LMFDB - Embedded newform 4600.2.e.m.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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
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			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.m.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{3} +5.12311i q^{7} -3.56155 q^{9} +O(q^{10})$	$q-2.56155i q^{3} +5.12311i q^{7} -3.56155 q^{9} -4.00000 q^{11} +0.561553i q^{13} -3.12311i q^{17} -4.00000 q^{19} +13.1231 q^{21} -1.00000i q^{23} +1.43845i q^{27} +8.56155 q^{29} +1.43845 q^{31} +10.2462i q^{33} -7.12311i q^{37} +1.43845 q^{39} +0.561553 q^{41} +9.12311i q^{43} -3.68466i q^{47} -19.2462 q^{49} -8.00000 q^{51} +4.24621i q^{53} +10.2462i q^{57} +6.24621 q^{59} +11.1231 q^{61} -18.2462i q^{63} +6.24621i q^{67} -2.56155 q^{69} +3.68466 q^{71} -16.5616i q^{73} -20.4924i q^{77} +10.2462 q^{79} -7.00000 q^{81} -12.0000i q^{83} -21.9309i q^{87} -10.0000 q^{89} -2.87689 q^{91} -3.68466i q^{93} -16.2462i q^{97} +14.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 16 * q^11 - 16 * q^19 + 36 * q^21 + 26 * q^29 + 14 * q^31 + 14 * q^39 - 6 * q^41 - 44 * q^49 - 32 * q^51 - 8 * q^59 + 28 * q^61 - 2 * q^69 - 10 * q^71 + 8 * q^79 - 28 * q^81 - 40 * q^89 - 28 * q^91 + 24 * 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
$3$	− 2.56155i	− 1.47891i	−0.673204	−	0.739457i	$-0.735083\pi$
$3$	− 2.56155i	− 1.47891i	0.673204	−	0.739457i	$-0.264917\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.12311i	1.93635i	0.250270	+	0.968176i	$0.419480\pi$
$7$	5.12311i	1.93635i	−0.250270	+	0.968176i	$0.580520\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	0.561553i	0.155747i	0.996963	+	0.0778734i	$0.0248130\pi$
$13$	0.561553i	0.155747i	−0.996963	+	0.0778734i	$0.975187\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.12311i	− 0.757464i	−0.925506	−	0.378732i	$-0.876360\pi$
$17$	− 3.12311i	− 0.757464i	0.925506	−	0.378732i	$-0.123640\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	13.1231	2.86370
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.43845i	0.276829i
$28$	0	0
$29$	8.56155	1.58984	0.794920	−	0.606714i	$-0.207513\pi$
$29$	8.56155	1.58984	0.794920	+	0.606714i	$0.207513\pi$
$30$	0	0
$31$	1.43845	0.258353	0.129176	−	0.991622i	$-0.458767\pi$
$31$	1.43845	0.258353	0.129176	+	0.991622i	$0.458767\pi$
$32$	0	0
$33$	10.2462i	1.78364i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.12311i	− 1.17103i	−0.810661	−	0.585516i	$-0.800892\pi$
$37$	− 7.12311i	− 1.17103i	0.810661	−	0.585516i	$-0.199108\pi$
$38$	0	0
$39$	1.43845	0.230336
$40$	0	0
$41$	0.561553	0.0876998	0.0438499	−	0.999038i	$-0.486038\pi$
$41$	0.561553	0.0876998	0.0438499	+	0.999038i	$0.486038\pi$
$42$	0	0
$43$	9.12311i	1.39126i	0.718400	+	0.695630i	$0.244875\pi$
$43$	9.12311i	1.39126i	−0.718400	+	0.695630i	$0.755125\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.68466i	− 0.537463i	−0.963215	−	0.268731i	$-0.913396\pi$
$47$	− 3.68466i	− 0.537463i	0.963215	−	0.268731i	$-0.0866044\pi$
$48$	0	0
$49$	−19.2462	−2.74946
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	4.24621i	0.583262i	0.956531	+	0.291631i	$0.0941979\pi$
$53$	4.24621i	0.583262i	−0.956531	+	0.291631i	$0.905802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.2462i	1.35714i
$58$	0	0
$59$	6.24621	0.813187	0.406594	−	0.913609i	$-0.366716\pi$
$59$	6.24621	0.813187	0.406594	+	0.913609i	$0.366716\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	0	0
$63$	− 18.2462i	− 2.29881i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.24621i	0.763096i	0.924349	+	0.381548i	$0.124609\pi$
$67$	6.24621i	0.763096i	−0.924349	+	0.381548i	$0.875391\pi$
$68$	0	0
$69$	−2.56155	−0.308375
$70$	0	0
$71$	3.68466	0.437289	0.218644	−	0.975805i	$-0.429837\pi$
$71$	3.68466	0.437289	0.218644	+	0.975805i	$0.429837\pi$
$72$	0	0
$73$	− 16.5616i	− 1.93838i	−0.246308	−	0.969192i	$-0.579218\pi$
$73$	− 16.5616i	− 1.93838i	0.246308	−	0.969192i	$-0.420782\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 20.4924i	− 2.33533i
$78$	0	0
$79$	10.2462	1.15279	0.576394	−	0.817172i	$-0.304459\pi$
$79$	10.2462	1.15279	0.576394	+	0.817172i	$0.304459\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 21.9309i	− 2.35124i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.87689	−0.301580
$92$	0	0
