LMFDB - Embedded newform 4600.2.e.u.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 $ x^10 + 28x^8 + 260x^6 + 897x^4 + 1056x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-3.36002i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.u.4049.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36002i q^{3} +1.90754i q^{7} -8.28974 q^{9} +O(q^{10})$	$q-3.36002i q^{3} +1.90754i q^{7} -8.28974 q^{9} -5.48021 q^{11} -1.04937i q^{13} +6.74222i q^{17} +1.55049 q^{19} +6.40939 q^{21} -1.00000i q^{23} +17.7736i q^{27} +3.38219 q^{29} +10.9327 q^{31} +18.4136i q^{33} -5.26201i q^{37} -3.52589 q^{39} +6.09801 q^{41} -0.403830i q^{47} +3.36128 q^{49} +22.6540 q^{51} +5.88332i q^{53} -5.20968i q^{57} -9.60111 q^{59} -7.09927 q^{61} -15.8130i q^{63} -13.7971i q^{67} -3.36002 q^{69} +0.478950 q^{71} +2.40383i q^{73} -10.4537i q^{77} +4.24037 q^{79} +34.8505 q^{81} -11.2620i q^{83} -11.3642i q^{87} -4.90495 q^{89} +2.00171 q^{91} -36.7340i q^{93} +12.3433i q^{97} +45.4295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 26 q^{9}+O(q^{10}) $ 10 * q - 26 * q^9	$ 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) $ 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.36002i	− 1.93991i	−0.243287	−	0.969954i	$-0.578226\pi$
$3$	− 3.36002i	− 1.93991i	0.243287	−	0.969954i	$-0.421774\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.90754i	0.720984i	0.932762	+	0.360492i	$0.117391\pi$
$7$	1.90754i	0.720984i	−0.932762	+	0.360492i	$0.882609\pi$
$8$	0	0
$9$	−8.28974	−2.76325
$10$	0	0
$11$	−5.48021	−1.65234	−0.826172	−	0.563418i	$-0.809486\pi$
$11$	−5.48021	−1.65234	−0.826172	+	0.563418i	$0.809486\pi$
$12$	0	0
$13$	− 1.04937i	− 0.291041i	−0.989355	−	0.145521i	$-0.953514\pi$
$13$	− 1.04937i	− 0.291041i	0.989355	−	0.145521i	$-0.0464858\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.74222i	1.63523i	0.575767	+	0.817614i	$0.304703\pi$
$17$	6.74222i	1.63523i	−0.575767	+	0.817614i	$0.695297\pi$
$18$	0	0
$19$	1.55049	0.355707	0.177853	−	0.984057i	$-0.443085\pi$
$19$	1.55049	0.355707	0.177853	+	0.984057i	$0.443085\pi$
$20$	0	0
$21$	6.40939	1.39864
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	17.7736i	3.42054i
$28$	0	0
$29$	3.38219	0.628058	0.314029	−	0.949413i	$-0.398321\pi$
$29$	3.38219	0.628058	0.314029	+	0.949413i	$0.398321\pi$
$30$	0	0
$31$	10.9327	1.96357	0.981784	−	0.190000i	$-0.0608489\pi$
$31$	10.9327	1.96357	0.981784	+	0.190000i	$0.0608489\pi$
$32$	0	0
$33$	18.4136i	3.20540i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.26201i	− 0.865069i	−0.901617	−	0.432534i	$-0.857619\pi$
$37$	− 5.26201i	− 0.865069i	0.901617	−	0.432534i	$-0.142381\pi$
$38$	0	0
$39$	−3.52589	−0.564594
$40$	0	0
$41$	6.09801	0.952350	0.476175	−	0.879351i	$-0.342023\pi$
$41$	6.09801	0.952350	0.476175	+	0.879351i	$0.342023\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.403830i	− 0.0589046i	−0.999566	−	0.0294523i	$-0.990624\pi$
$47$	− 0.403830i	− 0.0589046i	0.999566	−	0.0294523i	$-0.00937631\pi$
$48$	0	0
$49$	3.36128	0.480183
$50$	0	0
$51$	22.6540	3.17219
$52$	0	0
$53$	5.88332i	0.808136i	0.914729	+	0.404068i	$0.132404\pi$
$53$	5.88332i	0.808136i	−0.914729	+	0.404068i	$0.867596\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.20968i	− 0.690039i
$58$	0	0
$59$	−9.60111	−1.24996	−0.624979	−	0.780641i	$-0.714893\pi$
$59$	−9.60111	−1.24996	−0.624979	+	0.780641i	$0.714893\pi$
$60$	0	0
$61$	−7.09927	−0.908968	−0.454484	−	0.890755i	$-0.650176\pi$
$61$	−7.09927	−0.908968	−0.454484	+	0.890755i	$0.650176\pi$
$62$	0	0
$63$	− 15.8130i	− 1.99226i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.7971i	− 1.68559i	−0.538236	−	0.842794i	$-0.680909\pi$
$67$	− 13.7971i	− 1.68559i	0.538236	−	0.842794i	$-0.319091\pi$
$68$	0	0
$69$	−3.36002	−0.404499
$70$	0	0
$71$	0.478950	0.0568409	0.0284205	−	0.999596i	$-0.490952\pi$
$71$	0.478950	0.0568409	0.0284205	+	0.999596i	$0.490952\pi$
$72$	0	0
$73$	2.40383i	0.281347i	0.990056	+	0.140673i	$0.0449268\pi$
$73$	2.40383i	0.281347i	−0.990056	+	0.140673i	$0.955073\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.4537i	− 1.19131i
$78$	0	0
$79$	4.24037	0.477079	0.238540	−	0.971133i	$-0.423331\pi$
$79$	4.24037	0.477079	0.238540	+	0.971133i	$0.423331\pi$
$80$	0	0
$81$	34.8505	3.87228
$82$	0	0
$83$	− 11.2620i	− 1.23617i	−0.786113	−	0.618083i	$-0.787910\pi$
$83$	− 11.2620i	− 1.23617i	0.786113	−	0.618083i	$-0.212090\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 11.3642i	− 1.21837i
$88$	0	0
$89$	−4.90495	−0.519924	−0.259962	−	0.965619i	$-0.583710\pi$
$89$	−4.90495	−0.519924	−0.259962	+	0.965619i	$0.583710\pi$
