LMFDB - Embedded newform 2300.2.i.f.1057.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1057,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, 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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 $ x^16 + 74x^12 + 1357x^8 + 3177*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.5
Root			$0.600551 - 0.600551i$ of defining polynomial
Character	$\chi$	$=$	2300.1057
Dual form			2300.2.i.f.1793.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.600551 - 0.600551i) q^{3} +(1.01189 - 1.01189i) q^{7} +2.27868i q^{9} +O(q^{10})$	$q+(0.600551 - 0.600551i) q^{3} +(1.01189 - 1.01189i) q^{7} +2.27868i q^{9} +2.07874i q^{11} +(-0.221233 + 0.221233i) q^{13} +(-1.58132 + 1.58132i) q^{17} +8.07874 q^{19} -1.21538i q^{21} +(-4.78256 + 0.356571i) q^{23} +(3.17012 + 3.17012i) q^{27} +3.88637i q^{29} -3.40775 q^{31} +(1.24839 + 1.24839i) q^{33} +(1.10832 - 1.10832i) q^{37} +0.265724i q^{39} -2.60769 q^{41} +(4.22002 + 4.22002i) q^{43} +(3.57308 + 3.57308i) q^{47} +4.95215i q^{49} +1.89932i q^{51} +(-3.65059 - 3.65059i) q^{53} +(4.85169 - 4.85169i) q^{57} +8.97806i q^{59} +8.43077i q^{61} +(2.30577 + 2.30577i) q^{63} +(7.20840 - 7.20840i) q^{67} +(-2.65803 + 3.08631i) q^{69} -10.0942 q^{71} +(5.96091 - 5.96091i) q^{73} +(2.10345 + 2.10345i) q^{77} +5.02082 q^{79} -3.02840 q^{81} +(8.93344 + 8.93344i) q^{83} +(2.33396 + 2.33396i) q^{87} +13.8993 q^{89} +0.447728i q^{91} +(-2.04653 + 2.04653i) q^{93} +(-1.91424 + 1.91424i) q^{97} -4.73677 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) $ 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	0.600551	−	0.600551i	0.346728	−	0.346728i	−0.512161	−	0.858889i	$-0.671155\pi$
$3$	0.600551	−	0.600551i	0.346728	−	0.346728i	0.858889	+	0.512161i	$0.171155\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.01189	−	1.01189i	0.382459	−	0.382459i	−0.489528	−	0.871987i	$-0.662831\pi$
$7$	1.01189	−	1.01189i	0.382459	−	0.382459i	0.871987	+	0.489528i	$0.162831\pi$
$8$		0			0
$9$			2.27868i			0.759559i
$10$		0			0
$11$			2.07874i			0.626763i	0.949627	+	0.313381i	$0.101462\pi$
$11$			2.07874i			0.626763i	−0.949627	+	0.313381i	$0.898538\pi$
$12$		0			0
$13$	−0.221233	+	0.221233i	−0.0613591	+	0.0613591i	−0.737120	−	0.675761i	$-0.763815\pi$
$13$	−0.221233	+	0.221233i	−0.0613591	+	0.0613591i	0.675761	+	0.737120i	$0.263815\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.58132	+	1.58132i	−0.383525	+	0.383525i	−0.872371	−	0.488845i	$-0.837418\pi$
$17$	−1.58132	+	1.58132i	−0.383525	+	0.383525i	0.488845	+	0.872371i	$0.337418\pi$
$18$		0			0
$19$	8.07874			1.85339			0.926695	−	0.375815i	$-0.122637\pi$
$19$	8.07874			1.85339			0.926695	+	0.375815i	$0.122637\pi$
$20$		0			0
$21$		−	1.21538i		−	0.265219i
$22$		0			0
$23$	−4.78256	+	0.356571i	−0.997232	+	0.0743501i
$23$	−4.78256	+	0.356571i	−0.997232	+	0.0743501i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.17012	+	3.17012i	0.610089	+	0.610089i
$28$		0			0
$29$			3.88637i			0.721681i	0.932628	+	0.360840i	$0.117510\pi$
$29$			3.88637i			0.721681i	−0.932628	+	0.360840i	$0.882490\pi$
$30$		0			0
$31$	−3.40775			−0.612050			−0.306025	−	0.952023i	$-0.598999\pi$
$31$	−3.40775			−0.612050			−0.306025	+	0.952023i	$0.598999\pi$
$32$		0			0
$33$	1.24839	+	1.24839i	0.217316	+	0.217316i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.10832	−	1.10832i	0.182207	−	0.182207i	−0.610110	−	0.792317i	$-0.708875\pi$
$37$	1.10832	−	1.10832i	0.182207	−	0.182207i	0.792317	+	0.610110i	$0.208875\pi$
$38$		0			0
$39$			0.265724i			0.0425499i
$40$		0			0
$41$	−2.60769			−0.407253			−0.203627	−	0.979049i	$-0.565273\pi$
$41$	−2.60769			−0.407253			−0.203627	+	0.979049i	$0.565273\pi$
$42$		0			0
$43$	4.22002	+	4.22002i	0.643546	+	0.643546i	0.951426	−	0.307879i	$-0.0996192\pi$
$43$	4.22002	+	4.22002i	0.643546	+	0.643546i	−0.307879	+	0.951426i	$0.599619\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.57308	+	3.57308i	0.521187	+	0.521187i	0.917930	−	0.396743i	$-0.129860\pi$
$47$	3.57308	+	3.57308i	0.521187	+	0.521187i	−0.396743	+	0.917930i	$0.629860\pi$
$48$		0			0
$49$			4.95215i			0.707450i
$50$		0			0
$51$			1.89932i			0.265958i
$52$		0			0
$53$	−3.65059	−	3.65059i	−0.501447	−	0.501447i	0.410440	−	0.911888i	$-0.365375\pi$
$53$	−3.65059	−	3.65059i	−0.501447	−	0.501447i	−0.911888	+	0.410440i	$0.865375\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.85169	−	4.85169i	0.642623	−	0.642623i
$58$		0			0
$59$			8.97806i			1.16884i	0.811450	+	0.584422i	$0.198679\pi$
$59$			8.97806i			1.16884i	−0.811450	+	0.584422i	$0.801321\pi$
$60$		0			0
$61$			8.43077i			1.07945i	0.841842	+	0.539725i	$0.181472\pi$
$61$			8.43077i			1.07945i	−0.841842	+	0.539725i	$0.818528\pi$
$62$		0			0
$63$	2.30577	+	2.30577i	0.290500	+	0.290500i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.20840	−	7.20840i	0.880647	−	0.880647i	−0.112953	−	0.993600i	$-0.536031\pi$
$67$	7.20840	−	7.20840i	0.880647	−	0.880647i	0.993600	+	0.112953i	$0.0360311\pi$
$68$		0			0
$69$	−2.65803	+	3.08631i	−0.319989	+	0.371548i
$70$		0			0
$71$	−10.0942			−1.19796			−0.598979	−	0.800764i	$-0.704427\pi$
$71$	−10.0942			−1.19796			−0.598979	+	0.800764i	$0.704427\pi$
$72$		0			0
$73$	5.96091	−	5.96091i	0.697672	−	0.697672i	−0.266236	−	0.963908i	$-0.585780\pi$
$73$	5.96091	−	5.96091i	0.697672	−	0.697672i	0.963908	+	0.266236i	$0.0857800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.10345	+	2.10345i	0.239711	+	0.239711i
$78$		0			0
$79$	5.02082			0.564887			0.282443	−	0.959284i	$-0.408855\pi$
$79$	5.02082			0.564887			0.282443	+	0.959284i	$0.408855\pi$
$80$		0			0
$81$	−3.02840			−0.336489
$82$		0			0
$83$	8.93344	+	8.93344i	0.980572	+	0.980572i	0.999815	−	0.0192427i	$-0.00612552\pi$
