LMFDB - Embedded newform 4028.2.c.a.3497.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(3497,4028)]

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, 1]))

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

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

chi := DirichletCharacter("4028.3497");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.c (of order $2$, degree $1$, 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:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3497.9
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.74

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65407i q^{3} -1.30709i q^{5} -2.08185 q^{7} -4.04406 q^{9} +O(q^{10})$	$q-2.65407i q^{3} -1.30709i q^{5} -2.08185 q^{7} -4.04406 q^{9} -2.50490 q^{11} -0.453285 q^{13} -3.46909 q^{15} +6.25452 q^{17} +1.00000i q^{19} +5.52536i q^{21} +8.56428i q^{23} +3.29153 q^{25} +2.77101i q^{27} -5.29840 q^{29} +9.83780i q^{31} +6.64817i q^{33} +2.72115i q^{35} -0.222341 q^{37} +1.20305i q^{39} +3.55585i q^{41} -1.53908 q^{43} +5.28593i q^{45} +3.79991 q^{47} -2.66591 q^{49} -16.5999i q^{51} +(-0.187601 + 7.27769i) q^{53} +3.27412i q^{55} +2.65407 q^{57} -2.29705 q^{59} -0.204172i q^{61} +8.41912 q^{63} +0.592482i q^{65} +1.57136i q^{67} +22.7302 q^{69} +7.39944i q^{71} -10.7387i q^{73} -8.73593i q^{75} +5.21482 q^{77} +1.68652i q^{79} -4.77775 q^{81} -4.65621i q^{83} -8.17519i q^{85} +14.0623i q^{87} -3.61379 q^{89} +0.943671 q^{91} +26.1102 q^{93} +1.30709 q^{95} +19.5462 q^{97} +10.1300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) $ 82 * q - 8 * q^7 - 82 * q^9	$ 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) $ 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times$.

$n$	$2015$	$2281$	$2757$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	2.65407i		−	1.53233i	−0.642647	−	0.766163i	$-0.722164\pi$
$3$		−	2.65407i		−	1.53233i	0.642647	−	0.766163i	$-0.277836\pi$
$4$		0			0
$5$		−	1.30709i		−	0.584546i	−0.956335	−	0.292273i	$-0.905588\pi$
$5$		−	1.30709i		−	0.584546i	0.956335	−	0.292273i	$-0.0944117\pi$
$6$		0			0
$7$	−2.08185			−0.786865			−0.393432	−	0.919354i	$-0.628712\pi$
$7$	−2.08185			−0.786865			−0.393432	+	0.919354i	$0.628712\pi$
$8$		0			0
$9$	−4.04406			−1.34802
$10$		0			0
$11$	−2.50490			−0.755256			−0.377628	−	0.925957i	$-0.623260\pi$
$11$	−2.50490			−0.755256			−0.377628	+	0.925957i	$0.623260\pi$
$12$		0			0
$13$	−0.453285			−0.125719			−0.0628593	−	0.998022i	$-0.520022\pi$
$13$	−0.453285			−0.125719			−0.0628593	+	0.998022i	$0.520022\pi$
$14$		0			0
$15$	−3.46909			−0.895715
$16$		0			0
$17$	6.25452			1.51694			0.758472	−	0.651706i	$-0.225946\pi$
$17$	6.25452			1.51694			0.758472	+	0.651706i	$0.225946\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$			5.52536i			1.20573i
$22$		0			0
$23$			8.56428i			1.78578i	0.450278	+	0.892888i	$0.351325\pi$
$23$			8.56428i			1.78578i	−0.450278	+	0.892888i	$0.648675\pi$
$24$		0			0
$25$	3.29153			0.658305
$26$		0			0
$27$			2.77101i			0.533281i
$28$		0			0
$29$	−5.29840			−0.983889			−0.491944	−	0.870627i	$-0.663714\pi$
$29$	−5.29840			−0.983889			−0.491944	+	0.870627i	$0.663714\pi$
$30$		0			0
$31$			9.83780i			1.76692i	0.468506	+	0.883460i	$0.344793\pi$
$31$			9.83780i			1.76692i	−0.468506	+	0.883460i	$0.655207\pi$
$32$		0			0
$33$			6.64817i			1.15730i
$34$		0			0
$35$			2.72115i			0.459959i
$36$		0			0
$37$	−0.222341			−0.0365527			−0.0182763	−	0.999833i	$-0.505818\pi$
$37$	−0.222341			−0.0365527			−0.0182763	+	0.999833i	$0.505818\pi$
$38$		0			0
$39$			1.20305i			0.192642i
$40$		0			0
$41$			3.55585i			0.555330i	0.960678	+	0.277665i	$0.0895606\pi$
$41$			3.55585i			0.555330i	−0.960678	+	0.277665i	$0.910439\pi$
$42$		0			0
$43$	−1.53908			−0.234707			−0.117354	−	0.993090i	$-0.537441\pi$
$43$	−1.53908			−0.234707			−0.117354	+	0.993090i	$0.537441\pi$
$44$		0			0
$45$			5.28593i			0.787981i
$46$		0			0
$47$	3.79991			0.554274			0.277137	−	0.960830i	$-0.410614\pi$
$47$	3.79991			0.554274			0.277137	+	0.960830i	$0.410614\pi$
$48$		0			0
$49$	−2.66591			−0.380844
$50$		0			0
$51$		−	16.5999i		−	2.32445i
$52$		0			0
$53$	−0.187601	+	7.27769i	−0.0257690	+	0.999668i
$53$	−0.187601	+	7.27769i	−0.0257690	+	0.999668i
$54$		0			0
$55$			3.27412i			0.441482i
$56$		0			0
$57$	2.65407			0.351540
$58$		0			0
$59$	−2.29705			−0.299051			−0.149525	−	0.988758i	$-0.547775\pi$
$59$	−2.29705			−0.299051			−0.149525	+	0.988758i	$0.547775\pi$
$60$		0			0
$61$		−	0.204172i		−	0.0261416i	−0.999915	−	0.0130708i	$-0.995839\pi$
$61$		−	0.204172i		−	0.0261416i	0.999915	−	0.0130708i	$-0.00416068\pi$
$62$		0			0
$63$	8.41912			1.06071
$64$		0			0
$65$			0.592482i			0.0734884i
$66$		0			0
$67$			1.57136i			0.191972i	0.995383	+	0.0959858i	$0.0306004\pi$
$67$			1.57136i			0.191972i	−0.995383	+	0.0959858i	$0.969400\pi$
$68$		0			0
$69$	22.7302			2.73639
$70$		0			0
$71$			7.39944i			0.878152i	0.898450	+	0.439076i	$0.144694\pi$
$71$			7.39944i			0.878152i	−0.898450	+	0.439076i	$0.855306\pi$
$72$		0			0
$73$		−	10.7387i		−	1.25687i	−0.777862	−	0.628436i	$-0.783696\pi$
$73$		−	10.7387i		−	1.25687i	0.777862	−	0.628436i	$-0.216304\pi$
