# Properties

 Label 2300.2.c.b.1749.2 Level $2300$ Weight $2$ Character 2300.1749 Analytic conductor $18.366$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.c (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: $$18.3655924649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 92) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1749.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2300.1749 Dual form 2300.2.c.b.1749.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} -4.00000i q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} -4.00000i q^{7} -6.00000 q^{9} +2.00000 q^{11} +5.00000i q^{13} +4.00000i q^{17} +2.00000 q^{19} +12.0000 q^{21} -1.00000i q^{23} -9.00000i q^{27} +7.00000 q^{29} -3.00000 q^{31} +6.00000i q^{33} +2.00000i q^{37} -15.0000 q^{39} -9.00000 q^{41} +8.00000i q^{43} +9.00000i q^{47} -9.00000 q^{49} -12.0000 q^{51} -2.00000i q^{53} +6.00000i q^{57} -2.00000 q^{61} +24.0000i q^{63} +14.0000i q^{67} +3.00000 q^{69} -3.00000 q^{71} +3.00000i q^{73} -8.00000i q^{77} +6.00000 q^{79} +9.00000 q^{81} -8.00000i q^{83} +21.0000i q^{87} -12.0000 q^{89} +20.0000 q^{91} -9.00000i q^{93} -12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 4 q^{11} + 4 q^{19} + 24 q^{21} + 14 q^{29} - 6 q^{31} - 30 q^{39} - 18 q^{41} - 18 q^{49} - 24 q^{51} - 4 q^{61} + 6 q^{69} - 6 q^{71} + 12 q^{79} + 18 q^{81} - 24 q^{89} + 40 q^{91} - 24 q^{99}+O(q^{100})$$ 2 * q - 12 * q^9 + 4 * q^11 + 4 * q^19 + 24 * q^21 + 14 * q^29 - 6 * q^31 - 30 * q^39 - 18 * q^41 - 18 * q^49 - 24 * q^51 - 4 * q^61 + 6 * q^69 - 6 * q^71 + 12 * q^79 + 18 * q^81 - 24 * q^89 + 40 * q^91 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 5.00000i 1.38675i 0.720577 + 0.693375i $$0.243877\pi$$
−0.720577 + 0.693375i $$0.756123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 12.0000 2.61861
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 9.00000i − 1.73205i
$$28$$ 0 0
$$29$$ 7.00000 1.29987 0.649934 0.759991i $$-0.274797\pi$$
0.649934 + 0.759991i $$0.274797\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ 6.00000i 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −15.0000 −2.40192
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.00000i 1.31278i 0.754420 + 0.656392i $$0.227918\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.00000i 0.794719i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 24.0000i 3.02372i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.0000i 1.71037i 0.518321 + 0.855186i $$0.326557\pi$$
−0.518321 + 0.855186i $$0.673443\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 0 0
$$73$$ 3.00000i 0.351123i 0.984468 + 0.175562i $$0.0561742\pi$$
−0.984468 + 0.175562i $$0.943826\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 8.00000i − 0.911685i
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 21.0000i 2.25144i
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 20.0000 2.09657
$$92$$ 0 0
$$93$$ − 9.00000i − 0.933257i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ −12.0000 −1.20605
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.00000i 0.193347i 0.995316 + 0.0966736i $$0.0308203\pi$$
−0.995316 + 0.0966736i $$0.969180\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 20.0000i 1.88144i 0.339182 + 0.940721i $$0.389850\pi$$
−0.339182 + 0.940721i $$0.610150\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 30.0000i − 2.77350i
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 27.0000i − 2.43451i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 17.0000i − 1.50851i −0.656584 0.754253i $$-0.727999\pi$$
0.656584 0.754253i $$-0.272001\pi$$
$$128$$ 0 0
$$129$$ −24.0000 −2.11308
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ 1.00000 0.0848189 0.0424094 0.999100i $$-0.486497\pi$$
0.0424094 + 0.999100i $$0.486497\pi$$
$$140$$ 0 0
$$141$$ −27.0000 −2.27381
$$142$$ 0 0
$$143$$ 10.0000i 0.836242i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 27.0000i − 2.22692i
