LMFDB - Embedded newform 300.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,8,Mod(1,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("300.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	300.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$93.7155076452$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} +722.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} +722.000 q^{7} +729.000 q^{9} -3994.00 q^{11} -3030.00 q^{13} +20582.0 q^{17} -25320.0 q^{19} +19494.0 q^{21} -66652.0 q^{23} +19683.0 q^{27} -152664. q^{29} -123776. q^{31} -107838. q^{33} +337886. q^{37} -81810.0 q^{39} +396530. q^{41} -442852. q^{43} -170432. q^{47} -302259. q^{49} +555714. q^{51} +1.23943e6 q^{53} -683640. q^{57} -302354. q^{59} -2.83020e6 q^{61} +526338. q^{63} -3.74127e6 q^{67} -1.79960e6 q^{69} -1.00758e6 q^{71} -2.40464e6 q^{73} -2.88367e6 q^{77} +7.51783e6 q^{79} +531441. q^{81} +5.29963e6 q^{83} -4.12193e6 q^{87} -7.65025e6 q^{89} -2.18766e6 q^{91} -3.34195e6 q^{93} +1.00559e7 q^{97} -2.91163e6 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	722.000	0.795599	0.397799	−	0.917472i	$-0.369774\pi$
$7$	722.000	0.795599	0.397799	+	0.917472i	$0.369774\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3994.00	−0.904761	−0.452380	−	0.891825i	$-0.649425\pi$
$11$	−3994.00	−0.904761	−0.452380	+	0.891825i	$0.649425\pi$
$12$	0	0
$13$	−3030.00	−0.382508	−0.191254	−	0.981541i	$-0.561255\pi$
$13$	−3030.00	−0.382508	−0.191254	+	0.981541i	$0.561255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	20582.0	1.01605	0.508026	−	0.861341i	$-0.330375\pi$
$17$	20582.0	1.01605	0.508026	+	0.861341i	$0.330375\pi$
$18$	0	0
$19$	−25320.0	−0.846888	−0.423444	−	0.905922i	$-0.639179\pi$
$19$	−25320.0	−0.846888	−0.423444	+	0.905922i	$0.639179\pi$
$20$	0	0
$21$	19494.0	0.459339
$22$	0	0
$23$	−66652.0	−1.14226	−0.571131	−	0.820859i	$-0.693495\pi$
$23$	−66652.0	−1.14226	−0.571131	+	0.820859i	$0.693495\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−152664.	−1.16237	−0.581184	−	0.813772i	$-0.697410\pi$
$29$	−152664.	−1.16237	−0.581184	+	0.813772i	$0.697410\pi$
$30$	0	0
$31$	−123776.	−0.746226	−0.373113	−	0.927786i	$-0.621710\pi$
$31$	−123776.	−0.746226	−0.373113	+	0.927786i	$0.621710\pi$
$32$	0	0
$33$	−107838.	−0.522364
$34$	0	0
$35$	0	0
$36$	0	0
$37$	337886.	1.09664	0.548320	−	0.836269i	$-0.315268\pi$
$37$	337886.	1.09664	0.548320	+	0.836269i	$0.315268\pi$
$38$	0	0
$39$	−81810.0	−0.220841
$40$	0	0
$41$	396530.	0.898530	0.449265	−	0.893399i	$-0.351686\pi$
$41$	396530.	0.898530	0.449265	+	0.893399i	$0.351686\pi$
$42$	0	0
$43$	−442852.	−0.849413	−0.424707	−	0.905331i	$-0.639623\pi$
$43$	−442852.	−0.849413	−0.424707	+	0.905331i	$0.639623\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−170432.	−0.239447	−0.119723	−	0.992807i	$-0.538201\pi$
$47$	−170432.	−0.239447	−0.119723	+	0.992807i	$0.538201\pi$
$48$	0	0
$49$	−302259.	−0.367023
$50$	0	0
$51$	555714.	0.586618
$52$	0	0
$53$	1.23943e6	1.14355	0.571775	−	0.820411i	$-0.306255\pi$
$53$	1.23943e6	1.14355	0.571775	+	0.820411i	$0.306255\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−683640.	−0.488951
$58$	0	0
$59$	−302354.	−0.191661	−0.0958305	−	0.995398i	$-0.530551\pi$
$59$	−302354.	−0.191661	−0.0958305	+	0.995398i	$0.530551\pi$
$60$	0	0
$61$	−2.83020e6	−1.59648	−0.798238	−	0.602342i	$-0.794234\pi$
$61$	−2.83020e6	−1.59648	−0.798238	+	0.602342i	$0.794234\pi$
$62$	0	0
$63$	526338.	0.265200
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.74127e6	−1.51970	−0.759849	−	0.650099i	$-0.774727\pi$
$67$	−3.74127e6	−1.51970	−0.759849	+	0.650099i	$0.774727\pi$
$68$	0	0
$69$	−1.79960e6	−0.659485
$70$	0	0
$71$	−1.00758e6	−0.334099	−0.167050	−	0.985949i	$-0.553424\pi$
$71$	−1.00758e6	−0.334099	−0.167050	+	0.985949i	$0.553424\pi$
$72$	0	0
$73$	−2.40464e6	−0.723468	−0.361734	−	0.932281i	$-0.617815\pi$
$73$	−2.40464e6	−0.723468	−0.361734	+	0.932281i	$0.617815\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.88367e6	−0.719826
$78$	0	0
$79$	7.51783e6	1.71553	0.857764	−	0.514044i	$-0.171853\pi$
$79$	7.51783e6	1.71553	0.857764	+	0.514044i	$0.171853\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.29963e6	1.01735	0.508677	−	0.860957i	$-0.330135\pi$
$83$	5.29963e6	1.01735	0.508677	+	0.860957i	$0.330135\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.12193e6	−0.671093
$88$	0	0
$89$	−7.65025e6	−1.15030	−0.575149	−	0.818048i	$-0.695056\pi$
$89$	−7.65025e6	−1.15030	−0.575149	+	0.818048i	$0.695056\pi$
