LMFDB - Embedded newform 2600.2.k.f.2001.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(2001,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2600.2001");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.k (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:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60x^{16} + 1134x^{12} + 6924x^{8} + 3545x^{4} + 64 $ x^20 + 60x^16 + 1134x^12 + 6924x^8 + 3545x^4 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.14
Root			$-1.29975 - 1.29975i$ of defining polynomial
Character	$\chi$	$=$	2600.2001
Dual form			2600.2.k.f.2001.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.949078 q^{3} +3.85660i q^{7} -2.09925 q^{9} +O(q^{10})$	$q+0.949078 q^{3} +3.85660i q^{7} -2.09925 q^{9} +0.589684i q^{11} +(-1.38767 + 3.32782i) q^{13} +3.66022 q^{17} -5.94139i q^{19} +3.66022i q^{21} -3.51775 q^{23} -4.83959 q^{27} -5.33862 q^{29} +8.71673i q^{31} +0.559657i q^{33} +1.85660i q^{37} +(-1.31701 + 3.15836i) q^{39} -4.63292i q^{41} +6.30078 q^{43} -3.85660i q^{47} -7.87339 q^{49} +3.47383 q^{51} -10.7912 q^{53} -5.63884i q^{57} +10.3127i q^{59} -13.4874 q^{61} -8.09598i q^{63} -4.09598i q^{67} -3.33862 q^{69} -5.34145i q^{71} +6.09598i q^{73} -2.27418 q^{77} -11.1128 q^{79} +1.70460 q^{81} -8.77977i q^{83} -5.06677 q^{87} +0.413343i q^{89} +(-12.8341 - 5.35170i) q^{91} +8.27286i q^{93} +8.45713i q^{97} -1.23790i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{9}+O(q^{10}) $ 20 * q + 16 * q^9	$ 20 q + 16 q^{9} - 16 q^{29} + 24 q^{39} - 24 q^{49} - 60 q^{51} - 32 q^{61} + 24 q^{69} - 56 q^{79} + 44 q^{81} + 4 q^{91}+O(q^{100}) $ 20 * q + 16 * q^9 - 16 * q^29 + 24 * q^39 - 24 * q^49 - 60 * q^51 - 32 * q^61 + 24 * q^69 - 56 * q^79 + 44 * q^81 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	0.949078			0.547951			0.273975	−	0.961737i	$-0.411661\pi$
$3$	0.949078			0.547951			0.273975	+	0.961737i	$0.411661\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			3.85660i			1.45766i	0.684695	+	0.728830i	$0.259935\pi$
$7$			3.85660i			1.45766i	−0.684695	+	0.728830i	$0.740065\pi$
$8$		0			0
$9$	−2.09925			−0.699750
$10$		0			0
$11$			0.589684i			0.177797i	0.996041	+	0.0888983i	$0.0283346\pi$
$11$			0.589684i			0.177797i	−0.996041	+	0.0888983i	$0.971665\pi$
$12$		0			0
$13$	−1.38767	+	3.32782i	−0.384871	+	0.922970i
$13$	−1.38767	+	3.32782i	−0.384871	+	0.922970i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.66022			0.887734			0.443867	−	0.896093i	$-0.353606\pi$
$17$	3.66022			0.887734			0.443867	+	0.896093i	$0.353606\pi$
$18$		0			0
$19$		−	5.94139i		−	1.36305i	−0.731796	−	0.681524i	$-0.761317\pi$
$19$		−	5.94139i		−	1.36305i	0.731796	−	0.681524i	$-0.238683\pi$
$20$		0			0
$21$			3.66022i			0.798725i
$22$		0			0
$23$	−3.51775			−0.733502			−0.366751	−	0.930319i	$-0.619530\pi$
$23$	−3.51775			−0.733502			−0.366751	+	0.930319i	$0.619530\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.83959			−0.931379
$28$		0			0
$29$	−5.33862			−0.991357			−0.495679	−	0.868506i	$-0.665081\pi$
$29$	−5.33862			−0.991357			−0.495679	+	0.868506i	$0.665081\pi$
$30$		0			0
$31$			8.71673i			1.56557i	0.622291	+	0.782786i	$0.286202\pi$
$31$			8.71673i			1.56557i	−0.622291	+	0.782786i	$0.713798\pi$
$32$		0			0
$33$			0.559657i			0.0974237i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.85660i			0.305224i	0.988286	+	0.152612i	$0.0487684\pi$
$37$			1.85660i			0.305224i	−0.988286	+	0.152612i	$0.951232\pi$
$38$		0			0
$39$	−1.31701	+	3.15836i	−0.210890	+	0.505742i
$40$		0			0
$41$		−	4.63292i		−	0.723540i	−0.932267	−	0.361770i	$-0.882173\pi$
$41$		−	4.63292i		−	0.723540i	0.932267	−	0.361770i	$-0.117827\pi$
$42$		0			0
$43$	6.30078			0.960860			0.480430	−	0.877033i	$-0.340481\pi$
$43$	6.30078			0.960860			0.480430	+	0.877033i	$0.340481\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.85660i		−	0.562543i	−0.959628	−	0.281272i	$-0.909244\pi$
$47$		−	3.85660i		−	0.562543i	0.959628	−	0.281272i	$-0.0907562\pi$
$48$		0			0
$49$	−7.87339			−1.12477
$50$		0			0
$51$	3.47383			0.486434
$52$		0			0
$53$	−10.7912			−1.48229			−0.741145	−	0.671345i	$-0.765717\pi$
$53$	−10.7912			−1.48229			−0.741145	+	0.671345i	$0.765717\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	5.63884i		−	0.746883i
$58$		0			0
$59$			10.3127i			1.34260i	0.741185	+	0.671300i	$0.234264\pi$
$59$			10.3127i			1.34260i	−0.741185	+	0.671300i	$0.765736\pi$
$60$		0			0
$61$	−13.4874			−1.72688			−0.863439	−	0.504453i	$-0.831694\pi$
$61$	−13.4874			−1.72688			−0.863439	+	0.504453i	$0.831694\pi$
$62$		0			0
$63$		−	8.09598i		−	1.02000i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	4.09598i		−	0.500403i	−0.968194	−	0.250202i	$-0.919503\pi$
$67$		−	4.09598i		−	0.500403i	0.968194	−	0.250202i	$-0.0804969\pi$
$68$		0			0
$69$	−3.33862			−0.401923
$70$		0			0
$71$		−	5.34145i		−	0.633913i	−0.948440	−	0.316957i	$-0.897339\pi$
$71$		−	5.34145i		−	0.633913i	0.948440	−	0.316957i	$-0.102661\pi$
$72$		0			0
$73$			6.09598i			0.713480i	0.934204	+	0.356740i	$0.116112\pi$
$73$			6.09598i			0.713480i	−0.934204	+	0.356740i	$0.883888\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.27418			−0.259167
