LMFDB - Embedded newform 600.3.l.g.401.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(401,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("600.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.l (of order $2$, degree $1$, 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:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 30x^{8} - 216x^{6} + 1080x^{4} - 5184x^{2} + 46656 $ x^12 - 4x^10 + 30x^8 - 216x^6 + 1080x^4 - 5184*x^2 + 46656
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.5
Root			$2.38091 + 0.575548i$ of defining polynomial
Character	$\chi$	$=$	600.401
Dual form			600.3.l.g.401.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.26002 - 2.72256i) q^{3} +0.735748 q^{7} +(-5.82469 + 6.86097i) q^{9} +O(q^{10})$	$q+(-1.26002 - 2.72256i) q^{3} +0.735748 q^{7} +(-5.82469 + 6.86097i) q^{9} -10.9451i q^{11} +21.1901 q^{13} -7.03488i q^{17} -23.1529 q^{19} +(-0.927058 - 2.00312i) q^{21} -24.7483i q^{23} +(26.0187 + 7.21312i) q^{27} -32.3284i q^{29} -34.9482 q^{31} +(-29.7987 + 13.7911i) q^{33} -37.7818 q^{37} +(-26.6999 - 57.6912i) q^{39} +39.0848i q^{41} +22.6804 q^{43} -39.1076i q^{47} -48.4587 q^{49} +(-19.1529 + 8.86409i) q^{51} +60.9179i q^{53} +(29.1731 + 63.0352i) q^{57} +7.79696i q^{59} -11.1529 q^{61} +(-4.28551 + 5.04795i) q^{63} -33.3485 q^{67} +(-67.3787 + 31.1833i) q^{69} -96.9650i q^{71} -134.535 q^{73} -8.05284i q^{77} -121.049 q^{79} +(-13.1459 - 79.9261i) q^{81} -90.2345i q^{83} +(-88.0160 + 40.7344i) q^{87} -53.1846i q^{89} +15.5905 q^{91} +(44.0354 + 95.1486i) q^{93} +115.001 q^{97} +(75.0940 + 63.7519i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9}+O(q^{10}) $ 12 * q - 8 * q^9	$ 12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 + 4 * q^21 - 48 * q^31 + 128 * q^39 + 252 * q^49 + 48 * q^51 + 144 * q^61 - 268 * q^69 - 432 * q^79 - 188 * q^81 + 560 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−1.26002	−	2.72256i	−0.420007	−	0.907521i
$3$	−1.26002	−	2.72256i	−0.420007	−	0.907521i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.735748			0.105107			0.0525534	−	0.998618i	$-0.483264\pi$
$7$	0.735748			0.105107			0.0525534	+	0.998618i	$0.483264\pi$
$8$		0			0
$9$	−5.82469	+	6.86097i	−0.647188	+	0.762330i
$10$		0			0
$11$		−	10.9451i		−	0.995009i	−0.867461	−	0.497505i	$-0.834250\pi$
$11$		−	10.9451i		−	0.995009i	0.867461	−	0.497505i	$-0.165750\pi$
$12$		0			0
$13$	21.1901			1.63000			0.815002	−	0.579458i	$-0.196736\pi$
$13$	21.1901			1.63000			0.815002	+	0.579458i	$0.196736\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.03488i		−	0.413816i	−0.978360	−	0.206908i	$-0.933660\pi$
$17$		−	7.03488i		−	0.413816i	0.978360	−	0.206908i	$-0.0663401\pi$
$18$		0			0
$19$	−23.1529			−1.21857			−0.609287	−	0.792950i	$-0.708544\pi$
$19$	−23.1529			−1.21857			−0.609287	+	0.792950i	$0.708544\pi$
$20$		0			0
$21$	−0.927058	−	2.00312i	−0.0441456	−	0.0953867i
$22$		0			0
$23$		−	24.7483i		−	1.07601i	−0.842941	−	0.538006i	$-0.819178\pi$
$23$		−	24.7483i		−	1.07601i	0.842941	−	0.538006i	$-0.180822\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	26.0187	+	7.21312i	0.963654	+	0.267153i
$28$		0			0
$29$		−	32.3284i		−	1.11477i	−0.830254	−	0.557386i	$-0.811804\pi$
$29$		−	32.3284i		−	1.11477i	0.830254	−	0.557386i	$-0.188196\pi$
$30$		0			0
$31$	−34.9482			−1.12736			−0.563680	−	0.825993i	$-0.690615\pi$
$31$	−34.9482			−1.12736			−0.563680	+	0.825993i	$0.690615\pi$
$32$		0			0
$33$	−29.7987	+	13.7911i	−0.902992	+	0.417911i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−37.7818			−1.02113			−0.510565	−	0.859839i	$-0.670564\pi$
$37$	−37.7818			−1.02113			−0.510565	+	0.859839i	$0.670564\pi$
$38$		0			0
$39$	−26.6999	−	57.6912i	−0.684613	−	1.47926i
$40$		0			0
$41$			39.0848i			0.953288i	0.879096	+	0.476644i	$0.158147\pi$
$41$			39.0848i			0.953288i	−0.879096	+	0.476644i	$0.841853\pi$
$42$		0			0
$43$	22.6804			0.527451			0.263725	−	0.964598i	$-0.415049\pi$
$43$	22.6804			0.527451			0.263725	+	0.964598i	$0.415049\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	39.1076i		−	0.832076i	−0.909347	−	0.416038i	$-0.863418\pi$
$47$		−	39.1076i		−	0.832076i	0.909347	−	0.416038i	$-0.136582\pi$
$48$		0			0
$49$	−48.4587			−0.988953
$50$		0			0
$51$	−19.1529	+	8.86409i	−0.375547	+	0.173806i
$52$		0			0
$53$			60.9179i			1.14939i	0.818366	+	0.574697i	$0.194880\pi$
$53$			60.9179i			1.14939i	−0.818366	+	0.574697i	$0.805120\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	29.1731	+	63.0352i	0.511809	+	1.10588i
$58$		0			0
$59$			7.79696i			0.132152i	0.997815	+	0.0660759i	$0.0210480\pi$
$59$			7.79696i			0.132152i	−0.997815	+	0.0660759i	$0.978952\pi$
$60$		0			0
$61$	−11.1529			−0.182834			−0.0914171	−	0.995813i	$-0.529140\pi$
$61$	−11.1529			−0.182834			−0.0914171	+	0.995813i	$0.529140\pi$
$62$		0			0
$63$	−4.28551	+	5.04795i	−0.0680239	+	0.0801262i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−33.3485			−0.497739			−0.248869	−	0.968537i	$-0.580059\pi$
$67$	−33.3485			−0.497739			−0.248869	+	0.968537i	$0.580059\pi$
$68$		0			0
$69$	−67.3787	+	31.1833i	−0.976503	+	0.451933i
$70$		0			0
$71$		−	96.9650i		−	1.36570i	−0.730557	−	0.682852i	$-0.760739\pi$
$71$		−	96.9650i		−	1.36570i	0.730557	−	0.682852i	$-0.239261\pi$
$72$		0			0
$73$	−134.535			−1.84295			−0.921476	−	0.388436i	$-0.873016\pi$
$73$	−134.535			−1.84295			−0.921476	+	0.388436i	$0.873016\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	8.05284i		−	0.104582i
$78$		0			0
$79$	−121.049			−1.53227			−0.766134	−	0.642681i	$-0.777822\pi$
