LMFDB - Embedded newform 3600.2.w.j.1457.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(593,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

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

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

chi := DirichletCharacter("3600.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.w (of order $4$, degree $2$, 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:	$28.7461447277$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.1
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	3600.1457
Dual form			3600.2.w.j.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.23607 - 3.23607i) q^{7} +O(q^{10})$	$q+(-3.23607 - 3.23607i) q^{7} -4.57649i q^{11} +(4.23607 - 4.23607i) q^{13} +(1.74806 - 1.74806i) q^{17} +2.47214i q^{19} +(-2.82843 - 2.82843i) q^{23} -5.99070 q^{29} -1.52786 q^{31} +(-2.23607 - 2.23607i) q^{37} +7.07107i q^{41} +(2.47214 - 2.47214i) q^{43} +(1.74806 - 1.74806i) q^{47} +13.9443i q^{49} +1.08036 q^{59} +10.4721 q^{61} +(-1.52786 - 1.52786i) q^{67} +12.6491i q^{71} +(0.527864 - 0.527864i) q^{73} +(-14.8098 + 14.8098i) q^{77} -14.4721i q^{79} +(-1.08036 - 1.08036i) q^{83} -0.746512 q^{89} -27.4164 q^{91} +(-1.00000 - 1.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} + 16 q^{13} - 48 q^{31} - 16 q^{43} + 48 q^{61} - 48 q^{67} + 40 q^{73} - 112 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 16 * q^13 - 48 * q^31 - 16 * q^43 + 48 * q^61 - 48 * q^67 + 40 * q^73 - 112 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$	$1$

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
$7$	−3.23607	−	3.23607i	−1.22312	−	1.22312i	−0.966517	−	0.256601i	$-0.917397\pi$
$7$	−3.23607	−	3.23607i	−1.22312	−	1.22312i	−0.256601	−	0.966517i	$-0.582603\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.57649i		−	1.37986i	−0.723874	−	0.689932i	$-0.757640\pi$
$11$		−	4.57649i		−	1.37986i	0.723874	−	0.689932i	$-0.242360\pi$
$12$		0			0
$13$	4.23607	−	4.23607i	1.17487	−	1.17487i	0.193841	−	0.981033i	$-0.437905\pi$
$13$	4.23607	−	4.23607i	1.17487	−	1.17487i	0.981033	−	0.193841i	$-0.0620946\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.74806	−	1.74806i	0.423968	−	0.423968i	−0.462600	−	0.886567i	$-0.653083\pi$
$17$	1.74806	−	1.74806i	0.423968	−	0.423968i	0.886567	+	0.462600i	$0.153083\pi$
$18$		0			0
$19$			2.47214i			0.567147i	0.958951	+	0.283573i	$0.0915200\pi$
$19$			2.47214i			0.567147i	−0.958951	+	0.283573i	$0.908480\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	0.347801	−	0.937568i	$-0.386929\pi$
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	−0.937568	+	0.347801i	$0.886929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.99070			−1.11245			−0.556223	−	0.831033i	$-0.687750\pi$
$29$	−5.99070			−1.11245			−0.556223	+	0.831033i	$0.687750\pi$
$30$		0			0
$31$	−1.52786			−0.274412			−0.137206	−	0.990543i	$-0.543812\pi$
$31$	−1.52786			−0.274412			−0.137206	+	0.990543i	$0.543812\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.23607	−	2.23607i	−0.367607	−	0.367607i	0.498997	−	0.866604i	$-0.333702\pi$
$37$	−2.23607	−	2.23607i	−0.367607	−	0.367607i	−0.866604	+	0.498997i	$0.833702\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.07107i			1.10432i	0.833740	+	0.552158i	$0.186195\pi$
$41$			7.07107i			1.10432i	−0.833740	+	0.552158i	$0.813805\pi$
$42$		0			0
$43$	2.47214	−	2.47214i	0.376997	−	0.376997i	−0.493020	−	0.870018i	$-0.664107\pi$
$43$	2.47214	−	2.47214i	0.376997	−	0.376997i	0.870018	+	0.493020i	$0.164107\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.74806	−	1.74806i	0.254981	−	0.254981i	−0.568028	−	0.823009i	$-0.692293\pi$
$47$	1.74806	−	1.74806i	0.254981	−	0.254981i	0.823009	+	0.568028i	$0.192293\pi$
$48$		0			0
$49$			13.9443i			1.99204i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.08036			0.140651			0.0703256	−	0.997524i	$-0.477596\pi$
$59$	1.08036			0.140651			0.0703256	+	0.997524i	$0.477596\pi$
$60$		0			0
$61$	10.4721			1.34082			0.670410	−	0.741991i	$-0.266118\pi$
$61$	10.4721			1.34082			0.670410	+	0.741991i	$0.266118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.52786	−	1.52786i	−0.186658	−	0.186658i	0.607591	−	0.794250i	$-0.292136\pi$
$67$	−1.52786	−	1.52786i	−0.186658	−	0.186658i	−0.794250	+	0.607591i	$0.792136\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.6491i			1.50117i	0.660772	+	0.750587i	$0.270229\pi$
$71$			12.6491i			1.50117i	−0.660772	+	0.750587i	$0.729771\pi$
$72$		0			0
$73$	0.527864	−	0.527864i	0.0617818	−	0.0617818i	−0.675541	−	0.737323i	$-0.736090\pi$
$73$	0.527864	−	0.527864i	0.0617818	−	0.0617818i	0.737323	+	0.675541i	$0.236090\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.8098	+	14.8098i	−1.68774	+	1.68774i
$78$		0			0
$79$		−	14.4721i		−	1.62824i	−0.580695	−	0.814121i	$-0.697219\pi$
$79$		−	14.4721i		−	1.62824i	0.580695	−	0.814121i	$-0.302781\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.08036	−	1.08036i	−0.118585	−	0.118585i	0.645324	−	0.763909i	$-0.276722\pi$
