LMFDB - Embedded newform 3150.3.c.b.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$-1.28897 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.b.449.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} -12.1382i q^{11} +18.5830i q^{13} -3.74166i q^{14} +4.00000 q^{16} +10.9015 q^{17} -20.0000 q^{19} -17.1660i q^{22} -12.1382 q^{23} +26.2803i q^{26} -5.29150i q^{28} -41.8367i q^{29} +25.1660 q^{31} +5.65685 q^{32} +15.4170 q^{34} +38.0000i q^{37} -28.2843 q^{38} -60.6337i q^{41} -83.4980i q^{43} -24.2764i q^{44} -17.1660 q^{46} +16.9706 q^{47} -7.00000 q^{49} +37.1660i q^{52} +94.0424 q^{53} -7.48331i q^{56} -59.1660i q^{58} +58.2175i q^{59} +15.6680 q^{61} +35.5901 q^{62} +8.00000 q^{64} -132.664i q^{67} +21.8029 q^{68} -12.1382i q^{71} +76.9150i q^{73} +53.7401i q^{74} -40.0000 q^{76} -32.1147 q^{77} -33.6680 q^{79} -85.7490i q^{82} -60.5764 q^{83} -118.084i q^{86} -34.3320i q^{88} +4.77506i q^{89} +49.1660 q^{91} -24.2764 q^{92} +24.0000 q^{94} -188.413i q^{97} -9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 32 q^{16} - 160 q^{19} + 32 q^{31} + 208 q^{34} + 32 q^{46} - 56 q^{49} + 464 q^{61} + 64 q^{64} - 320 q^{76} - 608 q^{79} + 224 q^{91} + 192 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 32 * q^16 - 160 * q^19 + 32 * q^31 + 208 * q^34 + 32 * q^46 - 56 * q^49 + 464 * q^61 + 64 * q^64 - 320 * q^76 - 608 * q^79 + 224 * q^91 + 192 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	2.82843	0.353553
$9$	0	0
$10$	0	0
$11$	− 12.1382i	− 1.10347i	−0.834019	−	0.551736i	$-0.813965\pi$
$11$	− 12.1382i	− 1.10347i	0.834019	−	0.551736i	$-0.186035\pi$
$12$	0	0
$13$	18.5830i	1.42946i	0.699399	+	0.714731i	$0.253451\pi$
$13$	18.5830i	1.42946i	−0.699399	+	0.714731i	$0.746549\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	10.9015	0.641262	0.320631	−	0.947204i	$-0.396105\pi$
$17$	10.9015	0.641262	0.320631	+	0.947204i	$0.396105\pi$
$18$	0	0
$19$	−20.0000	−1.05263	−0.526316	−	0.850289i	$-0.676427\pi$
$19$	−20.0000	−1.05263	−0.526316	+	0.850289i	$0.676427\pi$
$20$	0	0
$21$	0	0
$22$	− 17.1660i	− 0.780273i
$23$	−12.1382	−0.527748	−0.263874	−	0.964557i	$-0.585000\pi$
$23$	−12.1382	−0.527748	−0.263874	+	0.964557i	$0.585000\pi$
$24$	0	0
$25$	0	0
$26$	26.2803i	1.01078i
$27$	0	0
$28$	− 5.29150i	− 0.188982i
$29$	− 41.8367i	− 1.44264i	−0.692600	−	0.721322i	$-0.743535\pi$
$29$	− 41.8367i	− 1.44264i	0.692600	−	0.721322i	$-0.256465\pi$
$30$	0	0
$31$	25.1660	0.811807	0.405903	−	0.913916i	$-0.366957\pi$
$31$	25.1660	0.811807	0.405903	+	0.913916i	$0.366957\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	15.4170	0.453441
$35$	0	0
$36$	0	0
$37$	38.0000i	1.02703i	0.858082	+	0.513514i	$0.171656\pi$
$37$	38.0000i	1.02703i	−0.858082	+	0.513514i	$0.828344\pi$
$38$	−28.2843	−0.744323
$39$	0	0
$40$	0	0
$41$	− 60.6337i	− 1.47887i	−0.673227	−	0.739435i	$-0.735092\pi$
$41$	− 60.6337i	− 1.47887i	0.673227	−	0.739435i	$-0.264908\pi$
$42$	0	0
$43$	− 83.4980i	− 1.94181i	−0.239455	−	0.970907i	$-0.576969\pi$
$43$	− 83.4980i	− 1.94181i	0.239455	−	0.970907i	$-0.423031\pi$
$44$	− 24.2764i	− 0.551736i
$45$	0	0
$46$	−17.1660	−0.373174
$47$	16.9706	0.361076	0.180538	−	0.983568i	$-0.442216\pi$
$47$	16.9706	0.361076	0.180538	+	0.983568i	$0.442216\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	37.1660i	0.714731i
$53$	94.0424	1.77439	0.887193	−	0.461399i	$-0.152652\pi$
$53$	94.0424	1.77439	0.887193	+	0.461399i	$0.152652\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	− 59.1660i	− 1.02010i
$59$	58.2175i	0.986738i	0.869820	+	0.493369i	$0.164235\pi$
$59$	58.2175i	0.986738i	−0.869820	+	0.493369i	$0.835765\pi$
$60$	0	0
$61$	15.6680	0.256852	0.128426	−	0.991719i	$-0.459008\pi$
$61$	15.6680	0.256852	0.128426	+	0.991719i	$0.459008\pi$
$62$	35.5901	0.574034
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 132.664i	− 1.98006i	−0.140856	−	0.990030i	$-0.544985\pi$
$67$	− 132.664i	− 1.98006i	0.140856	−	0.990030i	$-0.455015\pi$
$68$	21.8029	0.320631
$69$	0	0
$70$	0	0
$71$	− 12.1382i	− 0.170961i	−0.996340	−	0.0854803i	$-0.972758\pi$
$71$	− 12.1382i	− 0.170961i	0.996340	−	0.0854803i	$-0.0272425\pi$
$72$	0	0
$73$	76.9150i	1.05363i	0.849980	+	0.526815i	$0.176614\pi$
$73$	76.9150i	1.05363i	−0.849980	+	0.526815i	$0.823386\pi$
$74$	53.7401i	0.726218i
$75$	0	0
$76$	−40.0000	−0.526316
$77$	−32.1147	−0.417074
$78$	0	0
$79$	−33.6680	−0.426177	−0.213088	−	0.977033i	$-0.568352\pi$
$79$	−33.6680	−0.426177	−0.213088	+	0.977033i	$0.568352\pi$
$80$	0	0
$81$	0	0
$82$	− 85.7490i	− 1.04572i
$83$	−60.5764	−0.729836	−0.364918	−	0.931040i	$-0.618903\pi$
$83$	−60.5764	−0.729836	−0.364918	+	0.931040i	$0.618903\pi$
$84$	0	0
$85$	0	0
$86$	− 118.084i	− 1.37307i
$87$	0	0
$88$	− 34.3320i	− 0.390137i
$89$	4.77506i	0.0536523i	0.999640	+	0.0268262i	$0.00854006\pi$
$89$	4.77506i	0.0536523i	−0.999640	+	0.0268262i	$0.991460\pi$
$90$	0	0
$91$	49.1660	0.540286
$92$	−24.2764	−0.263874
$93$	0	0
$94$	24.0000	0.255319
