LMFDB - Embedded newform 1150.4.b.b.599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.b.599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +7.00000i q^{3} -4.00000 q^{4} -14.0000 q^{6} -20.0000i q^{7} -8.00000i q^{8} -22.0000 q^{9} +6.00000 q^{11} -28.0000i q^{12} +47.0000i q^{13} +40.0000 q^{14} +16.0000 q^{16} +132.000i q^{17} -44.0000i q^{18} -146.000 q^{19} +140.000 q^{21} +12.0000i q^{22} +23.0000i q^{23} +56.0000 q^{24} -94.0000 q^{26} +35.0000i q^{27} +80.0000i q^{28} +99.0000 q^{29} -253.000 q^{31} +32.0000i q^{32} +42.0000i q^{33} -264.000 q^{34} +88.0000 q^{36} +118.000i q^{37} -292.000i q^{38} -329.000 q^{39} +495.000 q^{41} +280.000i q^{42} +272.000i q^{43} -24.0000 q^{44} -46.0000 q^{46} -639.000i q^{47} +112.000i q^{48} -57.0000 q^{49} -924.000 q^{51} -188.000i q^{52} -342.000i q^{53} -70.0000 q^{54} -160.000 q^{56} -1022.00i q^{57} +198.000i q^{58} -240.000 q^{59} -370.000 q^{61} -506.000i q^{62} +440.000i q^{63} -64.0000 q^{64} -84.0000 q^{66} -698.000i q^{67} -528.000i q^{68} -161.000 q^{69} -357.000 q^{71} +176.000i q^{72} -259.000i q^{73} -236.000 q^{74} +584.000 q^{76} -120.000i q^{77} -658.000i q^{78} -542.000 q^{79} -839.000 q^{81} +990.000i q^{82} -1248.00i q^{83} -560.000 q^{84} -544.000 q^{86} +693.000i q^{87} -48.0000i q^{88} +828.000 q^{89} +940.000 q^{91} -92.0000i q^{92} -1771.00i q^{93} +1278.00 q^{94} -224.000 q^{96} -992.000i q^{97} -114.000i q^{98} -132.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 28 q^{6} - 44 q^{9} + 12 q^{11} + 80 q^{14} + 32 q^{16} - 292 q^{19} + 280 q^{21} + 112 q^{24} - 188 q^{26} + 198 q^{29} - 506 q^{31} - 528 q^{34} + 176 q^{36} - 658 q^{39} + 990 q^{41}+ \cdots - 264 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 28 * q^6 - 44 * q^9 + 12 * q^11 + 80 * q^14 + 32 * q^16 - 292 * q^19 + 280 * q^21 + 112 * q^24 - 188 * q^26 + 198 * q^29 - 506 * q^31 - 528 * q^34 + 176 * q^36 - 658 * q^39 + 990 * q^41 - 48 * q^44 - 92 * q^46 - 114 * q^49 - 1848 * q^51 - 140 * q^54 - 320 * q^56 - 480 * q^59 - 740 * q^61 - 128 * q^64 - 168 * q^66 - 322 * q^69 - 714 * q^71 - 472 * q^74 + 1168 * q^76 - 1084 * q^79 - 1678 * q^81 - 1120 * q^84 - 1088 * q^86 + 1656 * q^89 + 1880 * q^91 + 2556 * q^94 - 448 * q^96 - 264 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	7.00000i	1.34715i	0.739119	+	0.673575i	$0.235242\pi$
$3$	7.00000i	1.34715i	−0.739119	+	0.673575i	$0.764758\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−14.0000	−0.952579
$7$	− 20.0000i	− 1.07990i	−0.841698	−	0.539949i	$-0.818443\pi$
$7$	− 20.0000i	− 1.07990i	0.841698	−	0.539949i	$-0.181557\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−22.0000	−0.814815
$10$	0	0
$11$	6.00000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	6.00000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	− 28.0000i	− 0.673575i
$13$	47.0000i	1.00273i	0.865237	+	0.501364i	$0.167168\pi$
$13$	47.0000i	1.00273i	−0.865237	+	0.501364i	$0.832832\pi$
$14$	40.0000	0.763604
$15$	0	0
$16$	16.0000	0.250000
$17$	132.000i	1.88322i	0.336709	+	0.941609i	$0.390686\pi$
$17$	132.000i	1.88322i	−0.336709	+	0.941609i	$0.609314\pi$
$18$	− 44.0000i	− 0.576161i
$19$	−146.000	−1.76288	−0.881439	−	0.472297i	$-0.843425\pi$
$19$	−146.000	−1.76288	−0.881439	+	0.472297i	$0.843425\pi$
$20$	0	0
$21$	140.000	1.45479
$22$	12.0000i	0.116291i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	56.0000	0.476290
$25$	0	0
$26$	−94.0000	−0.709035
$27$	35.0000i	0.249472i
$28$	80.0000i	0.539949i
$29$	99.0000	0.633925	0.316963	−	0.948438i	$-0.397337\pi$
$29$	99.0000	0.633925	0.316963	+	0.948438i	$0.397337\pi$
$30$	0	0
$31$	−253.000	−1.46581	−0.732906	−	0.680330i	$-0.761836\pi$
$31$	−253.000	−1.46581	−0.732906	+	0.680330i	$0.761836\pi$
$32$	32.0000i	0.176777i
$33$	42.0000i	0.221553i
$34$	−264.000	−1.33164
$35$	0	0
$36$	88.0000	0.407407
$37$	118.000i	0.524299i	0.965027	+	0.262150i	$0.0844314\pi$
$37$	118.000i	0.524299i	−0.965027	+	0.262150i	$0.915569\pi$
$38$	− 292.000i	− 1.24654i
$39$	−329.000	−1.35082
$40$	0	0
$41$	495.000	1.88551	0.942756	−	0.333483i	$-0.108224\pi$
$41$	495.000	1.88551	0.942756	+	0.333483i	$0.108224\pi$
$42$	280.000i	1.02869i
$43$	272.000i	0.964642i	0.875995	+	0.482321i	$0.160206\pi$
$43$	272.000i	0.964642i	−0.875995	+	0.482321i	$0.839794\pi$
$44$	−24.0000	−0.0822304
$45$	0	0
$46$	−46.0000	−0.147442
$47$	− 639.000i	− 1.98314i	−0.129560	−	0.991572i	$-0.541356\pi$
$47$	− 639.000i	− 1.98314i	0.129560	−	0.991572i	$-0.458644\pi$
$48$	112.000i	0.336788i
$49$	−57.0000	−0.166181
$50$	0	0
$51$	−924.000	−2.53698
$52$	− 188.000i	− 0.501364i
$53$	− 342.000i	− 0.886364i	−0.896432	−	0.443182i	$-0.853849\pi$
$53$	− 342.000i	− 0.886364i	0.896432	−	0.443182i	$-0.146151\pi$
$54$	−70.0000	−0.176404
$55$	0	0
$56$	−160.000	−0.381802
$57$	− 1022.00i	− 2.37486i
$58$	198.000i	0.448253i
$59$	−240.000	−0.529582	−0.264791	−	0.964306i	$-0.585303\pi$
$59$	−240.000	−0.529582	−0.264791	+	0.964306i	$0.585303\pi$
$60$	0	0
$61$	−370.000	−0.776617	−0.388309	−	0.921529i	$-0.626941\pi$
$61$	−370.000	−0.776617	−0.388309	+	0.921529i	$0.626941\pi$
$62$	− 506.000i	− 1.03648i
$63$	440.000i	0.879917i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−84.0000	−0.156662
$67$	− 698.000i	− 1.27275i	−0.771380	−	0.636375i	$-0.780433\pi$
$67$	− 698.000i	− 1.27275i	0.771380	−	0.636375i	$-0.219567\pi$
$68$	− 528.000i	− 0.941609i
$69$	−161.000	−0.280900
$70$	0	0
$71$	−357.000	−0.596734	−0.298367	−	0.954451i	$-0.596442\pi$
