LMFDB - Embedded newform 4028.2.a.c.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.a (trivial)

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:	yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 27 x^{17} + 161 x^{16} + 253 x^{15} - 2103 x^{14} - 683 x^{13} + 14442 x^{12} + \cdots - 4088 $ x^19 - 5x^18 - 27x^17 + 161x^16 + 253x^15 - 2103x^14 - 683x^13 + 14442x^12 - 4144x^11 - 56325x^10 + 37245x^9 + 124233x^8 - 117486x^7 - 140498x^6 + 176987x^5 + 54044x^4 - 115318x^3 + 16295x^2 + 17097x - 4088
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-1.49075$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49075 q^{3} -0.450569 q^{5} +1.77312 q^{7} -0.777652 q^{9} +O(q^{10})$	$q+1.49075 q^{3} -0.450569 q^{5} +1.77312 q^{7} -0.777652 q^{9} -2.05386 q^{11} +1.50635 q^{13} -0.671688 q^{15} -1.71617 q^{17} +1.00000 q^{19} +2.64328 q^{21} -3.69103 q^{23} -4.79699 q^{25} -5.63155 q^{27} -6.49195 q^{29} -5.66313 q^{31} -3.06181 q^{33} -0.798912 q^{35} -6.57855 q^{37} +2.24559 q^{39} +11.3091 q^{41} -6.96277 q^{43} +0.350386 q^{45} -7.28510 q^{47} -3.85606 q^{49} -2.55838 q^{51} -1.00000 q^{53} +0.925407 q^{55} +1.49075 q^{57} +5.06214 q^{59} +8.91603 q^{61} -1.37887 q^{63} -0.678713 q^{65} +9.61198 q^{67} -5.50242 q^{69} +15.9979 q^{71} -14.2968 q^{73} -7.15113 q^{75} -3.64174 q^{77} +7.87998 q^{79} -6.06230 q^{81} -11.1161 q^{83} +0.773251 q^{85} -9.67790 q^{87} +14.2990 q^{89} +2.67093 q^{91} -8.44233 q^{93} -0.450569 q^{95} -13.9793 q^{97} +1.59719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9	$ 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9} - 3 q^{11} - 23 q^{13} - 18 q^{15} - 7 q^{17} + 19 q^{19} - 4 q^{21} - 6 q^{23} + 18 q^{25} - 17 q^{27} - 4 q^{29} - 30 q^{31} - 10 q^{33} - q^{35} - 31 q^{37} + 5 q^{39} - 15 q^{41} - 29 q^{43} + 6 q^{45} - 18 q^{47} + 23 q^{49} - 5 q^{51} - 19 q^{53} - 19 q^{55} - 5 q^{57} + 8 q^{59} - 4 q^{61} - 64 q^{63} - 26 q^{65} - 62 q^{67} + 3 q^{69} - 17 q^{71} + q^{73} - 40 q^{75} - 14 q^{77} - 28 q^{79} + 11 q^{81} + 4 q^{83} - 31 q^{85} - 20 q^{87} + 33 q^{89} - 29 q^{91} - 59 q^{93} - 5 q^{95} + 5 q^{97} - 48 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9 - 3 * q^11 - 23 * q^13 - 18 * q^15 - 7 * q^17 + 19 * q^19 - 4 * q^21 - 6 * q^23 + 18 * q^25 - 17 * q^27 - 4 * q^29 - 30 * q^31 - 10 * q^33 - q^35 - 31 * q^37 + 5 * q^39 - 15 * q^41 - 29 * q^43 + 6 * q^45 - 18 * q^47 + 23 * q^49 - 5 * q^51 - 19 * q^53 - 19 * q^55 - 5 * q^57 + 8 * q^59 - 4 * q^61 - 64 * q^63 - 26 * q^65 - 62 * q^67 + 3 * q^69 - 17 * q^71 + q^73 - 40 * q^75 - 14 * q^77 - 28 * q^79 + 11 * q^81 + 4 * q^83 - 31 * q^85 - 20 * q^87 + 33 * q^89 - 29 * q^91 - 59 * q^93 - 5 * q^95 + 5 * q^97 - 48 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.49075	0.860687	0.430344	−	0.902665i	$-0.358392\pi$
$3$	1.49075	0.860687	0.430344	+	0.902665i	$0.358392\pi$
$4$	0	0
$5$	−0.450569	−0.201501	−0.100750	−	0.994912i	$-0.532124\pi$
$5$	−0.450569	−0.201501	−0.100750	+	0.994912i	$0.532124\pi$
$6$	0	0
$7$	1.77312	0.670175	0.335088	−	0.942187i	$-0.391234\pi$
$7$	1.77312	0.670175	0.335088	+	0.942187i	$0.391234\pi$
$8$	0	0
$9$	−0.777652	−0.259217
$10$	0	0
$11$	−2.05386	−0.619263	−0.309632	−	0.950857i	$-0.600206\pi$
$11$	−2.05386	−0.619263	−0.309632	+	0.950857i	$0.600206\pi$
$12$	0	0
$13$	1.50635	0.417786	0.208893	−	0.977939i	$-0.433014\pi$
$13$	1.50635	0.417786	0.208893	+	0.977939i	$0.433014\pi$
$14$	0	0
$15$	−0.671688	−0.173429
$16$	0	0
$17$	−1.71617	−0.416231	−0.208116	−	0.978104i	$-0.566733\pi$
$17$	−1.71617	−0.416231	−0.208116	+	0.978104i	$0.566733\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.64328	0.576811
$22$	0	0
$23$	−3.69103	−0.769634	−0.384817	−	0.922993i	$-0.625735\pi$
$23$	−3.69103	−0.769634	−0.384817	+	0.922993i	$0.625735\pi$
$24$	0	0
$25$	−4.79699	−0.959398
$26$	0	0
$27$	−5.63155	−1.08379
$28$	0	0
$29$	−6.49195	−1.20553	−0.602763	−	0.797921i	$-0.705933\pi$
$29$	−6.49195	−1.20553	−0.602763	+	0.797921i	$0.705933\pi$
$30$	0	0
$31$	−5.66313	−1.01713	−0.508564	−	0.861024i	$-0.669823\pi$
$31$	−5.66313	−1.01713	−0.508564	+	0.861024i	$0.669823\pi$
$32$	0	0
$33$	−3.06181	−0.532992
$34$	0	0
$35$	−0.798912	−0.135041
$36$	0	0
$37$	−6.57855	−1.08151	−0.540754	−	0.841181i	$-0.681861\pi$
$37$	−6.57855	−1.08151	−0.540754	+	0.841181i	$0.681861\pi$
$38$	0	0
$39$	2.24559	0.359583
$40$	0	0
$41$	11.3091	1.76618	0.883090	−	0.469204i	$-0.155459\pi$
$41$	11.3091	1.76618	0.883090	+	0.469204i	$0.155459\pi$
$42$	0	0
$43$	−6.96277	−1.06181	−0.530906	−	0.847430i	$-0.678148\pi$
$43$	−6.96277	−1.06181	−0.530906	+	0.847430i	$0.678148\pi$
$44$	0	0
$45$	0.350386	0.0522325
$46$	0	0
$47$	−7.28510	−1.06264	−0.531320	−	0.847171i	$-0.678304\pi$
$47$	−7.28510	−1.06264	−0.531320	+	0.847171i	$0.678304\pi$
$48$	0	0