$90$	0	0
$91$	2.00171	0.209836
$92$	0	0
$93$	− 36.7340i	− 3.80914i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.3433i	1.25327i	0.779311	+	0.626637i	$0.215569\pi$
$97$	12.3433i	1.25327i	−0.779311	+	0.626637i	$0.784431\pi$
$98$	0	0
$99$	45.4295	4.56583
$100$	0	0
$101$	−1.44692	−0.143974	−0.0719870	−	0.997406i	$-0.522934\pi$
$101$	−1.44692	−0.143974	−0.0719870	+	0.997406i	$0.522934\pi$
$102$	0	0
$103$	− 7.76385i	− 0.764995i	−0.923957	−	0.382497i	$-0.875064\pi$
$103$	− 7.76385i	− 0.764995i	0.923957	−	0.382497i	$-0.124936\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.54197i	0.439089i	0.975603	+	0.219544i	$0.0704570\pi$
$107$	4.54197i	0.439089i	−0.975603	+	0.219544i	$0.929543\pi$
$108$	0	0
$109$	−12.4136	−1.18901	−0.594504	−	0.804093i	$-0.702652\pi$
$109$	−12.4136	−1.18901	−0.594504	+	0.804093i	$0.702652\pi$
$110$	0	0
$111$	−17.6805	−1.67815
$112$	0	0
$113$	9.27312i	0.872342i	0.899864	+	0.436171i	$0.143666\pi$
$113$	9.27312i	0.872342i	−0.899864	+	0.436171i	$0.856334\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.69896i	0.804219i
$118$	0	0
$119$	−12.8611	−1.17897
$120$	0	0
$121$	19.0327	1.73024
$122$	0	0
$123$	− 20.4894i	− 1.84747i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.4149i	− 1.10165i	−0.834621	−	0.550824i	$-0.814314\pi$
$127$	− 12.4149i	− 1.10165i	0.834621	−	0.550824i	$-0.185686\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.55434	0.834766	0.417383	−	0.908731i	$-0.362947\pi$
$131$	9.55434	0.834766	0.417383	+	0.908731i	$0.362947\pi$
$132$	0	0
$133$	2.95763i	0.256459i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.36558i	0.287541i	0.989611	+	0.143770i	$0.0459226\pi$
$137$	3.36558i	0.287541i	−0.989611	+	0.143770i	$0.954077\pi$
$138$	0	0
$139$	20.7361	1.75881	0.879406	−	0.476072i	$-0.157940\pi$
$139$	20.7361	1.75881	0.879406	+	0.476072i	$0.157940\pi$
$140$	0	0
$141$	−1.35688	−0.114270
$142$	0	0
$143$	5.75074i	0.480901i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.2940i	− 0.931510i
$148$	0	0
$149$	19.9399	1.63354	0.816769	−	0.576965i	$-0.195763\pi$
$149$	19.9399	1.63354	0.816769	+	0.576965i	$0.195763\pi$
$150$	0	0
$151$	23.7979	1.93664	0.968321	−	0.249709i	$-0.0803349\pi$
$151$	23.7979	1.93664	0.968321	+	0.249709i	$0.0803349\pi$
$152$	0	0
$153$	− 55.8912i	− 4.51854i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.3175i	− 1.06285i	−0.847105	−	0.531425i	$-0.821657\pi$
$157$	− 13.3175i	− 1.06285i	0.847105	−	0.531425i	$-0.178343\pi$
$158$	0	0
$159$	19.7681	1.56771
$160$	0	0
$161$	1.90754	0.150335
$162$	0	0
$163$	16.1529i	1.26519i	0.774483	+	0.632595i	$0.218010\pi$
$163$	16.1529i	1.26519i	−0.774483	+	0.632595i	$0.781990\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.6782i	1.60013i	0.599915	+	0.800064i	$0.295201\pi$
$167$	20.6782i	1.60013i	−0.599915	+	0.800064i	$0.704799\pi$
$168$	0	0
$169$	11.8988	0.915295
$170$	0	0
$171$	−12.8532	−0.982905
$172$	0	0
$173$	7.72058i	0.586985i	0.955961	+	0.293492i	$0.0948176\pi$
$173$	7.72058i	0.586985i	−0.955961	+	0.293492i	$0.905182\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	32.2599i	2.42480i
$178$	0	0
$179$	−9.88683	−0.738976	−0.369488	−	0.929236i	$-0.620467\pi$
$179$	−9.88683	−0.738976	−0.369488	+	0.929236i	$0.620467\pi$
$180$	0	0
$181$	−23.9883	−1.78304	−0.891519	−	0.452983i	$-0.850360\pi$
$181$	−23.9883	−1.78304	−0.891519	+	0.452983i	$0.850360\pi$
$182$	0	0
$183$	23.8537i	1.76332i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 36.9487i	− 2.70196i
$188$	0	0
$189$	−33.9040	−2.46615
$190$	0	0
$191$	6.72004	0.486245	0.243123	−	0.969996i	$-0.421828\pi$
$191$	6.72004	0.486245	0.243123	+	0.969996i	$0.421828\pi$
$192$	0	0
$193$	8.95007i	0.644240i	0.946699	+	0.322120i	$0.104395\pi$
$193$	8.95007i	0.644240i	−0.946699	+	0.322120i	$0.895605\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.65109i	0.473871i	0.971525	+	0.236935i	$0.0761429\pi$
$197$	6.65109i	0.473871i	−0.971525	+	0.236935i	$0.923857\pi$
$198$	0	0
$199$	10.9050	0.773032	0.386516	−	0.922283i	$-0.373678\pi$
$199$	10.9050	0.773032	0.386516	+	0.922283i	$0.373678\pi$
$200$	0	0
$201$	−46.3587	−3.26989
$202$	0	0
$203$	6.45168i	0.452819i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.28974i	0.576177i
$208$	0	0
$209$	−8.49700	−0.587750
$210$	0	0
$211$	−1.21874	−0.0839014	−0.0419507	−	0.999120i	$-0.513357\pi$
$211$	−1.21874	−0.0839014	−0.0419507	+	0.999120i	$0.513357\pi$
$212$	0	0
$213$	− 1.60928i	− 0.110266i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.8546i	1.41570i
$218$	0	0
$219$	8.07692	0.545787
