LMFDB - Embedded newform 6030.2.d.j.2411.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (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:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 519x^{12} + 3876x^{10} + 16111x^{8} + 36772x^{6} + 41293x^{4} + 16036x^{2} + 4 $ x^16 + 36x^14 + 519x^12 + 3876x^10 + 16111x^8 + 36772x^6 + 41293x^4 + 16036*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{67}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.11
Root			$-0.870493i$ of defining polynomial
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.j.2411.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.74812i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.74812i q^{7} +1.00000 q^{8} -1.00000 q^{10} -2.05953 q^{11} -2.27608i q^{13} +1.74812i q^{14} +1.00000 q^{16} -0.953894i q^{17} -2.84583 q^{19} -1.00000 q^{20} -2.05953 q^{22} -3.48078i q^{23} +1.00000 q^{25} -2.27608i q^{26} +1.74812i q^{28} +8.64013i q^{29} -3.23849i q^{31} +1.00000 q^{32} -0.953894i q^{34} -1.74812i q^{35} +6.47949 q^{37} -2.84583 q^{38} -1.00000 q^{40} +9.42263 q^{41} +8.13413i q^{43} -2.05953 q^{44} -3.48078i q^{46} +10.0128i q^{47} +3.94407 q^{49} +1.00000 q^{50} -2.27608i q^{52} -11.0991 q^{53} +2.05953 q^{55} +1.74812i q^{56} +8.64013i q^{58} +2.95009i q^{59} +5.06848i q^{61} -3.23849i q^{62} +1.00000 q^{64} +2.27608i q^{65} +(6.76390 - 4.60973i) q^{67} -0.953894i q^{68} -1.74812i q^{70} +10.3241i q^{71} -3.34901 q^{73} +6.47949 q^{74} -2.84583 q^{76} -3.60032i q^{77} +4.32175i q^{79} -1.00000 q^{80} +9.42263 q^{82} +12.4242i q^{83} +0.953894i q^{85} +8.13413i q^{86} -2.05953 q^{88} -6.33557i q^{89} +3.97886 q^{91} -3.48078i q^{92} +10.0128i q^{94} +2.84583 q^{95} +4.93178i q^{97} +3.94407 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.74812i			0.660728i	0.943854	+	0.330364i	$0.107172\pi$
$7$			1.74812i			0.660728i	−0.943854	+	0.330364i	$0.892828\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−2.05953			−0.620973			−0.310486	−	0.950578i	$-0.600492\pi$
$11$	−2.05953			−0.620973			−0.310486	+	0.950578i	$0.600492\pi$
$12$		0			0
$13$		−	2.27608i		−	0.631270i	−0.948881	−	0.315635i	$-0.897782\pi$
$13$		−	2.27608i		−	0.631270i	0.948881	−	0.315635i	$-0.102218\pi$
$14$			1.74812i			0.467206i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	0.953894i		−	0.231353i	−0.993287	−	0.115677i	$-0.963096\pi$
$17$		−	0.953894i		−	0.231353i	0.993287	−	0.115677i	$-0.0369036\pi$
$18$		0			0
$19$	−2.84583			−0.652879			−0.326440	−	0.945218i	$-0.605849\pi$
$19$	−2.84583			−0.652879			−0.326440	+	0.945218i	$0.605849\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	−2.05953			−0.439094
$23$		−	3.48078i		−	0.725793i	−0.931829	−	0.362897i	$-0.881788\pi$
$23$		−	3.48078i		−	0.725793i	0.931829	−	0.362897i	$-0.118212\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	2.27608i		−	0.446376i
$27$		0			0
$28$			1.74812i			0.330364i
$29$			8.64013i			1.60443i	0.597034	+	0.802216i	$0.296346\pi$
$29$			8.64013i			1.60443i	−0.597034	+	0.802216i	$0.703654\pi$
$30$		0			0
$31$		−	3.23849i		−	0.581649i	−0.956776	−	0.290825i	$-0.906070\pi$
$31$		−	3.23849i		−	0.581649i	0.956776	−	0.290825i	$-0.0939296\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	0.953894i		−	0.163591i
$35$		−	1.74812i		−	0.295487i
$36$		0			0
$37$	6.47949			1.06522			0.532611	−	0.846360i	$-0.321211\pi$
$37$	6.47949			1.06522			0.532611	+	0.846360i	$0.321211\pi$
$38$	−2.84583			−0.461655
$39$		0			0
$40$	−1.00000			−0.158114
$41$	9.42263			1.47157			0.735784	−	0.677216i	$-0.236814\pi$
$41$	9.42263			1.47157			0.735784	+	0.677216i	$0.236814\pi$
$42$		0			0
$43$			8.13413i			1.24044i	0.784427	+	0.620222i	$0.212957\pi$
$43$			8.13413i			1.24044i	−0.784427	+	0.620222i	$0.787043\pi$
$44$	−2.05953			−0.310486
$45$		0			0
$46$		−	3.48078i		−	0.513213i
$47$			10.0128i			1.46052i	0.683167	+	0.730262i	$0.260602\pi$
$47$			10.0128i			1.46052i	−0.683167	+	0.730262i	$0.739398\pi$
$48$		0			0
$49$	3.94407			0.563438
$50$	1.00000			0.141421
$51$		0			0
$52$		−	2.27608i		−	0.315635i
$53$	−11.0991			−1.52457			−0.762286	−	0.647240i	$-0.775923\pi$
$53$	−11.0991			−1.52457			−0.762286	+	0.647240i	$0.775923\pi$
$54$		0			0
$55$	2.05953			0.277708
$56$			1.74812i			0.233603i
$57$		0			0
$58$			8.64013i			1.13450i
$59$			2.95009i			0.384069i	0.981388	+	0.192035i	$0.0615086\pi$
$59$			2.95009i			0.384069i	−0.981388	+	0.192035i	$0.938491\pi$
$60$		0			0
$61$			5.06848i			0.648953i	0.945894	+	0.324476i	$0.105188\pi$
$61$			5.06848i			0.648953i	−0.945894	+	0.324476i	$0.894812\pi$
$62$		−	3.23849i		−	0.411288i
$63$		0			0
$64$	1.00000			0.125000
$65$			2.27608i			0.282313i
$66$		0			0
$67$	6.76390	−	4.60973i	0.826342	−	0.563168i
$67$	6.76390	−	4.60973i	0.826342	−	0.563168i
$68$		−	0.953894i		−	0.115677i
$69$		0			0
$70$		−	1.74812i		−	0.208941i
$71$			10.3241i			1.22524i	0.790376	+	0.612622i	$0.209885\pi$
$71$			10.3241i			1.22524i	−0.790376	+	0.612622i	$0.790115\pi$
$72$		0			0
$73$	−3.34901			−0.391972			−0.195986	−	0.980607i	$-0.562791\pi$
$73$	−3.34901			−0.391972			−0.195986	+	0.980607i	$0.562791\pi$
$74$	6.47949			0.753225
$75$		0			0
$76$	−2.84583			−0.326440
$77$		−	3.60032i		−	0.410295i
$78$		0			0
$79$			4.32175i			0.486235i	0.969997	+	0.243118i	$0.0781701\pi$
$79$			4.32175i			0.486235i	−0.969997	+	0.243118i	$0.921830\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	9.42263			1.04056
$83$			12.4242i			1.36373i	0.731477	+	0.681866i	$0.238831\pi$
