LMFDB - Embedded newform 684.5.h.c.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.h (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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 269x^{2} + 17592 $ x^4 + 269*x^2 + 17592
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$12.5228i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.c.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.8215 q^{5} +50.6430 q^{7} +O(q^{10})$	$q-27.8215 q^{5} +50.6430 q^{7} +26.1785 q^{11} +285.790i q^{13} +447.572 q^{17} +(-7.24940 - 360.927i) q^{19} -830.538 q^{23} +149.037 q^{25} -338.117i q^{29} -669.528i q^{31} -1408.97 q^{35} -623.907i q^{37} +1721.45i q^{41} +1361.32 q^{43} +3.46934 q^{47} +163.716 q^{49} -1178.03i q^{53} -728.325 q^{55} +4437.13i q^{59} +5916.84 q^{61} -7951.12i q^{65} +3673.66i q^{67} +7274.88i q^{71} +616.369 q^{73} +1325.76 q^{77} +674.916i q^{79} -5475.95 q^{83} -12452.1 q^{85} +3954.06i q^{89} +14473.3i q^{91} +(201.689 + 10041.5i) q^{95} +12827.0i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{5} + 24 q^{7}+O(q^{10}) $ 4 * q - 22 * q^5 + 24 * q^7	$ 4 q - 22 q^{5} + 24 q^{7} + 194 q^{11} + 1076 q^{17} - 654 q^{19} - 1090 q^{23} - 386 q^{25} - 4118 q^{35} + 4106 q^{43} - 5254 q^{47} - 1488 q^{49} + 926 q^{55} + 6078 q^{61} - 9856 q^{73} - 2822 q^{77} - 3868 q^{83} - 21862 q^{85} - 10354 q^{95}+O(q^{100}) $ 4 * q - 22 * q^5 + 24 * q^7 + 194 * q^11 + 1076 * q^17 - 654 * q^19 - 1090 * q^23 - 386 * q^25 - 4118 * q^35 + 4106 * q^43 - 5254 * q^47 - 1488 * q^49 + 926 * q^55 + 6078 * q^61 - 9856 * q^73 - 2822 * q^77 - 3868 * q^83 - 21862 * q^85 - 10354 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−27.8215			−1.11286			−0.556430	−	0.830894i	$-0.687829\pi$
$5$	−27.8215			−1.11286			−0.556430	+	0.830894i	$0.687829\pi$
$6$		0			0
$7$	50.6430			1.03353			0.516766	−	0.856127i	$-0.327136\pi$
$7$	50.6430			1.03353			0.516766	+	0.856127i	$0.327136\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	26.1785			0.216351			0.108176	−	0.994132i	$-0.465499\pi$
$11$	26.1785			0.216351			0.108176	+	0.994132i	$0.465499\pi$
$12$		0			0
$13$			285.790i			1.69107i	0.533923	+	0.845533i	$0.320717\pi$
$13$			285.790i			1.69107i	−0.533923	+	0.845533i	$0.679283\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	447.572			1.54869			0.774346	−	0.632762i	$-0.218079\pi$
$17$	447.572			1.54869			0.774346	+	0.632762i	$0.218079\pi$
$18$		0			0
$19$	−7.24940	−	360.927i	−0.0200814	−	0.999798i
$19$	−7.24940	−	360.927i	−0.0200814	−	0.999798i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−830.538			−1.57001			−0.785007	−	0.619486i	$-0.787341\pi$
$23$	−830.538			−1.57001			−0.785007	+	0.619486i	$0.787341\pi$
$24$		0			0
$25$	149.037			0.238459
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	338.117i		−	0.402041i	−0.979587	−	0.201021i	$-0.935574\pi$
$29$		−	338.117i		−	0.402041i	0.979587	−	0.201021i	$-0.0644258\pi$
$30$		0			0
$31$		−	669.528i		−	0.696699i	−0.937365	−	0.348350i	$-0.886742\pi$
$31$		−	669.528i		−	0.696699i	0.937365	−	0.348350i	$-0.113258\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1408.97			−1.15018
$36$		0			0
$37$		−	623.907i		−	0.455739i	−0.973692	−	0.227870i	$-0.926824\pi$
$37$		−	623.907i		−	0.455739i	0.973692	−	0.227870i	$-0.0731760\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1721.45i			1.02406i	0.858967	+	0.512030i	$0.171107\pi$
$41$			1721.45i			1.02406i	−0.858967	+	0.512030i	$0.828893\pi$
$42$		0			0
$43$	1361.32			0.736248			0.368124	−	0.929777i	$-0.380000\pi$
$43$	1361.32			0.736248			0.368124	+	0.929777i	$0.380000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.46934			0.00157055			0.000785275	−	1.00000i	$-0.499750\pi$
$47$	3.46934			0.00157055			0.000785275		1.00000i	$0.499750\pi$
$48$		0			0
$49$	163.716			0.0681867
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1178.03i		−	0.419379i	−0.977768	−	0.209689i	$-0.932755\pi$
$53$		−	1178.03i		−	0.419379i	0.977768	−	0.209689i	$-0.0672453\pi$
$54$		0			0
$55$	−728.325			−0.240769
$56$		0			0
$57$		0			0
$58$		0			0
$59$			4437.13i			1.27467i	0.770587	+	0.637335i	$0.219963\pi$
$59$			4437.13i			1.27467i	−0.770587	+	0.637335i	$0.780037\pi$
$60$		0			0
$61$	5916.84			1.59012			0.795060	−	0.606530i	$-0.207439\pi$
$61$	5916.84			1.59012			0.795060	+	0.606530i	$0.207439\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7951.12i		−	1.88192i
$66$		0			0
$67$			3673.66i			0.818370i	0.912451	+	0.409185i	$0.134187\pi$
$67$			3673.66i			0.818370i	−0.912451	+	0.409185i	$0.865813\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7274.88i			1.44314i	0.692340	+	0.721571i	$0.256580\pi$
$71$			7274.88i			1.44314i	−0.692340	+	0.721571i	$0.743420\pi$
$72$		0			0
$73$	616.369			0.115663			0.0578316	−	0.998326i	$-0.481581\pi$
$73$	616.369			0.115663			0.0578316	+	0.998326i	$0.481581\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1325.76			0.223606
$78$		0			0
$79$			674.916i			0.108142i	0.998537	+	0.0540712i	$0.0172198\pi$