$78$		0			0
$79$	−11.1128			−1.25028			−0.625142	−	0.780511i	$-0.714959\pi$
$79$	−11.1128			−1.25028			−0.625142	+	0.780511i	$0.714959\pi$
$80$		0			0
$81$	1.70460			0.189400
$82$		0			0
$83$		−	8.77977i		−	0.963705i	−0.876252	−	0.481852i	$-0.839964\pi$
$83$		−	8.77977i		−	0.963705i	0.876252	−	0.481852i	$-0.160036\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−5.06677			−0.543215
$88$		0			0
$89$			0.413343i			0.0438142i	0.999760	+	0.0219071i	$0.00697381\pi$
$89$			0.413343i			0.0438142i	−0.999760	+	0.0219071i	$0.993026\pi$
$90$		0			0
$91$	−12.8341	−	5.35170i	−1.34538	−	0.561011i
$92$		0			0
$93$			8.27286i			0.857856i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			8.45713i			0.858692i	0.903140	+	0.429346i	$0.141256\pi$
$97$			8.45713i			0.858692i	−0.903140	+	0.429346i	$0.858744\pi$
$98$		0			0
$99$		−	1.23790i		−	0.124413i
$100$		0			0
$101$	−11.4762			−1.14192			−0.570962	−	0.820977i	$-0.693430\pi$
$101$	−11.4762			−1.14192			−0.570962	+	0.820977i	$0.693430\pi$
$102$		0			0
$103$	−9.28280			−0.914661			−0.457331	−	0.889297i	$-0.651194\pi$
$103$	−9.28280			−0.914661			−0.457331	+	0.889297i	$0.651194\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.8788			−1.05169			−0.525846	−	0.850580i	$-0.676251\pi$
$107$	−10.8788			−1.05169			−0.525846	+	0.850580i	$0.676251\pi$
$108$		0			0
$109$			13.8138i			1.32313i	0.749890	+	0.661563i	$0.230106\pi$
$109$			13.8138i			1.32313i	−0.749890	+	0.661563i	$0.769894\pi$
$110$		0			0
$111$			1.76206i			0.167248i
$112$		0			0
$113$	9.43809			0.887861			0.443931	−	0.896061i	$-0.353584\pi$
$113$	9.43809			0.887861			0.443931	+	0.896061i	$0.353584\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.91307	−	6.98592i	0.269314	−	0.645849i
$118$		0			0
$119$			14.1160i			1.29401i
$120$		0			0
$121$	10.6523			0.968388
$122$		0			0
$123$		−	4.39700i		−	0.396464i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	20.1619			1.78908			0.894538	−	0.446992i	$-0.147505\pi$
$127$	20.1619			1.78908			0.894538	+	0.446992i	$0.147505\pi$
$128$		0			0
$129$	5.97994			0.526504
$130$		0			0
$131$	6.18960			0.540788			0.270394	−	0.962750i	$-0.412846\pi$
$131$	6.18960			0.540788			0.270394	+	0.962750i	$0.412846\pi$
$132$		0			0
$133$	22.9136			1.98686
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2.50691i			0.214179i	0.994249	+	0.107090i	$0.0341532\pi$
$137$			2.50691i			0.214179i	−0.994249	+	0.107090i	$0.965847\pi$
$138$		0			0
$139$	−9.71556			−0.824063			−0.412032	−	0.911170i	$-0.635181\pi$
$139$	−9.71556			−0.824063			−0.412032	+	0.911170i	$0.635181\pi$
$140$		0			0
$141$		−	3.66022i		−	0.308246i
$142$		0			0
$143$	−1.96236	−	0.818289i	−0.164101	−	0.0684288i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.47247			−0.616319
$148$		0			0
$149$			17.8973i			1.46621i	0.680118	+	0.733103i	$0.261929\pi$
$149$			17.8973i			1.46621i	−0.680118	+	0.733103i	$0.738071\pi$
$150$		0			0
$151$		−	1.40398i		−	0.114254i	−0.998367	−	0.0571272i	$-0.981806\pi$
$151$		−	1.40398i		−	0.114254i	0.998367	−	0.0571272i	$-0.0181940\pi$
$152$		0			0
$153$	−7.68372			−0.621192
$154$		0			0
$155$		0			0
$156$		0			0
$157$	18.6161			1.48573			0.742863	−	0.669443i	$-0.233467\pi$
$157$	18.6161			1.48573			0.742863	+	0.669443i	$0.233467\pi$
$158$		0			0
$159$	−10.2417			−0.812222
$160$		0			0
$161$		−	13.5666i		−	1.06920i
$162$		0			0
$163$			2.74619i			0.215098i	0.994200	+	0.107549i	$0.0343003\pi$
$163$			2.74619i			0.215098i	−0.994200	+	0.107549i	$0.965700\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.3304i		−	1.18630i	−0.805090	−	0.593152i	$-0.797883\pi$
$167$		−	15.3304i		−	1.18630i	0.805090	−	0.593152i	$-0.202117\pi$
$168$		0			0
$169$	−9.14873	−	9.23584i	−0.703748	−	0.710449i
$170$		0			0
$171$			12.4725i			0.953793i
$172$		0			0
$173$	12.5456			0.953825			0.476913	−	0.878951i	$-0.341756\pi$
$173$	12.5456			0.953825			0.476913	+	0.878951i	$0.341756\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			9.78757i			0.735679i
$178$		0			0
$179$	−8.82362			−0.659508			−0.329754	−	0.944067i	$-0.606966\pi$
$179$	−8.82362			−0.659508			−0.329754	+	0.944067i	$0.606966\pi$
$180$		0			0
$181$	7.88929			0.586406			0.293203	−	0.956050i	$-0.405279\pi$
$181$	7.88929			0.586406			0.293203	+	0.956050i	$0.405279\pi$
$182$		0			0
$183$	−12.8006			−0.946244
$184$		0			0
$185$		0			0
$186$		0			0
$187$			2.15837i			0.157836i
$188$		0			0
$189$		−	18.6644i		−	1.35763i
$190$		0			0
$191$	−16.2975			−1.17924			−0.589621	−	0.807680i	$-0.700723\pi$
$191$	−16.2975			−1.17924			−0.589621	+	0.807680i	$0.700723\pi$
$192$		0			0
$193$			18.8476i			1.35668i	0.734749	+	0.678339i	$0.237300\pi$
$193$			18.8476i			1.35668i	−0.734749	+	0.678339i	$0.762700\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			26.1727i			1.86473i	0.361522	+	0.932364i	$0.382257\pi$
$197$			26.1727i			1.86473i	−0.361522	+	0.932364i	$0.617743\pi$
$198$		0			0
$199$	3.02941			0.214749			0.107375	−	0.994219i	$-0.465756\pi$
$199$	3.02941			0.214749			0.107375	+	0.994219i	$0.465756\pi$
$200$		0			0
$201$		−	3.88740i		−	0.274196i
$202$		0			0
$203$		−	20.5890i		−	1.44506i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.38464			0.513268