$74$		0			0
$75$		−	8.73593i		−	1.00874i
$76$		0			0
$77$	5.21482			0.594284
$78$		0			0
$79$			1.68652i			0.189748i	0.995489	+	0.0948740i	$0.0302448\pi$
$79$			1.68652i			0.189748i	−0.995489	+	0.0948740i	$0.969755\pi$
$80$		0			0
$81$	−4.77775			−0.530861
$82$		0			0
$83$		−	4.65621i		−	0.511085i	−0.966798	−	0.255543i	$-0.917746\pi$
$83$		−	4.65621i		−	0.511085i	0.966798	−	0.255543i	$-0.0822541\pi$
$84$		0			0
$85$		−	8.17519i		−	0.886724i
$86$		0			0
$87$			14.0623i			1.50764i
$88$		0			0
$89$	−3.61379			−0.383061			−0.191531	−	0.981487i	$-0.561345\pi$
$89$	−3.61379			−0.383061			−0.191531	+	0.981487i	$0.561345\pi$
$90$		0			0
$91$	0.943671			0.0989236
$92$		0			0
$93$	26.1102			2.70750
$94$		0			0
$95$	1.30709			0.134104
$96$		0			0
$97$	19.5462			1.98462			0.992310	−	0.123777i	$-0.0395006\pi$
$97$	19.5462			1.98462			0.992310	+	0.123777i	$0.0395006\pi$
$98$		0			0
$99$	10.1300			1.01810
$100$		0			0
$101$		−	3.38621i		−	0.336940i	−0.985707	−	0.168470i	$-0.946117\pi$
$101$		−	3.38621i		−	0.336940i	0.985707	−	0.168470i	$-0.0538827\pi$
$102$		0			0
$103$		−	1.54565i		−	0.152297i	−0.997096	−	0.0761486i	$-0.975738\pi$
$103$		−	1.54565i		−	0.152297i	0.997096	−	0.0761486i	$-0.0242623\pi$
$104$		0			0
$105$	7.22212			0.704807
$106$		0			0
$107$	−15.0421			−1.45418			−0.727089	−	0.686543i	$-0.759127\pi$
$107$	−15.0421			−1.45418			−0.727089	+	0.686543i	$0.759127\pi$
$108$		0			0
$109$			14.4808i			1.38701i	0.720453	+	0.693504i	$0.243934\pi$
$109$			14.4808i			1.38701i	−0.720453	+	0.693504i	$0.756066\pi$
$110$		0			0
$111$			0.590108i			0.0560106i
$112$		0			0
$113$	13.6909			1.28793			0.643967	−	0.765054i	$-0.277288\pi$
$113$	13.6909			1.28793			0.643967	+	0.765054i	$0.277288\pi$
$114$		0			0
$115$	11.1943			1.04387
$116$		0			0
$117$	1.83311			0.169471
$118$		0			0
$119$	−13.0210			−1.19363
$120$		0			0
$121$	−4.72547			−0.429589
$122$		0			0
$123$	9.43746			0.850947
$124$		0			0
$125$		−	10.8377i		−	0.969357i
$126$		0			0
$127$			5.20322i			0.461711i	0.972988	+	0.230855i	$0.0741524\pi$
$127$			5.20322i			0.461711i	−0.972988	+	0.230855i	$0.925848\pi$
$128$		0			0
$129$			4.08482i			0.359648i
$130$		0			0
$131$	18.6150			1.62640			0.813200	−	0.581985i	$-0.197724\pi$
$131$	18.6150			1.62640			0.813200	+	0.581985i	$0.197724\pi$
$132$		0			0
$133$		−	2.08185i		−	0.180519i
$134$		0			0
$135$	3.62194			0.311727
$136$		0			0
$137$		−	8.01746i		−	0.684978i	−0.939522	−	0.342489i	$-0.888730\pi$
$137$		−	8.01746i		−	0.684978i	0.939522	−	0.342489i	$-0.111270\pi$
$138$		0			0
$139$			14.2069i			1.20501i	0.798115	+	0.602505i	$0.205831\pi$
$139$			14.2069i			1.20501i	−0.798115	+	0.602505i	$0.794169\pi$
$140$		0			0
$141$		−	10.0852i		−	0.849328i
$142$		0			0
$143$	1.13543			0.0949498
$144$		0			0
$145$			6.92546i			0.575129i
$146$		0			0
$147$			7.07549i			0.583577i
$148$		0			0
$149$	−7.16666			−0.587116			−0.293558	−	0.955941i	$-0.594839\pi$
$149$	−7.16666			−0.587116			−0.293558	+	0.955941i	$0.594839\pi$
$150$		0			0
$151$			15.3350i			1.24794i	0.781447	+	0.623971i	$0.214482\pi$
$151$			15.3350i			1.24794i	−0.781447	+	0.623971i	$0.785518\pi$
$152$		0			0
$153$	−25.2937			−2.04487
$154$		0			0
$155$	12.8588			1.03285
$156$		0			0
$157$			4.43039i			0.353583i	0.984248	+	0.176792i	$0.0565719\pi$
$157$			4.43039i			0.353583i	−0.984248	+	0.176792i	$0.943428\pi$
$158$		0			0
$159$	19.3155	+	0.497906i	1.53182	+	0.0394865i
$160$		0			0
$161$		−	17.8295i		−	1.40516i
$162$		0			0
$163$	−2.94804			−0.230909			−0.115454	−	0.993313i	$-0.536832\pi$
$163$	−2.94804			−0.230909			−0.115454	+	0.993313i	$0.536832\pi$
$164$		0			0
$165$	8.68972			0.676494
$166$		0			0
$167$		−	17.3097i		−	1.33947i	−0.742601	−	0.669734i	$-0.766408\pi$
$167$		−	17.3097i		−	1.33947i	0.742601	−	0.669734i	$-0.233592\pi$
$168$		0			0
$169$	−12.7945			−0.984195
$170$		0			0
$171$		−	4.04406i		−	0.309257i
$172$		0			0
$173$		−	6.66615i		−	0.506818i	−0.967359	−	0.253409i	$-0.918448\pi$
$173$		−	6.66615i		−	0.506818i	0.967359	−	0.253409i	$-0.0815519\pi$
$174$		0			0
$175$	−6.85246			−0.517997
$176$		0			0
$177$			6.09653i			0.458243i
$178$		0			0
$179$			14.9694i			1.11886i	0.828877	+	0.559431i	$0.188980\pi$
$179$			14.9694i			1.11886i	−0.828877	+	0.559431i	$0.811020\pi$
$180$		0			0
$181$		−	25.4141i		−	1.88901i	−0.328493	−	0.944506i	$-0.606541\pi$
$181$		−	25.4141i		−	1.88901i	0.328493	−	0.944506i	$-0.393459\pi$
$182$		0			0
$183$	−0.541886			−0.0400574
$184$		0			0
$185$			0.290619i			0.0213667i
$186$		0			0
$187$	−15.6669			−1.14568
$188$		0			0
$189$		−	5.76882i		−	0.419620i
$190$		0			0
$191$			12.5410i			0.907432i	0.891146	+	0.453716i	$0.149902\pi$
$191$			12.5410i			0.907432i	−0.891146	+	0.453716i	$0.850098\pi$
$192$		0			0
$193$		−	3.49657i		−	0.251689i	−0.992050	−	0.125844i	$-0.959836\pi$
$193$		−	3.49657i		−	0.251689i	0.992050	−	0.125844i	$-0.0401640\pi$
$194$		0			0
$195$	1.57249			0.112608
$196$		0			0
$197$	−6.06680			−0.432242			−0.216121	−	0.976367i	$-0.569341\pi$
$197$	−6.06680			−0.432242			−0.216121	+	0.976367i	$0.569341\pi$
$198$		0			0
$199$	15.9751			1.13244			0.566222	−	0.824253i	$-0.308405\pi$