$49$	−3.85606	−0.550865
$50$	0	0
$51$	−2.55838	−0.358245
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	0.925407	0.124782
$56$	0	0
$57$	1.49075	0.197455
$58$	0	0
$59$	5.06214	0.659035	0.329518	−	0.944149i	$-0.393114\pi$
$59$	5.06214	0.659035	0.329518	+	0.944149i	$0.393114\pi$
$60$	0	0
$61$	8.91603	1.14158	0.570790	−	0.821096i	$-0.306637\pi$
$61$	8.91603	1.14158	0.570790	+	0.821096i	$0.306637\pi$
$62$	0	0
$63$	−1.37887	−0.173721
$64$	0	0
$65$	−0.678713	−0.0841840
$66$	0	0
$67$	9.61198	1.17429	0.587145	−	0.809482i	$-0.300252\pi$
$67$	9.61198	1.17429	0.587145	+	0.809482i	$0.300252\pi$
$68$	0	0
$69$	−5.50242	−0.662414
$70$	0	0
$71$	15.9979	1.89860	0.949301	−	0.314369i	$-0.101793\pi$
$71$	15.9979	1.89860	0.949301	+	0.314369i	$0.101793\pi$
$72$	0	0
$73$	−14.2968	−1.67331	−0.836655	−	0.547731i	$-0.815492\pi$
$73$	−14.2968	−1.67331	−0.836655	+	0.547731i	$0.815492\pi$
$74$	0	0
$75$	−7.15113	−0.825741
$76$	0	0
$77$	−3.64174	−0.415015
$78$	0	0
$79$	7.87998	0.886567	0.443283	−	0.896382i	$-0.353814\pi$
$79$	7.87998	0.886567	0.443283	+	0.896382i	$0.353814\pi$
$80$	0	0
$81$	−6.06230	−0.673589
$82$	0	0
$83$	−11.1161	−1.22015	−0.610076	−	0.792343i	$-0.708861\pi$
$83$	−11.1161	−1.22015	−0.610076	+	0.792343i	$0.708861\pi$
$84$	0	0
$85$	0.773251	0.0838709
$86$	0	0
$87$	−9.67790	−1.03758
$88$	0	0
$89$	14.2990	1.51569	0.757846	−	0.652434i	$-0.226252\pi$
$89$	14.2990	1.51569	0.757846	+	0.652434i	$0.226252\pi$
$90$	0	0
$91$	2.67093	0.279990
$92$	0	0
$93$	−8.44233	−0.875429
$94$	0	0
$95$	−0.450569	−0.0462274
$96$	0	0
$97$	−13.9793	−1.41939	−0.709693	−	0.704511i	$-0.751166\pi$
$97$	−13.9793	−1.41939	−0.709693	+	0.704511i	$0.751166\pi$
$98$	0	0
$99$	1.59719	0.160524
$100$	0	0
$101$	4.83125	0.480727	0.240364	−	0.970683i	$-0.422733\pi$
$101$	4.83125	0.480727	0.240364	+	0.970683i	$0.422733\pi$
$102$	0	0
$103$	1.24787	0.122956	0.0614780	−	0.998108i	$-0.480419\pi$
$103$	1.24787	0.122956	0.0614780	+	0.998108i	$0.480419\pi$
$104$	0	0
$105$	−1.19098	−0.116228
$106$	0	0
$107$	−5.02694	−0.485972	−0.242986	−	0.970030i	$-0.578127\pi$
$107$	−5.02694	−0.485972	−0.242986	+	0.970030i	$0.578127\pi$
$108$	0	0
$109$	−14.8326	−1.42071	−0.710354	−	0.703845i	$-0.751465\pi$
$109$	−14.8326	−1.42071	−0.710354	+	0.703845i	$0.751465\pi$
$110$	0	0
$111$	−9.80700	−0.930839
$112$	0	0
$113$	0.761853	0.0716691	0.0358345	−	0.999358i	$-0.488591\pi$
$113$	0.761853	0.0716691	0.0358345	+	0.999358i	$0.488591\pi$
$114$	0	0
$115$	1.66306	0.155082
$116$	0	0
$117$	−1.17141	−0.108297
$118$	0	0
$119$	−3.04296	−0.278948
$120$	0	0
$121$	−6.78164	−0.616513
$122$	0	0
$123$	16.8590	1.52013
$124$	0	0
$125$	4.41422	0.394820
$126$	0	0
$127$	−12.3538	−1.09623	−0.548113	−	0.836404i	$-0.684654\pi$
$127$	−12.3538	−1.09623	−0.548113	+	0.836404i	$0.684654\pi$
$128$	0	0
$129$	−10.3798	−0.913889
$130$	0	0
$131$	−8.49464	−0.742180	−0.371090	−	0.928597i	$-0.621016\pi$
$131$	−8.49464	−0.742180	−0.371090	+	0.928597i	$0.621016\pi$
$132$	0	0
$133$	1.77312	0.153749
$134$	0	0
$135$	2.53740	0.218385
$136$	0	0
$137$	4.90163	0.418775	0.209387	−	0.977833i	$-0.432853\pi$
$137$	4.90163	0.418775	0.209387	+	0.977833i	$0.432853\pi$
$138$	0	0
$139$	7.61182	0.645626	0.322813	−	0.946463i	$-0.395371\pi$
$139$	7.61182	0.645626	0.322813	+	0.946463i	$0.395371\pi$
$140$	0	0
$141$	−10.8603	−0.914601
$142$	0	0
$143$	−3.09383	−0.258719
$144$	0	0
$145$	2.92507	0.242914
$146$	0	0
$147$	−5.74843	−0.474123
$148$	0	0
$149$	11.5011	0.942207	0.471103	−	0.882078i	$-0.343856\pi$
$149$	11.5011	0.942207	0.471103	+	0.882078i	$0.343856\pi$
$150$	0	0
$151$	2.29004	0.186361	0.0931806	−	0.995649i	$-0.470297\pi$
$151$	2.29004	0.186361	0.0931806	+	0.995649i	$0.470297\pi$
$152$	0	0
$153$	1.33458	0.107894
$154$	0	0
$155$	2.55163	0.204952
$156$	0	0
$157$	−18.5017	−1.47659	−0.738297	−	0.674476i	$-0.764370\pi$
$157$	−18.5017	−1.47659	−0.738297	+	0.674476i	$0.764370\pi$
$158$	0	0
$159$	−1.49075	−0.118224
$160$	0	0
$161$	−6.54463	−0.515789
$162$	0	0
$163$	−11.1430	−0.872786	−0.436393	−	0.899756i	$-0.643744\pi$
$163$	−11.1430	−0.872786	−0.436393	+	0.899756i	$0.643744\pi$
$164$	0	0
$165$	1.37955	0.107398
$166$	0	0
$167$	4.52475	0.350136	0.175068	−	0.984556i	$-0.443986\pi$
$167$	4.52475	0.350136	0.175068	+	0.984556i	$0.443986\pi$
$168$	0	0
$169$	−10.7309	−0.825455
$170$	0	0
$171$	−0.777652	−0.0594685
$172$	0	0
$173$	1.27564	0.0969850	0.0484925	−	0.998824i	$-0.484558\pi$
$173$	1.27564	0.0969850	0.0484925	+	0.998824i	$0.484558\pi$
$174$	0	0
$175$	−8.50562	−0.642964
$176$	0	0
$177$	7.54641	0.567223
$178$	0	0
$179$	−3.98289	−0.297695	−0.148848	−	0.988860i	$-0.547556\pi$
$179$	−3.98289	−0.297695	−0.148848	+	0.988860i	$0.547556\pi$
$180$	0	0
$181$	7.43481	0.552625	0.276313	−	0.961068i	$-0.410888\pi$
$181$	7.43481	0.552625	0.276313	+	0.961068i	$0.410888\pi$