$95$	0	0
$96$	0	0
$97$	− 188.413i	− 1.94240i	−0.238260	−	0.971201i	$-0.576577\pi$
$97$	− 188.413i	− 1.94240i	0.238260	−	0.971201i	$-0.423423\pi$
$98$	−9.89949	−0.101015
$99$	0	0
$100$	0	0
$101$	106.713i	1.05656i	0.849069	+	0.528282i	$0.177164\pi$
$101$	106.713i	1.05656i	−0.849069	+	0.528282i	$0.822836\pi$
$102$	0	0
$103$	− 131.498i	− 1.27668i	−0.769755	−	0.638340i	$-0.779621\pi$
$103$	− 131.498i	− 1.27668i	0.769755	−	0.638340i	$-0.220379\pi$
$104$	52.5607i	0.505391i
$105$	0	0
$106$	132.996	1.25468
$107$	−82.3793	−0.769900	−0.384950	−	0.922937i	$-0.625781\pi$
$107$	−82.3793	−0.769900	−0.384950	+	0.922937i	$0.625781\pi$
$108$	0	0
$109$	−33.8301	−0.310367	−0.155184	−	0.987886i	$-0.549597\pi$
$109$	−33.8301	−0.310367	−0.155184	+	0.987886i	$0.549597\pi$
$110$	0	0
$111$	0	0
$112$	− 10.5830i	− 0.0944911i
$113$	−28.5190	−0.252381	−0.126190	−	0.992006i	$-0.540275\pi$
$113$	−28.5190	−0.252381	−0.126190	+	0.992006i	$0.540275\pi$
$114$	0	0
$115$	0	0
$116$	− 83.6734i	− 0.721322i
$117$	0	0
$118$	82.3320i	0.697729i
$119$	− 28.8426i	− 0.242374i
$120$	0	0
$121$	−26.3360	−0.217653
$122$	22.1579	0.181622
$123$	0	0
$124$	50.3320	0.405903
$125$	0	0
$126$	0	0
$127$	129.668i	1.02101i	0.859875	+	0.510504i	$0.170541\pi$
$127$	129.668i	1.02101i	−0.859875	+	0.510504i	$0.829459\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	0	0
$131$	− 148.017i	− 1.12990i	−0.825124	−	0.564952i	$-0.808895\pi$
$131$	− 148.017i	− 1.12990i	0.825124	−	0.564952i	$-0.191105\pi$
$132$	0	0
$133$	52.9150i	0.397857i
$134$	− 187.615i	− 1.40011i
$135$	0	0
$136$	30.8340	0.226721
$137$	76.9573	0.561732	0.280866	−	0.959747i	$-0.409378\pi$
$137$	76.9573	0.561732	0.280866	+	0.959747i	$0.409378\pi$
$138$	0	0
$139$	−217.328	−1.56351	−0.781756	−	0.623585i	$-0.785676\pi$
$139$	−217.328	−1.56351	−0.781756	+	0.623585i	$0.785676\pi$
$140$	0	0
$141$	0	0
$142$	− 17.1660i	− 0.120887i
$143$	225.564	1.57737
$144$	0	0
$145$	0	0
$146$	108.774i	0.745029i
$147$	0	0
$148$	76.0000i	0.513514i
$149$	− 161.925i	− 1.08674i	−0.839492	−	0.543371i	$-0.817148\pi$
$149$	− 161.925i	− 1.08674i	0.839492	−	0.543371i	$-0.182852\pi$
$150$	0	0
$151$	−93.1660	−0.616993	−0.308497	−	0.951225i	$-0.599826\pi$
$151$	−93.1660	−0.616993	−0.308497	+	0.951225i	$0.599826\pi$
$152$	−56.5685	−0.372161
$153$	0	0
$154$	−45.4170	−0.294916
$155$	0	0
$156$	0	0
$157$	− 184.996i	− 1.17832i	−0.808017	−	0.589159i	$-0.799459\pi$
$157$	− 184.996i	− 1.17832i	0.808017	−	0.589159i	$-0.200541\pi$
$158$	−47.6137	−0.301353
$159$	0	0
$160$	0	0
$161$	32.1147i	0.199470i
$162$	0	0
$163$	− 86.9961i	− 0.533718i	−0.963736	−	0.266859i	$-0.914014\pi$
$163$	− 86.9961i	− 0.533718i	0.963736	−	0.266859i	$-0.0859858\pi$
$164$	− 121.267i	− 0.739435i
$165$	0	0
$166$	−85.6680	−0.516072
$167$	−60.5764	−0.362733	−0.181366	−	0.983416i	$-0.558052\pi$
$167$	−60.5764	−0.362733	−0.181366	+	0.983416i	$0.558052\pi$
$168$	0	0
$169$	−176.328	−1.04336
$170$	0	0
$171$	0	0
$172$	− 166.996i	− 0.970907i
$173$	162.572	0.939721	0.469860	−	0.882741i	$-0.344304\pi$
$173$	162.572	0.939721	0.469860	+	0.882741i	$0.344304\pi$
$174$	0	0
$175$	0	0
$176$	− 48.5528i	− 0.275868i
$177$	0	0
$178$	6.75295i	0.0379379i
$179$	− 223.091i	− 1.24632i	−0.782095	−	0.623159i	$-0.785849\pi$
$179$	− 223.091i	− 1.24632i	0.782095	−	0.623159i	$-0.214151\pi$
$180$	0	0
$181$	188.915	1.04373	0.521865	−	0.853028i	$-0.325237\pi$
$181$	188.915	1.04373	0.521865	+	0.853028i	$0.325237\pi$
$182$	69.5312	0.382040
$183$	0	0
$184$	−34.3320	−0.186587
$185$	0	0
$186$	0	0
$187$	− 132.324i	− 0.707616i
$188$	33.9411	0.180538
$189$	0	0
$190$	0	0
$191$	− 228.038i	− 1.19391i	−0.802273	−	0.596957i	$-0.796376\pi$
$191$	− 228.038i	− 1.19391i	0.802273	−	0.596957i	$-0.203624\pi$
$192$	0	0
$193$	− 134.000i	− 0.694301i	−0.937810	−	0.347150i	$-0.887149\pi$
$193$	− 134.000i	− 0.694301i	0.937810	−	0.347150i	$-0.112851\pi$
$194$	− 266.456i	− 1.37349i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	188.560	0.957157	0.478579	−	0.878045i	$-0.341152\pi$
$197$	188.560	0.957157	0.478579	+	0.878045i	$0.341152\pi$
$198$	0	0
$199$	−102.494	−0.515046	−0.257523	−	0.966272i	$-0.582906\pi$
$199$	−102.494	−0.515046	−0.257523	+	0.966272i	$0.582906\pi$
$200$	0	0
$201$	0	0
$202$	150.915i	0.747104i
$203$	−110.689	−0.545268
$204$	0	0
$205$	0	0
$206$	− 185.966i	− 0.902749i
$207$	0	0
$208$	74.3320i	0.357365i
$209$	242.764i	1.16155i
$210$	0	0
$211$	−84.5020	−0.400483	−0.200242	−	0.979747i	$-0.564173\pi$
$211$	−84.5020	−0.400483	−0.200242	+	0.979747i	$0.564173\pi$
$212$	188.085	0.887193
$213$	0	0
$214$	−116.502	−0.544402
$215$	0	0
$216$	0	0
$217$	− 66.5830i	− 0.306834i
$218$	−47.8429	−0.219463
$219$	0	0
$220$	0	0
$221$	202.582i	0.916660i
$222$	0	0
$223$	158.494i	0.710736i	0.934727	+	0.355368i	$0.115644\pi$
$223$	158.494i	0.710736i	−0.934727	+	0.355368i	$0.884356\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	−40.3320	−0.178460
$227$	−101.823	−0.448561	−0.224281	−	0.974525i	$-0.572003\pi$
