LMFDB - Embedded newform 208.4.f.b.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.f (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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.4.f.b.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +9.00000i q^{5} -15.0000i q^{7} -26.0000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +9.00000i q^{5} -15.0000i q^{7} -26.0000 q^{9} +48.0000i q^{11} +(26.0000 + 39.0000i) q^{13} +9.00000i q^{15} -45.0000 q^{17} +6.00000i q^{19} -15.0000i q^{21} -162.000 q^{23} +44.0000 q^{25} -53.0000 q^{27} -144.000 q^{29} +264.000i q^{31} +48.0000i q^{33} +135.000 q^{35} +303.000i q^{37} +(26.0000 + 39.0000i) q^{39} +192.000i q^{41} +97.0000 q^{43} -234.000i q^{45} -111.000i q^{47} +118.000 q^{49} -45.0000 q^{51} -414.000 q^{53} -432.000 q^{55} +6.00000i q^{57} -522.000i q^{59} +376.000 q^{61} +390.000i q^{63} +(-351.000 + 234.000i) q^{65} -36.0000i q^{67} -162.000 q^{69} +357.000i q^{71} -1098.00i q^{73} +44.0000 q^{75} +720.000 q^{77} +830.000 q^{79} +649.000 q^{81} -438.000i q^{83} -405.000i q^{85} -144.000 q^{87} -438.000i q^{89} +(585.000 - 390.000i) q^{91} +264.000i q^{93} -54.0000 q^{95} +852.000i q^{97} -1248.00i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 52 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 52 * q^9	$ 2 q + 2 q^{3} - 52 q^{9} + 52 q^{13} - 90 q^{17} - 324 q^{23} + 88 q^{25} - 106 q^{27} - 288 q^{29} + 270 q^{35} + 52 q^{39} + 194 q^{43} + 236 q^{49} - 90 q^{51} - 828 q^{53} - 864 q^{55} + 752 q^{61} - 702 q^{65} - 324 q^{69} + 88 q^{75} + 1440 q^{77} + 1660 q^{79} + 1298 q^{81} - 288 q^{87} + 1170 q^{91} - 108 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 52 * q^9 + 52 * q^13 - 90 * q^17 - 324 * q^23 + 88 * q^25 - 106 * q^27 - 288 * q^29 + 270 * q^35 + 52 * q^39 + 194 * q^43 + 236 * q^49 - 90 * q^51 - 828 * q^53 - 864 * q^55 + 752 * q^61 - 702 * q^65 - 324 * q^69 + 88 * q^75 + 1440 * q^77 + 1660 * q^79 + 1298 * q^81 - 288 * q^87 + 1170 * q^91 - 108 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.00000			0.192450			0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000			0.192450			0.0962250	+	0.995360i	$0.469323\pi$
$4$		0			0
$5$			9.00000i			0.804984i	0.915423	+	0.402492i	$0.131856\pi$
$5$			9.00000i			0.804984i	−0.915423	+	0.402492i	$0.868144\pi$
$6$		0			0
$7$		−	15.0000i		−	0.809924i	−0.914334	−	0.404962i	$-0.867285\pi$
$7$		−	15.0000i		−	0.809924i	0.914334	−	0.404962i	$-0.132715\pi$
$8$		0			0
$9$	−26.0000			−0.962963
$10$		0			0
$11$			48.0000i			1.31569i	0.753155	+	0.657843i	$0.228531\pi$
$11$			48.0000i			1.31569i	−0.753155	+	0.657843i	$0.771469\pi$
$12$		0			0
$13$	26.0000	+	39.0000i	0.554700	+	0.832050i
$13$	26.0000	+	39.0000i	0.554700	+	0.832050i
$14$		0			0
$15$			9.00000i			0.154919i
$16$		0			0
$17$	−45.0000			−0.642006			−0.321003	−	0.947078i	$-0.604020\pi$
$17$	−45.0000			−0.642006			−0.321003	+	0.947078i	$0.604020\pi$
$18$		0			0
$19$			6.00000i			0.0724471i	0.999344	+	0.0362235i	$0.0115328\pi$
$19$			6.00000i			0.0724471i	−0.999344	+	0.0362235i	$0.988467\pi$
$20$		0			0
$21$		−	15.0000i		−	0.155870i
$22$		0			0
$23$	−162.000			−1.46867			−0.734333	−	0.678789i	$-0.762505\pi$
$23$	−162.000			−1.46867			−0.734333	+	0.678789i	$0.762505\pi$
$24$		0			0
$25$	44.0000			0.352000
$26$		0			0
$27$	−53.0000			−0.377772
$28$		0			0
$29$	−144.000			−0.922073			−0.461037	−	0.887381i	$-0.652522\pi$
$29$	−144.000			−0.922073			−0.461037	+	0.887381i	$0.652522\pi$
$30$		0			0
$31$			264.000i			1.52954i	0.644302	+	0.764771i	$0.277148\pi$
$31$			264.000i			1.52954i	−0.644302	+	0.764771i	$0.722852\pi$
$32$		0			0
$33$			48.0000i			0.253204i
$34$		0			0
$35$	135.000			0.651976
$36$		0			0
$37$			303.000i			1.34629i	0.739509	+	0.673147i	$0.235058\pi$
$37$			303.000i			1.34629i	−0.739509	+	0.673147i	$0.764942\pi$
$38$		0			0
$39$	26.0000	+	39.0000i	0.106752	+	0.160128i
$40$		0			0
$41$			192.000i			0.731350i	0.930743	+	0.365675i	$0.119162\pi$
$41$			192.000i			0.731350i	−0.930743	+	0.365675i	$0.880838\pi$
$42$		0			0
$43$	97.0000			0.344008			0.172004	−	0.985096i	$-0.444976\pi$
$43$	97.0000			0.344008			0.172004	+	0.985096i	$0.444976\pi$
$44$		0			0
$45$		−	234.000i		−	0.775170i
$46$		0			0
$47$		−	111.000i		−	0.344490i	−0.985054	−	0.172245i	$-0.944898\pi$
$47$		−	111.000i		−	0.344490i	0.985054	−	0.172245i	$-0.0551020\pi$
$48$		0			0
$49$	118.000			0.344023
$50$		0			0
$51$	−45.0000			−0.123554
$52$		0			0
$53$	−414.000			−1.07297			−0.536484	−	0.843911i	$-0.680248\pi$
$53$	−414.000			−1.07297			−0.536484	+	0.843911i	$0.680248\pi$
$54$		0			0
$55$	−432.000			−1.05911
$56$		0			0
$57$			6.00000i			0.0139424i
$58$		0			0
$59$		−	522.000i		−	1.15184i	−0.817506	−	0.575920i	$-0.804644\pi$
$59$		−	522.000i		−	1.15184i	0.817506	−	0.575920i	$-0.195356\pi$
$60$		0			0
$61$	376.000			0.789211			0.394605	−	0.918851i	$-0.370881\pi$
$61$	376.000			0.789211			0.394605	+	0.918851i	$0.370881\pi$
$62$		0			0
$63$			390.000i			0.779927i
$64$		0			0
$65$	−351.000	+	234.000i	−0.669788	+	0.446525i
$66$		0			0
$67$		−	36.0000i		−	0.0656433i	−0.999461	−	0.0328216i	$-0.989551\pi$
$67$		−	36.0000i		−	0.0656433i	0.999461	−	0.0328216i	$-0.0104493\pi$
$68$		0			0
$69$	−162.000			−0.282645
$70$		0			0
$71$			357.000i			0.596734i	0.954451	+	0.298367i	$0.0964419\pi$
$71$			357.000i			0.596734i	−0.954451	+	0.298367i	$0.903558\pi$
$72$		0			0
$73$		−	1098.00i		−	1.76043i	−0.474578	−	0.880214i	$-0.657399\pi$
$73$		−	1098.00i		−	1.76043i	0.474578	−	0.880214i	$-0.342601\pi$
$74$		0			0
$75$	44.0000			0.0677424