$90$	0	0
$91$	−2.18766e6	−0.304323
$92$	0	0
$93$	−3.34195e6	−0.430834
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.00559e7	1.11872	0.559360	−	0.828925i	$-0.311047\pi$
$97$	1.00559e7	1.11872	0.559360	+	0.828925i	$0.311047\pi$
$98$	0	0
$99$	−2.91163e6	−0.301587
$100$	0	0
$101$	1.12987e7	1.09120	0.545599	−	0.838047i	$-0.316302\pi$
$101$	1.12987e7	1.09120	0.545599	+	0.838047i	$0.316302\pi$
$102$	0	0
$103$	−1.08623e7	−0.979472	−0.489736	−	0.871871i	$-0.662907\pi$
$103$	−1.08623e7	−0.979472	−0.489736	+	0.871871i	$0.662907\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.25151e7	−0.987625	−0.493812	−	0.869568i	$-0.664397\pi$
$107$	−1.25151e7	−0.987625	−0.493812	+	0.869568i	$0.664397\pi$
$108$	0	0
$109$	1.17133e7	0.866335	0.433167	−	0.901313i	$-0.357396\pi$
$109$	1.17133e7	0.866335	0.433167	+	0.901313i	$0.357396\pi$
$110$	0	0
$111$	9.12292e6	0.633146
$112$	0	0
$113$	−2.90032e7	−1.89091	−0.945455	−	0.325753i	$-0.894382\pi$
$113$	−2.90032e7	−1.89091	−0.945455	+	0.325753i	$0.894382\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.20887e6	−0.127503
$118$	0	0
$119$	1.48602e7	0.808370
$120$	0	0
$121$	−3.53514e6	−0.181408
$122$	0	0
$123$	1.07063e7	0.518767
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.85078e7	−1.66815	−0.834075	−	0.551651i	$-0.813998\pi$
$127$	−3.85078e7	−1.66815	−0.834075	+	0.551651i	$0.813998\pi$
$128$	0	0
$129$	−1.19570e7	−0.490409
$130$	0	0
$131$	1.54294e7	0.599652	0.299826	−	0.953994i	$-0.403071\pi$
$131$	1.54294e7	0.599652	0.299826	+	0.953994i	$0.403071\pi$
$132$	0	0
$133$	−1.82810e7	−0.673783
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.26181e7	−1.41603	−0.708014	−	0.706199i	$-0.750408\pi$
$137$	−4.26181e7	−1.41603	−0.708014	+	0.706199i	$0.750408\pi$
$138$	0	0
$139$	1.18260e6	0.0373497	0.0186749	−	0.999826i	$-0.494055\pi$
$139$	1.18260e6	0.0373497	0.0186749	+	0.999826i	$0.494055\pi$
$140$	0	0
$141$	−4.60166e6	−0.138245
$142$	0	0
$143$	1.21018e7	0.346078
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.16099e6	−0.211901
$148$	0	0
$149$	−8.63875e6	−0.213943	−0.106972	−	0.994262i	$-0.534115\pi$
$149$	−8.63875e6	−0.213943	−0.106972	+	0.994262i	$0.534115\pi$
$150$	0	0
$151$	−5.30161e7	−1.25311	−0.626554	−	0.779378i	$-0.715535\pi$
$151$	−5.30161e7	−1.25311	−0.626554	+	0.779378i	$0.715535\pi$
$152$	0	0
$153$	1.50043e7	0.338684
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.78454e7	−1.39917	−0.699587	−	0.714547i	$-0.746633\pi$
$157$	−6.78454e7	−1.39917	−0.699587	+	0.714547i	$0.746633\pi$
$158$	0	0
$159$	3.34645e7	0.660229
$160$	0	0
$161$	−4.81227e7	−0.908782
$162$	0	0
$163$	4.05638e7	0.733638	0.366819	−	0.930292i	$-0.380447\pi$
$163$	4.05638e7	0.733638	0.366819	+	0.930292i	$0.380447\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.02930e8	−1.71015	−0.855075	−	0.518504i	$-0.826489\pi$
$167$	−1.02930e8	−1.71015	−0.855075	+	0.518504i	$0.826489\pi$
$168$	0	0
$169$	−5.35676e7	−0.853687
$170$	0	0
$171$	−1.84583e7	−0.282296
$172$	0	0
$173$	−5.77293e7	−0.847686	−0.423843	−	0.905736i	$-0.639319\pi$
$173$	−5.77293e7	−0.847686	−0.423843	+	0.905736i	$0.639319\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.16356e6	−0.110656
$178$	0	0
$179$	−1.39435e6	−0.0181714	−0.00908568	−	0.999959i	$-0.502892\pi$
$179$	−1.39435e6	−0.0181714	−0.00908568	+	0.999959i	$0.502892\pi$
$180$	0	0
$181$	8.46573e7	1.06118	0.530590	−	0.847628i	$-0.321970\pi$
$181$	8.46573e7	1.06118	0.530590	+	0.847628i	$0.321970\pi$
$182$	0	0
$183$	−7.64153e7	−0.921726
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.22045e7	−0.919285
$188$	0	0
$189$	1.42111e7	0.153113
$190$	0	0
$191$	6.83324e7	0.709594	0.354797	−	0.934943i	$-0.384550\pi$
$191$	6.83324e7	0.709594	0.354797	+	0.934943i	$0.384550\pi$
$192$	0	0
$193$	1.37870e7	0.138045	0.0690223	−	0.997615i	$-0.478012\pi$
$193$	1.37870e7	0.138045	0.0690223	+	0.997615i	$0.478012\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.51565e7	0.234433	0.117216	−	0.993106i	$-0.462603\pi$
$197$	2.51565e7	0.234433	0.117216	+	0.993106i	$0.462603\pi$
$198$	0	0
$199$	−1.36320e8	−1.22623	−0.613117	−	0.789992i	$-0.710085\pi$
$199$	−1.36320e8	−1.22623	−0.613117	+	0.789992i	$0.710085\pi$
$200$	0	0
$201$	−1.01014e8	−0.877398
$202$	0	0
$203$	−1.10223e8	−0.924778
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.85893e7	−0.380754
$208$	0	0
$209$	1.01128e8	0.766231
$210$	0	0
$211$	1.96310e8	1.43864	0.719322	−	0.694677i	$-0.244453\pi$
$211$	1.96310e8	1.43864	0.719322	+	0.694677i	$0.244453\pi$
