LMFDB - Embedded newform 325.6.b.h.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$-7.11691i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.11691i q^{2} +3.22994i q^{3} -33.8842 q^{4} +26.2172 q^{6} -20.7248i q^{7} +15.2938i q^{8} +232.567 q^{9} +O(q^{10})$	\(q-8.11691i q^{2} +3.22994i q^{3} -33.8842 q^{4} +26.2172 q^{6} -20.7248i q^{7} +15.2938i q^{8} +232.567 q^{9} +134.772 q^{11} -109.444i q^{12} -169.000i q^{13} -168.222 q^{14} -960.156 q^{16} +2192.81i q^{17} -1887.73i q^{18} +1981.97 q^{19} +66.9401 q^{21} -1093.93i q^{22} +4227.53i q^{23} -49.3980 q^{24} -1371.76 q^{26} +1536.06i q^{27} +702.244i q^{28} -2547.33 q^{29} -1368.81 q^{31} +8282.90i q^{32} +435.305i q^{33} +17798.9 q^{34} -7880.36 q^{36} +14161.1i q^{37} -16087.5i q^{38} +545.861 q^{39} +11750.1 q^{41} -543.347i q^{42} -6627.63i q^{43} -4566.63 q^{44} +34314.5 q^{46} -14822.2i q^{47} -3101.25i q^{48} +16377.5 q^{49} -7082.67 q^{51} +5726.43i q^{52} -24261.4i q^{53} +12468.0 q^{54} +316.961 q^{56} +6401.66i q^{57} +20676.4i q^{58} -1438.11 q^{59} +16295.7 q^{61} +11110.5i q^{62} -4819.92i q^{63} +36506.5 q^{64} +3533.33 q^{66} +16700.2i q^{67} -74301.7i q^{68} -13654.7 q^{69} +27153.9 q^{71} +3556.83i q^{72} -63222.7i q^{73} +114944. q^{74} -67157.5 q^{76} -2793.12i q^{77} -4430.70i q^{78} +58165.8 q^{79} +51552.5 q^{81} -95374.1i q^{82} +121286. i q^{83} -2268.21 q^{84} -53795.9 q^{86} -8227.72i q^{87} +2061.17i q^{88} +49897.1 q^{89} -3502.50 q^{91} -143246. i q^{92} -4421.18i q^{93} -120311. q^{94} -26753.3 q^{96} -21395.4i q^{97} -132934. i q^{98} +31343.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 8.11691i	− 1.43488i	−0.696620	−	0.717440i	$-0.745314\pi$
$2$	− 8.11691i	− 1.43488i	0.696620	−	0.717440i	$-0.254686\pi$
$3$	3.22994i	0.207201i	0.994619	+	0.103601i	$0.0330364\pi$
$3$	3.22994i	0.207201i	−0.994619	+	0.103601i	$0.966964\pi$
$4$	−33.8842	−1.05888
$5$	0	0
$5$	0	0
$6$	26.2172	0.297309
$7$	− 20.7248i	− 0.159862i	−0.996800	−	0.0799312i	$-0.974530\pi$
$7$	− 20.7248i	− 0.159862i	0.996800	−	0.0799312i	$-0.0254700\pi$
$8$	15.2938i	0.0844869i
$9$	232.567	0.957068
$10$	0	0
$11$	134.772	0.335828	0.167914	−	0.985802i	$-0.446297\pi$
$11$	134.772	0.335828	0.167914	+	0.985802i	$0.446297\pi$
$12$	− 109.444i	− 0.219401i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−168.222	−0.229383
$15$	0	0
$16$	−960.156	−0.937652
$17$	2192.81i	1.84026i	0.391612	+	0.920131i	$0.371918\pi$
$17$	2192.81i	1.84026i	−0.391612	+	0.920131i	$0.628082\pi$
$18$	− 1887.73i	− 1.37328i
$19$	1981.97	1.25954	0.629772	−	0.776780i	$-0.283148\pi$
$19$	1981.97	1.25954	0.629772	+	0.776780i	$0.283148\pi$
$20$	0	0
$21$	66.9401	0.0331236
$22$	− 1093.93i	− 0.481873i
$23$	4227.53i	1.66635i	0.553006	+	0.833177i	$0.313481\pi$
$23$	4227.53i	1.66635i	−0.553006	+	0.833177i	$0.686519\pi$
$24$	−49.3980	−0.0175058
$25$	0	0
$26$	−1371.76	−0.397964
$27$	1536.06i	0.405507i
$28$	702.244i	0.169275i
$29$	−2547.33	−0.562457	−0.281229	−	0.959641i	$-0.590742\pi$
$29$	−2547.33	−0.562457	−0.281229	+	0.959641i	$0.590742\pi$
$30$	0	0
$31$	−1368.81	−0.255822	−0.127911	−	0.991786i	$-0.540827\pi$
$31$	−1368.81	−0.255822	−0.127911	+	0.991786i	$0.540827\pi$
$32$	8282.90i	1.42991i
$33$	435.305i	0.0695840i
$34$	17798.9	2.64055
$35$	0	0
$36$	−7880.36	−1.01342
$37$	14161.1i	1.70056i	0.526327	+	0.850282i	$0.323569\pi$
$37$	14161.1i	1.70056i	−0.526327	+	0.850282i	$0.676431\pi$
$38$	− 16087.5i	− 1.80729i
$39$	545.861	0.0574672
$40$	0	0
$41$	11750.1	1.09164	0.545821	−	0.837902i	$-0.316218\pi$
$41$	11750.1	1.09164	0.545821	+	0.837902i	$0.316218\pi$
$42$	− 543.347i	− 0.0475285i
$43$	− 6627.63i	− 0.546622i	−0.961926	−	0.273311i	$-0.911881\pi$
$43$	− 6627.63i	− 0.546622i	0.961926	−	0.273311i	$-0.0881189\pi$
$44$	−4566.63	−0.355602
$45$	0	0
$46$	34314.5	2.39102
$47$	− 14822.2i	− 0.978744i	−0.872075	−	0.489372i	$-0.837226\pi$
$47$	− 14822.2i	− 0.978744i	0.872075	−	0.489372i	$-0.162774\pi$
$48$	− 3101.25i	− 0.194283i
$49$	16377.5	0.974444
$50$	0	0
$51$	−7082.67	−0.381304
$52$	5726.43i	0.293681i
$53$	− 24261.4i	− 1.18639i	−0.805060	−	0.593193i	$-0.797867\pi$
$53$	− 24261.4i	− 1.18639i	0.805060	−	0.593193i	$-0.202133\pi$
$54$	12468.0	0.581853
$55$	0	0
$56$	316.961	0.0135063
$57$	6401.66i	0.260979i
$58$	20676.4i	0.807059i
$59$	−1438.11	−0.0537850	−0.0268925	−	0.999638i	$-0.508561\pi$
$59$	−1438.11	−0.0537850	−0.0268925	+	0.999638i	$0.508561\pi$
$60$	0	0
$61$	16295.7	0.560724	0.280362	−	0.959894i	$-0.409546\pi$
$61$	16295.7	0.560724	0.280362	+	0.959894i	$0.409546\pi$
$62$	11110.5i	0.367074i
$63$	− 4819.92i	− 0.152999i
$64$	36506.5	1.11409
$65$	0	0
$66$	3533.33	0.0998446
$67$	16700.2i	0.454502i	0.973836	+	0.227251i	$0.0729738\pi$
$67$	16700.2i	0.454502i	−0.973836	+	0.227251i	$0.927026\pi$
$68$	− 74301.7i	− 1.94862i
$69$	−13654.7	−0.345270
$70$	0	0
$71$	27153.9	0.639273	0.319636	−	0.947540i	$-0.396439\pi$
$71$	27153.9	0.639273	0.319636	+	0.947540i	$0.396439\pi$
$72$	3556.83i	0.0808597i
$73$	− 63222.7i	− 1.38856i	−0.719703	−	0.694282i	$-0.755722\pi$
$73$	− 63222.7i	− 1.38856i	0.719703	−	0.694282i	$-0.244278\pi$
$74$	114944.	2.44011
$75$	0	0
$76$	−67157.5	−1.33371
$77$	− 2793.12i	− 0.0536863i
$78$	− 4430.70i	− 0.0824586i
$79$	58165.8	1.04858	0.524288	−	0.851541i	$-0.324331\pi$
$79$	58165.8	1.04858	0.524288	+	0.851541i	$0.324331\pi$
$80$	0	0
$81$	51552.5	0.873046
$82$	− 95374.1i	− 1.56638i
$83$	121286.i	1.93249i	0.257632	+	0.966243i	$0.417058\pi$
$83$	121286.i	1.93249i	−0.257632	+	0.966243i	$0.582942\pi$
