LMFDB - Embedded newform 6300.2.d.b.3401.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(3401,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6300.3401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3401.1
Root			$-2.28825i$ of defining polynomial
Character	$\chi$	$=$	6300.3401
Dual form			6300.2.d.b.3401.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23607 - 1.41421i) q^{7} +O(q^{10})$	$q+(-2.23607 - 1.41421i) q^{7} +1.41421i q^{11} -1.74806i q^{13} +4.47214 q^{17} +1.74806i q^{19} -3.16228i q^{23} -2.08191i q^{29} +4.57649i q^{31} -1.52786 q^{37} -0.472136 q^{41} +0.472136 q^{43} +2.47214 q^{47} +(3.00000 + 6.32456i) q^{49} -8.81913i q^{53} -1.52786 q^{59} +3.49613i q^{61} +7.73877i q^{71} -16.5579i q^{73} +(2.00000 - 3.16228i) q^{77} +8.94427 q^{79} -5.52786 q^{83} -10.9443 q^{89} +(-2.47214 + 3.90879i) q^{91} +0.412662i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{37} + 16 q^{41} - 16 q^{43} - 8 q^{47} + 12 q^{49} - 24 q^{59} + 8 q^{77} - 40 q^{83} - 8 q^{89} + 8 q^{91}+O(q^{100}) $ 4 * q - 24 * q^37 + 16 * q^41 - 16 * q^43 - 8 * q^47 + 12 * q^49 - 24 * q^59 + 8 * q^77 - 40 * q^83 - 8 * q^89 + 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.23607	−	1.41421i	−0.845154	−	0.534522i
$7$	−2.23607	−	1.41421i	−0.845154	−	0.534522i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$		−	1.74806i		−	0.484826i	−0.970173	−	0.242413i	$-0.922061\pi$
$13$		−	1.74806i		−	0.484826i	0.970173	−	0.242413i	$-0.0779389\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$			1.74806i			0.401033i	0.979690	+	0.200517i	$0.0642621\pi$
$19$			1.74806i			0.401033i	−0.979690	+	0.200517i	$0.935738\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.16228i		−	0.659380i	−0.944089	−	0.329690i	$-0.893056\pi$
$23$		−	3.16228i		−	0.659380i	0.944089	−	0.329690i	$-0.106944\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.08191i		−	0.386602i	−0.981140	−	0.193301i	$-0.938081\pi$
$29$		−	2.08191i		−	0.386602i	0.981140	−	0.193301i	$-0.0619194\pi$
$30$		0			0
$31$			4.57649i			0.821962i	0.911644	+	0.410981i	$0.134814\pi$
$31$			4.57649i			0.821962i	−0.911644	+	0.410981i	$0.865186\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.52786			−0.251179			−0.125590	−	0.992082i	$-0.540082\pi$
$37$	−1.52786			−0.251179			−0.125590	+	0.992082i	$0.540082\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.472136			−0.0737352			−0.0368676	−	0.999320i	$-0.511738\pi$
$41$	−0.472136			−0.0737352			−0.0368676	+	0.999320i	$0.511738\pi$
$42$		0			0
$43$	0.472136			0.0720001			0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136			0.0720001			0.0360000	+	0.999352i	$0.488538\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.47214			0.360598			0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214			0.360598			0.180299	+	0.983612i	$0.442293\pi$
$48$		0			0
$49$	3.00000	+	6.32456i	0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.81913i		−	1.21140i	−0.795693	−	0.605700i	$-0.792893\pi$
$53$		−	8.81913i		−	1.21140i	0.795693	−	0.605700i	$-0.207107\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.52786			−0.198911			−0.0994555	−	0.995042i	$-0.531710\pi$
$59$	−1.52786			−0.198911			−0.0994555	+	0.995042i	$0.531710\pi$
$60$		0			0
$61$			3.49613i			0.447633i	0.974631	+	0.223817i	$0.0718517\pi$
$61$			3.49613i			0.447633i	−0.974631	+	0.223817i	$0.928148\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.73877i			0.918423i	0.888327	+	0.459211i	$0.151868\pi$
$71$			7.73877i			0.918423i	−0.888327	+	0.459211i	$0.848132\pi$
$72$		0			0
$73$		−	16.5579i		−	1.93796i	−0.247148	−	0.968978i	$-0.579493\pi$
$73$		−	16.5579i		−	1.93796i	0.247148	−	0.968978i	$-0.420507\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000	−	3.16228i	0.227921	−	0.360375i
$78$		0			0
$79$	8.94427			1.00631			0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427			1.00631			0.503155	+	0.864196i	$0.332173\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.52786			−0.606762			−0.303381	−	0.952869i	$-0.598115\pi$
$83$	−5.52786			−0.606762			−0.303381	+	0.952869i	$0.598115\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.9443			−1.16009			−0.580045	−	0.814584i	$-0.696965\pi$
$89$	−10.9443			−1.16009			−0.580045	+	0.814584i	$0.696965\pi$
$90$		0			0
$91$	−2.47214	+	3.90879i	−0.259150	+	0.409753i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			0.412662i			0.0418995i	0.999781	+	0.0209497i	$0.00666900\pi$