$212$	0	0
$213$	−2.72047e7	−0.192892
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.93663e7	−0.593697
$218$	0	0
$219$	−6.49252e7	−0.417694
$220$	0	0
$221$	−6.23635e7	−0.388649
$222$	0	0
$223$	−2.29880e8	−1.38814	−0.694071	−	0.719907i	$-0.744184\pi$
$223$	−2.29880e8	−1.38814	−0.694071	+	0.719907i	$0.744184\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.13844e8	1.21341	0.606703	−	0.794928i	$-0.292492\pi$
$227$	2.13844e8	1.21341	0.606703	+	0.794928i	$0.292492\pi$
$228$	0	0
$229$	−7.83835e7	−0.431321	−0.215660	−	0.976468i	$-0.569190\pi$
$229$	−7.83835e7	−0.431321	−0.215660	+	0.976468i	$0.569190\pi$
$230$	0	0
$231$	−7.78590e7	−0.415592
$232$	0	0
$233$	1.66949e8	0.864644	0.432322	−	0.901719i	$-0.357694\pi$
$233$	1.66949e8	0.864644	0.432322	+	0.901719i	$0.357694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.02981e8	0.990461
$238$	0	0
$239$	3.59643e7	0.170404	0.0852019	−	0.996364i	$-0.472846\pi$
$239$	3.59643e7	0.170404	0.0852019	+	0.996364i	$0.472846\pi$
$240$	0	0
$241$	−8.43354e7	−0.388106	−0.194053	−	0.980991i	$-0.562163\pi$
$241$	−8.43354e7	−0.388106	−0.194053	+	0.980991i	$0.562163\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.67196e7	0.323942
$248$	0	0
$249$	1.43090e8	0.587370
$250$	0	0
$251$	2.23657e8	0.892740	0.446370	−	0.894848i	$-0.352716\pi$
$251$	2.23657e8	0.892740	0.446370	+	0.894848i	$0.352716\pi$
$252$	0	0
$253$	2.66208e8	1.03347
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.66466e8	−1.34669	−0.673346	−	0.739328i	$-0.735143\pi$
$257$	−3.66466e8	−1.34669	−0.673346	+	0.739328i	$0.735143\pi$
$258$	0	0
$259$	2.43954e8	0.872486
$260$	0	0
$261$	−1.11292e8	−0.387456
$262$	0	0
$263$	1.98987e8	0.674495	0.337248	−	0.941416i	$-0.390504\pi$
$263$	1.98987e8	0.674495	0.337248	+	0.941416i	$0.390504\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.06557e8	−0.664125
$268$	0	0
$269$	1.55449e8	0.486918	0.243459	−	0.969911i	$-0.421718\pi$
$269$	1.55449e8	0.486918	0.243459	+	0.969911i	$0.421718\pi$
$270$	0	0
$271$	−4.81312e8	−1.46904	−0.734522	−	0.678585i	$-0.762593\pi$
$271$	−4.81312e8	−1.46904	−0.734522	+	0.678585i	$0.762593\pi$
$272$	0	0
$273$	−5.90668e7	−0.175701
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.43398e7	−0.0688079	−0.0344039	−	0.999408i	$-0.510953\pi$
$277$	−2.43398e7	−0.0688079	−0.0344039	+	0.999408i	$0.510953\pi$
$278$	0	0
$279$	−9.02327e7	−0.248742
$280$	0	0
$281$	2.74530e8	0.738104	0.369052	−	0.929409i	$-0.379682\pi$
$281$	2.74530e8	0.738104	0.369052	+	0.929409i	$0.379682\pi$
$282$	0	0
$283$	−7.98883e7	−0.209522	−0.104761	−	0.994497i	$-0.533408\pi$
$283$	−7.98883e7	−0.209522	−0.104761	+	0.994497i	$0.533408\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.86295e8	0.714869
$288$	0	0
$289$	1.32801e7	0.0323636
$290$	0	0
$291$	2.71510e8	0.645894
$292$	0	0
$293$	1.87993e8	0.436621	0.218311	−	0.975879i	$-0.429945\pi$
$293$	1.87993e8	0.436621	0.218311	+	0.975879i	$0.429945\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.86139e7	−0.174121
$298$	0	0
$299$	2.01956e8	0.436925
$300$	0	0
$301$	−3.19739e8	−0.675792
$302$	0	0
$303$	3.05065e8	0.630003
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.55766e8	1.88524	0.942621	−	0.333865i	$-0.108353\pi$
$307$	9.55766e8	1.88524	0.942621	+	0.333865i	$0.108353\pi$
$308$	0	0
$309$	−2.93283e8	−0.565499
$310$	0	0
$311$	9.95378e8	1.87641	0.938203	−	0.346086i	$-0.112490\pi$
$311$	9.95378e8	1.87641	0.938203	+	0.346086i	$0.112490\pi$
$312$	0	0
$313$	6.50637e8	1.19932	0.599658	−	0.800256i	$-0.295303\pi$
$313$	6.50637e8	1.19932	0.599658	+	0.800256i	$0.295303\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.49473e8	1.32144	0.660722	−	0.750631i	$-0.270250\pi$
$317$	7.49473e8	1.32144	0.660722	+	0.750631i	$0.270250\pi$
$318$	0	0
$319$	6.09740e8	1.05166
$320$	0	0
$321$	−3.37908e8	−0.570205
$322$	0	0
$323$	−5.21136e8	−0.860483
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.16259e8	0.500179
$328$	0	0
$329$	−1.23052e8	−0.190503
$330$	0	0
$331$	−7.88546e8	−1.19517	−0.597584	−	0.801806i	$-0.703873\pi$
$331$	−7.88546e8	−1.19517	−0.597584	+	0.801806i	$0.703873\pi$
$332$	0	0
$333$	2.46319e8	0.365547
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.68871e8	1.37899	0.689495	−	0.724290i	$-0.257832\pi$
$337$	9.68871e8	1.37899	0.689495	+	0.724290i	$0.257832\pi$
$338$	0	0
$339$	−7.83085e8	−1.09172
$340$	0	0
$341$	4.94361e8	0.675156
$342$	0	0
$343$	−8.12829e8	−1.08760
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.12592e8	0.658595	0.329298	−	0.944226i	$-0.393188\pi$