$84$	−2268.21	−0.0350740
$85$	0	0
$86$	−53795.9	−0.784337
$87$	− 8227.72i	− 0.116542i
$88$	2061.17i	0.0283731i
$89$	49897.1	0.667729	0.333865	−	0.942621i	$-0.391647\pi$
$89$	49897.1	0.667729	0.333865	+	0.942621i	$0.391647\pi$
$90$	0	0
$91$	−3502.50	−0.0443378
$92$	− 143246.i	− 1.76447i
$93$	− 4421.18i	− 0.0530067i
$94$	−120311.	−1.40438
$95$	0	0
$96$	−26753.3	−0.296278
$97$	− 21395.4i	− 0.230883i	−0.993314	−	0.115441i	$-0.963172\pi$
$97$	− 21395.4i	− 0.230883i	0.993314	−	0.115441i	$-0.0368283\pi$
$98$	− 132934.i	− 1.39821i
$99$	31343.5	0.321410
$100$	0	0
$101$	−107983.	−1.05330	−0.526648	−	0.850084i	$-0.676551\pi$
$101$	−107983.	−1.05330	−0.526648	+	0.850084i	$0.676551\pi$
$102$	57489.4i	0.547126i
$103$	40865.0i	0.379541i	0.981828	+	0.189771i	$0.0607744\pi$
$103$	40865.0i	0.379541i	−0.981828	+	0.189771i	$0.939226\pi$
$104$	2584.65	0.0234325
$105$	0	0
$106$	−196928.	−1.70232
$107$	− 74698.6i	− 0.630744i	−0.948968	−	0.315372i	$-0.897871\pi$
$107$	− 74698.6i	− 0.630744i	0.948968	−	0.315372i	$-0.102129\pi$
$108$	− 52048.0i	− 0.429383i
$109$	84724.3	0.683033	0.341517	−	0.939876i	$-0.389059\pi$
$109$	84724.3	0.683033	0.341517	+	0.939876i	$0.389059\pi$
$110$	0	0
$111$	−45739.6	−0.352359
$112$	19899.1i	0.149895i
$113$	92231.9i	0.679493i	0.940517	+	0.339747i	$0.110341\pi$
$113$	92231.9i	0.679493i	−0.940517	+	0.339747i	$0.889659\pi$
$114$	51961.7	0.374473
$115$	0	0
$116$	86314.1	0.595575
$117$	− 39303.9i	− 0.265443i
$118$	11673.0i	0.0771751i
$119$	45445.7	0.294188
$120$	0	0
$121$	−142888.	−0.887219
$122$	− 132271.i	− 0.804572i
$123$	37952.0i	0.226189i
$124$	46381.0	0.270885
$125$	0	0
$126$	−39122.9	−0.219535
$127$	− 330445.i	− 1.81799i	−0.416812	−	0.908993i	$-0.636853\pi$
$127$	− 330445.i	− 1.81799i	0.416812	−	0.908993i	$-0.363147\pi$
$128$	− 31267.3i	− 0.168681i
$129$	21406.9	0.113261
$130$	0	0
$131$	−73109.1	−0.372214	−0.186107	−	0.982529i	$-0.559587\pi$
$131$	−73109.1	−0.372214	−0.186107	+	0.982529i	$0.559587\pi$
$132$	− 14750.0i	− 0.0736811i
$133$	− 41076.0i	− 0.201354i
$134$	135554.	0.652156
$135$	0	0
$136$	−33536.4	−0.155478
$137$	− 137616.i	− 0.626421i	−0.949684	−	0.313211i	$-0.898595\pi$
$137$	− 137616.i	− 0.626421i	0.949684	−	0.313211i	$-0.101405\pi$
$138$	110834.i	0.495422i
$139$	−366876.	−1.61058	−0.805291	−	0.592880i	$-0.797991\pi$
$139$	−366876.	−1.61058	−0.805291	+	0.592880i	$0.797991\pi$
$140$	0	0
$141$	47875.0	0.202797
$142$	− 220406.i	− 0.917280i
$143$	− 22776.4i	− 0.0931420i
$144$	−223301.	−0.897397
$145$	0	0
$146$	−513173.	−1.99242
$147$	52898.4i	0.201906i
$148$	− 479838.i	− 1.80070i
$149$	426714.	1.57460	0.787302	−	0.616567i	$-0.211477\pi$
$149$	426714.	1.57460	0.787302	+	0.616567i	$0.211477\pi$
$150$	0	0
$151$	22573.4	0.0805663	0.0402832	−	0.999188i	$-0.487174\pi$
$151$	22573.4	0.0805663	0.0402832	+	0.999188i	$0.487174\pi$
$152$	30311.8i	0.106415i
$153$	509977.i	1.76125i
$154$	−22671.5	−0.0770334
$155$	0	0
$156$	−18496.0	−0.0608510
$157$	496911.i	1.60890i	0.594018	+	0.804451i	$0.297541\pi$
$157$	496911.i	1.60890i	−0.594018	+	0.804451i	$0.702459\pi$
$158$	− 472127.i	− 1.50458i
$159$	78363.0	0.245821
$160$	0	0
$161$	87614.9	0.266387
$162$	− 418447.i	− 1.25272i
$163$	− 129489.i	− 0.381737i	−0.981616	−	0.190868i	$-0.938870\pi$
$163$	− 129489.i	− 0.381737i	0.981616	−	0.190868i	$-0.0611303\pi$
$164$	−398141.	−1.15592
$165$	0	0
$166$	984469.	2.77289
$167$	− 413362.i	− 1.14694i	−0.819227	−	0.573469i	$-0.805597\pi$
$167$	− 413362.i	− 1.14694i	0.819227	−	0.573469i	$-0.194403\pi$
$168$	1023.77i	0.00279852i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	460942.	1.20547
$172$	224572.i	0.578808i
$173$	2949.45i	0.00749249i	0.999993	+	0.00374625i	$0.00119247\pi$
$173$	2949.45i	0.00749249i	−0.999993	+	0.00374625i	$0.998808\pi$
$174$	−66783.7	−0.167223
$175$	0	0
$176$	−129402.	−0.314890
$177$	− 4645.01i	− 0.0111443i
$178$	− 405010.i	− 0.958112i
$179$	431301.	1.00612	0.503058	−	0.864253i	$-0.332208\pi$
$179$	431301.	1.00612	0.503058	+	0.864253i	$0.332208\pi$
$180$	0	0
$181$	65154.9	0.147826	0.0739129	−	0.997265i	$-0.476451\pi$
$181$	65154.9	0.147826	0.0739129	+	0.997265i	$0.476451\pi$
$182$	28429.5i	0.0636195i
$183$	52634.3i	0.116183i
$184$	−64654.9	−0.140785
$185$	0	0
$186$	−35886.3	−0.0760582
$187$	295529.i	0.618012i
$188$	502239.i	1.03637i
$189$	31834.5	0.0648252
$190$	0	0
$191$	405278.	0.803840	0.401920	−	0.915675i	$-0.368343\pi$
$191$	405278.	0.803840	0.401920	+	0.915675i	$0.368343\pi$
$192$	117914.i	0.230841i
$193$	596684.i	1.15306i	0.817077	+	0.576529i	$0.195593\pi$
$193$	596684.i	1.15306i	−0.817077	+	0.576529i	$0.804407\pi$
$194$	−173665.	−0.331289
$195$	0	0
$196$	−554938.	−1.03182
$197$	240155.i	0.440886i	0.975400	+	0.220443i	$0.0707504\pi$
$197$	240155.i	0.440886i	−0.975400	+	0.220443i	$0.929250\pi$
$198$	− 254413.i	− 0.461185i
$199$	−438913.	−0.785680	−0.392840	−	0.919607i	$-0.628507\pi$
$199$	−438913.	−0.785680	−0.392840	+	0.919607i	$0.628507\pi$
$200$	0	0
$201$	−53940.9	−0.0941733
$202$	876484.i	1.51135i
$203$	52792.9i	0.0899157i
$204$	239990.	0.403756
$205$	0	0
$206$	331698.	0.544596
$207$	983186.i	1.59481i
$208$	162266.i	0.260058i
$209$	267114.	0.422990
$210$	0	0
$211$	−107094.	−0.165600	−0.0828000	−	0.996566i	$-0.526386\pi$
$211$	−107094.	−0.165600	−0.0828000	+	0.996566i	$0.526386\pi$
$212$	822078.i	1.25624i
$213$	87705.6i	0.132458i
$214$	−606321.	−0.905041
$215$	0	0
$216$	−23492.1	−0.0342600
$217$	28368.3i	0.0408964i
$218$	− 687699.i	− 0.980071i
