LMFDB - Embedded newform 4012.2.b.b.237.5

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			237.5
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.42

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.81557i q^{3} +1.77357i q^{5} -1.01302i q^{7} -4.92746 q^{9} +O(q^{10})$	$q-2.81557i q^{3} +1.77357i q^{5} -1.01302i q^{7} -4.92746 q^{9} +1.98267i q^{11} +5.96810 q^{13} +4.99361 q^{15} +(-3.47244 - 2.22309i) q^{17} +1.12852 q^{19} -2.85222 q^{21} +3.54324i q^{23} +1.85446 q^{25} +5.42691i q^{27} -3.47464i q^{29} +9.40563i q^{31} +5.58236 q^{33} +1.79665 q^{35} +4.75181i q^{37} -16.8036i q^{39} +10.0497i q^{41} -2.36274 q^{43} -8.73919i q^{45} -5.00336 q^{47} +5.97380 q^{49} +(-6.25928 + 9.77692i) q^{51} +11.5834 q^{53} -3.51640 q^{55} -3.17742i q^{57} +1.00000 q^{59} +7.35908i q^{61} +4.99160i q^{63} +10.5848i q^{65} +9.95061 q^{67} +9.97626 q^{69} +1.41124i q^{71} -13.9379i q^{73} -5.22136i q^{75} +2.00848 q^{77} +1.05486i q^{79} +0.497499 q^{81} -6.70479 q^{83} +(3.94281 - 6.15861i) q^{85} -9.78311 q^{87} -16.1957 q^{89} -6.04578i q^{91} +26.4823 q^{93} +2.00150i q^{95} +10.4135i q^{97} -9.76953i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.81557i		−	1.62557i	−0.582562	−	0.812786i	$-0.697950\pi$
$3$		−	2.81557i		−	1.62557i	0.582562	−	0.812786i	$-0.302050\pi$
$4$		0			0
$5$			1.77357i			0.793164i	0.917999	+	0.396582i	$0.129804\pi$
$5$			1.77357i			0.793164i	−0.917999	+	0.396582i	$0.870196\pi$
$6$		0			0
$7$		−	1.01302i		−	0.382884i	−0.981504	−	0.191442i	$-0.938684\pi$
$7$		−	1.01302i		−	0.382884i	0.981504	−	0.191442i	$-0.0613164\pi$
$8$		0			0
$9$	−4.92746			−1.64249
$10$		0			0
$11$			1.98267i			0.597797i	0.954285	+	0.298899i	$0.0966193\pi$
$11$			1.98267i			0.597797i	−0.954285	+	0.298899i	$0.903381\pi$
$12$		0			0
$13$	5.96810			1.65525			0.827627	−	0.561279i	$-0.189691\pi$
$13$	5.96810			1.65525			0.827627	+	0.561279i	$0.189691\pi$
$14$		0			0
$15$	4.99361			1.28935
$16$		0			0
$17$	−3.47244	−	2.22309i	−0.842191	−	0.539179i
$17$	−3.47244	−	2.22309i	−0.842191	−	0.539179i
$18$		0			0
$19$	1.12852			0.258899			0.129450	−	0.991586i	$-0.458679\pi$
$19$	1.12852			0.258899			0.129450	+	0.991586i	$0.458679\pi$
$20$		0			0
$21$	−2.85222			−0.622406
$22$		0			0
$23$			3.54324i			0.738817i	0.929267	+	0.369409i	$0.120440\pi$
$23$			3.54324i			0.738817i	−0.929267	+	0.369409i	$0.879560\pi$
$24$		0			0
$25$	1.85446			0.370891
$26$		0			0
$27$			5.42691i			1.04441i
$28$		0			0
$29$		−	3.47464i		−	0.645224i	−0.946531	−	0.322612i	$-0.895439\pi$
$29$		−	3.47464i		−	0.645224i	0.946531	−	0.322612i	$-0.104561\pi$
$30$		0			0
$31$			9.40563i			1.68930i	0.535318	+	0.844651i	$0.320192\pi$
$31$			9.40563i			1.68930i	−0.535318	+	0.844651i	$0.679808\pi$
$32$		0			0
$33$	5.58236			0.971763
$34$		0			0
$35$	1.79665			0.303690
$36$		0			0
$37$			4.75181i			0.781193i	0.920562	+	0.390597i	$0.127731\pi$
$37$			4.75181i			0.781193i	−0.920562	+	0.390597i	$0.872269\pi$
$38$		0			0
$39$		−	16.8036i		−	2.69073i
$40$		0			0
$41$			10.0497i			1.56950i	0.619814	+	0.784749i	$0.287208\pi$
$41$			10.0497i			1.56950i	−0.619814	+	0.784749i	$0.712792\pi$
$42$		0			0
$43$	−2.36274			−0.360315			−0.180157	−	0.983638i	$-0.557661\pi$
$43$	−2.36274			−0.360315			−0.180157	+	0.983638i	$0.557661\pi$
$44$		0			0
$45$		−	8.73919i		−	1.30276i
$46$		0			0
$47$	−5.00336			−0.729815			−0.364908	−	0.931044i	$-0.618899\pi$
$47$	−5.00336			−0.729815			−0.364908	+	0.931044i	$0.618899\pi$
$48$		0			0
$49$	5.97380			0.853400
$50$		0			0
$51$	−6.25928	+	9.77692i	−0.876475	+	1.36904i
$52$		0			0
$53$	11.5834			1.59111			0.795553	−	0.605884i	$-0.207180\pi$
$53$	11.5834			1.59111			0.795553	+	0.605884i	$0.207180\pi$
$54$		0			0
$55$	−3.51640			−0.474151
$56$		0			0
$57$		−	3.17742i		−	0.420860i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			7.35908i			0.942234i	0.882071	+	0.471117i	$0.156149\pi$
$61$			7.35908i			0.942234i	−0.882071	+	0.471117i	$0.843851\pi$
$62$		0			0
$63$			4.99160i			0.628882i
$64$		0			0
$65$			10.5848i			1.31289i
$66$		0			0
$67$	9.95061			1.21566			0.607830	−	0.794067i	$-0.292040\pi$
$67$	9.95061			1.21566			0.607830	+	0.794067i	$0.292040\pi$
$68$		0			0
$69$	9.97626			1.20100
$70$		0			0
$71$			1.41124i			0.167483i	0.996488	+	0.0837415i	$0.0266870\pi$
$71$			1.41124i			0.167483i	−0.996488	+	0.0837415i	$0.973313\pi$
$72$		0			0
$73$		−	13.9379i		−	1.63131i	−0.578536	−	0.815657i	$-0.696376\pi$
$73$		−	13.9379i		−	1.63131i	0.578536	−	0.815657i	$-0.303624\pi$
$74$		0			0
$75$		−	5.22136i		−	0.602911i
$76$		0			0
$77$	2.00848			0.228887
$78$		0			0
$79$			1.05486i			0.118681i	0.998238	+	0.0593407i	$0.0188999\pi$
$79$			1.05486i			0.118681i	−0.998238	+	0.0593407i	$0.981100\pi$
$80$		0			0
$81$	0.497499			0.0552776
$82$		0			0
$83$	−6.70479			−0.735947			−0.367973	−	0.929836i	$-0.619948\pi$
$83$	−6.70479			−0.735947			−0.367973	+	0.929836i	$0.619948\pi$
$84$		0			0
$85$	3.94281	−	6.15861i	0.427657	−	0.667995i
$86$		0			0
$87$	−9.78311			−1.04886
$88$		0			0
$89$	−16.1957			−1.71674			−0.858368	−	0.513034i	$-0.828521\pi$
