LMFDB - Embedded newform 2352.2.h.p.2255.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625 $ x^16 - 12x^14 + 97x^12 - 432x^10 + 1392x^8 - 2502x^6 + 3181x^4 - 1650*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.8
Root			$-1.75344 - 1.01235i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.p.2255.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.777403 + 1.54779i) q^{3} +3.40332i q^{5} +(-1.79129 - 2.40651i) q^{9} +O(q^{10})$	$q+(-0.777403 + 1.54779i) q^{3} +3.40332i q^{5} +(-1.79129 - 2.40651i) q^{9} +3.94748 q^{13} +(-5.26761 - 2.64575i) q^{15} -6.09632i q^{17} -4.38774i q^{19} -8.33639 q^{23} -6.58258 q^{25} +(5.11732 - 0.901703i) q^{27} -3.80848i q^{29} -4.89898i q^{31} -7.58258 q^{37} +(-3.06878 + 6.10985i) q^{39} +0.710314i q^{41} -9.66930i q^{43} +(8.19012 - 6.09632i) q^{45} -11.7894 q^{47} +(9.43581 + 4.73930i) q^{51} -1.00454i q^{53} +(6.79129 + 3.41105i) q^{57} -4.66442 q^{59} +1.70938 q^{61} +13.4345i q^{65} -3.46410i q^{67} +(6.48074 - 12.9030i) q^{69} -6.59649 q^{71} +14.3757 q^{73} +(5.11732 - 10.1884i) q^{75} -6.92820i q^{79} +(-2.58258 + 8.62150i) q^{81} -16.4539 q^{83} +20.7477 q^{85} +(5.89472 + 2.96073i) q^{87} -6.09632i q^{89} +(7.58258 + 3.80848i) q^{93} +14.9329 q^{95} +9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^9	$ 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) $ 16 * q + 8 * q^9 - 32 * q^25 - 48 * q^37 + 72 * q^57 + 32 * q^81 + 112 * q^85 + 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−0.777403	+	1.54779i	−0.448834	+	0.893615i
$3$	−0.777403	+	1.54779i	−0.448834	+	0.893615i
$4$		0			0
$5$			3.40332i			1.52201i	0.648746	+	0.761005i	$0.275294\pi$
$5$			3.40332i			1.52201i	−0.648746	+	0.761005i	$0.724706\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.79129	−	2.40651i	−0.597096	−	0.802170i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	3.94748			1.09483			0.547417	−	0.836860i	$-0.315611\pi$
$13$	3.94748			1.09483			0.547417	+	0.836860i	$0.315611\pi$
$14$		0			0
$15$	−5.26761	−	2.64575i	−1.36009	−	0.683130i
$16$		0			0
$17$		−	6.09632i		−	1.47858i	−0.673390	−	0.739288i	$-0.735162\pi$
$17$		−	6.09632i		−	1.47858i	0.673390	−	0.739288i	$-0.264838\pi$
$18$		0			0
$19$		−	4.38774i		−	1.00662i	−0.864107	−	0.503308i	$-0.832116\pi$
$19$		−	4.38774i		−	1.00662i	0.864107	−	0.503308i	$-0.167884\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.33639			−1.73826			−0.869129	−	0.494585i	$-0.835320\pi$
$23$	−8.33639			−1.73826			−0.869129	+	0.494585i	$0.835320\pi$
$24$		0			0
$25$	−6.58258			−1.31652
$26$		0			0
$27$	5.11732	−	0.901703i	0.984828	−	0.173533i
$28$		0			0
$29$		−	3.80848i		−	0.707218i	−0.935393	−	0.353609i	$-0.884954\pi$
$29$		−	3.80848i		−	0.707218i	0.935393	−	0.353609i	$-0.115046\pi$
$30$		0			0
$31$		−	4.89898i		−	0.879883i	−0.898027	−	0.439941i	$-0.854999\pi$
$31$		−	4.89898i		−	0.879883i	0.898027	−	0.439941i	$-0.145001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.58258			−1.24657			−0.623284	−	0.781996i	$-0.714202\pi$
$37$	−7.58258			−1.24657			−0.623284	+	0.781996i	$0.714202\pi$
$38$		0			0
$39$	−3.06878	+	6.10985i	−0.491398	+	0.978359i
$40$		0			0
$41$			0.710314i			0.110932i	0.998461	+	0.0554662i	$0.0176645\pi$
$41$			0.710314i			0.110932i	−0.998461	+	0.0554662i	$0.982335\pi$
$42$		0			0
$43$		−	9.66930i		−	1.47456i	−0.675590	−	0.737278i	$-0.736111\pi$
$43$		−	9.66930i		−	1.47456i	0.675590	−	0.737278i	$-0.263889\pi$
$44$		0			0
$45$	8.19012	−	6.09632i	1.22091	−	0.908786i
$46$		0			0
$47$	−11.7894			−1.71967			−0.859833	−	0.510575i	$-0.829433\pi$
$47$	−11.7894			−1.71967			−0.859833	+	0.510575i	$0.829433\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	9.43581	+	4.73930i	1.32128	+	0.663635i
$52$		0			0
$53$		−	1.00454i		−	0.137984i	−0.997617	−	0.0689918i	$-0.978022\pi$
$53$		−	1.00454i		−	0.137984i	0.997617	−	0.0689918i	$-0.0219782\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.79129	+	3.41105i	0.899528	+	0.451804i
$58$		0			0
$59$	−4.66442			−0.607256			−0.303628	−	0.952791i	$-0.598198\pi$
$59$	−4.66442			−0.607256			−0.303628	+	0.952791i	$0.598198\pi$
$60$		0			0
$61$	1.70938			0.218863			0.109432	−	0.993994i	$-0.465097\pi$
$61$	1.70938			0.218863			0.109432	+	0.993994i	$0.465097\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			13.4345i			1.66635i
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	6.48074	−	12.9030i	0.780189	−	1.55333i
$70$		0			0
$71$	−6.59649			−0.782859			−0.391429	−	0.920208i	$-0.628019\pi$
$71$	−6.59649			−0.782859			−0.391429	+	0.920208i	$0.628019\pi$
$72$		0			0
$73$	14.3757			1.68255			0.841274	−	0.540609i	$-0.181806\pi$
$73$	14.3757			1.68255			0.841274	+	0.540609i	$0.181806\pi$
$74$		0			0
$75$	5.11732	−	10.1884i	0.590897	−	1.17646i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	6.92820i		−	0.779484i	−0.920924	−	0.389742i	$-0.872564\pi$
$79$		−	6.92820i		−	0.779484i	0.920924	−	0.389742i	$-0.127436\pi$
$80$		0			0
$81$	−2.58258	+	8.62150i	−0.286953	+	0.957945i
$82$		0			0
$83$	−16.4539			−1.80605			−0.903023	−	0.429592i	$-0.858657\pi$
$83$	−16.4539			−1.80605			−0.903023	+	0.429592i	$0.858657\pi$
$84$		0			0
$85$	20.7477			2.25041