$182$	0	0
$183$	13.2916	0.982544
$184$	0	0
$185$	2.96409	0.217924
$186$	0	0
$187$	3.52477	0.257757
$188$	0	0
$189$	−9.98540	−0.726331
$190$	0	0
$191$	19.0830	1.38080	0.690400	−	0.723428i	$-0.257435\pi$
$191$	19.0830	1.38080	0.690400	+	0.723428i	$0.257435\pi$
$192$	0	0
$193$	6.17964	0.444820	0.222410	−	0.974953i	$-0.428608\pi$
$193$	6.17964	0.444820	0.222410	+	0.974953i	$0.428608\pi$
$194$	0	0
$195$	−1.01179	−0.0724561
$196$	0	0
$197$	−0.430276	−0.0306559	−0.0153279	−	0.999883i	$-0.504879\pi$
$197$	−0.430276	−0.0306559	−0.0153279	+	0.999883i	$0.504879\pi$
$198$	0	0
$199$	−8.04799	−0.570507	−0.285254	−	0.958452i	$-0.592078\pi$
$199$	−8.04799	−0.570507	−0.285254	+	0.958452i	$0.592078\pi$
$200$	0	0
$201$	14.3291	1.01070
$202$	0	0
$203$	−11.5110	−0.807913
$204$	0	0
$205$	−5.09551	−0.355886
$206$	0	0
$207$	2.87034	0.199502
$208$	0	0
$209$	−2.05386	−0.142069
$210$	0	0
$211$	−9.62775	−0.662802	−0.331401	−	0.943490i	$-0.607521\pi$
$211$	−9.62775	−0.662802	−0.331401	+	0.943490i	$0.607521\pi$
$212$	0	0
$213$	23.8489	1.63410
$214$	0	0
$215$	3.13721	0.213956
$216$	0	0
$217$	−10.0414	−0.681654
$218$	0	0
$219$	−21.3130	−1.44020
$220$	0	0
$221$	−2.58514	−0.173895
$222$	0	0
$223$	−16.2529	−1.08838	−0.544188	−	0.838963i	$-0.683162\pi$
$223$	−16.2529	−1.08838	−0.544188	+	0.838963i	$0.683162\pi$
$224$	0	0
$225$	3.73039	0.248693
$226$	0	0
$227$	22.7891	1.51257	0.756283	−	0.654244i	$-0.227013\pi$
$227$	22.7891	1.51257	0.756283	+	0.654244i	$0.227013\pi$
$228$	0	0
$229$	16.2751	1.07549	0.537746	−	0.843107i	$-0.319276\pi$
$229$	16.2751	1.07549	0.537746	+	0.843107i	$0.319276\pi$
$230$	0	0
$231$	−5.42894	−0.357198
$232$	0	0
$233$	−3.99945	−0.262013	−0.131006	−	0.991382i	$-0.541821\pi$
$233$	−3.99945	−0.262013	−0.131006	+	0.991382i	$0.541821\pi$
$234$	0	0
$235$	3.28244	0.214123
$236$	0	0
$237$	11.7471	0.763057
$238$	0	0
$239$	0.967421	0.0625773	0.0312886	−	0.999510i	$-0.490039\pi$
$239$	0.967421	0.0625773	0.0312886	+	0.999510i	$0.490039\pi$
$240$	0	0
$241$	14.0495	0.905005	0.452503	−	0.891763i	$-0.350531\pi$
$241$	14.0495	0.905005	0.452503	+	0.891763i	$0.350531\pi$
$242$	0	0
$243$	7.85725	0.504043
$244$	0	0
$245$	1.73742	0.111000
$246$	0	0
$247$	1.50635	0.0958466
$248$	0	0
$249$	−16.5714	−1.05017
$250$	0	0
$251$	15.8413	0.999891	0.499946	−	0.866057i	$-0.333353\pi$
$251$	15.8413	0.999891	0.499946	+	0.866057i	$0.333353\pi$
$252$	0	0
$253$	7.58088	0.476606
$254$	0	0
$255$	1.15273	0.0721866
$256$	0	0
$257$	7.83120	0.488497	0.244248	−	0.969713i	$-0.421459\pi$
$257$	7.83120	0.488497	0.244248	+	0.969713i	$0.421459\pi$
$258$	0	0
$259$	−11.6645	−0.724799
$260$	0	0
$261$	5.04848	0.312493
$262$	0	0
$263$	2.74495	0.169261	0.0846304	−	0.996412i	$-0.473029\pi$
$263$	2.74495	0.169261	0.0846304	+	0.996412i	$0.473029\pi$
$264$	0	0
$265$	0.450569	0.0276782
$266$	0	0
$267$	21.3163	1.30454
$268$	0	0
$269$	−18.1025	−1.10373	−0.551865	−	0.833933i	$-0.686084\pi$
$269$	−18.1025	−1.10373	−0.551865	+	0.833933i	$0.686084\pi$
$270$	0	0
$271$	−29.8225	−1.81159	−0.905793	−	0.423720i	$-0.860724\pi$
$271$	−29.8225	−1.81159	−0.905793	+	0.423720i	$0.860724\pi$
$272$	0	0
$273$	3.98170	0.240983
$274$	0	0
$275$	9.85236	0.594120
$276$	0	0
$277$	−16.7043	−1.00366	−0.501831	−	0.864966i	$-0.667340\pi$
$277$	−16.7043	−1.00366	−0.501831	+	0.864966i	$0.667340\pi$
$278$	0	0
$279$	4.40394	0.263657
$280$	0	0
$281$	−13.7319	−0.819176	−0.409588	−	0.912271i	$-0.634328\pi$
$281$	−13.7319	−0.819176	−0.409588	+	0.912271i	$0.634328\pi$
$282$	0	0
$283$	8.61090	0.511865	0.255933	−	0.966695i	$-0.417617\pi$
$283$	8.61090	0.511865	0.255933	+	0.966695i	$0.417617\pi$
$284$	0	0
$285$	−0.671688	−0.0397873
$286$	0	0
$287$	20.0523	1.18365
$288$	0	0
$289$	−14.0548	−0.826751
$290$	0	0
$291$	−20.8397	−1.22165
$292$	0	0
$293$	−23.3714	−1.36537	−0.682687	−	0.730711i	$-0.739189\pi$
$293$	−23.3714	−1.36537	−0.682687	+	0.730711i	$0.739189\pi$
$294$	0	0
$295$	−2.28085	−0.132796
$296$	0	0
$297$	11.5664	0.671153
$298$	0	0
$299$	−5.55998	−0.321542
$300$	0	0
$301$	−12.3458	−0.711601
$302$	0	0
$303$	7.20220	0.413756
$304$	0	0
$305$	−4.01729	−0.230029
$306$	0	0
$307$	−23.7029	−1.35279	−0.676397	−	0.736537i	$-0.736460\pi$
$307$	−23.7029	−1.35279	−0.676397	+	0.736537i	$0.736460\pi$
$308$	0	0
$309$	1.86026	0.105827
$310$	0	0
$311$	−34.4997	−1.95630	−0.978150	−	0.207901i	$-0.933337\pi$
$311$	−34.4997	−1.95630	−0.978150	+	0.207901i	$0.933337\pi$
$312$	0	0
$313$	−19.5659	−1.10593	−0.552965	−	0.833204i	$-0.686504\pi$
$313$	−19.5659	−1.10593	−0.552965	+	0.833204i	$0.686504\pi$
$314$	0	0
$315$	0.621275	0.0350049
$316$	0	0
$317$	−4.61504	−0.259206	−0.129603	−	0.991566i	$-0.541370\pi$
$317$	−4.61504	−0.259206	−0.129603	+	0.991566i	$0.541370\pi$
$318$	0	0
$319$	13.3336	0.746537
$320$	0	0
$321$	−7.49393	−0.418270
$322$	0	0