$199$	15.9751			1.13244			0.566222	+	0.824253i	$0.308405\pi$
$200$		0			0
$201$	4.17048			0.294163
$202$		0			0
$203$	11.0305			0.774187
$204$		0			0
$205$	4.64780			0.324616
$206$		0			0
$207$		−	34.6345i		−	2.40726i
$208$		0			0
$209$		−	2.50490i		−	0.173268i
$210$		0			0
$211$	−20.6975			−1.42487			−0.712437	−	0.701736i	$-0.752408\pi$
$211$	−20.6975			−1.42487			−0.712437	+	0.701736i	$0.752408\pi$
$212$		0			0
$213$	19.6386			1.34561
$214$		0			0
$215$			2.01171i			0.137197i
$216$		0			0
$217$		−	20.4808i		−	1.39033i
$218$		0			0
$219$	−28.5012			−1.92594
$220$		0			0
$221$	−2.83508			−0.190708
$222$		0			0
$223$	22.2613			1.49073			0.745363	−	0.666658i	$-0.232276\pi$
$223$	22.2613			1.49073			0.745363	+	0.666658i	$0.232276\pi$
$224$		0			0
$225$	−13.3111			−0.887409
$226$		0			0
$227$	−7.35108			−0.487908			−0.243954	−	0.969787i	$-0.578445\pi$
$227$	−7.35108			−0.487908			−0.243954	+	0.969787i	$0.578445\pi$
$228$		0			0
$229$	−17.8314			−1.17833			−0.589165	−	0.808012i	$-0.700543\pi$
$229$	−17.8314			−1.17833			−0.589165	+	0.808012i	$0.700543\pi$
$230$		0			0
$231$		−	13.8405i		−	0.910637i
$232$		0			0
$233$			2.32921i			0.152591i	0.997085	+	0.0762957i	$0.0243093\pi$
$233$			2.32921i			0.152591i	−0.997085	+	0.0762957i	$0.975691\pi$
$234$		0			0
$235$		−	4.96681i		−	0.323999i
$236$		0			0
$237$	4.47612			0.290755
$238$		0			0
$239$			16.6485i			1.07690i	0.842657	+	0.538451i	$0.180990\pi$
$239$			16.6485i			1.07690i	−0.842657	+	0.538451i	$0.819010\pi$
$240$		0			0
$241$	−6.85145			−0.441341			−0.220670	−	0.975348i	$-0.570825\pi$
$241$	−6.85145			−0.441341			−0.220670	+	0.975348i	$0.570825\pi$
$242$		0			0
$243$			20.9935i			1.34673i
$244$		0			0
$245$			3.48457i			0.222621i
$246$		0			0
$247$		−	0.453285i		−	0.0288418i
$248$		0			0
$249$	−12.3579			−0.783149
$250$		0			0
$251$		−	13.8228i		−	0.872490i	−0.899828	−	0.436245i	$-0.856308\pi$
$251$		−	13.8228i		−	0.872490i	0.899828	−	0.436245i	$-0.143692\pi$
$252$		0			0
$253$		−	21.4527i		−	1.34872i
$254$		0			0
$255$	−21.6975			−1.35875
$256$		0			0
$257$		−	4.38953i		−	0.273812i	−0.990584	−	0.136906i	$-0.956284\pi$
$257$		−	4.38953i		−	0.273812i	0.990584	−	0.136906i	$-0.0437158\pi$
$258$		0			0
$259$	0.462880			0.0287620
$260$		0			0
$261$	21.4271			1.32630
$262$		0			0
$263$			19.7817i			1.21979i	0.792482	+	0.609895i	$0.208788\pi$
$263$			19.7817i			1.21979i	−0.792482	+	0.609895i	$0.791212\pi$
$264$		0			0
$265$	9.51257	+	0.245211i	0.584352	+	0.0150632i
$266$		0			0
$267$			9.59125i			0.586975i
$268$		0			0
$269$	−5.26573			−0.321057			−0.160529	−	0.987031i	$-0.551320\pi$
$269$	−5.26573			−0.321057			−0.160529	+	0.987031i	$0.551320\pi$
$270$		0			0
$271$	9.99168			0.606951			0.303476	−	0.952839i	$-0.401853\pi$
$271$	9.99168			0.606951			0.303476	+	0.952839i	$0.401853\pi$
$272$		0			0
$273$		−	2.50456i		−	0.151583i
$274$		0			0
$275$	−8.24495			−0.497189
$276$		0			0
$277$			9.52825i			0.572497i	0.958155	+	0.286249i	$0.0924083\pi$
$277$			9.52825i			0.572497i	−0.958155	+	0.286249i	$0.907592\pi$
$278$		0			0
$279$		−	39.7847i		−	2.38185i
$280$		0			0
$281$	23.0320			1.37397			0.686987	−	0.726670i	$-0.258933\pi$
$281$	23.0320			1.37397			0.686987	+	0.726670i	$0.258933\pi$
$282$		0			0
$283$		−	5.62334i		−	0.334273i	−0.985934	−	0.167137i	$-0.946548\pi$
$283$		−	5.62334i		−	0.334273i	0.985934	−	0.167137i	$-0.0534521\pi$
$284$		0			0
$285$		−	3.46909i		−	0.205491i
$286$		0			0
$287$		−	7.40274i		−	0.436970i
$288$		0			0
$289$	22.1190			1.30112
$290$		0			0
$291$		−	51.8770i		−	3.04108i
$292$		0			0
$293$	10.6834			0.624128			0.312064	−	0.950061i	$-0.398980\pi$
$293$	10.6834			0.624128			0.312064	+	0.950061i	$0.398980\pi$
$294$		0			0
$295$			3.00245i			0.174809i
$296$		0			0
$297$		−	6.94110i		−	0.402763i
$298$		0			0
$299$		−	3.88206i		−	0.224505i
$300$		0			0
$301$	3.20413			0.184683
$302$		0			0
$303$	−8.98721			−0.516302
$304$		0			0
$305$	−0.266870			−0.0152810
$306$		0			0
$307$	5.86276			0.334605			0.167303	−	0.985906i	$-0.446494\pi$
$307$	5.86276			0.334605			0.167303	+	0.985906i	$0.446494\pi$
$308$		0			0
$309$	−4.10225			−0.233369
$310$		0			0
$311$	14.6489			0.830665			0.415333	−	0.909670i	$-0.363665\pi$
$311$	14.6489			0.830665			0.415333	+	0.909670i	$0.363665\pi$
$312$		0			0
$313$			8.47190i			0.478860i	0.970914	+	0.239430i	$0.0769606\pi$
$313$			8.47190i			0.478860i	−0.970914	+	0.239430i	$0.923039\pi$
$314$		0			0
$315$		−	11.0045i		−	0.620034i
$316$		0			0
$317$	4.93533			0.277196			0.138598	−	0.990349i	$-0.455740\pi$
$317$	4.93533			0.277196			0.138598	+	0.990349i	$0.455740\pi$
$318$		0			0
$319$	13.2720			0.743088
$320$		0			0
$321$			39.9228i			2.22827i
$322$		0			0
$323$			6.25452i			0.348011i
$324$		0			0
$325$	−1.49200			−0.0827613
$326$		0			0
$327$	38.4329			2.12535
$328$		0			0
$329$	−7.91083			−0.436138
$330$		0			0
$331$	−4.88081			−0.268274			−0.134137	−	0.990963i	$-0.542826\pi$
$331$	−4.88081			−0.268274			−0.134137	+	0.990963i	$0.542826\pi$
$332$		0			0
$333$	0.899161			0.0492737
$334$		0			0
$335$	2.05390			0.112216
$336$		0			0
$337$			34.2087i			1.86347i	0.363142	+	0.931734i	$0.381704\pi$