$208$		0			0
$209$	3.50354			0.242345
$210$		0			0
$211$	−8.82362			−0.607443			−0.303722	−	0.952761i	$-0.598229\pi$
$211$	−8.82362			−0.607443			−0.303722	+	0.952761i	$0.598229\pi$
$212$		0			0
$213$		−	5.06945i		−	0.347353i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−33.6170			−2.28207
$218$		0			0
$219$			5.78556i			0.390952i
$220$		0			0
$221$	−5.07919	+	12.1805i	−0.341663	+	0.819352i
$222$		0			0
$223$			3.27236i			0.219133i	0.993979	+	0.109567i	$0.0349463\pi$
$223$			3.27236i			0.219133i	−0.993979	+	0.109567i	$0.965054\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	2.12956i		−	0.141344i	−0.997500	−	0.0706718i	$-0.977486\pi$
$227$		−	2.12956i		−	0.141344i	0.997500	−	0.0706718i	$-0.0225143\pi$
$228$		0			0
$229$		−	2.62136i		−	0.173225i	−0.996242	−	0.0866123i	$-0.972396\pi$
$229$		−	2.62136i		−	0.173225i	0.996242	−	0.0866123i	$-0.0276041\pi$
$230$		0			0
$231$	−2.15837			−0.142011
$232$		0			0
$233$	10.3197			0.676066			0.338033	−	0.941134i	$-0.390238\pi$
$233$	10.3197			0.676066			0.338033	+	0.941134i	$0.390238\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−10.5469			−0.685094
$238$		0			0
$239$		−	1.76463i		−	0.114145i	−0.998370	−	0.0570723i	$-0.981823\pi$
$239$		−	1.76463i		−	0.114145i	0.998370	−	0.0570723i	$-0.0181766\pi$
$240$		0			0
$241$			22.9136i			1.47599i	0.674804	+	0.737997i	$0.264228\pi$
$241$			22.9136i			1.47599i	−0.674804	+	0.737997i	$0.735772\pi$
$242$		0			0
$243$	16.1366			1.03516
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19.7719	+	8.24470i	1.25805	+	0.524598i
$248$		0			0
$249$		−	8.33269i		−	0.528063i
$250$		0			0
$251$	8.79239			0.554971			0.277486	−	0.960730i	$-0.410499\pi$
$251$	8.79239			0.554971			0.277486	+	0.960730i	$0.410499\pi$
$252$		0			0
$253$		−	2.07436i		−	0.130414i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	24.3907			1.52145			0.760725	−	0.649074i	$-0.224843\pi$
$257$	24.3907			1.52145			0.760725	+	0.649074i	$0.224843\pi$
$258$		0			0
$259$	−7.16019			−0.444912
$260$		0			0
$261$	11.2071			0.693702
$262$		0			0
$263$	−4.14545			−0.255619			−0.127810	−	0.991799i	$-0.540795\pi$
$263$	−4.14545			−0.255619			−0.127810	+	0.991799i	$0.540795\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			0.392295i			0.0240080i
$268$		0			0
$269$	−10.2081			−0.622402			−0.311201	−	0.950344i	$-0.600731\pi$
$269$	−10.2081			−0.622402			−0.311201	+	0.950344i	$0.600731\pi$
$270$		0			0
$271$			3.82991i			0.232650i	0.993211	+	0.116325i	$0.0371115\pi$
$271$			3.82991i			0.232650i	−0.993211	+	0.116325i	$0.962889\pi$
$272$		0			0
$273$	−12.1805	−	5.07919i	−0.737200	−	0.307406i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−21.9080			−1.31632			−0.658162	−	0.752877i	$-0.728666\pi$
$277$	−21.9080			−1.31632			−0.658162	+	0.752877i	$0.728666\pi$
$278$		0			0
$279$		−	18.2986i		−	1.09551i
$280$		0			0
$281$		−	12.0345i		−	0.717920i	−0.933353	−	0.358960i	$-0.883131\pi$
$281$		−	12.0345i		−	0.717920i	0.933353	−	0.358960i	$-0.116869\pi$
$282$		0			0
$283$	−14.1180			−0.839226			−0.419613	−	0.907703i	$-0.637834\pi$
$283$	−14.1180			−0.839226			−0.419613	+	0.907703i	$0.637834\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	17.8673			1.05467
$288$		0			0
$289$	−3.60279			−0.211929
$290$		0			0
$291$			8.02648i			0.470521i
$292$		0			0
$293$			27.3166i			1.59585i	0.602755	+	0.797926i	$0.294070\pi$
$293$			27.3166i			1.59585i	−0.602755	+	0.797926i	$0.705930\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	2.85383i		−	0.165596i
$298$		0			0
$299$	4.88149	−	11.7064i	0.282304	−	0.677001i
$300$		0			0
$301$			24.2996i			1.40061i
$302$		0			0
$303$	−10.8918			−0.625718
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	25.0772i		−	1.43123i	−0.698493	−	0.715616i	$-0.746146\pi$
$307$		−	25.0772i		−	1.43123i	0.698493	−	0.715616i	$-0.253854\pi$
$308$		0			0
$309$	−8.81010			−0.501189
$310$		0			0
$311$	28.3022			1.60487			0.802434	−	0.596741i	$-0.203538\pi$
$311$	28.3022			1.60487			0.802434	+	0.596741i	$0.203538\pi$
$312$		0			0
$313$	24.0428			1.35898			0.679490	−	0.733685i	$-0.262201\pi$
$313$	24.0428			1.35898			0.679490	+	0.733685i	$0.262201\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	32.6514i		−	1.83388i	−0.399022	−	0.916941i	$-0.630650\pi$
$317$		−	32.6514i		−	1.83388i	0.399022	−	0.916941i	$-0.369350\pi$
$318$		0			0
$319$		−	3.14810i		−	0.176260i
$320$		0			0
$321$	−10.3248			−0.576275
$322$		0			0
$323$		−	21.7468i		−	1.21002i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			13.1104i			0.725007i
$328$		0			0
$329$	14.8734			0.819997
$330$		0			0
$331$			20.3679i			1.11952i	0.828654	+	0.559761i	$0.189107\pi$
$331$			20.3679i			1.11952i	−0.828654	+	0.559761i	$0.810893\pi$
$332$		0			0
$333$		−	3.89748i		−	0.213580i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.1805			0.663516			0.331758	−	0.943364i	$-0.392358\pi$
$337$	12.1805			0.663516			0.331758	+	0.943364i	$0.392358\pi$
$338$		0			0
$339$	8.95749			0.486504
$340$		0			0
$341$	−5.14012			−0.278353
$342$		0			0
$343$		−	3.36833i		−	0.181873i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.9841			−1.12649			−0.563243	−	0.826291i	$-0.690447\pi$