$220$	0	0
$221$	7.07505	0.475919
$222$	0	0
$223$	14.1542i	0.947835i	0.880569	+	0.473917i	$0.157160\pi$
$223$	14.1542i	0.947835i	−0.880569	+	0.473917i	$0.842840\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.89902i	0.325159i	0.986695	+	0.162580i	$0.0519815\pi$
$227$	4.89902i	0.325159i	−0.986695	+	0.162580i	$0.948019\pi$
$228$	0	0
$229$	−12.3946	−0.819056	−0.409528	−	0.912298i	$-0.634307\pi$
$229$	−12.3946	−0.819056	−0.409528	+	0.912298i	$0.634307\pi$
$230$	0	0
$231$	−35.1248	−2.31104
$232$	0	0
$233$	0.882062i	0.0577858i	0.999583	+	0.0288929i	$0.00919818\pi$
$233$	0.882062i	0.0577858i	−0.999583	+	0.0288929i	$0.990802\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.2477i	− 0.925490i
$238$	0	0
$239$	19.9506	1.29049	0.645247	−	0.763974i	$-0.276754\pi$
$239$	19.9506	1.29049	0.645247	+	0.763974i	$0.276754\pi$
$240$	0	0
$241$	−2.81877	−0.181573	−0.0907865	−	0.995870i	$-0.528938\pi$
$241$	−2.81877	−0.181573	−0.0907865	+	0.995870i	$0.528938\pi$
$242$	0	0
$243$	− 63.7777i	− 4.09134i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.62703i	− 0.103525i
$248$	0	0
$249$	−37.8406	−2.39805
$250$	0	0
$251$	17.9883	1.13541	0.567706	−	0.823231i	$-0.307831\pi$
$251$	17.9883	1.13541	0.567706	+	0.823231i	$0.307831\pi$
$252$	0	0
$253$	5.48021i	0.344538i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1705i	0.883930i	0.897032	+	0.441965i	$0.145718\pi$
$257$	14.1705i	0.883930i	−0.897032	+	0.441965i	$0.854282\pi$
$258$	0	0
$259$	10.0375	0.623701
$260$	0	0
$261$	−28.0375	−1.73548
$262$	0	0
$263$	− 6.88386i	− 0.424477i	−0.977218	−	0.212238i	$-0.931925\pi$
$263$	− 6.88386i	− 0.424477i	0.977218	−	0.212238i	$-0.0680754\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.4807i	1.00861i
$268$	0	0
$269$	23.3463	1.42345	0.711724	−	0.702459i	$-0.247915\pi$
$269$	23.3463	1.42345	0.711724	+	0.702459i	$0.247915\pi$
$270$	0	0
$271$	13.9519	0.847517	0.423759	−	0.905775i	$-0.360711\pi$
$271$	13.9519	0.847517	0.423759	+	0.905775i	$0.360711\pi$
$272$	0	0
$273$	− 6.72579i	− 0.407063i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 3.69490i	− 0.222005i	−0.993820	−	0.111003i	$-0.964594\pi$
$277$	− 3.69490i	− 0.222005i	0.993820	−	0.111003i	$-0.0354062\pi$
$278$	0	0
$279$	−90.6291	−5.42582
$280$	0	0
$281$	15.9582	0.951984	0.475992	−	0.879450i	$-0.342089\pi$
$281$	15.9582	0.951984	0.475992	+	0.879450i	$0.342089\pi$
$282$	0	0
$283$	− 8.21649i	− 0.488420i	−0.969722	−	0.244210i	$-0.921471\pi$
$283$	− 8.21649i	− 0.488420i	0.969722	−	0.244210i	$-0.0785286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.6322i	0.686628i
$288$	0	0
$289$	−28.4575	−1.67397
$290$	0	0
$291$	41.4738	2.43124
$292$	0	0
$293$	22.0878i	1.29038i	0.764021	+	0.645191i	$0.223222\pi$
$293$	22.0878i	1.29038i	−0.764021	+	0.645191i	$0.776778\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 97.4032i	− 5.65191i
$298$	0	0
$299$	−1.04937	−0.0606863
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.86168i	0.279296i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.2091i	− 0.868030i	−0.900906	−	0.434015i	$-0.857097\pi$
$307$	− 15.2091i	− 0.868030i	0.900906	−	0.434015i	$-0.142903\pi$
$308$	0	0
$309$	−26.0867	−1.48402
$310$	0	0
$311$	0.254818	0.0144494	0.00722471	−	0.999974i	$-0.497700\pi$
$311$	0.254818	0.0144494	0.00722471	+	0.999974i	$0.497700\pi$
$312$	0	0
$313$	3.57095i	0.201842i	0.994894	+	0.100921i	$0.0321790\pi$
$313$	3.57095i	0.201842i	−0.994894	+	0.100921i	$0.967821\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.3235i	− 0.692155i	−0.938206	−	0.346078i	$-0.887513\pi$
$317$	− 12.3235i	− 0.692155i	0.938206	−	0.346078i	$-0.112487\pi$
$318$	0	0
$319$	−18.5351	−1.03777
$320$	0	0
$321$	15.2611	0.851792
$322$	0	0
$323$	10.4537i	0.581661i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	41.7100i	2.30657i
$328$	0	0
$329$	0.770323	0.0424693
$330$	0	0
$331$	26.1082	1.43503	0.717517	−	0.696541i	$-0.245278\pi$
$331$	26.1082	1.43503	0.717517	+	0.696541i	$0.245278\pi$
$332$	0	0
$333$	43.6207i	2.39040i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.68246i	− 0.527437i	−0.964600	−	0.263718i	$-0.915051\pi$
$337$	− 9.68246i	− 0.527437i	0.964600	−	0.263718i	$-0.0849490\pi$
$338$	0	0
$339$	31.1579	1.69226
$340$	0	0
$341$	−59.9134	−3.24449
$342$	0	0
$343$	19.7646i	1.06719i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.9747i	− 1.34071i	−0.742040	−	0.670355i	$-0.766142\pi$
$347$	− 24.9747i	− 1.34071i	0.742040	−	0.670355i	$-0.233858\pi$
$348$	0	0
$349$	35.3019	1.88967	0.944835	−	0.327547i	$-0.106222\pi$
$349$	35.3019	1.88967	0.944835	+	0.327547i	$0.106222\pi$