$93$	− 3.68466i	− 0.382081i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 16.2462i	− 1.64955i	−0.565459	−	0.824776i	$-0.691301\pi$
$97$	− 16.2462i	− 1.64955i	0.565459	−	0.824776i	$-0.308699\pi$
$98$	0	0
$99$	14.2462	1.43180
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	2.24621i	0.221326i	0.993858	+	0.110663i	$0.0352974\pi$
$103$	2.24621i	0.221326i	−0.993858	+	0.110663i	$0.964703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	−8.24621	−0.789844	−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621	−0.789844	−0.394922	+	0.918715i	$0.629228\pi$
$110$	0	0
$111$	−18.2462	−1.73185
$112$	0	0
$113$	− 20.2462i	− 1.90460i	−0.305158	−	0.952302i	$-0.598709\pi$
$113$	− 20.2462i	− 1.90460i	0.305158	−	0.952302i	$-0.401291\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 1.43845i	− 0.129700i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16.8078i	− 1.49145i	−0.666255	−	0.745724i	$-0.732104\pi$
$127$	− 16.8078i	− 1.49145i	0.666255	−	0.745724i	$-0.267896\pi$
$128$	0	0
$129$	23.3693	2.05755
$130$	0	0
$131$	9.93087	0.867664	0.433832	−	0.900994i	$-0.357161\pi$
$131$	9.93087	0.867664	0.433832	+	0.900994i	$0.357161\pi$
$132$	0	0
$133$	− 20.4924i	− 1.77692i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.1231i	1.29205i	0.763315	+	0.646027i	$0.223571\pi$
$137$	15.1231i	1.29205i	−0.763315	+	0.646027i	$0.776429\pi$
$138$	0	0
$139$	−0.315342	−0.0267469	−0.0133735	−	0.999911i	$-0.504257\pi$
$139$	−0.315342	−0.0267469	−0.0133735	+	0.999911i	$0.504257\pi$
$140$	0	0
$141$	−9.43845	−0.794861
$142$	0	0
$143$	− 2.24621i	− 0.187838i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	49.3002i	4.06621i
$148$	0	0
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	−16.8078	−1.36780	−0.683898	−	0.729577i	$-0.739717\pi$
$151$	−16.8078	−1.36780	−0.683898	+	0.729577i	$0.739717\pi$
$152$	0	0
$153$	11.1231i	0.899250i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	10.8769	0.862594
$160$	0	0
$161$	5.12311	0.403757
$162$	0	0
$163$	0.315342i	0.0246995i	0.999924	+	0.0123497i	$0.00393114\pi$
$163$	0.315342i	0.0246995i	−0.999924	+	0.0123497i	$0.996069\pi$
$164$	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
$169$	12.6847	0.975743
$170$	0	0
$171$	14.2462	1.08944
$172$	0	0
$173$	− 10.4924i	− 0.797724i	−0.917011	−	0.398862i	$-0.869405\pi$
$173$	− 10.4924i	− 0.797724i	0.917011	−	0.398862i	$-0.130595\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 16.0000i	− 1.20263i
$178$	0	0
$179$	7.68466	0.574378	0.287189	−	0.957874i	$-0.407279\pi$
$179$	7.68466	0.574378	0.287189	+	0.957874i	$0.407279\pi$
$180$	0	0
$181$	3.12311	0.232139	0.116069	−	0.993241i	$-0.462971\pi$
$181$	3.12311	0.232139	0.116069	+	0.993241i	$0.462971\pi$
$182$	0	0
$183$	− 28.4924i	− 2.10622i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.4924i	0.913536i
$188$	0	0
$189$	−7.36932	−0.536039
$190$	0	0
$191$	−2.87689	−0.208165	−0.104082	−	0.994569i	$-0.533191\pi$
$191$	−2.87689	−0.208165	−0.104082	+	0.994569i	$0.533191\pi$
$192$	0	0
$193$	− 24.5616i	− 1.76798i	−0.467507	−	0.883990i	$-0.654848\pi$
$193$	− 24.5616i	− 1.76798i	0.467507	−	0.883990i	$-0.345152\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.4384i	− 0.814956i	−0.913215	−	0.407478i	$-0.866408\pi$
$197$	− 11.4384i	− 0.814956i	0.913215	−	0.407478i	$-0.133592\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	43.8617i	3.07849i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.56155i	0.247545i
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−14.2462	−0.980750	−0.490375	−	0.871512i	$-0.663140\pi$
$211$	−14.2462	−0.980750	−0.490375	+	0.871512i	$0.663140\pi$
$212$	0	0
$213$	− 9.43845i	− 0.646712i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.36932i	0.500262i
$218$	0	0
$219$	−42.4233	−2.86670
$220$	0	0
$221$	1.75379	0.117973
$222$	0	0
$223$	20.4924i	1.37227i	0.727472	+	0.686137i	$0.240695\pi$
$223$	20.4924i	1.37227i	−0.727472	+	0.686137i	$0.759305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.12311i	0.605522i	0.953067	+	0.302761i	$0.0979084\pi$
$227$	9.12311i	0.605522i	−0.953067	+	0.302761i	$0.902092\pi$
$228$	0	0
$229$	−0.246211	−0.0162701	−0.00813505	−	0.999967i	$-0.502589\pi$
$229$	−0.246211	−0.0162701	−0.00813505	+	0.999967i	$0.502589\pi$
$230$	0	0
$231$	−52.4924	−3.45375
$232$	0	0