$83$			12.4242i			1.36373i	−0.731477	+	0.681866i	$0.761169\pi$
$84$		0			0
$85$			0.953894i			0.103464i
$86$			8.13413i			0.877126i
$87$		0			0
$88$	−2.05953			−0.219547
$89$		−	6.33557i		−	0.671569i	−0.941939	−	0.335785i	$-0.890999\pi$
$89$		−	6.33557i		−	0.671569i	0.941939	−	0.335785i	$-0.109001\pi$
$90$		0			0
$91$	3.97886			0.417098
$92$		−	3.48078i		−	0.362897i
$93$		0			0
$94$			10.0128i			1.03275i
$95$	2.84583			0.291976
$96$		0			0
$97$			4.93178i			0.500747i	0.968149	+	0.250373i	$0.0805534\pi$
$97$			4.93178i			0.500747i	−0.968149	+	0.250373i	$0.919447\pi$
$98$	3.94407			0.398411
$99$		0			0
$100$	1.00000			0.100000
$101$	−9.03182			−0.898699			−0.449350	−	0.893356i	$-0.648344\pi$
$101$	−9.03182			−0.898699			−0.449350	+	0.893356i	$0.648344\pi$
$102$		0			0
$103$	2.72439			0.268442			0.134221	−	0.990951i	$-0.457147\pi$
$103$	2.72439			0.268442			0.134221	+	0.990951i	$0.457147\pi$
$104$		−	2.27608i		−	0.223188i
$105$		0			0
$106$	−11.0991			−1.07804
$107$			19.4998i			1.88512i	0.334041	+	0.942559i	$0.391588\pi$
$107$			19.4998i			1.88512i	−0.334041	+	0.942559i	$0.608412\pi$
$108$		0			0
$109$			6.86532i			0.657579i	0.944403	+	0.328789i	$0.106641\pi$
$109$			6.86532i			0.657579i	−0.944403	+	0.328789i	$0.893359\pi$
$110$	2.05953			0.196369
$111$		0			0
$112$			1.74812i			0.165182i
$113$	−5.22560			−0.491583			−0.245791	−	0.969323i	$-0.579048\pi$
$113$	−5.22560			−0.491583			−0.245791	+	0.969323i	$0.579048\pi$
$114$		0			0
$115$			3.48078i			0.324585i
$116$			8.64013i			0.802216i
$117$		0			0
$118$			2.95009i			0.271578i
$119$	1.66752			0.152862
$120$		0			0
$121$	−6.75832			−0.614393
$122$			5.06848i			0.458879i
$123$		0			0
$124$		−	3.23849i		−	0.290825i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	0.104575			0.00927952			0.00463976	−	0.999989i	$-0.498523\pi$
$127$	0.104575			0.00927952			0.00463976	+	0.999989i	$0.498523\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			2.27608i			0.199625i
$131$		−	0.347877i		−	0.0303942i	−0.999885	−	0.0151971i	$-0.995162\pi$
$131$		−	0.347877i		−	0.0303942i	0.999885	−	0.0151971i	$-0.00483757\pi$
$132$		0			0
$133$		−	4.97487i		−	0.431376i
$134$	6.76390	−	4.60973i	0.584312	−	0.398220i
$135$		0			0
$136$		−	0.953894i		−	0.0817957i
$137$	−23.1801			−1.98041			−0.990204	−	0.139628i	$-0.955409\pi$
$137$	−23.1801			−1.98041			−0.990204	+	0.139628i	$0.955409\pi$
$138$		0			0
$139$		−	1.63191i		−	0.138417i	−0.997602	−	0.0692086i	$-0.977953\pi$
$139$		−	1.63191i		−	0.138417i	0.997602	−	0.0692086i	$-0.0220474\pi$
$140$		−	1.74812i		−	0.147743i
$141$		0			0
$142$			10.3241i			0.866379i
$143$			4.68766i			0.392002i
$144$		0			0
$145$		−	8.64013i		−	0.717524i
$146$	−3.34901			−0.277166
$147$		0			0
$148$	6.47949			0.532611
$149$		−	16.0783i		−	1.31719i	−0.752498	−	0.658594i	$-0.771151\pi$
$149$		−	16.0783i		−	1.31719i	0.752498	−	0.658594i	$-0.228849\pi$
$150$		0			0
$151$	−10.1599			−0.826804			−0.413402	−	0.910549i	$-0.635660\pi$
$151$	−10.1599			−0.826804			−0.413402	+	0.910549i	$0.635660\pi$
$152$	−2.84583			−0.230828
$153$		0			0
$154$		−	3.60032i		−	0.290122i
$155$			3.23849i			0.260121i
$156$		0			0
$157$	18.2087			1.45321			0.726606	−	0.687054i	$-0.241097\pi$
$157$	18.2087			1.45321			0.726606	+	0.687054i	$0.241097\pi$
$158$			4.32175i			0.343820i
$159$		0			0
$160$	−1.00000			−0.0790569
$161$	6.08483			0.479552
$162$		0			0
$163$	−17.2241			−1.34909			−0.674547	−	0.738232i	$-0.735661\pi$
$163$	−17.2241			−1.34909			−0.674547	+	0.738232i	$0.735661\pi$
$164$	9.42263			0.735784
$165$		0			0
$166$			12.4242i			0.964305i
$167$			1.39206i			0.107721i	0.998548	+	0.0538605i	$0.0171526\pi$
$167$			1.39206i			0.107721i	−0.998548	+	0.0538605i	$0.982847\pi$
$168$		0			0
$169$	7.81947			0.601498
$170$			0.953894i			0.0731603i
$171$		0			0
$172$			8.13413i			0.620222i
$173$			11.6064i			0.882415i	0.897405	+	0.441207i	$0.145450\pi$
$173$			11.6064i			0.882415i	−0.897405	+	0.441207i	$0.854550\pi$
$174$		0			0
$175$			1.74812i			0.132146i
$176$	−2.05953			−0.155243
$177$		0			0
$178$		−	6.33557i		−	0.474871i
$179$	1.44735			0.108180			0.0540899	−	0.998536i	$-0.482774\pi$
$179$	1.44735			0.108180			0.0540899	+	0.998536i	$0.482774\pi$
$180$		0			0
$181$	15.9107			1.18263			0.591315	−	0.806441i	$-0.298609\pi$
$181$	15.9107			1.18263			0.591315	+	0.806441i	$0.298609\pi$
$182$	3.97886			0.294933
$183$		0			0
$184$		−	3.48078i		−	0.256607i
$185$	−6.47949			−0.476381
$186$		0			0
$187$			1.96458i			0.143664i
$188$			10.0128i			0.730262i
$189$		0			0
$190$	2.84583			0.206458
$191$	5.63422			0.407678			0.203839	−	0.979004i	$-0.434658\pi$
$191$	5.63422			0.407678			0.203839	+	0.979004i	$0.434658\pi$
$192$		0			0
$193$	−3.72089			−0.267835			−0.133918	−	0.990992i	$-0.542756\pi$
$193$	−3.72089			−0.267835			−0.133918	+	0.990992i	$0.542756\pi$
$194$			4.93178i			0.354082i
$195$		0			0
$196$	3.94407			0.281719
$197$	−17.3241			−1.23429			−0.617145	−	0.786849i	$-0.711711\pi$
$197$	−17.3241			−1.23429			−0.617145	+	0.786849i	$0.711711\pi$
$198$		0			0
$199$	−6.49845			−0.460663			−0.230331	−	0.973112i	$-0.573981\pi$
$199$	−6.49845			−0.460663			−0.230331	+	0.973112i	$0.573981\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	−9.03182			−0.635476
$203$	−15.1040			−1.06009
$204$		0			0
$205$	−9.42263			−0.658105
$206$	2.72439			0.189817
$207$		0			0
$208$		−	2.27608i		−	0.157818i
$209$	5.86109			0.405420
$210$		0			0
$211$	−6.16632			−0.424507			−0.212254	−	0.977215i	$-0.568080\pi$