$79$	−121.049			−1.53227			−0.766134	+	0.642681i	$0.777822\pi$
$80$		0			0
$81$	−13.1459	−	79.9261i	−0.162295	−	0.986742i
$82$		0			0
$83$		−	90.2345i		−	1.08716i	−0.839356	−	0.543582i	$-0.817068\pi$
$83$		−	90.2345i		−	1.08716i	0.839356	−	0.543582i	$-0.182932\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−88.0160	+	40.7344i	−1.01168	+	0.468212i
$88$		0			0
$89$		−	53.1846i		−	0.597579i	−0.954319	−	0.298790i	$-0.903417\pi$
$89$		−	53.1846i		−	0.597579i	0.954319	−	0.298790i	$-0.0965829\pi$
$90$		0			0
$91$	15.5905			0.171325
$92$		0			0
$93$	44.0354	+	95.1486i	0.473499	+	1.02310i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	115.001			1.18557			0.592787	−	0.805359i	$-0.298028\pi$
$97$	115.001			1.18557			0.592787	+	0.805359i	$0.298028\pi$
$98$		0			0
$99$	75.0940	+	63.7519i	0.758526	+	0.643958i
$100$		0			0
$101$		−	29.1802i		−	0.288913i	−0.989511	−	0.144457i	$-0.953857\pi$
$101$		−	29.1802i		−	0.288913i	0.989511	−	0.144457i	$-0.0461434\pi$
$102$		0			0
$103$	89.9481			0.873283			0.436641	−	0.899636i	$-0.356168\pi$
$103$	89.9481			0.873283			0.436641	+	0.899636i	$0.356168\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	153.586i		−	1.43538i	−0.696363	−	0.717689i	$-0.745200\pi$
$107$		−	153.586i		−	1.43538i	0.696363	−	0.717689i	$-0.254800\pi$
$108$		0			0
$109$	59.5623			0.546444			0.273222	−	0.961951i	$-0.411911\pi$
$109$	59.5623			0.546444			0.273222	+	0.961951i	$0.411911\pi$
$110$		0			0
$111$	47.6059	+	102.863i	0.428882	+	0.926697i
$112$		0			0
$113$		−	1.01796i		−	0.00900851i	−0.999990	−	0.00450426i	$-0.998566\pi$
$113$		−	1.01796i		−	0.00900851i	0.999990	−	0.00450426i	$-0.00143375\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−123.426	+	145.384i	−1.05492	+	1.24260i
$118$		0			0
$119$		−	5.17590i		−	0.0434949i
$120$		0			0
$121$	1.20473			0.00995643
$122$		0			0
$123$	106.411	−	49.2477i	0.865129	−	0.400388i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	209.731			1.65142			0.825712	−	0.564091i	$-0.190773\pi$
$127$	209.731			1.65142			0.825712	+	0.564091i	$0.190773\pi$
$128$		0			0
$129$	−28.5778	−	61.7488i	−0.221533	−	0.478672i
$130$		0			0
$131$			210.051i			1.60344i	0.597698	+	0.801721i	$0.296082\pi$
$131$			210.051i			1.60344i	−0.597698	+	0.801721i	$0.703918\pi$
$132$		0			0
$133$	−17.0347			−0.128080
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	168.688i		−	1.23130i	−0.788021	−	0.615648i	$-0.788894\pi$
$137$		−	168.688i		−	1.23130i	0.788021	−	0.615648i	$-0.211106\pi$
$138$		0			0
$139$	129.251			0.929866			0.464933	−	0.885346i	$-0.346078\pi$
$139$	129.251			0.929866			0.464933	+	0.885346i	$0.346078\pi$
$140$		0			0
$141$	−106.473	+	49.2763i	−0.755126	+	0.349478i
$142$		0			0
$143$		−	231.927i		−	1.62187i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	61.0590	+	131.932i	0.415367	+	0.897495i
$148$		0			0
$149$			83.9655i			0.563527i	0.959484	+	0.281763i	$0.0909193\pi$
$149$			83.9655i			0.563527i	−0.959484	+	0.281763i	$0.909081\pi$
$150$		0			0
$151$	9.04922			0.0599286			0.0299643	−	0.999551i	$-0.490461\pi$
$151$	9.04922			0.0599286			0.0299643	+	0.999551i	$0.490461\pi$
$152$		0			0
$153$	48.2661	+	40.9760i	0.315465	+	0.267817i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−162.054			−1.03219			−0.516097	−	0.856530i	$-0.672616\pi$
$157$	−162.054			−1.03219			−0.516097	+	0.856530i	$0.672616\pi$
$158$		0			0
$159$	165.853	−	76.7578i	1.04310	−	0.482754i
$160$		0			0
$161$		−	18.2085i		−	0.113096i
$162$		0			0
$163$	−136.172			−0.835411			−0.417705	−	0.908583i	$-0.637166\pi$
$163$	−136.172			−0.835411			−0.417705	+	0.908583i	$0.637166\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	140.931i		−	0.843898i	−0.906620	−	0.421949i	$-0.861346\pi$
$167$		−	140.931i		−	0.843898i	0.906620	−	0.421949i	$-0.138654\pi$
$168$		0			0
$169$	280.018			1.65691
$170$		0			0
$171$	134.859	−	158.851i	0.788646	−	0.928955i
$172$		0			0
$173$			132.351i			0.765032i	0.923949	+	0.382516i	$0.124942\pi$
$173$			132.351i			0.765032i	−0.923949	+	0.382516i	$0.875058\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	21.2277	−	9.82433i	0.119931	−	0.0555047i
$178$		0			0
$179$			92.2499i			0.515363i	0.966230	+	0.257681i	$0.0829585\pi$
$179$			92.2499i			0.515363i	−0.966230	+	0.257681i	$0.917042\pi$
$180$		0			0
$181$	115.896			0.640311			0.320156	−	0.947365i	$-0.396265\pi$
$181$	115.896			0.640311			0.320156	+	0.947365i	$0.396265\pi$
$182$		0			0
$183$	14.0529	+	30.3644i	0.0767917	+	0.165926i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−76.9974			−0.411751
$188$		0			0
$189$	19.1432	+	5.30704i	0.101287	+	0.0280796i
$190$		0			0
$191$		−	53.1183i		−	0.278106i	−0.990285	−	0.139053i	$-0.955594\pi$
$191$		−	53.1183i		−	0.278106i	0.990285	−	0.139053i	$-0.0444059\pi$
$192$		0			0
$193$	271.315			1.40578			0.702890	−	0.711299i	$-0.251893\pi$
$193$	271.315			1.40578			0.702890	+	0.711299i	$0.251893\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			64.3941i			0.326873i	0.986554	+	0.163437i	$0.0522580\pi$
$197$			64.3941i			0.326873i	−0.986554	+	0.163437i	$0.947742\pi$
$198$		0			0
$199$	−72.0308			−0.361964			−0.180982	−	0.983486i	$-0.557928\pi$
$199$	−72.0308			−0.361964			−0.180982	+	0.983486i	$0.557928\pi$
$200$		0			0
$201$	42.0198	+	90.7933i	0.209054	+	0.451708i
$202$		0			0
$203$		−	23.7855i		−	0.117170i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	169.797	+	144.151i	0.820276	+	0.696382i
$208$		0			0
$209$			253.411i			1.21249i
$210$		0			0
$211$	108.583			0.514613			0.257307	−	0.966330i	$-0.417165\pi$
$211$	108.583			0.514613			0.257307	+	0.966330i	$0.417165\pi$