$83$	8.93344	+	8.93344i	0.980572	+	0.980572i	−0.0192427	+	0.999815i	$0.506126\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.33396	+	2.33396i	0.250227	+	0.250227i
$88$		0			0
$89$	13.8993			1.47333			0.736663	−	0.676260i	$-0.236401\pi$
$89$	13.8993			1.47333			0.736663	+	0.676260i	$0.236401\pi$
$90$		0			0
$91$			0.447728i			0.0469347i
$92$		0			0
$93$	−2.04653	+	2.04653i	−0.212215	+	0.212215i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.91424	+	1.91424i	−0.194362	+	0.194362i	−0.797578	−	0.603216i	$-0.793886\pi$
$97$	−1.91424	+	1.91424i	−0.194362	+	0.194362i	0.603216	+	0.797578i	$0.293886\pi$
$98$		0			0
$99$	−4.73677			−0.476063
$100$		0			0
$101$	−3.47862			−0.346135			−0.173068	−	0.984910i	$-0.555368\pi$
$101$	−3.47862			−0.346135			−0.173068	+	0.984910i	$0.555368\pi$
$102$		0			0
$103$	−10.7171	−	10.7171i	−1.05598	−	1.05598i	−0.998337	−	0.0576474i	$-0.981640\pi$
$103$	−10.7171	−	10.7171i	−1.05598	−	1.05598i	−0.0576474	−	0.998337i	$-0.518360\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.76715	−	5.76715i	0.557532	−	0.557532i	−0.371072	−	0.928604i	$-0.621010\pi$
$107$	5.76715	−	5.76715i	0.557532	−	0.557532i	0.928604	+	0.371072i	$0.121010\pi$
$108$		0			0
$109$	14.7148			1.40942			0.704712	−	0.709493i	$-0.251076\pi$
$109$	14.7148			1.40942			0.704712	+	0.709493i	$0.251076\pi$
$110$		0			0
$111$		−	1.33121i		−	0.126353i
$112$		0			0
$113$	−11.2865	−	11.2865i	−1.06174	−	1.06174i	−0.997964	−	0.0637802i	$-0.979684\pi$
$113$	−11.2865	−	11.2865i	−1.06174	−	1.06174i	−0.0637802	−	0.997964i	$-0.520316\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.504119	−	0.504119i	−0.0466058	−	0.0466058i
$118$		0			0
$119$			3.20024i			0.293365i
$120$		0			0
$121$	6.67886			0.607169
$122$		0			0
$123$	−1.56605	+	1.56605i	−0.141206	+	0.141206i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	13.8483	+	13.8483i	1.22884	+	1.22884i	0.964404	+	0.264432i	$0.0851844\pi$
$127$	13.8483	+	13.8483i	1.22884	+	1.22884i	0.264432	+	0.964404i	$0.414816\pi$
$128$		0			0
$129$	5.06867			0.446272
$130$		0			0
$131$	−2.55984			−0.223655			−0.111827	−	0.993728i	$-0.535670\pi$
$131$	−2.55984			−0.223655			−0.111827	+	0.993728i	$0.535670\pi$
$132$		0			0
$133$	8.17480	−	8.17480i	0.708845	−	0.708845i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.2548	−	12.2548i	1.04699	−	1.04699i	0.0481544	−	0.998840i	$-0.484666\pi$
$137$	12.2548	−	12.2548i	1.04699	−	1.04699i	0.998840	−	0.0481544i	$-0.0153339\pi$
$138$		0			0
$139$			6.09418i			0.516902i	0.966024	+	0.258451i	$0.0832120\pi$
$139$			6.09418i			0.516902i	−0.966024	+	0.258451i	$0.916788\pi$
$140$		0			0
$141$	4.29163			0.361420
$142$		0			0
$143$	−0.459886	−	0.459886i	−0.0384576	−	0.0384576i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.97402	+	2.97402i	0.245293	+	0.245293i
$148$		0			0
$149$	−8.25815			−0.676534			−0.338267	−	0.941050i	$-0.609841\pi$
$149$	−8.25815			−0.676534			−0.338267	+	0.941050i	$0.609841\pi$
$150$		0			0
$151$	−2.22076			−0.180723			−0.0903616	−	0.995909i	$-0.528802\pi$
$151$	−2.22076			−0.180723			−0.0903616	+	0.995909i	$0.528802\pi$
$152$		0			0
$153$	−3.60331	−	3.60331i	−0.291310	−	0.291310i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.63411	+	7.63411i	−0.609268	+	0.609268i	−0.942755	−	0.333486i	$-0.891775\pi$
$157$	−7.63411	+	7.63411i	−0.609268	+	0.609268i	0.333486	+	0.942755i	$0.391775\pi$
$158$		0			0
$159$	−4.38473			−0.347732
$160$		0			0
$161$	−4.47862	+	5.20024i	−0.352964	+	0.409836i
$162$		0			0
$163$	0.282886	−	0.282886i	0.0221573	−	0.0221573i	−0.695941	−	0.718099i	$-0.745013\pi$
$163$	0.282886	−	0.282886i	0.0221573	−	0.0221573i	0.718099	+	0.695941i	$0.245013\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.61685	+	2.61685i	0.202498	+	0.202498i	0.801069	−	0.598572i	$-0.204265\pi$
$167$	2.61685	+	2.61685i	0.202498	+	0.202498i	−0.598572	+	0.801069i	$0.704265\pi$
$168$		0			0
$169$			12.9021i			0.992470i
$170$		0			0
$171$			18.4088i			1.40776i
$172$		0			0
$173$	2.34983	−	2.34983i	0.178654	−	0.178654i	−0.612115	−	0.790769i	$-0.709681\pi$
$173$	2.34983	−	2.34983i	0.178654	−	0.178654i	0.790769	+	0.612115i	$0.209681\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.39178	+	5.39178i	0.405271	+	0.405271i
$178$		0			0
$179$		−	10.2492i		−	0.766058i	−0.923736	−	0.383029i	$-0.874881\pi$
$179$		−	10.2492i		−	0.766058i	0.923736	−	0.383029i	$-0.125119\pi$
$180$		0			0
$181$		−	8.58326i		−	0.637989i	−0.947757	−	0.318994i	$-0.896655\pi$
$181$		−	8.58326i		−	0.637989i	0.947757	−	0.318994i	$-0.103345\pi$
$182$		0			0
$183$	5.06311	+	5.06311i	0.374276	+	0.374276i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.28714	−	3.28714i	−0.240379	−	0.240379i
$188$		0			0
$189$	6.41562			0.466668
$190$		0			0
$191$			19.7356i			1.42802i	0.700135	+	0.714011i	$0.253123\pi$
$191$			19.7356i			1.42802i	−0.700135	+	0.714011i	$0.746877\pi$
$192$		0			0
$193$	14.2132	−	14.2132i	1.02309	−	1.02309i	0.0233637	−	0.999727i	$-0.492562\pi$
$193$	14.2132	−	14.2132i	1.02309	−	1.02309i	0.999727	−	0.0233637i	$-0.00743758\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.75100	+	5.75100i	0.409741	+	0.409741i	0.881648	−	0.471907i	$-0.156434\pi$
$197$	5.75100	+	5.75100i	0.409741	+	0.409741i	−0.471907	+	0.881648i	$0.656434\pi$
$198$		0			0
$199$	−9.01006			−0.638707			−0.319353	−	0.947636i	$-0.603466\pi$
$199$	−9.01006			−0.638707			−0.319353	+	0.947636i	$0.603466\pi$
$200$		0			0
$201$		−	8.65803i		−	0.610690i
$202$		0			0
$203$	3.93258	+	3.93258i	0.276013	+	0.276013i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.812509	−	10.8979i	−0.0564733	−	0.757457i
$208$		0			0
$209$			16.7936i			1.16163i
$210$		0			0
$211$	−8.02082			−0.552176			−0.276088	−	0.961132i	$-0.589038\pi$
