LMFDB - Embedded newform 8280.2.p.a.1241.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.5
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.44

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.49780i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.49780i q^{7} +1.14935 q^{11} -2.26948 q^{13} +6.86922 q^{17} -6.79149i q^{19} +(-4.43998 + 1.81290i) q^{23} +1.00000 q^{25} +2.09393i q^{29} +5.90831 q^{31} +3.49780i q^{35} +2.76049i q^{37} -0.214944i q^{41} -9.72506i q^{43} -5.38952i q^{47} -5.23463 q^{49} +6.28889 q^{53} -1.14935 q^{55} -5.04036i q^{59} +2.18158i q^{61} +2.26948 q^{65} +6.63066i q^{67} -2.63143i q^{71} +7.54836 q^{73} -4.02019i q^{77} -5.09251i q^{79} +1.54002 q^{83} -6.86922 q^{85} +0.845460 q^{89} +7.93820i q^{91} +6.79149i q^{95} -5.15703i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	3.49780i		−	1.32205i	−0.750366	−	0.661023i	$-0.770123\pi$
$7$		−	3.49780i		−	1.32205i	0.750366	−	0.661023i	$-0.229877\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.14935			0.346541			0.173271	−	0.984874i	$-0.444566\pi$
$11$	1.14935			0.346541			0.173271	+	0.984874i	$0.444566\pi$
$12$		0			0
$13$	−2.26948			−0.629441			−0.314721	−	0.949184i	$-0.601911\pi$
$13$	−2.26948			−0.629441			−0.314721	+	0.949184i	$0.601911\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.86922			1.66603			0.833015	−	0.553250i	$-0.186612\pi$
$17$	6.86922			1.66603			0.833015	+	0.553250i	$0.186612\pi$
$18$		0			0
$19$		−	6.79149i		−	1.55808i	−0.626977	−	0.779038i	$-0.715708\pi$
$19$		−	6.79149i		−	1.55808i	0.626977	−	0.779038i	$-0.284292\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.43998	+	1.81290i	−0.925799	+	0.378016i
$23$	−4.43998	+	1.81290i	−0.925799	+	0.378016i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.09393i			0.388833i	0.980919	+	0.194416i	$0.0622813\pi$
$29$			2.09393i			0.388833i	−0.980919	+	0.194416i	$0.937719\pi$
$30$		0			0
$31$	5.90831			1.06116			0.530582	−	0.847634i	$-0.321974\pi$
$31$	5.90831			1.06116			0.530582	+	0.847634i	$0.321974\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.49780i			0.591237i
$36$		0			0
$37$			2.76049i			0.453821i	0.973916	+	0.226911i	$0.0728625\pi$
$37$			2.76049i			0.453821i	−0.973916	+	0.226911i	$0.927137\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.214944i		−	0.0335686i	−0.999859	−	0.0167843i	$-0.994657\pi$
$41$		−	0.214944i		−	0.0335686i	0.999859	−	0.0167843i	$-0.00534287\pi$
$42$		0			0
$43$		−	9.72506i		−	1.48306i	−0.670921	−	0.741529i	$-0.734101\pi$
$43$		−	9.72506i		−	1.48306i	0.670921	−	0.741529i	$-0.265899\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	5.38952i		−	0.786143i	−0.919508	−	0.393071i	$-0.871413\pi$
$47$		−	5.38952i		−	0.786143i	0.919508	−	0.393071i	$-0.128587\pi$
$48$		0			0
$49$	−5.23463			−0.747804
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.28889			0.863845			0.431923	−	0.901911i	$-0.357835\pi$
$53$	6.28889			0.863845			0.431923	+	0.901911i	$0.357835\pi$
$54$		0			0
$55$	−1.14935			−0.154978
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.04036i		−	0.656198i	−0.944643	−	0.328099i	$-0.893592\pi$
$59$		−	5.04036i		−	0.656198i	0.944643	−	0.328099i	$-0.106408\pi$
$60$		0			0
$61$			2.18158i			0.279323i	0.990199	+	0.139662i	$0.0446015\pi$
$61$			2.18158i			0.279323i	−0.990199	+	0.139662i	$0.955399\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.26948			0.281495
$66$		0			0
$67$			6.63066i			0.810063i	0.914303	+	0.405032i	$0.132740\pi$
$67$			6.63066i			0.810063i	−0.914303	+	0.405032i	$0.867260\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.63143i		−	0.312293i	−0.987734	−	0.156147i	$-0.950093\pi$
$71$		−	2.63143i		−	0.312293i	0.987734	−	0.156147i	$-0.0499072\pi$
$72$		0			0
$73$	7.54836			0.883469			0.441734	−	0.897146i	$-0.354363\pi$
$73$	7.54836			0.883469			0.441734	+	0.897146i	$0.354363\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	4.02019i		−	0.458144i
$78$		0			0
$79$		−	5.09251i		−	0.572952i	−0.958087	−	0.286476i	$-0.907516\pi$
$79$		−	5.09251i		−	0.572952i	0.958087	−	0.286476i	$-0.0924838\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.54002			0.169039			0.0845194	−	0.996422i	$-0.473065\pi$
$83$	1.54002			0.169039			0.0845194	+	0.996422i	$0.473065\pi$
$84$		0			0
$85$	−6.86922			−0.745072
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.845460			0.0896186			0.0448093	−	0.998996i	$-0.485732\pi$
$89$	0.845460			0.0896186			0.0448093	+	0.998996i	$0.485732\pi$
$90$		0			0
$91$			7.93820i			0.832150i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			6.79149i			0.696793i
$96$		0			0
$97$		−	5.15703i		−	0.523618i	−0.965120	−	0.261809i	$-0.915681\pi$