$211$	−6.16632			−0.424507			−0.212254	+	0.977215i	$0.568080\pi$
$212$	−11.0991			−0.762286
$213$		0			0
$214$			19.4998i			1.33298i
$215$		−	8.13413i		−	0.554743i
$216$		0			0
$217$	5.66127			0.384312
$218$			6.86532i			0.464978i
$219$		0			0
$220$	2.05953			0.138854
$221$	−2.17114			−0.146046
$222$		0			0
$223$	19.0226			1.27385			0.636924	−	0.770926i	$-0.280206\pi$
$223$	19.0226			1.27385			0.636924	+	0.770926i	$0.280206\pi$
$224$			1.74812i			0.116801i
$225$		0			0
$226$	−5.22560			−0.347602
$227$		−	1.77218i		−	0.117624i	−0.998269	−	0.0588120i	$-0.981269\pi$
$227$		−	1.77218i		−	0.117624i	0.998269	−	0.0588120i	$-0.0187313\pi$
$228$		0			0
$229$			27.9820i			1.84910i	0.381060	+	0.924550i	$0.375559\pi$
$229$			27.9820i			1.84910i	−0.381060	+	0.924550i	$0.624441\pi$
$230$			3.48078i			0.229516i
$231$		0			0
$232$			8.64013i			0.567252i
$233$	26.5276			1.73788			0.868940	−	0.494918i	$-0.164802\pi$
$233$	26.5276			1.73788			0.868940	+	0.494918i	$0.164802\pi$
$234$		0			0
$235$		−	10.0128i		−	0.653166i
$236$			2.95009i			0.192035i
$237$		0			0
$238$	1.66752			0.108089
$239$	−18.2867			−1.18287			−0.591434	−	0.806353i	$-0.701438\pi$
$239$	−18.2867			−1.18287			−0.591434	+	0.806353i	$0.701438\pi$
$240$		0			0
$241$	−4.73391			−0.304938			−0.152469	−	0.988308i	$-0.548722\pi$
$241$	−4.73391			−0.304938			−0.152469	+	0.988308i	$0.548722\pi$
$242$	−6.75832			−0.434441
$243$		0			0
$244$			5.06848i			0.324476i
$245$	−3.94407			−0.251977
$246$		0			0
$247$			6.47734i			0.412143i
$248$		−	3.23849i		−	0.205644i
$249$		0			0
$250$	−1.00000			−0.0632456
$251$	−3.47900			−0.219593			−0.109796	−	0.993954i	$-0.535020\pi$
$251$	−3.47900			−0.219593			−0.109796	+	0.993954i	$0.535020\pi$
$252$		0			0
$253$			7.16879i			0.450698i
$254$	0.104575			0.00656161
$255$		0			0
$256$	1.00000			0.0625000
$257$			8.64389i			0.539191i	0.962974	+	0.269596i	$0.0868900\pi$
$257$			8.64389i			0.539191i	−0.962974	+	0.269596i	$0.913110\pi$
$258$		0			0
$259$			11.3269i			0.703822i
$260$			2.27608i			0.141156i
$261$		0			0
$262$		−	0.347877i		−	0.0214919i
$263$		−	31.9308i		−	1.96894i	−0.175558	−	0.984469i	$-0.556173\pi$
$263$		−	31.9308i		−	1.96894i	0.175558	−	0.984469i	$-0.443827\pi$
$264$		0			0
$265$	11.0991			0.681810
$266$		−	4.97487i		−	0.305029i
$267$		0			0
$268$	6.76390	−	4.60973i	0.413171	−	0.281584i
$269$			25.1437i			1.53304i	0.642221	+	0.766520i	$0.278013\pi$
$269$			25.1437i			1.53304i	−0.642221	+	0.766520i	$0.721987\pi$
$270$		0			0
$271$			9.76317i			0.593070i	0.955022	+	0.296535i	$0.0958312\pi$
$271$			9.76317i			0.593070i	−0.955022	+	0.296535i	$0.904169\pi$
$272$		−	0.953894i		−	0.0578383i
$273$		0			0
$274$	−23.1801			−1.40036
$275$	−2.05953			−0.124195
$276$		0			0
$277$	−2.61183			−0.156930			−0.0784648	−	0.996917i	$-0.525002\pi$
$277$	−2.61183			−0.156930			−0.0784648	+	0.996917i	$0.525002\pi$
$278$		−	1.63191i		−	0.0978757i
$279$		0			0
$280$		−	1.74812i		−	0.104470i
$281$	23.0757			1.37658			0.688290	−	0.725435i	$-0.258362\pi$
$281$	23.0757			1.37658			0.688290	+	0.725435i	$0.258362\pi$
$282$		0			0
$283$	25.4464			1.51263			0.756316	−	0.654207i	$-0.226997\pi$
$283$	25.4464			1.51263			0.756316	+	0.654207i	$0.226997\pi$
$284$			10.3241i			0.612622i
$285$		0			0
$286$			4.68766i			0.277187i
$287$			16.4719i			0.972307i
$288$		0			0
$289$	16.0901			0.946476
$290$		−	8.64013i		−	0.507366i
$291$		0			0
$292$	−3.34901			−0.195986
$293$			17.0004i			0.993176i	0.867987	+	0.496588i	$0.165414\pi$
$293$			17.0004i			0.993176i	−0.867987	+	0.496588i	$0.834586\pi$
$294$		0			0
$295$		−	2.95009i		−	0.171761i
$296$	6.47949			0.376613
$297$		0			0
$298$		−	16.0783i		−	0.931393i
$299$	−7.92253			−0.458172
$300$		0			0
$301$	−14.2195			−0.819596
$302$	−10.1599			−0.584639
$303$		0			0
$304$	−2.84583			−0.163220
$305$		−	5.06848i		−	0.290220i
$306$		0			0
$307$	1.83174			0.104543			0.0522716	−	0.998633i	$-0.483354\pi$
$307$	1.83174			0.104543			0.0522716	+	0.998633i	$0.483354\pi$
$308$		−	3.60032i		−	0.205147i
$309$		0			0
$310$			3.23849i			0.183934i
$311$	−30.5527			−1.73248			−0.866242	−	0.499625i	$-0.833471\pi$
$311$	−30.5527			−1.73248			−0.866242	+	0.499625i	$0.833471\pi$
$312$		0			0
$313$			3.29341i			0.186155i	0.995659	+	0.0930773i	$0.0296704\pi$
$313$			3.29341i			0.186155i	−0.995659	+	0.0930773i	$0.970330\pi$
$314$	18.2087			1.02758
$315$		0			0
$316$			4.32175i			0.243118i
$317$		−	30.4328i		−	1.70927i	−0.519225	−	0.854637i	$-0.673779\pi$
$317$		−	30.4328i		−	1.70927i	0.519225	−	0.854637i	$-0.326221\pi$
$318$		0			0
$319$		−	17.7946i		−	0.996309i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	6.08483			0.339095
$323$			2.71462i			0.151046i
$324$		0			0
$325$		−	2.27608i		−	0.126254i
$326$	−17.2241			−0.953953
$327$		0			0
$328$	9.42263			0.520278
$329$	−17.5037			−0.965010
$330$		0			0
$331$		−	12.6957i		−	0.697821i	−0.937156	−	0.348911i	$-0.886552\pi$
$331$		−	12.6957i		−	0.697821i	0.937156	−	0.348911i	$-0.113448\pi$
$332$			12.4242i			0.681866i
$333$		0			0
$334$			1.39206i			0.0761702i
$335$	−6.76390	+	4.60973i	−0.369551	+	0.251857i
$336$		0			0
$337$		−	28.6725i		−	1.56189i	−0.624600	−	0.780945i	$-0.714738\pi$
$337$		−	28.6725i		−	1.56189i	0.624600	−	0.780945i	$-0.285262\pi$
$338$	7.81947			0.425323
$339$		0			0
$340$			0.953894i			0.0517321i
$341$			6.66977i			0.361188i
$342$		0			0
$343$			19.1316i			1.03301i
$344$			8.13413i			0.438563i
$345$		0			0
$346$			11.6064i			0.623962i