$211$	−8.02082			−0.552176			−0.276088	+	0.961132i	$0.589038\pi$
$212$		0			0
$213$	−6.06207	+	6.06207i	−0.415366	+	0.415366i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.44827	+	3.44827i	−0.234084	+	0.234084i
$218$		0			0
$219$		−	7.15967i		−	0.483805i
$220$		0			0
$221$		−	0.699680i		−	0.0470655i
$222$		0			0
$223$	−4.73162	+	4.73162i	−0.316853	+	0.316853i	−0.847557	−	0.530704i	$-0.821927\pi$
$223$	−4.73162	+	4.73162i	−0.316853	+	0.316853i	0.530704	+	0.847557i	$0.321927\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.74516	+	3.74516i	−0.248575	+	0.248575i	−0.820386	−	0.571811i	$-0.806241\pi$
$227$	−3.74516	+	3.74516i	−0.248575	+	0.248575i	0.571811	+	0.820386i	$0.306241\pi$
$228$		0			0
$229$	−6.17262			−0.407898			−0.203949	−	0.978982i	$-0.565378\pi$
$229$	−6.17262			−0.407898			−0.203949	+	0.978982i	$0.565378\pi$
$230$		0			0
$231$	2.52646			0.166229
$232$		0			0
$233$	−3.06590	+	3.06590i	−0.200854	+	0.200854i	−0.800366	−	0.599512i	$-0.795361\pi$
$233$	−3.06590	+	3.06590i	−0.200854	+	0.200854i	0.599512	+	0.800366i	$0.295361\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.01526	−	3.01526i	0.195862	−	0.195862i
$238$		0			0
$239$		−	13.4013i		−	0.866855i	−0.901188	−	0.433428i	$-0.857304\pi$
$239$		−	13.4013i		−	0.866855i	0.901188	−	0.433428i	$-0.142696\pi$
$240$		0			0
$241$		−	17.0140i		−	1.09597i	−0.836488	−	0.547985i	$-0.815395\pi$
$241$		−	17.0140i		−	1.09597i	0.836488	−	0.547985i	$-0.184605\pi$
$242$		0			0
$243$	−11.3291	+	11.3291i	−0.726759	+	0.726759i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.78729	+	1.78729i	−0.113722	+	0.113722i
$248$		0			0
$249$	10.7300			0.679984
$250$		0			0
$251$		−	14.0309i		−	0.885622i	−0.896615	−	0.442811i	$-0.853981\pi$
$251$		−	14.0309i		−	0.885622i	0.896615	−	0.442811i	$-0.146019\pi$
$252$		0			0
$253$	−0.741216	−	9.94168i	−0.0465999	−	0.625028i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.95547	+	2.95547i	0.184357	+	0.184357i	0.793251	−	0.608894i	$-0.208387\pi$
$257$	2.95547	+	2.95547i	0.184357	+	0.184357i	−0.608894	+	0.793251i	$0.708387\pi$
$258$		0			0
$259$		−	2.24300i		−	0.139374i
$260$		0			0
$261$	−8.85578			−0.548159
$262$		0			0
$263$	7.27245	+	7.27245i	0.448438	+	0.448438i	0.894835	−	0.446397i	$-0.147293\pi$
$263$	7.27245	+	7.27245i	0.448438	+	0.448438i	−0.446397	+	0.894835i	$0.647293\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	8.34725	−	8.34725i	0.510844	−	0.510844i
$268$		0			0
$269$		−	19.7989i		−	1.20716i	−0.797301	−	0.603581i	$-0.793740\pi$
$269$		−	19.7989i		−	1.20716i	0.797301	−	0.603581i	$-0.206260\pi$
$270$		0			0
$271$	−4.93641			−0.299866			−0.149933	−	0.988696i	$-0.547906\pi$
$271$	−4.93641			−0.299866			−0.149933	+	0.988696i	$0.547906\pi$
$272$		0			0
$273$	0.268884	+	0.268884i	0.0162736	+	0.0162736i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.42688	−	1.42688i	−0.0857331	−	0.0857331i	0.662940	−	0.748673i	$-0.269309\pi$
$277$	−1.42688	−	1.42688i	−0.0857331	−	0.0857331i	−0.748673	+	0.662940i	$0.769309\pi$
$278$		0			0
$279$		−	7.76516i		−	0.464888i
$280$		0			0
$281$			16.9572i			1.01158i	0.862656	+	0.505792i	$0.168800\pi$
$281$			16.9572i			1.01158i	−0.862656	+	0.505792i	$0.831200\pi$
$282$		0			0
$283$	−7.44855	−	7.44855i	−0.442771	−	0.442771i	0.450172	−	0.892942i	$-0.351363\pi$
$283$	−7.44855	−	7.44855i	−0.442771	−	0.442771i	−0.892942	+	0.450172i	$0.851363\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.63870	+	2.63870i	−0.155758	+	0.155758i
$288$		0			0
$289$			11.9989i			0.705817i
$290$		0			0
$291$			2.29920i			0.134782i
$292$		0			0
$293$	−0.915459	−	0.915459i	−0.0534817	−	0.0534817i	0.679860	−	0.733342i	$-0.262040\pi$
$293$	−0.915459	−	0.915459i	−0.0534817	−	0.0534817i	−0.733342	+	0.679860i	$0.762040\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.58983	+	6.58983i	−0.382381	+	0.382381i
$298$		0			0
$299$	0.979176	−	1.13695i	0.0566272	−	0.0657513i
$300$		0			0
$301$	8.54039			0.492260
$302$		0			0
$303$	−2.08909	+	2.08909i	−0.120015	+	0.120015i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10.1327	−	10.1327i	−0.578302	−	0.578302i	0.356133	−	0.934435i	$-0.384095\pi$
$307$	−10.1327	−	10.1327i	−0.578302	−	0.578302i	−0.934435	+	0.356133i	$0.884095\pi$
$308$		0			0
$309$	−12.8723			−0.732279
$310$		0			0
$311$	7.25028			0.411126			0.205563	−	0.978644i	$-0.434098\pi$
$311$	7.25028			0.411126			0.205563	+	0.978644i	$0.434098\pi$
$312$		0			0
$313$	−2.30943	−	2.30943i	−0.130536	−	0.130536i	0.638820	−	0.769356i	$-0.279423\pi$
$313$	−2.30943	−	2.30943i	−0.130536	−	0.130536i	−0.769356	+	0.638820i	$0.779423\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.3036	−	14.3036i	−0.803371	−	0.803371i	0.180250	−	0.983621i	$-0.442309\pi$
$317$	−14.3036	−	14.3036i	−0.803371	−	0.803371i	−0.983621	+	0.180250i	$0.942309\pi$
$318$		0			0
$319$	−8.07874			−0.452322
$320$		0			0
$321$		−	6.92694i		−	0.386624i
$322$		0			0
$323$	−12.7750	+	12.7750i	−0.710822	+	0.710822i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	8.83701	−	8.83701i	0.488688	−	0.488688i
$328$		0			0
$329$	7.23113			0.398665
$330$		0			0
$331$	−6.55486			−0.360288			−0.180144	−	0.983640i	$-0.557656\pi$
$331$	−6.55486			−0.360288			−0.180144	+	0.983640i	$0.557656\pi$
$332$		0			0
$333$	2.52551	+	2.52551i	0.138397	+	0.138397i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11.8672	+	11.8672i	−0.646450	+	0.646450i	−0.952133	−	0.305684i	$-0.901115\pi$
$337$	−11.8672	+	11.8672i	−0.646450	+	0.646450i	0.305684	+	0.952133i	$0.401115\pi$
$338$		0			0
$339$	−13.5562			−0.736274
$340$		0			0
$341$		−	7.08382i		−	0.383610i
$342$		0			0
$343$	12.0943	+	12.0943i	0.653030	+	0.653030i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.6790	−	18.6790i	−1.00274	−	1.00274i	−0.999996	−	0.00274604i	$-0.999126\pi$