$347$	5.12592e8	0.658595	0.329298	+	0.944226i	$0.393188\pi$
$348$	0	0
$349$	−1.20751e9	−1.52055	−0.760277	−	0.649599i	$-0.774937\pi$
$349$	−1.20751e9	−1.52055	−0.760277	+	0.649599i	$0.774937\pi$
$350$	0	0
$351$	−5.96395e7	−0.0736138
$352$	0	0
$353$	2.01922e8	0.244327	0.122164	−	0.992510i	$-0.461017\pi$
$353$	2.01922e8	0.244327	0.122164	+	0.992510i	$0.461017\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.01226e8	0.466713
$358$	0	0
$359$	−1.07611e9	−1.22751	−0.613757	−	0.789495i	$-0.710342\pi$
$359$	−1.07611e9	−1.22751	−0.613757	+	0.789495i	$0.710342\pi$
$360$	0	0
$361$	−2.52769e8	−0.282780
$362$	0	0
$363$	−9.54486e7	−0.104736
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.53073e8	0.161647	0.0808233	−	0.996728i	$-0.474245\pi$
$367$	1.53073e8	0.161647	0.0808233	+	0.996728i	$0.474245\pi$
$368$	0	0
$369$	2.89070e8	0.299510
$370$	0	0
$371$	8.94866e8	0.909807
$372$	0	0
$373$	−1.67576e8	−0.167198	−0.0835989	−	0.996499i	$-0.526641\pi$
$373$	−1.67576e8	−0.167198	−0.0835989	+	0.996499i	$0.526641\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.62572e8	0.444615
$378$	0	0
$379$	8.80517e8	0.830807	0.415404	−	0.909637i	$-0.363640\pi$
$379$	8.80517e8	0.830807	0.415404	+	0.909637i	$0.363640\pi$
$380$	0	0
$381$	−1.03971e9	−0.963107
$382$	0	0
$383$	7.41581e8	0.674470	0.337235	−	0.941421i	$-0.390508\pi$
$383$	7.41581e8	0.674470	0.337235	+	0.941421i	$0.390508\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.22839e8	−0.283138
$388$	0	0
$389$	1.60660e9	1.38383	0.691915	−	0.721979i	$-0.256767\pi$
$389$	1.60660e9	1.38383	0.691915	+	0.721979i	$0.256767\pi$
$390$	0	0
$391$	−1.37183e9	−1.16060
$392$	0	0
$393$	4.16594e8	0.346209
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.41280e9	1.13322	0.566609	−	0.823987i	$-0.308255\pi$
$397$	1.41280e9	1.13322	0.566609	+	0.823987i	$0.308255\pi$
$398$	0	0
$399$	−4.93588e8	−0.389009
$400$	0	0
$401$	1.33603e9	1.03469	0.517345	−	0.855777i	$-0.326920\pi$
$401$	1.33603e9	1.03469	0.517345	+	0.855777i	$0.326920\pi$
$402$	0	0
$403$	3.75041e8	0.285438
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.34952e9	−0.992197
$408$	0	0
$409$	−9.00104e8	−0.650521	−0.325260	−	0.945625i	$-0.605452\pi$
$409$	−9.00104e8	−0.650521	−0.325260	+	0.945625i	$0.605452\pi$
$410$	0	0
$411$	−1.15069e9	−0.817544
$412$	0	0
$413$	−2.18300e8	−0.152485
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.19303e7	0.0215639
$418$	0	0
$419$	1.21164e8	0.0804680	0.0402340	−	0.999190i	$-0.487190\pi$
$419$	1.21164e8	0.0804680	0.0402340	+	0.999190i	$0.487190\pi$
$420$	0	0
$421$	2.11866e9	1.38380	0.691901	−	0.721992i	$-0.256773\pi$
$421$	2.11866e9	1.38380	0.691901	+	0.721992i	$0.256773\pi$
$422$	0	0
$423$	−1.24245e8	−0.0798155
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.04340e9	−1.27015
$428$	0	0
$429$	3.26749e8	0.199808
$430$	0	0
$431$	−3.97856e8	−0.239362	−0.119681	−	0.992812i	$-0.538187\pi$
$431$	−3.97856e8	−0.239362	−0.119681	+	0.992812i	$0.538187\pi$
$432$	0	0
$433$	3.05052e9	1.80578	0.902892	−	0.429868i	$-0.141440\pi$
$433$	3.05052e9	1.80578	0.902892	+	0.429868i	$0.141440\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.68763e9	0.967368
$438$	0	0
$439$	5.19220e8	0.292904	0.146452	−	0.989218i	$-0.453215\pi$
$439$	5.19220e8	0.292904	0.146452	+	0.989218i	$0.453215\pi$
$440$	0	0
$441$	−2.20347e8	−0.122341
$442$	0	0
$443$	3.47660e8	0.189995	0.0949974	−	0.995478i	$-0.469716\pi$
$443$	3.47660e8	0.189995	0.0949974	+	0.995478i	$0.469716\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.33246e8	−0.123520
$448$	0	0
$449$	3.48886e9	1.81895	0.909475	−	0.415758i	$-0.136484\pi$
$449$	3.48886e9	1.81895	0.909475	+	0.415758i	$0.136484\pi$
$450$	0	0
$451$	−1.58374e9	−0.812954
$452$	0	0
$453$	−1.43143e9	−0.723482
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.53273e9	−1.73143	−0.865713	−	0.500542i	$-0.833134\pi$
$457$	−3.53273e9	−1.73143	−0.865713	+	0.500542i	$0.833134\pi$
$458$	0	0
$459$	4.05116e8	0.195539
$460$	0	0
$461$	4.10819e9	1.95298	0.976490	−	0.215564i	$-0.0691589\pi$
$461$	4.10819e9	1.95298	0.976490	+	0.215564i	$0.0691589\pi$
$462$	0	0
$463$	−1.34107e9	−0.627940	−0.313970	−	0.949433i	$-0.601659\pi$
$463$	−1.34107e9	−0.627940	−0.313970	+	0.949433i	$0.601659\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.10858e9	−0.503682	−0.251841	−	0.967769i	$-0.581036\pi$
$467$	−1.10858e9	−0.503682	−0.251841	+	0.967769i	$0.581036\pi$
$468$	0	0
$469$	−2.70120e9	−1.20907
$470$	0	0
