LMFDB - Embedded newform 1850.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.a (trivial)

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:	yes
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1791440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} + 13x - 4 $ x^5 - 9x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.53175$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} -4.67211 q^{7} -1.00000 q^{8} -0.653743 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} -4.67211 q^{7} -1.00000 q^{8} -0.653743 q^{9} +0.0451320 q^{11} -1.53175 q^{12} -5.26514 q^{13} +4.67211 q^{14} +1.00000 q^{16} -3.60861 q^{17} +0.653743 q^{18} -6.22000 q^{19} +7.15650 q^{21} -0.0451320 q^{22} +2.20164 q^{23} +1.53175 q^{24} +5.26514 q^{26} +5.59662 q^{27} -4.67211 q^{28} -4.20386 q^{29} -3.01051 q^{31} -1.00000 q^{32} -0.0691309 q^{33} +3.60861 q^{34} -0.653743 q^{36} -1.00000 q^{37} +6.22000 q^{38} +8.06487 q^{39} -7.38299 q^{41} -7.15650 q^{42} +5.54789 q^{43} +0.0451320 q^{44} -2.20164 q^{46} +4.28072 q^{47} -1.53175 q^{48} +14.8286 q^{49} +5.52749 q^{51} -5.26514 q^{52} -6.10215 q^{53} -5.59662 q^{54} +4.67211 q^{56} +9.52749 q^{57} +4.20386 q^{58} +13.5275 q^{59} -4.27299 q^{61} +3.01051 q^{62} +3.05436 q^{63} +1.00000 q^{64} +0.0691309 q^{66} -12.3751 q^{67} -3.60861 q^{68} -3.37235 q^{69} -4.49354 q^{71} +0.653743 q^{72} -7.03811 q^{73} +1.00000 q^{74} -6.22000 q^{76} -0.210861 q^{77} -8.06487 q^{78} -8.72499 q^{79} -6.61139 q^{81} +7.38299 q^{82} +0.880231 q^{83} +7.15650 q^{84} -5.54789 q^{86} +6.43926 q^{87} -0.0451320 q^{88} +9.97602 q^{89} +24.5993 q^{91} +2.20164 q^{92} +4.61135 q^{93} -4.28072 q^{94} +1.53175 q^{96} -0.240408 q^{97} -14.8286 q^{98} -0.0295047 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.53175	−0.884356	−0.442178	−	0.896927i	$-0.645794\pi$
$3$	−1.53175	−0.884356	−0.442178	+	0.896927i	$0.645794\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.53175	0.625334
$7$	−4.67211	−1.76589	−0.882946	−	0.469475i	$-0.844443\pi$
$7$	−4.67211	−1.76589	−0.882946	+	0.469475i	$0.844443\pi$
$8$	−1.00000	−0.353553
$9$	−0.653743	−0.217914
$10$	0	0
$11$	0.0451320	0.0136078	0.00680390	−	0.999977i	$-0.497834\pi$
$11$	0.0451320	0.0136078	0.00680390	+	0.999977i	$0.497834\pi$
$12$	−1.53175	−0.442178
$13$	−5.26514	−1.46029	−0.730143	−	0.683294i	$-0.760547\pi$
$13$	−5.26514	−1.46029	−0.730143	+	0.683294i	$0.760547\pi$
$14$	4.67211	1.24867
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.60861	−0.875217	−0.437608	−	0.899166i	$-0.644174\pi$
$17$	−3.60861	−0.875217	−0.437608	+	0.899166i	$0.644174\pi$
$18$	0.653743	0.154089
$19$	−6.22000	−1.42697	−0.713483	−	0.700672i	$-0.752884\pi$
$19$	−6.22000	−1.42697	−0.713483	+	0.700672i	$0.752884\pi$
$20$	0	0
$21$	7.15650	1.56168
$22$	−0.0451320	−0.00962217
$23$	2.20164	0.459073	0.229536	−	0.973300i	$-0.426279\pi$
$23$	2.20164	0.459073	0.229536	+	0.973300i	$0.426279\pi$
$24$	1.53175	0.312667
$25$	0	0
$26$	5.26514	1.03258
$27$	5.59662	1.07707
$28$	−4.67211	−0.882946
$29$	−4.20386	−0.780637	−0.390319	−	0.920680i	$-0.627635\pi$
$29$	−4.20386	−0.780637	−0.390319	+	0.920680i	$0.627635\pi$
$30$	0	0
$31$	−3.01051	−0.540704	−0.270352	−	0.962762i	$-0.587140\pi$
$31$	−3.01051	−0.540704	−0.270352	+	0.962762i	$0.587140\pi$
$32$	−1.00000	−0.176777
$33$	−0.0691309	−0.0120341
$34$	3.60861	0.618872
$35$	0	0
$36$	−0.653743	−0.108957
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	6.22000	1.00902
$39$	8.06487	1.29141
$40$	0	0
$41$	−7.38299	−1.15303	−0.576515	−	0.817087i	$-0.695588\pi$
$41$	−7.38299	−1.15303	−0.576515	+	0.817087i	$0.695588\pi$
$42$	−7.15650	−1.10427
$43$	5.54789	0.846046	0.423023	−	0.906119i	$-0.360969\pi$
$43$	5.54789	0.846046	0.423023	+	0.906119i	$0.360969\pi$
$44$	0.0451320	0.00680390
$45$	0	0
$46$	−2.20164	−0.324613
$47$	4.28072	0.624407	0.312204	−	0.950015i	$-0.398933\pi$
$47$	4.28072	0.624407	0.312204	+	0.950015i	$0.398933\pi$
$48$	−1.53175	−0.221089
$49$	14.8286	2.11837
$50$	0	0
$51$	5.52749	0.774003
$52$	−5.26514	−0.730143
$53$	−6.10215	−0.838194	−0.419097	−	0.907941i	$-0.637653\pi$
$53$	−6.10215	−0.838194	−0.419097	+	0.907941i	$0.637653\pi$
$54$	−5.59662	−0.761603
$55$	0	0
$56$	4.67211	0.624337
$57$	9.52749	1.26195
$58$	4.20386	0.551994
$59$	13.5275	1.76113	0.880565	−	0.473926i	$-0.157164\pi$
$59$	13.5275	1.76113	0.880565	+	0.473926i	$0.157164\pi$
$60$	0	0
$61$	−4.27299	−0.547100	−0.273550	−	0.961858i	$-0.588198\pi$
$61$	−4.27299	−0.547100	−0.273550	+	0.961858i	$0.588198\pi$
$62$	3.01051	0.382335
$63$	3.05436	0.384813
$64$	1.00000	0.125000
$65$	0	0
$66$	0.0691309	0.00850942
$67$	−12.3751	−1.51186	−0.755932	−	0.654650i	$-0.772816\pi$
$67$	−12.3751	−1.51186	−0.755932	+	0.654650i	$0.772816\pi$
$68$	−3.60861	−0.437608
$69$	−3.37235	−0.405984
$70$	0	0
$71$	−4.49354	−0.533285	−0.266642	−	0.963796i	$-0.585914\pi$
$71$	−4.49354	−0.533285	−0.266642	+	0.963796i	$0.585914\pi$
$72$	0.653743	0.0770443
$73$	−7.03811	−0.823748	−0.411874	−	0.911241i	$-0.635126\pi$
$73$	−7.03811	−0.823748	−0.411874	+	0.911241i	$0.635126\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−6.22000	−0.713483
$77$	−0.210861	−0.0240299
$78$	−8.06487	−0.913167