$83$	−1.08036	−	1.08036i	−0.118585	−	0.118585i	−0.763909	+	0.645324i	$0.776722\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.746512			−0.0791302			−0.0395651	−	0.999217i	$-0.512597\pi$
$89$	−0.746512			−0.0791302			−0.0395651	+	0.999217i	$0.512597\pi$
$90$		0			0
$91$	−27.4164			−2.87402
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	−	1.00000i	−0.101535	−	0.101535i	0.654515	−	0.756049i	$-0.272873\pi$
$97$	−1.00000	−	1.00000i	−0.101535	−	0.101535i	−0.756049	+	0.654515i	$0.772873\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			10.9799i			1.09254i	0.837610	+	0.546268i	$0.183952\pi$
$101$			10.9799i			1.09254i	−0.837610	+	0.546268i	$0.816048\pi$
$102$		0			0
$103$	−12.1803	+	12.1803i	−1.20016	+	1.20016i	−0.226049	+	0.974116i	$0.572581\pi$
$103$	−12.1803	+	12.1803i	−1.20016	+	1.20016i	−0.974116	+	0.226049i	$0.927419\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.2333	+	10.2333i	−0.989295	+	0.989295i	−0.999943	−	0.0106485i	$-0.996610\pi$
$107$	−10.2333	+	10.2333i	−0.989295	+	0.989295i	0.0106485	+	0.999943i	$0.496610\pi$
$108$		0			0
$109$		−	15.4164i		−	1.47662i	−0.674459	−	0.738312i	$-0.735623\pi$
$109$		−	15.4164i		−	1.47662i	0.674459	−	0.738312i	$-0.264377\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.16228	+	3.16228i	0.297482	+	0.297482i	0.840027	−	0.542545i	$-0.182539\pi$
$113$	3.16228	+	3.16228i	0.297482	+	0.297482i	−0.542545	+	0.840027i	$0.682539\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.3137			−1.03713
$120$		0			0
$121$	−9.94427			−0.904025
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.23607	−	7.23607i	−0.642097	−	0.642097i	0.308973	−	0.951071i	$-0.400015\pi$
$127$	−7.23607	−	7.23607i	−0.642097	−	0.642097i	−0.951071	+	0.308973i	$0.900015\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.41577i		−	0.211066i	−0.994416	−	0.105533i	$-0.966345\pi$
$131$		−	2.41577i		−	0.211066i	0.994416	−	0.105533i	$-0.0336549\pi$
$132$		0			0
$133$	8.00000	−	8.00000i	0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.81913	+	8.81913i	−0.753469	+	0.753469i	−0.975125	−	0.221656i	$-0.928854\pi$
$137$	−8.81913	+	8.81913i	−0.753469	+	0.753469i	0.221656	+	0.975125i	$0.428854\pi$
$138$		0			0
$139$		−	8.94427i		−	0.758643i	−0.925265	−	0.379322i	$-0.876157\pi$
$139$		−	8.94427i		−	0.758643i	0.925265	−	0.379322i	$-0.123843\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.3863	−	19.3863i	−1.62117	−	1.62117i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14.4760			−1.18592			−0.592959	−	0.805232i	$-0.702041\pi$
$149$	−14.4760			−1.18592			−0.592959	+	0.805232i	$0.702041\pi$
$150$		0			0
$151$	4.94427			0.402359			0.201180	−	0.979554i	$-0.435523\pi$
$151$	4.94427			0.402359			0.201180	+	0.979554i	$0.435523\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.708204	+	0.708204i	0.0565208	+	0.0565208i	0.734802	−	0.678281i	$-0.237275\pi$
$157$	0.708204	+	0.708204i	0.0565208	+	0.0565208i	−0.678281	+	0.734802i	$0.737275\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			18.3060i			1.44271i
$162$		0			0
$163$	2.47214	−	2.47214i	0.193633	−	0.193633i	−0.603631	−	0.797264i	$-0.706280\pi$
$163$	2.47214	−	2.47214i	0.193633	−	0.193633i	0.797264	+	0.603631i	$0.206280\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.32456	+	6.32456i	−0.489409	+	0.489409i	−0.908120	−	0.418711i	$-0.862482\pi$
$167$	−6.32456	+	6.32456i	−0.489409	+	0.489409i	0.418711	+	0.908120i	$0.362482\pi$
$168$		0			0
$169$		−	22.8885i		−	1.76066i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.99070	−	5.99070i	−0.455465	−	0.455465i	0.441699	−	0.897163i	$-0.354376\pi$
$173$	−5.99070	−	5.99070i	−0.455465	−	0.455465i	−0.897163	+	0.441699i	$0.854376\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.7295			1.02619			0.513095	−	0.858332i	$-0.328499\pi$
$179$	13.7295			1.02619			0.513095	+	0.858332i	$0.328499\pi$
$180$		0			0
$181$	−10.4721			−0.778388			−0.389194	−	0.921156i	$-0.627246\pi$
$181$	−10.4721			−0.778388			−0.389194	+	0.921156i	$0.627246\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−8.00000	−	8.00000i	−0.585018	−	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	18.3060i		−	1.32457i	−0.749251	−	0.662287i	$-0.769586\pi$
$191$		−	18.3060i		−	1.32457i	0.749251	−	0.662287i	$-0.230414\pi$
$192$		0			0
$193$	1.47214	−	1.47214i	0.105967	−	0.105967i	−0.652136	−	0.758102i	$-0.726127\pi$
$193$	1.47214	−	1.47214i	0.105967	−	0.105967i	0.758102	+	0.652136i	$0.226127\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.82998	−	3.82998i	0.272875	−	0.272875i	−0.557382	−	0.830256i	$-0.688194\pi$
$197$	3.82998	−	3.82998i	0.272875	−	0.272875i	0.830256	+	0.557382i	$0.188194\pi$
$198$		0			0
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	19.3863	+	19.3863i	1.36065	+	1.36065i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.3137			0.782586
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.94427	+	4.94427i	0.335639	+	0.335639i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	14.8098i		−	0.996217i
$222$		0			0
$223$	19.2361	−	19.2361i	1.28814	−	1.28814i	0.352228	−	0.935914i	$-0.385424\pi$