$76$		0			0
$77$	720.000			1.06561
$78$		0			0
$79$	830.000			1.18205			0.591027	−	0.806652i	$-0.298723\pi$
$79$	830.000			1.18205			0.591027	+	0.806652i	$0.298723\pi$
$80$		0			0
$81$	649.000			0.890261
$82$		0			0
$83$		−	438.000i		−	0.579238i	−0.957142	−	0.289619i	$-0.906471\pi$
$83$		−	438.000i		−	0.579238i	0.957142	−	0.289619i	$-0.0935286\pi$
$84$		0			0
$85$		−	405.000i		−	0.516805i
$86$		0			0
$87$	−144.000			−0.177453
$88$		0			0
$89$		−	438.000i		−	0.521662i	−0.965384	−	0.260831i	$-0.916003\pi$
$89$		−	438.000i		−	0.521662i	0.965384	−	0.260831i	$-0.0839965\pi$
$90$		0			0
$91$	585.000	−	390.000i	0.673897	−	0.449265i
$92$		0			0
$93$			264.000i			0.294360i
$94$		0			0
$95$	−54.0000			−0.0583188
$96$		0			0
$97$			852.000i			0.891830i	0.895075	+	0.445915i	$0.147122\pi$
$97$			852.000i			0.891830i	−0.895075	+	0.445915i	$0.852878\pi$
$98$		0			0
$99$		−	1248.00i		−	1.26696i
$100$		0			0
$101$	396.000			0.390133			0.195067	−	0.980790i	$-0.437508\pi$
$101$	396.000			0.390133			0.195067	+	0.980790i	$0.437508\pi$
$102$		0			0
$103$	−182.000			−0.174107			−0.0870534	−	0.996204i	$-0.527745\pi$
$103$	−182.000			−0.174107			−0.0870534	+	0.996204i	$0.527745\pi$
$104$		0			0
$105$	135.000			0.125473
$106$		0			0
$107$	612.000			0.552937			0.276469	−	0.961023i	$-0.410836\pi$
$107$	612.000			0.552937			0.276469	+	0.961023i	$0.410836\pi$
$108$		0			0
$109$		−	1083.00i		−	0.951675i	−0.879533	−	0.475838i	$-0.842145\pi$
$109$		−	1083.00i		−	0.951675i	0.879533	−	0.475838i	$-0.157855\pi$
$110$		0			0
$111$			303.000i			0.259094i
$112$		0			0
$113$	90.0000			0.0749247			0.0374623	−	0.999298i	$-0.488073\pi$
$113$	90.0000			0.0749247			0.0374623	+	0.999298i	$0.488073\pi$
$114$		0			0
$115$		−	1458.00i		−	1.18225i
$116$		0			0
$117$	−676.000	−	1014.00i	−0.534156	−	0.801234i
$118$		0			0
$119$			675.000i			0.519976i
$120$		0			0
$121$	−973.000			−0.731029
$122$		0			0
$123$			192.000i			0.140748i
$124$		0			0
$125$			1521.00i			1.08834i
$126$		0			0
$127$	−2086.00			−1.45750			−0.728750	−	0.684780i	$-0.759898\pi$
$127$	−2086.00			−1.45750			−0.728750	+	0.684780i	$0.759898\pi$
$128$		0			0
$129$	97.0000			0.0662044
$130$		0			0
$131$	1467.00			0.978415			0.489208	−	0.872167i	$-0.337286\pi$
$131$	1467.00			0.978415			0.489208	+	0.872167i	$0.337286\pi$
$132$		0			0
$133$	90.0000			0.0586766
$134$		0			0
$135$		−	477.000i		−	0.304101i
$136$		0			0
$137$		−	414.000i		−	0.258178i	−0.991633	−	0.129089i	$-0.958795\pi$
$137$		−	414.000i		−	0.258178i	0.991633	−	0.129089i	$-0.0412053\pi$
$138$		0			0
$139$	2419.00			1.47609			0.738046	−	0.674750i	$-0.235749\pi$
$139$	2419.00			1.47609			0.738046	+	0.674750i	$0.235749\pi$
$140$		0			0
$141$		−	111.000i		−	0.0662971i
$142$		0			0
$143$	−1872.00	+	1248.00i	−1.09472	+	0.729811i
$144$		0			0
$145$		−	1296.00i		−	0.742255i
$146$		0			0
$147$	118.000			0.0662073
$148$		0			0
$149$			930.000i			0.511333i	0.966765	+	0.255666i	$0.0822948\pi$
$149$			930.000i			0.511333i	−0.966765	+	0.255666i	$0.917705\pi$
$150$		0			0
$151$			1683.00i			0.907024i	0.891250	+	0.453512i	$0.149829\pi$
$151$			1683.00i			0.907024i	−0.891250	+	0.453512i	$0.850171\pi$
$152$		0			0
$153$	1170.00			0.618228
$154$		0			0
$155$	−2376.00			−1.23126
$156$		0			0
$157$	1874.00			0.952621			0.476310	−	0.879277i	$-0.341974\pi$
$157$	1874.00			0.952621			0.476310	+	0.879277i	$0.341974\pi$
$158$		0			0
$159$	−414.000			−0.206493
$160$		0			0
$161$			2430.00i			1.18951i
$162$		0			0
$163$			1194.00i			0.573750i	0.957968	+	0.286875i	$0.0926165\pi$
$163$			1194.00i			0.573750i	−0.957968	+	0.286875i	$0.907384\pi$
$164$		0			0
$165$	−432.000			−0.203825
$166$		0			0
$167$			2388.00i			1.10652i	0.833008	+	0.553260i	$0.186617\pi$
$167$			2388.00i			1.10652i	−0.833008	+	0.553260i	$0.813383\pi$
$168$		0			0
$169$	−845.000	+	2028.00i	−0.384615	+	0.923077i
$170$		0			0
$171$		−	156.000i		−	0.0697638i
$172$		0			0
$173$	−1566.00			−0.688213			−0.344106	−	0.938931i	$-0.611818\pi$
$173$	−1566.00			−0.688213			−0.344106	+	0.938931i	$0.611818\pi$
$174$		0			0
$175$		−	660.000i		−	0.285093i
$176$		0			0
$177$		−	522.000i		−	0.221672i
$178$		0			0
$179$	657.000			0.274338			0.137169	−	0.990548i	$-0.456200\pi$
$179$	657.000			0.274338			0.137169	+	0.990548i	$0.456200\pi$
$180$		0			0
$181$	−1222.00			−0.501826			−0.250913	−	0.968010i	$-0.580731\pi$
$181$	−1222.00			−0.501826			−0.250913	+	0.968010i	$0.580731\pi$
$182$		0			0
$183$	376.000			0.151884
$184$		0			0
$185$	−2727.00			−1.08375
$186$		0			0
$187$		−	2160.00i		−	0.844678i
$188$		0			0
$189$			795.000i			0.305967i
$190$		0			0
$191$	−1260.00			−0.477332			−0.238666	−	0.971102i	$-0.576710\pi$
$191$	−1260.00			−0.477332			−0.238666	+	0.971102i	$0.576710\pi$
$192$		0			0
$193$			342.000i			0.127553i	0.997964	+	0.0637764i	$0.0203145\pi$
$193$			342.000i			0.127553i	−0.997964	+	0.0637764i	$0.979686\pi$
$194$		0			0
$195$	−351.000	+	234.000i	−0.128901	+	0.0859338i
$196$		0			0
$197$		−	81.0000i		−	0.0292945i	−0.999893	−	0.0146472i	$-0.995337\pi$
$197$		−	81.0000i		−	0.0292945i	0.999893	−	0.0146472i	$-0.00466253\pi$
$198$		0			0
$199$	−1996.00			−0.711019			−0.355509	−	0.934673i	$-0.615693\pi$
$199$	−1996.00			−0.711019			−0.355509	+	0.934673i	$0.615693\pi$
$200$		0			0
$201$		−	36.0000i		−	0.0126331i
$202$		0			0
$203$			2160.00i			0.746809i
$204$		0			0
$205$	−1728.00			−0.588726
$206$		0			0
$207$	4212.00			1.41427
$208$		0			0
$209$	−288.000			−0.0953176
$210$		0			0
$211$	−2833.00			−0.924321			−0.462161	−	0.886796i	$-0.652926\pi$
$211$	−2833.00			−0.924321			−0.462161	+	0.886796i	$0.652926\pi$