$347$	−20.9841			−1.12649			−0.563243	+	0.826291i	$0.690447\pi$
$348$		0			0
$349$			14.8415i			0.794445i	0.917722	+	0.397222i	$0.130026\pi$
$349$			14.8415i			0.794445i	−0.917722	+	0.397222i	$0.869974\pi$
$350$		0			0
$351$	6.71577	−	16.1053i	0.358461	−	0.859635i
$352$		0			0
$353$			3.93760i			0.209577i	0.994495	+	0.104789i	$0.0334166\pi$
$353$			3.93760i			0.209577i	−0.994495	+	0.104789i	$0.966583\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			13.3972i			0.709055i
$358$		0			0
$359$		−	18.2781i		−	0.964681i	−0.875984	−	0.482340i	$-0.839787\pi$
$359$		−	18.2781i		−	0.964681i	0.875984	−	0.482340i	$-0.160213\pi$
$360$		0			0
$361$	−16.3001			−0.857900
$362$		0			0
$363$	10.1098			0.530629
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−25.0296			−1.30654			−0.653268	−	0.757126i	$-0.726603\pi$
$367$	−25.0296			−1.30654			−0.653268	+	0.757126i	$0.726603\pi$
$368$		0			0
$369$			9.72565i			0.506297i
$370$		0			0
$371$		−	41.6175i		−	2.16067i
$372$		0			0
$373$	21.3882			1.10744			0.553719	−	0.832704i	$-0.313208\pi$
$373$	21.3882			1.10744			0.553719	+	0.832704i	$0.313208\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.40826	−	17.7660i	0.381545	−	0.914993i
$378$		0			0
$379$			20.4784i			1.05191i	0.850514	+	0.525953i	$0.176291\pi$
$379$			20.4784i			1.05191i	−0.850514	+	0.525953i	$0.823709\pi$
$380$		0			0
$381$	19.1352			0.980325
$382$		0			0
$383$			12.7276i			0.650352i	0.945653	+	0.325176i	$0.105424\pi$
$383$			12.7276i			0.650352i	−0.945653	+	0.325176i	$0.894576\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13.2269			−0.672362
$388$		0			0
$389$	15.2680			0.774120			0.387060	−	0.922054i	$-0.373491\pi$
$389$	15.2680			0.774120			0.387060	+	0.922054i	$0.373491\pi$
$390$		0			0
$391$	−12.8757			−0.651154
$392$		0			0
$393$	5.87442			0.296325
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	2.96702i		−	0.148910i	−0.997224	−	0.0744552i	$-0.976278\pi$
$397$		−	2.96702i		−	0.148910i	0.997224	−	0.0744552i	$-0.0237218\pi$
$398$		0			0
$399$	21.7468			1.08870
$400$		0			0
$401$			15.8408i			0.791050i	0.918455	+	0.395525i	$0.129437\pi$
$401$			15.8408i			0.791050i	−0.918455	+	0.395525i	$0.870563\pi$
$402$		0			0
$403$	−29.0077	−	12.0960i	−1.44498	−	0.602544i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.09481			−0.0542677
$408$		0			0
$409$		−	19.5460i		−	0.966487i	−0.875486	−	0.483243i	$-0.839459\pi$
$409$		−	19.5460i		−	0.966487i	0.875486	−	0.483243i	$-0.160541\pi$
$410$		0			0
$411$			2.37925i			0.117360i
$412$		0			0
$413$	−39.7720			−1.95705
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9.22083			−0.451546
$418$		0			0
$419$	39.9411			1.95125			0.975625	−	0.219444i	$-0.0704244\pi$
$419$	39.9411			1.95125			0.975625	+	0.219444i	$0.0704244\pi$
$420$		0			0
$421$		−	11.4551i		−	0.558287i	−0.960249	−	0.279144i	$-0.909949\pi$
$421$		−	11.4551i		−	0.558287i	0.960249	−	0.279144i	$-0.0900506\pi$
$422$		0			0
$423$			8.09598i			0.393640i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	52.0154i		−	2.51720i
$428$		0			0
$429$	−1.86243	−	0.776620i	−0.0899192	−	0.0374956i
$430$		0			0
$431$		−	35.4522i		−	1.70767i	−0.520542	−	0.853836i	$-0.674270\pi$
$431$		−	35.4522i		−	1.70767i	0.520542	−	0.853836i	$-0.325730\pi$
$432$		0			0
$433$	−28.9479			−1.39115			−0.695574	−	0.718455i	$-0.744850\pi$
$433$	−28.9479			−1.39115			−0.695574	+	0.718455i	$0.744850\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			20.9003i			0.999799i
$438$		0			0
$439$	33.6681			1.60689			0.803446	−	0.595377i	$-0.202997\pi$
$439$	33.6681			1.60689			0.803446	+	0.595377i	$0.202997\pi$
$440$		0			0
$441$	16.5282			0.787058
$442$		0			0
$443$	−15.1485			−0.719727			−0.359863	−	0.933005i	$-0.617177\pi$
$443$	−15.1485			−0.719727			−0.359863	+	0.933005i	$0.617177\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			16.9860i			0.803408i
$448$		0			0
$449$			10.5311i			0.496996i	0.968632	+	0.248498i	$0.0799369\pi$
$449$			10.5311i			0.496996i	−0.968632	+	0.248498i	$0.920063\pi$
$450$		0			0
$451$	2.73196			0.128643
$452$		0			0
$453$		−	1.33249i		−	0.0626057i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			15.1073i			0.706692i	0.935493	+	0.353346i	$0.114956\pi$
$457$			15.1073i			0.706692i	−0.935493	+	0.353346i	$0.885044\pi$
$458$		0			0
$459$	−17.7140			−0.826817
$460$		0			0
$461$		−	14.9226i		−	0.695016i	−0.937677	−	0.347508i	$-0.887028\pi$
$461$		−	14.9226i		−	0.695016i	0.937677	−	0.347508i	$-0.112972\pi$
$462$		0			0
$463$			0.0959767i			0.00446041i	0.999998	+	0.00223021i	$0.000709898\pi$
$463$			0.0959767i			0.00446041i	−0.999998	+	0.00223021i	$0.999290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.55364			−0.349541			−0.174770	−	0.984609i	$-0.555918\pi$
$467$	−7.55364			−0.349541			−0.174770	+	0.984609i	$0.555918\pi$
$468$		0			0
$469$	15.7966			0.729417
$470$		0			0
$471$	17.6681			0.814105
$472$		0			0
$473$			3.71547i			0.170838i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	22.6535			1.03723
$478$		0			0
$479$		−	2.15464i		−	0.0984479i	−0.998788	−	0.0492240i	$-0.984325\pi$
$479$		−	2.15464i		−	0.0984479i	0.998788	−	0.0492240i	$-0.0156748\pi$
$480$		0			0
