LMFDB - Embedded newform 1040.2.k.e.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(961,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1040.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.k (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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196 $ x^8 - 2x^7 + 2x^6 + 6x^5 + 36x^4 - 52x^3 + 50x^2 + 140*x + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.3
Root			$-1.61013 - 1.61013i$ of defining polynomial
Character	$\chi$	$=$	1040.961
Dual form			1040.2.k.e.961.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.18501 q^{3} -1.00000i q^{5} -5.22025i q^{7} -1.59576 q^{9} +O(q^{10})$	$q-1.18501 q^{3} -1.00000i q^{5} -5.22025i q^{7} -1.59576 q^{9} -4.40526i q^{11} +(-1.18501 + 3.40526i) q^{13} +1.18501i q^{15} -0.624487 q^{17} +3.41076i q^{19} +6.18602i q^{21} -7.99552 q^{23} -1.00000 q^{25} +5.44600 q^{27} +6.96577 q^{29} +9.02974i q^{31} +5.22025i q^{33} -5.22025 q^{35} -0.850240i q^{37} +(1.40424 - 4.03524i) q^{39} +0.994498i q^{41} -5.00102 q^{43} +1.59576i q^{45} +3.97128i q^{47} -20.2510 q^{49} +0.740021 q^{51} +0.994498 q^{53} -4.40526 q^{55} -4.04177i q^{57} -5.78077i q^{59} -2.58476 q^{61} +8.33028i q^{63} +(3.40526 + 1.18501i) q^{65} -12.7818i q^{67} +9.47473 q^{69} -1.15628i q^{71} -12.0308i q^{73} +1.18501 q^{75} -22.9965 q^{77} -7.18052 q^{79} -1.66625 q^{81} +5.14976i q^{83} +0.624487i q^{85} -8.25448 q^{87} +14.4405i q^{89} +(17.7763 + 6.18602i) q^{91} -10.7003i q^{93} +3.41076 q^{95} -5.18052i q^{97} +7.02974i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 16 * q^9	$ 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−1.18501			−0.684163			−0.342082	−	0.939670i	$-0.611132\pi$
$3$	−1.18501			−0.684163			−0.342082	+	0.939670i	$0.611132\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	5.22025i		−	1.97307i	−0.163553	−	0.986535i	$-0.552296\pi$
$7$		−	5.22025i		−	1.97307i	0.163553	−	0.986535i	$-0.447704\pi$
$8$		0			0
$9$	−1.59576			−0.531921
$10$		0			0
$11$		−	4.40526i		−	1.32823i	−0.747628	−	0.664117i	$-0.768808\pi$
$11$		−	4.40526i		−	1.32823i	0.747628	−	0.664117i	$-0.231192\pi$
$12$		0			0
$13$	−1.18501	+	3.40526i	−0.328661	+	0.944448i
$13$	−1.18501	+	3.40526i	−0.328661	+	0.944448i
$14$		0			0
$15$			1.18501i			0.305967i
$16$		0			0
$17$	−0.624487			−0.151460			−0.0757302	−	0.997128i	$-0.524129\pi$
$17$	−0.624487			−0.151460			−0.0757302	+	0.997128i	$0.524129\pi$
$18$		0			0
$19$			3.41076i			0.782481i	0.920288	+	0.391241i	$0.127954\pi$
$19$			3.41076i			0.782481i	−0.920288	+	0.391241i	$0.872046\pi$
$20$		0			0
$21$			6.18602i			1.34990i
$22$		0			0
$23$	−7.99552			−1.66718			−0.833590	−	0.552383i	$-0.813718\pi$
$23$	−7.99552			−1.66718			−0.833590	+	0.552383i	$0.813718\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.44600			1.04808
$28$		0			0
$29$	6.96577			1.29351			0.646756	−	0.762697i	$-0.276125\pi$
$29$	6.96577			1.29351			0.646756	+	0.762697i	$0.276125\pi$
$30$		0			0
$31$			9.02974i			1.62179i	0.585192	+	0.810895i	$0.301019\pi$
$31$			9.02974i			1.62179i	−0.585192	+	0.810895i	$0.698981\pi$
$32$		0			0
$33$			5.22025i			0.908729i
$34$		0			0
$35$	−5.22025			−0.882383
$36$		0			0
$37$		−	0.850240i		−	0.139779i	−0.997555	−	0.0698893i	$-0.977735\pi$
$37$		−	0.850240i		−	0.139779i	0.997555	−	0.0698893i	$-0.0222646\pi$
$38$		0			0
$39$	1.40424	−	4.03524i	0.224858	−	0.646156i
$40$		0			0
$41$			0.994498i			0.155314i	0.996980	+	0.0776572i	$0.0247440\pi$
$41$			0.994498i			0.155314i	−0.996980	+	0.0776572i	$0.975256\pi$
$42$		0			0
$43$	−5.00102			−0.762648			−0.381324	−	0.924441i	$-0.624532\pi$
$43$	−5.00102			−0.762648			−0.381324	+	0.924441i	$0.624532\pi$
$44$		0			0
$45$			1.59576i			0.237882i
$46$		0			0
$47$			3.97128i			0.579270i	0.957137	+	0.289635i	$0.0935339\pi$
$47$			3.97128i			0.579270i	−0.957137	+	0.289635i	$0.906466\pi$
$48$		0			0
$49$	−20.2510			−2.89300
$50$		0			0
$51$	0.740021			0.103624
$52$		0			0
$53$	0.994498			0.136605			0.0683024	−	0.997665i	$-0.478242\pi$
$53$	0.994498			0.136605			0.0683024	+	0.997665i	$0.478242\pi$
$54$		0			0
$55$	−4.40526			−0.594004
$56$		0			0
$57$		−	4.04177i		−	0.535345i
$58$		0			0
$59$		−	5.78077i		−	0.752592i	−0.926500	−	0.376296i	$-0.877198\pi$
$59$		−	5.78077i		−	0.752592i	0.926500	−	0.376296i	$-0.122802\pi$
$60$		0			0
$61$	−2.58476			−0.330944			−0.165472	−	0.986214i	$-0.552915\pi$
$61$	−2.58476			−0.330944			−0.165472	+	0.986214i	$0.552915\pi$
$62$		0			0
$63$			8.33028i			1.04952i
$64$		0			0
$65$	3.40526	+	1.18501i	0.422370	+	0.146982i
$66$		0			0
$67$		−	12.7818i		−	1.56154i	−0.624816	−	0.780772i	$-0.714826\pi$
$67$		−	12.7818i		−	1.56154i	0.624816	−	0.780772i	$-0.285174\pi$
$68$		0			0
$69$	9.47473			1.14062
$70$		0			0
$71$		−	1.15628i		−	0.137225i	−0.997643	−	0.0686126i	$-0.978143\pi$
$71$		−	1.15628i		−	0.137225i	0.997643	−	0.0686126i	$-0.0218573\pi$
$72$		0			0
$73$		−	12.0308i		−	1.40809i	−0.710153	−	0.704047i	$-0.751374\pi$
$73$		−	12.0308i		−	1.40809i	0.710153	−	0.704047i	$-0.248626\pi$
$74$		0			0
$75$	1.18501			0.136833
$76$		0			0
$77$	−22.9965			−2.62070
$78$		0			0
$79$	−7.18052			−0.807872			−0.403936	−	0.914787i	$-0.632358\pi$
$79$	−7.18052			−0.807872			−0.403936	+	0.914787i	$0.632358\pi$
$80$		0			0
$81$	−1.66625			−0.185139