$$148$$ 0 0
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ 13.0000 1.05792 0.528962 0.848645i $$-0.322581\pi$$
0.528962 + 0.848645i $$0.322581\pi$$
$$152$$ 0 0
$$153$$ − 24.0000i − 1.94029i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ 5.00000i 0.391630i 0.980641 + 0.195815i $$0.0627352\pi$$
−0.980641 + 0.195815i $$0.937265\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −12.0000 −0.917663
$$172$$ 0 0
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 25.0000 1.86859 0.934294 0.356504i $$-0.116031\pi$$
0.934294 + 0.356504i $$0.116031\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ − 6.00000i − 0.443533i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ −36.0000 −2.61861
$$190$$ 0 0
$$191$$ 2.00000 0.144715 0.0723575 0.997379i $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ 0 0
$$193$$ − 17.0000i − 1.22369i −0.790979 0.611843i $$-0.790428\pi$$
0.790979 0.611843i $$-0.209572\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 9.00000i 0.641223i 0.947211 + 0.320612i $$0.103888\pi$$
−0.947211 + 0.320612i $$0.896112\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ −42.0000 −2.96245
$$202$$ 0 0
$$203$$ − 28.0000i − 1.96521i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000i 0.417029i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ − 9.00000i − 0.616670i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000i 0.814613i
$$218$$ 0 0
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ −20.0000 −1.34535
$$222$$ 0 0
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 24.0000 1.57908
$$232$$ 0 0
$$233$$ − 27.0000i − 1.76883i −0.466702 0.884414i $$-0.654558\pi$$
0.466702 0.884414i $$-0.345442\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 18.0000i 1.16923i
$$238$$ 0 0
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 10.0000i 0.636285i
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ − 2.00000i − 0.125739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7.00000i 0.436648i 0.975876 + 0.218324i $$0.0700590\pi$$
−0.975876 + 0.218324i $$0.929941\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −42.0000 −2.59973
$$262$$ 0 0
$$263$$ 10.0000i 0.616626i 0.951285 + 0.308313i $$0.0997645\pi$$
−0.951285 + 0.308313i $$0.900236\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 36.0000i − 2.20316i
$$268$$ 0 0
$$269$$ −13.0000 −0.792624 −0.396312 0.918116i $$-0.629710\pi$$
−0.396312 + 0.918116i $$0.629710\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 60.0000i 3.63137i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 11.0000i − 0.660926i −0.943819 0.330463i $$-0.892795\pi$$
0.943819 0.330463i $$-0.107205\pi$$
$$278$$ 0 0
$$279$$ 18.0000 1.07763
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ 26.0000i 1.54554i 0.634686 + 0.772770i $$0.281129\pi$$
−0.634686 + 0.772770i $$0.718871\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 36.0000i 2.12501i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 16.0000i − 0.934730i −0.884064 0.467365i $$-0.845203\pi$$
0.884064 0.467365i $$-0.154797\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 18.0000i − 1.04447i
$$298$$ 0 0
$$299$$ 5.00000 0.289157
$$300$$ 0 0
$$301$$ 32.0000 1.84445
$$302$$ 0 0
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8.00000i 0.456584i 0.973593 + 0.228292i $$0.0733141\pi$$
−0.973593 + 0.228292i $$0.926686\pi$$
$$308$$ 0 0
$$309$$ −24.0000 −1.36531
$$310$$ 0 0
$$311$$ 5.00000 0.283524 0.141762 0.989901i $$-0.454723\pi$$
0.141762 + 0.989901i $$0.454723\pi$$
$$312$$ 0 0
$$313$$ − 16.0000i − 0.904373i −0.891923 0.452187i $$-0.850644\pi$$
0.891923 0.452187i $$-0.149356\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ 0 0
$$319$$ 14.0000 0.783850
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 36.0000 1.98474
$$330$$ 0 0
$$331$$ 11.0000 0.604615 0.302307 0.953211i $$-0.402243\pi$$
0.302307 + 0.953211i $$0.402243\pi$$
$$332$$ 0 0
