LMFDB - Embedded newform 4600.2.e.p.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:	$6$
Coefficient field:	6.0.431642176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 19x^{4} + 97x^{2} + 64 $ x^6 + 19x^4 + 97x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$-3.07912i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.p.4049.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.07912i q^{3} -2.07912i q^{7} -6.48097 q^{9} +O(q^{10})$	$q-3.07912i q^{3} -2.07912i q^{7} -6.48097 q^{9} +5.07912 q^{11} +3.48097i q^{13} +6.48097i q^{17} +7.48097 q^{19} -6.40185 q^{21} -1.00000i q^{23} +10.7183i q^{27} +1.56009 q^{29} +0.0791189 q^{31} -15.6392i q^{33} -9.71833i q^{37} +10.7183 q^{39} -0.480973 q^{41} -8.00000i q^{43} +6.96195i q^{47} +2.67726 q^{49} +19.9557 q^{51} -11.7183i q^{53} -23.0348i q^{57} +11.5601 q^{59} +7.88283 q^{61} +13.4747i q^{63} -9.71833i q^{67} -3.07912 q^{69} -9.67726 q^{71} +13.2784i q^{73} -10.5601i q^{77} +12.3165 q^{79} +13.5601 q^{81} +4.59815i q^{83} -4.80371i q^{87} -8.31648 q^{89} +7.23736 q^{91} -0.243616i q^{93} +7.23736i q^{97} -32.9176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 20 q^{9}+O(q^{10}) $ 6 * q - 20 * q^9	$ 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) $ 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * 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
$3$	− 3.07912i	− 1.77773i	−0.458169	−	0.888865i	$-0.651495\pi$
$3$	− 3.07912i	− 1.77773i	0.458169	−	0.888865i	$-0.348505\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.07912i	− 0.785833i	−0.919574	−	0.392917i	$-0.871466\pi$
$7$	− 2.07912i	− 0.785833i	0.919574	−	0.392917i	$-0.128534\pi$
$8$	0	0
$9$	−6.48097	−2.16032
$10$	0	0
$11$	5.07912	1.53141	0.765706	−	0.643191i	$-0.222390\pi$
$11$	5.07912	1.53141	0.765706	+	0.643191i	$0.222390\pi$
$12$	0	0
$13$	3.48097i	0.965448i	0.875772	+	0.482724i	$0.160353\pi$
$13$	3.48097i	0.965448i	−0.875772	+	0.482724i	$0.839647\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.48097i	1.57187i	0.618311	+	0.785933i	$0.287817\pi$
$17$	6.48097i	1.57187i	−0.618311	+	0.785933i	$0.712183\pi$
$18$	0	0
$19$	7.48097	1.71625	0.858126	−	0.513438i	$-0.171629\pi$
$19$	7.48097	1.71625	0.858126	+	0.513438i	$0.171629\pi$
$20$	0	0
$21$	−6.40185	−1.39700
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10.7183i	2.06274i
$28$	0	0
$29$	1.56009	0.289702	0.144851	−	0.989453i	$-0.453730\pi$
$29$	1.56009	0.289702	0.144851	+	0.989453i	$0.453730\pi$
$30$	0	0
$31$	0.0791189	0.0142102	0.00710508	−	0.999975i	$-0.497738\pi$
$31$	0.0791189	0.0142102	0.00710508	+	0.999975i	$0.497738\pi$
$32$	0	0
$33$	− 15.6392i	− 2.72244i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.71833i	− 1.59768i	−0.601541	−	0.798842i	$-0.705446\pi$
$37$	− 9.71833i	− 1.59768i	0.601541	−	0.798842i	$-0.294554\pi$
$38$	0	0
$39$	10.7183	1.71631
$40$	0	0
$41$	−0.480973	−0.0751154	−0.0375577	−	0.999294i	$-0.511958\pi$
$41$	−0.480973	−0.0751154	−0.0375577	+	0.999294i	$0.511958\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.96195i	1.01550i	0.861503	+	0.507752i	$0.169523\pi$
$47$	6.96195i	1.01550i	−0.861503	+	0.507752i	$0.830477\pi$
$48$	0	0
$49$	2.67726	0.382466
$50$	0	0
$51$	19.9557	2.79435
$52$	0	0
$53$	− 11.7183i	− 1.60964i	−0.593521	−	0.804818i	$-0.702263\pi$
$53$	− 11.7183i	− 1.60964i	0.593521	−	0.804818i	$-0.297737\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 23.0348i	− 3.05103i
$58$	0	0
$59$	11.5601	1.50500	0.752498	−	0.658595i	$-0.228849\pi$
$59$	11.5601	1.50500	0.752498	+	0.658595i	$0.228849\pi$
$60$	0	0
$61$	7.88283	1.00929	0.504646	−	0.863326i	$-0.331623\pi$
$61$	7.88283	1.00929	0.504646	+	0.863326i	$0.331623\pi$
$62$	0	0
$63$	13.4747i	1.69765i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.71833i	− 1.18728i	−0.804730	−	0.593641i	$-0.797690\pi$
$67$	− 9.71833i	− 1.18728i	0.804730	−	0.593641i	$-0.202310\pi$
$68$	0	0
$69$	−3.07912	−0.370682
$70$	0	0
$71$	−9.67726	−1.14848	−0.574240	−	0.818687i	$-0.694702\pi$
$71$	−9.67726	−1.14848	−0.574240	+	0.818687i	$0.694702\pi$
$72$	0	0
$73$	13.2784i	1.55412i	0.629425	+	0.777061i	$0.283290\pi$
$73$	13.2784i	1.55412i	−0.629425	+	0.777061i	$0.716710\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.5601i	− 1.20343i
$78$	0	0
$79$	12.3165	1.38571	0.692856	−	0.721076i	$-0.256352\pi$
$79$	12.3165	1.38571	0.692856	+	0.721076i	$0.256352\pi$
$80$	0	0
$81$	13.5601	1.50668
$82$	0	0
$83$	4.59815i	0.504712i	0.967634	+	0.252356i	$0.0812054\pi$