$97$		−	5.15703i		−	0.523618i	0.965120	−	0.261809i	$-0.0843190\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			6.16311i			0.613253i	0.951830	+	0.306626i	$0.0992002\pi$
$101$			6.16311i			0.613253i	−0.951830	+	0.306626i	$0.900800\pi$
$102$		0			0
$103$			11.3588i			1.11921i	0.828759	+	0.559606i	$0.189048\pi$
$103$			11.3588i			1.11921i	−0.828759	+	0.559606i	$0.810952\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.6516			1.12640			0.563199	−	0.826321i	$-0.309570\pi$
$107$	11.6516			1.12640			0.563199	+	0.826321i	$0.309570\pi$
$108$		0			0
$109$		−	10.6861i		−	1.02354i	−0.859123	−	0.511769i	$-0.828990\pi$
$109$		−	10.6861i		−	1.02354i	0.859123	−	0.511769i	$-0.171010\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.40681			0.414557			0.207279	−	0.978282i	$-0.433539\pi$
$113$	4.40681			0.414557			0.207279	+	0.978282i	$0.433539\pi$
$114$		0			0
$115$	4.43998	−	1.81290i	0.414030	−	0.169054i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	24.0272i		−	2.20257i
$120$		0			0
$121$	−9.67900			−0.879909
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−18.8625			−1.67378			−0.836890	−	0.547371i	$-0.815629\pi$
$127$	−18.8625			−1.67378			−0.836890	+	0.547371i	$0.815629\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			5.34698i			0.467168i	0.972337	+	0.233584i	$0.0750453\pi$
$131$			5.34698i			0.467168i	−0.972337	+	0.233584i	$0.924955\pi$
$132$		0			0
$133$	−23.7553			−2.05985
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.4722			1.32188			0.660940	−	0.750439i	$-0.270158\pi$
$137$	15.4722			1.32188			0.660940	+	0.750439i	$0.270158\pi$
$138$		0			0
$139$	−13.8024			−1.17070			−0.585352	−	0.810779i	$-0.699044\pi$
$139$	−13.8024			−1.17070			−0.585352	+	0.810779i	$0.699044\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.60842			−0.218127
$144$		0			0
$145$		−	2.09393i		−	0.173891i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.92817			−0.157962			−0.0789811	−	0.996876i	$-0.525167\pi$
$149$	−1.92817			−0.157962			−0.0789811	+	0.996876i	$0.525167\pi$
$150$		0			0
$151$	2.79415			0.227384			0.113692	−	0.993516i	$-0.463732\pi$
$151$	2.79415			0.227384			0.113692	+	0.993516i	$0.463732\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.90831			−0.474567
$156$		0			0
$157$			17.2497i			1.37667i	0.725391	+	0.688337i	$0.241659\pi$
$157$			17.2497i			1.37667i	−0.725391	+	0.688337i	$0.758341\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.34117	+	15.5302i	0.499754	+	1.22395i
$162$		0			0
$163$	−17.7803			−1.39266			−0.696330	−	0.717722i	$-0.745185\pi$
$163$	−17.7803			−1.39266			−0.696330	+	0.717722i	$0.745185\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.2642i		−	1.10380i	−0.833911	−	0.551899i	$-0.813903\pi$
$167$		−	14.2642i		−	1.10380i	0.833911	−	0.551899i	$-0.186097\pi$
$168$		0			0
$169$	−7.84945			−0.603804
$170$		0			0
$171$		0			0
$172$		0			0
$173$			13.4111i			1.01963i	0.860285	+	0.509813i	$0.170285\pi$
$173$			13.4111i			1.01963i	−0.860285	+	0.509813i	$0.829715\pi$
$174$		0			0
$175$		−	3.49780i		−	0.264409i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	20.6075i		−	1.54027i	−0.637879	−	0.770137i	$-0.720188\pi$
$179$		−	20.6075i		−	1.54027i	0.637879	−	0.770137i	$-0.279812\pi$
$180$		0			0
$181$		−	5.70177i		−	0.423809i	−0.977290	−	0.211905i	$-0.932033\pi$
$181$		−	5.70177i		−	0.423809i	0.977290	−	0.211905i	$-0.0679666\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.76049i		−	0.202955i
$186$		0			0
$187$	7.89513			0.577349
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.04205			−0.0754001			−0.0377001	−	0.999289i	$-0.512003\pi$
$191$	−1.04205			−0.0754001			−0.0377001	+	0.999289i	$0.512003\pi$
$192$		0			0
$193$	5.09842			0.366992			0.183496	−	0.983020i	$-0.441259\pi$
$193$	5.09842			0.366992			0.183496	+	0.983020i	$0.441259\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	22.1612i		−	1.57892i	−0.613801	−	0.789460i	$-0.710360\pi$
$197$		−	22.1612i		−	1.57892i	0.613801	−	0.789460i	$-0.289640\pi$
$198$		0			0
$199$			9.46928i			0.671260i	0.941994	+	0.335630i	$0.108949\pi$
$199$			9.46928i			0.671260i	−0.941994	+	0.335630i	$0.891051\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.32415			0.514054
$204$		0			0
$205$			0.214944i			0.0150124i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	7.80579i		−	0.539938i
$210$		0			0
$211$	−6.15703			−0.423867			−0.211934	−	0.977284i	$-0.567976\pi$
$211$	−6.15703			−0.423867			−0.211934	+	0.977284i	$0.567976\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			9.72506i			0.663243i
$216$		0			0
$217$		−	20.6661i		−	1.40291i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.5896			−1.04867
$222$		0			0
$223$	1.98069			0.132637			0.0663184	−	0.997799i	$-0.478875\pi$