$350$	0	0
$351$	18.6510	0.995518
$352$	0	0
$353$	− 10.0326i	− 0.533980i	−0.963699	−	0.266990i	$-0.913971\pi$
$353$	− 10.0326i	− 0.533980i	0.963699	−	0.266990i	$-0.0860290\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	43.2135i	2.28710i
$358$	0	0
$359$	30.8691	1.62921	0.814603	−	0.580019i	$-0.196955\pi$
$359$	30.8691	1.62921	0.814603	+	0.580019i	$0.196955\pi$
$360$	0	0
$361$	−16.5960	−0.873473
$362$	0	0
$363$	− 63.9502i	− 3.35651i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 33.3234i	− 1.73947i	−0.493521	−	0.869734i	$-0.664291\pi$
$367$	− 33.3234i	− 1.73947i	0.493521	−	0.869734i	$-0.335709\pi$
$368$	0	0
$369$	−50.5509	−2.63158
$370$	0	0
$371$	−11.2227	−0.582653
$372$	0	0
$373$	4.62023i	0.239227i	0.992821	+	0.119613i	$0.0381655\pi$
$373$	4.62023i	0.239227i	−0.992821	+	0.119613i	$0.961835\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.54916i	− 0.182791i
$378$	0	0
$379$	−15.0020	−0.770600	−0.385300	−	0.922791i	$-0.625902\pi$
$379$	−15.0020	−0.770600	−0.385300	+	0.922791i	$0.625902\pi$
$380$	0	0
$381$	−41.7145	−2.13710
$382$	0	0
$383$	1.87882i	0.0960033i	0.998847	+	0.0480016i	$0.0152853\pi$
$383$	1.87882i	0.0960033i	−0.998847	+	0.0480016i	$0.984715\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.48539	0.0753121	0.0376560	−	0.999291i	$-0.488011\pi$
$389$	1.48539	0.0753121	0.0376560	+	0.999291i	$0.488011\pi$
$390$	0	0
$391$	6.74222	0.340968
$392$	0	0
$393$	− 32.1028i	− 1.61937i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.26722i	− 0.264354i	−0.991226	−	0.132177i	$-0.957803\pi$
$397$	− 5.26722i	− 0.264354i	0.991226	−	0.132177i	$-0.0421968\pi$
$398$	0	0
$399$	9.93769	0.497507
$400$	0	0
$401$	26.1490	1.30582	0.652910	−	0.757436i	$-0.273548\pi$
$401$	26.1490	1.30582	0.652910	+	0.757436i	$0.273548\pi$
$402$	0	0
$403$	− 11.4724i	− 0.571480i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.8369i	1.42939i
$408$	0	0
$409$	14.7503	0.729355	0.364677	−	0.931134i	$-0.381179\pi$
$409$	14.7503	0.729355	0.364677	+	0.931134i	$0.381179\pi$
$410$	0	0
$411$	11.3084	0.557803
$412$	0	0
$413$	− 18.3145i	− 0.901200i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 69.6737i	− 3.41194i
$418$	0	0
$419$	−1.10616	−0.0540394	−0.0270197	−	0.999635i	$-0.508602\pi$
$419$	−1.10616	−0.0540394	−0.0270197	+	0.999635i	$0.508602\pi$
$420$	0	0
$421$	9.16362	0.446607	0.223304	−	0.974749i	$-0.428316\pi$
$421$	9.16362	0.446607	0.223304	+	0.974749i	$0.428316\pi$
$422$	0	0
$423$	3.34764i	0.162768i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 13.5422i	− 0.655351i
$428$	0	0
$429$	19.3226	0.932904
$430$	0	0
$431$	−32.1712	−1.54963	−0.774817	−	0.632186i	$-0.782158\pi$
$431$	−32.1712	−1.54963	−0.774817	+	0.632186i	$0.782158\pi$
$432$	0	0
$433$	− 8.37861i	− 0.402650i	−0.979524	−	0.201325i	$-0.935475\pi$
$433$	− 8.37861i	− 0.402650i	0.979524	−	0.201325i	$-0.0645248\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.55049i	− 0.0741700i
$438$	0	0
$439$	15.3093	0.730673	0.365336	−	0.930876i	$-0.380954\pi$
$439$	15.3093	0.730673	0.365336	+	0.930876i	$0.380954\pi$
$440$	0	0
$441$	−27.8641	−1.32686
$442$	0	0
$443$	− 3.24129i	− 0.153998i	−0.997031	−	0.0769992i	$-0.975466\pi$
$443$	− 3.24129i	− 0.153998i	0.997031	−	0.0769992i	$-0.0245339\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 66.9984i	− 3.16892i
$448$	0	0
$449$	9.02215	0.425782	0.212891	−	0.977076i	$-0.431712\pi$
$449$	9.02215	0.425782	0.212891	+	0.977076i	$0.431712\pi$
$450$	0	0
$451$	−33.4184	−1.57361
$452$	0	0
$453$	− 79.9613i	− 3.75691i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.03526i	− 0.329096i	−0.986369	−	0.164548i	$-0.947384\pi$
$457$	− 7.03526i	− 0.329096i	0.986369	−	0.164548i	$-0.0526165\pi$
$458$	0	0
$459$	−119.834	−5.59336
$460$	0	0
$461$	−1.59617	−0.0743411	−0.0371705	−	0.999309i	$-0.511834\pi$
$461$	−1.59617	−0.0743411	−0.0371705	+	0.999309i	$0.511834\pi$
$462$	0	0
$463$	9.61655i	0.446919i	0.974713	+	0.223459i	$0.0717350\pi$
$463$	9.61655i	0.446919i	−0.974713	+	0.223459i	$0.928265\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.4764i	− 1.13263i	−0.824188	−	0.566317i	$-0.808368\pi$
$467$	− 24.4764i	− 1.13263i	0.824188	−	0.566317i	$-0.191632\pi$
$468$	0	0
$469$	26.3186	1.21528
$470$	0	0
$471$	−44.7470	−2.06183
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 48.7712i	− 2.23308i
$478$	0	0
$479$	13.2108	0.603618	0.301809	−	0.953368i	$-0.402410\pi$
$479$	13.2108	0.603618	0.301809	+	0.953368i	$0.402410\pi$
$480$	0	0
$481$	−5.52177	−0.251771
$482$	0	0
