LMFDB - Embedded newform 1300.2.f.f.701.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(701,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1300.701");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.f (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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.796594176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 $ x^8 - 2x^7 - 5x^6 - 2x^5 + 63x^4 - 64x^3 + 46x^2 - 16*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.8
Root			$-2.12039 + 1.80156i$ of defining polynomial
Character	$\chi$	$=$	1300.701
Dual form			1300.2.f.f.701.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.09557 q^{3} +4.37780i q^{7} +6.58258 q^{9} +O(q^{10})$	$q+3.09557 q^{3} +4.37780i q^{7} +6.58258 q^{9} -2.53326i q^{11} +(-2.44949 + 2.64575i) q^{13} -1.29217 q^{17} +5.36169i q^{19} +13.5518i q^{21} +1.80341 q^{23} +11.0901 q^{27} +7.58258 q^{29} -3.12359i q^{31} -7.84190i q^{33} -2.55040i q^{37} +(-7.58258 + 8.19012i) q^{39} -7.89495i q^{41} -4.38774 q^{43} +6.20520i q^{47} -12.1652 q^{49} -4.00000 q^{51} +6.19115 q^{53} +16.5975i q^{57} -10.4282i q^{59} +3.58258 q^{61} +28.8172i q^{63} -2.55040i q^{67} +5.58258 q^{69} -0.295164i q^{71} +2.55040i q^{73} +11.0901 q^{77} -11.1652 q^{79} +14.5826 q^{81} -0.723000i q^{83} +23.4724 q^{87} -11.3137i q^{89} +(-11.5826 - 10.7234i) q^{91} -9.66930i q^{93} -12.2197i q^{97} -16.6754i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 24 q^{29} - 24 q^{39} - 24 q^{49} - 32 q^{51} - 8 q^{61} + 8 q^{69} - 16 q^{79} + 80 q^{81} - 56 q^{91}+O(q^{100}) $ 8 * q + 16 * q^9 + 24 * q^29 - 24 * q^39 - 24 * q^49 - 32 * q^51 - 8 * q^61 + 8 * q^69 - 16 * q^79 + 80 * q^81 - 56 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	3.09557			1.78723			0.893615	−	0.448834i	$-0.148161\pi$
$3$	3.09557			1.78723			0.893615	+	0.448834i	$0.148161\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.37780i			1.65465i	0.561721	+	0.827327i	$0.310140\pi$
$7$			4.37780i			1.65465i	−0.561721	+	0.827327i	$0.689860\pi$
$8$		0			0
$9$	6.58258			2.19419
$10$		0			0
$11$		−	2.53326i		−	0.763808i	−0.924202	−	0.381904i	$-0.875269\pi$
$11$		−	2.53326i		−	0.763808i	0.924202	−	0.381904i	$-0.124731\pi$
$12$		0			0
$13$	−2.44949	+	2.64575i	−0.679366	+	0.733799i
$13$	−2.44949	+	2.64575i	−0.679366	+	0.733799i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.29217			−0.313397			−0.156698	−	0.987647i	$-0.550085\pi$
$17$	−1.29217			−0.313397			−0.156698	+	0.987647i	$0.550085\pi$
$18$		0			0
$19$			5.36169i			1.23006i	0.788505	+	0.615028i	$0.210855\pi$
$19$			5.36169i			1.23006i	−0.788505	+	0.615028i	$0.789145\pi$
$20$		0			0
$21$			13.5518i			2.95725i
$22$		0			0
$23$	1.80341			0.376036			0.188018	−	0.982166i	$-0.439794\pi$
$23$	1.80341			0.376036			0.188018	+	0.982166i	$0.439794\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	11.0901			2.13430
$28$		0			0
$29$	7.58258			1.40805			0.704024	−	0.710176i	$-0.251385\pi$
$29$	7.58258			1.40805			0.704024	+	0.710176i	$0.251385\pi$
$30$		0			0
$31$		−	3.12359i		−	0.561013i	−0.959852	−	0.280507i	$-0.909498\pi$
$31$		−	3.12359i		−	0.561013i	0.959852	−	0.280507i	$-0.0905025\pi$
$32$		0			0
$33$		−	7.84190i		−	1.36510i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	2.55040i		−	0.419283i	−0.977778	−	0.209642i	$-0.932770\pi$
$37$		−	2.55040i		−	0.419283i	0.977778	−	0.209642i	$-0.0672297\pi$
$38$		0			0
$39$	−7.58258	+	8.19012i	−1.21418	+	1.31147i
$40$		0			0
$41$		−	7.89495i		−	1.23298i	−0.787361	−	0.616492i	$-0.788553\pi$
$41$		−	7.89495i		−	1.23298i	0.787361	−	0.616492i	$-0.211447\pi$
$42$		0			0
$43$	−4.38774			−0.669124			−0.334562	−	0.942374i	$-0.608588\pi$
$43$	−4.38774			−0.669124			−0.334562	+	0.942374i	$0.608588\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.20520i			0.905122i	0.891733	+	0.452561i	$0.149490\pi$
$47$			6.20520i			0.905122i	−0.891733	+	0.452561i	$0.850510\pi$
$48$		0			0
$49$	−12.1652			−1.73788
$50$		0			0
$51$	−4.00000			−0.560112
$52$		0			0
$53$	6.19115			0.850419			0.425210	−	0.905095i	$-0.360200\pi$
$53$	6.19115			0.850419			0.425210	+	0.905095i	$0.360200\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			16.5975i			2.19839i
$58$		0			0
$59$		−	10.4282i		−	1.35764i	−0.734305	−	0.678819i	$-0.762492\pi$
$59$		−	10.4282i		−	1.35764i	0.734305	−	0.678819i	$-0.237508\pi$
$60$		0			0
$61$	3.58258			0.458702			0.229351	−	0.973344i	$-0.426340\pi$
$61$	3.58258			0.458702			0.229351	+	0.973344i	$0.426340\pi$
$62$		0			0
$63$			28.8172i			3.63063i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	2.55040i		−	0.311581i	−0.987790	−	0.155791i	$-0.950208\pi$
$67$		−	2.55040i		−	0.311581i	0.987790	−	0.155791i	$-0.0497925\pi$
$68$		0			0
$69$	5.58258			0.672063
$70$		0			0
$71$		−	0.295164i		−	0.0350295i	−0.999847	−	0.0175147i	$-0.994425\pi$
$71$		−	0.295164i		−	0.0350295i	0.999847	−	0.0175147i	$-0.00557540\pi$
$72$		0			0
$73$			2.55040i			0.298502i	0.988799	+	0.149251i	$0.0476862\pi$
$73$			2.55040i			0.298502i	−0.988799	+	0.149251i	$0.952314\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.0901			1.26384