$79$			674.916i			0.108142i	−0.998537	+	0.0540712i	$0.982780\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5475.95			−0.794883			−0.397441	−	0.917628i	$-0.630102\pi$
$83$	−5475.95			−0.794883			−0.397441	+	0.917628i	$0.630102\pi$
$84$		0			0
$85$	−12452.1			−1.72348
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3954.06i			0.499187i	0.968351	+	0.249594i	$0.0802971\pi$
$89$			3954.06i			0.499187i	−0.968351	+	0.249594i	$0.919703\pi$
$90$		0			0
$91$			14473.3i			1.74777i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	201.689	+	10041.5i	0.0223478	+	1.11264i
$96$		0			0
$97$			12827.0i			1.36327i	0.731693	+	0.681635i	$0.238731\pi$
$97$			12827.0i			1.36327i	−0.731693	+	0.681635i	$0.761269\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7784.44			−0.763105			−0.381553	−	0.924347i	$-0.624610\pi$
$101$	−7784.44			−0.763105			−0.381553	+	0.924347i	$0.624610\pi$
$102$		0			0
$103$			14831.5i			1.39801i	0.715116	+	0.699006i	$0.246374\pi$
$103$			14831.5i			1.39801i	−0.715116	+	0.699006i	$0.753626\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13332.9i			1.16454i	0.812994	+	0.582272i	$0.197836\pi$
$107$			13332.9i			1.16454i	−0.812994	+	0.582272i	$0.802164\pi$
$108$		0			0
$109$		−	14552.4i		−	1.22485i	−0.790529	−	0.612424i	$-0.790194\pi$
$109$		−	14552.4i		−	1.22485i	0.790529	−	0.612424i	$-0.209806\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			21313.5i			1.66916i	0.550889	+	0.834579i	$0.314289\pi$
$113$			21313.5i			1.66916i	−0.550889	+	0.834579i	$0.685711\pi$
$114$		0			0
$115$	23106.8			1.74721
$116$		0			0
$117$		0			0
$118$		0			0
$119$	22666.4			1.60062
$120$		0			0
$121$	−13955.7			−0.953192
$122$		0			0
$123$		0			0
$124$		0			0
$125$	13242.0			0.847489
$126$		0			0
$127$		−	25075.7i		−	1.55469i	−0.629072	−	0.777347i	$-0.716565\pi$
$127$		−	25075.7i		−	1.55469i	0.629072	−	0.777347i	$-0.283435\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4832.33			−0.281588			−0.140794	−	0.990039i	$-0.544965\pi$
$131$	−4832.33			−0.281588			−0.140794	+	0.990039i	$0.544965\pi$
$132$		0			0
$133$	−367.132	−	18278.4i	−0.0207548	−	1.03332i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11246.2			0.599188			0.299594	−	0.954067i	$-0.403149\pi$
$137$	11246.2			0.599188			0.299594	+	0.954067i	$0.403149\pi$
$138$		0			0
$139$	−36121.0			−1.86952			−0.934759	−	0.355282i	$-0.884385\pi$
$139$	−36121.0			−1.86952			−0.934759	+	0.355282i	$0.884385\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7481.55i			0.365864i
$144$		0			0
$145$			9406.92i			0.447416i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14354.0			0.646549			0.323275	−	0.946305i	$-0.395216\pi$
$149$	14354.0			0.646549			0.323275	+	0.946305i	$0.395216\pi$
$150$		0			0
$151$			25670.1i			1.12583i	0.826514	+	0.562916i	$0.190321\pi$
$151$			25670.1i			1.12583i	−0.826514	+	0.562916i	$0.809679\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			18627.3i			0.775329i
$156$		0			0
$157$	35480.1			1.43941			0.719707	−	0.694278i	$-0.244276\pi$
$157$	35480.1			1.43941			0.719707	+	0.694278i	$0.244276\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−42061.0			−1.62266
$162$		0			0
$163$	−5740.35			−0.216054			−0.108027	−	0.994148i	$-0.534453\pi$
$163$	−5740.35			−0.216054			−0.108027	+	0.994148i	$0.534453\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9999.93i		−	0.358562i	−0.983798	−	0.179281i	$-0.942623\pi$
$167$		−	9999.93i		−	0.358562i	0.983798	−	0.179281i	$-0.0573771\pi$
$168$		0			0
$169$	−53115.0			−1.85970
$170$		0			0
$171$		0			0
$172$		0			0
$173$			36092.8i			1.20595i	0.797761	+	0.602973i	$0.206017\pi$
$173$			36092.8i			1.20595i	−0.797761	+	0.602973i	$0.793983\pi$
$174$		0			0
$175$	7547.67			0.246454
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	35538.6i		−	1.10916i	−0.832130	−	0.554580i	$-0.812879\pi$
$179$		−	35538.6i		−	1.10916i	0.832130	−	0.554580i	$-0.187121\pi$
$180$		0			0
$181$			9397.52i			0.286851i	0.989661	+	0.143425i	$0.0458117\pi$
$181$			9397.52i			0.286851i	−0.989661	+	0.143425i	$0.954188\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			17358.0i			0.507174i
$186$		0			0
$187$	11716.8			0.335061
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−39080.5			−1.07126			−0.535628	−	0.844454i	$-0.679925\pi$
$191$	−39080.5			−1.07126			−0.535628	+	0.844454i	$0.679925\pi$
$192$		0			0
$193$			71446.1i			1.91807i	0.283291	+	0.959034i	$0.408574\pi$
$193$			71446.1i			1.91807i	−0.283291	+	0.959034i	$0.591426\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	70705.6			1.82189			0.910943	−	0.412532i	$-0.135355\pi$
$197$	70705.6			1.82189			0.910943	+	0.412532i	$0.135355\pi$
$198$		0			0
$199$	40473.6			1.02203			0.511017	−	0.859570i	$-0.329269\pi$
$199$	40473.6			1.02203			0.511017	+	0.859570i	$0.329269\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	17123.3i		−	0.415522i
$204$		0			0