$347$	−18.6790	−	18.6790i	−1.00274	−	1.00274i	−0.00274604	−	0.999996i	$-0.500874\pi$
$348$		0			0
$349$			6.81082i			0.364575i	0.983245	+	0.182287i	$0.0583501\pi$
$349$			6.81082i			0.364575i	−0.983245	+	0.182287i	$0.941650\pi$
$350$		0			0
$351$	−1.40267			−0.0748690
$352$		0			0
$353$	19.7271	−	19.7271i	1.04997	−	1.04997i	0.0512851	−	0.998684i	$-0.483668\pi$
$353$	19.7271	−	19.7271i	1.04997	−	1.04997i	0.998684	−	0.0512851i	$-0.0163317\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.92191	+	1.92191i	0.101718	+	0.101718i
$358$		0			0
$359$	−6.61596			−0.349177			−0.174589	−	0.984641i	$-0.555860\pi$
$359$	−6.61596			−0.349177			−0.174589	+	0.984641i	$0.555860\pi$
$360$		0			0
$361$	46.2660			2.43505
$362$		0			0
$363$	4.01099	−	4.01099i	0.210523	−	0.210523i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.7631	+	11.7631i	−0.614031	+	0.614031i	−0.943994	−	0.329963i	$-0.892964\pi$
$367$	−11.7631	+	11.7631i	−0.614031	+	0.614031i	0.329963	+	0.943994i	$0.392964\pi$
$368$		0			0
$369$		−	5.94209i		−	0.309333i
$370$		0			0
$371$	−7.38800			−0.383566
$372$		0			0
$373$	4.80799	+	4.80799i	0.248948	+	0.248948i	0.820539	−	0.571591i	$-0.193673\pi$
$373$	4.80799	+	4.80799i	0.248948	+	0.248948i	−0.571591	+	0.820539i	$0.693673\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.859795	−	0.859795i	−0.0442817	−	0.0442817i
$378$		0			0
$379$	19.1039			0.981304			0.490652	−	0.871356i	$-0.336759\pi$
$379$	19.1039			0.981304			0.490652	+	0.871356i	$0.336759\pi$
$380$		0			0
$381$	16.6332			0.852145
$382$		0			0
$383$	19.9020	+	19.9020i	1.01694	+	1.01694i	0.999854	+	0.0170893i	$0.00543997\pi$
$383$	19.9020	+	19.9020i	1.01694	+	1.01694i	0.0170893	+	0.999854i	$0.494560\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−9.61605	+	9.61605i	−0.488811	+	0.488811i
$388$		0			0
$389$	2.06979			0.104942			0.0524712	−	0.998622i	$-0.483290\pi$
$389$	2.06979			0.104942			0.0524712	+	0.998622i	$0.483290\pi$
$390$		0			0
$391$	6.99888	−	8.12658i	0.353949	−	0.410979i
$392$		0			0
$393$	−1.53732	+	1.53732i	−0.0775474	+	0.0775474i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−22.3884	−	22.3884i	−1.12364	−	1.12364i	−0.991189	−	0.132452i	$-0.957715\pi$
$397$	−22.3884	−	22.3884i	−1.12364	−	1.12364i	−0.132452	−	0.991189i	$-0.542285\pi$
$398$		0			0
$399$		−	9.81877i		−	0.491553i
$400$		0			0
$401$		−	13.8583i		−	0.692049i	−0.938225	−	0.346024i	$-0.887531\pi$
$401$		−	13.8583i		−	0.692049i	0.938225	−	0.346024i	$-0.112469\pi$
$402$		0			0
$403$	0.753908	−	0.753908i	0.0375549	−	0.0375549i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.30391	+	2.30391i	0.114201	+	0.114201i
$408$		0			0
$409$		−	5.16066i		−	0.255178i	−0.991827	−	0.127589i	$-0.959276\pi$
$409$		−	5.16066i		−	0.255178i	0.991827	−	0.127589i	$-0.0407239\pi$
$410$		0			0
$411$		−	14.7192i		−	0.726045i
$412$		0			0
$413$	9.08482	+	9.08482i	0.447035	+	0.447035i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.65987	+	3.65987i	0.179224	+	0.179224i
$418$		0			0
$419$	−19.0928			−0.932743			−0.466371	−	0.884589i	$-0.654439\pi$
$419$	−19.0928			−0.932743			−0.466371	+	0.884589i	$0.654439\pi$
$420$		0			0
$421$		−	30.3409i		−	1.47872i	−0.673309	−	0.739361i	$-0.735128\pi$
$421$		−	30.3409i		−	1.47872i	0.673309	−	0.739361i	$-0.264872\pi$
$422$		0			0
$423$	−8.14188	+	8.14188i	−0.395872	+	0.395872i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.53102	+	8.53102i	0.412845	+	0.412845i
$428$		0			0
$429$	−0.552370			−0.0266687
$430$		0			0
$431$			19.8583i			0.956539i	0.878213	+	0.478270i	$0.158736\pi$
$431$			19.8583i			0.956539i	−0.878213	+	0.478270i	$0.841264\pi$
$432$		0			0
$433$	−28.0750	−	28.0750i	−1.34920	−	1.34920i	−0.886534	−	0.462664i	$-0.846894\pi$
$433$	−28.0750	−	28.0750i	−1.34920	−	1.34920i	−0.462664	−	0.886534i	$-0.653106\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−38.6370	+	2.88064i	−1.84826	+	0.137800i
$438$		0			0
$439$			2.55486i			0.121937i	0.998140	+	0.0609684i	$0.0194189\pi$
$439$			2.55486i			0.121937i	−0.998140	+	0.0609684i	$0.980581\pi$
$440$		0			0
$441$	−11.2844			−0.537350
$442$		0			0
$443$	−19.8281	+	19.8281i	−0.942062	+	0.942062i	−0.998411	−	0.0563492i	$-0.982054\pi$
$443$	−19.8281	+	19.8281i	−0.942062	+	0.942062i	0.0563492	+	0.998411i	$0.482054\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−4.95944	+	4.95944i	−0.234574	+	0.234574i
$448$		0			0
$449$		−	1.06798i		−	0.0504010i	−0.999682	−	0.0252005i	$-0.991978\pi$
$449$		−	1.06798i		−	0.0504010i	0.999682	−	0.0252005i	$-0.00802241\pi$
$450$		0			0
$451$		−	5.42070i		−	0.255251i
$452$		0			0
$453$	−1.33368	+	1.33368i	−0.0626619	+	0.0626619i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.3561	−	10.3561i	0.484436	−	0.484436i	−0.422109	−	0.906545i	$-0.638710\pi$
$457$	10.3561	−	10.3561i	0.484436	−	0.484436i	0.906545	+	0.422109i	$0.138710\pi$
$458$		0			0
$459$	−10.0259			−0.467969
$460$		0			0
$461$	23.1042			1.07607			0.538036	−	0.842922i	$-0.319166\pi$
$461$	23.1042			1.07607			0.538036	+	0.842922i	$0.319166\pi$
$462$		0			0
$463$	−4.13412	+	4.13412i	−0.192129	+	0.192129i	−0.796615	−	0.604487i	$-0.793378\pi$
$463$	−4.13412	+	4.13412i	−0.192129	+	0.192129i	0.604487	+	0.796615i	$0.293378\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.1489	+	24.1489i	−1.11748	+	1.11748i	−0.125365	+	0.992111i	$0.540010\pi$
$467$	−24.1489	+	24.1489i	−1.11748	+	1.11748i	−0.992111	+	0.125365i	$0.959990\pi$
$468$		0			0
$469$		−	14.5882i		−	0.673622i
$470$		0			0
$471$			9.16935i			0.422501i
$472$		0			0
$473$	−8.77230	+	8.77230i	−0.403351	+	0.403351i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	8.31852	−	8.31852i	0.380879	−	0.380879i
$478$		0			0
$479$	−32.0241			−1.46322			−0.731609	−	0.681724i	$-0.761230\pi$