$212$		0			0
$213$			357.000i			0.114841i
$214$		0			0
$215$			873.000i			0.276921i
$216$		0			0
$217$	3960.00			1.23881
$218$		0			0
$219$		−	1098.00i		−	0.338794i
$220$		0			0
$221$	−1170.00	−	1755.00i	−0.356121	−	0.534181i
$222$		0			0
$223$		−	3507.00i		−	1.05312i	−0.850138	−	0.526561i	$-0.823481\pi$
$223$		−	3507.00i		−	1.05312i	0.850138	−	0.526561i	$-0.176519\pi$
$224$		0			0
$225$	−1144.00			−0.338963
$226$		0			0
$227$		−	228.000i		−	0.0666647i	−0.999444	−	0.0333324i	$-0.989388\pi$
$227$		−	228.000i		−	0.0666647i	0.999444	−	0.0333324i	$-0.0106120\pi$
$228$		0			0
$229$			5493.00i			1.58510i	0.609808	+	0.792549i	$0.291247\pi$
$229$			5493.00i			1.58510i	−0.609808	+	0.792549i	$0.708753\pi$
$230$		0			0
$231$	720.000			0.205076
$232$		0			0
$233$	3627.00			1.01980			0.509898	−	0.860235i	$-0.329683\pi$
$233$	3627.00			1.01980			0.509898	+	0.860235i	$0.329683\pi$
$234$		0			0
$235$	999.000			0.277309
$236$		0			0
$237$	830.000			0.227486
$238$		0			0
$239$			6075.00i			1.64418i	0.569357	+	0.822090i	$0.307192\pi$
$239$			6075.00i			1.64418i	−0.569357	+	0.822090i	$0.692808\pi$
$240$		0			0
$241$			210.000i			0.0561298i	0.999606	+	0.0280649i	$0.00893451\pi$
$241$			210.000i			0.0561298i	−0.999606	+	0.0280649i	$0.991065\pi$
$242$		0			0
$243$	2080.00			0.549103
$244$		0			0
$245$			1062.00i			0.276933i
$246$		0			0
$247$	−234.000	+	156.000i	−0.0602796	+	0.0401864i
$248$		0			0
$249$		−	438.000i		−	0.111474i
$250$		0			0
$251$	−7092.00			−1.78344			−0.891719	−	0.452589i	$-0.850501\pi$
$251$	−7092.00			−1.78344			−0.891719	+	0.452589i	$0.850501\pi$
$252$		0			0
$253$		−	7776.00i		−	1.93230i
$254$		0			0
$255$		−	405.000i		−	0.0994592i
$256$		0			0
$257$	−5805.00			−1.40897			−0.704486	−	0.709718i	$-0.748823\pi$
$257$	−5805.00			−1.40897			−0.704486	+	0.709718i	$0.748823\pi$
$258$		0			0
$259$	4545.00			1.09040
$260$		0			0
$261$	3744.00			0.887923
$262$		0			0
$263$	−792.000			−0.185691			−0.0928457	−	0.995681i	$-0.529596\pi$
$263$	−792.000			−0.185691			−0.0928457	+	0.995681i	$0.529596\pi$
$264$		0			0
$265$		−	3726.00i		−	0.863722i
$266$		0			0
$267$		−	438.000i		−	0.100394i
$268$		0			0
$269$	5472.00			1.24027			0.620137	−	0.784493i	$-0.287077\pi$
$269$	5472.00			1.24027			0.620137	+	0.784493i	$0.287077\pi$
$270$		0			0
$271$		−	2331.00i		−	0.522502i	−0.965271	−	0.261251i	$-0.915865\pi$
$271$		−	2331.00i		−	0.522502i	0.965271	−	0.261251i	$-0.0841351\pi$
$272$		0			0
$273$	585.000	−	390.000i	0.129692	−	0.0864611i
$274$		0			0
$275$			2112.00i			0.463121i
$276$		0			0
$277$	−1384.00			−0.300204			−0.150102	−	0.988671i	$-0.547960\pi$
$277$	−1384.00			−0.300204			−0.150102	+	0.988671i	$0.547960\pi$
$278$		0			0
$279$		−	6864.00i		−	1.47289i
$280$		0			0
$281$		−	4062.00i		−	0.862344i	−0.902270	−	0.431172i	$-0.858100\pi$
$281$		−	4062.00i		−	0.862344i	0.902270	−	0.431172i	$-0.141900\pi$
$282$		0			0
$283$	3764.00			0.790624			0.395312	−	0.918547i	$-0.370636\pi$
$283$	3764.00			0.790624			0.395312	+	0.918547i	$0.370636\pi$
$284$		0			0
$285$	−54.0000			−0.0112235
$286$		0			0
$287$	2880.00			0.592338
$288$		0			0
$289$	−2888.00			−0.587828
$290$		0			0
$291$			852.000i			0.171633i
$292$		0			0
$293$			4227.00i			0.842812i	0.906872	+	0.421406i	$0.138463\pi$
$293$			4227.00i			0.842812i	−0.906872	+	0.421406i	$0.861537\pi$
$294$		0			0
$295$	4698.00			0.927214
$296$		0			0
$297$		−	2544.00i		−	0.497030i
$298$		0			0
$299$	−4212.00	−	6318.00i	−0.814670	−	1.22200i
$300$		0			0
$301$		−	1455.00i		−	0.278621i
$302$		0			0
$303$	396.000			0.0750812
$304$		0			0
$305$			3384.00i			0.635303i
$306$		0			0
$307$		−	306.000i		−	0.0568871i	−0.999595	−	0.0284436i	$-0.990945\pi$
$307$		−	306.000i		−	0.0568871i	0.999595	−	0.0284436i	$-0.00905509\pi$
$308$		0			0
$309$	−182.000			−0.0335069
$310$		0			0
$311$	2106.00			0.383988			0.191994	−	0.981396i	$-0.438505\pi$
$311$	2106.00			0.383988			0.191994	+	0.981396i	$0.438505\pi$
$312$		0			0
$313$	10051.0			1.81507			0.907534	−	0.419979i	$-0.137963\pi$
$313$	10051.0			1.81507			0.907534	+	0.419979i	$0.137963\pi$
$314$		0			0
$315$	−3510.00			−0.627829
$316$		0			0
$317$			2154.00i			0.381643i	0.981625	+	0.190821i	$0.0611151\pi$
$317$			2154.00i			0.381643i	−0.981625	+	0.190821i	$0.938885\pi$
$318$		0			0
$319$		−	6912.00i		−	1.21316i
$320$		0			0
$321$	612.000			0.106413
$322$		0			0
$323$		−	270.000i		−	0.0465115i
$324$		0			0
$325$	1144.00	+	1716.00i	0.195254	+	0.292882i
$326$		0			0
$327$		−	1083.00i		−	0.183150i
$328$		0			0
$329$	−1665.00			−0.279010
$330$		0			0
$331$			10770.0i			1.78844i	0.447630	+	0.894219i	$0.352268\pi$
$331$			10770.0i			1.78844i	−0.447630	+	0.894219i	$0.647732\pi$
$332$		0			0
$333$		−	7878.00i		−	1.29643i
$334$		0			0
$335$	324.000			0.0528418
$336$		0			0
$337$	−2171.00			−0.350926			−0.175463	−	0.984486i	$-0.556142\pi$
$337$	−2171.00			−0.350926			−0.175463	+	0.984486i	$0.556142\pi$
$338$		0			0
$339$	90.0000			0.0144193
$340$		0			0
$341$	−12672.0			−2.01240
$342$		0			0
$343$		−	6915.00i		−	1.08856i
$344$		0			0
$345$		−	1458.00i		−	0.227525i
$346$		0			0
$347$	7047.00			1.09021			0.545105	−	0.838368i	$-0.316490\pi$
$347$	7047.00			1.09021			0.545105	+	0.838368i	$0.316490\pi$
$348$		0			0
$349$		−	6873.00i		−	1.05416i	−0.849814	−	0.527082i	$-0.823286\pi$
$349$		−	6873.00i		−	1.05416i	0.849814	−	0.527082i	$-0.176714\pi$
$350$		0			0
$351$	−1378.00	−	2067.00i	−0.209550	−	0.314326i
$352$		0			0
$353$			9318.00i			1.40495i	0.711709	+	0.702475i	$0.247922\pi$
$353$			9318.00i			1.40495i	−0.711709	+	0.702475i	$0.752078\pi$