$79$	−8.72499	−0.981638	−0.490819	−	0.871262i	$-0.663302\pi$
$79$	−8.72499	−0.981638	−0.490819	+	0.871262i	$0.663302\pi$
$80$	0	0
$81$	−6.61139	−0.734599
$82$	7.38299	0.815315
$83$	0.880231	0.0966179	0.0483090	−	0.998832i	$-0.484617\pi$
$83$	0.880231	0.0966179	0.0483090	+	0.998832i	$0.484617\pi$
$84$	7.15650	0.780839
$85$	0	0
$86$	−5.54789	−0.598245
$87$	6.43926	0.690361
$88$	−0.0451320	−0.00481108
$89$	9.97602	1.05746	0.528728	−	0.848791i	$-0.322669\pi$
$89$	9.97602	1.05746	0.528728	+	0.848791i	$0.322669\pi$
$90$	0	0
$91$	24.5993	2.57871
$92$	2.20164	0.229536
$93$	4.61135	0.478175
$94$	−4.28072	−0.441523
$95$	0	0
$96$	1.53175	0.156334
$97$	−0.240408	−0.0244097	−0.0122049	−	0.999926i	$-0.503885\pi$
$97$	−0.240408	−0.0244097	−0.0122049	+	0.999926i	$0.503885\pi$
$98$	−14.8286	−1.49792
$99$	−0.0295047	−0.00296533
$100$	0	0
$101$	−5.61775	−0.558987	−0.279494	−	0.960148i	$-0.590167\pi$
$101$	−5.61775	−0.558987	−0.279494	+	0.960148i	$0.590167\pi$
$102$	−5.52749	−0.547303
$103$	12.5663	1.23819	0.619095	−	0.785316i	$-0.287499\pi$
$103$	12.5663	1.23819	0.619095	+	0.785316i	$0.287499\pi$
$104$	5.26514	0.516289
$105$	0	0
$106$	6.10215	0.592693
$107$	17.9395	1.73427	0.867137	−	0.498070i	$-0.165958\pi$
$107$	17.9395	1.73427	0.867137	+	0.498070i	$0.165958\pi$
$108$	5.59662	0.538535
$109$	3.09936	0.296865	0.148433	−	0.988923i	$-0.452577\pi$
$109$	3.09936	0.296865	0.148433	+	0.988923i	$0.452577\pi$
$110$	0	0
$111$	1.53175	0.145387
$112$	−4.67211	−0.441473
$113$	−4.36184	−0.410328	−0.205164	−	0.978728i	$-0.565773\pi$
$113$	−4.36184	−0.410328	−0.205164	+	0.978728i	$0.565773\pi$
$114$	−9.52749	−0.892331
$115$	0	0
$116$	−4.20386	−0.390319
$117$	3.44204	0.318217
$118$	−13.5275	−1.24531
$119$	16.8598	1.54554
$120$	0	0
$121$	−10.9980	−0.999815
$122$	4.27299	0.386858
$123$	11.3089	1.01969
$124$	−3.01051	−0.270352
$125$	0	0
$126$	−3.05436	−0.272104
$127$	−17.6572	−1.56683	−0.783413	−	0.621502i	$-0.786523\pi$
$127$	−17.6572	−1.56683	−0.783413	+	0.621502i	$0.786523\pi$
$128$	−1.00000	−0.0883883
$129$	−8.49798	−0.748206
$130$	0	0
$131$	−10.3103	−0.900812	−0.450406	−	0.892824i	$-0.648721\pi$
$131$	−10.3103	−0.900812	−0.450406	+	0.892824i	$0.648721\pi$
$132$	−0.0691309	−0.00601707
$133$	29.0605	2.51987
$134$	12.3751	1.06905
$135$	0	0
$136$	3.60861	0.309436
$137$	−3.21863	−0.274986	−0.137493	−	0.990503i	$-0.543905\pi$
$137$	−3.21863	−0.274986	−0.137493	+	0.990503i	$0.543905\pi$
$138$	3.37235	0.287074
$139$	20.9400	1.77611	0.888055	−	0.459737i	$-0.152056\pi$
$139$	20.9400	1.77611	0.888055	+	0.459737i	$0.152056\pi$
$140$	0	0
$141$	−6.55699	−0.552198
$142$	4.49354	0.377089
$143$	−0.237626	−0.0198713
$144$	−0.653743	−0.0544786
$145$	0	0
$146$	7.03811	0.582478
$147$	−22.7137	−1.87340
$148$	−1.00000	−0.0821995
$149$	−21.9832	−1.80093	−0.900467	−	0.434924i	$-0.856775\pi$
$149$	−21.9832	−1.80093	−0.900467	+	0.434924i	$0.856775\pi$
$150$	0	0
$151$	14.1877	1.15458	0.577290	−	0.816539i	$-0.304110\pi$
$151$	14.1877	1.15458	0.577290	+	0.816539i	$0.304110\pi$
$152$	6.22000	0.504509
$153$	2.35910	0.190722
$154$	0.210861	0.0169917
$155$	0	0
$156$	8.06487	0.645706
$157$	19.9316	1.59072	0.795359	−	0.606139i	$-0.207283\pi$
$157$	19.9316	1.59072	0.795359	+	0.606139i	$0.207283\pi$
$158$	8.72499	0.694123
$159$	9.34696	0.741262
$160$	0	0
$161$	−10.2863	−0.810673
$162$	6.61139	0.519440
$163$	−8.65087	−0.677588	−0.338794	−	0.940861i	$-0.610019\pi$
$163$	−8.65087	−0.677588	−0.338794	+	0.940861i	$0.610019\pi$
$164$	−7.38299	−0.576515
$165$	0	0
$166$	−0.880231	−0.0683192
$167$	11.9640	0.925803	0.462901	−	0.886410i	$-0.346808\pi$
$167$	11.9640	0.925803	0.462901	+	0.886410i	$0.346808\pi$
$168$	−7.15650	−0.552136
$169$	14.7216	1.13243
$170$	0	0
$171$	4.06628	0.310956
$172$	5.54789	0.423023
$173$	5.33508	0.405619	0.202809	−	0.979218i	$-0.434993\pi$
$173$	5.33508	0.405619	0.202809	+	0.979218i	$0.434993\pi$
$174$	−6.43926	−0.488159
$175$	0	0
$176$	0.0451320	0.00340195
$177$	−20.7207	−1.55747
$178$	−9.97602	−0.747734
$179$	−5.27905	−0.394575	−0.197288	−	0.980346i	$-0.563213\pi$
$179$	−5.27905	−0.394575	−0.197288	+	0.980346i	$0.563213\pi$
$180$	0	0
$181$	17.7448	1.31896	0.659478	−	0.751723i	$-0.270777\pi$
$181$	17.7448	1.31896	0.659478	+	0.751723i	$0.270777\pi$
$182$	−24.5993	−1.82342
$183$	6.54515	0.483832
$184$	−2.20164	−0.162307
$185$	0	0
$186$	−4.61135	−0.338121
$187$	−0.162864	−0.0119098
$188$	4.28072	0.312204
$189$	−26.1480	−1.90199
$190$	0	0
$191$	−3.11649	−0.225501	−0.112751	−	0.993623i	$-0.535966\pi$
$191$	−3.11649	−0.225501	−0.112751	+	0.993623i	$0.535966\pi$
$192$	−1.53175	−0.110545
$193$	8.78278	0.632198	0.316099	−	0.948726i	$-0.397627\pi$
$193$	8.78278	0.632198	0.316099	+	0.948726i	$0.397627\pi$
$194$	0.240408	0.0172603
$195$	0	0
$196$	14.8286	1.05919
$197$	−7.00278	−0.498928	−0.249464	−	0.968384i	$-0.580254\pi$
$197$	−7.00278	−0.498928	−0.249464	+	0.968384i	$0.580254\pi$
$198$	0.0295047	0.00209681
$199$	17.3202	1.22780	0.613900	−	0.789384i	$-0.289600\pi$
$199$	17.3202	1.22780	0.613900	+	0.789384i	$0.289600\pi$
$200$	0	0
$201$	18.9556	1.33703
$202$	5.61775	0.395264
$203$	19.6409	1.37852
$204$	5.52749	0.387002
$205$	0	0
$206$	−12.5663	−0.875533
