LMFDB - Embedded newform 4012.2.b.b.237.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.4
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.43

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.13247i q^{3} -3.91603i q^{5} +2.98385i q^{7} -6.81236 q^{9} +O(q^{10})$	$q-3.13247i q^{3} -3.91603i q^{5} +2.98385i q^{7} -6.81236 q^{9} -5.81225i q^{11} +1.72232 q^{13} -12.2668 q^{15} +(-2.58285 + 3.21386i) q^{17} -0.658321 q^{19} +9.34682 q^{21} +9.22325i q^{23} -10.3353 q^{25} +11.9421i q^{27} +10.2249i q^{29} -5.64524i q^{31} -18.2067 q^{33} +11.6848 q^{35} -3.51339i q^{37} -5.39511i q^{39} +0.336674i q^{41} -2.85742 q^{43} +26.6774i q^{45} -4.71431 q^{47} -1.90336 q^{49} +(10.0673 + 8.09069i) q^{51} -3.06039 q^{53} -22.7609 q^{55} +2.06217i q^{57} +1.00000 q^{59} +4.36582i q^{61} -20.3271i q^{63} -6.74465i q^{65} +4.44979 q^{67} +28.8915 q^{69} +10.8686i q^{71} -3.72041i q^{73} +32.3750i q^{75} +17.3429 q^{77} +5.49978i q^{79} +16.9712 q^{81} -15.3146 q^{83} +(12.5856 + 10.1145i) q^{85} +32.0293 q^{87} -9.45135 q^{89} +5.13914i q^{91} -17.6835 q^{93} +2.57801i q^{95} -9.73000i q^{97} +39.5952i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	3.13247i		−	1.80853i	−0.426970	−	0.904266i	$-0.640419\pi$
$3$		−	3.13247i		−	1.80853i	0.426970	−	0.904266i	$-0.359581\pi$
$4$		0			0
$5$		−	3.91603i		−	1.75130i	−0.482944	−	0.875651i	$-0.660433\pi$
$5$		−	3.91603i		−	1.75130i	0.482944	−	0.875651i	$-0.339567\pi$
$6$		0			0
$7$			2.98385i			1.12779i	0.825847	+	0.563895i	$0.190698\pi$
$7$			2.98385i			1.12779i	−0.825847	+	0.563895i	$0.809302\pi$
$8$		0			0
$9$	−6.81236			−2.27079
$10$		0			0
$11$		−	5.81225i		−	1.75246i	−0.481894	−	0.876229i	$-0.660051\pi$
$11$		−	5.81225i		−	1.75246i	0.481894	−	0.876229i	$-0.339949\pi$
$12$		0			0
$13$	1.72232			0.477685			0.238842	−	0.971058i	$-0.423232\pi$
$13$	1.72232			0.477685			0.238842	+	0.971058i	$0.423232\pi$
$14$		0			0
$15$	−12.2668			−3.16729
$16$		0			0
$17$	−2.58285	+	3.21386i	−0.626433	+	0.779475i
$17$	−2.58285	+	3.21386i	−0.626433	+	0.779475i
$18$		0			0
$19$	−0.658321			−0.151029			−0.0755146	−	0.997145i	$-0.524060\pi$
$19$	−0.658321			−0.151029			−0.0755146	+	0.997145i	$0.524060\pi$
$20$		0			0
$21$	9.34682			2.03964
$22$		0			0
$23$			9.22325i			1.92318i	0.274491	+	0.961590i	$0.411491\pi$
$23$			9.22325i			1.92318i	−0.274491	+	0.961590i	$0.588509\pi$
$24$		0			0
$25$	−10.3353			−2.06706
$26$		0			0
$27$			11.9421i			2.29826i
$28$		0			0
$29$			10.2249i			1.89872i	0.314187	+	0.949361i	$0.398268\pi$
$29$			10.2249i			1.89872i	−0.314187	+	0.949361i	$0.601732\pi$
$30$		0			0
$31$		−	5.64524i		−	1.01392i	−0.861971	−	0.506958i	$-0.830770\pi$
$31$		−	5.64524i		−	1.01392i	0.861971	−	0.506958i	$-0.169230\pi$
$32$		0			0
$33$	−18.2067			−3.16938
$34$		0			0
$35$	11.6848			1.97510
$36$		0			0
$37$		−	3.51339i		−	0.577598i	−0.957390	−	0.288799i	$-0.906744\pi$
$37$		−	3.51339i		−	0.577598i	0.957390	−	0.288799i	$-0.0932559\pi$
$38$		0			0
$39$		−	5.39511i		−	0.863909i
$40$		0			0
$41$			0.336674i			0.0525797i	0.999654	+	0.0262899i	$0.00836928\pi$
$41$			0.336674i			0.0525797i	−0.999654	+	0.0262899i	$0.991631\pi$
$42$		0			0
$43$	−2.85742			−0.435752			−0.217876	−	0.975976i	$-0.569913\pi$
$43$	−2.85742			−0.435752			−0.217876	+	0.975976i	$0.569913\pi$
$44$		0			0
$45$			26.6774i			3.97684i
$46$		0			0
$47$	−4.71431			−0.687652			−0.343826	−	0.939033i	$-0.611723\pi$
$47$	−4.71431			−0.687652			−0.343826	+	0.939033i	$0.611723\pi$
$48$		0			0
$49$	−1.90336			−0.271909
$50$		0			0
$51$	10.0673	+	8.09069i	1.40971	+	1.13292i
$52$		0			0
$53$	−3.06039			−0.420377			−0.210189	−	0.977661i	$-0.567408\pi$
$53$	−3.06039			−0.420377			−0.210189	+	0.977661i	$0.567408\pi$
$54$		0			0
$55$	−22.7609			−3.06909
$56$		0			0
$57$			2.06217i			0.273141i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			4.36582i			0.558986i	0.960148	+	0.279493i	$0.0901664\pi$
$61$			4.36582i			0.558986i	−0.960148	+	0.279493i	$0.909834\pi$
$62$		0			0
$63$		−	20.3271i		−	2.56097i
$64$		0			0
$65$		−	6.74465i		−	0.836571i
$66$		0			0
$67$	4.44979			0.543628			0.271814	−	0.962350i	$-0.412376\pi$
$67$	4.44979			0.543628			0.271814	+	0.962350i	$0.412376\pi$
$68$		0			0
$69$	28.8915			3.47813
$70$		0			0
$71$			10.8686i			1.28987i	0.764238	+	0.644935i	$0.223115\pi$
$71$			10.8686i			1.28987i	−0.764238	+	0.644935i	$0.776885\pi$
$72$		0			0
$73$		−	3.72041i		−	0.435441i	−0.976011	−	0.217721i	$-0.930138\pi$
$73$		−	3.72041i		−	0.435441i	0.976011	−	0.217721i	$-0.0698622\pi$
$74$		0			0
$75$			32.3750i			3.73834i
$76$		0			0
$77$	17.3429			1.97640
$78$		0			0
$79$			5.49978i			0.618773i	0.950936	+	0.309387i	$0.100124\pi$
$79$			5.49978i			0.618773i	−0.950936	+	0.309387i	$0.899876\pi$
$80$		0			0
$81$	16.9712			1.88569
$82$		0			0
$83$	−15.3146			−1.68099			−0.840496	−	0.541817i	$-0.817736\pi$
$83$	−15.3146			−1.68099			−0.840496	+	0.541817i	$0.817736\pi$
$84$		0			0
$85$	12.5856	+	10.1145i	1.36510	+	1.09707i
$86$		0			0
$87$	32.0293			3.43390
$88$		0			0
$89$	−9.45135			−1.00184			−0.500921	−	0.865493i	$-0.667005\pi$