$354$		0			0
$355$	−3213.00			−0.480362
$356$		0			0
$357$			675.000i			0.100069i
$358$		0			0
$359$		−	4128.00i		−	0.606873i	−0.952852	−	0.303437i	$-0.901866\pi$
$359$		−	4128.00i		−	0.606873i	0.952852	−	0.303437i	$-0.0981341\pi$
$360$		0			0
$361$	6823.00			0.994751
$362$		0			0
$363$	−973.000			−0.140687
$364$		0			0
$365$	9882.00			1.41712
$366$		0			0
$367$	2536.00			0.360703			0.180352	−	0.983602i	$-0.442276\pi$
$367$	2536.00			0.360703			0.180352	+	0.983602i	$0.442276\pi$
$368$		0			0
$369$		−	4992.00i		−	0.704263i
$370$		0			0
$371$			6210.00i			0.869022i
$372$		0			0
$373$	−92.0000			−0.0127710			−0.00638550	−	0.999980i	$-0.502033\pi$
$373$	−92.0000			−0.0127710			−0.00638550	+	0.999980i	$0.502033\pi$
$374$		0			0
$375$			1521.00i			0.209451i
$376$		0			0
$377$	−3744.00	−	5616.00i	−0.511474	−	0.767211i
$378$		0			0
$379$			10182.0i			1.37998i	0.723817	+	0.689992i	$0.242386\pi$
$379$			10182.0i			1.37998i	−0.723817	+	0.689992i	$0.757614\pi$
$380$		0			0
$381$	−2086.00			−0.280496
$382$		0			0
$383$		−	579.000i		−	0.0772468i	−0.999254	−	0.0386234i	$-0.987703\pi$
$383$		−	579.000i		−	0.0772468i	0.999254	−	0.0386234i	$-0.0122973\pi$
$384$		0			0
$385$			6480.00i			0.857796i
$386$		0			0
$387$	−2522.00			−0.331267
$388$		0			0
$389$	2106.00			0.274495			0.137247	−	0.990537i	$-0.456174\pi$
$389$	2106.00			0.274495			0.137247	+	0.990537i	$0.456174\pi$
$390$		0			0
$391$	7290.00			0.942893
$392$		0			0
$393$	1467.00			0.188296
$394$		0			0
$395$			7470.00i			0.951535i
$396$		0			0
$397$		−	1974.00i		−	0.249552i	−0.992185	−	0.124776i	$-0.960179\pi$
$397$		−	1974.00i		−	0.249552i	0.992185	−	0.124776i	$-0.0398213\pi$
$398$		0			0
$399$	90.0000			0.0112923
$400$		0			0
$401$			11886.0i			1.48020i	0.672499	+	0.740098i	$0.265221\pi$
$401$			11886.0i			1.48020i	−0.672499	+	0.740098i	$0.734779\pi$
$402$		0			0
$403$	−10296.0	+	6864.00i	−1.27266	+	0.848437i
$404$		0			0
$405$			5841.00i			0.716646i
$406$		0			0
$407$	−14544.0			−1.77130
$408$		0			0
$409$		−	1254.00i		−	0.151605i	−0.997123	−	0.0758023i	$-0.975848\pi$
$409$		−	1254.00i		−	0.151605i	0.997123	−	0.0758023i	$-0.0241518\pi$
$410$		0			0
$411$		−	414.000i		−	0.0496864i
$412$		0			0
$413$	−7830.00			−0.932903
$414$		0			0
$415$	3942.00			0.466278
$416$		0			0
$417$	2419.00			0.284074
$418$		0			0
$419$	−5823.00			−0.678931			−0.339466	−	0.940618i	$-0.610246\pi$
$419$	−5823.00			−0.678931			−0.339466	+	0.940618i	$0.610246\pi$
$420$		0			0
$421$			7341.00i			0.849830i	0.905233	+	0.424915i	$0.139696\pi$
$421$			7341.00i			0.849830i	−0.905233	+	0.424915i	$0.860304\pi$
$422$		0			0
$423$			2886.00i			0.331731i
$424$		0			0
$425$	−1980.00			−0.225986
$426$		0			0
$427$		−	5640.00i		−	0.639201i
$428$		0			0
$429$	−1872.00	+	1248.00i	−0.210678	+	0.140452i
$430$		0			0
$431$		−	7485.00i		−	0.836519i	−0.908328	−	0.418260i	$-0.862640\pi$
$431$		−	7485.00i		−	0.836519i	0.908328	−	0.418260i	$-0.137360\pi$
$432$		0			0
$433$	−15203.0			−1.68732			−0.843660	−	0.536878i	$-0.819604\pi$
$433$	−15203.0			−1.68732			−0.843660	+	0.536878i	$0.819604\pi$
$434$		0			0
$435$		−	1296.00i		−	0.142847i
$436$		0			0
$437$		−	972.000i		−	0.106401i
$438$		0			0
$439$	1762.00			0.191562			0.0957809	−	0.995402i	$-0.469465\pi$
$439$	1762.00			0.191562			0.0957809	+	0.995402i	$0.469465\pi$
$440$		0			0
$441$	−3068.00			−0.331282
$442$		0			0
$443$	7317.00			0.784743			0.392372	−	0.919807i	$-0.371655\pi$
$443$	7317.00			0.784743			0.392372	+	0.919807i	$0.371655\pi$
$444$		0			0
$445$	3942.00			0.419930
$446$		0			0
$447$			930.000i			0.0984060i
$448$		0			0
$449$		−	5016.00i		−	0.527215i	−0.964630	−	0.263608i	$-0.915088\pi$
$449$		−	5016.00i		−	0.527215i	0.964630	−	0.263608i	$-0.0849124\pi$
$450$		0			0
$451$	−9216.00			−0.962227
$452$		0			0
$453$			1683.00i			0.174557i
$454$		0			0
$455$	3510.00	+	5265.00i	0.361651	+	0.542477i
$456$		0			0
$457$		−	9870.00i		−	1.01028i	−0.863037	−	0.505141i	$-0.831440\pi$
$457$		−	9870.00i		−	1.01028i	0.863037	−	0.505141i	$-0.168560\pi$
$458$		0			0
$459$	2385.00			0.242532
$460$		0			0
$461$			14541.0i			1.46907i	0.678570	+	0.734536i	$0.262600\pi$
$461$			14541.0i			1.46907i	−0.678570	+	0.734536i	$0.737400\pi$
$462$		0			0
$463$			2112.00i			0.211993i	0.994366	+	0.105997i	$0.0338033\pi$
$463$			2112.00i			0.211993i	−0.994366	+	0.105997i	$0.966197\pi$
$464$		0			0
$465$	−2376.00			−0.236956
$466$		0			0
$467$	−3276.00			−0.324615			−0.162307	−	0.986740i	$-0.551894\pi$
$467$	−3276.00			−0.324615			−0.162307	+	0.986740i	$0.551894\pi$
$468$		0			0
$469$	−540.000			−0.0531661
$470$		0			0
$471$	1874.00			0.183332
$472$		0			0
$473$			4656.00i			0.452607i
$474$		0			0
$475$			264.000i			0.0255014i
$476$		0			0
$477$	10764.0			1.03323
$478$		0			0
$479$			15453.0i			1.47404i	0.675870	+	0.737020i	$0.263768\pi$
$479$			15453.0i			1.47404i	−0.675870	+	0.737020i	$0.736232\pi$
$480$		0			0
$481$	−11817.0	+	7878.00i	−1.12018	+	0.746790i
$482$		0			0
$483$			2430.00i			0.228921i
$484$		0			0
$485$	−7668.00			−0.717909
$486$		0			0
$487$		−	3660.00i		−	0.340555i	−0.985396	−	0.170278i	$-0.945534\pi$
$487$		−	3660.00i		−	0.340555i	0.985396	−	0.170278i	$-0.0544665\pi$
$488$		0			0
$489$			1194.00i			0.110418i
$490$		0			0
$491$	−747.000			−0.0686591			−0.0343296	−	0.999411i	$-0.510930\pi$
$491$	−747.000			−0.0686591			−0.0343296	+	0.999411i	$0.510930\pi$
$492$		0			0
$493$	6480.00			0.591977
$494$		0			0
$495$	11232.0			1.01988
$496$		0			0
$497$	5355.00			0.483309
$498$		0			0
$499$		−	15804.0i		−	1.41780i	−0.705307	−	0.708902i	$-0.749191\pi$