$97$			0.412662i			0.0418995i	−0.999781	+	0.0209497i	$0.993331\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.47214			−0.843009			−0.421505	−	0.906826i	$-0.638498\pi$
$101$	−8.47214			−0.843009			−0.421505	+	0.906826i	$0.638498\pi$
$102$		0			0
$103$			3.49613i			0.344484i	0.985055	+	0.172242i	$0.0551011\pi$
$103$			3.49613i			0.344484i	−0.985055	+	0.172242i	$0.944899\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.16228i			0.305709i	0.988249	+	0.152854i	$0.0488466\pi$
$107$			3.16228i			0.305709i	−0.988249	+	0.152854i	$0.951153\pi$
$108$		0			0
$109$	8.94427			0.856706			0.428353	−	0.903612i	$-0.359094\pi$
$109$	8.94427			0.856706			0.428353	+	0.903612i	$0.359094\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.65841i		−	0.626370i	−0.949692	−	0.313185i	$-0.898604\pi$
$113$		−	6.65841i		−	0.626370i	0.949692	−	0.313185i	$-0.101396\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.0000	−	6.32456i	−0.916698	−	0.579771i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.4164			1.34694			0.673469	−	0.739216i	$-0.264804\pi$
$131$	15.4164			1.34694			0.673469	+	0.739216i	$0.264804\pi$
$132$		0			0
$133$	2.47214	−	3.90879i	0.214361	−	0.338935i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.99070i		−	0.511820i	−0.966701	−	0.255910i	$-0.917625\pi$
$137$		−	5.99070i		−	0.511820i	0.966701	−	0.255910i	$-0.0823751\pi$
$138$		0			0
$139$		−	21.5471i		−	1.82760i	−0.406168	−	0.913799i	$-0.633135\pi$
$139$		−	21.5471i		−	1.82760i	0.406168	−	0.913799i	$-0.366865\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.47214			0.206730
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	16.8918i		−	1.38383i	−0.721981	−	0.691913i	$-0.756768\pi$
$149$		−	16.8918i		−	1.38383i	0.721981	−	0.691913i	$-0.243232\pi$
$150$		0			0
$151$	−14.9443			−1.21615			−0.608074	−	0.793881i	$-0.708058\pi$
$151$	−14.9443			−1.21615			−0.608074	+	0.793881i	$0.708058\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	12.3941i		−	0.989155i	−0.869134	−	0.494577i	$-0.835323\pi$
$157$		−	12.3941i		−	0.989155i	0.869134	−	0.494577i	$-0.164677\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.47214	+	7.07107i	−0.352454	+	0.557278i
$162$		0			0
$163$	20.9443			1.64048			0.820241	−	0.572018i	$-0.193839\pi$
$163$	20.9443			1.64048			0.820241	+	0.572018i	$0.193839\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21.8885			−1.69379			−0.846893	−	0.531763i	$-0.821530\pi$
$167$	−21.8885			−1.69379			−0.846893	+	0.531763i	$0.821530\pi$
$168$		0			0
$169$	9.94427			0.764944
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.0000			−1.06440			−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000			−1.06440			−0.532200	+	0.846619i	$0.678635\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	17.5595i		−	1.31246i	−0.754563	−	0.656228i	$-0.772151\pi$
$179$		−	17.5595i		−	1.31246i	0.754563	−	0.656228i	$-0.227849\pi$
$180$		0			0
$181$		−	17.6383i		−	1.31104i	−0.755177	−	0.655521i	$-0.772449\pi$
$181$		−	17.6383i		−	1.31104i	0.755177	−	0.655521i	$-0.227551\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			6.32456i			0.462497i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.74962i			0.198955i	0.995040	+	0.0994776i	$0.0317172\pi$
$191$			2.74962i			0.198955i	−0.995040	+	0.0994776i	$0.968283\pi$
$192$		0			0
$193$	7.41641			0.533845			0.266922	−	0.963718i	$-0.413993\pi$
$193$	7.41641			0.533845			0.266922	+	0.963718i	$0.413993\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.99070i		−	0.426820i	−0.976963	−	0.213410i	$-0.931543\pi$
$197$		−	5.99070i		−	0.426820i	0.976963	−	0.213410i	$-0.0684570\pi$
$198$		0			0
$199$		−	14.3972i		−	1.02059i	−0.860000	−	0.510294i	$-0.829536\pi$
$199$		−	14.3972i		−	1.02059i	0.860000	−	0.510294i	$-0.170464\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.94427	+	4.65530i	−0.206647	+	0.326738i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.47214			−0.171001
$210$		0			0
$211$	−14.0000			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.47214	−	10.2333i	0.439357	−	0.694685i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	7.81758i		−	0.525867i
$222$		0			0
$223$			21.1344i			1.41526i	0.706581	+	0.707632i	$0.250236\pi$
$223$			21.1344i			1.41526i	−0.706581	+	0.707632i	$0.749764\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.52786			0.101408			0.0507039	−	0.998714i	$-0.483854\pi$
$227$	1.52786			0.101408			0.0507039	+	0.998714i	$0.483854\pi$
$228$		0			0
$229$		−	17.6383i		−	1.16557i	−0.812627	−	0.582785i	$-0.801963\pi$