$78$		0			0
$79$	−11.1652			−1.25618			−0.628089	−	0.778142i	$-0.716163\pi$
$79$	−11.1652			−1.25618			−0.628089	+	0.778142i	$0.716163\pi$
$80$		0			0
$81$	14.5826			1.62029
$82$		0			0
$83$		−	0.723000i		−	0.0793596i	−0.999212	−	0.0396798i	$-0.987366\pi$
$83$		−	0.723000i		−	0.0793596i	0.999212	−	0.0396798i	$-0.0126338\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	23.4724			2.51651
$88$		0			0
$89$		−	11.3137i		−	1.19925i	−0.800281	−	0.599625i	$-0.795316\pi$
$89$		−	11.3137i		−	1.19925i	0.800281	−	0.599625i	$-0.204684\pi$
$90$		0			0
$91$	−11.5826	−	10.7234i	−1.21418	−	1.12412i
$92$		0			0
$93$		−	9.66930i		−	1.00266i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	12.2197i		−	1.24072i	−0.784316	−	0.620362i	$-0.786986\pi$
$97$		−	12.2197i		−	1.24072i	0.784316	−	0.620362i	$-0.213014\pi$
$98$		0			0
$99$		−	16.6754i		−	1.67594i
$100$		0			0
$101$	−9.16515			−0.911967			−0.455983	−	0.889988i	$-0.650712\pi$
$101$	−9.16515			−0.911967			−0.455983	+	0.889988i	$0.650712\pi$
$102$		0			0
$103$	9.28672			0.915048			0.457524	−	0.889197i	$-0.348736\pi$
$103$	9.28672			0.915048			0.457524	+	0.889197i	$0.348736\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.5789			−1.02270			−0.511350	−	0.859373i	$-0.670854\pi$
$107$	−10.5789			−1.02270			−0.511350	+	0.859373i	$0.670854\pi$
$108$		0			0
$109$			8.48528i			0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$			8.48528i			0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$		0			0
$111$		−	7.89495i		−	0.749356i
$112$		0			0
$113$	11.0901			1.04327			0.521636	−	0.853168i	$-0.325322\pi$
$113$	11.0901			1.04327			0.521636	+	0.853168i	$0.325322\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−16.1240	+	17.4159i	−1.49066	+	1.61010i
$118$		0			0
$119$		−	5.65685i		−	0.518563i
$120$		0			0
$121$	4.58258			0.416598
$122$		0			0
$123$		−	24.4394i		−	2.20363i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.511238			0.0453651			0.0226825	−	0.999743i	$-0.492779\pi$
$127$	0.511238			0.0453651			0.0226825	+	0.999743i	$0.492779\pi$
$128$		0			0
$129$	−13.5826			−1.19588
$130$		0			0
$131$	−3.16515			−0.276541			−0.138270	−	0.990395i	$-0.544154\pi$
$131$	−3.16515			−0.276541			−0.138270	+	0.990395i	$0.544154\pi$
$132$		0			0
$133$	−23.4724			−2.03532
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.3923i			0.887875i	0.896058	+	0.443937i	$0.146419\pi$
$137$			10.3923i			0.887875i	−0.896058	+	0.443937i	$0.853581\pi$
$138$		0			0
$139$	7.16515			0.607740			0.303870	−	0.952713i	$-0.401721\pi$
$139$	7.16515			0.607740			0.303870	+	0.952713i	$0.401721\pi$
$140$		0			0
$141$			19.2087i			1.61766i
$142$		0			0
$143$	6.70239	+	6.20520i	0.560482	+	0.518905i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−37.6581			−3.10599
$148$		0			0
$149$		−	2.82843i		−	0.231714i	−0.993266	−	0.115857i	$-0.963039\pi$
$149$		−	2.82843i		−	0.231714i	0.993266	−	0.115857i	$-0.0369614\pi$
$150$		0			0
$151$		−	16.0851i		−	1.30898i	−0.756069	−	0.654492i	$-0.772882\pi$
$151$		−	16.0851i		−	1.30898i	0.756069	−	0.654492i	$-0.227118\pi$
$152$		0			0
$153$	−8.50579			−0.687652
$154$		0			0
$155$		0			0
$156$		0			0
$157$	22.4499			1.79170			0.895850	−	0.444356i	$-0.146567\pi$
$157$	22.4499			1.79170			0.895850	+	0.444356i	$0.146567\pi$
$158$		0			0
$159$	19.1652			1.51990
$160$		0			0
$161$			7.89495i			0.622210i
$162$		0			0
$163$		−	11.3060i		−	0.885555i	−0.896632	−	0.442777i	$-0.853993\pi$
$163$		−	11.3060i		−	0.885555i	0.896632	−	0.442777i	$-0.146007\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.723000i		−	0.0559474i	−0.999609	−	0.0279737i	$-0.991095\pi$
$167$		−	0.723000i		−	0.0559474i	0.999609	−	0.0279737i	$-0.00890547\pi$
$168$		0			0
$169$	−1.00000	−	12.9615i	−0.0769231	−	0.997037i
$170$		0			0
$171$			35.2937i			2.69898i
$172$		0			0
$173$	−3.60681			−0.274221			−0.137110	−	0.990556i	$-0.543782\pi$
$173$	−3.60681			−0.274221			−0.137110	+	0.990556i	$0.543782\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	32.2813i		−	2.42641i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−23.5826			−1.75288			−0.876440	−	0.481512i	$-0.840088\pi$
$181$	−23.5826			−1.75288			−0.876440	+	0.481512i	$0.840088\pi$
$182$		0			0
$183$	11.0901			0.819806
$184$		0			0
$185$		0			0
$186$		0			0
$187$			3.27340i			0.239375i
$188$		0			0
$189$			48.5504i			3.53152i
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$		−	0.190700i		−	0.0137269i	−0.999976	−	0.00686346i	$-0.997815\pi$
$193$		−	0.190700i		−	0.0137269i	0.999976	−	0.00686346i	$-0.00218472\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	22.8027i		−	1.62463i	−0.583222	−	0.812313i	$-0.698208\pi$
$197$		−	22.8027i		−	1.62463i	0.583222	−	0.812313i	$-0.301792\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$		−	7.89495i		−	0.556867i
$202$		0			0
$203$			33.1950i			2.32983i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	11.8711			0.825095
$208$		0			0
$209$	13.5826			0.939526
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		−	0.913701i		−	0.0626057i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	13.6745			0.928283
