LMFDB - Embedded newform 2070.3.c.b.91.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (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:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.13
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.37049i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.37049i q^{7} -2.82843 q^{8} -3.16228i q^{10} -12.6432i q^{11} -17.1953 q^{13} -1.93817i q^{14} +4.00000 q^{16} -17.1987i q^{17} +5.73865i q^{19} +4.47214i q^{20} +17.8802i q^{22} +(-22.4818 - 4.85478i) q^{23} -5.00000 q^{25} +24.3178 q^{26} +2.74098i q^{28} -19.0719 q^{29} +27.6139 q^{31} -5.65685 q^{32} +24.3226i q^{34} -3.06451 q^{35} +28.5685i q^{37} -8.11568i q^{38} -6.32456i q^{40} +40.8110 q^{41} +33.6014i q^{43} -25.2864i q^{44} +(31.7941 + 6.86570i) q^{46} +36.0593 q^{47} +47.1218 q^{49} +7.07107 q^{50} -34.3906 q^{52} -72.1605i q^{53} +28.2711 q^{55} -3.87633i q^{56} +26.9718 q^{58} -33.0861 q^{59} +102.589i q^{61} -39.0519 q^{62} +8.00000 q^{64} -38.4498i q^{65} +83.7272i q^{67} -34.3974i q^{68} +4.33387 q^{70} +53.3503 q^{71} -6.09068 q^{73} -40.4019i q^{74} +11.4773i q^{76} +17.3274 q^{77} -16.0407i q^{79} +8.94427i q^{80} -57.7155 q^{82} +119.789i q^{83} +38.4574 q^{85} -47.5196i q^{86} +35.7604i q^{88} +71.6620i q^{89} -23.5660i q^{91} +(-44.9636 - 9.70956i) q^{92} -50.9955 q^{94} -12.8320 q^{95} -47.4489i q^{97} -66.6402 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4}+O(q^{10}) $ 32 * q + 64 * q^4	$ 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			1.37049i			0.195784i	0.995197	+	0.0978922i	$0.0312100\pi$
$7$			1.37049i			0.195784i	−0.995197	+	0.0978922i	$0.968790\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$		−	12.6432i		−	1.14938i	−0.818370	−	0.574692i	$-0.805122\pi$
$11$		−	12.6432i		−	1.14938i	0.818370	−	0.574692i	$-0.194878\pi$
$12$		0			0
$13$	−17.1953			−1.32271			−0.661357	−	0.750071i	$-0.730019\pi$
$13$	−17.1953			−1.32271			−0.661357	+	0.750071i	$0.730019\pi$
$14$		−	1.93817i		−	0.138440i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	17.1987i		−	1.01169i	−0.862625	−	0.505844i	$-0.831181\pi$
$17$		−	17.1987i		−	1.01169i	0.862625	−	0.505844i	$-0.168819\pi$
$18$		0			0
$19$			5.73865i			0.302034i	0.988531	+	0.151017i	$0.0482548\pi$
$19$			5.73865i			0.302034i	−0.988531	+	0.151017i	$0.951745\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$			17.8802i			0.812737i
$23$	−22.4818	−	4.85478i	−0.977469	−	0.211077i
$23$	−22.4818	−	4.85478i	−0.977469	−	0.211077i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	24.3178			0.935301
$27$		0			0
$28$			2.74098i			0.0978922i
$29$	−19.0719			−0.657653			−0.328826	−	0.944390i	$-0.606653\pi$
$29$	−19.0719			−0.657653			−0.328826	+	0.944390i	$0.606653\pi$
$30$		0			0
$31$	27.6139			0.890770			0.445385	−	0.895339i	$-0.353067\pi$
$31$	27.6139			0.890770			0.445385	+	0.895339i	$0.353067\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			24.3226i			0.715371i
$35$	−3.06451			−0.0875574
$36$		0			0
$37$			28.5685i			0.772120i	0.922474	+	0.386060i	$0.126164\pi$
$37$			28.5685i			0.772120i	−0.922474	+	0.386060i	$0.873836\pi$
$38$		−	8.11568i		−	0.213570i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	40.8110			0.995391			0.497695	−	0.867352i	$-0.334180\pi$
$41$	40.8110			0.995391			0.497695	+	0.867352i	$0.334180\pi$
$42$		0			0
$43$			33.6014i			0.781428i	0.920512	+	0.390714i	$0.127772\pi$
$43$			33.6014i			0.781428i	−0.920512	+	0.390714i	$0.872228\pi$
$44$		−	25.2864i		−	0.574692i
$45$		0			0
$46$	31.7941	+	6.86570i	0.691175	+	0.149254i
$47$	36.0593			0.767218			0.383609	−	0.923496i	$-0.374681\pi$
$47$	36.0593			0.767218			0.383609	+	0.923496i	$0.374681\pi$
$48$		0			0
$49$	47.1218			0.961668
$50$	7.07107			0.141421
$51$		0			0
$52$	−34.3906			−0.661357
$53$		−	72.1605i		−	1.36152i	−0.732507	−	0.680760i	$-0.761650\pi$
$53$		−	72.1605i		−	1.36152i	0.732507	−	0.680760i	$-0.238350\pi$
$54$		0			0
$55$	28.2711			0.514020
$56$		−	3.87633i		−	0.0692202i
$57$		0			0
$58$	26.9718			0.465031
$59$	−33.0861			−0.560781			−0.280391	−	0.959886i	$-0.590464\pi$
$59$	−33.0861			−0.560781			−0.280391	+	0.959886i	$0.590464\pi$
$60$		0			0
$61$			102.589i			1.68179i	0.541196	+	0.840896i	$0.317972\pi$
$61$			102.589i			1.68179i	−0.541196	+	0.840896i	$0.682028\pi$
$62$	−39.0519			−0.629869
$63$		0			0
$64$	8.00000			0.125000
$65$		−	38.4498i		−	0.591536i
$66$		0			0
$67$			83.7272i			1.24966i	0.780761	+	0.624830i	$0.214832\pi$
$67$			83.7272i			1.24966i	−0.780761	+	0.624830i	$0.785168\pi$
$68$		−	34.3974i		−	0.505844i
$69$		0			0
$70$	4.33387			0.0619125
$71$	53.3503			0.751413			0.375706	−	0.926739i	$-0.377400\pi$
$71$	53.3503			0.751413			0.375706	+	0.926739i	$0.377400\pi$
$72$		0			0
$73$	−6.09068			−0.0834340			−0.0417170	−	0.999129i	$-0.513283\pi$
$73$	−6.09068			−0.0834340			−0.0417170	+	0.999129i	$0.513283\pi$
$74$		−	40.4019i		−	0.545971i
$75$		0			0
$76$			11.4773i			0.151017i
$77$	17.3274			0.225031
$78$		0			0
$79$		−	16.0407i		−	0.203047i	−0.994833	−	0.101524i	$-0.967628\pi$
$79$		−	16.0407i		−	0.203047i	0.994833	−	0.101524i	$-0.0323718\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	−57.7155			−0.703848
$83$			119.789i			1.44325i	0.692287	+	0.721623i	$0.256603\pi$
$83$			119.789i			1.44325i	−0.692287	+	0.721623i	$0.743397\pi$
$84$		0			0
$85$	38.4574			0.452440
$86$		−	47.5196i		−	0.552553i
$87$		0			0
$88$			35.7604i			0.406369i
$89$			71.6620i			0.805191i	0.915378	+	0.402596i	$0.131892\pi$