$323$	−1.71617	−0.0954900
$324$	0	0
$325$	−7.22593	−0.400822
$326$	0	0
$327$	−22.1118	−1.22279
$328$	0	0
$329$	−12.9173	−0.712155
$330$	0	0
$331$	28.8561	1.58607	0.793037	−	0.609174i	$-0.208499\pi$
$331$	28.8561	1.58607	0.793037	+	0.609174i	$0.208499\pi$
$332$	0	0
$333$	5.11582	0.280345
$334$	0	0
$335$	−4.33086	−0.236620
$336$	0	0
$337$	13.2152	0.719877	0.359938	−	0.932976i	$-0.382798\pi$
$337$	13.2152	0.719877	0.359938	+	0.932976i	$0.382798\pi$
$338$	0	0
$339$	1.13573	0.0616847
$340$	0	0
$341$	11.6313	0.629870
$342$	0	0
$343$	−19.2491	−1.03935
$344$	0	0
$345$	2.47922	0.133477
$346$	0	0
$347$	28.9618	1.55475	0.777375	−	0.629038i	$-0.216551\pi$
$347$	28.9618	1.55475	0.777375	+	0.629038i	$0.216551\pi$
$348$	0	0
$349$	5.49386	0.294080	0.147040	−	0.989131i	$-0.453025\pi$
$349$	5.49386	0.294080	0.147040	+	0.989131i	$0.453025\pi$
$350$	0	0
$351$	−8.48307	−0.452793
$352$	0	0
$353$	5.18294	0.275860	0.137930	−	0.990442i	$-0.455955\pi$
$353$	5.18294	0.275860	0.137930	+	0.990442i	$0.455955\pi$
$354$	0	0
$355$	−7.20816	−0.382569
$356$	0	0
$357$	−4.53631	−0.240087
$358$	0	0
$359$	14.4107	0.760568	0.380284	−	0.924870i	$-0.375826\pi$
$359$	14.4107	0.760568	0.380284	+	0.924870i	$0.375826\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−10.1098	−0.530625
$364$	0	0
$365$	6.44168	0.337173
$366$	0	0
$367$	15.8839	0.829133	0.414567	−	0.910019i	$-0.363933\pi$
$367$	15.8839	0.829133	0.414567	+	0.910019i	$0.363933\pi$
$368$	0	0
$369$	−8.79452	−0.457824
$370$	0	0
$371$	−1.77312	−0.0920557
$372$	0	0
$373$	34.6098	1.79203	0.896014	−	0.444027i	$-0.146450\pi$
$373$	34.6098	1.79203	0.896014	+	0.444027i	$0.146450\pi$
$374$	0	0
$375$	6.58051	0.339816
$376$	0	0
$377$	−9.77913	−0.503651
$378$	0	0
$379$	−23.0876	−1.18593	−0.592965	−	0.805228i	$-0.702043\pi$
$379$	−23.0876	−1.18593	−0.592965	+	0.805228i	$0.702043\pi$
$380$	0	0
$381$	−18.4165	−0.943508
$382$	0	0
$383$	10.2941	0.526004	0.263002	−	0.964795i	$-0.415287\pi$
$383$	10.2941	0.526004	0.263002	+	0.964795i	$0.415287\pi$
$384$	0	0
$385$	1.64086	0.0836257
$386$	0	0
$387$	5.41462	0.275240
$388$	0	0
$389$	−5.79233	−0.293683	−0.146841	−	0.989160i	$-0.546911\pi$
$389$	−5.79233	−0.293683	−0.146841	+	0.989160i	$0.546911\pi$
$390$	0	0
$391$	6.33442	0.320346
$392$	0	0
$393$	−12.6634	−0.638785
$394$	0	0
$395$	−3.55047	−0.178644
$396$	0	0
$397$	23.0832	1.15851	0.579256	−	0.815146i	$-0.303343\pi$
$397$	23.0832	1.15851	0.579256	+	0.815146i	$0.303343\pi$
$398$	0	0
$399$	2.64328	0.132330
$400$	0	0
$401$	38.8769	1.94142	0.970709	−	0.240257i	$-0.0772318\pi$
$401$	38.8769	1.94142	0.970709	+	0.240257i	$0.0772318\pi$
$402$	0	0
$403$	−8.53064	−0.424941
$404$	0	0
$405$	2.73148	0.135729
$406$	0	0
$407$	13.5114	0.669738
$408$	0	0
$409$	6.07439	0.300359	0.150180	−	0.988659i	$-0.452015\pi$
$409$	6.07439	0.300359	0.150180	+	0.988659i	$0.452015\pi$
$410$	0	0
$411$	7.30713	0.360434
$412$	0	0
$413$	8.97577	0.441669
$414$	0	0
$415$	5.00858	0.245861
$416$	0	0
$417$	11.3474	0.555682
$418$	0	0
$419$	28.5974	1.39707	0.698537	−	0.715574i	$-0.253835\pi$
$419$	28.5974	1.39707	0.698537	+	0.715574i	$0.253835\pi$
$420$	0	0
$421$	18.4634	0.899850	0.449925	−	0.893066i	$-0.351451\pi$
$421$	18.4634	0.899850	0.449925	+	0.893066i	$0.351451\pi$
$422$	0	0
$423$	5.66527	0.275455
$424$	0	0
$425$	8.23243	0.399331
$426$	0	0
$427$	15.8092	0.765059
$428$	0	0
$429$	−4.61214	−0.222676
$430$	0	0
$431$	−2.44836	−0.117934	−0.0589668	−	0.998260i	$-0.518781\pi$
$431$	−2.44836	−0.117934	−0.0589668	+	0.998260i	$0.518781\pi$
$432$	0	0
$433$	−17.5261	−0.842249	−0.421124	−	0.907003i	$-0.638364\pi$
$433$	−17.5261	−0.842249	−0.421124	+	0.907003i	$0.638364\pi$
$434$	0	0
$435$	4.36056	0.209073
$436$	0	0
$437$	−3.69103	−0.176566
$438$	0	0
$439$	−11.0511	−0.527442	−0.263721	−	0.964599i	$-0.584950\pi$
$439$	−11.0511	−0.527442	−0.263721	+	0.964599i	$0.584950\pi$
$440$	0	0
$441$	2.99867	0.142794
$442$	0	0
$443$	−39.6411	−1.88340	−0.941702	−	0.336448i	$-0.890775\pi$
$443$	−39.6411	−1.88340	−0.941702	+	0.336448i	$0.890775\pi$
$444$	0	0
$445$	−6.44269	−0.305413
$446$	0	0
$447$	17.1453	0.810945
$448$	0	0
$449$	−9.54640	−0.450522	−0.225261	−	0.974298i	$-0.572324\pi$
$449$	−9.54640	−0.450522	−0.225261	+	0.974298i	$0.572324\pi$
$450$	0	0
$451$	−23.2273	−1.09373
$452$	0	0
$453$	3.41389	0.160399
$454$	0	0
$455$	−1.20344	−0.0564181
$456$	0	0
$457$	23.2297	1.08664	0.543319	−	0.839526i	$-0.317167\pi$
$457$	23.2297	1.08664	0.543319	+	0.839526i	$0.317167\pi$
$458$	0	0
$459$	9.66468	0.451108
$460$	0	0
$461$	−16.4008	−0.763861	−0.381930	−	0.924191i	$-0.624741\pi$
$461$	−16.4008	−0.763861	−0.381930	+	0.924191i	$0.624741\pi$
$462$	0	0
$463$	26.9018	1.25023	0.625116	−	0.780532i	$-0.285052\pi$
$463$	26.9018	1.25023	0.625116	+	0.780532i	$0.285052\pi$
$464$	0	0
$465$	3.80385	0.176399