$223$	1.98069			0.132637			0.0663184	+	0.997799i	$0.478875\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.76597			−0.581818			−0.290909	−	0.956751i	$-0.593958\pi$
$227$	−8.76597			−0.581818			−0.290909	+	0.956751i	$0.593958\pi$
$228$		0			0
$229$		−	3.07244i		−	0.203033i	−0.994834	−	0.101516i	$-0.967631\pi$
$229$		−	3.07244i		−	0.203033i	0.994834	−	0.101516i	$-0.0323694\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.78421i		−	0.182400i	−0.995833	−	0.0911998i	$-0.970930\pi$
$233$		−	2.78421i		−	0.182400i	0.995833	−	0.0911998i	$-0.0290702\pi$
$234$		0			0
$235$			5.38952i			0.351574i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	20.8824i		−	1.35077i	−0.737466	−	0.675384i	$-0.763978\pi$
$239$		−	20.8824i		−	1.35077i	0.737466	−	0.675384i	$-0.236022\pi$
$240$		0			0
$241$		−	28.6330i		−	1.84441i	−0.386697	−	0.922207i	$-0.626384\pi$
$241$		−	28.6330i		−	1.84441i	0.386697	−	0.922207i	$-0.373616\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.23463			0.334428
$246$		0			0
$247$			15.4132i			0.980717i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.8077			−1.06089			−0.530445	−	0.847719i	$-0.677975\pi$
$251$	−16.8077			−1.06089			−0.530445	+	0.847719i	$0.677975\pi$
$252$		0			0
$253$	−5.10308	+	2.08365i	−0.320828	+	0.130998i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	11.1865i		−	0.697793i	−0.937161	−	0.348897i	$-0.886556\pi$
$257$		−	11.1865i		−	0.697793i	0.937161	−	0.348897i	$-0.113444\pi$
$258$		0			0
$259$	9.65564			0.599972
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.07687			−0.128065			−0.0640327	−	0.997948i	$-0.520396\pi$
$263$	−2.07687			−0.128065			−0.0640327	+	0.997948i	$0.520396\pi$
$264$		0			0
$265$	−6.28889			−0.386323
$266$		0			0
$267$		0			0
$268$		0			0
$269$			26.7656i			1.63192i	0.578105	+	0.815962i	$0.303792\pi$
$269$			26.7656i			1.63192i	−0.578105	+	0.815962i	$0.696208\pi$
$270$		0			0
$271$	−9.09674			−0.552588			−0.276294	−	0.961073i	$-0.589106\pi$
$271$	−9.09674			−0.552588			−0.276294	+	0.961073i	$0.589106\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.14935			0.0693083
$276$		0			0
$277$	−3.24885			−0.195205			−0.0976024	−	0.995225i	$-0.531117\pi$
$277$	−3.24885			−0.195205			−0.0976024	+	0.995225i	$0.531117\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.99929			0.238577			0.119289	−	0.992860i	$-0.461939\pi$
$281$	3.99929			0.238577			0.119289	+	0.992860i	$0.461939\pi$
$282$		0			0
$283$		−	22.6846i		−	1.34846i	−0.738521	−	0.674231i	$-0.764475\pi$
$283$		−	22.6846i		−	1.34846i	0.738521	−	0.674231i	$-0.235525\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.751832			−0.0443793
$288$		0			0
$289$	30.1862			1.77566
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.9677			−1.69231			−0.846157	−	0.532934i	$-0.821089\pi$
$293$	−28.9677			−1.69231			−0.846157	+	0.532934i	$0.821089\pi$
$294$		0			0
$295$			5.04036i			0.293461i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.0764	−	4.11435i	0.582736	−	0.237939i
$300$		0			0
$301$	−34.0163			−1.96067
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.18158i		−	0.124917i
$306$		0			0
$307$	20.9380			1.19500			0.597499	−	0.801870i	$-0.296161\pi$
$307$	20.9380			1.19500			0.597499	+	0.801870i	$0.296161\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			7.92949i			0.449640i	0.974400	+	0.224820i	$0.0721794\pi$
$311$			7.92949i			0.449640i	−0.974400	+	0.224820i	$0.927821\pi$
$312$		0			0
$313$			0.243765i			0.0137784i	0.999976	+	0.00688922i	$0.00219292\pi$
$313$			0.243765i			0.0137784i	−0.999976	+	0.00688922i	$0.997807\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	5.20753i		−	0.292484i	−0.989249	−	0.146242i	$-0.953282\pi$
$317$		−	5.20753i		−	0.292484i	0.989249	−	0.146242i	$-0.0467178\pi$
$318$		0			0
$319$			2.40665i			0.134747i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	46.6523i		−	2.59580i
$324$		0			0
$325$	−2.26948			−0.125888
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.8515			−1.03932
$330$		0			0
$331$	5.53264			0.304101			0.152051	−	0.988373i	$-0.451412\pi$
$331$	5.53264			0.304101			0.152051	+	0.988373i	$0.451412\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	6.63066i		−	0.362271i
$336$		0			0
$337$			0.369581i			0.0201324i	0.999949	+	0.0100662i	$0.00320422\pi$
$337$			0.369581i			0.0201324i	−0.999949	+	0.0100662i	$0.996796\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.79070			0.367737
$342$		0			0
$343$		−	6.17492i		−	0.333414i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.4159i			0.666519i	0.942835	+	0.333259i	$0.108149\pi$
$347$			12.4159i			0.666519i	−0.942835	+	0.333259i	$0.891851\pi$
$348$		0			0
$349$	−21.8444			−1.16931			−0.584653	−	0.811283i	$-0.698769\pi$
$349$	−21.8444			−1.16931			−0.584653	+	0.811283i	$0.698769\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.1375i		−	1.07181i	−0.844278	−	0.535906i	$-0.819970\pi$