$89$	−16.1957			−1.71674			−0.858368	+	0.513034i	$0.828521\pi$
$90$		0			0
$91$		−	6.04578i		−	0.633770i
$92$		0			0
$93$	26.4823			2.74608
$94$		0			0
$95$			2.00150i			0.205349i
$96$		0			0
$97$			10.4135i			1.05733i	0.848830	+	0.528665i	$0.177307\pi$
$97$			10.4135i			1.05733i	−0.848830	+	0.528665i	$0.822693\pi$
$98$		0			0
$99$		−	9.76953i		−	0.981875i
$100$		0			0
$101$	19.4250			1.93286			0.966431	−	0.256925i	$-0.0827095\pi$
$101$	19.4250			1.93286			0.966431	+	0.256925i	$0.0827095\pi$
$102$		0			0
$103$	8.62785			0.850128			0.425064	−	0.905163i	$-0.360252\pi$
$103$	8.62785			0.850128			0.425064	+	0.905163i	$0.360252\pi$
$104$		0			0
$105$		−	5.05861i		−	0.493670i
$106$		0			0
$107$		−	12.6904i		−	1.22683i	−0.789762	−	0.613414i	$-0.789796\pi$
$107$		−	12.6904i		−	1.22683i	0.789762	−	0.613414i	$-0.210204\pi$
$108$		0			0
$109$		−	10.4306i		−	0.999068i	−0.866294	−	0.499534i	$-0.833505\pi$
$109$		−	10.4306i		−	0.999068i	0.866294	−	0.499534i	$-0.166495\pi$
$110$		0			0
$111$	13.3791			1.26989
$112$		0			0
$113$			16.6706i			1.56824i	0.620610	+	0.784119i	$0.286885\pi$
$113$			16.6706i			1.56824i	−0.620610	+	0.784119i	$0.713115\pi$
$114$		0			0
$115$	−6.28418			−0.586003
$116$		0			0
$117$	−29.4076			−2.71873
$118$		0			0
$119$	−2.25203	+	3.51764i	−0.206443	+	0.322461i
$120$		0			0
$121$	7.06902			0.642638
$122$		0			0
$123$	28.2956			2.55133
$124$		0			0
$125$			12.1568i			1.08734i
$126$		0			0
$127$	19.6325			1.74210			0.871049	−	0.491196i	$-0.163440\pi$
$127$	19.6325			1.74210			0.871049	+	0.491196i	$0.163440\pi$
$128$		0			0
$129$			6.65247i			0.585718i
$130$		0			0
$131$		−	14.4265i		−	1.26045i	−0.776413	−	0.630224i	$-0.782963\pi$
$131$		−	14.4265i		−	1.26045i	0.776413	−	0.630224i	$-0.217037\pi$
$132$		0			0
$133$		−	1.14320i		−	0.0991284i
$134$		0			0
$135$	−9.62500			−0.828388
$136$		0			0
$137$	10.2949			0.879554			0.439777	−	0.898107i	$-0.355057\pi$
$137$	10.2949			0.879554			0.439777	+	0.898107i	$0.355057\pi$
$138$		0			0
$139$			12.6278i			1.07108i	0.844510	+	0.535540i	$0.179892\pi$
$139$			12.6278i			1.07108i	−0.844510	+	0.535540i	$0.820108\pi$
$140$		0			0
$141$			14.0873i			1.18637i
$142$		0			0
$143$			11.8328i			0.989506i
$144$		0			0
$145$	6.16251			0.511769
$146$		0			0
$147$		−	16.8197i		−	1.38726i
$148$		0			0
$149$	5.16778			0.423361			0.211681	−	0.977339i	$-0.432106\pi$
$149$	5.16778			0.423361			0.211681	+	0.977339i	$0.432106\pi$
$150$		0			0
$151$	15.5358			1.26428			0.632142	−	0.774853i	$-0.282176\pi$
$151$	15.5358			1.26428			0.632142	+	0.774853i	$0.282176\pi$
$152$		0			0
$153$	17.1103	+	10.9542i	1.38329	+	0.885595i
$154$		0			0
$155$	−16.6815			−1.33989
$156$		0			0
$157$	−5.90512			−0.471280			−0.235640	−	0.971840i	$-0.575719\pi$
$157$	−5.90512			−0.471280			−0.235640	+	0.971840i	$0.575719\pi$
$158$		0			0
$159$		−	32.6140i		−	2.58646i
$160$		0			0
$161$	3.58936			0.282881
$162$		0			0
$163$		−	10.5183i		−	0.823859i	−0.911216	−	0.411929i	$-0.864855\pi$
$163$		−	10.5183i		−	0.823859i	0.911216	−	0.411929i	$-0.135145\pi$
$164$		0			0
$165$			9.90069i			0.770767i
$166$		0			0
$167$		−	25.0986i		−	1.94219i	−0.238700	−	0.971093i	$-0.576721\pi$
$167$		−	25.0986i		−	1.94219i	0.238700	−	0.971093i	$-0.423279\pi$
$168$		0			0
$169$	22.6182			1.73986
$170$		0			0
$171$	−5.56072			−0.425239
$172$		0			0
$173$		−	7.25160i		−	0.551329i	−0.961254	−	0.275664i	$-0.911102\pi$
$173$		−	7.25160i		−	0.551329i	0.961254	−	0.275664i	$-0.0888978\pi$
$174$		0			0
$175$		−	1.87859i		−	0.142008i
$176$		0			0
$177$		−	2.81557i		−	0.211632i
$178$		0			0
$179$	−20.5865			−1.53870			−0.769352	−	0.638825i	$-0.779421\pi$
$179$	−20.5865			−1.53870			−0.769352	+	0.638825i	$0.779421\pi$
$180$		0			0
$181$			16.2432i			1.20735i	0.797230	+	0.603675i	$0.206298\pi$
$181$			16.2432i			1.20735i	−0.797230	+	0.603675i	$0.793702\pi$
$182$		0			0
$183$	20.7200			1.53167
$184$		0			0
$185$	−8.42766			−0.619614
$186$		0			0
$187$	4.40766	−	6.88471i	0.322320	−	0.503460i
$188$		0			0
$189$	5.49755			0.399888
$190$		0			0
$191$	−0.717439			−0.0519121			−0.0259560	−	0.999663i	$-0.508263\pi$
$191$	−0.717439			−0.0519121			−0.0259560	+	0.999663i	$0.508263\pi$
$192$		0			0
$193$			15.8853i			1.14345i	0.820447	+	0.571723i	$0.193725\pi$
$193$			15.8853i			1.14345i	−0.820447	+	0.571723i	$0.806275\pi$
$194$		0			0
$195$	29.8024			2.13419
$196$		0			0
$197$		−	10.5858i		−	0.754205i	−0.926172	−	0.377102i	$-0.876921\pi$
$197$		−	10.5858i		−	0.754205i	0.926172	−	0.377102i	$-0.123079\pi$
$198$		0			0
$199$		−	12.3598i		−	0.876163i	−0.898935	−	0.438082i	$-0.855658\pi$
$199$		−	12.3598i		−	0.876163i	0.898935	−	0.438082i	$-0.144342\pi$
$200$		0			0
$201$		−	28.0167i		−	1.97615i
$202$		0			0
$203$	−3.51986			−0.247046
$204$		0			0
$205$	−17.8238			−1.24487
$206$		0			0
$207$		−	17.4592i		−	1.21350i
$208$		0			0
$209$			2.23747i			0.154769i
$210$		0			0
$211$		−	15.7388i		−	1.08350i	−0.840538	−	0.541752i	$-0.817761\pi$
$211$		−	15.7388i		−	1.08350i	0.840538	−	0.541752i	$-0.182239\pi$
$212$		0			0
$213$	3.97344			0.272256
$214$		0			0
$215$		−	4.19048i		−	0.285788i
