LMFDB - Embedded newform 608.6.b.b.303.55

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,6,Mod(303,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("608.303");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	608.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$97.5133624463$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.55
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.42

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.49700i q^{3} -93.3850i q^{5} -95.0367i q^{7} +222.777 q^{9} +O(q^{10})$	$q+4.49700i q^{3} -93.3850i q^{5} -95.0367i q^{7} +222.777 q^{9} -300.630 q^{11} -276.831 q^{13} +419.952 q^{15} -1297.16 q^{17} +(-654.993 + 1430.76i) q^{19} +427.380 q^{21} +2226.45i q^{23} -5595.77 q^{25} +2094.60i q^{27} -3508.64 q^{29} +3472.71 q^{31} -1351.93i q^{33} -8875.01 q^{35} +5424.58 q^{37} -1244.91i q^{39} +19973.8i q^{41} -15101.0 q^{43} -20804.0i q^{45} +4008.50i q^{47} +7775.02 q^{49} -5833.31i q^{51} -3492.83 q^{53} +28074.3i q^{55} +(-6434.14 - 2945.50i) q^{57} -5272.90i q^{59} -5218.99i q^{61} -21172.0i q^{63} +25851.9i q^{65} -44164.6i q^{67} -10012.3 q^{69} +59386.9 q^{71} +67977.3 q^{73} -25164.2i q^{75} +28570.9i q^{77} -2688.59 q^{79} +44715.4 q^{81} +53937.2 q^{83} +121135. i q^{85} -15778.3i q^{87} -46866.4i q^{89} +26309.2i q^{91} +15616.8i q^{93} +(133612. + 61166.6i) q^{95} +81391.3i q^{97} -66973.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6168 q^{9}+O(q^{10}) $ 96 * q - 6168 * q^9	$ 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) $ 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			4.49700i			0.288483i	0.989543	+	0.144241i	$0.0460741\pi$
$3$			4.49700i			0.288483i	−0.989543	+	0.144241i	$0.953926\pi$
$4$		0			0
$5$		−	93.3850i		−	1.67052i	−0.549853	−	0.835261i	$-0.685316\pi$
$5$		−	93.3850i		−	1.67052i	0.549853	−	0.835261i	$-0.314684\pi$
$6$		0			0
$7$		−	95.0367i		−	0.733072i	−0.930404	−	0.366536i	$-0.880544\pi$
$7$		−	95.0367i		−	0.733072i	0.930404	−	0.366536i	$-0.119456\pi$
$8$		0			0
$9$	222.777			0.916778
$10$		0			0
$11$	−300.630			−0.749118			−0.374559	−	0.927203i	$-0.622206\pi$
$11$	−300.630			−0.749118			−0.374559	+	0.927203i	$0.622206\pi$
$12$		0			0
$13$	−276.831			−0.454315			−0.227158	−	0.973858i	$-0.572943\pi$
$13$	−276.831			−0.454315			−0.227158	+	0.973858i	$0.572943\pi$
$14$		0			0
$15$	419.952			0.481917
$16$		0			0
$17$	−1297.16			−1.08860			−0.544302	−	0.838889i	$-0.683206\pi$
$17$	−1297.16			−1.08860			−0.544302	+	0.838889i	$0.683206\pi$
$18$		0			0
$19$	−654.993	+	1430.76i	−0.416248	+	0.909251i
$19$	−654.993	+	1430.76i	−0.416248	+	0.909251i
$20$		0			0
$21$	427.380			0.211478
$22$		0			0
$23$			2226.45i			0.877592i	0.898587	+	0.438796i	$0.144595\pi$
$23$			2226.45i			0.877592i	−0.898587	+	0.438796i	$0.855405\pi$
$24$		0			0
$25$	−5595.77			−1.79065
$26$		0			0
$27$			2094.60i			0.552957i
$28$		0			0
$29$	−3508.64			−0.774718			−0.387359	−	0.921929i	$-0.626613\pi$
$29$	−3508.64			−0.774718			−0.387359	+	0.921929i	$0.626613\pi$
$30$		0			0
$31$	3472.71			0.649028			0.324514	−	0.945881i	$-0.394799\pi$
$31$	3472.71			0.649028			0.324514	+	0.945881i	$0.394799\pi$
$32$		0			0
$33$		−	1351.93i		−	0.216107i
$34$		0			0
$35$	−8875.01			−1.22461
$36$		0			0
$37$	5424.58			0.651422			0.325711	−	0.945469i	$-0.394396\pi$
$37$	5424.58			0.651422			0.325711	+	0.945469i	$0.394396\pi$
$38$		0			0
$39$		−	1244.91i		−	0.131062i
$40$		0			0
$41$			19973.8i			1.85567i	0.372992	+	0.927834i	$0.378332\pi$
$41$			19973.8i			1.85567i	−0.372992	+	0.927834i	$0.621668\pi$
$42$		0			0
$43$	−15101.0			−1.24548			−0.622739	−	0.782430i	$-0.713980\pi$
$43$	−15101.0			−1.24548			−0.622739	+	0.782430i	$0.713980\pi$
$44$		0			0
$45$		−	20804.0i		−	1.53150i
$46$		0			0
$47$			4008.50i			0.264690i	0.991204	+	0.132345i	$0.0422506\pi$
$47$			4008.50i			0.264690i	−0.991204	+	0.132345i	$0.957749\pi$
$48$		0			0
$49$	7775.02			0.462606
$50$		0			0
$51$		−	5833.31i		−	0.314043i
$52$		0			0
$53$	−3492.83			−0.170800			−0.0854001	−	0.996347i	$-0.527217\pi$
$53$	−3492.83			−0.170800			−0.0854001	+	0.996347i	$0.527217\pi$
$54$		0			0
$55$			28074.3i			1.25142i
$56$		0			0
$57$	−6434.14	−	2945.50i	−0.262303	−	0.120080i
$58$		0			0
$59$		−	5272.90i		−	0.197206i	−0.995127	−	0.0986028i	$-0.968563\pi$
$59$		−	5272.90i		−	0.197206i	0.995127	−	0.0986028i	$-0.0314373\pi$
$60$		0			0
$61$		−	5218.99i		−	0.179582i	−0.995961	−	0.0897909i	$-0.971380\pi$
$61$		−	5218.99i		−	0.179582i	0.995961	−	0.0897909i	$-0.0286199\pi$
$62$		0			0
$63$		−	21172.0i		−	0.672064i
$64$		0			0
$65$			25851.9i			0.758943i
$66$		0			0
$67$		−	44164.6i		−	1.20195i	−0.799267	−	0.600976i	$-0.794779\pi$
$67$		−	44164.6i		−	1.20195i	0.799267	−	0.600976i	$-0.205221\pi$
$68$		0			0
$69$	−10012.3			−0.253170
$70$		0			0
$71$	59386.9			1.39812			0.699061	−	0.715062i	$-0.253602\pi$
$71$	59386.9			1.39812			0.699061	+	0.715062i	$0.253602\pi$
$72$		0			0
$73$	67977.3			1.49299			0.746495	−	0.665391i	$-0.231735\pi$
$73$	67977.3			1.49299			0.746495	+	0.665391i	$0.231735\pi$
$74$		0			0
$75$		−	25164.2i		−	0.516570i
$76$		0			0
$77$			28570.9i			0.549157i
$78$		0			0
$79$	−2688.59			−0.0484681			−0.0242341	−	0.999706i	$-0.507715\pi$
$79$	−2688.59			−0.0484681			−0.0242341	+	0.999706i	$0.507715\pi$
$80$		0			0
$81$	44715.4			0.757259
$82$		0			0