$$333$$ − 12.0000i − 0.657596i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4.00000i 0.217894i 0.994048 + 0.108947i $$0.0347479\pi$$
−0.994048 + 0.108947i $$0.965252\pi$$
$$338$$ 0 0
$$339$$ −60.0000 −3.25875
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 8.00000i 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 0 0
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ 45.0000 2.40192
$$352$$ 0 0
$$353$$ 31.0000i 1.64996i 0.565159 + 0.824982i $$0.308815\pi$$
−0.565159 + 0.824982i $$0.691185\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 48.0000i 2.54043i
$$358$$ 0 0
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ − 21.0000i − 1.10221i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 0 0
$$369$$ 54.0000 2.81113
$$370$$ 0 0
$$371$$ −8.00000 −0.415339
$$372$$ 0 0
$$373$$ 34.0000i 1.76045i 0.474554 + 0.880227i $$0.342610\pi$$
−0.474554 + 0.880227i $$0.657390\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 35.0000i 1.80259i
$$378$$ 0 0
$$379$$ 32.0000 1.64373 0.821865 0.569683i $$-0.192934\pi$$
0.821865 + 0.569683i $$0.192934\pi$$
$$380$$ 0 0
$$381$$ 51.0000 2.61281
$$382$$ 0 0
$$383$$ 36.0000i 1.83951i 0.392488 + 0.919757i $$0.371614\pi$$
−0.392488 + 0.919757i $$0.628386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 48.0000i − 2.43998i
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ − 45.0000i − 2.26995i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 37.0000i − 1.85698i −0.371361 0.928488i $$-0.621109\pi$$
0.371361 0.928488i $$-0.378891\pi$$
$$398$$ 0 0
$$399$$ 24.0000 1.20150
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ − 15.0000i − 0.747203i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.00000i 0.198273i
$$408$$ 0 0
$$409$$ 11.0000 0.543915 0.271957 0.962309i $$-0.412329\pi$$
0.271957 + 0.962309i $$0.412329\pi$$
$$410$$ 0 0
$$411$$ 36.0000 1.77575
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 3.00000i 0.146911i
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ 0 0
$$423$$ − 54.0000i − 2.62557i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ 0 0
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 2.00000i − 0.0956730i
$$438$$ 0 0
$$439$$ 13.0000 0.620456 0.310228 0.950662i $$-0.399595\pi$$
0.310228 + 0.950662i $$0.399595\pi$$
$$440$$ 0 0
$$441$$ 54.0000 2.57143
$$442$$ 0 0
$$443$$ − 29.0000i − 1.37783i −0.724841 0.688916i $$-0.758087\pi$$
0.724841 0.688916i $$-0.241913\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 42.0000i − 1.98653i
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ −18.0000 −0.847587
$$452$$ 0 0
$$453$$ 39.0000i 1.83238i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 40.0000i − 1.87112i −0.353166 0.935561i $$-0.614895\pi$$
0.353166 0.935561i $$-0.385105\pi$$
$$458$$ 0 0
$$459$$ 36.0000 1.68034
$$460$$ 0 0
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.0000i 1.01804i 0.860755 + 0.509019i $$0.169992\pi$$
−0.860755 + 0.509019i $$0.830008\pi$$
$$468$$ 0 0
$$469$$ 56.0000 2.58584
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000i 0.549442i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −10.0000 −0.455961
$$482$$ 0 0
$$483$$ − 12.0000i − 0.546019i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 31.0000i 1.40474i 0.711810 + 0.702372i $$0.247876\pi$$
−0.711810 + 0.702372i $$0.752124\pi$$
$$488$$ 0 0
$$489$$ −15.0000 −0.678323
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 28.0000i 1.26106i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ −37.0000 −1.65635 −0.828174 0.560471i $$-0.810620\pi$$
−0.828174 + 0.560471i $$0.810620\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 0 0
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 36.0000i − 1.59882i
$$508$$ 0 0
$$509$$ 3.00000 0.132973 0.0664863 0.997787i $$-0.478821\pi$$
0.0664863 + 0.997787i $$0.478821\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ 0 0
$$513$$ − 18.0000i − 0.794719i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.0000i 0.791639i
$$518$$ 0 0