$483$	− 6.40939i	− 0.291637i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.7078i	1.07431i	0.843485	+	0.537153i	$0.180500\pi$
$487$	23.7078i	1.07431i	−0.843485	+	0.537153i	$0.819500\pi$
$488$	0	0
$489$	54.2739	2.45435
$490$	0	0
$491$	18.5930	0.839091	0.419546	−	0.907734i	$-0.362189\pi$
$491$	18.5930	0.839091	0.419546	+	0.907734i	$0.362189\pi$
$492$	0	0
$493$	22.8035i	1.02702i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.913618i	0.0409814i
$498$	0	0
$499$	−17.9121	−0.801858	−0.400929	−	0.916109i	$-0.631313\pi$
$499$	−17.9121	−0.801858	−0.400929	+	0.916109i	$0.631313\pi$
$500$	0	0
$501$	69.4792	3.10410
$502$	0	0
$503$	1.24890i	0.0556855i	0.999612	+	0.0278428i	$0.00886377\pi$
$503$	1.24890i	0.0556855i	−0.999612	+	0.0278428i	$0.991136\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 39.9803i	− 1.77559i
$508$	0	0
$509$	−15.8757	−0.703679	−0.351839	−	0.936060i	$-0.614444\pi$
$509$	−15.8757	−0.703679	−0.351839	+	0.936060i	$0.614444\pi$
$510$	0	0
$511$	−4.58541	−0.202847
$512$	0	0
$513$	27.5578i	1.21671i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.21307i	0.0973307i
$518$	0	0
$519$	25.9413	1.13870
$520$	0	0
$521$	26.5488	1.16312	0.581561	−	0.813503i	$-0.302442\pi$
$521$	26.5488	1.16312	0.581561	+	0.813503i	$0.302442\pi$
$522$	0	0
$523$	31.0258i	1.35666i	0.734757	+	0.678331i	$0.237296\pi$
$523$	31.0258i	1.35666i	−0.734757	+	0.678331i	$0.762704\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	73.7105i	3.21088i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	79.5907	3.45394
$532$	0	0
$533$	− 6.39904i	− 0.277173i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	33.2199i	1.43355i
$538$	0	0
$539$	−18.4205	−0.793427
$540$	0	0
$541$	22.4123	0.963579	0.481790	−	0.876287i	$-0.339987\pi$
$541$	22.4123	0.963579	0.481790	+	0.876287i	$0.339987\pi$
$542$	0	0
$543$	80.6013i	3.45893i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.3980i	1.85556i	0.373122	+	0.927782i	$0.378287\pi$
$547$	43.3980i	1.85556i	−0.373122	+	0.927782i	$0.621713\pi$
$548$	0	0
$549$	58.8511	2.51170
$550$	0	0
$551$	5.24406	0.223404
$552$	0	0
$553$	8.08870i	0.343966i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.2690i	0.901197i	0.892727	+	0.450599i	$0.148789\pi$
$557$	21.2690i	0.901197i	−0.892727	+	0.450599i	$0.851211\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−124.149	−5.24155
$562$	0	0
$563$	10.6629i	0.449389i	0.974429	+	0.224694i	$0.0721383\pi$
$563$	10.6629i	0.449389i	−0.974429	+	0.224694i	$0.927862\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	66.4789i	2.79185i
$568$	0	0
$569$	−31.8986	−1.33726	−0.668630	−	0.743596i	$-0.733119\pi$
$569$	−31.8986	−1.33726	−0.668630	+	0.743596i	$0.733119\pi$
$570$	0	0
$571$	−7.87477	−0.329549	−0.164774	−	0.986331i	$-0.552690\pi$
$571$	−7.87477	−0.329549	−0.164774	+	0.986331i	$0.552690\pi$
$572$	0	0
$573$	− 22.5795i	− 0.943271i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 40.7070i	− 1.69466i	−0.531070	−	0.847328i	$-0.678210\pi$
$577$	− 40.7070i	− 1.69466i	0.531070	−	0.847328i	$-0.321790\pi$
$578$	0	0
$579$	30.0724	1.24977
$580$	0	0
$581$	21.4828	0.891256
$582$	0	0
$583$	− 32.2418i	− 1.33532i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.9321i	0.946508i	0.880926	+	0.473254i	$0.156921\pi$
$587$	22.9321i	0.946508i	−0.880926	+	0.473254i	$0.843079\pi$
$588$	0	0
$589$	16.9510	0.698454
$590$	0	0
$591$	22.3478	0.919266
$592$	0	0
$593$	27.6250i	1.13442i	0.823572	+	0.567211i	$0.191978\pi$
$593$	27.6250i	1.13442i	−0.823572	+	0.567211i	$0.808022\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 36.6409i	− 1.49961i
$598$	0	0
$599$	18.2139	0.744199	0.372099	−	0.928193i	$-0.378638\pi$
$599$	18.2139	0.744199	0.372099	+	0.928193i	$0.378638\pi$
$600$	0	0
$601$	−24.0407	−0.980640	−0.490320	−	0.871543i	$-0.663120\pi$
$601$	−24.0407	−0.980640	−0.490320	+	0.871543i	$0.663120\pi$
$602$	0	0
$603$	114.375i	4.65770i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 5.52878i	− 0.224406i	−0.993685	−	0.112203i	$-0.964209\pi$
$607$	− 5.52878i	− 0.224406i	0.993685	−	0.112203i	$-0.0357907\pi$
$608$	0	0
$609$	21.6778	0.878428
$610$	0	0
$611$	−0.423765	−0.0171437
$612$	0	0
$613$	− 21.1205i	− 0.853050i	−0.904476	−	0.426525i	$-0.859738\pi$
$613$	− 21.1205i	− 0.853050i	0.904476	−	0.426525i	$-0.140262\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19.6523i	− 0.791174i	−0.918429	−	0.395587i	$-0.870541\pi$
$617$	− 19.6523i	− 0.791174i	0.918429	−	0.395587i	$-0.129459\pi$
$618$	0	0
$619$	−40.9931	−1.64765	−0.823826	−	0.566843i	$-0.808164\pi$
$619$	−40.9931	−1.64765	−0.823826	+	0.566843i	$0.808164\pi$