$86$		0			0
$87$	5.89472	+	2.96073i	0.631980	+	0.317423i
$88$		0			0
$89$		−	6.09632i		−	0.646209i	−0.946363	−	0.323104i	$-0.895273\pi$
$89$		−	6.09632i		−	0.646209i	0.946363	−	0.323104i	$-0.104727\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	7.58258	+	3.80848i	0.786276	+	0.394921i
$94$		0			0
$95$	14.9329			1.53208
$96$		0			0
$97$	9.89949			1.00514			0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949			1.00514			0.502571	+	0.864536i	$0.332388\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			11.6306i			1.15729i	0.815581	+	0.578643i	$0.196418\pi$
$101$			11.6306i			1.15729i	−0.815581	+	0.578643i	$0.803582\pi$
$102$		0			0
$103$			18.5734i			1.83010i	0.403345	+	0.915048i	$0.367847\pi$
$103$			18.5734i			1.83010i	−0.403345	+	0.915048i	$0.632153\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.59649			0.637706			0.318853	−	0.947804i	$-0.396702\pi$
$107$	6.59649			0.637706			0.318853	+	0.947804i	$0.396702\pi$
$108$		0			0
$109$	−4.41742			−0.423113			−0.211556	−	0.977366i	$-0.567853\pi$
$109$	−4.41742			−0.423113			−0.211556	+	0.977366i	$0.567853\pi$
$110$		0			0
$111$	5.89472	−	11.7362i	0.559502	−	1.11395i
$112$		0			0
$113$		−	12.4300i		−	1.16931i	−0.811280	−	0.584657i	$-0.801229\pi$
$113$		−	12.4300i		−	1.16931i	0.811280	−	0.584657i	$-0.198771\pi$
$114$		0			0
$115$		−	28.3714i		−	2.64565i
$116$		0			0
$117$	−7.07107	−	9.49964i	−0.653720	−	0.878242i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−1.09941	−	0.552200i	−0.0991309	−	0.0497902i
$124$		0			0
$125$		−	5.38601i		−	0.481739i
$126$		0			0
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$		0			0
$129$	14.9660	+	7.51695i	1.31768	+	0.661831i
$130$		0			0
$131$	7.12502			0.622516			0.311258	−	0.950325i	$-0.399250\pi$
$131$	7.12502			0.622516			0.311258	+	0.950325i	$0.399250\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.06878	+	17.4159i	0.264119	+	1.49892i
$136$		0			0
$137$			18.2475i			1.55899i	0.626407	+	0.779496i	$0.284525\pi$
$137$			18.2475i			1.55899i	−0.626407	+	0.779496i	$0.715475\pi$
$138$		0			0
$139$			0.511238i			0.0433627i	0.999765	+	0.0216813i	$0.00690192\pi$
$139$			0.511238i			0.0433627i	−0.999765	+	0.0216813i	$0.993098\pi$
$140$		0			0
$141$	9.16515	−	18.2475i	0.771845	−	1.53672i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	12.9615			1.07639
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1.00454i		−	0.0822948i	−0.999153	−	0.0411474i	$-0.986899\pi$
$149$		−	1.00454i		−	0.0822948i	0.999153	−	0.0411474i	$-0.0131013\pi$
$150$		0			0
$151$			15.8745i			1.29185i	0.763401	+	0.645925i	$0.223528\pi$
$151$			15.8745i			1.29185i	−0.763401	+	0.645925i	$0.776472\pi$
$152$		0			0
$153$	−14.6709	+	10.9203i	−1.18607	+	0.882851i
$154$		0			0
$155$	16.6728			1.33919
$156$		0			0
$157$	−1.70938			−0.136423			−0.0682116	−	0.997671i	$-0.521729\pi$
$157$	−1.70938			−0.136423			−0.0682116	+	0.997671i	$0.521729\pi$
$158$		0			0
$159$	1.55481	+	0.780929i	0.123304	+	0.0619317i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	4.18710i		−	0.327959i	−0.986464	−	0.163980i	$-0.947567\pi$
$163$		−	4.18710i		−	0.327959i	0.986464	−	0.163980i	$-0.0524331\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	2.58258			0.198660
$170$		0			0
$171$	−10.5591	+	7.85971i	−0.807478	+	0.601047i
$172$		0			0
$173$			3.40332i			0.258750i	0.991596	+	0.129375i	$0.0412970\pi$
$173$			3.40332i			0.258750i	−0.991596	+	0.129375i	$0.958703\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.62614	−	7.21953i	0.272557	−	0.542653i
$178$		0			0
$179$	−6.59649			−0.493045			−0.246522	−	0.969137i	$-0.579288\pi$
$179$	−6.59649			−0.493045			−0.246522	+	0.969137i	$0.579288\pi$
$180$		0			0
$181$	9.01400			0.670006			0.335003	−	0.942217i	$-0.391263\pi$
$181$	9.01400			0.670006			0.335003	+	0.942217i	$0.391263\pi$
$182$		0			0
$183$	−1.32888	+	2.64575i	−0.0982333	+	0.195580i
$184$		0			0
$185$		−	25.8059i		−	1.89729i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.0763			0.729095			0.364548	−	0.931185i	$-0.381224\pi$
$191$	10.0763			0.729095			0.364548	+	0.931185i	$0.381224\pi$
$192$		0			0
$193$	−18.7477			−1.34949			−0.674745	−	0.738051i	$-0.735747\pi$
$193$	−18.7477			−1.34949			−0.674745	+	0.738051i	$0.735747\pi$
$194$		0			0
$195$	−20.7938	−	10.4440i	−1.48907	−	0.747913i
$196$		0			0
$197$			1.00454i			0.0715702i	0.999360	+	0.0357851i	$0.0113932\pi$
$197$			1.00454i			0.0715702i	−0.999360	+	0.0357851i	$0.988607\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$	5.36169	+	2.69300i	0.378185	+	0.189950i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.41742			−0.168840
$206$		0			0
$207$	14.9329	+	20.0616i	1.03791	+	1.39438i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			15.8745i			1.09285i	0.837509	+	0.546423i	$0.184011\pi$
$211$			15.8745i			1.09285i	−0.837509	+	0.546423i	$0.815989\pi$
$212$		0			0
$213$	5.12813	−	10.2100i	0.351374	−	0.699575i
$214$		0			0
$215$	32.9077			2.24429
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−11.1757	+	22.2505i	−0.755185	+	1.50355i
$220$		0			0
$221$		−	24.0651i		−	1.61879i
$222$		0			0
$223$			1.02248i			0.0684701i	0.999414	+	0.0342350i	$0.0108995\pi$