$353$		−	20.1375i		−	1.07181i	0.844278	−	0.535906i	$-0.180030\pi$
$354$		0			0
$355$			2.63143i			0.139662i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.86978			0.309795			0.154898	−	0.987931i	$-0.450495\pi$
$359$	5.86978			0.309795			0.154898	+	0.987931i	$0.450495\pi$
$360$		0			0
$361$	−27.1244			−1.42760
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.54836			−0.395099
$366$		0			0
$367$			1.31525i			0.0686554i	0.999411	+	0.0343277i	$0.0109290\pi$
$367$			1.31525i			0.0686554i	−0.999411	+	0.0343277i	$0.989071\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	21.9973i		−	1.14204i
$372$		0			0
$373$			11.1498i			0.577312i	0.957433	+	0.288656i	$0.0932084\pi$
$373$			11.1498i			0.577312i	−0.957433	+	0.288656i	$0.906792\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.75213i		−	0.244747i
$378$		0			0
$379$		−	9.02061i		−	0.463357i	−0.972792	−	0.231679i	$-0.925578\pi$
$379$		−	9.02061i		−	0.463357i	0.972792	−	0.231679i	$-0.0744218\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	27.3555			1.39780			0.698901	−	0.715218i	$-0.253673\pi$
$383$	27.3555			1.39780			0.698901	+	0.715218i	$0.253673\pi$
$384$		0			0
$385$			4.02019i			0.204888i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.0550			−1.37174			−0.685871	−	0.727723i	$-0.740579\pi$
$389$	−27.0550			−1.37174			−0.685871	+	0.727723i	$0.740579\pi$
$390$		0			0
$391$	−30.4992	+	12.4532i	−1.54241	+	0.629786i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			5.09251i			0.256232i
$396$		0			0
$397$	−29.0479			−1.45787			−0.728936	−	0.684581i	$-0.759985\pi$
$397$	−29.0479			−1.45787			−0.728936	+	0.684581i	$0.759985\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.7018			−0.883986			−0.441993	−	0.897018i	$-0.645728\pi$
$401$	−17.7018			−0.883986			−0.441993	+	0.897018i	$0.645728\pi$
$402$		0			0
$403$	−13.4088			−0.667940
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.17276i			0.157268i
$408$		0			0
$409$	−19.6776			−0.972994			−0.486497	−	0.873682i	$-0.661725\pi$
$409$	−19.6776			−0.972994			−0.486497	+	0.873682i	$0.661725\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.6302			−0.867524
$414$		0			0
$415$	−1.54002			−0.0755965
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−25.3634			−1.23908			−0.619542	−	0.784963i	$-0.712682\pi$
$419$	−25.3634			−1.23908			−0.619542	+	0.784963i	$0.712682\pi$
$420$		0			0
$421$		−	10.7539i		−	0.524113i	−0.965052	−	0.262057i	$-0.915599\pi$
$421$		−	10.7539i		−	0.524113i	0.965052	−	0.262057i	$-0.0844007\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.86922			0.333206
$426$		0			0
$427$	7.63075			0.369278
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.8819			0.813175			0.406587	−	0.913612i	$-0.366719\pi$
$431$	16.8819			0.813175			0.406587	+	0.913612i	$0.366719\pi$
$432$		0			0
$433$			4.87090i			0.234081i	0.993127	+	0.117040i	$0.0373407\pi$
$433$			4.87090i			0.234081i	−0.993127	+	0.117040i	$0.962659\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.3123	+	30.1541i	0.588977	+	1.44246i
$438$		0			0
$439$	−19.0795			−0.910614			−0.455307	−	0.890335i	$-0.650470\pi$
$439$	−19.0795			−0.910614			−0.455307	+	0.890335i	$0.650470\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	28.7794i		−	1.36735i	−0.729787	−	0.683675i	$-0.760381\pi$
$443$		−	28.7794i		−	1.36735i	0.729787	−	0.683675i	$-0.239619\pi$
$444$		0			0
$445$	−0.845460			−0.0400786
$446$		0			0
$447$		0			0
$448$		0			0
$449$			13.7546i			0.649121i	0.945865	+	0.324560i	$0.105216\pi$
$449$			13.7546i			0.649121i	−0.945865	+	0.324560i	$0.894784\pi$
$450$		0			0
$451$		−	0.247046i		−	0.0116329i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	7.93820i		−	0.372149i
$456$		0			0
$457$			27.1644i			1.27070i	0.772226	+	0.635348i	$0.219143\pi$
$457$			27.1644i			1.27070i	−0.772226	+	0.635348i	$0.780857\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	9.59046i		−	0.446672i	−0.974742	−	0.223336i	$-0.928305\pi$
$461$		−	9.59046i		−	0.446672i	0.974742	−	0.223336i	$-0.0716947\pi$
$462$		0			0
$463$	28.8893			1.34260			0.671300	−	0.741186i	$-0.265736\pi$
$463$	28.8893			1.34260			0.671300	+	0.741186i	$0.265736\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.8209			0.963475			0.481738	−	0.876316i	$-0.340006\pi$
$467$	20.8209			0.963475			0.481738	+	0.876316i	$0.340006\pi$
$468$		0			0
$469$	23.1927			1.07094
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.1775i		−	0.513941i
$474$		0			0
$475$		−	6.79149i		−	0.311615i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.0450			0.824498			0.412249	−	0.911071i	$-0.364743\pi$
$479$	18.0450			0.824498			0.412249	+	0.911071i	$0.364743\pi$
$480$		0			0
