LMFDB - Embedded newform 6300.2.f.c.3149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(3149,6300)]

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, 1, 1]))

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

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

chi := DirichletCharacter("6300.3149");

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.f (of order $2$, degree $1$, not 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:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3149.1
Root			$0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	6300.3149
Dual form			6300.2.f.c.3149.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 2.23607i) q^{7} +O(q^{10})$	$q+(-1.41421 - 2.23607i) q^{7} -1.41421i q^{11} +1.74806 q^{13} +4.47214i q^{17} +1.74806i q^{19} +3.16228 q^{23} -2.08191i q^{29} -4.57649i q^{31} -1.52786i q^{37} -0.472136 q^{41} -0.472136i q^{43} +2.47214i q^{47} +(-3.00000 + 6.32456i) q^{49} +8.81913 q^{53} +1.52786 q^{59} -3.49613i q^{61} -7.73877i q^{71} +16.5579 q^{73} +(-3.16228 + 2.00000i) q^{77} -8.94427 q^{79} +5.52786i q^{83} +10.9443 q^{89} +(-2.47214 - 3.90879i) q^{91} +0.412662 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 32 q^{41} - 24 q^{49} + 48 q^{59} + 16 q^{89} + 16 q^{91}+O(q^{100}) $ 8 * q + 32 * q^41 - 24 * q^49 + 48 * q^59 + 16 * q^89 + 16 * 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
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$	1.74806			0.484826			0.242413	−	0.970173i	$-0.422061\pi$
$13$	1.74806			0.484826			0.242413	+	0.970173i	$0.422061\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.47214i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$			4.47214i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$		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
$23$	3.16228			0.659380			0.329690	−	0.944089i	$-0.393056\pi$
$23$	3.16228			0.659380			0.329690	+	0.944089i	$0.393056\pi$
$24$		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	$-0.865186\pi$
$31$		−	4.57649i		−	0.821962i	0.911644	−	0.410981i	$-0.134814\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	1.52786i		−	0.251179i	−0.992082	−	0.125590i	$-0.959918\pi$
$37$		−	1.52786i		−	0.251179i	0.992082	−	0.125590i	$-0.0400823\pi$
$38$		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
$43$		−	0.472136i		−	0.0720001i	−0.999352	−	0.0360000i	$-0.988538\pi$
$43$		−	0.472136i		−	0.0720001i	0.999352	−	0.0360000i	$-0.0114616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.47214i			0.360598i	0.983612	+	0.180299i	$0.0577065\pi$
$47$			2.47214i			0.360598i	−0.983612	+	0.180299i	$0.942293\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.81913			1.21140			0.605700	−	0.795693i	$-0.292893\pi$
$53$	8.81913			1.21140			0.605700	+	0.795693i	$0.292893\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.52786			0.198911			0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786			0.198911			0.0994555	+	0.995042i	$0.468290\pi$
$60$		0			0
$61$		−	3.49613i		−	0.447633i	−0.974631	−	0.223817i	$-0.928148\pi$
$61$		−	3.49613i		−	0.447633i	0.974631	−	0.223817i	$-0.0718517\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.73877i		−	0.918423i	−0.888327	−	0.459211i	$-0.848132\pi$
$71$		−	7.73877i		−	0.918423i	0.888327	−	0.459211i	$-0.151868\pi$
$72$		0			0
$73$	16.5579			1.93796			0.968978	−	0.247148i	$-0.0794932\pi$
$73$	16.5579			1.93796			0.968978	+	0.247148i	$0.0794932\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.16228	+	2.00000i	−0.360375	+	0.227921i
$78$		0			0
$79$	−8.94427			−1.00631			−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427			−1.00631			−0.503155	+	0.864196i	$0.667827\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.52786i			0.606762i	0.952869	+	0.303381i	$0.0981155\pi$
$83$			5.52786i			0.606762i	−0.952869	+	0.303381i	$0.901885\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.9443			1.16009			0.580045	−	0.814584i	$-0.303035\pi$
$89$	10.9443			1.16009			0.580045	+	0.814584i	$0.303035\pi$
$90$		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