$212$		0			0
$213$	−263.993	+	122.178i	−1.23940	+	0.573605i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−25.7130			−0.118493
$218$		0			0
$219$	169.518	+	366.281i	0.774053	+	1.67252i
$220$		0			0
$221$		−	149.069i		−	0.674522i
$222$		0			0
$223$	−83.6192			−0.374974			−0.187487	−	0.982267i	$-0.560034\pi$
$223$	−83.6192			−0.374974			−0.187487	+	0.982267i	$0.560034\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	51.6591i		−	0.227573i	−0.993505	−	0.113787i	$-0.963702\pi$
$227$		−	51.6591i		−	0.227573i	0.993505	−	0.113787i	$-0.0362980\pi$
$228$		0			0
$229$	280.974			1.22696			0.613480	−	0.789710i	$-0.289769\pi$
$229$	280.974			1.22696			0.613480	+	0.789710i	$0.289769\pi$
$230$		0			0
$231$	−21.9244	+	10.1467i	−0.0949106	+	0.0439253i
$232$		0			0
$233$		−	169.192i		−	0.726148i	−0.931760	−	0.363074i	$-0.881727\pi$
$233$		−	169.192i		−	0.726148i	0.931760	−	0.363074i	$-0.118273\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	152.525	+	329.564i	0.643564	+	1.39057i
$238$		0			0
$239$			1.12039i			0.00468782i	0.999997	+	0.00234391i	$0.000746090\pi$
$239$			1.12039i			0.00468782i	−0.999997	+	0.00234391i	$0.999254\pi$
$240$		0			0
$241$	153.254			0.635908			0.317954	−	0.948106i	$-0.397004\pi$
$241$	153.254			0.635908			0.317954	+	0.948106i	$0.397004\pi$
$242$		0			0
$243$	−201.040	+	136.499i	−0.827324	+	0.561725i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−490.611			−1.98628
$248$		0			0
$249$	−245.669	+	113.697i	−0.986623	+	0.456616i
$250$		0			0
$251$			126.692i			0.504751i	0.967629	+	0.252375i	$0.0812118\pi$
$251$			126.692i			0.504751i	−0.967629	+	0.252375i	$0.918788\pi$
$252$		0			0
$253$	−270.872			−1.07064
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	396.692i		−	1.54355i	−0.635897	−	0.771774i	$-0.719370\pi$
$257$		−	396.692i		−	1.54355i	0.635897	−	0.771774i	$-0.280630\pi$
$258$		0			0
$259$	−27.7979			−0.107328
$260$		0			0
$261$	221.804	+	188.303i	0.849824	+	0.721467i
$262$		0			0
$263$			394.431i			1.49974i	0.661587	+	0.749868i	$0.269883\pi$
$263$			394.431i			1.49974i	−0.661587	+	0.749868i	$0.730117\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−144.798	+	67.0137i	−0.542316	+	0.250987i
$268$		0			0
$269$		−	104.802i		−	0.389597i	−0.980843	−	0.194798i	$-0.937595\pi$
$269$		−	104.802i		−	0.389597i	0.980843	−	0.194798i	$-0.0624053\pi$
$270$		0			0
$271$	−335.355			−1.23747			−0.618736	−	0.785599i	$-0.712355\pi$
$271$	−335.355			−1.23747			−0.618736	+	0.785599i	$0.712355\pi$
$272$		0			0
$273$	−19.6444	−	42.4462i	−0.0719576	−	0.155481i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	167.790			0.605739			0.302870	−	0.953032i	$-0.402055\pi$
$277$	167.790			0.605739			0.302870	+	0.953032i	$0.402055\pi$
$278$		0			0
$279$	203.562	−	239.778i	0.729614	−	0.859421i
$280$		0			0
$281$			99.4601i			0.353950i	0.984215	+	0.176975i	$0.0566312\pi$
$281$			99.4601i			0.353950i	−0.984215	+	0.176975i	$0.943369\pi$
$282$		0			0
$283$	487.022			1.72093			0.860464	−	0.509512i	$-0.170174\pi$
$283$	487.022			1.72093			0.860464	+	0.509512i	$0.170174\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			28.7566i			0.100197i
$288$		0			0
$289$	239.511			0.828756
$290$		0			0
$291$	−144.903	−	313.097i	−0.497950	−	1.07593i
$292$		0			0
$293$		−	343.107i		−	1.17101i	−0.810667	−	0.585507i	$-0.800895\pi$
$293$		−	343.107i		−	1.17101i	0.810667	−	0.585507i	$-0.199105\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	78.9484	−	284.777i	0.265819	−	0.958845i
$298$		0			0
$299$		−	524.417i		−	1.75390i
$300$		0			0
$301$	16.6870			0.0554387
$302$		0			0
$303$	−79.4450	+	36.7677i	−0.262195	+	0.121346i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	364.627			1.18771			0.593854	−	0.804573i	$-0.297606\pi$
$307$	364.627			1.18771			0.593854	+	0.804573i	$0.297606\pi$
$308$		0			0
$309$	−113.337	−	244.889i	−0.366785	−	0.792522i
$310$		0			0
$311$		−	121.963i		−	0.392164i	−0.980587	−	0.196082i	$-0.937178\pi$
$311$		−	121.963i		−	0.392164i	0.980587	−	0.196082i	$-0.0628219\pi$
$312$		0			0
$313$	94.8060			0.302895			0.151447	−	0.988465i	$-0.451607\pi$
$313$	94.8060			0.302895			0.151447	+	0.988465i	$0.451607\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			511.282i			1.61288i	0.591318	+	0.806438i	$0.298608\pi$
$317$			511.282i			1.61288i	−0.591318	+	0.806438i	$0.701392\pi$
$318$		0			0
$319$	−353.837			−1.10921
$320$		0			0
$321$	−418.146	+	193.521i	−1.30264	+	0.602869i
$322$		0			0
$323$			162.878i			0.504265i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−75.0498	−	162.162i	−0.229510	−	0.495909i
$328$		0			0
$329$		−	28.7733i		−	0.0874569i
$330$		0			0
$331$	−182.682			−0.551909			−0.275954	−	0.961171i	$-0.588994\pi$
$331$	−182.682			−0.551909			−0.275954	+	0.961171i	$0.588994\pi$
$332$		0			0
$333$	220.067	−	259.220i	0.660863	−	0.778438i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−89.0617			−0.264278			−0.132139	−	0.991231i	$-0.542185\pi$
$337$	−89.0617			−0.264278			−0.132139	+	0.991231i	$0.542185\pi$
$338$		0			0
$339$	−2.77147	+	1.28265i	−0.00817541	+	0.00378364i
$340$		0			0
$341$			382.511i			1.12173i
$342$		0			0
$343$	−71.7050			−0.209053
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	17.7180i		−	0.0510605i	−0.999674	−	0.0255303i	$-0.991873\pi$
$347$		−	17.7180i		−	0.0510605i	0.999674	−	0.0255303i	$-0.00812742\pi$
$348$		0			0
$349$	−229.114			−0.656488			−0.328244	−	0.944593i	$-0.606457\pi$
$349$	−229.114			−0.656488			−0.328244	+	0.944593i	$0.606457\pi$
$350$		0			0
$351$	551.337	+	152.846i	1.57076	+	0.435460i