$223$	19.2361	−	19.2361i	1.28814	−	1.28814i	0.935914	−	0.352228i	$-0.114576\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.4667	+	20.4667i	−1.35842	+	1.35842i	−0.482558	+	0.875864i	$0.660292\pi$
$227$	−20.4667	+	20.4667i	−1.35842	+	1.35842i	−0.875864	+	0.482558i	$0.839708\pi$
$228$		0			0
$229$			19.8885i			1.31427i	0.753772	+	0.657136i	$0.228232\pi$
$229$			19.8885i			1.31427i	−0.753772	+	0.657136i	$0.771768\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.5579	−	16.5579i	−1.08474	−	1.08474i	−0.996060	−	0.0886844i	$-0.971734\pi$
$233$	−16.5579	−	16.5579i	−1.08474	−	1.08474i	−0.0886844	−	0.996060i	$-0.528266\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.6491			0.818203			0.409101	−	0.912489i	$-0.365842\pi$
$239$	12.6491			0.818203			0.409101	+	0.912489i	$0.365842\pi$
$240$		0			0
$241$	2.94427			0.189657			0.0948286	−	0.995494i	$-0.469770\pi$
$241$	2.94427			0.189657			0.0948286	+	0.995494i	$0.469770\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	10.4721	+	10.4721i	0.666326	+	0.666326i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			30.7000i			1.93777i	0.247513	+	0.968885i	$0.420387\pi$
$251$			30.7000i			1.93777i	−0.247513	+	0.968885i	$0.579613\pi$
$252$		0			0
$253$	−12.9443	+	12.9443i	−0.813799	+	0.813799i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.16228	−	3.16228i	0.197257	−	0.197257i	−0.601566	−	0.798823i	$-0.705456\pi$
$257$	3.16228	−	3.16228i	0.197257	−	0.197257i	0.798823	+	0.601566i	$0.205456\pi$
$258$		0			0
$259$			14.4721i			0.899255i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14.1421	−	14.1421i	−0.872041	−	0.872041i	0.120653	−	0.992695i	$-0.461501\pi$
$263$	−14.1421	−	14.1421i	−0.872041	−	0.872041i	−0.992695	+	0.120653i	$0.961501\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.3153			0.750875			0.375437	−	0.926848i	$-0.377493\pi$
$269$	12.3153			0.750875			0.375437	+	0.926848i	$0.377493\pi$
$270$		0			0
$271$	28.9443			1.75824			0.879120	−	0.476601i	$-0.158131\pi$
$271$	28.9443			1.75824			0.879120	+	0.476601i	$0.158131\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.1803	−	15.1803i	−0.912098	−	0.912098i	0.0843389	−	0.996437i	$-0.473122\pi$
$277$	−15.1803	−	15.1803i	−0.912098	−	0.912098i	−0.996437	+	0.0843389i	$0.973122\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	17.5595i		−	1.04751i	−0.851869	−	0.523755i	$-0.824531\pi$
$281$		−	17.5595i		−	1.04751i	0.851869	−	0.523755i	$-0.175469\pi$
$282$		0			0
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	−0.903705	−	0.428155i	$-0.859164\pi$
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	0.428155	+	0.903705i	$0.359164\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.8825	−	22.8825i	1.35071	−	1.35071i
$288$		0			0
$289$			10.8885i			0.640503i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.15143	−	8.15143i	−0.476212	−	0.476212i	0.427706	−	0.903918i	$-0.359322\pi$
$293$	−8.15143	−	8.15143i	−0.476212	−	0.476212i	−0.903918	+	0.427706i	$0.859322\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−23.9628			−1.38581
$300$		0			0
$301$	−16.0000			−0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.4721	+	10.4721i	0.597676	+	0.597676i	0.939694	−	0.342017i	$-0.111110\pi$
$307$	10.4721	+	10.4721i	0.597676	+	0.597676i	−0.342017	+	0.939694i	$0.611110\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			14.8098i			0.839789i	0.907573	+	0.419894i	$0.137933\pi$
$311$			14.8098i			0.839789i	−0.907573	+	0.419894i	$0.862067\pi$
$312$		0			0
$313$	7.47214	−	7.47214i	0.422350	−	0.422350i	−0.463662	−	0.886012i	$-0.653465\pi$
$313$	7.47214	−	7.47214i	0.422350	−	0.422350i	0.886012	+	0.463662i	$0.153465\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.81758	−	7.81758i	0.439079	−	0.439079i	−0.452623	−	0.891702i	$-0.649512\pi$
$317$	7.81758	−	7.81758i	0.439079	−	0.439079i	0.891702	+	0.452623i	$0.149512\pi$
$318$		0			0
$319$			27.4164i			1.53502i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.32145	+	4.32145i	0.240452	+	0.240452i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.3137			−0.623745
$330$		0			0
$331$	5.52786			0.303839			0.151919	−	0.988393i	$-0.451455\pi$
$331$	5.52786			0.303839			0.151919	+	0.988393i	$0.451455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.4164	−	14.4164i	−0.785312	−	0.785312i	0.195410	−	0.980722i	$-0.437396\pi$
$337$	−14.4164	−	14.4164i	−0.785312	−	0.785312i	−0.980722	+	0.195410i	$0.937396\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.99226i			0.378652i
$342$		0			0
$343$	22.4721	−	22.4721i	1.21338	−	1.21338i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.99226	−	6.99226i	0.375364	−	0.375364i	−0.494062	−	0.869426i	$-0.664489\pi$
$347$	6.99226	−	6.99226i	0.375364	−	0.375364i	0.869426	+	0.494062i	$0.164489\pi$
$348$		0			0
$349$		−	5.52786i		−	0.295900i	−0.988995	−	0.147950i	$-0.952733\pi$
$349$		−	5.52786i		−	0.295900i	0.988995	−	0.147950i	$-0.0472674\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.0540	+	20.0540i	1.06737	+	1.06737i	0.997560	+	0.0698078i	$0.0222386\pi$