$207$	−1.43930	−0.100038
$208$	−5.26514	−0.365071
$209$	−0.280721	−0.0194179
$210$	0	0
$211$	6.77438	0.466368	0.233184	−	0.972433i	$-0.425086\pi$
$211$	6.77438	0.466368	0.233184	+	0.972433i	$0.425086\pi$
$212$	−6.10215	−0.419097
$213$	6.88297	0.471613
$214$	−17.9395	−1.22632
$215$	0	0
$216$	−5.59662	−0.380802
$217$	14.0654	0.954825
$218$	−3.09936	−0.209915
$219$	10.7806	0.728487
$220$	0	0
$221$	18.9998	1.27807
$222$	−1.53175	−0.102804
$223$	10.2559	0.686784	0.343392	−	0.939192i	$-0.388424\pi$
$223$	10.2559	0.686784	0.343392	+	0.939192i	$0.388424\pi$
$224$	4.67211	0.312168
$225$	0	0
$226$	4.36184	0.290145
$227$	11.2744	0.748306	0.374153	−	0.927367i	$-0.377934\pi$
$227$	11.2744	0.748306	0.374153	+	0.927367i	$0.377934\pi$
$228$	9.52749	0.630973
$229$	−9.40494	−0.621496	−0.310748	−	0.950492i	$-0.600579\pi$
$229$	−9.40494	−0.621496	−0.310748	+	0.950492i	$0.600579\pi$
$230$	0	0
$231$	0.322987	0.0212510
$232$	4.20386	0.275997
$233$	3.22867	0.211517	0.105759	−	0.994392i	$-0.466273\pi$
$233$	3.22867	0.211517	0.105759	+	0.994392i	$0.466273\pi$
$234$	−3.44204	−0.225013
$235$	0	0
$236$	13.5275	0.880565
$237$	13.3645	0.868118
$238$	−16.8598	−1.09286
$239$	−17.8612	−1.15534	−0.577672	−	0.816269i	$-0.696039\pi$
$239$	−17.8612	−1.15534	−0.577672	+	0.816269i	$0.696039\pi$
$240$	0	0
$241$	26.6840	1.71887	0.859434	−	0.511248i	$-0.170816\pi$
$241$	26.6840	1.71887	0.859434	+	0.511248i	$0.170816\pi$
$242$	10.9980	0.706976
$243$	−6.66286	−0.427423
$244$	−4.27299	−0.273550
$245$	0	0
$246$	−11.3089	−0.721029
$247$	32.7492	2.08378
$248$	3.01051	0.191168
$249$	−1.34829	−0.0854447
$250$	0	0
$251$	−30.1552	−1.90338	−0.951690	−	0.307060i	$-0.900655\pi$
$251$	−30.1552	−1.90338	−0.951690	+	0.307060i	$0.900655\pi$
$252$	3.05436	0.192406
$253$	0.0993641	0.00624697
$254$	17.6572	1.10791
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.15376	0.196726	0.0983632	−	0.995151i	$-0.468639\pi$
$257$	3.15376	0.196726	0.0983632	+	0.995151i	$0.468639\pi$
$258$	8.49798	0.529061
$259$	4.67211	0.290311
$260$	0	0
$261$	2.74824	0.170112
$262$	10.3103	0.636970
$263$	−14.0332	−0.865322	−0.432661	−	0.901557i	$-0.642425\pi$
$263$	−14.0332	−0.865322	−0.432661	+	0.901557i	$0.642425\pi$
$264$	0.0691309	0.00425471
$265$	0	0
$266$	−29.0605	−1.78182
$267$	−15.2808	−0.935167
$268$	−12.3751	−0.755932
$269$	−30.1257	−1.83679	−0.918397	−	0.395660i	$-0.870516\pi$
$269$	−30.1257	−1.83679	−0.918397	+	0.395660i	$0.870516\pi$
$270$	0	0
$271$	14.8802	0.903910	0.451955	−	0.892041i	$-0.350727\pi$
$271$	14.8802	0.903910	0.451955	+	0.892041i	$0.350727\pi$
$272$	−3.60861	−0.218804
$273$	−37.6800	−2.28049
$274$	3.21863	0.194445
$275$	0	0
$276$	−3.37235	−0.202992
$277$	−0.147278	−0.00884908	−0.00442454	−	0.999990i	$-0.501408\pi$
$277$	−0.147278	−0.00884908	−0.00442454	+	0.999990i	$0.501408\pi$
$278$	−20.9400	−1.25590
$279$	1.96810	0.117827
$280$	0	0
$281$	−25.5247	−1.52268	−0.761339	−	0.648354i	$-0.775458\pi$
$281$	−25.5247	−1.52268	−0.761339	+	0.648354i	$0.775458\pi$
$282$	6.55699	0.390463
$283$	−0.0562691	−0.00334485	−0.00167242	−	0.999999i	$-0.500532\pi$
$283$	−0.0562691	−0.00334485	−0.00167242	+	0.999999i	$0.500532\pi$
$284$	−4.49354	−0.266642
$285$	0	0
$286$	0.237626	0.0140511
$287$	34.4942	2.03613
$288$	0.653743	0.0385222
$289$	−3.97793	−0.233996
$290$	0	0
$291$	0.368245	0.0215869
$292$	−7.03811	−0.411874
$293$	−13.7033	−0.800557	−0.400278	−	0.916394i	$-0.631087\pi$
$293$	−13.7033	−0.800557	−0.400278	+	0.916394i	$0.631087\pi$
$294$	22.7137	1.32469
$295$	0	0
$296$	1.00000	0.0581238
$297$	0.252586	0.0146565
$298$	21.9832	1.27345
$299$	−11.5919	−0.670377
$300$	0	0
$301$	−25.9204	−1.49403
$302$	−14.1877	−0.816411
$303$	8.60499	0.494344
$304$	−6.22000	−0.356742
$305$	0	0
$306$	−2.35910	−0.134861
$307$	17.5823	1.00348	0.501738	−	0.865020i	$-0.332694\pi$
$307$	17.5823	1.00348	0.501738	+	0.865020i	$0.332694\pi$
$308$	−0.210861	−0.0120149
$309$	−19.2484	−1.09500
$310$	0	0
$311$	0.353133	0.0200244	0.0100122	−	0.999950i	$-0.496813\pi$
$311$	0.353133	0.0200244	0.0100122	+	0.999950i	$0.496813\pi$
$312$	−8.06487	−0.456583
$313$	18.8717	1.06669	0.533346	−	0.845897i	$-0.320934\pi$
$313$	18.8717	1.06669	0.533346	+	0.845897i	$0.320934\pi$
$314$	−19.9316	−1.12481
$315$	0	0
$316$	−8.72499	−0.490819
$317$	−15.1979	−0.853601	−0.426800	−	0.904346i	$-0.640359\pi$
$317$	−15.1979	−0.853601	−0.426800	+	0.904346i	$0.640359\pi$
$318$	−9.34696	−0.524152
$319$	−0.189728	−0.0106228
$320$	0	0
$321$	−27.4788	−1.53372
$322$	10.2863	0.573232
$323$	22.4456	1.24890
$324$	−6.61139	−0.367300
$325$	0	0
$326$	8.65087	0.479127
$327$	−4.74745	−0.262535
$328$	7.38299	0.407658
$329$	−20.0000	−1.10264
$330$	0	0
$331$	−23.0210	−1.26535	−0.632674	−	0.774418i	$-0.718043\pi$
$331$	−23.0210	−1.26535	−0.632674	+	0.774418i	$0.718043\pi$
$332$	0.880231	0.0483090
$333$	0.653743	0.0358249
$334$	−11.9640	−0.654641
$335$	0	0
$336$	7.15650	0.390419
$337$	−34.0776	−1.85633	−0.928163	−	0.372173i	$-0.878613\pi$
$337$	−34.0776	−1.85633	−0.928163	+	0.372173i	$0.878613\pi$
$338$	−14.7216	−0.800752
$339$	6.68125	0.362876
$340$	0	0
$341$	−0.135870	−0.00735779
$342$	−4.06628	−0.219879
$343$	−36.5761	−1.97493
$344$	−5.54789	−0.299122