$223$			1.02248i			0.0684701i	−0.999414	+	0.0342350i	$0.989101\pi$
$224$		0			0
$225$	11.7913	+	15.8410i	0.786086	+	1.05607i
$226$		0			0
$227$	−16.4539			−1.09208			−0.546041	−	0.837759i	$-0.683866\pi$
$227$	−16.4539			−1.09208			−0.546041	+	0.837759i	$0.683866\pi$
$228$		0			0
$229$	15.2612			1.00849			0.504244	−	0.863561i	$-0.331771\pi$
$229$	15.2612			1.00849			0.504244	+	0.863561i	$0.331771\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.2475i		−	1.19544i	−0.801706	−	0.597718i	$-0.796074\pi$
$233$		−	18.2475i		−	1.19544i	0.801706	−	0.597718i	$-0.203926\pi$
$234$		0			0
$235$		−	40.1232i		−	2.61735i
$236$		0			0
$237$	10.7234	+	5.38601i	0.696558	+	0.349859i
$238$		0			0
$239$	−8.33639			−0.539236			−0.269618	−	0.962967i	$-0.586898\pi$
$239$	−8.33639			−0.539236			−0.269618	+	0.962967i	$0.586898\pi$
$240$		0			0
$241$	−22.8610			−1.47260			−0.736302	−	0.676653i	$-0.763430\pi$
$241$	−22.8610			−1.47260			−0.736302	+	0.676653i	$0.763430\pi$
$242$		0			0
$243$	−11.3365	−	10.6997i	−0.727240	−	0.686384i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	17.3205i		−	1.10208i
$248$		0			0
$249$	12.7913	−	25.4671i	0.810615	−	1.61391i
$250$		0			0
$251$	28.2433			1.78270			0.891351	−	0.453314i	$-0.149759\pi$
$251$	28.2433			1.78270			0.891351	+	0.453314i	$0.149759\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−16.1294	+	32.1131i	−1.01006	+	2.01100i
$256$		0			0
$257$		−	14.3236i		−	0.893481i	−0.894664	−	0.446740i	$-0.852585\pi$
$257$		−	14.3236i		−	0.893481i	0.894664	−	0.446740i	$-0.147415\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−9.16515	+	6.82209i	−0.567309	+	0.422277i
$262$		0			0
$263$	−26.7491			−1.64942			−0.824710	−	0.565556i	$-0.808661\pi$
$263$	−26.7491			−1.64942			−0.824710	+	0.565556i	$0.808661\pi$
$264$		0			0
$265$	3.41875			0.210012
$266$		0			0
$267$	9.43581	+	4.73930i	0.577462	+	0.290041i
$268$		0			0
$269$		−	3.40332i		−	0.207504i	−0.994603	−	0.103752i	$-0.966915\pi$
$269$		−	3.40332i		−	0.207504i	0.994603	−	0.103752i	$-0.0330848\pi$
$270$		0			0
$271$			19.5959i			1.19037i	0.803590	+	0.595184i	$0.202921\pi$
$271$			19.5959i			1.19037i	−0.803590	+	0.595184i	$0.797079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	−11.7894	+	8.77548i	−0.705815	+	0.525374i
$280$		0			0
$281$		−	10.6306i		−	0.634167i	−0.948398	−	0.317083i	$-0.897296\pi$
$281$		−	10.6306i		−	0.634167i	0.948398	−	0.317083i	$-0.102704\pi$
$282$		0			0
$283$		−	15.2082i		−	0.904032i	−0.892010	−	0.452016i	$-0.850705\pi$
$283$		−	15.2082i		−	0.904032i	0.892010	−	0.452016i	$-0.149295\pi$
$284$		0			0
$285$	−11.6089	+	23.1129i	−0.687650	+	1.36909i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−20.1652			−1.18619
$290$		0			0
$291$	−7.69590	+	15.3223i	−0.451142	+	0.898210i
$292$		0			0
$293$			1.98269i			0.115830i	0.998322	+	0.0579150i	$0.0184452\pi$
$293$			1.98269i			0.115830i	−0.998322	+	0.0579150i	$0.981555\pi$
$294$		0			0
$295$		−	15.8745i		−	0.924250i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−32.9077			−1.90310
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−18.0017	−	9.04165i	−1.03417	−	0.519429i
$304$		0			0
$305$			5.81755i			0.333112i
$306$		0			0
$307$		−	0.511238i		−	0.0291779i	−0.999894	−	0.0145890i	$-0.995356\pi$
$307$		−	0.511238i		−	0.0291779i	0.999894	−	0.0145890i	$-0.00464397\pi$
$308$		0			0
$309$	−28.7477	−	14.4391i	−1.63540	−	0.821409i
$310$		0			0
$311$	21.1183			1.19751			0.598754	−	0.800933i	$-0.295663\pi$
$311$	21.1183			1.19751			0.598754	+	0.800933i	$0.295663\pi$
$312$		0			0
$313$	−20.0325			−1.13231			−0.566153	−	0.824300i	$-0.691569\pi$
$313$	−20.0325			−1.13231			−0.566153	+	0.824300i	$0.691569\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27.8736i		−	1.56554i	−0.622314	−	0.782768i	$-0.713807\pi$
$317$		−	27.8736i		−	1.56554i	0.622314	−	0.782768i	$-0.286193\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−5.12813	+	10.2100i	−0.286224	+	0.569864i
$322$		0			0
$323$	−26.7491			−1.48836
$324$		0			0
$325$	−25.9846			−1.44136
$326$		0			0
$327$	3.43412	−	6.83723i	0.189907	−	0.378100i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			2.74110i			0.150665i	0.997158	+	0.0753323i	$0.0240017\pi$
$331$			2.74110i			0.150665i	−0.997158	+	0.0753323i	$0.975998\pi$
$332$		0			0
$333$	13.5826	+	18.2475i	0.744321	+	0.999959i
$334$		0			0
$335$	11.7894			0.644126
$336$		0			0
$337$	−28.7477			−1.56599			−0.782994	−	0.622029i	$-0.786309\pi$
$337$	−28.7477			−1.56599			−0.782994	+	0.622029i	$0.786309\pi$
$338$		0			0
$339$	19.2390	+	9.66311i	1.04492	+	0.524828i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	43.9129	+	22.0560i	2.36419	+	1.18746i
$346$		0			0
$347$	3.47981			0.186806			0.0934031	−	0.995628i	$-0.470225\pi$
$347$	3.47981			0.186806			0.0934031	+	0.995628i	$0.470225\pi$
$348$		0			0
$349$	−14.6709			−0.785313			−0.392657	−	0.919685i	$-0.628444\pi$
$349$	−14.6709			−0.785313			−0.392657	+	0.919685i	$0.628444\pi$
$350$		0			0
$351$	20.2005	−	3.55945i	1.07822	−	0.189989i
$352$		0			0
$353$			16.8683i			0.897811i	0.893579	+	0.448906i	$0.148186\pi$
$353$			16.8683i			0.897811i	−0.893579	+	0.448906i	$0.851814\pi$