$352$		0			0
$353$		−	183.760i		−	0.520566i	−0.965532	−	0.260283i	$-0.916184\pi$
$353$		−	183.760i		−	0.520566i	0.965532	−	0.260283i	$-0.0838158\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−14.0917	+	6.52174i	−0.0394726	+	0.0182682i
$358$		0			0
$359$			537.837i			1.49815i	0.662484	+	0.749076i	$0.269502\pi$
$359$			537.837i			1.49815i	−0.662484	+	0.749076i	$0.730498\pi$
$360$		0			0
$361$	175.056			0.484921
$362$		0			0
$363$	−1.51798	−	3.27995i	−0.00418177	−	0.00903567i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	153.740			0.418909			0.209455	−	0.977818i	$-0.432831\pi$
$367$	153.740			0.418909			0.209455	+	0.977818i	$0.432831\pi$
$368$		0			0
$369$	−268.160	−	227.657i	−0.726721	−	0.616957i
$370$		0			0
$371$			44.8202i			0.120809i
$372$		0			0
$373$	211.056			0.565833			0.282917	−	0.959145i	$-0.408698\pi$
$373$	211.056			0.565833			0.282917	+	0.959145i	$0.408698\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	685.040i		−	1.81708i
$378$		0			0
$379$	−699.345			−1.84524			−0.922618	−	0.385715i	$-0.873955\pi$
$379$	−699.345			−1.84524			−0.922618	+	0.385715i	$0.873955\pi$
$380$		0			0
$381$	−264.265	−	571.006i	−0.693610	−	1.49870i
$382$		0			0
$383$		−	186.008i		−	0.485662i	−0.970069	−	0.242831i	$-0.921924\pi$
$383$		−	186.008i		−	0.485662i	0.970069	−	0.242831i	$-0.0780760\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−132.106	+	155.609i	−0.341360	+	0.402092i
$388$		0			0
$389$			288.194i			0.740858i	0.928861	+	0.370429i	$0.120789\pi$
$389$			288.194i			0.740858i	−0.928861	+	0.370429i	$0.879211\pi$
$390$		0			0
$391$	−174.101			−0.445271
$392$		0			0
$393$	571.877	−	264.669i	1.45516	−	0.673457i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	64.3054			0.161978			0.0809892	−	0.996715i	$-0.474192\pi$
$397$	64.3054			0.161978			0.0809892	+	0.996715i	$0.474192\pi$
$398$		0			0
$399$	21.4641	+	46.3780i	0.0537947	+	0.116236i
$400$		0			0
$401$		−	659.774i		−	1.64532i	−0.568533	−	0.822661i	$-0.692489\pi$
$401$		−	659.774i		−	1.64532i	0.568533	−	0.822661i	$-0.307511\pi$
$402$		0			0
$403$	−740.553			−1.83760
$404$		0			0
$405$		0			0
$406$		0			0
$407$			413.526i			1.01603i
$408$		0			0
$409$	−217.863			−0.532672			−0.266336	−	0.963880i	$-0.585813\pi$
$409$	−217.863			−0.532672			−0.266336	+	0.963880i	$0.585813\pi$
$410$		0			0
$411$	−459.263	+	212.550i	−1.11743	+	0.517153i
$412$		0			0
$413$			5.73660i			0.0138901i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−162.859	−	351.895i	−0.390550	−	0.843872i
$418$		0			0
$419$		−	407.129i		−	0.971668i	−0.874051	−	0.485834i	$-0.838516\pi$
$419$		−	407.129i		−	0.971668i	0.874051	−	0.485834i	$-0.161484\pi$
$420$		0			0
$421$	−69.1949			−0.164359			−0.0821793	−	0.996618i	$-0.526188\pi$
$421$	−69.1949			−0.164359			−0.0821793	+	0.996618i	$0.526188\pi$
$422$		0			0
$423$	268.316	+	227.790i	0.634316	+	0.538510i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.20572			−0.0192171
$428$		0			0
$429$	−631.437	+	292.233i	−1.47188	+	0.681197i
$430$		0			0
$431$		−	452.663i		−	1.05026i	−0.851021	−	0.525132i	$-0.824016\pi$
$431$		−	452.663i		−	1.05026i	0.851021	−	0.525132i	$-0.175984\pi$
$432$		0			0
$433$	226.323			0.522686			0.261343	−	0.965246i	$-0.415835\pi$
$433$	226.323			0.522686			0.261343	+	0.965246i	$0.415835\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			572.994i			1.31120i
$438$		0			0
$439$	−188.642			−0.429709			−0.214855	−	0.976646i	$-0.568928\pi$
$439$	−188.642			−0.429709			−0.214855	+	0.976646i	$0.568928\pi$
$440$		0			0
$441$	282.257	−	332.474i	0.640038	−	0.753908i
$442$		0			0
$443$			499.705i			1.12800i	0.825774	+	0.564001i	$0.190739\pi$
$443$			499.705i			1.12800i	−0.825774	+	0.564001i	$0.809261\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	228.601	−	105.798i	0.511412	−	0.236685i
$448$		0			0
$449$		−	818.928i		−	1.82389i	−0.410310	−	0.911946i	$-0.634579\pi$
$449$		−	818.928i		−	1.82389i	0.410310	−	0.911946i	$-0.365421\pi$
$450$		0			0
$451$	427.787			0.948531
$452$		0			0
$453$	−11.4022	−	24.6371i	−0.0251704	−	0.0543864i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	311.602			0.681842			0.340921	−	0.940092i	$-0.389261\pi$
$457$	311.602			0.681842			0.340921	+	0.940092i	$0.389261\pi$
$458$		0			0
$459$	50.7434	−	183.038i	0.110552	−	0.398776i
$460$		0			0
$461$		−	7.18351i		−	0.0155825i	−0.999970	−	0.00779123i	$-0.997520\pi$
$461$		−	7.18351i		−	0.0155825i	0.999970	−	0.00779123i	$-0.00248005\pi$
$462$		0			0
$463$	−557.563			−1.20424			−0.602120	−	0.798406i	$-0.705677\pi$
$463$	−557.563			−1.20424			−0.602120	+	0.798406i	$0.705677\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			659.257i			1.41168i	0.708369	+	0.705842i	$0.249431\pi$
$467$			659.257i			1.41168i	−0.708369	+	0.705842i	$0.750569\pi$
$468$		0			0
$469$	−24.5361			−0.0523157
$470$		0			0
$471$	204.192	+	441.203i	0.433529	+	0.936738i
$472$		0			0
$473$		−	248.239i		−	0.524818i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−417.956	−	354.828i	−0.876218	−	0.743875i
$478$		0			0
$479$		−	30.3870i		−	0.0634384i	−0.999497	−	0.0317192i	$-0.989902\pi$
$479$		−	30.3870i		−	0.0634384i	0.999497	−	0.0317192i	$-0.0100982\pi$
$480$		0			0
$481$	−800.598			−1.66445
$482$		0			0
$483$	−49.5738	+	22.9431i	−0.102637	+	0.0475012i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	26.5618			0.0545416			0.0272708	−	0.999628i	$-0.491318\pi$
$487$	26.5618			0.0545416			0.0272708	+	0.999628i	$0.491318\pi$
$488$		0			0
$489$	171.580	+	370.737i	0.350878	+	0.758153i
$490$		0			0