$481$	−6.17844	−	2.57636i	−0.281712	−	0.117472i
$482$		0			0
$483$		−	12.8757i		−	0.585867i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			41.6555i			1.88759i	0.330529	+	0.943796i	$0.392773\pi$
$487$			41.6555i			1.88759i	−0.330529	+	0.943796i	$0.607227\pi$
$488$		0			0
$489$			2.60635i			0.117863i
$490$		0			0
$491$	13.2206			0.596638			0.298319	−	0.954466i	$-0.403574\pi$
$491$	13.2206			0.596638			0.298319	+	0.954466i	$0.403574\pi$
$492$		0			0
$493$	−19.5405			−0.880061
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.5999			0.924030
$498$		0			0
$499$		−	21.3534i		−	0.955911i	−0.878384	−	0.477956i	$-0.841378\pi$
$499$		−	21.3534i		−	0.955911i	0.878384	−	0.477956i	$-0.158622\pi$
$500$		0			0
$501$		−	14.5498i		−	0.650037i
$502$		0			0
$503$	39.7750			1.77348			0.886740	−	0.462268i	$-0.152964\pi$
$503$	39.7750			1.77348			0.886740	+	0.462268i	$0.152964\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8.68286	−	8.76554i	−0.385619	−	0.389291i
$508$		0			0
$509$			15.9107i			0.705229i	0.935769	+	0.352615i	$0.114707\pi$
$509$			15.9107i			0.705229i	−0.935769	+	0.352615i	$0.885293\pi$
$510$		0			0
$511$	−23.5098			−1.04001
$512$		0			0
$513$			28.7539i			1.26951i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.27418			0.100018
$518$		0			0
$519$	11.9068			0.522649
$520$		0			0
$521$	13.7669			0.603137			0.301568	−	0.953445i	$-0.402490\pi$
$521$	13.7669			0.603137			0.301568	+	0.953445i	$0.402490\pi$
$522$		0			0
$523$	6.04142			0.264173			0.132086	−	0.991238i	$-0.457832\pi$
$523$	6.04142			0.264173			0.132086	+	0.991238i	$0.457832\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			31.9052i			1.38981i
$528$		0			0
$529$	−10.6254			−0.461975
$530$		0			0
$531$		−	21.6490i		−	0.939485i
$532$		0			0
$533$	15.4175	+	6.42897i	0.667806	+	0.278470i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8.37431			−0.361378
$538$		0			0
$539$		−	4.64282i		−	0.199980i
$540$		0			0
$541$		−	20.4075i		−	0.877388i	−0.898637	−	0.438694i	$-0.855441\pi$
$541$		−	20.4075i		−	0.877388i	0.898637	−	0.438694i	$-0.144559\pi$
$542$		0			0
$543$	7.48756			0.321322
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10.8898			0.465616			0.232808	−	0.972523i	$-0.425209\pi$
$547$	10.8898			0.465616			0.232808	+	0.972523i	$0.425209\pi$
$548$		0			0
$549$	28.3133			1.20838
$550$		0			0
$551$			31.7188i			1.35127i
$552$		0			0
$553$		−	42.8575i		−	1.82249i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			10.2405i			0.433904i	0.976182	+	0.216952i	$0.0696116\pi$
$557$			10.2405i			0.433904i	−0.976182	+	0.216952i	$0.930388\pi$
$558$		0			0
$559$	−8.74343	+	20.9679i	−0.369808	+	0.886846i
$560$		0			0
$561$			2.04847i			0.0864863i
$562$		0			0
$563$	25.5180			1.07545			0.537727	−	0.843119i	$-0.319283\pi$
$563$	25.5180			1.07545			0.537727	+	0.843119i	$0.319283\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			6.57398i			0.276081i
$568$		0			0
$569$	8.25524			0.346078			0.173039	−	0.984915i	$-0.444641\pi$
$569$	8.25524			0.346078			0.173039	+	0.984915i	$0.444641\pi$
$570$		0			0
$571$	−20.2662			−0.848115			−0.424058	−	0.905635i	$-0.639395\pi$
$571$	−20.2662			−0.848115			−0.424058	+	0.905635i	$0.639395\pi$
$572$		0			0
$573$	−15.4676			−0.646167
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	4.26745i		−	0.177656i	−0.996047	−	0.0888280i	$-0.971688\pi$
$577$		−	4.26745i		−	0.177656i	0.996047	−	0.0888280i	$-0.0283122\pi$
$578$		0			0
$579$			17.8878i			0.743393i
$580$		0			0
$581$	33.8601			1.40475
$582$		0			0
$583$		−	6.36342i		−	0.263546i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.84276i		−	0.0760590i	−0.999277	−	0.0380295i	$-0.987892\pi$
$587$		−	1.84276i		−	0.0760590i	0.999277	−	0.0380295i	$-0.0121081\pi$
$588$		0			0
$589$	51.7895			2.13395
$590$		0			0
$591$			24.8399i			1.02178i
$592$		0			0
$593$			36.3238i			1.49164i	0.666148	+	0.745819i	$0.267942\pi$
$593$			36.3238i			1.49164i	−0.666148	+	0.745819i	$0.732058\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.87515			0.117672
$598$		0			0
$599$	17.2727			0.705745			0.352873	−	0.935671i	$-0.385205\pi$
$599$	17.2727			0.705745			0.352873	+	0.935671i	$0.385205\pi$
$600$		0			0
$601$	−32.1781			−1.31257			−0.656286	−	0.754512i	$-0.727874\pi$
$601$	−32.1781			−1.31257			−0.656286	+	0.754512i	$0.727874\pi$
$602$		0			0
$603$			8.59848i			0.350157i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.88535			0.279468			0.139734	−	0.990189i	$-0.455375\pi$
$607$	6.88535			0.279468			0.139734	+	0.990189i	$0.455375\pi$
$608$		0			0
$609$		−	19.5405i		−	0.791822i
$610$		0			0
$611$	12.8341	+	5.35170i	0.519211	+	0.216507i
$612$		0			0
$613$			23.1565i			0.935283i	0.883918	+	0.467642i	$0.154896\pi$
$613$			23.1565i			0.935283i	−0.883918	+	0.467642i	$0.845104\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			34.3256i			1.38190i	0.722904	+	0.690949i	$0.242807\pi$
$617$			34.3256i			1.38190i	−0.722904	+	0.690949i	$0.757193\pi$
$618$		0			0
$619$		−	35.0066i		−	1.40704i	−0.710678	−	0.703518i	$-0.751612\pi$
$619$		−	35.0066i		−	1.40704i	0.710678	−	0.703518i	$-0.248388\pi$
$620$		0			0
$621$	17.0245			0.683169
$622$		0			0
$623$	−1.59410			−0.0638662