$89$	−9.45135			−1.00184			−0.500921	+	0.865493i	$0.667005\pi$
$90$		0			0
$91$			5.13914i			0.538728i
$92$		0			0
$93$	−17.6835			−1.83370
$94$		0			0
$95$			2.57801i			0.264498i
$96$		0			0
$97$		−	9.73000i		−	0.987932i	−0.869481	−	0.493966i	$-0.835547\pi$
$97$		−	9.73000i		−	0.987932i	0.869481	−	0.493966i	$-0.164453\pi$
$98$		0			0
$99$			39.5952i			3.97946i
$100$		0			0
$101$	−6.03437			−0.600442			−0.300221	−	0.953870i	$-0.597060\pi$
$101$	−6.03437			−0.600442			−0.300221	+	0.953870i	$0.597060\pi$
$102$		0			0
$103$	−11.9289			−1.17539			−0.587696	−	0.809082i	$-0.699965\pi$
$103$	−11.9289			−1.17539			−0.587696	+	0.809082i	$0.699965\pi$
$104$		0			0
$105$		−	36.6024i		−	3.57203i
$106$		0			0
$107$		−	8.75201i		−	0.846088i	−0.906109	−	0.423044i	$-0.860962\pi$
$107$		−	8.75201i		−	0.846088i	0.906109	−	0.423044i	$-0.139038\pi$
$108$		0			0
$109$		−	13.2858i		−	1.27255i	−0.771463	−	0.636274i	$-0.780475\pi$
$109$		−	13.2858i		−	1.27255i	0.771463	−	0.636274i	$-0.219525\pi$
$110$		0			0
$111$	−11.0056			−1.04460
$112$		0			0
$113$			0.860415i			0.0809410i	0.999181	+	0.0404705i	$0.0128857\pi$
$113$			0.860415i			0.0809410i	−0.999181	+	0.0404705i	$0.987114\pi$
$114$		0			0
$115$	36.1185			3.36807
$116$		0			0
$117$	−11.7331			−1.08472
$118$		0			0
$119$	−9.58968	−	7.70683i	−0.879084	−	0.706484i
$120$		0			0
$121$	−22.7822			−2.07111
$122$		0			0
$123$	1.05462			0.0950921
$124$		0			0
$125$			20.8932i			1.86874i
$126$		0			0
$127$	1.07177			0.0951040			0.0475520	−	0.998869i	$-0.484858\pi$
$127$	1.07177			0.0951040			0.0475520	+	0.998869i	$0.484858\pi$
$128$		0			0
$129$			8.95078i			0.788072i
$130$		0			0
$131$		−	10.9683i		−	0.958303i	−0.877732	−	0.479152i	$-0.840944\pi$
$131$		−	10.9683i		−	0.958303i	0.877732	−	0.479152i	$-0.159056\pi$
$132$		0			0
$133$		−	1.96433i		−	0.170329i
$134$		0			0
$135$	46.7657			4.02495
$136$		0			0
$137$	−16.2010			−1.38415			−0.692073	−	0.721828i	$-0.743302\pi$
$137$	−16.2010			−1.38415			−0.692073	+	0.721828i	$0.743302\pi$
$138$		0			0
$139$			17.2007i			1.45894i	0.684010	+	0.729472i	$0.260234\pi$
$139$			17.2007i			1.45894i	−0.684010	+	0.729472i	$0.739766\pi$
$140$		0			0
$141$			14.7674i			1.24364i
$142$		0			0
$143$		−	10.0105i		−	0.837123i
$144$		0			0
$145$	40.0411			3.32524
$146$		0			0
$147$			5.96223i			0.491756i
$148$		0			0
$149$	11.9746			0.981000			0.490500	−	0.871441i	$-0.336814\pi$
$149$	11.9746			0.981000			0.490500	+	0.871441i	$0.336814\pi$
$150$		0			0
$151$	2.12731			0.173118			0.0865590	−	0.996247i	$-0.472413\pi$
$151$	2.12731			0.173118			0.0865590	+	0.996247i	$0.472413\pi$
$152$		0			0
$153$	17.5953	−	21.8940i	1.42250	−	1.77002i
$154$		0			0
$155$	−22.1069			−1.77567
$156$		0			0
$157$	−13.8654			−1.10658			−0.553290	−	0.832989i	$-0.686628\pi$
$157$	−13.8654			−1.10658			−0.553290	+	0.832989i	$0.686628\pi$
$158$		0			0
$159$			9.58659i			0.760266i
$160$		0			0
$161$	−27.5208			−2.16894
$162$		0			0
$163$		−	24.8869i		−	1.94929i	−0.223759	−	0.974644i	$-0.571833\pi$
$163$		−	24.8869i		−	1.94929i	0.223759	−	0.974644i	$-0.428167\pi$
$164$		0			0
$165$			71.2980i			5.55054i
$166$		0			0
$167$			5.21241i			0.403349i	0.979453	+	0.201674i	$0.0646383\pi$
$167$			5.21241i			0.403349i	−0.979453	+	0.201674i	$0.935362\pi$
$168$		0			0
$169$	−10.0336			−0.771817
$170$		0			0
$171$	4.48472			0.342955
$172$		0			0
$173$		−	9.30877i		−	0.707733i	−0.935296	−	0.353866i	$-0.884867\pi$
$173$		−	9.30877i		−	0.707733i	0.935296	−	0.353866i	$-0.115133\pi$
$174$		0			0
$175$		−	30.8390i		−	2.33121i
$176$		0			0
$177$		−	3.13247i		−	0.235451i
$178$		0			0
$179$	−24.2082			−1.80941			−0.904703	−	0.426042i	$-0.859908\pi$
$179$	−24.2082			−1.80941			−0.904703	+	0.426042i	$0.859908\pi$
$180$		0			0
$181$			9.13559i			0.679043i	0.940598	+	0.339522i	$0.110265\pi$
$181$			9.13559i			0.679043i	−0.940598	+	0.339522i	$0.889735\pi$
$182$		0			0
$183$	13.6758			1.01094
$184$		0			0
$185$	−13.7585			−1.01155
$186$		0			0
$187$	18.6798	+	15.0122i	1.36600	+	1.09780i
$188$		0			0
$189$	−35.6335			−2.59195
$190$		0			0
$191$	18.8385			1.36311			0.681554	−	0.731768i	$-0.261304\pi$
$191$	18.8385			1.36311			0.681554	+	0.731768i	$0.261304\pi$
$192$		0			0
$193$			13.3985i			0.964448i	0.876048	+	0.482224i	$0.160171\pi$
$193$			13.3985i			0.964448i	−0.876048	+	0.482224i	$0.839829\pi$
$194$		0			0
$195$	−21.1274			−1.51296
$196$		0			0
$197$		−	8.45154i		−	0.602148i	−0.953601	−	0.301074i	$-0.902655\pi$
$197$		−	8.45154i		−	0.602148i	0.953601	−	0.301074i	$-0.0973450\pi$
$198$		0			0
$199$			14.2354i			1.00912i	0.863376	+	0.504561i	$0.168345\pi$
$199$			14.2354i			1.00912i	−0.863376	+	0.504561i	$0.831655\pi$
$200$		0			0
$201$		−	13.9388i		−	0.983169i
$202$		0			0
$203$	−30.5097			−2.14136
$204$		0			0
$205$	1.31843			0.0920829
$206$		0			0
$207$		−	62.8321i		−	4.36713i
$208$		0			0
$209$			3.82633i			0.264673i
$210$		0			0
$211$		−	17.0329i		−	1.17260i	−0.810095	−	0.586298i	$-0.800585\pi$
$211$		−	17.0329i		−	1.17260i	0.810095	−	0.586298i	$-0.199415\pi$
$212$		0			0
$213$	34.0457			2.33277
$214$		0			0
