LMFDB - Embedded newform 275.4.b.f.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 80x^{8} + 2296x^{6} + 27417x^{4} + 110472x^{2} + 21904 $ x^10 + 80x^8 + 2296x^6 + 27417x^4 + 110472x^2 + 21904
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-4.78071i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.4.b.f.199.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.78071i q^{2} +5.99252i q^{3} -14.8552 q^{4} +28.6485 q^{6} -11.9641i q^{7} +32.7728i q^{8} -8.91034 q^{9} -11.0000 q^{11} -89.0202i q^{12} +22.4142i q^{13} -57.1967 q^{14} +37.8357 q^{16} +131.986i q^{17} +42.5978i q^{18} -99.0508 q^{19} +71.6949 q^{21} +52.5878i q^{22} -0.206447i q^{23} -196.392 q^{24} +107.156 q^{26} +108.403i q^{27} +177.729i q^{28} +163.714 q^{29} +217.432 q^{31} +81.3008i q^{32} -65.9178i q^{33} +630.987 q^{34} +132.365 q^{36} -17.8883i q^{37} +473.533i q^{38} -134.318 q^{39} -32.3119 q^{41} -342.753i q^{42} +490.270i q^{43} +163.407 q^{44} -0.986965 q^{46} +518.924i q^{47} +226.731i q^{48} +199.861 q^{49} -790.929 q^{51} -332.968i q^{52} +110.670i q^{53} +518.242 q^{54} +392.096 q^{56} -593.564i q^{57} -782.669i q^{58} -242.866 q^{59} -713.857 q^{61} -1039.48i q^{62} +106.604i q^{63} +691.362 q^{64} -315.134 q^{66} -571.193i q^{67} -1960.68i q^{68} +1.23714 q^{69} -113.267 q^{71} -292.017i q^{72} +767.158i q^{73} -85.5187 q^{74} +1471.42 q^{76} +131.605i q^{77} +642.135i q^{78} -470.874 q^{79} -890.185 q^{81} +154.474i q^{82} -1158.67i q^{83} -1065.04 q^{84} +2343.84 q^{86} +981.059i q^{87} -360.501i q^{88} -719.465 q^{89} +268.165 q^{91} +3.06682i q^{92} +1302.96i q^{93} +2480.83 q^{94} -487.197 q^{96} +510.721i q^{97} -955.480i q^{98} +98.0137 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41}+ \cdots + 682 q^{99}+O(q^{100}) $ 10 * q - 80 * q^4 - 84 * q^6 - 62 * q^9 - 110 * q^11 + 266 * q^14 + 416 * q^16 - 46 * q^19 + 564 * q^21 + 982 * q^24 + 644 * q^26 + 366 * q^29 - 2 * q^31 + 1300 * q^34 + 2676 * q^36 + 540 * q^39 + 188 * q^41 + 880 * q^44 + 2012 * q^46 - 454 * q^49 - 2256 * q^51 + 2410 * q^54 - 1670 * q^56 - 2348 * q^59 + 1280 * q^61 + 1898 * q^64 + 924 * q^66 + 3292 * q^69 - 1346 * q^71 + 1122 * q^74 + 2496 * q^76 - 208 * q^79 - 2438 * q^81 - 8706 * q^84 + 4994 * q^86 - 5642 * q^89 + 960 * q^91 + 3574 * q^94 - 3638 * q^96 + 682 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.78071i	− 1.69024i	−0.534579	−	0.845119i	$-0.679530\pi$
$2$	− 4.78071i	− 1.69024i	0.534579	−	0.845119i	$-0.320470\pi$
$3$	5.99252i	1.15326i	0.817005	+	0.576631i	$0.195633\pi$
$3$	5.99252i	1.15326i	−0.817005	+	0.576631i	$0.804367\pi$
$4$	−14.8552	−1.85690
$5$	0	0
$5$	0	0
$6$	28.6485	1.94929
$7$	− 11.9641i	− 0.645998i	−0.946399	−	0.322999i	$-0.895309\pi$
$7$	− 11.9641i	− 0.645998i	0.946399	−	0.322999i	$-0.104691\pi$
$8$	32.7728i	1.44837i
$9$	−8.91034	−0.330013
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 89.0202i	− 2.14149i
$13$	22.4142i	0.478199i	0.970995	+	0.239100i	$0.0768523\pi$
$13$	22.4142i	0.478199i	−0.970995	+	0.239100i	$0.923148\pi$
$14$	−57.1967	−1.09189
$15$	0	0
$16$	37.8357	0.591183
$17$	131.986i	1.88302i	0.336989	+	0.941509i	$0.390592\pi$
$17$	131.986i	1.88302i	−0.336989	+	0.941509i	$0.609408\pi$
$18$	42.5978i	0.557800i
$19$	−99.0508	−1.19599	−0.597995	−	0.801500i	$-0.704036\pi$
$19$	−99.0508	−1.19599	−0.597995	+	0.801500i	$0.704036\pi$
$20$	0	0
$21$	71.6949	0.745005
$22$	52.5878i	0.509626i
$23$	− 0.206447i	− 0.00187162i	−1.00000		0.000935809i	$-0.999702\pi$
$23$	− 0.206447i	− 0.00187162i	1.00000		0.000935809i	$-0.000297877\pi$
$24$	−196.392	−1.67035
$25$	0	0
$26$	107.156	0.808270
$27$	108.403i	0.772671i
$28$	177.729i	1.19956i
$29$	163.714	1.04831	0.524154	−	0.851624i	$-0.324382\pi$
$29$	163.714	1.04831	0.524154	+	0.851624i	$0.324382\pi$
$30$	0	0
$31$	217.432	1.25974	0.629869	−	0.776702i	$-0.283109\pi$
$31$	217.432	1.25974	0.629869	+	0.776702i	$0.283109\pi$
$32$	81.3008i	0.449128i
$33$	− 65.9178i	− 0.347721i
$34$	630.987	3.18275
$35$	0	0
$36$	132.365	0.612801
$37$	− 17.8883i	− 0.0794814i	−0.999210	−	0.0397407i	$-0.987347\pi$
$37$	− 17.8883i	− 0.0794814i	0.999210	−	0.0397407i	$-0.0126532\pi$
$38$	473.533i	2.02151i
$39$	−134.318	−0.551489
$40$	0	0
$41$	−32.3119	−0.123080	−0.0615399	−	0.998105i	$-0.519601\pi$
$41$	−32.3119	−0.123080	−0.0615399	+	0.998105i	$0.519601\pi$
$42$	− 342.753i	− 1.25924i
$43$	490.270i	1.73873i	0.494169	+	0.869366i	$0.335472\pi$
$43$	490.270i	1.73873i	−0.494169	+	0.869366i	$0.664528\pi$
$44$	163.407	0.559877
$45$	0	0
$46$	−0.986965	−0.00316348
$47$	518.924i	1.61049i	0.592944	+	0.805244i	$0.297965\pi$
$47$	518.924i	1.61049i	−0.592944	+	0.805244i	$0.702035\pi$
$48$	226.731i	0.681789i
$49$	199.861	0.582686
$50$	0	0
$51$	−790.929	−2.17161
$52$	− 332.968i	− 0.887969i
$53$	110.670i	0.286825i	0.989663	+	0.143413i	$0.0458076\pi$
$53$	110.670i	0.286825i	−0.989663	+	0.143413i	$0.954192\pi$
$54$	518.242	1.30600
$55$	0	0
$56$	392.096	0.935643
$57$	− 593.564i	− 1.37929i
$58$	− 782.669i	− 1.77189i
$59$	−242.866	−0.535906	−0.267953	−	0.963432i	$-0.586347\pi$
$59$	−242.866	−0.535906	−0.267953	+	0.963432i	$0.586347\pi$
$60$	0	0
$61$	−713.857	−1.49836	−0.749180	−	0.662366i	$-0.769552\pi$
$61$	−713.857	−1.49836	−0.749180	+	0.662366i	$0.769552\pi$
$62$	− 1039.48i	− 2.12926i
$63$	106.604i	0.213188i
$64$	691.362	1.35032
$65$	0	0
$66$	−315.134	−0.587732
$67$	− 571.193i	− 1.04153i	−0.853701	−	0.520764i	$-0.825647\pi$
$67$	− 571.193i	− 1.04153i	0.853701	−	0.520764i	$-0.174353\pi$