$82$		0			0
$83$			5.14976i			0.565260i	0.959229	+	0.282630i	$0.0912068\pi$
$83$			5.14976i			0.565260i	−0.959229	+	0.282630i	$0.908793\pi$
$84$		0			0
$85$			0.624487i			0.0677351i
$86$		0			0
$87$	−8.25448			−0.884973
$88$		0			0
$89$			14.4405i			1.53069i	0.643620	+	0.765345i	$0.277432\pi$
$89$			14.4405i			1.53069i	−0.643620	+	0.765345i	$0.722568\pi$
$90$		0			0
$91$	17.7763	+	6.18602i	1.86346	+	0.648471i
$92$		0			0
$93$		−	10.7003i		−	1.10957i
$94$		0			0
$95$	3.41076			0.349936
$96$		0			0
$97$		−	5.18052i		−	0.526002i	−0.964796	−	0.263001i	$-0.915288\pi$
$97$		−	5.18052i		−	0.526002i	0.964796	−	0.263001i	$-0.0847123\pi$
$98$		0			0
$99$			7.02974i			0.706516i
$100$		0			0
$101$	−3.62999			−0.361197			−0.180599	−	0.983557i	$-0.557804\pi$
$101$	−3.62999			−0.361197			−0.180599	+	0.983557i	$0.557804\pi$
$102$		0			0
$103$	−12.3205			−1.21397			−0.606987	−	0.794712i	$-0.707622\pi$
$103$	−12.3205			−1.21397			−0.606987	+	0.794712i	$0.707622\pi$
$104$		0			0
$105$	6.18602			0.603694
$106$		0			0
$107$	−12.9305			−1.25004			−0.625021	−	0.780608i	$-0.714909\pi$
$107$	−12.9305			−1.25004			−0.625021	+	0.780608i	$0.714909\pi$
$108$		0			0
$109$		−	0.440500i		−	0.0421923i	−0.999777	−	0.0210961i	$-0.993284\pi$
$109$		−	0.440500i		−	0.0421923i	0.999777	−	0.0210961i	$-0.00671560\pi$
$110$		0			0
$111$			1.00754i			0.0956313i
$112$		0			0
$113$	5.81601			0.547124			0.273562	−	0.961854i	$-0.411798\pi$
$113$	5.81601			0.547124			0.273562	+	0.961854i	$0.411798\pi$
$114$		0			0
$115$			7.99552i			0.745586i
$116$		0			0
$117$	1.89099	−	5.43398i	0.174822	−	0.502372i
$118$		0			0
$119$			3.25998i			0.298842i
$120$		0			0
$121$	−8.40627			−0.764207
$122$		0			0
$123$		−	1.17848i		−	0.106260i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−9.44152			−0.837799			−0.418900	−	0.908033i	$-0.637584\pi$
$127$	−9.44152			−0.837799			−0.418900	+	0.908033i	$0.637584\pi$
$128$		0			0
$129$	5.92623			0.521776
$130$		0			0
$131$	12.4295			1.08597			0.542985	−	0.839742i	$-0.317294\pi$
$131$	12.4295			1.08597			0.542985	+	0.839742i	$0.317294\pi$
$132$		0			0
$133$	17.8050			1.54389
$134$		0			0
$135$		−	5.44600i		−	0.468717i
$136$		0			0
$137$		−	9.61898i		−	0.821805i	−0.911679	−	0.410903i	$-0.865214\pi$
$137$		−	9.61898i		−	0.821805i	0.911679	−	0.410903i	$-0.134786\pi$
$138$		0			0
$139$	8.42950			0.714980			0.357490	−	0.933917i	$-0.383633\pi$
$139$	8.42950			0.714980			0.357490	+	0.933917i	$0.383633\pi$
$140$		0			0
$141$		−	4.70598i		−	0.396315i
$142$		0			0
$143$	15.0010	+	5.22025i	1.25445	+	0.436539i
$144$		0			0
$145$		−	6.96577i		−	0.578476i
$146$		0			0
$147$	23.9976			1.97928
$148$		0			0
$149$		−	11.9315i		−	0.977470i	−0.872432	−	0.488735i	$-0.837458\pi$
$149$		−	11.9315i		−	0.977470i	0.872432	−	0.488735i	$-0.162542\pi$
$150$		0			0
$151$			15.9558i			1.29846i	0.760591	+	0.649232i	$0.224909\pi$
$151$			15.9558i			1.29846i	−0.760591	+	0.649232i	$0.775091\pi$
$152$		0			0
$153$	0.996533			0.0805650
$154$		0			0
$155$	9.02974			0.725286
$156$		0			0
$157$	−15.6190			−1.24653			−0.623265	−	0.782010i	$-0.714194\pi$
$157$	−15.6190			−1.24653			−0.623265	+	0.782010i	$0.714194\pi$
$158$		0			0
$159$	−1.17848			−0.0934599
$160$		0			0
$161$			41.7386i			3.28946i
$162$		0			0
$163$			6.46922i			0.506709i	0.967374	+	0.253354i	$0.0815339\pi$
$163$			6.46922i			0.506709i	−0.967374	+	0.253354i	$0.918466\pi$
$164$		0			0
$165$	5.22025			0.406396
$166$		0			0
$167$		−	15.2223i		−	1.17794i	−0.808156	−	0.588968i	$-0.799534\pi$
$167$		−	15.2223i		−	1.17794i	0.808156	−	0.588968i	$-0.200466\pi$
$168$		0			0
$169$	−10.1915	−	8.07049i	−0.783964	−	0.620807i
$170$		0			0
$171$		−	5.44276i		−	0.416218i
$172$		0			0
$173$	19.4350			1.47762			0.738808	−	0.673916i	$-0.235389\pi$
$173$	19.4350			1.47762			0.738808	+	0.673916i	$0.235389\pi$
$174$		0			0
$175$			5.22025i			0.394614i
$176$		0			0
$177$			6.85024i			0.514896i
$178$		0			0
$179$	22.0020			1.64451			0.822255	−	0.569120i	$-0.192716\pi$
$179$	22.0020			1.64451			0.822255	+	0.569120i	$0.192716\pi$
$180$		0			0
$181$	−7.47473			−0.555592			−0.277796	−	0.960640i	$-0.589604\pi$
$181$	−7.47473			−0.555592			−0.277796	+	0.960640i	$0.589604\pi$
$182$		0			0
$183$	3.06295			0.226420
$184$		0			0
$185$	−0.850240			−0.0625109
$186$		0			0
$187$			2.75103i			0.201175i
$188$		0			0
$189$		−	28.4295i		−	2.06794i
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$		−	11.6300i		−	0.837145i	−0.908183	−	0.418572i	$-0.862531\pi$
$193$		−	11.6300i		−	0.837145i	0.908183	−	0.418572i	$-0.137469\pi$
$194$		0			0
$195$	−4.03524	−	1.40424i	−0.288970	−	0.100560i
$196$		0			0
$197$		−	14.0705i		−	1.00248i	−0.865308	−	0.501240i	$-0.832877\pi$
$197$		−	14.0705i		−	1.00248i	0.865308	−	0.501240i	$-0.167123\pi$
$198$		0			0
$199$	−22.0020			−1.55968			−0.779841	−	0.625977i	$-0.784700\pi$
$199$	−22.0020			−1.55968			−0.779841	+	0.625977i	$0.784700\pi$
$200$		0			0
$201$			15.1465i			1.06835i
$202$		0			0
$203$		−	36.3631i		−	2.55219i
$204$		0			0
$205$	0.994498			0.0694587
$206$		0			0
$207$	12.7589			0.886808
$208$		0			0
$209$	15.0253			1.03932
$210$		0			0
$211$	−5.62999			−0.387584			−0.193792	−	0.981043i	$-0.562079\pi$
$211$	−5.62999			−0.387584			−0.193792	+	0.981043i	$0.562079\pi$
$212$		0			0
$213$			1.37020i			0.0938845i