$215$			11.1897i			0.763134i
$216$		0			0
$217$	16.8446			1.14348
$218$		0			0
$219$	−11.6541			−0.787510
$220$		0			0
$221$	−4.44849	+	5.53529i	−0.299238	+	0.372344i
$222$		0			0
$223$	9.69697			0.649357			0.324678	−	0.945824i	$-0.394744\pi$
$223$	9.69697			0.649357			0.324678	+	0.945824i	$0.394744\pi$
$224$		0			0
$225$	70.4078			4.69385
$226$		0			0
$227$		−	17.6422i		−	1.17095i	−0.810690	−	0.585476i	$-0.800908\pi$
$227$		−	17.6422i		−	1.17095i	0.810690	−	0.585476i	$-0.199092\pi$
$228$		0			0
$229$	13.6189			0.899960			0.449980	−	0.893039i	$-0.351431\pi$
$229$	13.6189			0.899960			0.449980	+	0.893039i	$0.351431\pi$
$230$		0			0
$231$		−	54.3260i		−	3.57439i
$232$		0			0
$233$			1.42634i			0.0934428i	0.998908	+	0.0467214i	$0.0148773\pi$
$233$			1.42634i			0.0934428i	−0.998908	+	0.0467214i	$0.985123\pi$
$234$		0			0
$235$			18.4614i			1.20429i
$236$		0			0
$237$	17.2279			1.11907
$238$		0			0
$239$	10.1072			0.653783			0.326892	−	0.945062i	$-0.393999\pi$
$239$	10.1072			0.653783			0.326892	+	0.945062i	$0.393999\pi$
$240$		0			0
$241$		−	26.5467i		−	1.71003i	−0.518607	−	0.855013i	$-0.673549\pi$
$241$		−	26.5467i		−	1.71003i	0.518607	−	0.855013i	$-0.326451\pi$
$242$		0			0
$243$		−	17.3355i		−	1.11207i
$244$		0			0
$245$			7.45363i			0.476195i
$246$		0			0
$247$	−1.13384			−0.0721444
$248$		0			0
$249$			47.9724i			3.04013i
$250$		0			0
$251$	9.33762			0.589386			0.294693	−	0.955592i	$-0.404783\pi$
$251$	9.33762			0.589386			0.294693	+	0.955592i	$0.404783\pi$
$252$		0			0
$253$	53.6078			3.37029
$254$		0			0
$255$	31.6834	−	39.4239i	1.98409	−	2.46882i
$256$		0			0
$257$	−10.8648			−0.677729			−0.338864	−	0.940835i	$-0.610043\pi$
$257$	−10.8648			−0.677729			−0.338864	+	0.940835i	$0.610043\pi$
$258$		0			0
$259$	10.4834			0.651409
$260$		0			0
$261$		−	69.6560i		−	4.31160i
$262$		0			0
$263$	−1.53470			−0.0946337			−0.0473169	−	0.998880i	$-0.515067\pi$
$263$	−1.53470			−0.0946337			−0.0473169	+	0.998880i	$0.515067\pi$
$264$		0			0
$265$			11.9846i			0.736208i
$266$		0			0
$267$			29.6061i			1.81186i
$268$		0			0
$269$			19.1124i			1.16530i	0.812722	+	0.582651i	$0.197985\pi$
$269$			19.1124i			1.16530i	−0.812722	+	0.582651i	$0.802015\pi$
$270$		0			0
$271$	27.0407			1.64261			0.821303	−	0.570493i	$-0.193248\pi$
$271$	27.0407			1.64261			0.821303	+	0.570493i	$0.193248\pi$
$272$		0			0
$273$	16.0982			0.974307
$274$		0			0
$275$			60.0713i			3.62244i
$276$		0			0
$277$		−	31.4152i		−	1.88756i	−0.330578	−	0.943779i	$-0.607244\pi$
$277$		−	31.4152i		−	1.88756i	0.330578	−	0.943779i	$-0.392756\pi$
$278$		0			0
$279$			38.4574i			2.30239i
$280$		0			0
$281$	−19.9424			−1.18966			−0.594831	−	0.803851i	$-0.702781\pi$
$281$	−19.9424			−1.18966			−0.594831	+	0.803851i	$0.702781\pi$
$282$		0			0
$283$		−	7.80395i		−	0.463897i	−0.972728	−	0.231948i	$-0.925490\pi$
$283$		−	7.80395i		−	0.463897i	0.972728	−	0.231948i	$-0.0745100\pi$
$284$		0			0
$285$	8.07553			0.478353
$286$		0			0
$287$	−1.00459			−0.0592988
$288$		0			0
$289$	−3.65779	−	16.6018i	−0.215164	−	0.976578i
$290$		0			0
$291$	−30.4789			−1.78671
$292$		0			0
$293$	−8.42796			−0.492366			−0.246183	−	0.969223i	$-0.579176\pi$
$293$	−8.42796			−0.492366			−0.246183	+	0.969223i	$0.579176\pi$
$294$		0			0
$295$		−	3.91603i		−	0.228000i
$296$		0			0
$297$	69.4105			4.02761
$298$		0			0
$299$			15.8854i			0.918674i
$300$		0			0
$301$		−	8.52611i		−	0.491437i
$302$		0			0
$303$			18.9025i			1.08592i
$304$		0			0
$305$	17.0967			0.978954
$306$		0			0
$307$	29.3463			1.67488			0.837440	−	0.546529i	$-0.184051\pi$
$307$	29.3463			1.67488			0.837440	+	0.546529i	$0.184051\pi$
$308$		0			0
$309$			37.3670i			2.12574i
$310$		0			0
$311$		−	0.880200i		−	0.0499116i	−0.999689	−	0.0249558i	$-0.992056\pi$
$311$		−	0.880200i		−	0.0499116i	0.999689	−	0.0249558i	$-0.00794450\pi$
$312$		0			0
$313$		−	3.59647i		−	0.203285i	−0.994821	−	0.101642i	$-0.967590\pi$
$313$		−	3.59647i		−	0.203285i	0.994821	−	0.101642i	$-0.0324097\pi$
$314$		0			0
$315$	−79.6015			−4.48503
$316$		0			0
$317$			10.6419i			0.597707i	0.954299	+	0.298853i	$0.0966042\pi$
$317$			10.6419i			0.597707i	−0.954299	+	0.298853i	$0.903396\pi$
$318$		0			0
$319$	59.4298			3.32743
$320$		0			0
$321$	−27.4154			−1.53018
$322$		0			0
$323$	1.70034	−	2.11575i	0.0946097	−	0.117724i
$324$		0			0
$325$	−17.8007			−0.987403
$326$		0			0
$327$	−41.6174			−2.30145
$328$		0			0
$329$		−	14.0668i		−	0.775527i
$330$		0			0
$331$	19.5501			1.07457			0.537285	−	0.843401i	$-0.319450\pi$
$331$	19.5501			1.07457			0.537285	+	0.843401i	$0.319450\pi$
$332$		0			0
$333$			23.9345i			1.31160i
$334$		0			0
$335$		−	17.4255i		−	0.952057i
$336$		0			0
$337$			20.1471i			1.09748i	0.835992	+	0.548742i	$0.184893\pi$
$337$			20.1471i			1.09748i	−0.835992	+	0.548742i	$0.815107\pi$
$338$		0			0
$339$	2.69522			0.146384
$340$		0			0
$341$	−32.8116			−1.77685
$342$		0			0
$343$			15.2076i			0.821133i
$344$		0			0
$345$		−	113.140i		−	6.09126i
$346$		0			0
$347$		−	36.3318i		−	1.95040i	−0.221336	−	0.975198i	$-0.571042\pi$
$347$		−	36.3318i		−	1.95040i	0.221336	−	0.975198i	$-0.428958\pi$