$68$	− 1960.68i	− 3.49658i
$69$	1.23714	0.00215847
$70$	0	0
$71$	−113.267	−0.189329	−0.0946644	−	0.995509i	$-0.530178\pi$
$71$	−113.267	−0.189329	−0.0946644	+	0.995509i	$0.530178\pi$
$72$	− 292.017i	− 0.477980i
$73$	767.158i	1.22999i	0.788532	+	0.614994i	$0.210841\pi$
$73$	767.158i	1.22999i	−0.788532	+	0.614994i	$0.789159\pi$
$74$	−85.5187	−0.134342
$75$	0	0
$76$	1471.42	2.22084
$77$	131.605i	0.194776i
$78$	642.135i	0.932147i
$79$	−470.874	−0.670601	−0.335300	−	0.942111i	$-0.608838\pi$
$79$	−470.874	−0.670601	−0.335300	+	0.942111i	$0.608838\pi$
$80$	0	0
$81$	−890.185	−1.22110
$82$	154.474i	0.208034i
$83$	− 1158.67i	− 1.53230i	−0.642661	−	0.766151i	$-0.722170\pi$
$83$	− 1158.67i	− 1.53230i	0.642661	−	0.766151i	$-0.277830\pi$
$84$	−1065.04	−1.38340
$85$	0	0
$86$	2343.84	2.93887
$87$	981.059i	1.20897i
$88$	− 360.501i	− 0.436699i
$89$	−719.465	−0.856889	−0.428444	−	0.903568i	$-0.640938\pi$
$89$	−719.465	−0.856889	−0.428444	+	0.903568i	$0.640938\pi$
$90$	0	0
$91$	268.165	0.308916
$92$	3.06682i	0.00347541i
$93$	1302.96i	1.45281i
$94$	2480.83	2.72211
$95$	0	0
$96$	−487.197	−0.517962
$97$	510.721i	0.534596i	0.963614	+	0.267298i	$0.0861308\pi$
$97$	510.721i	0.534596i	−0.963614	+	0.267298i	$0.913869\pi$
$98$	− 955.480i	− 0.984878i
$99$	98.0137	0.0995025
$100$	0	0
$101$	−806.012	−0.794072	−0.397036	−	0.917803i	$-0.629961\pi$
$101$	−806.012	−0.794072	−0.397036	+	0.917803i	$0.629961\pi$
$102$	3781.20i	3.67054i
$103$	1851.44i	1.77114i	0.464506	+	0.885570i	$0.346232\pi$
$103$	1851.44i	1.77114i	−0.464506	+	0.885570i	$0.653768\pi$
$104$	−734.578	−0.692609
$105$	0	0
$106$	529.083	0.484803
$107$	− 1429.22i	− 1.29129i	−0.763637	−	0.645646i	$-0.776588\pi$
$107$	− 1429.22i	− 1.29129i	0.763637	−	0.645646i	$-0.223412\pi$
$108$	− 1610.35i	− 1.43477i
$109$	1027.59	0.902984	0.451492	−	0.892275i	$-0.350892\pi$
$109$	1027.59	0.902984	0.451492	+	0.892275i	$0.350892\pi$
$110$	0	0
$111$	107.196	0.0916629
$112$	− 452.669i	− 0.381903i
$113$	− 604.312i	− 0.503087i	−0.967846	−	0.251544i	$-0.919062\pi$
$113$	− 604.312i	− 0.503087i	0.967846	−	0.251544i	$-0.0809382\pi$
$114$	−2837.66	−2.33133
$115$	0	0
$116$	−2432.01	−1.94660
$117$	− 199.719i	− 0.157812i
$118$	1161.07i	0.905808i
$119$	1579.09	1.21643
$120$	0	0
$121$	121.000	0.0909091
$122$	3412.74i	2.53258i
$123$	− 193.630i	− 0.141943i
$124$	−3229.99	−2.33921
$125$	0	0
$126$	509.642	0.360337
$127$	− 1150.51i	− 0.803871i	−0.915668	−	0.401936i	$-0.868338\pi$
$127$	− 1150.51i	− 0.803871i	0.915668	−	0.401936i	$-0.131662\pi$
$128$	− 2654.80i	− 1.83323i
$129$	−2937.95	−2.00521
$130$	0	0
$131$	744.407	0.496482	0.248241	−	0.968698i	$-0.420148\pi$
$131$	744.407	0.496482	0.248241	+	0.968698i	$0.420148\pi$
$132$	979.223i	0.645685i
$133$	1185.05i	0.772607i
$134$	−2730.71	−1.76043
$135$	0	0
$136$	−4325.55	−2.72730
$137$	− 1216.87i	− 0.758862i	−0.925220	−	0.379431i	$-0.876120\pi$
$137$	− 1216.87i	− 0.758862i	0.925220	−	0.379431i	$-0.123880\pi$
$138$	− 5.91441i	− 0.00364832i
$139$	−2180.15	−1.33035	−0.665173	−	0.746689i	$-0.731642\pi$
$139$	−2180.15	−1.33035	−0.665173	+	0.746689i	$0.731642\pi$
$140$	0	0
$141$	−3109.67	−1.85731
$142$	541.498i	0.320010i
$143$	− 246.557i	− 0.144183i
$144$	−337.129	−0.195098
$145$	0	0
$146$	3667.56	2.07897
$147$	1197.67i	0.671990i
$148$	265.734i	0.147589i
$149$	2465.36	1.35551	0.677753	−	0.735289i	$-0.262954\pi$
$149$	2465.36	1.35551	0.677753	+	0.735289i	$0.262954\pi$
$150$	0	0
$151$	−1568.89	−0.845528	−0.422764	−	0.906240i	$-0.638940\pi$
$151$	−1568.89	−0.845528	−0.422764	+	0.906240i	$0.638940\pi$
$152$	− 3246.17i	− 1.73223i
$153$	− 1176.04i	− 0.621419i
$154$	629.164	0.329217
$155$	0	0
$156$	1995.32	1.02406
$157$	3091.06i	1.57130i	0.618672	+	0.785649i	$0.287671\pi$
$157$	3091.06i	1.57130i	−0.618672	+	0.785649i	$0.712329\pi$
$158$	2251.11i	1.13347i
$159$	−663.195	−0.330785
$160$	0	0
$161$	−2.46995	−0.00120906
$162$	4255.72i	2.06396i
$163$	− 2402.69i	− 1.15456i	−0.816547	−	0.577279i	$-0.804114\pi$
$163$	− 2402.69i	− 1.15456i	0.816547	−	0.577279i	$-0.195886\pi$
$164$	480.000	0.228547
$165$	0	0
$166$	−5539.29	−2.58995
$167$	− 3771.51i	− 1.74759i	−0.486290	−	0.873797i	$-0.661650\pi$
$167$	− 3771.51i	− 1.74759i	0.486290	−	0.873797i	$-0.338350\pi$
$168$	2349.64i	1.07904i
$169$	1694.60	0.771325
$170$	0	0
$171$	882.576	0.394692
$172$	− 7283.07i	− 3.22865i
$173$	1744.46i	0.766642i	0.923615	+	0.383321i	$0.125220\pi$
$173$	1744.46i	0.766642i	−0.923615	+	0.383321i	$0.874780\pi$
$174$	4690.16	2.04345
$175$	0	0
$176$	−416.193	−0.178248
$177$	− 1455.38i	− 0.618039i
$178$	3439.55i	1.44835i
$179$	503.463	0.210227	0.105113	−	0.994460i	$-0.466479\pi$
$179$	503.463	0.210227	0.105113	+	0.994460i	$0.466479\pi$
$180$	0	0
$181$	1994.94	0.819240	0.409620	−	0.912256i	$-0.365661\pi$
$181$	1994.94	0.819240	0.409620	+	0.912256i	$0.365661\pi$
$182$	− 1282.02i	− 0.522141i
$183$	− 4277.80i	− 1.72800i
$184$	6.76586	0.00271079
$185$	0	0
$186$	6229.10	2.45559
$187$	− 1451.85i	− 0.567751i
$188$	− 7708.73i	− 2.99052i
$189$	1296.94	0.499144
$190$	0	0
$191$	1385.75	0.524972	0.262486	−	0.964936i	$-0.415458\pi$
$191$	1385.75	0.524972	0.262486	+	0.964936i	$0.415458\pi$
$192$	4143.00i	1.55727i
$193$	1215.82i	0.453454i	0.973958	+	0.226727i	$0.0728025\pi$
$193$	1215.82i	0.453454i	−0.973958	+	0.226727i	$0.927198\pi$
$194$	2441.61	0.903594
$195$	0	0
$196$	−2968.98	−1.08199
$197$	− 1113.77i	− 0.402806i	−0.979508	−	0.201403i	$-0.935450\pi$
$197$	− 1113.77i	− 0.402806i	0.979508	−	0.201403i	$-0.0645501\pi$
$198$	− 468.576i	− 0.168183i