$219$	204206.	0.287712
$220$	0	0
$221$	370586.	0.510397
$222$	371264.i	0.505593i
$223$	865096.i	1.16494i	0.812853	+	0.582468i	$0.197913\pi$
$223$	865096.i	1.16494i	−0.812853	+	0.582468i	$0.802087\pi$
$224$	171662.	0.228588
$225$	0	0
$226$	748638.	0.974991
$227$	772248.i	0.994700i	0.867550	+	0.497350i	$0.165693\pi$
$227$	772248.i	0.994700i	−0.867550	+	0.497350i	$0.834307\pi$
$228$	− 216915.i	− 0.276346i
$229$	615081.	0.775075	0.387538	−	0.921854i	$-0.373326\pi$
$229$	615081.	0.775075	0.387538	+	0.921854i	$0.373326\pi$
$230$	0	0
$231$	9021.64	0.0111239
$232$	− 38958.2i	− 0.0475203i
$233$	1.38212e6i	1.66784i	0.551882	+	0.833922i	$0.313910\pi$
$233$	1.38212e6i	1.66784i	−0.551882	+	0.833922i	$0.686090\pi$
$234$	−319026.	−0.380879
$235$	0	0
$236$	48729.1	0.0569519
$237$	187872.i	0.217266i
$238$	− 368879.i	− 0.422125i
$239$	1.15404e6	1.30685	0.653424	−	0.756993i	$-0.273332\pi$
$239$	1.15404e6	1.30685	0.653424	+	0.756993i	$0.273332\pi$
$240$	0	0
$241$	−1.36874e6	−1.51802	−0.759011	−	0.651078i	$-0.774317\pi$
$241$	−1.36874e6	−1.51802	−0.759011	+	0.651078i	$0.774317\pi$
$242$	1.15981e6i	1.27305i
$243$	539774.i	0.586403i
$244$	−552167.	−0.593740
$245$	0	0
$246$	308053.	0.324555
$247$	− 334953.i	− 0.349335i
$248$	− 20934.2i	− 0.0216136i
$249$	−391748.	−0.400413
$250$	0	0
$251$	1.37571e6	1.37830	0.689149	−	0.724620i	$-0.257985\pi$
$251$	1.37571e6	1.37830	0.689149	+	0.724620i	$0.257985\pi$
$252$	163319.i	0.162008i
$253$	569752.i	0.559609i
$254$	−2.68219e6	−2.60859
$255$	0	0
$256$	914414.	0.872054
$257$	− 594799.i	− 0.561742i	−0.959745	−	0.280871i	$-0.909377\pi$
$257$	− 594799.i	− 0.561742i	0.959745	−	0.280871i	$-0.0906234\pi$
$258$	− 173758.i	− 0.162516i
$259$	293487.	0.271856
$260$	0	0
$261$	−592425.	−0.538310
$262$	593420.i	0.534083i
$263$	− 1.60634e6i	− 1.43202i	−0.698092	−	0.716008i	$-0.745967\pi$
$263$	− 1.60634e6i	− 1.43202i	0.698092	−	0.716008i	$-0.254033\pi$
$264$	−6657.46	−0.00587894
$265$	0	0
$266$	−333410.	−0.288918
$267$	161165.i	0.138354i
$268$	− 565874.i	− 0.481263i
$269$	764991.	0.644578	0.322289	−	0.946641i	$-0.395548\pi$
$269$	764991.	0.644578	0.322289	+	0.946641i	$0.395548\pi$
$270$	0	0
$271$	479475.	0.396591	0.198296	−	0.980142i	$-0.436459\pi$
$271$	479475.	0.396591	0.198296	+	0.980142i	$0.436459\pi$
$272$	− 2.10544e6i	− 1.72552i
$273$	− 11312.9i	− 0.00918685i
$274$	−1.11701e6	−0.898839
$275$	0	0
$276$	462678.	0.365600
$277$	− 218179.i	− 0.170849i	−0.996345	−	0.0854246i	$-0.972775\pi$
$277$	− 218179.i	− 0.170849i	0.996345	−	0.0854246i	$-0.0272247\pi$
$278$	2.97790e6i	2.31099i
$279$	−318340.	−0.244839
$280$	0	0
$281$	−824857.	−0.623179	−0.311590	−	0.950217i	$-0.600861\pi$
$281$	−824857.	−0.623179	−0.311590	+	0.950217i	$0.600861\pi$
$282$	− 388597.i	− 0.290989i
$283$	1.37912e6i	1.02361i	0.859101	+	0.511806i	$0.171023\pi$
$283$	1.37912e6i	1.02361i	−0.859101	+	0.511806i	$0.828977\pi$
$284$	−920087.	−0.676914
$285$	0	0
$286$	−184874.	−0.133648
$287$	− 243518.i	− 0.174512i
$288$	1.92633e6i	1.36852i
$289$	−3.38858e6	−2.38656
$290$	0	0
$291$	69106.1	0.0478392
$292$	2.14225e6i	1.47032i
$293$	− 2.22384e6i	− 1.51334i	−0.653799	−	0.756668i	$-0.726826\pi$
$293$	− 2.22384e6i	− 1.51334i	0.653799	−	0.756668i	$-0.273174\pi$
$294$	429371.	0.289711
$295$	0	0
$296$	−216577.	−0.143676
$297$	207017.i	0.136181i
$298$	− 3.46360e6i	− 2.25937i
$299$	714453.	0.462163
$300$	0	0
$301$	−137357.	−0.0873843
$302$	− 183226.i	− 0.115603i
$303$	− 348778.i	− 0.218244i
$304$	−1.90300e6	−1.18101
$305$	0	0
$306$	4.13944e6	2.52719
$307$	− 2.71603e6i	− 1.64471i	−0.568975	−	0.822355i	$-0.692660\pi$
$307$	− 2.71603e6i	− 1.64471i	0.568975	−	0.822355i	$-0.307340\pi$
$308$	94642.7i	0.0568474i
$309$	−131992.	−0.0786413
$310$	0	0
$311$	−2.07670e6	−1.21751	−0.608754	−	0.793359i	$-0.708330\pi$
$311$	−2.07670e6	−1.21751	−0.608754	+	0.793359i	$0.708330\pi$
$312$	8348.27i	0.00485523i
$313$	− 1.45231e6i	− 0.837910i	−0.908007	−	0.418955i	$-0.862396\pi$
$313$	− 1.45231e6i	− 0.837910i	0.908007	−	0.418955i	$-0.137604\pi$
$314$	4.03338e6	2.30858
$315$	0	0
$316$	−1.97090e6	−1.11032
$317$	− 1.58790e6i	− 0.887511i	−0.896148	−	0.443755i	$-0.853646\pi$
$317$	− 1.58790e6i	− 0.887511i	0.896148	−	0.443755i	$-0.146354\pi$
$318$	− 636065.i	− 0.352723i
$319$	−343308.	−0.188889
$320$	0	0
$321$	241272.	0.130691
$322$	− 711162.i	− 0.382234i
$323$	4.34609e6i	2.31789i
$324$	−1.74681e6	−0.924452
$325$	0	0
$326$	−1.05105e6	−0.547746
$327$	273655.i	0.141525i
$328$	179703.i	0.0922295i
$329$	−307188.	−0.156464
$330$	0	0
$331$	−2.91368e6	−1.46174	−0.730872	−	0.682515i	$-0.760886\pi$
$331$	−2.91368e6	−1.46174	−0.730872	+	0.682515i	$0.760886\pi$
$332$	− 4.10969e6i	− 2.04627i
$333$	3.29341e6i	1.62756i
$334$	−3.35522e6	−1.64572
$335$	0	0
$336$	−64272.9	−0.0310585
$337$	2.39665e6i	1.14956i	0.818309	+	0.574778i	$0.194912\pi$
$337$	2.39665e6i	1.14956i	−0.818309	+	0.574778i	$0.805088\pi$
$338$	231827.i	0.110375i
$339$	−297904.	−0.140792
$340$	0	0
$341$	−184477.	−0.0859124
$342$	− 3.74142e6i	− 1.72970i
$343$	− 687743.i	− 0.315639i
$344$	101362.	0.0461824
$345$	0	0
$346$	23940.4	0.0107508
$347$	529222.i	0.235947i	0.993017	+	0.117973i	$0.0376398\pi$
$347$	529222.i	0.235947i	−0.993017	+	0.117973i	$0.962360\pi$
$348$	278790.i	0.123404i
$349$	847145.	0.372301	0.186151	−	0.982521i	$-0.440399\pi$
$349$	847145.	0.372301	0.186151	+	0.982521i	$0.440399\pi$
$350$	0	0
$351$	259594.	0.112467
$352$	1.11630e6i	0.480203i
$353$	3.44913e6i	1.47324i	0.676308	+	0.736619i	$0.263579\pi$