$348$		0			0
$349$	−30.0188			−1.60687			−0.803436	−	0.595392i	$-0.796997\pi$
$349$	−30.0188			−1.60687			−0.803436	+	0.595392i	$0.796997\pi$
$350$		0			0
$351$			20.5681i			1.09784i
$352$		0			0
$353$	−3.99202			−0.212474			−0.106237	−	0.994341i	$-0.533880\pi$
$353$	−3.99202			−0.212474			−0.106237	+	0.994341i	$0.533880\pi$
$354$		0			0
$355$	42.5619			2.25895
$356$		0			0
$357$	−24.1414	+	30.0394i	−1.27770	+	1.58985i
$358$		0			0
$359$	−5.14723			−0.271661			−0.135830	−	0.990732i	$-0.543370\pi$
$359$	−5.14723			−0.271661			−0.135830	+	0.990732i	$0.543370\pi$
$360$		0			0
$361$	−18.5666			−0.977190
$362$		0			0
$363$			71.3647i			3.74567i
$364$		0			0
$365$	−14.5693			−0.762590
$366$		0			0
$367$			5.00098i			0.261049i	0.991445	+	0.130524i	$0.0416661\pi$
$367$			5.00098i			0.261049i	−0.991445	+	0.130524i	$0.958334\pi$
$368$		0			0
$369$		−	2.29355i		−	0.119397i
$370$		0			0
$371$		−	9.13176i		−	0.474097i
$372$		0			0
$373$	−19.2402			−0.996221			−0.498110	−	0.867114i	$-0.665973\pi$
$373$	−19.2402			−0.996221			−0.498110	+	0.867114i	$0.665973\pi$
$374$		0			0
$375$	65.4473			3.37968
$376$		0			0
$377$			17.6106i			0.906991i
$378$		0			0
$379$		−	13.5274i		−	0.694858i	−0.937706	−	0.347429i	$-0.887055\pi$
$379$		−	13.5274i		−	0.694858i	0.937706	−	0.347429i	$-0.112945\pi$
$380$		0			0
$381$		−	3.35728i		−	0.171999i
$382$		0			0
$383$	31.6922			1.61940			0.809698	−	0.586847i	$-0.199631\pi$
$383$	31.6922			1.61940			0.809698	+	0.586847i	$0.199631\pi$
$384$		0			0
$385$		−	67.9153i		−	3.46128i
$386$		0			0
$387$	19.4658			0.989502
$388$		0			0
$389$	−11.4870			−0.582412			−0.291206	−	0.956660i	$-0.594056\pi$
$389$	−11.4870			−0.582412			−0.291206	+	0.956660i	$0.594056\pi$
$390$		0			0
$391$	−29.6422	−	23.8222i	−1.49907	−	1.20474i
$392$		0			0
$393$	−34.3578			−1.73312
$394$		0			0
$395$	21.5373			1.08366
$396$		0			0
$397$		−	6.04883i		−	0.303582i	−0.988413	−	0.151791i	$-0.951496\pi$
$397$		−	6.04883i		−	0.303582i	0.988413	−	0.151791i	$-0.0485040\pi$
$398$		0			0
$399$	−6.15321			−0.308046
$400$		0			0
$401$			14.2256i			0.710393i	0.934792	+	0.355197i	$0.115586\pi$
$401$			14.2256i			0.710393i	−0.934792	+	0.355197i	$0.884414\pi$
$402$		0			0
$403$		−	9.72290i		−	0.484332i
$404$		0			0
$405$		−	66.4598i		−	3.30241i
$406$		0			0
$407$	−20.4207			−1.01222
$408$		0			0
$409$	30.8935			1.52759			0.763793	−	0.645461i	$-0.223335\pi$
$409$	30.8935			1.52759			0.763793	+	0.645461i	$0.223335\pi$
$410$		0			0
$411$			50.7491i			2.50327i
$412$		0			0
$413$			2.98385i			0.146826i
$414$		0			0
$415$			59.9723i			2.94393i
$416$		0			0
$417$	53.8807			2.63855
$418$		0			0
$419$			39.9490i			1.95164i	0.218582	+	0.975819i	$0.429857\pi$
$419$			39.9490i			1.95164i	−0.218582	+	0.975819i	$0.570143\pi$
$420$		0			0
$421$	−0.500758			−0.0244054			−0.0122027	−	0.999926i	$-0.503884\pi$
$421$	−0.500758			−0.0244054			−0.0122027	+	0.999926i	$0.503884\pi$
$422$		0			0
$423$	32.1156			1.56151
$424$		0			0
$425$	26.6945	−	33.2162i	1.29487	−	1.61122i
$426$		0			0
$427$	−13.0270			−0.630419
$428$		0			0
$429$	−31.3577			−1.51396
$430$		0			0
$431$		−	5.04039i		−	0.242787i	−0.992604	−	0.121394i	$-0.961264\pi$
$431$		−	5.04039i		−	0.242787i	0.992604	−	0.121394i	$-0.0387363\pi$
$432$		0			0
$433$	4.55396			0.218850			0.109425	−	0.993995i	$-0.465099\pi$
$433$	4.55396			0.218850			0.109425	+	0.993995i	$0.465099\pi$
$434$		0			0
$435$		−	125.428i		−	6.01380i
$436$		0			0
$437$		−	6.07186i		−	0.290456i
$438$		0			0
$439$		−	6.28646i		−	0.300036i	−0.988683	−	0.150018i	$-0.952067\pi$
$439$		−	6.28646i		−	0.300036i	0.988683	−	0.150018i	$-0.0479332\pi$
$440$		0			0
$441$	12.9664			0.617448
$442$		0			0
$443$	−30.8639			−1.46639			−0.733195	−	0.680018i	$-0.761972\pi$
$443$	−30.8639			−1.46639			−0.733195	+	0.680018i	$0.761972\pi$
$444$		0			0
$445$			37.0118i			1.75453i
$446$		0			0
$447$		−	37.5102i		−	1.77417i
$448$		0			0
$449$		−	11.8093i		−	0.557316i	−0.960390	−	0.278658i	$-0.910110\pi$
$449$		−	11.8093i		−	0.557316i	0.960390	−	0.278658i	$-0.0898896\pi$
$450$		0			0
$451$	1.95684			0.0921438
$452$		0			0
$453$		−	6.66373i		−	0.313089i
$454$		0			0
$455$	20.1250			0.943476
$456$		0			0
$457$	−21.8513			−1.02216			−0.511079	−	0.859534i	$-0.670754\pi$
$457$	−21.8513			−1.02216			−0.511079	+	0.859534i	$0.670754\pi$
$458$		0			0
$459$	−38.3803	−	30.8447i	−1.79144	−	1.43971i
$460$		0			0
$461$	−22.1769			−1.03288			−0.516441	−	0.856322i	$-0.672744\pi$
$461$	−22.1769			−1.03288			−0.516441	+	0.856322i	$0.672744\pi$
$462$		0			0
$463$	−24.9244			−1.15834			−0.579168	−	0.815208i	$-0.696623\pi$
$463$	−24.9244			−1.15834			−0.579168	+	0.815208i	$0.696623\pi$
$464$		0			0
$465$			69.2493i			3.21136i
$466$		0			0
$467$	−30.9741			−1.43331			−0.716656	−	0.697427i	$-0.754328\pi$
$467$	−30.9741			−1.43331			−0.716656	+	0.697427i	$0.754328\pi$
$468$		0			0
$469$			13.2775i			0.613098i
$470$		0			0
$471$			43.4330i			2.00129i
$472$		0			0
$473$			16.6080i			0.763638i
$474$		0			0
$475$	6.80395			0.312186
$476$		0			0
$477$	20.8485			0.954588