$205$		−	47893.2i		−	1.13964i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−189.778	−	9448.53i	−0.00434464	−	0.216307i
$210$		0			0
$211$			15128.1i			0.339798i	0.985461	+	0.169899i	$0.0543441\pi$
$211$			15128.1i			0.339798i	−0.985461	+	0.169899i	$0.945656\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−37874.1			−0.819341
$216$		0			0
$217$		−	33906.9i		−	0.720060i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			127912.i			2.61894i
$222$		0			0
$223$			54212.7i			1.09016i	0.838383	+	0.545081i	$0.183501\pi$
$223$			54212.7i			1.09016i	−0.838383	+	0.545081i	$0.816499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			44673.3i			0.866954i	0.901165	+	0.433477i	$0.142713\pi$
$227$			44673.3i			0.866954i	−0.901165	+	0.433477i	$0.857287\pi$
$228$		0			0
$229$	15643.5			0.298307			0.149154	−	0.988814i	$-0.452345\pi$
$229$	15643.5			0.298307			0.149154	+	0.988814i	$0.452345\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	31912.0			0.587817			0.293909	−	0.955834i	$-0.405044\pi$
$233$	31912.0			0.587817			0.293909	+	0.955834i	$0.405044\pi$
$234$		0			0
$235$	−96.5224			−0.00174780
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−63704.3			−1.11525			−0.557626	−	0.830092i	$-0.688288\pi$
$239$	−63704.3			−1.11525			−0.557626	+	0.830092i	$0.688288\pi$
$240$		0			0
$241$			87289.1i			1.50288i	0.659799	+	0.751442i	$0.270642\pi$
$241$			87289.1i			1.50288i	−0.659799	+	0.751442i	$0.729358\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4554.84			−0.0758823
$246$		0			0
$247$	103149.	−	2071.81i	1.69073	−	0.0339590i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14230.5			−0.225878			−0.112939	−	0.993602i	$-0.536026\pi$
$251$	−14230.5			−0.225878			−0.112939	+	0.993602i	$0.536026\pi$
$252$		0			0
$253$	−21742.2			−0.339674
$254$		0			0
$255$		0			0
$256$		0			0
$257$			89311.3i			1.35220i	0.736811	+	0.676099i	$0.236331\pi$
$257$			89311.3i			1.35220i	−0.736811	+	0.676099i	$0.763669\pi$
$258$		0			0
$259$		−	31596.5i		−	0.471021i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3477.95			0.0502820			0.0251410	−	0.999684i	$-0.491997\pi$
$263$	3477.95			0.0502820			0.0251410	+	0.999684i	$0.491997\pi$
$264$		0			0
$265$			32774.7i			0.466710i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	39297.9i		−	0.543081i	−0.962427	−	0.271540i	$-0.912467\pi$
$269$		−	39297.9i		−	0.543081i	0.962427	−	0.271540i	$-0.0875331\pi$
$270$		0			0
$271$	−61483.1			−0.837177			−0.418589	−	0.908176i	$-0.637475\pi$
$271$	−61483.1			−0.837177			−0.418589	+	0.908176i	$0.637475\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3901.55			0.0515908
$276$		0			0
$277$	−134140.			−1.74823			−0.874115	−	0.485718i	$-0.838558\pi$
$277$	−134140.			−1.74823			−0.874115	+	0.485718i	$0.838558\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	43582.6i		−	0.551951i	−0.961165	−	0.275975i	$-0.910999\pi$
$281$		−	43582.6i		−	0.551951i	0.961165	−	0.275975i	$-0.0890009\pi$
$282$		0			0
$283$	8204.37			0.102441			0.0512203	−	0.998687i	$-0.483689\pi$
$283$	8204.37			0.102441			0.0512203	+	0.998687i	$0.483689\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			87179.3i			1.05840i
$288$		0			0
$289$	116800.			1.39845
$290$		0			0
$291$		0			0
$292$		0			0
$293$			17438.2i			0.203127i	0.994829	+	0.101563i	$0.0323845\pi$
$293$			17438.2i			0.203127i	−0.994829	+	0.101563i	$0.967616\pi$
$294$		0			0
$295$		−	123448.i		−	1.41853i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	237360.i		−	2.65500i
$300$		0			0
$301$	68941.5			0.760935
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−164615.			−1.76958
$306$		0			0
$307$		−	102263.i		−	1.08503i	−0.840046	−	0.542515i	$-0.817472\pi$
$307$		−	102263.i		−	1.08503i	0.840046	−	0.542515i	$-0.182528\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	30122.3			0.311435			0.155717	−	0.987802i	$-0.450231\pi$
$311$	30122.3			0.311435			0.155717	+	0.987802i	$0.450231\pi$
$312$		0			0
$313$	30961.5			0.316033			0.158017	−	0.987436i	$-0.449490\pi$
$313$	30961.5			0.316033			0.158017	+	0.987436i	$0.449490\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	162737.i		−	1.61946i	−0.586806	−	0.809728i	$-0.699615\pi$
$317$		−	162737.i		−	1.61946i	0.586806	−	0.809728i	$-0.300385\pi$
$318$		0			0
$319$		−	8851.38i		−	0.0869821i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3244.63	−	161541.i	−0.0311000	−	1.54838i
$324$		0			0
$325$			42593.2i			0.403249i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	175.698			0.00162321
$330$		0			0
$331$		−	94272.9i		−	0.860460i	−0.902719	−	0.430230i	$-0.858432\pi$
$331$		−	94272.9i		−	0.860460i	0.902719	−	0.430230i	$-0.141568\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	102207.i		−	0.910732i
$336$		0			0
$337$			46995.7i			0.413807i	0.978361	+	0.206904i	$0.0663387\pi$
$337$			46995.7i			0.413807i	−0.978361	+	0.206904i	$0.933661\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	17527.2i		−	0.150732i
$342$		0			0
$343$	−113303.			−0.963058
$344$		0			0
$345$		0			0
$346$		0			0