$227$	−101.823	−0.448561	−0.224281	+	0.974525i	$0.572003\pi$
$228$	0	0
$229$	268.915	1.17430	0.587151	−	0.809478i	$-0.300250\pi$
$229$	268.915	1.17430	0.587151	+	0.809478i	$0.300250\pi$
$230$	0	0
$231$	0	0
$232$	− 118.332i	− 0.510052i
$233$	−26.2748	−0.112767	−0.0563836	−	0.998409i	$-0.517957\pi$
$233$	−26.2748	−0.112767	−0.0563836	+	0.998409i	$0.517957\pi$
$234$	0	0
$235$	0	0
$236$	116.435i	0.493369i
$237$	0	0
$238$	− 40.7895i	− 0.171385i
$239$	− 92.2733i	− 0.386081i	−0.981191	−	0.193040i	$-0.938165\pi$
$239$	− 92.2733i	− 0.386081i	0.981191	−	0.193040i	$-0.0618348\pi$
$240$	0	0
$241$	343.247	1.42426	0.712131	−	0.702047i	$-0.247730\pi$
$241$	343.247	1.42426	0.712131	+	0.702047i	$0.247730\pi$
$242$	−37.2447	−0.153904
$243$	0	0
$244$	31.3360	0.128426
$245$	0	0
$246$	0	0
$247$	− 371.660i	− 1.50470i
$248$	71.1802	0.287017
$249$	0	0
$250$	0	0
$251$	356.382i	1.41985i	0.704278	+	0.709924i	$0.251271\pi$
$251$	356.382i	1.41985i	−0.704278	+	0.709924i	$0.748729\pi$
$252$	0	0
$253$	147.336i	0.582356i
$254$	183.378i	0.721961i
$255$	0	0
$256$	16.0000	0.0625000
$257$	254.730	0.991169	0.495584	−	0.868560i	$-0.334954\pi$
$257$	254.730	0.991169	0.495584	+	0.868560i	$0.334954\pi$
$258$	0	0
$259$	100.539	0.388180
$260$	0	0
$261$	0	0
$262$	− 209.328i	− 0.798962i
$263$	261.979	0.996117	0.498059	−	0.867143i	$-0.334046\pi$
$263$	261.979	0.996117	0.498059	+	0.867143i	$0.334046\pi$
$264$	0	0
$265$	0	0
$266$	74.8331i	0.281328i
$267$	0	0
$268$	− 265.328i	− 0.990030i
$269$	− 93.6246i	− 0.348047i	−0.984742	−	0.174023i	$-0.944323\pi$
$269$	− 93.6246i	− 0.348047i	0.984742	−	0.174023i	$-0.0556768\pi$
$270$	0	0
$271$	1.16601	0.00430262	0.00215131	−	0.999998i	$-0.499315\pi$
$271$	1.16601	0.00430262	0.00215131	+	0.999998i	$0.499315\pi$
$272$	43.6058	0.160316
$273$	0	0
$274$	108.834	0.397204
$275$	0	0
$276$	0	0
$277$	32.0000i	0.115523i	0.998330	+	0.0577617i	$0.0183964\pi$
$277$	32.0000i	0.115523i	−0.998330	+	0.0577617i	$0.981604\pi$
$278$	−307.348	−1.10557
$279$	0	0
$280$	0	0
$281$	166.757i	0.593441i	0.954964	+	0.296721i	$0.0958930\pi$
$281$	166.757i	0.593441i	−0.954964	+	0.296721i	$0.904107\pi$
$282$	0	0
$283$	− 16.3399i	− 0.0577381i	−0.999583	−	0.0288691i	$-0.990809\pi$
$283$	− 16.3399i	− 0.0577381i	0.999583	−	0.0288691i	$-0.00919059\pi$
$284$	− 24.2764i	− 0.0854803i
$285$	0	0
$286$	318.996	1.11537
$287$	−160.422	−0.558961
$288$	0	0
$289$	−170.158	−0.588782
$290$	0	0
$291$	0	0
$292$	153.830i	0.526815i
$293$	368.921	1.25912	0.629558	−	0.776953i	$-0.283236\pi$
$293$	368.921	1.25912	0.629558	+	0.776953i	$0.283236\pi$
$294$	0	0
$295$	0	0
$296$	107.480i	0.363109i
$297$	0	0
$298$	− 228.996i	− 0.768443i
$299$	− 225.564i	− 0.754396i
$300$	0	0
$301$	−220.915	−0.733937
$302$	−131.757	−0.436280
$303$	0	0
$304$	−80.0000	−0.263158
$305$	0	0
$306$	0	0
$307$	− 192.664i	− 0.627570i	−0.949494	−	0.313785i	$-0.898403\pi$
$307$	− 192.664i	− 0.627570i	0.949494	−	0.313785i	$-0.101597\pi$
$308$	−64.2293	−0.208537
$309$	0	0
$310$	0	0
$311$	131.276i	0.422109i	0.977474	+	0.211055i	$0.0676898\pi$
$311$	131.276i	0.422109i	−0.977474	+	0.211055i	$0.932310\pi$
$312$	0	0
$313$	− 43.3281i	− 0.138428i	−0.997602	−	0.0692142i	$-0.977951\pi$
$313$	− 43.3281i	− 0.138428i	0.997602	−	0.0692142i	$-0.0220492\pi$
$314$	− 261.624i	− 0.833197i
$315$	0	0
$316$	−67.3360	−0.213088
$317$	251.724	0.794083	0.397042	−	0.917801i	$-0.370037\pi$
$317$	251.724	0.794083	0.397042	+	0.917801i	$0.370037\pi$
$318$	0	0
$319$	−507.822	−1.59192
$320$	0	0
$321$	0	0
$322$	45.4170i	0.141047i
$323$	−218.029	−0.675013
$324$	0	0
$325$	0	0
$326$	− 123.031i	− 0.377396i
$327$	0	0
$328$	− 171.498i	− 0.522860i
$329$	− 44.8999i	− 0.136474i
$330$	0	0
$331$	361.490	1.09212	0.546058	−	0.837748i	$-0.316128\pi$
$331$	361.490	1.09212	0.546058	+	0.837748i	$0.316128\pi$
$332$	−121.153	−0.364918
$333$	0	0
$334$	−85.6680	−0.256491
$335$	0	0
$336$	0	0
$337$	− 298.834i	− 0.886748i	−0.896337	−	0.443374i	$-0.853781\pi$
$337$	− 298.834i	− 0.886748i	0.896337	−	0.443374i	$-0.146219\pi$
$338$	−249.366	−0.737768
$339$	0	0
$340$	0	0
$341$	− 305.470i	− 0.895807i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	− 236.168i	− 0.686535i
$345$	0	0
$346$	229.911	0.664483
$347$	−206.120	−0.594006	−0.297003	−	0.954876i	$-0.595987\pi$
$347$	−206.120	−0.594006	−0.297003	+	0.954876i	$0.595987\pi$
$348$	0	0
$349$	−434.324	−1.24448	−0.622241	−	0.782826i	$-0.713778\pi$
$349$	−434.324	−1.24448	−0.622241	+	0.782826i	$0.713778\pi$
$350$	0	0
$351$	0	0
$352$	− 68.6640i	− 0.195068i
$353$	−185.439	−0.525324	−0.262662	−	0.964888i	$-0.584600\pi$
$353$	−185.439	−0.525324	−0.262662	+	0.964888i	$0.584600\pi$
$354$	0	0
$355$	0	0
$356$	9.55012i	0.0268262i
$357$	0	0
$358$	− 315.498i	− 0.881279i
$359$	516.767i	1.43946i	0.694254	+	0.719731i	$0.255735\pi$
$359$	516.767i	1.43946i	−0.694254	+	0.719731i	$0.744265\pi$
$360$	0	0
$361$	39.0000	0.108033
$362$	267.166	0.738028
$363$	0	0
$364$	98.3320	0.270143