$83$	53937.2			0.859395			0.429698	−	0.902973i	$-0.358620\pi$
$83$	53937.2			0.859395			0.429698	+	0.902973i	$0.358620\pi$
$84$		0			0
$85$			121135.i			1.81854i
$86$		0			0
$87$		−	15778.3i		−	0.223493i
$88$		0			0
$89$		−	46866.4i		−	0.627172i	−0.949560	−	0.313586i	$-0.898470\pi$
$89$		−	46866.4i		−	0.627172i	0.949560	−	0.313586i	$-0.101530\pi$
$90$		0			0
$91$			26309.2i			0.333045i
$92$		0			0
$93$			15616.8i			0.187233i
$94$		0			0
$95$	133612.	+	61166.6i	1.51892	+	0.695352i
$96$		0			0
$97$			81391.3i			0.878311i	0.898411	+	0.439156i	$0.144722\pi$
$97$			81391.3i			0.878311i	−0.898411	+	0.439156i	$0.855278\pi$
$98$		0			0
$99$	−66973.4			−0.686774
$100$		0			0
$101$			2539.86i			0.0247746i	0.999923	+	0.0123873i	$0.00394310\pi$
$101$			2539.86i			0.0247746i	−0.999923	+	0.0123873i	$0.996057\pi$
$102$		0			0
$103$	−92949.2			−0.863282			−0.431641	−	0.902045i	$-0.642065\pi$
$103$	−92949.2			−0.863282			−0.431641	+	0.902045i	$0.642065\pi$
$104$		0			0
$105$		−	39910.9i		−	0.353279i
$106$		0			0
$107$		−	168149.i		−	1.41983i	−0.704288	−	0.709914i	$-0.748734\pi$
$107$		−	168149.i		−	1.41983i	0.704288	−	0.709914i	$-0.251266\pi$
$108$		0			0
$109$	190552.			1.53620			0.768098	−	0.640332i	$-0.221203\pi$
$109$	190552.			1.53620			0.768098	+	0.640332i	$0.221203\pi$
$110$		0			0
$111$			24394.3i			0.187924i
$112$		0			0
$113$			200780.i			1.47919i	0.673050	+	0.739597i	$0.264984\pi$
$113$			200780.i			1.47919i	−0.673050	+	0.739597i	$0.735016\pi$
$114$		0			0
$115$	207917.			1.46604
$116$		0			0
$117$	−61671.7			−0.416506
$118$		0			0
$119$			123278.i			0.798025i
$120$		0			0
$121$	−70672.8			−0.438823
$122$		0			0
$123$	−89822.0			−0.535328
$124$		0			0
$125$			230733.i			1.32079i
$126$		0			0
$127$	−360363.			−1.98258			−0.991292	−	0.131684i	$-0.957962\pi$
$127$	−360363.			−1.98258			−0.991292	+	0.131684i	$0.957962\pi$
$128$		0			0
$129$		−	67909.4i		−	0.359299i
$130$		0			0
$131$	−145926.			−0.742943			−0.371472	−	0.928444i	$-0.621147\pi$
$131$	−145926.			−0.742943			−0.371472	+	0.928444i	$0.621147\pi$
$132$		0			0
$133$	135975.	+	62248.4i	0.666546	+	0.305140i
$134$		0			0
$135$	195604.			0.923727
$136$		0			0
$137$	−65742.5			−0.299257			−0.149629	−	0.988742i	$-0.547808\pi$
$137$	−65742.5			−0.299257			−0.149629	+	0.988742i	$0.547808\pi$
$138$		0			0
$139$	−220332.			−0.967253			−0.483627	−	0.875274i	$-0.660681\pi$
$139$	−220332.			−0.967253			−0.483627	+	0.875274i	$0.660681\pi$
$140$		0			0
$141$	−18026.2			−0.0763583
$142$		0			0
$143$	83223.7			0.340335
$144$		0			0
$145$			327654.i			1.29418i
$146$		0			0
$147$			34964.3i			0.133454i
$148$		0			0
$149$			311465.i			1.14933i	0.818390	+	0.574663i	$0.194867\pi$
$149$			311465.i			1.14933i	−0.818390	+	0.574663i	$0.805133\pi$
$150$		0			0
$151$	171779.			0.613093			0.306547	−	0.951856i	$-0.400826\pi$
$151$	171779.			0.613093			0.306547	+	0.951856i	$0.400826\pi$
$152$		0			0
$153$	−288977.			−0.998008
$154$		0			0
$155$		−	324299.i		−	1.08422i
$156$		0			0
$157$		−	60313.0i		−	0.195282i	−0.995222	−	0.0976409i	$-0.968870\pi$
$157$		−	60313.0i		−	0.195282i	0.995222	−	0.0976409i	$-0.0311297\pi$
$158$		0			0
$159$		−	15707.3i		−	0.0492729i
$160$		0			0
$161$	211594.			0.643338
$162$		0			0
$163$	−134718.			−0.397153			−0.198576	−	0.980085i	$-0.563632\pi$
$163$	−134718.			−0.397153			−0.198576	+	0.980085i	$0.563632\pi$
$164$		0			0
$165$	−126250.			−0.361012
$166$		0			0
$167$	459295.			1.27438			0.637192	−	0.770705i	$-0.280096\pi$
$167$	459295.			1.27438			0.637192	+	0.770705i	$0.280096\pi$
$168$		0			0
$169$	−294657.			−0.793598
$170$		0			0
$171$	−145917.	+	318741.i	−0.381607	+	0.833581i
$172$		0			0
$173$	−254781.			−0.647219			−0.323609	−	0.946191i	$-0.604896\pi$
$173$	−254781.			−0.647219			−0.323609	+	0.946191i	$0.604896\pi$
$174$		0			0
$175$			531803.i			1.31267i
$176$		0			0
$177$	23712.2			0.0568904
$178$		0			0
$179$			616670.i			1.43853i	0.694734	+	0.719267i	$0.255522\pi$
$179$			616670.i			1.43853i	−0.694734	+	0.719267i	$0.744478\pi$
$180$		0			0
$181$	−493543.			−1.11977			−0.559885	−	0.828571i	$-0.689155\pi$
$181$	−493543.			−1.11977			−0.559885	+	0.828571i	$0.689155\pi$
$182$		0			0
$183$	23469.8			0.0518062
$184$		0			0
$185$		−	506575.i		−	1.08821i
$186$		0			0
$187$	389964.			0.815493
$188$		0			0
$189$	199064.			0.405357
$190$		0			0
$191$			839278.i			1.66465i	0.554290	+	0.832324i	$0.312990\pi$
$191$			839278.i			1.66465i	−0.554290	+	0.832324i	$0.687010\pi$
$192$		0			0
$193$		−	85037.8i		−	0.164331i	−0.996619	−	0.0821653i	$-0.973816\pi$
$193$		−	85037.8i		−	0.164331i	0.996619	−	0.0821653i	$-0.0261835\pi$
$194$		0			0
$195$	−116256.			−0.218942
$196$		0			0
$197$			932798.i			1.71247i	0.516589	+	0.856233i	$0.327201\pi$
$197$			932798.i			1.71247i	−0.516589	+	0.856233i	$0.672799\pi$
$198$		0			0
$199$			158415.i			0.283573i	0.989897	+	0.141786i	$0.0452846\pi$
$199$			158415.i			0.283573i	−0.989897	+	0.141786i	$0.954715\pi$
$200$		0			0
$201$	198608.			0.346742
$202$		0			0
$203$			333450.i			0.567924i
$204$		0			0
$205$	1.86525e6			3.09994
$206$		0			0
$207$			496001.i			0.804557i
$208$		0			0
$209$	196910.	−	430130.i	0.311819	−	0.681136i
$210$		0			0
$211$			140861.i			0.217813i	0.994052	+	0.108907i	$0.0347350\pi$
$211$			140861.i			0.217813i	−0.994052	+	0.108907i	$0.965265\pi$
$212$		0			0
$213$			267063.i			0.403334i