$89$			71.6620i			0.805191i	−0.915378	+	0.402596i	$0.868108\pi$
$90$		0			0
$91$		−	23.5660i		−	0.258967i
$92$	−44.9636	−	9.70956i	−0.488735	−	0.105539i
$93$		0			0
$94$	−50.9955			−0.542505
$95$	−12.8320			−0.135074
$96$		0			0
$97$		−	47.4489i		−	0.489164i	−0.969629	−	0.244582i	$-0.921349\pi$
$97$		−	47.4489i		−	0.489164i	0.969629	−	0.244582i	$-0.0786507\pi$
$98$	−66.6402			−0.680002
$99$		0			0
$100$	−10.0000			−0.100000
$101$	25.2315			0.249817			0.124908	−	0.992168i	$-0.460136\pi$
$101$	25.2315			0.249817			0.124908	+	0.992168i	$0.460136\pi$
$102$		0			0
$103$		−	102.361i		−	0.993800i	−0.867808	−	0.496900i	$-0.834471\pi$
$103$		−	102.361i		−	0.993800i	0.867808	−	0.496900i	$-0.165529\pi$
$104$	48.6356			0.467650
$105$		0			0
$106$			102.050i			0.962740i
$107$			99.6919i			0.931700i	0.884864	+	0.465850i	$0.154251\pi$
$107$			99.6919i			0.931700i	−0.884864	+	0.465850i	$0.845749\pi$
$108$		0			0
$109$			70.6544i			0.648205i	0.946022	+	0.324103i	$0.105062\pi$
$109$			70.6544i			0.648205i	−0.946022	+	0.324103i	$0.894938\pi$
$110$	−39.9814			−0.363467
$111$		0			0
$112$			5.48196i			0.0489461i
$113$		−	117.729i		−	1.04185i	−0.853603	−	0.520924i	$-0.825587\pi$
$113$		−	117.729i		−	1.04185i	0.853603	−	0.520924i	$-0.174413\pi$
$114$		0			0
$115$	10.8556	−	50.2708i	0.0943967	−	0.437138i
$116$	−38.1438			−0.328826
$117$		0			0
$118$	46.7908			0.396532
$119$	23.5706			0.198073
$120$		0			0
$121$	−38.8511			−0.321084
$122$		−	145.083i		−	1.18921i
$123$		0			0
$124$	55.2277			0.445385
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−210.520			−1.65764			−0.828819	−	0.559516i	$-0.810987\pi$
$127$	−210.520			−1.65764			−0.828819	+	0.559516i	$0.810987\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$			54.3763i			0.418279i
$131$	−179.091			−1.36711			−0.683553	−	0.729901i	$-0.739566\pi$
$131$	−179.091			−1.36711			−0.683553	+	0.729901i	$0.739566\pi$
$132$		0			0
$133$	−7.86477			−0.0591336
$134$		−	118.408i		−	0.883643i
$135$		0			0
$136$			48.6452i			0.357685i
$137$			68.6261i			0.500921i	0.968127	+	0.250460i	$0.0805820\pi$
$137$			68.6261i			0.500921i	−0.968127	+	0.250460i	$0.919418\pi$
$138$		0			0
$139$	35.6149			0.256222			0.128111	−	0.991760i	$-0.459109\pi$
$139$	35.6149			0.256222			0.128111	+	0.991760i	$0.459109\pi$
$140$	−6.12902			−0.0437787
$141$		0			0
$142$	−75.4487			−0.531329
$143$			217.404i			1.52031i
$144$		0			0
$145$		−	42.6461i		−	0.294111i
$146$	8.61352			0.0589967
$147$		0			0
$148$			57.1369i			0.386060i
$149$			169.122i			1.13505i	0.823356	+	0.567525i	$0.192099\pi$
$149$			169.122i			1.13505i	−0.823356	+	0.567525i	$0.807901\pi$
$150$		0			0
$151$	51.9046			0.343739			0.171870	−	0.985120i	$-0.445019\pi$
$151$	51.9046			0.343739			0.171870	+	0.985120i	$0.445019\pi$
$152$		−	16.2314i		−	0.106785i
$153$		0			0
$154$	−24.5047			−0.159121
$155$			61.7465i			0.398364i
$156$		0			0
$157$			103.070i			0.656500i	0.944591	+	0.328250i	$0.106459\pi$
$157$			103.070i			0.656500i	−0.944591	+	0.328250i	$0.893541\pi$
$158$			22.6850i			0.143576i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	6.65343	−	30.8111i	0.0413257	−	0.191373i
$162$		0			0
$163$	56.7509			0.348165			0.174083	−	0.984731i	$-0.444304\pi$
$163$	56.7509			0.348165			0.174083	+	0.984731i	$0.444304\pi$
$164$	81.6220			0.497695
$165$		0			0
$166$		−	169.408i		−	1.02053i
$167$	309.988			1.85622			0.928108	−	0.372310i	$-0.121434\pi$
$167$	309.988			1.85622			0.928108	+	0.372310i	$0.121434\pi$
$168$		0			0
$169$	126.678			0.749575
$170$	−54.3870			−0.319924
$171$		0			0
$172$			67.2028i			0.390714i
$173$	−301.774			−1.74436			−0.872179	−	0.489186i	$-0.837294\pi$
$173$	−301.774			−1.74436			−0.872179	+	0.489186i	$0.837294\pi$
$174$		0			0
$175$		−	6.85245i		−	0.0391569i
$176$		−	50.5729i		−	0.287346i
$177$		0			0
$178$		−	101.345i		−	0.569356i
$179$	66.8168			0.373278			0.186639	−	0.982429i	$-0.440240\pi$
$179$	66.8168			0.373278			0.186639	+	0.982429i	$0.440240\pi$
$180$		0			0
$181$		−	140.236i		−	0.774785i	−0.921915	−	0.387392i	$-0.873376\pi$
$181$		−	140.236i		−	0.774785i	0.921915	−	0.387392i	$-0.126624\pi$
$182$			33.3273i			0.183117i
$183$		0			0
$184$	63.5881	+	13.7314i	0.345588	+	0.0746271i
$185$	−63.8810			−0.345303
$186$		0			0
$187$	−217.447			−1.16282
$188$	72.1185			0.383609
$189$		0			0
$190$	18.1472			0.0955116
$191$			345.209i			1.80738i	0.428190	+	0.903689i	$0.359151\pi$
$191$			345.209i			1.80738i	−0.428190	+	0.903689i	$0.640849\pi$
$192$		0			0
$193$	186.257			0.965062			0.482531	−	0.875879i	$-0.339718\pi$
$193$	186.257			0.965062			0.482531	+	0.875879i	$0.339718\pi$
$194$			67.1028i			0.345891i
$195$		0			0
$196$	94.2435			0.480834
$197$	279.423			1.41839			0.709196	−	0.705012i	$-0.249058\pi$
$197$	279.423			1.41839			0.709196	+	0.705012i	$0.249058\pi$
$198$		0			0
$199$		−	45.7328i		−	0.229813i	−0.993376	−	0.114907i	$-0.963343\pi$
$199$		−	45.7328i		−	0.229813i	0.993376	−	0.114907i	$-0.0366569\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	−35.6827			−0.176647
$203$		−	26.1379i		−	0.128758i
$204$		0			0
$205$			91.2562i			0.445152i
$206$			144.761i			0.702723i
$207$		0			0
$208$	−68.7812			−0.330679
$209$	72.5550			0.347153
$210$		0			0
$211$	307.575			1.45770			0.728851	−	0.684673i	$-0.240055\pi$
$211$	307.575			1.45770			0.728851	+	0.684673i	$0.240055\pi$
$212$		−	144.321i		−	0.680760i
$213$		0			0
$214$		−	140.986i		−	0.658811i
$215$	−75.1351			−0.349465
$216$		0			0
$217$			37.8445i			0.174399i
$218$		−	99.9204i		−	0.458350i