$71$	−357.000	−0.596734	−0.298367	+	0.954451i	$0.596442\pi$
$72$	176.000i	0.288081i
$73$	− 259.000i	− 0.415256i	−0.978208	−	0.207628i	$-0.933426\pi$
$73$	− 259.000i	− 0.415256i	0.978208	−	0.207628i	$-0.0665743\pi$
$74$	−236.000	−0.370736
$75$	0	0
$76$	584.000	0.881439
$77$	− 120.000i	− 0.177601i
$78$	− 658.000i	− 0.955177i
$79$	−542.000	−0.771896	−0.385948	−	0.922521i	$-0.626126\pi$
$79$	−542.000	−0.771896	−0.385948	+	0.922521i	$0.626126\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	990.000i	1.33326i
$83$	− 1248.00i	− 1.65043i	−0.564818	−	0.825216i	$-0.691054\pi$
$83$	− 1248.00i	− 1.65043i	0.564818	−	0.825216i	$-0.308946\pi$
$84$	−560.000	−0.727393
$85$	0	0
$86$	−544.000	−0.682105
$87$	693.000i	0.853993i
$88$	− 48.0000i	− 0.0581456i
$89$	828.000	0.986155	0.493078	−	0.869985i	$-0.335872\pi$
$89$	828.000	0.986155	0.493078	+	0.869985i	$0.335872\pi$
$90$	0	0
$91$	940.000	1.08284
$92$	− 92.0000i	− 0.104257i
$93$	− 1771.00i	− 1.97467i
$94$	1278.00	1.40229
$95$	0	0
$96$	−224.000	−0.238145
$97$	− 992.000i	− 1.03837i	−0.854660	−	0.519187i	$-0.826235\pi$
$97$	− 992.000i	− 1.03837i	0.854660	−	0.519187i	$-0.173765\pi$
$98$	− 114.000i	− 0.117508i
$99$	−132.000	−0.134005
$100$	0	0
$101$	−1542.00	−1.51916	−0.759578	−	0.650416i	$-0.774594\pi$
$101$	−1542.00	−1.51916	−0.759578	+	0.650416i	$0.774594\pi$
$102$	− 1848.00i	− 1.79391i
$103$	32.0000i	0.0306122i	0.999883	+	0.0153061i	$0.00487227\pi$
$103$	32.0000i	0.0306122i	−0.999883	+	0.0153061i	$0.995128\pi$
$104$	376.000	0.354518
$105$	0	0
$106$	684.000	0.626754
$107$	834.000i	0.753512i	0.926312	+	0.376756i	$0.122961\pi$
$107$	834.000i	0.753512i	−0.926312	+	0.376756i	$0.877039\pi$
$108$	− 140.000i	− 0.124736i
$109$	1192.00	1.04746	0.523729	−	0.851885i	$-0.324540\pi$
$109$	1192.00	1.04746	0.523729	+	0.851885i	$0.324540\pi$
$110$	0	0
$111$	−826.000	−0.706310
$112$	− 320.000i	− 0.269975i
$113$	− 132.000i	− 0.109890i	−0.998489	−	0.0549448i	$-0.982502\pi$
$113$	− 132.000i	− 0.109890i	0.998489	−	0.0549448i	$-0.0174983\pi$
$114$	2044.00	1.67928
$115$	0	0
$116$	−396.000	−0.316963
$117$	− 1034.00i	− 0.817037i
$118$	− 480.000i	− 0.374471i
$119$	2640.00	2.03368
$120$	0	0
$121$	−1295.00	−0.972953
$122$	− 740.000i	− 0.549151i
$123$	3465.00i	2.54007i
$124$	1012.00	0.732906
$125$	0	0
$126$	−880.000	−0.622195
$127$	− 89.0000i	− 0.0621848i	−0.999517	−	0.0310924i	$-0.990101\pi$
$127$	− 89.0000i	− 0.0621848i	0.999517	−	0.0310924i	$-0.00989862\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−1904.00	−1.29952
$130$	0	0
$131$	−1797.00	−1.19851	−0.599254	−	0.800559i	$-0.704536\pi$
$131$	−1797.00	−1.19851	−0.599254	+	0.800559i	$0.704536\pi$
$132$	− 168.000i	− 0.110777i
$133$	2920.00i	1.90373i
$134$	1396.00	0.899970
$135$	0	0
$136$	1056.00	0.665818
$137$	1836.00i	1.14496i	0.819917	+	0.572482i	$0.194019\pi$
$137$	1836.00i	1.14496i	−0.819917	+	0.572482i	$0.805981\pi$
$138$	− 322.000i	− 0.198627i
$139$	1027.00	0.626683	0.313342	−	0.949640i	$-0.398551\pi$
$139$	1027.00	0.626683	0.313342	+	0.949640i	$0.398551\pi$
$140$	0	0
$141$	4473.00	2.67159
$142$	− 714.000i	− 0.421955i
$143$	282.000i	0.164909i
$144$	−352.000	−0.203704
$145$	0	0
$146$	518.000	0.293630
$147$	− 399.000i	− 0.223871i
$148$	− 472.000i	− 0.262150i
$149$	−2310.00	−1.27008	−0.635042	−	0.772477i	$-0.719017\pi$
$149$	−2310.00	−1.27008	−0.635042	+	0.772477i	$0.719017\pi$
$150$	0	0
$151$	−2149.00	−1.15817	−0.579083	−	0.815268i	$-0.696589\pi$
$151$	−2149.00	−1.15817	−0.579083	+	0.815268i	$0.696589\pi$
$152$	1168.00i	0.623272i
$153$	− 2904.00i	− 1.53447i
$154$	240.000	0.125583
$155$	0	0
$156$	1316.00	0.675412
$157$	− 1832.00i	− 0.931271i	−0.884977	−	0.465635i	$-0.845826\pi$
$157$	− 1832.00i	− 0.931271i	0.884977	−	0.465635i	$-0.154174\pi$
$158$	− 1084.00i	− 0.545813i
$159$	2394.00	1.19407
$160$	0	0
$161$	460.000	0.225174
$162$	− 1678.00i	− 0.813803i
$163$	1217.00i	0.584802i	0.956296	+	0.292401i	$0.0944543\pi$
$163$	1217.00i	0.584802i	−0.956296	+	0.292401i	$0.905546\pi$
$164$	−1980.00	−0.942756
$165$	0	0
$166$	2496.00	1.16703
$167$	− 3048.00i	− 1.41234i	−0.708041	−	0.706172i	$-0.750421\pi$
$167$	− 3048.00i	− 1.41234i	0.708041	−	0.706172i	$-0.249579\pi$
$168$	− 1120.00i	− 0.514344i
$169$	−12.0000	−0.00546199
$170$	0	0
$171$	3212.00	1.43642
$172$	− 1088.00i	− 0.482321i
$173$	774.000i	0.340151i	0.985431	+	0.170076i	$0.0544012\pi$
$173$	774.000i	0.340151i	−0.985431	+	0.170076i	$0.945599\pi$
$174$	−1386.00	−0.603864
$175$	0	0
$176$	96.0000	0.0411152
$177$	− 1680.00i	− 0.713427i
$178$	1656.00i	0.697317i
$179$	1875.00	0.782928	0.391464	−	0.920193i	$-0.371969\pi$
$179$	1875.00	0.782928	0.391464	+	0.920193i	$0.371969\pi$
$180$	0	0
$181$	−1606.00	−0.659520	−0.329760	−	0.944065i	$-0.606968\pi$
$181$	−1606.00	−0.659520	−0.329760	+	0.944065i	$0.606968\pi$
$182$	1880.00i	0.765686i
$183$	− 2590.00i	− 1.04622i
$184$	184.000	0.0737210
$185$	0	0
$186$	3542.00	1.39630
$187$	792.000i	0.309715i
$188$	2556.00i	0.991572i
$189$	700.000	0.269405
$190$	0	0
$191$	2982.00	1.12969	0.564843	−	0.825199i	$-0.308937\pi$
$191$	2982.00	1.12969	0.564843	+	0.825199i	$0.308937\pi$
$192$	− 448.000i	− 0.168394i
$193$	1385.00i	0.516552i	0.966071	+	0.258276i	$0.0831543\pi$
$193$	1385.00i	0.516552i	−0.966071	+	0.258276i	$0.916846\pi$
$194$	1984.00	0.734242
$195$	0	0
$196$	228.000	0.0830904
$197$	− 957.000i	− 0.346109i	−0.984912	−	0.173054i	$-0.944636\pi$
$197$	− 957.000i	− 0.346109i	0.984912	−	0.173054i	$-0.0553636\pi$
$198$	− 264.000i	− 0.0947559i
$199$	358.000	0.127527	0.0637637	−	0.997965i	$-0.479690\pi$