$347$	143263.			1.18980			0.594902	−	0.803798i	$-0.297191\pi$
$347$	143263.			1.18980			0.594902	+	0.803798i	$0.297191\pi$
$348$		0			0
$349$	50339.8			0.413296			0.206648	−	0.978415i	$-0.433745\pi$
$349$	50339.8			0.413296			0.206648	+	0.978415i	$0.433745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	111315.			0.893318			0.446659	−	0.894704i	$-0.352614\pi$
$353$	111315.			0.893318			0.446659	+	0.894704i	$0.352614\pi$
$354$		0			0
$355$		−	202398.i		−	1.60602i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	198233.			1.53811			0.769053	−	0.639184i	$-0.220728\pi$
$359$	198233.			1.53811			0.769053	+	0.639184i	$0.220728\pi$
$360$		0			0
$361$	−130216.	+	5233.01i	−0.999193	+	0.0401548i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−17148.3			−0.128717
$366$		0			0
$367$	50895.1			0.377871			0.188936	−	0.981989i	$-0.439496\pi$
$367$	50895.1			0.377871			0.188936	+	0.981989i	$0.439496\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	59659.3i		−	0.433441i
$372$		0			0
$373$		−	55481.8i		−	0.398779i	−0.979920	−	0.199390i	$-0.936104\pi$
$373$		−	55481.8i		−	0.398779i	0.979920	−	0.199390i	$-0.0638959\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	96630.4			0.679878
$378$		0			0
$379$			20996.6i			0.146174i	0.997326	+	0.0730869i	$0.0232850\pi$
$379$			20996.6i			0.146174i	−0.997326	+	0.0730869i	$0.976715\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			34553.2i			0.235554i	0.993040	+	0.117777i	$0.0375768\pi$
$383$			34553.2i			0.235554i	−0.993040	+	0.117777i	$0.962423\pi$
$384$		0			0
$385$	−36884.6			−0.248842
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−103703.			−0.685315			−0.342657	−	0.939460i	$-0.611327\pi$
$389$	−103703.			−0.685315			−0.342657	+	0.939460i	$0.611327\pi$
$390$		0			0
$391$	−371726.			−2.43147
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	18777.2i		−	0.120347i
$396$		0			0
$397$	−99884.3			−0.633748			−0.316874	−	0.948468i	$-0.602633\pi$
$397$	−99884.3			−0.633748			−0.316874	+	0.948468i	$0.602633\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	211625.i		−	1.31607i	−0.752989	−	0.658034i	$-0.771388\pi$
$401$		−	211625.i		−	1.31607i	0.752989	−	0.658034i	$-0.228612\pi$
$402$		0			0
$403$	191344.			1.17816
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16332.9i		−	0.0985997i
$408$		0			0
$409$			59115.4i			0.353390i	0.984266	+	0.176695i	$0.0565406\pi$
$409$			59115.4i			0.353390i	−0.984266	+	0.176695i	$0.943459\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			224710.i			1.31741i
$414$		0			0
$415$	152349.			0.884593
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−266956.			−1.52059			−0.760294	−	0.649579i	$-0.774945\pi$
$419$	−266956.			−1.52059			−0.760294	+	0.649579i	$0.774945\pi$
$420$		0			0
$421$		−	136181.i		−	0.768338i	−0.923263	−	0.384169i	$-0.874488\pi$
$421$		−	136181.i		−	0.768338i	0.923263	−	0.384169i	$-0.125512\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	66704.7			0.369299
$426$		0			0
$427$	299647.			1.64344
$428$		0			0
$429$		0			0
$430$		0			0
$431$			62111.0i			0.334360i	0.985926	+	0.167180i	$0.0534660\pi$
$431$			62111.0i			0.334360i	−0.985926	+	0.167180i	$0.946534\pi$
$432$		0			0
$433$			175495.i			0.936027i	0.883721	+	0.468014i	$0.155030\pi$
$433$			175495.i			0.936027i	−0.883721	+	0.468014i	$0.844970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6020.90	+	299764.i	0.0315282	+	1.56970i
$438$		0			0
$439$		−	46581.0i		−	0.241702i	−0.992671	−	0.120851i	$-0.961438\pi$
$439$		−	46581.0i		−	0.241702i	0.992671	−	0.120851i	$-0.0385623\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−97027.2			−0.494408			−0.247204	−	0.968963i	$-0.579512\pi$
$443$	−97027.2			−0.494408			−0.247204	+	0.968963i	$0.579512\pi$
$444$		0			0
$445$		−	110008.i		−	0.555526i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	151069.i		−	0.749346i	−0.927157	−	0.374673i	$-0.877755\pi$
$449$		−	151069.i		−	0.749346i	0.927157	−	0.374673i	$-0.122245\pi$
$450$		0			0
$451$			45064.9i			0.221557i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	402669.i		−	1.94502i
$456$		0			0
$457$	−220171.			−1.05421			−0.527107	−	0.849799i	$-0.676723\pi$
$457$	−220171.			−1.05421			−0.527107	+	0.849799i	$0.676723\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	70055.5			0.329640			0.164820	−	0.986324i	$-0.447296\pi$
$461$	70055.5			0.329640			0.164820	+	0.986324i	$0.447296\pi$
$462$		0			0
$463$	−405846.			−1.89321			−0.946606	−	0.322393i	$-0.895513\pi$
$463$	−405846.			−1.89321			−0.946606	+	0.322393i	$0.895513\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−293552.			−1.34602			−0.673010	−	0.739633i	$-0.734999\pi$
$467$	−293552.			−1.34602			−0.673010	+	0.739633i	$0.734999\pi$
$468$		0			0
$469$			186045.i			0.845811i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	35637.4			0.159288
$474$		0			0
$475$	−1080.43	−	53791.4i	−0.00478859	−	0.238411i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	299532.			1.30548			0.652742	−	0.757580i	$-0.273618\pi$