$83$	4.59815i	0.504712i	−0.967634	+	0.252356i	$0.918795\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.80371i	− 0.515012i
$88$	0	0
$89$	−8.31648	−0.881545	−0.440772	−	0.897619i	$-0.645295\pi$
$89$	−8.31648	−0.881545	−0.440772	+	0.897619i	$0.645295\pi$
$90$	0	0
$91$	7.23736	0.758681
$92$	0	0
$93$	− 0.243616i	− 0.0252618i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.23736i	0.734842i	0.930055	+	0.367421i	$0.119759\pi$
$97$	7.23736i	0.734842i	−0.930055	+	0.367421i	$0.880241\pi$
$98$	0	0
$99$	−32.9176	−3.30835
$100$	0	0
$101$	2.43991	0.242780	0.121390	−	0.992605i	$-0.461265\pi$
$101$	2.43991	0.242780	0.121390	+	0.992605i	$0.461265\pi$
$102$	0	0
$103$	− 7.88283i	− 0.776718i	−0.921508	−	0.388359i	$-0.873042\pi$
$103$	− 7.88283i	− 0.776718i	0.921508	−	0.388359i	$-0.126958\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.6803i	− 1.61254i	−0.591546	−	0.806272i	$-0.701482\pi$
$107$	− 16.6803i	− 1.61254i	0.591546	−	0.806272i	$-0.298518\pi$
$108$	0	0
$109$	−5.32274	−0.509826	−0.254913	−	0.966964i	$-0.582047\pi$
$109$	−5.32274	−0.509826	−0.254913	+	0.966964i	$0.582047\pi$
$110$	0	0
$111$	−29.9239	−2.84025
$112$	0	0
$113$	2.20556i	0.207482i	0.994604	+	0.103741i	$0.0330813\pi$
$113$	2.20556i	0.207482i	−0.994604	+	0.103741i	$0.966919\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 22.5601i	− 2.08568i
$118$	0	0
$119$	13.4747	1.23522
$120$	0	0
$121$	14.7974	1.34522
$122$	0	0
$123$	1.48097i	0.133535i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.15824i	0.546455i	0.961949	+	0.273228i	$0.0880912\pi$
$127$	6.15824i	0.546455i	−0.961949	+	0.273228i	$0.911909\pi$
$128$	0	0
$129$	−24.6330	−2.16881
$130$	0	0
$131$	−7.19629	−0.628743	−0.314371	−	0.949300i	$-0.601794\pi$
$131$	−7.19629	−0.628743	−0.314371	+	0.949300i	$0.601794\pi$
$132$	0	0
$133$	− 15.5538i	− 1.34869i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.6012i	1.07659i	0.842757	+	0.538295i	$0.180931\pi$
$137$	12.6012i	1.07659i	−0.842757	+	0.538295i	$0.819069\pi$
$138$	0	0
$139$	8.75638	0.742707	0.371353	−	0.928492i	$-0.378894\pi$
$139$	8.75638	0.742707	0.371353	+	0.928492i	$0.378894\pi$
$140$	0	0
$141$	21.4367	1.80529
$142$	0	0
$143$	17.6803i	1.47850i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 8.24362i	− 0.679922i
$148$	0	0
$149$	−1.79745	−0.147253	−0.0736264	−	0.997286i	$-0.523457\pi$
$149$	−1.79745	−0.147253	−0.0736264	+	0.997286i	$0.523457\pi$
$150$	0	0
$151$	−19.3956	−1.57839	−0.789196	−	0.614142i	$-0.789502\pi$
$151$	−19.3956	−1.57839	−0.789196	+	0.614142i	$0.789502\pi$
$152$	0	0
$153$	− 42.0030i	− 3.39574i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.36380i	0.348269i	0.984722	+	0.174135i	$0.0557128\pi$
$157$	4.36380i	0.348269i	−0.984722	+	0.174135i	$0.944287\pi$
$158$	0	0
$159$	−36.0821	−2.86150
$160$	0	0
$161$	−2.07912	−0.163858
$162$	0	0
$163$	− 5.32274i	− 0.416909i	−0.978032	−	0.208454i	$-0.933157\pi$
$163$	− 5.32274i	− 0.416909i	0.978032	−	0.208454i	$-0.0668433\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	0.882827	0.0679098
$170$	0	0
$171$	−48.4840	−3.70766
$172$	0	0
$173$	9.23736i	0.702303i	0.936319	+	0.351152i	$0.114210\pi$
$173$	9.23736i	0.702303i	−0.936319	+	0.351152i	$0.885790\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 35.5949i	− 2.67548i
$178$	0	0
$179$	−2.96195	−0.221386	−0.110693	−	0.993855i	$-0.535307\pi$
$179$	−2.96195	−0.221386	−0.110693	+	0.993855i	$0.535307\pi$
$180$	0	0
$181$	10.8355	0.805397	0.402698	−	0.915333i	$-0.368072\pi$
$181$	10.8355	0.805397	0.402698	+	0.915333i	$0.368072\pi$
$182$	0	0
$183$	− 24.2722i	− 1.79425i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	32.9176i	2.40718i
$188$	0	0
$189$	22.2847	1.62097
$190$	0	0
$191$	7.76565	0.561903	0.280952	−	0.959722i	$-0.409350\pi$
$191$	7.76565	0.561903	0.280952	+	0.959722i	$0.409350\pi$
$192$	0	0
$193$	2.15824i	0.155353i	0.996979	+	0.0776767i	$0.0247502\pi$
$193$	2.15824i	0.155353i	−0.996979	+	0.0776767i	$0.975250\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0411i	0.857890i	0.903331	+	0.428945i	$0.141115\pi$
$197$	12.0411i	0.857890i	−0.903331	+	0.428945i	$0.858885\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−29.9239	−2.11067
$202$	0	0
$203$	− 3.24362i	− 0.227657i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.48097i	0.450459i
$208$	0	0
$209$	37.9968	2.62829
$210$	0	0
$211$	3.40185	0.234193	0.117097	−	0.993121i	$-0.462641\pi$
$211$	3.40185	0.234193	0.117097	+	0.993121i	$0.462641\pi$
$212$	0	0
$213$	29.7974i	2.04169i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.164498i	− 0.0111668i
$218$	0	0