$218$		0			0
$219$			7.89495i			0.533492i
$220$		0			0
$221$	3.16515	−	3.41875i	0.212911	−	0.229970i
$222$		0			0
$223$			2.55040i			0.170787i	0.996347	+	0.0853937i	$0.0272148\pi$
$223$			2.55040i			0.170787i	−0.996347	+	0.0853937i	$0.972785\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			26.9898i			1.79138i	0.444682	+	0.895688i	$0.353317\pi$
$227$			26.9898i			1.79138i	−0.444682	+	0.895688i	$0.646683\pi$
$228$		0			0
$229$			10.7234i			0.708621i	0.935128	+	0.354310i	$0.115284\pi$
$229$			10.7234i			0.708621i	−0.935128	+	0.354310i	$0.884716\pi$
$230$		0			0
$231$	34.3303			2.25877
$232$		0			0
$233$	−12.3823			−0.811191			−0.405596	−	0.914053i	$-0.632936\pi$
$233$	−12.3823			−0.811191			−0.405596	+	0.914053i	$0.632936\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−34.5625			−2.24508
$238$		0			0
$239$			30.2272i			1.95524i	0.210389	+	0.977618i	$0.432527\pi$
$239$			30.2272i			1.95524i	−0.210389	+	0.977618i	$0.567473\pi$
$240$		0			0
$241$		−	14.7325i		−	0.949001i	−0.880255	−	0.474501i	$-0.842629\pi$
$241$		−	14.7325i		−	0.949001i	0.880255	−	0.474501i	$-0.157371\pi$
$242$		0			0
$243$	11.8711			0.761529
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−14.1857	−	13.1334i	−0.902614	−	0.835659i
$248$		0			0
$249$		−	2.23810i		−	0.141834i
$250$		0			0
$251$	−15.1652			−0.957216			−0.478608	−	0.878029i	$-0.658858\pi$
$251$	−15.1652			−0.957216			−0.478608	+	0.878029i	$0.658858\pi$
$252$		0			0
$253$		−	4.56850i		−	0.287219i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.21362			−0.449973			−0.224987	−	0.974362i	$-0.572234\pi$
$257$	−7.21362			−0.449973			−0.224987	+	0.974362i	$0.572234\pi$
$258$		0			0
$259$	11.1652			0.693769
$260$		0			0
$261$	49.9129			3.08953
$262$		0			0
$263$	16.5003			1.01745			0.508727	−	0.860928i	$-0.330116\pi$
$263$	16.5003			1.01745			0.508727	+	0.860928i	$0.330116\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	35.0224i		−	2.14334i
$268$		0			0
$269$	−9.16515			−0.558809			−0.279405	−	0.960173i	$-0.590137\pi$
$269$	−9.16515			−0.558809			−0.279405	+	0.960173i	$0.590137\pi$
$270$		0			0
$271$			22.3323i			1.35659i	0.734791	+	0.678294i	$0.237280\pi$
$271$			22.3323i			1.35659i	−0.734791	+	0.678294i	$0.762720\pi$
$272$		0			0
$273$	−35.8547	−	33.1950i	−2.17003	−	2.00905i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−28.3714			−1.70467			−0.852336	−	0.522994i	$-0.824815\pi$
$277$	−28.3714			−1.70467			−0.852336	+	0.522994i	$0.824815\pi$
$278$		0			0
$279$		−	20.5613i		−	1.23097i
$280$		0			0
$281$		−	12.3712i		−	0.738001i	−0.929429	−	0.369001i	$-0.879700\pi$
$281$		−	12.3712i		−	0.738001i	0.929429	−	0.369001i	$-0.120300\pi$
$282$		0			0
$283$	−4.38774			−0.260824			−0.130412	−	0.991460i	$-0.541630\pi$
$283$	−4.38774			−0.260824			−0.130412	+	0.991460i	$0.541630\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	34.5625			2.04016
$288$		0			0
$289$	−15.3303			−0.901783
$290$		0			0
$291$		−	37.8270i		−	2.21746i
$292$		0			0
$293$			6.20520i			0.362512i	0.983436	+	0.181256i	$0.0580162\pi$
$293$			6.20520i			0.362512i	−0.983436	+	0.181256i	$0.941984\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	28.0942i		−	1.63019i
$298$		0			0
$299$	−4.41742	+	4.77136i	−0.255466	+	0.275935i
$300$		0			0
$301$		−	19.2087i		−	1.10717i
$302$		0			0
$303$	−28.3714			−1.62989
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	8.03260i		−	0.458445i	−0.973374	−	0.229222i	$-0.926382\pi$
$307$		−	8.03260i		−	0.458445i	0.973374	−	0.229222i	$-0.0736183\pi$
$308$		0			0
$309$	28.7477			1.63540
$310$		0			0
$311$	6.33030			0.358959			0.179479	−	0.983762i	$-0.442559\pi$
$311$	6.33030			0.358959			0.179479	+	0.983762i	$0.442559\pi$
$312$		0			0
$313$	3.87650			0.219113			0.109556	−	0.993981i	$-0.465057\pi$
$313$	3.87650			0.219113			0.109556	+	0.993981i	$0.465057\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	20.0616i		−	1.12677i	−0.826194	−	0.563386i	$-0.809498\pi$
$317$		−	20.0616i		−	1.12677i	0.826194	−	0.563386i	$-0.190502\pi$
$318$		0			0
$319$		−	19.2087i		−	1.07548i
$320$		0			0
$321$	−32.7477			−1.82780
$322$		0			0
$323$		−	6.92820i		−	0.385496i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			26.2668i			1.45256i
$328$		0			0
$329$	−27.1652			−1.49766
$330$		0			0
$331$			24.5704i			1.35051i	0.737585	+	0.675254i	$0.235966\pi$
$331$			24.5704i			1.35051i	−0.737585	+	0.675254i	$0.764034\pi$
$332$		0			0
$333$		−	16.7882i		−	0.919988i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23.4724			−1.27862			−0.639312	−	0.768947i	$-0.720781\pi$
$337$	−23.4724			−1.27862			−0.639312	+	0.768947i	$0.720781\pi$
$338$		0			0
$339$	34.3303			1.86457
$340$		0			0
$341$	−7.91288			−0.428506
$342$		0			0
$343$		−	22.6120i		−	1.22093i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.07310			0.111290			0.0556448	−	0.998451i	$-0.482279\pi$
$347$	2.07310			0.111290			0.0556448	+	0.998451i	$0.482279\pi$
$348$		0			0
$349$		−	32.1701i		−	1.72203i	−0.508581	−	0.861014i	$-0.669830\pi$
$349$		−	32.1701i		−	1.72203i	0.508581	−	0.861014i	$-0.330170\pi$