$214$		0			0
$215$			1.41021e6i			2.08060i
$216$		0			0
$217$		−	330035.i		−	0.475784i
$218$		0			0
$219$			305694.i			0.430702i
$220$		0			0
$221$	359094.			0.494569
$222$		0			0
$223$	−2964.04			−0.00399137			−0.00199568	−	0.999998i	$-0.500635\pi$
$223$	−2964.04			−0.00399137			−0.00199568	+	0.999998i	$0.500635\pi$
$224$		0			0
$225$	−1.24661e6			−1.64162
$226$		0			0
$227$		−	447740.i		−	0.576715i	−0.957523	−	0.288358i	$-0.906891\pi$
$227$		−	447740.i		−	0.576715i	0.957523	−	0.288358i	$-0.0931092\pi$
$228$		0			0
$229$		−	99233.7i		−	0.125046i	−0.998044	−	0.0625231i	$-0.980085\pi$
$229$		−	99233.7i		−	0.125046i	0.998044	−	0.0625231i	$-0.0199147\pi$
$230$		0			0
$231$	−128483.			−0.158422
$232$		0			0
$233$	1.17832e6			1.42192			0.710960	−	0.703232i	$-0.248261\pi$
$233$	1.17832e6			1.42192			0.710960	+	0.703232i	$0.248261\pi$
$234$		0			0
$235$	374334.			0.442170
$236$		0			0
$237$		−	12090.6i		−	0.0139822i
$238$		0			0
$239$		−	1.16687e6i		−	1.32138i	−0.750658	−	0.660690i	$-0.770264\pi$
$239$		−	1.16687e6i		−	1.32138i	0.750658	−	0.660690i	$-0.229736\pi$
$240$		0			0
$241$		−	1.02176e6i		−	1.13320i	−0.823994	−	0.566599i	$-0.808259\pi$
$241$		−	1.02176e6i		−	1.13320i	0.823994	−	0.566599i	$-0.191741\pi$
$242$		0			0
$243$			710072.i			0.771413i
$244$		0			0
$245$		−	726071.i		−	0.772794i
$246$		0			0
$247$	181323.	−	396080.i	0.189108	−	0.413086i
$248$		0			0
$249$			242555.i			0.247921i
$250$		0			0
$251$	786942.			0.788421			0.394211	−	0.919020i	$-0.371018\pi$
$251$	786942.			0.788421			0.394211	+	0.919020i	$0.371018\pi$
$252$		0			0
$253$		−	669336.i		−	0.657420i
$254$		0			0
$255$	−544744.			−0.524617
$256$		0			0
$257$		−	680976.i		−	0.643130i	−0.946887	−	0.321565i	$-0.895791\pi$
$257$		−	680976.i		−	0.643130i	0.946887	−	0.321565i	$-0.104209\pi$
$258$		0			0
$259$		−	515535.i		−	0.477539i
$260$		0			0
$261$	−781644.			−0.710245
$262$		0			0
$263$		−	702665.i		−	0.626411i	−0.949685	−	0.313205i	$-0.898597\pi$
$263$		−	702665.i		−	0.626411i	0.949685	−	0.313205i	$-0.101403\pi$
$264$		0			0
$265$			326179.i			0.285326i
$266$		0			0
$267$	210758.			0.180928
$268$		0			0
$269$	954364.			0.804144			0.402072	−	0.915608i	$-0.368290\pi$
$269$	954364.			0.804144			0.402072	+	0.915608i	$0.368290\pi$
$270$		0			0
$271$			1.30739e6i			1.08139i	0.841219	+	0.540695i	$0.181839\pi$
$271$			1.30739e6i			1.08139i	−0.841219	+	0.540695i	$0.818161\pi$
$272$		0			0
$273$	−118312.			−0.0960778
$274$		0			0
$275$	1.68225e6			1.34140
$276$		0			0
$277$			2.32222e6i			1.81846i	0.416290	+	0.909232i	$0.363330\pi$
$277$			2.32222e6i			1.81846i	−0.416290	+	0.909232i	$0.636670\pi$
$278$		0			0
$279$	773639.			0.595015
$280$		0			0
$281$			1.63016e6i			1.23158i	0.787909	+	0.615792i	$0.211164\pi$
$281$			1.63016e6i			1.23158i	−0.787909	+	0.615792i	$0.788836\pi$
$282$		0			0
$283$	−821287.			−0.609577			−0.304789	−	0.952420i	$-0.598586\pi$
$283$	−821287.			−0.609577			−0.304789	+	0.952420i	$0.598586\pi$
$284$		0			0
$285$	−275066.	+	600852.i	−0.200597	+	0.438183i
$286$		0			0
$287$	1.89824e6			1.36034
$288$		0			0
$289$	262758.			0.185060
$290$		0			0
$291$	−366016.			−0.253378
$292$		0			0
$293$	−2.23127e6			−1.51839			−0.759194	−	0.650864i	$-0.774407\pi$
$293$	−2.23127e6			−1.51839			−0.759194	+	0.650864i	$0.774407\pi$
$294$		0			0
$295$	−492410.			−0.329436
$296$		0			0
$297$		−	629698.i		−	0.414230i
$298$		0			0
$299$		−	616351.i		−	0.398703i
$300$		0			0
$301$			1.43515e6i			0.913024i
$302$		0			0
$303$	−11421.8			−0.00714704
$304$		0			0
$305$	−487376.			−0.299995
$306$		0			0
$307$			1.82839e6i			1.10719i	0.832786	+	0.553595i	$0.186745\pi$
$307$			1.82839e6i			1.10719i	−0.832786	+	0.553595i	$0.813255\pi$
$308$		0			0
$309$		−	417992.i		−	0.249042i
$310$		0			0
$311$		−	1.90568e6i		−	1.11724i	−0.829422	−	0.558622i	$-0.811330\pi$
$311$		−	1.90568e6i		−	1.11724i	0.829422	−	0.558622i	$-0.188670\pi$
$312$		0			0
$313$	−2.31801e6			−1.33738			−0.668688	−	0.743543i	$-0.733144\pi$
$313$	−2.31801e6			−1.33738			−0.668688	+	0.743543i	$0.733144\pi$
$314$		0			0
$315$	−1.97715e6			−1.12270
$316$		0			0
$317$	2.82404e6			1.57842			0.789211	−	0.614122i	$-0.210490\pi$
$317$	2.82404e6			1.57842			0.789211	+	0.614122i	$0.210490\pi$
$318$		0			0
$319$	1.05480e6			0.580355
$320$		0			0
$321$	756167.			0.409596
$322$		0			0
$323$	849628.	−	1.85592e6i	0.453130	−	0.989815i
$324$		0			0
$325$	1.54908e6			0.813517
$326$		0			0
$327$			856911.i			0.443166i
$328$		0			0
$329$	380954.			0.194036
$330$		0			0
$331$			2.30744e6i			1.15760i	0.815468	+	0.578802i	$0.196479\pi$
$331$			2.30744e6i			1.15760i	−0.815468	+	0.578802i	$0.803521\pi$
$332$		0			0
$333$	1.20847e6			0.597209
$334$		0			0
$335$	−4.12431e6			−2.00789
$336$		0			0
$337$			1.22825e6i			0.589130i	0.955631	+	0.294565i	$0.0951748\pi$
$337$			1.22825e6i			0.589130i	−0.955631	+	0.294565i	$0.904825\pi$
$338$		0			0
$339$	−902909.			−0.426722
$340$		0			0
$341$	−1.04400e6			−0.486199
$342$		0			0
$343$		−	2.33619e6i		−	1.07219i
$344$		0			0
$345$			935002.i			0.422926i
$346$		0			0
$347$	−884418.			−0.394307			−0.197153	−	0.980373i	$-0.563170\pi$
$347$	−884418.			−0.394307			−0.197153	+	0.980373i	$0.563170\pi$
$348$		0			0
$349$			1.51416e6i			0.665438i	0.943026	+	0.332719i	$0.107966\pi$
$349$			1.51416e6i			0.665438i	−0.943026	+	0.332719i	$0.892034\pi$