$479$	−32.0241			−1.46322			−0.731609	+	0.681724i	$0.761230\pi$
$480$		0			0
$481$			0.490396i			0.0223601i
$482$		0			0
$483$	0.433371	+	5.81265i	0.0197190	+	0.264485i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.63354	+	3.63354i	0.164651	+	0.164651i	0.784624	−	0.619972i	$-0.212856\pi$
$487$	3.63354	+	3.63354i	0.164651	+	0.164651i	−0.619972	+	0.784624i	$0.712856\pi$
$488$		0			0
$489$		−	0.339775i		−	0.0153652i
$490$		0			0
$491$	−23.8209			−1.07502			−0.537511	−	0.843257i	$-0.680635\pi$
$491$	−23.8209			−1.07502			−0.537511	+	0.843257i	$0.680635\pi$
$492$		0			0
$493$	−6.14558	−	6.14558i	−0.276783	−	0.276783i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.2142	+	10.2142i	−0.458170	+	0.458170i
$498$		0			0
$499$		−	19.7967i		−	0.886224i	−0.896466	−	0.443112i	$-0.853874\pi$
$499$		−	19.7967i		−	0.886224i	0.896466	−	0.443112i	$-0.146126\pi$
$500$		0			0
$501$	3.14310			0.140424
$502$		0			0
$503$	−19.9325	−	19.9325i	−0.888746	−	0.888746i	0.105657	−	0.994403i	$-0.466306\pi$
$503$	−19.9325	−	19.9325i	−0.888746	−	0.888746i	−0.994403	+	0.105657i	$0.966306\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	7.74838	+	7.74838i	0.344118	+	0.344118i
$508$		0			0
$509$		−	29.4429i		−	1.30503i	−0.757774	−	0.652517i	$-0.773713\pi$
$509$		−	29.4429i		−	1.30503i	0.757774	−	0.652517i	$-0.226287\pi$
$510$		0			0
$511$		−	12.0636i		−	0.533662i
$512$		0			0
$513$	25.6105	+	25.6105i	1.13073	+	1.13073i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−7.42748	+	7.42748i	−0.326660	+	0.326660i
$518$		0			0
$519$		−	2.82238i		−	0.123889i
$520$		0			0
$521$		−	6.98812i		−	0.306155i	−0.988214	−	0.153078i	$-0.951082\pi$
$521$		−	6.98812i		−	0.306155i	0.988214	−	0.153078i	$-0.0489184\pi$
$522$		0			0
$523$	−7.25025	−	7.25025i	−0.317031	−	0.317031i	0.530595	−	0.847626i	$-0.321969\pi$
$523$	−7.25025	−	7.25025i	−0.317031	−	0.317031i	−0.847626	+	0.530595i	$0.821969\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.38873	−	5.38873i	0.234737	−	0.234737i
$528$		0			0
$529$	22.7457	−	3.41064i	0.988944	−	0.148289i
$530$		0			0
$531$	−20.4581			−0.887806
$532$		0			0
$533$	0.576909	−	0.576909i	0.0249887	−	0.0249887i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.15515	−	6.15515i	−0.265614	−	0.265614i
$538$		0			0
$539$	−10.2942			−0.443403
$540$		0			0
$541$	44.5839			1.91681			0.958407	−	0.285406i	$-0.0921285\pi$
$541$	44.5839			1.91681			0.958407	+	0.285406i	$0.0921285\pi$
$542$		0			0
$543$	−5.15469	−	5.15469i	−0.221209	−	0.221209i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−26.8419	−	26.8419i	−1.14768	−	1.14768i	−0.987009	−	0.160666i	$-0.948636\pi$
$547$	−26.8419	−	26.8419i	−1.14768	−	1.14768i	−0.160666	−	0.987009i	$-0.551364\pi$
$548$		0			0
$549$	−19.2110			−0.819905
$550$		0			0
$551$			31.3970i			1.33756i
$552$		0			0
$553$	5.08053	−	5.08053i	0.216046	−	0.216046i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.5627	+	24.5627i	−1.04075	+	1.04075i	−0.0416200	+	0.999134i	$0.513252\pi$
$557$	−24.5627	+	24.5627i	−1.04075	+	1.04075i	−0.999134	+	0.0416200i	$0.986748\pi$
$558$		0			0
$559$	−1.86722			−0.0789749
$560$		0			0
$561$	−3.94819			−0.166693
$562$		0			0
$563$	20.4110	+	20.4110i	0.860221	+	0.860221i	0.991364	−	0.131142i	$-0.0418645\pi$
$563$	20.4110	+	20.4110i	0.860221	+	0.860221i	−0.131142	+	0.991364i	$0.541864\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−3.06441	+	3.06441i	−0.128693	+	0.128693i
$568$		0			0
$569$	−23.7317			−0.994884			−0.497442	−	0.867497i	$-0.665727\pi$
$569$	−23.7317			−0.994884			−0.497442	+	0.867497i	$0.665727\pi$
$570$		0			0
$571$			4.93631i			0.206578i	0.994651	+	0.103289i	$0.0329367\pi$
$571$			4.93631i			0.206578i	−0.994651	+	0.103289i	$0.967063\pi$
$572$		0			0
$573$	11.8523	+	11.8523i	0.495136	+	0.495136i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12.3564	−	12.3564i	−0.514403	−	0.514403i	0.401469	−	0.915872i	$-0.368500\pi$
$577$	−12.3564	−	12.3564i	−0.514403	−	0.514403i	−0.915872	+	0.401469i	$0.868500\pi$
$578$		0			0
$579$		−	17.0715i		−	0.709469i
$580$		0			0
$581$	18.0793			0.750057
$582$		0			0
$583$	7.58862	−	7.58862i	0.314288	−	0.314288i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.0744	+	23.0744i	0.952383	+	0.952383i	0.998917	−	0.0465334i	$-0.0148174\pi$
$587$	23.0744	+	23.0744i	0.952383	+	0.952383i	−0.0465334	+	0.998917i	$0.514817\pi$
$588$		0			0
$589$	−27.5303			−1.13437
$590$		0			0
$591$	6.90753			0.284138
$592$		0			0
$593$	−14.4965	+	14.4965i	−0.595299	+	0.595299i	−0.939058	−	0.343759i	$-0.888300\pi$
$593$	−14.4965	+	14.4965i	−0.595299	+	0.595299i	0.343759	+	0.939058i	$0.388300\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.41100	+	5.41100i	−0.221458	+	0.221458i
$598$		0			0
$599$		−	5.88348i		−	0.240393i	−0.992750	−	0.120196i	$-0.961648\pi$
$599$		−	5.88348i		−	0.240393i	0.992750	−	0.120196i	$-0.0383524\pi$
$600$		0			0
$601$	24.0420			0.980695			0.490347	−	0.871527i	$-0.336870\pi$
$601$	24.0420			0.980695			0.490347	+	0.871527i	$0.336870\pi$
$602$		0			0
$603$	16.4256	+	16.4256i	0.668903	+	0.668903i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25.8128	−	25.8128i	−1.04771	−	1.04771i	−0.998803	−	0.0489077i	$-0.984426\pi$
$607$	−25.8128	−	25.8128i	−1.04771	−	1.04771i	−0.0489077	−	0.998803i	$-0.515574\pi$
$608$		0			0
$609$	4.72343			0.191403
$610$		0			0
$611$	−1.58097			−0.0639591
$612$		0			0
$613$	−16.3616	−	16.3616i	−0.660838	−	0.660838i	0.294740	−	0.955578i	$-0.404767\pi$
$613$	−16.3616	−	16.3616i	−0.660838	−	0.660838i	−0.955578	+	0.294740i	$0.904767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.8486	−	27.8486i	1.12114	−	1.12114i	0.129573	−	0.991570i	$-0.458639\pi$
$617$	27.8486	−	27.8486i	1.12114	−	1.12114i	0.991570	−	0.129573i	$-0.0413605\pi$
$618$		0			0