$337$			34.2087i			1.86347i	−0.363142	+	0.931734i	$0.618296\pi$
$338$		0			0
$339$		−	36.3366i		−	1.97353i
$340$		0			0
$341$		−	24.6427i		−	1.33448i
$342$		0			0
$343$	20.1230			1.08654
$344$		0			0
$345$		−	29.7103i		−	1.59955i
$346$		0			0
$347$	−26.9740			−1.44804			−0.724021	−	0.689777i	$-0.757708\pi$
$347$	−26.9740			−1.44804			−0.724021	+	0.689777i	$0.757708\pi$
$348$		0			0
$349$		−	30.3551i		−	1.62487i	−0.583051	−	0.812435i	$-0.698141\pi$
$349$		−	30.3551i		−	1.62487i	0.583051	−	0.812435i	$-0.301859\pi$
$350$		0			0
$351$		−	1.25606i		−	0.0670433i
$352$		0			0
$353$			7.18357i			0.382343i	0.981557	+	0.191171i	$0.0612286\pi$
$353$			7.18357i			0.382343i	−0.981557	+	0.191171i	$0.938771\pi$
$354$		0			0
$355$	9.67170			0.513321
$356$		0			0
$357$			34.5585i			1.82903i
$358$		0			0
$359$			10.2781i			0.542455i	0.962515	+	0.271227i	$0.0874296\pi$
$359$			10.2781i			0.542455i	−0.962515	+	0.271227i	$0.912570\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$			12.5417i			0.658269i
$364$		0			0
$365$	−14.0364			−0.734700
$366$		0			0
$367$	−35.8456			−1.87112			−0.935562	−	0.353162i	$-0.885107\pi$
$367$	−35.8456			−1.87112			−0.935562	+	0.353162i	$0.885107\pi$
$368$		0			0
$369$		−	14.3801i		−	0.748597i
$370$		0			0
$371$	0.390557	−	15.1511i	0.0202767	−	0.786603i
$372$		0			0
$373$			14.7111i			0.761714i	0.924634	+	0.380857i	$0.124371\pi$
$373$			14.7111i			0.761714i	−0.924634	+	0.380857i	$0.875629\pi$
$374$		0			0
$375$	−28.7641			−1.48537
$376$		0			0
$377$	2.40169			0.123693
$378$		0			0
$379$			26.4095i			1.35656i	0.734802	+	0.678281i	$0.237275\pi$
$379$			26.4095i			1.35656i	−0.734802	+	0.678281i	$0.762725\pi$
$380$		0			0
$381$	13.8097			0.707491
$382$		0			0
$383$		−	2.53653i		−	0.129611i	−0.997898	−	0.0648053i	$-0.979357\pi$
$383$		−	2.53653i		−	0.129611i	0.997898	−	0.0648053i	$-0.0206426\pi$
$384$		0			0
$385$		−	6.81622i		−	0.347387i
$386$		0			0
$387$	6.22413			0.316390
$388$		0			0
$389$		−	14.4313i		−	0.731694i	−0.930675	−	0.365847i	$-0.880779\pi$
$389$		−	14.4313i		−	0.731694i	0.930675	−	0.365847i	$-0.119221\pi$
$390$		0			0
$391$			53.5655i			2.70892i
$392$		0			0
$393$		−	49.4054i		−	2.49217i
$394$		0			0
$395$	2.20442			0.110916
$396$		0			0
$397$			16.7864i			0.842484i	0.906948	+	0.421242i	$0.138406\pi$
$397$			16.7864i			0.842484i	−0.906948	+	0.421242i	$0.861594\pi$
$398$		0			0
$399$	−5.52536			−0.276614
$400$		0			0
$401$			3.85919i			0.192719i	0.995347	+	0.0963594i	$0.0307198\pi$
$401$			3.85919i			0.192719i	−0.995347	+	0.0963594i	$0.969280\pi$
$402$		0			0
$403$		−	4.45933i		−	0.222135i
$404$		0			0
$405$			6.24493i			0.310313i
$406$		0			0
$407$	0.556942			0.0276066
$408$		0			0
$409$	24.7538			1.22399			0.611997	−	0.790860i	$-0.290366\pi$
$409$	24.7538			1.22399			0.611997	+	0.790860i	$0.290366\pi$
$410$		0			0
$411$	−21.2789			−1.04961
$412$		0			0
$413$	4.78212			0.235313
$414$		0			0
$415$	−6.08606			−0.298753
$416$		0			0
$417$	37.7059			1.84647
$418$		0			0
$419$			38.6303i			1.88722i	0.331066	+	0.943608i	$0.392592\pi$
$419$			38.6303i			1.88722i	−0.331066	+	0.943608i	$0.607408\pi$
$420$		0			0
$421$			12.0274i			0.586182i	0.956085	+	0.293091i	$0.0946838\pi$
$421$			12.0274i			0.586182i	−0.956085	+	0.293091i	$0.905316\pi$
$422$		0			0
$423$	−15.3671			−0.747172
$424$		0			0
$425$	20.5869			0.998612
$426$		0			0
$427$			0.425055i			0.0205699i
$428$		0			0
$429$		−	3.01352i		−	0.145494i
$430$		0			0
$431$	25.1758			1.21268			0.606338	−	0.795207i	$-0.292638\pi$
$431$	25.1758			1.21268			0.606338	+	0.795207i	$0.292638\pi$
$432$		0			0
$433$	15.2435			0.732553			0.366277	−	0.930506i	$-0.380632\pi$
$433$	15.2435			0.732553			0.366277	+	0.930506i	$0.380632\pi$
$434$		0			0
$435$	18.3806			0.881284
$436$		0			0
$437$	−8.56428			−0.409685
$438$		0			0
$439$	−33.0364			−1.57674			−0.788370	−	0.615201i	$-0.789075\pi$
$439$	−33.0364			−1.57674			−0.788370	+	0.615201i	$0.789075\pi$
$440$		0			0
$441$	10.7811			0.513386
$442$		0			0
$443$			12.1411i			0.576840i	0.957504	+	0.288420i	$0.0931300\pi$
$443$			12.1411i			0.576840i	−0.957504	+	0.288420i	$0.906870\pi$
$444$		0			0
$445$			4.72354i			0.223917i
$446$		0			0
$447$			19.0208i			0.899652i
$448$		0			0
$449$	8.45171			0.398861			0.199430	−	0.979912i	$-0.436091\pi$
$449$	8.45171			0.398861			0.199430	+	0.979912i	$0.436091\pi$
$450$		0			0
$451$		−	8.90705i		−	0.419417i
$452$		0			0
$453$	40.7000			1.91225
$454$		0			0
$455$		−	1.23346i		−	0.0578254i
$456$		0			0
$457$			24.2391i			1.13386i	0.823766	+	0.566930i	$0.191869\pi$
$457$			24.2391i			1.13386i	−0.823766	+	0.566930i	$0.808131\pi$
$458$		0			0
$459$			17.3313i			0.808957i
$460$		0			0
$461$	−6.59518			−0.307168			−0.153584	−	0.988136i	$-0.549082\pi$
$461$	−6.59518			−0.307168			−0.153584	+	0.988136i	$0.549082\pi$
$462$		0			0
$463$		−	16.1675i		−	0.751366i	−0.926748	−	0.375683i	$-0.877408\pi$
$463$		−	16.1675i		−	0.751366i	0.926748	−	0.375683i	$-0.122592\pi$
$464$		0			0
$465$		−	34.1282i		−	1.58266i
$466$		0			0
$467$	5.68550			0.263094			0.131547	−	0.991310i	$-0.458006\pi$
$467$	5.68550			0.263094			0.131547	+	0.991310i	$0.458006\pi$