$219$	40.8858	2.76281
$220$	0	0
$221$	−22.5601	−1.51756
$222$	0	0
$223$	− 5.19629i	− 0.347969i	−0.984748	−	0.173985i	$-0.944336\pi$
$223$	− 5.19629i	− 0.347969i	0.984748	−	0.173985i	$-0.0556643\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.35453i	0.355393i	0.984085	+	0.177696i	$0.0568645\pi$
$227$	5.35453i	0.355393i	−0.984085	+	0.177696i	$0.943136\pi$
$228$	0	0
$229$	0.803708	0.0531105	0.0265553	−	0.999647i	$-0.491546\pi$
$229$	0.803708	0.0531105	0.0265553	+	0.999647i	$0.491546\pi$
$230$	0	0
$231$	−32.5158	−2.13938
$232$	0	0
$233$	− 14.1582i	− 0.927537i	−0.885956	−	0.463768i	$-0.846497\pi$
$233$	− 14.1582i	− 0.927537i	0.885956	−	0.463768i	$-0.153503\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 37.9239i	− 2.46342i
$238$	0	0
$239$	−10.9146	−0.706008	−0.353004	−	0.935622i	$-0.614840\pi$
$239$	−10.9146	−0.706008	−0.353004	+	0.935622i	$0.614840\pi$
$240$	0	0
$241$	−27.4367	−1.76735	−0.883675	−	0.468100i	$-0.844939\pi$
$241$	−27.4367	−1.76735	−0.883675	+	0.468100i	$0.844939\pi$
$242$	0	0
$243$	− 9.59815i	− 0.615721i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.0411i	1.65695i
$248$	0	0
$249$	14.1582	0.897242
$250$	0	0
$251$	−16.5190	−1.04267	−0.521336	−	0.853352i	$-0.674566\pi$
$251$	−16.5190	−1.04267	−0.521336	+	0.853352i	$0.674566\pi$
$252$	0	0
$253$	− 5.07912i	− 0.319321i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.7657i	1.23295i	0.787375	+	0.616474i	$0.211439\pi$
$257$	19.7657i	1.23295i	−0.787375	+	0.616474i	$0.788561\pi$
$258$	0	0
$259$	−20.2056	−1.25551
$260$	0	0
$261$	−10.1109	−0.625850
$262$	0	0
$263$	− 7.44292i	− 0.458950i	−0.973315	−	0.229475i	$-0.926299\pi$
$263$	− 7.44292i	− 0.458950i	0.973315	−	0.229475i	$-0.0737009\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.6074i	1.56715i
$268$	0	0
$269$	22.7564	1.38748	0.693741	−	0.720225i	$-0.255961\pi$
$269$	22.7564	1.38748	0.693741	+	0.720225i	$0.255961\pi$
$270$	0	0
$271$	−19.0411	−1.15666	−0.578331	−	0.815802i	$-0.696296\pi$
$271$	−19.0411	−1.15666	−0.578331	+	0.815802i	$0.696296\pi$
$272$	0	0
$273$	− 22.2847i	− 1.34873i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 18.3165i	− 1.10053i	−0.834990	−	0.550265i	$-0.814527\pi$
$277$	− 18.3165i	− 1.10053i	0.834990	−	0.550265i	$-0.185473\pi$
$278$	0	0
$279$	−0.512767	−0.0306986
$280$	0	0
$281$	26.0821	1.55593	0.777965	−	0.628308i	$-0.216252\pi$
$281$	26.0821	1.55593	0.777965	+	0.628308i	$0.216252\pi$
$282$	0	0
$283$	25.6422i	1.52427i	0.647417	+	0.762136i	$0.275849\pi$
$283$	25.6422i	1.52427i	−0.647417	+	0.762136i	$0.724151\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00000i	0.0590281i
$288$	0	0
$289$	−25.0030	−1.47077
$290$	0	0
$291$	22.2847	1.30635
$292$	0	0
$293$	− 30.8385i	− 1.80161i	−0.434229	−	0.900803i	$-0.642979\pi$
$293$	− 30.8385i	− 1.80161i	0.434229	−	0.900803i	$-0.357021\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	54.4397i	3.15891i
$298$	0	0
$299$	3.48097	0.201310
$300$	0	0
$301$	−16.6330	−0.958707
$302$	0	0
$303$	− 7.51277i	− 0.431597i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.88283i	− 0.107459i	−0.998556	−	0.0537293i	$-0.982889\pi$
$307$	− 1.88283i	− 0.107459i	0.998556	−	0.0537293i	$-0.0171108\pi$
$308$	0	0
$309$	−24.2722	−1.38080
$310$	0	0
$311$	1.60742	0.0911482	0.0455741	−	0.998961i	$-0.485488\pi$
$311$	1.60742	0.0911482	0.0455741	+	0.998961i	$0.485488\pi$
$312$	0	0
$313$	1.68654i	0.0953286i	0.998863	+	0.0476643i	$0.0151778\pi$
$313$	1.68654i	0.0953286i	−0.998863	+	0.0476643i	$0.984822\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.6773i	− 0.712026i	−0.934481	−	0.356013i	$-0.884136\pi$
$317$	− 12.6773i	− 0.712026i	0.934481	−	0.356013i	$-0.115864\pi$
$318$	0	0
$319$	7.92389	0.443653
$320$	0	0
$321$	−51.3606	−2.86667
$322$	0	0
$323$	48.4840i	2.69772i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.3893i	0.906332i
$328$	0	0
$329$	14.4747	0.798017
$330$	0	0
$331$	33.3893	1.83524	0.917622	−	0.397454i	$-0.130106\pi$
$331$	33.3893	1.83524	0.917622	+	0.397454i	$0.130106\pi$
$332$	0	0
$333$	62.9842i	3.45151i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 17.6392i	− 0.960869i	−0.877031	−	0.480435i	$-0.840479\pi$
$337$	− 17.6392i	− 0.960869i	0.877031	−	0.480435i	$-0.159521\pi$
$338$	0	0
$339$	6.79119	0.368847
$340$	0	0
$341$	0.401854	0.0217616
$342$	0	0
$343$	− 20.1202i	− 1.08639i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.7121i	1.05820i	0.848560	+	0.529100i	$0.177470\pi$
$347$	19.7121i	1.05820i	−0.848560	+	0.529100i	$0.822530\pi$
$348$	0	0