$350$		0			0
$351$		−	579851.i		−	0.251217i
$352$		0			0
$353$	−924516.			−0.394892			−0.197446	−	0.980314i	$-0.563265\pi$
$353$	−924516.			−0.394892			−0.197446	+	0.980314i	$0.563265\pi$
$354$		0			0
$355$		−	5.54585e6i		−	2.33559i
$356$		0			0
$357$	−554379.			−0.230216
$358$		0			0
$359$			3.57999e6i			1.46604i	0.680208	+	0.733019i	$0.261889\pi$
$359$			3.57999e6i			1.46604i	−0.680208	+	0.733019i	$0.738111\pi$
$360$		0			0
$361$	−1.61807e6	−	1.87428e6i	−0.653474	−	0.756949i
$362$		0			0
$363$		−	317816.i		−	0.126593i
$364$		0			0
$365$		−	6.34807e6i		−	2.49407i
$366$		0			0
$367$		−	1.35151e6i		−	0.523787i	−0.965097	−	0.261893i	$-0.915653\pi$
$367$		−	1.35151e6i		−	0.523787i	0.965097	−	0.261893i	$-0.0843469\pi$
$368$		0			0
$369$			4.44970e6i			1.70124i
$370$		0			0
$371$			331948.i			0.125209i
$372$		0			0
$373$	−4.87847e6			−1.81556			−0.907781	−	0.419443i	$-0.862225\pi$
$373$	−4.87847e6			−1.81556			−0.907781	+	0.419443i	$0.862225\pi$
$374$		0			0
$375$	−1.03760e6			−0.381025
$376$		0			0
$377$	971302.			0.351966
$378$		0			0
$379$			2.12271e6i			0.759088i	0.925174	+	0.379544i	$0.123919\pi$
$379$			2.12271e6i			0.759088i	−0.925174	+	0.379544i	$0.876081\pi$
$380$		0			0
$381$		−	1.62055e6i		−	0.571941i
$382$		0			0
$383$	−506796.			−0.176537			−0.0882687	−	0.996097i	$-0.528133\pi$
$383$	−506796.			−0.176537			−0.0882687	+	0.996097i	$0.528133\pi$
$384$		0			0
$385$	2.66809e6			0.917379
$386$		0			0
$387$	−3.36417e6			−1.14183
$388$		0			0
$389$			1.63640e6i			0.548296i	0.961688	+	0.274148i	$0.0883958\pi$
$389$			1.63640e6i			0.548296i	−0.961688	+	0.274148i	$0.911604\pi$
$390$		0			0
$391$		−	2.88805e6i		−	0.955351i
$392$		0			0
$393$		−	656231.i		−	0.214326i
$394$		0			0
$395$			251074.i			0.0809671i
$396$		0			0
$397$		−	2.66069e6i		−	0.847261i	−0.905835	−	0.423631i	$-0.860755\pi$
$397$		−	2.66069e6i		−	0.847261i	0.905835	−	0.423631i	$-0.139245\pi$
$398$		0			0
$399$	−279931.	+	611480.i	−0.0880275	+	0.192287i
$400$		0			0
$401$		−	4.45829e6i		−	1.38454i	−0.721636	−	0.692272i	$-0.756610\pi$
$401$		−	4.45829e6i		−	1.38454i	0.721636	−	0.692272i	$-0.243390\pi$
$402$		0			0
$403$	−961354.			−0.294863
$404$		0			0
$405$		−	4.17575e6i		−	1.26502i
$406$		0			0
$407$	−1.63079e6			−0.487991
$408$		0			0
$409$			4.62121e6i			1.36599i	0.730424	+	0.682994i	$0.239323\pi$
$409$			4.62121e6i			1.36599i	−0.730424	+	0.682994i	$0.760677\pi$
$410$		0			0
$411$		−	295644.i		−	0.0863306i
$412$		0			0
$413$	−501119.			−0.144566
$414$		0			0
$415$		−	5.03693e6i		−	1.43564i
$416$		0			0
$417$		−	990832.i		−	0.279036i
$418$		0			0
$419$	4.42550e6			1.23148			0.615740	−	0.787950i	$-0.288857\pi$
$419$	4.42550e6			1.23148			0.615740	+	0.787950i	$0.288857\pi$
$420$		0			0
$421$	−2.80313e6			−0.770793			−0.385396	−	0.922751i	$-0.625935\pi$
$421$	−2.80313e6			−0.770793			−0.385396	+	0.922751i	$0.625935\pi$
$422$		0			0
$423$			893001.i			0.242662i
$424$		0			0
$425$	7.25858e6			1.94930
$426$		0			0
$427$	−495996.			−0.131646
$428$		0			0
$429$			374257.i			0.0981808i
$430$		0			0
$431$	3.60636e6			0.935137			0.467569	−	0.883957i	$-0.345130\pi$
$431$	3.60636e6			0.935137			0.467569	+	0.883957i	$0.345130\pi$
$432$		0			0
$433$			5.37862e6i			1.37864i	0.724457	+	0.689320i	$0.242091\pi$
$433$			5.37862e6i			1.37864i	−0.724457	+	0.689320i	$0.757909\pi$
$434$		0			0
$435$	−1.47346e6			−0.373350
$436$		0			0
$437$	−3.18552e6	−	1.45831e6i	−0.797952	−	0.365297i
$438$		0			0
$439$	1.29172e6			0.319894			0.159947	−	0.987126i	$-0.448868\pi$
$439$	1.29172e6			0.319894			0.159947	+	0.987126i	$0.448868\pi$
$440$		0			0
$441$	1.73210e6			0.424107
$442$		0			0
$443$	−2.33522e6			−0.565352			−0.282676	−	0.959215i	$-0.591222\pi$
$443$	−2.33522e6			−0.565352			−0.282676	+	0.959215i	$0.591222\pi$
$444$		0			0
$445$	−4.37662e6			−1.04770
$446$		0			0
$447$	−1.40066e6			−0.331560
$448$		0			0
$449$		−	2.56046e6i		−	0.599379i	−0.954037	−	0.299690i	$-0.903117\pi$
$449$		−	2.56046e6i		−	0.599379i	0.954037	−	0.299690i	$-0.0968831\pi$
$450$		0			0
$451$		−	6.00471e6i		−	1.39011i
$452$		0			0
$453$			772488.i			0.176867i
$454$		0			0
$455$	2.45688e6			0.556360
$456$		0			0
$457$	−2.84015e6			−0.636138			−0.318069	−	0.948068i	$-0.603034\pi$
$457$	−2.84015e6			−0.636138			−0.318069	+	0.948068i	$0.603034\pi$
$458$		0			0
$459$		−	2.71702e6i		−	0.601951i
$460$		0			0
$461$		−	3.23395e6i		−	0.708730i	−0.935107	−	0.354365i	$-0.884697\pi$
$461$		−	3.23395e6i		−	0.708730i	0.935107	−	0.354365i	$-0.115303\pi$
$462$		0			0
$463$		−	8.68840e6i		−	1.88360i	−0.336180	−	0.941798i	$-0.609135\pi$
$463$		−	8.68840e6i		−	1.88360i	0.336180	−	0.941798i	$-0.390865\pi$
$464$		0			0
$465$	1.45837e6			0.312778
$466$		0			0
$467$	7.68880e6			1.63142			0.815711	−	0.578459i	$-0.196346\pi$
$467$	7.68880e6			1.63142			0.815711	+	0.578459i	$0.196346\pi$
$468$		0			0
$469$	−4.19726e6			−0.881117
$470$		0			0
$471$	271227.			0.0563354
$472$		0			0
$473$	4.53982e6			0.933009
$474$		0			0
$475$	3.66519e6	−	8.00622e6i	0.745353	−	1.62815i
$476$		0			0
$477$	−778123.			−0.156586
$478$		0			0
$479$			6.32466e6i			1.25950i	0.776797	+	0.629751i	$0.216843\pi$
$479$			6.32466e6i			1.25950i	−0.776797	+	0.629751i	$0.783157\pi$
$480$		0			0
$481$	−1.50170e6			−0.295951
$482$		0			0
$483$			951539.i			0.185592i
$484$		0			0
$485$	7.60073e6			1.46724