$620$	0	0
$621$	17.7736	0.713231
$622$	0	0
$623$	− 9.35641i	− 0.374857i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	28.5501i	1.14018i
$628$	0	0
$629$	35.4776	1.41458
$630$	0	0
$631$	−45.5595	−1.81369	−0.906847	−	0.421461i	$-0.861517\pi$
$631$	−45.5595	−1.81369	−0.906847	+	0.421461i	$0.861517\pi$
$632$	0	0
$633$	4.09499i	0.162761i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.52721i	− 0.139753i
$638$	0	0
$639$	−3.97037	−0.157065
$640$	0	0
$641$	8.97997	0.354688	0.177344	−	0.984149i	$-0.443250\pi$
$641$	8.97997	0.354688	0.177344	+	0.984149i	$0.443250\pi$
$642$	0	0
$643$	2.69616i	0.106326i	0.998586	+	0.0531630i	$0.0169303\pi$
$643$	2.69616i	0.106326i	−0.998586	+	0.0531630i	$0.983070\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.12638i	− 0.358795i	−0.983777	−	0.179398i	$-0.942585\pi$
$647$	− 9.12638i	− 0.358795i	0.983777	−	0.179398i	$-0.0574149\pi$
$648$	0	0
$649$	52.6161	2.06536
$650$	0	0
$651$	70.0718	2.74633
$652$	0	0
$653$	41.5497i	1.62596i	0.582289	+	0.812982i	$0.302157\pi$
$653$	41.5497i	1.62596i	−0.582289	+	0.812982i	$0.697843\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 19.9271i	− 0.777431i
$658$	0	0
$659$	−20.9256	−0.815145	−0.407573	−	0.913173i	$-0.633625\pi$
$659$	−20.9256	−0.815145	−0.407573	+	0.913173i	$0.633625\pi$
$660$	0	0
$661$	5.25688	0.204469	0.102234	−	0.994760i	$-0.467401\pi$
$661$	5.25688	0.204469	0.102234	+	0.994760i	$0.467401\pi$
$662$	0	0
$663$	− 23.7723i	− 0.923240i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.38219i	− 0.130959i
$668$	0	0
$669$	47.5584	1.83871
$670$	0	0
$671$	38.9055	1.50193
$672$	0	0
$673$	38.1900i	1.47212i	0.676918	+	0.736059i	$0.263315\pi$
$673$	38.1900i	1.47212i	−0.676918	+	0.736059i	$0.736685\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.83529i	0.185835i	0.995674	+	0.0929176i	$0.0296193\pi$
$677$	4.83529i	0.185835i	−0.995674	+	0.0929176i	$0.970381\pi$
$678$	0	0
$679$	−23.5454	−0.903591
$680$	0	0
$681$	16.4608	0.630780
$682$	0	0
$683$	− 50.3044i	− 1.92484i	−0.271559	−	0.962422i	$-0.587539\pi$
$683$	− 50.3044i	− 1.92484i	0.271559	−	0.962422i	$-0.412461\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	41.6460i	1.58889i
$688$	0	0
$689$	6.17375	0.235201
$690$	0	0
$691$	20.6298	0.784793	0.392396	−	0.919796i	$-0.371646\pi$
$691$	20.6298	0.784793	0.392396	+	0.919796i	$0.371646\pi$
$692$	0	0
$693$	86.6587i	3.29189i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	41.1141i	1.55731i
$698$	0	0
$699$	2.96375	0.112099
$700$	0	0
$701$	−22.9598	−0.867182	−0.433591	−	0.901110i	$-0.642754\pi$
$701$	−22.9598	−0.867182	−0.433591	+	0.901110i	$0.642754\pi$
$702$	0	0
$703$	− 8.15869i	− 0.307711i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.76006i	− 0.103803i
$708$	0	0
$709$	−11.9069	−0.447174	−0.223587	−	0.974684i	$-0.571777\pi$
$709$	−11.9069	−0.447174	−0.223587	+	0.974684i	$0.571777\pi$
$710$	0	0
$711$	−35.1516	−1.31829
$712$	0	0
$713$	− 10.9327i	− 0.409432i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 67.0343i	− 2.50344i
$718$	0	0
$719$	−24.5244	−0.914604	−0.457302	−	0.889311i	$-0.651184\pi$
$719$	−24.5244	−0.914604	−0.457302	+	0.889311i	$0.651184\pi$
$720$	0	0
$721$	14.8099	0.551549
$722$	0	0
$723$	9.47113i	0.352235i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 11.4034i	− 0.422928i	−0.977386	−	0.211464i	$-0.932177\pi$
$727$	− 11.4034i	− 0.422928i	0.977386	−	0.211464i	$-0.0678231\pi$
$728$	0	0
$729$	−109.743	−4.06454
$730$	0	0
$731$	0	0
$732$	0	0
$733$	35.8722i	1.32497i	0.749076	+	0.662484i	$0.230498\pi$
$733$	35.8722i	1.32497i	−0.749076	+	0.662484i	$0.769502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	75.6112i	2.78517i
$738$	0	0
$739$	17.1503	0.630883	0.315441	−	0.948945i	$-0.397847\pi$
$739$	17.1503	0.630883	0.315441	+	0.948945i	$0.397847\pi$
$740$	0	0
$741$	−5.46685	−0.200830
$742$	0	0
$743$	27.7242i	1.01710i	0.861031	+	0.508552i	$0.169819\pi$
$743$	27.7242i	1.01710i	−0.861031	+	0.508552i	$0.830181\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	93.3591i	3.41583i
$748$	0	0
$749$	−8.66400	−0.316576
$750$	0	0
$751$	−2.80173	−0.102236	−0.0511182	−	0.998693i	$-0.516279\pi$
$751$	−2.80173	−0.102236	−0.0511182	+	0.998693i	$0.516279\pi$
$752$	0	0
$753$	− 60.4411i	− 2.20260i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.6710i	1.51456i	0.653092	+	0.757278i	$0.273471\pi$
$757$	41.6710i	1.51456i	−0.653092	+	0.757278i	$0.726529\pi$
$758$	0	0
$759$	18.4136	0.668372
$760$	0	0
$761$	−35.4285	−1.28428	−0.642141	−	0.766587i	$-0.721954\pi$
$761$	−35.4285	−1.28428	−0.642141	+	0.766587i	$0.721954\pi$