$466$	0	0
$467$	−30.6374	−1.41773	−0.708866	−	0.705344i	$-0.750793\pi$
$467$	−30.6374	−1.41773	−0.708866	+	0.705344i	$0.750793\pi$
$468$	0	0
$469$	17.0432	0.786980
$470$	0	0
$471$	−27.5814	−1.27089
$472$	0	0
$473$	14.3006	0.657542
$474$	0	0
$475$	−4.79699	−0.220101
$476$	0	0
$477$	0.777652	0.0356062
$478$	0	0
$479$	−20.3572	−0.930145	−0.465073	−	0.885273i	$-0.653972\pi$
$479$	−20.3572	−0.930145	−0.465073	+	0.885273i	$0.653972\pi$
$480$	0	0
$481$	−9.90958	−0.451838
$482$	0	0
$483$	−9.75644	−0.443933
$484$	0	0
$485$	6.29865	0.286007
$486$	0	0
$487$	19.0270	0.862194	0.431097	−	0.902306i	$-0.358127\pi$
$487$	19.0270	0.862194	0.431097	+	0.902306i	$0.358127\pi$
$488$	0	0
$489$	−16.6115	−0.751196
$490$	0	0
$491$	−35.7862	−1.61501	−0.807504	−	0.589862i	$-0.799182\pi$
$491$	−35.7862	−1.61501	−0.807504	+	0.589862i	$0.799182\pi$
$492$	0	0
$493$	11.1413	0.501777
$494$	0	0
$495$	−0.719645	−0.0323456
$496$	0	0
$497$	28.3662	1.27240
$498$	0	0
$499$	28.4786	1.27488	0.637439	−	0.770501i	$-0.279994\pi$
$499$	28.4786	1.27488	0.637439	+	0.770501i	$0.279994\pi$
$500$	0	0
$501$	6.74529	0.301357
$502$	0	0
$503$	−29.3225	−1.30743	−0.653713	−	0.756743i	$-0.726790\pi$
$503$	−29.3225	−1.30743	−0.653713	+	0.756743i	$0.726790\pi$
$504$	0	0
$505$	−2.17681	−0.0968668
$506$	0	0
$507$	−15.9972	−0.710459
$508$	0	0
$509$	5.53636	0.245395	0.122698	−	0.992444i	$-0.460845\pi$
$509$	5.53636	0.245395	0.122698	+	0.992444i	$0.460845\pi$
$510$	0	0
$511$	−25.3498	−1.12141
$512$	0	0
$513$	−5.63155	−0.248639
$514$	0	0
$515$	−0.562250	−0.0247757
$516$	0	0
$517$	14.9626	0.658054
$518$	0	0
$519$	1.90166	0.0834737
$520$	0	0
$521$	41.6296	1.82383	0.911914	−	0.410382i	$-0.134605\pi$
$521$	41.6296	1.82383	0.911914	+	0.410382i	$0.134605\pi$
$522$	0	0
$523$	−21.7927	−0.952929	−0.476465	−	0.879194i	$-0.658082\pi$
$523$	−21.7927	−0.952929	−0.476465	+	0.879194i	$0.658082\pi$
$524$	0	0
$525$	−12.6798	−0.553391
$526$	0	0
$527$	9.71887	0.423360
$528$	0	0
$529$	−9.37628	−0.407664
$530$	0	0
$531$	−3.93659	−0.170833
$532$	0	0
$533$	17.0354	0.737884
$534$	0	0
$535$	2.26498	0.0979237
$536$	0	0
$537$	−5.93751	−0.256223
$538$	0	0
$539$	7.91981	0.341130
$540$	0	0
$541$	16.7738	0.721161	0.360581	−	0.932728i	$-0.382579\pi$
$541$	16.7738	0.721161	0.360581	+	0.932728i	$0.382579\pi$
$542$	0	0
$543$	11.0835	0.475637
$544$	0	0
$545$	6.68312	0.286273
$546$	0	0
$547$	10.8972	0.465930	0.232965	−	0.972485i	$-0.425157\pi$
$547$	10.8972	0.465930	0.232965	+	0.972485i	$0.425157\pi$
$548$	0	0
$549$	−6.93357	−0.295918
$550$	0	0
$551$	−6.49195	−0.276566
$552$	0	0
$553$	13.9721	0.594155
$554$	0	0
$555$	4.41873	0.187565
$556$	0	0
$557$	−11.3973	−0.482921	−0.241460	−	0.970411i	$-0.577626\pi$
$557$	−11.3973	−0.482921	−0.241460	+	0.970411i	$0.577626\pi$
$558$	0	0
$559$	−10.4884	−0.443610
$560$	0	0
$561$	5.25457	0.221848
$562$	0	0
$563$	−19.2183	−0.809956	−0.404978	−	0.914326i	$-0.632721\pi$
$563$	−19.2183	−0.809956	−0.404978	+	0.914326i	$0.632721\pi$
$564$	0	0
$565$	−0.343267	−0.0144414
$566$	0	0
$567$	−10.7492	−0.451423
$568$	0	0
$569$	19.8994	0.834227	0.417114	−	0.908854i	$-0.363042\pi$
$569$	19.8994	0.834227	0.417114	+	0.908854i	$0.363042\pi$
$570$	0	0
$571$	−29.6171	−1.23944	−0.619718	−	0.784825i	$-0.712753\pi$
$571$	−29.6171	−1.23944	−0.619718	+	0.784825i	$0.712753\pi$
$572$	0	0
$573$	28.4481	1.18844
$574$	0	0
$575$	17.7058	0.738384
$576$	0	0
$577$	30.2884	1.26092	0.630461	−	0.776221i	$-0.282866\pi$
$577$	30.2884	1.26092	0.630461	+	0.776221i	$0.282866\pi$
$578$	0	0
$579$	9.21233	0.382851
$580$	0	0
$581$	−19.7102	−0.817716
$582$	0	0
$583$	2.05386	0.0850623
$584$	0	0
$585$	0.527803	0.0218220
$586$	0	0
$587$	−28.7310	−1.18586	−0.592929	−	0.805255i	$-0.702028\pi$
$587$	−28.7310	−1.18586	−0.592929	+	0.805255i	$0.702028\pi$
$588$	0	0
$589$	−5.66313	−0.233345
$590$	0	0
$591$	−0.641436	−0.0263851
$592$	0	0
$593$	37.8789	1.55550	0.777751	−	0.628573i	$-0.216361\pi$
$593$	37.8789	1.55550	0.777751	+	0.628573i	$0.216361\pi$
$594$	0	0
$595$	1.37106	0.0562082
$596$	0	0
$597$	−11.9976	−0.491028
$598$	0	0
$599$	25.4176	1.03854	0.519268	−	0.854611i	$-0.326205\pi$
$599$	25.4176	1.03854	0.519268	+	0.854611i	$0.326205\pi$
$600$	0	0
$601$	−31.1664	−1.27130	−0.635652	−	0.771976i	$-0.719269\pi$
$601$	−31.1664	−1.27130	−0.635652	+	0.771976i	$0.719269\pi$
$602$	0	0
$603$	−7.47478	−0.304396
$604$	0	0
$605$	3.05560	0.124228
$606$	0	0
$607$	−10.1636	−0.412528	−0.206264	−	0.978496i	$-0.566131\pi$
$607$	−10.1636	−0.412528	−0.206264	+	0.978496i	$0.566131\pi$
$608$	0	0
$609$	−17.1601	−0.695361
$610$	0	0
$611$	−10.9739	−0.443956
$612$	0	0
$613$	1.66555	0.0672708	0.0336354	−	0.999434i	$-0.489291\pi$
$613$	1.66555	0.0672708	0.0336354	+	0.999434i	$0.489291\pi$
$614$	0	0
$615$	−7.59616	−0.306307
$616$	0	0