$619$	24.6450			0.990568			0.495284	−	0.868731i	$-0.335064\pi$
$619$	24.6450			0.990568			0.495284	+	0.868731i	$0.335064\pi$
$620$		0			0
$621$	−16.2916	−	14.0309i	−0.653761	−	0.563040i
$622$		0			0
$623$	14.0646	−	14.0646i	0.563486	−	0.563486i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	10.0854	+	10.0854i	0.402772	+	0.402772i
$628$		0			0
$629$			3.50522i			0.139762i
$630$		0			0
$631$			7.20644i			0.286884i	0.989659	+	0.143442i	$0.0458170\pi$
$631$			7.20644i			0.286884i	−0.989659	+	0.143442i	$0.954183\pi$
$632$		0			0
$633$	−4.81692	+	4.81692i	−0.191455	+	0.191455i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.09558	−	1.09558i	−0.0434085	−	0.0434085i
$638$		0			0
$639$		−	23.0014i		−	0.909920i
$640$		0			0
$641$			7.08382i			0.279794i	0.990166	+	0.139897i	$0.0446771\pi$
$641$			7.08382i			0.279794i	−0.990166	+	0.139897i	$0.955323\pi$
$642$		0			0
$643$	1.48488	+	1.48488i	0.0585581	+	0.0585581i	0.735779	−	0.677221i	$-0.236816\pi$
$643$	1.48488	+	1.48488i	0.0585581	+	0.0585581i	−0.677221	+	0.735779i	$0.736816\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.4116	−	16.4116i	−0.645207	−	0.645207i	0.306623	−	0.951831i	$-0.400801\pi$
$647$	−16.4116	−	16.4116i	−0.645207	−	0.645207i	−0.951831	+	0.306623i	$0.900801\pi$
$648$		0			0
$649$	−18.6630			−0.732587
$650$		0			0
$651$			4.14173i			0.162327i
$652$		0			0
$653$	13.8279	−	13.8279i	0.541126	−	0.541126i	−0.382733	−	0.923859i	$-0.625017\pi$
$653$	13.8279	−	13.8279i	0.541126	−	0.541126i	0.923859	+	0.382733i	$0.125017\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	13.5830	+	13.5830i	0.529923	+	0.529923i
$658$		0			0
$659$	9.23733			0.359835			0.179918	−	0.983682i	$-0.442417\pi$
$659$	9.23733			0.359835			0.179918	+	0.983682i	$0.442417\pi$
$660$		0			0
$661$		−	27.8712i		−	1.08406i	−0.840358	−	0.542032i	$-0.817655\pi$
$661$		−	27.8712i		−	1.08406i	0.840358	−	0.542032i	$-0.182345\pi$
$662$		0			0
$663$	−0.420193	−	0.420193i	−0.0163190	−	0.0163190i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.38577	−	18.5868i	−0.0536570	−	0.719683i
$668$		0			0
$669$			5.68316i			0.219724i
$670$		0			0
$671$	−17.5253			−0.676558
$672$		0			0
$673$	7.71983	−	7.71983i	0.297578	−	0.297578i	−0.542487	−	0.840064i	$-0.682517\pi$
$673$	7.71983	−	7.71983i	0.297578	−	0.297578i	0.840064	+	0.542487i	$0.182517\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.1034	−	24.1034i	0.926370	−	0.926370i	−0.0710989	−	0.997469i	$-0.522651\pi$
$677$	24.1034	−	24.1034i	0.926370	−	0.926370i	0.997469	+	0.0710989i	$0.0226506\pi$
$678$		0			0
$679$			3.87401i			0.148671i
$680$		0			0
$681$			4.49832i			0.172376i
$682$		0			0
$683$	−15.0781	+	15.0781i	−0.576948	+	0.576948i	−0.934061	−	0.357113i	$-0.883761\pi$
$683$	−15.0781	+	15.0781i	−0.576948	+	0.576948i	0.357113	+	0.934061i	$0.383761\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−3.70697	+	3.70697i	−0.141430	+	0.141430i
$688$		0			0
$689$	1.61527			0.0615367
$690$		0			0
$691$	30.1783			1.14804			0.574018	−	0.818843i	$-0.305384\pi$
$691$	30.1783			1.14804			0.574018	+	0.818843i	$0.305384\pi$
$692$		0			0
$693$	−4.79309	+	4.79309i	−0.182075	+	0.182075i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	4.12358	−	4.12358i	0.156192	−	0.156192i
$698$		0			0
$699$			3.68247i			0.139284i
$700$		0			0
$701$		−	38.8307i		−	1.46661i	−0.679898	−	0.733307i	$-0.737976\pi$
$701$		−	38.8307i		−	1.46661i	0.679898	−	0.733307i	$-0.262024\pi$
$702$		0			0
$703$	8.95385	−	8.95385i	0.337701	−	0.337701i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.51998	+	3.51998i	−0.132383	+	0.132383i
$708$		0			0
$709$	−8.02650			−0.301442			−0.150721	−	0.988576i	$-0.548159\pi$
$709$	−8.02650			−0.301442			−0.150721	+	0.988576i	$0.548159\pi$
$710$		0			0
$711$			11.4408i			0.429065i
$712$		0			0
$713$	16.2978	−	1.21510i	0.610356	−	0.0455060i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.04814	−	8.04814i	−0.300563	−	0.300563i
$718$		0			0
$719$		−	42.9550i		−	1.60195i	−0.598697	−	0.800976i	$-0.704315\pi$
$719$		−	42.9550i		−	1.60195i	0.598697	−	0.800976i	$-0.295685\pi$
$720$		0			0
$721$	−21.6890			−0.807741
$722$		0			0
$723$	−10.2178	−	10.2178i	−0.380004	−	0.380004i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.76508	−	1.76508i	0.0654633	−	0.0654633i	−0.673617	−	0.739080i	$-0.735260\pi$
$727$	1.76508	−	1.76508i	0.0654633	−	0.0654633i	0.739080	+	0.673617i	$0.235260\pi$
$728$		0			0
$729$			4.52216i			0.167487i
$730$		0			0
$731$	−13.3464			−0.493633
$732$		0			0
$733$	−27.5721	−	27.5721i	−1.01840	−	1.01840i	−0.999828	−	0.0185695i	$-0.994089\pi$
$733$	−27.5721	−	27.5721i	−1.01840	−	1.01840i	−0.0185695	−	0.999828i	$-0.505911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.9844	+	14.9844i	0.551956	+	0.551956i
$738$		0			0
$739$			13.3222i			0.490066i	0.969515	+	0.245033i	$0.0787988\pi$
$739$			13.3222i			0.490066i	−0.969515	+	0.245033i	$0.921201\pi$
$740$		0			0
$741$			2.14671i			0.0788615i
$742$		0			0
$743$	−5.26788	−	5.26788i	−0.193260	−	0.193260i	0.603843	−	0.797103i	$-0.293635\pi$
$743$	−5.26788	−	5.26788i	−0.193260	−	0.193260i	−0.797103	+	0.603843i	$0.793635\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−20.3564	+	20.3564i	−0.744802	+	0.744802i
$748$		0			0
$749$		−	11.6715i		−	0.426466i
$750$		0			0
$751$		−	29.2293i		−	1.06659i	−0.845928	−	0.533296i	$-0.820953\pi$
$751$		−	29.2293i		−	1.06659i	0.845928	−	0.533296i	$-0.179047\pi$
$752$		0			0
$753$	−8.42627	−	8.42627i	−0.307070	−	0.307070i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34.6648	+	34.6648i	−1.25991	+	1.25991i	−0.308781	+	0.951133i	$0.599921\pi$
$757$	−34.6648	+	34.6648i	−1.25991	+	1.25991i	−0.951133	+	0.308781i	$0.900079\pi$
$758$		0			0
$759$	−6.41562	−	5.52535i	−0.232872	−	0.200557i
$760$		0			0