$479$	299532.			1.30548			0.652742	+	0.757580i	$0.273618\pi$
$480$		0			0
$481$	178306.			0.770685
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	356867.i		−	1.51713i
$486$		0			0
$487$			103571.i			0.436697i	0.975871	+	0.218349i	$0.0700670\pi$
$487$			103571.i			0.436697i	−0.975871	+	0.218349i	$0.929933\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	317179.			1.31565			0.657827	−	0.753169i	$-0.271476\pi$
$491$	317179.			1.31565			0.657827	+	0.753169i	$0.271476\pi$
$492$		0			0
$493$		−	151332.i		−	0.622638i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			368422.i			1.49153i
$498$		0			0
$499$	−816.803			−0.00328032			−0.00164016	−	0.999999i	$-0.500522\pi$
$499$	−816.803			−0.00328032			−0.00164016	+	0.999999i	$0.500522\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	174475.			0.689601			0.344800	−	0.938676i	$-0.387947\pi$
$503$	174475.			0.689601			0.344800	+	0.938676i	$0.387947\pi$
$504$		0			0
$505$	216575.			0.849230
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	345595.i		−	1.33393i	−0.745090	−	0.666964i	$-0.767593\pi$
$509$		−	345595.i		−	1.33393i	0.745090	−	0.666964i	$-0.232407\pi$
$510$		0			0
$511$	31214.8			0.119541
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	412635.i		−	1.55579i
$516$		0			0
$517$	90.8221			0.000339790
$518$		0			0
$519$		0			0
$520$		0			0
$521$			64679.5i			0.238282i	0.992877	+	0.119141i	$0.0380140\pi$
$521$			64679.5i			0.238282i	−0.992877	+	0.119141i	$0.961986\pi$
$522$		0			0
$523$			187404.i			0.685135i	0.939493	+	0.342567i	$0.111296\pi$
$523$			187404.i			0.685135i	−0.939493	+	0.342567i	$0.888704\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	299662.i		−	1.07897i
$528$		0			0
$529$	409952.			1.46495
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−491972.			−1.73175
$534$		0			0
$535$		−	370941.i		−	1.29598i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4285.85			0.0147523
$540$		0			0
$541$	217204.			0.742119			0.371060	−	0.928609i	$-0.378995\pi$
$541$	217204.			0.742119			0.371060	+	0.928609i	$0.378995\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			404871.i			1.36309i
$546$		0			0
$547$			136963.i			0.457751i	0.973456	+	0.228875i	$0.0735049\pi$
$547$			136963.i			0.457751i	−0.973456	+	0.228875i	$0.926495\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−122036.	+	2451.14i	−0.401960	+	0.00807357i
$552$		0			0
$553$			34179.8i			0.111768i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−213586.			−0.688435			−0.344218	−	0.938890i	$-0.611856\pi$
$557$	−213586.			−0.688435			−0.344218	+	0.938890i	$0.611856\pi$
$558$		0			0
$559$			389053.i			1.24504i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	220639.i		−	0.696091i	−0.937478	−	0.348045i	$-0.886846\pi$
$563$		−	220639.i		−	0.696091i	0.937478	−	0.348045i	$-0.113154\pi$
$564$		0			0
$565$		−	592973.i		−	1.85754i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			335154.i			1.03519i	0.855626	+	0.517595i	$0.173173\pi$
$569$			335154.i			1.03519i	−0.855626	+	0.517595i	$0.826827\pi$
$570$		0			0
$571$	−124545.			−0.381991			−0.190996	−	0.981591i	$-0.561172\pi$
$571$	−124545.			−0.381991			−0.190996	+	0.981591i	$0.561172\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−123781.			−0.374384
$576$		0			0
$577$	215350.			0.646834			0.323417	−	0.946257i	$-0.395168\pi$
$577$	215350.			0.646834			0.323417	+	0.946257i	$0.395168\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−277318.			−0.821536
$582$		0			0
$583$		−	30839.2i		−	0.0907331i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−396441.			−1.15054			−0.575271	−	0.817963i	$-0.695103\pi$
$587$	−396441.			−1.15054			−0.575271	+	0.817963i	$0.695103\pi$
$588$		0			0
$589$	−241651.	+	4853.68i	−0.696559	+	0.0139907i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	160553.			0.456573			0.228286	−	0.973594i	$-0.426688\pi$
$593$	160553.			0.456573			0.228286	+	0.973594i	$0.426688\pi$
$594$		0			0
$595$	−630614.			−1.78127
$596$		0			0
$597$		0			0
$598$		0			0
$599$			39005.8i			0.108711i	0.998522	+	0.0543557i	$0.0173105\pi$
$599$			39005.8i			0.108711i	−0.998522	+	0.0543557i	$0.982690\pi$
$600$		0			0
$601$		−	297294.i		−	0.823071i	−0.911394	−	0.411535i	$-0.864993\pi$
$601$		−	297294.i		−	0.823071i	0.911394	−	0.411535i	$-0.135007\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	388268.			1.06077
$606$		0			0
$607$		−	638986.i		−	1.73426i	−0.498082	−	0.867130i	$-0.665962\pi$
$607$		−	638986.i		−	1.73426i	0.498082	−	0.867130i	$-0.334038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			991.504i			0.00265590i
$612$		0			0
$613$	620859.			1.65224			0.826118	−	0.563497i	$-0.190544\pi$
$613$	620859.			1.65224			0.826118	+	0.563497i	$0.190544\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−264977.			−0.696045			−0.348022	−	0.937486i	$-0.613147\pi$
$617$	−264977.			−0.696045			−0.348022	+	0.937486i	$0.613147\pi$
$618$		0			0