$199$	358.000	0.127527	0.0637637	+	0.997965i	$0.479690\pi$
$200$	0	0
$201$	4886.00	1.71459
$202$	− 3084.00i	− 1.07421i
$203$	− 1980.00i	− 0.684575i
$204$	3696.00	1.26849
$205$	0	0
$206$	−64.0000	−0.0216461
$207$	− 506.000i	− 0.169901i
$208$	752.000i	0.250682i
$209$	−876.000	−0.289924
$210$	0	0
$211$	−5380.00	−1.75533	−0.877665	−	0.479275i	$-0.840900\pi$
$211$	−5380.00	−1.75533	−0.877665	+	0.479275i	$0.840900\pi$
$212$	1368.00i	0.443182i
$213$	− 2499.00i	− 0.803890i
$214$	−1668.00	−0.532814
$215$	0	0
$216$	280.000	0.0882018
$217$	5060.00i	1.58293i
$218$	2384.00i	0.740664i
$219$	1813.00	0.559412
$220$	0	0
$221$	−6204.00	−1.88835
$222$	− 1652.00i	− 0.499437i
$223$	1040.00i	0.312303i	0.987733	+	0.156151i	$0.0499088\pi$
$223$	1040.00i	0.312303i	−0.987733	+	0.156151i	$0.950091\pi$
$224$	640.000	0.190901
$225$	0	0
$226$	264.000	0.0777036
$227$	3744.00i	1.09470i	0.836902	+	0.547352i	$0.184364\pi$
$227$	3744.00i	1.09470i	−0.836902	+	0.547352i	$0.815636\pi$
$228$	4088.00i	1.18743i
$229$	−2804.00	−0.809142	−0.404571	−	0.914507i	$-0.632579\pi$
$229$	−2804.00	−0.809142	−0.404571	+	0.914507i	$0.632579\pi$
$230$	0	0
$231$	840.000	0.239255
$232$	− 792.000i	− 0.224126i
$233$	− 4869.00i	− 1.36901i	−0.729009	−	0.684504i	$-0.760019\pi$
$233$	− 4869.00i	− 1.36901i	0.729009	−	0.684504i	$-0.239981\pi$
$234$	2068.00	0.577732
$235$	0	0
$236$	960.000	0.264791
$237$	− 3794.00i	− 1.03986i
$238$	5280.00i	1.43803i
$239$	2877.00	0.778651	0.389326	−	0.921100i	$-0.372708\pi$
$239$	2877.00	0.778651	0.389326	+	0.921100i	$0.372708\pi$
$240$	0	0
$241$	1622.00	0.433536	0.216768	−	0.976223i	$-0.430448\pi$
$241$	1622.00	0.433536	0.216768	+	0.976223i	$0.430448\pi$
$242$	− 2590.00i	− 0.687981i
$243$	− 4928.00i	− 1.30095i
$244$	1480.00	0.388309
$245$	0	0
$246$	−6930.00	−1.79610
$247$	− 6862.00i	− 1.76769i
$248$	2024.00i	0.518242i
$249$	8736.00	2.22338
$250$	0	0
$251$	−4752.00	−1.19499	−0.597497	−	0.801871i	$-0.703838\pi$
$251$	−4752.00	−1.19499	−0.597497	+	0.801871i	$0.703838\pi$
$252$	− 1760.00i	− 0.439959i
$253$	138.000i	0.0342924i
$254$	178.000	0.0439713
$255$	0	0
$256$	256.000	0.0625000
$257$	5073.00i	1.23130i	0.788018	+	0.615652i	$0.211107\pi$
$257$	5073.00i	1.23130i	−0.788018	+	0.615652i	$0.788893\pi$
$258$	− 3808.00i	− 0.918898i
$259$	2360.00	0.566190
$260$	0	0
$261$	−2178.00	−0.516532
$262$	− 3594.00i	− 0.847474i
$263$	1314.00i	0.308079i	0.988065	+	0.154039i	$0.0492283\pi$
$263$	1314.00i	0.308079i	−0.988065	+	0.154039i	$0.950772\pi$
$264$	336.000	0.0783309
$265$	0	0
$266$	−5840.00	−1.34614
$267$	5796.00i	1.32850i
$268$	2792.00i	0.636375i
$269$	−5265.00	−1.19336	−0.596678	−	0.802481i	$-0.703513\pi$
$269$	−5265.00	−1.19336	−0.596678	+	0.802481i	$0.703513\pi$
$270$	0	0
$271$	−2488.00	−0.557695	−0.278847	−	0.960335i	$-0.589952\pi$
$271$	−2488.00	−0.557695	−0.278847	+	0.960335i	$0.589952\pi$
$272$	2112.00i	0.470804i
$273$	6580.00i	1.45875i
$274$	−3672.00	−0.809612
$275$	0	0
$276$	644.000	0.140450
$277$	− 5465.00i	− 1.18542i	−0.805418	−	0.592708i	$-0.798059\pi$
$277$	− 5465.00i	− 1.18542i	0.805418	−	0.592708i	$-0.201941\pi$
$278$	2054.00i	0.443132i
$279$	5566.00	1.19436
$280$	0	0
$281$	8940.00	1.89792	0.948960	−	0.315396i	$-0.102137\pi$
$281$	8940.00	1.89792	0.948960	+	0.315396i	$0.102137\pi$
$282$	8946.00i	1.88910i
$283$	842.000i	0.176861i	0.996082	+	0.0884306i	$0.0281851\pi$
$283$	842.000i	0.176861i	−0.996082	+	0.0884306i	$0.971815\pi$
$284$	1428.00	0.298367
$285$	0	0
$286$	−564.000	−0.116608
$287$	− 9900.00i	− 2.03616i
$288$	− 704.000i	− 0.144040i
$289$	−12511.0	−2.54651
$290$	0	0
$291$	6944.00	1.39885
$292$	1036.00i	0.207628i
$293$	− 4032.00i	− 0.803932i	−0.915655	−	0.401966i	$-0.868327\pi$
$293$	− 4032.00i	− 0.803932i	0.915655	−	0.401966i	$-0.131673\pi$
$294$	798.000	0.158300
$295$	0	0
$296$	944.000	0.185368
$297$	210.000i	0.0410284i
$298$	− 4620.00i	− 0.898085i
$299$	−1081.00	−0.209083
$300$	0	0
$301$	5440.00	1.04172
$302$	− 4298.00i	− 0.818947i
$303$	− 10794.0i	− 2.04653i
$304$	−2336.00	−0.440720
$305$	0	0
$306$	5808.00	1.08504
$307$	1096.00i	0.203753i	0.994797	+	0.101876i	$0.0324846\pi$
$307$	1096.00i	0.203753i	−0.994797	+	0.101876i	$0.967515\pi$
$308$	480.000i	0.0888004i
$309$	−224.000	−0.0412392
$310$	0	0
$311$	−4653.00	−0.848384	−0.424192	−	0.905572i	$-0.639442\pi$
$311$	−4653.00	−0.848384	−0.424192	+	0.905572i	$0.639442\pi$
$312$	2632.00i	0.477589i
$313$	3440.00i	0.621215i	0.950538	+	0.310608i	$0.100532\pi$
$313$	3440.00i	0.621215i	−0.950538	+	0.310608i	$0.899468\pi$
$314$	3664.00	0.658508
$315$	0	0
$316$	2168.00	0.385948
$317$	− 3066.00i	− 0.543229i	−0.962406	−	0.271615i	$-0.912442\pi$
$317$	− 3066.00i	− 0.543229i	0.962406	−	0.271615i	$-0.0875576\pi$
$318$	4788.00i	0.844332i
$319$	594.000	0.104256
$320$	0	0
$321$	−5838.00	−1.01509
$322$	920.000i	0.159222i
$323$	− 19272.0i	− 3.31988i
$324$	3356.00	0.575446
$325$	0	0
$326$	−2434.00	−0.413518
$327$	8344.00i	1.41108i
$328$	− 3960.00i	− 0.666629i
$329$	−12780.0	−2.14159
$330$	0	0
$331$	1505.00	0.249916	0.124958	−	0.992162i	$-0.460120\pi$
$331$	1505.00	0.249916	0.124958	+	0.992162i	$0.460120\pi$
$332$	4992.00i	0.825216i
$333$	− 2596.00i	− 0.427207i
$334$	6096.00	0.998677
$335$	0	0
$336$	2240.00	0.363696
$337$	3268.00i	0.528247i	0.964489	+	0.264124i	$0.0850827\pi$
$337$	3268.00i	0.528247i	−0.964489	+	0.264124i	$0.914917\pi$
$338$	− 24.0000i	− 0.00386221i
$339$	924.000	0.148038
$340$	0	0
$341$	−1518.00	−0.241068
$342$	6424.00i	1.01570i
$343$	− 5720.00i	− 0.900440i