$617$	−11.3425	−0.456632	−0.228316	−	0.973587i	$-0.573322\pi$
$617$	−11.3425	−0.456632	−0.228316	+	0.973587i	$0.573322\pi$
$618$	0	0
$619$	16.1517	0.649190	0.324595	−	0.945853i	$-0.394772\pi$
$619$	16.1517	0.649190	0.324595	+	0.945853i	$0.394772\pi$
$620$	0	0
$621$	20.7862	0.834123
$622$	0	0
$623$	25.3538	1.01578
$624$	0	0
$625$	21.9960	0.879841
$626$	0	0
$627$	−3.06181	−0.122277
$628$	0	0
$629$	11.2899	0.450157
$630$	0	0
$631$	34.6923	1.38108	0.690539	−	0.723295i	$-0.257373\pi$
$631$	34.6923	1.38108	0.690539	+	0.723295i	$0.257373\pi$
$632$	0	0
$633$	−14.3526	−0.570465
$634$	0	0
$635$	5.56626	0.220890
$636$	0	0
$637$	−5.80856	−0.230144
$638$	0	0
$639$	−12.4408	−0.492151
$640$	0	0
$641$	26.6377	1.05213	0.526063	−	0.850446i	$-0.323668\pi$
$641$	26.6377	1.05213	0.526063	+	0.850446i	$0.323668\pi$
$642$	0	0
$643$	11.6494	0.459406	0.229703	−	0.973261i	$-0.426225\pi$
$643$	11.6494	0.459406	0.229703	+	0.973261i	$0.426225\pi$
$644$	0	0
$645$	4.67681	0.184149
$646$	0	0
$647$	−40.1191	−1.57724	−0.788622	−	0.614879i	$-0.789205\pi$
$647$	−40.1191	−1.57724	−0.788622	+	0.614879i	$0.789205\pi$
$648$	0	0
$649$	−10.3970	−0.408116
$650$	0	0
$651$	−14.9692	−0.586691
$652$	0	0
$653$	−37.0276	−1.44900	−0.724502	−	0.689273i	$-0.757930\pi$
$653$	−37.0276	−1.44900	−0.724502	+	0.689273i	$0.757930\pi$
$654$	0	0
$655$	3.82742	0.149550
$656$	0	0
$657$	11.1179	0.433751
$658$	0	0
$659$	29.9487	1.16663	0.583317	−	0.812245i	$-0.301755\pi$
$659$	29.9487	1.16663	0.583317	+	0.812245i	$0.301755\pi$
$660$	0	0
$661$	10.5271	0.409455	0.204727	−	0.978819i	$-0.434369\pi$
$661$	10.5271	0.409455	0.204727	+	0.978819i	$0.434369\pi$
$662$	0	0
$663$	−3.85381	−0.149670
$664$	0	0
$665$	−0.798912	−0.0309805
$666$	0	0
$667$	23.9620	0.927813
$668$	0	0
$669$	−24.2291	−0.936751
$670$	0	0
$671$	−18.3123	−0.706939
$672$	0	0
$673$	28.9154	1.11461	0.557303	−	0.830309i	$-0.311836\pi$
$673$	28.9154	1.11461	0.557303	+	0.830309i	$0.311836\pi$
$674$	0	0
$675$	27.0145	1.03979
$676$	0	0
$677$	−28.2330	−1.08508	−0.542542	−	0.840029i	$-0.682538\pi$
$677$	−28.2330	−1.08508	−0.542542	+	0.840029i	$0.682538\pi$
$678$	0	0
$679$	−24.7870	−0.951237
$680$	0	0
$681$	33.9730	1.30185
$682$	0	0
$683$	23.8658	0.913200	0.456600	−	0.889672i	$-0.349067\pi$
$683$	23.8658	0.913200	0.456600	+	0.889672i	$0.349067\pi$
$684$	0	0
$685$	−2.20852	−0.0843833
$686$	0	0
$687$	24.2622	0.925662
$688$	0	0
$689$	−1.50635	−0.0573873
$690$	0	0
$691$	40.7956	1.55194	0.775969	−	0.630771i	$-0.217261\pi$
$691$	40.7956	1.55194	0.775969	+	0.630771i	$0.217261\pi$
$692$	0	0
$693$	2.83201	0.107579
$694$	0	0
$695$	−3.42965	−0.130094
$696$	0	0
$697$	−19.4082	−0.735139
$698$	0	0
$699$	−5.96220	−0.225511
$700$	0	0
$701$	−30.6058	−1.15597	−0.577983	−	0.816049i	$-0.696160\pi$
$701$	−30.6058	−1.15597	−0.577983	+	0.816049i	$0.696160\pi$
$702$	0	0
$703$	−6.57855	−0.248115
$704$	0	0
$705$	4.89331	0.184293
$706$	0	0
$707$	8.56637	0.322172
$708$	0	0
$709$	−10.7141	−0.402376	−0.201188	−	0.979553i	$-0.564480\pi$
$709$	−10.7141	−0.402376	−0.201188	+	0.979553i	$0.564480\pi$
$710$	0	0
$711$	−6.12788	−0.229814
$712$	0	0
$713$	20.9028	0.782816
$714$	0	0
$715$	1.39398	0.0521321
$716$	0	0
$717$	1.44219	0.0538595
$718$	0	0
$719$	−48.7945	−1.81973	−0.909865	−	0.414905i	$-0.863815\pi$
$719$	−48.7945	−1.81973	−0.909865	+	0.414905i	$0.863815\pi$
$720$	0	0
$721$	2.21262	0.0824021
$722$	0	0
$723$	20.9443	0.778927
$724$	0	0
$725$	31.1418	1.15658
$726$	0	0
$727$	−32.4133	−1.20214	−0.601071	−	0.799196i	$-0.705259\pi$
$727$	−32.4133	−1.20214	−0.601071	+	0.799196i	$0.705259\pi$
$728$	0	0
$729$	29.9001	1.10741
$730$	0	0
$731$	11.9493	0.441960
$732$	0	0
$733$	−29.6655	−1.09572	−0.547860	−	0.836570i	$-0.684557\pi$
$733$	−29.6655	−1.09572	−0.547860	+	0.836570i	$0.684557\pi$
$734$	0	0
$735$	2.59006	0.0955360
$736$	0	0
$737$	−19.7417	−0.727195
$738$	0	0
$739$	−40.9934	−1.50797	−0.753983	−	0.656893i	$-0.771870\pi$
$739$	−40.9934	−1.50797	−0.753983	+	0.656893i	$0.771870\pi$
$740$	0	0
$741$	2.24559	0.0824939
$742$	0	0
$743$	−8.29854	−0.304444	−0.152222	−	0.988346i	$-0.548643\pi$
$743$	−8.29854	−0.304444	−0.152222	+	0.988346i	$0.548643\pi$
$744$	0	0
$745$	−5.18204	−0.189855
$746$	0	0
$747$	8.64448	0.316285
$748$	0	0
$749$	−8.91335	−0.325687
$750$	0	0
$751$	−30.7350	−1.12153	−0.560767	−	0.827973i	$-0.689494\pi$
$751$	−30.7350	−1.12153	−0.560767	+	0.827973i	$0.689494\pi$
$752$	0	0
$753$	23.6154	0.860594
$754$	0	0
$755$	−1.03182	−0.0375519
$756$	0	0
$757$	−13.1451	−0.477767	−0.238883	−	0.971048i	$-0.576781\pi$
$757$	−13.1451	−0.477767	−0.238883	+	0.971048i	$0.576781\pi$
$758$	0	0
$759$	11.3012	0.410208
$760$	0	0
$761$	27.7368	1.00546	0.502729	−	0.864444i	$-0.332329\pi$
$761$	27.7368	1.00546	0.502729	+	0.864444i	$0.332329\pi$
$762$	0	0
$763$	−26.3000	−0.952123
$764$	0	0
$765$	−0.601320	−0.0217408