$233$	12.5616i	0.822935i	0.911425	+	0.411467i	$0.134984\pi$
$233$	12.5616i	0.822935i	−0.911425	+	0.411467i	$0.865016\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 26.2462i	− 1.70487i
$238$	0	0
$239$	8.80776	0.569727	0.284863	−	0.958568i	$-0.408052\pi$
$239$	8.80776	0.569727	0.284863	+	0.958568i	$0.408052\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	0	0
$243$	22.2462i	1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.24621i	− 0.142923i
$248$	0	0
$249$	−30.7386	−1.94798
$250$	0	0
$251$	6.24621	0.394257	0.197129	−	0.980378i	$-0.436838\pi$
$251$	6.24621	0.394257	0.197129	+	0.980378i	$0.436838\pi$
$252$	0	0
$253$	4.00000i	0.251478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.05398i	− 0.564771i	−0.959301	−	0.282386i	$-0.908874\pi$
$257$	− 9.05398i	− 0.564771i	0.959301	−	0.282386i	$-0.0911258\pi$
$258$	0	0
$259$	36.4924	2.26753
$260$	0	0
$261$	−30.4924	−1.88743
$262$	0	0
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.6155i	1.56764i
$268$	0	0
$269$	23.3002	1.42064	0.710319	−	0.703880i	$-0.248551\pi$
$269$	23.3002	1.42064	0.710319	+	0.703880i	$0.248551\pi$
$270$	0	0
$271$	−2.24621	−0.136448	−0.0682238	−	0.997670i	$-0.521733\pi$
$271$	−2.24621	−0.136448	−0.0682238	+	0.997670i	$0.521733\pi$
$272$	0	0
$273$	7.36932i	0.446011i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 11.4384i	− 0.687270i	−0.939103	−	0.343635i	$-0.888342\pi$
$277$	− 11.4384i	− 0.687270i	0.939103	−	0.343635i	$-0.111658\pi$
$278$	0	0
$279$	−5.12311	−0.306712
$280$	0	0
$281$	30.4924	1.81903	0.909513	−	0.415676i	$-0.136455\pi$
$281$	30.4924	1.81903	0.909513	+	0.415676i	$0.136455\pi$
$282$	0	0
$283$	− 22.8769i	− 1.35989i	−0.733263	−	0.679945i	$-0.762004\pi$
$283$	− 22.8769i	− 1.35989i	0.733263	−	0.679945i	$-0.237996\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.87689i	0.169818i
$288$	0	0
$289$	7.24621	0.426248
$290$	0	0
$291$	−41.6155	−2.43955
$292$	0	0
$293$	12.8769i	0.752276i	0.926564	+	0.376138i	$0.122748\pi$
$293$	12.8769i	0.752276i	−0.926564	+	0.376138i	$0.877252\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.75379i	− 0.333869i
$298$	0	0
$299$	0.561553	0.0324754
$300$	0	0
$301$	−46.7386	−2.69397
$302$	0	0
$303$	− 0.630683i	− 0.0362318i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	5.75379	0.327322
$310$	0	0
$311$	13.9309	0.789947	0.394974	−	0.918692i	$-0.370754\pi$
$311$	13.9309	0.789947	0.394974	+	0.918692i	$0.370754\pi$
$312$	0	0
$313$	− 15.1231i	− 0.854808i	−0.904061	−	0.427404i	$-0.859428\pi$
$313$	− 15.1231i	− 0.854808i	0.904061	−	0.427404i	$-0.140572\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.7386i	1.16480i	0.812903	+	0.582399i	$0.197886\pi$
$317$	20.7386i	1.16480i	−0.812903	+	0.582399i	$0.802114\pi$
$318$	0	0
$319$	−34.2462	−1.91742
$320$	0	0
$321$	−30.7386	−1.71566
$322$	0	0
$323$	12.4924i	0.695097i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.1231i	1.16811i
$328$	0	0
$329$	18.8769	1.04072
$330$	0	0
$331$	10.5616	0.580515	0.290258	−	0.956949i	$-0.406259\pi$
$331$	10.5616	0.580515	0.290258	+	0.956949i	$0.406259\pi$
$332$	0	0
$333$	25.3693i	1.39023i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.7538i	0.858164i	0.903266	+	0.429082i	$0.141163\pi$
$337$	15.7538i	0.858164i	−0.903266	+	0.429082i	$0.858837\pi$
$338$	0	0
$339$	−51.8617	−2.81674
$340$	0	0
$341$	−5.75379	−0.311585
$342$	0	0
$343$	− 62.7386i	− 3.38757i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.4924i	− 0.885360i	−0.896680	−	0.442680i	$-0.854028\pi$
$347$	− 16.4924i	− 0.885360i	0.896680	−	0.442680i	$-0.145972\pi$
$348$	0	0
$349$	2.80776	0.150296	0.0751481	−	0.997172i	$-0.476057\pi$
$349$	2.80776	0.150296	0.0751481	+	0.997172i	$0.476057\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	− 34.8078i	− 1.85263i	−0.376750	−	0.926315i	$-0.622958\pi$
$353$	− 34.8078i	− 1.85263i	0.376750	−	0.926315i	$-0.377042\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 40.9848i	− 2.16915i
$358$	0	0
$359$	13.7538	0.725897	0.362949	−	0.931809i	$-0.381770\pi$
$359$	13.7538	0.725897	0.362949	+	0.931809i	$0.381770\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 12.8078i	− 0.672233i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.4924i	1.06970i	0.844948	+	0.534848i	$0.179631\pi$
$367$	20.4924i	1.06970i	−0.844948	+	0.534848i	$0.820369\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−21.7538	−1.12940