$97$	0.412662			0.0418995			0.0209497	−	0.999781i	$-0.493331\pi$
$97$	0.412662			0.0418995			0.0209497	+	0.999781i	$0.493331\pi$
$98$		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
$103$	−3.49613			−0.344484			−0.172242	−	0.985055i	$-0.555101\pi$
$103$	−3.49613			−0.344484			−0.172242	+	0.985055i	$0.555101\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.16228			0.305709			0.152854	−	0.988249i	$-0.451153\pi$
$107$	3.16228			0.305709			0.152854	+	0.988249i	$0.451153\pi$
$108$		0			0
$109$	−8.94427			−0.856706			−0.428353	−	0.903612i	$-0.640906\pi$
$109$	−8.94427			−0.856706			−0.428353	+	0.903612i	$0.640906\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.65841			0.626370			0.313185	−	0.949692i	$-0.398604\pi$
$113$	6.65841			0.626370			0.313185	+	0.949692i	$0.398604\pi$
$114$		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
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$		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
$133$	3.90879	−	2.47214i	0.338935	−	0.214361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.99070			−0.511820			−0.255910	−	0.966701i	$-0.582375\pi$
$137$	−5.99070			−0.511820			−0.255910	+	0.966701i	$0.582375\pi$
$138$		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
$143$		−	2.47214i		−	0.206730i
$144$		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
$157$	−12.3941			−0.989155			−0.494577	−	0.869134i	$-0.664677\pi$
$157$	−12.3941			−0.989155			−0.494577	+	0.869134i	$0.664677\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.47214	−	7.07107i	−0.352454	−	0.557278i
$162$		0			0
$163$		−	20.9443i		−	1.64048i	−0.572018	−	0.820241i	$-0.693839\pi$
$163$		−	20.9443i		−	1.64048i	0.572018	−	0.820241i	$-0.306161\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	21.8885i		−	1.69379i	−0.531763	−	0.846893i	$-0.678470\pi$
$167$		−	21.8885i		−	1.69379i	0.531763	−	0.846893i	$-0.321530\pi$
$168$		0			0
$169$	−9.94427			−0.764944
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		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	$0.227551\pi$
$181$			17.6383i			1.31104i	−0.755177	+	0.655521i	$0.772449\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	6.32456			0.462497
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	2.74962i		−	0.198955i	−0.995040	−	0.0994776i	$-0.968283\pi$
$191$		−	2.74962i		−	0.198955i	0.995040	−	0.0994776i	$-0.0317172\pi$
$192$		0			0
$193$		−	7.41641i		−	0.533845i	−0.963718	−	0.266922i	$-0.913993\pi$
$193$		−	7.41641i		−	0.533845i	0.963718	−	0.266922i	$-0.0860067\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.99070			−0.426820			−0.213410	−	0.976963i	$-0.568457\pi$
$197$	−5.99070			−0.426820			−0.213410	+	0.976963i	$0.568457\pi$
$198$		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
$203$	−4.65530	+	2.94427i	−0.326738	+	0.206647i
$204$		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
$217$	−10.2333	+	6.47214i	−0.694685	+	0.439357i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			7.81758i			0.525867i
$222$		0			0
$223$	−21.1344			−1.41526			−0.707632	−	0.706581i	$-0.750236\pi$
$223$	−21.1344			−1.41526			−0.707632	+	0.706581i	$0.750236\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1.52786i			0.101408i	0.998714	+	0.0507039i	$0.0161465\pi$
$227$			1.52786i			0.101408i	−0.998714	+	0.0507039i	$0.983854\pi$
$228$		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
$233$	27.1251			1.77702			0.888512	−	0.458853i	$-0.151740\pi$
$233$	27.1251			1.77702			0.888512	+	0.458853i	$0.151740\pi$
$234$		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	$-0.779893\pi$
$241$		−	19.7990i		−	1.27537i	0.770299	−	0.637683i	$-0.220107\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$			3.05573i			0.194431i
$248$		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
$253$		−	4.47214i		−	0.281161i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	16.4721i		−	1.02750i	−0.857939	−	0.513752i	$-0.828255\pi$
$257$		−	16.4721i		−	1.02750i	0.857939	−	0.513752i	$-0.171745\pi$
$258$		0			0