$345$	0	0
$346$	−5.33508	−0.286816
$347$	−13.8449	−0.743236	−0.371618	−	0.928386i	$-0.621197\pi$
$347$	−13.8449	−0.743236	−0.371618	+	0.928386i	$0.621197\pi$
$348$	6.43926	0.345181
$349$	11.3765	0.608970	0.304485	−	0.952517i	$-0.401516\pi$
$349$	11.3765	0.608970	0.304485	+	0.952517i	$0.401516\pi$
$350$	0	0
$351$	−29.4670	−1.57283
$352$	−0.0451320	−0.00240554
$353$	−19.2631	−1.02527	−0.512636	−	0.858606i	$-0.671331\pi$
$353$	−19.2631	−1.02527	−0.512636	+	0.858606i	$0.671331\pi$
$354$	20.7207	1.10129
$355$	0	0
$356$	9.97602	0.528728
$357$	−25.8250	−1.36681
$358$	5.27905	0.279007
$359$	14.6645	0.773961	0.386980	−	0.922088i	$-0.373518\pi$
$359$	14.6645	0.773961	0.386980	+	0.922088i	$0.373518\pi$
$360$	0	0
$361$	19.6884	1.03623
$362$	−17.7448	−0.932643
$363$	16.8461	0.884192
$364$	24.5993	1.28935
$365$	0	0
$366$	−6.54515	−0.342121
$367$	−0.868349	−0.0453275	−0.0226637	−	0.999743i	$-0.507215\pi$
$367$	−0.868349	−0.0453275	−0.0226637	+	0.999743i	$0.507215\pi$
$368$	2.20164	0.114768
$369$	4.82658	0.251262
$370$	0	0
$371$	28.5099	1.48016
$372$	4.61135	0.239087
$373$	−32.4555	−1.68048	−0.840240	−	0.542214i	$-0.817586\pi$
$373$	−32.4555	−1.68048	−0.840240	+	0.542214i	$0.817586\pi$
$374$	0.162864	0.00842148
$375$	0	0
$376$	−4.28072	−0.220761
$377$	22.1339	1.13995
$378$	26.1480	1.34491
$379$	−30.2511	−1.55389	−0.776947	−	0.629566i	$-0.783233\pi$
$379$	−30.2511	−1.55389	−0.776947	+	0.629566i	$0.783233\pi$
$380$	0	0
$381$	27.0465	1.38563
$382$	3.11649	0.159453
$383$	−14.4935	−0.740585	−0.370293	−	0.928915i	$-0.620743\pi$
$383$	−14.4935	−0.740585	−0.370293	+	0.928915i	$0.620743\pi$
$384$	1.53175	0.0781668
$385$	0	0
$386$	−8.78278	−0.447032
$387$	−3.62689	−0.184365
$388$	−0.240408	−0.0122049
$389$	36.9033	1.87107	0.935537	−	0.353229i	$-0.114916\pi$
$389$	36.9033	1.87107	0.935537	+	0.353229i	$0.114916\pi$
$390$	0	0
$391$	−7.94485	−0.401788
$392$	−14.8286	−0.748958
$393$	15.7927	0.796639
$394$	7.00278	0.352795
$395$	0	0
$396$	−0.0295047	−0.00148267
$397$	−22.9365	−1.15115	−0.575575	−	0.817749i	$-0.695222\pi$
$397$	−22.9365	−1.15115	−0.575575	+	0.817749i	$0.695222\pi$
$398$	−17.3202	−0.868185
$399$	−44.5135	−2.22846
$400$	0	0
$401$	27.4332	1.36995	0.684973	−	0.728568i	$-0.259814\pi$
$401$	27.4332	1.36995	0.684973	+	0.728568i	$0.259814\pi$
$402$	−18.9556	−0.945420
$403$	15.8508	0.789582
$404$	−5.61775	−0.279494
$405$	0	0
$406$	−19.6409	−0.974761
$407$	−0.0451320	−0.00223711
$408$	−5.52749	−0.273651
$409$	−21.3920	−1.05777	−0.528883	−	0.848695i	$-0.677389\pi$
$409$	−21.3920	−1.05777	−0.528883	+	0.848695i	$0.677389\pi$
$410$	0	0
$411$	4.93014	0.243186
$412$	12.5663	0.619095
$413$	−63.2019	−3.10996
$414$	1.43930	0.0707379
$415$	0	0
$416$	5.26514	0.258144
$417$	−32.0749	−1.57071
$418$	0.280721	0.0137305
$419$	23.0891	1.12797	0.563987	−	0.825784i	$-0.309267\pi$
$419$	23.0891	1.12797	0.563987	+	0.825784i	$0.309267\pi$
$420$	0	0
$421$	−1.83757	−0.0895576	−0.0447788	−	0.998997i	$-0.514258\pi$
$421$	−1.83757	−0.0895576	−0.0447788	+	0.998997i	$0.514258\pi$
$422$	−6.77438	−0.329772
$423$	−2.79849	−0.136067
$424$	6.10215	0.296346
$425$	0	0
$426$	−6.88297	−0.333481
$427$	19.9639	0.966120
$428$	17.9395	0.867137
$429$	0.363983	0.0175733
$430$	0	0
$431$	−15.9796	−0.769710	−0.384855	−	0.922977i	$-0.625749\pi$
$431$	−15.9796	−0.769710	−0.384855	+	0.922977i	$0.625749\pi$
$432$	5.59662	0.269267
$433$	25.2525	1.21356	0.606779	−	0.794870i	$-0.292461\pi$
$433$	25.2525	1.21356	0.606779	+	0.794870i	$0.292461\pi$
$434$	−14.0654	−0.675163
$435$	0	0
$436$	3.09936	0.148433
$437$	−13.6942	−0.655082
$438$	−10.7806	−0.515118
$439$	−9.30792	−0.444243	−0.222121	−	0.975019i	$-0.571298\pi$
$439$	−9.30792	−0.444243	−0.222121	+	0.975019i	$0.571298\pi$
$440$	0	0
$441$	−9.69410	−0.461624
$442$	−18.9998	−0.903729
$443$	9.52445	0.452520	0.226260	−	0.974067i	$-0.427350\pi$
$443$	9.52445	0.452520	0.226260	+	0.974067i	$0.427350\pi$
$444$	1.53175	0.0726936
$445$	0	0
$446$	−10.2559	−0.485629
$447$	33.6728	1.59267
$448$	−4.67211	−0.220736
$449$	5.43278	0.256389	0.128194	−	0.991749i	$-0.459082\pi$
$449$	5.43278	0.256389	0.128194	+	0.991749i	$0.459082\pi$
$450$	0	0
$451$	−0.333209	−0.0156902
$452$	−4.36184	−0.205164
$453$	−21.7320	−1.02106
$454$	−11.2744	−0.529132
$455$	0	0
$456$	−9.52749	−0.446166
$457$	−7.54789	−0.353076	−0.176538	−	0.984294i	$-0.556490\pi$
$457$	−7.54789	−0.353076	−0.176538	+	0.984294i	$0.556490\pi$
$458$	9.40494	0.439464
$459$	−20.1960	−0.942670
$460$	0	0
$461$	−35.4323	−1.65025	−0.825124	−	0.564952i	$-0.808895\pi$
$461$	−35.4323	−1.65025	−0.825124	+	0.564952i	$0.808895\pi$
$462$	−0.322987	−0.0150267
$463$	−20.9124	−0.971883	−0.485942	−	0.873991i	$-0.661523\pi$
$463$	−20.9124	−0.971883	−0.485942	+	0.873991i	$0.661523\pi$
$464$	−4.20386	−0.195159
$465$	0	0
$466$	−3.22867	−0.149565
$467$	−6.62523	−0.306579	−0.153289	−	0.988181i	$-0.548987\pi$
$467$	−6.62523	−0.306579	−0.153289	+	0.988181i	$0.548987\pi$
$468$	3.44204	0.159109
$469$	57.8180	2.66979
$470$	0	0
$471$	−30.5303	−1.40676
$472$	−13.5275	−0.622653
$473$	0.250387	0.0115128
$474$	−13.3645	−0.613852
$475$	0	0
$476$	16.8598	0.772769
$477$	3.98923	0.182654
$478$	17.8612	0.816952