$766$	0	0
$767$	7.62535	0.275335
$768$	0	0
$769$	−15.4980	−0.558872	−0.279436	−	0.960164i	$-0.590148\pi$
$769$	−15.4980	−0.558872	−0.279436	+	0.960164i	$0.590148\pi$
$770$	0	0
$771$	11.6744	0.420443
$772$	0	0
$773$	41.8050	1.50362	0.751811	−	0.659378i	$-0.229180\pi$
$773$	41.8050	1.50362	0.751811	+	0.659378i	$0.229180\pi$
$774$	0	0
$775$	27.1660	0.975830
$776$	0	0
$777$	−17.3890	−0.623826
$778$	0	0
$779$	11.3091	0.405189
$780$	0	0
$781$	−32.8575	−1.17573
$782$	0	0
$783$	36.5598	1.30654
$784$	0	0
$785$	8.33628	0.297535
$786$	0	0
$787$	−39.2552	−1.39930	−0.699649	−	0.714487i	$-0.746660\pi$
$787$	−39.2552	−1.39930	−0.699649	+	0.714487i	$0.746660\pi$
$788$	0	0
$789$	4.09204	0.145681
$790$	0	0
$791$	1.35085	0.0480308
$792$	0	0
$793$	13.4306	0.476936
$794$	0	0
$795$	0.671688	0.0238223
$796$	0	0
$797$	13.7185	0.485936	0.242968	−	0.970034i	$-0.421879\pi$
$797$	13.7185	0.485936	0.242968	+	0.970034i	$0.421879\pi$
$798$	0	0
$799$	12.5024	0.442304
$800$	0	0
$801$	−11.1197	−0.392894
$802$	0	0
$803$	29.3636	1.03622
$804$	0	0
$805$	2.94881	0.103932
$806$	0	0
$807$	−26.9864	−0.949967
$808$	0	0
$809$	−1.77388	−0.0623661	−0.0311831	−	0.999514i	$-0.509927\pi$
$809$	−1.77388	−0.0623661	−0.0311831	+	0.999514i	$0.509927\pi$
$810$	0	0
$811$	31.6253	1.11052	0.555258	−	0.831678i	$-0.312620\pi$
$811$	31.6253	1.11052	0.555258	+	0.831678i	$0.312620\pi$
$812$	0	0
$813$	−44.4580	−1.55921
$814$	0	0
$815$	5.02068	0.175867
$816$	0	0
$817$	−6.96277	−0.243597
$818$	0	0
$819$	−2.07705	−0.0725782
$820$	0	0
$821$	16.4757	0.575006	0.287503	−	0.957780i	$-0.407175\pi$
$821$	16.4757	0.575006	0.287503	+	0.957780i	$0.407175\pi$
$822$	0	0
$823$	−6.96616	−0.242825	−0.121413	−	0.992602i	$-0.538742\pi$
$823$	−6.96616	−0.242825	−0.121413	+	0.992602i	$0.538742\pi$
$824$	0	0
$825$	14.6874	0.511351
$826$	0	0
$827$	−45.3291	−1.57625	−0.788124	−	0.615516i	$-0.788948\pi$
$827$	−45.3291	−1.57625	−0.788124	+	0.615516i	$0.788948\pi$
$828$	0	0
$829$	3.01741	0.104799	0.0523995	−	0.998626i	$-0.483313\pi$
$829$	3.01741	0.104799	0.0523995	+	0.998626i	$0.483313\pi$
$830$	0	0
$831$	−24.9020	−0.863839
$832$	0	0
$833$	6.61763	0.229287
$834$	0	0
$835$	−2.03871	−0.0705525
$836$	0	0
$837$	31.8922	1.10236
$838$	0	0
$839$	33.0995	1.14272	0.571360	−	0.820699i	$-0.306416\pi$
$839$	33.0995	1.14272	0.571360	+	0.820699i	$0.306416\pi$
$840$	0	0
$841$	13.1454	0.453291
$842$	0	0
$843$	−20.4709	−0.705054
$844$	0	0
$845$	4.83502	0.166330
$846$	0	0
$847$	−12.0247	−0.413172
$848$	0	0
$849$	12.8367	0.440556
$850$	0	0
$851$	24.2816	0.832364
$852$	0	0
$853$	−5.39996	−0.184891	−0.0924456	−	0.995718i	$-0.529468\pi$
$853$	−5.39996	−0.184891	−0.0924456	+	0.995718i	$0.529468\pi$
$854$	0	0
$855$	0.350386	0.0119829
$856$	0	0
$857$	27.8463	0.951212	0.475606	−	0.879658i	$-0.342229\pi$
$857$	27.8463	0.951212	0.475606	+	0.879658i	$0.342229\pi$
$858$	0	0
$859$	7.77419	0.265252	0.132626	−	0.991166i	$-0.457659\pi$
$859$	7.77419	0.265252	0.132626	+	0.991166i	$0.457659\pi$
$860$	0	0
$861$	29.8930	1.01875
$862$	0	0
$863$	−45.6475	−1.55386	−0.776930	−	0.629587i	$-0.783224\pi$
$863$	−45.6475	−1.55386	−0.776930	+	0.629587i	$0.783224\pi$
$864$	0	0
$865$	−0.574763	−0.0195425
$866$	0	0
$867$	−20.9522	−0.711574
$868$	0	0
$869$	−16.1844	−0.549018
$870$	0	0
$871$	14.4790	0.490602
$872$	0	0
$873$	10.8711	0.367929
$874$	0	0
$875$	7.82693	0.264598
$876$	0	0
$877$	12.3435	0.416809	0.208404	−	0.978043i	$-0.433173\pi$
$877$	12.3435	0.416809	0.208404	+	0.978043i	$0.433173\pi$
$878$	0	0
$879$	−34.8411	−1.17516
$880$	0	0
$881$	46.3454	1.56142	0.780709	−	0.624895i	$-0.214858\pi$
$881$	46.3454	1.56142	0.780709	+	0.624895i	$0.214858\pi$
$882$	0	0
$883$	−10.1126	−0.340316	−0.170158	−	0.985417i	$-0.554428\pi$
$883$	−10.1126	−0.340316	−0.170158	+	0.985417i	$0.554428\pi$
$884$	0	0
$885$	−3.40018	−0.114296
$886$	0	0
$887$	−21.2547	−0.713662	−0.356831	−	0.934169i	$-0.616143\pi$
$887$	−21.2547	−0.713662	−0.356831	+	0.934169i	$0.616143\pi$
$888$	0	0
$889$	−21.9048	−0.734664
$890$	0	0
$891$	12.4511	0.417129
$892$	0	0
$893$	−7.28510	−0.243786
$894$	0	0
$895$	1.79457	0.0599858
$896$	0	0
$897$	−8.28856	−0.276747
$898$	0	0
$899$	36.7647	1.22617
$900$	0	0
$901$	1.71617	0.0571738
$902$	0	0
$903$	−18.4046	−0.612466
$904$	0	0
$905$	−3.34989	−0.111354
$906$	0	0
$907$	−26.9858	−0.896050	−0.448025	−	0.894021i	$-0.647872\pi$
$907$	−26.9858	−0.896050	−0.448025	+	0.894021i	$0.647872\pi$
$908$	0	0
$909$	−3.75703	−0.124613
$910$	0	0
$911$	−16.3678	−0.542291	−0.271145	−	0.962538i	$-0.587402\pi$
$911$	−16.3678	−0.542291	−0.271145	+	0.962538i	$0.587402\pi$
$912$	0	0
$913$	22.8310	0.755596
$914$	0	0
$915$	−5.98879	−0.197983
$916$	0	0
$917$	−15.0620	−0.497391
$918$	0	0
$919$	19.4116	0.640330	0.320165	−	0.947362i	$-0.396262\pi$