$214$		0			0
$215$			5.00102i			0.341067i
$216$		0			0
$217$	47.1375			3.19990
$218$		0			0
$219$			14.2565i			0.963366i
$220$		0			0
$221$	0.740021	−	2.12654i	0.0497792	−	0.143046i
$222$		0			0
$223$		−	14.9207i		−	0.999166i	−0.866266	−	0.499583i	$-0.833487\pi$
$223$		−	14.9207i		−	0.999166i	0.866266	−	0.499583i	$-0.166513\pi$
$224$		0			0
$225$	1.59576			0.106384
$226$		0			0
$227$		−	6.31928i		−	0.419425i	−0.977763	−	0.209713i	$-0.932747\pi$
$227$		−	6.31928i		−	0.419425i	0.977763	−	0.209713i	$-0.0672528\pi$
$228$		0			0
$229$			21.4350i			1.41646i	0.705980	+	0.708232i	$0.250507\pi$
$229$			21.4350i			1.41646i	−0.705980	+	0.708232i	$0.749493\pi$
$230$		0			0
$231$	27.2510			1.79298
$232$		0			0
$233$	24.8125			1.62552			0.812762	−	0.582596i	$-0.197963\pi$
$233$	24.8125			1.62552			0.812762	+	0.582596i	$0.197963\pi$
$234$		0			0
$235$	3.97128			0.259057
$236$		0			0
$237$	8.50895			0.552716
$238$		0			0
$239$		−	16.7192i		−	1.08148i	−0.841191	−	0.540738i	$-0.818145\pi$
$239$		−	16.7192i		−	1.08148i	0.841191	−	0.540738i	$-0.181855\pi$
$240$		0			0
$241$		−	12.1970i		−	0.785680i	−0.919607	−	0.392840i	$-0.871493\pi$
$241$		−	12.1970i		−	0.785680i	0.919607	−	0.392840i	$-0.128507\pi$
$242$		0			0
$243$	−14.3635			−0.921418
$244$		0			0
$245$			20.2510i			1.29379i
$246$		0			0
$247$	−11.6145	−	4.04177i	−0.739013	−	0.257171i
$248$		0			0
$249$		−	6.10249i		−	0.386730i
$250$		0			0
$251$	−3.92951			−0.248028			−0.124014	−	0.992280i	$-0.539577\pi$
$251$	−3.92951			−0.248028			−0.124014	+	0.992280i	$0.539577\pi$
$252$		0			0
$253$			35.2223i			2.21441i
$254$		0			0
$255$		−	0.740021i		−	0.0463419i
$256$		0			0
$257$	−8.38305			−0.522920			−0.261460	−	0.965214i	$-0.584204\pi$
$257$	−8.38305			−0.522920			−0.261460	+	0.965214i	$0.584204\pi$
$258$		0			0
$259$	−4.43846			−0.275793
$260$		0			0
$261$	−11.1157			−0.688046
$262$		0			0
$263$	−21.3710			−1.31779			−0.658897	−	0.752233i	$-0.728977\pi$
$263$	−21.3710			−1.31779			−0.658897	+	0.752233i	$0.728977\pi$
$264$		0			0
$265$		−	0.994498i		−	0.0610915i
$266$		0			0
$267$		−	17.1121i		−	1.04724i
$268$		0			0
$269$	−12.5110			−0.762809			−0.381404	−	0.924408i	$-0.624559\pi$
$269$	−12.5110			−0.762809			−0.381404	+	0.924408i	$0.624559\pi$
$270$		0			0
$271$			0.775265i			0.0470940i	0.999723	+	0.0235470i	$0.00749594\pi$
$271$			0.775265i			0.0470940i	−0.999723	+	0.0235470i	$0.992504\pi$
$272$		0			0
$273$	−21.0650	−	7.33047i	−1.27491	−	0.443660i
$274$		0			0
$275$			4.40526i			0.265647i
$276$		0			0
$277$	0.485544			0.0291735			0.0145867	−	0.999894i	$-0.495357\pi$
$277$	0.485544			0.0291735			0.0145867	+	0.999894i	$0.495357\pi$
$278$		0			0
$279$		−	14.4093i		−	0.862664i
$280$		0			0
$281$		−	2.04504i		−	0.121997i	−0.998138	−	0.0609985i	$-0.980572\pi$
$281$		−	2.04504i		−	0.121997i	0.998138	−	0.0609985i	$-0.0194285\pi$
$282$		0			0
$283$	32.2520			1.91718			0.958592	−	0.284783i	$-0.0919215\pi$
$283$	32.2520			1.91718			0.958592	+	0.284783i	$0.0919215\pi$
$284$		0			0
$285$	−4.04177			−0.239414
$286$		0			0
$287$	5.19153			0.306446
$288$		0			0
$289$	−16.6100			−0.977060
$290$		0			0
$291$			6.13894i			0.359871i
$292$		0			0
$293$		−	1.90079i		−	0.111045i	−0.998457	−	0.0555225i	$-0.982318\pi$
$293$		−	1.90079i		−	0.111045i	0.998457	−	0.0555225i	$-0.0176825\pi$
$294$		0			0
$295$	−5.78077			−0.336569
$296$		0			0
$297$		−	23.9910i		−	1.39210i
$298$		0			0
$299$	9.47473	−	27.2268i	0.547938	−	1.57456i
$300$		0			0
$301$			26.1066i			1.50476i
$302$		0			0
$303$	4.30156			0.247118
$304$		0			0
$305$			2.58476i			0.148003i
$306$		0			0
$307$			2.39873i			0.136903i	0.997654	+	0.0684515i	$0.0218058\pi$
$307$			2.39873i			0.136903i	−0.997654	+	0.0684515i	$0.978194\pi$
$308$		0			0
$309$	14.5998			0.830556
$310$		0			0
$311$	1.12104			0.0635681			0.0317841	−	0.999495i	$-0.489881\pi$
$311$	1.12104			0.0635681			0.0317841	+	0.999495i	$0.489881\pi$
$312$		0			0
$313$	−15.2961			−0.864584			−0.432292	−	0.901734i	$-0.642295\pi$
$313$	−15.2961			−0.864584			−0.432292	+	0.901734i	$0.642295\pi$
$314$		0			0
$315$	8.33028			0.469358
$316$		0			0
$317$			28.0308i			1.57436i	0.616721	+	0.787182i	$0.288461\pi$
$317$			28.0308i			1.57436i	−0.616721	+	0.787182i	$0.711539\pi$
$318$		0			0
$319$		−	30.6860i		−	1.71809i
$320$		0			0
$321$	15.3227			0.855232
$322$		0			0
$323$		−	2.12997i		−	0.118515i
$324$		0			0
$325$	1.18501	−	3.40526i	0.0657323	−	0.188890i
$326$		0			0
$327$			0.521995i			0.0288664i
$328$		0			0
$329$	20.7311			1.14294
$330$		0			0
$331$			4.33477i			0.238260i	0.992879	+	0.119130i	$0.0380106\pi$
$331$			4.33477i			0.238260i	−0.992879	+	0.119130i	$0.961989\pi$
$332$		0			0
$333$			1.35678i			0.0743511i
$334$		0			0
$335$	−12.7818			−0.698344
$336$		0			0
$337$	−15.0075			−0.817513			−0.408756	−	0.912644i	$-0.634037\pi$
$337$	−15.0075			−0.817513			−0.408756	+	0.912644i	$0.634037\pi$
$338$		0			0
$339$	−6.89201			−0.374322
$340$		0			0
$341$	39.7783			2.15412
$342$		0			0
$343$			69.1736i			3.73502i
$344$		0			0
$345$		−	9.47473i		−	0.510102i
$346$		0			0
$347$	−16.0065			−0.859275			−0.429638	−	0.903001i	$-0.641359\pi$
$347$	−16.0065			−0.859275			−0.429638	+	0.903001i	$0.641359\pi$
$348$		0			0
$349$			22.4855i			1.20362i	0.798638	+	0.601812i	$0.205554\pi$