$353$	3.44913e6i	1.47324i	−0.676308	+	0.736619i	$0.736421\pi$
$354$	−37703.1	−0.0159908
$355$	0	0
$356$	−1.69072e6	−0.707046
$357$	146787.i	0.0609562i
$358$	− 3.50083e6i	− 1.44366i
$359$	192156.	0.0786895	0.0393448	−	0.999226i	$-0.487473\pi$
$359$	192156.	0.0786895	0.0393448	+	0.999226i	$0.487473\pi$
$360$	0	0
$361$	1.45211e6	0.586451
$362$	− 528856.i	− 0.212112i
$363$	− 461519.i	− 0.183833i
$364$	118679.	0.0469485
$365$	0	0
$366$	427228.	0.166708
$367$	1.34308e6i	0.520518i	0.965539	+	0.260259i	$0.0838079\pi$
$367$	1.34308e6i	0.520518i	−0.965539	+	0.260259i	$0.916192\pi$
$368$	− 4.05909e6i	− 1.56246i
$369$	2.73268e6	1.04478
$370$	0	0
$371$	−502814.	−0.189658
$372$	149808.i	0.0561277i
$373$	− 3.94149e6i	− 1.46686i	−0.679765	−	0.733430i	$-0.737918\pi$
$373$	− 3.94149e6i	− 1.46686i	0.679765	−	0.733430i	$-0.262082\pi$
$374$	2.39878e6	0.886773
$375$	0	0
$376$	226688.	0.0826911
$377$	430498.i	0.155998i
$378$	− 258398.i	− 0.0930164i
$379$	769080.	0.275026	0.137513	−	0.990500i	$-0.456089\pi$
$379$	769080.	0.275026	0.137513	+	0.990500i	$0.456089\pi$
$380$	0	0
$381$	1.06732e6	0.376689
$382$	− 3.28960e6i	− 1.15341i
$383$	1.33597e6i	0.465371i	0.972552	+	0.232685i	$0.0747513\pi$
$383$	1.33597e6i	0.465371i	−0.972552	+	0.232685i	$0.925249\pi$
$384$	100992.	0.0349509
$385$	0	0
$386$	4.84323e6	1.65450
$387$	− 1.54137e6i	− 0.523155i
$388$	724967.i	0.244477i
$389$	−158354.	−0.0530584	−0.0265292	−	0.999648i	$-0.508446\pi$
$389$	−158354.	−0.0530584	−0.0265292	+	0.999648i	$0.508446\pi$
$390$	0	0
$391$	−9.27019e6	−3.06653
$392$	250473.i	0.0823278i
$393$	− 236138.i	− 0.0771232i
$394$	1.94932e6	0.632619
$395$	0	0
$396$	−1.06205e6	−0.340335
$397$	2.11342e6i	0.672992i	0.941685	+	0.336496i	$0.109242\pi$
$397$	2.11342e6i	0.672992i	−0.941685	+	0.336496i	$0.890758\pi$
$398$	3.56261e6i	1.12736i
$399$	132673.	0.0417207
$400$	0	0
$401$	−2.39852e6	−0.744874	−0.372437	−	0.928057i	$-0.621478\pi$
$401$	−2.39852e6	−0.744874	−0.372437	+	0.928057i	$0.621478\pi$
$402$	437833.i	0.135127i
$403$	231329.i	0.0709524i
$404$	3.65890e6	1.11531
$405$	0	0
$406$	428515.	0.129018
$407$	1.90852e6i	0.571098i
$408$	− 108321.i	− 0.0322152i
$409$	2.12794e6	0.629000	0.314500	−	0.949257i	$-0.398163\pi$
$409$	2.12794e6	0.629000	0.314500	+	0.949257i	$0.398163\pi$
$410$	0	0
$411$	444491.	0.129795
$412$	− 1.38468e6i	− 0.401889i
$413$	29804.6i	0.00859820i
$414$	7.98043e6	2.28837
$415$	0	0
$416$	1.39981e6	0.396584
$417$	− 1.18499e6i	− 0.333714i
$418$	− 2.16814e6i	− 0.606941i
$419$	3.35254e6	0.932908	0.466454	−	0.884545i	$-0.345531\pi$
$419$	3.35254e6	0.932908	0.466454	+	0.884545i	$0.345531\pi$
$420$	0	0
$421$	205253.	0.0564397	0.0282198	−	0.999602i	$-0.491016\pi$
$421$	205253.	0.0564397	0.0282198	+	0.999602i	$0.491016\pi$
$422$	869275.i	0.237616i
$423$	− 3.44717e6i	− 0.936724i
$424$	371048.	0.100234
$425$	0	0
$426$	711898.	0.190061
$427$	− 337726.i	− 0.0896386i
$428$	2.53110e6i	0.667882i
$429$	73566.6	0.0192991
$430$	0	0
$431$	−6.22797e6	−1.61493	−0.807464	−	0.589917i	$-0.799161\pi$
$431$	−6.22797e6	−1.61493	−0.807464	+	0.589917i	$0.799161\pi$
$432$	− 1.47485e6i	− 0.380224i
$433$	− 4.07150e6i	− 1.04360i	−0.853067	−	0.521801i	$-0.825260\pi$
$433$	− 4.07150e6i	− 1.04360i	0.853067	−	0.521801i	$-0.174740\pi$
$434$	230263.	0.0586814
$435$	0	0
$436$	−2.87081e6	−0.723251
$437$	8.37885e6i	2.09885i
$438$	− 1.65752e6i	− 0.412832i
$439$	−1.75484e6	−0.434588	−0.217294	−	0.976106i	$-0.569723\pi$
$439$	−1.75484e6	−0.434588	−0.217294	+	0.976106i	$0.569723\pi$
$440$	0	0
$441$	3.80887e6	0.932609
$442$	− 3.00801e6i	− 0.732358i
$443$	4.08504e6i	0.988979i	0.869184	+	0.494489i	$0.164645\pi$
$443$	4.08504e6i	0.988979i	−0.869184	+	0.494489i	$0.835355\pi$
$444$	1.54985e6	0.373106
$445$	0	0
$446$	7.02191e6	1.67154
$447$	1.37826e6i	0.326260i
$448$	− 756592.i	− 0.178101i
$449$	−4.32979e6	−1.01356	−0.506782	−	0.862074i	$-0.669165\pi$
$449$	−4.32979e6	−1.01356	−0.506782	+	0.862074i	$0.669165\pi$
$450$	0	0
$451$	1.58358e6	0.366604
$452$	− 3.12520e6i	− 0.719502i
$453$	72910.7i	0.0166934i
$454$	6.26826e6	1.42728
$455$	0	0
$456$	−97905.5	−0.0220493
$457$	4.37096e6i	0.979009i	0.872001	+	0.489504i	$0.162822\pi$
$457$	4.37096e6i	0.979009i	−0.872001	+	0.489504i	$0.837178\pi$
$458$	− 4.99256e6i	− 1.11214i
$459$	−3.36829e6	−0.746238
$460$	0	0
$461$	−3.13672e6	−0.687421	−0.343711	−	0.939076i	$-0.611684\pi$
$461$	−3.13672e6	−0.687421	−0.343711	+	0.939076i	$0.611684\pi$
$462$	− 73227.8i	− 0.0159614i
$463$	− 6.47218e6i	− 1.40313i	−0.712605	−	0.701566i	$-0.752485\pi$
$463$	− 6.47218e6i	− 1.40313i	0.712605	−	0.701566i	$-0.247515\pi$
$464$	2.44583e6	0.527389
$465$	0	0
$466$	1.12185e7	2.39316
$467$	2.32552e6i	0.493433i	0.969088	+	0.246717i	$0.0793517\pi$
$467$	2.32552e6i	0.493433i	−0.969088	+	0.246717i	$0.920648\pi$
$468$	1.33178e6i	0.281072i
$469$	346110.	0.0726577
$470$	0	0
$471$	−1.60500e6	−0.333366
$472$	− 21994.1i	− 0.00454413i
$473$	− 893218.i	− 0.183571i
$474$	1.52494e6	0.311751
$475$	0	0
$476$	−1.53989e6	−0.311510
$477$	− 5.64241e6i	− 1.13545i
$478$	− 9.36721e6i	− 1.87517i
$479$	1.16074e6	0.231150	0.115575	−	0.993299i	$-0.463129\pi$
$479$	1.16074e6	0.231150	0.115575	+	0.993299i	$0.463129\pi$
$480$	0	0
$481$	2.39323e6	0.471652
$482$	1.11099e7i	2.17818i
$483$	282991.i	0.0551957i
$484$	4.84163e6	0.939460
$485$	0	0
$486$	4.38129e6	0.841417
$487$	3.63836e6i	0.695158i	0.937651	+	0.347579i	$0.112996\pi$
$487$	3.63836e6i	0.695158i	−0.937651	+	0.347579i	$0.887004\pi$
$488$	249223.i	0.0473739i