$353$	20.0540	+	20.0540i	1.06737	+	1.06737i	0.0698078	+	0.997560i	$0.477761\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−33.1158			−1.74779			−0.873893	−	0.486119i	$-0.838412\pi$
$359$	−33.1158			−1.74779			−0.873893	+	0.486119i	$0.838412\pi$
$360$		0			0
$361$	12.8885			0.678344
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.29180	+	2.29180i	0.119631	+	0.119631i	0.764388	−	0.644757i	$-0.223041\pi$
$367$	2.29180	+	2.29180i	0.119631	+	0.119631i	−0.644757	+	0.764388i	$0.723041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.2918	+	11.2918i	−0.584667	+	0.584667i	−0.936182	−	0.351515i	$-0.885667\pi$
$373$	−11.2918	+	11.2918i	−0.584667	+	0.584667i	0.351515	+	0.936182i	$0.385667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−25.3770	+	25.3770i	−1.30698	+	1.30698i
$378$		0			0
$379$			16.9443i			0.870369i	0.900341	+	0.435184i	$0.143317\pi$
$379$			16.9443i			0.870369i	−0.900341	+	0.435184i	$0.856683\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.90879	−	3.90879i	−0.199730	−	0.199730i	0.600154	−	0.799884i	$-0.295106\pi$
$383$	−3.90879	−	3.90879i	−0.199730	−	0.199730i	−0.799884	+	0.600154i	$0.795106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.7974			0.953068			0.476534	−	0.879156i	$-0.341893\pi$
$389$	18.7974			0.953068			0.476534	+	0.879156i	$0.341893\pi$
$390$		0			0
$391$	−9.88854			−0.500085
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.23607	+	6.23607i	0.312979	+	0.312979i	0.846063	−	0.533083i	$-0.178967\pi$
$397$	6.23607	+	6.23607i	0.312979	+	0.312979i	−0.533083	+	0.846063i	$0.678967\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	27.5378i		−	1.37517i	−0.726104	−	0.687585i	$-0.758671\pi$
$401$		−	27.5378i		−	1.37517i	0.726104	−	0.687585i	$-0.241329\pi$
$402$		0			0
$403$	−6.47214	+	6.47214i	−0.322400	+	0.322400i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.2333	+	10.2333i	−0.507248	+	0.507248i
$408$		0			0
$409$		−	25.8885i		−	1.28011i	−0.768331	−	0.640053i	$-0.778912\pi$
$409$		−	25.8885i		−	1.28011i	0.768331	−	0.640053i	$-0.221088\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.49613	−	3.49613i	−0.172033	−	0.172033i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.40182			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$419$	5.40182			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−33.8885	−	33.8885i	−1.63998	−	1.63998i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$		0			0
$433$	16.4164	−	16.4164i	0.788922	−	0.788922i	−0.192395	−	0.981318i	$-0.561626\pi$
$433$	16.4164	−	16.4164i	0.788922	−	0.788922i	0.981318	+	0.192395i	$0.0616256\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.99226	−	6.99226i	0.334485	−	0.334485i
$438$		0			0
$439$		−	20.9443i		−	0.999616i	−0.866136	−	0.499808i	$-0.833404\pi$
$439$		−	20.9443i		−	0.999616i	0.866136	−	0.499808i	$-0.166596\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.4667	+	20.4667i	0.972402	+	0.972402i	0.999629	−	0.0272274i	$-0.00866783\pi$
$443$	20.4667	+	20.4667i	0.972402	+	0.972402i	−0.0272274	+	0.999629i	$0.508668\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−38.6938			−1.82608			−0.913038	−	0.407875i	$-0.866270\pi$
$449$	−38.6938			−1.82608			−0.913038	+	0.407875i	$0.866270\pi$
$450$		0			0
$451$	32.3607			1.52380
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.0000	−	13.0000i	−0.608114	−	0.608114i	0.334339	−	0.942453i	$-0.391487\pi$
$457$	−13.0000	−	13.0000i	−0.608114	−	0.608114i	−0.942453	+	0.334339i	$0.891487\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			9.48683i			0.441846i	0.975291	+	0.220923i	$0.0709069\pi$
$461$			9.48683i			0.441846i	−0.975291	+	0.220923i	$0.929093\pi$
$462$		0			0
$463$	10.6525	−	10.6525i	0.495063	−	0.495063i	−0.414834	−	0.909897i	$-0.636160\pi$
$463$	10.6525	−	10.6525i	0.495063	−	0.495063i	0.909897	+	0.414834i	$0.136160\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.57649	−	4.57649i	0.211775	−	0.211775i	−0.593246	−	0.805021i	$-0.702154\pi$
$467$	4.57649	−	4.57649i	0.211775	−	0.211775i	0.805021	+	0.593246i	$0.202154\pi$
$468$		0			0
$469$			9.88854i			0.456611i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.3137	−	11.3137i	−0.520205	−	0.520205i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−39.5980			−1.80928			−0.904639	−	0.426179i	$-0.859859\pi$
$479$	−39.5980			−1.80928			−0.904639	+	0.426179i	$0.859859\pi$
$480$		0			0
$481$	−18.9443			−0.863784
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	8.76393	+	8.76393i	0.397132	+	0.397132i	0.877220	−	0.480088i	$-0.159395\pi$
$487$	8.76393	+	8.76393i	0.397132	+	0.397132i	−0.480088	+	0.877220i	$0.659395\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			30.7000i			1.38547i	0.721191	+	0.692737i	$0.243595\pi$
$491$			30.7000i			1.38547i	−0.721191	+	0.692737i	$0.756405\pi$
$492$		0			0
$493$	−10.4721	+	10.4721i	−0.471641	+	0.471641i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	40.9334	−	40.9334i	1.83611	−	1.83611i
$498$		0			0
$499$			26.4721i			1.18506i	0.805550	+	0.592528i	$0.201870\pi$