$619$	−105833.			−0.276210			−0.138105	−	0.990418i	$-0.544101\pi$
$619$	−105833.			−0.276210			−0.138105	+	0.990418i	$0.544101\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			200246.i			0.515926i
$624$		0			0
$625$	−461561.			−1.18160
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	279243.i		−	0.705800i
$630$		0			0
$631$	108947.			0.273626			0.136813	−	0.990597i	$-0.456314\pi$
$631$	108947.			0.273626			0.136813	+	0.990597i	$0.456314\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			697643.i			1.73016i
$636$		0			0
$637$			46788.5i			0.115308i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			711723.i			1.73219i	0.499882	+	0.866093i	$0.333377\pi$
$641$			711723.i			1.73219i	−0.499882	+	0.866093i	$0.666623\pi$
$642$		0			0
$643$	712239.			1.72268			0.861338	−	0.508032i	$-0.169627\pi$
$643$	712239.			1.72268			0.861338	+	0.508032i	$0.169627\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	498438.			1.19070			0.595350	−	0.803467i	$-0.297013\pi$
$647$	498438.			1.19070			0.595350	+	0.803467i	$0.297013\pi$
$648$		0			0
$649$			116157.i			0.275776i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−130175.			−0.305281			−0.152641	−	0.988282i	$-0.548778\pi$
$653$	−130175.			−0.305281			−0.152641	+	0.988282i	$0.548778\pi$
$654$		0			0
$655$	134443.			0.313368
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	258996.i		−	0.596379i	−0.954507	−	0.298190i	$-0.903617\pi$
$659$		−	258996.i		−	0.596379i	0.954507	−	0.298190i	$-0.0963828\pi$
$660$		0			0
$661$			767174.i			1.75586i	0.478785	+	0.877932i	$0.341077\pi$
$661$			767174.i			1.75586i	−0.478785	+	0.877932i	$0.658923\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10214.2	+	508534.i	0.0230972	+	1.14994i
$666$		0			0
$667$			280819.i			0.631211i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	154894.			0.344024
$672$		0			0
$673$			63926.3i			0.141140i	0.997507	+	0.0705699i	$0.0224818\pi$
$673$			63926.3i			0.141140i	−0.997507	+	0.0705699i	$0.977518\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			466563.i			1.01797i	0.860777	+	0.508983i	$0.169978\pi$
$677$			466563.i			1.01797i	−0.860777	+	0.508983i	$0.830022\pi$
$678$		0			0
$679$			649598.i			1.40898i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			133694.i			0.286595i	0.989680	+	0.143298i	$0.0457706\pi$
$683$			133694.i			0.286595i	−0.989680	+	0.143298i	$0.954229\pi$
$684$		0			0
$685$	−312885.			−0.666813
$686$		0			0
$687$		0			0
$688$		0			0
$689$	336671.			0.709197
$690$		0			0
$691$	−185512.			−0.388522			−0.194261	−	0.980950i	$-0.562231\pi$
$691$	−185512.			−0.388522			−0.194261	+	0.980950i	$0.562231\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.00494e6			2.08051
$696$		0			0
$697$			770471.i			1.58596i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−77805.2			−0.158333			−0.0791667	−	0.996861i	$-0.525226\pi$
$701$	−77805.2			−0.158333			−0.0791667	+	0.996861i	$0.525226\pi$
$702$		0			0
$703$	−225185.	+	4522.95i	−0.455647	+	0.00915190i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−394227.			−0.788693
$708$		0			0
$709$	−425698.			−0.846856			−0.423428	−	0.905930i	$-0.639173\pi$
$709$	−425698.			−0.846856			−0.423428	+	0.905930i	$0.639173\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			556068.i			1.09383i
$714$		0			0
$715$		−	208148.i		−	0.407156i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−434230.			−0.839967			−0.419984	−	0.907532i	$-0.637964\pi$
$719$	−434230.			−0.839967			−0.419984	+	0.907532i	$0.637964\pi$
$720$		0			0
$721$			751113.i			1.44489i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	50391.8i		−	0.0958702i
$726$		0			0
$727$	571303.			1.08093			0.540465	−	0.841366i	$-0.318248\pi$
$727$	571303.			1.08093			0.540465	+	0.841366i	$0.318248\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	609290.			1.14022
$732$		0			0
$733$	257404.			0.479079			0.239540	−	0.970887i	$-0.423003\pi$
$733$	257404.			0.479079			0.239540	+	0.970887i	$0.423003\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			96170.9i			0.177055i
$738$		0			0
$739$	−239337.			−0.438248			−0.219124	−	0.975697i	$-0.570320\pi$
$739$	−239337.			−0.438248			−0.219124	+	0.975697i	$0.570320\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	591739.i		−	1.07190i	−0.844251	−	0.535948i	$-0.819954\pi$
$743$		−	591739.i		−	1.07190i	0.844251	−	0.535948i	$-0.180046\pi$
$744$		0			0
$745$	−399351.			−0.719519
$746$		0			0
$747$		0			0
$748$		0			0
$749$			675217.i			1.20359i
$750$		0			0
$751$		−	89204.9i		−	0.158164i	−0.996868	−	0.0790822i	$-0.974801\pi$
$751$		−	89204.9i		−	0.158164i	0.996868	−	0.0790822i	$-0.0251990\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	714181.i		−	1.25289i
$756$		0			0
$757$	−673763.			−1.17575			−0.587875	−	0.808952i	$-0.700035\pi$
$757$	−673763.			−1.17575			−0.587875	+	0.808952i	$0.700035\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	490950.			0.847750			0.423875	−	0.905721i	$-0.360670\pi$