$478$		0			0
$479$			31.6766i			1.44734i	0.690146	+	0.723670i	$0.257546\pi$
$479$			31.6766i			1.44734i	−0.690146	+	0.723670i	$0.742454\pi$
$480$		0			0
$481$		−	6.05118i		−	0.275910i
$482$		0			0
$483$			86.2080i			3.92260i
$484$		0			0
$485$	−38.1030			−1.73017
$486$		0			0
$487$		−	13.9104i		−	0.630341i	−0.949035	−	0.315171i	$-0.897938\pi$
$487$		−	13.9104i		−	0.630341i	0.949035	−	0.315171i	$-0.102062\pi$
$488$		0			0
$489$	−77.9573			−3.52535
$490$		0			0
$491$	36.8827			1.66449			0.832246	−	0.554406i	$-0.187055\pi$
$491$	36.8827			1.66449			0.832246	+	0.554406i	$0.187055\pi$
$492$		0			0
$493$	−32.8615	−	26.4094i	−1.48001	−	1.18942i
$494$		0			0
$495$	155.056			6.96924
$496$		0			0
$497$	−32.4304			−1.45470
$498$		0			0
$499$			31.6469i			1.41671i	0.705856	+	0.708355i	$0.250563\pi$
$499$			31.6469i			1.41671i	−0.705856	+	0.708355i	$0.749437\pi$
$500$		0			0
$501$	16.3277			0.729469
$502$		0			0
$503$			18.2068i			0.811800i	0.913917	+	0.405900i	$0.133042\pi$
$503$			18.2068i			0.811800i	−0.913917	+	0.405900i	$0.866958\pi$
$504$		0			0
$505$			23.6308i			1.05156i
$506$		0			0
$507$			31.4300i			1.39586i
$508$		0			0
$509$	−20.2981			−0.899696			−0.449848	−	0.893105i	$-0.648522\pi$
$509$	−20.2981			−0.899696			−0.449848	+	0.893105i	$0.648522\pi$
$510$		0			0
$511$	11.1012			0.491086
$512$		0			0
$513$		−	7.86175i		−	0.347105i
$514$		0			0
$515$			46.7141i			2.05847i
$516$		0			0
$517$			27.4007i			1.20508i
$518$		0			0
$519$	−29.1594			−1.27996
$520$		0			0
$521$		−	12.0908i		−	0.529707i	−0.964289	−	0.264854i	$-0.914676\pi$
$521$		−	12.0908i		−	0.529707i	0.964289	−	0.264854i	$-0.0853236\pi$
$522$		0			0
$523$	7.57044			0.331032			0.165516	−	0.986207i	$-0.447071\pi$
$523$	7.57044			0.331032			0.165516	+	0.986207i	$0.447071\pi$
$524$		0			0
$525$	−96.6022			−4.21606
$526$		0			0
$527$	18.1430	+	14.5808i	0.790322	+	0.635150i
$528$		0			0
$529$	−62.0683			−2.69862
$530$		0			0
$531$	−6.81236			−0.295631
$532$		0			0
$533$			0.579860i			0.0251165i
$534$		0			0
$535$	−34.2731			−1.48176
$536$		0			0
$537$			75.8315i			3.27237i
$538$		0			0
$539$			11.0628i			0.476509i
$540$		0			0
$541$		−	10.5176i		−	0.452185i	−0.974106	−	0.226092i	$-0.927405\pi$
$541$		−	10.5176i		−	0.452185i	0.974106	−	0.226092i	$-0.0725952\pi$
$542$		0			0
$543$	28.6170			1.22807
$544$		0			0
$545$	−52.0276			−2.22862
$546$		0			0
$547$		−	23.5505i		−	1.00695i	−0.864010	−	0.503474i	$-0.832055\pi$
$547$		−	23.5505i		−	1.00695i	0.864010	−	0.503474i	$-0.167945\pi$
$548$		0			0
$549$		−	29.7416i		−	1.26934i
$550$		0			0
$551$		−	6.73129i		−	0.286763i
$552$		0			0
$553$	−16.4105			−0.697846
$554$		0			0
$555$			43.0982i			1.82942i
$556$		0			0
$557$	−17.4459			−0.739205			−0.369602	−	0.929190i	$-0.620506\pi$
$557$	−17.4459			−0.739205			−0.369602	+	0.929190i	$0.620506\pi$
$558$		0			0
$559$	−4.92138			−0.208152
$560$		0			0
$561$	47.0251	−	58.5137i	1.98540	−	2.47045i
$562$		0			0
$563$	18.2805			0.770433			0.385217	−	0.922826i	$-0.374127\pi$
$563$	18.2805			0.770433			0.385217	+	0.922826i	$0.374127\pi$
$564$		0			0
$565$	3.36941			0.141752
$566$		0			0
$567$			50.6396i			2.12666i
$568$		0			0
$569$	20.6682			0.866454			0.433227	−	0.901285i	$-0.357375\pi$
$569$	20.6682			0.866454			0.433227	+	0.901285i	$0.357375\pi$
$570$		0			0
$571$		−	25.4837i		−	1.06646i	−0.845970	−	0.533230i	$-0.820978\pi$
$571$		−	25.4837i		−	1.06646i	0.845970	−	0.533230i	$-0.179022\pi$
$572$		0			0
$573$		−	59.0111i		−	2.46522i
$574$		0			0
$575$		−	95.3250i		−	3.97533i
$576$		0			0
$577$	−2.37458			−0.0988549			−0.0494274	−	0.998778i	$-0.515740\pi$
$577$	−2.37458			−0.0988549			−0.0494274	+	0.998778i	$0.515740\pi$
$578$		0			0
$579$	41.9705			1.74424
$580$		0			0
$581$		−	45.6964i		−	1.89581i
$582$		0			0
$583$			17.7878i			0.736694i
$584$		0			0
$585$			45.9470i			1.89967i
$586$		0			0
$587$	24.8290			1.02480			0.512401	−	0.858746i	$-0.328756\pi$
$587$	24.8290			1.02480			0.512401	+	0.858746i	$0.328756\pi$
$588$		0			0
$589$			3.71638i			0.153131i
$590$		0			0
$591$	−26.4742			−1.08900
$592$		0			0
$593$	−12.7367			−0.523034			−0.261517	−	0.965199i	$-0.584223\pi$
$593$	−12.7367			−0.523034			−0.261517	+	0.965199i	$0.584223\pi$
$594$		0			0
$595$	−30.1802	+	37.5535i	−1.23727	+	1.53954i
$596$		0			0
$597$	44.5920			1.82503
$598$		0			0
$599$	−25.4577			−1.04017			−0.520086	−	0.854114i	$-0.674100\pi$
$599$	−25.4577			−1.04017			−0.520086	+	0.854114i	$0.674100\pi$
$600$		0			0
$601$			33.2443i			1.35606i	0.735033	+	0.678032i	$0.237167\pi$
$601$			33.2443i			1.35606i	−0.735033	+	0.678032i	$0.762833\pi$
$602$		0			0
$603$	−30.3136			−1.23446
$604$		0			0
$605$			89.2159i			3.62714i
$606$		0			0
$607$			44.2653i			1.79667i	0.439307	+	0.898337i	$0.355224\pi$
$607$			44.2653i			1.79667i	−0.439307	+	0.898337i	$0.644776\pi$
$608$		0			0
$609$			95.5706i			3.87272i
$610$		0			0
$611$	−8.11953			−0.328481
$612$		0			0
$613$	−10.1077			−0.408246			−0.204123	−	0.978945i	$-0.565434\pi$
$613$	−10.1077			−0.408246			−0.204123	+	0.978945i	$0.565434\pi$
$614$		0			0
$615$		−	4.12993i		−	0.166535i