$479$	−29.4581	−1.34597	−0.672987	−	0.739655i	$-0.734989\pi$
$479$	−29.4581	−1.34597	−0.672987	+	0.739655i	$0.734989\pi$
$480$	0	0
$481$	5.26514	0.240070
$482$	−26.6840	−1.21542
$483$	15.7560	0.716923
$484$	−10.9980	−0.499907
$485$	0	0
$486$	6.66286	0.302233
$487$	−8.80654	−0.399063	−0.199531	−	0.979891i	$-0.563942\pi$
$487$	−8.80654	−0.399063	−0.199531	+	0.979891i	$0.563942\pi$
$488$	4.27299	0.193429
$489$	13.2510	0.599229
$490$	0	0
$491$	5.14258	0.232082	0.116041	−	0.993244i	$-0.462980\pi$
$491$	5.14258	0.232082	0.116041	+	0.993244i	$0.462980\pi$
$492$	11.3089	0.509844
$493$	15.1701	0.683227
$494$	−32.7492	−1.47345
$495$	0	0
$496$	−3.01051	−0.135176
$497$	20.9943	0.941723
$498$	1.34829	0.0604185
$499$	−12.9692	−0.580579	−0.290290	−	0.956939i	$-0.593752\pi$
$499$	−12.9692	−0.580579	−0.290290	+	0.956939i	$0.593752\pi$
$500$	0	0
$501$	−18.3259	−0.818739
$502$	30.1552	1.34589
$503$	−17.2091	−0.767314	−0.383657	−	0.923476i	$-0.625336\pi$
$503$	−17.2091	−0.767314	−0.383657	+	0.923476i	$0.625336\pi$
$504$	−3.05436	−0.136052
$505$	0	0
$506$	−0.0993641	−0.00441727
$507$	−22.5499	−1.00148
$508$	−17.6572	−0.783413
$509$	−12.1085	−0.536698	−0.268349	−	0.963322i	$-0.586478\pi$
$509$	−12.1085	−0.536698	−0.268349	+	0.963322i	$0.586478\pi$
$510$	0	0
$511$	32.8828	1.45465
$512$	−1.00000	−0.0441942
$513$	−34.8110	−1.53694
$514$	−3.15376	−0.139107
$515$	0	0
$516$	−8.49798	−0.374103
$517$	0.193197	0.00849681
$518$	−4.67211	−0.205281
$519$	−8.17201	−0.358711
$520$	0	0
$521$	−18.7936	−0.823362	−0.411681	−	0.911328i	$-0.635058\pi$
$521$	−18.7936	−0.823362	−0.411681	+	0.911328i	$0.635058\pi$
$522$	−2.74824	−0.120287
$523$	42.9985	1.88019	0.940096	−	0.340909i	$-0.110735\pi$
$523$	42.9985	1.88019	0.940096	+	0.340909i	$0.110735\pi$
$524$	−10.3103	−0.450406
$525$	0	0
$526$	14.0332	0.611875
$527$	10.8638	0.473233
$528$	−0.0691309	−0.00300853
$529$	−18.1528	−0.789252
$530$	0	0
$531$	−8.84350	−0.383775
$532$	29.0605	1.25993
$533$	38.8725	1.68375
$534$	15.2808	0.661263
$535$	0	0
$536$	12.3751	0.534525
$537$	8.08619	0.348945
$538$	30.1257	1.29881
$539$	0.669244	0.0288264
$540$	0	0
$541$	30.1751	1.29733	0.648664	−	0.761075i	$-0.275328\pi$
$541$	30.1751	1.29733	0.648664	+	0.761075i	$0.275328\pi$
$542$	−14.8802	−0.639161
$543$	−27.1805	−1.16643
$544$	3.60861	0.154718
$545$	0	0
$546$	37.6800	1.61255
$547$	31.6736	1.35426	0.677132	−	0.735862i	$-0.263223\pi$
$547$	31.6736	1.35426	0.677132	+	0.735862i	$0.263223\pi$
$548$	−3.21863	−0.137493
$549$	2.79344	0.119221
$550$	0	0
$551$	26.1480	1.11394
$552$	3.37235	0.143537
$553$	40.7641	1.73347
$554$	0.147278	0.00625724
$555$	0	0
$556$	20.9400	0.888055
$557$	0.708971	0.0300401	0.0150200	−	0.999887i	$-0.495219\pi$
$557$	0.708971	0.0300401	0.0150200	+	0.999887i	$0.495219\pi$
$558$	−1.96810	−0.0833163
$559$	−29.2104	−1.23547
$560$	0	0
$561$	0.249466	0.0105325
$562$	25.5247	1.07670
$563$	29.9399	1.26182	0.630908	−	0.775857i	$-0.282682\pi$
$563$	29.9399	1.26182	0.630908	+	0.775857i	$0.282682\pi$
$564$	−6.55699	−0.276099
$565$	0	0
$566$	0.0562691	0.00236517
$567$	30.8892	1.29722
$568$	4.49354	0.188545
$569$	21.1453	0.886456	0.443228	−	0.896409i	$-0.353833\pi$
$569$	21.1453	0.886456	0.443228	+	0.896409i	$0.353833\pi$
$570$	0	0
$571$	−19.5326	−0.817416	−0.408708	−	0.912665i	$-0.634020\pi$
$571$	−19.5326	−0.817416	−0.408708	+	0.912665i	$0.634020\pi$
$572$	−0.237626	−0.00993564
$573$	4.77368	0.199423
$574$	−34.4942	−1.43976
$575$	0	0
$576$	−0.653743	−0.0272393
$577$	−1.08303	−0.0450873	−0.0225436	−	0.999746i	$-0.507176\pi$
$577$	−1.08303	−0.0450873	−0.0225436	+	0.999746i	$0.507176\pi$
$578$	3.97793	0.165460
$579$	−13.4530	−0.559088
$580$	0	0
$581$	−4.11254	−0.170617
$582$	−0.368245	−0.0152642
$583$	−0.275402	−0.0114060
$584$	7.03811	0.291239
$585$	0	0
$586$	13.7033	0.566079
$587$	−1.16839	−0.0482244	−0.0241122	−	0.999709i	$-0.507676\pi$
$587$	−1.16839	−0.0482244	−0.0241122	+	0.999709i	$0.507676\pi$
$588$	−22.7137	−0.936698
$589$	18.7254	0.771566
$590$	0	0
$591$	10.7265	0.441230
$592$	−1.00000	−0.0410997
$593$	6.01305	0.246926	0.123463	−	0.992349i	$-0.460600\pi$
$593$	6.01305	0.246926	0.123463	+	0.992349i	$0.460600\pi$
$594$	−0.252586	−0.0103637
$595$	0	0
$596$	−21.9832	−0.900467
$597$	−26.5303	−1.08581
$598$	11.5919	0.474028
$599$	−11.9208	−0.487072	−0.243536	−	0.969892i	$-0.578307\pi$
$599$	−11.9208	−0.487072	−0.243536	+	0.969892i	$0.578307\pi$
$600$	0	0
$601$	21.7772	0.888309	0.444155	−	0.895950i	$-0.353504\pi$
$601$	21.7772	0.888309	0.444155	+	0.895950i	$0.353504\pi$
$602$	25.9204	1.05644
$603$	8.09015	0.329457
$604$	14.1877	0.577290
$605$	0	0
$606$	−8.60499	−0.349554
$607$	−5.16486	−0.209635	−0.104818	−	0.994491i	$-0.533426\pi$
$607$	−5.16486	−0.209635	−0.104818	+	0.994491i	$0.533426\pi$
$608$	6.22000	0.252254
$609$	−30.0849	−1.21910
$610$	0	0
$611$	−22.5386	−0.911813
$612$	2.35910	0.0953611
$613$	39.5817	1.59869	0.799344	−	0.600873i	$-0.205180\pi$
$613$	39.5817	1.59869	0.799344	+	0.600873i	$0.205180\pi$
$614$	−17.5823	−0.709565
$615$	0	0
$616$	0.210861	0.00849585
$617$	−44.9917	−1.81130	−0.905650	−	0.424027i	$-0.860616\pi$
$617$	−44.9917	−1.81130	−0.905650	+	0.424027i	$0.860616\pi$
$618$	19.2484	0.774283