$489$	418242.	0.0790962
$490$	0	0
$491$	8.45015e6	1.58183	0.790917	−	0.611924i	$-0.209604\pi$
$491$	8.45015e6	1.58183	0.790917	+	0.611924i	$0.209604\pi$
$492$	− 1.28597e6i	− 0.239508i
$493$	− 5.58581e6i	− 1.03507i
$494$	−2.71878e6	−0.501253
$495$	0	0
$496$	1.31427e6	0.239872
$497$	− 562760.i	− 0.102196i
$498$	3.17978e6i	0.574545i
$499$	6.28285e6	1.12955	0.564775	−	0.825245i	$-0.308963\pi$
$499$	6.28285e6	1.12955	0.564775	+	0.825245i	$0.308963\pi$
$500$	0	0
$501$	1.33514e6	0.237647
$502$	− 1.11665e7i	− 1.97769i
$503$	− 2.30761e6i	− 0.406670i	−0.979109	−	0.203335i	$-0.934822\pi$
$503$	− 2.30761e6i	− 0.406670i	0.979109	−	0.203335i	$-0.0651781\pi$
$504$	73714.8	0.0129264
$505$	0	0
$506$	4.62462e6	0.802971
$507$	− 92250.5i	− 0.0159385i
$508$	1.11969e7i	1.92503i
$509$	2.24701e6	0.384423	0.192212	−	0.981353i	$-0.438434\pi$
$509$	2.24701e6	0.384423	0.192212	+	0.981353i	$0.438434\pi$
$510$	0	0
$511$	−1.31028e6	−0.221979
$512$	− 8.42277e6i	− 1.41997i
$513$	3.04442e6i	0.510753i
$514$	−4.82793e6	−0.806033
$515$	0	0
$516$	−725355.	−0.119930
$517$	− 1.99762e6i	− 0.328690i
$518$	− 2.38221e6i	− 0.390081i
$519$	−9526.57	−0.00155245
$520$	0	0
$521$	−1.20692e7	−1.94797	−0.973987	−	0.226602i	$-0.927238\pi$
$521$	−1.20692e7	−1.94797	−0.973987	+	0.226602i	$0.927238\pi$
$522$	4.80866e6i	0.772410i
$523$	− 1.19375e6i	− 0.190835i	−0.995437	−	0.0954174i	$-0.969581\pi$
$523$	− 1.19375e6i	− 0.190835i	0.995437	−	0.0954174i	$-0.0304186\pi$
$524$	2.47724e6	0.394131
$525$	0	0
$526$	−1.30385e7	−2.05477
$527$	− 3.00154e6i	− 0.470780i
$528$	− 417961.i	− 0.0652456i
$529$	−1.14357e7	−1.77674
$530$	0	0
$531$	−334457.	−0.0514759
$532$	1.39183e6i	0.213209i
$533$	− 1.98576e6i	− 0.302767i
$534$	1.30816e6	0.198522
$535$	0	0
$536$	−255410.	−0.0383995
$537$	1.39308e6i	0.208468i
$538$	− 6.20936e6i	− 0.924892i
$539$	2.20722e6	0.327246
$540$	0	0
$541$	−9.83092e6	−1.44411	−0.722056	−	0.691834i	$-0.756803\pi$
$541$	−9.83092e6	−1.44411	−0.722056	+	0.691834i	$0.756803\pi$
$542$	− 3.89186e6i	− 0.569061i
$543$	210447.i	0.0306297i
$544$	−1.81629e7	−2.63140
$545$	0	0
$546$	−91825.6	−0.0131820
$547$	8.63291e6i	1.23364i	0.787104	+	0.616821i	$0.211580\pi$
$547$	8.63291e6i	1.23364i	−0.787104	+	0.616821i	$0.788420\pi$
$548$	4.66299e6i	0.663305i
$549$	3.78986e6	0.536651
$550$	0	0
$551$	−5.04873e6	−0.708440
$552$	− 208832.i	− 0.0291708i
$553$	− 1.20548e6i	− 0.167628i
$554$	−1.77094e6	−0.245148
$555$	0	0
$556$	1.24313e7	1.70541
$557$	− 3.22690e6i	− 0.440704i	−0.975420	−	0.220352i	$-0.929279\pi$
$557$	− 3.22690e6i	− 0.440704i	0.975420	−	0.220352i	$-0.0707207\pi$
$558$	2.58394e6i	0.351315i
$559$	−1.12007e6	−0.151606
$560$	0	0
$561$	−954544.	−0.128053
$562$	6.69529e6i	0.894187i
$563$	683340.i	0.0908586i	0.998968	+	0.0454293i	$0.0144656\pi$
$563$	683340.i	0.0908586i	−0.998968	+	0.0454293i	$0.985534\pi$
$564$	−1.62221e6	−0.214738
$565$	0	0
$566$	1.11942e7	1.46876
$567$	− 1.06842e6i	− 0.139567i
$568$	415285.i	0.0540102i
$569$	5.95105e6	0.770572	0.385286	−	0.922797i	$-0.374103\pi$
$569$	5.95105e6	0.770572	0.385286	+	0.922797i	$0.374103\pi$
$570$	0	0
$571$	−2.57116e6	−0.330019	−0.165009	−	0.986292i	$-0.552765\pi$
$571$	−2.57116e6	−0.330019	−0.165009	+	0.986292i	$0.552765\pi$
$572$	771761.i	0.0986263i
$573$	1.30903e6i	0.166556i
$574$	−1.97661e6	−0.250404
$575$	0	0
$576$	8.49023e6	1.06626
$577$	1.09233e6i	0.136588i	0.997665	+	0.0682940i	$0.0217556\pi$
$577$	1.09233e6i	0.136588i	−0.997665	+	0.0682940i	$0.978244\pi$
$578$	2.75047e7i	3.42443i
$579$	−1.92726e6	−0.238915
$580$	0	0
$581$	2.51364e6	0.308932
$582$	− 560928.i	− 0.0686435i
$583$	− 3.26975e6i	− 0.398422i
$584$	966913.	0.117315
$585$	0	0
$586$	−1.80507e7	−2.17146
$587$	1.10253e7i	1.32067i	0.750972	+	0.660334i	$0.229585\pi$
$587$	1.10253e7i	1.32067i	−0.750972	+	0.660334i	$0.770415\pi$
$588$	− 1.79242e6i	− 0.213794i
$589$	−2.71294e6	−0.322219
$590$	0	0
$591$	−775689.	−0.0913521
$592$	− 1.35969e7i	− 1.59454i
$593$	1.27015e7i	1.48327i	0.670806	+	0.741633i	$0.265948\pi$
$593$	1.27015e7i	1.48327i	−0.670806	+	0.741633i	$0.734052\pi$
$594$	1.68034e6	0.195403
$595$	0	0
$596$	−1.44589e7	−1.66732
$597$	− 1.41766e6i	− 0.162794i
$598$	− 5.79915e6i	− 0.663149i
$599$	2.40180e6	0.273508	0.136754	−	0.990605i	$-0.456333\pi$
$599$	2.40180e6	0.273508	0.136754	+	0.990605i	$0.456333\pi$
$600$	0	0
$601$	−6.52655e6	−0.737051	−0.368526	−	0.929618i	$-0.620137\pi$
$601$	−6.52655e6	−0.737051	−0.368526	+	0.929618i	$0.620137\pi$
$602$	1.11491e6i	0.125386i
$603$	3.88393e6i	0.434989i
$604$	−764880.	−0.0853102
$605$	0	0
$606$	−2.83100e6	−0.313154
$607$	− 1.25664e7i	− 1.38433i	−0.721740	−	0.692164i	$-0.756657\pi$
$607$	− 1.25664e7i	− 1.38433i	0.721740	−	0.692164i	$-0.243343\pi$
$608$	1.64165e7i	1.80103i
$609$	−170518.	−0.0186306
$610$	0	0
$611$	−2.50496e6	−0.271455
$612$	− 1.72802e7i	− 1.86496i
$613$	5.17046e6i	0.555748i	0.960617	+	0.277874i	$0.0896298\pi$
$613$	5.17046e6i	0.555748i	−0.960617	+	0.277874i	$0.910370\pi$
$614$	−2.20458e7	−2.35996
$615$	0	0
$616$	42717.4	0.00453579
$617$	7.58899e6i	0.802548i	0.915958	+	0.401274i	$0.131433\pi$
$617$	7.58899e6i	0.802548i	−0.915958	+	0.401274i	$0.868567\pi$
$618$	1.07137e6i	0.112841i
$619$	3.72082e6	0.390312	0.195156	−	0.980772i	$-0.437479\pi$
$619$	3.72082e6	0.390312	0.195156	+	0.980772i	$0.437479\pi$
$620$	0	0
$621$	−6.49373e6	−0.675717
$622$	1.68563e7i	1.74698i
$623$	− 1.03411e6i	− 0.106745i
$624$	−524111.	−0.0538843
$625$	0	0
$626$	−1.17882e7	−1.20230
$627$	862763.i	0.0876441i
$628$	− 1.68374e7i	− 1.70364i