$$519$$ −54.0000 −2.37034
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 12.0000i − 0.522728i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 45.0000i − 1.94917i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 75.0000i 3.23649i
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −15.0000 −0.644900 −0.322450 0.946586i $$-0.604506\pi$$
−0.322450 + 0.946586i $$0.604506\pi$$
$$542$$ 0 0
$$543$$ 54.0000i 2.31736i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 21.0000i − 0.897895i −0.893558 0.448948i $$-0.851799\pi$$
0.893558 0.448948i $$-0.148201\pi$$
$$548$$ 0 0
$$549$$ 12.0000 0.512148
$$550$$ 0 0
$$551$$ 14.0000 0.596420
$$552$$ 0 0
$$553$$ − 24.0000i − 1.02058i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 28.0000i 1.18640i 0.805056 + 0.593199i $$0.202135\pi$$
−0.805056 + 0.593199i $$0.797865\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ 0 0
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 36.0000i − 1.51186i
$$568$$ 0 0
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 6.00000i 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 33.0000i 1.37381i 0.726748 + 0.686904i $$0.241031\pi$$
−0.726748 + 0.686904i $$0.758969\pi$$
$$578$$ 0 0
$$579$$ 51.0000 2.11949
$$580$$ 0 0
$$581$$ −32.0000 −1.32758
$$582$$ 0 0
$$583$$ − 4.00000i − 0.165663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 19.0000i − 0.784214i −0.919920 0.392107i $$-0.871746\pi$$
0.919920 0.392107i $$-0.128254\pi$$
$$588$$ 0 0
$$589$$ −6.00000 −0.247226
$$590$$ 0 0
$$591$$ −27.0000 −1.11063
$$592$$ 0 0
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 30.0000i 1.22782i
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ 13.0000 0.530281 0.265141 0.964210i $$-0.414582\pi$$
0.265141 + 0.964210i $$0.414582\pi$$
$$602$$ 0 0
$$603$$ − 84.0000i − 3.42074i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ 0 0
$$609$$ 84.0000 3.40385
$$610$$ 0 0
$$611$$ −45.0000 −1.82051
$$612$$ 0 0
$$613$$ − 20.0000i − 0.807792i −0.914805 0.403896i $$-0.867656\pi$$
0.914805 0.403896i $$-0.132344\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −9.00000 −0.361158
$$622$$ 0 0
$$623$$ 48.0000i 1.92308i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 12.0000i 0.479234i
$$628$$ 0 0
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 0 0
$$633$$ 12.0000i 0.476957i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 45.0000i − 1.78296i
$$638$$ 0 0
$$639$$ 18.0000 0.712069
$$640$$ 0 0
$$641$$ −4.00000 −0.157991 −0.0789953 0.996875i $$-0.525171\pi$$
−0.0789953 + 0.996875i $$0.525171\pi$$
$$642$$ 0 0
$$643$$ − 10.0000i − 0.394362i −0.980367 0.197181i $$-0.936821\pi$$
0.980367 0.197181i $$-0.0631786\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17.0000i 0.668339i 0.942513 + 0.334169i $$0.108456\pi$$
−0.942513 + 0.334169i $$0.891544\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −36.0000 −1.41095
$$652$$ 0 0
$$653$$ − 3.00000i − 0.117399i −0.998276 0.0586995i $$-0.981305\pi$$
0.998276 0.0586995i $$-0.0186954\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 18.0000i − 0.702247i
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ 0 0
$$663$$ − 60.0000i − 2.33021i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 7.00000i − 0.271041i
$$668$$ 0 0
$$669$$ −48.0000 −1.85579
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ − 43.0000i − 1.65753i −0.559598 0.828764i $$-0.689045\pi$$
0.559598 0.828764i $$-0.310955\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 9.00000i 0.344375i 0.985064 + 0.172188i $$0.0550836\pi$$
−0.985064 + 0.172188i $$0.944916\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 12.0000i 0.457829i
$$688$$ 0 0
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 0 0
$$693$$ 48.0000i 1.82337i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.0000i − 1.36360i
$$698$$ 0 0
$$699$$ 81.0000 3.06370
$$700$$ 0 0
$$701$$ −12.0000 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$702$$ 0 0