$499$			26.4721i			1.18506i	−0.805550	+	0.592528i	$0.798130\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.2070	+	29.2070i	1.30228	+	1.30228i	0.926854	+	0.375422i	$0.122502\pi$
$503$	29.2070	+	29.2070i	1.30228	+	1.30228i	0.375422	+	0.926854i	$0.377498\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.9830			0.575460			0.287730	−	0.957712i	$-0.407099\pi$
$509$	12.9830			0.575460			0.287730	+	0.957712i	$0.407099\pi$
$510$		0			0
$511$	−3.41641			−0.151133
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.00000	−	8.00000i	−0.351840	−	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			6.24574i			0.273631i	0.990597	+	0.136816i	$0.0436867\pi$
$521$			6.24574i			0.273631i	−0.990597	+	0.136816i	$0.956313\pi$
$522$		0			0
$523$	−9.88854	+	9.88854i	−0.432396	+	0.432396i	−0.889443	−	0.457047i	$-0.848907\pi$
$523$	−9.88854	+	9.88854i	−0.432396	+	0.432396i	0.457047	+	0.889443i	$0.348907\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.67080	+	2.67080i	−0.116342	+	0.116342i
$528$		0			0
$529$		−	7.00000i		−	0.304348i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	29.9535	+	29.9535i	1.29743	+	1.29743i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	63.8158			2.74874
$540$		0			0
$541$	−25.3050			−1.08794			−0.543972	−	0.839103i	$-0.683080\pi$
$541$	−25.3050			−1.08794			−0.543972	+	0.839103i	$0.683080\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	0.915470	−	0.402387i	$-0.131819\pi$
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	−0.402387	+	0.915470i	$0.631819\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	14.8098i		−	0.630920i
$552$		0			0
$553$	−46.8328	+	46.8328i	−1.99153	+	1.99153i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.1235	−	26.1235i	1.10689	−	1.10689i	0.113333	−	0.993557i	$-0.463847\pi$
$557$	26.1235	−	26.1235i	1.10689	−	1.10689i	0.993557	−	0.113333i	$-0.0361527\pi$
$558$		0			0
$559$		−	20.9443i		−	0.885848i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.33540	−	1.33540i	−0.0562805	−	0.0562805i	0.678406	−	0.734687i	$-0.262671\pi$
$563$	−1.33540	−	1.33540i	−0.0562805	−	0.0562805i	−0.734687	+	0.678406i	$0.762671\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.3662			1.27302			0.636508	−	0.771270i	$-0.280378\pi$
$569$	30.3662			1.27302			0.636508	+	0.771270i	$0.280378\pi$
$570$		0			0
$571$	7.41641			0.310367			0.155184	−	0.987886i	$-0.450403\pi$
$571$	7.41641			0.310367			0.155184	+	0.987886i	$0.450403\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	12.0557	+	12.0557i	0.501887	+	0.501887i	0.912024	−	0.410137i	$-0.134519\pi$
$577$	12.0557	+	12.0557i	0.501887	+	0.501887i	−0.410137	+	0.912024i	$0.634519\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			6.99226i			0.290088i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.8825	+	22.8825i	−0.944460	+	0.944460i	−0.998537	−	0.0540767i	$-0.982778\pi$
$587$	−22.8825	+	22.8825i	−0.944460	+	0.944460i	0.0540767	+	0.998537i	$0.482778\pi$
$588$		0			0
$589$		−	3.77709i		−	0.155632i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.1359	−	22.1359i	−0.909014	−	0.909014i	0.0871785	−	0.996193i	$-0.472215\pi$
$593$	−22.1359	−	22.1359i	−0.909014	−	0.909014i	−0.996193	+	0.0871785i	$0.972215\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.6491			0.516829			0.258414	−	0.966034i	$-0.416800\pi$
$599$	12.6491			0.516829			0.258414	+	0.966034i	$0.416800\pi$
$600$		0			0
$601$	−30.8328			−1.25770			−0.628848	−	0.777528i	$-0.716473\pi$
$601$	−30.8328			−1.25770			−0.628848	+	0.777528i	$0.716473\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	30.6525	+	30.6525i	1.24415	+	1.24415i	0.958265	+	0.285880i	$0.0922860\pi$
$607$	30.6525	+	30.6525i	1.24415	+	1.24415i	0.285880	+	0.958265i	$0.407714\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	14.8098i		−	0.599142i
$612$		0			0
$613$	26.7082	−	26.7082i	1.07873	−	1.07873i	0.0821110	−	0.996623i	$-0.473834\pi$
$613$	26.7082	−	26.7082i	1.07873	−	1.07873i	0.996623	−	0.0821110i	$-0.0261662\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.5579	+	16.5579i	−0.666596	+	0.666596i	−0.956926	−	0.290330i	$-0.906235\pi$
$617$	−16.5579	+	16.5579i	−0.666596	+	0.666596i	0.290330	+	0.956926i	$0.406235\pi$
$618$		0			0
$619$		−	5.88854i		−	0.236681i	−0.992973	−	0.118340i	$-0.962243\pi$
$619$		−	5.88854i		−	0.236681i	0.992973	−	0.118340i	$-0.0377574\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.41577	+	2.41577i	0.0967856	+	0.0967856i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.81758			−0.311707
$630$		0			0
$631$	−8.36068			−0.332833			−0.166417	−	0.986056i	$-0.553220\pi$
$631$	−8.36068			−0.332833			−0.166417	+	0.986056i	$0.553220\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	59.0689	+	59.0689i	2.34039	+	2.34039i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	5.57804i		−	0.220319i	−0.993914	−	0.110160i	$-0.964864\pi$
$641$		−	5.57804i		−	0.220319i	0.993914	−	0.110160i	$-0.0351362\pi$
$642$		0			0
$643$	−20.0000	+	20.0000i	−0.788723	+	0.788723i	−0.981285	−	0.192562i	$-0.938320\pi$