$619$	30.0465	1.20767	0.603836	−	0.797108i	$-0.293638\pi$
$619$	30.0465	1.20767	0.603836	+	0.797108i	$0.293638\pi$
$620$	0	0
$621$	12.3217	0.494453
$622$	−0.353133	−0.0141594
$623$	−46.6091	−1.86735
$624$	8.06487	0.322853
$625$	0	0
$626$	−18.8717	−0.754265
$627$	0.429994	0.0171723
$628$	19.9316	0.795359
$629$	3.60861	0.143885
$630$	0	0
$631$	5.01448	0.199623	0.0998116	−	0.995006i	$-0.468176\pi$
$631$	5.01448	0.199623	0.0998116	+	0.995006i	$0.468176\pi$
$632$	8.72499	0.347061
$633$	−10.3767	−0.412435
$634$	15.1979	0.603587
$635$	0	0
$636$	9.34696	0.370631
$637$	−78.0747	−3.09343
$638$	0.189728	0.00751142
$639$	2.93762	0.116210
$640$	0	0
$641$	18.9133	0.747031	0.373515	−	0.927624i	$-0.378152\pi$
$641$	18.9133	0.747031	0.373515	+	0.927624i	$0.378152\pi$
$642$	27.4788	1.08450
$643$	−22.9344	−0.904444	−0.452222	−	0.891906i	$-0.649368\pi$
$643$	−22.9344	−0.904444	−0.452222	+	0.891906i	$0.649368\pi$
$644$	−10.2863	−0.405336
$645$	0	0
$646$	−22.4456	−0.883109
$647$	−10.4741	−0.411779	−0.205889	−	0.978575i	$-0.566009\pi$
$647$	−10.4741	−0.411779	−0.205889	+	0.978575i	$0.566009\pi$
$648$	6.61139	0.259720
$649$	0.610522	0.0239651
$650$	0	0
$651$	−21.5447	−0.844405
$652$	−8.65087	−0.338794
$653$	44.7251	1.75023	0.875115	−	0.483916i	$-0.160786\pi$
$653$	44.7251	1.75023	0.875115	+	0.483916i	$0.160786\pi$
$654$	4.74745	0.185640
$655$	0	0
$656$	−7.38299	−0.288257
$657$	4.60111	0.179506
$658$	20.0000	0.779681
$659$	15.1056	0.588431	0.294215	−	0.955739i	$-0.404942\pi$
$659$	15.1056	0.588431	0.294215	+	0.955739i	$0.404942\pi$
$660$	0	0
$661$	34.9654	1.36000	0.679998	−	0.733214i	$-0.261981\pi$
$661$	34.9654	1.36000	0.679998	+	0.733214i	$0.261981\pi$
$662$	23.0210	0.894736
$663$	−29.1030	−1.13027
$664$	−0.880231	−0.0341596
$665$	0	0
$666$	−0.653743	−0.0253320
$667$	−9.25537	−0.358369
$668$	11.9640	0.462901
$669$	−15.7094	−0.607361
$670$	0	0
$671$	−0.192848	−0.00744483
$672$	−7.15650	−0.276068
$673$	−35.2496	−1.35877	−0.679385	−	0.733782i	$-0.737754\pi$
$673$	−35.2496	−1.35877	−0.679385	+	0.733782i	$0.737754\pi$
$674$	34.0776	1.31262
$675$	0	0
$676$	14.7216	0.566217
$677$	23.2835	0.894858	0.447429	−	0.894320i	$-0.352340\pi$
$677$	23.2835	0.894858	0.447429	+	0.894320i	$0.352340\pi$
$678$	−6.68125	−0.256592
$679$	1.12321	0.0431049
$680$	0	0
$681$	−17.2695	−0.661769
$682$	0.135870	0.00520274
$683$	11.8131	0.452017	0.226008	−	0.974125i	$-0.427432\pi$
$683$	11.8131	0.452017	0.226008	+	0.974125i	$0.427432\pi$
$684$	4.06628	0.155478
$685$	0	0
$686$	36.5761	1.39648
$687$	14.4060	0.549624
$688$	5.54789	0.211511
$689$	32.1286	1.22400
$690$	0	0
$691$	14.1858	0.539652	0.269826	−	0.962909i	$-0.413034\pi$
$691$	14.1858	0.539652	0.269826	+	0.962909i	$0.413034\pi$
$692$	5.33508	0.202809
$693$	0.137849	0.00523646
$694$	13.8449	0.525547
$695$	0	0
$696$	−6.43926	−0.244080
$697$	26.6423	1.00915
$698$	−11.3765	−0.430607
$699$	−4.94552	−0.187057
$700$	0	0
$701$	19.2184	0.725870	0.362935	−	0.931815i	$-0.381775\pi$
$701$	19.2184	0.725870	0.362935	+	0.931815i	$0.381775\pi$
$702$	29.4670	1.11216
$703$	6.22000	0.234592
$704$	0.0451320	0.00170097
$705$	0	0
$706$	19.2631	0.724976
$707$	26.2468	0.987111
$708$	−20.7207	−0.778733
$709$	4.52525	0.169949	0.0849746	−	0.996383i	$-0.472919\pi$
$709$	4.52525	0.169949	0.0849746	+	0.996383i	$0.472919\pi$
$710$	0	0
$711$	5.70390	0.213913
$712$	−9.97602	−0.373867
$713$	−6.62805	−0.248222
$714$	25.8250	0.966478
$715$	0	0
$716$	−5.27905	−0.197288
$717$	27.3589	1.02174
$718$	−14.6645	−0.547273
$719$	0.915477	0.0341415	0.0170708	−	0.999854i	$-0.494566\pi$
$719$	0.915477	0.0341415	0.0170708	+	0.999854i	$0.494566\pi$
$720$	0	0
$721$	−58.7110	−2.18651
$722$	−19.6884	−0.732728
$723$	−40.8732	−1.52009
$724$	17.7448	0.659478
$725$	0	0
$726$	−16.8461	−0.625218
$727$	11.7429	0.435521	0.217760	−	0.976002i	$-0.430125\pi$
$727$	11.7429	0.435521	0.217760	+	0.976002i	$0.430125\pi$
$728$	−24.5993	−0.911710
$729$	30.0400	1.11259
$730$	0	0
$731$	−20.0202	−0.740473
$732$	6.54515	0.241916
$733$	−27.6776	−1.02230	−0.511148	−	0.859493i	$-0.670780\pi$
$733$	−27.6776	−1.02230	−0.511148	+	0.859493i	$0.670780\pi$
$734$	0.868349	0.0320514
$735$	0	0
$736$	−2.20164	−0.0811534
$737$	−0.558514	−0.0205731
$738$	−4.82658	−0.177669
$739$	−6.52187	−0.239911	−0.119956	−	0.992779i	$-0.538275\pi$
$739$	−6.52187	−0.239911	−0.119956	+	0.992779i	$0.538275\pi$
$740$	0	0
$741$	−50.1635	−1.84280
$742$	−28.5099	−1.04663
$743$	−43.0091	−1.57785	−0.788925	−	0.614489i	$-0.789362\pi$
$743$	−43.0091	−1.57785	−0.788925	+	0.614489i	$0.789362\pi$
$744$	−4.61135	−0.169060
$745$	0	0
$746$	32.4555	1.18828
$747$	−0.575445	−0.0210544
$748$	−0.162864	−0.00595489
$749$	−83.8152	−3.06254
$750$	0	0
$751$	−29.7068	−1.08402	−0.542008	−	0.840373i	$-0.682336\pi$
$751$	−29.7068	−1.08402	−0.542008	+	0.840373i	$0.682336\pi$
$752$	4.28072	0.156102
$753$	46.1902	1.68327
$754$	−22.1339	−0.806069
$755$	0	0
$756$	−26.1480	−0.950994
$757$	−32.7918	−1.19184	−0.595920	−	0.803044i	$-0.703212\pi$
$757$	−32.7918	−1.19184	−0.595920	+	0.803044i	$0.703212\pi$
$758$	30.2511	1.09877
$759$	−0.152201	−0.00552455
$760$	0	0
$761$	22.9411	0.831616	0.415808	−	0.909452i	$-0.363499\pi$
$761$	22.9411	0.831616	0.415808	+	0.909452i	$0.363499\pi$