$347$	33.8143			1.81524			0.907622	−	0.419787i	$-0.137895\pi$
$347$	33.8143			1.81524			0.907622	+	0.419787i	$0.137895\pi$
$348$		0			0
$349$	25.1364			1.34552			0.672760	−	0.739861i	$-0.265109\pi$
$349$	25.1364			1.34552			0.672760	+	0.739861i	$0.265109\pi$
$350$			1.74812i			0.0934411i
$351$		0			0
$352$	−2.05953			−0.109774
$353$	−20.6503			−1.09911			−0.549553	−	0.835459i	$-0.685202\pi$
$353$	−20.6503			−1.09911			−0.549553	+	0.835459i	$0.685202\pi$
$354$		0			0
$355$		−	10.3241i		−	0.547946i
$356$		−	6.33557i		−	0.335785i
$357$		0			0
$358$	1.44735			0.0764946
$359$			14.8848i			0.785588i	0.919627	+	0.392794i	$0.128491\pi$
$359$			14.8848i			0.785588i	−0.919627	+	0.392794i	$0.871509\pi$
$360$		0			0
$361$	−10.9012			−0.573749
$362$	15.9107			0.836245
$363$		0			0
$364$	3.97886			0.208549
$365$	3.34901			0.175295
$366$		0			0
$367$			6.63013i			0.346090i	0.984914	+	0.173045i	$0.0553606\pi$
$367$			6.63013i			0.346090i	−0.984914	+	0.173045i	$0.944639\pi$
$368$		−	3.48078i		−	0.181448i
$369$		0			0
$370$	−6.47949			−0.336852
$371$		−	19.4025i		−	1.00733i
$372$		0			0
$373$			3.80654i			0.197095i	0.995132	+	0.0985475i	$0.0314196\pi$
$373$			3.80654i			0.197095i	−0.995132	+	0.0985475i	$0.968580\pi$
$374$			1.96458i			0.101586i
$375$		0			0
$376$			10.0128i			0.516373i
$377$	19.6656			1.01283
$378$		0			0
$379$			6.74552i			0.346494i	0.984878	+	0.173247i	$0.0554259\pi$
$379$			6.74552i			0.346494i	−0.984878	+	0.173247i	$0.944574\pi$
$380$	2.84583			0.145988
$381$		0			0
$382$	5.63422			0.288272
$383$	14.0797			0.719438			0.359719	−	0.933061i	$-0.382873\pi$
$383$	14.0797			0.719438			0.359719	+	0.933061i	$0.382873\pi$
$384$		0			0
$385$			3.60032i			0.183489i
$386$	−3.72089			−0.189388
$387$		0			0
$388$			4.93178i			0.250373i
$389$			17.4269i			0.883577i	0.897119	+	0.441789i	$0.145656\pi$
$389$			17.4269i			0.883577i	−0.897119	+	0.441789i	$0.854344\pi$
$390$		0			0
$391$	−3.32030			−0.167915
$392$	3.94407			0.199205
$393$		0			0
$394$	−17.3241			−0.872775
$395$		−	4.32175i		−	0.217451i
$396$		0			0
$397$	32.7714			1.64475			0.822374	−	0.568948i	$-0.192649\pi$
$397$	32.7714			1.64475			0.822374	+	0.568948i	$0.192649\pi$
$398$	−6.49845			−0.325738
$399$		0			0
$400$	1.00000			0.0500000
$401$	34.8535			1.74050			0.870250	−	0.492610i	$-0.163957\pi$
$401$	34.8535			1.74050			0.870250	+	0.492610i	$0.163957\pi$
$402$		0			0
$403$	−7.37105			−0.367178
$404$	−9.03182			−0.449350
$405$		0			0
$406$	−15.1040			−0.749599
$407$	−13.3447			−0.661474
$408$		0			0
$409$		−	29.5388i		−	1.46060i	−0.683127	−	0.730299i	$-0.739381\pi$
$409$		−	29.5388i		−	1.46060i	0.683127	−	0.730299i	$-0.260619\pi$
$410$	−9.42263			−0.465351
$411$		0			0
$412$	2.72439			0.134221
$413$	−5.15712			−0.253765
$414$		0			0
$415$		−	12.4242i		−	0.609880i
$416$		−	2.27608i		−	0.111594i
$417$		0			0
$418$	5.86109			0.286675
$419$			1.15165i			0.0562619i	0.999604	+	0.0281310i	$0.00895555\pi$
$419$			1.15165i			0.0562619i	−0.999604	+	0.0281310i	$0.991044\pi$
$420$		0			0
$421$	−3.06597			−0.149426			−0.0747131	−	0.997205i	$-0.523804\pi$
$421$	−3.06597			−0.149426			−0.0747131	+	0.997205i	$0.523804\pi$
$422$	−6.16632			−0.300172
$423$		0			0
$424$	−11.0991			−0.539018
$425$		−	0.953894i		−	0.0462706i
$426$		0			0
$427$	−8.86033			−0.428781
$428$			19.4998i			0.942559i
$429$		0			0
$430$		−	8.13413i		−	0.392263i
$431$		−	28.8945i		−	1.39180i	−0.718139	−	0.695900i	$-0.755006\pi$
$431$		−	28.8945i		−	1.39180i	0.718139	−	0.695900i	$-0.244994\pi$
$432$		0			0
$433$		−	30.4897i		−	1.46524i	−0.680638	−	0.732620i	$-0.738297\pi$
$433$		−	30.4897i		−	1.46524i	0.680638	−	0.732620i	$-0.261703\pi$
$434$	5.66127			0.271750
$435$		0			0
$436$			6.86532i			0.328789i
$437$			9.90573i			0.473855i
$438$		0			0
$439$	−22.0069			−1.05033			−0.525166	−	0.851000i	$-0.675997\pi$
$439$	−22.0069			−1.05033			−0.525166	+	0.851000i	$0.675997\pi$
$440$	2.05953			0.0981845
$441$		0			0
$442$	−2.17114			−0.103270
$443$	−13.8066			−0.655973			−0.327987	−	0.944682i	$-0.606370\pi$
$443$	−13.8066			−0.655973			−0.327987	+	0.944682i	$0.606370\pi$
$444$		0			0
$445$			6.33557i			0.300335i
$446$	19.0226			0.900747
$447$		0			0
$448$			1.74812i			0.0825911i
$449$			20.1997i			0.953281i	0.879098	+	0.476640i	$0.158146\pi$
$449$			20.1997i			0.953281i	−0.879098	+	0.476640i	$0.841854\pi$
$450$		0			0
$451$	−19.4062			−0.913804
$452$	−5.22560			−0.245791
$453$		0			0
$454$		−	1.77218i		−	0.0831727i
$455$	−3.97886			−0.186532
$456$		0			0
$457$	27.8467			1.30261			0.651306	−	0.758815i	$-0.274221\pi$
$457$	27.8467			1.30261			0.651306	+	0.758815i	$0.274221\pi$
$458$			27.9820i			1.30751i
$459$		0			0
$460$			3.48078i			0.162292i
$461$			14.7964i			0.689138i	0.938761	+	0.344569i	$0.111975\pi$
$461$			14.7964i			0.689138i	−0.938761	+	0.344569i	$0.888025\pi$
$462$		0			0
$463$		−	5.66786i		−	0.263408i	−0.991289	−	0.131704i	$-0.957955\pi$
$463$		−	5.66786i		−	0.263408i	0.991289	−	0.131704i	$-0.0420448\pi$
$464$			8.64013i			0.401108i
$465$		0			0
$466$	26.5276			1.22887
$467$			10.5292i			0.487232i	0.969872	+	0.243616i	$0.0783337\pi$
$467$			10.5292i			0.487232i	−0.969872	+	0.243616i	$0.921666\pi$
$468$		0			0
$469$	8.05838	+	11.8241i	0.372101	+	0.545988i
$470$		−	10.0128i		−	0.461858i
$471$		0			0
$472$			2.95009i			0.135789i
$473$		−	16.7525i		−	0.770282i
$474$		0			0
$475$	−2.84583			−0.130576
$476$	1.66752			0.0764308
$477$		0			0
$478$	−18.2867			−0.836414
$479$			5.22686i			0.238821i	0.992845	+	0.119411i	$0.0381005\pi$