$616$		0			0
$617$		−	18.8646i		−	0.759459i	−0.925098	−	0.379730i	$-0.876017\pi$
$617$		−	18.8646i		−	0.759459i	0.925098	−	0.379730i	$-0.123983\pi$
$618$		0			0
$619$			12.9507i			0.520532i	0.965537	+	0.260266i	$0.0838102\pi$
$619$			12.9507i			0.520532i	−0.965537	+	0.260266i	$0.916190\pi$
$620$		0			0
$621$	−110.145			−4.41997
$622$		0			0
$623$		−	28.2014i		−	1.12987i
$624$		0			0
$625$	30.1419			1.20567
$626$		0			0
$627$	11.9859			0.478669
$628$		0			0
$629$	11.2915	+	9.07456i	0.450223	+	0.361826i
$630$		0			0
$631$	27.1397			1.08041			0.540207	−	0.841532i	$-0.318346\pi$
$631$	27.1397			1.08041			0.540207	+	0.841532i	$0.318346\pi$
$632$		0			0
$633$	−53.3552			−2.12068
$634$		0			0
$635$		−	4.19708i		−	0.166556i
$636$		0			0
$637$	−3.27820			−0.129887
$638$		0			0
$639$		−	74.0411i		−	2.92902i
$640$		0			0
$641$		−	5.00559i		−	0.197709i	−0.995102	−	0.0988545i	$-0.968482\pi$
$641$		−	5.00559i		−	0.197709i	0.995102	−	0.0988545i	$-0.0315178\pi$
$642$		0			0
$643$			29.5653i			1.16594i	0.812492	+	0.582972i	$0.198110\pi$
$643$			29.5653i			1.16594i	−0.812492	+	0.582972i	$0.801890\pi$
$644$		0			0
$645$	35.0515			1.38015
$646$		0			0
$647$	19.9091			0.782709			0.391354	−	0.920240i	$-0.372007\pi$
$647$	19.9091			0.782709			0.391354	+	0.920240i	$0.372007\pi$
$648$		0			0
$649$		−	5.81225i		−	0.228151i
$650$		0			0
$651$		−	52.7651i		−	2.06803i
$652$		0			0
$653$		−	11.8521i		−	0.463807i	−0.972739	−	0.231904i	$-0.925505\pi$
$653$		−	11.8521i		−	0.463807i	0.972739	−	0.231904i	$-0.0744954\pi$
$654$		0			0
$655$	−42.9521			−1.67828
$656$		0			0
$657$			25.3448i			0.988795i
$658$		0			0
$659$	29.3068			1.14163			0.570815	−	0.821079i	$-0.306627\pi$
$659$	29.3068			1.14163			0.570815	+	0.821079i	$0.306627\pi$
$660$		0			0
$661$	20.0032			0.778033			0.389017	−	0.921231i	$-0.372815\pi$
$661$	20.0032			0.778033			0.389017	+	0.921231i	$0.372815\pi$
$662$		0			0
$663$	17.3391	+	13.9347i	0.673395	+	0.541181i
$664$		0			0
$665$	−7.69239			−0.298298
$666$		0			0
$667$	−94.3070			−3.65158
$668$		0			0
$669$		−	30.3754i		−	1.17438i
$670$		0			0
$671$	25.3752			0.979601
$672$		0			0
$673$		−	35.9777i		−	1.38684i	−0.720534	−	0.693420i	$-0.756103\pi$
$673$		−	35.9777i		−	1.38684i	0.720534	−	0.693420i	$-0.243897\pi$
$674$		0			0
$675$		−	123.425i		−	4.75064i
$676$		0			0
$677$			6.04713i			0.232410i	0.993225	+	0.116205i	$0.0370730\pi$
$677$			6.04713i			0.232410i	−0.993225	+	0.116205i	$0.962927\pi$
$678$		0			0
$679$	29.0329			1.11418
$680$		0			0
$681$	−55.2635			−2.11770
$682$		0			0
$683$		−	8.85864i		−	0.338966i	−0.985533	−	0.169483i	$-0.945790\pi$
$683$		−	8.85864i		−	0.338966i	0.985533	−	0.169483i	$-0.0542098\pi$
$684$		0			0
$685$			63.4436i			2.42406i
$686$		0			0
$687$		−	42.6607i		−	1.62761i
$688$		0			0
$689$	−5.27097			−0.200808
$690$		0			0
$691$			1.00329i			0.0381668i	0.999818	+	0.0190834i	$0.00607481\pi$
$691$			1.00329i			0.0381668i	−0.999818	+	0.0190834i	$0.993925\pi$
$692$		0			0
$693$	−118.146			−4.48800
$694$		0			0
$695$	67.3585			2.55505
$696$		0			0
$697$	−1.08202	−	0.869579i	−0.0409846	−	0.0329376i
$698$		0			0
$699$	4.46797			0.168994
$700$		0			0
$701$	−23.9352			−0.904020			−0.452010	−	0.892013i	$-0.649293\pi$
$701$	−23.9352			−0.904020			−0.452010	+	0.892013i	$0.649293\pi$
$702$		0			0
$703$			2.31294i			0.0872342i
$704$		0			0
$705$	57.8297			2.17799
$706$		0			0
$707$		−	18.0056i		−	0.677172i
$708$		0			0
$709$		−	10.7276i		−	0.402883i	−0.979501	−	0.201442i	$-0.935437\pi$
$709$		−	10.7276i		−	0.402883i	0.979501	−	0.201442i	$-0.0645626\pi$
$710$		0			0
$711$		−	37.4665i		−	1.40510i
$712$		0			0
$713$	52.0675			1.94994
$714$		0			0
$715$	−39.2016			−1.46606
$716$		0			0
$717$		−	31.6606i		−	1.18239i
$718$		0			0
$719$		−	20.6924i		−	0.771697i	−0.922562	−	0.385848i	$-0.873909\pi$
$719$		−	20.6924i		−	0.771697i	0.922562	−	0.385848i	$-0.126091\pi$
$720$		0			0
$721$		−	35.5942i		−	1.32560i
$722$		0			0
$723$	−83.1569			−3.09264
$724$		0			0
$725$		−	105.678i		−	3.92477i
$726$		0			0
$727$	−49.3708			−1.83106			−0.915531	−	0.402248i	$-0.868229\pi$
$727$	−49.3708			−1.83106			−0.915531	+	0.402248i	$0.868229\pi$
$728$		0			0
$729$	−3.38916			−0.125525
$730$		0			0
$731$	7.38028	−	9.18335i	0.272970	−	0.339658i
$732$		0			0
$733$	32.4054			1.19692			0.598460	−	0.801153i	$-0.295779\pi$
$733$	32.4054			1.19692			0.598460	+	0.801153i	$0.295779\pi$
$734$		0			0
$735$	23.3483			0.861214
$736$		0			0
$737$		−	25.8633i		−	0.952686i
$738$		0			0
$739$	−10.1692			−0.374080			−0.187040	−	0.982352i	$-0.559889\pi$
$739$	−10.1692			−0.374080			−0.187040	+	0.982352i	$0.559889\pi$
$740$		0			0
$741$			3.55171i			0.130475i
$742$		0			0
$743$			46.6302i			1.71070i	0.518053	+	0.855348i	$0.326657\pi$
$743$			46.6302i			1.71070i	−0.518053	+	0.855348i	$0.673343\pi$
$744$		0			0
$745$		−	46.8930i		−	1.71803i
$746$		0			0
$747$	104.328			3.81718
$748$		0			0
$749$	26.1147			0.954210
$750$		0			0
$751$		−	40.3355i		−	1.47186i	−0.677056	−	0.735932i	$-0.736744\pi$
$751$		−	40.3355i		−	1.47186i	0.677056	−	0.735932i	$-0.263256\pi$