$219$		0			0
$220$	56.5422			0.257010
$221$			295.736i			1.33817i
$222$		0			0
$223$	−356.509			−1.59870			−0.799348	−	0.600869i	$-0.794821\pi$
$223$	−356.509			−1.59870			−0.799348	+	0.600869i	$0.794821\pi$
$224$		−	7.75267i		−	0.0346101i
$225$		0			0
$226$			166.494i			0.736698i
$227$			167.195i			0.736542i	0.929719	+	0.368271i	$0.120050\pi$
$227$			167.195i			0.736542i	−0.929719	+	0.368271i	$0.879950\pi$
$228$		0			0
$229$			281.250i			1.22817i	0.789242	+	0.614083i	$0.210474\pi$
$229$			281.250i			1.22817i	−0.789242	+	0.614083i	$0.789526\pi$
$230$	−15.3522	+	71.0937i	−0.0667485	+	0.309103i
$231$		0			0
$232$	53.9435			0.232515
$233$	−60.2115			−0.258419			−0.129209	−	0.991617i	$-0.541244\pi$
$233$	−60.2115			−0.258419			−0.129209	+	0.991617i	$0.541244\pi$
$234$		0			0
$235$			80.6309i			0.343110i
$236$	−66.1722			−0.280391
$237$		0			0
$238$	−33.3339			−0.140058
$239$	−44.8445			−0.187634			−0.0938170	−	0.995589i	$-0.529907\pi$
$239$	−44.8445			−0.187634			−0.0938170	+	0.995589i	$0.529907\pi$
$240$		0			0
$241$			0.284271i			0.00117955i	1.00000		0.000589774i	$0.000187731\pi$
$241$			0.284271i			0.00117955i	−1.00000		0.000589774i	$0.999812\pi$
$242$	54.9438			0.227040
$243$		0			0
$244$			205.179i			0.840896i
$245$			105.367i			0.430071i
$246$		0			0
$247$		−	98.6778i		−	0.399505i
$248$	−78.1038			−0.314935
$249$		0			0
$250$			15.8114i			0.0632456i
$251$		−	219.635i		−	0.875041i	−0.899208	−	0.437521i	$-0.855857\pi$
$251$		−	219.635i		−	0.875041i	0.899208	−	0.437521i	$-0.144143\pi$
$252$		0			0
$253$	−61.3801	+	284.242i	−0.242609	+	1.12349i
$254$	297.720			1.17213
$255$		0			0
$256$	16.0000			0.0625000
$257$	83.7888			0.326026			0.163013	−	0.986624i	$-0.447879\pi$
$257$	83.7888			0.326026			0.163013	+	0.986624i	$0.447879\pi$
$258$		0			0
$259$	−39.1528			−0.151169
$260$		−	76.8997i		−	0.295768i
$261$		0			0
$262$	253.273			0.966690
$263$			52.1022i			0.198107i	0.995082	+	0.0990536i	$0.0315815\pi$
$263$			52.1022i			0.198107i	−0.995082	+	0.0990536i	$0.968418\pi$
$264$		0			0
$265$	161.356			0.608890
$266$	11.1225			0.0418138
$267$		0			0
$268$			167.454i			0.624830i
$269$	317.806			1.18143			0.590717	−	0.806879i	$-0.298845\pi$
$269$	317.806			1.18143			0.590717	+	0.806879i	$0.298845\pi$
$270$		0			0
$271$	−249.647			−0.921207			−0.460604	−	0.887606i	$-0.652367\pi$
$271$	−249.647			−0.921207			−0.460604	+	0.887606i	$0.652367\pi$
$272$		−	68.7947i		−	0.252922i
$273$		0			0
$274$		−	97.0520i		−	0.354204i
$275$			63.2161i			0.229877i
$276$		0			0
$277$	−35.6857			−0.128829			−0.0644146	−	0.997923i	$-0.520518\pi$
$277$	−35.6857			−0.128829			−0.0644146	+	0.997923i	$0.520518\pi$
$278$	−50.3670			−0.181176
$279$		0			0
$280$	8.66774			0.0309562
$281$		−	237.163i		−	0.843996i	−0.906597	−	0.421998i	$-0.861329\pi$
$281$		−	237.163i		−	0.843996i	0.906597	−	0.421998i	$-0.138671\pi$
$282$		0			0
$283$			284.190i			1.00420i	0.864809	+	0.502102i	$0.167440\pi$
$283$			284.190i			1.00420i	−0.864809	+	0.502102i	$0.832560\pi$
$284$	106.701			0.375706
$285$		0			0
$286$		−	307.456i		−	1.07502i
$287$			55.9311i			0.194882i
$288$		0			0
$289$	−6.79461			−0.0235108
$290$			60.3107i			0.207968i
$291$		0			0
$292$	−12.1814			−0.0417170
$293$			551.696i			1.88292i	0.337124	+	0.941460i	$0.390546\pi$
$293$			551.696i			1.88292i	−0.337124	+	0.941460i	$0.609454\pi$
$294$		0			0
$295$		−	73.9828i		−	0.250789i
$296$		−	80.8038i		−	0.272986i
$297$		0			0
$298$		−	239.175i		−	0.802601i
$299$	386.581	+	83.4794i	1.29291	+	0.279195i
$300$		0			0
$301$	−46.0504			−0.152991
$302$	−73.4042			−0.243060
$303$		0			0
$304$			22.9546i			0.0755085i
$305$	−229.397			−0.752120
$306$		0			0
$307$	96.3675			0.313901			0.156950	−	0.987606i	$-0.449834\pi$
$307$	96.3675			0.313901			0.156950	+	0.987606i	$0.449834\pi$
$308$	34.6548			0.112516
$309$		0			0
$310$		−	87.3227i		−	0.281686i
$311$	−423.308			−1.36112			−0.680560	−	0.732693i	$-0.738263\pi$
$311$	−423.308			−1.36112			−0.680560	+	0.732693i	$0.738263\pi$
$312$		0			0
$313$		−	58.3745i		−	0.186500i	−0.995643	−	0.0932500i	$-0.970274\pi$
$313$		−	58.3745i		−	0.186500i	0.995643	−	0.0932500i	$-0.0297256\pi$
$314$		−	145.764i		−	0.464215i
$315$		0			0
$316$		−	32.0815i		−	0.101524i
$317$	−60.9213			−0.192181			−0.0960904	−	0.995373i	$-0.530634\pi$
$317$	−60.9213			−0.192181			−0.0960904	+	0.995373i	$0.530634\pi$
$318$		0			0
$319$			241.131i			0.755895i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−9.40937	+	43.5735i	−0.0292217	+	0.135321i
$323$	98.6972			0.305564
$324$		0			0
$325$	85.9765			0.264543
$326$	−80.2579			−0.246190
$327$		0			0
$328$	−115.431			−0.351924
$329$			49.4189i			0.150209i
$330$		0			0
$331$	245.050			0.740332			0.370166	−	0.928966i	$-0.379301\pi$
$331$	245.050			0.740332			0.370166	+	0.928966i	$0.379301\pi$
$332$			239.579i			0.721623i
$333$		0			0
$334$	−438.390			−1.31254
$335$	−187.220			−0.558865
$336$		0			0
$337$			570.626i			1.69325i	0.532188	+	0.846626i	$0.321370\pi$
$337$			570.626i			1.69325i	−0.532188	+	0.846626i	$0.678630\pi$
$338$	−179.150			−0.530029
$339$		0			0
$340$	76.9148			0.226220
$341$		−	349.128i		−	1.02384i
$342$		0			0
$343$			131.734i			0.384064i
$344$		−	95.0392i		−	0.276277i
$345$		0			0
$346$	426.773			1.23345
$347$	−72.5341			−0.209032			−0.104516	−	0.994523i	$-0.533329\pi$
$347$	−72.5341			−0.209032			−0.104516	+	0.994523i	$0.533329\pi$
$348$		0			0
$349$	436.542			1.25084			0.625418	−	0.780290i	$-0.284928\pi$
$349$	436.542			1.25084			0.625418	+	0.780290i	$0.284928\pi$