$643$	−20.0000	+	20.0000i	−0.788723	+	0.788723i	0.192562	+	0.981285i	$0.438320\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.3755	−	24.3755i	0.958299	−	0.958299i	−0.0408656	−	0.999165i	$-0.513012\pi$
$647$	24.3755	−	24.3755i	0.958299	−	0.958299i	0.999165	+	0.0408656i	$0.0130115\pi$
$648$		0			0
$649$		−	4.94427i		−	0.194080i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.3137	+	11.3137i	0.442740	+	0.442740i	0.892932	−	0.450192i	$-0.148644\pi$
$653$	11.3137	+	11.3137i	0.442740	+	0.442740i	−0.450192	+	0.892932i	$0.648644\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	35.5316			1.38411			0.692057	−	0.721843i	$-0.256705\pi$
$659$	35.5316			1.38411			0.692057	+	0.721843i	$0.256705\pi$
$660$		0			0
$661$	41.3050			1.60658			0.803288	−	0.595591i	$-0.203082\pi$
$661$	41.3050			1.60658			0.803288	+	0.595591i	$0.203082\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.9443	+	16.9443i	0.656085	+	0.656085i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	47.9256i		−	1.85015i
$672$		0			0
$673$	−19.4721	+	19.4721i	−0.750596	+	0.750596i	−0.974590	−	0.223995i	$-0.928090\pi$
$673$	−19.4721	+	19.4721i	−0.750596	+	0.750596i	0.223995	+	0.974590i	$0.428090\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.3060	+	18.3060i	−0.703555	+	0.703555i	−0.965172	−	0.261617i	$-0.915744\pi$
$677$	−18.3060	+	18.3060i	−0.703555	+	0.703555i	0.261617	+	0.965172i	$0.415744\pi$
$678$		0			0
$679$			6.47214i			0.248378i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.0649	−	15.0649i	−0.576441	−	0.576441i	0.357480	−	0.933921i	$-0.383636\pi$
$683$	−15.0649	−	15.0649i	−0.576441	−	0.576441i	−0.933921	+	0.357480i	$0.883636\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	20.3607			0.774557			0.387278	−	0.921963i	$-0.373415\pi$
$691$	20.3607			0.774557			0.387278	+	0.921963i	$0.373415\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.3607	+	12.3607i	0.468194	+	0.468194i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30.6212i		−	1.15655i	−0.815843	−	0.578274i	$-0.803727\pi$
$701$		−	30.6212i		−	1.15655i	0.815843	−	0.578274i	$-0.196273\pi$
$702$		0			0
$703$	5.52786	−	5.52786i	0.208487	−	0.208487i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	35.5316	−	35.5316i	1.33630	−	1.33630i
$708$		0			0
$709$			15.8885i			0.596707i	0.954455	+	0.298353i	$0.0964374\pi$
$709$			15.8885i			0.596707i	−0.954455	+	0.298353i	$0.903563\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.32145	+	4.32145i	0.161840	+	0.161840i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.32145			0.161163			0.0805815	−	0.996748i	$-0.474322\pi$
$719$	4.32145			0.161163			0.0805815	+	0.996748i	$0.474322\pi$
$720$		0			0
$721$	78.8328			2.93589
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9.12461	+	9.12461i	0.338413	+	0.338413i	0.855770	−	0.517357i	$-0.173084\pi$
$727$	9.12461	+	9.12461i	0.338413	+	0.338413i	−0.517357	+	0.855770i	$0.673084\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	8.64290i		−	0.319669i
$732$		0			0
$733$	−11.1803	+	11.1803i	−0.412955	+	0.412955i	−0.882767	−	0.469811i	$-0.844322\pi$
$733$	−11.1803	+	11.1803i	−0.412955	+	0.412955i	0.469811	+	0.882767i	$0.344322\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.99226	+	6.99226i	−0.257563	+	0.257563i
$738$		0			0
$739$		−	12.3607i		−	0.454695i	−0.973814	−	0.227347i	$-0.926995\pi$
$739$		−	12.3607i		−	0.454695i	0.973814	−	0.227347i	$-0.0730053\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−29.2070	−	29.2070i	−1.07150	−	1.07150i	−0.997239	−	0.0742626i	$-0.976340\pi$
$743$	−29.2070	−	29.2070i	−1.07150	−	1.07150i	−0.0742626	−	0.997239i	$-0.523660\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	66.2316			2.42005
$750$		0			0
$751$	−32.3607			−1.18086			−0.590429	−	0.807090i	$-0.701041\pi$
$751$	−32.3607			−1.18086			−0.590429	+	0.807090i	$0.701041\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10.7082	−	10.7082i	−0.389196	−	0.389196i	0.485204	−	0.874401i	$-0.338745\pi$
$757$	−10.7082	−	10.7082i	−0.389196	−	0.389196i	−0.874401	+	0.485204i	$0.838745\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	26.0447i		−	0.944121i	−0.881566	−	0.472060i	$-0.843510\pi$
$761$		−	26.0447i		−	0.944121i	0.881566	−	0.472060i	$-0.156490\pi$
$762$		0			0
$763$	−49.8885	+	49.8885i	−1.80609	+	1.80609i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.57649	−	4.57649i	0.165248	−	0.165248i
$768$		0			0
$769$			32.0000i			1.15395i	0.816762	+	0.576975i	$0.195767\pi$
$769$			32.0000i			1.15395i	−0.816762	+	0.576975i	$0.804233\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.81758	−	7.81758i	−0.281179	−	0.281179i	0.552400	−	0.833579i	$-0.313712\pi$
$773$	−7.81758	−	7.81758i	−0.281179	−	0.281179i	−0.833579	+	0.552400i	$0.813712\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−17.4806			−0.626309
$780$		0			0
$781$	57.8885			2.07141
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−30.4721	−	30.4721i	−1.08621	−	1.08621i	−0.995915	−	0.0902997i	$-0.971218\pi$