$624$		0			0
$625$		0			0
$626$		0			0
$627$	3.32514			0.132793
$628$		0			0
$629$			6.79558i			0.270957i
$630$		0			0
$631$		−	4.95144i		−	0.197114i	−0.995131	−	0.0985570i	$-0.968577\pi$
$631$		−	4.95144i		−	0.197114i	0.995131	−	0.0985570i	$-0.0314227\pi$
$632$		0			0
$633$	−8.37431			−0.332849
$634$		0			0
$635$		0			0
$636$		0			0
$637$	10.9257	−	26.2012i	0.432892	−	1.03813i
$638$		0			0
$639$			11.2130i			0.443581i
$640$		0			0
$641$	−1.02941			−0.0406594			−0.0203297	−	0.999793i	$-0.506472\pi$
$641$	−1.02941			−0.0406594			−0.0203297	+	0.999793i	$0.506472\pi$
$642$		0			0
$643$			8.30102i			0.327360i	0.986513	+	0.163680i	$0.0523365\pi$
$643$			8.30102i			0.327360i	−0.986513	+	0.163680i	$0.947663\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.13808			−0.241313			−0.120656	−	0.992694i	$-0.538500\pi$
$647$	−6.13808			−0.241313			−0.120656	+	0.992694i	$0.538500\pi$
$648$		0			0
$649$	−6.08125			−0.238710
$650$		0			0
$651$	−31.9052			−1.25046
$652$		0			0
$653$	−3.78421			−0.148088			−0.0740438	−	0.997255i	$-0.523590\pi$
$653$	−3.78421			−0.148088			−0.0740438	+	0.997255i	$0.523590\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	12.7970i		−	0.499258i
$658$		0			0
$659$	25.6634			0.999705			0.499853	−	0.866110i	$-0.333387\pi$
$659$	25.6634			0.999705			0.499853	+	0.866110i	$0.333387\pi$
$660$		0			0
$661$			10.8080i			0.420384i	0.977660	+	0.210192i	$0.0674089\pi$
$661$			10.8080i			0.420384i	−0.977660	+	0.210192i	$0.932591\pi$
$662$		0			0
$663$	−4.82055	+	11.5603i	−0.187215	+	0.448964i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	18.7800			0.727163
$668$		0			0
$669$			3.10572i			0.120074i
$670$		0			0
$671$		−	7.95328i		−	0.307033i
$672$		0			0
$673$	15.8803			0.612141			0.306071	−	0.952009i	$-0.400986\pi$
$673$	15.8803			0.612141			0.306071	+	0.952009i	$0.400986\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.340888			−0.0131014			−0.00655069	−	0.999979i	$-0.502085\pi$
$677$	−0.340888			−0.0131014			−0.00655069	+	0.999979i	$0.502085\pi$
$678$		0			0
$679$	−32.6158			−1.25168
$680$		0			0
$681$		−	2.02112i		−	0.0774493i
$682$		0			0
$683$		−	17.9483i		−	0.686771i	−0.939194	−	0.343386i	$-0.888426\pi$
$683$		−	17.9483i		−	0.686771i	0.939194	−	0.343386i	$-0.111574\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	2.48788i		−	0.0949185i
$688$		0			0
$689$	14.9747	−	35.9112i	0.570491	−	1.36811i
$690$		0			0
$691$			2.24959i			0.0855783i	0.999084	+	0.0427891i	$0.0136244\pi$
$691$			2.24959i			0.0855783i	−0.999084	+	0.0427891i	$0.986376\pi$
$692$		0			0
$693$	4.77407			0.181352
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	16.9575i		−	0.642311i
$698$		0			0
$699$	9.79421			0.370451
$700$		0			0
$701$	−6.89790			−0.260530			−0.130265	−	0.991479i	$-0.541583\pi$
$701$	−6.89790			−0.260530			−0.130265	+	0.991479i	$0.541583\pi$
$702$		0			0
$703$	11.0308			0.416035
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	44.2591i		−	1.66454i
$708$		0			0
$709$		−	35.4512i		−	1.33140i	−0.746220	−	0.665699i	$-0.768133\pi$
$709$		−	35.4512i		−	1.33140i	0.746220	−	0.665699i	$-0.231867\pi$
$710$		0			0
$711$	23.3285			0.874886
$712$		0			0
$713$		−	30.6633i		−	1.14835i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	1.67478i		−	0.0625457i
$718$		0			0
$719$	−24.2319			−0.903698			−0.451849	−	0.892095i	$-0.649235\pi$
$719$	−24.2319			−0.903698			−0.451849	+	0.892095i	$0.649235\pi$
$720$		0			0
$721$		−	35.8001i		−	1.33326i
$722$		0			0
$723$			21.7468i			0.808772i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−13.4972			−0.500585			−0.250292	−	0.968170i	$-0.580527\pi$
$727$	−13.4972			−0.500585			−0.250292	+	0.968170i	$0.580527\pi$
$728$		0			0
$729$	10.2011			0.377817
$730$		0			0
$731$	23.0622			0.852988
$732$		0			0
$733$		−	38.3706i		−	1.41725i	−0.705585	−	0.708625i	$-0.749316\pi$
$733$		−	38.3706i		−	1.41725i	0.705585	−	0.708625i	$-0.250684\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.41533			0.0889700
$738$		0			0
$739$			9.36168i			0.344375i	0.985064	+	0.172187i	$0.0550835\pi$
$739$			9.36168i			0.344375i	−0.985064	+	0.172187i	$0.944917\pi$
$740$		0			0
$741$	18.7650	+	7.82487i	0.689351	+	0.287454i
$742$		0			0
$743$			11.4643i			0.420585i	0.977639	+	0.210292i	$0.0674416\pi$
$743$			11.4643i			0.420585i	−0.977639	+	0.210292i	$0.932558\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			18.4309i			0.674353i
$748$		0			0
$749$		−	41.9551i		−	1.53301i
$750$		0			0
$751$	6.56686			0.239628			0.119814	−	0.992796i	$-0.461770\pi$
$751$	6.56686			0.239628			0.119814	+	0.992796i	$0.461770\pi$
$752$		0			0
$753$	8.34467			0.304097
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34.6035			−1.25768			−0.628842	−	0.777533i	$-0.716471\pi$
$757$	−34.6035			−1.25768			−0.628842	+	0.777533i	$0.716471\pi$
$758$		0			0
$759$		−	1.96873i		−	0.0714605i
$760$		0			0
$761$			40.4330i			1.46570i	0.680393	+	0.732848i	$0.261809\pi$
$761$			40.4330i			1.46570i	−0.680393	+	0.732848i	$0.738191\pi$
$762$		0			0
$763$	−53.2745			−1.92867
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−34.3188	−	14.3107i	−1.23918	−	0.516728i
$768$		0			0