$479$			5.22686i			0.238821i	−0.992845	+	0.119411i	$0.961900\pi$
$480$		0			0
$481$		−	14.7478i		−	0.672443i
$482$	−4.73391			−0.215624
$483$		0			0
$484$	−6.75832			−0.307196
$485$		−	4.93178i		−	0.223941i
$486$		0			0
$487$		−	31.1506i		−	1.41157i	−0.708426	−	0.705785i	$-0.750594\pi$
$487$		−	31.1506i		−	1.41157i	0.708426	−	0.705785i	$-0.249406\pi$
$488$			5.06848i			0.229439i
$489$		0			0
$490$	−3.94407			−0.178175
$491$			34.8070i			1.57082i	0.618978	+	0.785409i	$0.287547\pi$
$491$			34.8070i			1.57082i	−0.618978	+	0.785409i	$0.712453\pi$
$492$		0			0
$493$	8.24176			0.371190
$494$			6.47734i			0.291429i
$495$		0			0
$496$		−	3.23849i		−	0.145412i
$497$	−18.0478			−0.809554
$498$		0			0
$499$			40.0580i			1.79324i	0.442797	+	0.896622i	$0.353986\pi$
$499$			40.0580i			1.79324i	−0.442797	+	0.896622i	$0.646014\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	−3.47900			−0.155275
$503$	29.8216			1.32968			0.664840	−	0.746985i	$-0.268500\pi$
$503$	29.8216			1.32968			0.664840	+	0.746985i	$0.268500\pi$
$504$		0			0
$505$	9.03182			0.401911
$506$			7.16879i			0.318692i
$507$		0			0
$508$	0.104575			0.00463976
$509$		−	14.0788i		−	0.624031i	−0.950077	−	0.312015i	$-0.898996\pi$
$509$		−	14.0788i		−	0.624031i	0.950077	−	0.312015i	$-0.101004\pi$
$510$		0			0
$511$		−	5.85448i		−	0.258987i
$512$	1.00000			0.0441942
$513$		0			0
$514$			8.64389i			0.381266i
$515$	−2.72439			−0.120051
$516$		0			0
$517$		−	20.6218i		−	0.906946i
$518$			11.3269i			0.497677i
$519$		0			0
$520$			2.27608i			0.0998126i
$521$	−26.4068			−1.15690			−0.578451	−	0.815717i	$-0.696342\pi$
$521$	−26.4068			−1.15690			−0.578451	+	0.815717i	$0.696342\pi$
$522$		0			0
$523$	18.0872			0.790899			0.395449	−	0.918488i	$-0.370589\pi$
$523$	18.0872			0.790899			0.395449	+	0.918488i	$0.370589\pi$
$524$		−	0.347877i		−	0.0151971i
$525$		0			0
$526$		−	31.9308i		−	1.39225i
$527$	−3.08917			−0.134566
$528$		0			0
$529$	10.8842			0.473224
$530$	11.0991			0.482112
$531$		0			0
$532$		−	4.97487i		−	0.215688i
$533$		−	21.4466i		−	0.928957i
$534$		0			0
$535$		−	19.4998i		−	0.843050i
$536$	6.76390	−	4.60973i	0.292156	−	0.199110i
$537$		0			0
$538$			25.1437i			1.08402i
$539$	−8.12294			−0.349880
$540$		0			0
$541$		−	19.0008i		−	0.816908i	−0.912779	−	0.408454i	$-0.866068\pi$
$541$		−	19.0008i		−	0.816908i	0.912779	−	0.408454i	$-0.133932\pi$
$542$			9.76317i			0.419364i
$543$		0			0
$544$		−	0.953894i		−	0.0408979i
$545$		−	6.86532i		−	0.294078i
$546$		0			0
$547$			5.86641i			0.250830i	0.992104	+	0.125415i	$0.0400262\pi$
$547$			5.86641i			0.250830i	−0.992104	+	0.125415i	$0.959974\pi$
$548$	−23.1801			−0.990204
$549$		0			0
$550$	−2.05953			−0.0878188
$551$		−	24.5884i		−	1.04750i
$552$		0			0
$553$	−7.55496			−0.321270
$554$	−2.61183			−0.110966
$555$		0			0
$556$		−	1.63191i		−	0.0692086i
$557$		−	38.8051i		−	1.64423i	−0.569324	−	0.822113i	$-0.692795\pi$
$557$		−	38.8051i		−	1.64423i	0.569324	−	0.822113i	$-0.307205\pi$
$558$		0			0
$559$	18.5139			0.783055
$560$		−	1.74812i		−	0.0738717i
$561$		0			0
$562$	23.0757			0.973389
$563$	−20.0734			−0.845994			−0.422997	−	0.906131i	$-0.639022\pi$
$563$	−20.0734			−0.845994			−0.422997	+	0.906131i	$0.639022\pi$
$564$		0			0
$565$	5.22560			0.219843
$566$	25.4464			1.06959
$567$		0			0
$568$			10.3241i			0.433189i
$569$			2.28881i			0.0959520i	0.998848	+	0.0479760i	$0.0152771\pi$
$569$			2.28881i			0.0959520i	−0.998848	+	0.0479760i	$0.984723\pi$
$570$		0			0
$571$	21.4851			0.899125			0.449562	−	0.893249i	$-0.351580\pi$
$571$	21.4851			0.899125			0.449562	+	0.893249i	$0.351580\pi$
$572$			4.68766i			0.196001i
$573$		0			0
$574$			16.4719i			0.687525i
$575$		−	3.48078i		−	0.145159i
$576$		0			0
$577$			30.0002i			1.24892i	0.781055	+	0.624462i	$0.214682\pi$
$577$			30.0002i			1.24892i	−0.781055	+	0.624462i	$0.785318\pi$
$578$	16.0901			0.669259
$579$		0			0
$580$		−	8.64013i		−	0.358762i
$581$	−21.7190			−0.901057
$582$		0			0
$583$	22.8589			0.946718
$584$	−3.34901			−0.138583
$585$		0			0
$586$			17.0004i			0.702281i
$587$	−31.9392			−1.31827			−0.659135	−	0.752025i	$-0.729077\pi$
$587$	−31.9392			−1.31827			−0.659135	+	0.752025i	$0.729077\pi$
$588$		0			0
$589$			9.21619i			0.379747i
$590$		−	2.95009i		−	0.121453i
$591$		0			0
$592$	6.47949			0.266305
$593$	−39.3707			−1.61676			−0.808379	−	0.588662i	$-0.799655\pi$
$593$	−39.3707			−1.61676			−0.808379	+	0.588662i	$0.799655\pi$
$594$		0			0
$595$	−1.66752			−0.0683618
$596$		−	16.0783i		−	0.658594i
$597$		0			0
$598$	−7.92253			−0.323976
$599$	16.1798			0.661091			0.330545	−	0.943790i	$-0.392767\pi$
$599$	16.1798			0.661091			0.330545	+	0.943790i	$0.392767\pi$
$600$		0			0
$601$	32.3314			1.31883			0.659413	−	0.751781i	$-0.270805\pi$
$601$	32.3314			1.31883			0.659413	+	0.751781i	$0.270805\pi$
$602$	−14.2195			−0.579542
$603$		0			0
$604$	−10.1599			−0.413402
$605$	6.75832			0.274765
$606$		0			0
$607$	33.7197			1.36864			0.684321	−	0.729181i	$-0.260099\pi$
$607$	33.7197			1.36864			0.684321	+	0.729181i	$0.260099\pi$
$608$	−2.84583			−0.115414
$609$		0			0
$610$		−	5.06848i		−	0.205217i
$611$	22.7900			0.921986
$612$		0			0
$613$	12.6892			0.512511			0.256256	−	0.966609i	$-0.417511\pi$
$613$	12.6892			0.512511			0.256256	+	0.966609i	$0.417511\pi$
$614$	1.83174			0.0739232
$615$		0			0
$616$		−	3.60032i		−	0.145061i
$617$			0.291834i			0.0117488i	0.999983	+	0.00587439i	$0.00186989\pi$
$617$			0.291834i			0.0117488i	−0.999983	+	0.00587439i	$0.998130\pi$