$350$			9.69083i			0.0276881i
$351$		0			0
$352$			71.5209i			0.203184i
$353$	33.3967			0.0946081			0.0473040	−	0.998881i	$-0.484937\pi$
$353$	33.3967			0.0946081			0.0473040	+	0.998881i	$0.484937\pi$
$354$		0			0
$355$			119.295i			0.336042i
$356$			143.324i			0.402596i
$357$		0			0
$358$	−94.4932			−0.263948
$359$			348.492i			0.970730i	0.874312	+	0.485365i	$0.161313\pi$
$359$			348.492i			0.970730i	−0.874312	+	0.485365i	$0.838687\pi$
$360$		0			0
$361$	328.068			0.908775
$362$			198.324i			0.547855i
$363$		0			0
$364$		−	47.1320i		−	0.129483i
$365$		−	13.6192i		−	0.0373128i
$366$		0			0
$367$			42.8645i			0.116797i	0.998293	+	0.0583985i	$0.0185994\pi$
$367$			42.8645i			0.116797i	−0.998293	+	0.0583985i	$0.981401\pi$
$368$	−89.9272	−	19.4191i	−0.244367	−	0.0527694i
$369$		0			0
$370$	90.3414			0.244166
$371$	98.8954			0.266564
$372$		0			0
$373$			548.173i			1.46963i	0.678266	+	0.734817i	$0.262732\pi$
$373$			548.173i			1.46963i	−0.678266	+	0.734817i	$0.737268\pi$
$374$	307.516			0.822236
$375$		0			0
$376$	−101.991			−0.271253
$377$	327.947			0.869887
$378$		0			0
$379$		−	575.949i		−	1.51966i	−0.650125	−	0.759828i	$-0.725283\pi$
$379$		−	575.949i		−	1.51966i	0.650125	−	0.759828i	$-0.274717\pi$
$380$	−25.6640			−0.0675369
$381$		0			0
$382$		−	488.199i		−	1.27801i
$383$			485.343i			1.26721i	0.773655	+	0.633607i	$0.218426\pi$
$383$			485.343i			1.26721i	−0.773655	+	0.633607i	$0.781574\pi$
$384$		0			0
$385$			38.7453i			0.100637i
$386$	−263.407			−0.682402
$387$		0			0
$388$		−	94.8977i		−	0.244582i
$389$		−	34.6469i		−	0.0890666i	−0.999008	−	0.0445333i	$-0.985820\pi$
$389$		−	34.6469i		−	0.0890666i	0.999008	−	0.0445333i	$-0.0141801\pi$
$390$		0			0
$391$	−83.4958	+	386.657i	−0.213544	+	0.988893i
$392$	−133.280			−0.340001
$393$		0			0
$394$	−395.164			−1.00295
$395$	35.8682			0.0908055
$396$		0			0
$397$	−576.948			−1.45327			−0.726635	−	0.687024i	$-0.758917\pi$
$397$	−576.948			−1.45327			−0.726635	+	0.687024i	$0.758917\pi$
$398$			64.6760i			0.162502i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			627.475i			1.56477i	0.622792	+	0.782387i	$0.285998\pi$
$401$			627.475i			1.56477i	−0.622792	+	0.782387i	$0.714002\pi$
$402$		0			0
$403$	−474.828			−1.17823
$404$	50.4630			0.124908
$405$		0			0
$406$			36.9646i			0.0910457i
$407$	361.197			0.887463
$408$		0			0
$409$	22.7131			0.0555333			0.0277666	−	0.999614i	$-0.491160\pi$
$409$	22.7131			0.0555333			0.0277666	+	0.999614i	$0.491160\pi$
$410$		−	129.056i		−	0.314770i
$411$		0			0
$412$		−	204.723i		−	0.496900i
$413$		−	45.3442i		−	0.109792i
$414$		0			0
$415$	−267.857			−0.645439
$416$	97.2713			0.233825
$417$		0			0
$418$	−102.608			−0.245474
$419$		−	212.213i		−	0.506474i	−0.967404	−	0.253237i	$-0.918505\pi$
$419$		−	212.213i		−	0.506474i	0.967404	−	0.253237i	$-0.0814952\pi$
$420$		0			0
$421$			592.278i			1.40684i	0.710776	+	0.703419i	$0.248344\pi$
$421$			592.278i			1.40684i	−0.710776	+	0.703419i	$0.751656\pi$
$422$	−434.977			−1.03075
$423$		0			0
$424$			204.101i			0.481370i
$425$			85.9934i			0.202337i
$426$		0			0
$427$	−140.598			−0.329269
$428$			199.384i			0.465850i
$429$		0			0
$430$	106.257			0.247109
$431$			333.504i			0.773791i	0.922124	+	0.386895i	$0.126453\pi$
$431$			333.504i			0.773791i	−0.922124	+	0.386895i	$0.873547\pi$
$432$		0			0
$433$			61.7200i			0.142540i	0.997457	+	0.0712702i	$0.0227053\pi$
$433$			61.7200i			0.142540i	−0.997457	+	0.0712702i	$0.977295\pi$
$434$		−	53.5203i		−	0.123319i
$435$		0			0
$436$			141.309i			0.324103i
$437$	27.8599	−	129.015i	0.0637526	−	0.295229i
$438$		0			0
$439$	858.490			1.95556			0.977779	−	0.209640i	$-0.0672293\pi$
$439$	858.490			1.95556			0.977779	+	0.209640i	$0.0672293\pi$
$440$	−79.9628			−0.181734
$441$		0			0
$442$		−	418.234i		−	0.946232i
$443$	−707.452			−1.59696			−0.798479	−	0.602023i	$-0.794362\pi$
$443$	−707.452			−1.59696			−0.798479	+	0.602023i	$0.794362\pi$
$444$		0			0
$445$	−160.241			−0.360092
$446$	504.180			1.13045
$447$		0			0
$448$			10.9639i			0.0244730i
$449$	27.9221			0.0621873			0.0310936	−	0.999516i	$-0.490101\pi$
$449$	27.9221			0.0621873			0.0310936	+	0.999516i	$0.490101\pi$
$450$		0			0
$451$		−	515.983i		−	1.14409i
$452$		−	235.458i		−	0.520924i
$453$		0			0
$454$		−	236.449i		−	0.520813i
$455$	52.6952			0.115814
$456$		0			0
$457$			391.345i			0.856335i	0.903699	+	0.428167i	$0.140841\pi$
$457$			391.345i			0.856335i	−0.903699	+	0.428167i	$0.859159\pi$
$458$		−	397.748i		−	0.868444i
$459$		0			0
$460$	21.7112	−	100.542i	0.0471983	−	0.218569i
$461$	−358.584			−0.777841			−0.388920	−	0.921271i	$-0.627152\pi$
$461$	−358.584			−0.777841			−0.388920	+	0.921271i	$0.627152\pi$
$462$		0			0
$463$	−49.7419			−0.107434			−0.0537169	−	0.998556i	$-0.517107\pi$
$463$	−49.7419			−0.107434			−0.0537169	+	0.998556i	$0.517107\pi$
$464$	−76.2877			−0.164413
$465$		0			0
$466$	85.1519			0.182729
$467$			7.85966i			0.0168301i	0.999965	+	0.00841505i	$0.00267863\pi$
$467$			7.85966i			0.0168301i	−0.999965	+	0.00841505i	$0.997321\pi$
$468$		0			0
$469$	−114.747			−0.244664
$470$		−	114.029i		−	0.242616i
$471$		0			0
$472$	93.5816			0.198266
$473$	424.830			0.898161
$474$		0			0
$475$		−	28.6932i		−	0.0604068i
$476$	47.1413			0.0990363
$477$		0			0
$478$	63.4197			0.132677
$479$		−	567.601i		−	1.18497i	−0.805581	−	0.592486i	$-0.798147\pi$
$479$		−	567.601i		−	1.18497i	0.805581	−	0.592486i	$-0.201853\pi$
$480$		0			0
$481$		−	491.243i		−	1.02129i
$482$		−	0.402020i		−	0.000834067i
