	−0.356010	−	0.934482i	$-0.615863\pi$
$859$	−1.32272e10	−0.712020	−0.356010	+	0.934482i	$0.615863\pi$
$860$	0	0
$861$	−1.63310e10	−0.871973
$862$	0	0
$863$	−2.20566e10	−1.16816	−0.584078	−	0.811698i	$-0.698544\pi$
$863$	−2.20566e10	−1.16816	−0.584078	+	0.811698i	$0.698544\pi$
$864$	0	0
$865$	6.43678e9	0.338152
$866$	0	0
$867$	−2.26143e9	−0.117846
$868$	0	0
$869$	5.16971e8	0.0267237
$870$	0	0
$871$	−1.52786e9	−0.0783468
$872$	0	0
$873$	−2.12097e10	−1.07891
$874$	0	0
$875$	−1.65107e10	−0.833175
$876$	0	0
$877$	3.09262e10	1.54820	0.774102	−	0.633061i	$-0.218202\pi$
$877$	3.09262e10	1.54820	0.774102	+	0.633061i	$0.218202\pi$
$878$	0	0
$879$	−4.96363e9	−0.246512
$880$	0	0
$881$	3.43105e10	1.69049	0.845244	−	0.534381i	$-0.179455\pi$
$881$	3.43105e10	1.69049	0.845244	+	0.534381i	$0.179455\pi$
$882$	0	0
$883$	2.04637e9	0.100028	0.0500141	−	0.998749i	$-0.484073\pi$
$883$	2.04637e9	0.100028	0.0500141	+	0.998749i	$0.484073\pi$
$884$	0	0
$885$	−6.40435e9	−0.310580
$886$	0	0
$887$	2.39442e10	1.15204	0.576020	−	0.817436i	$-0.304605\pi$
$887$	2.39442e10	1.15204	0.576020	+	0.817436i	$0.304605\pi$
$888$	0	0
$889$	−7.93081e9	−0.378583
$890$	0	0
$891$	−1.72750e10	−0.818173
$892$	0	0
$893$	−1.98864e9	−0.0934492
$894$	0	0
$895$	3.30298e9	0.154002
$896$	0	0
$897$	1.70209e9	0.0787427
$898$	0	0
$899$	−1.16128e10	−0.533061
$900$	0	0
$901$	−2.51681e9	−0.114634
$902$	0	0
$903$	3.02254e10	1.36605
$904$	0	0
$905$	1.44173e10	0.646570
$906$	0	0
$907$	3.15992e10	1.40621	0.703105	−	0.711086i	$-0.251796\pi$
$907$	3.15992e10	1.40621	0.703105	+	0.711086i	$0.251796\pi$
$908$	0	0
$909$	−1.96148e10	−0.866186
$910$	0	0
$911$	−2.22131e10	−0.973410	−0.486705	−	0.873566i	$-0.661801\pi$
$911$	−2.22131e10	−0.973410	−0.486705	+	0.873566i	$0.661801\pi$
$912$	0	0
$913$	2.61920e10	1.13899
$914$	0	0
$915$	3.92072e9	0.169197
$916$	0	0
$917$	−2.26323e10	−0.969249
$918$	0	0
$919$	3.55547e10	1.51110	0.755548	−	0.655093i	$-0.227370\pi$
$919$	3.55547e10	1.51110	0.755548	+	0.655093i	$0.227370\pi$
$920$	0	0
$921$	−4.45503e10	−1.87907
$922$	0	0
$923$	2.94089e10	1.23104
$924$	0	0
$925$	−2.83020e10	−1.17576
$926$	0	0
$927$	−1.75608e10	−0.724043
$928$	0	0
$929$	7.16237e9	0.293091	0.146545	−	0.989204i	$-0.453185\pi$
$929$	7.16237e9	0.293091	0.146545	+	0.989204i	$0.453185\pi$
$930$	0	0
$931$	−1.52941e9	−0.0621155
$932$	0	0
$933$	5.11551e10	2.06207
$934$	0	0
$935$	−1.18899e10	−0.475706
$936$	0	0
$937$	−1.18833e10	−0.471897	−0.235948	−	0.971766i	$-0.575820\pi$
$937$	−1.18833e10	−0.471897	−0.235948	+	0.971766i	$0.575820\pi$