$229$		−	17.6383i		−	1.16557i	0.812627	−	0.582785i	$-0.198037\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	27.1251i		−	1.77702i	−0.458853	−	0.888512i	$-0.651740\pi$
$233$		−	27.1251i		−	1.77702i	0.458853	−	0.888512i	$-0.348260\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	28.2055i		−	1.82446i	−0.409679	−	0.912230i	$-0.634359\pi$
$239$		−	28.2055i		−	1.82446i	0.409679	−	0.912230i	$-0.365641\pi$
$240$		0			0
$241$			19.7990i			1.27537i	0.770299	+	0.637683i	$0.220107\pi$
$241$			19.7990i			1.27537i	−0.770299	+	0.637683i	$0.779893\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.05573			0.194431
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.5279			1.35883			0.679413	−	0.733756i	$-0.262234\pi$
$251$	21.5279			1.35883			0.679413	+	0.733756i	$0.262234\pi$
$252$		0			0
$253$	4.47214			0.281161
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.4721			−1.02750			−0.513752	−	0.857939i	$-0.671745\pi$
$257$	−16.4721			−1.02750			−0.513752	+	0.857939i	$0.671745\pi$
$258$		0			0
$259$	3.41641	+	2.16073i	0.212285	+	0.134261i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	19.3075i		−	1.19055i	−0.803521	−	0.595276i	$-0.797043\pi$
$263$		−	19.3075i		−	1.19055i	0.803521	−	0.595276i	$-0.202957\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−19.5279			−1.19063			−0.595317	−	0.803491i	$-0.702974\pi$
$269$	−19.5279			−1.19063			−0.595317	+	0.803491i	$0.702974\pi$
$270$		0			0
$271$		−	11.5687i		−	0.702751i	−0.936235	−	0.351376i	$-0.885714\pi$
$271$		−	11.5687i		−	0.702751i	0.936235	−	0.351376i	$-0.114286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	6.00000			0.360505			0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000			0.360505			0.180253	+	0.983620i	$0.442309\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6.40337i			0.381993i	0.981591	+	0.190996i	$0.0611719\pi$
$281$			6.40337i			0.381993i	−0.981591	+	0.190996i	$0.938828\pi$
$282$		0			0
$283$		−	1.49302i		−	0.0887511i	−0.999015	−	0.0443756i	$-0.985870\pi$
$283$		−	1.49302i		−	0.0887511i	0.999015	−	0.0443756i	$-0.0141298\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.05573	+	0.667701i	0.0623177	+	0.0394131i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−29.4164			−1.71852			−0.859262	−	0.511535i	$-0.829077\pi$
$293$	−29.4164			−1.71852			−0.859262	+	0.511535i	$0.829077\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.52786			−0.319685
$300$		0			0
$301$	−1.05573	−	0.667701i	−0.0608512	−	0.0384856i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			32.4481i			1.85191i	0.377633	+	0.925955i	$0.376738\pi$
$307$			32.4481i			1.85191i	−0.377633	+	0.925955i	$0.623262\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.3607			1.38137			0.690684	−	0.723157i	$-0.257310\pi$
$311$	24.3607			1.38137			0.690684	+	0.723157i	$0.257310\pi$
$312$		0			0
$313$			1.08036i			0.0610657i	0.999534	+	0.0305329i	$0.00972042\pi$
$313$			1.08036i			0.0610657i	−0.999534	+	0.0305329i	$0.990280\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.49768i		−	0.252615i	−0.991991	−	0.126307i	$-0.959687\pi$
$317$		−	4.49768i		−	0.252615i	0.991991	−	0.126307i	$-0.0403126\pi$
$318$		0			0
$319$	2.94427			0.164848
$320$		0			0
$321$		0			0
$322$		0			0
$323$			7.81758i			0.434982i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.52786	−	3.49613i	−0.304761	−	0.192748i
$330$		0			0
$331$	−11.8885			−0.653453			−0.326727	−	0.945119i	$-0.605946\pi$
$331$	−11.8885			−0.653453			−0.326727	+	0.945119i	$0.605946\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.4164			0.621891			0.310946	−	0.950428i	$-0.399354\pi$
$337$	11.4164			0.621891			0.310946	+	0.950428i	$0.399354\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.47214			−0.350486
$342$		0			0
$343$	2.23607	−	18.3848i	0.120736	−	0.992685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			22.9613i			1.23263i	0.787502	+	0.616313i	$0.211374\pi$
$347$			22.9613i			1.23263i	−0.787502	+	0.616313i	$0.788626\pi$
$348$		0			0
$349$		−	22.6274i		−	1.21122i	−0.795762	−	0.605609i	$-0.792930\pi$
$349$		−	22.6274i		−	1.21122i	0.795762	−	0.605609i	$-0.207070\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.2272i		−	0.961992i	−0.876723	−	0.480996i	$-0.840275\pi$
$359$		−	18.2272i		−	0.961992i	0.876723	−	0.480996i	$-0.159725\pi$
$360$		0			0
$361$	15.9443			0.839172
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$			2.82843i			0.147643i	0.997271	+	0.0738213i	$0.0235195\pi$