$350$		0			0
$351$	−27.1652	+	29.3417i	−1.44997	+	1.56614i
$352$		0			0
$353$			6.20520i			0.330270i	0.986271	+	0.165135i	$0.0528059\pi$
$353$			6.20520i			0.330270i	−0.986271	+	0.165135i	$0.947194\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	17.5112i		−	0.926791i
$358$		0			0
$359$			21.7419i			1.14749i	0.819032	+	0.573747i	$0.194511\pi$
$359$			21.7419i			1.14749i	−0.819032	+	0.573747i	$0.805489\pi$
$360$		0			0
$361$	−9.74773			−0.513038
$362$		0			0
$363$	14.1857			0.744556
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9.28672			−0.484763			−0.242381	−	0.970181i	$-0.577929\pi$
$367$	−9.28672			−0.484763			−0.242381	+	0.970181i	$0.577929\pi$
$368$		0			0
$369$		−	51.9691i		−	2.70541i
$370$		0			0
$371$			27.1036i			1.40715i
$372$		0			0
$373$	−14.6969			−0.760979			−0.380489	−	0.924785i	$-0.624244\pi$
$373$	−14.6969			−0.760979			−0.380489	+	0.924785i	$0.624244\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.5734	+	20.0616i	−0.956581	+	1.03323i
$378$		0			0
$379$			1.35261i			0.0694789i	0.999396	+	0.0347394i	$0.0110601\pi$
$379$			1.35261i			0.0694789i	−0.999396	+	0.0347394i	$0.988940\pi$
$380$		0			0
$381$	1.58258			0.0810778
$382$		0			0
$383$			18.6156i			0.951213i	0.879658	+	0.475607i	$0.157771\pi$
$383$			18.6156i			0.951213i	−0.879658	+	0.475607i	$0.842229\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−28.8826			−1.46819
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	−2.33030			−0.117848
$392$		0			0
$393$	−9.79796			−0.494242
$394$		0			0
$395$		0			0
$396$		0			0
$397$			25.1624i			1.26287i	0.775431	+	0.631433i	$0.217533\pi$
$397$			25.1624i			1.26287i	−0.775431	+	0.631433i	$0.782467\pi$
$398$		0			0
$399$	−72.6606			−3.63758
$400$		0			0
$401$		−	22.6274i		−	1.12996i	−0.825105	−	0.564980i	$-0.808884\pi$
$401$		−	22.6274i		−	1.12996i	0.825105	−	0.564980i	$-0.191116\pi$
$402$		0			0
$403$	8.26424	+	7.65120i	0.411671	+	0.381134i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.46084			−0.320252
$408$		0			0
$409$			6.71430i			0.332001i	0.986126	+	0.166000i	$0.0530853\pi$
$409$			6.71430i			0.332001i	−0.986126	+	0.166000i	$0.946915\pi$
$410$		0			0
$411$			32.1701i			1.58684i
$412$		0			0
$413$	45.6527			2.24642
$414$		0			0
$415$		0			0
$416$		0			0
$417$	22.1803			1.08617
$418$		0			0
$419$	−33.4955			−1.63636			−0.818180	−	0.574962i	$-0.805017\pi$
$419$	−33.4955			−1.63636			−0.818180	+	0.574962i	$0.805017\pi$
$420$		0			0
$421$		−	17.4377i		−	0.849861i	−0.905226	−	0.424930i	$-0.860299\pi$
$421$		−	17.4377i		−	0.849861i	0.905226	−	0.424930i	$-0.139701\pi$
$422$		0			0
$423$			40.8462i			1.98601i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			15.6838i			0.758993i
$428$		0			0
$429$	20.7477	+	19.2087i	1.00171	+	0.927403i
$430$		0			0
$431$		−	30.2272i		−	1.45599i	−0.685581	−	0.727997i	$-0.740452\pi$
$431$		−	30.2272i		−	1.45599i	0.685581	−	0.727997i	$-0.259548\pi$
$432$		0			0
$433$	9.79796			0.470860			0.235430	−	0.971891i	$-0.424350\pi$
$433$	9.79796			0.470860			0.235430	+	0.971891i	$0.424350\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			9.66930i			0.462546i
$438$		0			0
$439$	37.4955			1.78956			0.894780	−	0.446507i	$-0.147332\pi$
$439$	37.4955			1.78956			0.894780	+	0.446507i	$0.147332\pi$
$440$		0			0
$441$	−80.0780			−3.81324
$442$		0			0
$443$	25.2758			1.20089			0.600445	−	0.799666i	$-0.294990\pi$
$443$	25.2758			1.20089			0.600445	+	0.799666i	$0.294990\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	8.75560i		−	0.414126i
$448$		0			0
$449$		−	34.9986i		−	1.65168i	−0.563901	−	0.825842i	$-0.690700\pi$
$449$		−	34.9986i		−	1.65168i	0.563901	−	0.825842i	$-0.309300\pi$
$450$		0			0
$451$	−20.0000			−0.941763
$452$		0			0
$453$		−	49.7925i		−	2.33946i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			1.63670i			0.0765616i	0.999267	+	0.0382808i	$0.0121881\pi$
$457$			1.63670i			0.0765616i	−0.999267	+	0.0382808i	$0.987812\pi$
$458$		0			0
$459$	−14.3303			−0.668881
$460$		0			0
$461$			0.590327i			0.0274943i	0.999906	+	0.0137471i	$0.00437599\pi$
$461$			0.590327i			0.0274943i	−0.999906	+	0.0137471i	$0.995624\pi$
$462$		0			0
$463$		−	39.0188i		−	1.81336i	−0.421821	−	0.906679i	$-0.638609\pi$
$463$		−	39.0188i		−	1.81336i	0.421821	−	0.906679i	$-0.361391\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	31.4670			1.45612			0.728059	−	0.685515i	$-0.240423\pi$
$467$	31.4670			1.45612			0.728059	+	0.685515i	$0.240423\pi$
$468$		0			0
$469$	11.1652			0.515559
$470$		0			0
$471$	69.4955			3.20218
$472$		0			0
$473$			11.1153i			0.511082i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	40.7537			1.86598
$478$		0			0
$479$			0.295164i			0.0134864i	0.999977	+	0.00674318i	$0.00214644\pi$
$479$			0.295164i			0.0134864i	−0.999977	+	0.00674318i	$0.997854\pi$
$480$		0			0
$481$	6.74773	+	6.24718i	0.307670	+	0.284847i
$482$		0			0
$483$			24.4394i			1.11203i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	9.47860i		−	0.429517i	−0.976667	−	0.214758i	$-0.931104\pi$
$487$		−	9.47860i		−	0.429517i	0.976667	−	0.214758i	$-0.0688964\pi$
$488$		0			0