$349$			22.4855i			1.20362i	−0.798638	+	0.601812i	$0.794446\pi$
$350$		0			0
$351$	−6.45354	+	18.5450i	−0.344465	+	0.989860i
$352$		0			0
$353$		−	26.7598i		−	1.42428i	−0.702038	−	0.712140i	$-0.747726\pi$
$353$		−	26.7598i		−	1.42428i	0.702038	−	0.712140i	$-0.252274\pi$
$354$		0			0
$355$	−1.15628			−0.0613690
$356$		0			0
$357$		−	3.86309i		−	0.204457i
$358$		0			0
$359$			13.1563i			0.694362i	0.937798	+	0.347181i	$0.112861\pi$
$359$			13.1563i			0.694362i	−0.937798	+	0.347181i	$0.887139\pi$
$360$		0			0
$361$	7.36673			0.387723
$362$		0			0
$363$	9.96148			0.522842
$364$		0			0
$365$	−12.0308			−0.629719
$366$		0			0
$367$	0.258961			0.0135177			0.00675884	−	0.999977i	$-0.497849\pi$
$367$	0.258961			0.0135177			0.00675884	+	0.999977i	$0.497849\pi$
$368$		0			0
$369$		−	1.58698i		−	0.0826150i
$370$		0			0
$371$		−	5.19153i		−	0.269531i
$372$		0			0
$373$	−22.8830			−1.18484			−0.592419	−	0.805630i	$-0.701827\pi$
$373$	−22.8830			−1.18484			−0.592419	+	0.805630i	$0.701827\pi$
$374$		0			0
$375$		−	1.18501i		−	0.0611934i
$376$		0			0
$377$	−8.25448	+	23.7202i	−0.425127	+	1.22165i
$378$		0			0
$379$			12.0373i			0.618314i	0.951011	+	0.309157i	$0.100047\pi$
$379$			12.0373i			0.618314i	−0.951011	+	0.309157i	$0.899953\pi$
$380$		0			0
$381$	11.1882			0.573191
$382$		0			0
$383$		−	3.39183i		−	0.173315i	−0.996238	−	0.0866573i	$-0.972381\pi$
$383$		−	3.39183i		−	0.173315i	0.996238	−	0.0866573i	$-0.0276185\pi$
$384$		0			0
$385$			22.9965i			1.17201i
$386$		0			0
$387$	7.98044			0.405668
$388$		0			0
$389$	16.3610			0.829538			0.414769	−	0.909927i	$-0.363862\pi$
$389$	16.3610			0.829538			0.414769	+	0.909927i	$0.363862\pi$
$390$		0			0
$391$	4.99310			0.252512
$392$		0			0
$393$	−14.7290			−0.742981
$394$		0			0
$395$			7.18052i			0.361291i
$396$		0			0
$397$		−	14.3303i		−	0.719216i	−0.933103	−	0.359608i	$-0.882910\pi$
$397$		−	14.3303i		−	0.719216i	0.933103	−	0.359608i	$-0.117090\pi$
$398$		0			0
$399$	−21.0990			−1.05627
$400$		0			0
$401$		−	13.9890i		−	0.698577i	−0.937015	−	0.349289i	$-0.886423\pi$
$401$		−	13.9890i		−	0.698577i	0.937015	−	0.349289i	$-0.113577\pi$
$402$		0			0
$403$	−30.7486	−	10.7003i	−1.53170	−	0.533019i
$404$		0			0
$405$			1.66625i			0.0827968i
$406$		0			0
$407$	−3.74552			−0.185659
$408$		0			0
$409$		−	25.8755i		−	1.27946i	−0.768599	−	0.639731i	$-0.779046\pi$
$409$		−	25.8755i		−	1.27946i	0.768599	−	0.639731i	$-0.220954\pi$
$410$		0			0
$411$			11.3985i			0.562249i
$412$		0			0
$413$	−30.1771			−1.48492
$414$		0			0
$415$	5.14976			0.252792
$416$		0			0
$417$	−9.98900			−0.489163
$418$		0			0
$419$	−8.43846			−0.412246			−0.206123	−	0.978526i	$-0.566085\pi$
$419$	−8.43846			−0.412246			−0.206123	+	0.978526i	$0.566085\pi$
$420$		0			0
$421$		−	11.7005i		−	0.570246i	−0.958491	−	0.285123i	$-0.907965\pi$
$421$		−	11.7005i		−	0.570246i	0.958491	−	0.285123i	$-0.0920345\pi$
$422$		0			0
$423$		−	6.33721i		−	0.308126i
$424$		0			0
$425$	0.624487			0.0302921
$426$		0			0
$427$			13.4931i			0.652976i
$428$		0			0
$429$	−17.7763	−	6.18602i	−0.858247	−	0.298664i
$430$		0			0
$431$		−	2.97026i		−	0.143072i	−0.997438	−	0.0715361i	$-0.977210\pi$
$431$		−	2.97026i		−	0.143072i	0.997438	−	0.0715361i	$-0.0227901\pi$
$432$		0			0
$433$	−24.5020			−1.17749			−0.588746	−	0.808318i	$-0.700378\pi$
$433$	−24.5020			−1.17749			−0.588746	+	0.808318i	$0.700378\pi$
$434$		0			0
$435$			8.25448i			0.395772i
$436$		0			0
$437$		−	27.2708i		−	1.30454i
$438$		0			0
$439$	30.4535			1.45347			0.726734	−	0.686919i	$-0.241037\pi$
$439$	30.4535			1.45347			0.726734	+	0.686919i	$0.241037\pi$
$440$		0			0
$441$	32.3158			1.53885
$442$		0			0
$443$	7.49757			0.356220			0.178110	−	0.984011i	$-0.443002\pi$
$443$	7.49757			0.356220			0.178110	+	0.984011i	$0.443002\pi$
$444$		0			0
$445$	14.4405			0.684545
$446$		0			0
$447$			14.1389i			0.668749i
$448$		0			0
$449$		−	11.4570i		−	0.540690i	−0.962764	−	0.270345i	$-0.912862\pi$
$449$		−	11.4570i		−	0.540690i	0.962764	−	0.270345i	$-0.0871377\pi$
$450$		0			0
$451$	4.38102			0.206294
$452$		0			0
$453$		−	18.9077i		−	0.888361i
$454$		0			0
$455$	6.18602	−	17.7763i	0.290005	−	0.833365i
$456$		0			0
$457$			15.7005i			0.734437i	0.930135	+	0.367219i	$0.119690\pi$
$457$			15.7005i			0.734437i	−0.930135	+	0.367219i	$0.880310\pi$
$458$		0			0
$459$	−3.40096			−0.158743
$460$		0			0
$461$		−	27.0560i		−	1.26012i	−0.776545	−	0.630062i	$-0.783029\pi$
$461$		−	27.0560i		−	1.26012i	0.776545	−	0.630062i	$-0.216971\pi$
$462$		0			0
$463$		−	5.50877i		−	0.256014i	−0.991773	−	0.128007i	$-0.959142\pi$
$463$		−	5.50877i		−	0.256014i	0.991773	−	0.128007i	$-0.0408580\pi$
$464$		0			0
$465$	−10.7003			−0.496214
$466$		0			0
$467$	−6.73554			−0.311683			−0.155842	−	0.987782i	$-0.549809\pi$
$467$	−6.73554			−0.311683			−0.155842	+	0.987782i	$0.549809\pi$
$468$		0			0
$469$	−66.7241			−3.08103
$470$		0			0
$471$	18.5086			0.852830
$472$		0			0
$473$			22.0308i			1.01298i
$474$		0			0
$475$		−	3.41076i		−	0.156496i
$476$		0			0
$477$	−1.58698			−0.0726629
$478$		0			0
$479$		−	11.6057i		−	0.530280i	−0.964210	−	0.265140i	$-0.914582\pi$
$479$		−	11.6057i		−	0.530280i	0.964210	−	0.265140i	$-0.0854182\pi$
$480$		0			0
$481$	2.89528	+	1.00754i	0.132014	+	0.0459398i
$482$		0			0
$483$		−	49.4604i		−	2.25053i
$484$		0			0
