LMFDB - Embedded newform 650.6.a.p.1.2

Error: no document with id 270934971 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,6,Mod(1,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,20,-9,80,0,-36,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$104.249482878$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 601x^{3} + 1405x^{2} + 36840x - 60300 $ x^5 - x^4 - 601x^3 + 1405x^2 + 36840*x - 60300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.92606$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} -9.92606 q^{3} +16.0000 q^{4} -39.7042 q^{6} -124.276 q^{7} +64.0000 q^{8} -144.473 q^{9} +120.384 q^{11} -158.817 q^{12} -169.000 q^{13} -497.102 q^{14} +256.000 q^{16} +1234.94 q^{17} -577.893 q^{18} +2711.16 q^{19} +1233.57 q^{21} +481.535 q^{22} +1573.59 q^{23} -635.268 q^{24} -676.000 q^{26} +3846.08 q^{27} -1988.41 q^{28} -184.523 q^{29} -51.4763 q^{31} +1024.00 q^{32} -1194.94 q^{33} +4939.77 q^{34} -2311.57 q^{36} -12051.0 q^{37} +10844.6 q^{38} +1677.50 q^{39} -4439.57 q^{41} +4934.27 q^{42} -13909.2 q^{43} +1926.14 q^{44} +6294.35 q^{46} -27970.2 q^{47} -2541.07 q^{48} -1362.59 q^{49} -12258.1 q^{51} -2704.00 q^{52} +24170.2 q^{53} +15384.3 q^{54} -7953.64 q^{56} -26911.1 q^{57} -738.091 q^{58} -38786.4 q^{59} +53666.6 q^{61} -205.905 q^{62} +17954.5 q^{63} +4096.00 q^{64} -4779.75 q^{66} -10772.8 q^{67} +19759.1 q^{68} -15619.5 q^{69} +6297.35 q^{71} -9246.29 q^{72} +4254.45 q^{73} -48204.1 q^{74} +43378.5 q^{76} -14960.8 q^{77} +6710.02 q^{78} -49474.9 q^{79} -3069.45 q^{81} -17758.3 q^{82} -100785. q^{83} +19737.1 q^{84} -55636.8 q^{86} +1831.58 q^{87} +7704.56 q^{88} +55060.2 q^{89} +21002.6 q^{91} +25177.4 q^{92} +510.957 q^{93} -111881. q^{94} -10164.3 q^{96} -177285. q^{97} -5450.34 q^{98} -17392.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 20 q^{2} - 9 q^{3} + 80 q^{4} - 36 q^{6} + 42 q^{7} + 320 q^{8} + 4 q^{9} - 213 q^{11} - 144 q^{12} - 845 q^{13} + 168 q^{14} + 1280 q^{16} - 601 q^{17} + 16 q^{18} - 567 q^{19} - 3700 q^{21} - 852 q^{22}+ \cdots - 440284 q^{99}+O(q^{100}) $ 5 * q + 20 * q^2 - 9 * q^3 + 80 * q^4 - 36 * q^6 + 42 * q^7 + 320 * q^8 + 4 * q^9 - 213 * q^11 - 144 * q^12 - 845 * q^13 + 168 * q^14 + 1280 * q^16 - 601 * q^17 + 16 * q^18 - 567 * q^19 - 3700 * q^21 - 852 * q^22 - 636 * q^23 - 576 * q^24 - 3380 * q^26 - 5283 * q^27 + 672 * q^28 + 540 * q^29 - 10828 * q^31 + 5120 * q^32 + 2227 * q^33 - 2404 * q^34 + 64 * q^36 - 12708 * q^37 - 2268 * q^38 + 1521 * q^39 - 11461 * q^41 - 14800 * q^42 - 2020 * q^43 - 3408 * q^44 - 2544 * q^46 - 35062 * q^47 - 2304 * q^48 - 7285 * q^49 + 12859 * q^51 - 13520 * q^52 - 7718 * q^53 - 21132 * q^54 + 2688 * q^56 - 76689 * q^57 + 2160 * q^58 - 44686 * q^59 + 18972 * q^61 - 43312 * q^62 + 23698 * q^63 + 20480 * q^64 + 8908 * q^66 - 98667 * q^67 - 9616 * q^68 - 32718 * q^69 + 23928 * q^71 + 256 * q^72 + 17529 * q^73 - 50832 * q^74 - 9072 * q^76 - 74234 * q^77 + 6084 * q^78 - 67676 * q^79 + 26629 * q^81 - 45844 * q^82 + 13689 * q^83 - 59200 * q^84 - 8080 * q^86 - 64204 * q^87 - 13632 * q^88 - 287231 * q^89 - 7098 * q^91 - 10176 * q^92 + 38102 * q^93 - 140248 * q^94 - 9216 * q^96 - 78846 * q^97 - 29140 * q^98 - 440284 * q^99

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.92606	−0.636757	−0.318379	−	0.947964i	$-0.603138\pi$
$3$	−9.92606	−0.636757	−0.318379	+	0.947964i	$0.603138\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−39.7042	−0.450255
$7$	−124.276	−0.958607	−0.479304	−	0.877649i	$-0.659111\pi$
$7$	−124.276	−0.958607	−0.479304	+	0.877649i	$0.659111\pi$
$8$	64.0000	0.353553
$9$	−144.473	−0.594540
$10$	0	0
$11$	120.384	0.299976	0.149988	−	0.988688i	$-0.452077\pi$
$11$	120.384	0.299976	0.149988	+	0.988688i	$0.452077\pi$
$12$	−158.817	−0.318379
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−497.102	−0.677838
$15$	0	0
$16$	256.000	0.250000
$17$	1234.94	1.03639	0.518197	−	0.855262i	$-0.326604\pi$
$17$	1234.94	1.03639	0.518197	+	0.855262i	$0.326604\pi$
$18$	−577.893	−0.420404
$19$	2711.16	1.72294	0.861471	−	0.507807i	$-0.169544\pi$
$19$	2711.16	1.72294	0.861471	+	0.507807i	$0.169544\pi$
$20$	0	0
$21$	1233.57	0.610400
$22$	481.535	0.212115
$23$	1573.59	0.620257	0.310128	−	0.950695i	$-0.399628\pi$
$23$	1573.59	0.620257	0.310128	+	0.950695i	$0.399628\pi$
$24$	−635.268	−0.225128
$25$	0	0
$26$	−676.000	−0.196116
$27$	3846.08	1.01533
$28$	−1988.41	−0.479304
$29$	−184.523	−0.0407432	−0.0203716	−	0.999792i	$-0.506485\pi$
$29$	−184.523	−0.0407432	−0.0203716	+	0.999792i	$0.506485\pi$
$30$	0	0
$31$	−51.4763	−0.00962062	−0.00481031	−	0.999988i	$-0.501531\pi$
$31$	−51.4763	−0.00962062	−0.00481031	+	0.999988i	$0.501531\pi$
$32$	1024.00	0.176777
$33$	−1194.94	−0.191012
$34$	4939.77	0.732841
$35$	0	0
$36$	−2311.57	−0.297270
$37$	−12051.0	−1.44717	−0.723585	−	0.690236i	$-0.757507\pi$
$37$	−12051.0	−1.44717	−0.723585	+	0.690236i	$0.757507\pi$
$38$	10844.6	1.21830
$39$	1677.50	0.176605
$40$	0	0
$41$	−4439.57	−0.412459	−0.206230	−	0.978504i	$-0.566119\pi$
$41$	−4439.57	−0.412459	−0.206230	+	0.978504i	$0.566119\pi$
$42$	4934.27	0.431618
$43$	−13909.2	−1.14718	−0.573590	−	0.819143i	$-0.694450\pi$
$43$	−13909.2	−1.14718	−0.573590	+	0.819143i	$0.694450\pi$
$44$	1926.14	0.149988
$45$	0	0
$46$	6294.35	0.438588
$47$	−27970.2	−1.84693	−0.923465	−	0.383683i	$-0.874655\pi$
$47$	−27970.2	−1.84693	−0.923465	+	0.383683i	$0.874655\pi$
$48$	−2541.07	−0.159189
$49$	−1362.59	−0.0810725
$50$	0	0
$51$	−12258.1	−0.659931
$52$	−2704.00	−0.138675
$53$	24170.2	1.18193	0.590963	−	0.806699i	$-0.298748\pi$
$53$	24170.2	1.18193	0.590963	+	0.806699i	$0.298748\pi$
$54$	15384.3	0.717950
$55$	0	0