$489$		−	34.9986i		−	1.58269i
$490$		0			0
$491$	30.3303			1.36879			0.684394	−	0.729113i	$-0.260067\pi$
$491$	30.3303			1.36879			0.684394	+	0.729113i	$0.260067\pi$
$492$		0			0
$493$	−9.79796			−0.441278
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.29217			0.0579616
$498$		0			0
$499$			0.885491i			0.0396400i	0.999804	+	0.0198200i	$0.00630932\pi$
$499$			0.885491i			0.0396400i	−0.999804	+	0.0198200i	$0.993691\pi$
$500$		0			0
$501$		−	2.23810i		−	0.0999909i
$502$		0			0
$503$	−5.94960			−0.265280			−0.132640	−	0.991164i	$-0.542345\pi$
$503$	−5.94960			−0.265280			−0.132640	+	0.991164i	$0.542345\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.09557	−	40.1232i	−0.137479	−	1.78193i
$508$		0			0
$509$			11.9040i			0.527637i	0.964572	+	0.263819i	$0.0849820\pi$
$509$			11.9040i			0.527637i	−0.964572	+	0.263819i	$0.915018\pi$
$510$		0			0
$511$	−11.1652			−0.493917
$512$		0			0
$513$			59.4618i			2.62530i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	15.7194			0.691339
$518$		0			0
$519$	−11.1652			−0.490096
$520$		0			0
$521$	31.5826			1.38366			0.691829	−	0.722061i	$-0.256805\pi$
$521$	31.5826			1.38366			0.691829	+	0.722061i	$0.256805\pi$
$522$		0			0
$523$	−1.53371			−0.0670647			−0.0335323	−	0.999438i	$-0.510676\pi$
$523$	−1.53371			−0.0670647			−0.0335323	+	0.999438i	$0.510676\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.03620i			0.175820i
$528$		0			0
$529$	−19.7477			−0.858597
$530$		0			0
$531$		−	68.6445i		−	2.97892i
$532$		0			0
$533$	20.8881	+	19.3386i	0.904763	+	0.837648i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			30.8175i			1.32741i
$540$		0			0
$541$			19.6758i			0.845928i	0.906147	+	0.422964i	$0.139010\pi$
$541$			19.6758i			0.845928i	−0.906147	+	0.422964i	$0.860990\pi$
$542$		0			0
$543$	−73.0016			−3.13280
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−15.2082			−0.650255			−0.325127	−	0.945670i	$-0.605407\pi$
$547$	−15.2082			−0.650255			−0.325127	+	0.945670i	$0.605407\pi$
$548$		0			0
$549$	23.5826			1.00648
$550$		0			0
$551$			40.6554i			1.73198i
$552$		0			0
$553$		−	48.8788i		−	2.07854i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.65120i			0.324192i	0.986775	+	0.162096i	$0.0518254\pi$
$557$			7.65120i			0.324192i	−0.986775	+	0.162096i	$0.948175\pi$
$558$		0			0
$559$	10.7477	−	11.6089i	0.454580	−	0.491003i
$560$		0			0
$561$			10.1331i			0.427818i
$562$		0			0
$563$	−24.5230			−1.03352			−0.516761	−	0.856129i	$-0.672863\pi$
$563$	−24.5230			−1.03352			−0.516761	+	0.856129i	$0.672863\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			63.8396i			2.68101i
$568$		0			0
$569$	19.5826			0.820944			0.410472	−	0.911873i	$-0.365364\pi$
$569$	19.5826			0.820944			0.410472	+	0.911873i	$0.365364\pi$
$570$		0			0
$571$	17.4955			0.732162			0.366081	−	0.930583i	$-0.380699\pi$
$571$	17.4955			0.732162			0.366081	+	0.930583i	$0.380699\pi$
$572$		0			0
$573$	−37.1469			−1.55183
$574$		0			0
$575$		0			0
$576$		0			0
$577$			22.2704i			0.927129i	0.886063	+	0.463565i	$0.153430\pi$
$577$			22.2704i			0.927129i	−0.886063	+	0.463565i	$0.846570\pi$
$578$		0			0
$579$		−	0.590327i		−	0.0245332i
$580$		0			0
$581$	3.16515			0.131313
$582$		0			0
$583$		−	15.6838i		−	0.649557i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26.9898i		−	1.11399i	−0.830516	−	0.556994i	$-0.811954\pi$
$587$		−	26.9898i		−	1.11399i	0.830516	−	0.556994i	$-0.188046\pi$
$588$		0			0
$589$	16.7477			0.690078
$590$		0			0
$591$		−	70.5875i		−	2.90358i
$592$		0			0
$593$			15.8745i			0.651888i	0.945389	+	0.325944i	$0.105682\pi$
$593$			15.8745i			0.651888i	−0.945389	+	0.325944i	$0.894318\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	12.3823			0.506774
$598$		0			0
$599$	5.66970			0.231658			0.115829	−	0.993269i	$-0.463048\pi$
$599$	5.66970			0.231658			0.115829	+	0.993269i	$0.463048\pi$
$600$		0			0
$601$	7.66970			0.312853			0.156427	−	0.987690i	$-0.450002\pi$
$601$	7.66970			0.312853			0.156427	+	0.987690i	$0.450002\pi$
$602$		0			0
$603$		−	16.7882i		−	0.683669i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−21.9387			−0.890465			−0.445232	−	0.895415i	$-0.646879\pi$
$607$	−21.9387			−0.890465			−0.445232	+	0.895415i	$0.646879\pi$
$608$		0			0
$609$			102.758i			4.16395i
$610$		0			0
$611$	−16.4174	−	15.1996i	−0.664178	−	0.614909i
$612$		0			0
$613$			17.7019i			0.714973i	0.933918	+	0.357487i	$0.116366\pi$
$613$			17.7019i			0.714973i	−0.933918	+	0.357487i	$0.883634\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			35.2131i			1.41763i	0.705396	+	0.708813i	$0.250769\pi$
$617$			35.2131i			1.41763i	−0.705396	+	0.708813i	$0.749231\pi$
$618$		0			0
$619$			39.3028i			1.57971i	0.613291	+	0.789857i	$0.289845\pi$
$619$			39.3028i			1.57971i	−0.613291	+	0.789857i	$0.710155\pi$
$620$		0			0
$621$	20.0000			0.802572
$622$		0			0
$623$	49.5292			1.98434
$624$		0			0
$625$		0			0
$626$		0			0
$627$	42.0459			1.67915
$628$		0			0
$629$			3.29555i			0.131402i
$630$		0			0
$631$			30.8175i			1.22683i	0.789762	+	0.613413i	$0.210204\pi$
$631$			30.8175i			1.22683i	−0.789762	+	0.613413i	$0.789796\pi$