$367$			2.82843i			0.147643i	−0.997271	+	0.0738213i	$0.976481\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.4721	+	19.7202i	−0.647521	+	1.02382i
$372$		0			0
$373$	26.9443			1.39512			0.697561	−	0.716526i	$-0.254269\pi$
$373$	26.9443			1.39512			0.697561	+	0.716526i	$0.254269\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.63932			−0.187435
$378$		0			0
$379$	11.8885			0.610673			0.305337	−	0.952244i	$-0.401231\pi$
$379$	11.8885			0.610673			0.305337	+	0.952244i	$0.401231\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.9443			−0.865812			−0.432906	−	0.901439i	$-0.642512\pi$
$383$	−16.9443			−0.865812			−0.432906	+	0.901439i	$0.642512\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	19.7202i		−	0.999853i	−0.866068	−	0.499926i	$-0.833360\pi$
$389$		−	19.7202i		−	0.999853i	0.866068	−	0.499926i	$-0.166640\pi$
$390$		0			0
$391$		−	14.1421i		−	0.715199i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			13.7295i			0.689063i	0.938775	+	0.344531i	$0.111962\pi$
$397$			13.7295i			0.689063i	−0.938775	+	0.344531i	$0.888038\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	28.2055i		−	1.40851i	−0.709945	−	0.704257i	$-0.751280\pi$
$401$		−	28.2055i		−	1.40851i	0.709945	−	0.704257i	$-0.248720\pi$
$402$		0			0
$403$	8.00000			0.398508
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.16073i		−	0.107103i
$408$		0			0
$409$			35.9442i			1.77733i	0.458559	+	0.888664i	$0.348366\pi$
$409$			35.9442i			1.77733i	−0.458559	+	0.888664i	$0.651634\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.41641	+	2.16073i	0.168110	+	0.106322i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	22.8328			1.11546			0.557728	−	0.830024i	$-0.311673\pi$
$419$	22.8328			1.11546			0.557728	+	0.830024i	$0.311673\pi$
$420$		0			0
$421$	−19.8885			−0.969308			−0.484654	−	0.874706i	$-0.661055\pi$
$421$	−19.8885			−0.969308			−0.484654	+	0.874706i	$0.661055\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.94427	−	7.81758i	0.239270	−	0.378319i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			17.5595i			0.845809i	0.906174	+	0.422905i	$0.138989\pi$
$431$			17.5595i			0.845809i	−0.906174	+	0.422905i	$0.861011\pi$
$432$		0			0
$433$		−	0.255039i		−	0.0122564i	−0.999981	−	0.00612820i	$-0.998049\pi$
$433$		−	0.255039i		−	0.0122564i	0.999981	−	0.00612820i	$-0.00195068\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.52786			0.264434
$438$		0			0
$439$		−	13.0618i		−	0.623404i	−0.950180	−	0.311702i	$-0.899101\pi$
$439$		−	13.0618i		−	0.623404i	0.950180	−	0.311702i	$-0.100899\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	33.6073i		−	1.59673i	−0.602174	−	0.798365i	$-0.705699\pi$
$443$		−	33.6073i		−	1.59673i	0.602174	−	0.798365i	$-0.294301\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			15.5563i			0.734150i	0.930191	+	0.367075i	$0.119641\pi$
$449$			15.5563i			0.734150i	−0.930191	+	0.367075i	$0.880359\pi$
$450$		0			0
$451$		−	0.667701i		−	0.0314408i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.4721			−1.05120			−0.525601	−	0.850731i	$-0.676160\pi$
$457$	−22.4721			−1.05120			−0.525601	+	0.850731i	$0.676160\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.8885			0.740003			0.370002	−	0.929031i	$-0.379357\pi$
$461$	15.8885			0.740003			0.370002	+	0.929031i	$0.379357\pi$
$462$		0			0
$463$	−7.05573			−0.327907			−0.163954	−	0.986468i	$-0.552425\pi$
$463$	−7.05573			−0.327907			−0.163954	+	0.986468i	$0.552425\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	31.4164			1.45378			0.726889	−	0.686755i	$-0.240965\pi$
$467$	31.4164			1.45378			0.726889	+	0.686755i	$0.240965\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.667701i			0.0307009i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.3607			−1.11307			−0.556534	−	0.830825i	$-0.687869\pi$
$479$	−24.3607			−1.11307			−0.556534	+	0.830825i	$0.687869\pi$
$480$		0			0
$481$			2.67080i			0.121778i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.52786			−0.159863			−0.0799314	−	0.996800i	$-0.525470\pi$
$487$	−3.52786			−0.159863			−0.0799314	+	0.996800i	$0.525470\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			23.2163i			1.04774i	0.851799	+	0.523869i	$0.175512\pi$
$491$			23.2163i			1.04774i	−0.851799	+	0.523869i	$0.824488\pi$
$492$		0			0
$493$		−	9.31061i		−	0.419329i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.9443	−	17.3044i	0.490918	−	0.776209i
$498$		0			0
$499$	14.9443			0.668997			0.334499	−	0.942396i	$-0.391433\pi$