$56$	−7953.64	−0.338919
$57$	−26911.1	−1.09710
$58$	−738.091	−0.0288098
$59$	−38786.4	−1.45060	−0.725302	−	0.688430i	$-0.758300\pi$
$59$	−38786.4	−1.45060	−0.725302	+	0.688430i	$0.758300\pi$
$60$	0	0
$61$	53666.6	1.84663	0.923314	−	0.384045i	$-0.125469\pi$
$61$	53666.6	1.84663	0.923314	+	0.384045i	$0.125469\pi$
$62$	−205.905	−0.00680281
$63$	17954.5	0.569931
$64$	4096.00	0.125000
$65$	0	0
$66$	−4779.75	−0.135066
$67$	−10772.8	−0.293184	−0.146592	−	0.989197i	$-0.546830\pi$
$67$	−10772.8	−0.293184	−0.146592	+	0.989197i	$0.546830\pi$
$68$	19759.1	0.518197
$69$	−15619.5	−0.394953
$70$	0	0
$71$	6297.35	0.148256	0.0741280	−	0.997249i	$-0.476383\pi$
$71$	6297.35	0.148256	0.0741280	+	0.997249i	$0.476383\pi$
$72$	−9246.29	−0.210202
$73$	4254.45	0.0934407	0.0467204	−	0.998908i	$-0.485123\pi$
$73$	4254.45	0.0934407	0.0467204	+	0.998908i	$0.485123\pi$
$74$	−48204.1	−1.02330
$75$	0	0
$76$	43378.5	0.861471
$77$	−14960.8	−0.287559
$78$	6710.02	0.124878
$79$	−49474.9	−0.891901	−0.445951	−	0.895058i	$-0.647134\pi$
$79$	−49474.9	−0.891901	−0.445951	+	0.895058i	$0.647134\pi$
$80$	0	0
$81$	−3069.45	−0.0519814
$82$	−17758.3	−0.291653
$83$	−100785.	−1.60584	−0.802918	−	0.596089i	$-0.796721\pi$
$83$	−100785.	−1.60584	−0.802918	+	0.596089i	$0.796721\pi$
$84$	19737.1	0.305200
$85$	0	0
$86$	−55636.8	−0.811178
$87$	1831.58	0.0259435
$88$	7704.56	0.106057
$89$	55060.2	0.736823	0.368411	−	0.929663i	$-0.379902\pi$
$89$	55060.2	0.736823	0.368411	+	0.929663i	$0.379902\pi$
$90$	0	0
$91$	21002.6	0.265870
$92$	25177.4	0.310128
$93$	510.957	0.00612600
$94$	−111881.	−1.30598
$95$	0	0
$96$	−10164.3	−0.112564
$97$	−177285.	−1.91312	−0.956559	−	0.291538i	$-0.905833\pi$
$97$	−177285.	−1.91312	−0.956559	+	0.291538i	$0.905833\pi$
$98$	−5450.34	−0.0573269
$99$	−17392.2	−0.178348
$100$	0	0
$101$	2837.71	0.0276799	0.0138399	−	0.999904i	$-0.495594\pi$
$101$	2837.71	0.0276799	0.0138399	+	0.999904i	$0.495594\pi$
$102$	−49032.5	−0.466642
$103$	−26102.5	−0.242431	−0.121216	−	0.992626i	$-0.538679\pi$
$103$	−26102.5	−0.242431	−0.121216	+	0.992626i	$0.538679\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	96680.7	0.835748
$107$	23317.1	0.196886	0.0984429	−	0.995143i	$-0.468614\pi$
$107$	23317.1	0.196886	0.0984429	+	0.995143i	$0.468614\pi$
$108$	61537.3	0.507667
$109$	−107035.	−0.862902	−0.431451	−	0.902136i	$-0.641998\pi$
$109$	−107035.	−0.862902	−0.431451	+	0.902136i	$0.641998\pi$
$110$	0	0
$111$	119619.	0.921496
$112$	−31814.5	−0.239652
$113$	131107.	0.965898	0.482949	−	0.875649i	$-0.339566\pi$
$113$	131107.	0.965898	0.482949	+	0.875649i	$0.339566\pi$
$114$	−107644.	−0.775763
$115$	0	0
$116$	−2952.36	−0.0203716
$117$	24416.0	0.164896
$118$	−155145.	−1.02573
$119$	−153473.	−0.993494
$120$	0	0
$121$	−146559.	−0.910015
$122$	214666.	1.30576
$123$	44067.4	0.262636
$124$	−823.621	−0.00481031
$125$	0	0
$126$	71818.0	0.403002
$127$	311667.	1.71467	0.857336	−	0.514757i	$-0.172118\pi$
$127$	311667.	1.71467	0.857336	+	0.514757i	$0.172118\pi$
$128$	16384.0	0.0883883
$129$	138064.	0.730474
$130$	0	0
$131$	−230646.	−1.17427	−0.587134	−	0.809490i	$-0.699744\pi$
$131$	−230646.	−1.17427	−0.587134	+	0.809490i	$0.699744\pi$
$132$	−19119.0	−0.0955059
$133$	−336930.	−1.65162
$134$	−43091.1	−0.207313
$135$	0	0
$136$	79036.3	0.366420
$137$	−233169.	−1.06138	−0.530689	−	0.847567i	$-0.678067\pi$
$137$	−233169.	−1.06138	−0.530689	+	0.847567i	$0.678067\pi$
$138$	−62478.1	−0.279274
$139$	−30074.1	−0.132025	−0.0660125	−	0.997819i	$-0.521028\pi$
$139$	−30074.1	−0.132025	−0.0660125	+	0.997819i	$0.521028\pi$
$140$	0	0
$141$	277634.	1.17605
$142$	25189.4	0.104833
$143$	−20344.9	−0.0831983
$144$	−36985.2	−0.148635
$145$	0	0
$146$	17017.8	0.0660726
$147$	13525.1	0.0516235
$148$	−192816.	−0.723585
$149$	19221.5	0.0709288	0.0354644	−	0.999371i	$-0.488709\pi$
$149$	19221.5	0.0709288	0.0354644	+	0.999371i	$0.488709\pi$
$150$	0	0
$151$	342311.	1.22174	0.610869	−	0.791732i	$-0.290820\pi$
$151$	342311.	1.22174	0.610869	+	0.791732i	$0.290820\pi$
$152$	173514.	0.609152
$153$	−178416.	−0.616178
$154$	−59843.0	−0.203335
$155$	0	0
$156$	26840.1	0.0883023
$157$	−549352.	−1.77869	−0.889347	−	0.457233i	$-0.848841\pi$
$157$	−549352.	−1.77869	−0.889347	+	0.457233i	$0.848841\pi$
$158$	−197899.	−0.630670
$159$	−239915.	−0.752600
$160$	0	0
$161$	−195559.	−0.594582
$162$	−12277.8	−0.0367564
$163$	280982.	0.828342	0.414171	−	0.910199i	$-0.364072\pi$
$163$	280982.	0.828342	0.414171	+	0.910199i	$0.364072\pi$
$164$	−71033.1	−0.206230
$165$	0	0
$166$	−403141.	−1.13550
$167$	−652406.	−1.81020	−0.905101	−	0.425197i	$-0.860205\pi$
$167$	−652406.	−1.81020	−0.905101	+	0.425197i	$0.860205\pi$
$168$	78948.3	0.215809
$169$	28561.0	0.0769231
$170$	0	0
$171$	−391690.	−1.02436
$172$	−222547.	−0.573590
$173$	−474856.	−1.20628	−0.603139	−	0.797636i	$-0.706083\pi$
$173$	−474856.	−1.20628	−0.603139	+	0.797636i	$0.706083\pi$
$174$	7326.33	0.0183448
$175$	0	0
$176$	30818.2	0.0749939
$177$	384996.	0.923683
$178$	220241.	0.521012
$179$	−274864.	−0.641189	−0.320594	−	0.947217i	$-0.603883\pi$
$179$	−274864.	−0.641189	−0.320594	+	0.947217i	$0.603883\pi$
$180$	0	0
$181$	483212.	1.09633	0.548165	−	0.836370i	$-0.315326\pi$
$181$	483212.	1.09633	0.548165	+	0.836370i	$0.315326\pi$
$182$	84010.3	0.187998
$183$	−532698.	−1.17585
$184$	100710.	0.219294
$185$	0	0
$186$	2043.83	0.00433174
$187$	148667.	0.310893
$188$	−447523.	−0.923465
$189$	−477974.	−0.973307
$190$	0	0
$191$	517802.	1.02702	0.513511	−	0.858083i	$-0.328344\pi$
$191$	517802.	1.02702	0.513511	+	0.858083i	$0.328344\pi$
$192$	−40657.1	−0.0795946
$193$	−294358.	−0.568830	−0.284415	−	0.958701i	$-0.591799\pi$