$632$		0			0
$633$	61.9115			2.46076
$634$		0			0
$635$		0			0
$636$		0			0
$637$	29.7984	−	32.1860i	1.18066	−	1.27525i
$638$		0			0
$639$		−	1.94294i		−	0.0768614i
$640$		0			0
$641$	12.3303			0.487018			0.243509	−	0.969899i	$-0.421702\pi$
$641$	12.3303			0.487018			0.243509	+	0.969899i	$0.421702\pi$
$642$		0			0
$643$			35.7454i			1.40966i	0.709375	+	0.704831i	$0.248977\pi$
$643$			35.7454i			1.40966i	−0.709375	+	0.704831i	$0.751023\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.4895			1.27729			0.638646	−	0.769501i	$-0.279495\pi$
$647$	32.4895			1.27729			0.638646	+	0.769501i	$0.279495\pi$
$648$		0			0
$649$	−26.4174			−1.03697
$650$		0			0
$651$	42.3303			1.65906
$652$		0			0
$653$	−44.6302			−1.74651			−0.873257	−	0.487259i	$-0.837997\pi$
$653$	−44.6302			−1.74651			−0.873257	+	0.487259i	$0.837997\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			16.7882i			0.654970i
$658$		0			0
$659$	33.4955			1.30480			0.652399	−	0.757876i	$-0.273763\pi$
$659$	33.4955			1.30480			0.652399	+	0.757876i	$0.273763\pi$
$660$		0			0
$661$			8.48528i			0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$			8.48528i			0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$		0			0
$663$	9.79796	−	10.5830i	0.380521	−	0.411010i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	13.6745			0.529477
$668$		0			0
$669$			7.89495i			0.305237i
$670$		0			0
$671$		−	9.07561i		−	0.350360i
$672$		0			0
$673$	23.4724			0.904795			0.452398	−	0.891816i	$-0.350569\pi$
$673$	23.4724			0.904795			0.452398	+	0.891816i	$0.350569\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23.7421			−0.912483			−0.456242	−	0.889856i	$-0.650805\pi$
$677$	−23.7421			−0.912483			−0.456242	+	0.889856i	$0.650805\pi$
$678$		0			0
$679$	53.4955			2.05297
$680$		0			0
$681$			83.5490i			3.20160i
$682$		0			0
$683$			26.9898i			1.03274i	0.856367	+	0.516368i	$0.172716\pi$
$683$			26.9898i			1.03274i	−0.856367	+	0.516368i	$0.827284\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			33.1950i			1.26647i
$688$		0			0
$689$	−15.1652	+	16.3802i	−0.577746	+	0.624037i
$690$		0			0
$691$			20.0942i			0.764418i	0.924076	+	0.382209i	$0.124836\pi$
$691$			20.0942i			0.764418i	−0.924076	+	0.382209i	$0.875164\pi$
$692$		0			0
$693$	73.0016			2.77310
$694$		0			0
$695$		0			0
$696$		0			0
$697$			10.2016i			0.386413i
$698$		0			0
$699$	−38.3303			−1.44979
$700$		0			0
$701$	−14.8348			−0.560304			−0.280152	−	0.959956i	$-0.590385\pi$
$701$	−14.8348			−0.560304			−0.280152	+	0.959956i	$0.590385\pi$
$702$		0			0
$703$	13.6745			0.515742
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	40.1232i		−	1.50899i
$708$		0			0
$709$		−	15.1996i		−	0.570832i	−0.958404	−	0.285416i	$-0.907868\pi$
$709$		−	15.1996i		−	0.570832i	0.958404	−	0.285416i	$-0.0921318\pi$
$710$		0			0
$711$	−73.4955			−2.75629
$712$		0			0
$713$		−	5.63310i		−	0.210961i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			93.5705i			3.49446i
$718$		0			0
$719$	33.4955			1.24917			0.624585	−	0.780957i	$-0.285268\pi$
$719$	33.4955			1.24917			0.624585	+	0.780957i	$0.285268\pi$
$720$		0			0
$721$			40.6554i			1.51409i
$722$		0			0
$723$		−	45.6054i		−	1.69608i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	15.2082			0.564040			0.282020	−	0.959409i	$-0.408996\pi$
$727$	15.2082			0.564040			0.282020	+	0.959409i	$0.408996\pi$
$728$		0			0
$729$	−7.00000			−0.259259
$730$		0			0
$731$	5.66970			0.209701
$732$		0			0
$733$			33.0043i			1.21904i	0.792770	+	0.609521i	$0.208638\pi$
$733$			33.0043i			1.21904i	−0.792770	+	0.609521i	$0.791362\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.46084			−0.237988
$738$		0			0
$739$		−	33.5228i		−	1.23315i	−0.787294	−	0.616577i	$-0.788519\pi$
$739$		−	33.5228i		−	1.23315i	0.787294	−	0.616577i	$-0.211481\pi$
$740$		0			0
$741$	−43.9129	−	40.6554i	−1.61318	−	1.49351i
$742$		0			0
$743$			18.6156i			0.682940i	0.939893	+	0.341470i	$0.110925\pi$
$743$			18.6156i			0.682940i	−0.939893	+	0.341470i	$0.889075\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	4.75920i		−	0.174130i
$748$		0			0
$749$		−	46.3123i		−	1.69221i
$750$		0			0
$751$	4.83485			0.176426			0.0882131	−	0.996102i	$-0.471884\pi$
$751$	4.83485			0.176426			0.0882131	+	0.996102i	$0.471884\pi$
$752$		0			0
$753$	−46.9448			−1.71077
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−26.3264			−0.956851			−0.478426	−	0.878128i	$-0.658792\pi$
$757$	−26.3264			−0.956851			−0.478426	+	0.878128i	$0.658792\pi$
$758$		0			0
$759$		−	14.1421i		−	0.513327i
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$	−37.1469			−1.34481
$764$		0			0
$765$		0			0
$766$		0			0
$767$	27.5905	+	25.5438i	0.996234	+	0.922334i
$768$		0			0
$769$			42.8935i			1.54678i	0.633930	+	0.773390i	$0.281441\pi$
$769$			42.8935i			1.54678i	−0.633930	+	0.773390i	$0.718559\pi$
$770$		0			0
$771$	−22.3303			−0.804206
$772$		0			0
$773$			6.20520i			0.223186i	0.993754	+	0.111593i	$0.0355952\pi$
$773$			6.20520i			0.223186i	−0.993754	+	0.111593i	$0.964405\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	34.5625			1.23992