$259$	−3.41641	+	2.16073i	−0.212285	+	0.134261i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	19.3075			1.19055			0.595276	−	0.803521i	$-0.297043\pi$
$263$	19.3075			1.19055			0.595276	+	0.803521i	$0.297043\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	19.5279			1.19063			0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279			1.19063			0.595317	+	0.803491i	$0.297026\pi$
$270$		0			0
$271$			11.5687i			0.702751i	0.936235	+	0.351376i	$0.114286\pi$
$271$			11.5687i			0.702751i	−0.936235	+	0.351376i	$0.885714\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			6.00000i			0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$			6.00000i			0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6.40337i		−	0.381993i	−0.981591	−	0.190996i	$-0.938828\pi$
$281$		−	6.40337i		−	0.381993i	0.981591	−	0.190996i	$-0.0611719\pi$
$282$		0			0
$283$	1.49302			0.0887511			0.0443756	−	0.999015i	$-0.485870\pi$
$283$	1.49302			0.0887511			0.0443756	+	0.999015i	$0.485870\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.667701	+	1.05573i	0.0394131	+	0.0623177i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$			29.4164i			1.71852i	0.511535	+	0.859262i	$0.329077\pi$
$293$			29.4164i			1.71852i	−0.511535	+	0.859262i	$0.670923\pi$
$294$		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
$307$	32.4481			1.85191			0.925955	−	0.377633i	$-0.123262\pi$
$307$	32.4481			1.85191			0.925955	+	0.377633i	$0.123262\pi$
$308$		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
$313$	−1.08036			−0.0610657			−0.0305329	−	0.999534i	$-0.509720\pi$
$313$	−1.08036			−0.0610657			−0.0305329	+	0.999534i	$0.509720\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.49768			−0.252615			−0.126307	−	0.991991i	$-0.540313\pi$
$317$	−4.49768			−0.252615			−0.126307	+	0.991991i	$0.540313\pi$
$318$		0			0
$319$	−2.94427			−0.164848
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.81758			−0.434982
$324$		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
$337$			11.4164i			0.621891i	0.950428	+	0.310946i	$0.100646\pi$
$337$			11.4164i			0.621891i	−0.950428	+	0.310946i	$0.899354\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.47214			−0.350486
$342$		0			0
$343$	18.3848	−	2.23607i	0.992685	−	0.120736i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.9613			1.23263			0.616313	−	0.787502i	$-0.288626\pi$
$347$	22.9613			1.23263			0.616313	+	0.787502i	$0.288626\pi$
$348$		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
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		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
$367$	2.82843			0.147643			0.0738213	−	0.997271i	$-0.476481\pi$
$367$	2.82843			0.147643			0.0738213	+	0.997271i	$0.476481\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.4721	−	19.7202i	−0.647521	−	1.02382i
$372$		0			0
$373$		−	26.9443i		−	1.39512i	−0.716526	−	0.697561i	$-0.754269\pi$
$373$		−	26.9443i		−	1.39512i	0.716526	−	0.697561i	$-0.245731\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.63932i		−	0.187435i
$378$		0			0
$379$	−11.8885			−0.610673			−0.305337	−	0.952244i	$-0.598769\pi$
$379$	−11.8885			−0.610673			−0.305337	+	0.952244i	$0.598769\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			16.9443i			0.865812i	0.901439	+	0.432906i	$0.142512\pi$
$383$			16.9443i			0.865812i	−0.901439	+	0.432906i	$0.857488\pi$
$384$		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
$397$	13.7295			0.689063			0.344531	−	0.938775i	$-0.388038\pi$
$397$	13.7295			0.689063			0.344531	+	0.938775i	$0.388038\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			28.2055i			1.40851i	0.709945	+	0.704257i	$0.248720\pi$
$401$			28.2055i			1.40851i	−0.709945	+	0.704257i	$0.751280\pi$
$402$		0			0
$403$		−	8.00000i		−	0.398508i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.16073			−0.107103
$408$		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
$413$	−2.16073	−	3.41641i	−0.106322	−	0.168110i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−22.8328			−1.11546			−0.557728	−	0.830024i	$-0.688327\pi$