$491$			19.3354i			0.0393796i	0.999806	+	0.0196898i	$0.00626786\pi$
$491$			19.3354i			0.0393796i	−0.999806	+	0.0196898i	$0.993732\pi$
$492$		0			0
$493$	−227.426			−0.461311
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	71.3418i		−	0.143545i
$498$		0			0
$499$	−313.190			−0.627635			−0.313817	−	0.949483i	$-0.601608\pi$
$499$	−313.190			−0.627635			−0.313817	+	0.949483i	$0.601608\pi$
$500$		0			0
$501$	−383.693	+	177.576i	−0.765855	+	0.354443i
$502$		0			0
$503$			551.684i			1.09679i	0.836220	+	0.548394i	$0.184760\pi$
$503$			551.684i			1.09679i	−0.836220	+	0.548394i	$0.815240\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−352.829	−	762.368i	−0.695915	−	1.50368i
$508$		0			0
$509$		−	567.057i		−	1.11406i	−0.830492	−	0.557031i	$-0.811941\pi$
$509$		−	567.057i		−	1.11406i	0.830492	−	0.557031i	$-0.188059\pi$
$510$		0			0
$511$	−98.9842			−0.193707
$512$		0			0
$513$	−602.407	−	167.005i	−1.17428	−	0.325545i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−428.036			−0.827923
$518$		0			0
$519$	360.333	−	166.764i	0.694283	−	0.321319i
$520$		0			0
$521$			740.223i			1.42077i	0.703811	+	0.710387i	$0.251480\pi$
$521$			740.223i			1.42077i	−0.703811	+	0.710387i	$0.748520\pi$
$522$		0			0
$523$	2.11805			0.00404981			0.00202491	−	0.999998i	$-0.499355\pi$
$523$	2.11805			0.00404981			0.00202491	+	0.999998i	$0.499355\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			245.856i			0.466520i
$528$		0			0
$529$	−83.4771			−0.157802
$530$		0			0
$531$	−53.4947	−	45.4149i	−0.100743	−	0.0855271i
$532$		0			0
$533$			828.210i			1.55386i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	251.156	−	116.237i	0.467702	−	0.216456i
$538$		0			0
$539$			530.385i			0.984017i
$540$		0			0
$541$	6.61681			0.0122307			0.00611535	−	0.999981i	$-0.498053\pi$
$541$	6.61681			0.0122307			0.00611535	+	0.999981i	$0.498053\pi$
$542$		0			0
$543$	−146.032	−	315.535i	−0.268935	−	0.581096i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−230.092			−0.420644			−0.210322	−	0.977632i	$-0.567451\pi$
$547$	−230.092			−0.420644			−0.210322	+	0.977632i	$0.567451\pi$
$548$		0			0
$549$	64.9622	−	76.5197i	0.118328	−	0.139380i
$550$		0			0
$551$			748.495i			1.35843i
$552$		0			0
$553$	−89.0617			−0.161052
$554$		0			0
$555$		0			0
$556$		0			0
$557$			529.620i			0.950844i	0.879758	+	0.475422i	$0.157705\pi$
$557$			529.620i			0.950844i	−0.879758	+	0.475422i	$0.842295\pi$
$558$		0			0
$559$	480.598			0.859747
$560$		0			0
$561$	97.0184	+	209.630i	0.172938	+	0.373673i
$562$		0			0
$563$			131.530i			0.233623i	0.993154	+	0.116812i	$0.0372673\pi$
$563$			131.530i			0.233623i	−0.993154	+	0.116812i	$0.962733\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−9.67206	−	58.8055i	−0.0170583	−	0.103713i
$568$		0			0
$569$			172.534i			0.303224i	0.988440	+	0.151612i	$0.0484464\pi$
$569$			172.534i			0.303224i	−0.988440	+	0.151612i	$0.951554\pi$
$570$		0			0
$571$	197.935			0.346646			0.173323	−	0.984865i	$-0.444550\pi$
$571$	197.935			0.346646			0.173323	+	0.984865i	$0.444550\pi$
$572$		0			0
$573$	−144.618	+	66.9302i	−0.252387	+	0.116807i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	865.374			1.49978			0.749891	−	0.661562i	$-0.230106\pi$
$577$	865.374			1.49978			0.749891	+	0.661562i	$0.230106\pi$
$578$		0			0
$579$	−341.863	−	738.673i	−0.590437	−	1.27577i
$580$		0			0
$581$		−	66.3899i		−	0.114268i
$582$		0			0
$583$	666.753			1.14366
$584$		0			0
$585$		0			0
$586$		0			0
$587$			30.8886i			0.0526211i	0.999654	+	0.0263105i	$0.00837587\pi$
$587$			30.8886i			0.0526211i	−0.999654	+	0.0263105i	$0.991624\pi$
$588$		0			0
$589$	809.151			1.37377
$590$		0			0
$591$	175.317	−	81.1379i	0.296644	−	0.137289i
$592$		0			0
$593$		−	511.286i		−	0.862202i	−0.902304	−	0.431101i	$-0.858125\pi$
$593$		−	511.286i		−	0.862202i	0.902304	−	0.431101i	$-0.141875\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	90.7603	+	196.108i	0.152027	+	0.328490i
$598$		0			0
$599$			514.866i			0.859543i	0.902938	+	0.429772i	$0.141406\pi$
$599$			514.866i			0.859543i	−0.902938	+	0.429772i	$0.858594\pi$
$600$		0			0
$601$	853.532			1.42019			0.710093	−	0.704107i	$-0.248653\pi$
$601$	853.532			1.42019			0.710093	+	0.704107i	$0.248653\pi$
$602$		0			0
$603$	194.245	−	228.803i	0.322131	−	0.379441i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−88.0713			−0.145093			−0.0725464	−	0.997365i	$-0.523113\pi$
$607$	−88.0713			−0.145093			−0.0725464	+	0.997365i	$0.523113\pi$
$608$		0			0
$609$	−64.7576	+	29.9703i	−0.106334	+	0.0492123i
$610$		0			0
$611$		−	828.691i		−	1.35629i
$612$		0			0
$613$	403.346			0.657987			0.328994	−	0.944332i	$-0.393291\pi$
$613$	403.346			0.657987			0.328994	+	0.944332i	$0.393291\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			365.738i			0.592769i	0.955069	+	0.296384i	$0.0957810\pi$
$617$			365.738i			0.592769i	−0.955069	+	0.296384i	$0.904219\pi$
$618$		0			0
$619$	399.495			0.645389			0.322694	−	0.946503i	$-0.395411\pi$
$619$	399.495			0.645389			0.322694	+	0.946503i	$0.395411\pi$
$620$		0			0
$621$	178.512	−	643.917i	0.287460	−	1.03690i
$622$		0			0
$623$		−	39.1304i		−	0.0628097i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	689.927	−	319.303i	1.10036	−	0.509255i
$628$		0			0
$629$			265.790i			0.422560i
$630$		0			0
$631$	−715.762			−1.13433			−0.567165	−	0.823604i	$-0.691960\pi$
$631$	−715.762			−1.13433			−0.567165	+	0.823604i	$0.691960\pi$
$632$		0			0
$633$	−136.817	−	295.625i	−0.216141	−	0.467022i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1026.84			−1.61200
$638$		0			0