$365$	0	0
$366$	0	0
$367$	− 117.490i	− 0.320137i	−0.987106	−	0.160068i	$-0.948829\pi$
$367$	− 117.490i	− 0.320137i	0.987106	−	0.160068i	$-0.0511715\pi$
$368$	−48.5528	−0.131937
$369$	0	0
$370$	0	0
$371$	− 248.813i	− 0.670655i
$372$	0	0
$373$	402.664i	1.07953i	0.841816	+	0.539764i	$0.181487\pi$
$373$	402.664i	1.07953i	−0.841816	+	0.539764i	$0.818513\pi$
$374$	− 187.135i	− 0.500360i
$375$	0	0
$376$	48.0000	0.127660
$377$	777.451	2.06221
$378$	0	0
$379$	−398.834	−1.05233	−0.526166	−	0.850382i	$-0.676371\pi$
$379$	−398.834	−1.05233	−0.526166	+	0.850382i	$0.676371\pi$
$380$	0	0
$381$	0	0
$382$	− 322.494i	− 0.844225i
$383$	744.804	1.94466	0.972329	−	0.233614i	$-0.0750553\pi$
$383$	744.804	1.94466	0.972329	+	0.233614i	$0.0750553\pi$
$384$	0	0
$385$	0	0
$386$	− 189.505i	− 0.490945i
$387$	0	0
$388$	− 376.826i	− 0.971201i
$389$	535.162i	1.37574i	0.725834	+	0.687869i	$0.241454\pi$
$389$	535.162i	1.37574i	−0.725834	+	0.687869i	$0.758546\pi$
$390$	0	0
$391$	−132.324	−0.338425
$392$	−19.7990	−0.0505076
$393$	0	0
$394$	266.664	0.676812
$395$	0	0
$396$	0	0
$397$	− 94.3241i	− 0.237592i	−0.992919	−	0.118796i	$-0.962096\pi$
$397$	− 94.3241i	− 0.237592i	0.992919	−	0.118796i	$-0.0379035\pi$
$398$	−144.949	−0.364192
$399$	0	0
$400$	0	0
$401$	− 103.593i	− 0.258335i	−0.991623	−	0.129168i	$-0.958769\pi$
$401$	− 103.593i	− 0.258335i	0.991623	−	0.129168i	$-0.0412306\pi$
$402$	0	0
$403$	467.660i	1.16045i
$404$	213.426i	0.528282i
$405$	0	0
$406$	−156.539	−0.385563
$407$	461.252	1.13330
$408$	0	0
$409$	9.75689	0.0238555	0.0119277	−	0.999929i	$-0.496203\pi$
$409$	9.75689	0.0238555	0.0119277	+	0.999929i	$0.496203\pi$
$410$	0	0
$411$	0	0
$412$	− 262.996i	− 0.638340i
$413$	154.029	0.372952
$414$	0	0
$415$	0	0
$416$	105.121i	0.252696i
$417$	0	0
$418$	343.320i	0.821340i
$419$	− 339.411i	− 0.810051i	−0.914305	−	0.405025i	$-0.867263\pi$
$419$	− 339.411i	− 0.810051i	0.914305	−	0.405025i	$-0.132737\pi$
$420$	0	0
$421$	−599.320	−1.42356	−0.711782	−	0.702401i	$-0.752111\pi$
$421$	−599.320	−1.42356	−0.711782	+	0.702401i	$0.752111\pi$
$422$	−119.504	−0.283184
$423$	0	0
$424$	265.992	0.627340
$425$	0	0
$426$	0	0
$427$	− 41.4536i	− 0.0970810i
$428$	−164.759	−0.384950
$429$	0	0
$430$	0	0
$431$	710.978i	1.64960i	0.565424	+	0.824800i	$0.308712\pi$
$431$	710.978i	1.64960i	−0.565424	+	0.824800i	$0.691288\pi$
$432$	0	0
$433$	− 377.984i	− 0.872943i	−0.899718	−	0.436471i	$-0.856228\pi$
$433$	− 377.984i	− 0.872943i	0.899718	−	0.436471i	$-0.143772\pi$
$434$	− 94.1626i	− 0.216964i
$435$	0	0
$436$	−67.6601	−0.155184
$437$	242.764	0.555524
$438$	0	0
$439$	−528.146	−1.20307	−0.601533	−	0.798848i	$-0.705443\pi$
$439$	−528.146	−1.20307	−0.601533	+	0.798848i	$0.705443\pi$
$440$	0	0
$441$	0	0
$442$	286.494i	0.648177i
$443$	36.6438	0.0827174	0.0413587	−	0.999144i	$-0.486831\pi$
$443$	36.6438	0.0827174	0.0413587	+	0.999144i	$0.486831\pi$
$444$	0	0
$445$	0	0
$446$	224.144i	0.502566i
$447$	0	0
$448$	− 21.1660i	− 0.0472456i
$449$	397.612i	0.885550i	0.896633	+	0.442775i	$0.146006\pi$
$449$	397.612i	0.885550i	−0.896633	+	0.442775i	$0.853994\pi$
$450$	0	0
$451$	−735.984	−1.63189
$452$	−57.0381	−0.126190
$453$	0	0
$454$	−144.000	−0.317181
$455$	0	0
$456$	0	0
$457$	344.324i	0.753445i	0.926326	+	0.376722i	$0.122949\pi$
$457$	344.324i	0.753445i	−0.926326	+	0.376722i	$0.877051\pi$
$458$	380.303	0.830357
$459$	0	0
$460$	0	0
$461$	− 370.936i	− 0.804634i	−0.915500	−	0.402317i	$-0.868205\pi$
$461$	− 370.936i	− 0.804634i	0.915500	−	0.402317i	$-0.131795\pi$
$462$	0	0
$463$	− 78.3320i	− 0.169184i	−0.996416	−	0.0845918i	$-0.973041\pi$
$463$	− 78.3320i	− 0.169184i	0.996416	−	0.0845918i	$-0.0269586\pi$
$464$	− 167.347i	− 0.360661i
$465$	0	0
$466$	−37.1581	−0.0797385
$467$	399.758	0.856014	0.428007	−	0.903775i	$-0.359216\pi$
$467$	399.758	0.856014	0.428007	+	0.903775i	$0.359216\pi$
$468$	0	0
$469$	−350.996	−0.748392
$470$	0	0
$471$	0	0
$472$	164.664i	0.348864i
$473$	−1013.52	−2.14274
$474$	0	0
$475$	0	0
$476$	− 57.6851i	− 0.121187i
$477$	0	0
$478$	− 130.494i	− 0.273000i
$479$	703.328i	1.46833i	0.678973	+	0.734163i	$0.262425\pi$
$479$	703.328i	1.46833i	−0.678973	+	0.734163i	$0.737575\pi$
$480$	0	0
$481$	−706.154	−1.46810
$482$	485.425	1.00711
$483$	0	0
$484$	−52.6719	−0.108826
$485$	0	0
$486$	0	0
$487$	− 82.5098i	− 0.169425i	−0.996405	−	0.0847124i	$-0.973003\pi$
$487$	− 82.5098i	− 0.169425i	0.996405	−	0.0847124i	$-0.0269971\pi$
$488$	44.3157	0.0908109
$489$	0	0
$490$	0	0
$491$	− 184.203i	− 0.375158i	−0.982249	−	0.187579i	$-0.939936\pi$
$491$	− 184.203i	− 0.375158i	0.982249	−	0.187579i	$-0.0600641\pi$
$492$	0	0
$493$	− 456.081i	− 0.925114i
$494$	− 525.607i	− 1.06398i
$495$	0	0
$496$	100.664	0.202952
$497$	−32.1147	−0.0646170
$498$	0	0
$499$	−752.810	−1.50864	−0.754319	−	0.656508i	$-0.772033\pi$
$499$	−752.810	−1.50864	−0.754319	+	0.656508i	$0.772033\pi$
$500$	0	0
$501$	0	0
$502$	504.000i	1.00398i
$503$	−662.540	−1.31718	−0.658588	−	0.752504i	$-0.728846\pi$