$938$	0	0
$939$	−9.15698e9	−0.360930
$940$	0	0
$941$	−1.59820e9	−0.0625270	−0.0312635	−	0.999511i	$-0.509953\pi$
$941$	−1.59820e9	−0.0625270	−0.0312635	+	0.999511i	$0.509953\pi$
$942$	0	0
$943$	8.13623e8	0.0315960
$944$	0	0
$945$	1.14323e9	0.0440680
$946$	0	0
$947$	−3.17909e10	−1.21641	−0.608203	−	0.793782i	$-0.708109\pi$
$947$	−3.17909e10	−1.21641	−0.608203	+	0.793782i	$0.708109\pi$
$948$	0	0
$949$	−3.74998e10	−1.42428
$950$	0	0
$951$	−2.61044e10	−0.984196
$952$	0	0
$953$	1.26693e10	0.474161	0.237081	−	0.971490i	$-0.423809\pi$
$953$	1.26693e10	0.474161	0.237081	+	0.971490i	$0.423809\pi$
$954$	0	0
$955$	−1.25861e9	−0.0467604
$956$	0	0
$957$	1.30127e10	0.479929
$958$	0	0
$959$	2.29812e10	0.841408
$960$	0	0
$961$	1.17513e10	0.427125
$962$	0	0
$963$	4.29245e10	1.54887
$964$	0	0
$965$	−3.16261e10	−1.13292
$966$	0	0
$967$	2.08359e10	0.741003	0.370502	−	0.928832i	$-0.379186\pi$
$967$	2.08359e10	0.741003	0.370502	+	0.928832i	$0.379186\pi$
$968$	0	0
$969$	9.41891e9	0.332558
$970$	0	0
$971$	−5.10058e10	−1.78794	−0.893968	−	0.448131i	$-0.852090\pi$
$971$	−5.10058e10	−1.78794	−0.893968	+	0.448131i	$0.852090\pi$
$972$	0	0
$973$	−2.51243e10	−0.874379
$974$	0	0
$975$	−3.44259e10	−1.18951
$976$	0	0
$977$	2.28581e10	0.784169	0.392085	−	0.919929i	$-0.371754\pi$
$977$	2.28581e10	0.784169	0.392085	+	0.919929i	$0.371754\pi$
$978$	0	0
$979$	−3.37221e10	−1.14862
$980$	0	0
$981$	−1.31648e10	−0.445219
$982$	0	0
$983$	3.39199e10	1.13898	0.569491	−	0.821998i	$-0.307140\pi$
$983$	3.39199e10	1.13898	0.569491	+	0.821998i	$0.307140\pi$
$984$	0	0
$985$	−2.29365e10	−0.764715
$986$	0	0
$987$	1.46250e10	0.484156
$988$	0	0
$989$	−1.50585e9	−0.0494988
$990$	0	0
$991$	1.49529e9	0.0488053	0.0244026	−	0.999702i	$-0.492232\pi$
$991$	1.49529e9	0.0488053	0.0244026	+	0.999702i	$0.492232\pi$
$992$	0	0
$993$	−4.37469e10	−1.41783
$994$	0	0
$995$	3.98296e9	0.128182
$996$	0	0
$997$	−5.93902e10	−1.89794	−0.948969	−	0.315370i	$-0.897871\pi$
$997$	−5.93902e10	−1.89794	−0.948969	+	0.315370i	$0.897871\pi$
$998$	0	0
$999$	4.97176e9	0.157772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.8.a.j.1.1		6
4.3	odd	2		152.8.a.a.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.8.a.a.1.6	✓	6	4.3	odd	2