$762$	−27.0465	−0.979790
$763$	−14.4806	−0.524232
$764$	−3.11649	−0.112751
$765$	0	0
$766$	14.4935	0.523673
$767$	−71.2241	−2.57175
$768$	−1.53175	−0.0552723
$769$	10.8123	0.389902	0.194951	−	0.980813i	$-0.437545\pi$
$769$	10.8123	0.389902	0.194951	+	0.980813i	$0.437545\pi$
$770$	0	0
$771$	−4.83078	−0.173976
$772$	8.78278	0.316099
$773$	37.9989	1.36673	0.683363	−	0.730079i	$-0.260517\pi$
$773$	37.9989	1.36673	0.683363	+	0.730079i	$0.260517\pi$
$774$	3.62689	0.130366
$775$	0	0
$776$	0.240408	0.00863014
$777$	−7.15650	−0.256738
$778$	−36.9033	−1.32305
$779$	45.9222	1.64533
$780$	0	0
$781$	−0.202802	−0.00725683
$782$	7.94485	0.284107
$783$	−23.5274	−0.840801
$784$	14.8286	0.529593
$785$	0	0
$786$	−15.7927	−0.563309
$787$	−31.7730	−1.13258	−0.566292	−	0.824205i	$-0.691623\pi$
$787$	−31.7730	−1.13258	−0.566292	+	0.824205i	$0.691623\pi$
$788$	−7.00278	−0.249464
$789$	21.4953	0.765252
$790$	0	0
$791$	20.3790	0.724594
$792$	0.0295047	0.00104840
$793$	22.4979	0.798923
$794$	22.9365	0.813986
$795$	0	0
$796$	17.3202	0.613900
$797$	−9.02551	−0.319700	−0.159850	−	0.987141i	$-0.551101\pi$
$797$	−9.02551	−0.319700	−0.159850	+	0.987141i	$0.551101\pi$
$798$	44.5135	1.57576
$799$	−15.4475	−0.546492
$800$	0	0
$801$	−6.52175	−0.230435
$802$	−27.4332	−0.968698
$803$	−0.317643	−0.0112094
$804$	18.9556	0.668513
$805$	0	0
$806$	−15.8508	−0.558319
$807$	46.1450	1.62438
$808$	5.61775	0.197632
$809$	29.7376	1.04552	0.522759	−	0.852481i	$-0.324903\pi$
$809$	29.7376	1.04552	0.522759	+	0.852481i	$0.324903\pi$
$810$	0	0
$811$	15.7650	0.553585	0.276793	−	0.960930i	$-0.410729\pi$
$811$	15.7650	0.553585	0.276793	+	0.960930i	$0.410729\pi$
$812$	19.6409	0.689260
$813$	−22.7928	−0.799378
$814$	0.0451320	0.00158187
$815$	0	0
$816$	5.52749	0.193501
$817$	−34.5079	−1.20728
$818$	21.3920	0.747954
$819$	−16.0816	−0.561937
$820$	0	0
$821$	−4.27798	−0.149303	−0.0746513	−	0.997210i	$-0.523784\pi$
$821$	−4.27798	−0.149303	−0.0746513	+	0.997210i	$0.523784\pi$
$822$	−4.93014	−0.171958
$823$	13.8985	0.484471	0.242236	−	0.970217i	$-0.422119\pi$
$823$	13.8985	0.484471	0.242236	+	0.970217i	$0.422119\pi$
$824$	−12.5663	−0.437766
$825$	0	0
$826$	63.2019	2.19908
$827$	30.4944	1.06039	0.530197	−	0.847875i	$-0.322118\pi$
$827$	30.4944	1.06039	0.530197	+	0.847875i	$0.322118\pi$
$828$	−1.43930	−0.0500192
$829$	0.493020	0.0171233	0.00856165	−	0.999963i	$-0.497275\pi$
$829$	0.493020	0.0171233	0.00856165	+	0.999963i	$0.497275\pi$
$830$	0	0
$831$	0.225593	0.00782574
$832$	−5.26514	−0.182536
$833$	−53.5107	−1.85404
$834$	32.0749	1.11066
$835$	0	0
$836$	−0.280721	−0.00970894
$837$	−16.8487	−0.582376
$838$	−23.0891	−0.797598
$839$	−27.9308	−0.964278	−0.482139	−	0.876095i	$-0.660140\pi$
$839$	−27.9308	−0.964278	−0.482139	+	0.876095i	$0.660140\pi$
$840$	0	0
$841$	−11.3276	−0.390606
$842$	1.83757	0.0633268
$843$	39.0975	1.34659
$844$	6.77438	0.233184
$845$	0	0
$846$	2.79849	0.0962141
$847$	51.3837	1.76556
$848$	−6.10215	−0.209549
$849$	0.0861901	0.00295804
$850$	0	0
$851$	−2.20164	−0.0754711
$852$	6.88297	0.235807
$853$	−31.4029	−1.07521	−0.537607	−	0.843195i	$-0.680672\pi$
$853$	−31.4029	−1.07521	−0.537607	+	0.843195i	$0.680672\pi$
$854$	−19.9639	−0.683150
$855$	0	0
$856$	−17.9395	−0.613158
$857$	−39.7806	−1.35888	−0.679441	−	0.733731i	$-0.737777\pi$
$857$	−39.7806	−1.35888	−0.679441	+	0.733731i	$0.737777\pi$
$858$	−0.363983	−0.0124262
$859$	41.8673	1.42849	0.714247	−	0.699894i	$-0.246769\pi$
$859$	41.8673	1.42849	0.714247	+	0.699894i	$0.246769\pi$
$860$	0	0
$861$	−52.8364	−1.80066
$862$	15.9796	0.544267
$863$	8.52387	0.290156	0.145078	−	0.989420i	$-0.453657\pi$
$863$	8.52387	0.290156	0.145078	+	0.989420i	$0.453657\pi$
$864$	−5.59662	−0.190401
$865$	0	0
$866$	−25.2525	−0.858116
$867$	6.09319	0.206936
$868$	14.0654	0.477412
$869$	−0.393776	−0.0133579
$870$	0	0
$871$	65.1568	2.20775
$872$	−3.09936	−0.104958
$873$	0.157165	0.00531922
$874$	13.6942	0.463213
$875$	0	0
$876$	10.7806	0.364243
$877$	37.1163	1.25333	0.626664	−	0.779289i	$-0.284420\pi$
$877$	37.1163	1.25333	0.626664	+	0.779289i	$0.284420\pi$
$878$	9.30792	0.314127
$879$	20.9901	0.707977
$880$	0	0
$881$	29.8633	1.00612	0.503060	−	0.864252i	$-0.332208\pi$
$881$	29.8633	1.00612	0.503060	+	0.864252i	$0.332208\pi$
$882$	9.69410	0.326417
$883$	9.98952	0.336174	0.168087	−	0.985772i	$-0.446241\pi$
$883$	9.98952	0.336174	0.168087	+	0.985772i	$0.446241\pi$
$884$	18.9998	0.639033
$885$	0	0
$886$	−9.52445	−0.319980
$887$	−23.2703	−0.781340	−0.390670	−	0.920531i	$-0.627757\pi$
$887$	−23.2703	−0.781340	−0.390670	+	0.920531i	$0.627757\pi$
$888$	−1.53175	−0.0514022
$889$	82.4965	2.76684
$890$	0	0
$891$	−0.298385	−0.00999628
$892$	10.2559	0.343392
$893$	−26.6261	−0.891008
$894$	−33.6728	−1.12619
$895$	0	0
$896$	4.67211	0.156084
$897$	17.7559	0.592852
$898$	−5.43278	−0.181294
$899$	12.6558	0.422094
$900$	0	0
$901$	22.0203	0.733602
$902$	0.333209	0.0110946
$903$	39.7035	1.32125
$904$	4.36184	0.145073
$905$	0	0
$906$	21.7320	0.721998
$907$	34.0053	1.12913	0.564564	−	0.825389i	$-0.309044\pi$
$907$	34.0053	1.12913	0.564564	+	0.825389i	$0.309044\pi$
$908$	11.2744	0.374153
$909$	3.67256	0.121811
$910$	0	0