$344$	2176.00	0.341052
$345$	0	0
$346$	−1548.00	−0.240523
$347$	− 4164.00i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 4164.00i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 2772.00i	− 0.426997i
$349$	2911.00	0.446482	0.223241	−	0.974763i	$-0.428336\pi$
$349$	2911.00	0.446482	0.223241	+	0.974763i	$0.428336\pi$
$350$	0	0
$351$	−1645.00	−0.250153
$352$	192.000i	0.0290728i
$353$	9753.00i	1.47054i	0.677776	+	0.735269i	$0.262944\pi$
$353$	9753.00i	1.47054i	−0.677776	+	0.735269i	$0.737056\pi$
$354$	3360.00	0.504469
$355$	0	0
$356$	−3312.00	−0.493078
$357$	18480.0i	2.73968i
$358$	3750.00i	0.553614i
$359$	−3858.00	−0.567180	−0.283590	−	0.958946i	$-0.591525\pi$
$359$	−3858.00	−0.567180	−0.283590	+	0.958946i	$0.591525\pi$
$360$	0	0
$361$	14457.0	2.10774
$362$	− 3212.00i	− 0.466351i
$363$	− 9065.00i	− 1.31071i
$364$	−3760.00	−0.541422
$365$	0	0
$366$	5180.00	0.739789
$367$	− 7856.00i	− 1.11738i	−0.829375	−	0.558692i	$-0.811303\pi$
$367$	− 7856.00i	− 1.11738i	0.829375	−	0.558692i	$-0.188697\pi$
$368$	368.000i	0.0521286i
$369$	−10890.0	−1.53634
$370$	0	0
$371$	−6840.00	−0.957184
$372$	7084.00i	0.987334i
$373$	− 34.0000i	− 0.00471971i	−0.999997	−	0.00235986i	$-0.999249\pi$
$373$	− 34.0000i	− 0.00471971i	0.999997	−	0.00235986i	$-0.000751166\pi$
$374$	−1584.00	−0.219002
$375$	0	0
$376$	−5112.00	−0.701147
$377$	4653.00i	0.635654i
$378$	1400.00i	0.190498i
$379$	6064.00	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	6064.00	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	623.000	0.0837723
$382$	5964.00i	0.798808i
$383$	11868.0i	1.58336i	0.610937	+	0.791679i	$0.290793\pi$
$383$	11868.0i	1.58336i	−0.610937	+	0.791679i	$0.709207\pi$
$384$	896.000	0.119072
$385$	0	0
$386$	−2770.00	−0.365257
$387$	− 5984.00i	− 0.786005i
$388$	3968.00i	0.519187i
$389$	−8616.00	−1.12300	−0.561502	−	0.827475i	$-0.689776\pi$
$389$	−8616.00	−1.12300	−0.561502	+	0.827475i	$0.689776\pi$
$390$	0	0
$391$	−3036.00	−0.392678
$392$	456.000i	0.0587538i
$393$	− 12579.0i	− 1.61457i
$394$	1914.00	0.244736
$395$	0	0
$396$	528.000	0.0670025
$397$	− 3119.00i	− 0.394303i	−0.980373	−	0.197151i	$-0.936831\pi$
$397$	− 3119.00i	− 0.394303i	0.980373	−	0.197151i	$-0.0631690\pi$
$398$	716.000i	0.0901755i
$399$	−20440.0	−2.56461
$400$	0	0
$401$	7986.00	0.994518	0.497259	−	0.867602i	$-0.334340\pi$
$401$	7986.00	0.994518	0.497259	+	0.867602i	$0.334340\pi$
$402$	9772.00i	1.21240i
$403$	− 11891.0i	− 1.46981i
$404$	6168.00	0.759578
$405$	0	0
$406$	3960.00	0.484068
$407$	708.000i	0.0862267i
$408$	7392.00i	0.896957i
$409$	3475.00	0.420117	0.210058	−	0.977689i	$-0.432635\pi$
$409$	3475.00	0.420117	0.210058	+	0.977689i	$0.432635\pi$
$410$	0	0
$411$	−12852.0	−1.54244
$412$	− 128.000i	− 0.0153061i
$413$	4800.00i	0.571895i
$414$	1012.00	0.120138
$415$	0	0
$416$	−1504.00	−0.177259
$417$	7189.00i	0.844237i
$418$	− 1752.00i	− 0.205007i
$419$	−10992.0	−1.28161	−0.640805	−	0.767704i	$-0.721399\pi$
$419$	−10992.0	−1.28161	−0.640805	+	0.767704i	$0.721399\pi$
$420$	0	0
$421$	2012.00	0.232919	0.116459	−	0.993195i	$-0.462845\pi$
$421$	2012.00	0.232919	0.116459	+	0.993195i	$0.462845\pi$
$422$	− 10760.0i	− 1.24121i
$423$	14058.0i	1.61589i
$424$	−2736.00	−0.313377
$425$	0	0
$426$	4998.00	0.568436
$427$	7400.00i	0.838668i
$428$	− 3336.00i	− 0.376756i
$429$	−1974.00	−0.222158
$430$	0	0
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	560.000i	0.0623681i
$433$	5786.00i	0.642165i	0.947051	+	0.321082i	$0.104047\pi$
$433$	5786.00i	0.642165i	−0.947051	+	0.321082i	$0.895953\pi$
$434$	−10120.0	−1.11930
$435$	0	0
$436$	−4768.00	−0.523729
$437$	− 3358.00i	− 0.367586i
$438$	3626.00i	0.395564i
$439$	−2549.00	−0.277123	−0.138562	−	0.990354i	$-0.544248\pi$
$439$	−2549.00	−0.277123	−0.138562	+	0.990354i	$0.544248\pi$
$440$	0	0
$441$	1254.00	0.135407
$442$	− 12408.0i	− 1.33527i
$443$	1311.00i	0.140604i	0.997526	+	0.0703019i	$0.0223963\pi$
$443$	1311.00i	0.140604i	−0.997526	+	0.0703019i	$0.977604\pi$
$444$	3304.00	0.353155
$445$	0	0
$446$	−2080.00	−0.220832
$447$	− 16170.0i	− 1.71099i
$448$	1280.00i	0.134987i
$449$	14610.0	1.53561	0.767805	−	0.640684i	$-0.221349\pi$
$449$	14610.0	1.53561	0.767805	+	0.640684i	$0.221349\pi$
$450$	0	0
$451$	2970.00	0.310093
$452$	528.000i	0.0549448i
$453$	− 15043.0i	− 1.56022i
$454$	−7488.00	−0.774073
$455$	0	0
$456$	−8176.00	−0.839641
$457$	− 80.0000i	− 0.00818871i	−0.999992	−	0.00409436i	$-0.998697\pi$
$457$	− 80.0000i	− 0.00818871i	0.999992	−	0.00409436i	$-0.00130328\pi$
$458$	− 5608.00i	− 0.572150i
$459$	−4620.00	−0.469811
$460$	0	0
$461$	−2343.00	−0.236712	−0.118356	−	0.992971i	$-0.537762\pi$
$461$	−2343.00	−0.236712	−0.118356	+	0.992971i	$0.537762\pi$
$462$	1680.00i	0.169179i
$463$	− 3400.00i	− 0.341277i	−0.985334	−	0.170639i	$-0.945417\pi$
$463$	− 3400.00i	− 0.341277i	0.985334	−	0.170639i	$-0.0545831\pi$
$464$	1584.00	0.158481
$465$	0	0
$466$	9738.00	0.968035
$467$	1374.00i	0.136148i	0.997680	+	0.0680740i	$0.0216854\pi$
$467$	1374.00i	0.136148i	−0.997680	+	0.0680740i	$0.978315\pi$
$468$	4136.00i	0.408519i
$469$	−13960.0	−1.37444
$470$	0	0
$471$	12824.0	1.25456
$472$	1920.00i	0.187236i
$473$	1632.00i	0.158646i
$474$	7588.00	0.735292
$475$	0	0
$476$	−10560.0	−1.01684
$477$	7524.00i	0.722223i
$478$	5754.00i	0.550590i
$479$	−4536.00	−0.432683	−0.216341	−	0.976318i	$-0.569412\pi$
$479$	−4536.00	−0.432683	−0.216341	+	0.976318i	$0.569412\pi$
$480$	0	0
$481$	−5546.00	−0.525729
$482$	3244.00i	0.306556i
$483$	3220.00i	0.303344i
$484$	5180.00	0.486476