$199$	988.791	0.352229	0.176114	−	0.984370i	$-0.443647\pi$
$199$	988.791	0.352229	0.176114	+	0.984370i	$0.443647\pi$
$200$	0	0
$201$	3422.89	1.20115
$202$	3853.31i	1.34217i
$203$	− 1958.68i	− 0.677204i
$204$	11749.4	4.03247
$205$	0	0
$206$	8851.18	2.99365
$207$	1.83951i	0 0.000617658i
$208$	848.059i	0.282703i
$209$	1089.56	0.360605
$210$	0	0
$211$	798.252	0.260445	0.130223	−	0.991485i	$-0.458431\pi$
$211$	798.252	0.260445	0.130223	+	0.991485i	$0.458431\pi$
$212$	− 1644.03i	− 0.532607i
$213$	− 678.756i	− 0.218346i
$214$	−6832.70	−2.18259
$215$	0	0
$216$	−3552.66	−1.11911
$217$	− 2601.36i	− 0.813788i
$218$	− 4912.61i	− 1.52626i
$219$	−4597.21	−1.41850
$220$	0	0
$221$	−2958.37	−0.900458
$222$	− 512.473i	− 0.154932i
$223$	1836.54i	0.551497i	0.961230	+	0.275749i	$0.0889258\pi$
$223$	1836.54i	0.551497i	−0.961230	+	0.275749i	$0.911074\pi$
$224$	972.688	0.290136
$225$	0	0
$226$	−2889.04	−0.850337
$227$	− 2009.93i	− 0.587681i	−0.955854	−	0.293841i	$-0.905066\pi$
$227$	− 2009.93i	− 0.587681i	0.955854	−	0.293841i	$-0.0949335\pi$
$228$	8817.52i	2.56121i
$229$	1063.95	0.307022	0.153511	−	0.988147i	$-0.450942\pi$
$229$	1063.95	0.307022	0.153511	+	0.988147i	$0.450942\pi$
$230$	0	0
$231$	−788.644	−0.224627
$232$	5365.37i	1.51833i
$233$	1259.97i	0.354264i	0.984187	+	0.177132i	$0.0566819\pi$
$233$	1259.97i	0.354264i	−0.984187	+	0.177132i	$0.943318\pi$
$234$	−954.797	−0.266739
$235$	0	0
$236$	3607.82	0.995124
$237$	− 2821.72i	− 0.773378i
$238$	− 7549.16i	− 2.05605i
$239$	−6963.01	−1.88452	−0.942258	−	0.334887i	$-0.891302\pi$
$239$	−6963.01	−1.88452	−0.942258	+	0.334887i	$0.891302\pi$
$240$	0	0
$241$	−6364.98	−1.70126	−0.850632	−	0.525762i	$-0.823780\pi$
$241$	−6364.98	−1.70126	−0.850632	+	0.525762i	$0.823780\pi$
$242$	− 578.466i	− 0.153658i
$243$	− 2407.58i	− 0.635582i
$244$	10604.5	2.78231
$245$	0	0
$246$	−925.688	−0.239918
$247$	− 2220.15i	− 0.571922i
$248$	7125.85i	1.82456i
$249$	6943.38	1.76714
$250$	0	0
$251$	3733.69	0.938919	0.469459	−	0.882954i	$-0.344449\pi$
$251$	3733.69	0.938919	0.469459	+	0.882954i	$0.344449\pi$
$252$	− 1583.62i	− 0.395868i
$253$	2.27092i	0 0.000564314i
$254$	−5500.28	−1.35873
$255$	0	0
$256$	−7160.92	−1.74827
$257$	3819.22i	0.926990i	0.886099	+	0.463495i	$0.153405\pi$
$257$	3819.22i	0.926990i	−0.886099	+	0.463495i	$0.846595\pi$
$258$	14045.5i	3.38929i
$259$	−214.016	−0.0513448
$260$	0	0
$261$	−1458.75	−0.345955
$262$	− 3558.79i	− 0.839172i
$263$	5634.49i	1.32106i	0.750802	+	0.660528i	$0.229667\pi$
$263$	5634.49i	1.32106i	−0.750802	+	0.660528i	$0.770333\pi$
$264$	2160.31	0.503629
$265$	0	0
$266$	5665.38	1.30589
$267$	− 4311.41i	− 0.988217i
$268$	8485.20i	1.93401i
$269$	−4313.40	−0.977667	−0.488834	−	0.872377i	$-0.662577\pi$
$269$	−4313.40	−0.977667	−0.488834	+	0.872377i	$0.662577\pi$
$270$	0	0
$271$	2870.96	0.643537	0.321768	−	0.946818i	$-0.395723\pi$
$271$	2870.96	0.643537	0.321768	+	0.946818i	$0.395723\pi$
$272$	4993.78i	1.11321i
$273$	1606.99i	0.356261i
$274$	−5817.50	−1.28266
$275$	0	0
$276$	−18.3780	−0.00400806
$277$	− 3007.06i	− 0.652262i	−0.945325	−	0.326131i	$-0.894255\pi$
$277$	− 3007.06i	− 0.652262i	0.945325	−	0.326131i	$-0.105745\pi$
$278$	10422.7i	2.24860i
$279$	−1937.39	−0.415729
$280$	0	0
$281$	4149.04	0.880821	0.440411	−	0.897796i	$-0.354833\pi$
$281$	4149.04	0.880821	0.440411	+	0.897796i	$0.354833\pi$
$282$	14866.4i	3.13930i
$283$	3269.92i	0.686844i	0.939181	+	0.343422i	$0.111586\pi$
$283$	3269.92i	0.686844i	−0.939181	+	0.343422i	$0.888414\pi$
$284$	1682.61	0.351565
$285$	0	0
$286$	−1178.72	−0.243703
$287$	386.581i	0.0795093i
$288$	− 724.418i	− 0.148218i
$289$	−12507.3	−2.54575
$290$	0	0
$291$	−3060.51	−0.616529
$292$	− 11396.3i	− 2.28397i
$293$	3423.02i	0.682508i	0.939971	+	0.341254i	$0.110852\pi$
$293$	3423.02i	0.682508i	−0.939971	+	0.341254i	$0.889148\pi$
$294$	5725.74	1.13582
$295$	0	0
$296$	586.249	0.115118
$297$	− 1192.43i	− 0.232969i
$298$	− 11786.2i	− 2.29113i
$299$	4.62736	0.000895007 0
$300$	0	0
$301$	5865.62	1.12322
$302$	7500.43i	1.42914i
$303$	− 4830.05i	− 0.915772i
$304$	−3747.66	−0.707049
$305$	0	0
$306$	−5622.31	−1.05035
$307$	60.1063i	0.0111741i	0.999984	+	0.00558705i	$0.00177842\pi$
$307$	60.1063i	0.0111741i	−0.999984	+	0.00558705i	$0.998222\pi$
$308$	− 1955.01i	− 0.361680i
$309$	−11094.8	−2.04259
$310$	0	0
$311$	7623.50	1.39000	0.694998	−	0.719012i	$-0.255405\pi$
$311$	7623.50	1.39000	0.694998	+	0.719012i	$0.255405\pi$
$312$	− 4401.98i	− 0.798759i
$313$	− 273.026i	− 0.0493046i	−0.999696	−	0.0246523i	$-0.992152\pi$
$313$	− 273.026i	− 0.0493046i	0.999696	−	0.0246523i	$-0.00784787\pi$
$314$	14777.5	2.65587
$315$	0	0
$316$	6994.94	1.24524
$317$	− 4272.90i	− 0.757065i	−0.925588	−	0.378533i	$-0.876429\pi$
$317$	− 4272.90i	− 0.757065i	0.925588	−	0.378533i	$-0.123571\pi$
$318$	3170.54i	0.559105i
$319$	−1800.85	−0.316076
$320$	0	0
$321$	8564.65	1.48920
$322$	11.8081i	0.00204360i
$323$	− 13073.3i	− 2.25207i
$324$	13223.9	2.26747
$325$	0	0
$326$	−11486.6	−1.95148
$327$	6157.85i	1.04138i
$328$	− 1058.95i	− 0.178265i
$329$	6208.44	1.04037
$330$	0	0
$331$	−7137.75	−1.18528	−0.592638	−	0.805469i	$-0.701913\pi$
$331$	−7137.75	−1.18528	−0.592638	+	0.805469i	$0.701913\pi$
$332$	17212.4i	2.84533i
$333$	159.391i	0.0262299i
$334$	−18030.5	−2.95385
$335$	0	0
$336$	2712.63	0.440434
$337$	11179.0i	1.80700i	0.428591	+	0.903499i	$0.359010\pi$
$337$	11179.0i	1.80700i	−0.428591	+	0.903499i	$0.640990\pi$
$338$	− 8101.40i	− 1.30372i
$339$	3621.35	0.580191
$340$	0	0