$354$		0			0
$355$		−	22.4499i		−	1.19152i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.33639			0.439978			0.219989	−	0.975502i	$-0.429398\pi$
$359$	8.33639			0.439978			0.219989	+	0.975502i	$0.429398\pi$
$360$		0			0
$361$	−0.252273			−0.0132775
$362$		0			0
$363$	8.55144	−	17.0257i	0.448834	−	0.893615i
$364$		0			0
$365$			48.9251i			2.56086i
$366$		0			0
$367$		−	1.02248i		−	0.0533728i	−0.999644	−	0.0266864i	$-0.991504\pi$
$367$		−	1.02248i		−	0.0533728i	0.999644	−	0.0266864i	$-0.00849556\pi$
$368$		0			0
$369$	1.70938	−	1.27238i	0.0889866	−	0.0662373i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$	8.33639	+	4.18710i	0.430490	+	0.216221i
$376$		0			0
$377$		−	15.0339i		−	0.774285i
$378$		0			0
$379$			9.66930i			0.496679i	0.968673	+	0.248339i	$0.0798848\pi$
$379$			9.66930i			0.496679i	−0.968673	+	0.248339i	$0.920115\pi$
$380$		0			0
$381$	16.0851	+	8.07901i	0.824063	+	0.413900i
$382$		0			0
$383$	21.1183			1.07909			0.539547	−	0.841956i	$-0.318596\pi$
$383$	21.1183			1.07909			0.539547	+	0.841956i	$0.318596\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−23.2693	+	17.3205i	−1.18284	+	0.880451i
$388$		0			0
$389$			17.4527i			0.884885i	0.896797	+	0.442443i	$0.145888\pi$
$389$			17.4527i			0.884885i	−0.896797	+	0.442443i	$0.854112\pi$
$390$		0			0
$391$			50.8213i			2.57015i
$392$		0			0
$393$	−5.53901	+	11.0280i	−0.279406	+	0.556290i
$394$		0			0
$395$	23.5789			1.18638
$396$		0			0
$397$	5.71846			0.287001			0.143501	−	0.989650i	$-0.454164\pi$
$397$	5.71846			0.287001			0.143501	+	0.989650i	$0.454164\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			25.0696i			1.25192i	0.779856	+	0.625959i	$0.215292\pi$
$401$			25.0696i			1.25192i	−0.779856	+	0.625959i	$0.784708\pi$
$402$		0			0
$403$		−	19.3386i		−	0.963325i
$404$		0			0
$405$	−29.3417	−	8.78933i	−1.45800	−	0.436745i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4.83297			−0.238975			−0.119487	−	0.992836i	$-0.538125\pi$
$409$	−4.83297			−0.238975			−0.119487	+	0.992836i	$0.538125\pi$
$410$		0			0
$411$	−28.2433	−	14.1857i	−1.39314	−	0.699729i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	55.9977i		−	2.74882i
$416$		0			0
$417$	−0.791288	−	0.397438i	−0.0387495	−	0.0194626i
$418$		0			0
$419$	7.12502			0.348080			0.174040	−	0.984739i	$-0.444318\pi$
$419$	7.12502			0.348080			0.174040	+	0.984739i	$0.444318\pi$
$420$		0			0
$421$	−19.4955			−0.950150			−0.475075	−	0.879945i	$-0.657579\pi$
$421$	−19.4955			−0.950150			−0.475075	+	0.879945i	$0.657579\pi$
$422$		0			0
$423$	21.1183	+	28.3714i	1.02681	+	1.37946i
$424$		0			0
$425$			40.1295i			1.94657i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	38.2022			1.84013			0.920066	−	0.391762i	$-0.128134\pi$
$431$	38.2022			1.84013			0.920066	+	0.391762i	$0.128134\pi$
$432$		0			0
$433$	−12.1376			−0.583296			−0.291648	−	0.956526i	$-0.594204\pi$
$433$	−12.1376			−0.583296			−0.291648	+	0.956526i	$0.594204\pi$
$434$		0			0
$435$	−10.0763	+	20.0616i	−0.483122	+	0.961881i
$436$		0			0
$437$			36.5779i			1.74976i
$438$		0			0
$439$			14.6969i			0.701447i	0.936479	+	0.350723i	$0.114064\pi$
$439$			14.6969i			0.701447i	−0.936479	+	0.350723i	$0.885936\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	23.2693			1.10556			0.552778	−	0.833328i	$-0.313568\pi$
$443$	23.2693			1.10556			0.552778	+	0.833328i	$0.313568\pi$
$444$		0			0
$445$	20.7477			0.983537
$446$		0			0
$447$	1.55481	+	0.780929i	0.0735398	+	0.0369367i
$448$		0			0
$449$			24.8600i			1.17321i	0.809872	+	0.586607i	$0.199537\pi$
$449$			24.8600i			1.17321i	−0.809872	+	0.586607i	$0.800463\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−24.5704	−	12.3409i	−1.15442	−	0.579826i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.74773			0.128533			0.0642666	−	0.997933i	$-0.479529\pi$
$457$	2.74773			0.128533			0.0642666	+	0.997933i	$0.479529\pi$
$458$		0			0
$459$	−5.49707	−	31.1968i	−0.256581	−	1.45614i
$460$		0			0
$461$			8.78933i			0.409360i	0.978829	+	0.204680i	$0.0656153\pi$
$461$			8.78933i			0.409360i	−0.978829	+	0.204680i	$0.934385\pi$
$462$		0			0
$463$		−	5.48220i		−	0.254780i	−0.991853	−	0.127390i	$-0.959340\pi$
$463$		−	5.48220i		−	0.254780i	0.991853	−	0.127390i	$-0.0406599\pi$
$464$		0			0
$465$	−12.9615	+	25.8059i	−0.601074	+	1.19672i
$466$		0			0
$467$	−25.7827			−1.19308			−0.596541	−	0.802583i	$-0.703459\pi$
$467$	−25.7827			−1.19308			−0.596541	+	0.802583i	$0.703459\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.32888	−	2.64575i	0.0612314	−	0.121910i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			28.8826i			1.32523i
$476$		0			0
$477$	−2.41742	+	1.79941i	−0.110686	+	0.0823894i
$478$		0			0
$479$	−14.2500			−0.651101			−0.325550	−	0.945525i	$-0.605550\pi$
$479$	−14.2500			−0.651101			−0.325550	+	0.945525i	$0.605550\pi$
$480$		0			0
$481$	−29.9320			−1.36478
$482$		0			0
$483$		0			0
$484$		0			0
$485$			33.6911i			1.52984i
$486$		0			0
$487$		−	4.91010i		−	0.222498i	−0.993793	−	0.111249i	$-0.964515\pi$
$487$		−	4.91010i		−	0.222498i	0.993793	−	0.111249i	$-0.0354851\pi$
$488$		0			0
$489$	6.48074	+	3.25507i	0.293069	+	0.147199i