$349$	16.0348	0.858323	0.429162	−	0.903228i	$-0.358809\pi$
$349$	16.0348	0.858323	0.429162	+	0.903228i	$0.358809\pi$
$350$	0	0
$351$	−37.3102	−1.99147
$352$	0	0
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 41.4902i	− 2.19590i
$358$	0	0
$359$	13.1202	0.692457	0.346228	−	0.938150i	$-0.387462\pi$
$359$	13.1202	0.692457	0.346228	+	0.938150i	$0.387462\pi$
$360$	0	0
$361$	36.9650	1.94552
$362$	0	0
$363$	− 45.5631i	− 2.39144i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.47796i	0.181548i	0.995872	+	0.0907741i	$0.0289341\pi$
$367$	3.47796i	0.181548i	−0.995872	+	0.0907741i	$0.971066\pi$
$368$	0	0
$369$	3.11717	0.162274
$370$	0	0
$371$	−24.3638	−1.26491
$372$	0	0
$373$	− 21.5949i	− 1.11814i	−0.829120	−	0.559071i	$-0.811158\pi$
$373$	− 21.5949i	− 1.11814i	0.829120	−	0.559071i	$-0.188842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.43064i	0.279692i
$378$	0	0
$379$	−7.80070	−0.400695	−0.200347	−	0.979725i	$-0.564207\pi$
$379$	−7.80070	−0.400695	−0.200347	+	0.979725i	$0.564207\pi$
$380$	0	0
$381$	18.9619	0.971450
$382$	0	0
$383$	− 32.4274i	− 1.65696i	−0.560017	−	0.828481i	$-0.689205\pi$
$383$	− 32.4274i	− 1.65696i	0.560017	−	0.828481i	$-0.310795\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	51.8478i	2.63557i
$388$	0	0
$389$	−23.5538	−1.19423	−0.597113	−	0.802157i	$-0.703686\pi$
$389$	−23.5538	−1.19423	−0.597113	+	0.802157i	$0.703686\pi$
$390$	0	0
$391$	6.48097	0.327757
$392$	0	0
$393$	22.1582i	1.11774i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.370060i	− 0.0185728i	−0.999957	−	0.00928639i	$-0.997044\pi$
$397$	− 0.370060i	− 0.0185728i	0.999957	−	0.00928639i	$-0.00295599\pi$
$398$	0	0
$399$	−47.8921	−2.39760
$400$	0	0
$401$	11.3545	0.567018	0.283509	−	0.958970i	$-0.408501\pi$
$401$	11.3545	0.567018	0.283509	+	0.958970i	$0.408501\pi$
$402$	0	0
$403$	0.275411i	0.0137192i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 49.3606i	− 2.44671i
$408$	0	0
$409$	−4.88283	−0.241440	−0.120720	−	0.992687i	$-0.538520\pi$
$409$	−4.88283	−0.241440	−0.120720	+	0.992687i	$0.538520\pi$
$410$	0	0
$411$	38.8005	1.91389
$412$	0	0
$413$	− 24.0348i	− 1.18268i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 26.9619i	− 1.32033i
$418$	0	0
$419$	−1.51277	−0.0739035	−0.0369518	−	0.999317i	$-0.511765\pi$
$419$	−1.51277	−0.0739035	−0.0369518	+	0.999317i	$0.511765\pi$
$420$	0	0
$421$	22.2722	1.08548	0.542739	−	0.839901i	$-0.317387\pi$
$421$	22.2722	1.08548	0.542739	+	0.839901i	$0.317387\pi$
$422$	0	0
$423$	− 45.1202i	− 2.19382i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16.3893i	− 0.793135i
$428$	0	0
$429$	54.4397	2.62837
$430$	0	0
$431$	−16.8858	−0.813362	−0.406681	−	0.913570i	$-0.633314\pi$
$431$	−16.8858	−0.813362	−0.406681	+	0.913570i	$0.633314\pi$
$432$	0	0
$433$	− 24.3956i	− 1.17238i	−0.810175	−	0.586189i	$-0.800628\pi$
$433$	− 24.3956i	− 1.17238i	0.810175	−	0.586189i	$-0.199372\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.48097i	− 0.357863i
$438$	0	0
$439$	−40.0378	−1.91090	−0.955450	−	0.295152i	$-0.904630\pi$
$439$	−40.0378	−1.91090	−0.955450	+	0.295152i	$0.904630\pi$
$440$	0	0
$441$	−17.3513	−0.826251
$442$	0	0
$443$	5.22809i	0.248394i	0.992258	+	0.124197i	$0.0396355\pi$
$443$	5.22809i	0.248394i	−0.992258	+	0.124197i	$0.960365\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.53456i	0.261776i
$448$	0	0
$449$	−4.72459	−0.222967	−0.111484	−	0.993766i	$-0.535560\pi$
$449$	−4.72459	−0.222967	−0.111484	+	0.993766i	$0.535560\pi$
$450$	0	0
$451$	−2.44292	−0.115033
$452$	0	0
$453$	59.7213i	2.80595i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.2311i	1.55449i	0.629201	+	0.777243i	$0.283382\pi$
$457$	33.2311i	1.55449i	−0.629201	+	0.777243i	$0.716618\pi$
$458$	0	0
$459$	−69.4652	−3.24236
$460$	0	0
$461$	−28.9619	−1.34889	−0.674446	−	0.738324i	$-0.735618\pi$
$461$	−28.9619	−1.34889	−0.674446	+	0.738324i	$0.735618\pi$
$462$	0	0
$463$	− 4.39258i	− 0.204141i	−0.994777	−	0.102070i	$-0.967453\pi$
$463$	− 4.39258i	− 0.204141i	0.994777	−	0.102070i	$-0.0325467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.0729i	0.790038i	0.918673	+	0.395019i	$0.129262\pi$
$467$	17.0729i	0.790038i	−0.918673	+	0.395019i	$0.870738\pi$
$468$	0	0
$469$	−20.2056	−0.933006
$470$	0	0
$471$	13.4367	0.619129
$472$	0	0
$473$	− 40.6330i	− 1.86831i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	75.9462i	3.47734i
$478$	0	0
$479$	−27.0441	−1.23568	−0.617838	−	0.786306i	$-0.711991\pi$
$479$	−27.0441	−1.23568	−0.617838	+	0.786306i	$0.711991\pi$
$480$	0	0
$481$	33.8292	1.54248
$482$	0	0