$919$	19.4116	0.640330	0.320165	+	0.947362i	$0.396262\pi$
$920$	0	0
$921$	−35.3352	−1.16433
$922$	0	0
$923$	24.0984	0.793208
$924$	0	0
$925$	31.5572	1.03760
$926$	0	0
$927$	−0.970407	−0.0318723
$928$	0	0
$929$	31.5712	1.03582	0.517909	−	0.855436i	$-0.326711\pi$
$929$	31.5712	1.03582	0.517909	+	0.855436i	$0.326711\pi$
$930$	0	0
$931$	−3.85606	−0.126377
$932$	0	0
$933$	−51.4306	−1.68376
$934$	0	0
$935$	−1.58815	−0.0519381
$936$	0	0
$937$	44.6968	1.46018	0.730091	−	0.683350i	$-0.239478\pi$
$937$	44.6968	1.46018	0.730091	+	0.683350i	$0.239478\pi$
$938$	0	0
$939$	−29.1679	−0.951860
$940$	0	0
$941$	−34.0329	−1.10944	−0.554720	−	0.832037i	$-0.687175\pi$
$941$	−34.0329	−1.10944	−0.554720	+	0.832037i	$0.687175\pi$
$942$	0	0
$943$	−41.7421	−1.35931
$944$	0	0
$945$	4.49911	0.146356
$946$	0	0
$947$	41.1924	1.33857	0.669287	−	0.743004i	$-0.266600\pi$
$947$	41.1924	1.33857	0.669287	+	0.743004i	$0.266600\pi$
$948$	0	0
$949$	−21.5359	−0.699085
$950$	0	0
$951$	−6.87988	−0.223096
$952$	0	0
$953$	50.5466	1.63737	0.818683	−	0.574246i	$-0.194705\pi$
$953$	50.5466	1.63737	0.818683	+	0.574246i	$0.194705\pi$
$954$	0	0
$955$	−8.59822	−0.278232
$956$	0	0
$957$	19.8771	0.642535
$958$	0	0
$959$	8.69117	0.280652
$960$	0	0
$961$	1.07101	0.0345488
$962$	0	0
$963$	3.90921	0.125972
$964$	0	0
$965$	−2.78436	−0.0896316
$966$	0	0
$967$	11.7886	0.379095	0.189548	−	0.981872i	$-0.439298\pi$
$967$	11.7886	0.379095	0.189548	+	0.981872i	$0.439298\pi$
$968$	0	0
$969$	−2.55838	−0.0821871
$970$	0	0
$971$	−26.6282	−0.854540	−0.427270	−	0.904124i	$-0.640525\pi$
$971$	−26.6282	−0.854540	−0.427270	+	0.904124i	$0.640525\pi$
$972$	0	0
$973$	13.4966	0.432683
$974$	0	0
$975$	−10.7721	−0.344983
$976$	0	0
$977$	22.8179	0.730010	0.365005	−	0.931005i	$-0.381067\pi$
$977$	22.8179	0.730010	0.365005	+	0.931005i	$0.381067\pi$
$978$	0	0
$979$	−29.3682	−0.938612
$980$	0	0
$981$	11.5346	0.368272
$982$	0	0
$983$	−11.1265	−0.354881	−0.177440	−	0.984132i	$-0.556782\pi$
$983$	−11.1265	−0.354881	−0.177440	+	0.984132i	$0.556782\pi$
$984$	0	0
$985$	0.193869	0.00617718
$986$	0	0
$987$	−19.2566	−0.612943
$988$	0	0
$989$	25.6998	0.817207
$990$	0	0
$991$	−10.0376	−0.318856	−0.159428	−	0.987210i	$-0.550965\pi$
$991$	−10.0376	−0.318856	−0.159428	+	0.987210i	$0.550965\pi$
$992$	0	0
$993$	43.0173	1.36511
$994$	0	0
$995$	3.62618	0.114958
$996$	0	0
$997$	41.6858	1.32020	0.660102	−	0.751176i	$-0.270513\pi$
$997$	41.6858	1.32020	0.660102	+	0.751176i	$0.270513\pi$
$998$	0	0
$999$	37.0474	1.17213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.c.1.14	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.c.1.14	✓	19	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49075 q^{3} -0.450569 q^{5} +1.77312 q^{7} -0.777652 q^{9} +O(q^{10})\)	\(q+1.49075 q^{3} -0.450569 q^{5} +1.77312 q^{7} -0.777652 q^{9} -2.05386 q^{11} +1.50635 q^{13} -0.671688 q^{15} -1.71617 q^{17} +1.00000 q^{19} +2.64328 q^{21} -3.69103 q^{23} -4.79699 q^{25} -5.63155 q^{27} -6.49195 q^{29} -5.66313 q^{31} -3.06181 q^{33} -0.798912 q^{35} -6.57855 q^{37} +2.24559 q^{39} +11.3091 q^{41} -6.96277 q^{43} +0.350386 q^{45} -7.28510 q^{47} -3.85606 q^{49} -2.55838 q^{51} -1.00000 q^{53} +0.925407 q^{55} +1.49075 q^{57} +5.06214 q^{59} +8.91603 q^{61} -1.37887 q^{63} -0.678713 q^{65} +9.61198 q^{67} -5.50242 q^{69} +15.9979 q^{71} -14.2968 q^{73} -7.15113 q^{75} -3.64174 q^{77} +7.87998 q^{79} -6.06230 q^{81} -11.1161 q^{83} +0.773251 q^{85} -9.67790 q^{87} +14.2990 q^{89} +2.67093 q^{91} -8.44233 q^{93} -0.450569 q^{95} -13.9793 q^{97} +1.59719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9	\( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9} - 3 q^{11} - 23 q^{13} - 18 q^{15} - 7 q^{17} + 19 q^{19} - 4 q^{21} - 6 q^{23} + 18 q^{25} - 17 q^{27} - 4 q^{29} - 30 q^{31} - 10 q^{33} - q^{35} - 31 q^{37} + 5 q^{39} - 15 q^{41} - 29 q^{43} + 6 q^{45} - 18 q^{47} + 23 q^{49} - 5 q^{51} - 19 q^{53} - 19 q^{55} - 5 q^{57} + 8 q^{59} - 4 q^{61} - 64 q^{63} - 26 q^{65} - 62 q^{67} + 3 q^{69} - 17 q^{71} + q^{73} - 40 q^{75} - 14 q^{77} - 28 q^{79} + 11 q^{81} + 4 q^{83} - 31 q^{85} - 20 q^{87} + 33 q^{89} - 29 q^{91} - 59 q^{93} - 5 q^{95} + 5 q^{97} - 48 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9 - 3 * q^11 - 23 * q^13 - 18 * q^15 - 7 * q^17 + 19 * q^19 - 4 * q^21 - 6 * q^23 + 18 * q^25 - 17 * q^27 - 4 * q^29 - 30 * q^31 - 10 * q^33 - q^35 - 31 * q^37 + 5 * q^39 - 15 * q^41 - 29 * q^43 + 6 * q^45 - 18 * q^47 + 23 * q^49 - 5 * q^51 - 19 * q^53 - 19 * q^55 - 5 * q^57 + 8 * q^59 - 4 * q^61 - 64 * q^63 - 26 * q^65 - 62 * q^67 + 3 * q^69 - 17 * q^71 + q^73 - 40 * q^75 - 14 * q^77 - 28 * q^79 + 11 * q^81 + 4 * q^83 - 31 * q^85 - 20 * q^87 + 33 * q^89 - 29 * q^91 - 59 * q^93 - 5 * q^95 + 5 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more