$639$	665.274	+	564.791i	1.04112	+	0.883867i
$640$		0			0
$641$		−	627.158i		−	0.978406i	−0.872170	−	0.489203i	$-0.837288\pi$
$641$		−	627.158i		−	0.978406i	0.872170	−	0.489203i	$-0.162712\pi$
$642$		0			0
$643$	802.039			1.24734			0.623669	−	0.781688i	$-0.285641\pi$
$643$	802.039			1.24734			0.623669	+	0.781688i	$0.285641\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	19.9275i		−	0.0307998i	−0.999881	−	0.0153999i	$-0.995098\pi$
$647$		−	19.9275i		−	0.0307998i	0.999881	−	0.0153999i	$-0.00490214\pi$
$648$		0			0
$649$	85.3385			0.131492
$650$		0			0
$651$	32.3990	+	70.0054i	0.0497680	+	0.107535i
$652$		0			0
$653$			78.4638i			0.120159i	0.998194	+	0.0600795i	$0.0191354\pi$
$653$			78.4638i			0.120159i	−0.998194	+	0.0600795i	$0.980865\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	783.628	−	923.044i	1.19274	−	1.40494i
$658$		0			0
$659$			371.793i			0.564178i	0.959388	+	0.282089i	$0.0910274\pi$
$659$			371.793i			0.564178i	−0.959388	+	0.282089i	$0.908973\pi$
$660$		0			0
$661$	782.868			1.18437			0.592185	−	0.805802i	$-0.298266\pi$
$661$	782.868			1.18437			0.592185	+	0.805802i	$0.298266\pi$
$662$		0			0
$663$	−405.851	+	187.831i	−0.612143	+	0.283304i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−800.071			−1.19951
$668$		0			0
$669$	105.362	+	227.658i	0.157492	+	0.340297i
$670$		0			0
$671$			122.070i			0.181922i
$672$		0			0
$673$	−221.323			−0.328860			−0.164430	−	0.986389i	$-0.552579\pi$
$673$	−221.323			−0.328860			−0.164430	+	0.986389i	$0.552579\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			576.855i			0.852076i	0.904705	+	0.426038i	$0.140091\pi$
$677$			576.855i			0.852076i	−0.904705	+	0.426038i	$0.859909\pi$
$678$		0			0
$679$	84.6116			0.124612
$680$		0			0
$681$	−140.645	+	65.0916i	−0.206527	+	0.0955824i
$682$		0			0
$683$		−	1272.02i		−	1.86239i	−0.364516	−	0.931197i	$-0.618766\pi$
$683$		−	1272.02i		−	1.86239i	0.364516	−	0.931197i	$-0.381234\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−354.033	−	764.969i	−0.515332	−	1.11349i
$688$		0			0
$689$			1290.85i			1.87352i
$690$		0			0
$691$	1.61487			0.00233701			0.00116851	−	0.999999i	$-0.499628\pi$
$691$	1.61487			0.00233701			0.00116851	+	0.999999i	$0.499628\pi$
$692$		0			0
$693$	55.2503	+	46.9053i	0.0797263	+	0.0676844i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	274.957			0.394486
$698$		0			0
$699$	−460.637	+	213.186i	−0.658994	+	0.304987i
$700$		0			0
$701$			550.235i			0.784929i	0.919767	+	0.392465i	$0.128377\pi$
$701$			550.235i			0.784929i	−0.919767	+	0.392465i	$0.871623\pi$
$702$		0			0
$703$	874.758			1.24432
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	21.4693i		−	0.0303668i
$708$		0			0
$709$	430.539			0.607249			0.303624	−	0.952792i	$-0.401803\pi$
$709$	430.539			0.607249			0.303624	+	0.952792i	$0.401803\pi$
$710$		0			0
$711$	705.075	−	830.515i	0.991666	−	1.16809i
$712$		0			0
$713$			864.907i			1.21305i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	3.05033	−	1.41171i	0.00425429	−	0.00196892i
$718$		0			0
$719$		−	460.561i		−	0.640558i	−0.947323	−	0.320279i	$-0.896223\pi$
$719$		−	460.561i		−	0.640558i	0.947323	−	0.320279i	$-0.103777\pi$
$720$		0			0
$721$	66.1792			0.0917880
$722$		0			0
$723$	−193.103	−	417.243i	−0.267086	−	0.577100i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	413.275			0.568467			0.284233	−	0.958755i	$-0.408261\pi$
$727$	413.275			0.568467			0.284233	+	0.958755i	$0.408261\pi$
$728$		0			0
$729$	624.942	+	375.352i	0.857259	+	0.514886i
$730$		0			0
$731$		−	159.554i		−	0.218268i
$732$		0			0
$733$	−307.003			−0.418831			−0.209416	−	0.977827i	$-0.567156\pi$
$733$	−307.003			−0.418831			−0.209416	+	0.977827i	$0.567156\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			365.003i			0.495255i
$738$		0			0
$739$	988.511			1.33763			0.668817	−	0.743427i	$-0.266801\pi$
$739$	988.511			1.33763			0.668817	+	0.743427i	$0.266801\pi$
$740$		0			0
$741$	618.180	+	1335.72i	0.834251	+	1.80259i
$742$		0			0
$743$		−	652.187i		−	0.877775i	−0.898542	−	0.438887i	$-0.855373\pi$
$743$		−	652.187i		−	0.877775i	0.898542	−	0.438887i	$-0.144627\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	619.097	+	525.589i	0.828777	+	0.703599i
$748$		0			0
$749$		−	113.000i		−	0.150868i
$750$		0			0
$751$	554.876			0.738850			0.369425	−	0.929261i	$-0.379555\pi$
$751$	554.876			0.738850			0.369425	+	0.929261i	$0.379555\pi$
$752$		0			0
$753$	344.928	−	159.635i	0.458072	−	0.211999i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−547.156			−0.722795			−0.361397	−	0.932412i	$-0.617700\pi$
$757$	−547.156			−0.722795			−0.361397	+	0.932412i	$0.617700\pi$
$758$		0			0
$759$	341.305	+	737.467i	0.449677	+	0.971630i
$760$		0			0
$761$			1361.34i			1.78888i	0.447184	+	0.894442i	$0.352427\pi$
$761$			1361.34i			1.78888i	−0.447184	+	0.894442i	$0.647573\pi$
$762$		0			0
$763$	43.8229			0.0574350
$764$		0			0
$765$		0			0
$766$		0			0
$767$			165.218i			0.215408i
$768$		0			0
$769$	396.180			0.515188			0.257594	−	0.966253i	$-0.417070\pi$
$769$	396.180			0.515188			0.257594	+	0.966253i	$0.417070\pi$
$770$		0			0
$771$	−1080.02	+	499.840i	−1.40080	+	0.648301i
$772$		0			0
$773$			664.286i			0.859361i	0.902981	+	0.429681i	$0.141374\pi$
$773$			664.286i			0.859361i	−0.902981	+	0.429681i	$0.858626\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	35.0259	+	75.6815i	0.0450784	+	0.0974022i
$778$		0			0
$779$		−	904.927i		−	1.16165i
$780$		0			0
$781$	−1061.29			−1.35889
$782$		0			0