$485$	0	0
$486$	9856.00	0.919912
$487$	11455.0i	1.06586i	0.846158	+	0.532932i	$0.178910\pi$
$487$	11455.0i	1.06586i	−0.846158	+	0.532932i	$0.821090\pi$
$488$	2960.00i	0.274576i
$489$	−8519.00	−0.787817
$490$	0	0
$491$	−10395.0	−0.955437	−0.477719	−	0.878513i	$-0.658536\pi$
$491$	−10395.0	−0.955437	−0.477719	+	0.878513i	$0.658536\pi$
$492$	− 13860.0i	− 1.27003i
$493$	13068.0i	1.19382i
$494$	13724.0	1.24994
$495$	0	0
$496$	−4048.00	−0.366453
$497$	7140.00i	0.644412i
$498$	17472.0i	1.57217i
$499$	5497.00	0.493145	0.246573	−	0.969124i	$-0.420696\pi$
$499$	5497.00	0.493145	0.246573	+	0.969124i	$0.420696\pi$
$500$	0	0
$501$	21336.0	1.90264
$502$	− 9504.00i	− 0.844989i
$503$	7158.00i	0.634512i	0.948340	+	0.317256i	$0.102761\pi$
$503$	7158.00i	0.634512i	−0.948340	+	0.317256i	$0.897239\pi$
$504$	3520.00	0.311098
$505$	0	0
$506$	−276.000	−0.0242484
$507$	− 84.0000i	− 0.00735813i
$508$	356.000i	0.0310924i
$509$	−12801.0	−1.11472	−0.557362	−	0.830270i	$-0.688186\pi$
$509$	−12801.0	−1.11472	−0.557362	+	0.830270i	$0.688186\pi$
$510$	0	0
$511$	−5180.00	−0.448434
$512$	512.000i	0.0441942i
$513$	− 5110.00i	− 0.439789i
$514$	−10146.0	−0.870663
$515$	0	0
$516$	7616.00	0.649759
$517$	− 3834.00i	− 0.326149i
$518$	4720.00i	0.400357i
$519$	−5418.00	−0.458235
$520$	0	0
$521$	16788.0	1.41170	0.705850	−	0.708361i	$-0.250565\pi$
$521$	16788.0	1.41170	0.705850	+	0.708361i	$0.250565\pi$
$522$	− 4356.00i	− 0.365243i
$523$	19040.0i	1.59189i	0.605366	+	0.795947i	$0.293027\pi$
$523$	19040.0i	1.59189i	−0.605366	+	0.795947i	$0.706973\pi$
$524$	7188.00	0.599254
$525$	0	0
$526$	−2628.00	−0.217845
$527$	− 33396.0i	− 2.76044i
$528$	672.000i	0.0553883i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	5280.00	0.431511
$532$	− 11680.0i	− 0.951865i
$533$	23265.0i	1.89065i
$534$	−11592.0	−0.939391
$535$	0	0
$536$	−5584.00	−0.449985
$537$	13125.0i	1.05472i
$538$	− 10530.0i	− 0.843830i
$539$	−342.000	−0.0273302
$540$	0	0
$541$	−13339.0	−1.06005	−0.530026	−	0.847981i	$-0.677818\pi$
$541$	−13339.0	−1.06005	−0.530026	+	0.847981i	$0.677818\pi$
$542$	− 4976.00i	− 0.394350i
$543$	− 11242.0i	− 0.888472i
$544$	−4224.00	−0.332909
$545$	0	0
$546$	−13160.0	−1.03149
$547$	22975.0i	1.79587i	0.440130	+	0.897934i	$0.354932\pi$
$547$	22975.0i	1.79587i	−0.440130	+	0.897934i	$0.645068\pi$
$548$	− 7344.00i	− 0.572482i
$549$	8140.00	0.632799
$550$	0	0
$551$	−14454.0	−1.11753
$552$	1288.00i	0.0993133i
$553$	10840.0i	0.833569i
$554$	10930.0	0.838215
$555$	0	0
$556$	−4108.00	−0.313342
$557$	17964.0i	1.36653i	0.730169	+	0.683267i	$0.239441\pi$
$557$	17964.0i	1.36653i	−0.730169	+	0.683267i	$0.760559\pi$
$558$	11132.0i	0.844543i
$559$	−12784.0	−0.967273
$560$	0	0
$561$	−5544.00	−0.417233
$562$	17880.0i	1.34203i
$563$	12636.0i	0.945904i	0.881088	+	0.472952i	$0.156812\pi$
$563$	12636.0i	0.945904i	−0.881088	+	0.472952i	$0.843188\pi$
$564$	−17892.0	−1.33580
$565$	0	0
$566$	−1684.00	−0.125060
$567$	16780.0i	1.24285i
$568$	2856.00i	0.210977i
$569$	10302.0	0.759020	0.379510	−	0.925188i	$-0.376093\pi$
$569$	10302.0	0.759020	0.379510	+	0.925188i	$0.376093\pi$
$570$	0	0
$571$	12380.0	0.907333	0.453666	−	0.891172i	$-0.350116\pi$
$571$	12380.0	0.907333	0.453666	+	0.891172i	$0.350116\pi$
$572$	− 1128.00i	− 0.0824546i
$573$	20874.0i	1.52186i
$574$	19800.0	1.43978
$575$	0	0
$576$	1408.00	0.101852
$577$	− 1913.00i	− 0.138023i	−0.997616	−	0.0690115i	$-0.978015\pi$
$577$	− 1913.00i	− 0.138023i	0.997616	−	0.0690115i	$-0.0219845\pi$
$578$	− 25022.0i	− 1.80065i
$579$	−9695.00	−0.695873
$580$	0	0
$581$	−24960.0	−1.78230
$582$	13888.0i	0.989134i
$583$	− 2052.00i	− 0.145772i
$584$	−2072.00	−0.146815
$585$	0	0
$586$	8064.00	0.568465
$587$	− 16767.0i	− 1.17896i	−0.807784	−	0.589479i	$-0.799333\pi$
$587$	− 16767.0i	− 1.17896i	0.807784	−	0.589479i	$-0.200667\pi$
$588$	1596.00i	0.111935i
$589$	36938.0	2.58405
$590$	0	0
$591$	6699.00	0.466261
$592$	1888.00i	0.131075i
$593$	− 16722.0i	− 1.15799i	−0.815330	−	0.578997i	$-0.803444\pi$
$593$	− 16722.0i	− 1.15799i	0.815330	−	0.578997i	$-0.196556\pi$
$594$	−420.000	−0.0290115
$595$	0	0
$596$	9240.00	0.635042
$597$	2506.00i	0.171799i
$598$	− 2162.00i	− 0.147844i
$599$	4200.00	0.286490	0.143245	−	0.989687i	$-0.454246\pi$
$599$	4200.00	0.286490	0.143245	+	0.989687i	$0.454246\pi$
$600$	0	0
$601$	−19915.0	−1.35166	−0.675832	−	0.737056i	$-0.736215\pi$
$601$	−19915.0	−1.35166	−0.675832	+	0.737056i	$0.736215\pi$
$602$	10880.0i	0.736604i
$603$	15356.0i	1.03706i
$604$	8596.00	0.579083
$605$	0	0
$606$	21588.0	1.44712
$607$	− 24044.0i	− 1.60777i	−0.594785	−	0.803885i	$-0.702763\pi$
$607$	− 24044.0i	− 1.60777i	0.594785	−	0.803885i	$-0.297237\pi$
$608$	− 4672.00i	− 0.311636i
$609$	13860.0	0.922226
$610$	0	0
$611$	30033.0	1.98855
$612$	11616.0i	0.767237i
$613$	3452.00i	0.227447i	0.993512	+	0.113723i	$0.0362778\pi$
$613$	3452.00i	0.227447i	−0.993512	+	0.113723i	$0.963722\pi$
$614$	−2192.00	−0.144075
$615$	0	0
$616$	−960.000	−0.0627914
$617$	16374.0i	1.06838i	0.845363	+	0.534192i	$0.179384\pi$
$617$	16374.0i	1.06838i	−0.845363	+	0.534192i	$0.820616\pi$
$618$	− 448.000i	− 0.0291605i
$619$	12760.0	0.828542	0.414271	−	0.910154i	$-0.364037\pi$
$619$	12760.0	0.828542	0.414271	+	0.910154i	$0.364037\pi$
$620$	0	0
$621$	−805.000	−0.0520186
$622$	− 9306.00i	− 0.599898i
$623$	− 16560.0i	− 1.06495i
$624$	−5264.00	−0.337706
$625$	0	0
$626$	−6880.00	−0.439265
$627$	− 6132.00i	− 0.390572i
$628$	7328.00i	0.465635i
$629$	−15576.0	−0.987370
$630$	0	0
$631$	29420.0	1.85609	0.928044	−	0.372470i	$-0.121489\pi$