$216$		0			0
$217$	9.52805			0.646806
$218$		0			0
$219$	−39.2433			−2.65182
$220$		0			0
$221$	−20.7239	−	13.2676i	−1.39404	−	0.892478i
$222$		0			0
$223$	−14.7639			−0.988666			−0.494333	−	0.869273i	$-0.664588\pi$
$223$	−14.7639			−0.988666			−0.494333	+	0.869273i	$0.664588\pi$
$224$		0			0
$225$	−9.13777			−0.609184
$226$		0			0
$227$			3.24334i			0.215268i	0.994191	+	0.107634i	$0.0343275\pi$
$227$			3.24334i			0.215268i	−0.994191	+	0.107634i	$0.965673\pi$
$228$		0			0
$229$	−4.52390			−0.298948			−0.149474	−	0.988766i	$-0.547758\pi$
$229$	−4.52390			−0.298948			−0.149474	+	0.988766i	$0.547758\pi$
$230$		0			0
$231$		−	5.65501i		−	0.372073i
$232$		0			0
$233$		−	13.1725i		−	0.862957i	−0.902123	−	0.431479i	$-0.857992\pi$
$233$		−	13.1725i		−	0.862957i	0.902123	−	0.431479i	$-0.142008\pi$
$234$		0			0
$235$		−	8.87380i		−	0.578863i
$236$		0			0
$237$	2.97005			0.192925
$238$		0			0
$239$	−8.11549			−0.524947			−0.262474	−	0.964939i	$-0.584538\pi$
$239$	−8.11549			−0.524947			−0.262474	+	0.964939i	$0.584538\pi$
$240$		0			0
$241$			28.9264i			1.86331i	0.363341	+	0.931656i	$0.381636\pi$
$241$			28.9264i			1.86331i	−0.363341	+	0.931656i	$0.618364\pi$
$242$		0			0
$243$			14.8800i			0.954552i
$244$		0			0
$245$			10.5949i			0.676886i
$246$		0			0
$247$	6.73509			0.428544
$248$		0			0
$249$			18.8778i			1.19633i
$250$		0			0
$251$	0.609374			0.0384634			0.0192317	−	0.999815i	$-0.493878\pi$
$251$	0.609374			0.0384634			0.0192317	+	0.999815i	$0.493878\pi$
$252$		0			0
$253$	−7.02508			−0.441663
$254$		0			0
$255$	−17.3400	−	11.1013i	−1.08588	−	0.695188i
$256$		0			0
$257$	−4.50186			−0.280818			−0.140409	−	0.990094i	$-0.544842\pi$
$257$	−4.50186			−0.280818			−0.140409	+	0.990094i	$0.544842\pi$
$258$		0			0
$259$	4.81366			0.299106
$260$		0			0
$261$			17.1212i			1.05977i
$262$		0			0
$263$	−7.19803			−0.443850			−0.221925	−	0.975064i	$-0.571234\pi$
$263$	−7.19803			−0.443850			−0.221925	+	0.975064i	$0.571234\pi$
$264$		0			0
$265$			20.5440i			1.26201i
$266$		0			0
$267$			45.6001i			2.79068i
$268$		0			0
$269$		−	20.3740i		−	1.24223i	−0.783721	−	0.621113i	$-0.786681\pi$
$269$		−	20.3740i		−	1.24223i	0.783721	−	0.621113i	$-0.213319\pi$
$270$		0			0
$271$	−3.58302			−0.217653			−0.108826	−	0.994061i	$-0.534709\pi$
$271$	−3.58302			−0.217653			−0.108826	+	0.994061i	$0.534709\pi$
$272$		0			0
$273$	−17.0223			−1.03024
$274$		0			0
$275$			3.67678i			0.221718i
$276$		0			0
$277$			3.80453i			0.228592i	0.993447	+	0.114296i	$0.0364613\pi$
$277$			3.80453i			0.228592i	−0.993447	+	0.114296i	$0.963539\pi$
$278$		0			0
$279$		−	46.3459i		−	2.77466i
$280$		0			0
$281$	−27.4803			−1.63934			−0.819668	−	0.572839i	$-0.805842\pi$
$281$	−27.4803			−1.63934			−0.819668	+	0.572839i	$0.805842\pi$
$282$		0			0
$283$		−	18.4731i		−	1.09811i	−0.835787	−	0.549055i	$-0.814988\pi$
$283$		−	18.4731i		−	1.09811i	0.835787	−	0.549055i	$-0.185012\pi$
$284$		0			0
$285$	5.63537			0.333811
$286$		0			0
$287$	10.1805			0.600935
$288$		0			0
$289$	7.11572	+	15.4391i	0.418571	+	0.908184i
$290$		0			0
$291$	29.3200			1.71877
$292$		0			0
$293$	25.0094			1.46106			0.730532	−	0.682878i	$-0.239272\pi$
$293$	25.0094			1.46106			0.730532	+	0.682878i	$0.239272\pi$
$294$		0			0
$295$			1.77357i			0.103261i
$296$		0			0
$297$	−10.7598			−0.624346
$298$		0			0
$299$			21.1464i			1.22293i
$300$		0			0
$301$			2.39349i			0.137959i
$302$		0			0
$303$		−	54.6926i		−	3.14201i
$304$		0			0
$305$	−13.0518			−0.747346
$306$		0			0
$307$	28.2474			1.61217			0.806083	−	0.591802i	$-0.201583\pi$
$307$	28.2474			1.61217			0.806083	+	0.591802i	$0.201583\pi$
$308$		0			0
$309$		−	24.2924i		−	1.38194i
$310$		0			0
$311$			21.4590i			1.21683i	0.793620	+	0.608414i	$0.208194\pi$
$311$			21.4590i			1.21683i	−0.793620	+	0.608414i	$0.791806\pi$
$312$		0			0
$313$			0.372065i			0.0210303i	0.999945	+	0.0105152i	$0.00334714\pi$
$313$			0.372065i			0.0210303i	−0.999945	+	0.0105152i	$0.996653\pi$
$314$		0			0
$315$	−8.85294			−0.498806
$316$		0			0
$317$		−	13.6019i		−	0.763960i	−0.924171	−	0.381980i	$-0.875242\pi$
$317$		−	13.6019i		−	0.763960i	0.924171	−	0.381980i	$-0.124758\pi$
$318$		0			0
$319$	6.88906			0.385713
$320$		0			0
$321$	−35.7308			−1.99430
$322$		0			0
$323$	−3.91871	−	2.50880i	−0.218043	−	0.139593i
$324$		0			0
$325$	11.0676			0.613919
$326$		0			0
$327$	−29.3681			−1.62406
$328$		0			0
$329$			5.06848i			0.279435i
$330$		0			0
$331$	−31.6998			−1.74238			−0.871189	−	0.490949i	$-0.836650\pi$
$331$	−31.6998			−1.74238			−0.871189	+	0.490949i	$0.836650\pi$
$332$		0			0
$333$		−	23.4144i		−	1.28310i
$334$		0			0
$335$			17.6481i			0.964218i
$336$		0			0
$337$			15.4649i			0.842426i	0.906962	+	0.421213i	$0.138396\pi$
$337$			15.4649i			0.842426i	−0.906962	+	0.421213i	$0.861604\pi$
$338$		0			0
$339$	46.9373			2.54929
$340$		0			0
$341$	−18.6483			−1.00986
$342$		0			0
$343$		−	13.1427i		−	0.709637i
$344$		0			0
$345$			17.6936i			0.952591i
$346$		0			0
$347$		−	10.1605i		−	0.545446i	−0.962093	−	0.272723i	$-0.912076\pi$
$347$		−	10.1605i		−	0.545446i	0.962093	−	0.272723i	$-0.0879242\pi$