$787$	−30.4721	−	30.4721i	−1.08621	−	1.08621i	−0.0902997	−	0.995915i	$-0.528782\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	20.4667i		−	0.727712i
$792$		0			0
$793$	44.3607	−	44.3607i	1.57529	−	1.57529i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.66925	+	1.66925i	−0.0591280	+	0.0591280i	−0.736052	−	0.676924i	$-0.763312\pi$
$797$	−1.66925	+	1.66925i	−0.0591280	+	0.0591280i	0.676924	+	0.736052i	$0.263312\pi$
$798$		0			0
$799$		−	6.11146i		−	0.216208i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.41577	−	2.41577i	−0.0852505	−	0.0852505i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.2272			0.640833			0.320416	−	0.947277i	$-0.396177\pi$
$809$	18.2272			0.640833			0.320416	+	0.947277i	$0.396177\pi$
$810$		0			0
$811$	13.8885			0.487693			0.243846	−	0.969814i	$-0.421591\pi$
$811$	13.8885			0.487693			0.243846	+	0.969814i	$0.421591\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.11146	+	6.11146i	0.213813	+	0.213813i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	7.32611i		−	0.255683i	−0.991795	−	0.127841i	$-0.959195\pi$
$821$		−	7.32611i		−	0.255683i	0.991795	−	0.127841i	$-0.0408049\pi$
$822$		0			0
$823$	13.1246	−	13.1246i	0.457495	−	0.457495i	−0.440337	−	0.897832i	$-0.645141\pi$
$823$	13.1246	−	13.1246i	0.457495	−	0.457495i	0.897832	+	0.440337i	$0.145141\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.81758	−	7.81758i	0.271844	−	0.271844i	−0.557998	−	0.829842i	$-0.688430\pi$
$827$	7.81758	−	7.81758i	0.271844	−	0.271844i	0.829842	+	0.557998i	$0.188430\pi$
$828$		0			0
$829$		−	36.3607i		−	1.26286i	−0.775433	−	0.631429i	$-0.782469\pi$
$829$		−	36.3607i		−	1.26286i	0.775433	−	0.631429i	$-0.217531\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24.3755	+	24.3755i	0.844560	+	0.844560i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.8021			−0.752692			−0.376346	−	0.926479i	$-0.622820\pi$
$839$	−21.8021			−0.752692			−0.376346	+	0.926479i	$0.622820\pi$
$840$		0			0
$841$	6.88854			0.237536
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	32.1803	+	32.1803i	1.10573	+	1.10573i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			12.6491i			0.433606i
$852$		0			0
$853$	28.1246	−	28.1246i	0.962968	−	0.962968i	−0.0363700	−	0.999338i	$-0.511579\pi$
$853$	28.1246	−	28.1246i	0.962968	−	0.962968i	0.999338	+	0.0363700i	$0.0115795\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.3755	−	24.3755i	0.832651	−	0.832651i	−0.155228	−	0.987879i	$-0.549611\pi$
$857$	24.3755	−	24.3755i	0.832651	−	0.832651i	0.987879	+	0.155228i	$0.0496112\pi$
$858$		0			0
$859$		−	4.00000i		−	0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$		−	4.00000i		−	0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.9597	−	21.9597i	−0.747517	−	0.747517i	0.226495	−	0.974012i	$-0.427273\pi$
$863$	−21.9597	−	21.9597i	−0.747517	−	0.747517i	−0.974012	+	0.226495i	$0.927273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−66.2316			−2.24675
$870$		0			0
$871$	−12.9443			−0.438600
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.29180	+	7.29180i	0.246226	+	0.246226i	0.819420	−	0.573194i	$-0.194296\pi$
$877$	7.29180	+	7.29180i	0.246226	+	0.246226i	−0.573194	+	0.819420i	$0.694296\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	27.5378i		−	0.927771i	−0.885895	−	0.463885i	$-0.846455\pi$
$881$		−	27.5378i		−	0.927771i	0.885895	−	0.463885i	$-0.153545\pi$
$882$		0			0
$883$	−13.5279	+	13.5279i	−0.455249	+	0.455249i	−0.897092	−	0.441843i	$-0.854325\pi$
$883$	−13.5279	+	13.5279i	−0.455249	+	0.455249i	0.441843	+	0.897092i	$0.354325\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.6166	−	27.6166i	0.927274	−	0.927274i	−0.0702554	−	0.997529i	$-0.522381\pi$
$887$	27.6166	−	27.6166i	0.927274	−	0.927274i	0.997529	+	0.0702554i	$0.0223814\pi$
$888$		0			0
$889$			46.8328i			1.57072i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.32145	+	4.32145i	0.144612	+	0.144612i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.15298			0.305269
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	25.5279	+	25.5279i	0.847639	+	0.847639i	0.989838	−	0.142199i	$-0.0454174\pi$
$907$	25.5279	+	25.5279i	0.847639	+	0.847639i	−0.142199	+	0.989838i	$0.545417\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	30.4450i		−	1.00869i	−0.863503	−	0.504344i	$-0.831734\pi$
$911$		−	30.4450i		−	1.00869i	0.863503	−	0.504344i	$-0.168266\pi$
$912$		0			0
$913$	−4.94427	+	4.94427i	−0.163632	+	0.163632i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.81758	+	7.81758i	−0.258159	+	0.258159i
$918$		0			0
$919$		−	22.4721i		−	0.741287i	−0.928775	−	0.370644i	$-0.879137\pi$
$919$		−	22.4721i		−	0.741287i	0.928775	−	0.370644i	$-0.120863\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	53.5825	+	53.5825i	1.76369	+	1.76369i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.08191			−0.0683054			−0.0341527	−	0.999417i	$-0.510873\pi$
$929$	−2.08191			−0.0683054			−0.0341527	+	0.999417i	$0.510873\pi$
$930$		0			0
$931$	−34.4721			−1.12978
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	19.0000	+	19.0000i	0.620703	+	0.620703i	0.945711	−	0.325008i	$-0.105367\pi$