$499$		−	15804.0i		−	1.41780i	0.705307	−	0.708902i	$-0.250809\pi$
$500$		0			0
$501$			2388.00i			0.212950i
$502$		0			0
$503$	12078.0			1.07064			0.535319	−	0.844650i	$-0.320191\pi$
$503$	12078.0			1.07064			0.535319	+	0.844650i	$0.320191\pi$
$504$		0			0
$505$			3564.00i			0.314051i
$506$		0			0
$507$	−845.000	+	2028.00i	−0.0740193	+	0.177646i
$508$		0			0
$509$		−	16110.0i		−	1.40287i	−0.712731	−	0.701437i	$-0.752542\pi$
$509$		−	16110.0i		−	1.40287i	0.712731	−	0.701437i	$-0.247458\pi$
$510$		0			0
$511$	−16470.0			−1.42581
$512$		0			0
$513$		−	318.000i		−	0.0273685i
$514$		0			0
$515$		−	1638.00i		−	0.140153i
$516$		0			0
$517$	5328.00			0.453240
$518$		0			0
$519$	−1566.00			−0.132447
$520$		0			0
$521$	3915.00			0.329212			0.164606	−	0.986359i	$-0.447365\pi$
$521$	3915.00			0.329212			0.164606	+	0.986359i	$0.447365\pi$
$522$		0			0
$523$	−16184.0			−1.35311			−0.676555	−	0.736392i	$-0.736528\pi$
$523$	−16184.0			−1.35311			−0.676555	+	0.736392i	$0.736528\pi$
$524$		0			0
$525$		−	660.000i		−	0.0548662i
$526$		0			0
$527$		−	11880.0i		−	0.981975i
$528$		0			0
$529$	14077.0			1.15698
$530$		0			0
$531$			13572.0i			1.10918i
$532$		0			0
$533$	−7488.00	+	4992.00i	−0.608520	+	0.405680i
$534$		0			0
$535$			5508.00i			0.445106i
$536$		0			0
$537$	657.000			0.0527964
$538$		0			0
$539$			5664.00i			0.452627i
$540$		0			0
$541$		−	7923.00i		−	0.629642i	−0.949151	−	0.314821i	$-0.898055\pi$
$541$		−	7923.00i		−	0.629642i	0.949151	−	0.314821i	$-0.101945\pi$
$542$		0			0
$543$	−1222.00			−0.0965765
$544$		0			0
$545$	9747.00			0.766084
$546$		0			0
$547$	14389.0			1.12473			0.562367	−	0.826888i	$-0.309891\pi$
$547$	14389.0			1.12473			0.562367	+	0.826888i	$0.309891\pi$
$548$		0			0
$549$	−9776.00			−0.759981
$550$		0			0
$551$		−	864.000i		−	0.0668015i
$552$		0			0
$553$		−	12450.0i		−	0.957374i
$554$		0			0
$555$	−2727.00			−0.208567
$556$		0			0
$557$		−	10383.0i		−	0.789842i	−0.918715	−	0.394921i	$-0.870772\pi$
$557$		−	10383.0i		−	0.789842i	0.918715	−	0.394921i	$-0.129228\pi$
$558$		0			0
$559$	2522.00	+	3783.00i	0.190822	+	0.286232i
$560$		0			0
$561$		−	2160.00i		−	0.162558i
$562$		0			0
$563$	16425.0			1.22954			0.614770	−	0.788706i	$-0.289249\pi$
$563$	16425.0			1.22954			0.614770	+	0.788706i	$0.289249\pi$
$564$		0			0
$565$			810.000i			0.0603132i
$566$		0			0
$567$		−	9735.00i		−	0.721043i
$568$		0			0
$569$	12213.0			0.899817			0.449908	−	0.893075i	$-0.351457\pi$
$569$	12213.0			0.899817			0.449908	+	0.893075i	$0.351457\pi$
$570$		0			0
$571$	6383.00			0.467811			0.233906	−	0.972259i	$-0.424849\pi$
$571$	6383.00			0.467811			0.233906	+	0.972259i	$0.424849\pi$
$572$		0			0
$573$	−1260.00			−0.0918626
$574$		0			0
$575$	−7128.00			−0.516971
$576$		0			0
$577$		−	6426.00i		−	0.463636i	−0.972759	−	0.231818i	$-0.925533\pi$
$577$		−	6426.00i		−	0.463636i	0.972759	−	0.231818i	$-0.0744674\pi$
$578$		0			0
$579$			342.000i			0.0245476i
$580$		0			0
$581$	−6570.00			−0.469139
$582$		0			0
$583$		−	19872.0i		−	1.41169i
$584$		0			0
$585$	9126.00	−	6084.00i	0.644981	−	0.429987i
$586$		0			0
$587$		−	21330.0i		−	1.49980i	−0.661551	−	0.749901i	$-0.730101\pi$
$587$		−	21330.0i		−	1.49980i	0.661551	−	0.749901i	$-0.269899\pi$
$588$		0			0
$589$	−1584.00			−0.110811
$590$		0			0
$591$		−	81.0000i		−	0.00563772i
$592$		0			0
$593$			12084.0i			0.836813i	0.908260	+	0.418407i	$0.137411\pi$
$593$			12084.0i			0.836813i	−0.908260	+	0.418407i	$0.862589\pi$
$594$		0			0
$595$	−6075.00			−0.418573
$596$		0			0
$597$	−1996.00			−0.136836
$598$		0			0
$599$	−2394.00			−0.163299			−0.0816496	−	0.996661i	$-0.526019\pi$
$599$	−2394.00			−0.163299			−0.0816496	+	0.996661i	$0.526019\pi$
$600$		0			0
$601$	−21971.0			−1.49121			−0.745604	−	0.666389i	$-0.767839\pi$
$601$	−21971.0			−1.49121			−0.745604	+	0.666389i	$0.767839\pi$
$602$		0			0
$603$			936.000i			0.0632121i
$604$		0			0
$605$		−	8757.00i		−	0.588467i
$606$		0			0
$607$	15406.0			1.03017			0.515083	−	0.857141i	$-0.327761\pi$
$607$	15406.0			1.03017			0.515083	+	0.857141i	$0.327761\pi$
$608$		0			0
$609$			2160.00i			0.143724i
$610$		0			0
$611$	4329.00	−	2886.00i	0.286633	−	0.191088i
$612$		0			0
$613$			9630.00i			0.634506i	0.948341	+	0.317253i	$0.102760\pi$
$613$			9630.00i			0.634506i	−0.948341	+	0.317253i	$0.897240\pi$
$614$		0			0
$615$	−1728.00			−0.113300
$616$		0			0
$617$		−	14748.0i		−	0.962289i	−0.876641	−	0.481144i	$-0.840221\pi$
$617$		−	14748.0i		−	0.962289i	0.876641	−	0.481144i	$-0.159779\pi$
$618$		0			0
$619$			3672.00i			0.238433i	0.992868	+	0.119217i	$0.0380383\pi$
$619$			3672.00i			0.238433i	−0.992868	+	0.119217i	$0.961962\pi$
$620$		0			0
$621$	8586.00			0.554822
$622$		0			0
$623$	−6570.00			−0.422506
$624$		0			0
$625$	−8189.00			−0.524096
$626$		0			0
$627$	−288.000			−0.0183439
$628$		0			0
$629$		−	13635.0i		−	0.864329i
$630$		0			0
$631$			19875.0i			1.25390i	0.779059	+	0.626950i	$0.215697\pi$
$631$			19875.0i			1.25390i	−0.779059	+	0.626950i	$0.784303\pi$
$632$		0			0
$633$	−2833.00			−0.177886
$634$		0			0
$635$		−	18774.0i		−	1.17327i
$636$		0			0
$637$	3068.00	+	4602.00i	0.190830	+	0.286245i
$638$		0			0
$639$		−	9282.00i		−	0.574633i
$640$		0			0
$641$	1710.00			0.105368			0.0526840	−	0.998611i	$-0.483222\pi$
$641$	1710.00			0.105368			0.0526840	+	0.998611i	$0.483222\pi$
$642$		0			0
$643$		−	16452.0i		−	1.00903i	−0.863404	−	0.504513i	$-0.831672\pi$
$643$		−	16452.0i		−	1.00903i	0.863404	−	0.504513i	$-0.168328\pi$
$644$		0			0
$645$			873.000i			0.0532936i
$646$		0			0