$761$	490950.			0.847750			0.423875	+	0.905721i	$0.360670\pi$
$762$		0			0
$763$		−	736979.i		−	1.26592i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.26809e6			−2.15555
$768$		0			0
$769$	−572323.			−0.967807			−0.483904	−	0.875121i	$-0.660781\pi$
$769$	−572323.			−0.967807			−0.483904	+	0.875121i	$0.660781\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			233033.i			0.389995i	0.980804	+	0.194997i	$0.0624698\pi$
$773$			233033.i			0.389995i	−0.980804	+	0.194997i	$0.937530\pi$
$774$		0			0
$775$		−	99784.2i		−	0.166134i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	621317.	−	12479.5i	1.02385	−	0.0205646i
$780$		0			0
$781$			190445.i			0.312226i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−987111.			−1.60187
$786$		0			0
$787$		−	876845.i		−	1.41571i	−0.706359	−	0.707854i	$-0.749664\pi$
$787$		−	876845.i		−	1.41571i	0.706359	−	0.707854i	$-0.250336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1.07938e6i			1.72513i
$792$		0			0
$793$			1.69097e6i			2.68900i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	376141.i		−	0.592153i	−0.955164	−	0.296076i	$-0.904322\pi$
$797$		−	376141.i		−	0.592153i	0.955164	−	0.296076i	$-0.0956783\pi$
$798$		0			0
$799$	1552.78			0.00243230
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16135.6			0.0250239
$804$		0			0
$805$	1.17020e6			1.80579
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−979635.			−1.49681			−0.748406	−	0.663241i	$-0.769180\pi$
$809$	−979635.			−1.49681			−0.748406	+	0.663241i	$0.769180\pi$
$810$		0			0
$811$			583716.i			0.887483i	0.896155	+	0.443741i	$0.146349\pi$
$811$			583716.i			0.887483i	−0.896155	+	0.443741i	$0.853651\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	159705.			0.240438
$816$		0			0
$817$	−9868.77	−	491338.i	−0.0147849	−	0.736100i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	702550.			1.04230			0.521148	−	0.853466i	$-0.325504\pi$
$821$	702550.			1.04230			0.521148	+	0.853466i	$0.325504\pi$
$822$		0			0
$823$	−366360.			−0.540889			−0.270444	−	0.962736i	$-0.587171\pi$
$823$	−366360.			−0.540889			−0.270444	+	0.962736i	$0.587171\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			718634.i			1.05074i	0.850873	+	0.525372i	$0.176074\pi$
$827$			718634.i			1.05074i	−0.850873	+	0.525372i	$0.823926\pi$
$828$		0			0
$829$		−	421812.i		−	0.613775i	−0.951746	−	0.306888i	$-0.900712\pi$
$829$		−	421812.i		−	0.613775i	0.951746	−	0.306888i	$-0.0992876\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	73274.9			0.105600
$834$		0			0
$835$			278213.i			0.399029i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	172794.i		−	0.245473i	−0.992439	−	0.122737i	$-0.960833\pi$
$839$		−	172794.i		−	0.245473i	0.992439	−	0.122737i	$-0.0391671\pi$
$840$		0			0
$841$	592958.			0.838363
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.47774e6			2.06959
$846$		0			0
$847$	−706758.			−0.985154
$848$		0			0
$849$		0			0
$850$		0			0
$851$			518178.i			0.715517i
$852$		0			0
$853$	−395245.			−0.543211			−0.271606	−	0.962409i	$-0.587555\pi$
$853$	−395245.			−0.543211			−0.271606	+	0.962409i	$0.587555\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.07871e6i			1.46874i	0.678750	+	0.734370i	$0.262522\pi$
$857$			1.07871e6i			1.46874i	−0.678750	+	0.734370i	$0.737478\pi$
$858$		0			0
$859$	888609.			1.20427			0.602136	−	0.798393i	$-0.294316\pi$
$859$	888609.			1.20427			0.602136	+	0.798393i	$0.294316\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			510125.i			0.684944i	0.939528	+	0.342472i	$0.111264\pi$
$863$			510125.i			0.684944i	−0.939528	+	0.342472i	$0.888736\pi$
$864$		0			0
$865$		−	1.00416e6i		−	1.34205i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			17668.3i			0.0233967i
$870$		0			0
$871$	−1.04990e6			−1.38392
$872$		0			0
$873$		0			0
$874$		0			0
$875$	670616.			0.875907
$876$		0			0
$877$			1.24926e6i			1.62425i	0.583485	+	0.812124i	$0.301689\pi$
$877$			1.24926e6i			1.62425i	−0.583485	+	0.812124i	$0.698311\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	977935.			1.25996			0.629982	−	0.776610i	$-0.283062\pi$
$881$	977935.			1.25996			0.629982	+	0.776610i	$0.283062\pi$
$882$		0			0
$883$	465114.			0.596538			0.298269	−	0.954482i	$-0.403591\pi$
$883$	465114.			0.596538			0.298269	+	0.954482i	$0.403591\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.12281e6i		−	1.42711i	−0.700599	−	0.713555i	$-0.747084\pi$
$887$		−	1.12281e6i		−	1.42711i	0.700599	−	0.713555i	$-0.252916\pi$
$888$		0			0
$889$		−	1.26991e6i		−	1.60682i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25.1507	−	1252.18i	−3.15389e−5	−	0.00157023i
$894$		0			0
$895$			988738.i			1.23434i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−226379.			−0.280102
$900$		0			0
$901$		−	527256.i		−	0.649489i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	261453.i		−	0.319225i
$906$		0			0
$907$		−	637072.i		−	0.774415i	−0.921993	−	0.387207i	$-0.873440\pi$