$752$		0			0
$753$		−	29.2498i		−	1.06592i
$754$		0			0
$755$		−	8.33061i		−	0.303182i
$756$		0			0
$757$	−34.3459			−1.24832			−0.624162	−	0.781295i	$-0.714560\pi$
$757$	−34.3459			−1.24832			−0.624162	+	0.781295i	$0.714560\pi$
$758$		0			0
$759$		−	167.925i		−	6.09528i
$760$		0			0
$761$	−29.3644			−1.06446			−0.532229	−	0.846601i	$-0.678645\pi$
$761$	−29.3644			−1.06446			−0.532229	+	0.846601i	$0.678645\pi$
$762$		0			0
$763$	39.6428			1.43517
$764$		0			0
$765$	−85.7375	−	68.9038i	−3.09985	−	2.49122i
$766$		0			0
$767$	1.72232			0.0621893
$768$		0			0
$769$	11.4053			0.411287			0.205644	−	0.978627i	$-0.434071\pi$
$769$	11.4053			0.411287			0.205644	+	0.978627i	$0.434071\pi$
$770$		0			0
$771$			34.0337i			1.22569i
$772$		0			0
$773$	13.0912			0.470857			0.235428	−	0.971892i	$-0.424351\pi$
$773$	13.0912			0.470857			0.235428	+	0.971892i	$0.424351\pi$
$774$		0			0
$775$			58.3452i			2.09582i
$776$		0			0
$777$		−	32.8390i		−	1.17809i
$778$		0			0
$779$		−	0.221640i		−	0.00794107i
$780$		0			0
$781$	63.1712			2.26044
$782$		0			0
$783$	−122.107			−4.36376
$784$		0			0
$785$			54.2974i			1.93796i
$786$		0			0
$787$			31.1097i			1.10894i	0.832203	+	0.554471i	$0.187079\pi$
$787$			31.1097i			1.10894i	−0.832203	+	0.554471i	$0.812921\pi$
$788$		0			0
$789$			4.80740i			0.171148i
$790$		0			0
$791$	−2.56735			−0.0912844
$792$		0			0
$793$			7.51933i			0.267019i
$794$		0			0
$795$	37.5414			1.33146
$796$		0			0
$797$	−46.7194			−1.65489			−0.827443	−	0.561550i	$-0.810205\pi$
$797$	−46.7194			−1.65489			−0.827443	+	0.561550i	$0.810205\pi$
$798$		0			0
$799$	12.1763	−	15.1511i	0.430768	−	0.536008i
$800$		0			0
$801$	64.3860			2.27497
$802$		0			0
$803$	−21.6240			−0.763093
$804$		0			0
$805$			107.772i			3.79847i
$806$		0			0
$807$	59.8689			2.10749
$808$		0			0
$809$			14.2007i			0.499272i	0.968340	+	0.249636i	$0.0803109\pi$
$809$			14.2007i			0.499272i	−0.968340	+	0.249636i	$0.919689\pi$
$810$		0			0
$811$			5.21477i			0.183115i	0.995800	+	0.0915577i	$0.0291846\pi$
$811$			5.21477i			0.183115i	−0.995800	+	0.0915577i	$0.970815\pi$
$812$		0			0
$813$		−	84.7042i		−	2.97070i
$814$		0			0
$815$	−97.4577			−3.41379
$816$		0			0
$817$	1.88110			0.0658114
$818$		0			0
$819$		−	35.0097i		−	1.22334i
$820$		0			0
$821$		−	13.8961i		−	0.484977i	−0.970154	−	0.242488i	$-0.922036\pi$
$821$		−	13.8961i		−	0.484977i	0.970154	−	0.242488i	$-0.0779636\pi$
$822$		0			0
$823$			8.60683i			0.300015i	0.988685	+	0.150008i	$0.0479298\pi$
$823$			8.60683i			0.300015i	−0.988685	+	0.150008i	$0.952070\pi$
$824$		0			0
$825$	188.172			6.55129
$826$		0			0
$827$			24.5573i			0.853941i	0.904265	+	0.426971i	$0.140419\pi$
$827$			24.5573i			0.853941i	−0.904265	+	0.426971i	$0.859581\pi$
$828$		0			0
$829$	−24.2845			−0.843435			−0.421718	−	0.906727i	$-0.638573\pi$
$829$	−24.2845			−0.843435			−0.421718	+	0.906727i	$0.638573\pi$
$830$		0			0
$831$	−98.4072			−3.41371
$832$		0			0
$833$	4.91610	−	6.11714i	0.170333	−	0.211946i
$834$		0			0
$835$	20.4120			0.706385
$836$		0			0
$837$	67.4161			2.33024
$838$		0			0
$839$		−	9.72375i		−	0.335701i	−0.985812	−	0.167851i	$-0.946317\pi$
$839$		−	9.72375i		−	0.335701i	0.985812	−	0.167851i	$-0.0536826\pi$
$840$		0			0
$841$	−75.5492			−2.60515
$842$		0			0
$843$			62.4688i			2.15154i
$844$		0			0
$845$			39.2920i			1.35168i
$846$		0			0
$847$		−	67.9788i		−	2.33578i
$848$		0			0
$849$	−24.4456			−0.838972
$850$		0			0
$851$	32.4049			1.11082
$852$		0			0
$853$			1.98274i			0.0678877i	0.999424	+	0.0339438i	$0.0108067\pi$
$853$			1.98274i			0.0678877i	−0.999424	+	0.0339438i	$0.989193\pi$
$854$		0			0
$855$		−	17.5623i		−	0.600619i
$856$		0			0
$857$			11.0838i			0.378617i	0.981918	+	0.189308i	$0.0606245\pi$
$857$			11.0838i			0.378617i	−0.981918	+	0.189308i	$0.939375\pi$
$858$		0			0
$859$	−12.7295			−0.434326			−0.217163	−	0.976135i	$-0.569680\pi$
$859$	−12.7295			−0.434326			−0.217163	+	0.976135i	$0.569680\pi$
$860$		0			0
$861$			3.14683i			0.107244i
$862$		0			0
$863$	−44.6932			−1.52137			−0.760687	−	0.649119i	$-0.775138\pi$
$863$	−44.6932			−1.52137			−0.760687	+	0.649119i	$0.775138\pi$
$864$		0			0
$865$	−36.4534			−1.23945
$866$		0			0
$867$	−52.0047	+	11.4579i	−1.76617	+	0.389131i
$868$		0			0
$869$	31.9661			1.08437
$870$		0			0
$871$	7.66395			0.259683
$872$		0			0
$873$			66.2843i			2.24338i
$874$		0			0
$875$	−62.3421			−2.10755
$876$		0			0
$877$		−	15.0084i		−	0.506797i	−0.967362	−	0.253399i	$-0.918452\pi$
$877$		−	15.0084i		−	0.506797i	0.967362	−	0.253399i	$-0.0815484\pi$
$878$		0			0
$879$			26.4003i			0.890460i
$880$		0			0
$881$			32.7194i			1.10234i	0.834391	+	0.551172i	$0.185819\pi$
$881$			32.7194i			1.10234i	−0.834391	+	0.551172i	$0.814181\pi$
$882$		0			0
$883$	−19.2174			−0.646717			−0.323358	−	0.946277i	$-0.604812\pi$
$883$	−19.2174			−0.646717			−0.323358	+	0.946277i	$0.604812\pi$
$884$		0			0
$885$	−12.2668			−0.412346
$886$		0			0
$887$		−	13.4229i		−	0.450698i	−0.974278	−	0.225349i	$-0.927648\pi$
$887$		−	13.4229i		−	0.450698i	0.974278	−	0.225349i	$-0.0723523\pi$
$888$		0			0