$471$	−1.83183e9	−0.807814
$472$	0	0
$473$	1.76875e9	0.768516
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.03542e8	0.381183
$478$	0	0
$479$	3.03817e9	1.26310	0.631550	−	0.775335i	$-0.282419\pi$
$479$	3.03817e9	1.26310	0.631550	+	0.775335i	$0.282419\pi$
$480$	0	0
$481$	−1.02379e9	−0.419474
$482$	0	0
$483$	−1.29931e9	−0.524686
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.79818e7	0.0109780	0.00548901	−	0.999985i	$-0.498253\pi$
$487$	2.79818e7	0.0109780	0.00548901	+	0.999985i	$0.498253\pi$
$488$	0	0
$489$	1.09522e9	0.423566
$490$	0	0
$491$	4.06715e9	1.55062	0.775309	−	0.631582i	$-0.217594\pi$
$491$	4.06715e9	1.55062	0.775309	+	0.631582i	$0.217594\pi$
$492$	0	0
$493$	−3.14213e9	−1.18103
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.27473e8	−0.265809
$498$	0	0
$499$	−4.38379e9	−1.57942	−0.789710	−	0.613481i	$-0.789769\pi$
$499$	−4.38379e9	−1.57942	−0.789710	+	0.613481i	$0.789769\pi$
$500$	0	0
$501$	−2.77911e9	−0.987355
$502$	0	0
$503$	−4.26965e9	−1.49591	−0.747953	−	0.663752i	$-0.768963\pi$
$503$	−4.26965e9	−1.49591	−0.747953	+	0.663752i	$0.768963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.44633e9	−0.492877
$508$	0	0
$509$	−2.48939e9	−0.836722	−0.418361	−	0.908281i	$-0.637395\pi$
$509$	−2.48939e9	−0.836722	−0.418361	+	0.908281i	$0.637395\pi$
$510$	0	0
$511$	−1.73615e9	−0.575590
$512$	0	0
$513$	−4.98374e8	−0.162984
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.80705e8	0.216642
$518$	0	0
$519$	−1.55869e9	−0.489412
$520$	0	0
$521$	−1.70747e9	−0.528958	−0.264479	−	0.964391i	$-0.585200\pi$
$521$	−1.70747e9	−0.528958	−0.264479	+	0.964391i	$0.585200\pi$
$522$	0	0
$523$	−2.94813e9	−0.901137	−0.450568	−	0.892742i	$-0.648779\pi$
$523$	−2.94813e9	−0.901137	−0.450568	+	0.892742i	$0.648779\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.54756e9	−0.758205
$528$	0	0
$529$	1.03766e9	0.304763
$530$	0	0
$531$	−2.20416e8	−0.0638870
$532$	0	0
$533$	−1.20149e9	−0.343695
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.76476e7	−0.0104912
$538$	0	0
$539$	1.20722e9	0.332068
$540$	0	0
$541$	3.06917e9	0.833355	0.416677	−	0.909054i	$-0.363195\pi$
$541$	3.06917e9	0.833355	0.416677	+	0.909054i	$0.363195\pi$
$542$	0	0
$543$	2.28575e9	0.612673
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.73829e9	−0.976602	−0.488301	−	0.872675i	$-0.662383\pi$
$547$	−3.73829e9	−0.976602	−0.488301	+	0.872675i	$0.662383\pi$
$548$	0	0
$549$	−2.06321e9	−0.532159
$550$	0	0
$551$	3.86545e9	0.984396
$552$	0	0
$553$	5.42787e9	1.36487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.80359e8	0.240376	0.120188	−	0.992751i	$-0.461650\pi$
$557$	9.80359e8	0.240376	0.120188	+	0.992751i	$0.461650\pi$
$558$	0	0
$559$	1.34184e9	0.324908
$560$	0	0
$561$	−2.21952e9	−0.530749
$562$	0	0
$563$	−3.89470e9	−0.919802	−0.459901	−	0.887970i	$-0.652115\pi$
$563$	−3.89470e9	−0.919802	−0.459901	+	0.887970i	$0.652115\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.83700e8	0.0883999
$568$	0	0
$569$	−8.31489e8	−0.189219	−0.0946093	−	0.995514i	$-0.530160\pi$
$569$	−8.31489e8	−0.189219	−0.0946093	+	0.995514i	$0.530160\pi$
$570$	0	0
$571$	−1.48429e9	−0.333651	−0.166826	−	0.985986i	$-0.553352\pi$
$571$	−1.48429e9	−0.333651	−0.166826	+	0.985986i	$0.553352\pi$
$572$	0	0
$573$	1.84497e9	0.409684
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.33013e9	−1.15511	−0.577554	−	0.816353i	$-0.695993\pi$
$577$	−5.33013e9	−1.15511	−0.577554	+	0.816353i	$0.695993\pi$
$578$	0	0
$579$	3.72249e8	0.0797001
$580$	0	0
$581$	3.82633e9	0.809405
$582$	0	0
$583$	−4.95027e9	−1.03464
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.04566e9	−1.84590	−0.922948	−	0.384926i	$-0.874227\pi$
$587$	−9.04566e9	−1.84590	−0.922948	+	0.384926i	$0.874227\pi$
$588$	0	0
$589$	3.13401e9	0.631970
$590$	0	0
$591$	6.79225e8	0.135350
$592$	0	0
$593$	5.41450e9	1.06627	0.533134	−	0.846031i	$-0.321014\pi$
$593$	5.41450e9	1.06627	0.533134	+	0.846031i	$0.321014\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.68064e9	−0.707967
$598$	0	0
$599$	−1.81841e9	−0.345699	−0.172850	−	0.984948i	$-0.555297\pi$
$599$	−1.81841e9	−0.345699	−0.172850	+	0.984948i	$0.555297\pi$
$600$	0	0
$601$	−9.04386e9	−1.69939	−0.849695	−	0.527274i	$-0.823214\pi$
$601$	−9.04386e9	−1.69939	−0.849695	+	0.527274i	$0.823214\pi$
$602$	0	0
$603$	−2.72739e9	−0.506566
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.86937e8	0.124668	0.0623342	−	0.998055i	$-0.480146\pi$
$607$	6.86937e8	0.124668	0.0623342	+	0.998055i	$0.480146\pi$