$341$	−2391.75	−0.379825
$342$	− 4219.34i	− 0.667123i
$343$	− 6494.82i	− 1.02241i
$344$	−16067.5	−2.51832
$345$	0	0
$346$	8339.78	1.29581
$347$	− 2203.72i	− 0.340928i	−0.985364	−	0.170464i	$-0.945473\pi$
$347$	− 2203.72i	− 0.340928i	0.985364	−	0.170464i	$-0.0545266\pi$
$348$	− 14573.8i	− 2.24494i
$349$	6539.45	1.00300	0.501502	−	0.865156i	$-0.332781\pi$
$349$	6539.45	1.00300	0.501502	+	0.865156i	$0.332781\pi$
$350$	0	0
$351$	−2429.76	−0.369491
$352$	− 894.309i	− 0.135417i
$353$	10811.6i	1.63014i	0.579360	+	0.815072i	$0.303303\pi$
$353$	10811.6i	1.63014i	−0.579360	+	0.815072i	$0.696697\pi$
$354$	−6957.75	−1.04463
$355$	0	0
$356$	10687.8	1.59116
$357$	9462.72i	1.40286i
$358$	− 2406.91i	− 0.355333i
$359$	611.969	0.0899680	0.0449840	−	0.998988i	$-0.485676\pi$
$359$	611.969	0.0899680	0.0449840	+	0.998988i	$0.485676\pi$
$360$	0	0
$361$	2952.06	0.430392
$362$	− 9537.22i	− 1.38471i
$363$	725.095i	0.104842i
$364$	−3983.65	−0.573627
$365$	0	0
$366$	−20451.0	−2.92073
$367$	11822.2i	1.68151i	0.541419	+	0.840753i	$0.317887\pi$
$367$	11822.2i	1.68151i	−0.541419	+	0.840753i	$0.682113\pi$
$368$	− 7.81108i	− 0.00110647i
$369$	287.910	0.0406179
$370$	0	0
$371$	1324.07	0.185289
$372$	− 19355.8i	− 2.69772i
$373$	5067.95i	0.703508i	0.936093	+	0.351754i	$0.114415\pi$
$373$	5067.95i	0.703508i	−0.936093	+	0.351754i	$0.885585\pi$
$374$	−6940.86	−0.959634
$375$	0	0
$376$	−17006.6	−2.33258
$377$	3669.52i	0.501300i
$378$	− 6200.28i	− 0.843672i
$379$	10985.2	1.48885	0.744423	−	0.667708i	$-0.232724\pi$
$379$	10985.2	1.48885	0.744423	+	0.667708i	$0.232724\pi$
$380$	0	0
$381$	6894.48	0.927074
$382$	− 6624.89i	− 0.887327i
$383$	− 8682.54i	− 1.15837i	−0.815195	−	0.579187i	$-0.803370\pi$
$383$	− 8682.54i	− 1.15837i	0.815195	−	0.579187i	$-0.196630\pi$
$384$	15908.9	2.11419
$385$	0	0
$386$	5812.48	0.766444
$387$	− 4368.47i	− 0.573803i
$388$	− 7586.87i	− 0.992693i
$389$	12420.5	1.61888	0.809440	−	0.587203i	$-0.199771\pi$
$389$	12420.5	1.61888	0.809440	+	0.587203i	$0.199771\pi$
$390$	0	0
$391$	27.2481	0.00352429
$392$	6550.02i	0.843944i
$393$	4460.87i	0.572573i
$394$	−5324.62	−0.680838
$395$	0	0
$396$	−1456.02	−0.184766
$397$	− 5449.11i	− 0.688875i	−0.938809	−	0.344437i	$-0.888070\pi$
$397$	− 5449.11i	− 0.688875i	0.938809	−	0.344437i	$-0.111930\pi$
$398$	− 4727.13i	− 0.595350i
$399$	−7101.43	−0.891018
$400$	0	0
$401$	−11555.1	−1.43899	−0.719495	−	0.694498i	$-0.755627\pi$
$401$	−11555.1	−1.43899	−0.719495	+	0.694498i	$0.755627\pi$
$402$	− 16363.8i	− 2.03023i
$403$	4873.56i	0.602406i
$404$	11973.5	1.47451
$405$	0	0
$406$	−9363.90	−1.14464
$407$	196.771i	0.0239645i
$408$	− 25921.0i	− 3.14529i
$409$	−570.365	−0.0689553	−0.0344776	−	0.999405i	$-0.510977\pi$
$409$	−570.365	−0.0689553	−0.0344776	+	0.999405i	$0.510977\pi$
$410$	0	0
$411$	7292.11	0.875166
$412$	− 27503.5i	− 3.28883i
$413$	2905.66i	0.346194i
$414$	8.79419	0.00104399
$415$	0	0
$416$	−1822.30	−0.214773
$417$	− 13064.6i	− 1.53424i
$418$	− 5208.87i	− 0.609507i
$419$	12273.2	1.43098	0.715492	−	0.698620i	$-0.246202\pi$
$419$	12273.2	1.43098	0.715492	+	0.698620i	$0.246202\pi$
$420$	0	0
$421$	733.949	0.0849655	0.0424828	−	0.999097i	$-0.486473\pi$
$421$	733.949	0.0849655	0.0424828	+	0.999097i	$0.486473\pi$
$422$	− 3816.21i	− 0.440214i
$423$	− 4623.79i	− 0.531481i
$424$	−3626.98	−0.415429
$425$	0	0
$426$	−3244.94	−0.369056
$427$	8540.62i	0.967938i
$428$	21231.4i	2.39780i
$429$	1477.50	0.166280
$430$	0	0
$431$	−16442.1	−1.83756	−0.918782	−	0.394765i	$-0.870826\pi$
$431$	−16442.1	−1.83756	−0.918782	+	0.394765i	$0.870826\pi$
$432$	4101.50i	0.456790i
$433$	3747.82i	0.415955i	0.978134	+	0.207978i	$0.0666881\pi$
$433$	3747.82i	0.415955i	−0.978134	+	0.207978i	$0.933312\pi$
$434$	−12436.4	−1.37550
$435$	0	0
$436$	−15265.1	−1.67675
$437$	20.4488i	0.00223844i
$438$	21978.0i	2.39760i
$439$	−1917.83	−0.208504	−0.104252	−	0.994551i	$-0.533245\pi$
$439$	−1917.83	−0.208504	−0.104252	+	0.994551i	$0.533245\pi$
$440$	0	0
$441$	−1780.83	−0.192294
$442$	14143.1i	1.52199i
$443$	− 9764.08i	− 1.04719i	−0.851967	−	0.523595i	$-0.824590\pi$
$443$	− 9764.08i	− 1.04719i	0.851967	−	0.523595i	$-0.175410\pi$
$444$	−1592.42	−0.170209
$445$	0	0
$446$	8779.98	0.932162
$447$	14773.8i	1.56325i
$448$	− 8271.49i	− 0.872302i
$449$	9623.06	1.01145	0.505724	−	0.862695i	$-0.331225\pi$
$449$	9623.06	1.01145	0.505724	+	0.862695i	$0.331225\pi$
$450$	0	0
$451$	355.431	0.0371099
$452$	8977.18i	0.934184i
$453$	− 9401.63i	− 0.975115i
$454$	−9608.89	−0.993321
$455$	0	0
$456$	19452.8	1.99772
$457$	16210.3i	1.65927i	0.558303	+	0.829637i	$0.311453\pi$
$457$	16210.3i	1.65927i	−0.558303	+	0.829637i	$0.688547\pi$
$458$	− 5086.45i	− 0.518939i
$459$	−14307.6	−1.45495
$460$	0	0
$461$	−3932.92	−0.397341	−0.198671	−	0.980066i	$-0.563662\pi$
$461$	−3932.92	−0.397341	−0.198671	+	0.980066i	$0.563662\pi$
$462$	3770.28i	0.379674i
$463$	768.960i	0.0771849i	0.999255	+	0.0385925i	$0.0122874\pi$
$463$	768.960i	0.0771849i	−0.999255	+	0.0385925i	$0.987713\pi$
$464$	6194.23	0.619742
$465$	0	0
$466$	6023.56	0.598790
$467$	− 2635.60i	− 0.261159i	−0.991438	−	0.130579i	$-0.958316\pi$
$467$	− 2635.60i	− 0.261159i	0.991438	−	0.130579i	$-0.0416838\pi$
$468$	2966.86i	0.293041i
$469$	−6833.78	−0.672825
$470$	0	0
$471$	−18523.3	−1.81212
$472$	− 7959.40i	− 0.776188i
$473$	− 5392.97i	− 0.524247i
$474$	−13489.9	−1.30719
$475$	0	0
$476$	−23457.7	−2.25878
$477$	− 986.111i	− 0.0946560i
$478$	33288.1i	3.18528i
$479$	−687.247	−0.0655555	−0.0327778	−	0.999463i	$-0.510435\pi$