$481$		−	6.26488i		−	0.285654i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			5.15703i			0.234169i
$486$		0			0
$487$	−18.6141			−0.843486			−0.421743	−	0.906715i	$-0.638581\pi$
$487$	−18.6141			−0.843486			−0.421743	+	0.906715i	$0.638581\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			16.7448i			0.755681i	0.925871	+	0.377841i	$0.123333\pi$
$491$			16.7448i			0.755681i	−0.925871	+	0.377841i	$0.876667\pi$
$492$		0			0
$493$			14.3837i			0.647807i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.20422			−0.412866
$498$		0			0
$499$	19.1846			0.858819			0.429409	−	0.903110i	$-0.358722\pi$
$499$	19.1846			0.858819			0.429409	+	0.903110i	$0.358722\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	42.5929			1.89913			0.949563	−	0.313577i	$-0.101527\pi$
$503$	42.5929			1.89913			0.949563	+	0.313577i	$0.101527\pi$
$504$		0			0
$505$		−	6.16311i		−	0.274255i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			0.713893i			0.0316428i	0.999875	+	0.0158214i	$0.00503631\pi$
$509$			0.713893i			0.0316428i	−0.999875	+	0.0158214i	$0.994964\pi$
$510$		0			0
$511$		−	26.4027i		−	1.16799i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	11.3588i		−	0.500527i
$516$		0			0
$517$		−	6.19444i		−	0.272431i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.2085			0.534863			0.267432	−	0.963577i	$-0.413825\pi$
$521$	12.2085			0.534863			0.267432	+	0.963577i	$0.413825\pi$
$522$		0			0
$523$		−	31.4039i		−	1.37320i	−0.727037	−	0.686598i	$-0.759103\pi$
$523$		−	31.4039i		−	1.37320i	0.727037	−	0.686598i	$-0.240897\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	40.5855			1.76793
$528$		0			0
$529$	16.4268	−	16.0985i	0.714208	−	0.699934i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			0.487812i			0.0211295i
$534$		0			0
$535$	−11.6516			−0.503741
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.01641			−0.259145
$540$		0			0
$541$	−17.4478			−0.750139			−0.375069	−	0.926997i	$-0.622381\pi$
$541$	−17.4478			−0.750139			−0.375069	+	0.926997i	$0.622381\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			10.6861i			0.457740i
$546$		0			0
$547$	18.2321			0.779547			0.389774	−	0.920911i	$-0.372553\pi$
$547$	18.2321			0.779547			0.389774	+	0.920911i	$0.372553\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.2209			0.605831
$552$		0			0
$553$	−17.8126			−0.757468
$554$		0			0
$555$		0			0
$556$		0			0
$557$	33.8465			1.43412			0.717062	−	0.697009i	$-0.245486\pi$
$557$	33.8465			1.43412			0.717062	+	0.697009i	$0.245486\pi$
$558$		0			0
$559$			22.0708i			0.933497i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.0485			−0.929233			−0.464616	−	0.885512i	$-0.653808\pi$
$563$	−22.0485			−0.929233			−0.464616	+	0.885512i	$0.653808\pi$
$564$		0			0
$565$	−4.40681			−0.185396
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.3063			0.767439			0.383720	−	0.923450i	$-0.374643\pi$
$569$	18.3063			0.767439			0.383720	+	0.923450i	$0.374643\pi$
$570$		0			0
$571$		−	31.6123i		−	1.32293i	−0.749974	−	0.661467i	$-0.769934\pi$
$571$		−	31.6123i		−	1.32293i	0.749974	−	0.661467i	$-0.230066\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.43998	+	1.81290i	−0.185160	+	0.0756032i
$576$		0			0
$577$	−32.9871			−1.37327			−0.686636	−	0.727002i	$-0.740913\pi$
$577$	−32.9871			−1.37327			−0.686636	+	0.727002i	$0.740913\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	5.38668i		−	0.223477i
$582$		0			0
$583$	7.22812			0.299358
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	16.4136i		−	0.677463i	−0.940883	−	0.338732i	$-0.890002\pi$
$587$		−	16.4136i		−	0.677463i	0.940883	−	0.338732i	$-0.109998\pi$
$588$		0			0
$589$		−	40.1262i		−	1.65337i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	33.4105i		−	1.37200i	−0.727600	−	0.686002i	$-0.759364\pi$
$593$		−	33.4105i		−	1.37200i	0.727600	−	0.686002i	$-0.240636\pi$
$594$		0			0
$595$			24.0272i			0.985019i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	14.7817i		−	0.603965i	−0.953313	−	0.301983i	$-0.902352\pi$
$599$		−	14.7817i		−	0.603965i	0.953313	−	0.301983i	$-0.0976485\pi$
$600$		0			0
$601$	−38.0613			−1.55255			−0.776275	−	0.630394i	$-0.782893\pi$
$601$	−38.0613			−1.55255			−0.776275	+	0.630394i	$0.782893\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.67900			0.393507
$606$		0			0
$607$	24.5393			0.996020			0.498010	−	0.867171i	$-0.334064\pi$
$607$	24.5393			0.996020			0.498010	+	0.867171i	$0.334064\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			12.2314i			0.494830i
$612$		0			0
$613$		−	36.0618i		−	1.45652i	−0.685300	−	0.728261i	$-0.740329\pi$
$613$		−	36.0618i		−	1.45652i	0.685300	−	0.728261i	$-0.259671\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.1151			0.407219			0.203609	−	0.979052i	$-0.434733\pi$