$483$		0			0
$484$	−77.7022			−0.160542
$485$	106.099			0.218761
$486$		0			0
$487$	374.905			0.769825			0.384913	−	0.922953i	$-0.374232\pi$
$487$	374.905			0.769825			0.384913	+	0.922953i	$0.374232\pi$
$488$		−	290.166i		−	0.594603i
$489$		0			0
$490$		−	149.012i		−	0.304106i
$491$	412.426			0.839971			0.419985	−	0.907531i	$-0.362035\pi$
$491$	412.426			0.839971			0.419985	+	0.907531i	$0.362035\pi$
$492$		0			0
$493$			328.012i			0.665339i
$494$			139.551i			0.282493i
$495$		0			0
$496$	110.455			0.222692
$497$			73.1161i			0.147115i
$498$		0			0
$499$	327.529			0.656370			0.328185	−	0.944613i	$-0.393563\pi$
$499$	327.529			0.656370			0.328185	+	0.944613i	$0.393563\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$			310.611i			0.618748i
$503$		−	636.770i		−	1.26595i	−0.774174	−	0.632973i	$-0.781835\pi$
$503$		−	636.770i		−	1.26595i	0.774174	−	0.632973i	$-0.218165\pi$
$504$		0			0
$505$			56.4193i			0.111721i
$506$	86.8045	−	401.979i	0.171550	−	0.794426i
$507$		0			0
$508$	−421.040			−0.828819
$509$	−125.717			−0.246988			−0.123494	−	0.992345i	$-0.539410\pi$
$509$	−125.717			−0.246988			−0.123494	+	0.992345i	$0.539410\pi$
$510$		0			0
$511$		−	8.34722i		−	0.0163351i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−118.495			−0.230535
$515$	228.887			0.444441
$516$		0			0
$517$		−	455.905i		−	0.881828i
$518$	55.3704			0.106893
$519$		0			0
$520$			108.753i			0.209140i
$521$		−	668.477i		−	1.28306i	−0.767096	−	0.641532i	$-0.778299\pi$
$521$		−	668.477i		−	1.28306i	0.767096	−	0.641532i	$-0.221701\pi$
$522$		0			0
$523$		−	421.745i		−	0.806396i	−0.915113	−	0.403198i	$-0.867899\pi$
$523$		−	421.745i		−	0.806396i	0.915113	−	0.403198i	$-0.132101\pi$
$524$	−358.182			−0.683553
$525$		0			0
$526$		−	73.6837i		−	0.140083i
$527$		−	474.922i		−	0.901180i
$528$		0			0
$529$	481.862	+	218.288i	0.910893	+	0.412643i
$530$	−228.192			−0.430550
$531$		0			0
$532$	−15.7295			−0.0295668
$533$	−701.757			−1.31662
$534$		0			0
$535$	−222.918			−0.416669
$536$		−	236.816i		−	0.441821i
$537$		0			0
$538$	−449.445			−0.835400
$539$		−	595.771i		−	1.10533i
$540$		0			0
$541$	769.598			1.42255			0.711273	−	0.702916i	$-0.248119\pi$
$541$	769.598			1.42255			0.711273	+	0.702916i	$0.248119\pi$
$542$	353.054			0.651392
$543$		0			0
$544$			97.2904i			0.178843i
$545$	−157.988			−0.289886
$546$		0			0
$547$	406.464			0.743079			0.371539	−	0.928417i	$-0.378830\pi$
$547$	406.464			0.743079			0.371539	+	0.928417i	$0.378830\pi$
$548$			137.252i			0.250460i
$549$		0			0
$550$		−	89.4011i		−	0.162547i
$551$		−	109.447i		−	0.198634i
$552$		0			0
$553$	21.9837			0.0397535
$554$	50.4672			0.0910960
$555$		0			0
$556$	71.2297			0.128111
$557$		−	753.052i		−	1.35198i	−0.736912	−	0.675989i	$-0.763717\pi$
$557$		−	753.052i		−	1.35198i	0.736912	−	0.675989i	$-0.236283\pi$
$558$		0			0
$559$		−	577.786i		−	1.03361i
$560$	−12.2580			−0.0218894
$561$		0			0
$562$			335.399i			0.596796i
$563$		−	79.5409i		−	0.141280i	−0.997502	−	0.0706402i	$-0.977496\pi$
$563$		−	79.5409i		−	0.141280i	0.997502	−	0.0706402i	$-0.0225042\pi$
$564$		0			0
$565$	263.250			0.465929
$566$		−	401.905i		−	0.710079i
$567$		0			0
$568$	−150.897			−0.265665
$569$		−	210.524i		−	0.369989i	−0.982740	−	0.184994i	$-0.940773\pi$
$569$		−	210.524i		−	0.369989i	0.982740	−	0.184994i	$-0.0592267\pi$
$570$		0			0
$571$			243.033i			0.425626i	0.977093	+	0.212813i	$0.0682626\pi$
$571$			243.033i			0.425626i	−0.977093	+	0.212813i	$0.931737\pi$
$572$			434.808i			0.760154i
$573$		0			0
$574$		−	79.0986i		−	0.137802i
$575$	112.409	+	24.2739i	0.195494	+	0.0422155i
$576$		0			0
$577$	−343.119			−0.594661			−0.297330	−	0.954775i	$-0.596096\pi$
$577$	−343.119			−0.594661			−0.297330	+	0.954775i	$0.596096\pi$
$578$	9.60903			0.0166246
$579$		0			0
$580$		−	85.2922i		−	0.147056i
$581$	−164.170			−0.282565
$582$		0			0
$583$	−912.342			−1.56491
$584$	17.2270			0.0294984
$585$		0			0
$586$		−	780.216i		−	1.33143i
$587$	664.216			1.13154			0.565772	−	0.824562i	$-0.308578\pi$
$587$	664.216			1.13154			0.565772	+	0.824562i	$0.308578\pi$
$588$		0			0
$589$			158.466i			0.269043i
$590$			104.627i			0.177335i
$591$		0			0
$592$			114.274i			0.193030i
$593$	−614.904			−1.03694			−0.518469	−	0.855097i	$-0.673498\pi$
$593$	−614.904			−1.03694			−0.518469	+	0.855097i	$0.673498\pi$
$594$		0			0
$595$			52.7055i			0.0885807i
$596$			338.245i			0.567525i
$597$		0			0
$598$	−546.708	−	118.058i	−0.914228	−	0.197421i
$599$	−606.007			−1.01170			−0.505849	−	0.862622i	$-0.668821\pi$
$599$	−606.007			−1.01170			−0.505849	+	0.862622i	$0.668821\pi$
$600$		0			0
$601$	−281.807			−0.468896			−0.234448	−	0.972129i	$-0.575328\pi$
$601$	−281.807			−0.468896			−0.234448	+	0.972129i	$0.575328\pi$
$602$	65.1252			0.108181
$603$		0			0
$604$	103.809			0.171870
$605$		−	86.8737i		−	0.143593i
$606$		0			0
$607$	2.23403			0.00368044			0.00184022	−	0.999998i	$-0.499414\pi$
$607$	2.23403			0.00368044			0.00184022	+	0.999998i	$0.499414\pi$
$608$		−	32.4627i		−	0.0533926i
$609$		0			0
$610$	324.416			0.531829
$611$	−620.049			−1.01481
$612$		0			0
$613$		−	24.5222i		−	0.0400035i	−0.999800	−	0.0200018i	$-0.993633\pi$
$613$		−	24.5222i		−	0.0400035i	0.999800	−	0.0200018i	$-0.00636719\pi$
$614$	−136.284			−0.221961
$615$		0			0
$616$	−49.0093			−0.0795606
$617$		−	1084.85i		−	1.75827i	−0.476570	−	0.879137i	$-0.658120\pi$
$617$		−	1084.85i		−	1.75827i	0.476570	−	0.879137i	$-0.341880\pi$
$618$		0			0
$619$		−	817.729i		−	1.32105i	−0.750804	−	0.660525i	$-0.770334\pi$