$193$	−294358.	−0.568830	−0.284415	+	0.958701i	$0.591799\pi$
$194$	−709139.	−1.35278
$195$	0	0
$196$	−21801.4	−0.0405362
$197$	−124861.	−0.229225	−0.114613	−	0.993410i	$-0.536563\pi$
$197$	−124861.	−0.229225	−0.114613	+	0.993410i	$0.536563\pi$
$198$	−69569.0	−0.126111
$199$	−142136.	−0.254433	−0.127216	−	0.991875i	$-0.540604\pi$
$199$	−142136.	−0.254433	−0.127216	+	0.991875i	$0.540604\pi$
$200$	0	0
$201$	106931.	0.186687
$202$	11350.8	0.0195726
$203$	22931.7	0.0390567
$204$	−196130.	−0.329965
$205$	0	0
$206$	−104410.	−0.171425
$207$	−227341.	−0.368768
$208$	−43264.0	−0.0693375
$209$	326379.	0.516841
$210$	0	0
$211$	313639.	0.484981	0.242490	−	0.970154i	$-0.422036\pi$
$211$	313639.	0.484981	0.242490	+	0.970154i	$0.422036\pi$
$212$	386723.	0.590963
$213$	−62507.9	−0.0944030
$214$	93268.3	0.139219
$215$	0	0
$216$	246149.	0.358975
$217$	6397.25	0.00922240
$218$	−428142.	−0.610164
$219$	−42229.9	−0.0594991
$220$	0	0
$221$	−208705.	−0.287444
$222$	478477.	0.651596
$223$	−692260.	−0.932196	−0.466098	−	0.884733i	$-0.654341\pi$
$223$	−692260.	−0.932196	−0.466098	+	0.884733i	$0.654341\pi$
$224$	−127258.	−0.169459
$225$	0	0
$226$	524430.	0.682993
$227$	−308536.	−0.397412	−0.198706	−	0.980059i	$-0.563674\pi$
$227$	−308536.	−0.397412	−0.198706	+	0.980059i	$0.563674\pi$
$228$	−430578.	−0.548548
$229$	−382705.	−0.482254	−0.241127	−	0.970494i	$-0.577517\pi$
$229$	−382705.	−0.482254	−0.241127	+	0.970494i	$0.577517\pi$
$230$	0	0
$231$	148501.	0.183105
$232$	−11809.4	−0.0144049
$233$	−893271.	−1.07794	−0.538969	−	0.842326i	$-0.681186\pi$
$233$	−893271.	−1.07794	−0.538969	+	0.842326i	$0.681186\pi$
$234$	97664.0	0.116599
$235$	0	0
$236$	−620582.	−0.725302
$237$	491091.	0.567925
$238$	−613893.	−0.702506
$239$	494660.	0.560160	0.280080	−	0.959977i	$-0.409639\pi$
$239$	494660.	0.560160	0.280080	+	0.959977i	$0.409639\pi$
$240$	0	0
$241$	1.35585e6	1.50372	0.751861	−	0.659322i	$-0.229156\pi$
$241$	1.35585e6	1.50372	0.751861	+	0.659322i	$0.229156\pi$
$242$	−586235.	−0.643477
$243$	−904131.	−0.982235
$244$	858666.	0.923314
$245$	0	0
$246$	176270.	0.185712
$247$	−458185.	−0.477858
$248$	−3294.48	−0.00340140
$249$	1.00040e6	1.02253
$250$	0	0
$251$	1.54382e6	1.54672	0.773362	−	0.633965i	$-0.218574\pi$
$251$	1.54382e6	1.54672	0.773362	+	0.633965i	$0.218574\pi$
$252$	287272.	0.284965
$253$	189434.	0.186062
$254$	1.24667e6	1.21246
$255$	0	0
$256$	65536.0	0.0625000
$257$	−538940.	−0.508988	−0.254494	−	0.967074i	$-0.581909\pi$
$257$	−538940.	−0.508988	−0.254494	+	0.967074i	$0.581909\pi$
$258$	552255.	0.516523
$259$	1.49765e6	1.38727
$260$	0	0
$261$	26658.6	0.0242235
$262$	−922583.	−0.830333
$263$	−548461.	−0.488941	−0.244471	−	0.969657i	$-0.578614\pi$
$263$	−548461.	−0.488941	−0.244471	+	0.969657i	$0.578614\pi$
$264$	−76475.9	−0.0675328
$265$	0	0
$266$	−1.34772e6	−1.16787
$267$	−546531.	−0.469177
$268$	−172364.	−0.146592
$269$	−1.67327e6	−1.40989	−0.704947	−	0.709260i	$-0.749029\pi$
$269$	−1.67327e6	−1.40989	−0.704947	+	0.709260i	$0.749029\pi$
$270$	0	0
$271$	150037.	0.124101	0.0620503	−	0.998073i	$-0.480236\pi$
$271$	150037.	0.124101	0.0620503	+	0.998073i	$0.480236\pi$
$272$	316145.	0.259098
$273$	−208473.	−0.169294
$274$	−932678.	−0.750508
$275$	0	0
$276$	−249913.	−0.197476
$277$	−1.59009e6	−1.24515	−0.622574	−	0.782561i	$-0.713913\pi$
$277$	−1.59009e6	−1.24515	−0.622574	+	0.782561i	$0.713913\pi$
$278$	−120297.	−0.0933558
$279$	7436.95	0.00571985
$280$	0	0
$281$	1.11166e6	0.839862	0.419931	−	0.907556i	$-0.362054\pi$
$281$	1.11166e6	0.839862	0.419931	+	0.907556i	$0.362054\pi$
$282$	1.11053e6	0.831590
$283$	960225.	0.712701	0.356350	−	0.934352i	$-0.384021\pi$
$283$	960225.	0.712701	0.356350	+	0.934352i	$0.384021\pi$
$284$	100758.	0.0741280
$285$	0	0
$286$	−81379.4	−0.0588301
$287$	551730.	0.395387
$288$	−147941.	−0.105101
$289$	105227.	0.0741110
$290$	0	0
$291$	1.75974e6	1.21819
$292$	68071.2	0.0467204
$293$	−1.01821e6	−0.692898	−0.346449	−	0.938069i	$-0.612613\pi$
$293$	−1.01821e6	−0.692898	−0.346449	+	0.938069i	$0.612613\pi$
$294$	54100.4	0.0365033
$295$	0	0
$296$	−771265.	−0.511652
$297$	463006.	0.304576
$298$	76886.1	0.0501542
$299$	−265936.	−0.172028
$300$	0	0
$301$	1.72857e6	1.09969
$302$	1.36924e6	0.863899
$303$	−28167.3	−0.0176254
$304$	694056.	0.430735
$305$	0	0
$306$	−713665.	−0.435703
$307$	3.27466e6	1.98299	0.991494	−	0.130152i	$-0.0415464\pi$
$307$	3.27466e6	1.98299	0.991494	+	0.130152i	$0.0415464\pi$
$308$	−239372.	−0.143779
$309$	259095.	0.154370
$310$	0	0
$311$	2.37790e6	1.39409	0.697047	−	0.717026i	$-0.254497\pi$
$311$	2.37790e6	1.39409	0.697047	+	0.717026i	$0.254497\pi$
$312$	107360.	0.0624392
$313$	−424354.	−0.244832	−0.122416	−	0.992479i	$-0.539064\pi$
$313$	−424354.	−0.244832	−0.122416	+	0.992479i	$0.539064\pi$
$314$	−2.19741e6	−1.25773
$315$	0	0
$316$	−791598.	−0.445951
$317$	−2.30415e6	−1.28784	−0.643921	−	0.765092i	$-0.722694\pi$
$317$	−2.30415e6	−1.28784	−0.643921	+	0.765092i	$0.722694\pi$
$318$	−959659.	−0.532168
$319$	−22213.5	−0.0122220
$320$	0	0
$321$	−231447.	−0.125368
$322$	−782234.	−0.420433
$323$	3.34812e6	1.78564
$324$	−49111.2	−0.0259907
$325$	0	0
$326$	1.12393e6	0.585726
$327$	1.06244e6	0.549459
$328$	−284132.	−0.145826
$329$	3.47601e6	1.77048
$330$	0	0
$331$	−2.03936e6	−1.02311	−0.511557	−	0.859250i	$-0.670931\pi$
$331$	−2.03936e6	−1.02311	−0.511557	+	0.859250i	$0.670931\pi$
$332$	−1.61256e6	−0.802918
$333$	1.74105e6	0.860401
$334$	−2.60962e6	−1.28001
$335$	0	0
$336$	315793.	0.152600
$337$	−1.87281e6	−0.898293	−0.449147	−	0.893458i	$-0.648272\pi$
$337$	−1.87281e6	−0.898293	−0.449147	+	0.893458i	$0.648272\pi$
$338$	114244.	0.0543928
$339$	−1.30138e6	−0.615042
$340$	0	0