$499$	14.9443			0.668997			0.334499	+	0.942396i	$0.391433\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0557			−0.671302			−0.335651	−	0.941986i	$-0.608956\pi$
$503$	−15.0557			−0.671302			−0.335651	+	0.941986i	$0.608956\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	26.3607			1.16842			0.584208	−	0.811604i	$-0.301405\pi$
$509$	26.3607			1.16842			0.584208	+	0.811604i	$0.301405\pi$
$510$		0			0
$511$	−23.4164	+	37.0246i	−1.03588	+	1.63787i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			3.49613i			0.153760i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.3607			−1.85586			−0.927928	−	0.372761i	$-0.878411\pi$
$521$	−42.3607			−1.85586			−0.927928	+	0.372761i	$0.878411\pi$
$522$		0			0
$523$		−	27.4589i		−	1.20070i	−0.799739	−	0.600348i	$-0.795029\pi$
$523$		−	27.4589i		−	1.20070i	0.799739	−	0.600348i	$-0.204971\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			20.4667i			0.891543i
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$			0.825324i			0.0357487i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.94427	+	4.24264i	−0.385257	+	0.182743i
$540$		0			0
$541$	27.8885			1.19902			0.599511	−	0.800366i	$-0.295362\pi$
$541$	27.8885			1.19902			0.599511	+	0.800366i	$0.295362\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−22.8328			−0.976261			−0.488130	−	0.872771i	$-0.662321\pi$
$547$	−22.8328			−0.976261			−0.488130	+	0.872771i	$0.662321\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.63932			0.155040
$552$		0			0
$553$	−20.0000	−	12.6491i	−0.850487	−	0.537895i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	15.1437i		−	0.641659i	−0.947137	−	0.320829i	$-0.896038\pi$
$557$		−	15.1437i		−	0.641659i	0.947137	−	0.320829i	$-0.103962\pi$
$558$		0			0
$559$		−	0.825324i		−	0.0349075i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.8885			−0.922492			−0.461246	−	0.887272i	$-0.652597\pi$
$563$	−21.8885			−0.922492			−0.461246	+	0.887272i	$0.652597\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			11.2349i			0.470991i	0.971875	+	0.235496i	$0.0756714\pi$
$569$			11.2349i			0.470991i	−0.971875	+	0.235496i	$0.924329\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	36.3569i		−	1.51356i	−0.653671	−	0.756779i	$-0.726772\pi$
$577$		−	36.3569i		−	1.51356i	0.653671	−	0.756779i	$-0.273228\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.3607	+	7.81758i	0.512807	+	0.324328i
$582$		0			0
$583$	12.4721			0.516543
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.8328			0.777313			0.388657	−	0.921383i	$-0.372939\pi$
$587$	18.8328			0.777313			0.388657	+	0.921383i	$0.372939\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$		0			0
$592$		0			0
$593$	23.8885			0.980985			0.490492	−	0.871445i	$-0.336817\pi$
$593$	23.8885			0.980985			0.490492	+	0.871445i	$0.336817\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.41421i			0.0577832i	0.999583	+	0.0288916i	$0.00919776\pi$
$599$			1.41421i			0.0577832i	−0.999583	+	0.0288916i	$0.990802\pi$
$600$		0			0
$601$			1.33540i			0.0544722i	0.999629	+	0.0272361i	$0.00867059\pi$
$601$			1.33540i			0.0544722i	−0.999629	+	0.0272361i	$0.991329\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	25.9659i		−	1.05392i	−0.849889	−	0.526962i	$-0.823331\pi$
$607$		−	25.9659i		−	1.05392i	0.849889	−	0.526962i	$-0.176669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	4.32145i		−	0.174827i
$612$		0			0
$613$	14.9443			0.603593			0.301797	−	0.953372i	$-0.402414\pi$
$613$	14.9443			0.603593			0.301797	+	0.953372i	$0.402414\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	15.8114i		−	0.636543i	−0.948000	−	0.318271i	$-0.896898\pi$
$617$		−	15.8114i		−	0.636543i	0.948000	−	0.318271i	$-0.103102\pi$
$618$		0			0
$619$			22.0571i			0.886551i	0.896385	+	0.443275i	$0.146184\pi$
$619$			22.0571i			0.886551i	−0.896385	+	0.443275i	$0.853816\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.4721	+	15.4775i	0.980455	+	0.620094i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.83282			−0.272442
$630$		0			0
$631$	−8.94427			−0.356066			−0.178033	−	0.984025i	$-0.556973\pi$
$631$	−8.94427			−0.356066			−0.178033	+	0.984025i	$0.556973\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	11.0557	−	5.24419i	0.438044	−	0.207782i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.8949i		−	0.746302i	−0.927771	−	0.373151i	$-0.878277\pi$
$641$		−	18.8949i		−	0.746302i	0.927771	−	0.373151i	$-0.121723\pi$
$642$		0			0
$643$			10.6460i			0.419838i	0.977719	+	0.209919i	$0.0673200\pi$