$348$		0			0
$349$	2.33941			0.125226			0.0626128	−	0.998038i	$-0.480057\pi$
$349$	2.33941			0.125226			0.0626128	+	0.998038i	$0.480057\pi$
$350$		0			0
$351$			32.3884i			1.72876i
$352$		0			0
$353$	19.7266			1.04994			0.524970	−	0.851121i	$-0.324076\pi$
$353$	19.7266			1.04994			0.524970	+	0.851121i	$0.324076\pi$
$354$		0			0
$355$	−2.50292			−0.132841
$356$		0			0
$357$	9.90418	+	6.34075i	0.524185	+	0.335588i
$358$		0			0
$359$	−8.02116			−0.423341			−0.211670	−	0.977341i	$-0.567890\pi$
$359$	−8.02116			−0.423341			−0.211670	+	0.977341i	$0.567890\pi$
$360$		0			0
$361$	−17.7265			−0.932971
$362$		0			0
$363$		−	19.9034i		−	1.04466i
$364$		0			0
$365$	24.7199			1.29390
$366$		0			0
$367$			23.3745i			1.22014i	0.792349	+	0.610069i	$0.208858\pi$
$367$			23.3745i			1.22014i	−0.792349	+	0.610069i	$0.791142\pi$
$368$		0			0
$369$		−	49.5195i		−	2.57788i
$370$		0			0
$371$		−	11.7342i		−	0.609209i
$372$		0			0
$373$	−6.07516			−0.314560			−0.157280	−	0.987554i	$-0.550273\pi$
$373$	−6.07516			−0.314560			−0.157280	+	0.987554i	$0.550273\pi$
$374$		0			0
$375$	34.2285			1.76755
$376$		0			0
$377$		−	20.7370i		−	1.06801i
$378$		0			0
$379$		−	0.180751i		−	0.00928458i	−0.999989	−	0.00464229i	$-0.998522\pi$
$379$		−	0.180751i		−	0.00928458i	0.999989	−	0.00464229i	$-0.00147769\pi$
$380$		0			0
$381$		−	55.2766i		−	2.83191i
$382$		0			0
$383$	−19.5919			−1.00110			−0.500549	−	0.865708i	$-0.666868\pi$
$383$	−19.5919			−1.00110			−0.500549	+	0.865708i	$0.666868\pi$
$384$		0			0
$385$			3.56217i			0.181545i
$386$		0			0
$387$	11.6423			0.591812
$388$		0			0
$389$	26.1382			1.32526			0.662630	−	0.748947i	$-0.269440\pi$
$389$	26.1382			1.32526			0.662630	+	0.748947i	$0.269440\pi$
$390$		0			0
$391$	7.87696	−	12.3037i	0.398355	−	0.622225i
$392$		0			0
$393$	−40.6188			−2.04895
$394$		0			0
$395$	−1.87087			−0.0941339
$396$		0			0
$397$			9.85946i			0.494832i	0.968909	+	0.247416i	$0.0795815\pi$
$397$			9.85946i			0.494832i	−0.968909	+	0.247416i	$0.920419\pi$
$398$		0			0
$399$	−3.21878			−0.161140
$400$		0			0
$401$		−	8.62190i		−	0.430557i	−0.976553	−	0.215279i	$-0.930934\pi$
$401$		−	8.62190i		−	0.430557i	0.976553	−	0.215279i	$-0.0690660\pi$
$402$		0			0
$403$			56.1337i			2.79622i
$404$		0			0
$405$			0.882348i			0.0438442i
$406$		0			0
$407$	−9.42128			−0.466995
$408$		0			0
$409$	21.4765			1.06194			0.530972	−	0.847389i	$-0.321827\pi$
$409$	21.4765			1.06194			0.530972	+	0.847389i	$0.321827\pi$
$410$		0			0
$411$		−	28.9861i		−	1.42978i
$412$		0			0
$413$		−	1.01302i		−	0.0498472i
$414$		0			0
$415$		−	11.8914i		−	0.583726i
$416$		0			0
$417$	35.5547			1.74112
$418$		0			0
$419$			27.4699i			1.34199i	0.741461	+	0.670996i	$0.234133\pi$
$419$			27.4699i			1.34199i	−0.741461	+	0.670996i	$0.765867\pi$
$420$		0			0
$421$	12.3256			0.600712			0.300356	−	0.953827i	$-0.402895\pi$
$421$	12.3256			0.600712			0.300356	+	0.953827i	$0.402895\pi$
$422$		0			0
$423$	24.6539			1.19871
$424$		0			0
$425$	−6.43949	−	4.12263i	−0.312361	−	0.199977i
$426$		0			0
$427$	7.45487			0.360766
$428$		0			0
$429$	33.3161			1.60851
$430$		0			0
$431$		−	33.6273i		−	1.61977i	−0.586587	−	0.809886i	$-0.699529\pi$
$431$		−	33.6273i		−	1.61977i	0.586587	−	0.809886i	$-0.300471\pi$
$432$		0			0
$433$	20.2399			0.972665			0.486333	−	0.873774i	$-0.338334\pi$
$433$	20.2399			0.972665			0.486333	+	0.873774i	$0.338334\pi$
$434$		0			0
$435$		−	17.3510i		−	0.831917i
$436$		0			0
$437$			3.99860i			0.191279i
$438$		0			0
$439$		−	33.9267i		−	1.61923i	−0.586959	−	0.809616i	$-0.699675\pi$
$439$		−	33.9267i		−	1.61923i	0.586959	−	0.809616i	$-0.300325\pi$
$440$		0			0
$441$	−29.4357			−1.40170
$442$		0			0
$443$	16.4757			0.782782			0.391391	−	0.920224i	$-0.371994\pi$
$443$	16.4757			0.782782			0.391391	+	0.920224i	$0.371994\pi$
$444$		0			0
$445$		−	28.7241i		−	1.36165i
$446$		0			0
$447$		−	14.5503i		−	0.688205i
$448$		0			0
$449$			20.0182i			0.944720i	0.881406	+	0.472360i	$0.156598\pi$
$449$			20.0182i			0.944720i	−0.881406	+	0.472360i	$0.843402\pi$
$450$		0			0
$451$	−19.9252			−0.938242
$452$		0			0
$453$		−	43.7421i		−	2.05518i
$454$		0			0
$455$	10.7226			0.502683
$456$		0			0
$457$	−36.5676			−1.71056			−0.855280	−	0.518166i	$-0.826615\pi$
$457$	−36.5676			−1.71056			−0.855280	+	0.518166i	$0.826615\pi$
$458$		0			0
$459$	12.0645	−	18.8447i	0.563124	−	0.879593i
$460$		0			0
$461$	38.8086			1.80750			0.903748	−	0.428065i	$-0.140805\pi$
$461$	38.8086			1.80750			0.903748	+	0.428065i	$0.140805\pi$
$462$		0			0
$463$	−16.6910			−0.775697			−0.387848	−	0.921723i	$-0.626782\pi$
$463$	−16.6910			−0.775697			−0.387848	+	0.921723i	$0.626782\pi$
$464$		0			0
$465$			46.9681i			2.17809i
$466$		0			0
$467$	34.8224			1.61139			0.805694	−	0.592332i	$-0.201793\pi$
$467$	34.8224			1.61139			0.805694	+	0.592332i	$0.201793\pi$
$468$		0			0
$469$		−	10.0801i		−	0.465457i
$470$		0			0
$471$			16.6263i			0.766100i
$472$		0			0
$473$		−	4.68453i		−	0.215395i
$474$		0			0
$475$	2.09278			0.0960235
$476$		0			0
$477$	−57.0769			−2.61337
$478$		0			0
$479$			29.0259i			1.32623i	0.748519	+	0.663114i	$0.230765\pi$