$485$	−5.18052			−0.235235
$486$		0			0
$487$			24.6318i			1.11618i	0.829782	+	0.558088i	$0.188465\pi$
$487$			24.6318i			1.11618i	−0.829782	+	0.558088i	$0.811535\pi$
$488$		0			0
$489$		−	7.66606i		−	0.346671i
$490$		0			0
$491$	−12.3500			−0.557349			−0.278675	−	0.960386i	$-0.589895\pi$
$491$	−12.3500			−0.557349			−0.278675	+	0.960386i	$0.589895\pi$
$492$		0			0
$493$	−4.35004			−0.195916
$494$		0			0
$495$	7.02974			0.315963
$496$		0			0
$497$	−6.03607			−0.270755
$498$		0			0
$499$		−	14.2323i		−	0.637124i	−0.947902	−	0.318562i	$-0.896800\pi$
$499$		−	14.2323i		−	0.637124i	0.947902	−	0.318562i	$-0.103200\pi$
$500$		0			0
$501$			18.0385i			0.805900i
$502$		0			0
$503$	−11.2775			−0.502839			−0.251420	−	0.967878i	$-0.580897\pi$
$503$	−11.2775			−0.502839			−0.251420	+	0.967878i	$0.580897\pi$
$504$		0			0
$505$			3.62999i			0.161532i
$506$		0			0
$507$	12.0770	+	9.56357i	0.536359	+	0.424733i
$508$		0			0
$509$		−	17.4570i		−	0.773768i	−0.922128	−	0.386884i	$-0.873551\pi$
$509$		−	17.4570i		−	0.773768i	0.922128	−	0.386884i	$-0.126449\pi$
$510$		0			0
$511$	−62.8036			−2.77827
$512$		0			0
$513$			18.5750i			0.820106i
$514$		0			0
$515$			12.3205i			0.542905i
$516$		0			0
$517$	17.4945			0.769406
$518$		0			0
$519$	−23.0306			−1.01093
$520$		0			0
$521$	−2.00531			−0.0878544			−0.0439272	−	0.999035i	$-0.513987\pi$
$521$	−2.00531			−0.0878544			−0.0439272	+	0.999035i	$0.513987\pi$
$522$		0			0
$523$	17.1760			0.751056			0.375528	−	0.926811i	$-0.377461\pi$
$523$	17.1760			0.751056			0.375528	+	0.926811i	$0.377461\pi$
$524$		0			0
$525$		−	6.18602i		−	0.269980i
$526$		0			0
$527$		−	5.63896i		−	0.245637i
$528$		0			0
$529$	40.9283			1.77949
$530$		0			0
$531$			9.22473i			0.400319i
$532$		0			0
$533$	−3.38652	−	1.17848i	−0.146686	−	0.0510458i
$534$		0			0
$535$			12.9305i			0.559035i
$536$		0			0
$537$	−26.0725			−1.12511
$538$		0			0
$539$			89.2109i			3.84258i
$540$		0			0
$541$		−	16.6155i		−	0.714357i	−0.934036	−	0.357178i	$-0.883739\pi$
$541$		−	16.6155i		−	0.714357i	0.934036	−	0.357178i	$-0.116261\pi$
$542$		0			0
$543$	8.85759			0.380116
$544$		0			0
$545$	−0.440500			−0.0188689
$546$		0			0
$547$	−1.07151			−0.0458144			−0.0229072	−	0.999738i	$-0.507292\pi$
$547$	−1.07151			−0.0458144			−0.0229072	+	0.999738i	$0.507292\pi$
$548$		0			0
$549$	4.12466			0.176036
$550$		0			0
$551$			23.7586i			1.01215i
$552$		0			0
$553$			37.4841i			1.59399i
$554$		0			0
$555$	1.00754			0.0427676
$556$		0			0
$557$			27.4998i			1.16520i	0.812758	+	0.582602i	$0.197965\pi$
$557$			27.4998i			1.16520i	−0.812758	+	0.582602i	$0.802035\pi$
$558$		0			0
$559$	5.92623	−	17.0297i	0.250653	−	0.720281i
$560$		0			0
$561$		−	3.25998i		−	0.137636i
$562$		0			0
$563$	3.50794			0.147842			0.0739209	−	0.997264i	$-0.476449\pi$
$563$	3.50794			0.147842			0.0739209	+	0.997264i	$0.476449\pi$
$564$		0			0
$565$		−	5.81601i		−	0.244681i
$566$		0			0
$567$			8.69826i			0.365292i
$568$		0			0
$569$	14.5273			0.609016			0.304508	−	0.952510i	$-0.401508\pi$
$569$	14.5273			0.609016			0.304508	+	0.952510i	$0.401508\pi$
$570$		0			0
$571$	−12.6826			−0.530749			−0.265375	−	0.964145i	$-0.585496\pi$
$571$	−12.6826			−0.530749			−0.265375	+	0.964145i	$0.585496\pi$
$572$		0			0
$573$	9.48004			0.396034
$574$		0			0
$575$	7.99552			0.333436
$576$		0			0
$577$		−	16.2292i		−	0.675630i	−0.941213	−	0.337815i	$-0.890312\pi$
$577$		−	16.2292i		−	0.675630i	0.941213	−	0.337815i	$-0.109688\pi$
$578$		0			0
$579$			13.7816i			0.572744i
$580$		0			0
$581$	26.8830			1.11530
$582$		0			0
$583$		−	4.38102i		−	0.181443i
$584$		0			0
$585$	−5.43398	−	1.89099i	−0.224667	−	0.0781827i
$586$		0			0
$587$		−	34.8413i		−	1.43805i	−0.694983	−	0.719027i	$-0.744588\pi$
$587$		−	34.8413i		−	1.43805i	0.694983	−	0.719027i	$-0.255412\pi$
$588$		0			0
$589$	−30.7983			−1.26902
$590$		0			0
$591$			16.6736i			0.685860i
$592$		0			0
$593$			5.43360i			0.223131i	0.993757	+	0.111566i	$0.0355865\pi$
$593$			5.43360i			0.223131i	−0.993757	+	0.111566i	$0.964413\pi$
$594$		0			0
$595$	3.25998			0.133646
$596$		0			0
$597$	26.0725			1.06708
$598$		0			0
$599$	13.6080			0.556007			0.278003	−	0.960580i	$-0.410327\pi$
$599$	13.6080			0.556007			0.278003	+	0.960580i	$0.410327\pi$
$600$		0			0
$601$	1.46904			0.0599232			0.0299616	−	0.999551i	$-0.490461\pi$
$601$	1.46904			0.0599232			0.0299616	+	0.999551i	$0.490461\pi$
$602$		0			0
$603$			20.3967i			0.830618i
$604$		0			0
$605$			8.40627i			0.341764i
$606$		0			0
$607$	−13.2355			−0.537213			−0.268607	−	0.963250i	$-0.586563\pi$
$607$	−13.2355			−0.537213			−0.268607	+	0.963250i	$0.586563\pi$
$608$		0			0
$609$			43.0904i			1.74611i
$610$		0			0
$611$	−13.5232	−	4.70598i	−0.547090	−	0.190384i
$612$		0			0
$613$		−	5.63203i		−	0.227475i	−0.993511	−	0.113738i	$-0.963718\pi$
$613$		−	5.63203i		−	0.227475i	0.993511	−	0.113738i	$-0.0362823\pi$
$614$		0			0
$615$	−1.17848			−0.0475211
$616$		0			0
$617$			23.1251i			0.930982i	0.885053	+	0.465491i	$0.154122\pi$
$617$			23.1251i			0.930982i	−0.885053	+	0.465491i	$0.845878\pi$
$618$		0			0
$619$			22.7323i			0.913687i	0.889547	+	0.456843i	$0.151020\pi$
$619$			22.7323i			0.913687i	−0.889547	+	0.456843i	$0.848980\pi$
$620$		0			0
$621$	−43.5436			−1.74734
$622$		0			0
$623$	75.3830			3.02016