$372$	0	0
$373$	24.7386i	1.28092i	0.767992	+	0.640459i	$0.221256\pi$
$373$	24.7386i	1.28092i	−0.767992	+	0.640459i	$0.778744\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.80776i	0.247612i
$378$	0	0
$379$	14.2462	0.731779	0.365889	−	0.930658i	$-0.380765\pi$
$379$	14.2462	0.731779	0.365889	+	0.930658i	$0.380765\pi$
$380$	0	0
$381$	−43.0540	−2.20572
$382$	0	0
$383$	− 33.6155i	− 1.71767i	−0.512250	−	0.858837i	$-0.671188\pi$
$383$	− 33.6155i	− 1.71767i	0.512250	−	0.858837i	$-0.328812\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 32.4924i	− 1.65168i
$388$	0	0
$389$	−7.61553	−0.386123	−0.193061	−	0.981187i	$-0.561842\pi$
$389$	−7.61553	−0.386123	−0.193061	+	0.981187i	$0.561842\pi$
$390$	0	0
$391$	−3.12311	−0.157942
$392$	0	0
$393$	− 25.4384i	− 1.28320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.93087i	0.197285i	0.995123	+	0.0986423i	$0.0314500\pi$
$397$	3.93087i	0.197285i	−0.995123	+	0.0986423i	$0.968550\pi$
$398$	0	0
$399$	−52.4924	−2.62791
$400$	0	0
$401$	−28.7386	−1.43514	−0.717569	−	0.696487i	$-0.754745\pi$
$401$	−28.7386	−1.43514	−0.717569	+	0.696487i	$0.754745\pi$
$402$	0	0
$403$	0.807764i	0.0402376i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.4924i	1.41232i
$408$	0	0
$409$	2.31534	0.114486	0.0572431	−	0.998360i	$-0.481769\pi$
$409$	2.31534	0.114486	0.0572431	+	0.998360i	$0.481769\pi$
$410$	0	0
$411$	38.7386	1.91084
$412$	0	0
$413$	32.0000i	1.57462i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.807764i	0.0395564i
$418$	0	0
$419$	9.12311	0.445693	0.222846	−	0.974854i	$-0.428465\pi$
$419$	9.12311	0.445693	0.222846	+	0.974854i	$0.428465\pi$
$420$	0	0
$421$	26.4924	1.29116	0.645581	−	0.763692i	$-0.276615\pi$
$421$	26.4924	1.29116	0.645581	+	0.763692i	$0.276615\pi$
$422$	0	0
$423$	13.1231i	0.638067i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	56.9848i	2.75769i
$428$	0	0
$429$	−5.75379	−0.277796
$430$	0	0
$431$	−33.6155	−1.61920	−0.809602	−	0.586980i	$-0.800317\pi$
$431$	−33.6155	−1.61920	−0.809602	+	0.586980i	$0.800317\pi$
$432$	0	0
$433$	− 24.7386i	− 1.18886i	−0.804146	−	0.594431i	$-0.797377\pi$
$433$	− 24.7386i	− 1.18886i	0.804146	−	0.594431i	$-0.202623\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−21.3002	−1.01660	−0.508301	−	0.861179i	$-0.669726\pi$
$439$	−21.3002	−1.01660	−0.508301	+	0.861179i	$0.669726\pi$
$440$	0	0
$441$	68.5464	3.26411
$442$	0	0
$443$	15.6847i	0.745201i	0.927992	+	0.372600i	$0.121534\pi$
$443$	15.6847i	0.745201i	−0.927992	+	0.372600i	$0.878466\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 10.8769i	− 0.514459i
$448$	0	0
$449$	−16.7386	−0.789945	−0.394972	−	0.918693i	$-0.629246\pi$
$449$	−16.7386	−0.789945	−0.394972	+	0.918693i	$0.629246\pi$
$450$	0	0
$451$	−2.24621	−0.105770
$452$	0	0
$453$	43.0540i	2.02285i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.876894i	− 0.0410194i	−0.999790	−	0.0205097i	$-0.993471\pi$
$457$	− 0.876894i	− 0.0410194i	0.999790	−	0.0205097i	$-0.00652890\pi$
$458$	0	0
$459$	4.49242	0.209688
$460$	0	0
$461$	23.4384	1.09164	0.545819	−	0.837903i	$-0.316219\pi$
$461$	23.4384	1.09164	0.545819	+	0.837903i	$0.316219\pi$
$462$	0	0
$463$	− 10.2462i	− 0.476182i	−0.971243	−	0.238091i	$-0.923478\pi$
$463$	− 10.2462i	− 0.476182i	0.971243	−	0.238091i	$-0.0765216\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.3693i	1.63670i	0.574722	+	0.818348i	$0.305110\pi$
$467$	35.3693i	1.63670i	−0.574722	+	0.818348i	$0.694890\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−5.12311	−0.236060
$472$	0	0
$473$	− 36.4924i	− 1.67792i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 15.1231i	− 0.692439i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	− 13.1231i	− 0.597122i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.05398i	− 0.138389i	−0.997603	−	0.0691944i	$-0.977957\pi$
$487$	− 3.05398i	− 0.138389i	0.997603	−	0.0691944i	$-0.0220429\pi$
$488$	0	0
$489$	0.807764	0.0365284
$490$	0	0
$491$	10.5616	0.476636	0.238318	−	0.971187i	$-0.423404\pi$
$491$	10.5616	0.476636	0.238318	+	0.971187i	$0.423404\pi$
$492$	0	0
$493$	− 26.7386i	− 1.20425i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.8769i	0.846744i
$498$	0	0
$499$	−0.946025	−0.0423499	−0.0211749	−	0.999776i	$-0.506741\pi$
$499$	−0.946025	−0.0423499	−0.0211749	+	0.999776i	$0.506741\pi$