$486$		0			0
$487$	−1.68772e6			−0.322462			−0.161231	−	0.986917i	$-0.551546\pi$
$487$	−1.68772e6			−0.322462			−0.161231	+	0.986917i	$0.551546\pi$
$488$		0			0
$489$		−	605828.i		−	0.114572i
$490$		0			0
$491$	4.39190e6			0.822146			0.411073	−	0.911602i	$-0.365154\pi$
$491$	4.39190e6			0.822146			0.411073	+	0.911602i	$0.365154\pi$
$492$		0			0
$493$	4.55126e6			0.843362
$494$		0			0
$495$			6.25431e6i			1.14727i
$496$		0			0
$497$		−	5.64394e6i		−	1.02492i
$498$		0			0
$499$	−5.21260e6			−0.937137			−0.468568	−	0.883427i	$-0.655230\pi$
$499$	−5.21260e6			−0.937137			−0.468568	+	0.883427i	$0.655230\pi$
$500$		0			0
$501$			2.06545e6i			0.367638i
$502$		0			0
$503$		−	1.33359e6i		−	0.235018i	−0.993072	−	0.117509i	$-0.962509\pi$
$503$		−	1.33359e6i		−	0.235018i	0.993072	−	0.117509i	$-0.0374909\pi$
$504$		0			0
$505$	237185.			0.0413865
$506$		0			0
$507$		−	1.32507e6i		−	0.228939i
$508$		0			0
$509$	−960938.			−0.164400			−0.0821998	−	0.996616i	$-0.526195\pi$
$509$	−960938.			−0.164400			−0.0821998	+	0.996616i	$0.526195\pi$
$510$		0			0
$511$		−	6.46034e6i		−	1.09447i
$512$		0			0
$513$	−2.99687e6	−	1.37195e6i	−0.502777	−	0.230167i
$514$		0			0
$515$			8.68007e6i			1.44213i
$516$		0			0
$517$		−	1.20507e6i		−	0.198284i
$518$		0			0
$519$		−	1.14575e6i		−	0.186711i
$520$		0			0
$521$			1.08312e7i			1.74816i	0.485782	+	0.874080i	$0.338535\pi$
$521$			1.08312e7i			1.74816i	−0.485782	+	0.874080i	$0.661465\pi$
$522$		0			0
$523$			3.86911e6i			0.618524i	0.950977	+	0.309262i	$0.100082\pi$
$523$			3.86911e6i			0.618524i	−0.950977	+	0.309262i	$0.899918\pi$
$524$		0			0
$525$	−2.39152e6			−0.378683
$526$		0			0
$527$	−4.50464e6			−0.706535
$528$		0			0
$529$	1.47928e6			0.229832
$530$		0			0
$531$		−	1.17468e6i		−	0.180794i
$532$		0			0
$533$		−	5.52937e6i		−	0.843058i
$534$		0			0
$535$	−1.57026e7			−2.37186
$536$		0			0
$537$	−2.77316e6			−0.414992
$538$		0			0
$539$	−2.33740e6			−0.346546
$540$		0			0
$541$		−	9.20084e6i		−	1.35156i	−0.737105	−	0.675779i	$-0.763807\pi$
$541$		−	9.20084e6i		−	1.35156i	0.737105	−	0.675779i	$-0.236193\pi$
$542$		0			0
$543$		−	2.21946e6i		−	0.323034i
$544$		0			0
$545$		−	1.77947e7i		−	2.56625i
$546$		0			0
$547$			1.36673e7i			1.95306i	0.215380	+	0.976530i	$0.430901\pi$
$547$			1.36673e7i			1.95306i	−0.215380	+	0.976530i	$0.569099\pi$
$548$		0			0
$549$		−	1.16267e6i		−	0.164637i
$550$		0			0
$551$	2.29813e6	−	5.02003e6i	0.322475	−	0.704413i
$552$		0			0
$553$			255515.i			0.0355306i
$554$		0			0
$555$	2.27807e6			0.313931
$556$		0			0
$557$			4.93456e6i			0.673923i	0.941518	+	0.336962i	$0.109399\pi$
$557$			4.93456e6i			0.673923i	−0.941518	+	0.336962i	$0.890601\pi$
$558$		0			0
$559$	4.18045e6			0.565839
$560$		0			0
$561$			1.75367e6i			0.235255i
$562$		0			0
$563$			4.87905e6i			0.648730i	0.945932	+	0.324365i	$0.105151\pi$
$563$			4.87905e6i			0.648730i	−0.945932	+	0.324365i	$0.894849\pi$
$564$		0			0
$565$	1.87499e7			2.47103
$566$		0			0
$567$		−	4.24961e6i		−	0.555125i
$568$		0			0
$569$		−	3.03918e6i		−	0.393528i	−0.980451	−	0.196764i	$-0.936957\pi$
$569$		−	3.03918e6i		−	0.393528i	0.980451	−	0.196764i	$-0.0630433\pi$
$570$		0			0
$571$	−715082.			−0.0917837			−0.0458919	−	0.998946i	$-0.514613\pi$
$571$	−715082.			−0.0917837			−0.0458919	+	0.998946i	$0.514613\pi$
$572$		0			0
$573$	−3.77423e6			−0.480222
$574$		0			0
$575$		−	1.24587e7i		−	1.57146i
$576$		0			0
$577$	−7.57765e6			−0.947534			−0.473767	−	0.880650i	$-0.657106\pi$
$577$	−7.57765e6			−0.947534			−0.473767	+	0.880650i	$0.657106\pi$
$578$		0			0
$579$	382415.			0.0474065
$580$		0			0
$581$		−	5.12601e6i		−	0.629998i
$582$		0			0
$583$	1.05005e6			0.127949
$584$		0			0
$585$			5.75921e6i			0.695783i
$586$		0			0
$587$	−7.04413e6			−0.843785			−0.421893	−	0.906646i	$-0.638634\pi$
$587$	−7.04413e6			−0.843785			−0.421893	+	0.906646i	$0.638634\pi$
$588$		0			0
$589$	−2.27460e6	+	4.96862e6i	−0.270157	+	0.590130i
$590$		0			0
$591$	−4.19479e6			−0.494017
$592$		0			0
$593$	−1.54904e7			−1.80895			−0.904473	−	0.426531i	$-0.859736\pi$
$593$	−1.54904e7			−1.80895			−0.904473	+	0.426531i	$0.859736\pi$
$594$		0			0
$595$	1.15123e7			1.33312
$596$		0			0
$597$	−712393.			−0.0818058
$598$		0			0
$599$	−8.16729e6			−0.930059			−0.465030	−	0.885295i	$-0.653956\pi$
$599$	−8.16729e6			−0.930059			−0.465030	+	0.885295i	$0.653956\pi$
$600$		0			0
$601$			5.00996e6i			0.565781i	0.959152	+	0.282891i	$0.0912933\pi$
$601$			5.00996e6i			0.565781i	−0.959152	+	0.282891i	$0.908707\pi$
$602$		0			0
$603$		−	9.83886e6i		−	1.10192i
$604$		0			0
$605$			6.59979e6i			0.733063i
$606$		0			0
$607$	1.30779e7			1.44068			0.720338	−	0.693623i	$-0.243987\pi$
$607$	1.30779e7			1.44068			0.720338	+	0.693623i	$0.243987\pi$
$608$		0			0
$609$	−1.49952e6			−0.163836
$610$		0			0
$611$		−	1.10968e6i		−	0.120252i
$612$		0			0
$613$			3.44034e6i			0.369785i	0.982759	+	0.184893i	$0.0591937\pi$
$613$			3.44034e6i			0.369785i	−0.982759	+	0.184893i	$0.940806\pi$
$614$		0			0
$615$			8.38803e6i			0.894278i
$616$		0			0
$617$	−1.31680e7			−1.39254			−0.696269	−	0.717781i	$-0.745158\pi$
$617$	−1.31680e7			−1.39254			−0.696269	+	0.717781i	$0.745158\pi$
$618$		0			0
$619$	−1.10190e7			−1.15589			−0.577945	−	0.816076i	$-0.696145\pi$
$619$	−1.10190e7			−1.15589			−0.577945	+	0.816076i	$0.696145\pi$
$620$		0			0