$647$	−25902.0			−1.57390			−0.786950	−	0.617017i	$-0.788341\pi$
$647$	−25902.0			−1.57390			−0.786950	+	0.617017i	$0.788341\pi$
$648$		0			0
$649$	25056.0			1.51546
$650$		0			0
$651$	3960.00			0.238410
$652$		0			0
$653$	18108.0			1.08518			0.542589	−	0.839999i	$-0.317444\pi$
$653$	18108.0			1.08518			0.542589	+	0.839999i	$0.317444\pi$
$654$		0			0
$655$			13203.0i			0.787609i
$656$		0			0
$657$			28548.0i			1.69523i
$658$		0			0
$659$	32904.0			1.94500			0.972502	−	0.232894i	$-0.0748195\pi$
$659$	32904.0			1.94500			0.972502	+	0.232894i	$0.0748195\pi$
$660$		0			0
$661$			15318.0i			0.901363i	0.892685	+	0.450682i	$0.148819\pi$
$661$			15318.0i			0.901363i	−0.892685	+	0.450682i	$0.851181\pi$
$662$		0			0
$663$	−1170.00	−	1755.00i	−0.0685355	−	0.102803i
$664$		0			0
$665$			810.000i			0.0472338i
$666$		0			0
$667$	23328.0			1.35422
$668$		0			0
$669$		−	3507.00i		−	0.202673i
$670$		0			0
$671$			18048.0i			1.03835i
$672$		0			0
$673$	−7729.00			−0.442691			−0.221346	−	0.975195i	$-0.571045\pi$
$673$	−7729.00			−0.442691			−0.221346	+	0.975195i	$0.571045\pi$
$674$		0			0
$675$	−2332.00			−0.132976
$676$		0			0
$677$	19242.0			1.09236			0.546182	−	0.837667i	$-0.316081\pi$
$677$	19242.0			1.09236			0.546182	+	0.837667i	$0.316081\pi$
$678$		0			0
$679$	12780.0			0.722314
$680$		0			0
$681$		−	228.000i		−	0.0128296i
$682$		0			0
$683$		−	22518.0i		−	1.26153i	−0.775973	−	0.630767i	$-0.782740\pi$
$683$		−	22518.0i		−	1.26153i	0.775973	−	0.630767i	$-0.217260\pi$
$684$		0			0
$685$	3726.00			0.207829
$686$		0			0
$687$			5493.00i			0.305052i
$688$		0			0
$689$	−10764.0	−	16146.0i	−0.595175	−	0.892763i
$690$		0			0
$691$			9168.00i			0.504728i	0.967632	+	0.252364i	$0.0812081\pi$
$691$			9168.00i			0.504728i	−0.967632	+	0.252364i	$0.918792\pi$
$692$		0			0
$693$	−18720.0			−1.02614
$694$		0			0
$695$			21771.0i			1.18823i
$696$		0			0
$697$		−	8640.00i		−	0.469531i
$698$		0			0
$699$	3627.00			0.196260
$700$		0			0
$701$	1170.00			0.0630389			0.0315195	−	0.999503i	$-0.489965\pi$
$701$	1170.00			0.0630389			0.0315195	+	0.999503i	$0.489965\pi$
$702$		0			0
$703$	−1818.00			−0.0975351
$704$		0			0
$705$	999.000			0.0533681
$706$		0			0
$707$		−	5940.00i		−	0.315978i
$708$		0			0
$709$		−	1662.00i		−	0.0880363i	−0.999031	−	0.0440181i	$-0.985984\pi$
$709$		−	1662.00i		−	0.0880363i	0.999031	−	0.0440181i	$-0.0140159\pi$
$710$		0			0
$711$	−21580.0			−1.13827
$712$		0			0
$713$		−	42768.0i		−	2.24639i
$714$		0			0
$715$	−11232.0	−	16848.0i	−0.587487	−	0.881230i
$716$		0			0
$717$			6075.00i			0.316423i
$718$		0			0
$719$	−30960.0			−1.60586			−0.802930	−	0.596073i	$-0.796727\pi$
$719$	−30960.0			−1.60586			−0.802930	+	0.596073i	$0.796727\pi$
$720$		0			0
$721$			2730.00i			0.141013i
$722$		0			0
$723$			210.000i			0.0108022i
$724$		0			0
$725$	−6336.00			−0.324570
$726$		0			0
$727$	8372.00			0.427098			0.213549	−	0.976932i	$-0.431498\pi$
$727$	8372.00			0.427098			0.213549	+	0.976932i	$0.431498\pi$
$728$		0			0
$729$	−15443.0			−0.784586
$730$		0			0
$731$	−4365.00			−0.220855
$732$		0			0
$733$			2739.00i			0.138018i	0.997616	+	0.0690091i	$0.0219837\pi$
$733$			2739.00i			0.138018i	−0.997616	+	0.0690091i	$0.978016\pi$
$734$		0			0
$735$			1062.00i			0.0532959i
$736$		0			0
$737$	1728.00			0.0863659
$738$		0			0
$739$			6756.00i			0.336297i	0.985762	+	0.168148i	$0.0537788\pi$
$739$			6756.00i			0.336297i	−0.985762	+	0.168148i	$0.946221\pi$
$740$		0			0
$741$	−234.000	+	156.000i	−0.0116008	+	0.00773388i
$742$		0			0
$743$			29643.0i			1.46366i	0.681490	+	0.731828i	$0.261332\pi$
$743$			29643.0i			1.46366i	−0.681490	+	0.731828i	$0.738668\pi$
$744$		0			0
$745$	−8370.00			−0.411615
$746$		0			0
$747$			11388.0i			0.557785i
$748$		0			0
$749$		−	9180.00i		−	0.447837i
$750$		0			0
$751$	−18128.0			−0.880826			−0.440413	−	0.897795i	$-0.645168\pi$
$751$	−18128.0			−0.880826			−0.440413	+	0.897795i	$0.645168\pi$
$752$		0			0
$753$	−7092.00			−0.343223
$754$		0			0
$755$	−15147.0			−0.730140
$756$		0			0
$757$	−6410.00			−0.307761			−0.153881	−	0.988089i	$-0.549177\pi$
$757$	−6410.00			−0.307761			−0.153881	+	0.988089i	$0.549177\pi$
$758$		0			0
$759$		−	7776.00i		−	0.371872i
$760$		0			0
$761$			28290.0i			1.34758i	0.738921	+	0.673792i	$0.235336\pi$
$761$			28290.0i			1.34758i	−0.738921	+	0.673792i	$0.764664\pi$
$762$		0			0
$763$	−16245.0			−0.770784
$764$		0			0
$765$			10530.0i			0.497664i
$766$		0			0
$767$	20358.0	−	13572.0i	0.958390	−	0.638926i
$768$		0			0
$769$			27960.0i			1.31114i	0.755136	+	0.655568i	$0.227571\pi$
$769$			27960.0i			1.31114i	−0.755136	+	0.655568i	$0.772429\pi$
$770$		0			0
$771$	−5805.00			−0.271157
$772$		0			0
$773$			5649.00i			0.262847i	0.991326	+	0.131423i	$0.0419547\pi$
$773$			5649.00i			0.262847i	−0.991326	+	0.131423i	$0.958045\pi$
$774$		0			0
$775$			11616.0i			0.538399i
$776$		0			0
$777$	4545.00			0.209847
$778$		0			0
$779$	−1152.00			−0.0529842
$780$		0			0
$781$	−17136.0			−0.785114
$782$		0			0
$783$	7632.00			0.348334
$784$		0			0
$785$			16866.0i			0.766845i
$786$		0			0
$787$		−	756.000i		−	0.0342420i	−0.999853	−	0.0171210i	$-0.994550\pi$
$787$		−	756.000i		−	0.0342420i	0.999853	−	0.0171210i	$-0.00545006\pi$
$788$		0			0
$789$	−792.000			−0.0357363
$790$		0			0
$791$		−	1350.00i		−	0.0606833i
$792$		0			0
$793$	9776.00	+	14664.0i	0.437775	+	0.656663i
$794$		0			0
$795$		−	3726.00i		−	0.166223i
$796$		0			0
$797$	31194.0			1.38638			0.693192	−	0.720753i	$-0.256204\pi$
$797$	31194.0			1.38638			0.693192	+	0.720753i	$0.256204\pi$
$798$		0			0
$799$			4995.00i			0.221164i