$631$	29420.0	1.85609	0.928044	+	0.372470i	$0.121489\pi$
$632$	4336.00i	0.272906i
$633$	− 37660.0i	− 2.36469i
$634$	6132.00	0.384121
$635$	0	0
$636$	−9576.00	−0.597033
$637$	− 2679.00i	− 0.166634i
$638$	1188.00i	0.0737200i
$639$	7854.00	0.486228
$640$	0	0
$641$	−13692.0	−0.843684	−0.421842	−	0.906669i	$-0.638616\pi$
$641$	−13692.0	−0.843684	−0.421842	+	0.906669i	$0.638616\pi$
$642$	− 11676.0i	− 0.717780i
$643$	27398.0i	1.68036i	0.542307	+	0.840180i	$0.317551\pi$
$643$	27398.0i	1.68036i	−0.542307	+	0.840180i	$0.682449\pi$
$644$	−1840.00	−0.112587
$645$	0	0
$646$	38544.0	2.34751
$647$	6417.00i	0.389920i	0.980811	+	0.194960i	$0.0624577\pi$
$647$	6417.00i	0.389920i	−0.980811	+	0.194960i	$0.937542\pi$
$648$	6712.00i	0.406902i
$649$	−1440.00	−0.0870954
$650$	0	0
$651$	−35420.0	−2.13244
$652$	− 4868.00i	− 0.292401i
$653$	14583.0i	0.873931i	0.899478	+	0.436965i	$0.143947\pi$
$653$	14583.0i	0.873931i	−0.899478	+	0.436965i	$0.856053\pi$
$654$	−16688.0	−0.997787
$655$	0	0
$656$	7920.00	0.471378
$657$	5698.00i	0.338356i
$658$	− 25560.0i	− 1.51434i
$659$	9624.00	0.568889	0.284444	−	0.958693i	$-0.408191\pi$
$659$	9624.00	0.568889	0.284444	+	0.958693i	$0.408191\pi$
$660$	0	0
$661$	−24586.0	−1.44672	−0.723362	−	0.690469i	$-0.757404\pi$
$661$	−24586.0	−1.44672	−0.723362	+	0.690469i	$0.757404\pi$
$662$	3010.00i	0.176717i
$663$	− 43428.0i	− 2.54390i
$664$	−9984.00	−0.583516
$665$	0	0
$666$	5192.00	0.302081
$667$	2277.00i	0.132183i
$668$	12192.0i	0.706172i
$669$	−7280.00	−0.420719
$670$	0	0
$671$	−2220.00	−0.127723
$672$	4480.00i	0.257172i
$673$	14339.0i	0.821289i	0.911795	+	0.410645i	$0.134696\pi$
$673$	14339.0i	0.821289i	−0.911795	+	0.410645i	$0.865304\pi$
$674$	−6536.00	−0.373527
$675$	0	0
$676$	48.0000	0.00273100
$677$	− 14658.0i	− 0.832131i	−0.909335	−	0.416066i	$-0.863409\pi$
$677$	− 14658.0i	− 0.832131i	0.909335	−	0.416066i	$-0.136591\pi$
$678$	1848.00i	0.104678i
$679$	−19840.0	−1.12134
$680$	0	0
$681$	−26208.0	−1.47473
$682$	− 3036.00i	− 0.170461i
$683$	16797.0i	0.941024i	0.882394	+	0.470512i	$0.155931\pi$
$683$	16797.0i	0.941024i	−0.882394	+	0.470512i	$0.844069\pi$
$684$	−12848.0	−0.718210
$685$	0	0
$686$	11440.0	0.636707
$687$	− 19628.0i	− 1.09004i
$688$	4352.00i	0.241161i
$689$	16074.0	0.888782
$690$	0	0
$691$	8132.00	0.447693	0.223846	−	0.974624i	$-0.428139\pi$
$691$	8132.00	0.447693	0.223846	+	0.974624i	$0.428139\pi$
$692$	− 3096.00i	− 0.170076i
$693$	2640.00i	0.144712i
$694$	8328.00	0.455514
$695$	0	0
$696$	5544.00	0.301932
$697$	65340.0i	3.55083i
$698$	5822.00i	0.315711i
$699$	34083.0	1.84426
$700$	0	0
$701$	19668.0	1.05970	0.529850	−	0.848091i	$-0.322248\pi$
$701$	19668.0	1.05970	0.529850	+	0.848091i	$0.322248\pi$
$702$	− 3290.00i	− 0.176885i
$703$	− 17228.0i	− 0.924276i
$704$	−384.000	−0.0205576
$705$	0	0
$706$	−19506.0	−1.03983
$707$	30840.0i	1.64053i
$708$	6720.00i	0.356713i
$709$	−18200.0	−0.964055	−0.482028	−	0.876156i	$-0.660100\pi$
$709$	−18200.0	−0.964055	−0.482028	+	0.876156i	$0.660100\pi$
$710$	0	0
$711$	11924.0	0.628952
$712$	− 6624.00i	− 0.348659i
$713$	− 5819.00i	− 0.305643i
$714$	−36960.0	−1.93725
$715$	0	0
$716$	−7500.00	−0.391464
$717$	20139.0i	1.04896i
$718$	− 7716.00i	− 0.401056i
$719$	11880.0	0.616202	0.308101	−	0.951354i	$-0.400307\pi$
$719$	11880.0	0.616202	0.308101	+	0.951354i	$0.400307\pi$
$720$	0	0
$721$	640.000	0.0330580
$722$	28914.0i	1.49040i
$723$	11354.0i	0.584038i
$724$	6424.00	0.329760
$725$	0	0
$726$	18130.0	0.926815
$727$	2554.00i	0.130292i	0.997876	+	0.0651462i	$0.0207514\pi$
$727$	2554.00i	0.130292i	−0.997876	+	0.0651462i	$0.979249\pi$
$728$	− 7520.00i	− 0.382843i
$729$	11843.0	0.601687
$730$	0	0
$731$	−35904.0	−1.81663
$732$	10360.0i	0.523110i
$733$	6308.00i	0.317860i	0.987290	+	0.158930i	$0.0508044\pi$
$733$	6308.00i	0.317860i	−0.987290	+	0.158930i	$0.949196\pi$
$734$	15712.0	0.790110
$735$	0	0
$736$	−736.000	−0.0368605
$737$	− 4188.00i	− 0.209317i
$738$	− 21780.0i	− 1.08636i
$739$	−9557.00	−0.475724	−0.237862	−	0.971299i	$-0.576447\pi$
$739$	−9557.00	−0.475724	−0.237862	+	0.971299i	$0.576447\pi$
$740$	0	0
$741$	48034.0	2.38134
$742$	− 13680.0i	− 0.676831i
$743$	19128.0i	0.944466i	0.881474	+	0.472233i	$0.156552\pi$
$743$	19128.0i	0.944466i	−0.881474	+	0.472233i	$0.843448\pi$
$744$	−14168.0	−0.698151
$745$	0	0
$746$	68.0000	0.00333734
$747$	27456.0i	1.34480i
$748$	− 3168.00i	− 0.154858i
$749$	16680.0	0.813717
$750$	0	0
$751$	−18448.0	−0.896374	−0.448187	−	0.893940i	$-0.647930\pi$
$751$	−18448.0	−0.896374	−0.448187	+	0.893940i	$0.647930\pi$
$752$	− 10224.0i	− 0.495786i
$753$	− 33264.0i	− 1.60984i
$754$	−9306.00	−0.449476
$755$	0	0
$756$	−2800.00	−0.134702
$757$	5602.00i	0.268967i	0.990916	+	0.134484i	$0.0429376\pi$
$757$	5602.00i	0.268967i	−0.990916	+	0.134484i	$0.957062\pi$
$758$	12128.0i	0.581146i
$759$	−966.000	−0.0461971
$760$	0	0
$761$	4005.00	0.190777	0.0953884	−	0.995440i	$-0.469591\pi$
$761$	4005.00	0.190777	0.0953884	+	0.995440i	$0.469591\pi$
$762$	1246.00i	0.0592360i
$763$	− 23840.0i	− 1.13115i
$764$	−11928.0	−0.564843
$765$	0	0
$766$	−23736.0	−1.11960
$767$	− 11280.0i	− 0.531026i
$768$	1792.00i	0.0841969i
$769$	−41726.0	−1.95667	−0.978334	−	0.207032i	$-0.933620\pi$
$769$	−41726.0	−1.95667	−0.978334	+	0.207032i	$0.933620\pi$
$770$	0	0
$771$	−35511.0	−1.65875
$772$	− 5540.00i	− 0.258276i
$773$	34116.0i	1.58741i	0.608303	+	0.793705i	$0.291850\pi$
$773$	34116.0i	1.58741i	−0.608303	+	0.793705i	$0.708150\pi$
$774$	11968.0	0.555789
$775$	0	0