$500$	0	0
$501$	−20.4924	−0.915534
$502$	0	0
$503$	− 25.6155i	− 1.14214i	−0.820901	−	0.571070i	$-0.806528\pi$
$503$	− 25.6155i	− 1.14214i	0.820901	−	0.571070i	$-0.193472\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 32.4924i	− 1.44304i
$508$	0	0
$509$	8.56155	0.379484	0.189742	−	0.981834i	$-0.439235\pi$
$509$	8.56155	0.379484	0.189742	+	0.981834i	$0.439235\pi$
$510$	0	0
$511$	84.8466	3.75339
$512$	0	0
$513$	− 5.75379i	− 0.254036i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.7386i	0.648204i
$518$	0	0
$519$	−26.8769	−1.17976
$520$	0	0
$521$	−17.8617	−0.782537	−0.391269	−	0.920277i	$-0.627964\pi$
$521$	−17.8617	−0.782537	−0.391269	+	0.920277i	$0.627964\pi$
$522$	0	0
$523$	23.8617i	1.04340i	0.853129	+	0.521701i	$0.174702\pi$
$523$	23.8617i	1.04340i	−0.853129	+	0.521701i	$0.825298\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.49242i	− 0.195693i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−22.2462	−0.965403
$532$	0	0
$533$	0.315342i	0.0136590i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 19.6847i	− 0.849456i
$538$	0	0
$539$	76.9848	3.31597
$540$	0	0
$541$	33.6847	1.44822	0.724108	−	0.689686i	$-0.242252\pi$
$541$	33.6847	1.44822	0.724108	+	0.689686i	$0.242252\pi$
$542$	0	0
$543$	− 8.00000i	− 0.343313i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.946025i	0.0404491i	0.999795	+	0.0202245i	$0.00643811\pi$
$547$	0.946025i	0.0404491i	−0.999795	+	0.0202245i	$0.993562\pi$
$548$	0	0
$549$	−39.6155	−1.69075
$550$	0	0
$551$	−34.2462	−1.45894
$552$	0	0
$553$	52.4924i	2.23220i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.75379i	− 0.328539i	−0.986416	−	0.164269i	$-0.947473\pi$
$557$	− 7.75379i	− 0.328539i	0.986416	−	0.164269i	$-0.0525266\pi$
$558$	0	0
$559$	−5.12311	−0.216684
$560$	0	0
$561$	32.0000	1.35104
$562$	0	0
$563$	39.2311i	1.65339i	0.562649	+	0.826696i	$0.309782\pi$
$563$	39.2311i	1.65339i	−0.562649	+	0.826696i	$0.690218\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 35.8617i	− 1.50605i
$568$	0	0
$569$	20.7386	0.869409	0.434704	−	0.900573i	$-0.356853\pi$
$569$	20.7386	0.869409	0.434704	+	0.900573i	$0.356853\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	7.36932i	0.307858i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.4233i	− 1.34980i	−0.737910	−	0.674900i	$-0.764187\pi$
$577$	− 32.4233i	− 1.34980i	0.737910	−	0.674900i	$-0.235813\pi$
$578$	0	0
$579$	−62.9157	−2.61469
$580$	0	0
$581$	61.4773	2.55051
$582$	0	0
$583$	− 16.9848i	− 0.703440i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.1771i	− 1.49319i	−0.665280	−	0.746594i	$-0.731688\pi$
$587$	− 36.1771i	− 1.49319i	0.665280	−	0.746594i	$-0.268312\pi$
$588$	0	0
$589$	−5.75379	−0.237081
$590$	0	0
$591$	−29.3002	−1.20525
$592$	0	0
$593$	− 26.9848i	− 1.10813i	−0.832472	−	0.554067i	$-0.813075\pi$
$593$	− 26.9848i	− 1.10813i	0.832472	−	0.554067i	$-0.186925\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	61.4773i	2.51610i
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	18.1771	0.741459	0.370729	−	0.928741i	$-0.379108\pi$
$601$	18.1771	0.741459	0.370729	+	0.928741i	$0.379108\pi$
$602$	0	0
$603$	− 22.2462i	− 0.905936i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	112.354	4.55282
$610$	0	0
$611$	2.06913	0.0837081
$612$	0	0
$613$	− 3.12311i	− 0.126141i	−0.998009	−	0.0630705i	$-0.979911\pi$
$613$	− 3.12311i	− 0.126141i	0.998009	−	0.0630705i	$-0.0200893\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.6155i	1.43383i	0.697162	+	0.716914i	$0.254446\pi$
$617$	35.6155i	1.43383i	−0.697162	+	0.716914i	$0.745554\pi$
$618$	0	0
$619$	28.9848	1.16500	0.582500	−	0.812831i	$-0.302075\pi$
$619$	28.9848	1.16500	0.582500	+	0.812831i	$0.302075\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	− 51.2311i	− 2.05253i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 40.9848i	− 1.63678i
$628$	0	0
$629$	−22.2462	−0.887015
$630$	0	0
$631$	−31.3693	−1.24879	−0.624396	−	0.781108i	$-0.714655\pi$
$631$	−31.3693	−1.24879	−0.624396	+	0.781108i	$0.714655\pi$
$632$	0	0
$633$	36.4924i	1.45044i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 10.8078i	− 0.428219i
$638$	0	0
$639$	−13.1231	−0.519142
$640$	0	0
$641$	23.7538	0.938218	0.469109	−	0.883140i	$-0.344575\pi$
$641$	23.7538	0.938218	0.469109	+	0.883140i	$0.344575\pi$
$642$	0	0