$479$			29.0259i			1.32623i	−0.748519	+	0.663114i	$0.769235\pi$
$480$		0			0
$481$			28.3593i			1.29307i
$482$		0			0
$483$		−	10.1061i		−	0.459844i
$484$		0			0
$485$	−18.4690			−0.838636
$486$		0			0
$487$		−	14.7029i		−	0.666250i	−0.942883	−	0.333125i	$-0.891897\pi$
$487$		−	14.7029i		−	0.666250i	0.942883	−	0.333125i	$-0.108103\pi$
$488$		0			0
$489$	−29.6151			−1.33924
$490$		0			0
$491$	−19.6682			−0.887613			−0.443807	−	0.896123i	$-0.646372\pi$
$491$	−19.6682			−0.887613			−0.443807	+	0.896123i	$0.646372\pi$
$492$		0			0
$493$	−7.72445	+	12.0655i	−0.347892	+	0.543402i
$494$		0			0
$495$	17.3269			0.778787
$496$		0			0
$497$	1.42960			0.0641265
$498$		0			0
$499$			11.1454i			0.498936i	0.968383	+	0.249468i	$0.0802558\pi$
$499$			11.1454i			0.498936i	−0.968383	+	0.249468i	$0.919744\pi$
$500$		0			0
$501$	−70.6669			−3.15717
$502$		0			0
$503$		−	13.7333i		−	0.612338i	−0.951977	−	0.306169i	$-0.900953\pi$
$503$		−	13.7333i		−	0.612338i	0.951977	−	0.306169i	$-0.0990472\pi$
$504$		0			0
$505$			34.4516i			1.53308i
$506$		0			0
$507$		−	63.6833i		−	2.82827i
$508$		0			0
$509$	22.5571			0.999826			0.499913	−	0.866076i	$-0.333365\pi$
$509$	22.5571			0.999826			0.499913	+	0.866076i	$0.333365\pi$
$510$		0			0
$511$	−14.1194			−0.624604
$512$		0			0
$513$			6.12436i			0.270397i
$514$		0			0
$515$			15.3021i			0.674290i
$516$		0			0
$517$		−	9.92001i		−	0.436282i
$518$		0			0
$519$	−20.4174			−0.896225
$520$		0			0
$521$		−	6.46984i		−	0.283449i	−0.989906	−	0.141724i	$-0.954735\pi$
$521$		−	6.46984i		−	0.283449i	0.989906	−	0.141724i	$-0.0452647\pi$
$522$		0			0
$523$	36.0518			1.57643			0.788217	−	0.615397i	$-0.211004\pi$
$523$	36.0518			1.57643			0.788217	+	0.615397i	$0.211004\pi$
$524$		0			0
$525$	−5.28932			−0.230845
$526$		0			0
$527$	20.9096	−	32.6605i	0.910836	−	1.42271i
$528$		0			0
$529$	10.4454			0.454149
$530$		0			0
$531$	−4.92746			−0.213834
$532$		0			0
$533$			59.9775i			2.59792i
$534$		0			0
$535$	22.5073			0.973075
$536$		0			0
$537$			57.9627i			2.50128i
$538$		0			0
$539$			11.8441i			0.510160i
$540$		0			0
$541$			25.2692i			1.08641i	0.839601	+	0.543203i	$0.182789\pi$
$541$			25.2692i			1.08641i	−0.839601	+	0.543203i	$0.817211\pi$
$542$		0			0
$543$	45.7340			1.96264
$544$		0			0
$545$	18.4993			0.792424
$546$		0			0
$547$			23.7132i			1.01390i	0.861975	+	0.506951i	$0.169228\pi$
$547$			23.7132i			1.01390i	−0.861975	+	0.506951i	$0.830772\pi$
$548$		0			0
$549$		−	36.2616i		−	1.54761i
$550$		0			0
$551$		−	3.92119i		−	0.167048i
$552$		0			0
$553$	1.06859			0.0454412
$554$		0			0
$555$			23.7287i			1.00723i
$556$		0			0
$557$	−14.1225			−0.598391			−0.299195	−	0.954192i	$-0.596718\pi$
$557$	−14.1225			−0.598391			−0.299195	+	0.954192i	$0.596718\pi$
$558$		0			0
$559$	−14.1011			−0.596412
$560$		0			0
$561$	−19.3844	−	12.4101i	−0.818410	−	0.523955i
$562$		0			0
$563$	3.11376			0.131229			0.0656146	−	0.997845i	$-0.479099\pi$
$563$	3.11376			0.131229			0.0656146	+	0.997845i	$0.479099\pi$
$564$		0			0
$565$	−29.5664			−1.24387
$566$		0			0
$567$		−	0.503974i		−	0.0211649i
$568$		0			0
$569$	39.6305			1.66140			0.830699	−	0.556723i	$-0.187941\pi$
$569$	39.6305			1.66140			0.830699	+	0.556723i	$0.187941\pi$
$570$		0			0
$571$		−	10.7523i		−	0.449972i	−0.974362	−	0.224986i	$-0.927766\pi$
$571$		−	10.7523i		−	0.449972i	0.974362	−	0.224986i	$-0.0722336\pi$
$572$		0			0
$573$			2.02000i			0.0843869i
$574$		0			0
$575$			6.57079i			0.274021i
$576$		0			0
$577$	42.7902			1.78138			0.890689	−	0.454614i	$-0.150223\pi$
$577$	42.7902			1.78138			0.890689	+	0.454614i	$0.150223\pi$
$578$		0			0
$579$	44.7262			1.85876
$580$		0			0
$581$			6.79206i			0.281782i
$582$		0			0
$583$			22.9661i			0.951159i
$584$		0			0
$585$		−	52.1564i		−	2.15640i
$586$		0			0
$587$	−21.5944			−0.891296			−0.445648	−	0.895208i	$-0.647027\pi$
$587$	−21.5944			−0.891296			−0.445648	+	0.895208i	$0.647027\pi$
$588$		0			0
$589$			10.6144i			0.437359i
$590$		0			0
$591$	−29.8050			−1.22601
$592$		0			0
$593$	−25.2159			−1.03549			−0.517747	−	0.855534i	$-0.673229\pi$
$593$	−25.2159			−1.03549			−0.517747	+	0.855534i	$0.673229\pi$
$594$		0			0
$595$	−6.23877	−	3.99412i	−0.255765	−	0.163743i
$596$		0			0
$597$	−34.7999			−1.42427
$598$		0			0
$599$	−21.3818			−0.873635			−0.436817	−	0.899550i	$-0.643894\pi$
$599$	−21.3818			−0.873635			−0.436817	+	0.899550i	$0.643894\pi$
$600$		0			0
$601$		−	25.1080i		−	1.02418i	−0.858932	−	0.512089i	$-0.828872\pi$
$601$		−	25.1080i		−	1.02418i	0.858932	−	0.512089i	$-0.171128\pi$
$602$		0			0
$603$	−49.0313			−1.99671
$604$		0			0
$605$			12.5374i			0.509717i
$606$		0			0
$607$			6.83691i			0.277502i	0.990327	+	0.138751i	$0.0443087\pi$
$607$			6.83691i			0.277502i	−0.990327	+	0.138751i	$0.955691\pi$
$608$		0			0
$609$			9.91044i			0.401591i
$610$		0			0
$611$	−29.8606			−1.20803
$612$		0			0
$613$	18.3054			0.739346			0.369673	−	0.929162i	$-0.379470\pi$
$613$	18.3054			0.739346			0.369673	+	0.929162i	$0.379470\pi$
$614$		0			0
$615$			50.1843i			2.02362i
$616$		0			0