$911$	−29.2297	−0.968424	−0.484212	−	0.874951i	$-0.660894\pi$
$911$	−29.2297	−0.968424	−0.484212	+	0.874951i	$0.660894\pi$
$912$	9.52749	0.315487
$913$	0.0397266	0.00131476
$914$	7.54789	0.249662
$915$	0	0
$916$	−9.40494	−0.310748
$917$	48.1707	1.59074
$918$	20.1960	0.666568
$919$	43.0238	1.41923	0.709613	−	0.704592i	$-0.248870\pi$
$919$	43.0238	1.41923	0.709613	+	0.704592i	$0.248870\pi$
$920$	0	0
$921$	−26.9317	−0.887430
$922$	35.4323	1.16690
$923$	23.6591	0.778748
$924$	0.322987	0.0106255
$925$	0	0
$926$	20.9124	0.687225
$927$	−8.21510	−0.269819
$928$	4.20386	0.137998
$929$	10.7786	0.353635	0.176817	−	0.984244i	$-0.443420\pi$
$929$	10.7786	0.353635	0.176817	+	0.984244i	$0.443420\pi$
$930$	0	0
$931$	−92.2340	−3.02285
$932$	3.22867	0.105759
$933$	−0.540912	−0.0177087
$934$	6.62523	0.216784
$935$	0	0
$936$	−3.44204	−0.112507
$937$	−36.5755	−1.19487	−0.597435	−	0.801918i	$-0.703813\pi$
$937$	−36.5755	−1.19487	−0.597435	+	0.801918i	$0.703813\pi$
$938$	−57.8180	−1.88782
$939$	−28.9067	−0.943336
$940$	0	0
$941$	58.9579	1.92197	0.960986	−	0.276596i	$-0.0892064\pi$
$941$	58.9579	1.92197	0.960986	+	0.276596i	$0.0892064\pi$
$942$	30.5303	0.994730
$943$	−16.2547	−0.529325
$944$	13.5275	0.440282
$945$	0	0
$946$	−0.250387	−0.00814079
$947$	−9.63990	−0.313255	−0.156627	−	0.987658i	$-0.550062\pi$
$947$	−9.63990	−0.313255	−0.156627	+	0.987658i	$0.550062\pi$
$948$	13.3645	0.434059
$949$	37.0566	1.20291
$950$	0	0
$951$	23.2794	0.754887
$952$	−16.8598	−0.546430
$953$	−31.4809	−1.01977	−0.509884	−	0.860243i	$-0.670312\pi$
$953$	−31.4809	−1.01977	−0.509884	+	0.860243i	$0.670312\pi$
$954$	−3.98923	−0.129156
$955$	0	0
$956$	−17.8612	−0.577672
$957$	0.290616	0.00939430
$958$	29.4581	0.951747
$959$	15.0378	0.485596
$960$	0	0
$961$	−21.9368	−0.707639
$962$	−5.26514	−0.169755
$963$	−11.7278	−0.377923
$964$	26.6840	0.859434
$965$	0	0
$966$	−15.7560	−0.506941
$967$	−25.5439	−0.821438	−0.410719	−	0.911762i	$-0.634722\pi$
$967$	−25.5439	−0.821438	−0.410719	+	0.911762i	$0.634722\pi$
$968$	10.9980	0.353488
$969$	−34.3810	−1.10448
$970$	0	0
$971$	43.7224	1.40312	0.701559	−	0.712611i	$-0.252488\pi$
$971$	43.7224	1.40312	0.701559	+	0.712611i	$0.252488\pi$
$972$	−6.66286	−0.213711
$973$	−97.8341	−3.13642
$974$	8.80654	0.282180
$975$	0	0
$976$	−4.27299	−0.136775
$977$	44.6598	1.42879	0.714396	−	0.699741i	$-0.246701\pi$
$977$	44.6598	1.42879	0.714396	+	0.699741i	$0.246701\pi$
$978$	−13.2510	−0.423719
$979$	0.450237	0.0143896
$980$	0	0
$981$	−2.02619	−0.0646912
$982$	−5.14258	−0.164106
$983$	−9.17840	−0.292745	−0.146373	−	0.989230i	$-0.546760\pi$
$983$	−9.17840	−0.292745	−0.146373	+	0.989230i	$0.546760\pi$
$984$	−11.3089	−0.360514
$985$	0	0
$986$	−15.1701	−0.483114
$987$	30.6350	0.975123
$988$	32.7492	1.04189
$989$	12.2144	0.388397
$990$	0	0
$991$	16.1246	0.512216	0.256108	−	0.966648i	$-0.417560\pi$
$991$	16.1246	0.512216	0.256108	+	0.966648i	$0.417560\pi$
$992$	3.01051	0.0955839
$993$	35.2624	1.11902
$994$	−20.9943	−0.665899
$995$	0	0
$996$	−1.34829	−0.0427223
$997$	−8.99979	−0.285026	−0.142513	−	0.989793i	$-0.545518\pi$
$997$	−8.99979	−0.285026	−0.142513	+	0.989793i	$0.545518\pi$
$998$	12.9692	0.410532
$999$	−5.59662	−0.177069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bd.1.2		5
5.2	odd	4		370.2.b.d.149.4	✓	10
5.3	odd	4		370.2.b.d.149.7	yes	10
5.4	even	2		1850.2.a.be.1.4		5
15.2	even	4		3330.2.d.p.1999.6		10
15.8	even	4		3330.2.d.p.1999.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.d.149.4	✓	10	5.2	odd	4
370.2.b.d.149.7	yes	10	5.3	odd	4
1850.2.a.bd.1.2		5	1.1	even	1	trivial
1850.2.a.be.1.4		5	5.4	even	2
3330.2.d.p.1999.1		10	15.8	even	4
3330.2.d.p.1999.6		10	15.2	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} -4.67211 q^{7} -1.00000 q^{8} -0.653743 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} -4.67211 q^{7} -1.00000 q^{8} -0.653743 q^{9} +0.0451320 q^{11} -1.53175 q^{12} -5.26514 q^{13} +4.67211 q^{14} +1.00000 q^{16} -3.60861 q^{17} +0.653743 q^{18} -6.22000 q^{19} +7.15650 q^{21} -0.0451320 q^{22} +2.20164 q^{23} +1.53175 q^{24} +5.26514 q^{26} +5.59662 q^{27} -4.67211 q^{28} -4.20386 q^{29} -3.01051 q^{31} -1.00000 q^{32} -0.0691309 q^{33} +3.60861 q^{34} -0.653743 q^{36} -1.00000 q^{37} +6.22000 q^{38} +8.06487 q^{39} -7.38299 q^{41} -7.15650 q^{42} +5.54789 q^{43} +0.0451320 q^{44} -2.20164 q^{46} +4.28072 q^{47} -1.53175 q^{48} +14.8286 q^{49} +5.52749 q^{51} -5.26514 q^{52} -6.10215 q^{53} -5.59662 q^{54} +4.67211 q^{56} +9.52749 q^{57} +4.20386 q^{58} +13.5275 q^{59} -4.27299 q^{61} +3.01051 q^{62} +3.05436 q^{63} +1.00000 q^{64} +0.0691309 q^{66} -12.3751 q^{67} -3.60861 q^{68} -3.37235 q^{69} -4.49354 q^{71} +0.653743 q^{72} -7.03811 q^{73} +1.00000 q^{74} -6.22000 q^{76} -0.210861 q^{77} -8.06487 q^{78} -8.72499 q^{79} -6.61139 q^{81} +7.38299 q^{82} +0.880231 q^{83} +7.15650 q^{84} -5.54789 q^{86} +6.43926 q^{87} -0.0451320 q^{88} +9.97602 q^{89} +24.5993 q^{91} +2.20164 q^{92} +4.61135 q^{93} -4.28072 q^{94} +1.53175 q^{96} -0.240408 q^{97} -14.8286 q^{98} -0.0295047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Introduction

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