$468$		0			0
$469$		−	3.27132i		−	0.151056i
$470$		0			0
$471$	11.7585			0.541805
$472$		0			0
$473$	3.85524			0.177264
$474$		0			0
$475$			3.29153i			0.151026i
$476$		0			0
$477$	0.758671	−	29.4314i	0.0347371	−	1.34757i
$478$		0			0
$479$			19.1053i			0.872944i	0.899718	+	0.436472i	$0.143772\pi$
$479$			19.1053i			0.872944i	−0.899718	+	0.436472i	$0.856228\pi$
$480$		0			0
$481$	0.100784			0.00459535
$482$		0			0
$483$	−47.3208			−2.15317
$484$		0			0
$485$		−	25.5486i		−	1.16010i
$486$		0			0
$487$	−37.1183			−1.68199			−0.840996	−	0.541042i	$-0.818030\pi$
$487$	−37.1183			−1.68199			−0.840996	+	0.541042i	$0.818030\pi$
$488$		0			0
$489$			7.82430i			0.353827i
$490$		0			0
$491$		−	11.8714i		−	0.535751i	−0.963454	−	0.267875i	$-0.913678\pi$
$491$		−	11.8714i		−	0.535751i	0.963454	−	0.267875i	$-0.0863215\pi$
$492$		0			0
$493$	−33.1389			−1.49250
$494$		0			0
$495$		−	13.2407i		−	0.595127i
$496$		0			0
$497$		−	15.4045i		−	0.690987i
$498$		0			0
$499$		−	14.1698i		−	0.634328i	−0.948371	−	0.317164i	$-0.897269\pi$
$499$		−	14.1698i		−	0.634328i	0.948371	−	0.317164i	$-0.102731\pi$
$500$		0			0
$501$	−45.9412			−2.05250
$502$		0			0
$503$		−	20.3518i		−	0.907442i	−0.891144	−	0.453721i	$-0.850096\pi$
$503$		−	20.3518i		−	0.907442i	0.891144	−	0.453721i	$-0.149904\pi$
$504$		0			0
$505$	−4.42606			−0.196957
$506$		0			0
$507$			33.9575i			1.50811i
$508$		0			0
$509$		−	9.57681i		−	0.424485i	−0.977217	−	0.212242i	$-0.931923\pi$
$509$		−	9.57681i		−	0.424485i	0.977217	−	0.212242i	$-0.0680766\pi$
$510$		0			0
$511$			22.3564i			0.988988i
$512$		0			0
$513$	−2.77101			−0.122343
$514$		0			0
$515$	−2.02029			−0.0890248
$516$		0			0
$517$	−9.51839			−0.418619
$518$		0			0
$519$	−17.6924			−0.776610
$520$		0			0
$521$	34.6415			1.51767			0.758836	−	0.651282i	$-0.225769\pi$
$521$	34.6415			1.51767			0.758836	+	0.651282i	$0.225769\pi$
$522$		0			0
$523$	32.3559			1.41482			0.707412	−	0.706801i	$-0.249862\pi$
$523$	32.3559			1.41482			0.707412	+	0.706801i	$0.249862\pi$
$524$		0			0
$525$			18.1869i			0.793740i
$526$		0			0
$527$			61.5307i			2.68032i
$528$		0			0
$529$	−50.3470			−2.18900
$530$		0			0
$531$	9.28943			0.403127
$532$		0			0
$533$		−	1.61181i		−	0.0698154i
$534$		0			0
$535$			19.6614i			0.850035i
$536$		0			0
$537$	39.7297			1.71446
$538$		0			0
$539$	6.67783			0.287635
$540$		0			0
$541$	−12.4846			−0.536754			−0.268377	−	0.963314i	$-0.586487\pi$
$541$	−12.4846			−0.536754			−0.268377	+	0.963314i	$0.586487\pi$
$542$		0			0
$543$	−67.4506			−2.89458
$544$		0			0
$545$	18.9276			0.810770
$546$		0			0
$547$	35.5565			1.52029			0.760143	−	0.649756i	$-0.225129\pi$
$547$	35.5565			1.52029			0.760143	+	0.649756i	$0.225129\pi$
$548$		0			0
$549$			0.825685i			0.0352394i
$550$		0			0
$551$		−	5.29840i		−	0.225720i
$552$		0			0
$553$		−	3.51107i		−	0.149306i
$554$		0			0
$555$	0.771321			0.0327408
$556$		0			0
$557$			25.0462i			1.06124i	0.847610	+	0.530620i	$0.178041\pi$
$557$			25.0462i			1.06124i	−0.847610	+	0.530620i	$0.821959\pi$
$558$		0			0
$559$	0.697642			0.0295071
$560$		0			0
$561$			41.5811i			1.75555i
$562$		0			0
$563$		−	28.3221i		−	1.19364i	−0.802377	−	0.596818i	$-0.796431\pi$
$563$		−	28.3221i		−	1.19364i	0.802377	−	0.596818i	$-0.203569\pi$
$564$		0			0
$565$		−	17.8952i		−	0.752857i
$566$		0			0
$567$	9.94655			0.417716
$568$		0			0
$569$		−	34.2559i		−	1.43608i	−0.696000	−	0.718042i	$-0.745039\pi$
$569$		−	34.2559i		−	1.43608i	0.696000	−	0.718042i	$-0.254961\pi$
$570$		0			0
$571$		−	27.0406i		−	1.13161i	−0.824538	−	0.565807i	$-0.808565\pi$
$571$		−	27.0406i		−	1.13161i	0.824538	−	0.565807i	$-0.191435\pi$
$572$		0			0
$573$	33.2845			1.39048
$574$		0			0
$575$			28.1896i			1.17559i
$576$		0			0
$577$	−5.95976			−0.248108			−0.124054	−	0.992275i	$-0.539590\pi$
$577$	−5.95976			−0.248108			−0.124054	+	0.992275i	$0.539590\pi$
$578$		0			0
$579$	−9.28013			−0.385669
$580$		0			0
$581$			9.69352i			0.402155i
$582$		0			0
$583$	0.469922	−	18.2299i	0.0194622	−	0.755005i
$584$		0			0
$585$		−	2.39604i		−	0.0990639i
$586$		0			0
$587$	−24.5551			−1.01350			−0.506749	−	0.862094i	$-0.669153\pi$
$587$	−24.5551			−1.01350			−0.506749	+	0.862094i	$0.669153\pi$
$588$		0			0
$589$	−9.83780			−0.405359
$590$		0			0
$591$			16.1017i			0.662335i
$592$		0			0
$593$	20.0382			0.822868			0.411434	−	0.911439i	$-0.365028\pi$
$593$	20.0382			0.822868			0.411434	+	0.911439i	$0.365028\pi$
$594$		0			0
$595$			17.0195i			0.697732i
$596$		0			0
$597$		−	42.3989i		−	1.73527i
$598$		0			0
$599$	10.0469			0.410507			0.205254	−	0.978709i	$-0.434198\pi$
$599$	10.0469			0.410507			0.205254	+	0.978709i	$0.434198\pi$
$600$		0			0
$601$			6.67065i			0.272102i	0.990702	+	0.136051i	$0.0434410\pi$
$601$			6.67065i			0.272102i	−0.990702	+	0.136051i	$0.956559\pi$
$602$		0			0
$603$		−	6.35466i		−	0.258782i
$604$		0			0
$605$			6.17660i			0.251114i
$606$		0			0
$607$	−15.2465			−0.618835			−0.309418	−	0.950926i	$-0.600134\pi$
$607$	−15.2465			−0.618835			−0.309418	+	0.950926i	$0.600134\pi$
$608$		0			0
$609$		−	29.2756i		−	1.18631i
$610$		0			0
$611$	−1.72244			−0.0696826