$490$		0			0
$491$	−10.0763			−0.454737			−0.227369	−	0.973809i	$-0.573012\pi$
$491$	−10.0763			−0.454737			−0.227369	+	0.973809i	$0.573012\pi$
$492$		0			0
$493$	−23.2177			−1.04567
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	22.8027i		−	1.02079i	−0.859940	−	0.510395i	$-0.829499\pi$
$499$		−	22.8027i		−	1.02079i	0.859940	−	0.510395i	$-0.170501\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.2500			−0.635378			−0.317689	−	0.948195i	$-0.602907\pi$
$503$	−14.2500			−0.635378			−0.317689	+	0.948195i	$0.602907\pi$
$504$		0			0
$505$	−39.5826			−1.76140
$506$		0			0
$507$	−2.00770	+	3.99728i	−0.0891652	+	0.177525i
$508$		0			0
$509$		−	29.2092i		−	1.29468i	−0.762203	−	0.647338i	$-0.775882\pi$
$509$		−	29.2092i		−	1.29468i	0.762203	−	0.647338i	$-0.224118\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−3.95644	−	22.4535i	−0.174681	−	0.991345i
$514$		0			0
$515$	−63.2113			−2.78542
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−5.26761	−	2.64575i	−0.231222	−	0.116136i
$520$		0			0
$521$			16.8683i			0.739015i	0.929228	+	0.369508i	$0.120474\pi$
$521$			16.8683i			0.739015i	−0.929228	+	0.369508i	$0.879526\pi$
$522$		0			0
$523$			32.7591i			1.43246i	0.697866	+	0.716229i	$0.254133\pi$
$523$			32.7591i			1.43246i	−0.697866	+	0.716229i	$0.745867\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−29.8658			−1.30097
$528$		0			0
$529$	46.4955			2.02154
$530$		0			0
$531$	8.35532	+	11.2250i	0.362590	+	0.487122i
$532$		0			0
$533$			2.80395i			0.121452i
$534$		0			0
$535$			22.4499i			0.970596i
$536$		0			0
$537$	5.12813	−	10.2100i	0.221295	−	0.440592i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	31.4955			1.35410			0.677048	−	0.735939i	$-0.263259\pi$
$541$	31.4955			1.35410			0.677048	+	0.735939i	$0.263259\pi$
$542$		0			0
$543$	−7.00752	+	13.9518i	−0.300721	+	0.598727i
$544$		0			0
$545$		−	15.0339i		−	0.643982i
$546$		0			0
$547$			35.9361i			1.53652i	0.640139	+	0.768259i	$0.278877\pi$
$547$			35.9361i			1.53652i	−0.640139	+	0.768259i	$0.721123\pi$
$548$		0			0
$549$	−3.06199	−	4.11363i	−0.130682	−	0.175566i
$550$		0			0
$551$	−16.7106			−0.711897
$552$		0			0
$553$		0			0
$554$		0			0
$555$	39.9421	+	20.0616i	1.69545	+	0.851568i
$556$		0			0
$557$		−	27.8736i		−	1.18104i	−0.807022	−	0.590521i	$-0.798922\pi$
$557$		−	27.8736i		−	1.18104i	0.807022	−	0.590521i	$-0.201078\pi$
$558$		0			0
$559$		−	38.1694i		−	1.61439i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.2433			1.19031			0.595157	−	0.803610i	$-0.297090\pi$
$563$	28.2433			1.19031			0.595157	+	0.803610i	$0.297090\pi$
$564$		0			0
$565$	42.3032			1.77971
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	32.6866i		−	1.37029i	−0.728405	−	0.685147i	$-0.759738\pi$
$569$		−	32.6866i		−	1.37029i	0.728405	−	0.685147i	$-0.240262\pi$
$570$		0			0
$571$		−	16.5975i		−	0.694584i	−0.937757	−	0.347292i	$-0.887101\pi$
$571$		−	16.5975i		−	0.694584i	0.937757	−	0.347292i	$-0.112899\pi$
$572$		0			0
$573$	−7.83335	+	15.5960i	−0.327243	+	0.651531i
$574$		0			0
$575$	54.8749			2.28844
$576$		0			0
$577$	−21.0900			−0.877988			−0.438994	−	0.898490i	$-0.644665\pi$
$577$	−21.0900			−0.877988			−0.438994	+	0.898490i	$0.644665\pi$
$578$		0			0
$579$	14.5745	−	29.0175i	0.605698	−	1.20593i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	32.3303	−	24.0651i	1.33669	−	0.994969i
$586$		0			0
$587$	2.20382			0.0909614			0.0454807	−	0.998965i	$-0.485518\pi$
$587$	2.20382			0.0909614			0.0454807	+	0.998965i	$0.485518\pi$
$588$		0			0
$589$	−21.4955			−0.885705
$590$		0			0
$591$	−1.55481	−	0.780929i	−0.0639562	−	0.0321231i
$592$		0			0
$593$		−	15.7442i		−	0.646537i	−0.946307	−	0.323269i	$-0.895218\pi$
$593$		−	15.7442i		−	0.646537i	0.946307	−	0.323269i	$-0.104782\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	15.1652	+	7.61697i	0.620668	+	0.311742i
$598$		0			0
$599$	26.7491			1.09294			0.546469	−	0.837479i	$-0.315972\pi$
$599$	26.7491			1.09294			0.546469	+	0.837479i	$0.315972\pi$
$600$		0			0
$601$	12.1376			0.495103			0.247551	−	0.968875i	$-0.420374\pi$
$601$	12.1376			0.495103			0.247551	+	0.968875i	$0.420374\pi$
$602$		0			0
$603$	−8.33639	+	6.20520i	−0.339484	+	0.252695i
$604$		0			0
$605$		−	37.4365i		−	1.52201i
$606$		0			0
$607$			3.87650i			0.157342i	0.996901	+	0.0786712i	$0.0250677\pi$
$607$			3.87650i			0.157342i	−0.996901	+	0.0786712i	$0.974932\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−46.5385			−1.88275
$612$		0			0
$613$	−44.2432			−1.78697			−0.893483	−	0.449098i	$-0.851745\pi$
$613$	−44.2432			−1.78697			−0.893483	+	0.449098i	$0.851745\pi$
$614$		0			0
$615$	1.87931	−	3.74166i	0.0757813	−	0.150878i
$616$		0			0
$617$		−	25.0696i		−	1.00927i	−0.863334	−	0.504633i	$-0.831628\pi$
$617$		−	25.0696i		−	1.00927i	0.863334	−	0.504633i	$-0.168372\pi$
$618$		0			0
$619$		−	19.0847i		−	0.767078i	−0.923525	−	0.383539i	$-0.874705\pi$
$619$		−	19.0847i		−	0.767078i	0.923525	−	0.383539i	$-0.125295\pi$
$620$		0			0
$621$	−42.6600	+	7.51695i	−1.71189	+	0.301645i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−14.5826			−0.583303
$626$		0			0
$627$		0			0
$628$		0			0