$778$		0			0
$779$	42.3303			1.51664
$780$		0			0
$781$	−0.747727			−0.0267558
$782$		0			0
$783$	84.0917			3.00519
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	8.03260i		−	0.286331i	−0.989699	−	0.143166i	$-0.954272\pi$
$787$		−	8.03260i		−	0.286331i	0.989699	−	0.143166i	$-0.0457282\pi$
$788$		0			0
$789$	51.0780			1.81843
$790$		0			0
$791$			48.5504i			1.72625i
$792$		0			0
$793$	−8.77548	+	9.47860i	−0.311627	+	0.336595i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.8322			1.23382			0.616911	−	0.787033i	$-0.288384\pi$
$797$	34.8322			1.23382			0.616911	+	0.787033i	$0.288384\pi$
$798$		0			0
$799$		−	8.01816i		−	0.283662i
$800$		0			0
$801$		−	74.4733i		−	2.63139i
$802$		0			0
$803$	6.46084			0.227998
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−28.3714			−0.998721
$808$		0			0
$809$	22.7477			0.799767			0.399884	−	0.916566i	$-0.369050\pi$
$809$	22.7477			0.799767			0.399884	+	0.916566i	$0.369050\pi$
$810$		0			0
$811$			12.0760i			0.424045i	0.977265	+	0.212023i	$0.0680051\pi$
$811$			12.0760i			0.424045i	−0.977265	+	0.212023i	$0.931995\pi$
$812$		0			0
$813$			69.1311i			2.42453i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	23.5257i		−	0.823060i
$818$		0			0
$819$	−76.2432	−	70.5875i	−2.66415	−	2.46653i
$820$		0			0
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$	−39.7031			−1.38396			−0.691981	−	0.721916i	$-0.743262\pi$
$823$	−39.7031			−1.38396			−0.691981	+	0.721916i	$0.743262\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			32.4720i			1.12916i	0.825377	+	0.564581i	$0.190962\pi$
$827$			32.4720i			1.12916i	−0.825377	+	0.564581i	$0.809038\pi$
$828$		0			0
$829$	−40.2432			−1.39770			−0.698852	−	0.715267i	$-0.746305\pi$
$829$	−40.2432			−1.39770			−0.698852	+	0.715267i	$0.746305\pi$
$830$		0			0
$831$	−87.8258			−3.04664
$832$		0			0
$833$	15.7194			0.544645
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	34.6410i		−	1.19737i
$838$		0			0
$839$			0.762282i			0.0263169i	0.999913	+	0.0131585i	$0.00418859\pi$
$839$			0.762282i			0.0263169i	−0.999913	+	0.0131585i	$0.995811\pi$
$840$		0			0
$841$	28.4955			0.982602
$842$		0			0
$843$		−	38.2958i		−	1.31898i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			20.0616i			0.689325i
$848$		0			0
$849$	−13.5826			−0.466153
$850$		0			0
$851$		−	4.59941i		−	0.157666i
$852$		0			0
$853$		−	39.0188i		−	1.33598i	−0.744171	−	0.667989i	$-0.767155\pi$
$853$		−	39.0188i		−	1.33598i	0.744171	−	0.667989i	$-0.232845\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−44.3605			−1.51533			−0.757663	−	0.652646i	$-0.773659\pi$
$857$	−44.3605			−1.51533			−0.757663	+	0.652646i	$0.773659\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$	106.991			3.64624
$862$		0			0
$863$			6.20520i			0.211228i	0.994407	+	0.105614i	$0.0336807\pi$
$863$			6.20520i			0.211228i	−0.994407	+	0.105614i	$0.966319\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−47.4561			−1.61169
$868$		0			0
$869$			28.2843i			0.959478i
$870$		0			0
$871$	6.74773	+	6.24718i	0.228638	+	0.211678i
$872$		0			0
$873$		−	80.4371i		−	2.72238i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	20.5939i		−	0.695407i	−0.937605	−	0.347703i	$-0.886962\pi$
$877$		−	20.5939i		−	0.695407i	0.937605	−	0.347703i	$-0.113038\pi$
$878$		0			0
$879$			19.2087i			0.647892i
$880$		0			0
$881$	4.41742			0.148827			0.0744134	−	0.997227i	$-0.476292\pi$
$881$	4.41742			0.148827			0.0744134	+	0.997227i	$0.476292\pi$
$882$		0			0
$883$	−31.7367			−1.06802			−0.534012	−	0.845477i	$-0.679316\pi$
$883$	−31.7367			−1.06802			−0.534012	+	0.845477i	$0.679316\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	40.2425			1.35121			0.675605	−	0.737264i	$-0.263883\pi$
$887$	40.2425			1.35121			0.675605	+	0.737264i	$0.263883\pi$
$888$		0			0
$889$			2.23810i			0.0750635i
$890$		0			0
$891$		−	36.9415i		−	1.23759i
$892$		0			0
$893$	−33.2704			−1.11335
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−13.6745	+	14.7701i	−0.456577	+	0.493160i
$898$		0			0
$899$		−	23.6849i		−	0.789934i
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$		−	59.4618i		−	1.97877i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−39.7031			−1.31832			−0.659159	−	0.752003i	$-0.729088\pi$
$907$	−39.7031			−1.31832			−0.659159	+	0.752003i	$0.729088\pi$
$908$		0			0
$909$	−60.3303			−2.00103
$910$		0			0
$911$	17.6697			0.585423			0.292712	−	0.956201i	$-0.405442\pi$
$911$	17.6697			0.585423			0.292712	+	0.956201i	$0.405442\pi$
$912$		0			0
$913$	−1.83155			−0.0606155
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	13.8564i		−	0.457579i
$918$		0			0
$919$	−13.6697			−0.450922			−0.225461	−	0.974252i	$-0.572389\pi$
$919$	−13.6697			−0.450922			−0.225461	+	0.974252i	$0.572389\pi$
$920$		0			0
$921$		−	24.8655i		−	0.819347i
$922$		0			0
$923$	0.780929	+	0.723000i	0.0257046	+	0.0237978i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	61.1305			2.00779
$928$		0			0
$929$		−	43.9510i		−	1.44198i	−0.692943	−	0.720992i	$-0.743686\pi$
$929$		−	43.9510i		−	1.44198i	0.692943	−	0.720992i	$-0.256314\pi$