$612$		0			0
$613$			31.2223i			1.26106i	0.776165	+	0.630529i	$0.217162\pi$
$613$			31.2223i			1.26106i	−0.776165	+	0.630529i	$0.782838\pi$
$614$		0			0
$615$		−	12.3356i		−	0.497418i
$616$		0			0
$617$			45.3249i			1.82471i	0.409400	+	0.912355i	$0.365738\pi$
$617$			45.3249i			1.82471i	−0.409400	+	0.912355i	$0.634262\pi$
$618$		0			0
$619$	−42.8681			−1.72301			−0.861507	−	0.507746i	$-0.830479\pi$
$619$	−42.8681			−1.72301			−0.861507	+	0.507746i	$0.830479\pi$
$620$		0			0
$621$	−23.7317			−0.952320
$622$		0			0
$623$	7.52337			0.301418
$624$		0			0
$625$	2.29179			0.0916716
$626$		0			0
$627$	−6.64817			−0.265502
$628$		0			0
$629$	−1.39064			−0.0554483
$630$		0			0
$631$			20.2430i			0.805860i	0.915231	+	0.402930i	$0.132008\pi$
$631$			20.2430i			0.805860i	−0.915231	+	0.402930i	$0.867992\pi$
$632$		0			0
$633$			54.9325i			2.18337i
$634$		0			0
$635$	6.80105			0.269891
$636$		0			0
$637$	1.20842			0.0478792
$638$		0			0
$639$		−	29.9238i		−	1.18377i
$640$		0			0
$641$		−	35.0702i		−	1.38519i	−0.721326	−	0.692596i	$-0.756467\pi$
$641$		−	35.0702i		−	1.38519i	0.721326	−	0.692596i	$-0.243533\pi$
$642$		0			0
$643$	6.83784			0.269658			0.134829	−	0.990869i	$-0.456951\pi$
$643$	6.83784			0.269658			0.134829	+	0.990869i	$0.456951\pi$
$644$		0			0
$645$	5.33921			0.210231
$646$		0			0
$647$	27.6321			1.08633			0.543164	−	0.839627i	$-0.317226\pi$
$647$	27.6321			1.08633			0.543164	+	0.839627i	$0.317226\pi$
$648$		0			0
$649$	5.75389			0.225860
$650$		0			0
$651$	−54.3574			−2.13043
$652$		0			0
$653$	−14.1210			−0.552596			−0.276298	−	0.961072i	$-0.589108\pi$
$653$	−14.1210			−0.552596			−0.276298	+	0.961072i	$0.589108\pi$
$654$		0			0
$655$		−	24.3314i		−	0.950706i
$656$		0			0
$657$			43.4280i			1.69429i
$658$		0			0
$659$		−	5.59965i		−	0.218131i	−0.994035	−	0.109066i	$-0.965214\pi$
$659$		−	5.59965i		−	0.218131i	0.994035	−	0.109066i	$-0.0347859\pi$
$660$		0			0
$661$	24.5421			0.954576			0.477288	−	0.878747i	$-0.341620\pi$
$661$	24.5421			0.954576			0.477288	+	0.878747i	$0.341620\pi$
$662$		0			0
$663$			7.52449i			0.292227i
$664$		0			0
$665$	−2.72115			−0.105522
$666$		0			0
$667$		−	45.3770i		−	1.75701i
$668$		0			0
$669$		−	59.0829i		−	2.28428i
$670$		0			0
$671$			0.511431i			0.0197436i
$672$		0			0
$673$	−12.5229			−0.482724			−0.241362	−	0.970435i	$-0.577594\pi$
$673$	−12.5229			−0.482724			−0.241362	+	0.970435i	$0.577594\pi$
$674$		0			0
$675$			9.12085i			0.351062i
$676$		0			0
$677$			5.99773i			0.230511i	0.993336	+	0.115256i	$0.0367687\pi$
$677$			5.99773i			0.230511i	−0.993336	+	0.115256i	$0.963231\pi$
$678$		0			0
$679$	−40.6923			−1.56163
$680$		0			0
$681$			19.5102i			0.747634i
$682$		0			0
$683$	−28.6282			−1.09543			−0.547714	−	0.836665i	$-0.684502\pi$
$683$	−28.6282			−1.09543			−0.547714	+	0.836665i	$0.684502\pi$
$684$		0			0
$685$	−10.4795			−0.400401
$686$		0			0
$687$			47.3257i			1.80559i
$688$		0			0
$689$	0.0850368	−	3.29887i	0.00323965	−	0.125677i
$690$		0			0
$691$			33.4698i			1.27325i	0.771174	+	0.636625i	$0.219670\pi$
$691$			33.4698i			1.27325i	−0.771174	+	0.636625i	$0.780330\pi$
$692$		0			0
$693$	−21.0891			−0.801107
$694$		0			0
$695$	18.5696			0.704384
$696$		0			0
$697$			22.2401i			0.842405i
$698$		0			0
$699$	6.18187			0.233820
$700$		0			0
$701$			17.3582i			0.655609i	0.944746	+	0.327805i	$0.106309\pi$
$701$			17.3582i			0.655609i	−0.944746	+	0.327805i	$0.893691\pi$
$702$		0			0
$703$		−	0.222341i		−	0.00838575i
$704$		0			0
$705$	−13.1822			−0.496471
$706$		0			0
$707$			7.04957i			0.265126i
$708$		0			0
$709$			37.1326i			1.39454i	0.716807	+	0.697272i	$0.245603\pi$
$709$			37.1326i			1.39454i	−0.716807	+	0.697272i	$0.754397\pi$
$710$		0			0
$711$		−	6.82038i		−	0.255784i
$712$		0			0
$713$	−84.2537			−3.15533
$714$		0			0
$715$		−	1.48411i		−	0.0555025i
$716$		0			0
$717$	44.1862			1.65016
$718$		0			0
$719$		−	3.82742i		−	0.142739i	−0.997450	−	0.0713693i	$-0.977263\pi$
$719$		−	3.82742i		−	0.142739i	0.997450	−	0.0713693i	$-0.0227369\pi$
$720$		0			0
$721$			3.21780i			0.119837i
$722$		0			0
$723$			18.1842i			0.676278i
$724$		0			0
$725$	−17.4398			−0.647699
$726$		0			0
$727$	−48.8649			−1.81230			−0.906150	−	0.422957i	$-0.860992\pi$
$727$	−48.8649			−1.81230			−0.906150	+	0.422957i	$0.860992\pi$
$728$		0			0
$729$	41.3848			1.53277
$730$		0			0
$731$	−9.62620			−0.356038
$732$		0			0
$733$	−20.6903			−0.764215			−0.382107	−	0.924118i	$-0.624802\pi$
$733$	−20.6903			−0.764215			−0.382107	+	0.924118i	$0.624802\pi$
$734$		0			0
$735$	9.24827			0.341128
$736$		0			0
$737$		−	3.93609i		−	0.144988i
$738$		0			0
$739$			4.02663i			0.148122i	0.997254	+	0.0740609i	$0.0235959\pi$
$739$			4.02663i			0.148122i	−0.997254	+	0.0740609i	$0.976404\pi$
$740$		0			0
$741$	−1.20305			−0.0441951
$742$		0			0
$743$	51.2798			1.88127			0.940637	−	0.339415i	$-0.110229\pi$
$743$	51.2798			1.88127			0.940637	+	0.339415i	$0.110229\pi$
$744$		0			0
$745$			9.36744i			0.343196i
$746$		0			0
$747$			18.8300i			0.688954i
$748$		0			0
$749$	31.3155			1.14424
$750$		0			0
$751$	2.49747			0.0911341			0.0455671	−	0.998961i	$-0.485491\pi$