$643$	− 44.3542i	− 1.74916i	−0.484884	−	0.874579i	$-0.661138\pi$
$643$	− 44.3542i	− 1.74916i	0.484884	−	0.874579i	$-0.338862\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.43845i	− 0.0565512i	−0.999600	−	0.0282756i	$-0.990998\pi$
$647$	− 1.43845i	− 0.0565512i	0.999600	−	0.0282756i	$-0.00900160\pi$
$648$	0	0
$649$	−24.9848	−0.980741
$650$	0	0
$651$	18.8769	0.739844
$652$	0	0
$653$	35.7926i	1.40067i	0.713813	+	0.700337i	$0.246967\pi$
$653$	35.7926i	1.40067i	−0.713813	+	0.700337i	$0.753033\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	58.9848i	2.30122i
$658$	0	0
$659$	−14.2462	−0.554954	−0.277477	−	0.960732i	$-0.589498\pi$
$659$	−14.2462	−0.554954	−0.277477	+	0.960732i	$0.589498\pi$
$660$	0	0
$661$	10.4924	0.408108	0.204054	−	0.978960i	$-0.434588\pi$
$661$	10.4924	0.408108	0.204054	+	0.978960i	$0.434588\pi$
$662$	0	0
$663$	− 4.49242i	− 0.174471i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.56155i	− 0.331505i
$668$	0	0
$669$	52.4924	2.02947
$670$	0	0
$671$	−44.4924	−1.71761
$672$	0	0
$673$	4.56155i	0.175835i	0.996128	+	0.0879175i	$0.0280212\pi$
$673$	4.56155i	0.175835i	−0.996128	+	0.0879175i	$0.971979\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.7386i	0.797050i	0.917157	+	0.398525i	$0.130478\pi$
$677$	20.7386i	0.797050i	−0.917157	+	0.398525i	$0.869522\pi$
$678$	0	0
$679$	83.2311	3.19411
$680$	0	0
$681$	23.3693	0.895514
$682$	0	0
$683$	18.5616i	0.710238i	0.934821	+	0.355119i	$0.115560\pi$
$683$	18.5616i	0.710238i	−0.934821	+	0.355119i	$0.884440\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.630683i	0.0240621i
$688$	0	0
$689$	−2.38447	−0.0908411
$690$	0	0
$691$	6.24621	0.237617	0.118809	−	0.992917i	$-0.462093\pi$
$691$	6.24621	0.237617	0.118809	+	0.992917i	$0.462093\pi$
$692$	0	0
$693$	72.9848i	2.77247i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.75379i	− 0.0664295i
$698$	0	0
$699$	32.1771	1.21705
$700$	0	0
$701$	38.9848	1.47244	0.736219	−	0.676744i	$-0.236610\pi$
$701$	38.9848	1.47244	0.736219	+	0.676744i	$0.236610\pi$
$702$	0	0
$703$	28.4924i	1.07461i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.26137i	0.0474386i
$708$	0	0
$709$	40.7386	1.52997	0.764986	−	0.644047i	$-0.222746\pi$
$709$	40.7386	1.52997	0.764986	+	0.644047i	$0.222746\pi$
$710$	0	0
$711$	−36.4924	−1.36857
$712$	0	0
$713$	− 1.43845i	− 0.0538703i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 22.5616i	− 0.842577i
$718$	0	0
$719$	−38.7386	−1.44471	−0.722354	−	0.691524i	$-0.756940\pi$
$719$	−38.7386	−1.44471	−0.722354	+	0.691524i	$0.756940\pi$
$720$	0	0
$721$	−11.5076	−0.428565
$722$	0	0
$723$	47.3693i	1.76168i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.1231i	1.37682i	0.725322	+	0.688410i	$0.241691\pi$
$727$	37.1231i	1.37682i	−0.725322	+	0.688410i	$0.758309\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	28.4924	1.05383
$732$	0	0
$733$	− 11.1231i	− 0.410841i	−0.978674	−	0.205421i	$-0.934144\pi$
$733$	− 11.1231i	− 0.410841i	0.978674	−	0.205421i	$-0.0658562\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.9848i	− 0.920329i
$738$	0	0
$739$	−27.5464	−1.01331	−0.506655	−	0.862149i	$-0.669118\pi$
$739$	−27.5464	−1.01331	−0.506655	+	0.862149i	$0.669118\pi$
$740$	0	0
$741$	−5.75379	−0.211371
$742$	0	0
$743$	13.7538i	0.504578i	0.967652	+	0.252289i	$0.0811833\pi$
$743$	13.7538i	0.504578i	−0.967652	+	0.252289i	$0.918817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	42.7386i	1.56372i
$748$	0	0
$749$	61.4773	2.24633
$750$	0	0
$751$	49.6155	1.81050	0.905248	−	0.424883i	$-0.139685\pi$
$751$	49.6155	1.81050	0.905248	+	0.424883i	$0.139685\pi$
$752$	0	0
$753$	− 16.0000i	− 0.583072i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	24.8769i	0.904166i	0.891976	+	0.452083i	$0.149319\pi$
$757$	24.8769i	0.904166i	−0.891976	+	0.452083i	$0.850681\pi$
$758$	0	0
$759$	10.2462	0.371914
$760$	0	0
$761$	−11.3002	−0.409631	−0.204816	−	0.978801i	$-0.565660\pi$
$761$	−11.3002	−0.409631	−0.204816	+	0.978801i	$0.565660\pi$
$762$	0	0
$763$	− 42.2462i	− 1.52942i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.50758i	0.126651i
$768$	0	0
$769$	−38.4924	−1.38807	−0.694036	−	0.719940i	$-0.744169\pi$
$769$	−38.4924	−1.38807	−0.694036	+	0.719940i	$0.744169\pi$
$770$	0	0
$771$	−23.1922	−0.835248
$772$	0	0
$773$	4.24621i	0.152726i	0.997080	+	0.0763628i	$0.0243307\pi$