$629$	−3.10527e7	−3.12948
$630$	0	0
$631$	1.80988e7	1.80958	0.904788	−	0.425862i	$-0.140029\pi$
$631$	1.80988e7	1.80958	0.904788	+	0.425862i	$0.140029\pi$
$632$	889575.i	0.0885910i
$633$	− 345909.i	− 0.0343125i
$634$	−1.28888e7	−1.27347
$635$	0	0
$636$	−2.65527e6	−0.260295
$637$	− 2.76779e6i	− 0.270262i
$638$	2.78660e6i	0.271033i
$639$	6.31511e6	0.611827
$640$	0	0
$641$	3.95349e6	0.380046	0.190023	−	0.981780i	$-0.439144\pi$
$641$	3.95349e6	0.380046	0.190023	+	0.981780i	$0.439144\pi$
$642$	− 1.95838e6i	− 0.187526i
$643$	− 2.15047e6i	− 0.205119i	−0.994727	−	0.102559i	$-0.967297\pi$
$643$	− 2.15047e6i	− 0.205119i	0.994727	−	0.102559i	$-0.0327032\pi$
$644$	−2.96876e6	−0.282072
$645$	0	0
$646$	3.52768e7	3.32589
$647$	− 7.72603e6i	− 0.725598i	−0.931867	−	0.362799i	$-0.881821\pi$
$647$	− 7.72603e6i	− 0.725598i	0.931867	−	0.362799i	$-0.118179\pi$
$648$	788432.i	0.0737610i
$649$	−193816.	−0.0180625
$650$	0	0
$651$	−91628.2	−0.00847377
$652$	4.38763e6i	0.404213i
$653$	7.07138e6i	0.648965i	0.945892	+	0.324482i	$0.105190\pi$
$653$	7.07138e6i	0.648965i	−0.945892	+	0.324482i	$0.894810\pi$
$654$	2.22123e6	0.203072
$655$	0	0
$656$	−1.12819e7	−1.02358
$657$	− 1.47035e7i	− 1.32895i
$658$	2.49342e6i	0.224507i
$659$	−8.83103e6	−0.792133	−0.396066	−	0.918222i	$-0.629625\pi$
$659$	−8.83103e6	−0.792133	−0.396066	+	0.918222i	$0.629625\pi$
$660$	0	0
$661$	−8.94111e6	−0.795954	−0.397977	−	0.917395i	$-0.630288\pi$
$661$	−8.94111e6	−0.795954	−0.397977	+	0.917395i	$0.630288\pi$
$662$	2.36500e7i	2.09743i
$663$	1.19697e6i	0.105755i
$664$	−1.85492e6	−0.163270
$665$	0	0
$666$	2.67323e7	2.33535
$667$	− 1.07689e7i	− 0.937253i
$668$	1.40064e7i	1.21447i
$669$	−2.79421e6	−0.241376
$670$	0	0
$671$	2.19620e6	0.188307
$672$	554458.i	0.0473637i
$673$	− 2.00280e7i	− 1.70451i	−0.523123	−	0.852257i	$-0.675233\pi$
$673$	− 2.00280e7i	− 1.70451i	0.523123	−	0.852257i	$-0.324767\pi$
$674$	1.94534e7	1.64948
$675$	0	0
$676$	967766.	0.0814524
$677$	1.44506e6i	0.121175i	0.998163	+	0.0605877i	$0.0192975\pi$
$677$	1.44506e6i	0.121175i	−0.998163	+	0.0605877i	$0.980703\pi$
$678$	2.41806e6i	0.202019i
$679$	−443417.	−0.0369095
$680$	0	0
$681$	−2.49432e6	−0.206103
$682$	1.49738e6i	0.123274i
$683$	3.37739e6i	0.277032i	0.990360	+	0.138516i	$0.0442332\pi$
$683$	3.37739e6i	0.277032i	−0.990360	+	0.138516i	$0.955767\pi$
$684$	−1.56186e7	−1.27645
$685$	0	0
$686$	−5.58235e6	−0.452904
$687$	1.98668e6i	0.160596i
$688$	6.36356e6i	0.512542i
$689$	−4.10018e6	−0.329044
$690$	0	0
$691$	−1.36877e7	−1.09052	−0.545262	−	0.838265i	$-0.683570\pi$
$691$	−1.36877e7	−1.09052	−0.545262	+	0.838265i	$0.683570\pi$
$692$	− 99939.8i	− 0.00793366i
$693$	− 649590.i	− 0.0513814i
$694$	4.29565e6	0.338555
$695$	0	0
$696$	125833.	0.00984626
$697$	2.57657e7i	2.00891i
$698$	− 6.87620e6i	− 0.534207i
$699$	−4.46417e6	−0.345579
$700$	0	0
$701$	2.12015e7	1.62957	0.814784	−	0.579765i	$-0.196856\pi$
$701$	2.12015e7	1.62957	0.814784	+	0.579765i	$0.196856\pi$
$702$	− 2.10710e6i	− 0.161377i
$703$	2.80669e7i	2.14194i
$704$	4.92005e6	0.374143
$705$	0	0
$706$	2.79963e7	2.11392
$707$	2.23792e6i	0.168382i
$708$	157392.i	0.0118005i
$709$	−4.77971e6	−0.357097	−0.178549	−	0.983931i	$-0.557140\pi$
$709$	−4.77971e6	−0.357097	−0.178549	+	0.983931i	$0.557140\pi$
$710$	0	0
$711$	1.35275e7	1.00356
$712$	763115.i	0.0564144i
$713$	− 5.78668e6i	− 0.426291i
$714$	1.19146e6	0.0874648
$715$	0	0
$716$	−1.46143e7	−1.06536
$717$	3.72747e6i	0.270780i
$718$	− 1.55971e6i	− 0.112910i
$719$	1.56246e7	1.12716	0.563580	−	0.826062i	$-0.309424\pi$
$719$	1.56246e7	1.12716	0.563580	+	0.826062i	$0.309424\pi$
$720$	0	0
$721$	846921.	0.0606743
$722$	− 1.17866e7i	− 0.841487i
$723$	− 4.42095e6i	− 0.314536i
$724$	−2.20772e6	−0.156530
$725$	0	0
$726$	−3.74611e6	−0.263778
$727$	− 1.10204e7i	− 0.773327i	−0.922221	−	0.386664i	$-0.873627\pi$
$727$	− 1.10204e7i	− 0.773327i	0.922221	−	0.386664i	$-0.126373\pi$
$728$	− 53566.4i	− 0.00374597i
$729$	1.07838e7	0.751543
$730$	0	0
$731$	1.45332e7	1.00593
$732$	− 1.78347e6i	− 0.123024i
$733$	− 3.38982e6i	− 0.233033i	−0.993189	−	0.116516i	$-0.962827\pi$
$733$	− 3.38982e6i	− 0.233033i	0.993189	−	0.116516i	$-0.0371727\pi$
$734$	1.09016e7	0.746880
$735$	0	0
$736$	−3.50162e7	−2.38273
$737$	2.25072e6i	0.152635i
$738$	− 2.21809e7i	− 1.49913i
$739$	−2.83527e7	−1.90978	−0.954891	−	0.296956i	$-0.904029\pi$
$739$	−2.83527e7	−1.90978	−0.954891	+	0.296956i	$0.904029\pi$
$740$	0	0
$741$	1.08188e6	0.0723825
$742$	4.08129e6i	0.272137i
$743$	− 2.16151e7i	− 1.43643i	−0.695820	−	0.718216i	$-0.744959\pi$
$743$	− 2.16151e7i	− 1.43643i	0.695820	−	0.718216i	$-0.255041\pi$
$744$	67616.5	0.00447837
$745$	0	0
$746$	−3.19927e7	−2.10477
$747$	2.82072e7i	1.84952i
$748$	− 1.00138e7i	− 0.654401i
$749$	−1.54812e6	−0.100832
$750$	0	0
$751$	1.11361e7	0.720500	0.360250	−	0.932856i	$-0.382691\pi$
$751$	1.11361e7	0.720500	0.360250	+	0.932856i	$0.382691\pi$
$752$	1.42317e7i	0.917721i
$753$	4.44347e6i	0.285585i
$754$	3.49431e6	0.223838
$755$	0	0
$756$	−1.07869e6	−0.0686422
$757$	4.00933e6i	0.254291i	0.991884	+	0.127146i	$0.0405816\pi$
$757$	4.00933e6i	0.254291i	−0.991884	+	0.127146i	$0.959418\pi$
$758$	− 6.24255e6i	− 0.394629i
$759$	−1.84027e6	−0.115952
$760$	0	0
$761$	3.18411e7	1.99309	0.996543	−	0.0830812i	$-0.0264761\pi$
$761$	3.18411e7	1.99309	0.996543	+	0.0830812i	$0.0264761\pi$
$762$	− 8.66334e6i	− 0.540503i
$763$	− 1.75590e6i	− 0.109191i
$764$	−1.37325e7	−0.851170
$765$	0	0
$766$	1.08439e7	0.667752
$767$	243040.i	0.0149173i