$483$	6.40185i	0.291294i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 28.0821i	− 1.27252i	−0.771474	−	0.636261i	$-0.780480\pi$
$487$	− 28.0821i	− 1.27252i	0.771474	−	0.636261i	$-0.219520\pi$
$488$	0	0
$489$	−16.3893	−0.741151
$490$	0	0
$491$	−7.48398	−0.337747	−0.168874	−	0.985638i	$-0.554013\pi$
$491$	−7.48398	−0.337747	−0.168874	+	0.985638i	$0.554013\pi$
$492$	0	0
$493$	10.1109i	0.455373i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.1202i	0.902514i
$498$	0	0
$499$	−21.7183	−0.972246	−0.486123	−	0.873890i	$-0.661589\pi$
$499$	−21.7183	−0.972246	−0.486123	+	0.873890i	$0.661589\pi$
$500$	0	0
$501$	−24.6330	−1.10052
$502$	0	0
$503$	7.27541i	0.324395i	0.986758	+	0.162197i	$0.0518581\pi$
$503$	7.27541i	0.324395i	−0.986758	+	0.162197i	$0.948142\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.71833i	− 0.120725i
$508$	0	0
$509$	−22.7151	−1.00683	−0.503414	−	0.864045i	$-0.667923\pi$
$509$	−22.7151	−1.00683	−0.503414	+	0.864045i	$0.667923\pi$
$510$	0	0
$511$	27.6074	1.22128
$512$	0	0
$513$	80.1835i	3.54019i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	35.3606i	1.55516i
$518$	0	0
$519$	28.4429	1.24851
$520$	0	0
$521$	−16.5568	−0.725368	−0.362684	−	0.931912i	$-0.618140\pi$
$521$	−16.5568	−0.725368	−0.362684	+	0.931912i	$0.618140\pi$
$522$	0	0
$523$	− 24.8037i	− 1.08459i	−0.840188	−	0.542295i	$-0.817555\pi$
$523$	− 24.8037i	− 1.08459i	0.840188	−	0.542295i	$-0.182445\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.512767i	0.0223365i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−74.9206	−3.25128
$532$	0	0
$533$	− 1.67425i	− 0.0725200i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.12018i	0.393565i
$538$	0	0
$539$	13.5981	0.585714
$540$	0	0
$541$	−21.8292	−0.938512	−0.469256	−	0.883062i	$-0.655478\pi$
$541$	−21.8292	−0.938512	−0.469256	+	0.883062i	$0.655478\pi$
$542$	0	0
$543$	− 33.3638i	− 1.43178i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 29.0759i	− 1.24319i	−0.783337	−	0.621597i	$-0.786484\pi$
$547$	− 29.0759i	− 1.24319i	0.783337	−	0.621597i	$-0.213516\pi$
$548$	0	0
$549$	−51.0884	−2.18040
$550$	0	0
$551$	11.6710	0.497202
$552$	0	0
$553$	− 25.6074i	− 1.08894i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.0348i	− 0.933645i	−0.884351	−	0.466822i	$-0.845399\pi$
$557$	− 22.0348i	− 0.933645i	0.884351	−	0.466822i	$-0.154601\pi$
$558$	0	0
$559$	27.8478	1.17784
$560$	0	0
$561$	101.357	4.27931
$562$	0	0
$563$	− 1.40185i	− 0.0590811i	−0.999564	−	0.0295406i	$-0.990596\pi$
$563$	− 1.40185i	− 0.0590811i	0.999564	−	0.0295406i	$-0.00940442\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 28.1930i	− 1.18400i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−18.3575	−0.768239	−0.384120	−	0.923283i	$-0.625495\pi$
$571$	−18.3575	−0.768239	−0.384120	+	0.923283i	$0.625495\pi$
$572$	0	0
$573$	− 23.9114i	− 0.998912i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.5128i	− 0.479283i	−0.970861	−	0.239641i	$-0.922970\pi$
$577$	− 11.5128i	− 0.479283i	0.970861	−	0.239641i	$-0.0770299\pi$
$578$	0	0
$579$	6.64547	0.276176
$580$	0	0
$581$	9.56009	0.396619
$582$	0	0
$583$	− 59.5188i	− 2.46502i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.1232i	1.65606i	0.560683	+	0.828031i	$0.310539\pi$
$587$	40.1232i	1.65606i	−0.560683	+	0.828031i	$0.689461\pi$
$588$	0	0
$589$	0.591886	0.0243882
$590$	0	0
$591$	37.0759	1.52510
$592$	0	0
$593$	− 18.7912i	− 0.771662i	−0.922570	−	0.385831i	$-0.873915\pi$
$593$	− 18.7912i	− 0.771662i	0.922570	−	0.385831i	$-0.126085\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 43.1077i	− 1.76428i
$598$	0	0
$599$	27.4777	1.12271	0.561355	−	0.827575i	$-0.310280\pi$
$599$	27.4777	1.12271	0.561355	+	0.827575i	$0.310280\pi$
$600$	0	0
$601$	−19.5283	−0.796576	−0.398288	−	0.917260i	$-0.630396\pi$
$601$	−19.5283	−0.796576	−0.398288	+	0.917260i	$0.630396\pi$
$602$	0	0
$603$	62.9842i	2.56492i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	45.9239i	1.86399i	0.362467	+	0.931997i	$0.381935\pi$
$607$	45.9239i	1.86399i	−0.362467	+	0.931997i	$0.618065\pi$
$608$	0	0
$609$	−9.98748	−0.404713
$610$	0	0
$611$	−24.2343	−0.980417
$612$	0	0
$613$	− 12.7091i	− 0.513314i	−0.966503	−	0.256657i	$-0.917379\pi$
$613$	− 12.7091i	− 0.513314i	0.966503	−	0.256657i	$-0.0826211\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.3102i	1.94490i	0.233120	+	0.972448i	$0.425107\pi$
$617$	48.3102i	1.94490i	−0.233120	+	0.972448i	$0.574893\pi$
$618$	0	0
$619$	−0.677265	−0.0272216	−0.0136108	−	0.999907i	$-0.504333\pi$