$773$	4.24621i	0.152726i	−0.997080	+	0.0763628i	$0.975669\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 93.4773i	− 3.35348i
$778$	0	0
$779$	−2.24621	−0.0804789
$780$	0	0
$781$	−14.7386	−0.527390
$782$	0	0
$783$	12.3153i	0.440114i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 46.2462i	− 1.64850i	−0.566226	−	0.824250i	$-0.691597\pi$
$787$	− 46.2462i	− 1.64850i	0.566226	−	0.824250i	$-0.308403\pi$
$788$	0	0
$789$	20.4924	0.729550
$790$	0	0
$791$	103.723	3.68798
$792$	0	0
$793$	6.24621i	0.221809i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.1231i	0.960750i	0.877063	+	0.480375i	$0.159499\pi$
$797$	27.1231i	0.960750i	−0.877063	+	0.480375i	$0.840501\pi$
$798$	0	0
$799$	−11.5076	−0.407109
$800$	0	0
$801$	35.6155	1.25841
$802$	0	0
$803$	66.2462i	2.33778i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 59.6847i	− 2.10100i
$808$	0	0
$809$	10.4924	0.368894	0.184447	−	0.982842i	$-0.440951\pi$
$809$	10.4924	0.368894	0.184447	+	0.982842i	$0.440951\pi$
$810$	0	0
$811$	−8.31534	−0.291991	−0.145996	−	0.989285i	$-0.546639\pi$
$811$	−8.31534	−0.291991	−0.145996	+	0.989285i	$0.546639\pi$
$812$	0	0
$813$	5.75379i	0.201794i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 36.4924i	− 1.27671i
$818$	0	0
$819$	10.2462	0.358032
$820$	0	0
$821$	−31.7538	−1.10821	−0.554107	−	0.832445i	$-0.686940\pi$
$821$	−31.7538	−1.10821	−0.554107	+	0.832445i	$0.686940\pi$
$822$	0	0
$823$	17.4384i	0.607866i	0.952693	+	0.303933i	$0.0982999\pi$
$823$	17.4384i	0.607866i	−0.952693	+	0.303933i	$0.901700\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−50.4924	−1.75367	−0.876837	−	0.480787i	$-0.840351\pi$
$829$	−50.4924	−1.75367	−0.876837	+	0.480787i	$0.840351\pi$
$830$	0	0
$831$	−29.3002	−1.01641
$832$	0	0
$833$	60.1080i	2.08262i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.06913i	0.0715196i
$838$	0	0
$839$	−9.61553	−0.331965	−0.165982	−	0.986129i	$-0.553080\pi$
$839$	−9.61553	−0.331965	−0.165982	+	0.986129i	$0.553080\pi$
$840$	0	0
$841$	44.3002	1.52759
$842$	0	0
$843$	− 78.1080i	− 2.69018i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.6155i	0.880160i
$848$	0	0
$849$	−58.6004	−2.01116
$850$	0	0
$851$	−7.12311	−0.244177
$852$	0	0
$853$	− 22.9848i	− 0.786986i	−0.919328	−	0.393493i	$-0.871267\pi$
$853$	− 22.9848i	− 0.786986i	0.919328	−	0.393493i	$-0.128733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.1771i	1.16747i	0.811945	+	0.583733i	$0.198409\pi$
$857$	34.1771i	1.16747i	−0.811945	+	0.583733i	$0.801591\pi$
$858$	0	0
$859$	−38.4233	−1.31099	−0.655493	−	0.755201i	$-0.727539\pi$
$859$	−38.4233	−1.31099	−0.655493	+	0.755201i	$0.727539\pi$
$860$	0	0
$861$	7.36932	0.251146
$862$	0	0
$863$	− 18.4233i	− 0.627136i	−0.949566	−	0.313568i	$-0.898476\pi$
$863$	− 18.4233i	− 0.627136i	0.949566	−	0.313568i	$-0.101524\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 18.5616i	− 0.630383i
$868$	0	0
$869$	−40.9848	−1.39032
$870$	0	0
$871$	−3.50758	−0.118850
$872$	0	0
$873$	57.8617i	1.95832i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 40.7386i	− 1.37565i	−0.725878	−	0.687823i	$-0.758567\pi$
$877$	− 40.7386i	− 1.37565i	0.725878	−	0.687823i	$-0.241433\pi$
$878$	0	0
$879$	32.9848	1.11255
$880$	0	0
$881$	24.1080	0.812217	0.406109	−	0.913825i	$-0.366885\pi$
$881$	24.1080	0.812217	0.406109	+	0.913825i	$0.366885\pi$
$882$	0	0
$883$	40.4924i	1.36268i	0.731968	+	0.681339i	$0.238602\pi$
$883$	40.4924i	1.36268i	−0.731968	+	0.681339i	$0.761398\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33.4384i	− 1.12275i	−0.827560	−	0.561377i	$-0.810272\pi$
$887$	− 33.4384i	− 1.12275i	0.827560	−	0.561377i	$-0.189728\pi$
$888$	0	0
$889$	86.1080	2.88797
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	14.7386i	0.493210i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.43845i	− 0.0480284i
$898$	0	0
$899$	12.3153	0.410740
$900$	0	0
$901$	13.2614	0.441800
$902$	0	0
$903$	119.723i	3.98415i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	0	0
$909$	−0.876894	−0.0290848
$910$	0	0
$911$	−20.4924	−0.678944	−0.339472	−	0.940616i	$-0.610248\pi$
$911$	−20.4924	−0.678944	−0.339472	+	0.940616i	$0.610248\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	50.8769i	1.68010i
$918$	0	0
$919$	35.8617	1.18297	0.591485	−	0.806316i	$-0.298542\pi$
$919$	35.8617	1.18297	0.591485	+	0.806316i	$0.298542\pi$