$503$	−662.540	−1.31718	−0.658588	+	0.752504i	$0.728846\pi$
$504$	0	0
$505$	0	0
$506$	208.365i	0.411788i
$507$	0	0
$508$	259.336i	0.510504i
$509$	− 949.115i	− 1.86467i	−0.361601	−	0.932333i	$-0.617770\pi$
$509$	− 949.115i	− 1.86467i	0.361601	−	0.932333i	$-0.382230\pi$
$510$	0	0
$511$	203.498	0.398235
$512$	22.6274	0.0441942
$513$	0	0
$514$	360.243	0.700862
$515$	0	0
$516$	0	0
$517$	− 205.992i	− 0.398437i
$518$	142.183	0.274485
$519$	0	0
$520$	0	0
$521$	714.344i	1.37110i	0.728025	+	0.685551i	$0.240439\pi$
$521$	714.344i	1.37110i	−0.728025	+	0.685551i	$0.759561\pi$
$522$	0	0
$523$	232.000i	0.443595i	0.975093	+	0.221797i	$0.0711923\pi$
$523$	232.000i	0.443595i	−0.975093	+	0.221797i	$0.928808\pi$
$524$	− 296.035i	− 0.564952i
$525$	0	0
$526$	370.494	0.704361
$527$	274.346	0.520581
$528$	0	0
$529$	−381.664	−0.721482
$530$	0	0
$531$	0	0
$532$	105.830i	0.198929i
$533$	1126.76	2.11399
$534$	0	0
$535$	0	0
$536$	− 375.231i	− 0.700057i
$537$	0	0
$538$	− 132.405i	− 0.246106i
$539$	84.9674i	0.157639i
$540$	0	0
$541$	165.668	0.306225	0.153113	−	0.988209i	$-0.451070\pi$
$541$	165.668	0.306225	0.153113	+	0.988209i	$0.451070\pi$
$542$	1.64899	0.00304241
$543$	0	0
$544$	61.6680	0.113360
$545$	0	0
$546$	0	0
$547$	295.676i	0.540541i	0.962784	+	0.270270i	$0.0871131\pi$
$547$	295.676i	0.540541i	−0.962784	+	0.270270i	$0.912887\pi$
$548$	153.915	0.280866
$549$	0	0
$550$	0	0
$551$	836.734i	1.51857i
$552$	0	0
$553$	89.0771i	0.161080i
$554$	45.2548i	0.0816874i
$555$	0	0
$556$	−434.656	−0.781756
$557$	−76.8426	−0.137958	−0.0689790	−	0.997618i	$-0.521974\pi$
$557$	−76.8426	−0.137958	−0.0689790	+	0.997618i	$0.521974\pi$
$558$	0	0
$559$	1551.64	2.77575
$560$	0	0
$561$	0	0
$562$	235.830i	0.419626i
$563$	1016.33	1.80521	0.902605	−	0.430470i	$-0.141652\pi$
$563$	1016.33	1.80521	0.902605	+	0.430470i	$0.141652\pi$
$564$	0	0
$565$	0	0
$566$	− 23.1081i	− 0.0408270i
$567$	0	0
$568$	− 34.3320i	− 0.0604437i
$569$	586.533i	1.03081i	0.856946	+	0.515406i	$0.172359\pi$
$569$	586.533i	1.03081i	−0.856946	+	0.515406i	$0.827641\pi$
$570$	0	0
$571$	951.644	1.66663	0.833314	−	0.552800i	$-0.186441\pi$
$571$	951.644	1.66663	0.833314	+	0.552800i	$0.186441\pi$
$572$	451.129	0.788686
$573$	0	0
$574$	−226.871	−0.395245
$575$	0	0
$576$	0	0
$577$	− 148.672i	− 0.257664i	−0.991666	−	0.128832i	$-0.958877\pi$
$577$	− 148.672i	− 0.257664i	0.991666	−	0.128832i	$-0.0411227\pi$
$578$	−240.640	−0.416332
$579$	0	0
$580$	0	0
$581$	160.270i	0.275852i
$582$	0	0
$583$	− 1141.51i	− 1.95799i
$584$	217.549i	0.372515i
$585$	0	0
$586$	521.733	0.890330
$587$	−332.564	−0.566548	−0.283274	−	0.959039i	$-0.591421\pi$
$587$	−332.564	−0.566548	−0.283274	+	0.959039i	$0.591421\pi$
$588$	0	0
$589$	−503.320	−0.854533
$590$	0	0
$591$	0	0
$592$	152.000i	0.256757i
$593$	217.251	0.366359	0.183180	−	0.983079i	$-0.441361\pi$
$593$	217.251	0.366359	0.183180	+	0.983079i	$0.441361\pi$
$594$	0	0
$595$	0	0
$596$	− 323.849i	− 0.543371i
$597$	0	0
$598$	− 318.996i	− 0.533438i
$599$	172.179i	0.287444i	0.989618	+	0.143722i	$0.0459072\pi$
$599$	172.179i	0.287444i	−0.989618	+	0.143722i	$0.954093\pi$
$600$	0	0
$601$	−418.000	−0.695507	−0.347754	−	0.937586i	$-0.613055\pi$
$601$	−418.000	−0.695507	−0.347754	+	0.937586i	$0.613055\pi$
$602$	−312.421	−0.518972
$603$	0	0
$604$	−186.332	−0.308497
$605$	0	0
$606$	0	0
$607$	627.158i	1.03321i	0.856224	+	0.516605i	$0.172804\pi$
$607$	627.158i	1.03321i	−0.856224	+	0.516605i	$0.827196\pi$
$608$	−113.137	−0.186081
$609$	0	0
$610$	0	0
$611$	315.364i	0.516144i
$612$	0	0
$613$	279.328i	0.455674i	0.973699	+	0.227837i	$0.0731653\pi$
$613$	279.328i	0.455674i	−0.973699	+	0.227837i	$0.926835\pi$
$614$	− 272.468i	− 0.443759i
$615$	0	0
$616$	−90.8340	−0.147458
$617$	−358.380	−0.580843	−0.290422	−	0.956899i	$-0.593796\pi$
$617$	−358.380	−0.580843	−0.290422	+	0.956899i	$0.593796\pi$
$618$	0	0
$619$	983.644	1.58909	0.794543	−	0.607208i	$-0.207710\pi$
$619$	983.644	1.58909	0.794543	+	0.607208i	$0.207710\pi$
$620$	0	0
$621$	0	0
$622$	185.652i	0.298476i
$623$	12.6336	0.0202787
$624$	0	0
$625$	0	0
$626$	− 61.2752i	− 0.0978836i
$627$	0	0
$628$	− 369.992i	− 0.589159i
$629$	414.256i	0.658594i
$630$	0	0
$631$	−298.996	−0.473845	−0.236922	−	0.971529i	$-0.576139\pi$
$631$	−298.996	−0.473845	−0.236922	+	0.971529i	$0.576139\pi$
$632$	−95.2274	−0.150676
$633$	0	0
$634$	355.992	0.561502
$635$	0	0
$636$	0	0
$637$	− 130.081i	− 0.204209i
$638$	−718.169	−1.12566
$639$	0	0
$640$	0	0
$641$	− 311.957i	− 0.486672i	−0.969942	−	0.243336i	$-0.921758\pi$
$641$	− 311.957i	− 0.486672i	0.969942	−	0.243336i	$-0.0782418\pi$
$642$	0	0
$643$	604.000i	0.939347i	0.882840	+	0.469673i	$0.155628\pi$
$643$	604.000i	0.939347i	−0.882840	+	0.469673i	$0.844372\pi$
$644$	64.2293i	0.0997350i
$645$	0	0
$646$	−308.340	−0.477306
$647$	−179.600	−0.277588	−0.138794	−	0.990321i	$-0.544323\pi$
$647$	−179.600	−0.277588	−0.138794	+	0.990321i	$0.544323\pi$
$648$	0	0
$649$	706.656	1.08884