$930$		0			0
$931$		−	65.2258i		−	2.13769i
$932$		0			0
$933$	19.5959			0.641542
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−7.75301			−0.253280			−0.126640	−	0.991949i	$-0.540419\pi$
$937$	−7.75301			−0.253280			−0.126640	+	0.991949i	$0.540419\pi$
$938$		0			0
$939$	12.0000			0.391605
$940$		0			0
$941$		−	20.8564i		−	0.679900i	−0.940443	−	0.339950i	$-0.889590\pi$
$941$		−	20.8564i		−	0.679900i	0.940443	−	0.339950i	$-0.110410\pi$
$942$		0			0
$943$		−	14.2378i		−	0.463647i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			35.3640i			1.14918i	0.818443	+	0.574588i	$0.194838\pi$
$947$			35.3640i			1.14918i	−0.818443	+	0.574588i	$0.805162\pi$
$948$		0			0
$949$	−6.74773	−	6.24718i	−0.219040	−	0.202792i
$950$		0			0
$951$		−	62.1022i		−	2.01380i
$952$		0			0
$953$	52.1135			1.68812			0.844061	−	0.536247i	$-0.180158\pi$
$953$	52.1135			1.68812			0.844061	+	0.536247i	$0.180158\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	59.4618i		−	1.92213i
$958$		0			0
$959$	−45.4955			−1.46912
$960$		0			0
$961$	21.2432			0.685264
$962$		0			0
$963$	−69.6363			−2.24400
$964$		0			0
$965$		0			0
$966$		0			0
$967$			44.5010i			1.43106i	0.698584	+	0.715528i	$0.253814\pi$
$967$			44.5010i			1.43106i	−0.698584	+	0.715528i	$0.746186\pi$
$968$		0			0
$969$		−	21.4468i		−	0.688969i
$970$		0			0
$971$	48.6606			1.56159			0.780797	−	0.624785i	$-0.214814\pi$
$971$	48.6606			1.56159			0.780797	+	0.624785i	$0.214814\pi$
$972$		0			0
$973$			31.3676i			1.00560i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	58.7388i		−	1.87922i	−0.342245	−	0.939611i	$-0.611187\pi$
$977$		−	58.7388i		−	1.87922i	0.342245	−	0.939611i	$-0.388813\pi$
$978$		0			0
$979$	−28.6606			−0.915997
$980$		0			0
$981$			55.8550i			1.78331i
$982$		0			0
$983$			58.7388i			1.87348i	0.350029	+	0.936739i	$0.386172\pi$
$983$			58.7388i			1.87348i	−0.350029	+	0.936739i	$0.613828\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−84.0917			−2.67667
$988$		0			0
$989$	−7.91288			−0.251615
$990$		0			0
$991$	−25.4955			−0.809890			−0.404945	−	0.914341i	$-0.632709\pi$
$991$	−25.4955			−0.809890			−0.404945	+	0.914341i	$0.632709\pi$
$992$		0			0
$993$			76.0593i			2.41367i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−30.4164			−0.963296			−0.481648	−	0.876365i	$-0.659962\pi$
$997$	−30.4164			−0.963296			−0.481648	+	0.876365i	$0.659962\pi$
$998$		0			0
$999$		−	28.2843i		−	0.894875i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.f.f.701.8		8
5.2	odd	4		260.2.d.a.129.1	✓	8
5.3	odd	4		260.2.d.a.129.8	yes	8
5.4	even	2	inner	1300.2.f.f.701.1		8
13.12	even	2	inner	1300.2.f.f.701.7		8
15.2	even	4		2340.2.j.d.649.5		8
15.8	even	4		2340.2.j.d.649.2		8
20.3	even	4		1040.2.f.e.129.2		8
20.7	even	4		1040.2.f.e.129.7		8
65.8	even	4		3380.2.c.d.2029.7		8
65.12	odd	4		260.2.d.a.129.2	yes	8
65.18	even	4		3380.2.c.d.2029.8		8
65.38	odd	4		260.2.d.a.129.7	yes	8
65.47	even	4		3380.2.c.d.2029.1		8
65.57	even	4		3380.2.c.d.2029.2		8
65.64	even	2	inner	1300.2.f.f.701.2		8
195.38	even	4		2340.2.j.d.649.7		8
195.77	even	4		2340.2.j.d.649.4		8
260.103	even	4		1040.2.f.e.129.1		8
260.207	even	4		1040.2.f.e.129.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.d.a.129.1	✓	8	5.2	odd	4
260.2.d.a.129.2	yes	8	65.12	odd	4
260.2.d.a.129.7	yes	8	65.38	odd	4
260.2.d.a.129.8	yes	8	5.3	odd	4
1040.2.f.e.129.1		8	260.103	even	4
1040.2.f.e.129.2		8	20.3	even	4
1040.2.f.e.129.7		8	20.7	even	4
1040.2.f.e.129.8		8	260.207	even	4
1300.2.f.f.701.1		8	5.4	even	2	inner
1300.2.f.f.701.2		8	65.64	even	2	inner
1300.2.f.f.701.7		8	13.12	even	2	inner
1300.2.f.f.701.8		8	1.1	even	1	trivial
2340.2.j.d.649.2		8	15.8	even	4
2340.2.j.d.649.4		8	195.77	even	4
2340.2.j.d.649.5		8	15.2	even	4
2340.2.j.d.649.7		8	195.38	even	4
3380.2.c.d.2029.1		8	65.47	even	4
3380.2.c.d.2029.2		8	65.57	even	4
3380.2.c.d.2029.7		8	65.8	even	4
3380.2.c.d.2029.8		8	65.18	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.09557 q^{3} +4.37780i q^{7} +6.58258 q^{9} +O(q^{10})\)	\(q+3.09557 q^{3} +4.37780i q^{7} +6.58258 q^{9} -2.53326i q^{11} +(-2.44949 + 2.64575i) q^{13} -1.29217 q^{17} +5.36169i q^{19} +13.5518i q^{21} +1.80341 q^{23} +11.0901 q^{27} +7.58258 q^{29} -3.12359i q^{31} -7.84190i q^{33} -2.55040i q^{37} +(-7.58258 + 8.19012i) q^{39} -7.89495i q^{41} -4.38774 q^{43} +6.20520i q^{47} -12.1652 q^{49} -4.00000 q^{51} +6.19115 q^{53} +16.5975i q^{57} -10.4282i q^{59} +3.58258 q^{61} +28.8172i q^{63} -2.55040i q^{67} +5.58258 q^{69} -0.295164i q^{71} +2.55040i q^{73} +11.0901 q^{77} -11.1652 q^{79} +14.5826 q^{81} -0.723000i q^{83} +23.4724 q^{87} -11.3137i q^{89} +(-11.5826 - 10.7234i) q^{91} -9.66930i q^{93} -12.2197i q^{97} -16.6754i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 24 q^{29} - 24 q^{39} - 24 q^{49} - 32 q^{51} - 8 q^{61} + 8 q^{69} - 16 q^{79} + 80 q^{81} - 56 q^{91}+O(q^{100}) \) 8 * q + 16 * q^9 + 24 * q^29 - 24 * q^39 - 24 * q^49 - 32 * q^51 - 8 * q^61 + 8 * q^69 - 16 * q^79 + 80 * q^81 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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