$920$	0	0
$921$	30.7386	1.01287
$922$	0	0
$923$	2.06913i	0.0681063i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−13.0540	−0.428287	−0.214144	−	0.976802i	$-0.568696\pi$
$929$	−13.0540	−0.428287	−0.214144	+	0.976802i	$0.568696\pi$
$930$	0	0
$931$	76.9848	2.52308
$932$	0	0
$933$	− 35.6847i	− 1.16826i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.3693i	1.35148i	0.737142	+	0.675738i	$0.236175\pi$
$937$	41.3693i	1.35148i	−0.737142	+	0.675738i	$0.763825\pi$
$938$	0	0
$939$	−38.7386	−1.26419
$940$	0	0
$941$	−60.6004	−1.97552	−0.987758	−	0.155995i	$-0.950142\pi$
$941$	−60.6004	−1.97552	−0.987758	+	0.155995i	$0.950142\pi$
$942$	0	0
$943$	− 0.561553i	− 0.0182867i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 35.1922i	− 1.14359i	−0.820395	−	0.571797i	$-0.806246\pi$
$947$	− 35.1922i	− 1.14359i	0.820395	−	0.571797i	$-0.193754\pi$
$948$	0	0
$949$	9.30019	0.301897
$950$	0	0
$951$	53.1231	1.72263
$952$	0	0
$953$	− 13.5076i	− 0.437553i	−0.975775	−	0.218777i	$-0.929793\pi$
$953$	− 13.5076i	− 0.437553i	0.975775	−	0.218777i	$-0.0702066\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	87.7235i	2.83570i
$958$	0	0
$959$	−77.4773	−2.50187
$960$	0	0
$961$	−28.9309	−0.933254
$962$	0	0
$963$	42.7386i	1.37723i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.177081i	0.00569454i	0.999996	+	0.00284727i	$0.000906315\pi$
$967$	0.177081i	0.00569454i	−0.999996	+	0.00284727i	$0.999094\pi$
$968$	0	0
$969$	32.0000	1.02799
$970$	0	0
$971$	3.36932	0.108127	0.0540633	−	0.998538i	$-0.482783\pi$
$971$	3.36932	0.108127	0.0540633	+	0.998538i	$0.482783\pi$
$972$	0	0
$973$	− 1.61553i	− 0.0517915i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.1080i	1.53911i	0.638581	+	0.769555i	$0.279522\pi$
$977$	48.1080i	1.53911i	−0.638581	+	0.769555i	$0.720478\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	29.3693	0.937690
$982$	0	0
$983$	− 37.1231i	− 1.18404i	−0.805922	−	0.592022i	$-0.798330\pi$
$983$	− 37.1231i	− 1.18404i	0.805922	−	0.592022i	$-0.201670\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 48.3542i	− 1.53913i
$988$	0	0
$989$	9.12311	0.290098
$990$	0	0
$991$	−12.4924	−0.396835	−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924	−0.396835	−0.198417	+	0.980118i	$0.563580\pi$
$992$	0	0
$993$	− 27.0540i	− 0.858532i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 48.7386i	− 1.54357i	−0.635885	−	0.771784i	$-0.719365\pi$
$997$	− 48.7386i	− 1.54357i	0.635885	−	0.771784i	$-0.280635\pi$
$998$	0	0
$999$	10.2462	0.324176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.m.4049.1		4
5.2	odd	4		4600.2.a.r.1.1		2
5.3	odd	4		920.2.a.f.1.2	✓	2
5.4	even	2	inner	4600.2.e.m.4049.4		4
15.8	even	4		8280.2.a.bb.1.2		2
20.3	even	4		1840.2.a.k.1.1		2
20.7	even	4		9200.2.a.bx.1.2		2
40.3	even	4		7360.2.a.bm.1.2		2
40.13	odd	4		7360.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.f.1.2	✓	2	5.3	odd	4
1840.2.a.k.1.1		2	20.3	even	4
4600.2.a.r.1.1		2	5.2	odd	4
4600.2.e.m.4049.1		4	1.1	even	1	trivial
4600.2.e.m.4049.4		4	5.4	even	2	inner
7360.2.a.bj.1.1		2	40.13	odd	4
7360.2.a.bm.1.2		2	40.3	even	4
8280.2.a.bb.1.2		2	15.8	even	4
9200.2.a.bx.1.2		2	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)

\(f(q)\)	\(=\)	\(q-2.56155i q^{3} +5.12311i q^{7} -3.56155 q^{9} +O(q^{10})\)	\(q-2.56155i q^{3} +5.12311i q^{7} -3.56155 q^{9} -4.00000 q^{11} +0.561553i q^{13} -3.12311i q^{17} -4.00000 q^{19} +13.1231 q^{21} -1.00000i q^{23} +1.43845i q^{27} +8.56155 q^{29} +1.43845 q^{31} +10.2462i q^{33} -7.12311i q^{37} +1.43845 q^{39} +0.561553 q^{41} +9.12311i q^{43} -3.68466i q^{47} -19.2462 q^{49} -8.00000 q^{51} +4.24621i q^{53} +10.2462i q^{57} +6.24621 q^{59} +11.1231 q^{61} -18.2462i q^{63} +6.24621i q^{67} -2.56155 q^{69} +3.68466 q^{71} -16.5616i q^{73} -20.4924i q^{77} +10.2462 q^{79} -7.00000 q^{81} -12.0000i q^{83} -21.9309i q^{87} -10.0000 q^{89} -2.87689 q^{91} -3.68466i q^{93} -16.2462i q^{97} +14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 16 * q^11 - 16 * q^19 + 36 * q^21 + 26 * q^29 + 14 * q^31 + 14 * q^39 - 6 * q^41 - 44 * q^49 - 32 * q^51 - 8 * q^59 + 28 * q^61 - 2 * q^69 - 10 * q^71 + 8 * q^79 - 28 * q^81 - 40 * q^89 - 28 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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