$419$	−22.8328			−1.11546			−0.557728	+	0.830024i	$0.688327\pi$
$420$		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
$427$	−7.81758	+	4.94427i	−0.378319	+	0.239270i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	17.5595i		−	0.845809i	−0.906174	−	0.422905i	$-0.861011\pi$
$431$		−	17.5595i		−	0.845809i	0.906174	−	0.422905i	$-0.138989\pi$
$432$		0			0
$433$	0.255039			0.0122564			0.00612820	−	0.999981i	$-0.498049\pi$
$433$	0.255039			0.0122564			0.00612820	+	0.999981i	$0.498049\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5.52786i			0.264434i
$438$		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
$443$	33.6073			1.59673			0.798365	−	0.602174i	$-0.205699\pi$
$443$	33.6073			1.59673			0.798365	+	0.602174i	$0.205699\pi$
$444$		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
$457$		−	22.4721i		−	1.05120i	−0.850731	−	0.525601i	$-0.823840\pi$
$457$		−	22.4721i		−	1.05120i	0.850731	−	0.525601i	$-0.176160\pi$
$458$		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
$463$			7.05573i			0.327907i	0.986468	+	0.163954i	$0.0524248\pi$
$463$			7.05573i			0.327907i	−0.986468	+	0.163954i	$0.947575\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.4164i			1.45378i	0.686755	+	0.726889i	$0.259035\pi$
$467$			31.4164i			1.45378i	−0.686755	+	0.726889i	$0.740965\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.667701			−0.0307009
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.3607			1.11307			0.556534	−	0.830825i	$-0.312131\pi$
$479$	24.3607			1.11307			0.556534	+	0.830825i	$0.312131\pi$
$480$		0			0
$481$		−	2.67080i		−	0.121778i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	3.52786i		−	0.159863i	−0.996800	−	0.0799314i	$-0.974530\pi$
$487$		−	3.52786i		−	0.159863i	0.996800	−	0.0799314i	$-0.0254701\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	23.2163i		−	1.04774i	−0.851799	−	0.523869i	$-0.824488\pi$
$491$		−	23.2163i		−	1.04774i	0.851799	−	0.523869i	$-0.175512\pi$
$492$		0			0
$493$	9.31061			0.419329
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−17.3044	+	10.9443i	−0.776209	+	0.490918i
$498$		0			0
$499$	−14.9443			−0.668997			−0.334499	−	0.942396i	$-0.608567\pi$
$499$	−14.9443			−0.668997			−0.334499	+	0.942396i	$0.608567\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			15.0557i			0.671302i	0.941986	+	0.335651i	$0.108956\pi$
$503$			15.0557i			0.671302i	−0.941986	+	0.335651i	$0.891044\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−26.3607			−1.16842			−0.584208	−	0.811604i	$-0.698595\pi$
$509$	−26.3607			−1.16842			−0.584208	+	0.811604i	$0.698595\pi$
$510$		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
$517$	3.49613			0.153760
$518$		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
$523$	27.4589			1.20070			0.600348	−	0.799739i	$-0.295029\pi$
$523$	27.4589			1.20070			0.600348	+	0.799739i	$0.295029\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.4667			0.891543
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.825324			−0.0357487
$534$		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
$547$		−	22.8328i		−	0.976261i	−0.872771	−	0.488130i	$-0.837679\pi$
$547$		−	22.8328i		−	0.976261i	0.872771	−	0.488130i	$-0.162321\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.63932			0.155040
$552$		0			0
$553$	12.6491	+	20.0000i	0.537895	+	0.850487i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.1437			−0.641659			−0.320829	−	0.947137i	$-0.603962\pi$
$557$	−15.1437			−0.641659			−0.320829	+	0.947137i	$0.603962\pi$
$558$		0			0
$559$		−	0.825324i		−	0.0349075i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			21.8885i			0.922492i	0.887272	+	0.461246i	$0.152597\pi$
$563$			21.8885i			0.922492i	−0.887272	+	0.461246i	$0.847403\pi$
$564$		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
$577$	−36.3569			−1.51356			−0.756779	−	0.653671i	$-0.773228\pi$
$577$	−36.3569			−1.51356			−0.756779	+	0.653671i	$0.773228\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.3607	−	7.81758i	0.512807	−	0.324328i
$582$		0			0