$619$		−	817.729i		−	1.32105i	0.750804	−	0.660525i	$-0.229666\pi$
$620$			123.493i			0.199182i
$621$		0			0
$622$	598.648			0.962457
$623$	−98.2121			−0.157644
$624$		0			0
$625$	25.0000			0.0400000
$626$			82.5540i			0.131875i
$627$		0			0
$628$			206.141i			0.328250i
$629$	491.340			0.781144
$630$		0			0
$631$		−	393.978i		−	0.624371i	−0.950021	−	0.312185i	$-0.898939\pi$
$631$		−	393.978i		−	0.624371i	0.950021	−	0.312185i	$-0.101061\pi$
$632$			45.3701i			0.0717881i
$633$		0			0
$634$	86.1558			0.135892
$635$		−	470.737i		−	0.741319i
$636$		0			0
$637$	−810.272			−1.27201
$638$		−	341.010i		−	0.534499i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$			803.224i			1.25308i	0.779389	+	0.626540i	$0.215529\pi$
$641$			803.224i			1.25308i	−0.779389	+	0.626540i	$0.784471\pi$
$642$		0			0
$643$		−	590.715i		−	0.918686i	−0.888259	−	0.459343i	$-0.848085\pi$
$643$		−	590.715i		−	0.918686i	0.888259	−	0.459343i	$-0.151915\pi$
$644$	13.3069	−	61.6222i	0.0206628	−	0.0956866i
$645$		0			0
$646$	−139.579			−0.216066
$647$	−328.095			−0.507102			−0.253551	−	0.967322i	$-0.581599\pi$
$647$	−328.095			−0.507102			−0.253551	+	0.967322i	$0.581599\pi$
$648$		0			0
$649$			418.315i			0.644553i
$650$	−121.589			−0.187060
$651$		0			0
$652$	113.502			0.174083
$653$	−291.306			−0.446104			−0.223052	−	0.974806i	$-0.571602\pi$
$653$	−291.306			−0.446104			−0.223052	+	0.974806i	$0.571602\pi$
$654$		0			0
$655$		−	400.459i		−	0.611388i
$656$	163.244			0.248848
$657$		0			0
$658$		−	69.8888i		−	0.106214i
$659$		−	500.076i		−	0.758841i	−0.925224	−	0.379420i	$-0.876123\pi$
$659$		−	500.076i		−	0.758841i	0.925224	−	0.379420i	$-0.123877\pi$
$660$		0			0
$661$			803.159i			1.21507i	0.794294	+	0.607533i	$0.207841\pi$
$661$			803.159i			1.21507i	−0.794294	+	0.607533i	$0.792159\pi$
$662$	−346.553			−0.523494
$663$		0			0
$664$		−	338.815i		−	0.510264i
$665$		−	17.5862i		−	0.0264453i
$666$		0			0
$667$	428.771	+	92.5900i	0.642835	+	0.138816i
$668$	619.976			0.928108
$669$		0			0
$670$	264.769			0.395177
$671$	1297.06			1.93303
$672$		0			0
$673$	−908.196			−1.34947			−0.674737	−	0.738058i	$-0.735743\pi$
$673$	−908.196			−1.34947			−0.674737	+	0.738058i	$0.735743\pi$
$674$		−	806.987i		−	1.19731i
$675$		0			0
$676$	253.356			0.374787
$677$			968.574i			1.43069i	0.698774	+	0.715343i	$0.253729\pi$
$677$			968.574i			1.43069i	−0.698774	+	0.715343i	$0.746271\pi$
$678$		0			0
$679$	65.0282			0.0957706
$680$	−108.774			−0.159962
$681$		0			0
$682$			493.742i			0.723962i
$683$	−455.566			−0.667008			−0.333504	−	0.942749i	$-0.608231\pi$
$683$	−455.566			−0.667008			−0.333504	+	0.942749i	$0.608231\pi$
$684$		0			0
$685$	−153.453			−0.224019
$686$		−	186.300i		−	0.271574i
$687$		0			0
$688$			134.406i			0.195357i
$689$			1240.82i			1.80090i
$690$		0			0
$691$	47.1105			0.0681772			0.0340886	−	0.999419i	$-0.489147\pi$
$691$	47.1105			0.0681772			0.0340886	+	0.999419i	$0.489147\pi$
$692$	−603.548			−0.872179
$693$		0			0
$694$	102.579			0.147808
$695$			79.6373i			0.114586i
$696$		0			0
$697$		−	701.896i		−	1.00702i
$698$	−617.364			−0.884475
$699$		0			0
$700$		−	13.7049i		−	0.0195784i
$701$		−	364.053i		−	0.519334i	−0.965698	−	0.259667i	$-0.916387\pi$
$701$		−	364.053i		−	0.519334i	0.965698	−	0.259667i	$-0.0836127\pi$
$702$		0			0
$703$	−163.944			−0.233207
$704$		−	101.146i		−	0.143673i
$705$		0			0
$706$	−47.2300			−0.0668980
$707$			34.5795i			0.0489102i
$708$		0			0
$709$			853.680i			1.20406i	0.798472	+	0.602031i	$0.205642\pi$
$709$			853.680i			1.20406i	−0.798472	+	0.602031i	$0.794358\pi$
$710$		−	168.709i		−	0.237618i
$711$		0			0
$712$		−	202.691i		−	0.284678i
$713$	−620.809	−	134.059i	−0.870700	−	0.188021i
$714$		0			0
$715$	−486.130			−0.679902
$716$	133.634			0.186639
$717$		0			0
$718$		−	492.842i		−	0.686410i
$719$	−825.637			−1.14831			−0.574157	−	0.818745i	$-0.694670\pi$
$719$	−825.637			−1.14831			−0.574157	+	0.818745i	$0.694670\pi$
$720$		0			0
$721$	140.285			0.194571
$722$	−463.958			−0.642601
$723$		0			0
$724$		−	280.472i		−	0.387392i
$725$	95.3596			0.131531
$726$		0			0
$727$		−	1157.96i		−	1.59279i	−0.604778	−	0.796394i	$-0.706738\pi$
$727$		−	1157.96i		−	1.59279i	0.604778	−	0.796394i	$-0.293262\pi$
$728$			66.6547i			0.0915586i
$729$		0			0
$730$			19.2604i			0.0263841i
$731$	577.900			0.790561
$732$		0			0
$733$		−	937.923i		−	1.27957i	−0.768555	−	0.639784i	$-0.779024\pi$
$733$		−	937.923i		−	1.27957i	0.768555	−	0.639784i	$-0.220976\pi$
$734$		−	60.6195i		−	0.0825879i
$735$		0			0
$736$	127.176	+	27.4628i	0.172794	+	0.0373136i
$737$	1058.58			1.43634
$738$		0			0
$739$	276.870			0.374655			0.187327	−	0.982298i	$-0.440017\pi$
$739$	276.870			0.374655			0.187327	+	0.982298i	$0.440017\pi$
$740$	−127.762			−0.172651
$741$		0			0
$742$	−139.859			−0.188489
$743$		−	953.464i		−	1.28326i	−0.767013	−	0.641631i	$-0.778258\pi$
$743$		−	953.464i		−	1.28326i	0.767013	−	0.641631i	$-0.221742\pi$
$744$		0			0
$745$	−378.169			−0.507610
$746$		−	775.234i		−	1.03919i
$747$		0			0
$748$	−434.894			−0.581408
$749$	−136.627			−0.182412
$750$		0			0
$751$			101.682i			0.135396i	0.997706	+	0.0676980i	$0.0215654\pi$
$751$			101.682i			0.135396i	−0.997706	+	0.0676980i	$0.978435\pi$
$752$	144.237			0.191805
$753$		0			0
$754$	−463.788			−0.615103
$755$			116.062i			0.153725i
$756$		0			0
$757$		−	979.734i		−	1.29423i	−0.762391	−	0.647116i	$-0.775975\pi$
$757$		−	979.734i		−	1.29423i	0.762391	−	0.647116i	$-0.224025\pi$