$608$	0	0
$609$	−2.97603e9	−0.533921
$610$	0	0
$611$	5.16409e8	0.0915903
$612$	0	0
$613$	5.77667e9	1.01290	0.506449	−	0.862270i	$-0.330958\pi$
$613$	5.77667e9	1.01290	0.506449	+	0.862270i	$0.330958\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.13316e9	−1.56539	−0.782696	−	0.622405i	$-0.786156\pi$
$617$	−9.13316e9	−1.56539	−0.782696	+	0.622405i	$0.786156\pi$
$618$	0	0
$619$	−7.67435e8	−0.130054	−0.0650271	−	0.997884i	$-0.520713\pi$
$619$	−7.67435e8	−0.130054	−0.0650271	+	0.997884i	$0.520713\pi$
$620$	0	0
$621$	−1.31191e9	−0.219828
$622$	0	0
$623$	−5.52348e9	−0.915176
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.73046e9	0.442384
$628$	0	0
$629$	6.95437e9	1.11424
$630$	0	0
$631$	−3.85030e9	−0.610087	−0.305044	−	0.952338i	$-0.598671\pi$
$631$	−3.85030e9	−0.610087	−0.305044	+	0.952338i	$0.598671\pi$
$632$	0	0
$633$	5.30036e9	0.830601
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.15845e8	0.140389
$638$	0	0
$639$	−7.34526e8	−0.111366
$640$	0	0
$641$	5.19408e9	0.778942	0.389471	−	0.921039i	$-0.372658\pi$
$641$	5.19408e9	0.778942	0.389471	+	0.921039i	$0.372658\pi$
$642$	0	0
$643$	−4.44257e9	−0.659016	−0.329508	−	0.944153i	$-0.606883\pi$
$643$	−4.44257e9	−0.659016	−0.329508	+	0.944153i	$0.606883\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.83945e9	0.412163	0.206081	−	0.978535i	$-0.433929\pi$
$647$	2.83945e9	0.412163	0.206081	+	0.978535i	$0.433929\pi$
$648$	0	0
$649$	1.20760e9	0.173407
$650$	0	0
$651$	−2.41289e9	−0.342771
$652$	0	0
$653$	8.73229e9	1.22725	0.613624	−	0.789599i	$-0.289711\pi$
$653$	8.73229e9	1.22725	0.613624	+	0.789599i	$0.289711\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.75298e9	−0.241156
$658$	0	0
$659$	−6.74019e9	−0.917430	−0.458715	−	0.888583i	$-0.651690\pi$
$659$	−6.74019e9	−0.917430	−0.458715	+	0.888583i	$0.651690\pi$
$660$	0	0
$661$	−7.84424e9	−1.05644	−0.528221	−	0.849107i	$-0.677141\pi$
$661$	−7.84424e9	−1.05644	−0.528221	+	0.849107i	$0.677141\pi$
$662$	0	0
$663$	−1.68381e9	−0.224386
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.01754e10	1.32773
$668$	0	0
$669$	−6.20675e9	−0.801444
$670$	0	0
$671$	1.13038e10	1.44443
$672$	0	0
$673$	5.34502e7	0.00675922	0.00337961	−	0.999994i	$-0.498924\pi$
$673$	5.34502e7	0.00675922	0.00337961	+	0.999994i	$0.498924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.16539e10	1.44348	0.721741	−	0.692163i	$-0.243342\pi$
$677$	1.16539e10	1.44348	0.721741	+	0.692163i	$0.243342\pi$
$678$	0	0
$679$	7.26039e9	0.890053
$680$	0	0
$681$	5.77378e9	0.700561
$682$	0	0
$683$	−9.96640e9	−1.19692	−0.598461	−	0.801152i	$-0.704221\pi$
$683$	−9.96640e9	−1.19692	−0.598461	+	0.801152i	$0.704221\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.11635e9	−0.249023
$688$	0	0
$689$	−3.75546e9	−0.437417
$690$	0	0
$691$	4.45822e9	0.514030	0.257015	−	0.966407i	$-0.417261\pi$
$691$	4.45822e9	0.514030	0.257015	+	0.966407i	$0.417261\pi$
$692$	0	0
$693$	−2.10219e9	−0.239942
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.16138e9	0.912954
$698$	0	0
$699$	4.50761e9	0.499202
$700$	0	0
$701$	−5.00766e9	−0.549063	−0.274531	−	0.961578i	$-0.588523\pi$
$701$	−5.00766e9	−0.549063	−0.274531	+	0.961578i	$0.588523\pi$
$702$	0	0
$703$	−8.55527e9	−0.928732
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.15765e9	0.868155
$708$	0	0
$709$	3.67293e9	0.387036	0.193518	−	0.981097i	$-0.438010\pi$
$709$	3.67293e9	0.387036	0.193518	+	0.981097i	$0.438010\pi$
$710$	0	0
$711$	5.48050e9	0.571843
$712$	0	0
$713$	8.24992e9	0.852386
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.71037e8	0.0983827
$718$	0	0
$719$	1.60618e10	1.61155	0.805776	−	0.592220i	$-0.201749\pi$
$719$	1.60618e10	1.61155	0.805776	+	0.592220i	$0.201749\pi$
$720$	0	0
$721$	−7.84259e9	−0.779267
$722$	0	0
$723$	−2.27706e9	−0.224073
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−7.74536e9	−0.747603	−0.373801	−	0.927509i	$-0.621946\pi$
$727$	−7.74536e9	−0.747603	−0.373801	+	0.927509i	$0.621946\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−9.11478e9	−0.863049
$732$	0	0
$733$	−2.55716e9	−0.239825	−0.119912	−	0.992784i	$-0.538261\pi$
$733$	−2.55716e9	−0.239825	−0.119912	+	0.992784i	$0.538261\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.49426e10	1.37496
$738$	0	0
$739$	−1.16806e10	−1.06465	−0.532327	−	0.846539i	$-0.678682\pi$
$739$	−1.16806e10	−1.06465	−0.532327	+	0.846539i	$0.678682\pi$
$740$	0	0
$741$	2.07143e9	0.187028
$742$	0	0
$743$	6.07204e9	0.543092	0.271546	−	0.962425i	$-0.412465\pi$