$479$	−687.247	−0.0655555	−0.0327778	+	0.999463i	$0.510435\pi$
$480$	0	0
$481$	400.952	0.0380080
$482$	30429.2i	2.87554i
$483$	− 14.8012i	− 0.00139436i
$484$	−1797.48	−0.168809
$485$	0	0
$486$	−11510.0	−1.07428
$487$	− 5600.80i	− 0.521142i	−0.965455	−	0.260571i	$-0.916089\pi$
$487$	− 5600.80i	− 0.521142i	0.965455	−	0.260571i	$-0.0839109\pi$
$488$	− 23395.1i	− 2.17018i
$489$	14398.2	1.33151
$490$	0	0
$491$	−6268.30	−0.576139	−0.288070	−	0.957609i	$-0.593013\pi$
$491$	−6268.30	−0.576139	−0.288070	+	0.957609i	$0.593013\pi$
$492$	2876.41i	0.263574i
$493$	21607.9i	1.97398i
$494$	−10613.9	−0.966683
$495$	0	0
$496$	8226.68	0.744736
$497$	1355.13i	0.122306i
$498$	− 33194.3i	− 2.98689i
$499$	5676.45	0.509244	0.254622	−	0.967041i	$-0.418049\pi$
$499$	5676.45	0.509244	0.254622	+	0.967041i	$0.418049\pi$
$500$	0	0
$501$	22600.9	2.01543
$502$	− 17849.7i	− 1.58700i
$503$	15475.8i	1.37183i	0.727680	+	0.685917i	$0.240599\pi$
$503$	15475.8i	1.37183i	−0.727680	+	0.685917i	$0.759401\pi$
$504$	−3493.71	−0.308774
$505$	0	0
$506$	10.8566	0.000953825 0
$507$	10154.9i	0.889540i
$508$	17091.1i	1.49271i
$509$	17896.1	1.55841	0.779204	−	0.626770i	$-0.215623\pi$
$509$	17896.1	1.55841	0.779204	+	0.626770i	$0.215623\pi$
$510$	0	0
$511$	9178.32	0.794569
$512$	12995.9i	1.12177i
$513$	− 10737.4i	− 0.924107i
$514$	18258.6	1.56683
$515$	0	0
$516$	43644.0	3.72348
$517$	− 5708.17i	− 0.485580i
$518$	1023.15i	0.0867850i
$519$	−10453.7	−0.884139
$520$	0	0
$521$	−11719.0	−0.985453	−0.492726	−	0.870184i	$-0.664000\pi$
$521$	−11719.0	−0.985453	−0.492726	+	0.870184i	$0.664000\pi$
$522$	6973.85i	0.584745i
$523$	1263.95i	0.105676i	0.998603	+	0.0528380i	$0.0168267\pi$
$523$	1263.95i	0.105676i	−0.998603	+	0.0528380i	$0.983173\pi$
$524$	−11058.3	−0.921918
$525$	0	0
$526$	26936.9	2.23290
$527$	28697.9i	2.37211i
$528$	− 2494.05i	− 0.205567i
$529$	12167.0	0.999996
$530$	0	0
$531$	2164.02	0.176856
$532$	− 17604.2i	− 1.43466i
$533$	− 724.246i	− 0.0588566i
$534$	−20611.6	−1.67032
$535$	0	0
$536$	18719.6	1.50851
$537$	3017.01i	0.242446i
$538$	20621.1i	1.65249i
$539$	−2198.48	−0.175687
$540$	0	0
$541$	12341.8	0.980801	0.490401	−	0.871497i	$-0.336850\pi$
$541$	12341.8	0.980801	0.490401	+	0.871497i	$0.336850\pi$
$542$	− 13725.2i	− 1.08773i
$543$	11954.7i	0.944798i
$544$	−10730.6	−0.845716
$545$	0	0
$546$	7682.54	0.602165
$547$	− 6068.31i	− 0.474337i	−0.971469	−	0.237168i	$-0.923781\pi$
$547$	− 6068.31i	− 0.474337i	0.971469	−	0.237168i	$-0.0762193\pi$
$548$	18076.8i	1.40913i
$549$	6360.71	0.494478
$550$	0	0
$551$	−16216.0	−1.25376
$552$	40.5446i	0.00312625i
$553$	5633.56i	0.433207i
$554$	−14375.9	−1.10248
$555$	0	0
$556$	32386.6	2.47032
$557$	− 21379.0i	− 1.62631i	−0.582046	−	0.813156i	$-0.697748\pi$
$557$	− 21379.0i	− 1.62631i	0.582046	−	0.813156i	$-0.302252\pi$
$558$	9262.10i	0.702681i
$559$	−10989.0	−0.831461
$560$	0	0
$561$	8700.22	0.654766
$562$	− 19835.4i	− 1.48880i
$563$	− 18838.1i	− 1.41018i	−0.709120	−	0.705088i	$-0.750907\pi$
$563$	− 18838.1i	− 1.41018i	0.709120	−	0.705088i	$-0.249093\pi$
$564$	46194.8	3.44885
$565$	0	0
$566$	15632.6	1.16093
$567$	10650.2i	0.788831i
$568$	− 3712.08i	− 0.274218i
$569$	673.926	0.0496528	0.0248264	−	0.999692i	$-0.492097\pi$
$569$	673.926	0.0496528	0.0248264	+	0.999692i	$0.492097\pi$
$570$	0	0
$571$	−9663.99	−0.708276	−0.354138	−	0.935193i	$-0.615226\pi$
$571$	−9663.99	−0.708276	−0.354138	+	0.935193i	$0.615226\pi$
$572$	3662.65i	0.267733i
$573$	8304.16i	0.605430i
$574$	1848.13	0.134390
$575$	0	0
$576$	−6160.27	−0.445621
$577$	− 19082.2i	− 1.37678i	−0.725339	−	0.688392i	$-0.758317\pi$
$577$	− 19082.2i	− 1.37678i	0.725339	−	0.688392i	$-0.241683\pi$
$578$	59793.8i	4.30293i
$579$	−7285.82	−0.522951
$580$	0	0
$581$	−13862.4	−0.989864
$582$	14631.4i	1.04208i
$583$	− 1217.37i	− 0.0864811i
$584$	−25141.9	−1.78147
$585$	0	0
$586$	16364.5	1.15360
$587$	− 6451.54i	− 0.453635i	−0.973937	−	0.226817i	$-0.927168\pi$
$587$	− 6451.54i	− 0.453635i	0.973937	−	0.226817i	$-0.0728321\pi$
$588$	− 17791.7i	− 1.24782i
$589$	−21536.8	−1.50663
$590$	0	0
$591$	6674.29	0.464541
$592$	− 676.815i	− 0.0469881i
$593$	− 8852.64i	− 0.613043i	−0.951864	−	0.306521i	$-0.900835\pi$
$593$	− 8852.64i	− 0.613043i	0.951864	−	0.306521i	$-0.0991651\pi$
$594$	−5700.67	−0.393773
$595$	0	0
$596$	−36623.5	−2.51704
$597$	5925.35i	0.406212i
$598$	− 22.1221i	− 0.00151277i
$599$	13815.2	0.942361	0.471180	−	0.882037i	$-0.343828\pi$
$599$	13815.2	0.942361	0.471180	+	0.882037i	$0.343828\pi$
$600$	0	0
$601$	−10772.5	−0.731148	−0.365574	−	0.930782i	$-0.619127\pi$
$601$	−10772.5	−0.731148	−0.365574	+	0.930782i	$0.619127\pi$
$602$	− 28041.8i	− 1.89850i
$603$	5089.52i	0.343717i
$604$	23306.2	1.57006
$605$	0	0
$606$	−23091.1	−1.54787
$607$	− 23470.0i	− 1.56938i	−0.619886	−	0.784692i	$-0.712821\pi$
$607$	− 23470.0i	− 1.56938i	0.619886	−	0.784692i	$-0.287179\pi$
$608$	− 8052.91i	− 0.537152i
$609$	11737.4	0.780994
$610$	0	0
$611$	−11631.3	−0.770134
$612$	17470.3i	1.15392i
$613$	− 13362.8i	− 0.880455i	−0.897886	−	0.440227i	$-0.854898\pi$
$613$	− 13362.8i	− 0.880455i	0.897886	−	0.440227i	$-0.145102\pi$
$614$	287.351	0.0188869
$615$	0	0
$616$	−4313.05	−0.282107
$617$	20357.5i	1.32830i	0.747600	+	0.664149i	$0.231206\pi$
$617$	20357.5i	1.32830i	−0.747600	+	0.664149i	$0.768794\pi$
$618$	53040.9i	3.45246i
$619$	6655.61	0.432168	0.216084	−	0.976375i	$-0.430672\pi$
$619$	6655.61	0.432168	0.216084	+	0.976375i	$0.430672\pi$
$620$	0	0
$621$	22.3794	0.00144614
$622$	− 36445.7i	− 2.34942i