$629$			46.2258i			1.84314i
$630$		0			0
$631$		−	8.37420i		−	0.333372i	−0.986010	−	0.166686i	$-0.946693\pi$
$631$		−	8.37420i		−	0.333372i	0.986010	−	0.166686i	$-0.0533066\pi$
$632$		0			0
$633$	−24.5704	−	12.3409i	−0.976584	−	0.490507i
$634$		0			0
$635$	35.3683			1.40355
$636$		0			0
$637$		0			0
$638$		0			0
$639$	11.8162	+	15.8745i	0.467442	+	0.627986i
$640$		0			0
$641$		−	32.6866i		−	1.29104i	−0.763742	−	0.645521i	$-0.776640\pi$
$641$		−	32.6866i		−	1.29104i	0.763742	−	0.645521i	$-0.223360\pi$
$642$		0			0
$643$		−	17.0397i		−	0.671981i	−0.941865	−	0.335991i	$-0.890929\pi$
$643$		−	17.0397i		−	0.671981i	0.941865	−	0.335991i	$-0.109071\pi$
$644$		0			0
$645$	−25.5826	+	50.9341i	−1.00731	+	2.00553i
$646$		0			0
$647$	−2.46060			−0.0967361			−0.0483681	−	0.998830i	$-0.515402\pi$
$647$	−2.46060			−0.0967361			−0.0483681	+	0.998830i	$0.515402\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.4255i		−	0.447112i	−0.974691	−	0.223556i	$-0.928233\pi$
$653$		−	11.4255i		−	0.447112i	0.974691	−	0.223556i	$-0.0717666\pi$
$654$		0			0
$655$			24.2487i			0.947476i
$656$		0			0
$657$	−25.7510	−	34.5952i	−1.00464	−	1.34969i
$658$		0			0
$659$	46.5385			1.81288			0.906442	−	0.422330i	$-0.138788\pi$
$659$	46.5385			1.81288			0.906442	+	0.422330i	$0.138788\pi$
$660$		0			0
$661$	−36.1176			−1.40481			−0.702406	−	0.711776i	$-0.747891\pi$
$661$	−36.1176			−1.40481			−0.702406	+	0.711776i	$0.747891\pi$
$662$		0			0
$663$	37.2476	+	18.7083i	1.44658	+	0.726570i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			31.7490i			1.22933i
$668$		0			0
$669$	−1.58258	−	0.794877i	−0.0611859	−	0.0307317i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	3.16515			0.122008			0.0610038	−	0.998138i	$-0.480570\pi$
$673$	3.16515			0.122008			0.0610038	+	0.998138i	$0.480570\pi$
$674$		0			0
$675$	−33.6851	+	5.93553i	−1.29654	+	0.228459i
$676$		0			0
$677$			29.2092i			1.12260i	0.827612	+	0.561301i	$0.189699\pi$
$677$			29.2092i			1.12260i	−0.827612	+	0.561301i	$0.810301\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	12.7913	−	25.4671i	0.490163	−	0.975900i
$682$		0			0
$683$	10.0763			0.385559			0.192779	−	0.981242i	$-0.438250\pi$
$683$	10.0763			0.385559			0.192779	+	0.981242i	$0.438250\pi$
$684$		0			0
$685$	−62.1022			−2.37280
$686$		0			0
$687$	−11.8641	+	23.6211i	−0.452644	+	0.901200i
$688$		0			0
$689$		−	3.96538i		−	0.151069i
$690$		0			0
$691$		−	23.9837i		−	0.912381i	−0.889882	−	0.456191i	$-0.849213\pi$
$691$		−	23.9837i		−	0.912381i	0.889882	−	0.456191i	$-0.150787\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.73991			−0.0659984
$696$		0			0
$697$	4.33030			0.164022
$698$		0			0
$699$	28.2433	+	14.1857i	1.06826	+	0.536552i
$700$		0			0
$701$		−	32.6866i		−	1.23456i	−0.786745	−	0.617278i	$-0.788235\pi$
$701$		−	32.6866i		−	1.23456i	0.786745	−	0.617278i	$-0.211765\pi$
$702$		0			0
$703$			33.2704i			1.25482i
$704$		0			0
$705$	62.1022	+	31.1919i	2.33890	+	1.17476i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.0780			−0.641379			−0.320689	−	0.947184i	$-0.603915\pi$
$709$	−17.0780			−0.641379			−0.320689	+	0.947184i	$0.603915\pi$
$710$		0			0
$711$	−16.6728	+	12.4104i	−0.625278	+	0.465427i
$712$		0			0
$713$			40.8398i			1.52946i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.48074	−	12.9030i	0.242028	−	0.481870i
$718$		0			0
$719$	2.46060			0.0917649			0.0458824	−	0.998947i	$-0.485390\pi$
$719$	2.46060			0.0917649			0.0458824	+	0.998947i	$0.485390\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	17.7722	−	35.3839i	0.660955	−	1.31594i
$724$		0			0
$725$			25.0696i			0.931063i
$726$		0			0
$727$			43.0683i			1.59732i	0.601785	+	0.798658i	$0.294456\pi$
$727$			43.0683i			1.59732i	−0.601785	+	0.798658i	$0.705544\pi$
$728$		0			0
$729$	25.3739	−	9.22860i	0.939773	−	0.341800i
$730$		0			0
$731$	−58.9472			−2.18024
$732$		0			0
$733$	17.4993			0.646351			0.323175	−	0.946339i	$-0.395250\pi$
$733$	17.4993			0.646351			0.323175	+	0.946339i	$0.395250\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	24.9717i		−	0.918599i	−0.888281	−	0.459300i	$-0.848100\pi$
$739$		−	24.9717i		−	0.918599i	0.888281	−	0.459300i	$-0.151900\pi$
$740$		0			0
$741$	26.8085	+	13.4650i	0.984833	+	0.494650i
$742$		0			0
$743$	−21.5294			−0.789836			−0.394918	−	0.918716i	$-0.629227\pi$
$743$	−21.5294			−0.789836			−0.394918	+	0.918716i	$0.629227\pi$
$744$		0			0
$745$	3.41875			0.125253
$746$		0			0
$747$	29.4736	+	39.5964i	1.07838	+	1.44876i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			8.94630i			0.326455i	0.986588	+	0.163228i	$0.0521905\pi$
$751$			8.94630i			0.326455i	−0.986588	+	0.163228i	$0.947809\pi$
$752$		0			0
$753$	−21.9564	+	43.7146i	−0.800137	+	1.59305i
$754$		0			0
$755$	−54.0260			−1.96621
$756$		0			0
$757$	22.7477			0.826780			0.413390	−	0.910554i	$-0.364345\pi$
$757$	22.7477			0.826780			0.413390	+	0.910554i	$0.364345\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			14.3236i			0.519230i	0.965712	+	0.259615i	$0.0835956\pi$
$761$			14.3236i			0.519230i	−0.965712	+	0.259615i	$0.916404\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−37.1652	−	49.9296i	−1.34371	−	1.80521i