$743$	6.07204e9	0.543092	0.271546	+	0.962425i	$0.412465\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.86343e9	0.339118
$748$	0	0
$749$	−9.03592e9	−0.785753
$750$	0	0
$751$	2.22230e10	1.91453	0.957264	−	0.289214i	$-0.0933939\pi$
$751$	2.22230e10	1.91453	0.957264	+	0.289214i	$0.0933939\pi$
$752$	0	0
$753$	6.03875e9	0.515424
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.62098e10	1.35814	0.679068	−	0.734075i	$-0.262384\pi$
$757$	1.62098e10	1.35814	0.679068	+	0.734075i	$0.262384\pi$
$758$	0	0
$759$	7.18762e9	0.596676
$760$	0	0
$761$	1.72440e10	1.41837	0.709187	−	0.705021i	$-0.249062\pi$
$761$	1.72440e10	1.41837	0.709187	+	0.705021i	$0.249062\pi$
$762$	0	0
$763$	8.45699e9	0.689255
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.16133e8	0.0733119
$768$	0	0
$769$	−6.61571e8	−0.0524607	−0.0262304	−	0.999656i	$-0.508350\pi$
$769$	−6.61571e8	−0.0524607	−0.0262304	+	0.999656i	$0.508350\pi$
$770$	0	0
$771$	−9.89459e9	−0.777513
$772$	0	0
$773$	9.81593e9	0.764369	0.382185	−	0.924086i	$-0.375172\pi$
$773$	9.81593e9	0.764369	0.382185	+	0.924086i	$0.375172\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.58675e9	0.503730
$778$	0	0
$779$	−1.00401e10	−0.760954
$780$	0	0
$781$	4.02427e9	0.302280
$782$	0	0
$783$	−3.00489e9	−0.223698
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.03612e10	−0.757700	−0.378850	−	0.925458i	$-0.623680\pi$
$787$	−1.03612e10	−0.757700	−0.378850	+	0.925458i	$0.623680\pi$
$788$	0	0
$789$	5.37264e9	0.389420
$790$	0	0
$791$	−2.09403e10	−1.50441
$792$	0	0
$793$	8.57550e9	0.610665
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.68949e9	−0.468046	−0.234023	−	0.972231i	$-0.575189\pi$
$797$	−6.68949e9	−0.468046	−0.234023	+	0.972231i	$0.575189\pi$
$798$	0	0
$799$	−3.50783e9	−0.243290
$800$	0	0
$801$	−5.57703e9	−0.383433
$802$	0	0
$803$	9.60412e9	0.654565
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.19714e9	0.281123
$808$	0	0
$809$	2.60306e10	1.72848	0.864240	−	0.503080i	$-0.167800\pi$
$809$	2.60306e10	1.72848	0.864240	+	0.503080i	$0.167800\pi$
$810$	0	0
$811$	7.26792e9	0.478451	0.239225	−	0.970964i	$-0.423107\pi$
$811$	7.26792e9	0.478451	0.239225	+	0.970964i	$0.423107\pi$
$812$	0	0
$813$	−1.29954e10	−0.848153
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.12130e10	0.719358
$818$	0	0
$819$	−1.59480e9	−0.101441
$820$	0	0
$821$	−2.54672e10	−1.60613	−0.803065	−	0.595891i	$-0.796799\pi$
$821$	−2.54672e10	−1.60613	−0.803065	+	0.595891i	$0.796799\pi$
$822$	0	0
$823$	2.63976e10	1.65069	0.825345	−	0.564629i	$-0.190981\pi$
$823$	2.63976e10	1.65069	0.825345	+	0.564629i	$0.190981\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.49307e10	0.917931	0.458965	−	0.888454i	$-0.348220\pi$
$827$	1.49307e10	0.917931	0.458965	+	0.888454i	$0.348220\pi$
$828$	0	0
$829$	6.48355e9	0.395250	0.197625	−	0.980278i	$-0.436677\pi$
$829$	6.48355e9	0.395250	0.197625	+	0.980278i	$0.436677\pi$
$830$	0	0
$831$	−6.57175e8	−0.0397263
$832$	0	0
$833$	−6.22109e9	−0.372915
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.43628e9	−0.143611
$838$	0	0
$839$	−9.69950e9	−0.566999	−0.283500	−	0.958972i	$-0.591495\pi$
$839$	−9.69950e9	−0.566999	−0.283500	+	0.958972i	$0.591495\pi$
$840$	0	0
$841$	6.05642e9	0.351099
$842$	0	0
$843$	7.41231e9	0.426144
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.55237e9	−0.144328
$848$	0	0
$849$	−2.15698e9	−0.120968
$850$	0	0
$851$	−2.25208e10	−1.25265
$852$	0	0
$853$	1.38797e10	0.765697	0.382849	−	0.923811i	$-0.374943\pi$
$853$	1.38797e10	0.765697	0.382849	+	0.923811i	$0.374943\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.36817e9	0.291336	0.145668	−	0.989334i	$-0.453467\pi$
$857$	5.36817e9	0.291336	0.145668	+	0.989334i	$0.453467\pi$
$858$	0	0
$859$	−1.91636e10	−1.03158	−0.515788	−	0.856716i	$-0.672501\pi$
$859$	−1.91636e10	−1.03158	−0.515788	+	0.856716i	$0.672501\pi$
$860$	0	0
$861$	7.72996e9	0.412730
$862$	0	0
$863$	−2.43413e10	−1.28916	−0.644580	−	0.764537i	$-0.722968\pi$
$863$	−2.43413e10	−1.28916	−0.644580	+	0.764537i	$0.722968\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.58561e8	0.0186852
$868$	0	0
$869$	−3.00262e10	−1.55214
$870$	0	0
$871$	1.13361e10	0.581297
$872$	0	0
$873$	7.33078e9	0.372907
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.55474e10	0.778320	0.389160	−	0.921170i	$-0.372765\pi$
$877$	1.55474e10	0.778320	0.389160	+	0.921170i	$0.372765\pi$
$878$	0	0
$879$	5.07581e9	0.252083
$880$	0	0
$881$	−2.80993e10	−1.38446	−0.692228	−	0.721679i	$-0.743371\pi$