$768$	2.95351e6i	0.180690i
$769$	2.53683e7	1.54695	0.773474	−	0.633828i	$-0.218517\pi$
$769$	2.53683e7	1.54695	0.773474	+	0.633828i	$0.218517\pi$
$770$	0	0
$771$	1.92117e6	0.116394
$772$	− 2.02182e7i	− 1.22095i
$773$	2.50569e7i	1.50827i	0.656720	+	0.754134i	$0.271943\pi$
$773$	2.50569e7i	1.50827i	−0.656720	+	0.754134i	$0.728057\pi$
$774$	−1.25112e7	−0.750664
$775$	0	0
$776$	327217.	0.0195066
$777$	947947.i	0.0563289i
$778$	1.28534e6i	0.0761325i
$779$	2.32883e7	1.37497
$780$	0	0
$781$	3.65958e6	0.214686
$782$	7.52453e7i	4.40010i
$783$	− 3.91284e6i	− 0.228080i
$784$	−1.57249e7	−0.913690
$785$	0	0
$786$	−1.91671e6	−0.110663
$787$	− 2.15591e7i	− 1.24078i	−0.784294	−	0.620389i	$-0.786975\pi$
$787$	− 2.15591e7i	− 1.24078i	0.784294	−	0.620389i	$-0.213025\pi$
$788$	− 8.13747e6i	− 0.466846i
$789$	5.18839e6	0.296715
$790$	0	0
$791$	1.91149e6	0.108625
$792$	479361.i	0.0271550i
$793$	− 2.75398e6i	− 0.155517i
$794$	1.71545e7	0.965663
$795$	0	0
$796$	1.48722e7	0.831941
$797$	− 2.48604e6i	− 0.138631i	−0.997595	−	0.0693157i	$-0.977918\pi$
$797$	− 2.48604e6i	− 0.138631i	0.997595	−	0.0693157i	$-0.0220816\pi$
$798$	− 1.07690e6i	− 0.0598642i
$799$	3.25024e7	1.80114
$800$	0	0
$801$	1.16044e7	0.639062
$802$	1.94686e7i	1.06881i
$803$	− 8.52063e6i	− 0.466319i
$804$	1.82774e6	0.0997183
$805$	0	0
$806$	1.87767e6	0.101808
$807$	2.47088e6i	0.133557i
$808$	− 1.65146e6i	− 0.0889897i
$809$	2.67078e7	1.43472	0.717360	−	0.696703i	$-0.245350\pi$
$809$	2.67078e7	1.43472	0.717360	+	0.696703i	$0.245350\pi$
$810$	0	0
$811$	3.53377e6	0.188663	0.0943313	−	0.995541i	$-0.469929\pi$
$811$	3.53377e6	0.188663	0.0943313	+	0.995541i	$0.469929\pi$
$812$	− 1.78885e6i	− 0.0952100i
$813$	1.54868e6i	0.0821741i
$814$	1.54913e7	0.819457
$815$	0	0
$816$	6.80046e6	0.357531
$817$	− 1.31358e7i	− 0.688495i
$818$	− 1.72723e7i	− 0.902540i
$819$	−814567.	−0.0424343
$820$	0	0
$821$	−8.58501e6	−0.444511	−0.222256	−	0.974988i	$-0.571342\pi$
$821$	−8.58501e6	−0.444511	−0.222256	+	0.974988i	$0.571342\pi$
$822$	− 3.60789e6i	− 0.186240i
$823$	1.58052e7i	0.813396i	0.913563	+	0.406698i	$0.133320\pi$
$823$	1.58052e7i	0.813396i	−0.913563	+	0.406698i	$0.866680\pi$
$824$	−624981.	−0.0320663
$825$	0	0
$826$	241921.	0.0123374
$827$	− 1.82089e7i	− 0.925808i	−0.886408	−	0.462904i	$-0.846808\pi$
$827$	− 1.82089e7i	− 0.925808i	0.886408	−	0.462904i	$-0.153192\pi$
$828$	− 3.33145e7i	− 1.68872i
$829$	−1.13685e7	−0.574534	−0.287267	−	0.957850i	$-0.592747\pi$
$829$	−1.13685e7	−0.574534	−0.287267	+	0.957850i	$0.592747\pi$
$830$	0	0
$831$	704705.	0.0354001
$832$	− 6.16960e6i	− 0.308993i
$833$	3.59128e7i	1.79323i
$834$	−9.61846e6	−0.478840
$835$	0	0
$836$	−9.05093e6	−0.447896
$837$	− 2.10257e6i	− 0.103738i
$838$	− 2.72123e7i	− 1.33861i
$839$	2.63011e7	1.28994	0.644970	−	0.764208i	$-0.276870\pi$
$839$	2.63011e7	1.28994	0.644970	+	0.764208i	$0.276870\pi$
$840$	0	0
$841$	−1.40223e7	−0.683642
$842$	− 1.66602e6i	− 0.0809841i
$843$	− 2.66424e6i	− 0.129123i
$844$	3.62881e6	0.175351
$845$	0	0
$846$	−2.79804e7	−1.34409
$847$	2.96132e6i	0.141833i
$848$	2.32947e7i	1.11242i
$849$	−4.45448e6	−0.212094
$850$	0	0
$851$	−5.98666e7	−2.83374
$852$	− 2.97183e6i	− 0.140257i
$853$	2.59081e7i	1.21916i	0.792723	+	0.609582i	$0.208663\pi$
$853$	2.59081e7i	1.21916i	−0.792723	+	0.609582i	$0.791337\pi$
$854$	−2.74129e6	−0.128621
$855$	0	0
$856$	1.14242e6	0.0532896
$857$	2.09566e7i	0.974697i	0.873208	+	0.487348i	$0.162036\pi$
$857$	2.09566e7i	0.974697i	−0.873208	+	0.487348i	$0.837964\pi$
$858$	− 597133.i	− 0.0276919i
$859$	−3.29521e7	−1.52370	−0.761852	−	0.647751i	$-0.775710\pi$
$859$	−3.29521e7	−1.52370	−0.761852	+	0.647751i	$0.775710\pi$
$860$	0	0
$861$	786550.	0.0361592
$862$	5.05518e7i	2.31723i
$863$	− 2.33805e7i	− 1.06863i	−0.845286	−	0.534314i	$-0.820570\pi$
$863$	− 2.33805e7i	− 1.06863i	0.845286	−	0.534314i	$-0.179430\pi$
$864$	−1.27230e7	−0.579836
$865$	0	0
$866$	−3.30480e7	−1.49744
$867$	− 1.09449e7i	− 0.494498i
$868$	− 961238.i	− 0.0433044i
$869$	7.83911e6	0.352142
$870$	0	0
$871$	2.82234e6	0.126056
$872$	1.29575e6i	0.0577074i
$873$	− 4.97588e6i	− 0.220971i
$874$	6.80103e7	3.01159
$875$	0	0
$876$	−6.91935e6	−0.304653
$877$	− 6.24910e6i	− 0.274359i	−0.990546	−	0.137179i	$-0.956196\pi$
$877$	− 6.24910e6i	− 0.274359i	0.990546	−	0.137179i	$-0.0438036\pi$
$878$	1.42439e7i	0.623581i
$879$	7.18290e6	0.313565
$880$	0	0
$881$	8.57115e6	0.372048	0.186024	−	0.982545i	$-0.440440\pi$
$881$	8.57115e6	0.372048	0.186024	+	0.982545i	$0.440440\pi$
$882$	− 3.09162e7i	− 1.33818i
$883$	− 2.98999e7i	− 1.29053i	−0.763958	−	0.645265i	$-0.776747\pi$
$883$	− 2.98999e7i	− 1.29053i	0.763958	−	0.645265i	$-0.223253\pi$
$884$	−1.25570e7	−0.540449
$885$	0	0
$886$	3.31579e7	1.41907
$887$	− 3.63685e7i	− 1.55209i	−0.630680	−	0.776043i	$-0.717224\pi$
$887$	− 3.63685e7i	− 1.55209i	0.630680	−	0.776043i	$-0.282776\pi$
$888$	− 699531.i	− 0.0297697i
$889$	−6.84843e6	−0.290627
$890$	0	0
$891$	6.94782e6	0.293194
$892$	− 2.93131e7i	− 1.23353i
$893$	− 2.93772e7i	− 1.23277i
$894$	1.11872e7	0.468144
$895$	0	0
$896$	−648011.	−0.0269657
$897$	2.30764e6i	0.0957608i
$898$	3.51445e7i	1.45434i
$899$	3.48680e6	0.143889
$900$	0	0
$901$	5.32007e7	2.18326
$902$	− 1.28537e7i	− 0.526033i
$903$	− 443654.i	− 0.0181061i
$904$	−1.41057e6	−0.0574083
$905$	0	0
$906$	591809.	0.0239531
$907$	− 2.51577e7i	− 1.01544i	−0.861523	−	0.507718i	$-0.830489\pi$
$907$	− 2.51577e7i	− 1.01544i	0.861523	−	0.507718i	$-0.169511\pi$
$908$	− 2.61670e7i	− 1.05327i