$643$			10.6460i			0.419838i	−0.977719	+	0.209919i	$0.932680\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.00000			0.314512			0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000			0.314512			0.157256	+	0.987558i	$0.449735\pi$
$648$		0			0
$649$		−	2.16073i		−	0.0848159i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	32.1142i		−	1.25673i	−0.777920	−	0.628364i	$-0.783725\pi$
$653$		−	32.1142i		−	1.25673i	0.777920	−	0.628364i	$-0.216275\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	0.588890i		−	0.0229399i	−0.999934	−	0.0114699i	$-0.996349\pi$
$659$		−	0.588890i		−	0.0229399i	0.999934	−	0.0114699i	$-0.00365108\pi$
$660$		0			0
$661$			2.16073i			0.0840425i	0.999117	+	0.0420213i	$0.0133797\pi$
$661$			2.16073i			0.0840425i	−0.999117	+	0.0420213i	$0.986620\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.58359			−0.254918
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.94427			−0.190872
$672$		0			0
$673$	−34.4721			−1.32880			−0.664402	−	0.747376i	$-0.731314\pi$
$673$	−34.4721			−1.32880			−0.664402	+	0.747376i	$0.731314\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.4164			−1.43803			−0.719015	−	0.694995i	$-0.755407\pi$
$677$	−37.4164			−1.43803			−0.719015	+	0.694995i	$0.755407\pi$
$678$		0			0
$679$	0.583592	−	0.922740i	0.0223962	−	0.0354115i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	15.9690i		−	0.611037i	−0.952186	−	0.305519i	$-0.901170\pi$
$683$		−	15.9690i		−	0.611037i	0.952186	−	0.305519i	$-0.0988298\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.4164			−0.587318
$690$		0			0
$691$			25.8685i			0.984084i	0.870571	+	0.492042i	$0.163749\pi$
$691$			25.8685i			0.984084i	−0.870571	+	0.492042i	$0.836251\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.11146			−0.0799771
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7.89639i			0.298243i	0.988819	+	0.149121i	$0.0476445\pi$
$701$			7.89639i			0.298243i	−0.988819	+	0.149121i	$0.952356\pi$
$702$		0			0
$703$		−	2.67080i		−	0.100731i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.9443	+	11.9814i	0.712473	+	0.450607i
$708$		0			0
$709$	0.944272			0.0354629			0.0177314	−	0.999843i	$-0.494356\pi$
$709$	0.944272			0.0354629			0.0177314	+	0.999843i	$0.494356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.4721			0.541986
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.4721			−0.838069			−0.419035	−	0.907970i	$-0.637631\pi$
$719$	−22.4721			−0.838069			−0.419035	+	0.907970i	$0.637631\pi$
$720$		0			0
$721$	4.94427	−	7.81758i	0.184134	−	0.291142i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			16.9706i			0.629403i	0.949191	+	0.314702i	$0.101904\pi$
$727$			16.9706i			0.629403i	−0.949191	+	0.314702i	$0.898096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.11146			0.0780950
$732$		0			0
$733$			16.5579i			0.611580i	0.952099	+	0.305790i	$0.0989206\pi$
$733$			16.5579i			0.611580i	−0.952099	+	0.305790i	$0.901079\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	3.05573			0.112407			0.0562034	−	0.998419i	$-0.482100\pi$
$739$	3.05573			0.112407			0.0562034	+	0.998419i	$0.482100\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			46.2564i			1.69698i	0.529210	+	0.848491i	$0.322489\pi$
$743$			46.2564i			1.69698i	−0.529210	+	0.848491i	$0.677511\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.47214	−	7.07107i	0.163408	−	0.258371i
$750$		0			0
$751$	34.0000			1.24068			0.620339	−	0.784334i	$-0.286995\pi$
$751$	34.0000			1.24068			0.620339	+	0.784334i	$0.286995\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	18.4721			0.671381			0.335691	−	0.941972i	$-0.391030\pi$
$757$	18.4721			0.671381			0.335691	+	0.941972i	$0.391030\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.8885			−0.430959			−0.215480	−	0.976508i	$-0.569132\pi$
$761$	−11.8885			−0.430959			−0.215480	+	0.976508i	$0.569132\pi$
$762$		0			0
$763$	−20.0000	−	12.6491i	−0.724049	−	0.457929i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			2.67080i			0.0964372i
$768$		0			0
$769$			16.9706i			0.611974i	0.952036	+	0.305987i	$0.0989864\pi$
$769$			16.9706i			0.611974i	−0.952036	+	0.305987i	$0.901014\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	0.825324i		−	0.0295703i
$780$		0			0
$781$	−10.9443			−0.391617
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			4.32145i			0.154043i	0.997029	+	0.0770216i	$0.0245410\pi$
$787$			4.32145i			0.154043i	−0.997029	+	0.0770216i	$0.975459\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−9.41641	+	14.8886i	−0.334809	+	0.529379i
$792$		0			0