$617$	10.1151			0.407219			0.203609	+	0.979052i	$0.434733\pi$
$618$		0			0
$619$			9.87070i			0.396737i	0.980128	+	0.198369i	$0.0635643\pi$
$619$			9.87070i			0.396737i	−0.980128	+	0.198369i	$0.936436\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	2.95725i		−	0.118480i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			18.9624i			0.756080i
$630$		0			0
$631$		−	38.3318i		−	1.52597i	−0.646418	−	0.762983i	$-0.723734\pi$
$631$		−	38.3318i		−	1.52597i	0.646418	−	0.762983i	$-0.276266\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.8625			0.748537
$636$		0			0
$637$	11.8799			0.470699
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.63412			−0.380525			−0.190262	−	0.981733i	$-0.560934\pi$
$641$	−9.63412			−0.380525			−0.190262	+	0.981733i	$0.560934\pi$
$642$		0			0
$643$		−	5.42458i		−	0.213925i	−0.994263	−	0.106962i	$-0.965888\pi$
$643$		−	5.42458i		−	0.213925i	0.994263	−	0.106962i	$-0.0341124\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			33.8394i			1.33036i	0.746682	+	0.665182i	$0.231646\pi$
$647$			33.8394i			1.33036i	−0.746682	+	0.665182i	$0.768354\pi$
$648$		0			0
$649$		−	5.79312i		−	0.227400i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			40.5072i			1.58517i	0.609762	+	0.792585i	$0.291265\pi$
$653$			40.5072i			1.58517i	−0.609762	+	0.792585i	$0.708735\pi$
$654$		0			0
$655$		−	5.34698i		−	0.208924i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.6844			−0.688887			−0.344444	−	0.938807i	$-0.611932\pi$
$659$	−17.6844			−0.688887			−0.344444	+	0.938807i	$0.611932\pi$
$660$		0			0
$661$			34.4692i			1.34069i	0.742047	+	0.670347i	$0.233855\pi$
$661$			34.4692i			1.34069i	−0.742047	+	0.670347i	$0.766145\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.7553			0.921191
$666$		0			0
$667$	−3.79608	−	9.29699i	−0.146985	−	0.359981i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.50740i			0.0967971i
$672$		0			0
$673$	40.0309			1.54308			0.771538	−	0.636183i	$-0.219488\pi$
$673$	40.0309			1.54308			0.771538	+	0.636183i	$0.219488\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.3539			−0.705396			−0.352698	−	0.935737i	$-0.614736\pi$
$677$	−18.3539			−0.705396			−0.352698	+	0.935737i	$0.614736\pi$
$678$		0			0
$679$	−18.0383			−0.692246
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	5.43977i		−	0.208147i	−0.994570	−	0.104073i	$-0.966812\pi$
$683$		−	5.43977i		−	0.208147i	0.994570	−	0.104073i	$-0.0331877\pi$
$684$		0			0
$685$	−15.4722			−0.591162
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.2725			−0.543740
$690$		0			0
$691$	−14.0283			−0.533661			−0.266831	−	0.963743i	$-0.585976\pi$
$691$	−14.0283			−0.533661			−0.266831	+	0.963743i	$0.585976\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.8024			0.523555
$696$		0			0
$697$		−	1.47650i		−	0.0559264i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.1416			−0.836278			−0.418139	−	0.908383i	$-0.637318\pi$
$701$	−22.1416			−0.836278			−0.418139	+	0.908383i	$0.637318\pi$
$702$		0			0
$703$	18.7478			0.707088
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.5574			0.810748
$708$		0			0
$709$		−	24.8740i		−	0.934164i	−0.884214	−	0.467082i	$-0.845305\pi$
$709$		−	24.8740i		−	0.934164i	0.884214	−	0.467082i	$-0.154695\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−26.2327	+	10.7112i	−0.982424	+	0.401137i
$714$		0			0
$715$	2.60842			0.0975496
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	13.4112i		−	0.500155i	−0.968226	−	0.250077i	$-0.919544\pi$
$719$		−	13.4112i		−	0.500155i	0.968226	−	0.250077i	$-0.0804561\pi$
$720$		0			0
$721$	39.7307			1.47965
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2.09393i			0.0777665i
$726$		0			0
$727$			32.9287i			1.22126i	0.791917	+	0.610629i	$0.209083\pi$
$727$			32.9287i			1.22126i	−0.791917	+	0.610629i	$0.790917\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	66.8036i		−	2.47082i
$732$		0			0
$733$		−	29.9092i		−	1.10472i	−0.833606	−	0.552360i	$-0.813727\pi$
$733$		−	29.9092i		−	1.10472i	0.833606	−	0.552360i	$-0.186273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7.62093i			0.280721i
$738$		0			0
$739$	−17.9206			−0.659219			−0.329610	−	0.944117i	$-0.606917\pi$
$739$	−17.9206			−0.659219			−0.329610	+	0.944117i	$0.606917\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.25594			−0.192822			−0.0964108	−	0.995342i	$-0.530736\pi$
$743$	−5.25594			−0.192822			−0.0964108	+	0.995342i	$0.530736\pi$
$744$		0			0
$745$	1.92817			0.0706429
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	40.7549i		−	1.48915i
$750$		0			0
$751$			28.3889i			1.03593i	0.855403	+	0.517963i	$0.173310\pi$
$751$			28.3889i			1.03593i	−0.855403	+	0.517963i	$0.826690\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.79415			−0.101689