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−17.8050			−0.711064
$628$		0			0
$629$			0.530964i			0.0211709i
$630$		0			0
$631$		−	31.7938i		−	1.26569i	−0.774278	−	0.632846i	$-0.781887\pi$
$631$		−	31.7938i		−	1.26569i	0.774278	−	0.632846i	$-0.218113\pi$
$632$		0			0
$633$	6.67157			0.265171
$634$		0			0
$635$			9.44152i			0.374675i
$636$		0			0
$637$	23.9976	−	68.9599i	0.950818	−	2.73229i
$638$		0			0
$639$			1.84515i			0.0729930i
$640$		0			0
$641$	−38.3720			−1.51560			−0.757802	−	0.652484i	$-0.773727\pi$
$641$	−38.3720			−1.51560			−0.757802	+	0.652484i	$0.773727\pi$
$642$		0			0
$643$		−	23.0723i		−	0.909884i	−0.890521	−	0.454942i	$-0.849660\pi$
$643$		−	23.0723i		−	0.909884i	0.890521	−	0.454942i	$-0.150340\pi$
$644$		0			0
$645$		−	5.92623i		−	0.233345i
$646$		0			0
$647$	38.9126			1.52981			0.764906	−	0.644142i	$-0.222785\pi$
$647$	38.9126			1.52981			0.764906	+	0.644142i	$0.222785\pi$
$648$		0			0
$649$	−25.4658			−0.999618
$650$		0			0
$651$	−55.8582			−2.18926
$652$		0			0
$653$	−32.0526			−1.25431			−0.627157	−	0.778893i	$-0.715781\pi$
$653$	−32.0526			−1.25431			−0.627157	+	0.778893i	$0.715781\pi$
$654$		0			0
$655$		−	12.4295i		−	0.485661i
$656$		0			0
$657$			19.1982i			0.748995i
$658$		0			0
$659$	10.6805			0.416055			0.208027	−	0.978123i	$-0.433296\pi$
$659$	10.6805			0.416055			0.208027	+	0.978123i	$0.433296\pi$
$660$		0			0
$661$		−	7.27098i		−	0.282809i	−0.989952	−	0.141404i	$-0.954838\pi$
$661$		−	7.27098i		−	0.282809i	0.989952	−	0.141404i	$-0.0451617\pi$
$662$		0			0
$663$	−0.876928	+	2.51996i	−0.0340571	+	0.0978671i
$664$		0			0
$665$		−	17.8050i		−	0.690449i
$666$		0			0
$667$	−55.6949			−2.15652
$668$		0			0
$669$			17.6811i			0.683592i
$670$		0			0
$671$			11.3865i			0.439572i
$672$		0			0
$673$	−16.3140			−0.628857			−0.314429	−	0.949281i	$-0.601813\pi$
$673$	−16.3140			−0.628857			−0.314429	+	0.949281i	$0.601813\pi$
$674$		0			0
$675$	−5.44600			−0.209617
$676$		0			0
$677$	38.6990			1.48733			0.743663	−	0.668555i	$-0.233087\pi$
$677$	38.6990			1.48733			0.743663	+	0.668555i	$0.233087\pi$
$678$		0			0
$679$	−27.0436			−1.03784
$680$		0			0
$681$			7.48837i			0.286955i
$682$		0			0
$683$		−	16.8172i		−	0.643493i	−0.946826	−	0.321747i	$-0.895730\pi$
$683$		−	16.8172i		−	0.643493i	0.946826	−	0.321747i	$-0.104270\pi$
$684$		0			0
$685$	−9.61898			−0.367523
$686$		0			0
$687$		−	25.4006i		−	0.969093i
$688$		0			0
$689$	−1.17848	+	3.38652i	−0.0448967	+	0.129016i
$690$		0			0
$691$		−	29.2288i		−	1.11192i	−0.831210	−	0.555958i	$-0.812351\pi$
$691$		−	29.2288i		−	1.11192i	0.831210	−	0.555958i	$-0.187649\pi$
$692$		0			0
$693$	36.6970			1.39400
$694$		0			0
$695$		−	8.42950i		−	0.319749i
$696$		0			0
$697$		−	0.621051i		−	0.0235240i
$698$		0			0
$699$	−29.4030			−1.11212
$700$		0			0
$701$	−24.3700			−0.920443			−0.460221	−	0.887804i	$-0.652230\pi$
$701$	−24.3700			−0.920443			−0.460221	+	0.887804i	$0.652230\pi$
$702$		0			0
$703$	2.89996			0.109374
$704$		0			0
$705$	−4.70598			−0.177238
$706$		0			0
$707$			18.9495i			0.712668i
$708$		0			0
$709$		−	46.2585i		−	1.73728i	−0.495447	−	0.868638i	$-0.664996\pi$
$709$		−	46.2585i		−	1.73728i	0.495447	−	0.868638i	$-0.335004\pi$
$710$		0			0
$711$	11.4584			0.429724
$712$		0			0
$713$		−	72.1974i		−	2.70382i
$714$		0			0
$715$	5.22025	−	15.0010i	0.195226	−	0.561006i
$716$		0			0
$717$			19.8124i			0.739906i
$718$		0			0
$719$	20.0615			0.748168			0.374084	−	0.927395i	$-0.377957\pi$
$719$	20.0615			0.748168			0.374084	+	0.927395i	$0.377957\pi$
$720$		0			0
$721$			64.3160i			2.39525i
$722$		0			0
$723$			14.4535i			0.537533i
$724$		0			0
$725$	−6.96577			−0.258702
$726$		0			0
$727$	−45.9976			−1.70595			−0.852977	−	0.521948i	$-0.825206\pi$
$727$	−45.9976			−1.70595			−0.852977	+	0.521948i	$0.825206\pi$
$728$		0			0
$729$	22.0196			0.815540
$730$		0			0
$731$	3.12307			0.115511
$732$		0			0
$733$			12.2005i			0.450634i	0.974285	+	0.225317i	$0.0723418\pi$
$733$			12.2005i			0.450634i	−0.974285	+	0.225317i	$0.927658\pi$
$734$		0			0
$735$		−	23.9976i		−	0.885163i
$736$		0			0
$737$	−56.3070			−2.07410
$738$		0			0
$739$		−	31.4578i		−	1.15720i	−0.815613	−	0.578598i	$-0.803600\pi$
$739$		−	31.4578i		−	1.15720i	0.815613	−	0.578598i	$-0.196400\pi$
$740$		0			0
$741$	13.7632	+	4.78951i	0.505605	+	0.175947i
$742$		0			0
$743$			28.4493i			1.04370i	0.853037	+	0.521851i	$0.174758\pi$
$743$			28.4493i			1.04370i	−0.853037	+	0.521851i	$0.825242\pi$
$744$		0			0
$745$	−11.9315			−0.437138
$746$		0			0
$747$		−	8.21780i		−	0.300673i
$748$		0			0
$749$			67.5006i			2.46642i
$750$		0			0
$751$	−46.2946			−1.68931			−0.844657	−	0.535307i	$-0.820196\pi$
$751$	−46.2946			−1.68931			−0.844657	+	0.535307i	$0.820196\pi$
$752$		0			0
$753$	4.65649			0.169692
$754$		0			0
$755$	15.9558			0.580691
$756$		0			0
$757$	−32.6840			−1.18792			−0.593960	−	0.804495i	$-0.702436\pi$
$757$	−32.6840			−1.18792			−0.593960	+	0.804495i	$0.702436\pi$
$758$		0			0
$759$		−	41.7386i		−	1.51501i
$760$		0			0
$761$		−	45.4515i		−	1.64761i	−0.566870	−	0.823807i	$-0.691846\pi$
$761$		−	45.4515i		−	1.64761i	0.566870	−	0.823807i	$-0.308154\pi$
$762$		0			0
$763$	−2.29952			−0.0832482
$764$		0			0
$765$		−	0.996533i		−	0.0360297i
$766$		0			0
$767$	19.6850	+	6.85024i	0.710784	+	0.247348i