$618$		0			0
$619$	−14.2548			−0.572948			−0.286474	−	0.958088i	$-0.592483\pi$
$619$	−14.2548			−0.572948			−0.286474	+	0.958088i	$0.592483\pi$
$620$			3.23849i			0.130061i
$621$		0			0
$622$	−30.5527			−1.22505
$623$	11.0754			0.443725
$624$		0			0
$625$	1.00000			0.0400000
$626$			3.29341i			0.131631i
$627$		0			0
$628$	18.2087			0.726606
$629$		−	6.18074i		−	0.246442i
$630$		0			0
$631$			4.66515i			0.185717i	0.995679	+	0.0928583i	$0.0296003\pi$
$631$			4.66515i			0.185717i	−0.995679	+	0.0928583i	$0.970400\pi$
$632$			4.32175i			0.171910i
$633$		0			0
$634$		−	30.4328i		−	1.20864i
$635$	−0.104575			−0.00414993
$636$		0			0
$637$		−	8.97700i		−	0.355682i
$638$		−	17.7946i		−	0.704497i
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	−1.99005			−0.0786021			−0.0393011	−	0.999227i	$-0.512513\pi$
$641$	−1.99005			−0.0786021			−0.0393011	+	0.999227i	$0.512513\pi$
$642$		0			0
$643$	−21.7379			−0.857259			−0.428630	−	0.903480i	$-0.641003\pi$
$643$	−21.7379			−0.857259			−0.428630	+	0.903480i	$0.641003\pi$
$644$	6.08483			0.239776
$645$		0			0
$646$			2.71462i			0.106805i
$647$	−47.5872			−1.87084			−0.935422	−	0.353532i	$-0.884981\pi$
$647$	−47.5872			−1.87084			−0.935422	+	0.353532i	$0.884981\pi$
$648$		0			0
$649$		−	6.07581i		−	0.238496i
$650$		−	2.27608i		−	0.0892751i
$651$		0			0
$652$	−17.2241			−0.674547
$653$	−4.70388			−0.184077			−0.0920386	−	0.995755i	$-0.529338\pi$
$653$	−4.70388			−0.184077			−0.0920386	+	0.995755i	$0.529338\pi$
$654$		0			0
$655$			0.347877i			0.0135927i
$656$	9.42263			0.367892
$657$		0			0
$658$	−17.5037			−0.682365
$659$		−	4.89899i		−	0.190838i	−0.995437	−	0.0954188i	$-0.969581\pi$
$659$		−	4.89899i		−	0.190838i	0.995437	−	0.0954188i	$-0.0304190\pi$
$660$		0			0
$661$			11.6492i			0.453100i	0.974000	+	0.226550i	$0.0727446\pi$
$661$			11.6492i			0.453100i	−0.974000	+	0.226550i	$0.927255\pi$
$662$		−	12.6957i		−	0.493434i
$663$		0			0
$664$			12.4242i			0.482152i
$665$			4.97487i			0.192917i
$666$		0			0
$667$	30.0744			1.16449
$668$			1.39206i			0.0538605i
$669$		0			0
$670$	−6.76390	+	4.60973i	−0.261312	+	0.178089i
$671$		−	10.4387i		−	0.402982i
$672$		0			0
$673$			8.52209i			0.328502i	0.986419	+	0.164251i	$0.0525208\pi$
$673$			8.52209i			0.328502i	−0.986419	+	0.164251i	$0.947479\pi$
$674$		−	28.6725i		−	1.10442i
$675$		0			0
$676$	7.81947			0.300749
$677$	−14.6821			−0.564281			−0.282140	−	0.959373i	$-0.591044\pi$
$677$	−14.6821			−0.564281			−0.282140	+	0.959373i	$0.591044\pi$
$678$		0			0
$679$	−8.62137			−0.330858
$680$			0.953894i			0.0365802i
$681$		0			0
$682$			6.66977i			0.255399i
$683$	8.93286			0.341806			0.170903	−	0.985288i	$-0.445331\pi$
$683$	8.93286			0.341806			0.170903	+	0.985288i	$0.445331\pi$
$684$		0			0
$685$	23.1801			0.885665
$686$			19.1316i			0.730447i
$687$		0			0
$688$			8.13413i			0.310111i
$689$			25.2623i			0.962418i
$690$		0			0
$691$	25.0816			0.954150			0.477075	−	0.878863i	$-0.341697\pi$
$691$	25.0816			0.954150			0.477075	+	0.878863i	$0.341697\pi$
$692$			11.6064i			0.441207i
$693$		0			0
$694$	33.8143			1.28357
$695$			1.63191i			0.0619020i
$696$		0			0
$697$		−	8.98819i		−	0.340452i
$698$	25.1364			0.951426
$699$		0			0
$700$			1.74812i			0.0660728i
$701$	4.18605			0.158105			0.0790525	−	0.996870i	$-0.474811\pi$
$701$	4.18605			0.158105			0.0790525	+	0.996870i	$0.474811\pi$
$702$		0			0
$703$	−18.4395			−0.695461
$704$	−2.05953			−0.0776216
$705$		0			0
$706$	−20.6503			−0.777185
$707$		−	15.7887i		−	0.593796i
$708$		0			0
$709$	−25.1400			−0.944154			−0.472077	−	0.881557i	$-0.656496\pi$
$709$	−25.1400			−0.944154			−0.472077	+	0.881557i	$0.656496\pi$
$710$		−	10.3241i		−	0.387456i
$711$		0			0
$712$		−	6.33557i		−	0.237436i
$713$	−11.2725			−0.422157
$714$		0			0
$715$		−	4.68766i		−	0.175309i
$716$	1.44735			0.0540899
$717$		0			0
$718$			14.8848i			0.555494i
$719$			28.7253i			1.07127i	0.844448	+	0.535637i	$0.179928\pi$
$719$			28.7253i			1.07127i	−0.844448	+	0.535637i	$0.820072\pi$
$720$		0			0
$721$			4.76256i			0.177367i
$722$	−10.9012			−0.405702
$723$		0			0
$724$	15.9107			0.591315
$725$			8.64013i			0.320886i
$726$		0			0
$727$		−	22.6468i		−	0.839923i	−0.907542	−	0.419962i	$-0.862044\pi$
$727$		−	22.6468i		−	0.839923i	0.907542	−	0.419962i	$-0.137956\pi$
$728$	3.97886			0.147467
$729$		0			0
$730$	3.34901			0.123952
$731$	7.75910			0.286981
$732$		0			0
$733$		−	27.6334i		−	1.02066i	−0.859977	−	0.510332i	$-0.829522\pi$
$733$		−	27.6334i		−	1.02066i	0.859977	−	0.510332i	$-0.170478\pi$
$734$			6.63013i			0.244723i
$735$		0			0
$736$		−	3.48078i		−	0.128303i
$737$	−13.9305	+	9.49390i	−0.513136	+	0.349712i
$738$		0			0
$739$			19.3561i			0.712027i	0.934481	+	0.356014i	$0.115864\pi$
$739$			19.3561i			0.712027i	−0.934481	+	0.356014i	$0.884136\pi$
$740$	−6.47949			−0.238191
$741$		0			0
$742$		−	19.4025i		−	0.712289i
$743$		−	21.7206i		−	0.796851i	−0.917201	−	0.398425i	$-0.869557\pi$
$743$		−	21.7206i		−	0.796851i	0.917201	−	0.398425i	$-0.130443\pi$
$744$		0			0
$745$			16.0783i			0.589065i
$746$			3.80654i			0.139367i
$747$		0			0
$748$			1.96458i			0.0718320i
$749$	−34.0881			−1.24555
$750$		0			0
$751$	−34.4649			−1.25764			−0.628820	−	0.777551i	$-0.716462\pi$
$751$	−34.4649			−1.25764			−0.628820	+	0.777551i	$0.716462\pi$
$752$			10.0128i			0.365131i
$753$		0			0
$754$	19.6656			0.716179
$755$	10.1599			0.369758
$756$		0			0
$757$		−	50.3768i		−	1.83098i	−0.402344	−	0.915488i	$-0.631805\pi$