$341$	−6196.91	−0.00288595
$342$	−1.56676e6	−0.724331
$343$	2.25804e6	1.03632
$344$	−890189.	−0.405589
$345$	0	0
$346$	−1.89943e6	−0.852967
$347$	501716.	0.223684	0.111842	−	0.993726i	$-0.464325\pi$
$347$	501716.	0.223684	0.111842	+	0.993726i	$0.464325\pi$
$348$	29305.3	0.0129717
$349$	−1.52260e6	−0.669147	−0.334573	−	0.942370i	$-0.608592\pi$
$349$	−1.52260e6	−0.669147	−0.334573	+	0.942370i	$0.608592\pi$
$350$	0	0
$351$	−649988.	−0.281603
$352$	123273.	0.0530287
$353$	1.87049e6	0.798949	0.399475	−	0.916744i	$-0.369193\pi$
$353$	1.87049e6	0.798949	0.399475	+	0.916744i	$0.369193\pi$
$354$	1.53998e6	0.653142
$355$	0	0
$356$	880964.	0.368411
$357$	1.52338e6	0.632614
$358$	−1.09946e6	−0.453389
$359$	−1.70160e6	−0.696820	−0.348410	−	0.937342i	$-0.613278\pi$
$359$	−1.70160e6	−0.696820	−0.348410	+	0.937342i	$0.613278\pi$
$360$	0	0
$361$	4.87427e6	1.96853
$362$	1.93285e6	0.775223
$363$	1.45475e6	0.579458
$364$	336041.	0.132935
$365$	0	0
$366$	−2.13079e6	−0.831454
$367$	−314731.	−0.121976	−0.0609880	−	0.998139i	$-0.519425\pi$
$367$	−314731.	−0.121976	−0.0609880	+	0.998139i	$0.519425\pi$
$368$	402839.	0.155064
$369$	641399.	0.245224
$370$	0	0
$371$	−3.00376e6	−1.13300
$372$	8175.31	0.00306300
$373$	2.16445e6	0.805519	0.402759	−	0.915306i	$-0.368051\pi$
$373$	2.16445e6	0.805519	0.402759	+	0.915306i	$0.368051\pi$
$374$	594668.	0.219834
$375$	0	0
$376$	−1.79009e6	−0.652988
$377$	31184.3	0.0113001
$378$	−1.91190e6	−0.688232
$379$	1.67898e6	0.600410	0.300205	−	0.953875i	$-0.402945\pi$
$379$	1.67898e6	0.600410	0.300205	+	0.953875i	$0.402945\pi$
$380$	0	0
$381$	−3.09362e6	−1.09183
$382$	2.07121e6	0.726215
$383$	−2.42699e6	−0.845417	−0.422709	−	0.906266i	$-0.638921\pi$
$383$	−2.42699e6	−0.845417	−0.422709	+	0.906266i	$0.638921\pi$
$384$	−162629.	−0.0562819
$385$	0	0
$386$	−1.17743e6	−0.402224
$387$	2.00951e6	0.682044
$388$	−2.83656e6	−0.956559
$389$	−3.41549e6	−1.14440	−0.572202	−	0.820113i	$-0.693911\pi$
$389$	−3.41549e6	−1.14440	−0.572202	+	0.820113i	$0.693911\pi$
$390$	0	0
$391$	1.94329e6	0.642830
$392$	−87205.5	−0.0286635
$393$	2.28940e6	0.747724
$394$	−499445.	−0.162087
$395$	0	0
$396$	−278276.	−0.0891739
$397$	−3.57792e6	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$397$	−3.57792e6	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$398$	−568546.	−0.179911
$399$	3.34439e6	1.05168
$400$	0	0
$401$	−4.70140e6	−1.46004	−0.730022	−	0.683423i	$-0.760490\pi$
$401$	−4.70140e6	−1.46004	−0.730022	+	0.683423i	$0.760490\pi$
$402$	427725.	0.132008
$403$	8699.50	0.00266828
$404$	45403.3	0.0138399
$405$	0	0
$406$	91726.6	0.0276172
$407$	−1.45075e6	−0.434116
$408$	−784520.	−0.233321
$409$	−4.96552e6	−1.46776	−0.733882	−	0.679277i	$-0.762293\pi$
$409$	−4.96552e6	−1.46776	−0.733882	+	0.679277i	$0.762293\pi$
$410$	0	0
$411$	2.31445e6	0.675840
$412$	−417640.	−0.121216
$413$	4.82020e6	1.39056
$414$	−909366.	−0.260758
$415$	0	0
$416$	−173056.	−0.0490290
$417$	298518.	0.0840679
$418$	1.30552e6	0.365462
$419$	−2.71480e6	−0.755445	−0.377723	−	0.925919i	$-0.623293\pi$
$419$	−2.71480e6	−0.755445	−0.377723	+	0.925919i	$0.623293\pi$
$420$	0	0
$421$	883315.	0.242890	0.121445	−	0.992598i	$-0.461247\pi$
$421$	883315.	0.242890	0.121445	+	0.992598i	$0.461247\pi$
$422$	1.25456e6	0.342933
$423$	4.04094e6	1.09807
$424$	1.54689e6	0.417874
$425$	0	0
$426$	−250032.	−0.0667530
$427$	−6.66945e6	−1.77019
$428$	373073.	0.0984429
$429$	201944.	0.0529771
$430$	0	0
$431$	−5.47099e6	−1.41864	−0.709321	−	0.704886i	$-0.750998\pi$
$431$	−5.47099e6	−1.41864	−0.709321	+	0.704886i	$0.750998\pi$
$432$	984597.	0.253834
$433$	−5.10026e6	−1.30729	−0.653646	−	0.756801i	$-0.726761\pi$
$433$	−5.10026e6	−1.30729	−0.653646	+	0.756801i	$0.726761\pi$
$434$	25589.0	0.00652122
$435$	0	0
$436$	−1.71257e6	−0.431451
$437$	4.26624e6	1.06867
$438$	−168920.	−0.0420722
$439$	4.45494e6	1.10327	0.551633	−	0.834087i	$-0.314005\pi$
$439$	4.45494e6	1.10327	0.551633	+	0.834087i	$0.314005\pi$
$440$	0	0
$441$	196857.	0.0482009
$442$	−834821.	−0.203253
$443$	938918.	0.227310	0.113655	−	0.993520i	$-0.463744\pi$
$443$	938918.	0.227310	0.113655	+	0.993520i	$0.463744\pi$
$444$	1.91391e6	0.460748
$445$	0	0
$446$	−2.76904e6	−0.659162
$447$	−190794.	−0.0451644
$448$	−509033.	−0.119826
$449$	4.55776e6	1.06693	0.533464	−	0.845823i	$-0.320890\pi$
$449$	4.55776e6	1.06693	0.533464	+	0.845823i	$0.320890\pi$
$450$	0	0
$451$	−534452.	−0.123728
$452$	2.09772e6	0.482949
$453$	−3.39780e6	−0.777950
$454$	−1.23414e6	−0.281013
$455$	0	0
$456$	−1.72231e6	−0.387882
$457$	294965.	0.0660662	0.0330331	−	0.999454i	$-0.489483\pi$
$457$	294965.	0.0660662	0.0330331	+	0.999454i	$0.489483\pi$
$458$	−1.53082e6	−0.341005
$459$	4.74969e6	1.05229
$460$	0	0
$461$	−3.53147e6	−0.773933	−0.386966	−	0.922094i	$-0.626477\pi$
$461$	−3.53147e6	−0.773933	−0.386966	+	0.922094i	$0.626477\pi$
$462$	594006.	0.129475
$463$	6.20665e6	1.34556	0.672782	−	0.739841i	$-0.265099\pi$
$463$	6.20665e6	1.34556	0.672782	+	0.739841i	$0.265099\pi$
$464$	−47237.8	−0.0101858
$465$	0	0
$466$	−3.57308e6	−0.762217
$467$	3.68433e6	0.781746	0.390873	−	0.920445i	$-0.372173\pi$
$467$	3.68433e6	0.781746	0.390873	+	0.920445i	$0.372173\pi$
$468$	390656.	0.0824479
$469$	1.33879e6	0.281048
$470$	0	0
$471$	5.45290e6	1.13260
$472$	−2.48233e6	−0.512866
$473$	−1.67444e6	−0.344126
$474$	1.96436e6	0.401583
$475$	0	0
$476$	−2.45557e6	−0.496747
$477$	−3.49195e6	−0.702703
$478$	1.97864e6	0.396093
$479$	3.55198e6	0.707347	0.353673	−	0.935369i	$-0.384932\pi$
$479$	3.55198e6	0.707347	0.353673	+	0.935369i	$0.384932\pi$
$480$	0	0
$481$	2.03662e6	0.401373
$482$	5.42338e6	1.06329
$483$	1.94113e6	0.378605
$484$	−2.34494e6	−0.455007
$485$	0	0