$768$		0			0
$769$		−	7.73349i		−	0.278877i	−0.990231	−	0.139438i	$-0.955470\pi$
$769$		−	7.73349i		−	0.278877i	0.990231	−	0.139438i	$-0.0445297\pi$
$770$		0			0
$771$	9.93396			0.357763
$772$		0			0
$773$			39.6013i			1.42436i	0.701998	+	0.712179i	$0.252292\pi$
$773$			39.6013i			1.42436i	−0.701998	+	0.712179i	$0.747708\pi$
$774$		0			0
$775$		−	9.02974i		−	0.324358i
$776$		0			0
$777$	5.25960			0.188687
$778$		0			0
$779$	−3.39199			−0.121531
$780$		0			0
$781$	−5.09371			−0.182267
$782$		0			0
$783$	37.9356			1.35571
$784$		0			0
$785$			15.6190i			0.557465i
$786$		0			0
$787$			25.0129i			0.891612i	0.895130	+	0.445806i	$0.147083\pi$
$787$			25.0129i			0.891612i	−0.895130	+	0.445806i	$0.852917\pi$
$788$		0			0
$789$	25.3248			0.901586
$790$		0			0
$791$		−	30.3610i		−	1.07951i
$792$		0			0
$793$	3.06295	−	8.80176i	0.108769	−	0.312560i
$794$		0			0
$795$			1.17848i			0.0417965i
$796$		0			0
$797$	46.8036			1.65787			0.828934	−	0.559347i	$-0.188948\pi$
$797$	46.8036			1.65787			0.828934	+	0.559347i	$0.188948\pi$
$798$		0			0
$799$		−	2.48001i		−	0.0877365i
$800$		0			0
$801$		−	23.0436i		−	0.814206i
$802$		0			0
$803$	−52.9986			−1.87028
$804$		0			0
$805$	41.7386			1.47109
$806$		0			0
$807$	14.8256			0.521885
$808$		0			0
$809$	47.7453			1.67864			0.839318	−	0.543641i	$-0.182955\pi$
$809$	47.7453			1.67864			0.839318	+	0.543641i	$0.182955\pi$
$810$		0			0
$811$			6.79728i			0.238685i	0.992853	+	0.119342i	$0.0380786\pi$
$811$			6.79728i			0.238685i	−0.992853	+	0.119342i	$0.961921\pi$
$812$		0			0
$813$		−	0.918694i		−	0.0322200i
$814$		0			0
$815$	6.46922			0.226607
$816$		0			0
$817$		−	17.0573i		−	0.596758i
$818$		0			0
$819$	−28.3667	−	9.87143i	−0.991214	−	0.344936i
$820$		0			0
$821$			6.28852i			0.219471i	0.993961	+	0.109735i	$0.0350003\pi$
$821$			6.28852i			0.219471i	−0.993961	+	0.109735i	$0.965000\pi$
$822$		0			0
$823$	49.5250			1.72633			0.863167	−	0.504918i	$-0.168477\pi$
$823$	49.5250			1.72633			0.863167	+	0.504918i	$0.168477\pi$
$824$		0			0
$825$		−	5.22025i		−	0.181746i
$826$		0			0
$827$		−	21.2907i		−	0.740352i	−0.928962	−	0.370176i	$-0.879297\pi$
$827$		−	21.2907i		−	0.740352i	0.928962	−	0.370176i	$-0.120703\pi$
$828$		0			0
$829$	26.3317			0.914537			0.457269	−	0.889329i	$-0.348828\pi$
$829$	26.3317			0.914537			0.457269	+	0.889329i	$0.348828\pi$
$830$		0			0
$831$	−0.575372			−0.0199594
$832$		0			0
$833$	12.6465			0.438175
$834$		0			0
$835$	−15.2223			−0.526789
$836$		0			0
$837$			49.1760i			1.69977i
$838$		0			0
$839$		−	24.7519i		−	0.854529i	−0.904127	−	0.427264i	$-0.859477\pi$
$839$		−	24.7519i		−	0.854529i	0.904127	−	0.427264i	$-0.140523\pi$
$840$		0			0
$841$	19.5220			0.673172
$842$		0			0
$843$			2.42339i			0.0834659i
$844$		0			0
$845$	−8.07049	+	10.1915i	−0.277633	+	0.350599i
$846$		0			0
$847$			43.8828i			1.50783i
$848$		0			0
$849$	−38.2188			−1.31167
$850$		0			0
$851$			6.79811i			0.233036i
$852$		0			0
$853$			34.3344i			1.17559i	0.809011	+	0.587793i	$0.200003\pi$
$853$			34.3344i			1.17559i	−0.809011	+	0.587793i	$0.799997\pi$
$854$		0			0
$855$	−5.44276			−0.186138
$856$		0			0
$857$	8.58148			0.293138			0.146569	−	0.989200i	$-0.453177\pi$
$857$	8.58148			0.293138			0.146569	+	0.989200i	$0.453177\pi$
$858$		0			0
$859$	33.7620			1.15194			0.575972	−	0.817469i	$-0.304624\pi$
$859$	33.7620			1.15194			0.575972	+	0.817469i	$0.304624\pi$
$860$		0			0
$861$	−6.15198			−0.209659
$862$		0			0
$863$			1.58822i			0.0540638i	0.999635	+	0.0270319i	$0.00860557\pi$
$863$			1.58822i			0.0540638i	−0.999635	+	0.0270319i	$0.991394\pi$
$864$		0			0
$865$		−	19.4350i		−	0.660810i
$866$		0			0
$867$	19.6830			0.668468
$868$		0			0
$869$			31.6320i			1.07304i
$870$		0			0
$871$	43.5252	+	15.1465i	1.47480	+	0.513219i
$872$		0			0
$873$			8.26688i			0.279792i
$874$		0			0
$875$	5.22025			0.176477
$876$		0			0
$877$			38.7441i			1.30830i	0.756367	+	0.654148i	$0.226972\pi$
$877$			38.7441i			1.30830i	−0.756367	+	0.654148i	$0.773028\pi$
$878$		0			0
$879$			2.25244i			0.0759729i
$880$		0			0
$881$	5.86675			0.197656			0.0988279	−	0.995105i	$-0.468491\pi$
$881$	5.86675			0.197656			0.0988279	+	0.995105i	$0.468491\pi$
$882$		0			0
$883$	−15.9505			−0.536776			−0.268388	−	0.963311i	$-0.586491\pi$
$883$	−15.9505			−0.536776			−0.268388	+	0.963311i	$0.586491\pi$
$884$		0			0
$885$	6.85024			0.230268
$886$		0			0
$887$	−22.5495			−0.757139			−0.378569	−	0.925573i	$-0.623584\pi$
$887$	−22.5495			−0.757139			−0.378569	+	0.925573i	$0.623584\pi$
$888$		0			0
$889$			49.2871i			1.65304i
$890$		0			0
$891$			7.34027i			0.245908i
$892$		0			0
$893$	−13.5451			−0.453268
$894$		0			0
$895$		−	22.0020i		−	0.735447i
$896$		0			0
$897$	−11.2276	+	32.2639i	−0.374879	+	1.07726i
$898$		0			0
$899$			62.8991i			2.09780i
$900$		0			0
$901$	−0.621051			−0.0206902
$902$		0			0
$903$		−	30.9364i		−	1.02950i
$904$		0			0
$905$			7.47473i			0.248468i
$906$		0			0
$907$	22.7486			0.755354			0.377677	−	0.925937i	$-0.376723\pi$
$907$	22.7486			0.755354			0.377677	+	0.925937i	$0.376723\pi$
$908$		0			0
$909$	5.79260			0.192129
$910$		0			0
$911$	23.2551			0.770475			0.385238	−	0.922817i	$-0.374119\pi$
$911$	23.2551			0.770475			0.385238	+	0.922817i	$0.374119\pi$
$912$		0			0
$913$	22.6860			0.750797
$914$		0			0