$751$	2.49747			0.0911341			0.0455671	+	0.998961i	$0.485491\pi$
$752$		0			0
$753$	−36.6867			−1.33694
$754$		0			0
$755$	20.0441			0.729480
$756$		0			0
$757$	19.3987			0.705057			0.352529	−	0.935801i	$-0.385322\pi$
$757$	19.3987			0.705057			0.352529	+	0.935801i	$0.385322\pi$
$758$		0			0
$759$	−56.9368			−2.06668
$760$		0			0
$761$		−	17.6878i		−	0.641184i	−0.947217	−	0.320592i	$-0.896118\pi$
$761$		−	17.6878i		−	0.641184i	0.947217	−	0.320592i	$-0.103882\pi$
$762$		0			0
$763$		−	30.1468i		−	1.09139i
$764$		0			0
$765$			33.0610i			1.19532i
$766$		0			0
$767$	1.04122			0.0375963
$768$		0			0
$769$			14.2075i			0.512334i	0.966633	+	0.256167i	$0.0824597\pi$
$769$			14.2075i			0.512334i	−0.966633	+	0.256167i	$0.917540\pi$
$770$		0			0
$771$	−11.6501			−0.419569
$772$		0			0
$773$			38.9354i			1.40041i	0.713943	+	0.700204i	$0.246908\pi$
$773$			38.9354i			1.40041i	−0.713943	+	0.700204i	$0.753092\pi$
$774$		0			0
$775$			32.3814i			1.16317i
$776$		0			0
$777$		−	1.22851i		−	0.0440727i
$778$		0			0
$779$	−3.55585			−0.127402
$780$		0			0
$781$		−	18.5349i		−	0.663229i
$782$		0			0
$783$		−	14.6819i		−	0.524689i
$784$		0			0
$785$	5.79089			0.206686
$786$		0			0
$787$			13.1136i			0.467448i	0.972303	+	0.233724i	$0.0750912\pi$
$787$			13.1136i			0.467448i	−0.972303	+	0.233724i	$0.924909\pi$
$788$		0			0
$789$	52.5018			1.86912
$790$		0			0
$791$	−28.5024			−1.01343
$792$		0			0
$793$			0.0925482i			0.00328648i
$794$		0			0
$795$	0.650805	−	25.2470i	0.0230817	−	0.895418i
$796$		0			0
$797$			8.23733i			0.291781i	0.989301	+	0.145891i	$0.0466047\pi$
$797$			8.23733i			0.291781i	−0.989301	+	0.145891i	$0.953395\pi$
$798$		0			0
$799$	23.7666			0.840802
$800$		0			0
$801$	14.6144			0.516375
$802$		0			0
$803$			26.8994i			0.949259i
$804$		0			0
$805$	−23.3047			−0.821384
$806$		0			0
$807$			13.9756i			0.491964i
$808$		0			0
$809$		−	13.7929i		−	0.484934i	−0.970160	−	0.242467i	$-0.922043\pi$
$809$		−	13.7929i		−	0.484934i	0.970160	−	0.242467i	$-0.0779566\pi$
$810$		0			0
$811$	5.67657			0.199331			0.0996657	−	0.995021i	$-0.468223\pi$
$811$	5.67657			0.199331			0.0996657	+	0.995021i	$0.468223\pi$
$812$		0			0
$813$		−	26.5186i		−	0.930046i
$814$		0			0
$815$			3.85334i			0.134977i
$816$		0			0
$817$		−	1.53908i		−	0.0538456i
$818$		0			0
$819$	−3.81626			−0.133351
$820$		0			0
$821$			5.19244i			0.181217i	0.995887	+	0.0906086i	$0.0288812\pi$
$821$			5.19244i			0.181217i	−0.995887	+	0.0906086i	$0.971119\pi$
$822$		0			0
$823$	−19.0819			−0.665152			−0.332576	−	0.943077i	$-0.607918\pi$
$823$	−19.0819			−0.665152			−0.332576	+	0.943077i	$0.607918\pi$
$824$		0			0
$825$			21.8826i			0.761855i
$826$		0			0
$827$			34.3906i			1.19588i	0.801541	+	0.597939i	$0.204014\pi$
$827$			34.3906i			1.19588i	−0.801541	+	0.597939i	$0.795986\pi$
$828$		0			0
$829$			0.581066i			0.0201813i	0.999949	+	0.0100906i	$0.00321200\pi$
$829$			0.581066i			0.0201813i	−0.999949	+	0.0100906i	$0.996788\pi$
$830$		0			0
$831$	25.2886			0.877252
$832$		0			0
$833$	−16.6740			−0.577719
$834$		0			0
$835$	−22.6253			−0.782982
$836$		0			0
$837$	−27.2606			−0.942265
$838$		0			0
$839$	−14.0616			−0.485459			−0.242729	−	0.970094i	$-0.578043\pi$
$839$	−14.0616			−0.485459			−0.242729	+	0.970094i	$0.578043\pi$
$840$		0			0
$841$	−0.926936			−0.0319633
$842$		0			0
$843$		−	61.1284i		−	2.10538i
$844$		0			0
$845$			16.7235i			0.575308i
$846$		0			0
$847$	9.83772			0.338028
$848$		0			0
$849$	−14.9247			−0.512215
$850$		0			0
$851$		−	1.90419i		−	0.0652749i
$852$		0			0
$853$		−	11.7348i		−	0.401791i	−0.979613	−	0.200896i	$-0.935615\pi$
$853$		−	11.7348i		−	0.401791i	0.979613	−	0.200896i	$-0.0643852\pi$
$854$		0			0
$855$	−5.28593			−0.180775
$856$		0			0
$857$	−20.6004			−0.703695			−0.351847	−	0.936057i	$-0.614446\pi$
$857$	−20.6004			−0.703695			−0.351847	+	0.936057i	$0.614446\pi$
$858$		0			0
$859$	−49.9817			−1.70535			−0.852677	−	0.522439i	$-0.825022\pi$
$859$	−49.9817			−1.70535			−0.852677	+	0.522439i	$0.825022\pi$
$860$		0			0
$861$	−19.6474			−0.669580
$862$		0			0
$863$	51.2841			1.74573			0.872866	−	0.487960i	$-0.162259\pi$
$863$	51.2841			1.74573			0.872866	+	0.487960i	$0.162259\pi$
$864$		0			0
$865$	−8.71323			−0.296259
$866$		0			0
$867$		−	58.7052i		−	1.99373i
$868$		0			0
$869$		−	4.22456i		−	0.143308i
$870$		0			0
$871$		−	0.712272i		−	0.0241344i
$872$		0			0
$873$	−79.0462			−2.67531
$874$		0			0
$875$			22.5625i			0.762752i
$876$		0			0
$877$	32.8110			1.10795			0.553974	−	0.832534i	$-0.313111\pi$
$877$	32.8110			1.10795			0.553974	+	0.832534i	$0.313111\pi$
$878$		0			0
$879$		−	28.3543i		−	0.956367i
$880$		0			0
$881$			49.0431i			1.65230i	0.563447	+	0.826152i	$0.309475\pi$
$881$			49.0431i			1.65230i	−0.563447	+	0.826152i	$0.690525\pi$
$882$		0			0
$883$		−	32.7210i		−	1.10115i	−0.834786	−	0.550575i	$-0.814409\pi$
$883$		−	32.7210i		−	1.10115i	0.834786	−	0.550575i	$-0.185591\pi$
$884$		0			0
$885$	7.96869			0.267864
$886$		0			0
$887$		−	6.98069i		−	0.234389i	−0.993109	−	0.117194i	$-0.962610\pi$
$887$		−	6.98069i		−	0.234389i	0.993109	−	0.117194i	$-0.0373900\pi$
$888$		0			0
$889$		−	10.8323i		−	0.363304i