$619$	−0.677265	−0.0272216	−0.0136108	+	0.999907i	$0.504333\pi$
$620$	0	0
$621$	10.7183	0.430112
$622$	0	0
$623$	17.2909i	0.692747i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 116.997i	− 4.67239i
$628$	0	0
$629$	62.9842	2.51135
$630$	0	0
$631$	44.3986	1.76748	0.883740	−	0.467978i	$-0.155017\pi$
$631$	44.3986	1.76748	0.883740	+	0.467978i	$0.155017\pi$
$632$	0	0
$633$	− 10.4747i	− 0.416332i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.31949i	0.369251i
$638$	0	0
$639$	62.7181	2.48109
$640$	0	0
$641$	16.8798	0.666713	0.333356	−	0.942801i	$-0.391819\pi$
$641$	16.8798	0.666713	0.333356	+	0.942801i	$0.391819\pi$
$642$	0	0
$643$	26.7564i	1.05517i	0.849503	+	0.527584i	$0.176902\pi$
$643$	26.7564i	1.05517i	−0.849503	+	0.527584i	$0.823098\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.5568i	1.27994i	0.768399	+	0.639971i	$0.221054\pi$
$647$	32.5568i	1.27994i	−0.768399	+	0.639971i	$0.778946\pi$
$648$	0	0
$649$	58.7151	2.30477
$650$	0	0
$651$	−0.506507	−0.0198516
$652$	0	0
$653$	9.87356i	0.386382i	0.981161	+	0.193191i	$0.0618837\pi$
$653$	9.87356i	0.386382i	−0.981161	+	0.193191i	$0.938116\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 86.0571i	− 3.35741i
$658$	0	0
$659$	13.5128	0.526383	0.263191	−	0.964744i	$-0.415225\pi$
$659$	13.5128	0.526383	0.263191	+	0.964744i	$0.415225\pi$
$660$	0	0
$661$	−26.7687	−1.04118	−0.520590	−	0.853807i	$-0.674288\pi$
$661$	−26.7687	−1.04118	−0.520590	+	0.853807i	$0.674288\pi$
$662$	0	0
$663$	69.4652i	2.69780i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.56009i	− 0.0604070i
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	40.0378	1.54564
$672$	0	0
$673$	0.803708i	0.0309807i	0.999880	+	0.0154903i	$0.00493092\pi$
$673$	0.803708i	0.0309807i	−0.999880	+	0.0154903i	$0.995069\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.6422i	− 1.29298i	−0.762924	−	0.646488i	$-0.776237\pi$
$677$	− 33.6422i	− 1.29298i	0.762924	−	0.646488i	$-0.223763\pi$
$678$	0	0
$679$	15.0473	0.577463
$680$	0	0
$681$	16.4872	0.631792
$682$	0	0
$683$	11.8736i	0.454329i	0.973856	+	0.227165i	$0.0729455\pi$
$683$	11.8736i	0.454329i	−0.973856	+	0.227165i	$0.927054\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.47471i	− 0.0944162i
$688$	0	0
$689$	40.7912	1.55402
$690$	0	0
$691$	27.9114	1.06180	0.530899	−	0.847435i	$-0.321854\pi$
$691$	27.9114	1.06180	0.530899	+	0.847435i	$0.321854\pi$
$692$	0	0
$693$	68.4397i	2.59981i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.11717i	− 0.118071i
$698$	0	0
$699$	−43.5949	−1.64891
$700$	0	0
$701$	−26.3668	−0.995861	−0.497930	−	0.867217i	$-0.665906\pi$
$701$	−26.3668	−0.995861	−0.497930	+	0.867217i	$0.665906\pi$
$702$	0	0
$703$	− 72.7026i	− 2.74203i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.07286i	− 0.190785i
$708$	0	0
$709$	−11.8007	−0.443184	−0.221592	−	0.975139i	$-0.571125\pi$
$709$	−11.8007	−0.443184	−0.221592	+	0.975139i	$0.571125\pi$
$710$	0	0
$711$	−79.8227	−2.99359
$712$	0	0
$713$	− 0.0791189i	− 0.00296302i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.6074i	1.25509i
$718$	0	0
$719$	12.9589	0.483287	0.241643	−	0.970365i	$-0.422314\pi$
$719$	12.9589	0.483287	0.241643	+	0.970365i	$0.422314\pi$
$720$	0	0
$721$	−16.3893	−0.610371
$722$	0	0
$723$	84.4807i	3.14187i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 19.0318i	− 0.705850i	−0.935652	−	0.352925i	$-0.885187\pi$
$727$	− 19.0318i	− 0.705850i	0.935652	−	0.352925i	$-0.114813\pi$
$728$	0	0
$729$	11.1264	0.412090
$730$	0	0
$731$	51.8478	1.91766
$732$	0	0
$733$	30.5220i	1.12736i	0.825994	+	0.563679i	$0.190614\pi$
$733$	30.5220i	1.12736i	−0.825994	+	0.563679i	$0.809386\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 49.3606i	− 1.81822i
$738$	0	0
$739$	6.18702	0.227593	0.113797	−	0.993504i	$-0.463699\pi$
$739$	6.18702	0.227593	0.113797	+	0.993504i	$0.463699\pi$
$740$	0	0
$741$	80.1835	2.94562
$742$	0	0
$743$	− 16.2722i	− 0.596968i	−0.954415	−	0.298484i	$-0.903519\pi$
$743$	− 16.2722i	− 0.596968i	0.954415	−	0.298484i	$-0.0964809\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 29.8005i	− 1.09034i
$748$	0	0
$749$	−34.6803	−1.26719
$750$	0	0
$751$	−17.5949	−0.642047	−0.321023	−	0.947071i	$-0.604027\pi$
$751$	−17.5949	−0.642047	−0.321023	+	0.947071i	$0.604027\pi$
$752$	0	0
$753$	50.8640i	1.85359i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.2496i	1.49924i	0.661866	+	0.749622i	$0.269765\pi$
$757$	41.2496i	1.49924i	−0.661866	+	0.749622i	$0.730235\pi$
$758$	0	0
$759$	−15.6392	−0.567667
$760$	0	0
$761$	26.8067	0.971743	0.485871	−	0.874030i	$-0.338502\pi$