304.8.a.j.1.1		6	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-65.0909 q^{3} +165.215 q^{5} +774.961 q^{7} +2049.82 q^{9} +O(q^{10})\)	\(q-65.0909 q^{3} +165.215 q^{5} +774.961 q^{7} +2049.82 q^{9} +3411.22 q^{11} -10405.3 q^{13} -10754.0 q^{15} -21097.0 q^{17} +6859.00 q^{19} -50442.9 q^{21} +2513.10 q^{23} -50828.9 q^{25} +8929.03 q^{27} -58605.6 q^{29} +198151. q^{31} -222039. q^{33} +128035. q^{35} +556808. q^{37} +677289. q^{39} +323753. q^{41} -599201. q^{43} +338662. q^{45} -289931. q^{47} -222978. q^{49} +1.37322e6 q^{51} +119297. q^{53} +563585. q^{55} -446458. q^{57} +595531. q^{59} -364583. q^{61} +1.58853e6 q^{63} -1.71911e6 q^{65} +146835. q^{67} -163580. q^{69} -2.82634e6 q^{71} +3.60391e6 q^{73} +3.30850e6 q^{75} +2.64356e6 q^{77} +151550. q^{79} -5.06416e6 q^{81} +7.67819e6 q^{83} -3.48554e6 q^{85} +3.81469e6 q^{87} -9.88564e6 q^{89} -8.06369e6 q^{91} -1.28978e7 q^{93} +1.13321e6 q^{95} -1.03471e7 q^{97} +6.99239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 29 q^{3} - 43 q^{5} + 908 q^{7} - 235 q^{9}+O(q^{10}) \) 6 * q - 29 * q^3 - 43 * q^5 + 908 * q^7 - 235 * q^9	\( 6 q - 29 q^{3} - 43 q^{5} + 908 q^{7} - 235 q^{9} - 3935 q^{11} + 6449 q^{13} - 8422 q^{15} - 6920 q^{17} + 41154 q^{19} - 54471 q^{21} + 43633 q^{23} - 48003 q^{25} - 8705 q^{27} - 134097 q^{29} + 95448 q^{31} - 212700 q^{33} + 202017 q^{35} - 57516 q^{37} + 683975 q^{39} - 399254 q^{41} + 359107 q^{43} - 668039 q^{45} + 590285 q^{47} - 1140196 q^{49} + 2380541 q^{51} - 2926531 q^{53} + 1799465 q^{55} - 198911 q^{57} + 2933563 q^{59} - 4738005 q^{61} + 2126109 q^{63} - 7312896 q^{65} + 4389837 q^{67} - 5697861 q^{69} + 751308 q^{71} - 7310018 q^{73} + 8992379 q^{75} - 12493471 q^{77} + 2495758 q^{79} - 18423454 q^{81} + 4583082 q^{83} - 13253407 q^{85} + 4493999 q^{87} - 21914280 q^{89} + 8775919 q^{91} - 18623792 q^{93} - 294937 q^{95} - 21931958 q^{97} + 2897895 q^{99}+O(q^{100}) \) 6 * q - 29 * q^3 - 43 * q^5 + 908 * q^7 - 235 * q^9 - 3935 * q^11 + 6449 * q^13 - 8422 * q^15 - 6920 * q^17 + 41154 * q^19 - 54471 * q^21 + 43633 * q^23 - 48003 * q^25 - 8705 * q^27 - 134097 * q^29 + 95448 * q^31 - 212700 * q^33 + 202017 * q^35 - 57516 * q^37 + 683975 * q^39 - 399254 * q^41 + 359107 * q^43 - 668039 * q^45 + 590285 * q^47 - 1140196 * q^49 + 2380541 * q^51 - 2926531 * q^53 + 1799465 * q^55 - 198911 * q^57 + 2933563 * q^59 - 4738005 * q^61 + 2126109 * q^63 - 7312896 * q^65 + 4389837 * q^67 - 5697861 * q^69 + 751308 * q^71 - 7310018 * q^73 + 8992379 * q^75 - 12493471 * q^77 + 2495758 * q^79 - 18423454 * q^81 + 4583082 * q^83 - 13253407 * q^85 + 4493999 * q^87 - 21914280 * q^89 + 8775919 * q^91 - 18623792 * q^93 - 294937 * q^95 - 21931958 * q^97 + 2897895 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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