$757$		−	50.3768i		−	1.83098i	0.402344	−	0.915488i	$-0.368195\pi$
$758$			6.74552i			0.245008i
$759$		0			0
$760$	2.84583			0.103229
$761$		−	1.92509i		−	0.0697844i	−0.999391	−	0.0348922i	$-0.988891\pi$
$761$		−	1.92509i		−	0.0697844i	0.999391	−	0.0348922i	$-0.0111088\pi$
$762$		0			0
$763$	−12.0014			−0.434481
$764$	5.63422			0.203839
$765$		0			0
$766$	14.0797			0.508719
$767$	6.71463			0.242451
$768$		0			0
$769$		−	37.1821i		−	1.34082i	−0.741990	−	0.670411i	$-0.766118\pi$
$769$		−	37.1821i		−	1.34082i	0.741990	−	0.670411i	$-0.233882\pi$
$770$			3.60032i			0.129747i
$771$		0			0
$772$	−3.72089			−0.133918
$773$		−	18.0490i		−	0.649178i	−0.945855	−	0.324589i	$-0.894774\pi$
$773$		−	18.0490i		−	0.649178i	0.945855	−	0.324589i	$-0.105226\pi$
$774$		0			0
$775$		−	3.23849i		−	0.116330i
$776$			4.93178i			0.177041i
$777$		0			0
$778$			17.4269i			0.624783i
$779$	−26.8152			−0.960756
$780$		0			0
$781$		−	21.2628i		−	0.760844i
$782$	−3.32030			−0.118734
$783$		0			0
$784$	3.94407			0.140859
$785$	−18.2087			−0.649896
$786$		0			0
$787$		−	29.6582i		−	1.05720i	−0.848871	−	0.528600i	$-0.822717\pi$
$787$		−	29.6582i		−	1.05720i	0.848871	−	0.528600i	$-0.177283\pi$
$788$	−17.3241			−0.617145
$789$		0			0
$790$		−	4.32175i		−	0.153761i
$791$		−	9.13499i		−	0.324803i
$792$		0			0
$793$	11.5363			0.409665
$794$	32.7714			1.16301
$795$		0			0
$796$	−6.49845			−0.230331
$797$		−	9.90025i		−	0.350685i	−0.984508	−	0.175342i	$-0.943897\pi$
$797$		−	9.90025i		−	0.350685i	0.984508	−	0.175342i	$-0.0561033\pi$
$798$		0			0
$799$	9.55119			0.337897
$800$	1.00000			0.0353553
$801$		0			0
$802$	34.8535			1.23072
$803$	6.89740			0.243404
$804$		0			0
$805$	−6.08483			−0.214462
$806$	−7.37105			−0.259634
$807$		0			0
$808$	−9.03182			−0.317738
$809$	10.8964			0.383096			0.191548	−	0.981483i	$-0.438649\pi$
$809$	10.8964			0.383096			0.191548	+	0.981483i	$0.438649\pi$
$810$		0			0
$811$			18.8817i			0.663025i	0.943451	+	0.331512i	$0.107559\pi$
$811$			18.8817i			0.663025i	−0.943451	+	0.331512i	$0.892441\pi$
$812$	−15.1040			−0.530047
$813$		0			0
$814$	−13.3447			−0.467732
$815$	17.2241			0.603333
$816$		0			0
$817$		−	23.1484i		−	0.809860i
$818$		−	29.5388i		−	1.03280i
$819$		0			0
$820$	−9.42263			−0.329053
$821$		−	49.6059i		−	1.73126i	−0.500686	−	0.865629i	$-0.666919\pi$
$821$		−	49.6059i		−	1.73126i	0.500686	−	0.865629i	$-0.333081\pi$
$822$		0			0
$823$	−23.0571			−0.803720			−0.401860	−	0.915701i	$-0.631636\pi$
$823$	−23.0571			−0.803720			−0.401860	+	0.915701i	$0.631636\pi$
$824$	2.72439			0.0949085
$825$		0			0
$826$	−5.15712			−0.179439
$827$		−	9.61752i		−	0.334434i	−0.985920	−	0.167217i	$-0.946522\pi$
$827$		−	9.61752i		−	0.334434i	0.985920	−	0.167217i	$-0.0534780\pi$
$828$		0			0
$829$	−14.3843			−0.499589			−0.249794	−	0.968299i	$-0.580363\pi$
$829$	−14.3843			−0.499589			−0.249794	+	0.968299i	$0.580363\pi$
$830$		−	12.4242i		−	0.431250i
$831$		0			0
$832$		−	2.27608i		−	0.0789088i
$833$		−	3.76222i		−	0.130353i
$834$		0			0
$835$		−	1.39206i		−	0.0481743i
$836$	5.86109			0.202710
$837$		0			0
$838$			1.15165i			0.0397832i
$839$			55.8455i			1.92800i	0.265902	+	0.964000i	$0.414330\pi$
$839$			55.8455i			1.92800i	−0.265902	+	0.964000i	$0.585670\pi$
$840$		0			0
$841$	−45.6518			−1.57420
$842$	−3.06597			−0.105660
$843$		0			0
$844$	−6.16632			−0.212254
$845$	−7.81947			−0.268998
$846$		0			0
$847$		−	11.8144i		−	0.405947i
$848$	−11.0991			−0.381143
$849$		0			0
$850$		−	0.953894i		−	0.0327183i
$851$		−	22.5537i		−	0.773130i
$852$		0			0
$853$	−9.37478			−0.320986			−0.160493	−	0.987037i	$-0.551308\pi$
$853$	−9.37478			−0.320986			−0.160493	+	0.987037i	$0.551308\pi$
$854$	−8.86033			−0.303194
$855$		0			0
$856$			19.4998i			0.666490i
$857$	−54.1585			−1.85002			−0.925009	−	0.379945i	$-0.875943\pi$
$857$	−54.1585			−1.85002			−0.925009	+	0.379945i	$0.875943\pi$
$858$		0			0
$859$	−4.30465			−0.146873			−0.0734364	−	0.997300i	$-0.523397\pi$
$859$	−4.30465			−0.146873			−0.0734364	+	0.997300i	$0.523397\pi$
$860$		−	8.13413i		−	0.277372i
$861$		0			0
$862$		−	28.8945i		−	0.984151i
$863$			44.3876i			1.51097i	0.655165	+	0.755486i	$0.272599\pi$
$863$			44.3876i			1.51097i	−0.655165	+	0.755486i	$0.727401\pi$
$864$		0			0
$865$		−	11.6064i		−	0.394628i
$866$		−	30.4897i		−	1.03608i
$867$		0			0
$868$	5.66127			0.192156
$869$		−	8.90080i		−	0.301939i
$870$		0			0
$871$	−10.4921	−	15.3952i	−0.355512	−	0.521645i
$872$			6.86532i			0.232489i
$873$		0			0
$874$			9.90573i			0.335066i
$875$		−	1.74812i		−	0.0590973i
$876$		0			0
$877$	−14.9018			−0.503198			−0.251599	−	0.967832i	$-0.580956\pi$
$877$	−14.9018			−0.503198			−0.251599	+	0.967832i	$0.580956\pi$
$878$	−22.0069			−0.742697
$879$		0			0
$880$	2.05953			0.0694269
$881$		−	31.7612i		−	1.07006i	−0.844833	−	0.535031i	$-0.820300\pi$
$881$		−	31.7612i		−	1.07006i	0.844833	−	0.535031i	$-0.179700\pi$
$882$		0			0
$883$		−	0.608330i		−	0.0204719i	−0.999948	−	0.0102360i	$-0.996742\pi$
$883$		−	0.608330i		−	0.0204719i	0.999948	−	0.0102360i	$-0.00325826\pi$
$884$	−2.17114			−0.0730232
$885$		0			0
$886$	−13.8066			−0.463843
$887$			19.7947i			0.664642i	0.943166	+	0.332321i	$0.107832\pi$
$887$			19.7947i			0.664642i	−0.943166	+	0.332321i	$0.892168\pi$
$888$		0			0
$889$			0.182810i			0.00613124i
$890$			6.33557i			0.212369i
$891$		0			0
$892$	19.0226			0.636924
$893$		−	28.4949i		−	0.953545i
$894$		0			0
$895$	−1.44735			−0.0483794
$896$			1.74812i			0.0584007i
$897$		0			0