$766$		0			0
$767$	−18.4127			−0.664844
$768$		0			0
$769$	37.0031			1.33437			0.667183	−	0.744894i	$-0.267500\pi$
$769$	37.0031			1.33437			0.667183	+	0.744894i	$0.267500\pi$
$770$		0			0
$771$	22.1699	+	11.1352i	0.798428	+	0.401025i
$772$		0			0
$773$			3.40332i			0.122409i	0.998125	+	0.0612044i	$0.0194942\pi$
$773$			3.40332i			0.122409i	−0.998125	+	0.0612044i	$0.980506\pi$
$774$		0			0
$775$			32.2479i			1.15838i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.11667			0.111666
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−3.43412	−	19.4892i	−0.122725	−	0.696488i
$784$		0			0
$785$		−	5.81755i		−	0.207637i
$786$		0			0
$787$		−	26.8377i		−	0.956660i	−0.878180	−	0.478330i	$-0.841242\pi$
$787$		−	26.8377i		−	0.956660i	0.878180	−	0.478330i	$-0.158758\pi$
$788$		0			0
$789$	20.7948	−	41.4019i	0.740316	−	1.47395i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.74773			0.239619
$794$		0			0
$795$	−2.65775	+	5.29150i	−0.0942607	+	0.187670i
$796$		0			0
$797$			17.0166i			0.602759i	0.953504	+	0.301379i	$0.0974470\pi$
$797$			17.0166i			0.602759i	−0.953504	+	0.301379i	$0.902553\pi$
$798$		0			0
$799$			71.8722i			2.54266i
$800$		0			0
$801$	−14.6709	+	10.9203i	−0.518369	+	0.385849i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	5.26761	+	2.64575i	0.185429	+	0.0931349i
$808$		0			0
$809$		−	39.2990i		−	1.38168i	−0.723008	−	0.690840i	$-0.757241\pi$
$809$		−	39.2990i		−	1.38168i	0.723008	−	0.690840i	$-0.242759\pi$
$810$		0			0
$811$		−	21.1296i		−	0.741962i	−0.928641	−	0.370981i	$-0.879022\pi$
$811$		−	21.1296i		−	0.741962i	0.928641	−	0.370981i	$-0.120978\pi$
$812$		0			0
$813$	−30.3303	−	15.2339i	−1.06373	−	0.534277i
$814$		0			0
$815$	14.2500			0.499157
$816$		0			0
$817$	−42.4264			−1.48431
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	27.8736i		−	0.972795i	−0.873738	−	0.486397i	$-0.838311\pi$
$821$		−	27.8736i		−	0.972795i	0.873738	−	0.486397i	$-0.161689\pi$
$822$		0			0
$823$			8.37420i			0.291906i	0.989291	+	0.145953i	$0.0466249\pi$
$823$			8.37420i			0.291906i	−0.989291	+	0.145953i	$0.953375\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10.0763			0.350387			0.175194	−	0.984534i	$-0.443945\pi$
$827$	10.0763			0.350387			0.175194	+	0.984534i	$0.443945\pi$
$828$		0			0
$829$	23.1561			0.804246			0.402123	−	0.915586i	$-0.368272\pi$
$829$	23.1561			0.804246			0.402123	+	0.915586i	$0.368272\pi$
$830$		0			0
$831$	−1.55481	+	3.09557i	−0.0539357	+	0.107384i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.41742	−	25.0696i	−0.152689	−	0.866533i
$838$		0			0
$839$	−2.46060			−0.0849493			−0.0424747	−	0.999098i	$-0.513524\pi$
$839$	−2.46060			−0.0849493			−0.0424747	+	0.999098i	$0.513524\pi$
$840$		0			0
$841$	14.4955			0.499843
$842$		0			0
$843$	16.4539	+	8.26424i	0.566701	+	0.284636i
$844$		0			0
$845$			8.78933i			0.302362i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	23.5390	+	11.8229i	0.807857	+	0.405760i
$850$		0			0
$851$	63.2113			2.16686
$852$		0			0
$853$	10.7850			0.369271			0.184635	−	0.982807i	$-0.440890\pi$
$853$	10.7850			0.369271			0.184635	+	0.982807i	$0.440890\pi$
$854$		0			0
$855$	−26.7491	−	35.9361i	−0.914799	−	1.22899i
$856$		0			0
$857$			34.7435i			1.18682i	0.804902	+	0.593408i	$0.202218\pi$
$857$			34.7435i			1.18682i	−0.804902	+	0.593408i	$0.797782\pi$
$858$		0			0
$859$		−	27.8602i		−	0.950576i	−0.879830	−	0.475288i	$-0.842344\pi$
$859$		−	27.8602i		−	0.950576i	0.879830	−	0.475288i	$-0.157656\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.4622			−1.24119			−0.620595	−	0.784132i	$-0.713109\pi$
$863$	−36.4622			−1.24119			−0.620595	+	0.784132i	$0.713109\pi$
$864$		0			0
$865$	−11.5826			−0.393819
$866$		0			0
$867$	15.6765	−	31.2114i	0.532400	−	1.05999i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	13.6745i		−	0.463341i
$872$		0			0
$873$	−17.7328	−	23.8232i	−0.600166	−	0.806294i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.5826			0.661257			0.330628	−	0.943761i	$-0.392739\pi$
$877$	19.5826			0.661257			0.330628	+	0.943761i	$0.392739\pi$
$878$		0			0
$879$	−3.06878	−	1.54135i	−0.103507	−	0.0519885i
$880$		0			0
$881$		−	21.1302i		−	0.711895i	−0.934506	−	0.355948i	$-0.884158\pi$
$881$		−	21.1302i		−	0.711895i	0.934506	−	0.355948i	$-0.115842\pi$
$882$		0			0
$883$		−	42.1413i		−	1.41817i	−0.705124	−	0.709084i	$-0.749109\pi$
$883$		−	42.1413i		−	1.41817i	0.705124	−	0.709084i	$-0.250891\pi$
$884$		0			0
$885$	24.5704	+	12.3409i	0.825923	+	0.414835i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			51.7290i			1.73105i
$894$		0			0
$895$		−	22.4499i		−	0.750419i
$896$		0			0
$897$	25.5826	−	50.9341i	0.854177	−	1.70064i
$898$		0			0
$899$	−18.6577			−0.622269
$900$		0			0
$901$	−6.12397			−0.204019
$902$		0			0
$903$		0			0
$904$		0			0
$905$			30.6775i			1.01976i
$906$		0			0
$907$			11.8383i			0.393084i	0.980495	+	0.196542i	$0.0629713\pi$
$907$			11.8383i			0.393084i	−0.980495	+	0.196542i	$0.937029\pi$
$908$		0			0
$909$	27.9891	−	20.8337i	0.928340	−	0.691011i
$910$		0			0
$911$	−21.5294			−0.713300			−0.356650	−	0.934238i	$-0.616081\pi$