$758$			814.515i			1.07456i
$759$		0			0
$760$	36.2944			0.0477558
$761$	113.842			0.149595			0.0747974	−	0.997199i	$-0.476169\pi$
$761$	113.842			0.149595			0.0747974	+	0.997199i	$0.476169\pi$
$762$		0			0
$763$	−96.8312			−0.126908
$764$			690.418i			0.903689i
$765$		0			0
$766$		−	686.378i		−	0.896055i
$767$	568.925			0.741754
$768$		0			0
$769$		−	683.546i		−	0.888876i	−0.895810	−	0.444438i	$-0.853403\pi$
$769$		−	683.546i		−	0.888876i	0.895810	−	0.444438i	$-0.146597\pi$
$770$		−	54.7941i		−	0.0711612i
$771$		0			0
$772$	372.514			0.482531
$773$			1180.64i			1.52735i	0.645600	+	0.763676i	$0.276607\pi$
$773$			1180.64i			1.52735i	−0.645600	+	0.763676i	$0.723393\pi$
$774$		0			0
$775$	−138.069			−0.178154
$776$			134.206i			0.172945i
$777$		0			0
$778$			48.9981i			0.0629796i
$779$			234.200i			0.300642i
$780$		0			0
$781$		−	674.520i		−	0.863662i
$782$	118.081	−	546.816i	0.150999	−	0.699253i
$783$		0			0
$784$	188.487			0.240417
$785$	−230.473			−0.293596
$786$		0			0
$787$			838.063i			1.06488i	0.846467	+	0.532442i	$0.178725\pi$
$787$			838.063i			1.06488i	−0.846467	+	0.532442i	$0.821275\pi$
$788$	558.846			0.709196
$789$		0			0
$790$	−50.7253			−0.0642092
$791$	161.346			0.203978
$792$		0			0
$793$		−	1764.05i		−	2.22453i
$794$	815.927			1.02762
$795$		0			0
$796$		−	91.4656i		−	0.114907i
$797$		−	49.2134i		−	0.0617483i	−0.999523	−	0.0308741i	$-0.990171\pi$
$797$		−	49.2134i		−	0.0617483i	0.999523	−	0.0308741i	$-0.00982911\pi$
$798$		0			0
$799$		−	620.172i		−	0.776185i
$800$	28.2843			0.0353553
$801$		0			0
$802$		−	887.383i		−	1.10646i
$803$			77.0058i			0.0958977i
$804$		0			0
$805$	68.8957	+	14.8775i	0.0855847	+	0.0184814i
$806$	671.509			0.833138
$807$		0			0
$808$	−71.3654			−0.0883235
$809$	639.240			0.790161			0.395080	−	0.918647i	$-0.370717\pi$
$809$	639.240			0.790161			0.395080	+	0.918647i	$0.370717\pi$
$810$		0			0
$811$	1108.15			1.36640			0.683201	−	0.730231i	$-0.260587\pi$
$811$	1108.15			1.36640			0.683201	+	0.730231i	$0.260587\pi$
$812$		−	52.2758i		−	0.0643790i
$813$		0			0
$814$	−510.810			−0.627531
$815$			126.899i			0.155704i
$816$		0			0
$817$	−192.827			−0.236018
$818$	−32.1212			−0.0392680
$819$		0			0
$820$			182.512i			0.222576i
$821$	503.173			0.612879			0.306439	−	0.951890i	$-0.400862\pi$
$821$	503.173			0.612879			0.306439	+	0.951890i	$0.400862\pi$
$822$		0			0
$823$	−1222.86			−1.48586			−0.742931	−	0.669368i	$-0.766565\pi$
$823$	−1222.86			−1.48586			−0.742931	+	0.669368i	$0.766565\pi$
$824$			289.522i			0.351362i
$825$		0			0
$826$			64.1264i			0.0776348i
$827$			875.358i			1.05847i	0.848474	+	0.529237i	$0.177522\pi$
$827$			875.358i			1.05847i	−0.848474	+	0.529237i	$0.822478\pi$
$828$		0			0
$829$	−176.619			−0.213051			−0.106525	−	0.994310i	$-0.533973\pi$
$829$	−176.619			−0.213051			−0.106525	+	0.994310i	$0.533973\pi$
$830$	378.807			0.456394
$831$		0			0
$832$	−137.562			−0.165339
$833$		−	810.432i		−	0.972908i
$834$		0			0
$835$			693.155i			0.830125i
$836$	145.110			0.173577
$837$		0			0
$838$			300.114i			0.358131i
$839$			1034.37i			1.23287i	0.787407	+	0.616433i	$0.211423\pi$
$839$			1034.37i			1.23287i	−0.787407	+	0.616433i	$0.788577\pi$
$840$		0			0
$841$	−477.262			−0.567493
$842$		−	837.608i		−	0.994784i
$843$		0			0
$844$	615.150			0.728851
$845$			283.261i			0.335220i
$846$		0			0
$847$		−	53.2451i		−	0.0628632i
$848$		−	288.642i		−	0.340380i
$849$		0			0
$850$		−	121.613i		−	0.143074i
$851$	138.694	−	642.270i	0.162977	−	0.754724i
$852$		0			0
$853$	612.490			0.718042			0.359021	−	0.933329i	$-0.383111\pi$
$853$	612.490			0.718042			0.359021	+	0.933329i	$0.383111\pi$
$854$	198.835			0.232828
$855$		0			0
$856$		−	281.971i		−	0.329406i
$857$	1497.60			1.74749			0.873743	−	0.486388i	$-0.161686\pi$
$857$	1497.60			1.74749			0.873743	+	0.486388i	$0.161686\pi$
$858$		0			0
$859$	464.073			0.540248			0.270124	−	0.962826i	$-0.412935\pi$
$859$	464.073			0.540248			0.270124	+	0.962826i	$0.412935\pi$
$860$	−150.270			−0.174733
$861$		0			0
$862$		−	471.646i		−	0.547153i
$863$	−1150.00			−1.33256			−0.666282	−	0.745700i	$-0.732115\pi$
$863$	−1150.00			−1.33256			−0.666282	+	0.745700i	$0.732115\pi$
$864$		0			0
$865$		−	674.787i		−	0.780101i
$866$		−	87.2852i		−	0.100791i
$867$		0			0
$868$			75.6891i			0.0871994i
$869$	−202.807			−0.233379
$870$		0			0
$871$		−	1439.71i		−	1.65294i
$872$		−	199.841i		−	0.229175i
$873$		0			0
$874$	−39.3998	+	182.455i	−0.0450799	+	0.208759i
$875$	15.3226			0.0175115
$876$		0			0
$877$	−30.2402			−0.0344814			−0.0172407	−	0.999851i	$-0.505488\pi$
$877$	−30.2402			−0.0344814			−0.0172407	+	0.999851i	$0.505488\pi$
$878$	−1214.09			−1.38279
$879$		0			0
$880$	113.084			0.128505
$881$		−	402.743i		−	0.457143i	−0.973527	−	0.228572i	$-0.926594\pi$
$881$		−	402.743i		−	0.457143i	0.973527	−	0.228572i	$-0.0734055\pi$
$882$		0			0
$883$	734.792			0.832154			0.416077	−	0.909329i	$-0.363405\pi$
$883$	734.792			0.832154			0.416077	+	0.909329i	$0.363405\pi$
$884$			591.473i			0.669087i
$885$		0			0
$886$	1000.49			1.12922
$887$	−87.1915			−0.0982993			−0.0491496	−	0.998791i	$-0.515651\pi$
$887$	−87.1915			−0.0982993			−0.0491496	+	0.998791i	$0.515651\pi$
$888$		0			0
$889$		−	288.516i		−	0.324540i
$890$	226.615			0.254624
$891$		0			0
$892$	−713.018			−0.799348
$893$			206.931i			0.231726i
$894$		0			0
$895$			149.407i			0.166935i
$896$		−	15.5053i		−	0.0173051i
$897$		0			0
$898$	−39.4878			−0.0439730
$899$	−526.650			−0.585817
$900$		0			0
$901$	−1241.07			−1.37743