$486$	−3.61652e6	−0.694545
$487$	−2.39824e6	−0.458217	−0.229108	−	0.973401i	$-0.573581\pi$
$487$	−2.39824e6	−0.458217	−0.229108	+	0.973401i	$0.573581\pi$
$488$	3.43466e6	0.652882
$489$	−2.78904e6	−0.527452
$490$	0	0
$491$	−7.64714e6	−1.43151	−0.715757	−	0.698350i	$-0.753918\pi$
$491$	−7.64714e6	−1.43151	−0.715757	+	0.698350i	$0.753918\pi$
$492$	705079.	0.131318
$493$	−227875.	−0.0422259
$494$	−1.83274e6	−0.337897
$495$	0	0
$496$	−13177.9	−0.00240516
$497$	−782607.	−0.142119
$498$	4.00160e6	0.723036
$499$	−1.00417e6	−0.180533	−0.0902666	−	0.995918i	$-0.528772\pi$
$499$	−1.00417e6	−0.180533	−0.0902666	+	0.995918i	$0.528772\pi$
$500$	0	0
$501$	6.47582e6	1.15266
$502$	6.17528e6	1.09370
$503$	6.21382e6	1.09506	0.547531	−	0.836785i	$-0.315568\pi$
$503$	6.21382e6	1.09506	0.547531	+	0.836785i	$0.315568\pi$
$504$	1.14909e6	0.201501
$505$	0	0
$506$	757738.	0.131566
$507$	−283498.	−0.0489813
$508$	4.98667e6	0.857336
$509$	6.93843e6	1.18704	0.593522	−	0.804818i	$-0.297737\pi$
$509$	6.93843e6	1.18704	0.593522	+	0.804818i	$0.297737\pi$
$510$	0	0
$511$	−528724.	−0.0895729
$512$	262144.	0.0441942
$513$	1.04273e7	1.74936
$514$	−2.15576e6	−0.359909
$515$	0	0
$516$	2.20902e6	0.365237
$517$	−3.36715e6	−0.554034
$518$	5.99059e6	0.980946
$519$	4.71345e6	0.768106
$520$	0	0
$521$	−3.32358e6	−0.536428	−0.268214	−	0.963359i	$-0.586433\pi$
$521$	−3.32358e6	−0.536428	−0.268214	+	0.963359i	$0.586433\pi$
$522$	106634.	0.0171286
$523$	5.25138e6	0.839497	0.419748	−	0.907641i	$-0.362118\pi$
$523$	5.25138e6	0.839497	0.419748	+	0.907641i	$0.362118\pi$
$524$	−3.69033e6	−0.587134
$525$	0	0
$526$	−2.19384e6	−0.345734
$527$	−63570.3	−0.00997075
$528$	−305904.	−0.0477529
$529$	−3.96016e6	−0.615282
$530$	0	0
$531$	5.60359e6	0.862443
$532$	−5.39089e6	−0.825812
$533$	750287.	0.114396
$534$	−2.18612e6	−0.331758
$535$	0	0
$536$	−689458.	−0.103656
$537$	2.72832e6	0.408282
$538$	−6.69310e6	−0.996946
$539$	−164033.	−0.0243198
$540$	0	0
$541$	−8.53307e6	−1.25346	−0.626732	−	0.779234i	$-0.715608\pi$
$541$	−8.53307e6	−1.25346	−0.626732	+	0.779234i	$0.715608\pi$
$542$	600146.	0.0877524
$543$	−4.79640e6	−0.698096
$544$	1.26458e6	0.183210
$545$	0	0
$546$	−833891.	−0.119709
$547$	550812.	0.0787109	0.0393554	−	0.999225i	$-0.487470\pi$
$547$	550812.	0.0787109	0.0393554	+	0.999225i	$0.487470\pi$
$548$	−3.73071e6	−0.530689
$549$	−7.75339e6	−1.09790
$550$	0	0
$551$	−500270.	−0.0701981
$552$	−999650.	−0.139637
$553$	6.14852e6	0.854983
$554$	−6.36034e6	−0.880453
$555$	0	0
$556$	−481186.	−0.0660125
$557$	4.49923e6	0.614470	0.307235	−	0.951634i	$-0.400596\pi$
$557$	4.49923e6	0.614470	0.307235	+	0.951634i	$0.400596\pi$
$558$	29747.8	0.00404454
$559$	2.35066e6	0.318170
$560$	0	0
$561$	−1.47568e6	−0.197963
$562$	4.44666e6	0.593872
$563$	4.45441e6	0.592269	0.296134	−	0.955146i	$-0.404302\pi$
$563$	4.45441e6	0.592269	0.296134	+	0.955146i	$0.404302\pi$
$564$	4.44214e6	0.588023
$565$	0	0
$566$	3.84090e6	0.503955
$567$	381458.	0.0498298
$568$	403031.	0.0524164
$569$	1.11716e7	1.44655	0.723274	−	0.690561i	$-0.242636\pi$
$569$	1.11716e7	1.44655	0.723274	+	0.690561i	$0.242636\pi$
$570$	0	0
$571$	−165900.	−0.0212939	−0.0106469	−	0.999943i	$-0.503389\pi$
$571$	−165900.	−0.0212939	−0.0106469	+	0.999943i	$0.503389\pi$
$572$	−325518.	−0.0415992
$573$	−5.13973e6	−0.653964
$574$	2.20692e6	0.279581
$575$	0	0
$576$	−591763.	−0.0743175
$577$	1.14786e7	1.43532	0.717659	−	0.696394i	$-0.245213\pi$
$577$	1.14786e7	1.43532	0.717659	+	0.696394i	$0.245213\pi$
$578$	420908.	0.0524044
$579$	2.92182e6	0.362207
$580$	0	0
$581$	1.25251e7	1.53937
$582$	7.03896e6	0.861392
$583$	2.90970e6	0.354549
$584$	272285.	0.0330363
$585$	0	0
$586$	−4.07285e6	−0.489953
$587$	−1.47023e7	−1.76113	−0.880563	−	0.473930i	$-0.842835\pi$
$587$	−1.47023e7	−1.76113	−0.880563	+	0.473930i	$0.842835\pi$
$588$	216402.	0.0258117
$589$	−139560.	−0.0165758
$590$	0	0
$591$	1.23938e6	0.145961
$592$	−3.08506e6	−0.361792
$593$	3.89089e6	0.454373	0.227186	−	0.973851i	$-0.427047\pi$
$593$	3.89089e6	0.454373	0.227186	+	0.973851i	$0.427047\pi$
$594$	1.85202e6	0.215368
$595$	0	0
$596$	307545.	0.0354644
$597$	1.41086e6	0.162012
$598$	−1.06375e6	−0.121642
$599$	4.56164e6	0.519462	0.259731	−	0.965681i	$-0.416366\pi$
$599$	4.56164e6	0.519462	0.259731	+	0.965681i	$0.416366\pi$
$600$	0	0
$601$	−2.07826e6	−0.234700	−0.117350	−	0.993091i	$-0.537440\pi$
$601$	−2.07826e6	−0.234700	−0.117350	+	0.993091i	$0.537440\pi$
$602$	6.91430e6	0.777601
$603$	1.55638e6	0.174310
$604$	5.47697e6	0.610869
$605$	0	0
$606$	−112669.	−0.0124630
$607$	−4.20281e6	−0.462986	−0.231493	−	0.972837i	$-0.574361\pi$
$607$	−4.20281e6	−0.462986	−0.231493	+	0.972837i	$0.574361\pi$
$608$	2.77622e6	0.304576
$609$	−227621.	−0.0248696
$610$	0	0
$611$	4.72696e6	0.512246
$612$	−2.85466e6	−0.308089
$613$	−5.73233e6	−0.616141	−0.308071	−	0.951363i	$-0.599683\pi$
$613$	−5.73233e6	−0.616141	−0.308071	+	0.951363i	$0.599683\pi$
$614$	1.30986e7	1.40218
$615$	0	0
$616$	−957489.	−0.101667
$617$	−7.42348e6	−0.785046	−0.392523	−	0.919742i	$-0.628398\pi$
$617$	−7.42348e6	−0.785046	−0.392523	+	0.919742i	$0.628398\pi$
$618$	1.03638e6	0.109156
$619$	1.60192e7	1.68041	0.840203	−	0.542271i	$-0.182436\pi$
$619$	1.60192e7	1.68041	0.840203	+	0.542271i	$0.182436\pi$
$620$	0	0
$621$	6.05215e6	0.629768
$622$	9.51158e6	0.985773
$623$	−6.84264e6	−0.706323
$624$	429441.	0.0441512
$625$	0	0
$626$	−1.69742e6	−0.173122
$627$	−3.23966e6	−0.329102
$628$	−8.78963e6	−0.889347
$629$	−1.48823e7	−1.49984
$630$	0	0
$631$	−1.04270e7	−1.04252	−0.521260	−	0.853398i	$-0.674538\pi$
$631$	−1.04270e7	−1.04252	−0.521260	+	0.853398i	$0.674538\pi$
$632$	−3.16639e6	−0.315335