$937$	19.0000	+	19.0000i	0.620703	+	0.620703i	−0.325008	+	0.945711i	$0.605367\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	2.49458i		−	0.0813209i	−0.999173	−	0.0406604i	$-0.987054\pi$
$941$		−	2.49458i		−	0.0813209i	0.999173	−	0.0406604i	$-0.0129462\pi$
$942$		0			0
$943$	20.0000	−	20.0000i	0.651290	−	0.651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.16073	+	2.16073i	−0.0702142	+	0.0702142i	−0.741342	−	0.671128i	$-0.765810\pi$
$947$	−2.16073	+	2.16073i	−0.0702142	+	0.0702142i	0.671128	+	0.741342i	$0.265810\pi$
$948$		0			0
$949$		−	4.47214i		−	0.145172i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	34.7851	+	34.7851i	1.12680	+	1.12680i	0.990695	+	0.136104i	$0.0434581\pi$
$953$	34.7851	+	34.7851i	1.12680	+	1.12680i	0.136104	+	0.990695i	$0.456542\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	57.0786			1.84316
$960$		0			0
$961$	−28.6656			−0.924698
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−34.0689	−	34.0689i	−1.09558	−	1.09558i	−0.994921	−	0.100661i	$-0.967904\pi$
$967$	−34.0689	−	34.0689i	−1.09558	−	1.09558i	−0.100661	−	0.994921i	$-0.532096\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	28.5393i		−	0.915870i	−0.888986	−	0.457935i	$-0.848589\pi$
$971$		−	28.5393i		−	0.915870i	0.888986	−	0.457935i	$-0.151411\pi$
$972$		0			0
$973$	−28.9443	+	28.9443i	−0.927911	+	0.927911i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.0463	−	27.0463i	0.865287	−	0.865287i	−0.126659	−	0.991946i	$-0.540425\pi$
$977$	27.0463	−	27.0463i	0.865287	−	0.865287i	0.991946	+	0.126659i	$0.0404254\pi$
$978$		0			0
$979$			3.41641i			0.109189i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.7264	−	11.7264i	−0.374013	−	0.374013i	0.494923	−	0.868937i	$-0.335196\pi$
$983$	−11.7264	−	11.7264i	−0.374013	−	0.374013i	−0.868937	+	0.494923i	$0.835196\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.9845			−0.444682
$990$		0			0
$991$	22.8328			0.725308			0.362654	−	0.931924i	$-0.381871\pi$
$991$	22.8328			0.725308			0.362654	+	0.931924i	$0.381871\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−25.6525	−	25.6525i	−0.812422	−	0.812422i	0.172574	−	0.984996i	$-0.444791\pi$
$997$	−25.6525	−	25.6525i	−0.812422	−	0.812422i	−0.984996	+	0.172574i	$0.944791\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.w.j.1457.1		8
3.2	odd	2	inner	3600.2.w.j.1457.2		8
4.3	odd	2		1800.2.s.e.1457.4		8
5.2	odd	4		720.2.w.e.593.4		8
5.3	odd	4	inner	3600.2.w.j.593.1		8
5.4	even	2		720.2.w.e.17.2		8
12.11	even	2		1800.2.s.e.1457.3		8
15.2	even	4		720.2.w.e.593.2		8
15.8	even	4	inner	3600.2.w.j.593.2		8
15.14	odd	2		720.2.w.e.17.4		8
20.3	even	4		1800.2.s.e.593.4		8
20.7	even	4		360.2.s.b.233.3	yes	8
20.19	odd	2		360.2.s.b.17.1	✓	8
40.19	odd	2		2880.2.w.m.2177.3		8
40.27	even	4		2880.2.w.m.2753.1		8
40.29	even	2		2880.2.w.o.2177.4		8
40.37	odd	4		2880.2.w.o.2753.2		8
60.23	odd	4		1800.2.s.e.593.3		8
60.47	odd	4		360.2.s.b.233.1	yes	8
60.59	even	2		360.2.s.b.17.3	yes	8
120.29	odd	2		2880.2.w.o.2177.2		8
120.59	even	2		2880.2.w.m.2177.1		8
120.77	even	4		2880.2.w.o.2753.4		8
120.107	odd	4		2880.2.w.m.2753.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.s.b.17.1	✓	8	20.19	odd	2
360.2.s.b.17.3	yes	8	60.59	even	2
360.2.s.b.233.1	yes	8	60.47	odd	4
360.2.s.b.233.3	yes	8	20.7	even	4
720.2.w.e.17.2		8	5.4	even	2
720.2.w.e.17.4		8	15.14	odd	2
720.2.w.e.593.2		8	15.2	even	4
720.2.w.e.593.4		8	5.2	odd	4
1800.2.s.e.593.3		8	60.23	odd	4
1800.2.s.e.593.4		8	20.3	even	4
1800.2.s.e.1457.3		8	12.11	even	2
1800.2.s.e.1457.4		8	4.3	odd	2
2880.2.w.m.2177.1		8	120.59	even	2
2880.2.w.m.2177.3		8	40.19	odd	2
2880.2.w.m.2753.1		8	40.27	even	4
2880.2.w.m.2753.3		8	120.107	odd	4
2880.2.w.o.2177.2		8	120.29	odd	2
2880.2.w.o.2177.4		8	40.29	even	2
2880.2.w.o.2753.2		8	40.37	odd	4
2880.2.w.o.2753.4		8	120.77	even	4
3600.2.w.j.593.1		8	5.3	odd	4	inner
3600.2.w.j.593.2		8	15.8	even	4	inner
3600.2.w.j.1457.1		8	1.1	even	1	trivial
3600.2.w.j.1457.2		8	3.2	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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.23607 - 3.23607i) q^{7} +O(q^{10})\)	\(q+(-3.23607 - 3.23607i) q^{7} -4.57649i q^{11} +(4.23607 - 4.23607i) q^{13} +(1.74806 - 1.74806i) q^{17} +2.47214i q^{19} +(-2.82843 - 2.82843i) q^{23} -5.99070 q^{29} -1.52786 q^{31} +(-2.23607 - 2.23607i) q^{37} +7.07107i q^{41} +(2.47214 - 2.47214i) q^{43} +(1.74806 - 1.74806i) q^{47} +13.9443i q^{49} +1.08036 q^{59} +10.4721 q^{61} +(-1.52786 - 1.52786i) q^{67} +12.6491i q^{71} +(0.527864 - 0.527864i) q^{73} +(-14.8098 + 14.8098i) q^{77} -14.4721i q^{79} +(-1.08036 - 1.08036i) q^{83} -0.746512 q^{89} -27.4164 q^{91} +(-1.00000 - 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} + 16 q^{13} - 48 q^{31} - 16 q^{43} + 48 q^{61} - 48 q^{67} + 40 q^{73} - 112 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 16 * q^13 - 48 * q^31 - 16 * q^43 + 48 * q^61 - 48 * q^67 + 40 * q^73 - 112 * q^91 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more