$761$	−45.6748			−1.65571			−0.827856	−	0.560941i	$-0.810440\pi$
$761$	−45.6748			−1.65571			−0.827856	+	0.560941i	$0.810440\pi$
$762$		0			0
$763$	14.8898	−	14.8898i	0.539047	−	0.539047i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.98625	−	1.98625i	−0.0717192	−	0.0717192i
$768$		0			0
$769$	7.42467			0.267740			0.133870	−	0.990999i	$-0.457259\pi$
$769$	7.42467			0.267740			0.133870	+	0.990999i	$0.457259\pi$
$770$		0			0
$771$	3.54982			0.127844
$772$		0			0
$773$	−16.6112	−	16.6112i	−0.597463	−	0.597463i	0.342174	−	0.939637i	$-0.388837\pi$
$773$	−16.6112	−	16.6112i	−0.597463	−	0.597463i	−0.939637	+	0.342174i	$0.888837\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.34704	−	1.34704i	−0.0483248	−	0.0483248i
$778$		0			0
$779$	−21.0669			−0.754799
$780$		0			0
$781$		−	20.9831i		−	0.750836i
$782$		0			0
$783$	−12.3202	+	12.3202i	−0.440289	+	0.440289i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	20.3805	−	20.3805i	0.726486	−	0.726486i	−0.243432	−	0.969918i	$-0.578273\pi$
$787$	20.3805	−	20.3805i	0.726486	−	0.726486i	0.969918	+	0.243432i	$0.0782734\pi$
$788$		0			0
$789$	8.73496			0.310973
$790$		0			0
$791$	−22.8414			−0.812147
$792$		0			0
$793$	−1.86517	−	1.86517i	−0.0662340	−	0.0662340i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.1910	−	15.1910i	0.538093	−	0.538093i	−0.384876	−	0.922968i	$-0.625756\pi$
$797$	15.1910	−	15.1910i	0.538093	−	0.538093i	0.922968	+	0.384876i	$0.125756\pi$
$798$		0			0
$799$	−11.3003			−0.399777
$800$		0			0
$801$			31.6721i			1.11908i
$802$		0			0
$803$	12.3912	+	12.3912i	0.437275	+	0.437275i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−11.8903	−	11.8903i	−0.418558	−	0.418558i
$808$		0			0
$809$			39.5141i			1.38924i	0.719377	+	0.694620i	$0.244428\pi$
$809$			39.5141i			1.38924i	−0.719377	+	0.694620i	$0.755572\pi$
$810$		0			0
$811$	39.0075			1.36974			0.684870	−	0.728665i	$-0.259859\pi$
$811$	39.0075			1.36974			0.684870	+	0.728665i	$0.259859\pi$
$812$		0			0
$813$	−2.96457	+	2.96457i	−0.103972	+	0.103972i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	34.0924	+	34.0924i	1.19274	+	1.19274i
$818$		0			0
$819$	−1.02023			−0.0356496
$820$		0			0
$821$	32.9360			1.14947			0.574737	−	0.818338i	$-0.305104\pi$
$821$	32.9360			1.14947			0.574737	+	0.818338i	$0.305104\pi$
$822$		0			0
$823$	32.2794	−	32.2794i	1.12519	−	1.12519i	0.134242	−	0.990949i	$-0.457140\pi$
$823$	32.2794	−	32.2794i	1.12519	−	1.12519i	0.990949	−	0.134242i	$-0.0428598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.56569	−	1.56569i	0.0544443	−	0.0544443i	−0.679360	−	0.733805i	$-0.737743\pi$
$827$	1.56569	−	1.56569i	0.0544443	−	0.0544443i	0.733805	+	0.679360i	$0.237743\pi$
$828$		0			0
$829$		−	2.31606i		−	0.0804402i	−0.999191	−	0.0402201i	$-0.987194\pi$
$829$		−	2.31606i		−	0.0804402i	0.999191	−	0.0402201i	$-0.0128059\pi$
$830$		0			0
$831$	−1.71383			−0.0594522
$832$		0			0
$833$	−7.83092	−	7.83092i	−0.271325	−	0.271325i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−10.8030	−	10.8030i	−0.373405	−	0.373405i
$838$		0			0
$839$	26.8964			0.928567			0.464283	−	0.885687i	$-0.346312\pi$
$839$	26.8964			0.928567			0.464283	+	0.885687i	$0.346312\pi$
$840$		0			0
$841$	13.8961			0.479177
$842$		0			0
$843$	10.1837	+	10.1837i	0.350745	+	0.350745i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.75827	−	6.75827i	0.232217	−	0.232217i
$848$		0			0
$849$	−8.94647			−0.307042
$850$		0			0
$851$	−4.90542	+	5.69581i	−0.168156	+	0.195250i
$852$		0			0
$853$	24.8693	−	24.8693i	0.851509	−	0.851509i	−0.138810	−	0.990319i	$-0.544328\pi$
$853$	24.8693	−	24.8693i	0.851509	−	0.851509i	0.990319	+	0.138810i	$0.0443278\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.01214	+	9.01214i	0.307849	+	0.307849i	0.844075	−	0.536226i	$-0.180150\pi$
$857$	9.01214	+	9.01214i	0.307849	+	0.307849i	−0.536226	+	0.844075i	$0.680150\pi$
$858$		0			0
$859$			53.9780i			1.84171i	0.389910	+	0.920853i	$0.372506\pi$
$859$			53.9780i			1.84171i	−0.389910	+	0.920853i	$0.627494\pi$
$860$		0			0
$861$			3.16935i			0.108011i
$862$		0			0
$863$	−31.0496	+	31.0496i	−1.05694	+	1.05694i	−0.0586626	+	0.998278i	$0.518684\pi$
$863$	−31.0496	+	31.0496i	−1.05694	+	1.05694i	−0.998278	+	0.0586626i	$0.981316\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	7.20594	+	7.20594i	0.244727	+	0.244727i
$868$		0			0
$869$			10.4370i			0.354050i
$870$		0			0
$871$			3.18948i			0.108071i
$872$		0			0
$873$	−4.36194	−	4.36194i	−0.147629	−	0.147629i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.92956	+	7.92956i	0.267762	+	0.267762i	0.828198	−	0.560436i	$-0.189366\pi$
$877$	7.92956	+	7.92956i	0.267762	+	0.267762i	−0.560436	+	0.828198i	$0.689366\pi$
$878$		0			0
$879$	−1.09956			−0.0370872
$880$		0			0
$881$			8.94209i			0.301267i	0.988590	+	0.150633i	$0.0481313\pi$
$881$			8.94209i			0.301267i	−0.988590	+	0.150633i	$0.951869\pi$
$882$		0			0
$883$	−3.16706	+	3.16706i	−0.106580	+	0.106580i	−0.758386	−	0.651806i	$-0.774012\pi$
$883$	−3.16706	+	3.16706i	−0.106580	+	0.106580i	0.651806	+	0.758386i	$0.274012\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.2948	+	20.2948i	0.681431	+	0.681431i	0.960323	−	0.278891i	$-0.0899669\pi$
$887$	20.2948	+	20.2948i	0.681431	+	0.681431i	−0.278891	+	0.960323i	$0.589967\pi$
$888$		0			0
$889$	28.0259			0.939959
$890$		0			0
$891$		−	6.29524i		−	0.210898i
$892$		0			0
$893$	28.8659	+	28.8659i	0.965962	+	0.965962i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.0947494	−	1.27084i	−0.00316359	−	0.0424321i
$898$		0			0
$899$		−	13.2438i		−	0.441705i
$900$		0			0
$901$	11.5455			0.384636
$902$		0			0
$903$	5.12894	−	5.12894i	0.170681	−	0.170681i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.1815	+	22.1815i	−0.736523	+	0.736523i	−0.971903	−	0.235380i	$-0.924367\pi$