$623$	8607.71i	0.553549i
$624$	−5082.01	−0.326031
$625$	0	0
$626$	−1305.26	−0.0833365
$627$	6529.21i	0.415871i
$628$	− 45918.4i	− 2.91775i
$629$	2361.00	0.149665
$630$	0	0
$631$	−10352.0	−0.653104	−0.326552	−	0.945179i	$-0.605887\pi$
$631$	−10352.0	−0.653104	−0.326552	+	0.945179i	$0.605887\pi$
$632$	− 15431.9i	− 0.971277i
$633$	4783.54i	0.300361i
$634$	−20427.5	−1.27962
$635$	0	0
$636$	9851.90	0.614235
$637$	4479.74i	0.278640i
$638$	8609.36i	0.534244i
$639$	1009.25	0.0624809
$640$	0	0
$641$	27221.5	1.67736	0.838679	−	0.544626i	$-0.183328\pi$
$641$	27221.5	1.67736	0.838679	+	0.544626i	$0.183328\pi$
$642$	− 40945.1i	− 2.51710i
$643$	− 4336.49i	− 0.265964i	−0.991118	−	0.132982i	$-0.957545\pi$
$643$	− 4336.49i	− 0.265964i	0.991118	−	0.132982i	$-0.0424552\pi$
$644$	36.6916	0.00224511
$645$	0	0
$646$	−62499.8	−3.80653
$647$	17128.4i	1.04078i	0.853928	+	0.520391i	$0.174214\pi$
$647$	17128.4i	1.04078i	−0.853928	+	0.520391i	$0.825786\pi$
$648$	− 29173.9i	− 1.76861i
$649$	2671.52	0.161582
$650$	0	0
$651$	15588.7	0.938511
$652$	35692.5i	2.14390i
$653$	− 19633.3i	− 1.17659i	−0.808647	−	0.588294i	$-0.799800\pi$
$653$	− 19633.3i	− 1.17659i	0.808647	−	0.588294i	$-0.200200\pi$
$654$	29438.9	1.76017
$655$	0	0
$656$	−1222.54	−0.0727627
$657$	− 6835.64i	− 0.405911i
$658$	− 29680.8i	− 1.75848i
$659$	5824.46	0.344293	0.172146	−	0.985071i	$-0.444930\pi$
$659$	5824.46	0.344293	0.172146	+	0.985071i	$0.444930\pi$
$660$	0	0
$661$	11332.2	0.666826	0.333413	−	0.942781i	$-0.391800\pi$
$661$	11332.2	0.666826	0.333413	+	0.942781i	$0.391800\pi$
$662$	34123.5i	2.00340i
$663$	− 17728.1i	− 1.03846i
$664$	37973.0	2.21934
$665$	0	0
$666$	762.000	0.0443347
$667$	− 33.7983i	− 0.00196203i
$668$	56026.6i	3.24511i
$669$	−11005.5	−0.636021
$670$	0	0
$671$	7852.42	0.451773
$672$	5828.85i	0.334603i
$673$	7347.26i	0.420826i	0.977613	+	0.210413i	$0.0674809\pi$
$673$	7347.26i	0.420826i	−0.977613	+	0.210413i	$0.932519\pi$
$674$	53443.5	3.05425
$675$	0	0
$676$	−25173.7	−1.43228
$677$	− 9906.82i	− 0.562407i	−0.959648	−	0.281204i	$-0.909266\pi$
$677$	− 9906.82i	− 0.562407i	0.959648	−	0.281204i	$-0.0907337\pi$
$678$	− 17312.6i	− 0.980661i
$679$	6110.29	0.345348
$680$	0	0
$681$	12044.5	0.677750
$682$	11434.3i	0.641995i
$683$	− 10572.3i	− 0.592295i	−0.955142	−	0.296148i	$-0.904298\pi$
$683$	− 10572.3i	− 0.592295i	0.955142	−	0.296148i	$-0.0957021\pi$
$684$	−13110.9	−0.732904
$685$	0	0
$686$	−31049.9	−1.72812
$687$	6375.76i	0.354076i
$688$	18549.7i	1.02791i
$689$	−2480.59	−0.137160
$690$	0	0
$691$	9051.54	0.498317	0.249158	−	0.968463i	$-0.419846\pi$
$691$	9051.54	0.498317	0.249158	+	0.968463i	$0.419846\pi$
$692$	− 25914.4i	− 1.42358i
$693$	− 1172.64i	− 0.0642785i
$694$	−10535.4	−0.576249
$695$	0	0
$696$	−32152.1	−1.75104
$697$	− 4264.71i	− 0.231761i
$698$	− 31263.2i	− 1.69532i
$699$	−7550.41	−0.408559
$700$	0	0
$701$	27429.8	1.47790	0.738950	−	0.673760i	$-0.235322\pi$
$701$	27429.8	1.47790	0.738950	+	0.673760i	$0.235322\pi$
$702$	11616.0i	0.624527i
$703$	1771.85i	0.0950590i
$704$	−7604.98	−0.407136
$705$	0	0
$706$	51686.9	2.75533
$707$	9643.18i	0.512969i
$708$	21620.0i	1.14764i
$709$	31443.4	1.66556	0.832779	−	0.553605i	$-0.186748\pi$
$709$	31443.4	1.66556	0.832779	+	0.553605i	$0.186748\pi$
$710$	0	0
$711$	4195.65	0.221307
$712$	− 23578.9i	− 1.24109i
$713$	− 44.8881i	− 0.00235775i
$714$	45238.5	2.37116
$715$	0	0
$716$	−7479.05	−0.390371
$717$	− 41726.0i	− 2.17334i
$718$	− 2925.65i	− 0.152067i
$719$	−8076.20	−0.418903	−0.209452	−	0.977819i	$-0.567168\pi$
$719$	−8076.20	−0.418903	−0.209452	+	0.977819i	$0.567168\pi$
$720$	0	0
$721$	22150.7	1.14415
$722$	− 14112.9i	− 0.727465i
$723$	− 38142.3i	− 1.96200i
$724$	−29635.2	−1.52125
$725$	0	0
$726$	3466.47	0.177208
$727$	29837.6i	1.52217i	0.648654	+	0.761084i	$0.275332\pi$
$727$	29837.6i	1.52217i	−0.648654	+	0.761084i	$0.724668\pi$
$728$	8788.53i	0.447424i
$729$	−9607.51	−0.488112
$730$	0	0
$731$	−64708.8	−3.27406
$732$	63547.7i	3.20873i
$733$	30839.8i	1.55402i	0.629491	+	0.777008i	$0.283264\pi$
$733$	30839.8i	1.55402i	−0.629491	+	0.777008i	$0.716736\pi$
$734$	56518.4	2.84214
$735$	0	0
$736$	16.7843	0.000840596 0
$737$	6283.12i	0.314032i
$738$	− 1376.41i	− 0.0686538i
$739$	19272.1	0.959318	0.479659	−	0.877455i	$-0.340760\pi$
$739$	19272.1	0.959318	0.479659	+	0.877455i	$0.340760\pi$
$740$	0	0
$741$	13304.3	0.659575
$742$	− 6329.98i	− 0.313182i
$743$	− 6929.41i	− 0.342147i	−0.985258	−	0.171074i	$-0.945276\pi$
$743$	− 6929.41i	− 0.342147i	0.985258	−	0.171074i	$-0.0547236\pi$
$744$	−42701.8	−2.10420
$745$	0	0
$746$	24228.4	1.18909
$747$	10324.2i	0.505679i
$748$	21567.5i	1.05426i
$749$	−17099.3	−0.834172
$750$	0	0
$751$	7785.84	0.378308	0.189154	−	0.981947i	$-0.439425\pi$
$751$	7785.84	0.378308	0.189154	+	0.981947i	$0.439425\pi$
$752$	19633.9i	0.952093i
$753$	22374.2i	1.08282i
$754$	17542.9	0.847316
$755$	0	0
$756$	−19266.3	−0.926861
$757$	19392.0i	0.931065i	0.885031	+	0.465532i	$0.154137\pi$
$757$	19392.0i	0.931065i	−0.885031	+	0.465532i	$0.845863\pi$
$758$	− 52517.2i	− 2.51650i
$759$	−13.6085	−0.000650802 0
$760$	0	0
$761$	40527.1	1.93050	0.965248	−	0.261336i	$-0.0841630\pi$
$761$	40527.1	1.93050	0.965248	+	0.261336i	$0.0841630\pi$
$762$	− 32960.6i	− 1.56697i
$763$	− 12294.1i	− 0.583326i
$764$	−20585.7	−0.974821
$765$	0	0
$766$	−41508.7	−1.95793
$767$	− 5443.65i	− 0.256270i
$768$	− 42912.0i	− 2.01621i
$769$	−447.221	−0.0209716	−0.0104858	−	0.999945i	$-0.503338\pi$
$769$	−447.221	−0.0209716	−0.0104858	+	0.999945i	$0.503338\pi$