$915$		−	3.06295i		−	0.101258i
$916$		0			0
$917$		−	64.8851i		−	2.14269i
$918$		0			0
$919$	17.4711			0.576317			0.288159	−	0.957583i	$-0.406957\pi$
$919$	17.4711			0.576317			0.288159	+	0.957583i	$0.406957\pi$
$920$		0			0
$921$		−	2.84251i		−	0.0936640i
$922$		0			0
$923$	3.93743	+	1.37020i	0.129602	+	0.0451006i
$924$		0			0
$925$			0.850240i			0.0279557i
$926$		0			0
$927$	19.6606			0.645738
$928$		0			0
$929$		−	23.2366i		−	0.762367i	−0.924499	−	0.381184i	$-0.875517\pi$
$929$		−	23.2366i		−	0.762367i	0.924499	−	0.381184i	$-0.124483\pi$
$930$		0			0
$931$		−	69.0713i		−	2.26372i
$932$		0			0
$933$	−1.32843			−0.0434910
$934$		0			0
$935$	2.75103			0.0899682
$936$		0			0
$937$	26.0395			0.850674			0.425337	−	0.905035i	$-0.360156\pi$
$937$	26.0395			0.850674			0.425337	+	0.905035i	$0.360156\pi$
$938$		0			0
$939$	18.1259			0.591516
$940$		0			0
$941$		−	41.7891i		−	1.36229i	−0.732150	−	0.681143i	$-0.761483\pi$
$941$		−	41.7891i		−	1.36229i	0.732150	−	0.681143i	$-0.238517\pi$
$942$		0			0
$943$		−	7.95152i		−	0.258937i
$944$		0			0
$945$	−28.4295			−0.924812
$946$		0			0
$947$			41.7202i			1.35573i	0.735188	+	0.677863i	$0.237094\pi$
$947$			41.7202i			1.35573i	−0.735188	+	0.677863i	$0.762906\pi$
$948$		0			0
$949$	40.9678	+	14.2565i	1.32987	+	0.462786i
$950$		0			0
$951$		−	33.2166i		−	1.07712i
$952$		0			0
$953$	54.4336			1.76328			0.881638	−	0.471926i	$-0.156441\pi$
$953$	54.4336			1.76328			0.881638	+	0.471926i	$0.156441\pi$
$954$		0			0
$955$			8.00000i			0.258874i
$956$		0			0
$957$			36.3631i			1.17545i
$958$		0			0
$959$	−50.2135			−1.62148
$960$		0			0
$961$	−50.5362			−1.63020
$962$		0			0
$963$	20.6341			0.664923
$964$		0			0
$965$	−11.6300			−0.374383
$966$		0			0
$967$		−	28.1033i		−	0.903741i	−0.892084	−	0.451870i	$-0.850757\pi$
$967$		−	28.1033i		−	0.903741i	0.892084	−	0.451870i	$-0.149243\pi$
$968$		0			0
$969$			2.52403i			0.0810836i
$970$		0			0
$971$	−6.88304			−0.220887			−0.110444	−	0.993882i	$-0.535227\pi$
$971$	−6.88304			−0.220887			−0.110444	+	0.993882i	$0.535227\pi$
$972$		0			0
$973$		−	44.0041i		−	1.41071i
$974$		0			0
$975$	−1.40424	+	4.03524i	−0.0449716	+	0.129231i
$976$		0			0
$977$		−	36.8543i		−	1.17907i	−0.807741	−	0.589537i	$-0.799310\pi$
$977$		−	36.8543i		−	1.17907i	0.807741	−	0.589537i	$-0.200690\pi$
$978$		0			0
$979$	63.6141			2.03312
$980$		0			0
$981$			0.702934i			0.0224429i
$982$		0			0
$983$			39.7223i			1.26694i	0.773766	+	0.633472i	$0.218371\pi$
$983$			39.7223i			1.26694i	−0.773766	+	0.633472i	$0.781629\pi$
$984$		0			0
$985$	−14.0705			−0.448323
$986$		0			0
$987$	−24.5664			−0.781957
$988$		0			0
$989$	39.9857			1.27147
$990$		0			0
$991$	48.9515			1.55500			0.777498	−	0.628886i	$-0.216489\pi$
$991$	48.9515			1.55500			0.777498	+	0.628886i	$0.216489\pi$
$992$		0			0
$993$		−	5.13672i		−	0.163009i
$994$		0			0
$995$			22.0020i			0.697511i
$996$		0			0
$997$	1.42399			0.0450983			0.0225491	−	0.999746i	$-0.492822\pi$
$997$	1.42399			0.0450983			0.0225491	+	0.999746i	$0.492822\pi$
$998$		0			0
$999$		−	4.63041i		−	0.146500i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.k.e.961.3		8
4.3	odd	2		520.2.k.b.441.5	✓	8
12.11	even	2		4680.2.g.k.2521.8		8
13.12	even	2	inner	1040.2.k.e.961.4		8
20.3	even	4		2600.2.f.f.649.6		8
20.7	even	4		2600.2.f.e.649.3		8
20.19	odd	2		2600.2.k.c.2001.3		8
52.31	even	4		6760.2.a.bc.1.3		4
52.47	even	4		6760.2.a.bd.1.3		4
52.51	odd	2		520.2.k.b.441.6	yes	8
156.155	even	2		4680.2.g.k.2521.1		8
260.103	even	4		2600.2.f.e.649.6		8
260.207	even	4		2600.2.f.f.649.3		8
260.259	odd	2		2600.2.k.c.2001.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.b.441.5	✓	8	4.3	odd	2
520.2.k.b.441.6	yes	8	52.51	odd	2
1040.2.k.e.961.3		8	1.1	even	1	trivial
1040.2.k.e.961.4		8	13.12	even	2	inner
2600.2.f.e.649.3		8	20.7	even	4
2600.2.f.e.649.6		8	260.103	even	4
2600.2.f.f.649.3		8	260.207	even	4
2600.2.f.f.649.6		8	20.3	even	4
2600.2.k.c.2001.3		8	20.19	odd	2
2600.2.k.c.2001.4		8	260.259	odd	2
4680.2.g.k.2521.1		8	156.155	even	2
4680.2.g.k.2521.8		8	12.11	even	2
6760.2.a.bc.1.3		4	52.31	even	4
6760.2.a.bd.1.3		4	52.47	even	4

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.18501 q^{3} -1.00000i q^{5} -5.22025i q^{7} -1.59576 q^{9} +O(q^{10})\)	\(q-1.18501 q^{3} -1.00000i q^{5} -5.22025i q^{7} -1.59576 q^{9} -4.40526i q^{11} +(-1.18501 + 3.40526i) q^{13} +1.18501i q^{15} -0.624487 q^{17} +3.41076i q^{19} +6.18602i q^{21} -7.99552 q^{23} -1.00000 q^{25} +5.44600 q^{27} +6.96577 q^{29} +9.02974i q^{31} +5.22025i q^{33} -5.22025 q^{35} -0.850240i q^{37} +(1.40424 - 4.03524i) q^{39} +0.994498i q^{41} -5.00102 q^{43} +1.59576i q^{45} +3.97128i q^{47} -20.2510 q^{49} +0.740021 q^{51} +0.994498 q^{53} -4.40526 q^{55} -4.04177i q^{57} -5.78077i q^{59} -2.58476 q^{61} +8.33028i q^{63} +(3.40526 + 1.18501i) q^{65} -12.7818i q^{67} +9.47473 q^{69} -1.15628i q^{71} -12.0308i q^{73} +1.18501 q^{75} -22.9965 q^{77} -7.18052 q^{79} -1.66625 q^{81} +5.14976i q^{83} +0.624487i q^{85} -8.25448 q^{87} +14.4405i q^{89} +(17.7763 + 6.18602i) q^{91} -10.7003i q^{93} +3.41076 q^{95} -5.18052i q^{97} +7.02974i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 16 * q^9	\( 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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