$881$	−2.80993e10	−1.38446	−0.692228	+	0.721679i	$0.743371\pi$
$882$	0	0
$883$	−6.35021e9	−0.310403	−0.155201	−	0.987883i	$-0.549603\pi$
$883$	−6.35021e9	−0.310403	−0.155201	+	0.987883i	$0.549603\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.19974e9	−0.442632	−0.221316	−	0.975202i	$-0.571035\pi$
$887$	−9.19974e9	−0.442632	−0.221316	+	0.975202i	$0.571035\pi$
$888$	0	0
$889$	−2.78026e10	−1.32718
$890$	0	0
$891$	−2.12258e9	−0.100529
$892$	0	0
$893$	4.31534e9	0.202784
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.45280e9	0.252259
$898$	0	0
$899$	1.88961e10	0.867389
$900$	0	0
$901$	2.55099e10	1.16191
$902$	0	0
$903$	−8.63296e9	−0.390169
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.81905e10	−0.809504	−0.404752	−	0.914427i	$-0.632642\pi$
$907$	−1.81905e10	−0.809504	−0.404752	+	0.914427i	$0.632642\pi$
$908$	0	0
$909$	8.23674e9	0.363732
$910$	0	0
$911$	8.15926e9	0.357550	0.178775	−	0.983890i	$-0.442787\pi$
$911$	8.15926e9	0.357550	0.178775	+	0.983890i	$0.442787\pi$
$912$	0	0
$913$	−2.11667e10	−0.920462
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.11400e10	0.477083
$918$	0	0
$919$	2.54889e10	1.08329	0.541647	−	0.840606i	$-0.317801\pi$
$919$	2.54889e10	1.08329	0.541647	+	0.840606i	$0.317801\pi$
$920$	0	0
$921$	2.58057e10	1.08844
$922$	0	0
$923$	3.05297e9	0.127796
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.91863e9	−0.326491
$928$	0	0
$929$	−1.68343e10	−0.688874	−0.344437	−	0.938809i	$-0.611930\pi$
$929$	−1.68343e10	−0.688874	−0.344437	+	0.938809i	$0.611930\pi$
$930$	0	0
$931$	7.65320e9	0.310827
$932$	0	0
$933$	2.68752e10	1.08334
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.50534e9	0.377467	0.188733	−	0.982028i	$-0.439562\pi$
$937$	9.50534e9	0.377467	0.188733	+	0.982028i	$0.439562\pi$
$938$	0	0
$939$	1.75672e10	0.692425
$940$	0	0
$941$	2.70484e10	1.05823	0.529113	−	0.848551i	$-0.322525\pi$
$941$	2.70484e10	1.05823	0.529113	+	0.848551i	$0.322525\pi$
$942$	0	0
$943$	−2.64295e10	−1.02636
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.13125e10	0.815472	0.407736	−	0.913100i	$-0.366318\pi$
$947$	2.13125e10	0.815472	0.407736	+	0.913100i	$0.366318\pi$
$948$	0	0
$949$	7.28605e9	0.276733
$950$	0	0
$951$	2.02358e10	0.762936
$952$	0	0
$953$	−4.54903e10	−1.70252	−0.851262	−	0.524741i	$-0.824162\pi$
$953$	−4.54903e10	−1.70252	−0.851262	+	0.524741i	$0.824162\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.64630e10	0.607179
$958$	0	0
$959$	−3.07702e10	−1.12659
$960$	0	0
$961$	−1.21921e10	−0.443146
$962$	0	0
$963$	−9.12353e9	−0.329208
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.08975e10	0.387556	0.193778	−	0.981045i	$-0.437926\pi$
$967$	1.08975e10	0.387556	0.193778	+	0.981045i	$0.437926\pi$
$968$	0	0
$969$	−1.40707e10	−0.496800
$970$	0	0
$971$	−4.95413e10	−1.73660	−0.868301	−	0.496038i	$-0.834788\pi$
$971$	−4.95413e10	−1.73660	−0.868301	+	0.496038i	$0.834788\pi$
$972$	0	0
$973$	8.53840e8	0.0297154
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.66182e10	−1.25622	−0.628110	−	0.778124i	$-0.716171\pi$
$977$	−3.66182e10	−1.25622	−0.628110	+	0.778124i	$0.716171\pi$
$978$	0	0
$979$	3.05551e10	1.04074
$980$	0	0
$981$	8.53898e9	0.288778
$982$	0	0
$983$	1.72791e10	0.580207	0.290104	−	0.956995i	$-0.406310\pi$
$983$	1.72791e10	0.580207	0.290104	+	0.956995i	$0.406310\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.32240e9	−0.109987
$988$	0	0
$989$	2.95170e10	0.970253
$990$	0	0
$991$	−1.96802e10	−0.642350	−0.321175	−	0.947020i	$-0.604078\pi$
$991$	−1.96802e10	−0.642350	−0.321175	+	0.947020i	$0.604078\pi$
$992$	0	0
$993$	−2.12907e10	−0.690031
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.89071e10	−1.24336	−0.621678	−	0.783273i	$-0.713549\pi$
$997$	−3.89071e10	−1.24336	−0.621678	+	0.783273i	$0.713549\pi$
$998$	0	0
$999$	6.65061e9	0.211049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.8.a.f.1.1		1
5.2	odd	4		60.8.d.a.49.1	✓	2
5.3	odd	4		60.8.d.a.49.2	yes	2
5.4	even	2		300.8.a.b.1.1		1
15.2	even	4		180.8.d.a.109.1		2
15.8	even	4		180.8.d.a.109.2		2
20.3	even	4		240.8.f.b.49.1		2
20.7	even	4		240.8.f.b.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.8.d.a.49.1	✓	2	5.2	odd	4
60.8.d.a.49.2	yes	2	5.3	odd	4
180.8.d.a.109.1		2	15.2	even	4
180.8.d.a.109.2		2	15.8	even	4
240.8.f.b.49.1		2	20.3	even	4
240.8.f.b.49.2		2	20.7	even	4
300.8.a.b.1.1		1	5.4	even	2
300.8.a.f.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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