$633$	−3.11320e6	−0.308815
$634$	−9.21660e6	−0.910642
$635$	0	0
$636$	−3.83863e6	−0.376300
$637$	230277.	0.0224855
$638$	−88854.1	−0.00864223
$639$	−909799.	−0.0881441
$640$	0	0
$641$	5.56044e6	0.534520	0.267260	−	0.963624i	$-0.413882\pi$
$641$	5.56044e6	0.534520	0.267260	+	0.963624i	$0.413882\pi$
$642$	−925787.	−0.0886489
$643$	−3.35787e6	−0.320285	−0.160142	−	0.987094i	$-0.551195\pi$
$643$	−3.35787e6	−0.320285	−0.160142	+	0.987094i	$0.551195\pi$
$644$	−3.12894e6	−0.297291
$645$	0	0
$646$	1.33925e7	1.26264
$647$	1.16363e7	1.09284	0.546419	−	0.837512i	$-0.315991\pi$
$647$	1.16363e7	1.09284	0.546419	+	0.837512i	$0.315991\pi$
$648$	−196445.	−0.0183782
$649$	−4.66925e6	−0.435146
$650$	0	0
$651$	−63499.5	−0.00587243
$652$	4.49571e6	0.414171
$653$	−3.20815e6	−0.294423	−0.147211	−	0.989105i	$-0.547030\pi$
$653$	−3.20815e6	−0.294423	−0.147211	+	0.989105i	$0.547030\pi$
$654$	4.24976e6	0.388526
$655$	0	0
$656$	−1.13653e6	−0.103115
$657$	−614654.	−0.0555543
$658$	1.39040e7	1.25192
$659$	1.08724e6	0.0975242	0.0487621	−	0.998810i	$-0.484472\pi$
$659$	1.08724e6	0.0975242	0.0487621	+	0.998810i	$0.484472\pi$
$660$	0	0
$661$	7.81571e6	0.695769	0.347884	−	0.937537i	$-0.386900\pi$
$661$	7.81571e6	0.695769	0.347884	+	0.937537i	$0.386900\pi$
$662$	−8.15743e6	−0.723450
$663$	2.07162e6	0.183032
$664$	−6.45025e6	−0.567749
$665$	0	0
$666$	6.96420e6	0.608395
$667$	−290363.	−0.0252712
$668$	−1.04385e7	−0.905101
$669$	6.87142e6	0.593582
$670$	0	0
$671$	6.46059e6	0.553944
$672$	1.26317e6	0.107904
$673$	1.58854e7	1.35195	0.675975	−	0.736924i	$-0.263723\pi$
$673$	1.58854e7	1.35195	0.675975	+	0.736924i	$0.263723\pi$
$674$	−7.49123e6	−0.635189
$675$	0	0
$676$	456976.	0.0384615
$677$	1.74735e7	1.46524	0.732621	−	0.680637i	$-0.238297\pi$
$677$	1.74735e7	1.46524	0.732621	+	0.680637i	$0.238297\pi$
$678$	−5.20552e6	−0.434900
$679$	2.20322e7	1.83393
$680$	0	0
$681$	3.06255e6	0.253055
$682$	−24787.6	−0.00204068
$683$	9.62616e6	0.789590	0.394795	−	0.918769i	$-0.370816\pi$
$683$	9.62616e6	0.789590	0.394795	+	0.918769i	$0.370816\pi$
$684$	−6.26703e6	−0.512179
$685$	0	0
$686$	9.03214e6	0.732792
$687$	3.79875e6	0.307078
$688$	−3.56076e6	−0.286795
$689$	−4.08476e6	−0.327807
$690$	0	0
$691$	−5.38851e6	−0.429312	−0.214656	−	0.976690i	$-0.568863\pi$
$691$	−5.38851e6	−0.429312	−0.214656	+	0.976690i	$0.568863\pi$
$692$	−7.59770e6	−0.603139
$693$	2.16143e6	0.170965
$694$	2.00686e6	0.158168
$695$	0	0
$696$	117221.	0.00917241
$697$	−5.48262e6	−0.427470
$698$	−6.09039e6	−0.473158
$699$	8.86666e6	0.686384
$700$	0	0
$701$	−2.33991e6	−0.179848	−0.0899238	−	0.995949i	$-0.528662\pi$
$701$	−2.33991e6	−0.179848	−0.0899238	+	0.995949i	$0.528662\pi$
$702$	−2.59995e6	−0.199124
$703$	−3.26722e7	−2.49339
$704$	493092.	0.0374970
$705$	0	0
$706$	7.48197e6	0.564943
$707$	−352658.	−0.0265341
$708$	6.15993e6	0.461841
$709$	−4.25213e6	−0.317681	−0.158840	−	0.987304i	$-0.550776\pi$
$709$	−4.25213e6	−0.317681	−0.158840	+	0.987304i	$0.550776\pi$
$710$	0	0
$711$	7.14780e6	0.530271
$712$	3.52385e6	0.260506
$713$	−81002.5	−0.00596726
$714$	6.09354e6	0.447326
$715$	0	0
$716$	−4.39783e6	−0.320594
$717$	−4.91003e6	−0.356686
$718$	−6.80639e6	−0.492726
$719$	−5.05655e6	−0.364781	−0.182390	−	0.983226i	$-0.558383\pi$
$719$	−5.05655e6	−0.364781	−0.182390	+	0.983226i	$0.558383\pi$
$720$	0	0
$721$	3.24390e6	0.232396
$722$	1.94971e7	1.39196
$723$	−1.34582e7	−0.957506
$724$	7.73140e6	0.548165
$725$	0	0
$726$	5.81900e6	0.409739
$727$	−5.03627e6	−0.353405	−0.176703	−	0.984264i	$-0.556543\pi$
$727$	−5.03627e6	−0.353405	−0.176703	+	0.984264i	$0.556543\pi$
$728$	1.34416e6	0.0939992
$729$	9.72033e6	0.677427
$730$	0	0
$731$	−1.71771e7	−1.18893
$732$	−8.52317e6	−0.587927
$733$	−4.12919e6	−0.283861	−0.141930	−	0.989877i	$-0.545331\pi$
$733$	−4.12919e6	−0.283861	−0.141930	+	0.989877i	$0.545331\pi$
$734$	−1.25892e6	−0.0862500
$735$	0	0
$736$	1.61135e6	0.109647
$737$	−1.29687e6	−0.0879482
$738$	2.56560e6	0.173399
$739$	−1.44286e7	−0.971881	−0.485940	−	0.873992i	$-0.661523\pi$
$739$	−1.44286e7	−0.971881	−0.485940	+	0.873992i	$0.661523\pi$
$740$	0	0
$741$	4.54798e6	0.304279
$742$	−1.20150e7	−0.801154
$743$	−1.53445e7	−1.01972	−0.509860	−	0.860258i	$-0.670303\pi$
$743$	−1.53445e7	−1.01972	−0.509860	+	0.860258i	$0.670303\pi$
$744$	32701.2	0.00216587
$745$	0	0
$746$	8.65780e6	0.569588
$747$	1.45608e7	0.954735
$748$	2.37867e6	0.155446
$749$	−2.89774e6	−0.188736
$750$	0	0
$751$	−2.17168e7	−1.40506	−0.702530	−	0.711654i	$-0.747946\pi$
$751$	−2.17168e7	−1.40506	−0.702530	+	0.711654i	$0.747946\pi$
$752$	−7.16036e6	−0.461732
$753$	−1.53241e7	−0.984887
$754$	124737.	0.00799039
$755$	0	0
$756$	−7.64759e6	−0.486654
$757$	−4.93082e6	−0.312737	−0.156369	−	0.987699i	$-0.549979\pi$
$757$	−4.93082e6	−0.312737	−0.156369	+	0.987699i	$0.549979\pi$
$758$	6.71592e6	0.424554
$759$	−1.88034e6	−0.118476
$760$	0	0
$761$	2.14880e6	0.134504	0.0672520	−	0.997736i	$-0.478577\pi$
$761$	2.14880e6	0.134504	0.0672520	+	0.997736i	$0.478577\pi$
$762$	−1.23745e7	−0.772040
$763$	1.33019e7	0.827184
$764$	8.28483e6	0.513511
$765$	0	0
$766$	−9.70796e6	−0.597800
$767$	6.55490e6	0.402325
$768$	−650514.	−0.0397973
$769$	−296609.	−0.0180871	−0.00904354	−	0.999959i	$-0.502879\pi$
$769$	−296609.	−0.0180871	−0.00904354	+	0.999959i	$0.502879\pi$
$770$	0	0
$771$	5.34955e6	0.324102
$772$	−4.70973e6	−0.284415
$773$	−4.37418e6	−0.263298	−0.131649	−	0.991296i	$-0.542027\pi$
$773$	−4.37418e6	−0.263298	−0.131649	+	0.991296i	$0.542027\pi$
$774$	8.03804e6	0.482278
$775$	0	0
$776$	−1.13462e7	−0.676390
$777$	−1.48657e7	−0.883352
$778$	−1.36620e7	−0.809216
$779$	−1.20364e7	−0.710643
$780$	0	0
$781$	758099.	0.0444732