$$703$$ 4.00000i 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 8.00000i − 0.300871i
$$708$$ 0 0
$$709$$ 40.0000 1.50223 0.751116 0.660171i $$-0.229516\pi$$
0.751116 + 0.660171i $$0.229516\pi$$
$$710$$ 0 0
$$711$$ −36.0000 −1.35011
$$712$$ 0 0
$$713$$ 3.00000i 0.112351i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 9.00000i 0.336111i
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 0 0
$$723$$ − 6.00000i − 0.223142i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 30.0000i − 1.11264i −0.830969 0.556319i $$-0.812213\pi$$
0.830969 0.556319i $$-0.187787\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −32.0000 −1.18356
$$732$$ 0 0
$$733$$ − 28.0000i − 1.03420i −0.855924 0.517102i $$-0.827011\pi$$
0.855924 0.517102i $$-0.172989\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 28.0000i 1.03139i
$$738$$ 0 0
$$739$$ 17.0000 0.625355 0.312678 0.949859i $$-0.398774\pi$$
0.312678 + 0.949859i $$0.398774\pi$$
$$740$$ 0 0
$$741$$ −30.0000 −1.10208
$$742$$ 0 0
$$743$$ − 48.0000i − 1.76095i −0.474093 0.880475i $$-0.657224\pi$$
0.474093 0.880475i $$-0.342776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 48.0000i 1.75623i
$$748$$ 0 0
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 24.0000i 0.874609i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 42.0000i − 1.52652i −0.646094 0.763258i $$-0.723599\pi$$
0.646094 0.763258i $$-0.276401\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −51.0000 −1.84875 −0.924374 0.381487i $$-0.875412\pi$$
−0.924374 + 0.381487i $$0.875412\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ −21.0000 −0.756297
$$772$$ 0 0
$$773$$ 52.0000i 1.87031i 0.354239 + 0.935155i $$0.384740\pi$$
−0.354239 + 0.935155i $$0.615260\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 24.0000i 0.860995i
$$778$$ 0 0
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 0 0
$$783$$ − 63.0000i − 2.25144i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 36.0000i − 1.28326i −0.767014 0.641631i $$-0.778258\pi$$
0.767014 0.641631i $$-0.221742\pi$$
$$788$$ 0 0
$$789$$ −30.0000 −1.06803
$$790$$ 0 0
$$791$$ 80.0000 2.84447
$$792$$ 0 0
$$793$$ − 10.0000i − 0.355110i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 10.0000i − 0.354218i −0.984191 0.177109i $$-0.943325\pi$$
0.984191 0.177109i $$-0.0566745\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ 72.0000 2.54399
$$802$$ 0 0
$$803$$ 6.00000i 0.211735i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 39.0000i − 1.37287i
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 5.00000 0.175574 0.0877869 0.996139i $$-0.472021\pi$$
0.0877869 + 0.996139i $$0.472021\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000i 0.559769i
$$818$$ 0 0
$$819$$ −120.000 −4.19314
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ − 45.0000i − 1.56860i −0.620381 0.784301i $$-0.713022\pi$$
0.620381 0.784301i $$-0.286978\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 33.0000 1.14476
$$832$$ 0 0
$$833$$ − 36.0000i − 1.24733i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 27.0000i 0.933257i
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 20.0000 0.689655
$$842$$ 0 0
$$843$$ 36.0000i 1.23991i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 28.0000i 0.962091i
$$848$$ 0 0
$$849$$ −78.0000 −2.67695
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3.00000i 0.102478i 0.998686 + 0.0512390i $$0.0163170\pi$$
−0.998686 + 0.0512390i $$0.983683\pi$$
$$858$$ 0 0
$$859$$ −13.0000 −0.443554 −0.221777 0.975097i $$-0.571186\pi$$
−0.221777 + 0.975097i $$0.571186\pi$$
$$860$$ 0 0
$$861$$ −108.000 −3.68063
$$862$$ 0 0
$$863$$ − 7.00000i − 0.238283i −0.992877 0.119141i $$-0.961986\pi$$
0.992877 0.119141i $$-0.0380142\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.00000i 0.101885i
$$868$$ 0 0
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ −70.0000 −2.37186
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 6.00000i 0.202606i 0.994856 + 0.101303i $$0.0323011\pi$$