$761$	26.8067	0.971743	0.485871	+	0.874030i	$0.338502\pi$
$762$	0	0
$763$	11.0666i	0.400638i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.2404i	1.45300i
$768$	0	0
$769$	26.8798	0.969311	0.484655	−	0.874705i	$-0.338945\pi$
$769$	26.8798	0.969311	0.484655	+	0.874705i	$0.338945\pi$
$770$	0	0
$771$	60.8608	2.19185
$772$	0	0
$773$	− 41.0441i	− 1.47625i	−0.674662	−	0.738126i	$-0.735711\pi$
$773$	− 41.0441i	− 1.47625i	0.674662	−	0.738126i	$-0.264289\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	62.2153i	2.23196i
$778$	0	0
$779$	−3.59815	−0.128917
$780$	0	0
$781$	−49.1520	−1.75880
$782$	0	0
$783$	16.7216i	0.597580i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 21.4840i	− 0.765821i	−0.923785	−	0.382911i	$-0.874922\pi$
$787$	− 21.4840i	− 0.765821i	0.923785	−	0.382911i	$-0.125078\pi$
$788$	0	0
$789$	−22.9176	−0.815889
$790$	0	0
$791$	4.58563	0.163046
$792$	0	0
$793$	27.4399i	0.974420i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.2692i	1.35556i	0.735263	+	0.677781i	$0.237058\pi$
$797$	38.2692i	1.35556i	−0.735263	+	0.677781i	$0.762942\pi$
$798$	0	0
$799$	−45.1202	−1.59624
$800$	0	0
$801$	53.8989	1.90442
$802$	0	0
$803$	67.4427i	2.38000i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 70.0696i	− 2.46657i
$808$	0	0
$809$	27.2026	0.956391	0.478195	−	0.878253i	$-0.341291\pi$
$809$	27.2026	0.956391	0.478195	+	0.878253i	$0.341291\pi$
$810$	0	0
$811$	38.5095	1.35225	0.676126	−	0.736786i	$-0.263657\pi$
$811$	38.5095	1.35225	0.676126	+	0.736786i	$0.263657\pi$
$812$	0	0
$813$	58.6297i	2.05623i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 59.8478i	− 2.09381i
$818$	0	0
$819$	−46.9051	−1.63900
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 40.1457i	− 1.39939i	−0.714441	−	0.699696i	$-0.753319\pi$
$823$	− 40.1457i	− 1.39939i	0.714441	−	0.699696i	$-0.246681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.851033i	0.0295933i	0.999891	+	0.0147967i	$0.00471009\pi$
$827$	0.851033i	0.0295933i	−0.999891	+	0.0147967i	$0.995290\pi$
$828$	0	0
$829$	2.83851	0.0985856	0.0492928	−	0.998784i	$-0.484303\pi$
$829$	2.83851	0.0985856	0.0492928	+	0.998784i	$0.484303\pi$
$830$	0	0
$831$	−56.3986	−1.95645
$832$	0	0
$833$	17.3513i	0.601186i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.848022i	0.0293119i
$838$	0	0
$839$	10.2343	0.353329	0.176664	−	0.984271i	$-0.443469\pi$
$839$	10.2343	0.353329	0.176664	+	0.984271i	$0.443469\pi$
$840$	0	0
$841$	−26.5661	−0.916073
$842$	0	0
$843$	− 80.3100i	− 2.76602i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 30.7657i	− 1.05712i
$848$	0	0
$849$	78.9554	2.70974
$850$	0	0
$851$	−9.71833	−0.333140
$852$	0	0
$853$	− 17.7213i	− 0.606767i	−0.952869	−	0.303384i	$-0.901884\pi$
$853$	− 17.7213i	− 0.606767i	0.952869	−	0.303384i	$-0.0981163\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.6835i	0.604058i	0.953299	+	0.302029i	$0.0976639\pi$
$857$	17.6835i	0.604058i	−0.953299	+	0.302029i	$0.902336\pi$
$858$	0	0
$859$	26.6042	0.907722	0.453861	−	0.891072i	$-0.350046\pi$
$859$	26.6042	0.907722	0.453861	+	0.891072i	$0.350046\pi$
$860$	0	0
$861$	3.07912	0.104936
$862$	0	0
$863$	53.5949i	1.82439i	0.409755	+	0.912196i	$0.365614\pi$
$863$	53.5949i	1.82439i	−0.409755	+	0.912196i	$0.634386\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	76.9872i	2.61462i
$868$	0	0
$869$	62.5568	2.12210
$870$	0	0
$871$	33.8292	1.14626
$872$	0	0
$873$	− 46.9051i	− 1.58750i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.9650i	0.876774i	0.898786	+	0.438387i	$0.144450\pi$
$877$	25.9650i	0.876774i	−0.898786	+	0.438387i	$0.855550\pi$
$878$	0	0
$879$	−94.9554	−3.20277
$880$	0	0
$881$	29.9239	1.00816	0.504081	−	0.863657i	$-0.331831\pi$
$881$	29.9239	1.00816	0.504081	+	0.863657i	$0.331831\pi$
$882$	0	0
$883$	4.75939i	0.160166i	0.996788	+	0.0800832i	$0.0255186\pi$
$883$	4.75939i	0.160166i	−0.996788	+	0.0800832i	$0.974481\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.8037i	1.30290i	0.758691	+	0.651451i	$0.225839\pi$
$887$	38.8037i	1.30290i	−0.758691	+	0.651451i	$0.774161\pi$
$888$	0	0
$889$	12.8037	0.429423
$890$	0	0
$891$	68.8733	2.30734
$892$	0	0
$893$	52.0821i	1.74286i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 10.7183i	− 0.357875i
$898$	0	0
$899$	0.123433	0.00411671
$900$	0	0
$901$	75.9462	2.53013
$902$	0	0
$903$	51.2148i	1.70432i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 51.0729i	− 1.69585i	−0.530119	−	0.847923i	$-0.677853\pi$
$907$	− 51.0729i	− 1.69585i	0.530119	−	0.847923i	$-0.322147\pi$