$756$		0			0
$757$		−	14.9278i		−	0.542560i	−0.962500	−	0.271280i	$-0.912553\pi$
$757$		−	14.9278i		−	0.542560i	0.962500	−	0.271280i	$-0.0874470\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.31868i			0.156552i	0.996932	+	0.0782760i	$0.0249415\pi$
$761$			4.31868i			0.156552i	−0.996932	+	0.0782760i	$0.975058\pi$
$762$		0			0
$763$	−37.3777			−1.35316
$764$		0			0
$765$		0			0
$766$		0			0
$767$			11.4390i			0.413038i
$768$		0			0
$769$		−	39.4553i		−	1.42279i	−0.702791	−	0.711396i	$-0.748063\pi$
$769$		−	39.4553i		−	1.42279i	0.702791	−	0.711396i	$-0.251937\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.3919			0.409736			0.204868	−	0.978790i	$-0.434323\pi$
$773$	11.3919			0.409736			0.204868	+	0.978790i	$0.434323\pi$
$774$		0			0
$775$	5.90831			0.212233
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.45979			−0.0523025
$780$		0			0
$781$		−	3.02443i		−	0.108222i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	17.2497i		−	0.615668i
$786$		0			0
$787$			14.2530i			0.508066i	0.967196	+	0.254033i	$0.0817572\pi$
$787$			14.2530i			0.508066i	−0.967196	+	0.254033i	$0.918243\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	15.4141i		−	0.548064i
$792$		0			0
$793$		−	4.95107i		−	0.175818i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.51124			−0.0535308			−0.0267654	−	0.999642i	$-0.508521\pi$
$797$	−1.51124			−0.0535308			−0.0267654	+	0.999642i	$0.508521\pi$
$798$		0			0
$799$		−	37.0218i		−	1.30974i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.67569			0.306158
$804$		0			0
$805$	−6.34117	−	15.5302i	−0.223497	−	0.547366i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			17.0530i			0.599552i	0.954010	+	0.299776i	$0.0969120\pi$
$809$			17.0530i			0.599552i	−0.954010	+	0.299776i	$0.903088\pi$
$810$		0			0
$811$	2.94918			0.103560			0.0517798	−	0.998659i	$-0.483511\pi$
$811$	2.94918			0.103560			0.0517798	+	0.998659i	$0.483511\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17.7803			0.622816
$816$		0			0
$817$	−66.0477			−2.31072
$818$		0			0
$819$		0			0
$820$		0			0
$821$			0.584640i			0.0204041i	0.999948	+	0.0102020i	$0.00324746\pi$
$821$			0.584640i			0.0204041i	−0.999948	+	0.0102020i	$0.996753\pi$
$822$		0			0
$823$	53.0488			1.84917			0.924583	−	0.380980i	$-0.124413\pi$
$823$	53.0488			1.84917			0.924583	+	0.380980i	$0.124413\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.3788			0.673867			0.336933	−	0.941528i	$-0.390610\pi$
$827$	19.3788			0.673867			0.336933	+	0.941528i	$0.390610\pi$
$828$		0			0
$829$	28.1047			0.976115			0.488058	−	0.872811i	$-0.337706\pi$
$829$	28.1047			0.976115			0.488058	+	0.872811i	$0.337706\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−35.9578			−1.24587
$834$		0			0
$835$			14.2642i			0.493634i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−37.0072			−1.27763			−0.638815	−	0.769360i	$-0.720575\pi$
$839$	−37.0072			−1.27763			−0.638815	+	0.769360i	$0.720575\pi$
$840$		0			0
$841$	24.6155			0.848809
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.84945			0.270029
$846$		0			0
$847$			33.8552i			1.16328i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.00449	−	12.2565i	−0.171552	−	0.420147i
$852$		0			0
$853$	−31.6759			−1.08456			−0.542280	−	0.840198i	$-0.682439\pi$
$853$	−31.6759			−1.08456			−0.542280	+	0.840198i	$0.682439\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			18.9652i			0.647839i	0.946085	+	0.323919i	$0.105001\pi$
$857$			18.9652i			0.647839i	−0.946085	+	0.323919i	$0.894999\pi$
$858$		0			0
$859$	6.86619			0.234272			0.117136	−	0.993116i	$-0.462629\pi$
$859$	6.86619			0.234272			0.117136	+	0.993116i	$0.462629\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.3621i			1.23778i	0.785477	+	0.618891i	$0.212418\pi$
$863$			36.3621i			1.23778i	−0.785477	+	0.618891i	$0.787582\pi$
$864$		0			0
$865$		−	13.4111i		−	0.455991i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	5.85306i		−	0.198552i
$870$		0			0
$871$		−	15.0482i		−	0.509887i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			3.49780i			0.118247i
$876$		0			0
$877$	−13.7438			−0.464095			−0.232048	−	0.972704i	$-0.574543\pi$
$877$	−13.7438			−0.464095			−0.232048	+	0.972704i	$0.574543\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−21.0224			−0.708264			−0.354132	−	0.935195i	$-0.615224\pi$
$881$	−21.0224			−0.708264			−0.354132	+	0.935195i	$0.615224\pi$
$882$		0			0
$883$	−9.40804			−0.316606			−0.158303	−	0.987391i	$-0.550602\pi$
$883$	−9.40804			−0.316606			−0.158303	+	0.987391i	$0.550602\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			29.3452i			0.985315i	0.870223	+	0.492657i	$0.163974\pi$
$887$			29.3452i			0.985315i	−0.870223	+	0.492657i	$0.836026\pi$