$782$	7.77317e6	0.454549
$783$	−709690.	−0.0413680
$784$	−348822.	−0.0202681
$785$	0	0
$786$	9.15762e6	0.528720
$787$	3.18884e7	1.83525	0.917625	−	0.397447i	$-0.130104\pi$
$787$	3.18884e7	1.83525	0.917625	+	0.397447i	$0.130104\pi$
$788$	−1.99778e6	−0.114613
$789$	5.44406e6	0.311337
$790$	0	0
$791$	−1.62934e7	−0.925916
$792$	−1.11310e6	−0.0630554
$793$	−9.06966e6	−0.512163
$794$	−1.43117e7	−0.805637
$795$	0	0
$796$	−2.27418e6	−0.127216
$797$	−3.17291e7	−1.76934	−0.884671	−	0.466216i	$-0.845617\pi$
$797$	−3.17291e7	−1.76934	−0.884671	+	0.466216i	$0.845617\pi$
$798$	1.33776e7	0.743652
$799$	−3.45416e7	−1.91415
$800$	0	0
$801$	−7.95473e6	−0.438071
$802$	−1.88056e7	−1.03241
$803$	512167.	0.0280300
$804$	1.71090e6	0.0933436
$805$	0	0
$806$	34798.0	0.00188676
$807$	1.66090e7	0.897760
$808$	181613.	0.00978632
$809$	−2.63959e7	−1.41796	−0.708981	−	0.705227i	$-0.750845\pi$
$809$	−2.63959e7	−1.41796	−0.708981	+	0.705227i	$0.750845\pi$
$810$	0	0
$811$	3.04865e7	1.62763	0.813814	−	0.581125i	$-0.197387\pi$
$811$	3.04865e7	1.62763	0.813814	+	0.581125i	$0.197387\pi$
$812$	366906.	0.0195283
$813$	−1.48927e6	−0.0790220
$814$	−5.80299e6	−0.306966
$815$	0	0
$816$	−3.13808e6	−0.164983
$817$	−3.77100e7	−1.97652
$818$	−1.98621e7	−1.03787
$819$	−3.03431e6	−0.158070
$820$	0	0
$821$	−2.32779e7	−1.20528	−0.602638	−	0.798015i	$-0.705884\pi$
$821$	−2.32779e7	−1.20528	−0.602638	+	0.798015i	$0.705884\pi$
$822$	9.25782e6	0.477891
$823$	9.66183e6	0.497233	0.248616	−	0.968602i	$-0.420024\pi$
$823$	9.66183e6	0.497233	0.248616	+	0.968602i	$0.420024\pi$
$824$	−1.67056e6	−0.0857124
$825$	0	0
$826$	1.92808e7	0.983274
$827$	1.66692e7	0.847523	0.423761	−	0.905774i	$-0.360710\pi$
$827$	1.66692e7	0.847523	0.423761	+	0.905774i	$0.360710\pi$
$828$	−3.63746e6	−0.184384
$829$	−1.71154e7	−0.864970	−0.432485	−	0.901641i	$-0.642363\pi$
$829$	−1.71154e7	−0.864970	−0.432485	+	0.901641i	$0.642363\pi$
$830$	0	0
$831$	1.57833e7	0.792857
$832$	−692224.	−0.0346688
$833$	−1.68271e6	−0.0840230
$834$	1.19407e6	0.0594450
$835$	0	0
$836$	5.22207e6	0.258420
$837$	−197982.	−0.00976815
$838$	−1.08592e7	−0.534180
$839$	−4.89613e6	−0.240131	−0.120066	−	0.992766i	$-0.538310\pi$
$839$	−4.89613e6	−0.240131	−0.120066	+	0.992766i	$0.538310\pi$
$840$	0	0
$841$	−2.04771e7	−0.998340
$842$	3.53326e6	0.171749
$843$	−1.10345e7	−0.534788
$844$	5.01823e6	0.242490
$845$	0	0
$846$	1.61638e7	0.776456
$847$	1.82137e7	0.872346
$848$	6.18757e6	0.295481
$849$	−9.53126e6	−0.453817
$850$	0	0
$851$	−1.89633e7	−0.897617
$852$	−1.00013e6	−0.0472015
$853$	2.59891e6	0.122298	0.0611490	−	0.998129i	$-0.480524\pi$
$853$	2.59891e6	0.122298	0.0611490	+	0.998129i	$0.480524\pi$
$854$	−2.66778e7	−1.25171
$855$	0	0
$856$	1.49229e6	0.0696097
$857$	−5.01734e6	−0.233357	−0.116679	−	0.993170i	$-0.537225\pi$
$857$	−5.01734e6	−0.233357	−0.116679	+	0.993170i	$0.537225\pi$
$858$	807777.	0.0374605
$859$	−2.55874e7	−1.18316	−0.591580	−	0.806246i	$-0.701496\pi$
$859$	−2.55874e7	−1.18316	−0.591580	+	0.806246i	$0.701496\pi$
$860$	0	0
$861$	−5.47651e6	−0.251765
$862$	−2.18840e7	−1.00313
$863$	−1.15679e7	−0.528721	−0.264361	−	0.964424i	$-0.585161\pi$
$863$	−1.15679e7	−0.528721	−0.264361	+	0.964424i	$0.585161\pi$
$864$	3.93839e6	0.179488
$865$	0	0
$866$	−2.04010e7	−0.924394
$867$	−1.04449e6	−0.0471907
$868$	102356.	0.00461120
$869$	−5.95597e6	−0.267549
$870$	0	0
$871$	1.82060e6	0.0813147
$872$	−6.85027e6	−0.305082
$873$	2.56129e7	1.13743
$874$	1.70650e7	0.755661
$875$	0	0
$876$	−675679.	−0.0297495
$877$	−1.09871e7	−0.482376	−0.241188	−	0.970478i	$-0.577537\pi$
$877$	−1.09871e7	−0.482376	−0.241188	+	0.970478i	$0.577537\pi$
$878$	1.78198e7	0.780127
$879$	1.01068e7	0.441208
$880$	0	0
$881$	2.80039e7	1.21557	0.607784	−	0.794102i	$-0.292059\pi$
$881$	2.80039e7	1.21557	0.607784	+	0.794102i	$0.292059\pi$
$882$	787429.	0.0340832
$883$	−2.79370e7	−1.20581	−0.602904	−	0.797814i	$-0.705990\pi$
$883$	−2.79370e7	−1.20581	−0.602904	+	0.797814i	$0.705990\pi$
$884$	−3.33929e6	−0.143722
$885$	0	0
$886$	3.75567e6	0.160732
$887$	−1.68485e7	−0.719040	−0.359520	−	0.933137i	$-0.617060\pi$
$887$	−1.68485e7	−0.719040	−0.359520	+	0.933137i	$0.617060\pi$
$888$	7.65563e6	0.325798
$889$	−3.87325e7	−1.64370
$890$	0	0
$891$	−369512.	−0.0155932
$892$	−1.10762e7	−0.466098
$893$	−7.58315e7	−3.18215
$894$	−763176.	−0.0319360
$895$	0	0
$896$	−2.03613e6	−0.0847297
$897$	2.63970e6	0.109540
$898$	1.82310e7	0.754433
$899$	9498.54	0.000391975 0
$900$	0	0
$901$	2.98488e7	1.22494
$902$	−2.13781e6	−0.0874888
$903$	−1.71579e7	−0.700238
$904$	8.39087e6	0.341496
$905$	0	0
$906$	−1.35912e7	−0.550094
$907$	−3.17618e7	−1.28200	−0.640998	−	0.767543i	$-0.721479\pi$
$907$	−3.17618e7	−1.28200	−0.640998	+	0.767543i	$0.721479\pi$
$908$	−4.93658e6	−0.198706
$909$	−409973.	−0.0164568
$910$	0	0
$911$	1.96774e7	0.785548	0.392774	−	0.919635i	$-0.371516\pi$
$911$	1.96774e7	0.785548	0.392774	+	0.919635i	$0.371516\pi$
$912$	−6.88924e6	−0.274274
$913$	−1.21329e7	−0.481712
$914$	1.17986e6	0.0467159
$915$	0	0
$916$	−6.12328e6	−0.241127
$917$	2.86636e7	1.12566
$918$	1.89988e7	0.744079
$919$	2.01620e7	0.787491	0.393745	−	0.919220i	$-0.371179\pi$
$919$	2.01620e7	0.787491	0.393745	+	0.919220i	$0.371179\pi$
$920$	0	0
$921$	−3.25045e7	−1.26268
$922$	−1.41259e7	−0.547253
$923$	−1.06425e6	−0.0411188
$924$	2.37602e6	0.0915526
$925$	0	0
$926$	2.48266e7	0.951458
$927$	3.77111e6	0.144135
$928$	−188951.	−0.00720244
$929$	−6.24509e6	−0.237410	−0.118705	−	0.992930i	$-0.537874\pi$
$929$	−6.24509e6	−0.237410	−0.118705	+	0.992930i	$0.537874\pi$
$930$	0	0
$931$	−3.69418e6	−0.139683
$932$	−1.42923e7	−0.538969
$933$	−2.36031e7	−0.887699
$934$	1.47373e7	0.552778