$617$			6.06759i			0.244272i	0.992513	+	0.122136i	$0.0389744\pi$
$617$			6.06759i			0.244272i	−0.992513	+	0.122136i	$0.961026\pi$
$618$		0			0
$619$			1.30249i			0.0523513i	0.999657	+	0.0261757i	$0.00833292\pi$
$619$			1.30249i			0.0523513i	−0.999657	+	0.0261757i	$0.991667\pi$
$620$		0			0
$621$	−19.2289			−0.771628
$622$		0			0
$623$			16.4065i			0.657311i
$624$		0			0
$625$	−12.2887			−0.491548
$626$		0			0
$627$	6.29978			0.251589
$628$		0			0
$629$	10.5637	−	16.5004i	0.421203	−	0.657914i
$630$		0			0
$631$	−44.7455			−1.78129			−0.890644	−	0.454700i	$-0.849746\pi$
$631$	−44.7455			−1.78129			−0.890644	+	0.454700i	$0.849746\pi$
$632$		0			0
$633$	−44.3138			−1.76132
$634$		0			0
$635$			34.8195i			1.38177i
$636$		0			0
$637$	35.6522			1.41259
$638$		0			0
$639$		−	6.95381i		−	0.275089i
$640$		0			0
$641$			18.9106i			0.746922i	0.927646	+	0.373461i	$0.121829\pi$
$641$			18.9106i			0.746922i	−0.927646	+	0.373461i	$0.878171\pi$
$642$		0			0
$643$			3.31783i			0.130842i	0.997858	+	0.0654211i	$0.0208391\pi$
$643$			3.31783i			0.130842i	−0.997858	+	0.0654211i	$0.979161\pi$
$644$		0			0
$645$	−11.7986			−0.464570
$646$		0			0
$647$	8.23766			0.323856			0.161928	−	0.986803i	$-0.448229\pi$
$647$	8.23766			0.323856			0.161928	+	0.986803i	$0.448229\pi$
$648$		0			0
$649$			1.98267i			0.0778266i
$650$		0			0
$651$		−	26.8269i		−	1.05143i
$652$		0			0
$653$			41.8275i			1.63684i	0.574622	+	0.818419i	$0.305149\pi$
$653$			41.8275i			1.63684i	−0.574622	+	0.818419i	$0.694851\pi$
$654$		0			0
$655$	25.5864			0.999741
$656$		0			0
$657$			68.6787i			2.67941i
$658$		0			0
$659$	−26.8807			−1.04712			−0.523561	−	0.851988i	$-0.675397\pi$
$659$	−26.8807			−1.04712			−0.523561	+	0.851988i	$0.675397\pi$
$660$		0			0
$661$	−46.5871			−1.81203			−0.906014	−	0.423247i	$-0.860890\pi$
$661$	−46.5871			−1.81203			−0.906014	+	0.423247i	$0.860890\pi$
$662$		0			0
$663$	−37.3560	+	58.3497i	−1.45079	+	2.26611i
$664$		0			0
$665$	2.02755			0.0786250
$666$		0			0
$667$	12.3115			0.476703
$668$		0			0
$669$			41.5690i			1.60715i
$670$		0			0
$671$	−14.5906			−0.563265
$672$		0			0
$673$		−	50.8104i		−	1.95860i	−0.202417	−	0.979299i	$-0.564880\pi$
$673$		−	50.8104i		−	1.95860i	0.202417	−	0.979299i	$-0.435120\pi$
$674$		0			0
$675$			10.0640i			0.387363i
$676$		0			0
$677$			40.6993i			1.56420i	0.623151	+	0.782102i	$0.285852\pi$
$677$			40.6993i			1.56420i	−0.623151	+	0.782102i	$0.714148\pi$
$678$		0			0
$679$	10.5490			0.404835
$680$		0			0
$681$	9.13187			0.349934
$682$		0			0
$683$			20.6882i			0.791614i	0.918334	+	0.395807i	$0.129535\pi$
$683$			20.6882i			0.791614i	−0.918334	+	0.395807i	$0.870465\pi$
$684$		0			0
$685$			18.2587i			0.697630i
$686$		0			0
$687$			12.7374i			0.485962i
$688$		0			0
$689$	69.1310			2.63368
$690$		0			0
$691$			22.4370i			0.853545i	0.904359	+	0.426772i	$0.140349\pi$
$691$			22.4370i			0.853545i	−0.904359	+	0.426772i	$0.859651\pi$
$692$		0			0
$693$	−9.89669			−0.375944
$694$		0			0
$695$	−22.3963			−0.849542
$696$		0			0
$697$	22.3414	−	34.8970i	0.846240	−	1.32182i
$698$		0			0
$699$	−37.0881			−1.40280
$700$		0			0
$701$	−6.48705			−0.245013			−0.122506	−	0.992468i	$-0.539093\pi$
$701$	−6.48705			−0.245013			−0.122506	+	0.992468i	$0.539093\pi$
$702$		0			0
$703$			5.36250i			0.202250i
$704$		0			0
$705$	−24.9849			−0.940984
$706$		0			0
$707$		−	19.6779i		−	0.740062i
$708$		0			0
$709$			43.5525i			1.63565i	0.575469	+	0.817824i	$0.304820\pi$
$709$			43.5525i			1.63565i	−0.575469	+	0.817824i	$0.695180\pi$
$710$		0			0
$711$		−	5.19780i		−	0.194933i
$712$		0			0
$713$	−33.3264			−1.24808
$714$		0			0
$715$	−20.9862			−0.784840
$716$		0			0
$717$			22.8498i			0.853340i
$718$		0			0
$719$			38.1879i			1.42417i	0.702094	+	0.712084i	$0.252249\pi$
$719$			38.1879i			1.42417i	−0.702094	+	0.712084i	$0.747751\pi$
$720$		0			0
$721$		−	8.74015i		−	0.325500i
$722$		0			0
$723$	81.4444			3.02895
$724$		0			0
$725$		−	6.44357i		−	0.239308i
$726$		0			0
$727$	38.8269			1.44001			0.720006	−	0.693968i	$-0.244139\pi$
$727$	38.8269			1.44001			0.720006	+	0.693968i	$0.244139\pi$
$728$		0			0
$729$	43.3883			1.60697
$730$		0			0
$731$	8.20448	+	5.25259i	0.303454	+	0.194274i
$732$		0			0
$733$	−22.1228			−0.817125			−0.408562	−	0.912730i	$-0.633970\pi$
$733$	−22.1228			−0.817125			−0.408562	+	0.912730i	$0.633970\pi$
$734$		0			0
$735$	29.8308			1.10033
$736$		0			0
$737$			19.7288i			0.726719i
$738$		0			0
$739$	−42.5717			−1.56603			−0.783013	−	0.622006i	$-0.786318\pi$
$739$	−42.5717			−1.56603			−0.783013	+	0.622006i	$0.786318\pi$
$740$		0			0
$741$		−	18.9632i		−	0.696629i
$742$		0			0
$743$			6.67140i			0.244750i	0.992484	+	0.122375i	$0.0390510\pi$
$743$			6.67140i			0.244750i	−0.992484	+	0.122375i	$0.960949\pi$
$744$		0			0
$745$			9.16541i			0.335795i
$746$		0			0
$747$	33.0376			1.20878
$748$		0			0
$749$	−12.8556			−0.469732
$750$		0			0
$751$			11.7081i			0.427233i	0.976918	+	0.213617i	$0.0685243\pi$
$751$			11.7081i			0.427233i	−0.976918	+	0.213617i	$0.931476\pi$
$752$		0			0
$753$		−	1.71574i		−	0.0625250i
$754$		0			0
$755$			27.5537i			1.00278i