$621$	−4.66351e6			−0.485271
$622$		0			0
$623$	−4.45403e6			−0.459762
$624$		0			0
$625$	4.06021e6			0.415765
$626$		0			0
$627$	1.93429e6	+	885505.i	0.196496	+	0.0899544i
$628$		0			0
$629$	−7.03653e6			−0.709140
$630$		0			0
$631$		−	6.53113e6i		−	0.653003i	−0.945197	−	0.326501i	$-0.894130\pi$
$631$		−	6.53113e6i		−	0.653003i	0.945197	−	0.326501i	$-0.105870\pi$
$632$		0			0
$633$	−633452.			−0.0628354
$634$		0			0
$635$			3.36526e7i			3.31195i
$636$		0			0
$637$	−2.15237e6			−0.210169
$638$		0			0
$639$	1.32300e7			1.28177
$640$		0			0
$641$		−	1.26861e7i		−	1.21950i	−0.792594	−	0.609750i	$-0.791270\pi$
$641$		−	1.26861e7i		−	1.21950i	0.792594	−	0.609750i	$-0.208730\pi$
$642$		0			0
$643$	−8.74310e6			−0.833946			−0.416973	−	0.908919i	$-0.636909\pi$
$643$	−8.74310e6			−0.833946			−0.416973	+	0.908919i	$0.636909\pi$
$644$		0			0
$645$	−6.34172e6			−0.600216
$646$		0			0
$647$		−	2.39445e6i		−	0.224877i	−0.993659	−	0.112438i	$-0.964134\pi$
$647$		−	2.39445e6i		−	0.224877i	0.993659	−	0.112438i	$-0.0358661\pi$
$648$		0			0
$649$			1.58519e6i			0.147730i
$650$		0			0
$651$	1.48416e6			0.137255
$652$		0			0
$653$		−	4.14995e6i		−	0.380855i	−0.981701	−	0.190428i	$-0.939013\pi$
$653$		−	4.14995e6i		−	0.380855i	0.981701	−	0.190428i	$-0.0609874\pi$
$654$		0			0
$655$			1.36273e7i			1.24110i
$656$		0			0
$657$	1.51438e7			1.36874
$658$		0			0
$659$		−	1.08924e7i		−	0.977036i	−0.872554	−	0.488518i	$-0.837538\pi$
$659$		−	1.08924e7i		−	0.977036i	0.872554	−	0.488518i	$-0.162462\pi$
$660$		0			0
$661$	3.44356e6			0.306552			0.153276	−	0.988183i	$-0.451018\pi$
$661$	3.44356e6			0.306552			0.153276	+	0.988183i	$0.451018\pi$
$662$		0			0
$663$			1.61484e6i			0.142675i
$664$		0			0
$665$	5.81307e6	−	1.26980e7i	0.509743	−	1.11348i
$666$		0			0
$667$		−	7.81180e6i		−	0.679887i
$668$		0			0
$669$		−	13329.3i		−	0.00115144i
$670$		0			0
$671$			1.56898e6i			0.134528i
$672$		0			0
$673$			1.24306e7i			1.05792i	0.848645	+	0.528962i	$0.177419\pi$
$673$			1.24306e7i			1.05792i	−0.848645	+	0.528962i	$0.822581\pi$
$674$		0			0
$675$		−	1.17209e7i		−	0.990150i
$676$		0			0
$677$	1.41359e7			1.18536			0.592681	−	0.805437i	$-0.298069\pi$
$677$	1.41359e7			1.18536			0.592681	+	0.805437i	$0.298069\pi$
$678$		0			0
$679$	7.73516e6			0.643865
$680$		0			0
$681$	2.01349e6			0.166372
$682$		0			0
$683$			1.02525e7i			0.840967i	0.907300	+	0.420484i	$0.138140\pi$
$683$			1.02525e7i			0.840967i	−0.907300	+	0.420484i	$0.861860\pi$
$684$		0			0
$685$			6.13937e6i			0.499916i
$686$		0			0
$687$	446254.			0.0360736
$688$		0			0
$689$	966927.			0.0775971
$690$		0			0
$691$	2.17104e6			0.172971			0.0864854	−	0.996253i	$-0.472436\pi$
$691$	2.17104e6			0.172971			0.0864854	+	0.996253i	$0.472436\pi$
$692$		0			0
$693$			6.36493e6i			0.503455i
$694$		0			0
$695$			2.05757e7i			1.61582i
$696$		0			0
$697$		−	2.59091e7i		−	2.02009i
$698$		0			0
$699$			5.29892e6i			0.410199i
$700$		0			0
$701$			1.47483e7i			1.13357i	0.823866	+	0.566785i	$0.191813\pi$
$701$			1.47483e7i			1.13357i	−0.823866	+	0.566785i	$0.808187\pi$
$702$		0			0
$703$	−3.55306e6	+	7.76130e6i	−0.271153	+	0.592306i
$704$		0			0
$705$			1.68338e6i			0.127558i
$706$		0			0
$707$	241380.			0.0181616
$708$		0			0
$709$			579292.i			0.0432795i	0.999766	+	0.0216397i	$0.00688868\pi$
$709$			579292.i			0.0432795i	−0.999766	+	0.0216397i	$0.993111\pi$
$710$		0			0
$711$	−598955.			−0.0444345
$712$		0			0
$713$			7.73180e6i			0.569582i
$714$		0			0
$715$		−	7.77185e6i		−	0.568538i
$716$		0			0
$717$	5.24742e6			0.381195
$718$		0			0
$719$			2.45890e7i			1.77386i	0.461906	+	0.886929i	$0.347166\pi$
$719$			2.45890e7i			1.77386i	−0.461906	+	0.886929i	$0.652834\pi$
$720$		0			0
$721$			8.83359e6i			0.632848i
$722$		0			0
$723$	4.59485e6			0.326908
$724$		0			0
$725$	1.96335e7			1.38725
$726$		0			0
$727$			1.45708e7i			1.02246i	0.859443	+	0.511232i	$0.170811\pi$
$727$			1.45708e7i			1.02246i	−0.859443	+	0.511232i	$0.829189\pi$
$728$		0			0
$729$	7.67265e6			0.534720
$730$		0			0
$731$	1.95884e7			1.35583
$732$		0			0
$733$		−	1.87450e6i		−	0.128862i	−0.997922	−	0.0644311i	$-0.979477\pi$
$733$		−	1.87450e6i		−	0.128862i	0.997922	−	0.0644311i	$-0.0205233\pi$
$734$		0			0
$735$	3.26514e6			0.222938
$736$		0			0
$737$			1.32772e7i			0.900404i
$738$		0			0
$739$	1.58113e7			1.06502			0.532508	−	0.846425i	$-0.321250\pi$
$739$	1.58113e7			1.06502			0.532508	+	0.846425i	$0.321250\pi$
$740$		0			0
$741$	1.78117e6	+	815408.i	0.119168	+	0.0545543i
$742$		0			0
$743$	1.94622e7			1.29336			0.646680	−	0.762761i	$-0.276157\pi$
$743$	1.94622e7			1.29336			0.646680	+	0.762761i	$0.276157\pi$
$744$		0			0
$745$	2.90861e7			1.91997
$746$		0			0
$747$	1.20160e7			0.787875
$748$		0			0
$749$	−1.59804e7			−1.04084
$750$		0			0
$751$	−2.18798e7			−1.41561			−0.707804	−	0.706409i	$-0.750314\pi$
$751$	−2.18798e7			−1.41561			−0.707804	+	0.706409i	$0.750314\pi$
$752$		0			0
$753$			3.53887e6i			0.227446i
$754$		0			0
$755$		−	1.60415e7i		−	1.02419i
$756$		0			0
$757$		−	2.59831e7i		−	1.64797i	−0.566608	−	0.823987i	$-0.691745\pi$
$757$		−	2.59831e7i		−	1.64797i	0.566608	−	0.823987i	$-0.308255\pi$
$758$		0			0
$759$	3.01000e6			0.189654
$760$		0			0
$761$	−1.42185e7			−0.890006			−0.445003	−	0.895529i	$-0.646797\pi$
$761$	−1.42185e7			−0.890006			−0.445003	+	0.895529i	$0.646797\pi$
$762$		0			0