$935$	0	0
$936$	1.56262e6	0.0582995
$937$	3.44012e7	1.28004	0.640021	−	0.768358i	$-0.278926\pi$
$937$	3.44012e7	1.28004	0.640021	+	0.768358i	$0.278926\pi$
$938$	5.35517e6	0.198731
$939$	4.21217e6	0.155898
$940$	0	0
$941$	1.22495e7	0.450967	0.225483	−	0.974247i	$-0.427604\pi$
$941$	1.22495e7	0.450967	0.225483	+	0.974247i	$0.427604\pi$
$942$	2.18116e7	0.800867
$943$	−6.98605e6	−0.255831
$944$	−9.92931e6	−0.362651
$945$	0	0
$946$	−6.69777e6	−0.243334
$947$	1.04126e7	0.377298	0.188649	−	0.982045i	$-0.439589\pi$
$947$	1.04126e7	0.377298	0.188649	+	0.982045i	$0.439589\pi$
$948$	7.85745e6	0.283962
$949$	−719002.	−0.0259158
$950$	0	0
$951$	2.28711e7	0.820043
$952$	−9.82229e6	−0.351253
$953$	1.10546e7	0.394286	0.197143	−	0.980375i	$-0.436834\pi$
$953$	1.10546e7	0.394286	0.197143	+	0.980375i	$0.436834\pi$
$954$	−1.39678e7	−0.496886
$955$	0	0
$956$	7.91457e6	0.280080
$957$	220493.	0.00778242
$958$	1.42079e7	0.500170
$959$	2.89773e7	1.01744
$960$	0	0
$961$	−2.86265e7	−0.999907
$962$	8.14649e6	0.283813
$963$	−3.36869e6	−0.117057
$964$	2.16935e7	0.751861
$965$	0	0
$966$	7.76450e6	0.267714
$967$	−5.56761e6	−0.191471	−0.0957354	−	0.995407i	$-0.530520\pi$
$967$	−5.56761e6	−0.191471	−0.0957354	+	0.995407i	$0.530520\pi$
$968$	−9.37976e6	−0.321739
$969$	−3.32337e7	−1.13702
$970$	0	0
$971$	2.43458e7	0.828661	0.414330	−	0.910127i	$-0.364016\pi$
$971$	2.43458e7	0.828661	0.414330	+	0.910127i	$0.364016\pi$
$972$	−1.44661e7	−0.491118
$973$	3.73748e6	0.126560
$974$	−9.59298e6	−0.324008
$975$	0	0
$976$	1.37387e7	0.461657
$977$	2.31597e7	0.776242	0.388121	−	0.921608i	$-0.373124\pi$
$977$	2.31597e7	0.776242	0.388121	+	0.921608i	$0.373124\pi$
$978$	−1.11562e7	−0.372965
$979$	6.62836e6	0.221029
$980$	0	0
$981$	1.54638e7	0.513030
$982$	−3.05886e7	−1.01223
$983$	2.54095e7	0.838710	0.419355	−	0.907822i	$-0.362256\pi$
$983$	2.54095e7	0.838710	0.419355	+	0.907822i	$0.362256\pi$
$984$	2.82032e6	0.0928560
$985$	0	0
$986$	−911500.	−0.0298582
$987$	−3.45031e7	−1.12737
$988$	−7.33097e6	−0.238929
$989$	−2.18874e7	−0.711546
$990$	0	0
$991$	6.88244e6	0.222617	0.111308	−	0.993786i	$-0.464496\pi$
$991$	6.88244e6	0.222617	0.111308	+	0.993786i	$0.464496\pi$
$992$	−52711.7	−0.00170070
$993$	2.02428e7	0.651475
$994$	−3.13043e6	−0.100493
$995$	0	0
$996$	1.60064e7	0.511264
$997$	−4.06394e7	−1.29482	−0.647411	−	0.762141i	$-0.724148\pi$
$997$	−4.06394e7	−1.29482	−0.647411	+	0.762141i	$0.724148\pi$
$998$	−4.01669e6	−0.127656
$999$	−4.63492e7	−1.46936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.p.1.2	yes	5
5.2	odd	4		650.6.b.m.599.9		10
5.3	odd	4		650.6.b.m.599.2		10
5.4	even	2		650.6.a.o.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.6.a.o.1.4	✓	5	5.4	even	2
650.6.a.p.1.2	yes	5	1.1	even	1	trivial
650.6.b.m.599.2		10	5.3	odd	4
650.6.b.m.599.9		10	5.2	odd	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -9.92606 q^{3} +16.0000 q^{4} -39.7042 q^{6} -124.276 q^{7} +64.0000 q^{8} -144.473 q^{9} +120.384 q^{11} -158.817 q^{12} -169.000 q^{13} -497.102 q^{14} +256.000 q^{16} +1234.94 q^{17} -577.893 q^{18} +2711.16 q^{19} +1233.57 q^{21} +481.535 q^{22} +1573.59 q^{23} -635.268 q^{24} -676.000 q^{26} +3846.08 q^{27} -1988.41 q^{28} -184.523 q^{29} -51.4763 q^{31} +1024.00 q^{32} -1194.94 q^{33} +4939.77 q^{34} -2311.57 q^{36} -12051.0 q^{37} +10844.6 q^{38} +1677.50 q^{39} -4439.57 q^{41} +4934.27 q^{42} -13909.2 q^{43} +1926.14 q^{44} +6294.35 q^{46} -27970.2 q^{47} -2541.07 q^{48} -1362.59 q^{49} -12258.1 q^{51} -2704.00 q^{52} +24170.2 q^{53} +15384.3 q^{54} -7953.64 q^{56} -26911.1 q^{57} -738.091 q^{58} -38786.4 q^{59} +53666.6 q^{61} -205.905 q^{62} +17954.5 q^{63} +4096.00 q^{64} -4779.75 q^{66} -10772.8 q^{67} +19759.1 q^{68} -15619.5 q^{69} +6297.35 q^{71} -9246.29 q^{72} +4254.45 q^{73} -48204.1 q^{74} +43378.5 q^{76} -14960.8 q^{77} +6710.02 q^{78} -49474.9 q^{79} -3069.45 q^{81} -17758.3 q^{82} -100785. q^{83} +19737.1 q^{84} -55636.8 q^{86} +1831.58 q^{87} +7704.56 q^{88} +55060.2 q^{89} +21002.6 q^{91} +25177.4 q^{92} +510.957 q^{93} -111881. q^{94} -10164.3 q^{96} -177285. q^{97} -5450.34 q^{98} -17392.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 20 q^{2} - 9 q^{3} + 80 q^{4} - 36 q^{6} + 42 q^{7} + 320 q^{8} + 4 q^{9} - 213 q^{11} - 144 q^{12} - 845 q^{13} + 168 q^{14} + 1280 q^{16} - 601 q^{17} + 16 q^{18} - 567 q^{19} - 3700 q^{21} - 852 q^{22}+ \cdots - 440284 q^{99}+O(q^{100}) \) 5 * q + 20 * q^2 - 9 * q^3 + 80 * q^4 - 36 * q^6 + 42 * q^7 + 320 * q^8 + 4 * q^9 - 213 * q^11 - 144 * q^12 - 845 * q^13 + 168 * q^14 + 1280 * q^16 - 601 * q^17 + 16 * q^18 - 567 * q^19 - 3700 * q^21 - 852 * q^22 - 636 * q^23 - 576 * q^24 - 3380 * q^26 - 5283 * q^27 + 672 * q^28 + 540 * q^29 - 10828 * q^31 + 5120 * q^32 + 2227 * q^33 - 2404 * q^34 + 64 * q^36 - 12708 * q^37 - 2268 * q^38 + 1521 * q^39 - 11461 * q^41 - 14800 * q^42 - 2020 * q^43 - 3408 * q^44 - 2544 * q^46 - 35062 * q^47 - 2304 * q^48 - 7285 * q^49 + 12859 * q^51 - 13520 * q^52 - 7718 * q^53 - 21132 * q^54 + 2688 * q^56 - 76689 * q^57 + 2160 * q^58 - 44686 * q^59 + 18972 * q^61 - 43312 * q^62 + 23698 * q^63 + 20480 * q^64 + 8908 * q^66 - 98667 * q^67 - 9616 * q^68 - 32718 * q^69 + 23928 * q^71 + 256 * q^72 + 17529 * q^73 - 50832 * q^74 - 9072 * q^76 - 74234 * q^77 + 6084 * q^78 - 67676 * q^79 + 26629 * q^81 - 45844 * q^82 + 13689 * q^83 - 59200 * q^84 - 8080 * q^86 - 64204 * q^87 - 13632 * q^88 - 287231 * q^89 - 7098 * q^91 - 10176 * q^92 + 38102 * q^93 - 140248 * q^94 - 9216 * q^96 - 78846 * q^97 - 29140 * q^98 - 440284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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