$783$	233.189	−	841.141i	0.297814	−	1.07425i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	957.551			1.21671			0.608355	−	0.793665i	$-0.291830\pi$
$787$	957.551			1.21671			0.608355	+	0.793665i	$0.291830\pi$
$788$		0			0
$789$	1073.86	−	496.991i	1.36104	−	0.629900i
$790$		0			0
$791$		−	0.748964i		−	0.000946857i
$792$		0			0
$793$	−236.330			−0.298021
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	643.860i		−	0.807854i	−0.914791	−	0.403927i	$-0.867645\pi$
$797$		−	643.860i		−	0.807854i	0.914791	−	0.403927i	$-0.132355\pi$
$798$		0			0
$799$	−275.117			−0.344326
$800$		0			0
$801$	364.898	+	309.784i	0.455553	+	0.386746i
$802$		0			0
$803$			1472.50i			1.83375i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−285.329	+	132.052i	−0.353567	+	0.163633i
$808$		0			0
$809$		−	675.342i		−	0.834786i	−0.908726	−	0.417393i	$-0.862944\pi$
$809$		−	675.342i		−	0.834786i	0.908726	−	0.417393i	$-0.137056\pi$
$810$		0			0
$811$	808.662			0.997117			0.498559	−	0.866856i	$-0.333863\pi$
$811$	808.662			0.997117			0.498559	+	0.866856i	$0.333863\pi$
$812$		0			0
$813$	422.554	+	913.025i	0.519747	+	1.12303i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−525.116			−0.642737
$818$		0			0
$819$	−90.8101	+	106.966i	−0.110879	+	0.130606i
$820$		0			0
$821$		−	1348.63i		−	1.64267i	−0.570445	−	0.821336i	$-0.693229\pi$
$821$		−	1348.63i		−	1.64267i	0.570445	−	0.821336i	$-0.306771\pi$
$822$		0			0
$823$	−1433.75			−1.74211			−0.871053	−	0.491188i	$-0.836563\pi$
$823$	−1433.75			−1.74211			−0.871053	+	0.491188i	$0.836563\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.5276i		−	0.0332861i	−0.999861	−	0.0166431i	$-0.994702\pi$
$827$		−	27.5276i		−	0.0332861i	0.999861	−	0.0166431i	$-0.00529789\pi$
$828$		0			0
$829$	−374.884			−0.452212			−0.226106	−	0.974103i	$-0.572600\pi$
$829$	−374.884			−0.452212			−0.226106	+	0.974103i	$0.572600\pi$
$830$		0			0
$831$	−211.419	−	456.818i	−0.254415	−	0.549721i
$832$		0			0
$833$			340.901i			0.409245i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−909.304	−	252.085i	−1.08639	−	0.301177i
$838$		0			0
$839$		−	1173.68i		−	1.39890i	−0.714681	−	0.699451i	$-0.753428\pi$
$839$		−	1173.68i		−	1.39890i	0.714681	−	0.699451i	$-0.246572\pi$
$840$		0			0
$841$	−204.123			−0.242715
$842$		0			0
$843$	270.786	−	125.322i	0.321217	−	0.148662i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	0.886377			0.00104649
$848$		0			0
$849$	−613.659	−	1325.95i	−0.722802	−	1.56178i
$850$		0			0
$851$			935.034i			1.09875i
$852$		0			0
$853$	112.279			0.131629			0.0658143	−	0.997832i	$-0.479036\pi$
$853$	112.279			0.131629			0.0658143	+	0.997832i	$0.479036\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			619.631i			0.723023i	0.932368	+	0.361511i	$0.117739\pi$
$857$			619.631i			0.723023i	−0.932368	+	0.361511i	$0.882261\pi$
$858$		0			0
$859$	1227.29			1.42875			0.714373	−	0.699765i	$-0.246712\pi$
$859$	1227.29			1.42875			0.714373	+	0.699765i	$0.246712\pi$
$860$		0			0
$861$	78.2916	−	36.2339i	0.0909310	−	0.0420835i
$862$		0			0
$863$		−	975.318i		−	1.13015i	−0.825040	−	0.565074i	$-0.808848\pi$
$863$		−	975.318i		−	1.13015i	0.825040	−	0.565074i	$-0.191152\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−301.788	−	652.082i	−0.348083	−	0.752113i
$868$		0			0
$869$			1324.90i			1.52462i
$870$		0			0
$871$	−706.656			−0.811316
$872$		0			0
$873$	−669.844	+	789.017i	−0.767290	+	0.903799i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	991.857			1.13097			0.565483	−	0.824760i	$-0.308690\pi$
$877$	991.857			1.13097			0.565483	+	0.824760i	$0.308690\pi$
$878$		0			0
$879$	−934.131	+	432.322i	−1.06272	+	0.491834i
$880$		0			0
$881$		−	1024.25i		−	1.16260i	−0.813690	−	0.581299i	$-0.802545\pi$
$881$		−	1024.25i		−	1.16260i	0.813690	−	0.581299i	$-0.197455\pi$
$882$		0			0
$883$	−1170.22			−1.32527			−0.662637	−	0.748941i	$-0.730563\pi$
$883$	−1170.22			−1.32527			−0.662637	+	0.748941i	$0.730563\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	540.358i		−	0.609197i	−0.952481	−	0.304599i	$-0.901478\pi$
$887$		−	540.358i		−	0.609197i	0.952481	−	0.304599i	$-0.0985223\pi$
$888$		0			0
$889$	154.309			0.173576
$890$		0			0
$891$	−874.800	+	143.883i	−0.981818	+	0.161485i
$892$		0			0
$893$			905.453i			1.01395i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1427.76	+	660.777i	−1.59170	+	0.736652i
$898$		0			0
$899$			1129.82i			1.25675i
$900$		0			0
$901$	428.550			0.475638
$902$		0			0
$903$	−21.0260	−	45.4315i	−0.0232846	−	0.0503118i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	682.738			0.752744			0.376372	−	0.926469i	$-0.377172\pi$
$907$	682.738			0.752744			0.376372	+	0.926469i	$0.377172\pi$
$908$		0			0
$909$	200.205	+	169.966i	0.220247	+	0.186981i
$910$		0			0
$911$			1326.30i			1.45587i	0.685647	+	0.727934i	$0.259519\pi$
$911$			1326.30i			1.45587i	−0.685647	+	0.727934i	$0.740481\pi$
$912$		0			0
$913$	−987.626			−1.08174
$914$		0			0
$915$		0			0
$916$		0			0
$917$			154.545i			0.168533i
$918$		0			0
$919$	−688.107			−0.748756			−0.374378	−	0.927276i	$-0.622144\pi$
$919$	−688.107			−0.748756			−0.374378	+	0.927276i	$0.622144\pi$
$920$		0			0
$921$	−459.437	−	992.719i	−0.498846	−	1.07787i
$922$		0			0
$923$		−	2054.69i		−	2.22610i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−523.920	+	617.132i	−0.565178	+	0.665730i
$928$		0			0
$929$			637.084i			0.685774i	0.939377	+	0.342887i	$0.111405\pi$
$929$			637.084i			0.685774i	−0.939377	+	0.342887i	$0.888595\pi$
$930$		0			0
$931$	1121.96			1.20511
$932$		0			0
$933$	−332.052	+	153.676i	−0.355897	+	0.164712i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−358.254			−0.382341			−0.191171	−	0.981557i	$-0.561228\pi$