$907$	−22.1815	+	22.1815i	−0.736523	+	0.736523i	0.235380	+	0.971903i	$0.424367\pi$
$908$		0			0
$909$		−	7.92664i		−	0.262910i
$910$		0			0
$911$			15.9190i			0.527421i	0.964602	+	0.263710i	$0.0849463\pi$
$911$			15.9190i			0.527421i	−0.964602	+	0.263710i	$0.915054\pi$
$912$		0			0
$913$	−18.5703	+	18.5703i	−0.614586	+	0.614586i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.59028	+	2.59028i	−0.0855387	+	0.0855387i
$918$		0			0
$919$	14.9788			0.494103			0.247052	−	0.969002i	$-0.420538\pi$
$919$	14.9788			0.494103			0.247052	+	0.969002i	$0.420538\pi$
$920$		0			0
$921$	−12.1704			−0.401028
$922$		0			0
$923$	2.23317	−	2.23317i	0.0735057	−	0.0735057i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	24.4207	−	24.4207i	0.802082	−	0.802082i
$928$		0			0
$929$			17.0629i			0.559814i	0.960027	+	0.279907i	$0.0903036\pi$
$929$			17.0629i			0.559814i	−0.960027	+	0.279907i	$0.909696\pi$
$930$		0			0
$931$			40.0071i			1.31118i
$932$		0			0
$933$	4.35416	−	4.35416i	0.142549	−	0.142549i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	8.93924	−	8.93924i	0.292032	−	0.292032i	−0.545850	−	0.837883i	$-0.683793\pi$
$937$	8.93924	−	8.93924i	0.292032	−	0.292032i	0.837883	+	0.545850i	$0.183793\pi$
$938$		0			0
$939$	−2.77386			−0.0905214
$940$		0			0
$941$			12.2294i			0.398667i	0.979932	+	0.199334i	$0.0638778\pi$
$941$			12.2294i			0.398667i	−0.979932	+	0.199334i	$0.936122\pi$
$942$		0			0
$943$	12.4714	−	0.929827i	0.406126	−	0.0302793i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.3973	+	15.3973i	0.500344	+	0.500344i	0.911545	−	0.411201i	$-0.134890\pi$
$947$	15.3973	+	15.3973i	0.500344	+	0.500344i	−0.411201	+	0.911545i	$0.634890\pi$
$948$		0			0
$949$			2.63751i			0.0856171i
$950$		0			0
$951$	−17.1801			−0.557103
$952$		0			0
$953$	9.84524	+	9.84524i	0.318919	+	0.318919i	0.848352	−	0.529433i	$-0.177595\pi$
$953$	9.84524	+	9.84524i	0.318919	+	0.318919i	−0.529433	+	0.848352i	$0.677595\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−4.85169	+	4.85169i	−0.156833	+	0.156833i
$958$		0			0
$959$		−	24.8010i		−	0.800865i
$960$		0			0
$961$	−19.3872			−0.625394
$962$		0			0
$963$	13.1415	+	13.1415i	0.423478	+	0.423478i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	41.9208	+	41.9208i	1.34808	+	1.34808i	0.887743	+	0.460339i	$0.152272\pi$
$967$	41.9208	+	41.9208i	1.34808	+	1.34808i	0.460339	+	0.887743i	$0.347728\pi$
$968$		0			0
$969$			15.3441i			0.492924i
$970$		0			0
$971$			29.7953i			0.956176i	0.878312	+	0.478088i	$0.158670\pi$
$971$			29.7953i			0.956176i	−0.878312	+	0.478088i	$0.841330\pi$
$972$		0			0
$973$	6.16665	+	6.16665i	0.197694	+	0.197694i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.8187	−	19.8187i	0.634057	−	0.634057i	−0.315026	−	0.949083i	$-0.602013\pi$
$977$	19.8187	−	19.8187i	0.634057	−	0.634057i	0.949083	+	0.315026i	$0.102013\pi$
$978$		0			0
$979$			28.8930i			0.923425i
$980$		0			0
$981$			33.5303i			1.07054i
$982$		0			0
$983$	−5.47392	−	5.47392i	−0.174591	−	0.174591i	0.614402	−	0.788993i	$-0.289397\pi$
$983$	−5.47392	−	5.47392i	−0.174591	−	0.174591i	−0.788993	+	0.614402i	$0.789397\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	4.34266	−	4.34266i	0.138228	−	0.138228i
$988$		0			0
$989$	−21.6872	−	18.6777i	−0.689613	−	0.593918i
$990$		0			0
$991$	−54.3876			−1.72768			−0.863839	−	0.503768i	$-0.831947\pi$
$991$	−54.3876			−1.72768			−0.863839	+	0.503768i	$0.831947\pi$
$992$		0			0
$993$	−3.93653	+	3.93653i	−0.124922	+	0.124922i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−29.2716	−	29.2716i	−0.927040	−	0.927040i	0.0704733	−	0.997514i	$-0.477549\pi$
$997$	−29.2716	−	29.2716i	−0.927040	−	0.927040i	−0.997514	+	0.0704733i	$0.977549\pi$
$998$		0			0
$999$	7.02702			0.222325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.i.f.1057.5	yes	16
5.2	odd	4		2300.2.i.e.1793.4	yes	16
5.3	odd	4		2300.2.i.e.1793.5	yes	16
5.4	even	2	inner	2300.2.i.f.1057.4	yes	16
23.22	odd	2		2300.2.i.e.1057.5	yes	16
115.22	even	4	inner	2300.2.i.f.1793.4	yes	16
115.68	even	4	inner	2300.2.i.f.1793.5	yes	16
115.114	odd	2		2300.2.i.e.1057.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.i.e.1057.4	✓	16	115.114	odd	2
2300.2.i.e.1057.5	yes	16	23.22	odd	2
2300.2.i.e.1793.4	yes	16	5.2	odd	4
2300.2.i.e.1793.5	yes	16	5.3	odd	4
2300.2.i.f.1057.4	yes	16	5.4	even	2	inner
2300.2.i.f.1057.5	yes	16	1.1	even	1	trivial
2300.2.i.f.1793.4	yes	16	115.22	even	4	inner
2300.2.i.f.1793.5	yes	16	115.68	even	4	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.600551 - 0.600551i) q^{3} +(1.01189 - 1.01189i) q^{7} +2.27868i q^{9} +O(q^{10})\)	\(q+(0.600551 - 0.600551i) q^{3} +(1.01189 - 1.01189i) q^{7} +2.27868i q^{9} +2.07874i q^{11} +(-0.221233 + 0.221233i) q^{13} +(-1.58132 + 1.58132i) q^{17} +8.07874 q^{19} -1.21538i q^{21} +(-4.78256 + 0.356571i) q^{23} +(3.17012 + 3.17012i) q^{27} +3.88637i q^{29} -3.40775 q^{31} +(1.24839 + 1.24839i) q^{33} +(1.10832 - 1.10832i) q^{37} +0.265724i q^{39} -2.60769 q^{41} +(4.22002 + 4.22002i) q^{43} +(3.57308 + 3.57308i) q^{47} +4.95215i q^{49} +1.89932i q^{51} +(-3.65059 - 3.65059i) q^{53} +(4.85169 - 4.85169i) q^{57} +8.97806i q^{59} +8.43077i q^{61} +(2.30577 + 2.30577i) q^{63} +(7.20840 - 7.20840i) q^{67} +(-2.65803 + 3.08631i) q^{69} -10.0942 q^{71} +(5.96091 - 5.96091i) q^{73} +(2.10345 + 2.10345i) q^{77} +5.02082 q^{79} -3.02840 q^{81} +(8.93344 + 8.93344i) q^{83} +(2.33396 + 2.33396i) q^{87} +13.8993 q^{89} +0.447728i q^{91} +(-2.04653 + 2.04653i) q^{93} +(-1.91424 + 1.91424i) q^{97} -4.73677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) \) 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Introduction

L-functions

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