$756$		0			0
$757$	20.0015			0.726965			0.363483	−	0.931601i	$-0.381588\pi$
$757$	20.0015			0.726965			0.363483	+	0.931601i	$0.381588\pi$
$758$		0			0
$759$			19.7796i			0.717955i
$760$		0			0
$761$	−14.6525			−0.531153			−0.265576	−	0.964090i	$-0.585562\pi$
$761$	−14.6525			−0.531153			−0.265576	+	0.964090i	$0.585562\pi$
$762$		0			0
$763$	−10.5663			−0.382527
$764$		0			0
$765$	−19.4280	+	30.3463i	−0.702422	+	1.09717i
$766$		0			0
$767$	5.96810			0.215496
$768$		0			0
$769$	−40.0552			−1.44443			−0.722214	−	0.691670i	$-0.756875\pi$
$769$	−40.0552			−1.44443			−0.722214	+	0.691670i	$0.756875\pi$
$770$		0			0
$771$			12.6753i			0.456490i
$772$		0			0
$773$	−39.8881			−1.43467			−0.717337	−	0.696726i	$-0.754639\pi$
$773$	−39.8881			−1.43467			−0.717337	+	0.696726i	$0.754639\pi$
$774$		0			0
$775$			17.4423i			0.626547i
$776$		0			0
$777$		−	13.5532i		−	0.486219i
$778$		0			0
$779$			11.3412i			0.406342i
$780$		0			0
$781$	−2.79802			−0.100121
$782$		0			0
$783$	18.8566			0.673879
$784$		0			0
$785$		−	10.4731i		−	0.373802i
$786$		0			0
$787$		−	45.1506i		−	1.60945i	−0.593651	−	0.804723i	$-0.702314\pi$
$787$		−	45.1506i		−	1.60945i	0.593651	−	0.804723i	$-0.297686\pi$
$788$		0			0
$789$			20.2666i			0.721510i
$790$		0			0
$791$	16.8876			0.600453
$792$		0			0
$793$			43.9197i			1.55964i
$794$		0			0
$795$	57.8432			2.05149
$796$		0			0
$797$	−5.95140			−0.210810			−0.105405	−	0.994429i	$-0.533614\pi$
$797$	−5.95140			−0.210810			−0.105405	+	0.994429i	$0.533614\pi$
$798$		0			0
$799$	17.3739	+	11.1229i	0.614644	+	0.393501i
$800$		0			0
$801$	79.8035			2.81972
$802$		0			0
$803$	27.6344			0.975195
$804$		0			0
$805$			6.36597i			0.224371i
$806$		0			0
$807$	−57.3646			−2.01933
$808$		0			0
$809$		−	20.8156i		−	0.731837i	−0.930647	−	0.365918i	$-0.880755\pi$
$809$		−	20.8156i		−	0.731837i	0.930647	−	0.365918i	$-0.119245\pi$
$810$		0			0
$811$			8.74033i			0.306915i	0.988155	+	0.153457i	$0.0490408\pi$
$811$			8.74033i			0.306915i	−0.988155	+	0.153457i	$0.950959\pi$
$812$		0			0
$813$			10.0883i			0.353810i
$814$		0			0
$815$	18.6550			0.653455
$816$		0			0
$817$	−2.66639			−0.0932852
$818$		0			0
$819$			29.7903i			1.04096i
$820$		0			0
$821$		−	47.2515i		−	1.64909i	−0.565799	−	0.824544i	$-0.691432\pi$
$821$		−	47.2515i		−	1.64909i	0.565799	−	0.824544i	$-0.308568\pi$
$822$		0			0
$823$		−	13.6782i		−	0.476791i	−0.971168	−	0.238395i	$-0.923379\pi$
$823$		−	13.6782i		−	0.476791i	0.971168	−	0.238395i	$-0.0766214\pi$
$824$		0			0
$825$	10.3522			0.360419
$826$		0			0
$827$		−	7.48432i		−	0.260255i	−0.991497	−	0.130128i	$-0.958461\pi$
$827$		−	7.48432i		−	0.260255i	0.991497	−	0.130128i	$-0.0415387\pi$
$828$		0			0
$829$	−20.8066			−0.722642			−0.361321	−	0.932442i	$-0.617674\pi$
$829$	−20.8066			−0.722642			−0.361321	+	0.932442i	$0.617674\pi$
$830$		0			0
$831$	10.7120			0.371594
$832$		0			0
$833$	−20.7437	−	13.2803i	−0.718726	−	0.460136i
$834$		0			0
$835$	44.5140			1.54047
$836$		0			0
$837$	−51.0436			−1.76432
$838$		0			0
$839$		−	21.0983i		−	0.728394i	−0.931322	−	0.364197i	$-0.881343\pi$
$839$		−	21.0983i		−	0.728394i	0.931322	−	0.364197i	$-0.118657\pi$
$840$		0			0
$841$	16.9269			0.583686
$842$		0			0
$843$			77.3728i			2.66486i
$844$		0			0
$845$			40.1149i			1.38000i
$846$		0			0
$847$		−	7.16103i		−	0.246056i
$848$		0			0
$849$	−52.0123			−1.78506
$850$		0			0
$851$	−16.8368			−0.577159
$852$		0			0
$853$			11.1600i			0.382112i	0.981579	+	0.191056i	$0.0611911\pi$
$853$			11.1600i			0.382112i	−0.981579	+	0.191056i	$0.938809\pi$
$854$		0			0
$855$		−	9.86231i		−	0.337284i
$856$		0			0
$857$		−	14.3642i		−	0.490673i	−0.969438	−	0.245337i	$-0.921102\pi$
$857$		−	14.3642i		−	0.490673i	0.969438	−	0.245337i	$-0.0788985\pi$
$858$		0			0
$859$	−8.16076			−0.278442			−0.139221	−	0.990261i	$-0.544460\pi$
$859$	−8.16076			−0.278442			−0.139221	+	0.990261i	$0.544460\pi$
$860$		0			0
$861$		−	28.6639i		−	0.976864i
$862$		0			0
$863$	−2.83413			−0.0964750			−0.0482375	−	0.998836i	$-0.515360\pi$
$863$	−2.83413			−0.0964750			−0.0482375	+	0.998836i	$0.515360\pi$
$864$		0			0
$865$	12.8612			0.437294
$866$		0			0
$867$	43.4700	−	20.0348i	1.47632	−	0.680418i
$868$		0			0
$869$	−2.09145			−0.0709475
$870$		0			0
$871$	59.3862			2.01223
$872$		0			0
$873$		−	51.3121i		−	1.73665i
$874$		0			0
$875$	12.3151			0.416325
$876$		0			0
$877$			36.3538i			1.22758i	0.789470	+	0.613790i	$0.210356\pi$
$877$			36.3538i			1.22758i	−0.789470	+	0.613790i	$0.789644\pi$
$878$		0			0
$879$		−	70.4158i		−	2.37507i
$880$		0			0
$881$		−	53.9608i		−	1.81799i	−0.416811	−	0.908993i	$-0.636852\pi$
$881$		−	53.9608i		−	1.81799i	0.416811	−	0.908993i	$-0.363148\pi$
$882$		0			0
$883$	53.5487			1.80206			0.901029	−	0.433759i	$-0.142813\pi$
$883$	53.5487			1.80206			0.901029	+	0.433759i	$0.142813\pi$
$884$		0			0
$885$	4.99361			0.167858
$886$		0			0
$887$			30.1683i			1.01295i	0.862254	+	0.506475i	$0.169052\pi$
$887$			30.1683i			1.01295i	−0.862254	+	0.506475i	$0.830948\pi$
$888$		0			0
$889$		−	19.8880i		−	0.667022i
$890$		0			0
$891$			0.986375i			0.0330448i