$762$	0	0
$763$	− 23.6795i	− 0.857255i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.0751i	0.363790i
$768$	0	0
$769$	−23.8731	−0.860885	−0.430442	−	0.902618i	$-0.641642\pi$
$769$	−23.8731	−0.860885	−0.430442	+	0.902618i	$0.641642\pi$
$770$	0	0
$771$	47.6131	1.71474
$772$	0	0
$773$	11.3403i	0.407881i	0.978983	+	0.203941i	$0.0653750\pi$
$773$	11.3403i	0.407881i	−0.978983	+	0.203941i	$0.934625\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 33.7262i	− 1.20992i
$778$	0	0
$779$	9.45490	0.338757
$780$	0	0
$781$	−2.62475	−0.0939208
$782$	0	0
$783$	60.1139i	2.14829i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.1906i	0.933595i	0.884364	+	0.466797i	$0.154592\pi$
$787$	26.1906i	0.933595i	−0.884364	+	0.466797i	$0.845408\pi$
$788$	0	0
$789$	−23.1299	−0.823446
$790$	0	0
$791$	−17.6889	−0.628944
$792$	0	0
$793$	7.44973i	0.264547i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.48133i	0.158737i	0.996845	+	0.0793684i	$0.0252903\pi$
$797$	4.48133i	0.158737i	−0.996845	+	0.0793684i	$0.974710\pi$
$798$	0	0
$799$	2.72271	0.0963224
$800$	0	0
$801$	40.6608	1.43668
$802$	0	0
$803$	− 13.1735i	− 0.464882i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 78.4440i	− 2.76136i
$808$	0	0
$809$	32.4608	1.14126	0.570630	−	0.821207i	$-0.306699\pi$
$809$	32.4608	1.14126	0.570630	+	0.821207i	$0.306699\pi$
$810$	0	0
$811$	11.3500	0.398554	0.199277	−	0.979943i	$-0.436141\pi$
$811$	11.3500	0.398554	0.199277	+	0.979943i	$0.436141\pi$
$812$	0	0
$813$	− 46.8786i	− 1.64411i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−16.5936	−0.579829
$820$	0	0
$821$	48.2070	1.68244	0.841218	−	0.540697i	$-0.181839\pi$
$821$	48.2070	1.68244	0.841218	+	0.540697i	$0.181839\pi$
$822$	0	0
$823$	− 37.7179i	− 1.31476i	−0.753558	−	0.657381i	$-0.771664\pi$
$823$	− 37.7179i	− 1.31476i	0.753558	−	0.657381i	$-0.228336\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.89736i	0.309392i	0.987962	+	0.154696i	$0.0494397\pi$
$827$	8.89736i	0.309392i	−0.987962	+	0.154696i	$0.950560\pi$
$828$	0	0
$829$	−30.9058	−1.07340	−0.536702	−	0.843772i	$-0.680330\pi$
$829$	−30.9058	−1.07340	−0.536702	+	0.843772i	$0.680330\pi$
$830$	0	0
$831$	−12.4149	−0.430670
$832$	0	0
$833$	22.6625i	0.785208i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	194.313i	6.71646i
$838$	0	0
$839$	−1.34387	−0.0463954	−0.0231977	−	0.999731i	$-0.507385\pi$
$839$	−1.34387	−0.0463954	−0.0231977	+	0.999731i	$0.507385\pi$
$840$	0	0
$841$	−17.5608	−0.605543
$842$	0	0
$843$	− 53.6198i	− 1.84676i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36.3056i	1.24748i
$848$	0	0
$849$	−27.6076	−0.947489
$850$	0	0
$851$	−5.26201	−0.180379
$852$	0	0
$853$	− 26.3940i	− 0.903713i	−0.892091	−	0.451857i	$-0.850762\pi$
$853$	− 26.3940i	− 0.903713i	0.892091	−	0.451857i	$-0.149238\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.5048i	0.597953i	0.954260	+	0.298976i	$0.0966452\pi$
$857$	17.5048i	0.597953i	−0.954260	+	0.298976i	$0.903355\pi$
$858$	0	0
$859$	−11.7199	−0.399877	−0.199938	−	0.979808i	$-0.564074\pi$
$859$	−11.7199	−0.399877	−0.199938	+	0.979808i	$0.564074\pi$
$860$	0	0
$861$	39.0845	1.33200
$862$	0	0
$863$	26.6578i	0.907443i	0.891144	+	0.453722i	$0.149904\pi$
$863$	26.6578i	0.907443i	−0.891144	+	0.453722i	$0.850096\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	95.6177i	3.24735i
$868$	0	0
$869$	−23.2381	−0.788299
$870$	0	0
$871$	−14.4782	−0.490576
$872$	0	0
$873$	− 102.323i	− 3.46311i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19.9574i	− 0.673912i	−0.941520	−	0.336956i	$-0.890603\pi$
$877$	− 19.9574i	− 0.673912i	0.941520	−	0.336956i	$-0.109397\pi$
$878$	0	0
$879$	74.2154	2.50322
$880$	0	0
$881$	−20.7297	−0.698402	−0.349201	−	0.937048i	$-0.613547\pi$
$881$	−20.7297	−0.698402	−0.349201	+	0.937048i	$0.613547\pi$
$882$	0	0
$883$	2.31093i	0.0777690i	0.999244	+	0.0388845i	$0.0123804\pi$
$883$	2.31093i	0.0777690i	−0.999244	+	0.0388845i	$0.987620\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.6924i	0.392592i	0.980545	+	0.196296i	$0.0628913\pi$
$887$	11.6924i	0.392592i	−0.980545	+	0.196296i	$0.937109\pi$
$888$	0	0
$889$	23.6820	0.794270
$890$	0	0
$891$	−190.988	−6.39835
$892$	0	0
$893$	− 0.626134i	− 0.0209528i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.52589i	0.117726i
$898$	0	0
$899$	36.9765	1.23323
$900$	0	0
$901$	−39.6666	−1.32149
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 32.5061i	− 1.07935i	−0.841875	−	0.539673i	$-0.818548\pi$
$907$	− 32.5061i	− 1.07935i	0.841875	−	0.539673i	$-0.181452\pi$
$908$	0	0
$909$	11.9946	0.397836