$776$	−7936.00	−0.367121
$777$	16520.0i	0.762743i
$778$	− 17232.0i	− 0.794084i
$779$	−72270.0	−3.32393
$780$	0	0
$781$	−2142.00	−0.0981393
$782$	− 6072.00i	− 0.277665i
$783$	3465.00i	0.158147i
$784$	−912.000	−0.0415452
$785$	0	0
$786$	25158.0	1.14167
$787$	− 1652.00i	− 0.0748252i	−0.999300	−	0.0374126i	$-0.988088\pi$
$787$	− 1652.00i	− 0.0748252i	0.999300	−	0.0374126i	$-0.0119116\pi$
$788$	3828.00i	0.173054i
$789$	−9198.00	−0.415028
$790$	0	0
$791$	−2640.00	−0.118670
$792$	1056.00i	0.0473779i
$793$	− 17390.0i	− 0.778735i
$794$	6238.00	0.278814
$795$	0	0
$796$	−1432.00	−0.0637637
$797$	− 12486.0i	− 0.554927i	−0.960736	−	0.277463i	$-0.910506\pi$
$797$	− 12486.0i	− 0.554927i	0.960736	−	0.277463i	$-0.0894938\pi$
$798$	− 40880.0i	− 1.81345i
$799$	84348.0	3.73469
$800$	0	0
$801$	−18216.0	−0.803534
$802$	15972.0i	0.703231i
$803$	− 1554.00i	− 0.0682932i
$804$	−19544.0	−0.857293
$805$	0	0
$806$	23782.0	1.03931
$807$	− 36855.0i	− 1.60763i
$808$	12336.0i	0.537103i
$809$	−5490.00	−0.238589	−0.119294	−	0.992859i	$-0.538063\pi$
$809$	−5490.00	−0.238589	−0.119294	+	0.992859i	$0.538063\pi$
$810$	0	0
$811$	−14785.0	−0.640162	−0.320081	−	0.947390i	$-0.603710\pi$
$811$	−14785.0	−0.640162	−0.320081	+	0.947390i	$0.603710\pi$
$812$	7920.00i	0.342288i
$813$	− 17416.0i	− 0.751299i
$814$	−1416.00	−0.0609715
$815$	0	0
$816$	−14784.0	−0.634245
$817$	− 39712.0i	− 1.70055i
$818$	6950.00i	0.297067i
$819$	−20680.0	−0.882317
$820$	0	0
$821$	−12486.0	−0.530773	−0.265386	−	0.964142i	$-0.585500\pi$
$821$	−12486.0	−0.530773	−0.265386	+	0.964142i	$0.585500\pi$
$822$	− 25704.0i	− 1.09067i
$823$	− 39805.0i	− 1.68592i	−0.537973	−	0.842962i	$-0.680810\pi$
$823$	− 39805.0i	− 1.68592i	0.537973	−	0.842962i	$-0.319190\pi$
$824$	256.000	0.0108230
$825$	0	0
$826$	−9600.00	−0.404391
$827$	− 15024.0i	− 0.631724i	−0.948805	−	0.315862i	$-0.897706\pi$
$827$	− 15024.0i	− 0.631724i	0.948805	−	0.315862i	$-0.102294\pi$
$828$	2024.00i	0.0849503i
$829$	−14618.0	−0.612430	−0.306215	−	0.951962i	$-0.599063\pi$
$829$	−14618.0	−0.612430	−0.306215	+	0.951962i	$0.599063\pi$
$830$	0	0
$831$	38255.0	1.59693
$832$	− 3008.00i	− 0.125341i
$833$	− 7524.00i	− 0.312955i
$834$	−14378.0	−0.596966
$835$	0	0
$836$	3504.00	0.144962
$837$	− 8855.00i	− 0.365679i
$838$	− 21984.0i	− 0.906235i
$839$	10152.0	0.417743	0.208871	−	0.977943i	$-0.433021\pi$
$839$	10152.0	0.417743	0.208871	+	0.977943i	$0.433021\pi$
$840$	0	0
$841$	−14588.0	−0.598139
$842$	4024.00i	0.164699i
$843$	62580.0i	2.55678i
$844$	21520.0	0.877665
$845$	0	0
$846$	−28116.0	−1.14261
$847$	25900.0i	1.05069i
$848$	− 5472.00i	− 0.221591i
$849$	−5894.00	−0.238259
$850$	0	0
$851$	−2714.00	−0.109324
$852$	9996.00i	0.401945i
$853$	− 22306.0i	− 0.895361i	−0.894194	−	0.447680i	$-0.852250\pi$
$853$	− 22306.0i	− 0.895361i	0.894194	−	0.447680i	$-0.147750\pi$
$854$	−14800.0	−0.593028
$855$	0	0
$856$	6672.00	0.266407
$857$	− 1731.00i	− 0.0689963i	−0.999405	−	0.0344982i	$-0.989017\pi$
$857$	− 1731.00i	− 0.0689963i	0.999405	−	0.0344982i	$-0.0109833\pi$
$858$	− 3948.00i	− 0.157089i
$859$	12649.0	0.502419	0.251210	−	0.967933i	$-0.419172\pi$
$859$	12649.0	0.502419	0.251210	+	0.967933i	$0.419172\pi$
$860$	0	0
$861$	69300.0	2.74302
$862$	− 19584.0i	− 0.773821i
$863$	− 16143.0i	− 0.636749i	−0.947965	−	0.318374i	$-0.896863\pi$
$863$	− 16143.0i	− 0.636749i	0.947965	−	0.318374i	$-0.103137\pi$
$864$	−1120.00	−0.0441009
$865$	0	0
$866$	−11572.0	−0.454079
$867$	− 87577.0i	− 3.43053i
$868$	− 20240.0i	− 0.791464i
$869$	−3252.00	−0.126947
$870$	0	0
$871$	32806.0	1.27622
$872$	− 9536.00i	− 0.370332i
$873$	21824.0i	0.846083i
$874$	6716.00	0.259922
$875$	0	0
$876$	−7252.00	−0.279706
$877$	− 4094.00i	− 0.157633i	−0.996889	−	0.0788167i	$-0.974886\pi$
$877$	− 4094.00i	− 0.157633i	0.996889	−	0.0788167i	$-0.0251142\pi$
$878$	− 5098.00i	− 0.195956i
$879$	28224.0	1.08302
$880$	0	0
$881$	30396.0	1.16239	0.581196	−	0.813764i	$-0.302585\pi$
$881$	30396.0	1.16239	0.581196	+	0.813764i	$0.302585\pi$
$882$	2508.00i	0.0957469i
$883$	− 21148.0i	− 0.805987i	−0.915203	−	0.402994i	$-0.867970\pi$
$883$	− 21148.0i	− 0.805987i	0.915203	−	0.402994i	$-0.132030\pi$
$884$	24816.0	0.944177
$885$	0	0
$886$	−2622.00	−0.0994219
$887$	− 5031.00i	− 0.190445i	−0.995456	−	0.0952223i	$-0.969644\pi$
$887$	− 5031.00i	− 0.190445i	0.995456	−	0.0952223i	$-0.0303562\pi$
$888$	6608.00i	0.249718i
$889$	−1780.00	−0.0671533
$890$	0	0
$891$	−5034.00	−0.189276
$892$	− 4160.00i	− 0.156151i
$893$	93294.0i	3.49604i
$894$	32340.0	1.20986
$895$	0	0
$896$	−2560.00	−0.0954504
$897$	− 7567.00i	− 0.281666i
$898$	29220.0i	1.08584i
$899$	−25047.0	−0.929215
$900$	0	0
$901$	45144.0	1.66922
$902$	5940.00i	0.219269i
$903$	38080.0i	1.40335i
$904$	−1056.00	−0.0388518
$905$	0	0
$906$	30086.0	1.10325
$907$	538.000i	0.0196957i	0.999952	+	0.00984785i	$0.00313472\pi$
$907$	538.000i	0.0196957i	−0.999952	+	0.00984785i	$0.996865\pi$
$908$	− 14976.0i	− 0.547352i
$909$	33924.0	1.23783
$910$	0	0
$911$	3078.00	0.111941	0.0559707	−	0.998432i	$-0.482175\pi$
$911$	3078.00	0.111941	0.0559707	+	0.998432i	$0.482175\pi$
$912$	− 16352.0i	− 0.593716i
$913$	− 7488.00i	− 0.271431i
$914$	160.000	0.00579029
$915$	0	0
$916$	11216.0	0.404571
$917$	35940.0i	1.29427i
$918$	− 9240.00i	− 0.332206i
$919$	−20288.0	−0.728226	−0.364113	−	0.931355i	$-0.618628\pi$
$919$	−20288.0	−0.728226	−0.364113	+	0.931355i	$0.618628\pi$
$920$	0	0
$921$	−7672.00	−0.274485
$922$	− 4686.00i	− 0.167381i
$923$	− 16779.0i	− 0.598361i
$924$	−3360.00	−0.119628
$925$	0	0