$769$		−	9.52918i		−	0.343631i	−0.985129	−	0.171816i	$-0.945037\pi$
$769$		−	9.52918i		−	0.343631i	0.985129	−	0.171816i	$-0.0549633\pi$
$770$		0			0
$771$	23.1487			0.833680
$772$		0			0
$773$		−	4.70473i		−	0.169217i	−0.996414	−	0.0846087i	$-0.973036\pi$
$773$		−	4.70473i		−	0.169217i	0.996414	−	0.0846087i	$-0.0269640\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.79558			−0.243790
$778$		0			0
$779$	−27.5260			−0.986220
$780$		0			0
$781$	3.14977			0.112708
$782$		0			0
$783$	25.8367			0.923330
$784$		0			0
$785$		0			0
$786$		0			0
$787$			24.8675i			0.886430i	0.896415	+	0.443215i	$0.146162\pi$
$787$			24.8675i			0.886430i	−0.896415	+	0.443215i	$0.853838\pi$
$788$		0			0
$789$	−3.93436			−0.140067
$790$		0			0
$791$			36.3990i			1.29420i
$792$		0			0
$793$	18.7160	−	44.8834i	0.664626	−	1.59386i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.3616			0.685824			0.342912	−	0.939368i	$-0.388587\pi$
$797$	19.3616			0.685824			0.342912	+	0.939368i	$0.388587\pi$
$798$		0			0
$799$		−	14.1160i		−	0.499389i
$800$		0			0
$801$		−	0.867710i		−	0.0306590i
$802$		0			0
$803$	−3.59470			−0.126854
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−9.68833			−0.341045
$808$		0			0
$809$	3.50590			0.123261			0.0616304	−	0.998099i	$-0.480370\pi$
$809$	3.50590			0.123261			0.0616304	+	0.998099i	$0.480370\pi$
$810$		0			0
$811$			7.32489i			0.257212i	0.991696	+	0.128606i	$0.0410502\pi$
$811$			7.32489i			0.257212i	−0.991696	+	0.128606i	$0.958950\pi$
$812$		0			0
$813$			3.63488i			0.127481i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	37.4354i		−	1.30970i
$818$		0			0
$819$	26.9419	+	11.2346i	0.941427	+	0.392568i
$820$		0			0
$821$			30.8633i			1.07714i	0.842582	+	0.538569i	$0.181035\pi$
$821$			30.8633i			1.07714i	−0.842582	+	0.538569i	$0.818965\pi$
$822$		0			0
$823$	49.4309			1.72305			0.861526	−	0.507714i	$-0.169509\pi$
$823$	49.4309			1.72305			0.861526	+	0.507714i	$0.169509\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	31.1108i		−	1.08183i	−0.841078	−	0.540914i	$-0.818078\pi$
$827$		−	31.1108i		−	1.08183i	0.841078	−	0.540914i	$-0.181922\pi$
$828$		0			0
$829$	−0.440726			−0.0153070			−0.00765352	−	0.999971i	$-0.502436\pi$
$829$	−0.440726			−0.0153070			−0.00765352	+	0.999971i	$0.502436\pi$
$830$		0			0
$831$	−20.7924			−0.721280
$832$		0			0
$833$	−28.8183			−0.998497
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	42.1854i		−	1.45814i
$838$		0			0
$839$		−	21.0334i		−	0.726153i	−0.931759	−	0.363076i	$-0.881726\pi$
$839$		−	21.0334i		−	0.726153i	0.931759	−	0.363076i	$-0.118274\pi$
$840$		0			0
$841$	−0.499104			−0.0172105
$842$		0			0
$843$		−	11.4217i		−	0.393385i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			41.0816i			1.41158i
$848$		0			0
$849$	−13.3991			−0.459855
$850$		0			0
$851$		−	6.53107i		−	0.223882i
$852$		0			0
$853$		−	31.6034i		−	1.08208i	−0.840997	−	0.541040i	$-0.818031\pi$
$853$		−	31.6034i		−	1.08208i	0.840997	−	0.541040i	$-0.181969\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−29.1751			−0.996602			−0.498301	−	0.867004i	$-0.666043\pi$
$857$	−29.1751			−0.996602			−0.498301	+	0.867004i	$0.666043\pi$
$858$		0			0
$859$	−34.7221			−1.18470			−0.592351	−	0.805680i	$-0.701800\pi$
$859$	−34.7221			−1.18470			−0.592351	+	0.805680i	$0.701800\pi$
$860$		0			0
$861$	16.9575			0.577910
$862$		0			0
$863$			23.2447i			0.791258i	0.918410	+	0.395629i	$0.129473\pi$
$863$			23.2447i			0.791258i	−0.918410	+	0.395629i	$0.870527\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−3.41933			−0.116127
$868$		0			0
$869$		−	6.55302i		−	0.222296i
$870$		0			0
$871$	13.6307	+	5.68388i	0.461857	+	0.192591i
$872$		0			0
$873$		−	17.7536i		−	0.600870i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			14.3892i			0.485890i	0.970040	+	0.242945i	$0.0781134\pi$
$877$			14.3892i			0.485890i	−0.970040	+	0.242945i	$0.921887\pi$
$878$		0			0
$879$			25.9256i			0.874449i
$880$		0			0
$881$	17.9750			0.605593			0.302797	−	0.953055i	$-0.402080\pi$
$881$	17.9750			0.605593			0.302797	+	0.953055i	$0.402080\pi$
$882$		0			0
$883$	−31.2125			−1.05038			−0.525191	−	0.850984i	$-0.676006\pi$
$883$	−31.2125			−1.05038			−0.525191	+	0.850984i	$0.676006\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	53.2656			1.78848			0.894242	−	0.447585i	$-0.147716\pi$
$887$	53.2656			1.78848			0.894242	+	0.447585i	$0.147716\pi$
$888$		0			0
$889$			77.7563i			2.60786i
$890$		0			0
$891$			1.00518i			0.0336747i
$892$		0			0
$893$	−22.9136			−0.766774
$894$		0			0
$895$		0			0
$896$		0			0
$897$	4.63292	−	11.1103i	0.154689	−	0.370963i
$898$		0			0
$899$		−	46.5354i		−	1.55204i
$900$		0			0
$901$	−39.4983			−1.31588
$902$		0			0
$903$			23.0622i			0.767464i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	41.5858			1.38084			0.690418	−	0.723411i	$-0.257427\pi$
$907$	41.5858			1.38084			0.690418	+	0.723411i	$0.257427\pi$
$908$		0			0
$909$	24.0914			0.799061
$910$		0			0
$911$	34.6225			1.14710			0.573548	−	0.819172i	$-0.305567\pi$
$911$	34.6225			1.14710			0.573548	+	0.819172i	$0.305567\pi$
$912$		0			0
$913$	5.17729			0.171343
$914$		0			0
$915$		0			0
$916$		0			0
$917$			23.8708i			0.788284i