−0.994856 + 0.101303i $$0.967699\pi$$
$$878$$ 0 0
$$879$$ 48.0000 1.61900
$$880$$ 0 0
$$881$$ 36.0000 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$882$$ 0 0
$$883$$ − 12.0000i − 0.403832i −0.979403 0.201916i $$-0.935283\pi$$
0.979403 0.201916i $$-0.0647168\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 39.0000i − 1.30949i −0.755849 0.654746i $$-0.772776\pi$$
0.755849 0.654746i $$-0.227224\pi$$
$$888$$ 0 0
$$889$$ −68.0000 −2.28065
$$890$$ 0 0
$$891$$ 18.0000 0.603023
$$892$$ 0 0
$$893$$ 18.0000i 0.602347i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 15.0000i 0.500835i
$$898$$ 0 0
$$899$$ −21.0000 −0.700389
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ 0 0
$$903$$ 96.0000i 3.19468i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 38.0000i − 1.26177i −0.775877 0.630885i $$-0.782692\pi$$
0.775877 0.630885i $$-0.217308\pi$$
$$908$$ 0 0
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −46.0000 −1.52405 −0.762024 0.647549i $$-0.775794\pi$$
−0.762024 + 0.647549i $$0.775794\pi$$
$$912$$ 0 0
$$913$$ − 16.0000i − 0.529523i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 60.0000i 1.98137i
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −24.0000 −0.790827
$$922$$ 0 0
$$923$$ − 15.0000i − 0.493731i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 48.0000i − 1.57653i
$$928$$ 0 0
$$929$$ −15.0000 −0.492134 −0.246067 0.969253i $$-0.579138\pi$$
−0.246067 + 0.969253i $$0.579138\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 0 0
$$933$$ 15.0000i 0.491078i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 38.0000i − 1.24141i −0.784046 0.620703i $$-0.786847\pi$$
0.784046 0.620703i $$-0.213153\pi$$
$$938$$ 0 0
$$939$$ 48.0000 1.56642
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ 9.00000i 0.293080i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 3.00000i − 0.0974869i −0.998811 0.0487435i $$-0.984478\pi$$
0.998811 0.0487435i $$-0.0155217\pi$$
$$948$$ 0 0
$$949$$ −15.0000 −0.486921
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 50.0000i 1.61966i 0.586665 + 0.809829i $$0.300440\pi$$
−0.586665 + 0.809829i $$0.699560\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 42.0000i 1.35767i
$$958$$ 0 0
$$959$$ −48.0000 −1.55000
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 13.0000i − 0.418052i −0.977910 0.209026i $$-0.932971\pi$$
0.977910 0.209026i $$-0.0670293\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −14.0000 −0.449281 −0.224641 0.974442i $$-0.572121\pi$$
−0.224641 + 0.974442i $$0.572121\pi$$
$$972$$ 0 0
$$973$$ − 4.00000i − 0.128234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 6.00000i − 0.191370i −0.995412 0.0956851i $$-0.969496\pi$$
0.995412 0.0956851i $$-0.0305042\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 108.000i 3.43768i
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 33.0000i 1.04722i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 22.0000i 0.696747i 0.937356 + 0.348373i $$0.113266\pi$$
−0.937356 + 0.348373i $$0.886734\pi$$
$$998$$ 0 0
$$999$$ 18.0000 0.569495
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2300.2.c.b.1749.2 2
5.2 odd 4 2300.2.a.h.1.1 1
5.3 odd 4 92.2.a.a.1.1 1
5.4 even 2 inner 2300.2.c.b.1749.1 2
15.8 even 4 828.2.a.c.1.1 1
20.3 even 4 368.2.a.g.1.1 1
20.7 even 4 9200.2.a.b.1.1 1
35.13 even 4 4508.2.a.d.1.1 1
40.3 even 4 1472.2.a.b.1.1 1
40.13 odd 4 1472.2.a.n.1.1 1
60.23 odd 4 3312.2.a.q.1.1 1
115.68 even 4 2116.2.a.a.1.1 1
460.183 odd 4 8464.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.a.a.1.1 1 5.3 odd 4
368.2.a.g.1.1 1 20.3 even 4
828.2.a.c.1.1 1 15.8 even 4
1472.2.a.b.1.1 1 40.3 even 4
1472.2.a.n.1.1 1 40.13 odd 4
2116.2.a.a.1.1 1 115.68 even 4
2300.2.a.h.1.1 1 5.2 odd 4
2300.2.c.b.1749.1 2 5.4 even 2 inner
2300.2.c.b.1749.2 2 1.1 even 1 trivial
3312.2.a.q.1.1 1 60.23 odd 4
4508.2.a.d.1.1 1 35.13 even 4
8464.2.a.s.1.1 1 460.183 odd 4
9200.2.a.b.1.1 1 20.7 even 4