$800$		0			0
$801$			11388.0i			0.502341i
$802$		0			0
$803$	52704.0			2.31617
$804$		0			0
$805$	−21870.0			−0.957536
$806$		0			0
$807$	5472.00			0.238691
$808$		0			0
$809$	17055.0			0.741189			0.370594	−	0.928795i	$-0.379154\pi$
$809$	17055.0			0.741189			0.370594	+	0.928795i	$0.379154\pi$
$810$		0			0
$811$			35520.0i			1.53795i	0.639280	+	0.768974i	$0.279232\pi$
$811$			35520.0i			1.53795i	−0.639280	+	0.768974i	$0.720768\pi$
$812$		0			0
$813$		−	2331.00i		−	0.100556i
$814$		0			0
$815$	−10746.0			−0.461860
$816$		0			0
$817$			582.000i			0.0249224i
$818$		0			0
$819$	−15210.0	+	10140.0i	−0.648938	+	0.432625i
$820$		0			0
$821$		−	1095.00i		−	0.0465478i	−0.999729	−	0.0232739i	$-0.992591\pi$
$821$		−	1095.00i		−	0.0465478i	0.999729	−	0.0232739i	$-0.00740899\pi$
$822$		0			0
$823$	2554.00			0.108174			0.0540868	−	0.998536i	$-0.482775\pi$
$823$	2554.00			0.108174			0.0540868	+	0.998536i	$0.482775\pi$
$824$		0			0
$825$			2112.00i			0.0891278i
$826$		0			0
$827$		−	21522.0i		−	0.904950i	−0.891777	−	0.452475i	$-0.850541\pi$
$827$		−	21522.0i		−	0.904950i	0.891777	−	0.452475i	$-0.149459\pi$
$828$		0			0
$829$	−13124.0			−0.549838			−0.274919	−	0.961467i	$-0.588651\pi$
$829$	−13124.0			−0.549838			−0.274919	+	0.961467i	$0.588651\pi$
$830$		0			0
$831$	−1384.00			−0.0577743
$832$		0			0
$833$	−5310.00			−0.220865
$834$		0			0
$835$	−21492.0			−0.890732
$836$		0			0
$837$		−	13992.0i		−	0.577819i
$838$		0			0
$839$			23424.0i			0.963869i	0.876207	+	0.481935i	$0.160066\pi$
$839$			23424.0i			0.963869i	−0.876207	+	0.481935i	$0.839934\pi$
$840$		0			0
$841$	−3653.00			−0.149781
$842$		0			0
$843$		−	4062.00i		−	0.165958i
$844$		0			0
$845$	−18252.0	−	7605.00i	−0.743063	−	0.309609i
$846$		0			0
$847$			14595.0i			0.592078i
$848$		0			0
$849$	3764.00			0.152156
$850$		0			0
$851$		−	49086.0i		−	1.97726i
$852$		0			0
$853$			31077.0i			1.24743i	0.781653	+	0.623714i	$0.214377\pi$
$853$			31077.0i			1.24743i	−0.781653	+	0.623714i	$0.785623\pi$
$854$		0			0
$855$	1404.00			0.0561588
$856$		0			0
$857$	−19422.0			−0.774146			−0.387073	−	0.922049i	$-0.626514\pi$
$857$	−19422.0			−0.774146			−0.387073	+	0.922049i	$0.626514\pi$
$858$		0			0
$859$	−1744.00			−0.0692718			−0.0346359	−	0.999400i	$-0.511027\pi$
$859$	−1744.00			−0.0692718			−0.0346359	+	0.999400i	$0.511027\pi$
$860$		0			0
$861$	2880.00			0.113996
$862$		0			0
$863$			19179.0i			0.756501i	0.925703	+	0.378251i	$0.123474\pi$
$863$			19179.0i			0.756501i	−0.925703	+	0.378251i	$0.876526\pi$
$864$		0			0
$865$		−	14094.0i		−	0.554000i
$866$		0			0
$867$	−2888.00			−0.113128
$868$		0			0
$869$			39840.0i			1.55521i
$870$		0			0
$871$	1404.00	−	936.000i	0.0546185	−	0.0364123i
$872$		0			0
$873$		−	22152.0i		−	0.858799i
$874$		0			0
$875$	22815.0			0.881472
$876$		0			0
$877$		−	29217.0i		−	1.12496i	−0.826812	−	0.562479i	$-0.809848\pi$
$877$		−	29217.0i		−	1.12496i	0.826812	−	0.562479i	$-0.190152\pi$
$878$		0			0
$879$			4227.00i			0.162199i
$880$		0			0
$881$	−15633.0			−0.597831			−0.298916	−	0.954280i	$-0.596625\pi$
$881$	−15633.0			−0.597831			−0.298916	+	0.954280i	$0.596625\pi$
$882$		0			0
$883$	−30589.0			−1.16580			−0.582900	−	0.812544i	$-0.698082\pi$
$883$	−30589.0			−1.16580			−0.582900	+	0.812544i	$0.698082\pi$
$884$		0			0
$885$	4698.00			0.178442
$886$		0			0
$887$	25884.0			0.979819			0.489910	−	0.871773i	$-0.337030\pi$
$887$	25884.0			0.979819			0.489910	+	0.871773i	$0.337030\pi$
$888$		0			0
$889$			31290.0i			1.18046i
$890$		0			0
$891$			31152.0i			1.17130i
$892$		0			0
$893$	666.000			0.0249573
$894$		0			0
$895$			5913.00i			0.220838i
$896$		0			0
$897$	−4212.00	−	6318.00i	−0.156783	−	0.235175i
$898$		0			0
$899$		−	38016.0i		−	1.41035i
$900$		0			0
$901$	18630.0			0.688852
$902$		0			0
$903$		−	1455.00i		−	0.0536206i
$904$		0			0
$905$		−	10998.0i		−	0.403962i
$906$		0			0
$907$	12305.0			0.450475			0.225237	−	0.974304i	$-0.427684\pi$
$907$	12305.0			0.450475			0.225237	+	0.974304i	$0.427684\pi$
$908$		0			0
$909$	−10296.0			−0.375684
$910$		0			0
$911$	−29772.0			−1.08276			−0.541378	−	0.840779i	$-0.682097\pi$
$911$	−29772.0			−1.08276			−0.541378	+	0.840779i	$0.682097\pi$
$912$		0			0
$913$	21024.0			0.762095
$914$		0			0
$915$			3384.00i			0.122264i
$916$		0			0
$917$		−	22005.0i		−	0.792442i
$918$		0			0
$919$	−47644.0			−1.71015			−0.855076	−	0.518502i	$-0.826490\pi$
$919$	−47644.0			−1.71015			−0.855076	+	0.518502i	$0.826490\pi$
$920$		0			0
$921$		−	306.000i		−	0.0109479i
$922$		0			0
$923$	−13923.0	+	9282.00i	−0.496513	+	0.331008i
$924$		0			0
$925$			13332.0i			0.473896i
$926$		0			0
$927$	4732.00			0.167658
$928$		0			0
$929$		−	21924.0i		−	0.774277i	−0.922022	−	0.387138i	$-0.873464\pi$
$929$		−	21924.0i		−	0.774277i	0.922022	−	0.387138i	$-0.126536\pi$
$930$		0			0
$931$			708.000i			0.0249235i
$932$		0			0
$933$	2106.00			0.0738985
$934$		0			0
$935$	19440.0			0.679953
$936$		0			0
$937$	32398.0			1.12956			0.564779	−	0.825242i	$-0.308961\pi$
$937$	32398.0			1.12956			0.564779	+	0.825242i	$0.308961\pi$
$938$		0			0
$939$	10051.0			0.349310
$940$		0			0
$941$		−	2097.00i		−	0.0726464i	−0.999340	−	0.0363232i	$-0.988435\pi$
$941$		−	2097.00i		−	0.0726464i	0.999340	−	0.0363232i	$-0.0115646\pi$
$942$		0			0
$943$		−	31104.0i		−	1.07411i
$944$		0			0
$945$	−7155.00			−0.246299
$946$		0			0
$947$			20016.0i			0.686835i	0.939183	+	0.343417i	$0.111585\pi$
$947$			20016.0i			0.686835i	−0.939183	+	0.343417i	$0.888415\pi$
$948$		0			0
$949$	42822.0	−	28548.0i	1.46476	−	0.976509i
$950$		0			0
$951$			2154.00i			0.0734471i