$898$			20.1997i			0.674071i
$899$	27.9809			0.933216
$900$		0			0
$901$			10.5873i			0.352715i
$902$	−19.4062			−0.646157
$903$		0			0
$904$	−5.22560			−0.173801
$905$	−15.9107			−0.528888
$906$		0			0
$907$	23.9386			0.794867			0.397433	−	0.917631i	$-0.369901\pi$
$907$	23.9386			0.794867			0.397433	+	0.917631i	$0.369901\pi$
$908$		−	1.77218i		−	0.0588120i
$909$		0			0
$910$	−3.97886			−0.131898
$911$		−	14.7524i		−	0.488768i	−0.969679	−	0.244384i	$-0.921414\pi$
$911$		−	14.7524i		−	0.488768i	0.969679	−	0.244384i	$-0.0785858\pi$
$912$		0			0
$913$		−	25.5881i		−	0.846841i
$914$	27.8467			0.921086
$915$		0			0
$916$			27.9820i			0.924550i
$917$	0.608132			0.0200823
$918$		0			0
$919$			31.5176i			1.03967i	0.854267	+	0.519835i	$0.174007\pi$
$919$			31.5176i			1.03967i	−0.854267	+	0.519835i	$0.825993\pi$
$920$			3.48078i			0.114758i
$921$		0			0
$922$			14.7964i			0.487294i
$923$	23.4984			0.773461
$924$		0			0
$925$	6.47949			0.213044
$926$		−	5.66786i		−	0.186258i
$927$		0			0
$928$			8.64013i			0.283626i
$929$	−48.7982			−1.60102			−0.800508	−	0.599322i	$-0.795437\pi$
$929$	−48.7982			−1.60102			−0.800508	+	0.599322i	$0.795437\pi$
$930$		0			0
$931$	−11.2242			−0.367857
$932$	26.5276			0.868940
$933$		0			0
$934$			10.5292i			0.344525i
$935$		−	1.96458i		−	0.0642485i
$936$		0			0
$937$		−	21.4098i		−	0.699427i	−0.936857	−	0.349714i	$-0.886279\pi$
$937$		−	21.4098i		−	0.699427i	0.936857	−	0.349714i	$-0.113721\pi$
$938$	8.05838	+	11.8241i	0.263115	+	0.386072i
$939$		0			0
$940$		−	10.0128i		−	0.326583i
$941$	26.1441			0.852275			0.426137	−	0.904658i	$-0.359874\pi$
$941$	26.1441			0.852275			0.426137	+	0.904658i	$0.359874\pi$
$942$		0			0
$943$		−	32.7981i		−	1.06805i
$944$			2.95009i			0.0960173i
$945$		0			0
$946$		−	16.7525i		−	0.544672i
$947$		−	35.5310i		−	1.15460i	−0.816531	−	0.577301i	$-0.804106\pi$
$947$		−	35.5310i		−	1.15460i	0.816531	−	0.577301i	$-0.195894\pi$
$948$		0			0
$949$			7.62261i			0.247440i
$950$	−2.84583			−0.0923310
$951$		0			0
$952$	1.66752			0.0540447
$953$			3.01683i			0.0977246i	0.998806	+	0.0488623i	$0.0155595\pi$
$953$			3.01683i			0.0977246i	−0.998806	+	0.0488623i	$0.984440\pi$
$954$		0			0
$955$	−5.63422			−0.182319
$956$	−18.2867			−0.591434
$957$		0			0
$958$			5.22686i			0.168872i
$959$		−	40.5216i		−	1.30851i
$960$		0			0
$961$	20.5122			0.661684
$962$		−	14.7478i		−	0.475489i
$963$		0			0
$964$	−4.73391			−0.152469
$965$	3.72089			0.119780
$966$		0			0
$967$	−3.61700			−0.116315			−0.0581574	−	0.998307i	$-0.518523\pi$
$967$	−3.61700			−0.116315			−0.0581574	+	0.998307i	$0.518523\pi$
$968$	−6.75832			−0.217221
$969$		0			0
$970$		−	4.93178i		−	0.158350i
$971$		−	33.0657i		−	1.06113i	−0.847645	−	0.530564i	$-0.821980\pi$
$971$		−	33.0657i		−	1.06113i	0.847645	−	0.530564i	$-0.178020\pi$
$972$		0			0
$973$	2.85279			0.0914561
$974$		−	31.1506i		−	0.998131i
$975$		0			0
$976$			5.06848i			0.162238i
$977$		−	8.06883i		−	0.258145i	−0.991635	−	0.129072i	$-0.958800\pi$
$977$		−	8.06883i		−	0.258145i	0.991635	−	0.129072i	$-0.0411999\pi$
$978$		0			0
$979$			13.0483i			0.417026i
$980$	−3.94407			−0.125989
$981$		0			0
$982$			34.8070i			1.11074i
$983$	53.1123			1.69402			0.847010	−	0.531577i	$-0.178400\pi$
$983$	53.1123			1.69402			0.847010	+	0.531577i	$0.178400\pi$
$984$		0			0
$985$	17.3241			0.551992
$986$	8.24176			0.262471
$987$		0			0
$988$			6.47734i			0.206072i
$989$	28.3131			0.900305
$990$		0			0
$991$		−	29.5769i		−	0.939542i	−0.882788	−	0.469771i	$-0.844337\pi$
$991$		−	29.5769i		−	0.939542i	0.882788	−	0.469771i	$-0.155663\pi$
$992$		−	3.23849i		−	0.102822i
$993$		0			0
$994$	−18.0478			−0.572441
$995$	6.49845			0.206015
$996$		0			0
$997$	12.2690			0.388563			0.194282	−	0.980946i	$-0.437762\pi$
$997$	12.2690			0.388563			0.194282	+	0.980946i	$0.437762\pi$
$998$			40.0580i			1.26802i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.j.2411.11	yes	16
3.2	odd	2		6030.2.d.i.2411.11	yes	16
67.66	odd	2		6030.2.d.i.2411.6	✓	16
201.200	even	2	inner	6030.2.d.j.2411.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.i.2411.6	✓	16	67.66	odd	2
6030.2.d.i.2411.11	yes	16	3.2	odd	2
6030.2.d.j.2411.6	yes	16	201.200	even	2	inner
6030.2.d.j.2411.11	yes	16	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.74812i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.74812i q^{7} +1.00000 q^{8} -1.00000 q^{10} -2.05953 q^{11} -2.27608i q^{13} +1.74812i q^{14} +1.00000 q^{16} -0.953894i q^{17} -2.84583 q^{19} -1.00000 q^{20} -2.05953 q^{22} -3.48078i q^{23} +1.00000 q^{25} -2.27608i q^{26} +1.74812i q^{28} +8.64013i q^{29} -3.23849i q^{31} +1.00000 q^{32} -0.953894i q^{34} -1.74812i q^{35} +6.47949 q^{37} -2.84583 q^{38} -1.00000 q^{40} +9.42263 q^{41} +8.13413i q^{43} -2.05953 q^{44} -3.48078i q^{46} +10.0128i q^{47} +3.94407 q^{49} +1.00000 q^{50} -2.27608i q^{52} -11.0991 q^{53} +2.05953 q^{55} +1.74812i q^{56} +8.64013i q^{58} +2.95009i q^{59} +5.06848i q^{61} -3.23849i q^{62} +1.00000 q^{64} +2.27608i q^{65} +(6.76390 - 4.60973i) q^{67} -0.953894i q^{68} -1.74812i q^{70} +10.3241i q^{71} -3.34901 q^{73} +6.47949 q^{74} -2.84583 q^{76} -3.60032i q^{77} +4.32175i q^{79} -1.00000 q^{80} +9.42263 q^{82} +12.4242i q^{83} +0.953894i q^{85} +8.13413i q^{86} -2.05953 q^{88} -6.33557i q^{89} +3.97886 q^{91} -3.48078i q^{92} +10.0128i q^{94} +2.84583 q^{95} +4.93178i q^{97} +3.94407 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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