$650$	0	0
$651$	0	0
$652$	− 173.992i	− 0.266859i
$653$	481.892	0.737966	0.368983	−	0.929436i	$-0.379706\pi$
$653$	481.892	0.737966	0.368983	+	0.929436i	$0.379706\pi$
$654$	0	0
$655$	0	0
$656$	− 242.535i	− 0.369718i
$657$	0	0
$658$	− 63.4980i	− 0.0965016i
$659$	− 877.408i	− 1.33142i	−0.746209	−	0.665711i	$-0.768128\pi$
$659$	− 877.408i	− 1.33142i	0.746209	−	0.665711i	$-0.231872\pi$
$660$	0	0
$661$	−521.644	−0.789175	−0.394587	−	0.918858i	$-0.629112\pi$
$661$	−521.644	−0.789175	−0.394587	+	0.918858i	$0.629112\pi$
$662$	511.224	0.772242
$663$	0	0
$664$	−171.336	−0.258036
$665$	0	0
$666$	0	0
$667$	507.822i	0.761353i
$668$	−121.153	−0.181366
$669$	0	0
$670$	0	0
$671$	− 190.181i	− 0.283429i
$672$	0	0
$673$	659.992i	0.980672i	0.871534	+	0.490336i	$0.163126\pi$
$673$	659.992i	0.980672i	−0.871534	+	0.490336i	$0.836874\pi$
$674$	− 422.615i	− 0.627025i
$675$	0	0
$676$	−352.656	−0.521681
$677$	−1016.28	−1.50115	−0.750573	−	0.660787i	$-0.770222\pi$
$677$	−1016.28	−1.50115	−0.750573	+	0.660787i	$0.770222\pi$
$678$	0	0
$679$	−498.494	−0.734159
$680$	0	0
$681$	0	0
$682$	− 432.000i	− 0.633431i
$683$	−235.114	−0.344238	−0.172119	−	0.985076i	$-0.555061\pi$
$683$	−235.114	−0.344238	−0.172119	+	0.985076i	$0.555061\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	− 333.992i	− 0.485454i
$689$	1747.59i	2.53642i
$690$	0	0
$691$	−50.9803	−0.0737776	−0.0368888	−	0.999319i	$-0.511745\pi$
$691$	−50.9803	−0.0737776	−0.0368888	+	0.999319i	$0.511745\pi$
$692$	325.143	0.469860
$693$	0	0
$694$	−291.498	−0.420026
$695$	0	0
$696$	0	0
$697$	− 660.996i	− 0.948344i
$698$	−614.227	−0.879982
$699$	0	0
$700$	0	0
$701$	141.530i	0.201898i	0.994892	+	0.100949i	$0.0321879\pi$
$701$	141.530i	0.201898i	−0.994892	+	0.100949i	$0.967812\pi$
$702$	0	0
$703$	− 760.000i	− 1.08108i
$704$	− 97.1056i	− 0.137934i
$705$	0	0
$706$	−262.251	−0.371460
$707$	282.336	0.399344
$708$	0	0
$709$	55.4980	0.0782765	0.0391382	−	0.999234i	$-0.487539\pi$
$709$	55.4980	0.0782765	0.0391382	+	0.999234i	$0.487539\pi$
$710$	0	0
$711$	0	0
$712$	13.5059i	0.0189690i
$713$	−305.470	−0.428429
$714$	0	0
$715$	0	0
$716$	− 446.182i	− 0.623159i
$717$	0	0
$718$	730.818i	1.01785i
$719$	1009.03i	1.40338i	0.712484	+	0.701688i	$0.247570\pi$
$719$	1009.03i	1.40338i	−0.712484	+	0.701688i	$0.752430\pi$
$720$	0	0
$721$	−347.911	−0.482540
$722$	55.1543	0.0763910
$723$	0	0
$724$	377.830	0.521865
$725$	0	0
$726$	0	0
$727$	− 365.182i	− 0.502313i	−0.967946	−	0.251157i	$-0.919189\pi$
$727$	− 365.182i	− 0.502313i	0.967946	−	0.251157i	$-0.0808109\pi$
$728$	139.062	0.191020
$729$	0	0
$730$	0	0
$731$	− 910.251i	− 1.24521i
$732$	0	0
$733$	353.077i	0.481688i	0.970564	+	0.240844i	$0.0774242\pi$
$733$	353.077i	0.481688i	−0.970564	+	0.240844i	$0.922576\pi$
$734$	− 166.156i	− 0.226371i
$735$	0	0
$736$	−68.6640	−0.0932935
$737$	−1610.30	−2.18494
$738$	0	0
$739$	−329.684	−0.446121	−0.223061	−	0.974805i	$-0.571605\pi$
$739$	−329.684	−0.446121	−0.223061	+	0.974805i	$0.571605\pi$
$740$	0	0
$741$	0	0
$742$	− 351.875i	− 0.474224i
$743$	−112.061	−0.150822	−0.0754112	−	0.997153i	$-0.524027\pi$
$743$	−112.061	−0.150822	−0.0754112	+	0.997153i	$0.524027\pi$
$744$	0	0
$745$	0	0
$746$	569.453i	0.763342i
$747$	0	0
$748$	− 264.648i	− 0.353808i
$749$	217.955i	0.290995i
$750$	0	0
$751$	−144.826	−0.192844	−0.0964222	−	0.995341i	$-0.530740\pi$
$751$	−144.826	−0.192844	−0.0964222	+	0.995341i	$0.530740\pi$
$752$	67.8823	0.0902690
$753$	0	0
$754$	1099.48	1.45820
$755$	0	0
$756$	0	0
$757$	78.1699i	0.103263i	0.998666	+	0.0516314i	$0.0164421\pi$
$757$	78.1699i	0.103263i	−0.998666	+	0.0516314i	$0.983558\pi$
$758$	−564.036	−0.744111
$759$	0	0
$760$	0	0
$761$	1465.50i	1.92576i	0.269928	+	0.962880i	$0.413000\pi$
$761$	1465.50i	1.92576i	−0.269928	+	0.962880i	$0.587000\pi$
$762$	0	0
$763$	89.5059i	0.117308i
$764$	− 456.076i	− 0.596957i
$765$	0	0
$766$	1053.31	1.37508
$767$	−1081.86	−1.41050
$768$	0	0
$769$	−729.320	−0.948401	−0.474200	−	0.880417i	$-0.657263\pi$
$769$	−729.320	−0.948401	−0.474200	+	0.880417i	$0.657263\pi$
$770$	0	0
$771$	0	0
$772$	− 268.000i	− 0.347150i
$773$	−434.559	−0.562172	−0.281086	−	0.959683i	$-0.590695\pi$
$773$	−434.559	−0.562172	−0.281086	+	0.959683i	$0.590695\pi$
$774$	0	0
$775$	0	0
$776$	− 532.913i	− 0.686743i
$777$	0	0
$778$	756.834i	0.972794i
$779$	1212.67i	1.55671i
$780$	0	0
$781$	−147.336	−0.188650
$782$	−187.135	−0.239303
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 15.3517i	− 0.0195066i	−0.999952	−	0.00975331i	$-0.996895\pi$
$787$	− 15.3517i	− 0.0195066i	0.999952	−	0.00975331i	$-0.00310462\pi$
$788$	377.120	0.478579
$789$	0	0
$790$	0	0
$791$	75.4543i	0.0953910i
$792$	0	0
$793$	291.158i	0.367160i
$794$	− 133.394i	− 0.168003i
$795$	0	0
$796$	−204.988	−0.257523
$797$	1043.48	1.30927	0.654633	−	0.755947i	$-0.272823\pi$
$797$	1043.48	1.30927	0.654633	+	0.755947i	$0.272823\pi$
$798$	0	0
$799$	185.004	0.231544
$800$	0	0
$801$	0	0
$802$	− 146.502i	− 0.182671i