$793$	6.11146			0.217024
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.8328			−0.879623			−0.439812	−	0.898090i	$-0.644955\pi$
$797$	−24.8328			−0.879623			−0.439812	+	0.898090i	$0.644955\pi$
$798$		0			0
$799$	11.0557			0.391124
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.4164			0.826347
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			30.2086i			1.06208i	0.847348	+	0.531038i	$0.178198\pi$
$809$			30.2086i			1.06208i	−0.847348	+	0.531038i	$0.821802\pi$
$810$		0			0
$811$			9.40802i			0.330360i	0.986263	+	0.165180i	$0.0528206\pi$
$811$			9.40802i			0.330360i	−0.986263	+	0.165180i	$0.947179\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$			0.825324i			0.0288744i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	52.9936i		−	1.84949i	−0.380588	−	0.924745i	$-0.624278\pi$
$821$		−	52.9936i		−	1.84949i	0.380588	−	0.924745i	$-0.375722\pi$
$822$		0			0
$823$	2.11146			0.0736007			0.0368004	−	0.999323i	$-0.488283\pi$
$823$	2.11146			0.0736007			0.0368004	+	0.999323i	$0.488283\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	24.4543i		−	0.850359i	−0.905109	−	0.425180i	$-0.860211\pi$
$827$		−	24.4543i		−	0.850359i	0.905109	−	0.425180i	$-0.139789\pi$
$828$		0			0
$829$			5.81448i			0.201945i	0.994889	+	0.100973i	$0.0321954\pi$
$829$			5.81448i			0.201945i	−0.994889	+	0.100973i	$0.967805\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.4164	+	28.2843i	0.464851	+	0.979992i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.8885			−0.755676			−0.377838	−	0.925872i	$-0.623332\pi$
$839$	−21.8885			−0.755676			−0.377838	+	0.925872i	$0.623332\pi$
$840$		0			0
$841$	24.6656			0.850539
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−20.1246	−	12.7279i	−0.691490	−	0.437337i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			4.83153i			0.165623i
$852$		0			0
$853$			11.5687i			0.396106i	0.980191	+	0.198053i	$0.0634619\pi$
$853$			11.5687i			0.396106i	−0.980191	+	0.198053i	$0.936538\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.94427			−0.100574			−0.0502872	−	0.998735i	$-0.516014\pi$
$857$	−2.94427			−0.100574			−0.0502872	+	0.998735i	$0.516014\pi$
$858$		0			0
$859$			30.0323i			1.02469i	0.858780	+	0.512345i	$0.171223\pi$
$859$			30.0323i			1.02469i	−0.858780	+	0.512345i	$0.828777\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	42.7603i		−	1.45558i	−0.685802	−	0.727788i	$-0.740548\pi$
$863$		−	42.7603i		−	1.45558i	0.685802	−	0.727788i	$-0.259452\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			12.6491i			0.429092i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	33.7771			1.14057			0.570286	−	0.821446i	$-0.306833\pi$
$877$	33.7771			1.14057			0.570286	+	0.821446i	$0.306833\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.8328			0.432349			0.216174	−	0.976355i	$-0.430642\pi$
$881$	12.8328			0.432349			0.216174	+	0.976355i	$0.430642\pi$
$882$		0			0
$883$	−20.4721			−0.688942			−0.344471	−	0.938797i	$-0.611942\pi$
$883$	−20.4721			−0.688942			−0.344471	+	0.938797i	$0.611942\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.3050			−0.581043			−0.290522	−	0.956868i	$-0.593829\pi$
$887$	−17.3050			−0.581043			−0.290522	+	0.956868i	$0.593829\pi$
$888$		0			0
$889$	−8.94427	−	5.65685i	−0.299981	−	0.189725i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			4.32145i			0.144612i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.52786			0.317772
$900$		0			0
$901$		−	39.4404i		−	1.31395i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−45.1935			−1.50063			−0.750313	−	0.661083i	$-0.770097\pi$
$907$	−45.1935			−1.50063			−0.750313	+	0.661083i	$0.770097\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			45.1760i			1.49675i	0.663277	+	0.748374i	$0.269165\pi$
$911$			45.1760i			1.49675i	−0.663277	+	0.748374i	$0.730835\pi$
$912$		0			0
$913$		−	7.81758i		−	0.258724i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−34.4721	−	21.8021i	−1.13837	−	0.719969i
$918$		0			0
$919$	−2.94427			−0.0971226			−0.0485613	−	0.998820i	$-0.515464\pi$
$919$	−2.94427			−0.0971226			−0.0485613	+	0.998820i	$0.515464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.5279			0.445275
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	37.0557			1.21576			0.607880	−	0.794029i	$-0.292020\pi$
$929$	37.0557			1.21576			0.607880	+	0.794029i	$0.292020\pi$
$930$		0			0
$931$	−11.0557	+	5.24419i	−0.362337	+	0.171871i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	21.5471i		−	0.703912i	−0.936017	−	0.351956i	$-0.885517\pi$