$926$	6800.00	0.241320
$927$	− 704.000i	− 0.0249433i
$928$	3168.00i	0.112063i
$929$	−28911.0	−1.02103	−0.510516	−	0.859868i	$-0.670546\pi$
$929$	−28911.0	−1.02103	−0.510516	+	0.859868i	$0.670546\pi$
$930$	0	0
$931$	8322.00	0.292957
$932$	19476.0i	0.684504i
$933$	− 32571.0i	− 1.14290i
$934$	−2748.00	−0.0962712
$935$	0	0
$936$	−8272.00	−0.288866
$937$	− 14810.0i	− 0.516352i	−0.966098	−	0.258176i	$-0.916879\pi$
$937$	− 14810.0i	− 0.516352i	0.966098	−	0.258176i	$-0.0831214\pi$
$938$	− 27920.0i	− 0.971877i
$939$	−24080.0	−0.836870
$940$	0	0
$941$	−2544.00	−0.0881318	−0.0440659	−	0.999029i	$-0.514031\pi$
$941$	−2544.00	−0.0881318	−0.0440659	+	0.999029i	$0.514031\pi$
$942$	25648.0i	0.887109i
$943$	11385.0i	0.393157i
$944$	−3840.00	−0.132396
$945$	0	0
$946$	−3264.00	−0.112179
$947$	11145.0i	0.382433i	0.981548	+	0.191216i	$0.0612432\pi$
$947$	11145.0i	0.382433i	−0.981548	+	0.191216i	$0.938757\pi$
$948$	15176.0i	0.519930i
$949$	12173.0	0.416388
$950$	0	0
$951$	21462.0	0.731812
$952$	− 21120.0i	− 0.719016i
$953$	− 4386.00i	− 0.149083i	−0.997218	−	0.0745417i	$-0.976251\pi$
$953$	− 4386.00i	− 0.149083i	0.997218	−	0.0745417i	$-0.0237494\pi$
$954$	−15048.0	−0.510689
$955$	0	0
$956$	−11508.0	−0.389326
$957$	4158.00i	0.140448i
$958$	− 9072.00i	− 0.305953i
$959$	36720.0	1.23644
$960$	0	0
$961$	34218.0	1.14860
$962$	− 11092.0i	− 0.371747i
$963$	− 18348.0i	− 0.613973i
$964$	−6488.00	−0.216768
$965$	0	0
$966$	−6440.00	−0.214496
$967$	− 56381.0i	− 1.87496i	−0.348033	−	0.937482i	$-0.613150\pi$
$967$	− 56381.0i	− 1.87496i	0.348033	−	0.937482i	$-0.386850\pi$
$968$	10360.0i	0.343991i
$969$	134904.	4.47238
$970$	0	0
$971$	43782.0	1.44699	0.723497	−	0.690327i	$-0.242534\pi$
$971$	43782.0	1.44699	0.723497	+	0.690327i	$0.242534\pi$
$972$	19712.0i	0.650476i
$973$	− 20540.0i	− 0.676755i
$974$	−22910.0	−0.753679
$975$	0	0
$976$	−5920.00	−0.194154
$977$	− 3714.00i	− 0.121619i	−0.998149	−	0.0608093i	$-0.980632\pi$
$977$	− 3714.00i	− 0.121619i	0.998149	−	0.0608093i	$-0.0193682\pi$
$978$	− 17038.0i	− 0.557071i
$979$	4968.00	0.162184
$980$	0	0
$981$	−26224.0	−0.853484
$982$	− 20790.0i	− 0.675596i
$983$	− 4662.00i	− 0.151266i	−0.997136	−	0.0756331i	$-0.975902\pi$
$983$	− 4662.00i	− 0.151266i	0.997136	−	0.0756331i	$-0.0240978\pi$
$984$	27720.0	0.898050
$985$	0	0
$986$	−26136.0	−0.844158
$987$	− 89460.0i	− 2.88505i
$988$	27448.0i	0.883843i
$989$	−6256.00	−0.201142
$990$	0	0
$991$	51440.0	1.64889	0.824443	−	0.565945i	$-0.191489\pi$
$991$	51440.0	1.64889	0.824443	+	0.565945i	$0.191489\pi$
$992$	− 8096.00i	− 0.259121i
$993$	10535.0i	0.336675i
$994$	−14280.0	−0.455668
$995$	0	0
$996$	−34944.0	−1.11169
$997$	− 686.000i	− 0.0217912i	−0.999941	−	0.0108956i	$-0.996532\pi$
$997$	− 686.000i	− 0.0217912i	0.999941	−	0.0108956i	$-0.00346825\pi$
$998$	10994.0i	0.348706i
$999$	−4130.00	−0.130798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.b.599.2		2
5.2	odd	4		230.4.a.c.1.1	✓	1
5.3	odd	4		1150.4.a.e.1.1		1
5.4	even	2	inner	1150.4.b.b.599.1		2
15.2	even	4		2070.4.a.j.1.1		1
20.7	even	4		1840.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.c.1.1	✓	1	5.2	odd	4
1150.4.a.e.1.1		1	5.3	odd	4
1150.4.b.b.599.1		2	5.4	even	2	inner
1150.4.b.b.599.2		2	1.1	even	1	trivial
1840.4.a.a.1.1		1	20.7	even	4
2070.4.a.j.1.1		1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +7.00000i q^{3} -4.00000 q^{4} -14.0000 q^{6} -20.0000i q^{7} -8.00000i q^{8} -22.0000 q^{9} +6.00000 q^{11} -28.0000i q^{12} +47.0000i q^{13} +40.0000 q^{14} +16.0000 q^{16} +132.000i q^{17} -44.0000i q^{18} -146.000 q^{19} +140.000 q^{21} +12.0000i q^{22} +23.0000i q^{23} +56.0000 q^{24} -94.0000 q^{26} +35.0000i q^{27} +80.0000i q^{28} +99.0000 q^{29} -253.000 q^{31} +32.0000i q^{32} +42.0000i q^{33} -264.000 q^{34} +88.0000 q^{36} +118.000i q^{37} -292.000i q^{38} -329.000 q^{39} +495.000 q^{41} +280.000i q^{42} +272.000i q^{43} -24.0000 q^{44} -46.0000 q^{46} -639.000i q^{47} +112.000i q^{48} -57.0000 q^{49} -924.000 q^{51} -188.000i q^{52} -342.000i q^{53} -70.0000 q^{54} -160.000 q^{56} -1022.00i q^{57} +198.000i q^{58} -240.000 q^{59} -370.000 q^{61} -506.000i q^{62} +440.000i q^{63} -64.0000 q^{64} -84.0000 q^{66} -698.000i q^{67} -528.000i q^{68} -161.000 q^{69} -357.000 q^{71} +176.000i q^{72} -259.000i q^{73} -236.000 q^{74} +584.000 q^{76} -120.000i q^{77} -658.000i q^{78} -542.000 q^{79} -839.000 q^{81} +990.000i q^{82} -1248.00i q^{83} -560.000 q^{84} -544.000 q^{86} +693.000i q^{87} -48.0000i q^{88} +828.000 q^{89} +940.000 q^{91} -92.0000i q^{92} -1771.00i q^{93} +1278.00 q^{94} -224.000 q^{96} -992.000i q^{97} -114.000i q^{98} -132.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 28 q^{6} - 44 q^{9} + 12 q^{11} + 80 q^{14} + 32 q^{16} - 292 q^{19} + 280 q^{21} + 112 q^{24} - 188 q^{26} + 198 q^{29} - 506 q^{31} - 528 q^{34} + 176 q^{36} - 658 q^{39} + 990 q^{41}+ \cdots - 264 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 28 * q^6 - 44 * q^9 + 12 * q^11 + 80 * q^14 + 32 * q^16 - 292 * q^19 + 280 * q^21 + 112 * q^24 - 188 * q^26 + 198 * q^29 - 506 * q^31 - 528 * q^34 + 176 * q^36 - 658 * q^39 + 990 * q^41 - 48 * q^44 - 92 * q^46 - 114 * q^49 - 1848 * q^51 - 140 * q^54 - 320 * q^56 - 480 * q^59 - 740 * q^61 - 128 * q^64 - 168 * q^66 - 322 * q^69 - 714 * q^71 - 472 * q^74 + 1168 * q^76 - 1084 * q^79 - 1678 * q^81 - 1120 * q^84 - 1088 * q^86 + 1656 * q^89 + 1880 * q^91 + 2556 * q^94 - 448 * q^96 - 264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more