$770$	0	0
$771$	−22886.8	−1.06906
$772$	− 18061.3i	− 0.842019i
$773$	13950.3i	0.649105i	0.945868	+	0.324553i	$0.105214\pi$
$773$	13950.3i	0.649105i	−0.945868	+	0.324553i	$0.894786\pi$
$774$	−20884.4	−0.969864
$775$	0	0
$776$	−16737.8	−0.774292
$777$	− 1282.50i	− 0.0592140i
$778$	− 59378.8i	− 2.73629i
$779$	3200.52	0.147202
$780$	0	0
$781$	1245.94	0.0570848
$782$	− 130.265i	− 0.00595689i
$783$	17747.0i	0.809996i
$784$	7561.90	0.344474
$785$	0	0
$786$	21326.2	0.967785
$787$	24643.3i	1.11619i	0.829778	+	0.558093i	$0.188467\pi$
$787$	24643.3i	1.11619i	−0.829778	+	0.558093i	$0.811533\pi$
$788$	16545.3i	0.747972i
$789$	−33764.8	−1.52352
$790$	0	0
$791$	−7230.02	−0.324993
$792$	3212.19i	0.144116i
$793$	− 16000.6i	− 0.716515i
$794$	−26050.7	−1.16436
$795$	0	0
$796$	−14688.7	−0.654054
$797$	− 973.804i	− 0.0432797i	−0.999766	−	0.0216398i	$-0.993111\pi$
$797$	− 973.804i	− 0.0432797i	0.999766	−	0.0216398i	$-0.00688871\pi$
$798$	33949.9i	1.50603i
$799$	−68490.7	−3.03258
$800$	0	0
$801$	6410.67	0.282784
$802$	55241.7i	2.43223i
$803$	− 8438.74i	− 0.370855i
$804$	−50847.7	−2.23042
$805$	0	0
$806$	23299.1	1.01821
$807$	− 25848.1i	− 1.12751i
$808$	− 26415.3i	− 1.15011i
$809$	−10386.8	−0.451396	−0.225698	−	0.974197i	$-0.572466\pi$
$809$	−10386.8	−0.451396	−0.225698	+	0.974197i	$0.572466\pi$
$810$	0	0
$811$	1009.30	0.0437007	0.0218503	−	0.999761i	$-0.493044\pi$
$811$	1009.30	0.0437007	0.0218503	+	0.999761i	$0.493044\pi$
$812$	29096.6i	1.25750i
$813$	17204.3i	0.742166i
$814$	940.705	0.0405058
$815$	0	0
$816$	−29925.4	−1.28382
$817$	− 48561.6i	− 2.07951i
$818$	2726.75i	0.116551i
$819$	−2389.44	−0.101946
$820$	0	0
$821$	−25309.9	−1.07591	−0.537955	−	0.842973i	$-0.680803\pi$
$821$	−25309.9	−1.07591	−0.537955	+	0.842973i	$0.680803\pi$
$822$	− 34861.5i	− 1.47924i
$823$	13675.7i	0.579230i	0.957143	+	0.289615i	$0.0935272\pi$
$823$	13675.7i	0.579230i	−0.957143	+	0.289615i	$0.906473\pi$
$824$	−60676.8	−2.56526
$825$	0	0
$826$	13891.1	0.585150
$827$	− 28106.0i	− 1.18179i	−0.806747	−	0.590897i	$-0.798774\pi$
$827$	− 28106.0i	− 1.18179i	0.806747	−	0.590897i	$-0.201226\pi$
$828$	− 27.3264i	− 0.00114693i
$829$	−971.943	−0.0407201	−0.0203601	−	0.999793i	$-0.506481\pi$
$829$	−971.943	−0.0407201	−0.0203601	+	0.999793i	$0.506481\pi$
$830$	0	0
$831$	18019.9	0.752228
$832$	15496.3i	0.645720i
$833$	26378.9i	1.09721i
$834$	−62458.2	−2.59323
$835$	0	0
$836$	−16185.6	−0.669607
$837$	23570.2i	0.973362i
$838$	− 58674.4i	− 2.41870i
$839$	−17858.3	−0.734847	−0.367423	−	0.930054i	$-0.619760\pi$
$839$	−17858.3	−0.734847	−0.367423	+	0.930054i	$0.619760\pi$
$840$	0	0
$841$	2413.24	0.0989479
$842$	− 3508.80i	− 0.143612i
$843$	24863.2i	1.01582i
$844$	−11858.2	−0.483621
$845$	0	0
$846$	−22105.0	−0.898329
$847$	− 1447.65i	− 0.0587271i
$848$	4187.29i	0.169566i
$849$	−19595.1	−0.792111
$850$	0	0
$851$	−3.69298	−0.000148759 0
$852$	10083.1i	0.405446i
$853$	− 3010.35i	− 0.120835i	−0.998173	−	0.0604176i	$-0.980757\pi$
$853$	− 3010.35i	− 0.120835i	0.998173	−	0.0604176i	$-0.0192432\pi$
$854$	40830.3	1.63605
$855$	0	0
$856$	46839.7	1.87026
$857$	− 39487.8i	− 1.57395i	−0.616984	−	0.786976i	$-0.711645\pi$
$857$	− 39487.8i	− 1.57395i	0.616984	−	0.786976i	$-0.288355\pi$
$858$	− 7063.49i	− 0.281053i
$859$	2338.21	0.0928741	0.0464370	−	0.998921i	$-0.485213\pi$
$859$	2338.21	0.0928741	0.0464370	+	0.998921i	$0.485213\pi$
$860$	0	0
$861$	−2316.60	−0.0916950
$862$	78605.2i	3.10592i
$863$	− 25637.9i	− 1.01127i	−0.862748	−	0.505635i	$-0.831258\pi$
$863$	− 25637.9i	− 1.01127i	0.862748	−	0.505635i	$-0.168742\pi$
$864$	−8813.23	−0.347028
$865$	0	0
$866$	17917.2	0.703063
$867$	− 74950.2i	− 2.93592i
$868$	38643.8i	1.51112i
$869$	5179.61	0.202194
$870$	0	0
$871$	12802.9	0.498058
$872$	33677.0i	1.30785i
$873$	− 4550.69i	− 0.176423i
$874$	97.7596	0.00378349
$875$	0	0
$876$	68292.6	2.63401
$877$	48054.8i	1.85028i	0.379624	+	0.925141i	$0.376053\pi$
$877$	48054.8i	1.85028i	−0.379624	+	0.925141i	$0.623947\pi$
$878$	9168.62i	0.352421i
$879$	−20512.5	−0.787111
$880$	0	0
$881$	16092.9	0.615418	0.307709	−	0.951481i	$-0.400438\pi$
$881$	16092.9	0.615418	0.307709	+	0.951481i	$0.400438\pi$
$882$	8513.65i	0.325022i
$883$	− 39499.4i	− 1.50539i	−0.658369	−	0.752696i	$-0.728753\pi$
$883$	− 39499.4i	− 1.50539i	0.658369	−	0.752696i	$-0.271247\pi$
$884$	43947.2	1.67206
$885$	0	0
$886$	−46679.3	−1.77000
$887$	− 17421.1i	− 0.659461i	−0.944075	−	0.329730i	$-0.893042\pi$
$887$	− 17421.1i	− 0.659461i	0.944075	−	0.329730i	$-0.106958\pi$
$888$	3513.11i	0.132762i
$889$	−13764.8	−0.519299
$890$	0	0
$891$	9792.04	0.368177
$892$	− 27282.2i	− 1.02408i
$893$	− 51399.9i	− 1.92613i
$894$	70629.1	2.64227
$895$	0	0
$896$	−31762.1	−1.18426
$897$	27.7295i	0.00103218i
$898$	− 46005.1i	− 1.70959i
$899$	35596.6	1.32059
$900$	0	0
$901$	−14606.9	−0.540097
$902$	− 1699.21i	− 0.0627246i
$903$	35149.8i	1.29536i
$904$	19805.0	0.728656
$905$	0	0
$906$	−44946.5	−1.64818
$907$	21501.9i	0.787164i	0.919290	+	0.393582i	$0.128764\pi$
$907$	21501.9i	0.787164i	−0.919290	+	0.393582i	$0.871236\pi$
$908$	29857.9i	1.09127i
$909$	7181.84	0.262054
$910$	0	0
$911$	−20432.4	−0.743092	−0.371546	−	0.928415i	$-0.621172\pi$
$911$	−20432.4	−0.743092	−0.371546	+	0.928415i	$0.621172\pi$
$912$	− 22457.9i	− 0.815413i
$913$	12745.4i	0.462006i
$914$	77497.0	2.80457
$915$	0	0
$916$	−15805.2	−0.570109
$917$	− 8906.12i	− 0.320726i
$918$	68400.7i	2.45922i
$919$	−34960.0	−1.25487	−0.627434	−	0.778670i	$-0.715895\pi$