$937$		−	21.5471i		−	0.703912i	0.936017	−	0.351956i	$-0.114483\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.83282			0.157545			0.0787726	−	0.996893i	$-0.474900\pi$
$941$	4.83282			0.157545			0.0787726	+	0.996893i	$0.474900\pi$
$942$		0			0
$943$			1.49302i			0.0486196i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			48.9272i			1.58992i	0.606662	+	0.794960i	$0.292508\pi$
$947$			48.9272i			1.58992i	−0.606662	+	0.794960i	$0.707492\pi$
$948$		0			0
$949$	−28.9443			−0.939571
$950$		0			0
$951$		0			0
$952$		0			0
$953$			55.9194i			1.81141i	0.423910	+	0.905704i	$0.360657\pi$
$953$			55.9194i			1.81141i	−0.423910	+	0.905704i	$0.639343\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.47214	+	13.3956i	−0.273580	+	0.432567i
$960$		0			0
$961$	10.0557			0.324378
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	29.8885			0.961151			0.480575	−	0.876953i	$-0.340428\pi$
$967$	29.8885			0.961151			0.480575	+	0.876953i	$0.340428\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−32.0000			−1.02693			−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000			−1.02693			−0.513464	+	0.858111i	$0.671638\pi$
$972$		0			0
$973$	−30.4721	+	48.1807i	−0.976892	+	1.54460i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	28.4605i		−	0.910532i	−0.890355	−	0.455266i	$-0.849544\pi$
$977$		−	28.4605i		−	0.910532i	0.890355	−	0.455266i	$-0.150456\pi$
$978$		0			0
$979$		−	15.4775i		−	0.494664i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.583592			−0.0186137			−0.00930685	−	0.999957i	$-0.502963\pi$
$983$	−0.583592			−0.0186137			−0.00930685	+	0.999957i	$0.502963\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.49302i		−	0.0474754i
$990$		0			0
$991$	1.05573			0.0335363			0.0167682	−	0.999859i	$-0.494662\pi$
$991$	1.05573			0.0335363			0.0167682	+	0.999859i	$0.494662\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			20.7217i			0.656264i	0.944632	+	0.328132i	$0.106419\pi$
$997$			20.7217i			0.656264i	−0.944632	+	0.328132i	$0.893581\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.d.b.3401.1		4
3.2	odd	2		6300.2.d.a.3401.1		4
5.2	odd	4		6300.2.f.c.3149.6		8
5.3	odd	4		6300.2.f.c.3149.4		8
5.4	even	2		1260.2.d.a.881.4	yes	4
7.6	odd	2		6300.2.d.a.3401.2		4
15.2	even	4		6300.2.f.a.3149.5		8
15.8	even	4		6300.2.f.a.3149.3		8
15.14	odd	2		1260.2.d.b.881.4	yes	4
20.19	odd	2		5040.2.f.b.881.1		4
21.20	even	2	inner	6300.2.d.b.3401.2		4
35.13	even	4		6300.2.f.a.3149.8		8
35.27	even	4		6300.2.f.a.3149.2		8
35.34	odd	2		1260.2.d.b.881.3	yes	4
60.59	even	2		5040.2.f.d.881.1		4
105.62	odd	4		6300.2.f.c.3149.1		8
105.83	odd	4		6300.2.f.c.3149.7		8
105.104	even	2		1260.2.d.a.881.3	✓	4
140.139	even	2		5040.2.f.d.881.2		4
420.419	odd	2		5040.2.f.b.881.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.d.a.881.3	✓	4	105.104	even	2
1260.2.d.a.881.4	yes	4	5.4	even	2
1260.2.d.b.881.3	yes	4	35.34	odd	2
1260.2.d.b.881.4	yes	4	15.14	odd	2
5040.2.f.b.881.1		4	20.19	odd	2
5040.2.f.b.881.2		4	420.419	odd	2
5040.2.f.d.881.1		4	60.59	even	2
5040.2.f.d.881.2		4	140.139	even	2
6300.2.d.a.3401.1		4	3.2	odd	2
6300.2.d.a.3401.2		4	7.6	odd	2
6300.2.d.b.3401.1		4	1.1	even	1	trivial
6300.2.d.b.3401.2		4	21.20	even	2	inner
6300.2.f.a.3149.2		8	35.27	even	4
6300.2.f.a.3149.3		8	15.8	even	4
6300.2.f.a.3149.5		8	15.2	even	4
6300.2.f.a.3149.8		8	35.13	even	4
6300.2.f.c.3149.1		8	105.62	odd	4
6300.2.f.c.3149.4		8	5.3	odd	4
6300.2.f.c.3149.6		8	5.2	odd	4
6300.2.f.c.3149.7		8	105.83	odd	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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.23607 - 1.41421i) q^{7} +O(q^{10})\)	\(q+(-2.23607 - 1.41421i) q^{7} +1.41421i q^{11} -1.74806i q^{13} +4.47214 q^{17} +1.74806i q^{19} -3.16228i q^{23} -2.08191i q^{29} +4.57649i q^{31} -1.52786 q^{37} -0.472136 q^{41} +0.472136 q^{43} +2.47214 q^{47} +(3.00000 + 6.32456i) q^{49} -8.81913i q^{53} -1.52786 q^{59} +3.49613i q^{61} +7.73877i q^{71} -16.5579i q^{73} +(2.00000 - 3.16228i) q^{77} +8.94427 q^{79} -5.52786 q^{83} -10.9443 q^{89} +(-2.47214 + 3.90879i) q^{91} +0.412662i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{37} + 16 q^{41} - 16 q^{43} - 8 q^{47} + 12 q^{49} - 24 q^{59} + 8 q^{77} - 40 q^{83} - 8 q^{89} + 8 q^{91}+O(q^{100}) \) 4 * q - 24 * q^37 + 16 * q^41 - 16 * q^43 - 8 * q^47 + 12 * q^49 - 24 * q^59 + 8 * q^77 - 40 * q^83 - 8 * q^89 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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