$803$	933.610	1.16265
$804$	0	0
$805$	0	0
$806$	661.371i	0.820560i
$807$	0	0
$808$	301.830i	0.373552i
$809$	− 1041.31i	− 1.28716i	−0.765378	−	0.643581i	$-0.777448\pi$
$809$	− 1041.31i	− 1.28716i	0.765378	−	0.643581i	$-0.222552\pi$
$810$	0	0
$811$	502.316	0.619379	0.309689	−	0.950838i	$-0.399775\pi$
$811$	502.316	0.619379	0.309689	+	0.950838i	$0.399775\pi$
$812$	−221.379	−0.272634
$813$	0	0
$814$	652.308	0.801362
$815$	0	0
$816$	0	0
$817$	1669.96i	2.04402i
$818$	13.7983	0.0168684
$819$	0	0
$820$	0	0
$821$	− 23.1137i	− 0.0281531i	−0.999901	−	0.0140765i	$-0.995519\pi$
$821$	− 23.1137i	− 0.0281531i	0.999901	−	0.0140765i	$-0.00448085\pi$
$822$	0	0
$823$	600.664i	0.729847i	0.931038	+	0.364923i	$0.118905\pi$
$823$	600.664i	0.729847i	−0.931038	+	0.364923i	$0.881095\pi$
$824$	− 371.933i	− 0.451375i
$825$	0	0
$826$	217.830	0.263717
$827$	1309.21	1.58308	0.791540	−	0.611118i	$-0.209280\pi$
$827$	1309.21	1.58308	0.791540	+	0.611118i	$0.209280\pi$
$828$	0	0
$829$	621.919	0.750204	0.375102	−	0.926984i	$-0.377608\pi$
$829$	621.919	0.750204	0.375102	+	0.926984i	$0.377608\pi$
$830$	0	0
$831$	0	0
$832$	148.664i	0.178683i
$833$	−76.3102	−0.0916089
$834$	0	0
$835$	0	0
$836$	485.528i	0.580775i
$837$	0	0
$838$	− 480.000i	− 0.572792i
$839$	− 1190.30i	− 1.41871i	−0.704851	−	0.709355i	$-0.748986\pi$
$839$	− 1190.30i	− 1.41871i	0.704851	−	0.709355i	$-0.251014\pi$
$840$	0	0
$841$	−909.308	−1.08122
$842$	−847.567	−1.00661
$843$	0	0
$844$	−169.004	−0.200242
$845$	0	0
$846$	0	0
$847$	69.6784i	0.0822649i
$848$	376.170	0.443596
$849$	0	0
$850$	0	0
$851$	− 461.252i	− 0.542011i
$852$	0	0
$853$	− 137.012i	− 0.160623i	−0.996770	−	0.0803117i	$-0.974408\pi$
$853$	− 137.012i	− 0.160623i	0.996770	−	0.0803117i	$-0.0255916\pi$
$854$	− 58.6242i	− 0.0686466i
$855$	0	0
$856$	−233.004	−0.272201
$857$	−466.141	−0.543922	−0.271961	−	0.962308i	$-0.587672\pi$
$857$	−466.141	−0.543922	−0.271961	+	0.962308i	$0.587672\pi$
$858$	0	0
$859$	−23.9843	−0.0279211	−0.0139606	−	0.999903i	$-0.504444\pi$
$859$	−23.9843	−0.0279211	−0.0139606	+	0.999903i	$0.504444\pi$
$860$	0	0
$861$	0	0
$862$	1005.47i	1.16644i
$863$	0.114603	0.000132796 0	6.63982e−5	−	1.00000i	$-0.499979\pi$
$863$	0.114603	0.000132796 0	6.63982e−5		1.00000i	$0.499979\pi$
$864$	0	0
$865$	0	0
$866$	− 534.550i	− 0.617264i
$867$	0	0
$868$	− 133.166i	− 0.153417i
$869$	408.669i	0.470275i
$870$	0	0
$871$	2465.30	2.83042
$872$	−95.6858	−0.109731
$873$	0	0
$874$	343.320	0.392815
$875$	0	0
$876$	0	0
$877$	997.304i	1.13718i	0.822622	+	0.568589i	$0.192510\pi$
$877$	997.304i	1.13718i	−0.822622	+	0.568589i	$0.807490\pi$
$878$	−746.912	−0.850697
$879$	0	0
$880$	0	0
$881$	935.649i	1.06203i	0.847362	+	0.531015i	$0.178189\pi$
$881$	935.649i	1.06203i	−0.847362	+	0.531015i	$0.821811\pi$
$882$	0	0
$883$	1549.47i	1.75478i	0.479774	+	0.877392i	$0.340719\pi$
$883$	1549.47i	1.75478i	−0.479774	+	0.877392i	$0.659281\pi$
$884$	405.164i	0.458330i
$885$	0	0
$886$	51.8222	0.0584900
$887$	894.493	1.00845	0.504224	−	0.863573i	$-0.331779\pi$
$887$	894.493	1.00845	0.504224	+	0.863573i	$0.331779\pi$
$888$	0	0
$889$	343.069	0.385905
$890$	0	0
$891$	0	0
$892$	316.988i	0.355368i
$893$	−339.411	−0.380080
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	562.308i	0.626179i
$899$	− 1052.86i	− 1.17115i
$900$	0	0
$901$	1025.20	1.13785
$902$	−1040.84	−1.15392
$903$	0	0
$904$	−80.6640	−0.0892301
$905$	0	0
$906$	0	0
$907$	− 135.838i	− 0.149766i	−0.997192	−	0.0748831i	$-0.976142\pi$
$907$	− 135.838i	− 0.149766i	0.997192	−	0.0748831i	$-0.0238584\pi$
$908$	−203.647	−0.224281
$909$	0	0
$910$	0	0
$911$	− 1242.01i	− 1.36335i	−0.731655	−	0.681675i	$-0.761252\pi$
$911$	− 1242.01i	− 1.36335i	0.731655	−	0.681675i	$-0.238748\pi$
$912$	0	0
$913$	735.289i	0.805355i
$914$	486.948i	0.532766i
$915$	0	0
$916$	537.830	0.587151
$917$	−391.617	−0.427063
$918$	0	0
$919$	388.162	0.422374	0.211187	−	0.977446i	$-0.432267\pi$
$919$	388.162	0.422374	0.211187	+	0.977446i	$0.432267\pi$
$920$	0	0
$921$	0	0
$922$	− 524.583i	− 0.568962i
$923$	225.564	0.244382
$924$	0	0
$925$	0	0
$926$	− 110.778i	− 0.119631i
$927$	0	0
$928$	− 236.664i	− 0.255026i
$929$	621.694i	0.669207i	0.942359	+	0.334604i	$0.108602\pi$
$929$	621.694i	0.669207i	−0.942359	+	0.334604i	$0.891398\pi$
$930$	0	0
$931$	140.000	0.150376
$932$	−52.5495	−0.0563836
$933$	0	0
$934$	565.344	0.605293
$935$	0	0
$936$	0	0
$937$	1262.00i	1.34685i	0.739255	+	0.673426i	$0.235178\pi$
$937$	1262.00i	1.34685i	−0.739255	+	0.673426i	$0.764822\pi$
$938$	−496.383	−0.529193
$939$	0	0
$940$	0	0
$941$	672.410i	0.714569i	0.933996	+	0.357285i	$0.116297\pi$
$941$	672.410i	0.714569i	−0.933996	+	0.357285i	$0.883703\pi$
$942$	0	0
$943$	735.984i	0.780471i
$944$	232.870i	0.246684i
$945$	0	0
$946$	−1433.33	−1.51515
$947$	−1159.75	−1.22465	−0.612327	−	0.790605i	$-0.709766\pi$
$947$	−1159.75	−1.22465	−0.612327	+	0.790605i	$0.709766\pi$
$948$	0	0
$949$	−1429.31	−1.50612
$950$	0	0
$951$	0	0