$583$		−	12.4721i		−	0.516543i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.8328i			0.777313i	0.921383	+	0.388657i	$0.127061\pi$
$587$			18.8328i			0.777313i	−0.921383	+	0.388657i	$0.872939\pi$
$588$		0			0
$589$	8.00000			0.329634
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	23.8885i		−	0.980985i	−0.871445	−	0.490492i	$-0.836817\pi$
$593$		−	23.8885i		−	0.980985i	0.871445	−	0.490492i	$-0.163183\pi$
$594$		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	$-0.991329\pi$
$601$		−	1.33540i		−	0.0544722i	0.999629	−	0.0272361i	$-0.00867059\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25.9659			−1.05392			−0.526962	−	0.849889i	$-0.676669\pi$
$607$	−25.9659			−1.05392			−0.526962	+	0.849889i	$0.676669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4.32145i			0.174827i
$612$		0			0
$613$		−	14.9443i		−	0.603593i	−0.953372	−	0.301797i	$-0.902414\pi$
$613$		−	14.9443i		−	0.603593i	0.953372	−	0.301797i	$-0.0975864\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.8114			−0.636543			−0.318271	−	0.948000i	$-0.603102\pi$
$617$	−15.8114			−0.636543			−0.318271	+	0.948000i	$0.603102\pi$
$618$		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
$623$	−15.4775	−	24.4721i	−0.620094	−	0.980455i
$624$		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
$637$	−5.24419	+	11.0557i	−0.207782	+	0.438044i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			18.8949i			0.746302i	0.927771	+	0.373151i	$0.121723\pi$
$641$			18.8949i			0.746302i	−0.927771	+	0.373151i	$0.878277\pi$
$642$		0			0
$643$	−10.6460			−0.419838			−0.209919	−	0.977719i	$-0.567320\pi$
$643$	−10.6460			−0.419838			−0.209919	+	0.977719i	$0.567320\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			8.00000i			0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$			8.00000i			0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$		0			0
$649$		−	2.16073i		−	0.0848159i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	32.1142			1.25673			0.628364	−	0.777920i	$-0.283725\pi$
$653$	32.1142			1.25673			0.628364	+	0.777920i	$0.283725\pi$
$654$		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	$-0.986620\pi$
$661$		−	2.16073i		−	0.0840425i	0.999117	−	0.0420213i	$-0.0133797\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	6.58359i		−	0.254918i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.94427			−0.190872
$672$		0			0
$673$			34.4721i			1.32880i	0.747376	+	0.664402i	$0.231314\pi$
$673$			34.4721i			1.32880i	−0.747376	+	0.664402i	$0.768686\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	37.4164i		−	1.43803i	−0.694995	−	0.719015i	$-0.744593\pi$
$677$		−	37.4164i		−	1.43803i	0.694995	−	0.719015i	$-0.255407\pi$
$678$		0			0
$679$	−0.583592	−	0.922740i	−0.0223962	−	0.0354115i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.9690			0.611037			0.305519	−	0.952186i	$-0.401170\pi$
$683$	15.9690			0.611037			0.305519	+	0.952186i	$0.401170\pi$
$684$		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	$-0.836251\pi$
$691$		−	25.8685i		−	0.984084i	0.870571	−	0.492042i	$-0.163749\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	2.11146i		−	0.0799771i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7.89639i		−	0.298243i	−0.988819	−	0.149121i	$-0.952356\pi$
$701$		−	7.89639i		−	0.298243i	0.988819	−	0.149121i	$-0.0476445\pi$
$702$		0			0
$703$	2.67080			0.100731
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.9814	+	18.9443i	0.450607	+	0.712473i
$708$		0			0
$709$	−0.944272			−0.0354629			−0.0177314	−	0.999843i	$-0.505644\pi$
$709$	−0.944272			−0.0354629			−0.0177314	+	0.999843i	$0.505644\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	14.4721i		−	0.541986i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.4721			0.838069			0.419035	−	0.907970i	$-0.362369\pi$
$719$	22.4721			0.838069			0.419035	+	0.907970i	$0.362369\pi$
$720$		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
$727$	16.9706			0.629403			0.314702	−	0.949191i	$-0.398096\pi$
$727$	16.9706			0.629403			0.314702	+	0.949191i	$0.398096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.11146			0.0780950