$890$		0			0
$891$	11.9678			0.400936
$892$		0			0
$893$			3.79991i			0.127159i
$894$		0			0
$895$	19.5662			0.654027
$896$		0			0
$897$	−10.3032			−0.344015
$898$		0			0
$899$		−	52.1246i		−	1.73845i
$900$		0			0
$901$	−1.17335	+	45.5185i	−0.0390901	+	1.51644i
$902$		0			0
$903$		−	8.50397i		−	0.282994i
$904$		0			0
$905$	−33.2184			−1.10422
$906$		0			0
$907$	−25.9484			−0.861602			−0.430801	−	0.902447i	$-0.641769\pi$
$907$	−25.9484			−0.861602			−0.430801	+	0.902447i	$0.641769\pi$
$908$		0			0
$909$			13.6940i			0.454202i
$910$		0			0
$911$	4.40624			0.145985			0.0729926	−	0.997332i	$-0.476745\pi$
$911$	4.40624			0.145985			0.0729926	+	0.997332i	$0.476745\pi$
$912$		0			0
$913$			11.6633i			0.386000i
$914$		0			0
$915$			0.708292i			0.0234154i
$916$		0			0
$917$	−38.7536			−1.27976
$918$		0			0
$919$		−	29.5169i		−	0.973674i	−0.873493	−	0.486837i	$-0.838151\pi$
$919$		−	29.5169i		−	0.973674i	0.873493	−	0.486837i	$-0.161849\pi$
$920$		0			0
$921$		−	15.5601i		−	0.512724i
$922$		0			0
$923$		−	3.35406i		−	0.110400i
$924$		0			0
$925$	−0.731842			−0.0240628
$926$		0			0
$927$			6.25069i			0.205300i
$928$		0			0
$929$	48.6037			1.59464			0.797318	−	0.603560i	$-0.206252\pi$
$929$	48.6037			1.59464			0.797318	+	0.603560i	$0.206252\pi$
$930$		0			0
$931$		−	2.66591i		−	0.0873716i
$932$		0			0
$933$		−	38.8792i		−	1.27285i
$934$		0			0
$935$			20.4780i			0.669703i
$936$		0			0
$937$	−17.9349			−0.585906			−0.292953	−	0.956127i	$-0.594638\pi$
$937$	−17.9349			−0.585906			−0.292953	+	0.956127i	$0.594638\pi$
$938$		0			0
$939$	22.4850			0.733770
$940$		0			0
$941$	32.1933			1.04947			0.524735	−	0.851265i	$-0.324164\pi$
$941$	32.1933			1.04947			0.524735	+	0.851265i	$0.324164\pi$
$942$		0			0
$943$	−30.4533			−0.991696
$944$		0			0
$945$	−7.54034			−0.245287
$946$		0			0
$947$	−44.7050			−1.45272			−0.726358	−	0.687316i	$-0.758789\pi$
$947$	−44.7050			−1.45272			−0.726358	+	0.687316i	$0.758789\pi$
$948$		0			0
$949$			4.86770i			0.158012i
$950$		0			0
$951$		−	13.0987i		−	0.424754i
$952$		0			0
$953$	23.4496			0.759606			0.379803	−	0.925067i	$-0.375992\pi$
$953$	23.4496			0.759606			0.379803	+	0.925067i	$0.375992\pi$
$954$		0			0
$955$	16.3921			0.530436
$956$		0			0
$957$		−	35.2247i		−	1.13865i
$958$		0			0
$959$			16.6911i			0.538985i
$960$		0			0
$961$	−65.7823			−2.12201
$962$		0			0
$963$	60.8313			1.96026
$964$		0			0
$965$	−4.57032			−0.147124
$966$		0			0
$967$	−26.0149			−0.836584			−0.418292	−	0.908313i	$-0.637371\pi$
$967$	−26.0149			−0.836584			−0.418292	+	0.908313i	$0.637371\pi$
$968$		0			0
$969$	16.5999			0.533266
$970$		0			0
$971$	3.79036			0.121638			0.0608192	−	0.998149i	$-0.480629\pi$
$971$	3.79036			0.121638			0.0608192	+	0.998149i	$0.480629\pi$
$972$		0			0
$973$		−	29.5765i		−	0.948180i
$974$		0			0
$975$			3.95987i			0.126817i
$976$		0			0
$977$			20.2873i			0.649047i	0.945878	+	0.324523i	$0.105204\pi$
$977$			20.2873i			0.649047i	−0.945878	+	0.324523i	$0.894796\pi$
$978$		0			0
$979$	9.05219			0.289309
$980$		0			0
$981$		−	58.5612i		−	1.86971i
$982$		0			0
$983$	−35.8774			−1.14431			−0.572156	−	0.820145i	$-0.693893\pi$
$983$	−35.8774			−1.14431			−0.572156	+	0.820145i	$0.693893\pi$
$984$		0			0
$985$			7.92983i			0.252665i
$986$		0			0
$987$			20.9959i			0.668306i
$988$		0			0
$989$		−	13.1811i		−	0.419135i
$990$		0			0
$991$	48.8474			1.55169			0.775845	−	0.630924i	$-0.217324\pi$
$991$	48.8474			1.55169			0.775845	+	0.630924i	$0.217324\pi$
$992$		0			0
$993$			12.9540i			0.411082i
$994$		0			0
$995$		−	20.8808i		−	0.661966i
$996$		0			0
$997$	−9.54833			−0.302399			−0.151199	−	0.988503i	$-0.548314\pi$
$997$	−9.54833			−0.302399			−0.151199	+	0.988503i	$0.548314\pi$
$998$		0			0
$999$		−	0.616109i		−	0.0194928i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.9	✓	82
53.52	even	2	inner	4028.2.c.a.3497.74	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.9	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.74	yes	82	53.52	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.65407i q^{3} -1.30709i q^{5} -2.08185 q^{7} -4.04406 q^{9} +O(q^{10})\)	\(q-2.65407i q^{3} -1.30709i q^{5} -2.08185 q^{7} -4.04406 q^{9} -2.50490 q^{11} -0.453285 q^{13} -3.46909 q^{15} +6.25452 q^{17} +1.00000i q^{19} +5.52536i q^{21} +8.56428i q^{23} +3.29153 q^{25} +2.77101i q^{27} -5.29840 q^{29} +9.83780i q^{31} +6.64817i q^{33} +2.72115i q^{35} -0.222341 q^{37} +1.20305i q^{39} +3.55585i q^{41} -1.53908 q^{43} +5.28593i q^{45} +3.79991 q^{47} -2.66591 q^{49} -16.5999i q^{51} +(-0.187601 + 7.27769i) q^{53} +3.27412i q^{55} +2.65407 q^{57} -2.29705 q^{59} -0.204172i q^{61} +8.41912 q^{63} +0.592482i q^{65} +1.57136i q^{67} +22.7302 q^{69} +7.39944i q^{71} -10.7387i q^{73} -8.73593i q^{75} +5.21482 q^{77} +1.68652i q^{79} -4.77775 q^{81} -4.65621i q^{83} -8.17519i q^{85} +14.0623i q^{87} -3.61379 q^{89} +0.943671 q^{91} +26.1102 q^{93} +1.30709 q^{95} +19.5462 q^{97} +10.1300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 82 * q - 8 * q^7 - 82 * q^9	\( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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