$763$		−	1.81094e7i		−	1.12614i
$764$		0			0
$765$			2.69861e7i			1.66720i
$766$		0			0
$767$			1.45970e6i			0.0895935i
$768$		0			0
$769$	1.06124e7			0.647142			0.323571	−	0.946204i	$-0.395117\pi$
$769$	1.06124e7			0.647142			0.323571	+	0.946204i	$0.395117\pi$
$770$		0			0
$771$	3.06235e6			0.185532
$772$		0			0
$773$	2.24785e7			1.35306			0.676531	−	0.736414i	$-0.263483\pi$
$773$	2.24785e7			1.35306			0.676531	+	0.736414i	$0.263483\pi$
$774$		0			0
$775$	−1.94325e7			−1.16218
$776$		0			0
$777$	2.31836e6			0.137762
$778$		0			0
$779$	−2.85777e7	−	1.30827e7i	−1.68727	−	0.772419i
$780$		0			0
$781$	−1.78535e7			−1.04736
$782$		0			0
$783$		−	7.34919e6i		−	0.428386i
$784$		0			0
$785$	−5.63233e6			−0.326223
$786$		0			0
$787$			3.06719e7i			1.76524i	0.470088	+	0.882619i	$0.344222\pi$
$787$			3.06719e7i			1.76524i	−0.470088	+	0.882619i	$0.655778\pi$
$788$		0			0
$789$	3.15988e6			0.180709
$790$		0			0
$791$	1.90815e7			1.08435
$792$		0			0
$793$			1.44478e6i			0.0815867i
$794$		0			0
$795$	−1.46682e6			−0.0823114
$796$		0			0
$797$	−2.18608e7			−1.21905			−0.609524	−	0.792767i	$-0.708640\pi$
$797$	−2.18608e7			−1.21905			−0.609524	+	0.792767i	$0.708640\pi$
$798$		0			0
$799$		−	5.19965e6i		−	0.288142i
$800$		0			0
$801$		−	1.04408e7i		−	0.574977i
$802$		0			0
$803$	−2.04360e7			−1.11843
$804$		0			0
$805$		−	1.97597e7i		−	1.07471i
$806$		0			0
$807$			4.29178e6i			0.231981i
$808$		0			0
$809$	−2.05190e7			−1.10226			−0.551131	−	0.834419i	$-0.685803\pi$
$809$	−2.05190e7			−1.10226			−0.551131	+	0.834419i	$0.685803\pi$
$810$		0			0
$811$		−	8.82513e6i		−	0.471160i	−0.971855	−	0.235580i	$-0.924301\pi$
$811$		−	8.82513e6i		−	0.471160i	0.971855	−	0.235580i	$-0.0756990\pi$
$812$		0			0
$813$	−5.87933e6			−0.311962
$814$		0			0
$815$			1.25807e7i			0.663453i
$816$		0			0
$817$	9.89108e6	−	2.16060e7i	0.518428	−	1.13245i
$818$		0			0
$819$			5.86108e6i			0.305329i
$820$		0			0
$821$			2.37349e7i			1.22894i	0.788942	+	0.614468i	$0.210629\pi$
$821$			2.37349e7i			1.22894i	−0.788942	+	0.614468i	$0.789371\pi$
$822$		0			0
$823$		−	2.05277e7i		−	1.05643i	−0.849110	−	0.528215i	$-0.822861\pi$
$823$		−	2.05277e7i		−	1.05643i	0.849110	−	0.528215i	$-0.177139\pi$
$824$		0			0
$825$			7.56509e6i			0.386972i
$826$		0			0
$827$			3.05316e7i			1.55234i	0.630526	+	0.776168i	$0.282839\pi$
$827$			3.05316e7i			1.55234i	−0.630526	+	0.776168i	$0.717161\pi$
$828$		0			0
$829$	−2.25540e6			−0.113982			−0.0569911	−	0.998375i	$-0.518151\pi$
$829$	−2.25540e6			−0.113982			−0.0569911	+	0.998375i	$0.518151\pi$
$830$		0			0
$831$	−1.04430e7			−0.524595
$832$		0			0
$833$	−1.00854e7			−0.503595
$834$		0			0
$835$		−	4.28913e7i		−	2.12889i
$836$		0			0
$837$			7.27392e6i			0.358885i
$838$		0			0
$839$	1.74935e7			0.857969			0.428984	−	0.903312i	$-0.358872\pi$
$839$	1.74935e7			0.857969			0.428984	+	0.903312i	$0.358872\pi$
$840$		0			0
$841$	−8.20059e6			−0.399812
$842$		0			0
$843$	−7.33082e6			−0.355290
$844$		0			0
$845$			2.75166e7i			1.32572i
$846$		0			0
$847$			6.71652e6i			0.321688i
$848$		0			0
$849$		−	3.69333e6i		−	0.175852i
$850$		0			0
$851$			1.20776e7i			0.571683i
$852$		0			0
$853$			3.59976e7i			1.69395i	0.531631	+	0.846976i	$0.321579\pi$
$853$			3.59976e7i			1.69395i	−0.531631	+	0.846976i	$0.678421\pi$
$854$		0			0
$855$	2.97657e7	+	1.36265e7i	1.39252	+	0.637484i
$856$		0			0
$857$			1.66793e7i			0.775758i	0.921710	+	0.387879i	$0.126792\pi$
$857$			1.66793e7i			0.775758i	−0.921710	+	0.387879i	$0.873208\pi$
$858$		0			0
$859$	−3.05812e7			−1.41407			−0.707036	−	0.707178i	$-0.749968\pi$
$859$	−3.05812e7			−1.41407			−0.707036	+	0.707178i	$0.749968\pi$
$860$		0			0
$861$			8.53639e6i			0.392434i
$862$		0			0
$863$	2.23026e7			1.01936			0.509682	−	0.860363i	$-0.329763\pi$
$863$	2.23026e7			1.01936			0.509682	+	0.860363i	$0.329763\pi$
$864$		0			0
$865$			2.37927e7i			1.08119i
$866$		0			0
$867$			1.18162e6i			0.0533864i
$868$		0			0
$869$	808269.			0.0363083
$870$		0			0
$871$			1.22262e7i			0.546065i
$872$		0			0
$873$			1.81321e7i			0.805216i
$874$		0			0
$875$	2.19281e7			0.968234
$876$		0			0
$877$	4.16206e7			1.82730			0.913648	−	0.406505i	$-0.133253\pi$
$877$	4.16206e7			1.82730			0.913648	+	0.406505i	$0.133253\pi$
$878$		0			0
$879$		−	1.00340e7i		−	0.438029i
$880$		0			0
$881$	−7.80431e6			−0.338762			−0.169381	−	0.985551i	$-0.554177\pi$
$881$	−7.80431e6			−0.338762			−0.169381	+	0.985551i	$0.554177\pi$
$882$		0			0
$883$	−2.87613e7			−1.24139			−0.620693	−	0.784054i	$-0.713149\pi$
$883$	−2.87613e7			−1.24139			−0.620693	+	0.784054i	$0.713149\pi$
$884$		0			0
$885$		−	2.21437e6i		−	0.0950367i
$886$		0			0
$887$	−1.02058e7			−0.435550			−0.217775	−	0.975999i	$-0.569880\pi$
$887$	−1.02058e7			−0.435550			−0.217775	+	0.975999i	$0.569880\pi$
$888$		0			0
$889$			3.42478e7i			1.45338i
$890$		0			0
$891$	−1.34428e7			−0.567276
$892$		0			0
$893$	−5.73521e6	−	2.62554e6i	−0.240669	−	0.110177i
$894$		0			0
$895$	5.75877e7			2.40310
$896$		0			0
$897$	2.77173e6			0.115019
$898$		0			0
$899$	−1.21845e7			−0.502814
$900$		0			0
$901$	4.53075e6			0.185934
$902$		0			0
$903$	−6.45389e6			−0.263392
$904$		0			0
$905$			4.60895e7i			1.87060i
$906$		0			0
$907$		−	2.99715e7i		−	1.20974i	−0.796325	−	0.604868i	$-0.793226\pi$
$907$		−	2.99715e7i		−	1.20974i	0.796325	−	0.604868i	$-0.206774\pi$
$908$		0			0