$732$		0			0
$733$	−16.5579			−0.611580			−0.305790	−	0.952099i	$-0.598921\pi$
$733$	−16.5579			−0.611580			−0.305790	+	0.952099i	$0.598921\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−3.05573			−0.112407			−0.0562034	−	0.998419i	$-0.517900\pi$
$739$	−3.05573			−0.112407			−0.0562034	+	0.998419i	$0.517900\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−46.2564			−1.69698			−0.848491	−	0.529210i	$-0.822489\pi$
$743$	−46.2564			−1.69698			−0.848491	+	0.529210i	$0.822489\pi$
$744$		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
$757$			18.4721i			0.671381i	0.941972	+	0.335691i	$0.108970\pi$
$757$			18.4721i			0.671381i	−0.941972	+	0.335691i	$0.891030\pi$
$758$		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
$763$	12.6491	+	20.0000i	0.457929	+	0.724049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.67080			0.0964372
$768$		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
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		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
$787$	4.32145			0.154043			0.0770216	−	0.997029i	$-0.475459\pi$
$787$	4.32145			0.154043			0.0770216	+	0.997029i	$0.475459\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−9.41641	−	14.8886i	−0.334809	−	0.529379i
$792$		0			0
$793$		−	6.11146i		−	0.217024i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	24.8328i		−	0.879623i	−0.898090	−	0.439812i	$-0.855045\pi$
$797$		−	24.8328i		−	0.879623i	0.898090	−	0.439812i	$-0.144955\pi$
$798$		0			0
$799$	−11.0557			−0.391124
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	23.4164i		−	0.826347i
$804$		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	$-0.947179\pi$
$811$		−	9.40802i		−	0.330360i	0.986263	−	0.165180i	$-0.0528206\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.825324			0.0288744
$818$		0			0
$819$		0			0
$820$		0			0
$821$			52.9936i			1.84949i	0.380588	+	0.924745i	$0.375722\pi$
$821$			52.9936i			1.84949i	−0.380588	+	0.924745i	$0.624278\pi$
$822$		0			0
$823$		−	2.11146i		−	0.0736007i	−0.999323	−	0.0368004i	$-0.988283\pi$
$823$		−	2.11146i		−	0.0736007i	0.999323	−	0.0368004i	$-0.0117166\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.4543			−0.850359			−0.425180	−	0.905109i	$-0.639789\pi$
$827$	−24.4543			−0.850359			−0.425180	+	0.905109i	$0.639789\pi$
$828$		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
$833$	−28.2843	−	13.4164i	−0.979992	−	0.464851i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.8885			0.755676			0.377838	−	0.925872i	$-0.376668\pi$
$839$	21.8885			0.755676			0.377838	+	0.925872i	$0.376668\pi$
$840$		0			0
$841$	24.6656			0.850539
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−12.7279	−	20.1246i	−0.437337	−	0.691490i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	4.83153i		−	0.165623i
$852$		0			0
$853$	−11.5687			−0.396106			−0.198053	−	0.980191i	$-0.563462\pi$
$853$	−11.5687			−0.396106			−0.198053	+	0.980191i	$0.563462\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	2.94427i		−	0.100574i	−0.998735	−	0.0502872i	$-0.983986\pi$
$857$		−	2.94427i		−	0.100574i	0.998735	−	0.0502872i	$-0.0160137\pi$
$858$		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
$863$	42.7603			1.45558			0.727788	−	0.685802i	$-0.240548\pi$
$863$	42.7603			1.45558			0.727788	+	0.685802i	$0.240548\pi$
$864$		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
$877$			33.7771i			1.14057i	0.821446	+	0.570286i	$0.193167\pi$
$877$			33.7771i			1.14057i	−0.821446	+	0.570286i	$0.806833\pi$
$878$		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
$883$			20.4721i			0.688942i	0.938797	+	0.344471i	$0.111942\pi$
$883$			20.4721i			0.688942i	−0.938797	+	0.344471i	$0.888058\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	17.3050i		−	0.581043i	−0.956868	−	0.290522i	$-0.906171\pi$
$887$		−	17.3050i		−	0.581043i	0.956868	−	0.290522i	$-0.0938288\pi$
$888$		0			0
$889$	8.94427	−	5.65685i	0.299981	−	0.189725i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.32145			−0.144612