$911$	−21.5294			−0.713300			−0.356650	+	0.934238i	$0.616081\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−9.00433	−	4.52259i	−0.297674	−	0.149512i
$916$		0			0
$917$		0			0
$918$		0			0
$919$			47.0514i			1.55208i	0.630682	+	0.776042i	$0.282775\pi$
$919$			47.0514i			1.55208i	−0.630682	+	0.776042i	$0.717225\pi$
$920$		0			0
$921$	0.791288	+	0.397438i	0.0260738	+	0.0130960i
$922$		0			0
$923$	−26.0395			−0.857100
$924$		0			0
$925$	49.9129			1.64113
$926$		0			0
$927$	44.6972	−	33.2704i	1.46805	−	1.09274i
$928$		0			0
$929$		−	27.6404i		−	0.906851i	−0.891294	−	0.453425i	$-0.850202\pi$
$929$		−	27.6404i		−	0.906851i	0.891294	−	0.453425i	$-0.149798\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−16.4174	+	32.6866i	−0.537482	+	1.07011i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−11.0801			−0.361973			−0.180986	−	0.983486i	$-0.557929\pi$
$937$	−11.0801			−0.361973			−0.180986	+	0.983486i	$0.557929\pi$
$938$		0			0
$939$	15.5734	−	31.0061i	0.508218	−	1.01185i
$940$		0			0
$941$		−	18.4372i		−	0.601036i	−0.953776	−	0.300518i	$-0.902840\pi$
$941$		−	18.4372i		−	0.601036i	0.953776	−	0.300518i	$-0.0971596\pi$
$942$		0			0
$943$		−	5.92146i		−	0.192829i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	50.0184			1.62538			0.812689	−	0.582697i	$-0.198003\pi$
$947$	50.0184			1.62538			0.812689	+	0.582697i	$0.198003\pi$
$948$		0			0
$949$	56.7477			1.84211
$950$		0			0
$951$	43.1424	+	21.6690i	1.39899	+	0.702666i
$952$		0			0
$953$			32.8963i			1.06561i	0.846237	+	0.532807i	$0.178863\pi$
$953$			32.8963i			1.06561i	−0.846237	+	0.532807i	$0.821137\pi$
$954$		0			0
$955$			34.2929i			1.10969i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.00000			0.225806
$962$		0			0
$963$	−11.8162	−	15.8745i	−0.380772	−	0.511549i
$964$		0			0
$965$		−	63.8045i		−	2.05394i
$966$		0			0
$967$		−	29.7309i		−	0.956082i	−0.878338	−	0.478041i	$-0.841347\pi$
$967$		−	29.7309i		−	0.956082i	0.878338	−	0.478041i	$-0.158653\pi$
$968$		0			0
$969$	20.7948	−	41.4019i	0.668026	−	1.33002i
$970$		0			0
$971$	2.20382			0.0707240			0.0353620	−	0.999375i	$-0.488742\pi$
$971$	2.20382			0.0707240			0.0353620	+	0.999375i	$0.488742\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	20.2005	−	40.2186i	0.646933	−	1.28803i
$976$		0			0
$977$		−	31.8917i		−	1.02031i	−0.860084	−	0.510153i	$-0.829589\pi$
$977$		−	31.8917i		−	1.02031i	0.860084	−	0.510153i	$-0.170411\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	7.91288	+	10.6306i	0.252639	+	0.339408i
$982$		0			0
$983$	11.7894			0.376025			0.188012	−	0.982167i	$-0.439795\pi$
$983$	11.7894			0.376025			0.188012	+	0.982167i	$0.439795\pi$
$984$		0			0
$985$	−3.41875			−0.108931
$986$		0			0
$987$		0			0
$988$		0			0
$989$			80.6071i			2.56316i
$990$		0			0
$991$			4.03620i			0.128214i	0.997943	+	0.0641071i	$0.0204199\pi$
$991$			4.03620i			0.128214i	−0.997943	+	0.0641071i	$0.979580\pi$
$992$		0			0
$993$	−4.24264	−	2.13094i	−0.134636	−	0.0676234i
$994$		0			0
$995$	33.3456			1.05713
$996$		0			0
$997$	45.6603			1.44608			0.723039	−	0.690807i	$-0.242745\pi$
$997$	45.6603			1.44608			0.723039	+	0.690807i	$0.242745\pi$
$998$		0			0
$999$	−38.8024	+	6.83723i	−1.22765	+	0.216320i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.p.2255.8	yes	16
3.2	odd	2	inner	2352.2.h.p.2255.11	yes	16
4.3	odd	2	inner	2352.2.h.p.2255.10	yes	16
7.6	odd	2	inner	2352.2.h.p.2255.9	yes	16
12.11	even	2	inner	2352.2.h.p.2255.5	✓	16
21.20	even	2	inner	2352.2.h.p.2255.6	yes	16
28.27	even	2	inner	2352.2.h.p.2255.7	yes	16
84.83	odd	2	inner	2352.2.h.p.2255.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.h.p.2255.5	✓	16	12.11	even	2	inner
2352.2.h.p.2255.6	yes	16	21.20	even	2	inner
2352.2.h.p.2255.7	yes	16	28.27	even	2	inner
2352.2.h.p.2255.8	yes	16	1.1	even	1	trivial
2352.2.h.p.2255.9	yes	16	7.6	odd	2	inner
2352.2.h.p.2255.10	yes	16	4.3	odd	2	inner
2352.2.h.p.2255.11	yes	16	3.2	odd	2	inner
2352.2.h.p.2255.12	yes	16	84.83	odd	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.777403 + 1.54779i) q^{3} +3.40332i q^{5} +(-1.79129 - 2.40651i) q^{9} +O(q^{10})\)	\(q+(-0.777403 + 1.54779i) q^{3} +3.40332i q^{5} +(-1.79129 - 2.40651i) q^{9} +3.94748 q^{13} +(-5.26761 - 2.64575i) q^{15} -6.09632i q^{17} -4.38774i q^{19} -8.33639 q^{23} -6.58258 q^{25} +(5.11732 - 0.901703i) q^{27} -3.80848i q^{29} -4.89898i q^{31} -7.58258 q^{37} +(-3.06878 + 6.10985i) q^{39} +0.710314i q^{41} -9.66930i q^{43} +(8.19012 - 6.09632i) q^{45} -11.7894 q^{47} +(9.43581 + 4.73930i) q^{51} -1.00454i q^{53} +(6.79129 + 3.41105i) q^{57} -4.66442 q^{59} +1.70938 q^{61} +13.4345i q^{65} -3.46410i q^{67} +(6.48074 - 12.9030i) q^{69} -6.59649 q^{71} +14.3757 q^{73} +(5.11732 - 10.1884i) q^{75} -6.92820i q^{79} +(-2.58258 + 8.62150i) q^{81} -16.4539 q^{83} +20.7477 q^{85} +(5.89472 + 2.96073i) q^{87} -6.09632i q^{89} +(7.58258 + 3.80848i) q^{93} +14.9329 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^9	\( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) \) 16 * q + 8 * q^9 - 32 * q^25 - 48 * q^37 + 72 * q^57 + 32 * q^81 + 112 * q^85 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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