$908$	0	0
$909$	−15.8130	−0.524483
$910$	0	0
$911$	−30.4933	−1.01029	−0.505143	−	0.863035i	$-0.668560\pi$
$911$	−30.4933	−1.01029	−0.505143	+	0.863035i	$0.668560\pi$
$912$	0	0
$913$	23.3545i	0.772922i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.9619i	0.494087i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−5.79745	−0.191032
$922$	0	0
$923$	− 33.6863i	− 1.10880i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	51.0884i	1.67796i
$928$	0	0
$929$	17.3257	0.568439	0.284220	−	0.958759i	$-0.408266\pi$
$929$	17.3257	0.568439	0.284220	+	0.958759i	$0.408266\pi$
$930$	0	0
$931$	20.0285	0.656409
$932$	0	0
$933$	− 4.94943i	− 0.162037i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 47.5631i	− 1.55382i	−0.629612	−	0.776909i	$-0.716786\pi$
$937$	− 47.5631i	− 1.55382i	0.629612	−	0.776909i	$-0.283214\pi$
$938$	0	0
$939$	5.19304	0.169469
$940$	0	0
$941$	14.2754	0.465365	0.232683	−	0.972553i	$-0.425250\pi$
$941$	14.2754	0.465365	0.232683	+	0.972553i	$0.425250\pi$
$942$	0	0
$943$	0.480973i	0.0156626i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.1900i	1.17602i	0.808854	+	0.588009i	$0.200088\pi$
$947$	36.1900i	1.17602i	−0.808854	+	0.588009i	$0.799912\pi$
$948$	0	0
$949$	−46.2218	−1.50042
$950$	0	0
$951$	−39.0348	−1.26579
$952$	0	0
$953$	− 37.7942i	− 1.22427i	−0.790752	−	0.612137i	$-0.790310\pi$
$953$	− 37.7942i	− 1.22427i	0.790752	−	0.612137i	$-0.209690\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 24.3986i	− 0.788695i
$958$	0	0
$959$	26.1993	0.846020
$960$	0	0
$961$	−30.9937	−0.999798
$962$	0	0
$963$	108.104i	3.48362i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.6390i	1.62844i	0.580557	+	0.814220i	$0.302835\pi$
$967$	50.6390i	1.62844i	−0.580557	+	0.814220i	$0.697165\pi$
$968$	0	0
$969$	149.288	4.79582
$970$	0	0
$971$	2.82623	0.0906981	0.0453490	−	0.998971i	$-0.485560\pi$
$971$	2.82623	0.0906981	0.0453490	+	0.998971i	$0.485560\pi$
$972$	0	0
$973$	− 18.2056i	− 0.583644i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.8448i	0.826848i	0.910539	+	0.413424i	$0.135667\pi$
$977$	25.8448i	0.826848i	−0.910539	+	0.413424i	$0.864333\pi$
$978$	0	0
$979$	−42.2404	−1.35001
$980$	0	0
$981$	34.4965	1.10139
$982$	0	0
$983$	41.3668i	1.31940i	0.751531	+	0.659698i	$0.229316\pi$
$983$	41.3668i	1.31940i	−0.751531	+	0.659698i	$0.770684\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 44.5694i	− 1.41866i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	54.4745	1.73044	0.865219	−	0.501394i	$-0.167179\pi$
$991$	54.4745	1.73044	0.865219	+	0.501394i	$0.167179\pi$
$992$	0	0
$993$	− 102.810i	− 3.26257i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 15.5128i	− 0.491294i	−0.969359	−	0.245647i	$-0.921000\pi$
$997$	− 15.5128i	− 0.491294i	0.969359	−	0.245647i	$-0.0790004\pi$
$998$	0	0
$999$	104.164	3.29561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.p.4049.1		6
5.2	odd	4		4600.2.a.x.1.1		3
5.3	odd	4		920.2.a.h.1.3	✓	3
5.4	even	2	inner	4600.2.e.p.4049.6		6
15.8	even	4		8280.2.a.bj.1.1		3
20.3	even	4		1840.2.a.s.1.1		3
20.7	even	4		9200.2.a.ce.1.3		3
40.3	even	4		7360.2.a.cc.1.3		3
40.13	odd	4		7360.2.a.by.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.3	✓	3	5.3	odd	4
1840.2.a.s.1.1		3	20.3	even	4
4600.2.a.x.1.1		3	5.2	odd	4
4600.2.e.p.4049.1		6	1.1	even	1	trivial
4600.2.e.p.4049.6		6	5.4	even	2	inner
7360.2.a.by.1.1		3	40.13	odd	4
7360.2.a.cc.1.3		3	40.3	even	4
8280.2.a.bj.1.1		3	15.8	even	4
9200.2.a.ce.1.3		3	20.7	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)

\(f(q)\)	\(=\)	\(q-3.07912i q^{3} -2.07912i q^{7} -6.48097 q^{9} +O(q^{10})\)	\(q-3.07912i q^{3} -2.07912i q^{7} -6.48097 q^{9} +5.07912 q^{11} +3.48097i q^{13} +6.48097i q^{17} +7.48097 q^{19} -6.40185 q^{21} -1.00000i q^{23} +10.7183i q^{27} +1.56009 q^{29} +0.0791189 q^{31} -15.6392i q^{33} -9.71833i q^{37} +10.7183 q^{39} -0.480973 q^{41} -8.00000i q^{43} +6.96195i q^{47} +2.67726 q^{49} +19.9557 q^{51} -11.7183i q^{53} -23.0348i q^{57} +11.5601 q^{59} +7.88283 q^{61} +13.4747i q^{63} -9.71833i q^{67} -3.07912 q^{69} -9.67726 q^{71} +13.2784i q^{73} -10.5601i q^{77} +12.3165 q^{79} +13.5601 q^{81} +4.59815i q^{83} -4.80371i q^{87} -8.31648 q^{89} +7.23736 q^{91} -0.243616i q^{93} +7.23736i q^{97} -32.9176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 20 q^{9}+O(q^{10}) \) 6 * q - 20 * q^9	\( 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) \) 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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