$892$		0			0
$893$	−5.64637			−0.188949
$894$		0			0
$895$		−	36.5115i		−	1.22044i
$896$		0			0
$897$	59.5393			1.98796
$898$		0			0
$899$	32.6812			1.08998
$900$		0			0
$901$	−40.2228	−	25.7510i	−1.34002	−	0.857891i
$902$		0			0
$903$	6.73906			0.224262
$904$		0			0
$905$	−28.8085			−0.957626
$906$		0			0
$907$			44.7746i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$907$			44.7746i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$908$		0			0
$909$	−95.7161			−3.17470
$910$		0			0
$911$			36.0936i			1.19583i	0.801558	+	0.597917i	$0.204005\pi$
$911$			36.0936i			1.19583i	−0.801558	+	0.597917i	$0.795995\pi$
$912$		0			0
$913$		−	13.2934i		−	0.439947i
$914$		0			0
$915$			36.7484i			1.21486i
$916$		0			0
$917$	−14.6143			−0.482605
$918$		0			0
$919$	30.6556			1.01124			0.505618	−	0.862757i	$-0.331265\pi$
$919$	30.6556			1.01124			0.505618	+	0.862757i	$0.331265\pi$
$920$		0			0
$921$		−	79.5328i		−	2.62070i
$922$		0			0
$923$			8.42240i			0.277227i
$924$		0			0
$925$			8.81203i			0.289738i
$926$		0			0
$927$	−42.5134			−1.39632
$928$		0			0
$929$		−	31.0848i		−	1.01986i	−0.860216	−	0.509929i	$-0.829672\pi$
$929$		−	31.0848i		−	1.01986i	0.860216	−	0.509929i	$-0.170328\pi$
$930$		0			0
$931$	6.74153			0.220945
$932$		0			0
$933$	60.4194			1.97804
$934$		0			0
$935$	12.2105	+	7.81728i	0.399326	+	0.255653i
$936$		0			0
$937$	−28.1705			−0.920291			−0.460146	−	0.887843i	$-0.652203\pi$
$937$	−28.1705			−0.920291			−0.460146	+	0.887843i	$0.652203\pi$
$938$		0			0
$939$	1.04758			0.0341863
$940$		0			0
$941$			47.1930i			1.53845i	0.638980	+	0.769223i	$0.279357\pi$
$941$			47.1930i			1.53845i	−0.638980	+	0.769223i	$0.720643\pi$
$942$		0			0
$943$	−35.6085			−1.15957
$944$		0			0
$945$			9.75028i			0.317177i
$946$		0			0
$947$			39.9220i			1.29729i	0.761091	+	0.648645i	$0.224664\pi$
$947$			39.9220i			1.29729i	−0.761091	+	0.648645i	$0.775336\pi$
$948$		0			0
$949$		−	83.1831i		−	2.70024i
$950$		0			0
$951$	−38.2972			−1.24187
$952$		0			0
$953$	38.3805			1.24326			0.621632	−	0.783309i	$-0.286470\pi$
$953$	38.3805			1.24326			0.621632	+	0.783309i	$0.286470\pi$
$954$		0			0
$955$		−	1.27243i		−	0.0411748i
$956$		0			0
$957$		−	19.3967i		−	0.627005i
$958$		0			0
$959$		−	10.4289i		−	0.336767i
$960$		0			0
$961$	−57.4659			−1.85374
$962$		0			0
$963$			62.5315i			2.01505i
$964$		0			0
$965$	−28.1736			−0.906940
$966$		0			0
$967$	24.7990			0.797481			0.398740	−	0.917064i	$-0.369447\pi$
$967$	24.7990			0.797481			0.398740	+	0.917064i	$0.369447\pi$
$968$		0			0
$969$	−7.06370	+	11.0334i	−0.226919	+	0.354444i
$970$		0			0
$971$	−45.9030			−1.47310			−0.736548	−	0.676385i	$-0.763545\pi$
$971$	−45.9030			−1.47310			−0.736548	+	0.676385i	$0.763545\pi$
$972$		0			0
$973$	12.7922			0.410099
$974$		0			0
$975$		−	31.1616i		−	0.997970i
$976$		0			0
$977$	−35.8438			−1.14674			−0.573372	−	0.819295i	$-0.694365\pi$
$977$	−35.8438			−1.14674			−0.573372	+	0.819295i	$0.694365\pi$
$978$		0			0
$979$		−	32.1106i		−	1.02626i
$980$		0			0
$981$			51.3963i			1.64096i
$982$		0			0
$983$			17.9493i			0.572495i	0.958156	+	0.286248i	$0.0924080\pi$
$983$			17.9493i			0.572495i	−0.958156	+	0.286248i	$0.907592\pi$
$984$		0			0
$985$	18.7746			0.598208
$986$		0			0
$987$	14.2707			0.454241
$988$		0			0
$989$		−	8.37176i		−	0.266207i
$990$		0			0
$991$		−	31.3742i		−	0.996633i	−0.866995	−	0.498317i	$-0.833952\pi$
$991$		−	31.3742i		−	0.996633i	0.866995	−	0.498317i	$-0.166048\pi$
$992$		0			0
$993$			89.2531i			2.83236i
$994$		0			0
$995$	21.9209			0.694941
$996$		0			0
$997$			35.8920i			1.13671i	0.822783	+	0.568356i	$0.192420\pi$
$997$			35.8920i			1.13671i	−0.822783	+	0.568356i	$0.807580\pi$
$998$		0			0
$999$	−25.7877			−0.815886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.5	✓	46
17.16	even	2	inner	4012.2.b.b.237.42	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.5	✓	46	1.1	even	1	trivial
4012.2.b.b.237.42	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)

\(f(q)\)	\(=\)	\(q-2.81557i q^{3} +1.77357i q^{5} -1.01302i q^{7} -4.92746 q^{9} +O(q^{10})\)	\(q-2.81557i q^{3} +1.77357i q^{5} -1.01302i q^{7} -4.92746 q^{9} +1.98267i q^{11} +5.96810 q^{13} +4.99361 q^{15} +(-3.47244 - 2.22309i) q^{17} +1.12852 q^{19} -2.85222 q^{21} +3.54324i q^{23} +1.85446 q^{25} +5.42691i q^{27} -3.47464i q^{29} +9.40563i q^{31} +5.58236 q^{33} +1.79665 q^{35} +4.75181i q^{37} -16.8036i q^{39} +10.0497i q^{41} -2.36274 q^{43} -8.73919i q^{45} -5.00336 q^{47} +5.97380 q^{49} +(-6.25928 + 9.77692i) q^{51} +11.5834 q^{53} -3.51640 q^{55} -3.17742i q^{57} +1.00000 q^{59} +7.35908i q^{61} +4.99160i q^{63} +10.5848i q^{65} +9.95061 q^{67} +9.97626 q^{69} +1.41124i q^{71} -13.9379i q^{73} -5.22136i q^{75} +2.00848 q^{77} +1.05486i q^{79} +0.497499 q^{81} -6.70479 q^{83} +(3.94281 - 6.15861i) q^{85} -9.78311 q^{87} -16.1957 q^{89} -6.04578i q^{91} +26.4823 q^{93} +2.00150i q^{95} +10.4135i q^{97} -9.76953i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 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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