$937$	−358.254			−0.382341			−0.191171	+	0.981557i	$0.561228\pi$
$938$		0			0
$939$	−119.458	−	258.115i	−0.127218	−	0.274883i
$940$		0			0
$941$		−	124.973i		−	0.132809i	−0.997793	−	0.0664043i	$-0.978847\pi$
$941$		−	124.973i		−	0.132809i	0.997793	−	0.0664043i	$-0.0211527\pi$
$942$		0			0
$943$	967.282			1.02575
$944$		0			0
$945$		0			0
$946$		0			0
$947$			378.826i			0.400027i	0.979793	+	0.200014i	$0.0640987\pi$
$947$			378.826i			0.400027i	−0.979793	+	0.200014i	$0.935901\pi$
$948$		0			0
$949$	−2850.81			−3.00402
$950$		0			0
$951$	1392.00	−	644.226i	1.46372	−	0.677419i
$952$		0			0
$953$			6.82139i			0.00715780i	0.999994	+	0.00357890i	$0.00113920\pi$
$953$			6.82139i			0.00715780i	−0.999994	+	0.00357890i	$0.998861\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	445.842	+	963.344i	0.465875	+	1.00663i
$958$		0			0
$959$		−	124.112i		−	0.129418i
$960$		0			0
$961$	260.374			0.270941
$962$		0			0
$963$	1053.75	+	894.589i	1.09423	+	0.928960i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1193.53			1.23426			0.617130	−	0.786861i	$-0.288295\pi$
$967$	1193.53			1.23426			0.617130	+	0.786861i	$0.288295\pi$
$968$		0			0
$969$	443.445	−	205.229i	0.457631	−	0.211795i
$970$		0			0
$971$		−	1256.06i		−	1.29357i	−0.762672	−	0.646785i	$-0.776113\pi$
$971$		−	1256.06i		−	1.29357i	0.762672	−	0.646785i	$-0.223887\pi$
$972$		0			0
$973$	95.0964			0.0977353
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	551.251i		−	0.564228i	−0.959381	−	0.282114i	$-0.908964\pi$
$977$		−	551.251i		−	0.564228i	0.959381	−	0.282114i	$-0.0910357\pi$
$978$		0			0
$979$	−582.110			−0.594597
$980$		0			0
$981$	−346.932	+	408.656i	−0.353652	+	0.416570i
$982$		0			0
$983$			1598.65i			1.62630i	0.582053	+	0.813151i	$0.302250\pi$
$983$			1598.65i			1.62630i	−0.582053	+	0.813151i	$0.697750\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−78.3371	+	36.2550i	−0.0793689	+	0.0367325i
$988$		0			0
$989$		−	561.300i		−	0.567543i
$990$		0			0
$991$	−1734.40			−1.75015			−0.875077	−	0.483984i	$-0.839189\pi$
$991$	−1734.40			−1.75015			−0.875077	+	0.483984i	$0.839189\pi$
$992$		0			0
$993$	230.183	+	497.363i	0.231806	+	0.500869i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1118.15			1.12151			0.560757	−	0.827980i	$-0.310510\pi$
$997$	1118.15			1.12151			0.560757	+	0.827980i	$0.310510\pi$
$998$		0			0
$999$	−983.032	−	272.525i	−0.984016	−	0.272798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.l.g.401.5		12
3.2	odd	2	inner	600.3.l.g.401.6		12
4.3	odd	2		1200.3.l.y.401.8		12
5.2	odd	4		120.3.c.a.89.2	yes	12
5.3	odd	4		120.3.c.a.89.11	yes	12
5.4	even	2	inner	600.3.l.g.401.8		12
12.11	even	2		1200.3.l.y.401.7		12
15.2	even	4		120.3.c.a.89.12	yes	12
15.8	even	4		120.3.c.a.89.1	✓	12
15.14	odd	2	inner	600.3.l.g.401.7		12
20.3	even	4		240.3.c.e.209.2		12
20.7	even	4		240.3.c.e.209.11		12
20.19	odd	2		1200.3.l.y.401.5		12
40.3	even	4		960.3.c.j.449.11		12
40.13	odd	4		960.3.c.k.449.2		12
40.27	even	4		960.3.c.j.449.2		12
40.37	odd	4		960.3.c.k.449.11		12
60.23	odd	4		240.3.c.e.209.12		12
60.47	odd	4		240.3.c.e.209.1		12
60.59	even	2		1200.3.l.y.401.6		12
120.53	even	4		960.3.c.k.449.12		12
120.77	even	4		960.3.c.k.449.1		12
120.83	odd	4		960.3.c.j.449.1		12
120.107	odd	4		960.3.c.j.449.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.c.a.89.1	✓	12	15.8	even	4
120.3.c.a.89.2	yes	12	5.2	odd	4
120.3.c.a.89.11	yes	12	5.3	odd	4
120.3.c.a.89.12	yes	12	15.2	even	4
240.3.c.e.209.1		12	60.47	odd	4
240.3.c.e.209.2		12	20.3	even	4
240.3.c.e.209.11		12	20.7	even	4
240.3.c.e.209.12		12	60.23	odd	4
600.3.l.g.401.5		12	1.1	even	1	trivial
600.3.l.g.401.6		12	3.2	odd	2	inner
600.3.l.g.401.7		12	15.14	odd	2	inner
600.3.l.g.401.8		12	5.4	even	2	inner
960.3.c.j.449.1		12	120.83	odd	4
960.3.c.j.449.2		12	40.27	even	4
960.3.c.j.449.11		12	40.3	even	4
960.3.c.j.449.12		12	120.107	odd	4
960.3.c.k.449.1		12	120.77	even	4
960.3.c.k.449.2		12	40.13	odd	4
960.3.c.k.449.11		12	40.37	odd	4
960.3.c.k.449.12		12	120.53	even	4
1200.3.l.y.401.5		12	20.19	odd	2
1200.3.l.y.401.6		12	60.59	even	2
1200.3.l.y.401.7		12	12.11	even	2
1200.3.l.y.401.8		12	4.3	odd	2

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.26002 - 2.72256i) q^{3} +0.735748 q^{7} +(-5.82469 + 6.86097i) q^{9} +O(q^{10})\)	\(q+(-1.26002 - 2.72256i) q^{3} +0.735748 q^{7} +(-5.82469 + 6.86097i) q^{9} -10.9451i q^{11} +21.1901 q^{13} -7.03488i q^{17} -23.1529 q^{19} +(-0.927058 - 2.00312i) q^{21} -24.7483i q^{23} +(26.0187 + 7.21312i) q^{27} -32.3284i q^{29} -34.9482 q^{31} +(-29.7987 + 13.7911i) q^{33} -37.7818 q^{37} +(-26.6999 - 57.6912i) q^{39} +39.0848i q^{41} +22.6804 q^{43} -39.1076i q^{47} -48.4587 q^{49} +(-19.1529 + 8.86409i) q^{51} +60.9179i q^{53} +(29.1731 + 63.0352i) q^{57} +7.79696i q^{59} -11.1529 q^{61} +(-4.28551 + 5.04795i) q^{63} -33.3485 q^{67} +(-67.3787 + 31.1833i) q^{69} -96.9650i q^{71} -134.535 q^{73} -8.05284i q^{77} -121.049 q^{79} +(-13.1459 - 79.9261i) q^{81} -90.2345i q^{83} +(-88.0160 + 40.7344i) q^{87} -53.1846i q^{89} +15.5905 q^{91} +(44.0354 + 95.1486i) q^{93} +115.001 q^{97} +(75.0940 + 63.7519i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9}+O(q^{10}) \) 12 * q - 8 * q^9	\( 12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 + 4 * q^21 - 48 * q^31 + 128 * q^39 + 252 * q^49 + 48 * q^51 + 144 * q^61 - 268 * q^69 - 432 * q^79 - 188 * q^81 + 560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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