$902$			729.710i			0.808991i
$903$		0			0
$904$			332.988i			0.368349i
$905$	313.577			0.346494
$906$		0			0
$907$		−	1514.40i		−	1.66968i	−0.550490	−	0.834842i	$-0.685559\pi$
$907$		−	1514.40i		−	1.66968i	0.550490	−	0.834842i	$-0.314441\pi$
$908$			334.390i			0.368271i
$909$		0			0
$910$	−74.5222			−0.0818925
$911$			1384.87i			1.52016i	0.649830	+	0.760080i	$0.274840\pi$
$911$			1384.87i			1.52016i	−0.649830	+	0.760080i	$0.725160\pi$
$912$		0			0
$913$	1514.52			1.65884
$914$		−	553.446i		−	0.605520i
$915$		0			0
$916$			562.500i			0.614083i
$917$		−	245.442i		−	0.267658i
$918$		0			0
$919$		−	1089.53i		−	1.18556i	−0.805364	−	0.592780i	$-0.798030\pi$
$919$		−	1089.53i		−	1.18556i	0.805364	−	0.592780i	$-0.201970\pi$
$920$	−30.7043	+	142.187i	−0.0333743	+	0.154551i
$921$		0			0
$922$	507.115			0.550016
$923$	−917.374			−0.993905
$924$		0			0
$925$		−	142.842i		−	0.154424i
$926$	70.3456			0.0759672
$927$		0			0
$928$	107.887			0.116258
$929$	−914.187			−0.984055			−0.492027	−	0.870580i	$-0.663744\pi$
$929$	−914.187			−0.984055			−0.492027	+	0.870580i	$0.663744\pi$
$930$		0			0
$931$			270.415i			0.290457i
$932$	−120.423			−0.129209
$933$		0			0
$934$		−	11.1152i		−	0.0119007i
$935$		−	486.226i		−	0.520028i
$936$		0			0
$937$			1594.66i			1.70188i	0.525263	+	0.850940i	$0.323967\pi$
$937$			1594.66i			1.70188i	−0.525263	+	0.850940i	$0.676033\pi$
$938$	162.277			0.173003
$939$		0			0
$940$			161.262i			0.171555i
$941$			77.9212i			0.0828068i	0.999143	+	0.0414034i	$0.0131829\pi$
$941$			77.9212i			0.0828068i	−0.999143	+	0.0414034i	$0.986817\pi$
$942$		0			0
$943$	−917.505	−	198.129i	−0.972964	−	0.210105i
$944$	−132.344			−0.140195
$945$		0			0
$946$	−600.801			−0.635096
$947$	−102.848			−0.108604			−0.0543018	−	0.998525i	$-0.517293\pi$
$947$	−102.848			−0.108604			−0.0543018	+	0.998525i	$0.517293\pi$
$948$		0			0
$949$	104.731			0.110359
$950$			40.5784i			0.0427141i
$951$		0			0
$952$	−66.6678			−0.0700292
$953$		−	401.943i		−	0.421766i	−0.977511	−	0.210883i	$-0.932366\pi$
$953$		−	401.943i		−	0.421766i	0.977511	−	0.210883i	$-0.0676339\pi$
$954$		0			0
$955$	−771.911			−0.808284
$956$	−89.6890			−0.0938170
$957$		0			0
$958$			802.710i			0.837902i
$959$	−94.0515			−0.0980724
$960$		0			0
$961$	−198.474			−0.206529
$962$			694.722i			0.722165i
$963$		0			0
$964$			0.568543i			0.000589774i
$965$			416.483i			0.431589i
$966$		0			0
$967$	893.378			0.923865			0.461933	−	0.886915i	$-0.347156\pi$
$967$	893.378			0.923865			0.461933	+	0.886915i	$0.347156\pi$
$968$	109.888			0.113520
$969$		0			0
$970$	−150.047			−0.154687
$971$		−	502.813i		−	0.517830i	−0.965900	−	0.258915i	$-0.916635\pi$
$971$		−	502.813i		−	0.517830i	0.965900	−	0.258915i	$-0.0833649\pi$
$972$		0			0
$973$			48.8098i			0.0501643i
$974$	−530.195			−0.544349
$975$		0			0
$976$			410.357i			0.420448i
$977$		−	274.182i		−	0.280637i	−0.990106	−	0.140318i	$-0.955187\pi$
$977$		−	274.182i		−	0.280637i	0.990106	−	0.140318i	$-0.0448126\pi$
$978$		0			0
$979$	906.039			0.925474
$980$			210.735i			0.215036i
$981$		0			0
$982$	−583.258			−0.593949
$983$			1541.49i			1.56815i	0.620664	+	0.784076i	$0.286863\pi$
$983$			1541.49i			1.56815i	−0.620664	+	0.784076i	$0.713137\pi$
$984$		0			0
$985$			624.809i			0.634324i
$986$		−	463.879i		−	0.470465i
$987$		0			0
$988$		−	197.356i		−	0.199753i
$989$	163.128	−	755.420i	0.164942	−	0.763822i
$990$		0			0
$991$	−1777.97			−1.79412			−0.897061	−	0.441907i	$-0.854302\pi$
$991$	−1777.97			−1.79412			−0.897061	+	0.441907i	$0.854302\pi$
$992$	−156.208			−0.157467
$993$		0			0
$994$		−	103.402i		−	0.104026i
$995$	102.262			0.102776
$996$		0			0
$997$	1971.04			1.97698			0.988488	−	0.151302i	$-0.0483465\pi$
$997$	1971.04			1.97698			0.988488	+	0.151302i	$0.0483465\pi$
$998$	−463.196			−0.464124
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.13		32
3.2	odd	2		690.3.c.a.91.19	✓	32
23.22	odd	2	inner	2070.3.c.b.91.4		32
69.68	even	2		690.3.c.a.91.22	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.19	✓	32	3.2	odd	2
690.3.c.a.91.22	yes	32	69.68	even	2
2070.3.c.b.91.4		32	23.22	odd	2	inner
2070.3.c.b.91.13		32	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.37049i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.37049i q^{7} -2.82843 q^{8} -3.16228i q^{10} -12.6432i q^{11} -17.1953 q^{13} -1.93817i q^{14} +4.00000 q^{16} -17.1987i q^{17} +5.73865i q^{19} +4.47214i q^{20} +17.8802i q^{22} +(-22.4818 - 4.85478i) q^{23} -5.00000 q^{25} +24.3178 q^{26} +2.74098i q^{28} -19.0719 q^{29} +27.6139 q^{31} -5.65685 q^{32} +24.3226i q^{34} -3.06451 q^{35} +28.5685i q^{37} -8.11568i q^{38} -6.32456i q^{40} +40.8110 q^{41} +33.6014i q^{43} -25.2864i q^{44} +(31.7941 + 6.86570i) q^{46} +36.0593 q^{47} +47.1218 q^{49} +7.07107 q^{50} -34.3906 q^{52} -72.1605i q^{53} +28.2711 q^{55} -3.87633i q^{56} +26.9718 q^{58} -33.0861 q^{59} +102.589i q^{61} -39.0519 q^{62} +8.00000 q^{64} -38.4498i q^{65} +83.7272i q^{67} -34.3974i q^{68} +4.33387 q^{70} +53.3503 q^{71} -6.09068 q^{73} -40.4019i q^{74} +11.4773i q^{76} +17.3274 q^{77} -16.0407i q^{79} +8.94427i q^{80} -57.7155 q^{82} +119.789i q^{83} +38.4574 q^{85} -47.5196i q^{86} +35.7604i q^{88} +71.6620i q^{89} -23.5660i q^{91} +(-44.9636 - 9.70956i) q^{92} -50.9955 q^{94} -12.8320 q^{95} -47.4489i q^{97} -66.6402 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 32 * q + 64 * q^4	\( 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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