$910$	0	0
$911$	10.7518	0.356224	0.178112	−	0.984010i	$-0.443001\pi$
$911$	10.7518	0.356224	0.178112	+	0.984010i	$0.443001\pi$
$912$	0	0
$913$	61.7181i	2.04257i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.2253i	0.601853i
$918$	0	0
$919$	24.2758	0.800785	0.400392	−	0.916344i	$-0.368874\pi$
$919$	24.2758	0.800785	0.400392	+	0.916344i	$0.368874\pi$
$920$	0	0
$921$	−51.1029	−1.68390
$922$	0	0
$923$	− 0.502594i	− 0.0165431i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	64.3603i	2.11387i
$928$	0	0
$929$	−8.18842	−0.268653	−0.134327	−	0.990937i	$-0.542887\pi$
$929$	−8.18842	−0.268653	−0.134327	+	0.990937i	$0.542887\pi$
$930$	0	0
$931$	5.21163	0.170804
$932$	0	0
$933$	− 0.856194i	− 0.0280305i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26.2312i	− 0.856936i	−0.903557	−	0.428468i	$-0.859053\pi$
$937$	− 26.2312i	− 0.856936i	0.903557	−	0.428468i	$-0.140947\pi$
$938$	0	0
$939$	11.9985	0.391556
$940$	0	0
$941$	23.6292	0.770291	0.385145	−	0.922856i	$-0.374151\pi$
$941$	23.6292	0.770291	0.385145	+	0.922856i	$0.374151\pi$
$942$	0	0
$943$	− 6.09801i	− 0.198579i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.9026i	0.516765i	0.966043	+	0.258383i	$0.0831895\pi$
$947$	15.9026i	0.516765i	−0.966043	+	0.258383i	$0.916810\pi$
$948$	0	0
$949$	2.52249	0.0818836
$950$	0	0
$951$	−41.4071	−1.34272
$952$	0	0
$953$	20.9451i	0.678478i	0.940700	+	0.339239i	$0.110169\pi$
$953$	20.9451i	0.678478i	−0.940700	+	0.339239i	$0.889831\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	62.2784i	2.01318i
$958$	0	0
$959$	−6.41998	−0.207312
$960$	0	0
$961$	88.5236	2.85560
$962$	0	0
$963$	− 37.6517i	− 1.21331i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.0732i	0.452563i	0.974062	+	0.226281i	$0.0726569\pi$
$967$	14.0732i	0.452563i	−0.974062	+	0.226281i	$0.927343\pi$
$968$	0	0
$969$	35.1248	1.12837
$970$	0	0
$971$	−11.4404	−0.367139	−0.183569	−	0.983007i	$-0.558765\pi$
$971$	−11.4404	−0.367139	−0.183569	+	0.983007i	$0.558765\pi$
$972$	0	0
$973$	39.5550i	1.26808i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.9119i	− 1.40487i	−0.711750	−	0.702433i	$-0.752097\pi$
$977$	− 43.9119i	− 1.40487i	0.711750	−	0.702433i	$-0.247903\pi$
$978$	0	0
$979$	26.8802	0.859094
$980$	0	0
$981$	102.906	3.28552
$982$	0	0
$983$	− 43.0420i	− 1.37283i	−0.727212	−	0.686413i	$-0.759184\pi$
$983$	− 43.0420i	− 1.37283i	0.727212	−	0.686413i	$-0.240816\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.58830i	− 0.0823865i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	43.5288	1.38274	0.691369	−	0.722501i	$-0.257008\pi$
$991$	43.5288	1.38274	0.691369	+	0.722501i	$0.257008\pi$
$992$	0	0
$993$	− 87.7240i	− 2.78384i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.4448i	− 0.837517i	−0.908098	−	0.418758i	$-0.862465\pi$
$997$	− 26.4448i	− 0.837517i	0.908098	−	0.418758i	$-0.137535\pi$
$998$	0	0
$999$	93.5250	2.95900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.u.4049.1		10
5.2	odd	4		920.2.a.j.1.1	✓	5
5.3	odd	4		4600.2.a.be.1.5		5
5.4	even	2	inner	4600.2.e.u.4049.10		10
15.2	even	4		8280.2.a.bs.1.3		5
20.3	even	4		9200.2.a.cu.1.1		5
20.7	even	4		1840.2.a.v.1.5		5
40.27	even	4		7360.2.a.cp.1.1		5
40.37	odd	4		7360.2.a.co.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.1	✓	5	5.2	odd	4
1840.2.a.v.1.5		5	20.7	even	4
4600.2.a.be.1.5		5	5.3	odd	4
4600.2.e.u.4049.1		10	1.1	even	1	trivial
4600.2.e.u.4049.10		10	5.4	even	2	inner
7360.2.a.co.1.5		5	40.37	odd	4
7360.2.a.cp.1.1		5	40.27	even	4
8280.2.a.bs.1.3		5	15.2	even	4
9200.2.a.cu.1.1		5	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.36002i q^{3} +1.90754i q^{7} -8.28974 q^{9} +O(q^{10})\)	\(q-3.36002i q^{3} +1.90754i q^{7} -8.28974 q^{9} -5.48021 q^{11} -1.04937i q^{13} +6.74222i q^{17} +1.55049 q^{19} +6.40939 q^{21} -1.00000i q^{23} +17.7736i q^{27} +3.38219 q^{29} +10.9327 q^{31} +18.4136i q^{33} -5.26201i q^{37} -3.52589 q^{39} +6.09801 q^{41} -0.403830i q^{47} +3.36128 q^{49} +22.6540 q^{51} +5.88332i q^{53} -5.20968i q^{57} -9.60111 q^{59} -7.09927 q^{61} -15.8130i q^{63} -13.7971i q^{67} -3.36002 q^{69} +0.478950 q^{71} +2.40383i q^{73} -10.4537i q^{77} +4.24037 q^{79} +34.8505 q^{81} -11.2620i q^{83} -11.3642i q^{87} -4.90495 q^{89} +2.00171 q^{91} -36.7340i q^{93} +12.3433i q^{97} +45.4295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 26 q^{9}+O(q^{10}) \) 10 * q - 26 * q^9	\( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) \) 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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