$909$			565823.i			0.0227128i
$910$		0			0
$911$	−3.93768e7			−1.57197			−0.785985	−	0.618246i	$-0.787844\pi$
$911$	−3.93768e7			−1.57197			−0.785985	+	0.618246i	$0.787844\pi$
$912$		0			0
$913$	−1.62151e7			−0.643788
$914$		0			0
$915$		−	2.19173e6i		−	0.0865434i
$916$		0			0
$917$			1.38684e7i			0.544631i
$918$		0			0
$919$			1.74439e7i			0.681326i	0.940185	+	0.340663i	$0.110652\pi$
$919$			1.74439e7i			0.681326i	−0.940185	+	0.340663i	$0.889348\pi$
$920$		0			0
$921$	−8.22226e6			−0.319405
$922$		0			0
$923$	−1.64402e7			−0.635188
$924$		0			0
$925$	−3.03547e7			−1.16646
$926$		0			0
$927$	−2.07069e7			−0.791438
$928$		0			0
$929$	−2.13025e7			−0.809824			−0.404912	−	0.914356i	$-0.632698\pi$
$929$	−2.13025e7			−0.809824			−0.404912	+	0.914356i	$0.632698\pi$
$930$		0			0
$931$	−5.09258e6	+	1.11242e7i	−0.192559	+	0.420625i
$932$		0			0
$933$	8.56982e6			0.322305
$934$		0			0
$935$		−	3.64168e7i		−	1.36230i
$936$		0			0
$937$	−333669.			−0.0124156			−0.00620778	−	0.999981i	$-0.501976\pi$
$937$	−333669.			−0.0124156			−0.00620778	+	0.999981i	$0.501976\pi$
$938$		0			0
$939$		−	1.04241e7i		−	0.385810i
$940$		0			0
$941$	4.98749e7			1.83615			0.918076	−	0.396405i	$-0.129743\pi$
$941$	4.98749e7			1.83615			0.918076	+	0.396405i	$0.129743\pi$
$942$		0			0
$943$	−4.44706e7			−1.62852
$944$		0			0
$945$		−	1.85896e7i		−	0.677158i
$946$		0			0
$947$	−3.12985e7			−1.13409			−0.567047	−	0.823685i	$-0.691914\pi$
$947$	−3.12985e7			−1.13409			−0.567047	+	0.823685i	$0.691914\pi$
$948$		0			0
$949$	−1.88183e7			−0.678288
$950$		0			0
$951$			1.26997e7i			0.455347i
$952$		0			0
$953$			6.44799e6i			0.229981i	0.993367	+	0.114991i	$0.0366838\pi$
$953$			6.44799e6i			0.229981i	−0.993367	+	0.114991i	$0.963316\pi$
$954$		0			0
$955$	7.83760e7			2.78083
$956$		0			0
$957$			4.74344e6i			0.167422i
$958$		0			0
$959$			6.24795e6i			0.219377i
$960$		0			0
$961$	−1.65695e7			−0.578762
$962$		0			0
$963$		−	3.74598e7i		−	1.30167i
$964$		0			0
$965$	−7.94125e6			−0.274518
$966$		0			0
$967$		−	3.65275e7i		−	1.25619i	−0.778139	−	0.628093i	$-0.783836\pi$
$967$		−	3.65275e7i		−	1.25619i	0.778139	−	0.628093i	$-0.216164\pi$
$968$		0			0
$969$	8.34609e6	+	3.82078e6i	0.285544	+	0.130720i
$970$		0			0
$971$			1.46193e7i			0.497598i	0.968555	+	0.248799i	$0.0800358\pi$
$971$			1.46193e7i			0.497598i	−0.968555	+	0.248799i	$0.919964\pi$
$972$		0			0
$973$			2.09396e7i			0.709066i
$974$		0			0
$975$			6.96623e6i			0.234685i
$976$		0			0
$977$		−	1.70811e7i		−	0.572505i	−0.958154	−	0.286252i	$-0.907590\pi$
$977$		−	1.70811e7i		−	0.572505i	0.958154	−	0.286252i	$-0.0924096\pi$
$978$		0			0
$979$			1.40894e7i			0.469825i
$980$		0			0
$981$	4.24506e7			1.40835
$982$		0			0
$983$	−4.08627e7			−1.34879			−0.674394	−	0.738372i	$-0.735595\pi$
$983$	−4.08627e7			−1.34879			−0.674394	+	0.738372i	$0.735595\pi$
$984$		0			0
$985$	8.71094e7			2.86071
$986$		0			0
$987$			1.71315e6i			0.0559761i
$988$		0			0
$989$		−	3.36217e7i		−	1.09302i
$990$		0			0
$991$	4.47322e7			1.44689			0.723446	−	0.690381i	$-0.242557\pi$
$991$	4.47322e7			1.44689			0.723446	+	0.690381i	$0.242557\pi$
$992$		0			0
$993$	−1.03765e7			−0.333948
$994$		0			0
$995$	1.47936e7			0.473715
$996$		0			0
$997$		−	3.59648e7i		−	1.14588i	−0.819597	−	0.572941i	$-0.805802\pi$
$997$		−	3.59648e7i		−	1.14588i	0.819597	−	0.572941i	$-0.194198\pi$
$998$		0			0
$999$			1.13623e7i			0.360208i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.55		96
4.3	odd	2		152.6.b.b.75.93	yes	96
8.3	odd	2	inner	608.6.b.b.303.56		96
8.5	even	2		152.6.b.b.75.3	✓	96
19.18	odd	2	inner	608.6.b.b.303.41		96
76.75	even	2		152.6.b.b.75.4	yes	96
152.37	odd	2		152.6.b.b.75.94	yes	96
152.75	even	2	inner	608.6.b.b.303.42		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.3	✓	96	8.5	even	2
152.6.b.b.75.4	yes	96	76.75	even	2
152.6.b.b.75.93	yes	96	4.3	odd	2
152.6.b.b.75.94	yes	96	152.37	odd	2
608.6.b.b.303.41		96	19.18	odd	2	inner
608.6.b.b.303.42		96	152.75	even	2	inner
608.6.b.b.303.55		96	1.1	even	1	trivial
608.6.b.b.303.56		96	8.3	odd	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:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.49700i q^{3} -93.3850i q^{5} -95.0367i q^{7} +222.777 q^{9} +O(q^{10})\)	\(q+4.49700i q^{3} -93.3850i q^{5} -95.0367i q^{7} +222.777 q^{9} -300.630 q^{11} -276.831 q^{13} +419.952 q^{15} -1297.16 q^{17} +(-654.993 + 1430.76i) q^{19} +427.380 q^{21} +2226.45i q^{23} -5595.77 q^{25} +2094.60i q^{27} -3508.64 q^{29} +3472.71 q^{31} -1351.93i q^{33} -8875.01 q^{35} +5424.58 q^{37} -1244.91i q^{39} +19973.8i q^{41} -15101.0 q^{43} -20804.0i q^{45} +4008.50i q^{47} +7775.02 q^{49} -5833.31i q^{51} -3492.83 q^{53} +28074.3i q^{55} +(-6434.14 - 2945.50i) q^{57} -5272.90i q^{59} -5218.99i q^{61} -21172.0i q^{63} +25851.9i q^{65} -44164.6i q^{67} -10012.3 q^{69} +59386.9 q^{71} +67977.3 q^{73} -25164.2i q^{75} +28570.9i q^{77} -2688.59 q^{79} +44715.4 q^{81} +53937.2 q^{83} +121135. i q^{85} -15778.3i q^{87} -46866.4i q^{89} +26309.2i q^{91} +15616.8i q^{93} +(133612. + 61166.6i) q^{95} +81391.3i q^{97} -66973.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 96 * q - 6168 * q^9	\( 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) \) 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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