$889$			3.19800i			0.107257i
$890$		0			0
$891$		−	98.6409i		−	3.30459i
$892$		0			0
$893$	3.10353			0.103856
$894$		0			0
$895$			94.8001i			3.16882i
$896$		0			0
$897$	49.7604			1.66145
$898$		0			0
$899$	57.7222			1.92514
$900$		0			0
$901$	7.90453	−	9.83568i	0.263338	−	0.327674i
$902$		0			0
$903$	−26.7078			−0.888780
$904$		0			0
$905$	35.7753			1.18921
$906$		0			0
$907$		−	55.4105i		−	1.83988i	−0.392063	−	0.919938i	$-0.628239\pi$
$907$		−	55.4105i		−	1.83988i	0.392063	−	0.919938i	$-0.371761\pi$
$908$		0			0
$909$	41.1083			1.36348
$910$		0			0
$911$			20.2262i			0.670125i	0.942196	+	0.335063i	$0.108757\pi$
$911$			20.2262i			0.670125i	−0.942196	+	0.335063i	$0.891243\pi$
$912$		0			0
$913$			89.0121i			2.94587i
$914$		0			0
$915$		−	53.5549i		−	1.77047i
$916$		0			0
$917$	32.7277			1.08076
$918$		0			0
$919$	10.3389			0.341049			0.170524	−	0.985353i	$-0.445454\pi$
$919$	10.3389			0.341049			0.170524	+	0.985353i	$0.445454\pi$
$920$		0			0
$921$		−	91.9263i		−	3.02907i
$922$		0			0
$923$			18.7192i			0.616151i
$924$		0			0
$925$			36.3119i			1.19393i
$926$		0			0
$927$	81.2642			2.66907
$928$		0			0
$929$			17.5865i			0.576993i	0.957481	+	0.288497i	$0.0931554\pi$
$929$			17.5865i			0.576993i	−0.957481	+	0.288497i	$0.906845\pi$
$930$		0			0
$931$	1.25302			0.0410662
$932$		0			0
$933$	−2.75720			−0.0902666
$934$		0			0
$935$	58.7881	−	73.1505i	1.92258	−	2.39228i
$936$		0			0
$937$	−10.8067			−0.353040			−0.176520	−	0.984297i	$-0.556484\pi$
$937$	−10.8067			−0.353040			−0.176520	+	0.984297i	$0.556484\pi$
$938$		0			0
$939$	−11.2658			−0.367647
$940$		0			0
$941$			1.38382i			0.0451112i	0.999746	+	0.0225556i	$0.00718029\pi$
$941$			1.38382i			0.0451112i	−0.999746	+	0.0225556i	$0.992820\pi$
$942$		0			0
$943$	−3.10523			−0.101120
$944$		0			0
$945$			139.542i			4.53930i
$946$		0			0
$947$		−	33.2227i		−	1.07959i	−0.841795	−	0.539797i	$-0.818501\pi$
$947$		−	33.2227i		−	1.07959i	0.841795	−	0.539797i	$-0.181499\pi$
$948$		0			0
$949$		−	6.40773i		−	0.208004i
$950$		0			0
$951$	33.3353			1.08097
$952$		0			0
$953$	−23.2514			−0.753186			−0.376593	−	0.926379i	$-0.622905\pi$
$953$	−23.2514			−0.753186			−0.376593	+	0.926379i	$0.622905\pi$
$954$		0			0
$955$		−	73.7723i		−	2.38721i
$956$		0			0
$957$		−	186.162i		−	6.01777i
$958$		0			0
$959$		−	48.3414i		−	1.56102i
$960$		0			0
$961$	−0.868756			−0.0280244
$962$		0			0
$963$			59.6219i			1.92129i
$964$		0			0
$965$	52.4691			1.68904
$966$		0			0
$967$	1.52229			0.0489535			0.0244768	−	0.999700i	$-0.492208\pi$
$967$	1.52229			0.0489535			0.0244768	+	0.999700i	$0.492208\pi$
$968$		0			0
$969$	−6.62753	−	5.32628i	−0.212907	−	0.171105i
$970$		0			0
$971$	8.14779			0.261475			0.130738	−	0.991417i	$-0.458265\pi$
$971$	8.14779			0.261475			0.130738	+	0.991417i	$0.458265\pi$
$972$		0			0
$973$	−51.3243			−1.64538
$974$		0			0
$975$			55.7600i			1.78575i
$976$		0			0
$977$	49.5093			1.58394			0.791972	−	0.610558i	$-0.209055\pi$
$977$	49.5093			1.58394			0.791972	+	0.610558i	$0.209055\pi$
$978$		0			0
$979$			54.9336i			1.75569i
$980$		0			0
$981$			90.5077i			2.88969i
$982$		0			0
$983$			14.3649i			0.458170i	0.973406	+	0.229085i	$0.0735734\pi$
$983$			14.3649i			0.458170i	−0.973406	+	0.229085i	$0.926427\pi$
$984$		0			0
$985$	−33.0965			−1.05454
$986$		0			0
$987$	−44.0638			−1.40257
$988$		0			0
$989$		−	26.3547i		−	0.838030i
$990$		0			0
$991$		−	8.10592i		−	0.257493i	−0.991678	−	0.128746i	$-0.958905\pi$
$991$		−	8.10592i		−	0.257493i	0.991678	−	0.128746i	$-0.0410953\pi$
$992$		0			0
$993$		−	61.2401i		−	1.94339i
$994$		0			0
$995$	55.7463			1.76728
$996$		0			0
$997$			35.5735i			1.12662i	0.826245	+	0.563312i	$0.190473\pi$
$997$			35.5735i			1.12662i	−0.826245	+	0.563312i	$0.809527\pi$
$998$		0			0
$999$	41.9573			1.32747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.4	✓	46
17.16	even	2	inner	4012.2.b.b.237.43	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.4	✓	46	1.1	even	1	trivial
4012.2.b.b.237.43	yes	46	17.16	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.13247i q^{3} -3.91603i q^{5} +2.98385i q^{7} -6.81236 q^{9} +O(q^{10})\)	\(q-3.13247i q^{3} -3.91603i q^{5} +2.98385i q^{7} -6.81236 q^{9} -5.81225i q^{11} +1.72232 q^{13} -12.2668 q^{15} +(-2.58285 + 3.21386i) q^{17} -0.658321 q^{19} +9.34682 q^{21} +9.22325i q^{23} -10.3353 q^{25} +11.9421i q^{27} +10.2249i q^{29} -5.64524i q^{31} -18.2067 q^{33} +11.6848 q^{35} -3.51339i q^{37} -5.39511i q^{39} +0.336674i q^{41} -2.85742 q^{43} +26.6774i q^{45} -4.71431 q^{47} -1.90336 q^{49} +(10.0673 + 8.09069i) q^{51} -3.06039 q^{53} -22.7609 q^{55} +2.06217i q^{57} +1.00000 q^{59} +4.36582i q^{61} -20.3271i q^{63} -6.74465i q^{65} +4.44979 q^{67} +28.8915 q^{69} +10.8686i q^{71} -3.72041i q^{73} +32.3750i q^{75} +17.3429 q^{77} +5.49978i q^{79} +16.9712 q^{81} -15.3146 q^{83} +(12.5856 + 10.1145i) q^{85} +32.0293 q^{87} -9.45135 q^{89} +5.13914i q^{91} -17.6835 q^{93} +2.57801i q^{95} -9.73000i q^{97} +39.5952i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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