$909$	−2.51132e7	−1.00807
$910$	0	0
$911$	−4.68717e7	−1.87118	−0.935588	−	0.353092i	$-0.885130\pi$
$911$	−4.68717e7	−1.87118	−0.935588	+	0.353092i	$0.885130\pi$
$912$	− 6.14659e6i	− 0.244707i
$913$	1.63460e7i	0.648983i
$914$	3.54787e7	1.40476
$915$	0	0
$916$	−2.08415e7	−0.820712
$917$	1.51517e6i	0.0595031i
$918$	2.73401e7i	1.07076i
$919$	879544.	0.0343533	0.0171767	−	0.999852i	$-0.494532\pi$
$919$	879544.	0.0343533	0.0171767	+	0.999852i	$0.494532\pi$
$920$	0	0
$921$	8.77264e6	0.340785
$922$	2.54604e7i	0.986367i
$923$	− 4.58901e6i	− 0.177302i
$924$	−305691.	−0.0117788
$925$	0	0
$926$	−5.25341e7	−2.01332
$927$	9.50388e6i	0.363247i
$928$	− 2.10992e7i	− 0.804261i
$929$	−2.05492e7	−0.781187	−0.390593	−	0.920563i	$-0.627730\pi$
$929$	−2.05492e7	−0.781187	−0.390593	+	0.920563i	$0.627730\pi$
$930$	0	0
$931$	3.24597e7	1.22736
$932$	− 4.68320e7i	− 1.76605i
$933$	− 6.70761e6i	− 0.252269i
$934$	1.88761e7	0.708018
$935$	0	0
$936$	601105.	0.0224265
$937$	− 3.29842e7i	− 1.22732i	−0.789571	−	0.613659i	$-0.789697\pi$
$937$	− 3.29842e7i	− 1.22732i	0.789571	−	0.613659i	$-0.210303\pi$
$938$	− 2.80934e6i	− 0.104255i
$939$	4.69087e6	0.173616
$940$	0	0
$941$	1.67975e7	0.618400	0.309200	−	0.950997i	$-0.399939\pi$
$941$	1.67975e7	0.618400	0.309200	+	0.950997i	$0.399939\pi$
$942$	1.30276e7i	0.478341i
$943$	4.96737e7i	1.81906i
$944$	1.38081e6	0.0504316
$945$	0	0
$946$	−7.25017e6	−0.263403
$947$	1.07215e7i	0.388492i	0.980953	+	0.194246i	$0.0622259\pi$
$947$	1.07215e7i	0.388492i	−0.980953	+	0.194246i	$0.937774\pi$
$948$	− 6.36590e6i	− 0.230059i
$949$	−1.06846e7	−0.385118
$950$	0	0
$951$	5.12881e6	0.183893
$952$	695036.i	0.0248551i
$953$	− 1.79930e7i	− 0.641758i	−0.947120	−	0.320879i	$-0.896022\pi$
$953$	− 1.79930e7i	− 0.641758i	0.947120	−	0.320879i	$-0.103978\pi$
$954$	−4.57989e7	−1.62924
$955$	0	0
$956$	−3.91036e7	−1.38380
$957$	− 1.10886e6i	− 0.0391380i
$958$	− 9.42158e6i	− 0.331673i
$959$	−2.85206e6	−0.100141
$960$	0	0
$961$	−2.67555e7	−0.934555
$962$	− 1.94256e7i	− 0.676764i
$963$	− 1.73725e7i	− 0.603664i
$964$	4.63786e7	1.60740
$965$	0	0
$966$	2.29702e6	0.0791992
$967$	− 4.83225e7i	− 1.66182i	−0.556409	−	0.830908i	$-0.687821\pi$
$967$	− 4.83225e7i	− 1.66182i	0.556409	−	0.830908i	$-0.312179\pi$
$968$	− 2.18529e6i	− 0.0749585i
$969$	−1.40376e7	−0.480269
$970$	0	0
$971$	4.13992e7	1.40911	0.704553	−	0.709652i	$-0.251148\pi$
$971$	4.13992e7	1.40911	0.704553	+	0.709652i	$0.251148\pi$
$972$	− 1.82898e7i	− 0.620931i
$973$	7.60345e6i	0.257471i
$974$	2.95322e7	0.997468
$975$	0	0
$976$	−1.56464e7	−0.525764
$977$	− 2.77983e7i	− 0.931713i	−0.884860	−	0.465856i	$-0.845746\pi$
$977$	− 2.77983e7i	− 0.931713i	0.884860	−	0.465856i	$-0.154254\pi$
$978$	− 3.39483e6i	− 0.113494i
$979$	6.72472e6	0.224242
$980$	0	0
$981$	1.97041e7	0.653709
$982$	− 6.85891e7i	− 2.26974i
$983$	− 2.08694e7i	− 0.688853i	−0.938813	−	0.344426i	$-0.888073\pi$
$983$	− 2.08694e7i	− 0.688853i	0.938813	−	0.344426i	$-0.111927\pi$
$984$	−580430.	−0.0191100
$985$	0	0
$986$	−4.53395e7	−1.48520
$987$	− 992202.i	− 0.0324196i
$988$	1.13496e7i	0.369904i
$989$	2.80185e7	0.910866
$990$	0	0
$991$	5.53538e7	1.79045	0.895227	−	0.445611i	$-0.147014\pi$
$991$	5.53538e7	1.79045	0.895227	+	0.445611i	$0.147014\pi$
$992$	− 1.13377e7i	− 0.365802i
$993$	− 9.41101e6i	− 0.302875i
$994$	−4.56787e6	−0.146638
$995$	0	0
$996$	1.32741e7	0.423990
$997$	− 3.35323e6i	− 0.106838i	−0.998572	−	0.0534189i	$-0.982988\pi$
$997$	− 3.35323e6i	− 0.106838i	0.998572	−	0.0534189i	$-0.0170119\pi$
$998$	− 5.09973e7i	− 1.62077i
$999$	−2.17523e7	−0.689590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.3		18
5.2	odd	4		325.6.a.i.1.8	yes	9
5.3	odd	4		325.6.a.h.1.2	✓	9
5.4	even	2	inner	325.6.b.h.274.16		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.2	✓	9	5.3	odd	4
325.6.a.i.1.8	yes	9	5.2	odd	4
325.6.b.h.274.3		18	1.1	even	1	trivial
325.6.b.h.274.16		18	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.11691i q^{2} +3.22994i q^{3} -33.8842 q^{4} +26.2172 q^{6} -20.7248i q^{7} +15.2938i q^{8} +232.567 q^{9} +O(q^{10})\)	\(q-8.11691i q^{2} +3.22994i q^{3} -33.8842 q^{4} +26.2172 q^{6} -20.7248i q^{7} +15.2938i q^{8} +232.567 q^{9} +134.772 q^{11} -109.444i q^{12} -169.000i q^{13} -168.222 q^{14} -960.156 q^{16} +2192.81i q^{17} -1887.73i q^{18} +1981.97 q^{19} +66.9401 q^{21} -1093.93i q^{22} +4227.53i q^{23} -49.3980 q^{24} -1371.76 q^{26} +1536.06i q^{27} +702.244i q^{28} -2547.33 q^{29} -1368.81 q^{31} +8282.90i q^{32} +435.305i q^{33} +17798.9 q^{34} -7880.36 q^{36} +14161.1i q^{37} -16087.5i q^{38} +545.861 q^{39} +11750.1 q^{41} -543.347i q^{42} -6627.63i q^{43} -4566.63 q^{44} +34314.5 q^{46} -14822.2i q^{47} -3101.25i q^{48} +16377.5 q^{49} -7082.67 q^{51} +5726.43i q^{52} -24261.4i q^{53} +12468.0 q^{54} +316.961 q^{56} +6401.66i q^{57} +20676.4i q^{58} -1438.11 q^{59} +16295.7 q^{61} +11110.5i q^{62} -4819.92i q^{63} +36506.5 q^{64} +3533.33 q^{66} +16700.2i q^{67} -74301.7i q^{68} -13654.7 q^{69} +27153.9 q^{71} +3556.83i q^{72} -63222.7i q^{73} +114944. q^{74} -67157.5 q^{76} -2793.12i q^{77} -4430.70i q^{78} +58165.8 q^{79} +51552.5 q^{81} -95374.1i q^{82} +121286. i q^{83} -2268.21 q^{84} -53795.9 q^{86} -8227.72i q^{87} +2061.17i q^{88} +49897.1 q^{89} -3502.50 q^{91} -143246. i q^{92} -4421.18i q^{93} -120311. q^{94} -26753.3 q^{96} -21395.4i q^{97} -132934. i q^{98} +31343.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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