$894$		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
$907$		−	45.1935i		−	1.50063i	−0.661083	−	0.750313i	$-0.729903\pi$
$907$		−	45.1935i		−	1.50063i	0.661083	−	0.750313i	$-0.270097\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	45.1760i		−	1.49675i	−0.663277	−	0.748374i	$-0.730835\pi$
$911$		−	45.1760i		−	1.49675i	0.663277	−	0.748374i	$-0.269165\pi$
$912$		0			0
$913$	7.81758			0.258724
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−21.8021	−	34.4721i	−0.719969	−	1.13837i
$918$		0			0
$919$	2.94427			0.0971226			0.0485613	−	0.998820i	$-0.484536\pi$
$919$	2.94427			0.0971226			0.0485613	+	0.998820i	$0.484536\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	13.5279i		−	0.445275i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−37.0557			−1.21576			−0.607880	−	0.794029i	$-0.707980\pi$
$929$	−37.0557			−1.21576			−0.607880	+	0.794029i	$0.707980\pi$
$930$		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
$937$	−21.5471			−0.703912			−0.351956	−	0.936017i	$-0.614483\pi$
$937$	−21.5471			−0.703912			−0.351956	+	0.936017i	$0.614483\pi$
$938$		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
$943$	−1.49302			−0.0486196
$944$		0			0
$945$		0			0
$946$		0			0
$947$	48.9272			1.58992			0.794960	−	0.606662i	$-0.207492\pi$
$947$	48.9272			1.58992			0.794960	+	0.606662i	$0.207492\pi$
$948$		0			0
$949$	28.9443			0.939571
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−55.9194			−1.81141			−0.905704	−	0.423910i	$-0.860657\pi$
$953$	−55.9194			−1.81141			−0.905704	+	0.423910i	$0.860657\pi$
$954$		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
$967$			29.8885i			0.961151i	0.876953	+	0.480575i	$0.159572\pi$
$967$			29.8885i			0.961151i	−0.876953	+	0.480575i	$0.840428\pi$
$968$		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
$973$	−48.1807	+	30.4721i	−1.54460	+	0.976892i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28.4605			−0.910532			−0.455266	−	0.890355i	$-0.650456\pi$
$977$	−28.4605			−0.910532			−0.455266	+	0.890355i	$0.650456\pi$
$978$		0			0
$979$		−	15.4775i		−	0.494664i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			0.583592i			0.0186137i	0.999957	+	0.00930685i	$0.00296250\pi$
$983$			0.583592i			0.0186137i	−0.999957	+	0.00930685i	$0.997037\pi$
$984$		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
$997$	20.7217			0.656264			0.328132	−	0.944632i	$-0.393581\pi$
$997$	20.7217			0.656264			0.328132	+	0.944632i	$0.393581\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.d.a.881.3	✓	4	5.2	odd	4
1260.2.d.a.881.4	yes	4	105.62	odd	4
1260.2.d.b.881.3	yes	4	15.2	even	4
1260.2.d.b.881.4	yes	4	35.27	even	4
5040.2.f.b.881.1		4	420.167	even	4
5040.2.f.b.881.2		4	20.7	even	4
5040.2.f.d.881.1		4	140.27	odd	4
5040.2.f.d.881.2		4	60.47	odd	4
6300.2.d.a.3401.1		4	35.13	even	4
6300.2.d.a.3401.2		4	15.8	even	4
6300.2.d.b.3401.1		4	105.83	odd	4
6300.2.d.b.3401.2		4	5.3	odd	4
6300.2.f.a.3149.2		8	3.2	odd	2
6300.2.f.a.3149.3		8	35.34	odd	2
6300.2.f.a.3149.5		8	7.6	odd	2
6300.2.f.a.3149.8		8	15.14	odd	2
6300.2.f.c.3149.1		8	1.1	even	1	trivial
6300.2.f.c.3149.4		8	105.104	even	2	inner
6300.2.f.c.3149.6		8	21.20	even	2	inner
6300.2.f.c.3149.7		8	5.4	even	2	inner

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 2.23607i) q^{7} +O(q^{10})\)	\(q+(-1.41421 - 2.23607i) q^{7} -1.41421i q^{11} +1.74806 q^{13} +4.47214i q^{17} +1.74806i q^{19} +3.16228 q^{23} -2.08191i q^{29} -4.57649i q^{31} -1.52786i q^{37} -0.472136 q^{41} -0.472136i q^{43} +2.47214i q^{47} +(-3.00000 + 6.32456i) q^{49} +8.81913 q^{53} +1.52786 q^{59} -3.49613i q^{61} -7.73877i q^{71} +16.5579 q^{73} +(-3.16228 + 2.00000i) q^{77} -8.94427 q^{79} +5.52786i q^{83} +10.9443 q^{89} +(-2.47214 - 3.90879i) q^{91} +0.412662 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 32 q^{41} - 24 q^{49} + 48 q^{59} + 16 q^{89} + 16 q^{91}+O(q^{100}) \) 8 * q + 32 * q^41 - 24 * q^49 + 48 * q^59 + 16 * q^89 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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