$952$		0			0
$953$	24993.0			0.849531			0.424765	−	0.905304i	$-0.360357\pi$
$953$	24993.0			0.849531			0.424765	+	0.905304i	$0.360357\pi$
$954$		0			0
$955$		−	11340.0i		−	0.384245i
$956$		0			0
$957$		−	6912.00i		−	0.233473i
$958$		0			0
$959$	−6210.00			−0.209105
$960$		0			0
$961$	−39905.0			−1.33950
$962$		0			0
$963$	−15912.0			−0.532458
$964$		0			0
$965$	−3078.00			−0.102678
$966$		0			0
$967$		−	40959.0i		−	1.36210i	−0.732236	−	0.681051i	$-0.761523\pi$
$967$		−	40959.0i		−	1.36210i	0.732236	−	0.681051i	$-0.238477\pi$
$968$		0			0
$969$		−	270.000i		−	0.00895113i
$970$		0			0
$971$	48933.0			1.61723			0.808617	−	0.588335i	$-0.200216\pi$
$971$	48933.0			1.61723			0.808617	+	0.588335i	$0.200216\pi$
$972$		0			0
$973$		−	36285.0i		−	1.19552i
$974$		0			0
$975$	1144.00	+	1716.00i	0.0375767	+	0.0563651i
$976$		0			0
$977$		−	47388.0i		−	1.55177i	−0.630876	−	0.775884i	$-0.717304\pi$
$977$		−	47388.0i		−	1.55177i	0.630876	−	0.775884i	$-0.282696\pi$
$978$		0			0
$979$	21024.0			0.686343
$980$		0			0
$981$			28158.0i			0.916428i
$982$		0			0
$983$		−	16803.0i		−	0.545201i	−0.962127	−	0.272600i	$-0.912116\pi$
$983$		−	16803.0i		−	0.545201i	0.962127	−	0.272600i	$-0.0878837\pi$
$984$		0			0
$985$	729.000			0.0235816
$986$		0			0
$987$	−1665.00			−0.0536956
$988$		0			0
$989$	−15714.0			−0.505234
$990$		0			0
$991$	57526.0			1.84397			0.921985	−	0.387226i	$-0.126567\pi$
$991$	57526.0			1.84397			0.921985	+	0.387226i	$0.126567\pi$
$992$		0			0
$993$			10770.0i			0.344185i
$994$		0			0
$995$		−	17964.0i		−	0.572359i
$996$		0			0
$997$	−25000.0			−0.794140			−0.397070	−	0.917788i	$-0.629973\pi$
$997$	−25000.0			−0.794140			−0.397070	+	0.917788i	$0.629973\pi$
$998$		0			0
$999$		−	16059.0i		−	0.508593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.f.b.129.2		2
4.3	odd	2		13.4.b.a.12.1	✓	2
8.3	odd	2		832.4.f.e.129.1		2
8.5	even	2		832.4.f.c.129.1		2
12.11	even	2		117.4.b.a.64.2		2
13.12	even	2	inner	208.4.f.b.129.1		2
20.3	even	4		325.4.d.a.324.1		2
20.7	even	4		325.4.d.b.324.2		2
20.19	odd	2		325.4.c.b.51.2		2
52.3	odd	6		169.4.e.d.147.2		4
52.7	even	12		169.4.c.b.146.1		2
52.11	even	12		169.4.c.b.22.1		2
52.15	even	12		169.4.c.c.22.1		2
52.19	even	12		169.4.c.c.146.1		2
52.23	odd	6		169.4.e.d.147.1		4
52.31	even	4		169.4.a.b.1.1		1
52.35	odd	6		169.4.e.d.23.1		4
52.43	odd	6		169.4.e.d.23.2		4
52.47	even	4		169.4.a.c.1.1		1
52.51	odd	2		13.4.b.a.12.2	yes	2
104.51	odd	2		832.4.f.e.129.2		2
104.77	even	2		832.4.f.c.129.2		2
156.47	odd	4		1521.4.a.d.1.1		1
156.83	odd	4		1521.4.a.i.1.1		1
156.155	even	2		117.4.b.a.64.1		2
260.103	even	4		325.4.d.b.324.1		2
260.207	even	4		325.4.d.a.324.2		2
260.259	odd	2		325.4.c.b.51.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.b.a.12.1	✓	2	4.3	odd	2
13.4.b.a.12.2	yes	2	52.51	odd	2
117.4.b.a.64.1		2	156.155	even	2
117.4.b.a.64.2		2	12.11	even	2
169.4.a.b.1.1		1	52.31	even	4
169.4.a.c.1.1		1	52.47	even	4
169.4.c.b.22.1		2	52.11	even	12
169.4.c.b.146.1		2	52.7	even	12
169.4.c.c.22.1		2	52.15	even	12
169.4.c.c.146.1		2	52.19	even	12
169.4.e.d.23.1		4	52.35	odd	6
169.4.e.d.23.2		4	52.43	odd	6
169.4.e.d.147.1		4	52.23	odd	6
169.4.e.d.147.2		4	52.3	odd	6
208.4.f.b.129.1		2	13.12	even	2	inner
208.4.f.b.129.2		2	1.1	even	1	trivial
325.4.c.b.51.1		2	260.259	odd	2
325.4.c.b.51.2		2	20.19	odd	2
325.4.d.a.324.1		2	20.3	even	4
325.4.d.a.324.2		2	260.207	even	4
325.4.d.b.324.1		2	260.103	even	4
325.4.d.b.324.2		2	20.7	even	4
832.4.f.c.129.1		2	8.5	even	2
832.4.f.c.129.2		2	104.77	even	2
832.4.f.e.129.1		2	8.3	odd	2
832.4.f.e.129.2		2	104.51	odd	2
1521.4.a.d.1.1		1	156.47	odd	4
1521.4.a.i.1.1		1	156.83	odd	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +9.00000i q^{5} -15.0000i q^{7} -26.0000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +9.00000i q^{5} -15.0000i q^{7} -26.0000 q^{9} +48.0000i q^{11} +(26.0000 + 39.0000i) q^{13} +9.00000i q^{15} -45.0000 q^{17} +6.00000i q^{19} -15.0000i q^{21} -162.000 q^{23} +44.0000 q^{25} -53.0000 q^{27} -144.000 q^{29} +264.000i q^{31} +48.0000i q^{33} +135.000 q^{35} +303.000i q^{37} +(26.0000 + 39.0000i) q^{39} +192.000i q^{41} +97.0000 q^{43} -234.000i q^{45} -111.000i q^{47} +118.000 q^{49} -45.0000 q^{51} -414.000 q^{53} -432.000 q^{55} +6.00000i q^{57} -522.000i q^{59} +376.000 q^{61} +390.000i q^{63} +(-351.000 + 234.000i) q^{65} -36.0000i q^{67} -162.000 q^{69} +357.000i q^{71} -1098.00i q^{73} +44.0000 q^{75} +720.000 q^{77} +830.000 q^{79} +649.000 q^{81} -438.000i q^{83} -405.000i q^{85} -144.000 q^{87} -438.000i q^{89} +(585.000 - 390.000i) q^{91} +264.000i q^{93} -54.0000 q^{95} +852.000i q^{97} -1248.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 52 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 52 * q^9	\( 2 q + 2 q^{3} - 52 q^{9} + 52 q^{13} - 90 q^{17} - 324 q^{23} + 88 q^{25} - 106 q^{27} - 288 q^{29} + 270 q^{35} + 52 q^{39} + 194 q^{43} + 236 q^{49} - 90 q^{51} - 828 q^{53} - 864 q^{55} + 752 q^{61} - 702 q^{65} - 324 q^{69} + 88 q^{75} + 1440 q^{77} + 1660 q^{79} + 1298 q^{81} - 288 q^{87} + 1170 q^{91} - 108 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 52 * q^9 + 52 * q^13 - 90 * q^17 - 324 * q^23 + 88 * q^25 - 106 * q^27 - 288 * q^29 + 270 * q^35 + 52 * q^39 + 194 * q^43 + 236 * q^49 - 90 * q^51 - 828 * q^53 - 864 * q^55 + 752 * q^61 - 702 * q^65 - 324 * q^69 + 88 * q^75 + 1440 * q^77 + 1660 * q^79 + 1298 * q^81 - 288 * q^87 + 1170 * q^91 - 108 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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