$952$	− 81.5791i	− 0.0856923i
$953$	−163.104	−0.171148	−0.0855740	−	0.996332i	$-0.527272\pi$
$953$	−163.104	−0.171148	−0.0855740	+	0.996332i	$0.527272\pi$
$954$	0	0
$955$	0	0
$956$	− 184.547i	− 0.193040i
$957$	0	0
$958$	994.656i	1.03826i
$959$	− 203.610i	− 0.212315i
$960$	0	0
$961$	−327.672	−0.340970
$962$	−998.653	−1.03810
$963$	0	0
$964$	686.494	0.712131
$965$	0	0
$966$	0	0
$967$	887.012i	0.917282i	0.888622	+	0.458641i	$0.151664\pi$
$967$	887.012i	0.917282i	−0.888622	+	0.458641i	$0.848336\pi$
$968$	−74.4893	−0.0769518
$969$	0	0
$970$	0	0
$971$	− 1416.32i	− 1.45862i	−0.684183	−	0.729310i	$-0.739841\pi$
$971$	− 1416.32i	− 1.45862i	0.684183	−	0.729310i	$-0.260159\pi$
$972$	0	0
$973$	574.996i	0.590952i
$974$	− 116.687i	− 0.119801i
$975$	0	0
$976$	62.6719	0.0642130
$977$	−339.051	−0.347032	−0.173516	−	0.984831i	$-0.555513\pi$
$977$	−339.051	−0.347032	−0.173516	+	0.984831i	$0.555513\pi$
$978$	0	0
$979$	57.9606	0.0592039
$980$	0	0
$981$	0	0
$982$	− 260.502i	− 0.265277i
$983$	−487.887	−0.496324	−0.248162	−	0.968718i	$-0.579827\pi$
$983$	−487.887	−0.496324	−0.248162	+	0.968718i	$0.579827\pi$
$984$	0	0
$985$	0	0
$986$	− 644.996i	− 0.654154i
$987$	0	0
$988$	− 743.320i	− 0.752348i
$989$	1013.52i	1.02479i
$990$	0	0
$991$	−937.474	−0.945988	−0.472994	−	0.881066i	$-0.656827\pi$
$991$	−937.474	−0.945988	−0.472994	+	0.881066i	$0.656827\pi$
$992$	142.360	0.143509
$993$	0	0
$994$	−45.4170	−0.0456911
$995$	0	0
$996$	0	0
$997$	461.012i	0.462399i	0.972906	+	0.231200i	$0.0742650\pi$
$997$	461.012i	0.462399i	−0.972906	+	0.231200i	$0.925735\pi$
$998$	−1064.63	−1.06677
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.b.449.6		8
3.2	odd	2	inner	3150.3.c.b.449.1		8
5.2	odd	4		3150.3.e.e.701.4		4
5.3	odd	4		126.3.b.a.71.1	✓	4
5.4	even	2	inner	3150.3.c.b.449.4		8
15.2	even	4		3150.3.e.e.701.2		4
15.8	even	4		126.3.b.a.71.4	yes	4
15.14	odd	2	inner	3150.3.c.b.449.7		8
20.3	even	4		1008.3.d.a.449.2		4
35.3	even	12		882.3.s.i.863.3		8
35.13	even	4		882.3.b.f.197.2		4
35.18	odd	12		882.3.s.e.863.4		8
35.23	odd	12		882.3.s.e.557.1		8
35.33	even	12		882.3.s.i.557.2		8
40.3	even	4		4032.3.d.j.449.3		4
40.13	odd	4		4032.3.d.i.449.3		4
45.13	odd	12		1134.3.q.c.1079.3		8
45.23	even	12		1134.3.q.c.1079.2		8
45.38	even	12		1134.3.q.c.701.3		8
45.43	odd	12		1134.3.q.c.701.2		8
60.23	odd	4		1008.3.d.a.449.3		4
105.23	even	12		882.3.s.e.557.4		8
105.38	odd	12		882.3.s.i.863.2		8
105.53	even	12		882.3.s.e.863.1		8
105.68	odd	12		882.3.s.i.557.3		8
105.83	odd	4		882.3.b.f.197.3		4
120.53	even	4		4032.3.d.i.449.2		4
120.83	odd	4		4032.3.d.j.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.b.a.71.1	✓	4	5.3	odd	4
126.3.b.a.71.4	yes	4	15.8	even	4
882.3.b.f.197.2		4	35.13	even	4
882.3.b.f.197.3		4	105.83	odd	4
882.3.s.e.557.1		8	35.23	odd	12
882.3.s.e.557.4		8	105.23	even	12
882.3.s.e.863.1		8	105.53	even	12
882.3.s.e.863.4		8	35.18	odd	12
882.3.s.i.557.2		8	35.33	even	12
882.3.s.i.557.3		8	105.68	odd	12
882.3.s.i.863.2		8	105.38	odd	12
882.3.s.i.863.3		8	35.3	even	12
1008.3.d.a.449.2		4	20.3	even	4
1008.3.d.a.449.3		4	60.23	odd	4
1134.3.q.c.701.2		8	45.43	odd	12
1134.3.q.c.701.3		8	45.38	even	12
1134.3.q.c.1079.2		8	45.23	even	12
1134.3.q.c.1079.3		8	45.13	odd	12
3150.3.c.b.449.1		8	3.2	odd	2	inner
3150.3.c.b.449.4		8	5.4	even	2	inner
3150.3.c.b.449.6		8	1.1	even	1	trivial
3150.3.c.b.449.7		8	15.14	odd	2	inner
3150.3.e.e.701.2		4	15.2	even	4
3150.3.e.e.701.4		4	5.2	odd	4
4032.3.d.i.449.2		4	120.53	even	4
4032.3.d.i.449.3		4	40.13	odd	4
4032.3.d.j.449.2		4	120.83	odd	4
4032.3.d.j.449.3		4	40.3	even	4

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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} -12.1382i q^{11} +18.5830i q^{13} -3.74166i q^{14} +4.00000 q^{16} +10.9015 q^{17} -20.0000 q^{19} -17.1660i q^{22} -12.1382 q^{23} +26.2803i q^{26} -5.29150i q^{28} -41.8367i q^{29} +25.1660 q^{31} +5.65685 q^{32} +15.4170 q^{34} +38.0000i q^{37} -28.2843 q^{38} -60.6337i q^{41} -83.4980i q^{43} -24.2764i q^{44} -17.1660 q^{46} +16.9706 q^{47} -7.00000 q^{49} +37.1660i q^{52} +94.0424 q^{53} -7.48331i q^{56} -59.1660i q^{58} +58.2175i q^{59} +15.6680 q^{61} +35.5901 q^{62} +8.00000 q^{64} -132.664i q^{67} +21.8029 q^{68} -12.1382i q^{71} +76.9150i q^{73} +53.7401i q^{74} -40.0000 q^{76} -32.1147 q^{77} -33.6680 q^{79} -85.7490i q^{82} -60.5764 q^{83} -118.084i q^{86} -34.3320i q^{88} +4.77506i q^{89} +49.1660 q^{91} -24.2764 q^{92} +24.0000 q^{94} -188.413i q^{97} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 32 q^{16} - 160 q^{19} + 32 q^{31} + 208 q^{34} + 32 q^{46} - 56 q^{49} + 464 q^{61} + 64 q^{64} - 320 q^{76} - 608 q^{79} + 224 q^{91} + 192 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 32 * q^16 - 160 * q^19 + 32 * q^31 + 208 * q^34 + 32 * q^46 - 56 * q^49 + 464 * q^61 + 64 * q^64 - 320 * q^76 - 608 * q^79 + 224 * q^91 + 192 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more