$918$		0			0
$919$	47.3494			1.56191			0.780956	−	0.624586i	$-0.214732\pi$
$919$	47.3494			1.56191			0.780956	+	0.624586i	$0.214732\pi$
$920$		0			0
$921$		−	23.8003i		−	0.784245i
$922$		0			0
$923$	17.7754	+	7.41218i	0.585083	+	0.243975i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	19.4869			0.640034
$928$		0			0
$929$			9.83607i			0.322711i	0.986896	+	0.161355i	$0.0515866\pi$
$929$			9.83607i			0.322711i	−0.986896	+	0.161355i	$0.948413\pi$
$930$		0			0
$931$			46.7789i			1.53312i
$932$		0			0
$933$	26.8610			0.879389
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.96076			0.0640554			0.0320277	−	0.999487i	$-0.489804\pi$
$937$	1.96076			0.0640554			0.0320277	+	0.999487i	$0.489804\pi$
$938$		0			0
$939$	22.8185			0.744654
$940$		0			0
$941$		−	21.2254i		−	0.691927i	−0.938248	−	0.345964i	$-0.887552\pi$
$941$		−	21.2254i		−	0.691927i	0.938248	−	0.345964i	$-0.112448\pi$
$942$		0			0
$943$			16.2975i			0.530718i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	53.5500i		−	1.74014i	−0.492927	−	0.870071i	$-0.664073\pi$
$947$		−	53.5500i		−	1.74014i	0.492927	−	0.870071i	$-0.335927\pi$
$948$		0			0
$949$	−20.2863	−	8.45922i	−0.658521	−	0.274598i
$950$		0			0
$951$		−	30.9887i		−	1.00488i
$952$		0			0
$953$	−1.13504			−0.0367676			−0.0183838	−	0.999831i	$-0.505852\pi$
$953$	−1.13504			−0.0367676			−0.0183838	+	0.999831i	$0.505852\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	2.98780i		−	0.0965817i
$958$		0			0
$959$	−9.66814			−0.312201
$960$		0			0
$961$	−44.9815			−1.45102
$962$		0			0
$963$	22.8373			0.735921
$964$		0			0
$965$		0			0
$966$		0			0
$967$			21.3213i			0.685647i	0.939400	+	0.342823i	$0.111383\pi$
$967$			21.3213i			0.685647i	−0.939400	+	0.342823i	$0.888617\pi$
$968$		0			0
$969$		−	20.6394i		−	0.663033i
$970$		0			0
$971$	37.2206			1.19447			0.597233	−	0.802068i	$-0.296267\pi$
$971$	37.2206			1.19447			0.597233	+	0.802068i	$0.296267\pi$
$972$		0			0
$973$		−	37.4691i		−	1.20120i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			22.7798i			0.728789i	0.931245	+	0.364395i	$0.118724\pi$
$977$			22.7798i			0.728789i	−0.931245	+	0.364395i	$0.881276\pi$
$978$		0			0
$979$	−0.243742			−0.00779002
$980$		0			0
$981$		−	28.9987i		−	0.925857i
$982$		0			0
$983$		−	2.48902i		−	0.0793873i	−0.999212	−	0.0396937i	$-0.987362\pi$
$983$		−	2.48902i		−	0.0793873i	0.999212	−	0.0396937i	$-0.0126382\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	14.1160			0.449318
$988$		0			0
$989$	−22.1646			−0.704793
$990$		0			0
$991$	−35.9068			−1.14062			−0.570308	−	0.821431i	$-0.693176\pi$
$991$	−35.9068			−1.14062			−0.570308	+	0.821431i	$0.693176\pi$
$992$		0			0
$993$			19.3307i			0.613443i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−15.0609			−0.476983			−0.238491	−	0.971145i	$-0.576653\pi$
$997$	−15.0609			−0.476983			−0.238491	+	0.971145i	$0.576653\pi$
$998$		0			0
$999$		−	8.98520i		−	0.284279i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.k.f.2001.14		20
5.2	odd	4		520.2.f.a.129.4	✓	10
5.3	odd	4		520.2.f.b.129.7	yes	10
5.4	even	2	inner	2600.2.k.f.2001.7		20
13.12	even	2	inner	2600.2.k.f.2001.13		20
20.3	even	4		1040.2.f.g.129.4		10
20.7	even	4		1040.2.f.f.129.7		10
65.12	odd	4		520.2.f.b.129.4	yes	10
65.38	odd	4		520.2.f.a.129.7	yes	10
65.64	even	2	inner	2600.2.k.f.2001.8		20
260.103	even	4		1040.2.f.f.129.4		10
260.207	even	4		1040.2.f.g.129.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.4	✓	10	5.2	odd	4
520.2.f.a.129.7	yes	10	65.38	odd	4
520.2.f.b.129.4	yes	10	65.12	odd	4
520.2.f.b.129.7	yes	10	5.3	odd	4
1040.2.f.f.129.4		10	260.103	even	4
1040.2.f.f.129.7		10	20.7	even	4
1040.2.f.g.129.4		10	20.3	even	4
1040.2.f.g.129.7		10	260.207	even	4
2600.2.k.f.2001.7		20	5.4	even	2	inner
2600.2.k.f.2001.8		20	65.64	even	2	inner
2600.2.k.f.2001.13		20	13.12	even	2	inner
2600.2.k.f.2001.14		20	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:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.949078 q^{3} +3.85660i q^{7} -2.09925 q^{9} +O(q^{10})\)	\(q+0.949078 q^{3} +3.85660i q^{7} -2.09925 q^{9} +0.589684i q^{11} +(-1.38767 + 3.32782i) q^{13} +3.66022 q^{17} -5.94139i q^{19} +3.66022i q^{21} -3.51775 q^{23} -4.83959 q^{27} -5.33862 q^{29} +8.71673i q^{31} +0.559657i q^{33} +1.85660i q^{37} +(-1.31701 + 3.15836i) q^{39} -4.63292i q^{41} +6.30078 q^{43} -3.85660i q^{47} -7.87339 q^{49} +3.47383 q^{51} -10.7912 q^{53} -5.63884i q^{57} +10.3127i q^{59} -13.4874 q^{61} -8.09598i q^{63} -4.09598i q^{67} -3.33862 q^{69} -5.34145i q^{71} +6.09598i q^{73} -2.27418 q^{77} -11.1128 q^{79} +1.70460 q^{81} -8.77977i q^{83} -5.06677 q^{87} +0.413343i q^{89} +(-12.8341 - 5.35170i) q^{91} +8.27286i q^{93} +8.45713i q^{97} -1.23790i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{9}+O(q^{10}) \) 20 * q + 16 * q^9	\( 20 q + 16 q^{9} - 16 q^{29} + 24 q^{39} - 24 q^{49} - 60 q^{51} - 32 q^{61} + 24 q^{69} - 56 q^{79} + 44 q^{81} + 4 q^{91}+O(q^{100}) \) 20 * q + 16 * q^9 - 16 * q^29 + 24 * q^39 - 24 * q^49 - 60 * q^51 - 32 * q^61 + 24 * q^69 - 56 * q^79 + 44 * q^81 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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