$907$		−	637072.i		−	0.774415i	0.921993	−	0.387207i	$-0.126560\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	311275.i		−	0.375066i	−0.982258	−	0.187533i	$-0.939951\pi$
$911$		−	311275.i		−	0.375066i	0.982258	−	0.187533i	$-0.0600491\pi$
$912$		0			0
$913$	−143352.			−0.171974
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−244724.			−0.291030
$918$		0			0
$919$	444591.			0.526416			0.263208	−	0.964739i	$-0.415219\pi$
$919$	444591.			0.526416			0.263208	+	0.964739i	$0.415219\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.07909e6			−2.44045
$924$		0			0
$925$		−	92985.0i		−	0.108675i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−526124.			−0.609617			−0.304808	−	0.952414i	$-0.598592\pi$
$929$	−526124.			−0.609617			−0.304808	+	0.952414i	$0.598592\pi$
$930$		0			0
$931$	−1186.85	−	59089.7i	−0.00136929	−	0.0681730i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−325978.			−0.372877
$936$		0			0
$937$	1.26151e6			1.43685			0.718426	−	0.695603i	$-0.244863\pi$
$937$	1.26151e6			1.43685			0.718426	+	0.695603i	$0.244863\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			243694.i			0.275211i	0.990487	+	0.137605i	$0.0439405\pi$
$941$			243694.i			0.275211i	−0.990487	+	0.137605i	$0.956059\pi$
$942$		0			0
$943$		−	1.42973e6i		−	1.60779i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.01262e6			−1.12914			−0.564569	−	0.825386i	$-0.690958\pi$
$947$	−1.01262e6			−1.12914			−0.564569	+	0.825386i	$0.690958\pi$
$948$		0			0
$949$			176152.i			0.195594i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.58910e6i			1.74971i	0.484386	+	0.874854i	$0.339043\pi$
$953$			1.58910e6i			1.74971i	−0.484386	+	0.874854i	$0.660957\pi$
$954$		0			0
$955$	1.08728e6			1.19216
$956$		0			0
$957$		0			0
$958$		0			0
$959$	569540.			0.619280
$960$		0			0
$961$	475253.			0.514610
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.98774e6i		−	2.13454i
$966$		0			0
$967$	854289.			0.913591			0.456795	−	0.889572i	$-0.348997\pi$
$967$	854289.			0.913591			0.456795	+	0.889572i	$0.348997\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	800970.i		−	0.849528i	−0.905304	−	0.424764i	$-0.860357\pi$
$971$		−	800970.i		−	0.849528i	0.905304	−	0.424764i	$-0.139643\pi$
$972$		0			0
$973$	−1.82928e6			−1.93221
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.32952e6i		−	1.39286i	−0.717627	−	0.696428i	$-0.754772\pi$
$977$		−	1.32952e6i		−	1.39286i	0.717627	−	0.696428i	$-0.245228\pi$
$978$		0			0
$979$			103511.i			0.108000i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.40685e6i		−	1.45593i	−0.685615	−	0.727964i	$-0.740467\pi$
$983$		−	1.40685e6i		−	1.45593i	0.685615	−	0.727964i	$-0.259533\pi$
$984$		0			0
$985$	−1.96714e6			−2.02751
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.13063e6			−1.15592
$990$		0			0
$991$			4828.18i			0.00491627i	0.999997	+	0.00245814i	$0.000782450\pi$
$991$			4828.18i			0.00491627i	−0.999997	+	0.00245814i	$0.999218\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.12604e6			−1.13738
$996$		0			0
$997$	1.85065e6			1.86180			0.930900	−	0.365274i	$-0.119025\pi$
$997$	1.85065e6			1.86180			0.930900	+	0.365274i	$0.119025\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.h.c.37.2		4
3.2	odd	2		76.5.c.b.37.4	yes	4
12.11	even	2		304.5.e.d.113.1		4
19.18	odd	2	inner	684.5.h.c.37.1		4
57.56	even	2		76.5.c.b.37.1	✓	4
228.227	odd	2		304.5.e.d.113.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.c.b.37.1	✓	4	57.56	even	2
76.5.c.b.37.4	yes	4	3.2	odd	2
304.5.e.d.113.1		4	12.11	even	2
304.5.e.d.113.4		4	228.227	odd	2
684.5.h.c.37.1		4	19.18	odd	2	inner
684.5.h.c.37.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-27.8215 q^{5} +50.6430 q^{7} +O(q^{10})\)	\(q-27.8215 q^{5} +50.6430 q^{7} +26.1785 q^{11} +285.790i q^{13} +447.572 q^{17} +(-7.24940 - 360.927i) q^{19} -830.538 q^{23} +149.037 q^{25} -338.117i q^{29} -669.528i q^{31} -1408.97 q^{35} -623.907i q^{37} +1721.45i q^{41} +1361.32 q^{43} +3.46934 q^{47} +163.716 q^{49} -1178.03i q^{53} -728.325 q^{55} +4437.13i q^{59} +5916.84 q^{61} -7951.12i q^{65} +3673.66i q^{67} +7274.88i q^{71} +616.369 q^{73} +1325.76 q^{77} +674.916i q^{79} -5475.95 q^{83} -12452.1 q^{85} +3954.06i q^{89} +14473.3i q^{91} +(201.689 + 10041.5i) q^{95} +12827.0i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{5} + 24 q^{7}+O(q^{10}) \) 4 * q - 22 * q^5 + 24 * q^7	\( 4 q - 22 q^{5} + 24 q^{7} + 194 q^{11} + 1076 q^{17} - 654 q^{19} - 1090 q^{23} - 386 q^{25} - 4118 q^{35} + 4106 q^{43} - 5254 q^{47} - 1488 q^{49} + 926 q^{55} + 6078 q^{61} - 9856 q^{73} - 2822 q^{77} - 3868 q^{83} - 21862 q^{85} - 10354 q^{95}+O(q^{100}) \) 4 * q - 22 * q^5 + 24 * q^7 + 194 * q^11 + 1076 * q^17 - 654 * q^19 - 1090 * q^23 - 386 * q^25 - 4118 * q^35 + 4106 * q^43 - 5254 * q^47 - 1488 * q^49 + 926 * q^55 + 6078 * q^61 - 9856 * q^73 - 2822 * q^77 - 3868 * q^83 - 21862 * q^85 - 10354 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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