$919$	−34960.0	−1.25487	−0.627434	+	0.778670i	$0.715895\pi$
$920$	0	0
$921$	−360.188	−0.0128867
$922$	18802.1i	0.671601i
$923$	− 2538.80i	− 0.0905369i
$924$	11715.5	0.417111
$925$	0	0
$926$	3676.18	0.130461
$927$	− 16496.9i	− 0.584498i
$928$	13310.1i	0.470824i
$929$	−3925.52	−0.138635	−0.0693176	−	0.997595i	$-0.522082\pi$
$929$	−3925.52	−0.138635	−0.0693176	+	0.997595i	$0.522082\pi$
$930$	0	0
$931$	−19796.4	−0.696887
$932$	− 18717.1i	− 0.657833i
$933$	45684.0i	1.60303i
$934$	−12600.1	−0.441420
$935$	0	0
$936$	6545.34	0.228570
$937$	− 10607.3i	− 0.369826i	−0.982755	−	0.184913i	$-0.940800\pi$
$937$	− 10607.3i	− 0.369826i	0.982755	−	0.184913i	$-0.0592003\pi$
$938$	32670.4i	1.13723i
$939$	1636.11	0.0568611
$940$	0	0
$941$	−4209.94	−0.145845	−0.0729224	−	0.997338i	$-0.523233\pi$
$941$	−4209.94	−0.145845	−0.0729224	+	0.997338i	$0.523233\pi$
$942$	88554.5i	3.06291i
$943$	6.67070i	0 0.000230358i
$944$	−9189.00	−0.316818
$945$	0	0
$946$	−25782.2	−0.886103
$947$	6133.98i	0.210483i	0.994447	+	0.105242i	$0.0335616\pi$
$947$	6133.98i	0.210483i	−0.994447	+	0.105242i	$0.966438\pi$
$948$	41917.3i	1.43609i
$949$	−17195.3	−0.588179
$950$	0	0
$951$	25605.4	0.873094
$952$	51751.1i	1.76183i
$953$	− 15745.6i	− 0.535203i	−0.963530	−	0.267602i	$-0.913769\pi$
$953$	− 15745.6i	− 0.535203i	0.963530	−	0.267602i	$-0.0862311\pi$
$954$	−4714.31	−0.159991
$955$	0	0
$956$	103437.	3.49936
$957$	− 10791.7i	− 0.364519i
$958$	3285.53i	0.110804i
$959$	−14558.7	−0.490223
$960$	0	0
$961$	17485.5	0.586939
$962$	− 1916.84i	− 0.0642425i
$963$	12734.9i	0.426142i
$964$	94553.2	3.15908
$965$	0	0
$966$	−70.7603	−0.00235681
$967$	− 12077.7i	− 0.401649i	−0.979627	−	0.200824i	$-0.935638\pi$
$967$	− 12077.7i	− 0.401649i	0.979627	−	0.200824i	$-0.0643620\pi$
$968$	3965.51i	0.131670i
$969$	78342.1	2.59723
$970$	0	0
$971$	−11513.5	−0.380519	−0.190260	−	0.981734i	$-0.560933\pi$
$971$	−11513.5	−0.380519	−0.190260	+	0.981734i	$0.560933\pi$
$972$	35765.1i	1.18021i
$973$	26083.5i	0.859401i
$974$	−26775.8	−0.880854
$975$	0	0
$976$	−27009.3	−0.885806
$977$	14847.9i	0.486208i	0.970000	+	0.243104i	$0.0781657\pi$
$977$	14847.9i	0.486208i	−0.970000	+	0.243104i	$0.921834\pi$
$978$	− 68833.5i	− 2.25056i
$979$	7914.11	0.258362
$980$	0	0
$981$	−9156.17	−0.297996
$982$	29966.9i	0.973812i
$983$	− 11752.3i	− 0.381322i	−0.981656	−	0.190661i	$-0.938937\pi$
$983$	− 11752.3i	− 0.381322i	0.981656	−	0.190661i	$-0.0610631\pi$
$984$	6345.79	0.205586
$985$	0	0
$986$	103301.	3.33650
$987$	37204.2i	1.19982i
$988$	32980.8i	1.06200i
$989$	101.215	0.00325424
$990$	0	0
$991$	−32455.8	−1.04036	−0.520179	−	0.854058i	$-0.674135\pi$
$991$	−32455.8	−1.04036	−0.520179	+	0.854058i	$0.674135\pi$
$992$	17677.4i	0.565783i
$993$	− 42773.1i	− 1.36693i
$994$	6478.51	0.206726
$995$	0	0
$996$	−103145.	−3.28142
$997$	− 35177.9i	− 1.11745i	−0.829354	−	0.558723i	$-0.811291\pi$
$997$	− 35177.9i	− 1.11745i	0.829354	−	0.558723i	$-0.188709\pi$
$998$	− 27137.5i	− 0.860744i
$999$	1939.14	0.0614130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.b.f.199.2		10
5.2	odd	4		275.4.a.h.1.5	yes	5
5.3	odd	4		275.4.a.g.1.1	✓	5
5.4	even	2	inner	275.4.b.f.199.9		10
15.2	even	4		2475.4.a.bh.1.1		5
15.8	even	4		2475.4.a.bl.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.g.1.1	✓	5	5.3	odd	4
275.4.a.h.1.5	yes	5	5.2	odd	4
275.4.b.f.199.2		10	1.1	even	1	trivial
275.4.b.f.199.9		10	5.4	even	2	inner
2475.4.a.bh.1.1		5	15.2	even	4
2475.4.a.bl.1.5		5	15.8	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.78071i q^{2} +5.99252i q^{3} -14.8552 q^{4} +28.6485 q^{6} -11.9641i q^{7} +32.7728i q^{8} -8.91034 q^{9} -11.0000 q^{11} -89.0202i q^{12} +22.4142i q^{13} -57.1967 q^{14} +37.8357 q^{16} +131.986i q^{17} +42.5978i q^{18} -99.0508 q^{19} +71.6949 q^{21} +52.5878i q^{22} -0.206447i q^{23} -196.392 q^{24} +107.156 q^{26} +108.403i q^{27} +177.729i q^{28} +163.714 q^{29} +217.432 q^{31} +81.3008i q^{32} -65.9178i q^{33} +630.987 q^{34} +132.365 q^{36} -17.8883i q^{37} +473.533i q^{38} -134.318 q^{39} -32.3119 q^{41} -342.753i q^{42} +490.270i q^{43} +163.407 q^{44} -0.986965 q^{46} +518.924i q^{47} +226.731i q^{48} +199.861 q^{49} -790.929 q^{51} -332.968i q^{52} +110.670i q^{53} +518.242 q^{54} +392.096 q^{56} -593.564i q^{57} -782.669i q^{58} -242.866 q^{59} -713.857 q^{61} -1039.48i q^{62} +106.604i q^{63} +691.362 q^{64} -315.134 q^{66} -571.193i q^{67} -1960.68i q^{68} +1.23714 q^{69} -113.267 q^{71} -292.017i q^{72} +767.158i q^{73} -85.5187 q^{74} +1471.42 q^{76} +131.605i q^{77} +642.135i q^{78} -470.874 q^{79} -890.185 q^{81} +154.474i q^{82} -1158.67i q^{83} -1065.04 q^{84} +2343.84 q^{86} +981.059i q^{87} -360.501i q^{88} -719.465 q^{89} +268.165 q^{91} +3.06682i q^{92} +1302.96i q^{93} +2480.83 q^{94} -487.197 q^{96} +510.721i q^{97} -955.480i q^{98} +98.0137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41}+ \cdots + 682 q^{99}+O(q^{100}) \) 10 * q - 80 * q^4 - 84 * q^6 - 62 * q^9 - 110 * q^11 + 266 * q^14 + 416 * q^16 - 46 * q^19 + 564 * q^21 + 982 * q^24 + 644 * q^26 + 366 * q^29 - 2 * q^31 + 1300 * q^34 + 2676 * q^36 + 540 * q^39 + 188 * q^41 + 880 * q^44 + 2012 * q^46 - 454 * q^49 - 2256 * q^51 + 2410 * q^54 - 1670 * q^56 - 2348 * q^59 + 1280 * q^61 + 1898 * q^64 + 924 * q^66 + 3292 * q^69 - 1346 * q^71 + 1122 * q^74 + 2496 * q^76 - 208 * q^79 - 2438 * q^81 - 8706 * q^84 + 4994 * q^86 - 5642 * q^89 + 960 * q^91 + 3574 * q^94 - 3638 * q^96 + 682 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more