$888$		0			0
$889$			65.9775i			2.21281i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−36.6029			−1.22487
$894$		0			0
$895$			20.6075i			0.688831i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			12.3716i			0.412615i
$900$		0			0
$901$	43.1998			1.43919
$902$		0			0
$903$		0			0
$904$		0			0
$905$			5.70177i			0.189533i
$906$		0			0
$907$			33.2308i			1.10341i	0.834039	+	0.551706i	$0.186023\pi$
$907$			33.2308i			1.10341i	−0.834039	+	0.551706i	$0.813977\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.7974			0.589655			0.294828	−	0.955550i	$-0.404738\pi$
$911$	17.7974			0.589655			0.294828	+	0.955550i	$0.404738\pi$
$912$		0			0
$913$	1.77002			0.0585790
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.7027			0.617617
$918$		0			0
$919$			21.1691i			0.698304i	0.937066	+	0.349152i	$0.113530\pi$
$919$			21.1691i			0.698304i	−0.937066	+	0.349152i	$0.886470\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			5.97198i			0.196570i
$924$		0			0
$925$			2.76049i			0.0907642i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	20.9372i		−	0.686926i	−0.939166	−	0.343463i	$-0.888400\pi$
$929$		−	20.9372i		−	0.686926i	0.939166	−	0.343463i	$-0.111600\pi$
$930$		0			0
$931$			35.5510i			1.16514i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7.89513			−0.258198
$936$		0			0
$937$			21.9856i			0.718239i	0.933292	+	0.359120i	$0.116923\pi$
$937$			21.9856i			0.718239i	−0.933292	+	0.359120i	$0.883077\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.7018			0.414066			0.207033	−	0.978334i	$-0.433619\pi$
$941$	12.7018			0.414066			0.207033	+	0.978334i	$0.433619\pi$
$942$		0			0
$943$	0.389672	+	0.954347i	0.0126895	+	0.0310778i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.14111i			0.0370811i	0.999828	+	0.0185405i	$0.00590197\pi$
$947$			1.14111i			0.0370811i	−0.999828	+	0.0185405i	$0.994098\pi$
$948$		0			0
$949$	−17.1309			−0.556091
$950$		0			0
$951$		0			0
$952$		0			0
$953$	8.90552			0.288478			0.144239	−	0.989543i	$-0.453927\pi$
$953$	8.90552			0.288478			0.144239	+	0.989543i	$0.453927\pi$
$954$		0			0
$955$	1.04205			0.0337200
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	54.1187i		−	1.74758i
$960$		0			0
$961$	3.90808			0.126067
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.09842			−0.164124
$966$		0			0
$967$	−9.05405			−0.291158			−0.145579	−	0.989347i	$-0.546505\pi$
$967$	−9.05405			−0.291158			−0.145579	+	0.989347i	$0.546505\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.6254			0.918634			0.459317	−	0.888272i	$-0.348094\pi$
$971$	28.6254			0.918634			0.459317	+	0.888272i	$0.348094\pi$
$972$		0			0
$973$			48.2781i			1.54772i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.7705			0.824472			0.412236	−	0.911077i	$-0.364748\pi$
$977$	25.7705			0.824472			0.412236	+	0.911077i	$0.364748\pi$
$978$		0			0
$979$	0.971728			0.0310566
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−15.2902			−0.487681			−0.243841	−	0.969815i	$-0.578407\pi$
$983$	−15.2902			−0.487681			−0.243841	+	0.969815i	$0.578407\pi$
$984$		0			0
$985$			22.1612i			0.706115i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.6306	+	43.1790i	0.560619	+	1.37301i
$990$		0			0
$991$	2.97551			0.0945202			0.0472601	−	0.998883i	$-0.484951\pi$
$991$	2.97551			0.0945202			0.0472601	+	0.998883i	$0.484951\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	9.46928i		−	0.300196i
$996$		0			0
$997$	−26.3652			−0.834993			−0.417497	−	0.908679i	$-0.637092\pi$
$997$	−26.3652			−0.834993			−0.417497	+	0.908679i	$0.637092\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.5	✓	48
3.2	odd	2		8280.2.p.b.1241.5	yes	48
23.22	odd	2		8280.2.p.b.1241.44	yes	48
69.68	even	2	inner	8280.2.p.a.1241.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.5	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.44	yes	48	69.68	even	2	inner
8280.2.p.b.1241.5	yes	48	3.2	odd	2
8280.2.p.b.1241.44	yes	48	23.22	odd	2

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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.49780i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.49780i q^{7} +1.14935 q^{11} -2.26948 q^{13} +6.86922 q^{17} -6.79149i q^{19} +(-4.43998 + 1.81290i) q^{23} +1.00000 q^{25} +2.09393i q^{29} +5.90831 q^{31} +3.49780i q^{35} +2.76049i q^{37} -0.214944i q^{41} -9.72506i q^{43} -5.38952i q^{47} -5.23463 q^{49} +6.28889 q^{53} -1.14935 q^{55} -5.04036i q^{59} +2.18158i q^{61} +2.26948 q^{65} +6.63066i q^{67} -2.63143i q^{71} +7.54836 q^{73} -4.02019i q^{77} -5.09251i q^{79} +1.54002 q^{83} -6.86922 q^{85} +0.845460 q^{89} +7.93820i q^{91} +6.79149i q^{95} -5.15703i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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